Solid State Physics: Advances in Research and Applications, Vol. 54

SOLID STATE PHYSICS VOLUME 54

Founding Editors FREDERICK SEITZ DAVID TURNBULL

SOLID STATE PHYSICS Advances in Research and Applications

Editors HENRY EHRENREICH

FRANS SPAEPEN

Division of Engineering and Applied Sciences Harvard University Cambridge, Massachusetts

VOLUME 54

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Contents

vii

CONTRIBUTORS PREFACE. .

xi

QuasipartlcleCalculations in Solids

WILFRIED G. AULBUR. LARSJONSSON. AND JOHNW. WILKINS I . Many-Body Effects in Computational Solid State Physics

2 12 111. GWA Calculations: Numerical Considerations . . . . . . . . . . . . . . . . . . . . 89 133 IV . Semiconductors and Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 172 VI . GWA Calculations and Optical Response . . . . . . . . . . . . . . . . . . . . . . . VII . Excited States within Density Functional Theory . . . . . . . . . . . . . . . . . . . 195 207 Appendix: Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.. Quasiparticle Calculations in the GW Approximation . . . . . . . . . . . . . . . .

The Surfactant Effect in Semiconductor Thin-Film Growth

DANIELKANDELAND EFTHIMIOS KAXIRAS I. I1. 111. IV . V.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . Theoretical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . The Diffusion-De-Exchange-Passivation Model . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

219 223 233 242 260

The Two-Dlmenslonal Physics of Josephsondunction Arrays

R . S. NEWROCK. C. J . LOBB.U . GEIGENMULLER AND M . OCTAVIO I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . TheBasics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Classical Arrays: T = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Classical Arrays: T > O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Classical Arrays: Zero Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Classical Arrays: Nonzero Frequency Response . . . . . . . . . . . . . . . . . . . VII . Classical Arrays: Finite-Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII . Classical Arrays: Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX . Classical Arrays: Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X . Classical Arrays: Nonconventional Dynamics . . . . . . . . . . . . . . . . . . . . . XI . Classical Arrays: Strongly Driven . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI1 . Quantum Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI11. Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments and Apologia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

266 271 283 300 308 324 334 342 357 383 395 431 467 468

vi

CONTENTS

Appendix A: Correlation Functions: Vortices and Spin Waves . . . . . . . . . . . . . . . Appendix B Vortex-Pair Density: The Dilute Limit . . . . . . . . . . . . . . . . . . . . . Appendix C Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix D Current-Induced Vortex Unbinding . . . . . . . . . . . . . . . . . . . . . . . Appendix E: The Capacitance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix F: Offset Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix G Phase Correlation Function in the Absence of Coupling . . . . . . . . . . . Appendix H:Conductivity from Derivatives of the Partition Function . . . . . . . . . . . Appendix I: The Green’s Function for Gaussian Coarse Graining . . . . . . . . . . . . .

469 475 476 492 495 499 500 502 505

Contributors to Volume 54 Numbers in parentheses indicate the pages on which the authors’ contributions begin.

WILFREDG . AULBUR(l), Department of Physics, Ohio State University, Columbus, OH 43210-1106 U. GEIGENM~L (266), E R Department of Electrical Engineering, Delft University of Technology, Delft, The Netherlands LARSJONSON (l), Department of Physics, Ohio State University, Columbus, OH 43210-1106 DANIEL KANDEL(219), Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel EITHIMIOSKAXIRAS (219), Department of Physics and Division of Engineering and Applied Sciences, Harvard University, Cambridge, M A 02138 C. J . LOBB (266), Center for Superconductivity Research, Department of Physics, University of Maryland, College Park, M D 20742-411I R. S. NEWROCK (266), Physics Department, University of Cincinnati, Cincinnati, OH 45221-0011 M . OCTAVIO(266), Centro de Fisica, Instituto Venezolano de Investigationes CientiJicas, Apartado 21827, Caracas IOZOA, Venezuela

JOHN W. WILKINS(l), Department of Physics, Ohio State University, Columbus, OH 43210-1106

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Preface

This volume of the Solid State Physics series is perhaps the longest of the fifty-four published thus far. This fact is not meant to set a precedent for the future. It results simply from the fact that two of the three articles, concerning the calculation of quasiparticles states in solids and the physics of Josephson junction arrays respectively, represent truly comprehensive accounts of these subjects. The length of the remaining article on the effects of surfactants on semiconductor thin-film growth is perhaps more typical of what is ordinarily published in the series. The Coulomb interaction among the loz3 or so electrons in a solid presents a formidable challenge to calculations of quasi-particle and collective excitations. Many schemes have been devised for dealing with this problem over the years of which the local density (LDA) and the so-called GW approximations are among the most important. The article by Aulbur, Jonsson and Wilkins presents a comprehensive review, in the best sense of the term, of quasiparticle calculations in solids, and, in particular, the GW approximation. It provides a pedagogical discussion of the physics of the GWA, of its numerical implementation including new parallel algorithms for performing them, its applications to semiconductors, insulators and metals, and its use in calcultions of optical properties. Detailed comparisons are made among all published GWA calculations for five prototypical semiconductors. The relationship to experiment, for example, direct and inverse photoemission for a wide range of materials, is clearly delineated. The copious and scholarly compilation of over 600 references provides ready access to much of the literature. The review should thus be important for both practitioners and those wanting an overview of the method, in particular, a didactic introduction and an objective assessment of its successes and limitations. It relates directly to the earlier article by L. Hedin and S. Lunqvist in Vol. 23 in which this approach was first formulated, and that by N. D. Lang in Vol. 28 and J. Callaway and N. H. March concerning density functional theory in Vol. 38 of this series. ix

X

PREFACE

Kandel and Kaxiras review the effect of overlayers, commonly referred to as “surfactants”, on the epitaxial growth of semiconductors. Such overlayers make it possible to lower substantially the temperature required for growth of perfect epitaxial films. As device dimensions decrease, processing temperatures must be lowered to minimize diffusional broadening of interfaces and dopant profiles. The authors give an exhaustive review of the experiments, listing all known overlayers for homo- or heteroepitaxial growth of Si, Ge, Si-Ge and the 111-V compounds. First-principles calculations and kinetic Monte Carlo modeling of the atomistic processes are reviewed. The authors argue that, more than the suppression of diffusion or its relation to exchange, it is the passivation of the island edges that determines the efficiency of an overlayer. The final article by Newrock, Lobb, Geigenmuller and Octavio on Josephson-junction arrays is an almost book-length survey of the fascinating and rich two-dimensional physics that well controlled lower dimensional systems have shown to exhibit. As is well known, an entirely different but also well-controlled two dimensional semiconductor structure embedded in a MOSFET exhibits the quantum Hall effects, which were entirely unexpected at the time of their first observation. This also is true, perhaps to a somewhat lesser extent, for the systems under discussion here. The present overview is complete and pedagogically oriented. It begins with a generally accessible introduction to superconductors and single Josephson junctions. The presence of vortices is one of the natural consequences of arranging such junctions in a two-dimensional lattice. Much of the physics of these arrays, usually consisting of periodically ordered islands of superconductors coupled by Josephson junctions, is a result of their existence. Large arrays have proven to be very useful model systems for studying a wide variety of other physical problems, for example, phase transitions in frustrated and random systems, the dynamics of coupled non-linear systems and macroscopic quantum effects. Many of these are discussed here. They can be divided into classical and quantum arrays depending on the relative magnitude of the coupling energy between neighboring superconductors E , and the energy cost E , to place a charge on an island in these arrays. Classical arrays ( E , < < E j ) are physical representations of the XY model, a two-dimensional system of spins free to rotate in the XY plane. Thus, as the authors point out, these arrays, whose parameters are known, can serve as models that enable doing “statistical mechanics on a chip”. They can be used for studying Kosterlitz-Thouless phase transition, the effects of disorder on phase transitions, and for investigating dimensional crossover effects in phase transitions. These remarks should help in setting the broad orientation of this review into perspective. Given its lucid, unhurried exposition, this quasi-text, will

PREFACE

xi

without doubt serve as a key introduction and general reference to this important field. Its usefulness is further enhanced by the extensive referencing supplemented by an additional bibliography to several topics, such as chaos and turbulence, not explicitly discussed in the article. HENRYEHRENREICH FUNS SPAEPEN

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SOLID STATE PHYSICS VOLUME 54

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SOLID STATE PHYSICS. VOL . 54

Quasiparticle Calculations in Solids WILFRIED G. AULBUR.LARSJONSSON.

AND

JOHN W . WILKINS

Department of Physics Ohio State University Columbus. Ohio

I . Many-Body Effects in Computational Solid State Physics . . . . . . . . . . . . . . 1. Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Kohn-Sham Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Quasiparticle Calculations in the GW Approximation . . . . . . . . . . . . . . . . 4. The Quasiparticle Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . The Hedin Equations and the GWA . . . . . . . . . . . . . . . . . . . . . . . 6. Separation of the Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Determination of the Single-Particle Green Function . . . . . . . . . . . . . 8. Determination of the Dynamically Screened Interaction . . . . . . . . . . . 9. Early Quasiparticle Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 10. Local-Field Effects and the Nonlocality of the Self-Energy . . . . . . . . . . 11. Energy Dependence of the Self-Energy . . . . . . . . . . . . . . . . . . . . . . 12. Core-Polarization Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Self-Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. Vertex Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I11. GWA Calculations: Numerical Considerations . . . . . . . . . . . . . . . . . . . . 15. Different Implementations of the GWA . . . . . . . . . . . . . . . . . . . . . 16. Plane Waves: Numerical Details . . . . . . . . . . . . . . . . . . . . . . . . . 17. Parallel GWA Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. GWA Calculations for Five Prototypical Semiconductors . . . . . . . . . . IV . Semiconductors and Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. Bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. Superlattices-Interfaces-Schottky Barriers . . . . . . . . . . . . . . . . . . . 21. Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. Atoms and Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26. Bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27. Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 . Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . GWA Calculations and Optical Response . . . . . . . . . . . . . . . . . . . . . . .

2 2 5

9 12 16 18 24 28 29 42 51 57 63 69 79 89 90 102 113 120 133 133 140 147 152 153 157 158 163 163 169 171 172

1 ISBN O-LZ-M)7754-I ISSN 0081-1947/00$30.00

Copyright C)Zoo0 by Academic Press All rights of reproduction in any form reserved

2

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

29. Overestimation of Optical Constants within DFT 30. The “Scissors Operator” and its Limitations . . . 31. Local-Field Effects in Optical Response . . . . . . 32. Density-Polarization Functional Theory . . . . . VII. Excited States within Density Functional Theory . . . . 33. Functionals Based on Ground-State Densities . . 34. Functionals Based on Excited-State Densities . . . 35. Time-Dependent Density Functional Theory . . . 36. Monte-Carlo Calculations . . . . . . . . . . . . . . Appendix: Density Functional Theory . . . . . . . . . . . . . . 1. Universal Density Functionals . . . . . . . . . . . . 2. The Kohn-Sham System . . . . . . . . . . . . . . . . 3. The Band-Gap Discontinuity . . . . . ; . . . . . . . 4. The Exchange-Correlation Hole . . . . . . . . . . . 5. Coupling-Constant Averages . . . . . . . . . . . . . 6. Local Approximations . . . . . . . . . . . . . . . . .

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175 179 184 191 195 196 201 203 205 207 208 209 21 1 213 215 217

1. Many-Body Effects in Computational Solid State Physics 1. INTRODUCTIONAND OVERVIEW

During the last decade, quasiparticle calculations have been used successfully to describe the electronic excited-state properties of solids such as single-particle band structures and absorption spectra. Under the assumption that electronic and ionic degrees of freedom can be decoupled, the problem is that of N electrons in a solid described by the following Hamiltonian:

’

Here ri is the coordinate of electron i and Yxl is an external potential that accounts for the interaction with the nuclei. The last term in the above equation is the Coulomb interaction between the electrons, which correlates the electrons’ motion. As a consequence, an exact description of the many-electron problem requires the solution of an equation with 3N coupled spatial degrees of freedom. For macroscopic systems, the number of electrons N is on the order of Avogadro’s number ( % loz3),so a solution to the N-electron problem must be approximate. We use atomic units throughout this article unless otherwise noted. In these units, energy is measured in Hartree and h = e = me = 4m0 = 1; E~ is the permittivity of vacuum.

QUASIPARTICLE CALCULATIONS IN SOLIDS

3

A successful approximation for the determination of excited states is based on the quasiparticle concept and the Green function method. The Coulomb repulsion between electrons leads to a depletion of negative charge around a given electron, and the ensemble of this electron and its surrounding positive screening charge forms a quasiparticle. The mathematical description of quasiparticles is based on the single-particle Green function G, whose exact determination requires complete knowledge of the quasiparticle self-energy Z2 The self-energy Z is a non-Hermitian, energydependent, nonlocal operator that describes exchange and correlation effects beyond the Hartree approximation. A determination of the self-energy can only be approximate, and a working scheme for the quantitative calculation of excitation energies in metals, semiconductors, and insulators is the so-called dynamically screened interaction or the GW approximation (GWA).3,4In this approximation, the self-energy C is expanded linearly in terms of the screened interaction W:

which explains the name of the approximation. The GWA for the computation of quasiparticle energies was proposed by in 1965. However, not until the mid-eighties was the approach applied to large-scale, numerical electronic structure calculation^.^^^ The resulting ab-initio band structures compare favorably with experiment. Several reviews of quasiparticle calculations in the GWA have been published. An early review of bulk and surface calculations in the GWA was done by Hybertsen and Louie.’ Bechstedt’ discussed the physics of the GWA in relation to model approaches for the calculation of the dielectric response as well as the self-energy. Godby’ reviewed quasiparticle calculations for jellium, simple metals, and semiconductors. Mahan“ examined different GW approximations resulting from the inclusion of self-energy and

*

The self-energy I:is related to G via Dyson’s equation; see Section 11.14, Eq. (2.5). L. Hedin, Phys. Rev. 139, A796 (1965). L. Hedin and S. Lundqvist, in Solid State Physics, vol. 23, eds. F. Seitz, D. Turnbull, and H. Ehrenreich, Academic, New York (1969), 1 . M. S. Hybertsen and S. G. Louie, Phys. Rev. Lett. 55, 1418 (1985). R. W. Godby, M. Schliiter, and L. J. Sham, Phys. Rev. Lett. 56, 2415 (1986). M. S. Hybertsen and S. G. Louie, Comm. Cond. Mat. Phys. 13, 223 (1987). * F. Bechstedt, in ~esrkiipeiprooble~eelAdvances in Solid State Physics, vol. 32, ed. U. Rossler Vieweg, Braunschweig/Wiesbaden (1992), 161. R. W. Godby, “Unoccupied Electronic States,” Topics in Applied Physics, vol. 69, eds. J. E. Inglesfield and J. Fuggle, Springer, New York (1992). l o G. D. Mahan, Comm. Cond. Matt. Phys. 16,333 (1994).

’

’

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WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

vertex diagrams beyond the random phase approximation. Mahan and Plummer " discussed many-body effects in photoemission spectra of simple sp-bonded metals. Pollmann et al." concentrated on GWA calculations for semiconductor surfaces. FaridI3 considered mathematical aspects of quasiparticle calculations. Recently Aryasetiawan and G u n n a r ~ s o n 'fo~ cused on strongly correlated d and f electron systems and local-orbital basis

function^.'^ The aim of the present review is to discuss (1) the physics and extensions of the GWA (Section 11), (2) numerical aspects of GWA calculations (Section 111), (3) applications of the GWA to semiconductors and insulators (Section IV) and metals (Section V), .and (4) the relevance of GWA calculations to optical response (Section VI). Particular importance is given to semiconducting and insulating systems in a plane-wave basis. We compare all published first-principles GWA calculations for five prototypical semiconductors (Si, Ge, GaAs, Sic, GaN) and show that differences between published quasiparticle calculations for the lowest conduction-band state can be as large as 0.5 to 1.0eV. We also present for the first time parallel algorithms both for reciprocal and real-space/imaginary-time GWA calculations. In addition, Section VII gives a brief overview of alternative methods to determine excited states within density functional theory. The remainder of this section introduces the quasiparticle concept and defines the Green function and the spectral function (Section 1.2). The latter is important since it can be related to photoemission experiments. A short introduction to density functional theory is also necessary, since most current quasiparticle calculations start from density functional theory wave functions and energies, and measure their success by the degree of improvement of excited-state properties over the corresponding density functional description. Section 1.3 provides the basic equations and physics of density functional theory that are relevant for quasiparticle calculations. A more detailed overview of the basic concepts of density functional theory is given in the appendix.

G. D. Mahan and E. W. Plummer, to appear in Handbook of Surfaces, vol. 2, eds. K. Horn and M. Schemer. l 2 J. Pollmann, P. Kriiger, M. Rohlfing, M. Sabisch, and D. Vogel, Appl. SurJ Sci. 104-105, 1 (1996). l 3 B. Farid, to be published in Electron Correlation in the Solid Slate, ed. N. H. March, World Scientific/Imperial College Press, London, UK. l4 F. Aryasetiawan and 0. Gunnarsson, Rep. Prog. Phys. 61, 237 (1998). We call basis sets that contain functions explicitly centered at atomic sites (e.g., LMTO, Gaussian orbitals, etc.) local-orbital basis sets in this article.

QUASIPARTICLECALCULATIONS IN SOLIDS

5

2. QUASIPARTICLES Dejnition of Quasiparticles. The excitations of a system of strongly interacting particles can often be described in terms of weakly interacting quasiparticles. In a solid, an electron, or “bare” particle, repels the other electrons via the Coulomb potential and, in effect, surrounds itself with a positively charged polarization cloud. The positive screening charge and the bare electron form a quasiparticle that weakly interacts with other quasiparticles via a screened rather than the bare Coulomb potential. The quasiparticle lifetime is finite since quasiparticles are only approximate eigenstates of the N-electron Hamiltonian in Eq. (1.1). The residual interaction between the quasiparticles leads to a complex energy whose imaginary part is inversely proportional to the quasiparticle lifetime. That the quasiparticle concept works well in solid state systems-in spite of strong interactions between the bare particles -is demonstrated by the success of one-particle theories such as density functional theory in the local density approximation or GWA in the description of the structural and electronic properties of solids. The energy difference between the quasiparticle and the bare particle is usually described by the self-energy which must account for all exchange and correlation effects beyond the Hartree approximation. The self-energy is a nonlocal, energy-dependent, and in general non-Hermitian operator, whose properties will be discussed in more detail in Sections 11.10 and 11.11. An exact determination of the self-energy for real systems is not possible, since it contains all the complexities of the many-body system. Instead, practical approximations to the self-energy-such as the dynamically screened interaction or GWA, which is the topic of the present review (see Section 11)-must be used. The central equation that governs the behavior of quasiparticles is the so-called quasiparticle equation. Neglecting spin degrees of freedom, this equation can be written as

[

-

2

V2 + V,

+ V,,,

I S Yi(r) +

C(r, r’; Ei)Yi(r’)dr’= EiYi(r)

(1.3)

Here V, is the electrostatic or Hartree potential of the electrons, that is, with n as the electron density,

l6 The self-energy equals the energy of the bare particle interacting with itself via the polarization cloud that the particle generates in the many-body system.

6

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

V,,, is the external potential from the ions, and Y iand E i are the quasiparticle wave function and energy, respectively. We will come back to the quasiparticle equation and define Yiand Eiin Section 11.4. The Green Function. Quasiparticle properties such as energies, lifetimes, and expectation values of single-particle operators, such as the density and the total energy of a many-body system, are determined by the singleparticle Green function (see, for instance, standard textbooks such as Refs. 17, 18, 19, 20, 21, and 22). The Green function G is also called the single-particle propagator. With IN, 0) as the ground state of the N-electron Hamiltonian in Eq. (l.l), Y(rt) = exp(ifit)Y(r) exp( - ifit) as the fermion annihilation operator23 in the Heisenberg representation, Y t(rt) as the corresponding creation operator, and T as the time-ordering operator, the single-particle Green function is defined as G(rt, r'r')

= -i ( N ,

OlT[Y(rt)Y t(r'tr)] IN, 0 )

For t > t' (t' > t), G describes the propagation of a particle (hole) added to the many-body system described by fi, that is, G describes the dynamics of the N + N f 1 excitations in an N-electron system. G is a function of only six spatial degrees of freedom and hence much more manageable than the N-electron wave function, which depends on 3N spatial degrees of freedom. Many of the complexities of the ground-state wave function are eliminated by taking the expectation values. The imaginary part of the Green function determines the spectral function A, A(r, r'; E ) = n- 'IIm G(r, r'; E)I, which is closely connected to photoemission spectra. Quasiparticles are " A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover, New York (1975). " A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York (1971). l 9 P. Fulde, Electron Correlations in Molecules and Solids, Springer Series in Solid-state Sciences, vol. 100, Springer Verlag, Berlin (1991), Chap. 9. ' O J. C. Inkson, Many-body Theory of Solids, Plenum Press, New York (1984). " G. D. Mahan, Many-Particle Physics, 2nd. ed., Plenum Press, New York (1993). R. D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, Dover, New York (1992). 2 3 %(rt) destroys an electron at point r and time t.

''

QUASIPARTICLECALCULATIONS IN SOLIDS

7

identified with narrow peaks in the interacting spectral function which contain a significant amount of spectral strength. The peak position and width determine the quasiparticle energy and inverse lifetime; the area under the peak equals the quasiparticle weight. Figure 1 shows a schematic picture of an interacting and a noninteracting spectral function. The interacting spectral function has an approximate pole at I? = B + ir. If Z is the quasiparticle weight and $(E) a smooth function at I?, then the Green function can be expressed as

Assuming Z to be real and neglecting the smooth background Im $(E), one obtains the corresponding spectral function as A(E) 2z n - l z

r

(1.8)

( E - Elz + r2'

Fourier transformation of Eq. (1.7) into the time domain leads to an

EE'CL

E

FIG. 1. Schematic representation of the spectral function A ( E ) (Eq. (1.8)) for a noninteracting and an interacting many-body system. The differences between the two cases are (1) the real part E of the quasiparticle energy E = E + iT is shifted with respect to the bare energy E; (2) the quasiparticle acquires a finite lifetime ljr due to interaction compared to the infinite lifetime of the noninteracting particle; and (3) the spectral weight Z (shaded area) of the quasiparticle peak is less than unity due to redistribution of spectral weight into the incoherent background (Im+(E)), whereas the spectral weight of the bare particle is unity. The chemical potential is p.

8

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

exponential decay of G cc exp( - r t ) , which identifies r as the inverse quasiparticle lifetime.24The smaller r is, the longer the quasiparticle lives and the sharper the corresponding peak in A(E). The quasiparticle weight Z equals the area under the L ~ r e n t z i a n it ; ~(1) ~ must be on the order of unity for the quasiparticle to be clearly identifiable in the spectrum, and (2) is less than unity since A(E) is positive definite and normalized to unity. Connection to Experiment. Quasiparticle energies and lifetimes are measured experimentally by direct or inverse photoemission, which removes or adds one electron to the system and corresponds directly to the definition of the Green function in Eq. (1.5). In direct photoemission, a photon with energy hv impinges on a sample. An electron from an occupied band with energy E,, absorbs the photon energy and becomes a photoelectron whose kinetic energy Ekinallows the determination of the initial-state (valence) energy. From the schematic picture of the direct photoemission process in Fig. 2 and measuring the valence- and conduction-band energies with respect to the vacuum level, we obtain the energy balance Ekin= E,, + hv, which gives EVE. Inverse photoemission, the complementary process to direct photoemission, probes the energy of unoccupied states E,, by injecting an electron into a solid. The electron loses its kinetic energy &in via photon emission before it comes to rest at a point of lower energy in the conduction band. With hv as the energy of the emitted photon, a simplified energy balance of the process reads Ekin= Ec, + hv. This relation gives E,, since Ekinand hv are measured. A schematic description of inverse photoemission is given in Fig. 2. The photocurrent in photoemission experiments is closely related to the single-particle spectral function. The intrinsic photoemission spectrum, that is, the photoemission spectrum that takes only many-electron scattering into account, is expected to be reliable for the determination of the quasiparticle peak, or the quasiparticle energy.26*27.28 However, the intrinsic spectrum neglects matrix element effects, phonon and defect scattering, the inhomogeneous surface potential, and other complications. For the determination of, for example, the quasiparticle lifetime, these additional effects must be included (see Refs. 26,29, and 30 and references therein). Quasiparticle lifetimes in Si are discussed in connection with Fig. 15. Typical values of Z for semiconductors are listed in Table 9. 2 6 L. Hedin, Nucl. Instr. Meth. Phys. Res. A m , 169 (1991). ” L. Hedin, Int. J . Quant. Chem. 56,445 (1995). S. Hufner, Photoelectron Spectroscopy, Springer Series in Solid-state Sciences, Springer, Berlin (1996). 2 9 J. Fraxedas, M. K. Kelly, and M. Cardona, Phys. Rev. 843, 2159 (1991). 30 N. V. Smith, P. Thiry, and Y. Petro5, Phys. Rev. B47, 15476 (1993). 24

”

’*

9

QUASIPARTICLE CALCULATIONS IN SOLIDS

-P

photon -> electron: electron o photon:

photoelectrunspectroscopy Inverse photocmlrJion

I

%

hv

c

VB

\

photoekctron spectroscopy

inverse photoemlulon

N->N-1

N->N+1

FIG.2. (a) Schematic representation of an (inverse) photoemission experiment; (b) schematic representation of the excitation process in a photoemission (left) and inverse photoemission (right) experiment. The energy of the incoming (outgoing) photon is hv; the electron kinetic energy Eki,, E,,, is the band gap; E , the electronegativity; and E+ the photothreshold energy of the sample. Direct photoemission measures the quasiparticle properties of occupied bands and decreases the total number of electrons N in the system. An incoming photon ejects a valence-band (VB) electron out of the sample. The electron energy is measured and the valence-band energy can be obtained via E,,, = E,, + hv. Inverse photoemission measures the properties of quasiparticles in unoccupied bands. An incoming electron of energy E,,, impinges on the sample and loses energy via emission of a photon hv. The electron reaches an energetically lower available conduction state leading to E,,, = E,, hv.

+

3. KOHN-SHAM PARTICLES

All GWA calculations start from a suitably chosen one-particle Hamiltonian whose eigenfunctions and eigenvalues are used to construct the singleparticle propagator G, the screened interaction W and the self-energy C, as will be detailed in Section 11. The independent-particle Hamiltonian of

10

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

choice31 is a density functional (DFT) Hamiltonian in the local density approximation (LDA).32v33Density functional theory describes groundstate properties and in particular the ground-state density, in principle, exactly, but it does not describe excited states accurately. The local density approximation to DFT is a practical starting point for quasiparticle calculations since (1) it describes the density of metals, semiconductors, and insulators accurately; and (2) exchange and correlation are described by a local potential. Point (1) suggests that LDA ground-state wave functions are good approximations for quasiparticle wave functions. This is often the case not only for ground states but also for excited states, as is further discussed in Section 11.7. Point (2) leads to computationally efficient quasiparticle calculations compared to quasiparticle calculations based on a nonlocal independent-particle Hamiltonian such as Hartree-Fock (see Section 11.7). The Kohn-Sham formulation of density functional theory34 maps the problem of N interacting electrons onto a system of N noninteracting, fictitious particles- the Kohn-Sham particles -which move in an effective potential V,,. The Kohn-Sham potential V,, is constructed to ensure that the ground-state density of the noninteracting, fictitious system equals the ground-state density n,(r) of the interacting system. Let mi(r) denote an orbital of the fictitious particles and E~ the corresponding eigenvalue of the Kohn-Sham Hamiltonian. With V,,, and V, as defined earlier, the relevant single-particle equations are3’

[

-

2

Vz

1

+ V, + V,,, + V,, mi = E ~ @ ~ ,

N

no = i=l

IQil2.

(1.9)

Here V, is the exchange-correlation potential, which is obtained as a functional derivative of the exchange-correlation energy Ex,:

(1.10)

3 1 Other independent-particle Hamiltonians such as Hartree and Hartree-Fock or empirical pseudopotential Hamiltonians have been used as well but to a much lesser extent. See, for instance, Sections 11.7 and 11.9. ” P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). 3 3 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 34 See the appendix for a brief review of the Kohn-Sham theory and other important concepts of density functional theory. 3 5 In a crystalline system, i stands for the combination of a band index n, a spin index s, and a wave vector k.

QUASIPARTICLE CALCULATIONS IN SOLIDS

11

and all potentials are evaluated at the ground-state density no@). The exchange-correlation energy, which contains all Coulomb correlation effects beyond the Hartree approximation and a part of the kinetic energy of the interacting electrons, is not known explicitly for real systems.36 Practical use of the Kohn-Sham equations requires good approximations to the exchange-correlation energy and, via Eq. (l.lO), to the exchange-correlation potential. The local density approximation replaces the inhomogeneous exchangecorrelation energy density per particle at a point r by the exchangecorrelation energy density per particle of a homogeneous electron gas, U,h:m(r), evaluated at the local density. The total exchange-correlation energy is then obtained as the integral over all local contributions: (1.11) The resulting exchange-correlation potential is local and energy independent. Despite its simplicity, the local density approximation and extensions that use gradient corrections to the local density -the generalized gradient approximations -successfully describe ground-state properties of atoms, molecules, and solids. This success and also several failures of LDA are reviewed extensively in the literature; see, for instance, Refs. 37 and 38. Structural properties of solids such as the lattice constant, the bulk modulus, and the cohesive energy are generally determined to within a few percent of the experimental value. The Band-Gap Problem. In the derivation of Kohn-Sham theory,33 the eigenvalues E~ in Eq. (1.9) enter as Lagrange parameters that ensure the orthogonality of the orbitals (Di of the fictitious particles. As a consequence, the E;S and (D;s must be considered as mathematical tools that contain no relevant physical information besides the fact that the square of the eigenfunctions sums up to the exact local ground-state density. In particular, there is no formal justification that links the eigenvalues gi to the energy dispersion of quasiparticles in a solid. A notable exception is the highest . and K ~ h identified n ~ ~ E~ with the chemical occupied eigenvalue E ~ Sham potential p of a metal. For semiconductors and insulators, Perdew, Parr, 36 For a formal expression of the exchange-correlation potential, see, for instance, L. J. Sham, Phys. Rev. B 32, 3876 (1985), and references therein. 3 7 R. 0.Jones and 0. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989). 38 Theory of the Inhomogeneous Electron Gas, eds. S. Lundqvist and N. H. March, Plenum Press, New York and London (1983). 39 L. J. Sham and W. Kohn, Phys. Rev. 145, 561 (1966).

12

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Levy, and Balduz4' proved that E~ equals the negative of the ionization energy.41 Perhaps the most prominent discrepancy between LDA band structures and experiment is the fact that LDA underestimates the band gap of semiconductors and insulators by about 0.5 to 2.0eV. In the case of Ge, LDA leads to a semimetal rather than to an indirect-gap semiconductor. Figure 3 demonstrates the underestimation of experimental band gaps in LDA for all semiconductors and insulators for which ab-initio GWA calculations have been reported. Figure 3 also shows that GWA calculations largely correct the LDA band-gap underestimation and are in good agreement with experiment. In spite of the bind-gap underestimation, LDA wave functions are often good approximations to quasiparticle wave function^.'^ In the absence of quasiparticle calculations, LDA energies are routinely used to interpret experimental spectra. LDA energy dispersions are often in fair agreement with experiment, and in some'cases the LDA band gap can be empirically adjusted to fit the experimental gap. This approach implies an interpretation of the LDA exchange-correlation potential as an approximate self-energy that neglects nonlocal, energy-dependent, and lifetime effects. Although the LDA band structure cannot claim quantitative accuracy for the determination of the electronic structure of solids, LDA generally provides a qualitative understanding. II. Quasiparticle Calculations in the GW Approximation

This section has four purposes. (1) The basic equations that govern the dynamically screened interaction approximation are introduced (Sections 11.4 and 11.5) but not derived. We refer the reader to standard textbooks, for example, Ref. 20, and the review articles by Hedin and Lundqvist4 and Aryasetiawan and G u n n a r ~ s o n in ' ~ particular for a derivation of the Hedin equations (Section 11.5). (2) Useful separations of the self-energy as well as basis-set-independent details about the evaluation of the single-particle propagator G and the screened interaction W are described in Sections 11.6, 11.7, and 11.8. (3) Section 11.9 gives a historical overview of early GWA calculations and related approaches. (4) The physics of the self-energy operator is analyzed in Sections 11.10, 11.11, 11.12, 11.13, and 11.14, with Sections 11.13 and 11.14 focusing on the consistency of GWA calculations and on extensions to the GWA, respectively. An overview of important equations for the GWA and symbols used in this article is given in Tables 1 and 2. 40

41

J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, Phys. Rev. Lett. 49, 1691 (1982). For a more detailed description, see the appendix.

-2

0

2

4

6

0

Expt., indirect gap Expt.. direct gap

a

1

Si; 42 Ge; 43 dlamond; 44 3c Sic; 42 UCI; 44 gcN.45

Bdf46

B P 46 BAS: 46

AIN; 47 AIP 48 A&; 43

AISb; 48

GaN; 42 GaP 48 GaAs; 43 GaSb; 48 InP 48 InAs: 48 InSb; 48

ZnS; 51

2% 51 ZnTe; 51 CdS; 52 CdSe; 51 CdTe; 51

% !

z

NIO; 55 CaCuO . 5 4 u 56 Z h * ; 57

6:

sno,; 58

FIG. 3. Comparison of characteristic direct and indirect LDA, GWA, and experimental energy gaps for all semiconductors and insulators for which first-principles GWA calculations have been reported. GWA corrects most of the LDA band gap underestimation over more than one order of magnitude in the experimental band gap. The values for MnO, ZnO, and CaCuO, are from model-GWA calculations, which are accurate to within 0.4eV. The discrepancy between GWA and experiment for LiO, results from the neglect of excitonic effects. The experimental value for BAS is tentative. The references for the LDA, GWA, and experimental values are listed after the element symbols.

e W

14

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS TABLE1. DEFINITIONS OF SYMBOLS AND QUANTITIES SYMBOL

DEFINITION (spatial time) Coordinate of first particle Chemical potential, Fermi energy Fermion annihilation or creation operator Quantum number, band index n, and wave vector k for solids Single-particle energy N-electron total energy for state i ith quasiparticle energy ith quasihole energy Wave function of single-particle Hamiltonian ith excited (i = 0 ground) state of N-electron system Quasiparticle many-body amplitude Quasihole many-body amplitude Single-particle Green function Spectral function Bare Coulomb interaction Electron density Density fluctuation operator Hartree potential External perturbation potential Total induced potential Irreducible polarizability Full polarizability Dielectric matrix Dynamically screened interaction Non-Hermitian, nonlocal, energy-dependent self-energy =exchange + energy-dependent correlation self-energy =Coulomb hole + screened-exchange self-energy Vertex function Fluctuation potential Energy of electron-hole excitation

W. G. Aulbur and J. W. Wilkins, unpublished. E. L. Shirley, X. Zhu, and S . G. Louie, Phys. Rev. B56, 6648 (1997). 44 M. S. Hybertsen and S . G. Louie, Phys. Rev. B34, 5390 (1986). 4 5 J. L. Corkill and M. L. Cohen, Phys. Rev. 848,17622 (1993). 46 M. P. Surh, S. G. Louie, and M. L. Cohen, Phys. Rev. B43,9126 (1991). 47 A. Rubio, J. L. Corkill, M. L. Cohen, E. L. Shirley, and S . G. Louie, Phys. Rev. B48, 11810 (1993). 48 X. Zhu and S . G. Louie, Phys. Rev. B43, 14142 (1991). 49 A. Rubio and M. L. Cohen, Phys. Rev. 851,4343 (1995). 5 0 S. Massidda, R. Resta, M. Posternak, and A. Baldereschi, Phys. Rev. B 52, R16977 (1995). 0. Zakharov, A. Rubio, X. Blase, M. L. Cohen, and S . G. Louie, Phys. Rev. B50, 10780 (1994). .52 M. Rohlfing, P. Kriiger, and J. Pollmann, Phys. Rev. Lett. 75, 3489 (1995). 42

43

15

QUASIPARTICLE CALCULATIONS IN SOLIDS TABLE2. IMPORTANT EQUATIONS FOR

THE

GWA

=jcm

1Yi(r)YT(r') E-Ei

Interacting Green function (Eq. (2.2))

G(r, r'; E)

Independent-particle Green function (Eq. (2.4))

Go(r, r'; E ) = C

Dyson's equation (Eq. (2.5))

G(r, r'; E ) = Go(r, r'; E )

[

A(r, r'; E')

dE'

mi(r)@f(r') E-Ei

i

+ Quasiparticle equation (Eq. (1.3))

=

Sj

,,

Go(r, r,; E)X(r r,; E)G(r ,, r'; E)dr,dr,

- V2 + V,

I S

+ Kx, Yi(r)+

X(r, r'; Ei)Yi(r')dr'

First-order perturbation theory for E i (Eq. (2.6)) Quasiparticle weight Zi (Eq. (2.7)) Dynamically screened interaction approximation (GWA) (Eqs. (2.1 1)-(2.13)) Independent-particle polarizability 0%. (2.21))

Dielectric matrix in RPA (Eq. (2.23)) Dynamically screened interaction 0%. (2.16)) occ

Energy-dependent correlation contribution to the self-energy (Eq. (2.25))

XC(r,r';E) =

C C

mfo

V,(r) V:(r')Qi (r)@:(r') E +E,,-E~-~S ""OSE

+

V, (r) V:(r')@i(r)@:(r')

Lo

E-E,,-Ei+i6

Static Coulomb-hole self-energy (Eq. (2.32))

XCoH(r,r') = 46(r -r')[W(r, r'; E = 0)- dr, r')]

Static screened-exchange self-energy (Eq. (2.33))

XSEX(r,r')=

OEE

-

1 @i(r)@f(r')W(r,r'; E = 0)

Note: Equation numbers within parentheses equal those in the text.

16

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

4. THEQUASIPARTICLE EQUATION

The physical relevance of the Green function6’ (Eq. (1.5)) can be made clear by expressing G in terms of quasiparticle wave functions and energies via its spectral function A. Consider a complete set of eigenstates of the many-body Hamiltonian for a system with N + 1 or N - 1 particles. Let p be the chemical potential and denote the quantum number of the (N + 1)-particle or (N - 1)-particle states with i. With EN,i as the energy of the N-electron system in state i (i = 0 for the ground state), one defines the quasiparticle amplitude Yi(r) and the quasiparticle energy Ei as4 Yi(r) = (N,Ol’&)lN

+ l,i),

Yi(r) = ( N - 1, &r)IN, o ) ,

Ei = EN+l,i- E N , O for Ei 2 p, (2.1) for E~ < p. E~ = E ~- ,E,- ~

As mentioned above, the quasiparticle amplitudes correspond to a nonlocal, energy-dependent, non-Hermitian Hamiltonian. They fulfill the completeness relation but are unnormalized and linearly d e ~ e n d e n tIntroduction .~ of the complete set of eigenstates in the definition of G given in Eq. (1.5) and Fourier transformation to energy space61 identifies the quasiparticle energies as the poles of the Green function,

The integral defines the interacting spectral weight function, A(r, r’; E ) =

1Yi(r)Y?(r’)6(E - Ei),

(2.3)

i

and the integration contour C runs infinitesimally above the real ,Y-axis for E‘ < p and infinitesimally below for E’ > p. 53

54

U. Schonberger and F. Aryasetiawan, Phys. Rev. B52,8788 (1995). S. Massidda, A. Continenza, M. Posternak, and A. Baldereschi, Phys. Rev. B 55, 13494

(1997).

F. Aryasetiawan and 0. Gunnarsson, Phys. Rev. Lett. 74, 3221 (1995). S. Albrecht, G. Onida, and L. Reining, Phys. Rev. B55, 10278 (1997). ” B. Kralik, E. K. Chang, and S. G. Louie, Phys. Rev. B57,7027 (1998). ’13 M. Palummo, L. Reining, M. Meyer, and C. M. Bertoni, 2 n d International Conference on the Physics of Semiconductors, vol. 1, ed. D. J. Lockwood, World Scientific, Singapore (1995), ”

56

161.

See Sections 11.7 and 11, and VI.30 for further discussions of this point. Sections 11.4 and 11.5 follow Ref. 4 closely. However, we do not consider the spin degrees of freedom explicitly. We add infinitesimal convergence factors f is to Ei for Ei 2 p. 59

6o

QUASIPARTICLE CALCULATIONS IN SOLIDS

17

Neglect of exchange and correlation effects beyond the Hartree approximation leads to the noninteracting single-particle Hamiltonian fro, whose corresponding Green function Go(r, r’; E) describes the propagation of a particle in a system of N + 1 noninteracting particles. With the complete set of orthonormalized single-particle wave functions Oi(r) and real, independent-particle energies q, we write the spectral representation of Go as follows: Go(r, r‘; E ) =

(r’) 1Qi(r)@.T E - E ~

Ao(r, r‘; E’) E - E dE.

(2.4)

The”independent-particle spectral function can be expressed either as in Eq. (2.3), with Y i being replaced by mi, or in the basis of the orthonormalized Qi. In the latter case, A’ reduces to a &function, A? = 6(E - q). The time development of the interacting Green function is determined by the quasiparticle self-energy via Dyson’s equation.” With G and Go as in Eqs. (2.2) and (2.4), we have” G(r, r’; E ) = Go(r, r’; E )

+

ss

Go(r, rl; E)C(r,, r,; E)G(r,, r’; E)dr,dr,,

(2.5)

where C(r, r’; E ) is the nonlocal, energy-dependent, non-Hermitian selfenergy introduced in the previous section. As mentioned earlier, C accounts for all exchange and correlation effects beyond the Hartree approximation. The above equation can also be written symbolically as G - ’ = (Go)-’ - C. Instead of determining the quasiparticle energies indirectly as poles of the Green function, it is more convenient to obtain these energies as solutions of the quasiparticle equation. The quasiparticle equation can be derived by inserting the spectral representation of the interacting Green function into Dyson’s equation and is given in Eq. (1.3). It is formally similar to the single-particle equation of Kohn-Sham theory (Eq. (1.9)), but the solutions of the quasiparticle equation, that is, the quasiparticle energy Ei and wave function Y i ,are physically meaningful rather than mere mathematical tools as is the case in DFT. Whenever possible one chooses as a starting point a suitable independentparticle Hamiltonian whose wave functions mi(r) are nearly identical with the quasiparticle wave functions Yi(r), which choice allows the determination of quasiparticle energies via first-order perturbation theory. Assume go= -V2/2 + V, V,,, + V,, as in Eq. (1.9) with (Qi1Yi)z 1. Then the quasiparticle energy must be determined self-consistently from

+

18

WILFRIED G . AULBUR, LARS JONSSON, A N D JOHN W. WILKINS

where the quasiparticle weight Zi is defined as

The linear expansion of the self-energy with respect to energy needed to derive the approximate relationship in Eq. (2.6) is well justified, as will be detailed in Section 11.1 1. Most current quasiparticle calculations determine quasiparticle energies via Eq. (2.6) once a suitable approximation for the self-energy has been found.

5. THEHEDINEQUATIONS AND THE GWA In principle, the exact self-energy can be obtained from a closed set of integro-differential equations -the Hedin equation^^,^ -that, in conjunction with the Dyson equation, link the single-particle propagator G, the self-energy X, and the screened interaction W to the irreducible polarizability P and the vertex function r, which will be defined now. Consider the application of a small perturbation SV,,, to the many-body system. The irreducible polarizability P is defined as the change in the density n upon a change in the total (external Hartree) field 6V = SV,,, SV,:

+

+

641) P(1,2) = SV(2) * Here 1 is a short notation for a combined space and time coordinate. Similarly, the vertex function r is given by the variation of the inverse Green function with respect to a change in the total potential or, alternatively, by the variation of the self-energy with respect to SV:

+

With the above definitions, 1 = (r ,t1 S), 6 > 0 infinitesimal, and u(l,2) as bare Coulomb interaction, Hedin's equations are +

Z(1,2) = i

s

G(1,4)W(1+,3)r(4, 2; 3)d(3,4),

J

P(1,2) = - i

s

(2.10) G(2,3)G(4,2)r(3, 4; l)d(3,4),

QUASIPARTICLE CALCULATIONS IN SOLIDS

19

From these equations, the quasiparticle self-energy can be determined iteratively, as is shown schematically in Fig. 4,panel (a). The simplest, consistent version of the Hedin equations sets the vertex function to unity and expresses the self-energy as the product of the self-consistent single-particle propagator G and the self-consistent dynamically screened interaction W! The GWA is consistent in the sense that it is a particle- and energy-conserving approximation- in other words, a conserying approximation in the Baym-Kadanoff ~ e n s e . ~The **~ ~ correGWA sponds to the first iteration of the Hedin equations; that is, higher-order vertex corrections are not included (for a discussion of vertex corrections see Section 11.14) and can be interpreted as the first-order term of an expansion of the self-energy C in terms of the screened interaction. The equations G. Baym and L. Kadano5, P h p . Rev. 124,287 (1961). G. Baym, Phys. Rev. 127, 1391 (1962).

FIG. 4. (a) Schematic representation of the iterative determination of the self-energy Z using Hedin’s equations (Eqs. (2.10)) in conjunction with Dyson’s equation (Eq. (2.5)). Entries in boxes symbolize the mathematical relations that link C, G,r, P, and W Starting with Z = 0 leads to an RPA screened interaction W and subsequently to C = GWRPA.(b) Schematic representation of the self-consistent determination of the self-energy in the GWA. P, U: and Z are constructed starting from an LDA or Hartree-Fock independent-particle propagator. Subsequently, Z updates the quasiparticle wave functions and energies and a new Green function G is determined. This process is repeated until self-consistency is reached. Most practical applications either determine only the quasiparticle energies self-consistently or do not update quasiparticle energies and wave functions at all. Self-consistency of GWA calculations is discussed in Section 11.13.

20

WILFRIED G. AULBUR, LARS JONSSON, A N D JOHN W. WILKINS

governing the GWA are

C(l,2) = iG(l,2)W(1+,2), W( 1, 2) = U( 1, 2)

+

s

W( 1,3)P(3,4)~(4, 2)d(3, 4),

P(1, 2) = -iG(l, 2)G(2, 1).

(2.11 ) (2.12) (2.13)

The dynamically screened-interaction approximation, or G WA, in principle requires a self-consistent determination of the single-particle propagator G and the screened interaction W as shown schematically in Fig. 4, panel (b). In practice, such a determination is computationally expensive and has been done only rarely as discussed in Section 11.13. Further approximations for the determination of G and W are described in Sections 11.7 and 11.8. The dynamically screened interaction W introduces energy-dependent correlation effects. In particular, we analyze in Section 11.6 the self-energy in terms of its bare-exchange and energy-dependent correlation or polarizati01-1~~ contribution. The bare-exchange contribution to C results from the bare Coulomb interaction, u(r, r’) = l/lr - r’J,whereas the energy-dependent correlation contribution results from W - u. Dynamic effects in the screening process are important since moving quasiparticles drag their polarization cloud behind them. As a consequence, dynamic screening is less efficient than static screening and directly affects quasiparticle energies (about a 20% effect, as noted, for example, in Ref. 44; see Section 11.1 1). The dynamically screened interaction approximation can be applied to weakly polarizable materials with a dielectric constant close to unity and to strongly polarizable solids such as the semiconductors Si and GaAs with dielectric constants of order ten. In the former case, the GWA reduces to Hartree-Fock theory, which is known to work well for systems such as atoms or large-band-gap, low-polarizability solids like rare-gas solids and ionic crystal^.^' In the latter case, the GWA roughly halves the HartreeFock gap and is close to experiment. The energy-dependent correlation correction to Hartree-Fock theory systematically lowers conduction-band energies and raises valence-band energies, as will be shown in detail in Section 11.6. 64 Both names are used interchangeably in the literature. We will use the name “energydependent correlation contribution” for the W - u contribution to the self-energy throughout this article. U. von Barth and L. Hedin, Nuouo Cimento 23B, 1 (1974).

‘’

QUASIPARTICLE CALCULATIONS IN SOLIDS

21

Limitations o f t h e GWA. Several important failures of the GWA have been pointed out: (1) For core-level spectra in atoms, strong electronelectron interaction breaks down the quasiparticle picture when single- and double-core holes are nearly degenerate.66 (2) s .+ d promotion energies for the second half of the iron series show large discrepancies with experiment because of the onset of strong 3d electron pairs (see Ref. 67 and Section IV.25a). (3) Exchange splittings in Ni are about a factor two larger than experiment because of the existence of strong 3d hole pairs (see Refs. 68 and 69 and Section V.26~).(4) The GWA satellite spectrum is poor. In the case of Ni, the 6-eV satellite is missing (Refs. 68, 69, 70, and 71) since the GWA does not capture strong correlations between 3d holes. The GWA plasmon satellite of a core electron is 50% too low at 1.50,,, where wpf is the plasmon frequency, rather than at o ~ Even ~ for. the~homogeneous ~ electron gas, the GWA yields a single plasmon satellite (Refs. 27, 73, 74, and 75) rather than a series of shake-up peaks.76 The same observation holds for nearly-freeelectron metals such as Na and Al.77 The neglect of vertex corrections (see Section 11.14) such as short-ranged particle-particle interactions -that is, ladder diagrams -causes the failures of RPA-based GWA calculations. Improvements for absorption spectra can be obtained by going beyond RPA-based GWA calculations and including vertex corrections in the dielectric matrix and the self-energy, as is detailed in Sections II.9b, 11.14, and IV.24. Improvement of satellite spectra and exchange splittings has recently been obtained by ab-initio cumulant expans i o n ~and ~ ~by an ab-initio T-matrix approach.’l For a review of these approaches, see, for instance, Ref. 14.

66 G . Wendin, Breakdown of the One-Electron Pictures in Photoelectron Spectra, Springer, Berlin (1981), 24. 67 E. L. Shirley and R. M. Martin, Phys. Rev. B47, 15404 (1993). 6 8 F. Aryasetiawan, Phys. Rev, B 4 6 , 13051 (1992). 6 9 F. Aryasetiawan and U. von Barth, Phys. Scripta T45, 270 (1992). 70 A. Liebsch, Phys. Rev. B 2 3 , 5203 (1981). 7 ’ M. Springer, F. Aryasetiawan, and K. Karlsson, Phys. Rev. Lett. 80, 2389 (1998), and references therein. 72 L. Hedin, B. I. Lundqvist, and S. Lundqvist, J . Res. Natl. Bur. Stand. Sect. A 74A, 417 (1970). 7 3 L. Hedin, Phys. Scripta 21, 477 (1980). 7 4 D. C. Langreth, Phys. Rev. E l , 471 (1970). 7 5 P. Minnhagen, J . Phys. C: Sol. State Phys. 8, 1535 (1975). 76 The term shake-up spectra corresponds to excitations created by a sudden change in a quantum-mechanical system. Consider, for example, the creation of a core hole in a solid via a photoemission process. The electron cloud will contract around the core hole to screen it, which leads to excitation or ionization of the residual ion- the so-called shake-up and shake-off processes. 7 7 F. Aryasetiawan, L. Hedin, and K. Karlsson, Phys. Rev. Lett. 77, 2268 (1996).

22

WILFRIED G . AULBUR, LARS JONSSON, AND JOHN W. WILKINS

The Dielectric Matrix. The screened interaction W can also be expressed in terms of the inverse dielectric matrix E - ’ , which describes screening in a solid when local fields due to density inhomogeneities and many-body effects are taken into account. Rather than using the integral equation, Eq. (2.12), W can be determined as a convolution of the inverse dielectric matrix with the bare Coulomb interaction in real space:

W(r, r’; o)=

s

E-

‘(r, r”;w)u(r”, r‘)dr”.

(2.14)

The definition of the spatial Fourier transform as (2.15)

with q as a vector in the reciprocal-space Brillouin zone and G,G’ as reciprocal lattice vectors, allows the transformation of the above convolution in real space into a simple multiplication in reciprocal space: (2.16)

The off-diagonal elements, EGGs(q;a),G # G’, describe screening caused by an inhomogeneous density distribution, that is, the local-field effects” (see Sections 11.10 and VI.31). Expressing the inverse dielectric matrix in terms of density fluctua t i o n ~ allows ~ ~ . ~a ~simple interpretation of corrections to Hartree-Fock theory by the energy-dependent correlation contribution to the self-energy (see also next section). Let IN, 0) (IN, m)) be the ground (excited) state of an N-electron system; A’, the density fluctuation operator: A’(rt) = W ( r t ) 4+(rt) - ( N , OIW(rt)+(rt)lN, 0);

(2.17)

n,(r) = ( N , mlA’(r)lN,0), a density fluctuation; and E, = EN,,- E,,,, an electron-hole energy. The inverse dielectric matrix is given as27*79 E-’(T,

r’; t ) = 6(r - r’)s(t) - i

J

u(r - r”)(N, OlT[A’(r”t)A’(r’O)]lN, 0)dr” (2.18)

7 8 Screening etTects due to inhomogeneous density distributions are in this article simply called local-field efects. Modifications of screening due to exchange and correlation beyond the Hartree approximation are called many-body local-field efects. 79 Ref. 21, Chapter 5.

23

QUASIPARTICLE CALCULATIONS IN SOLIDS

or, after Fourier transform, as27 E-

‘(r, r‘; o)= 6(r - r’)

+

s

u(r

-

r”)

C

2~, n, (r”)n%(r’)

,,,+o 0 - ( E m - i6)2

dr“. (2.19)

In actual calculations the time-ordered dielectric matrix rather than its inverse is determined from the irreducible polarizability P. The irreducible polarizability is connected to the dielectric matrix via E(r,

s

r’; 0)= 6(r - r’) -

u(r, r”)P(r”, r’; w)dr”.

(2.20)

To describe screening in solids in the time-dependent Hartreee or random phase approximation (RPA), P is replaced by the independent-particle polarizability Po, which can be obtained via the Adler-Wiser formalism.80981,82 If Qi(r) and E~ are as in Eq. (2.4) and fi are the corresponding Fermi factors, the independent-particle polarizability is given by

(2.21)

In reciprocal space one finds for the independent-particle polarizability with the factor two accounting for spin: 2

P&,(q; o)= - C (il exp(i(q

v ii’

+ G ) r)li’)(i’l +

exp( - i(q

( E ~ .- E~

-o

+ G ) .r’)li)

+ i6

and for the RPA dielectric matrix: EgL!(q;

0)= 6

4n:

~ -~1qs ~

+

G 1 2 p:G(q;

4.

To be consistent with the RPA-based GWA, the energies S. L. Adler, Phys. Rev. 126,413 (1962). M. S. Hybertsen and S. G. Louie, Phys. Rev. B35, 5585 (1987).

’’ N. Wiser, Phys. Rev. 129, 62 (1963).

(2.23) E~

should in

24

WILFRIED G . AULBUR, LARS JONSSON, AND JOHN W. WILKINS

principle be calculated from a Hartree Hamiltonian. However, an LDA spectrum is generally used instead for convenience, as described further in Section 11.7. 6. SEPARATION OF THE SELF-ENERGY

The energy-dependent GWA self-energy is a product of the propagator G and the screened interaction W in real space as a function of time (see Eq. (2.1 l)), which turns into a convolution in frequency space: eiE”G(r,r’; E

2x

+ E‘)W(r, r‘; E’)dE‘.

(2.24)

This equation is used below in conjunction with Eq. (2.4) for the independent-particle propagator to derive three possible, approximate (Go is used instead of G ) ways of separating the full self-energy C into physically meaningful pieces. Energy-Dependent Correlation + Bare Exchange. The energy-dependent correlation contribution Zc(E) describes self-energy effects beyond the bare-exchange or Hartree-Fock contribution Cx.Subtracting the HartreeFock exchange potential from the self-energy operator and using Eq. (2.19) for the inverse dielectric matrix leads to an expression of Cc(E) in terms of fluctuation potentials, V,(r) = 1u(r, r’)n,(r’)dr’, electron-hole energies, E , = EN,, - EN,,, and single-particle energies, E ~ : ” exp(iE‘G)[ W ( r , r’; E’) - u(r - r’)]Go(r,r’;E

2x

+ E’)dE‘

=?so OCC

V, ( r )V,*(r’)@ (r)@T(r’) E+E,-EEi-i6

+YCc i

,+o

V, ( r )V,* (r’)@ (r)@T(r’) E-&,-~~+i6 ’

(2.25)

The expectation value of Cc with respect to single-particle orbitals Ok= I k ) and 0,= Il) equals

Energy-dependent correlations decrease the Hartree-Fock band gap by raising the valence-band energy and lowering the conduction-band energy,

25

QUASIPARTICLE CALCULATIONS IN SOLIDS

as can be seen by considering the above equations. For a valence electron in the highest occupied state, Iub), the largest contribution to (ublCC(E = E J u b ) comes from the sum over the occupied states. For these states IE,, - E J is small compared to the electron-hole energy E,, which is on the scale of the Hartree-Fock energy gap. Hence, the polarization contribution to the self-energy shifts the valence-band maximum upward in energy. For the conduction-band minimum, Icb), the largest contribution to (cblCC(E = E,,)lcb) comes from the unoccupied states-that is, the second term in Eq. (2.26) -and hence leads to a reduction in the conduction-band energy. The correlation contribution to the self-energy of an electron in a highly excited Rydberg state reduces via Eq. (2.26) to the classical Coulomb energy of the Rydberg electron in the field of the induced core d i p ~ l e . ~ Assume *~~.~’ that the density fluctuations react instantaneously to the presence of the highly excited electron. The Fourier transform of the density fluctuations is therefore independent of frequency, which translates to E - E~ = 0 in Eq. (2.25). Summing over i and using completeness gives the following expression for the self-energy: CC(r,r‘; E

= 0)

z -6(r

-

r’)

1 K(r)J‘,*(r) r n + ~

Em

a

--

2r4

6(r - r’), (2.27)

where c1 is the dipole polarizability of the ion core around which the Rydberg electron cycles. The last approximate equality follows from a dipole expansion of the Coulomb potential inside the expression of the fluctuation potential in which one keeps only the lowest order, that is, dipole terms. The above expression equals the classical Coulomb energy of the Rydberg electron caused by the field of the induced dipole, which is adiabatically switched on in the ion core. The GWA recovers the relevant classical limit for this special case. Further examples whose essential physics is contained in the GWA are (1) the energy loss per unit time of a fast electron in an electron gas and (2) the self-energy shift of a core electron in a solid. These limiting cases are detailed in Refs. 4, 26, and 27 and are not discussed here. Coulomb Hole Screened Exchange (COHSEX). The COHSEX approximation is a physically motivated separation of the self-energy into a Coulomb-hole (COH) part and a screened-exchange (SEX) part whose static limit (1) has been used extensively to correct Hartree-Fock band structures; (2) produces direct band gaps to within 20% of experiment but gives a less reliable account of indirect band gaps (see Section 11.9); and (3) allows an examination of local-field effects on, and the energy dependence of, the self-energy. Consider the convolution in energy space of G and W given by Eq. (2.24). This convolution can be determined formally by

+

26

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

introducing the spectral function B(r, r’; E ) of the screened interaction W in analogy to the spectral representation of the Green function in Eq. (2.2). One can express the screened interaction in terms of B(r, r’; E ) - B is an antisymmetric function of energy -as4 2E’B(r, r’; E‘) dE‘, E 2 - (E’ - i6)2

W(r, r’; E ) = u(r, r’) +

(2.28)

where 6 = 0’. The real part of the self-energy C can then be written as the sum of a Coulomb-hole contribution CCoH arising from the poles of the screened interaction and a screened-exchange contribution CSEXarising from the poles of the Green function: ReC

+ CSEX.

(2.29)

= CCoH

Using the independent-particle Green function G o (Eq. (2.4)), one can show that ( P stands for principal part) CCoH(r,r’; E ) =

:j

1Oi(r)@?(r’)P i

dE

B(r, r‘; E’) E-q-E’

(2.30)

OCC

CSEX(r,r’; E ) = -

1Oi(r)@?(r’)Re(W(r, r’; E

-

q)),

(2.31)

i

which illustrates the interpretation of CSEXas a dynamically screened exchange interaction. The static COHSEX approximation assumes, in contrast to the GWA, that the screened interaction is instantaneous, that is, that retardation effects can be neglected. Hence, W is proportional to a &function in time and equal to a constant in energy space. In the static COHSEX approximation, the Coulomb-hole contribution to the self-energy equals the interaction of an electron with the induced potential due to rearrangement of the other electrons. The screened-exchange contribution equals the statically screened-exchange energy of the quasiparticle. To see this, one takes the limit of E - E~ + 0. In Eq. (2.31) this limit implies the neglect of E - E, in comparison to the poles of the screened interaction W which are given by the plasmon energies. This approximation should be reasonable for states close to the Fermi energy, which is verified by actual calculations (see Section II.9a). One finds CCoH(r,r’)

= +S(r - r’)[W(r,

r’; E = 0) - u(r, r’)],

(2.32)

QUASIPARTICLE CALCULATIONS IN SOLIDS

27

With these equations, the interpretation of CCoHand CSEXis clear. The factor 1/2 in CCoHresults from the adiabatic turn-on of the interaction. Core + Vulence. Under the assumption that one can energetically and spatially separate core electrons from valence electrons in a material, the self-energy equals the sum of the bare core-exchange potential, the screened core-polarization potential, and the valence electron self-energy,all of which will be defined below. Separating the Green function, the screened interaction, and the irreducible polarizability into core and valence contributions,

G

=

G,

+ G,,

w = w, + w,,

P = P, + P,,

(2.34)

the self-energy X in the GWA can be expressed as C = GW = G,W

+ G,W, + G,W, x G,W + G,W,P,W, + G,W,.

(2.35)

Here, we assume that the core polarizability is negligible in comparison to the valence polarizability, and we expand W in terms of the core polarizability: W = W , + W , P , W , + W , P , W , P , W , + - ~ ~ ~ WW,P,W,. u+

(2.36)

The three terms in Eq. (2.35) can be identified as the exchange potential from the core (G,W x G,u due to the large energy denominators involved, compare Refs. 83,84, and 85), as the screened polarization potential from the core ( G , W,P, W,), and as the self-energy of the valence electrons ( G , W,). A local density approximation of the bare core-exchange and screened core-polarization potential leads in general to small errors except for deviations on the order of 0.3 to 0.4eV in the band gap of materials with large, soft cores whose lowest conduction-band state is localized on the ionic cores. The nonlocal bare core-exchange and the screened, energy-dependent core-polarization potential are generally small, as discussed, for example, by Hybertsen and Louie.44Estimates for atomic Na4 or solid AlS6 indicate that both terms contribute approximately 1 eV, relative to the bottom of the valence band, to the quasiparticle energies. In GWA calculations based on the LDA the proper core-valence terms are replaced by an LDA exchangecorrelation potential, which leads to a much smaller error. For states that are localized on large, soft cores, such as the r2, conduction-band state in 83 84

” 86

W. Brinkman and B. Goodman, Phys. Rev. 149, 597 (1966). L. Hedin, Arkiu Fysik 30, 231 (1965). J. C. Phillips, Phys. Rev. 123,420 (1961). G. Arbman and U. von Barth, J . Phys. F 5, 1155 (1975).

28

WILFRIED G . AULBUR, LARS JONSSON, A N D J O H N W. WILKINS

Ge, the local, energy-independent approximation to the core-valence exchange and correlation breaks down and leads to states that are systematially too low in energy. Better treatments of core-valence exchange and correlation via (1) use of core-polarization potentials and (2) explicit inclusion of core states in the valence band are described in Section 11.12.

7. DETERMINATION OF THE SINGLE-PARTICLE GREEN FUNCTION Most current GWA calculations do not attempt a numerically expensive, self-consistent calculation of G and W but determine good approximations for the single-particle propagator and the screened interaction separately. That is, these calculations adopt a “best G, best W’ philosophy. Once the “best” G and the “best” W are chosen the self-energy is determined via Eq. (2.11) without further iteration. The main task is to find a single-particle Hamiltonian fi0 whose wave functions and energies result in a good single-particle Green function (Eq. (2.4)) and a good screened interaction (Eqs. (2.14), (2.20), and (2.22)). The common choice for fi0 is LDA or Hartree-Fock. Although this strategy does not correspond to a consistent determination of self-energy corrections starting from Hartree theory,” it has been applied with considerable success for the determination of band structures of semiconducting and metallic materials (see Sections IV and V). Calculated quasiparticle corrections to an LDA band structure generally agree well with experiment and have the additional advantage that the LDA potential is local -allowing numerically inexpensive calculations -and that the LDA wave functions are close to quasiparticle wave functions. Many-body corrections to the LDA Hamiltonian are determined by the expectation value of the operator C(r, r’; E ) - Vx.(r)6(r - r’). An overlap close to unity between LDA and quasiparticle wave functions has, for example, been reported for Si.44*90As a consequence, the self-energy operator is for all practical purposes diagonal in the LDA basis, which can be motivated by Eq. (2.26). Diagonal matrix elements (klClk) contain sums over positive definite matrix elements l(kl Vmli)I2, whereas off-diagonal matrix elements (klCII), k # 1, contain sums over terms proportional to (kl Vmli)(il V,*l1) -that is, over terms with varying phases- which potenThe determination of Z starting from Hartree theory requires self-consistency -the very complication that the “best G, best W” philosophy tries to avoid-since Hartree band structures and wave functions for semiconductors and insulators are qualitatively wrong. See Refs. 88 and 89. R. Daling and W. van Haeringen, Phys. Rev. B40, 11659 (1989). 89 R. Del Sole, L. Reining, and R. W. Godby, Phys. Rev. B49,8024 (1994). 90 F. Gygi and A. Baldereschi, Phys. Rev. B34, 4405 (1986).

QUASIPARTICLE CALCULATIONS IN SOLIDS

29

tially cancel.*' Note, however, that overlaps small compared to unity have been observed, for instance, in transition metals68 and transition-metal oxides (see Refs. 50, 54, 55, and 91). The disadvantage of using an independent-particle Hamiltonian in the LDA is that density functional theory is nonperturbative. Systematic improvements of GWA calculations based on LDA Hamiltonians are only feasible once self-consistency is achieved. In contrast to LDA, Hartree-Fock calculations provide a good singleparticle basis only for weakly polarizable materials and are numerically expensive due to the nonlocality of the exchange kernel. Hartree-Fock Hamiltonians were used as independent-particle Hamiltonians in the 1970s and 1980s to study self-energy corrections in rare earth and ionic solids (see Section II.9a). Current uses of Hartree-Fock wave functions and energies as input for GWA calculations are limited to the study of trans-polyacetylene (see Section IV.25b).

OF THE DYNAMICALLY SCREENED INTERACTION 8. DETERMINATION

The determination of the dynamically screened interaction and the dielectric matrix in reciprocal space as a function of frequency is numerically expensive and can be drastically simplified by (1) modeling EGG,(q; w = 0) via model dielectric matrices, and (2) modeling the frequency dependence of EGC'(q; w) via plasmon-pole models. The computational efficiency of model dielectric functions and matrices is offset by a loss in accuracy, which limits the use of model dielectric functions to systems whose quasiparticle energy shifts are larger than the accuracy of the model. The approximation of the imaginary part of the inverse dielectric matrix as a function of frequency by a simple pole, which is the common approximation made in plasmon-pole models (see Fig. 5), is less severe. As discussed below and in connection with Table 21, plasmon-pole models are accurate to within a few hundredths of an eV for states close to the Fermi level and to within a few tenths of an eV for states whose energy is on the order of the plasmon energy of the given material. This statement is true if the static dielectric matrix, which is an input parameter to all plasmon-pole models, is determined from firstprinciples using D F T rather than from model dielectric matrices. This review focuses on the results of accurate first-principles calculations of the self-energy Z since these calculations form a consistent framework for the discussion of self-energy effects. Results using model dielectric functions 91 S. Massidda, A. Continenza, M. Posternak, and A. Baldereschi, Phys. Rev. Lett. 74, 2323 (1995).

30

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Om2

t

0.0

-

-0.2 e n

3 v

-

0

3

GPP

i

.- 1

f-cw 8-8-cur

-2

w

.

-0.4 0

‘

I

Y

w

3

M

W W

-

---- CPP Model

s

Walter and &hen

n

n

.. ’.

4-0

2

*

4

8

8

1

0

1

0.6 I 0.4 0.2

0.0 -0.2 -0.4 ---*

-0.6 -0.4 0

10

M

90

40

50

0

CFT Model 4

8

12

16

fiw (eV)

fi w’ (ev)

FIG.5. Left panel: Numerical results obtained using the empirical pseudopotential technique for various elements of the real ( E , ) and imaginary ( E J parts of the dielectric matrix EGG(q;CO) as a function of frequency for Si (solid line). The real part of the Hybertsen-Louie plasmon-pole model is plotted for comparison (dashed line). The model replaces the peaked structure in E ~ ( C O ) by a &function, is constructed to describe the limits w -P 0 and w + co correctly, and breaks down for intermediate frequencies. Right panel: Real part of the inverse dielectric function of Si derived from the numerical, empirical-pseudopotential-based results by Walter and Cohen9’ in comparison to the Hybertsen-Louie plasmon-pole model. The average behavior of the inverse dielectric function is captured rather well by the plasmon-pole model for energies below the plasmon energy of Si (up,= 16.7 eV). (Adapted from Ref. 44.)

QUASIPARTICLE CALCULATIONS IN SOLIDS

31

will be used as supplementary material or in cases where no first-principles calculations exist. Similar to model dielectric functions, models for the self-energy C or the band-gap correction A are based on well-founded physical insights into the effects of correlation in solids, but their accuracy is often limited, leading to semiquantitative results. A review of simplified GWA calculations based on models for either the dielectric function or the self-energycan be found in Ref. 14. a. Model Dielectric Functions Models for the static dielectric matrix must describe (1) plasmon excitations, which dominate screening for small reciprocal lattice vectors, and (2) electron-hole excitations, which fulfill the f-sum rule at large reciprocal lattice vectors where screening is less effective. Plasmons are collective coherent excitations of the electron gas which result from screening and can be visualized as macroscopic density fluctuations. In Fig. 6 areas in (9, o)-space in which the imaginary part of the dielectric function of the 92

J. P. Walter and M. L. Cohen, Phys. Rev. B5,3101 (1972).

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

t

..

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

$?

wE

I

ubz Excitatiom... I

4, 2kF - Reciprocal lattice vector q --+-

FIG. 6. Spectrum of excitation energies versus wave vector transfer q for a homogeneous, interacting electron gas. The upper (lower) boundary of the single-particle excitations is O.S[(k, + q)2 - kb] ( O . S [ ( - k , + q)2 - kb]). In simple plasmon-pole models this spectrum is replaced by a single mode wp,(q)indicated by the dotted line. The real plasmon mode (full line) is the dominant excitation for small q-vectors (0 ,< q < 4,). Beyond qc, the plasmon is heavily damped (dashed line). (Adapted from Ref. 93.)

32

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

interacting homogeneous electron gas is nonzero are shown schematically. Besides the continuum of particle-hole excitations, the imaginary part is nonzero at the plasmon energy o,,(q).For small wave vectors q, 0 d q d q,, where q, is a cut-off vector, the plasmon is the dominant excitation and exhausts the f-sum rule to more than 90%.93 Close to q, the plasmon oscillator strength drops rapidly to zero and particle-hole excitations become important. For large wave vectors ( q 2 2k,) the spectrum converges towards the independent-particle spectrum since screening becomes less and less effective. A good model should therefore interpolate between the plasmon (0 d q d 4,) and the particle-hole ( q 2 2k,) excitation channels. Model dielectric matrices for real crystals must also capture important features that determine screening, such as density inhomogeneities and the existence of an energy gap in the excitation spectrum. Nonzero off-diagonal matrix elements are needed to account for local-field effects (see Sections 11.10 and VI.31). Also, the dynamical response of an electron to fluctuations in the density n(r) from its mean value is qualitatively different in a metal and a semiconductor or insulator. In a metal the Coulomb potential of a particle is screened very effectively by the electron gas, and the long-range Coulomb potential is turned into a short-range interaction that depends on the local electron density. In a semiconductor or insulator the Coulomb potential is not completely screened by the electron gas. For example, consider the case of an additional electron localized in a bond (i.e., an N + 1- rather than the original N-electron system). This electron will induce dipoles in neighboring bonds,94 which will in turn interact with and screen the electron. This screening mechanism is nonlocal in real space.9s Clearly, model dielectric matrices must account for the gap in the excitation spectrum and the qualitatively different screening of insulators. In the context of COHSEX calculations (see Section II.9), static model dielectric functions are often used. Fair agreement between experimental and calculated direct gaps is achieved due to a significant error cancellation between the neglect of local-field effects and the neglect of the energy dependence of dielectric screening. The model dielectric function used by Lundqvist9' for the determination of self-energy corrections of jellium and simple metals (e.g., potassium; see, 93 94

A. Overhauser, Phys. Rev. 8 3 , 1888 (1971). Ref. 19, 56-57.

9 5 There are simplified tight-binding models that sum all dipole contributions to get an efective screening potential that depends only on the local electron density; see, for instance, Refs. 96 and 97. 96 W. Hanke and L. J. Sham, Phys. Rev. B38, 13361 (1988). 97 P. A. Sterne and J. C. Inkson, J. Phys. C Sol. State Phys. 17, 1497 (1984). 98 B. I . Lundqvist, Phys. Kondens. Muter. 6, 206 (1967).

33

QUASIPARTICLE CALCULATIONS IN SOLIDS

for instance, Ref. 99) (1) reduces to a single plasmon in the limit q 0; (2) describes independent-particle excitations for large q; (3) reduces to Thomas-Fermi screening for intermediate q; and (4)fulfills important sum rules like the f-sum rule. L u n d q v i ~ t ' ssingle-plasmon-pole-model ~~ dielectric function is -+

&(q,0)= 1 -

wz

+

Wpzl - w2(q)

(2.37)

'

where wpl = (47cn0(0))"2 is the plasmon frequency of the system and no(0)is the G = 0 component of the unperturbed density. With uF as Fermi velocity, 4 q 2 / 3 +(q2/2)2. the plasmon dispersion is approximated by w2(q)= The continuous spectrum of density fluctuations is substituted by a single plasmon pole, and electron-hole excitations in particular are neglected. In the case of the homogeneous electron gas, L u n d q v i ~ t ~ ~ *shows ' ~ ~ ~that '~' electron-hole excitations are of minor importance for the determination of the self-energy C. In inhomogeneous semiconductors such as Si, quasiparticle energies determined using plasmon-pole models (e.g., Ref. 42) differ, by no more than 50meV for the direct gap at r, from those determined by taking the full ~ ~ *21 ~ shows ~ ~ that frequency dependence of &GG'((I; w) into a c c o ~ n t . ' Table the valence-band width in Si is about 0.3 eV smaller when the full frequency dependence is taken into account (compare calculations by Fleszar and Hankelo2 and Rieger et a1.1°3 with plasmon-pole model calculations). The effect of particle-hole excitations on quasiparticle energies is not significantly larger than typical numerical uncertainties of GWA calculations (see Sections 111.15, 111.16, and 111.18). A modification of the Lundqvist model for s e m i c o n d u ~ t o r(1) s~~ reduces ~ to the static dielectric constant for q -+ 0; (2) reproduces free-electron behavior at large q; (3) describes a modified Thomas-Fermi screening at intermediate q ; and (4) allows for the analytic evaluation of the static Coulomb-hole contribution to the self-energy. Capellini et u1.'04 suggested the following expression for the dielectric function of a semiconductor:

+

M. Schreiber and H. Bross, J . Phys. F : Met. Phys. 13, 1895 (1983). B. I. Lundqvist, Phys. Kondens. M a t . 6, 193 (1967). B. 1. Lundqvist, Phys. Kondens. M a t . 7, 117 (1968). A. Fleszar and W. Hanke, Phys. Rev. B 56, 10228 (1997). 103 M. M. Rieger, L. Steinbeck, I. D. White, H. N. Rojas, and R. W. Godby, Cond-Mat/ 9805246. G . Cappellini, R. Del Sole, L. Reining, and F. Bechstedt, Phys. Rev. B 47, 9892 (1993). 99

loo

34

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

where the Thomas-Fermi wave vector k,, depends on the electron density n and ~(0) is the static RPA dielectric constant. The parameter a is taken as a fitting parameter to optimize agreement with first-principles dielectric constants and turns out to be approximately constant for small- and medium-gap semiconductors.104As is shown in Fig. 7 for Si and GaAs, the model by Cappellini et al. gives a better description of the full RPA dielectric function than the Levine-Louie model, which is introduced below. However, in contrast to the Levine-Louie model, the model of Cappellini et al. does not reduce to the RPA dielectric function in the case of metallic screening.

\.

..-

FIG. 7. Static dielectric function E(q. w = 0) as a function of the reciprocal lattice vector q for Si and GaAs. The model dielectric function of Cappellini et (solid line) compares better with RPA results of Walter and C ~ h e n "(closed ~ boxes for q along the (111) direction, open boxes for q along the (100) direction) than the Levine-Louie model dielectric function (dashed line).lo6 RPA results by Baldereschi and Tosatti (stars)'" for large q seem to favor the Levine-Louie model over the model by Cappellini et al. The parameters a and ~(0) are the same as in Eq. (2.38). (Adapted from Ref. 104.)

QUASIPARTICLE CALCULATIONS IN SOLIDS

35

For an application of the model in the context of GWA see, for example, Ref 108. COHSEX calculations for insulators in the 1970s and early 1980s (see Section II.9a) often relied on the Penn dielectric function'09 and its variations, which describe the dielectric response of an isotropic, threedimensional insulating electron gas. Perm'" used the Ehrenreich-Cohen formula"0 for the dielectric response function. With Egapas an average optical gap adjusted to fit E ( q = 0) with the experimental dielectric constant, E F as the Fermi energy, and k, as the Fermi vector, the numerical results can be fitted with an interpolation formula:"'*''2 (2.39) where the factor F is given by (2.40)

Modifications of the Penn model based on a more complicated band structure have been suggested"'*"2 and applied to static COHSEX calculations (see Section II.9a). In the same spirit as the Penn model, the Levine-Louie dielectric function106 modifies the RPA dielectric function by an ad-hoc introduction of an energy gap in the spectrum of the homogeneous electron gas. This model dielectric function (1) interpolates smoothly between the screening properties of a metal and an insulator; (2) reproduces well the numerical results of Walter and Cohen"' for the static diagonal dielectric matrix in Si, as shown in Fig. 7; (3) fulfills important sum rules such as the f-sum rule; (4) has the correct long-wavelength behavior; and (5) neglects local-field effects. Since semiconductors and insulators cannot absorb light with energies below the fundamental band gap, Levine and Louie introduced the band gap in an ad-hoc fashion into the imaginary part of the RPA dielectric

lo'

lo*

log

'I2

J. P. Walter and M. L. Cohen, Phys. Rev. 8 2 , 1821 (1970). Z. H. Levine and S. G. Louie, Phys. Rev. B25, 6310 (1982). A. Baldereschi and E. Tosatti, Phys. Rev. B 17, 4710 (1978). F. Bechstedt, R. Del Sole, G. Cappellini, and L. Reining, Sol. Srure Comm. 84,765 (1992). D. R. Penn, Phys. Rev. 128, 2093 (1962). H. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959). J. L. Fry, Phys. Rev. 179, 892 (1969). N. 0. Lipari, J . Chem. Phys. 53, 1040 (1970).

36

WILFRIED G. AULBUR, LARS JONSSON, A N D J O H N W. WILKINS

function via

sgn(o-)

= sgn(o).

(2.41)

This approach results in a diagonal model dielectric function that depends on only two parameters: the average electron density given by the electron gas parameter rs, which determines the plasmon frequency opl,and the lowest excitation frequency ogap. In the long-wavelength limit ( q 0) one finds -+

E(q

-+

0,o) = 1

of, + oiap - o2.

(2.42)

For a known dielectric constant the above equation at o = 0 defines ogap. To include local-field effects -in particular the variation of the screening hole depth as a function of the location of an added electron-and the correct symmetry of the screening potential under exchange of r and r’, Hybertsen and Louie’I3 considered a screening potential that is the sum of the local Levine-Louie potentials evaluated at r and r’ (see Eq. (2.45) below). They evaluated the homogeneous Levine-Louie screened potential,

at the local density determined by r,(r’) and used the Levine-Louie gap parameter to ensure correct long-range screening:

Here ~ ( 0is) the static dielectric constant and an input parameter to the model. Hybertsen and Louie’ explicitly symmetrized the screening potential: WHL(r,r’) = l[WLL(r- r‘; rs(r’)) ‘I3

‘I4

+ WLL(r’- r; rs(r))],

M. S. Hybertsen and S. G. Louie, Phys. Rev. B37, 2733 (1988). M. Rohlfing, P. Kriiger, and J. Pollmann, Phys. Rev. B48, 17791 (1993).

(2.45)

31

QUASIPARTICLE CALCULATIONS IN SOLIDS

which leads to the following expression in reciprocal space:

[EE&]

-

+

;l

+

(q ;o = O)u(q G ) = - u(q G )

+ u(q+G')

s

J

[E~J -

'(Iq + GI; rs(r'))e'(G-G).dr' r'

1.

[ ~ ~ ~ ] - l ( l q + Gr,(r))e'(G'-G)'rdr 'I;

(2.46)

The diagonal part of the Hybertsen-Louie dielectric matrix is given as the average over the local Levine-Louie screening response at different points in the crystal. The Hybertsen-Levine-Louie model has been applied to a variety of bulk semiconductors and interfaces (see Sections IV. 19 to IV.22). It generally reproduces results of full RPA calculations for quasiparticle energies to within 0.1 to 0.4 eV for states close to the band gap.' 1 3 s 1 l 4 b. Plasmon-Pole Models The imaginary part of the important elements of the dielectric matrix has a peaked structure as a function of frequency, which plasmon-pole models approximate by a &function characterized by two parameters: the effective strength and the effective frequency of the plasmon excitation. Together with the independent-particle propagator (Eq. (2.4)), whose frequency dependence is straightforward, plasmon-pole models allow an analytic evaluation of the convolution of G and W in frequency space (Eq. (2.24)). As a consequence, the numerically expensive integration in Eq. (2.24) is avoided. Plasmon-pole models give a good description of both the low-energy behavior of the dielectric matrix -via reproduction of the static limit -and the high-energy behavior of the dielectric matrix -via reproduction of the first frequency moment. As a consequence, energy integrations of the screened interaction are sufficiently accurate for the determination of quasiparticle energies and effective masses." The effective strength and frequency of the plasmon excitation are determined by forcing the model to reproduce the static dielectric matrix in the zero-frequency limit and by using Johnson's sum rule.' l 7 The latter sum rule connects the first 6+1

l 5 N . H. March, Electron Correlation in Molecules and Condensed Phases, Plenum Press, New York (1996), 132. D. L. Johnson, Phys. Rev. B 9 , 4475 (1974); M. Taut, J. Phys. C: Sol. State Phys. 18, 2677 (1985); and Ref. 44. 'I7 The use of Johnson's sum rule in conjunction with nonlocal pseudopotentials is not justified, as pointed out for instance in Refs. 44 and 118. However, band-gap errors due to this procedure are only about 15 meV for Si, as reported in Ref. 119. G . E. Engel and B. Farid, Phys. Rev. B46, 15812 (1992). R. T. M. Ummels, P. A. Bobbert, W. van Haeringen, Phys. Rev. B57 11962 (1998).

38

WILFRIED G. AULBUR, LARS JONSSON, A N D JOHN W. WILKINS

frequency moment of the full polarizability

x = 6n/61/,,,,'20 (2.47)

with the ground-state density no(G)of the crystal, X&!AQ)

x

=

- 5 (9

+ G ).(4 + G')no(G- G').

(2.48)

In RPA, x is related to the irreducible polarizability P = Po in matrix notation via x(q, o)= P(q, 0)[1 - u(q)P(q, o)]-'. Quasiparticle energies in jellium calculated with the RPA dielectric function or with the Lundqvist plasmon-pole r n ~ d e l ~ ~are ~ 'very ~ ~ close ~ ' ~ ' (see previous subsection), which indicates that the neglect of electron-hole excitations in plasmon-pole models is justified for energies close to the Fermi energy. Similar observations hold for semiconductors, as discussed above. The plasmon-pole assumption, that the only possible quasiparticle decay mechanism -described by the imaginary part of the dielectric matrix -is due to quasiparticle scattering off plasmon excitations, results in finite quasiparticle lifetimes only at plasmon frequencies. Plasmon-pole models fail to give good descriptions of systems whose imaginary self-energy is large.' l 5 In particular, low-lying valence states and hence the valence band width are not described accurately. For example, Table 21 shows that taking into account the full frequency dependence of the dielectric matrix leads to a valence-band narrowing of about 0.3 eV in Si compared to a plasmon-pole model calculation (compare the frequency-dependent results of Refs. 102 and 103 with results using plasmon-pole approximations such as Ref. 42; see also Ref. 89). Hybertsen and approximated each matrix element of the dielectric matrix by a plasmon-pole model, which results in (1) N 2 plasmon-pole parameters ( N = size of the dielectric matrix), (2) a nontrivial extension of the model to systems without inversion symmetry,'2' and (3) unphysical solutions with imaginary plasmon-pole energies for some off-diagonal matrix elements. Rather than concentrate on the details of the HybertsenLouie plasmon-pole model, we consider two alternative plasmon-pole models suggested by von der Linden and Horsch'22 and Engel and G. E. Engel and B. Farid, Phys. Rev. B47, 15931 (1993). S.B. Zhang, D. Tomanek, M. L. Cohen, S. G. Louie, and M. S. Hybertsen, Phys. Rev. 840, 3162 (1989). 12*

W. von der Linden and P. Horsch, Phys. Rev. 837,8351 (1988).

39

QUASIPARTICLE CALCULATIONS IN SOLIDS

Farid.' 2o Both models give identical expressions for the expectation value of the self-energy operator provided that appropriately scaled plasmon-pole eigenvalues and eigenvectors are introduced (see below and Sections 111.15 and 111.16). Von der Linden and Horsch. Von der Linden and Horsch'22 considered an N-parameter plasmon-pole model derived from the eigenvalue decomposition of the symmetrized dielectric matrix E under the assumption that the energy dependence of E is contained in its eigenvalues only. Define the following symmetrized, Hermitian dielectric matrix, (2.49) whose inverse 2-l has the eigenvalue d e c o m p ~ s i t i o n , ' ~ ~ * ' ~ ~

where p numbers the real, positive eigenvalues &'(a) and the corresponding eigenvectors IOw(o)). Assume that the frequency dependence of the inverse dielectric matrix is solely contained in the eigenvalues (lOw(o)) = IOw(o = 0))) and is of the

where the plasmon pole strength, zw = 1 - h,'(O), is determined by comparison with the static dielectric matrix. With the definition of the scaled plasmon-pole eigenvectors as (2.52)

and the "first-moment" matrix, LGG'(q)

2

= (q $. G ) .(q + G')no(G- G') = -xg&(q), 71

(2.53)

one can express the plasmon-pole frequencies using Johnson's sum rule' lZ3

R. Hott, Phys. Rev. B44, 1057 (1991).

l6

40

WILFRIED G. AULBUR, LARS JBNSSON, A N D JOHN W. WILKINS

as

(2.54) Subsequently, the energy integration of Eq. (2.24) can be done and the result is given in Sections 111.15 and 111.16 for a plane-wave basis. Engel and Farid. Engel and Farid'a' derived an N-parameter plasmonpole model whose eigenvalues and eigenvectors are frequency dependent by explicitly constructing an approximation X to the full polarizability x, which approximation is exact in the static and the high-frequency limits. With X, x, and L-' (the inverse of the first-moment matrix L in Eq. (2.53)) as matrices in the reciprocal lattice vectors G and G', the approximation X to the full polarizability x can be expressed as"'

The Engel-Farid plasmon-pole model can be obtained by diagonalizing the above equation, which is equivalent to the solution of the following generalized eigenvalue problem: (2.56)

The eigenvectors x, are normalized as follows: (2.57) Defining the scaled eigenvectors y, as (2.58) one finds for the spectral representation of

X (2.59)

41

QUASIPARTICLE CALCULATIONS IN SOLIDS

and its inverse

These equations define the Engel-Farid plasmon-pole model. For future reference in the determination of the self-energy using the Engel-Farid plasmon-pole model, we define scaled plasmon-pole eigenvectors as (2.61)

An interpretation of the plasmon-pole eigenvalues as plasmon energies, as suggested by Engel and Farid, leads to an agreement between theory and experiment to within 10% for plasmon energies o,(O) and to within 30% for the small q-vector dispersion coefficient a, o,(q) z o,(O) alql', in the case of the semiconductors Si, Ge, GaAs, Sic, and GaN. This is seen in Table 3. March and Tosi131 showed that the plasmon frequencies of a system are the zeros of the determinant of the inverse full polarizability x- ':

+

(2.62) TABLE3. THEORETICAL AND EXPERIMENTAL PLASMON ENERGIES AND DISPERSION COEFFICIENTS Si, Ge, GaAs, Sic, AND GaN. THETHEORETICAL VALUESOF ENGELAND FARIDFOR Si'*O AND BACKES et a[. FOR SicL3' ARE LISTED I N PARENTHESES. AGREEMENTBETWEEN THE DIFFERENT CALCULATIONS IS WITHIN A FEW PERCENT. THE THEORETICAL AND EXPERIMENTAL PLASMON ENERGIES AGREE TO WITHIN 10% AND THE DISPERSION COEFFICIENTS TO WITHIN 30%. FOR

a

%(O)

THEORY'

Si

EXPT.~

Ge

15.7 (15.9) 14.2

GaAs

14.1

15.7

Sic GaN

22.5 (22.4) 19.4

22.1

16.7' 15.9-16.5

THEORY' 0.36 (0.34) 0.38 [lo01 0.35 [l 1 13 0.25 [ l l O ] 0.49

EXPT.~

0.41 0.38'

0.44' 0.61 [loo] 0.57 [llO] 0.53 [lll]'

"Ref. 42 except for values in parentheses; bRef. 124 for ~ ~ ( and 0 ) Ref. 125 for a unless otherwise noted; 'Ref. 126; 'Ref. 127; 'Ref. 128; /Ref. 129.

42

WILFRIED G . AULBUR, LARS JONSSON, AND JOHN W. WILKINS

This observation in conjunction with Eq. (2.60) led Engel and FaridI2’ to interpret the eigenenergies of Eq. (2.56) as plasmon energies and their small q-vector dispersion coefficient as plasmon dispersion coefficient. 9. EARLYQUASIPARTICLE CALCULATIONS This section reviews the history of GWA calculations and outlines some related approaches. As in the section on the model dielectric matrices, we concentrate on work done on semiconductors and insulators. We focus our survey on the time span between 1970 and 1986. Before 1970 most of the GWA work was done on the homogeneous electron gas. The only work on semiconductors is the paper by Brinkman and Goodman.83 These papers are thoroughly reviewed in Refs. 4 and 132. In 1985, first-principles quasiparticle calculations were done by Hybertsen and Louie’ and in 1986 by Godby, Schluter, and Sham.6 First-principles GWA calculations are discussed at length starting in Section 11.10. a. Static COHSEX Calculations General Considerations. The error cancellation between the neglect of dynamical effects in the static COHSEX approximation and the neglect of local-field effects due to the use of a diagonal model dielectric function- the common characteristic of all quasiparticle calculations on bulk insulators in the 1970s and early 1980s-results in direct quasiparticle band gaps that agree well with experiment, as shown in Table 4. Indirect band gaps, however, differ significantly from experimental values. In the case of silithe first few conduction bands at r are at 3.24 and 4.41 eV in comparison to experimental values of 3.4 and 4.2 eV. The indirect band gap is 0.68 eV and hence underestimates the experimental value of 1.17 eV by about 0.5 eV. As a consequence, the static COHSEX approximation cannot replace the dynamically screened interaction approximation if one is interested in the full band structure. H. Raether, Excitation of Plasmons and Interband Transitions by Electrons. Springer Tracts in Modem Physics, vol. 88, Springer-Verlag, Berlin (1980), Table 5.1. Ref. 124, Table 7.4. J. Stiebling and H. Raether, Phys. Rev. Lett 40,1293 (1978). 12’ H. Watanabe, J. Phys. SOC.Jpn. 11, 112 (1956). 12* C. von Festenberg, Z . Phys. 214,464 (1968). 12’ R. Manzke, J. Phys. C: Sol. State Phys. 13, 911 (1980). I 3 O W. H. Backes, P. A. Bobbert, and W. van Haenngen, Phys. Rev. B 51,4950 (1995). 13’ N. H. March and M. P. Tosi, Proc. R SOC.Lond. ,4330, 373 (1972). L. Hedin and B. I. Lundqvist, J . Phys. C Sol. State Phys. 4, 2064 (1971).

’’’

”’

QUASIPARTICLE CALCULATIONS IN SOLIDS

43

TABLE4. COMPARISON OF TECHNICAL ASPECTSOF COHSEX CALCULATIONS BY LIPARIet a/. REFS. 133, 134, 135, AND 136, -,I3’ BRENFX,’~*.’~~ BARONIet aL,140.141 AND GYGIAND BALDERESCHl,90 WITH GWA CALCULATIONSBY HYBERTSEN AND LOUIE’ AND GODBY, SCHLmR, AND SHAM.^ COREELECTRONS ARE EITHERTREATEDON AN LDA-PSEUDOPOTENTIAL (PSP) OR HARTREE-FOCK (HF, NEGLECTOF CORE-VALENCE POLARIZATION) LEVEL,OR TREATED ON THE S m LEVELAS THE VALENCE ELECTRONS. MOST COH_SEX CALCULATIONS START FROM A HARTREE-FOCK INDEPENDENT-PARTICLE HAMILTONIAN H o , WHICH IS APPROPRIATE THE S ~ DIELECTRIC L CONSTANTS OF THE MATERIALS. GWA CALCULATIONS USE CONSIDERING LDA AS AN INDEPENDENT-PARTICLE HAMILTONIAN INSTEAD.AGREEMENT WITH EXPERIMENT FOR THE DIRECTBANDGAPIS IN GENERAL WITHINA FEW TENTHSOF AN eV. VALUES FOR THE BAND GAP AND THE DIELECTRIC CONSTANT ARE TAKENFROM THE ORIGINALPAPERSWITH THE EXCEPTION OF THE DIELECTRICCONSTANT OF LiFI4’ AND NaBr AND NaC1.’43 OTHER ABBREVIATIONSUSED ARE EPSP = EMPIRICAL P S P MB = MIXEDBASIS;(0)PW = (ORTHOGONA~IZED) PLANEWAVE;GO = GAUSSIAN ORBITAL;LCAO = LINEARCOMBINATION OF ATOMICORBITALS.

FEATURE

CORE

I?

BASIS

LiparP

HF

HF

MB/OPW

Kane“ Brenef

EPSP valence

EPSP HF

PW LCAO

Baroni

valence

HF

GO + OPW

Gygi” Hybertsenb Godby*

LDAIPSP LDAIPSP LDAIPSP

LDA LDA LDA

PSPIPW PSPIPW PSPIPW

DIRECTGAP (eV) Ehhp/E;:; (MATERIAL, EXP. DIELECTRIC CONSTANT) 13.7/14.3 (ArJ.67) 7.5/7.1-7.4 (NaBr,2.60) 8.6118.75 (NaQ2.33) NIA 7.617.3- 7.6 (C,5.70) 17.9113.6 (LiF, 1.96) 5.2414.99 (LiH,3.61) 14.62p4.15 (Ar, 1.67) 3.24/3.4 (Si,l1.7) 3.313.4 (Si,l1.7) 3.2713.4 (Si,ll.7)

“Static COHSEX calculation neglecting local fields and based on a model dielectric function. bGWAcalculation including energy dependence (plasmon-pole model in Ref. 5) and local fields, and based on first-principles dielectric matrix.

Hartree-Fock calculations offer a convenient single-particle basis for low-polarizability solids-the focus of much of the early work in the COHSEX approximation -but fail to give a good description of polarizable solids such as Si for which empirical pseudopotential or LDA wave functions and energies work better. Table 5 shows that in the case of Si,90 self-consistent COHSEX and LDA densities are very similar whereas Hartree-Fock calculations concentrate too much density in the bonds. Core-valence exchange and correlation potentials can be determined either via effective (e.g., core-polarization) potentials or via treatment of the core electrons on the same level of approximation as the valence electrons.

44

WILFRIED G. AULBUR, LARS JONSSON, A N D J O H N W. WILKINS

TABLE5. SELECTEDFOURIER COEFFICIENTS OF THE VALENCEDENSITYI N Si OBTAINED BY HARTREE,LDA, DIAGONALCOHSEX (STATIC,N o LOCAL FIELDS),AND HARTREE-FOCK CALCULATIONS, USINGTHE SINGLEMEAN-VALUE P O ~ TECHNIQUE. T ALL VALUESARE GIVENI N ELECTRONS PER UNIT CELL. THEDEGREEOF DENSITY LOCALIZATION IS LOWESTIN THE HARTREE AND HIGHESTIN THE HARTREE-FOCK CALCULATION. THELDA DENSITY LIES BETWEENTHESE T w o EXTREMES AND IS CLOSEST TO THE DENSITY OBTAINED FROM A SELF-CONSISTENT COHSEX CALCULATION. (ADAPTEDFROM REF. 90).

G

HARTREE

LDA

COHSEX

(1,1,1) (2,2,0) (3,1,1) (2,2,2) (4,O 30)

- 1.42 0.14 0.30 0.20 0.34

- 1.67 0.13 0.36 0.31 0.40

- 1.59

- 1.17

0.14 0.35 0.29 0.38

0.16 0.42 0.43 0.43

'

HARTREE-FOCK

Explicit treatment of core electrons using a linear-combination-of-atomicorbitals basis' 3 8 * 139 or a basis set of localized, contracted Gaussian-type orbital^'^^^'^^ -which allow the analytic evaluation of screened-exchange two-electron integrals -has been the method of choice in static COHSEX calculations. Alternatively, unscreened core-core and core-valence exchange potentials from Eq. (2.35) are taken into account in static COHSEX calculations, but self-energy contributions due to core-valence polarization are neglected (see Refs. 133, 134, 135, and 136). Recent GWA calculation^^*^ extend earlier COHSEX work by using a nondiagonal, frequency-dependent'44 first-principles dielectric matrix for the determination of quasiparticle corrections to the single-particle spectrum. The treatment of core electrons is (1) at LDA l e ~ e l , (2) ~ . implicit ~ via core-polarization potential^,^^ and (3) explicit by treating core electrons on the same level as valence electron^.^^*^^*'^^ The consequences of the GWA A. B. Kunz and N. 0. Lipari, Phys. Rev. B4, 1374 (1971). N. 0. Lipari and W. B. Fowler, Sol. State Comm. 8, 1395 (1970). N. 0. Lipari and W. B. Fowler, Phys. Rev. B2, 3354 (1970). N. 0. Lipari and A. B. Kunz, Phys. Rev. 8 3 , 4 9 1 (1971). 13' E. 0. Kane, Phys. Rev. B5, 1493 (1972). 13' N. Brener, Phys. Rev. B 11, 929 (1975). 1 3 9 N. Brener, Phys. Rev. E l l , 1600 (1975). S. Baroni, G. Grosso, and G. Pastori Parravicini, Phys. Rev. B29, 2891 (1984). 14' S. Baroni, G. Pastori Parravicini, and G. Pezzica, Phys. Rev. B32, 4077 (1985). 1 4 2 Handbook of Optical Constants ofsolids, ed. E. D. Palik, Academic Press, Orlando (1985). 1 4 3 M. E. Lines, Phys. Rev. B41, 3383 (1990). 144 Hybertsen and Louie used a plasmon-pole model; Godby and collaborators determined the full frequency dependence of the screened interaction. 1 4 5 F. Aryasetiawan and 0. Gunnarsson, Phys. Rev. B54, 17564 (1996). 133

134

QUASIPARTICLE CALCULATIONS IN SOLIDS

45

extensions compared to static COHSEX calculations are discussed in the following sections. Applications. Work on bulk insulators was done by Lipari and collaborators (NaCI, NaBr, solid Ar; Refs. 133, 134, 135, and 136) and Kane (Si; Ref. 137) in the early seventies; by Brener in the mid-seventies (LiF, diamond; Refs. 138 and 139);by Louis in the late seventies (AX, A = Li’, Na’, X = F-, C1-, Br-, I - ; Ref. 146); and by Baroni and collaborators (solid Ar, LiH; Refs. 140 and 141) as well as by Gygi and Baldereschi (Si; Ref. 90) in the mid-eighties. Calculations on potassium and other metals were, for instance, reported by Bross and collaborators (see Ref. 99 and references therein). The starting Hamiltonian for potassium is defined by a “parametrized Thomas-Fermilike” approximation to the static ~ e l f - e n e r g y : ~ ~ . ’ ~ ’

Modeling the frequency dependence of the screened interaction by Lundqvist’s plasmon-pole model9* (see Section 11.8) and core-valence polarization using the dipolar core polarizability of potassium leads to quasiparticle energies that agree to within 20% with e ~ p e r i m e n t The .~~ inclusion of core polarization reduces self-energy shifts by 0.1 to 0.2 eV.99 Other applications of the static COHSEX approximation include Gadzuk‘s determination of polarization energies of core holes in atoms or molecules that are embedded in or adsorbed on the surface of a free electron ’ ~ ~ a dynamic COHSEX gas (Ref. 148 and references therein). H o d g e ~ used approximation to explore the image potential of a charged particle trapped at the surface of a metal, and Cooper and Linderberg15’ used the COHSEX approximation within the Pariser-Parr-Pople model for the description of n-electron systems in hydrocarbons. Theoretical issues, such as the importance of vertex corrections, have been addressed by several author^.'^'*^^^*'^^ We review these works in more detail in Section 11.14, where we discuss vertex corrections to the GWA. E. Louis, Phys. Rev. B20, 2537 (1979). This Harniltonian can be derived within the framework of DFT; see Section VII.33. The notation in the equation is the same as used elsewhere in this article. 14’ J. W. Gadzuk, Phys. Rev. B 14, 2267 (1976). 14’ C. H. Hodges, J. Phys. C: Sol. State Phys. 8, 1849 (1975). I. L. Cooper and J. Linderberg, Molec. Phys. 25, 265 (1973). 1 5 1 L. W. Beeferrnan and H. Ehrenreich, Phys. Rev. B 2 , 364 (1970). P. Minnhagen, J. Phys. C Sol. State Phys. 7, 3013 (1974). H. Suehiro, Y. Ousaka, and H. Yasuhara, J. Phys. C: Sol. State Phys. 17, 6685 (1984). 146

14’

46

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Several alternative or more general approaches for the determination of quasiparticle energies have been suggested in the literature. The approaches by Hanke and collaborator^,'^^^'^^^^^^ by Fulde and collaborators,"7~'58~'59 by Pickett and Wang,'60*'61 and by Sterne and Wang16' are discussed in some detail in the paragraphs below, since they either extend RPA-based GWA calculations or correspond to interesting alternative pictures for 'the description of exchange and correlation in solids. Other approaches, such as the electronic polaron model by K ~ n z or ' ~ by ~ H e r m a n ~ o n , 'can ~ ~ be identified as limiting cases of GWA calculation^.^^ Their physics is contained in the general discussion of the GWA approach, and we do not discuss these references separately. b. Excitonic Efects An extension of the COHSEX approximation' 54,1 that includes excitonic correlations in the irreducible polarizability gives a better description of photoabsorption experiments-in which photons create interacting electronhole pairs- than do RPA-based approaches. In contrast to RPA-based GWA calculations whose fundamental excitations are noninteracting electron-hole pairs and plasmons, GWA calculations based on a description of screening in the time-dependent Hartree-Fock approximation include the short-range, attractive interaction between electrons and holes via ladder diagrams. The determination of the influence of excitonic correlations on density fluctuations and hence on the dielectric matrix requires the solution of the equation of motion of the two-particle Green function, that is, the Bethe Salpeter equation.' 5 4 * 1 5 5 ~ 15 6 The inverse dielectric matrix is given in terms of the density-density correlation function,'65 and the latter can be written as a two-particle Green function:'67 W. Hanke and L. J. Sham, Phys. Rev. Lett. 33, 582 (1974). W. Hanke and L. J. Sham, Phys. Rev. B 12,4501 (1975). 1 5 6 G. Strinati, H. J. Mattausch, and W. Hanke, Phys. Rev. B25,2867 (1982). "' W. Borrmann and P. Fulde, Phys. Rev. B35, 9569 (1987). S. Horsch, P. Horsch, and P. Fulde, Phys. Rev. B28,5977 (1983). S. Horsch, P. Horsch, and P. Fulde, Phys. Rev. B29, 1870 (1984). W. E. Pickett and C. S. Wang, Phys. Rev. E M ,4719 (1984). 16' W. E. Pickett and C. S. Wang, Int. J. Quant. Chem. Quant. Chem. Symp. 20, 299 (1986). P. A. Sterne and C. S. Wang, Phys. Rev. B37, 10436 (1988). A. B. Kunz, Phys. Rev. B 6 , 606 (1972). 164 J. Hermanson, Phys. Rev. B 6, 2427 (1972). 16' To be consistent with Hanke et aL's notation we do not consider density fluctuation operators A'(rt) = h(rt) - (A(rt)) as in Eqs. (2.17) and (2.18), but density operators h(rt). For the definition of the physically relevant retarded dielectric matrix, the two approaches are equivalent, since c-numbers always commute. For the time-ordered dielectric matrix, the approaches differ only in the limit w + 0. The formulation using the density fluctuation operator avoids a divergence due to the bosonic nature of the density-density correlation function (see, for instance, Refs. 18, 558ff., and 166).

QUASIPARTICLE CALCULATIONS IN SOLIDS

( N , OlT[A(rt)n*(r’O)](N,0)

= - iG(rt,

r’0, r’O+, rt’).

47 (2.64)

Let Go( 1, l’, 2,2’) be the noninteracting electron-hole propagator, GO(l,l’,2,2’) = G(l’, 2’)G(2, l),

(2.65)

and I be the irreducible electron-hole interaction, whose approximation is shown in Fig.

Here - W,, is a static approximation to the screened interaction between an electron and a hole and u is the unscreened Coulomb repulsion. Choosing W,, = 0 leads back to the RPA results. Solution of the Bethe-Salpeter equation -

G(1, l’,2,2’) = GO(l,l’, 2,2’)

+ -determines

GO(l, l’,3,3‘)1(3, 3’,4,4’)G(4,4‘,2,2’)d(3,3’, 4,4‘) (2.67)

the influence of excitonic correlations on the frequency

166 H. Stolz, Einfihrung in die Vielelektronentheorie der Kristalle, Bertelsmann Universitatsverlag, Diisseldorf (1975), Chap. 6. Generally, a two-particle Green function is defined as

G ( r , t , , rzt,, r3t3,r4f4) = ( - i ) * ( N , 16’

OlTC~(r,tl)Y(r2t2)Yt(r4r4)~t(r3r3)llN, 0).

x

W. Hanke and L. J. Sham, Phys. Rev. B21,4656 (1980).

,IITs=

4‘

3’

_ - _ _ - - _Coulomb repulsion

‘VVL Coulomb attraction FIG. 8. The irreducible electron-hole interaction in the time-dependent Hartree-Fock approximation is given by the sum of a screened Coulomb attraction and an unscreened Coulomb repulsion. The first term generates an infinite sum of ladder diagrams, whereas the second term generates an infinite sum of polarization bubbles (compare Eq. (2.66)). (From Ref. 168.)

48

WILFRIED G. AULBUR, LARS JbNSSON, A N D JOHN W. WILKINS

dependence of the dielectric matrix (via Eqs. (2.64) and (2.18)) and on the quasiparticle spectrum (via Eqs. (1.3) and (2.1 1)). The inclusion of vertex corrections in the irreducible polarizability but not in the self-energy is motivated by two observations. (1) An exact Wardidentity-based relation between the self-energy and the vertex function is expected to be better observed when the irreducible polarizability is determined in the time-dependent Hartree-Fock rather than the time-dependenl Hartree approximation.' 5 6 This Ward-identity-based relation results from charge conservation. (2) Hanke and collaborators suggested that the evaluation of the GWA self-energy with an interaction screened using the time-dependent Hartree-Fock approximation is more appropriate than an evaluation of C using an RPA-screened interaction. The choice of screening in the time-dependent Hartree-Fock approximation can be motivated a$ follows: Choosing C = 0 in Fig. 4 as the starting point for an iterative solution leads to l-(1,2; 3) = 6(1,2)6(1,3) and E = E~~~ (see Eqs. (2.10) and (2.23)). The next iteration yields C(1,2) = iG(1,2)W(1+,2) and subsequent13

x G(4,6)G(7,5)r ( 6 , 7 ; 3)d(4,5,6,7) = 6(1,2)6(1,3)

+i

s

W(1+,2)G(1,6)G(7,2)r(6,7; 3)d(6,7), (2.68

where contributions due to SW/SC are neglected since they are difficult tc handle and their contributions are estimated to be negligible for Si anc diamond.'69 Using the above expression for the vertex function in thc definition of the polarizability as well as the dielectric matrix, one finds tha the dielectric matrix includes not only RPA bubbles, corresponding tc 6(1,2)6(1,3) in the above equation, but also an infinite series of screenec ladder diagrams resulting from the second term in Eq. (2.68). Note tha Mahan" used Ward identities to justify the simultaneous inclusion of iden tical vertex corrections in C and P in contrast to the approach suggested b! Hanke et al. Mahan's approach is discussed in more detail in Section 11.14 Inclusion of excitonic effects via the solution of the Bethe-Salpete equation for several semiconductors leads to (see Refs. 156, 169, 170, 171 1 6 9 W. Hanke, N. Meskini, and H. Weiler, in Electronic Structure, Dynamics, and Quantun Structural Properties of Condensed Matter, eds. J. T. Devreese and P. Van Camp, Plenum, Nev York (1985), 113. 170 H. J. Mattausch, W. Hanke, and G. Strinati, Phys. Rev. B27,3735 (1983). N . Meskini, H. J. Mattausch, and W. Hanke, Sol. State Comm. 48, 807 (1983).

"'

QUASIPARTICLE CALCULATIONS IN SOLIDS

49

and 172): (1) band gaps and valence band widths in good agreement with experiment, (2) an increase in oscillator strength at the El peak of the frequency spectrum of E by 50- loo%, in good agreement with experiment, and (3) a larger El peak intensity than E , peak intensity for Si, in contrast with experiment (see Fig. 39 for a definition of the E l and E , peaks in the imaginary part of the Si dielectric function). Practical calculations use a matrix representation of the Bethe-Salpeter equation with respect to semiempirical tight-binding valence and conduction bands, which are expressed in a Gaussian-orbital basis c. Local Approach Local approaches to the correlation problem in solids (for a review see Ref. 19) are based on the physical picture that adding an electron or hole to an insulator leads to a polarization of neighboring bonds whose induced dipoles will act on and change the energy of the original particle. A local description of correlation effects based on quantum chemistry methods starts from a Hartree-Fock wave function and includes local correlations in the form of one- and two-particle1 7 3 excitations in the many-body wave function I Y ) via a projection operator exp($:

I'r)

= exp(~)l%).

s

(2.69)

The operator depends on parameters that characterize ground- and excited-state correlations and that can be determined variationally. Bond polarizations are interatomic correlation effects that dominate dielectric screening and hence quasiparticle energy shifts of occupied and unoccupied states in covalent semiconductors. These interatomic correlation effects can be described by projector methods based on a minimal basis set of Gaussian orbital^.'^^.'^^,'^^ Other important effects, like the relaxation of electronic orbitals in the neighborhood of an added electron or the change of ground-state correlations due to the presence of an extra electron, require a larger basis set. Each of these two partially canceling effects is estimated to amount to band-gap corrections on the order of 1 to 2eV based on molecular calculations.157~158~159 Local projector methods determine ground-state properties of a variety of semiconductors with better accuracy than LDA,' 74.1 7 5 - 17 6 but applica17*

173

"' 176

G. Strinati, H. .I. Mattausch, and W. Hanke, Phys. R K KLett 45, 290 (1980). Excitonic correlations are not included, in contrast to Refs. 156 and 169. M. Albrecht, B. Paulus, and H. Stoll, Phys. Rev. B56, 7339 (1997). B. Paulus, P. Fulde, and H. Stoll, Phys. Rev. B51, 10572 (1995). B. Paulus, P. Fulde, and H. Stoll, Phys. Rev. B54, 2556 (1996).

50

WILFRIED G. AULBUR, LARS JONSSON, A N D JOHN W. WILKINS

tions to excited states are limited and remain difficult due to the extended nature of these ~ t a t e s . ' ~ The ~ . ' ~calculations ~ scale as the number of basis functions to the fifth power,'79 compared to standard LDA and GWA calculations that scale as the number of basis functions to the third and fourth power, respectively. Applications of projector methods to excitedstate proper tie^'^^*'^^*'^^ have been reported for Si, Egap= 3.0eV (3.33.4 eV experiment); diamond, Egap= 7.2 eV (7.3 eV experiment); and Ge, Egap= 0.44eV (0.9 eV experiment). d. Quasiparticle Local Density Approximation (QPLDA) The self-energy is a short-ranged operator39 and -as a ground-state property -a functional of the ground-state density.39 For systems with slowly varying density, a possible approximation of I: is given by the self-energy I:,,of the homogeneous electron gas evaluated at the local density:39 C(r, r'; E )

%

I:,,(p(r); E - p - p,,(n(r)); n(r))6(r - r').

(2.70)

Here p(r) is a local momentum and the chemical potentials of the inhomogeneous (p) and the homogeneous @,,) system are introduced to line up the Fermi levels. Calculations on metals using this approximation are reported by, among others, Arbman and von BarthS6 for A1 and by Sacchetti"' for Cu. For practical calculations [Refs. 160, 161, 162, and 1811 in semiconductors the homogeneous self-energy is evaluated in the GWA with the further assumptions that (1) the Levine-Louie dielectric function describes screening, and (2) the Green function equals the Green function of a fictitious, insulating free-electron gas. Although this ansatz is guided by physical intuition, it lacks rigorous justification86*'60 and may elude systematic improvements.'60 Advantages of the QPLDA are that (1) it is not limited to minimal basis sets, in contrast to the early work of Hanke et u1.154*155,156 and Fulde et a1.,'57*'58*159 and (2) it leads to only a 30% increase in computation time compared to LDA and is therefore very efficient compared to the other methods discussed above. Important physics of the many-body self-energy is not captured in the QPLDA for two reasons: (1) the energy-dependence of the QPLDA self177

'"

179

"'

J. Grifenstein, H. Stoll, and P. Fulde, Phys. Rev. BS5, 13588 (1997). B. Paulus, private communication. G. Stollhoff and P. Fulde, Z . Phys. B29,231 (1978). F. Sacchetti, J. Phys. l? Met. Phys. 12, 281 (1982). C. S. Wang and W. E. Pickett, Phys. Rev. Lett. 51, 597 (1983).

QUASIPARTICLE CALCULATIONS IN SOLIDS

51

energy increases the LDA band gaps and achieves satisfactory agreement with experiment; and (2) local-field effects are not taken into account. In more accurate GWA calculations static, inhomogeneous screening effects open the LDA band gap of semiconductors, and dynamic effects reduce the gaps to within a few tenths of an eV of the experimental values (see next sections). Nevertheless, QPLDA band gaps generally agree with experiment to within 0.4 eV. For example, the direct band gap of Si is 2.99 eV compared to experimental values of 3.3 to 3.4eV, and the indirect gap equals 0.81 eV compared to an experimental value of 1.17 eV.162

10. LOCAL-FIELD EFFECTS AND

THE

NONLOCALITY OF THE SELF-ENERGY

The response of an inhomogeneous density distribution to an external electric field gives rise to microscopic fields that vary on the length scale of a bond length. These so-called local fields are described by the off-diagonal matrix elements of the dielectric matrix (Eq. (2.15)) whose diagonal elements account for homogeneous screening (see Section VI.31 for a detailed discussion). Local-field effects are often important in semiconductors and insulators but rarely in metals, In contrast to local operators-which depend only on one real-space variable r and act multiplicatively on a wave function -nonlocal operators such as the self-energy depend on two independent real-space variables and act on wave functions via a nontrivial real-space integral; that is, they probe the wave function not only for r = r’ but for all r’ within their range. Figure 9 shows schematically that fX(r, r’; E)cD(r’)dr‘will be large and negative for the highest valence-band wave function cDo and much smaller in absolute value for the lowest conduction-band wave function cDc of Si. The extra node of the conduction-band wave function leads to canceling positive and negative contributions to the integral and contributes to the discontinuous jump of self-energy corrections at the band gap (see below). The combined effect of local fields and the nonlocality of the self-energy operator on the band gap of solids is significant and cannot be clearly separated into two distinct contributions. For instance, local-field effects modify the range of the self-energy. Despite the interdependence of localfield effects and the self-energy nonlocality, the next two paragraphs discuss separately the dominant influence of local fields and of nonlocality on self-energy corrections.

a. Local Fields A real-space analysis of the static screening potential, Wscr(r,r’) = W(r, r‘; E = 0) - v(r, r’), around an extra electron centered at a bond center or at

52

WILFRIED G. AULBUR, LARS JBNSSON, AND JOHN W. WILKINS 2-

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

r' (BOND LENGTHS) FIG.9. Plot of the self-energy Z(r, r'; E = midgap) of Si where r is chosen to lie at the bond center and r' is varied along the [111] direction. Atomic positions are indicated by closed circles, and Yu,(Yc) denotes the real part of the highest-lying valence (lowest-lying conduction) band wave function close to the r point calculated in the LDA. The range of nonlocality of the self-energy is about one bond length. It follows from the figure that (Y,,lZIYv) is large and negative, whereas (Y,lZ(Y,) has large positive and negative contributions due to the nonlocality of the self-energy and the nodal structure of the conduction band wave function. Hence, the self-energy pulls the valence band deeper in energy with respect to the conduction band and the band gap opens up in comparison to the LDA. (Taken from Ref. 6.)

an interstitial site44 shows that (1) local fields contribute more than a third of the screening potential in the region near the center of the bond; (2) the local-field contribution is much smaller (roughly one-tenth) in the lowdensity, interstitial regions; (3) local fields are responsible for the anisotropy of the screening potential; and (4) local fields are short-ranged and their effect is negligible beyond roughly one bond length. Figure 10 shows the results for WScr evaluated at E = 0 in analogy to the static COHSEX approximation to separate the energy dependence of C from local-field effects. The screening potential due to an electron at site r evaluated at the same site determines the Coulomb-hole contribution to the self-energy, which consequently is constant when local fields are neglected and much deeper in the bonding region than in the interstitial region when local-field effects are

53

QUASIPARTICLE CALCULATIONS IN SOLIDS

FIG.10. Contour plot in the Si (110) plane of the screening potential W"' (in Rydberg) in response to a single electron (indicated by a cross) at (a) a bond-centered or (c) an interstitial site. Panels (b) and (d) show the corresponding local-field (LF) contributions. Local fields (1) contribute between 1/3 (bond center) and 1/10 (interstitial) to the total screening potential; (2) are responsible for the asymmetry of the screening potential; and (3) are short-ranged and become negligible after one bond length. (Adapted from Ref. 44.)

taken into account. Local fields are more important for valence than for conduction bands since the valence-band density is concentrated in the bonding region and the conduction-band density is concentrated in the interstitial region. The Coulomb-hole and screened-exchangecontributions to the self-energy can be expressed in terms of the screening potential WScr as follows (see Eqs. (2.32) and (2.33):

1 F o H ( r , r') = -6(r 2 SEX

- r')Wscr(r, r),

Wscr(r, r')

(2.71)

1,

+Ir - r'(

(2.72)

54

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

where the exchange density n, is defined as occ

n,(r, r') = C Qi(r)Qf(r').

(2.73)

i

Neglect of local fields leads to homogeneous screening; Wscr(r,r) is constant and does not contribute to dispersion within a band gap or a given band. Based on the above remarks about the strength of WScrin bonding and antibonding regions, one expects both an increase in the fundamental gap due to CCoHupon inclusion of local-field effects and a strong local-field effect on the expectation value of CCoHfor occupied but not for unoccupied states. Both expectations are confirmed by actual calculations, as shown in Tables 6 and 7. Local-field effects are somewhat weaker for the screened-exchange contribution to the self-energy, since XSEX is dominated by the bare Coulomb interaction when Ir - r'I is less than a typical bond length, that is, within the range of the local-field effects. Table 7 shows that for occupied states local-field effects in XSEX amount to about a third of the local-field effects in Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, eds. K.-H. Hellwege, 0. Madelung, M. Schulz, and H. Weiss, New Series, Group 111, vol. 17a, Springer-Verlag, New York (1982). lE3 G.Baldini and B. Bosacchi, Phys. Stat. Solid. 38, 325 (1970). BONDED SEMICONDUCTORS TABLE6. DIRECTAND INDIRECTBANDGAPSFOR THE COVALENTLY DIAMOND AND LiCI, USINGLDA, THE STATIC COHSEX Si AND Ge AND FOR THE INSULATORS APPROXIMATIONWITHAND WITHOUTLOCALFIELDS (LF), AND THE GWA IN COMPARISONWITH EXPERIMENT. THELDA BANDGAP UNDERESTIMATES EXPERIMENT BY AT LEAST0.7 TO 3.4eV. THE COHSEX APPROXIMATION WITHOUT LOCAL FIELDS LEADS(WITHTHE EXCEPTION OF si) TO AN INCREASE IN THE FUNDAMENTAL BANDGAPBUT STILL UNDERESTIMATES THE E X P E R I ~ A L VALUES.THE INCLUSION OF LOCAL FIELDS LEADS TO A FURTHER BAND-GAPINCREASE AND TO AN OVERE~TIMATION OF E X P E R IBY ~ ABOUT0.4 TO 1.1eV. DYNAMICEFFECTS (GWA) REDUCE THIS OvwEsnMATION AND REPRODUCE EXPERIMENT TO WITHIN0.1 TO 0.3eV. NOTE THAT RELATIVISTIC EFFECTSAND CORE ELECTRONSCAN HAVE A LARGE INFLUENCE ON SEMICONDUCTOR GAPS,AS SHOWNIN TABLE33. (FROMREF44.) COHSEX

Diamond Si Ge LiCl

LDA

No LF

LF

GWA

EXPT.'

3.9 0.52 0.07 6.0

5.1 0.50 0.33 8.2

6.6 1.70 1.09 10.4

5.6 1.29 0.75 9.1

5.48 1.17 0.744 9.4b

"Ref. 182 unless otherwise noted; bRef. 183.

QUASIPARTICLE CALCULATIONS IN SOLIDS

55

TABLE7. STATIC COH AND SEX CONTRIFIUTIONS TO THE VALENCE-BANDAND CONDUCTION-BAND-MINIMUM SELF-ENERGIES, WITHAND MAXIMUM WITHOUTLOCAL FIELDS,FOR Si IN COMPARISON TO THE GWA SELFENERGY. LOCAL-FIELD EFFECTSAFFECT OCCUPIED STATES SIGNIFICANTLY AND LEADTO ABOUT AN 1.6-eV DECREASE IN ZcoH FOR l-!z5m WHEREAS ZSEXIS LESSSENSITIVE TO LOCAL FIELDS (0.6 eV FOR rz,,, 0.3 eV FOR X , J . DYNAMICCORRECTIONS S m THE CONDUCTION AND VALENCE SELFAND FOR ENERGIES UPWARDBY ABOUT 10 TO 20%. IN ABSOLUTETERMS THE TOTALSELF-ENERGY, THE DYNAMIC CORRECTION IS LARGER FOR THE VALENCE BANDSAND LEADSTO A BAND-GAPDECREASE. (ADAPTED FROM REF.44.) COHSEX

Si

No LF

LF

GWA

- 8.72 - 8.72

- 10.30 - 8.70

- 8.41 - 7.40

ri5"

- 4.44

- 3.85

XI,

-2.37

- 2.08

- 3.56 - 1.65

xCOH

r25v

XI, zSEX

XCoH and that local-field effects for unoccupied bands in XSEX are even smaller. Since local-field effects in X S E X and CCoHare of opposite sign and amount to at least several tenths of an eV, they must be included in the determination of both terms to ensure quantitatively accurate results. The strong effect of local fields on XCoH for the valence band with respect to the conduction band, and the relative insensitivity of XSEX to local fields, leads to a band-gap increase of 0.8 to 4.4eV compared to the LDA or the static COHSEX approximation without local fields. Table 6 demonstrates that:44 (1) the LDA gives band gaps that are at least 0.5 to 2.0eV smaller than experiment (see also Fig. 3); (2) the COHSEX approximation without local fields gives band gaps that are in better agreement with experiment, although significant deviations from experiment remain; and (3) local fields dramatically open up the band gap and require inclusion of dynamic screening to achieve quantitative agreement with experiment. Local-field effects on the self-energy shifts in simple metals are less than the numerical uncertainties of GWA calculations (see Section V.26b and Ref. 184) since their density is relatively homogeneous compared to covalently bonded semiconductors. J. E. Northrup, M. S. Hybertsen, and S. G. Louie, Phys. Rev. E39, 8198 (1989).

56

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

b. Nonlocality of the Self-Energy The screened-exchange contribution to the self-energy is proportional to the nonlocal exchange density (Eq. (2.72)) and hence sensitive to the nodal structure of wave functions, which results in a discontinuity of the expectation value of CSEXupon crossing the band gap.185 This discontinuity, discussed in connection with Fig. 9 and Table 7, shows a 0.3-eV sensitivity to the inclusion of local-field effects.’86 The Coulomb-hole self-energy (Eq. (2.71)) does not experience any nonlocal effects upon crossing the band gap because it is a local operator. Figure 11 shows that the nonlocality of the self-energy X(r,r’;E) of common semiconductors in real space is,’dominated by a spherical hole

Ia6

R. W. Godby, M. Schluter, and L. J. Sham, Phys. Rev. B37, 10159 (1988). The discontinuity of ZSEXequals 1.77 eV with and 2.07 eV without local fields for Si.

FIG. 11. Contour plots of self-energy Z(r, r’; E = midgap) in eV a.u.-3 for r fixed at the bond center and r’ shown in the (110) plane for (a) Si, (b) GaAs, (c) AIAs, and (d) diamond. For silicon, the corresponding plots with r fixed at the tetrahedral interstitial site are also shown (panel (e)). For comparison, the self-energy operator of jellium with rr = 2 (the average density of silicon) is shown (panel (f)). Godby, Schliiter, and Sham suggested modeling the approximate form of the nonlocality of the self-energy (not its depth) using the results from jellium calculations. (From Ref. 185.)

QUASIPARTICLECALCULATIONS IN SOLIDS

57

centered around r = r’ whose shape-but not its strength or depth-is roughly independent of the location of r and can be approximated by the corresponding hole of the jellium self-energy.’” The nonlocality hole of the self-energy mirrors the nonlocality of the short-ranged screened interaction W rather than the long-ranged single-particle propagator G. The average radius of the first nodal surface of the nonlocality in jellium is 2r, (2.1 A for Si). The extent of the nonlocality is on the scale of a typical wavelength of a conduction-band wave function, and the hole accounts for more than 99% of the self-energy expectation value in Si. The nonlocality of the self-energy contributes significantly to the step-like structure in the self-energy corrections, at the band gap shown in Fig. 12, panels (a) and (b) for Si.IE5Panel (c) of the same figure shows that this step-like structure is absent in jellium, which indicates that although the overall form (not the depth) of the nonlocality hole is well reproduced by jellium at the appropriate density, the interaction between the wave functions and the nonlocality is not captured in the homogeneous electron gas.185 Evaluation of the self-energy operator of jellium at the local density of Si leads to an underestimation of the self-energy nonlocality radius by about -30% in the bonding region and an overestimation by about 80% in the interstitial region, as shown in Table 8. This result questions the validity of basic assumptions in the QPLDA.18’ In addition, the nonlocality hole is too anisotropic, and it therefore seems unlikely that QPLDA (see Section II.9d) can give quasiparticle energies of an accuracy comparable to the full GWA results. OF THE SELF-ENERGY 11. ENERGYDEPENDENCE

Table 6 shows that the dynamically screened interaction approximation reduces the 0.8- to 2.2-eV overestimation of direct band gaps in the static COHSEX approximation to a discrepancy of 0.1 to 0.3 eV between theory and e ~ p e r i m e n t A .~~ quantitative prediction of quasiparticle energies requires a correct treatment of the energy dependence of the self-energy. A simple interpretation of the dynamical effects can be obtained by realizing that the electron drags a polarization cloud behind it. Screening and the associated screening energy will depend on the velocity of the electron. The strongest screening and therefore the lowest energy is obtained in the static case. Including energy dependence will lead to a screening energy that is not as low and hence remedies the overestimation of the self-energy in the COHSEX approximation. The band-gap reduction results from the linear behavior of the self-energy as a function of energy close to the quasiparticle energy. Figure 13 shows

58

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

2

-

(2) 51

c

0 f;

=

W -

w

-2

-5

0

I

OW OUASIPARTICLE EN€RCr ( r V )

FIG. 12. (a) The real part of the matrix elements of the GWA self-energy operator of silicon

(Z(E))and the LDA exchange-correlation potential (V’D”) plotted against the quasiparticle energy E. (b) The differences (Z(E)- VkD”)(essentially the quasiparticle energy correction) and (Z(0) - V k D A )of silicon plotted against the quasiparticle energy. The energy dependence of the self-energy must be taken into account to obtain the correct dispersion for the quasiparticle energies. (c) For comparison, (Z(E)- V;””) and (Z(0) - V i D ” ) are plotted against the quasiparticle energy E for jellium at the average density of silicon (r, = 2). The Fermi energy E , is aligned with the middle of the band gap. Both are relatively featureless and

59

QUASIPARTICLE CALCULATIONS IN SOLIDS NONLOCALITY EVALUATED TABLE8. RANGE rnloe OF THE SELF-ENERGY BONDCENTERED(BC) AND AN INTERSTITIAL (I) SITEIN Si, GaAs, COMPARED TO THE CORRESPONDING VALUE OF THE AlAs, AND DIAMOND JELLIUM SELF-ENERGY EVALUATED AT THE FERMIENERGY AND AT (1) THE AVERAGEDENSITY OF THE SEMICONDUCTOR n(rs), AND (2) THE LOCAL DENSITY n(r) AT A POINTr IN THE SEMICONDUCTOR. rnlocEQUALS THE SPHERICALLY AVERAGED DISTANCETO THE FIRST ZERO IN THE OSCILLATORY FUNCTION Z(r, r’; E = MIDGAP) WITH r FIXEDAT A BC OR I SITE. THE SELF-ENERGY OF JELLIUM AT THE AVERAGEDENSITY OF THE SEMICONDUCTOR UNDERESTIMATESTHE RADIUS rnloc OF THE FULL ON AVERAGE BY ABOUT 10%. THE “LOCAL-DENSITY” CALCULATION rnlorIN JELLIUM CALCULATIONS SEVERELY UNDERESTIMATE (OVERESTIMATE) HIGH (LOW) DENSITY REGIONS (-30% ON AVERAGEFOR A BC SITE, + 80% ON AVERAG~FOR AN 1 SITE). (ADAPTED FROM REF. 185.) AT A

rnrocIN

JELLIUM (a.u.)

r“l0C

MATERIAL

SITE

(a.u.)

nlrJ

n(r)

Si

BC I BC I BC I BC I

4.1 4.7 4.2 4.3 4.3 4.6 2.8 3.5

3.9 3.9

2.8 8.8 2.8

GaAs AlAs C

4.0 4.0 4.0 4.0 2.7 2.7

9.0

2.8 9.0

2.2 4.7

that the expectation values of the self-energy operator in diamond, Si, Ge, and LiCl have a negative slope of about -0.2.44 This slope translates into nearly constant quasiparticle weights Ziof about 0.8, as shown in Table 9 (Eq. (2.7)), and to about a 20% reduction in the quasiparticle shifts via Eq. (2.6). However, the quasiparticle weights Ziare close enough to unity that quasiparticles are well-defined excitations for energies close to the band gap, although dynamical effects are nonnegligible. This statement is no longer valid in d and f electron systems, as discussed in Section V.26~.The curves of X ( E ) bend upward for hole and downward for electron states, which

clearly do not share the sharp discontinuities present in (a) and (b). (d) The real parts of the matrix elements of the bare-exchange (Hartree-Fock) self-energy operator of Si (Z,) and the statically screened exchange self-energy operator ( X s s x ) . ( VhD”) is subtracted as in (b). The nonlocality of Z, has too large a range due to the neglect of screening which leads to a jump in the self-energy corrections at the Fermi level that is too large in comparison with experiment. (e) The real parts of the matrix elements in silicon and jellium (r, = 2) of the frequency derivative of the self-energy ( d Z ( o = midgap)/aw). (Taken from Ref. 185.)

60

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Energy (eV) FIG. 13. Matrix elements of the electron self-energy operator evaluated in the GWA drawn as a function of energy for selected states near the band gap are displayed for (a) diamond, (b) Si, (c) Ge, and (d) LiCI. The quasiparticle energies of these states are indicated on the energy axis. The self-energy shows an approximately linear dependence on energy close to the quasiparticle energy for all states and all materials considered. The slope of (Z(E))is roughly material and state independent and amounts to about -0.2. This indicates that dynamical effects are important for a quantitative description of quasiparticle energies in solids. (Taken from Ref. 44.)

61

QUASIPARTICLE CALCULATIONS IN SOLIDS

TABLE9. QUASIPARTICLE WEIGHTZ, (EQ.(2.7)) FOR THE VALENCE-BAND MAXIMUM (VBM) CONDUCTION-BAND MINIMUM(CBM) AT r, X, AND L FOR Si, Ge, GaAs, Sic, GaN, DIAMOND, AND LICI. Zi FOR THE VBM AND CBM OF DIAMOND AND LiCl IS TAKEN FROM REF. 44; ALL OTHER VALUES ARE FXOM REF. 42. FOR THE SP-BONDED MATERIALS SHOWN IN THE TABLE,ziIS EQUAL TO 0.8 TO WITHIN 10% AND IS THEREFORE APPROXIMATELY A MATERIALAND STATE-INDEPENDENT CONSTANT.NOTE, HOWEVER,THAT COVALENTLY BONDEDSEMICONDUCTORS, WHICH HAVELARGERDIELECTRIC CONSTANTS, SHOW A STRONGER THAN DO IONICINSULATORS. DYNAMIC RENORMALIZATION AND THE

MATERIAL k-POINT Si

r X L

Ge

r

X L GaAs

r

X L

VBM

CBM

0.79 0.78 0.78 0.79 0.77 0.78 0.79 0.78 0.79

0.79 0.80 0.80 0.79 0.80 0.80 0.80 0.81 0.81

MATERIAL k - m w Sic

r X L

GaN

r

X L C LiCl

VBM

CBM

0.82 0.82 0.82 0.82 0.83 0.82 0.86 0.83

0.82 0.85 0.83 0.85 0.87 0.86 0.86 0.87

agrees with calculations of the energy dependence of the self-energy of j e l l i ~ mCore . ~ electrons can qualitatively change the energy dependence of the self-energy, as discussed for the case of Ag in Section V.26a. Besides band-gap reduction, the energy dependence of C is important in changing the dispersion of bands, which follows from the comparison of the expectation value ( C ( E ) - V,,) with the expectation value evaluated at the mid-gap energy (C(0) - Kc).1s5For Si, Fig. 12 panels (a)-(c) show thatls5 the expectation value ( C ( E ) ) has a pronounced state dependence, which is largely canceled by subtracting the expectation value of the exchangecorrelation potential; that ( C ( E ) - V $ P A ) is dominated by a jump at the energy gap and a flat dispersion for both occupied and unoccupied states; that (Z(0) - V$?’) shows a larger jump at the band gap than ( C ( E ) - V$PA) and a linear dispersion for both occupied and unoccupied states; and that, with the exception of the jump at the Fermi energy, the energy dependence can be modeled by the energy dependence of the jellium self-energy. Hence, the effect of the energy dependence of the self-energy, besides the reduction in the band gap, is a significant change-from linear to flat -in the dispersion of the bands. Self-energy corrections in small- and medium-gap semiconductors align the theoretical and experimental band gaps by an approximately rigid shift -the so-called scissors shift. Although the scissors-shift approach works well in Si, it breaks down in wide-band-gap materials such as Sic and

62

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

0.5

r

‘

1

2

3

5 6 Wave Vector

4

7

8

9

I

10

FIG. 14. Self-energy correction A(k) = (Z(E)- V;””) to the direct band gap at the point and at 10 special k-points in the fcc Brillouin zone for Si, SIC, and GaN. The standard deviation from the average self-energy correction (the “scissors” shift) amounts to 0.03 eV for Si, 0.13 eV for SIC, and 0.17eV for GaN. This corresponds to 4.5%/10.4%/11.7% of the average selfenergy correction (0.73 eV/1.20 eV/1.47 eV for Si/SiC/GaN). Although the scissors approximation works well for covalently bonded Si, it clearly breaks down for more ionic materials such as Sic and GaN.

GaN, as shown in Fig. 14. The increased dispersion in the self-energy correction for S i c and GaN may indicate that the LDA wave functions approximate less well the quasiparticle wave functions as the band gap increases. A review of applications of and extensions to the scissors-shift approach for the determination of optical response functions is given in Section VI.30. A related observation is that the deviation of the highest occupied eigenvalue in LDA from the highest occupied DFT eigenvalue, which equals the highest occupied quasiparticle eigenvalue (see Section 1.3), increases with increasing ionicity of the material under c ~ n s i d e r a t i o nThis .~~ deviation amounts to about 0.1 eV for covalent semiconductors such as Si

QUASIPARTICLE CALCULATIONS IN SOLIDS

63

and Ge. For more ionic materials such as diamond and LiCI, the deviation is as large as 1.5 eV. A simplified model of the self-energy of an insulator approximates its nonlocality and energy dependence by the corresponding quantities of the jellium self-energy evaluated at the local density in the i n s u l a t ~ r . Figure '~~ 12, panel (e) shows that the variation of the self-energy with respect to energy can be modeled by results obtained from the homogeneous electron gas. Assuming that the energy dependence of the self-energy factors out completely and considering the observations in the previous section, Godby, Schliiter, and Sham' 8 5 suggested the following physically appealing approximation for the self-energy:

where g(lr - r'l) and h ( E ) are functions that describe the nonlocality and energy dependence of the self-energy of jellium. The factor (f(r) + f(r'))/2 accounts for local-field effects. To the best of our knowledge applications of this formula have been limited to model calculations. 12. CORE-POLARIZATION EFFECTS Accounting for the hybridization of valence orbitals with semicore orb i t a l ~ and ' ~ ~ the relaxation of the latter (e.g., d orbitals in II-VI materials) in calculations leads to better agreement with experiment for the structural properties of II-VI (Refs. 188, 189, 190, and 191) and of some III-V (GaAs, G a N Refs. 192, 193, and 194) materials. In general, the equilibrium lattice parameters are increased and the cohesive energies reduced. Density distributions are changed to such a degree that bond and interface dipoles can change ~ i g n . ' ~ In ~ .general, ' ~ ~ semicore states should not be treated via a frozen-core or pseudopotential approach. I S 7 We define semicore orbitals loosely as d and f orbitals that have a significant overlap with and are close in energy to valence electrons. IS8 G. E. Engel and R. J. Needs, Phys. Rev. B41, 7876 (1990). '13' V. Fiorentini, M. Methfessel, and M. Schemer, Phys. Rev. B 47, 13353 (1993). A. Nazzal and A. Qteish, Phys. Rev. B 53, 8262 (1996). S.-H. Wei and A. Zunger, Phys. Rev. B 37, 8958 (1988). G. B. Bachelet and N. E. Christensen, Phys. Rev. 831, 879 (1985). N. E. Christensen and I. Gorczyca, Phys. Rev. B 50,4397 (1994). A. Garcia and M. L. Cohen, Phys. Rev. B 47, 6751 (1993). W. G. Aulbur, Z. H. Levine, J. W. Wilkins, and D. C. Allan, Phys. Rev. B51,10691 (1995). D. Cociorva, W. G. Aulbur, and J. W. Wilkins, unpublished.

64

WlLFRlED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Semicore states also significantly affect the electronic and optical properties of solids, since, for example, (1) band gaps decrease upon inclusion of semicore states as valence states by about 0.5 eV;'91*'92 (2) the calculated dielectric constant increases by 10-50% upon inclusion of semicore state^,'^'*'^^*'^^ and (3) for higher-order optical response functions, such as the coefficient of second-harmonic generation, inclusion of semicore states can give an effect sufficiently large to switch their sign.'9s-200Since quasiparticle calculations depend crucially on an accurate description of screening, a careful treatment of core states is necessary in 11-VI semiconductors, some 111-V materials, alkali metals, alkali earths, and noble metals. The LDA description of exchange and correlation between valence, semicore, and same-shell core electrons (e.g., Cd 5s, Cd 4d, and Cd 4p, 4s; Ref 52) is based on a local, energy-independent functional of the total density of the system. However, exchange and correlation effects between core and valence electrons are nonlocal and energy dependent. Dynamic correlation effects occur when fluctuating core dipoles interact with fluctuating valence densities. These correlations modify the effective interaction between valence electrons. Rather than interacting via the bare Coulomb potential u, valence electrons experience an interaction potential, W, = EC u, where EC describes the screening by core dipoles.20' The errors due to the LDA can be eliminated either by a core-polarization-potential (CPP) approach20'*202*203which is computationally efficient, since core electrons can still be treated on a frozen-core or pseudopotential level -or by treatment of core electrons as valence elect r o n ~ . ' ~ * ~The ~ , 'most ~ ' important error introduced by the pseudopotential approximation is the total neglect of core relaxation, that is, changes in the core orbitals due to a change in the chemical environment (e.g., crystal-field effects) and to hybridization of core and valence orbitals. These effects must be estimated in the C P P approach, whereas they are included on an LDA level when core electrons are treated explicitly. Also, the semicore d states of 11-VI materials energetically overlap the valence band and hence cannot be eliminated via a core-polarization potential. Explicit inclusion of the d electrons in the valence band amounts to a significant increase in computational cost for plane-wave-based quasiparticle calculations204 such as the H. R. Philipp and H. Ehrenreich, Phys. Rev. 129, 1550 (1963). T. Tomoyose, J . Phys. SOC.Jpn. 63, 1149 (1994). 19' S. H. Wemple and M. DiDomenico, Jr., Phys. Rev. B 3 , 1338 (1971). ,O0 B. F. Levine, Phys. Rev. B 7 , 2591 (1973). E. L. Shirley and R. M. Martin, Phys. Rev. B47, 15413 (1993). ,02 W. Miiller, J. Flesch, and W. Meyer, J . Chem. Phys. 80, 3297 (1984). 203 W. Miiller and W. Meyer, J . Chem. Phys. 80, 3311 (1984). ,04 Only one plane-wave calculation has been reported so far for ZrO, that includes the Zr 4s, 4p, and 4d core and semicore states in addition to the 5s electrons as valence electrons (see Ref. 57). 19'

19*

QUASIPARTICLE CALCULATIONS IN SOLIDS

65

one used in Ref. 43. Other approaches that describe semicore states via only nonlinear core corrections205have been shown to give band structures in good agreement with experiment for 11-VI materials.51 However, this agreement must be considered coincidental, since it relies upon error cancellation between core-relaxation and core-valence exchange and polarization effects. a. Core-Polarization Po ten tial Dynamical intershell correlation between semicore and valence electrons leads to induced polarization of the ion cores by the valence electrons. It can be taken into account in the valence-electron Hamiltonian by adding the energy contribution of all induced core dipoles in the electric field of the valence electrons and of the core dipoles, excluding s e l f - i n t e r a c t i ~ n . ~ ~ ~ + ~ ~ Core-polarization functions are approximated by static, atomic polarizabilities aJ ( J = Jth core at location RJ).201*202One obtains in a point dipole picture," (2.75) where Ej is the electric field at R, (2.76) and ri is the location of the ith valence electron and 2, the atomic number of the Jth core. The modification of the original valence Hamiltonian Vcpp can now be written aszo1

(2.77)

(t;

Besides the standard -a/(2r4) term see below), the electric fields due to the ions and valence electrons introduce (1) an additional term f i - I in the ion-ion interaction of the total energy of the system; (2) an additional local potential k - I ,which is felt by every electron; and (3) an additional '05

S. G. Louie. S. Froven. and M. L. Cohen. Phvs. Rev. B26. 1738 (1982).

66

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS A

interaction V,- between two valence electrons due to core-polarization effects.z0’~202~z03 The first two effects are easy to incorporate into standard band structure calculations. The contributions due to the valence electrons need further modifications, since valence electrons cannot be described as point charges. For instance, the single-electron potential for one valence electron far away from a core of polarizability a reduces correctly to the classical result - 0 1 / ( 2 r ~ ) .Ho ’ ~ ~wever, valence electrons can penetrate the core where the classical result diverges. Accordingly, a cut-off function has to be i n t r o d u ~ e d , and ~ ~ ~a ’typical ~ ~ choice is

(2.79) The sum over 1 in the above equations is a sum over angular momenta, and is the corresponding projection operator. For each atomic species one has to determine the parameters 01, r I , and Fe-,. Shirley et al.43used the experimental core polarizabilityZo7 and varied the rl (1 = 0, 1, 2) to obtain the correct removal energy for one valence electron of angular momentum 1 outside the ion core (rl = rz for 12 3) and Fe-e = 0.5(r0 rl). The two-electron potential is an effective interaction between two valence electrons due to core-polarization effects. This term screens the bare Coulomb interaction, which screening has to be taken into account in quasiparticle calculations. The effective interaction between valence electrons can be expressed as4943

k-e

+

(2.80) where { x J } are the self-consistent density-response functions of isolated cores.43 The effective interaction W, replaces the bare Coulomb interaction in the determination of the valence-valence self-energy in the GWA and in the plasmon-pole sum Shirley et al.43 used this modified GWA approach and evaluated the self-energy corrections using LDA wave functions and energies. Self-consistency in the quasiparticle energies is achieved and core relaxation effects are M. Born and W. Heisenberg, Z . Phys. 23, 388 (1924). Calculations of a in the time-dependent Hartree-Fock approximation, that is, including excitonic effects, have been reported as well (Refs. 67 and 201). ’06 ’07

67

QUASIPARTICLE CALCULATIONS IN SOLIDS

added a posteriori by comparison of LDA all-electron and pseudopotential calculations. Core-polarization-potential calculations give accurate quasiparticle energies for materials with semicore states that do not energetically overlap with the valence bands and, in particular, describe Ge correctly as an indirect rather than a direct emi icon duct or.^^ Table 10 shows that for Ge and GaAs, the fundamental band gaps are significantly improved compared to GWA calculations that treat core-valence interactions on an LDA level.2l o 208 "Crystal and Solid State Physics", in Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, vol. 17a, ed. by 0. Madelung, Springer, Berlin (1984). 2 0 9 As cited in Ref. 43. 210 Note that the correct ordering of the lowest conduction bands in Ge can also be achieved in calculations that have no self-interaction errors (Refs. 211 and 212). The energies of the lowest-lying conduction bands in these methods deviate significantly from experiment (,!IcB at L - r - X = 0.10 eV - 0.12 eV - 0.53 eV (Ref. 21 l), 1.01 eV - 1.28 eV - 1.34 eV (Ref. 212) versus experimental values of 0.74 eV - 0.90 eV - 1.2 eV given in Table 10.

TABLE10. BANDENERGYDIFFERENCES IN Ge AND GaAs IN ev. R e x r ~ T sFOR LDA FULLPOTENTIAL CALCULATIONS, QUASIPARTICLE (QP) CALCULATIONS WITH AN LDA TREATMENT OF CORE-VALENCEINTERACTIONS (CVI), QUASIPARTICLECALCULATIONS WITH A COREPOLARIZATION-POTENTIAL-BASED (CPP) TREATMENTOF CORE-VALENCE INTERACTIONS, AND EXPERIMENTARE SHOWN.ALL RESULTSINCLUDECORE-RELAXATION EFFECTS(ADDELI A POSTERIORI FOR THE QUASIPARTICLE CALCULATIONS). FUNDAMENTAL BAND GAPS IN THE D ~ R E N TAPPROACHES ARE PRINTED IN BOLDFACE.OVERALL AGRE~MENT BETWEEN EXPERIMENT AND QUASIPARTICLE CALCULATIONS IS IMPROVED UPON INCLUSION OF COREPOLARIZATION EFFECTS. IN PARTICULAR, Ge IS PREDICTED TO BE AN INDIRECT RATHERTHAN A DIRECT GAPSEMICONDUCTOR. (ADAPTED FROM REF. 43.) QP QUANTITY

LDA

CVI IN LDA

CVI IN CPP

0.53

ExF-T."

-0.26 0.55 -0.05 0.60

1.28 0.70 0.58

0.85 1.09 0.73 0.36

0.89 l.lOb 0.744 0.36

0.13 1.21 0.70 0.51 0.21

1.02 2.07 1.56 0.52 0.26

1.42 1.95 1.75 0.20 0.33

152 2.01 1.84 0.17 0.40

GaAs '8"

--t

'8"

'6r x6c

'SO +

L6c

L6c --* x 6 c

x,,

+

x7c

"Ref. 208 unless noted otherwise; bRef. 209.

68

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

b. Explicit Treatment of Core Electrons Local-orbital basis function^'^ such as Gaussian orbitals (Refs. 52, 140, 141, 213, and 214) or LMT0'45*215allow the explicit treatment of core electrons on the same level of approximation as the valence electrons. They include relaxation and hybridization effects on the LDA level and exchange and dynamic screening effects of core and semicore electrons on the GWA level. The advantage of local-orbital basis functions compared to plane-wave methods is the drastically reduced number of basis functions. Including semicore d states as valence electrons in non-self-consistent GWA calculations for 11-VI materials leads to disappointing First, the LDA d levels are shifted upwards in energy, that is, away from experiment; second, the self-energy operator is nondiagonal in the basis of LDA states. Unphysical self-interactions in LDA -that is, incomplete cancellations between Hartree and exchange potentials -shift the d bands up in energy and lead to a too strong hybridization with the anion p bands. As a consequence of the unphysical p-d mixing, the LDA states are not good approximations to quasiparticle states and the self-energy is nondiagonal. An iterative determination of quasiparticle wave functions and energies in principle eliminates the self-interaction error contained in the wave functions and energies. However, self-consistency alone is not sufficient, since the exchange interaction with the 4s and 4p electrons must be taken into account as well.52 The localized semicore d states strongly overlap with the wave functions of the same-shell s and p electrons, which leads to large exchange energies between the corresponding state^.^' Treatment of the s, p , and d electrons as valence electrons changes the non-self-consistent self-energy operator qualitatively5' in three ways: (1) quasiparticle energy corrections lower the semicore d states and lead to improved agreement with experiment; (2) the self-energy becomes diagonal in the LDA basis; and (3) in the case of CdS, the exchange between 4s and 5s electrons increases the band gap by 0.85 eV. The energy of the semicore 4d level is unchanged (E4d= - 7.4 eV) regardless of whether 4d, 4d and 4p, or 4d, 4p, and 4s electrons are treated as valence electrons." This suggests that hybridization between cation d and anion p orbitals is not affected significantly by the s and p core electrons. Remaining self-interaction errors in the wave function may cause errors in the non-selfconsistent GWA calculations on the order of about 1 eV. 'I1 212 '13

'14

'I5

M. M. Rieger and P. Vogl, Phys. Rev. B52, 16567 (1995). M. Stadele and A. Gorling, private communication. M. Rohlfing, P. Kruger, and J. Pollmann, Phys. Rev. B56, R7065 (1997). M. Rohlfing, P. Kriiger, and J. Pollmann, Phys. Rev. B57, 6485 (1998). F. Aryasetiawan, Physica B 237-238, 321 (1997).

QUASIPARTICLE CALCULATIONS IN SOLIDS

69

As shown earlier, nonlocal exchange and inclusion of screening effects due to shallow core states are important and affect quasiparticle properties.43.197*198 Exchange can have qualitative effects on the energy dependence of the self-energy, as demonstrated by Horsch, von der Linden, and Lukas216 and discussed in Section V.26a. Moreover, the results of Ref. 52 clearly illustrate the qualitative importance of exchange for the determination of band structures. The qualitative accuracy of the results in Ref. 52 is less certain, however, because of technical difficulties that are detailed in Section 111.15a. Approximate self-consistent GWA calculations improve the position of the d bands in comparison to experiment,213 although (1) the discrepancy between theory and experiment for energy gaps is increased upon selfconsistency -that is, the improvement of quasiparticle energies is not uniform; (2) vertex corrections have not been included although they may be important; and (3) self-consistency increases discrepancy with experiment for all other known calculations (see the next section) with the exception of total energies and charge conservation. The most drastic example of point (1) is CdS:213The 4d level equals -7.2 eV in LDA,’17 -9.1 eV in a partially self-consistent GWA calculation, and -9.2/-9.5 eV in experiment, as shown in Table 11. The corresponding values for the band gap are 2.45 eV, 3.21 eV, and 2.50/2.55 eV. Partial self-consistency leads to an overestimation of the direct band gap by about 0.7eV. Vertex corrections (point (2)) increase calculated RPA core polarizabilities by about 30%67and may be important for semicore levels as well. Semicore states in ZnSe, GaAs, and Ge have been determined within ~ ~summarized in standard GWA by Aryasetiawan and G u n n a r ~ s o n . ’As Table 11, standard GWA calculations describe semicore levels within a 5- 15% range. The remaining discrepancies with experiment are partially due to the lack of self-consistency, vertex corrections, and self-interaction errors in the LDA wave functions. 13. SELF-CONSISTENCY General Remarks. Most current GWA calculations have three characteristics: (1) the Green function is calculated in an appropriate single-particle basis following the “best G, best W ” philosophy (Hartree-Fock, LDA, or in some cases empirical pseudopotentials122~1 (2) the screened interaction is determined in RPA using the chosen single-particle wave functions 239224);

216

2’7

P. Horsch, W. von der Linden, and W.-D. Lukas, Sol. State Comm. 62, 359 (1987). The quoted value in Ref. 52 is slightly diBerent and equals -7.4 eV.

70

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

TABLE 11. CALCULATED BINDING ENERGIES IN eV OF THE SEMICORE STATESCd 4d IN Cumc CdS, Ge 3d IN Ge, AND Si 2p IN Si. THEZEROOF THE ENERGY SCALE IS GIVEN RESPECTIVEVALENCE-BAND MAXIMUM. THE EXPERIMENTAL DATAARE WEIGHTED AVERAGESOF SPIN-ORBIT-SPLIT LEVELS AND ARE T m AS QUOTED IN REF.213. THES ~ LGLDA S AND GQPSCORRESPOND TO GWA CALCULATIONS USING AN LDA PROPAGATOR OR A PROPAGATOR CALCULATED m A RENORMALIZED SPECTRALFUNCTION (APPROXIMATELY SELF-CONSISTENT GWA CALCULATION; SEE REF. 213). ALL VALUESARE TAKEN FROM REF.213 mTHE FOLLOWING EXCEPTIONS: IN THE CASEOF Ge WE ALSO LISTLDA, GLDAWA, AND SLATER-TRANSITION-STATE (STS) RESULTS OBTAINED BY AN LMTO FORCdS, WE LISTTHE LDA A M ) SELF-INTERACTIONAND RELAXATION-CORRECTED PSEUDOPOTENTIAL (SIRC-PSP) &.SULTs OF REF. 218. BY THE

GWA LDA Cd 4d Ge 3d Si 2p

-7.2 -6.8” -24.6 -24.4‘ - 89.4

GLDA

-8.1 -27.7 -28.5‘ -94.8

’

GQPS

-9.1 -9.7“Vb - 30.0 -30.1‘*’ - 100.4

Em. -9.2,’ -9.5’

- 29.5,’ -29.7h -99.0,’ - 100.0*

“Ref. 218; bSIRC-PSP ‘Ref. 145; dSTS; ‘Ref. 219; fRefs. 220 and 221; 8Ref. 222; hRef. 223.

and energies; and (3) in the majority of cases the frequency dependence of the dielectric response is further approximated by plasmon-pole models. This approach, which we call the “standard” GWA, or GoWoA in what follows, works well in practical applications and generally leads to good agreement with experiment for quasiparticle energies. However, from a principle point of view it has several shortcomings. A consistent approach to GWA calculations evaluates the RPA polarizability and the GWA self-energy using Hartree wave functions and energies and iterates the calculation until self-consistency in G and W is achieved. D. Vogel, P. Kriiger, and J. Pollmann, Phys. Rev. B54, 5495 (1996). A. P. J. Stampfl, P. Hofmann, 0.SchalT, and A. M. Bradshaw, Phys. Rev. B 55,9679 (1997). 220 L. Ley, R. A. Pollak, F. R. McFeely, S. P. Kowalczyk, and D. A. Shirley, Phys. Rev. B9, 600 (1974). N. G. Stoffel, Phys. Rev. B28,3306 (1983). 2 2 2 Zahlenwerte und Funktionen aus Naturwissenschajien und Technik, eds. A. Goldmann and E. E. Koch, Landolt-Bornstein, New Series, Group 111, vol. 23a, Springer-Verlag,Berlin (1989). 223 W. Monch, in Semiconductor Surfaces and Interfaces, eds. by G. Ertl, R. Gomer, and D. Mills, Springer Series in Surface Sciences, vol. 26, Springeer-Verlag,Berlin (1993). 224 W. von der Linden and P. Horsch, Phys. Scripta 38,617 (1988). ’18

’”

71

QUASIPARTICLE CALCULATIONS IN SOLIDS

Non-self-consistent results for quasiparticle energies obtained starting from Hartree theory differ from, for example, LDA-based “best G, best W” approaches, since Hartree wave functions and energies are qualitatively different from the corresponding LDA quantities.88 Self-consistencyeliminates the dependency of the final results on the starting wave functions and energies, as shown explicitly for a 1D semiconducting wire by de Groot, Bobbert, and van H a e ~ i n g e n . ~ ~ ’ An approximate procedure for incorporating self-consistency into a standard GWA calculation is to shift the energy spectrum of the independent-particle Green function so that the Fermi levels of the interacting system and the noninteracting system are aligned.3.75This approximation (1) leads to an increase of fundamental band gaps by 0.1 to 0.2eV in semiconductors and insulator^;^^ (2) improves the satellite spectrum of a Hubbard cluster significantly;z26(3) leads to improved charge conservation in GWA calculations;2z7and (4)is necessary for systems like NiO to obtain reasonable agreement with e ~ p e r i m e n t A . ~ slightly ~ modified version for insulators (see, for instance, Ref. 46) shifts the independent-particle valence (u) and conduction (c) bands in a linear, isotropic manner until they agree with the corresponding quasiparticle energies: E,k

= (1

+

A,)E,k

+ B,,

E,k

= (1

+ Ac)Eck + B,.

(2.81)

Here A iand Bi,i = u, c, are fitting parameters determined via the band gaps at two high-symmetry points. Note, however, that Ricezz8advised against using a shifted self-energy since the energy dependence of the self-energy is largely canceled by other self-consistencyeffects. In constrast to the energies, the independent-particle wave functions are left unchanged and not replaced by the interacting spectral function. The corresponding assumption that no significant quasiparticle weight is transferred to the satellite spectrum contradicts experiment since 10-50% of the total spectral weight resides in the incoherent background and the plasmon satellites. Fleszar and Hanke”’ examined the quasiparticle energy shifts in Si as a function of energy (- 15 eV < E < 70 eV) for a non-self-consistent GWA calculation that (1) takes the frequency dependence of the screened interaction fully into account and (2) accounts for finite lifetime effects, that is, the finite imaginary part of the self-energy.In most GWA calculations, point (1) is treated approximately via plasmon-pole models and point (2) is neglected. Figure 15 (bottom panel) shows that the neglect of the imaginary part of the

’” H. J. de Groot, P. A. Bobbert, and W. van Haeringen, Phys. Rev. B52,11OOO (1995). 226

T. J. Pollehn, A. Schindlmayr,and R. W. Godby, J . Phys. Cond. Mat. 10, 1273 (1998).

”*

T. M. Rice, Ann. Phys. 31, lOO(1965).

’*’A. Schindlmayr,Phys. Rev. B56,3528 (1997).

72

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS 4

u J 1.6

5

Y

4 41B 4

1 1.8

1

0.8

0 4.6 -1

- 1 0 0

i

o

m

5

o

a

m

6

0

7

0

ELDA(ev)

FIG. 15. Top panel: Silicon quasiparticle energy shifts E,, - EL,, as a function of the LDA energy E,,,. The overall shape of the energy shifts is similar to the shape of quasiparticle energy shifts in jellium4~z29 (not shown) and Ag (see Fig. 42 and Section V.26a). Bottom panel: Diagonal matrix elements of the imaginary part of the Si self-energy with respect to LDA orbitals calculated at the energy of the corresponding quasiparticle peak. The shape of the imaginary Si self-energy is similar to the shape of the non-self-consistent decay rate of jellium (see Fig. 18). In both figures, the full circles are the result of a standard (RPA) GWA calculation, the empty circles have the (time-dependent) LDA vertex function included (GWT calculations, see Section 11.14). The self-energy is calculated along the L-T-X line. (From Ref. 102.)

QUASIPARTICLE CALCULATIONS IN SOLIDS

73

self-energy is only justified in the immediate vicinity of the band gap. As can be seen in Fig. 15 (top panel), quasiparticle energy shifts arelo2 linearly decreasing for occupied states, roughly constant for conduction band states with 0 IE < 25 eV, and roughly linearly increasing for energies E 2 25 eV. Equation (2.81) does not capture the behavior of quasiparticle corrections over an extended energy range. The overall shape of the energy dependence of quasiparticle corrections in Si is reminiscent of the behavior in j e l l i ~ m ~ ~ ~ and Ag (see Section V.26a and Ref. 216). A truly self-consistent GWA calculation not only eliminates the dependency of results on the initial starting point of the iteration but is also charge and energy conserving in a Baym-Kadanoff ~ e n s e . ~ It' . has ~ ~ five disadvantages: (1) the self-consistent fundamental band gap of Si ( x 1.9 eV) exceeds experiment (1.17eV) by as much as LDA (x0.5eV) falls below experiment;230 (2) the band width of simple metals is larger than independentparticle results and contradicts experiment; 2 3 0 * 2 3 (3) plasmon-pole peaks in the spectral function are strongly suppressed, in contrast to experiment (Refs. 230, 231, 232, and 233); (4) the screened interaction W does not fulfill the f-sum rule and is only a mathematical tool to determine the self-energy-that is, it has no physical meaning;231*232 and ( 5 ) selfconsistent GWA calculations are computationally very demanding. These points will be discussed in more detail below. The increasing discrepancy between theory and experiment upon full self-consistency clearly indicates that missing physics in the form of vertex corrections must be included in dynamically screened interaction calculations and that inclusion of self-consistency alone is not justified due to the increase in computational cost and the loss of predictive power. The success of standard GWA calculations seems to result from error cancellation between the lack of vertex corrections and the omission of self-consistency. Applications. Most fully self-consistent GWA calculations have been reported for model systems: a one-dimensional semiconducting wire,225a Hubbard cluster,234and the homogeneous electron Self-consistent calculations for real solids were reported very recently by Schone and E g u i l ~ z . ~Partially ~' or approximate self-consistent calculations for the same model ~ y ~ t e m and~for ~the ~semiconductors ~ * ~ ~ Si ~and*C' ~l 9 have ~ ~ 19232

K. W.-K. Shung, B. E. Sernelius, and G. D. Mahan, Phys. Rev. B36,4499 (1987). W. D. Schone and A. G. Eguiluz, Phys. Rev. Lett. 81, 1662 (1998). 2 3 1 B. Holm and U . von Barth, Phys. Rev. B57, 2108 (1998). 232 E. L. Shirley, Phys. Rev. B54, 7758 (1996). 2 3 3 A description of (multiple) satellites that improves upon the GWA can be obtained by cumulant expansions (Ref. 77) or by a T-matrix approach (Ref. 71). For a review. see Ref. 14. 234 A. Schindlmayr, T. J. Pollehn, and R. W. Godby, Phys. Rev. B58, 12684 (1998). 2 3 5 A. G. Eguiluz and W.-D. Schone, Mol. Phys. 94, 87 (1998). 229

230

74

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

been reported as well (see below). In particular, the calculations by Shirley232 and by von Barth and Holm'" determined the propagator selfconsistently but calculated the screened interaction from the independentparticle wave functions and energies. We call this approach the GWOA approach, in contrast to the non-self-consistent GoWoA and fully selfconsistent GWA approaches. Results. Fully self-consistent GWA calculations in jellium find spectral weight functions whose plasmon satellite weight is significantly reduced,231*235 as shown in Fig. 16. This trend is in clear contradiction to experiments on simple metals, which show a narrow plasmon satellite at wpl below the quasiparticle energy and smaller plasmon satellites further down in energy. Note that the spectral function of Ref. 235 has discernible plasmon peaks rather than a featureless background, as in the spectral function of Ref. 231. This difference may result from technical differences in the calculations (finite T versus T = 0 approach, fitting of the spectral function versus numerical evaluation, Loss of plasmon spectral weight translates into an increase in quasiparticle weight, as shown in Fig. 17. Also, the broad and featureless plasmon spectrum of Ref. 231 reduces inelastic quasiparticle-plasmon collisions. Hence, the quasiparticle decay rate of self-consistent GWA calculations shown in Fig. 18 does not increase strongly due to the onset of plasmon production -which is what is physically expected-in contrast to the GOW'A and the GW'A decay rates. Figure 19 shows that the self-consistent GWA band width of j e l l i ~ m ~ ~ ' is larger rather than smaller than the independent-electron band ~ i d t h . ~The ~ ~ latter * ~is ~already ~ . ~too~ large ~ in comparison to the experimental band width of simple metals. To explain the discrepancy between theory and experiment, several combinations of vertex corrections' 84.232.240 and the inclusion of surface effect^^^'.^^' have been suggested, as discussed in the next section and Section V.26b. A generally accepted explanation for this discrepancy has not yet emerged. Total energy calculations of jellium in the energy-conserving self-consistent GWA agree with Monte-Carlo data to within 25% (I, = 2) and 1% (I, = 4).231Schindlmayr, Pollehn, and G ~ d b noted y ~ ~ in this ~ context that the total energy of a finite, two-leg Hubbard ladder is always raised by

236

H. J. de Groot, R. T. M. Ummels, P. A. Bobbert, and W. van Haeringen, Phys. Rev. 854,

2374 (1996). 237

239 240

241

U. von Barth and B. Holm, Phys. Rev. 854,8411 (1996). A. G. Eguiluz, private communication. B. Holm and U. von Barth, private communication. G. D. Mahan and B. E.Sernelius, Phys. Rev. Left. 62,2718 (1989). K. W.-K. Shung and G. D. Mahan, Phyx Rev. Left. 57, 1076 (1986).

75

I

-20

10

0

-10

-20

-ray (ev) FIG. 16. Fully self-consistent one-particle spectral function A(k, o) for the homogeneous electron gas evaluated at k = k,, r, = 4, T = 0 (Ref. 231, upper panel), and at k = 0.99kF,r, = 5, finite T (Ref. 235, lower panel). The results in the upper panel are compared with a partially self-consistent spectral function obtained from a GW'A calculation.237 Those in the lower panel are compared with results obtained after the first and third iteration toward selfconsistency. Both calculations show that self-consistency (1) sharpens and increases the weight of the quasiparticle peaks (see also Fig. 17), and (2) reduces the spectral weight of the plasmon satellite peaks. The two self-consistent calculations differ in the shape of the plasmon peaks.

76

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

FIG. 17. The quasiparticle weight Z , for jellium at rs = 4 determined by a fully selfconsistent GWA calculation (solid line), a partially self-consistent GW’A calculation (dashed line), and a standard G’W’A calculation (dotted line). Self-consistency leads to a systematic increase in quasiparticle weight and to a loss of structure due to the suppression of plasmon satellites in the fully self-consistent result. (From Ref. 231.)

self-consistent calculations relative to standard GWA calculations. This is true not only for RPA-based GWA calculations but also for GWA calculations that include vertex corrections and for a Bethe-Goldstone approach based on the T - m a t r i ~ . ’Depending ~~ on the approximation used and the parameters of the model, the self-consistent energy can even be in worse agreement with experiment than the non-self-consistent energy. Schindlmayr, Pollehn, and G ~ d b y ’interpreted ~~ this fact as an indication that the excellent total energies obtained for the electron gas with self-consistent GWA may be fortuitous. The qualitative changes in the Hubbard-ladder spectral function at self-consistency (broadening of high-energy plasmons, sharpening of quasiparticle features) agree with the findings of Holm and von BarthZ3land of Eguiluz and S ~ h O n e . ’ ~ ~ The effects of self-consistency for real materials are qualitatively similar to those observed in j e l l i ~ r n . ’ A ~ ~self-consistent GWA c a l ~ u l a t i o n ’overes~~ timates the valence band width of potassium by more than 1 eV (2.64eV compared to 1.60 eV in experiment, 2.21 eV in LDA, and 2.04 eV in standard GWA). Self-consistency increases the quasiparticle weight by 20% (0.72 compared to 0.60 non-self-consistently). Results for Si are equally dra-

QUASIPARTICLE CALCULATIONS IN SOLIDS

77

FIG.18. Decay rate r, of jelliurn at r, = 4 determined by a fully self-consistent GWA calculation (solid line), a partially self-consistent GW’A calculation (dashed line), and a standard G’W’A calculation (dotted line). The two latter results show a region with a much larger decay rate (shorter lifetime) caused by inelastic collision with plasmons. This physically correct feature is absent from the fully self-consistent GWA result. (From Ref. 231.)

ma ti^:'^' The direct band gap is 4.02 eV in self-consistent GWA compared to 3.4 eV in experiment (2.57 eV in LDA, 3.27 eV in standard GWA), and the indirect band gap is 1.91 eV compared to 1.17eV experimentally (0.53 eV in LDA, 1.34eV in standard GWA). The self-consistent occupied band width (13.1 eV) is marginally in better agreement with experimentz4’ (12.5 f 0.6 eV) than the result of a standard GWA calculation (11.65eV, 11.93 eV in LDA). Total Energy and Density. Early total-energy calculations for jelliumz4’ and bulk Si244based on the Galitskii-Migdal were reported by Levin ef al.243and Farid ef al.744 In addition, Rieger and G ~ d b y recently ’~~ determined the density of Si and Ge in the GWA using the real-space/ For a discussion of the experimental value of 12.5 eV, see Section III.18a. Y. Levin, C.D. Wu, and Y . Bar-Yam, Comp. Mat. Sci. 3, 505 (1995). 244 B. Farid, R. W. Godby, and R. J. Needs, in 20th International Conference on the Physics of Semiconductors, eds. E. M . Anastassakis and J. D. Joannopoulos, vol. 3, World Scientific, Singapore (1990), 1759. 245 V. Galitskii and A. Migdal, Zh. Eksp. n o r . Fiz. 34, 139 (1958) [Sox Phys. JETP 7, 96 (1958)l. 246 M.M. Rieger and R. W. Godby, Phys. Rev. 858, 1343 (1998). 242 243

78

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

4.2-

ur' 4.6 Y

e

-0.4

-1.4

0.0

-

'

02

0.4

0.6

0.8

1.0

I

12

WkF FIG. 19. Quasiparticle dispersion Ek for a fully self-consistent GWA calculation (solid line), a partially self-consistent GW'A calculation (dashed line), and a standard G'W'A calculation (dotted line), and for noninteracting electrons (dashed-dotted line). Only the standard G'W'A calculation leads to band-width narrowing in comparison to the free-electron result. However, this observed band-width narrowing is too small to explain experimentally observed narrow band widths in simple metals. Self-consistency increases the band width in comparison to free electrons and hence leads to increased deviation from experiment. Data shown correspond to r, = 4 and are taken from Ref. 231.

imaginary-time algorithm. In the latter work, the Hartree potential of Eq. (1.3) is iterated to self-consistency in the presence of a non-self-consistent GWA self-energy. With Go as the LDA Green function, AV, as the change in the Hartree potential due to a change in density, and A as a constant shift that aligns the Fermi level of the noninteracting and interacting Green functions (see Eq. 2.81), the interacting Green function can be obtained via Dyson's equation247(Eq. (2.5)):

+ Go(io)[C(io) + AV,

G(io) = Go(io)

- V, - A] G(io),

(2.82)

from which the density follows via 3

ro

n(r) = -

do ImG(r, r; a). R

(2.83)

-00

247 All quantities are to be interpreted as matrices with respect to plane-wave coefficients (see Section III.15b).

QUASIPARTICLE CALCULATIONS IN SOLIDS

79

Both equations must be solved self-consistently, since a change in the density yields a change in the Hartree potential. Within this approach248 Rieger and Godby found that (1) density is conserved to within 0.3% in Si and 0.05% in Ge; (2) LDA and GWA structure factors differ by less than 0.1YOand agree well with experiment; and (3) a self-consistent Hartree potential leads to band structure changes of less than 0.1 eV.

14. VERTEXCORRECTIONS As follows from Eq. (2.9), vertex corrections to the self-energy describe the linear response of the self-energy to a change in the total electric potential of the system. Vertex corrections in dielectric screening describe exchange and correlation effects between an electron and other electrons in the screening density. For example, if screening is provided by a spin-up electron, other spin-up electrons cannot get too close and are less likely to help with the screening." As a consequence, the screening is weakened and the interaction is strengthened. Such short-ranged vertex corrections improve the description of quasiparticles and of low-energy satellites. Other, long-ranged, vertex corrections improve the description of the plasmon satellites, as discussed in detail in Refs. 14 and 249. Vertex corrections increase correlation functions such as the densitydensity response function, whereas self-energy insertions in the Green function, that is, self-consistency corrections, reduce correlation functions. DuBois250*25'was the first to notice that, within the context of the GWA, vertex-correction diagrams and self-energy diagrams cancel each other to a large degree. Because of this cancellation, discrepancies between experiment and self-consistent GWA calculations, such as those described in the previous section, are to be expected. In principle, a consistent treatment of vertex corrections and self-consistency is required for a quantitative description of experiment. M a h a n " ~suggested ~ ~ ~ on the basis of the work of Baym and Kadanoff and Ward,253 that a consistent procedure to include vertex corrections in the polarizability P can be obtained as follows. The vertex diagrams associated with a self-energy Z -X governs the equation of motion of the interacting Green function G- that should be included in the 62963

248 Special care must be taken to account for core charges that are missing in the pseudopotential approach of Ref. 246. 249 C. Verdozzi, R. W. Godby, and S. Holloway, Phys. Rev. Lett. 74, 2327 (1995). D. F. DuBois, Ann. Phys. 7, 174 (1959). D. F. DuBois, Ann. Phys. 8, 24 (1959). 2 5 2 G. D. Mahan, Int. J . Mod. Phys. 8 6 , 3381 (1992). 2 5 3 J. C. Ward, Phys. Rev. 78, 182 (1950).

80

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

determination of correlation functions are obtained by attaching an external interaction line to the internal Green function lines of the self-energy diagrams. As a consequence, a consistent conserving approximation for, for example, the inclusion of ladder diagrams in P requires the use of the self-consistent single-particle propagator G rather than the independentparticle propagator As discussed below, most of the self-consistent or vertex-corrected calculations are at variance with Mahan’s suggestion in one way or another. Early work on vertex corrections in the homogeneous electron gas is not reviewed here (for a general review, see Ref. 10). The present focus is on work related to the effect of vertex corrections255on the electronic properties of insulator^^^^"^ as well as on recent work on the homogeneous electron gas and simple metals (see Refs. 232, 240, 256, 257, and 258). Generally, vertex corrections have been studied for excitations close to the Fermi level. Little is known about vertex corrections of high-energy excitations. This is unfortunate, since the physics of high- and low-energy excitations can differ significantly. An example is the energy dependence of the self-energy of Ag discussed in Section V.26a. Two possibilities to include vertex corrections beyond the standard GWA approach are being studied in the literature: (1) an iterative evaluation of Hedin’s equations (Eq. (2.10)) starting from Hartree theory,z59 and (2) an expansion of the self-energy C and the polarizability P to order n and n - 1 in the screened interaction N respectively, with n = 2 being the case studied so far (Refs. 3, 119, 232, and 236). Method (1) reformulates the BetheSalpeter equation for the vertex function and has been applied to a two-dimensional, 3 x 3 Hubbard cluster. Method (2) is the method of choice for most of the current work on vertex corrections although the expansion of C in terms of W is only asymptotically or conditionally ~onvergent,’~’and Minnhagen75.’52 found unphysical results-a negative spectral density- by extending the GWA approach for the self-energy of the homogeneous electron gas to second order in W Method (2) is based on the expectation that higher-order contributions to C and P are negligible3 or are canceled by self-consistency effects, which is empirically supported by the good agreement between standard GWA calculations and experiment. In what follows, we discuss both methods starting with method 2 5 4 This assumes a bare interaction between the electrons. Screened interactions have propagators in internal lines, which lead to additional vertex corrections (Ref. 10). 2 5 5 For a discussion of vertex corrections see also B. Farid, Phil. Mag. B76, 145 (1997). 2 5 6 M. Hindgren and G O . Almbladh, Phys. Rev. B56, 12832 (1997). J. E. Northrup, M. S. Hybertsen, and S. G. Louie, Phys. Rev. Lett. 59, 819 (1987). ”* M. P. Surh, J. E. Northrup, and S. G. Louie, Phys. Rev. B38, 5976 (1988). A. Schindlmayrand R. W. Godby, Phys. Rev. Lett. 80, 1702 (1998).

’” ’”

81

QUASIPARTICLE CALCULATIONS IN SOLIDS

(1). In addition, we discuss approaches that include vertex corrections on an LDA level as well as techniques that combine vertex-correction and selfconsistency diagrams. Iterative Solution of Hedin's Equations. Schindlmayr and G ~ d b y ' ~ ~ solved Hedin's equations iteratively starting from Hartree theory. The key feature of their approach is the replacement of the implicit Bethe-Salpeter equation for the vertex function (compare Eq. (2.10), n = order of the iteration)

x

P+ ')(6, 7; 3)d(4, 5, 6, 71,

(2.84)

by an expression that depends only on the self-energy of the nth iteration, the vertex function of the first iteration, and G('!

x F1)(6,7; 3)d(4, 5, 6, 7),

(2.85)

A further advantage of Eq. (2.85) is that the functional derivative 6ZC'")/6G'o) can in principle be calculated at all levels of iteration.259For n = 1, a vertex correction r(')is obtained that contains not only terms to second order in the RPA screened interaction but also terms of zeroth and first order. Numerical results for a 3 x 3 Hubbard model suggest that an iterative evaluation of Hedin's equations may improve upon the plasmon satellite spectrum compared to standard GWA calculations. Vertex Corrections to Second Order in the Interaction. The second-order contribution in the bare Coulomb interaction to the self-energy and the energy gap of Si is negligible compared to the first-order contribution.88 In a bond-orbital approximation199260 the second-order terms can be written as the sum of two partially canceling contributions:260 (1) the partial blocking of ground-state correlations within a bond upon addition/removal of a particle, and (2) exchange enhancements of the polarization cloud of the electron (hole) for the bonds adjacent to the bond under consideration. This suggests that second- and higher-order effects have a negligible impact on band gaps and can be safely neglected. R. Daling, P. Unger, P. Fulde, and W. van Haeringen, Phys. Rev. B 43, 1851 (1991).

82

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

However, vertex corrections evaluated with a screened rather than a bare interaction lead to a band-gap narrowing261 of -0.26 eV at the r point of Si, which amounts to -40% of the standard GWA quasiparticle correction of about 0.7 eV.262Although higher-order corrections are included in the self-energy, they are neglected in the polarizability, in conflict with Mahan's suggestion." The second-order contribution partially cancels the additional Effects due to the band-gap increase of 0.7 eV due to self-con~istency.~~~ consistent use of vertex corrections not only in the self-energy but also in the polarizability are discussed in the paragraph Ertex Corrections and Partial Self-Consistency below. LDA Vertex Corrections. Extension of a standard GWA (GWRPA) calculation for SiS9by including LDA mean-field vertex corrections in the polarizability (GWK,) or in the polarizability and the self-energy (GWT) shows that relative energies are modified by including vertex corrections only in the polarizability and that cancellations between vertex corrections to the polarizability and to the self-energy occur for relative but not for absolute energy shifts. The GWT approximation uses an LDA exchangecorrelation potential263as zeroth-order self-energy approximation and can be expressed in a GWA-like form using an effective electron-electron interaction tt and a dielectric matrix P9

X ( 1 , 2) = iG(1, 2)@(1, 2),

(2.86)

where

tt=

u[1 - P O ( U

+ Kxc)]-I= U / P , (2.87)

Table 12 shows that these vertex corrections included only in the screening reduce band gaps by 0.1 to 0.2 eV and valence band widths by about 0.5 eV in Si. Subsequent inclusion of vertex corrections in the self-energy causes a band-gap increase that cancels the effect of the vertex corrections in the screening. Note the qualitative difference between this band-gap increase" and the band-gap decrease obtained by including dynamical vertex corrections in X,119*262 as discussed in the paragraph Vertex Corrections to Second 261 We take the corrected value of Ref. 119 rather than the original value of 0.12 eV quoted in Ref. 262. 262 P. A. Bobbert and W. van Haeringen, Phys. Rev. B49, 10326 (1994). 263 Note that because of the use of the LDA, the diagrammatic structure of these vertex corrections is not well defined.

83

QUASIPARTICLE CALCULATIONS IN SOLIDS

TABLE12. QUASIPARTICLE CORRECTIONS (GWA MINUSLDA ENERGY VALUES IN eV) FOR Si CALCULATED WITHINGWRPA (NO VERTEX CORRECTIONS FOR THE DIELECTRIC MATRIX E AND SELF-ENERGY Z), GWK, (VERTEX CORRECTIONS IN E BUT NOT IN Z), AND GWT (VERTEX CORRECTIONS M E AND Z; CORRECTIONS LISTEDIN FIRSTF m Rows, LDA GAP = 0.52 eV). THE LAST Row IS AN ABSOLUTEQUASIPARTICLE CORRECTION TO THE VALENCE-BAND MAXIMUM. LDA VERTEXCORRECTIONS IN THE POLARIZABILITY REDUCEFUNDAMENTAL BANDGAPS BY ABOUT 0.1 eV TO 0.2eV AND THE VALENCE-BAND WIDTHBY ABOUT 0.5 eV. INTO ACCOUNT FOR BOTHTHE POLARIZABILITY AND TAKINGVERTEX CORRECTIONS THE SELF-ENERGY DOES NOT HAVEA SIGNIFICANT EFFECTON RELATIVE ENERGY DIFFERENCFSCOMPARED TO RPA-BAs~, GWA, BUT SHIFTS THE ABSOLUTE ENERGY SCALEBY ABOUT 0.4 eV. (ADAPTED FROM REF.89.)

Direct gap at r Direct gap at X Direct gap at L Valence bandwidth Minimum gap Valence-band maximum

GWRPA

GWKX

GWT

0.64 0.78 0.68 - 0.56 0.63 -0.36

0.56 0.57 0.58 - 1.01 0.59 -0.44

0.65 0.73 0.72 -0.48 0.66 0.01

Order in the Interaction above. Absolute energies between GWRPA and GWT are shifted by about 0.4 eV. Whereas in GWRPA the highest occupied state is shifted by - 0.36 eV, it is shifted by only + 10 meV in GWT. Since the highest occupied state is believed to be well described by LDA, GWT could turn out to be a better starting point, for example, for valence-band offset calculations at interfaces. Vertex Corrections and the Alkali Metal Band Width. In jellium and simple metals, band-width corrections due to vertex corrections to the polarizability P and to the self-energy C cancel, leading to a reproduction of RPA results, as shown in Fig. 20, and an overestimation of the experimental band width. The discrepancy with experim.ent can be eliminated by (1) inclusion of vertex corrections only in the polarizability-that is, use of a GWK,, approach -and self-consistent determination of quasiparticle energies;'84*258(2) careful treatment of surface and finite lifetime effects (Refs. 229,240,241, and 265); and (3) inclusion of vertex corrections in both P and C in a partially self-consistent calculation.232 Approach (1) is motivated by Ward-identity-like arguments by Strinati et al.156(see Section II.9b). Mahan and c o l l a b ~ r a t o r sand ~ ~Ting, ~ ~ ~Lee, ~ ~ and ~ ~ ~Quinn266 ~ empha264

P. Vashishta and K. S . Singwi, Phys. Rev. B6, 875 (1972).

265

K.W.-K. Shung and G. D. Mahan, Phys. Rev. B38,3856 (1988).

266

C. S. Tin& T. K. Lee, J. J. Quinn, Phys. Rev. Leu. 34, 870 (1975).

84

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

-1.0

0

1

2

5

4

5

6

rs

FIG. 20. Band narrowing relative to the independent-particle band width versus electron density parameter rr for the homogeneous electron gas and several approaches to the determination of the self-energy. A positive value corresponds to an actual narrowing of the bands. The two dashed lines correspond to a standard (RPA) GWA calculation (in which vertex corrections are neglected in the dielectric function E and the self-energy C) and a GWA calculation in which vertex corrections are included only in E (I'= 1). The solid line corresponds to a calculation that includes vertex corrections in both E and Z using a Vashista-Singwi many-body local-field factor.264This curve is universal in the sense that it is hardly a5ected by a change of the many-body local-field factor. The r = 1 calculation leads to band narrowing that is about twice as large as the RPA band narrowing for metallic densities (rs z 2 - 6) and compares favorably with experiments for simple metals. Adding consistent vertex corrections to the self-energy cancels the effect of vertex corrections in the dielectric screening, giving results in close agreement with RPA. (From Ref. 240.)

sized that a consistent way of handling vertex corrections requires the inclusion of identical vertex corrections in P and C. Both contributions cancel to a large degree as shown in Fig. 20, leading to a band width nearly identical to the one obtained in RPA and too large in comparison with experiment. Vertex corrections are neglected in approach (2).229 The required band-width narrowing is obtained from a treatment of the inhomogeneous surface potential and the finite imaginary part of the selfenergy only. Shirley232 combined results from partially self-consistent GWOA calculations with static vertex corrections in P and dynamic second-

85

QUASIPARTICLE CALCULATIONS IN SOLIDS

order corrections in 2, and s ~ g g e s t e d ~that ~ ~ dynamic * ~ ~ ’ vertex corrections for C are more appropriate than the static corrections considered by Mahan et al. In spite of the different physics contained in the respective approximations, all three approaches agree with the experimental band widths (for example, 0.7 eV band-width narrowing in sodium compared to the freeelectron result; see Section V.26b for more details). We estimate that missing self-consistency effects could increase the band width of all of the above calculations by 0.4 to 0.7 eV based on the results of Holm and von BarthZ3’ for fully self-consistent GWA calculations in jellium with r, = 4 (average Na density). This suggests that the interpretation of alkali metal photoemission spectra is a difficult, not yet fully understood, problem.268 Vertex Corrections and Partial Self-Consistency. Table 13 shows that the inclusion of dynamic vertex corrections for C to second order in WZ3’ virtually cancels the results of partial self-consistency. Vertex corrections decrease the quasiparticle weight and the band width and lead to a Fermi energy in worse agreement with the exact solution for the homogeneous

’‘’ E. L. Shirley, private communication. 2 6 8 For a critical discussion of Mahan’s photoemission theory for the determination of the alkali metal photoemission spectra, see, for instance, Ref. 11.

TABLE13. QUASIPARTICLE WEIGHTOF JELLIUMAT k = 0 AND k = k, ( k , = FERMIWAVE GWA VECTOR),BAND WIDTH w , AND FERMIENERGYE, FOR A NON-SELF-CONSISTENT CALCULATION, A PARTIALLY SELF-CONSISTENT GW’A CALCULATION (DATAIN PARENTHESES), AND A GW’A CALCULATION THAT INCLUDES DYNAMIC VERTEX CORRECTIONS TO SECOND ORDER IN W IN THE SELF-ENERGY (DATAUNDER“HIGHERORDER”)EVALUATED AT SEVERAL VALUES OF r,. EXACTVALUES FOR E , ARE GIVENAS WELL.THE SCREENED INTERACTION CONTAINS A STATICVERTEXCORRECTION. SELF-CONSISTENCY IMPROVES E , EXCEPTFOR rs = 2, WHERETHE PLASMON-POLE MODELOF REF.232 IS ONLYMARGINALLY VALIDBUT INCREASES Z AND w IN CONTRADICTION TO EXPERIMENTS ON SIMPLE METALS.DYNAMIC VERTEXCORRECTIONS ALMOST COMPLETELY CANCEL SELF-CONSISTENCY EFFECTS. (ADAPTED FROM REF.232.) HIGHERORDER

GWA

2 3 4

5

0.6,O.S (0.6,O.g) 0.5,0.7 (0.6,0.7) 0.5,0.6 (0.6,0.7) 0.4,0.6 (0.5,0.7)

11.9 (13.3) 5.0 (6.0) 2.6 (3.3) 1.6 (2.3)

EXACT EF

EF

z(o),Z(k,)

W

0.10 (0.13) - 0.05 ( - 0.04) - 0.09 ( - 0.08) -0.10 ( - 0.08)

0.5.0.8

11.6

0.5.0.7

5.0

-0.06

-0.04

0.5,0.6

2.7

-0.09

-0.08

0.4.0.6

1.7

-0.09

-0.08

EF

0.08

0.10

86

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

electron gas. Overall, the results agree with those of a non-self-consistent GWK,, calculation that takes vertex corrections for E into account via the following many-body local-field factor f ( q ):

(2.88) where K J n ) is defined as in Eq. (2.87) and where Vxc(n)is the exchangecorrelation potential of a homogeneous electron gas with density n and Fermi momentum k, .269 Inclusion of lowest-order vertex and self-consistency effects beyond the RPA polarizability for Si and diamond”’ leads to (1) a cancellation of the constant contributions to the polarizability at k + O of vertex and selfconsistency corrections, (2) a significant modification of the dielectric constant, (3) an increase in fundamental band gaps compared to RPA screening by 0.1 eV (0.2 ev) for Si (diamond), (4) a significant increase in the absolute value of the diagonal polarizability matrix elements compared to RPA by vertex corrections, and ( 5 ) a significant decrease in the same matrix elements by self-consistency effects, leading to large cancellations and an overall increase in the matrix elements by < 15% compared to RPA. The diagrams that are calculated for the dielectric polarizability and the selfenergy are shown in Fig. 21. The solid lines in these diagrams correspond to a single-particle propagator constructed from LDA wave functions and energies. The wavy lines correspond to the screened interaction whose frequency depedence is modeled by the Engel-Farid plasmon-pole m0de1.~” The results of adding the lowest-order vertex (v) and self-energy (SC) corrections to the RPA polarizability are shown graphically in Fig. 22. Both the vertex and the self-consistency corrections to the RPA polarizability are finite for k + 0 and only their sum tends to zero proportional to k2. This proportionality for the head of the static polarizability is necessary to ensure the correct screening behavior of a semiconductor in the long-wavelength limit’ 19,272 and provides a formal motivation for an inclusion of both effects

269 The local-field factor f(q) is obtained by fitting the dielectric constant to quantum Monte-Carlo data, Ref. 270. A similar approximation has been used by Hindgren and Almbladh, Ref. 256, who showed it to be very reliable. ”* C. Bowen, G. Sugiyama, and B. J. Alder, P h p . Rev. SSO, 14838 (1994). This approximation affects the self-energy corrections to the lowest conduction and highest valence band by less than 20 meV, as shown in Table 21. Also, de Groot et al. (Ref. 236) studied the lowest-order vertex and self-consistency corrections to the RPA polarizability and the GWA self-energy for a 1D semiconducting wire. We do not discuss that model system but concentrate on the results for Si. 272 W. Kohn, Phys. Rev. 110,857 (1958).

+

0' i

+

+o V

SCI

Qsc4

SC2

SC3

SC4

+%

FIG.21. Left panel: RPA potarizability and corrections to first order in the screened interaction W SCl-SC4 denote corrections due to selfenergy insertions in the Green function, that is, due to lowest-order self-consistency effects, V corresponds to a ladder diagram ,' and the that is the lowest-order vertex correction. The solid directed line denotes the LDA Green function. The cross denotes - V wavy line denotes the RPAdynamically-screened interaction. Right panel GWA self-energy plus lowest-order self-consistency (SC1-SC4)and vertex (V)corrections. The diagrams SC3 and SC4 are selfenergy corrections due to lowest-order corrections to the valence density. The wavy line denotes the dynamically screened interaction, which can be evaluated either in RPA or in RPA V SC.The notation is otherwise identical to the one used in the left panel. (From Ref. 119.)

+ +

2 xU r

88

WILFRIED G. AULBUR, LARS JQNSSON, AND JOHN W. WILKINS

0.03

0.01

4 - 0 4

-0.-

0.0 1.0 2 0 3 . 0

4.0 6.0 6.0

[Zrua]

1-

a a.0 e.0

4-07 0.0 4.0

at.0

1-

4.0

6.0

[2rr/orl

FIG. 22. Cancellation between vertex (V) and self-consistency (SC) corrections to the diagonal elements of the RPA static polarizability PGc(k;w = 0) (in Rydberg am) for silicon and diamond. Lowest-order vertex and self-consistency corrections as considered in Ref. 119 largely cancel, leading to an overall enhancement of the RPA polarizability by < 15%. The constant contributions of the vertex and self-consistency corrections to the head element of the RPA polarizability matrix for k + 0 cancel, which ensures the correct screening behavior in semiconductors. (From Ref. 119.)

in the polarizability. The effect of the vertex and self-consistency corrections on the dielectric constant within the approximations by Ummels et ul.'19 amounts to -20% in the case of Si and - 5 % in the case of diamond273 and is therefore at least as important as local-field effects due to an inhomogeneous density distribution. Inclusion of vertex and self-consistency corrections in the self-energy leads to an increase of fundamental band gaps by 0.4 eV (0.3 eV) in Si and 0.7 eV (0.5 eV) in diamond using RPA + V + SC (RPA) screening.' Band gaps determined using different combinations of vertex and self-consistency corrections for the polarizability and the self-energy are shown in Table 14. The influence of V + SC corrections to the self-energy operator leads to a significant increase in fundamental band gaps. The results of Ummels et Silicon: E ~ 12.8, ~ E 5.3, E~~~ = 5.5, 5.7.

273

~~

=~

~

10.4, + ~P

P+ = ~

11.4, ~ 11.7; = diamond cRP"=5.6, E

~

~

~

+

~

89

QUASIPARTICLE CALCULATIONS IN SOLIDS

TABLE 14. INFLUENCEOF LOWEST-ORDER VERTEX (V) AND SELF-CONSISTENCY (SC) TO THE SELF-ENERGY (GWA VERSUS GWA + V + SC) AND TO THE DIELECTRIC CORRECTIONS BANDGAPSAT r, X, AND L SCREENING (RPA VERSUS RPA + V + SC) ON THE FUNDAMENTAL IN Si AND AT r IN DIAMOND. LOWEST-ORDER CORRECTIONS TO THE DIELECTRIC SCREENING LEAD TO A SMALLINCREASE IN FUNDAMENTAL BANDGAPS( x 0.1 eV IN Si, x 0.2 eV IN DIAMOND). TO THE SELF-ENERGY LEADTO A SUBSTANTIAL INCREASE IN THE LOWEST-ORDER CORRECTIONS FUNDAMENTAL BANDGAP( x 0.3 eV IN Si, x 0.5 eV IN DIAMOND), RESULTING IN A DISCREPANCY OF ABOUT 0.3 eV IN Si AND 1eV IN DIAMOND BETWEEN CALCULATION AND EXPERIMENT. Wmm THE APPROXIMATION OF THE CALCULATION (LDA GREENFUNCTION, PLASMON-POLE MODEL, ETC.) THE EFFECTS OF VERTEX AND SELF-CONSISTENCY CORRECTIONS DO NOT CANCEL. ENERGIES ARE IN eV. (ADAPTED FROM REF. 119.) DIAMOND

Si

METHOD (SELF-ENERGY/SCREENING) LDA GWA/RPA GWA/RPA+V+SC GWA + V + SC/RPA GWA+V+ SC/RPA+V+SC EXPT.”.~

r

X

L

r

2.5 3.3 3.4 3.6 3.1 3.4

3.4 4.2 4.3 4.5 4.6 4.3

2.6 3.4 3.4 3.1 3.8 3.5

5.5

1.6 1.8 8.1 8.4 1.3

“Ref. 182; bRef. 274.

al.’ l 9 suggest that the cancellation between lowest-order vertex and selfconsistency contributions is far from complete if one determines these corrections to the GWA self-energy using a noninteracting LDA Green function. In fact, the V contribution cancels the SC contribution to the self-energy by only 35% (45%) in the case of Si and only - 10% (15%) in the case of diamond for RPA V SC (RPA) screening. In the latter case, even the term “cancellation” is therefore inappropriate.

+ +

’’

111. GWA Calculations: Numerical Considerations

This section (1) describes various reciprocal-space and real-space/imaginary-time implementations of the GWA; (2) gives numerical details for an implementation of the GWA in a plane-wave basis; (3) describes parallel algorithms of GWA calculations in reciprocal space as well as in real space/imaginary time; and (4) compares GWA calculations for five prototypical semiconductors. 274

D. E. Aspnes and A. A. Studna, Phys. Rev. B27,985 (1983).

90

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

15. DIFFWENT IMPLEMENTATIONS OF THE GWA Quasiparticle calculations can be performed either in reciprocal space as a function of frequency or in real space as a function of imaginary time. Both approaches are described in this section; subsection a contains the reciprocalspace approach; subsection b, the real-space approach. The reciprocal-space method is widely used and has been implemented using plane waves as well as a variety of local-orbital-basis sets. Subsection a contains a comprehensive comparison of the advantages and weaknesses of the different basis sets. a. Reciprocal-Space Approach

Early calculation^^.^ in the dynamically screened potential approximation determined the expectation value of the self-energy operator using a planewave basis set in reciprocal and frequency space, since this allows both a straightforward evaluation of matrix elements that occur in the self-energy and a systematiccontrol over convergence. In addition, a GWA calculation in reciprocal and frequency space parallels the experimentalsituation in (inverse) photoemission, which determines the band structure of a solid as a function of the reciprocal wave vector k and of the frequency w of the quasiparticle. a. 1. PLANE WAVES. Pseudopotentialsin conjunction with a plane-wave basis set are widely used in computationalcondensed matter theory because of their ease of use and systematic convergence proper tie^."^ Mostly, plane-wavebased GWA calculations are applied to sp-bonded bulk solids and to their interfaces, surfaces, defects, and clusters, as discussed in Sections IV and V. The two main disadvantages of plane-wave basis sets are that (1) the number of plane waves Npw increases with the system volume V and the energy cut-off E,,, needed for a converged description as N,, x (1/21)VE:i? (in Hartree atomic units); and (2) plane waves have no direct physical interpretation in contrast to local-orbital basis sets discussed below. Reciprocal-space GWA calculations scale as N $ , (see below), which makes planewave-based GWA calculations prohibitively expensive for large- V systems such as complicated defect structures and for large-E,,, systems such as d and f electron materials. Detailed Formulas. A Bloch wave function of wave vector k and band n can be expressed in terms of its Fourier components cnk(G),where G is a reciprocal lattice vector, as 1

@nk(r)

J

*’’

1 c,,(G)

=V

G

W. E. F’ickett, Comp. Phys. Rep. 9, 115 (1989).

exp(i(k + G) * r).

(3.1)

91

QUASIPARTICLE CALCULATIONS IN SOLIDS

Define the following matrix element between occupied valence (n unoccupied conduction ( n = c) states:

= u)

and

(3.2) The independent-particle polarizability given by Eq. (2.22) is

where the factor 2 accounts for spin degeneracy. The frequency convolution in Eq. (2.24), together with the independentparticle Green function (Eq. (2.4)) expressed in terms of Bloch wave functions and the Fourier component of the screened interaction WGG.(q; w), allows the determination of the matrix elements of the self-energy with respect to Bloch states @mk = Im,k) and @,, = I/, k) as the following sum over occupied and unoccupied states: 1 (4k I W ) I L k) = -

v

occ+unocc BZ

c n

1 2 ME&, q

GG'

exp(iw6) (-27Ci) E

q)CM"d(kk, q>l*

+0-

WGG~(q9 0) do. - i6Sgn(P - E,k-q)

Enk- q

(3.4) Here, the matrix elements M;.(k, q) generalize Eq. (3.2) to arbitrary states n and m. The remaining frequency integration is usually performed analytically using the plasmon-pole models presented in Section II.8b rather than numerically as in Refs. 102, 103, 185, and 276. The plasmon-pole model (PPM) is accurate to within a few tens of meV for states close to the Fermi level and to within a few tenths of an eV for low-lying valence states, as is demonstrated in Table 21. Splitting the self-energy into a bare exchange part C x and an energy-dependent correlation contribution C c ( E ) , as detailed in 276

H. N. Rojas, R. W. Godby, and R. J. Needs, Phys. Rev. Left. 74, 1827 (1995).

92

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Section 11.6 gives (3.5)

(3.6) where the matrix elements are defined as277

P;m(k, 4) =

c CM&rn(k,:q)I*@,(G),

(3.7)

G

and ci = 27tzpq0,(q) for the von der Linden-Horsch model and ct = 1 for the Engel-Farid plasmon-pole model. The N & scaling of the reciprocal-space GWA algorithm follows from Eqs. (3.2), (3.6), and (3.7). The construction of the matrix elements in Eq. (3.2) scales as the number of plane waves used to describe the LDA wave functions. The matrix elements in Eq. (3.7) are determined in N , operations, where N , is the number of plane waves used to describe the plasmon-pole eigenvectors. The self-energy itself (Eq. (3.6)) must be determined via a sum over LDA bands and a sum over plasmon-pole bands. The number of LDA and plasmon-pole bands as well as N , and N p , increase linearly with system size and are all of the same order. Hence, the overall algorithm scales as the size of the system or the number of plane waves to the fourth power. An approximate evaluation of the ~elf-energy"~via Taylor expansions around a given set of energies reduces the GWA scaling to N;w and allows larger GWA calculations based on plane waves. Other appro ache^^^^.^^^ permit the determination of the density-density response function with an O ( N ; , ) effort, but have not been extended to the calculation of the self-energy. a.2. LOCAL-ORBITAL BASISSETS. The number of basis functions of localorbital basis sets,' s which is needed to describe bulk semiconductors, is smaller by about one order of magnitude than the corresponding number of plane waves.281 Although the overall scaling of the reciprocal-space 277 The scaled plasmon-pole eigenvectors for the von der Linden-Horsch and the Engel-Farid plasmon-pole models are defined in Eqs. (2.52) and (2.61), respectively. 2 7 8 L. Reining, G. Onida, and R. W. Godby, Phys. Rev. B56,R4301 (1997). X. Blase, A. Rubio, S. G. Louie, and M. L. Cohen, Phys. Rev. BS2, R2225 (1995). A. A. Quong and A. G. Eguiluz, Phys. Rev. Lett. 70, 3955 (1993). The description of bulk Si requires, for example, 40-60 Gaussian orbitals compared to about 450 plane waves (Ref. 114).

'''

QUASIPARTICLE CALCULATIONS IN SOLIDS

93

GWA algorithm still scales as the number of basis functions to the fourth power, the strong reduction of the number of basis functions results in significant savings in computation time and allows the study of materials that are not accessible in plane-wave-based approaches such as N i 0 5 5 or Si,H, clusters.’” Unfortunately, the matrix element construction is computationally more intense in a local-orbital than in a plane-wave basis, which partially reduces gains due to the smaller number of basis function^."^ Linearized Augmented Plane Waves (LAP W ) . LAPW is an all-electron method that has been applied to GWA calculations in Si’85 and Ni6’ In Si, 45 basis functions per Si atom are needed, which corresponds to a reduction by a factor of five compared to plane-wave-based calculations. Although LAPW calculations allow systematic convergence, they do not seem to offer significant computational savings compared to plane-wave-based calculations, which may have inhibited a widespread use of LAPW for GWA calculations.’86 Linearized Mufin-Tin Orbitals (LMTO). LMTO is an all-electron method that has been used, for instance, for GWA calculations of transitionmetal oxide5’ and the Compton profile of alkali It is extensively reviewed in Ref. 14. GWA calculations for transition-metal oxides are prohibitively expensive for plane-wave-based methods, since an accurate description of the transition-metal d and oxygen 2 p electrons requires a very large energy cut-off and several thousand plane waves. In LMTO, the description of transition-metal oxides uses only 50- 100 basis functions per atom.55 Aryasetiawan, Gunnarsson, and collaborator^'^ used LMTO in the atomic sphere approximation and included so-called combined correct i o n ~ ’ ’ ~only in the energies, not in the wave f ~ n c t i o n s . ’This ~ ~ approach has two disadvantages. First, omission of the combined-correction term leads to discontinuous wave functions in the interstitial region and in the overlapping region of the muffin-tin sphere^.*^^*^^' This results in errors in M. Rohlfing and S. G. Louie, Phys. Rev. Lett. 80, 3320 (1998). In the case of bulk Si, Gaussian-orbital-based calculations take as much time as plane-wave-based calculations (Ref. 284). For the Si(OO1) (2 x 1) reconstructed surface, a speed-up of a factor of five compared to plane waves is achieved (Ref. 284). 284 M. Rohlfing, P. Kriiger, and J. Pollmann, Phys. Rev. B52, 1905 (1995). 2 8 5 N. Hamada, M. Hwang, and A. J. Freeman, Phys. Rev. B41, 3620 (1990). 2 8 6 F. Aryasetiawan and 0. Gunnarsson, Phys. Rev. B49, 16214 (1994). 2 8 7 Y. Kubo, J . Phys. Soc. Jpn. 65, 16 (1996). 2 8 8 Y. Kubo, J . Phys. Soc. Jpn. 66, 2236 (1997). 2 8 y 0. K. Andersen, Phys. Rev. B 12, 3060 (1975). F. Aryasetiawan, private communication. 291 H. L. Skriver, The LMTO method: Mujin-Tin Orbitals and Electronic Structure, Springer Series in Solid-state Sciences, Vol. 41, Springer, Berlin (1984). 2 y 2 M. Alouani, private communication. 282

283

94

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

the determination of the matrix elements of the momentum operator and hence, for instance, in the dielectric properties and subsequently the selfenergy operator. Second, the atomic sphere approximation assumes a spherically symmetric potential. This shape approximation can affect properties such as the dielectric constant. Christensen and G o r ~ z y c a , 'for ~~ instance, determined the dielectric constant of GaN within the atomic sphere approximation, including combined-correction terms, to be 4.78 -a 20% underestimation compared to the full-potential LMTO value of 5.96.293 An estimate of the influence of the above approximations on quasiparticle energies is difficult. Insight can be gained by comparing all-electron LMTO results for GaAsZ9' with pseudopotential results that include core-polarization potentials and core-relaxation effects,43 as well as with experiments. Shirley and collaborators obtained quasiparticle corrections of 1.29/0.74/ 1.05 eV for the lowest conduction band at r, X , and L, compared to values of 1.19/1.59/1.42 eV using scalar relativistic LMT0.290This can be inferred from Table 23. LMTO quasiparticle corrections are larger than the best plane-wave results by 0.1 to 0.8 eV. In addition, LMTO in the atomic sphere approximation overestimates the low-lying experimental conduction-band energies by 0.3 to 1.1 eV, again as shown in Table 23. Note that the LMTO calculation of Ref. 290 chooses a trial energy around which it makes an expansion. This trial energy is chosen to accurately describe the valence and semicore states of GaAs with less emphasis placed on unoccupied states. The occupied states are indeed accurately described in LMTO, while the description of the unoccupied states is substantially worse. Choosing several reference energies can improve the description of unoccupied states.290Note that Kubo's calculation^^^^.^^^ avoid the above mentioned problems by evaluating the GWA based on a full-potential LMTO calculation.294 Gaussian Orbitals. Baroni and c o l l a b ~ r a t o r s ' ~use ~ ~ 'a~ ~contracted Gaussian-orbital basis set to determine the electronic properties of core and valence electrons in the COHSEX approximation (see Section 9). More recently, Rohlfing, Kruger, and P01lman~~" 14,284 developed an approach for a GWA calculation that is based on pseudopotentials in conjunction with an uncontracted Gaussian-orbital basis set. The merit of Gaussian-orbital-based GWA calculations is the reduced number of basis functions and hence their greater efficiency and applicability compared to plane-wave calculations, as exemplified by (1) a factor of five speed-up compared to a plane-wave calculation of the Si(OO1) (2 x 1) M. Alouani and J. Wills, as presented in Ref. 542. In contrast to most other GWA calculations, Kubo also used a linear tetrahedron method for the evaluation of Brillouin zone integrals. 293

294

95

QUASIPARTICLE CALCULATIONS IN SOLIDS

reconstructed ~ u r f a c e ; ” (2) ~ an explicit treatment of the Cd n = 4 core and semicore electron^;^' and (3) the inclusion of excitonic effects in Si,H, clusters.’ The main disadvantage of Gaussian orbitals is that while Gaussian basis sets can systematically converge, constructing them is difficult. Table 15 demonstrates the problems in establishing the energies of low-lying conduction states of small-gap and medium-gap semiconductors. For bulk semiconductors, 20 Gaussians per atom are suggested to be adequate for convergence of quasiparticle energies of low-lying conduction and valence states to within 0.1eV.”4 While this is true for Si”4 and for large-gap materials with strongly localized densities such as diamond, Sic, and GaN,295good convergence for more “metallic,” small- and medium-gap semiconductors such as Ge and GaAs requires at least 30 Gaussians per atom. Tables 15 and 22 show that in the case of Ge (1) the lowest plane wave conduction-band energies at r, X , and L for a well-~onverged’~~ M. Rohlfing, diploma thesis, University of Munster, Germany (1993). The LDA band structure of Ref. 42 is calculated at 10 special k-points and a 10-Hartree cut-off. Comparison with a calculation using 60 special k-points and an energy cut-off of 18 Hartree leads to shifts in the eight lowest eigenvalues at r, X , and L by less than 0.03 eV (rZc: 0.05 eV). 295

296

TABLE15. CONVERGENCE OF LDA AND QUASIPARTICLE ENERGIES FOR SELECTED STATES I N Ge GaAs FOR A GAUSSIAN-ORBITAL-BASED METHOD,”4’295 IN COMPARISON TO A PLANEWAVE (Pw) FORBOTHMATERIALS AT LEAST60 GAUSSIAN ORBITALS (GO) ARE NECESSARY TO OBTAIN A CONVERGED LDA BAND STRUCTURE. THEGWA BAND STRUCTURE MAY REQUIREMO RE THAN60 GAUSSIANS FOR COMPLETE CONVERGENCE. A 40-GAUSSIAN BASIS SETIS USEDIN THE LITERATURE,’ l 4 BUT THAT SETIS INSUFFICIENT TO OBTAIN BANDSTRUCTURES ACCURATETO WITHIN0.1 eV FOR SMALL-AND MEDIUM-GAP SEMICONDUCTORS. NOTE THAT IMPROVING CONVERGENCE OF THE GAUSSIAN-ORBITAL CALCULATION MOVESTHE RESULTS SYSTEMATICALLY TOWARD THE PLANE-WAVE RESULTS.ALL ENERGIES ARE IN eV. AND

PW

Ge

Xl, LI,

L;, GaAs

Xl, x3c

Ll,

Ll,

60 GO

40 GO

LDA

G WA

LDA

G WA

LDA

GWA

0.70 0.12 7.07 0.47 1.42 1.62 0.97 7.74

1.15 0.61 7.56 1.16 2.00 2.24 1.62 8.40

0.79 0.18 7.46 0.52 1.47 1.66 0.98 8.08

1.36 0.76 8.1 1 1.21 2.14 2.37 1.66 8.86

1.03 0.33 8.33 0.57 1.80 1.85 1.13 8.88

1.74 0.98 9.20 1.32 2.65 2.72 1.92 9.92

96

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

and a 40-Gaussian-orbital calculation differ by up to 0.3 eV in LDA and up to 0.6 eV in GWA; (2) higher-lying states, such as the L;, state, differ by up to 1.3 eV in LDA and 1.6 eV in GWA; and (3) the Gaussian-orbital results converge systematically towards the plane-wave results when 60 Gaussians rather than 40 Gaussians are used. Similar conclusions can be drawn from Tables 15 and 23 for GaAs. In particular, the difficulty in describing lowlying conduction states at X and L with only 40 Gaussians may result from an enhanced d character of these states.297 The problem ofproperly treating d states near the valence band is illustrated by the work on CdS5’ As pointed out in Section 11.12,semicore d states modify dynamic screening in a ~ o l i d . ~ ~ ~In~ addition, ’ ~ ’ ~ ’ the exchange interaction between all electrons of a shell ( n = 4 in CdS) affects the energies of semicore and valence electron states5’ if the corresponding wave functions overlap strongl ~ . ’ ~Quantitative ’ results for CdS are difficult to obtain since CdS seems to be sensitive to the pseudopotential used. As shown in Table 16, treating the Cd 4d electrons’96 as core electrons, via nonlinear core correction^,'^^ or as valence electrons gives LDA energy gaps of 1.72 eV, 1.60 eV, and 0.84 eV, respectively, for scalar-relativistic Troullier-Martins pseudopotentials in the KleinmanBylander form.304Semi-relativistic,nonlocal ab-initio pseudo potential^^ give an energy gap of 1.36 eV when nonlinear core corrections are used (E,,, = 12.5 Hartree); and scalar relativistic Bachelet-Hamann-Schliiter pseudopotentials301give an energy gap of 1.65 eV when the d electrons are treated as core electrons and of 0.3 eV when the d electrons are treated as valence electrons.305 Using the scalar-relativisticBachelet-Hamann-Schliiter Cd pseudopotential in Refs. 52 and 302 gives an LDA energy gap of 2.15 eV and 2.18 eV, respectively, when the d electrons are treated as core electrons. An energy gap of 0.78 eV is obtained when they are treated as valence electron^.^' The deviation in the fundamental LDA energy gap between the calculations of Refs. 51,196, and 301 and those of Refs. 52 and 302 using different Cd’ pseudopotentials amount to at least 0.5 eV and are on the order of the self-energy shifts observed upon inclusion of 4p and 4s electrons as valence electron^.^' Good quasiparticle band gaps are obtained by treating the core and semicore electrons in II-VI materials via nonlinear core correction^.^' However, the success of this approach is based on a fortuitous cancellation of errors due to the neglect of wave-function relaxation and hybridization and to the neglect of dynamic screening by semicore electrons and exchange interaction with core and semicore electrons.

’

+

’” S. G. Louie, Phys. Rev. B22, 1933 (1980). 298 This observation agrees with the determination of the influence of core charges on the self-energy of Ag (Ref. 216) and on the self-energy of Ni (Ref. 68), as detailed in Section V.26. 2 9 9 As in Ref. 182, vol. 17b. 300 M. Cardona, M. Weinstein, and G. A. Wol5, Phys. Rev. 140, A633 (1965).

97

QUASIPARTICLE CALCULATIONS IN SOLIDS

TABLE16. LDA AND GWA (BOLDFACE) ENERGYGAPS AND LDA LATTICECONSTANTS (PERCENT DEVIATION FROM EXPERIMENT IN PARENTHESES) IN COMPARISON WITH EXPERIMENT FOR CUBIC CdS. COCIORVAet al. 196 USED SCALAR-RELATIVISTIC TROULLIER-MARTINS IN THE KLEINMAN-BYLANDER FORM (E,,, = 25 HARTREE FOR d ELECTRONS IN PSEUDOPOTENTIALS CORE,45 HARTREE FOR d ELECTRONS I N VALENCE). CHANGet USEDSCALAR-RELATIVISTIC BACHELET-HAMANN-SCHL~ER PSEUDOPOTENTIALS (E,,, = 6.75 HARTREE FOR d ELECTRONS IN IN VALENCE) A N D SHOWED THAT THE NEGLECT OF CORE, 9.5 HARTREEFOR d ELECTRONS RELATIVISTIC EFFECTS(NUMBERS IN PARENTHESES) INCREASES FUNDAMENTAL BANDGAPSBY ABOUT 0.3 ev. ZAKHAROV et al.” USEDSEMIRELATIVISTIC, NONLOCAL PSEUDOPOTENTIALS WITH NONLINEAR CORECORRECTIONS (NLCC)”’ (E,,, = 12.5 HARTREE). ROHLFING et AND FLESZAR302 USED SCALAR-RELATIVISTIC BACHELET-HAMANN-SCHL~ER PSEUDOPOTENTIALS. THEDIFFERENCE OF ABOUT 0.5 eV IN THE FUNDAMENTAL LDA BANDGAPBETWEEN DIFFERENT Cd2+ CALCULATIONS IS OF THE SAMEORDER AS THE GWA BAND-GAPINCREASE UPON INCLUSION OF 4s AND 4p ELECTRONS AS VALENCE ELECTRONS IN GWA CALCULATIONS. OF THE 4d ELECTRONS AS VALENCEELECTRONS LEADSTO A REPRODUCTION OF THE INCLUSION EXPERIMENTAL LATTICE CONSTANT TO WITHIN-3.6%52 AND -0.7%.196 THISDIFFERENCE IN THE EQUILIBRIUM STRUCTURE LEADSTO DIFFERENCES IN THE FUNDAMENTAL BANDGAPOF 0.2 TO 0.3 eV.196-303 THELDA AND GWA ENERGY GAPSARE DETERMINED AT THE EXPERIMENTAL LATTICE CONSTANT IN ALL CALCULATIONS. ________~~ ~

E,,,(W d ELECTRONS

a d 4

COCIORVA CHANG ZAKHAROV ROHLFING

FLESZAR COCIORVA ROHLFINC

~~

Core NLCC

1.72

1.65 (1.93)

1.60

2.15

2.18

3.70

3.83

136

2.79 Valence Expt.“

0.84

0.3 (0.67)

0.78 1.50 2.50 2.55b

5.05 5.05 (-13.2%) (-13.2%) 5.33 ( - 8.4%) 5.78 5.61 (-0.7%) (-3.6%) 5.818

“Ref. 299 unless noted otherwise; bRef. 300.

b. Real-Spacellmaginary-Time Approach The basic idea of the space-time approach is to choose the representation (e.g., reciprocal space and frequency or real space and time) that minimizes the computations necessary to evaluate the basic GWA quantities: G , E, W K. J. Chang, S. Froyen, and M. L. Cohen, Phys. Rev. B28,4736 (1983). A. Fleszar, private communication. 3 0 3 M. Rohlfing, P. Kriiger, and J. Pollmann, private communication. 304 The energy cut-offs used in the calculations are 25 Hartree, 25 Hartree, and 45 Hartree, respectively. 305 The energy cut-off used for the core case is 6.75 Hartree and 9.5 Hartree for the valence case. Neglect of scalar relativistic effects increases the gap by about 0.3 eV to 1.93 eV (core) and 0.67 eV (valence) (Ref. 301). 301

302

98

WILFRIED G . AULBUR, LARS JONSSON, A N D JOHN W. WILKINS

and C.For example, the determination of the self-energy in reciprocal and frequency space, Eqs. (3.5) and (3.6), involves numerically expensive convolutions that lead to an overall scaling of the algorithm proportional306 to N& and to N:-the square of the number of mesh points used for the representation of the frequency dependence of the screened interaction. In real space and as a function of time, the self-energy is a simple product (compare Eq. (2.1 1)): C(r, r’; t) = iG(r, r’; t) W(r, r’; T),

(3.8)

which eliminates two convolutions in reciprocal space and one in the frequency domain, and leads to an algorithm whose dominant parts scale as O(N$,) and O ( N , ) . Similarly, the dielectric matrix and the screened interaction require convolutions in real space (Eqs. (2.20) and (2.14)) but multiplications in reciprocal space (Eqs. (2.23) and (2.16)). A representation change from reciprocal to real or frequency to time space using Fast Fourier Transforms (FFTs) and vice versa scales as N,, log(N,,) or N , log(N,) and is computationally very efficient. This allows the efficient determination of C, E, and W as products rather than as convolutions. The evaluation of operators in real space and the extensive use of FFTs require operators that are short-ranged in real space and quantities that can be represented by an equidistant numerical grid. Relevant quantities in the GWA, namely, G, C, and Po, are proportional to Ir - r’l-’ for Ir - r’I -, co and indeed short-ranged (compare for example Figs. 9 and 11). The screened interaction decays proportionally to Ir - r’1-l for Ir - r’I -+ co, but this long-ranged tail can be taken into account e~plicitly,”~ allowing the use of a finite cut-off for Ir - r’I in numerical work. Although G, X, and W have branch cuts on the real-frequency axis, they can be analytically continued to the imaginary-frequency (and imaginary-time) axis along which they are much smoother and can be accurately discretized by an equidistant Fourier grid, as shown in Fig. 23. The central task of the real-space/imaginary-time algorithm is the same as that of the reciprocal-space/frequency-space algorithm: determination of G -, Po -, E W -, X and subsequent calculation of quasiparticle shifts and, if desired, the solution of Dyson’s equation. Here, we will describe in detail the construction of G and Po and the Fourier transformation of Po, since their construction is used to illustrate the parallel real-space/imaginary-time algorithm in Section 111.17. For further details and convergence studies we refer the reader to the work by Rieger et 306 All real-space/imaginary-time algorithms implemented so far are based on pseudopotentials and use plane waves as basis functions.

QUASIPARTICLE CALCULATIONS IN SOLIDS

99

2 h

2

v

L3

.-

o

0 -

9 -2

20

-z

10

A

3

0 -

9

0 -10

-60 -40 -20

0 0

20

40

60

(W

FIG. 23. Energy-dependent correlation contribution to the self-energy of Si as a function of imaginary (upper panel) and real (lower panel) frequency. The expectation value of ZC has significantly less structure along the imaginary axis compared to the real axis, allowing the use of Fast Fourier Transforms with equidistant frequency and time grids to switch between imaginary time and frequency. (From Ref. 103.)

b.1. THEGREEN FUNCTION.Analytic continuation of the spectral representation (Eq. (2.4)) of the noninteracting Green function from real to imaginary energy and Fourier transformation results in3’’ G’(r, r’; ir) =

i XF@nk(r)@n*k(r’)exp(q,kr),r > 0 - i X:Pcc@nk(r)@:k(r’) exp(Enkr), r < 0,

(3.9)

In this approach, r denotes a point in the irreducible part of the real-space 307 The index i has been replaced by the band index n and the reciprocal vector k in the first Brillouin zone. Spin is disregarded. The Fermi energy, which is at zero in this subsection, lies in the band gap; that is, if' < 0 and E;Y z 0.

100

WILFRIED G . AULBUR, LARS JONSSON, AND JOHN W. WILKINS

unit cell while r’ denotes a point in the “interaction cell” outside of which Go is set to zero. Denote as unk(r) the periodic part of the Bloch wave function Onk(r),and define for a given r a vector r” in the real-space unit cell and a real-space lattice vector L, such that308 r’ = r

+ r” + L = r + x;

x = r”

+ L.

(3.10)

Then occ(unocc)

Go(r, r

+ x; ir) = i sgn(.s) C

u,,k(r)t&(r

+ r”) exp(

-

ik .x) exp(snkc).

nk

(3.1 1) Note that T is positive or negative depending on whether the sum is over occupied or unoccupied states, respectively. Finally, write the sum over k-points in the full Brillouin zone (BZ) as a sum over k-points in the irreducible Brillouin zone (IBZ) and over symmetry operations S that generate the k-points in the full BZ. One obtains the following formula for Go, which is implemented in the real-space/imaginary-time program: Go@, r

+ x; ir) = isgn(T)

c c c unk(S-’r)z&(S-’(r + r”))

occ(unocc) IBZ n

k

S

x exp( -iSk.x) exp(cnkr).

(3.12)

Assuming an interaction cell that is of constant size for a given material, the determination of the independent-particle propagator scales as N , x Nband x N,, where N , is the number of frequencies or times considered; Nband, the number of bands in the summation of the above equation; and N,, the number of r-points in the real-space unit cell. Since Nband and N , increase linearly with the size of the system, the overall scaling of the method is proportional to the system size squared and hence of order O(N2,,). Convergence considerations. Convergence parameters important for the real-space approach are (1) the number of bands in Eq. (3.12), (2) the realspace grid spacing Ar and the size of the interaction cell, and (3) the time spacing AT and the maximum sampling time rmax.Convergence to within 20meV for Si requires 145 bands, a Ar = 0.32 a.u., an interaction cell radius309 Rmax= 18 a.u., a time spacing AT = 0.15 a.u., and T~~~ z 20 a.u. 308 The r and r‘ meshes are not offset, which consequently requires special treatment of the Coulomb divergency; see Ref. 103. 309 Assume a spherical interaction cell for simplicity.

QUASIPARTICLE CALCULATIONS IN SOLIDS

101

This corresponds to 170 r-points in the irreducible wedge of the real-space % 0.75 x lo6 points in the interacunit cell and to Nr.% (471/3)(R~~JAr)~ tion cell.310The propagator Go(r, r’; r) requires 0.8 MW storage for every r ~ 130 time points. Storage requirements for the and each of the r m a x / A = algorithm are therefore large despite the finite range of the interaction cell and may become a bottleneck of the calculation for larger systems.311 POLARIZABILITY. b.2. THE INDEPENDENT-PARTICLE The independent-particle p ~ l a r i z a b i l i t y ~can ~ ’ be determined in real space and imaginary time by an O(N,,) operation (compare Eq. (2.13)):

Po(r, r’;

iT) =

- iGo(r, r’; ir)Go(r,

r’; -ir).

(3.13)

However, the dielectric matrix and the screened interaction require convolutions in real space (Eqs. (2.20) and (2.14)) but multiplications in reciprocal space (Eqs. (2.23) and (2.16)). It is easier to construct these quantities in reciprocal space and as a function of imaginary frequency, which requires a Fourier transformation of Po(r, r’; ir). The spatial Fourier transforms require special care since Po is not translationally invariant in the 6dimensional space spanned by r and r‘. In the following equations, we do not exhibit the time/frequency dependence for simplicity and define Po(k, r, r’)-where both r and r’ are in the real-space unit cell-as .,

IC(r1

Po(k, r, r’) =

exp( - ik. (r - r’ - R))Po(r, r’ R

+ R).

(3.14)

The sum over lattice vectors R is limited to vectors in the interaction cell of r (ZC(r)). The quantity Po(k, r, r’) is periodic in r and r’ separately, and its Fourier transform can be determined by two successive 3-dimensional F F T s . ~ The Fourier transform of the independent-particle polarizability is given as (V, is the unit cell volume)314 P&(k)

=

v,1 fvc

fvc

exp( -iG.r)Po(k, r, r’)exp(iG’.r’)drdr’.

(3.15)

3 ’ 0 Materials such as GaN with larger density variations than Si will require an even smaller Ar to obtain similar convergence. The range of the interaction cell should not increase, although the dielectric constant of GaN is less than half of the dielectric constant of Si since the short-range Green function, not the long-range screened interaction, determines the range. 311 L. Steinbeck, private communication. Note that Steinbeck, Rieger, and Godby recently developed algorithms that reduce memory requirements by an order of magnitude. 3 1 2 We discuss the case of an RPA-based GWA calculation for simplicity. 3 1 3 This so-called “mixed-space’’ representation of the nonlocal operators in the GWA was introduced by Godby et al. (Ref. 185) and discussed in detail by Blase et al. (Ref. 279). 314 A slightly different notation was used in Ref. 103.

102

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Once Po is determined in reciprocal space and imaginary time, the determination of the dielectric matrix and the screened interaction reduces to matrix multiplications and inversions that scale as the third power of the matrix size. The actual computation is not dominated by this part of the algorithm since the dimension of the dielectric matrix can be chosen to be much smaller than the number of points in the real-space unit cell. The self-energy must be calculated in real space and imaginary time to avoid convolutions. Therefore, the screened interaction W must be Fouriertransformed from reciprocal space and imaginary frequency to real space and imaginary time, with careful consideration of the long-ranged part of W103 Expectation values of the self-energy operator (@'nkIC(i~)l@nk) can then be determined after Fourier transformation of C from imaginary time to imaginary frequency. The analytic structure of G and W dictates that the self-energy has poles in the second and fourth quadrant of the complex plane. One can therefore analytically continue the self-energy from the negative (positive) imaginary-frequency axis to the negative (positive) realfrequency axis without crossing any branch cuts. This analytic continuation is achieved by fitting the self-energy along the imaginary-frequency axis to a multipole function.'03 16. PLANEWAVES:NUMERICAL DETAILS

This section has a threefold purpose: (1) Detailed information on the use of symmetry and the integration of the Coulomb divergence in actual GWA implementations (see subsections a and b) is presented. (2) The effects of different numerical parameters on quasiparticle energies in plane-wavebased GWA calculations are specified (see subsections c and d). (3) Details about a particular GWA implementation are provided (see subsections e and f). a. Use of Symmetry The special k-points technique3' reduces by symmetry the summation over a uniform mesh of k-points needed to integrate the independent-particle polarizability (Eq. (3.3)) and the self-energy (Eqs. (3.5) and (3.6)) over the Brillouin zone to the summation over a smaller set of special k-points. With S as an element of the little group L, of the external wave vector,316 that is, Sq = q, and T~ as its nonsymmorphic translation, the matrix elements of Eq. H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976). Umklapp processes are neglected since they are only important for k-points on the Brillouin zone boundary. 315 316

103

QUASIPARTICLE CALCULATIONS IN SOLIDS

(3.2) can be written ass1 ME(S-'k, q)

= exp(i(q

+ S G ) . Q M % ( k , 9).

(3.16)

With BZq as the irreducible part of the Brillouin zone defined by the little group of q, the static independent-particle polarizability (Eq. (3.3)) is given by8'

where the contribution from each k-point in the BZq is multiplied by an appropriate weight. The use of symmetry reduces considerably the computational demand for q-points along high-symmetry directions. For instance, 256 k-points in the full zincblende (cubic) Brillouin zone can be reduced to 40 for q parallel to qx and to 60 for q parallel to qL. Care must be taken to add up the phase factors in the above equation correctly. The external wave vector k for Cx and Cc(E) determines the Brillouin zone BZ, that is relevant for the integration of the self-energy in reciprocal space. Application of the elements of the little group k to Eqs. (3.5) and (3.6) leads to a cancellation of phase factors and the following expressions for C x and Cc(E):

(3.19) where ct = 2nzPqo,(q) for the von der Linden-Horsch and Engel-Farid plasmon-pole model.

ct =

1 for the

b. Integration of the Coulomb Singularity The expectation value of the self-energy contains sums over umklapp processes whose G = 0 contribution has an integrable l/lqI2 divergency for q+O. Consider the case317 of Cx and let f(q) be a smooth function that 317

The case of Z C ( E ) is analogous and will not be discussed here.

104

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

diverges as l/lqlz for q -,0 and whose integral over the Brillouin zone is known analytically. Rewrite the expectation value of C x as9'

(3.20) The term in square brackets is no longer singular at q = 0 and can be integrated using special points. The last term is integrated analytically. Possible choices for f(q) for fcc symmetry are (a = lattice constant, V, = unit cell v~lurne)~' (a/a2 f(q) = 3 - cos(aqx/2)cos(aqJ2) -cos(aq$2) cos(aqz/2)- cos(aqz/2)cos(aqx/2)' (3.21) and with 0 as the step function and b = lqLl = fix/u,"'"

Choosing either Eq. (3.21) or (3.22) for f(q) leads to deviations in the valence and low-lying conduction quasiparticle energies of Si and Ge at r, X , and L of less than 30meV. Functions f(q) appropriate for symmetries other than fcc have been suggested by Wenzien, Cappellini, and Be~hstedt.~~ c. Convergence Quasiparticle band structures can vary by a few hundreds to a fev tens of meV because of differences in (1) the number of q-points used in the integration of Eqs. (3.5) and (3.6); (2) the energy cut-off used in the expansion of the LDA wave functions; (3) the number of single-particle and plasmon-pole bands kept in the summations in Eq. (3.6); (4) the number of G-vectors kept in the summation in Eq. (3.7); and ( 5 ) the degree of convergence of the plasmon-pole band structure. To determine reliable 318 319

W. H. Backes, private communication. B. Wenzien, G. Cappellini, and F. Bechstedt, Phys. Rev. 851, 14701 (1995).

QUASIPARTICLE CALCULATIONS IN SOLIDS

105

quasiparticle band structures, all of these numerical cut-offs have to be checked systematically. Convergence with respect to the numerical cut-offs mentioned in items 1 through 4 is generally smooth,320as can be inferred from Figs. 24-27 and Table 17. Details of the calculations are given in the captions. A plot of the logarithm of the relative deviation of the quantity to 320 Similar conclusions hold for the convergence properties of the plasmon-pole parameters, point (3,which are not presented here. ' 2 1 C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York (1978), 369.

-1

-2

K

Si risV=o.o

c

.-0 .-iii -3 al 5 U al >

.-c -m E c 0

0 -4

-

-5

-6

40 number of k-points in IBZ

20

so

FIG.24. Convergence of valence and conduction band energies at the point in Si as a function of the number of k-points used in the integration of the self-energy (Eqs. (3.18) and (3.19)). Convergence is smooth, with the equivalent of 10 k-points in the IBZ being necessary to converge the low-lying and r2, to better than 1%. The top of the valence band is set to zero in each case. The converged value for each energy is determined by Shanks iteration.321 All other numerical cut-offs are given in Table 18.

106

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Si

-5 number of reciprocal lattice vectors in sum

FIG. 25. Convergence of valence and conduction band energies at the r point in Si as a function of the number of G-vectors in the internal sum over G-vectors in Eq. (3.7). Convergence is smooth, and about 100 G-vectors are sufficient to converge the energies to less than 1%. The top of the valence band is set to zero in each case. The converged value for each energy is approximated by the energy calculated with 283 G-vectors corresponding to a cut-off in reciprocal space of 3.9 Hartree. All other numerical parameters are identical to those in Table 18.

be converged E from its converged value Em as a function of the convergence parameter N is roughly linear.322 In other words, with 01 being a constant log,,

(y) x -uN

3

E

= E,(1 -

(3.23)

322 Note that fitting schemes for convergence plots are not unique, as discussed, for example, in S. Wei, D. C. Allan, and J. W. Wilkins, Phys. Rev. B46, 12411 (1992) (see Fig. 3 and discussion thereof).

107

QUASIPARTICLE CALCULATIONS IN SOLIDS -1

-2

c

.-0 .-z -3 a a .->

-

V

m -

g! c

0

-4

0 rn

-

-5

-6

40

80

120

number of plasmon pole bands

FIG. 26. Convergence of valence- and conduction-band energies at the r point in Si as a function of the number of plasmon-pole bands included in the sum of Eq. (3.19). Convergence The top is smooth, and 40 plasmon-pole bands are sufficient to converge results to within 1YO. of the valence band is set to zero in each case. The converged value for each energy is approximated by the energy calculated with 130 plasmon-pole bands. The sum over reciprocal G-vectors is cut off at 2.7 Hartree. All other numerical parameters are identical to those in Table 18.

The numerical cut-offs used for Si as well as cut-offs resulting from convergence studies of the other four materials considered in Section 111.18 are summarized in Table 18. Convergence properties differ from material to material and have to be repeated for every material under consideration. Convergence studies as a function of the energy cut-off and the number of k-points are costly, since a new plasmon-pole band structure has to be generated each time. Often convergence studies can be limited to two and ten special k-points. An appropriate energy cut-off can be determined by choosing the cut-off large

108

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS -1

-2

.s iii -3 .U a, a,

.c

-m

p!

c

i

-

-4

-5

Y

R

"25

75 125 number of bands in sum over states

1

J

FIG. 27. Convergence of valence- and conduction-band energies at the r point in Si as a function of the number of bands included in the sum over states of Eq. (3.19). Convergence is smooth with the exception of the Y2,state, which is converged with as little as 40 bands, 60 bands are necessary for good convergence ( < 1 YO)of all energies. The top of the valence band is set to zero in each case. The converged value for each energy is approximated by the energy calculated with 300 bands. The maximum number of bands available at an energy cut-off of 8.5 Hartree is about 310. The sum over reciprocal G-vectors is cut off at 2.7 Hartree. All other numerical parameters are identical to those in Table 18.

enough to converge LDA valence-band and low-lying conduction-band energies to within 50 meV. For example, for Si an increase in energy cut-off from 8.5 to 18 Hartree changes the LDA energies of the valence and the four lowest conduction bands at r, X , and L by less than 80meV; the corresponding quasiparticle energies (shown in Table 17), by less than 0.1 eV. d. Choice of Pseudopotentials and Plasmon-Pole Models Differences in pseudopotentials can influence LDA band structures by about 0.1 eV for states close to the band gap. Hybertsen and collaborator^^*^^ used

109

QUASIPARTICLE CALCULATIONS IN SOLIDS

TABLE17. CONVERGENCE OF VALENCE- AND CONDUCTION-BAND ENERGIES AT THE r POINT IN si AS A FUNCTION OF THE ENERGYCUT-OFF USED TO DETERMINE THE SELF-CONSISTENT GROUND-STATE DENSITY,AS WELL AS THE WAVE FUNCTIONS USED TO CALCULATE THE INDEPENDENT-PARTICLE POLARIZABILITY AND THE EXPECTATION VALUES OF THE SELF-ENERGY OPERATOR. DEVIATIONS BETWEENTHE DIFFERENT EIGENVALUES ARE LESSTHAN 0.08eV. ALL CALCULATIONAL PARAMETERS ARE GIVEN IN TABLE 18 EXCEPT FOR THE CUT-OFF IN RECIPROCAL SPACE FOR THE DETERMINATION OF Z, WHICHIS 2.7 a.u. FOR THE PRESENT STUDY. THECUT-OFF ENERGYE,,, IS GIVEN IN HARTREE. E,,,

r,

8.5 12.0 18.0

- 11.90 -11.88 - 11.88

c

Y25"

r;

0.00 0.00 0.00

3.25 3.17 3.23

5c

r2,

l-k

3.86 3.80 3.83

8.32 8.26 8.33

TABLE18. PARAMETERS USED TO DETERMINE THE SELF-CONSISTENT DENSITY n(r), THE INDEPENDENT-PARTICLE POLARIZABILITY Po (SEE EQ. (3.17)), AND THE SELF-ENERGY Z (SEE EQS. (3.18) AND (3.19)) FOR s i , Ge, G a A s , Sic, AND GaN. ALL WAVE-FUNCTION COMPONENTS ARE KEPTIN THE CALCULATION. THEVON DER LINDEN-HORSCH PLASMON-POLE MODEL AND THE GYGI-BALDERESCHI SCHEMEFOR THE INTEGRATIONOF THE COULOMB SINGULARITY ARE USED. FOR THE LATTERSCHEME, THE PLASMON-POLE BAND STRUCTURE IS DETERMINED AT q = O.00125qx, USING THE EQUIVALENT OF 60 SPECIAL k-POINTS IN THE IRREDUCIBLE BRILLOUIN ZONE. TROULLIER-MARTINS PSEUDOPOTENTIALS WITH NONLINEAR CORECORRECTIONS FOR GaAs AND GaN ARE USEDTO MODEL THE IONIC POTENTIAL. THE ENERGYCUT-OFFE,,, IS IN HARTREE;THE CUT-OFF VALUEG,,, FOR SUMSOVERRECIPROCAL LATTICEVECTORSIS IN ATOMIC UNITS.N,, EQUALS THE NUMBER OF UNOCCUPIED BANDSFOR P o AND THE NUMBER OF OCCUPIED AND UNOCCUPIED BANDSFOR Z. Nkpt IS THE EQUIVALENT NUMBEROF k-POINTS IN THE IRREDUCIBLE BRILLOLIIN ZONE OBTAINED USING THE FULLSPACE-GROUP SYMMETRYOF THE CRYSTAL, AND N,,, IS THE NUMBER OF PLASMON-POLE BANDSKEPT.ALL CALCULATIONS ARE DONEAT THE EXPERIMENTAL LATTICECONSTANT. Si

Ge

GaAs

SIC

GaN

10 8.5 10 8.5 3.4 146 10 8.5 3.0 60 60

28 10 10 10 3.4 146 28 10 3.0 60 100

28 10 10 10 3.1 146 10 10 3.0 60 100

10 25 10 25 4.5 146 10 25 4.5 100 150

10 25 10 25 4.5 196 10 25 4.5 80 150

110

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Kerker pseudo potential^;^^^ Godby et u1.,6*185Rohlfing et u1.,52*114and used Bachelet-Hamann-Schliiter pseudo potential^;^^^ and Backes et dL3' Ref. 42 used soft Troullier-Martins pseudo potential^.^^ Deviations for well-converged LDA and GWA calculations from different authors of about 0.1 to 0.2eV are to be expected. Table 19 estimates the influence of the pseudopotential choice on GWA calculations by comparing quasiparticle energies obtained with a Troullier-Martins p s e u d o p ~ t e n t i awith l ~ ~ ~energies obtained for a generalized norm-conserving p s e u d o p ~ t e n t i a l Deviations .~~~ between the LDA and GWA calculations using both pseuodopotentials do not exceed 0.2 eV for both the valence and low-lying conduction bands. Corresponding deviations for self-energy shifts are 60 meV or less. Table 20 compares Si quasiparticle energies obtained using the von der Linden-Horsch and the Engel-Farid plasmon-pole models and shows that the choice of different models changes quasiparticle energies by 0.1 eV or less. This agrees with the findings of Northrup et who found only small differences in the band structure of nearly-free-electron metals G. P. Kerker, J . Phys. C 13, L189 (1980). G. B. Bachelet, D. R. Hamann, and M. Schliiter, Phys. Rev. B26,4199 (1982). 325 N. Troullier and J. L. Martins, Phys. Rev. B43, 8861, 1993 (1991). 3 2 h These pseudopotentials are based on the Teter '93 pseudopotentials (M. Teter, Phys. Rev. B48, 5031 (1993)) and were provided by D. C. Allan. They do include core charges, but they do not take hardness into account, which leads to a much lower plane-wave kinetic energy cut-off. In the case of Si we have for instance E,,, = 11 Hartrees for pseudopotentials without hardness conservation, versus E,,, = 25 Hartrees for pseudopotentials with hardness conservation. 323 324

TABLE19. COMPARISON OF LDA AND GWA ENERGIES AND THE QUASIPARTICLE SHIFTAT THE r POINT IN si FOR A TROULLIER-MARTINS PSEUDOPOTENTIAL325 (TM) AND A GENERALIZED NORM-CONSERVING326(GNC) PSEUDOPOTENTIAL.DEVIATIONS IN THE GWA BANDSTRUCTURE ARE LARGEST (0.18eV) FOR THE r2,STATE,WHICH IS THE MOST DIFFICULTLOW-LYING CONDUCTION-BAND STATETO CONVERGE. THECHOICEOF THE PSEUDOPOTENTIAL AFFECTSTHE QUASIPARTICLE SHIFTSSIGNIFICANTLY LESSTHAN THE ABSOLUTEENERGIES (60-meV MAXIMUM ALL NUMERICAL PARAMETERS ARE GIVENIN TABLE 18 EXCEPTFOR G,,,, WHICH DEVIATION). EQUALS 3.1 a.u. r~ THE PRESENT STUDY. ALL ENERGIES ARE IN eV.

PSP TM

GNC

LDA GWA GWA -LDA LDA GWA GWA-LDA

- 11.98 - 11.92

0.06 - 12.07 - 12.01 0.06

0.00 0.00 0.00 0.00

0.00 0.00

2.52 3.19 0.67 2.54 3.21 0.67

3.15 3.82 0.67 3.27 4.00 0.73

7.64 8.29 0.65 7.70 8.36 0.66

111

QUASIPARTICLE CALCULATIONS IN SOLIDS

TABLE20. COMPARISON OF QUASIPARTICLE ENERGIES CALCULATED WITH THE VON DER LINDEN(EF) PLASMON-POLE MODELSFOR HIGH-SYMMETRY HORSCH'" (LH) AND ENGEL-FARID'~' DEVIATION BETWEEN THE Two SETSOF QUASIPARTICLE ENERGIES POINTSIN Si. THE TYPICAL CLOSETO THE BAND GAPIS ON THE ORDEROF A FEW TENS OF meV. THELARGEST DISCREPANCY OF O.lOeV OCCURSFOR THE LOW-LYING r,"STATE,FOR WHICH THE PLASMON-POLE MODELIS AN INSUFFICIENT APPROXIMATION. ALLCALCULATIONAL DETAILS ARE IDENTICAL TO THOSE GIVEN IN TABLE18 EXCEPTFOR G ,,,, WHICH EQUALS3.1a.u. IN THE PRESENT STUDY.MINOR DIFFERENCES WITH THE LH Si BANDSTRUCTURE ENERGIES IN TABLE 21 RESULT FROM A BETTER CONVERGENCE IN THE LATTER. ALL ENERGIES ARE IN eV.

rl" r 2 5 v r15c

r;, rlc

LH

EF

-11.92 0.00 3.19 3.82 8.29

-11.82 0.00 3.21 3.81 8.28

XI,

X4, XI, X,,

LH

EF

-1.92 -2.98 1.26 10.66

-7.86 -2.96 1.30 10.60

LH

EF

L;,

-9.67

L,, L;, L,, L,,

-7.15

-9.59 -7.01 -1.25 2.10 4.10 8.18

L;,

-1.26 2.08 4.08 8.17

using the Hybertsen-Louie model compared to an earlier calculation by Lundqvist9' that used a different plasmon-pole model. e. Object Orientation The common computing paradigm in most scientific applications is procedure-oriented programming. The central task of a software engineer using the procedure-oriented paradigm is to identify the procedures, that is, the data manipulations, required to solve a problem and the corresponding best algorithms. The structure element of a procedure-oriented code is the procedure or subroutine with its input and output data. This way of programming leads in general to many global and few local data. The global data are either passed explicitly from subroutine to subroutine or-as in Fortran -via COMMON blocks. This violation of the software engineering principle of data can lead to severe problems once a complex piece of software has to be debugged, modified, or extended beyond the realm of its initial use.328

3 2 7 B. Stroustrup, The C++ Programming Languuge, 2nd ed., Addison-Wesley, New York (1991) 14ff. 3 2 8 For example, it took a graduate student at the University of Illinois, Urbana-Champaign, two years to include d basis functions into an O ( N ) tight-binding code. The 20,000 lines of O ( N ) code were originally implemented for s and p functions only (D. Drabold, private communication).

112

WILFRIED G. AULBUR, LARS JONSSON, A N D JOHN W. WILKINS

An object-oriented programming paradigm329 conceives the task of solving a computational problem as identifying interacting classes or objects. A class is an abstract or a user-defined data type. It comprises data and all operations necessary to manipulate it. For instance, the abstract data type VECTOR is the ensemble of the operations that create, destroy, and manipulate (scalar and vector multiplication, addition, etc.) a vector, an integer corresponding to the vector length, and a pointer to its first element. Classes consist of a header file that declares the class type- that is, its data members and member functions-and a file containing the actual implementation of the class functions. A software user should be able to use a class efficiently in a particular application, by looking only at the header file. Details of the representation of the data as well as algorithms used for a particular numerical problem are hidden from the user. Moreover, unauthorized user access to class data is prohibited. This principle of data or information hiding reduces the use of global variables and leads to the design of reusable, local program modules with encapsulated data. Classes as program modules should be written in a reusable and general way. For instance, a class VECTOR should not be limited to vector-vector operations but should also allow matrix-vector operations. The principle of inheritance can increase the reusability of code considerably. The basic idea is that common features of two different classes are incorporated into a base class whose properties are then inherited by the so-called derived classes. For example, consider a base class HUMAN with private data AGE and WEIGHT and member functions EAT, DRINK, and SLEEP. A derived class STUDENT would contain all the functionalities of HUMAN but, on top of them, the member functions ATTEND -LECTURE, DO -HOMEWORK, and GOOF -OFF. The reciprocal-space GWA code of Ref. 42 is written in C + + in an object-oriented style. Data abstraction at a low level is achieved through extensive use of an in-house C + + library that defines the abstract data types VECTOR and MATRIX.330At a higher level, the central classes of the code are symmetry,selfenergy,polarizability,and wavefunct ion. To describe data encapsulation and code reusability, we consider symmetry and wavefunction as examples. The class symmetry can currently handle zincblende materials such as GaAs and elemental semiconductors such as Si. However, it provides the full functionality (apply a particular symmetry operation, find its inverse and apply the inverse, determine the little group of a vector, etc.) irrespective of the space group 3 2 9 M. A. Ellis and B. Stroustrup, The Annotated C++ Reference Manual, Addison-Wesley, New York (1990). 3 3 0 This library was written and is maintained by W. Wenzel, University of Dortmund, Germany, and M. M. Steiner, Ohio State University.

QUASIPARTICLE CALCULATIONS IN SOLIDS

113

under consideration. Treating a system with, for example, wurtzite symmetry therefore requires only hardcoding of the corresponding symmetry operations in the implementation of symmetry (about 150 lines out of a total of 10,000 lines of code). The class wavefunction contains the k-point at which the wave function is defined, the number of bands and plane waves kept, and the wave-function coefficients and energies. Moreover, it contains basic operations such as time reversal, convolution, and Fourier transform. These basic operations are written in a general manner such that they do not apply only to the construction of the matrix elements of the independent-particle polarizability (see Eq. (3.2)) from the wave functions. They can be used as well for the plasmon-pole eigenvectors (also of type wavefunc tion) and the construction of the self-energy expectation values (see Eq. (3.7)).

f. Eficiency Since C + + is less well established in the scientific community than is Fortran, an interface with existing Fortran codes such as low-level BLAS subroutines is a must. Efficient code results, for example, from interpreting wave functions as matrices rather than as linear arrays and by performing convolutions as in Eqs. (3.2) and (3.7) via BLAS3 matrix multiplications. As a consequence, major portions of the code run at about 270 MFlops on a CRAY-YMP8 (peak performance 333 MFlops). The overall code performance is 170 MFlops and the code uses memory very efficiently according to the standard CRAY performance tools. About 20% of the program is documentation to make it easy to maintain and use. A good test of the reliability of a code is a “standardized” test calculation and comparison with an independently written code. To eliminate subtle differences in pseudopotentials, parametrization of exchange-correlation potentials, and the like, which plague input to self-energy calculations using different LDA codes, one can generate wave functions and energies using an empirical Si p ~ e u d o p o t e n t i a instead. l ~ ~ ~ The corresponding wave functions and energies are, in addition, easy to generate. Results based on empirical pseudopotentials between two independent codes from Refs. 42 and 302 agree to within nine significant digits. GWA CALCULATIONS 17. PARALLEL The use of local-orbital basis functions or recent, real-space algorithms can cut down on the computational cost of quasiparticle calculations, as 331

M. L. Cohen and T. K. Bergstresser, Phys. Rev. 141, 789 (1966)

114

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

discussed in Section 111.15. Another way to speed up quasiparticle calculations is the use of massively parallel computers. This section describes parallel algorithms for reciprocal-space and real-space quasiparticle calculations. a. Reciprocal-Space Approach The basic idea for a parallel determination of the independent-particle polarizability (Eq. (3.17)) and the self-energy (Eqs. (3.18) and (3.19)) is to distribute the conduction bands over n p e s processing elements (PEs) during the course of the calculation. The principle will be explained for the case of the independent-particle polarizabiliiy; the case of the self-energy can be handled analogously. Let PEO denote the PE that does all the 1 / 0 between the file system and the parallel machine; Nvblcbthe number of valence/conduction bands; N, the number of G-vectors; and N , , the number of plane waves. For each k-vector PEO reads the energies and the G-vectors. Then it reads the first N,,/npes conduction bands (assume mod(Ncb/npes)= 0 for simplicity) and sends the information about k-point, energies, G-vectors, and the first block of conduction bands to PE1. Subsequently, PEO reads the second block of conduction bands and sends it together with the k-point, energies, and G-vectors to PE2, and so on. Once all conduction bands are distributed, PEO reads the valence-band information for a point k - q in the BZ, and broadcasts that information to all PEs. Then all PEs -including PEO -calculate the following expression: (3.24) Finally, the results of this calculation are added up via a single-node accumulation on PEO. For a graphical description of the algorithm see Fig. 28. The important features of this parallel implementation are as follows. 1. Storage requirements. The storage needed for complex wave functions on each PE is proportional to ( N u b+ N,,/npes)N,,. The memory requirements per node for wave functions increase approximately linearly with system size since N,, << N,, and since an increasing number of processors can compensate for increasing system size. For the matrix elements M g the storage requirements are proportional to N,,(N,,/npes)N,, which leads to a quadratic memory increase per node with system size. The use of massively parallel computers leads to a gain in memory per node by one order in system size for both wave functions and matrix elements.

QUASIPARTICLE CALCULATIONS IN SOLIDS

115

mp over q-points ... .... ............... loop over k-points . . . . . . . . . . . . . . . . . . . . .

'

I

' '

I I

' '

I II

$

1

read CBs i*ncb/npes. from disk and send to PEi

read VB for k-point in BZq and broadcast to all PEs calculate

single-node accumulation of partial sums a t PEO to calculate Plw(k.q)

FIG. 28. Schematic diagram of a parallel calculation of the independent-particle polarizability in reciprocal space (Eq. (3.24)). The same scheme applies to the calculation of the self-energy in reciprocal space (Eqs. (3.18) and (3.19)). PEO manages disk IjO and distributes the valence and blocks of the conduction-band wave functions over all PEs. Each PE calculates its partial contribution to the independent-particle polarizability. The partial contributions are summed up at the end of the calculation via a single-node accumulation on PEO.

2. Scalability. The increase in the volume of the system by a factor of two leads to twice as many conduction bands. The corresponding matrix elements can be calculated efficiently by using twice as many processors, leading to a scalable algorithm. Moreover, scalability applies to memory usage as well, as discussed under storage requirements. 3. Load balance. The load balance is excellent. Not counting the 1/0 overhead, all PEs have exactly the same workload if mod(Ncb,npes)=O. If mod(N,,,npes) # 0, then the last n p e s - 1 PEs calculate the

116

WILFRIED G . AULBUR, LARS JONSSON, AND JOHN W. WILKINS

matrix elements M E for N,,/npes conduction bands, while PEO calculates M g for the remaining conduction bands. This distribution of workload compensates partially for the extra work that PEO has to do to assemble P&,(q,O) at the end of the calculation (compare Fig. 28). 4. Communication and portability. The algorithm is coarse-grain parallel

and portable. PEO reads and sends the input information and wave function data to all other processors. These calculate the matrix elements for every k-point and send the results of their calculation back to PEO. PEO then assembles the total polarizability. Therefore, for a fixed k-point, communication occurs only at the beginning and at the end of the computation. The use of either MPI or PVM communication primitives in the coarse-grain parallel code ensures portability to different platforms.332The use of symmetry as described in Eq. (3.17) does not require any interprocessor communication since the matrix elements for all G-vectors, and hence for their images under symmetry operations of the little group of q, are created and used locally on each processor.

5. Object orientation-Use of C + + . The central object in the code is wavef unc t ion. As described in Section III.17e, this class comprises the k-point at which the wave function is defined, the number of bands and plane waves kept, and the wave function coefficients and energies. Sending a wave function w v f n c t n from processor my-pe to processor y o u r -pe requires only the following statements:

The actual sending of the data and the communication primitive used in this process are hidden from the user. This feature makes a change from PVM to, for example, MPI easy, since the change is done in only one function. Switching between PVM and MPI can be conveniently achieved via preprocessor directives. During the course of the code development the parallelization scheme changed twice. Parallelization schemes used are (1) the embarrassingly parallel approach of distributing k-vectors over P E s , (2) ~ ~a ~parallelization 332 333

Fully optimized PVM executables are no longer being created for most new platforms. W. G. Aulbur and J. W. Wilkins, Bull. Am. Phys. Soc. 40,No. 1,251 (1995).

QUASIPARTICLE CALCULATIONS IN SOLIDS

117

over reciprocal G - v e c t o r ~ and , ~ ~ ~(3) a parallelization over conduction bands. Changing the code to accommodate new parallelization schemes profited immensely from the flexibility that an object-oriented programming style offers compared to a procedure-oriented programming style. b. Real-Spaceflmaginary- Time Approach For the reciprocal-space GWA calculation, scalability can be attained by parallelizing over the conduction bands. In real space, the r-vectors in the irreducible real-space unit cell are an appropriate variable for a scalable parallel i m p l e m e n t a t i ~ nThe . ~ ~ ~number of vectors in the irreducible realspace unit cell is directly proportional to system size and hence the number of atoms in the system. A system that increases its size by a factor of two can therefore be handled by using twice as many processors. The important operations in the program are the assembly of the Green function (Eq. (3.12)) and of the independent-particle polarizability (Eq. (3.13)) in real space and imaginary time, and the subsequent transformation of the independent-particle polarizability to reciprocal space and imaginary freq ~ e n c y . ~ lAll * subsequent operations in the code are either the exact inversion (for instance, one-to-all broadcast and single-node accumulation) or slight modifications of these basic operations. It is sufficient to discuss parallelism only up to the construction of the screened interaction in reciprocal space and imaginary frequency, which is shown schematically in Figs. 29 and 30. Using managed 110, PEO starts by reading input information about the lattice, k-points, LDA energies, etc., and by broadcasting it to all other PEs. Then, in a small serial portion of the program, PEO reads the Fourier coefficients for each band and FFTs them to obtain the periodic parts of the Bloch wave functions unk(r)and U:k(r’). These wave functions are written to direct-access files. For u,k(r) the record index is the F F T index of the r-vector in the real-space unit cell. The record contains unk(r)for all bands and all k-points. For U n k ( r ’ ) the record index is the number of the special k-point, and the record contains all r’-points in the real-space unit cell for all bands. Next an external loop over imaginary time starts in which the nonin~ for all r-vectors in the teracting propagator at imaginary times f i and irreducible real-space unit cell is determined. Each P E has a list of r-vectors for which it calculates Go(r, r’; fi T ) for a particular time i.s and all r’ in the 334 This parallelization scheme was later adapted for the parallelization of the real-space/ imaginary-time code; see below. 335 This algorithm was developed by W. G. Aulbur in collaboration with L. Steinbeck, M. M. Rieger, and R. W. Godby.

118

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

read u ,#I for all bands and all k-points for PEI

read and broadcast u*Jr’) for one particular k-point. all bands and all r’

0

calculate C (r. r’. tit)

0

wrlte P 11. r’,

+ato dlsk

for each PEI

dlstrlbutePo ( 1 , r’. +Id over PEs

do 3D FFT to obtaln 4

,

1 ,

I,

Polk. r. G’. 4 redlstrlbute using MPILGATHER do 3D FFT to obtain P o (k. G. G’. 10 write @lk. G. 0’.Ul to dlsk

FIG. 29. Schematic diagram of the communication structure of the parallel implementation of the real-space GWA code. We list all communication and calculational steps needed to determine the independent-particle polarizability Po in real space and imaginary time, and to transform it to reciprocal space. The distribution of the unlr(r)and of Po(r,r‘; ir) over all PEs is described in the text. The redistribution step using MPI -GATHER is detailed in Fig. 30. The Fourier transformation from imaginary time to imaginary frequency and assembly of the screened interaction via parallel matrix operations is also described in the text.

interaction cell according to Eq. (3.12). For example, if the number nviuc of r-points in the irreducible real-space unit cell is 55 and the number of processors is 4, then PEO calculates Go for the first 13 r-vectors, PE1 for the following 14, and so on. To calculate Go at a point ri, each PE needs u,,k(ri) for all bands and all k-points, as well as U,*k(r’) for all r’, all bands, and all k-points. Let rank denote the number of a PE. For each i and each node, PEO reads the corresponding wave function information unk(ri)in real space and sends it to PE rank.At the end PEO reads the information that it needs for its own

119

QUASIPARTICLE CALCULATIONS I N SOLIDS of G-vectors

First block of r-vectors in irred. real-space cell

PEO

Images under space group symmetries Second block of r-vectors

PE 1

PE2

Third block of r-vectors

MPI-GATHER Fourth block of r-vectors

, I

I

I I

PE3

FIG. 30. Communication structure for the redistribution of the independent-particle polarizability over PEs. After the first 3D FFT each PE has the independent-particle polarizability Po(k, r, G’,ir) for one particular k-point, a subset of r-vectors in the irreducible real-space unit cell as well as their images under symmetry operations of the space group, all G’-vectors, and one particular time i t As indicated by the arrows in the figure, MPI -GATHER is used to assemble Po for all r-vectors and a block of G’-vectors on each PE (first block on PEO, second on PEI, and so on). After the MPI _GATHER step described in the figure, Po is reordered according to its r argument to have standard FFT ordering, and the second 3D FFT is performed on each node. The different length of the r-blocks in the picture corresponds to a different number of r-points on each PE. The total number of G‘-vectors kept after performing the first FFT is nG and n p e s is the number of PEs used. In the above example mod(n,.,npes) = 2.

calculations. Subsequently in a loop over k-points, PEO reads U,k(r’) for a particular k-point and broadcasts this wave function for all bands and all r’ in the real-space unit cell to all PEs, as shown in Fig. 29. After each PE has finished the loop over k-points and has calculated its noninteracting propagator at fiz, it sends the product Go(ri, r’; iz)Go(ri, r’; - ir)- that is; the independent-particle polarizability up to a constant factor -to PEO and PEO writes the result to disk. This communication is the inverse operation of the distribution of U,k(ri) described above. A graphical description of this communication step is given in Fig. 29.

+

120

WILFRIED G. AULBUR, LARS JONSSON, A N D JOHN W. WILKINS

In the next step, PEO reads the independent-particle polarizability for different vectors ri in the irreducible real-space unit cell and distributes Po(ri,r’; iz) over the different processors in an analogous way as for the distribution of unk(ri).Within an external loop over special k-points, each PE performs the first 3D FFT over r’ on node, turning Po(ri,r’;iz) into Po(k, ri, G’; iz). Two points are important for the 3D FFT: (1) Only those G’-vectors in Po(k, ri, G’; iz) with IG’I < G,,, are kept. In the case of Si, G,,, = 3.4 Hartree, which corresponds to 169 G’-vectors. (2) After the first 3D FFT each PE contains all G’-vectors and its chunk of r-vectors in the irreducible real-space unit cell as well as their images under symmetry operations of the space group of the crystal. Since the distribution of the r-vectors is not contiguous in any coordinate, parallel FFTs to perform the next 3D FFT cannot be used. Instead, a “block-column’’ gather operation using the MPI -GATHER function is performed, which is illustrated in Fig. 30. The basic idea is to assemble all r-vectors and a block of G’-vectors on one PE such that a local 3D FFT on Po(k, r, G’; iz) can be done after reordering the r-vectors in FFT order. Although the Green function is constructed with close to perfect load balancing by parallelizing over the r-vectors in the irreducible real-space unit cell, the load balancing for the construction of the independent-particle polarizability in reciprocal space is less favorable since now the symmetry of the r-vectors in the irreducible real-space unit cell is important as well. However, larger systems have many r-vectors in the irreducible real-space unit cell, and low symmetry and differences in symmetries should average out. After all of these steps, one obtains Po(k, G, G’; iz) for all G-vectors and a block of G’-vectors, which is written to disk via PEO for all k-points and all times ir. The next step is the Fourier transform with respect to imaginary time. To do this Fourier transform, PEO reads quadratic submatrices in (G,G’) of Po(k, G, G’; iz) for all times iz and distributes these submatrices over a quadratic grid of processors. The time Fourier transform is done locally. Further matrix operations such as the construction of the RPA dielectric matrix and of the screened interaction are done using ScaLapack routines for the multiplication and inversion of distributed matrices. 18. GWA CALCULATIONS FOR FIVE PROTOTYPICAL SEMICONDUCTORS

This section summarizes all quasiparticle calculations for five prototypical semiconductors and is intended to serve as a reference for 336 For GaAs, we include unpublished data provided by F. Aryasetiawan (Ref. 290). For Ge, GaAs, and SIC, we include unpublished 60-Gaussian orbital data by Rohlfing, Kriiger, and Pollmann (Ref. 303).

121

QUASIPARTICLE CALCULATIONS IN SOLIDS

readers with a detailed interest in the subject. In addition, the accuracy of quasiparticle calculations is assessed by comparison of independent calculations for several materials and is shown to be state- and material-dependent and often significantly larger than the commonly quoted kO.1 eV. All quasiparticle calculations yield good results for Si, suggesting that tests of quasiparticle approaches cannot be limited to this Relativistic effects and the treatment of core electrons have significant effects on LDA band gaps, as demonstrated in Table 33. These effects have to be considered in standard GWA calculations to achieve good agreement with experiment, as demonstrated by the work of Shirley et ~

1

.

~

~

a. Silicon Silicon is a prototypical semiconductor that has been studied in seven independent GWA calculations based on (1) pseudopotentials and plane waves (Refs. 5,6,42,102, and 103), (2) pseudopotentials and Gaussian orbitals,114and (3) LAPW.285 In contrast to all other calculations, the pseudopotential calculations of Rieger et d 1 0 3 and of Fleszar and HankeIo2 avoid the plasmon-pole approximation by explicitly computing the frequency-dependent interaction. Comparison of both the LDA and GWA energy variations between the different calculations for the lowest conduction-band states of Si listed in Table 21 reveals 1. All GWA calculations for the lowest conduction band lie within 0.2 eV of the range of experimental energies observed. 2. Between all calculations, the variations in both LDA and GWA energies of the lowest conduction band are 0.03 eV and 0.14 eV at r, 0.05 eV and 0.30 eV at X , and 0.1 eV and 0.17 eV at L, respectively. 3. Comparison of plane-wave calculations without an update of the Green function, disregarding354Ref. 6, gives variations for the lowest conduction-band energy of 30 meV and 40 meV at r, 50 meV and 40 meV at X , and 50 meV and 50 meV at L, for LDA and GWA, respectively. 4. An update of the energies in the Green function (see Section 11.13) increases the fundamental gap by about 0.1 eV. Four main conclusions can be drawn from the above observations: 1. LDA and GWA calculations for the lowest direct band gaps in the prototypical semiconductor Si are accurate to within 0.1 eV for LDA and 0.1 to 0.3 eV for GWA.

33' 338

This point will be discussed in more detail in Sections VI.29 and 30. E. L. Shirley and S. G. Louie, Pbys. Rev. Lett. 71, 133 (1993).

9

~

~

122

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

OF Si LDA (ROMAN), GWA (BOLDFACE), AND EXPERIMENTAL (ITALIC) TABLE21. COMPARISON ENERGIES AT THE r, X, AND L POINTS FOR ALL QUASIPARTICLE CALCULATIONS AVAILABLE IN THE LITERATURE. QUASIPARTICLE ENERGIES OF DIFFERENT CALCULATIONS FOR BOTHCONDUCTIONAND VALENCE-BAND STATES VARYBETWEEN 0.1 AND 0.6 eV BECAUSE OF DIFFERENT TECHNIQUES, DIFFERENT DEGREES OF CONVERGENCE, ETC. FOREASEOF COMPARISON, THE LDA AND GWA ABSOLUTE ENERGYS m s OF THE VALENCE-BAND TOPSARE SETTO ZERO.THE SUBSTANTIAL GWA VALENCE-BANDMAXIMUM WITH RESPECTTO THE LDA VALENCE-BAND MAXIMUM ARE GIVENIN BRACKETS IN THE ri5" COLUMN. UNLESSNOTEDOTHERWISE, ALLCALCULATIONS USE PSEUDOPOTENTIALS (PSP) AND PLANEWAVES,A PLASMON-POLE MODEL (PPM), AND DETERMINE SELF-ENERGY S m s IN FIRST-ORDER PERTURBATION THEORY (PT). CALCULATIONS MODELTAKETHE FREQUENCY DEPENDENCE OF THE THAT Do NOT USE A PLASMON-POLE SCREENED INTERACTION FULLY INTO ACCOUNT. THIS FREQUENCY DEPENDENCE LEADSTO A VALENCE-BAND-WIDTH DECREASE OF ABOUT 0.3 eV BUT AFFECTS THE LOWESTDIRECTGAPSBY ONLYA FEW TENSOF meV. QUASIPARTICLE ENERGIES DETERMINED WITH AN UPDATED GREEN FUNCTION (GF)- WHOSEENERGIES ARE GIVENBY THE QUASIPARTICLE ENERGIES OBTAINED IN THE FIRST ITERATION-AREGIVENIN PARENTHESES. IN THE CASEOF REF 102, WE GIVETHE RESULTS OF STANDARDGWA AND GWT (IN BRACKETS)CALCULATIONS (SEE TEXT). OF SHIRLEY et a1.353 IS NOT INCLUDED IN THIS TABLESINCETHE GWA THE CALCULATION VALUESWITHOUTCOREPOLARIZATION AGREEWITH THE RESULTS OF HYBERTSEN et al.44 THIS IS BECAUSE CORE POLARIZATION IS A SMALLEFFECTIN Si. GO STANDSFOR GAUSSIAN LINEARAUGMENTEDPLANEWAVE.ALLENERGIES ARE ORBITALS; FLAPW FOR FULLPOTENTIAL GIVENIN eV.

REF.

I-1"

Expt.

-12.5(6)" -12.4(15)~ -11.4' -11.24 A u l b ~ r ~ ~ - 11.96 -11.90 (- 12.04) Rohlfing'14 - 11.91 - 11.95 HamadazS5 - 11.95 - 12.21 H y b e r t ~ e n ' . ~ ~ - 11.93 - 11.84 (- 12.04) Godby6

r25u

Fleszar

O2

-11.89 -11.57 (- 11.58) - 11.93 -11.57 [- 11.721

r;,

3.34-3.36' 3.05'

4.15(5)' 4.1' 4.21(21f 3.18 3.86 (3.92) 3.24 3.89 3.17 4.19 (4.08)

[ - 0.49'1 [( - 0.53')]

2.56 3.25 (3.30) 2.57 3.36 2.55 3.30 2.57 3.27 (3.35) 2.51 3.30 2.58 3.24 (3.32)

[-0.631 [ -0.061

2.55 3.23 C3.221

0.00

[ -0.651

[ -0.401" [ -0.311 [ -0.201"

C0.071

Rieger"'

r15c

3.56 4.27 3.28 3.94 (4.02) 3.26 3.96 [3.99]

COMMENTS

PSP/GO FLAPW Update GF

2nd order PT No PPM Update G F No PPM

123

QUASIPARTICLE CALCULATIONS IN SOLIDS TABLE 21. Continued XI"

Expt.

Aulbur4'

- 7.82

- 7.90 ( - 7.99) - 7.77

Ha~nada~'~

- 7.95 - 7.82 -8.11

Fleszar

O'

Expt. Aulb~r~~

-2.9' -2.5(3)" -3.3(2)' - 2.87 - 2.96 ( - 2.99) - 2.78 - 2.93 - 2.84 -3.03 ( -2.99)

- 7.78

- 2.82

- 7.67

- 2.80

( - 7.68)

( - 2.81)

- 7.77

- 2.83

- 7.57 [ - 7.681

- 2.83 [-2.881

- 9.3(4)"

- 9.62 -9.65

-6.8(2)" -6.4(4)' - 7.00

( - 9.79)

-7.13 (-7.21) - 6.94 - 7.14 - 6.98 - 7.31 (-7.18)

-9.57 - 9.39 ( - 9.40) - 9.58 - 9.35 [ - 9.471

- 6.96 -6.86 ( - 6.88) - 6.96 -6.78 [ -6.883

( - 9.76)

- 9.58 -9.70 - 9.63

- 9.92

Godby6 Rieger"' Fleszar

O'

x4,

X4"

0.66 1.31 (1.37) 0.65 1.43 0.65 1.14 (1.44) C3.531" C4.271" 0.61 1.34 (1.42) 0.65 1.35 1.341

9.99 10.72 (10.79) 10.03 10.76

10.1 1 10.54 (10.63)

c

-1.2(2)e -1.5' - 1.21 - 1.25 (- 1.26) - 1.17 - 1.25 -1.19 - 1.26 - 1.21 - 1.26 (- 1.27) - 1.22 - 1.19 - 1.17 - 1.17 (- 1.17) - 1.19 - 1.20 [- 1.221

2.06(3) 2.40 (15) li 1.46 2.13 (2.18) 1.47 2.19 1.43 2.15 1.51 2.18 (2.27) 1.53 2.30 1.46 2.14 (2.22) 1.50 2.18 C2.181

3.9(1)' 4.15 (10) Ir 3.36 4.13 (4.19) 3.32 4.25 3.35 4.08 3.37 4.14 (4.24) 3.37 4.11 3.33 4.05 (4.14) 3.33 4.06 [4.05]

7.55 8.23 (8.30) 7.77 8.56

7.71 8.29 (8.39)

"Ref. 339; 'Ref 340; 'Ref. 341; dRef. 342; 'Ref. 343; IRef. 344; @Ref.345; *Ref, 346; 'Ref. 347; 'Ref. 348; 'Ref. 349; 'Ref. 311; '"Ref. 350, gap at X ; "estimated from Fig. 1 of Ref. 284; "estimated from Fig. 1 of Ref. 44; PRef. 351; 4Ref. 352.

124

WILFRIED G. AULBUR, LARS JdNSSON, AND JOHN W. WILKINS

2. The accuracy of converged plane-wave calculations for the same transitions as above is 50meV in LDA and 50meV in GWA and comparable to chemical accuracy.

3. Plasmon-pole approximations do not affect the lowest direct band gaps in Si by more than 50meV. 4. GWA improves significantly upon the LDA description of excited states and gives agreement with experiment within the experimental and computational uncertainties.

Conclusions 1,2, and 4 are state and material dependent. Take the F2, state in Si, which is particularly difficult to converge, as an example. Here, the deviations between all calculations are 0.39 eV in LDA and 0.41 eV in GWA. The deviations for well-converged plane-wave calculations are 0.1 eV in both LDA and GWA, well outside chemical accuracy. In addition, materials that are less “benign” than Si- that is, in which, for example, core electrons or relativistic effects are relevant -exhibit larger variations in LDA and GWA energies, as will be shown below. Exceptions to conclusion 4 will be discussed in Sections IV and V. The plasmon-pole approximation is expected to break down for quasiparticle energies of the order of the Si plasmon energy, wpl = 16.7 eV (see Table 3). Comparing plane-wave calculations, with the exception of Ref. 6, for the lowest valence-band energies at r, X,and L shows that the plasmon-pole 339 L. Ley, S. P. Kowalcyzk, R. A. Pollak, and D. A. Shirley, Phys. Rev. Lett. 29, 1088 (1972), as presented by J. R. Chelikowsky and M. L. Cohen, Phys. Rev. 8 10, 5095 (1974). 340 W. D. Grobman and D. E. Eastman, Phys. Rev. Lett. 29, 1508 (1972). 341 M. Welkowsky and R. Braunstein, Phys. Rev. 8 5 , 497 (1972). J. E. Ortega and F. J. Himpsel, Phys. Rev. B47,2130 (1993). 343 W. E. Spicer and R. C. Eden, in Proceedings of the Ninth International Conference on the Physics of Semiconductors, Moskau, 1968, ed. by S. M. Ryvkin, Nauka, Leningrad (1968), vol. 1, 61. 344 D. E. Aspnes and A. A. Studna, Sol. State Comm. 11, 1375 (1972). 345 Ref. 341, as presented by F. Szmulowicz, Phys. Rev. 823, 1652 (1981). 346 R. Hulthen and N. G. Nilsson, Sol. State Comm. 18, 1341 (1976). 347 A. L. Wachs, T. Miller, T. C. Hsieh, A. P. Shapiro, and T.-C. Chiang, Phys. Rev. B32,2326 (1985), as presented in Ref. 44. 348 F. J. Himpsel, P. Heimann, and D. E. Eastman, Phys. Rev. 824, 2003 (1981). 349 D. Straub, L. Ley, and F. J. Himpsel, Phys. Rev. Lett. 54, 142 (1985). 350 R. W. Godby, M. Schliiter, and L. J. Sham, Phys. Rev. 836, 6497 (1987). 3 5 1 D. H. Rich, T. Miller, G. E. Franklin, and T. C. Chiang, Phys. Rev. 839, 1438 (1989). 3 5 2 J. R. Chelikowsky,T. J. Wagener, J. H. Weaver, and A. Jin, Phys. Rev. B40,9644 (1989). 353 E.L. Shirley, X.Zhu, and S. G. Louie, Phys. Rev. Lett. 69, 2955 (1992). 354 Ref. 6 used 169 plane waves, which corresponds to an energy cut-off of about 5.5 Hartree that is significantlylower than the 8.5-Hartree cut-off used in the other calculations.

QUASIPARTICLE CALCULATIONS IN SOLIDS

125

approximation leads to an artificial widening of the valence band width. The widening amounts to about 0.25eV and shows very little variation ( <20 meV) between r, X, and L. Taking the full frequency dependence into account improves agreement with experiment at the L point but leads to disagreement with experiment at the r point if the experimental value of 12.5eV is used, as shown in Table 21. The accuracy of the commonly quoted experimental value of 12.5 f 0.6 eV has been q u e s t i ~ n e d . ~Recent ~.~~~ experiments find a valence band width at r of 11.4eV3" and ll.2eV,352in good agreement with the calculations by Fleszar and Hanke102and Rieger et al.' O 3 Fleszar and Hanke"' compared standard GWA calculations for Si with GW'T calculations (see, for example, Section 11.14) that included LDA vertex corrections in the dielectric matrix and the self-energy. Inclusion of vertex corrections (see also Section 11.14, Eqs. (2.86) and (2.87)) (1) shifts absolute energies by about 0.5 eV, (2) affects relative energies of valence and conduction bands close to the Fermi level by less than 50 meV, and (3) leads to larger shifts of up to 0.15 eV for the relative energies of low-lying valence states. Absolute energies depend on the choice of the exchange-correlation potential,44 the degree of convergence,'02 and the choice of the self-energy approximation (GWA versus GWT; Refs. 89 and 102), as discussed in Section 11.14. As a result, absolute energies fluctuate from +0.07eV6 to -0.65eV for the risvstate. Absolute energies agree to within 0.16eV (- 0.49 eV to - 0.65 eV) for well-converged, plane-wave-based GWA calculations whose LDA exchange correlation is based on the Ceperly-Alder data for the homogeneous electron gas.355 For these calculations, the main self-energy correction is in the valence band and not in the conduction band."' In density functional theory the highest occupied valence energy is given and, under the assumption that LDA is a good approximation to DFT, one would expect small GWA corrections to the highest occupied LDA energy. Indeed, GWA c a l c ~ l a t i o n sbased ~ * ~ ~on an LDA calculation using RPA correlation356or GWT calculations consistent with the Ceperley-Alder exchange correlation' O' shift the highest occupied state of Si by only 0.1 to 0.2 eV. These calculations may be the most appropriate, for example, for the determination of valence-band offsets.357Note that the self-energy corrections to the highest occupied LDA state increase with an

"' D. M. Ceperley and B. I. Alder, Phys. Rev. Lett. 45, 566 (1980). U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972). The difference between relative energies in GWA and GWT for the lowest direct band gaps is less than 0.02 eV (Ref. 102), as can be seen in Table 21. 356

357

126

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

increasing energy gap and ionicity of the material and exceed, for example, a negative correction of 1 eV in the case of LiCl even with the use of RPA ~orrelation.~~ Figure 31 compares the results of an LDA and GWA calculation with photoemission and inverse photoemission experiments. The size of a typical experimental resolution is 0.27 eV in energy and 0.1 8- in momentum.342 In numerical calculations the momentum is well defined, while the energy uncertainty is estimated to be about kO.1 to 0.2eV from the above discussions. The agreement between theory and experiment is good along

15

10

z-

5

zi

> B 9 s

-m

0

e

>r

F

E

w

-5

-10

-1 5

L

A

r

A

X

Wave vector

FIG. 31. Comparison of LDA and GWA band structures along L - r - X with photoemission and inverse photoemission experiments for Si. Agreement between theory and experiment is within the experimental and theoretical uncertainties. Data are taken from Ref. 342 (full diamonds), Ref. 347 (open circles), Ref. 348 (full triangle), Ref. 351 (open triangles), and Ref. 358 (open diamonds). The typical experimental resolution is 0.27 eV in energy and 0.1 k'in momentum (compare Ref. 342). The theoretical error in energy is estimated to be about +0.1 to 0.2 eV (compare Table 21).

QUASIPARTICLE CALCULATIONS IN SOLIDS

127

the T L direction. The lowest conduction band shows the largest discrepancy between theory and experiment and seems to lie slightly higher than the calculation. Along the T X direction agreement between experiment and theory is satisfactory. In part due to the large experimental momentum uncertainty, theory and experiment still agree to within the respective uncertainties. However, the lowest valence band as well as the first two conduction bands -the A, and the A; bands -show less dispersion than the theoretical prediction. b. Germanium and Gallium Arsenide Ge and GaAs are more difficult theoretically due to the importance of the extended core and relativistic effects. The core can influence the valence energies in several ways: core relaxation, core-valence exchange, corevalence correlation, and core polarization. As shown in Section 11.12, an improved treatment of exchange and correlation effects due to core electrons beyond LDA increases the direct band gaps of Ge and GaAs obtained in standard GWA calculations by 0.32 eV and 0.40 eV, respectively. This leads to good agreement with experiment (Ge (GaAs): 0.85 (1.42) eV from theory versus 0.89 (1.52)eV from e ~ p e r i m e n t ~As ~ ) .shown in Table 22, scalarrelativistic GWA calculations for Ge that treat core-valence exchange and correlation on an LDA level deviate significantly from experiment (e.g., direct gap of 0.48 eV in contrast to 0.89 eV in experiment), and they agree to within 0.1-0.3 eV with each other if an update of quasiparticle energies44 and insufficient convergence (Ref. 114; see Section III.15a) are accounted for. An update of the energy spectrum of the Green function is bound to affect the l-‘z5”- r;, gap significantly because the second iteration deals with a semiconductor rather than a semimetal. The results for GaAs are summarized in Table 23. Due to the neglect of core-valence interactions the experimental band gap at r is underestimated by about 0.4 eV in standard GWA calculations.353The quasiparticle energies vary by 0.3 to 1.5 eV between the different calculations due to different approximations, as detailed in the caption of Table 23 and discussed in Sections III.l5a, 111.16, and III.18a. In particular, deviations of 0.65 eV and 0.3 eV with the 40-Gaussian-orbital calculation of Rohlfing et u Z . ” ~ for the lowest conduction-band state at X and L reduce to 0.14eV and 0.04eV when 60 Gaussian orbitals are used, as shown in Table 15. Also, deviations between pseudopotential results that include core-polarization and core-relaxation effects43 and scalar-relativistic, all-electron LMTO calculat i o n are ~ ~likely ~ ~ to result from the neglect of combined corrections in the LMTO wave functions and from the atomic sphere approximation. 358

D. Straub, L. Ley, and F. J. Himpsel, Phys. Rev. B33, 2607 (1986).

128

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

TABLE22. Ge ENERGIES AT HIGH SYMMETRY POINTS;THE NOTATION IS THE SAME AS IN TABLE21. BOTHGWA VALENCE- AND CONDUCTION-BAND ENERGIES VARY BY 0.1 TO 1.6eV B m D ~ ~ E R ECALCULATIONS NT BECAUSE OF A D~PFERENT TRE~TMENT OF COREVALENCE POLARIZATION AND OF CORERELAXATION, A USEOF UPDATED QUASIPARTICLE ENERGIES RATHER THAN LDA ENERGIES IN THE GREENFUNCTION, AND DIFFERENTDEGREES OF CONVERGENCE (SEE TABLE15). ALL CALCULATIONS USE PLASMON-POLE MODELS. THECHOSEN BASISSET IS PLANE WAVES EXCEPTFOR THE GAUSSIAN ORBITALCALCULATION BY ROHLFING et THE QUASIPARTICLE ENERGIES OF HYBWTSEN AND AND SHIRLEY, ZHU,AND L ~ u I USE E~~ INCLUDE CORERELAXATIONAND UPDATED GREENFUNCTIONS. THE DATAIN PARENTHESES CORE-POLARIZATION POTENTIALS, WHICH ARE NECESSARY TO DETER^ THE QUASIPARTICLE GAPIN QUANTITATIVE AGREEMENTWITH EXPERIMENT. HYBERTSEN AND L o r n INCLUDE SPMORBIT(SO) COUPLING TO FIRSTORDER. ALL ENERGIES ARE IN eV. REF.

Expt."

r1.

- 12.6

G" r;, 0.00

0.90 0.8Y

0.00

- 0.05 0.48

- 12.9(2)b

Aulbur4' Rohlfing114 Hybert~en~~

- 12.74 - 1259 - 12.19 - 1284 - 1286

0.00

3.25(1) 3.006' 3.206'

2.59 3.10

0.01 0.65 0.71

0.00 0.00 0.00 - 0.30

2.53 3.21 3.04 3.26

- 0.26

Shirley43

COMMENTS

r15c

053

Update GF, SO 1st order Update GF, w/o (w) core polarization

(0.85)

XI"

Expt.'

- 9.3(2)b

-3.15(20) -3.5(2)'

1.3(2)

Aulbur4'

- 8.88

- 3.02 - 3.08 - 3.02 - 3.16 - 3.22

0.70

-887

Rohlfing114 Hybert~en~~ Shirley43

-8.91 - 9.06 -9.13

9.50

1.15

10.04

1.03

9.54

1.74 1.23

10.19

0.55 1.28 (1-09)

Expt."

-10.6(5)

-7.7(2)

-1.4(3)

0.744

Aulbur4'

- 10.61

-7.51

3.77

7.01

-7.63

0.61

4.30

756

Rohlfing114

- 1.37 - 1.39 - 1.40 - 1.47 - 1.61 - 1.43

0.12

- 10.60 - 10.71

Hybertsen" Shirley4,

- 1082

- 1.63 - 7.81

- 10.89

-7.82

4.3(2) 4.2(1)'

7.8(6) 7.9(1)'

0.33

3.80

8.33

0.98 0.75

457 4.33 4.43

9.20 7.61

-0.05 0.70 (0.73)

"Ref. 182 unless otherwise noted; 'Ref. 347;'Ref. 359;"Ref. 349.

129

QUASIPARTICLE CALCULATIONS IN SOLIDS

TABLE23. GaAs ENERGIES AT HIGHSYMMETRY POINTS, SAMENOTATION AS IN TABLE21. THE LDA AND GWA ENERGIES AT r, X,AND L FOR THE LOWEST CONDUCTION BAND VARY BETWEEN THE LISTED CALCULATIONS BY 0.4 TO 0.6 eV IN LDA AND 0.3 TO 1.0 eV IN GWA BECAUSE OF (1) OF CORERELAXATION AND POLARIZATION IN REFS.43 AND 290, (2) INSUFFICIENT INCLUSION CONVERGENCE (REF.114; SEE TABLE15), (3) USE OF THE ATOMIC SPHERE APPROXIMATION AND AND (4) UPDATE OF GREENFUNCTIONS IN REFS. 43 AND NEGLECT OF COMBINED 121 IN CONTRASTTO REFS. 42, 114,290, AND 360. THE EFFECTSOF THESE APPROXIMATIONS ARE DISCUSSEDIN SECTIONS 111.15 AND 111.18 IN THE TEXT. ALL CALCULATIONS USE PLANE WAVES OF THE COMMENTS IN THE TABLE, SEE TABLES UNLESSNOTEDOTHERWISE. FORAN EXPLANATION ARE GIVENIN eV. SPIN-ORBIT SPLIITNGIS NEGLECTED IN THE 21 AND 22. ALL ENERGIES EXPERIMENTAL DATA.MOSTOF THE EXPERIMENTAL DATAARE AT ROOMTEMPERATURE. TI"

REF.

r15.

COMMENTS

TIC

Expt." A~lbur~~

-13.21 - 12.62 - 12.46

0.00

1.52

4.61

0.00 0.00

0.47 1.16

3.80 4.47

Rohlfing' l4

- 12.69 - 1269

0.00 0.00

0.51 132

3.73 4.60

0.00 0.00

0.38 1.29

ZhangIz'

PSP/GO

SO 1st order; update GF

-0.34

-0.34

G~dby~~'

0.00 0.00 -0.34 - 0.34

0.56 1.47

3.70 4.52

0.13 1.02 (1.42)

Shirley43

A r y a s e t i a ~ a n ~ ~ ~ - 12.85 - 1297

0.00 0.00

0.04 1.23

SO 1st order;

w/o (w) core polarization

3.93 5.61

LMTO

-10.86

-6.81

-2.91

1.90

2.47

- 10.29

-6.78 -6.98 -6.19 -7.16

-2.57 -2.68 -2.56 -2.71

1.42 2.00

1.62 2.24

10.19 10.93

1.80 2.65

1.85 2.72

10.33 11.20

'

-2.11 - 2.79 - 2.79 - 2.87

1.32 2.05

G~dby~~'

-2.66 -2.13 - 2.13 -2.80

1.38 2.08

Expt." Aulbur41

-10.12

Rohlfing'14 Zhang

-10.37 - 10.27

1.55 230

130

WILFRIED G. AULBUR, LARS JONSSON, A N D JOHN W. WILKINS

TABLE 23. Continued

Shirley43

1.21 2.07 (1.95)

AryasetiawanZgo

Expt." Aulbur4' Rohlfing114

- 10.37

-7.01

-2.72

1.29

1.59

10.34

-10.50

-6.88

-2.45

2.88

3.38

11.50

-11.35

-6.81

-1.41

1.74

5.45b

8.6b 8.40

'

-11.01

-6.56

-1.09

0.97

-10.84

-6.74

-1.12

1.62

4.68 5.38

-11.08

-6.59

-1.10

1.13

4.67

8.88

-11.02

-6.91

-1.17

1.92

5.65

9.92

-1.16 - 1.19 - 1.37

0.88

Zhanglzl

7.74

1.69

- 1.40

GodbyJ6'

- 1.07

1.04

4.57

- 1.11

1.82

5.41

- 1.28 - 1.32

Shirley43

0.70 1.55 (1.75)

Aryasetiawan'"

-11.11

-6.84

-1.12

0.72

4.66

8.26

-11.27

-6.59

-1.01

2.14

6.51

9.51

"Ref. 361 unless noted otherwise; bRef. 342.

c. Silicon Carbide and Gallium Nitride Numerical and experimental data on the quasiparticle band structure of Sic and GaN are scarce. To the best of our knowledge only three GWA calculations for S i c have been published so far: one based on Gaussian orbitals"4 and two based on plane waves.42~130 Table 24 demonstrates that agreement between the three calculations is unsatisfactory. In particular, the

359

360 361

D. E. Aspnes, Phys. Rev. B 12, 2297 (1975). R. W. Godby, M. Schliiter, and L. J. Sham, Phys. Rev. B35,4170 (1987). as in Ref. 182, vol. 22a.

131

QUASIPARTICLE CALCULATIONS IN SOLIDS

TABLE24. CUBICS i c ENERGIES AT HIGHSYMMETRY POINTS,SAMENOTATION AS IN TABLE 21. AVERAGEDIFFERENCES BETWEEN THE DIFFERENTCALCULATIONS OF THE LDA AND GWA BAND AND THE QUASIPARTICLE SHIITSARE 0.19 eV, 0.37 eV, AND 0.29 eV, RESPECTIVELY, STRUCTURES AND SIGNIFICANTLY LARGERTHAN THE COMMONLY ACCEPTED UNCERTAINTY OF 0.1 eV IN QUASIPARTICLE ENERGIES. RESULTS OF PLANE-WAVE CALCULATIONS4'" 30 AND OF A GAUSSIANARE COMPARED. ALL ENERGIES ARE GIVEN IN ev. COMPARISON WITH ORBITAL EXPERIMENT IS GIVEN IN TABLE 25. REF. Aulbur4' Rohlfing'14 Backes'30

rl"

Rohlfing''4 Backes' 30

Aul bur4' Rohlfing'14 Backesl3'

TIC

r15c

- 15.34 - 16.08 - 15.44

0.00 0.00

6.21

7.16

7.19

8.18

0.00

6.41

7.16

- 16.44

0.00

7.35

8.35

- 15.07

0.00 0.00

6.51

7.24

7.81

8.66

- 16.13

Aulbur4'

rl5"

XI"

X3"

X,"

- 10.22

-7.82

-3.20

1.31

4.16

13.78

- 10.96

-8.44

-3.53

2.19

5.23

15.23

-10.31

-7.89

-3.22

1.31

4.33

14.05

-11.24

-8.64

-3.62

2.34

5.59

15.78

-10.12

-7.10

-3.06

1.22

4.14

13.93

-11.19

-8.38

-3.42

2.37

555

16.05

-11.71

-8.56

7.12

9.90

-9.19

-1.06 -1.21

5.33

-12.46

6.30

8.25

11.32

-11.80

-8.63

-1.06

5.46

7.20

10.31

-12.75

-9.42

-1.21

6.53

8.57

12.04

-11.57

-8.45

-0.98

5.46

7.12

10.14

-12.65

-9.15

-1.11

6.76

8.68

12.08

average deviation in the LDA band structure between all three approaches is 0.19 eV (maximum 0.41 eV for the L,, state); the average deviation in the quasiparticle energies is 0.36 eV (maximum 0.82 eV for the X,, state); and the average deviation in the quasiparticle shifts is 0.29 eV (maximum 0.67 eV for the X,, state). It is not understood what causes the significant differences in the LDA and quasiparticle band structures of these three independent calculations.362 All three, however, give a good account of experimentally

'''

Determination of self-consistent energy eigenvalues at r, X, and L with 40 Hartree rather than 25 Hartree changes the LDA energies of Ref. 42 by less than 0.01 eV for the lowest 10 states. Convergence tests indicate that the corresponding quasiparticle energies are converged on the level of 0.1 eV.

132

WILFRIED G. AULBUR, LARS JONSSON, AND J O H N W. WILKINS

accessible interband transitions and energies, as shown in Table 25. Average deviations from experiment excluding the L,, state range from 0.22 eV4' to 0.24 eV (Ref. 114, 0.20 eV if 60 Gaussians are used) to 0.29 eV,130 All three calculations place the L,, state between 6.30 and 6.76eV, in marked contrast to experiment (4.2 eV). The difference between theory and experiment is well beyond theoretical and experimental uncertainties. The agreement of the three GWA calculations for the L,, state strongly suggests that the experimental 4.2-eV transition cannot be interpreted as an indirect transition between quasiparticle-like r15, and L,, states.l14 GaN is a very important material for optical devices such as blue-lightemitting diodes and blue lasers.365However, two obstacles impede theoretical p r o g r e ~ s : ' * The ~ * ~Ga ~ ~ 3d electrons interact strongly with the N 2s electrons, and GaN band structures are sensitive to the choice of the N pseudopotential. In GaN the 3d electrons in LDA lie at the bottom of the W. R. L. Lambrecht, B. Segall, M.Yoganathan, W. Suttrop, R. P. Devaty, W. J. Choyke, J. A. Edmond, J. A. Powell, and M.Alouani, Phys. Rev. BSO, 10722 (1994). 364 R. G. Humphreys, D. Bimberg, and W. J. Choyke, Sol. State Comm. 39, 163 (1981).

TABLE25. COMPARISON OF CUBICS i c QUASIPARTICLE ENERGIESFOR PLANE-WAVE CALCULAGAUSSIAN ORBITAL APPROACH"^ WITH EXPERIMENTAL ENERGIESAT HIGHPOINTS.AGREEMENT WITH EXPERIMENT IS GOODFOR ALL THREE GWA CALCULASYMMETRY TIONS.AVERAGE DEVIATIONS ARE 0.22 eV,4* 0.24 eV (REF. 11% 0.20 eV IF 60 GAUSSIANS ARE USED), AND 0.29eV.13' INCREASING THE CONVERGENCE OF THE GAUSSIAN-ORBITAL-BASED CALCULATION BY KEEPING 60 (REF. 295; NUMBERSIN PARENTHESES) RATHERTHAN 40 (REF.114) GAUSSIANSMom THE QUASIPARTICLE ENERGIES CONSISTENTLY TOWARDSTHE CONVERGED RESULTS OF THE PLANE-WAVE CALCULATION OF REF.42. E X P ~ M E N TVALW AL ARE TAKEN FROM REF. 361 UNLESSO~OWWISE NOTED.ALL ENERGIES ARE GIVENIN eV. T I O N S ~ ' . ' ~AND ~ A

~~

EXPT.' x3, - Xlc L3" - XI, X5" - X I , X5" - x 3 c L3" - L3c L3" - LIC

r1.

rl5C

X5" XI, LIC

K,

3.10 3.55 6.0 8.3 9.7 7Sb 7.4b 7.75 - 3.4b 2.39, 2.417' 4.2

~~

ALJLBIJR~~ 3.04 3.40 5.72 8.76 9.46 7.51 7.19 8.18 -3.53 2.19 6.30 4.00

"Ref. 361 unless otherwise noted; bRef. 363; 'Ref. 364.

RoHLmc114

BACKES130

3.25 (3.09) 3.55 (3.49) 5.96 (5.83) 9.21 (8.92) 9.78 (9.70) 7.74 (7.64) 7.35 (7.29) 8.35 (8.42) - 3.62 (- 3.52) 2.34 (2.31) 6.53 (6.46)

3.18 3.48 5.79 8.97 9.79 7.87 7.81 8.66 - 3.42 2.37 6.76

QUASIPARTICLE CALCULATIONS IN SOLIDS

133

valence band and hybridize with the s bands. In LDA, d electrons are underbound in comparison to experiment, leading to a strong, unphysical s-d hybridization. A proper treatment of the Ga 3d electrons requires a self-consistent determination of the wave functions and the inclusion of exchange with the 3s and 3p electrons, as explained in Section III.15a. A computationally cheaper, surprisingly accurate approach to the determination of the quasiparticle band gap is the treatment of the 3d electrons via nonlinear core corrections,205 which has been adopted in the plane-wave calculations of Refs. 42,47, and 367. Variations in the quasiparticle energies of the lowest-lying conduction band for the three different calculations amount to about 0.2 to 0.7 eV and are significantly larger than the generally quoted GWA uncertainty of f O . l eV. In part, these variations may result from the choice of different pseudopotentials, as specified in Table 26, and the use of a model dielectric matrix in Ref. 47 compared to RPA dielectric matrices in Refs. 42 and 367.

IV. Semiconductors and Insulators

This section gives a comprehensive overview of applications of quasiparticle calculations to semiconductors and insulators. These applications are grouped according to bulk materials, superlattices and interfaces, surfaces, defects, pressure dependence, and excitons. The final section discusses quasiparticle calculations for atoms and molecules. 19. BULK

The technical details of GWA calculations in bulk semiconductors and insulators and results for five prototypical semiconductors were extensively discussed in Section 111. Figure 3 shows an overview of-to the best of our knowledge -all GWA calculations for semiconductors and insulators published so far. This section is therefore limited to three examples of the relevance of GWA to describe the electronic structure of materials: (1) band-gap narrowing in n-type Si, (2) transition-metal oxides, and (3) solid C,, and related systems. 365 S. Nakamura and G . Fasol, The Blue Laser Diode: GaN Based Light Emitters and Lasers, Springer-Verlag,Berlin and New York (1997). 366 A. F. Wright and J. S. Nelson, Phys. Rev. B50, 2159 (1994). M. Palummo, L. Reining, R. W. Godby, C. M. Bertoni, and N. Bornsen, Europhys. Lett. 26, 607 (1994).

134

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

AT HIGH-SYMMETRY POINTS, SAME NOTATION AS IN TABLE 21. THE TABLE26. GaN ENERGIES DIFFERENTCALCULATIONS OF THE LDA AND GWA ENERGIES FOR BOTHTHE VALENCE-BAND AND THE CONDUCTION-BAND MINIMUM DIFFERBY 0.1 TO 0.4 eV AND BY 0.2 TO 0.7 eV, MAXIMUM RESPECTIVELY. TAKINGALL STATESINTO ACCOUNT LEADSTO A VARIATION OF 0.1 TO 0.8eV FOR LDA AND 0.1 TO 1.3eV FOR GWA. ALL CALCULATIONS ARE PSEUDOPOTENTIAL-PLANEWAVE-BASED AND USEA FIRST-PRINCIPLES RPA DIELECTRIC MATRIX, WITH THE EXCEPTION OF RUBIOet aL4' WHO USEDA MODELDIELECTRIC MATRIX.THE MODELDIELECTRIC MATRIXIS USUALLY ACCURATE TO WITHIN0.1 TO 0.4 eV AND CANNOT ACCOUNT FOR ALL THE DIFFERENCES IN THE DATA.FURTHER DEVIATIONS MAY RESULTFROM THE DIFFERENT CHOICE OF THE NITRCGENPSEUDOPOTENTIAL. BHS STANDS FOR BACHELET-HAMANN-SCHLmR324 PSEUDOPOTENTIALS AS IMPLEMENTED BY STUMPF, GONZE AND sCHEFFLER;370 HSC STANDS FOR HAMANN-SCHL~ER-CHIANG PSEUDOPOTENTIALS;3 AND TM STANDS FOR TROULLIER-MARTINS PSEUDOPOTENTIALS.32s'372 ALL ENERGIESARE IN ev.

rl"

r*

rls"

rlc

0.0

3.2" 3.3*

- 15.5

0.0 0.0

2.0 3.1

10.5 12.0

E

- 16.7

Palurn~no~~~

0.0 0.0

1.9 2.8

10.2 11.3

E~'",

Rubio4'

10.6 12.2

E'""''

REF. Expt. Aulbur4'

COMMENTS

5e

0.0 0.0

2.1 3.1

-12.4 -13.5

-6.1 -6.8

-2.4 -2.7

3.3 4.4

6.7 8.1

Pal~rnrno~~~

-6.2 -6.8

-2.6 -2.9

3.2 4.0

6.6 7.7

Rubio4'

-13.0 -14.8

-6.5 -6.9

-2.8 -3.0

3.2 4.7

Aulbur4'

-13.2 -14.3

-6.8 -7.6

-0.8 -0.9

Palu~nrno~~~

-7.0 -7.5

R ~ b i o ~ ~

-7.4 -7.8

- 13.8

-15.5

"Ref. 368; "Ref. 369.

~

,

BHS

, update GF, HSC(Ga), TM(N)

- 16.3

- 17.8

Aulbur4'

~ TM~

11.9 14.0

14.2 16.1

6.9 8.4

12.2 14.5

14.6 16.7

4.8 6.1

8.9 10.8

10.5 11.9

-0.9 -1.0

4.7 5.7

8.9 10.4

10.3 11.3

-1.0 -1.1

5.0 6.2

9.1 11.2

10.6 12.3

QUASIPARTICLE CALCULATIONS IN SOLIDS

135

a. Band-Gap Narrowing in Si Two competing processes influence the band-gap modification of Si upon n-type doping:373(1) Electrons fill up conduction bands to some new Fermi level and effectively increase the photoexcitation gap of valence electrons. (2) The conduction electrons form a low-density, “metallic” gas of carriers and increase screening of the electron-electron interaction, which reduces the band gap. A quantitative understanding of these effects on the band gap of doped semiconductors is important since it affects the performance of semiconductor devices. Quasiparticle calculation^^^^*^ 75 have so far considered band-gap narrowing in doped Si and find that (1) LDA is inadequate for the description of band-gap narrowing, (2) the modification of screening due to excess carriers dominates band-gap narrowing in the GWA calculation, and (3) the energy dependence of the intrinsic dielectric matrix -omitted in all model calculations (for a review of model calculations see Ref. 375) -modifies band-gap narrowing by up to a factor of two for common dopant concen~~~*~~~ trations in Si. A direct comparison of GWA ~ a l c u l a t i o n s with experiment376*377-378 is inappropriate since the self-energy calculations do not consider electron-donor scattering.379With AVH and AVxc as changes in the electrostatic and LDA exchange-correlation potentials upon introduction of additional electrons in the system, the LDA contribution to bandgap narrowing can be calculated in first-order perturbation theory as AEnk =

s

@:k(r)[AVH(r)

+ AVxc(r)]@nk(r)dr~

(4.1)

3 6 8 T. Lei, T. D. Moustakas, R. J. Graham, Y. He, and S. J. Berkowitz, J. Appl. fhys. 11,4933 (1992); T. Lei, M. Fanciulli, R. J. Molnar, T. D. Moustakas, R. J. Graham, and J. Scanlon, Appl. fhys. Lett 59, 944 (1991); C. R. Eddy, T. M. Moustakas, and J. Scanlon, J. Appl. fhys. 73, 448 (1993). 369 M. J. Paisley, Z. Sitar, J. B. Posthill, and R. F. Davis, J . Vac. Sci. Echnol. A 7 , 701 (1989); Z . Sitar, M. J. Paisley, J. Ruan, J. W. Choyke, and R. F. Davis, J. Mat. Sci. Leu. 11, 261 (1992). 3 7 0 R. Stumpf, X. Gonze, and M. Schemer, Research Report of the Fritz-Haber Institute, Berlin, Germany, April 1990. 3 7 1 D. R. Hamann, M. Schliiter, and C. Chiang, Phys. Rev. Leu. 43, 1494 (1979). 3 7 2 N. Troullier and J. L. Martins, Sol. State Comm. 7 4 , 613 (1990). 3 7 3 Ref. 20, 210. 3 7 4 A. Oschlies, R. W. Godby, and R. J. Needs, fhys. Rev. B45, 13741 (1992). 375 A. Oschlies, R. W. Godby, and R. J. Needs, fhys. Rev. B51, 1527 (1995). 3 7 6 P. E. Schmid, fhys. Rev. B 2 3 , 5531 (1981). 3’7 D. D. Tang, I E E E Trans. Electron. Deu. 27, 563 (1980). 3 7 8 J. Wagner, Phys. Rev. B 3 2 , 1323 (1985). 3 7 9 Charge neutrality is achieved by considering a homogeneous, positive background charge (Refs. 374 and 375).

136

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

which even for as high electron concentrations as n = 4 x lo2’ cm-3 affects the band gap by less than 5 meV.375 In a GWA calculation contributions to band-gap narrowing result from a change in the screening of the electron-electron interaction due to the presence of additional charge carriers, and a change in the pole structure of the Green function in the doped case. In first-order perturbation theory and neglecting the unimportant Hartree contribution, the quasiparticle bandgap change equals

where (* denotes convolution in frequency space) AX(r,

E ) = Gdoped * Wdoped - Gintrinsic * wintrinsic -

Gdoped * A W + AG * w n t r i n s i c .

(4.3)

The first term describes the change in the screening of the electron-electron interaction and turns out to be the dominant contribution to band-gap narrowing, equaling about -0.1 eV at a carrier concentration of n = lo2’ cm-3 and about -0.25 eV at n = lo2’ ~ m - The ~ . effect of screening becomes stronger with increasing carrier concentrations. The second term stems from the changed pole structure of the Green function and also increases with increasing concentration. However, its absolute contribution to the band-gap narrowing is negligible and amounts to less than 10meV for concentrations smaller than cm-j. First-principles GWA calculations identify the neglect of frequency dependence of the intrinsic dielectric matrix as an unphysical assumption common to many model calculations of carrier-induced band-gap narrowing.37 5 The frequency dependence of the intrinsic dielectric matrix increases band-gap narrowing by about 30% of the dominant GdoPedAW contribution at n = lo2’ cmV3and by about a factor of two at n = 1021~ m - and ~ , should not be neglected. Other effects, such as local fields in the intrinsic dielectric matrix or the description of the conduction-band states either as LDA states or as plane waves, are unimportant since the main contributions to the modification of screening arise from the long-wavelength component. b. Transition-Metal Oxides The large discrepancy between local spin density (LSDA) functional calculations and experiment indicates that strongly correlated transition-metal

QUASIPARTICLE CALCULATIONS IN SOLIDS

137

oxides pose a challenge to perturbation theory calculations such as the GWA. Transition-metal oxides such as NiO are antiferromagnetic insulators whose 3d onsite repulsion is about 8 eV and thus on the same scale as the 3d band width, which is indicative of strong correlations between the 3d electrons. The band gap in NiO is a charge transfer gap; that is, if a hole with a small binding energy is created on the Ni site, it is filled with a high probability by an oxygen 2 p electron. Experimentally, the hole in the highest valence band has a strong oxygen 2 p character, in contrast to local spin density calculations that show a predominant Ni 3d character. In addition, the LSDA severely underestimates the band gap and magnetic moment in transition-metal oxides such as NiO (Eizp = 0.2 eV versus Eiig = 4.0 eV; pLSDAM lpB versus pexp= 1 . 7 - 1 . 9 ~ ~since ) the LSDA is unable to properly describe localized states in these materials. Accurate treatment of the transition-metal d and oxygen 2 p states is difficult and has limited the number of transition-metal-oxide quasiparticle calculations. Aryasetiawan and Gunnarssod5 reported an LMTO study of NiO, and Massidda et a1.54*91*3a0 used a model-GWA approach3a1 to determine the band structure of MnO, NiO, and CaCuO,. The large discrepancy between the LSDA and the experimental gap, for instance, in the case of NiO suggests that the initial LSDA system and, in particular, the LSDA wave functions are not good approximations to the quasiparticle system and wave functions. This suspicion is confirmed in actual calculation^.^^^^^ Self-consistency is generally used to construct a better basis set than the LSDA basis set. Aryasetiawan and G u n n a r ~ s o n ~ ~ used an approximate self-consistency scheme implemented via a modification of the LSDA one-particle Hamiltonian, whereas Massidda et expanded the quasiparticle wave functions in a basis of occupied and unoccupied LSDA wave functions and determined the expansion coefficients self-consistently. The basic physics behind the model-GWA calculation of Massidda et al. is the separation of the screened interaction W into a short-ranged part wIEG that describes the screening of an inhomogeneous electron gas and a long-range part 6 W that accounts for the incomplete screening in insulat o r ~ The . ~ ~short-ranged ~ contribution WIEGis approximated by the local, energy-independent Kohn-Sham exchange-correlation p~tential.~’ The long-ranged screened interaction 6 W must decay as l/lr - r’l for large Ir - r‘l and accounts for the increase of, for example, the NiO LSDA gap of 0.3 eV by 3.4 & 0.4 eV.54 380

S. Massidda, A. Continenza, M. Posternak, and A. Baldereschi, Physica B237-238, 324

(1997). 381

F. Gygi and A. Baldereschi, Phys. Rev. Lett. 62, 2160 (1989).

138

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Table 27 summarizes GWA results for transition-metal oxides and shows that fundamental energy gaps are in fair agreement with experiment and magnetic moments are in good agreement with experiment. Both quantities show significant improvement over LSDA and their accuracy is comparable to or better than LDA + U or self-interaction-corrected calculations. The inability of RPA-based GWA to reproduce the satellites in photoemission spectra is attributed to the lack of short-ranged correlations in GWA and is reminiscent of the failure to reproduce the 6-eV satellite in Ni (see Sections 11.5 and V.26~and Ref. 68). While self-consistency in GWA calculations increases the oxygen 2 p character of the highest valence band,54 that increase is not enough to eliminate the doininance of the Ni 3d state and to achieve agreement with e ~ p e r i m e n t . ~ ~ c. C,, and Related Systems Quasiparticle calculations describe the electronic structure of solid c60 well (see Ref. 394 for an overview of experimental and theoretical results for the C,, band structure). Estimates of on-molecule and nearest-neighborTABLE27. ENERGY GAPS(eV) AND MAGNETIC MOMENTS (pB) FOR TRANSITION-METAL OXIDES IN ab-initio GWA (ARYASETIAWAN AND G U N N A R ~ ~ MODEL ~ N ~ ~ GWA ), (MASSIDDA OBTAINED AND cOLLABORATORS50'54),A N D LOCALSPIN DENSITY (LSDA) FUNCTIONAL CALCULATIONS IN COMPARISON TO EXPERIMENT, LDA + U CALCULATIONS, AND SELF-INTERACTION-CORRECTED WITH EXPERIMENT IS BETTER THAN OR AT LEASTAS (SIC) CALCULATIONS. GWA AGREEMENT GCOD AS IN LDA+U OR SIC. BECAUSE OF THE LARGEGWA CORRECTIONS TO LSDA r~ THE TRANSITION METALS, SELF-CONSISTENCY IN THE GWA WAVEFUNCTIONS IS REQUIREDTO GET VALUES IN PARENTHESES FOR THE MAGNETIC MOMENTS GOODAGREEMENTWITH EXPERIMENT. THE ORBITAL CONTRIBUTION. (ADAFTED FROM REF. 54.) INCLUDE

ARYASETIAWAN COMPOUND MnO NiO

LSDA

0.2

CaCuO,

SICb

1.o 0.3

ENERGY GAPS(eV) 4.2 3.5 3.7 3.1

3.98 2.54

0.0

1.4

LSDA

5.5

MnO 1.o

LDA+U"

GWA

CaCuO,

NiO

MASSIDDA

1.6

GWA

2.1

MAGNETIC MOMENTS (pB) 4.29 4.52 4.61 (4.52) 1.12 1.56 1.59 (1.83) 0.42

0.66

4.49 (4.49) 1.53 (1.80)

EXPT. 3.8-4.2"' 4.3' 4.0r 1 9 4.58h 1.77' 1.64' 1.90h 0.51'

"Ref. 382; bRefs. 383 and 384; 'Ref. 385; dRef. 386; 'Ref. 387; IRef. 388; @Ref.389; hRef. 390; 'Ref. 391; 'Ref. 392; 'Ref. 393.

QUASIPARTICLE CALCULATIONS IN SOLIDS

139

molecule Hubbard-U parameters of 1.0 and 0.5 eV, respectively, are of the same order as the measured band width of about 0.5eV of the highest occupied (Hlu) and two lowest unoccupied (T,, and 7J multiband complexes. In spite of these intermediate to strong electronic correlations, GWA calculations for face-centered cubic C,, describe the fundamental energy gaps quantitatively we11.338*394,395 Self-energy corrections double the LDA fundamental gap of 1.04eV to a quasiparticle gap of 2.15 eV that compares well with a gap of 1.85 & 0.1 eV obtained in microwave conductivity experiments396 or of 2.3-2.7 eV deduced from direct and indirect photoemission experiment^.^^'^^^^.^^^ The H,-T,, peak-to-peak distance in the experimental density of states equals 3.5-3.7 eV3979398,399 and is reproduced reasonably well by a quasiparticle value of 3.0eV, which corrects an LDA result of 1.6- 1.7 eV. Quasiparticle calculations for perfectly ordered, crystalline C , , lead to about a 30% increase in the LDA band width, as shown in Fig. 32, resulting in GWA band widths of 0.9 eV, 0.7 eV, and 0.8 eV for H , , TI,, and Tlg, respectively. Since LDA overestimates the experimental band width, further physics is needed to explain the experimental band-width narrowing within a quasiparticle framework. Four causes for a lack of dispersion in angle-resolved photoemission are (1) the multiband nature of spectra of the H , , T,,, and Tlg bands338*394 the system, (2) orientational disorder, (3) integration over reciprocal lattice V. I. Anisimov, J. Zaanen, and 0. K. Andersen, Phys. Rev. B44, 943 (1991). A. Svane and 0. Gunnarsson, Phys. Rev. Lett. 65, 1148 (1990). 3 8 4 Z. Szotek, W. M. Temmerman, and H. Winter, Phys. Rev. B47, 4029 (1993). I. A. Drabkin, L. T. Emel'yanova, R. N. Iskenderov, and Y.M. Ksendzov, Pis. Tverd Tela (Leningrad) 10, 3082 (1968) [Sov. Phys. Sol. State 10, 2428 (1969)l. 386 J. van Elp, J. L. Wieland, H. Eskes, P. Kuiper, G. A. Sawatzky, F. M. F. de Groot, and T. S. Turner, Phys. Rev. B44, 6090 (1991). 387 A. Fujimori and F. Minami, Phys. Rev. B30, 957 (1984). 3 8 8 S. Hufner, J. Osterwalder, T. Riesterer, and F. Hulliger, Sol. State Comm. 52, 793 (1984). 389 Y. Tokura, S. Koshihara, T. Arima, H. Takagi, S. Ishibashi, T. Ido, and S. Uchida, Phys. Rev. B41, 11657 (1990). 390 A. K. Cheetham and D. A. 0. Hope, Phys. Rev. B27, 6964 (1983), and references therein. 3 9 1 B. E. F. Fender, A. J. Jacobson, and F. A. Wegwood, J . Chem. Phys. 48, 990 (1968). 392 H. A. Alperin, J. Phys. SOC.Jpn. Suppl. B 17, 12 (1962). 393 D. Vaknin, E. Caignol, P. K. Davies, J. E. Fischer, D. C. Johnston, and D. P. Goshorn, Phys. Rev. B39, 9122 (1989). 394 S. G. Louie and E. L. Shirley, J . Phys. Chem. Solids 54, 1767 (1993). 395 Undoped C,, is a band insulator since, for example, direct photoemission creates only one hole in an otherwise filled band. This hole can travel freely without having to pay the large Coulomb interaction of Li z 0.5 to 1.OeV (0.Gunnarsson, private communication). 396 T. Rabenau, A. Simon, R. K. Kremer, and E. Sohmen, 2. Phys. BW,69 (1993). 397 R. W. Lof, M. A. van Veenendaal, B. Koopmans, H. T. Jonkman, and G. A. Sawatzky, Phys. Rev. Lett. 68, 3924 (1992). 383

140

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

4

j

-

2

0 L

r

x w

FIG.32. Band structure of the face-centered cubic Fm3 structure of solid C,, as obtained in LDA (a) and GWA (b). GWA leads to a band gap in good agreement with experiment and to a 30% increase in band width of the T,g,TI, and If, bands compared to LDA. (From Ref. 338.)

vectors perpendicular to the sample surface, and (4) finite experimental resolution. calculation^^^^*^^^ of the photoemission spectra that are based on a Slater-Koster parametrization of the quasiparticle energies and that include orientational disorder of the C,, molecules agree with experiment. These calculations prove that a lack of dispersion of the H,,TI,, and Tlg band complexes of disordered molecular C,, solid in experiment cannot be interpreted as a sign of strong electron-electron correlation^.^^^*^^^ Further Applications. Band-width renormalization in A3C,, (A = K, Rb) due to the T,, plasmon was examined by Gunnarsson within the framework of the GWA.400 Quasiparticle calculations on periodically repeated BN sheets with interlayer distances varying from 5.5 8, to 13.5 8, can be used to deduce the band structure of BN nanotubes via zone folding, as shown by Blase et al.401*402The layered structures have a calculated indirect band gap of about 5.5 eV between the top of the valence band at K and the bottom of the conduction band at r, which is relatively independent of the interlayer distance.

20. SUPERLATTICES-INTERFACES-SCHOTTKY BARRIERS a. Superlattices Simple superlattices consist of periodically repeated units of n layers of a 398 T. Takahashi, S. Suzuki, T. Morikawa, H.Katayama-Yoshida, S. Hasegawa, H.Inokuchi, K. Seki, K. Kikuchi, S. Suzuki, K. Ikemoto, and Y. Achiba, Phys. Rev. Lett. 68, 1232 (1992). 399 J. H. Weaver, P. J. Benning, F. Stepniak, and D. M. Poirier, J. Phys. Chem. Solih 53,1707 (1992). 400 0. Gunnarsson, J . Phys. Cond. Mat. 9, 5635 (1997). 401 X. Blase, A. Rubio, S. G. Louie, and M. L. Cohen, Europhys. Lett. 28, 335 (1994). 402 X. Blase, A. Rubio, S. G. Louie, and M. L. Cohen, Phys. Rev. B51,6868 (1995).

QUASIPARTICLE CALCULATIONS IN SOLIDS

141

material A stacked onto m layers of a material B and are designated by A,,B,. The superstructure of these artificial compound semiconductors gives rise to a variety of new physical phenomena such as confinement, built-in strain, Brillouin zone folding, and two-dimensional electron gas behavior.403,404*405 Variation of superlattice characteristics permits fabrication of semiconductor devices with custom-tailored electronic properties. Molecular beam epitaxy (MBE) allows the growth of superlattices with monolayer precision.404 Quasiparticle calculations for superlattices A,, B, are identical to bulk calculations with the sole exception that the unit cell contains n layers of material A and m layers of material B and is therefore a factor (n + m)/2 larger than the unit cell of the corresponding binary semiconductor AB. The first application of GWA for a semiconductor superlattice was for Si,Ge,,4°6*407 a superlattice based on the indirect-gap semiconductors Si and Ge, since this structure could potentially be a pseudo-direct-gap material suitable for optoelectronic applications and is compatible with Si-based chip technology. However, for Si substrates GWA finds two indirect band edges at 0.85 and 0.95 eV in comparison to values of 0.78 and 0.90 eV obtained in photocurrent experiments.407Direct, zone-folded transitions are predicted at 1.24, 1.34, 1.76, and 1.86eV and explain features observed in electroreflectance measurements at 1.1-1.25 and 1.8 eV.,07 For a Ge substrate, Si,Ge, is predicted to be approximately a direct-gap semiconductor. Confinement of states in either the Si or the Ge region of the superlattice should not have a significant quantitative effect on the expectation value of the self-energy operator since the self-energy effects in Si and Ge are very similar, as shown in Section 111.18. Figure 33 shows the qualitative agreement between quasiparticle shifts for bulk Si, tetragonally strained Ge, and Si,Ge, on a silicon substrate. This agreement occurs after all quasiparticle energies have been aligned at the top of the valence band. Further Applications. Band-gap variation and variation of direct-indirect band-gap transitions due to a change in superlattice period or in concentration in the case of ordered alloys have been studied in GaN/AlN,,O* GaAs, -,N,, and AlAs, -,N, systems.49At the interface of (GaAs),(AlAs),, 403 G. Bastard, Wave Mechanics Applied ro Semiconducfor Hererostrucfures, Halsted Press, New York (1988). 404 J. H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press, Cambridge (1998). 405 T. P. Pearsall, “Strained-Layer Superlattices: Physics,” Semiconductors and Semimetals, vol. 32, eds. R. K. Williardson and A. C. Beer, Academic Press, New York (1990). 406 M. S. Hybertsen and M. Schliiter, Phys. Rev. 836, 9683 (1987). 407 M. S. Hybertsen, M. Schliiter, R. People, S. A. Jackson, D. V. Lang, T. P. Pearsall, J. C. Bean, J. M. Vandenberg, and J. Bevk, Phys. Rev. B 37, 10195 (1988). 408 A. Rubio, J. L. Corkill, and M. L. Cohen, Phys. Rev. 849, 1952 (1994).

142

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

0 0

0.5

0

si (CUBIC) Ge(TETR1 Si4Ge4 (001)

Y

4

P

I I

-0.5 -4

I

-2

I

0

:

I

2

I

4

E''(eV) FIG. 33. The calculated self-energy correction EQP - E~~~ is plotted against the quasiparticle energy EQP for a Si,Ge, superlattice, cubic Si, and tetragonally distorted Ge. No significant quantitative difference between the self-energy corrections in, the superlattice and the bulk materials is found since self-energy effects in Si and Ge are very close. All values are aligned with respect to the valence-band edge. (From Ref. 406.)

(001) (n = 1,2) super lattice^,^^^*^^^*^^^ electrons in excited states whose density is located on cation sites accumulate in the GaAs rather than the AlAs region of the superlattice, because of the relative repulsive character of the A1 versus the Ga pseudopotential. This local bonding effect causes the GWA direct gap in 1 x 1 and 2 x 2 superlattices to be lower in energy than the pseudodirect gap, in contrast to effective-mass model calculations. Effective-mass calculations miss the local bonding effect and consequently show an inverted ordering of bands.41' b. Interfaces An interface between two dissimilar semi-infinite semiconductors A and B is characterized by the difference in the valence-band energy between the two bulk regions, the valence-band offset AEu, and the corresponding conduction-band offset AEc,413 as shown in Fig. 34. Interfaces play an essential role in heterojunction devices, and the band offsets AEu and AEc determine their transport proper tie^.^'^ The qualitative effects of interface S. B. Zhang, M. S. Hybertsen, M. L. Cohen, S. G. Louie, and D. Tomanek, Phys. Rev. Lett. 63,1495 (1989). S.B. Zhang, M. L. Cohen, S. G. Louie, D. Tomanek, and M. S. Hybertsen, Phys. Rev. B41, ,09

10058 (1990).

S. B. Zhang, M. L. Cohen, and S. G. Louie, Phys. Rev. B43,9951 (1991). X. Zhu, S. B. Zhang, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 63,2112 (1989). For a review, see, for instance, M. S. Hybertsen, Mat. Sci. Eng. B 14,254 (1992). See, for instance, F. Capasso and G. Margaritondo (eds.), Heterojunction Band Discontinuities: Physics and Applications, North Holland, New York (1987). 4'2

143

QUASIPARTICLE CALCULATIONS IN SOLIDS

FIG. 34. Schematic representation of the volume-averaged dipole density n(z) (dashed line), the Hartree potential V ( z ) (solid line), and the interface conduction-band offset, AEc, and valence-band offset, AE,,, of two semiconductors A and B. In the semiconductor bulk regions, V ( z ) equals the average Hartree potential ( which serves as an energy reference for the J bulk conduction-band minimum (Ec.,qnJ of each bulk valence-band maximum ( E u , A ( Band semiconductor. The dipole potential Vdipolc equals the difference between the average Hartree is the energy gap of the bulk semiconductors. potentials. Egap,AfBl

orientation, defects, polarity, and strain on band offsets can be studied using density functional theory. However, a more accurate prediction of valenceand conduction-band offsets requires a quasiparticle approach.413 Since the sum of the valence- and conduction-band offsets equals the known band-gap discontinuity between the semiconductors A and B, AE,,, = AE” AE,, self-energy corrections of only the valence-band offset are discussed in this subsection. Within a few monolayers of the sharp interface, charge density rearrangement leads to the formation of an interface dipole that causes a potential step between the bulk materials.41 The dipole contribution can be combined with the bulk-derived valence-band edge positions to obtain a band offset for a particular interface. Let the volume-a~eraged,~’electrostatic (Hartree) potential ( VH)A(B) define an absolute energy level with are measured. respect to which the quasiparticle valence-band edges EV,A(B) ) (VH)LjD”). The (LDA) dipole potential is given as V&:c) = ( V H ) y D ”Denote the difference in the quasiparticle corrections to the bulk LDA band structure of the semiconductors A and B as AX. The valence-band offset is

+

144

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

then given as

AEv = &,A

+

- E ~ . B hipole =

ELDA U.A

- ELDA v,B

+ AX + Vdipole.

(4~~)

Under the assumption that LDA describes the interface dipole correctly, = Vk;ie, and ignoring possible complications related to that is, hipole absolute energy shifts in GWA,416AEv is determined from an LDA supercell calculation, which gives Ef,3A- Ef,$A and the interface dipole, and from bulk GWA calculations, which determine AX. This implies that interface GWA calculations are computationally less demanding than quasiparticle calculations of superlattices since they require consideration of only the bulk rather than the generally much larger superlattice unit cell. Neglect of the difference between the real and the LDA interface dipole potential is considered to be a good approximation, since LDA describes semiconductor densities to within a few percent. Density distributions at semiconductor interfaces such as GaAs-A1As are smooth because of the similar average valence electron densities and screening properties of the constituent materials (average valence electron density = 0.177 electrons/A3 (GaAs), 0.176 electrons/A3 (AlAs); dielectric constant = 10.8 (GaAs), 8.2 (AlAs)). More inhomogeneous systems such as metal-vaccum interfaces require a self-consistent treatment of quasiparticle wave functions, as discussed in Section V.28. Quasiparticle corrections to the valence-band offset at a GaAs-A1As i n t e r f a ~ e ~amount l ~ . ~ ~to~about 0.1 eV or 30% of the valence-band offset of 0.41 eV determined in LDA. Self-energy corrections are crucial to obtain a theoretical valence-band offset to 0.53 f 0.05 eV, in good agreement with experimental values of 0.53-0.56 eV.418*419 As in the case of GaAs-AlAs, Table 28 shows that quasiparticle effects After performing an in-plane average of, for example, the potential v(z) = l/sJsV(x,y,z)dxdy (S is the in-plane area of the surface unit cell), an additional “running” average over the extension of a unit cell in the growth direction (z) is performed. With a as the lattice constant in the z direction, one defines V(z) = I/a J:?$z T(z’)dz’, which reduces to a constant in the bulk and shows a smooth transition at the interface. For a discussion of these kinds of averages, see, for instance, S. Baroni, R. Resta, A. Baldereschi, and M. Peressi, in Spectroscopy of Semiconductor Microstructures, eds. G. Fasol, A. Fasolino, and P. Lugli, NATO AS1 Series B, vol. 206, Plenum Press, New York (1989), 251. 416 The absolute energy level in GWA can be shifted by convergence, the choice of the exchange-correlation potential, and the choice of vertex corrections, as detailed in Section 11.14. The valence-band offset is determined with the implicit assumption that the difference in self-energy corrections is physically meaningful and independent of technical details provided that both semiconductors are treated on the same level of approximation. 417 S. B. Zhang, D. Tomhek, S. G. Louie, M. L. Cohen, and M. S. Hybertsen, Sol. State Comm. 66, 585 (1988). 418 P. Dawson, K. J. Moore, and C. T. Foxon, in Quantum Well and Superlattice Physics, Proceedings of the SPIE 792, eds. G. H. Dohler and J. N. Schulman, SPIE, Washington (1987), 208. 415

145

QUASIPARTICLE CALCULATIONS IN SOLIDS

TABLE 28. VALENCE-BAND OFFSETM eV AT THE NONPOLAR, ZINCBLENDE OF GaN/AlN( 110) INTERFACE, USINGTHE AVERAGELATTICECONSTANT AlN AND GaN AND PSEUDOPOTENTIALS WITH THE Ga 3d ELECTRONS TREATED EITHERAS CORE OR AS VALENCE ELECTRONS'96OR LMT0.420.421 OF THE Ga 3d ELECTRONS AS VALENCE ELECTRONS LEADSTO INCLUSION AGREEMENT BETWEEN THE THREE LDA CALCULATIONS AND CHANGES THE SIGN OF THE INTERFACED1POLE.lg6 THE SELF-ENERGY CORRECTION OF 0.25 eV EQUALS ABOUT 30% OF THE LDA VALENCE-BAND OFFSET AND HAS BEEN ESTIMATED FROM BULK CALCULATIONS THAT TREATTHE Ga 3d VIA NONLINEAR CORE CORRECTIONS'96 (SEE SECTION III.15a ELECTRONS FOR A DISCUSIONOF THE TREATMENT OF d ELECTRONS IN GWA). THE EXPERIMENTAL VALUESREFeR TO WURTZITE INTERFACES.

LDA Ga 3d CORE Cociorva' 96 Albanesi4'' Ke4" Expt.

0.62

Ga 3d VALENCE 0.93 0.85 0.81 0.70 f 0.24" 1.36f0.07'

GWA 1.18

"Ref. 422; *Ref. 423.

increase the valence-band offset at a zincblende GaN-AlN (110) nonpolar interface by about 30%, or 0.25 eV.Ig6The size of the self-energy correction for the nonpolar interface as well as the large discrepancy between experimental valence-band offsets for the polar wurtzite GaN-AlN (OOO1) interface (0.70 & 0.24 eV422versus 1.36 f 0.07 eV423in comparison to a theoretical value of 1.18 eV) suggests that self-energy corrections are important in understanding GaN-AlN interfaces. Treatment of the Ga 3d electrons as valence rather than core electrons increases the LDA valence-band offset by For about 50% and switches the sign of the LDA interface dip01e.I~~ practical purposes, one can treat the Ga 3d electrons via nonlinear core corrections205in the GWA-that is, one can rely upon error cancellation between the neglect of wave function relaxation and the neglect of screening and exchange (see Section III.15a for a discussion of the GWA treatment of d electrons). This approach gives good quasiparticle energies in comparison with experiment even for 11-VI semiconductors.51 D. J. Wolford, in Proceedings of the 18th International Conference on the Physics of Semiconductors, World Scientific, Singapore (1987), 1115. 420 E. A. Albanesi, W. R. L. Lambrecht, and B. Segall, J. Vac. Sci. Technol. B 12,2470 (1994). 421 S.-H. Ke, K.-M. Zhang, and X.-D. Xie, J. Appl. Phys. 80,2918 (1996). 422 G. Martin, A. Botchkarev, A. Rockett, and H.Morkw, Appl. Phys. Lett. 68, 2541 (1996). 423 J. R. Waldrop and R. W. Grant, Appl. Phys. Lett. 68,2879 (1996). 419

146

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Quantitatively accurate predictions of transport properties require an accuracy of quasiparticle calculations that is better than the relevant energy scale of about 25meV set by room temperature. As discussed in Section 111.18a, only a few independent quasiparticle calculations agree to within 50 meV for quasiparticle energies or even quasiparticle energy corrections. Differences on the order of a few tenths of an eV between quasiparticle calculations and also between theory and experiment are the norm, indicating that the determination of self-energy corrections to band offsets pushes the accuracy limits of the GWA. Nevertheless, a consistent improvement over LDA of calculated valence-band offsets compared to experiment can be achieved.41 Further Applications. An overview of valence-band offset calculations in the GWA and explicit calculations for the In,X, -,As-In,Y, - y Z (X = Ga, Al; Y = Ga, Al; Z = P, As) system is given in Ref. 413. These alloy calculations are based on the virtual crystal approximation, which may not capture local bond-length fluctuations in the real alloy that could systematically change the interface dipole. c. Schottky Barriers At a metal/insulator interface such as Al/GaAs(l lo), alignment of the metal and semiconductor Fermi levels leads to the creation of a 100-1000-A-thick space charge layer on the semiconductor side, which, via Poisson’s equation, bends the valence and conduction bands. Electrons (holes) that flow from the metal into the n-type (p-type) semiconductor must overcome the so-called Schottky barrier if the metal Fermi level is pinned inside the semiconducting gap E, (for a recent review of Schottky barriers see Ref. 424). The n-type and p-type Schottky barriers B, and B, are defined in terms of the conduction-band minimum E, and the valence-band maximum E, of the bulk semiconductor and the bulk metal Fermi energy E, as B, = E, - E , B, = E , - E ,

* B,

+ B, = E,.

(4.5)

In analogy with Eq. (4.3) and neglecting density rearrangements beyond LDA, we obtain with AX as the quasiparticle correction to E, - E , and E , - E,, respectively, Bn(,) = Bkg:

+ AX

where B t i t is the LDA Schottky barrier. 424 J.-G. Li, Mat. Chem. Phys. 47, 126 (1997),and references therein.

(4.6)

QUASIPARTICLE CALCULATIONS IN SOLIDS

147

In contrast to the LDA Schottky barriers, the bulk quasiparticle corrections do not depend upon the atomic structure of the interface and equal 0.22 eV in the case of Al/GaAs(l 10).42s*426*427 This correction amounts to 20-60% of the LDA barrier depending on the particular interface structure chosen and establishes the importance of many-body corrections. Self-energy effects lead not only to corrections to the Schottky barrier height but also to semiconductor band-gap narrowing in the 1-lo-A vicinity of the metal-semiconductor i n t e r f a ~ e . ~Classically, ~ ” ~ ~ ~ an electron in the semiconductor experiences an additional energy-lowering, electrostatic potential whose effect can be described by the interaction of the electron with its image charge in the metal. With E as the dielectric constant of the semiconductor and z as the electron’s distance from the interface, the energy lowering of the conduction band equals 1/(4~z).Analogously, the valence band is bent upward by the same amount. Quasiparticle calculations for Al/GaAs(llO) give a significant band-gap reduction of 0.4eV compared to a bulk GWA gap value of 1.1 eV (core-polarization effects are omitted; see Refs. 425, 426) and suggest that the narrowing of the gap is mainly due to the bending of the conduction band rather than to the equal distribution of the gap between the valence and conduction bands as in the classical case.42s,426 21. SURFACES The theoretical determination of surface structures and reconstructions relies on two techniques:428 (1) calculation of surface-state bands for proposed surface geometries and comparison with spectroscopic data, and (2) total energy minimization over some set of geometries to find optimal positions for atoms near the surface. While the second approach is a reliable tool because of the accuracy of LDA in determining densities, the first approach suffers from the LDA band-gap underestimation, which translates into similar albeit smaller errors for surface states. Reliable empirical corrections to LDA are difficult to construct since quasiparticle corrections for surface states can show strong dispersion rather than simple “scissorsshift”-like behavior, as discussed below. LDA surface-state energies disagree with experiment4” since (1) band gaps between empty and occupied surface-state energies are too small, (2) the dispersion of LDA surface band states is too small in some cases, too 425 J. P. A. Charlesworth, R. W. Godby, R. J. Needs, and L. J. Sham, Mat. Sci. Eng. B 14, 262 (1 992). 426 J. P. A. Charlesworth, R. W. Godby, and R. J. Needs, Phys. Rev. Lett. 70, 1685 (1993). 4 2 7 R. J. Needs, J. P. A. Charlesworth, and R. W. Godby, Europhys. Lett. 25, 31 (1994). 4 2 8 M. S. Hybertsen and S. G. Louie, Phys. Rev. B38,4033 (1988).

148

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

large in others, and (3) the placement of occupied surface-state energies is in some cases too high by 0.5 to 1.0eV relative to the bulk valence-band maximum. Two missing physical effects must be considered to improve upon these LDA failures.428 First, the inclusion of local fields in the dielectric matrix is crucial for the quasiparticle approach since these local fields describe the strongly inhomogeneous screening (bulk-like versus vacuum) at the surface. Second, the nonlocality of the self-energy operator is more sensitive to the localization properties of surface states than the only density-dependent LDA. This nonlocality leads to a modified dispersion of the quasiparticle energies throughout the surface Brillouin zone. Quasiparticle corrections to the LDA surface band gap lead to improved agreement with experiment and depend strongly on the character of the occupied and unoccupied surface states.429 The occupied and unoccupied surface states of the GaAs(ll1) (2 x 1) surface430 derive their character from the corresponding bulk valence and conduction bands and are localized on the As and Ga atoms, respectively. Consequently, the quasiparticle corrections to the surface states are substantial and amount to a roughly rigid shift by 0.7eV of the unoccupied surface states with respect to the occupied surface states, which leads to a quasiparticle surface gap of 2.0 eV,430in good agreement with the experimental value of 1.9 eV.431The quasiparticle corrections for surfaces states of the Ge(ll1):As (1 x 1)428,432 surface are different from the bulk corrections for Ge valence and conduction bands since (1) the occupied surface state is an As lone pair, that is, a filled As dangling bond, and not derived from bulk Ge states, and (2) the unoccupied surface states derive their character from both Ge valence and conduction bands. The bulk self-energy corrections amount to -0.1 eV for occupied and +0.6 eV for unoccupied states. In contrast, the self-energy shift of the As lone pair equals about +0.1 eV and that of the unoccupied surface state is intermediate between the bulk Ge occupied and unoccupied shifts and equals +0.45 eV.428,432 Quasiparticle corrections often improve the dispersion of occupied and unoccupied LDA surface states and agree well with experiment. Self-energy corrections can either increase or decrease the surface-state dispersion, as shown by the following examples. For the Ge(l1l):As (1 x 1) s ~ r f a ~ e ~ ~ ~ quasiparticle corrections increase the band width of the occupied As lonepair band by 0.5eV (LDA 1.18eV, GWA 1.64eV), in excellent agreement with experiment (1.62 eV; Refs. 433,434, and 439, and modify the unphysi429

X. Blase, X. Zhu, and S. G. Louie, Phys. Rev. B49,4973 (1994).

430

X.Zhu, S. B. Zhang, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 63,2112(1989).

431

432

J. van Laar, A. Huijser, and T. L. van Rooy, J. Vuc. Sci. Technol. 14,894 (1977). M.S. Hybertsen and S. G. Louie, Phys. Rev. Lett. 58, 1551 (1987).

QUASIPARTICLE CALCULATIONS IN SOLIDS

149

r

cal, flat LDA dispersion into a parabolic dispersion at in the surface Brillouin zone, as shown in Fig. 35. For the Si(ll1):H (1 x 1) s ~ r f a ~ e ~ ~ ~ the occupied surface band a’ between M and I? has an LDA band width of 0.42 eV but none in GWA and experiment. Note that unreconstructed (1 x 1) surfaces that result from passivation with As or H are excellent systems for quasiparticle calculations since the complications of surface reconstructions are eliminated and supercell calculations can be based on a small 1 x 1 unit cell rather than 2 x 1, 4 x 2, or even larger unit cells. Quasiparticle calculations bring the alignment of occupied states with respect to the valence-band maximum into agreement with experiment. Large self-energycorrections of -0.5 to -0.8 eV of occupied surfaces states are obtained for strongly localized surface states of the Si(ll1):H (1 x 1) surface and are shown in Table 29. Strongly localized LDA states are underbound since the Hartree and exchange self-interactions cancel only R. D. Bringans, R. I. G. Uhrberg, R. Z. Bachrach, and J. E. Northrup, Phys. Rev. Left. 55, 533 (1985). 434 R. D. Bringans, R. I. G. Uhrberg, R. Z. Bachrach, and J. E. Northrup, J. Vuc. Sci. Technol. A 4 , 1380 (1986). 4 3 5 R. D. Bringans, R. I. G. Uhrberg, and R. Z. Bachrach, Phys. Rev. 834,2373 (1986). 436 K. Hricovini, G. Giinther, P. Thiry, A. Taleb-Ibrahimi, G. Indlekofer, J. E. Bonnet, P. Dumas, Y. Petroff, X.Blase, X. Zhu, S. G. Louie, Y. J. Chabal, and P. A. Thiry, Phys. Rev. Lett. 70, 1992 (1993). 433

5 ,

0

-%

-I

P:

-8

bl

P

-

r

it

FIG. 35. Left panel: LDA and quasiparticle (QP) Ge( 1 1 1):As (1 x 1) surface-state energies in comparison to the projected bulk quasiparticle band structure (shaded) along high-symmetry directions in the surface Brillouin zone. Quasiparticle corrections (1) open up the gap between the surface states; (2) eliminate the unphysical, flat LDA dispersion at and (3) improve the position of the occupied surface bands with respect to the valence-band maximum in comparison to experiment. Right panel: Calculated occupied quasiparticle surface-state energies compared to angle-resolved photoemission data (Ref. 433). Agreement between theory and experiment is within the quoted theoretical uncertainty of kO.1 eV. (Adapted from Ref. 432.)

r;

150

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

TABLE29. LDA, GWA, AND EXPERIMENTAL ENERGIESOF Si( 11 l):H( 1 x 1) SURFACESTATESAT I? AND (eV) WITH RESPECT TO THE VALENCE-BAND MAXIMUM.THEGWA CORRECTS LDA BY -0.5 TO -0.8eV. LDA UNDERBINDS STRONGLY ENERGIES LOCALIZEDSTATESBECAUSE OF UNPHYSICAL SELF-INTERACTIONS, WHICH ARE ELIMINATED IN THE GWA. (ADAPTED FROM REF.429.) k-POINT

LDA

GWA

EXPT.~

I?

- 3.22 - 4.29 - 7.85 - 3.87

- 3.82 -4.76 - 8.47 -4.63

- 3.80

M

-4.78 - 8.64 -4.76

“Ref. 436.

partially in LDA. These self-interaction errors are largely eliminated in GWA, leading to the large self-energy corrections mentioned above. GWA calculations cannot always eliminate the discrepancies between L D A and experiment in the dispersion and position of occupied surface states. A case in point is the free Ge(001) (2 x 1) surface4j7 whose band structure is shown in Fig. 36. Two conclusions follow from Fig. 36: (1) a significant discrepancy of up to 0.8 eV remains between quasiparticle calcu437

M. Rohlfing, P. Kriiger, and J. Pollmann, Phys. Rev. B54, 13759 (1996).

r

J

,

K

J’

r

[oto]

J;

FIG. 36. GWA (solid line), LDA (dashed line), and experimental photoemission data (diamonds: Refs. 438, 439, 440, circles: Refs. 441, 442) for occupied and unoccupied states of the clean Ge(001) (2 x 1) surface. Clear discrepancies with experiment exist for the dispersion of the unoccupied Ddown state, of the occupied D,, state between JK/2 and J’, and for the dispersion of the occupied states at I-. The projected bulk GWA band structure is shadowed, and energies are measured with respect to the valence-band maximum. Open symbols represent weak features in the experimental spectra. (From Ref. 437.)

QUASIPARTICLE CALCULATIONS IN SOLIDS

151

lations and experiment along the JK/2 to J' direction in the surface Brillouin zone, and (2) the 0.6-eV experimental band width of the Ddownstate is strongly overestimated in quasiparticle theory (z1.0eV, Ref. 437). Similar discrepancies exist for the sulphur-terminated Ge(001):S (2 x 1) surface.437 Here the energies of the occupied quasiparticle D surface band and the occupied B surface band are underbound by 0.9 and 0.4eV, respectively, compared to experiment. Quasiparticle calculations for surface states help explain and analyze surface reconstructions and can be especially valuable when LDA total energy minimizations lead to contradicting results with very different electronic properties. For example, the Pandey n-bonded-chain model is favored for the explanation of the C( 111) (2 x 1) surface reconstruction, but the precise position of the atoms in the surface has been a subject of discussion. Vanderbilt and suggested a surface geometry of slightly buckled but undimerized chains based on a linear-combination-of-atomicorbitals LDA calculation. Iarlori et a1.444found a dimerized surface with no buckling based on LDA molecular dynamics simulations. Model GWA calculations by KreB, Fiedler, and B e ~ h s t e d t ,whose ~ ~ ~ results , ~ ~ ~are shown in Fig. 37, support the dimerized model. The quasiparticle calculations show that the Vanderbilt-Louie model leads to a metallic surface and that the model by Iarlori et al. opens up a surface gap of 1.7 k 0.3 eV, in good agreement with the experimental value of about 2.0 eV.447,448 L. Kipp, R. Manzke, and M. Skibowski, Surf: Sci. 269/270, 854 (1992). L. Kipp, R. Manzke, and M. Skibowski, Sol. State Comm. 93, 603 (1995). 440 M. Skibowski and L. Kipp, J. Electron Spectrosc. Relat. Phenom. 68, 77 (1994). 4 4 1 E. Landemark, R. I. G. Uhrberg, P. Kriiger, and J. Pollmann, Surt Sci. Lett. 236, L359 (1990). 442 E. Landemark, C. J. Karlsson, L. S. 0. Johansson, and R. I. G. Uhrberg, Phys. Rev. B49, 16523 (1994). 443 D. Vanderbilt and S. G. Louie, Phys. Rev. B29, 7099 (1984). 444 S. Iarlori, G. Galli, F. Gygi, M. Parinello, and E. Tosatti, Phys. Rev. Lett. 69, 2947 (1992). 445 C. KreB, M. Fiedler, and F. Bechstedt, Europhys. Lett. 28, 433 (1994). 446 C. KreB, M. Fiedler, W. G. Schmidt, and F. Bechstedt, Surf. Sci. 331-333, 1152 (1995). 447 S. V. Pepper, Surf. Sci. 123, 47 (1982). 448 Further applications: Several other surfaces have been considered in first-principles GWA calculations, such as the Ge(ll1) (2 x 1) surface (Ref. 449); the Ge(001):H (2 x 1) surface (Ref. 437); the Si(l1l):As (1 x 1) surface (Refs. 428 and 450); the Si(ll1) (2 x 1) surface (Ref. 451); the Si(OO1) (2 x 1) surface (Refs. 284 and 452); the Si(001) c(4 x 2) surface (Ref. 453); and the p-SiC(001) (2 x 1 ) surface (Ref. 454). 449 X. Zhu and S. G. Louie, Phys. Rev. B43, 12146 (1991). 4 5 0 R. S. Becker, B. S. Swartzentruber, J. S. Vickers, M. S. Hybertsen, and S . G. Louie, Phys. Rev. Left. 60, 1 16 (1988). 4 5 ' J. E. Northrup, M. S. Hybertsen, and S. G. Louie, Phys. Rev. Lett. 66, 500 (1991). 4 5 2 M. Rohlfing, P. Kriiger, and J. Pollmann, Phys. Rev. B52, 13753 (1995). 453 J. E. Northrup, Phys. Rev. B47, 10032 (1993). 438

439

152

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

r

J

K

J’

r

J

K

J’

FIG. 37. Quasiparticle band structure of the C(111) (2 x 1) surface for slightly buckled, undimerized rr-bonded chains (left and dimerized but unbuckled r-bonded chains (right Only the latter case (b) leads to semiconducting rather than metallic behavior and a surface gap of 1.7 eV445in good agreement with the experimental gap of about 2.0 eV.447 The projected bulk quasiparticle band structure is shown in black. (Adapted from Ref. 445.)

22. DEFECTS Defects in semiconductors affect the transport, electronic, and optical properties by introducing defect levels within the band gap. An LDA description of defect levels is insufficient since the underestimation of the bulk band gap by 30-100% translates into a corresponding uncertainty for the defect level position. Also, quasiparticle shifts depend on the character of the defect level and cannot be estimated from the shifts of the bulk band edges in a simple way. Indeed, we have discussed examples of strongly dispersive and state-dependent quasiparticle shifts at semiconductor surfaces, in transition metals, and even in large-gap bulk insulators (Sections 111.18, IV.19, IV.20, IV.21, and V.26). Quasiparticle calculations can determine defect levels accurately but are computationally challenging. To date, applications are limited to the Fcenter defect or halogen vacancy in LiC14” and an oxygen vacancy in zirconia, ZrO,.” We confine the discussion to the halogen vacancy in LiCl since this system is an electronically and structurally simple point defect. This neutral vacancy contains one bound electron in a highly localized 454

M.Sabisch, P. Kriiger, A. Mazur, M.Rohlfing, and J. Pollmann, Phys. Reo. B53, 13121

( 1996).

QUASIPARTICLE CALCULATIONS IN SOLIDS

153

(within one to two lattice constants) 1s state and shows structural relaxations relative to the ideal crystal structure that are mostly confined to nearest-neighbor alkali-metal at0ms.4~~ In this case, a theoretical description of the F-center using a supercell approach with relatively small unit cells seems appropriate. The fundamental electronic F-center transitions are (1) transitions from the localized 1s electron to the conduction band critical points at L, A, and X, and (2) an intra-defect transition from the localized 1s to a localized 2 p level. For the first transition, the experimental values of 4.5 eV, 5.0 eV, and 5.8eV for the L, A, and X transitions are well reproduced by the quasiparticle values 4.5 eV, 5.0 eV, and 5.7 eV, which correct LDA results of 1.8 eV, 2.2 eV, and 2.8 eV by an almost constant shift of 2.8 eV.455For the second, because of the localized nature of the 1s and 2 p orbitals involved in the 1s -,2p intra-defect excitation, excitonic effects must be included in the determination of the quasiparticle transition energy. These effects lower the transition energy by about 1eV, leading to a quasiparticle value of 3.4 eVa 1.0-eV correction to the LDA value of 2.4eV-in good agreement with experiments ranging from 3.1 to 3.3 eV.455 Besides excitonic effects, quasiparticle defect calculations are challenging because of problems related to finite-size effects, the treatment of the localization and spin of the defect electron, the possible need for a selfconsistent treatment of quasiparticle defect states, and so forth. In the case of the 2p defect state even the determination of the corresponding LDA state is difficult since the band gap underestimation in LDA causes this state to be resonant and hybridize with bulk bands. For a detailed discussion of technical problems occurring in GWA defect calculations, we refer the reader to Ref. 455. 23. PRESSURE Quasiparticle calculations can describe isostructural metal-insulator transitions due to pressure-induced overlap of conduction and valence bands. Examples discussed below include solid Xe,456*457 solid molecular hydrogen,457*458 and diamond.459 Other possible scenarios for metal-insulator transitions include structural transformations, as is the case for Si under and simultaneous magnetic and metal-insulator transitions, as

455

456 457

4s8 4s9 460

M. P. Surh, H. Cacham, and S. G. Louie, Phys. Rev. 851, 7464 (1995). H. Chacham, X. Zhu, and S. G. Louie, Europhys. Lerr. 14,65 (1991). H. Chacham, X. Zhu, and S. G. Louie, Phys. Rev. 846,6688 (1992). H. Chacham and S. G. Louie, Phys. Rev. Leu. 66,64 (1991). M. P. Surh, S. G. Louie, and M. L. Cohen, Phys. Rev. 845, 8239 (1992). For a review, see F. Siringo, R. Pucci, and N. H. March, High Press. Res. 2, 109 (1989).

154

WILFRIED G. AULBUR, LARS JBNSSON, AND JOHN W. WILKINS OF Si BANDGAPS(eV/Mbar) AT TABLE30. THE PRESSURE DERIVATIVES EQUILIBRIUM r~ LDA, GWA, AND EXPERIMENT. INPARTICULAR, THE DATA OF ZHU et al.462 SHOWTHAT LDA PRESSURE DERIVATIVES ARE IN GOOD AGREEMENT WITH EXPERIMENT AND WITH GWA. THE THEORETICAL RESULTS OF REF. 463 ARE ACCURATE TO WITHIN f0.3eV/Mbar.463 (ADAPTED FROM REFS.462 and 463.)

GODBY463

z H U4 6 2

STATE 0.85 X , ,

EXPT. - 1.6"

LDA

GWA

LDA

GWA

-1.41

-1.32

-1.3

- 1.8

-1.73 3.95 0.59 11.9

-1.68 4.06 0.53 12.2

-0.9

- 1.6

- 1.44.b

XI, LIC r15c

r;,

0.5

0.6

"Ref. 182; bRef. 464.

in NiI,.461 To date, quasiparticle calculations for structural or magnetic metal-insulator transitions have not been attempted. The simplest estimate of the metal-insulator transition pressure, at which band overlap occurs, is given by the negative ratio of the energy gap EBap and the variation of the energy gap with respect to pressure, dE,,,/dP. Assuming linear behavior, a theory that predicts Egap and dE,,,/dP in agreement with experiment will predict correct transition pressures. Table 30 indicates that the derivative of energy gaps with respect to pressure evaluated at the equilibrium volume is rather well described in LDA. The success of LDA in the determination of dE,,,/dP can be understood in terms of a simple two-band semiconductor which shows that the screened exchange term is not very sensitive to pressure and that the dependence of the self-energy on the density and hence on pressure is mainly due to its Coulomb-hole term. The Coulomb-hole term can be approximated by a local potential, for instance, an LDA exchange-correlation potential (see Section 11.10). Note that with dE,,,/dP being rather well described in LDA, the LDA band-gap underestimation leads to a systematic underestimation of the metal-insulator transition pressure. 4 6 1 M. P. Pasternak, R. D. Taylor, A. Chen, C. Meade, L. M. Falicov, A. Giesekus, R. Jeanloz, and P. Yu,Phys. Rev. Lett. 65, 790 (1990). 462 X.Zhu, S. Fahy, and S. G. Louie, Phys. Rev. B39,7840 (1989). 463 R. W. Godby and R. J. Needs, Phys. Rev. Lett. 62, 1169 (1989). 464 B. Welber, C. K. Kim, M. Cardona, and S. Rodriguez, Sol. State Comm. 17, 1021 (1975). 465 L. Brey and C. Tejedor, Sol. State Comm. 55, 1093 (1985).

QUASIPARTICLE CALCULATIONS IN SOLIDS

155

The increased band gap in GWA calculations compared to LDA leads to a higher transition pressure in better agreement with experiment. For example, the transition pressure of solid Xe457is 128 GPa in comparison to 104 GPa in LDA and experimental values of 132 & 5 GPa466 and 150GPa.467Note that spin-orbit coupling must be taken into account in the cal~ulations.4~'An LDA calculation without spin orbit predicts a transition pressure of 123 GPa and hence agrees with experiment, although it does not describe the physics correctly. The high-pressure behavior of diamond is of particular interest because of the widespread use of diamondanvil cells. In diamond, the minimum band gap decreases under application of anisotropic pressure along the [OOl] direction, and the pressure coefficient for the minimum gap is very sensitive to the degree of anisotropy of the stress.459The experimental geometry and hence the stress anisotropy in diamond-anvil cells is not fully known, which may explain the underestimation of the experimental diamond metallization pressure of 700-900 GPa468 by GWA calculations459that predict a metallization pressure of 400 GPa for an idealized geometry. The self-energy correction to the LDA band structure of solid molecular hydrogen shows a nonlinear density dependence and is larger for molecular hydrogen at equilibrium than for molecular hydrogen under large pressure (at high densities).457Figure 38 depicts the density dependence of the band gap of solid molecular hydrogen for LDA, Hartree-Fock, and GWA. For low densities (atmospheric pressure) the self-energy correction to LDA is large since this regime is dominated by exchange and since GWA is close to Hartree-Fock, which is known to be accurate for molecular solids. For high densities, that is, large pressures, the electron density of solid molecular hydrogen becomes more uniform and consequently the corrections to LDA become smaller. Since the corrections to the LDA band structure are nonconstant, it follows that dE,,/dP will be different in LDA and quasiparticle calculations. Note that LDA calculations for solid, molecular hydrogen are difficult457since (1) the exact experimental structure at metallization is unknown; (2) the large zero-point motion energy of the H, molecules makes accurate total energy calculations difficult since it is larger by an order of magnitude than the differences in electronic energies between different molecular orientations; and (3) in contrast to the total energy the minimum band gap is very sensitive to the orientation of H, molecule vibration. Disorder in the vibration direction of the H, molecules increases K. A. Goettel, J. H. Eggert, I. F. Silvera, and W. C. Moss,Phys. Rev. Lett. 62,665 (1989). R. Reichlin, K. E. Brister, A. K. McMahan, M. Ross, S. Martin, Y. K. Vohra, and A. L. Ruoff, Phys. Rev. Lett. 62,669 (1989). 468 A. L. Ruoff, H. Luo, and Y. K. Vohra, J. Appl. Php. 69,6413 (1991). 466 467

156

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

a

E

6

rn

Density (mol/cma) FIG. 38. Minimum band gap of onentationally ordered solid molecular hydrogen in the hcp structure as a function of density and hence pressure for Hartree-Fock (HF), GWA (GW), and LDA. GWA is close to Hartree-Fock for low densities and close to LDA for high densities. Lines are drawn as a guide to the eye. (From Ref. 457.)

the minimum band gap and is (1) negligible at zero pressure,"69 and (2) crucial at high densities exemplified by a metal-insulator transition pressure of 151 GPa for hcp H,, whose molecules are perfectly aligned along the c-axis and of 300 GPa for orientationally disordered H2.457 Experimentally, metallization of solid hydrogen-defined by a finite T + 0 DC conductivity-remains to be observed (Refs. 470, 471, 472, and 473). Optical experiments have failed to detect evidence for gap closure below about 200 GPa,"70.471 and a metal-insulator transition pressure of more than 300 GPa seems likely.473

The LDA and GWA gaps for hcp H, with all molecules aligned along the c-axis are 8.4 and 15.4 eV. When disorder is included via a virtual crystal approximation, the gaps change to 8.3 and 15.3 eV compared to an experimental value of 14.5+ 1 eV (Ref. 457). 470 N. H. Chen, E. Sterer, and I. F. Silvera, Phys. Rev. Left. 76, 1663 (1996). 471 R. J. Hemley, H.-K. Mao, A. F. Goncharov, M. Hanfland, and V. Struzhkin, Phys. Rev. Left 76, 1667 (1996). 472 R. J. Hemley and N. W. Ashcroft, Physics Today 51.26 (1998). 473 M. Ross, Phys. Rev. 854, R9589 (1996). 469

QUASIPARTICLE CALCULATIONS IN SOLIDS

157

24. EXCITONS Interacting electron-hole pairs are created in absorption spectroscopy, and their electron-hole attraction leads to the following modifications of the optical absorption spectra of semiconductors and insulators: (1) the energies of low-lying excited states are decreased; and (2) the corresponding oscillator strength is increased. Semi-empirical tight-binding results of Hanke, Sham, and collaborators (see Section II.9b) for Si show about a 1-eV shift of the energies of low-lying excited states to lower energies and an oscillator strength increase of the El peak at about 3.5eV by 50%, in good agreement with experiment. However, the strength of the theoretical El peak is predicted to be larger than that of the 4.2 eV E, peak, in contrast to experiment. Recent implementations that include excitonic correlations extend the Hanke-Sham approach by using first-principles rather than semi-empirical wave functions and band energies. Applications to bulk semiconductors and wide-gap insulators (Si, Ge, GaAs, diamond, Li,O, MgO, LiF; Refs. 56,474, 475, 476, and 477) confirm Hanke and Sham’s conclusions regarding the shift of low-lying excited states to lower energies and find in addition the correct ordering (El < E,) of the two main peaks in the Si absorption spectrum. Figure 39 compares theoretical absorption spectra of Si and diamond with and without excitonic correlations with experimental res u l t ~and ~ shows ~ ~ that , ~ good ~ ~agreement between theory and experiment is obtained and that excitonic effects are more important in large-gap insulators than in small- and medium-gap semiconductors, since screening is less effective in the former. Core-hole excitons have been studied in LiF, NaF, KF, graphite, diamond, and h-BN.479 Excitonic binding energies in clusters such as Na, (Ref. 480) or Si,H, (Ref. 282) amount to a few eV compared to binding energies of a few meV in bulk, since screening is inefficient compared to the bulk and electrons and holes are confined in a small region. In contrast to bulk excitons, which at low energies can be described by singly excited electronic states, excitons in clusters require the consideration of several excited electronic states.262*480 Note also that dynamic screening of the electron-hole interaction results in smaller changes (a few tenths of an eV) in the final excitation energies.282 474 475

476 477 478 479 480

S.Albrecht, L. Reining, R. Del Sole, and G. Onida, Phys. Rev. Lett. 80, 4510 (1998). L. X. Benedict, E. L. Shirley, and R. B. Bohm, Phys. Rev. B57, R9385 (1998). L. X. Benedict, E. L. Shirley, and R. B. Bohm, Phys. Rev. Lett. 80,4514 (1998). M. Rohlfing and S. G. Louie, Phys. Rev. Lett. 81, 2312 (1998). Handbook of Optical Constants of Solidr, ed. E. D. Palik, Academic Press, Boston (1991). E. L. Shirley, Phys Rev. Lett. 80, 794 (1998). G. Onida, L. Reining, R. W. Godby, R. Del Sole, and W. Andreoni, Phys. Rev. Lett. 75,

818 (1995).

158

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

2

2.5

3

3.5

4

4.5

5

5.5

6

Eaasy (eV)

2 S , ] , ,

;‘ ‘

a

E2

E,

6

8 10 12 14 16 1% 20 hiD (eV)

6

8 10 12 14 16 18 20

(eV)

FIG.39. Imaginary part of the dielectric function E ~ ( w )= Im(&(o)) as a function of ho for Si (upper panel, adapted from Ref. 474) and diamond (lower panel, adapted from Ref. 475). For Si, the dots correspond to experiment274 and the solid line includes local fields and electron-hole attraction in the Hamiltonian; the long-dashed line includes only local fields, the short-dashed line only RPA screening with quasiparticle shifts. The low-energy experimental peak is commonly referred to as the E , peak, and the high-energy experimental peak is referred to as the E , peak. For diamond, the left panel contains theoretical results, neglecting the electron-hole interaction (dashed lines) and experimental results from Refs. 142 and 478 (solid lines). The right panel contains theoretical results including the electron-hole interaction (dashed line) and the same experimental results (solid line). Excitonic effects improve agreement between theory and experiment significantly for peak position and height and are larger in large-gap materials such as diamond than in medium-gap semiconductors such as Si.

25. ATOMSAND MOLECULES

This section describes quasiparticle calculations of atoms and molecules. Atoms are applications of GWA to “zero-dimensional’’ systems. GWA calculations for molecules have been reported only for quasi-one-dimen-

QUASIPARTICLE CALCULATIONS IN SOLIDS

159

sional and quasi-two-dimensional systems. Understanding whether GWA captures the most important physics of these systems also gives important insights into the usefulness of quasiparticle calculations in strongly correlated solids such as NiO and high-T, cuprates, where correlations are often strongly localized, that is, “atomic-like.’’ Extensions to quasiparticle calculations to include vertex corrections valid for atoms and molecules may therefore be of relevance for solids and are the focus of this section. Other aspects, such as the physics of core-valence correlations in atoms and their relevance for solids were discussed in Section 11.12. a. Atoms Assessment of the usefulness of GWA for atoms requires the comparison of theoretical and experimental results for some key quantities, such as the first and second ionization potentials and transfer energies.67The GWA ionization potentials of major group elements such as B, Al, and Ga deviate on average only 3% from experimental values, compared to a 12% deviation in H a r t r e e - F ~ c k No . ~ ~ systematic improvement compared to the local spin density approximation (LSDA)481*482 is obtained. For s + p promotion energies the error is 20% in GWA, 40% in Hartree-Fock, and 5% in LSDA, as shown in Table 31. In absolute terms, deviations between GWA and experiment vary between 0 and 0.8eV. Hence, taking the dynamically screened interaction into account leads to a significantly improved agreement with experiment compared to Hartree-Fock but not compared to LSDA. Absolute errors exceed chemical accuracy by more than one order of magnit~de.~’ Figure 40 shows the s + d promotion energy in the iron series and demonstrate^^^.^^^ that (1) GWA agrees with experiment quantitatively for the first half of the iron series, in contrast to Hartree-Fock and the LSDA; (2) GWA reproduces experimental trends qualitatively for the second half of the iron series, in contrast to Hartree-Fock, but is inferior by about a factor of three to the LSDA; and (3) GWA cannot compete in accuracy with A. Gorling, private communication. J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. E 4 6 , 6671 (1992). 4 8 3 W. C. Martin and W. L. Wiese, in Atomic, Molecular, and Optical Physics Handbook, ed. G . W. F. Drake, AIP, Woodbury, N Y (1996), 135-153. 484 A. A. Radzig and B. M. Smirnov, Reference Data on Atoms, Molecules, and Ions, Springer Series in Chemical Physics, vol. 31, Springer-Verlag, Berlin (1985). 4 8 5 F. R. Vukajlovic, E. L. Shirley, and R. M. Martin, Phys. Rev. B 43, 3994 (1991). 486 D. M. Bylander and L. Kleinman, Phys. Rev. E41,7868 (1990). 481 482

160

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

TABLE 31. FIRST AND SECOND IONIZATION POTENTIALS AND S~-PROMOTION ENERGIES IN GWA, HARTREE-FOCK (HF), LOCALSPIN DENSITY FUNCTIONAL TWRY (LSDA), AND EXPERIMENT FOR THE GROUP-111ELEMENTS BORON, ALUMINUM, AND GALLIUM.GWA IMPROVES UPON HARTREE-FOCK BUT ERRORSOF ABOUT 20% IN THE s + p PROMOTION ENERGIESREMAIN COMPARED TO AN ERROROF ABOUT 5% IN LSDA. IMPROVEMENT OF GWA IONIZATION POTENTLUS o m LSDA IS MARGINAL.ALL ENERGIES ARE IN eV. (ADAPTED FROM REF. 67.) PROMOTION ENERGY

IONIZATION POTENTLUS S Z p ( 2 P ) +s2(

1s)

sZ( 1s) +42s)

S Z p ( 2 P ) +sp(3P)

S Z p ( 2 P ) +spZ(4P)

10.4 12.0 12.7 12.9

2.0 2.8 3.2 3.6

B HF GWA LSDA" Expt.' A1 HF GWA LSDA" Expt.' Ga HF GWA LSDA" Expt.'

7.5 8.5 8.6, 8.6b 8.3

23.8 24.9 24.6 25.2

5.3 6.0 6.0, 6.p 6.0

17.8 18.9 18.9 18.8

8.5 10.0 10.7 10.6

2.1 3.0 3.6 3.6

5.3 6.0 6.1 6.0

19.0 20.0 20.8 20.5

9.5 11.2 12.3 12.0

2.9 4.1 5.0 4.8

:

"Ref. 481 unless otherwise noted; bRef. 482; 'as cited in Ref. 67; for further reference, see Refs. 483 and 484.

the results of a screened-exchange local spin density approach4" for the entire iron series. The breakdown of GWA for the second half of the iron series stems from the formation of strongly correlated pairs of electrons with opposite spin.67 These strong, localized two-body correlations are not included in the standard dynamically screened interaction approximation, and their inclusion would require vertex corrections. b. Molecules Besides "zero-dimensional" atoms, quasiparticle calculations have been done for quasi-one-dimensional trans-polyacetylene (trans-(CH)J4" -the simplest prototype of a conducting polymer -and a quasi-two-dimensional

"' This approach modifies the LSDA by considering a nonlocal, screened Hartree-Fock operator as part of the exchange-correlation potential (Ref. 486). This technique includes nonlocality effects of the quasiparticle self-energy but omits its energy dependence, in contrast to GWA. 488 E. C. Ethridge, J. L. Fry, and M. Zaider, Phys. Rev. B53, 3662 (1996).

161

QUASIPARTICLE CALCULATIONS IN SOLIDS

0.._.__. 0HF D-OGWA 4-4 Exp. B - - V LSD A- - 4 LSDSX

Ce

Sc

Ti

..a,

R

......

Q\

f

'0

'\

V Cr Mn Fe Co Element of iron series

Ni

Cu

FIG.40. The neutral s - + d promotion energies (eV) for the iron series elements from experiment, Hartree-Fock (HF), and GWA6' and from the local spin-density approximation (LSDA) and the screened-exchange local spin-density approximation (LSDSX)486from Ref. 485. GWA improves upon Hartree-Fock for all elements and upon the local spin-density approximation for the first half of the series, but cannot compete in accuracy with the screened-exchange local spin-density results. The figure combines data from Refs. 67 and 485.

hypothetical polysilane c0mpound.4~~ The polysilane compound consists of planar Si layers stacked in the (1 11) direction and terminated by hydrogen. Replacement of 50% of all hydrogen in polysilane by OH groups gives siloxene, a compound suggested to be responsible for visible luminescence in porous silic0n.4~~ These calculations are discussed below. Calculations on BN sheets have been mentioned in connection with the quasiparticle calculations of solid C,, (see Section IV.19c). C. G. Van de Walle and J. E. Northrup, P h p . Rev. Lett. 70, 1116 (1993). M. S. Brandt, H. D. Fuchs, M. Stutzmann, J. Weber, and M. Cardona, Sol. State Comm. 81, 307 (1992). 489

490

162

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Duns-Polyacetylene. In trans-(CH),,488 GWA leads to a 68% reduction of the initial Hartree-Fock band gap of 5.83 eV to predict a gap of 1.86 eV, in good agreement with the experimental range of gap energies from 1.4 to 1.8 eV (see Refs. 491, 492, 493, 494, 495, 496). This statement holds only if quasiparticle energies are determined self-consistently. Self-consistency in the quasiparticle energies is necessary to achieve quantitative agreement with experiment, since the first iteration reduces the Hartree-Fock gap only to 3.1 eV.488 The theoretical electron-energy-loss spectrum shows a lowenergy plasmon peak whose location and linear dispersion with respect to the reciprocal wave vector k agree well with e ~ p e r i m e n t . ~ ~ A ~second ,~~',~~~ plasmon peak at 13.2eV deviates significantly from the experimental peak at 22.5eV. This discrepancy may be due to the limited Gaussian basis set used in Ref. 488, which cannot describe high-energy states accurately, and to experimental difficulties related to the production of high-quality crystalline t r a n ~ - ( C H ) , . ~ ~ ~ Polysilane. A two Si-layer polysilane compound489 exhibits strong quantum confinement effects with strong quasi-direct optical transitions. The direct GWA gap at r equals 3.0eV and is only 0.2eV larger than the indirect gap. Self-energy corrections of 0.9 eV are a substantial fraction of the band gap. Siloxene is formed by substitution of OH for H for 50% of all hydrogen atoms in the planar polysilane. Siloxene has a direct quasiparticle gap of 1.7 f 0.3 eV489with strong optical transition^.^^' The siloxene band gap corresponds reasonably well to emission in the red, which emission is observed in porous silicon.498 C. R. Fincher, Jr., M. Ozaki, M. Tanaka, D. Peebles, L. Lauchlan, A. J. Heeger, and A. G. MacDiarmid, Phys. Rev. B20, 1589 (1979). 492 J. Fink and G. Leising, Phys. Rev. B34, 5320 (1986). 493 H. Fritzsche, N. Niicker, B. Scheerer, and J. Fink, Synth. Met. 28,D237 (1989). 494 D. Moses, A. Feldblum, E. Ehrenfreund, A. J. Heeger, T.-C. Chung, and A. G. MacDiarmid, Phys. Rev. B26,3361 (1982). 495 J. J. Ritsko, E. J. Mele, A. J. Heeger, A. G. MacDiarmid, and M. Ozaki, Phys. Rev. Letr. 44, 1351 (1980). 496 H. Zscheile, R. Griindler, U. Dahms, J. Frohner, and G. Lehmann, Phys. Stat. Solidi. B 121, K161 (1984). 497 The same quasiparticle correction is assumed to be valid for the polysilane and the siloxene compound based on the similarity of the highest occupied and lowest unoccupied state in the two materials. This approximation is crude, since quasiparticle shifts are determined by a sum over a range of occupied and unoccupied states. 498 For recent reviews on porous silicon, see, for instance, P. D. J. Calcott, Mat. Sci. Eng. B 51, 132 (1998); A. G. Cullis, J. Appl. Phys. 82,909 (1997); P.M. Fauchet, J. Lumin.70, 294 (1996); B. Hamilton, Semicond. Sci. Technol. 10,1187 (1995); G. C. John and V. A. Singh, Phys. Rep. 263, 93 (1995); and M. H. Ludwig, Crit. Rev. Sol. State Mat. Sci. 21, 265 (1996). 491

QUASIPARTICLE CALCULATIONS IN SOLIDS

163

V. Metals Quasiparticle calculations for metals have been reported for bulk (Section V.26), clusters (Section V.27), and surfaces (Section V.28), and are reviewed in detail in Ref. 14. This section concentrates on a few important aspects such as the importance of core-valence exchange for the energy-dependence of the self-energy, the band width of alkali metals, and quasiparticle corrections in Ni. The reader is referred to Ref. 14 for a more detailed discussion of some of the quasiparticle applications. Regarding surfaces in particular, we discuss only the recent work of White et ~ 1 on an . A1~ (111)~ metal surface since the extensive work on jellium surfaces was reviewed in Ref.’ 14.

26. BULK a Core- Valance Exchange Inverse photoemission spectra of transition and noble metals for energies up to 70eV above the Fermi energy E , show peaks that can be identified with similar structures in a LDA density of states.500 However, for energies larger than about 10 eV the theoretical peak positions underestimate experiment by an amount that grows linearly with energy (prefactor of about 0.05 to 0.10; Ref. 500). Since GWA self-energy corrections within the jellium model decrease as the inverse square root of the energy E, rather than increase linearly with E, improvements beyond the jellium model must be considered. The nonlocal exchange interaction between valence electrons (e.g., 5s electrons in Ag) and electrons in closed shells (e.g., the 4s, 4p, and 4d electrons in Ag) is neglected in jellium-based GWA calculations, even though it contributes about 1- 10eV to the conduction-band electron self-energy for the Ag 5s electron,216 as shown in Fig. 41. As shown in both Fig. 41 and Fig. 42, nonlocal exchange of the Ag 5s electron with the Ag 4s, 4p, and 4d electrons216is essential to obtain qualitative agreement with experiment and leads to a self-energy that (1) is approximately constant for E < lOeV, (2) increases linearly with energy for lOeV < E < 70eV, and (3) decreases as E - ’ for E 2 70eV.’16 More than ’ s 5

499 500 501

I. D. White, R. W. Godby, M. M. Rieger, and R. J. Needs, Phys. Rev. Lett. 80,4265(1998). W. Speier, R. Zeller, and J. C. Fuggle, Phys. Rev. B32, 3597 (1985). G. Materlik, J. E. Miiller, and J. W. Wilkins, Phys. Rev. Left. 50, 267 (1983).

~

164

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS 0

c

-

-2

0

20

40

80

00

Energy (eV)

100

'

FIG.41. Self-energy contribution to the energy of the Ag 5s photoelectron arising from the exchange interaction with electrons in the filled valence shell (n = 4) for different values of the angular momentum 1 (dashed lines). Comparison of the sum of the s, p, and d contributions (dashed-dotted line) with the solution that takes the effect of dynamic screening into account (solid line) shows that exchange is dominant. Energies are measured from the Fermi level. (From Ref. 216.)

qualitative or semi-quantitative agreement with experiment cannot be expected since Ref. 216 models (1) dielectric screening by fitting the (2) the disperexperimental electron energy loss function Im[l/e(q = 0; o)]; sion of plasmon poles; and (3) the 4s, 4p, and 4d electrons using atomic Roothaan-Hartree-Fock wave function^.^^^*^^^ b. Alkali Metals The weakness of the effective crystal potential in alkali metals seems to description of the conduction elecpermit a nearly-free-electr~n-model~~~ trons and seems to offer an experimentally accessible system whose manybody corrections are weak, allowing the use of perturbation theory such as GWA, and not complicated by the effects of a complex band structure. E. Clementi and C. Roetti, Atomic Data and Nuclear Data Tables 14, 177 (1974). Figure 42 shows that screening of the nonlocal exchange part leads to moderate quantitative changes in the results, as does the inclusion of lifetime effects, that is, the imaginary part of the selfenergy. N. W. Ashcroft and N. D. Mermin, Solid Srare Physics, Holt, Rinehart, and Winston, New York (1976) 2858. '02

165

QUASIPARTICLE CALCULATIONS IN SOLIDS

0

100

50

150

Energy (eV) FIG.42. Self-energy of a 5s photoelectron in Ag relative to the Fermi energy as a function of energy. The different curves are for (1) the bare exchange due to the 5s conduction electrons (dotted line), (2) the screened exchange due to the 5s conduction electrons (dashed line), (3) the screened exchange due to the 5s conduction electrons and including the effects of a finite imaginary part of the self-energy (dash-dotted line), (4) same as in (3) but adding the bare exchange with the core electrons (solid line a), and ( 5 ) same as in (3) but adding the screened exchange with the core electrons (solid line b). Experimental data from Ref. 500 are given as squares. Inclusion of exchange with the core electrons changes the energy dependence of the self-energy qualitatively. Screening and a nonzero imaginary part of the self-energy affect the results quantitatively. (From Ref. 216.)

However, the experimental determination of the occupied band width Aw of Na via photoemission505~506 contradicts nearly-free-electron theory in two important ways: (1) the measured band width is about 0.6 eV smaller than the nearly-free-electron value of 3.2 eV, and (2) sharp, nondispersive peaks in the energy gap can be identified for photon energies of about 35eV, which peaks have no equivalent in nearly-free-electron theory. Selfenergy corrections in GWA for jellium at rs = 3.95 (average density of Na) narrow the band width by only 0.3 eV, raising concern about the validity of GWA for this simple, weakly correlated solid. Several improvements and extensions to GWA are considered in the literature that all reproduce experimentally observed Aw’s but correspond to different physics. Differences between these approaches were discussed in Section 11.14. Here, we give a short summary of the basic assumptions and results of the different techniques. E. Jensen and E. W. Plummer, Phys. Rev. Len. 55, 1912 (1985). L-W. Lyo and E. W. Plummer, Phys. Rev. Leu. 60,1558 (1988).

166

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN

w. WILKINS

Self-energy effects and surface-induced momentum-nonconserving excitations each account for about half of the observed Na band-width correction.Zz9*z41~26’ The latter effect also provides a possible explanation for the experimentally observed nondispersive peaks around 35 eV in the photoemission spectrum shown in Fig. 43. The inclusion of surface effects is important, according to the work of Mahan et u1.,229~241~265 since the mean-free path of the photoelectrons is only 5-6A in Na; that is, photoelectrons probe only the first two to three layers of the material. Mahan et ul. used Rayleigh-Schrodinger perturbation theory -suggesting that it is more accurate than GWA for alkali metals -and the surface potential by Lang and KohnSo7(see Section 11.14). ,

507

N. D. Lang and W. Kohn, Phys. Rev. B1, 4555 (1970).

I

a0 26.0

I

I

I

I

360 46.0 66.0 06.0 76.0 Photon energy (eV)

FIG. 43. Photoemission peak positions as a function of photon energy for the Na (110) surface. The quasiparticle (QP) bands include the real and imaginary part of the self-energy and are represented by solid curves. The QP bands are narrower than the nearly-free-electron bands (NFE, dashed curves) by 0.37 eV. The full theory (solid dots) also includes surface effects and reproduces the sharp peaks in the “gap” at photon energies of about 35 eV. Experimental data are denoted by crosses and taken from Ref. 505. (From Ref. 229.)

167

QUASIPARTICLE CALCULATIONS IN SOLIDS

A “best G, best W ’ approach that goes beyond standard RPA-based GWA calculations by (1) including LDA vertex corrections in the dielectric screening but not in the self-energy and (2) determining the quasiparticle energies self-consistently leads to agreement between theoretical and experimental band ~ i d t h s , ’ ~as~shown . ~ ~ in~ Table * ~ ~32.~ Point (1) alone leads to a band-width reduction of 0.57 eV in Na; points (1) and (2) together lead to a band-width reduction of 0.71 eV, compared to a standard GWA value of 0.31 eV, and reduce the band width from the LDA value of 3.16eV to 2.45eV.’” This approach does not address the existence of a nearly dispersionless peak in the photoemission spectrum. Northrup et u1.184*257.258 motivated the inclusion of vertex corrections in the screening in terms of a better fulfillment of a Ward identity related to charge conservation.’ s6 Their approach contradicts the arguments of Mahan’ and others.266 Results of partially self-consistent GWA calculations of the band-width reduction for j e l l i ~ mcan ~ ~be~ extended to alkali metals and give good H. J. Levinson, F. Greuter, and E. W. Plummer, Phys. Rev. B27,727 (1983). R. S. Crisp and S. E. Williams, Philos. Mug. 5, 1205 (1960). ’I0 E. W. Plummer, Phys. Scr. T17,186 (1987). ’I1 The results of Refs. 257 and 184 differ by 70 meV, leading to 2.52 eV for the quasiparticle band width as used in Table 32. ’08

TABLE32. COMPARISON BETWEEN EXPERIMENTAL AND THEORETICALOCCUPIEDBANDWIDTH WITH (1) VERTEXCORRECTIONS IN THE DIELECTRICMATRIX E ONLY AND USEOF (eV) OBTAINED SELF-CONSISTENT QUASIPARTICLE E N E R G I E S ; ~ *(2) ~ ~ ’PARTIAL ~~ SELF-CONSISTENCY, STATIC VERTEXCORRECTIONS IN E, AND DYNAMIC VERTEXCORRECTIONS IN THE SELF-ENERGY 2;’” AND (3) NONZEROIMAGINARY PART OF I: AND SURFACE EFFECTS.229~265 AGREEMENT WITH EXPERIMENT IS GENERALLY GOOD. HOWEVER,SELF-CONSISTENCY EFFECTSNOT TAKENINTO ACCOUNT IN THE ABOVECALCULATIONS MAYWIDENTHE Na BANDWIDTHBY AN ESTIMATED 0.4 TO 0.7 eV BASEDON RESULTSOF REF. 231, AND SHIFTTHE THEORETICAL RESULTSTOWARD THE NEARLY-FREE-ELECTRON AND LDA BANDWIDTHVALUES. THERESULTS FOR REF. 265 ARE PRESENTED AS AN AVERAGE OVER THE RANGEOF BANDWIDTHSGIVEN. r, IS THE WIGNER-SEITZ GASWITH IDENTICAL AVERAGEDENSITY AS THE SIMPLE METALS. RADIUSOF AN ELECTRON NORTHRUP et

A1 Li Na K

Ul.184’258

SHIRLEYz3’

r,

LDA

GWA

GWA

2.1 3.3 4.0 4.9

3.5 3.2 2.3

10.0 2.9 2.5 1.6

10.2 3.1 2.7 1.9

“Ref. 508; ’Ref. 509; ‘Ref. 506; dRef. 510.

SHUNG et

RSPT

EXPT.

2.5 1.5

10.6“ 3.0‘ 2.65 f 0.05‘ 1.4d

168

WILFRIED G. AULBUR, LARS JoNSSON, AND JOHN W. WILKINS

agreement with experiment if (1) static vertex corrections similar to the Hubbard local-field factor are included in the dielectric function, and ( 2 ) dynamical vertex corrections to second order in the screened interaction W are included in the self-energy. In contrast to Mahan,” Shirley suggested that a dynamical vertex correction to the self-energy in conjunction with a static vertex correction in the screening is more appropriate’” for the determination of Aw. Results of Shirley’s c a l ~ u l a t i o n ’are ~ ~ given in Table 32. Estimates based on the partially and fully self-consistent GWA calculations shown in Fig. 19 for the band-width reduction Aw of jellium at rs = 4 suggest that (1) non-self-consistent results for Aw increase by about 0.7 eV upon inclusion of self-consistency,and (2) partially self-consistent results for Aw increase by about 0.4 eV upon inclusion of full self-consistency. This estimate suggests that all published results for Aw miss physical effects that are as large as the difference between the Jensen-Lyo-Plummer experimental datasos*s06and nearly-free-electron theory.’I4 Consequently, a quantitative analysis and unified physical understanding of the Na photoemission experiments require further work, in particular the establishment of a consistent set of vertex corrections, self-consistency, surface, and lifetime effects.’

’

c. d and f Electron Metals The LDA band structure of the transition metal Ni has four discrepancies with experiment? (1) The experimental 3d band width of 3.3 eV is about 30% smaller than its LDA value of 4.5eV; (2) the experimental exchange splitting of 0.25-0.30 eV is about half of the LDA value; ( 3 ) a 6-eV satellite in the photoemission spectrum is absent in LDA; and (4) the bottom of the 3d band cannot be described by sharp LDA excitations since these states have quasiparticle widths of about 2 eV, indicating strong interactions between 3d electrons. Quasiparticle calculations of the d electron metal Ni and the f electron metal Gd can be found in Refs. 68 and 516, respectively, and are extensively reviewed in Ref. 14. Here, the discussion is limited to a ’I2 The appropriate choice of vertex corrections is a topic of current discussions (Refs. 232 and 267). ’13 Note that both Shirley and Northrup er al. use plasmon-pole models, which may affect the accuracy of their calculated valence band widths. ’14 See also Ref. 11 for a critical discussion of the theoretical determination of the alkali metal bandwidth. Further Applications: Recently, Kubo reported Compton profile studies of Li and Na using full-potential LMTO (Refs. 287 and 288). ’I6 F. Aryasetiawan and K. Karlsson, Phys. Rev. 854, 5353 (1996).

’”

QUASIPARTICLE CALCULATIONS IN SOLIDS

169

short description of the successes and failures of GWA in the case of Ni6' Quasiparticle calculations for Ni lead to a significant improvement of the LDA band structure and in particular reduce the 3d band width by about 1 eV, in agreement with experiment. Experimental quasiparticle lifetimes are reproduced as well. Self-energy shifts in Ni are strongly state dependent because the LDA exchange-correlation potential is a better approximation for the self-energy of, for example, free-electron-like s states than for the self-energy of strongly correlated d states. The different character of the s-like and d-like states is also reflected in the quasiparticle weight, which is about 0.7 for s-like but only about 0.5 for d-like states. The 6-eV satellite is not reproduced in GWA, and the exchange splitting is only marginally improved since both features result from strong hole-hole interactions that require the inclusion of vertex corrections. A common explanation of the photoemission process starts with the creation of a 3d hole, which introduces a strong perturbation because of its localized nature and excites another 3d electron to an empty state just above the Fermi level. The two holes scatter repeatedly and form a bound state at 6eV. A first-principles T-matrix calculation7' includes the neglected hole-hole interaction via ladder diagrams and leads to a 6-eV satellite and an improved exchange splitting. 27. CLUSTERS The total energy per atom of, for example, Na clusters as a function of the number of Na atoms N is a smooth function except for small kinks at N = 8, 18, 20, 34, and so on (for a review on metal clusters, see, for instance, Ref. 517). This shell structure is reminiscent of the behavior observed in nuclei and atoms and results from the fact that the electrons in the alkali cluster can be reasonably well described by an effective one-particle, spherical potential. The properties of the metal clusters-such as their total energy and hence their stability, their ionization potential, and their electron affinity-change in an abrupt way whenever one shell of electrons is filled up, leading to an abrupt change as a function of cluster size. The spherical, one-particle potential of choice is the so-called jellium-sphere-background model in which the positive ion cores in the cluster are replaced with a constant-density sphere. The sphere radius is given by T , N ' / ~with , r, being the Wigner-Seitz radius corresponding to the average metal density. As shown in Fig. 44, LDA calculations of the ionization potential and electron affinity do not reflect the experimental shell structure, that is, the dependence on cluster size of these q u a n t i t i e ~ . ~This ~ ~ *discrepancy '~~ arises 517

W. A. De Heer, W. D. Knight, M. Y. Chou, and M. L. Cohen, Sol. State Phys. 40,93(1987).

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WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

21 0

10

20

50

40

Number of atoms per cluster, n

J'/>!

4

............x......)<

X....''

-/ 0

1 0

10

20

50

Number of atoms per cluster, n

40

FIG. 44. The absolute value of the quasiparticle energies of the highest-occupied (top) and lowest-unoccupied (bottom) states in potassium as a function of cluster size in LDA (open circles), GWA (filled circles), and experiment (triangles, data from Refs. 517, 518, 519, 520, and 521). The crosses represent LDA total-energy differences (ASCF) between positive (KJ) and neutral (top) clusters or between neutral (K,,) and negative (Kn-, bottom) clusters. The LDA calculation is not possible for small (n = 2 and 8) clusters. GWA tracks experiment and the ASCF calculations closely for the highest-occupied states in contrast to LDA. The same is true for the lowest-unoccupied state although the improvement over LDA is less dramatic. (Adapted from Ref. 523.)

QUASIPARTICLE CALCULATIONS IN SOLIDS

171

from LDA’s incomplete cancellation between the Hartree and exchange self-interaction -the so-called self-interaction error -and increases with decreasing system size, which conceals the strong size dependence of the ionization potential in p a r t i c ~ l a r . ~ ’ ~Self-interaction ,~’~ errors lead to an underbinding of the occupied and an overbinding of the unoccupied states in strongly localized systems. The size dependence of the ionization potential and the electron affinity are well described in GWA because self-interaction errors are small in standard GWA.524 Remaining quantitative differences with experiment are attributed to the jellium-sphere-background model and finite temperature e f f e ~ t s . ~ ’ ’ ~The ~ ’ ~ quasiparticle results for the ionization potential are systematically above the corresponding LDA results and eliminate the LDA underbinding of occupied states because of self-interaction errors. Similarly, the absolute value of the GWA electron affinity is below the corresponding LDA value since GWA corrects for the overbinding of unoccupied states within LDA.

28. SURFACES In classical electrostatics, an electron at a location z outside a metal surface induces a surface charge and in turn experiences an attractive image potential ym(z) whose asymptotic form for large z is given as - 1/(4(z - zo)), where zo is the effective edge of the metal. On a microscopic level, the rearrangement of charges at the metal surface is due to long-range exchange and correlation effects, which are absent in LDA because of the exponential, ather than inverse power-law decay of the LDA exchange correlation potential outside the metal surface.525This severe LDA failure leads to a poor description of surface states and to an absence of image states and resonances in LDA. Discrepancies between LDA results and the results of surface-sensitive experimental techniques such as low-energy electron difP. Fayet, J. P. Wolf, and L. Woste, Phys. Rev. B33, 6792 (1986). A. Herrmann, E. Schumacher, and L. Woste, J . Chem. Phys. 68, 2327 (1978). 5 2 0 M. M. Kappes, M. Schar, P. Radi, and E. Schumacher, J . Chem. Phys. 84, 1863 (1986). 5 2 1 K. M. McHugh, J. G. Eaton, G. H. Lee, H. W. Sarkas, L. H. Kidder, J. T. Snodgrass, M. R. Manaa, and K. H. Bowen, J. Chem. Phys. 91,3792 (1989). 5 2 2 S. Saito, S. B. Zhang, S. G. Louie, and M. L. Cohen, Phys. Rev. B40, 3643 (1989). 5 2 3 S. Saito, S. B. Zhang, S. G. Louie, and M. L. Cohen, J. Phys. Cond. Mar. 2, 9041 (1990). 5 2 4 What self-interaction errors there are in GWA arise from the input wave functions and energies used for the construction of the self-energy operator. Errors in the wave functions are small since the overlap between quasiparticle and LDA wave functions is larger than about 95% (Refs. 522 and 523). Errors in the energies can be eliminated by a self-consistent determination of quasiparticle energies. 5 2 5 N. D. Lang and W. Kohn, Phys. Rev. B7, 3541 (1973). 518

’I9

172

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

fra~tion,’~~*’~’ scanning tunneling m i c r o s ~ o p y , ’ ~ and ~ * ’ inverse ~~ and twophoton photoemission experiment^^^^*'^^ are therefore expected. The nonlocal, energy-dependent, many-body effects that cause the asymptotic inverse power-law behavior of the image potential are contained in the GWA self-energy. To obtain a local, state-dependent potential Koc(r) from the nonlocal self-energy that can be compared with the local image potential and the LDA exchange correlation potential Vk:A, one defines532

Koc(r) =

s

X(r, r’; E)Y(r’)dr’/Y(r).

(5.1)

For an Al(111) surface, Fig. 45 shows that (1) outside the metal surface &(r) has the correct asymptotic, image-like behavior, and (2) inside the metal surface self-energy corrections to LDA for states close to the Fermi energy are small, resulting in nearly identical values of Koc(r) and V:FA(r). Essential for the above successful applications of the dynamically screened interaction approximation is a self-consistent evaluation of quasiparticle states.499The difference between the exponential LDA decay and the GWA inverse power-law decay leads to an increased weight of the quasiparticle states in the near-surface region, demonstrated in Fig. 46. Note that results similar to the ones discussed here for the Al( 111)surface were obtained earlier than those of Ref. 499 by Deisz, Eguiluz, Hanke, and c ~ l l a b o r a t o r s ~ ~ ~ ~ ’ ~ ~ ~ ’ for the case of a jellium surface. This work was reviewed in Ref. 14. VI. GWA Calculations and Optical Response

For semiconductors and insulators, the lessons learned from the GWA regarding the band structure have led to a new level of accuracy in calculations of optical response. The reason is the crucial role played by the 526

P. J. Jennings and R. 0. Jones, Phys. Rev. B34,6695 (1986).

”’ J. Rundgren and G. Malmstrom, Phys. Rev. Lett. 38, 836 (1977). G. Binnig, N. Garcia, H. Rohrer, J. M. Soler, and F. Flores, Phys. Rev. B 30,4816 (1984). G. Binnig, K. H. Frank, H. Fuchs, N. Garcia, B. Reihe, H. Rohrer, F. Salvan, and A. R. Williams, Phys. Rev. Lett. 55, 991 (1985). 530 U.Hofer, I. L. Shumay, C. Re&, U. Thomann, W. Wallauer, and T. Fauster, Science 277, 1480 (1997). 5 3 ’ F. Passek and M. Donath, Phys. Rev. L e r r . 69, 1101 (1992). 5 3 2 J. J. Deisz, A. G. Eguiluz, and W. Hanke, Phys. Rev. Lett. 71,2793 (1993). 533 J. Deisz and A. G. Eguiluz, J . Phys. C o d . Mat. 5, A95 (1993). 5 3 4 A. G. Eguiluz, M. Heinrichsmeier, A. Fleszar, and W. Hanke, Phys. Rev. Lett. 68, 1359 (1992). 528

529

173

QUASIPARTICLE CALCULATIONS IN SOLIDS

0.00

-0.10

-0.20

-0.30 I ”

-0.401 . ’ -10.0 -5.0

.

’

.

’

’

5.0 10.0 Distance outside surface (a.u.) 0.0

15.0

FIG.45. Surface averaged effective local potential (Eq. (5.1)) at the Al(111) surface compared with the exponentially decaying LDA exchange correlation potential Vkfl”. The local potential calculated from the GWA self-energy is virtually identical to Vkfl” in the A1 region, and crosses over to the classical image form (best fit shown) in the vacuum, in contrast to KtflA.(From Ref. 499.)

4.00

I

1 I

-5.0

.

0.0 5.0 10.0 15.0 Distance outside surface (a.u.)

FIG. 46. Surface-state quasiparticle wave function (full line) of the Al(111) surface at r 1.66 eV below the vacuum level in comparison to its LDA counterpart (dashed line). Since the local potential resulting from the self-energy decays as l/z, z being the distance from the surface, rather than exponentially as does the LDA exchange-correlation potential, the quasiparticle state has weight transferred into the vacuum relative to the LDA state. The quasiparticle state is obtained by an iterative solution of the energy-dependent quasiparticle equation. (From Ref. 499.)

174

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

band gap in determining the optical polarizability of nonconducting materials. Efforts to correct the too small LDA band gaps are therefore directly connected to efforts to accurately calculate optical response: dielectric constant, optical absorption, second-harmonic generation, and so on. In Section VI.29, we discuss in detail the overestimate of optical constants in LDA calculations. In Section VI.30 we introduce the widely used scissorsoperator a p p r o a ~ h , in ~ ~which ~ ~ ~the ~ ’self-energy corrections to the LDA band structure are approximated by a rigid shift of all the conduction-band energies while the valence-band energies are unaffected. The inhomogeneous electron density in crystalline materials generates electric fields that vary over distances of the order of the lattice constant. These so-called local fields80,82must be considered when accurate calculations of optical response are performed. Typically, as shown below, the local-field effects are of order 5-10% in the dielectric constant and 10-30% in the second-harmonic coefficient. In Section VL3 1, we outline the theory of local-field effects in optical response and review the numerical results for a range of semiconductors and insulators. Finally, in Section V1.32 we discuss the possibility of obtaining correct optical response coefficientsfrom density functional theory without the introduction of quasiparticle corrections. Recent work by Gonze, Ghosez, and GodbyS3’ suggests that previous optical response calculations neglected an important exchange-correlation effect specific to systems with a band gap. The KohnSham electrons are acted upon not only by the physical electric field- the external optical field plus the induced Hartree field -but also by a fictitious field derived from the induced change in the exchange-correlation potential. This fictitious exchange-correlation electric field counteracts the unphysical Kohn-Sham energy spectrum to produce the correct induced density or polarization. The discussion below has some overlap with the previous discussion concerning the calculation of the dielectric matrix for use in GWA calculations. However, the focus is different. Previously, we were concerned only with methods to obtain a dielectric matrix suitable for calculations of the screened Coulomb interaction in the GWA. In particular, a prominent role was played by the need to perform an integration over all frequencies (see Eq. (2.24)). In the discussion below, we are concerned with ways to calculate optical response coefficients with high accuracy (within 5% of experiment). In particular, we discuss calculations of the linear and nonlinear electric susceptibilities in the long-wavelength limit. 535

536 537

Z. H. Levine and D. C. Allan, Phys. Rev. Lett. 66,41 (1991). Z. H. Levine and D. C. Allan, Phys. Rev. B43,4187 (1991). X. Gonze, P. Ghosez, and R. W. Godby, Phys. Rev. Lett. 14, 4035 (1995).

QUASIPARTICLE CALCULATIONS IN SOLIDS

175

29. OVERESTIMATION OF OPTICAL CONSTANTS WITHIN DFT In terms of the induced macroscopic polarization density P, the linear and second-order susceptibilities x and x‘’) are defined by

where E is the total internal electric field. The scalar product symbolizes the fact that in general the susceptibilities are tensors. However, in many common materials only a few tensor components are independent -for crystals with cubic symmetry in particular, x = x,, = xyy = x,, are the only nonzero components. We use a simple scalar notation for all tensor properties below, and only when needed do we add explicit functional dependencies on time, position, frequency, or momentum. In an insulator with cubic symmetry, the linear susceptibility, in the independent-particle approximation, can be written as (I/ is the total volume)538

x=v4

c I(ckIxl~k)l’

c,u,k

&ck

- &uk

9

(6.2)

where c, u represent summations over conduction and valence bands, k represents a summation over the first Brillouin zone, and E ~ , ” are the corresponding single-particle energies. The matrix element is of the position operator x in some direction in real space. Any electronic band structure can be used in Eq. (6.2), but the most common choice is an LDA band structure. Another possible choice would be, for example, one based on empirical pseudo potential^^^^ for which the pseudopotential parameters have been adjusted to reproduce an experimental band structure. Since this section focuses on the influence of GWA self-energy corrections in optical response, we assume LDA wave functions throughout. A major problem with LDA calculations of the dielectric constant, E = 1 4nx, is the underestimation of the band gap. It leads to a roughly equal relative overestimation of x and therefore E, as shown in Fig. 47. However, in these sp-bonded materials the error due to the use of LDA wave functions seems less severe. This conclusion is supported by the fact that GWA calculations give reasonable corrections to the gap without changing the wave functions dramatically. For higher-order response, the error due to the too small gap becomes even worse, since for each order the power of

+

C. Aversa and J. E. Sipe, Phys. Rev. B52, 14636 (1995). M. L. Cohen and J. R. Chelikowski, Electronic Structure and Optical Properties of Semiconductors, Springer Series in Solid-state Sciences, vol. 75, Springer, New York (1989). 538

539

176

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Ge

-

30

Se 00

GaAS

J

x 0It20

-

I

a

9"

0

si

F

AlAs 0

GaP 0 0

10 -

AIP

sic 0 0 GaN

OO

0 warn

OC 0

5 10 Direct band gap (eV)

Bh 0 1

FIG. 47. Percent deviation of the LDA dielectric constants from experiment for 13 semiconductors and insulators. The numerical and experimentaldata are from Refs. 536, 540, 541, 542, and 543). The calculations use scalar relativistic pseudopotentials.

the energy differences increases by one in the denominator of the formulas for the response coefficients. Hence, the primary focus in optical-response theory of semiconductors over the last few decades has been on the best way to correct the LDA band gap without worrying about changes to the wave functions. However, before discussing how gap corrections can be introduced with a reasonable computational effort, we want to discuss in some detail how large the LDA error actually is. When the results of a particular publication are considered, it is important to note exactly how the LDA band structure was obtained. In Table 33, we

177

QUASIPARTICLE CALCULATIONS IN SOLIDS

TABLE33. DIRECT LDA BANDGAP (IN eV) AT r WITH DIFFERENT LEVELS OF APPROXIMATION: NR -NONRELATIVISTIC; SR-PP -SCALARRELATIVISTIC PSEUDOPOTENTIAL (WITH ATOMIC 3d CORE);FR-PP- FULLY RELATIVISTICPSEUDOPOTENTIAL; FR-CR -FULLY RELATIVISTICWITH CORE RELAXATION. THE TAFIULATED VALUES REFLECT MEDIANS OF A MULTITUDE OF PUBLISHED VALUES.TYPICALLY, A PARTICULAR REFERENCEWILL HAVEVALUESWITHIN0.15eV OF THE TABULATED ONES.HOWEVER,DESPITEVARIATIONS IN ABSOLUTE VALUES THE RELATIVETRENDS AGREEBETWEEN DEWRENT REFERENCES. E,,(eV) NR Si Ge GaAs

2.7a*b O.Tb l.lo*b

SR-PP

2.Wd 00 0.6d.g.h.i

FR-PP

FR-CR

EXPT.

2.W' -0.1'*' 0 5c.e.g.h

2.6'~' -0.3'" 0.2e.J.#.l-"

3.4" 0.9" 1.5"

"Ref. 544; bRef, 545; 'Ref. 114; dRef. 185; 'Ref. 43; 'Ref. 546; ORef. 192; "Ref. 121; 'Ref. 541; 'Ref. 547; 'Ref. 548; 'Ref. 549; "Ref. 550; "Ref. 551.

show for Si, Ge, and GaAs how different approximations affect the LDA gap. The table shows, from left to right, the importance of scalar relativistic effects with frozen core states, spin-orbit interactions (fully relativistic),and relaxation and hybridization of core states. In GaAs, all these effects are important if we aim for an accuracy of 0.1 eV for band-gap predictions. With a 1.5-eV band gap, a 0.1-eV error corresponds to a 5-10% error in the dielectric constant. The commonly used scalar relativistic pseudopotentials typically give a gap of 0.4-0.5 eV in GaAs (including a 0.1-eV spin-orbit correction), which is about 0.3 eV larger than a fully relativistic all-electron calculation. In contrast to GaAs, the band gap of silicon is not sensitive to the level of approximation, which shows that silicon is not a good test case for the evaluation of accurate methods. Note also the large effects in Table 33 for Ge and GaAs from the neglect of relativistic effects even for these relatively light atoms. J. Chen, Z. H. Levine, and J. W. Wilkins, Phys. Rev. B50, 11514 (1994). Z. H. Levine and D. C. Allan, Phys. Rev. 844, 12781 (1991). 542 J. Chen, Z. H. Levine, and J. W. Wilkins, Appl. Phys. Lett. 66, 1129 (1995). '43 H. Zhong, Z. H.Levine, D. C. Allan, and J. W. Wilkins, Phys. Rev. 848, 1384 (1993). '44 C. S. Wang and B. M. Klein, Phys. Rev. B24,3393 (1981). "' M.-Z. Huang and W. Y.Ching, Phys. Rev. B 47, 9449 (1993). 546 M. Alouani and J. Wills, Phys. Rev. B 54, 2480 (1996). 547 S. Bei der Kellen and A. J. Freeman, Phys. Rev. B54, 11187 (1996). '48 S.-H. Wei and A. Zunger, Phys. Rev. B39, 3279 (1989). 549 H. Krakauer, S.-H. Wei, B. M. Klein, and C. S . Wang, Bull. Am. Phys. SOC.29,391 (1984). ''O B. I. Min, S. M d d a , and A. J. Freeman, Phys. Rev. 838, 1970 (1988). Crystd and Wid State Physics, Landholt-Bornstein, Numerical Data and Functional Relation&@ in Sciawv and Technology, vol. 17% ed. 0. Madelung, Springer, Berlin (1984). '40

541

'"

178

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

This emphasis on the level of approximations made is equally important when the general accuracy of the GWA is evaluated. Table 34 compares the underestimation of the most accurate LDA gaps (the difference between the last two columns in Table 33) with the gap corrections from various approximate GWA schemes: first-order perturbation theory, self-consistently updated band energies, and inclusion of core polarization. The most accurate values (last column) show that GWA corrections including corepolarization effects added onto fully relativistic, all-electron LDA energies do indeed come within 0.1eV of experiment. However, other levels of approximation yield different conclusions. For example, pseudopotential calculations that neglect spin-orbit interactions (Table 33, column 2) combined with perturbative GWA corrections (Table 34, column 2), which are commonly used, typically underestimate the band gap by 0.1-0.3 eV for Si, Ge, and GaAs. The use of LDA theoretical lattice constants calculated within a pseudopotential approximation can yield gaps half an eV larger than the gaps obtained at the experimental lattice constant. For example, one calculation552found a gap of 1.0eV in GaAs with a theoretical lattice constant obtained with a scalar relativistic pseudopotential. That approach gave a 1.7% too small lattice constant (5.55A), which generated a 0.5-eV increase in the LDA band gap compared to calculations at the experimental i ~ ~ ~that the band gaps lattice constant (5.65A). Similarly, F i ~ r e n t i n found at the theoretical lattice constant from pseudopotential calculations are increased by 0.02 eV in Si, 0.45 eV in Ge, 0.61 eV in GaAs, and 0.27 eV in

’” A. Dal Corso, F. Mauri, and A. Rubio, Phys. Rev. B53, 15638 (1996). ’” V. Fiorentini, Phys. Rev. B46, 2086 (1992). TABLE 34. GWA CORRECTION (INev) TO THE LDA DIRECT GAPAT r WITH DIR+”T LEVELSOF APPROXIMATIONIN THE GWA CALCULATION: PT-PERTURBATION THEORY WITH LDA BAND STRUCTURE; UE -SELF-CONSISTBNTLY UPDATED BANDENERGIES; CP -INCLUSION OF CORE-POLARIZATION EFFECTS. THE FIRST COLUMNIS THE LDA GAP ERRORGIVENBY THE DIFFERENCE BETWEEN THE LASTTwo COLUMNS IN TABLE 33. THEGWA VALUES ARE FROM TABLES21,22, AND 23 IN SECTION 111.18. AGWA 8.P

4 , P

Si Ge GaAs

LDA

PT

UE

CP

-0.8

0.7 0.6 0.7

0.7 0.8 0.9

0.7 1.1 1.3

- 1.2 - 1.3

QUASIPARTICLE CALCULATIONS IN SOLIDS

179

AlAs, compared to gaps obtained at the experimental lattice constants. Such large overestimations of the LDA band gap can lead to the conclusion that LDA gaps are good enough without self-energy corrections. However, such a claim is countered by the fact that fully relativistic LDA all-electron calculations give lattice constants in much better agreement with experiment and much smaller gaps. For example, Alouani and Wills546 obtained a theoretical lattice constant in GaAs of 5.62A and a direct gap of 0.29 eV. Therefore, conclusions concerning the need for quasiparticle corrections based on pseudopotential calculations at the theoretical lattice constant must be considered with scepticism. The inclusion of gradient corrections (GGA) to LDA does not alter this conclusion, although GGA typically overestimates the lattice ~ o n s t a n t . ’ ~ ~ . ~ ~ ~ AND 30. THE“SCISSORS OPERATOR”

ITS

LIMITATIONS

Ideally, one would like to add an energy- and momentum-dependent self-energy to the denominator in Eq. (6.2), but this is in general too computationally expensive. An often used simplification, called the “scissorsoperator approach”, is to add a constant, energy- and momentum-independent shift A to the conduction-band energies, leaving the valence energies and all wave functions ~ n c h a n g e d .That ~ ~ ~is,. a~ term ~~

is added to the LDA Hamiltonian. In the next section, we investigate the consequences of doing this. But we first discuss what value to choose for A. The discussion above concerning the band-gap problem suggests that one should choose A = - EkF. However, such a choice often leads to an overcorrection of the too high LDA dielectric constant. This overcorrection can be understood by considering the band and k-space summations in Eq. (6.2). The most important regions in k-space when calculating E are those where valence and conduction bands are approximately parallel, leading to a large joint density of states. The effect of the joint density of states on the dielectric constant is best seen when E is written in terms of the imaginary part ~ ~ ( via the Kramers-Kronig relation:

554 555

A. Dal Corso, S. Baroni, R. Resta, Phys. Rev. B49, 5323 (1994). A. Dal Corso, A. Pasquarello, A. Baldereschi, and R. Car, Phys. Rev. B 53, 1180 (1996).

0 )

180

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

where

The position matrix elements are the same in LDA and in the scissorsoperator approach, since the wave functions are the same by construction. The only effect of the scissors shift A in the expression for E ~ ( w )is that A is added inside the 6 function. As shown in Fig. 39 and in Figs. 48 and 49, for most semiconductors the dominant structure in &'(a) is not the region around the band gap but the one that consists of a couple of peaks a few eV higher up in energy. To shift the main peaks in ~ ' ( 0to) their experimental positions requires a A smaller than the gap mismatch. Because of both band-structure and

50

40 n

3 w-

30

Y II

20 10

-0

n

2

4

8

8

1

0

1

0

Energy [eV]

0

2

4

6

8

Energy [eV] FIG. 48. Imaginary part ~ ~ (of 0dielectric ) functions for GaAs and GaP from Ref. 556. The solid curve was obtained with a FLAPW (full-potential linearized augmented plane-wave) band structure with a scissors shift chosen to fit the experimental band gap. This choice of scissors shift puts the dominant peaks in the absorption spectra about 0.5 eV too high compared to experiment (dashed lines-GaAs, Ref. 197; Gap, Ref. 274).

QUASIPARTICLE CALCULATIONS IN SOLIDS

Energy (OW

181

EnWY (*v)

FIG. 49. Effects of quasiparticle weights and vertex corrections on the imaginary part of the dielectric constants for diamond and silicon (from Ref. 564): short dashed line-experiment (diamond, Ref. 197; Si, Ref. 565); dotted line- LDA, long-dashed line-self-energy corrected without quasiparticle-weight correction; dash-dotted line -self-energy corrected with weight factor Z , Z , for each matrix element; solid line- self-energy corrected including vertex corrections. The close agreement between the solid and long-dashed curve shows that the strong effect of the weight factor Z,Z, is almost canceled by vertex corrections. Since the dotted LDA curves look almost like solid lines, note that the LDA curves are farthest to the left and have the highest peak in both materials.

many-electron effects, the mismatch between the LDA and the experimental regions of strong optical absorption is not related to the band-gap mismatch in an obvious way. Figure 48 shows the results by Hughes and SipeSs6for GaAs and Gap. They used a scissors shift to fit the LDA band gaps to the experimental gaps, which puts the dominant absorption peaks about 0.5 eV too high. As discussed in Sections II.9b and IV.24, inclusion of excitonic effects will strongly improve the absorption spectrum in regard to both peak position and the peak heights (see Refs. 154, 155, 156,474, 475, and 476). A systematic first-principles argument for what A to choose is therefore hard to find and the scissors-shift approach must be considered largely empirical. Early calculations found good agreement with experiment by using LDA gaps plus an averaged GWA correction for A, which suggests a close to ab-initio method for optical calculations.535~536 Table 35 shows the results of Levine and Allan for the dielectric constant of Si, Ge, GaAs, Gap, AlAs, and Alp. Leaving the discussion of local-field corrections to the next section, we see that the scissors approximation brings the too large LDA values to within 5% of experiment. These good results are obtained in large part because of the particular approximation made to the band structure. For

'" J. P. L. Hughes and J. E. Sipe, Phys. Rev. E 53, 10751 (1996).

182

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS TABLE35. DIELECTRIC CONSTANTS FOR SMALL-AND MEDIUMIN LDA AND WITHIN THE SCISSORS-SHIFT GAPSEMICONDUCTORS APPROACH WITHOUT(No LF) AND WITH (LF) LOCAL-FIELD CORRECTIONS.536'54'THE VALUES INCLUDING LOCAL-FIELD ARE ALL WITHIN5% OF EXPERIMENT. CORRECTIONS ____

Si Ge GaAs GaP AlAs AlP

~~

~

LDA

N o LF

LF

EXPT.

13.5 22.0 13.7 10.4 9.5 8.3

11.7 16.6 11.7 9.3 8.7 7.8

11.2 16.0 11.2 8.8 8.1 7.2

11.4" 15.3" 1O.Sb 9.0b 8.2b 7.4b

'

"Ref. 557; bRef. 558.

GaAs, Levine and Allan found good agreement with experiment by using the average GWA correction of 0.8 eV from the semirelativistic pseudopotential calculation by Godby, Schluter, and Sham."' This correction is added to a semirelativistic pseudopotential LDA band structure (neglecting Ga 3d core relaxation and spin-orbit corrections). Their LDA gap is 0.5 eV at the experimental lattice constant, which yields a 1.3-eV gap after the scissors correction. This gap is 0.2eV too small compared to experiment, but this underestimation of the gap is what is needed to get the dominant contribution from the strong peaks in the LDA absorption spectrum to end up close to the experimental peak structures. For large band-gap materials (E,,, > 4 eV), GWA-based scissors-operator corrections frequently lead to an underestimation of E that is as large as the overestimation by LDA. This effect was first seen by Chen et al. for diamond and Table 36 shows the results of Chen et al. for diamond, Sic, and GaN. Here the LDA values again overestimate the dielectric constant but by only 4-6%. The average GWA corrections to the direct gaps are of the order of 1-2 eV, which when used as scissors corrections give dielectric constants well below experiment. Similar results were obtained by Adolph et aLS6l and by Gavrilenko and B e ~ h s t e d t , ~who ~ ' concluded that to obtain an E in agreement with experiment a shift of OSA,,, is needed in Si; 0.45A,,, in Sic; and 0.2A,,, in diamond. Another important consideration is the effect on the optical response of the quasiparticle weight Z that multiplies the single-particle Green function (see Eqs. (1.7), (1.8), (2.7), and Section 11.11). Within the independent55'

H. H. Li, J . Chem. Phys. Re$ Data 9, 561 (1980). Scripta 3, 193 (1971).

"* B. Monemar, Phys

QUASIPARTICLE CALCULATIONS IN SOLIDS

183

TABLE36. DIELECTRIC CONSTANTS FOR LARGE-GAP INSULALDA AND WITHINTHE SCISSORS-SHIFT APPROACH. THE C AND Sic VALUESARE FROM REF. 540, AND THE GaN VALUESARE FROM REFS.542 (LDA) AND 560 (Scrss). THE SCISSORS SHIITS A, DERIVEDFROM AVERAGED GWA ARE SHOWN WITHIN PARENTHESIS. THE CORRECTIONS, GWA-BASEDSCISSORS-SHIFT APPROACHSYSTEMATICALLY OVERCORRECTS THE SOMEWHAT Too LARGELDA VALUES.

TORS IN

C Sic GaN

LDA

Scrss (A)

EXPT.

5.9 6.9 5.5

5.2 (1.8 eV) 6.1 (1.2 eV) 4.8 (1.5 eV)

5.7" 6.5" 5.35 0.20b

*

"Ref. 182; bRef. 559.

quasiparticle approximation, the susceptibility in Eq. (6.2) should be multiplied by the product of the electron and hole quasiparticle weights Z,Z,, leading to a drastic reduction of about a factor of two in the dielectric and even larger reductions in higher-order response. In practice, the multiplication by Z,Z, is seldom made in optical response calculations, which accounts for their success. This practice of disregarding the reduction due to quasiparticle weights has recently been put on a more solid foundation by the calculations of ' ~ ~ showed that when vertex corrections in the form of Bechstedt et ~ 1 . They excitonic electron-hole interactions are considered the weight factor Z , Z , is replaced by a combined weight factor Z,, N 0.9. Hence the correction due to 2, and 2, is almost canceled by the effect of vertex corrections, which are also neglected in the GWA-based scissors-operator approach. Figure 49 shows the imaginary part of the dielectric function of diamond and silicon obtained by Bechstedt et al. from LDA, GWA with and without the Z,Z, correction, and GWA plus vertex corrections. A similar cancellation between quasiparticle weight and vertex corrections for interband transitions in metals was discussed by Beeferman and Ehrenreich.' 5 1

A. S. Barker and M. Ilegems, Phys. Rev. B7, 743 (1973). J. Chen, Calculation of Linear and Nonlinear Optical Susceptibilities in Wide Gap Semiconductors, Ph. D. thesis, Ohio State University at Columbus, September 1996. 561 B. Adolph, V. I. Gavrilenko, K. Tenelsen, F. Bechstedt, and R. Del Sole, Phys. Rev. B53, 9797 (1996). 5 6 2 V. I. Gavrilenko and F. Bechstedt, Phys. Rev. B55,4343 (1997). 563 R. Del Sole and R. Girlanda, Phys. Rev. B54, 14376 (1996). 564 F. Bechstedt, K. Tenelsen, B. Adolph, and R. Del Sole, Phys. Rev. Lett. 78, 1528 (1997). 559

560

184

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

In summary, to choose A from the gap mismatch leads to corrections too large in most materials, with increasing error as the band gap increases;The conclusion is that we lack a first-principles method to choose a correct scissors shift. However, there is no doubt that some scissors shift is needed, since the best LDA band structures yield gap values well below the experimental gaps. Most authors today use a scissors shift in optical response calculations, but the method of choosing the shift varies, and the shift is largely an empirical way to include quasiparticle and other manyelectron effects.

31. LOCAL-FIELD EFFECTSIN OPTICAL RELSPONSE In semiconductors and insulators, the inhomgeneity of the density distribution leads to internal electric fields that vary over distances comparable to the lattice constant. As shown below, such local fields give rise to about 10% corrections in E and 10-30% corrections in x ( ~ )and , can lead to dominant corrections in properties such as the optical rotatory power. a. Definitions and Gauge In the notation of Section 11, an external potential Kxt(q;a)of wave vector q and frequency o yields a total internal electrical potential Kot given by Kot.c(q;

0 ) = &lo1 K x t h ; 0).

(6.6)

Note that only the G = 0 component of the inverse dielectric matrix E;,$ appears on the right-hand side of Eq. (6.6), since the external optical field has no local-field component. It is important to distinguish the dielectric matrix EGG,(q; a) from the dielectric function E(q; a).The dielectric function E(q; a) is defined by E(q; a)= &(q; a)/K:,t,o(q;a). When local fields are disregarded, we have E(q; o)= b ( q ; o),but when local-field effects are included, the dielectric function is given by the inverse of the G, G = 0 component of the inverse dielectric matrix: E(%

a)= l/CGil(q; 011.

(6.7)

For independent electrons in the time-dependent Hartree or random phase approximation (RPA) the dielectric matrix is that given in Section 11.5 in terms of the independent-particle polarizability Po:

185

QUASIPARTICLE CALCULATIONS IN SOLIDS

where P&(q; o)is given by the Adler-Wiserso*82formula: 2 Pgd.(q; o)= -

(ilexp(i(q + G)*r)Ii')(i'(exp(-i(q + G ) - r ) l i ) v1 ii' A(1

-.&,I

+ o + i6

E~ - E ~ ,

+

A 4 -A) E~, E~ - o

+ i6

which is an extension to include local-field effects of the Ehrenreich and Cohen formula for the dielectric constant in a solid."' In the scissors-operator approach, the conduction-band energies in the denominator of the Adler-Wiser formula for P&(q; o) are shifted by A. Without local fields, the static susceptibility x, defined by E = 1 + 47q, is obtained by taking the limit 4 + 0 in Eq. (6.8): x = - limqd0P&(q,O)/q2. In numerical implementations, two alternative forms of the Adler-Wiser formalism have been used: (1) the length-gauge formulation of Sipe and C O W O ~ ~ based ~ ~ on S matrix ~ ~ ~elements * ~ ~of ~the position operator, which was used in Eq. (6.2); and (2) the velocity-gauge formulation developed by Levine and Allan (see Refs. 535, 536, 541, 567, and 568). These two formulations differ by the way the matrix elements of type (cklexp(iq.r)luk) are evaluated. The length-gauge formulation can be obtained by an expansion in q of the exponential exp(iq-r): (cklexp(iq * r) luk')

= iq. (cklrl uk)6k,k,.

(6.10)

The q-independent term in the expansion, (ckluk'), is zero because of the orthogonality of orbitals from different bands. When Eq. (6.10) is used in the Adler-Wiser formula, we obtain the expression for x given in Eq. (6.2), if we identify the x-direction with the direction along q and if we anticipate that the intraband matrix elements do not contribute when all the bandindices and k vectors are summed over. The direct expansion in Eq. (6.10) is valid only for interband matrix elements, since the position operator is not well defined for intraband matrix elements. A detailed derivation of Eq. (6.2) requires a more careful treatment of intraband matrix element^.^^^.^^^ Further, when the Hamiltonian has nonlocal components, because of pseudopotential terms or self-energy

'" A. D. Papadopoulos and E. Anastassakis, Phys. Rev. 843,5090 (1991). 566

567

J. E. Sipe and E. Ghahramani, Phys. Rev. B48, 11705 (1993). Z. H. Levine, Phys. Rev. B42, 3567 (1990). Z. H. Levine, Int. J. Quunt. Chem. S28, 411 (1994).

186

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

corrections, additional complications arise. These points have been discussed extensively in recent In the velocity-gauge formulation, the exponential is not expanded in terms of q. Instead the matrix element (cklexp(iq r)luk),which is taken between the full Bloch states integrated over all space, is reformulated in terms of the periodic part of the Bloch states and the integration limited to the volume R of a unit cell in the crystal: 1 V

1 R

- (ck(exp(iq.r)luk) = - (ckluk

+ q)&+,,k*

(6.11)

Now the expansion in q can be performed by k - p perturbation theory, which yields

Instead of a position operator, the velocity operator v k = i[& r] = V k H k appears, and each matrix element also yields an extra energy denominator. Hence, in the velocity gauge the formula for x will contain two velocity matrix elements and three energy denominators. When the scissors-operator approach is used, it is crucial to recognize the different origins of the three energy denominators in the velocity gauge. Only one energy denominator should be shifted by A; the other two should remain unshifted, and the velocity matrix element should be taken with respect to the LDA Hamiltonian without the shift. Since the wave functions are unchanged, the overlap (ckluk q) must be the same in LDA and in the scissors-operator approach. This requires

+

(6.13)

where Hi = H t D A+ A&lck)(ckl. Eq. (6.13) must be true for all u, c, and k, so in the velocity gauge the velocity matrix elements in the scissors-shift approach are given by the LDA velocity matrix elements scaled by (&uk

- &ck + A)/(Euk

- %k)*

b. Numerical Results The numerical results below for local-field effects in semiconductors and insulators are all based on an extension to the Adler-Wiser expression in Eq. 569

0.Pulci, G. Onida, A. I. Shkrebtii, R. Del Sole, and B. Adolph, Phys. Rev B55,6685 (1997).

QUASIPARTICLE CALCULATIONS IN SOLIDS

187

(6.9), which extension treats the local fields in the time-dependent localdensity approximation570 rather than in RPA (time-dependent Hartree approximation). In this extension, exchange and correlation effects in the induced electron density are calculated self-consistently within LDA. For further details, see the references cited in the following discussion. Columns two and three in Table 35 show typical local-field corrections to the dielectric constant. Invariably a reduction of E by 5-10% is obtained. These results are consistent with the effect of local-field corrections on the absorption spectrum shown in Fig. 39. The weight in the dominant peaks in E ~ ( W ) shifts somewhat to higher frequencies, causing a reduction in E. In second-order response, the local fields give a larger correction -generally reducing x(') but occasionally increasing it. Table 37 shows the effect of both the scissors correction and the local fields on x(') for those of the previously discussed materials that lack inversion symmetry, which is a prerequisite for a nonzero second-order response. The LDA values are too large by as much as a factor of two, with the severest errors for the smallergap materials. The local-field corrections are of the order of 10-20% and are negative except for the zzz component in Sic, which shows a 13% increase. Despite the limited experimental data, we can see a trend similar to that found for &-the scissors-operator approach works well for the smaller-gap materials, but an overcorrection is obtained for the larger-gap materials, here exemplified by Sic and GaN. ''O

A. Zangwill and P. Soven, Phys. Rev. Lett. 45, 204 (1980).

TABLE37. SECOND-HARMONIC COEFFICIENT d = x(')/2 IN pm/V IN LDA AND WITHINTHE S c r s s o ~ s - S mAPPROACH WITHOUT (No LF) AND WITH (LF) LOCAL-FIELD THE SECOND COLUMN SHOWS WHICH TENSORCOMPONENT IS BEINGTABULATED. TIil3 LOCALFIELDCORRECTION IS TYPICALLY OF ORDER - 15%, BUT IS 4- 13% (ABSOLUTE VALUE) IN THE s i c ZZZ COMPONENT.

cow.

LDA

NoLF

LF

Em.

XYZ

65 205 21 35 6.6 -4.1 3.2 5.4

46 106 17 24 4.9 -2.7 -27 4.2

38 95 14 22 4.4 -3.1 -2.1 3.5

37 f 2" 81 k 5"

GaP GaAs AIP AlAs SIC

xxz

GaN

xxz

XYZ

XYZ

XY*

zzz ZZZ

"Ref.571; bRef.572.

-

f2.6b k5.4b

188

WILFRIED G . AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Hughes and Sipe556made an important observation: It is not clear that the experimental values for x(') are actually correct. Hughes and Sipe found that their theoretical values for the second-harmonic coefficient with a scissors-correctedband structure were quite far from experimental values in, for example, GaAs and Gap. However, they found close agreement with recent measurements of the linear electro-optic coefficient given by x'*)(o;o,0) in contrast to the second-harmonic coefficient ~'')(2o;o,a).The second-harmonic (SH) coefficient gives the polarization at 2w induced by a field at o,while the electro-optic (EO) coefficient gives the polarization at o from a field at o and a static field. Formally, these two coefficients should be equal in the static limit, while the experimental values disagree by almost a factor of two. For example, in GaAs x ~ = 162~ _+ 1 0, ~ m ~/ V ~ ~ ' and xg&xpl=99.8 pm/V,556*574 while some of the theoretical values are 172pm/V ( p s e ~ d o p o t e n t i a l ) ,96.5 ~ ~ ~pm/V (FLAPW -full-potential linearized augmented plane wave),556and 104.8 pm/V (LMTO -linearized muffin-tin orbitals).576All of these calculations were done with scissorsshifted LDA band structures. Clearly, both more experimental and theoretical work is called for. The theory for local-field corrections to nonlinear response was correctly worked out only recently by Chen et a1.573Earlier work had invariably assumed that only the linear local fields are important in nonlinear response. However, Chen et al. showed that the nonlinear local fields are of equal importance. For example, for second-harmonic generation we must consider local fields both at the driving frequency o and at the second harmonic 2 0 . It can also be shown analytically that for scalar potentials the dominant term of the linear local field is exactly twice the dominant term of the nonlinear local field, leading to a 50% increase in the local-field correction compared to previously published results. The typical correction due to local fields is therefore about - 15% for second-harmonic generation, but with large variation in strength and an occasional change of sign. Besides the work of Chen et a1.,573several other authors discuss nonlinear response in semiconductors, although Chen et al. were the only ones to consider the full effect of local-field corrections. The plane-wave pseudo5 7 1 B. F. Levine and C. G. Bethea, Appl. Phys. Lett 20,272 (1972), as revised by D. A. Roberts, IEEE J. Quantum Electron. 28, 2057 (1992). 5 7 2 J. Miragliotta, D. K. Wickenden, T. J. Kistenmacher, and W. A. Bryden, J. Opt. Soc. Am. B 10, 1447 (1993). 573 J. Chen, L. Jonsson, J. W. Wilkins, and Z. H. Levine, Phys. Rev. B56, 1787 (1997). 5 7 4 S. Adachi, GaAs and related Materials: Bulk Semiconducting and Superlattice Properties, World Scientific, Teaneck, NJ (1994). 5 7 5 Z. H. Levine, Phys. Rev. B49,4532 (1994). 5 7 6 S. N. Rashkeev, W. R. L. Lambrecht, and B. Segall, Phys. Rev. B57, 3905 (1998).

~

QUASIPARTICLE CALCULATIONS IN SOLIDS

189

potential method, used by Chen et al., was developed by Levine and Allan (Refs. 536, 541, 567, and 568), who showed how to include a scissors shift in the velocity gauge. Their method is mostly applied in the longwavelength limit,540*542 although frequency dependence below the gap can be obtained.575 Sipe and coworker^^^^,^^^ made a detailed comparison between the velocity-gauge and length-gauge approaches and found that a simpler formulation than the Levine-Allan approach can be obtained when working in the length gauge. The length-gauge formulation was used by Hughes and Sipe for GaAs and and by Hughes, Wang, and Sipe for GaN and ) They calculated both the real and imaginary part of x ( ~ for frequencies up to well above the absorption threshold within a scissorsshift approach using an LDA FLAPW (full-potential linearized augmented plane wave) band structure. However, they included no local-field corrections. The formulation of Aversa and Sipe was also implemented by Rashkeev et al.576using an LMTO (linearized muffin-tin orbital) band structure. They too studied the frequency-dependent x") and added BN and S i c to the materials studied by Hughes et al. The article by Rashkeev et al. also includes a detailed discussion of the strengths and weaknesses of the scissors-shift approach. Adolph and Bechstedt 78 cal) Gap, GaAs, InP, InAs, and S i c culated the frequency-dependent x ( ~ for with a plane-wave-pseudopotential method at the theoretical lattice constants. They used scissors shifts for the 111-V compounds but discussed a more sophisticated momentum- and band-dependent self-energy shift in Sic. The optical response calculations described so far have been done by sum-over-states methods, but an alternative exists. Dal Corso et ul.552used the so-called 2n + 1 theorem579 to derive a different and more efficient numerical method for second-order response within time-dependent density functional theory. The 2n + 1 theorem in this context expresses secondorder response functions (derivable from third-order derivatives of the total energy) in terms of first-order changes in the wave function. The calculations by Dal Corso et al. were performed at theoretical lattice constants far from the experimental ones, which strongly affected their values. Nevertheless, their method should lead to the same results as others at the experimental lattice constants. The relatively modest local-field corrections we have seen up to this point 1calculated . ~ the~local-field ~ corrections are not a general rule. Jonsson et ~ 577 578 579

J. P. L. Hughes, Y. Wang, and J. E. Sipe, Phys. Rev. B55, 13630 (1997). B. Adolph and F. Bechstedt, Phys. Rev. B57, 6519 (1998). X. Gonze and J.-P. Vigneron, Phys. Rev. B39, 13120 (1989).

190

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

TABLE 38. LOCAL-FIELD(LF) CORRECTIONS FOR SELENIUM AND a-QUARTZ (sic),) IN DIELECTRICCONSTANT E , SECOND-HARMONIC COEFFICIENT d, AND OPTICAL ROTATORY POWER p.543*583 MODESTLOCAL-FIELD CORRECTIONS FOR E DO NOT IMPLY MODESTCORRECTIONS IN THE OTHER OPTICALPROPERTIES. NOTE ALSOTHAT FOR BOTH d AND p THE SIGN OF THE LOCAL-FIELD CORRECTION DIFFERSBETWEEN Se AND QUARTZ.FORTHE SECOND-HARMONIC COEFFICIENT d, THERE ARE MOREEXPERIMENTAL RESULTS THAN QUOTEDIN THE TABLE.FOR SELENIUMTHESE EXPERIMENTALVALUESVARY SUBSTANTIALLY, WHILE FOR QUARTZTHE DIFFERENT VALUES ARE IN GOODAGREEMENT. WE REFER TO REF. 543 FOR A MORE DETAILED DISCUSSION AND REFERENCES. THE QUOTEDEXPERIMENTAL VALUES FOR THE ROTATORY POWER ARE THE ZERO-FREQUENCY LIMITSGMN IN REF. 583, WHICH WERE EXTRAPOLATED FROM THE DATAOF REFS 584 AND 585 (Se) AND REF. 586 (QUARTZ).

Se a-quartz

9.0 2.42

7.9 7.3 f 1.1" 2.30 2.35b

78 0.35

111 0.33

97+25' 0.34'

21 0.7

-55 5.6

k56k30 4.6f0.1

"Ref. 142; *Ref. 580; 'Ref. 581; dRef. 582.

to the optical rotatory power of a-quartz and selenium and found that the local fields dominate the response. The optical rotatory power is the ability of crystals with a chiral structure to rotate the plane of polarization of light passing through. Table 38 shows the effect of local fields on the dielectric constants, the second-order susceptibilities, and the rotatory powers of selenium and a-quartz. For E in both materials and for x") in quartz, the corrections are modest reductions. For x(2) in Se there is a 33% positive correction. Finally, in the rotatory power the local fields dominate the response -in a-quartz the local-field correction increases the rotatory power by a factor of eight, while in Se the local fields change the sign and more than double the absolute value. The calculations for the rotatory power cannot be made with the scalar theory of optical response described above, since the physical effect is a rotation of polarization and requires a vector description. Therefore, a generalization to vector response in terms of induced currents and vector F. F. Martens, Ann. Phys. (Leipzig) 6, 603 (1901), cited in D. E. Gray, American Institute of Physics Handbook, 2nd ed., McGraw-Hill, New York (1963). G. W. Day, Appl. Phys. Lett. 18, 347 (1971). 5 8 2 B. F. Levine and C. G. Bethea, Appl. Phys. Lett. 20, 272 (1972). 583 L. Jonsson, Z. H. Levine, and J. W. Wilkins, Phys. Rev. Lett. 76, 1372 (1996). 584 W. Henrion and F. Eckart, Z. Naturforsch. 19A, 1024 (1964). 5 8 5 J. E. Adams and W. Haas, in The Physics of Selenium and Tellurium, ed. W. C. Cooper, Pergamon, New York (1969), 293. 5 8 6 A. Carvallo, C. R Acad. Sci. 126, 728 (1898), cited in D. E. Gray, American Institute of Physics Handbook, 2nd ed., McGraw-Hill, New York (1963).

QUASIPARTICLE CALCULATIONS IN SOLIDS

191

potentials has to be made.543*583*587 H owever, for wavelengths much longer than the lattice constant one can use a hybrid scheme that treats the slowly varying fields by vector theory but uses scalar theory for the local fields (see Refs. 583, 588, 589, 590), thereby greatly simplifying the treatment and avoiding the use of time-dependent current-density functional theory. 32. DENSITY-POLARIZATION FUNCTIONAL THEORY In the static limit, the induced response to a weak perturbing potential can be obtained from a ground-state calculation and should therefore be obtained exactly by Kohn-Sham perturbation theory. This fact has always been the strong, and just, argument made by opponents to the scissorsoperator approach, which artificially adds a nonlocal potential to a theory that should be able to give the correct low-frequency optical response by using an appropriate Kohn-Sham potential. However, there is no doubt that LDA does not suffice, because of the band-gap problem, and there are indications that even in exact Kohn-Sham theory a large gap mismatch exists (see Appendix, Section 3c). a. Divergence of K,, A solution to this apparent paradox was recently presented by Gonze, Ghosez, and G ~ d b y . ’ ~They ’ pointed out that in an infinite sample the exchange-correlation potential in Kohn-Sham theory can be divergent in the long-wavelength limit. In a system with a gap, within which a finite electric field can exist and a finite polarization can be induced, great care has to be applied when the limit of infinite sample size is taken within density functional theory. The discovery of Gonze, Ghosez, and Godby has important consequences for many situations where bulk polarization occurs (see Refs. 591, 592, 593, 594, 595, 596, and 597), not least for ferroelectrics, but here we discuss only the necessary revision of linear response theory. 58’ H. Zhong, Z. H. Levine, D. C. Allan, and J. W. Wilkins, Phys. Rev. Lett. 69, 379 (1992); 70, 1032(E) (1993). 5 8 8 R. Del Sole and E. Fiorino, Sol. State Comm. 38, 169 (1981). 5 8 9 W. L. Mochan and R. G. Barrera, Phys. Rev. B32,4984, (1985); 4989 (1985). S. T. Chui, H. Ma, R. V. Kasowski, and W. Y. Hsu, Phys. Rev. B47, 6293 (1993). 5 9 1 R. Resta, Phys. Rev. Lett. 77, 2265 (1996). ”* W. G. Aulbur, L. Jonsson, and J. W. Wilkins, Phys. Rev. 854, 8540 (1996). 593 X. Gonze, P. Ghosez, and R. W. Godby, Phys. Rev. Lett. 78, 294 (1997). 594 R. M. Martin and G. Ortiz, Phys. Rev. B56, 1124 (1997). 5 9 5 P. Ghosez, X. Gonze, and R. W. Godby, Phys. Rev. B56, 12811 (1997). 5 9 6 D. Vanderbilt, Phys. Rev. Lett. 79, 3966 (1997). 5 9 7 G. Ortiz, I. Souza, and R. M. Martin, Phys. Rev. Lett. 80, 353 (1998).

192

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Let us first state unequivocally that there are no flaws in the original work by Hohenberg and Kohn3’ and Kohn and Sham.33Their work addressed finite systems. The only question is how the limit of infinite extent should be taken. This limit can be approached in two ways: (1) The infinite-volume limit can be taken for the finite-size theory in a weak constant field, with careful consideration of surface effects; or (2) the long-wavelength limit can be taken in the infinite-sample results. Here we adopt the latter approach and focus on the long-wavelength limit of optical response in the type of bulk formulations we use in the previous sections. In the appendix, we define the exchange-correlation potential Vxc(r)= &Ex,[n] /6n(r), which is the additional potential besides the physical potential felt by the Kohn-Sham electrons. By “physical potential” we mean the electrical potential that would act on a weak test charge inside the sample. The physical potential consists of the external potential and the Hartree potential. The fictitious Kohn-Sham electrons are, in addition, acted upon by the exchange-correlation potential, which by construction is necessary to yield the same density in the Kohn-Sham system as for the electrons in the real, interacting system. In linear response, the exchange-correlation potential can be expanded to first order in the density change. The exchange-correlation kernel K,,, which is the response function giving the induced exchange-correlation potential in terms of the induced density, is given by the second functional derivative of the exchange-correlation energy:

6V,,(r)

=

s

Kxc(r,r’)&n(r’)dr’;

KXE(r, r’) =

(6.14)

where no is the ground-state density. For the Kohn-Sham electrons this induced potential must be added to the induced Hartree potential. The total induced potential acting on the Kohn-Sham electrons is therefore (6.15)

When local-field effects are disregarded, we can write in momentum space: (6.16)

The key point of the work by Gonze, Ghosez, and Godby is that the exchange-correlation kernel in an insulator should be expected to have a

QUASIPARTICLECALCULATIONS IN SOLIDS

193

l/q2 divergence just like the Coulomb potential in the Hartree term. This assertion has been shown to be true in model calculations.595 In a different language, the l/q2 divergence of K,, means that the Kohn-Sham electrons feel a macroscopic potential different from the physical electrical potential. In LDA, K,, goes to a constant as q + 0 and is unimportant compared to the l/q2 divergence of the Coulomb potential. This finiteness of K,, in LDA is due to the metallic character of a homogeneous electron gas, which is used to compute the LDA kernel. Hence, despite its success for many material properties, LDA completely fails to describe the divergence in K,, expected in insulators. The discussion above concerning potentials and density can be recast in terms of electric fields and polarization. Instead of the long-wavelength component of the density, we can consider the induced polarization P and a corresponding exchange-correlation electric field felt by the Kohn-Sham electrons besides the physical field. In general, for longitudinal, scalar potentials we can define E = - V V = - iqV and 6n = - V . P = -iq.P. Then an induced exchange-correlation potential, SV,, = K,,6n a 6n/q2, is equivalent to an exchange-correlation electric field directly proportional to the induced polarization, Ex,a P. The latter formulation has given this version of Kohn-Sham theory for infinite insulators the name “densitypolarization functional theory.” The problem with the band-gap mismatch between the quasiparticle and the Kohn-Sham spectrum has now been put in a different light than in our discussion in the previous sections, where the focus was on how to adjust the gap. Concerning optical response at long wavelengths, the Kohn-Sham picture is very different from the quasiparticle picture. The quasiparticles are driven by the physical electrical field and need a correct band structure to give results close to experiments. On the other hand, the fictitious KohnSham electrons inherently will not have the correct band structure and must therefore be driven by a fictitious additional electric field that exactly compensates for the spectral differences and ensures that the correct density, or polarization, is induced as guaranteed by density functional theory in the static limit. b. Real Materials The above discussion demonstrates only a qualitative way in which densityfunctional theory can give correct response, but there is no recipe yet that yields a numerical expression for the exchange-correlation field. However, as discussed by Aulbur et the fact that this field is proportional to the induced polarization allows us to gain some important insight about its strength and qualitative behavior.

194

WILFRIED G . AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Let us define the parameter y to be the assumed constant of proportionality between the exchange-correlation electric field and the polarization, Ex, = yP.From the definition of the susceptibility, x, P = xE,, where E, is the physical (Hartree + external) field, we obtain Ex,= yxE,. Further, in the Kohn-Sham picture we have P = xKs(E, + EJ. That is, the KohnSham electrons are driven by the physical field plus the exchange-correlation field, and the induced polarization is given by their response function xKS, which is the independent-particle susceptibility given by the KohnSham band structure. By construction, the polarization must be the same in both pictures. This yields

Hence, by elimination of E, from the second equality:

Because of the too small gap in the Kohn-Sham spectrum, we expect and therefore y to be negative. Further from the assumption that the main difference between the quasiparticle spectrum and the Kohn-Sham spectrum resides in the energy differences, not the wave functions, we obtain from Eq. (6.2) y N - A/(xEgap).An induced exchange-correlation potential approximately proportional to the gap mismatch was also discussed by Godby and Shams9* in an analysis of band bending at doped Schottky (metal-semiconductor) interfaces. Aulbur et ~ 1 . considered ~ ~ ’ the consequences of Eq. (6.18) and its extension to nonlinear response for most semiconductors and insulators for which data exist. They showed that y is remarkably material independent and of order -0.25 in most small- and medium-gap materials, as shown in Fig. 50. For large-band-gap materials the value for y goes down to about -0.1. These numbers were obtained by using the experimental values for x and the LDA susceptibility for xKS.The latter approximation is motivated by the observation that LDA, being derived from a metallic system, completely ignores any effect of Ex, but describes ground-state properties well. Hence, it is reasonable to assume as a first approximation that xLDais close to xKS. In summary, the Kohn-Sham electrons in density functional theory are fictitious particles that, besides the physical potential, are acted upon by a fictitious exchange-correlation potential. Formally, this extra potential derives from the divergence of the exchange-correlation kernel in exact Kohn-Sham theory for infinite insulators. This divergence is absent in LDA.

xKS> x

598

R. W. Godby and L. J. Sham, Phys. Rev. B 49,

1849 (1994).

QUASIPARTICLE CALCULATIONS IN SOLIDS 0.1

195

1

\ \ \

AIN

\

\

0.0

\

n

k \ I <

-0.1

>2 -0.2

\ \ \

Sic

II

w. >

&

---

-0.3

seC&-

Ge\

--

-0.4

3

4 5 Average bond length (bohr)

6

FIG. 50. The y parameter defined by E , = yP and approximated by y = I/xLDA- l/xexp,for 11 semiconductors and insulators (from Ref. 592). The solid line is a linear fit to the data. The broken curves show how much y varies when xLoA is changed by * 5 % . For small- and medium-gap materials with a bond length above 5 bohr, the value for y is almost material independent. For larger-gap materials, y gradually becomes lower in absolute value as the bond length decreases.

All calculations of optical response in semiconductors and insulators to date have neglected the existence of a macroscopic exchange-correlation potential. When this extra potential is considered in optical-response calculations, agreement with experiment can be obtained without adjustments to the energy spectrum. VII. Excited States within Density Functional Theory

Within density functional theory, there are several methods to directly calculate excited states. Some of these methods yield band gaps in semiconductors without need for quasiparticle theory. This section gives a short description of these alternatives to GWA calculations. To structure the presentation, we classify the methods according to the density on which the density functionals are based. For time-independent

196

WILFRIED G. AULBUR, LARS JONSSON, A N D JOHN W. WILKINS

functionals either the ground-state density or the excited-state density can be used to construct a density functional theory for excited states. These two cases are discussed in Sections VII.33 and VII.34, respectively. Time-dependent density functional theory, outlined in Section VII.35, can also be used to find excited-state properties by analysis of various response functions. Finally, Monte-Carlo methods, as discussed in Section VII.36, can yield excited states and generate highly accurate densities for the purpose of detailed analysis of density functional theories. Most of the work with these new methods has been done for atoms and small molecules rather than for extended systems. For a review of results for atoms and molecules, see the recent article by nag^.'^^ Here, we give a brief description of the ideas behind the methods and focus the discussion on the band gaps in semiconductors, even though that aspect represents only a small fraction of the published work. The material in this section deals with extensions to excited states of conventional density functional theory, which is constructed to give only the ground-state density and energy. In the main text, we assume that the reader is familiar with standard Hohenberg-Kohn and Kohn-Sham theory. However, for the reader who needs a reminder, we give a short review in the appendix of the density functional concepts central to the discussion below. BASEDON GROUND-STATE DENSITIES 33. FUNCTIONALS The ASCF Method. With a slight extension of the original HohenbergKohn formulation the theory is valid not only for ground states but for any state that is the lowest of its symmetry class.600 The excitation energies between two such states of different symmetry can therefore be calculated as an energy difference between two separate calculations. This approach is called the ASCF method,37 since a difference between two self-consistent energy calculations is taken. The band gap in a semiconductor, with N electrons in the fully occupied valence orbitals, is equal to the difference between the energies of the highest occupied Kohn-Sham orbital in the (N + 1)-electron ground state and the highest occuped Kohn-Sham orbital in the N-electron ground state: Egap= E ~ + ~ -, EN,N, ~ + where ~ the notation EN,M refers to the Mth KohnSham orbital in the N-electron system. The difference between the energies and E N , N + (lowest unoccupied orbital in the N-electron system) EN+ l , N + is, by definition, equal to the discontinuity AXc of the Kohn-Sham po599

6oo

A. Nagy, Phys. Rep. 298, l

(1998). 0. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976).

QUASIPARTICLE CALCULATIONS IN SOLIDS

197

tential on addition of an extra electron to the N-electron system: Egap= E N , N + - EN,N A,.. (see the appendix, Section 3). Since the discontinuity Axc is believed to be of the order eV in semiconductors, practical applications of the ASCF method require a density functional that captures the discontinuity. Local-density approximations, including gradient corrections, therefore cannot be used to calculate the band gap in semiconductors. When a single electron is added to a macroscopic system, the density does not change; therefore, local-density approximations give E ~ + - E ~ , 1.~ +In practice, only nonlocal, orbitaldependent functionals give rise to a potential discontinuity. Exact Exchange. Functionals based on an exact treatment of exchange yield Kohn-Sham potentials that include the exchange part Ax of Ax. = Ax + Ac. Krieger, Li, and Iafrate (KLI)601 showed how the KohnSham potential for exact exchange can be calculated in practice in finite systems. Earlier, Sharp and Horton602 and Talman and Shadwick603 showed in principle how the exact-exchange potential -the so-called optimized effective potential (OEP) -can be derived, but KLI demonstrated approximations that lead to computationally simpler equations with little loss of accuracy. Exact-exchange methods (EXX) for extended systems were developed by Bylander and Kleinman,604~605*606 who used the simplified KLI version, by Kotani and and by Gorling.608 Stadele et aL609 applied the formally exact formulation of Gorling to several semiconductors. Table 39 collects the exact-exchange values for the direct band gaps of Si, Ge, diamond, and GaAs. For completeness, we also include the corresponding experimental, LDA, GWA, and Hartree-Fock gaps and gaps obtained from other methods discussed below. The values related to exact-exchange methods are in rows 6-8: EXX-KS are the Kohn-Sham gaps E N , N + - E ~ , N ; EXX+A, are the ASCF gaps E N , N + ~- EN,N Ax; and KLI-KS are the Kohn-Sham gaps obtained with the approximation suggested by Krieger, Li, and Iafrate. The Kohn-Sham gaps of Stadele et al. for Si and C agree well with experiment, while the KLI values of Bylander and Kleinman and

+

+

601 J. B. Krieger, Y. Li, and G. J. Iafrate, Phys. Rev. A 45, 101 (1992); 46,5453 (1992); Y. Li, J. B. Krieger, and G. J. Iafrate, Phys. Rev. A47, 165 (1993). 6 0 2 R. T. Sharp and G. K. Horton. Phys. Rev. 90, 317 (1953). 6 0 3 J. D. Talman and W. F. Shadwick, Phys. Rev. A 14,36 (1976). 604 D. M. Bylander and L. Kleinman, Phys. Rev. B52, 14566 (1995). 6 0 5 D. M. Bylander and L. Kleinman, Phys. Rev. B54, 7891 (1996). 606 D. M. Bylander and L. Kleinman, Phys. Rev. B55,9432 (1997). '07 T. Kotani and H. Akai, Phys. Rev. B 5 4 , 16502 (1996). 6 0 8 A. Gorling, Phys. Rev. A 53, 7024 (1996). M. Stadele, J. A. Majewski, P. Vogl, and A. Gorling, Phys. Rev. Lett. 79, 2089 (1997).

198

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

FUNCTIONAL TABLE39. DIRECTBANDGAPSIN eV AT r FOR DENSITY METHODS THAT GO BEYOND LDA COMPARED TO EXPERIMENT, GWA, AND HARTREE-FOCK: EXX-KS-KOHN-SHAM O R B ~ A GAP L WITH EXACT EXCHANGE; EXX + A,-FuLL BAND GAP WITH EXACT EXCHANGE INCLUDINGTHE GAP DISCONTINUITYAx; KLI-KSKOHN-SHAM GAPWITH THE SIMPLIFTED EXACT-EXCHANGE METHOD OF KRIEGER, LI, AND IAFRATE;sX-LDA-GENERALIZEDNONLOCAL KOHN-SHAM FORMULATION WITH SCREENED-EXCHANGE AND LOCALDENSITYAPPROXIMATION OF CORRELATION; FRITSCHE -FmscHE's GENERALIZED DENSITYFUNCTIONAL THEORYFOR EXCITEDSTATES; MONTE-CARLO BAND-STRUCTURE METHOD.FOR QMC -QUANTUM Si, Ge, AND GaAs, THE EXPERIMENTAL AND LDA VALUESARE FROM COLUMNS 5 AND 4 IN TABLE33, WHILE THE GWA VALUESARE THE suhl OF THE LDA GAP AND THE GWA CORRECTION IN THE LASTCOLUMN OF TABLE34. FOR DIAMOND,THE VALUES ARE FROM TABLE 14 COMPLEMENTED WITH THE EXPERIMENTAL VALUE (6.5eV) QUOTED IN REF. 614 TO ILLUSTRATE THE RATHERLARGEEXPERIMENTAL UNCERTAINTY. THE HARTREE-FOCK GAPSARE FROM REF. 615 FOR Si, Ge, AND C, AND FROM REF.616 FOR GaAs. METHOD

Si

Expt.

3.4

LDA GWA HF EXX-KS

2.6 3.3 8.7 3.3" 2.9' 9.6" 2.9' 3.4' 3.7J 3.2# 3.Ih 3.9'

EXX + Ax KLI-KS sX-LDA Fritsche QMC

Ge 0.9 -0.3 0.8 7.9 1.6'

C

GaAs

6.5 7.3 5.5 7.6 14.6 6.3" 5.9' 15.6"

1.5

1.3' 0.3'

0.2 1.5 9.1

1.9' 1.1' 6.5' 6.4O

"Ref. 609; 'Ref. 607; 'Ref. 604, 'Ref. 605; 'Ref. 610; 'Ref. 611 with Gaussian exchange-correlation hole; #Ref. 61 1 with Lorentzian exchange-correlation hole; hRef. 612- Diffusion MC; 'Ref. 613 -Variational MC.

the EXX values of Kotani and Akai, who used a nonrelativistic KKR band structure within the atomic-sphere approximation, deviate by up to 0.7 eV from experiment. The ASCF gaps, which include the band-gap discontinuity of the exchange potential, are close to the Hartree-Fock values and overestimate the experimental gaps by 5-8 eV. The good agreement between experiment and the EXX Kohn-Sham gaps, together with the large values of Ax (5- 10eV), imply that Ac is equally large but negative.

199

QUASIPARTICLE CALCULATIONS IN SOLIDS

Generalized Kohn-Sham Schemes. A different solution to the band-gap problem was proposed by Seidl et a1.610 The Kohn-Sham equations are generalized so that more of the gap is given by the Kohn-Sham gap and less by the discontinuity. Formally, this redistribution is achieved by modifying the original partition of the total-energy functional into kinetic, Hartree, external, and exchange-correlation functionals. Part of the exchange-correlation energy is added to the kinetic energy before the functional is defined through the constrained search approach. In such a formulation, the Kohn-Sham potential is nonlocal. One generalized Kohn-Sham theory discussed by Seidl et al. is based on exact inclusion of exchange. In the original Kohn-Sham formulation, for nondegenerate ground states, the kinetic-energy functional is defined by minimizing the kinetic energy operator ? over all Slater determinants @ that yield the given density n:

To include exchange, Seidl et al. defined a new functional Go that, besides also includes the Coulomb interaction ?caul, but with the minimum still taken over Slater determinants @ with N orbitals ai:

where E , is the Hartree energy and E x is the exchange energy: @: (r)@y (r‘)Qj(r)mi(r’)

drdr’.

(7.3)

i<j

Whereas in the standard formulation, as shown in the appendix, the exchange-correlation energy is defined by E,,[n] = F[n] - E,[n] - Torn], ‘lo

A. Seidl, A. Gorling, P. Vogl, J. A. Majewski, and M. Levy, Phys. Rev. B 53,3764 (1996).

‘” L. Fritsche and Y. M. Gu, Phys. Rev. B48,4250 (1993).

‘12 A. J. Williamson, R. Q. Hood, R. J. Needs, and G. Rajagopal, Phys. Rev. B57, 12140 (1998). ‘13

P. R. C. Kent, R. Q. Hood, M. D. Towler, R. J. Needs, and G. Rajagopal, Phys. Rev. B57,

15293 (1998). ‘14

‘15 616

0. Madelung (ed.), Semiconductors-Busic Data, Springer, New York (1996). A. Svane, Phys. Rev. B35,5496 (1987). R. Padjen, D. Paquet, and F. Bonnouvrier, Int. J. Quunt. Chem. Symp. 21,45 (1987).

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WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

where F[n] is the Hohenberg-Kohn functional, Seidl et al. defined a new functional G,[n] = F[n] - Go[n]. By construction, G, is expected to be dominated by correlation, even though some exchange and kinetic energy remain in G,. The “exact”-exchange generalized Kohn-Sham equations are similar to the Hartree-Fock equations but not identical to them because of the formally exact treatment of correlations. However, if G , is neglected, the Hartree-Fock equations are obtained. Note, therefore, that with this generalization the Kohn-Sham equations are nonlocal. We also know that in this case the Kohn-Sham band gap will typically be 5-10eV larger than experiment, since this is the result obtained in Hartree-Fock calculations. However, by using screened exchange, Seidl et al. found better agreement with experiment, even though the gaps in Ge and GaAs are now half an eV too small. In screened exchange, an extra factor, exp(-kk,FIr - r’l), is introduced in the exchange integral in Eq. (7.3), where k,, is the ThomasFermi screening constant for the average density. In Table 39, we have included the sX-LDA gaps of Seidl et al., which are calculated with the screened-exchange functional and a local density approximation for G,. Coupling-Constant Perturbation Theory. The excited-state theory of Gorling6’ gives all excited-state properties as functionals of the exact ground-state density of the N-electron system. In the standard formulation of Kohn-Sham theory, a formal connection between the ground state in the noninteracting Kohn-Sham system and the exact ground state is obtained if the Coulomb coupling constant ez is adiabatically switched off. See the appendix, Section 5, for further details. During the switching, the density is kept fixed by the help of a coupling-constant-dependent potential chosen so that the density is independent of the coupling constant and equal to the exact ground-state density. In the limit of no coupling this external potential is by construction equal to the Kohn-Sham potential for the noninteracting system. In the method of GOrling6l7 the adiabatic connection is assumed to exist not just between the ground states of the Kohn-Sham and the interacting system, but also between all of the excited states of the two systems. Perturbation theory to infinite order in the coupling constant then gives all the excited-state properties of the interacting system as functions of the complete set of Kohn-Sham orbitals and energies, including the orbitals unoccupied in the ground state. These orbitals and energies are all determined by the Kohn-Sham potential and are in that sense functionals of the exact ground-state density, since the Kohn-Sham potential is uniquely determined by the ground-state density. In the Kohn-Sham system, the excited states are given by all possible Slater determinants that can be formed from the complete set of single-particle 6’7

A. Coding, Phys. Rev. A 54, 3912 (1996).

QUASIPARTICLE CALCULATIONS IN SOLIDS

20 1

Kohn-Sham orbitals. By construction, the ground-state determinant yields the correct ground-state density, while the excited Kohn-Sham determinants yield densities and energies that are not equal to the exact excited-state energies and densities. However, by the assumed adiabatic connection between the Kohn-Sham and interacting systems, the exact energies and densities are obtained as an infinite perturbation series in the coupling constant. For example, in the coupling-constant expansion of the band gap the zeroth-order term is the Kohn-Sham gap, the first-order correction is the exchange contribution, and the higher-order terms are corrections due to correlation. No calculations for semiconductors using this method have been published to date, and the convergence properties of the perturbation expansion are not known.

34. FUNCTIONALS BASED ON EXCITED-STATE DENSITIES In the appendix, the universal functional F [ n ] that appears in the Hohenberg-Kohn theory as refined by Levy618*619 and Lieb620 is defined by a constrained minimization of the kinetic and Coulomb energies over all N-particle wave functions Y that yield the given density n:

That is, that functional is defined for a very large set of densities and, in particular, for any well-behaved excited-state density. The total energy E [ n ] is obtained by adding the energy of the interaction with the external potential: E [ n ] = F [ n ] + 1T/ext(r)n(r)dr. What is the relationship between the exact energy Ei of the’ ith excited state and the energy E [ n i ] obtained from the exact density,of the ith excited state? From the minimization in the constrained search in Eq. (7.4), it follows that the functional value of E[ni] must be lower or’equal to the exact excited-state energy: E [ n i ] < E i . E r n , ] is by construction the lowest energy any wave function can have for the given density ni; however, the wave function that minimizes E [ n i ] does not have to be equal to the excited-state wave function, even though they both yield the same density. Perdew and Levy6” showed that the equal sign in the relation E [ n i ] < Ei holds if and only if the density ni yields an extremum for E [ n i ] . However, all excited states do not yield an extremum for ECn,]. Perdew and Levy took M. Levy, Proc. Nut. Acad. Sci. U S A 76, 6062 (1979). M. Levy, Phys. Rev. A 26, 1200 (1982). ‘’O E. H. Lieb, in Physics as Natural Philosophy, eds. A. Shimony and H. Feshbach, MIT Press, Cambridge (1982); Int. J. Quant. Chem 24, 243 (1983); in Density Functional Methods in Physics, NATO AS1 Series B123, eds. R. M. Dreizler and J. da Providencia, Plenum, New York (1985). J. P. Perdew and M. Levy, Phys. Rev. B31, 6264 (1985). 618

‘I9

202

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

as an example two noninteracting electrons in an attractive Coulomb potential, and showed that, with the ground-state configuration ls’, the excited-state configuration 2s’ yields an extremum of E [n] while the configuration ls2s does not. That is, they found E[n,,,] = EZS2and E [ n l s z s ] < Elszs.There is no mechanism in general to identify which excited state can be obtained from the ground-state functional, so no practical approach is obtained from these considerations. Theophilou and collaborators622~623 demonstrated that excited-state energies can be obtained from an extension of density functional theory to the sum of the lowest-lying states. For the ground state, the Rayleigh-Ritz variational principle states that the ground-state wave function is the one that minimizes the expectation value of the Hamiltonian. An extension of this principle says that the sum of the N lowest energies is obtained as the minimum of the sum of the expectation values of the Hamiltonian fi with respect to N orthogonal states:

Based on this extended variational principle a density functional theory can be constructed in complete analogy with the standard Hohenberg-Kohn formulation. The density obtained from this theory is the sum of the densities of the N lowest eigenstates, and the energy is the sum of the corresponding energies. The excited-state energy of a given state can be found with this extended formulation if the energies of all the lower-lying states are known. For example, to obtain the energy of the first excited state, the ground-state energy is first calculated and then the sum of the ground-state and excited-state energies. From these two calculations, the excited-state energy is obtained by subtraction. This so-called ensemble density functional theory can be generalized to arbitrary mixtures of states, that is, sums of states with unequal ~ e i g h t s .Ho ~ wever, ~ ~ * the ~ ~functional ~ itself is different for every mixture, and it is not clear how to obtain these functionals in practice. This method has been used for atoms and molecules599but, as far as we know, not for semiconductors. Valone and C a p i t a n F used a variational formulation for the operator (fi - U)’, where U is a free parameter, to construct a functional RJn, U, V,,,] that, when minimized, yields the exact excited-state energy 622

623 624

625 626

A. K. Theophilou, J. Phys. C 12, 5419 (1979). N. Hadjisavvas and A. Theophilou, Phys. Rev. A 32, 720 (1985). E. K. U. Gross, L. N. Oliveira, and W. Kohn, Phys. Rev. A 37,2805 (1988). E. K. U. Gross, L. N. Oliveira, and W. Kohn, Phys. Rev. A37, 2809 (1988). S. M. Valone and J. F. Capitani, Phys. Rev. A 23, 2127 (1981).

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closest to U . A scan of U over a range of energies can then in principle yield all excited states in that range. However, as indicated in the functional dependence of R,, this functional is implicitly and nontrivially dependent on the external potential and is not universal as is the Hohenberg-Kohn functional. The potential dependence makes this theory impractical for application to real systems. F r i t s ~ h e ~ ~ formulated ’.~~* a generalized Kohn-Sham theory for excited states based on a formal partitioning of any excited state Y i into a sum Y i = m i + qi, where mi is a Slater determinant that yields the exact excited-state density ni, while the remainder Pi gives no contribution to ni. By adiabatic switching of the coupling constant, a mapping between an excited state in a noninteracting Kohn-Sham system and an excited state in the interacting system can be obtained. During the switching of the coupling constant the density is kept fixed to the exact excited-state density of a given excited state in the interacting system. The Kohn-Sham potential in this formalism is therefore different for each excited state, and the noninteracting system is in general in an excited state described by a Slater determinant with some excited-state orbitals occupied. This Slater determinant is identified with the wave function mi in the partitioning above, while the second part qi remains unknown. The formalism of Fritsche can be used to derive an expression for the band gap in semiconductors, which for silicon and diamond yields direct band gaps in good agreement with experiment. However, the expression for the band gap depends explicitly on the exchange-correlation energy density and the exchange-correlation potential. To obtain numerical values, Fritsche and Gu61 modeled the shape of the equal-spin exchange-correlation hole (see the appendix, Section 4), from which the energy density and potential can be obtained, and neglected the opposite-spin correlation hole. In Table 39, we have included the gap values for silicon and diamond obtained by assuming either a Gaussian or Lorentzian shape of the exchange-correlation hole.

35. TIME-DEPENDENT DENSITY FUNCTIONAL THEORY For finite systems, a practical method for obtaining excited-state energies is to find the poles in the frequency-dependent linear-response functions.629 Within density functional theory this approach requires an extension of

”’ L. Fritsche, Phys. Rev. B33, 3976 (1986). 629

L. Fritsche, Physica B 172, 7 (1991). M. Petersilka, U. J. Gossman, and E. K. U. Gross, Phys. Rev. Lett. 76, 1212 (1996).

204

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Kohn-Sham theory to time-dependent external potentials. Runge and Gross630showed that such a generalization can be obtained under general circumstances. However, the obtained density functionals depend on the initial state, and the time dependence of the external potential must be such that it can be expanded in a Taylor series in time. For a review of time-dependent density functional theory see, for example, Ref. 63 1. As for the time-independent case, the exchange and correlation effects in linear response are contained in an exchange-correlation kernel K,(q; a), which is now frequency dependent. For numerical applications some approximation must be chosen for K,,(q; o).A useful kernel at low frequencies and for slowly varying densities is given by the adiabatic local-density a p p r o ~ i m a t i o n , ~in’ ~which the frequency dependence is neglected and the static LDA kernel is used. However, Vignale and K ~ h n ~showed ~ ’ that local-density approximations fail at finite frequencies. In particular, for interacting electrons in a harmonic potential exact constraint^^^^.^^^ on the form of K,,(q; o)are obtained, which constraints can only be satisfied by nonlocal kernels. Thus, a central issue in time-dependent density functional theory is to find improved approximations to K,,(q; a). Many of the theories described in Section VII.33 and VII.34 can be generalized to include time dependence, each of them corresponding to a particular approximation to K,(q; o).Examples are the nonadiabatic local-density kernel of Gross and K ~ h n which , ~ ~violates ~ the harmonicpotential constraints, and exact-exchange The coupling-constant perturbation theory of Gorling can also be extended to time-dependent potentials, from which theory formally exact correlation can be systematically included in K,,(q; o).638 For semiconductors, as we saw in Section VI.32 concerning densitypolarization functional theory, the divergence of K,,(q; o)as q + 0 plays a central role in optical response. This aspect of K,,(q; o)has yet to be fully understood in the static limit, and nothing is known at present concerning the finite frequency aspects of this problem. The calculations of optical coefficients in Section VI.3 1 are obtained by using time-dependent density E. Runge and E. K. U. Gross, Phys. Rev. Leu. 52, 997 (1984). E. K. U. Gross and W. Kohn, Adv. Quantum Chem. 21, 255 (1990). 632 G. Vignale and W. Kohn, Phys. Rev. Lett. 77, 2037 (1996). 6 3 3 J. F. Dobson, Phys. Rev. Lett 73, 2244 (1994). 6 3 4 G. Vignale, Phys. Rev. Lett. 74, 3233 (1995). 6 3 5 E. K. U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850 (1985). 6 3 6 C. A. Ullrich, U. J. Gossman, and E. K. U. Gross, Phys. Rev. Lett. 74, 872 (1995). 637 S. J. A. van Gisbergen, F. Kootstra, P. R. T. Schipper, 0. V. Gritsenko, J. G. Snijders, and E. J. Baerends, Phys. Rev. A 57, 2556 (1998). 6 3 8 A. Gorling, Phys. Rev. A 57, 3433 (1998). 630

631

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205

functional theory with a static LDA kernel to describe the effect of exchange and correlation on the local fields. The need for a scissors-operator correction to obtain a reasonable absorption spectrum demonstrates that this simple kernel is inadequate for the description of semiconductor gaps. Vignale and K ~ h n developed ~ ~ ’ a time-dependent formalism based on functionals of both density and current. They showed that in this formalism local approximations are more promising. Further, the theory can be cast in a form where the electron gas is described by a continuum fluid model in which the viscosity and elasticity coefficients can be obtained from parameters of the homogeneous electron gas.639,640On e of the successes of this method is a derivation of the damping of collective modes in an inhomogeneous electron gas. Applications to quantum wells give dampings of the correct order of magnitude, although these applications also reveal some fundamental difficulties.641 36. MONTE-CARLO CALCULATIONS In quantum Monte-Carlo approaches, statistical sampling methods are used -or ’ to propato calculate matrix elements -variational Monte C a r 1 0 ~ ~ gate the imaginary-time Schrodinger equation by numerically simulated or Green function Monte C a r 1 0 . ~In~both ~ of these diffusion methods, a real-space many-electron wave-function representation allows the generation of highly accurate wave functions, including strong correlation effects. In variational Monte Carlo, a parametrized form of the wave function is chosen and the total energy is then minimized by numerical integration of the matrix elements of the Hamiltonian. In diffusion Monte Carlo, the mathematical equivalence between the imaginary-time Schrodinger equation and a real-time diffusion equation is used to propagate the Schrodinger equation by simulated numerical diffusion of a collection of fictitious particles (“walkers”) whose density corresponds to the wave function. Propagation of the imaginary-time Schrodinger equation exponentially projects out the ground state from any initial state that is not orthogonal to the ground state. In this way, an exact numerical wave function for the ground state can be obtained from a reasonably accurate initial guess. Often a variational Monte-Carlo calculation generates an

640 641

642 643

644

G. Vignale, C. A. Ullrich, and S. Conti, Phys. Rev. Lett. 79, 4878 (1997). S. Conti, R. Nifosi, and M. P. Tosi, J. Phys. Cond. Mat. 9, L475 (1997). C. A. Ullrich and G. Vignale, Phys. Rev. E58, 15756 (1998). W. L. McMillan, Phys. Rev. 138, A442 (1965). D. Ceperley, G. Chester, and M. Kalos, Phys. Rev. E16, 3081 (1977). M. H. Kalos, Phys. Rev. A 128, 1791 (1962).

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WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

initial wave function for a more accurate, and more computationally expensive, diffusion Monte-Carlo calculation. Application of diffusion Monte Carlo is straightforward only for bosons, while for fermions the antisymmetry of the wave function is difficult to handle. If the antisymmetry is not strictly enforced, the numerical simulation will in general lead to a bosonic ground state even for a fermionic initial wave function. The common way to avoid this problem is to force the wave function to keep the nodal structure of the initial wave function. This “fixed-node” approximation355ensures that the obtained wave function is antisymmetric. However, the exact ground state cannot be found this way, since it is generally impossible to predict the correct nodal structure. The wave function found is instead the one with the lowest energy of all states with the given node structure. In practice, the nodal structure of the initial wave function is given by a Slater determinant of single-particle orbitals obtained from, for example, a Hartree-Fock or LDA calculation. Despite the fixed-node approximation, diffusion Monte-Carlo calculations usually generate highly accurate wave functions. Diffusion Monte Carlo can be used to obtain the lowest energy eigenstate of a given symmetry. Mostly indirect band gaps in solids have been c o m p ~ t e d , ~ ~ since ’ . ~ the ~ ~difference ~ ~ ~ ’ in total crystal momentum between the excited state and the ground state ensures their orthogonality. However, Williamson et d 6 1 2 showed that, although in principle diffusion Monte Carlo is not useful for calculations of direct gaps, in practice such calculations can be done within the fixed-node approximation. To obtain the direct gap at r, for example, the highest valence-band orbital in the ground-state wave function is replaced by a conduction-band orbital. Without the fixed-node approximation this initial wave function would again generate the ground state. However, within the fixed-node approximation this does not happen, since the nodal structure is not the same in the initial state and the ground state because of the difference between the valence- and conduction-band orbitals. Williamson et al. obtained a 3.7-eV gap in silicon (see Table 39) by implementing this form of diffusion Monte Carlo. The fact that Monte-Carlo methods can generate highly accurate densities makes these methods particularly helpful in analyzing detailed features of density functional theory. For example, Hood et used variational L. Mitas and R. M. Martin, Phys. Rev. Lett. 72, 2439 (1994). L. Mitas, Comp. Phys. Comm. %, 107 (1996). 647 L. Mitas, Physicu B 237-238, 318 (1997). 648 R. Q. Hood, M. Y. Chou, A. J. Williamson, G. Rajagopal, R. J. Needs, and W. M. C. Foulkes, Phys. Rev. Lett. 78, 3350 (1997). 645 646

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207

Monte Carlo to calculate the exact form of the exchange correlation hole in silicon, including the dependence on coupling constant. Other examples are the work by Knorr and Godby 649 and Engel et a1.,650who analyzed the Kohn-Sham potential for model semiconductors with the help of exact densities obtained from diffusion Monte-Carlo simulations. Also, Kent et uL613 used variational Monte Carlo to calculate both density matrices and the full band structure of silicon (see Table 39 for their value of the gap).

ACKNOWLEDGMENTS It is a pleasure to acknowledge helpful comments from M. Alouani, F. Aryasetiawan, J. Chen, A. G. Eguiluz, A. Fleszar, R. W. Godby, A. Gorling, 0. Gunnarsson, P. Kruger, M. Rohlfing, L. J. Sham, E. L. Shirley, J. E. Sipe, and U. von Barth. Also, we thank F. Aryasetiawan and 0. Gunnarsson, A. Gorling, and M. Rohlfing, P. Kruger, and J. Pollman for allowing us to use unpublished data; A. Fleszar, A. Gorling, and L. Steinbeck for a critical reading of the manuscript; A. G. Eguiluz, B. Farid, and G. D. Mahan for sending us material prior to publication; and J. Chen for providing pseudopotentials. WGA gratefully acknowledges continued collaboration with L. Steinbeck, M. M. Rieger, and R. W. Godby on the real-space/imaginary-time GWA. Appendix: Density Functional Theory

This appendix gives a short overview of density functional theory. The purpose is to remind the reader of those concepts and ideas that are built upon in the main text of this review. We shall neither attempt a complete review of density functional theory nor follow a historical sequence of presentation. For further review see, for example, Refs. 651, 652, 653, 654, and 655. W. Knorr and R. W. Godby, Phys. Rev. B50, 1779 (1994). G. E. Engel, Y. Kwon, and R. M. Martin, Phys. Rev. B51, 13538 (1995). 6 5 1 Density Functional Methods in Physics, eds. R. M. Dreizler and J. da Providencia, NATO AS1 Series B123, Plenum Press, New York (1985). "' R. M. Dreizler and E. K. U. Gross, Density Funcrional Theory, Springer, New York (1990). 653 Density Functional Theory, eds. E. K. U. Gross and R. M. Dreizler, NATO AS1 Series B 337, Plenum Press, New York (1995). 6 5 4 R. G . Parr and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press, New York (1989). 6 5 5 Density Functional Theory of Many-Fermion Systems, ed. S . B. Trickey, Advances in Quantum Chemistry, vol. 21, Academic Press, San Diego (1990). 649

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WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

1. UNIVERSAL DENSITY FUNCTIONALS

The goal of density functional theory, as introduced by Hohenberg and K ~ h n , ~is' to find the exact ground-state density and total energy of a system of N interacting electrons in an external potential V,,,. The electrons are assumed to interact pairwise via the Coulomb interaction. The Hamiltonian I? for this system is

where ri is the coordinate for the ith electron. To simplify the notation, we denote the first term in Eq. (A.l), the kinetic-energy operator, by 2 and the second term, the Coulomb-interaction operator, by pcOul.These two terms are the same in all N-electron systems considered; only the external potential differs between systems. This invariance of the kinetic and Coulomb operators in the Hamiltonian is the basis for the idea of universal functionals. Following the formulation by Levy61* p 6 1 9 and Lieb,620 we define the density functional,

where the minimization is over all N-electron wave functions Y that yield the given electron density n(r). That is, F [ n ] is the lowest expectation value, or energy, of the Hamiltonian ? + pcoulthat can be obtained by any N-electron wave function, given that the density is equal to n. This functional is universal in the sense that it does not include any dependence on the external potential. The total energy functional E [ n ] for a specific system is obtained by addition of the interaction with the external potential:

The above formulation is valid for an integer number of electrons. However, the definition of the functionals can be extended to noninteger electron numbers by the inclusion of mixtures between states with different total number of electrons in the minimization in Eq. (A.2).40*656For example, 656

N. D. Mermin, Phys. Rev. 137, A1441 (1965).

QUASIPARTICLE CALCULATIONS IN SOLIDS

209

+

fractional electron numbers between N and N 1 are obtained from mixtures of N - and ( N 1)-electron wave functions. When a fractional total number of electrons is allowed, the ground-state energy and density are obtained by minimizing E [ n ] under the constraint that I,n(r)dr = N :

+

where p is a Lagrange multiplier. In terms of density functional derivatives, we can therefore write

where n,(r) is the ground-state density. It can further be shown that if N is ” a noninteger pN is the chemical potential of the N-electron ~ y s t e m . ~ For integer N , the chemical potential is in general undefined and p N as a function of N is discontinuous. According to the theory of Hohenberg and Kohn3’ there is a unique ground state belonging to each external potential. This one-to-one mapping between the density and the external potential means that the exact ground-state wave function is also a functional of the density. A given density corresponds to a particular external potential, and that potential gives, for nondegenerate ground states, a unique ground-state wave function of the Hamiltonian in Eq. (A.1). Since the ground-state wave function is a functional of the density, so are all expectation values of this wave function.

2. THEKOHN-SHAM SYSTEM Kohn and Sham33 showed how a practical computational method can be constructed from the density functionals of Hohenberg and Kohn. The idea is to associate the physical system of N interacting electrons with a fictitious system of N noninteracting electrons. We call this fictitious system the “Kohn-Sham system” and the noninteracting electrons the “Kohn-Sham electrons.” The connection between the two systems is that they are defined to have exactly the same ground-state density and chemical potential. The external potentials in the two systems are not the same. On the contrary, the external potential in the Kohn-Sham system -the KohnSham potential -must be precisely that single-particle potential that yields a ground-state density for the Kohn-Sham system that is equal to the ground-state density of the interacting system.

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WILFRIED G. AULBUR, LARS JONSSON, A N D JOHN W. WILKINS

For nondegenerate ground states, the wave function of the Kohn-Sham electrons is a Slater determinant of the N lowest orbitals Oi of the Kohn-Sham equations:

where the notation VKs(no;r) means that V,, is a local single-particle potential that is an implicit functional of the exact ground-state density no. The Kohn-Sham equations must be solved self-consistently so that both equations in Eq. (A.6) are fulfilled. Formally, the Kohn-Sham potential can be obtained from the functional F [ n ] . The first step is to define another universal functional To[n]:

where Y is any N-electron wave function that yields the density n. That is, To is the lowest kinetic energy any wave function can give under the constraint that it must yield the density n. The total-energy functional of the Kohn-Sham system is

When the minimization in Eq. (A.7) is extended to include mixed states and fractional occupation, T,,[n] has a well-defined functional derivative, and we obtain, in analogy with Eq. (AS),

Comparing Eqs. (A.9) and (AS), we see that VKSmust be defined by

It is customary to split the universal functional F[n]

- T,[n] into a Hartree

piece, (A. 11)

21 1

QUASIPARTICLE CALCULATIONS IN SOLIDS

and a remainder Ex,[n], which is called the “exchange-correlation energy”:

FCnl

-

ToCnl

= EHCnl

+ ExcCnI.

(A.12)

The Kohn-Sham potential can then be expressed in terms of E,,[n] as

where V, is the ground-state Hartree potential, (A.14) and V,, is the so-called exchange-correlation potential, (A.15) In systems with a fractional number of electrons, the Kohn-Sham orbitals have fractional occupation number^.^" Specifically, a continuous change from the (N - 1)-electron system to the N-electron system is obtained by a variation between 0 and 1 of the occupation number of the highest occupied orbital in the N-electron system. One of the most important conclusions from such a formulation is that the energy E N , N of the highest occupied Kohn-Sham orbital in the N-electron system (integer N) is equal to the chemical potential p i :40*658 EN,N =

Pi,

(A.16)

where p i = pN-&,6 infinitesimal. That is, by the chemical potential for integer N we mean, by definition, the chemical potential of a system with a fractional occupation of the Nth Kohn-Sham orbital. The notation E ~ refers to the energy of the Mth Kohn-Sham orbital in the N-electron system. 3. THEBAND-GAP DISCONTINUITY

Let us first define the ionization energy I and the electron affinity A for the N-electron system: (A.17) I = E N - 1 - EN; A = EN - EN+l, J. F. Janak, Phys. Rev. B18, 7165 (1978). J. P. Perdew, in Density Functional Methou5 in Physics, eds. R. M . Dreizler and J. da Providencia, NATO AS1 Series B123, Plenum Press, New York (1985), 265. 657

,

~

212

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

where EM denotes the ground-state energy of the M-electron system. In a macroscopic semiconductor with N electrons in the fully occupied valence bands, we can associate I with minus the chemical potential p; of the N-electron system, and A with minus the chemical potential pi+1 of the (N + 1)-electron system (one (fractional) electron in the lowest conduction state). Hence, from Eq. (A.16) we obtain:

I = -p-N

- - E N,N;

A =

-pi + I

= -EN+~,N+~.

(A.18)

is given by the difference between I and A: The quasiparticle band gap EBap EBap= I - A = & N + I , N + I - E N , N = A x c

+ EN,N+I - E N ~ N ,

(A.19)

where we have defined Axc = cN+l , N + - E N , N + 1, that is, the energy difference between the highest occupied Kohn-Sham orbital in the (N + 1)electron system and the lowest unoccupied Kohfi-Sham orbital in the Nelectron system. This difference is in general finite659*660 and believed to be of the order of eV in semiconductors and insulators.6 The last two terms in Eq. (A.19), E N , N + I - E N , N , give the band gap in the N-electron Kohn-Sham system. As the notation suggests, Axc is due to the exchange-correlation potential V,, in Eq. (A.15) and is caused by a discontinuous jump in Vxcon addition of an electron:659*660v661

where the explicit r dependence indicates that the difference between the exchange-correlation potentials in the (N + 1)-electron system and the N-electron system is a position-independentconstant Axe. Note that since the potential only changes by a constant, there is no difference between the Kohn-Sham wave functions in the (N 1)-electron system and the Nelectron system. The discontinuity Axc originates solely from the exchange-correlation potential. The external potential is by construction unchanged; the Hartree potential has a simple, direct dependence on the density, which only changes infinitesimally, while the discontinuity in the kinetic energy To is exactly equal to the Kohn-Sham gap E N , N + - E N , N . This discontinuity in the kinetic

+

~~~~

J. P. Perdew and M. Levy, Phys. Rev. Lett. 51, 1884 (1983). 660 L. J. Sham and M. Schliiter, Phys. Rev. Lett. 51, 1888 (1983). 6 6 1 Strictly, similarly to the chemical potential, the exchange-correlation potentials at integer N must also be defined by their limits as the electron number approaches N from below in a system with fractional occupation of the Nth Kohn-Sham orbital. 659

QUASIPARTICLE CALCULATIONS IN SOLIDS

213

energy is due to the qualitative difference between the lowest conductionband orbital and the highest valence-band orbital. However, the definition of the kinetic energy in terms of the squared gradient of the orbital means that, since the orbitals in the (N + 1)-electron system and the N-electron system are the same, the discontinuity in the kinetic energy is the same whether we calculate it with the orbitals in the N-electron system or from the lowest conduction-band orbital in the (N 1)-electron system and the highest valence-band orbital in the N-electron system.662 Numerical implementations of Kohn-Sham theory, using LDA or GGA (see Section A.6), typically obtain Kohn-Sham band gaps about 0.5-2 eV smaller than the experimental gaps, see Fig. 3, Section 1.3. Even though some of this discrepancy can be due to the LDA or GGA approximations, there is evidence that a significant part is due to Axc.6 This inability of Kohn-Sham theory to give band gaps close to experiment is often referred to as the “band-gap problem” and is the main reason that quasiparticle calculations are needed as a complement to Kohn-Sham theory when unoccupied bands are considered.

+

4. THEEXCHANGE-CORRELATION HOLE In terms of a normalized N-electron wave function Y(rl, r2,. . . ,r,), we define the density n(r) by

n(r)

=N

I

IY(r, rz,. . . ,rN)12dr2... dr,.

(A.21)

Similarly, we can define a pair-distribution function n(r, r’) by n(r, r’) = N(N

-

1)

J”

IY(r, r’, r3,. . . ,rN)12dr3... d r N .

(A.22)

Note that the total integral over both coordinates in n(r, r’) is equal to twice the number of unique electron pairs. The exchange-correlation hole is described by the function g(r, r’), which is defined through n(r, r’) = n(r)n(r’)(l

+ g(r, r’)).

(A.23)

Formally, by applying the theory for fractionally occupied orbitals to the noninteracting Kohn-Sham system, one can prove that the discontinuity in the kinetic energy To as the electron number passes through N is exactly equal to the Kohn-Sham band gap. The intuitive argument in the text relying on the infinitesimal difference between the Kohn-Sham orbitals in the N- and (N + 1)-electron systems is not necessary.

214

WILFRIED G. AULBUR, LARS JONSSON, A N D JOHN W. WILKINS

If we put g = 0, the pair-distribution function is equal to the product of the densities at the two points r and r’, which implies that the electrons are uncorrelated. However, g cannot be equal to zero for all r and r’. The integral of n(r, r’) over r’ yields ( N - l)n(r), while the integral of n(r)n(r’) yields Nn(r). Hence, we arrive at the important sum rule,

That is, if one electron is taken to be at r, the total number of remaining electrons in all of space is exactly N - 1, not N . The sum rule for g subtracts out this one electron from the density distribution n(r’)(l + g(r, r’)) as a function of r‘ for fixed r. Though the sum rule prevents g from being zero in all of space, g typically approaches zero when r and r’ are far apart, which means that the electrons are uncorrelated at large distances. The exchange-correlation hole is therefore often of finite extent, and for a given electron in r it describes the redistribution of the other electrons in the vicinity of r caused by exchange and Coulomb repulsion. For example, Ref. 648 presents a Monte-Carlo calculation of the exchange-correlation hole in silicon and considers the coupling-constant dependence discussed in the next section. For electrons with spin 1/2, each spatial coordinate is associated with a spin index that has two values +(1/2) or ‘‘7” and “1”. The density then has two nonzero components and the pair-distribution function is a two-by-two matrix in the spin indices. The sum rule in Eq. (A.24) is thus modified to n

(A.25)

(A.26) and the sum rules for the other two g components are obtained by interchange of 7 and 1 in Eqs. (A.25) and (A.26). These two relations can be understood by extending the motivation for the sum rule for spinless particles to fermions with N f and N1 particles of each spin, respectively. If the particle in r is assumed to have spin up, there are N T - 1 more particles with spin up in the distribution described by nf(r’)(l + gTT(r,r’)), which gives Eq. (A.25). However, there are still N1 particles of opposite spin left in the distribution nT(r‘)(l + gT1(r,r‘)), which gives Eq. (A.26).

QUASIPARTICLE CALCULATIONS IN SOLIDS

215

The pair-distribution function plays a unique role in an interacting electron system because the exact Coulomb energy of the system is equal to the Coulomb energy of the distribution n(r, r’):

From this equation we see directly that g = 0 gives the Hartree energy, while,the exchange and correlation effects are described by g. For a system of noninteracting electrons described by a single Slater determinant, there are no correlation effects, only exchange. The sum rule for g describes the important role of exchange to subtract out the self-interaction piece from the Hartree term, which piece is unphysically included in the Hartree term when defined in terms of an integral over the total densities. For spin one-half electrons, gfT describes both exchange and Coulomb correlation among the equal-spin electrons, and the sum rule for gTTensures that the self-interaction is subtracted out, while gT1describes Coulomb correlation among electrons with opposite spin. 5. COUPLING-CONSTANT AVERAGES

A formal relationship between the physical, interacting electron system and the noninteracting Kohn-Sham system can be obtained by a couplingconstant integration in which the electron-electron interaction is multiplied by a parameter h that is varied between 0 and l.37 In addition-and this additional constraint is what makes the procedure useful -we require that the electron density should be the same for all h, which is achieved by introducing a fictitious single-particle potential K(r).663 The h dependence of V,(r) is implicitly given by the requirement that nJr) = n,(r) for all h. The Hamiltonian A, for a given h is

(A.28)

By construction, we must recover the interacting system for h = 1 and the 663

J. Harris and R. 0. Jones, J. Phys. F4, 1170 (1974).

.

216

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

Kohn-Sham system for h = 0, which implies

v,=,= K,,;

v,=, = V K S .

(A.29)

Denoting the ground-state wave function for a given h by Y,, the total ground-state energy E , is E,

=

(Ynl~IY,)

+~

+

~ , l ~ ~ c o u l K l ~ ( r ,b o~ ( W .

From the Hellmann-Feynman theorem, dE,/dh have

=

(A.30)

(Y,ldH,/dhlY,),

we

(A.31)

which after integration over h from 0 to 1, together with the relations in Eq. (A.29), yields

+

But E L = O= ToCnol + f v VKsWn0(r)dr and E x e l = F[no3 f v Kx,(r)no(r)dr, so the single-particle potentials can be eliminated from Eq. (A.32), and we obtain a relation between the universal functionals F [ n o ] and To[no] at the ground-state density no: FCnol

=

Torn01

+

~

~

~

l

~

c

o

u

l

l

~

*

~

~

~

We now define the average exchange-correlation hole, ij = f: g,dh, and reach the important conclusion that the exchange-correlation energy E x , in Kohn-Sham theory is equal to the Coulomb energy of the couplingconstant-averaged pair-distribution function: (A.34)

In the interacting system the functional F[nO] is equal to the sum of the

QUASIPARTICLE CALCULATIONS IN SOLIDS

217

kinetic energy, the Hartree energy, and the Coulomb energy of the exchange-correlation hole g, while the Kohn-Sham system is described by the kinetic energy To[nO],the Hartree energy, and the Coulomb energy of the averaged exchange-correlation hole @.Hence, the effect of the couplingconstant average is to incorporate the difference between To and the total physical kinetic energy into the Kohn-Sham exchange-correlation functional E,. The Kohn-Sham construction is made specifically to obtain a set of single-particle equations by which the exact density can be found. However, the partitioning of F[n] into a sum of To[n],E H [ n ] , and a remainder E,,[n] is not a physical partitioning, so the exchange-correlation energy contains both ,kinetic and exchange-correlation energy. 6. LOCALAPPROXIMATIONS

For practical implementations of the Kohn-Sham scheme, an approximation must be chosen for the exchange-correlation energy E,,[n] or the exchange-correlation potential 6E,, [n]/6n. The most commonly used approximation is the local density approximation (LDA),33 in which the local exchange-correlation energy per electron is approximated by the exchangecorrelation energy per electron of a homogeneous electron gas with the local density. In general, we can define an exchange-correlation energy per electron Uxc(r) by

(A.35) In LDA, it is assumed that U is a function ( N . B . not functional) of the local density; that is, u L D A ( r ) = ULDA(n(r)). There are several parametrizations for example, Refs. 664 and 665 -for ULDA(n)based on accurate calculations for the homogeneous electron gas3’ Historically, Slater’s density functional theory666 based on local exchange preceded the formal foundation in Kohn-Sham theory. Even earlier, the first de fucto density functional theory was the Thomas-Fermi equation, in which the kinetic energy is also treated in a local a p p r ~ x i m a t i o n . ~ ~ ~ . ~ ~ ~ The local density approximation is by construction exact for homogeneous systems and is the zeroth-order term in an expansion of ExC[n]in terms 664 665

666 667

668

S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980). J. P. Perdew and A. Zunger, Phys. Rev. B23, 5048 (1981). J. C. Slater, Phys. Rev. 81, 385 (1951); ibid. 82, 538 (1951). E. Fermi, Z. Phys. 48, 73 (1928). L. H. Thomas, Proc. Curnb. Phil. SOC.23, 542 (1927).

218

WILFRIED G. AULBUR, LARS JONSSON, AND JOHN W. WILKINS

of the spatial gradients of n. Attempts to include more terms in the gradient expansion turned out to be difficult. A straightforward gradient expansion (GEA)33 yields a s" that violates the exchange-correlation-hole sum rule, which leads to severe problem^.^^^.^^' More sophisticated so-called generalized gradient approximations (GGA) (Refs. 669,670,671, and 672) assume that U is a function of the density and its gradient Vn-that is, UGGA(r)= UGGA(n(r), Vn(r)) -but without limiting U G G , to be an actual gradient expansion of U(r). Instead, the various GGA exchange-correlation energies that have been proposed are specifically constructed to obey the important sum rule for J and several other exact relations that it has been shown a good U(r) should fulfill. However, although GGA often yields better results than LDA-for example, for atomic total energies and cohesive energies in crystals, there are several cases in which GGA gives worse results than LDA -for example, for bulk moduli and phonon f r e q u e n c i e ~ . For ~~~*~~~ lattice constants, both LDA and GGA give about 1% errors, but LDA gives too small values while GGA gives too large v a l ~ e s . ~ ~ ~ * ~ ~ ~

D. C. Langreth and J. P. Perdew, Phys. Rev. B21,5469 (1980). J. P. Perdew, Phys. Rev. 833.8822 (1986); ibid. 34,7406(E) (1986). 671 J. P. Perdew, in Density Fwrerbnul l'kory, eds. E K. U. Gross and R. M. Dreizler, NATO AS1 Series 9337, Plenum Press, New York (1995), 51. 6 7 2 K.Burke, J. P. Perdew, and M. E m r h d , J. Chem Phys. 109,3760 (1998). 673 1.-H. Lee and R. M. Martin, Phys. Rev. B56,7197 (1997). 669

670

SOLID STATE PHYSICS, VOL. 54

The Surfactant Effect in Semiconductor Thin-Film Growth DANIEL KANDEL Department of Physics of Complex Systems Weizmann Institute of Science Rehovot 76100. Israel

EFTHIMIOS KAXIRAS Department of Physics and Division of Engineering and Applied Sciences Harvard University, Cambridge MA 02138. USA

I. Introduction . . . . . . . . . . . . . . . . 11. Experimental Observations . . . . . . . . . . 1. Group-IV Films on GroupIV Substrates . . 2. 111-V Films on 111-V Substrates . . . . . . 3. Mixed Film and Substrate Systems . . . . 111. Theoretical Models . . . . . . . . . . . . . 4. Microscopic Models . . . . . . . . . . . 5. Macroscopic Models . . . . . . . . . . IV. The Diffusion-De-Exchange-Passivation Model . 6. General Considerations . . . . . . . . . 7. First-Principles Calculations . . . . . . . 8. De-Exchange and Generalized Diffusion . . 9. Island-Edge Passivation . . . . . . . . . 10. Kinetic Monte-Carlo Simulations . . . . . V. Discussion . . . . . . . . . . . . . . . . .

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

.

.

. .. . . . . .

. . . . . . . . . .

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

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

. .

.

.

.

.

219 223 223 225 23 1 233 233 239 242 242 244 249 25 1 252 260

1. Introduction

Progress in the fields of electronic and optical devices relies on the ability of the semiconductor industry to fabricate components of ever-increasing complexity and decreasing size. The drive for miniaturization has actually provided the impetus for much fundamental and applied research in recent 219

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years. Nanoscale structures in 2, 1, or 0 dimensions, referred to as quantum wells, wires, and dots, respectively, are at the forefront of exploratory work for next-generation devices. In most cases these structures must be fabricated through epitaxial growth of semiconductor thin films, in either homoepitaxial (material A on substrate A) or heteroepitaxial (material A on substrate B) mode. There are usually two important requirements in this process. First, the film must be of high-quality crystalline material; second, a relatively low temperature must be maintained during growth. The need for the first requirement is self-evident, since highly defected crystals typically perform poorly in electronic applications; the imperfections, usually in the form of dislocations, grain boundaries, or point defects, act as electronic traps and degrade the electronic properties to an unacceptable level. The second requirement arises from the need to preserve the characteristics of the substrate during growth, such as doping profiles and sharp interfaces between layers, which can be degraded because of atomic diffusion when the growth temperature is high. These two requirements seem to be incompatible: In order to improve crystal quality, atoms need to have sufficient surface mobility so that they can find the proper crystalline sites to be incorporated into a defect-free crystal. On the other hand, excessive atomic mobility in the bulk must be avoided, which can be achieved only by maintaining lower than typical growth temperatures. These problems are exacerbated in the case of heteroepitaxial growth, where the presence of strain makes smooth, layer-by-layer growth problematic even when the temperature is high, since the equilibrium structure involves strain-relieving defects. Early studies of the effects of contaminants and impurities on growth modes indicated that it is indeed possible to alter the mode of growth by the inclusion of certain elements, different from either the growing film or the substrate. For a discussion of epitaxial growth modes and the effect of contaminants, see the authoritative review by Kern et al.’ A breakthrough in the quest for controlled growth of semiconductor films was reported in 1989 when Copel et al.’ demonstrated that the use of a single layer of As can improve the heteroepitaxial growth of Ge on Si, which is otherwise difficult because of the presence of strain. Growth in this system typically proceeds in the Stranski-Krastanov mode; that is, it begins with a few (approximately three) wetting layers but quickly reverts to three-dimensional (3D) island growth. The eventual coalescence of the islands unavoidR. Kern, G. Le Lay, and J. J. Metois, “Basic Mechanisms in the Early Stages of Epitaxy,” in Current Topics in Materials Science, Vol. 3, ed. E. Kaldis, North-Holland Publishing Company (1979). M. Copel, M. C. Reuter, E. Kaxiras, and R. M. Tromp, Phys. Rev. Lett. 63, 632 (1989).

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ably produces highly defective material. In the experiments of Cope1 et a/., the monolayer of As was first deposited to the Si substrate and continued to float during the growth of the Ge overlayers. The presence of the As monolayer in the system led to a drastic change in both the thermodynamics (balance of surface and interface energies) and the kinetics (surface and bulk mobility of deposited atoms), making it possible to grow a Ge film in a layer-by-layer fashion to unprecedented thicknesses for this system (several tens of layers). This remarkable behavior was termed the “surfactant effect” in semiconductor growth. Since then, a large number of experiments have confirmed this behavior in a variety of semiconductor systems (for early reviews see Refs. 3, 4, and 5). We digress momentarily to justify the terminology. The typical meaning of the word surfactant is unlike its meaning here, which has led to some debate about the appropriateness of the term in the context of semiconductor thin-film growth. As defined in scientific dictionaries, surfactant is used commonly in chemistry to describe “a substance that lowers the surface or interfacial tension of the medium in which it is dissolved,6” or “a material that improves the emulsifying, dispersing, wetting, or other surface modifying properties of liquids7” While these physical situations and the effect itself are different from the systems considered in the present review, we adopt the term surfactant to describe the effect of adsorbate layers in semiconductor thin-film growth for two reasons. First, for a reason of substance: There are indeed some similarities between adsorbate layers in semiconductor thin-film growth and the classical systems to which the term applies; namely, in both cases the presence of this extra layer reduces the surface tension and changes the kinetics of atoms or clusters of atoms (small islands in semiconductor surfaces, molecules in classical systems) at the surface. Second, for a practical reason: The term surfactant has essentially become accepted by virtue of its wide use in semiconductor growth, and the need for consistency with existing literature forces it upon us. These reasons, we feel, justify and legitimize the use of the term in the present context. Surfactants have been used to modify the growth mode of several systems, including metal layers in homoepitaxy and heteroepitaxy. In our view, the physics relevant to such systems is significantly different from that in the E. Tournie and K. H. Ploog, Thin Solid Films 231,43 (1993).

K.H. Ploog and E. Tournie, Di$us. Defect Data B, Sol. State Phenom. 32-33, 129 (1993). E. Tournie, K.H. Ploog, N. Grandjean,and J. Massies, I E E E 6 f h ConJ ZPRM (March 1994), 49. L. L. Schram, The Language of Colloid and Interfacial Science, University of Calgary,Alberta, Canada. See http://www.sti.nasa.gov/thesaurus/S/word15290.html, Official Scientific Thesaurus of NASA.

’

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DANIEL KANDEL AND EFTHIMIOS KAXIRAS OF METAL-ON-METAL GROWTH MEDIATED BY TABLE1. EXAMPLES SURFACTANTS

FILM/SUBSTRATE SURFACTANT Ag/Ag(llU Cu/Cu( 111) Fe/Au( 111) Fe/Cu( 11 1) Cu/Ru(OoO1) Co/Cu( 111)

Sb

0 Au

c+o 0 Sb Pb

REFERENCE

10, 11, 12, 13, 14, 15 16 17 18 19, 20, 21 22 23,24

case of semiconductors. Supporting this view is the fact that typically a small fraction of a monolayer is needed to produce the surfactant effect in metal growth, whereas in semiconductors typically a full monolayer of the adsorbate species (or what is the equivalent to full substrate coverage, depending on the surface reconstruction) is required. Presumably, in the case of metals a small amount of the adsorbate is sufficient to induce the required changes in surface kinetics'.' by altering nucleation rates or step-edge barriers (for some representative examples of metal-on-metal growth mediated by surfac-

*

V. Fiorentini, S. Oppo, and M. Schemer, Appl. Phys. A 60,399 (1995). M. Breeman, G. T. Barkema, M. H. Lanelaar, and D. 0. Boerma, Thin Solid Films 272, 195

(1996). l o H. A. Van der Vegt, H. M. van Pinxteren, M. Lohmeier, E. Vlieg, and J. M. C. Thornton, Phys. Rev. Lett. 68, 3335 (1992). G. Rosenfeld, R. Servaty, C. Teichert, B. Poelsema, and G. Comsa, Phys. Rev. Lett. 71, 895 (1993). l 2 J. Vrijmoeth, H. A. Van der Vegt, J. A. Meyer, E. Vlieg, and R. J. Behm, Phys. Rev. Lett. 72, 3843 (1994). l 3 J. A. Meyer, R. J. Behm, G. Rosenfeld, B. Poelsema, and G. Comsa, Phys. Rev. Lett. 73, 364 (1994). I4 J. A. Meyer, J. Vrijmoeth, H. A. van der Vegt, E. Vlieg, and R. J. Behm, Phys. Rev. 51, B14790 (1995). l 5 J. A. Meyer, H. A. van der Vegt, and J. Vrijmoeth, E. Vlieg, and R. J. Behm, Suif Sci. 355, L375 (1996). l6 W. Wulfhekel, N. N. Lipkin, J. Kliewer, G. Rosenfeld, L. C. Jorritsma, B. Poelsema, and G. Comsa, Suif Sci. 348,227 (1996). A. M. Begley, S. K. Kim, J. Quinn, F. Jona, H. Over, and P. M. Marcus, Phys. Rev. B48, 1779 (1993). N. Memmel, and E. Bertel, Phys. Rev. Lett. 75, 495 (1995). l9 H. Wolter, M. Schmidt, and K. Wandelt, SurJ Sci. 298, 173 (1993). 2o M. Schmidt, H. Wolter, M. Nohlen, and K. Wandelt, J . Vac. Sci. 7'echnol. A12, 1818 (1993). M. Schmidt, H. Wolter, and K. Wandelt, SurJ Sci. 307-309, 507 (1994).

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tants see Table 1). In semiconductors, on the other hand, the entire surface must be covered by the adsorbate in order for the required changes in energetics and kinetics to be obtained. Because of this fundamental difference, in the present article we will concentrate on the surfactant effect in semiconductor systems. In the following we first review the available information on the subject, from both the experimental (Section 11) and the theoretical (Section 111) points of view. We then present some theoretical arguments that we have advanced in an effort to create a comprehensive picture of the phenomenon (Section IV). Finally, we discuss our views on important issues that remain for future research on surfactants and comment on prospects for their use in the fabrication of electronic and optical semiconductor devices (Section V). II. Experimental Observations A wide range of systems have been studied where the surfactant effect was demonstrated. We classify these in three categories: growth of group-IV layers on group-IV substrates; growth of 111-V compounds on 111-V substrates; and mixed systems, including growth of elemental and compound systems on various substrates. This categorization has been inspired by the substrate features and the nature of the deposited species, which together determine the growth processes.

1. GROUP-IV FILMS ON GROUP-IV SUBSTRATES In the first category, the substrate is either Si or Ge (in different crystallographic orientations), on which combinations of different group-IV elements are deposited (Si, Ge, C). In these systems the deposited species are mostly in the form of single group-IV atoms. The adsorbate layers consist of monovalent (H), trivalent (Ga, In), tetravalent (Sn, Pb), pentavalent (As, Sb, Bi), and hexavalent (Te) elements, or noble metals (Au). These adsorbates remove the usual reconstruction of the surface (the different versions of the (2 x 1) reconstruction for the Si and Ge(100) surfaces, the (7 x 7) and the c(2 x 8) for the Si and Ge(ll1) surfaces, respectively) and produce simpler V. Scheuch, K. Potthast, B. Voigthder, and H. P. Bonze], Surj Sci. 318, 115 (1994). J. Camarero, L. Spendeler, G. Schmidt, K. Heinz, J. J. de Miguel, and R. Miranda, Phys. Rev. Lett. 73, 2448 (1994). 24 J. Camarero, T. Graf, J. J. de Miguel, R. Miranda, W. Kuch, M. Zharnikov, A. Dittschar, C. M. Schnieder, and J. Kirschner, Phys. Reo. Lett. 76, 4428 (1996). 22

23

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DANIEL KANDEL A N D EFTHIMIOS KAXIRAS

reconstructions that are chemically passivated. Characteristic examples are the (1 x 1) reconstruction (induced by H or As on the (111) surfaces); the ($ x $1 reconstruction on the (111) surfaces with either one adsorbate atom per unit cell (induced by Ga or In) or three adsorbate atoms per unit cell (induced by Sn, Sb, Pb, Au); the (2 x 1) reconstruction of the (100) surfaces (induced by the trivalent and pentavalent elements or by H in the monohydride phase); and the (1 x 1) reconstruction of the (100) surfaces (induced by Te or by H in the dihydride phase). In these reconstructions the dangling bonds of the substrate atoms are saturated by the additional electrons of the adsorbate atoms, producing low-energy, chemically unreactive surfaces. One important issue in these systems is the strain induced by the deposition of atoms with a different covalent radius than that of the substrate atoms. The normal growth mode in strained systems involves the formation of 3D islands that relieve the strain by relaxation at the island edges, either right from the initial stages of deposition (the so-called Volmer-Weber or 3D-island growth) or after the formation of a wetting layer (the Stranski-Krastanov growth). When surfactants are employed, it is possible to induce layer-by-layer growth in strained systems by avoiding the formation of 3D islands for film thicknesses much beyond what is obtained under normal conditions. The reduction of strain-induced islanding was, in fact, one of the early intended results of surfactant use and remains a goal pursued in several experimental studies. Even when surfactants are used, however, and the 3D-island mode of growth is suppressed, the strain in the heteroepitaxial film is still present and is usually relieved by the introduction of a network of misfit dislocations. The mechanism by which this happens is not known and remains to be analyzed by atomistic models. It is natural to expect that diffusion of group-IV adatoms on the adsorbate-covered surfaces will be relatively easy because of the chemical passivation by the surfactant layer. Such a situation may lead to a substantial increase of the diffusion length of adatoms on top of the surfactant layer. Indeed, it has been reported experimentally that in Ge on Si heteroepitaxy certain elements, like Ga, In, Sn, and Pb, lead to an increase in the width of the depleted zone around islands.” At the same time, however, it has also been found that other elements, like As, Sb, Bi, and Te, lead to a decrease in the width of the depleted zone.” These observations were interpreted as indicative that the former type of surfactant (group-I11 and group-IV atoms) enhance the diffusion length while the latter type (group-V and group-VI atoms) reduce the diffusion length. Moreover, this interpretation has been

25

B. Voigtlander, A. Zinner, A. Weber, and H. P. Bonze], Phys. Rev. B51, 7583 (1995).

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frequently invoked as an explanation of the suppression of 3D islanding in heteroepitaxy by group-V and group-VI surfactants. Since it is generally easier for group-V and group-VI elements to provide a chemically passive surface, we argue that the above interpretation may not be unique. In fact, we show in Section IV that even if it were true, it would not explain the surfactant effect either in homoepitaxy or in heteroepitaxy. We propose an alternative interpretation of the experimental results, according to which the diffusion length is mostly irrelevant. Instead, the essential question is whether or not the surfactant layer passivates island edges. Some surfactants (group-I11 and group-IV elements) cannot passivate island edges, which then act as strong sinks of newly deposited atoms, while other surfactants (group-V and group-VI elements) passivate island edges as well as terraces, so that island edges do not act as adatom sinks and the width of the depleted zone is reduced. We show that this interpretation is consistent with the experimentally observed surface morphologies and island densities in the presence of surfactants. It also explains why group-V and group-VI adsorbates suppress 3D islanding in heteroepitaxy. The systems studied experimentally that belong to this category are listed in Table 2. The relative simplicity of the surface reconstruction induced by the surfactant and the fact that the deposited species is mostly single group-IV atoms make the systems in this category the easiest to analyze from a microscopic point of view. Indeed, most atomistic scale models of the surfactant effect address systems in this category. 2. 111-vFILMS ON 111-vSUBSTRATES The second category consists of 111-V substrates on which combinations of other 111-V systems are deposited. The deposited species in this case are more complicated, since at least two types of atoms have to be supplied with different chemical identities. Under usual conditions the group-I11 species is deposited as single atoms, whereas the group-V species is deposited as molecules (dimers or tetramers) that have to react with the group-111 atoms and become incorporated in the growing film. This is already a significant complication in growth dynamics and makes the construction of detailed atomistic growth models considerably more difficult. Moreover, the usual surface reconstructions of these substrates are more complicated and depend on deposition conditions (temperature and relative flux of group-111 to group-V atoms). In the presence of surfactants both the surface reconstructions and the atomic motion are altered, but much less is known about the atomic-level details. The surfactants used in these systems include H, Be, B, In, Sn, Pb, As, Sb, and Te. In certain cases, the surfactant species is the same

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TABLE2. SYSTEMS M THE FIRSTCATEGORY OF SURFACTANT-MEDIATED SEMICONDUCTOR ON GROUP-IVSUBSTRATES GROWTH:GROUP-IVFILMS FILM/SUBSTRATE Si/Si( 100) Si/Si(111)

Ge/Si( 100)

Ge/Si( 111)

SURFACTANT

H Ga In Sn As Sb Au H B In Sn Pb As Sb

Bi Te H Ga In As Sb Bi AU

Si/Ge(100) Si/Ge( 111) Si, - xC,/Si( 100) Gel -,Si,/Si(l00)

Si/SiGe(100) Gel -,C,/Si(l00) Si/Si, -,Ge,(100) Sn/Ge( 100) Si, Ge,/Si( 100) Si, --x -,Ge,C,/Si(100)

H Sb Ge Sb

H Sn As Sb As Sb Sb H Sb Sb Sb

REFERENCE

,

26 25, 21, 28, 29, 30, 31 25,32 33, 34, 35 25, 28, 29, 30 25, 28, 29, 30 36,31 38, 39,40, 41, 42, 43, 44, 45 46 41, 48, 49 50 2, 51, 52, 53, 54, 55, 56, 51 51, 53, 58, 59, 60, 61,62, 63, 64,65, 66, 61, 41, 48, 68, 69, 10, 11, 12, 13, 14, 15, 16, 11, 18, 19, 80 19, 81, 82, 83 69, 10, 84, 85 86 25, 39, 81, 88, 89 25,90 25, 91, 92, 93 29, 13, 94, 95, 96, 91, 98, 99, 100, 101 25, 102, 103, 104 105 31 106 43, 69 101 108, 109 110,111 49, 112 113 114, 115 51 51 116 111 118 119 120

26

M. Cope1 and R. M. Tromp. Phys. Rev. Lett. 72, 1236 (1994).

”

H. Nakahara and M. Ichikawa, Appl. Phys. Lett. 61, 1531 (1992).

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M. Horn-von Hoegen, M. Pook, A. Al-Falou, B. H. Muller, and M. Henzler, Surf: Sci. 284, 53 (1993). 9 ' M. Horn-von Hoegen, A. Al-Falou, H. Pietsch, B. H. Muller, and M. Henzler, Surf: Sci. 298, 29 (1993). 98 M. Horn-von Hoegen, Appl. Phys. A59, 503 (1994). 9 9 M. Horn-von Hoegen, B. H. Muller, and A. Al-Falou, Phys. Rev. 850, 11640 (1994). l o o M. Horn-von Hoegen and M. Henzler, Physica Stat. Sol. A146, 337 (1994). ''I M. I. Larsson, W.-X. Ni, K. Joelsson, and G. V. Hansson, Appl. Phys. Lett. 65, 1409 (1994). B. Voigtlander and A. Zinner, Surf: Sci. 351, L233 (1996). '03 I. Davoli, R. Gunnella, R. Bernardini, and M. De Crescenzi, J . Electron Spectroscopy and Related Phenomena 83, 137 (1997). D. Reinking, M. Kammler, M. Horn-von Hoegen, and K. R. Hofmann, Appl. Phys. Lett. 71,924 (1997). 'H. Minoda, S. Sakamoto, and K. Yagi, Surf: Sci. 372, 1 (1997). H. Akazawa, J . Cryst. Growth 173, 343 (1997). lo' D.-S. Lin, H. Hong, T. Miller, and T.-C. Chiang, Surf: Sci. 312, 213 (1994). ' 0 8 P. 0. Pettersson, C. C. Ahn, T. C. McGill, E. T. Croke, and A. T. Hunter, Appl. Phys. Lett. 67, 2530 (1995). P. 0. Pettersson, C. C. Ahn, T. C. McGill, E. T. Croke, and A. T. Hunter, J . Vac. Sci. Technol. 814, 3030 (1997). R. Chelly, J. Werckmann, T. Angot, P. Louis, D. Bolmont, and J. J. Koulmann, Thin Solid Films 294, 84 (1997). ' I 1 H. Wado, T. Shimizu, M. Ishida, and T. Nakamura, J . Cryst. Growth 147, 320 (1995). A. Wakahara, K. K. Vong, T. Hasegawa, A. Fujuhara, and A. Saski, J . Cryst. Growth 151, 52 (1995). M.-H. Xie, A. K. Lees, J. M. Fernandez, J. Zhang, and B. A. Joyce, J . Cryst. Growth 173, 336 (1997). M. Li, Q. Cui, S. F. Cui, L. Zhang, J. M. Zhou, Z. H. Mai, C. Dong, H. Chen, and F. Wu, J . Appl. Phys. 78, 1681 (1995). S. Nilsson, H. P. Zeindl, D. Kruger, J. Klatt, et al., Materials Research Society Symposia Proceedings, eds. A. Zangwill, D. Jesson, D. Chambliss, and D. Clarke, Materials Research Society, Pittsburgh (1996). 'I6 H. J. Osten, E. Bugiel, and P. Zaumseil, J . Cryst. Growth 142, 322 (1994). H. Ohtani, S. Mokler, M. H. Xie, J. Zhang, and B. A. Joyce, Surf: Sci. 284, 305 (1993). W. Dondl, P. Schittenhelm, and G. Abstreiter, Thin Solid Films 294, 308 (1997). 'I9 H. Presting, U. Menczigar, and H. Kibbel, J . Vac. Sci. Technol. El l , 1110 (1992). 12' E. T. Croke, A. T. Hunter, P. 0. Pettersson, C. C. Ahn, and T. C. McGill, Thin Solid Films 294, 105 (1997). 96

'''

as one of the atoms in the growing film (such as In in growth of InAs on GaAs), or one of the atoms in the substrate (such as Sb in growth of InAs on AlSb). Strain effects are important in these systems as well. Building high-quality 111-V heterostructures has been one of the goals of many technologically oriented studies, and the use of surfactants has been beneficial in reducing the problems associated with strain. However, the more complex nature of these systems has prevented detailed analysis of the type afforded in group-IV systems. A compilation of experimental results for this category is given in Table 3.

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TABLE 3. SYSTEMS IN THE SECONDCATEGORY OF SURFACTANT-MEDIATED SEMICONDUCTOR GROWTH:111-V FILMS ON 111-V SUBSTRATES FILM/~UBSTRATE

SURFACTANT

REFERENCE ~~~~

GaAs/GaAs( 100)

InAs/GaAs( 100) AlGa As/AIGaAs( 100) InGaAs/GaAs( 100) GaAs/InP( 100) InGaAs/InP( 100) GaAs/GaAs( 111) GaAs/GaInAs( 100) InAs/GaInAs( 100) GaN/GaN( 100) GaN/AIGaN( 100) InAs/AIInAs(100) InAs/AISb( 100) InAs/InPSb( 100) InAs/InGaAs( 100)

I*’

H Sn Pb Te H In Be Sb

Sn Te H H In Te In As SiKH,), In Sb Sb

In

~

~

121, 122, 123 124 124 124, 125 126, 127 128 129 130, 131 132 133, 134, 135, 136 127 137 138 139 140 141 142, 143 144, 145 146 146 147, 148, 149

Y. Okada, T. Sugaya, S. Ohta, T. Fujita, and M. Kawabe, Jap. J. Appl. Phys. 34,238 (1995). Y.Okada and J. S. Harris Jr., J. Vac. Sci. Technol. B14, 1725 (1996). M. Kawabe, J. Cryst. Growth 150, 370 (1995).

J. Massies and N. Grandjean, Phys. Rev. B48, 8502 (1993). N. Grandjean and J. Massies, Phys. Rev. 853, 13231 (1996). J. C. Yong, Y. Okada, M. Kawabe, J. Cryst. Growth 150,497 (1995). Y. J. Chun, Y. Okada, and M. Kawabe, Jap. J. Appl. Phys. 35, L1689 (1996). J. Behrend, M. Wassermeier, and K. H. Ploog, J. Cryst. Growth 167, 440 (1996). J. E. Cunningham, K. W. Goosen, W. Jan, M. D. William, J. Vac. Sci. Technol. B13, 646 (1995). R. Kaspi, D. C. Reynolds, K. R. Evans, E. N. Taylor, Proceedings of the 2lst International Symposium on Compound Semiconductors, San Diego (1994). 1 3 1 R. Kaspi, K. R. Evans, D. C. Reynolds, J. Brown, and M. Skowronski, Materials Research Society Symposia Proceedings, eds. E. A. Fitzgerald, J. Hoyt, K.-Y. Cheng, J. Bean, Materials Research Society, Pittsburgh (1995), 79. 13’ G. S. Petrich, A. M. Dabiran, J. E. MacDonald, and P. I. Cohen, J. Vac. Sci. Technol. 9, 2150 (1991). 1 3 3 J. Massies, N. Grandjean, and V. H. Etgens, Appl. Phys. Lett. 61,99 (1992). 134 N. Grandjean, J. Massies, and V. H. Etgens, Phys. Rev. Lett. 69, 796 (1992). N. Grandjean, J. Massies, C. Delamarre, L. P. Wang, A. Dubon, and J. Y. Laval, Appl. Phys. Lett. 63, 66 (1993).

”’

SURFACTANT EFFECT IN SEMICONDUCTOR THIN-FILM GROWTH

23 1

3. MIXEDFILMAND SUBSTRATE SYSTEMS

The final category consists of mixed systems in which group-IV films are grown on 111-V substrates (for example, Si on GaAs) or vice versa (for example, GaN on Si). In these systems, in addition to the usual strain effects, one has to consider polarity effects, caused by the fact that at the interface different types of atoms are brought together and their dangling bonds contain different amounts of electronic charge that do not add up to the proper value for the formation of covalent bonds. It is possible that the surfactant layer plays an important role in reducing polarity problems as well as in modifying the energetics and in suppressing strain effects, as it does in the previous two categories. Since the substrate and the thin film are rather different for systems in this category, we include in the same category a number of odd systems that involve the presence of insulating buffer layers (like CaF, in the growth of Ge on Si substrates) and the growth of metal layers (like In on Si, In on GaAs, Fe on Ge, Fe and Au on GaAs, and Ag on Si) or silicide layers (like CoSi, on Si), as well as the growth of technologically important semiconductors on insulators (like GaN on sapphire). All these cases are important for device applications, and it is interesting to study how surfactants can be employed to improve the quality of growth. However, the complexity of the structures involved and the several different species of atoms present make

C. W. Synder, B. G. Orr, N. Grandjean, and J . Massies, Phys. Rev. Lett. 70, 1030 (1993). R. R. Lapierre, B. J. Robinson, and D. A. Thompson, J . Vac. Sci. Technol. B15, 1707 (1997). 13' M. Ilg, D. Eissler, C . Lange, and K. Ploog, Appl. Phys. A56, 397 (1993). 139 C. Delamarre, J. Y. Laval, L. P. Wang, A. Dubon, and G. Schiffmacher, J . Cryst. Growth 117,6 (1997). E. Tournie and K. H. Ploog, J . Cryst. Growth 135,97 (1994). 141 G. Feuillet, H. Hamaguchi, K. Ohta, P. Hacke, H. Okumura, and S . Yoshida, Appl. Phys. Lett. 70, 1025 (1997). 14* S. Tanaka, S. Iwai, and Y. Aoyagi, Appl. Phys. Lett. 69, 4096 (1996). 143 S. Tanaka, H. Hirayama, S. Iwai, and Y. Aoyagi, Materials Research Society Symposia Proceedings, eds. F. A. Ponce, T. D. Moustakas, I. Akasaki, and B. A. Monemar, Materials Research Society, Pittsburgh (1997), 135. 144 E. Tournie, 0. Brandt, K. H. Ploog, and M. Hohenstein, Appl. Phys. ,456, 91 (1993). 145 E. Tournie and K. H. Ploog, Appl. Phys. Lett. 62, 858 (1993). 146 J. Tuemmler, J. Woitok, J. Hermans, J. Geurts, P. Schneider, D. Moulin, M. Behet, and K. Heime, J . Cryst. Growth 170,772 (1997). 147 E. Tournie, 0. Brandt, C. Giannini, K. H. Ploog, and M. Hohenstein, J . Cryst. Growth 127, 765 (1992). 14* E. Tournie, N. Grandjean, A. Trampert, J. Massies, and K. H. Ploog, J . Cryst. Growth 150, 460 (1995). 149 K. H. Ploog, A. Trampert, and E. Tournie, Physica Stat. Sol. ,4146,353 (1994). 136 13'

232

DANIEL KANDEL AND EFTHIMIOS KAXIRAS TABLE4. SYSTEMS IN THE THIRDCATEGORY OF SURFACTANT-MEDIATED SEMICONDUCTOR GROWTH: MIXEDFILMSAND SUBSTRATES FILM/SUBSTRATE Si/GaAs( 1 1 l), (31 1) Si/GaAs( 100) Ge/InP( 100) Ge/GaF2/Si(100), (1 11) GaAs/Ge( 100) CoSi,/Si( 100) Ag/Si( 1 11) In/Si( 1 11) In/GaAs( 100) Fe/Ge( 100) Fe,Au/GaAs( 100) MnS b/GaAs( 100) GaN/Si( 11 1) GaN/SiC( 100) GaN/AI,O,

SURFACTANT REFEFSNCE H As Sb B H As Sb H ' Sb S Fe,Au H As

As Bi

150, 151 152, 153 154, 155 156 157 158 159 160 161 162 163 164 165 166 167

Z. Peng and Y. Horikoshi, Appl. Phys. Lett. 70, 604 (1997). J. M. Zhang, M. Cardona, Z. L. Peng, and Y. Horiskoshi, Appl. Phys. Lett. 71,1813 (1997). 1 5 2 A. R. Avery, D. M. Holmes, J. L. Sudijono, T. S. Jones, M. R. Fahy, and B. A. Joyce, J . Cryst. Growth 150,202 (1995). 1 5 3 J. L. Sudijono, A. R. Avery, B. A. Joyce, and T. S. Jones, J. Mat. Sci. 7, 333 (1996). 154 D. Rioux and H. Hochst, J. Vac. Sci. Tech. A10, 759 (1991). 1 5 5 D. Rioux and H. Hochst, Phys. Rev. B46,6857 (1992). 1 5 6 C.-C. Cho, H.-Y. Liu, L. K. Magel, and J. M. Anthony, Appl. Phys. Lett. 63, 3291 (1993). 15' Y. Okada, J. S. Harris, Jr., A. Sutoh, and M. Kawabe, J. Cryst. Growth 175, 1039 (1997). 15' V. Scheuch, B. Voigtlhder, and H. P. Bonze], Su$ Sci. 381, L546 (1997). 1 5 9 K.-H. Park, J. S. Ha, S.-J. Park, and E.-H. Lee, Su$ Sci. 380, 258 (1997). lL0 P. Leisenberger, H. Ofner, M. G. Ramsey, and F. P. Netzer, SurJ Sci. 383, 25 (1997). 1 6 1 S. Schintke, U. Resch-Esser, N. Esser, A. Krost, W. Richter, and B. 0. Fimland, SurJ Sci. 377-379, 953 (1997). I L 2 G. W. Anderson, A. P. Ma, and P. R Norton, J . Appl. Phys. 79, 5641 (1996). K. Sano, and T. Miyagawa, Appl. Su$ Sci. 60-61,813 (1992). H. Ikekame, Y. Yanase, T. Ibarashi, T. Saito, Y. Morishita, and K. Sato, J. Cryst. Growth 173, 218 (1997). 1 6 5 G. Mendoza-Diaz, K. S. Stevens, A. F. Schwartzman, and R. Beresford, J. Cryst. Growth 178, 45 (1997). 166 G. Feuillet, P. Hacke, H. Okumura, H. Hamaguchi, K. Ohta, K. Balakrishnan, and S. Yosida, Materials Research Society Symposia Proceedings, eds. F. A. Ponce, T. D. Moustakas, I. Akasaki, B. A. Monemar, Materials Research Society, Pittsburgh, (1997), 257. 16' R. Klockenbrink, Y. Kim, M. S. H. Leung, C. Kisielowski, J. Kruger, G. S. Sudhir, M. Rubin, and E. R. Weber, Materials Research Society Symposia Proceedings, eds. C. R. Abernathy, H. Amano, J. C. Zolper, Materials Research Society, Pittsburgh (1997), 75. 150

15'

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233

the detailed analysis of systems in this category rather difficult. We tabulate the experimental information for systems in this category in Table 4.

111. Theoretical Models

Following the experimental observations, a number of theoretical models have been studied in order to understand and explain the surfactant effect in semiconductor growth. We divide these models in three categories. In the first category we place models that have concentrated on the microscopic aspects, attempting to understand the atomic-scale features and processes involved in this phenomenon; models of this type typically employ sophisticated quantum-mechanical calculations of the total energy in order to evaluate the relative importance of the various structures and to determine the relevant activation energies involved in the kinetic processes. In the second category we place models that are more concerned with the macroscopic aspects of the surfactant effect, such as island morphologies and distributions as well as the effects of strain, without attempting to explain the details of the atomistic processes (although these may be taken into account heuristically). In the third category we place models that attempt to combine both aspects -that is, they try to use realistic descriptions of the atomistic processes as the basis for macroscopic models. Evidently, this last type is the most desirable but also the most difficult to construct. We review these three categories in turn.

4. MICROSCOPIC MODELS Initial attempts at understanding the microscopic aspects of surfactantmediated growth focused on the thermodynamic aspects; that is, they strived to justify why it is reasonable to expect the surfactant layer to float on top of the growing film. This was investigated by calculating energy differences between configurations with the surfactant layer buried below layers of the newly deposited atoms and configurations with the surfactant on top of the newly deposited atoms.’ These energetic comparisons, based on first-principles calculations employing density functional theory, established that there is a strong thermodynamic incentive for keeping the surfactant layer on top of the growing film. In a similar vein, calculations by Kaxiras’ 68 established that certain surfactants are more likely to lead to layer-by-layer growth than others, E. Kaxiras, Europhys. Lett. 21, 685 (1993).

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DANIEL KANDEL A N D EFTHIMIOS KAXIRAS

while a simplistic analysis of their chemical nature does not reveal such differences. These calculations used different types of surfactants on the same substrate and considered the relative energies of the various surface reconstructions induced by the surfactant layer. Specifically, three types of group-V atoms were considered as surfactants-p, As, and Sb- on the Si(l11) surface for Ge or Si growth. The similar chemical nature of the three elements would argue for very similar surfactant behavior; however, the total energy calculations indicated that the three elements give significantly different reconstructions, some of which would lead to relatively easy floating of the surfactant layer while others would hamper this process. This is related to the manner in which, in a given reconstruction, the surfactant atoms are bonded to the substrate. For instance, in the energetically preferred reconstructions, the P and As atoms are bonded to the substrate with three strong covalent bonds each, while Sb atoms are bonded to the substrate with only one strong covalent bond per adsorbate atom. Based on these comparisons, Kaxiras proposed that Sb would work well on this substrate as a surfactant while P and As would not.'683169This fact was subsequently verified e~perimentally.~' A study of the same system by Nakamura et ~ 2 1 . ' ~ ' (the Si(ll1) substrate with Sb as surfactant for Ge growth), based on the discrete variational approach and the cluster method to model the surface, reported that the presence of the surfactant strengthens the bonds between the Ge atoms on the surface. This effect, it was argued, leads to nucleation of stress-relieving dislocations at the surfaces, which is beneficial for layered growth of defect-free films. In this analysis neither the defects themselves nor any exchange and nucleation mechanisms were considered explicitly. Moreover, the bond-strengthening arguments are of a chemical nature, which may be useful in a local description of chemical stability but sheds little light on the dynamics of atoms during surfactant-mediated growth. The chemical nature of Sb bonding on the Si(100) and the Ge(100) substrates was also investigated by Jenkins and Sri~astava."~~' 72 In this work, first-principles density functional theory calculations were employed to determine the structure and the nature of bonding of Sb dimers in the (2 x 1) reconstruction, which, though interesting in itself, provides little direct insight into the process of surfactant-mediated growth. The theoretical models considered so far addressed the problem of surfactant-mediated growth by considering what happens at the microscopic

I7O

E. Kaxiras, Mat. Sci. Eng. B30, 175 (1995). J. Nakamura, H. Konogi, and T. Osaka, Jap. J . Appl. Phys. 35, L441 (1996). S. J. Jenkins and G. P. Srivastava, Surf Sci. 325-354, 411 (1996). S. J. Jenkins and G. P. Srivastava, Phys. Rev. B56, 9221 (1997).

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level, but for entire monolayers- that is, by imposing the periodicity of the reconstructed surface in the presence of the surfactant. Subsequent atomistic models studied the equilibrium configurations and dynamics of individual adsorbed atoms or dimers, which is more appropriate for understanding the nature of growth on the surfactant-covered surface. A first example was an attempt by Yu et a1.173*174*175 to justify how newly deposited Ge atoms on the As-covered Si(100) surface exchange place with the surfactant atoms in order to become embedded below the surfactant layer. In these calculations, based on total energy comparisons obtained from density functional theory, the metastable and stable positions of Ge dimers were established, indicating possible paths through which the newly deposited Ge atoms can be incorporated under the surfactant As layer. However, no explicit pathways were determined, and therefore no activation energies that might be relevant to growth kinetics were established. Furthermore, even though specific mechanisms for growth of needle-like islands by appending Ge dimers to a seed were discussed, the lack of calculated energy barriers for the exchange process and a large number of unproven assumptions involved in the proposed mechanisms means that they are of little help in understanding the claimed that Ge dimers are surfactant effect. For instance, Yu et actually situated between, instead of on top of, As dimer rows, while in their proposed island growth mechanism they employed configurations that involve Ge dimers on top of the As dimer rows. A similar type of analysis by O h n ~ , ' ~ also ~ , 'using ~ ~ density functional theory calculations of the total energy, was reported for Si-on-Si(100) homoepitaxy using As as surfactant. Ways of incorporating the newly deposited Si dimers below the As layer were considered by studying stable and metastable positions and the rebonding that follows the exchange process. Again, though, actual exchange pathways and the corresponding activation energies relevant to growth kinetics were not considered. In this study it was shown explicitly how exchange of isolated Si dimers on top of the As layer is not exothermic, while the presence of two Si dimers leads to an energetically preferred configuration after exchange. This fact was used to argue that the Si dimer interactions are responsible for both their mutual repulsion and the initiation of the exchange. It appears, however, that these two effects- the strong repulsion of ad-dimers and the requirement of their B. D. Yu and A. Oshiyama, Phys. Rev. Lett. 72, 3190 (1994). B. D. Yu, T. Ide, and A. Oshiyama, Phys. Rev. 850, 14631 (1994). '15 B. D. Yu and A. Oshiyama, Proceedings of the 22nd International Conference on the Physics of Semiconductors, vol. 1, ed. D. J. Lockwood, World Scientific, Singapore (1999, 668. 176 T. Ohno, Phys. Rev. Lett. 73, 460 (1994). 177 T. Ohno, Thin Solid Films 272, 331 (1996). 173

174

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DANIEL KANDEL AND EFTHIMIOS KAXIRAS

presence at neighboring sites for the initiation of exchange-are incompatible as far as growth is concerned. Both Yu et al.173.174*175 and Ohno'76*177 dealt with mechanisms in which the basic unit involved in the exchange process is a deposited dimer, as was originally suggested by Tromp and Reuter.5 4 An interesting microscopic study of surfactant mechanisms was reported by Kim et al.'78~'79*'80 In this work, first-principles molecular dynamics simulations were employed to investigate the effect of Sb atoms at step edges on the Si(100) surface for Si homoepitaxy. This study examined the effect of Sb dimers on the step-edge barriers (also referred to as Schwoebel-Ehrlich barriers,'8'.'82 for which we adopt here the acronym SEB, which is both descriptive and referential). These are extra barriers to adatom attachment to the step edge when the adatom arrives from the upper terrace, compared to the barriers for diffusion on the flat terraces. The authors found that the presence of Sb at the step edge gives a significant SEB for the attachment of a single Si atom but a much smaller SEB for attachment of a Si dimer by the push-over mechanism (in which the Si dimer at the upper terrace pushes the Sb dimer at the step edge over by one lattice constant and thus becomes incorporated in the bulk). This relative suppression of the SEB for dimer attachment leads to layer-by-layer growth as opposed to 3D island growth; consequently, Kim et al. argued, the presence of the surfactant Sb dimer at the step edge leads to layered growth. This is an interesting suggestion, but it remains to be proven correct for the system under consideration. Specifically, it is not clear whether a configuration with Sb dimers only at step edges of the Si(100) surface is stable. Typically, an entire monolayer is needed for the surfactant effect in similar systems, and the precise coverage is a crucial aspect of the effect. If the surfactant coverage is different from that assumed in the model of Kim et al., the atomic processes at the step edge may be very different, leading to a different picture of the effect. Furthermore, kinetic Monte-Carlo studies are required to establish that the calculated energy barriers can actually lead to the predicted mode of growth, 178 E. Kim, C. W. Oh, Y. H. Lee, K. Y. Lim, and H. J. Lee, Proceedings of the 23rd International Conference on the Physics of Semiconductors, vol. 2, eds. M. Schemer and R. Zimmermann, World Scientific, Singapore (1996), 1019. C. W. Oh, E. Kim, and Y. H. Lee, Phys. Rev. Lett. 76, 776 (1996). E. Kim, C. W. Oh, Y. H. Lee, Material Research Society Symposia Proceedings, eds. S . M. Prokes, 0. J. Glembocki, S. K. Brierly, J. M. Gibson, and J. M. Woodall, Materials Research Society, Pittsburgh (1997), 135. R. L. Schwoebel and E. J. Shipsey, J. Appl. Phys. 37,3682 (1996); R. L. Schwoebel, J. Appl. Phys. 40,614 (1969). G. Ehrlich and F. G. Huda, J. Chem. Phys. 44, 1039 (1966); S. C. Wang and G. Ehrlich, Phys. Rev. Lett. 70, 41 (1993).

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since the density of Si adatoms (determined by flux, diffusion rate, and attachment-detachment rates) also influence the growth process. Another detailed study of activation energies for diffusion and exchange This processes in surfactant-mediated epitaxy was reported by KO et study was also based on first-principles calculations of the energetics and addressed Si epitaxy on Si(100) with As acting as surfactant. In this work it was established that the exchange of a Si adatom with a sublayer As site involves an energy barrier of 0.1 eV, which is considerably lower than the energy barrier for diffusion (of order 0.5 eV) or the energy barrier for dimer exchange (of order 1.0 eV) which had been invoked as a possible mechanism in ,earlier studies of the same system (see Refs. 54, 173, 174, 175, 176, and 177). This is a very interesting suggestion, but falls short of providing a complete picture of the surfactant effect. Specifically, it is not clear how a single exchange step of the type investigated by KO et al. can lead to a configuration that will nucleate the next layer of the crystal. This process may well involve additional important steps with different activation barriers, so that the barrier calculated, though important and interesting, may not be the determining step in the growth process. In fact, KO et al. found that exchange of two individual Si atoms at neighboring sites leads to the formation of a protruding As dimer, which acts as a seed for further growth. This protruding As dimer binds additional Si adatoms and leads to the formation of a Si dimer, which eventually undergoes site exchange with a neighboring As dimer with an energy barrier of 1.1 eV. It then appears that it is this last step that is the determining one in growth, which leads back to the dimer-exchange picture discussed earlier, albeit now with a more detailed picture of how this process may be initiated by the barrierless exchange of single Si adatoms. Two separate studies of growth on 111-V surfaces addressed the surfactant ~ dimer-exchange effect in these systems. In the first, by Miwa et ~ l . , ' *the mechanism on 111-V surfaces, using Te as the surfactant, was investigated using first-principles calculations. The authors found that InAs growth on GaAs(100) proceeds by complete dimer exchange between the In and Te layers on the As-terminated surface, while on the In-terminated surface the exchange between the Te layer and an overlayer of As is only partial. The second study, by Shiraishi and examined the equilibrium conY.-J. KO,J.-Y. Yi, s.-J.Park, E.-H. Lee, and K. J. Chang, Phys. Rev. Lett. 76, 3160 (1996). R. H. Miwa, A. C. Ferraz, W. N. Rodrigues, and H. Chacham, Proceedings of the 23rd International Conference on the Physics of Semiconductors, vol. 2, eds. M. Scheffler and R. Zimmermann, World Scientific, Singapore (1996), 1107. K. Shiraishi and T. Ito, Proceedings ofthe 23rd International Conference on the Physics of Semiconductors, vol. 2, eds. M. Scheffler and R. Zimmermann, World Scientific, Singapore (1996), 1103.

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figurations of adatoms on the GaAs( 100) surface with different As coverages, using first-principles total-energy calculations. This study concluded that preadsorbed Ga atoms create a "self-surfactant" effect by significantly influencing the adsorption energy of As dimers at various sites on the surface. In this analysis, energy barriers for diffusion and exchange mechanisms were not taken into account; consequently, the interpretation of actual growth processes was limited. The most detailed study of actual atomistic mechanisms for diffusion and exchange was reported by Schroeder et ~ 1 . ' ~ ' This work examined the motion of Si adatoms on the As-covered Si(111) surface, using first-principles total-energy calculations. The authors reported a very interesting pathway for exchange between the additional Si atom and an As surfactant atom with an energy barrier of only 0.27 eV. This is comparable to the diffusion barrier for the Si atom on top of the As layer, calculated to be 0.25 eV. The Si atom can undergo the reverse of the exchange process, and by so doing it can get on top of the As surfactant layer, a process that involves an energy barrier of 1.1 eV, according to the results of Schroeder et al. This leads to a rather complex sequence of events, with Si adatoms arriving at the surfactant covered substrate, diffusing, exchanging, undergoing the opposite of the exchange process and diffusing again, with the relevant energy barriers. The possibility of the opposite of the exchange process was first explicitly introduced in the work of Kandel and Kaxiras,'" where it was called de-exchange. We adopt this term in the following as more descriptive of the opposite of the exchange process, since this undoes the effect of an exchange step rather than repeat it, as the term re-exchange (used in the work of Schroeder et al.) might suggest. The de-exchange process was shown by Kandel and Kaxiras to be crucial in maintaining the layer-by-layer growth mode in the presence of the surfactant (see more details below). They found that the de-exchange process has a higher activation energy (1.6 eV) than either exchange (0.8 ev) or diffusion (0.5 eV), although this was established by considering the exchange or de-exchange of entire monolayers of newly deposited atoms on top of the surfactant layer. Schroeder et al. showed that the same energy ordering also is valid for individual adatoms on top of the surfactant layer, but the actual barriers for individual adatoms are lower (1.1 eV, 0.27 eV, and 0.25 eV for de-exchange, exchange, and diffusion, respectively). This establishes unequivocally the importance of the de-exchange process. K. Shirasihi and T. Ito, Phys. Rev. B57, 6301 (1998).

'" K. Schroeder, B. Engels, P. Richard, and S. Bliigel, Phys. Rev. Lett. 80,2873 (1998). 18'

D. Kandel and E. Kaxiras, Phys. Rev. Lett. 75,2742 (1995); E. Kaxiras and D. D. Kandel,

Appl. SurJ Sci. 102, 3 (1996).

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What is lacking from the work of Schroeder et al. is a sequence of steps that can actually lead to the formation of the next layer of deposited material. Specifically, even after the single Si adatom has exchanged positions with a surfactant As atom, the system is not in a configuration from which the repeated sequence of similar steps can lead to the formation of a new layer. In the system studied by Schroeder et al. this process may be quite complicated, since the Si(11 1) surface consists of double layers, the formation of which may involve additional energy barriers that supersede the one determined for the exchange of a single adatom. 5. MACROSCOPIC MODELS

There is a debate in the literature on whether the suppression of 3D islanding by surfactants in heteroepitaxy is an equilibrium effect or a kinetic one. While most researchers in the field take the kinetic approach, there has been some effort to explain the surfactant effect using thermodynamic considerations. According to the thermodynamic approach, the equilibrium state of the newly deposited material in the presence of a surfactant layer is a smooth flat film. The underlying assumption behind kinetic models is that even with surfactants the true equilibrium state of the system is that of 3D islands. The role of surfactants in this case is to induce layer-by-layer growth kinetically and to make the approach to equilibrium longer than realistic time scales. We will first give examples of the thermodynamic approach to the surfactant effect and then elaborate on some kinetic models. Kern and Miiller'89 calculated the free energy of the formation of a crystal of material A stretched to be coherent with a substrate of material B. They took into account effects of surface energy as well as surface stress and obtained the equilibrium shape of the crystal by minimizing its free energy with respect to its height and width. In their view, surfactants may reduce surface stress and surface energy, and hence lead to flatter islands and perhaps even to wetting of the substrate by the deposited material (which happens when the equilibrium island height vanishes). Kern and Muller viewed such surfactant-induced wetting as a transition from 3D growth to 2D layer-by-layer growth. This Kern-Muller criterion may indicate whether the effect of a certain surfactant is in the right direction to suppress 3D islanding. However, the researchers did not consider the possibility of 3D growth when the deposited material wets the substrate (Stranski-Krastanov growth mode). They also ignored strain relaxation, which reduces the cost of 3D-island formation. Thus, the R. Kern and P. Miiller, J. Cryst. Growth 146, 193 (1995).

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question of whether the surfactant effect is a purely thermodynamic one is left unanswered. A different equilibrium argument was proposed by Eaglesham et al.?' who argued that surfactants change the surface energy anisotropy, leading to the suppression of 3D islanding. They examined experimentally islands of Ge on Si(100) films with and without surfactants, and found that their equilibrium shape changes radically in the presence of surfactants and depends strongly of the specific surfactant used. For example, Sb as a surfactant favors (100) facets, whereas In favors (31 1) facets. Eaglesham et al. advanced the idea that if the surfactant favors facets in the same orientation as the substrate, the equilibrium shape of the islands generated will be flat. This will lead to earlier coalescence of islands and will enhance layer-by-layer growth. The mechanism proposed by Eaglesham et al. may have a significant impact on the growth mode, but it cannot be the main explanation of the surfactant effect, since the equilibrium morphologies observed in their experiments include 3D islands. Therefore, in their explanation of the surfactant effect they supplemented the equilibrium consideration with a kinetic one- that is, the reduction of the diffusion length induced by surfactants. It seems quite difficult to explain the surfactant effect on the basis of thermodynamics alone. For this reason most researchers in the field make the assumption that surfactants suppress 3D islanding kinetically. Markov's work 190.19 1,192 is an example of such a kinetic model. He developed an atomistic theory of nucleation in the presence of surfactants. The main results of this work are expressions for the nucleation rate and saturation density of islands. These quantities depend crucially on the difference between the energy barrier for adatom diffusion on top of the surfactant layer and the barrier for diffusion on a clean surface. If this difference is positive, surfactants decrease the diffusion length for adatoms and the saturation density of islands rises sharply. Such an anomalously high island density in the presence of surfactants has been seen experimentally in various systems, and it is viewed by many researchers (see Refs. 25, 28, 29, 52, 54, 70, 125, 126, 183, and 193) as the main mechanism by which surfactants change the growth mode of the film and suppress 3D islanding in heteroepitaxy. We will show in Section IV that this mechanism does not explain the surfactant effect. An entirely different approach was taken by B a r a b a ~ i . ' ~ ~ .Rather '~' than

I9O 191

19' 193

I. Markov, Phys. Rev. BSO, 11271 (1994). I. Markov, Phys. Rev. B53,4148 (1996). I. Markov, Mat. Chem. Phys. 49, 93 (1997). A. Sakai, T. Tatsumi, and K. Ishida, Phys. Rev. 847, 6803 (1993).

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look at the kinetics of the system on the atomic length scale, he viewed the growing film on a much coarser scale. He represented the local height of the film and the local width of the surfactant layers as continuous fluctuating ~ ~the basis fields, in the spirit of the KPZ model of kinetic r 0 ~ g h e n i n g . lOn of the relevant symmetries of the system, he wrote down a set of coupled differential equations that describe the dynamics of these two fields. The quantity of interest in this approach is the width of the film surface and its dependence on system size. Typically, such a theory would predict a rough surface where the width diverges with system size. Barabasi found that surfactants can induce a flat phase where the surface width does not diverge with system size, and he associated this phase with a layer-by-layer growth mode. It is interesting to see that a theory on such a macroscopic length scale can capture effects that depend critically on processes that occur on an atomic scale. The drawback of this theory is that it is not clear what role lattice mismatch and strain play in the kinetics of the system. Also, in the rough phase the model predicts a self-similar structure for the surface. Experimentally, however, the morphology of a surface with 3D islands is not self-similar, and it is not clear whether this continuum theory can describe the experimental morphologies. Barabasi and Kaxiras19’ extended this model to include two different dynamical fields -one representing the surfactant layer; the other the surface film layer. This allowed an investigation of whether subsurface diffusion, which had been neglected in the previous model, could change the behavior. It was found that subsurface diffusion essentially always leads to roughening and, if operative in real systems, would prevent layer-by-layer growth. Most models of surfactant-mediated epitaxial growth emphasize the significance of adatom diffusion for the determination of the growth mode. Another atomic process of importance is attachment and detachment of adatoms from island edges. In fact, in Section IV we develop a model according to which surfactants suppress 3D islanding by passivating island edges, thus suppressing adatom detachment. It is therefore of interest to investigate the influence of island-edge passivation on surface morphology. Kaxiras first introduced the idea of island-edge passivation by the surfactant’98*’99 and carried out kinetic Monte-Carlo simulations on a very A.-L. Barabasi, Phys. Rev. Lett. 70,4102 (1993). A.-L. Barabasi, Fractals 1, 846 (1993). M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. 56, 889 (1986). 19’ A.-L. Barabasi and E. Kaxiras, Europhys. Lett. 36, 129 (1996). 19’ E. Kaxiras, Thin Solid Films 272, 386 (1996). 199 E. Kaxiras “Simulation of Semiconductor Growth Mechanisms in the Presence of Adsorbate Layers,” in Computational Materials Science, ed. C. Y . Fong, World Scientific, Singapore (1997). 194 195

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simple model to show that it can lead to morphologies compatible with experimental observations. Kandel also carried out a study of island-edge passivation effects.200 He investigated a simple model of submonolayer homoepitaxial growth in the framework of rate equation theory, using the critical island approximation (only islands of more than i* atoms are stable, while smaller islands decay). The main result of this work is that the island density scales with flux, F, as FX with x = 2i*/(i* + 3) when island edges are passivated, while x = i*/(i* + 2) without island-edge passivation. This conclusion is important because the exponent x can be measured experimentally and one can learn from its value whether island-edge passivation is operative or not in the experimental system at hand. For example, a value of x > 1 can occur only if the surfactants passivate island edges. Kandel's theory relies on a somewhat oversimplified picture of submonolayer growth, and the conclusions are yet to be verified with a more rigorous theory or by detailed simulations of the growth process. IV. The Diffusion-De-Exchange-Passivation Model

To our knowledge the only attempt to construct a comprehensive model that includes both the microscopic aspects of atomic motion and a realistic description of the large length-scale evolution of the surface morphology was reported by the present authors.lS8 The work of Zhang and Lagally"' was another attempt to link the microscopics and the macroscopics of the effect of surfactants on thin-film growth. However, their work discussed homoepitaxial growth of metals, a subject interesting in its own right but beyond the scope of the present review article.

6. GENERAL CONSIDERATIONS Before we embark on the construction of the theoretical we briefly review the relevant experimental information since we do not claim that a single model captures every type of surfactant-mediated growth mode. We focus here on growth of elemental or compound semiconductors, in which a single species of atom controls the diffusion and exchange (or deexchange) processes, and the surfactant produces a chemically passivated surface. We take as the canonical case a group-IV substrate (examples are Si(100) and Si(ll1)) and a group-I11 or groupV surfactant (these are zoo D.U e ! , Phys. Lett. 78,499 (1997). zol Z. Zhang and M.G. Lagally, Phys. Reu

Lett. 72,693 (1994).

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actually the systems that have been studied most extensively experimentally, as is evident from Tables 2 through 4). It appears that a full monolayer of surfactant coverage is required for growth of high-quality semiconductor crystals. This is different from the case of surfactant effects in the growth of metals, where a small amount of surfactant (typically a few percent of a monolayer coverage) is sufficient. The most direct evidence on this issue was provided by the experiments of Wilk et ~ l . , ~ who ' studied homoepitaxial growth of Si on Si(111) using Au as a surfactant. These authors reported that the density of defects in the film correlates well with the surfactant coverage, with the minimum defect density corresponding to full monolayer coverage by the surfactant. This is a physically appealing result that can be interpreted as evidence that the better the passivation of the surface by the surfactant, the more effective the surfactant in promoting high-quality growth. In the following we will assume that full monolayer coverage of the substrate is the standard condition for successful surfactant-mediated growth of semiconductors. The model we will now describe assumes that the surfactant effect is kinetic. As with all other kinetic models of surfactant-mediated growth, the underlying idea is that at equilibrium the heteroepitaxial system generates 3D islands even in the presence of surfactants. Surfactants make the approach to equilibrium very slow, so that 3D islands are not generated during the growth of the film. This means that surfactants kinetically suppress one or more microscopic processes, which are essential for the growth of 3D islands. The most important ingredient of any explanation of the surfactant effect is the identification of these processes. Almost all the explanations found in the literature identify the relevant process as adatom diffusion (see Refs. 25, 28, 29, 52, 54, 70, 125, 126, 183, and 193). The idea is that the energy barrier for exchange of an adatom with a surfactant atom, E,,, is smaller than the barrier for diffusion of an adatom on top of the surfactant layer, Ed. An adatom therefore diffuses a very short distance before it exchanges, and after exchange it cannot diffuse (once it is underneath the surfactant layer). This suppressed diffusion mechanism explains the surfactant effect in the following way. The reduced diffusion length makes the density of islands nucleating on the surface very high. As a result, island coalescence occurs before any second-layer islands nucleate on top of existing first-layer islands. This is how, according to this mechanism, 3D islanding is suppressed. As mentioned in Section 11, support for this hypothesis comes from various experiments, particularly those of Voigtlander et ~ l . , ~ who ' studied the effects of various surfactants on submonolayer homoepitaxial growth of Si on Si(ll1). The results were correlated with studies of the effect of the same surfactants on heteroepitaxial growth of Ge on Si(ll1). Voigtlander et

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al. found that generally there are two types of surfactants. Group-I11 and group-IV elements tend to significantly decrease the island density in submonolayer homoepitaxy and lead to 3D islanding in heteroepitaxy. On the other hand, group-V and group-VI elements drastically increase the island density in submonolayer homoepitaxy and suppress 3D islanding in heteroepitaxy. If one interprets an increase in the island density as an indication of suppression of diffusion, these results confirm the mechanism discussed above. Despite the appeal of the suppressed diffusion hypothesis, we propose that it may not be the entire story. Our concerns arise from the fact that group-V and group-VI elements chemically passivate the surface more efficiently than do group-I11 and group-IV elements. Intuitively, this should lead to faster diffusion on surfaces covered by the former elements, but the experimental results are consistent with the latter elements enhancing diffusion and the former ones suppressing it. To clarify this issue, we decided to examine more carefully the microscopic processes involved in the kinetics of surfactantmediated epitaxy. Our investigation led to an entirely different explanation of the influence of surfactants on epitaxial growth modes. A schematic representation of the possible atomic processes is shown in Fig. 1. The simplest process is, of course, diffusion of adatoms on top of the surfactant layer (Fig. l(a)). A second important process is the exchange of adsorbed atoms with the surfactant atoms, so that the former can be buried under the surfactant layer and become part of the bulk. This process can take place either on a terrace or at a step (Fig. l(b)). From thermodynamic considerations, we must also consider the process by which atoms deexchange and become adatoms that can diffuse on top of the surfactant layer (Fig. l(c)). Again, this process can take place on terraces or at surface steps. Finally, we have to consider separately surfactants that cannot passivate step edges, in which case both the exchange (Fig. l(d)) and de-exchange processes (Fig. l(e)) will be different than at passivated steps, since they no longer involve actual exchange events between adatoms and surfactant atoms. We refer to our model as diffusion-de-exchange-passivation(DDP), since these are the three processes that determine the behavior in surfactant-mediated epitaxy: Diffusion is always present; de-exchangeobviously implies also the presence of exchange; and passivation (always present on terraces) may or may not be present at island edges, but its presence or absence is a crucial element.

7. FIRST-PRINCIPLES CALCULATIONS In order to evaluate the relative contributions of these processes and their influence on the growth mode, the corresponding activation energies must be calculated. This is a d a c u l t task because very little is known about the

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A

n

A

FIG. 1. Schematic illustration of important mechanisms in surfactant-mediated growth on a substrate (represented by white circles) with a full monolayer surfactant coverage (represented by continuous shaded area). (a) Diffusion on terraces and steps for surfactant that passivates step edges. (b) Exchange at terraces and passivated steps. (c) De-exchange at terraces and passivated steps. (d) Diffusion on terrace and exchange at nonpassivated steps. (e) De-exchange at terrace and at nonpassivated steps.

atomic configurations involved. We therefore begin by considering two idealized processes that involve entire monolayers; then we discuss how the corresponding activation energies can be representative and relevant for growth mechanisms, and obtain their values from first-principles calculations. The first process we consider is diffusion on a surface covered by a surfactant monolayer. The representative system we chose to study consists

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FIG. 2. Representative surface diffusion pathway, (a) top and (b) side views. The dark circles represent the substrate atoms; the light circles, the surfactant atoms. The smaller gray circle represents an extra atom deposited on top of the surfactant layer, at different positions. The geometries correspond to a Ge adatom on a Si(ll1) surface (the substrate) covered by a monolayer of Sb (the surfactant) in a (2 x I) chain reconstruction.

of a Si(ll1) substrate, covered by a bilayer of Ge, with Sb as the surfactant. In this case, it was known that the structure of the Sb layer is a chain geometry with a periodicity of (2 x l), as shown in Fig. 2.16*An additional Ge atom was then placed on top of the Sb layer, and the energy optimized for a fixed position of the Ge atom along the direction parallel to the Sb chains. All other atomic coordinates, including those of the Ge atom perpendicular to the Sb chain and vertical with respect to the surface, were allowed to relax in order to obtain the minimum energy configuration. The energy and forces were computed in the framework of density functional theory and local density approximation (DFT/LDA), a methodology that is known to provide accurate energetic comparisons for this type of system

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(see in particular the reviews by Kaxiras202.203on the application of such calculations to semiconductor growth phenomena). By considering several positions of the extra Ge atom along the chain direction and calculating the corresponding total energy of the system, we obtained a measure of the activation energy for diffusion in this direction. We found that the activation energy for diffusion along this path is 0.5 eV. We next considered a possible exchange mechanism in the same system, through which the newly deposited Ge atoms can interchange positions with the surfactant atoms and become buried under them. To this end, we modeled the system by a full monolayer of Ge deposited on top of the surfactant layer (Fig. 3(a)). We studied a concerted exchange type of motion for the Ge-Sb interchange. In the final configuration (Fig. 3(e)) the Ge layer was below the Sb layer and the system was now ready for the deposition of the next Ge layer on top of the surfactant. The middle configuration, Fig. 3(c), corresponds to a metastable structure, in which half of the newly deposited Ge layer has interchanged position with the Sb surfactant layer. The configurations between the initial and middle geometries and the middle and final geometries, Fig. 3(b) and Fig. 3(d), respectively, correspond to the saddle-point geometries that determine the activation energy for the exchange. From our DFT/LDA calculations we found that the energy difference between structures 3(a) and 3(b) is 0.8 eV, and the energy difference between structures 3(c) and 3(d) is the same to within the accuracy of the results. Similarly, the energy difference between structures 3(c) and 3(b) and structures 3(e) and 3(d) is 1.6 eV. These two numbers correspond to the exchange activation energy (0.8 eV, going from 3(a) to 3(c) through 3(b), or going from 3(c) to 3(e) through 3(d)) and the de-exchange activation energy (1.6 eV, going from 3(c) to 3(a) through 3(b), or going from 3(e) to 3(c) through 3(d)) for this hypothetical process. We discuss next why these calculations give reasonable estimates for the activation energies involved in surfactant-mediated growth. As far as the diffusion process is concerned, it is typical for semiconductor surfaces to exhibit anisotropic diffusion constants depending on the surface reconstruction, with the fast diffusion direction along channels of atoms that are bonded strongly among themselves. This is precisely the pathway we examined in Fig. 2. As for the exchange process, it is believed that the only way in which atoms can exchange positions in the bulk is through a concerted exchange type of motion, as first proposed by Pandey for selfdiffusion in bulk Si.204 This motion involves the breaking of the smallest 'O'

'03 '04

E. Kaxiras, Sutf Rev. Lett. 3, 1295 (1996). E. Kaxiras, Comp. Mat. Sci. 6, 158 (1996). K. C. Pandey, Phys. Rev. Lett. 57, 2287 (1986).

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FIG.3. Representative exchange pathway. The color scheme is the same as in Fig. 2. (a) Structure with one layer of newly deposited atoms on top of the surfactant layer. The geometries depicted in (b), (c), and (d) are the intermediate structures during a concerted exchange that brinks the surfactant layer on top of the newly deposited layer, shown as the final configuration in (e). Structure (c) is metastable, while structures (b) and (d) are saddle-point configurations. Solid lines linking the atoms correspond to covalent bonds, while dashed lines correspond to broken bonds. The geometries correspond to the same physical system as in Fig. 2.

possible number of covalent bonds during the exchange, which keeps the activation energy relatively low. In the case of bulk Si, the activation energy for concerted exchange is 4.5 eV.’04 In the present case the activation energy is only 0.8 eV because, unlike in bulk Si, the initial configuration (Fig. 3(a)) is not optimal, having the pentavalent Sb atoms as fourfold coordinated (they would prefer threefold coordination) and the newly deposited Ge atoms as threefold coordinated (they would prefer fourfold coordination). In the final configuration (Fig. 3(e)), which has lower energy than the initial one, all atoms are coordinated properly (threefold for Sb, fourfold for Ge). While we have argued that the above described atomic processes are physically plausible, we have not established their uniqueness or their supremacy over other possible atomic motions. In fact, the calculations of

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Schroeder et al.,lE7discussed in Section 111.5, are much more realistic concerning the exchange of single adatoms with surfactant atoms on terraces. However, those calculations refer to a single event, and the formation of an additional substrate layer could (and probably does) involve additional steps in the exchange process because of the double-layer nature of the Si(ll1) substrate. In our calculations, the structure of the layer below the surfactant is compatible with the lower half of the substrate double layer, so that the process of exchange can proceed with very similar steps to complete the double-layer growth. In this sense, we feel that the barriers we obtained are not too far from realistic values. To keep our discussion general we will consider the two sets of energy barriers as corresponding to a range of physical systems: The first set is suggested by our results (Ed = 0.5 eV, E,, = 0.8 eV, Ed,-,, = 1.6 eV); the second, by the results of Schroeder et al. ( E d = 0.25 eV, E,, = 0.27 eV, Ede-ex = 1.1 ev). 8. DE-EXCHANGE AND GENERALIZED DIFFVSION It is clear that diffusion and exchange processes affect the morphology of the growing film. What about de-exchange processes? The energy barrier for de-exchange is significantly larger than the other two energy barriers. Are de-exchange events frequent enough to have any effect on the growth mode? Or, to be more quantitative, suppose an adatom exchanged with a surfactant atom; will it de-exchange before another adatom exchanges in its vicinity? To answer this question we assume that the time scale associated with a process with energy barrier E is v - l exp(E/kT), where v = 1013, sec-I is the basic attempt rate, k is the Boltzmann constant, and T is the temperature. Even at the fairly low temperature of 350°C and with the large de-exchange barrier of 1.6 eV, the time it takes an atom to de-exchange is only 0.9 seconds. The time it takes to grow a layer at a typical flux of 0.3 layers/minute is 200 seconds. Therefore, an atom will de-exchange quite a few times before it will interact with additional atoms in the same layer. We conclude that de-exchange processes can influence the growth mode and should not be ignored. The above discussion changes our view of diffusion in surfactant-mediated epitaxy. The effect of diffusion cannot be understood simply by comparing Ed with E,,, because after an adatom exchanges it may still continue to diffuse on top of the surfactant layer by de-exchanging with a surfactant atom. It is instructive to compare the effective diffusion constant, D,, which corresponds to this complex diffusion process, with the bare diffusion constant of an adatom on a surface (without surfactants), D = ,az exp( - ELb)/kT). Here a is the lattice constant and ELb)is the energy barrier for bare diffusion.

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To calculate D, we consider the case E,, > Ed (a similar calculation can be done for the opposite case and yields an identical result). An effective diffusion hop consists of a de-exchange event followed by several microscopic diffusion hops and finally an exchange event. We calculate D, from the expression D,, = A’/T,,,, where ,~T is the average time it takes to carry out an effective diffusion hop. A is the average distance an atom travels during such a hop, and it obeys the relation A = a & , where n is the average number of microscopic diffusion hops the atom carries out between the de-exchange and exchange events. n is easily calculated as the ratio between the time for an exchange event and the time for a microscopic diffusion hop. This leads to the result n = exp[(E,, - Ed)/kT]. z, is the time it takes to carry out a de-exchange event followed by an exchange event. Therefore, T~,, = v-’[exp(Ede-,,/kT) + exp(E,/kT)]. The final expression for the effective diffusion constant is

Clearly, a comparison of Ed with E,, does not tell us much about the magnitude of D., Passivation of the surface by the surfactant implies E$b)> Ed and Ede-ex> Eex.Thus, both the numerator and the denominator are larger than 1, and the question is which is the larger of the two. For the values of energy barriers calculated from first principles (both ours and those of Schroeder ef d.),Ed,-,, - E, x 0.8 eV. The denominator of Eq. (4.1) is therefore a very large number (between lo3 and lo6 for typical temperatures). The numerator is much smaller, and therefore D >> D,that is, diffusion is suppressed. This is not necessarily the case for all surfactants. For example, for surfactants that are less efficient in passivating the surface, Ede-ex- E, may be comparable or even smaller than Eib) - Ed, which would lead to diffusion enhancement. This may be the case for group-I11 and group-IV surfactants, which enhance diffusion according to experiments (see Refs. 33, 37, 50, and 205). It would be interesting to check this possibility with DFT/LDA calculations. One conclusion is that effective diffusion can be suppressed by surfactants even if E,, > Ed; in other words, a surfactant can enhance diffusion on top of the surfactant layer and, at the same time, suppress effective diffusion, which takes into account deexchange

processes.

205

S. Iwanari,

Y.Kimura, and K.Takayanagi, J . Cryst. Growth 119,241 (1992).

COLORRATE 1. Kinetic Monte-Carlo simulations of surfactant-mediated homoepitaxy in the DDP model, on a substrate of size 100 x 100 at a temperature. of 600°C. A total of 0.15 monolayer of new material has been deposited. (a) Simulations with IEP, with the activation energies Ed = 0.5 eV, E , = 0.8 ev, Edc-, = 1.6 ev, and Ehr = 3 ev. (b) Simulations without EP, with the activation ener= 1.6 eV, and E&, = 1.6 eV. (c) Simulations with EP, with the gies Ed = 0.5 eV, E , = 0.8 eV, = 1.1 eV, and E&( = 2.5 eV. (d) Simulations withactivation energies Ed = 0.5 eV, E, = 0.3 eV, out IEP, with the activation energies Ed = 0.5 eV, E, = 0.3 eV, Eh-ex = 1.1 eV and Ehr = 1.6 eV. IEP clearly increases the island density and significantly affects the island shape.

COLORPLATE2. Kinetic Monte-Carlo simulations of homoepitaxial surfactant-mediated growth in the DDP model, with IEP on a substrate of size 300 x 300. The activation energies were Ed = 0.5 eV, E, = 0.8 eV, Eh-- = 1.6 eV, and E&f = 3 eV. A total of 0.15 monolayer of new material has been deposited at (a) 600°C. (b) 700"C, and (c) 850°C. The high density of small islands at low temperature is evident, as well as the decrease of the island density with increasing temperature.

COLORRATE 3. Kinetic Monte-Carlo simulations of surfactant mediated heteroepitaxy in the DDP model, on a substrate of size 100 x 100 at a temperature of 300°C. A total of 1 monolayer of new material has been deposited. (a) Simulations with IEP, with the activation energies Ed = 0.5 eV, E,, = 0.8 eV, and Edr-cr= 1.6 eV. (b) Simulations without IEP, with the activation energies Ed = 0.5 eV, E , = 0.8 eV, and Ede-- = 1.6 eV. (c) Simulations with IEP,with the activation energies Ed = 0.5 eV, E , = 0.3 eV, and Ede-er = 1.1 eV. (d) Simulations without IEP, with the activation energies Ed = 0.5 eV, E, = 0.3 eV, EdC-- = 1.1 eV. Different colors indicate different surface heights. The surfactant that passivates island edges suppresses 3D islanding completely and induces layer-by-layer growth. Without IEP, 3D islands form on the film. They reach a height of 7 layers after deposition of 1 layer of material.

COLOR PLATE 4. Kinetic Monte Car10 simulations of heteroepitaxial surfactant-mediatedgrowth in the DDP model, on a substrate of size 300 x 300 with IEP. The activation energies were Ed = 0.5 eV, EeX= 0.8 eV, and Edr-- = 1.6 eV. A total of 1 monolayer of new material has been deposited at (a) 300°C. (b) 350"C, (c) 400°C and (d) 450°C. The different colors indicate surface heights. The transition between layer-by-layer growth and 3D island growth takes place somewhere between 350°C and 400°C.

COLORPLATE 5. Kinetic Monte-Car10 simulations of heteroepitaxid surfactant-mediated growth in the DDP model, on a vicinal substrate of size 300 x 300 with IEP. The activation energies were Ed = 0.5 eV, Ea = 0.8 eV, and Eh-- = 1.6 eV. A total of 1 monolayer of new material has been deposited at (a) 300"C, (b) 350°C, (c) 4OO"C, and (d) 450°C. The different colors indicate surface heights.

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9. ISLAND-EDGE PASSIVATION

At this stage, it is tempting to claim that we have reached a much better understanding of the surfactant effect. However, we show next that this is not so. In fact, a more careful analysis shows that suppression of diffusion has nothing to do with the explanation of the surfactant effect, and that two surfactants, which lead to the same value of D,, may induce very different growth modes. This happens because different surfactants may vary drastically in their ability to passivate steps or island edges. We will see that the issue of island-edge passivation is crucially important to the morphology of the growing film and its surface. It is especially important for the ability of the surfactant to suppress 3D islanding in heteroepitaxy. To understand the role of island edges in the determination of growth modes, we have to understand the reason for the formation of 3D islands in heteroepitaxial growth. The islands form because their presence facilitates strain relaxation much more efficiently than do flat layers. For example, Tersoff and Tromp206calculated the elastic energy per unit volume, E e l , of a strained rectangular island of lateral dimensions s and t (measured in units of the lattice constant). They showed that

Eel

- (e+ y).

The energy of a narrow island is thus smaller than the energy of a wide one. Therefore, after a monolayer-high island has grown beyond a certain width, it is beneficial to grow another layer on top of it rather than to make it wider. The film then tends to grow in narrow and fairly tall 3D islands. The kinetic process that prevents the island from continuing to grow laterally is the detachment of atoms from the island edges. If such detachment processes are suppressed, the island will not reach its equilibrium shape, but will tend to be too wide and flat. It is quite obvious that surfactants that passivate island edges also suppress detachment events. Hence, they may change the growth mode from 3D islanding to layer by layer. Suppression of diffusion may not be sufficient to suppress 3D islanding, since detachment of atoms from island edges may lead to islanding even with very little diffusion. Passivation of island edges, on the other hand, can change the growth mode even if diffusion is not enhanced. We now use our knowledge of the chemical nature of different surfactants to speculate about their ability to passivate island edges: Group-V atoms '06

J. Tersoff and R. M. Tromp, Phys. Rev. Lett. 70, 2782 (1993).

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(especially As and Sb) should be effective in passivating steps and island edges on the (111) and (100) surfaces of tetravalent semiconductors such as Si and Ge. This is because group-V atoms prefer to have threefold coordination, in which they form three strong covalent bonds with their neighbors, using three of their valence electrons while the other two valence electrons remain in a low-energy lone-pair state. This is precisely what is needed for passivation of both terrace and step geometries on the (111) and (100) surfaces of the diamond lattice which are characterized by threefold coordinated atoms. On the other hand, it is expected that elements with the same valence as that of the substrate, or noble metals, will not be effective in passivating step edges. In the case of the tetravalent semiconductors Si and Ge, for example, the elements Sn and Pb have the same valence, and while they can form full passivating layers on top of the substrate, they clearly cannot passivate the step geometries because they have exactly the same valence as that of the substrate atoms and hence can only form similar structures. Analogously, certain noble metals can form a passivating monolayer on the semiconductor surface, but their lack of strong covalent bonding cannot affect the step structure. We note that not all noble metals behave in a similar manner; some of them form complex structures in which they intermix with the surface atoms of the substrate (such as Ag on the Si(ll1) surface), in which case it is doubtful that they will exhibit good surfactant behavior. 10. KINETICMONTE-CARLO SIMULATIONS We have given a plausibility argument that surfactants suppress 3D islanding in heteroepitaxy by limiting atom detachment from island edges, not by suppressing diffusion. The complexity of the growth process does not allow us to give a more rigorous argument. However, our ansatz can be tested quite easily by carrying out kinetic Monte-Carlo (KMC) simulations of homoepitaxial and heteroepitaxial growth, in which all the relevant microscopic processes occur randomly with rates determined by the corresponding activation energies. Accordingly, we consider a system in which the processes examined above are operative and the activation energies corresponding to them are the ones obtained from the DFT/LDA calculations for the hypothetical cases illustrated in Fig. 2 and Fig. 3. For simplicity, our simulation was carried out on a cubic lattice. Atoms land on the surfactant-covered surface with a flux of 0.3 layerslsecond (a typical value of the flux in experiments) and diffuse on top of the surfactant. They can exchange with surfactant atoms and become buried underneath the surfactant layer. A buried atom can de-exchange with a surfactant atom

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and float on top of the surfactant layer again. This can happen provided the buried atom does not have lateral bonds with other atoms underneath the surfactant layer. If it is bonded laterally, we consider this atom as part of an island edge. An atom attached to an island underneath the surfactant layer can detach from the island edge and float on top of the surfactant layer. This detachment process is of major importance, as discussed above; however, it involves the breaking of lateral bonds between the detaching atom and the island edge, and this will be taken into account in the activation barrier for detachment. Also, we did not allow simultaneous breaking of two or more lateral bonds, so an atom attached to an island edge by more than a single lateral bond cannot detach. A diffusing atom can attach to a step or an island edge. The activation barriers for attachment and detachment processes depend on whether or not the surfactant passivates steps and island edges. Barriers for detachment from an island edge also depend on whether or not the island is strained. We now describe the results of the simulations we did under various conditions. In each case we give a detailed list of the activation energy values. First, we studied homoepitaxial growth; that is, we considered a system without lattice mismatch and hence no strain effects. To investigate the influence of island-edge passivation (IEP) on surface morphology, we carried out KMC simulations of a surface of size 100 x 100 at a temperature of 600°C, and deposited on it 0.15 of a layer. The values of the activation energies used were Ed = 0.5 eV, E,, = 0.8 eV, and Ed,-,, = 1.6 eV. The energy barrier for detachment from an island edge (provided only one lateral bond is broken) was Edet = 3eV for a surfactant that passivates island-edges and Ed,, = 1.6eV for a surfactant that does not. Typical morphologies are shown in Fig. 4(a) with IEP and Fig. 4(b) without IEP. Evidently, there is a marked difference between the growth process with and without IEP. First, surfactants that passivate island edges lead to a significantly higher island density in submonolayer growth. Second, with IEP the island edges are very rough, while without it the islands are faceted. As discussed above, experimental results indicatez5 that surfactants that suppress 3D islanding also increase the island density in homoepitaxy. The rough island edges are also observed experimentally. This gives strong support to the IEP ansatz. We note that the high density of islands induced by surfactants that passivate island edges is not a result of suppression of diffusion. Rather, it arises from the fact that adatoms can cross passivated island edges without attaching to them and then nucleate on a flat part of the surface, thus generating more islands. In Figs. 4(c) and 4(d) we present results from similar simulations with another set of activation barriers: Ed = 0.5 eV, E,, = 0.3 eV, Ed,-,, = 1.1 eV, Edet = 2.5 with IEP, and Ed,, = 1.6 without IEP. Although these barriers are

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FIG. 4. Kinetic Monte-Carlo simulations of surfactant-mediated homoepitaxy in the DDP model, on a sustrate of size 100 x 100 at a temperature of 600°C. A total of 0.15 monolayer of new material has been deposited. (a) Simulations with IEP, with the activation energies Ed = 0.5 eV, E,, = 0.8 eV, Ede-ex = 1.6 eV, and Ed,, = 3 eV. (b) Simulations without IEP, with the activation energies Ed = 0.5 eV, E,, = 0.8eV, Ede-ex = 1.6eV, and Ede, = 1.6eV. (c) Simulations with IEP, with the activation energies Ed = 0.5 eV, E,, = 0.3 eV, Ede-ex = 1.1 eV, and Ede, = 2.5 eV. (d) Simulations without IEP, with the activation energies Ed = 0.5 eV, E,, = 0.3eV, Ede-ex = 1.1eV, and Ede, = 1.6 eV. IEP clearly increases the island density and significantly affects the island shape. (See also color Plate 1.)

very different from the ones we used to produce Figs. 4(a) and 4(b), the morphologies are very similar. The change in the energy barriers has not influenced the island densities, nor has it affected the shape of the islands significantly. The only noticeable effect is that the shape of the islands in Fig. 4(c) is more fractal-like than that of the ones in Fig. 4(d). Note that in the first set of energy barriers Ed < E,,, whereas in the second set the opposite holds. Thus, the relation between Ed and E,, does not have a significant influence on the growth morphology. Based on these results, we expect the energy barriers we calculated using DFT/LDA and those of Schroeder et al. (see above) to lead to similar growth morphologies; that is, the difference between these two sets of activation barriers is irrelevant for the determination of the growth mode. The results of Fig. 4 support our generalized diffusion analysis. The two sets of energy barriers we used give the same

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value for the effective diffusion constant, D,, according to Eq. (4.1). This is the reason the final surface morphologies are so similar. It is also important to check the temperature dependence of the growth process in the case of surfactants with IEP. To this end we performed K M C simulations of a 300 x 300 lattice with activation energies identical to the ones used for Fig. 4(a). The resulting surface morphologies after deposition of 0.15 of a layer are shown in Fig. 5 for three different temperatures: 600°C,

FIG. 5. Kinetic Monte-Carlo simulations of homoepitaxial surfactant-mediated growth in the D D P model, with IEP on a substrate of size 300 x 300. The activation energies were Ed = OSeV, E,, = 0.8eV, EdP--FX = 1.6eV, and Edp, = 3eV. A total of 0.15 monolayer of new material has been deposited at (a) 600"C, (b) 7 W C , and ( c ) 850'C. The high density of small islands at low temperature is evident, as well as the decrease of the island density with increasing temperature. (See also color Plate 2.)

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DANIEL KANDEL A N D EFTHIMIOS KAXIRAS

700°C, and 850°C. At all three temperatures IEP leads to a high density of compact islands with rough edges. The island density decreases with temperature. All of these observations are consistent with experimental results.25 Finally, we consider the effects of strain in surfactant-mediated heteroepitaxial growth. Strain is difficult to include in an atomistic calculation selfconsistently. Here we will rely on the theory developed by Tersoff and Tromp’” for the elastic energy of strained islands on a substrate (see Eq. (4.2)). In analogy with this theory, we postulate that the effect of strain is to alter the strength of the bonds in elastically strained islands according to the expression of Eq. (4.2), which depends on the island size through the values of s and t. The most important consequence of this effect is a change in the activation energy for detachment of atoms from island edges, Eder,since this process involves the breaking of a lateral bond that is strongly affected by strain. Ederwill now depend on the island size. The other barriers, having to do with processes that take place on top of the surfactant (diffusion and exchange on terraces and island edges), will be unaffected to the lowest order by the presence of strain. Therefore, the only important change in the kinetics comes from an island-size-dependent detachment rate, given by Ins

Eder

= &O -k

&1

lnt

(7-k 7).

(4.3)

where E~ = for surfactants that passivate island edges, and E~ = 0 when there is no IEP. In our simulations we took the value = 3.0eV, which is a reasonable number for the typical strength of bonds and the amount of strain involved in the systems of interest (4% for the case of Ge on Si). As was done in the case of homoepitaxial growth, we first studied the effects of passivation of island edges on surface morphology. We simulated a system of size 100 x 100 at a temperature of 300”C, and deposited on it one layer. The values of the activation energies used were Ed = 0.5 eV, E,, = 0.8 eV, and Edeeex= 1.6eV. The results are shown in Figs. 6(a) and 6(b) for the cases with and without IEP, respectively. Different surface heights are represented by different colors; white is the initially flat substrate. The system without IEP shows clear 3D islanding (Fig. 6(b)). Most of the substrate is exposed even after deposition of a full layer, and the deposited material is assembled in faceted tall islands (up to seven layers high). The surfactant that passivates island edges, on the other hand, suppresses 3D islanding completely (Fig. 6(a)). Most of the surface is covered by one layer (blue), with some small one-layer-high islands and holes in it.

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257 (b)

FIG. 6. Kinetic Monte-Car10 simulations of surfactant-mediated heteroepitaxy in the DDP model, on a substrate of size 100 x 100 at a temperature of 300°C. A total of 1 monolayer of new material has been deposited. (a) Simulations with IEP, with the activation energies Ed = O.SeV, E,, = 0.8eV, and E d e - e x= 1.6eV. (b) Simulations without IEP, with the activation energies Ed = 0.5 eV, E,, = 0.8 eV, and Ede-ex = 1.6eV. (c) Simulations with IEP, with the activation energies Ed = O.SeV, E,, = 0.3 eV, and Ede-ex = 1.1 eV. (d) Simulations without = 1.1. Different colors IEP, with the activation energies Ed = 0.5 eV, E,, = 0.3 eV, and indicate different surface heights. The surfactant that passivates island edges suppresses 3D islanding completely and induces layer-by-layer growth. Without IEP, 3D islands form on the film. They reach a height of 7 layers after deposition of 1 layer of material. (See also color Plate 3.)

We also checked the influence of changes in the values of the activation barriers, by repeating the simulations with Ed = 0.5 eV, E,, = 0.3 eV, and Ede-ex = 1.1 eV. The results are presented in Figs. 6(c) and 6(d) for the cases with and without IEP, respectively. The system without IEP does not show any change. In the case with IEP the densities of islands and holes decreased and their sizes increased accordingly, but the growth mode remained layer

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DANIEL KANDEL AND EFTHIMIOS KAXIRAS

by layer and 3D islanding was entirely suppressed. These results, together with the results on homoepitaxy, demonstrate convincingly that island-edge passivation, and not suppression of diffusion, is responsible for the surfactant effect. To study the effect of temperature on the growth mode in heteroepitaxial growth with IEP, we simulated a system of size 300 x 300, with the activation barriers Ed = 0.5 eV, E,, = 0.8 eV, and EdePex= 1.6 eV. The resulting morphologies at temperatures of 300"C, 350"C, 400"C, and 450°C are shown in Fig. 7. In the first two cases, growth is essentially indistinguish-

FIG.7. Kinetic Monte-Carlo simulations of heteroepitaxial surfactant-mediated growth in the DDP model, on a substrate of size 300 x 300 with IEP. The activation energies were Ed = 0.5 eV, E,, = 0.8 eV, and Ede-ex= 1.6 eV. A total of 1 monolayer of new material has been deposited at (a) 300"C, (b) 350"C, (c) 4 W C , and (d) 450°C. The diferent colors indicate surface heights. The transition between layer-by-layer growth and 3D island growth takes place somewhere between 350°C and 400°C. (See also color Plate 4.)

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able from the case of homoepitaxy, with a high density of small islands. However, at T = 400°C,despite the small rise of only 50"C,a dramatically different growth mode is evident, with a large number of tall 3D islands and a substantial amount of the substrate left uncovered. This trend is even more evident at the higher temperature of 450°C.We also carried out simulations of heteroepitaxial growth on vicinal surfaces, with exactly the same parameters as those of Fig. 7, but we started from a system with atomic steps present on the substrate. Fig. 8 shows the results of these K M C simulations for the same temperatures as in Fig. 7. Again, the surfactant suppressed 3D islanding at low temperatures but not at high temperatures. This is precisely the type of abrupt transition from layer-by-layer growth at low temperature

FIG. 8. Kinetic Monte-Carlo simulations of heteroepitaxial surfactant-mediated growth in the DDP model, on a oicind substrate of size 300 x 300 with IEP. The activation energies were Ed = 0.5 eV, E,, = 0.8 eV, and Ede-ex = 1.6 eV. A total of 1 monolayer of new material has been deposited at (a) 300"C, (b) 350"C, (c) 400"C, and (d) 450°C. The different colors indicate surface. heights. (See also color Plate 5.)

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DANIEL KANDEL A N D EFTHIMIOS KAXIRAS

to 3D island growth at higher temperature observed experimentally for the strained heteroepitaxial systems, such as Ge/Si with Sb as a surfactant. V. Discussion

We have provided a critical review of the literature on surfactant-mediated semiconductor epitaxy, with an emphasis on comparisons between experimental observations and model calculations. Our main goal has been to arrive at a consistent explanation of the mechanism by which surfactants suppress 3D islanding in heteroepitaxial growth. A vast number of experimental articles on the subject exist, and we have gathered most of them in tables according to the relevant combination of deposit-surfactant-substrate materials. The most important message, to be taken from these experimental studies is that in semiconductor epitaxy surfactants can be divided into two categories: the first category is surfactants that lead to an anomalously high island density in submonolayer homoepitaxy and also suppress 3D islanding in heteroepitaxy; the second category is surfactants that lead to step flow growth in homoepitaxy and are inefficient in suppressing 3D islanding in heteroepitaxy. Explanations of the surfactant effect have focused on the relation between the activation energy for adatom diffusion on top of the surfactant layer and the barrier for exchange of an adatom with a surfactant atom. The rationale was that surfactants of the first category suppress 3D islanding and increase island density because they suppress diffusion. Suppression of diffusion was associated with relatively easy exchange processes. Surfactants of the second category, on the other hand, were thought to enhance diffusion, and exchange processes were expected to be relatively difficult. Several first-principles calculations of the diffusion and exchange barriers have been carried out for various systems. In a typical calculation, specific paths for diffusion and exchange were proposed and total energies of the system in relaxed configurations along these paths were calculated. This allows fairly accurate estimates of the relevant energy barriers. Different studies arrived at different conclusions about the barriers mainly because the paths proposed were different. Thus, the main deficiency of these microscopic calculations is their inability to predict the correct kinetic path for the process under consideration. We propose a new scenario for the explanation of the surfactant effect. According to our ansatz, neither the relation between diffusion and exchange nor the suppression of diffusion is relevant for the explanation of the surfactant effect. Instead, we argue that the efficiency of a surfactant is determined by its ability to passivate island edges. Surfactants that passivate

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261

island edges also lead to an anomalously high density of islands and suppress 3D islanding. We supply ample evidence for this scenario. The most convincing evidence comes from kinetic Monte-Carlo simulations of the growth process and from the comparison of the results with experimental observations. Using realistic activation energies we show that a surfactant that suppresses diffusion, but does not passivate island edges, does not suppress 3D islanding. It also does not lead to a very high density of islands. Moreover, the islands generated in the growth process mediated by such a surfactant are faceted and do not have the rough edges observed in experiments. By contrast, island-edge passivation does lead to suppression of 3D islanding and to islands of a shape consistent with the experimentally observed shape. The temperature dependence of the island density, as well as the abrupt transition from layer-by-layer to 3D growth as the temperature is raised, is predicted correctly by simulations with IEP. The evidence we provide for the validity of our scenario, although convincing, is far from a rigorous proof. In fact, it is based on a very simplified model that fails to take into account various aspects of the experimental system that may be important. For example, the use of an isotropic square lattice is not appropriate for all cases (the substrates with diamond lattice and (1 11) orientation have a hexagonal surface lattice, whereas those with (100) orientation have a square lattice but exhibit strong anisotropy in the directions parallel and perpendicular to the surface dimers). Nor do we account for the fact that in some cases the film grows in bilayers (as in (111) substrates) rather than in monolayers (as in (100) substrates). Finally, we do not properly treat the issue of critical island size, which may be very large in Si homoepitaxy or in heteroepitaxy of Ge on Si. For these reasons there is room for further discussion and more detailed modeling of surfactant-mediated thin-film growth. There are various unresolved issues in surfactant-mediated epitaxial growth, which we have not discussed. Perhaps the most important is the issue of strain relaxation. Heteroepitaxial films grown in the layer-by-layer growth mode are initially highly strained and this strain energy must somehow relax after growth of a few layers. Indeed, dislocations appear in the film during surfactant-mediated heteroepitaxy. In some cases these dislocations d o not thread the film and hence do not harm its epitaxial quality, but in other cases they do. It is therefore very important to study strain relaxation in the films. Some experimental studies have been carried out, but their description is beyond the scope of the present article. To the best of our knowledge, there has not been any detailed theoretical work on the problem. Another important issue is that some of the surfactant layer inevitably becomes trapped in the growing film. This leads to unintended doping if the

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surfactant is not isoelectronic with the deposited material. This could be beneficial if high levels of doping are desired, or detrimental if a film of high purity is desired. In any case, controlling the amount of the incorporated surfactant by carefully adjusting external conditions (such as flux rate, temperature, and surface preparation) is highly desirable.’07 A better understanding of the surfactant effect, along the lines proposed here for the DDP model, will probably go a long way toward controlling the electronic properties of the film, which are strongly influenced by surfactant incorporation, strain relaxation defects, and surface morphology. Further research in these directions is necessary and essential before surfactant-mediated growth can become useful in practical applications. ,

This work was supported by the Office of Naval Research, Grant # NOOO14-95-1-0350, and by The Israeli Science Foundation founded by

The Israeli Academy of Sciences and Humanities., DK is the incumbent of the Ruth Epstein Recu Career Development Chair.

*07

M.-H. Xie, J. Zhang, A. Lees, J. M. Fernandez, and B. A. Joyce, SurJ Sci. 367,231 (1996).

SOLID STATE PHYSICS, VOL. 54

The Two-Dimensional Physics of Josephson Junction Arrays R. S. NEWROCK Physics Department University of Cincinnati Cincinnati, Ohio 45221-001 I

C. J. LOBB Center f o r Superconductivity Research Department of Physics University of Maryland College Park, Maryland 20742-4111

U. GEIGENM~LLER* Department of Electrical Engineering Delft University of Technology Mekelweg 4 2 6 2 8 C D De/ft, The Netherlands

M. OCTAVIO'~ Centro de Fisica Instituto Venezolano de Investigationes Cient$cas Apartado 21827 Caracas I02OA Venexela

' Current address: Roccade Finance, DeBrand 10,3823 LH Amersfoort, Postbus 800,3800 AV Amersfoort, The Netherlands. Current address: Bancaracus, Caracas, Venezuela. Adjunct Professor Physics, University of Cincinnati.

*

263

264

R. S. NEWROCK ET AL.

Contents I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 I1. The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 1. Superconductivity and the Josephson Equations . . . . . . . . . . . . . . . . 271 275 2. TheRCSJModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 3. The Washboard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Classical Arrays: T = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 283 4. Zero Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Nonzero Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 289 a . The Ground State for B = 0 . . . . . . . . . . . . . . . . . . . . . . . . . b . The Ground State for Small Magnetic Field . . . . . . . . . . . . . . . . 291 c. Vortex Depinning and the Critical Current . . . . . . . . . . . . . . . . 292 d . Vortex Motion Above the Critical Current . . . . . . . . . . . . . . . . . 296 IV. Classical Arrays: T > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 6. Single Junctions at Nonzero Temperature . . . . . . . . . . . . . . . . . . . . 301 7. Arrays at Nonzero Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . 302 304 8. Estimates of the Transition Temperature . . . . . . . . . . . . . . . . . . . . V . Classical Arrays: Zero Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 9. The Resistivity Above the Transition Temperature . . . . . . . . . . . . . . 309 309 a . The Free-Vortex Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 b. TheResistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c. Experimental Determinations of Resistance . . . . . . . . . . . . . . . . 311 313 10. The Current-Voltage Characteristics . . . . . . . . . . . . . . . . . . . . . . . a . Effects of Applied Currents on Bound Pairs of Vortices . . . . . . . . . 313 b. Experimental Current-Voltage Characteristics . . . . . . . . . . . . . . . 316 318 11. General Current-Voltage Characteristics . . . . . . . . . . . . . . . . . . . . 318 a . Scale Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b. Schematic Current-Voltage Characteristics . . . . . . . . . . . . . . . . 320 VI . Classical Arrays: Nonzero Frequency Response . . . . . . . . . . . . . . . . . . . 324 326 12. Dynamics of Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Array Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 VII . Classical Arrays: Finite-Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 334 14. Incomplete Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. Finite-Size-Induced Free Vortices . . . . . . . . . . . . . . . . . . . . . . . . 335 339 16. Size Effects due to Residual Fields . . . . . . . . . . . . . . . . . . . . . . . . 342 VIII . Classical Arrays: Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX . Classical Arrays: Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 358 17. Bond and Site Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 18. AC Susceptibility and Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . 19. Positional Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 378 20. Positional Disorder -Fully Frustrated . . . . . . . . . . . . . . . . . . . . .

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265

21. Dynamics of Disordered Arrays: Random Disorder . . . . . . . . . . . . . .

380 381 383 383 388 395 395 399 401 413 418 423 424 431 431 431 435 435 436 437 438 438 442 442 444 448 449 452 455 455 459 463 467 468 469 475 476 492 495 499 500 502 505

22. Dynamics of Disordered Arrays: Correlated Disorder . . . . . . . . . . . . . X . Classical Arrays: Nonconventional Dynamics . . . . . . . . . . . . . . . . . . . . 23. Nonconventional Vortex Dynamics . . . . . . . . . . . . . . . . . . . . . . . 24. Ballistic Motion of Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI . Classical Arrays: Strongly Driven . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. Shapiro Steps -Single Josephson Junctions . . . . . . . . . . . . . . . . . . 26. Shapiro Steps -Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 . Fractional Giant Shapiro Steps . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Subharmonic Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29. Simulations: Inductance and Symmetry Breaking . . . . . . . . . . . . . . . 30. Effects of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 . Emission of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI1. Quantum Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32. Single Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a . Quantization and Commutation Relations . . . . . . . . . . . . . . . . . b. Phase Delocalization in Single Junctions . . . . . . . . . . . . . . . . . . i. E , > > E , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii. E , >> E , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c. Historic Note and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 33. The Superconducting Phase Transition in Arrays . . . . . . . . . . . . . . . a . Quantum Corrections to the Kosterlitz-Thouless Transition . . . . . . b. Mean-Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i. The Importance of Dimensionality . . . . . . . . . . . . . . . . . . ii. Self-Consistent Mean-Field Theory . . . . . . . . . . . . . . . . . . ... 111. Re-Entrance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv . Influence of Offset Charges . . . . . . . . . . . . . . . . . . . . . . . v . Variational Improvement of the Mean-Field Theory . . . . . . . . 34. The Resistance of Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a . The Resistance as Control Parameter . . . . . . . . . . . . . . . . . . . . b . Ginzburg-Landau Description . . . . . . . . . . . . . . . . . . . . . . . . c. Resistance at the T = 0 Phase Boundary . . . . . . . . . . . . . . . . . . XI11 . Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments and Apologia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Correlation Functions: Vortices and Spin Waves . . . . . . . . . . . . . . . Appendix B: Vortex-Pair Density: The Dilute Limit . . . . . . . . . . . . . . . . . . . . . Appendix C: Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix D: Current-Induced Vortex Unbinding . . . . . . . . . . . . . . . . . . . . . . . Appendix E: The Capacitance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix F: Offset Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix G: Phase Correlation Function in the Absence of Coupling . . . . . . . . . . . Appendix H: Conductivity from Derivatives of the Partition Function . . . . . . . . . . . Appendix I: The Green’s Function for Gaussian Coarse Graining . . . . . . . . . . . . .

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1. Introduction

Since the late 1970s and early 1980s,’ large arrays of Josephson junctions have been used to study the physics of low-dimensional systems. More recently large arrays have also proven to be extremely useful model systems for studying a wide variety of other physics, including phase transitions in frustrated and random systems, dynamics in coupled nonlinear systems, and macroscopic quantum effects. This review is meant to be a coherent discussion of the essential physics of these arrays. Josephson junction arrays consist of islands of superconductor, usually arranged on an ordered lattice, coupled by Josephson junctions. They can be divided into classical and quantum arrays depending on the ratio of the Josephson coupling energy, E,, to the charging energy,’ E,. Josephson junction arrays can also be divided into overdamped and underdamped arrays, referring to the fact that the equation of motion for a single Josephson junction is identical to a damped pendulum. For reasons that will be explained later, the dividing line between classical and quantum arrays occurs when E,/E, = 1. The dividing line between underdamped and overdamped arrays is determined when another dimensionless parameter, /?,,is equal to 1. The McCumber parameter, /?,,defines the amount of damping in terms of the junction capacitance and resistance. Figure 1 shows the “phase space” for Josephson junction arrays3 in terms of the junction resistance and capacitance. Classical two-dimensional arrays ( E , << E,) can be shown to be isomorphic to a two-dimensional XY spin system -they are physical representations of the XY model, which is a two-dimensional lattice of spins free to rotate in the XY plane. Since they are systems in which the various parameters are well known and easily controlled, they serve as models for investigating statistical mechanics -indeed, we can literally do “statistical mechanics on a chip.” Classical two-dimensional arrays are used in zero magnetic field for studying phase transitions such as the Kosterlitz-Thouless-Berezinskii transition, for studying the effects of disorder on phase transitions, and for

J. L. Berchier and D. H. Sanchez, Revue De Physique Appliquie 14, 757 (1979); D. Sanchez and J. L. Berchier, J . Low Temp. Phys. 43, 65 (1981). E , refers to the coupling energy of the two superconductors that make up a single junction in the array, and E , is the energy cost to place a charge on an island in the array. H. S. J. van der Zant, Ph.D. thesis, Delft University of Technology (1991).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

A

\ \

\

W

267

regime II underdamped classical

underdkped quantum \

lo-’

-

\

regime I V \ overdamped quantum \ I

10-2

I

\

FIG. 1. The “phase space” for Josephson-junction arrays. The lines indicate the phase boundaries between classical and quantum arrays (E,/4EC = 1) and underdamped and over= 1). E , was calculated assuming a charge transfer of 2e; E , and were damped arrays calculated assuming a critical current of 0.1 milliamp. C is the capacitance and r, the normal state resistance of the junction. (From Ref. 3).

(s,

s,

studying dimensional crossover effects in phase transitions. Arrays in zero magnetic field are used as disorder-resistant voltage standards and recently were shown to be coherent emitters of very-high-frequency (hundreds of gigahertz) radiation. In a magnetic field, classical two-dimensional Josephson junction arrays provide a realization for frustrated XY magnets, but ones in which the frustration can be tuned by varying the applied field. This allows the effects of frustration on the phase transition to be studied. There is also a rich structure when magnetic fields and high-frequency radiation are applied. Classical Josephson junction arrays have brought physicists and applied mathematicians together in the world of nonlinear dynamics. They have become archetypes for many-degrees-of-freedom nonlinear systems, putting

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them in the forefront of research into nonlinear dynamics and chaos. An N x N Josephson junction array is represented by N 2 coupled nonlinear differential equations -differential equations that also represent other systems. The response of junction arrays to various driving currents often aids the applied mathematician in generating solutions to the equations, and mathematicians’ solutions to the equations often suggest experiments to physicists. Because Josephson junction arrays are artificial, they are very well characterized, leading to an interesting synergy between experiments, computer simulations, and applied mathematics. The discrete nature of an array and the well-known physics of a Josephson junction allow easy simulations of array behavior. These simulations can be compared to experiments or can suggest experiments. This is of considerable benefit, for example, in studying disorder-one knows exactly what type of disorder and how much is introduced. Arrays with disorder can be shown to be the limiting case of an extreme inhomogeneous type-I1 superconductor, allowing the study of such superconductors in samples in which the disorder is nearly exactly known. Vortices play a fundamental role in Josephson junction arrays, and vortex dynamics are dominant in classical arrays. Vortices occur naturally in the zero applied field state as part of the phase transition. With an applied magnetic field, Josephson junction arrays are used to study flux flow, flux pinning, vortex lattice melting, and vortex glasses. In underdamped arrays vortices have nonnegligible mass. Ballistic motion of vortices has been reported, and they probably should be treated as quantum-mechanical objects. Vortices also play fundamental and poorly understood roles in other areas of physics. Turbulence is a prime example. Vortices are seen in normal fluid turbulence and in spatially varying chaos, and vortex-like structures have been observed in discrete hydrodynamics and in chemical-reaction chaos. At present the role of vortices in turbulence is far from understood. An important question is whether they control the dynamical instabilities which determine the onset of turbulence. Josephson junction arrays driven by external currents are excellent systems in which to investigate such questions. Quantum arrays are obtained when the charging energy dominates the coupling energy. This is achieved experimentally by using junctions with low capacitance and high resistance. Such arrays allow us to study quantum effects in macroscopic systems. For example, a single charge on one island can form a bound pair with an “anticharge” on another island, and a superconductor-insulator transition of the Kosterlitz-Thouless type may occur. Other such transitions have been predicted, and the vortex dynamics near the predicted transition temperature are of considerable interest. In

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such a system, frustration can be induced not only by applying a magnetic field but also by inducing charges on the junction electrodes. One can investigate other such charge-vortex dynamics in such arrays. Other interesting physics includes a possible quantized Hall conductivity in an appropriate limit (the quantum Hall effect) and DC voltages and currents that may show plateaus -giant and inverse giant Shapiro steps. Finally, granular superconductors can be considered highly disordered Josephson junction arrays and should exhibit similar behavior. A superconductor-insulator quantum phase transition is believed to be observed in both systems. Josephson junction arrays are model systems for experimental investigations of all these interesting topics for a number of reasons: Arrays can be fabricated easily using modern nanolithograpic techniques. Transport measurements, magnetic measurements, and other types of experiments are relatively simple. Many of the important parameters can be easily varied, creating different experimental situations and even universality classes and “rich” phase diagrams. Among those parameters are DC currents, RF currents, external magnetic fields, junction capacitances and resistances, loop inductances, shunt resistances, and disorder. Vortices are natural thermal excitations in these systems and can also be introduced by external fields. The dynamic response has a natural interaction between length and time scales. The nonlinearities are simple to see and are determined by the relatively well-understood physics of the junctions. Arrays are naturally discrete, and their responses to external stimuli are amenable to computer simulation. The first experiments on two-dimensional Josephson junction arrays were those of Sanchez and Berchier.’ The use of large Josephson junction arrays (about 1 million junctions) to study the two-dimensional superconducting phase transition and to serve as a model system for granular materials was introduced by Resnick et aL4 Since then a variety of two-dimensional arrays D. J. Resnick, J. C. Garland, J. T. Boyd, S. Shoemaker, and R. S. Newrock, Phys. Reu. Lett. 47, 1542 (1981).

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has been studied, including square and triangular proximity-coupled arrays; square and triangular tunnel junction arrays; disordered arrays: Euclidian wire networks; exotic wire networks (Sierpinski gaskets and carpets, Penrose tiles, and other quasiperiodic structures); and ladder arrays. In this review, we limit our interest to the first three. In most of this article we take a “tutorial” approach, which is somewhat at variance with the normal review article in at least two ways. First, we view our primary audience as researchers unfamiliar with the field, and our main concern is to convey to that audience the essential physics. A certain dichotomy exist, in that while the basic ideas are relatively simple, completely correct results require sophisticated techniques, e.g., the renormalization group. In order to maintain a simple physical narrative, particularly in the early sections of this article, we rely considerably on physical arguments, with careful, more mathematical considerations relegated to the appendices. Second, we have not undertaken a critical review of the literature; we have not cited and summarized every paper. We prefer to tell a coherent story that allows the reader to quickly grasp the essentials, and reference only a few papers in the text. Other germane papers are listed in a bibliography under appropriate, albeit rough, topical headings. This review is divided into twelve sections. Section I1 is a discussion of the basics of superconductivity needed to understand the rest of the article, including a description of Josephson junctions, the RCSJ model of a junction, and the “washboard” model. Section I11 is an introduction to classical arrays and their behavior in and out of small magnetic fields at zero temperature. Section IV is a discussion of the thermodynamics of classical arrays, including the Kosterlitz-Thouless-Berezinskii transition. Section V is a discussion of the response of Josephson junction arrays to small DC currents, and Section VI is a discussion of AC current effects. Section VII presents finite-size effects, including size-induced free vortices. Section VIII is a discussion of arrays in small magnetic fields at nonzero temperatures, and Section IX presents disordered arrays, including bond, site, and positional disorder. Section X is a discussion of nonconventional dynamics, and Section XI a discussion of the dynamical properties of arrays when they are strongly driven by DC and AC currents. Section XI1 is an introduction to quantum effects, mostly concentrating on phase transitions. We do not discuss several very interesting areas, including the role of two-dimensional arrays in studies of turbulence and of granular metals, the possibilities for observing such things as the quantum Hall effect, or the use of arrays as model systems for nonlinear mechanical and biological systems. These are very interesting areas, and we list several useful references in the bibliography. There have been a number of other review articles that complement this

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work. Perhaps the most extensive review of the Kosterlitz-Thouless-Berezinskii transition and the Coulomb gas model is that by Minnhagen,’ “The Two-Dimensional Coulomb Gas, Vortex Unbinding, and Superfluid-Superconducting films.” Others include “Superconducting Weak Links” by Likharey6 “Static and Dynamic Interactions between Josephson-Junctions’’ by Bindslev-Hansen and Lindelof;’ “Mutual Phase Locking in Josephson“Non-Equilibrium Coherent Vortex States Junction Arrays” by Jain et d.;* and Subharmonic Giant Shapiro Steps in Josephson-Junction Arrays” by Dominguez and Jose;’ and “Vortex Dynamics in Classical JosephsonJunction Arrays: Models and Recent Experimental Developments” by Ciria and Giovannella.” II. The Basics 1. SUPERCONDUCTIVITY AND THE JOSEPHSON EQUATIONS

A Josephson junction is a device that consists of two superconductors weakly coupled by a region which may be either nonsuperconducting or a weaker superconductor.’ When the nonsuperconducting region is an insulator, the device is called a superconductor-insulator-superconductor (SIS) junction (Fig. 2(a)), and when it is a normal metal, it is a superconductor-normal-superconductor (SNS) junction (Fig. 2(b)). Josephson junctions can also be fabricated by making a constriction in a film (a microbridge) (Fig. 2(c)), or by coupling through a weaker superconductor6 (SS’S). For the purposes of this review, we will mainly consider arrays of SIS and SNS junctions, because two-dimensional arrays have been made exclusively of these types. An electron micrograph of a single SNS junction is shown in Fig. 3. In the superconducting regions there is a complex order parameter

whose phase is &v) and whose magnitude squared lY(r)I2 gives the local P. Minnhagen, Rev. Mod. Phys. 59, 1001 (1987). K. K. Likharev, Ret.. M o d . Phys. 51, 101 (1979). J. Bindslev-Hansen and P. E. Lindelof, Rev. M o d . Phys. 56, 431 (1984). A. K. Jain, K. K. Likharev, J. E. Lukens, J. E. Sauvageau. Phys. Reports 109, 309 (1984). D. Dominguez and J. V. Jose, Intl. J . M o d . Phys. 88, 3749 (1994). l o J. C. Ciria and C. Giovannella, J . Phys. Cond. M a t t . 10, 1453 (1998). I ’ B. D. Josephson, Phys. Lett 1, 252 (1962). l 2 M. Tinkham, Introduction to Superconductiuity, McGraw-Hill, New York (1996).

’

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-0.6pm

+

- 6.Opm f

FIG. 2. Schematic diagrams of various types of Josephson junctions. (a) A superconductorinsulator-superconductor (SIS) tunnel junction, where the barrier is a nonconductor; (b) a superconductor-normal-metal-superconductor (SNS) junction; and (c) a microbridge, where the barrier is a small constriction.

density” of superconducting “particles” n,*(r):

The superconducting “particles” are Cooper pairs with charge q* = & 2e (depending on whether the carriers in the normal state are electrons (-) or holes (+)) and mass in* = 2m,, twice the mass of the e1ectr0n.l~ The Cooper-pair current density in a superconductor flows without dissipation. The standard quantum-mechanical equation for the current holds for the supercurrent J,:

l3

B. Cabrera, H. Gutfreund, and W. A. Little, Phys. Rev. 825, 6644 (1982).

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273

FIG.3. An electron microscope picture of a superconductor-normal-metal-superconductor junction consisting of a gold normal region (light colored) underlying two niobium “banks” (dark region). The gap distance is about 0.45 micrometers and the junction width is 1.5 micrometers.

where A is the magnetic vector potential. This expression can be written in a useful form:

where Y = lYlei4. Equation (2.4) shows that supercurrents are caused by a gauge-invariant combination of phase gradients and vector potentials. Supercurrents can also flow through Josephson junctions, either by tunneling through an insulating barrier or by diffusing through a normal

274

R. S . NEWROCK ET AL.

metal. Whatever the mechanism for current transport, the supercurrent is is given by the Josephson equation,'

is

= i,

sgn(q*) sin

+2

- 4' -

4*h

112

A .dr) .

Here 4' and 42 are the phases of the order parameters of the two superconductors coupled by the weak link, and i, is the maximum supercurrent the junction can carry. (Note that we will use i for currents in a single junction and I for currents in an array.) Wk explicitly include the sgn(q*) term in Eq. (2.5) to reflect the fact that the same phase difference causes currents in hole-like and electron-like superconductors to flow in opposite directions. We will drop the sgnq* in subsequent equations and assume q* = +2e- i.e., the carriers are holes. In the limit where the gauge-invariant phase difference yZl,

in Eq. (2.5) is small, the sine may be replaced by its argument and Eq. (2.5) can be seen to be a finite-difference version of Eq. (2.4). The sine in Eq. (2.5) ensures that changes in the phases 4 of 27t will not have any measurable consequences. A second equation is needed to describe voltages. This is the second Josephson equation,'

which states that the voltage V2, across a Josephson junction is proportional to the time derivative of the gauge-invariant phase difference. This result is a consequence of the quantum-mechanical principle that the time derivative of the phase is proportional to the energy of a state; thus, the time derivative of a phase difference is proportional to the voltage in a charged system. It will be useful to have an expression for the energy stored in the supercurrents in a Josephson junction. The time integral of the voltage times the current gives this energy. Combining Eqs. (2.5) and (2.7) in this way, we

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

275

find the energy stored in the supercurrent, E,

=

-

hi, - c o s ( Y ~ ~ ) -E,cos(~~,), 2e

(2.8)

where E , is called the Josephson coupling energy. Equation (2.5) represents only the supercurrent that flows through a junction. There are other “channels” that allow current flow. Any junction will have a mutual capacitance C because it consists of two separate pieces of metal. When there is a time-varying voltage across the junction, there will be a displacement current id,

.

Id =

c-dV = hCd2yZl -

dt

~

2e dt2 ’

where the second equality holds because of Eq. (2.7). The capacitive channel can also store energy, (2.10)

Finally, a normal current can flow across the junction by tunneling of normal carriers from one electrode to the other (if the barrier is an insulator), by the flow of normal carriers in the barrier (if it is a normal metal), or by an external shunting resistor (which is often added to tunnel junctions to create nonhysteretic behavior). This current is often approximated by an ohmic relation, (2.11)

where R , is the resistance and we have again used Eq. (2.7) for the voltage. Equation (2.11) represents a dissipative channel and thus does not lead to any energy storage analogous to Eqs. (2.8) and (2.10).

2. THERCSJ MODEL The “resistively-capacitively” shunted junction (RCSJ) model combines the channels described above for the supercurrent, the displacement current, and the normal current into a circuit model.’4*’ An equivalent circuit for l4 Is

W. C. Stewart, Appl. Phys. Lett. 12, 277 (1968). D. E. McCumber, J . Appl. Phys. 39, 3113 (1968).

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R. S . NEWROCK ET AL.

FIG.4. Equivalent circuit diagram for the RCSJ model. From left to right are the supercurrent, capacitive, and resistive channels.

this model is shown in Fig. 4. We will first discuss this model at zero temperature so that the effects of noise can be neglected. Since the channels are parallel, the currents add; thus Eqs. (2.5), (2.9), and (2.1 1) combine to give h 4 2 , hCd2YZl (2.12) i = i, sin(y,,) + 2eR, dt 2e dt2 ’ ~

+--

where i is the total current flowing through the device. The analogous equation for the Hamiltonian is obtained by adding Eqs. (2.8) and (2.10), (2.13)

Equations (2.7), (2.12), and (2.13) are good semiclassical descriptions of the behavior of small Josephson junctions, but they do have limitations. In the first instance, if the junctions are too large, the gauge-invariant phase differences y z l will depend on position along the junction, and this leads to a wave-like equation for yZl (see, for example, the texts by Barone and PaternoI6 and by Tinkham.” We have implicitly assumed that y,, is uniform in the junction. This means that the junction is small compared to l 6 A. Barone and G . Paterno, Physics and Applications of the Josephson Effect, John Wiley, New York (1982).

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a characteristic length called the Josephson penetration depth, which is of order 1 mm for typical junction parameters. We have also assumed that the normal current channel, represented by the resistance R, in Eq. (2.11), is truly ohmic, which is correct in SNS and shunted SIS junctions and microbridges but incorrect in detail for unshunted SIS junctions. Finally, our equations are classical in nature. This means that we have implicitly assumed that the fractional uncertainties in canonically conjugate variables are small. Equations (2.7), (2.12), and (2.13) may be viewed as describing a system with coordinate y z l and momentum ACV/2e. This leads to an uncertainty relationship of the form 2e AVAyB-. C

(2.14)

Thus, if the capacitance is small, the Hamiltonian of Eq. (2.13) must be treated quantum mechanically. For the moment, we will assume the classical limit. Quantum effects will be dealt with in Section XII. There is a large body of literature dealing with the RCSJ model for Josephson junctions, and we will not review it in detail here (see Ref. 16 for more on this subject). We will, however, briefly show how the model is used to qualitatively describe the current-voltage (IV) characteristics of single junctions, because the model is very useful for understanding the properties of arrays. To start, consider the case where DC current i, < i, is applied to a junction at temperature T = 0. There is a solution to Eqs. (2.5) and (2.6), yzl

= sin-'(i,,/iC),

(2.15)

that is constant in time. From Eq. (2.7) we see that this is a zero-voltage solution, so the RCSJ model correctly predicts the existence of a lossless supercurrent through the barrier. In the opposite limit, where i, >> i, and thus the time-averaged voltage ( V ) is large, Eq. (2.7) says that dy,,/dt is large, so the sine and second derivative terms in Eq. (2.12) will have time-averaged values of zero. The remaining terms then reduce to ( V ) = iR,, so that in the high-current limit a junction is ohmic. When a constant current is sent through a junction, the voltage measured on conventional voltmeters is this average voltage ( V) since the high-frequency components implied by Eq. (2.7) are beyond the range of most amplifiers. In SNS junctions, the capacitive term in Eqs. (2.12) and (2.13) may be neglected because the RC time constant is very short. When the capacitive

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term can be neglected, the junction is said to be overdamped, which considerably simplifies the analysis. For example, when the junction is biased with a constant current, Eq. (2.12) may be integrated directly, yielding the simple form14*l

< ic i,, > i, iDC

= R,(iiC

-

i;)'''

(2.16) '

The resulting average voltage ( V ) versus the current i,, curve is shown in Fig. 5. It is seen that there is no average voltage for iD, < i,, with a smooth crossover to ohmic behavior at higher currents. When i,, > i, part of the

3.0 I

I

10

I

I

I

I

I

I

I

I

I

I

I

I

I

864-

"ct

2-

0

FIG. 5. (a) The normalized time-averaged voltage versus the normalized current for an overdamped (C = 0) junction, from the RCSJ model. (b) The derivative of the normalized time-averaged voltage versus the normalized current for the IV characteristic in (a).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

5

0

10

15

279

20

z FIG. 6. The instantaneous normalized voltage from the RCSJ model (with C = 0) as a function of the reduced time (T = t(h/2ei,R0)for three values of the external DC current; from bottom to top: ,,i = 1.2ic, i,, = 1.7ic, and ,,i = 4i,.

current has to flow through the resistive channel, and there will be an average voltage across the junction. The instantaneous voltage will be time-dependent since, from Eqs. (2.5) and (2.7), yZ1 will change in time. For currents slightly larger than i,, the voltage waveform is a succession of pulses, as shown in the lowest curve in Fig. 6. At high currents the voltage waveform has a substantial D C component and an almost sinusoidal AC component, as shown in the top curve in Fig. 6. 3. THE WASHBOARD MODEL

The current-voltage characteristics of Josephson junctions can be qualitatively understood from a simple mechanical analogue. This analogue will also be very useful in understanding array properties. Equations (2.12) and (2.13) can be viewed as describing a “phase particle” with a “position” given by y 2 , and a “velocity” given by dy,,/dt. The particle moves in a potential of the form

u,=

-

(2.17)

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R. S. NEWROCK ET AL.

+

with a total current i = i, iA,. The DC current has been integrated with respect to time (Eq. 2.12) and included in the potential energy. The particle has a “mass” M , given by M

=(;)

2

c,

(2.18)

and there is a viscous damping force acting on the particle: (2.19) In this analogy an applied AC current is interpreted as an external timedependent force, F,,,

h 2e

= - iA,.

(2.20)

This “tilted washboard” model for U , is shown in Fig. 7 for various values of the DC current. The analogy is summarized in Table 1. The current-voltage characteristic when ,i = 0, described earlier for the overdamped case, can readily be understood in terms of this model. When . . I = I,,= 0, the washboard is horizontal, as shown in Fig. 7(a). The particle will settle down into one of the minima in the potential, where y Z l is an integer multiple of 2n. This gives ( V ) = 0 from Eq. (2.7) and i = 0 from Eq. (2.12). As i is increased, the washboard is “tilted” by the second term in the potential energy, Eq. (2.17). If the tilt is not too great, there will be relative minima in the tilted potential so that the particle will assume a new static equilibrium position given by Eq. (2.15). This situation is shown in the washboard model in Fig. 7(b). Once ,,i > i,, the potential no longer has minima, only inflection points, as in Fig. 7(c), so that the particle “rolls down the washboard.” These “running states” have ( d y , , / d t ) # 0, so there is a nonzero average voltage, as given by the time average of Eq. (2.7). For high currents the damping dominates and the average velocity ( d y 2 , / d t ) = 2eiR,/h. For intermediate current values the particle has a nonzero average velocity, which is modulated by the periodic potential as it rolls down the washboard, as in Fig. 7(c), leading to the waveforms shown in Fig. 6(a) and 6(b). This mechanical analogue also allows us to understand the main difference between the current-voltage curve for overdamped (SNS) and underdamped (SIS) junctions: The curve for an SIS junction is hysteretic. In both

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

28 1

2

0

-2 -4 -6

-2-8

4z

-10

-12 -14 -16 -18

On:

2n:

4n:

6X

y21 FIG. 7. Washboard picture of the RCSJ model for a Josephson junction. The normalized washboard potential energy (Eq. (2.1 7)) is plotted versus the gauge invariant phase difference across a single junction. Curve (a) is for,,i = 0, with the equilibrium position of yZ1 a multiple of 2n. Curve (b) is for 0 < i,, < i,, for which, since there are stable minima, ( V ) = 0 (from Eq. 2.7). Curve (c) shows the potential for ,,i > ic, which has no time-independent solutions.

cases, when the current is increased from zero, the particle remains trapped in its minimum until i,, > i,. When the current is decreased from above i,, the overdamped and underdamped cases differ significantly. In the overdamped case the particle has no mass, and thus no inertia, so it becomes immediately retrapped at a current i, = i, as soon as a minimum appears in the tilted washboard potential. In contrast, in the underdamped case it is necessary to reduce the current to a retrapping current i , < i,, since the

TABLE1. RCSJ MODELANALOGIES Mechanical mass damping position

Junction

282

R. S. NEWROCK ET AL.

particle now has a mass and therefore a nonzero “momentum,” and can “overshoot” the minimum. At or below the current i, the particle is stopped and the junction returns to the zero voltage state. This leads to a hysteretic current-voltage curve for an underdamped junction, as shown in Fig. 8. The McCumber parameter, pc, is a measure of the degree of damping in a junction, (2.21)

When j?, << 1 the junction is said to be overdamped and the phase-particle moves as a massless particle. In the underdamped opposite limit, the energy stored in the capacitor must be taken into account. The terms overand underdamped arise because Eq. (2.12) is mathematically identical to the equation of motion of a damped pendulum.

FIG. 8. Normalized voltage versus normalized current for an underdamped (C # 0, B, > 1) junction (A is the superconducting energy gap). As the applied current is increased, the junction switches to the voltage state at i, as expected, but when the current is reduced the junction does not return to the zero voltage state until the applied current is less than a retrapping current .,i The dotted lines show the normal state resistance and the dynamic resistance at retrapping.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

283

111. Classical Arrays: T = 0 4. ZEROMAGNETIC FIELD

Consider now the simplest two-dimensional Josephson array, a square lattice with lattice constant a, on which there are Josephson junctions in each vertical and horizontal bond, as in Fig. 9. The array size is M by N , where N is the number of rows ofjunctions along the current direction and M is the number of columns perpendicular to the current. Each of the junctions is connected to two superconducting “islands,” which are the nodes of the lattice. The total external current I = Mi is fed into leads connected to the islands on the left and extracted from the islands on the right. Each lead carries a current i, so that current flow into and out of the array is uniform. A real sample is shown in Fig. 10, an electron micrograph of an SNS array. If a current I flows uniformly into the edge of an array at zero temperature and zero magnetic field, one possible situation is for the current to flow through the array uniformly from left to right. If this occurs, each horizontal junction has a current i = Z/M, each vertical junction has i = 0, and the total voltage across the array will be ( y o , ) = N‘C/;where V is the voltage across any one junction in the array. The current-voltage curve will thus be a rescaled version of a single junction curve.

V A

V A

V A

V

A

*

V

V

V

A

FIG.9. Josephson-junction array schematic. The crosses represent the individual junctions connecting superconducting islands, which are represented by the squares. The lattice constant is a and the array has M x N junctions, where M is the number of rows and N the number of columns. The arrows represent current injection and extraction.

284

R. S. NEWROCK ET AL.

FIG. 10. Electron micrograph of a portion of an SNS array made from niobium crosses (dark regions) on a gold underlayer (light regions). The crosses are 10 pm center to center; the cross arms are 1.5 pm wide; and the gaps between crossanns, which are the junctions, are of the order of 0.45pm.

While the above solution is correct for a lattice of linear circuit elements such as resistors, it must be used with care on Josephson junctions. From Eq. (2.7), when ( V ) # 0 we see that the phases will advance with time, so that each i will vary with time from Eq. (2.5). Whether or not the uniform solution described above will occur depends on the stability of this solution for this system of coupled nonlinear circuit elements. In addition, if i, is too large, the array equations lead to an analogue of the Meissner effect in superconductors, leading to a “bunching up” of the current at the edges of

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

285

the array, even for i < i,. If i, is small, the uniform solution has been shown to be stable at zero temperature, where the array will behave like a single junction with critical current i, = Mi, and resistance NRJM. Experimental curves’ for ( V ) versus I and d( V ) / d I versus I for an SNS array are shown in Fig. 11. Data are shown for both zero magnetic field, f = @/Q0= 0 (a is the external flux through a plaquette of the array and (Do(=h/2e) is the flux quantum), and a small field f = 0.04. Here f is referred to as the frustration for reasons that will become clear in the next section. Figure 1 1 may be compared to Fig. 5. The single-junction and the array curves in zero field are qualitatively similar, although the array curves

’

”

M. S. Rzchowski, S. P. Benz, M. Tinkham, and C. J. Lobb, Phys. Rev. 842, 2041 (1990).

a

1

2

3

4

CURRENT ( mA )

1

O

f

2 3 CURRENT ( mA )

4

FIG. 11. The dynamic resistance (d( V ) / d l ) versus current at 2.5 K for normalized flux per unit cell f = @/m0 = 0 and 0.04. The peak at I,, = 2.1 mA indicates the single junction critical current. The inset shows ( V ) versus I obtained by direct integration of the data. The application of a small magnetic field introduces a nonzero differential resistance at low currents (see arrows), indicting depinning of field-induced vortices. (From Ref. 17, Fig. 8.)

286

R. S. NEWROCK ET AL.

are much more rounded than the single-junction curves. This difference is due to both thermal fluctuations and inhomogeneity in the junctions in the arrays, because it is impossible to make all the junctions with identical properties. Were this the only physics associated with Josephson arrays, there would be no need for this review. However, when H # 0 or T # 0, vortices will be present in the system and these lead to a rich variety of interesting effects. 5. NONZERO MAGNETIC FIELD

Many of the interesting properties of superconductors and junctions occur because of the vector potential term in the current-phase relations -Eq. (2.4) for a superconductor and Eq. (2.5) for a junction. For a superconductor in a small applied field, this vector-potential term leads to the Meissner effect by causing a current to flow that opposes the external field, leading to no magnetic field inside the superconductor. At higher fields, when screening is inadequate to exclude the field completely, type-I1 superconductors allow field to enter a bit at a time in the form of vortices." These vortices consist of supercurrents that circulate around a small core of normal material and have a large impact on the electrodynamic properties of superconductors. As mentioned after Eq. (2.6), when applied fields are small the currentphase relationship for a junction, Eq. (2.5), is a finite-difference version of the differential current-phase relationship in a superconductor, Eq. (2.4). Thus, for small fields and currents, where the derivative and difference are each small, the equations are formally equivalent. This suggests that many of the effects that occur in superconductors should occur in arrays. Of particular interest to this review are vortices in Josephson junction arrays. To understand what an array vortex is, consider the blow-up of one cell or plaquette of an array, shown in Fig. 12. Each point in each island in the figure is characterized by a phase, which can only be defined modulo 27c (see Eq. (2.1)). Consider a closed path in the array, such as the one around the single plaquette shown in Fig. 12. Around any such path we sum the phase differences across the junctions and across the superconducting islands. From Eq. (2.5) the phase difference across the junction connecting superconducting island i to island j can be written as

where

4i and 4 j are the phases at the edge of the two superconducting

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

287

FIG. 12. Schematic of a single-array plaquette showing the ith and jth islands; the phases +i and + j are the phases at the adjacent edges of the two islands.

islands. The phase difference across the superconducting island from point j to any point r along the dotted lines shown in Fig. 12 is given by Eq. (2.4), (3.2)

To sum around the closed path, we add the contribution for each junction, given by Eq. (3.1), and for each island, given by Eq. (3.2). Around this path the total phase difference has to change by 27cn, where n is an integer, because the phase is defined modulo 2n. Thus,

The first term is the sum of the phase differences across the junctions traversed by the path. The second term is the sum of all of the phase differences across all of the islands traversed by the path. The last term is the integral of the vector potential around the entire path, which is the total flux enclosed by the closed path, It is customary to assume that in the islands one can always find a path with J, = 0 because of the Meissner effect. This assumption will be correct as long as the dimensions of the islands are all larger than the London penetration depth, or the currents flowing in the

288

R. S. NEWROCK ET AL.

islands are much less than the island critical current. This makes the second sum in Eq. (3.3) equal to zero. With these simplifications we obtain

where (Do= h/2e = 2 x 10-’5Tm2, or, using Eqs. (2.5) and (2.6),

While the total phase difference around any closed path is constrained to be 2 x 4 for each physical situation the equilibrium value of n will be that which minimizes the free energy of the system. To calculate the energy (which equals the free energy when T = 0), we add the individual junction energies of Eqs. (2.8) and (2.10) to obtain

where we have used Q0 = h/2e for notational convenience. Here E j is the Josephson coupling energy of Eq. (2.8) and ( i j ) means summing over nearest neighbors. The first term in the Hamiltonian is the energy associated with the charging energy of Eq. (2.10). If the gauge-invariant phase differences do not depend on time, we may neglect this first term. Thus, for calculating static properties only, we may use the simpler Hamiltonian, H=-CE,cos

(3.7)


Equations (3.5) and (3.7) enable us to understand the possible static phase configurations allowed by the system, including the existence of vortices as natural excitations in two-dimensional Josephson arrays. In what follows we will consider arrangements of the phases that are solutions to both Eqs. (3.5) and (3.7) and determine which will have the lowest energy. Among other things, this will allow us to show why vortices occur in two-dimensional Josephson arrays in the presence of a magnetic field.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

a. The Ground State for B

289

=0

When the applied magnetic field B = 0, the total flux in the array Qtotal We can thus choose a gauge where A = 0 and first consider the case n Then Eq. (3.5) tells us that

= 0. = 0.

To satisfy Eq. (3.8) and simultaneously minimize the energy of the system, Eq. (3.7), all of the phases must be equal, as shown schematically in Fig. 13, whefe the phase is represented by a unit vector whose angle with respect to the x-axis is the phase. This is analogous to the ground state of a ferromagnet, where all of the magnetic moments in the system point in the same direction. In this case, since all the phases are equal, the energy will be

Eo,o = - 2 M N E J ,

(3.9)

where the notation is Consider now the same case with B = 0 and A = 0, Qtotal = 0, and choose n = 1. The configuration for this case will have

FIG. 13. Schematic of the uniform phase configuration state (the f = @/m0 = 0 ground state). The angle between the arrow on an island and the +x-axis represents the phase of the superconducting order parameter of that island. The junctions connecting the islands are omitted for simplicity.

290

R. S. NEWROCK ET AL.

to be like that of Fig. 14(a) since, according to Eq. (3.5), (3.10)

and the phase has to change by 2n: as we go around any closed loop containing the center plaquette in the array. In going around the smallest closed loop around the center of Fig. 14(a), for example, all the phase differences between adjacent islands will be equal to 4 2 . These phase differences will induce circulating supercurrents, forming a whirlpool of current. This is a vortex. It is possible to calculate'* the energy of such an

(b)

FIG. 14. Phase configuration for a vortex. (a) low-energy state; (b) high-energy state. The dots indicate the center of the vortex. (From Ref. 18, Fig. 2.)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

isolated vortex in a large square array of length or width L N a 4 00 to find E l , o E nE,ln

(3 -

- 2MNEJ.

291

= Ma =

(3.11)

For the particular case we are discussing-zero applied magnetic field the vortex configuration, while possible, will not be energetically favorable because the energy of this state will be larger than that of the n = 0 ground-state energy, Eq. (3.9). b. The Ground State for Small Magnetic Field For = 0, the presence of a vortex raises the energy of the array. We will now show that in the presence of a magnetic field the existence of a vortex in the system might be energetically favored. Consider, for example, the case of a uniform applied magnetic field such that (Dlotal = BMNa2 = (Do (that is, one flux quantum in the entire array, (Dtotal/(Do = 1). If we impose the uniform-phase solution, with n = 1 in this case, we will get a higher energy than in the previous section because the presence of a quantum of flux and equal phases on all of the islands (Eq. (3.5)) requires that (3.12) Thus, the energy of the system will be larger than the energy found earlier for the n = 0 configuration in zero magnetic field. The energy in this case can be shown to be Eo,l Z nEj In

(k)

-

2MNE,.

(3.13)

But consider now the n = 1 configuration of a vortex. In this case (3.14) junctions

The phase differences bj - Sican adjust to cancel the vector-potential term, leading, according to Eq. (3.7), to a configuration with a vortex in the sample and with a lower energy than that of the uniform-phase energy of "

C. J. Lobb, D. Abraham, and M. Tinkham, Phys. Rev. 827, 150 (1983).

292

R. S . NEWROCK ET AL.

Eq. (3.13). The phases can adjust themselves to completely cancel the vector-potential term in Eq. (3.14); this is the phase configuration shown in Fig. 14(a), leading, when a flux quantum is present, to an n = 1 energy, E l , l = -2MNE,.

(3.15)

c. Vortex Depinning and the Critical Current In the previous discussion we assumed that the vortex center was located in the center of a plaquette, as in Fig. 14(a). However, a vortex solution can also be constructed with the center in any other location. In particular, there is the solution, shown in Fig. 14(b), with the vortex center located at one of the junctions between two islands. While both Fig. 14(a) and Fig. 14(b) are valid vortex configurations, Fig. 14(a) is a local minimum in the energy while Fig. 14(b) is a saddle point and thus unstable. The energy difference between these two configurations has been calculated’8 and found to be E,,

%

0.199 E j

(3.16)

in the limit of large lattice size and for a square lattice. This energy represents a barrier for vortex motion. In order for the vortex to move from the minimum in one unit cell to an adjacent minimum, it must traverse the saddle point. For a triangular lattice of junctions, the barrier is found to have a much smaller value,18 0.043 E,. The full two-dimensional vortex potential may be approximately calculated by finding the energy of a single junction for a vortex center located at all points in the vicinity of that j u n ~ t i 0 n . lThis ~ results in the “egg crate” potential of Fig. 15, which has as its fundamental period the lattice spacing a. The equilibrium position of the vortex will be any of the lowest energy positions that are equivalent to those described in Fig 14(a). We note that the potential shown in Fig. 15 is quite close to a sinusoidal potential if motion is confined to a principle (2 or 9) direction. If a uniform external current is imposed on the array, there will be a “Lorentz force” acting on the vortex transverse to the current. The physical origin of this force can be seen with the aid of Fig. 16. The superposition of the circulating vortex current and the externally applied current leads to a current gradient in the direction perpendicular to the applied current. This yields a force analogous to a Bernoulli force in fluids or to the Lorentz force on a current-carrying wire in a magnetic field. The force has a magnitude given by

F , = mojZd=

(A)(!). 2e a

(3.17)

293

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

FIG. 15. The egg crate pinning potential for a vortex in an array.

A

A

A

A

A

A

A

~

A

~

~

)

~

~

-

-

F

~

-

+

+

-

-

L

~

-

Y

c

Y

Y

~

~

~

Y

Y

Y

Y

.

I

A

A A A A ~ - I - - ~ - + - - L Y Y ~ Y Y Y ~

N

N

~

~

A

~

-

-

w

~

Y

Y

~

~

~

r r r f f ~ m ~ ~ ~ ~ ~ ~ K Y , - C - V ~ \ L \ ~ I I L ~

frrf

f

f

+\\\..

\ \ \ \ U '

f l r j f f

f f U f ) r

+ f # i \ ' '

'~+tf+y--#+\+k''

f ! f f t t k - - - H / + + + \ \ ' /

f

t

t f

t

\

-

M

/

#

f

f

t

\

. \

-

-

,

/

+ 4

4 \

+ &

b +

\

. \

/

t

t

t

l

t

\

-

*

l

l

\

t

4

\

/

/

#

l

I

*

.

.

.

O

,

l

\

\

\

/

,

'

'

<

>

.

,

>

.

\

,

\

,

, , .

,

FIG. 16. The superposition of the circulating vortex current and an externally applied transport current. (In this figure arrows indicate the magnitude and direction of circulating current, not phases. The transport current is moving to the right.) The combination yields a current gradient perpendicular to the transport current, leading to a force analogous to a Bernoulli force in liquids or the Lorentz force on a current-carrying wire in a magnetic field.

294

R. S. NEWROCK ET AL.

The vortex will have to overcome the barrier E,, (Eq. (3.16)) in order to be depinned from the lattice. Thus, the critical current, when there is a field-induced vortex present, will be determined by the barrier height E,,, together with the shape of the potential, in a manner analogous to the washboard model discussed in Section 11.3 -the external current can be made large enough to overcome the force associated with the egg crate potential. There are differences, however, between a vortex moving in the array lattice and the phase evolving in a single junction. The vortex potential is a two-dimensional potential, and the vortex is a real object moving in space. This is in contrast to the washboard potential for a single junction, a one-dimensional mechanical model in which the phase difference across the junction is only analogous to the position of a particle moving in a periodic potential. Since the potential of the vortex in the x direction is given approximately by Vp,(x) = - %2C O S ( T ) ,

(3.18)

the pinning force is given by (3.19) The critical current is exceeded when the Lorentz force of Eq. (3.17) exceeds the maximum restoring force as calculated from Eq. (3.19). This leads to a vortex-depinning current (normalized to the number of junctions in the width of the array) of (3.20) The earlier result for the barrier height, E,, (3.20), yields

I=.,

= 0.1 Mi,.

= 0.199 E,,

when used in Eq. (3.21)

It is possible to measure the depinning critical current” and thus infer the value of E,,. The transition to the finite voltage state is shown in Fig. 17. For now we are interested only in the case of a very small external l9

S. P. Benz, M. S. Rzchowski, M. Tinkham, and C. J. Lobb, Phys. Rev. B42, 6165 (1990).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

11

I

I :

1

295

/L\

.

.

.

.

1

.

.

.

.

,

FIG. 17. The dynamic resistance versus current for a 1000 x 1000 array at T = 2.09 K. for three different perpendicular magnetic fields. The vortex depinning current, for which the dynamic resistance is no longer zero, is about 0.7 mA. (From Ref. 19, Fig. 1.)

magnetic field. When there is considerable rounding due to thermal fluctuations, it follows from studies of the effects of thermal fluctuations in single junctions” that the peak in dV/dI represents a very good estimate of the nonfluctuating critical current of an array, I,. The depinning current is that current for which the differential resistance is no longer zero, which, from the figure, occurs for I z 0.7mA. Since the peak in dV/dI occurs for I = 7 mA, ICTv x 0.1 I,, which is consistent with the value for the depinning current, Eq. (3.21). One can obtain a similar measure of the depinning critical current by measuring the appearance of a voltage as a function of both magnetic field and temperature,’l as shown in Fig. 18. This measurement would only be correct for low values of the magnetic field because the estimate of I,,v was obtained by assuming that a single vortex was present in the array. At higher fields interactions between vortices have to be taken into account in V. U. Ambegaokar and B. I. Halperin, Phys. Rev. Lett. 22, 1364 (1969); Errarurn, Phys. Reu. Lett. 23, 274 (1969). H. S. J. van der Zant, F. C. Fritschy, T. P. Orlando, and J. E. Mooij, Phys. Rev. 847, 295 (1993). 2o

296

R. S. NEWROCK ET AL.

FIG. 18. The depinning current normalized to the temperature-dependentcritical current as a function of temperature for f = 0.1 (squares), and the depinning current versus frustration for a fixed temperature, T = 10 mK (circles). The dashed lines give the expected depinning current O.lIc. (From Ref. 21.)

the problem. The figure shows that the critical current is essentially independent of temperature up to 1 K. At the same time, the depinning currents were higher than the expected value of 0.1 I , by about a factor of 2. These high values of the depinning current I,," are most likely due to screening effects. d. Vortex Motion above The Critical Current In the previous section, we were able to ignore the vortex mass because the vortex is stationary up to the depinning current. Once the depinning current is exceeded, however, the motion of the vortex will be determined by its mass, by the damping that the vortex feels as it moves, and by the form of the potential. Since the vortex potential of Eq. (3.18) is analogous to the potential for the phase difference of a single junction when the vortex motion is along a principle direction, the full equation of motion for a massive vortex will also be quite similar to that of the single junction discussed earlier. We now calculate the various terms in such an equation." Note that arrays as well as junctions can be overdamped or underdamped, depending

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

297

on whether or not p, is much greater than or much less than 1. When p, << 1, vortices can move in an array as massless particles and the capacitance of the junctions can be neglected. This is the case for SNS arrays. When p, >> 1, the arrays are underdamped, dissipation is low, and the vortex inertia is important -the electrical energy stored in the junction capacitors must be taken into account. When a vortex moves through the lattice, it creates phase differences that vary in time and voltages are created via the Josephson relation, Eq. (2.7). The voltages induce currents in the resistive channels of the junctions, which is a dissipative process. The power dissipated is proportional to the voltage squared and thus to the vortex velocity squared. This results in a viscous force on the vortex FD

=

-v,,

(3.22)

where v, is the vortex velocity. The power dissipated by the vortex has to be equal to the sum of the power dissipated across all of the junctions in the array, which for a uniform array will be (3.23) Consider a vortex that moves from one minimum energy position to an adjacent one. As the vortex moves across the junction, it will increase the gauge-invariant phase difference by 7c, as can be seen from Fig. 19(c) and (d). From the Josephson relation this implies an average voltage drop of

6)

(v,), where ( v , ) is the average vortex velocity. The square lattice of

resistive channels presents an effective resistance" average coefficient of viscosity will be given by

of RJ2, so that the

(3.24) where we have ignored the differences between (vLf) and ( v , ) ' . For a triangular lattice the effective resistance is R,/3 and qA = (3/2)qn. Next, we consider the mass of a vortex. As the vortex moves, the voltages generated also induce charge on all of the capacitances in the array. This 22

S. Kirkpatrick, Rev. Mod. Phys. 45, 574 (1973).

298

R. S . NEWROCK ET AL.

charge stores energy, with the amount of energy being stored proportional to the square of the vortex velocity. Thus, in order to move, the vortex must acquire energy proportional to its velocity squared. This energy is not dissipated as heat and so acts like a kinetic energy of the vortex. Equating this stored electrical energy to the kinetic energy of the vortex gives (3.25) where we have assumed that the capacitance C is the same for all junctions. Here, the mass of the vortex is M , . In analogy to Eq. (3.24), this leads to Mvn -

>(;.

2

(3.26)

For the triangular lattice a similar argument leads to M , , = (2/3) MVo. Combining Eqs. (3.17), (3.19), (3.22), and (3.26), we can write an equation of motion for a vortex moving in the 9 direction: (3.27) where the external current flowing in each junction in the 2 direction is i. Substituting the coefficients yl, E,,, and M , , Eq. (3.27) becomes

This is isomorphic to the RCSJ equation for a single junction, Eq. (2.12), if we associate ,.i 2R,, C/2, and 2zyla in the array with i,, R,, C, and y in a single junction. Thus, the IV characteristics of an array containing a single vortex at T = 0 will be rescaled versions of the IV characteristics of a single junction at T = 0. We now use Eq. (3.28) to derive two very important results concerning the electrical response of an array containing a single vortex (which may be induced by an external magnetic field or, as we will see below, by thermal fluctuations) and the response to a small density of vortices. We specifically consider the case C = 0, although the calculation has been done without this assumption.21 This simplifies the equation by reducing the order of the differential equation, Eq. (3.28). We imagine that the current i is sufficiently

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

299

large that the sine term in Eqs. (3.27) and (3.28) is unimportant. Taking the time average of what remains in Eq. (3.28) yields (3.29)

As Fig. 19 shows, when a vortex crosses the width W of a sample, the phase

mmmmmm

000000

FIG. 19. Schematic drawing of a vortex moving through an array. (a) All of the phases are (b) The vortex is just below the figure. (c) The vortex aligned and the vortex is at y = -a. has entered the sample. (d) The vortex has moved upward one lattice spacing from its position in (c). (e) The vortex is just above the figure. (f) The vortex at y = + a .It is seen that in moving through the array the vortex causes a phase change of 2n between the ends of the sample, +left - h l g h l .

300

R. S. NEWROCK ET AL.

difference between different ends of the sample advances by 271. If the vortex transit time is z, for the average voltage due to the motion of one vortex ( V , ) , Eq. (2.7) yields (3.30)

Since ( v , )

=

WIT,the last two equations yield

(3.31)

Using Eq. (3.24) to eliminate q and putting i/a = I/W into Eq. (3.31), we have a relation between the total voltage and total current in a sample containing one vortex,

(Ii . ) 2

(V,)

= 2R0

(3.32)

If there are a small number of weakly interacting free vortices per unit area nf in the sample, the total number of vortices in the sample will be nfLW Multiplying this by the right-hand side of Eq. (3.32), we get the total average voltage ( V ) as a function of nf and I , L ( V ) = 2R0(nfa2)- I . W

(3.33)

This is an important result. It allows comparison between a measured response of the array, voltage as a function of current, and a key microscopic parameter- the density of free vortices. Notice that everything in Eq. (3.33) either can be determined from the design of the sample (L, W and a) or is a measured normal-state property (Ro), so that a measurement of ( V ) and I determines n,.. A great deal of the remainder of this review will rely on Eq. (3.33). IV. Classical Arrays: T > 0

In the previous section we showed that at T = 0 Josephson junctions and Josephson junction arrays have a stable, time-independent state (dy, / dt = 0) for currents smaller than characteristic critical currents. This means,

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

301

from Eq. (2.7), that the voltage across the sample is zero: There exists a zero resistance state and a nonzero critical current. For larger currents, the phases evolve in time, leading to nonzero average voltages across the sample. In this section we look at how nonzero temperatures modify this picture. For a single junction we show that thermal fluctuations lead to a nonvanishing resistance for T > 0; there is no true zero-resistance state. For arrays the behavior is more complex and very rich. In zero magnetic field, in the thermodynamic limit, the system undergoes a true phase transition, from a resistive high-temperature state to a zero-resistance (but with zero critical current!) low-temperature state. This phase transition is of the KosJerlitz-Thouless-Berezinskii type.23*24 6. SINGLEJUNCTIONS

AT

NONZERO TEMPERATURE

To explore the nonzero temperature behavior of single junctions, we return to the washboard model (Fig. 7) and consider only the overdamped case for now (p, = 0, applicable to SNS junctions and arrays). At T = 0, when ,,i = 0, the washboard is not tilted (Fig. 7(a)) and the phase rests at one of the minima of the potential. For temperatures greater than zero, thermal fluctuations can feed energy into the system and cause the phase to move out of its minimum into an adjacent one.2o The energy is lost to dissipation. Since dy2,/dt is then not zero, there will be a time-varying voltage across the junction (Eq. (2.7)). As the washboard is not tilted at ,i = 0, thermal fluctuations will result in a random walk that is just as likely to cause the phase to move to the left as to the right, resulting in a zero average DC voltage, ( V ) cc ( d ~ ~ ~ / = d t0.) When a small positive current is applied, the washboard tilts, sloping downward toward positive y (Fig. 7(b)); the barrier to motion will be different in the two directions, and the random walk will tend to move the phase in a preferred direction, in this case to the right with greater y. This will lead to a nonzero time-averaged voltage, ( V ) cc (dy2,/dt) > 0. The energy the system gains from the thermal fluctuations, which enables it to go over the barrier, is rapidly dissipated by the damping forces, and the system immediately settles into the next minimum. The next series of fluctuations could push the phase over the next barrier, but there is also a (lower) probability that they could return it to the first minimum. The net result of many fluctuations is a diffusive process: The phase of the junction proceeds along the positive y-axis by a random walk over the series of 23 24

J. M. Kosterlitz and D. Thouless, J . Phys. C6, 1181 (1973). V. L. Berezinskii, Zh. Eksp. Eor. Fiz. 59, 907 (1970); Sou. Phys. J E T P 32,493 (1971).

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R. S. NEWROCK ET AL

barriers with a net motion in one direction. Each jump over a barrier results in a change of phase of 271, known as a “phase slip,” and the phase of the junction moves via a series of phase slips in the positive y direction. These thermally induced phase slips cause a nonzero average voltage. For low currents, i,, z 0, the height of the energy barrier E , over which the phase must jump is (from Eq. (2.17))

A reasonable guess for the apparent resistance of the junction in this case can be made by multiplying the only Gharacteristic resistance in the problem -the junction normal-state resistance R, -by the probability of a phase slip occurring, which we obtain by using the barrier energy of Eq. (4.1) in a Boltzmann factor, _ _E*

R ( T ) g Roe

k ~ T =

Qo -~

Roe

4

X~BT.

(4.2)

According to this expression, the resistance of a single junction is never zero except at T = 0, but is instead exponentially small at low temperatures. The known exact result’’ is essentially the same.2s This presents us with an interesting situation. At T = 0, the resistance is zero and the current, up to ic(0),is all supercurrent. However, for T > 0, the resistance is nonzero and, as soon as a current is applied, a small, but nonzero, voltage appears. 7. ARRAYSAT NONZERO TEMPERATURES

Earlier we discussed the comparison between a single junction and an array at T = 0 (Section 11.3). Based on that discussion it might appear that a 2 5 Ambegaokar and Halperin (Ref. 20) showed that this activated resistance is highly nonlinear in the temperature. It increases as the current approaches the critical current and has a nonzeo limiting value R , for small current given by

where I , is the modified Bessel function and R , is the normal-state resistance. When the argument of the Bessel function is much greater than 1 (low temperatures), the resistance has the exponential dependence

indicated in Eq. (4.2).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

303

two-dimensional array of Josephson junctions should have a temperaturedependent resistance similar to that of a single junction, Eq. (4.2)- that is, a nonzero resistance, decreasing exponentially as the temperature goes to zero. However, this simple picture, essentially a phase-slip model, does not apply to a two-dimensional array, a result that is easy to demonstrate. In an array, in order for a thermally activated phase slip to occur resulting in a nonzero voltage and dissipation, all M junctions in any column perpendicular to the current flow must slip at the same time. This means that the barrier against such phase slipping in an array is M times larger than that of a single junction. In the thermodynamic limit, M + 00, the barrier is infinite. This suggests (incorrectly) that an array of Josephson junctions should have zero resistance at any temperature below the transition temperature of the islands. A more subtle analysis leads to another conclusion. Low-energy longwavelength (thermal) excitations of the array, in which a small gradient occurs in the phase configuration shown in Fig. 13, can occur. These “phase waves,” analogous to spin waves in an XY magnet, can be shown to destroy long-range phase coherence at nonzero temperature^^^*^^^^' in two-dimensional systems, leading to zero critical current at nonzero temperatures. (See Appendix A for a more detailed discussion of this issue.) The Hamiltonian, Eq. (3.7), has no conventional long-range order except at T = 0, when all of the phases are equal. One can show that at T = 0 (Appendix A) the two-dimensional two-point phase-phase correlation function is a constant not equal to zero in the limit I -+ co. In other words, at T = 0 we have a state with long-range order. At high temperatures, in contrast, one can show that long wavelength fluctuations (“phase waves” or spin waves) cause the correlations to decay exponentially. This exponential dependence corresponds to short-range order, indicating a disordered phase at high temperatures. The new physics occurs at low, but nonzero, temperatures. Here the correlations decay algebraically- that is, they exist at any finite distance but are zero at infinite distances. One could imagine some sort of smooth crossover between the two states (i.e., no phase transition), but the possibility also exists for a phase transition from a high-temperature disordered state to a more ordered-, but not infinite-range, low-temperature state. This latter in fact occurs: The spin-wave picture, using two-point correlation functions, tends to ignore other possible thermal excitations (in particular, vortices) of certain two-dimensional systems, and the key to understanding the physics of phase transitions in two-dimensional arrays lies in understanding the crucial role played by thermally excited vortices. 26

N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).

’’ F. Wegner, Z . Phys. 206, 465 (1967).

304

R. S. NEWROCK ET AL.

When vortices are taken into account properly, one can define a new quasi-long-range order, called topological long-range order. One can show (Appendix A and below) that a critical temperature exists where the system changes from the high-temperature state with no long-range order to a low-temperature state with quasi-long-range order. The existence of this phase transition depends on the existence of thermally generated vortices. In the previous section we described vortices in an array, showing that at zero temperature a vortex state would not exist unless there were an applied field. However, in a two-dimensional system at nonzero temperatures there is thermal energy sufficient to induce vortices. Kosterlitz and ThoulessZ3 showed that, in many two-dimensional systems,’* thermally generated bound pairs of vortices of opposite circulation will occur at temperatures greater than zero. At the critical temperature, now called the KosterlitzThouless temperature, TKT,a phase transition occurs and free vortices appear. This is the famous Kosterlitz-Thouless transition. 8. ESTIMATES OF THE TRANSITION TEMPERATURE We begin by showing that the thermal generation of bound pairs of vortices is energetically favored, that there are no free vortices present at sufficiently low temperatures, and that free vortices will appear as thermal fluctuations above the Kosterlitz-Thouless temperature. From Eq. (3.9), the ground-state energy of the array with all the phases equal, and from Eq. (3.11), the energy of the system with a single vortex present, the energy required to add a single vortex to a two-dimensional array in zero magnetic field is

where L is the size of the system and a is the lattice spacing. The expression is valid for L large and A, >> L, where 1, is the penetration depth into the 0ne can also calculate array for magnetic fields perpendicular to the energy of two vortices of opposite circulation, bound together a distance For example, vortices in thin He4 films and screw dislocations in two-dimensional crystals. J. Pearl, Appl. Phys. Lett. 5, 65 (1964); J. Pearl, Proc. 9th Intl. Conf: Low Temp. Phys., eds. F. J. Milford, D. 0. Edwards, and M. Yaqub, New York, Plenum (1965), 566; J. Pearl, J . A p p l . Phys. 37, 2856 (1966). 30 M. Tinkham and C. J. Lobb, “Physical Properties of the New Superconductors,” in Solid State Physics, Vol. 42, eds. H. Ehrenreich and D. Turnbull, Academic Press, San Diego (1989), 91. 28 29

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

305

r apart; this has been shown to be’8929932 E,

= 2nE,

In

(:).

(4.4)

Equation (4.4) is valid for two vortices whose separation r is less than AL. In general, pairs with r << L occur since E , << E,, so that at any nonzero temperature it is much more probable that bound pairs of vortices will be thermally generated than single ones will be. An example of a bound pair of vortices is shown in Fig. 20. We will now use the energies of the vortices to estimate the critical temperature TKTin two ways. The change in the free energy caused by the introduction of a single free vortex in an array is AF

= E, -

TAS,,

(4.5)

where ASu is the positional entropy of the vortex in the array, ASu = k , In

($).

This leads to AF

= (nE, -

(3

2k,T) In - .

(4.7)

The probability of finding a single free vortex in a system of size L is proportional to the exponential of the free energy change of Eq. (4.7),

3 1 In a thin superconducting film,the region in which the diamagnetic shielding currents can flow is severely restricted, resulting in a perpendicular penetration depth much longer than in the bulk material. This penetration depth can be shown to be LL = L2/2d, where 1 is the bulk penetration depth and d is the film thickness (Ref. 29). A suitably altered definition also works where L3D,array= [@,,a/2np,i,(T)]”2 (Ref. 30). However, for our for arrays: LI = (L3D,array)2/ar purposes it is adequate to note that the logarithmic dependence of the interaction energy on vortex separation is only valid for separations less than some scale length, which for now we take to be very large. 32 The free energy of a pair of vortices has an additional term, -2pc where - p c is the core energy. In order for the Kosterlitz-Thouless theory to be valid, pc must be large. This ensures (from Appendix B) that the mean distance between vortex pairs is large-i.e., that we have a dilute gas of vortex pairs so that the interactions between pairs is sufficiently low that calculations to lowest order suffice.

306

R. S. NEWROCK ET AL.

FIG. 20. A bound pair of vortices. The arrows indicate the phases of the islands. (From Ref. 3, Fig. l.B.l.)

In the thermodynamic limit ( L = a),PI is zero at low temperatures and nonzero at high temperatures. That is, in a sufficiently large system, free vortices will not be present at sufficiently low temperatures but will be present for sufficiently high temperatures. For the large system, an abrupt change occurs when the exponent in the second part of Eq. (4.8) reaches zero; this defines the Kosterlitz-Thouless transition temperature TKTand gives us our first estimate for it:

(4.9) This is the lowest temperature at which free vortices can appear in an infinite array. Note that if EJ is temperature dependent, Eq. (4.9) is an implicit

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

307

expression for TKT.We can rewrite this in a useful form, using Eq. (2.8), as (4.10) Additional insight into the meaning of Eq. (4.9) is obtained by asking where the free vortices come from. As discussed above, the thermal energy necessary to form a bound pair of vortices is lower than the energy needed to form isolated vortices. Furthermore, since the energy to form a pair is bounded, as seen in Eq. (4.4), one expects pairs of vortices to be present for all T > 0. At high temperatures, there will be sufficient thermal energy to unbind many of these pairs to create free vortices. To see when this will occur we can look at the polarizability of the system of vortex dipoles, since it is the polarizability that will diverge at the critical temperature. The important thing to calculate is thus the mean-square separation of the two bound vortices as a function of temperature,

:j ( r 2 )=

exp[ -2nE,/k,T]r22nrdr

ja

= a2

exp[ - 2nEJ/k,T]2nrdr

2nE, - 2kBT 2nE, - 4k,T'

(4.11 )

At zero temperature, the RMS separation between the two vortices is just a ; essentially the mean separation is the smallest possible meaningful length in the array, the lattice spacing. The mean-square separation increases as the temperature increases until, significantly, at the same temperature given by Eq. (4.9) it diverges -i.e., the vortex pair becomes unbound. This unbinding of bound pairs of vortices is the source of free vortices above TKT.(Notice that this does not imply that all pairs will unbind at TKT.) While the essential physics of the above is correct, the exact relationships are not. The pair interaction energy, given by Eq. (4.4),is correct if there are only two vortices in the system. Since thermal fluctuations create many pairs, two vortices which are separated by some large distance are very likely to have several smaller, polarizable vortex pairs between them. As shown by Kosterlitz and T h o u l e ~ s ,these ~ ~ other vortices weaken the attractive interaction potential, especially for those pairs separated by large distances, in a manner analogous to the screening of the Coulomb interaction between the electrons in a metal. This screening for the superconducting case was discussed by Kadin et al.33 and is discussed in some detail in 33

A. M. Kadin, K. Epstein, and A. M. Goldman, Phys. Rev. 827, 6691 (1983).

308

R. S . NEWROCK ET AL.

Appendix C. For our current purposes it is sufficient to note that the main effect of this screening is to reduce the value of the coupling energy E j which appears in Eqs. (4.4) through (4.11) by introducing into the interaction potential a length-dependent “dielectric” constant e(r) such that Eq. (4.4) becomes

s

”=‘

EPM =

2nEJ(T) 1 - dr’, e(r’) r’

(4.12)

where E,( T ) is the “bare” or unrenormalized coupling energy, hi,( T)/2e. This is often written as Ep(r, T ) = 2nE3T) ln(r/a), where EJ is the renormalized coupling energy. For infinite samples, vortex pairs of all sizes modify Ep(r) and the interaction is fully renormalized. This renormalization, however, leaves the essential features of the Kosterlitz-Thouless transition -e.g., the vortex unbinding transition preserved -although some of the constants may differ. V. Classical Arrays: Zero Frequency

Complicating the role vortices play in determining the DC response of arrays -the current-voltage (IV) characteristics and the electrical resistance (R(T) = (d( V)/dI)I+o) -are the relationships among several pertinent length scales. These relationships determine the form of the IV characteristics and whether or not we are in the regime where we can observe a true Kosterlitz-Thouless transition. First we will discuss the case where ,lL >> L >> a and where the relations among the other length scales place us in the regime where the Kosterlitz-Thouless transition is observable. In the preceding section we showed by Eqs. (4.9) and (4.11) that there will be thermally generated free vortices present in all samples at temperatures above TKT;this leads to a free vortex density n,-, =O

n/

= { 20

T < TKT T 2 TKT’

We showed in Section 111.5 that when an external current is applied to a Josephson junction array, the motion of free vortices results in dissipation and an electrical resistance. Thus, from Eq. (5.1) there will be resistance in large arrays above the Kosterlitz-Thouless temperature. From Eq. (3.33), reproduced here, 1

(V)

L

= 21R, - a 2 n f ,

W

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

309

current flowing in an array produces a voltage proportional to the density of free vortices. The voltage, then, is directly related to n,., a thermodynamic property, and all calculations of the resistivity, and indeed, all calculations of all the general current-voltage characteristics, require a determination of the free-vortex density, including its temperature dependence. There are other contributions to the free-vortex density in an array. One obvious source is an external magnetic field. An external field will create free vortices, and they will contribute to the dissipation. Another way of creating additional free vortices is to apply an electric current that will “push” the members of a bound pair in opposite directions, effectively reducing the interaction potential and assisting in thermally dissociating pairs of bound vortices. The free-vortex density is therefore a function of IT; I , and H . In this section we will not consider contributions from applied magnetic fields and will indicate the free-vortex density as n,.(T) or n,(T, I ) . 9. THERESISTIVITY ABOVE

THE

TRANSITION TEMPERATURE

a. IThe Free-Vortex Density The density of free vortices above TKTcan be determined by finding the mean distance between unbound vortices or, since these vortices are essentially thermal fluctuations, the mean size of the fluctuations associated with the phase transition. This latter is the correlation length 5 , . It is identical to the correlation length for an excitation in the lattice XY model (since the array Hamiltonian (Eq. (3.7)) is isomorphic to the XY spin Hamiltonian). It has been calculated by K ~ s t e r l i t zand ~ ~ Tobochnik and Chester35 (see also Appendix C):

where c1 and c2 are constants of order 1 and temperature defined by’

?

is a dimensionless

(5.4)

This dimensionless temperature is very useful for SNS arrays in that it eliminates the temperature dependence of the critical current from the problem, allowing us to focus only on the transition. It is of less importance 34

35

J. Kosterlitz, J . Phys. C7, 1046 (1974). J. Tobochnik and G. V. Chester, Phys. Rev. B20, 3761 (1979).

3 10

R. S. NEWROCK ET AL.

in SIS arrays where the critical current has little temperature dependence near T K T . The prefactor in Eq. (5.3) is essentially a, the lattice spacing, as one might intuitively expect. < + diverges as T approaches TKT,yielding an infinite average distance between free vortices at T‘,-that is, there are no free vortices in large samples below TKT.5 , may also be thought of as the size of a fluctuation-all phases less than 5 , from the center of the fluctuation are correlated (as part of the fluctuation or vortex). This implies that two vortices of opposite sign a distance r >> < + apart are uncorrelated-that is, “free”-while two vortices a distance r << 5 , apart are bound. The area of a fluctuation or vortex is <$,,and therefore the density of free vortices, solely because of thermal effects, n,(T 0), is proportional to t i 2 and n,(T)

= b,a

- 2e[62/(? -

?;(r)]’’‘

9

where b,

= 4c,

(5.5)

(Eq. (5.3)).

b. The Resistance Equation (5.5) can be inserted into Eq. (5.2) to obtain the resistance above the Kosterlitz-Thouless temperature (note again that this resistance is the slope of the IV characteristic in the limit of zero current, so that effects of the transport current in breaking bound pairs of vortices do not contribute):

We see the dominant factor in determining the resistance is the free-vortex density. Below the transition temperature there are no free vortices and there is no resistance. What has happened to the exponential dependence of the resistance on temperature that is seen in single junctions, Eq. (4.2)? A more careful derivation of these results’* shows that there is a temperaturedependent prefactor that multiplies Eq. (5.6): I,

2

( E )

2

(5.7)

where I , is the modified Bessel function. This is similar to the prefactor for the resistance of a single junction, when the calculation is done in detail (see

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

31 1

Ref. 25), and this prefactor can be shown to lead to a similar exponential dependence on temperature in the appropriate low-temperature limit. However, below TKT this temperature dependence doesn’t matter since the density of free vortices goes to zero, Eq. (5.6), and the resistance follows. Above TKT,although the argument of the Bessel function varies from 0 to 2/(1071), the prefactor itself varies less than 1 % over this range. We see, therefore, that in a large array the exponential temperature dependence of the resistance of the single junctions is almost completely suppressed, a result traceable to the much smaller activation energy in the lattice. The other junctions in the lattice act to suppress the temperature dependence of the resistance of a particular junction in favor of the exponential factor of Eq. (5.6), which in turn is a reflection of the free-vortex density in the sample. We find, then, in sufficiently large samples, that the resistance [ ( d ( V ) / dl)l+,,] should be strictly zero for T < TKTbecause the free vortices vanish, and that there can be electrical resistance in an array only if two energy barriers are overcome. The first is the barrier to nucleation of a free vortex, and this is infinite for temperatures below the transition temperature. The second is the barrier to vortex motion and this is relatively small. In contrast, in a single junction the only barrier to a finite electrical resistance is the barrier opposing phase slips- there is no cost for nucleation. c. Experimental Determinations of Resistance The first measurements of the electrical resistance of an array as a function of the temperature were by Resnick et al.4 using a triangular array of lead disks covered by a thin layer of tin. They found a good fit to Eq. (5.6) in the “tail” of the data. More recent data are shown in Fig. 21, which shows typical R versus T data for a square array of niobium crosses on a gold underlayer (Fig. 10). All the data shown were taken at I = 10 PA. The onset of superconductivity in the niobium crosses occurs at 9.2 K. After the initial drop, as the temperature decreases further there is a long steady decline in the resistance. This is due to proximity effects reducing the resistance of the normal region between the crosses. Then there is a rapid drop in the resistance toward zero, followed by a small “tail,” which is a manifestation of the Kosterlitz-Thouless transition. It is due to the presence of free vortices and the resultant flux-flow resistance. In the figure a line is fit to the data in the “tail” according to Eq. (5.6). The fit is quite good, at least on the scale shown. Expanding the scale reveals that the fit is good at higher temperatures but becomes less so close to TKT. The reason lies in the way the data are taken, contrasted to the way the theory is constructed. Equation (5.6) is for the resistance R = (d( V)/dI),+o,

312

R. S. NEWROCK ET AL.

TEMPERATURE ( K ) FIG. 21. R ( T ) versus T for an Nb-Au 300 x 300 array. The line through the data points on the “tail” of the figure is the fit to Eq. (5.6). The niobium crosses become superconducting at about 9.2K and the array resistance drops as the temperature is lowered because of the proximity eflect. The jump in the resistance at the niobium transition is due to current redistribution in the leads. (R.S. Newrock, unpublished.)

whereas the data are essentially ( V ) / I for fixed I . The problem is that electrical currents exert forces on vortices, Eq. (3.17). This force pushes the two members of a bound pair apart and helps to create additional free vortices. At low currents these current-induced free vortices are of little importance for T >> TKT,where they are overwhelmed by the large number of thermally unbound vortices. But, near TKT,where there are few thermally unbound vortices, they are of considerable importance and can easily dominate the measurement. To obtain good fits to the theory, one needs to reduce the current to values as close as possible to zero as T approaches TKT.This creates an obvious difficulty. Just as the number of free vortices is shrinking toward zero, resulting in very weak signals, we must reduce the measurement current substantially (see Eq. (5.2)). This makes an exact determination of TKTdifficult from resistance data alone. The electrical resistance is not the best way to observe the KosterlitzThouless transition. We must instead look closely at the details of the current-voltage characteristics.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

3 13

10. THECURRENT-VOLTAGE CHARACTERISTICS The central point of the previous subsection was the determination of the free-vortex density and the resistivity above the transition. The resistivity is zero below TKTas there are no free vortices (in sufficiently large samples); that is, R = 0 for I -,0. Since a nonzero current will unbind vortex pairs, nf(7: I) will not be zero and for finite currents there will be dissipation even below T K T . We now consider the effects of nonzero current. The details are complicated by renormalization effects (Appendix C) but the essential physics is very easily obtained. We will show that for any current, no matter how small, some pairs of vortices will be unbound. Unlike a three-dimensional superconductor, the critical current of a two-dimensional superconductor is thus zero. This sensitivity to small perturbation is characteristic of quasi-long-range order in two dimension^.^^ a. Eflects of Applied Currents on Bound Pairs of Vortices The vortices that form a bound pair have opposite circulation, and therefore the forces exerted on them by an electric current push them apart. Even a small current will unbind the most weakly bound pairs of vortices.36 We start by considering an applied current density j , here a twodimensional quantity, the current per width, j = IfMa, where I is the array current and M a the array width. The Lorentz force on a vortex due to a current density j is

FL = k 0 , j

x

1,

where "+" is for counter-clockwise circulating current and "-"is for clockwise, leading to an additional potential energy for a pair oriented as shown in Fig. 22 of

Using the result of Eq. (4.4), we may write the total potential energy of a single pair of vortices, when a current is flowing in the system, as Up = 2nEJ In

(:)

-. j 0 , r .

(5.10)

36 The most weakly bound vortices are those with the greatest separation. These are also the ones most likely to have other, smaller pairs between them, reducing the interaction energy. It is the importance of these weakly bound vortices and interaction effects that leads to the need to look carefully at renormalizationeffects for exact results.

314

R. S . NEWROCK ET AL.

r/a FIG. 22. The potential energy of a bound pair of vortices when a transport current is applied. rc indicates the critical vortex-antivortex separation when the potential is at a maximum.

(Note that we are only considering a single vortex pair here and therefore do not use the renormalized interaction potential.) This potential energy is plotted in Fig. 22. The maximum of this potential, U , , is located at r,

With j

= I / M a = i/a,

=

[:]

(5.1 1)

a.

where i is the current per junction, we have

[i

1

U, = U p ( r c )= 2nE, In' - 1 .

(5.12)

U, is the energy barrier that a vortex must overcome if it is to escape from the potential well shown in Fig. 22. At all nonzero temperatures thermal energy is available to assist the vortices in escaping over the barrier. The available thermal energy becomes more effective as the current is increased because the barrier height is reduced, as seen from Eq. (5.12). We next determine escape and recombination rates and use these to determine the free-vortex density due to current-induced unbinding. We confine ourselves to the small current limit, where the first term in the bracket of Eq. (5.12) dominates the second. Also, in the small current limit, r, is very large, and we start with essentially no free vortices due to current-induced unbinding (above or below TKT).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

3 15

The escape rate Re over the energy barrier is determined in the usual way as 2eiCR0

=

,-Up/kBT

('"h")(:) C

0

-ZffEl/k~T

(5.13)

The prefactor is suggested by choosing a characteristic overdamped rate. This expression gives us the rate of generation of pairs of free vortices when an external current is applied. There will also be recombination of pairs, which in the usual way is proportional to the square of the free-vortex density. In equilibrium we have Re - anj(T

-= TKT,I) = 0,

(5.14)

where c1 is a proportionality constant. The free-vortex density due to current unbinding is then

When this expression is put into Eq. (3.33), which relates the voltage to the free-vortex density, we obtain

which can be written as Va

(5.17)

where (5.18) The last equality used Eq. (4.9) and assumes that E , is either independent of temperature or not very temperature dependent near the KosterlitzThouless transition. We summarize the behavior above and below TKT using Eqs. (5.6), (5.17), and (5.18). Below TKT, a(T) is greater than 3-a nonohmic, power-law response. From the previous section's results we can state that above TKTin the low current limit, a(T) = 1 and the ohmic response is described by Eq. (5.6). The exponent a(T)jumps suddenly from 1 to 3 at TKT.These results for the IV characteristics and a(T) are displayed schematically in Fig. 23. Figure 23(a) shows the expected behavior of R(T) in the low current limit. Figure 23(b) shows the expected behavior of the power-law exponent a( T ) , also in the low current limit.

316

R. S. NEWROCK ET AL.

vortices I

-

I

vortices

’

I

1

b

TRKT

FIG. 23. Schematic drawing of R ( T ) (the array resistance) and a ( T ) (the IV exponent) as a function of the temperature in the low current limit. R ( T ) varies with temperature according to Eq. (5.6); a ( T ) varies according to Eq. (5.18).

Thus, we have several interesting results. First, we can directly probe the Kosterlitz-Thouless phase transition using the current-voltage characteristics of the array. Second, we have a remarkable jump in a(T),which is a “smoking gun” for the phase tran~ition.~’(We note that the jump is “universal”; see Appendix D.) Third, current-voltage curves can be used to determine n , ( T ) above and nf(Z, 7‘) below TKT. When there is more than one pair of vortices present, renormalization effects are important, and these reduce E, and U,, reducing rc as well. This enhances the effectiveness of the current in unbinding pairs and changes Eq. (5.17) somewhat (Appendix D). The primary result of Eq. (5.17) at TKT is unchanged-a(T) jumps from 1 to 3 at the Kosterlitz-Thouless temperature; however, the temperature dependence of a( 7’)is changed. b. Experimental Current- Voltage Characteristics. Resnick et al.4 published the IV characteristics for their lead-tin arrays, but they did not observe the universal jump in the exponent. Abraham et al.38 37 38

D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Let?. 39, 1201 (1977). D. Abraham, C. J. Lobb, M. Tinkham, and T. M. Klapwijk, Phys. Rev. 826, 5268 (1982).

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317

measured SNS arrays consisting of lead squares on a copper film and found the first clear indication of the jump in a( T ) . Figure 24(a) shows more recent data from Herbert et al.39Note the two lines in Fig. 24(a), which have slopes of one and three. The higher temperature curves, on the left-hand side of the figure, are clearly ohmic. As the temperature decreases, the higher-current 39 S. T. Herbert, Y. Jun, R. S. Newrock, C. J. Lobb, K. Ravindran, H.-K. Shin, D. B. Mast, and S. Elhamri, Phys. Rev. 857, 1154 (1998).

6 -

5 -

--c m

4 -

3 -

2 -

1 I

I

I

,

2.0

2.5

3.0

3.5

J 40

Temperature (K)

FIG. 24. (a) Current-voltage characteristic curves for a square array of niobium crosses on a gold underlayer (see Fig. 10). (From Ref. 39, Fig. 2(a).) (b) The power-law exponent a ( T ) versus T for the data in (a).

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R. S. NEWROCK ET AL.

IV data at each temperature are no longer ohmic, but the lower-current end of the curves remains so. Eventually, the entire curve appears to have power-law behavior. Fig 24(b) shows the exponents a( T ) determined from the IV curves in Fig. 24(a). These were determined by choosing a threshold voltage (50nV) and determining the slope of the IV curve near that threshold. The “jump” in a(T) from one to three is clearly visible. It was found that when a higher threshold voltage was used the “sharp” jump in a ( T ) disappears, showing the need to examine only low-current IV data. Herbert et al. chose TKTto be at the jump in a(T). They found that when this value for TKTwas used in Eq. (5.6) for the resistance, a very good fit was obtained. They note, however, that the.values for a(T) versus T below TKTdo not have the correct slope and are generally below the predicted value, Eq. (5.18), even when properly renormalized values for the coupling energy are used. Others found similar results -reasonable agreement with Eq. (5.6), with the jump in a ( T ) seen but agreement with the theoretical temperature ~ ~ *general ~ ~ * ~pattern * apdependence of a( T ) generally i n ~ o r r e c t . ~ *The pears to be that Eq. (5.6) is correct when the appropriate value of the Kosterlitz-Thouless temperature is used, that the jump in a ( T ) at TKTis observed, but that the details of a ( T ) versus T are incorrect. Recently Minnhagen et al.43 derived a different expression for a( T ) , one that may provide a better fit to the data.

11. GENERAL CURRENT-VOLTAGE CHARACTERISTICS A close inspection of IV data generally shows that near, but above, the transition temperature only the low-current portions of the IV curves are linear; the high-current portions tend to have a power-law behavior. As the transition temperature is approached and the limits of voltage sensitivity are reached, the linear term disappears, even slightly above TKT,leaving only the power-law term. Thus, we have so far described only part of the story and we need to examine the IV curves more closely. a. Scale Lengths Up to this point we have limited our discussion to a special case, the one where the Kosterlitz-Thouless transition can be observed: very low currents, 40 41 42

43

D. Kimhi, F. Leyvraz, and D. Ariosa, Phys. Rev. B29, 1487 (1984). R. F. Voss and R. A. Webb, Phys. Rev. B25, 3446 (1982). B. J. van Wees, H. S. J. van der Zant, and J. E. Mooij, Phys. Rev. B35, 7291 (1987). P. Minnhagen, 0. Westrnan, A. Jonsson, and P. Olsson, Phys. Rev. k t t . 74, 3672 (1995).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

319

large specimens, and AL >> L. We note that a “pure” Kosterlitz-Thouless transition will occur only in infinite systems, in the sense that A* >> L -, 00. However, if a finite system shows the appropriate effects, we may consider the transition to be Kosterlitz-Thouless-like. We now need to look at the transition more closely, examining the relationships among various scale lengths, to determine what will be observed at all currents and at temperatures above and below TKT. To fully understand the role vortices play in determining the currentvoltage characteristics, it is important to understand the relationships among several pertinent length scales. These relationships determine the form of the IV characteristics and whether we are in a regime where we can observe the Kosterlitz-Thouless transition. There are six of these lengths, displayed in Table 2, five of which we encountered earlier. The new length introduced in Table 2, 5 - is related to the typical size of the bound pairs below TKT.For T close to TKT,both 5 , and 5 - diverge exponentially, Fig. 25 (see, for example, Eq. (5.3) and Appendix C for precise temperature dependences and derivations of these two lengths). We now consider “large” currents. In the previous section we worked with small currents, by which we meant that the number of current-induced free vortices is much smaller than the number of free vortices present with zero current (above T K T ) . As the current is increased, however, current-induced vortices dominate, and we can then use Eq. (5.16) above and below T K T . Below TKT the power-law exponent, a ( T ) 3 3. Above TKT,for currents sufficiently large that current-induced free vortices dominate the solely thermally induced ones, a ( T ) d 3. That is, for sufficiently large currents, power-law behavior is observed at all temperatures and the ohmic response above TKTis hidden. Thus, the scale length r, is important both above and below T K T . At an arbitrary temperature and current, if a bound pair is much closer together than rc (before the current is applied), it will remain bound after the current is applied, unless it is unbound by an unusually large and unlikely thermal TABLE2. SCALELENGTHS Length L

A,

5, 5rc a

Description Sample Size Perpendicular penetration depth Correlation length; typical distance between free vortices. T > TKT. Correlation length; typical size of bound pairs below TKT. Location of the maximum in the vortex-vortex interaction potential with an external current Array lattice parameter

320

R. S. NEWROCK ET AL. length

FIG. 25. The correlation lengths 5 , and 5 - versus reduced temperature, T/TKT.Curves are drawn assuming that the constants in 5 , and c- are 1. The upper dashed line is the smaller of L or I,. The lower dashed line is rc = (ic/i)a. rc increases/decreases as the current decreases/increases.

fluctuation. If the pair is separated by a distance much greater than rc, it will likely immediately unbind when the current is applied. If it is separated by a distance less than but close to rc, thermal energy will play an important role in breaking the pair -i.e., moving the vortex over the barrier shown in Fig. 22 and Eq. (5.12). b. Schematic Current- Voltage Characteristics

To understand and illustrate the complete transport properties of twodimensional arrays, we will develop a set of schematic IV curves. We begin by noting that the correlation length 5 , diverges as T approaches T K T from - above: 5 , c 1aecz(T - TKJ)-”’.+ (5.19) T + TK+~.

-

When the temperature is close to, but less than, TKT,another correlation length, 5 - , is a measure of the typical separation between the two vortices in a pair.44*45*46 It diverges in a similar manner as T approaches TKTfrom V. U. Ambegaokar, B. I. Halperin, D. R. Nelson, and E. D. Siggia, Phys. Rev. Lett. 40, 783 (1978). 4 5 V. U. Ambegaokar, B. I. Halperin, D. R. Nelson, and E. D. Siggia, Phys. Rev. B t l , 1806 ( 1980). 46 J. E. Mooij, Percolation, Localizzation and Superconductivity, eds. A. M. Goldman and S. A. Wolf, Plenum, New York (1983), 325. 44

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

321

below (Appendix C):

Figure 25 shows these correlation lengths, normalized to the lattice parameter a, as a function of the reduced temperature (in plotting the curves in the figure we assume the constants in Eqs. (5.19) and (5.20) are one). Clearly, to observe the transition, both (+, the average separation between free vortices, and <-,the average separation of the bound pairs of vortices, must be much less than the sample size L. Thus, in the discussion to follow, since ( - diverges as T -+ TKTfrom below, when we refer to T < TKTwe also imply that T is sufficiently far below TKTthat L >> 5- and similarly for T > TKT. The (s are strongly temperature dependent near TKT,whereas r, is not, because in the vicinity of TKT,i,(T) does not have a strong temperature variation. Furthermore, r, is current dependent. Thus, above or below TKT the relative size of the correlation lengths compared to r, can be changed by varying the temperature or by changing the current, as is indicated in Fig. 25. Finally, as a pedagogical aid to understanding the current-voltage characteristics, we will assume that these lengths are “sharp,” in the sense that we will ignore their statistical nature and assume that at temperatures above TKT all vortex pairs separated by distances greater than 5 , or r, are unbound, and closer vortex pairs are bound. For temperatures below TKT any pair of vortices separated by distances greater than r, are unbound. This is done solely for pedagogical purposes. We can then make the following arguments. T > TKTAND 5 , << Y, This implies either temperatures well away from TKT or small currents. Since 5 + , the average separation distance between free vortices, is small, most of the vortices in the sample will be free. Figure 26(a) is a schematic diagram showing this. Here we have sketched the “number density” of pairs of vortices n f ( r ) , with a particular separation r, versus r. From the figure we see that, under our assumption of “sharpness,” all vortices with separations between a and 5, are bound in pairs (by the definition of 5 , ) and all vortices with separations greater than 5 , are unbound. Since all of the vortex pairs with separations on the order of r, or greater are already unbound, the transport current plays no real role in unbinding vortices and the dominant unbinding mechanism is thermal. Since < + << rc implies i << (a/<+ ) i , (see Eq. (5.1 l)), where i is the transport current per junction, we see that small currents must be used above TKTin order to measure the intrinsic, purely thermally generated free-vortex population. Notice that this requirement becomes more and more stringent as T TKTand +(T ) -+ 00. -+

<

322

R. S. NEWROCK ET AL.

a

rc

r

E+

A

A

a

C r

€-

FIG.26. A schematic diagram showing the density of vortex pairs of a given size as a function of that size. (a) T > TKTand t +<< rc. (b) T > TKTand 5 , >> rc. (c) T < TKTand 5 - << rc. (d) T < TKTand c- >> r,.

T > TKTAND 5 , >> rc This implies either temperatures sufficiently close to TKTor large currents (see Fig. 25). Figure 26(b) is a schematic plot of vortex separation similar to Fig. 26(a). Here we see, again with our assumption of “sharpness,” that all vortices with separations between a and 5 , are bound when there is no transport current. In this example, the vortices with

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

323

separations r, < r < 5 , before the current is applied are bound, implying that the transport current will be a particularly effective unbinding mechanism: When the current is applied the overwhelming majority of the free vortices will be current generated and the IV curves will be nonlinear in this regime, as given by Eq. (5.17). Based on the above we can draw a schematic current-voltage characteristic for this regime: A+ >> L >> a and T TKT.This is shown in Fig. 27(a), where we plot In V versus In I . For very small currents, r, is very large; the free-vortex density is thermally determined and is proportional to 1/<:. This results in a linear IV curve, V a I . As the current increases, r, decreases until it becomes approximately equal to <+(T).At this point current unbinding becomes important and power-law IVs are observed, V a where a(T) < 3 (because we are above TKT;see Eq.(5.18)). As the current increases further, r, decreases until it is of the order of a, the lattice spacing. At this point all pairs are unbound and the IV characteristic reverts to linear behavior.

=-

In V +

/ I

FIG. 27. Schematic current-voltage characteristics for 1, >> L >> a, as described in the text. (a) T > TKT,which implies a(T) < 3. (b) T iTKT,which implies a(T) > 3.

324

R. S. NEWROCK ET AL.

This schematic IV illustrates an important point mentioned earlier. It is possible to observe power-law IVs abooe the Kosterlitz-Thouless temperature. As the temperature is lowered, the signature jump in a ( T ) can be observed only at currents sufficiently low that one is in the thermal unbinding regime, 5 , << rc, so that a(T > TKT)= 1. T
<-

T < TKTA N D >> rc This situation is shown in Fig. 26(d). Here all the pairs with separations larger than rc are unbound, and this constitutes the majority of the pairs. This is an ohmic region. Again, based on the previous discussion we can create a schematic IV curve for this region, T < TKT),Fig. 27(b). There is no ohmic tail-there are no free vortices for T < TKT.As the current increases, rc decreases and more and more vortex pairs unbind. This is the “classic” Kosterlitz-Thouless regime (Section IV.8) where I/ cc la(,)and a( T ) 2 3. As the current increases, rc decreases until it becomes approximately equal to ( - ( T ) . At this point there are few bound pairs left and the IVs become linear. These descriptions support our earlier comments as to when we will observe the classic Kosterlitz-Thouless transition: To observe it one must have samples for which I., > L >> a, and one must use measuring currents sufficiently small that one is in the power-law regime below T K T , 5 - << rc, and the ohmic regime above TKT,5 , << r,. VI. Classical Arrays: Nonzero Frequency Response

The studies of the Kosterlitz-Thouless phase transition thus far presented have been based on measurements of electrical resistivity and currentvoltage characteristics, which are zero-frequency properties. But detailed insight into the system of vortices is best provided by time-dependent studies. The two-coil mutual inductance t e c h n i q ~ e , ~ ’ + which ~~*~~*~~ A. T. Fiory and A. F. Hebard, AIP Conference Proc. 58, 293 (1980). A. F. Hebard and A. T. Fiory, Phys. Rev. Lett. 44, 291 (1980). 49 A. T. Fiory and A. F. Hebard, Magnetic Response of Superconductors and Other Spin Systems, eds. R. A. Hein, T. A. Francavilla, and D. H. Langenberg, Plenum Press, New York (1992). 5 0 B. Jeanneret, J. L. Gavilano, G. A. Racine, Ch. Leemann, and P. Martinoli, Appl. Phys. Lett. 55, 2336 (1989). 47

48

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

325

measures the complex sheet impedance, allows us to extract the dynamical properties of the vortices. The experimental arrangement is shown schematically in Fig. 28. An alternating current is applied to a drive coil, which induces a voltage in the pick-up coil. Without a sample this is just a simple transformer. The sample alters the response in the secondary. For example, if the sample were just a normal metal film, the eddy currents induced in it would create an additional component to the magnetic field, which can be detected in the pick-up coil. In actual use, the pick-up coil consists of two separate coils, counterwound and separated by a short distance. This serves to cancel the direct pick-up from the primary when no sample is present. When a sample is present, the two coils are no longer in balance and a signal results. To fully appreciate this section the reader should be familiar with Appendix C. However, a simplified picture of the AC response of an array is instructive. We consider an array at T = 0, i.e., one with no vortices present, and ask what the response is if an AC current I,, << I, of frequency o is applied. If the currents are small, we may approximate the sine term in the Josephson current equation, Eq. (2.5), by its argument. When this is combined with the Josephson voltage equation, Eq. (2.7), the supercurrent response of a junction is described by

FIG. 28. Schematic of the experimental arrangement for the two-coil mutual inductance technique. An alternating current Idrlvr induces a voltage I/p,ck.upin the pick-up coil. Vp,ck.up is affected by the electromagnetic response of the sample. Bucking coils are used to make a differential measurement.

326

R. S. NEWROCK ET AL.

This is formally the same as the equation of an inductor with inductance

Thus, at T = 0 and for small currents, the supercurrent channel acts like an inductor. The RCSJ model considered in Section 11.2, with the Josephson channel replaced by this inductance, would completely determine the response if no vortices were present. We consider for simplicity C = 0. At temperatures below T, (where i, # 0), the response is given by a parallel R-L circuit, with L = D0/27cic. The parameters of the model (R and L) can be determined by measuring the frequency dependence of the pick-up voltage As T + T,, i, + 0, and L + 0 0 ; above T, the response as a function of Idrive. would be purely ohmic-eddy currents would result as in a normal metal film. This grossly oversimplifies the physics because it ignores the vortexunbinding transition. In that case, raising the temperature has two additional effects. First, the presence of bound vortex pairs and eventually free vortices will increase the measured losses and thus the effective real part of the impedance in any AC measurement. Second, vortices will renormalize the coupling energy and thus the effective i, (as shown in Appendix C), causing the measured inductance to increase.

12. DYNAMICS OF VORTICES From the frequency dependence of mutual inductance data, the temperature dependence of 5 , can be determined and used to verify the qualitative behavior of the Kosterlitz-Thouless dielectric constant and to provide a quantitative value for comparison with theory. The dynamical properties of arrays strongly depend on the dynamical properties of the bound pairs of vortices. Since the response of a bound pair of vortices depends on the “dielectric” constant, we must consider screening effects from the outset.’l The dynamics of bound pairs was first worked out (for liquid helium films) by Ambegaokar et a1.44,45and Ambegaokar and Teitel,” and for arrays by S h e n ~ y . ’ ~They ~ ’ ~ found that the response of the 5 1 Recall that we use the term dielectric constant to describe the screening of a particular pair of vortices by the other, smaller pairs because of the Coulomb gas analogy, wherein the vortices are treated as effective charges. 5 2 V. U. Ambegaokar and S. Teitel, Phys. Rev. B19, 1667 (1979). 5 3 S. R. Shenoy, J. Phys. C: Sol. St. Phys. 18,5143 (1985). 5 4 S. R. Shenoy, J. Phys. C: Sol. St. Phys. 18,5163 (1985).

327

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

vortices to an external stimulation is determined primarily by those vortex pairs of a characteristic size, set by the frequency (6.3)

where D is the vortex diffusion constant: D=

2R0a2k,T

a:

This is correct for square arrays, and it may be obtained in the usual manner from the vortex viscosity, q, Eq. (3.24), and D = k,T/q. This characteristic size arises because the response of a bound pair of vortices, of size r, to an oscillating drive field is characterized by a relaxation time, proportional to r2/D, which determines the pair’s diffusive relaxation towards equilibrium with the drive field. In the dynamical theories, the bound-pair contribution to the response function, cb(w,T ) ,is obtained by integrating over a continuous distribution of these relaxation times. This procedure singles out a characteristic scale length given by Eq. (6.3).There is nothing special about the factor of 14; it enters because of a certain approximation to be discussed below. To understand this frequency response we start by considering the dynamics of a single pair of vortices, of opposite sign, separated by a distance r. We consider the Langevin equation for each vortex, assuming that it is driven by the oscillating screening supercurrent created as a response to the AC magnetic field (at frequency w ) from the primary coil near the sample, and by thermal diffusion. We subtract the Langevin equations for the two vortices to obtain an equation in the relative position,

where U p is the vortex pair binding energy, Eq. (5.10), reproduced here, U,(r)

= 27cE,(T)

ln(r/a) - j Q 0 r .

The Nyquist noise source qth, varies as q’(t)qj(r’)= 4D6(t and j are Cartesian coordinates.

(6.6)

where i

328

R. S. NEWROCK ET AL.

As mentioned above, we must include the screening of our particular pair by smaller pairs. We accomplish this by introducing the Kosterlitz-Thouless dielectric constant into the interaction energy (Eq. (4.12) and Appendix C):

The static dielectric constant is used because the screening is provided by pairs of small separation and it is assumed4’ that they relax to the applied fields much more rapidly than the pair considered in Eq. (6.5). Ambegaokar and Teite15’ and S h e n ~ y ’ ~ calculated ,’~ the time-dependent polarization per area to linear order in the drive. Below the transition, T < TKT,the contribution of the bound pairs to the dynamical dielectric constant is

g(r, w, T ) is a response function for pairs of separation r. Below the Kosterlitz-Thouless temperature two length scales are pertinent to this response function: t;-(T), the characteristic size of a vortex pair, and the vortex diffusion length, rD = (2D/w)”’. For ( - ( T ) << rD, Ambegaokar et a1.44*45show that g(r, w, T ) can be approximated as

(

g ( r , w , T ) z 1 -~

.

The physics of this response function is that the diffusion length determines the crossover point in the response of pairs of different sizes: Smaller pairs ( ( - ( T )<< rD) can equilibrate to the driving force; larger pairs ( ( - ( T )>> rD) cannot. Again, there is nothing magical about the factor of 14. It is just that the relaxation time must be adjusted so that the approximation for g(r, w, T ) matches the cut-off length of the full solution. This matching requires that 2D/r2 be replaced by 14Dfr’. From Eq. (6.9) an estimate for the dielectric constant is obtained from Eq. (6.8) by noting that E(r) is a slowly varying function of Inr for those separations for which g(r, w, T ) passes from its low- to high-frequency behavior. Then, to simplify evaluation of Eq. (6.8), Eq. (6.9) is approximated by Re g(r, w, T ) z 0 ( 1 4 D / r 2 - w )

(6.10)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

329

and

where 0 is the Heaviside function. It then follows, for T < T,,, R e q , ( q T ) = E(T

=

Ja)

(6.12)

(because &(a) = 1) and

1 Im E ~ ( o , T ) = --71 4

(6.13)

Equation (6.12) is the “inductive” response of the system, and Eq. (6.13) is the dissipative response. These equations hold for o < D / a 2 ; i.e., the frequency should not be large enough to probe length scales where the idea of a low pair density is inapplicable. These equations hold for T < T,,. Unlike the D C case, there is no sudden change in the response as the temperature approaches the KosterlitzThouless temperature from below. At TKTbound pairs of infinite separation unbind, but smaller pairs are still bound and still give an inductive response to the drive currents; the response through TKTis smooth. Above T,, we need to consider a different correlation length, t + ( T ) the , mean separation of the free vortices. For T > TKTwe have for the bound pairs ( r D<< t + ( T ) )

(6.14) To this we must add the contribution of the free vortices. We needn’t solve Eq. (6.14), however; free vortices dominate the response for t + ( T )= r D ,and for practical purposes Eqs. (6.8) to (6.13) are sufficient to determine the response of the bound vortices even above T K T . For the free vortices, Ambegaokar et al. made a simple Debye-Huckel approximation for a vortex diffusing in the average drive current and obtained a Drude term in the usual manner: E~ =

1

+ i-. 41t

WR”

(6.15)

Thus, for T > TKTthe contributions from vortex-antivortex pairs and free

330

R. S. NEWROCK ET AL.

vortices can be written as &(a, T ) = E ~ ( wT, )

4ni + -,WRV

(6.16)

where Rv = ~R,,(u/W)~ is the flux-flow resistance due to a single vortex; see Eq. (3.32). At temperatures sufficiently far above the Kosterlitz-Thouless transition temperature, we have an inductive response similar to that below the transition temperature but with an additional dissipative term. What will be observed experimentally?Below TKTthe polarization picture is valid for all 0 < o < D/a2. For o > 0, unlike the zero-frequency case, nothing special occurs at TKT.However, above TKT,as can be seen from Eqs. (6.12) and (6.13), there will be a crossover at a temperature T, such that (6.17) where is the reduced temperature introduced earlier, Eq. (5.4). At high temperatures 5 , is much less than r,. Pairs separated by distances greater than 5 , are unbound, there is no pair-derived inductive response to the AC field, and the dissipation is primarily due to free vortices. As the temperature drops towards TKT,5 , increases and eventually, at some temperature T,, 5 , equals r w . At this point many pairs with separations rw exist and, from Eqs. (6.12) and (6.13), we get a rapid change in the response. We note that changing the frequency will make r , larger or smaller, increasing or decreasing T,, and that as o 40, r, -, co and T, -,T K T . 13. ARRAY IMPEDANCE Experimentally we measure the complex sheet impedance of an array Z,(o, T). This is related to the overall dielectric constant of the system,

Z,(O, T ) = ~ ~ L , E T), (W,

(6.18)

where L,, Eq. (6.2), is the inductance of the array in the absence of vortices. Combining Eqs. (6.12), (6.13), and (6.15), we have E(O,

T )= ~'(o T ,)

E'(o,

T ) = c(r,),

&"(O,T )

=('.")I

+ iE"(o, T),

and

(6.19)

+-.4n 4

dr

r=ro

oRv

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

33 1

Implicit in these expressions is the assumption that the normal screening currents are small; this is correct if oL, << R,. Leemann et a1.,55*56measured the complex sheet conductance Go = (Z0)-l of Pb-Cu SNS arrays using the technique developed by Hebard and F i ~ r y . ' ~ ,The ~ * interaction between the screening currents and vortices induces a signal in the pick-up coil, 6V a oZZ,Go; (6.20) whqe A is a geometry-dependent constant. (This result is valid near the transition where the screening is weak. At low temperatures, T << TKT, the assumption of weak screening is no longer valid.) Since ic(T)is a monotonically decreasing function of temperature, any unusual behavior of the voltage near TKT must arise from the dielectric function. The real and imaginary parts of Eq. (6.20) are proportional to the dissipation and the kinetic inductance respectively. The dissipation, proportional to Im(l/E), depends on E". Below TKT,E" depends on the number of pairs with separation r,-the number of pairs on the length scale determined by the frequency and the diffusion constant. Those are the pairs that are the most out of phase with the drive. As the temperature rises the number of such pairs increases smoothly, with no discontinuity at TKT, until it is cut off at < + ( T ) ,Eq. (6.14). Thus, as T increases from below TKT the dissipative term should rise to a maximum near T, . On the high temperature side, E" is dominated by the free-vortex term. Since p v a nf a l / t : ( T ) , and 5 + ( T )decreases rapidly with temperature, the dissipation drops off rapidly from its peak. The kinetic inductance, proportional to Re (1/E) = E'/(E'' + E"'), drops off smoothly from a maximum at low temperatures to zero at high temperatures. This is what is observed e ~ p e r i m e n t a l l y . ~In ' , ~ Fig. ~ 29, a plot of the induced voltage versus temperature, we see the peak in the dissipation and the drop in the kinetic inductance. Leemann et al. used data similar to those in Fig. 29 to check if the temperature dependence of the correlation length + followed theory. The

<

Ch. Leemann, Ph. Lerch, G. A. Racine, and P. Martinoli, Phys. Rev. Lett. 56, 1291 (1986). Ch. Leemann, Ph. Lerch, R. Theron, and P. Martinoli, Helv. Phys. Acta. 60,128 (1987). 5 7 A. F. Hebard and A. T. Fiory, Inhomogeneous Superconductors, eds. D. U. Gubser, T. L. Francavilla, S. A. Wolf, and J. R. Leibowitz, American Institute of Physics Conf. Proc. #58, AIP, New York (1980), 293. A. F. Hebard and A. T. Fiory, Physica (Amsterdam) 109&110, B,C,1637 (1982). 55

56

332

R. S. NEWROCK E T AL. I

I

3

4

I

40

-Lt 30 c

u

3 -

>

20

v;) Y

2

T[KI

6

-

FIG. 29. The temperature dependence of the signal voltage from the secondary coil for an SNS array. (From Ref. 56, Fig. 3.)

precise behavior near T, was not treated in the dynamical theory, so they followed an ad-hoc procedure first used by Hebard and F i ~ r y . ~T*0 *i s~ ~ determined by extrapolating the steep portion of the kinetic inductance curve (Fig. 29) to zero. T o cast this in terms of a reduced temperature, the temperature dependence of the critical current is needed. To obtain it they measured the low-temperature critical current, fitting those data to the d e G e n n e ~expression ~~ for the critical current, (6.21) to determine ic(0) and tN(T). These fitting parameters were then used in Eq. (6.21) to calculate ic(T ) ,which they used to determine the reduced temperatures Tw.rw was determined by direct calculation from Eq. (6.3) using the vortex diffusion constant, Eq. (6.4). If the scaling parameter, C, = ln(rJu), is introduced, Eq. (6.17) can be written as e;2

= b-2[Tw -

The calculated values of C, and 59

TKTJ

Yw are displayed in

P. G. deGennes, Rev. Mod. Phys. 36, 225 (1964).

(6.22) Fig. 30, plotted as C’;

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

333

versus To.Note that Leemann et al.’s data is for two magnetic fields, f = 0 and 1. Since the Hamiltonian Eq. (3.7) is periodic in the frustration f, with period 1, they anticipated that the transition would be Kosterlitz-Thoulesslike for f = 1 and made measurements and performed the above analysis at both fields. The solid lines represent fits to Eq. (6.22) for f = 0 and 1, confirming the linear relationship between eL2 and Tu,and the expected temperature dependence for 5 + ( T ) is verified. When the Kosterlitz dielectric constant is included, the universal Kosterlitz-Thouless prediction (Eq. (4.10) and Appendix C) can be written as (6.23) where E, is the static Kosterlitz dielectric constant at TKT(and infinite scale, Appendix C). By extrapolating the fitted lines in Fig. 30 to zero frequency, Tu(o= 0) = TKTand TKTcan be determined, and, using the definition of ?: we find that Eq. (5.4) becomes (6.24)

3.0

3.5

T[KI

-

FIG. 30. The dependence of the scale parameter L, = ln(ru/u) on the dimensionless temperature To.The solid lines are fits to Eq. (6.22). The upper axis shows the real temperature for zero magnetic field. (From Ref. 55, Fig. 3.)

334

R. S. NEWROCK ET AL.

which, by comparison to Eq. (6.23), leads directly to a value E, = 1.81. This is in good agreement with theoretical calculations (Appendix C ) . VII. Classical Arrays: Finite-Size Effects

All our discussions to this point have assumed the thermodynamic limit, L + co and ,Il >> L. It turns out that in arrays these conditions are often not met, and this leads to interesting physics. For example, at a given temperature, 1, depends on i,(T), 1, a l/ic(T) a l/E,(T), and one can vary AL by varying the coupling between islands in different samples or by varying the temperature in the same sample. In the latter case it is possible to switch from A1 > L to A1 < L by changing the temperature. To observe the Kosterlitz-Thouless transition it is clearly important to choose the coupling strength E , such that 1, > L over a wide range of temperatures. There are also limits on L. L must be large enough so that the number of free vortices created thermally by nucleation from the edges is very small, but it must also be small enough so that 1, > L over the temperature range of interest. It also must be small enough that residual field effects do not need to be considered, as discussed below. These conditions are not easy to meet. The effects that result from violating any of these conditions are called finite-size effects. We will consider three types of finite-size effects. The first occurs when the sample size affects the completeness of the renormalization. The second occurs when free vortices can be thermally generated because the free energy to create them (Eq. (4.7)) is sufficiently small when L is small. The third is a remnant magnetic field effect, arising from our inability to create a truly field-free region. 14. INCOMPLETE RENORMALIZATION The details of renormalization in the Kosterlitz-Thouless transition and the effects on finite-sized arrays are complicated and are discussed in detail in Appendix C. For our purposes the essential point is that for a finite sample only vortex pairs separated by distances less than L can participate in the screening, and therefore the transition cannot be fully renormalized (i.e., we cannot include pairs with infinite separation). This broadens and tends to wash out the universal jump. Thus, the abrupt jump in a ( T ) is lost (see Section V.10) but power-law IVs are still observed. Figure 31 (from Ref. 33) shows a ( T ) - 1 versus T/TKTfor various values of a scaling parameter t (=ln(L/a) = In M for an M-wide array). a ( T ) here has its properly

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

335

a ( T )- I I-

1.05

FIG. 31. The temperature dependence of the IV exponent a(T) or the reduced stiffness constant, xK(L, T) = a ( T )- 1 for fixed values of scale f. f = 0 corresponds to the bare unrenormalized interaction; the curve for L = co,to the fully renormalized case. (After Ref. 33, Fig. 6 . )

renormalized value. Notice that the transition, which is infinitely sharp for infinite specimens, f = co, broadens for finite values o f f . For array widths between 300 and 1000junctions, f varies from about 6 to 7; from Fig. 31 we then expect the jump in a ( T ) to occur over a temperature range of about 1% to 2 % of TKT.This cannot be readily observed in arrays because other effects dominate. Even for arrays as small as 20 junctions wide, where f = 3, the width of the transition is only 3% of TKT.Thus, size-induced renormalization effects should not be readily observable in two-dimensional arrays. In actual fact, any effects of incomplete normalization of the IV characteristics are overwhelmed by the effects of thermally generated size-induced free vortices,39 discussed below.

15. FINITE-SIZE-INDUCED FREEVORTICES Thermally generated free vortices occur when the restrictions on AI and L are not adequately met. Unlike the renormalization effects discussed above, they create measurable deviations in the power-law behavior. Such devi-

336

R. S. NEWROCK ET AL.

ations have been observed, but have usually been ascribed to remnant magnetic fields or instrument noise. Herbert et al.39 showed that such deviations indicate the presence of finite-size-induced free vortices (see also the work of Simkin and Kosterlitz6'). In the strict sense they destroy the Kosterlitz-Thouless transition. If L is finite, then the energy to create a single vortex, Eq. (4.3), is finite at all temperatures and can easily be of order of k,T. Thus, there will exist a finite density of free vortices below TKT.The probability of finding a free vortex is easily calculated from the free energy, Eq. (4.7):

For T < TKT,Pf = 0 for infinite specimens. But for finite specimens, P , is never zero. Herbert et al.39 and Repaci et a1.61 obtained the vortex density when finite-size effects are important: (7.2) where Y = min{L, I.,} and c1 is a constant. This expression is valid for both finite-size limits, 1, >> L > a and L >> A, > a. In the latter case vortices separated by distances greater than A, can exist in an array. These will not be bound by a logarithmic potential. Thus, ,Il, not L, is the relevant length scale, and we replace AE,(L) [Eq. (4.7)] with AEv(Al) = nEj In Al/a. Finite-size-induced free vortices do not have a profound effect on experimental observations far above TKTbecause there are already a large number of free vortices obtained from the breaking of equilibrium pairs. This can be seen with the use of Fig. 25. The upper dashed line in the figure represents the smaller of the lengths AL and L. In the region where l + ( T )<< 9, temperatures are sufficiently far above TKTthat nf is large. 5 , is small in this regime so that all but the smallest pairs are unbound. The effects of finite-sized-induced free vortices are not observable, since nearly all vortex pairs are unbound in any case and the IV curves for temperatures in this region will follow the predicted Kosterlitz-Thouless behavior. The situation worsens as T decreases toward TKT,where 5 , grows quickly, eventually meeting and exceeding the cut-off length scale M. V. Simkin and J. M. Kosterlitz, Phys. Rev. 855, 11646 (1997). J. M. Repaci, M. C. Kwon, Qi Li, X. Jiang, T. Venkatessan, R. E. Glover 111, C. J. Lobb, and R. S. Newrock, Phys. Rev. B54,R9674 (1996). 6o

6'

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

337

min(L, AI}. For “infinite” samples, vortices separated by less than 5, should be bound while vortices separated by distances greater than this length remain unbound. The existence of the cut-off, however, means that the free-vortex density will have a temperature dependence as given in Eq. (7.2). In practice, however, the IV curves in this region will strongly resemble the ordinary Kosterlitz-Thouless behavior elucidated above: an ohmic region at the lowest currents and power-law behavior with a slope less than 3 at higher currents. The major difference is that the ohmic region will have a higher resistance resulting from the presence of the finite-size-induced vortices. While the IVs could in principle be analyzed to extract the excess resistance, this would require an exact knowledge of the free-vortex density, which is difficult to obtain. The most striking effect of finite-size-induced free vortices occurs at and below TKT.Here, in an infinite sample, the free-vortex density is zero (Eq. (5.1)) while it is nonzero (Eq.(7.2)) in a finite sample. This means that the IV characteristics will have a low-current, ohmic tail below TKT.The observed tail may not, in fact, have a slope of 1 due to limits on the experimental sensitivity, but may appear as a deviation from pure powerlaw behavior toward ohmic behavior. However, it must eventually have a slope of 1 at a sufficiently low current. Figure 32 shows the IV characteristics for a narrow (300 x 75) SNS array . ~ ~ array clearly shows a linear of niobium crosses on a gold ~ n d e r l a y e rThe tail below the temperature where a slope of 3 can be seen, indicative of the presence of finite-size-induced vortices. Since the Kosterlitz-Thouless transition involves the establishment of quasi-long-range order and the disappearance of all free vortices, the presence of free vortices destroys the phase transition in the stricr sense, even though the vortex unbinding mechanism is still in evidence. In this type of sample, TKT is a crossover temperature separating a low-temperature regime, where free vortices are due to finitesize effects, from a high-temperature regime, where most free vortices are not due to finite-size effects. Herbert et al.39 also note that the free vortices can obscure the signature of the vortex unbinding transition, the universal jump in a( T ) ,by modifying the apparent slopes of the IV characteristics. Finite-size-induced free vortices will contribute a flux-flow voltage of the form V = a2a2R,nJI, where a 2 is a dimensionless constant, in addition to the usual Kosterlitz-Thouless power-law characteristics, Eq. (5.17). The total voltage signal can be approximately written as the sum of the two contributions,

+

V = a1I“(*) a2a2R,nJ*I,

(7.3)

338

R. S. NEWROCK ET AL.

FIG. 32. The current-voltage characteristics as a function of temperature for an SNS array with a length of 300 islands and a width of 75 islands. The total current has been divided by 75 to give amperes/junction. The solid line indicates a slope of 3. (From Ref. 39, Fig. 2.)

where a l is temperature dependent with units of volts/(amp)"(T). Using this equation, and the fact that the free-vortex density [Eq. (7.2)] and the coupling energy E , are connected via a measurable quantity-the critical current -Herbert et al. were able to generate simulated IV curves that reproduced the features of the IVs of finite-sized arrays, showing very good qualitative agreement with the data. This revealed, however, an important consequence of finite sample size on array electrical transport measurements: the masking of the true value of the IV exponent in the power-law region. IV power-law behavior is typically observed in experiments over only one or two decades, and the presence of a significant ohmic tail can easily modify the observed slope of the IV curve, leading to misidentification of u(T) and thus the nominal vortex unbinding temperature.

TWO-DIMENSIONAL PHYSICS O F JOSEPHSON JUNCTION ARRAYS

16. SIZEEFFECTSDUE TO

RESIDUAL

339

FIELDS

Van der Zant et a1.62 identified a different form of finite-size effect, one due to the small residual magnetic field remaining in a cryostat after proper precautions for magnetic shielding have been taken. Careful r n e a ~ u r e m e n t s , 4 ’ * of ~ ~ the * ~ ~resistance near TKT show an exponential decrease in the resistance below TKT,contrary to Eq. (5.6). A typical result6’ is shown in Fig. 33, where the resistance of an aluminum SIS array is plotted as a function of the reduced temperature, t = T / T K T . In the figure the dashed line is the fit to Eq. (5.6) for the resistance above T K T . The deviation from the theory is clearly seen. This behavior is ascribed62 to thermally activated crossings of the sample by single residual field vortices. The reasons are simple and worth looking into, as this is a size effect that in the presence of a small magnetic field places a limit on L opposite to the earlier one -L cannot be too large. H. S. J. van der Zant, H. A. Rijken, and J. E. Mooij, J. Low Temp. Phys. 79, 289 (1990). J. P. Carini, Phys. Reo. 838, 63 (1988). 64 H. S. J. van der Zant, C. J. Muller, H. A. Rijken, B. J. van Wees, and J. E. Mooij, Physica 8152, 56 (1988). 62

lo-’

oc‘

\

CT

10-~

1o

-~

lo-’ 0

1

2

3

T

4

5

FIG. 33. The normalized resistance in nominally zero field versus the normalized temperature. The dashed line is a fit to Eq. (5.6). The inset shows the same data as a function of temperature. (From Ref. 62, Fig. 6.)

340

R. S. NEWROCK ET AL.

Experimentalists shield stray magnetic fields in a variety of ways (mu metal shields, superconducting shields, bucking coils. . .). This shielding is gauss. For an array generally effective down to fields on the order of In this small with a plaquette size a = 10 pm, this implies f = 5 x magnetic field, an array that is perhaps 100 junctions wide will have a current circulating along the outer edges of the array on the order of 0.1 i,, analogous to the Meissner effect in bulk superconductors.” This has a substantial effect on free vortices nucleated along the edges, and van der Zant et a1.62found that the existence of the circulating currents significantly modifies what are thought to be “zero-field data. To find the size of the energy barrier for vortex penetration, van der Zant et al. modeled the barrier by using an analogy between Josephson junction arrays and thin-film s u p e r c ~ n d u c t o r sThe . ~ ~ energy of a single vortex in a long array can be written as

where i, are currents from the vortex and i, are currents from external sources, including the shielding currents due to an external field, and the sum is from the edge of the vortex to its core. If y is the position of the vortex relative to the center of the array in units of a, then

The first term is the energy of the vortex in zero field, the second is the additional energy when a field is present, the third is due to transport currents, and the last represents the core energy. This is the energy barrier that a single vortex at the edge of an M-junction-wide array must overcome to enter the array. ) Fig. 34. The data of Fig. 33 are plotted as ln(R) versus l/z (= T K T / Tin The dotted line in this figure is Eq. (5.6). Van der Zant et al. examined this exponential behavior for a series of frustrations from nominally 0 to 0.1, also shown in Fig. 34. The slope of the exponential tail decreases with increasing frustration; that is, the deviation from Eq. (5.6) becomes more pronounced as the frustration increases -an unsurprising result. The fits to the data in 65

B. B. Goodman and J. Matricon, J. Phys. (Paris) 27, C3 (1966).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

341

lo-’

1o

-~

lo-’ 0

1

2

4

3

5

1/T FIG. 34. The data from Fig. 33, plotted versus the inverse reduced temperature for various values of j : The dotted line is Eq. (5.6). (From Ref. 62, Fig. 8.)

the figure are of the form

where q = Ebar(T)/EJand where Ebaris the energy barrier against a vortex entering the array from an edge. The square data points in Fig. 35 show the values of q needed to fit the data in Fig. 34 as a function of the frustration. The inset of Fig. 35 shows Us,Eq. (7.4), for zero transport current and an array width M = 63. For f = 0.0009, U,(y = 0) becomes negative- the array can now lower its energy by introducing a vortex. Van der Zant et al. call this the critical frustration. For their aluminum SIS arrays, with no applied.field (i.e., f = “O”), they find an energy barrier q = 11.5, while for zero field Eq. (7.4) yields q = 16.5. The lower barrier energy is interpreted as indicating the presence of a residual magnetic field. From Fig. 35, using the solid line, q = 11.5 corresponds to a critical frustration of 2.5 x lop4, or a residual field of 0.1 mG for van der Zant et al.’s specimens, an experimentally reasonable value. It follows from Eq. (7.4) that the energy barrier for a single vortex entering the array is 5E, nE, ln(2M/7c) and that a small residual field will reduce 1 027cln(2M/n), this considerably. For frustrations 0 < f < ,Ll E ~ * M - ~ ( + there will be no field-induced vortices in the array. A transport current will

+

342

R. S. NEWROCK ET AL.

-

u-

15

Y

=

10

5

f

10-l

FIG. 35. The energy barrier (in units of E,) for vortex penetration as a function of the frustration. The data (squares) are from fits to the data in Fig. 34. The inset shows the energy of a single vortex as a function of position, where zero is the center of the array, for five different frustrations. The solid line is the maximum barrier energy determined from the inset. (From Ref. 62, Fig. 9.)

reduce the barrier, leading, for sufficiently large currents, to vortices entering and to dissipation. Above “permanent” vortices enter the array, reducing the edge currents and slowing down the decrease of the barrier. Notice that increasing the size of the array to decrease the overall influence of the edge currents does not work. Larger M leads to stronger edge currents in the presence of residual fields. For fixed f > 1/nM2, the barrier for a single vortex from the residual field entering the sample decreases with increasing M . For van der Zant et al.’s samples, for instance, with a plaquette size a = 7 pm and a residual field of 0.1 mG, they calculate that the lowering of the energy barrier becomes important for M > 100. This size effect, then, in a manner opposite to that of the previous subsection, becomes smaller as the array width decreases. VIM. Classical Arrays: Magnetic Fields

For very small magnetic fields, an applied field creates a small density of free vortices, nf = B/m0 << l/a2, which manifest themselves as a linear tail on the

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

343

IV characteristics below TKT (see Eq. (5.2)). When the magnetic field becomes larger, more and more vortices enter the array and eventually the interaction between vortices can no longer be ignored. To consider this case, we return to the general Hamiltonian, Eq. (3.7). We make a gauge choice to include the external field explicitly:

where 9 is the unit vector in the y direction and we have again assumed that the self-field of the array is much smaller than the external field. When this gauge choice is used, the integral of the vector potential appearing in the Hamiltonian, Eq. (3.7), is zero for junctions that are parallel to the x direction and nonzero for junctions that are parallel to the y direction. When the integral in the y direction is evaluated, the potential energy part of the Ham,iltonian can be written as

H

=

-

1,E j C O S ( ~-~qhi) - 1 E , C O S ( ~ ~ - +i


-

27rfn( j

-

i)).

(8.2)

(ii)

In this equation we have divided the Hamiltonian into two sums: one over those junctions parallel to the x-axis and one over those junctions parallel to the y-axis. In Eq. (8.2), n is an integer that gives the x position of the islands in the second sum, x = nu, and f is again the frustration parameter, measuring the external field in terms of how many flux quanta there are per unit cell of the array: Bu2

f =-,

(8.3)

@o

Since the sum in Eq. (8.2) is over nearest neighbors, we note that (j-i)= f l . Equation (8.2) is at first sight intimidating. We can develop a feeling for what is going on by examining the second sum. The new term, 27rfn( j - i), makes it energetically advantageous for 4j # 4i. The ferromagnetic ground state found for f = 0 in Section 11, where all the phases have the same value, generally will not be the ground state when f # 0. This is one of the most attractive features of Josephson junction arrays: One can choose from a variety of Hamiltonians by changing the external magnetic field, which allows the study of a large number of different spin Hamiltonian systems. This Hamiltonian (Eq. (8.2)) is frustrated, which can be seen by considering a single plaquette, as shown in Fig. 36(a). We choose a plaquette and a value for f such that fn is an integer on the left-hand side of the plaquette.

344

R. S . NEWROCK ET AL.

FIG. 36. A single plaquette. (a) All the phases are equal. This configuration minimizes the energy only for integer f: (b) The minimum energy configuration for a fully frustrated (f= 1/2) array.

When this is the case, the left, top, and bottom junctions minimize their individual energies by having equal phases: 41

= 4 2 = 43 = 44.

(8.4)

The right junction will minimize its energy by making

44 = 43 + 2Xf:

(8.5)

Equations (8.4) and (8.5) cannot both be true for noninteger f, so it is not possible to simultaneously minimize all of the junctions’ energies. This is the definition of frustration. The concept of frustration was first introduced by Toulouse66 and plays a fundamental role in statistical physics, being central to the understanding of spin glasses, quasicrystals and other incommensurate systems, antiferromagnets, etc. Josephson junction arrays provide an extraordinarily useful system for studying frustration, as they are tunable 66

G . Toulouse, Commun. Phys. 2, 115 (1977).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

345

and can be studied in ordered and disordered systems, including quasiperiodic and fractal ones. Frustrated Josephson junction arrays are a physical realization of the frustrated XY model. Since ( j - i) = k 1, changing f by an integer will not change Eq. (8.2), so the properties of Josephson junction arrays should be periodic in A with period 1. One example6’ of this periodicity is shown in Fig. 37, where the measured resistance of an array is plotted as a function off: The resistance is found to have sharp minima for integer f and shallow minima at other rational values f = p / q , p and q integers. Since the cosine is an even function, changing f to -f will not change the Hamiltonian, so array properties do not depend on the sign of the field, 6’

M. Tinkham, D. Abraham, and C. J. Lobb, Phys. Ret.. BZS, 6578 (1983)

/

6.84 K

I

698 U

-203 -150 -100 -50

0

50

100 150 200

B(mG) FIG. 37. Frustration or magnetic field dependence of the resistance at various temperatures for a two-dimensional Pb-Cu 1000 x 1000 array. (From Ref. 67, Fig. 2.)

346

R. S. NEWROCK ET AL.

as can also be seen from Fig. 37. The periodicity and evenness in f together imply that we need only consider the interval 0 < f < 1/2; all other values off can be mapped back onto this interval. Because f = 1/2 is the case of maximum frustration (see Eqs. (8.4) and (8.5)), it has received the most attention. From Fig. 36(b) and Eq. (8.2), a compromise that minimizes the energy is to make 3 0 , &=n/4, +3 =n/2, and +4 = 344. This choice of phases makes the total energy -4E,/$ with a counter-clockwise current,

.

I = -

i,

$’

:

circulating around the plaquette, as shown in Fig. 36(b). The ground state for the full lattice for f = 1/2 is constructed by alternating plaquettes with clockwise and counter-clockwise currents, as shown in Fig. 38. The ground-state energy per junction for f = 1/2, E,(f = 1/2), is higher than the ground-state energy for f = 0:

(see Eq. (3.9)), so that the f = 1/2 state is less stable against thermal fluctuations than the f = 0 state. There are a number of methods for finding ground states for f = p/q,

FIG. 38. The ground state for the fully frustrated array, j = 1/2. The arrows indicate the direction of currents; all currents have a magnitude

iJd.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

347

where p and q are integer^.^*.^^.^^*^' Figure 39 shows the ground-state energy as a function off The function is not continuous. The higher-energy states will be less stable against thermal fluctuations and thus should have a higher resistance than the more stable states at high temperatures. This is the reason for the similarity between the experimental resistance data of Fig. 37 and the calculated ground-state energies of Fig. 39. Although a good deal is known about the ground states of the Hamiltonian of Eq. (8.2), there is no simple unifying picture of the behavior at nonzero temperature. In particular, except for integer f, where the Kosterlitz-Thouless transition occurs, the precise nature of the phase transition is not entirely clear. A? approximate but quite enlightening approach was taken by Tinkham et al.67 They simplified the problem by making the approximation that the

69 70 71

S. Teitel and C. Jayaprakash,Phys. Rev. Lett. 51, 1999 (1983). T. C. Halsey, Phys. Rev. 831, 5728 (1985). W. Y. Shih and D. Stroud, Phys. Rev. B28,6575 (1983). W. Y. Shih and D. Stroud, Phys. Rev. B32, 158 (1985).

I

FIG.39. The ground-state energy of a square Josephson-junction array as a function of the frustration. Here J is the Josephson coupling energy, E,. (After Ref. 70, Fig. 1.)

348

R. S. NEWROCK ET AL.

R(T) versus T curve was the same for all except for a shift in T,(f). This ignores the fact that the shape of the R(T) versus T curve depends on the nature of the phase transition. For example, when f = 0 the KosterlitzThouless transition determines the density of free vortices and the resistance as a function of temperature, as discussed in Section IV.8. The details are presumably different when f # 0. Tinkham et al. further assumed that the fractional shift in (see Eq. (5.4)) is equal to the fractional shift in the ground-state energy as f is varied. (This will be true only if the type of phase transition that occurs does not change when f is varied). When these ideas were combined, they obtained

To use Eq. (8.8), AR(f), R, T,(O), and aR/aT are measured. By using Eq. (5.4) and the measured ic(T), T-dependent quantities such as dR/dT can be converted to ?-dependent quantities. Equation (8.8) is useful because it allows quantitative comparison between experimentally measured properties and calculable ground-state energies. It gives good agreement with data in spite of the many approximations made to obtain it (see Fig. 40). To go beyond this basic approach, we must address the fundamental question of what type of phase transition occurs when a magnetic field is applied to an array. Outside of the zero-field limit, the situation is not clear. For general f the problem has proven to be very difficult and little if any progress has been made. Even the “simple” case of full frustration, f = 1/2, which has been extensively studied both theoretically and experimentally, remains controversial. One suggestion is that a vortex-unbinding transition still occurs, as can be seen schematically in Fig. 41. Figure 41(a) shows the and ground state for f = 1/2, Fig. 38, redrawn schematically with representing plaquettes with clockwise and counter-clockwise currents. A vortex-like excitation can be created by replacing one of the - plaquettes plaquette, as shown in Fig. 41(b). Such an excitation behaves very with a much like a vortex excitation in the f = 0 case. Most important, the energy to create this vortex is logarithmic in the sample size:

+

+

(8.9) Thus, when a vortex and an antivortex are present, as shown schematically in Fig. 41(c), they bind and unbind in the same manner as described in

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

349

AR mfl

T (K) FIG.40. The comparison of the observed temperature dependences of the amplitude of the periodic resistance change A R and d R / d T (From Ref. 67, Fig. 3.)

Section IV.8, with a pair energy logarithmic in the separation E, =E,ln(:): 271

Jz

(8.10)

this leads to a Kosterlitz-Thouless transition for f = 1/2. This approach may oversimplify the problem. Because the f = 1/2 ground state has lower symmetry than the f = 0 ground state, thermal excitations are present in the former that are not present in the latter. For example, the f = 1/2 ground state shown schematically in Fig. 41(a) has a double degeneracy not present in the f = 0 ground state shown in Fig. 13.The superlattice of + and currents in Fig. 41(a) would have the same energy if all of the currents were reversed, interchanging the + and - plaquettes. In equilibrium at T = 0, one of the two degenerate ground states will occur. As the temperature is increased from T = 0, thermal fluctuations will cause regions of the “wrong” type to form. Such a region, called an antiphase domain, is shown schematically in Fig. 41(d).

350

R. S. NEWROCK ET AL.

+-+-+-+ -+-+-++-+-+-+ -+-+-++-+-+-+ -+-+-+-

+-+-+-+ -+-+-+-

+ - +[+!+ _-_ -+

-+-+-++-+-+-+ -+-+-+-

(a)

+-+-+-+

- +:+;+ - + I___

+-+-+-+ -+-+-+-+-+-+FIG. 41. Schematic states for the fully frustrated case, f = 1/2. The pluses and minuses represent clockwise and counterclockwise currents circulating around a plaquette. (a) The ground state with no excitations (vortices) present. (b) One vortex-like excitation present, created by reversing the current direction on one plaquette (dotted lines). (c) A vortex and antivortex present. (d) An anti-phase domain.

The presence of antiphase domains suggests that a phase transition may occur that does not depend on vortex unbinding. If the system starts at T = 0 in a single domain, as the temperature increases larger and larger antiphase domains will occur as thermal fluctuations. If these are the dominant fluctuations, as they are in an king system, the phase transition to a disordered state occurs when an antiphase domain has infinite extent. Such a transition would be of the Ising type, with critical behavior and electrical consequences different from the Kosterlitz-Thouless transition. This fully frustrated case (f = 1/2) has been extensively studied because of these interesting symmetries and because of the interplay between the order parameters associated with those symmetries. Unfortunately, in spite

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

351

of all the work the controversy about the nature of the phase transition and whether or not there are two phase transitions (Kosterlitz-Thouless and Ising) is unresolved. The models discussed include the antiferromagnetic XY spin system on a triangular l a t t i ~ e , ~and ’ . ~ the ~ XY spin system on a square interest is the fully frustrated XY system on a l a t t i ~ e . ~ Of ~ . ~particular ’ square lattice, f = 1/2, because of the interactions between the XY model and the Ising symmetry associated with the antiferromagnetic arrangement of plaquette ~ ~ r r e n t ~ . Th ~ e~ ground * ~ ~ state * ~ of ~ *the~ model ~ . ~ has ~ plaquettes with alternating clockwise and counter-clockwise circulating currents in a checkerboard pattern that adds the discrete symmetry of the Ising model to the rotational symmetry of the XY model. At low temperatures, therefore, the Hamiltonian has the topological quasi-long-range order of the XY model and the ordinary long-range order of the Ising model. As the temperature is raised both the antiferromagnetic XY and the Ising order should disappear. h ave explored generMany theoretical treatments79.80’81’8’~83’84’85’86.87 alized versions of Eq. (8.2) where the relative strength of the couplings that cause the XY ordering and the Ising ordering are varied. (In a real array, the relative strengths are fixed since there is only one energy, E,, in the problem.) In the generalized models rich phase diagrams result: First- and second-order phase transitions have been proposed, as well as separate Ising and Kosterlitz-Thouless-type transitions and simultaneous Ising and Kosterlitz-Thouless transitions, depending on the relative coupling strengths. Interestingly, the case that corresponds to the fully frustrated array is perhaps the most difficult to analyze theoretically. The first numerical study of this system7’ found a sudden loss of XY order as the temperature is raised, followed by an increase in the specific heat with S. Miyashita and H. Shiba, J . Phys. SOC. Jpn. 53, 1145 (1984). D. H. Lee, J. D. Joannopoulos, J. W. Negele, and S. D. Landau, Phys. Rev. 860,128 (1986). 74 J. Villain, J . Phys. C10, 4793 (1977). 7 5 S. Teitel and C. Jayaprakash, Phys. Rev. 827,598 (1983). 76 T. C. Halsey, J . Phys. C18, 2437 (1985). 7 7 S. E. Korshunov and G. V. Uimin, J . Stat. Phys. 43, 1 (1986). ” E. Granato, J. M. Kosterlitz, and M. P. Nightingale, Physica 8222,266 (1996). 7 9 E. Granato, J . Phys. C20, L215 (1987). ‘O E. Granato, J. M. Kosterlitz, J. Lee, and M. P. Nightingale, Phys. Rev. Lett. 66, 1090 (1991). E. Granato and J. Kosterlitz, J . Phys. C19, L59 (1986). 8 2 E. Granato and J. M. Kosterlitz, J . Appl. Phys. 64,5636 (1988). 8 3 E. Granato, J. M. Kosterlitz, and J. Poulter, Phys. Rev. B33,4767 (1986). 84 J. Lee, E. Granato, and J. M. Kostelitz, Phys. Rev. 844,4819 (1991). 8 5 M. Y. Choi and S. Doniach, Phys. Rev. 831,4516 (1985). 86 M. Y. Choi and D. Stroud, Phys. Rev. 832,5773 (1985). 87 H. Yosefin and E. Domany, Phys. Rev. 832, 1778 (1985). 72 73

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R. S. NEWROCK ET AL.

lattice size -indicative of an Ising-type transition. The investigators were not able to determine the exact critical behavior of the model and speculated that either (a) as the Ising transition at TI is approached (from below) the king excitations interact with the Kosterlitz-Thouless excitations and the Kosterlitz-Thouless transition occurs before the loss of Ising order -that is, TKT< T,; or (b) as T, is approached from below the Ising excitations give rise to a large jump in the kinetic inductance (actually in the helicity modulus, defined below), and TI = TKT. Since that time, a large number of numerical studies of the frustrated XY model have been published but no consensus has been reached -indeed, the results are often conflicting. What does appear to be correct is that, if there is not a single transition, then two transitions occur very close to one ano~~er~72.80,84.88.89,90,91 .92.93,94,95

O l ~ s o nproposed ~~ that the finite-size scalings used in most of the papers mentioned were not quite sufficient. He presented new MonteCarlo work that appears to distinguish between the two possibilities, and argued that first a Kosterlitz-Thouless transition occurs and then an Ising transition at T, > TKT.That is, two distinct transitions should be present and observable. For the case of nonzero f, a theoretical prediction for the IV characteristics appears to be lacking. If the transition is Kosterlitz-Thouless-like, data may be interpreted in the standard way. This has been done for f = 1/2 by van Wees et al.,42 who concluded that the transition is Kosterlitz-Thoulesslike. In the limit of small current, array current-voltage characteristics are ohmic (Eq. (5.6)) or power law (Eq. (5.17)) for T above and below T,, respectively. This indicates the presence of free vortices above T, and only bound vortices below T,, just as it did in the zero-field case. Data comparing the f = 0 and f = 1/2 cases are shown in Fig. 42. Lerch et aL9' studied the f = 1/2 phase transition of square arrays using the inductance technique discussed in Section VI. Figure 43 shows their data "

Y. M. M. Knops, B. Nienhuis, H. J. F. Knops, and H. W. J. Blote, Phys. Rev. 850, 1061

( 1994). 89

90 91

92

93 y4

95 y6 9'

J. M. Thijssen and H. J. F. Knops, Phys. Rev. 842, 2438 (1990). J. Lee, J. M. Kosterlitz, and E. Granato, Phys. Rev. 843, 11531 (1991). E. Granato and M. P. Nightingale, Phys. Rev. 848, 7438 (1993). G. Rarnirez-Santiago and J. V. Jose, Phys. Rev. Lett. 68, 1224 (1992). G. Crest, Phys. Rev. 839, 9267 (1989). J.-R. Lee, Phys. Rev. 849, 3317 (1994). S. Lee and K. C. Lee, Phys. Rev. 849, 15184 (1994). P. Olsson, Phys. Rev. Lett. 75, 2758 (1995). Ph. Lerch, Ch. Leernann, R. Theron, and P. Martinoli, Phys. Rev. 841, 11579 (1990).

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353

T(K) 5.5

4.5

4.0

3.5

3.1

14

12 10

a

8 6 4

2

0

1

2

3

4

5

6

7

1/z FIG.42. The power-law exponent a ( T ) of the 1V characteristic curves as a function of the inverse normalized temperature T,,/T at f = 0 and ,f = 1/2. B: data at a voltage threshold of 3x volts. A:data at lo-’ volts. V:lo-’ volts. The drawn lines connect the data points. The dashed lines are Monte-Carlo simulations from Ref. 75. (From Ref. 42, Fig. 4.)

for ,f = 0 and f = 1/2 on Pb-Cu arrays. As the signals for both frustration values are quite similar, except for the expected overall shift to lower temperatures, it appears to be reasonable to analyze the data for full frustration within the Kosterlitz-Thouless framework. (Lerch et al. noted, however, that the Re(6V) data for f = 1/2 are wider than at f = 0, indicating that an additional dissipative mechanism is likely at work.) To determine how well the experimental results for f = 1/2 fit the Kosterlitz-Thouless theory, Lerch et al. compared measured and theoretical values of l + ( T )and what is called the helicity modulus, r. The helicity modulus is a measure of the phase ordering in a system, corresponding to the free energy increment associated with a twist (4-+ 4 + 64) in the phase. For our purposes, r is inversely proportional to the sheet inductance of the array, and we use it because it is the quantity that is generally calculated by

3 54

R. S. NEWROCK ET AL.

3

4

5

T(K) FIG.43. Real (Re(dV)) and imaginary (Im(6V)) part of the detected voltage as a function of temperature in zero field (f= 0) and at full frustration (f = 1/2). (From Ref. 97, Fig. 1.)

theorists.

can be written as98*99*100

r=F,, E'(

T )'

(8.11)

where F,, is a temperature-dependent quantity arising from spin waves whose value depends on the frustration. Figure 44 shows rf=o(?)and rf=l,2(?). The junction critical current needed to calculate the reduced temperatures was found in the same manner as in the section above with a correction due to H a l ~ e y ~ ~ .relating ~~.'~' T. Ohta and D. Jasnow, Phys. Rev. B20, 139 (1979). P. Minnhagen, Phys. Rev. B32, 7548 (1985). loo P. Martinoli, Ph. Lerch, Ch. Leemann, and H. Beck, Jpn. J . Appl. Phys. 26, Suppl. 26-3,

98

99

1999 (1987).

T. C. Halsey, Phys. Rev. Lett. 55, 1018 (1985).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

355

T(K)

3.8

3.2 3.4 3.5

3.9

4.0

1.o

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

t FIG. 44. The helicity modulus r(?)versus the reduced temperature 7for f = 0 and f = 1/2 at 8 kHz. The linear variation at low temperatures is due to spin waves. The solid lines are obtained from the Kosterlitz-Thouless scaling equations; the dashed line represents the universal jump. (From Ref. 97, Fig. 2.)

array and single junction critical currents for f = 1/2. The low-linear temperature portion of the curves is due to the ad-hoc inclusion of spin waves. The drop-off at the high temperatures is due to the vortices. The solid lines in the figure are from numerical solutions to the Kosterlitz-Thouless scaling equations.46 The dashed line in the figure is the universal jump in the kinetic inductance. The numerical calculations were performed for two different scale lengths, infinite scale and scales up to t,, stopping the renormalization (Appendix C) when I , = <+(T).As can be seen from the figure, the fits are quite good up to the critical temperature, but they diverge at that point. Lerch et al. speculate that the divergence is caused by the extra dissipation mentioned earlier in this section. Figure 45 shows the data plotted as in Fig. 3 0 t i 2 versus a reduced temperature, T, and corrected for spin waves. We see that the free-vortex

356

R. S. NEWROCK ET AL.

0.5

,

N

3 d

0 1

2

0

1

2

3

4

T FIG. 45. f i 2versus reduced temperature Fig. 3.)

T( = T/T,,)

for f = 0 and 1/2. (From Ref. 97,

correlation length has the expected temperature dependence for both f = 0 and 112. From the data for r and e,, Lerch et al. extracted values for the critical temperatures and E, and compared them with values inferred from MonteThey found that the Carlo calculations of Teitel and Jayapraka~h.~~.’’ agreement is quite good between experiment and simulation for the absolute numbers and that the ratio of the critical temperatures is nearly equal to the ratio predicted by simulations. Thus, from the work of Martinoli et al.,’” it appears that vortex unbinding plays a significant role in the transition in the fully frustrated system. Little experimental evidence was seen for the Ising-like transition related to the degenerate ground state- the mechanism responsible for the P. Martinoli, R. Theron, J.-B. Simond, R. Meyer, Y. Jaccard, and Ch. Leemann, Physica Scripta T49, 176 (1993).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

357

f

= 1/2 transition appears to be vortex unbinding, a conclusion strengthened by additional work on this system by Martinoli et al. Similarly, no evidence was found to support Halsey's''' hypothesis of the breaking up of quarter-vortex pairs sitting on domain wall corners. Other values of f have been studied, but the literature is much less extensive. Possibly because extremely high-quality samples are needed to see field effects for f = 1/3, 1/4, 1/5, 2/5, etc., little is known either theoretically or experimentally for fields other than f = 0 and f = 1/2. Halsey predicted that the low-temperature state for irrational f is a glassy state, but experimental study of irrational fields is hampered by sample imperfections and the difficulty of interpreting the data.

IX. Classical Arrays: Disorder Many phase transitions in naturally occurring systems are affected by randomness- no natural material is completely free of defects or impurities, and some systems, such as spin glasses, are dominated by randomness. For these reasons, phase transitions in disordered systems are actively studied. Josephson junction arrays provide a nearly ideal vehicle for these studies because well-defined disorder can be easily incorporated into the samples. In the previous sections, we assumed that the arrays are ideal-that all of the critical currents i, [Eq. (2.5)], and thus the coupling energies E, [Eq. (2.8)], are identical, and that the lattice parameter a is the same everywhere in the sample. But there is disorder in real arrays where critical currents can vary by perhaps + 5 % and lattice spacings by perhaps +2% in the best samples. It is therefore important to understand what effect irregularities, both intrinsic and deliberately introduced, will have. To understand what types of randomness can be built into arrays, we return to a slightly generalized version of the Hamiltonian, Eq. (3.7):

This equation contains two differences from Eq. (3.7). The first is explicit: The coupling energy is allowed to depend on the indices i and j , reflecting the fact that the critical currents of the different junctions may be different. Such disorder is called bond disorder, since the junctions form bonds between superconducting islands. A related disorder is site disorder, where entire islands are randomly removed from the sample; this is an extreme type of correlated bond disorder, since all the bonds attached to a missing

358

R. S. NEWROCK ET AL.

island have zero coupling energy. The second difference between Eqs. (3.7) and (9.1) is implicit: The integral of the vector potential will be modified if the positions of the islands themselves are altered, randomly changing the plaquette areas. Disorder of this type is called positional disorder. These types of disorder are discussed separately in the sections to follow. 17. BOND AND SITE DISORDER

A. B. Harris'03 considered weak bond disorder theoretically for systems that undergo phase transitions to states with true long-range order. He found that replacing uniform coupling energies by energies drawn from a Gaussian distribution can leave the phase transition intact or can destroy the long-range order, depending on whether or not the heat capacity diverges at the critical temperature in the uniform case. If the heat capacity does not diverge, the essential aspects of the phase transition remain unchanged. The correlation length, for example, will still diverge in the same way at a critical temperature T,, although T, may be altered. Although Harris's argument is strictly valid only for systems with true long-range order, it can be plausibly applied to two-dimensional Josephsonjunction arrays. This may be the reason that the weak disorder present in all real samples does not seem to destroy the Kosterlitz-Thouless transition. Much stronger disorder is obtained when bonds or islands are completely removed from the sample so that the coupling energy will be zero for the affected bonds. D. C . Harris et al.'04 made arrays in which random fractions of the islands were removed to see whether such percolative disorder'05 destroyed the Kosterlitz-Thouless transition. This type of disorder is probably important in granular films near the metal-insulator transition. Disordered Josephson arrays provide a very well-characterized model for this system. If p is the probability that a site is occupied (has an island), then above a critical percolation threshold p , there will be a path of filled sites (islands connected by junctions) across the specimen. The sites connected to the superconducting path are known as the infinite cluster. The theoretical value'05 of p , is 0.591; this value is for an infinite two-dimensional square system. A scale length 5, characterizes a percolating system. It is a measure of the size of isolated clusters of islands for p < p , and a measure of the size of the holes in the infinite cluster for p > p , . All the samples used by D. C . Io3 Io4

A. B. Harris, J . Phys. C: Sol. St. Phys. 7 , 1671 (1974). D. C. Harris, S. T. Herbert, D. Stroud, and J. C. Garland, Phys. Rev. Lett. 67, 3606 (1991).

l o ' D. Stauffer, Introduction to Percolation Theory, Taylor and Francis, London and Philadelphia (1985).

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359

Harris et al. had p > p,. As p approaches p , from above, 5, diverges and the superconductivity is lost. Near pc

5,

= 4 P - P,)

-

”.

(9.2)

In two dimensions the critical exponent v = 4/3. A useful picture of a percolating system is the nodes and links model.’06 In this picture the random network is approximated by a uniform network with nodes separated by links with a diverging length 5,. Between each node is a quite complex random structure. The essential feature of this structure, sometimes referred to as a “blob,” is that there is at least one point where it is singly connected, so that cutting a single bond will completely separate two adjacent nodes. The picture thus takes a random lattice of lattice spacing a and replaces it with a rescaled uniform l a t t i ~ e ’ ~ ~of ”length ~~”~~ scale 5,. In this picture a percolative array of Josephson junctions has a rescaled correlation length given by

5

+

tpe[C/(7.

-

Z,)I”’~

(9.3)

This equation is equivalent to Eq. (5.3) with a replaced by 5,. D. C . Harris et al., using a theoretical argument by Ebner and Stroud,’” argued that, just as lengths are rescaled by percolation effects, the Josephson coupling energy E , should also be renormalized. This leads to

where t is the conductivity exponent for percolation,”’ t = 1.30. Harris et al. experimentally investigated arrays in which loo%, go%, 8O%, 70%, and 60% of the sites were filled. They experimentally determined the critical percolation fraction to be p , = 0.5847, close to the theoretical value for a square lattice. Figure 46 shows’04 R versus T data typical for a series of five such arrays. In the figure the data for each array are normalized to their values just below the island superconducting transition. It is clear that the transition to the zero resistance state broadens with increasing disorder. Figure 47 shows typical IV curves, in this case for a 70% site-filled sample.

lo’

Io9 ‘lo

‘‘I

P. G. deGennes, J . Phys. Lett. (Paris) 37, 250 (1981). P. G. deGennes, J. Phys. Left. 37, L1 (1976). A. Coniglio, Phys. Rev. Lett. 46, 250 (1981). J. P. Straley, J . Phys. C15, 2333 (1982). C. Ebner and D. Stroud, Phys. Rev. 828, 5053 (1983). D. J. Frank and C. J. Lobb, Phys. Rev. 837, 302 (1988)

360

R. S. NEWROCK ET AL.

z

Q \

Q

(n),

FIG. 46. The normalized resistive transition for 100% (O),90% (A), 80% 70% (*), and 60% (0) site-disordered arrays with a measuring current of 1 PA. R , is the resistance at T = 8.85 K. The inset shows a portion of a disordered array. (From Ref. 104, Fig. 1.)

Harris et al. define the TKT(p)to be that temperature at which the IV exponent a ( T ) = 3. A plot of the reduced temperature, T K T ( p ) / f K Tversus (l) p - p,, is shown in Fig. 48. From the figure we see that the reduced transition temperature scales as a power of ( p - p,) in qualitative agreement with Eq. (9.4) but with a power of 1.16 instead of 1.30.This 11% discrepancy is typical of percolation experiments on finite lattices. To further check that the transition in these very disordered arrays is Kosterlitz-Thouless-like, D. C . Harris et al. plotted eighty different IV curves for the 70% to 100% site-filled samples, Fig. 49(b). The data were plotted versus reduced coordinates to demonstrate that universal twodimensional scaling is indeed taking place. As shown by M. P. A. Fisher"' and Koch et a1.,Il3 one signature of a superconducting transition is a universal collapse of the IV characteristics into two branches, one above and M. P. A. Fisher, Phys. Rev. Lett. 62, 1415 (1989). R. H. Koch, V. Foglietti, W. J. Gallagher, G. Koren, A. Gupta, and M. P. A. Fisher, Phys. Reo. Lett. 63, 1151 (1989). 'Iz

'I3

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

361

FIG. 47. Log voltage versus log current for a 70% array; the lines have slopes of 1 and 3. The unlabeled curves correspond to T = 6.80 K, 6.40 K, 6.20 K, and 6.00 K, respectively. Note that TKTis labeled T, in this figure. (From Ref. 104, Fig. 3a.).

FIG. 48. Reduced dimensionless transition temperature (see text) as a function of sample disorder, p - p , . Note that YKTis labeled 2 in this figure. (From Ref. 104, Fig. 2.)

362

R. S. NEWROCK ET AL.

0.96

0.90

1.00

1.06

1.10

T/Tc ,

1 6 1 .

.

,

.

.

.

. . .

11

1 -7

-5

-3

-1

1

3

5

log J(VT) €J FIG.49. (a) The IV exponent a ( T ) versus temperature for the 100% (O), 90% (O), 80% (A), and 70% (0) arrays. The inset shows the calculated a(T) versus T/T,, as a function of e,. (b) Condensation of IV curves for the 70% to 100% arrays, plotted in reduced variables z and < + to illustrate universal scaling. Twenty temperatures are shown for each array. Note that TKTis labeled T, in these figures. (From Ref. 104, Fig. 4.)

one below the transition temperature. The important scaling parameters are the dynamical exponent z associated with the correlation time, T cc (5 +)z, and the coherence length 5 , itself. D. C. Harris et al. treated z and the constant in the exponent of the coherence length, Eq. (5.3), as fitting parameters to obtain the best universal curve. The optimal fit occurred at z = 2.0 f 0.1, as predicted by the theory. The importance of this figure is that it demonstrates, for samples with 70% or more of the sites filled, that a Kosterlitz-Thouless transition is taking place and

TWO-DIMENSIONALPHYSICS OF JOSEPHSON JUNCTIONARRAYS

363

that the IV characteristics do show the expected behavior for a KT transition. One therefore concludes that the main effect of site disorder is to depress the vortex unbinding temperature and broaden the resistive transition. This can be seen directly from Fig. 49(a), where a ( T ) is plotted versus T/ TKT. The 100% sample shows the expected jump from one to three. A similar jump is observed for the disordered arrays, but the jump is not as pronounced and the linear growth of a ( T ) below TKTis less rapid. The results of Figs. 46 and 49(b) imply that Eq. (9.4) is quite plausible, implying that highly disordered arrays continue to undergo a KosterlitzThouless transition. Eq. (9.4) leads to the result:

This indicates that the reduction of TKTfrom its value in a fully ordered array can be viewed as a direct measure of percolative disorder in arrays. 18. AC SUSCEPTIBILITY AND DISORDER Eichenberger et al.' l 4 used the AC susceptibility method discussed earlier (Section VI) to probe site-diluted arrays very near the percolation limit. At low frequencies, in samples near the percolation threshold, p 2 p c , they observed the Kosterlitz-Thouless transition in much the same manner as it was observed via the AC susceptibility in ordered samples. In percolative samples, Eichenberger et al. found that the critical temperature depended on p in a manner consistent with the universal predicti~n.''~ To understand their results we first consider the perfect ( p = 1) array. At very low temperatures ( T << TKT)there are very few vortices present and the array can be considered a lattice of bonds, each of which is an inductor LJ = @.,/2ni, in parallel with a resistor R,(T); see Eq. (6.2). The sheet admittance for an SNS triangular array (as used by Eichenberger et al.) is then

where L = LJ/$ and R = R,/$. SNS arrays are generally manufactured by placing superconducting islands on an underlayer of normal metal. When such a system is site A. L. Eichenberger, J. Affolter, M. Willemin, M. Mombelli, H. Beck, P. Martinoli, and S. E. Korshunov, Phys. Rev. Lett. 77, 3905 (1996).

364

R. S. NEWROCK ET AL.

disordered by removing islands, the kinetic inductance associated with the junctions is certainly removed. However, the effect on the resistance is much smaller as the underlayer is still present. Thus, if one randomly site-dilutes an array, the randomness injected into the resistor network is not as important as that injected into the inductor network. For T << TKT,then, where vortex excitations can be ignored, a randomly site-disordered proximity-coupled array can be modeled by a two-component impedance network, with each bond having one of two admittances:

or

and each admittance appearing in the array with probability p and 1 - p , respectively. An AC susceptibility measurement done at frequency o sets a length scale r, (from Eq. (6.3)). At p = p c , 5, diverges and ro < 5, at all frequencies. In this limit the sheet conductivity can be shown to be G(o) cc o-',a result that reflects the dynamic scaling due to the self-similar nature of the system. This follows from the fact that at p,, G cc (GlG,)'i2. (This result, with a constant of proportionality 1, is exact for bond percolation on a square lattice." l 6 Scaling and universality arguments allow its use here.) Combining this result with Eq. (9.7) in the limit oe << 1, and comparing it to Eq. (9.6) yields '3'

where cL and cR are constants of order unity and e = LJR,. If we are above the percolation threshold, p > p,, these equations do not hold at all frequencies but only at frequencies such that ro < <,. That is, at high frequencies the length being sampled is smaller than the percolation correlation length, and the experiment never samples a length long enough to see the periodicity of the underlying lattice. This implies that a crossover frequency o,exists such that for o < o,we enter a "homogeneous" regime 'I5 'I6

A. M. Dykhne, Zh. Eksp. Eor. Fiz. 59, 110 (1970); Sov. Phys. JETP 32, 63 (1971). J. P. Straley, Phys. Rev. 815, 5733 (1977).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

365

where L and R are length-scale independent. Eichenberger et al. used general scaling arguments near p , to show that in this regime, as o approaches zero,

and

where Lo and R , are the sheet inductance and resistance as Q approaches zer,o, ck and c i are again constants of order 1 depending on the lattice structure, and t is the percolation conductivity exponent. They also find the crossover frequency to be

At low frequencies the entire specimen is sampled and the phase dynamics are characterized by phonon-like modes of the phase. At high frequencies, localized “fracton-like” modes are dominant. Eichenberger et al.’ l 4 tested these predictions and looked for disorder effects on the phase transition as well as in site-diluted triangular SNS arrays (Pb-Cu). They used two arrays with p = 0.55 and 0.51 -much closer to p , (=0.50 here) than the arrays of D. C. Harris et aI.lo4 Figure 50 displays the inverse kinetic inductance as a function of temperature for a very low frequency, o = 0.5 Hz, well below the crossover frequency. Both arrays exhibit a drop in the kinetic inductance consistent with a KosterlitzThouless transition at a temperature TKT(p).More interesting, perhaps, is that for temperatures below the transition (the Kosterlitz-Thouless transition is indicated by the dashed line in the figure) the temperature dependences of the two kinetic inductances are equal, indicating that the junction kinetic inductances, L,( T )= L ,(O)f( T ) ,differ only in L,(O). Since L,(O) cc R, and R , is known,’” the ratio of the kinetic inductances can be extracted, L- ‘ ( T , O . 5 5 ) / K 1 ( T , 0.51) z 4.1, and matched to the result given above, Eq. (9.9), by choosing t = 1.4. This may be compared to the theoretical value of f = 1.3 for two-dimensional percolation, confirming the interesting results obtained by D. C. Harris et a l l o 4 that even in extremely site-diluted arrays the Kosterlitz-Thouless transition still exists, albeit with a lowered transition temperature. In other words, disorder has “minimal” effect on the inductance (and thus on the effective superfluid density). ‘ I 7 K. K. Likharev, Dynumics of Josephson Junctions and Circuits, Gordon and Breach, New York (1979).

366

R. S. NEWROCK ET AL.

10"

-

d

--Q

'01 109

4 1O8

-

1o-2

L

E p4.55

1o

il

-~

FIG. 50. Semi-log plots of the temperature dependence of (a) the inverse sheet kinetic inductance at 0.5 Hz, and (b) the normalized dissipative component of the mutual inductance change at 317Hz for two disordered arrays with different percolation fractions. In (a) the dashed line is the universal prediction for the K T transition. In (b) the slashed area is below the sensitivity threshold of the mutual inductance measurements. (From Ref. 114, Fig. 1.)

However, the same is not quite true if one looks at the dissipation. The dissipation, or rather the dissipative component of the mutual inductance change, is plotted in Fig. 50(b) for a frequency of 317 Hz, still well below the crossover frequency. One can see the enormous increase in the dissipation as one moves closer to p,. In this experiment the change in the mutual inductance is proportional to R - 3 or, from Eq. (9.10), to (p - p J 3 ' , a rapid growth that is confirmed by Fig. 50(b).

TWO-DIMENSIONAL PHYSICS O F JOSEPHSON JUNCTION ARRAYS

367

Figure 51 shows the logarithm of the inverse sheet inductance as a function of the frequency for three temperatures far below the transition temperature. These data are for p = 0.51. L-'(w) is seen to be independent of frequency at low frequencies with a sudden crossover to power-law ) u z 0.5. behavior at about 1 kHz. L - ' ( w ) is found to vary as d - "with This is consistent with Eq (9.8). The resistive data show a similar effect, although the low-frequency data is not as good as it was taken at the limits of the experiment's sensitivity. From the resistive data, u = 0.7.

-

n I

$ 10'O

U d

k

1o9

n

c:

..........

Y

% 10-~

. O O ........................

A

O 0

n .? .? O

A

.

3

0

...........A ...

,.'

A

lo6

1

I

1oo

I

I

,,,,,,I

I

10'

1

,,,,,,I

1o2

,

,

,,,,,,,I

o3

1

0

,,,,,,,I

1o4

0/2n[Hzl FIG. 51. The frequency dependence of the inverse sheet kinetic inductance L-' and of the sheet resistance R at three different temperatures well below the critical temperature for a disordered array with p = 0.51. The dashed lines are guides to the eye to identify the low-frequency plateaus of R(w). (From Ref. 114, Fig. 2.)

368

R. S. NEWROCK ET AL.

19. POSITIONAL DISORDER In ordered arrays, when the applied field is commensurate with the underlying lattice (f= p / q , p , q integers), the transition temperature varies strongly with q but relatively weakly with p (Section VIII). In incommensurate systems it is believed68v759118 that there is no phase transition for T > 0, although a Monte-Carlo calculation”’ suggests that in this case the arrays undergo a spin-glass transition. In a sense, then, the system’s behavior is highly discontinuous as f varies. It is therefore interesting to investigate the behavior of the system when different values o f f are mixed together. This can be achieved with positional disorder, which introduces random frustration, random uncorrelated phase shifts, without pinning the phase angles themselves. Arrays with positional disorder are also very interesting because they are a limiting case of very inhomogeneous superconductors and therefore akin to granular supercondlictors. Granato and Kosterlitz’ l 9 considered arrays with positional disorder, as shown in Fig. 52.120 Here the centers of the superconducting islands are moved a random distance from their “official” lattice sites. They found a and glassy state but found a critical value for the applied field J(=@/@J predicted a re-entrant transition. This hypothesis was tested by Forrester et a1.,121and the area has seen much theoretical activity. We will consider it in some detail. We place the original locations of the island centers at Y and move the centers a distance u,, determined by a Gaussian distribution:

1

P(u,) = 2xA2

(3)

(9.12)

The “static” Hamiltonian, Eq. (3.7) can be split into two parts: a spin- (or “phase-”) wave part and a vortex part, H = H,, + H , . In the ordered case the spin-wave portion of the Hamiltonian does not contribute to the critical This remains correct in the disordered behavior other than to change TKT.98 system, and the equilibrium statistics of the vortices are decoupled from the spin waves. The vortex part of the Hamiltonian can be rewritten as a sum over variables on a lattice whose sites are the original centers of the plaquettes, the so-called “dual lattice” of the array.12*We can then write the

’” ‘I9 ‘’O

’”

M. Y. Choi and D. Stroud, Phys. Rev. B32, 7532 (1985). E. Granato and J. Kosterlitz, Phys. Rev. B33, 6533 (1986). M. G. Forrester, S. P. Benz, and C. J. Lobb, Phys. Rev. B41, 8749 (1990). M. G. Forrester, H. J. Lee, M. Tinkham, and C. J. Lobb, Phys. Rev. 837,5966 (1988). J. V. Jose, L. P.Kadanoff, S. Kirkpatrick, and D. R. Nelson, Phys. Rev. B16, 1217 (1977).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

369

FIG. 52. Schematic of a positionally disordered array. The crosses indicate the island sites in an ordered array; the dots indicate the sites in the positionally disordered array. The lines represent junctions. (From Ref. 120, Fig. 1.)

vortex portion of the Hamiltonian as HV

= 2n2EJ(T) R

1*R

( M R - fR)G(R

- R’)(MK

-fRh

(9.13)

where f R is the frustration or flux through the plaquette at R and the M R are integers, 0, 1, +2,. . . . This is the Coulomb gas Hamiltonian where the “vortex charges” are defined on the dual lattice sites R. The lattice Green’s function G is given by’23

+

(9.14)

and is valid for R >> a. F. Spitzer, Principles ofRnndorn Walk, Van Nostrand. Princeton, NJ (1964), 148

370

R. S. NEWROCK ET AL.

An examination of Eqs. (9.13) and (9.14) shows that the Hamiltonian can be viewed as a set of vortices (or charges) of strength

interacting via a logarithmic potential. We note first that in the original Hamiltonian, Eq. (3.7), y = $i- $ j - $ij

and $.. = IJ 2= @o

s'

A-dl.

i

As discussed earlier, the sum of the $ij around a plaquette must be equal to 2n(m - f ) where m = 0, f 1, f 2 , k.. . . The M,s in Eq. (9.13) are analogous to these ms and hence for a sum around a plaquette &(MR - fR) = 0. Second, in the continuum limit, a + 0, the original lattice and the dual lattice are indistinguishable, and we cannot distinguish between r and R. This means that the area of a plaquette A,, to lowest order in u, + U,,is A, = Ao(l + V,-u,), where A , is the area of the undisplaced plaquette. From this it follows that, to the lowest order in the displacement, the flux in the plaquette at R is

wheref, is the average flux per plaquette. If we put Eqs. (9.14) and (9.15) into Eq. (9.13), the latter can be rewritten as119

(9.16)

Granato and Kosterlitz used renormalization techniques to obtain the solution to this Hamiltonian; the physics, however, is quite accessible. Equation (9.16) represents a gas of fractional charges, q a (EJ)'IZ(M,- f,), interacting with a quenched dipole distribution, pR a Sou,. The first term is the vortex-vortex interaction, the second term is the vortex self-energy, and the third term is the vortex-quenched dipole interaction. By quenched we mean that the dipoles are frozen in place and do not change over time.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

371

When f, is an integer -that is, when the average number of flux quanta per plaquette is an integer -we have a gas of vortices perturbed by a random dipole distribution. We can understand the physical nature of the dipoles by examining Fig. 53. In Fig. 53(a) we show two adjacent plaquettes, each threaded by an integer flux f,. In the Coulomb gas analogy, both plaquettes are uncharged. We add positional disorder, Fig. 53(b), by moving the boundary between the plaquettes a distance 6 to the right. Since the magnetic field doesn’t change, this change in the plaquette area changes the net flux through each plaquette and, in the Coulomb gas analogy, creates “charges” ,f,d/a on these two plaquettes. These charges create a dipole of strength (f,S/a)a = f,S. The first two terms in the Hamiltonian, Eq. (9.16), form the Hamiltonian originally discussed by Kosterlitz and T h o ~ l e s sand ~ ~ discussed in detail

(b)

FIG. 53. (a) Two adjacent plaquettes of equal areas-“uncharged.” (b) The same two plaquettes “disordered” by moving the boundary between them a distance 6. The plaquettes are now “charged.” (c) The phase diagram predicted by Granato and Kosterlitz for the positionally disordered lattice. (After Ref. 121, Fig. 1.)

372

R. S. NEWROCK ET AL.

earlier. Granato and Kosterlitz investigated the disordered Hamiltonian by applying the results of Rubinstein et al.,lZ4 who had solved the similar problem of integer charges perturbed by a random background of dipoles. For large disorder, Rubinstein et al. found that the phase transition is always destroyed. For small disorder, Rubinstein et al. and Granato and Kosterlitz found a phase boundary in the temperature-magnetic-field plane, inside of which there are no free vortices and outside of which free vortices exist. Defining the effective disorder as foA, they predicted two transition temperatures at a given frustration, K-(f,A) and T,+(f,A), as shown in Fig. 53(c). Below T-(f,A), the background of quenched dipoles weakens the interaction between the mobile vortices so that some are unbound, leading to a disordered state. In the intermediate range, T,-(foA) < T T,+(f,A), the increasing density of thermally generated vortex pairs is adequate for screening the quenched dipoles and all vortices are bound. For T > T,+(S,A), thermal energy is sufficient to split bound pairs and we return to the case for T > TKTin the perfect array. Granato and Kosterlitz also showed that for a fixed value of the disorder A, the two critical temperatures merge when the frustration reaches a critical value:

-=

(9.17)

For f > f, there is no ordered region and the array is not superconducting. Thus, for values of the frustration less than f,, a re-entrant phase transition will be seen. In contrast to the case of the uniform array, the transition is not universal, depending on the frustration. These predictions were tested experimentally by Forrester et al.' using SNS arrays. They created positional disorder by moving the center of the plaquette a random distance in a random direction while keeping the location of the junctions unchanged. (Figure 54 shows their arrays; Fig. 54(a) is an ordered array; Fig. 54(b), a positionally disordered array.) By this procedure they varied the plaquette areas but did not introduce any (deliberate) disorder into the coupling energies. Figure 55 shows the resistance R ( f , ) versus magnetic field for two samples, one with and one without disorder. Both traces show singlejunction oscillations as well as the oscillations due to the field threading the plaquettes. The inset in the figure defines A R ( f , ) for the oscillations due to the plaquettes. Forrester et al. compensated AR(f,) for single-junction effects lZ4

M. Rubinstein, B. Shraiman, and D. R. Nelson, Phys. Rev. 827, 1800 (1983).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

FIG. 54. Sections of the lithographic masks used to prepare arrays with (a) A* A* = 0.10. (From Ref. 121, Fig. 2.)

=0

373

and (b)

and plotted the corrected AR(f,) versus f , for several values of the disorder as measured by A*, Fig. 56(a). Figure 56(b) shows a plot of the critical fields as a function of 1/A*; it shows the linear dependence predicted by Eq. (9.17). The experimental critical fields were taken to be where the array oscillations disappeared; they found that fc,exp= 0.95/A*, which is much larger than the theoretical prediction, Eq. (9.17). This disagreement had two causes: first, for experimental reasons they used a uniform rather than a Gaussian distribution. They argued that the two distributions mainly differ in their standard was derived. Second, they pointed out deviations, from which A* = A/$ that their measurement was sensitive to short-range order, which is more robust than the long-range order considered theoretically. fr,,heo measures the point of destruction of phase coherence over long distances, while fc,exp measures it over distances L a. This short-range phase coherence is much more robust and less easily destroyed, leading to a larger critical field.

-

374

R. S. NEWROCK ET AL.

FIG. 55. The resistance versus magnetic field, R(f,), for A* = 0.10 (upper trace) and A* = 0, lower trace), showing oscillations due to collective behavior, modulated by single-junction effects. The inset shows the definition of the oscillation amplitude, AR(f,). (From Ref. 121, Fig. 3.)

Forrester et al. quantified this by looking at the structure of R(f,)at rational values off = p / q , or coherence over a length scale L qa. Although the Kosterlitz and Granato theory is only valid for integer f, Forrester et al.3 data at f, = n + 1/2 andf, = n 1/3 suggest that it is possible to define critical fields ff,exp(q)for these higher-order effects. They found

-

+

(9.18) where c 1 and c2 are constants. Extrapolating their data to q = 0 yields fC,,,,(.o)A = 0.06, which is in excellent agreement with Kosterlitz and Granato’s theory, Eq. (9.17). In a second paper, mostly concerned with disorder in the fully frustrated system, Granato and Kosterlitz”’ point out that Forrester et al.’s extrapo-

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

375

r4

0, 0.b

,d

0.6

,

j

\ \ -

20 -

4,

.

IO0

I

0

-

I

-

10

5

l/A*

I

I

1s

,

20

.

1

FIG. 56. (a) The amplitude of the resistance oscillations versus f , for various values of A*. The lines are least-square fits, whose extrapolations to zero define the experimental critical fields .f,. (b) Values o f f , from (a) plotted versus l/A*. The line is a least-square fit constrained to have zero intercept. (From Ref. 121, Fig. 4.)

lation of the experimental critical field to large q values is necessary for a valid comparisons with theory. However, they note that such an extrapolation makes sense only if L,theo is independent of q and therefore ignores possible effects such as domain walls, which could presumably arise because the correlation function decays algebraically and true long-range order does not exist, and fractional vortex charges (e.g., those suggested by H a l ~ e y , ~ ~ Korshunov and U i m i r ~ and , ~ ~ G r a n a t ~ Forrester .~~ et al.'s conjecture can be correct only be if a finite amount of disorder decouples such defects from the vortices, which then unbind first. Should this be correct, the breakdown of superconductivity for f, = n p / q will be governed by the same mechanism as for f, = n. This is discussed further below.

+

E. Granato and J. Kosterlitz, Phys. Rev. Lett. 62, 823 (1989).

376

R. S. NEWROCK ET AL.

Forrester et al.'" found no experimental evidence for the predicted re-entrance. They performed Monte-Carlo simulations' 2o of positional disorder in an XY spin system and saw no clear evidence for re-entrance. They suggested that finite-size effects and vortex pinning due to disorder were blocking the re-entrance. Monte-Carlo calculations were also done by Chakabarti and Dasgupta,lZ6with the same result. For small average f they found that the system exhibits a transition from the normal state to a Kosterlitz-Thouless-like phase, with critical behavior qualitatively similar to the Kosterlitz-Thouless transition in a finite-ordered system. Thus, experiments"' and Monte-Carlo simulations' 2 0 * 126 agree with the predicted critical values for disorder and the higher-temperature branch of the phase boundary of Fig. 53. However, 'the re-entrance predicted by the lower-temperature branch of the phase boundary was not seen in either the experiment or the simulations. The Granato and Kosterlitz analysis, which leads to the re-entrant picture, is based on the model and calculations used by Rubinstein et al.lZ4 In turn, their work is based on considerations of the renormalization of the vortex interaction to lowest order in the presence of random disorder. However, several recent papers have cast doubt on the validity of this approach. Korshunov and Nattermann'27*'28~'29 showed that higher-order corrections to the work of Rubinstein et al. lead to new divergences in each order of the expansion used, leading to suppression of the domain of stability of the ordered phase. As Korshunov and Natermann pointed out, this means that the ordered phase -vortices bound in pairs -completely disappears, or else the correct description needs to be based on a different approach. On the other hand, Ozeki and N i ~ h i m o r i 'showed ~~ that if an ordered phase of the XY model with random disorder does exist, the phase diagram cannot include a re-entrant transition into a disordered phase. Korshunov and Nattermann'283129and Nattermann et a l l 3 ' used several approaches to demonstrate that if the disorder is sufficiently weak, the phase order will be stable at zero temperature and at sufficiently low temperatures. Nattermann et a l l 3 ' and Cha and Fertig13' suggested a phase diagram without a re-entrance line, and Nattermann et al. went further, suggesting that some sort of freezing phenomenon below a temperature T,(f,A) is preempting the re-entrant transition, which occurs at T-(f,A) < T,( f,A).

lZ7

13' 13'

A. Chakabarti and C. Dasgupta, Phys. Rev. 837, 7557 (1988). S. E. Korshunov, Phys. Rev. 848, 1124 (1993). S. E. Korshunov and T. Nattermann, Physica 8222, 280 (1996). S. E. Korshunov and T. Nattermann, Phys. Rev. 853, 2746 (1996). T. Ozeki and H. Nishimori, J . Phys. A26, 3399 (1993). T. Nattermann, S. Scheidl, S. E. Korshunov, and M. S. Li, J . Physique 15, 565 (1995). M.-C. Cha and H. A. Fertig, Phys. Rev. Lett. 74, 4867 (1995).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

377

Tang' 3 3 expanded on these ideas, extending the work in two directions. (Tang's paper is extraordinarily clear and very well written; we refer the reader to it for most of the details. Here we will briefly review his conclusions.) He first studied the equilibrium behavior of a single vortex in a background of quenched dipoles. He showed that there is a glass transition at a temperature given by

T, = EJ(~f0A/2)'".

(9.19)

Below T, the entropy of the vortex goes to zero, indicating that the vortex is localized at the lowest energy site. The free energy of the system is proportional to the logarithm of the system size at all temperatures, and setting the prefactor of the logarithm to zero generates the phase transition shown in Fig. 57. Tang then examined in detail a dilute gas of bound vortex-antivortex pairs. The freezing line T = T* in Fig. 57 can be shown to be related to the loss of entropy of the pair over an area in which the pair is isolated from other pairs of comparable size. If the center of the pair is fixed, the two vortices of the pair freeze at T,. In the ordered phase T* < because the pair explores an area much larger than its size and hence has a lower freezing temperature. Tang found that the freezing of the pairs is not associated with a singularity in the free energy of the entire system and there is thus no real phase transition at T*. Disorder also generates random zero-field polarization of the pairs, enhancing the effective disorder seen by vortices separated by a large distance. This effect, also noted by Scheidl (see note 22, Ref. 133), shifts the critical strength of the disorder from n/8 to a smaller value. The results on the dilute gas of vortex pairs are then used to generate a set of renormalization group recursion relations to obtain the average large-scale properties of the system. Aside from several minor differences, these are identical in form to those of Nattermann et al.,I3l and to the extent that the assumptions used and the simplifications that result are valid, a phase diagram of the kind shown in Fig. 57 is produced. Tang also indicated a drawback of the renormalization-group description. The renormalized or effective disorder is always assumed to have a Gaussian distribution, but his analysis shows that it is the tail of the distribution that dominates renormalization effects at low temperatures- putting a limit on how well renormalization-group predictions can be trusted insofar as the detailed shape of the phase boundary is concerned. There appears to be little doubt, however, that the ordered phase exists down to T = 0 when f is far 133

L.-H. Tang, Phys. Rev. B54, 3350 (1996).

378

R. S. NEWROCK ET AL.

t

Disordered

0 FIG. 57. The phase diagram proposed by Tang (Ref. 133), Nattermann et al. (Ref. 131) and Cha and Fertig (Ref. 132). T+ is the phase boundary for the order-disorder transition. There is no re-entrance, but vortex-pair excitations "freeze out" below T,. (After Ref. 133, Fig. l(b).)

below f,. Thus, the experiments and simulations of Forrester et al'zo~'z'. correctly produced the phase diagram prior to the full theory.

20. POSITIONAL DISORDER -FULLY FRUSTRATED The effects of positional disorder on fully frustrated specimens (average

f, = n + 1/2) have been studied by several authors, including Choi et al.'34

and Granato and K o s t e r l i t ~ . ' ~ ~ Choi et al. performed both renormalization-group and Monte-Carlo studies. The renormalization calculations showed that the phase transition remains to at least moderate disorder but the transition temperature decreases with the effective disorder, LA. Unlike the renormalization calculations in the unfrustrated system, there is no re-entrant transition. MonteCarlo simulations confirmed the renormalization-group calculations. They also obtained helicity modulus and specific heat results that suggest that the phase transition remains to moderate disorder, but for large disorder no clear evidence of specific heat peaks was found, indicating that the phase transition may be destroyed. 134

M. Y. Choi, J. S. Chung, and D. Stroud, Phys. Rev. 835, 1669 (1987).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

379

As mentioned earlier, Granato and Kosterlitz’ 2 5 found that, at least for the fully frustrated case, positional disorder decouples vortices from other possible excitations of the system. (They believe that this is also true for other values of f, but cannot demonstrate that.) The essential physics of their argument is based on two ideas. First, because of the underlying symmetries, the ordered system can have Ising order coexist with XY This leads to one of the two possible scenarios described earlier as the temperature is raised: Either an XY transition occurs followed by an Ising transition, or a single transition occurs in which Ising disorder triggers X Y disorder, yielding a simultaneous loss of both Ising order and superconductivity. Second, positional disorder has a much greater effect on the XY order than it does on the Ising order. This means that increasing the positional disorder will decrease the XY transition temperature. Eventually, the XY transition temperature will go below the Ising transition temperature and allow a double transition. At this point the existence or absence of superconductivity is controlled by the vortices. This brings us back to the system discussed in Section VIII -logarithmically interacting vortices in a random quenched dipole background, leading to a critical field L A d ( 3 2 ~ ) - ’ / ~When . the X Y degrees of freedom become disordered, the Ising order persists but does not qualitatively affect the superconducting transition. Figure 58 shows two possible phase diagrams for f, = n 1/2. In Fig. 58(a) we see the two separate phase transitions, Kosterlitz-Thouless and

+

f oA,

f,A

f oA

FIG. 58. Possible phase diagrams for a positionally disordered Josephson-junction array for average flux per plaquette f . = n 1/2. S denotes regions where both superconductivity and Ising (chiral) order exist; I denotes a normal region with king (chiral) order; and D is the disordered phase. In (a) two separate transitions occur and in (b) a single transition occurs for zero disorder (A = 0). In both cases phase coherence is possible only for ,&A < (32n)”’. (After Ref. 125, Fig. 1.)

+

380

R. S. NEWROCK ET AL.

Ising, and in Fig. 58(b) we see a single transition. In both cases superconductivity is possible for effective disorder less than ( 3 2 ~ ) - ' / ~Note . also that in both cases a reentrant transition is depicted, a result that Forrester et al.'s experiments and the later work of Korshunov et al. and Tang appears to rule out. The fully frustrated ordered array is believed to lie very close to the bifurcation point shown'35 in Fig. %(a) so that even a small amount of disorder, well below LA, will induce the bifurcation. They conclude that the estimate for L A should therefore be good for both integer and half-integer average field per plaquette, f,. There are also implications for the AC dynamics of fully frustrated arrays -Granato and K ~ s t e r l i t z 'point ~ ~ out that the crossover from inductive to resistive behavior may well be described by the dynamical theory of the unfrustrated case. As discussed earlier, the work of Martinoli and co-workers appears to bear this out. 21. DYNAMICS OF DISORDERED ARRAYS:RANDOMDISORDER To this point we have considered the effects of disorder on the phase transition. A number of investigators have also studied the effects of disorder on vortex dynamics in arrays. For very large driving forces it is intuitive that the randomness is irrelevant and that interactions between vortices dominate the dynamics. Such systems display homogeneous vortex flow. For very small driving forces the dynamics should be dominated by the disorder and the flow should be inhomogeneous. Examples of physical systems to which this might apply include' 3 7 the dynamics of charge density waves, magnetic bubble memories, fluid flow in porous media, and pinning in type-I1 superconductors. The large research effort in high temperature superconductors and the need to understand critical currents and flux-flow in those materials have led to much study of this problem in superconductors. The very nature of most experimental systems does not allow for control of the disorder; studying disordered Josephson junction arrays therefore holds particular promise in this area. Also, recently developed imaging techniques (for example, those of Lachenmann et al.' 38) allow direct observation of vortex dynamics in these arrays. 3691

J. M. Thijssen and H. J. F. Knops, Phys. Rev. B37, 7738 (1988). D. S. Fisher, Phys. Rev. Lett. 50, 1486 (1983). 1 3 7 D. S. Fisher, Non-Linearity in Condensed Matter, eds. A. R. Bishop et a/., Springer-Verlag, New York (1987). 1 3 * S. G. Lachenmann, T. Doderer, D. Hoffman, R. P. Heubener, P. A. A. Booi, and S. P. Benz, Phys. Rev. B50, 3158 (1994). 135

136

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

381

Dominguez and co-workers' 39,140*141numerically studied vortex motion with positional disorder, varying the effective disorder, f,A. If the average frustration, f = n p / q , is small (p/q << l), the effective vortex concentration is small. In the absence of disorder (A = 0) the vortices form a periodic lattice as discussed earlier. For very weak disorder, 0 Q LA Q 0.1, a vortex lattice still exists but is distorted. For intermediate values of the disorder, 0.2 < foA Q 0.5, and no transport current, Dominguez et al. found that the equilibrium vortex configuration is random -there is no crystalline order. For transport currents well below a critical current I=,, all of the vortices are pinned by the random potentials and there is no dissipation. For I 2 I,, some vortices depin and move- but only during a short transient time after the current is applied-to a new, but again stable, configuration. At I,, the simulations show that a channel for vortices through the specimen opens up. Through this channel vortices can move without pinning. As I increases further, more and more channels open, until, for sufficiently large I , all of the vortices move with a random inhomogeneous motion. In this regime the IV characteristics are shown to be very nonlinear and essentially identical to the so-called plastic-flow regime in disordered type-I1 superconductors. Above a higher characteristic current, the voltage grows linearly with the current and we enter the flux-flow regime- all of the vortices move with the same velocity. Dominguez et al. found that the transition from the plastic-flow to the flux-flow regime appears to be a dynamical critical phenomenon with a diverging correlation length.

+

22. DYNAMICS OF DISORDERED ARRAYS: CORRELATED DISORDER

Leath and c o - ~ o r k e r s , ' ~ ~ ~whose ' ~ ~ interest ~ ' ~ ~ ~was ~ ~ fracture ~ and breakdown in heterogeneous materials, studied the effects of correlated defects in Josephson junction arrays. They considered single defects consisting of a row of missing bonds and funnel or bow tie defects (Fig. 59). Numerical simulations done on long linear defects (Fig. 59(a)) show that just above a critical current I , the ends of the defect, where the current is D. Dominguez, Phys. Rev. Lett. 72, 3096 (1994). D. Dominguez, N. Gronbech-Jensen,and A. R. Bishop, in Proceedings of the Workshop on Macroscopic Quantum Phenomena and Coherent Superconducting Arrays, eds. C. Giovannella and M. Tinkham, World Scientific, Singapore (1995), 278. 141 D. Dominguez, Physica 8222, 293 (1996). W. Xia and P. L. Leath, Phys. Rev. Lett. 63, 1428 (1989). 143 P. L. Leath and W. Xia, Phys. Rev. 844, 9619 (1991). 144 Y. Cai, P. L. Leath, and 2. Yu, Phys. Rev. 849,4015 (1994). 14' P. L. Leath and Y. Cai, Phys. Rev. 851, 15638 (1995). 139

140

382

R. S. NEWROCK ET AL.

A

FIG. 59. The (a) linear defect and (b) funnel or bow tie defect used in the work of Leath and co-workers. (From Ref. 143, Figs. 8 and 11.)

concentrated as it passes around the defect, become sources of vortices and antivortices. These progressively depin and move across the sample toward the edges, generating a voltage drop across the specimen. As the current is increased the rate of generation of vortices increases until a second critical current is reached. At this point the vortices begin to be created on the two rows adjacent to the defect as well. This produces a sudden increase in the voltage with a very complicated time dependence. For a linear defect n bonds long, they found that the critical current varies as n-’’’. The effects of current focusing were investigated using the funnel or bow tie defects shown in Fig. 59(b). It was expected, because of the funnel, to find vortices depinning in the center of the funnel before depinning occurred at the edges. They found that the junctions in the center of the funnel do reach the local critical current first, however, the current through them then remains fixed until the supercurrent at the junctions at the outer funnel corners also reaches the local critical current. With a further increase in the

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

383

applied current, vortices then depin at these corners and the voltage state starts. Vortices flow along a “critical path” from the corner of the funnel to the edges of the sample. Leath’46 pointed out that in ordinarily (randomly) disordered samples this sort of self-organized criticality occurs as well. One region of the specimen after another becomes critical and excludes further increases in the current, until, eventually, a critical path is formed across the array. The critical path in the junction arrays is analogous to a critical crack in a mechanical system.

X. Classical Arrays: Nonconventional Dynamics One of the important underpinnings of the dynamics of the KosterlitzThouless phase transition is the concept of the vortices as a ‘‘gas’’ of interacting “charges.” This is the so-called Coulomb gas analogy (see, for example, Ref. 5). We assumed in the earlier sections of this article that we can treat vortices as point particles, with a well-defined mass, moving in a viscous medium and that neither the mass nor the viscosity depends on the vortex motion- that is, on the dynamics. In the sections below we discuss the possibility of nonconventional dynamics for the vortices, which means a departure from the simple point-particle picture. We also discuss the effects that occur in the underdamped system where the vortex mass is nonnegligible.

23. NONCONVENTIONAL VORTEXDYNAMICS Measurements by Theron et al.14’ in underdamped arrays, and simulations and calculations by Hagenaars et al.’48 for overdamped arrays, indicate that the simple point-mass picture may not be completely correct. In this section we discuss the measurements of Theron et al. The work of Hagenaars et al. is discussed below in the section on ballistic vortices. If the temperature is not too far below the transition temperature and the applied field is small (f < 0.05), the vortices in a Josephson junction array should behave very much like a gas of free (unpinned) and independent (noninteracting) charges. It should be possible to describe the response of these free vortices by a Drude-like dielectric constant. P. L. Leath, Physica B222,320 (1996). R. Theron, J.-B. Simond, Ch. Leemann, H. Beck, P. Martinoli, and P. Minnhagen, Phys. Rev. Lett. 71, 1246 (1993). 14* T. J. Hagenaars, J. E. Van Himbergen, J. V. Jose, and P. H. E. Tiesinga, Phys. Reo. 853, 2719 (1994). 146

14’

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R. S. NEWROCK ET AL.

Figure 60 shows the real (R) and imaginary ( L ) parts of the sheet impedance'49 Z ( w ) = R(w) + i w y w ) of a triangular SNS array over a narrow field range, If1 c 0.05. For w not too low, R(w) increases with w, nearly saturating at the top of the experimentally accessible field range. The temperature of the experiment was sufficiently far below TKT,and the sample '49

This differs from the convention used for zero field defined by Eq. (9.6).

250

200

Y

100

50

0 I

-4

-2

0

2

J

FIG. 60. The sheet resistance and sheet inductance of a triangular Josephson-junction array in the as a function of the applied field for If1 < 0.05. The inset shows the inductive data, Uf), interval If1 < 0.005. (Note that f is given in percent in this figure.) The reduced temperature (see text) is 5 = 0.5 << T~~ = 1.5. (From Ref. 147, Fig. 2.)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

385

sufficiently large, that there were essentially no thermally generated free vortices present, only those due to the applied field. The data in the figure are not consistent with the frequency-independent flux-flow resistance prediction, R ( f ) cc RoRf (see Eq.(3.33)), where RON is the normal-state sheet resistance of the array. Notice also the growing curvature of L(f)in the limited interval If1 < 0.005 (inset, Fig. 60); L(f)is seen to depart appreciably from linearity. This shows that even in the absence of pinning, vortex correlations are important, probably because of the long-range logarithmic interaction. To investigate this discrepancy, Theron et al. examined the response of the system near f = 0, investigating the dielectric constant defined by Z(w; f ) = iwL,,E(u, f), where LkO is the sheet kinetic inductance. If we view the vortices as a gas of “charges,” the Drude model gives E(0,

f)

=

1 + 7c%,

(10.1)

1 0

where gu,o is a frequency-independent conductivity proportional to the vortex d e n ~ i t y . ~Th ~e, Drude ~ ~ , ~model ~ thus yields a dielectric constant that is proportional to f/o at low frequency. Theron et al. extracted Re( 1 / ~ and ) Im( 1 / ~ from ) impedance data taken at four frequencies for I f I < 0.05; the results are shown in Fig. 61, plotted versus f/o. Within their experimental accuracy the data do appear to scale with f / w over nearly four decades of f / w . However, it is apparent that for large f / w , Re(l/E) scales as ( f / w ) - ’ ,in contrast to the (f / w ) - 2 predicted by the Drude model, Eq. (10.1). This response of the vortex medium at low frequency implies that Re(l/E) is nonanalytical at w = 0, pointing to anomalous vortex diffusion. Theron et al. were able to extract some interesting results from these data. They approximated the dependence of Re(l/e) in Fig. 61 as (10.2)

where the characteristic frequency w, is proportional to f: Assuming that this is a correct form at low frequency and a reasonable extrapolation at high frequency, they employed the Kramers-Kronig relations to obtain the imaginary part of the dielectric constant: (10.3)

386

R. S. NEWROCK ET AL. 7 " " " I

I O ~ 10'

' ' """1

' ' """I

' '""''1

10'

' '

IO,~

I

10'

27Cf/O) I s ] FIG. 61. The real and imaginary parts of the inverse complex vortex dielectric constant versus the ratio of frustration to frequency. The solid curve is Minnhagen and Westman's (1994) model, Ref. 150. The dashed curve is a Drude model, and the dotted straight line is an f / w fit to the Re(l/e) data at large f / w values. (From Ref. 147, Fig. 3.)

The dielectric constant given by Eqs. (10.2) and (10.3) can be put back into a Drude form if is replaced by a,(o) -a frequency-dependent complex vortex conductivity, cr = q 2 p : o u ( 4= n E J P u ( 4 n u t

(10.4)

where p , ( o ) is a complex vortex mobility, nu = 4f/(a2fi) is the vortex

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

density in a triangular array, and frequencies, w << w o ,

(7tEJ)’/’

387

is the vortex “charge.” At low

(10.5) where po = 0 0 / 2 n 2 E ,is the mobility of an isolated vortex. The result is that the unconventional scaling properties of &(Lo)imply that for a given ~ “ ( wvanishes ) as l/ln w as (0 + 0. The vortices become anomalously slow at low frequencies. Theron et al. believed this to be a general property of two-dimensional superconductors, since it emerges from an essentially pinning-free system (the triangular array). This speculation was corroborated by simulations of the dynamics of thermally nucleated free vortices ’ by the above TKTin square arrays by Minnhagen and W e ~ t m a n ’ ~and work of Tiesingha et al.15’ Theron et al. offered some speculative ideas concerning the origin of such “sluggish” vortices. One unlikely possibility is diffusion in disordered systems- this does not obey classical laws, and the transport properties are unusual.15’ However, careful measurements of Theron et al.’s array as a function of field show an exceedingly rich variety of structure out to substantial fields. Such structure is indicative of a very uniform system, and it is very unlikely that disorder-induced effects exist. Nonstandard pinning is another possibility, one source of which might be the sample edges in a finite-size specimen. The authors pointed out that the mutual inductance technique is particularly immune to such size effects. It should be noted, however, that the vortices involved can be quite large and if ,Il FZ L, the sample edges might very well affect vortex motion. The most plausible explanation is the coupling of the vortices to spin waves, which would cause the vortices to lose energy, reducing their mobility. This has been studied with respect to the motion of massive vortices in underdamped arrays’ 5 3 - 1 5 4 but could well be relevant for the viscosity in the underdamped system. Capezzali et al.ls5 found that such considerations were able to reproduce the l/ln o dependence of the mobility. Holmlund and Minnhagen’ 5 6 performed simulations of vortex motion assuming that the physics was given by the Langevin equation, Eq. (6.5).

15’

P. Minnhagen and 0. Westman, Pliysica (Amstrrdum) ZZOC, 237 (1994). P. H. E. Tiesinga, T. J. Haganaars, J. E. Van Himbergen, and J . V . JosC, Phys. Rev. L e f t

78, 519 (1997). 1 5 * S. Havlin and D. Ben Avraham, Adv. Phys. 36, 695 (1987). U. Eckern and A. Schmid, Phys. Rev. B39, 6441 (1989). 154 U. Geigenmuller, C. J. Lobb, and C. B. Whan, Phys. Rev. 847, 348 (1993). 15’ M. Capezzali, H. Beck, and S . R . Shenoy, Phys. Rev. Lett. 78, 523 (1997). IS‘ K. Holmlund and P. Minnhagen, Phys. Rev. 854, 523 (1996).

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They obtained the vortex density correlation function from which they extracted the complex frequency-dependent dielectric constant. They found that the density correlation function goes to zero as l/t for long times. (This result was also found from simulations of the XY model using timedependent Ginzburg-Landau dynamics.43) Translating this result into the frequency of the dielectric constant, they found

for small w. This means that the real part of the conductivity diverges as In o for small w, as found by Theron et ,al. Capezzali et al.'55 pointed out that Minnhagen et al.'s result is actually for intermediate frequencies and not for low frequencies.

24. BALLISTIC MOTIONOF VORTICES In an underdamped array, the mass of a vortex should be nonnegligible, Eq. (3.26). This was shown experimentally on an aluminum tunnel junction array by van der Zant et al.15' Since the mass is nonzero, if the vortices are driven by a current and the current is suddenly switched off, their inertia should keep them moving -i.e., ballistic vortex motion should be possible. Such motion has been theoretically predicted.' 5 3 * 15 8 ,1 5 9 3 To observe ballistic motion a vortex must first be accelerated to a sufficiently high velocity u , , ~such that the vortex translational kinetic energy is much greater than its potential energy, which is due to the lattice egg crate potential (Section I11 and Fig. 15). A transport current can be used to accelerate vortices and, from Eq. (3.27), assuming the potential is negligible, the vortex velocity will be u, = Q0i/qa.To observe ballistic motion, after the vortices are accelerated they must enter a relatively force-free region where their motion can be detected by measuring the induced voltage, Eq. (3.30). The criterion that the kinetic energy of the vortex must at least be equal to or greater than the potential energy implies that

(10.7)

Is' H. S. J. van der Zant, F. C. Fritschy, T. P. Orlando, and J. E. Mooij, Phys. Rev. Lett. 66, 2531 (1991). l S 8 U. Eckern, Application of Statistical and Field Theory Methods to Condensed Matter Physics, Vol. 218 of NATO Advanced Studies Institute Series, eds. D. Baeriswyl et al., Plenum, New York (1989), 311. T. P. Orlando, J. E. Mooij, and H. S. J. van der Zant, Phys. Rev. B43, 10218 (1991).

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389

where we have used Eqs. (3.16) and (3.26). For a square array, a1 = 0.199 and a2 = 1, and for a triangular array, a1 = 0.043 and a2 = 3/2. This is the minimum velocity a vortex must have if we are to observe ballistic motion. When self-field effects are important this minimum value for the vortex velocity can increase significantly because E,, increases, although calculations also indicate that the dynamical mass of a vortex may be an order of magnitude larger than that derived earlier.’60 On the other hand, the velocity cannot be too large. In addition to ohmic losses, fast-moving vortices lose energy in two ways: by “Josephson plasma” oscillationsl 54.1 5 7 , 1 6 1 , 1 6 2 and by row switching. When row switching occurs the,phases of all the junctions in a single row rotate continuously, with a monotonically increasing phase difference along the row. Simulations indicate that the maximum velocity a vortex can have if we are to avoid row switching is163

Plasma oscillations are excited when a moving vortex excites the oscillatory modes of the junctions. The single junction equation of motion, Eq. (2.12), also represents the equation of motion for a pendulum. A vortex crossing a junction is the same as rotating the pendulum over the top. Since the pendulum is underdamped, it rings for many oscillations. This coupling of the vortex to plasma oscillations causes the vortex to lose energy rapidly. Geigenmuller et aI.ls4 and Eckern and Sonin,’62 using a continuum approximation, showed that below a threshold velocity Vu,th the coupling leads to a threshold velocity of vu,th

0.ln J.l

vu,min.

(10.9)

When all of this is taken properly into account, ballistic motion for a square lattice should not be possible, but it should be possible in a triangular lattice for vortex velocities up to about 50% greater than v , , , ~ ~ . Fazio et al.164 did a more realistic calculation by treating the charge transferred across the junction as discrete and found a stiffer spectrum and,

“’ 162

163

R. Fazio and G. Schon, Phys. Rev. 843,5307 (1991). P. A. Bobbert, Phys. Rev. 845,7540 (1992). U.Eckern and E. B. Sonin, Phys. Rev. 847,505 (1993). H. S.J. van der Zant, Physica 8222,344 (1996). R. A. Fazio, A. van Otterlo, and G. Schon, Europhys. Lett. 25,453 (1994).

390

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possibly, a wider window for ballistic motion, (10.10)

To search for ballistic vortices van der Zant et al.165constructed a clever sample consisting of two triangular arrays of aluminum tunnel junctions (Fig. 62). A transport current flows through array 1 and no current flows through array 2 or through the narrow channel connecting the two arrays. A low density of vortices is generated by a small applied magnetic field. The current in array 1 accelerates the vortices ,to a suitable velocity, and some of them are driven into the narrow channel and pass into array 2. Superconducting banks confine the vortices to the narrow channel. Voltage probes, located at sites indicated by V, to V , , are used to detect vortex motion in array 2. At high temperatures is large and the vortices injected 1 6 5 H. S. J. van der Zant, F. C. Fritschy, T. P. Orlando, and J. E. Mooij, Europhys. Lett. 18, 343 (1992).

FIG. 62. Van der Zant et d ' s sample for searching for ballistic vortices. The vortices are created by a small applied magnetic field. They are accelerated in array 1 by a current, launched through the channel and into array 2. No current is applied to the channel or to array 2. The voltage probes VI to V12 are used to detect vortices in array 2. (From Ref. 165, Fig. 1.)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

391

into array 2 move viscously. To understand what is happening, we examine the vortex equation of motion, Eq. (3.27). With no applied current, strong damping, and high-energy vortices (i.e., neglecting the potential term), (10.11) and

The last equality, perhaps an unsurprising result, is for square lattices only; for a triangular lattice the time constant is (4/3) ROC.A vortex mean free path can be defined as (10.13) where p = 1 for a square array and 8/27 for a triangular array. /3, is the McCumber parameter introduced earlier, Eq. (2.21). In an underdamped SIS array at high temperatures, R , z R,, the normal-state resistance, (2a/74(1/lC(0))z 0.1, and 8, varies from 0.1 to 3 or so. Typically, therefore, ,Ifree z a, the vortices will move diffusively in array 2, and voltages will be detected between all the voltage probes. Figure 63 shows voltages measured across several of the pairs on the sample. To create a low density of vortices a small magnetic field, f = 0.019, was applied. We observe that at the higher temperatures, just below the transition, there is a substantial voltage between probes V, and V,, indicating that the vortices injected into array 2 are diffusing and passing out of the array well away from the ballistic path. Note also that curve (a), (Vz-Vll), which displays the voltage across the channel and is a measure of the number of vortices passing though the channel into array 2, is much higher than any of the other curves in this temperature region -in particular, Vz- V, is much larger than V,- V,, two probes that are only 15 cells apart and directly astride the ballistic path. Clearly, ballistic motion is not observed. As the temperature drops, quasiparticles freeze out and the junction resistances rise rapidly. fir can now be orders of magnitude larger (as high as lo7) and I , >> a. Ballistic motion is now theoretically possible. From Fig. 63 we see that curve (d), which displays the voltage between probes V, and V,, a voltage due to those vortices that leave the ballistic path, is falling rapidly as the temperature is lowered; no vortices are leaking out the corners

392

R. S. NEWROCK ET AL.

lo2

lo-' 0

500

T (mK)

1000

FIG. 63. Voltages measured between various pairs of the probes shown in Fig. 62 as a function of the temperature. Curve (a) is the channel voltage (V2-Vll), (b) is V3-Vl0, (c) is VS-V8, and (d) is V3-V4. All voltages were measured at J = 0.019. In the inset the voltages in the main figure are plotted as the ratios of the voltage across probes V3-V10 (solid squares) and V5-V8 (circles) to the voltage across the channel. The dashed line connecting the squares gives the ratio of the voltage across V3-V10 to the channel when the current direction in array 1 is reversed. (From Ref. 165, Fig. 4.)

of the sample (at least those corners opposite to the channel). Furthermore, as may be seen in detail in the inset, the voltage across probes V, and V, is essentially equal to the channel voltage- it would appear that every vortex that passes through the channel moves along a ballistic path and leaves array 2 nearly opposite the channel; i.e, it is very possible that ballistic motion of vortices is observed. Van der Zant et al. varied the applied field as well. They observed possible ballistic motion for 0.01 < f < 0.025. For f < 0.01 they believed that the depinning currents must so be large that they cause row switching. For f > 0.025 the ratio V,-V,, to the channel voltage still increases, but not all the vortices appear to reach the other side of the array. The authors believe that at these higher vortex densities vortex-vortex interactions become more important. One interesting result from this experiment is that the researchers were able to extract an effective vortex mass from the data, MEff = 7 x kg, to be compared with the calculated value for a triangular array, 1.8 x kg (Eq. (3.26). This would appear to support the idea that the

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

393

effective dynamical mass of vortices is significantly larger than simple theories would predict. The problem of ballistic motion was investigated theoretically by several author^.'^^^'^^^'^^^'^^^'^^ Bobbert16' simulated an array of junctions, focusing on the dependence of the vortex motion on the damping for both square and triangular arrays. He found that the vortex velocity saturated in the limit of low damping, indicating a finite vortex viscosity, in direct contrast to ballistic motion. In the wake of the moving vortex, the island phases oscillate at the plasma frequency (4E,e2/C)/h. Van der Zant et al.ls7 had already postulated this type of excitation as the source of the unexpectedly high vortex viscosity they found for vortex motion in underdamped arrays. Bobbert's simulations showed that in spite of the vortex inertia, when the current is stepped sharply from a finite value to zero, vortex motion ceases after at most one lattice cell. In addition, for sufficiently high vortex velocities, the aforementioned row switching was found. A possible explanation of the van der Zant et al. experiment was that the vortices push one another through the central row of the array because of their repulsive interaction. The fact that van der Zant et al. tried to bend the vortex beam by applying a current to array 2, with inconclusive results, supports this scenario. In all our discussions to this point we have used a classical equation of motion for a single point-like vortex, developed using simple physical parameters and arguments. Haganaars et al.' 67 posed the question somewhat differently. They developed the full dynamics and simulations for an array and extracted an equation of motion from their numerical results. 50, in normalized units: They found, for /3,

-=

dx 4 B C )

d2x

M(Bc)-jp

+

dx

= i,

+ id sin 2ax.

(10.14)

1+WC)';T;

Here i, is the (normalized) Lorentz force on the vortex, i d is the (normalized) depinning current, M(8J is a vortex effective mass, and A and B are functions of BC. For moderately damped arrays they found neghgible ~

166 P. A. Bobbert, U. Geigenmiiller, R. Fazio, and G. Schon, Macroscopic Quantum Phenomena, ed. T. D. Clark et al., World Scientific, Singapore (1991), 119. 16' T. J. Hagenaars, P. H. E. Tiesinga, J. E. van Himbergen, and J. V. Jose, Phys. Rev. B50, 1143 (1996). J. E. van Himbergen, T. J. Hagenaars, J. V. Jose, and P. H. E. Tiesinga, Physica 8222,299 (1996).

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hysteresis in the IV characteristics, indicating that the vortex mass is very small and the dynamics of a single vortex is that of an essentially massless particle. This is the underdamped case. The picture is one of a single, low-mass vortex, moving through a nonlinear viscous medium. For larger values of bc the nonlinearity is reduced and eventually the damping is dominated by coupling to plasma oscillations, as shown by Geigenmuller et al.’ s4 The simulations that generated the results above were done with periodic boundary conditions. In that case as the vortex moves it “sees” its own tail (van Himbergen et a1.’68)-an effect that becomes negligible as the size of the simulated arrays is increased. When free boundary conditions are used the vortex sees an infinite number of image vortices as it approaches the boundary. The authors also studied simulations for free boundaries with several values of 8,. and they looked at the vortex motion as a function of the applied current. They found two types of behavior. For small b, and currents greater than the depinning current, a vortex accelerates as it moves toward a boundary because of its interaction with its own image. When it reaches the boundary it leaves the array. The simulations indicate that the vortex motion is effectively almost massless. For p, > 2.5 a current range opens up where the vortex is reflected as an antivortex at the boundary. This antivortex is then reflected as a vortex from the opposite free boundary. The vortex/antivortex never escapes from the array but produces a nonzero time-averaged voltage. This behavior is interpreted as being a result of the inertia, or kinetic energy, carried by the vortex. Its image is a potential well from which the reflected antivortex must escape. The Lorentz force itself is insufficient to pull it out of the well; it needs a minimum kinetic energy to escape; i.e., a sufficiently large16’ M V and v,. The essential point is that whether or not simulations reveal vortex inertia depends on the specific dynamical parameters and boundary conditions considered. With RCSJ dynamics and moderate values of pc, the nonlinear viscosity found can create interesting effects. For example, van Himbergen et that suddenly changing the current gives rise to oscillations in the junction phases near the vortex center (spin waves). Instead of allowing the vortex to continue in motion, the energy stored in the capacitors leads to junction oscillations, allowing the vortex to oscillate back and forth between adjacent plaquettes until the energy is dissipated -thus, the distance traveled by the vortex after the current is turned off is zero. They concluded that ballistic motion is seemingly impossible and that van der Zant et al.’s results cannot be easily associated with single-vortex dynamics -at least in certain p, regions.

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395

Ballistic vortex motion might still be possible in the quantum regime, however. If we consider the vortex sitting in the potential well of the egg crate potential, we can calculate its oscillation frequency, wp. For quantum vortices in the lowest energy state, a finite energy h o p is needed to excite the system and, if the vortex's kinetic energy is kept below this, excitation is impossible and the vortex can move without any damping. Note, however, that in van der Zant et al.3 samples the ratio of the Josephson energy to the charging energy is on the order of 250, which is well into the classical regime. Ballistic vortex motion remains an important unresolved question. It is difficult to find fault with either the experiments, which strongly support the possibility, or the theory and simulations, which are less optimistic. XI. Classical Arrays: Strongly Driven In Sections IV and X, we discussed both the resistance and the nonlinear current-voltage characteristics of Josephson junction arrays for DC currents of a magnitude sufficient to make the effects measurable, but sufficiently small not to change the interaction between vortices in the arrays. In this section we describe the properties of arrays when they are driven by large DC and AC currents.

25. SHAPIRO STEPS-SINGLEJOSEPHSON JUNCTIONS

If we bias a single Josephson junction with a constant voltage, the phase difference y will change linearly with time, Eq. (2.7). If we put this time-dependent y into the Josephson equation for the supercurrents, Eq. (2.5), it yields an oscillating supercurrent, (11.1)

where yo is a constant of integration. Thus, a junction biased with a DC voltage V,, oscillates at the Josephson frequency W , = 2eV2/,,/h,equal to 483.6 MHz/pK The oscillating supercurrent causes the junction to emit high-frequency radiation. Because this emitted power is very small for a typical junction, and because the impedance mismatch to free space is large, the effect is difficult to measure directly. However, it can be probed by mixing the Josephson oscillations with an external radio frequency (RF) signal and observing the effect of the mixing on the IV characteristics.

396

R. S. NEWROCK ET AL.

Considering only the voltage-biased case for now (because it can be understood analytically- the current-biased case is similar but not solvable analytically), we apply a voltage of the form

v = v,, + v,,

cos W R F t .

(11.2)

Integrating the Josephson voltage relation (Eq. (2.7)) yields (11.3) Putting this phase difference into the Josephson supercurrent relation (Eq. (2.5)) and using the standard expansion for the sine of a sine in terms of Bessel functions, we obtain' i , = i c ~ ( - l ) " J , , 2el/,, - ) s i n ( y o + ( ~ - n2e wvDC ,,)t], n hw,,

(11.4)

where n = 1,2,3,. . . and J,(x) is a Bessel function. Inspection of Eq. (1 1.4) reveals that the junction's response to a combined DC and R F voltage bias is an oscillating supercurrent, with no DC component, unless V,, = n(hwR,)/2e. That is, there is a DC current whenever V', = V,, where

V,=-nhw,, 2e

n = 0 , 1 , 2 , 3 ,...

(11.5)

These voltage steps are known as S h a p i r ~ steps. ' ~ ~ When we include the normal current flowing in the resistive channel, we see that the total DC current flowing through the junction on the nth Shapiro step is

K

- - icJ,

R

-

(11.6)

Figure 64 shows a schematic IV curve showing voltage-biased Shapiro steps. The height'" of the nth step is in = 2icJ,

(2:;)-

(11.7)

S. Shapiro, Phys. Rev. Lett. 11, 80 (1963). Note that when we plot V versus I we refer to the step width; for I versus V we refer to the step height. 169

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

397

I

FIG. 64. The IV characteristics for an RF voltage biased single junction, showing the Shapiro steps. The spikes in the current occur at voltages V = nhwR,/2e.

Since the Bessel functions J,(x) vary as x" for small x, as V,, increases from zero, the n = 1 step appears first. As V,, increases further, the Bessel functions become oscillatory with an envelope decreasing as (V,,)'" for large V,,. Thus, once a given step appears its height oscillates as V,, increases, eventually decreasing toward zero. In the laboratory junctions cannot be easily voltage-biased. Instead they are current-biased, with the drive current varying as i =,,i

+ iRFsin(wR,t).

(11.8)

In this case numerical methods are required to solve Eq. (2.12) i,

+ ,i

sin(wR,t)

= i,

sin y

h dY +2eR, dt '

(11.9)

where we take C = 0 for convenience. R ~ s s e r 'showed ~~ that the results look very similar to the voltage-biased case, with Shapiro steps occurring at the same values as given by Eq. (11.5). However, there is a nonzero

'"

P. Russer, J . Appl. Phys. 43, 2008 (1972)

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R. S. NEWROCK ET AL

time-averaged DC current even for V,, # V,. The step heights oscillate with ,i but now in a “Bessel-function-like” manner. A simple way to understand the physics of these steps is to return to the washboard model discussed in Section 11. Recall that in this analogue one describes the motion of a phase particle with position y and velocity dy/dt, moving in a potential of the form

u,

=

-

hi, -c0s(y2,) 2e

-

hi,, -721. 2e

( 11.10)

In this analogue, a DC current tilts the washboard. A potential cannot be time dependent, but an RF current i,, may be viewed as an external force of the form F = (hiR,/2e) sin(w,,r) acting on the phase particle. Consider first the case of no R F current, i,, = 0, as discussed in Section I1 and shown in Fig. 7. As the DC current increases and the washboard is tilted, the static equilibrium position of the phase particle changes as long as i < i,. Once i, is exceeded, however, the particle begins to move down the washboard with velocity dyldt. As the particle moves, it gains velocity on the sharp drops in the potential and slows on the almost flat parts of the potential, as shown in Figs. 7(b) and 7(c), and the velocity of the particle oscillates as it goes downhill. This oscillation has a DC component and a component at the characteristic frequency wJ = 2eV/h, the Josephson frequency, as well as higher harmonics. When the system is driven by an AC current, there are two competing frequencies: the external RF drive and the Josephson frequency. The Shapiro steps occur whenever nwRF= w J - that is, whenever the natural oscillations of the phase particle on the washboard are equal in frequency to the harmonics of the RF frequency. The Josephson frequency will then lock to the external drive frequency much as coupled oscillators will lock to the same frequency. In terms of the washboard model, the steps occur when the particle executes a 2rcn motion in one R F cycle. If the tilt of the washboard is increased slightly, the particle prefers to maintain its oscillation locked to that of the external driving force. This means that there will be a range of D C current for which the Josephson oscillations will lock to the external drive frequency, leading to constant voltage steps at ( V ) = hw/2e. Since the device is nonlinear, locking , the Josephson will occur not only for wRF= w J ,but for multiples of w R Fas frequency locks to harmonics of the R F frequency. In the washboard model this corresponds to the particle moving 27cn in one R F ~ e r i 0 d . l ’ ~ M. Octavio, J. U. Free, S. P. Benz, R. S. Newrock, D. B. Mast, and C. J. Lobb, Phys. Rev.

B44, 4601 (1991).

TWO-DIMENSIONAL PHYSICS O F JOSEPHSON JUNCTION ARRAYS

399

The range of stability of the steps becomes smaller on the higher-order steps. The range of stability, or the width in current, over which this entrainment or locking takes place depends on the type of biasing, on the characteristic frequency of the junction, and on the external drive frequency.

26. SHAPIRO STEPS ARRAYS ~

If at zero temperature and magnetic field an ideal M x N array (where all of the E, have the same value) is R F and D C biased, the system may respond just like a single junction with an applied current Z J M . There should be a total average voltage drop across the array of N times the average DC voltage across each junction. When the individual junctions synchronize to the external R F drive (at frequency v R F )and to each other, giunt Shapiro steps occur at voltages

nhv

(V)=N-

2e

n

= 0, 1,2, ...

(11.11)

Such giant Shapiro steps are observed in arrays. The first observation' 73 was in Pb-Cu-Pb arrays, over a wide range of frequencies (0.5 to 100 MHz). Figure 65 shows the results of Benz et al.,174.175plotted as the dynamic resistance d( V)/dZ versus the normalized voltage. The inset in the figure shows the IV curve itself where the giant steps are clearly seen. The steps in the IV curve correspond to the dips in the dynamic resistance. Note that the synchronization of the junctions is not perfect, resulting in steps with a nonzero slope. If the slope were truly zero, the dips in the dynamic resistance would go all the way to zero. Similar giant steps have been observed in various types of arrays and array geometries.'74,'75.176.'77*'78.179 While the arguments above indicate that giant Shapiro steps are to be expected in arrays, one should not lose sight of the fact that the argument assumes that all the junctions are identical and that the currents distribute themselves uniformly across the array. It is yet unclear whether their Ch. Leemann, Ph. Lerch, and P. Martinoli, Physicu 126B, 475 (1984). S. P. Benz, M. S. Rzchowski, M. Tinkham, and C. J. Lobb, Phys. Rev. Lett. 64. 693 (1990). 1 7 5 S. P. Benz, M. S. Rzchowski, M. Tinkham, and C. J. Lobb, Physica 8165-166, 1645 (1990). 1 7 6 H.-C. Lee, D. B. Mast, R. S. Newrock, L. Bortner, K. Brown, F. P. Esposito, D. C. Harris, and J. C. Garland, Physicu 8165-166, 1571 (1990). 1 7 7 S. E. Hebboul and J. C. Garland, Phys. Rev. 843, 13703 (1991). 1 7 8 S. E. Hebboul and J. C. Garland, Phys. Reo. 847, 5190 (1993). ' 7 9 L. L. Sohn, M. S. Rzchowski, J. U. Free, S. P. Benz, M. Tinkham, and C. J. Lobb, Phys. Rev. 844, 925 (1991). 173 174

R. S. NEWROCK ET AL. -l

1

I

3

I

1

2

3 2eV/Nhv

4

, 5

6

FIG. 65. The dynamic resistance versus normalized voltage in zero magnetic field for a lo00 x lo00 SNS array. The data were taken at 3.0 K, where I, = 0.79 mA, with vRF = 0.73 MHz (0= 1) and I,, zz I,. The inset shows an IV curve taken with the same parameters. (From Ref. 174, Fig. 1.)

observation in real microfabricated arrays implies that the coupling of the junctions stabilizes this solution or whether what is observed reflects the intrinsic disorder of the array. If the description summarized by Eq. (11.1 1) were complete, then the giant Shapiro step widths, AZ, should be very much like those of single junctions, both in their frequency and power dependence. Giant step widths have been measured as a function of the RF current and are found to be in qualitative agreement with sir nu la ti on^.'^^^'^^^'^^^^^^^^^^ The main difficulty is that real arrays always have a finite amount of bond disorder, leading to variations in E,. Li et al.ls3 simulated arrays with different levels of disorder and found that, considered together with Johnson noise, it improves the agreement between experiment and simulations.

K. H. Lee, D. Stroud, and J. S. Chung, Phys. Rev. Lett. 64,962 (1990). J. U. Free, S. P. Benz, M. S. Rzchowski, M. Tinkham, C. J. Lobb, and M. Octavio. Phys. Reo. B41, 7267 (1990). K.-H. Lee and D. Stroud, Phys. Rev. B43, 5280 (1991). R. R. Li, K. Ravindran, H. C. Lee, R. S. Newrock, and D. B. Mast, Physica 8194-196, 1723 (1994). lSo

lS1

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

401

27. FRACTIONAL GIANT SHAPIRO STEPS While the giant Shapiro steps observed in zero field were expected to occur if an array could be fabricated with sufficient uniformity, it was truly surprising when “fractional” giant Shapiro steps were observed’ 7 4 * 17 6 at voltages

1 nhv V=-Nn = 0 , 1 , 2 , ... 4 2e

(11.12)

in the presence of a magnetic field corresponding to f = p / q flux quanta per unit cell, where p and q are integers. These steps are shown in Fig. 66, where the dynamic resistance d( V)/dZ of the same sample as in Fig. 65 is plotted as a function of reduced voltage for magnetic fields f = 0, 1/2, and 1/3, at the same temperature, RF amplitude, and frequency. We see the giant steps as well as the novel fractional giant steps given by Eq. (11.12). A simple e ~ p l a n a t i o n can ’ ~ ~be given for the origin of the giant fractional steps. As an example, consider the f = 1/2 ground state. At T = 0, the f = 1/2 ground state was shown in Fig. 41(a); it corresponds to a checkerboard superlattice of “+” and “ - ” vortices. Under the influence of an applied current, the circulating currents in the superlattice feel a force that

ot! 0

1

L

I

1

2

3

2 eV/Nhv FIG. 66. The dynamic resistance versus normalized voltage for different magnetic fields, 1/2, and 1/3. The data are at T = 2.1 K, I , = 7.9 mA, and with the same RF frequency v, = 0.73 MHz (a = 0.1) and amplitude I,, = 0.751, for each curve. (From Ref. 174, Fig. 2.)

f

= 0,

402

R. S. NEWROCK ET AL.

causes them to interchange plaquettes: Plaquettes with a clockwise current become plaquettes with a counter-clockwise current and vice versa. When this interchange occurs, the average phase change per junction can be shown to be Ay = 7c. This is plausible, as two such interchanges bring the system back to its starting point, a change of 27c. If the R F drive locks to the motion of the vortices, the lowest-order locking corresponds to a phase shift of 7c per junction per R F period that, from the Josephson voltage relation, Eq. (2.7), leads to Eq. (11.12) with n = 1 and q = 2. As the DC current is increased, states where the phase advances by 27c,3n, 47c,. . . ,n7c per R F period occur, leading to Eq. (1 l.12).'84 This picture was generalized to any f = p / q by noting that the ground state is very often a q x q superlattice of vortices. (There are exceptionssee Kolahchi and S t r a l e ~ ' ' and ~ Straley and Barnettls6.) In this case, the vortex superlattice must advance q steps through the array lattice before the configuration returns to its starting point, so that the smallest step corresponds to a phase change of 2nlq in one RF period. This is reflected in Eq. ( 11.12). The earliest computer simulations were done by K.-H. Lee et al.,'" who demonstrated that the fractional giant steps could arise from a model of resistively shunted junctions, and Free et al.,'" who performed simulations that also confirmed the physical picture described above. Both sets of authors considered an overdamped junction array, using the RCSJ model with C = 0, as is appropriate for SNS arrays. The current between any pair of nodes i and j is (see Eq. (2.12)) iji

=

i, sin yji

h 4ji +2eR0 d t ' ~

(11.13)

where (see Eq. (2.7)) (11.14) At each node Kirchoffs equation for current conservation must be satisfied, leading to a large number of coupled nonlinear equations. K.-H. Lee et al. included a Langevin noise current term to simulate the effects of temperaWe note that Straley (J. P. Straley, Phys. Rev. 838, 11225 (1988)) studied the motion of vortex superlattices with f = p / q and only a DC drive current. Interestingly, he found motion of the superlattice with a prominent frequency v = 2eVq/Nhp, a nearly correct result. M. R. Kolahchi and J. P. Straley, Phys. Rev. 843, 7651 (1991). J. P. Straley and G. M. Barnett, Phys. Rev. 848, 3309 (1993).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

403

ture, whereas Free et al. assumed T = 0. The resulting approximately N M differential equations are solved numerically. Figure 67 shows simulated IV characteristics for an 8 x 8 array for normalized frequency R = 1 and f = 1/2, with and without f R F / Z C = 1.6. The normalized or reduced frequency is defined as

(11.15)

and ,the particular value of I,, was chosen to maximize the width of the half-’stepfor the parameters chosen. We can see the one-half fractional giant step between the n = 0 and n = 1 giant steps. While it is useful to compare simulated and experimental IV characteristics, the power of numerical simulations is that they can provide detailed dynamical information not obtainable from experiment. For example, simulations show that the moving vortex picture outlined above is correct, as can be seen from Fig. 68 (from Free et a1.181). This figure shows the instantaneous voltage across two adjacent junctions in an array. It can be

1.1

I

I

I

I

I

I

1 .o

0 .9

$ k

0.8 0.7

‘i; 0.6 $ 0.’5 A 0.4

0.3 0.2

0.1

FIG. 67. Simulated 1V characteristics for an 8 x 8 SNS array with (upper curve) and without (lower curve) RF current for f = 1/2, R = 1, and I R F / I c = 1.6. (From Ref. 172, Fig. 1.)

404

R. S. NEWROCK ET AL.

1.5

-

1.0

-

0.5

-

0.0

-

L

.c

2 rJ

-0.5 i

100.0

100.5

101.5

101.0

102.0

102.5

V t

FIG. 68. The normalized instantaneous voltage versus the normalized time, across two adjacent junctions parallel to the current in a 4 x 5 array. The data is for the n = 1, q = 2 fractional giant Shapiro step, with f = 1/2, R = 1, and lm/lc = 0.65. Time is normalized to the external RF current’s period. (From Ref, 181, Fig. 2.)

seen that each junction advances 271 every two RF periods on the half-step, in agreement with the model, and that the junctions are one-half period out of phase. Octavio et a1.17* studied the details of the RF amplitude and frequency dependence of the giant and fractional giant Shapiro step widths. They found the behavior in general to be substantively different from the Shapiro steps in single junctions. The variation of the fractional step widths has a pronounced frequency dependence, including a sharp roll-off at high frequencies, which was not observed for the integer giant step widths that saturate at high frequencies. Octavio et al. explain this difference between the giant and the fractional step widths in terms of the importance of the ordered vortex superlattice in determining the step widths. At high frequencies the motion of the vortex superlattice is much less important and the system appears to revert to single-junction-like behavior. To understand these effects completely, it is useful to make comparisons to single-junction behavior, shown in Fig. 69. Here are plotted the normalized step widths Ain/ic as a function of the normalized RF current amplitude i R F / i c .The step width is defined to be the range of current over which the voltage is constant. This implies that the width of the zeroth step, Ai,,, is twice the critical current in the presence of the RF power. These step widths were calculated by numerically solving Eqs. (1 1.13) and (11.14) using

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS I

I

I

405

l

Ai,

. . . . ... . . . .. . ... . . ... . . . . .. . . . . . . . . . . . . .-2.u 1.0' .: ... ... .. .. .. ... ... ... .... . .. . .. . .... .... ... I.

a

A i,

0.0

0

0

1

1

2

2

3

3

4

4

5

5

6

7

8

9

10

6

klic

FIG. 69. The dependence of the zeroth (Ai,,) and first step (Ail) step widths for a single Josephson junction for a normalized frequency of (a) R = 0.3 and (b) R = 1.0.The step widths are normalized to the critical current of the single junction, and the zero for Ail/Aic has been moved upward by one in both figures (From Ref. 172, Fig. 2.)

Kirchhoffs law. Figure 69(a) shows the normalized step widths as a function of reduced R F amplitude for the zeroth and first step for normalized frequency R = 0.3. Figure 69(b) shows R = 1.0. For R = 1 the step widths are qualitatively similar to the Bessel function dependence of Eq. (1 1.7), but they are quantitatively different. In fact, they agree with the Bessel function behavior only at high frequencies (R > 1); for lower frequencies the step widths are smaller than those predicted by Eq. (11.7).

406

R. S. NEWROCK ET AL.

Figures 70 and 71 show Octavio et al.3 results for the normalized step widths of an array versus R F amplitude for the zeroth, first, and half-steps, for three different frequencies. In contrast to the single junction (Fig. 69), the R F current dependence is no longer Bessel-like. For example, for the zeroth step, the step width at the second maximum is smaller than that at the third, a behavior observed in the simulations over a wide range of frequencies. As the frequency increases past R = 1, the second maximum disappears entirely and the overall behavior reverts to being Bessel-like. Furthermore, Fig. 71 shows that as the frequency increases, the step width of the half-step very rapidly decreases (note the different vertical scales in Fig. 71(a) and (b)), and for frequencies R = 10 and (presumably) beyond, the half-step is unobservable. This rapid roll-off was also seen in an analytic model of the system.’87 Octavio et al.’ 7 2 also saw anomalies in the first step similar to those of the zeroth step. The experimental dependence of the step widths on RF amplitude has not been extensively studied, but there are a small number of results. The dependence on RF amplitude has been experimentally verified, and the fit of the measured step widths to the RCSJ simulations is reasonably good.’88 Ravindran et al.189and Hui’” investigated the frequency dependence of the step widths. In Fig. 72 we plot, for the first giant step, the maximum normalized step width-that is, we sweep the amplitude of the R F current at each frequency and record the maximum of each Bessel-like oscillation as a function of the R F drive frequency. As in the simulations of Octavio et al., the maximum normalized step width increases with frequency, becoming independent of frequency for R z 1. The frequency dependence of the giant fractional step at f = 1/2 is displayed in Fig. 73. Again, it rolls off at higher frequencies in agreement with the simulations. Kvale and Hebboul’” developed a slightly different model for the fractional steps. They made the approximation that the vortices move as a rigid lattice. This is not required by Benz et a l . ’ ~ model, ’ ~ ~ where it is assumed only that the motion is periodic, but it is consistent with it. Next, they decomposed the system into two subsystems, a phase-locked array without vortices and a lattice of vortices created by the external field. They assumed that these subsystems are completely independent and noninteracting. With these two assumptions the total phase difference across the system is the simple sum of the phase differences across the two subsystems. Setting M. S. Rzchowski, L. Sohn, and M. Tinkham, Phys. Rev. 843, 8682 (1991). R. S. Newrock, unpublished. K. Ravindran, L. B. Gomez, R. R. Li, S. T. Herbert, P. Lukens, Y. Jun, S. Elhamri, R. S. Newrock, and D. B. Mast, Phys. Rev. 853, 5141 (1996). I9O F. C. Hui, Ph.D. thesis, Ohio State University (1993). 19’ M. Kvale and S. Hebboul, Phys. Reo. 843, 3720 (1991).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS 0.7*.'

I

I

I

407

'

t

0

0.6-

(a)

0

0.5-

p 0.44

R = l f=1/2

0

0

-0

0.30

0.20.1 -

0 0 .

m 0

0.0

0

0

0 = 3 f=1/2

$:I

1

n = 1 o f = 112

mm,mo,

4 0.3

0.2

0

0.0

0

.*..

.*

0.1

10

\

20

-0:

30

.

'

40

*I:

50

u 60

70

D

lrf&

FIG. 70. The simulated R F current amplitude dependence of the zeroth step for an 8 x 8 array with f = 1/2 for three different frequencies: (a) R = 1, (b) R = 3, and (c) R = 10. (From Ref. 172, Fig. 6.)

408

R. S. NEWROCK ET AL.

0

R = 3 f=1/2

<$ 0 . 0 6 -

Q

#a*a a

a

a

0.04-

I

a

l a

a

0.02 -

aa*aa

a

0.00a 0

m a *

I

4

. * I 8

.

1

12

T - .

16-

t'

20

ill I i s

FIG. 71. The normalized step width as a function of the RF current for the half-step for 3. For i2 = 10 (not shown) the fractional step widths were immeasurably small. (From Ref. 172, Fig. 7.) = 1/2 and (a) R = 1 and (b) R =

the sum of three forces (pinning, viscous damping, and the Lorentz force, Section 111) to zero ( M , = 0 in the overdamped limit), results in an equation of motion for the vortex lattice moving collectively that is of the same form as the RCSJ equation for the phase. The average change of phase is found by averaging this RCSJ-like equation for the moving vortex lattice. From the change of phase, Kvale and Hebboul obtained the average voltage and recovered Eq. (11.12) in a slightly different form: Nho <w> =

kq

+ mp

(11.16)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

0.6. 0.5 -

I

”

CT -

f

- . . 0

1’7

,* ,,

X

0.2 0.1

-

0.1

;

s-u-

!j.cw,

02

-

--3:

0’

_._._._,_._..._

03

0.3-

I

.*................ 0 ...................................

.....’”””

0.4 -

I

I

409

.

80%

90%

- =.*

b

n

- d ,

.; 0

0.0 0.0

- oa‘ - 0.4. -0.0. .0.0 * 1.0 . . i-a . 1.4. . 1.0.

n

0.0

I

I

I

I

FIG. 72. The normalized frequency dependence of the maximum step width for the first step for a Nb-Au array at 1.9 K and zero applied field. The lines are drawn to guide the eye. The inset shows corresponding plots for 10% and 20% site-disordered samples measured at T = 3.4 K. (From Ref. 189, Fig. 1.)

where k and m are integers that arise from the averaging process. If p and q are relatively prime, kq and mp can generate all integers and we obtain Eq. ( 11.12). After a full R F cycle the system’s state is identical to the original state except for a lattice translation- the system “maps” onto itself. Since the differential equations are continuous, the time evolution is continuous and we have a topological mapping of the system onto itself. If the full system maps onto itself, each subsystem must also. The phase of the full system mapping onto itself implies topological quantization of the phase in multiples of 271, and since each subsystem is characterized by its own phase, the phase change of each subsystem can only change by multiples of 271, in particular, 2m71 and 2kn, where k and m are the winding numbers of the topological mapping. The Shapiro step voltage, Eq. (11.16), is a function only of k and m, so, at a particular p/q, the step voltage is a topological invariant. This implies that the step is very stable against small changes in the mapping-i.e., a transient or a small change in ZDC. Kvale and Hebboul point out that this form of topological invariant is different from the topological invariants usually considered. In physics one usually deals with invariants found in Hamiltonian systems for which either

410

R. S. NEWROCK ET AL.

5,

4 . 0 n

#

Q rJ

I

3 -

0 d

W

X

ii n n

e

0

2 -

40

80

120

160

v

hC

a

Y

”

0

30

60

90

120

150

180

v(Mhz) FIG. 73. The maximum step width versus frequency. The main plot shows the first step for f = 0 and f = 1/2, and the one-half step for f = 1/2. The inset shows an enlarged view of the one-half step width. (From Ref. 190, Fig. 7.)

an adiabatic transformation has taken place (e.g., the integral quantum Hall effect) or there are minimum energy configurations of the order parameter taken as topological transformations in a lower dimensional space (e.g., vortices, instantons, solitons). Here we also have order parameter configurations but in a dissipative system. Instead of Hamilton’s principle or minimum energy providing the local stability needed to define the topological classes, the stability in the dynamical system is provided by the mapping onto the limit cycle. Kvale and Hebboul believe the process is quite general. For any system demonstrating mode locking at various commensurations, (e.g., charge density waves), they expect similar topological invariants to occur.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

41 1

We next return to the experiment of Sohn et al.,179 discussed briefly above. They injected current along the [l 13 direction of the array rather than the [lo] of previous experiments. They observed only giant Shapiro steps- no fractional steps (or subharmonic steps; see the next section) were observed, at any field between f = 0 and 112 over a fairly wide frequency range (0.18 < R < 0.73), and at various R F power levels and temperatures, ~ ~ absence of the contrary to the theoretical predictions of H a 1 ~ e y . lThe fractional steps was not due to boundary effects, pinning, or sample homogeneities. Numerical simulations identical to those of Free et al."' (with the obvious exception of the current injection direction) were performed. For a wide range of frequencies (0.1 < R < lS), over a wide amplitude range, simulations on 8 x 8 and 4 x 4 arrays at f = 112, and 6 x 6 at f = 113, showed only giant steps. In a second, more comprehensive study, Sohn et al.'93 extended their measurements to a current injection direction 15" off the [ l l ] axis in square arrays and to triangular arrays with currents injected along two directions [l O i ] and [2ii]. They also performed more extensive simulations. Figure 74 summarizes their findings. They found giant and fractional giant steps in the 15" square lattice, but the fractional steps were considerably weaker than in the [lo] case. In the triangular arrays, the [1011-oriented arrays showed both giant and fractional giant steps in a field, while the [2ii]-oriented arrays showed only giant steps at the appropriate voltages. Sohn et al.,193using the moving vortex lattice model of Benz et al.,'74 were able to provide a phenomenological explanation of when fractional steps should and should not appear. They also used simulations and the pendulum model to demonstrate how the phases of the individual junctions evolve per cycle. They concluded that in the [11] square arrays and the [2TT] triangular arrays, where there are no junctions perpendicular to the transport currents, Kirchhoffs voltage law constrains the time evolution of the phase difference so that the only periodic solution allowed is one in which the phase differences correspond to integer giant Shapiro steps. In the [lo] square and [loll triangular arrays, where junctions perpendicular to the transport current do exist, they allow the phase difference of the other junctions to evolve more freely per R F cycle and the allowed periodic solutions correspond to fractional giant steps. However, the most important conclusion reached by Sohn et al.,193 has to do with the role of symmetry. In the square [lo] array, current injected 19*

193

T. C. Halsey, Phys. Rev. B41, 11634 (1990). L. L. Sohn, M. S. Rzchowski, J. U. Free, M. Tinkham, and C. J. Lobb, Phys. Rev. B45,

3003 (1992).

412

R. S. NEWROCK ET AL.

into the arrays breaks the symmetry- the current goes directly into junctions parallel to the current but not into those perpendicular to it (see Fig. 74). In the [113 square array the symmetry is preserved as the current feeds into all junctions equally. As seen from the figure, a similar situation obtains in the triangular arrays. In the [lo%] case, junctions exist that are perpendicular to the current direction and the symmetry is broken; the symmetry is preserved in the [2TT] arrays. Thus, they found that whenever the symmetry is broken both giant and fractional giant steps are observed but whenever the symmetry is preserved only giant steps are observed. We will return to the role of symmetry breaking in arrays several times in the following sections.

h

Y

+++ +++ +++

Geometry

Fractional Steps

[lo1

strong

[ioi]

weak

[2ii]

absent

FIG. 74. The strengths of the fractional giant Shapiro steps produced by various geometries. The black crosses and asterisks are superconducting islands. The applied current flows horizontally, and “geometry” indicates the direction of the current with respect to the array unit cell. All array types show integer giant Shapiro steps. (From Ref. 193, Table I.)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

41 3

28. SUBHARMONIC STEPS Noninteger Shapiro steps have also been observed in zero magnetic field; these have been named subharmonic Shapiro steps. They were first observed by Lee et a1.176*194and were subsequently seen by Sohn et al.193 and Hebboul and Garland.'77s'88 All of this work was done on SNS arrays. The previous sections suggest that a magnetic field is needed to create noninteger steps. In addition, single overdamped Josephson junctions do not exhibit subharmonic steps in zero field as long as the current-phase relationship is purely sinusoidal. In a single junction a nonsinusoidal current-phase relation can generate subharmonic steps in a form similar to Eq. (11.12), but since subharmonic steps have not been observed in RF-biased single SNS junctions, they must be a consequence of the interaction between junctions in the array. The early theories of the dynamics of overdamped junction arrays, which did not include the array self-inductance, do not predict subharmonic modes except as boundary condition artifacts"' or in special staircase geometries. '9 Figure 75 shows the dynamic resistance of an SNS array in zero fieldsubharmonic steps at reduced voltages of 1/2, 3/2, and 5/2 are clearly seen. H.-C. Lee et al.' and Hebboul and Garland'77.'88 also observed subharmonic steps in magnetic fields at values of l/q incommensurate with the field. Figure 76 shows subharmonic steps in a magnetic field f = 1/3, where the fractional steps at reduced voltages of 1/3 and 2/3 and the subharmonic steps at 1/2, 3/2, and 5/2 are seen. H.-C. Lee et al. postulated that the subharmonic steps come from edge-nucleated vortices created by the self-field of the array bias current. Unlike the uniform field of a simple external magnet, the self-field of a uniform current sheet, approximately representing the array, is sharply peaked at the edges of the array and asymmetric with respect to the array center axis. The magnitudes of the array currents used in their experiment were approximately sufficient to produce a self-field corresponding to f = 1/2 at the array edges. This introduces a nonuniform fluxoid lattice that is most dense at the edges, falling to zero at the center line. If the edge field corresponds to f = 1/2, 3/2, 5/2, etc., that region of the array is fully frustrated. The current loops for each of the plaquettes enclose a 2n (or -2n) center of phase. There are two limiting cases: strong screening and weak screening. In the weak screening limit, Al >> a, screening currents are negligible and clockwise and counter-clockwise current loops contain identical flux f: In the strong 767194

l g 4 H.-C. Lee, R. S. Newrock, D. B. Mast, S. E. Hebboul, J. C. Garland, and C. J. Lobb, Phys. Rev. 844, 921 (1991).

414

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\

0

113 112 213

1

413 312

2eV/Nhv FIG. 75. The dynamic resistance of an array as a function of the normalized voltage for vRF= 90 MHz and I , , = 31, in zero magnetic field. The data show the giant Shapiro steps at normalized voltages of 1, 2, and 3 and the subharmonic structure at 1/2, 3/2, and 5/2. (From Ref. 176, Fig. 2.)

0

113 112 213

1

413 312

2eV / Nhv FIG.76. The dynamic resistance of an array in a magnetic field f = 1/3. The zeroth and first integer giant Shapiro steps are seen, fractional giant steps are seen at 1/3 and 2/3, and subharmonic structure is seen at 1/2 and 3/2, which appears to co-exist with the field-induced fractional structure. (From Ref. 176, Fig. 3.)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

415

screening limit, plaquettes of one polarity each contain a flux quantum while neighboring plaquettes of opposite circulation are free of flux. The flux patterns on opposite sides of the array are identical, but one contains vortices and the other contains antivortices. H.-C. Lee et al.194speculated that the dynamical behavior of this f = 1/2 state corresponded to a back and forth alteration of the flux lattice, with the f = 1/2 subharmonic step occurring when this alternation frequency is locked to the R F drive current. In the weak screening regime this has the further physical significance of a lattice of magnetic vortices moving sideways across the array. Vortex-antivortex pairs, created at opposite edges of the sample, are drawn to the center of the sample, where they annihilate. In the case of Lee et al.'s experiment, AL was on the order of 22pm, compared with a = 10pm, placing the specimen in the intermediate area between the two screening limits. The screening is insufficient to constrain the flux to a single plaquette, so the plaquette-to-plaquette variation is less than a flux quantum. The dynamical state of the system corresponds to cyclic motion of concentric rings of quantized loops of flux that are created in phase with the R F current at the transverse edges of the sample. The radii of the rings are reduced with the R F current until they disappear at the center axis of the array. This is to be contrasted with the dynamical state of vortices created by an external field. In that case vortices of a single polarity are nucleated at one edge of the array and move at a constant velocity across the sample until they disappear at the other edge. H.-C. Lee et al. tested this picture by placing a planar conducting strip of the same size as the array just above the array. This made it possible to decrease or enhance the antisymmetric DC self-field of the array by varying the direction and magnitude of the current in the film, I , . As seen in Fig. 77, the amplitude of the subharmonic step is strongly influenced by the magnetic field created by the film. If I,, is too small to create subharmonic steps, a limited range of I , will generate subharmonic steps. Conversely, no value of I , can generate subharmonic steps if I,, is too large. The ability to create or destroy the subharmonic peaks with an external antisymmetric magnetic field indicates that the self-field of the combined DC and R F currents is an important factor in the dynamic behavior of the subharmonic state. We again note the importance of symmetry breaking: The symmetry of the system is broken by the inherently antisymmetric nature of the array self-field. Hebboul and Garland' 7 7 3 1 7 8 measured the R F spectral response of arrays to an imposed AC current. In addition to the expected harmonic peaks, narrow-bandwidth peaks at subharmonics of the bias frequency were found. These turned out to be associated with subharmonic giant Shapiro steps on

416

R. S. NEWROCK ET AL.

30

I

n

I

I

I

-

v)

t .-

c

3

d 20 L 0

W

-

a

2

z

E"

10 -

-

0

0 -4

-2

0

2

4

DC Current la FIG.77. The amplitude of the dynamic resistance for the 1/2 subharmonic step as a function of the DC current through a planar conducting film, as described in the text. The amplitude of the subharmonic step is strongly decreased by the anti-symmetricmagnetic field produced by the conducting film. The line is a guide to the eye. (From Ref. 176, Fig. 5.)

the array IV-the 2eVlNhv = n/q = 1/2 peak structure in the RF spectral response, for example, only appeared when the array was DC biased on the n/q = 1/2 or 3/2 subharmonic Shapiro step. They observed weaker subharmonic responses in the RF spectrum when the array was biased on the 1/3 subharmonic Shapiro step, suggesting that the array's spectral response is characterized by subharmonic structures that decrease in size as q increases. The strengths of the peaks showed nearly the same dependence on the RF amplitude as the integer giant steps, and the temperature dependence was similar as well. All of this suggests a subtle mechanism for the origin of the subharmonic steps. When a magnetic field was applied to the array, the response to different RF power levels was essentially similar to the response of the fractional steps. The oscillations shifted toward higher I,, as the RF frequency increased. This behavior is similar to the behavior of single Josephson junctions, a result that suggests that the applied field enhances the phase locking of the junctions and causes the array properties to be governed by single junction properties. Noting that it is possible to show that the properties of a single 2 x 2 plaquette of junctions are very well approximated by a DC SQUID,

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

417

Hebboul and Garland constructed such a model to explain subharmonics -in essence they attempted to explain subharmonics by examining the simpler system of two junctions in parallel coupled by an inductor. They modeled the SQUID using RCSJ equations and, in the limit of small loop inductance, found that when the SQUID was biased with both a DC and an RF current, the characteristics were identical to an effective hypothetical single junction with a nonsinusoidal current-phase relation. For f an integer, ideal junction behavior emerges, but when f = 1/2, subharmonic steps appear in the SQUID response -a direct result of the nonlinear coupling of the two junctions in the SQUID. However, the subharmonic step was not present in zero field. They found that such zero field subharmonic steps can be produced when the two junctions of the SQUID have different critical currents. This is an asymmetric DC SQUID. To obtain acceptable fits, however, they noted that the critical currents of the two junction in the SQUID must differ by about 4:l. They suggested that the subharmonic steps are caused by the combined effects of plaquette self-inductances and critical current variations. Note that this does not account for the results of Fig. 77. In a series of papers, Dominguez et a1.195,196,197*198 studied the effects of defects in the array lattice. They started from the work of Xia and Leath14’ and Leath and Xia143 (Section IX), which showed that when defects are present in arrays, vortices are nucleated by a sufficiently strong DC bias current. Dominguez et a1.195r196,197.198 added an AC current to the system, an extra component that changes the system’s response in a fundamental and co-workers called an way, creating what Dominguez195*1969197.’98 “axisymmetric coherent vortex state,” which is essentially an oscillating pattern of rows of vortices and antivortices symmetric about the central axis of the array. They found that these patterns of vortex motion nearly always lead to subharmonic giant Shapiro steps in the IV characteristics. The defects act as nucleating centers for vortex pairs, breaking the translational invariance of the array, which, as we have speculated, leads to subharmonic Shapiro steps. Thus, we have three possible explanations for the subharmonic steps: self-fields,critical current asymmetries, and defects. H.-C. Lee et al’s demonstrates experimentally that self-fields are important because the magnetic fields from an appropriately chosen current sheet field can completely D. Dominguez, J. V. Jose, A. Karma, and C. Wieko, Phys. Rev. Lett. 67, 2367 (1991). D. Dominguez and J. V. Jose, in 4th Intl. Workshop on Instabilities and Non-equilibrium Structures, Valparaiso, Chile (1991). D. Dominguez and J. V. Jose, Phys. Rev. Lett. 69, 514 (1992). D. Dominguez and J. V. Jose, Phys. Rev. 848, 13717 (1993). 195

‘91

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eliminate the subharmonic steps. Hebboul and Garland’ 7 7 * 1 7 8 showed that large critical current asymmetries will result in a subharmonic step. However, in order to obtain a 4:l change in critical currents, the junction gaps in typical arrays have to vary by nearly 20%, and electron micrographs of typical SNS junction arrays indicate that variations in the junction gaps are much less than that. Defects are also unlikely because the defect density necessary doesn’t exist in these arrays. Nevertheless, all three approaches demonstrate an important point: Symmetry breaking appears to be the key to the appearance of the subharmonic steps.

29. SIMULATIONS: INDUCTANCEAND SYMMETRY BREAKING

As discussed in Section 27, the initial theoretical and computational work 1 7 2 , 1 8 0 , 1 8 2 on the fractional giant Shapiro steps were simulations involving solutions of coupled RCSJ equations in the overdamped (8, = 0) and AL/u >> 1 limits. While these theories are successful in explaining the integer and fractional giant Shapiro steps in these limits (that is, in SNS arrays), the explanation of the subharmonic steps is partial at best. One key to understanding the subharmonic effects appears to lie in understanding the role that inductance plays in two-dimensional arrays. Self and mutual geometric inductances must be introduced into the coupled RCSJ equations that are used to simulate the properties of arrays. When one examines the coupled differential equations, the inductances (or capacitances as well) form a matrix. We will refer to the inductance or capacitance matrix in the discussions to follow. When inductive effects are considered, the perpendicular penetration depth, A, becomes an important scale parameter. It determines how rapidly a magnetic excitation decays, establishing the interaction range between vortices when their self-fields are considered. Indeed, as shown by Phillips et a1.l” the ability of an array to phase-lock to an external AC drive is limited by how small AL is relative to the sample size. (We note that most workers in this area compare AL to the plaquette size a and not to L). Note that A* depends on the critical current and, through it, the temperature, so it is easily possible to move between various AJu regimes in a single experiment. Dominguez, JosC, and co-workers extensively studied all the steps giant, fractional giant, and subharmonic-in arrays in the weak and strong screening limits, AJa >> 1 and I,/a << 1, and in the intermediate regime. Their work is carefully discussed in a recent review,’ and we will only touch on the important points. 199

J. R. Phillips, H. S. J. van der Zant, J. White, and T. P. Orlando, Phys. Rev. B50, 9387

(1994).

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419

Dominguez and Jose introduced the self-magnetic field into their RCSJ

simulation^^.'^^ of overdamped arrays. They used a local screening model for the inductance matrix, including only the self and mutual nearestneighbor inductances. They found that the inclusion of the self-magnetic field affects the formation of the fractional giant Shapiro steps and creates subharmonic steps. In zero magnetic field, the inclusion of the asymmetric self-field generates the subharmonic steps, proposed and shown experimentally by H.-C. Lee et al.194 These subharmonic steps are produced by a family of oscillating coherent vortex states, whose precise structure also depends on AJa. Phillips et a1.199*200 also included inductive effects in their calculations, and their work agrees with that of Dominguez and in that inductive effects do generate subharmonic Shapiro steps and do affect the other giant Shapiro steps. That is, self-fields lead to subharmonic steps. However, they included the full inductance matrix, not just the truncated nearest-neighbor model. (Dominguez and Jost, in a later work,' also used the full inductance matrix in simulations and obtained results similar to those of Phillips et al.) Using the full inductance matrix turns out to be very important in that the truncated matrix yields anomalously strong screening that destroys phase locking prematurely. Figure 78 shows an IV characteristic for an overdamped array in zero field, with normalized frequency R = 0.6 and I,, = I,. Three values of the penetration depth are shown, AJa = 0.1, 1, and 20. The integer giant Shapiro step occurs for all AJa, and at high A,/a it is the only structure. At lower AJa (Al less than the sample size) 1/2 and 3/2 steps become observable. For still smaller AJa many higher-order steps are observable. The width of the integer steps decreases as AJa decreases. Phillips et al. also looked at the same array for I,, = 21, and R = 2. The results were essentially the same -the integer steps always appear, and their widths decrease as AJa is reduced. The subharmonic half-step is present to higher 1.Ja as well, but the higher-order ( n > 1) steps disappear. Interestingly, Dominguez and J0st9 artificially suppressed the self-field of the array while maintaining the vortex-vortex interaction via the inductance matrix, they found that the subharmonic steps disappear- the self-field is clearly responsible for the subharmonic structure in these simulations. By examining the currents and fluxes carefully, an approximate description of the vortex motion can be obtained. In general, for an N x N array, Phillips et al. found that the subharmonic l/q Shapiro steps result from N vortices and antivortices moving across the array and annihilating in the zoo J. R. Phillips, H. S. J. van der Zant, J. White, and T. P. Orlando, Phys. Rec.. 847, 5219. (1993).

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R. S. NEWROCK ET AL.

i XL

1.5

/

=1

2

X I = 20

2.5

I

3

IDC Kc

FIG. 78. Simulated zero field IV characteristics of an 11 x 11 plaquette SNS array, biased with an AC and DC current. f = 0, 8, = 0, R = 0.6, and I,, = 1.01,. The vertical axis is the normalized time-averaged DC voltage; the horizontal axis is the normalized DC current injected at the edge of the array. (a) I,/a = 0.1, (b) I J a = 1, and (c) I,/u = 20. Successive curves are displaced along the horizontal axis by 0.75 units. (From Ref. 199, Fig. 1.)

interior-a process that occurs with a period vac/q. Similar results, but differing in detail because of the truncated inductance matrix, were found by Dominguez and J o s C . ~ * ' ~ 'However, the details of this motion depend very strongly on LJa, that is, on the screening. On the integer steps, on each A C cycle, one column of N vortices and one column of N antivortices move from the outer edges of the array into the center, where they annihilate. This occurs for all values of LJa. On the f = 1/2 step (q = 2), what occurs depends strongly on AJa. For LJa > N no subharmonic steps are observed. For N > AJa > 1, on every other A C cycle columns of m isolated vortices (m x N/2) move across and annihilate, with antivortices moving in from the opposite edge. On the alternate cycles N - rn vortices and antivortices cross the array so the net effect is to move N vortices (antivortices) across the array in a time twice the A C period. For AJa < 1 a single column of N vortices moves across the array and annihilates, with a column of N antivortices moving in from the opposite edge-exactly as for the q = 1 steps except that the period is twice as long. Phillips et al. found that the mechanics for subharmonic steps is essentially

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

421

similar. Dominguez and Jose’ found very similar results, which they called axisymmetric coherent vortex states. Phillips et al. also simulated larger arrays and found that this motion of linear columns of vortices is the dominant effect. As mentioned, H.-C. Lee et al.’94 proposed that the subharmonic halfstep was due to nucleation of an f = 1/2 vortex state at the edges of the saw array as a result of the array bias currents. Hebboul and Garland177,188 subharmonic steps with array bias currents that they felt were too small to generate such an f = 1/2 vortex state at the sample edges, and attributed the subharmonic steps to variations in the critical currents. Phillips et al.3 results indicate that the self-field does induce subharmonic steps even for very weak bias currents. However, they found no evidence for any special vortex state near the sample edges; in fact, they found that vortices moving in the array do not always nucleate at the edges. It is probably correct to say that self-fields are one of several types of symmetry-breaking mechanisms, all of which appear to lead to a subharmonic response. This conclusion was also reached by Dominguez and Jose.’ In fact, it appears that although many symmetry-breaking mechanisms are possible, in an ordered array the origin of the subharmonic Shapiro steps is the array self-field. Phillips et al. also explored the screening of transport currents by geometrical inductances. Figure 79(a) shows how the IV characteristics in the presence of an AC current change when N is increased for fixed LJa. The figure shows the simulated IV curve for a 63 x 63 array with parameters similar to the 11 x 11 array shown in Fig. 78. The subharmonic steps at 1/2 and 2/3 are clearly visible, but the shape of the curve near the integer steps is clearly very different from that of Fig. 78 -the rising side of the step is quite rounded and the step width is substantially reduced- the region of DC bias current where the array is phase-locked has been considerably decreased. To probe this further they investigated the current distribution across an array. Figure 79(b) shows these current distributions, normalized to what would occur for LJa = co, which is a uniform current distribution. The bold solid curve is for A,/a = 2, without vortices or an applied field and includes the full inductance matrix. The self-fields cause the current to bunch up near the edges of the array, implying a distribution of currents in the junctions across the width of the array. Since the current is injected uniformly into the ends of the array, this also implies a distribution in junction currents along the length of the array. The net result is that different junctions carry different currents, reducing the array’s ability to phase-lock to an AC drive current. The degree of inhomogeneity in the transport currents is found to increase with decreasing AJa, and we expect the ability of the array to phase-lock to decrease as AJa becomes small. (A similar size effect was discussed in Section VII). Thus, self-fields are very

422

R. S. NEWROCK ET AL.

:.s

g

1.

s

\

1530.5,

1.

f2

1

Is

0.

FIG. 79. (a) The simulated IV characteristics of a 63 x 63 SNS junction array, Ilia = 1, = 0.6, I,, = I , and 8, = 0. (b) The distribution of current in a cross-section of the 63 x 63 array, for I,/a = 2. The bold solid curve is for the full inductance matrix. The light solid curve is for a truncated inductance matrix, one including only self-inductances in calculating the induced fields. The light dashed curve is also a truncated inductance matrix, which includes both self and nearest-neighbor mutual inductances. All curves are normalized to the uniform junction current that would flow if I,/a = co. (c) The width of the integer giant Shapiro step as affected by self-fields. AI/Ic is the width of the first integer giant Shapiro step for an N x N array. R = 0.6, I,, = I,, and p, = 0. N varied from 7 to 80, and I,/a varied from I to 5 for small arrays, and l,/a = 1 for large arrays ( N > 31). (d) The zero field IV characteristics for an 11 x 1 1 SIS array with ,f = 0, R = 1.0, I,, = I , , p, = 0.7, and I , / a = 1. The dashed curve is for an array in free space. The solid curve is for an array with an ideal ground plane located near the array. (From Ref. 199, Fig. 5.)

R

important in determining the IV characteristics of large arrays even when weak induced fields are present. The expected effects on the step widths are shown in Fig. 79(c), where the normalized step width AI/Z, is plotted versus Na/A,. AI is the range of DC bias currents, for fixed l2 and I,,, over which the array remains on a

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

423

Shapiro step. AI/Ic decreases with increasing N a / l , , as expected from the arguments presented above. One way to reduce the effects of self-fields is to introduce a ground plane where the image currents assist in canceling the self-magnetic field. Figure 79(d) shows a simulation for an underdamped array. In this simulation a ground plane was placed near the array, and, as can be seen in the figure, the results without the ground plane (dashed line) are markedly different from the results with it (solid line). With the ground plane the integer steps are sharper and wider -that is, better phase locking is taking place. In addition, the subharmonic steps are smaller. Phillips et al. also looked at what happens when the inductance matrix is truncated. Figure 79(b) shows what happens. The light solid curve is the current distribution across the sample when only self-inductances contribute. The dashed line is the current distribution when self and nearestneighbor inductances are included. We see that as N -+ co for fixed AJa, the regions of the sample more than a few Al in from the edges carry essentially zero transport current. Calculating induced fields with a truncated inductance matrix results in simulations with unrealistically strong screening. Using the truncated inductance matrix also affects the vortex configurations one sees in the simulations. In the truncated model, where no current exists past a few penetration depths into the specimen, vortices in the interior will be unaffected by transport currents- the Lorentz force will be very weak. Vortices will be quickly pulled into the array from the edges, but they will then move sluggishly toward the center, bunching up in the interior before annihilating. When the full matrix is used vortices are quickly driven across the array -the columns of vortices and antivortices quickly annihilate.

30. EFFECTS OF DISORDER H.-C. Lee et al.’O1 studied the effects of bond disorder resulting from nonidentical junction gaps. They concentrated on the Shapiro steps themselves. Bond disorder creates nonidentical junction critical currents and junction resistances. These result in a decrease in the range of DC bias currents over which phase locking to the AC drive current can occur for the Shapiro steps, and over which coherent emission of radiation occurs for the spectral power density. The net result is a decrease in the Shapiro step width, nonzero step slopes and rounding of the step edges for the driven array, and ’01

H.-C. Lee, R. S. Newrock, and D. B. Mast, Helo. Phys. Acta 65, 371 (1992).

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R. S. NEWROCK E T AL.

a reduction in the emitted power and a broadening of the line width for the emitting array. Lee et al. performed a series of simulations (without inductance) on a 15 x 4 array with a uniform distribution of junction lengths to a maximum value (r. As the amount of disorder increases, the rounding of the steps increases and the slope increases (from zero). For (T = 0.05 (a 5% variation in gap distances), the step rounding and step slopes are in good agreement with experimental observations. Ravindran et al.lS9 measured the frequency dependence and the step widths of the giant Shapiro steps in site-disordered SNS arrays in which lo%, 20%, and 30% of the islands had been removed. The inset in Fig. 72 shows the maximum normalized step widths of the first integer giant Shapiro step as a function of the frequency for two of the site-disordered samples-the one with 90% (10% removed) and 80% (20% removed) of the sites present. The Bessel-function-like behavior expected for perfectly ordered samples is preserved in the disordered samples, but as the disorder increases the maximum step width rapidly decreases. Indeed, in the 30% disordered specimen, no Shapiro steps are discernable in the IV curve. There is also no saturation of the step widths with frequency, as occurs in the ordered specimen; instead, the maximum step width rolls off at frequencies R > 0.5. In the disordered samples the steps themselves are, in general, very broad and smeared out, with rounded edges and nonzero slopes. The locations of the steps are in the correct place for the 10% disordered array but not for the 20% disordered array, a result attributed to the nonuniform current distributions present in such a highly disordered array. These effects are likely due to vortices generated near the corners of the defects (see, for example, the work of Leath and co-worke r ~ ~ ~ ~ , The ~ subsequent ~ ~ , ~ motion ~ ~ of * these ~ ~ vortices ~ * disrupts ~ ~ ~ ) . phase locking, causing different voltage drops across different junctions, and the Shapiro steps for individual junctions tend to occur at different current levels. 3 1. EMISSION OF RADIATION

When a Josephson junction is biased well above i,, such that V > i,R,, the time-dependent voltage is nearly sinusoidal. The maximum power into a matched load (RL= R,) is P, = i:R,/8. Unfortunately, single-junction sources emit very little power and what is emitted is very difficult to match to typical load impedances (50 a).To overcome this TilleyZo2proposed using 'O'

D. R. Tilley, Phys. Lett. A33, 205 (1970).

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series (one-dimensional) arrays of phase-locked junctions. These would emit coherent power to a matched load, R, = N R , at N times the power of a single junction, P , = N P , = NizR,/8. Doing so, however, depends strongly on being able to manufacture N very nearly identical junctions. Onedimensional arrays have been extensively reviewed by Jain et a1.* Two-dimensional arrays should, in principle, have a number of advantages over one-dimensional arrays. First, the in-phase solution implies that each row acts like a linear array of junctions and the total power emitted by a two-dimensional M x N array will be M times that of a linear array. Second, the array resistance depends on the array length and width, and with M x N junctions the ratio M J N can be chosen so as to optimize the impedance matching, M J N = R J R , , resulting in a maximum power, P,, = M N P , . The third advantage lies in the fact that in a two-dimensional array the parallel channels compensate for possible nonuniformities in the junctions. For comparable amounts of disorder, this leads to narrower line widths and higher emitted powers than a series array, as shown by Octavio et a1.203 Significant emission from two-dimensional SIS arrays has been meared.^^^*^^^*^^^ The arrays, 10 x 10 shunted SIS junctions designed to be in the nonhysteretic regime, had a characteristic frequency, 2i,R,eJh z 150 Ghz. The emitted radiation was measured using a single SIS junction as a detector, connected to the array via a blocking capacitor. Figure 80 shows '03 '04

'05 '06

M. Octavio, C. B. Whan, and C. J. Lobb, Appl. Phys. Lett. 60,766 (1992). S. P. Benz and C. J. Burroughs, Appl. Phys. Lett. 58, 2162 (1991). P. A. A. Booi and S. P. Benz, Appl. Phys. Lett. 64, 2163 (1994). F. C. Cawthorne, P. Barbara, and C. J. Lobb, I E E E 'l?uns. Appl. Supercon. 7 , 3403 (1997).

FIG. 80. Schematic circuit layout for two-dimensional array emission and detection. The dielectric forming the DC-blocking capacitor is the SiO film between the two top plates. The detector junction is connected between the center plate and ground. The array junctions are depicted as black circles. Dimensions are not to scale.

426

R . S. NEWROCK ET AL.

a schematic diagram of the array and detector. The arrays were constructed in a manner that minimized their size with respect to the expected wavelength, allowing them to be considered as a lumped circuit element. Since they are square, M = N , they are impedance-matched to the detector junction. Note the ground plane under the array that minimizes inductive effects. Figure 81 shows the IV characteristics of the detector junction for four different values of the array D C bias current. Curve (a) is the detector IV in the absence of an array D C bias current. No Shapiro steps are seen. The substantial capacitance between the blocking capacitor’s lower plate and the ground plane is in parallel with the detecto? junctions, so the effective p, > 1 and the IV curve is quite hysteretic. Hysteretic regions may also be caused by resonances in the coupling circuits. As the array bias current is increased, Shapiro steps begin to appear in the IV curve of the detector junction. The first step appears at V =

FIG.81. Detector junction IV curves for four array current biases, showing the Shapiro steps for four different emission frequencies. (a) I , = 0, (b) I , = 2.1 mA, v = 100Ghz, (c) I , = 2.5 mA, v = 150 Ghz, (d) I , = 2.8 mA, and v = 205 Ghz. Each nth step is labeled. (From Ref. 204, Fig. 2.)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

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0.125 mV, corresponding to a frequency of 60 GHz. Steps were observed up to 4.35 mV, corresponding to frequencies of 210 GHz. Figure Sl(b), (c), and (d) show examples for 100, 200, and 205 GHz, respectively. Note that while the n = 1 step is only observed up to 210GHz, harmonics are seen to voltages of the order of 1.2mV, proving that the limitation is from the emission from the array and is not a limitation of the detector junction. Using comparisons to circuit simulations, Benz and bur rough^^'^ estimated the power coupled into the detector junction to be on the order of 0.4pW. A simple estimate confirms this result: from Fig. 81(c) we can see that the critical current has been reduced to nearly zero and that the n = 3 step is larger than both the n = 2 and n = 4 steps, implying that the third step is quite close to its maximum. Using the junction parameters and the power dependence of the step widths, Eq. (11.4), P = ( i R F ( t ) ) 2 R=, 0.43 pW, which is in very good agreement with the simulations. There are two requirements for optimal operation of Josephson junction arrays as oscillators. First, all of the junctions should oscillate at the same frequency. From the Josephson relation (Eq. 2.7), this requires that all of the junctions have the same voltage across them. Second, all of the junctions should be in phase; otherwise, there will be significant cancellation of the high-frequency signals from different junctions. These requirements seem straightforward, but are actually quite subtle. If all of the junctions are identical in an array, one possible solution is for all of the junctions to oscillate at the same frequency and in phase. However, since this is a system of coupled nonlinear differential equations, there is no guarantee that this is the only solution. The stability of the in-phase solution is an important issue. One can gain much insight into the problem following the analysis of Hadley et al.207*208 We consider a series array of resistively shunted junctions for simplicity, and drive the array with a DC current source, as shown in Fig. 82(a). There are N junction in series, and the individual junctions have resistance R,,ij and critical current ic,ij. Since the same DC current ,i flows through each junction, the RSJ model, Eq. (2.12), gives

,,i

= ic,ijsin yij

+ 2eR,,ij dYij , dt ~

-

(11.17)

where we allow for variations in the junction critical currents and resistances. This straightforward equation contains much important information. First, the equations for all of the junctions decouple-we have N indepen'07

'08

P. Hadley, M. R. Beasley, and K. Weisenfeld, Appl. Phys. Lett. 52, 1619 (1988). P. Hadley, M. R. Beasley, and K. Weisenfeld, Phys. Rev. 838, 8712 (1988).

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R. S . NEWROCK ET AL.

V

n

V

n

V

n

U

n

V

n

RL (b)

FIG. 82. (a) A series array of identical Josephson junctions driven by a current source. (b) A series array of Josephson junctions with a load driven by a current source.

dent current-biased junctions. Next, consider the case where all of the junctions are identical, R,,ij = R, and i,,ji = i,. In this case, all junctions obey the same equation and all oscillate at the same fundamental frequency w,. The steady-state solution is not unique, however. Without loss of generality, we can write

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

429

as the long-time solution to Eq. (11.17) after transients have died out. An equally good solution is Y(t) = %(t - 47) + f ( t - t o ) ,

(11.19)

where to is an arbitrary constant. This, and the fact that the equations decouple, shows that even in the ideal case of identical junctions, where the oscillation frequencies are identical, the junctions will not be in phase; any noise, or any difference in initial conditions, will cause dephasing. (Note that this argument still holds where there is capacitance present.) The situation improves when a more realistic circuit model is used. Consider, for example, that the oscillator is coupled to a load, which is always the case for a real oscillator. For simplicity, we model the load as a resistor, R,. This circuit model is shown in Fig, 82(b). Kirchoffs laws now give two equations: iDc - i, = ic,ijsin y i j

h dyij +2eR0,ij dt

(11.20)

and 1,

1 R,

=-

c --.2eh dy,,dt IY

(11.21)

j=l

When these equations are combined, they give

1 R,

lDC - -

1 -h- Jd=y .i. j=l

2e dt

..sinyij+--. h dyij 2eR0,ij dt

‘J’

(11.22)

Thus, the load causes the N independent equations of Eq. (1 1.17) to be coupled; each y i j now depends on all of the others. The coupling does not guarantee an in-phase solution; indeed, much work has been done to see at what frequencies, and for what specific loads, the in-phase solution is stable. The details are beyond the scope of this review, but can be found in the literature.207,208.209.210

Although a load is certainly one way to stabilize the phase-coherent solution, it is not necessarily the only way. Building on the work described above, Cawthorne and co-workers,211.212studied many arrays. For P. Hadley, Ph.D. thesis, Stanford University (1989). K. Weisenfeld, S. P. Benz, and P. A. A. Booi, J . Appl. Phys. 76, 3835 (1994). 2 1 1 F. C. Cawthorne, Ph.D. thesis, University of Maryland (1988). F. C. Cawthorne, P. Barbara, S. V. Shitov, C. J. Lobb, K. Wiesenfeld, and A. Zwangil, 1999, to appear in Phys. Rev. B. *09

’lo

”’

430

R. S. NEWROCK E T AL.

example, they studied 10 x 10 arrays that differed only in having ground planes either above or below the array. This did not significantly affect the load, but drastically changed the operating frequency range, from 50 to 270GHz for a ground plane on the top, to 100 to 400GHz for a ground plane on the bottom. They also studied different size arrays with different load impedances, and with ground planes on the bottom and the top, and found that the two arrays emitted over the same frequency range. This implied that, at least in some arrays, the distance between the ground plane and the array was somehow changing the stability. To test this idea, they studied a series array model without a load, but included the ground plane. They modeled the ground plane and the wires between junctions as a transmission line and included a shunt resistor and inductor in series around each junction. The resulting equations are more complicated than Eqs. ( 1 1.20) and (1 1.21):

,,i

+ iT(xij,t ) - is,,,

=C

d'y.. d dt2

1

dy,,

.

.

+ R(V)+ z,,~, sin yij; dt ~

h _ dYij dis.ij _ - Ls 7 + is.ijRs; 2e dt

I/ (x' T

IJ 7

t ) - I/T (XI t) = IJ

arT(x,

ax

t,

=

h dy.. 2e dt '

--'I.

-c, aVT(x? at

t)f

(11.23)

(1 1.24)

(1 1.25)

(11.26)

and

(11.27) In these equations x i j is the location of a junction, ZT(x, t ) is the current flowing in the circuit at time t and position x ; VT(x,t ) is the voltage along the transmissions lines that connect the junctions; is.ij is the current in the shunt of the ij junction; and C , and LT are the capacitance and inductance per unit length of the transmissions lines. Dissipation in the transmission lines is modeled by R T , the resistance per unit length of the transmission line. R( V ) represents the intrinsic junction resistance. When these equations were numerically simulated, stable phase-locked solutions were discovered over a frequency range in good agreement with

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

43 1

experiment. More work needs to be done. In particular, the transmission line model needs to be combined with an external load, and the analysis should be extended to two-dimensional arrays. XII. Quantum Arrays 32. SINGLE JUNCTIONS

In the preceding sections we treated the Josephson phase as a classical variable. This is appropriate as long as all the capacitances in the array are large, in the sense that even the addition of many Cooper pairs at any point in the array does not give rise to an appreciable charging energy. In granular films (which often are modeled by Josephson arrays), as well as in arrays fabricated with modern nanolithographic techniques, the charging energy is not necessarily small compared to the characteristic energies E , and k,T. In that case the number of electrons one can place on a superconducting island becomes important, the uncertainty in the number of Cooper pairs is no longer large, and the phase may no longer be a well-defined variable. The quantum properties of the Josephson phase then become important. a. Quantization and Commutation Relations In this section we examine several ways of introducing the quantum nature of Josephson arrays when the charging energy dominates the Josephson energy or is the same order of magnitude. A simple way to introduce the quantum dynamics of the Josephson phase is to apply the canonical quantization rules to the Hamiltonian H ; i.e., the sum of the potential energy, Eq. (2.10), of the Josephson phase and its kinetic energy, Eq. (2.8). Writing the former in terms of the charge Q = V C , the Hamiltonian becomes

H

Q2

=-

2c

- E,cos~.

(12.1)

We first need to find the appropriate pair of conjugate variables and cast the classical equations of motion for the phase difference and the charge (in the absence of a magnetic field and dissipation) into Hamiltonian form. For the phase we have d y - 2eV - 2eQ 8 Q’dH dt h he d(hQ/2e) 2c - a(hQ/2e) ’

(12.2)

432

R. S. NEWROCK ET AL.

and for the charge we have

h _Q - h . . a -d- _ icsiny= - - E dt 2e

aY

2e

J

aH cosy= --

aY

(12.3)

In the last parts of Eqs. (12.2) and (12.3) we recognize Hamilton’s equations where

hQ 2e

( 12.4)

is the generalized canonical momentum conjugate to y. The canonical quantization rule for the transition to quantum mechanics now tells us to make y and hQ/2e operators, with the commutator equal to ih, so that [y, Q] = i2e.

(12.5)

This simple way of introducing quantum dynamics may be perceived as somewhat suspicious because phase is rather elusive. Converting charge and phase to canonically conjugate variables is nowhere near as straightforward and obvious as, for example, converting position and velocity into the canonically conjugate variables position and momentum. A more satisfactory method is based on the very instructive treatment of the Josephson effect suggested by Ferrell and Prange.2’3 We consider two separate pieces of superconductor, both in the ground state, containing N , + N and N , - N Cooper pairs, respectively. For N = 0 both pieces are electrically neutral. We denote the wave function of the total system by IN). If the superconductors are large, the total energy of the system E o ( N ) is only very weakly dependent on N over a wide interval around N = 0. For simplicity, let us ignore for the moment all dependence of E , on N . Then the ground state of the total system is highly degenerate. Connecting the superconductors by a tunnel junction lifts this degeneracy. To lowest contributing order of degenerate perturbation theory, the tunnel junction couples states differing by a single Cooper pair, and the matrix elements of the Hamiltonian can be written as214

R. A. Ferrell and R. E. Prange, Phys. Rev. Lett. 10, 479 (1963). The coupling also adds a constant to the diagonal matrix element, but this is not important and we omit it. 213

214

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

433

As in the tight-binding model of a solid, this Hamiltonian can be diagonalized by a Fourier transformation to new eigenstates:

(12.7)

This defines the phase difference y as the variable conjugate to the Cooperpair number N . The energy eigenvalues E(y) = E,-,

-

E, cos y

(12.8)

show the familiar phase dependence of the Josephson energy. Here we have used the normalization C, ( N I N ) = 1. We note that the charge uncertainty is infinite in the states Iy): All charge eigenstates contribute with equal amplitude. The energy degeneracy of the charge eigenstates IN) in the absence of Josephson coupling is only approximately valid, as it is spoiled (primarily) by the electrostatic charging energy,”’ (2eN)’/2C, required to move N Cooper pairs across the junction. Including this energy (and choosing the energy scale such that E,(N = 0) vanishes) gives matrix elements216 (NIH”’)

(2eN)’ 2c

= -~ N , N ,

Ef

(~N,N*1 +

+ ~ N , N ’ -1).

(12.9)

In these matrix elements we recognize the charge representation of the Hamiltonian, Eq. (12.1). The eigenstates, lYj), of this Hamiltonian are no longer identical to the phase eigenstates Iy). When expanding the eigenstates of the Hamiltonian, lYj),in charge eigenstates, (12.10)

the coefficients ujN rapidly decay for N -+ co. This implies that the charge uncertainty is no longer infinitely large while that of the phase no longer ’15 P. W. Anderson, in Lectures on the Many-Body-Problem, Vol. 2, ed. E. R. Caianiello, Academic Press, New York (1964), 113. 216 It was assumed here that the matrix element of the Josephson coupling is not affected by charging effects. This assumption is correct if the superconductor’s energy gap A is large compared to the typical charging energy. Otherwise, the situation becomes more complicated; see, e.g., K. A. Matveev, M. Gisselfalt, L. I. Glazman, M. Jonson, and R. I. Shekhter, Phys. Rev. Lett. 70, 2940 (1993).

434

R. S. NEWROCK ET AL.

vanishes. The extent of the phase and charge uncertainties clearly depends on the ratio of the charging energy E = -e2 -2c

(12.11)

to the Josephson energy E,. We illustrate this with a simple example below. When the charge uncertainty is very large compared to 2e, the phase uncertainty is small and the regime of the classical Josephson effect is recovered. In the phase representation, the charge operator is given by Q = - i2eapy. On first sight it would seem that Q and y satisfy the canonical commutation relation Eq. (12.5). Closer inspection reveals that this commutation relation cannot be correct. For example, sandwiching it between two charge eigenstates yields, using the left-hand side of Eq. (12.5),

while the right-hand side of Eq. (12.5) gives (Nli2elN‘)

=

i2ehN,,#.

(12.13)

Comparing these equations for N = N ’ gives 0 = 1. The paradox arises because y itself is not an observable; the two diferent numbers y and y + 2n describe the same physical state. Only 271 periodic functions of y qualify as observables. A thorough discussion of the situation was given by Carruthers and Nieto,’” who recommended that Cexp(ir), Q1=

-

2e exp(iy)

(12.14)

be used instead of the “hazardous” formula, Eq. (12.5). Note that Eq. (12.14) reduces to Eq. (12.5) when y << 1. The Heisenberg uncertainty relation then takes the form (12.15) where ((Q)) = ( ( Q - ( Q ) ) ’ ) , etc. If the uncertainty of exp(iy) in the complex plane is small compared to 2n, Eq. (12.15) coincides with the usual uncertainty relation ((Q))((y)) Z e2.

’” P. Carruthers and M. M. Nieto, Rev. Mod. Phys. 40,411 (1968).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

435

+

The issue of “compact phase” (y equivalent to y 271) versus “extended phase” (all phase values in the interval ( - as, as) are distinct) gave rise to some controversy in the 1980s. The way we introduced the phase here makes it clear that the size of the phase domain depends on the size of the charge quantum. If the charge can only change by multiples of 2e, phases differing by multiples of 271 are equivalent. If quasiparticles also need to be included, and the charge can change in units of e, only phases differing by multiples of 471 are equivalent. Finally, if there is a possibility for a continuous change of charge-e.g., due to a current source or an ohmic shunt resistor-the phase must be treated as extended. Further discussion of this topic can be fouad in the work of A n d e r s ~ n , ” ~ *Likharev ”~ and Zorin,’” and Schon and’Zaikin.’” There are yet other ways to introduce the quantum properties of the Josephson phase besides the canonical quantization and the Ferrell and Prange procedure, the two discussed here. Starting, for example, from the quantized electromagnetic field, and using Gauss’s law as well as the Josephson relation between phase velocity and voltage, one can obtain the quantum properties of charge and phase from those of the electromagnetic field. This approach was worked out by Srivastava and Wid~rn.’~’*’’~ b. Phase Delocalization in Single Junctions To get an idea of how strongly the electrostatic energy delocalizes the phase, we now consider the ground state of the Hamiltonian, Eq. (12.1). In phase representation, the stationary Schrodinger equation can be solved exactly in terms of Mathieu f~nctions’’~(only 271 periodic functions are acceptable). For our purposes it is more illuminating to discuss approximations valid in the two limiting cases.

(i) E , >> E , In this limit we expect the phase to be strongly delocalized. Since here the charging energy is dominant, the ground state can be found by restricting H to the space spanned by the charge states N = 0 and N = 1 . (The other charge states are too costly in energy.) By exact diagonalization P. W. Anderson, Rev. Mod. Phys. 38, 298 (1966). P. W. Anderson, in Progress in Low Temperature Physics, Vol. 5, C . J. Gorter, North Holland, Amsterdam (1967), 1 . ’O K . K. Likharev and A. B. Zorin, J . Low Temp. Phys. 59, 347 (1985). G. Schon and A. D. Zaikin, Physica 8152, 203 (1988). Y. N. Srivastava and A. Widom, Physics Rep. 148, 1 (1987). 2 2 3 A. Widom, in Macroscopic Quantum Phenomena, eds. T. D. Clark et al., World Scientific, Singapore (1991) 55. 224 M. Abramowitz and I. A. Stegun, Handbook ofMathernatical Functions, Dover, New York (1970). ’I8

*19

’” ”’

436

R. S. NEWROCK ET AL.

of the matrix elements of those states, one obtains for the ground state of Eq. (12.9) (12.16)

where

and the ground-state energy E , is given by

The expectation value of sin y vanishes because of symmetry, so that ((cosy))

+ ((sin y ) )

=

1 - (cos Y ) ~ .

(12.19)

For the ground-state expectation value of the cosine one finds (cosy)

=

2a ~

1 + 2a2

(12.20)

= 4 3EC+ 0 ( ( 5 ) ? The average charge vanishes, and the charge uncertainty is easily seen to be

When Ec is very large, a is zero, and the uncertainty in the charge is very small. (ii) E , >> Ec In the opposite limit, when the Josephson energy is dominant, the delocalization of the wave function is weak so that one may use a second-order Taylor expansion of the potential near y = 0 and neglect the periodicity condition. This yields the quantum harmonic oscillator, and, for the ground-state wave function,

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

437

where and (QZ>

=

e2E

(12.24)

We see that when E, is very small, the charge uncertainty is very large, consequently, the phase uncertainty is very small. In Fig. 83 the phase and charge uncertainties are plotted versus the ratio EJ/4E,. (We use 4E, because (2e)2/2C is the natural variable for a superconductor.) In the figure the two tractable limits discussed above are connected by exact numerical results over the rest of the range. The dependence of the phase uncertainty on E,/E, could be quite directly demonstrated experimentally in a setup where the strength of the Josephson coupling could be changed in sit^.^^^ c. Historic Note and Outlook Although the quantum character of the Josephson phase was known in the early sixties, for a long time quantum fluctuations of the phase were 225

W. J. Elion, M. Matters, U. Geigenmiiller, and J. E. Mooij, Nature 371, 594 (1994).

I

I

-4

I

I

-2

I

I

I

I

I

I

0

2

4

k(E,/4E,) FIG. 83. The phase and charge uncertainties of a single junction as a function of the ratio E,/4EC. The solid line gives the charge undertainty 6Q = the dashed line, the phase uncertainty 6y = J(((coszy)) + ((sin’y)))/((cosy)’ + (sin Y ) ~ ) .

m;

438

R. S. NEWROCK ET AL.

generally thought to be unimportant in practice, mainly because of the masking of quantum fluctuations by thermal fluctuations. The artificially fabricated junctions of the sixties and seventies generally did not have capacitances below 1 pF. The elementary charging energy E , of a 1-pF capacitor corresponds to a temperature of about 1 mK, and even a dilution refrigerator typically cannot cool below 3 mK. In 1977, however, AbelesZz6 pointed out that in granular superconductors the grain size may well be small enough for the charging energy E , to be larger than both E , and k,T in an experimentally accessible temperature range. Based on qualitative reasoning, he suggested that the Josephson coupling would be quenched unless E , << E,. While this suggestion turned out to be too pessimistic, his note sparked a wealth of further research on the influence of quantum effects on the thermodynamics of Josephson arrays, in particular on the superconducting phase transition. This is the subject of the next section, in which we will concentrate on the quasi-classical treatment and mean-field theory. Magnetic-field effects and the influence of disorder will not be discussed. We note that some work has been done on magnetic-field effects, but very little has been done on disorder in the quantum regime. Another, less widely studied aspect of quantum effects in Josephson arrays is their influence on the dynamics of vortices -for example, the possibility of tunneling through the egg crate pinning potential. For a discussion of this issue we refer the reader to $imanek.227 The very large amount of work done on the quantum properties of single junctions and assemblies of few junctions falls outside the range of the present review. Discussion can be found in the reviews by Schon and Zaikin,”* Averin and L i k h a r e ~ , ” ~and Sondhi, Girvin, Carini, and Shahar.230 33. THESUPERCONDUCTING PHASE TRANSITION IN ARRAYS a. Quantum Corrections to the Kosterlitz- Thouless Transition In classical statistical mechanics the partition function can be factored into a kinetic part and a configurational part, the latter depending on the potential energy above. Because of this, the kinetic part has no influence on B. Abeles, Phys. Rev. B15,2828 (1977). E. Simanek, Inhomogeneous Superconductors, Oxford University Press, New York (1994). 2 2 8 G. Schon and A. D. Zaikin, Physics Rep. 198, 237 (1990). 2 2 9 D. V. Averin and K. K. Likharev, in Quantum Efects in Small Disordered Systems, eds. B. L. Altshuler, P. A. Lee, and R. A. Webb, North Holland, Amsterdam (1991), Ch. 6. 230 S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Rev. M o d . Phys. 69, 315 (1997). 226

227

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

439

phase transitions. For the same reason the charging energy does not influence the Kosterlitz-Thouless transition in the classical limit. In the quantum limit, when position and momentum (or phase and charge) do not commute, such a factorization of the partition function is generally impossible. In the quasi-classical regime, however, where quantum effects are weak, an approximate factorization can be achieved. The quantum corrections then show up as an additional terms in the configurational part of the partition function. Technically, the procedure (dating back to work by Wigner and by Uhlenbeck and Gropper in the early 1930s) amounts to an expansion in the small parameter characterizing the commutator of the canonical variables: h in the usual mechanical context, e in the present case. The application of the quasi-classical expansion to Josephson arrays is due to Jose,231and Jose and Rojas.232*233 We base our treatment on the array Hamiltonian (12.25) which corresponds to the classical Hamiltonian introduced in Eq. (3.6) but in which the phases and charges are now operators. Note that we have switched from the phase difference to the island phases 4 (canonically conjugate with the charge on an island). We have also dropped the vector potential term A , for the moment. Eq. (12.25) is the appropriate generalization for the single-junction Hamiltonian, Eq. (12.1). The phase and charge operator belonging to a particular island are canonically conjugate; all other operators commute. We also allow for a more general capacitance matrix C than just the nearest-neighbor capacitances used earlier.234In particular, a “self-capacitance” C, i.e., the capacitance the islands have with respect to a ground plane (or with respect to “infinity”- the cryostat wall, etc.)-will generally be included. This leads to an exponential screening of electrostatic interactions with screening length -

(12.26) J. V. Jose Phys. Reo. 829, 2836 (1984). J. V. Jose and C. Rojas, Physica 8203, 481 (1994). 2 3 3 Earlier, less systematic treatments of quantum corrections to the Kosterlitz-Thouless transition were given by Maekawa et (11. (Ref. 259); E. Simanek, Phys. Rev. Lett. 45, 1442 (1980); and Yu E. Lozovik and S. G. Akopov, J . Phys. C14, L31 (1981). The results of these treatments differ from each other as well as from the quasi-classical expansion discussed here. 2 3 4 In Eq. (12.25) and the rest of this article, C,; always means [C- I],,, 231

232

’

440

R. S. NEWROCK ET AL.

where this result holds for the Coo+ C,, approximation, where nn = nearest neighbor. The properties of the capacitance matrix are discussed in Appendix E. A convenient starting point for the quasi-classical approximation is to write the partition function as a trace over charge eigenstates:

Here v is the number of islands. We are looking for a small correction to the classical limit; we assume that E, is important but not dominant. We denote with Q the charge eigenvalues and with $J the phases on the v different islands. In the $J representation the charge eigenvectors lQ) are proportional to the exponentials e'+'Q.Since we are interested in corrections to the classical limit, it is sensible to split off the classical Boltzmann weight in the integrand, defining a correction factor ~ ( 4by )

H, is the Josephson potential, - E,(T) C O S ( ~4i). ~ The correction factor x can be expanded in powers of the phase-charge commutator, i.e., in powers of the charge e. If phase and charge commute, we are back in the classical case, so x equals unity to order eo. The first-order term is odd in the charges and vanishes upon the summation over Q so the lowest nonvanishing quantum correction is of order e2. Explicitly one finds (see, for example, Section 33 of the textbook by Landau and L i f ~ h i t z ' ~ ~ ) fZn

fZn

(12.29)

If the temperature is large compared to the unit-charging energy, the sum over charges can be replaced by an integral. Then the charge correlation function contained in Eq. (12.29) becomes easy to evaluate, and we obtain, after a partial integration, the desired factorized form of the partition 235

L. D. Landau and E. M. Lifshitz, Statistical Physics, Pergamon, Oxford (1968).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

441

function:

(12.30) (Note that the terms in the leading bracket contain the classical charge terms.) Because of the sinusoidal character of the Josephson term, double phase derivatives reproduce it, (12.31) The quantity E d -= e2(Ci' - C;')

(i, j nearest neighbors)

(12.32)

is the energy of a singly charged dipole with charges positioned on neighboring sites.236Since we are considering the region where the temperature is large compared to the charging energy, we may make the following approximation: EdP2

1+-

(12.33)

3

Thus, the quasi-classical approximation reproduces the classical partition function, except that in its potential part the temperature is renormalized as,237

p

- Ed8/3).

(12.34)

236 If we have, for example, + e on the Ith island and --e on the neighboring mth, and Qj = ( S j , - ajm),then, summing to get the energy, X(l/2)C;'QiQj yields E d , a dipole energy. 237 The quasi-classical result has also been used as input for the Kosterlitz-Thouless renormalization equations (see Refs. 231 and 232). This led to the prediction of a second phase transition at temperatures below T,,, a quantum-induced transition (QUIT) by Jose. Of course, the quasi-classical approximation is not expected to be valid at low temperatures. Therefore, very extensive Monte-Carlo calculations were performed (L. Jacobs, J. V. Jose, and M. A. Novotny, Phys. Rev. Lett. 53, 2177 (1984); L. Jacobs, J. V. Jose, M. A. Novotny, and A. M. Goldman, Europhys. Lett. 3, 1295 (1987); L. Jacobs, J. V. Jose, M. A. Novotny, and A. M. Goldman, Phys. Rev. B38, 4562 (1988); and L. Jacobs and J. V. Jose, Physica B152, 148 (1988)). These qualitatively confirmed the existence of a second phase transition. The character and physical significance of that phase transition are not yet quite clear. Mikalopas et al. (J. Mikalopas, M. Jarrell, F. J. Pinski, W. Chung, and M. Novotnov, Phys. Rev. B50, 1321 (1994)) also studied quantum Josephson arrays with local charging energy. No QUIT was found, although metastable states associated with the phase boundary were. They speculated that these metastable states could be misidentified as a QUIT.

442

R. S. NEWROCK ET AL.

This makes it possible to find the suppression of the Kosterlitz-Thouless phase-transition temperature T K T by small quantum fluctuations:

TKT(Ed)

=

TKT(Ed

2

= O)

(

1+

-/)

1-kBTKT(Ed

= O)

.

(12.35)

They This approach was tested experimentally by the Delft fabricated junctions that were much smaller than those described in earlier sections in order to make capacitances small. Electron-beam lithography was used to fabricate these arrays, which had a lattice spacing of 4 pm and a junction area of 0.01 (pm)2, leading to a hearest-neighbor capacitance of about 1.1 fF. To vary the ratio of the Josephson coupling energy to the charging energy, a series of arrays were made with the same capacitances, but with critical currents varying from about 8.9 nA to 322 nA. This series of samples thus had a ratio of charging energy (Eq. (12.11)) to Josephson coupling energy, E J E , , ranging from 0.26 to 10. Samples with low E,/E, became superconducting, with the transition temperature decreasing as the ratio increased. The two samples with the lowest ratios, 0.55 and 0.26, became insulating at the lowest temperatures. Assuming that the high-ratio samples underwent a Kosterlitz-Thouless transition, the measured critical temperatures were in remarkably good agreement with the quasi-classical formula, Eq. (12.39, as obtained by Josk and Rojas232*239 and shown in Fig. 84.238The agreement with the data is better than might have been expected, given that the quasi-classical approach is only meant to be a first-order correction. For samples outside this regime, another approach is needed. b. Mean-Field Theory With decreasing ratio k, TIEd the (i) THEIMPORTANCE OF DIMENSIONALITY quasi-classical approximation discussed above becomes unreliable; on the other hand, the quality of the “next best” approximation, mean-field theory, improves. This is a question of dimensionality. It is well known in the theory of phase transitions that mean-field theories, which are characterized by the neglect of correlations between different sites, work better the higher the dimension of the system. Above the so-called upper critical dimension, they even give quantitatively correct results. On a coarse length scale, the Josephson array can be modeled by a Ginzburg-Landau Hamiltonian (we will come back to that in b.5 below) 238 239

H. S. J. van der Zant, L. J. Geerligs, and J. E. Mooij, Europhys. Lett. 19, 541 (1992). JosC and Rojas also corroborated the quasi-classical results by Monte-Carlo calculations.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

443

19 Y

0.0 0.0

0.2

0.4

0.6

0.8

1.o

Ec/ E J

FIG.84. A comparison of the quasi-classical formula, Eq. (12.35), line, derived by Jose and Rojas (Ref. 232), to the experimental data of van der Zant et al. (Ref. 265), circles. The self-capacitance C, is negligible in comparison with the nearest neighbor capacitance C, so that Ed = e2/4C.

with an upper critical dimension equal to 4. Therefore, neither a twodimensional nor a three-dimensional array can be described quite correctly within mean-field theory. In fact, mean-field theories predict a low-temperature phase with nonvanishing order parameter (ei4) even in two dimensions, whereas we know from the work by PeierlsZ4' and by Mermin and Wagnerz6 that this type of phase order is impossible in two dimensions. Thus, in the classical case mean-field theory seems to be of little use because it gives misleading results. Quantum theory changes the picture somewhat, as can be seen especially well in a path-integral formulation of the problem, where the partition function takes the form of a path integral over the exponent of the negative Euclidean The latter is given by an integral of the Hamiltonian over imaginary time, from 0 to Bh. The Hamiltonian itself contains a sum over the spatial dimensions. The integration over imaginary time may, in 240 R. E. Peierls, Helu. Phys. Acta. 7 , Suppl. 11, 81 (1934); R. E. Peierls, Ann. Inst. Henri PoincarC 5, 177 (1935); R. E. Peierls, Surprises in Theoretical Physics, Princeton University Press, Princeton (1979). This last reference is usually more readily available than Peierls' original articles on the matter. 241 R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, New York (1965).

444

R. S. NEWROCK ET AL.

some respects, be treated just like an additional spatial dimension. (Of course, the interaction in the direction of imaginary time is usually somewhat different from that in the spatial directions). Thus, the quantum statistical mechanics of a system with n spatial dimensions corresponds to the classical statistical mechanics of an n + 1 dimensional system. For high temperatures the “system size,” Bh, in the time direction is small and the influence of the extra dimension becomes negligible. At T = 0, on the other hand, the system is infinitely large in the time direction. The additional dimension is still important for small, but nonzero, temperatures, when the size in the time direction is finite but large. The extra dimension supplied by quantum mechanics is not sufficient to make mean-field theory quantitatively correct for a two-dimensional array, but it does at least remove the contradiction with the Mermin-Wagner theorem at low temperatures. We may therefore hope to find qualitatively correct results from the mean-field theory, which is much simpler than the more correct renormalization-group approach and Monte-Carlo methods.

THEORYIn the context of phase transi(ii) SELF-CONSISTENT MEAN-FIELD tions in Josephson arrays and granular superconductors, mean-field theory was first used by Simanek.242The approximation consists in replacing the Josephson coupling of the phase of a given island i to its neighbors by an average coupling, in the sense that

Ci C O S (-~ ~

$j)

[cos $icos 4j + sin +i sin 4j] z z(cos 4 ) M F cos 4i.

= i

(12.36)

The summation in this formula is meant to extend over the z nearest neighbors of island j . Since the phase differences matter only in the Hamil= 0. It tonian, a constant may be added to all phases such that (sin 4 ) M F is possible, but more cumbersome, to avoid this The expectation value (cos 4 ) M F is calculated self-consistently, using the mean-field Hamiltonian HMF

= H, - zEj(cos 4 ) M F

1i cos

4 i 9

(12.37)

whose charging part (12.38)

242 243

E. Simanek, Sol. State Cornrn. 31, 419 (1979). C. J. Lobb and R. S. Newrock, J . Low Temp. Phys. 105, 133 (1996).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

445

is the same as in Eq. (12.25) but in which the different islands are not coupled by the Josephson term. The evaluation of (cos 4)MFbecomes particularly easy if only selfcharging is taken into account ( C , # 0, C = 0; see Appendix E). Then the capacitance matrix is diagonal, and different islands are completely uncoupled in the mean-field approximation. The eigenfunctions of the array are products of eigenfunctions $, describing the individual islands, which satisfy Mathieu's equation

This is to be solved with (cos

4)MFgiven self-consistently by

For high temperatures or low E, only the trivial solution (cos 4)MF= 0 exists. The effective Josephson coupling zE,(cos 4)MFof the mean-field theory then vanishes and with it the superconductivity of the array. For low temperatures or high E,, the solution of the coupled equations (12.39) and (12.40) is shown in Fig. 85. In the classical limit of vanishing charging energy, the self-consistency equation takes the form

which predicts a (spurious) phase transition at k,T," = zE,/2 (spurious because the mean-field theory is invalid in two dimensions). In the more general case that the nearest-neighbor capacitance is not zero, the capacitance coupling complicates the mean-field treatment. If one is interested only in the phase boundary,244however, it suffices to evaluate the self-consistency condition in perturbation theory with respect to the order 244 The phase boundary is where the phase transition occurs in the appropriate variable plane. We do not care what the size of the order parameter is, only where it stops being zero. In terms of Fig. 85, for example, we are interested in where the order parameter, (cos 4), increases from zero; that is, we merely want the line in the variable plane shown.

446

R. S. NEWROCK ET AL.

FIG. 85. Mean-field results for (cos 4) with a diagonal capacitance matrix (C = 0, C, # 0, and monopole energy Em = e2/2C,). Dimensionality and lattice structure only enter through the number z of nearest neighbors.

parameter (cos

4 ) M FTo . this end we use the interaction

repre~entation~~~

(12.42)

where T indicates ordering with respect to (imaginary) time z and H,,(z) erH0HMFeCrHo. Expanding the self-consistency condition we obtain

=

See, for example, A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Prentice Hall, Englewood Cliffs, NJ (1963), Sec. 245

12.1.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

= (COS

::J

4)MF~EJ

(cos 4 j ( z )cos &J0

+ O((cos 4)&).

447

(12.43)

Here (...), = Tr(e-pHo...)/Tr(e-PHo)denotes the thermal average in the absence of Josephson coupling. The explicit form of the correlation function,

is derived in Appendix G. In Eq. (12.44) (12.45) is the energy of a single charge sitting on island i in the array-that is, it is a singly charged m ~ n o p o l e ~ ~ ~ - a n = d C j C , j ' Q j is the voltage on island i. Substituting the correlation function, Eq. (12.44), into the self-consistency condition, Eq. (12.43), and performing the time integral (see Appendix G for details), one finds for the phase boundary the implicit e q u a t i ~ n ~ ~ ' . ~ ~ ~ 1

(12.46)

Since the Boltzmann factor in the thermal average of Eq. (12.46) is a Gaussian, one might think that the average is easy to calculate. The simplicity is spoiled, however, since we need to sum over discrete charge values rather than integrate over all charge values. At zero temperature, only the uncharged configuration contributes, the voltages vanish, and the mean-field phase transition occurs at 4 EJ = - E m . Z

(12.47)

The monopole energy Em diverges logarithmically with C/C,. This signals a breakdown of the result of Eq. (12.46) as CJC + 0. In b.5 a variationally 246 In the literature the monopole energy is frequently denoted by E,. Note also that when we place a charge on a single island in the array, in principle all of the capacitances in the array become charged. 2 4 7 K. B. Efetov, Zh. Eksp. Eor. Fiz. 78, 2017 (1980); Sou. Phys. JETP 51, 1015 (1980). 2 4 8 The effective potential term in Efetov's (Ref. 247) mean-field Hamiltonian is a factor of 2 smaller than that used by Simanek (Ref. 242). We believe that this is incorrect, as the value used by Simanek also follows from a variational approach (see Section 33.b.5).

448

R. S. NEWROCK ET AL.

improved version of the mean-field theory will be presented that does not suffer from this shortcoming.249 (iii) RE-ENTRANCEFor a nondiagonal capacitance matrix and nonzero temperature, the evaluation of Eq. (12.46) is difficult, even numerically. EfetovZ4’ treated the low-temperature region k,T << Em by including, besides the uncharged configuration, the lowest excited states corresponding to two adjacent islands charged by plus and minus 2e, respectively. These thermally excited dipoles act like a dielectric, which reduces the strength of the Coulomb interaction in the array and thus tends to enlarge (cos 4 ) . At low temperatures, this effect can outweigh the additional phase disorder due to thermal fluctuations, and can lead to a remarkable re-entrant bulge in the phase diagram: For E,/E, slightly below the value corresponding to the T = 0 phase boundary, the system first becomes superconducting with increasing temperature and then loses the superconductivity again upon further heating. From the physical interpretation of its origin it is clear that such re-entrant behavior can be expected to occur only for a long-range electrostatic interaction, i.e., for a large electrostatic screening length, A = a m ; otherwise, there is no room for the thermally excited dipoles within the interaction range. Indeed, for shortranged electrostatic interactions only a minuscule re-entrance is fo~nd.’~~.’~~ The importance of the range of the electrostatic interaction for re-entrant behavior was stressed by Fishman and Stro~d,’~’who calculated the mean-field phase boundary for C,/C down to 0.4 for low temperatures. Figure 86 shows the result for the mean-field phase boundary for large ranges of C,/C and temperature, obtained by a Monte-Carlo e v a l ~ a t i o n ’ ~ ~ of the charge autocorrelation function. Figure 87 shows resistance versus temperature for the arrays corresponding to the experimental points of Fig. 84. Near the separatrix that lies between samples turning superconducting and those turning insulating for 249 An alternative version of the mean-field theory, which also remains sensible for C , = 0, was proposed by Fazekas et (I/. (P. Fazekas, B. Miihlschlegel, and M. Schroter, Z . Phys. B57, 193 (1984)) and by Ferrell and Mirashem (Ref. 289). These authors considered small clusters of islands and treated both the Josephson coupling and the electrostatic coupling to islands outside the cluster in an average way. Mirashem and Ferrell (B. Mirashem and R. Ferrell, Physica C152, 361 (1988)) found the T = 0 phase boundary (for z = 4) at E , = ez/2(C, + 4C). This agrees with Eq. (12.46) for C = 0, but for C , = 0 it is a factor of 2 smaller than the result of the variationally enhanced mean-field theory (cf. Section 33.b.5). 2 5 0 E. Simanek, Phys. Rev. 823, 5762 (1981). 2 5 1 P. Fazekas, 2.Phys. 845, 215 (1982). 2 5 2 R. S. Fishman and D. Stroud, Phys. Rev. B37, 1499 (1988). 2 5 3 U. Geigenmiiller, unpublished.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

449

1.5

$

1.0

\

h

Q

&?

0.5

0.0 0.6

0.8

1.0 1.2 E,/Ern

1.4

1.6

FIG. 86. Mean-field phase boundary for C,/C = 00 (full line), 1 (triangles). 0.1 (circles), and 0.01 (squares). The data for finite C,/C were obtained by a Monte-Carlo evaluation of the charge autocorrelation. The dashed line is obtained by treating the charges as continuous instead of discrete variables. The larger the range of the interaction, the more the discreteness of the charges is smeared out and the longer the phase boundary follows the continuous charge approximation. The latter thus provides an upper limit for the re-entrant bulge.

T -+ 0, there is “quasi-re-entrant’’ behavior: the resistance first drops by two orders of magnitude as the temperature is lowered but then rises again. This dip in the resistance might be related to the re-entrant phase boundary predicted by Efetov. Similar results for arrays were obtained before by Geerligs et al.254

(iv) INFLUENCE OF OFFSETCHARGESFor the single junction discussed earlier in this section, the expectation value (cos 4) does not vanish for any positive E,. If one thinks in terms of ordering of the Josephson phases, it therefore appears strange that a phase transition to a state with (exp(i4i)) = 0 should exist in arrays, since the ordered phase is usually more stable in higher dimensions. This puzzle disappears when one thinks in terms of ordering of the charges, the superconducting phase being that of charge disorder (that is, a system in the superconducting state has the phase ordered; in the phase-disordered state it is not a superconductor). ~~

2 5 4 L. Geerligs, M. Peters, L. E. M. de Groot, A. Verbruggen, and J. E. Mooij, Phys. Rev. Lett. 63,326 (1989).

450

R. S. NEWROCK ET AL.

FIG.87. The resistance versus temperature for a quantum array, corresponding to the experimental points in Fig. 84. (From Ref. 239, Fig. 1.) The energy ratios are (from top to bottom) &/E, = 2.3, 0.9, 0.85, 0.63, 0.55, and 0.23.

In the charge picture, it is also understandable that ofset charges favor s u p e r c o n d u c t i ~ i t y .Offset ~ ~ ~ ~ ~charges ~~ are effective charges qi on the islands with magnitude smaller than 2e, so they cannot be eliminated by Cooper-pair tunneling. They change the charging part of the Hamiltonian into (1 2.48) i,j

In Appendix F we discuss how these fractional charges can be generated by charged impurities-in which case the offset charges are random- or by the application of a “gate voltage” between the array and ground. When 2 5 5 M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S . Fisher, Phys. Rev. 840,546 (1989). 2 5 6 C. Bruder, R. Fazio, A. Kampf, A. van Otterlo, and G. Schon, Physica Scripta 42, 159 (1992).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

451

generated by an externally controlled gate voltage, offset charges can be useful (and necessary) for the operation of practical devices made from several junction systems (the “electron turnstile” and the “single electron pump,” for instance; see E s t e ~ e ~ ~ In ’ ) .arrays they can play the role of a chemical potential for charges. Offset charges generally frustrate the system’s attempts to minimize the charging energy. This becomes especially clear for a diagonal capacitance matrix and uniform offset charge of magnitude e. Then the states with zero and one Cooper pair on the islands have equal energy, a degeneracy that hinders charge order so much that superconductivity persists even for infinitesimal EJE,,,. In Fig. 88 the influence of uniform offset charges on the mean-field phase boundary is shown for a diagonal (C = 0) capacitance matrix. For a nondiagonal capacitance matrix, the phase diagram becomes much more complicated as a function of uniform offset charge. In this case, at zero temperature, an intricate set of lobe-like insulating regions develops in the plane spanned by E J E , and offset charge256because of commensurability effects between the charge distribution and the lattice structure.258 257

D. Esthe, in Single Charge nnneling, eds. H. Grabert and M. H. Devoret, Plenum, New

York (1992) 109. 2 5 8 The situation is reminiscent of commensurability between the vortex configuration and the lattice structure in a classical, magnetically frustrated array.

1.5

5

4

1.0

\ L 4

19

Y

0.5

0.0 0.0

0.5

1.o EdEm

1.5

FIG.88. Dependence of the mean-field phase boundary on the offset charge, for a diagonal capacitance matrix. The curves correspond (from right to left) to offset charge 0,0.2e, 0.4e, 0.6e, 0.8e. and e.

452

R. S. NEWROCK ET AL

In terms of the enhancement of superconductivity by offset charges, one can understand a re-entrant bulge of the phase boundary found by Simanek242 and by Maekawa et al.259 for a diagonal capacitance matrix, where the re-entrance due to Efetov's screening effect is absent. These authors admitted as charges all integer multiples of e, not just multiples of the Cooper-pair charge, 2e. This means that both 271 and 471 periodic eigenfunctions of Eq. (12.39) are taken into account, as was pointed out by SimanekZ5' and Imry and Strongin.260 Since the Josephson term in Eq. (12.25) can only change the charges in units of 2e, oddly charged states play the role of offset charges for the manifold of evenly charged states, and vice versa. At zero temperature, odd charges &re completely suppressed electrostatically in favor of the evenly charged state Qi= 0. They appear as the temperature is increased and, by acting like offset charges, enhance superconductivity, thus giving rise to a re-entrant bulge. However, the authors neglected the fact that creation of every pair of odd charges requires the pair-breaking energy 2A on top of the electrostatic energy, which for A >> E m effectively eliminates odd charges at low temperature. Therefore, the reentrant bulge found by Simanek242and by Maekawa et al.259is not correct in general. In contrast to uniform offset charges due to a gate voltage, those generated by impurities are distributed randomly. They introduce disorder in an artificially fabricated, regular array even if its junction parameters are very uniform. Although this type of disorder is probably a generic feature, it is difficult to detect experimentally. For theoretical work including disorder see Fisher et al.255and Serrensen et THEORY The self(v) VARIATIONAL IMPROVEMENT OF THE MEAN-FIELD consistent mean-field theory described earlier fails in the limit of low self-capacitance, C,/C -+ 0. The essence of a mean-field theory, however, is not the self-consistency argument used above to determine the expectation value (cos 4 ) . The important point is to replace the interacting system by a noninteracting one, with a coupling to an effective mean field representing the actual interaction. The most realistic value of this mean field can be determined using the variational principle for the free energy. Frequently e.g., for the Weiss theory of ferromagnetism and the BCS theory of superconductivity- one obtains the same result as follows from the selfconsistency argument. This is also the case for our junction arrays if the capacitance matrix is diagonal.262However, as pointed out by Kissner and 259

260 26' 262

S. Maekawa, H. Fukuyama, and S. Kobayashi, Sol. State Comm. 37,45 (1980). Y. Imry and M. Strongin, Phys. Rev. 824,6353 (1981). E. S. Ssrensen, M. Wallin, S. M. Girvin, and A. P. Young, Phys. Rev. Lett. 69,828 (1992). E. Simanek, Phys. Rev. B32, 500 (1985).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

453

E ~ k e r n , ’the ~ ~two approaches are not equivalent for a general capacitance matrix, and the unphysical singularity in the limit C,/C -,0 disappears when the mean-field theory is based on the variational principle. The variational principle is based on the Gibbs-Bogoliubov inequality,

d

FTrial

+ ( H - HTrial)Trial,

(12.49)

~). for the free energy F = -k,Tln Trexp(-/?H) (cf. e.g. F e ~ n m a n ’ ~ This basic inequality of statistical mechanics holds for an arbitrary trial Hamiltonian HTrial;the average (...)Trialdenotes Tr[exp(. . . -/?HTrial)/Tr[exp( - b H T r i a l l and FTrial

- k,T

In Tr exp( - /?HTriaJ.

The best possible effective external potential is one that minimizes the right-hand side of the Gibbs-Bogoliubov inequality (Eq. 12.49). As a “test function” to be optimized with the aid of the variational principle, we take the mean-field Hamiltonian 1 =-CQiC,’Qj 2 ij

H ~ ~ i ~ l

-

(12.50)

ECCOS~~. i

The prefactor E of the potential term is to be determined variationally. This gives

yielding for the variational parameter E the implicit equation (12.52) where j is any nearest neighbor of i. 263 264

J. G. Kissner and U. Eckern, Z . Phys. B91, 155 (1993). R. P. Feynman, Statistical Mechanics, W. A. Benjamin, Reading, Mass. (1972), 67

454

R. S. NEWROCK ET AL.

If the capacitance matrix is diagonal, the mean-field Hamiltonian leaves different sites completely uncoupled. We then have (cos(4j - 4i))Trial

=

(cos

4i cos 4 j ) T r i a l + (sin 4i sin 4j)Trial

= (cos 4i)+rial,

(12.53) where we used the equivalence of all sites and again chose (sin 4i)= 0 (one possible realization of the spontaneous symmetry breaking). For a diagonal capacitance matrix, then, = zEJ(cos 4i)Trial

(12.54)

holds, which shows that in this case our earlier mean-field Hamiltonian Eq. (12.37) is optimal in the sense of the variational principle. In general, however, different islands are coupled electrostatically. As we did earlier, we can find the phase boundary by taking the limit E + 0 (i.e., let the effective coupling vanish) of Eq. (12.52). This gives253

4Em

+"(( 2

1 4Ed - 2 4 6 - vj) 4E, - 2ev 1

4Em-2evj4Ed+2e(K-vj (12.55)

wherej is any nearest neighbor of i, and the monopole energy Emand the dipole energy Ed are given in Eqs. (12.45) and (12.32), respectively. At zero temperature, the formula reduces to263

4 EJ=

2Em 1 + 2Em/Ed'

(12.56)

In contrast to the self-consistent mean-field theory, the variationally improved theory predicts the finite value E, = Ed for the Josephson energy at the T = 0 phase boundary of a square lattice (z = 4) in the limit C, -,0, Em-, co. This agrees within 10% with the experimental result by van der Zant et al.265(see Fig. 84). The predictions for E , at the T = 0 phase boundary of the variationally enhanced and the self-consistent mean-field theories are compared in Fig. 89. 26s H. S. J. van der Zant, W. J. Elion, L. J. Geerligs, and J. E. Mooij, Phys. Rev. 854, 10081 (1996).

TWO-DIMENSIONAL PHYSICS O F JOSEPHSON JUNCTION ARRAYS

455

We conclude by mentioning that the variational principle has also been applied to the present problem with a trial Hamiltonian quadratic in the phases, the so-called “self-consistent harmonic approximation” (SCHA)266*2673268,269 This has the advantage of simplicity, but it of course sacrifices the periodic character of the Josephson potential. Near the phase boundary, where fluctuations of the Josephson phase are large, the results of the SCHA are therefore less reliable than that of the variationally improved mean-field theory with the sinusoidal potential.

34. THERESISTANCEOF ARRAYS a. The Resistance as Control Parameter Prior to experiments on the quantum phase transition in regular, lithographically manufactured Josephson junction arrays, a number of studies of thin granular j l m s were conducted, which showed qualitatively the same 266

267

E. Simanek, Phys. Rev. B22, 459 (1980). S. G. Akopov and Yu. E. Lozovik, Sov.J. Low Temp. Phys. 7, 258 (1981) ( F i z . Nizk. Temp.

7, 521 (1981)). 2 6 8 D. M. Wood and D. Stroud, Phys. Rev. B25, 1600 (1982). 269 R. S. Fishman and D. Stroud, Phys. Rev. B38, 290 (1988).

3

I

I

I

I

600

800

4

6

C .4 .-6 CI

2 4 ’

2

\

4

5

4

.d

z

1

0

I

c v)

4 ’ 0 0

200

400

1000

c/c, FIG. 89. Comparison of the Josephson energy at the T = 0 phase transition according to self-consistent and variationally improved mean-field theory. The former diverges logarithmically for C / C , + c ~ .

456

R. S. NEWROCK ET AL.

temperature dependence of the resistance.’ 7 0 , 2 7 1 * 2 3,2 74*2 7 6 sufficiently thick films become superconducting at low temperature, but very thin films stay resistive and become insulating as T + 0. This was sometimes considered as a manifestation of the quantum-fluctuation-induced phase transition discussed above and sometimes as being related to electron (not Cooper-pair) localization; at other times it was interpreted along the lines of percolation theory. An especially extensive study of various film materials (Jaeger et al.275 indicated that the normal-state sheet resistance might be the control parameter that decides whether a film becomes superconducting or insulating at T = 0. Figure 90 shows examples of the experimental results. The critical value of this resistance was found to lie in the vicinity of ’9’

R,

E h/4e2 TZ

6.5kS1

5 9 2

(12.57)

independent of the detailed microscopic structure and the material of the film. This is exciting not only because of the possible universality of the critical resistance but also because the resistance is one of the few properties that can be measured easily (in contrast to the capacitance matrix, for instance). Orr et al.277suggested that the films should be thought of as arrays of resistively shunted junctions, and that it is the value of the shunt resistance that matters for the phase transition. It is known that a resistive shunt tends to suppress quantum fluctuations of the Josephson phase; the matter was reviewed by Schon and Zaikin.”* A qualitative, “hand-waving’’ argument for a single junction goes as follows. The energy scale for quantum fluctuations is E = (2e)2/2C. Quantum-mechanically, this energy corresponds to a time uncertainty 6 t = A/E, which has to be compared to the RC discharge time via the shunt. If the discharge time is shorter than 6t, i.e., for R < h/47ce2, it dominates the quantum-mechanical uncertainty and quantum fluctuations are suppressed. The idea that dissipation might be crucial in determining the T = 0 phase boundary gave rise to a wave of theoretical papers that tried to

’’’ A. F. Hebard and J. M . Vandenberg, Phys. Rev. Lett. 44, 50 (1980). 271 272

2’3

S. Kobayashi, Y. Tada, and W. Sasaki, Physica B107, 129 (1981). A. E. White, R. C. Dynes, and J. P. Garno, Phys. Rev. B33, 3549 (1986). N. Yoshikawa, T. Akeyoshi, M. Kojima, and M. Sugahara, Jap. J . App. Phys. 26, 949

(1987). 274S.Kobayashi and F. Komori, J . Phys. SOC.Jap. 57, 1884 (1988). 2 7 5 H. M. Jaeger, D. B. Haviland, B. G. Orr, and A. M. Goldman, Phys. Rev. 840, 182 (1989). 2 7 6 R. P. Barber, Jr., and R. E. Glover 111, Phys. Rev. 842, 6754 (199Oj. 2 7 7 B. G . Orr, H. M. Jaeger, A. M. Goldman, and C. G. Kuper, Phys. Rev. Lett. 56, 378 (1986).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON J U N C T I O N ARRAYS

0

5 10 T (K)

457

IS

Ca 2 0

I

*-

s

lo

I

T (K)

0

5

I0

i

T (K)

FIG.90. Temperature dependence of the resistance for granular films of aluminum. indium, gallium, and lead. (From Ref. 275, Fig. 2.)

458

R. S. NEWROCK ET AL.

put it on a more solid footing and extended it in various directions. 2 7 8 . 2 79.2 8 0 , 2 8 1.282.28 3 2 84.28 5 , 2 8 6 . 28 7 These theories will not be reviewed in detail here because the then necessary discussion of the quantum description of dissipative systems would take us too far astray. Also, their applicability is limited to systems with a dissipative shunt, either ohmic or due to quasiparticle tunneling. The high-quality SIS junction arrays used by van der Zant et al.,238 for example, do not fall into this category. Nevertheless, these arrays qualitatively show the same behavior as thin films (see Fig. 87). An interesting alternative link between the normal-state sheet resistance and the superconductor-insulator phase transition at zero temperature was suggested by Chakravarty et a1.288and by Ferrell and M i r h a ~ h e m . ’The ~~ authors recalled that the virtual tunneling of quasiparticles leads to an increase in the junction capacitance C by,290,291

6C=-

3nA 32AR, ’

(12.58)

where A is the superconducting gap and R , is the junctions’ normal-state resistance (this differs from the previously used R, in that the latter includes a possible shunt resistance). Using the Ambegaokar-Baratoff formula292 formula,

IEAA E , = - (at T = 0), 4R, e2

(12.59)

M. P. A. Fisher, Phys. Rev. Lett. 57, 885 (1986). M. P. A. Fisher, Phys. Rev. 836, 1917 (1987). 280 E. Simanek and R. Brown, Phys. Rev. 834, 3495 (1986). 2 8 1 S. Chakravarty, G. L. Ingold, S. Kivelson, and A. Luther, Phys. Rev. Lett. 56, 2303 (1986). 2 8 2 W. Zwerger, Sol. State Comm. 62, 285 (1987). 2 8 3 W. Zwerger, Physica 8152, 236 (1988). 284 A. Kampf and G. Schon, Phys. Rev. 836, 3651 (1987). 2 8 5 A. Kampf and G. Schon, Physica 8152,239 (1988). 286 A. D. Zaikin, Physica 8152, 251 (1988). 2 8 7 G. T. Zimanyi, Physica 8152, 233 (1988). 288 S. Chakravarty, G. L. Ingold, S. Kivelson, and G. Zimanyi, Phys. Rev. 837, 3283 (1988). 2 8 9 R. A. Ferrell and B. Mirhashem, Phys. Rev. 837, 648 (1988). 2 9 0 V. U. Arnbegaokar, U. Eckern, and G. Schon, Phys. Rev. Lett. 48, 1745 (1982). 2 9 1 A. I. Larkin and Yu. N. Ovchinnikov, Phys. Rev. 828, 6281 (1983). 2 9 2 V. U. Ambegaokar and A. Baratoff, Phys. Rev. Lett. 10, 486 (1963); Erratum, Phys. Rev. Lett. 11, 104 (1963). 278

279

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

459

we can express the Josephson energy in terms of 6C and R,/R,, (12.60) If 6C is much larger than C and C,, the monopole energy E m and the dipole energy Ed satisfy Em >> Ed and Ed cc 1/6C, so that the right-hand side of Eq. (12.56) for the T = 0 phase boundary becomes equal to Ed, and 6C drops out of the equation. The phase boundary can then be expressed purely in terms of the resistance ratio R,/R,. For a square lattice ( Z = 4) one has R,

= $RQ

z 4kR

(for z

= 4).

(12.61)

While this is close to the “critical” R , of many experiments, its general validity is not clear. On the one hand, the capacitance C is not always dominated by 6C; for instance, van der Zant, Geerligs, and M ~ o i j ~ ~ ’ estimated 6C/C z 0.1 for their lithographically fabricated arrays. On the other hand, Eq. (12.58)for the capacitance renormalization has been derived in the so-called adiabatic limit, A << e2/2C,and it may be trusted quantitatively only as long as 6C << C.293 After the initial enthusiasm for the normal-state sheet resistance as control parameter had calmed down, Fisher, Grinstein, and G i r ~ i n ~ ~ ~ concluded on the basis of scaling arguments that instead the resistance right at the T = 0 superconductor-insulator transition should have a universal value. This new idea has been worked out in various ways. In this review, we discuss in some detail only a comparatively simple, approximate way to calculate the resistance at the phase boundary. To this end, first a GinzburgLandau description of the array is derived. b. Ginzburg-Landau Description We now want to calculate the resistivity using a Ginzburg-Landau formulation. A continuous phase transition is characterized by an order parameter whose average is nonzero in one phase but which tends to zero continuously as the phase boundary is approached and vanishes identically in the other phase. The idea is to take the trace of e P B Hover all variables but the order parameter. The result can then be used to define an effective free energy

’” U. Geigenmiiller and M. Ueda, Phys. Rev. B50,9369 (1994). 294

M. P. A. Fisher, G. Grinstein, and S. M. Girvin, Phys. Rev. Lett. 64, 587 (1990).

460

R. S. NEWROCK ET AL.

functional that depends only on the order parameter. Since the latter is small near the phase boundary (at least its average is, and one usually assumes that essentially all contributing values of the order parameter are), one can simplify further by expanding in powers of the order parameter. In the mean-field theory, we introduced as an order parameter cos +i (making a specificchoice of spontaneous symmetry breaking). The Josephson interaction - E, C O S (-~4j) ~ of site j with site i was approximated by - E,(cos 4i) cos 4 j . To obtain the Ginzburg-Landau description, we do not want to replace the order parameter cos 4iwith its average, but rather keep it as variable. A convenient, standard way of achieving this is the so-called “Hubbard-Stratonovich transformation,” first applied to the present problem by D ~ n i a c h . ~ ~ ~ As a preparatory step we write the Josephson coupling in the form H,

= - E,

c COS(+~ 4 . -

3

-

(0)

1 A..) u = - - E,C rijeidiJ(j(j, (12.62) 2 ij

where

r..=

1 if i and j are neighboring islands

(12.63)

(12.64) Here we have reintroduced the vector potential A in the usual fashion (see Eq. (3.7)) via its line integrals, A i j = (271/@,)J { A.dr. This is done for technical reasons. Later on we want to calculate the conductivity from a formula that includes derivatives with respect to A , and we will need to express the electric field in terms of the vector potential. We also introduce the square root W of the matrix coupling the ( in Eq. (12.62), 1 (12.65) K~ = EJTijeiAiJ,

c wj 1

so that

H,

=

-

C
(12.66)

ijm

Now the well-known formula296

J

J

S. Doniach, Phys. Rev. 824, 5063 (1981). This formula also holds for operator-valued (, 5’ if, as in the present case, these operators commute and have a common complete set of eigenstates. 295

296

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

461

for Gaussian integrals is needed. Application of this formula to the partition function yields Z = Z,(T exp[ - J ! f l d ~ H J ( z ) / h ] ) yields

,

Z with

= const. x

Z,

s

D Re $0 Im $eCBF[IL1

(12.68)

(12.69) where Z , = Tre-PHo,i.e., the partition function for H , , and T is a timeordering operator (not the temperature, that is in /3). The right-hand side of Eq. (12.68) contains a path integral, indicated by 10, that is, separate integrations for each point of time in (0,hB). The free energy functional F still contains a thermal average (...)o with respect to the charging Hamiltonian. The progress we made is that the quantity averaged now corresponds to (the time-ordered exponent of) a system with no mutual Josephson interaction between sites, but only interaction with an “external field” $!. This reduction of an interacting system to a noninteracting system in an external field is also at the heart of mean-field theory. In contrast to the latter, however, the expression Eq. (12.68) is still exact. The price we have to pay is the extra functional integration over the order parameter field $. The next step in the Ginzburg-Landau procedure is to expand the free energy in powers of the order parameter field $ up to quartic terms. On the mean-field level one works with the most probable $-the one that minimizes the free energy functional. To locate the phase boundary on this level of description it suffices to include quadratic terms only:

(12.70)

462

R. S. NEWROCK ET AL.

For sufficiently small E, this functional is positive definite, and the minimum of Eq. (12.70) is attained at $ = 0. The phase transition to superconductivity occurs with increasing E,, as the first eigenvalue of the quadratic form (Eq. (12.70)) becomes equal to zero. In the absence of a magnetic field (Amfl= 0), the free energy will be minimum for uniform $ for symmetry reasons. The condition for the phase transition then becomes (12.71) This is equivalent to the relation (Eq. (12:43)) for the mean-field phase boundary found earlier. If one wants to study the system in the superconducting phase away from the phase transition, or study renormalization by fluctuations, one needs to go beyond the quadratic approximation for F and take at least the term quartic in $ into account. The validity of Eq. (12.70) is restricted to small values of C/C,, since the Ginzburg-Landau energy functional obtained by a straightforward Taylor expansion in powers of $ does not follow from the Gibbs-Bogoliubov variational principle. This not only is true for the quadratic term, Eq. (12.70) (see also Subsection 33b.9, but holds afortiori for the quartic one, which becomes negative263for C/C, larger than about 2. Using the variational principle, these authors also derived an effective Ginzburg-Landau free energy with properly behaved coefficients for all capacitance ratios. Here we follow D ~ n i a c h ~and ~ ’ Bruder et al.296 and use Eq. (12.70), keeping in mind its limited validity, in order to calculate the conductivity of the array in Gaussian approximation. We are interested in the longwavelength, low-frequency behavior of the system -in particular, the conductivity-in at most weak electric and magnetic fields. For $ that varies slowly in space297we can apply a continuum approximation and write

-

-

1{$*(r, z) 1 2 r

d = a2=,asy

[

2n

i-A(r).d

e%

$(r

+ d, z) - 2$(r, z’)

297 More precisely, fields for which $*(r,, 5 ) exp(id,,J$(r,, z’) changes little for lr, - r,l of order a. In regions where lAla is large (and these exist even for small magnetic fields if the array is large), this means that the gauge-invariant gradient, (V + i2sA/@,)$, must be small, rather than just V$.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS 2R

+ e -i- *o

A(r) .d

1+

$(r - d, r')

463

I

4$*(r, r)$(r, r')

$*(r, T')

+ -a41+*(r, r)$(r, r') (12.72)

If $ contains only low-frequency components, one can further approximate:

(12.73) where (12.74) In the spirit of Ginzburg-Landau theory we keep at most second-order derivatives and arrive at the so-called coarse-grained form of the free energy functional:

(12.75) c. Resistance at the T = 0 Phase Boundary The considerably simplified free energy functional, Eq. (12.79, can now be used to calculate the conductivity. The general formula linking the conductivity tensor to variational derivatives of the partition function with respect to the vector potential (which for this purpose has formally been given a dependence on time) is derived in Appendix H. Applying it to our GinzburgLandau partition function, we get for the wave vector and frequencydependent conductivity tensor om, on the positive imaginary frequency axis

464

(we calculate the real axis)

R. S . NEWROCK ET AL. 0

on the imaginary axis and later analytically continue it to

Here we abbreviated: (1 2.77)

which has the familiar structure of a current density. The evaluation of the correlation functions in Eq. (12.76) is simple, because the statistical weight exp( -BF,,) is Gaussian. Some details are provided in Appendix I. The resulting conductivity reads, in the limit A -,0, k -,0,

x

s

1

d ' k ' v ; [G(k',o,)- G(k',o, - w,)]G(k',o,),(12.78)

with the Fourier-transformed correlation function

Here we have specialized to the case of vanishing offset charges, when g 1 = 0 (see Appendix F). After the frequency summation in Eq. (12.78) has been done using the standard contour integration method (see, e.g., Rick-

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

465

a y ~ e n ~one ~ ~can ) , perform the analytic continuation to real frequencies. For T = 0 the remaining k integration is elementary and one obtains2s9

8 h

with the cut-off frequency (12.81) In the first part of the last equation we considered only nearest-neighbor and self-capacitances; in the second part we set the self-capacitance to zero and so it refers to the case of a diagonal capacitance matrix. Equation (12.80) is a very interesting result as it gives us a “universal” conductivity at the phase boundary. The real and imaginary parts of the conductivity are plotted in Fig. 91. As expected, at T = 0 the DC conductivity vanishes 298

G . Rickayzen, Green’s Functions und Condensed Mutter, Academic Press, London (1980).

b

-0.5

4

~

-1.0

I

1

-2

-1

,.... I.

I

I

I

0

1

2

0/ w c

FIG. 91. Real (solid line) and imaginary (dashed line) part of the frequency-dependent conductivity at T = 0, calculated in the insulating phase using the Gaussian approximation for the free energy and normalized with u* given by Eq. (12.82). (After Ref. 300, Fig. 2.)

466

R. S. NEWROCK ET AL.

below the phase transition we are considering. Above the phase transition the system is superconducting (then, of course, the quadratic approximation of F does not suffice). The DC sheet conductivity a* right at the phase transition, however, has the finite value g*

n

0.39

8RQ

RQ

= -x-

(12.82)

(corresponding to a resistance of 16.4 kR), which contains only the resistance quantum and a numerical factor and no parameters specific to the array. In this sense, the result is universal. For low frequencies, the conductivity, a M -i(2e2/3h)o,, behaves like that of a capacitor, whose capacitance diverges as the phase transition is approached. For 10) > o,,when the energy quanta provided by the AC electric field can overcome the charging energy barrier (renormalized by the Josephson coupling) that suppresses the free motion of Cooper pairs, the real part of the conductivity is nonzero. This value (Eq. (12.82)) for a* was first obtained by Cha et al.,299who also calculated the lowest-order correction to the Gaussian approximation in a formal expansion in the inverse dimensionality of the order parameter. They found a correction factor (1 - 32/9n2) x 0.64, yielding a* x 0.254/ RQ x 1/25 kR. Cha et al. also performed a Monte-Carlo calculation of a* with finite-size scaling, with the result a* = (0.285 f 0.02)/RQ( x 1/23 kR). Other generalizations include disorder and long-range Coulomb interactions. Including disorder, Serrensen et a1.261found by Monte-Carlo methods (and using the Villain approximation) o* = (0.14 f 0.03)/RQ( x 1/46 kR) for a short-ranged Coulomb interaction, and a* = (0.55 & O.l)/RQ( x 1/12 kR) for a long-ranged, l/r Coulomb interaction (different from the usual capacitance matrix model (Eq. (E.l)) in arrays). Within the Gaussian approximation, the calculation of the conductivity for the free-energy functional has also been extended in various directions. Introducing dissipation in a phenomenological way, van Otterlo et al.300 found a* M 0.117/RQ( x 1/55 kR). This is again independent of the phenomenological damping constant, as long as the latter does not vanish. Recently, Wagenblast and S ~ h o n ~investigated ~' the system with a microscopic, Caldeira-Leggett-type oscillator model for the dissipation. Once more parameter independence of a* was found but again a different value for a*RQ. 299 M.-C. Cha, M. P. A. Fisher, S. M. Girvin, M. Wallin, and A. P. Young, Phys. Reo. 844, 6883 (1991). 300 A. van Otterlo, K.-H. Wagenblast, R. Fazio, and G. Schon, Phys. Reo. 848,3316 (1993). 301 K.-H. Wagenblast and G. Schon, Phys. Reo. Lett. 74, 1695 (1995).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

467

In other work, Fazio and Zappala302 obtained yet another value for CT* E expansion, and Batrouni et al.303obtained a value (=0.118/RQ)from a world-line Monte-Carlo calculation. Runge and Zimanyi304 obtained a value similar to that of Ssrensen et a1.261from an exact diagonalization. Wagenblast et al.305obtained a continuous sequence of universality classes with different universal CT*,depending on the microscopics of the films, as modeled by a local damping of the Caldeira-Leggett type. The conclusion is that universality of the conductivity at the phase transition exists within classes of systems, but that there are manifold possible universality classes. The experiments with granular films typically gave critical conductances higher than the theoretical results for o*. For arrays there are at present too few measurements in the close vicinity of the separatrix for a detailed comparison, but a value CT*x 1/50 kR does not appear unreasonable (see Fig. 87). This is close to (some of) the theoretical results. ( =0.315/RQ)using the

XIII. Afterword

Over the past decades, condensed matter physicists have become increasingly interested in studying systems that are in one way or another artificial. Starting from the study of pure materials, the field has broadened to include equilibrium alloys and composites, out-of-equilibrium materials such as glasses and other alloys, and, finally, artificially fabricated materials including multilayers and heterostructures, monolayers, quantum wells, quantum dots, nanotubes, and mesoscopic systems. Josephson junction arrays fall near the extreme end of the list. They are, after all, just integrated circuits containing up to millions of Josephson junctions. We hope that this review has made clear that, while just integrated circuits, the nature of the individual junctions and the way they interact lead to a wide range of interesting physics. One of the first, and still one of the most striking, examples was that a chip with a million junctions undergoes a transition from a low-temperature state, where the junctions are correlated over long distances, to a high-temperature state, where correlations are R. Fazio and D. Zappala, Phys. Rev. 853, R8883 (1996). G. G. Batrouni, B. Larson, R. T. Scalettar, J. Tobochnik, and J. Wang, Phys. Rev. 848, 9628 (1993). 304 K. J. Runge and G. T. Zimanyi, Phys. Rev. 849, 15212 (1994). ' O 5 K.-H. Wagenblast, A. van Otterlo, G. Schon, and G. T. Zimanyi, Phys. Rev. Lett. 78, 1779 (1997). 302

303

468

R. S . NEWROCK ET AL.

short-ranged. This transition is an ordering of the phases of the superconducting order parameters in the system. The word “phase” acquires an interesting double meaning in this context. In this limit, the phase is a semi-classical variable, so while the underlying superconductivity is a quantum effect, the phase transition is quite classical. When the parameters of the array are changed, and the junctions are made small, the semi-classical variables become fully quantum-mechanical. In this limit, charge and phase are both important, and issues related to localization, superconductor-insulator transitions, and fundamental questions regarding quantum systems coupled to a dissipative environment come into play. The third area covered in this review was the dynamics of arrays. Individual junctions are nonlinear oscillators. (The equations describing them are the same as the pendulum equations when linear approximations are not made.) In this context, Josephson junction arrays are systems of large numbers of coupled nonlinear oscillators, and they have rich and often surprising behavior. This last area may also lead to practical applications: Arrays are potentially powerful enough to make useful far-infrared oscillators. Active research continues in most of the areas covered in this review. In addition, there has been considerable continuing research in areas we have not covered, including one-dimensional arrays (see, for example, the work of Jim Lukens, Kostya Likharev, and their coworkers), “natural” arrays in high-T, superconductors (Paul Miiller, Alexey Ustinov, their co-workers, and others); coupled long junctions (Neils Pederson, Gianni Costabile, their co-workers, and others); small arrays in interesting geometries (Terry Orlando, Herre van der Zant, Hans Mooij, their co-workers, and others); arrays of thin superconducting wires (Paul Chaikin, Alan Goldman, their coworkers, and others); and granular superconductors (too numerous to list). These are all are closely related to two-dimensional arrays, but were not covered here because of time limitations. All of these topics deserve reviews of their own. Acknowledgments and Apologia We are extremely grateful to a large number of people who have contributed their time to reading, correcting, and providing figures and advice. These include Paola Barbara, Brian Bedard, Fred Cawthorne, Said Elhamri, Paul Esposito, Richard Gass, Luis Gomez, Art Hebard, Steve Herbert, Jenny Holzer, Aaron Nielsen, Anne van Otterlo, Frank Pinski, Hugo Romero, Doug Strachan, and Branimir Vasilic. We acknowledge useful conversations

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

469

with Bert Halperin, Mark Jarrell, Mike Kosterlitz, Piero Martinoli, Petter Minnhagen, David Nelson, Subadhra Shenoy, Joe Straley, and Mike Tinkham. Special thanks go to Paula Davis and, especially, to Angelica Govaert for assistance with the mechanical details of producing this manuscript. We are also grateful for the support offered by the National Science Foundation (DMR-9732800 [CJL] and DMR 9801825 [RSN]), the AFOSR (F496209810072 [CJL]), the ONR (N00014-95-1-0083 [RSN]), and the U S . Air Force (LA1 OLA 96-5000-5529 Sub AF [RSN] and UTC 97-8402-18-15C1 Sub AF [RSN]). This review was begun early in the present decade. A number of factors have made the writing unusually protracted. The authors are spread over three continents, and two of them, UG and MO, responded to the siren call of industry in the middle of writing. The plain truth is, however, that most of the delay came from the breadth of the subject: It took only a few months before we realized that we had severely underestimated the work involved. At times it seemed that new work was appearing faster than we could review it, and we had to impose rather arbitrary time cut-offs on the various areas. In any case,

‘. . . so there ain’t nothing more to write about, and I am rotten glad of it, because if I’d a knowed what a trouble it was to make a book I wouldn’t a tackled it and aint’t agoing to no more.’ Mark Twain, Huckleberry Finn Appendix A. Correlation Functions: Vortices and Spin Waves

In this appendix we closely follow the excellent treatment of the subject in Le B e l l a ~ , ~adapted ’~ to array language. Perhaps the easiest way to understand the correlation functions involved in the Kosterlitz-Thouless phase transition is from the point of view of an array of spins. We begin with the Hamiltonian, Eq. (3.7), assuming zero magnetic field, B = 0 and A = 0. We represent the phase of the ith island which has unit length and makes an angle 4i with the x-axis. by a vector, Si, Sican be thought of as a spin, transforming the two-dimensional Josephsoncoupled array into a two-dimensional lattice of spins. These spins are free to rotate about an axis perpendicular to the xy plane. (Note that both the angle of the spin and the phase of the superconducting order parameter of 306 M. Le Bellac, Quantum Stutisricd Field Theory, Oxford Science Publications, Clarendon Press, Oxford (1991).

470

R. S. NEWROCK ET AL.

an island can only vary between 0 and 2n.) We further assume only nearest-neighbor interactions. This gives us a Hamiltonian H

=

-E,

1 COS(C)~- C)i) = - E , 1 S i . S j , (ii)

(ii)

which is invariant under rotation in the plane (it has O(2) symmetry). E,, the Josephson junction coupling energy, now represents the spin-spin interaction energy. The rightmost term, the Hamiltonian of a lattice of classical interacting spins confined to the XY plane, with only nearestneighbor interactions, is known as the XY Hamiltonian. It is clearly isomorphic to the Hamiltonian for an array of Josephson junctions in zero magnetic field. We will consider excitations and fluctuations of the spins of this Hamiltonian. The question, then, is the existence of long-range order in such a system and the effects upon that order of spin waves and vortices. Historically, there were several theoretical predictions about the luck of long-range order in two-dimensional systems. PeierlsZ4' argued that the thermal motion of long-wavelength phonons would destroy conventional long-range order in a two-dimensional crystal. He showed that in two dimensions the mean-square deviation of an atom from its equilibrium lattice position increases logarithmically with the system size, resulting in Bragg peaks in an X-ray diffraction pattern that are not sharp. The absence of this long-range order in two dimensions was rigorously shown by M e r m i ~ ~ Mermin .~'~ and Wagnerz6 showed that there is no spontaneous magnetization in a two-dimensional magnet if the spins have more than one degree of freedom. Similarly, Hohenberg3'* showed that the average value of the order parameter in a two-dimensional Bose fluid is zero. On the other hand, for the XY Hamiltonian WegnerZ7and Berezinskiiz4 found that the magnetization is proportional to H q (0 < q < 1) at low temperatures. They also indicated the possible existence of a high-temperature regime where the magnetization is proportional to H . In all of these cases two-point correlation functions were used to determine the presence or absence of long-range order. There was the possibility that a phase transition could still exist; that is, there is the somewhat subtle point that the absence of long-range order does not necessarily imply the absence of a phase transition. Such a phase transition would be from a disordered high-temperature state to an ordered, but not infinite-range, low-temperature state. Kosterlitz and Thouless showed that this is indeed correct by considering 307

308

N. D. Mermin, Phys. Rev. 176, 250 (1968). P. C. Hohenberg, Phys. Rev. 158,383 (1967).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

471

a very different aspect of two-dimensional systems, defining a "quasi-longrange order". They called this topological long-range order and applied it to two-dimensional crystals, neutral superfluids, and XY magnets, but not to two-dimensional superconductors or the isotropic two-dimensional Heisenberg magnet. Beasley et al.309 and Doniach and Huberman310 demonstrated that the theory can be extended to superconductors in certain cases. As discussed in Section 111.5, a vortex in the XY model is a configuration of the angles of the spins such that the sum of the angles between the spins around a closed loop is +2n (see Fig. 14). Vortices are short-wavelength fluctuations, as opposed to spin waves, which are long-wavelength fluctuations. The inclusion of vortices creates a change in the two-point spin-spin correlation function, making a phase transition possible. From Eq. (A.l) we can write a partition function: Z

=Jozxnd4,exp(a I

gj)).

C O S ( ~ -~

~ B < iT. j )

At high temperatures the exponential can be expanded in powers of E,/kBT; we consider only the terms with the lowest power. For the correlation function we have, for two spins located at sites 0 and 1,

The second equality comes about because Eq. (A.l) is invariant under 4i +-4i. Next we note that

J

d4=2n 0

and

J

d4ei4 = 0. 0

This tells us that if the integral in the partition function, Eq. (A.2), is to be nonzero, we need a factor of e-j4 for every eiO.For every nonzero term in Eq. (A.2), we can find a path through the lattice of spins going from spin 0 to spin 1 (Fig. 92), and along that path we see that

Every parenthesis in this equation comes from a factor of 309 310

C O S ( ~-~

M. R. Beasley, J. E. Mooij, and T. P. Orlando, Phys. Rev. Lett. 42, 1165 (1979). S. Doniach and B. A. Huberman, Phys. Rev. Lett. 42, 1169 (1979).

4j)

472

f g

d

0

h

e

a

FIG. 92. A path through a lattice o f spins. (After Ref. 306.)

associated with the energy of a junction. Each such term contributes a factor of order (EJ/kBT)N,where N is the number of junctions along the path between spin 0 and spin 1. The dominant term in the high-temperature limit comes from the shortest path between the two spins. For large r = r, - r I , this path length is approximately r/a and

We see that the correlation function has an exponential dependence, with a correlation length

this exponential dependence means that we have short-range order, indicating a disordered phase at high temperatures. At low temperatures, we can make the (reasonable) assumption that the dominant fluctuations have long wavelengths. That means that the phase changes little from island to island, that we can write C O S (-~4j) ~ % 1(1/2)(4 - 4j)z,and that

We use the notation ~3,,4~ = 4i+p - 4i, where p = 1 or 2 corresponds to motion through the lattice in the x or y directions and 4i+,,is a nearest

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

neighbor of

4,. We then rewrite the Hamiltonian

where, when we go to the continuum limit, i

-+

473

as

x, 4,

-+

4(x) and C(6,4,)’ + B

[Vd(x)]’. The correlation function is

Note that the integral’s limits are now f co.This is a very important point, and we will revisit it because it leads directly to the Kosterlitz-Thouless phase transition. The integral in Eq. (A.lO) is Gaussian, and therefore the correlation function G,,, corresponding to H = (1/2)E (6pq5i)2,becomes306 elk (x,-x,)

(2.)’

4

-

2 cos k , u

-

2 cos k,a’

(A. 11)

Expanding the denominator in small k shows that the integrand varies as dZk/k2-that is, it diverges logarithmically. It can be regularized by subtracting 1 from the exponential, defining G(Y):

The Gaussian integral in Eq. (A.lO) yields (A.13)

where Y is a vector between the two islands being considered, 0 and 1. If we move to the continuum limit, C(Y) satisfies the two-dimensional Poisson equation -V2G(r)

= 6(Y).

(A.14)

The solution to this equation can be found by looking at a simple electrostatics problem- the electric field from a line of charge with a charge

474

R. S . NEWROCK ET AL.

density 1 per unit length. The potential in the plane from such a charge distribution is (A. 15) and from Gauss’s law,

E(r) =

~

1 2ne,r

(A.16)

we obtain the potential

1

V(r) = - -In 2KE0

r

+ constant.

(A.17)

Thus, the solution for Eq. (A.14) must be (A.18) The integral can be evaluated directly for r >> a; the constant is then - 1/2n(y + 3/2 In 2) = 1/4, where y is Euler’s constant. We thus obtain for the correlation function, for large r, (A.19) a power-law correlation function with a temperature-dependent exponent. The order with such an algebraically decaying correlation function is called quasi-long-range order because, although true long-range order (all spins aligned) is not allowed (except at T = 0), the spins in areas of various sizes can be aligned; that is, there are islands of “magnetization” of all sizes. The correlations decay very slowly-much more slowly than in the high-temperature limit, Eq. (A.6). For example, at I = lOOOa and kBT/2aE,= 0.1, the correlation function has only decreased to 0.5. This is unlike a three-dimensional magnet, where the correlations last to infinite distance; in the two-dimensional system correlations are infinite range only at zero temperature. We must then have a transition where the correlation function changes from its low-temperature power-law behavior, Eq. (A.19), with quasi-longrange order, to its high-temperature exponential behavior, Eq. (A.6), with

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

475

short-range order. One might imagine a broad crossover from long-range order to quasi-long-range order -that is, no phase transition. This was shown by Kosterlitz and Thouless not to be the case. The question to be asked is why the low-temperature expansion fails at some temperature. The reason is that as the temperature rises, the fluctuations become sufficiently large that, since the integral over the phases is between plus and minus infinity, periodicities in the phase become important. This is where Kosterlitz and Thouless’s contribution entered- they realized that the quasilong-range order at low temperatures is destroyed by topological excitations that depend on a periodicity in the phase- that is, vortices. At low temperatures the only important contributions are from long-wavelength fluctuations -spin waves. Correlations decrease algebraically, and the system is somewhat ordered with islands of “magnetization” of all sizes, but there is no spontaneous “magnetization.” Spontaneous magnetization means that, at a large separation, (So.S,) z ( S o ) @ , ) = (magnetization)2 # 0, which is a result that would be inconsistent with the lowtemperature correlation function of Eq. (A.19). As discussed in Appendices B and C, there are no free vortices, but pairs of vortices closely bound. As the temperature increases, the vortices in a pair begin to separate, until, at the transition, the separation diverges, producing free vortices that destabilize the quasi-ordering of the spin waves, and the correlation function now decreases exponentially. Appendix B. Vortex-Pair Density: The Dilute Limit The Kosterlitz-Thouless theory is only valid in the very dilute vortex pair density limit. This is needed to make certain that the lowest energy configuration is one of “dipoles”- bound vortex-antivortex pairs. As we will see, the core potential pc must be large to ensure this. By a dilute density of pairs we mean that ( r 2 ) / ( d 2 ) << 1, where r is the mean-square separation of the two vortices in a pair and d is the mean distance between pairs. We determined the mean separation between the two vortices of a pair in Section IV and obtained Eq. (4.11), rewritten here as, ( r 2 ) = a’

2 ~ / 3 E, 2 2 ~ / 3 E, 4’

where /3 = l/kBT To calculate (d)’ we first calculate the probability of finding a pair in a given area, using the energy of a vortex-antivortex pair,

476

R. S. NEWROCK ET AL.

Eq. (4.4), but including the core potentials of the vortices. The probability is found by summing the Boltzmann factor over all values of r 1 and r2 in the area

P will be of order 1 when the area of integration is (d)’, which leads to, with r = (rl - r 2 )

(Kosterlitz and T h o ~ l e s spoint ~ ~ out that this calculation is “intractable” beyond first order in ezBPc). From Eqs. (B.l) and (B.3), (I2)

(d2)

7ce2P”c nPE, - 2

To obtain a dilute pair density (small ( r 2 / d 2 ) ) ,Eq. (B.4) shows that we need Ipc( to be large (note that pc is negative); this is true as long as nPE, < 2. For the two-dimensional XY model, p, = -n2J (Kosterlitz and ThoulesZ3). For a junction array, then, pc = -7c2E,/2 and

This is much less than one (i.e., we are in the dilute limit), as can be seen by inserting typical array parameters until one is very close to TKTwhere ( r 2 ) diverges.

Appendix C. Scaling In the discussion that follows we draw heavily on arguments by Kosterlitz and T h o u l e s ~ Young,311 ,~~ Mooij,46 and Binney et We recommend Young as being especially clear. A. P. Young, J . Phys. C11, L453 (1978). J. H. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman, The Theory of Critical Phenomena, Oxford University Press, Oxford (1 992). 3’1

312

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

477

THEKOSTERLITZ-THOULES RECURSION RELATIONS As discussed in Section IV, the vortex-vortex interaction depends on the logarithm of the separation between them, Eq. (4.7). To place the emphasis on the statistical mechanics of the vortices we will employ a simplified Hamiltonian, that considers only vortex energies and neglects spin waves. That the Hamiltonian can be factored into a spin-wave-dependent portion and a vortex portion has been shown by a number of authors (see, for example, Kosterlitz and T h o ~ l e s s , ~ and ~ ) , it is easily shown that the neglected spin-wave portions of the Hamiltonian are not crucial to the arguments (Binney et al.312)We consider vortices at locations r i , defined by their vorticity, mi = f 1. In this approximation the vortex portion of the Hamiltonian can be written as

where in the first sum we count each vortex pair interaction only once. Here C , indicates the core energy of a single vortex, not a capacitance. C , reflects the fact that the energy of a single vortex is only asymptotically logarithmic in the sample size. We impose the constraint C m i = 0 because, in the thermodynamic limit, an excess vortex of one sign or another costs infinite energy. The interaction potential between two vortices can be changed by two important processes. First, as the temperature increases the number of spin waves excited also increases, as does the average momentum k of these waves. These can alter the vortex-vortex interaction. Ohta and J a s n ~ w ~ ~ looked at such spin waves from the perspective of their effect on the Kosterlitz-Thouless transition. They found that the important features of the transition are retained, although some of the features may have somewhat modified parameters. For the rest of this appendix we ignore the effects of spin waves. Second, and more important, we must consider the effect of the presence of other vortex-antivortex pairs on the interaction potential between a particular vortex pair. If we consider a vortex pair with a relatively large separation, smaller vortex pairs located in the field of the larger pair will alter (screen) the interaction potential. Because this is very similar to the manner in which the electric dipoles in a polarizable material (a dielectric) screen the interaction between two charges, it is useful (and makes the following calculations easier to understand physically) to consider a twodimensional Coulomb gas; such a gas has a Hamiltonian similar to Eq. (C.l)

478

R. S. NEWROCK ET AL.

(see, e.g., Ref. 5). We imagine a gas of “charges” qi, which interact logarithmically for r > a and which have a “hard core” repulsive potential for r < a. The Coulomb gas Hamiltonian is

where the 2n appears (instead of 4n) because the problem is two-dimensional and on1 vortex pairs are counted. We then identify Gj = q j / 2 n ~- 2nE by comparison to Eq. (C.1). This analogy allows us to exploit our understanding of simple electrostatics. Equations (C.l) and (C.2) are general but, unfortunately, not tractable. To make progress two approximations must be made. First, we assume that the dominant energies in the problem come from pairs of opposite charges; this implies that the average separation between the charges of a pair is much smaller than the average distance between pairs. For this to be true the system must be in the dilute limit-that is, the core chemical potential has to be large, as discussed in Appendix B. This approximation allows us to rewrite Eq. (C.2) as

G-d

Here p is the number of pairs of charges in the system. While this is a very informative approximation, it is much too drastic as it fails to take into account the screening of the interaction between the two members of a large pair by the smaller pairs. The rapid divergence of the mean-square separation between the members of a bound pair of vortices (Eq. (4.11)) suggested to Kosterlitz and Thouless that the important part of the interaction between different pairs of vortices can be accounted for by introducing a “dielectric” constant E (named by following the electrostatic analogy). Thus, the force between opposite charges of separation r can be written as

where the dielectric constant e(r) depends only on r and is to be determined self-consistently. This leads to an effective potential between two charges separated by a distance r,

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

479

and a modification of Eq. (C.3),

where ri = lry) - &)I. (We note again that this only works in the dilute pair density limit, since we must assume that the lowest energy state of the system is one of dipoles, or bound pairs of vortices; see Appendix B.) The “field” from a pair of charges separated by a distance on the order of a, the lattice parameter, does not extend much beyond several a. Since we are dealing with a dilute pair density, it is very unlikely that there will be another pair of vortices in the vicinity, and the dielectric constant for such a pair will not be affected by the polarizability of the other pairs; that is, E = 1 for a closely bound pair. Only the energy of a vortex-antivortex pair whose separation is greater than the mean separation will be significantly affected by other, smaller, pairs lying inside its field range. Therefore, E will become larger and larger as the separation increases. We now do the statistical mechanics using Eqs. (C.5) and (C.6). The grand partition function is

As usual, B = l/k,T while s(p) indicates a sum over all states of the system with p pairs of charges. The term (p!)’ arises because there are p indistinguishable positive charges and p indistinguishable negative charges. The second sum can be converted into an integral, yielding

The u4 terms come from converting the sum over a lattice with lattice spacing a to an integral. Every charge’s coordinate is integrated over the plane without overlapping the other charges. Keeping only the first two terms in the partition sum yields

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R. S . NEWROCK ET AL.

where we now have

The two terms in Eq. (C.9) correspond to (1) a system with no pairs and (2) a system with one pair screened by smaller pairs. Switching variables from r 1 and r2 to r1 and r = Irl = Irl - r2Jgives

ZcG = 1

:1

+ aA"

2 ~ r d r e - ~ ~ ~ ~ ,

((2.1 1)

where A is the area of the sample. Equation (C.11) is now used to determine E(r) self-consistently. We do this by first determining the pair density n p = (P)/A,

from which we define a density differential dnp(r), 27L dnp(r) = -e-OHccrdr. a4

(C.13)

This gives us the density of pairs of separation r in a range dr. Next we calculate the response of a single pair of size r to an external electric field E - that is, we calculate the polarization of this pair, P(r). We do this by assuming that the pair may rotate in response to thermal fluctuations and to the external electric field, but that it does not stretch. If 8 is the angle between the electric field direction and a line joining the two charges, the torque on the dipole is 2qE sin 8 and the dependence of the potential energy on the angle is - rqE cos 8. The polarization is then given by

a

P(r) = - ( q r cos 8 ) E + 0 aE

(C.14)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

48 1

or

where the last term comes from letting E approach zero before doing the integral; that is, we invert the order of integration and the limit. With Eqs. (C.13) and (C.15) we can calculate the susceptibility X(r):

's'

X(r) = EO

*'=a

P(r')dn(r') =

-

aI T qE,2

s'

r f 3epBHcc;dr'.

(C.16)

In two dimensions E = 1 + X. Combining this with Eqs. (CS), (C.9), (C.lO), and (C.16) gives an implicit equation for E(r):

The Coulomb gas language was very helpful in deriving Eq. (C.17), and we can now proceed to convert Eq. (C.17) into two coupled nonlinear differential equations. However, before doing that let us put things back into the array language. We do this by remembering that Lj2 = 27cEJ, so that Eq. (C.4) becomes (C.18) and Eq. (C.5) becomes (C.19) or (C.20)

482

R. S. NEWROCK ET AL.

where we have introduced the scaling length G malized coupling strength K(r):

= ln(r/a)

and the “renor-

(C.21) From a similar derivation in a slightly different context, K(r) is also known as the “reduced stiffness constant,” in the sense of the stiffness of a spin system’s response to a small twist in the phase. Defining K O= PE,, Eq. (C.21) becomes (C.22) and Eq. (C.17) becomes

or C=lnr/a 1 1 &re-PW.+ L.’a)d/‘. K V ) - KO +4n3Je=0

(C.24)

From statistical mechanics we know that the quantity ezBrcis the square of the fugacity, so we can define yo = e-BcO and identify the constant Cowith the chemical potential. It then makes sense to break Eq. (C.24) into two pieces:. y2(G)

[,, 2n J;

= y,2e

-

K(l.Wb.1

(C.25)

and (C.26) y(G) can be thought of as a scale-dependent fugacity: y2(G) is proportional

to the probability of finding a pair of vortices separated by a distance r = a e C. From these last two equations the two famous Kosterlitz-Thoulessz3 relations follow:

* dG

= y(G)[2 - nK(G)]

(C.27)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

483

and dK -'(8) = 4n3y2(8), d8

(C.28)

which are to be solved subject to the boundary conditions

(C.29) This derivation follows the approach first taken by Kosterlitz and T h o ~ l e s s with , ~ ~ the important improvements of Young,31' who showed that Kosterlitz and Thouless had made an unnecessary approximation. While some of the literature seems to suggest that this is a renormalizationgroup approach, it is actually a clever mean-field theory with lengthdependent interactions built in. As was first pointed out by Young, the same final results, Eqs. (C.27) and (C.28), may be obtained from a renormalization-group calculation, as was done by K ~ s t e r l i t z We . ~ ~ note that many papers use the convention C , = pc, in contrast to the C , = -pc implicit in Eq. (C.29). The most interesting behavior is found to occur very near the phase transition, where K ( 8 ) = 2/n, so it is useful to change variables to 2 1 x(8) = -- 1. = K(4

(C.30)

With this we obtain the Kosterlitz-Thouless scaling equations (C.31) and (C.32) These are valid for y(8) << 1, i.e., in the dilute limit. These equations can be used to relate x ( t ) to y(8) by dividing Eq. (C.31) by Eq. (C.32) and integrating, x(8) - ln[l

+ x(8)]

- 2n2y2(t)

+ C ( T )= 0,

(C.33)

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R. S. NEWROCK ET AL.

where C(T),a constant of integration, does not depend on G but does depend on the only other parameter in the problem, the temperature. Equation (C.33) defines the “trajectories” of the system, a set of curves as shown in a plot of y versus x, Fig. 93. With the aid of these trajectories we can gain insight into the behavior of the system. Before we examine the trajectories, however, it is worthwhile to examine the fugacity. We examine Eq. (C.25) more closely. As mentioned, y Z ( t ) is proportional to the probability of finding a pair of vortices with separation ae‘. If we ignore the effects of interactions between vortex pairs of all sizes (i.e., assume E(G) = 1 for all e), and use, in Eq. (C.25), K ( d ) = K O= BE,, we obtain (C.34) Now let us “scale the system upward” (that is, “renormalize” it) by letting r -+sr, so that t + f‘ = t‘ In s. Then

+

pc/3

T + 2t + 2111s - 2-GTK T

- 2-111s

TKT T

1

2

(y)

= y(t)s

(C.35)

0

J

- 0.2

-0.1

0

0.1

x

FIG. 93. Trajectories that show the vortex fugacity as a function of the reduced stiffness constant at constant temperature. One possible initial scale, G = 0, is indicated for the starting conditions. e increases in the directions of the arrows. The trajectory indicated by C = 0 is the critical trajectory. (From Ref. 46, Fig. 1.)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

485

We see that for T < TKT,y(t‘) decreases with increasing length scale; thus, the probability of finding a bound pair with large separation decreases with increasing scale. This implies that below the transition there are no bound pairs of infinite separation present; that is, there are no free vortices. On the other hand, for T > TKT,increasing the scale increases the number of vortex pairs of large separation, indicating that there are many pairs with infinite separation, i.e., unbound pairs. This simple scaling description, while providing significant insight, does not take into account the interactions between vortex pairs, i.e., when ~ ( t ‘#) 1 at all scales. For that we must return to Eq. (C.33) and Fig. 93 and examine the trajectories. Moving along any of the trajectories shown in Fig. 93 is the same as the rescaling we just did. Along the trajectories, t varies and, if we know x and y at t‘ = 0 (scale zero), the trajectories show us how x and y develop with increasing scale. We may start, for example, on a trajectory for which C ( T ) < 0, beginning at t‘ = 0. We see that y(t‘) + 0 as t‘ + CCI. This means that the probability of finding a bound pair of vortices with infinite separation is zero, implying that there are no free vortices. This region must the point where therefore be T < TKT.Note that for small x, x,(x(C = a)), the trajectory crosses the x-axis, equals ( - 2C)’’’. Thus, for different temperatures, for C ( T ) < 0, along the y = 0-axis we have a line of fixed points, a characteristic of the Kosterlitz-Thouless transition. For C ( T ) > 0, y(8) first decreases but then turns away from the origin, increasing for x > 0 until it approaches infinity for infinite separation. x, = 00 means that total screening has occurred, and y , = co means that there are many free vortices present at large distances to do the screening. Therefore, C ( T ) = 0 must define a “critical trajectory,” where T = TKT. Note that although the trajectories are universal, the t‘ corresponding to a particular point on a trajectory depends on nonuniversal system properties; in particular, it depends on the core potential46 pc. A relationship between C and T can be obtained from the initial conditions. The critical trajectory approaches the point (x, y) = (0,O) on a straight line, 271y + x = 0. For t‘ = 0 we have, from Eqs. (C.29) and (C.30),

(C.37) Together, these define the t‘= 0 curve shown in Fig. 93. The critical temperature is where the line for t‘ = 0 intersects the critical trajectory (or

486

R. S. NEWROCK ET AL.

isotherm, since each value of C corresponds to a particular temperature). Combining Eqs. (C.33) and (C.36) yields

At TKT,x + 0 for large t.If we define E, (C.21) and (C.30),

=~ ( = t00) at

TKT,then, from Eqs.

(C.39) which is an implicit equation for the transition temperature, which reduces to Eq. (4.9) for E, = 1. Just above the critical temperature, the interaction at t = 00 is zero; that is, the two vortices of an infinitely separated pair are completely screened from one another. Using EJ(TKT) = hi,(TKT)/2e, we also obtain (C.40) This is the expression used in Section IV to obtain an experimental value for E,. At TKTthe fully renormalized interaction potential 2nE,( TKT)/&, is equal to 4kBTKTwhich is a universal result that holds for all two-dimensional systems with logarithmic interaction potential^.^^ This result is unchanged when spin waves are con~idered.'~ These results have led to an interesting nomenclature, derived from the application of the Kosterlitz-Thouless theory to thin superfluid helium films. For those, the interaction 2nE, must be replaced by one suitable for interactions between vortices in liquid helium, 2nh2n,/m (see Lobb et al.," for an elementary discussion of this connection), but the general behavior is the same as detailed above. In helium films the interaction between vortices is therefore proportional to the superfluid density. The temperature implicitly defined by Eq. (C.39) is known as the temperature at which the jump in the superfluid density occurs, because K jumps from 0 to 2/n at this point. To actually calculate the critical temperature from Eq. (C.39) we must know the core potential, which determines E, as a function of the temperature. For the XY p, = -n2J/2, which for a Josephson array 313

See, for example, Ref. 23.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

487

becomes -n2E,/2. When combined with Eq. (C.38) this yields (C.41) which yields E, = 1.175. Tobochnik and Chester35 obtained 1.75 from a Monte-Carlo calculation. Experimentally Leemann et a1.55956 obtained E, = 1.81. OF THE SCALE LENGTHS 1. TEMPERATURE DEPENDENCES

Above and below TKT we introduced two temperature-dependent scale lengths (Section IV), <+ and t-. The temperature dependencies of these are determined as follows. First we assume both y(/) and x(/) small, to simplify Eqs. (C.31) to (C.33), yielding differential equations and trajectories that are analytically solvable. The trajectories become x’(/)

-

+ 2 c = 0,

47c’y’(/)

(C.42)

and the differential equations are (C.43) and (C.44) Combining Eq. (C.42) and (C.43) yields -dx(e) - 2(X’(/)

d/

+ 2C).

(C.45)

The solution to Eq. (C.45) can be put back into Eq. (C.42) to yield closedform solutions for x(t) and y(8).

T < TKT: x(8) = - 12C1’/’ coth[212C1’/’/ 2 ~ y ( / )= 12C1’/’ csch[212C1’/’/

+ tanh- ’(

-

12C1’/’/xo)]

+ tanh- ’( - 12Cl’/’/xo)].

(C.46)

488

T

=

R. S. NEWROCK ET AL.

TKT: (C.47)

x(/) = 27cy(/)(x,’ - 2/)-’. T > TKT:

+ tan-’(

x(t)

=

-12C1’/2 c0t[212C1”~/

-12C1”2/x,)]

27cy(/)

=

1 2 ~ 1 ’C’ ~ S C [ ~ ~ ~ C ) ’ / ~ /tan-’(-12CI”2/x,)1.

+

We then use Eq. (C.22) to obtain

(C.48)

:

(C.49) and from Eq. (C.30) we have

+

+ +

1 x(/) 1 - 12C1’P’

&(/) = E , 1 x(/) 1 x, ~

(C.50)

where, since x, is the value of x where a trajectory for C(T) < 0 crosses the x-axis, X,

-

1n(l

+ x,)

=

-C(T).

(C.51)

Then, from Eq. (C.46), dropping the constant term since / is large, &(/) = &,

1 [l - (2C1’/2coth 212C(”2/]. 1 - 12c1”2

(C.52)

Using the large argument expansion for the hyperbolic cotangent yields &(/) = E ,

1 c1 1 - )2C1”2

We see that ~ ( approaches t) define 8- by

E,

exponentially as / becomes large. We can

TWO-DIMENSIONALPHYSICS OF JOSEPHSON JUNCTION ARRAYS

489

from which 1 412c11'2

/- =-=-

5-

1

=ln-.

4x,

a

(C.55)

where the last equality defines t-. When / = L,E(C) has undergone most of its asymptotic approach to E,. This means that there are not very many vortex pairs with C > C- to further renormalize the interaction. Thus, /- or 5 - is the length scale characterizing the separation between vortex pairs for T < TKT. Next we write the trajectory equation, Eq. (C.33), as a function of the reduced temperature difference,

where the reduced temperature difference r is defined as

T t=_r-l.

(C.57)

Note that in arrays the interaction strength is dependent on temperature and is not a constant. This is normally taken into account by introducing ) ) a similarly scaled the scaled temperature18 .?. = T ( E J ( T ) / E J ( T K Tand reduced temperature. However, we will assume that we are sufficiently close to the transition temperature that EJ(TKT) E,(T) and we do not need to scale the temperatures. For more precise results, both T and t in the equations to follow must be replaced by scaled temperatures. From Eqs. (C.36) and (C.39) we have

-

x,(t)

1

=

-(r

+ 1)

-

1

(C.58)

Ec

and (C.59) At the critical isotherm C ( t ) = 0. Expanding C ( t ) around t K T (=0) yields, for t << 1, C(t) = 0

+ Bt,

(C.60)

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R. S. NEWROCK ET AL.

where

Using Eqs. (C.58) and (C.59) in Eq. (C.61) we obtain

(C.62) Sufficiently close to TKT,x, is small. For small x, expanding Eq. (C.51) to second order yields, x, x (2Blt1)”’ (using Eq. (C.60)). From Eq. (C.59,

(C.63) In the literature a different constant, b

= n2/(8B), is

often used, leading to

(C.64) Above the transition temperature we can identify another scale length, 8,. From the flow diagram, Fig. 93, we see that the behaviors above and below TKTare markedly different. Below TKT,y ( t ) monotonically decreases as t increases. By contrast, above TKT,y ( t ) at first decreases and then increases toward infinity. This means that the probability of finding a pair of vortices with large separation increases as t increases. This increase in the pair density with larger t eventually invalidates two of our starting assumptions: that we have a dilute pair concentration, and that the only effect of smaller pairs is to renormalize the interaction of the larger pairs. Thus, we have to cut off the renormalization at some value of t, which we will call t,. Although the choice is somewhat arbitrary, we can define t+as the point at which, for x > 0, y ( t ) has increased to its initial value y(0):

y ( t + )is found by integrating y, starting from t = 0. A simple expression for f, is not easily obtained unless x, is small. From Eq. (C.48) for y ( t ) , derived in the limit of small x and y, we see that Eq. (C.65) holds if the arguments

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

49 1

of the sine terms are equal for t = 0 and t = t+- that is, if 2 m t + = n - 2 arctan

t+ = {l -:arctan(-?)](:-).

(2p).

$i%

(C.66)

(C.67)

If t is extremely small, we can neglect the arctan term and (C.68)

This is valid if (n/2)(t/b)'lz << x , << 1. Note that for similar values o f t , t- is 1/2n times smaller than t + . As mentioned, t , is essentially the point at which we have to cut off the renormalization because of the proliferation of free vortices. If we identify the average separation between free vortices as 5 , we can relate the two by

t+ = ln-.5 , a

(C.69)

Finally, we can write the two correlation lengths as they appear in Section V:

and (C.71)

where Eq. (C.70) is only valid for temperatures such that ((2.72)

From Eqs. (C.58), (C.59), and (C.62), we can calculate that x , = 0.149,

492 B

R. S . NEWROCK ET AL.

= 0.240, and

b

=

5.14. Thus, from Eq. (C.72) we see that we need

T

t = -- 1 << 0.05.

TK T

(C.73)

For temperatures very near the transition, the temperature range for the Equation (C.71) is valid over validity of Eq. (C.70) is TKTd T << (1.05)TKT. a much wider temperature range. This means, of course, that the formula for the resistivity, Eq. (5.6), is only valid over the same temperature range. Experimentally, however, it appears to be valid over a considerably wider range. Appendix D. Current-Induced Vortex Unbinding

In Section IV.8 we showed that a transport current exerts a force on a vortex perpendicular to the current direction. The forces on the two vortices of a bound pair are equal and opposite and tend to “stretch” the pair and rotate it towards the position of lowest energy. When the vortex pair is oriented in this manner, the transport current makes it easier to thermally unbind it. We can rewrite the pair potential of Eq. (5.10) as

or

where 8 is the angle between r and a line perpendicular to the current. (Note that here, unlike in Appendix C, we make the simple assumptionz3that the r dependence of the dielectric constant does not, to first order, contribute to the integral of the force between two vortices.) Inspection of Eq. (D.2) shows that the energy is a minimum for 8 = 0 and that a saddle point exists in the 8 = 0 direction, at position t,:

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

493

(Note that in Section IV rJ was identified as r,.) EJ(T)/z(CJ)is approximately equal to the fully renormalized interaction potential, and since i,(T) is a measured quantity and therefore also nearly fully renormalized,

Comparing the last two equations, e'J = i,/i and YJ = ln(i,/i). Therefore, rJ = a(i,/i) as found in Section IV. (Note that for the range of currents normally used in experiuients, r'J 2 3 tv 4 arid is ccrtairily rivt largc.) Tlic energy at the saddle point can be written as

. ~ ~ the dissipation in this regime for a superfluid Ambegaokar et ~ 2 1 calculated helium film, essentially using the generation-recombination model we used earlier. Halperin and Nelson3I4 converted those results to superconducting films. For voltages in arrays we have

where I ,

= Mi,

is the critical current of the array. They found that a ( T )=

3 3

+ ftI' CJ >> e+ ft;' eJ << e-,

(D.7)

where C, = In r J / a = ln(i,/i) and C- = In(( -/a); see Appendix C. When the current is very small, C, >> C- and large length scales are being probed by the current. In this limit, Eq. (C.64) gives /- = (1/2~c)(b/Ifl)'/~, from which we find

where

a ( T )= 3

+ 7r

(;)-

ll 2 .

B. I. Halperin and D. Nelson, J. Low Temp. Phys. 36, 599 (1979)

494

R. S. NEWROCK ET AL.

This result agrees with Eq. (5.18) in predicting a(TKT)= 3. It differs in that it predicts a square-root cusp for T M TKT(or t small). This is the correct result very near TKT;the other approach is approximate. By probing large length scales with 8, >> /-, the preceding result is insensitive to the scale-dependent dielectric constant ~ ( t )Essentially . ~ (>tL)z E, = constant. When shorter length scales are probed, however, the barrier height at rJ depends on & ( I J ) , and changes in rJ change E significantly. In the small-distance or high-current limit (t,<< t-)Eqs. (D.6) and (D.7) predict that 3 + 1 / 2 In ic/i

T/ = 21,R, @In))-'('>

, 4

(D.lO)

so that the exponent and prefactors depend logarithmically on current. For thin superconducting films, Kadin et aI.j3 took a slightly different approach, albeit also based on Ambegaokar et ~ 1 For. T ~< TKT, ~ assuming simple classical escape over the saddle point and equating the escape rate to the recombination rate, they determined that the current-dependent resistance R ( l ) depended exponentially on the pair interaction at the saddle point U ,

Therefore, a(T) =

dln V =I+-dlnl

~

=

1 +zK(IJ, T ) = 1 +xK(tJ, T), (D.12)

where K ( t ) is the reduced stiffness constant n'[x(t) + 13. Thus, measurements of a ( T ) can directly probe the vortex interactions. Kadin et al.33 calculated K ( t , T ) as a function of T for different values of d , Fig. 31. (In the figure n K ( T ) = a ( T ) - 1.) For an infinite specimen / = co, ZK jumps from 0 to 2, the universal jump.37 (In this case t, < t- and t, < t, = In( W/a)always.) However, most, if not all, experimental arrays have widths of 1000 junctions or less, for an em,,M 7, and we see from Fig. 31 that we will never see a really sharp jump in a ( T ) in an array. If we substitute Eq. (C.21) for K(8) and assume (incorrectly, it turns out for most measurements) that t, M co,then we can assume ~ ( tM, )E, near the transition temperature, with the result that (D.13)

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

495

If we use the result TKT= 7tEJ(TKT)/2k&, we obtain a(T‘,) = 3, as expected. The results detailed above have been widely accepted, although discrepancies between theory and experiment exist. Recently Minnhagen et al.43 offered a different expression for a(T): (D.14) At TKTthis yields u(TKT)= 3, as it should. The Minnhagen et al. result is not yet confirmed for arrays. The physics behind the two results is similar. Both are based on the transport current acting on both members of a vortex pair, increasing their separation and reducing the energy barrier. Increasing currents affect smaller and more tightly bound pairs and lead to a higher density of free vortices. In the model of Ambegaokar et al., the motion of these free vortices before they recombine is the main contribution to the dissipation. Minnhagen et ul.’s model suggests that the vortices are never really free-as soon as the pair of vortices is stretched over the barrier they rapidly recombine with other vortices. The main contribution to the dissipation is just the stretching over the barrier, We note that Minnhagen’s model would seem to imply a fairly high density of vortices, which may not be the appropriate limit for the Kosterlitz-Thouless transition. Appendix E. The Capacitance Matrix

In the literature the capacitance matrix C, which governs the electrostatic interaction between charges on the various islands, is often modeled by a sum of the contributions from the nearest-neighbor capacitances C between adjacent islands, and the self-capacitance C , of the islands with respect to ground : if i a n d j are neighboring islands, (E.1) otherwise. The number z of nearest neighbors depends on the spatial dimension as well as on the type of array. Unless stated otherwise, we consider square arrays in two dimensions. The simple model (Eq. (E.l)) for the capacitance matrix requires that electric field lines emanating from a superconducting island terminate either on neighboring islands or on a ground plane. This appears to be reasonable for regular arrays that are not too large. The form of Eq.

496

R. S. NEWROCK ET AL.

(E. 1) guarantees that the capacitance matrix is positive definite; neglecting this basic requirement may lead to unphysical results.2 It should also be kept in mind that in many real systems the capacitance matrix may be much more complicated, containing next-nearest-neighbor and further couplings. The off-diagonal matrix elements C', = (C-l)ijin two dimensions can be well-approximated by -1

c-

-[log(r/24

2nc

1 l(r)

=

~

2nc

+ y]

(r << A)

Ko(r/A) % C

(r >> A)

8nr

in the spatial continuum limit3'' (the fundamental solution of the Helmholtz equation) and where r = Iri - r j l . Here K O denotes a modified Bessel function, y = 0.5772.. . is Euler's constant, and

3 1 5 In three dimensions the continuum limit is given by the Yukawa potential (a/4nCr)exp( - r / I ) .

1.o

I I

I

I

10

20

I

I

I

0.8

1

h

S 0.6 4

l

0 x

0.4

0 0.2 0.0

0

30 ?-/a

40

50

FIG. 9 Comparison of C-'(x, y, x + r, y ) (circles) with its continuum approximation (sc line) in an infinite two-dimensional square array. Also shown are the logarithmic (short dashes) and exponentially damped (long dashes) asymptotes for small and large distances, respectively. The ratio of self-capacitance to junction capacitance is C,/C = 4 x corresponding to a screening length I = 50a. One sees that the continuum approximation is already very satisfactory for r = a, and that the logarithmic short distance behavior is valid for r smaller than about L/2.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

497

is the electrostatic screening length in the array. As demonstrated in Fig. 94, the continuum approximation is quite good for nearest-neighbor sites, r = a. For large distances the interaction is exponentially screened, whereas for r small compared to A, the charges interact logarithmically. In principle, this logarithmic interaction opens the possibility to experimentally realize ~.~'~.~' a two-dimensional Coulomb gas of electrical ~ h a r g e s , ' ~ ~ , ~in~ contradistinction to the Coulomb gas of vortices discussed in Section IV and Appendix C. The latter will of course still be present when E , dominates E,, leading to competition and duality of the two different Coulomb it is difficult to reduce the ratio C,/C below 1/1000 g a ~ e s . ' ~ In ' . ~practice, ~~ in lithographically manufactured arrays, so that Kosterlitz-Thouless behavior of the charge Coulomb gas will be blurred by rather strong finite-size effects. A short-ranged electrostatic interaction (A < a) is sometimes also modeled by assuming that the inverse capacitance matrix has the same structure as C (Eq. (E. 1)) with only diagonal and nearest-neighbor elements nonvanishing.2 S 1,269,300 Doniach 2 9 5 used an inverse capacitance matrix with only nearest-neighbor elements nonzero; this corresponds to a rather strange capacitance matrix, with off-diagonal elements growing logarithmically with distance. The energy

of a singly charged monopole in the system is determined by the diagonal matrix element C;'. In an infinite array, this matrix element can be expressed in terms of K , the complete elliptical integral of the first kind,318 1

ln(32C/C0) (CJC

-+

0).

It is plotted in Fig. 95. In the limit C, = 0, the charging energy depends on voltage diferences between the islands: a nonzero but uniform voltage costs no energy. This of course reflects the fact that the electrostatic potential is defined up to an arbitrary constant, which for nonzero C, was fixed by A. Widom and S. Badjou, Phys. Rev. B37, 7915 (1988). J. E. Mooij, B. J. van Wees, L. J. Geerligs, M. Peters, R. Fazio, and G. Schon, Phys. Rev. Lett. 65, 645 (1990). 3 1 8 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, San Diego (1980). 316 3'7

498

R. S. NEWROCK ET AL.

1o-2

I

I

1oo

10'

I

1o-2

lo-'

1o2

FIG.95. The dependence of C,' on the capacitance ratio CJC in an infinite twodimensional square array (solid line). The asymptotic curves ln(32C/Co)/4n (short dashes) and C/C, (long dashes) are also shown.

implicitly choosing the potential of the ground plane equal to zero. Vanishing C, thus implies that the capacitance matrix has one zero eigenvalue and that its inverse diverges. Configurations carrying a net charge are then excluded, since their electrostatic energy is infinite.319 After the monopole energy, the dipole energy, Ed

= - eZ[C;

- C,;

'1,

with i and j nearest neighbors, is the most important characteristic energy; it is one that enters various formulas for the boundary of the superconducting phase. For C, ( i and j nearest neighbors), an analytical expression can be given in the infinite array:

'

c : L1 [ "

4C+2C0 K ( 2KC 4 c + c, 1

-

2co

4c+c,

.('

1

+ CJ4C

)

1 2'1+C,/4C'1+CO/4C

)],

(in.n.1)

(E.7)

where II(+,n, k) is the elliptic integral of the third kind.318 A simple symmetry argument suffices to show that Ed = e2/4C for C, = 0. 319 When restricting the capacitance matrix to the subspace of neutral configurations, it is invertible even for C, = 0, but then the inverse matrix elements grow logarithmically with the system size.

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

499

Appendix F. Offset Charges

In a regular array with randomly placed charged impurities, there are three types of sites: (a) the islands of the array, between which Cooper pairs can be exchanged by tunneling; (b) sites with fixed impurity charges; and (c) the ground plane. The situation is sketched in Fig. 96(a) and modeled by the capacitance network in Fig. 96(b). Charges and voltages on the various types of sites are related by a capacitance matrix. In block matrix notation we have

The blocks C,,,

Cab,

etc., are the capacitance matrices C used in Appendix

superconducting islands

5.

impurity

1 I

0 conducting plane

-

FIG. 96. (a) Drawing of a sample with a gate and an offset charge residing on an impurity in the substrate. (b) Capacitor network model of the sample in (a).

500

R. S. NEWROCK ET AL.

E. The charging energy can be found by integrating

starting from a situation in which all type-a sites are uncharged. To this end the voltages V, are needed as a function of the island charges Q,, the impurity charges Qb,and the gate voltage V,:

The term independent of Q, defines the offset charges 4,.Since these are not real charges on the islands, they may take noninteger values. The coupling c a b is usually very small, so that one may approximate

The charging energy is then given by E

=i(Q,

+ q , , ) . C - ' . ( Q ,+ 4,)

- +q,*C-'.q,.

(F.5)

The last term on the right-hand side is constant and may be omitted. Appendix G. Phase Correlation Function in the Absence of Coupling To evaluate the correlation function (ei41(') e p i 4 m c 0 ) ) o for E , = 0, we first need to calculate e'@i"'.Repeated application of the commutation relation, Eq. (12.14), shows that for any power of Q,, and consequently for any analytic function f ( Q , ) ,

holds; the operator exp(i4,) increases the charge on island 1 by one Cooper

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

501

pair. Therefore,

= exp

(F c,

l(Qk

+ qk

-

1

es,,,)

eidt(Ol.

(G.2)

For generality, offset charges qk (see Appendix F) are included here. Using (G.2)the correlation function becomes

where I { n k } ) denotes a charge eigenstate with nk Cooper pairs on island k, and = CkCL1(Qk + qk). In the same way one can see that (ei4r(d eidm(Ol ), = 0, because

Writing the exponential functions in terms of sines and cosines gives

Because H , is translationally invariant in the phases, the sine-sine term equals the cosine-cosine term. For vanishing offset charges, H , is even in the phases, the mixed sine-cosine terms vanish, and

For the bosonic We also calculate the Fourier transform of (er4~(T1-'~~(01)0.

502

R. S. NEWROCK ET AL.

Matsubara frequencies, w,

= 27cp/hB, one

has

[2eni -

ihw,

+ qi]C;

'[2enj

+ qj]

+ 2 e C C;'[2enj + q j ] - 4E, i

At zero temperature only the uncharged configuration contributes, so 6= 0. Then the Taylor expansion needed in Eq. (12.81) reads

Appendix H. Conductivity from Derivatives of the Partition Function

We demonstrate here how conductivity can be expressed in terms of derivatives of the partition function. In order to probe the conductivity, a small electric field E is applied, which we write as the time derivative of a small additional vector potential, E = -(8AE/8t). As we are only interested in the linear response, we expand both the operator Z,(r) for the Josephson current in direction m ( = x or y ) at site r,

2e h

= - E,

sin(4(r) - 4(r

+ a;,)

- A,m - A&&))

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

2e h

= -E j

sin(+(r) - 4(r

+ a;,)

2e h

- - E j cos(4(r) - +(r

503

- Ar,,)

+ a;,)

- Ar,,)AEm(t)

+ O(E2),

(H.l)

+ O(E2),

(H.2)

as well as the Hamiltonian. H

=H , - Ej

1COS(+(Y)- 4(r + a;,)

-A,,

-

A&,(t))

r,m

2e

- H‘O) -

E , sin(+(r) - 4(r

+ a2,)

h

- Ar,,) 2e A;,(t)

r.m

to first order in E. Here A , , = (2e/h)A.;,,,a corresponds to a static magnetic field and A t , = (2e/h)AE(v).;,a to the perturbation by the electric field; the vector potentials are assumed to vary slowly on the scale of the lattice constant. The second term in the last piece of Eq. (H.l) (the “diamagnetic” current) already contains the electric field linearly and must therefore be averaged with the equilibrium density matrix corresponding to vanishing electric field. The first term (the “paramagnetic” current), which will be called Z,,,(O), is found back in the linearized Hamiltonian. (In this appendix, a superscript I means Ith order in the applied electric field.) With the aid of Kubo’s formula (see, e.g., Ref. 235), one can express (Z(’))(’) in terms of an equilibrium correlation function. For the average of the (sheet) current density j we obtain

This expression for the conductivity r~ can be linked to derivatives of the partition function by an analytic continuation to imaginary frequencies. The Fourier transform of the retarded Green’s function,

in Eq. (H.3), has, as its analytic continuation to the upper half of the

504

R. S. NEWROCK ET AL.

complex frequency plane, the Matsubara Fourier transform of the imaginary time Green's function, -1

-(TZ!,?(r,

h

t)ZLo)(r',0))")

(see e.g. Ref. 298). Thus, the formula for the conductivity becomes

1

dteiW@(TIg)(r, t)ZLo)(r',0))") .

(H.4)

This expression can now be written in terms of derivatives of the partition function Z with respect to the vector potential, which has formally been given a dependence on imaginary time. Taking a double functional derivative of 2, one obtains (for E = 0)

-(:yEJ(cos(4(r)

- 4(r

1

+ drn) - Ar,,))(0)6r.r.Srn,,S(t1 - t2) .

(H.5)

Combining Eqs.(H.4) and (H.5) and taking a spatial Fourier transform, we obtain

In the continuum limit

u2 1 -+ r

d2r and

6 6 + u2SArn(r,t) 6Arn(r,t) '

so that we arrive at the desired result:

505

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

Appendix 1. The Green’s Function for Gaussian Coarse Graining

For a quadratic free energy functional like F,,, the correlation function (I&, t ) $ * ( r , t’))immediately follows from the formula for the moments of a Gauss distribution. Defining G-’ by

bF,,[Ic/]

=

j jr j

! 2

=!f 2

d2r

dt

d2r‘

lo’”

dt’$(r, t ) G - ‘ ( r - r’, t - t’)$*(r’, t’)

d z r ~ ~ * d t ~ d 2 r ~ ~ ~ d t . tl )RGe- ’(( rI -(r r’ , , 7 - t ’ ) Re$(r’,

+ Im $(r, t ) G - ‘ ( r

-

r’, t

-

t’)Im $(r’, t’)],

t’)

(1.1)

we have (Re $(r, t )Re $(r’, 7’)) = (Im Ic/(r, t )Im $(r’, T’))

= G(r - r‘, t - t’) (1.2)

and (Re $(r, t )Im $(r’, t ’ ) )= 0. Therefore, ( $ ( r , t)$*(r’, t ’ ) )= 2G(r

-

r’, t

-

t’)

and

Because the statistical weight is Gaussian, the averages of higher powers of $ can be written as sums of products of pair averages (Wick’s theorem). Using translational invariance, for the correlation function of the current density defined in Eq. (12.77) one finds

a2G(r - r‘, t - T‘) dx,dx,

- G(r - r‘, T - t’)

for vanishing vector potential. Using Eq. (1.6), the expression, Eq. (12.76) for

506

R. S. NEWROCK ET AL.

the conductivity tensor, becomes

To proceed we need the explicit form of G, whose inverse can be read off Eq. (12.75). In the absence of offset charges and vector potentials,

so that

Since the integral f d2kG(k,w,) diverges logarithmically, the two terms (the diamagnetic and paramagnetic contribution) on the right-hand side of Eq. (1.7) do not exist separately, but the divergences of both terms cancel. To facilitate the calculation, one may temporarily introduce a convergence factor exp( - I k ) into G and take the limit I + 0 at the end. Now, since

a

- G(k, wv)e-'k =

ak

2 -A&G(k, wv)e-ak- -(u2goEJ)k[G(k, ~ , ) ] ~ e - ' (1.10) ~, ha2

the application of Gauss's theorem to (a/ak,)[k,G(k, w,)e-u] gives

s

d2kd,,,G(k, m,)e-Ak = I

+

h

d2kk,k,G(k, w,)2e-ak.

(1.11)

With the aid of this formula we obtain the relation Eq. (12.78) from Eq. (1.7).

TWO-DIMENSIONAL PHYSICS OF JOSEPHSON JUNCTION ARRAYS

507

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DISORDERED ARRAYS S. Alexander, Phys. Rev. B27, 1541 (1983). S. P. Benz, M. G. Forrester, M. Tinkham, and C. J. Lobb, Phys. Rev. 838, 2869 (1988). J. Bosch, R. Gross, and R. P.Huebener, Appl. Phys. Lett. 47, 1004 (1985). D.-X. Chen, J. J. Moreno, and A. Hernando, Phys. Rev. B52, R9859 (1985). M. B. Cohn, M. S. Rzchowski, S. P.Benz, and C. J. Lobb, Phys. Rev. B43, 12823 (1991). S. Datta, S. Das, D. Sahdev, R. Mehrotra, and S. Shenoy, Phys. Rev. B54,3545 (1996). S. Datta, S. Das, D. Sahdev, and R. Mehrotra, Modern Phys. Lett. B10, 451 (1996). S. Datta, S. Das, D. Sahdev, and R. Mehrotra, Phys. Rev. 854, 15438 (1996). A. Davidson and C. C. Tsuei, Physica B108, 1243 (1981). M. P. A. Fisher, Phys. Rev. B36, 1917 (1987). M. G. Forrester, Ph.D. thesis, Harvard University (1988). M. G. Forrester, H. J. Lee, M. Tinkham, and C. J. Lobb, Jpn. J. App. Phys. 26, Suppl. 26-3, 1423 (1987). A. Giannelli, C. Giovannella, and M. Bernaschi, Europhys. Lett. 22, 29 (1993). A. Giannelli, C. Giovannella, and M. Bernaschi, Physica B194-196, 1695 (1994). D. C. Harris, S. T. Herbert, and J. C. Garland, Physica B165-166, 1643 (1990). S. John, T. C. Lubensky, and J. Wang, Phys. Rev. B38, 2533 (1988). Y.-H. Li and S. Teitel, Phys. Rev. Lett. 67, 2894 (1991). A. Majhofer, L. Mankiewicz, and J. Skalski, Phys. Rev. B42, 1022 (1990). T. Onogi, R. Sugano, and Y. Murayama, Sol. State Comm. 78, 103 (1991). Y. Park, J. Kim, and D. Kim, Phys. Rev. 838, 741 (1988). S. Scheidl, Phys. Rev. 855,457 (1997).

FRUSTRATION AND MAGNETIC FIELD EFFECTS C. Auletta, G. Raiconi, R. De Luca, and S. Pace, Phys. Rev. 849, 12311 (1994). M. Benakli, H. Zheng, and M. Gabay, Phys. Rev. B55,278 (1997). K.A. Benedict, J . Phys. Cond. M a t t . 3, 5955 (1991). B. Berge, H. T. Diep, A. Ghazali, P.Lallemand, Phys. Rev. B34, 3177 (1986). R. K. Brown and J. C. Garland, Phys. Rev. 833, 7827 (1986). J. P. Carini, S. Nagel, R. Levi-Setti, Y.-L. Wang, J. Chabola, and G. Crow, Sol. State Comm. 65, 977 (1988). P.Chandra, L. B. Ioffe, and D. Sherrington, Phys. Rev. Lett. 75, 713 (1995). D.-X. Chen, A. Sanchez, and A. Hernando, Phys. Rev. B50, 13735 (1994). D.-X. Chen, A. Sanchez, and A. Hernando, Physica C244, 123 (1995).

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J. S. Chung, M. Y. Choi, and D. Stroud, Phys. Rev. B38, 11476 (1988). E. K. F. Dang, G. J. Halasz, and B. L Gyorffy, Phys. Rev. 846, 8353 (1992). H. Eikmans and J. E. van Himbergen, Phys. Rev. B41, 8927 (1990). H. Eikmans, J. E. van Himbergen, H. J. F. Knops, and J. M. Thijssen, Phys. Rev. B39, 11759 (1989). F. Falo, A. R. Bishop, and P. S. Lomdahl, Phys. Rev. 841, 10983 (1990). 0. Foda, Nuclear Physics B300, 61 1 (1988). P. Gandit, J. Chaussy, B. Pannetier, A. Vareille, and A. Tissier, Europhys. Lett. 3, 623 (1987). E. Granato, Phys. Rev. 854,R9655 (1996). N. Grenbech-Jensen, A. R. Bishop, F. Falo, and P. S. Lomdahi, Phys. Rev. 846,1 1 149 (1992). J. V. Jose, G. Ramirez-Santiago, and H. S. J. van der Zant, Physica B194-196, 1671 (1994). S. E. Korshunov, A. Vallat, and H. Beck, Phys. Rev. B51,3071 (1995). S. E. Korshunov, J. Star. Phys. 43, 17 (1986). J. M.' Kosterlitz and E. Granato, Phys. Rev. B34, 2026 (1986). H. J. Lee, M. G. Forrester, M. Tinkham, and C. J. Lobb, Jpn. J . Appl. Phys. 26, Suppl. 26-3, 1385 (1987). M. S. Li and M. Cieplak, Phys. Rev. B50,955 (1994). Y.-H. Li and S. Teitel, Phys. Rev. Lett. 65, 2595 (1990). Y.-L. Lin and F. Nori, Phys. Rev. B50, 15953 (1994). S. Magri and Y.-C. Zhang, Phys. Rev. B38, 815 (1988). A. Majhofer, T. Wolf, and W. Dieterich, Phys. Rev. B44, 9634 (1991). K. K. Mon and S. Teitel, Phys. Rev. Lett. 62, 673 (1989). D. B. Nicolaides, J. Phys. A24, L231 (1991). Q. Niu and F. Nori, Phys. Rev. B39, 2134 (1989). F. Nori, Q. Niu, E. Fradkin, and S. J. Chang, Phys. Rev. 836, 8338 (1987). B. Pannetier, A. Bezryadin, and A. Eichenberger, Physica B222, 253 (1996). M. V. Simkin, Phys. Rev. B55, 78 (1997). J. Straley, A. Morozov, and E. Kolomeisky, Phys. Rev. Lett. 79, 2534 (1997). D. Stroud and W. Y. Shih, Materials Science Forum 4, 177 (1985). D. Stroud and S. Kivelson, Phys. Rev. 835, 3478 (1987). R. Sugano, T. Onogi, and Y. Murayama, Physica C185-189, 1643 (1991). R. Thtron, S. E. Korshunov, J. B. Simond, Ch. Leemann, and P. Martinoli, Phys. Rev. Lett. 72, 562 (1994). S. Valfells and G. 0. Zimmerman, Physica B194-196, 1403 (1994). A. Vallat and H. Beck, Phys. Rev. Lett. 68, 3096 (1992). H. S. J. van der Zant, H. A. Rijken, and J. E. Mooij, J. L o w Temp. Phys. 82, 67 (1991). J. Villain, J. Phys. C10, 1717 (1977). Y. Y. Wang, R. Rammal, and B. Pannetier, J. Low Temp. Phys. 68, 301 (1987). R. A. Webb, R. Voss, G. Grinstein, and P. M. Horn, Phys. Rev. Lett. 51, 690 (1983).

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510

R. S. NEWROCK ET AL.

M. Y. Choi, Phys. Rev. 846, 564 (1992). H. Eikmans and J. E. van Himbergen, Phys. Rev. 844, 6937 (1991). T. Halsey and S . A. Langer, Physica 8169, 707 (1991). S. E. Hebboul, D. C. Harris, and J. C. Garland, Physica 8165-166, 1629 (1990). J. Kim and H. J. Lee, Phys. Rev. 847, 582 (1993). V. V. Kurin and D. A. Ryudyk, Physica C205, 85 (1993) S. G. Lachenmann, T. Doderer, and R. P. Huebener, Phys. Rev. 853, 14541 (1996). S.-J. Lee and T. Halsey, Phys. Rev. 847, 5133 (1993). L. L. Sohn and M. Octavio, Phys. Rev. 849, 9236 (1994). D. Stroud, I.-J. Hwang, and S. Ryu, Physica 8222, 331 (1996).

QUANTUM ARRAYS J. K. Bhattacharjee, R. A. Ferrell, and B. Mirhashem, Phys. Rev. 841, 11615 (1990). Ya. M. Blanter and G. Schon, Phys. Rev. 853, 14534 (1996). C. D. Chen, P. Delsing, D. B. Haviland, Y. Harada, and T. Claeson, Phys. Rev. 851, 15645 (1995). R. Fazio, U. Geigenmiiller, and G. Schon, Proceedings of the Adriatico Research Conference on Fluctuations in Mesoscopic and Macroscopic Systems (1990). R. Fazio, A. van Otterlo, G. Schon, H. S. J. van der Zant, and J. E. Mooij, Helv. Phys. Acta. 65, 228 (1997). R. A. Ferrell and B. Mirhashem, Phys. Rev. Lett. 63, 1753 (1989). R. S. Fishman and D. Stroud, Phys. Rev. 845, 1676 (1987). E. Granato and J. M. Kosterlitz, Phys. Rev. Lett. 65, 1267 (1990). A. Kampf and G. Schon, Phys. Rev. 837, 5954 (1988). A. Kanda and S . Kobayashi, J. Phys. Soc. Jpn. 64, 19 (1995). S. Katsumoto and S . Kobayashi, Physica 8194-196, 1123 (1994). S. Kobayashi, A. Kanda, and R. Yamada, Jpn. J. Appl. Phys. 34,4548 (1995). S . Kobayashi, R. Yamada, and S . Katsumoto, Physica 8194-196, 1125 (1994). T. K. Kopec, and J. V . Jose, Phys. Rev. 852, 16140 (1995). S. E. Korshunov, Physica 8152, 261 (1988). P. Lafarge, M. Matters, and J. E. Mooij, Phys. Rev. 854, 7380 (1996). G. Luciano, U. Eckern, and J. G . Kissner, Europhys. Lett. 32, 669 (1995). G. Luciano, U. Eckern, J. G. Kissner, and A. Tagliacozzo, J. Phys. Cond. Matt. 8, 1241 (1996). B. Mirhashem and R. A. Ferrell, Physica C152, 361 (1988). Yu. V. Nazarov and A. A. Odintsov, Physica 8194-196, 1737 (1994) A. J. Rimberg, T. R. Ho, and J. Clarke, Phys. Rev. Lett. 74,4714 (1995). A. J. Rimberg, T. R. Ho, C.Kurdak, and J. Clarke, Phys. Rev. Lett. 78, 2632 (1997). E. Roddick and D. Stroud, Phys. Rev. 848, 16600 (1993). E. Roddick and D. Stroud, Physica C235-240, 3335 (1994). C. Rojas and J. V . Jose, Phys. Rev. 854, 12361 (1996). E. Simanek, Sol. State Cornm. 48, 1023 (1983). E. Simanek, Phys. Lett. 101A, 161 (1984). E. Simanek, Phys. Lett. A177, 367 (1993). T. S. Tighe, A. T. Johnson, and M. Tinkham, Phys. Rev. 844, 10286 (1991). H. S. J. van der Zant, F. C. Fritschy, W. J. Elion, L. J. Geerligs, and J. E. Mooij, Phys. Rev. Lett. 69, 2971 (1992). A. van Otterlo, P. Bobbert, G . Schon, H. S. J. van der Zant, and J. E. Mooij, Helv. Phys. Acta. 65, 379 (1992).

TWO-DIMENSIONAL PHYSICS O F JOSEPHSON JUNCTION ARRAYS

5 11

A. van Otterlo, R. Fazio, and G. Schon, Physica 8194-196, 1153 (1994). A. van Otterlo, R. Fazio, and G. Schon, Physica 8203, 504 (1994). A. van.Oudenaarden and J . E. Mooij, Phys. Rev. Lett. 76, 4947 (1996). B. J. van Wees, Phys. Rev. 844, 2264 (1991). K. H. Wagenblast, R. Fazio, A. van Otterlo, G. Schon, D. Zappala, and G. T. Zimanyi, Physica 8222, 336 (1996). R. Yagi and S. Kobayashi, J . Pbys. SOC. J p n . 64, 375 (1995). R. Yagi and S. Kobayashi, J p n . J . A p p l . Phys. 34, 4569 (1995). W. Zwerger, Europhys. L r r r . 9, 421 (1989).

DYNAMICAL EFFECTS, CHAOS,

AND

TURBULENCE

R. Bhagavatula, C. Ebner, and C. Jayaprakash, Phys. Rev. 845, 4774 (1992). R. Bhagavatula, C. Ebner, and C. Jayaprakash, Phys. Rev. 850, 9376 (1994). Y. Brairnan and K. Wiesenfeld, Phys. Reu. 849, 15223 (1994). H. Chat6 and P. Mannville, Europbys. Lerr. 6, 591 (1988). A. A. Chernikov and G. Schmidt, Phys. Rev. E50, 3436 (1994). A. J. Chorin and J. Akao, Physica 052,403 (1991). A. Chudnovsky, Phys. Rev. 851, 15351 (1995). J. S. Chung, K. H. Lee, and D. Stroud, Phys. Reu. 840, 6570 (1989). A. E. Duwel, E. Trias, T. P. Orlando, H. S. J. van der Zant, S. Watanabe, and S. Strogatz, J . Appl. Phys. 79, 7864 (1996). I. F. Marino, Phys. Rev. 855, 551 (1997). R. Mehrotra and S. R. Shenoy, Europhys. Lett. 9, 11 (1989). R. Mehrotra and S. R. Shenoy, Phys. Rev. 846, 1088 (1992). R. Mehrotra, Physica 8222, 326 (1996). S. Nichols and K. Wiesenfeld, Phys. Rev. ,445, 8430 (1992). S. Nichols and K. Wiesenfeld, Phys. Rev. E48, 2569 (1993). S. Nichols and K. Wiesenfeld, Phys. Rev. E50, 205 (1994). M. Octavio, C. B. Whan, U. Geigenmiiller, and C. J. Lobb, Phys. Rev. 847, 1141 (1993). S. Watanabe and S. H. Strogatz, Physira 074, 197 (1994). K. Wiesenfeld, Phys. Rev. B45, 431 (1992). K. Wiesenfeld, Physica 8222, 315 (1996).

MISCELLANEOUS ARTICLES K. A. Benedict, J . Phys. Cond. Matt. 1, 4895 (1989). R. M. Bradley and S . Doniach, Phys. Rev. 830, 1138 (1984). M. Y. Choi, Phys. Rev. 850, 10088 (1994). M. Y. Choi, Phys. Rev. B50, 13875 (1994). M. Y. Choi, Physica B222, 358 (1996). T. D. Clark, Phys. Rev. 88, 137 (1973). T. D. Clark and D. R. Tilley, Phys. Lett. A28, 62 (1968). W. J. Elion, J. J. Wachters, L. L. Sohn, and J. E. Mooij, Physica 8194-196, 1001 (1994). W. J. Elion, J. J. Wachters, L. L. Sohn, and J. E. Mooij, Physica 8203, 497 (1994). M. Franz and S. Teitel, Phys. Rev. Lett. 73, 480 (1994). E. Granato, Phys. Rev. 845, 2557 (1992). B. Giovannini and L. Weiss, Helu. Phys. Acta. 51, 716 (1978).

512

R. S. NEWROCK ET AL.

B. Giovannini and L. Weiss, Sol. State Comm.27, 1005 (1978). M. E. Gouvea and A. S. T. Pires, Phys. Rev. 854, 14907 (1996). E. Granato, J. Appl. Phys. 75, 6960 (1994). B. Horovitz, Phys. Rev. 851, 3989 (1995). K. H. Lee and D. Stroud, Phys. Rev. 845,2417 (1992). K. H. Lee and D. Stroud, Phys. Rev. 846, 5699 (1992). Y.-H. Li and S. Teitel, Phys. Rev. 847, 359 (1993). A. A. Odintsov and Yu. V. Nasarov, Phys. Rev. 851, 1133 (1995). G. Parisi, J. Math Phys. 37, 5158 (1996). J. P. Peng, Phys. Lett. A129, 124 (1988). J. R. Phillips, H. S. J. van der Zant, and T. P. Orlando, Phys. Rev. B50, 9380 (1994). S. R. Shenoy and B. Chattopadhyay, Phys. Rev. 851,9129 (1995). L. L. Sohn, M. S. Rzchowski, J. U. Free, and M. Tinkham, Phys. Rev. 847, 967 (1993). L. L. Sohn, M. T. Tuominen, M. S. Rzchowski, J. U. Free, and M. Tinkham, Phys. Rev. 847, 975 (1993). A. Stern, Phys. Rev. 850, 10092 (1994). H. S. J. van der Zant, C. J. Muller, L. J. Geerligs, C. J. P. M. Harmans, and J. E. Mooij, Phys. Rev. 838, 5154 (1988). B. J. van Wees, Phys. Rev. Lett. 65, 255 (1990). L. Weiss and B. Giovannini, Hero. Phys. Acta 55, 468 (1982). W. Yu and D. Stroud, Phys. Rev. 846, 14005 (1992). W. Yu and D. Stroud, Phys. Rev. 850, 13632 (1994).

Author Index Numbers in parentheses are reference numbers and indicate that an author’s name is not cited in the text.

A

Abe, D., 228(91) Abeles, B., 438 Abraham, D., 290(18), 291(18), 292(18), 316, 345(67)

Abramowitz, M., 435(224) Abrikosov, A. A,, 6(17), 446(245) Abstreiter, G., 227(47), 229(118) Abukawa, T., 228(76) Achiba, Y., 140(398) Adachi, S., 188(574) Adams, J. E., 190(585) Adler, S. L., 23, 185, 186 Adolph, B., 182, 183(561, 564). 186(569), 189 Affolter, J., 363( 114) Ahn, C. C., 229(108, 109, 120) Aizawa, N., 228(93) Akai, H., 197, 198 Akazawa, H., 227(42), 229(106) Akeyoshi, T., 456(273) Akopov, S. G., 455(267) Albanesi, E. A., 145(420) Albrecht, M., 49(174) Albrecht, S., 16(56), 157(474) Alder, B. J., 86(270), 125 Al-Falou, A,, 228(71, 72), 229(96, 97, 99) Allan, D. C., 63(195), 64(195), 106(322), 174(535, 536), 177(541, 543), 181, 182, 185, 189, 191(587) Almbladh, C.-O., SO(256) Alouani, M., 93(292), 94(293), 132(363), 177(546) Alperin, H. A,, 139(392) Ambegaokar, V. U., 295(20), 302(25), 320(44, 45), 326, 328, 329, 385(45), 458, 493, 494,495 Anastassakis, E., 185(565) Andersen, 0. K., 93(289), 139(382) Anderson, G. W., 232( 162) Anderson, L., 228(72) Anderson, P. W., 433(215), 435 Andreoni, W., 157(480)

Angot, T., 229(110) Anisimov, V. I., 139(382) Anthony, J. M., 232(136) Aono, M., 228(79) Aoyagi, Y., 231(142, 143) Arbman, G., 27(86), 50 Arima, T., 139(389) Ariosa, D., 318(40) Aryasetiawan, F., 4, 12, 16(53, 55), 21(68, 69, 71, 77), 29(55, 68), 31(14), 44(68, 145), 64(68, 145), 68(145,215), 69, 71(55), 79(14), 93, 120(336), 137, 138(55), 163(14), 168(516) Ashcroft, N. W., 156(472), 164(504) Aspnes, D. E., 89(274), 124(344), 130(359) Aulbur, W. G., 14(42), 33(42), 63(195, 196), 64(195), 96(196), 116(333), 117(335), 130(42), 133(42), 191(592), 193, 194 Averin, D. V., 438 Aversa, C., 175(538), 189 Avery, A. R., 232(152, 153) Azuma, H., 228(83)

B

Bachelet, G. B., 63(192), 64(192), 96, 110 Backrach, R. Z., 149(433-435) Backes, W. H., 42(130), 104(318), 110 Badjou, S., 497(316) Baerends, E. J., 204(637) Bailes, A. A. 111, 228(77, 78) Balakrishnan, K., 232( 166) Baldereschi, A., 14(50), 16(54), 28(90), 29(50, 54, 91), 35(107), 42(90), 43(90), 45, 104(90), 137(54, 380, 381), 138(54), 144(415), 179(555) Balduz, J. L., 12 Barabasi, A.-L., 240-241 Baratoff, A,, 458 Barbara, P., 425(206), 429(212) Barber, R. P., Jr., 456(276) Barkema, G. T., 222(9)

513

514

AUTHOR INDEX

Barker, A. S., 183(559) Barone, A,, 276 Baroni, S., 44(140, 141), 45, 68(140, 141), 94, 144(41S), 179(554) Barnett, G. M., 402 Barrera, R. G., 191(589) Bar-Yam, Y., 77(243) Bastard, G., 141(403) Batrouni, G. G., 467 Baym, G., 19, 73, 79 Bean, J. C., 141(407) Beasley, M. R., 427(207, 208), 471 Bechstedt, F., 3, 33(104), 34(104), 35(108), 104, 151, 182, 183, 189 Beck, H., 354(100), 363(114), 383(147), 387(155) Becker, R. S., 151(450) Beeferman, L. W., 45(151), 183 Begley, A. M., 222(17) Behet, M.,231(146) Behm, R. J., 222(12-15) Behrend, J., 230(128) Bei der Kellen, S., 177(547) Ben Avraham, D., 387(152) Bender, C. M., lOS(321) Benedict, L. X., 157(475, 476) Bennett, M. R., 228(85) Benning, P. J., 140(399) Benz, S. P., 285(17), 294(19), 368(120), 380(138), 398(172), 399,400(181), 406,411, 425(204, 205), 427, 429(210) Berbezier, I., 227(31) Berchier, J. L., 266(1), 269 Beresford, R., 232(165) Berezinskii, V. L., 301, 470 Bergstresser, T. K., 113(331) Berkowitz, S. J., 135(368) Bernardini, R., 229(103) Bertel, E., 222(18) Bertoni, C. M., 16(58), 133(367) Bethea, C. G., 188(571), 190(582) Bevk, J., 141(407) Bindslev-Hansen, J., 271 Binney, J. H., 476, 477 Binning, G., 172(528, 529) Birmberg, D., 132(364) Bishop, A. R., 381(140) Blase, X., 14(51), 92(279), 140(401, 402), 148(429), 149(436) Blote, H.W.J., 352(88)

Bliigel, S., 238(187) Bobbert, P. A,, 37(119), 42(130), 71, 73(119, 236), 80(119, 236), 82(119, 262), 86(119), 389(161), 393 Boerma, D. O., 222(9) Bohm, R. B., 157(475, 476) Bolmont, D., 229(110) Bonnet, J. E., 149(436) Bonnouvrier, F., 199(616) Bonzel, H. P., 223(22), 224(25), 232(158) Booi, P.A.A., 380(138), 425(205), 429(210) Booth, K. H., 227(43) Bormann, W., 46( 157), 49( 157), 50( 157) Born,,'M., 66(206) Bornsen, N., 133(367) Bortner, L., 399(176) Boshart, M. A,, 228(77, 78) Botchkarev, A., 145(422) Bowen, C., 86(270) Bowen, K. H., 171(521) Boyd, J. T., 269(4) Bradshaw, A. M., 70(219) Brandt, M. S., 161(490) Brandt, O., 231(144, 147) Braunstein, R., 124(341) Breeman, M., 222(9) Brener, N., 44(138, 139), 45 Brey, L., 154(465) Bringans, R. D., 149(433-435) Brinkman, W., 27(83), 42 Brister, K. E., 155(467) Bross, H., 33(99), 45 Brown, J., 230(131) Brown, K.,399(176) Brown, R., 458(280) Bruder, C., 450(256), 462 Bryden, W. A., 188(572) Bugiel, E., 227(45, 61-64), 228(70, 74, 75), 229(116) Burke, K., 218(672) Burroughs, C. J., 425(204), 427 Bylander, D. M., 96, 159(486), 197 C

Cacham, H., 153(455-458) Cafolla, A. A., 228(85) Cai, Y., 381(144, 145) Caignol, E., 139(393) Cairns, J. W., 228(85)

515

A U T H O R INDEX Calcott, P.D.J., 162(498) Camarero, J., 223(23, 24) Cao, 228(65, 68, 69, 86) Capasso, F., 142(414) Capezzali, M., 387, 388 Capitani, J. F., 202 Cappellini, G., 33--34, 35(108), 104 Car, R., 179(555) Carbrera, B., 272(13) Cardona, M., 8(29), 96(300), 154(464), 161(490), 232(151)

Carini, J. P., 339(63), 438 Carruthers, P., 434 Carvallo, A,, 190(586) Cawthorne, F. C., 425(206), 429 Ceperley, D. M., 125, 205(643) Cerullo, M., 227(52) Cha, M.-C., 376,466 Chabal, Y. J., 149(436) Chacham, H., 237(184) Chaikin, P., 468 Chakabarti, A., 376 Chakravarty, S., 458 Chang, E. K., 16(57) Chang, K. J., 97(301), 237(183) Chang, T.-L., 227(50) Charlesworth, J.P.A., 147(425-427) Cheetham, A. K., 139(390) Chelikowsky, J. R., 124(352), 175(539) Chelly, R., 229( 110) Chen, A,, 154(461) Chen, H., 229( 114) Chen, J., 177(540, 542), 182, 183(560), 188, 189

Chen, N. H., 156(470) Chervinsky, J. F., 227(37) Chester, G., 205(643), 309, 487 Chevary, J. A., 159(482) Chiang, C., 135(371) Chiang, T.-C., 124(347, 351), 229(107) Ching, W. Y.,177(545) Cho, C.-C., 232(136) Choi, M. Y., 351(85, 86), 368(118), 378 Chou, M. Y., 169(517), 206(648) Choyke, J. W., 135(369) Choyke, W. J., 132(363, 364) Christensen, N. E., 63(192, 193), 64(192), 94 Chui, S. T., 191(590) Chun, Y. J., 230(127) Chung, J. S., 378(134), 400(180)

Chung, T.-C., 162(494) Ciria, J. C., 271 Clementi, E., 164(502) Cociorva, D., 63(196), 96(196) Cohen, M. L., 14(45,46,47,49,51), 31(92), 35, 38(121), 63(194), 65(205), 71(46), 92(279), 96(205), 97(301), 113(331), 140(401,402), 141(408), 142(409-412), 144(417), 148(430), 153(459), 169(517), 171(522, 523), 175(539), 185 Cohen, P. I., 230(132) Comsa, G., 222(11, 13, 16) Coniglio, A., 359(108) Conti, S., 205(639, 640) Continenza, A., 16(54), 29(54, 91), 137(54, 380), 138(54) Cooper, I. L., 45 Copel, M., 220, 221, 226(26), 227(30, 51, 53), 228(73, 87, 94, 95) Corkill, J. L., 14(45, 47), 133(47), 141(408) Costabile, G., 468 Crisp, R. S., 167(509) Croke, E. T., 229(108, 109, 120) Cui, S. F., 229( 114) Cui, Q., 229( 114) Cullis, A. G., 162(498) Cunningham, J. E., 230(129)

D

Dabiran, A. M., 230(132) Dahlheimer, B., 228(72) Dahms, U., 162(496) Dal Corso, A,, 178(552), 179(554, 555), 189 Daling, R., 28(88), 71(88), 81(260) Dasgupta, C., 376 Davies, J. H., 141(404) Davies, P. K., 139(393) Davis, R. F., 135(369) Davoli, I., 229(103) Dawson, P., 144(418) Day, G. W., 190(581) De Crescenzi, M., 229(103) deGennes, P. G., 332, 359(106, 107) de Groot, F.M.F., 139(386) de Groot, H. J., 71, 73(236), 80(236) de Groot, L.E.M., 449(254) De Heer, W. A., 169(517) Deisz, J., 172 Delamarre, C., 230(135), 231(139)

516

AUTHOR INDEX

Del Sole, R., 28(89), 33(104), 34(104), 35(108), 38(89), 80(89), 125(89), 157(474,480), 183(561, 563, 564), 186(569), 191(588) de Miguel, J. J., 223(23, 24) de Providencia, J., 207(651) Derrien, J., 227(31) Devaty, R. P., 132(363) DiDomenico, M., Jr., 64( 199) Dietrich, B., 227(61, 64), 228(74, 75) Dittschar, A,, 223(24) Dobson, J. F., 204(633) Doderer, T., 38q 138) Domany, E., 351(87) Dominguez, D., 271, 381,417,418-419,420, 42 1 Donath, M., 172(531) Dondl, W., 227(47), 229(118) Dong, C., 229(114) Doniach, S., 351(85), 460, 462, 471, 497 Dowrick, N. J., 476(312) Drabkin, I. A., 139(385) Drake, G.W.F., 159(483) Dreizler, R. M., 207(651, 652) DuBois, D. F., 79 Dubon, A., 231(139) Dumas, P., 149(436) Dunscombe, C. J., 228(85) Dykhne, A. M., 364(115) Dynes, R. C., 456(272) Dzaloshinski, I. E., 6(17), 446(245)

E Eaglesham, D. J., 227(46, 52), 240 Eastman, D. E., 124(340, 348) Eaton, J. G., 171(521) Ebner, C., 359 Eckart, F., 190(584) Eckern, U., 387(153), 388(158), 389, 453, 458(290) Eddy, C. R., 135(368) Eden, R. C., 124(343) Edmond, J. A,, 132(363) Efetov, K. B., 447(247), 448, 449, 452 Eggert, J. H., 155(466) Eguiluz, A. G., 73(230, 235), 74(235, 238), 76, 92(280), 172 Ehrenfreund, E., 162(494) Ehrenreich, H., 35, 45(151), 64(197), 69(197), 183, 185

Ehrlich, G., 236 Eichenberger, A. L., 363, 365 Eissler, D., 231(138) Elharnri, S., 317(39), 406(189) Elion, W. J., 437(225), 454(265) Ellis, M. A., 112(329) Emel’yanova, L. T., 139(385) Emzerhof, M., 218(672) Engel, G. E., 37(118), 38, 40-42, 63(188), 86, 110,207 Engels, B., 238(187) Epstein, K., 307(33) Eskes, ,H., 139(386) Esposito, F. P., 399(176) Esser, N., 232(161) Esteve, D., 451(257) Etgens, V. H., 230(133, 134) Ethridge, E. C., 160(488) Evans, K. R., 230(130, 131)

F Fahy, M. R., 232(152) Fahy, S., 154(462) Falicov, L. M., 154(461) Falta, J., 227(30), 228(87, 88) Fanciulli, M., 135(368) Farid, B., 4, 37(118), 38(120), 39, 40-42, 77, 80(255), 86, 110 Fasol, G., 133(365) Fauchet, P. M., 162(498) Fauster, T., 172(530) Fayet, P., 171(518) Fazekas, P., 448(251) Fazio, R. A,, 389, 393(166), 450(256), 466(300), 467, 497(317) Feldblum, A., 162(494) Fender, B. E. F., 139(391) Fermi, E., 217 Fernandez, J. M., 229(113) Ferraz, A. C., 237(184) Ferrell, R. A., 432, 458 Fertig, H. A., 376 Fetter, A. L., 6(18) Feuillet, G., 231(141), 232(166) Feynman, R. P., 443(241), 453 Fiedler, M., 151 Fimland, B. O., 232(161) Fincher, C. R., Jr., 162(491) Fink, J., 162(492, 493)

AUTHOR INDEX Finney, M. S., 227(58, 59) Fiolhais, C., 159(482) Fiorentini, V., 63(189), 178, 222(8) Fiorino, E., 191(588) Fiory, A. T., 324(47-49), 331, 332 Fischer, J. E., 139(393) Fisher, A. J., 476(312) Fisher, D. S., 380( 136, 137), 45q255) Fisher, M.P.A., 360, 450(255), 452, 458(278, 279), 459, 466(299) Fishman, R. S., 448,455(269) Flesch, J., 64(202), 65(202), 66(202) Fleszar, A., 33, 38(102), 66(102), 71, 73(102), 91(102), 97(302), 121, 125, 172(534) Flores, F., 172(528) Foglietti, V., 360( 113) Forrester, M. G., 368, 372, 374, 375-376, 378 Foulkes, W.M.C., 206(648) Fowler, W. B., 44(134, 135), 45(134, 135) Foxon, C. T., 144(418) Frank, D. J., 359(111) Frank, K. H., 172(529) Franklin, G. E., 124(351) Fraxedas, J., 8(29) Free, J. U., 398(172), 399(179), 400(181), 403. 41 1 Freeman, A. J., 93(285), 177(547, 550) Fritsche, L., 199(61 I), 203 Fritschy, F. C., 295(21), 298(21), 388(157), 390( 165) Fritzsche, H., 162(493) Frohner, J., 162(496) Froyen, S., 65(205), 96(205), 97(301) Fry, J. L., 35(111), 160(488) Fuchs, H. D., 161(490), 172(529) Fuggle, J. C., 163(500) Fujimori, A,, 139(387) Fujita, S., 228(89) Fujita, T., 230(121) Fujuhara, A., 229(112) Fukuyama, H., 452(259) Fulde, P., 6(19), 17(19), 32(19), 46, 49(157, 159, 175, 175), 50(157, 159, 177, 179), 50, 81(260) G Gadzuk, J. W., 45 Galitskii, V., 77

517

Gallagher, W. J., 360( 11 3) Gallas, B., 227(31) Galli, G., 151(444) Garcia, A., 63(194) Garcia, N., 172(528, 529) Garland, J. C., 269(4), 358(104), 399(176178), 413, 415, 417, 418, 421 Garno, J. P., 456(272) Gavilano, J. L., 324(50) Gavrilenko, V. I., 182, 183(561, 562) Geerligs, L. J., 442(238), 449, 454(265), 459, 497(3 17) Geigenmuller, U., 387( 154), 389, 393(166), 394, 437(225), 448(253), 459(293) Geurts, J., 231(146) Ghahramani, E., 185(566) Ghosez, P., 174(537), 191 Giannini, C., 231(147) Giesekus, A,, 154(461) Giovannella, C., 271 Girlanda, R., 183(563) Girvin, S. M., 438, 452(261), 459, 466(299) Glover, R. E. 111, 336(61), 456(276) Godby, R. W., 3, 28(89), 33(103), 38(89, 103), 42, 44(6), 56(185), 57(185), 61(185), 63, 71(226), 73(234), 74, 76, 77, 79, 80(89, 259), 81, 90(6), 91(103, 185, 276), 92(278), 110, 117(335), 124(350), 125(6, 89, 103), 130(360), 133(367), 135(374, 379, 147(425-427), 154(463), 157(480), 163(499), 174(537), 182, 191, 192, 194, 207 Goettel, K. A., 155(466) Goldman, A. M., 307(33), 456(275, 277), 468 Golovchenko, J. A., 227(37) Gomez, L. B., 406(189) Goncharov, A. F., 156(471) Gonze, X., 135(370), 174(537), 189(579), 191 Goodman, B., 27(83), 42, 340(65) Goosen, K. W., 230(129) Gorczyca, I., 63(193), 94 Gorkov, L. P., 6(17), 446(245) Gorling, A,, 68(212), 159(481), 197, 199(610), 200,204 Goshorn, D. P., 139(393) Gradshteyn, I. S., 497(318) Graf, T., 223(24) Graffenstein, J., 50(177) Graham, R. J., 135(368) Granato, E., 351(78-84), 352(90, 91), 368, 370, 372, 374-376,378-380

518

AUTHOR INDEX

Grandjean, N., 221(5), 230(124, 125, 133135), 231(136, 148) Grant, R. W., 145(423) Crest, G., 352(93) Greuter, F., 167(508) Grinstein, G., 450(255), 459 Gritsenko, 0. V., 204(637) Grobman, W. D., 124(340) Gronbech-Jensen, N., 381(140) Cropper, 439 Gross, E.K.U., 202(624, 625), 203(629), 204, 207(652,653) Grossman, U. J., 203(629), 204(636) Grosso, G., 44(140), 45(140), 68(140) Griindler, R., 162(496) Gu, Y. M., 199(611), 203 Gunnarsson, O., 4, 11(37), 12, 16(55), 29(55), 31(14), 44(145), 64(145), 68(145), 69,71(55), 79(14), 93, 137, 138(55), 139(383), 140(400), 163(14), 196(600), 215(37) Gunnella, R., 229( 103) Giinther, G., 149(436) Gupta, A., 360(1 13) Gutfreund, H., 272(13) Gygi, F., 28(90), 42(90), 43(90), 45, 104(90), 137(38l), 15l(444) H

Ha, J. S., 232(159) Ha, Y. H., 227(44) Haas, W., 190(585) Hacke, P., 231(141), 232(166) Hadjisavvas, N., 202(623) Hadley, P., 427, 429(209) Hagenaars, T. J., 383, 387(151), 393 Halperin, B. I., 295(20), 302(25), 320(44,45), 385(45), 493 Halsey, T. C., 347(69), 351(76), 354, 357, 375, 41 1 Hamada, N., 93(285) Hamaguchi, H., 231(141), 232(166) Hamann, D. R., 96, 110, 135(371) Hamilton, B., 162(498) Hanfland, M., 156(471) Hanke, W., 32(96), 33, 38(102), 46, 47(168), 48, 49(156, 169), 50, 66(102), 71, 73(102), 91(102), 121, 125, 157, 172 Hansson, G. V., 229(101) Harris, A. B., 358, 360

Harris, D. C., 358-359, 360, 362, 365, 399( 176) Harris, J., 215(663) Harris, J. S., Jr., 230(122), 232(157) Hasegawa, S., 140(398), 228(91) Hasegawa, T., 227(48), 229(112) Haviland, D. B., 456(275) Havlin, S., 387(152) He, Y., 135(368) Hebard, A. F., 324(47-49), 331, 332, 456(270) Hebboul, S. E., 399(177, 178), 406, 408-410, 413,415, 417, 418,421 Hedin, L., 3, 8(26, 27), 12, 18-24, 22(27), 23(27), 24(27), 25(4, 26, 27), 26(4), 27(84), 29(27), 42(4, 132), 46(65), 61(4), 66(4), 71(3), 80(3), 125(356), 163(132) Heeger, A. J., 162(491, 494, 495) Heimann, P., 124(348) Heime, K., 231(146) Heinrichsmeier, M., 172(534) Heinz, K., 223(23) Heisenberg, W., 66(206) Hemley, R. J., 156(471, 472) Henrion, W., 190(584) Henzler, M., 227(55), 228(71, 72), 229(96, 97, 1 Herbert, S. T., 317, 335(39), 336, 337, 338, 358(104), 406(189) Hermann, A,, 171(519) Hermans, J., 231(146) Hermanson, J., 46 Heubener, R. P., 380(138) Hibbs, A. R., 443(241) Higuchi, S., 228(70, 86) Hille, A., 228(88) Himpsel, F. J., 124(342, 348, 349), 127(358) Hindgren, M., SO(256) Hioki, T., 228(83) Hirayama, H., 231(143) Hochst, H., 232(154, 155) Hodges, C. H., 45 Hofer, U., 172(530) Hoffman, D., 380(138) Hofmann, K. R., 229( 104) Hofmann, P., 70(219) Hohenberg, P., 10(32), 192,201,202, 203, 208, 209,470 Hohenstein, M., 231(144, 147) Holloway, S., 79(249) Holm, B., 73(231, 237), 74, 76, 80(237), 85

w

519

AUTHOR INDEX Holmes, D. M., 232(152) Holmlund, K., 387 Homma, Y., 228(93) Hong, H., 229(107) Hood, R. Q., 199(612, 613). 206 Hope, D. A. O., 139(390) Horikoshi, Y., 232(150, 151) Horn-von Hoegen, M., 227(30, 51, 56), 228(71-73, 94, 9 3 , 229(96-100, 104) Horsch, P., 38, 39, 46(158, 159), 49(158, 159), SO(158, 159), 69, 70(224), 73(216), 92, 110, 163(216) Horsch, S., 46(158, 159), 49(158, 159), SO(158, 159) Horton, G. K., 197 Hott, R., 39(123), 69(123) Hricovini, K., 149(436) Hsieh, T. C., 124(347) Hsu, W. Y., 191(590) Hu, D., 228(80) Huang, D., 228(80) Huang, I.-$ 227(50) Huang, M.-Z., 177(545) Huberman, B. A,, 471 Huda, F. G., 236(182) Hiifner, S., 8(28), 139(388) Hughes, J.P.L., 181, 188, 189 Hui, F. C., 406 Huijser, A,, 148(431) Hulliger, F., 139(388) Hulthen, R., 124(346) Humphreys, R. G., 132(364) Hunter, A. T., 229(108, 109, 120) Hwang, M., 93(285) Hybertsen, M. S., 3, 14(44), 23(81), 27, 28(44), 36, 37, 38, 42, 44(5, 144), 52(44), SS(44, 184), 57(44), 59(44), 62(44), 71(44), 80(257), 90(5), 103(81), 108, 111, 126(44), 127(44), 141(406, 407), 142(409, 410, 413), 144(417), 147(428), 148(432), 151(450,451)

I Iafrate, G. J., 197 Iarlori, S., 151 Ibarashi, T., 232(164) Ichikawa, M., 226(27), 228(89) Ide, T., 227(57), 2 3 3 174) Ido, T., 139(389) Ikeda, Y., 227(40)

Ikekame, H., 232(164) Ikemoto, K., 140(398) Ikarashi, N., 227(41) Ilegems, M., 183(559) Ilg, M., 231(138) Imry, Y., 452 Indlekofer, G., 149(436) Ingold, G. L., 458(281, 288) Inkson, J. C., 6(20), 32(97) Inokuchi, H., 140(398) Ishibashi, S., 139(389) Ishida, K., 240( 193) Ishida, M., 229(111) Iskenderov, R. N., 139(385) Ito, T., 237, 238(186) Iwai, S., 231(142, 143) Iwanari, S., 227(33-35), 250(205)

J Jaccard, Y., 356(102) Jackson, K. A., 159(482) Jackson, S. A,, 141(407) Jacobson, A. J., 139(391) Jacobson, D. C., 227(46) Jaeger, H. M., 4.56 Jain, A. K., 271, 425 Jan, W., 230(129) Janak, J. F., 21 l(657) Jasnow, D., 354(98), 477 Jayaprakash, C., 347(68), 351(75), 356 Jeanloz, R., 154(461) Jeanneret, B., 324(50) Jenkins, S. J., 234 Jennings, P. J., 172(526) Jensen, E., 165(505), 168 Jiang, X., 228(80), 336(61) Jiang, Z., 228(80) Jin, A., 124(352) Joannopoulos, J. D., 351(73) Joelsson, K., 229(101) Johansson, L.S.O., 151(442) John, G. C., 162(498) Johnson, D. L., 37-38, 39 Johnston, D. C., 139(393) Jona, F., 222(17) Jones, R. O., 11(37), 172(526), 215(663) Jones, T. S., 232(152, 153) Jonkman, H. T., 139(397) Jonsson, A., 318(43)

520

AUTHOR INDEX

Jonsson, L., 188(573), 189, 190(583), 191(592) Jorritsma, L. C., 222(16) Jose, J. V., 271, 352(92), 368(122), 383(148), 387(151), 393(167, 168), 417(195-198), 418-419,420,421,439,442 Josephson, B. D., 271(11), 274(11) Joyce, B. A,, 227(39), 229(113, 117), 232(152, 153) Jun, Y., 317(39), 406(189) Jusko, O., 227(55, 56)

K Kadano5, L., 19, 73, 79, 368(122) Kadin, A. M., 307, 494 Kahng, S.-J., 227(43, 44) Kahng, Y., 227(43) Kalos, M., 205(643, 644) Kamrnler, M., 229(104) Kampf, A,, 450(256), 458(284, 285) Kandel, D., 238, 242 Kane, E. O., 44(137), 45 Kappes, M. M., 171(520) Kardar, M., 241(196) Karlsson, C. J., 151(442) Karlsson, K., 21(71, 77), 168(516) Karma, A,, 417(195) Kasowski, R. V., 191(590) Kaspi, R., 230(130, 131) Katayama, M., 228(79) Katayama-Yoshida, H., 140(398) Kawabe, M., 230(121, 123, 126,127), 232(157) Kawano, A., 228(83) Kaxiras, E., 220(2), 233, 234, 238, 241, 247 Ke, S.-H., 145(421) Kelly, M. K., 8(29) Kent, P.R.C., 199(613), 207 Kerker, G. P., 110 Kern, R., 220, 239 Kibbel, H., 229(119) Kidder, L. H., 171(521) Kikuchi, K., 140(398) Kim, C. K., 154(464) Kim, E., 236 Kim, S. K., 222(17), 227(44) Kim, Y., 232(167) Kimhi, D., 318(40) Kimura, Y., 250(205) Kipp, L., 151(438-440) Kirkpatrick, S., 297(22), 368(122)

Kirschner, J., 223(24) Kisielowski, C., 232(167) Kissinger, W., 228(74, 75) Kissner, J. G., 452-453 Kistenmacher, T. J., 188(572) Kitani, T., 227(40) Kivelson, S., 458(281, 288) Klapwijk, T. M., 316(38) Klatt, J., 227(45, 60-64), 228(70), 229(115) Klein, B. M., 177(544, 549) Kleinman, L., 96, 159(486), 197 Kliewer, J., 222(16) Klockenbrink, R., 232(167) Knight,'W. D., 169(517) Knops, H.J.F., 352(88, 89), 380(135) Knops, Y.M.M., 352(88) Knorr, W., 207 KO, Y.J., 237 Kobayashi, S., 452(259), 456(271, 274) Koch, R. H., 360 Kohler, U., 227(55, 56), 228(72) Kohn, W., 10(33), 11, 50(39), 86(272), 137, 166, 171(525), 191, 192, 193, 194, 196, 197, 199-2 17 Kojima, M., 456(273) Kolahchi, M. R., 402 Komori, F., 456(274) Kono, S., 228(76) Konogi, HH., 234(170) Konomi, I., 228(83) Koopmans, B., 139(397) Kootstra, F., 204(637) Koren, G., 360(113) Korshunov, S. E., 351(77), 363(114), 375, 376, 380 Koshihara, S., 139(389) Kosterlitz, J. M., 301, 304-311, 316, 318, 319, 324, 326, 328, 333, 334, 336, 347, 349, 350-353, 355, 358, 360, 362, 363, 365, 371, 374, 378-380,383,438-442,469,470-471, 475-478, 482,483,485, 486 Kotani, T., 197, 198 Koulmann, J. J., 229(110) Kowalczyk, S. P., 70(220), 124(339) Krakauer, H., 177(549) Kralik, B., 16(57) Kremer, R. K., 139(396) KreD, C., 151 Krieger, J. B., 197 Krost, A., 232(161)

AUTHOR INDEX Kruger, D., 227(45), 228(66), 229(115) Kruger, J., 232(167) Kriiger, P., 4(12), 14(52), 36(114), 37(114), 44(52), 64(52), 68(52, 213, 214), 69(213), 70(218), 93(284), 94, 95, 96(52), 97(303), 120(336), 130(114), 150(437), 151(441,452), 152(454) Ksendzov, Y. M., 139(385) Kubo, Y., 93c287, 288), 94, 168(515) Kuch, W., 223(24) Kuiper, P., 139(386) Kuk, Y., 227(43, 44) Kunz, A. B., 44(133, 136), 45(133, 136), 46 Kuper, C. G., 456(277) Kurps, R., 228(66) Kusumi, Y., 228(89) Kvale, M., 406, 408-410 Kwon, M. C., 336(61) Kwon, Y., 207(650) Kyoya, K., 228(67, 81, 82)

L Lachenrnann, S. G., 380 Lagally, M. G., 242 Lambrecht, W. R. L., 132(363), 145(420), 188(576) Landau, L. D., 440 Landau, S. D., 351(73) Landemark, E., 151(441, 442) Lanelaar, M. H., 222(9) Lang, C., 231(138) Lang, D. V., 141(407) Lang, N. D., 166, 171(525) Langreth, D. C., 21(74), 218(669) Lapierre, R. R., 231(137) Larkin, A. I., 458(291) Larson, B., 467(303) Larsson, M. I., 229(101) Lauchlan, L., 162(491) Laval, J. Y., 230(135), 231(139) Leath, P. L., 381, 383,417,424 Le Bellac, M., 469 Lee, D. H., 351(73) Lee, E.-H., 232(159), 237(183) Lee, G. H., 171(521) Lee, H.-C., 399(176), 400(183), 413,415, 417, 419,421,423-424 Lee, H. J., 236(178), 368(121) Lee, L H . , 218(673)

521

Lee, J., 227(43), 351(80, 84), 352(90, 94) Lee, K. C., 352(95) Lee, K.-H., 400(180, 182) Lee, S., 352(95) Lee, T. K., 83(266) Lee, Y. H., 236(178-180) Leemann, Ch., 324(50), 331, 352(97), 354(100), 356(102), 383(147), 399(173), 487 Lees, A. K., 229( 113) . LeGoues, F. K., 228(87, 94, 95) Lehrnann, G., 162(496) Lei, T., 135(368) Leisenberger, P., 232( 160) Leising, G., 162(492) Le Lay, G., 220( 1) Lerch, Ph., 331(55, 56), 352, 353, 354(100), 356, 399(173) Leung, M.S.H., 232(167) Levin, Y., 77 Levine, B. F., 64(200), 188(571), 190(582) Levine, Z. H., 35, 37, 63( 199, 64( 199, 174(535, 536), 177(540-543), 181, 182, 185, 188(573, 575), 189, 190(583), 191(587) Levinson, H. J., 167(508) Levy, M., 12, 199(610), 201, 208, 212(659) Ley, L., 70(220), 124(339, 349), 127(358) Leyvraz, F., 318(40) Li, H. H., 182(557) Li, J.-G., 146(424) Li, M., 228(80), 229(114) Li, M. S., 376(131) Li, Qi, 336(61) Li, R. R., 400,406(189) Li, Y., 197 Lieb, E. H., 201, 208 Liebsch, A,, 21(70) Lifshitz, E. M., 440 Likharev, K. K., 271, 365(117), 435, 438, 468 Liliental-Weber, Z., 227(48, 49) Lim, K. Y., 236(178) Lin, D.-S., 229(107) Lin, X. W., 227(48, 49) Lindelof, P. E., 271 Linderberg, J., 45 Lines, M. E., 44( 143) Lipari, N. O., 35(112), 44(133-136), 45 Lipkin, N. N., 222(16) Lippert, G., 227(60,61,63,64), 228(66,70, 74, 75) Little, W. A,, 272(13)

522

AUTHOR INDEX

Liu, H.-Y., 232(136) Liu, X., 228(80) Lobb, C. J., 285(17), 290(18), 291(18), 292(18), 294(19), 304(30), 316(38), 317(39), 336(61), 345(67), 359(11l), 368(120, 121), 387(154), 398(172), 399(174, 175, 179), 400(181), 411(193), 413(194), 425(203,206), 429(212), 444(243), 486 Lof, R. W., 139(397) Lohmeier, M., 222(10) Louie, S. G., 3, 14(43, 44, 46-48, 51), 16(57), 23(81), 27, 28(44), 35-38, 42, 44(5, 43, 144), 52(44), 55(44, 184), 57(44), 59(44), 62(44), 65(43, 205), 69(43), 71(44, 46), 80(257, 258), 83(184, 258), 90(5), 92(279), 93(282), 95(282), 96(43, 205, 297), 103(81), 111, 121(338), 124(353), 126(44), 127(44), 139(394), 140(401,402), 142(409-412), 144(417), 147(428), 148(429, 430, 432), 149(436), 151, 153(455-459), 154(462), 157(477), 171(522, 523) Louis, E., 45 Louis, P., 229( 110) Lozovik, Y. E., 455(267) Ludwig, M. H., 162(498) Lukas, W.-D., 69, 73(216), 163(216) Lukens, J. E., 271(8), 468 Lukens, P., 406(189) Lundqvist, B. I., 21(72), 32, 33,38,42(132), 45, 163(132), 196(600) Lundqvist, S., 3(4), 12, 21(72), 42(4), 61(4), 66(4), 111 Luo, H., 155(468) Luther, A,, 458(281) Lutjering, G., 227(47) LYO,I.-W., 165(506), 168

M Ma, A. P., 232(162) Ma, H., 191(590) MacDiarmid, A. G., 162(491, 494, 495) MacDonald, J. E., 227(58, 59), 228(85), 230( 132) Madelung, O., 199(614) Maehashi, K., 228(91) Maekawa, S., 452 Magel, L. K., 232( 136) Mahan, G . D., 3-4, 6(21), 48, 73(229), 74(240, 241), 79, 80, 82, 83(229, 240, 241, 265),

85, 166, 167, 168 Mai, Z. H., 229(114) Majewski, J. A,, 197(609), 199(610) Malmstrom, G., 172(527) Manaa, M. R., 171(521) Manzke, R.,42(129), 151(438,439) Mao, H.-K., 156(471) Mao, M., 228(80) March, N. H., 37(115), 41, 42(131) Marcus, P. M., 222(17) Margaritondo, G., 142(414) Markov, I., 240 Martens, F. F., 190(580) Martin, G., 145(422) Martin, R. M., 21(67), 64(201), 65(201), 66(67, 201), 159(67, 485), 191(594, 597), 206(645), 207(650), 218(673) Martin, S., 155(467) Martin, W. C., 159(483) Martinez, R. E., 227(37) Martinoli, P., 324(50), 331(55, 56), 352(97), 354(100), 356, 363(114), 380, 383(147), 399( 173) Martins, J. L., 96, 110, 135(372) Maruno, S., 228(89) Massidda, S., 14(50), 16(54), 29(50, 54,91), 137, 138(54), 177(550) Massies, J., 221(5), 230(124, 125, 133-135), 231(136, 148) Mast, D. B., 317(39), 398(172), 399(176), 400(183), 406(189), 413(194), 423(201) Materlik, G., 163(501), 228(88) Matricon, J., 340(65) Matsuhata, H., 228(67, 81, 82) Matsuyama, T., 227(40) Mattausch, H. J., 46(156), 48(156, 170, 171), 49(156, 172) Matters, M., 437(225) Mattuck, R. D., 6(22) Mauri, F., 178(552) Mazur, A., 152(454) McConville, C. F., 228(79) McCumber, D. E., 275(15) McFeely, F. R., 70(220) McGill, T. C., 229(108, 109, 120) McHugh, K. M., 171(521) McMahan, A. K., 155(467) McMillan, W. L., 205(642) Meade, C., 154(461) Mele, E. J., 162(495)

523

AUTHOR INDEX Memmel, N., 222(18) Menczigar, U., 229(119) Mendoza-Diaz, G., 232(165) Mermin, N. D., 164(504), 208(656), 303(26), 443,470 Meskini, N., 48(169, 171), 49(169) Methfessel, M., 63(189) Metois, J. J., 220(1) Meyer, J. A,, 222( 12- 15) Meyer, M., 16(58) Meyer, R., 356(102) Meyer, W., 64(202,203), 65(202,203), 66(202, 203) Migdal, A,, 77 Miki, K., 228(67, 81, 82) Miller, T., 124(347, 351), 229(107) Min, B. I., 177(550) Minami, F., 139(387) Minnhagen, P., 21(75), 45(152), 71(75), 80, 271, 318, 354(99), 383(147), 387, 388, 495 Minoda, H., 227(31, 36), 228(90), 229(105) Miragliotta, J., 188(572) Miranda, R., 223(23, 24) Mirhashem, B., 458 Mitas, L., 206(645-647) Miwa, R. H., 237 Miyagawa, T., 232( 163) Miyashita, S., 351(72) Mochan, W. L..,191(589) Mokler, S., 227(39), 229( 1 17) Molnar, R. J., 135(368) Mombelli, M., 363( 114) Monemar, B., 182(558) Monkhorst, H. J., 102(315) Mooij, J. E., 295(21), 298(21), 318(42), 320(46), 339(62, 64), 388(157, 159), 390( 165), 437(225), 442(238), 449(254), 454(265), 459, 468, 471(309), 476, 497(3 17) Moore, K. J., 144(418) Morikawa, T., 140(398) Morishita, Y., 232(164) MorkoG, H., 145(422) Moses, D., 162(494) Moss, W. C., 155(466) Moulin, D., 231(146) Moustakas, T. D., 135(368) Muller, C. J., 339(64) Muller, B., 227(55, 56), 228(71, 72), 229(96, 97, 99)

Miiller, J. E., 163(501) Miiller, P., 239, 468 Miiller, W., 64(202, 203), 65(202, 203), 66(202, 203) N Nagy, A, 196(599), 202(599) Nakahara, H., 226(27) Nakamura, J., 234 Nakamura, S., 133(365) Nakamura, T., 229( 1 11) Nakanishi, Y., 228(84) Nakashima, H., 228(91) Nakayama, T., 228(79) Nattermann, T., 376 Nazzal, A., 63(190) Needs, R. J., 63(188), 77(244), 91(276), 135(374, 375), 147(425-427), 154(463), 163(499), 199(612,613), 206(648) Negele, J. W., 351(73) Nelson, D. R., 316(37), 320(44, 45), 368(122), 372( 124), 385(45), 493 Nelson, J. S., 133(366) Netzer, F. P., 232(160) Neman, M.E.J., 476(312) Newrock, R. S., 269(4), 317(39), 336(61), 398( 172), 399( 176), 400( 183), 406( 188, 189), 413(194), 423(201), 444(243) Ni, W.-X., 229(101) Nienhuis, B., 352(88) Nieto, M. M., 434 Nifosi, R., 205(640) Nightingale, M. P., 351(78, 80), 352(91) Nilsson, N. G., 124(346) Nilsson, S., 229( 115) Nishimori, H., 376 Noda, S., 228(83) Nohlen, M., 222(20) Norris, C., 227(58, 59) Northrup, J. E., 55(184), 80(257, 258), 83(184, 258), 110, 149(433,434), 151(451,453), 161(489), 167, 168(513) Norton, P. R., 232(162) Ncker, N., 162(493) Nusair, M., 217(664)

0 Octavio, M., 398(172), 400(181), 404,406,425

524

AUTHOR INDEX

Ofner, H., 232(160) Oh, C. W., 236(178-180) Ohno, T., 235,236 Ohta, K., 231(141), 232(166) Ohta, S.,230(121) Ohta, T., 354(98), 477 Ohtani, H., 229(117) Ohtani, N., 227(39) Okada, Y., 230(121, 122, 126, 127), 232(157) Okumura, H., 231(141), 232(166) Oliveira, L. N., 202(624, 625) Olsson, P., 318(43), 352 Onida, G., 16(56), 92(278), 157(474,480), 186(569) Oppo, S.,222(8) Orlando, T. P., 295(21), 298(21), 388(157, 159), 390(165), 418(199), 419(200), 468, 47 l(309) Orr, B. G., 231(136), 456 Orszag, S. A., 105(321) Ortega, J. E., 124(342) Ortiz, G., 191(594, 597) Osaka, T., 234(170 Oschlies, A,, 135(374, 375) Oshiyama, A,, 227(41), 235(173-175) Osten, H. J., 227(45, 60-64), 228(66, 70, 74, 79, 229(116) Osterwalder, J., 139(388) Ousaka, Y., 45(153) Ovchinnikov, Y. N., 458(291) Over, H., 222(17) Overhauser, A,, 32(93) Oyanagi, H., 228(67) Ozaki, M., 162(491,495) Ozeki, T., 376

P Pack, J. D., 102(315) Padjen, R., 199(616) Paisley, M. J., 135(369) Palummo, M., 16(58), 133(367) Pandey, K. C., 247 Papadopoulos, A. D., 185(565) Paquet, D., 199(616) Parinello, M., 151(444) Parisi, G., 241(196) Park, J.-Y., 227(43, 44) Park, K.-H., 232(159) Park, S.-J., 232(159), 237(183)

Parr, R. G., 11, 12(40), 207(654) Parravicini, G. Pastori, 44(140, 141), 45(140, 141), 68(140, 141) Pasquarello, A., 179(555) Passek, F., 172(531) Pasternak, M. P., 154(461) Paterno, G., 276 Paulus, B., 49(174-176), SO(178) Pearl, J., 304(29) Pearsall, T. P., 141(405, 407) Pederson, M. R., 159(482) Pederson, N., 468 Peebles, D., 162(491) Peierls; R. E., 443, 470 Peng, Z., 232(150, 151) Penn, D. R., 35 People, R., 141(407) Pepper, S. V., 151(447) Perdew, J. P., 11, 12(40), 159(482), 201, 21 1(40,658), 212(659), 217(665), 218(669672) Peressi, M., 144(415) Peters, M., 449(254), 497(317) Petersilka, M., 203(629) Petrich, G. S., 230(132) Petroff, Y., 8(30), 149(436) Pettersson, P. O., 229(108, 109, 120) Pezzica, G., 44(141), 45(141), 68(141) Philipp, H. R., 64(197), 69(197) Phillips, J. C., 27(85), 418,419-423 Pianetta, P., 228(65, 68, 69, 86) Pickett, W. E., 46, SO(160, 161, 181), 90(275) Pietsch, G. J., 227(55) Pietsch, H., 229(97) Ploog, K. H., 221(3-5), 230(128), 231(138, 140, 144, 145, 147-149) Plummer, E. W., 4, 165(505, 506), 167(508, 510), 168 Poelsema, B., 222(11, 13, 16) Poirier, D. M., 140(399) Pollak, R. A,, 70(220), 124(339) Pollehn, T. J., 71(226), 73(234), 74, 76 Pollmann, J., 4, 14(52), 36(114), 37(114), 44(52), 64(52), 68(52, 213, 214), 69(213), 70(218), 93(284), 94, 95, 96(52), 97(303), 120(336), 130(114), 150(437), 151(441,452), 152(454) Pook, M., 227(56), 229(96) Posternak, M., 14(50), 16(54), 29(50, 54, 91), 137(54, 380), 138(54)

525

AUTHOR INDEX Posthill, J. B., 135(369) Potthast, K., 223(22) Poulter, J., 351(83) Powell, J. A,, 132(363) Prange, R. E., 432 Presting, H., 229( 119) Pulci, O., 186(569)

Q Qteish, A., 63(190) Quinn, J. J., 83(266), 222(17) Quong, A. A., 92(280)

R Rabenau, T., 139(396) Racine, G. A., 324(50), 331(55) Radi, P., 171(520) Radzig, A. A,, 159(484) Raether, H., 42(124, 126) Rajagopal, G., 199(612, 613), 206(648) Ramirez-Santiago, G., 352(92) Ramsey, M. G., 232(160) Rashkeev, S. N., 188(576), 189 Ravindran, K., 317(39), 400(183), 406, 424 Reichlin, R., 155(467) Reihe, B., 172(529) Reining, L., 16(56, 58), 28(89), 33(104), 34(104), 35(108), 38(89), 80(89), 92(278), 125(89), 133(367), 157(474, 480) Reinking, D., 229(104) Repaci, J. M., 336 Resch-Esser, U., 232(161) Resnick, D. J., 269, 311, 316 Resta, R., 14(50), 29(50), 144(415), 179(554), 191(591) ReuD, C., 172(530) Reuter, M. C., 220(2), 227(51, 54), 228(73,94), 236 Reynolds, D. C., 230(130, 131) Rice, T. M., 71 Rich, D. H., 124(351) Richard, P., 238(187) Richter, W., 232(161) Rickayzen, G., 465(298) Rieger, M. M., 33, 38(103), 68(211), 77, 79, 91(103), 98, 102(103), 117(335), 121, 125, 163(499) Riesterer, T., 139(388)

Rijken, H. A., 339(62, 64) Rioux, D., 232( 154, 155) Ritsko, J. J., 162(495) Robinson, B. J., 231(137) Rockett, A., 145(422) Rodrigues, W. N., 237(184) Rodriguez, S., 154(464) Roeser, D., 228(66) Roetti, C., 164(502) Rohlfing, M., 4(12), 14(52), 36(114), 37(114), 44(52), 64(52), 68(52, 213, 214), 69(213), 93(282, 284), 94, 95( 114, 282, 295), 96(52), 97(303), 110, 120(336), 127, 130(114), 150(437), 151(452), 152(454), 157(477) Rohrer, H., 172(528, 529) Rojas, C., 439, 442 Rojas, H. N., 33(103), 38(103), 91(103, 276) Rosenfeld, G., 222(11, 13, 16) Ross, M., 155(467), 156(473) Ruan, J., 135(369) Rubin, M., 232(167) Rubinstein, M., 372, 376 Rubio, A., 14(47, 49, 51), 92(279), 133(47), 140(401, 402), 141(408), 178(552) Rundgren, J., 172(527) Runge, E., 204 Runge, K. J., 467 Ruoff, A. L., 155(467, 468) Russer, P., 397(171) Ryzhik, I. M., 497(318) Rzchowski, M. S., 285(17), 294(19), 399(174, 175, 179), 400(181), 406(187), 411(193)

S Sabisch, M., 4(12), 152(454) Sacchetti, F., 50 Saito, S., 171(522, 523) Saito, T., 232(164) Sakai, A,, 227(38, 41), 240(193) Sakamoto, K., 228(67, 81, 82) Sakamoto, S., 229(105) Sakamoto, T., 228(67, 81, 82) Salvan, F., 172(529) Sanchez, D. H., 266(1), 269 Sano, K., 232(163) Sarkas, H. W., 171(521) Sasaki, A., 227(48), 229( 112) Sasaki, M., 228(76) Sasaki, W., 456(271)

526

AUTHOR INDEX

Sato, K., 227(40), 232(164) Sato, S., 228(76) Sauvageau, J. E., 271(8) Sawatzky, G. A,, 139(386, 397) Scalettar, R. T., 467(303) Scanlon, J., 135(368) Schaff, O., 70(219) Schar, M., 171(520) Scheerer, B., 162(493) Schemer, M., 63(189), 135(370), 222(8) Scheidl, S., 376(131) Schell-Sorokin, A. J., 228(92) Scheuch, V., 223(22), 232(158) Schiffmacher, G., 231(139) Schindlrnayr, A,. 71(226,227), 73(234), 74,76, 80(259), 81 Schintke, S., 232( 161) Schipper, P.R.T., 204(637) Schittenhelm, P., 229( 118) Schliiter, M., 3(6), 42, 44(6), 56(185), 57(185), 61(185), 63, 90(6), 91(185), 96, 110, 124(350), 130(360), 135(37l), 141(406,407), 182, 212(6, 660), 213(6) Schmid, A,, 387( 153) Schmid, P. E., 135(376) Schmidt, G., 223(23) Schmidt, J., 228(88) Schmidt, M., 222(19-21) Schmidt, W. G., 151(446) Schneider, P., 231( 146) Schnieder, C. M., 223(24) Schon, G., 389(160, 164), 393(166), 435, 438, 450(256), 456, 458(284, 285, 290), 466, 467(305), 497(3 17) Schonberger, U., 16(53) Schone, W. D., 73(230, 235), 74(235), 76 Schorer, R.,227(47) Schram, L. L., 221(6) Schreiber, M., 33(99), 45(99) Schroeder, K., 238, 239, 249, 254 Schumacher, E., 171(519, 520) Schwartzrnan, A. F., 232(165) Schwoebel, R. L., 236 Segall, B., 132(363), 145(420), 188(576) Seiberling, L. E., 228(77, 78) Seidl, A., 199, 200 Seki, K., 140(398) Sernelius, B. E., 73(229), 74(240), 80(240), 83(229, 240) Servaty, R., 222( 1 1 )

Shadwick, W. F., 197 Shahar, D., 438 Sham, L. J., 3(6), 10(33), 11, 32(96), 42, 44(6), 46(154, I%), 47(168), 50(39), 56(185), 57(185), 61(185), 63, 90(6), 91(185), 124(350), 130(360), 137, 147(425), 157, 182, 191, 192, 193, 194. 196, 197, 199-217 Shapiro, A. P., 124(347) Shapiro, S., 396 Sharp, R. T., 197 Shenoy, S. R., 326, 328, 385(53, 54), 387(155) Shiba,, H., 351(72) Shih, W. Y., 347(70, 71) Shima, M.,227(36) Shimizu, T., 229(111) Shin, H.-K., 317(39) Shipsey, E. J., 236(181) Shiraishi, K., 237, 238(186) Shirley, D. A,, 70(220), 124(339) Shirley, E. L., 14(43, 47), 21(67), 44(43), 64(201), 65(43, 201), 66(43, 67, 201), 67(43), 69(43), 73(232), 74, 80(232), 84, 85(232, 267), 94, 96(43), 121, 124(353), 133(47), 139(394), 157(475,476, 479), 159(67, 485), 168 Shitov, S. V., 429(212) Shkrebtii, A. I., 186(569) Shoemaker, S., 269(4) Shraiman, B., 372(124) Shumay, I. L., 172(530) Shung, K. W.-K., 73(229), 74(241), 83(229, 241, 265) Siggia, E. D., 320(44, 4 9 , 385(45) Silvera, I. F., 155(466), 156(470) Simanek, E., 438,444,448(250), 452,455(266), 458(280) Simkin, M. V., 336 Simon, A., 139(396) Simond, J.-B., 356(102), 383(147) Singh, D. J., 159(482) Singh, V. A., 162(498) Singwi, K. S., 83(264) Sipe, J. E., 175(538), 181, 185, 188, 189 Sitar, Z., 135(369) Skibowski, M., 151(438-440) Skowronski, M., 230(131) Skriver, H. L., 93(291) Slater, J. C., 217 Smirnov, B. M., 159(484) Smith, N. V., 8(30)

527

AUTHOR INDEX Snijders, J. G., 204(637) Snodgrass, J. T., 171(521) Sohmen, E., 139(396) Sohn, L. L., 399(179), 406(187), 411, 413 Soler, J. M., 172(528) Sondhi, S. L., 438 Sonin, E. B., 389 Serensen, E. S., 452, 467 Souza, I., 191(597) Soven, P., 187(570), 204(570) Spaepen, F., 227(37) Speier, W., 163(500) Spendeler, L., 223(23) Spicer, W. E., 124(343) Spitzer, F., 369(123) Springer, M., 21(71) Srivastava, G. P., 234 Srivastava, Y. N., 435 Stadele, M., 68(212), 197 Stampfl, A.P.J., 70(219) Stauffier, D., 358(105) Stegun, I. A,, 435(224) Steinbeck, L., 33(103), 38(103), 91(103), 101(311), 117(335) Steiner, M. M., 112(330) Stepniak, F., 140(399) Sterer, E., 156(470) Sterne, P. A,, 32(97), 46, 50(162), 51(162) Stevens, K. S., 232(165) Stewart, W. C., 275(14( Stiebling, J., 42(126) Stoffel, N. G., 70(221) Stoll, H., 49(174-176), 50(177) Stollhoff, G., 50(179) Stolz, H., 47(166) Straley, J. P., 359(109), 364(116), 402 Straub, D., 124(349), 127(358) Strinati, G., 46(156), 48(156, 170), 49(156, 172) Strongin, M., 452 Stroud, D., 347(70,71), 351(86), 358(104), 359, 368(118), 378(134), 400(180, 182), 448, 455(268, 269) Stroustrup, B., 11 1(327), 112(329) Struzhkin, V., 156(471) Studna, A. A,, 89(274), 124(344) Stumpf, R., 135(370) Stutzmann, M., 161(490) Sudhir, G. S., 232(167) Sudijono, J. L., 232(152, 153)

Suehiro, H., 45(153) Sugahara, M., 456(273) Sugaya, T., 230( 121) Sugiyama, G., 86(270) Sun, J., 228(80) Surh, M. P., 14(46), 71(46), 80(258), 83(258), 153(455, 459) Sutoh, A,, 232(157) Suttrop, W., 132(363) Suzuki, S., 140(398), 228(76) Svane, A,, 139(383), 199(615) Swartzentruber, B. S., 151(450) Synder, C. W., 231( 136) Szotek, Z., 139(384) T Tada, Y., 456(271) Takagi, H., 139(389) Takahashi, T., 140(398) Takayanagi, K., 227(33-35), 250(205) Taleb-Ibrahimi, A., 149(436) Talman, J. D., 197 Tanaka, M., 162(491) Tanaka, S., 231(142, 143) Tang, D. D., 135(377) Tang, L.-H., 377, 380 Tanishiro, Y., 227(32), 228(90) Tatsumi, T., 227(38, 41), 240(193) Taylor, E. N., 230(130) Taylor, R. D., 154(461) Teichert, C., 222( 11) Teitel, S., 326, 328, 347(68), 351(75), 356 Tejedor, C., 154(465) Temmerman, W. M., 139(384) Tenelsen, K., 183(561, 564) Terry, J., 228(65, 68, 69, 86) Tersoff, J., 251, 256 Theophilou, A. K., 202 Thiron, R., 331(55, 56), 352(97), 356(102), 383,385, 387, 388 Thijssen, J. M., 352(89), 380(135) Thiry, P., 8(30), 149(436) Thiry, P. A., 149(436) Thomann, U., 172(530) Thomas, L. H., 217 Thompson, D. A., 231(137) Thornton, J.M.C., 222(10), 227(58, 59) Thouless. D., 301, 304-311, 316. 318. 319. 324, 326, 328, 333, 334, 336, 347, 349,

528

AUTHOR INDEX

350-353, 355, 358, 360, 362, 363, 365, 371, 374, 378-380, 383,438442,469, 470-471, 475-478,482,483,485,486 Tiesinga, P.H.E., 383(148), 387, 393(167, 168) Tilly, D. R., 424-425 Ting, C. S., 83(266) Tinkham, M., 271(12), 272(12), 276, 285(17), 290( 18), 29 1(18), 292( 18), 294( 19), 304(30), 316(38), 345(67), 347, 348, 368(121), 396(12), 399(174, 175, 179), 400(181), 406(187), 411(193) Tobochnik, J., 309, 467(303), 487 Tokura, Y., 139(389) Tomanek, D., 38(121), 142(409, 410), 144(417) Tomita, M., 228(93) Tomoyose, T., 64(198), 69( 198) Tosatti, E., 35(107), 151(444) Tosi, M. P., 41, 42(131), 205(640) Toulouse, G., 344 Townie, E., 221(3-5), 231(140, 144, 145, 147-149) Towler, M. D., 199(613) Trampert, A., 231(148, 149) Trickey, S. B., 207(655) Tromp, R. M., 220(2), 226(26), 227(30, 51, 53, 54), 228(73, 87, 92, 94, 95), 236, 251, 256 Troullier, N., 96, 110, 135(372) Tsang, J. C., 228(73) Tsong, T. T., 227(50) Tuemmler, J., 231(146) Turner, T. S., 139(386)

U Uchida, S., 139(389) Ueda, M., 459(293) Uhlenbeck, 439 Uhrberg, R.I.G., 149(433-435), 151(441,442) Uimin, G. V., 351(77), 375 Ullrich, C. A., 204(636), 205(639, 640) Ummels, R. T. M., 37(119), 73(119, 236), SO(119, 236), 86(119), 88-89 Unger, P., 81(260) Unterwald, F. C., 227(46) Ustinov, A., 468

V Vaknin, D., 139(393)

Valone, S. M., 202 Vandenberg, J. M., 141(407), 456(270) Vanderbilt, D., 151, 191(596) van der Veen, J. F., 227(58) Van der Vegt, H. A., 222(10, 12, 14, 15) van der Zant, H.S.J., 266(3), 295(21), 298(21), 318(42), 339-342, 388, 389(163), 390, 392-395,418(199), 419(200), 442(238), 454, 458,459,468 Van de Walle, C. G., 161(489) van Elp, J., 139(386) van Gisbergen, S.J.A., 204(637) van Haeringen, W., 28(88), 37(119), 42(130), 71, 73(119, 236), SO(119, 236), 81(260), 82(119, 262), 86(119) van Himbergen, J. E., 383(148), 387(151), 393(167, 168), 394 van Laar, J., 148(431) van Otterlo, A., 389(164), 450(256), 466, 467(305) van Pinxteren, H. M., 222(10) van Rooy, T. L., 148(431) van Silfhout, R. G., 227(58, 59) van Veenendaal, M. A., 139(397) van Wees, B. J., 318(42), 339(64), 352, 497(317) Vashishta, P., 83(264) Venkatessan, T., 336(61) Verbruggen, A,, 449(254) Verdozzi, C., 79(249) Vickers, J. S., 151(450) Vignale, G., 204, 205 Vigneron, J.-P., 189(579) Villain, J., 351(74) Vlieg, E., 222(10, 12, 14, 15) Vogel, D., 4(12), 70(218) Vogl, P., 68(211), 197(609), 199(610) Vohra, Y. K., 155(467,468) Voigtlander, B., 223(22), 224(25), 227(28, 29), 229(102), 232(158), 243-244 von Barth, U., 20(65), 21(69), 27(86), 46(65), 50, 73(231, 237), 74, 76, 80(237), 85, 125(356) von der Linden, W., 38, 39, 69, 70(224), 73(216), 92, 110, 163(216) von Festenberg, C., 42(128) Vong, K. K., 229(112) Vosko, S. H., 159(482), 217(664) Voss, R. F., 318(41) Vrijmoeth, J., 222(12, 14, 15)

529

A U T H O R INDEX Vukajlovic, F. R., 159(485) W

Wachs, A. L., 124(347) Wado, H., 229(111) Wagenblast, K.-H., 466, 467 Wagener, T. J., 124(352) Wagner, H., 303(26), 443, 470 Wagner, J., 135(378) Wakahara, A., 227(48), 229(112) Waldrop, J. R., 145(423) Walecka, J. D., 6(18) Wallauer, W., 172(530) Wallin, M., 452(261), 466(299) Walter, J. P., 31(92). 35 Wandelt, K., 222(19-21) Wang, C. S., 46, 50(160-162, 181), 51(162), 177(544, 549), 189 Wang, J., 467(303) Wang, L. P., 231(139) Wang, X., 228(80) Ward, J. C., 79, 167 Washburn, J., 227(48, 49) Wassermeier, M., 230( 128) Watanabe, H., 42(127), 228(89) Weaver, J. H., 124(352), 140(399) Webb, R. A., 318(41) Weber, A., 224(25) Weber, E. R., 227(48, 49), 232(167) Weber, J., 161(490) Wegner, F., 303(27), 470 Wegscheider, W., 227(47) Wegwood, F. A., 139(391 Wei, S.-H., 63(191), 64(191), 106(322), 177(548, 549) Weichman, P. B., 450(255) Weiler, H., 48(169), 49(169) Weinstein, M., 96(300) Weisenfeld, K., 427(207, 208), 429(210, 212) Welber, B., 154(464) Welkowsky, M., 124(341) Wemple, S. H., 64(199) Wendin, G., 21(66) Wenzel, W., 112(330) Wenzien, B., 104 Werckmann, J., 229( 110) Westman, O., 318(43), 387 Whan, C. B., 387(154), 425(203) White, A. E., 456(272)

White, I. D., 33(103), 38(103), 91(103), 163 White, J., 418(199), 419(200) Wickenden, D. K., 188(572) Widom, A,, 435, 497(316) Wieko, C., 417(195) Wieland, J. L., 139(386) Wiese, W. L., 159(483) Wigner, 439 Wigren, C., 228(86) Wilhelm, J., 227(47) Wilk. G . D., 227(37) Wilk, L., 217(664) Wilkins, J. W., 14(42), 33(42), 63(195, 196), 64(195), 96(196), 106(322), 116(333), 133(42), 163(501), 177(540, 542, 543), 188(573), 190(583), 191(587, 592) Willemin, M., 363(114) Williams, A. A,, 227(58, 59) Williams, A. R., 172(529) Williams, M. D., 230(129) Williams, R. H., 228(85) Williams, S. E., 167(509) Williamson, A. J., 199(612), 206 Wills, J., 94(293), 177(546) Winter, H., 139(384) Wiser, N., 23, 185, 186 Woitok, J., 231(146) Wolf, J. P., 171(518) Wolff, G. A., 96(300) Wolford, D. J., 145(419) Wolter, H., 222(19-21) Wood, D. M., 455(268) Woste, L., 171(518, 519) Wright, A. F., 133(366) Wu, C. D., 77(243) Wu, F., 229(114) Wulfhekel, W., 222(16)

X Xia, W., 381(142, 143). 417 Xie, M.-H., 227(39), 229(113, 117) Xie, X.-D., 145(421) Xu, A,, 228(80) Y Yagi, K., 227(32, 36), 228(90), 229(105) Yamada, M., 228(76) Yamagmoto, N., 227(32), 228(90)

530

AUTHOR INDEX

Yamaguchi, H., 228(67) Yanase, Y., 232(164) Yang, W., 207(654) Yang, X., 228(65, 68, 69 Yasuda, H., 228(91) Yasuda, Y., 227(40) Yasuhara, H., 45(153) Yeom, H. W., 228(76) Yi, J.-Y., 237(183) Yoganathan, M., 132(363) Yong, J. C., 230(126) Yosefin, H., 351(87) Yoshida, S., 231(141), 232(166) Yoshikawa, N., 456(273) Young, A. P., 452(261), 466(299), 476, 483 Yu, B. D., 235, 236 Yu, P., 154(461) Yu, Z., 381(144) Z

Zaanen, J., 139(382) Zaider, M., 160(488) Zaikin, A. D., 435, 438, 456, 458(286) Zaima, S., 227(40) Zakharov, O., 14(51) Zangwill, A,, 187(570), 204(570)

Zappala, D., 467 Zaumseil, P., 229( 116) Zeindl, H. P., 229( 115) Zeller, R., 163(500) Zhang, J., 227(39), 229(113, 117), 232(151) Zhang, K.-M., 145(421) Zhang, L., 229(114) Zhang, S. B., 38(121), 142(409-412), 144(417), 148(430), 171(522, 523) Zhang, X., 228(80) Zhang, Y.-C., 241(196) Zhang, Z., 242 Zharnikgv, M., 223(24) Zhong, H., 177(543), 191(587) Zhou, J. M., 229(114) Zhu, H., 228(80) Zhu, X., 14(43,48), 44(43), 65(43), 69(43), 124(353), 142(412), 148(429,430), 149(436), 151(449), 153(456,457), 154(462) Ziminyi, G., 458(281, 287, 288), 467 Zinner, A., 224(25), 227(28,29), 229(102) Zorin, A. B., 435 Zscheile, H., 162(496) Zunger, A., 63(191), 64(191), 177(548), 217(665) Zwangil, A,, 429(212) Zwerger, W., 458(282,283)

Subject Index

A ab-inito band structures, 3 ab-inito cumulant expansions, 21 ab-inito T-matrix approach, 21 AC susceptibility. disorder and, 363-367 Alkali metals, 164-168 Antiphase domain, 349-350 Atoms, quasiparticle applications, 159- 160

B Ballistic motion of vortices, 388-395 Band-gap discontinuity, 21 1-213 narrowing in Si, 135-136 problem, 11-12, 213 Baym-Kadanoff approach, 19 Bethe-Salpeter equation, 46, 47, 48,49, 80, 81 Block wave function, 90 Bond and site disorder, 358-363, 423-424 Bulk materials, quasiparticle applications, 133-140 Bulk metals, quasiparticle applications, 163169 C Capacitance matrix, 418-423,440, 495-498 Clusters, quasiparticle applications, 169- 171 C,, and related systems, 138-140 Convergence considerations plane waves, 104-108 real-space/imaginary-time approach and, 100-101 Cooper pairs, 272, 432, 433 Core-polarization effects, 63-69 Core-polarization-potential (CPP), 64, 65-67 Core valance exchange, 163-164 Correlated disorder, 381 -383 Correlation functions, 469-475 Coulomb hole + screened exchange (COHSEX) calculations, 25-27, 42-46, 94 Coulomb interaction, 2 Coulomb singularity, integration of, 103-104 53 1

Coupling-constant perturbation theory, 200-201, 204 averages, 215-217 Coupling energy, Josephson, 442 Critical current depinning, 294 vortex depinning and, 292-296 vortex motion above, 296-300 Critical trajectory, 485 Current-voltage (IV) characteristics, 277-279, 280 effects of applied currents on bound pairs Of, 313-316 experimental, 3 16-31 8 scale lengths, 318-320 schematic, 320-324

D Debye-Hiickel approximation, 329 De-exchange passivation model, 242-260 Defects, quasiparticle applications, 152- 153 Density-density correlation function, 46, 79, 92 Density functional (DFT) Hamiltonian, 10 Density functional theory, 5 band-gap discontinuity, 21 1-213 coupling-constant perturbation theory, 200-201, 204, 215-217 exchange-correlation hole, 213-21 5 Kohn-Sham formulation of, 9-12, 199201, 209-211 local approximations, 2 17- 2 18 overestimation of optical constants within, 175-179 role of, 10 surfactants and, 233, 234, 235, 246 universal, 208-209 Density functional theory, excited states and ensemble, 202 excited-state densities, 201 -203 ground-state densities, 196-201 Monte Carlo calculations, 205-207 time-dependent, 203-205

532

SUBJECT INDEX

Density-polarization functional theory, 191-195, 204 Dielectric functions, models for, 31 -37 Dielectric matrix, 22-24 Diffusion-de-exchange passivation model, 242-260 Diffusion Monte Carlo, 205, 206 Dilute vortex-pair density limit, 475-476 Disorder AC susceptibility and, 363-367 bond and site, 358-363,423-424 correlated, 381-383 positional, 368-378 positional, fully frustrated, 378-380 random, 380-381 Dynamically screened interaction determination of, 29-42 equation, 15 Dyson’s equation, 15, 17, 18, 78

E Egg crate potential, 292 Energy-dependent correlation effects, 20, 24-25 Exact-exchange methods (EXX), 197-200 Exchange-correlation energy, 10- 11, 192, 21 1 Exchange-correlation hole, 2 13-2 15 equal-spin, 203 Exchange-correlation kernel, 204 Exchange-correlation potential, 21 1 Excited-state densities, 201-203 Excitonic effects, 46-49 Excitons, quasiparticle applications, 157-158

F Fast Fourier Transforms (FFTs), 98 Finite-size effects incomplete renormalization, 334-335 induced free vortices, 335-342 residual magnetic field effects, 339-342 First-order perturbation theory, 15, 135 Fixed-node approximation, 206 Fractional giant Shapiro steps, 401-412 Free-vortex density, 309-310 Frustration, definition of, 344-346

G Gallium arsenide, GWA calculations for, 127-130 Gallium nitride, GWA calculations for, 130-133 Gaussian coarse graining, 505-506 Gaussian orbitals, 94-96 Generalized gradient approximations (GGAs), 218 Germanium, GWA calculations for, 127-130 Giant Shapiro steps, 399-412 GibbsrBogoEubov inequality, 453 Ginzburg-Landau description, 388, 459-463 Green function, 3, 369 for Gaussian coarse graining, 505-506 independent-particle, equation for, 15 interacting equation, 15 Monte Carlo, 205 real-space/imaginary-timeapproach, 99101 single-particle, defined, 6, 7 single-particle, determination of, 28-29 two-particle, 46-47 Ground-state densities, 10, 11, 196-201 GW approximation (GWA), 3 limitations of, 21 GW approximation (GWA), numerical issues calculations for five prototypical semiconductors, 120- 133 parallel calculations, 113- 120 plane waves, 102- 113 real-space/imaginary-time approach, 97102 reciprocal-space approach, 90-97 GW approximation (GWA), quasiparticle calculations in COHSEX calculations, 25-27, 42-46 core-polarization effects, 63-69 dynamically screened interaction, determination of, 29-42 energy dependence of self-energy, 57-63 equations for, 15 excitonic effects, 46-49 Hedin equations, 18-24 local approaches, 49-50 local-field effects, 51-56 nonlocality of self-energy, 56-57 quasiparticle equation, use of, 16-18

533

SUBJECT INDEX quasiparticle local density approximation, 50-51 self-consistency, 69-79 separation of self-energy, 24-28 single-particle Green function, determination of, 28-29 symbols and quantities defined, 14 vertex corrections, 79-89 H

Hamiltonian, one-particle, 9- 10 Hartree approximation, 5 Hartree-Fock, 10, 20, 28-29, 43, 200 Hedin equations, 18-24, 81 Hellmann-Feynman theorem, 216 Hohenberg-Kohn theory, 196, 200, 202 Hubbard-Stratonovich transformation, 460 I

Imaginary-time approach, 97- 102 Impedance, array, 330-334 Independent-particle polarizability equation, 15, 91 real-space/imaginary-time approach and, 101-102 Inductance and symmetry breaking, 418-423 Interfaces, quasiparticle applications, 142146 Inverse capacitance matrix, 497 Island-edge passivation, 251 -252 J Johnson’s sum rule, 37-38, 39 Josephson equations, superconductivity and, 271-275 supercurrents, 274 voltages, 274 Josephson junction arrays RCSJ model, 275-279 washboard model, 279-282 Josephson junction arrays, classical applications, 267-268 current-voltage characteristics, 313-324 disorder, 357-383 finite-size effects, 334-342 impedance, 330-334

inductance and symmetry breaking, 418423 magnetic fields, 342-357 nonconventional dynamics, 383-395 nonzero frequency response, 324-334 radiation emission, 424-431 resistivity above transition temperature, 309- 3 12 Shapiro steps, 269, 395-412 subharmonic steps, 413-418 vortices, role of, 268 zero frequency, 308-324 Josephson junction arrays, classical (T = 0) egg crate potential, 292 Lorentz force, 292, 294 nonzero magnetic field, 286-300 vortex depinning and critical current, 292-296 vortex motion above critical current, 296-300 zero magnetic field, 283-286 Josephson junction arrays, classical (T > 0) arrays at nonzero temperature, 302-304 estimates of transition temperature, 304-308 single junctions at nonzero temperature, 301-302 Josephson junction arrays, overdamped, 266, 278, 280-282 Josephson junction arrays, quantum applications, 268 correction to Kosterlitz-Thouless transition, 438-442 mean-field theory, 438, 442-455 phase delocalization in single junctions, 435-437 quantization and commutation relations, 431-435 resistance of arrays, 455-467 single junctions, 43 1-438 superconducting phase transition in, 438455 Josephson junction arrays, underdamped, 266, 280-282 Josephson penetration depth, 277 K

Kinetic Monte Carlo simulations, 252-260

534

SUBJECT INDEX

Kleinman-Bylander form, 96 Kohn-Sham system/potential, 9- 12, 209-21 1 generalized, 199-201 Kosterlitz-Thouless-Berezinskii transition, 267, 301 Kosterlitz-Thouless transition temperature, 304 quantum corrections to, 438-442 resistivity above, 309-312 Kramers-Kronig relation, 179-180, 385

L Ladder diagrams, 21, 46, 80 Levine-Louie dielectric function, 35-36, 50 Levine-Louie screened potential, 36-37 Linearized augmented plane waves (LAPW), 93 Linearized muffin-tin orbitals (LMTO), 9394, 189 Local density approximation (LDA), 10, 1112, 92 band-gap narrowing in Si, 135 defects, 152, 153 density functional theory and, 217-218 dielectric constant and, 175 interfaces, 143- 145 pressure, 154, 155 quasiparticle (QPLDA), 50-51 Schottky barriers, 146-147 surfaces, 147-151 surfactants and, 246 vertex corrections, 82 wave functions, 175-179 Local-field effects, 22, 36, 51 -56 in optical response, 184- 191 Local-orbital basis functions, 68 sets, 92-96 use of term, 4n. 15 Local spin density approximation (LSDA), 136-138, 159 Lorentz force, 292, 294, 393, 394, 423

M Magnetic fields, 342-357 Many-body effects background information, 2 Kohn-Sham particles, 9-12

quasiparticles, 5-9 Many-body local-field effects, 22n78 Mean-field theory, 438,483 importance of dimensionality, 442-444 influence of offset charges, 449-452 re-entrance, 448-449 self-consistent, 444-448 variational improvement of, 452-455 Meissner effect, 284, 286, 340 Metals, quasiparticle applications bulk, 163-169 clusters, 169-171 surfaces, 171-172 Molecules, quasiparticle applications, 159160 Monte Carlo calculations, 205-207 kinetic simulations, 252-260 Mutual inductance technique, two-coil, 324325

N Ni, d and f electron, 168-169 Nonzero frequency response, 324-334 Nonzero magnetic field, 286-300 0

Offset charges, 449-452, 499-500 Optical response, GWA calculations and, 172-173 density-polarization functional theory, 191-195 local-field effects, 184- 191 overestimation of constants within DFT, 175- 179 scissors operator, 179- 184 Optimized effective potential (OEP), 197 P

Parallel GWA calculations, 113-120 Partition function, 502-504 Penn dielectric function, 35 Phase correlation function, 500-502 Phase slip, 302 Phase waves, 303 Photoemission, measurement of quasiparticles with direct or inverse, 8 Plane waves, basis sets, 90-92

535

SUBJECT INDEX convergence, 104-108 efficiency, 1 13 integration of Coulomb singularity, 103 104 object orientation, I 1 1 - 113 pseudopotentials and plasmon-pole models, choice of, 108-1 11 use of symmetry, 102-103 Plasmon-pole models (PPM), 37-42, 91, 108- 1 11 Polarity effects, 231 Polysilane, 162 Positional disorder, 368-378 fully frustrated, 378-380 Pressure, quasiparticle applications, 153- 156 Procedure-oriented paradigm, I 1 1 - 113 Pseudopotentials, choice of, 96, 108- 11 1

Q Quantum arrays. See Josephson junction arrays, quantum Quantum Hall effect, 269 Quasiparticle calculations See also GW approximation (GWA), quasiparticle calculations in in metals, 163-172 in semiconductors and insulators, 133- 162 Quasiparticle equation, 5, 15, 16-18 Quasiparticle local density approximation (QPLDA), 50-51 Quasiparticles applications in semiconductors and insulators, 133-162 defined, 5-8 reviews of calculations of, 3-4

R Radiation emission, 424-431 Random disorder, 380-381 Random phase approximation (RPA), 23, 184, 187 Rate equation theory, 242 Rayleigh-Ritz variational principles, 202 RCSJ (resistively-capacitively shunted junction) model, 275-279 Real-space/imaginary-time approach, 97- 102 parallel algorithms for, 117-120 Reciprocal-space approach, 90-97

parallel algorithms for, 114-1 17 Re-entrance, 448-449 Renormalization, incomplete, 334-335 Roothaan-Hartree-Fock wave functions, 164

S Scale lengths, 318-320 temperature dependences of, 487-492 Scaling, 476-492 Schottky barriers, quasiparticle applications, 146-147 Schrodinger equation, 205,435 Schwoebel-Ehrlich barriers, 236 Scissors operator, 179- 184, 205 Screening currents, 331,421-422, 477 Self-consistency GWA calculations and, 69-79 vertex corrections and partial, 85-89 Self-energy defined, 5 determination of, 3 effects, 166 energy dependence of, 57-63 equations, 15 nonlocality of, 56-57 separation of, 24-28 Self-interaction error, 171 Semiconductors and insulators, quasiparticle applications atoms, 159-160 bulk materials, 133-140 defects, 152-153 excitons, 157- 158 interfaces, 142-146 molecules, 160- 162 pressure, 153- 156 Schottky barriers, 146- 147 superlattices, 140- 142 surfaces, 147-151 Shake-up spectra, use of term, 21n76 Shapiro steps, 269, 395-399 fractional giant, 401-412 giant, 399-412 subharmonic, 413-418 Silicon band-gap narrowing in, 135- 136 GWA calculations for, 121-127 Silicon carbide, GWA calculations for, 130133

536

SUBJECT INDEX

Single-particle propagator, 6 Slater determinants, 199, 200-201, 203, 206, 210,215 Spectral function, 6-7 single-particle, 8 Spin waves, 303, 469-475, 477 Strain-induced islanding, 224 Stranski-Krastanov growth mode, 220, 224, 239 Subharmonic steps, 413-418 Superconductivity, Josephson equations and, 271-275 Superconductor-insulator-superconductor (SIS) junction, 271 Superconductor-normal-superconductor (SNS) junction, 271 Superlattices, quasiparticle applications, 140142 Surface-induced momentum-nonconserving excitations, 166 Surfaces, quasiparticle applications, 147- 151, 171-172 Surfactant effect, defined, 221 Surfactant effect, semiconductor thin-film growth and anisotropy, 240 de-exchange, 238, 249-250 density functional theory and, 233, 234, 235,246 diffusion-de-exchange passivation model, 242-260 dimer-exchange mechanism, 237 equilibrium configurations, 237-238 equilibrium versus kinetic effects, 239 first-principles calculations, 244-249 IV films on IV substrates, 223-225, 226 island-edge passivation, 251 -252 kinetic model, 240-242 kinetic Monte Carlo simulations, 252-260 local density approximation and, 246 macroscopic models, 239-242 mixed films and substrates, 231-233 microscopic models, 233-239 re-exchange, 238 thermodynamic approach, 239-240 111-V films on 111-V substrates, 225, 229230 unresolved issues, 261-262 Surfactants applications, 221-222

defined, 221 Symmetry, plane waves and, 102-103

T Thomas-Fermi screening, 33-34, 217 3D-island growth, 224, 225 critical island approximation, 242 kinetically, 240-242 Time-dependent density functional theory, 203-205 Topological long-range order, 304,471 Transition-metal oxides, 136-138 Trans-polyacetylene, 162 V

Variational Monte Carlo, 205-206 Vertex corrections, 79-89 alkali metal band width and, 83-85 local density approximation (LDA), 82 partial self-consistency and, 85-89 to second order in interaction, 81-82 Volmer-Weber growth, 224 Vortex/vortices ballistic motion of, 388-395 current-induced unbinding, 315, 492-495 density correlation function, 388 depinning and critical current, 292-296 dilute density limit, 475-476 dynamics of, 326-330 effects of applied currents on bound pairs Of, 313-316 finite-size effects, 334-342 free-vortex density, 309-310 kinetic energy, 298 motion above critical current, 296-300 nonconventional dynamics, 383-395 spin waves, 303,469-475,477 viscosity, 297

W Ward-identity-based relation, 48 Washboard model, 279-282

Z Zero frequency, classical arrays and, 308-324 Zero magnetic field, 283-286

Errata. The following corrected Acknowledgments apply to the article by Wilfried G. Aulbur, Lars Jonsson, and John W. Wilkins: It is a pleasure to acknowledge helpful comments from M. Alouani, F. Aryasetiawan, J. Chen, A. G. Eguiluz, A. Fleszar, R. W. Godby, A. Gorling, 0. Gunnarsson, P. Kriiger, J. P. Perdew, M. Rohlfing, L. J. Sham, E. L. Shirley, J. E. Sipe, and U. von Barth. Also. we thank F. fuyasetiawan and 0. Gunnarsson, A. Gorling, and M. Rohlfing, P. Kruger. and J. Pollmann for allowing us to use unpublished data; A. Fleszar, A. Gorling, and L. Steinbeck for a critical reading of the manuscript; A. G. Eguiluz, B. Farid, and G. D. Mahan for sending us material prior to publication; and J . Chen for providing pseudopotentials. WGA gratefully acknowledges continued collaboration with L. Steinbeck, M. M. Rieger, and R. W. Godby on the real-spacehmaginary-time GWA. The support of the National Science Foundation and the Department of Energy, Basic Energy Sciences, Division of Materials Science, has provided the intellectual continuity that enabled the writing of this review.

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