Solid State Physics: Advances in Research and Applications, Vol. 38

SOLID STATE PHYSICS VOLUME 38 Contributors to This Volume J. Callaway J. T. Devreese N. H. March F. M. Peeters G. A...

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SOLID STATE PHYSICS VOLUME 38

Contributors to This Volume

J. Callaway J. T. Devreese N. H. March

F. M. Peeters G. A. Samara Jai Singh Alexander Wokaun

SOLID STATE PHYSICS Advances in Research and Applications

Editors

HENRY EHRENREICH DAVID TURNBULL Division of Applied Sciences Hurvard University, Cambridge, Massachusetts Con.vultiny Editor

FREDERICK SEITZ The Rockefeller University, New York, New York

VOLUME 38 1984

ACADEMIC PRESS, INC. (Harcourt Brace Jovanovich, Publishers)

Orlando San Diego San Francisco New York London Toronto Montreal Sydney Tokyo

0

COPYRIGHT1984, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM O R BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS,INC.

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United Kingdoin Edition published by ACADEMIC PRESS, INC. ( L O N D O N ) LTD. 24/28 Oval Road, London N W l 7DX

LIBRARY OF CONGRESS CATALOG CARDNUMBER:55-12200 IBSN 0-12-607738-X PRINTED IN THE UNITED STATES OF AMERICA

84 85 86 87

9 8 7 6 5 4 3 2 1

Contents

CONTRIBUTORSTOVOLUME38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FOREWORD ................................................................ PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUPPLEMENTS ..............................................................

vii ix xi ...

Xlll

High-pressure Studies of Ionic Conductivity in Solids

G . A . SAMARA

I. I1. 111. IV .

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

4 20 75

Theory of Polaron Mobility

F . M . PEETERSA N D J . T . DEVREESE I. 11. III . IV . V. VI . VII .

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Boltzmann Equation for the Polaron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relaxation-Time Approximation for Linear Conductivity . . . . . . . . . . . . . . . . . . The Drifted Maxwellian Approach for Nonlinear Direct Current Conductivity The Polaron Impedance Function for Quantum Frequencies . . . . . . . . . . . . . . . The Polaron Mass and the Polaron Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AppendixA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..............................

AppendixC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82 85 91 97 107 117 127 129 130 132

Density Functional Methods: Theory and Applications

J . CALLAWAY AND N . H . MARCH

I. I1. I11. IV . V. VI . VII .

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hohenberg-Kohn Theorem and Its Extensions . .................. Principles of Calculational Procedures . . . . . . . . . . . . .................. Beyond the Local Density Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Response Theory . . . . . . . . . . . . . . . . . . . . . ................ Some Further Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

136 143 155 185 195 207 218

vi

CONTENTS Surface-Enhanced Electromagnetic Processes ALEXANDER WOKAUN

1. Introduction and Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Surface-Enhanced Raman Scattering: Experimental Observations . . . . . . . . . . . 111. Surface-Enhanced Raman Scattering: Theoretical Models . . . . . . . . . . . . . IV . Evidence for the Electromagnetic Model ............................ V. Extensions of the Electromagnetic Model ................................ VI . Particle Dipolar Interactions . . . ................... VII . Enhanced-Surface Second-Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . VIII . Enhanced Absorption by Adsorbed Dyes . . . . . . . . . . . . . . . ........ IX . Contributions to the Enhancement by Other Mechanisms . . . . . . . . . . . . . . . . . . X . Applications of Surface-Enhanced Phenomena; Summary .................

224 226

245 255 269 275 283 287

The Dynamics of Excitons

JAI SINGH

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Theory of Excitons ................................................... 111. Exciton-Phonon Interactions .......................................... IV . Composite Exciton-Phonon States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Exciton Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

295 301 328 336 341

AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

371 380

Contributors to Volume 38

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

J. CALLAWAY, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803 (1 35)

J. T. DEVREESE, Department of Physics and Institute ,for Applied Mathematics, University of Antwerp, B-2610 Antwerp, Belgium, and Department of Physics, Eindhoven University of Technology, Eindhoven, The Netherlands (81) N. H. MARCH,Theoretical Chemistry Department, University o f O.-cford, O.:ford O X 1 3TG, England (135)

F. M . PEETERS,Department of Physics, University of’ Antwerp, B-2610 Antwerp, Belgium (8 1) G. A. SAMARA, Sandia National Laboratories, Albuquerque, New Mexico 87185 (1) JAI SINGH,*Research School of Chemistry, Australian National University, Canberra A C T 2600, Australia (295) ALEXANDER WOKAUN,ETH Zentrum, Physical Chemistry Laboratory, CH-8092 Zurich, Switzerland (223)

* Present address : Department of Physics, National University of Singapore, Kent Ridge. Singapore 05 1 1.

.,..

Frederick Seitz

Foreword

This volume marks the transition of Frederick Seitz’s role in this publication from that of active coeditor to founding editor. Seitz’s treatise on “The Modern Theory of Solids,” published in 1940, played a major, perhaps the major, part in distinguishing and delineating the then fledgling discipline of solid state physics. It became a prime source of instruction and inspiration to a whole generation of solid state scientists. As the field grew and flourished following World War 11, Seitz felt a pressing need to revise and expand the treatise. However, it seemed that the scope and magnitude of the task required had become too large for any one person to encompass in a single volume. Paraphrasing remarks in his biographical notes [see “The Beginnings of Solid State Physics” (N. F. Mott, ed.), Proc. R . SOC. London, Ser. A 371, 84-99 (1980)], this situation contrasted sharply with that in the 1930s, when the total volume ofexperimental and theoretical literature dealing with the properties of solids was sufficiently limited that it was possible to become familiar with essentially every significant paper in the areas covered in”The Modern Theory of Solids.” Faced with the new situation, he concluded that the need for coverage might better be met by a multivolume work consisting of comprehensive and in-depth treatments of basic topics by a number of experts, and one of us (D.T.) accepted his invitation to join in the planning and editing of such a series. The late Kurt Jacoby, then vice president of Academic Press and a former Akademische-Verlag editor and refugee from Germany, enthusiastically encouraged the project and arranged to have the volumes published by Academic Press, beginning in 1955. Later, in 1967, the other o f u s (H.E.) accepted an invitation to share in the editorial responsibilities. Seitz’s broad perspective and extraordinary profundity as an author and editor reflect a scientific career distinguished by major contributions to all of the principal branches of condensed phase science. He was one of the small group who pioneered the development of modern solid state theory. His paper, in collaboration with Wigner, deriving from first principles the cohesive energy and other electronic properties of metallic sodium, stands as a major landmark in this development. His later research ranged over the entire field of solid state science, and the resulting contributions, especially those on the nature and operation of point and line imperfections in crystals, decisively influenced the modern development of solid state chemistry and physical metallurgy as well as solid state physics. As may be apparent from his versatility in the practice of research, he perceived condensed phase science a5 a unified whole, embracing the more ix

X

FOREWORD

applied fields of metallurgy and ceramics as well as the physics of solids. This catholicity of outlook motivated his effective promotion of interdisciplinary cooperation in the broad professional community as well as within the universities with which he was affiliated. It has been central to the editorial policy of this series. To implement this policy the editors have sought contributions, from experts distributed throughout the international community, over the whole spectrum, experimental and theoretical, of condensed phase science, using significance, depth, and timeliness as the principal criteria of acceptability. The scientific community is deeply indebted to Frederick Seitz both for his pioneering research on solids and for his notable achievements as a scientific statesman. It has been a high privilege for us to have been associated with him as coeditors of this series.

DAVID TURNBULL HENRYEHRENREICH

Preface

In this volume, Samara reviews the experience on the dependence of solid state ionic conductivity on hydrostatic pressure and its theoretical interpretation. He shows that the pressure results permit evaluation of the elastic volume relaxations in the formation and motion of point structural defects and often lead to identification of the dominant transport mechanism from among the various competing ones. In the second article, Peeters and Devreese review the theory of the polaron and its mobility. They present a comprehensive contemporary treatment of the role of polarons in charge transport processes. This subject has been previously discussed in Volume 2 1 of this series in an article by Appel. In the following article, Callaway and March present a critical review of the foundations, achievements, and limitations of density functional methods in determining inhomogeneous electron distributions in atoms, molecules, and solids. Some applications of these methods are described. Application of the methods to surface problems had been reviewed by N . D. Lang in Volume 28 of this series. The recent discovery of the tremendous enhancement of Raman scattering at roughened surfaces has stimulated much experimental and theoretical investigation. The fourth article in this volume, by Wokaun, reviews these investigations and presents evidence that the enhancement reflects a general electromagnetic phenomenon. Wokaun proposes a general mechanism involving localized surface plasmons, which may make a major contribution to the enhancement. The theory of excitons was reviewed in Supplement 5 (1963) to this series by R. S. Knox. More recent contributions to the theory are reviewed in the final article in this volume, by Singh. He presents a treatment directed at unifying the physical and chemical approaches, as exemplified by the Wannier and Frenkel theories, and describes various radiationless processes of excitons in inorganic as well as organic crystals. HENRYEHRENREICH DAVIDTURNBULL

This Page Intentionally Left Blank

Supplements

Supplement I : T. F? DASA N D E. L. HAHN Nuclear Quadrupole Resonance Spectroscopy, 1958 Supplement 2: WILLIAM Low Paramagnetic Resonance in Solids, 1960 Supplement 3: A. A. MARADUDIN, E. W. MONTROLL, G. H. WEISS,A N D I. €? IPATOVA,Theory of Lattice Dynamics in the Harmonic Approximation, 1971 (Second Edition) Supplement 4: ALBERT C. BEER Galvanomagnetic Effects in Semiconductors, 1963 Supplement 5: R. S. KNOX Theory of Excitons, 1963 Supplement 6: S. AMELINCKX The Direct Observation of Dislocations, 1964 Supplement 7: J . W. CORBETT Electron Radiation Damage in Semiconductors and Metals, 1966 Supplement 8: JORDAN J. MARKHAM F-Centers in Alkali Halides, 1966 Supplement 9: ESTHER M. CONWELL High Field Transport in Semiconductors. 1967 Supplement 10: C. B. DUKE Tunneling in Solids, 1969 Supplement 11 : MANUELCARDONA Optical Modulation Spectroscopy of Solids, 1969 Supplement 12: A. A. ABRIKOSOV An Introduction to the Theory of Normal Metals, 1971 A N D I? A. WOLFF Supplement 13: P M. PLATZMAN Waves and Interactions in Solid State Plasmas, 1973

Supplement 14: L. LIEBERT. Guest Editor Liquid Crystals, 1978 Supplement 15: ROBERTM. WHITEA N D THEODORE H. GEBALLE Long Range Order in Solids, 1979 ...

Xlll


SOLID STATE PHYSICS. VOLUME

38

High-pressure Studies of Ionic Conductivity in Solids G. A. SAMARA Sundio Nutioncrl Loboraioricjs, Albuquerque. N i w Mesico

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . .......... 1 . LatticeDefects ..................................................... 2. Temperature Dependence of the Conductivity. . . . . . . . . . . . . . . . . . . . .

Pressure Dependence of the Conductivity: Activation Volume. ............ 4. Model Calculations of Lattice-Defect Properties. . . . . . . . . . . . . . . . . . . . . . . . . 111. Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . 5. Alkali Halides.. ................ 6. The Silver Halides AgCl and AgBr 7. The Thallous Halides TICI, TIBr, a .............................. 8. Fluorites and Other RX,-Type Halides ................................ 9 . Fast Ion Conductors . . . . . ........... IV. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.

I 4 4

5 8 I1 20 23 32 36 44 65

15

1. Introduction

There has been a great deal of renewed interest in the past several years in the study of ionic charge transport (or conductivity) in solids. This interest derives not only from the need to develop advanced solid state batteries but also from the challenging physics involved, especially in the case of fast ion (or superionic) conductors. Ionic charge transport and, more generally, ionic diffusion in solids result from the existence and motion of crystalline defects. These can be either extrinsic defects, which are associated with the presence of chemical impurities (dopants) and lattice imperfections, or intrinsic defects. Any solid has at a given temperature an equilibrium concentration of intrinsic defects (usually vacancies or interstitials) which is controlled by a Boltzmann factor. Because ionic conduction and diffusion are generally sensitive to this thermal equilibrium concentration of intrinsic defects, they are called activated processes. I Copyright c 1984 by Academic Press. Inc All rights of reproduction in any form reserved. ISBN 0-12-M)7738-X

2

G . A. SAMARA

The situation is quite interesting in the case of fast ion (superionic) conductors. These materials are characterized either by the availability of a very large number of normally vacant lattice sites (i.e., a defect structure) or by an essentially complete disorder of the mobile ionic species. These mobile ions are distributed randomly over a large number of sites, and the magnitude of the conductivity indicates that nearly all of them must contribute to the conductivity. The activation energy for ionic motion in the superionic regime is usually small (- 0.1 eV). Ionic conductivity and diffusion have been studied extensively in a variety of ionic crystals at atmospheric pressure, and in many cases the mechanisms for the transport processes in terms of point defects have been established.’.2 Experimental results have generally been successfully interpreted in terms of absolute reaction-rate the0ry.l-j. In this theory the elementary diffusive jump is likened to a transition, in thermal equilibrium, between a ground state corresponding to the equilibrium lattice position of the mobile species and an excited state corresponding to the saddle-point position. The basic assumption underlying the use of this theory in diffusive processes is that there exists a well-defined transition (excited) state whose lifetime is sufficiently long compared to lattice thermal relaxation time that it makes sense to define the thermodynamic properties of the excited state. Although the theory has been criticized on this assumption and on other count^,^ its general success in interpreting experimental results provides strong support for the usefulness and, perhaps, validity of the equilibrium statistical mechanical treatment of the diffusion process in many systems. There has been a considerable number of studies of the effects of hydrostatic pressure on ionic conductivity and other related ionic transport processes. Both the formation and the motion of lattice defects which determine the conductivity normally depend exponentially on pressure. Most ‘of the early pressure work was on NaCl and the silver halides, whereas recent work has extended these studies to a broader range of materials that includes fast ion conductors. The results are reviewed briefly in Part 111. In all of these studies pressure is found to be a complementary variable to temperature in

’ See. e. g.. A . B. Lidiard, in “Handbuch der Physik” ( S . Fliigge. ed.), Vol. 20, p. 246. Springer*

Verlag, Berlin and New York, 1957; see also C. Kittel, “Introduction to Solid State Physics,” Chapter 18. Wiley, New York, 1966. L. W. Barr and A . B. Lidiard, in “Physical Chemistry: An Advanced Treatise” (W. Jost, ed.), Vol. 10. p. 151. Academic Press, New York, 1970. C. Zener, in “Imperfections in Nearly Perfect Crystals” (W. Shockley, ed.), p. 289. Wiley, New York, 1950; A c f u Crystallogr. 2, 163 (1949); C. Wert and C. Zener, Phys. Rev. [2]76, I I69 ( 1 949). D. Lazarus and N . H. Nachtrieb, in “Solids Under Pressure” (W. Paul and D. M. Warschauer, eds.), Chapter 3. McGraw-Hill. New York, 1963.

HIGH-PRESSURE STUDIES OF IONIC CONDUCTIVITY IN SOLIDS

3

trying to understand the mechanisms of ionic conduction. In some cases, as we shall see later, pressure turns out to be an essential variable. In ionic conductivity and other defect-dominated properties in general, it is important to know the elastic volume relaxation associated with the formation of lattice defects as well as the lattice relaxation accompanying the diffusive motion of these defects. Measurements of the hydrostatic pressure dependence of the ionic conductivity give, in principle, direct information about these volume relaxations, and this information can in turn be used to understand better the mechanisms of ionic transport and to test the validity of proposed models. These aspects are prominently featured in present review article. Pressure studies also provide a better understanding of the nature of the energy barriers associated with ionic transport and are, in addition, important to the understanding of the phase transitions observed in many ionic conductors. In the present article we shall review and discuss hydrostatic pressure studies performed on a variety of ionic conductors. There appear to be no earlier reviews of the subject except for a very brief recent account by the a ~ t h o r Two . ~ early reviews by Lazarus and Nachtrieb4 and Keyes,6 are relevant; however, at the time these reviews were published, very little information on pressure work on ionic conductivity was available. The emphasis of the present review is on experimental results and their interpretation in terms of relevant concepts and theory. It is not our intention to give a complete summary of all of the available literature; rather, we shall dwell on examples which illustrate special features or show systematic trends or both. Much of the available pressure work has been on relatively simple materials and crystal structures because these are amenable to theoretical treatment. Some of the materials investigated (e.g., PbF, and the thallous halides) are especially interesting because, am*ong other properties, they possess large dielectric constants and also exhibit relatively soft, low-lying phonon modes. These factors are important in relation to ionic conduction because the larger the dielectric constant of an ionic crystal, the lower the energy of formation of lattice defects. Also, physically, ionic transport occurs by hopping motion across an energy barrier, and this barrier might be expected to become smaller the “softer” the lattice. We shall examine the evidence for the connection between these properties and the transport properties. This review is organized as follows. We begin Part I1 with a brief theoretical background giving some of the concepts and results necessary for the G . A. Samara, in “High Pressure Science and Technology” ( B . Vodar and P. Marteau. eds.), p. 454. Pergamon. Oxford, 1980. R. W. Keyes, in “Solids Under Pressure” (W. Paul and D. M. Wdrschauer, eds.), Chapter 4. McGraw-Hill, New York, 1963; J . Chem. Phys. 19,467 (1959).

4

G. A. SAMARA

analysis and interpretation of the results to be presented in later sections. Part I11 reviews the pressure results and their interpretation. For this purpose it was found most convenient to divide the substances of interest into several groupings as follows: alkali halides (both NaCl and CsCl types), silver halides, thallium halides, fluorites and related structures, and fast ion conductors. Finally, Part IV provides some overall evaluation of the results, concluding remarks, and suggestions for future work. Throughout this review the pressure unit used is the giga pascal (GPa), or lo9 P. This is an SI unit of pressure: 1 GPa = 10 kilobar (kbar) = 10’O dyn cm-’.

II. Theoretical Background 1. LATTICE DEFECTS

The simplest and, for our present consideration, most important lattice imperfections are vacancies and interstitials. A lattice vacancy is known as a Schottky defect. A Schottky defect is formed in a perfect crystal by moving an ion from a lattice site in the interior to a lattice site on the surface of the crystal. At any given temperature, a certain equilibrium number of lattice vacancies is always present in a crystal because the entropy is increased by the presence of disorder in the lattice. To keep the crystal electrostatically neutral on a local scale, it is usually energetically favorable in ionic crystals to form roughly equal numbers of separated positive and negative ion vacancies (so-called Schottky pairs). It is easy to show that the concentration n of such pairs is’

n = N exp(- AG;/2kT), where N is the number of ions (sites)per unit volume, and AGF is the Gibbs free energy of formation of a pair. Another type of lattice defect is the Frenkel defect. In this case an ion is moved from a lattice site to an interstitial position, a normally unoccupied lattice position. The concentration of Frenkel defects is again easily shown to be given by’

n

=

(NN’)’’’ exp( -AG:/2kT),

where N is the number of lattice sites, N‘ is the number of interstitial sites (both per unit volume), and AG; is the Gibbs free energy for the formation of the interstitial.

HIGH-PRESSURE STUDIES

OF IONIC CONDUCTIVITY IN SOLIDS

5

Equations (1.1) and (1.2) are obtained in the limit n 0 and d'B/ap' < 0, Eqs. (4.17) and (4.18) indicate that pf > fi and ti, > K . These inequalities are attributed to higher order anharmonicities. An interesting feature of Eqs. (4.17) and (4.18) is that the vacancy properties pf and tif are determined solely by bulk properties. This is a rather surprising result. On the basis of the known variation of the bulk modulus with pressure, Varotsos et al. have estimated bounds on the ratio tif/K at p + 0. For the alkali halides, t i f / ~falls in the range 3-9. The results of the Varotsos et al. model are compared with available experimental results in Section 5.

b. Motional Energies and Volumes In principle, calculations of the energies and elastic relaxations associated with the motion of defects in ionic crystals follow the general procedures outlined earlier for the formation of such defects. These quantities are evaluated for the saddle-point configuration relative to the ground-state configuration. In practice, very few such calculations have been made. The results, primarily on NaC1-type alkali halides, give values of the motional energies which agree reasonably well with experimental values.' Here, choice of the appropriate repulsive potential is all important, and unfortunately there is no a priori procedure for making this choice.


17

There are apparently no explicit lattice statics model calculations for motional elastic relaxations or volumes, AVm. However, AVm has been generally estimated from phenomenological models such as the macroscopic strain-energy model and from a more microscopic dynamical model as well as from an empirical model. We shall now review briefly these models. (i) Strain-Energy Model. It has been common practice to compare experimentally deduced values of A V with those calculated from a strain-energy model of lattice imperfection^.^,^ In this rather simple continuum model,3 it is assumed that the work required to create or to move a lattice defect goes into elastically straining the lattice, and this can be treated by using ordinary elastic theory. Using this model it can be shown that3g6 AV/AG

= -K

+ (8In CISP), = - ~ [ 1 + (8 In C/8 In V),],

(4.19)

where C is an effective elastic shear modulus and K , = -(S In V/SP),, is the isothermal compressibility. For a cubic crystal a suitable choice for C is6,'

c = $44 + 3(Cll

-

(4.20)

Cl,),

and (SIn C/SP), can be evaluated from the pressure dependences of the Ci,s, where such data are available. In the absence of such data, (S In C/S In V ) , can be estimated by using the Gruneisen approximation. In this approximation all of the vibrational frequencies of the lattice have the same volume dependence, and this can be shown to lead to6

(8In C/S In V ) , = (8 In C/S In V),

=

d In C/d In V

=

-(2y

+ $),

(4.21)

where y is the elastic Gruneisen parameter which is given by Eq. (3.5). Equation (4.19)can then be rewritten as AV

=

2(y - $ ) K AG.

(4.22)

In Eq. (4.22),K is known and y is known or is given by Eq. (3.5), and AG is related to the measured activation enthalpy A H by the thermodynamic relation AH

=

AG -I- T AS.

(4.23)

In the Griieneisen approximation,6 AS/AV

= U/K.

(4.24)

Using Eqs. (4.23)and (4.24),Eq. (4.22) becomes AV=

2(y - $ ) K A H 1 24y - $)T .

+

(4.25)

18

G . A . SAMARA

All the quantities on the right-hand side of Eq. (4.25) are generally known, and A V can thus be calculated. We shall compare the A V values calculated from Eq. (4.25) with experimentally deduced values later on (Part 111). (ii) Dynnrrziciil Model. The strain-energy model is a static model. In principle, a more realistic view of the motion of lattice defects should reflect the dynamical nature of the problem. In the dynamical approach the parameters which determine diffusive motion are defined in terms of the normal coordinates of the crystal. The atomic displacements causing diffusion are treated as a superposition of phonons. Phonons (especially short-wavelength transverse optic modes in ionic crystals) should be effective in moving mobile species toward saddle points. A diffusive jump occurs when the normally random phases of the displacements of the phonons coincide. Detailed dynamical theories of diffusion are quite complicated, but several approximate treatments have been given.",17 In the context of these treatments it can be shown that the activation volume for diffusive motion is related to the pressure dependence of the relevant phonon frequency vi by' AV,, = (8In v2/JP)T AGm,

(4.26)

where AGm is given by AG,,

=

vZ~',

(4.27)

and q is a fluctuation parameter. Given that the mode Gruneisen parameter is defined by 7 , = K - ' ( J In vJdP),, Eq. (4.26) can be rewritten as

j',

AVm = 2yiti AG,,,.

(4.28)

This has the same form as Eq. (4.22). Van Vechtenl' proposed a simple ballistic model for vacancy motion in elemental crystals. In this model the enthalpy for vacancy migration is given by AH,,,

-

+Mi2,

(4.29)

where M is the mass and i is the velocity of the mobile atom and is given by i = d/z,,.

(4.30)

Here, d is the distance that the mobile atom jumps, and 5, is the vibrational period of the butting (to the vacancy) atoms in the appropriate migration A. Rice, Phys. Rrr. [2] 112,804 (1958). P. Flynn, "Point Defects and Diffusion," Chapters 7. 9. and 12, and references therein. Oxford Univ. Press (Clarendon), London and New York, 1972. J . A. Van Vechten. Phj..~.Re(,. B : Solid Stute [3] 12, 1247 (1975).

" S.

" C.

''

19


mode. In the Debye approximation,

z,

=

h/kO,,

(4.31)

where h and k are the Planck and Boltzmann constants, respectively, and OD is the Debye temperature. Substituting Eqs. (4.30) and (4.31) into Eq. (4.29) leads to AH, = i M ( F dkOD/h)2,

(4.32)

where the factor F x 1.0 is inserted as an empirically adjusted, structuredependent parameter used to account for the approximations made in the model. An expression similar to Eq. (4.32) was earlier derived from AH, for vacancy migration by Glyde'' who also showed that the formation energy for vacancies is also proportional to 0: and has the same form as Eq. (4.32). (iii) Empirical Model. By analogy to their treatment of the defect-formation process, Varotsos and Alexopoulos20,2 have used a macroscopic semiempirical model to treat the migration energies and activation volumes for vacancy motion in ionic solids. The Gibbs free energy for motion is written in the form AG,

=

[(a, - a , ) e 2 / t d ]

+ C,BQ

(4.33)

where a, and a, are Madelung constants for a normal lattice site and for the saddle point, respectively; e is the electronic charge; t is the static dielectric constant, d is the nearest cation-anion distance; C , is, as in Eq. (4.14), a parameter (assumed constant) dependent on the type of lattice and defect created; B is the bulk modulus; and Q is the mean volume per atom. We note that AG, for the migration of the defect through the saddle-point configuration consists of two terms. The first term on the right-hand side of Eq. (4.33) is the electrostatic energy and the second term is the dilatation (or deformation) energy. Varotsos and Alexopoulos have argued that the first term is considerably smaller than the second so that to a fairly good approximation, AG, x C,BQ,

(4.34)

where, by analogy with Eq. (4.15), C, x AH;/B"R". Equations (4.33) and (4.34) yield motional energies for the alkali halides which agree well with experimental results.

l9 2o

''

H . R. Clyde, J . Phys. Chem. Solids 28,2061 (1967). P. Varotsos and K. Alexopoulos, Phys. Rev. B : Solid State [3] 15,2348 (1977). P. Varotsos and K. Alexopoulos, Phys. Status Solidi A 47, K133 (1978); 55, K63 (1979)

20

G. A . SAMARA

Using Eq. (4.33), the activation volume for motion of the vacancy is

Thus, AV, can be calculated from macroscopic quantities and knowledge of the migration path. Using the known quantities in Eq. (4.35) for NaCl, Varotsos and Alexopoulos find AV, = 1.4 + 8.6 = 10.0 cm3/mole, which is in fairly good agreement with the experimental result to be discussed in Section 5. Thus, here again, the first term on the right-hand side of Eq. (4.35) is relatively small and, to a reasonable approximation, AV, z C,R[(dB/dp)

-

I].

(4.36)

By analogy with Eq. (4.18) for AVf, Varotsos and Alexopoulos define the compressibility K , of AV,,, as (4.37) K , = -(d In AV,/dp),. Varotsos and Alexopoulos point to a connection between the migration volumes for the cation and anion vacancy motion. Based on Eq. (4.36) they show that AV:/AV; z AH:/AH;. (4.38) By invoking the Gruneisen approximation they also showed that the dynamical model [Eq. (4.26)] leads to the same result. Without this approximation the dynamical model yields AV;/AV;

z AG:/AG;.

(4.39)

Equations (4.38) and (4.39) can be quite useful for estimating either AV: or AV; when only one of these quantities is known. This is, e.g., the case for the NaC1-type alkali halides where it has been possible to measure only AV:. We shall make use of these approximations in Part 111.

111. Results and Discussion

Figure 2 presents l o g o T versus T - ' plots for a number of solid ionic conductors showing the very large differences in conductivity among the various materials, each of which typifies a class of ionic conductors to be described later. Each class exhibits a different mechanism for ionic transport, and the pressure dependences of the conductivities of the different classes are different. Pressure studies have been reported on each of the materials in Fig. 2 as well as on other crystals of each class. In Part 111 we shall review and discuss primarily the highlights of this work. In a few cases we shall give


21

details of some of the experimental data to illustrate certain features or important results. For other details the reader is referred to the original papers. For the present purposes the materials to be discussed are grouped according to the following classes: (1) Alkali halides of both the NaCl and CsCl types. These are among the simplest and most widely investigated ionic solids. Results are reviewed for NaC1, NaBr, KCI, KBr, RbCI, and CsCl.

T (K) 800

500

400

300

250

I

I

I

I

I

u - RbAg4I5

\-

1.o

2.0

3.0

4.0

T - ~( l o 3 K - ~ )

'

FIG.2. Atmospheric pressure log c ~ T vs. T - plots for the various crystals or crystal classes discussed in the text showing the large variations in the magnitudes and temperature dependences of the ionic conductivities of the different materials. This summary figure does not show the detailed features (i.e., various stages) of c(T) for many of the materials presented. For clarity of presentation, the figure is divided into parts (a) and (b). The data are taken from the various sources cited in Section 111 of the text.

22

G . A . SAMARA

T(K) +4

1000

500

400

I

I

I

300 1

+2

Y

7

0

\

E,

2

F

I

C

v

c b

z -2

0)

-

PbCIk

-4

-6

1 .o

2.0

\

3.0

T - ~(lo3 K - ~ ) FIG.2. (Continued)

(2) Silver halides AgCl and AgBr. (3) Thallous halides TlCl, TlBr, and TlI. (4) Fluorites and related structures. In this class we include a variety of RX,-type compounds, where R is a divalent metal ion and X is a monovalent halogen ion. Included are (a) the important ionic conductor PbF,; (b)SrCl,; (c) the alkaline earth fluorides CaF,, SrF, , BaF, ; ( d ) CdF, ; ( e )a-PbC1, and a-PbBr,; and ( f ' )PbI,. Crystals of types ( u ) - ( f ) have the cubic fluorite structure at normal conditions, and some are known to also exist in an orthorhombic modification, especially at high pressure. Crystals of type (e) crystallize in the same orthorhombic modification as crystal types (u)-(e), whereas PbO, crystallizes in a hexagonal layered structure.


23

( 5 ) Fast ion conductors. Here we include a group of compounds, namely, AgI, RbAgJ,, Na-/3-A1203, Na, +xZr2SixP3-x012,Ag3SBr, and Ag3SI, which are related only by the fact that they exhibit high conductivities at relatively low temperatures with low activation energies. They are a11 generally referred to as fast ion, or superionic, conductors. In all of these materials the ionic conductivity is found to be weakly dependent on pressure, but there are interesting and important differences among the members.

5. ALKALI HALIDES a.

NaCl and Isomorphs

Sodium chloride (NaCl) is one of the simplest and most widely studied ionic conductors. It has the cubic NaC1-type crystal structure (Fig. 3a) and is the prototype of many alkali halides having this structure. In this facecentered cubic structure each cation is surrounded by six anions and vice versa, so that the coordination number is 6. There are four molecules per unit cell. NaCl is a poor conductor at low temperatures but becomes a relatively good conductor above -800 K (see Fig. 2). Schottky defects are dominant in this crystal and its isomorphs with charge transport taking place primarily by cation vacancies.'P2 Table I summarizes the migration and formation enthalpies for some of these crystals. NaCl has been by far the most widely studied solid ionic conductor at high pressure. The work is primarily due to Lazarus and co-workers' 1 9 2 2 , 2 3 who also investigated the isomorphous crystals NaBr, KCl, KBr, and RbCl. Measurements of the pressure dependence of the conductivity in both the intrinsic and extrinsic regimes were carried out. It is generally found that CJ decreases exponentially with pressure as expected; however, in several of

(a)

(b)

(C)

FIG. 3. Arrangement of the ions in three simple and important cubic crystal structures: (a) NaCI-type lattice, (b) CsC1-type lattice; and (c) RX,(fluorite)-type lattice.

B. Pierce, Phys. Rev. [2] 123, 744 (1961). M . Beyeler and D. Lazarus, 2.Nuturjorsch., A 26A, 291 (1971)

"C. 23

24

G . A . SAMARA

AND

TABLEI. MIGRATION ENTHALPIES FOR CATION A N D ANION VACANCIES CATION INTERSTITIALS AND FORMATION ENTHALPIES FOR INTRINSIC DEFECTS FOR SEVERAL ALKALI. SILVER, AND THALLOUS HALIDES' Migration enthalpy (eV)

Halide

Cation vacancies (AH:, .I)

Anion vacancies (AH;.")

NaCI" KClh NaBrh KB? CsCl AgCl AgBr TIC1 TIBrY

0.66-0.76 0.71 0.80 0.62-0.67 0.60*; 0.62' 0.27-0.34h; 0.32d 0.3Ih; 0.36'; 0.33/ 0.5"; 0.54g 0.56

0.90-1 .I0 0.95- 1.04 1.18 0.87-0.95 0.34h; 0.2-0.3'

Cation interstitials (AH2.J

0.06-0.16h; 0.15d 0.15'; 0.15"; 0.20'

0 . 2 h ; 0. l o g 0.25

Formation enthalpy for intrinsic defects (AH,) 2.18-2.38 2.26-2.3 I 1.72 2.30-2.53 1.86h;2.1-2.3' 1.24'; 144h; lSd 1.06'; 1.2e; 1.07' 1.3"; 1.3hY 1.1

The dominant intrinsic defects are Schottky pairs for all but AgCl and AgBr. in which Frenkel pairs are dominant. Data taken from a compilation by L. W. Barr and A. B. Lidiard, in "Physical Chemistry: An Advanced Treatise" (W. Jost, ed.), Vol. 10, p. 151. Academic Press, New York, 1970. ' Data from G. A. Samara, Phq's. Rev. Br Condens. Marter [3] 22, 6476 (1980). Data from A. E. Abey and C. T. Tomizuka, J. Phys. Chern. Soiids 27, 1149 (1966). ' Data from S. W. Kurnick, J. Chem. Phys. 20,218 (1952). Data from S. Lansiart and M. Beyeler, J . P h p . Chem. Solids 36, 703 (1975). Data from G . A. Samara, P h p . Rev. B : Condens. Matter [3]23, 575 (1981).

the crystals the data in the intrinsic regime revealed marked deviations from exponential behavior at pressures above a few tenths of a GPa." This is illustrated in Fig. 4 for NaCl. Analysis of this effect, including possible contributions from anions and extrinsic cation vacancies, indicated that the deviation is due to suppression of intrinsic vacancy formation by pressure and to the presence of divalent cation impurities.' ' At the higher pressures, as the intrinsic conductivity is suppressed, the contribution of the extrinsic cation vacancies becomes more pronounced, leading to higher 0. In this regard, it is worth noting that measurements of the pressure dependence of the self-diffusion of Na23 in NaCl indicate that the ionic conductivity data in Ref. 11 are not influenced appreciably by anion motiont4; rather, the conductivity is dominated by cation motion. Analysis of the pressure results has yielded the activation volumes for the motion of cation vacancies and for the formation of Schottky pairs. These 24

G. Martin. D. Lazarus, and J. L. Mitchell, Phys. Rro. Br Solid Stute 131 8, 1726 (1973).


25

FIG.4. Pressure dependence of the conductivity of NaCl in the intrinsic regime (stage 11) showing deviation from exponential behavior in the pressure range covered; T = 993.5 K . [After D. M. Yoon and D. Lazarus, Phys. Reo. B : Solid State [3] 5, 4935 (1972).]

volumes are summarized in Table 11. It is found that the values of the activation volumes of motion agree quite well (generally smaller by < 30%) with predictions of the strain-energy model when experimentally determined values of (a In C/aP), are used in Eq. (4.19)." By estimating (8In C/aP), from the macroscopic Griineisen parameter [Eq. (3.6)], Keyes6 had earlier concluded that the strain-energy model does not work well for the alkali halides. In comparing the experimental results with the dynamical model (Section 4,b) there is a question as to what is the relevant phonon mode to use in Eq. (4.28). Short-wavelength optical phonons can be expected to be most effective in moving mobile species toward saddle points; however, there appear to be no data on the pressure dependence of the frequencies of these modes. Use of experimentally determined 8 In v/aP data for the longwavelength transverse optic (TO)-modes, at least for KCl and KBr where data are available, yields" values of AV,,, which are larger by a factor of 3-4 than the experimental AV, values (Table I). This large disagreement is not understood; it could reflect a difficulty with the model for these crystals or simply the use of an inappropriate value of a In v/aP in Eq. (4.28). We suspect the former because it is not likely that there will be a very large difference in a In v/aP between short- and long-wavelength phonons in the alkali halides. have calculated AVf for NaCl and KCl and AVA for Varotsos et the Na' vacancy in NaCl by using their empirical model results Eqs. (4.16) and (4.35). Their results, given in Table 11, are in quite good agreement with the experimental results. These authors have also calculated the compressibilities of AV,. and AV,,, as defined by Eqs. (4.18) and (4.37), respectively. The results based on initial (i.e., P + O ) slopes are tif = 2.5 x lo-' and 2.9 x lo-' kbar-' for NaCl and KCl, respectively, and ti; = 2.2 x lo-' ~

1

.

~

~

3

'

~

26

G . A. SAMARA

kbar-' for the Na+ vacancy in NaCl. It is interesting to note that these compressibilities are about an order of magnitude larger than the bulk compressibilities of these crystals. This is indicative of the local softness of the lattice associated with the formation and motion of Schottky defects in these crystals. It is also interesting to note that K~ and K , are quite comparable in magnitude in NaCI. There are no pressure results on NaCl and its isomorphs which allow determination of the activation volume AV; associated with the motion of anion vacancies. In the absence of such results, AV; can be estimated from AH; in Table I by using the results of the dynamical and empirical models, TABLE 11. MIGRATION ACTIVATION VOLUMES FOR CATION A N D ANIONVACANCIES, FORMATION ACTIVATION VOLUMES FOR SCHOTTKY DEFECTS IN SEVERAL MONOVALENT HALIDES. AND RATIOS OF FORMATION TO MOLAR VOLUMES ACTIVATION VOLUMES

Migration activation volume (cm'/mole) Cation vacancies Halide (AV&) NaCI"

Formation activation volume of Schottky defects,

Anion vacancies AK (AVmJ (cm3/mole) A Y / V ,

+

7 1 5 + 1

55

9

1.9

k 0.3 a(P) data

10

NaBr" KCI"

KBr"

CsCI' TICld

TIBrd

+ +

8 1 9+1 8 1 6+1 23 k 2 11 1 9 + 1 2 24 18 k 2 17 +_ 1 15.3 9.0-14.6 14 k 1 18.0 10.5-17.0

+ +

44 f 9

1.2

k 0.3

61 f 9

1.5

k 0.2

54 k 9

1.1 & 0.2

5.5-9.0 80-87 1.8-2.0 4.5 0.1 41.6 f 3.0 1.2 0.1 3.1 38.0 1.9-3.0 6.3 0.7 45.4 k 3.0 1.2 k 0.1 8.0 35.0 4.6-7.5

+

+

Method

Strain-energy model [Eq. (4.19)] Varotsos and Alexopoulos model' u(P) data Strain-energy model [Eq. (4.19)] u(P) data Strain-energy model [Eq. (4.19)] Dynamical model [Eq. (4.28)] a(P) data Strain-energy model [Eq. (4. I9)] Dynamicdl model [Eq. (4.28)] a(P) data a(P) data Strain-energy model [Eq. (4.19)] Dynamical model [Eq. (4.28)] a(P) data Strain-energy model [Eq. (4. I9)] Dynamical model [Eq. (4.28)]

" Data from D. M. Yoon and D. Lazarus, Phys. Rev. B: Solid State [3] 5,4935 (1972). ' P.Varotsos and K. Alexopoulos, Phys. Status Solidi A 47, K133 (1978); 55, K63 (1979). Data from G. A. Samara, Phys. Rev. B: Condens. Mutter [3] 22,6476 (1980). Data from G. A. Samara, Phys. Rea. B: Condens. Matter [3] 23, 575 (1981).

27


Eqs. (4.39) and (4.38), respectively. As we shall see later, these expressions are found to be satisfactory for those materials where both AV; and AV; have been determined directly from experimental data. An important result of this work on NaCl and its isomorphs is the finding that the Schottky defect formation volumes are substantially larger than the molar volumes, especially for NaCl and KCl. This result was in disagreement with early model calculations which predicted an inward relaxation (i.e., AVf/VM < 1.0) of the lattice upon vacancy formation.8i'2 However, model calculations by Lidiard and Gillang discussed in Section 4,a have corrected this discrepancy. The results of all of the model calculations are compared with the experimental results in Table 111. It is seen that the results of Refs. 9 and 13 are in reasonably close agreement with the experimental results. As can be seen in Table 111, the empirical model of Varotsos et a/.' yields AVf > V , for both NaCl and KCl, and furthermore, as noted earlier, the magnitudes of the AVf values are in close agreement with experiments. RbCl undergoes a pressure-induced transition from the NaCl to the CsCl structure at -0.6 GPa. The conductivity increases by over an order of

'

TABLE! 111. COMPARISON OF THEORETICAL AND EXPERIMENTAL VALUES OF THE FORMATION ACTIVATION VOLUMES A & FOR SCHOTTKY DEFECTS IN SEVERAL ALKALIHALIDES~

Halide _____

NaCl NaBr KCI KBr

Experimentalb

Varotsos et al.'

T

=

0K

T

= 2T 3 mp

'

Lidiard'

~~

1.8g

1.9 f 0.3 1.2 f 0.3 1.5 k 0.2 1.1 f 0.2

~

1.3"; 1.4h 1.2g; 1.3h

1.57 1.47 1.57 1.54

1.94 1.86 1.90 1.87

1.52 1.43 1.50 1.48

~

The results are expressed as the ratio of AV, to the molar volume VM. Data from D. M. Yoon and D. Lazarus, Phys. Reu. B : Solid State [3] 5,4935 (1972), and references therein. ' Data from P. Varotsos, W. Ludwig, and K. Alexopoulos, Phys. Rev. B: Condens. Mutter [3] 18,2683 (1978). Data from M. J. Gillan, Philos. Mag. [Pnrt] A 43,291 (1981). 'The melting points are 1074, 1028, 1049, and 1003 K for NaCI, NaBr, KC1, and KBr, respectively. Data from A. B. Lidiard, Philos. Mug. [Part] A 43,301 (1981). All values were calculated at 300 K. At 300 K. At high temperature.

28

G. A. SAMARA

magnitude at the transition.22 The conductivity data in the CsCl phase d o not extend over sufficient temperature and pressure ranges to allow determination of the activation energies and volumes in this phase. Pressure results on another crystal, CsCI, having this structure are discussed in Section 5,b. Finally, we note that the pressure dependence of bound cation vacancy motion in NaCl has been studied by nuclear magnetic resonance (NMR) technique^.^^ The measurements were performed at -373 K on NaCl samples doped with the divalent impurities C d 2 + ,Mn 2 + ,and Hg". At this temperature there is essentially complete association between the Na+ vacancy and the divalent impurity, and the measured NMR relaxation time probes the relaxation of the nuclear magnetization associated with bound Na+ vacancy jumps among sites nearest and next nearest the impurities. The activation volume for this motion is calculated from the pressure dependence of the relaxation time and is found to be 7.4 & 1 cm3/mol. This value is, within experimental uncertainty, the same as that the AV; for the motion of the free N a + vacancy (Table 11) in NaCl determined from conductivity data. As we shall see later, a similar result is found for the motion of bound and free F - vacancies in CaF, and SrF,. That the two activation volumes are about the same in NaCl is intuitively reasonable, because reference to the details of the crystal structure (Fig. 3a) suggests that the lattice relaxation associated with vacancy motion should not, to first order, be different whether the vacancy is free or bound. This is also reflected by the fact that the activation enthalpies for free and bound vacancies in the alkali and silver halides are found to be the same within experimental uncertainties.2 These results also suggest that the substitutional divalent impurities employed in these studies d o not cause large local lattice perturbations in the crystals in question. b. Cesium Chloride Cesium chloride (CsCI) and its isomorphs CsBr and CsI crystallize at normal conditions in the cubic CsC1-type structure shown in Fig. 3b. In this simplest structure for an ionic crystal. the cations are on a simple cubic lattice and so are the anions, but the two lattices are so displaced that each ion is at the body center of a cube of ions of the opposite charge. The coordination number is 8, making this a tightly packed structure with very little space in interstitial positions. Consequently, the dominant defects in ionic crystals having this structure (including the cesium halides) tend to be of the Schottky type. 2s

R. E. Kost and R. A. Hultsch, Phys. Rer. B : Solid

Slute

[3] 10, 3480 (1974).


29

Ionic transport in the cesium halides has been investigated at atmospheric pressure by diffusion and ionic conductivity but only CsCl appears to have been investigated under pressure.29 Thus, in what follows we consider only CsCl. The atmospheric pressure diffusion and/or ionic conductivity studies on this crystal have led to the following conc l u s i o n ~ :~(1) ~ the ~ ~conductivity ’ is essentially completely ionic; (2) Schottky defects are dominant; (3) diffusive jumps are predominantly single-vacancy near-neighbor jumps; and (4)the mobility of the C1- ion vacancy is much larger than that of the Cs’ ion vacancy. Figure 5 shows the temperature dependence of the ionic conductivity a expressed as log a T versus 1/T at atmospheric pressure P = 0 and at 0.40 GPa. Over the temperature range of the measurements and for the samples in Fig. 5, the conductivity at atmospheric pressure exhibited two regimes. In regime I the conduction is extrinsic and the magnitude of a is sample dependent. The P = 0 data in Fig. 5 are on two samples which apparently have somewhat different impurity contents. For both samples El = 0.62 k 0.02 eV, a value that corresponds to the enthalpy of motion for the Cs’ vacancy AH: in C S C ~In. ~regime ~ I1 the conduction is intrinsic, and the activation energy E l , = 1.33 0.02 eV. Table I summarizes the formation and motion enthalpies for Schottky defects in CsCl. At atmospheric pressure, CsCl transforms from the CsCl structure to the lower density NaCl structure at 740 K.27 The transition is first order and is accompanied by a factor of lo2 decrease inconductivity (Fig. 5). Since pressure favors the CsCl phase, the transition temperature is expected to increase with pressure, but the effect does not appear to have been measured. Pressure causes a large suppression of the conductivity as shown in Fig. 5. This is associated with increases in the activation energies with pressure. Measurements at various pressurei up to 0.8 GPa yielded dE,,/dP = (37.5 f 1.5) x eV/GPa and dE,/dP = (7 f 2) x lo-’ eV/GPa.29 The large increase in El, with pressure is due mainly to the increase in Schottky defect formation enthalpy. Note that the 0.4 GPa isobar in Fig. 5 exhibits a third conduction regime I, in which a decreases with decreasing T faster than it does in regime I. Regime I, is believed to be due to association of Cs+ vacancies and divalent cation impurities to form neutral bound pairs which do not contribute to the cond~ctivity.~’ The association enthalpy is estimated to be AH, ‘v 0.36 eV.29

+

P. J. Harvey and I. M. Hoodless, Philos. Mug. [8] 16, 543 (1967). J. Arends and H. Nijboer, Solid State Commun. 5, 163 (1967). 28 D. W.Lynch, P h p . Rec. [2] 118,468 (1960). 29 G . A. Samara, Phys. ReRea. B: Condens. Matter [3] 22,6476 (1980). 26 27

30

G . A. SAMARA

At constant T, log F decreases linearly with pressure. This is illustrated in the inset in Fig. 5. The 500 K and 670-Kdata correspond to the pressure dependences of CJ in the extrinsic (I) and intrinsic (11) regimes, respectively, whereas the 570-Kdata show the pressure-induced transition from intrinsic (regime 11) to extrinsic conduction (regime 1). From the measured pressure dependences of (r as determined both from t~ versus P isotherms and (r versus T isobars, the activation volumes A V for regimes I and I1 have been determined by using Eq. (3.3).29The results are shown in Fig. 6.These volumes are

-81 1.2

I

1

1

I

1.6

2.0

2.4

2.8

T"

(lo3 K-'I

FIG.5. Temperature dependence of the ionic conductivity of CsCl at 0 and 0.40 GPd showing A = data taken on increasing temperature; @,A = data the various conduction regimes (0, taken on decreasing temperature t o show reversability of results). Also shown at atmospheric pressure is the change in conductivity a t the structural transition to the NaCl structure. The inset shows the pressure dependence of the ionic conductivity at constant temperature in regimes I and 11. [After G . A. Samara, Phys. Rev. B: Condens. Matter [3] 22,6476 (1980).]


31

nearly temperature independent over the temperature ranges covered. The width of the shaded bands of A V values in Fig. 6 represents the estimated uncertainties in the various quantities in Eq. (3.3).29 The calculated A V values in regime I simply represent the motional activation volumes for the Cs' vacancy A V L , since transport in this regime is dominated by the mobility of this vacancy. In regime I1 the conductivity is intrinsic and is dominated by the mobility of the C1- vacancies so that the calculated A V values represent 3 AVf AV;, where AVc is the formation volume for Schottky pairs and AV; is the motional activation volume for C1- vacancies. To obtain AV, one needs to know AV;. Unfortunately, there appears to be no direct measure of AV;. However, one can estimate AV; from either Eq. (4.38) or Eq. (4.39), as discussed earlier. Such estimates yield29 AV; = 5.5-9.0 cm3/mole, where the spread is largely due to the spread in AH; which is estimated to fall in the range 0.2-0.3 eV. This range of AV; leads to AVc = 80-87 cm3/mole. The activation volumes for CsCl are summarized in Table 11. The most important feature of the results in the large value of the formation volume AVf for Schottky defects in CsCI. This value is about 80-100%

+

Lii4w

CI-VACANCY MOTION

m

5w

700

TEMPERATURE IKI

FIG.6. The activation volume of CsCl in the intrinsic (11) and in the extrinsic (I) regimes dominated by Cs vacancy motion. Also depicted (dashed lines) is the estimated activation volume for C1- vacancy motion. The width of the hatched bands represents the estimated uncertainties. [After G. A. Samara. Phys. Rev. B : Condens. Mutter [3] 22,6476 (1980).] +

32

G. A.

SAMARA

larger than the molar volume V,. One can very readily confirm that this conclusion is not materially affected by any realistic uncertainties in the estimated value of AV; discussed above. As we shall see later in Section 7, qualitatively similar results (i.e., AVf > V,) are obtained for the two crystals TlCl and TlBr which also have the CsCl structure. Thus the relaxation of the lattice associated with vacancy formation for these CsC1-type materials is outward as is true of alkali halides having the NaCl structure (Section 5,a). These results are in qualitatizw disagreement with the early model calculations which yielded inward relaxation, but they are in qualitative agreement with the model calculations of Gillan' and Lidiard.13 Although these latter calculations were specifically performed on the NaCl structure, they are expected to hold for other ionic crystal types. Finally, reference to Tables I and I1 shows that AH, and AV,,, for C1vacancies are considerably smaller than the corresponding values for Cs+ vacancies. Two factors contribute to this difference.'g First, the polarizability of the Cs' ion is larger than that of the C1- ion (3.34 versus 2.96 A3),30and it is known that the polarization contributes a positive term to AH, and thereby also to AV,,,." Second, the C1- ion is larger than the Cs+ ion (1.81 versus 1.69 A in radius). Reference to the CsCl structure shows that this size difference allows less space for near-neighbor Cs+ ion jumps than it does for near-neighbor C1- ion jumps, and this leads to larger AH,,, and AV, for Cs' ion motion. In the case of alkali halides with the NaCl structure, the opposite situation is true. Specifically, AH", for cation vacancies is smaller than that for anion vacancies (Table I). Here, again, the explanation for this difference relates to size and polarizability differences between the ions.

6. SILVER HALIDES AGCL AND AGBR The ionic conductivities of AgCl and AgBr have been extensively investigated at 1 bar. These crystals have the cubic NaCl structure, and unlike NaCl and its isomorphous alkali halides where the dominant lattice defects are Schottky pairs, here Frenkel disorder in the cation sublattice is the dominant lattice defect.'.' The reason for this difference between the alkali and silver halides is not understood; it is certainly not due to relative ionic size considerations because the size of the Ag' ion (1.26 A) is intermediate between the sizes of the Na' (0.95A) and K + (1.33A) ions. In AgCl and AgBr the larger sizes of the halogen ions (compared to Ag') make it energetically unfavorable for these ions to move into interstitial positions. Charge transport occurs by the motion of both Ag vacancies and interstitials, but +

30

J. R.Tessman, A. H. Kahn, and W. Shockley, Phys. Rev. [2] 92,890 (1953)


33

the mobility of interstitials is much higher than that of vacancies.' Interstitials move by an interstitialcy mechanism.2 In this mechanism the interstitial does not migrate by pushing through a cube face (Fig. 3a) to a new interstitial position (the direct mechanism); rather the Ag+ interstitial pushes one of its neighboring Ag+ ions off its lattice position into an interstitial position and the first interstitial occupies the normal lattice site so vacated (the indirect mechanism). Table I gives a summary of the enthalpies for the migration and formation of Frenkel disorder in AgCl and AgBr. The effects of hydrostatic pressure on the ionic conductivity of pure and doped samples of AgCl have been investigated by Abey and Tomizuka3 and 103

102

- lo1 Y

r

'Q E

10

v

I-

. -b v)

1

10-1

10-2 0.0

0.2

0.4

0.6

0.8

1 .o

PRESSURE (GPa)

FIG. 7. Pressure dependence of the ionic conductivity of AgBr at various temperatures. Results for T I 357 K represent the response in the extrinsic regime (stage I), whereas results for T 2 392 K represent the response in the intrinsic regime (stage 11). The 392 K isotherm shows evidence for the onset of a gradual pressure-induced transition from intrinsic to extrinsic conduction above 0.4 GPa. [After S. Lansiart and M. Beyeler, J . Phys. Chem. Solids36, 703 (1975).]

3'

A. E. Abey and C. T. Tomizuka, J . P h v . Chrm. Solids 27, I149 (1966)

34

G . A. SAMARA

by Murin et Similar studies were performed on AgBr by K ~ r n i c and k~~ by Lansiart and B e ~ e l e rFor . ~ ~both crystals G decreases exponentially with pressure over the pressure ranges covered ( I 0.8 GPa), and the enthalpies of the various conduction regimes increase with pressure. Typical results are shown in Fig. 7. In Fig. 7 the data at T I 357 K are in the extrinsic regime where Ag+ vacancy motion dominates, whereas the data at T 2 402 K correspond to the intrinsic regime where the mobility of Ag+ interstitials dominates. The pressure results have yielded the activation volumes for Ag' vacancy and Ag+ interstitial motion as well as the formation for Ag' Frenkel defects. These volumes are given in Table IV. It is seen that the activation enthalpies and volumes for Ag interstitial motion are considerably smaller than the corresponding values for Ag' vacancy motion for both crystals. The ratio Q, of the mobility of the interstitial to that of the vacancy increases with pressure for both crystals. For example, for AgC1,d = 8 at 1 bar and 488 K and increases to 10 at 6 kbar and 527 K.31 The formation volumes for Ag' Frenkel pairs are considerably smaller than the molar volumes, as is expected for Frenkel disorder. Examination of the results in Table IV shows considerable disagreement in the A I/ values reported by the various authors. For AgBr there is good agreement in the AVf values reported by K u r n i ~ k ~ ~ and by Lansiart and B e ~ e l e r ,but ~ ~ the motional volumes AVm disagree somewhat. This disagreement may be partly due to the fact that Kurnick's samples were heavily doped (-0.1% Cd2+),and he assumed in the analysis of his data that all Ag+ vacancies are free,34 whereas, in fact, some are probably bound at this dopant concentration. To get the correct value of G , this assumption underestimates the mobility of the vacancies.34 In the case AgCl, the large value of AV[ reported by Murin et aL3*most likely relates to the fact that this value was determined at too high a temperature (629 I(;the melting point is 728 K) where the strong possibility exists that mixed Frenkel and Schottky disorder set in, as described later for AgBr. No account of this possibility was taken in the analysis by Murin et al. The differences between the AV,,, values reported by Abey and Tomizuka3' and those reported by Murin et al. may also be due to the possibility that the Murin et al. values were deduced from data obtained in mixed extrinsic-intrinsic regimes. All of the Murin et al. data were obtained in the temperature range 473-633 K, and reference to Abey and Tomizuka's work (their Fig. 4) suggests that the above-mentioned possibility is the most likely explanation for the differ+

32

33 34

A. N. Murin, I. V. Murin, and V. P Sivkov, Sor. Phys.-So/id State (Engl. Trunsl.) 15, 98 (1973). S. W. Kurnick. J . Chrm. Phys. 20,218 (1952). and references therein. S. Lansiart and M. Beyeler. J . Phys. Chem. Solids 36,703 (1975).


35

ences. In view of this, we feel that Abey and Tomizuka’s results on AgCl are the more reliable set. The pressure results on AgCl and AgBr have yielded additional insight. In the case of AgBr, Kurnick’s results33 show that in the intrinsic range above 600 K (i.e., just below the melting point of 670 K), (8In o/dPI is larger than expected from a simple Frenkel disorder mechanism. The results strongly suggest that mixed Frenkel and Schottky disorder set in at these high temperatures, and Kurnick interpreted his results accordingly. Abey and Tomizuka’s31 results on AgCl did not extend as far in relative temperature as Kurnick’s results on AgBr, but Murin et d.’s results32 on AgCl did,

-

N

TABLEIV. MIGRATION ACTIVATION VOLUMEFOR AG+ INTERSTITIALS AND A c t VACANCIES, FORMATION ACTIVATION VOLUMES FOR FRENKEL DEFECTS I N AGCL A N D AGBR, A N D RATIOS OF FORMATION ACTIVATION VOLUMES TO MOLARVOLUMES Migration activation volume (cm’/mole)

Halide AgCl

AgBr

Interstitials ( A Vmi) 3.3 5.2 I .2 1.1 2.6 2.6 3.6 1.5(2.0)” 1.4(1 .9)h 3.1(4.I ) h

Formation activation volume. A V, Vacancies A Vm;” (cm3/mole) 4.7 9.4 2.5 2.2 5.3 7.4 5.5

3.5(3.2) 3.2(3.0) 7.0(6.4)

A &/ VM

Method

0.64 0.85 0.44 0.39

a(P)data” a(P) datah Strain-energy model [Eq. (4.19)]‘ Strain-energy model [Eq. (4.25)l‘ Dynamical model [Eq. (4.28)y 16.0 0.55 a(P)data’ 14.0 0.48 a(P) dataH 1 I .0(9-.9)h 0.38(0.34) Strain-energy model [Eq. (4.19)]‘ 1 0.4(9.3)h 0.36(0.32) Strain-energy model [Eq. (4.25)‘ Dynamical model [Eq. (4.28)]‘

16.7 22 11.4 10.1

“ Data from A. E. Abey and C. T. Tomizuka, J. PhFs. Chem. Solids 27,1149 (1966). ” Data from A. N. Murin. I. V. Murin, and V. P. Sivkov, Sot:. Phjs.-Solid Slate (Enyl. Trans/.)15, 98 ( 1 973). Based on (a In C/dP), values calculated from the elastic data of K. F. Loje and D. E. Schuele, J . Phys. Chem. Solids 31, 2051 (1970). by using Eq. (4.20). Based on the thermodynamic elastic Criineisen parameters 7 = 2.03(2.39) for AgCI(AgBr) [K. F. Loje and D. E. Schuele, J. Phys. Chem. Solids 31,2051 (1970)l. ‘Based on microscopic mode ys = 5.0(5.6) for AgCI(AgBr) for the q z 0 T O phonon given in R. P. Lowndes. Phys. R w . B : Solid Sture [3] 6, 1490 (1972). Data from S. W. Kurnick, J . C h m . Phys. 20,218 (1952). Data from S. Lansiart and M . Beyeler, J. Phys. Chem. Solids 36, 703 (1975). Values in parentheses are based on AHs of S. Lansiart and M. Beyeler, J . Phys. Chem. Solids 36,703 (1975). whereas other values are based on A H s from S. W.Kumick, J. Chem. Phys. 20, 218 (1952).

36

G. A. SAMARA

and as noted before, they suggest that a similar situation exists for this crystal. Abey and Tomizuka’s results yielded values for the association (binding) enthalpy and volume of vacancy-divalent impurity complexes in AgC1. These values are AH, = 0.16 eV and AVa = 0.98 cm3/mole. The strain-energy model yields values of AV,,, and AV, for both crystals, which are of the correct order of magnitude but are generally 30-60% smaller than the experimental values deduced from the o(P)data (Table IV). The discrepancy is larger for AV,,, than it is for AVr. We have carried out the strain-energy model calculations using either Eqs. (4.19) and (4.20), based on the directly measured pressure dependences of the elastic ons st ants,^^ or Eq. (4.22), based on the macroscopic elastic Gruneisen parameters given by Loje and S ~ h u e l eAs . ~ seen ~ in Table IV, both approaches yield very nearly the same values for AV. To test the validity of the dynamical model [Eq. (4.28)], we need the microscopic Gruneisen parameters yi . The only directly measured yi’s known are those for the long-wavelength transverse optical (TO) modes of both crystals. The values are yi = 5.0 for AgCl and 5.6 for AgBr.36 Use of these values in Eq. (4.28) yields AV, as shown in Table IV. It is seen here that AV,,, is in much closer agreement with the experimental results than are the results of the strain-energy model. 7. THALLIUM HALIDES TLCL,TLBR,TLI Thallous chloride (TICI)and bromide (TlBr) have the cubic CsCl structure (Fig. 2a), and thallous iodide (TII) transforms from an orthorhombic phase (space group Amam-D;;) to the CsCl structure with increasing temperature and/or p r e ~ s u r e7., 3~8 These crystals have low melting temperatures (-700730 K) and relatively high ionic conductivities (Figs. 2, 8, and 9).They also have relatively large dielectric constant^.^' This is important to ionic transport because the larger the dielectric constant of an ionic crystal, the lower the energy of formation of lattice defects.’* Also of interest is the fact that earlier studies have shown the existence in these crystals of weakly soft optic and acoustic p h o n o n ~ . ~Physically, ’ ionic transport occurs by hopping motion across an energy barrier, and this barrier can be expected to be smaller the “softer,” or more anharmonic, the lattice. K. F. Loje and D. E. Schuele, J . Phys. Chem. Solids 31,2051 (1970). R. P. Lowndes. Phjs. Rea. B: Solid S i d e (31 6, 1490 (1972). 3 7 G. A. Samara, Phys. Reir. [2] 165,959 (1968). 3 R G. A. Samara, L. C. Walters, and D. A. Northrup, J . Phys. Chem. Solids 28, 1875 (1967). 39 G . A. Samara, Phys. Rev, B: Condens. Matter [3] 23, 575 (1981). and references therein.

35 36


37

11111111111111 - 3 1 ,\

\

TIC1 ( T - 6 2 5 K I TlBr ( T z 6 0 0 K 1 t TlBr I T ~ M O K I

-c-

0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

PRESSURE ( GPa I

FIG.8. Pressure dependences of the conductivities of TIC1 and TlBr near 600 K showing the change (turnaround) in conductivity with pressure. [After G. A. Samara, Phys. Rev. B : Condens. Matter [3] 23, 575 (1981).]

Earlier, diffusion and/or ionic conductivity studies at atmospheric pressure on TlC14' and T1Br41 have led to the following conclusions: (1) the conductivity is strictly ionic at 1 bar; (2) Schottky defects are predominant; ( 3 ) diffusive jumps are predominantly near-neighbor; (4) the mobility of the anion vacancy is much larger than that of the T1' vacancy. For TlCl the ratio of the mobilities of the two ions is in the range 30-100, depending on the temperature4'; and

40

41

R. J. Friauf. Z. Naturforsch. A 26A, 1210 (1971); J . Phy.r. Chrm. Solids 18, 203 (1961); and references therein. P. Herrmann, Z . Phjs. Chem. (Leipziq)227,338 (1964).

38

G. A. SAMARA

( 5 ) the intrinsic conductivity is relatively high, and polyvalent dopants are not sufficiently soluble in both crystals to give separable extrinsic regimes in the conductivity.

Thus, there was no information from conductivity data on the formation and motion energies of the individual defects. Pressure, as pointed out later, suppresses the intrinsic conductivity sufficiently and makes it possible to evaluate directly these energies.39 In the case of TlI, atmospheric pressure studies of the transport processes revealed a particularly high electronic -3

--"

-4

E

'2 D

-5

z? -6

,

I

0.8 +1

I

I

1.6 P (GPa)

,

l

2.4

0

0.8

L6 P (GPa)

0

-1

Y

-'35

d

I

c D

-2

-3

0,

8 -

-4

-5

FIG.9. Temperature dependence of the conductivity of a TlBr sample at various pressures. The inset on the left is a log u vs. P plot showing by arrows where the isobars in the lower portion of the figure were taken. The highest isobar (2.0 GPa) is in the electronic conduction regime. The inset on the right shows the pressure dependence of the activation energies in the intrinsic (11) and extrinsic (1) regimes. [After G . A. Samara, Phys. Reu. B : Condens. Mutter [3] 23, 575 (1981).]


39

conductivity in the orthorhombic phase, whereas in the cubic phase the conductivity is predominantly ionic with both the T1’ and I- ions contributing to the transport p r o ~ e s s . ~The ’,~~ fact that both ions contribute significantly to the transport makes it impossible to evaluate the formation and motion energies and volumes from conductivity measurements alone. Consequently, the discussion here in Section 7 deals primarily with TlCl and TlBr. The effects of combined pressure and temperature on the ionic conductivities of these crystals have been reported.39 The results on TlCl and TlBr have allowed determination of the lattice volume relaxations and energies associated with the formation and motion of Schottky defects in these crystals. The association energy of T1+ vacancies and divalent impurities was also determined for TIBr. These were the first such determinations of these properties from conductivity measurements for the thallous halides, and they were made possible by the large pressure-induced suppression of the intrinsic conductivities of these materials which results in observation of extrinsic conduction regimes. Another result was the observation of a pressure-induced transition from ionic to electronic conduction in these crystals. The effects ofpressure on the conductivities of TlCl and TlBr are illustrated in Fig. 8. At pressures below -0.5 G P a and at temperatures in the range 350-700 K, the logarithms of the conductivities of TlCl and TlBr decrease linearly with pressure.39 However, at higher pressures there is a marked change in the response at temperatures near 600 K. Specifically, there is a turnaround in the conductivity with pressure. This is a result of a pressureinduced transition from ionic conduction at low pressure to electronic conduction at high p r e ~ s u r e . ~The ” turnaround occurs sooner for TlBr than for TlC1, in agreement with expectation on the basis of what is known about the pressure dependence of the electronic structure.39 In the ionic regime, o decreases with pressure as a result bf a suppression of both the formation of defects and of their mobility, whereas in the electronic regime, o increases with pressure as a result of the decrease in the band gap.39 In TI1 the ionicelectronic transition occurs at lower pressures and temperature^.^' The effects of pressure on the conductivity of TlBr in the ionic regime are illustrated in Fig. 9. Qualitatively similar results are observed for TlCl.39 There are many features in these data that are noteworthy. At high temperatures (stage II), the conduction is ionic and intrinsic. There is a large suppression of c with pressure. This is associated with a large increase in activation energy (Fig. 9, upper right inset), which is largely due to the increase in Schottky defect formation energy.39 Stage I is an extrinsic regime which becomes observable at high pressure because of the relatively large 42

43

A. Schiraldi, A . Magistris. and E. Pezzati, Z . Nuturjorsch. A 29A, 782 (1974). 2. Morlin, Phys. StutuslSolidilA 8 , 565 (1971).

40

G . A. SAMARA

suppression of the intrinsic component of the conductivity. The pressure dependence of the activation energy E, in stage I is shown in the upper right inset, where E, corresponds to the mobility activation energy of the T1+ ion in T1Br.3y Stage I has never been seen before in conductivity data. It is observed only at high pressure, and this illustrates the usefulness of pressure as a complementary variable to temperature in the study of ionic transport processes. The faster drop in B with decreasing T in stage I, in Fig. 9 is believed to be due to the association (or binding) of the TI’ vacancies and divalent cation impurities to form neutral bound pairs which do not contribute to ~ 7The . ~ association enthalpy A H , was found to be -0.36 eV.39 This was the first evaluation of this quantity, again illustrating the usefulness of pressure in such studies. The upper left inset in Fig. 9 is a log B versus pressure plot showing by arrows where the log OT versus T-’ isobars in the lower portion of Fig. 9 were taken. Data at the lower pressures (0, 0.4, and 0.8 GPa) are clearly in the ionic regime. However, the highest isobar (2.0GPa) is clearly in the electronic conduction regime, and it can be seen that the activation energy at this pressure at high temperature is markedly different from those at lower pressures. In fact, E = 0.4eV as shown by the solid triangle in the upper right inset. This energy is believed to be associated with the ionization of an electronic impurity state within the b a n d g a ~ . ~ ~ The pressure dependences of the conductivities of TIC1 and TlBr in the various ionic conduction regimes are summarized in Fig. 10 as the initial (i.e., low pressure) logarithmic pressure derivatives of o versus T. The results were obtained from both a(P)isotherms and a(T) isobars. Qualitatively similar results have been reported for cubic T1I in the ionic conduction regime,3y but these are complicated by the aforementioned fact that both T1+ and 1- contribute to B in the intrinsic regime. Table I gives the motional enthalpies for positive and negative ion vacancies as well as the formation enthalpies for Schottky defects in TlCl and TlBr. Some general observations about the results in Table I should be made. The magnitudes of AH,,, and AHf for both crystals are remarkably small compared with other ionic crystals (particularly the alkali halides) where Schottky defects are dominant. The anomalously small values of AHf for Schottky defects in TlCl and TlBr are most likely manifestations of the large static dielectric constants (t N- 30 compared with c ‘v 5-6 for the alkali halides) and relatively soft phonons3’ in these materials. In a dielectric continuum model the formation energy for a vacancy is” of the order of (Ze)’/2cR, where Z e is the defect charge and R is the defect radius. This result should be adequate for slowly varying fields far from the defect, but it

~


41

cannot be expected to describe adequately the behavior very near the defect. To do so it is necessary to take into account both the details of the shortrange interactions around the defect as well as the continuum energy away from it. In any case, however, the main point for the present consideration is that large t leads to low formation energies. The relatively low values of A H , for both crystals may be a manifestations of the soft and strongly anharmonic nature of their lattices. We shall return to this point later. Finally, we note that A H ; is much smaller than AH: for both crystals and that A H , for the C1- ion is smaller than A H , for the Br- ion. A similar situation is seen to be true for the motion volumes AV,,, (as discussed later). These effects are undoubtedly due to the fact that the polarizability of the Tl' ion (5.2 A3) is larger than that of the Br- ion (4.2 A3), which, in turn, is larger than that of the C1- ion (3.0 A3).39It is known that the polarization energy contributes a positive term to A H , .' 7340

1

,

,

500

6W

1

MO

400

TEMPERATURE IKI

FIG. 10. Variation with temperature of the logarithmic pressure derivatives of the ionic conductivities of TlCl and TlBr in the intrinsic (II), extrinsic (I), and association (I,) conduction regimes. Typical error bars are shown. [After G. A. Samara, P h p . Rev. B: Condens. Marrer [3] 23, 575 (1981).]

42

G . A. SAMARA

The activation volumes in the various conduction regimes of TlCl and TlBr, calculated by using Eq. (3.4), are shown in Fig. 11, where the widths of the shaded bands of the A V values represent the estimated uncertainties in (a In R/dP), and the ranges of the values of (a In v/dP),. We note that the values for A V are only weakly temperature dependent. In the intrinsic regime the calculated A V in Fig. 11 represents AV, AV;, whereas in the extrinsic regimes AV represents the elastic volume relaxation associated with the motion of the individual species. Table I1 summarizes A V for cation and anion motion as well as for the formation of Schottky defects in TIC1 and TIBr. We note that despite the fact that the T1+ ion is smaller than the C1- and Br- ions (ionic radii are 1.49, 1.81, 1.97 A, respectively), the volume relaxation associated with the motion of the Tl' ion is much larger than those associated with the motions of C1- and Br- ions as is true of AH,,, in Table I. A similar situation obtains in CsCl (Section 5,b and Tables I and 11), but the opposite appears to be true for those crystals having the NaCl structure, at least with regard to the

+

40

r-----t

z F? c

a

1

"

c

a

0

Mo

400

5M1

600

TEMPERATURE (KI

FIG.1 1 . Temperature dependence of the activation volumes of TlCl and TlBr in the intrinsic (II), extrinsic (I). and association (I,) conduction regimes. The widths of the shaded bands represent the ranges of estimated uncertainties in the two logarithmic pressure derivatives in Eq. (3.4). [After G. A. Samara, Phys. Reu. B : Condens. Mutter [3] 23, 575 (1981).]


43

fact that for these crystals, AH; > AH: (Table I). These results and differences can be qualitatively explained in terms of ionic polarizability and size effects as well as details of the crystal structure^.^^ We shall come back to this point later. One of the most important features of the results in Table I1 is the large value of the formation volume AVf for Schottky defects in these crystal^.'^*^^ This value is 15-20'j/, larger than the molar volume V , for both TlBr and TlCI, respectively. As we have already seen (Section 5,b), a similar result is found for C S C ~where , ~ ~ AV,/V, E 1.8-2.0. Thus for the three crystals TlCl, TlBr, and CsCl having the CsCl structure, the relaxation of the lattice associated with vacancy formation is outward, as is true of alkali halides having the NaCl structure (Section 5,a). These results are in qualitative disagreement with the early model calculations which yielded inward relaxation,'"' but they agree with recent model c a l ~ u l a t i o n s . ~Although ~' the latter calculations were made for NaC1-type crystals, they are expected to hold for other ionic crystals. Table I1 also shows the activation volumes calculated from the strainenergy and dynamical models. The strain-energy model results were obtained using Eqs. (4.19) and (4.20) with the experimentally determined pressure derivatives of the elastic constants.39 These results are in satisfactory agreement with A V values deduced from the g ( P ) data. Somewhat poorer agreement with experimental results is obtained if we resort to the Gruneisen approximation [i.e., Eq. (4.21)] for (a In C/dP),.39 A similar finding is ohtained for NaC1-type alkali halides' ' (Section 5,a). The dynamical model results in Table I1 were calculated using Eq. (4.28), and the range of AVvalues given reflects the dispersion in transverse optic ) the Brillouin zone.39These T O modes should mode yi values ( ~ 2 - 5 across be effective in moving mobile speiies toward saddle points. The calculated AV,,, values range from 0.25 to 1.2 times those deduced from the o(P) data. Intuitively, we would expect short-wavelength TO modes to be most effective in moving ions toward saddle points; however, the calculated yi values for these modes are in the range 2-3, and they are responsible for the low end of the range of the calculated AV,,, values in Table 11. An interesting feature of the calculated results is that the experimentally determined values of y i for the long-wavelength T O phonons in both crystals yield AV: = 17.0(14.6) cm3/mole and AV; = 7.7(3.0) cm3/mole for TlBr(TlCl), respectively, and these values are in rather close agreement with those deduced from the a(P) data. The interesting point is that this T O mode is the soft optic mode of the system, which suggests a possible relationship between soft phonons and ionic transport. In this soft mode, the ionic displacements correspond to the vibration of the T1+ ions against the negative halogen ions. Reference to the CsCl crystal structure indicates that near-neighbor diffusive

-

44

G . A. SAMARA

jumps in this lattice can be affected by the degree of anharmonicity (or softness) of this mode.

8. FLUORITES AND OTHER RX,-TYPE HALIDES RX,-type ionic crystals (R is a divalent metal cation and X is a monovalent halogen anion) crystallize in a variety of crystal structures, of which the best known is the cubic fluorite structure (space group Fm3m-0;) shown in Fig. 3c. This is a relatively simple structure in which the cations are situated in the center of alternate cubes of the anion sublattice; i.e., each R 2 + ion is surrounded by eight X - ions, and each X- ion is tetrahedrally coordinated to four R 2 + ions. It is seen that the structure is relatively open, containing a large number of “voids” and thus can tolerate a high concentration of interstitial ions. Crystals with the fluorite structure have been model systems for the study of a wide variety of solid state phenomena. Considerable recent interest has centered around the study of the ionic conductivity of these materials, especially with regard to the high value and unusual behavior of the conductivity at high temperature. This behavior is associated with an orderdisorder-like transition, accompanied by a broad specific heat anomaly, which occurs in these materials well below their melting points.44 At temperature!: above the transition temperature, the conductivity is very high and is comparable to that of molten salts; this is presumably due to the existence of extensive disorder in the anion sublattice. Below the transition, anion Frenkel defects are the dominant intrinsic disorder in these material^.^ This can be essentially anticipated on the basis of the crystal structure shown in Fig. 3c. The effects of pressure on the ionic conductivity of several crystals having the fluorite structure have been investigated. These crystals include PbF, , CaF,, SrF,, BaF,, CdF,, and SrCl,, and the results are reviewed here in Section 8. Because the fluorite structure is an open structure, it can be expected to be unstable at sufficiently high pressure. Indeed, a pressure-induced phase transition to a more densely packed structure occurs in these materials. The transition has been observed in PbF2,45 BaF,,46 and SrCl,?’ and is expected to occur in the other materials. The high-pressure phase has the W. Hayes. Contemp Phys. 19, 469 (1979). G. A. Samara. J . Phys. Chem. So/& 40, 509 (1979). 4h G . A. Samara, Phys. Rer. B ; Solid State [3] 2,4194 (1970). 47 J . Oberschmidt and D. Lazarus, Phys. Reo. B : Condens. Matrer [ 3 ] 21,2952 (1980) 44

45


45

orthorhombic PbC1,-type structure (space group Prnnb-V;:), and it is about 10% more dense than the fluorite phase. In the case of PbF,, the orthorhombic phase can be recovered at atmospheric pressure. The effects of pressure on the ionic conductivities of the orthorhombic phases of PbF, and SrC1, have been reported and so have similar results on the isomorphous crystals PbCI, and PbBr,. These latter crystals exist in the orthorhombic phase at ambient conditions. Pressure studies have also been reported on PbI,. Unlike the other lead dihalides, this material crystallizes in a layered hexagonal structure (space group P h - D : , ) . The results for all of these materials are reviewed briefly here. a. Lead Fluoride (PbF,)

We start this section by reviewing the temperature and pressure dependences of the ionic conductivity of PbF,. We shall review the results on this material in somewhat more detail than for any other material in this section because of the following considerations: PbF, has attracted considerable recent attention, primarily centered around its high ionic conduct i ~ i t y , ~ which ~ , ~ ’ is much higher than that of any of the other RX,-type crystals to be discussed (Fig. 2). The conductivity versus temperature and pressure responses of this crystal exhibit a variety of interesting features (as discussed later), and some of these features are common to many of the other crystals to be discussed. The material has been more thoroughly investigated under pressure than the other RX,-type crystals, and finally, it exhibits the cubic-orthorhombic transition already mentioned. The orthorhombic phase ’ transforms to the cubic phase at -610 K at atmospheric p r e ~ su r e . ~The cubic phase, on the other hand, transforms to the orthorhombic modification at high pressure, where the transition pressure is -0.4 GPa at 300 K.4’ Both phases exhibit relatively high ionic conductivity, and the cubic phase becomes a “superionic” conductor at high temperature, where its conductivity attains at 2800 K what appears to be one of the highest values for any known solid ionic conductor (Fig. 2). Like most ionic crystals having the fluorite structure, PbF, is known to be a fluorine ion conductor and Frenkel defects are the dominant defects4’ Detailed investigations have been reported on the effects of temperature and hydrostatic pressure on the conductivity in both phases with some emphasis on the behavior near the phase transition^.^^.^' Some of this work was motivated, in part, by the large dielectric constants and the soft, low-lying phonon modes of the crystal and the possible connection between these properties and high ionic cond~ctivity.~’ Figure 12 shows the temperature dependence of the ionic conductivity of PbF, samples in both phases at atmospheric pressure. It is seen that in each

46

G. A. SAMARA

case o ( T )exhibits a number of different activated stages. The orthorhombic phase transforms to the cubic phase at -610 K. Results on samples with both monovalent and trivalent impurities substituted for the Pb2+ ion (which introduce F ion vacancies and interstitials, respectively) make it quite certain that the explanation for the o ( T )data in the various stages is as follows45:

Cubic phase. In stage I the conduction is extrinsic, and the mobile charge carrier is an F ion vacancy or F ion interstitial depending on the dopant. In stage I1 the conduction is intrinsic by F ion vacancies, and in stage I11 the conduction is intrinsic by F ion interstitials. Finally, in stage IV the F ion sublattice had been presumed to exhibit liquidlike characteristics because the F ions are disordered and in continuous motion, thus accounting for the very high conductivity. This is the phenomenon referred to as sublattice TEMPERATURE (K)

FIG.12. Temperature dependences of the ionic conductivities of cubic and orthorhombic PbF, samples measured at atmospheric pressure; ( 0 )cubic phase, ( A ) orthorhombic phase. I n each case the conductivity exhibits a number of different stages as indicated by the Roman numerals and as discussed in the text. [After G . A. Samara, J . Phys. Chem. Solids40,509 (1979).]


47

melting, and its lattice dynamical origin has been attributed to the vanishing of a zone-boundary optic phonon at the X point of the Brillouin zone of the cubic fluorite s t r ~ c t u r e . ~An ’ alternative interpretation of the high conductivity in this stage is based on a “unique” high mobility of anion vacancies nor well underin this regime, 48d but the situation is neither stood. We should point out that the experimental situation in stage IV is also not clear-cut. The data often exhibit some lack of reproducibility which has been attributed to sample contamination by impurities from the ambient atmosphere and to surface-layer effects. Orthorhombic phuse. In stage I1 the conduction is extrinsic by either F ion vacancies or interstitials depending on the dopant, and in stage 111 the conduction is intrinsic by F ion vacancies. Table V summarizes for the two phases the formation enthalpies for Frenkel defects AH, and the enthalpies for F vacancy motion and F interstitial motion. The cubic-orthorhombic transition in PbF, can be induced either by increasing pressure at constant temperature or by increasing temperature at finite pressure, and the transition pressure is strongly temperature deen dent.^^.^' The work has shown that G can either increase or decrease at the transition depending on the temperature and impurity content, and that the magnitude of the change is strongly temperature and impurity dependent.45 These features are illustrated in Fig. 13a where the transition is induced with increasing temperature at 0.12 G P a and in Fig. 13b where the transition is induced with increasing pressure at 280 K. The transition is a reconstructive first-order transition and involves the formation of a large number of crystalline defects. The features of the transition in Fig. 13b are typical of the response observed at -and below room temperature, a characteristic of which is the overshoot in the value of G at the end of the transition over the value expected (on the basis of an extrapolation of the high-temperature data) for the orthorhombic phase. This overshoot is probably associated with these defects, and the relatively large decrease in G with increasing pressure in the orthorhombic phase just above the transition is most likely due to some pressure “annealing” of these defects. With increasing pressure o relaxes to the value expected for the orthorhombic phase. When the transition takes place at temperatures 2 3 5 0 K (Fig. 13a), there is only a very small undershoot (overshoot) in CT below (above) its value in the orthorhombic phase, followed by rapid relaxation to this value with either increasing temperature or pressure. For this reason most of the results on L. L. Boyer, Phjs. Reti. Lett. 45, 1858 (1980). J. Schoonrnan, SoCiolidSiurc. Ionics 1, 121 (1980). 48b C. R.A. Catlow, Comments Solid State Phys. 9, I57 (1980). 48

48a

48

G . A. SAMARA

TABLEV. ENTHALPIES FOR THE FORMATION OF INTRINSIC DEFECTS A N D FOR THE MOTION OF ANION VACANCIES AND INTERSTITIALSFOR SEVERAL RX Z- Ty pCRYSTALS'' ~ Enthalpy (eV)

Crystal and phase Cubic PbF, SrCI, CaF,

Anion vacancy motion (AH;,")

BaF,

0.26'; 0.25' 0.37'; 0.34-0.56d 0.52"; 0.52-0.87' 0.36-0.47" 0.96'; 0.94- 1.Od 0.47''; 0.56y 0.51'; 0.56d; O.6lg

CdF,

0.60"; 0.39-0.5Id

SrF,

Anion interstitial motion

Formation of intrinsic defects (AHj)

0.52'

0.94h 2.50' 3.00"; 2.3-2.8'

0.82"; 0.53-1.64d 0.77- 1.08 0.94"; 0.71-1.0d 0.86-0.97y 0.79"; 0.62-0.79' 0.72-0.77'

2.28'; 2.3." I .90'; 1.9'

2.3h

0.40h Orthorhombic PbF, SrC1, PbCI, PbBr, Hexagonal Pbl,

0.36'; 0.33' 0.44' 0.32'; 0.20-0.38' 0.30'; 023-0.29' 0.20'

0.41"

1.12' 1.32' 2.40' 2.10'

2.10'

" Frenkel defects are the dominant intrinsic disorder for all these crystals except for orthorhombic PbCI, and PbBr,, in which Schottky disorder is dominant. Data from G. A. Samara, J . Phys. Chem. Solids 40,509 (1979). ' Data from J . Oberschmidt and D. Lazarus, Phys. Rev. B: Condens. Mutter [3] 21, 2952 ( 1980). Data from a compilation by A. B. Lidiard. in "Crystals with the Fluorite Structure" (W. Hayes, ed.). p. 101. Oxford Univ. Press (Clarendon), London and New York, 1974. Data from J . Oberschmidt and D. Lazarus, P h p . Rev. B: Condens. Murter [3] 21, 5823 (1980). Data from a compilation by J. J . Fontanella, M.C. Wintergill, and C. G. Andeen, Phys. Status Solidi B 97, 303 (1980). Data from a compilation by L. W. Barr and A. B. Lidiard. in "Physical Chemistry: An Advanced Treatise" (W. Jost,ed.), Vol. 10, p. 151. Academic Press, New York, 1970. Data from G. A. Samara. to be published. Data from J. Oberschmidt and D. Lazarus, Phys. Reo. B : Condens. Mutter 131 21, 5813 (1980). and references therein.

' I'

the orthorhombic phase to be discussed were obtained on samples transformed at 2400 K. The conductivity decreases with pressure in both phases. Isotherms of log (T versus P are essentially linear over the pressure ranges covered (up to 1 GPa). A summary of the initial pressure derivatives expressed as

-


430

TEMPERATURE (K) 400 370 340

49

315

1 0.0

0.2

0.4 0.6 0.8 PRESSURE (GPa)

1.0

FIG. 13. (a) Temperature dependence of the ionic conductivity of a PbF, sample measured at 0.12 G P a ; and (b) pressure dependence of the conductivity of a PbF, sample at 280 K. The results show the responses in both the cubic and orthorhombic phases as well as the various behaviors at the transition. [After G. A. Samara, J. Phys. Cbem. Solids 40, 509 (1979).]

(8 In o/8P), is given in Fig. 14. Nofe that this quantity generally decreases with temperature, and its magnitude as well as its temperature coefficient vary in the various stages. The activation volumes calculated using Eq. (3.4) are shown in Fig. 15, and Table VI summarizes the activation volumes for the various transport processes in PbF, and the other RX, materials. A number of comments on the PbF, results in Table VI and Fig. 15 should be made. For the cubic phase, the increase in volume associated with F vacancy motion is rather small (AV& = 1.9 cm3/mole) and is about onehalf that associated with F interstitial motion, which is also relatively small (AV;,i = 3.5 cm3/mole). The small magnitude of AV& is qualitatively consistent with what can be expected from consideration of the open crystal structure ofcubic PbF, (Fig. 3c). In the orthorhombic phase the two motional activation volumes AV& and AV;,i are comparable in magnitude, although they exhibit somewhat different temperature dependences. The near equivalence of these two AV,,, terms is a manifestation of the more closely packed nature of the orthorhombic PbF, lattice compared with that of cubic PbF,.

50

G . A. SAMARA

A hard-sphere model suggests that AV,,, is approximately equal to the volume of the diffusing species for vacancy and interstitial motion. In the present case, A V in stage I of the cubic phase is 5 1/3 the molar volume of the F ion. Relatively small A V reflects partly the fact that real ions are not hard spheres and, more important, the relatively open nature of the fluorite structure and the highly anharmonic (and soft) nature of the lattice vibrations of the PbF, lattice. The results in stage I1 suggest that A V decreases with increasing T, and this is what one would intuitively expect-as the lattice softens with temperature, a lower A V is needed for diffusive motion.

1

-180t-

-80 -

-

(a 1 I

240

280

320

360 T

I 400

IKI

-

-c 62

-1M-

L

-80

-

250

3n

450 TEMPERATURE (KI

550

650

FIG. 14. Temperature dependence of the logarithmic pressure derivative of the ionic conductivity of PbF, in the (a) cubic and (b) orthorhombic phases. The symbols represent various samples. The nature of the conduction mechanism and mobile charge species in the various conduction states are indicated (V, is the F ion vacancy, and Fi is the F ion interstitial). The lines through the data points are guides to the eye. Typical error bars are shown. [After G. A. Samara, J. Phys. Chem. Solids 40, 509 (1979).]


51

The formation volumes AV, for Frenkel pairs in both phases are small compared with the molar volumes V , (Table VI). This is expected for Frenkel disorder. Before these pressure results some uncertainty existed as to the type of crystalline disorder in the orthorhombic phase.45 The small AVf definitely rules out Schottky disorder in favor of Frenkel disorder as the dominant disorder in this phase. The larger AVf of the orthorhombic phase compared with the cubic phase is a manifestation of the more closely packed nature of the orthorhombic phase. It is also interesting to note that for cubic PbF,, AVf and AV;,i are nearly comparable in magnitude, a result that can also be anticipated from consideration of the crystal structure in Fig. 3c. Figure 15 and Table VI also compare A V deduced from the o(P)data with values calculated from the strain-energy model. The results,45 based on the

!

0l250

350

450

550

650

TEMPERATURE IKI

FIG. 15. Temperature dependence of the activation volume of PbF, in the (a) cubic and (b) orthorhombic phases. The nature of the conduction mechanism and mobile charge species in the various conduction stages are indicated. The various symbols (which represent the same samples as in Fig. 14) depict activation volumes calculated using by Eq. (3.3), and the dashed lines represent calculations based on the strain-energy model [Eq. (4.25)]. Results from a dynamical model are in better agreement with Eq. (3.3) than those based on the strain-energy model (see text). Typical error bars are indicated. [After G. A. Samara, J. Phys. Chem. Solids 40,509 (1 979).]

52

G . A. SAMARA

TABLE V1. ACTIVATION VOLUMESFOR THE MOTION OF ANIONVACANCIES A N D INTERSTITIALS A N D FOR THE FORMATION OF INTRINSICDEFECTS FOR SEVERAL RX,-TYPECOMPOUNDS" Activation volume (cm'lmole)

Crystal and phase Cubic PbF,

Anion vacancy motion (Av,.,)

Anion interstitial motion (AVii)

1.9 f 0.2 1.75 f 0.25 I .2

3.5 f 0.2 2.3

I .5

2.9

BaF,

I .8

2.8

3.5'; 2.0'

5.6'; 4.6'

3.3 k 0.3 3.2 7.5': 3.9' 3.0 2.0

3.1

+

SrCI, 4.3 0.3 PbCI, 4.7 k 0.2 PbBr, 4.4 f 0.6 Hexagonal PbI, 2.2 f 0.5"

0.12 0.40; 0.22; 0.19' 0.27 0.42

8.4; 5.0d; 5.0' 5.6

0.29; 0.17d.' 0.19

5.2; 4.6'; 3.6' 1.4

0.14; 0.12d; 0.10'' 0.20

6.8 f 1 . 1 7.0 & 0.5 6.4

0.28 0.29 0.26

3.1

k 0.3

- *

Method

u( P ) datah

a(P ) data'

Strain-energy model [Eq. (4.231 Dynamical model [Eq. (4.28)l u ( P )data' u(P ) data u ( P )dataq Strain-energy model [Eq. (4.25)]" Dynamical model [Eq. (4.28)] u ( P )data' a( P ) dataV Dynamical model [Eq. (4.28)1 u ( P ) data' Strain-energy model [Eq. (4.25)]* Dynamical model [Eq. (4.28)] u ( P )data' u( P ) data' Strain-energy model [Eq. (4.25)Ih

'

5.6 & 0.9 45.3 1.6 51.6 2.3

**

0.12 0.95 0.95

u(P ) datah u( P)data' Strain-energy model [Eq. ( 4 . W a ( P )data' u(P)data"' u ( P )data'"

9.2 & 1.6

0.12

a(P) data"'

7.0

10.0

1.5

0.13 0.16-0.19 0.12

5.8'; 6.4'

CdF, 3.1 f 0.2 3.2 0.2 I .2 Orthorhombic PbF, 3.7 0.3

AV,IVM

7.4': 6.4k

+ 0.3

3.1': 2.9'

4.0 f 0.6 (5-6) 0.8 3.6

6.0 9.8; 5 S d ; 4.6" 6.6 10.3

SKI, 3.5 f 0.3 CdF, 2.1 f 0.7 3.2

SrF,

Formation 01' intrinsic defects (AV,)

1

0.25 0.35

I .7

" Frenkel defects are the dominant intrinsic disorder for all these crystals except orthorhombic PbCI, and PbBr,, in which Schottky disorder is dominant.


53

use of the macroscopic Griineisen approximation in Eq. (4.25),are shown by the dashed lines in Fig. 15. It is seen that in both phases, A V calculated from the strain-energy model is smaller (- 10-60%) but of the same order of magnitude as that deduced from the pressure dependence of the conductivity via Eq. (3.4). This agreement between the two sets of values of A V can be regarded as quite satisfactory in view of the approximate nature of the model and the assumptions made concerning the values of y, K, and fi used.45 Use of a larger value of y than the 1.74 used (which is likely the case, at least for orthorhombic PbFJ would render the numerical agreement closer. In terms of the dynamical model [Eq. (4.28)], the only optic modes in PbF, for which yi can be estimated are the long-wavelength TO modes, and for these yi z 2, which is about the same value as the elastic Griineisen parameter.45 Thus Eq. (4.28)yields AV,,, values which are of the same magnitude as those deduced from the strain-energy model but which [because of the absence of the 1/3 factor in Eq. (4.28) compared with Eq. (4.22)] are in closer agreement with the experimental results. In fact, it can be readily seen in Table VI that the AV,,, values calculated from Eq. (4.28)are in remarkably good agreement with the experimentally deduced values. I'

Data from G . A. Samara, J . Phys. Chem. Solids 40, 509 (1979). Data from J. Oberschmidt and D. Lazarus, Phys. Rra. B : Condens. Mailer [3] 21, 2952

(1980).

Value as re-evaluated by J. J. Fontanella, M. C. Wintersgill, A. V. Chadwick, R. Saghafian, and C. G . Andeen, J . Ph-vs. C 14, 2451 (1981), by assuming that the mobility of F - vacancies dominates the conduction process at the low-temperature end of stage 11. See text for details. Same as footnote ( d ) , assuming that the mobility of F- vacancies dominates at the hightemperature end of stage 11. See text for details. Data from J. Oberschmidt and D. Lazarus. Phys. Rev. B ; Condens. Marirr [3] 21, 5823 (1980).

Data from M . Lallemand, Ph.D. thesis, University of Paris, 1972 (unpublished); quoted in J . Obserschmidt and D. Lazarus, Phj.s. Re[).B : Condens. Marter [3], 21,5823 (1980). Based on the use of thermal Griineisen parameters = 1.90 for CaF,, 1.57 for BaF,, and 1.96 for CaF,. Based on values of (q = 0)s,,.j of 3.2, 3.1, and 2.4 for CaF,, SrF,. and BaF,, respectively, from R. P. Lowndes, J . Phys. C 4, 3083 (1971). Based on the use of 7 for the Raman-active mode, namely, isR = 1.76, 1.70, and 1.86 for CaF,, SrF,, and BaF,, respectively, from J. J. Fontanella, M. C. Wintersgill, A . V. Chadwick, F. Saghafian, and C. G. Andeen, J . Phys. C 14,2451 (1981). Based on the use of ;'s ( = yTO) deduced from dielectric constant data, namely. y = 2.6 for all three crystals. Data from J. J. Fontanella, M. C. Wintersgill, A. V. Chadwick. R. Saghafian, and C. G. Andeen, J . Phys. C 14,2451 (1981). Data from G. A. Samara, to be published. Data from J . Oberschmidt and D. Lazarus, Phys, Reo. B : Condens. Mairer [3] 21, 5813 J

' '

(1980).

" This is just the activation volume for motion in the low-pressure phase. It is not known which species dominates the mobility.

54

G . A. SAMARA

b. Strontium Chloride (SrCl,)

SrCI, crystallizes at normal conditions in the cubic fluorite structure but transforms at high pressure into the orthorhombic (Pmnb- Vi!) structure.47 Its pressure-temperature phase diagram is qualitatively similar to that of PbF, .47 Studies of the ionic conductivity of the cubic phase at atmospheric pressure have established that Frenkel defects are the dominant intrinsic disorder, the conductivity is overwhelmingly anionic, and the mobility of anion vacancies is much larger than that of anion interstitial^.^^'^^ In all of these features the material is similar to cubic PbF,. Oberschmidt and L a z a r ~ investigated s~~ the effects of temperature and pressure on the ionic conductivities of both the cubic and orthorhombic phases of SrCl,. The results allowed determination of the activation energies and volumes for the motion of C1- vacancies and for the formation of Frenkel defects in both phases. Values of A H and A V are summarized and compared with those of other crystals in Tables V and VI. In obtaining these quantities for the orthorhombic phase, it was presumed47 that anion vacancies are more mobile than anion interstitials as is true of orthorhombic PbF,. This appears reasonable but apparently has not been directly confirmed. It was also presumed47 that Frenkel disorder is dominant in the orthorhombic phase, and this is strongly supported by the small value of the defect-formation volume AVf found in this phase (Table VI). The changes in conductivity at the cubic-orthorhombic transition have also been in~estigated.~’The results are qualitatively similar to those for PbF, .45 c. Alkaline Earth Fluorides CaF,, SrF,, and BuF,

These crystals have the cubic fluorite structure at normal conditions. It is known that they transform (or can be expected to transform) to the orthorhombic (Pmnb-V;,6) structure at high pressure,46 but nothing is known about the ionic conductivities in this phase. Thus, in what follows we summarize the results on the cubic phase. Early studies of the ionic conductivity and diffusion in these materials at atmospheric pressure have established that Frenkel defects are the dominant intrinsic disorder and that extrinsic point defects can be formed by doping the crystals with aliovalent i m p ~ r i t i e s . ~ ,The ~ ’ conductivity versus temperature responses exhibit features qualitatively similar to those of cubic E. Barsisand A. Taylor, J . Chem. Phys. 45,1154 (1966); M . Pailloux. A. Gervais, M. Jacquet, and M. Bathier, C . R. Hebd. Seances Acad. Sci., Ser. B 274,991 (1972). 5 0 A. B. Lidiard, in “Crystals with the Fluorite Structure” (W. Hayes, ed.),p. 101. Oxford Univ. Press (Ciarendon), London and New York, 1974.

49


55

PbF,; however, the features are shifted to much higher temperatures (Fig. 2). The motion enthalpies for both F- vacancies and interstitials and the formation enthalpies for Frenkel pairs‘are given in Table V. The data come from a wide variety of sources employing a variety of experimental techniques. We note that there is a large spread in some of the enthalpies (especially the motion enthalpies of CaF, and SrF,) reported by various authors. This spread is troublesome. Its cause is not understood, but its magnitude suggests difficulties in experimental results and/or interpretations. Despite this difficulty, it is generally agreed’.’’ that in CaF, and BaF,, vacancies are more mobile than interstitials in the extrinsic regime (stage 1) and that the conductivity in the intrinsic regime (stage 11) is dominated by vacancy motion (at least at the low-temperature end of this stage, as discussed later). In the case of SrF, , on the other hand, there has been some concern’ that the mobilities of vacancies and interstitials may be more nearly equal and that both species may contribute significantly to the intrinsic conductivity. The effects of pressure on the ionic conductivities of these crystals have been reported by Oberschmidt and Lazarus5 and by Lallemand.5’ The most detailed work appears to be that of Oberschmidt and Lazarus.” Analysis of the a(P) data in both the extrinsic and intrinsic regimes yielded5’s5’ the activation volumes given in Table VI for the motion of Fvacancies and for the formation of Frenkel defects in these crystals. The small values of AVf compared with the molar volumes are consistent with the dominance of Frenkel disorder. There is considerable quantitative disagreement in A V values reported by the two groups of authors. This is expecially true in the case of AVf for both CaF, and SrF,, where the OberSchmidt and Lazarus values are considerably larger than those of Lallemand. One possible explanation for the discrepancy may be that Lallemand’s data did not extend to sufficiently high temperatures. One also cannot ignore the possibility of sample contamination from the pressure fluid at the high temperatures necessary for the measurements. Alternatively, much of this discrepancy in AVf can be removed by a different interpretation of the Oberschmidt and Lazarus data, as was first considered by these same authors53and more recently discussed by Fontanella et u l 53a Oberschmidt and Lazarus abandoned this different interpretation because it was not

J . Oberschmidt and D. Lazarus. Phys. Rer. B . Condens. Matter [3] 21, 5823 (1980). M. Lallemand. Ph. D. Thesis, University of Paris, 1972 (unpublished), quoted in Ref. 51. ’ 3 J . M. Oberschmidt and D. Lazarus, in “Fast Ion Transport in Solids” (P. Vashishta. J . N. Mundy. and G . K. Shenoy, eds.), p. 691. Elsevier/North-Holland, Amsterdam, 1979; also J . M. Oberschmidt, private communication. 5 3 a J . J . Fontanella, M. C. Wintersgill, A . V. Chadwick, R. Saghafian, and C. G . Andeen, J . PIIJX.C 14, 2451 (1981). ” 52

56

G . A. SAMARA

totally consistent with all of their AH-and A V results on the fluorites and because of concern with sample contamination. The possibility of this alternative interpretation of the data can best be pointed out with reference to the Oberschmidt and Lazarus results of activation volume versus temperature. This is illustrated in Fig. 16 for the results on SrF,. The situation is qualitatively similar for CaF, and BaF,. Note the strong temperature dependence of AV. The primary consideration relating to the difference between the two interpretations has to do with the nature of the conduction in the intrinsic conduction regime (stage 11). Oberschmidt and Lazarus' assumed that in stage I1 (whose temperature extent is noted in Fig. 16) the conduction is dominated by the mobility of F- vacancies, and they took A V at the hightemperature end of this regime to be representative of the intrinsic value. The formation volume AVf was then calculated from AVintrinsic = AVf AV;,". However, there is considerable e v i d e n ~ e ~ that ~ , ' ~F-~ interstitial motion contributes significantly in the next higher temperature regime (stage HI), and it is also likely that this may be the case for the high-temperature

+

10.0

-I

STAGE I

~

I

STAGE II {

'

I

Z

E II

'

9.0

SrFp

8.0 INTRINSIC

= P

m' 5 "

7.0

6.0

9

5 .O

4.0

n1

I

I 36-

EXTRINSIC

3 0 _.

600

800

1000

1200

T (K)

FIG.16. The temperature dependence of the activation volume of SrF,. The symbols represent various samples. Also indicated is the approximate temperature range of conduction stages I , 11, and 111. [After J. Oberschmidt and M. Lazarus, Phys. Reu. B : Condens. Matter [3] 21,5823 (1980).]

57


TABLE VII. COMPARISON OF IN THE

FORMATION VOLUMES ( A V O FOR FRENKEL PAIRS ALKALINE EARTHFLUORIDES“

THE

Experimental, T

%

1000-1200 K Gillan’s theory

Halide

Oberschmidt and Lazarus‘

Fontanella et u I . ~

Fontanella rt a / . ’

Lallemand’

CaF, SrF, BaF,

0.40 0.29 0.14

0.22 0.17 0.12

0.19 17 0.10

0.27 0.19 0.07

~

T

=

0

0.39 0.19 0.12

T

=

ZT 3 mp

0.28 0.2s 0.19

a Expressed a s the ratio of A Vf to the molar volume VM.Experimental values deduced from a(P) data are compared with values calculated by M. J. Gillan, Philos. Mag. [Part] A 43,291 (1981) using the “volume derivation method,” Eq. (4.10). ” The melting points are 1633, 1463, and 1553 K for CaF,, SrF,, and BaF,. respectively. ‘ Data from J. Oberschmidt and D . Lazarus, P h p . Reo. B : Condens. Mutter [3] 21, 5823 ( I 980). Re-evaluated by J . J . Fontanella. M. C. Wintersgill, A. V. Chadwick, R. Saghafian, and C. G . Andeen, J. Phys. C 14, 2451 (1981) from the data of Oberschmidt and Lazarus (1980) by the method discussed in the text. Re-evaluated as in footnote ( d )by a second method discussed in the text. Data from M. Lallemand, Ph.D. Thesis, University of Paris, 1972 (unpublished), quoted in Oberschmidt and Lazarus (1980).

end of stage 11, especially since the transition between stages I1 and 111 in the fluorites is fairly broad. On this basis, it may be argued that it is more correct to take AV at the lower temperature end of stage I1 as characteristic of intrinsic conduction dominated by vacancy motion.53bThere is, of course, some arbitrariness in this (see Fig. 16), but, nevertheless, by doing so Fontanella et al. have recalculated AV,. for the three alkaline earth fluorides using the Oberschmidt and Lazarus data51 for the intrinsic A V and their for the activation volume for the motion of free vacancies. The new results, shown in Table VII, are in closer agreement with Lallemand’s results. In support of the alternative interpretation of the Oberschmidt and Lazarus data,51 Fontanella et a1.53aperformed a second evaluation of AVf. By assuming that conduction at the high-temperature end of stage I1 is dominated by interstitial rather than by vacancy motion, these authors used 53b

We note that in the isomorphous crystal PbF,, the transition from conduction regime I1 to regime 111 has been interpreted as a transition from intrinsic conduction by F- vacancy motion to intrinsic conduction by F - interstitial motion.45 This may give some support for the alternative interpretation.

58

G . A. SAMAKA

the high-temperature A V of Oberschmidt and Lazarus and their own values of the activation volumes for the motion of F- interstitials, AV;*i (see the following discussion) to evaluate AVf via the expression AVintrinsic = AVf + AVn;,i. The results, shown in Table VII, are in fairly close agreement with those evaluated by the first method (based on the A V data at the lowtemperature end of stage 11) and with Lallemand’s results. We should caution here that it is unlikely that conduction at the high-temperature end of stage I1 is completely dominated by F- interstitial motion for all three crystals. The F - vacancy motion must also contribute, and this is the most likely explanation for the finding that, for CaF, and BaF,, the Fontanella et al. values for AVf evaluated by their second method are smaller than those evaluated by their first method. For SrF,, both methods yield the same AVf (within experimental uncertainties), and this finding may indeed indicate the dominance of interstitial motion at high temperatures in stage I1 for this crystal (see also previous and subsequent discussions in the text). The experimental values of AVf for SrF, in Table VI are based on the assumption by Oberschmidt and Lazarus” and by Lallernand5, that the intrinsic conductivity is dominated by F - vacancy motion rather than by F - interstitial motion. Some support for this assumption (at low temperatures) comes from dielectric relaxation m e a s ~ r e m e n t sat~ high ~ pressure on SrF, which yielded a higher activation volume for the motion of bound interstitials than for vacancies. Specifically, AV,, is found to be 4.7 L 0.1 cm3/mole for bound F- interstitials compared with the AV,,, = 3.3 & 0.3 cm3/mole for F- ion vacancies determined from the o(P) data (Table VI). The dielectric relaxation measurements were performed at relatively low temperatures (300-360 K) on Er3+-dopedSrF, samples.54 For local charge neutrality, the substitution of Er3+ for SrZ+creates an F-- interstitial which, at the temperature of the experiments, remains associated (bound) to the substitutional Er3 ion. The bound charged pair forms an electric dipole, and the dielectric relaxation measurements probe the local migration of the F - interstitial around the substitutional Er3+ ion. The activation energy for this local (or bound) migration is found to be comparable in magnitude to that for the migration of the free interstitial in SrF,.54 We recall here that a similar result is found for the motion of bound and free cation vacancies in NaCl (Section 5). Dielectric relaxation measurements have been extended533 to include bound vacancy and interstitial point-defect motion in CaF,, SrF,, and BaF,. The F- interstitials were introduced by doping the crystals with +

54

C. Andeen. L. M. Hayden. and J . Fontanella. Phys. Rev. B : Condens. Matter [3] 21, 794 (1980).

55

J . J . Fontanelkd, M. C. Wintergill, and C. Andeen, Phys. Status Solidi B 97,303 (1980).


59

trivalent rare-earth ions (as noted earlier), and F - vacancies were introduced by doping with monovalent alkali metal ions. The results for the motional activation volumes are shown in Table VIII. One feature of these results is that the activation volumes for bound interstitial motion are larger than those for bound vacancy motion. The differences must be attributed to differences in the activation energies and appropriate mode y's for the various processes [Eqs. (4.28) and (4.39)]. Fontanella et find that AVm in Table VIII agrees well with values calculated from the dynamical model [Eq. (4.28)], using for interstitial motion y z yTo of the long-wavelength (q = 0) T O mode and for vacancy motion y z yR, where yR is associated with the long-wavelength Raman-active mode. The justification for these TABLE VIII. EXPERIMENTAL ACTIVATION A N D VOLUMESFOR BOUND(OR ENTHALPIES ASSOCIATED) F- VACANCY A N D INTERSTITIAL MOTIONI N THE ALKALINE EARTHFLUORIDES AS DEDUCED FROM DIELECTRIC RELAXATION MEASUREMENTS" SrF,

BaF,

0.59 3.0

0.60 2.2

0.41 2.9

0.70 4.7

0.57 4.4

7.2

6.0

5.7

CaF, Vacancies Bound AH,,, (eV) A V, (cm3/mol) Free

0.50 I .7

Interstitiuls Bound A H , (eV) AVm (cm3/mol) Free AV,,, (cm3/mol)

Also listed are experimental values of AV, for free F - vacancy motion and estimates [based o n Eq. (8.1)] of A V, for free F- interstitial motion. Free activation enthalpies are given in Table V. Data from J. J. Fontanella, M. C. Wintersgill, A. V. Chadwick, R. Saghafian. and C. G. Andeen, J. Phys. C 14, 2451 (1 981). 'From a(P) data; J. Oberschmidt and D. Lazarus, Phys. Rev. B : Condens. Matter [ 3 ] 21, 5823 (1980). From u(P) data; M. Lallemand, Ph.D. Thesis, University of Paris, 1972 (unpublished).

60

G . A. SAMARA

choices of mode y s is as follows. Reference t o Fig. 3c shows that direct interstitial motion (i.e., direct jump of an interstitial to an equivalent interstitial site-the dominant interstitial jump process in the alkaline earth fluorides) occur when the interstitial vibrates along the (110) directions against the cube of surrounding F- ions. This is the same motion as that associated with the q = 0 T O mode, which involves the vibration of the cation against the surrounding cube of F- ions. Vacancy motion, on the other hand, should be dominated by the mode involving (100) planes of fluorine ions vibrating against each other. This is just the motion associated with the Raman-active mode in these crystals. It should be cautioned that the above-mentioned “good agreement” between theory and experiment involves the use of y s associated with longwavelength modes. However, point-defect motion (especially bound motion) is a local phenomenon which should be controlled by short-wavelength modes. The agreement in question perhaps suggests that there is little dispersion in the ys across the Brillouin zone. It should also be kept in mind that the theoretical result [Eq. (4.28)] is only approximate, and overemphasis on the quality of agreement between it and experimentally deduced results can be misleading. On the basis of the apparent validity of the dynamical theory, Fontanella et used a variation of Eq. (4.39) to estimate At‘,,,, for free F- interstitial motion from their results for bound F interstitial motion. Specifically, the relationship is

(A ‘L,i/AGi,ihree

(A‘i.i/AGi,j

)bound.

(8.1)

The free energies for motion of free interstitials are known from earlier diffusion measurements, and those for bound interstitials were determined from the dielectric relaxation measurements.’ 3a The results are given in Table VIII. ~ ~ that the For the case of F- vacancy motion, Fontanella et ~ 1 . ’find activation enthalpies for bound and free vacancies are approximately equal when the size of the substitutional alkali ion is close to that of the host cation. For these cases they then postulate that the activation volumes for bound and free vacancy motion should be approximately equal. This is supported by the experimental evidence comparing their results for bound vacancies with those for free vacancies as determined from conductivity data (Table VIII). Again, caution should be exercised in connection with this approximation because there are features in the Fontanella et al. data which are not obviously interpretable in terms of ionic size alone. Also, as noted before in this review, the polarizability of the ion is bound to be a consideration in these matters.


61

The accuracy of the dielectric relaxation measurements is such that the results allowed determination of the thermal expansivity and compressibility of A V,,, , as defined by equations analogous to Eqs. (4.17) and (4.18), for the motion of the bound F - interstitials and vacancies in the alkaline earth fluorides.5 3 a , 5 4 , 5 5 Both of these quantities are found to be much larger (more than an order of magnitude) than the corresponding bulk properties. For example, for the F- interstitial in SrF,, B,, rr - 1 x K - ' and ti, N 0.2 G P a - ' , compared with the bulk values /j N 5 x K - ' and ti = 1.45 x lo-' GPa-'. Presumably similar differences hold for free F- interstitial and vacancy motion in these crystals. I t should be noted that these results are qualitatively similar to that found for N a + vacancy motion in NaCl (Section 5 ) and are in agreement with the empirical model results of Varotsos and Alexopoulos2' on free point defects. The negative sign of 8, has also been observed for free defects in PbF,45 (Figs. 14 and 15), but is not the general lule (e.g., Figs. 6 and 11). It can be intuitively rationalized as follows: as the lattice softens with increasing temperature, a smaller lattice relaxation, or AVm, is needed for diffusive motion. More formally, the sign of 8, derives from the balance among several contributions, as can be readily deduced from Eqs. (4.19) and (4.28). The validity of the strain-energy model apparently has not been tested for the alkaline earth fluorides. Consequently, we have examined this issue by using the known pressure dependences of the elastic constants of CaF, and BaF256via Eqs. (4.19)and (4.20). All of the parameters in these equations are known56 or are given in Table V. The calculated AV,,, and AVf values are found to be much smaller than those deduced from the a(P)data. The agreement is closest for CaF,, where the calculated AV,,, for F- vacancy motion and AV, are about a factor of 3 smaller than the experimentally deduced values. Considerably better agrekment between model and experimental results is obtained by using values of the thermal Gruneisen parameters directly in Eq. (4.25). These values are y = 1.90 and 1.57 for CaF, and BaF,, re~pectively.~'The results are shown in Table VI. Thus, for the alkaline earth fluorides, the situation is opposite that for NaCl and its isomorphs where direct use of the elastic constants data via Eqs. (4.19) and (4.20) leads to closer agreement with experiments than does use of the thermal Gruneisen parameter via Eq. (4.25)(Section 5). The reason for this difference is not clear. With respect to the dynamical model (Section 4), the question again is what is the appropriate mode and mode y to use in Eq. (4.28). We have already discussed this earlier in this section in connection with the Fontanella et results. There are some disagreements in the experimentally available

'' C. Wong and D. E. Schuele, J . Phys. Chew. Solids 29, 1309 (1968).

62

G. A . SAMARA

q = 0 optic mode y ~ , ~ but, ~ ~nevertheless, 9 ~ ~ -the~ model ~ yields activation volumes for the motion of both F- vacancies and interstitials which are in remarkably close agreement with the experimentally deduced values (Table VI). Finally, Table VII compares the formation volume for Frenkel pairs in these crystals calculated by Gillan’ by using the “volume derivative m e t h o d [Eq. (4. lo)], with the experimentally determined values from o (P ) data. Despite the fact that the calculated values exhibit a somewhat peculiar temperature dependence (compare CaF, with SrF, and BaF,) and that there are quantitative differences in the experimental results (as discussed earlier), it is seen that Gillan’s theoretical results agree closely with the experimental data in both magnitude and trend for the three crystals.

d. Cadmium Fluoride (CdF,)

-

This material crystallizes in the cubic fluorite structure. At temperatures below 570 K, conduction is predominantly ionic (stages I and II), but at higher temperatures a significant electronic contribution Data in this higher temperature regime also become irreproducible, which further complicates interpretation. In the ionic regime, anion Frenkel disorder is believed to be the dominant d e f e ~ t . ~ The effects of temperature and pressure on the ionic conductivity of nominally pure CdF, have been r e p ~ r t e d . ~ Below 560 K, the data are quite reproducible, and two distinct conduction regimes have been observed.58 In regime I, E = 0.4 eV, and the conduction is believed to be extrinsic and dominated by F- vacancy motion. In regime 11, E = 1.55 eV, and analogy with the alkaline earth fluorides suggests that conduction is intrinsic and dominated by F- vacancy motion. On this basis the formation enthalpy for Frenkel pairs is AHf = 2.3 eV. The a(P)data yield the activation volumes given in Table VI. There is good agreement between the results of Oberschmidt and Lazarussl and those of Samara.” The relatively small magnitude of AVf supports the presumption that Frenkel disorder is dominant in this crystal. The thermal elastic Griineisen parameter of CdF, is y = 1.96.59 Use of this quantity in the strain-energy model result [Eq. (4.25)] yields AV shown in Table VI. It is seen that the calculated value of AV, is in reasonable agreement with the experimental value deduced from the a(P)data. However, the calculated At‘;,” is about less than one-half of the experimental value. The ‘qS8

-

’’ R.P. Lowndes. J . Phys. C 4,3083 (1971). 58

59

G . A. Samara, to be published. S. Alterovitz and D. Gerlich, Phys. Rcw. Bc Solid State [3] 1,4136 (1970).


63

dynamic model cannot be tested for this crystal because of the apparent lack of microscopic mode gammas. e. Leud Chloride and Leud Bromide ( P b C l , und PbBr,)

These two materials crystallize in the orthorhombic (Pmnb- Vi,")structure to which cubic PbF, and SrCl, transform at high pressure. However, unlike orthorhombic PbF, and SrCl,, where Frenkel defects are the dominant intrinsic disorder, the dominant intrinsic defects in PbCl, and PbBr, are believed to be of the Schottky type, consisting of one P b 2 +vacancy and two X - halogen vacancies.60 However, this had not been definitive, and it had been suggested6' that the results of doping studies could have equally been interpreted as indicating dominance of anion Frenkel defects. As we shall see, the pressure results clearly favor Schottky disorder. It is believed that the halogen ion vacancy is much more mobile than the Pb2+ ion vacancy.60 The temperature dependences of the ionic conductivities along the orthorhombic c axis of nominally pure PbCl, and PbBr, at atmospheric pressure are shown in Fig. 2. A low-temperature extrinsic regime (I) and a higher temperature intrinsic regime (11) are evident. The conductivity in both regimes is dominated by the mobility of the anion vacancies.60The enthalpies for anion-vacancy motion and for the formation of Schottky defects are given in Table V. These activation enthalpies are found to be essentially independent of crystal orientation.60 We note that AH,,, for anion vacancy motion in these crystals is about equal to that for F- vacancy motion in othorhombic PbF, (Table V). However, the formation enthalpies are about twice as large as that for PbF,, reflecting the differences in defect mechanisms as well as the larger dielectric constant of PbF, . The pressure dependences of the ionic conductivities of PbCI, and PbBr, in regimes I and I1 have been reported by Oberschmidt and Lazarus.60 It was found that log CJ decreases linearly with pressure over the modest pressure range covered (to -0.4 GPa). The results allowed evaluation of the motional and formation activation volumes given in Table VI. The relatively large values of AV,/V, for both crystals compared with the corresponding ratios for orthorhombic PbF, and SrCl, (also listed in Table VI) argue strongly in favor of Schottky defects as the dominant intrinsic disorder in these materials. The fact that the deduced AV, values for PbCI, and PbBr, are slightly smaller than the molar volumes, is, according to Oberschmidt and Lazarus,60 probably not significant due to errors in the measured quantities which are multiplied by a factor of 3 in calculating AV,. Another 6o

J. Oberschmidt and D. Lazarus, Phys. R ~ PB. ; Condens. Murrer [ 3 ] 21, 5813 (1980). and references therein.

64

G . A. SAMARA

possible explanation that AVf is slightly less than the V , is the presence of a small number of anion Frenkel defects at the temperatures of interest. This would yield a lower AVf than the case where these latter defects were not present. The activation volumes for the motion of halogen ion vacancies are comparable in magnitude to those found in orthorhombic PbF, and ScCl,, and all of these values are larger than those found for the cubic fluorites (Table VI). The difference between the two structures reflects the more densely packed nature of the orthorhombic phase. Apparent lack of information on the Griineisen parameters for these two crystals precludes testing the model results for AV. An interesting feature of the data in Fig. 2b is the sharp rise in D of PbCl, at -763 K. This feature has been attributed6' to a diffuse order-disorder phase transition similar to that seen in the cubic fluorites discussed earlier. However, the transition appears too sharp (in fact, may be first order), and the magnitude of the AD jump is very unlike that of the diffuse transitions seen in the fluorites (e.g., the transition in PbF, at -750 K shown in Fig. 12). An alternative explanation for the transition has been suggested by Oberschmidt and Lazarus.60 They attribute it to a first-order transition from orthorhombic to cubic as seen in PbF, and SrCl,. Both the increase in, and the magnitude of, ACTat the transition support this suggestion (cf. Figs. 2 and 12).

f: Lead Iodide (Pbl,) At normal conditions lead iodide has the layered hexagonal Cd1,-type ( P 5 w 1 l - D : ~structure ) (phase I). It undergoes two pressure-induced structural phase transitions at -0.40 GPa (to phase 11) and at 1.5 GPa ( t o phase The structures of phases I1 and I11 are unknown. Oberschmidt and L a z a r d o have investigated the temperature and pressure dependences of the ionic conductivities in both phase I and 11. In interpreting their data, the authors presumed that the electronic conductivity was negligible. In phase I both extrinsic and intrinsic regimes are observed. In the extrinsic regime the conductivity can be dominated by the motion of either Pb2+ or I- ions or both, depending on the temperature. In the intrinsic regime (565-625 K) the mobility of the Pb2+ defects is believed to be dominant. The activation enthalpies and volumes are given in Table V and VI. Previous to these results there was a question as to the nature of the

-

6'

F. E. A. Melo. K. W. Garrett. J . Mendes Filho, and J . E. Moreira, Solid State Cornrnun. 31, 29 ( 1979).



65

intrinsic defects in hexagonal PbI,;60 however, baring any strong contributions from electronic conductivity (which is not likely), the pressure results strongly indicate the dominance of Frenkel disorder in this phase as evidenced by the small ratio AV,/V, in Table VI. The data in phase 11 were quite complicated; nevertheless, Oberschmidt and Lazarus6' were able to evaluate AH and A V in presumably the intrinsic regime of this phase. The values are 2.2 & 0.2 eV and 1.4 f 0.6 cm3/mole, respectively. This very small A V ensures that Frenkel disorder is the dominant intrinsic disorder in this phase also.

9. FASTION CONDUCTORS a. Silver Iodide ( A g l )

At normal conditions AgI can exist in either the hexagonal wurtzite structure (B-AgI) or the cubic zinc blende structure (y-AgI), which is the less stable of the two. /3-AgI transforms to the superionic, body-centered cubic phase (a-AgI) at 420 K. The Ag' ions are randomly disordered in the a phase. The P + a transition is first order and is accompanied by -6% decrease in volume.62 Thus, its transition temperature should decrease with pressure, and this is found to be the case as seen in Fig. 17, which shows the temperature-pressure phase diagram for AgL6j Note that two additional phases are induced by pressure. The effects of pressure on the ionic conductivities of the a, P, and y phases have been r e p ~ r t e d . ~ In ~ -the ~ ' superionic a phase the activation energy is =0.1 eV, and rs decreases slightly with pressure.64 The corresponding activation volume is 0.56 k 0.1 cm3/mple at 435 K and increases to 0.8 f 0.1 cm3/mole at 623 K (Fig. 18). These are very small A V values which are, for example, 4-6 times smaller than the AV,,, values for Ag' interstitial mobility in AgBr and AgCl (Table IV) despite the fact that the activation energies for the three crystals are comparable (cf. Table I). This suggests that the mechanism for interstitial diffusion in the superionic phase of AgI is different from that in normal ionic crystals. The situation is qualitatively similar for the other fast ion conductors to be discussed later. G. Burley, ACIUCrystulloyr. 23, 1 (1967). A. Bassett and T. Takahashi, A m . Mineral. 50, 1576 (1965). 64 P. C. Allen and D. Lazarus, Phys. Rec. B : Solid Siure [3] 17, 1913 (1978). R. N. Schock and E. Hinze. J . Phys. Chem. Solids 36,713 (1975). H. Hoshino and M. Shimoji, J . Phys. Chem. Solids 33,2303 (1972). 6 7 M. Hara, T. Mori, and M. Ishiguro, Jpn. J . Appl. Phys. 12,343 (1973). b2

"W.

'' ''

66

G . A. SAMARA 160

120

-

0

0c

80

,8 IWURTZITEI

Y (ZINC BLEND1

40

0

0.0

0.2 P ICPa)

0.4

FIG. 17. Temperature-pressure phase diagram for AgI. (After W . A. Bassett and T. Takahashi, Am. Minrrul. SO, 1576 (1965).)

-

As noted, A V , for a-AgI increases by 50% between 435 and 623 K. Allen and L a z a r ~ suggest s ~ ~ that this effect could be due to some mode softening with increasing T ; the lattice is able to relax outward more as the ions move through the saddle point. No indication is given as to which mode is involved, but in support of their point they note that the mean atomic displacement of the Ag+ and I- ions, as deduced from X-ray measurements, increases from 0.2 to 0.4 A over the same temperature range. In the nonsuperionic p- and y-AgI phases, 0 increases with pressure, leading to relatively large and negative A V (Fig. 18). For the p phase, the Allen and L a z a r ~ results s ~ ~ for AV (measured along the c axis) differ in two respects from those of Schock and H i n ~ eas , ~shown ~ in Fig. 18. The sudden increase in A V to less negative values at T > 340 K in the case of the Shock and Hinze results is not understood, but Allen and Lazarus suggested that it may be sample dependent and due to varying levels of impurities which could lead to different conduction mechanisms. The variation in the magnitude of A V between the two sets of authors is also not understood. We note that Hoshino and Shimoji66 reported large and negative A V for P-AgI with a rather anomalous T dependence. However, their results are on polycrystalline (pressed powder) samples, and this makes interpretation difficult since the fl phase is anisotropic. The A V values reported in Fig. 18 for the p phase are in the intrinsic regime, AVf, and AVm is expected to be small and and therefore A V = AV,,, possibly positive (although this has not been confirmed). In any case, however, it is clear that AVf is large and negative for this phase. A negative AVf

+

67


+6 0

+3

-

E .

0

___

v

SCHOCK AND HlNZE HOSHINO AND SHlMOJl

0

r/

AT 623

"r-t -e+

/

/

m

5

ALLEN AND LAZARUS

fl

-3

(C-AXIS)

>

-2

-6

fl

(C-AXIS)

-9

-12 280

I

300

I 320

I

I

I

I

I

340

360

380

400

420

I1

440620

6

640

T (K)

FIG. 18. The temperature dependences of the activation volumes in the various phases of Agl .

is predicted by the strain-energy model since it is known64*65 that the volume thermal expansivity, and thereby the macroscopic Griineisen parameter, for this phase are negative. A negative thermal expansitivity (or y) implies mode softening under pressure as can be seen, for example, from Eq. (3.5). The results on y-AgI are on pressed-powder sample^,^^,^' but this phase is isotropic. The conductivity increases with pressure. The resulting negative AV and its T dependence shown in Fig. 18 are attributed to the negative thermal expansivity (and thereby negative y) and its T dependence.66

b. Rubidium Silver Iodide (RbAg41,) This crystal exhibits two temperature-induced transitions at 1 bar.64 The high-temperature cubic a phase is a fast ion conductor. It transforms on cooling at 208 K to a rhombohedra1 p phase. This phase is also a fast ion conductor. The conductivity exhibits a small change in slope at this secondorder order-disorder transition. The p phase transforms to the low-temperature y phase at 122 K via a first-order phase transition accompanied by a factor of 100 decrease in conductivity. In both the a and /3 phases, (i increases initially with pressure (nonlinearly on log (i versus P plot) at low pressures. The effects are so small (Fig. 19) that here the pressure dependences of the lattice parameter and the jump

68

G. A. SAMARA 1.0

t

rn

I

r

0.2

0.4

I

0.5

[r

U v 0

-c

0.4

0.3 0.2 0.1

0.0 0.0

PRESSURE (GPa)

FIG.19. Pressure dependencesof the ionic conductivitiesof polycrystalline a- and /l-RbAg,I,. [After P. C . Allen and D. Lazarus, Phys. Rev. B : Solid Srair [3] 17, 1913 (1978).]

frequency in Eq. (3.3) have to be accurately accounted for in calculating A V from o(P)data. Consequently, the uncertainties in the calculated A V values are large. The AV values are64 -0.4 0.2 cm3/mole at -290 K for-the u phase and -0.2 f 0.1 cm3/mole at 170 K for the p phase. The results in Fig. 19 show that with increasing pressure, o turns around and ultimately decreases with pressure in both phases leading to positive, but still small, AV. For they phase (which is not superionic), o decreases with pressure, but the a(P)dependence is nonexponential. The initial slope yields AV = 9 k 1 ~m~/mole.~~ The key features of these results are the very small and negative values of A V in the superionic ci and B phases. We recall that a very small A V is also obtained for u-AgI (Table IX), but in the latter case A V is positive. Estimates of AV from the strain-energy model by using a y = 1.05 determined from elastic constants yield A V = 1.4 cm3/mole for cl-RbAg,I, and 2.3 cm3/mole for P-RbAgJ, .64 These values have the wrong sign and are 4 and 10 times, respectively, larger than the experimentally determined values. Similar elastic data are not available for a-AgI to make such a comparison. In any case, however, the RbAg41, results emphasize the unique

-



69

TABLEIx. SUMMARY OF ACTIVATION ENERGIES A N D ACTIVATION VOLUMES BASEDON INITIALPRESSURE DERIVATIVES OF u FOR SEVERAL FASTION (SUPERIONIC) CONDUCTORS Fast ion conductor

E (eV)

a-AgI"

0.1

a-RbAg,I j' B-RbAg,l j'' PbF," Na B-AI,O,' K B-Al,O,' LiB-AI,03C N a , + xZr,Sixp, - .o, Z d x = 1.8 x = 2.0 (NASICON) x = 2.3 Ag,SBr' P-Ag,SI"

0.12 0.17 ~

-

0.24 0.29 0.25 ~

AV (cm3/mole) 0.56 0.8 -0.4 -0.2 -2.0

f 0.1 at 435 K f 0.1 at 623 K i 0.2 at 290 K f 0.1 at 170 K at -800 K -0 >O . (3.2) The number of phonons with wave vector k at the inverse temperature B = l/kT,where k is the Boltzmann constant, is IZ(Ok) =

l/{e

- 1).

(3.3)

89

THEORY OF POLARON MOBILITY

The transition rate between the polaron states I P, n) and I P’,n’) is T,(P, nlP’,n’) = I (P, nl e-ik.rlP’, n’)

I ’.

(3.4)

The scattering processes involved in the Boltzmann equation [Eq. (3.l)] are schematically represented in Fig. 1. Some remarks about the validity of the polaron Boltzmann equation44 [Eq. (3.1)] are in order.

( I ) In the derivation of the usual Boltzmann equation, one assumes that the electron propagates as a free particle between two successive collisions which are well separated from each other in space and time. Interference between successive collisions is therefore not taken into account. It must be kept in mind that the usual Boltzmann equation approach to charge transport is a weak coupling theory in the sense that in zeroth order, the electron is a free electron. To describe polaron transport for arbitrary coupling we consider the propagation of a polaron between two successive collisions as described by the Feynman polaron Hamiltonian. In doing so, correlations between successive electron collisions with the phonons are taken into account approximately. In this approach it is the polaron, i.e., electron plus phonon cloud, which is scattered. In a collision, an LO phonon is emitted or absorbed; the polaron momentum is changed, and there is also the possibility that the polaron internal state changes (Fig. I). In other words, the virtual interaction between the electron and the phonon field is taken into account by supposing that in zeroth-order approximation the eigenstates of the Feynman polaron Hamiltonian are well-defined states. The dissipative nature of this electron-field interaction is described by the Boltzmann equation (3.1). Kadanoff and Revzen’ have shown by using Green’s function techniques that Eq. (3.1) is valid at least for low temperatures ( k T -

‘v‘

+=

TI Br K I A + RbI

t

L

GaAs InP

5\

GI

cQFz

fRb’ KBr

lo2 LLCLLLUL 103

FIG.9. The velocity and electric field values of 30 polar semiconductors and ionic crystals beyond which no stationary state exists. according to the TF theory at zero temperature.

to account for the velocity dependence of v and w. But at the present moment no theory is available which expresses the parameters (u, w) as a function of the average electron velocity. In Ref. 19 the influence of the velocity dependence of the parameters (v. w) was estimated from the nonparabolicity of the polaron energy spectrum as given in Ref. 15. Indeed, this nonparabolicity can be described by introducing a velocity-dependent polaron mass which, with the help of Eq. (2.5), gives an idea of the velocity dependence of v/w. However, for low carrier concentrations and weak coupling, studies of the Boltzmann equation (both analytical” and with the use of Monte Carlo techniques” ) have revealed that the polaron distribution at low temperature is certainly not Maxwellian; rather it has an elongated shape with (at least for relatively small electric fields) a discontinuity at the critical velocity for LO phonon emission. As shown previously, the electron velocity distribution function in the T F theory is a drifted Maxwellian for every value of the electron-phonon coupling, temperature, and electric field strength [Eq. (7.21)]. The electron velocity distribution function at small temperature and finite electric field (non-ohmic region) will be strongly anisotropic due to the inelastic nature of the electron-phonon scattering. Only for materials with a high carrier concentration will electron-electron correlations be dominant. A thermalization of the electron distribution then occurs, resulting in a Maxwell-Boltzmanntype velocity distribution function. The results of such studies are compared

106

F. M. PEETERS A N D J. T. DEVREESE

10’

Fic. 10. The velocity-electric field characteristic for a = 0.02 and three values of the temperature ( T IT’ = 0,0.27 I , 1.057) as obtained from the TF theory (--)and a Monte Carlo simulation:(o)T/T,=0;(0)T/TD-0.271;(A) TIT,= 1.057.

in Fig. 10with the TF result for ct = 0.02 (corresponding to the material InSb) and various values of the temperature (for InSb, T / T , = 0.271 and 1.057 correspond with 77 K and 141.9 K, respectively). At T = 0, both the result of the analytical method” and that of the Monte Carlo calculationz2 start at I/ = VL0/2 for E = 0 as is also the case for Schockley’s “streaming-motion“ model.” The Monte Carlo result shows the onset of the instability region at whereas the TF theory gives EfE, x 3 x lo-’. EfE, x 6 x In Section IV of Ref. 30, Thornber and Feynman also derived an equation which expresses the conservation of energy. In the present approach this would correspond to taking the second moment of the Boltzmann equation (BE) explicitly:

C P2 (BE) = 0.

(8.1)

P,n

Thornber and Feynman concluded that Eqs. (7.18) and (8.1) cannot be satisfied simultaneously at low temperature. This disagreement confirms the previous remarks concerning the electron velocity distribution function. At higher temperature (kT > hwL0/2), this discrepancy was not too serious.

’’ W. Shockley, Bell Sysr. Tech. J . 30,990 (1951).


107

V. The Polaron Impedance Function for Quantum Frequencies

9. REDERIVATION OF THE FEYNMAN-HELLWARTH-IDDINGS-PLATZMAN RESULT At high frequencies of the external field (o> mL0; this means infrared frequencies), the Boltzmann equation is no longer valid. Indeed, the influence of the electric field during the collisions can no longer be neglected. During a collision, the electron state will change under the action of both the phonon field and the external field. As in Part IV we shall rely on the equation expressing the conservation of momentum. First, Eq. (4.6)should be adapted in such a way that the influence of the external electric field during a collision is taken into account. In Eq. (4.6) the electron correlation is governed by H, [e.g., Eq. (4.5)], which is equivalent to choosing the electric field during a collision as equal to zero. What is the influence of a highly oscillating electric field on the motion of the electron? At high frequencies the electron will oscillate around its average position with a frequency equal to that of the external field:

+

r(t) = rAeiW' ro(r),

(9.1)

where rAis the amplitude of the electron oscillation and r&) describes the position vector of the electron in the absence of an electric field. In the Feynman polaron picture, ro(t)is given by Eq. (2.8). The time evolution of the electron position coordinate as given by Eq. (9.1) follows from the Hamiltonian HE(f)= HF

+ eE(t)*r,

(9.2)

which describes a Feynman polaron model in an external electric field E(r) = E0pior, and HE([) is the Hamiltonian which corresponds to the LagrangianZ9of FHIP. The transformation [Eq. (9.1)] formally eliminates the electric field from the problem. After this transformation, the expectation values in Eq. (4.6) are those corresponding to the equilibrium situation. In Eq. (4.6) the expectation values were defined by [Eqs. (4.4)and (4.5)]

where A [ r ( t ) ] is an operator which is a functional of the position coordinate r(t). The evolution operator is

uF(z) = eiHFr.

(9.4)

108

F. M . PEETERS A N D J. T. DEVREESE

To account for the influence of the electric field during the collisions. we replace the expectation values [Eq. (9.3)] by (~[r(t')])

=

C j ' ( ~ , nt)(P,nJA[u,(t', ,

r)r(r)u;'(r', r)]JP,n), (9.5)

P,n

where now the evolution operator is given by (9.6) With the help of Eqs. (9.1) and (9.2), one can write Eq. (9.5) as (~[r(t')]) =

C f(p,n,r)(P,nl~ x [rAeiO'+ u,(t' - t)r(t)u; ' ( t '

P.n

-

t)]IP,n).

(9.7)

As mentioned before, after the transformation we are back in the equilibrium

situation, which means that the polaron distribution function j'(P,n, t) is the equilibrium distribution as given by Eq. (5.3). Finally, Eq. (9.7) reduces to ( A [r(t')])

=

Tr(e -'IHFA [r,e'""

+ u,(t'

- t)r(t)u;

'(r'

-

t)])/Tr(r-BH").

(9.8)

The expectation values in Eq. (4.6) are the polaron momentum

(P)

=

((p

+ p,))

=

Tr[e-OHF(i

=

icu(r)e'"',

+ i(orAeiW'+ mrir)]/Tr(e-BHF) (9.9)

and the electron density-density correlation function

+ ?))exp(ik-r(t)))= exp(ik-r,exp(iwt)[l

(exp(-ik-r(t

- e x p ( i o ~ ) ] ).

x Tr[exp( -PHF)uf.(z)exp(-ik-r(t))

(9,lO)

x u; '(7)exp(ik- r(t))]/Tr[exp( -pH,)].

The resulting trace in Eq. (9.10) is the equilibrium electron density-density correlation function which is given by Eq. (A.9). One then obtains (exp( - i k - r ( r =

+ z))exp(ik*r(t)))

exp(ik-r, exp(iwt)[l

-

exp(iwr)]) exp( - k 2 D (

-T)),

(9.1 1 )

and. similarly,

- +

(exp(ik r(t))exp( - ik r(t =

exp{ik.r, exp(iwt)[I

5))) -

exp(ioz)]} exp(-k'D(z)).

(9.12)

109


Inserting Eqs. (9.9), (9.1I), and (9.12) into Eq. (4.6) results in

1v k ( 'k Re -02rA exp(iwt) = -eior, exp(iot)Z(w)- 2 1 h

dr JOW

x exp{ik-r, exp(iwt)[l - exp(iwr)]} x exp(io,r)[(I

+ n(ok))exp(-k2D(z)) - n(wk)exp(-k2D(-?))], (9.13)

where use has been made of [see Eq. (5.10)] iorAeiot= [ Z ( o ) ] - ' E ( t ) .

(9.14)

The linear ac response is obtained by expanding Eq. (9.13) for small electric fields (which is equivalent to making an expansion in the small electron oscillation amplitude r, ). :

x Im[(l

7

+ n(cy,))e'"' + n(ok)e-'"'

]e-kzD'r).

(9.15)

In going from Eq. (9.13) to Eq. (9.15), one should remember that the electric field is complex. In Eq. (9.13), rAeiWt and r,e'"('+') have to be considered as real quantities. Use was also made of the properties Re iA = -1m A ; Im A = -1m A * ; and D(t) = D*( - t ) . Eq. (9.15 ) is the final FHIP result [see Eqs. (41),(Ma), and (35b) of Ref. 291. Note that Thornber and Feynman obtained the same expression [Eq. (20a) in Ref. 301. It is convenient to introduce a meinory function X(o)by eZ(w) = -io + iX(w),

(9.16)

which, in the FHIP approximation: and for LO phonons, equals

10. DISCUSSION OF THE VALIDITY OF THE FHIP RESULT AND NUMERICAL ANALYSIS

To get an idea of the validity of the result [Eq. (9.15)], and to see the difference with the result for [ZRTA(o)]-'[Eq. (5.12)] obtained by a relaxation

2 10

F. M. PEETERS A N D J . T. DEVREESE

nn

nh

nh

nh

nn

FIG. 1 I . Scattering processes taken into account in the FHIP theory. A polaron in the nth internal state and having momentum P can scatter by emission (absorption) of an LO phonon and emission or absorption of a radiation quantum h e into the nth internal state and obtains a momentum P - hk for a phonon absorption P hk.

+

time approach, it is convenient to study the electron velocity distribution function. In Parts 111and IV, a polaron distribution function was chosen from which the final results for the response function were then obtained. This is no longer possible here in Part V. indeed, here we started with the same type of equation [Eq. (4.6)] as in Part IV, but the expectation values were defined in.a different way. In the present section, the influence of the electric field is directly taken into account in the electron evolution. Of course, it is indirectly contained in the electron distribution function. This is the main difference with the foregoing sections, and it is reflected in various scattering processes (Fig. 11). Because the electron distribution function is defined as an expectation value, it is still possible to calculate this function in the same approximation as before. The electron velocity distribution function is defined by [Eq. (4.6)]

.f(v,, 0

=

(S(P - .a)>

(10.1)

Using Eq. (9.Q one finds .f'(vo; t ) = Tr(e-'jHFS(i - rAiweiw'- vo))/Tr(e-8HF).

(10.2)

111


A similar calculation as presented in Ref. 19 then results in

f ( v o , t ) = (P,/27r)3i2 exp[-)/3,(v0

- r,zw exp(iot))*],

(10.3)

where the electron temperature Be is given by Eq. (7.22). Consequently, the electron distribution function is a drifted Maxwellian distribution which oscillates with the electric field. In the present section we are interested only in the linear response to a small electric field; the electron velocity distribution under these circumstances becomes

-

,f(vo, t ) = ( 8, /27~)~’’ [1 + BevO r, io exp(iwr)]exp( - :f?,vi ),

(1 0.4)

or, using Eq. (9.14), f’(U0

9

4 = f O ( V 0 ) ;1 + [Pe/Z(~)IV,E(d3 *

1

(1 0.5)

with the equilibrium electron distribution function Jo(vo) = (Pe/2n)3’2exp(-+BevC ).

( 10.6)

In the case of a constant electric field, lim,,,, rAiweio‘= V (with V the average velocity), and Eq. (10.3) defines the same electron distribution function as that found in Part TV. In the RTA, one can calculate the electron velocity distribution function from Eqs. (5.3) and (5.9). It has the same structure as Eq. (10.5). The nonperturbed electron distribution function, as calculated from fb(P. n, r) [Eq. (5.3)], becomes equal to the first term on the right-hand side of Eq. (10.5). For the deviation from equilibrium, as calculated from f l (P,n, t ) [Eq. (5.9)], no simple explicit expression is obtained. However, it takes the form g(vo,w)vo E(t), where g(vo,o)is a function of which only an integral representation is known. The second term pn the right-hand side of Eq. (10.5) has a similar structure. In this case, however, the velocity and frequency dependence of the function g(vo, w ) can be explicitly separated: g(v,, w ) = Jb (VO ) P e /‘(a). The electron distribution function as calculated from Eqs. (5.3) and (5.9) becomes equal to that defined by Eq. (10.5)if a constant relaxation time can be defined. Indeed, if in Eq. (5.9), A(P,n) is taken equal to a constant A, the calculated deviation from equilibrium becomes equal to the second term on the right-hand side of Eq. (10.5)if, furthermore, one takes Z(w) = e-’(A iw). In fact, this is the response function for a particle in a dispersive medium which can be described by a friction coefficient (cf. simple Brownian motion theory). The result for Z ( w ) is therefore consistent with the assumption of a constant relaxation time. It is not surprising that the electron distribution function as calculated from Eqs. (5.3) and (5.9) is identical to that following

-

+

112

F. M. PEETERS A N D J. T . DEVREESE

from Eq. (10.5)if a well-defined relaxation time can be defined. In that case, the electron distribution is known to be a displaced Maxwellian. In Fig. 12a-d, the optical absorption of a polaron, Re[l/ZFH,P(~)]is plotted as a function of the frequency for c1 = 1,3,5,and 7 for various values of the lattice temperature. For zero temperature, the optical absorption of a

FIG.12. Absorption of polarons as a function of the frequency as given by the FHIP theory for various values of the temperature and for four values of the electron-phonon coupling strength: (a) (x = 1 ; (b) G( = 3 ; ( c ) a = 5 ; and (d) (x = 7 .

THEORY OF POLARON MOBILITY 0.6

0.5

0.4 z

0 I-

% a3 v) 0

m

a

0.2

03

WlWLO

(d 1 FIG.12. (Continued).

113

114

F. M. PEETERS AND J . T. DEVREESE

t-

0.8

0.4 -

0

2

1

4

I

w 1 WLO

I

6

I

8

(a) FIG. 13. Real and imaginary part of the memory function as function of the frequency. resulting from the FHIP theory for various values of the temperature and two values of the electron-phonon coupling: (a) a = 1 and (b) a = 7.

polaron in the FHIP approximation has been obtained first in Ref. 56. As is ~ e l l - k n o w n at , ~T ~ = 0 there is no absorption for o < oLo (and o # 0). If the temperature is increased, one sees that for small and intermediate c1 there are two peaks: one situated at o = 0 and the other at some o > oLo. The latter peak appears at higher frequencies when the temperature is increased. The peak at o = 0 is due to a brownian-type motion of the polaron, whereas the peak at o > oLois the phonon emission peak. At sufficiently high temperature, only one broad peak is observed at w = 0. For strong electronphonon coupling, the relaxed excited states (RESs) come into play. The importance of relaxed excited polaron states at strong coupling has been 56

J . T. Devreese, J . De Sitter, and M. Goovaerts, Phys. Rev. B: Solid State [3] 5, 2367 (1972).

115


-3 W

40

E -

20

0

l -40 0

l 4

l

!

l

8 W/WLO

!

12

!

J 16

l 20

(b) FIG. 13. ( C o n l i n u d ) .

shown in Ref. 56a: they were identified in the optical absorption in the FHIP approximation in Ref. 56. The RESs are responsible for additional peaks in the polaron absorption spectrum which correspond to transitions of the polaron to an excited internal state which is relaxed. The difference between the small and large electron-phonon coupling regime is clearly seen in the behavior of the memory function (Figs. 13a,b). The position and the width of the peaks in the absorption spectrum are indicated in Fig. 14a and b for a = 5 and 7, respectively. At low temperature the peaks first become more pronounced with increasing temperature. This is a consequence of an increased effective electron-phonon interaction (Section 12). Increasing the temperature further causes the peaks to start to broaden, and at sufficiently high temperature some of the RESs peaks can no longer be resolved. Note also that the peaks are nonsymmetrical about their peak position. E. Kartheuser, R. Evrard, and J. Devreese, Phys. Rev. Leu. 22,94 (1969).

116


. 3

FIG.14. The position and width at half-height of the peaks appearing in the absorption spectrum ofthe FHIP theory as a function of the temperature for (a) a = 5 and (b) a = 7. The dashed lines with an arrow indicate that the peak did not attain its half-height value.

In earlier work5' the polaron absorption spectrum in the FHIP approximation was also calculated for various values of the temperature. However, in Ref. 57 the parameters of the Feynman model correspond to zero temperature, whereas in the present numerical analysis the temperature dependence of o and w was taken into account.

'' W. Huybrechts, J . De Sitter, and J. T. Devreese, Solid Stuie Commun. 13, 163 (1973).


117

VI. The Polaron Mass and the Polaron Mobility

11. PHYSICAL PICTURE OF

POLARON

A

In Parts I11 and V we calculated the impedance function Z ( w ) of polarons in two different approximations and obtained the corresponding optical absorption. From this function we can further obtain two important quantities: the polaron mass and the polaron mobility. To clarify this consider the following simplified physical picture of a polaron. The electron interacting with the LO phonons is influenced in two ways:

(1) The electron effective mass increases; a particle with mass m* (polaron = electron + phonon cloud) moves through the crystal. The inertia of the electron is determined by this renormalized mass m*. (2) In moving through the crystal, the electron feels friction (the electron can dissipate energy by emitting LO phonons). We describe this dissipation effect by a collision time T. According to this picture, the electron is governed by the equation of motion rn*i.(t) = -(m/~)r(t),

(1 1.1)

which gives the impedance function58 (1 1.2)

To the collision time T , a mobility p = er/m corresponds. The mobility can also be written as p = Re[Z(o)]-'(,=,. We expect that the previous quasi-particle picture of the polaron is valid in the limit of o -+ 0. The polaron impedance function can be written as [electron band mass = 1; Eq. (9.16)] ~/Z(W =)i e / [ o - C(o)],

(11.3)

with Z(W) the memory function. Comparing Eqs. (11.3) and (11.2) in the limit of zero frequency, we obtain for the polaron mass

m*/m

=

1 - lim [Re C(w)]/o, w-0

(11.4)

and for the mobility, l/p = -(m/e) lim Im Z(o).

(11.5)

W-+O

58

P. C. Martin, "Measurementsand Correlation Functions." Gordon & Breach, New York,

1968.

118


We later discuss both quantities separately for various approximate calculations of the impedance function Z(o)[or, equivalently, of the memory function C(o) = o - i e Z ( o ) ] . 12. THE TEMPERATURE DEPENDENCE OF THE POLARON MASS

A considerable number of publications have been devoted to the calculation of the polaron mass, and several techniques were used. Most of the results, however, are restricted to the zero-temperature case (e.g., Refs. 27, 59-62). In those works the dependence of the polaron mass on the electronphonon coupling constant was studied. This behavior is reasonably well understood (e.g., the exact values of the polaron mass in the limit of small and strong electron-phonon coupling are well known). The question of the temperature dependence of the polaron mass is not so well resolved.63 In one of the first publication^^^ on this subject, it was found that the polaron mass decreases with increasing temperature. For a > I ) AND HIGH-(/3 R , , the polaron mass becomes smaller than the bare electron mass m ;this, of course, is hard to understand physically. Consider the limit of small electron-phonon coupling (a > 1 (low temperature), R , x exp(-B/2)/& which can be neglected, and thus (m*/m)FHIP> 1. However, for p > 1 : /+HIP

=(~~T/~~~,o)PRTA.

(13.5)

Although in calculating pRTA no similar separation between emission and absorption processes can be made, it turns out that in Eq. (5.13) the absorption contribution to the collision frequency A(P, 0) provides the dominant contribution to pRTAfor B >> I . Consequently, pRTAis determined by the absorption processes at low temperature. This already indicates that p R T A is the correct mobility result. Indeed, for a decreasing electric field (ohmic regime), the polaron needs an arbitrary large time before it gains enough energy to emit an LO phonon. For the absorption process, no threshold exists and the electron can always relax via such a process. It has been pointed out by Feynman et aL2' and, later, implicitly or explicitly by others1*2,4.8" that the pRTA result is correct for B >> 1. Feynman et ~ 1attributed . ~ ~the discrepancy between pFHIP and PRTA to an interchange of two limits in calculating the impedance. They argued that in their theory, at small electron-phonon coupling, one takes limo+o limg+o,whereas the correct order is lima-o limwd0(a is the electron-phonon coupling constant "O

F. M .Peeters and J . T. Devreese. Phys. S/utus Solidi B 115, 539 (1983)

126

F. M. PEETERS A N D J. T. DEVREESE

0

2

6

6

8

1

0

INVERSE TEMPERATURE ITJT)

FIG. 18. The same a s Fig. 17 but for small electron-phonon coupling. i.e.. G( = 0.02. The results o f a Monte Carlo calculation (0) and of an analytic solution of the Boltzmann equation (+. Devreese and Brosens14) are also given; (-) RTA; (---) FHIP.

and o is the frequency of the external field). Later, in the article of Thornber and Feynman, the argument given by FHIP about the interchange of the limits a 0, w -+0 was questioned. We have analyzed and confirmed" the argument of FHIP. Here we shall generalize this argument to arbitrary values of the electron-phonon coupling u. For arbitrary coupling, one should replace the coupling constant a, appearing in the previous limits, by a parameter which characterizes the difference between the Frohlich Hamiltonian H [given by Eq. (1.1)] and the effective Hamiltonian Her = HF + Hph,where HFis the Hamiltonian of the Feynman polaron and Hphis the free phonon Hamiltonian, so that I V = H - He*. For arbitrary values of u, the order of the limits limu+o 1irni+' used to calculate the mobility is equivalent to calculating the resistivity to lowest order in the scattering potential AV. This is done in Appendix B, where the FHIP result is obtained again [the calculation in Appendix B is a generalization to arbitrary ci of a similar calculation done by one of us (J.T.D) and other^'^]. It has been proved several times32,81-83that the correct way to evaluate the mobility (and resistivity) consists of taking the order of limits limi+o lim,,,, which leads to the summation of an infinite number of divergent terms.81Such a procedure was followed in Appendix C, where it is shown that this procedure leads to the results pRrA. These arguments are supported by Fig. 18 in which the linear mobility results are compared for various theories in the limit of small electron--+

"' L. van Hove, Physicu (Arnsferdarn)23,441 (1957). 83

W. M. Kenkre and M. Dresden. Ph-ys. Rev. A [3] 6,769 (1972) K. M. Van Vliet, J . Marh. Phj's. 20, 2573 (1979).


127

phonon coupling. (The particular case of CI = 0.02, which corresponds to InSb, was chosen.) In this limit (i.e., M oLo). starting from the first moment of the Boltzmann equation and redefining the expectation values appearing in it, it was possible to obtain an equation which also expresses conservation of momentum at these high frequencies. The difference between the equation obtained by taking the first moment of the Boltzmann equation and the redefined equation is clearly seen in the difference in the resulting scattering mechanisms (compare Figs. 1 and 11). Further, we showed that the FHIP result can also be obtained by a calculation of the resistivity to lowest order in the scattering potential. This discussion and the relation between the various results are schematized in Fig. 19. To compare the different theories we calculated the electron distribution function corresponding to each of the approaches. Although it is the polaron POLARONBDLTZMANNE Q U A T I O N f ( G , A , t ) polaron distribution

first moment Boltzmann equation

other expectation expect iation value, D M linearizal: i o n linearization

intermediate c o u p 1 1ng

A.

Weyland

Y. Osaka LIO

C I I L.P.

Kadanoff

Iddings-Platrman

Thornber-Feynman E (”)

-7-

‘ri ‘IFHIP

FIG.19. Scheme of the present work which gives the interrelation between the various polaron mobility theories; DM, drifted Maxwellian.

129


distribution function which is the key quantity, i t is still instructive to compute the electron distribution function. It turns out that in FHIP and in TF, the electron distribution is a drifted Maxwellian. The electron temperature is determined by the effective electron-phonon interaction (as given by the parameters of the Feynman model). Such a distribution corresponds in fact to a constant relaxation time. On the other hand, in the result of Osaka the corresponding distribution function contains a relaxation time which depends on the polaron state. The + ( k T / h o , , )controversy in the low-temperature polaron theories was discussed. Three arguments have been given to support the conclusion that the mobility result p,, as obtained by Osaka, is correct (at least at low temper at ure). We conclude with a suggestion. The F H I P result for the impedance function is generally believed to be a good approximation for quantum frequencies. Although the generalized Osaka result is valid in the limit of classical frequencies, it would be interesting to,construct a theory which is valid over the whole frequency region. Such a theory should reproduce the FHIP result in the limit of high frequencies and the generalized Osaka result in the low-frequency limit.

Appendix A

In Appendix A we calculate [Eq. (5.8)] s (k, t) = (p, & k . r W t , P,n

-ik.r(O)

IPnh

(A.1)

for n = 0, and the Fourier transform of the electron density-density correlation function S(k, t ) = ( e - i k . r ( r ~ ~ + Oe i k . r ( r d >. (A.2) The time evolution of the electron coordinate is governed by the Feynman polaron Hamiltonian H , and is given by [Eq. (2.8)] r(t,

+ t ) = R(t,) + (P/M)t + a(c(t,)e-"" + c+(tO)ei"').

From Eqs. (A.1) and (A.2) one notes that the key operator is

-

exp( - ik r ( t , =

+ t ) ) exp(ik

k2 exp[-i-t 2 M

-

*

r(t,))

u2k2(1 - exp(-iut))

k-P

x exp[ iak*c+(t,)(l- exp(iut))]exp[ iak.c(t,)(l - exp(-iut))],

(A.3)

130

F. M . PEETERS A N D J . T. DEVREESE

where use was made of the relation

which is valid if the operators A and B commute with [ A . B ] From Eq. (A.4) one finds

[

SP,,,(k,t ) = exp - i

kZ t - u2 k 2( 1 2M

-

~

exp( -iot))

] (

exp i ’LPt), ~

(A.6)

where P now indicates a c number. For a generalization of Eq. (A.6) to arbitrary values of n. we refer to Ref. 84. If one makes use of Eqs. (A.4) and (2.2). the expectation value [Eq. (A.?)] S(k, t ) = ~ ~ ( ~ ~ - ~ ~ I ~ ~ - i ek i,k .rr ( i oI ) )/Tr(e-”H‘), ( l + ~ )

(A.7)

reduces to

Slk, t )

=

exp[ - i ( k 2 / 2 M ) t - u2k2(1

-

exp( - iur))]

- Pt/M))/Tr(e-ppP”2M)]

x [Tr(exp( - / ? P 2 / 2 M )exp( -ik x (Tr(exp(-/?~oc+c)exp[ x exp[ iuk.c(l

-

-

iak c t ( 1

-

exp(iot))]

exp(-iut))]/Tr[exp(-/?uctc)]).

(A.8)

The first trace can easily be performed by expressing Tr in the momentum representation. The last trace with the oscillator variables c f , c j has a standard form [ e g , Eq. (19) of Ref. 851. The final result for Eq. (A.8) is

D(t) = (2M)-

’ [ - i t + ( t 2 / b ) ]+ a2 [ 1

-

+ 4n(u) sin2 i u t ] .

err“

(A.1Q)

and n(u) = (e”’ - I ) - ’ . Note that D ( t ) has the following properties: D(t)* = D( - c ) ; D ( t ) = D(i/? - t ) ; D ( i f )E .A’; and D(r + i / ? / 2 ),2, ~ which are valid for t E 9. Appendix B

To obtain clearer insight into the mobility result P ~ , , ,as~ obtained from the calculations of Feynman et ~ 7 1 and . ~ ~Thornber and Feynman,”’ we present here an alternative derivation of pFHIPvia the calculation of the resistivity to lowest order in the scattering potential. The frequency-depenH4

J . Schwinger, Phy.s. Rer. (21 91, 728 (1953).

’‘ P. M . PlatTman. in “Polarons and Excitons” ( G . G . Kuper and G . D. Whitfield. eds.), p. 123 Oliver & Boyd, Edinburgh, 1963.

131


dent resistivity is also calculated. The FHIP expression [Eq. (9.15)] for the ac conductivity is obtained. Appendix B is a generalization to arbitrary a of a similar calculation performed in Ref. 13 for a

&[@I = F [ p ] +

s

u(r)p(r) d3r. (4.12)

So, E,[p] is a minimum relative to all density functions associated with some other external potential u’. We have now established the Hohenberg-Kohn theorem, which may be summarized as follows :

(1) The ground-state energy of a system of identical spinless fermions is a unique functional of the particle density.lgb (2) This functional attains its minimum value with respect to variation of the particle density subject to the normalization condition’ when ihe density has its correct values. Questions have arisen as to whether the phrase used in the sentence following Eq. (4.12) (“relative to all density functions associated with some other external potential u”’) is a significant restriction. As it stands, the functional F [ p ] appears to be defined only for those p which can be obtained via solution of the full many-body Schrodinger equation containing some external, potential u. In other words, F [ p ] is defined only for densities that are “u-representable.” One’s intuition is that this is not a serious problem in that any reasonable (i.e., nonpathological) p should be derivable from some u; however this intuition is incorrect according to Levy.” The problem of Iyb

2o

The expression “unique functional” means that the functional does not depend on the external potential except through the term d3r. M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76,6062 (1979): Phys. Reo. A . 26, 1200 (1982).

DENSITY FUNCTIONAL METHODS : THEORY AND APPLICATIONS

147

u-representability can be bypassed by using a result of Levy.2o He shows that it is possible to consider the more general class of F , which can be obtained from some (trial) antisymmetrical N-body wave function which need not be the solution of a Schrodinger equation containing a symmetrical potential u. The functional F [ p ] can be replaced by a more general functional Q [ p ] , which agrees with F [ p ] if p is u-representable: (4.13) where Yo is a wave function yielding the density p . One searches in principle IT Li 1 I)o)for all wave functions to find the value of Q [ p ] by evaluating (I)o yielding p, and one chooses the minimum value obtained in this process. Reiss and Munch” have extended the basic result by showing that the ground-state energy of a system which is contained in a finite volume is a unique functional of the density in any finite portion of the volume. The essential point of the argument is that the density is an analytic function of position except possibly on a set of points of measure zero. Gilbert” observed that the restriction to local external potentials is essential. If a nonlocal potential u(r, r’) is considered as part of H , then the ground-state energy is a functional of the density matrix rather than simply the (diagonal) density. It is interesting to consider whether there is a form of the HohenbergKohn theorem that applies to the momentum density. Is the ground-state energy a unique functional of the momentum density in the ground state? The answer would be yes if the conditions of the theorem were met; that is, if the external potential which has to be considered were a local potential in momentum space. Unfortunately, this is not the case in realistic problems. In particular, the most important type of external potential, that of the interaction between an electron and a point nucleus, is a nonlocal operator in momentum space. This defeats the intended application.22a

+

5. SPINAND RELATIVISTIC GENERALIZATIONS The Hohenberg-Kohn theorem just stated is, of course, important in formally completing the original Thomas-Fermi theory, but it is still significantly limited. In particular, it does not apply to a system of particles with spin (i.e., to real interacting fermions), and it refers only to the ground-state energy and the ground-state density. We shall now consider how the theorem has been extended to remove these limitations, at least in part. We shall show how spin can be included by means of an extension to relativistic

*’ J. Reiss and W. Munch, Theor. Chim. A c m 58, 295 (1981). 22

T. L. Gilbert, Phvs. Re?. B. 12, 211 I (1975). For additional discussion, see G. A. Henderson, Phys. Reti, A 23, (1981)

148

J . CALLAWAY A N D N. H . MARCH

quantum mechanics. The situation with regard to excited states is more complicated, but some conclusions can be obtained (this problem is discussed in Section 11). Finally, there is an extension to systems at finite, temperature. The extension to include spin has been made by several a ~ t h o r s . The ~~-~~ essential point is that instead of just considering the particle density p(r), we must now involve the spin-density vector s(r) as well. Similarly, instead of just considering a system subject to an external potential v(r), there will be a static external position-dependent magnetic field B(r) as well. The conclusion is rather obvious: the ground-state energy is a unique functional of the charge and spin densities which attains its minimum value when these quantities are correct. We derive this result according to the procedure of Rajagopal and Callasay,26which yields more: By starting from a relativistic point of view, one derives that the ground-state energy is a functional of the four-vector current density j p ( r ) , provided that this vector is constrained by an equation of continuity, which attains its minimum value when j , is correct. The result involving spin densities only is then obtained in the nonrelativistic limit. We begin with the Schrodinger equation for quantum electrodynamics in Fock space: [ih(a/at) - H ] ( Y ) = 0.

(5.1)

The Hamiltonian contains four parts: H

=

H,

+ H , + H , + H,,,.

(5.2)

The operator H o describes noninteracting Dirac and electromagnetic fields:

Ho

= Hem

+

s

d3rlCl+(x)h(x)+(x),

(5.3)

where x is used to denote coordinates and time. In Eqs. (5.2) and (5.3), Hem describes free electromagnetic field and t,bt are destruction and creation operators for the free Dirac field, respectively; and h(x) is the Hamiltonian of this field:

+

h(x) = c a - p + pmc'.

(5.4)

The electromagnetic field is described in the radiation gage so that the

23

J . C. Stoddart and N. H . March, Ann. Phjs. ( N . Y . )64, 174 (1971).

'' M. M . Pant and A . K. Rajagopal, Solid State Commun. 10, I189 (1972) 25

U. von Barth and L. Hedin. J . Phys. C 5 , 1629 (1972). K . Rajagopal and J . Callaway, Phys. Rrr. B . 7, 1912 (1973).

"A.


149

Coulomb interaction between electrons appears explicitly:

s

1

H, = d3r d3r’++(x)+(x)___ ++(x’)+(x‘) d3x. Ir - r‘I e2 2

(5.5)

The interaction between matter and the tranverse radiation field is contained in H I :

HI =

‘J

+ -C

j ( x ) A ( x ) d3r,

where j refers to the space components of the current operator j ” ( 4= c$+(4Y0Y”W,

(5.7)

which satisfies the continuity equation a j P

=

0.

(5.8)

Finally, we have the interaction between the current and a (classical)external electromagnetic field AL,(x): (5.9)

C

In principle, operator products in these equations must be normal ordered,” but we shall not indicate this with an explicit notation. Now we wish to consider the ground-state energy for the Hamiltonian [Eq. (5.2)]. There is an immediate technical problem because relativistic Hamiltonians are not bounded from below and therefore do not have a ground state. However, physical systems do have ground states. Therefore, in the conventional way, we suppose that all negative energy states are filled and proceed from this point. Let IG) be the ground state in this sense. We describe the current density in the ground state as JJX) =

(Gl&Ax)lG).

(5.10)

The expectation value is taken with respect to the Fock space operator so that J,(x) is an ordinary function of position. The four components of J , are not all independent since the equation of continuity must be satisfied:

d,J”(x) = 0.

(5.11)

’’ See, for example, S. S.Schweber. “An Introduction to Relativistic Quantum Field Theory. Harper & Row, New York, 1961.

150

J . CALLAWAY A N D N. H. MARCH

It is now possible to repeat the arguments of Section 4 to show that the ground-state energy of the system is a unique functional of J , ( x ) . As before, the proof is by reductio ad absurdum. Suppose that there should exist another external field A:;, which gave rise to the same ground-state current density J,(x). [We do not consider that external fields which differ by a gage transformation A‘ = A d,x are different for this purpose. It is required that ’x = 0. The Lagrangian, the energy momentum vector, and hence the Hamiltonian are unaffected by such a transformation.] The ground-state vectors ( G ) and IG’) must be different because they satisfy different Schrodinger equations. Let H , H’, E, and E’ be the Hamiltonians and the energies pertaining to IG) and IG’), respectively. Then, according to the minimum property of the ground-state energy,

+

E‘

=

(G’IH’IG’) < (GIH’IG)

=

(GIH

‘J

- -

c

d3rj,(X)[A;;t(X) -

A:,,(x)IlG).

(5.12)

However, the external fields are not quantized, so (GI

J

s

d 3 r j , ( X ) ~ : x , ( x )= ~~) d3r~,(x)~:,,(x),

“s

E‘ < E - C

(5.13)

d3rJ,(x)(A;~,,,(x) - A&,(x)).

If we now also assume that J , ( x ) = ( G ’ ~ j ~ , ( x ) ~ Gthen ’ ) , we may repeat the argument with primed and unprimed quantities interchanged. Then,

“s

E < E‘ - C

d3rJB(x)(AExt(x) - A;;,(x)).

+

(5.14)

+

The addition of Eqs. (5.13) and (5.14) leads to E’ E < E E‘, which is a contradiction. The argument indicates that Afx, must be a unique functional of J , ( x ) . However, H is determined if A&, is given, so the ground state must be a unique functional of J,. We therefore have

E

=

E[J].

(5.15)

The continuity equation [Eq. (5.11)] may be integrated over all threedimensional space. In the usual way,27 we derive the condition

s

Jo(x)d 3 r = (GI

s

j o ( x )d3rlG) = constant.

(5.16)

DENSITY FUNCTIONAL METHODS : THEORY A N D APPLICATIONS

15 1

Equation (5.16) is the familiar condition that the total number of particles in a system must be constant. It can now be shown that E [ J ] will possess a minimum for the correct J p ( x )(and specified &,). Let us define F[J]

=

(GIHo

E[J]

=

F[J]

so that

+ H , + H,IG),

‘S

- -

J,(x)A;,,(x) d3r.

(5.17) (5.18)

C

Since E [ J ] is a unique function of J , so is F [ J ] . Now we follow the procedure of Section 4. Let IG‘) be the ground-state vector associated with a different external field A ; . We construct the energy functional referring to the state vector IG’),E[G’],but we use A instead of A‘: E[G‘]

‘S

= F [ J ‘ ] - - J~(x)A:!,(x) (1%.

(5.19)

C

However, since IG’) is the wrong vector to be associated with A, E[G’] is too large:

E[G‘]> E [ J ]

=

‘J

F[J] - C

n

J , ( x ) A ~ , , ( xd3r. )

(5.20)

Hence, similar to the nonrelativistic case, the energy functional E [ J ] is a minimum with respect to all current density functionals associated with other external potentials. The condition that the current density satisfy the equation of continuity has to be imposed so that the number of particles remain constant. Now we wish to investigate the physical content of this theory by using the Gordon decomposition.’’ The components of the current are

where k

28

=

1, 2, 3.

See, for example, G . Baym, “Lectures on Quantum Mechanics.” Benjamin, New York, 1969.

152

J. CALLAWAY AND N. H. MARCH

We have used the notation 3 = $+yo, and tkjfis the Levi-Civitta notation 1, t Z l 3 = - 1, etc., and a summation over repeated indices is understood. The following functions are introduced: c l Z 3=

(1) the particle density p, P(X) =

(5.22a)

(GI$+(x)$(x)lG);

(2) the spin density vector, (5.22b)

sl(x) = (Gl$(x)a‘$(x)(G);

(3) the ordinary (plus polarization) current vector jf(x), JLX) =

(2m)- l(Gl$(x)(P/ - eAf/c)$(x) - [(Pl

+ eA,/c)3(x)I$(x)lG);

(5.22c)

(4) a vector without classical analog,

g’(x) = (-i/2m)(GI~(x)a’ll/(x)(G).

(5.22d)

The contribution to ( G l H , I G ) from the spin density can be transformed into

in which s(x) is the ordinary spin vector, and Bext(x)= V x A,,,. Thus, the ground-state energy functional can be reexpressed by substituting Eqs. (5.21)-(5.23) into (5.18) to obtain p(x)V,,,(x) e

e + 2m s(x)-B,,,(x) -

+-.Aex,(x) 2mc e ddtg

-C J(x).AeXt(x)

1

,

(5.24)

and V,,, = -e4ex,(x),where 4 is the scalar potential (zeroth component of A ) . This result generalizes the results of Section 4 by describing in a more comprehensive way the interaction between a system and an external field.

6. FINITE-TEMPERATURE THEORY There is a remarkable extension of the Hohenberg-Kohn theorem to . ~ ~ essential result is that the finite temperature that is due to M e r m i r ~ The grand potential of a system at finite temperature is a functional of the density

’’ N . D. Mermin, Phys. Rec. 137,A1441 (1965)


153

in the system at that temperature. Furthermore, a minimum principle applies: If the grand potential is computed with an “incorrect” density, the result must be larger than if it is computed with the correct density. Our discussion here follows primarily that of Kohn and Va~hishta.~’ The demonstration of this result requires some of the apparatus of statistical physics. The properties of the grand potential are discussed by R e i ~ h l . ~This ’ object, which we denote by Q, is useful in describing systems at fixed temperature T and constant volume V but having a varying number of particles. An equilibrium state of a system at fixed T and I/ with fixed chemical potential p is a state of minimum Q. To define Q in a quantum system, it is desirable to consider the density matrix operator 3, which is used to calculate thermal averages of physical quantities. This operator has the property tr j3 = 1,

(6.1)

and is given formally by

p

= ,-SW-

&Nl/tr[e-S(H - aN)],

(6.2)

in which B = l/k,T, where k , is Boltzmann’s constant as usual; N is the number operator; and H is the Hamiltonian. The average value of the operator 0 representing a physical quantity (0) is

(0)

=

tr(p0).

(6.3)

The grand potential is given by

SZ = -(I/fl)ln t r [ e o ( ~ ~ - ~ ) ] . It is this object which is determined by the density

(6.4)

(6.5)

Mermin29 introduces an object that we denote by o [ p ] , which is a function of a (trial) density matrix f i T : w[&] = tr{fiT(H - pN

+ D-’

In &)>.

(6.6)

This quantity has the property that, if it is evaluated with the correct density matrix p , it yields 0:

QCj31 = wCfi1.

(6.7)

W. Kohn and P. Vashishta, in “Theory of the Inhomogeneous Electron Gas” (S. Lundqvist and N. H. March, eds.), Chapter 2. Plenum, New York (1983). 3 1 L. E. Reichl, “A Modern Course in Statistical Physics,” p. 39. Univ. of Texas Press, Austin, 1980.

30

154


Mermin further shows that for fir # a ,

W T l

2

Q[PI>

i.e., that w attains its minimum value, which is a,when it is evaluated with the correct density matrix. As in Section 4, we are considering a system governed by a Hamiltonian H , where H = T + U + V, which contains an external potential I/ given by Eq. (4.3).We note that from Eq. (4.4), V

-

puN

=

s

(u(r) - p)t+bt(r)$(r)d3r.

It can now be established that u(r) - p is uniquely determined by the density distribution p(r). To see this, we suppose that p(r) is the density corresponding to u(r) - p . Moreover, we suppose that there is another pair of quantities u’(r) - p’ which gives rise to the same density. This situation shall be shown to be impossible. Associated with the Hamiltonian H ‘ containing V’, there is a density matrix given by Eq. (6.2) but containing H ’ and p’. For this a’, we consider the functional w‘[$’] (note that p(r) = p’(r), but # a’):

a’

w’[a’]

+ p-’

= tr[P’(H’

-

=

- p’) -

1

[(Lw

p‘N

ln(a’)] -

p)]p(r)

d3r

+ o[a’],

where w[a’] differs from w’[$’] in that it is constructed with the “original” u(r) - p and a‘: ( ~ [ p ’ ]= tr[a’(H - p N

+ fi-’

In a’)].

However, by Eq. (6.8), w[a’] > oi[a], so

s s

w’[a’] >

[(u’(r) - p ’ ) - (u(r) - p ) ] p ( r )d3r + w[a].

(6.10)

The argument may be repeated with primed and unprimed quantities interchanged, leading to w[a] >

[(u(r) - p ) - (u’(r)- p’)]p(r) d3r

+ w‘[a].

(6.1 1)

+

The addition of Eqs. (6.10) and (6.11) leads to a contradiction: w’[a’] o[a] 2 w[a] + W‘[$‘]. Hence, we see that the assumption that p ( r ) = p ’ ( r ) must be false. We know from this that a specification of p ( r ) determines ~ ( r-) p and

DENSITY FUNCTIONAL METHODS: THEORY AND APPLICATIONS

155

thus, in principle, defines the operator H - pN. Thus we may say that the complete density matrix i?is a functional of the diagonal p, as are all averages based on the grand canonical ensemble (including finite-temperature N-particle Green’s functions). In particular, we may use the diagonal density p instead of the density matrix p in the functionals: w = w [ p ] , etc. Furthermore, we may separate out the explicit V - pN terms in Eq. (6.6) to write for a trial density p : r

(6.12) where F is a new functional = tr{ij(T

+U+

In f i ) }

=

F[p]

+ U [ p ] - TS[p].

(6.13)

Here, F , U , and S are functionals of density representing the kinetic energy, interaction energy, and entropy. Equations (6.12) and (6.13) correspond to the expression for the grand potential:

f2 = E

-

TS

-

pN,

(6.14)

where E is (here) the internal energy. Finally, we see that it follows from Eq. (6.8) that 4PTl

’wcP1,

(6.15)

for trial densities which differ from the correct density. Thus, the grand potential attains an absolute minimum when evaluated with the correct density. Other extensions of the original Hohenberg-Kohn theorem have been given by R a j a g ~ p a l .The ~ ~ result of greatest generality is the relativistic extension of Mermin’s result. This theorem shall be stated without proof: In a grand canonical ensemble with fixed temperature and chemical potential, the components of the four-current density defined by J, =

tr(Pj,)?

(6.16)

are determined by a static external vector potential A,(r). The correct J,,, which is divergenceless, minimizes the grand potential for given .4Cxt(r). 111. Principles of Calculational Procedures

In Part 11, we have been concerned with the foundations of density functional theory. Of course, whereas the theorems discussed there have been a 32

A. K. Rajagopal. Ado. Chern. Phys. 41, 59 (1979)

156


prime reason for a revival of interest in descriptions based on the electron density, when we come to apply the theory we must naturally ask what useful approximations to the functionals can be made which will reduce the many-electron problem to a fully practical procedure. It is not surprising in view of the tradition of Hartree and Hartree-Fock single-particle methods, that one wishes, as far as possible, to solve single-particle equations. This then is the topic we must tackle next: What basis does the formal theory developed so far provide for such single-particle equations?

7. SINGLE-PARTICLE EQUATIONS The minimum property of the ground-state energy as a functional of the density can in fact be exploited to derive a set of effective Schrodinger equations for single-particle functions from which the density and the groundstate energy can be determined, as pointed out by Kohn and Sham.33 The similar properties of the grand potential for a system at T > 0 can be exploited in parallel fashion. In Section 7, we shall consider the derivation of these equations. We shall first investigate a system of spinless fermions at T = 0 K, then the inclusion of spin, and finally, we shall consider systems at finite temperature. We begin by separating the functional F [ p ] of Eq. (4.10) into three parts: the kinetic energy, the ordinary electrostatic energy, and everything else e.g., the exchange and correlation energy which are collectively denoted as E x c : (7.1) This is inserted into Eq. (4.10) to give EG [ p ] . Unfortunately, we do not know exactly either 5 or E x c . However, we bypass this problem for the moment and vary the density, subject to the condition that the number of particles remain constant,

We obtain by varying E , ,

33

s

6p(r) d 3 r

=

0.

W. Kohn and L . J . Sham, Pkvs. Rec. 140, A1133 (1965).

(7.4


157

The constraint of Eq. (7.2) is imposed, and we obtain

where p is a constant (a Lagrange multiplier). [If Eq. (7.4) is satisfied, we substitute p for the square bracket in Eq. (7.3) and take the constant outside the integral which then vanishes because of Eq. (7.2)]. This quantity is the chemical potential [cf. Eq. (2.6)]. However, Eq. (7.4) is not satisfactory for most purposes because we do not have sufficiently accurate expressions for 9-[p] (see, for example, ref. 22). However, apart from this complication, we may note that Eq. (7.4) is exactly what would be obtained if we considered a collection of noninteracting electrons moving in an external effective potential (cf. Eq. (2.7)]: V&)

=

V(r)

+ e2

1 ~

’(“) d3r’ Ir - r‘)

+ Vxc(r),

(7.5)

in which

So, the effects of exchange and correlation appear in an “exchange-correlation potential” V,, which may be nonlocal. This establishes a result referred to earlier in Part I. The difficulties with the kinetic energy are substantially alleviated by the following procedure. Suppose the system of interest contains N electrons. Then we introduce a set of N single-particle functions such that the exact density p(r) can be expressed as

Equation (7.7) is characteristic of this theory. We shall discuss this further subsequently, but at this point it is sufficient to note that this step can be justified rigorously. The intent is to carry out the variation of the density by varying the orbitals ui.In fact, it is sufficient to regard ui as complex and to write the result of variation with respect to uz from which we find an equation satisfied by ui.[Variation with respect to ui leads to the corresponding equation for u:]. The procedure is quite straightforward except for the case of the kinetic energy where some comment is required. We must write the kinetic energy functional in either of the equivalent forms: uT(r)(-V2)ui(r) d3r,

Vu:(r)-Vu,(r) d 3 r = i= 1

i= 1

(7.8)

158

J. CALLAWAY A N D N. H. MARCH

where the N functions ui are the same as those in Eq. (7.7). We believe that Eq. (7.8) is an approximation (this point shall also be discussed subsequently), because there has been no proof that Eq. (7.8) holds for the exact F [ p ] . Formally, at this point, the complication can be ignored, and the difference between F [ p ] and F s [ p ] can be absorbed into E,,[p]. We suppose this to have been done and do not alter the notation. The variation is carried out and yields -V2

+ u(r) + e2

s

~

p ( r ’ ) d3r‘ Ir - r‘(

1

+ Vx,(r)

ui(r) = ciui(r),

(7.9)

where V,, is given by Eq. (7.6). We have obtained a Schrodinger-like equation for the one-electron function ui(r). Since the operator in square brackets in Eq. (7.9) is Hermitian, the functions uimay be taken to be orthonormal. The significance of the eigenvalue .ci is discussed in Section 9. The ground-state density may be constructed from the solutions of Eq. (7.9) via the use of Eq. (7.7) in which the N functions with the smallest ci are included. The total energy may then be determined. The result is

where the sum over i includes the N smallest t i . The functions uiintroduced by Eq. (7.7) are often considered to be singleparticle wave functions in the Hartree-Fock sense. This view has no real justification because no assumption concerning a decomposition of the many-body wave function into some combination of one-particle functions has been introduced. However, the ansatz Eq. (7.8) leads to a remarkable consequence with regard to construction of the momentum density that is close to what would be expected if the ui were treated as one-particle wave functions. As shown by Lam and P l a t ~ m a nthe , ~ ~momentum density function N,, (the number of electrons which have momentum p) is given by N

N,

=

C J(pJui)J2+ correction,

(7.1 1)

i

(PI

in which represents a momentum eigenfunction (a plane wave). The dominant first term is just what would be obtained in a Hartree-Fock approximation in which the one-electron functions were the u i . The correction, which is not discussed here, involves the difference between the momentum distribution functions in interacting and noninteracting electron L. Lam and P. L. Platzman, Phys. Reu. B. 9, 5122 (1974). This result [Eq. (7.1 I)] has been generalized to general operator expectation values by G . E. W. Bauer, Phys. Rev. B 27, 5912 (1983).

34 34a


159

We now consider the question of spin dependence. As a result of the considerations of Section 5, the ground-state energy depends on not just the charge density p, but on the four-current density J of which the charge density is a component. In the nonrelativistic limit, a simple result survives. The exchange correlation functional Ex, depends on the spin densities. Let us choose some axis of quantization by which we distinguish between up and down spin. We denote the spin densities by po, p-,. Of course, P

= Po

+ P-o.

(7.12)

(7.13) and we generalize Eq. (7.7) to

(7.14) i =1

where N is the number of electrons of spin r ~ . Then we may repeat the argument that leads to Eq. (7.6), the modification being that we have a spindependent exchange and correlation potential V,,, :

(7.15) so that Eq. (7.9) becomes

Now the energies cia depend on spin. Equation (7.10) is similarly generalized to

E,

=

1 i

tia

-

52

( r - r’l

d3r d3r’ + Ex, -

puVx,(r)d3r. (7.17)

We include the N lowest values of ti, in the sums-this condition determines net spin (or magnetization) if any. We postpone a discussion of specific forms for E x , and V,,, until Section 8. The argument leading to the single-particle equation [Eq. (7.9)] at zero temperature can be generalized to yield similar equations at finite temperature. For this purpose we employ the minimum property of the grand potential from Section 6. The argument is based on that of Kohn and Vashishta.jo

160


We consider the decomposition of the grand potential according to Eqs. (6.13) and (6.14) (we no longer need to distinguish between o and Q).

Qbl = F S b l - 7-UPl +

s

[u(r)

-

P l m d3r (7.18)

in which .FJp ) denotes the kinetic energy of noninteracting single particles, and SJp] is the entropy of noninteracting single particles. The ordinary electrostatic interaction has been separated out, and the remainder of Q lumped into Qxc; which is the exchange and correlation contribution to the ground potential (this, however, includes kinetic energy and entropy contributions). The combination of kinetic energy and entropy terms represents the Gibbs free energy of a system of noninteracting fermions: (7.19) G s b l = F J P l - 73s[Pl. We then invoke the stationary property of Eq. (7.18) and find on varying [we do not need the condition of Eq. (7.2) because of the presence of p in Eq. (7.1811,

in which V:,[Pl

=

~Qxc[p1/6p(r)

(7.21)

is a temperature-dependent exchange and correlation potential. We see that compared to Eq. (7.4), the most obvious change is the presence of the freeenergy term G, where r was present before. It is important to realize that Eq. (7.20) is formally the same as would be obtained for a system of noninteracting particles in an external potential equal to the effective potential V,,,: (7.22) where V,,,

=

u(r)

+ e2

~

p(r') d3r'

Ir

- r'I

+ VL.(p(r)).

(7.23)

In the case of a system of noninteracting particles, the grand potential is 52

=

-k,T

1ln(1 + e-pf'i-p)), i

(7.24)


161

in which ti is the energy of the ith state. The occupation probabilities, here denoted n,, can be obtained from the relation between the number ofparticles and the grand potential

(g)o,T= 1 m

N = -

n,.

(7.25)

i= 1

We shall see in Section (10) that n, is expressed by the usual Ferm-Dirac formula [cf. Eq. (3.1)]: n, = {exp[(c, - p)/k,T]

+ 1}-'.

(7.26)

Likewise, the entropy of this system is =

-k,

1 [n, In n, + (1

-

n,) ln(1 - n,)].

(7.27)

i

P.0

We can now proceed to derive single-particle equations. In most circumstances, we are concerned with a system in which the number N of particles is fixed. In this case, it is appropriate to vary the Helmholtz free-energy A[p]: (7.28) 4 p l = QCPl + " An infinite set of single-particle functions u, is introduced. The generalization of Eq. (7.7) is rn

(7.29) i

where ni is given by Eq. (7.26). We can now substitute Eq. (7.18) into Eq, (7.28) and carry out the variation as before, noting that the entropy does not depend explicitly on the u,. The result is quite similar to Eq. (7.9): (7.30)

These equations must be solved self-consistently. Some specific approximation to V:, is, of course, required. Given this, the charge density can be reconstructed by using Eqs. (7.26) and (7.29) after an iteration, and an iterative process invoked that leads to self-consistency. We now wish to comment on the key steps in the derivation of the zerotemperature equations; specifically, Eqs. (7.7) and (7.8). Problems arise when we compare these equations with known results for the one-body reduceddensity matrix pl. We are only concerned here with systems of N fermions in a definite quantum state whose wave function is Y at T = 0 K. Then, in a position representation, the elements of p 1 are

r

P ~ ( T ' r) , =

N

J W',

r2, . . ., rN)\Y*(r,r2, . . ., rN)d 3 r 2 ,. . ., d3r,.

(7.31)

162


There is an exact expression for p1in terms of orthonormal functions, called natural orbitals, introduced by L ~ w d i nand , ~ ~here denoted : a,

(7.32) Note the following differences in comparing of Eqs. (7.32) and (7.7): The sum in Eq. (7.32) involves an infinite set of functions (instead of N ) , and occupation numbers ni are present (ni=< 1). In general, it is not possible exactly to express p1 for r # r’ in terms of a finite set of single-particle functions. An expression in terms of N such functions is possible only if the many-body wave function is a single Slater determinant. The kinetic energy functional has a simple form in terms of natural orbitals: S[p]

=

s

[V, V,!pl(r’,

mJ-

1 ni

d3r =

&+(r)(-V’)q$(r) d 3 r (7.33)

i= I

Again, the sum involves an infinite set of functions. However, although use of natural orbitals appears at first glance to give an expression close to (but not actually the same as) that desired for the derivation of single-particle equations, it has not yet been established that the natural orbitals are eigenfunctions of any relatively simple Hamiltonian. We conclude this discussion by showing that in spite of the previous remarks, a representation of the charge density by the sum of squares of N functions in accord with Eq. (7.7) is always possible. The proof involves the explicit construction of one such set of functions which is sufficient to establish an existence theorem. The proof is that of H a ~ - r i m a nOur . ~ ~demonstration is limited to one dimension; however, the extension of three dimensions is quite straightforward. Consider a charge density defined on the one-dimensional interval x 1 5 x 6 x2 and satisfying

j’

p ( x ) dx = N .

(7.34)

Since p(x) 2 0, we can define a real function

and adopt the convention that p ( x ) >= 0 throughout. [This convention is not necessary, and we may equally well allow p ( x ) to change sign at each zero

35

P. 0 . Lowdin, Pliys. Rep. 97, 1974 (1955).

’‘J . E. Harriman, Phys. Rev. A [3] 24,680 (1981).


163

of p ( x ) . ] In addition, we define a new function f ( x ) by (7.36) which obeys df/dx = (2n/n)p(x). We now introduce orbitals Wk(X)

(7.37)

= n - 1 /2p(x)@X),

where k is some integer, k = 0, +2, . . ., and we take These functions are orthonormal:

11

different values of k.

Further, we obviously have (7.39) This completes the proof of existence. The functions wk introduced here are complex. This is, however, not necessary because a set of real functions can be found if desired. However, the existence of functions satisfying Eq. (7.7) does not solve similar problems associated with the kinetic energy..For this purpose, it is required that there exist a set of functions ui which satisfy Ref. (5) and such that s[p]

=

J C

u*(r)(-V2)ui(r) d3r.

(7.40)

i=l

The combination of Eqs. (7.7) and (7.40) imposes more stringent conclusions on the functions than Eq. (7.7) alone, since the derivatives of the functions are constrained as well. The existence of a set of n functions obeying both Eq. (7.7) and Eq. (7.40) has not (to our knowledge) been proved.

8. THEEXCHANGE-CORRELATION FUNCTIONAL

The exchange-correlation functional plays an essential role in density functional theory, but at first sight it may appear to be somewhat mysterious.

164


It is therefore of interest to give an interpretable expression for it and to relate it to other quantities which appear in many-body t h e ~ r y . ~ ~ - ~ ’ Consider the Hamiltonian for a system of N electrons, H = T + V’, where T is the kinetic energy, and V contains both the interaction with an external potential as well as the electron-electron interaction. We have to allow the strength of the electron-electron interaction in V‘ to vary. This is accomplished by replacing e2 by Ae’, where 2 can vary between 0 and 1. In addition, we suppose that the external potential (now denoted by ud) can be chosen to depend on A in such a way that the particle density in the system is independent of 1. (We are not aware of a specific proof that such a va always exists). Let P(r) = Q+(r)t,b(r) be the electron-density operator whose expectation value is p(r). Then the potential part of the Hamiltonian is V; = le2

s

1 d 3 r d3r‘ ____ fi(r)[fi(r’)- 6(r - r’)] ) r - r‘(

+

s

d 3 r oa(r)fi(r). (8.1)

Let the ground-state energy of the Hamiltonian with v; be E,, and we denote expectation values in the ground state for a given 2 by ( Because the ground state for any iis stationary, a change in the wave functions affects the energy only in second order. Therefore the derivative of E , with respect to ican be calculated as the expectation value of the derivative of the Hamiltonian with respect to Hence,

We have used the fact that = p by assumption. We may integrate Eq. (8.2) over A from 0 to 1. When I = 1, we have the ground-state energy of the system of interest. At I = 0, the energy is just the free-particle kinetic energy which we write as ,Fs[p]. Hence,

E

=

%[p]

+ Jol

x (B(r)[P(r’)-

dl

-

r’)l)n.

(8.3)

J. Harris and R. 0. Jones, J . Pliys. F 4 , 1170 (1974). 0. Gunnarsson and B. 1. Lundqvist, Phvs. Rev. B. 13, 4274 l(1976). 39 D. C. Langreth and J . P. Perdew, Phjs. Re!>.B. 14, 2884 (1977). 39a This statement is equivalent to the Hellman-Feynman theorem which applies in density functional theory. For an explicit discussion, see J . C. Slater, J . Chrm. Phvs. 57,2389 (1972). 37

38


165

We have to compare this Eq. (8.3) with Eq. (7.1)where, as discussed in Section 7, the kinetic energy contribution is Ys. We then find that E&] is given by the difference between the last term in Eq. (8.3) and the ordinary electrostatic interaction. Note that (P(r)P(r’))2

- p(r)p(r’)= (“r)

x {(CP(r)

-

p(r)l”r)

- p(r’)I)2

- p(r)l[P(r) - ml), - p(r)

w - r’)>.

(8.4)

(8.5)

We define the pair correlation function q,(r, r’) which refers to a situation in which the electron-electron interaction has been reduced by a factor A by P(r)P(r)gA(r>r’) = c a ( r , P v ) > A - P(r) 6(r - rl).

(8.6)

Equation (8.4) enables us to cast Eq. (8.6)in the alternative forms: Ex&]

=e2

2

J d3r d3r‘TF!$y ~

Jol dA(qL(r,r’) - 1)

(8.7)

In Eq. (8.8) we have introduced a function ph which is called the exchangecorrelation hole. The definition is pdr, r’) = p ( ~ ‘ ) Wr‘), ,

(8.9a)

where h(r, r‘) is a symmetric function defined by n r

h(r, f )

= h(r’, r) =

J

dA[g,(r, r’) - 11.

(8.9b)

OL

The symmetry follows from Eq. (8.6). It also follows that

s

ph(r, r’) d3r‘ = - 1.

(8.9~)

The formula states that the exchange-correlation hole corresponds to the removal of one electron. The exchange-correlation functional can now be interpreted pictorially as the change in the electrostatic energy of the charge distribution produced by the presence of this “hole” around each electron, or equivalently, as the interaction energy between the normal density distribution and the hole. A formal expression for Q x c , the exchange-correlation

166


contribution to the grand potential of Section 6, has been given by Fetter and Wale~ka.~’ a. Local Density Approximations to Exchange-Correlation Potential

To implement the previous formally exact one-particle scheme, it is obviously necessary to make approximations to the exchange-correction contribution to the potential energy, namely VJr). Returning to the discussion of the Thomas-Fermi-Dirac theory, the simplest possible choice is the Dirac-Slater potential in Eq. (2.7). Many variations on this have been used, and it is not fruitful here to discuss these at length. At the level of a local density approximation, the obvious generalization of the DiracSlater potential, which used the exchange energy density of a uniform electron gas locally, is still to utilize the uniform electron fluid as a basis, but to account for correlations. This would then allow one to write for the exchange-correlation energy Ex, the local density approximation I-

(8.10a) or the local spin density approximation (8.lob) where cfc( p) is the exchange-correlation energy density for the uniform interacting electron assembly of density p. This problem is reviewed by Singwi and Tosi41 Many fairly accurate approximations to tfc(p(r))are now available, and it is not our purpose here to go into detail. We mention just three of those that have proved useful, although many more sophisticated fitting procedures for jellium now exist:

(1) W i g n e r ’ ~interpolation ~~ formula fits between the high-density result and the correlation energy of the extreme low-density Wigner electron crystal. (2) Nozieres and Pines’43 formula includes the many-electron highdensity correlation energy of Gell-Mann and B r ~ e c k n e r which , ~ ~ has the A. L. Fetter and J . D. Walecka, “Quantum Theory of Many Body Systems,” p. 252. McGrawHill. New York. 1971. 4 ’ K . S. Singwi and M. P. Tosi. S o / i d S / a f ePhjs. 36, I (1981). 4 2 E. P. Wigner. Phjs. Re(:.46, 1002 (1934); Truns. Furuday SOC.34,678 (1938). 43 P. Nozieres and D. Pines. Phvs. R e f . 111,442 (1958). 4 4 M. Cell-Mann and K . A. Brueckner. Phys. Rev. 106,364 (1957). 4”


167

form A In r, + B, where A and B are known constants, whereas r , is the mean interelectronic spacing. (3) Kim and Gordon’s45 numerical interpolation curve. For present purposes, we can say that &( p) is known to useful accuracy, and we take this over into the local density inhomogeneous gas. This is not to say that this represents the “best possible” local density approximation. However, it is clearly well suited to treating electronic charge clouds where the electron density varies by but a small fraction of itself over a de Broglie wavelength of a characteristic electron. This condition is reasonably well obeyed in the conduction band of the nearly free electron metals, say, Na and K. When there are strong density gradients, due to directional chemical bonding, for example, much greater care is called for in using Eqs. (8.10a) or (8.10b) as a starting approximation. We shall discuss ways of transcending the approximation Eqs. (8.10a) and (8.10b) for the exchange correlation potential in Part IV; however, we wish to mention here some work on the construction of these potentials. Von Barth and Hedin2’ developed a parametrized exchange correlation potential which allows for spin polarization and which has been widely employed. Other authors, such as Gunnarson and L u n d q v i ~ and t ~ ~ R a j a g ~ p a have l~~ developed potentials with the same functional form but different parametrizations. Experience with regard to applications, as discussed briefly in Part VI, indicates that the differences between these potentials are not large. We shall discuss here a local spin density approximation potential proposed by Painter.46 The fact that progress here is possible stems from the Monte Carlo computer simulation of jellium by Ceperley and Alder.47These workers obtained accurate correlation energies for the para- and ferroto magnetic states of jellium, and these were employed by Vosko et produce a new correlation energy density of increased accuracy and proper limiting behavior in the metallic density regime (r, 5 6 Bohr radii). In Painter’s work, the correlation potential in the local spin density approximation is derived from the correlation energy density given by Vosko et al. Suppose we have n , electrons with positive spin and, n- with negative spin. Introducing the relative magnetization [ = ( n , - n-)/n, the quantity calculated by Vosko et al., Ac,(r,, [), is defined by

Vosko et al. calculated Acc(rs,[) in the random phase approximation, this 45

46 47 4x

R.G . Gordon and Y . S. Kim. J . Chem. Phys. 56,3122 (1972). G. S. Painter, Phys. Rcc;. B . 24, 4264 (1981). D. M . Ceperley and B. J . Alder, Phys. Rcc, L e f t .45, 566 (1980). S. H . Vosko, L. Wilk, and M. Nusair, Cun. J . Phys. 58, 1200 (1980)

168


being the correction to the paramagnetic correlation energy density cz(rs). They introduced an interpolation expression for obtaining the correlation energy spin dependence. Carrying over this same representation for the local spin density approximation, the polarization dependence of the correlation energy density becomes where Blks) =

f(r)

=

Cf”(0)/%trs)lA-Ec(rs, 1) - 1,

[(i

+ 1)4/3+ (1 - 1 p 3 - 21/2(21/3 - 11,

(8.13) (8.14)

and CI, is the spin stiffness which is also parameterized by Painter (as is E:). Following Painter and putting x = r:/’, the correlation potential for an electron of given spin can then be written as

where F denotes the ferromagnetic state (t: is parametrized by Painter). Painter has applied this formula to hydrogen in atomic and molecular form. This is obviously a very difficult test of such local density methods which were originally designed to treat systems in the statistical limit- of many electrons (cf. Section 1). Painter emphasizes that since his modified treatment of correlation primarily affects the region of small r s , which is dominated by exchange, correlation corrections are small compared with errors in the exchange energy. Thus in light atoms the improved correlation model leads to a reduced cancellation of error between exchange and correlation, emphasizing the need for improved treatment of exchange. For more homogeneous systems, Painter emphasizes that his work should offer real improvement over existing procedures. The way in which these local density approximations based on correlations in jellium can be transcended is discussed in Part IV of this article. A temperature-dependent exchange correlation potential has been derived by Gupta and R a j a g ~ p aaccording l~~ to Eq. (7.30) from a calculation of the

‘’U. Gupta and A. K. Rajagopal, Phys. Reu. A [3] 21,2064 (1980); 22,2792 (1980); Phys. Rep. 87, 259 (1982).


169

ground potential of an interacting electron gas. Exchange is treated exactly, whereas correlation is included in a “ring-diagram approximation,” similar to that employed by von Barth and Hedin2’ in the zero-temperature problem. Relativistic exchange potentials, including both Coulomb and transverse photon interactions have been derived by Rajagopalso and by MacDonald and V o ~ k o . ~A’ relativistic correlation potential was obtained by Ramana and R a j a g ~ p a l . Correlation ’~ tends to be significantly smaller than exchange in the high-density situations in which relativistic effects may be important.

9. SIGNIFICANCE OF THE ONE-ELECTRON EIGENVALUES An important and somewhat controversial question concerns the significance of the single-particle eigenvalue in the local density method. In the case of the Hartree-Fock approximation, Koopmann’s theorem shows that the eigenvalue ci associated with a single-particle function ui is the negative of the energy required to remove an electron occupying uir provided that relaxation can be neglected. The eigenvalue difference c j - ci correspondingly gives the energy of the transition from i toj, again neglecting relaxation. These results do not apply in density functional theory; instead, one has according to Slaters3 and Janak54 the relation ti

= aE,/ani

(9.1)-

in which E , is the total energy of the system, and niis the occupation number of ui.Although plausible in appearance, this relation seems to raise serious conceptual problems in that we are accustomed to regarding the occupation number of the local density theory as either zero or unity. The purpose of Section 9 is to derive Eq. (9.1) in a fashion that makes its significance clear. Our conclusion is that Eq. (9.1) applies to an infinite system in a statistical sense (or one may consider an ensemble of similar systems).54a Our discussion is based on the finite-temperature extension of the original Hohenberg-Kohn theorem due to M e r m i ~ ~ ,as~ ’discussed previously in Sections 6 and 7 and is closely related to that of Slater.53We begin by considering a large system at some finite temperature T containing closely spaced energy levels. The grand potential R of this system has previously

’”A. K. Rajagopal, 1.P h p C 11,2943

(1978).

A. H . MacDonald and S. H. Vosko, J. Phys. C 12,2977 (1979). 5 2 M. V. Ramana and A. K. Rajagopal, Phys. Rec. A [3] 24, 1689 (1981). 5 3 J. C. Slater, “The Self-consistent Field for Molecules and Solids.” McGraw-Hill, New York, 51

1974.

J. F. Janak, P h ~ i s Rec. . B: Condens. Mutter [3] 18, 7165 (1978). 5 4 a For a different approach, see J . P. Perdew, R. G. Parr, M. Levy, and J . L. Baldur, Phys. Rer. Lerf. 49, 1691 (1982). 54

170


been shown to be a functional of the charge density. We can express the grand potential as 0 = -'.

(9.7) This is the familiar Fermi-Dirac distribution law. The single-particle levels of local density theory are to be occupied in accord with the rules of Fermi statistics with the occupation numbers determined by the single-particle energies t i , Let us now consider the T = 0 limit. In this case,

where Ex, refers to the ground state, and we consider systems with a fixed number of particles. Equation (9.8) holds because any contribution from the entropy to SZ,, vanishes in the limit (and the treatment of the kinetic energy is consistent with regard to 9,and Ex=).Therefore, Eq. (9.5) is formally exact in this case. Of course, it is apparent from Eq. (9.7) that in the T = 0 limit, the occupation numbers niare either zero (if c j > p) or 1 (if t i < p). In this way, the usual form of local density is recovered with fixed occupation numbers, either 0 or 1. The only exception could arise if we have some levels whose energies are exactly equal to the Fermi energy p. Since the Fermi energy is determined by substitution of Eq. (9.7) into Eq. (7.25), there remains the possibility that the occupation number of levels with ti = p might turn out to be different from 1 or zero. Since the occupation number should be the same for all levels of the same energy, fractional occupancy at T = 0 is possible for such levels.

10. THETRANSITION STATE ~~ We have seen in Section 9 that it is quite useful to adopt S l a t e r ' ~point of view that the total energy of a system in certain states should be regarded as a continuous and differentiable function of the occupation numbers of one-electron orbitals. The states involved are those which, as in the band theory of solids, can be reasonably well specified in terms of occupancy of one-electron orbitals. In a finite system they may be averages. For example, in the case of a free atom we are really considering averages over states of various L and S for definite configurations and specified M , . At present, we know how to apply density functional methods only to states (or to averages

172

J . CALLAWAY AND N. H. MARCH

over states) that can be described in terms of specified occupancy of singleparticle orbitals. Here, we continue this line of argument by expanding the total energy for such states in terms of the occupation numbers. Of course, physical systems have occupation numbers of unity or zero, except in the exceptional case in which there is a nontrivial degeneracy exactly at the Fermi energy. However, the expansion is useful as we shall see. Our basic expression for the total energy is Eq. (9.3)in which ni is a continuous variable. In addition, we must understand that the orbitals themselves are functions of the ni because the density appears in the single-particle equation. Also, note that it is not necessary to develop a special notation for the spin-polarized situation: the designation of spin (as or 1) can be incorporated into the orbital index i. Our purpose in Section 10 is to develop a reasonable approximation in which total energy differences between physical states of a system (with unit differences of occupation numbers) may be related to eigenvalues from a single-particle equation. Suppose then that we start from a reference state of a system in which the occupation numbers are n!’), and we use the notation dn, = ni - .Io), where ni is the variable occupation number. We thus expand ET as

(10.1)

We have included terms through third order in n,. Slaters3 discusses the numerical estimation of these derivatives in some detail in the case for which the X u exchange correlation potential is useful. Use of this expansion may take us beyond the consideration of ground-state properties for which the local density method was derived. This extension is discussed to some extent later in Section 10 and in more detail in Section 11; for the present, we simply use Eq. (10.1). Let us consider other differences in total energy between two states. In A, one electron occuplies the single-particle state j, whereas k is empty; in B, j is empty and k is occuplied; and all other occupation numbers are unchanged. We choose A as the reference state and calculate the energy of B according to Eq. (10.1):


173

The notation indicates that all quantities refer to state A. We see from this that although the difference of single-particle eigenvalues appears, there are obvious corrections of second, third, and, of course, higher order. In practice, these corrections seem to be important. Now let us consider a different choice of reference state, the so-called transition state (denoted T) in which both single-particle states j and k have occupancies 1/2. This transition state is, obviously, not physically realizable, but it is a useful mathematical construct. The energy difference EB - EA is now recalculated by using T as a reference state. The result is now

EB - EA

= €k(T) -

Ej(T)

Now all eigenvalues and derivatives refer to the transition state T. Comparison of Eqs. (10.3) and (10.2)shows that the second-derivative terms have disappeared from Eq. (10.3) and that the coefficient multiplying the thirdderivative terms has been reduced by a factor of 1/4. In other words, the energy difference between states A and B ought to be reasonably well approximated by the difference of single-particle eigenvalues in the fictitious transition state. Experience shows that t h s is the case. The transition-state procedure seems to be most interesting for finite systems. It is of course possible for the single-particle state k to be a continuum state. In this way we see that the ionization energy of an atom or molecule is given by the (negative of the) eigenvalue of the uppermost single-particle level in a transition state in which that level is half occupied. Likewise, the electron affinity is determined by the eigenvalue of the level into which the electron is added in'a (different) transition state in which the new level is half occupied. The arguments concerning ionization energies and electron affinities can be interpreted as relating the ground-state energies of systems with N - 1, N, and, N 1 electrons, respectively. As such, they fall within the conceptual range of ordinary density functional theories which are restricted to ground states. However, the application of the transition-state approach to the discussion of excitations within an N-particle system seems to be an extension of these methods without formal justification but also without any obvious change in the mathematical development. In Section 11, we discuss another argument that places the transition state on a firmer foundation. Before proceeding, a further remark about the transition state is useful. The single-particle eigenvalues in the transition state will be different from those in either the ground or the excited states for finite systems (atoms, molecules, clusters, localized states in solids) for which the Coulomb and

+

I74


exchange correlation potentials are altered when the transition occupation numbers are introduced. However, it appears that the eigenvalues of infinitely extended states, for example, band states in solids, are not altered because the alteration of the density produced by the change of occupation numbers is of order 1/N and hence insignificant for sufficiently large N . 1 1 . EXCITED STATES One of the most important and controversial questions in density functional theory concerns the extent to which excited states can be studied by these methods. We have already encountered this problem in our discussion of the transition state. From the most general point of view, the Hamiltonian, and hence all of its eigenstates, is specified in principle if the ground-state density is known: the energies of all states are functionals of the groundstate density. However, we do not at this time know how to extract useful information from such a general statement. Our naive hope would be that the energy of an arbitrary state of a system would be determined by the charge density in that state. This cannot be proven (except in circumstances discussed later), and is probably false. The difficulty is that any variational treatment of an excited state requires that it be orthogonal to lower states, so that in a sense all lower states must be known in order to determine the particular excited state. The exception noted earlier arises in those cases where the orthogonality problem has a trivial solution by reasons of symmetry. If a particular excited state is orthogonal by reasons of symmetry to all lower lying states, the density functional formalism will give the correct value for its energy (of Gunnarsson and Lundqvist3*).This follows in a rather straightforward manner because all we really require of the general variational theorem of quantum mechanics to make the density variation work is that convergence to the state of interest occur, and this is guaranteed if the state of interest is orthogonal to all lower states. This type of excited states is, in fact, quite important-we can frequently distinguish low-lying excited states in solids, for example, by the (total) wave vectors. This argument has the interesting and important implication that the energy functional can depend on the conserved quantum numbers which describe the states of the system. As an example, in the case of an atom for which L-S coupling is applicable, the functional can depend on the eigenvalues of L2, S2, L,, S , , total parity II, etc. In the case of a solid, the exact functional may depend on the total wave vector. However, our present knowledge of the functional is quite limited and is based on results for the


175

ground state of a free-electron gas with a uniform background of positive charge. We do not know how to obtain functionals that depend on these quantum numbers, and in practice we are reduced to using a single approximate functional in all cases. An exception exists in spin-polarized systems: we do allow the functional to depend on the degree of spin polarization or, in other words, on the total S z . An additional problem arises when we consider deriving a single-particle equation from which to construct charge and spin densities for states of this type (cf. Kohn and Vashishtaj'). One really needs to constrain the variational process so as not to go outside the subspace of states of the required symmetry. The charge and spin densities considered should be those which can be derived from many-body wave functions of the correct symmetry. We can consider this the s- (for symmetry) representability problem. Unfortunately, we do not know how to do this. In practice, one is guided by analogy with Hartree-Fock theory; i.e., the single-particle functions used in constructing densities are taken to have the same symmetry as one would use in constructing a Hartree-Fock wave function of the same symmetry. We shall discuss an example of this calculation in Section 12. A different attack on the excited-state problem has been made by T h e ~ p h i l o uwhich ~ ~ succeeds in relating the rather intuitive approach of Slater's transition state to more fundamental principles. We shall rephrase the essential aspects of the argument of Theophilou here. The original paper should be consulted if a less condensed treatment is desired. The basic claim is that if one wishes to discuss the M lowest energy eigenstates, it is necessary to consider a subspace of Fock space (denoted by a subscript S) spanned by the M-state vectors of lowest energy. In such a subspace there is a mathematical object which is determined. independently of the particular basis in this space: It is the trace of an operator. It turns out to be possible to say that the trace of the Hamiltonian matrix, which is of course the sum of the eigenvalues for the exact states, is specified in principle by (or is a unique functional of) the trace of the density operator (not the density matrix) in this space. , d1 is Let the M lowest eigenstates of H be denoted by 41,. . ., 4 M where the ground state. The space of interest is the subspace spanned by these functions. We are using a second quantized (Fock space) formalism in which the density operator j3 is (11.1) where $+and $ are fermion field operators. The Hamiltonian of the system 55

A. K . Theophilou, J . Phys. C 12,5419 (1979)

176


is supposed to have the form

H

=

Ho

+

s

$(r)V(r) d3r,

(11.2)

where V(r) is a local external potential and

The operation of taking the trace shall refer, unless otherwise specified to matrix elements on vectors in the subspace S: (11.4)

where the Ei are the exact energies of the many-body states $i. The operation of taking the trace refers (as in Section 6) to the Fock space and not to coordinate space. By applying the trace operation to Eq. (11.2), we may therefore write tr,(H) = tr,(Ho)

+

s

trs(p(r))V(r)d3r.

(11.5)

We shall denote N-' tr,@) as ps or as p ( r ) . It is an ordinary function of position. We may now proceed exactly as in the proof of the original argument of the Hohenberg-Kohn theorem (Section 7) to decide that tr,(H) is a functional of the density p ( r ) and write (11.6) Hence the sum of the first N eigenvalues is a functional of the trace of .the density operator. We need a minimum property of the trace so that we can ultimately find some one-particle equation. This is provided by the following remark. The object tr,(H) attains its minimum value if the subspace S' coincides with the actual space S spanned by the exact eigenstates. The argument is quite simple: If S' differs from S, it must contain vectors which have components on exact eigenstates which have higher energy that those in S. It is, of course, necessary for this argument that S' and S have the same dimensionality. Thus, tr,,(H)

2 tr,(H).

(11.7)

Hence the functional E&&] attains its minimum value when S' coincides with S (and therefore pk with ps). It must be stressed here that the functional E, is not the functional E , of the ground-state calculation. The connection with Slater's transition-state approach arises in the fol-


177

lowing way. Suppose we have a system of N electrons, and for simplicity, let n = 2; i.e., we consider the ground state and the first excited state. Then we fit the quantity p by the sum of squares of N 1 normalized one-electron functions

+

1

N-1

C

Iui(r)J2+

1

2[IuN(r)12 + luN+

(11.8) I(Y)~’]. 2 i= 1 The functions u are different from those involved in the ground state. It is evident from Eq. (11.8) that this is just Slater’s transition-state density. Of course, it integrates to the correct number N of electrons. We may now proceed in principle to use the variational property of tr,(H) to find a set of single-particle equations satisfied by the ui.Unfortunately, we know much less about the functional E J p ] when excited states are included than we do about the ground state. In fact, for practical purposes, we are reduced to using at this time the same functional that we use for the ground state. In principle, we may proceed as follows. We solve the Hohenberg-KohnSham problem for the ground-state energy E , . Then we determine the transition-state density [from Eq. ( 1 1.8)] that minimizes the functional E s [ p ] (for the case n = 2). This gives E, + E,. Then E , is found by subtraction. This process is continued until the desired number of excited states has been found. Unfortunately, our knowledge of the functionals involved is adequate only for crude approximations. Slater’s approach corresponds to a particular guess as to the functional. An alternative approach to these ideas has been given by KatrielS6 He proposes to consider the N-particle Hamiltonian as the Hamiltonian of a single superparticle with 3N degrees of freedom. If we are interested in M excited states of the system, we cpnsider a sum of M such superparticle Hamiltonians with no interaction among them. The lowest completely antisymmetric state of this M-superparticle system corresponds to the occupancy of the M lowest states of the original Hamiltonian. The Hohenberg-Kohn theorem can be applied to states of definite symmetry; in this case to antisymmetric states, as discussed earlier. The energy of the lowest state of this symmetry is the sum of the energies of the M states mentioned; the density is likewise the sum of the densities of these states. -

tr(P) = ~ ( r=)

12. THEATOMIC MULTIPLET PROBLEM Some of the limitations of our current understanding of how to apply density functional theory come into view clearly when we consider the 56

J . Katriel, J . Phys. C 13, L375 (1980)

178


calculation of multiplet structure in atoms following Ziegler et a/.,57von Barth,s8 and Lannoo rt aLS9In an atom, the conserved quantum numbers include, if spin-orbit coupling is neglected, the total angular momentum L and its z component M , the total spin S and z component M,,and the parity n. The exchange-correlation functional ought to depend on these quantum numbers. However, our knowledge of the functional is based on a freeelectron gas for which Ex, is an ordinary function of charge and spin densities only. We do not know how Ex, depends on the other quantities relevant to the description of an atom. A procedure which leads, at least in some cases, to a solution of this difficulty has been suggested in the papers just cited. It is of interest here insofar as it illustrates how the original ideas of density functional theory can be extended with reasonable success to calculations beyond the apparent range of validity of the methods as stated by the fundamental theorems. Our previous discussions indicate a method of calculating the energies of states of a system in the density functional approach which correspond in the Hartree-Fock approximation to single Slater determinants. The occupation numbers of the Hartree-Fock wave function are used to define charge and spin densities in term of combinations of the orbitals of the density functional methods. However, single Slater determinants need not correspond to states of definite values of the conserved quantum numbers, L, S, . .., etc. We shall use the term “properly symmetric states” to refer to simultaneous eigenstates of H,L2, Sz, L,, S,, and n. If a single Slater determinant is not a properly symmetric state, it can still be expressed as a linear combination of these states. Then the expectation value of the Hamiltonian with that determinantal function corresponds to some weighted average of the energies of properly symmetrized states. It may be possible to construct enough different single determinantal functions whose average energies correspond to combinations with differing weights so that it is possible to solve for the energies of properly symmetric states in terms of the average energies of single determinants. This procedure, which has been described in relation to the Hartree-Fock approximation, can be carried over into density functional theory. In this case, one simply substitutes for the average energies of single determinants the energies of the corresponding configuration calculated by density functional methods, and the configuration is defined by the same set of occupation numbers as the determinantal functions. It is assumed that the same relation between energies of states of T. Ziegler, A. Rauk, and E. J . Barends, Theor. Chirn. ACIU43,261 (1977). U.von Barth, Phys. Reo. A [3] 20, 1693 (1979). 5 9 M . Lannoo, G . A . Baraff, and M. Schluter, Phjxs. Rev. B . 24, 943 (1981). 57

5M


179

definite configurations and properly symmetric states holds in density functional theory as in the Hartree-Fock approximation. The energies of the properly symmetric states in t h s approximation can then be determined. It must be observed that the procedure does not have a firm foundation in the fundamental principles of density functional theory. Instead, we have a rather intuitive application of the ideas presented in our previous discussion: that the energy of a state in the density functional method is a function of the occupation numbers and that for a given set of occupation numbers, the energy can be computed from the charge density for those occupation numbers. This approach is, in principle, oversimplified. However, some practical success has been obtained as we shall see. It is useful to illustrate these ideas in relation to a specific example. Let us consider the 2lS and 23S states of the helium atom. In the case of the 23S state, we can construct single determinantal wave functions for the states with M , = f 1 but not for k, = 0. We use the symbol ISM,) to denote the wave functions. The function for the state with M , = 1 can be written as (12.1)

in which u is an up spinor ( p is used to represent a down spinor); u l s and u2, are the single-particle spatial wave functions for the 1s and 2s levels. The function for the 110) state is more complicated: 110) = +ru1s(1)u2s(2)- ~2,(1)~ls(2)IC.(l)P(2) + P(l)u(2)1.

(12.2)

The application of density functional theory to the 23S state appears to be justified in principle because it is the lowest or ground state of the triplet series. However, it is a degenerate ground state, and we can see that there is a difficulty: The spin densities in this approximation associated with 111) and 110) are different. In the usual formulation of density functional theory, we would therefore obtain different energies for (lo) and 111). However, application of the ideas presented by Levy2’ in a different context indicates that we should be justified in choosing the densities associated with the ( 1 1 ) state. The wave function for the 2lS state, denoted by (OO), is in the same sort of approximation:

again not a single determinant. We ignore the possible difference of the single-particle orbitals in Eqs. (12.2) and (12.3). The single-particle spin for both density is in fact the same as for the 110) state, t[luls(’

+Iu~,~~]

180


spin up and spin down. Hence a straightforward application of density functional theory would give 100) the same energy as IlO), which is incorrect. \ However, application of density functional theory to this state is not justified in principle because it is an excited state in a situation where there is a lower state of the same symmetry-the normal ground state 1’s. Moreover, in the present state of knowledge of the exchange-correlation functional, we have difficulty distinguishing among states of differing S but identical M , . The way out of this problem, as was discussed previously, is through the construction of a single determinantal function with M , = 0 but with no definite value of S. We call this the “mixed” state M and denote it by I MO) :

=

1

fi

~

[IlO)

+ loo)].

(12.4)

The average energy of this state can be calculated in the Hartree-Fock approximation. We denote this as E ( M ) :

+

(12.5)

E(2lS) = 2E(M) - E(23S).

(12.6)

E ( M ) = ( M O I H I M O= ) $[~(23s) E ( ~ ~ s ) I , in which the energy of the triplet state is ~ ( 2 3 s= ) (iippi) Hence we have the relation

In the density functional method, the state 111) is characterized by having one electron of spin up in both the 1s and 2s states, whereas the mixedstate has a single electron of spin up in the 1s state and an electron of spin down in the 2s state. We can calculate the energies of these states through a selfconsistent solution of the single-particle equations. The results can then be substituted into Eq. (12.6) to find E(2lS). Of course, it must be observed that no formal justification has been presented for studying the mixed state according to density functional theory: if does not correspond to any single eigenstate of the many-electron Hamiltonian. The calculation just described was actually performed by von Barth.58 He found that the energy of the triplet state in the local density approximation is 19.08 eV (measured with respect to the true ground state 1’s). The experimental value is 19.82 eV. Von Barth then found a self-consistent solution for the mixed state. This turns out to be 19.56 eV above the ground state. The energy of the singlet state according to Eq. (9.6) is found to be 20.04 eV, and the singlet-triplet splitting is 0.96 eV. The corresponding experimental


18 1

results are 20.62 and 0.82 eV. These local density values are an improvement over the Hartree-Fock results, which are 18.71 and 18.83 eV for the triplet and singlet states, respectively. Use of different orbitals for the Is and 2s states in the Hartree-Fock calculation does not improve these results significantly (in this case they are 18.70 and 18.82 eV). It would be possible to reformulate this calculation in a fashion which, although approximate, is more consistent with our previous discussion of excited states. The first step is a calculation of the energy of the 1’s ground state E,, which is essential in either this or the previous approach. The artificial mixed state I M ) is regarded now as the first excited state of the system, which is describable by a single Slater determinant and is not orthogonal to the ground state by reason of symmetry. So we attempt to describe this in the manner of T h e ~ p h i l o uby~ ~writing one-half of the sum of the densities of the ground state and I M ) as p(r)=

IUlSt

l2

+ +[lUlS,

I2

+ U2s1 I”.

(12.7)

The functions u l s need not be the same as those in the ground-state calculation. The single-particle equation can be solved with this density from E ( M ) , then subtract E,, and substitute the result which we compute E , into Eq. (12.6). Of course, the calculation for E(23S)also has to be performed. It would be quite interesting to compare the results of this approach with that of von Barth.58 Incidentally, we note that Eq. (12.7) is a density of the transition-state type, so it would be expected that

+

E(M) - E,

62sl

- 6lsl.

(12.8)

A discussion of helium, for example, serves to illustrate the problems of multiplet-structure calculations in density functional theory. Von Barth58 applied this method to a calculation of the energies of the ’P, ‘D, and ‘S states of the ground configuration of the carbon and silicon atoms, and of the 4S, 2D, and 2P states of nitrogen. The results are in all cases an improvement over the Hartree-Fock values. It must be mentioned that this procedure does not necessarily lead to a unique expression for the multiplet energies. It is possible to obtain an overdetermined set of equations, leading to consistency relations which may be only approximately satisfied. Lannoo et al.59 extended this procedure (with some modifications which we do not describe) to the multiplets associated with a configuration A:T: in Td symmetry relating to the neutral vacancy in crystalline silicon. In addition, they observed that this procedure could be extended to enable the performance of a kind of configuration interaction calculation. This is possible within the present framework when the matrix elements of the many-body Hamiltonian among functions of the same symmetry belonging to different configurations can be approximated by differences of the average

182


energies of single Slater determinants, which in turn are estimated from corresponding density functional calculations. However, a fundamental basis for configuration interaction within density functional theory is not available at the present time. 13. THESELF-INTERACTION CORRECTION A major source of inaccuracy in local density calculations in probably incomplete cancellation of self-interaction contributions contained in both the electrostatic and the exchange-correlation terms, as discussed by Perdew and Zunger.60To understand the origin of this problem, it is useful to consider the arguments of Slater.’ Slater was concerned with the task of solving the Hartree-Fock equations-a formidable problem even for simple systems at the time. A major source of difficulty in a numerical solution is that the Hartree-Fock Hamiltonian is nonlocal; or we may say that the effective potential is state dependent. Let us look at these equations, which are written in a notation consistent with our usage with regard to the Kohn-Sham equations [Eq. (7.9)]:

uj*(r’)ui(r’)

1

d3r‘ uj(r)

=

tiui(r).

(13.1)

We suppose here that uj is a spinor and that the integration includes summation over spin components. The equation for each electron state involves a slightly different effettive potential in that the term j = i is not included in either sum. In spite of this, the Hartree-Fock Hamiltonian is Hermitian, and the functions ui are orthogonal. The exclusion of i = j simply means that the electron in state i does not interact with itself. Equation (13.1) contains the ordinary and exchange interactions of an electron in state i with all the other electrons. Slater approached Eq. (13.1) in the following way. The term with i = j may be included in both summations since it would, in fact, cancel. The “self-Coulomb and the “self-exchange” terms are exactly equal in magnitude but opposite in sign. Now the electrostatic potential term in the first square bracket includes all electrons and is independent of the state considered. Slater then proceeded to simplify Eq. (13.1)by replacing the actual exchange term by the result for a free electron gas of the same density. The replacement 6”

J . P. Perdew and A. Zunger, Phys. Reo. B : 23,5048 (1981).


183

of the exchange interaction of the Hartree-Fock equation by a local exchange correlation potential (whose form is now substantially modified from that suggested by Slater) gives the single-particle equation of local density theory. But the exact cancellation between self-Coulomb and self-exchange terms has been lost in this replacement, except in the case of the free electron gas-a system where the density is constant. In systems where the density is not constant, we have included the exact electrostatic interaction of the electron considered with all electrons, including itself, but we have in practice made approximations in the exchange-correlation term, leading to imperfect cancellation. Perhaps the most obvious manifestation of this difficulty is in the behavior of the total potential electrostatic plus exchange-correlation in an atom a large distance from the nucleus. Since the electrostatic potential refers to a neutral and (usually) a spherically symmetric charge distribution, t h s potential decays exponentially with distance. Likewise, the exchangecorrelation potential, which typically has a dominant p ‘ I 3 dependence, decays exponentially at large r, although more slowly than the electrostatic potential. On the other hand, physical understanding, the Hartree-Fock equations, and many-body theory as developed by Gunnarson and Lundq v i ~ and t ~ Perdew ~ and Zunger6’ all tell us that the large-distance behavior should be - e 2 / r . Consider as the clearest example the case of a single electron outside a closed-shell core as in an alkali metal. Latter6’ considered this problem in the context of Slater’s p’I3 exchange potential. He suggested that at large distances where the magnitude of the total potential was less than that of - e 2 / r , the latter function should be used instead. This “Latter correction” was adopted by Herman and Skillman62 in their atomic structure calculations. Perdew and Zunger6’ have proposed a less ad hoc self-interaction correction, which not only leads to the correct behavior of the potential at large r, but also has some effect for all values of r. Consider a decomposition of the charge density in the standard pna = way into contributions from individual spin orbitals: p ( r ) = C,, It is required that the exchange correlation energy associated with a single orbital must exactly cancel its self-Coulomb energy. Let the exchange correlation energy associated with spin densities ( p t , p 1 ) be denoted by Ex, [ p t , p i ] , and let the Coulomb energy be V [ p ] .The requirement is then (13.2) U b n u l + E x c C ~ n o 01 , =0

Iu,,,(Y)~~.

62

xno

R. Latter, Phys. Reu. 99, 510 (1955). F. Herman and S. Skillman, “Atomic Structure calculations.” Prentice-Hall, Englewood Cliffs, New Jersey, 1963.

184


for all orbitals. In Eq. (13.2), (13.3) However, this condition is not satisfied by many of the approximate expressions for Ex, in current use. Suppose then that E x , [ p ] is some approximate functional that does not satisfy Eq. (13.2) but seems to be useful otherwise. Then we define a self-interaction corrected form of Ex,which we denote as E$: E:CPt>

P l l = E,,CPt,

P11 -

c (UCPnul + Ex,CP"u7 01).

(13.4)

nu

The effect of this correction on the exchange-correlation potential can be found in a straightforward manner by the variational procedure used in Section 7. Let Vx,be the potential associated with Ex,.The corrected potential will depend on the orbital that is being calculated; that is, [V$(r)Inu will appear in the equation for u,,(r). We find

This potential will lead to the proper behavior at large r. The use of V:: gets favorable effects in atomic calculations. Two important examples are (1) Reasonable values are obtained for the binding energies of some atomic negative ions. The examples given by Perdew and Zunger are H-, 0-, F-, and C - ; the H- and 0-are predicted to be unstable if an uncorrected potential is employed. (2) The eigenvalue of the last bound orbital agrees rather well in many cases with the observed first ionization potential. There are, however, some problems which indicate that the proposed solution is not final. The major difficulties, which are discussed by Perdew and Zunger, are as follows: (1) Because the corrected potential depends on the orbital, the individual orbitals are no longer exactly orthogonal. The situation here is akin to that when the Hartree equations are solved. (2) There is no clear way to apply the correction to the calculation of extended (i.e., Bloch) states in solids without obtaining a trivial answer. The last point is rather vexing and deserves some comment. The selfinteraction correction to the energy eigenvalue of a state extended throughand vanishes if C2 is allowed to become out a large volume C2 is of order


185

infinite. It is true that the number of occupied states grows as Q does so that the total self-interaction correction is of order W3,but this is small compared to the ordinary contributions, which are of order R. Hence we may conclude there is no self-interaction correction in extended systems.62a There is, however, another way to approach this problem which leaves the issue somewhat in doubt. It is possible to adopt a point of view analogous to that of the early Wigner-Seitz calculation of cohesive energies (e.g., Wigner and S e i t ~ by ~ ~the ) cellular method. The energies of states are evaluated by calculations made within a single atomic cell, the various wave vectors entering via the k dependence of the boundary conditions on the surface of the cell. The self-interaction correction could be applied within a single atomic cell, where it would not only be finite, but probably quite significant in many We would like to note here another proposal which eliminates self-interargue in reference to the action effects in a different manner. Stoll et correlation energy that the most important contribution is the interaction of electrons of opposite spins. They propose a correlation-only functional of the form -UP1=

J d3r

P(r)tc(Pt(r),Purl) -

c U

d3r P(r)~c(Pu(r)201,

(13.6)

J

in which tc is the correlation energy per particle in the usual local (spin) density approximation. An exchange contribution must be added to this. The second term in Eq. (13.6) eliminates the self-interaction. The main difficulty here is that the assumption that correlation need be considered only between electrons of antiparallel spin may not be adequate.

IV. Beyond the Local Density Approximation

In Part I of this article we approached the density functional theory via the Thomas-Fermi-Dirac theory. In that theory, as we emphasized, uniform electron gas relations for kinetic and exchange energy density were employed locally. In the full theory set out in Parts I1 and 111, no fundamental

For a different view, see R. A. Heaton, J. G . Harrison, and C. C. Lin, Phys. Rev. B 28, 5992 (1983). 6 3 E. P. Wigner and F. Seitz, Phys. Rev. 46, 509 (1934). 6 3 a The essential problem is that the self-interaction correction is not invariant under a unitary transformation of the orbital basis, as is required by Eq. (7.7). 64 H. Stoll, C. M. E. Pavlidou, and H. Preuss, Theor. Chim.Acta 49, 143 (1978). 62a

186


assumption of this kind has to be made, but in practice, the single-particle kinetic energy is treated exactly by solving single-particle equations. In these equations, the potential energy is of the form of the sum of the Hartree term V,(r) and the exchange-correlation term Vxc(r).In calculating this latter contribution to the potential energy, the most common procedure is again to use the knowledge that we have of the exchange-correlation energy of the uniform interacting electron fluid (see Singwi and Tosi41 for full details) in Eq. (8.10). This procedure, originally proposed by Slater’ as a simplification of the Hartree-Fock method, was fully justified by the work of Kohn and Sham33 in the presence of correlation and removes a major part of the error involved in the Thomas-Fermi approximation to the kinetic energy, although some error remains in the correlation kinetic energy which now enters V&). This error, plus that made in the potential energy of exchange-correlation interactions, will obviously be largest in the presence of strongly varying electron densities in space. The original work on this so-called inhomogeneity correction was done by von Wei~sacker,~’ who was concerned solely with the single-particle kinetic energy. His work allows one to write this in the form

where the coefficient A was simply its value 1 obtained earlier in Part I in the elementary discussion for a single particle. Actually, it was later shown by Kirsnits66 that in the spirit of Eq. (14.1), in which the density is slowly varying, the correct value of R is 1/9. The status of these two choices of 1was clarified by Jones and Young.67 We shall outline their argument in Section 17, but their conclusion, reached by studying the response of a uniform electron gas to a perturbation, was that the linear response function given by the gradient expansion [Eq. (14.1)] agrees with the exact response function in the limit of long wavelength with the Kirsnits choice of 1. For rapidly varying spatial disturbances, however, the von Weizsacker value of I is correct (see Fig. 1). We note that, for some atoms, the convergence of Eq. (IV.l) has been studied numerically by using densities of Hartree-Fock quality by Wang et ~ 1 They . use ~ T4 ~ as given by H o d g e ~ The . ~ ~numerical convergence is fair, but the term T4 is of a magnitude that is too large to allow Eq. (14.1) to answer questions with a high level of chemical accuracy. 65

C. F. von Weizsacker, 2. Phys. 96,431 (1935). (Engl. Trans/.)5,64 (1957). W . Jones and W. H . Young, J . Phys. C 4, 1322 (1971). W. P. Wang, R . G. Parr, D. R. Murphy, and G . A. Henderson, Chem. Phys. Lett. 43,409

’‘D. A. Kirsnits, Sou. Phys.-JETP 6’ 68

(1976). b9

C. H. Hodges, Can. J. Phys. 51, 1428 (1973).


187

14. PARTIAL SUMMATIONS OF GRADIENT SERIES Hohenberg and Kohn” (see also Kohn and Sham33)pointed out that the gradient series [Eq. (14.1)] for the kinetic energy could be partially summed into the form (with h = m = 1):

‘S

+8

-

d3r

s

d3r‘K(r‘, p(r))

where K is a known function recorded in the papers just cited. Kohn and Sham wrote down a formal generalization of Eq. (14.1) for the exchange energy but did not give an explicit form for the corresponding kernel. This can be obtained by using the density matrix perturbation theory of March and Murray” (see also Stoddart and March7’) in conjunction with the density matrix expression for the exchange energy (e = 1) (14.3) The result may be written [cf. Eq. (14.2)for kinetic energy)] as

where B(r’, p(r)) is given by Stoddart et aL7’ The corresponding exchange potential takes the form [refinding Eq. (2.7)]

(14.5) We emphasize that this argument has been based on the use of a first-order density matrix obtained from a one-body potential V(r), which is designed, ’O

”

’’

N . H . March and A. M . Murray, Proc. R. Soc. London, Ser. A 261, 119 (1961). J . C. Stoddart and N . H . March, Proc. R. Soc. London, Ser. A 299,219 (1967). J . C. Stoddart. A. M. Beattie, and N. H. March, Int. J . Quantum Chem. 4, 35 (1971)

188


in principle (cf. Section 31), to yield the exact particle density. An alternative approach is to construct the density matrix PI@, r’) in Eq. (14.3) by solution of the Hartree-Fock equations. This problem has been treated by Beattie et who show that Eq. (14.5) is regained, although with a different kernel B. One point which follows from both these methods of calculating B is that B(r, k,) falls off slowly with distance, in fact, like Y - ~ . This stems from the fact that the Fourier transform B(k, k,) has a small k expansion of the form B(k, k,)

=

E2(k, k,)[a,

+ a,k2 + a2k2In k + ...I,

(14.6)

where part of B has been represented by a power series. It is the k 2 In k-type singularity at k = 0 that leads to the long range of B(r, k,) in r space. Equation (14.6) means that if we treat exchange and correlation energies separately, neither will have a gradient e ~ p a n s i o n . ’ ~ - ’ ~ When correlations are included, to screen the exchange terms, there is some agreement among workers in the field that B drops off exponentially with distance, although we know of no proof. What this implies is that in k space, B has a power series expansion in integral powers of k2. ~ combined exchange-correlation contributions to Stoddart et ~ 1 . ’ have allow B to be estimated for Be metal. They have compared the local density approximation V,O, to the exchange-correlation potential V,, with the correction calculated from Eq. (14.5)at each point in the unit cell of Be. The maximum correction from the partial summation of the gradient series is > 30% to V:c and usually substantially less. It would be of interest to assess how this correction affects the electronic structure of Be and especially the Fermi surface, but to our knowledge this has not yet been done. 15. LENGTHSCALES

AND

LOCALDENSITY APPROXIMATION

Although the previous approaches have enabled corrections to the local density approximation to be calculated (for example, we referred to the correction to the potential energy in the unit cell of Be metal from the inhomogeneity terms), the underlying physical reasons why one must sum inhomogeneity corrections to all orders are not very clear. Therefore, although connected to the previous arguments, we want to refer now to the A. M. Beattie. J . C. Stoddart, and N . H . March, Proc. R. SOC.London, Ser. A 326,97 (1971). D. J . W. Geldart and M. Rasolt, Phys. Rev. B. 13, 1477 (1976); Phys. Reo. Lett. 35, 1234 (1975). 7 5 A. Sjolander, G. Niklasson, and K. S. Singwi, Phys. Rer. B. 11, l l e (1975). 76 J . C. Stoddart, P. Stoney, N . H . March, and I. B. Ortenburger, Nuouo Cimrnto Soc. I t d . Fis B [ 1 11 23B, 15 ( 1974). 73

74


189

work of Langreth and Meh1,77who have proposed a correction to the local density approximation that appears relatively straightforward to use. Their discussion starts out from a qualitative consideration of length scales. Then, refining Eq. (8.10), they use the form E x , = E:,

+y

s

Z(k,)(Vk,)2dr

=

+IVk,/kf(,

+ . . .,

(15.1)

where k, = ( 3 7 ~ ~ is ~ )the " ~local Fermi wave number and Z is a slowly varying function of k,. The choice y = e2/16z3 makes Z agree with other workers' definitions (e.g.. ref. 68). Langreth and Mehl then focus on a length 5, which measures the scale over which the density varies. They take the definition of 5 from

5-'

(15.2)

thus assuming that there is only one important length scale for a given region of the electronic charge cloud. They point out that for the local density approximation to be valid, one must have k f 5 >> 1,

qTF5

>> 1,

(15.3)

where q7; is the Thomas-Fermi screening length. For systems of interest in the present article, kf and qTF are not very different. Although the inequalities of Eq. (15.3) are not particularly strongly satisfied, it seems reasonable to proceed. For example, at the position of the first Bohr orbit in hydrogen, k, 4. There is an additional implicit assumption to discuss before tackling the second term on the right-hand side of Eq. (15.1). One must consider Ex, as a sum over excitations of varying size, say, il (see Nozieres and Pines43; Hubbard7' for jellium; see Langreth and P e r d e and ~ ~ also ~ Peuckert" for inhomogeneous systems). If we put k = il-', then it is necessary to assume in using the inequalities Eq. (1 5.3) that the typical k which contributes is not very different from k, or qTF. This does seem to be the case in practice (cf. Langreth and Perdew"). Turning to the second term on the right-hand side of Eq. (15.1), one needs to ask whether this will be a good approximation to E x , - E:, = AE. If the N

77

D. C. Langreth and M. J. Mehl, Phys. Rev. Lrtr. 47,446 (1981).

J. Hubbard, Proc. R. SOC.London, Ser. A 243,336 (1957). '' D.C. Langreth and J. P. Perdew, Solid State Commun. 31, 567 (1979). 7R

V. Peuckert, J . Phys. C 9,4173 (1976). D. C. Langreth and J . P. Perdew, Solid State Commun. 17, 581 (1975); Phys. Rev. B. 15, 2884 (1977).

190


assumption of a single 5 in a given region holds, then the inequalities Eq. (15.3) are appropriate to this case also. Why, then, if the local density approximation is good, does not the addition of the first gradient correction in Eq. (15.1) improve the theory? The answer lies in the fact that now, in this second term, although a part of the gradient contribution is distributed normally in k, so that k, is a typical value, a significant fraction is concentrated in a narrow range about k 0. Langreth and P e r d e ~ ~have ~ . ~argued ’ that one must require a stronger inequality, namely, k, >> 40; this results from a detailed numerical calculation. Langreth and Meh177proceed to discuss how to circumvent this stronger inequality requirement while still retaining Eq. (15.3).Let the error AE in the local density correction be decomposed into Fourier components as

-

AE

s:1

=

Z(k,) =

AE(k)dk,

(15.4)

z(k, k,) dk.

(15.5)

A relation such as Eq. (15.4) can also be written for the gradient correction in Eq. (15.1). The true E ( k ) approaches zero quickly as k + 0 when k t becomes less than unity, that is, when 1 becomes greater than (. This is a consequence of sum rules. However, the Fourier decomposition of the gradient correction in Eq. (15.1) does not have this property; i.e., the higher order gradient terms omitted in Eq. (15.1) are essential to restore the correct behavior of A E ( k ) when k t < 1. In contrast, these higher order terms make little contribution when k t > 1, provided that Eq. (15.3) is obeyed. Thus, one can circumvent qtTF> 40 by recognizing that the true A E ( k ) goes to the zero in the region k t < 1. Therefore, Langreth and Mehl propose to write

z(k,k , )

= Z g r a d ( k k,)e(k -

5-’),

(15.6)

and they attempt to restore quantitative reliability to the simple form [Eq. (15.6)] by using t-l

=

3 f l V k f / k , ( = flVP/PI,

(15.7)

where f is a fitting parameter. Actually, the value finally adopted by Langreth and Mehl, namely, f = 0.15, is not very different from the value f = 1/6 implied by the simplest considerations. The quantity zgrad(k,k , ) has been calculated in the random phase approxi-

*’

D. Langreth and J. P. Perdew. Phys. Reu. B . 21, 5469 (1980).

19 1


mation as Zgrad(k

kf 1 = z,(k k f )

+ Z A k , kf

( 1 5.8)

)9

where

2k,z,(k,kf) = -4x0(1 - x)

+ y b ( x - 1) + 6S’(x - l),

x = k/2k,. (15.9)

Although it would be cumbersome to apply the cutoff procedure to z,, it turns out to be unnecessary because z, vanishes as k 0. Thus, the cutoff in Eq. (15.6) is only to be applied to z,. Langreth and Mehl use the fitted form --+

‘

~

(

~

k3 f )

=(43/q1F)

exp(-2fik/qTF).

(15.10)

They then find

z(k, 1 = 2 exp( -2&/q& so that E x , = E:,

+ (4.28 x

s

-

d3rlVy12/p4’3{2e-F- +},

(15.1 1) (15.12)

where F = 0.2621VpI/p7’6.That a term like should appear in the energy density was first deduced by Herman rt aLS3on dimensional grounds. Obviously, since the factor in curly brackets in Eq. ( 1 5.12) is dimensionless, it could not be detected by such an argument.

16. THEONE-PARTICLE GREEN’S FUNCTION AND THE SELF-ENERGY The discussion of excited states according to the ideas of Theophilou” as presented in Section 1 1 has gfeater relevance to finite systems such as atoms, molecules, and clusters than to solids. In the case of an infinite solid, the excited states of interest, such as the situation in a metal in which an electron has been excited above the Fermi surface and a hole has been formed inside it, are not exact eigenstates of the Hamiltonian; rather, they are more like resonances superimposed on a background. The description of this sort of excitation is normally provided through Green’s functions. The /-particle Green’s function is the ground-state expectation value of a time-ordered product of 1 creation and 1 annihilation operators: Gl(l, 2, . . ., I ; l’, 2‘, . . ., 1’) =

x3

(GI T{$(1)$(2). . . $(0$+(1’)$’(2’). . . $ ‘ ( 1 ‘ ) ] [ G ) ,

F. Herman. J . P. van Dyke. and I . B. Ortenburger. Phys. Rev. Lett. 22,807 (1969).

(16.1)

192

J. CALLAWAY A N D N. H . MARCH

in which the numeral 1 denotes coordinates, spin, and time relative to electron 1, and T is the time-ordering operator. Because G, is a ground-state expectation value, it is a functional of the ground-state density. The problem is as always to compute G, and to interpret the results. In particular, the one-particle Green’s function describes the propagation of an extra electron or hole in an N-particle system. The Fourier transform of this function with respect to time has poles at (real) energies of isolated eigenstates and resonances at the positions of approximate eigenstates embedded in a continuum. The two-particle Green’s function describes systems with N - 2, N , or N 2 particles. Our discussion is based on the studies of Sham and KohnE4and Hedin and L ~ n d q v i s t We . ~ ~ shall only be concerned with the energy-dependent single-particle Green’s function (the Fourier transform just mentioned). The computation of this quantity is based on the Dyson equation

+

[E

-

h(r)]G(r, r’, E) -

in which

s

d3r, C(r, rz, E)G(r,, r’, E) = 6(r - r’), (16.2)

h(r)

=

-V2

+ u(r),

(1 6.3)

u(r) is a one-body external potential, if such is present, and C is the selfenergy operator. Like G, it is a functional of the density. It is convenient in the following to separate out of C the ordinary electrostatic potential, thus defining a new operator M , the mass operator,

C(r, r’, E)

=

4(r)S(r - r’) + M(r, r’, E),

(1 6.4)

where @(r)is given by (16.5) The single-particle Green’s function has a biorthogonal expansion in terms of left and right eigenfunctions xn. For notational convenience, the functions are assumed to be discrete. The expression for G is (16.6)

DENSITY FUNCTlONAL METHODS : THEORY AND APPLlCATlONS

I93

Note particularly that the mass operator and therefore the eigenvalues depend on E . One must therefore solve a transcendental equation E

=

c,(E)

(16.8)

to locate the poles of the Green’s function. The energy dependence of the mass operator is a characteristic feature of many-body theory. Approximations are required for practical calculations. Sham and Kohns4 show that the mass operator is a short-range object. They propose to relate M in an inhomogeneous “real” system to that for a homogeneous system (a free-electron gas) via M(r, r‘?E ,

= Mh(r

- r‘,

-

p

+ ph?Po),

( 16.9)

where p is the chemical potential in a free-electron gas whose density is po = p(ro), where ro = i ( r r’). Equation (16.9) gives a density-dependent

+

approximation to M . It is still somewhat inconvenient because it is not local. A local approximation is generated by supposing that the function on which M , acts is a plane wave with a position-dependent momentum p(r) (as in the WKB approximation); exp i[p(r)*r]. The effect of M on such a function is approximately equivalent to that of a local but energy-dependent potential w(r, E)6(r - r), where w(r) = Mh(p(r)? - p

+ ph(p), p).

(16.10)

The function Mh(p(r),. . .) is intended to be formed from a Fourier transform of Mh in which p is kept constant. After the calculation, we allow p and p to depend on position. It is still necessary to specify the local momentum p(r). For this purpose, Sham and Kohn use a relation valid for a system with slowly varying density p

=

z’ + d) + ph(p).

(16.11)

When the difference between t and E is neglected and the derivatives of p(r) are ignored, from Eq. (16.7) we then obtain

IP(r)12

+ Mh(p(r),E

-

P

+ ph(p),p ) = E

-p

+ ph(p).

(16.12)

We now have an equation for x containing a local energy-dependent potential [Nr)

The index

has been suppressed.

J . Sham and W. Kohn, Ph~7s.Rev. 145, 561 (1966). L. Hedin and B. 1. Lundqvist. J. Ph,..s. C 4, 2064 (1971)

“ L. ‘5

I?

+ d)(d + w(r, E)]x(r, E ) = c(E)x(r,E ) .

(16.13)

194


Now consider an elementary excitation at the Fermi surface. This is an exact eigenstate. Then, with E = p, Eq. (16.12) reduces to (p(r)12+ Mh(P(r)> ph(P), P )

=

ph(p).

(16.14)

Equation (16.14) is satisfied by P(r) = k,(r)

(always assuming slowly varying density). The quantity k, is the radius vector to the Fermi surface. Then the local potential w(r, p) becomes w(r,

= ph(p)

-

Ikf(p)(’ =

uxc(p(r)),

(16.15)

in which u,, is the exchange potential introduced in Section (7) and Eq. (7.4) has been used. If the Green’s function is to have a pole at E = p, t ( p ) = p. Equation (16.13) becomes

[Nr)

+ 4(f)+ uxc(r)lx(r?PI = Px(L P)-

(16.16)

This is, however, the same equation as that satisfied by the single-particle functions u in Section (7) for the case t = p. The Fermi energy is determined by requiring that the volume enclosed by the Fermi surface contain the proper number of particles. This argument, given originally by Sham and K0ht-1,~~ shows explicitly that in the case of a system with slowly varying density, the Fermi surface is given correctly, in principle, by a one-electron calculation using Eq. (7.9) (providing of course that one actually knew the correct uxc). Away from the Fermi energy, w(r, E ) departs from u,, . If one continues to consider a system of slowly varying density, it is possible to obtain an explicit result for the leading (first-order) correction to w. This calculation is given by Sham and K ~ h n , and ’ ~ we do not go into details here. The result is w(r, E ) = ux,(r)

+ ( E - pu)[l - m*(p(r))],

(16.17)

where rn*(p(r)) is the effective mass ratio in a free electron gas in which the (local) density is p. For an attempt to incorporate these ideas into band calculation with some divergencies in detail, see Arbman and von Barth.’6 One approximate qualitative consequence has attracted i n t e r e ~ t . ” ~ Sup’~ pose the second term in Eq. (16.17) is treated in first-order perturbation theory for a state whose energy is different from p. Let tk be the eigenvalue 86

G. Arbman and 0 . von Barth, Nuouo Cimcnfo SOC.Ifal. Fis. B [ l l ] 23B, 37 (1974)

’’J. F. Janak, A . R. Williams, and V. L. Moruzzi, Phys. Reu. B. 11, 1522 (1975). D. G. Laurent, J . Callaway, and C. S. Wang, Phys. Rev. B. 20, 1134 (1979).


195

obtained when u,, is used, and let E, be that resulting from Eqs. (16.13) and (16.8). Then, Ek = ‘,6 A(E - p), where 2 is a constant, the expectation value of 1 - m*(p). This leads to

+

Ek

- p = ([k -

-

A)

(16.18)

Eq. (16.18) implies that consideration of the self-energy leads (in a crude approximation) to a uniform expansion or contraction of the band structure , from the Fermi energy. The sign of A determines calculated from ZI,away which shall occur.

V. Linear Response Theory

In Part V of this article, we shall review fairly briefly some applications of linear response theory that have been directly motivated by the existence of a one-body potential V(r) in which all electrons can be considered to move. Since this potential is a functional of the ground-state electron density, it is clear that problems which can be tackled in terms of a small perturbation of a ground-state density by a change in the one-body potential energy V ( r ) lie within the scope of linear response theory. Two problems of this kind are treated first in a common framework: (1) the perturbation of a uniform electron gas by a test charge, and (2) perturbation of a perfect crystal by the freezing in of a phonon. Following these two problems, which are treated immediately in Section 17, Sections 18 and 19 treat the linear response to a magnetic field in para- and ferromagnetic phases, respectively. 17. DIELECTRIC FUNCTION AND KOHNANOMALY

The density matrix perturbation theory of March and Murray” allows us to write the density change from a uniform electron gas as Ap(r)

=

F(r, r’) AV(r’) d3r‘,

(17.1)

where F is the appropriate linear response function describing the change in the system induced by a change in the one-body potential denoted by AV(r). The result of March and Murray is that

j , ( x ) = (sin x - x cos x)/x2,

196


FIG. 1. Schematic form of tesponse function for a uniform electron gas. (1) Exact result [Eq. (17.4)], (2) von Weizsacker result (A = I), and (3) Kirsnits result (A = 119). The wave number k is plotted in units of the Fermi sphere diameter 2k,.

where the fact that F = F ( ( r - r’l) is a consequence of the translational invariance of the unperturbed system. In general, of course, this will not be true for real crystals, as referred to later. Before going on to develop the form of AV, let us refer to the argument concerning the gradient expansion [Eq. (14.1)]. As Jones and Young67point out, the response function F , calculated with this expansion, omitting T4, is given in terms of the parameter 1 by F,(k)

=

-(kf/~’)(l

+ 3 1 ~ ’ ) ~ ~ , = k/2k,,

(17.3)

whereas the exact form is given by the Fourier transform of Eq. (17.2) as (17.4) Figure 1 shows the schematic form of these response functions, and as seen by comparing curves 1 and 3, the Kirsnits choice of 1 is found to be correct in the long-wavelength limit. By contrast at large k, it is seen that curve 2, with the von Weizsacker choice of A, is the value leading to the correct limit. This confirms the conclusions reached in Part IV. Following Jones and M a r ~ h ,one ~ ~can, ~now ~ formally treat the dielectric

90

W. Jones and N. H. March, Proc. R.SOC. London, Ser. A 317,359 (1970). See also W. Jones and N . H. March, “Theoretical Solid State Physics,” Vol. I, p. 265. Wiley (Interscience), New York, 1973.


197

function of the uniform interacting electron gas by expressing the change in the one-body potential as a functional of the density. This shall be accomplished in Eq. (17.6), but to avoid repetition we shall proceed immediately to treat the linear response of a periodic crystal to “freezing in” a phonon. To be more precise, we shall discuss the way in which the phonon dispersion relations in metals (and, briefly, semiconductors) can be calculated in principle from one-body potential theory within the adiabatic and harmonic framework. The point to be reemphasized is that the phonon problem then reduces to one of calculating the electron density change when ions are moved from their lattice sites by small displacements. Suppose we start out as before with a periodic one-body potential V(r) which we take to generate the exact ground-state density p(r) in the metallic lattice. Next, we remove the nuclei from the lattice sites by small displacements such that the one-body potential energy changes to I/ AV, where the corresponding density change is given by Eq. (17.1). Of course, in the periodic crystal, the response function F must be calculated from the onebody potential V(r) in terms of the Bloch wave functions $nk and corresponding eigenvalues Erik. This can be done most readily from the Green’s function built from the previous wave functions as follows:

+

(17.5) as discussed, for example, by Stoddart et aL91 and by Flores et aL9* Of course, given the one-body potential energy V(r), which involves the first functional derivative of the exchange and correlation energy [Eq. (7.6)] with respect to the electron density, the calculation of F is a soluble one-body problem. Naturally, however, to calculate the change in V as the phonon is introduced, one needs a second fcnctional derivative of the exchange and correlation energy, which is essentially the quantity U introduced next. One can then write for the change in the one-body potential the following result: ,

AV(r)

=

AV,,,,,,,(r)

+

s

U(r, r’) Ap(r’) d3r‘.

(17.6)

a. Dielectric Function of Interacting Uniform Electron Gas

The simplest example for illustrating the use of Eqs. (17.1) and (17.6), is to calculate the dielectric function t ( k ) of the interacting uniform electron gas, following Jones and March,89 in terms of F ( k ) in Eq. (17.4) and the Fourier 91

92

J . C. Stoddart, N . H . March, and M. J . Stott, Phys. Rev. 186, 683 (1969). F. Flores. N. H . March, Y. Ohmura, and A. M. Stoneham, J . Phys. Chrm. Solids 40, 531 (1979).

198


transform U ( k ) of the now translationally invariant quantity U in Eq. (17.6). The result is

=

U(r

-

r‘)

(17.7) We regain the simple Lindhard theory of the dielectric function when we put U ( k )as the Fourier transform of the bare Coulomb interaction between electrons. Improved theories must screen U ( k ) ,and for a detailed discussion of the various approximations to this screening, we refer the reader to the review of Singwi and Tosi4’ and to the other references given there. It should be mentioned here that the self-consistent perturbation calculation of Singhal and C a l l a ~ a yleads ~ ~ to an expression for the frequencyand wave-vector-dependent dielectric matrix for a periodic system. The calculation is discussed in detail here, since it is quite similar to the calculation of the magnetic susceptibility which is described in Section 18. The expression is L= 1-

v,xp wx1-1 -

(17.8)

in which x is a matrix generalizing the response function previously denoted as F to a crystal,

where W is a matrix representing dV,,/ap, ( 17.10 )

(17.11) The matrix indices in these equations refer to reciprocal lattice vectors K s ) . The matrix expression [l - Wx1-l occurring in Eq. (17.8) represents the explicit result of the local density approximation for the so-called exchange correlation correction to the dielectric function. The quantities (1. . . I) in Eq. (17.9) are matrix elements between Bloch functions: thus, Ink) represents a Bloch state of wave vector k in band n, N J k ) is its occupation number, Y is the volume of the crystal, and l2 is the volume of a unit cell. Returning to the phonon problem, we can write the density change for ” S.

P. Singhal and J . Callaway, Phys. Rev. B . 14, 2347 (1976)


199

displacements u1 to first order as follows:

A m

=

c

UI -R,(r),

(17.12)

1

where the vector field R(r) can be determined by an integral equation derived by Jones and March,89 when F and U are given. The dynamical matrix can be set up and the phonon dispersion relations obtained (see, for example, . ~applications ~ to the nearly free-electron metal A1 and to Claesson et ~ 7 1 for covalently bonded semiconductors). All the previous discussion follows from one-body potential theory. Although we do not of course know V(r) exactly, because of the present need to make approximations to the exchange and correlation energy, in principle the treatment present here demonstrates that the Fermi surface mapped out by the K ~ h anomaly n ~ ~ in the phonon dispersion relations in metals can be generated exactly by one-body potential theory. It is occasionally questioned whether this would also be true of the Fermi surface mapped out by studying the de Haas-van Alphen effect, but to date there has been no experimental evidence that a different Fermi surface would result from application of these two techniques. Until such differences are demonstrated from experiment, there is no need to qualify the statement that the Fermi surface of a metal can be mapped out in principle exactly from one-body potential theory (see the argument in Section 16).

18. SPINSUSCEPTIBILITY : PARAMAGNETS

In Section 18, we consider the calculation of the spin susceptibility of a paramagnetic crystal. This problem was discussed by Kohn and Sham,33 and Pant and Tongg6applied that formalism to Na. We shall present here a simplified version of the discussion due to Callaway and C h a t t e ~ - j e eAn .~~ alternative, variational approach has been given by Vosko and P e r d e ~ . ~ ' We consider the Hamiltonian of the one-particle equation for electrons of spin CJ in the presence of a magnetic field B, which we take to define the z axis H , = -V2 Vo(r) V,,, ig@a,, (18.1)

+

+

+

in which V , is a periodic potential which contains the interaction of the A. Claesson, W. Jones, G. G. Chell, and N. H. March, Inr. J . Quantum Chem., Symp. 7,629 (1973). 9 5 W. Kohn, Phys. Rec. Lerr. 2,393 (1959). 96 M. M . Pant and B. Y . Tong, Phys. Letr. A 36A, 133 (1971). 9 7 J. Callaway and A. K . Chatterjee, J. Phys. F8, 2569 (1978). 98 S. H. Vosko and J. P. Perdew, Can. J . Phys. 53, 1385 (1975). 94

200

J. CALLAWAY AND N . H. MARCH

electron with the nuclei of the system as well as the electrostatic potential of the electron distribution; V,,, is the exchange correlation potential for electrons of spin 0 ; g is the electron g factor; and ,uBthe Bohr magneton. The magnetic field B is assumed to be weak enough so that we need consider only linear response. Diamagnetic (orbital) effects of B are ignored. We consider a magnetic field which is periodic in space and time and which has the form

B

=

B , cos[p;r

- ot],

with ps = p

+ K,,

(18.2)

where p is a wave vector in the Brillouin zone, and K , is a reciprocal lattice vector. An arbitrary wave vector can be expressed as a combination of a vector in the zone and a reciprocal lattice vector. The response must be calculated self-consistently. The Hamiltonian depends on the spin densities p , , so the first-order change in these quantities must be taken into account. If

P,

=

do'+ dP,,

(18.3)

where pbo) is the unperturbed spin density, the Hamiltonian [Eq. (18.1)] is expressed as

H

= Ho(pb0')

+ HI + H e x , ,

(1 8.4)

in which H, is the unperturbed Hamiltonian, H , contains terms linear in dp,, and He,, contains the external field. In the present case of perturbation around the paramagnetic state, we can assume that dp, = -dp-, (in first order) so that (18.5) It is convenient to express this as a Fourier sum: dp,

=

C.i [dp,(p,,

w)ei(pJ'r-wf) + d p , ( ~ , w~),* e - ' ( P j . r - w f )

1.

This form must be used to calculate the change in the exchange-correlation potential. The unperturbed potential and its derivatives are periodic in the crystal, and it turns out to be useful to introduce a Fourier representation of a particular combination of derivatives involving a matrix C whose elements are on a basis of plane waves whose wave vectors are reciprocal lattice vectors


201

in which the integral is taken over the unit cell whose volume is Q. In addition, we introduce the induced "magnetization, " ( 1 8.7)

where the quantity M differs from the physical magnetization by a factor -fgpB. We can then introduce a Fourier representation of the terms in HI and write the total perturbation

H,

=

H,

+ H,,,

(18.8)

in Eq. ( 1 8.4) as

where C.C. indicates complex conjugate. We are now in a position to carry out a straightforward perturbation calculation using the operator in Eq. (18.9). In particular, if we consider a single term of Eq. (18.9) having the form ~ C J , P ~ ( ~ ~ " - " " c.c., where b is a constant, the induced Fourier components of the magnetization are

+

S M ( p , ,Q) = /3x::'(P,

+

0) C.C.3

(18.10)

where xi:,) is a matrix element of the non-self-consistent susceptibility. The calculation of this quantity is entirely conventional and need not be described here :

x (Ik

+ plei(Pm"'(nk)..

(18.11 )

The initial factor of 2 allows for spin directions; N,(k) is the occupation number of the Bloch state Ink) (wave vector k and band n; 1 or zero at T = 0), and (. . .) denotes a matrix elemeiit between Bloch functions. We can now determine the magnetization self-consistently. The quantity can be determined from Eq. (18.9). We then solve for 6 M . The result can be interpreted in terms of a self-consistent susceptibility matrix x, which depends on wave vector and frequency

x

=

[I

-

xOcl-lxO.

(18.12)

In order to have something with which this result may be compared, we may consider the free electron gas in which exchange only (no correlation) is considered. Then,

v,, = -4(3pu/47c)l'3.

(18.13)

202


In this case, Eq. (18.12) becomes in the limit p = 0, o

x = xo/[l

=

0,

- (4/9~)'/~/~rJ,

(18.14)

where rs = ( 3 / 4 ~ p ) "and ~ xo now is the susceptibility with all interactions neglected. This agrees with the Hartree-Fock result.99 Equation (18.12) is presumably an exact result within the framework of the local density approximation. It does have the defect, in application to real solids, of requiring the inverse of an infinite matrix. This inevitably raises questions of convergence in practice because the matrix to be inverted will in general have to be truncated.99a Moreover, it is not clear how Eq. (1 8.12) can be applied to disordered systems. These considerations make it important to note the existence of a different, variationally based formula due to Vosko and Perdew9' which is free of many of these problems and which although approximate has met with considerable success in application. We present their result without derivation in a form due to Janak.'" It concerns only the static, uniform susxoo(p, 0).The result is ceptibility denoted simply by x = limp,w-ro

(18.15) in which N ( E , ) is the density of states at the Fermi energy, (18.16) 1

y ( r ) is the average of the wave functions of states at the Fermi energy, ( 1 8.17)

and K ( r ) is a measure of the electron interaction quite similar to the object called C. Suppose the exchange correlation functional is written in terms of p and M (instead of p r and p l ) . Then,

~ ( r6 () r - r ' ) = 6 2 E , , [ p , M]/[Grn(r) 6rn(r')].

( 18.18)

Janak has used Eq. (18.18) to compute the uniform susceptibilities of 32 metallic elements.

19. SPINSUSCEPTIBILITY : FERROMAGNETS The calculation of the spin susceptibility of a paramagnet discussed in Section 18 can be directly generalized to ferromagnets. The direct generalization is to the longitudinal susceptibility; i.e., the magnetic field is taken along


203

(parallel or opposite) to the direction of spin alignment. The analysis leads to many complications, as seen from Rajagopal’” and Callaway and Chatterjee.97 Application of a longitudinal magnetic field to a ferromagnet leads to a change in the charge density (as well as in the spin density). Application of an electrostatic potential will lead to a change in the magnetization. Thus, the calculation of the dielectric function and the magnetic susceptibility are coupled. We do not discuss this complicated situation further here. On the other hand, the transverse susceptibility is both simpler and more interesting. The simplicity results from the fact that the application of a small field perpendicular to the magnetization tends to rotate that magnetization without (in first order) changing its magnitude. There is therefore no change in charge density or in the Fermi energy. The interest results from the fact that the susceptibility matrix has a structure which reveals the existence of spin waves as a collective excitation. Although the treatment of electronelectron interactions in the theory based on the work of Callaway and Wang”’ using the local density approximation is restricted to the level of a self-consistent field, the results resemble those obtained from more complicated many-body theories. A generalization of the effective Hamiltonian of local density theory is necessary if the rotation of the magnetization is to be described self-consistently. The exchange “field must also rotate. In order to incorporate this effect, we must write the spin dependence of the Hamiltonian in the form a.A (where A is a unit vector in the direction of the magnetization) rather than simply oz (as is adequate in the longitudinal case). This change is demanded by rotational symmetry. Specifically, the Hamiltonian which acts on the Kohn-Sham functions is H

=

-V2

+ V&) + V,(r)a.A + *gp,a-B,

(19.1)

in which V , contains the interaction of the electrons with the nuclei, the electrostatic potential of the electron distribution, and the spin average of the exchange correlation potential. The function V, is a difference potential (19.2)

Our convention is that f (up) represents the direction of majority spin alignment in the ferromagnetic ground state. This will be the z direction. C. Herring, “Exchange Interactions among Itinerant Electrons.” Academic Press, New York, 1966. 99a For an approximate solution to this problem, see J. Callaway, A. K. Chatterjee, S. P. Singhal, and A. Ziegler, Phys. Rev. B 28, 3818 (1983). l o o J. F. Janal, Phys. Rev. B. 16, 255 (1977). l o ’ A. K. Rajagopal, Phys. Rev. B. 17, 2980 (1978). l o 2 J . Callaway and C. S. Wang, J . Phys. F 5, 21 19 (1975). 99

204


Finally, B is the external field, which will be assumed to have the form

B

=

B,,[A cos(p;r

-

at)

+j

sin(p;r

- ot)]e -ql'l.

(19.3)

As in Section 18, p, = p + K,, where p is in the Brillouin zone, K , is a reciprocal lattice vector, and v] is an infinitesimal positive quantity. The interaction Hamiltonian can be written as

H1 1 - 29PFP.B = +gaB~,[a- ei(ps.r

- w f )+

a+e - i(p,.r

(19.4)

-wt)],-sltl

The induced magnetization is calculated by first-order time-dependent perturbation theory. We do this first by holding the vector A fixed (not self-consistently). This calculation is described in detail by Callaway and Wang.'" We find it useful to study in particular the components M,, which are given by M,(r, t)

= -L 2gpB

1Nn(k)$?(k, r, t)a+$L(k?

r?

t,

k

= )B,

1(X'$)T)jse+'(~j.r--wf)

(19.5)

j

In the first line of Eq. (19.5), $; is a perturbed spinor (Kohn-Sham) Bloch function for a state of wave vector k in band n, and N,(k) is its occupation number. The second line is the definition of the susceptibility component x$", which is in fact a matrix on the reciprocal lattice vector basis. Only terms of first order in B, are retained in this calculation. We find that

+

x (nkla+e-i(pj'r)lIk p)(lk

[x!"'(P,

0)lj.s= [X")+(P,

+ p(a-ei(ps'r)Ink)

a)]*

(i9.6) (19.7)

It is now necessary to consider the rotation of the vector A in Eq. (19.1), which may be described qualitatively as a rotation of the exchange field. The change in A is, to first order,

SA = M i l ( M x A + M,j),

(19.8)

where M, is the magnetization in the ground state (19.9) (19.10) The change in the exchange-correlation potential is then SV,

=

V,(r)a-SA = *gpBA(r)[M+a-

+ M-a,],

(19.11)


205

in which A(r) is given by (19.12)

A(r) = vf(r)/[g~BMO(r)l.

We have previously written Eq. (19.5) for M , . If this equation is used explicitly, we have to iterate. Then at each step, the magnetization must be substituted into Eq. (19.11) to enable the reconstruction of V,. The final result amounts to using the self-consistent susceptibility x in place of x") in Eq. (19.5)and in the relation between induced magnetization and magnetic field which we use in Eq. (19.11). The latter equation is expressed in Fourier components in the form (possible because A is periodic in the crystal),

in which h.c. indicates Hermitian conjugate, and -

r

(19.14) The integral is taken over the unit cell. Equation (19.18) indicates that the system responds to an effective Hamiltonian Hf':

1

h l j ( ~-(p, + a))jsc-ei(pl"-mf)

Finally, the relation between the magnetization and the effective field is described by the non-self-consistent susceptibility x(O). Thus,

C CX'+"'-(P, I

m)InAC61s + C '1j(x+ -(P, a ) ) j s I > =

[x+ -(P,0)Ins.

(19.16)

The solution of Eq. (19.16) is, in matrix form,

x

I-' .

= [1 - x(O)A]-'.x ( 0 ) = xo[l - Ax(0)

(19.17)

This result resembles what one finds in applying the random phase approximation to the Hubbard modello3

X(P, 4 = X'O'(P, 4 [ 1 - IX0(P, 4 ] - 1 ,

(19.18)

where I is the interaction constant of the Hubbard model. The quantities in Eq. (19.18) are normally scalars, whereas Eq. (19.17) contains matrices in the reciprocal lattice vectors; the more important difference, however, is that we have in Eq. (19.17) a rather different quantity describing the interactionthe matrix A. The spin-wave spectrum is determined by the poles of the self-consistent susceptibility x, or equivalently, from det[l - x(o)A] = 0. '03

T. Izuyama, D. Kim. and R. Kubo, J. Phys. SOC.Jpn. 18, 1025 (1963).

(19.19)

206


Since the quantities in Eq. (19.19) are matrices, this equation may lead to optical spin-wave modes. It follows from very general considerations that there must be a spin wave mode of energy w = 0 when p = 0. This is not an obvious property of Eq. (19.19), but it is possible to transform this equation to a form in which the required result is apparent. A subsequent calculation leads to an explicit, but approximate, expression for the spin-wave stiffness coefficient D.'O4-Io7 The results of Callaway et u1.'07 given next without proof, again in matrix form : (19.20) where M, is a matrix representing the magnetization density, x"' is the non-self-consistent susceptibility previously introduced, and A is given by (19.21a)

+

x (nkle-iPt'ro+)lk p)(lk

+ p(eiPU"o-(2ipu-V)lnk)

(19.21b)

Equation (19.20) for the susceptibility does not contain explicit reference to the exchange-correlation potential. All the quantities involved, M,, A, and X'O), are determined from the energies and wave functions of the electron states, Of course, the exchange-correlation potential is explicitly involved in their determination. The spin-wave energies are now determined from det[A(p, w)]

=

0.

(19.22)

The property memtioned before, that there ought to exist a spin-wave mode with w = 0 if p = 0, now follows immediately on examination of Eqs. (19.21a) and (19.21b). For if o = 0 and p = 0, one entire column of the matrix A, that referring to the K , = 0 reciprocal lattice vector vanishes so that Eq. (19.22) is satisfied under these conditions. In the case of a long-wavelength spin wave, further calculation shows that w = Dp2

+ higher order terni.

(19.23)

The constant D is known as the spin-wave stiffness. The explicit formula for C. S. Wang and J. Callaway, Solid State Commun. 20,255 (1976). D. M. Edwards and M. A. Rahman, J . Phys. F 8 , 1539 (1978). ' 0 6 K . L. Liu and S. H. Vosko, J. Phys. F S , 1539 (1978). l o ' J . Callaway, C. S. Wang, and D. G. Laurent, Phys. Rev. 5. 24,6491 (1981). '04


207

D referred to previously is, however, only approximate

(19.24) Equation (19.24) seems to lead to reasonable results in the case of nickel, where it has been evaluated by using wave functions from a band calculation.' O7

VI. Some Further Applications

20. NONMAGNETICALLY ORDERED SOLIDS

The purpose of Fictions 20 and 21 is to assess in a general way the practical adequacy of density functional theory as we presently understand how to use it; i.e., in the local density approximation with exchange and correlation potentials derived from calculations for these quantities in a free electron gas. It would extend this review far beyond reasonable bounds to attempt to analyze the successes and failures of band- and cohesive energy calculations element by element. We shall therefore restrict ourselves to general remarks supported by a few specific examples with characteristic properties. In Section 20 we shall discuss primarily metals which are not magnetically ordered, semiconductors, and insulators. Because severe and unexpected problems arise with regard to band structures of ferromagnetic metals, we have decided to reserve that topic to Section 21. At the outset, it should be observed that the band structures produced by different exchange-correlation potentials for the same material are not greatly different. In fact, quite reasonable band structures are often obtained in nonmagnetic metals with an exhange-only potential of the Kohn-Sham form. A clear example of this is furnished by the case of chromium, where Laurent et d l o S have compared the results of calculations with the KohnSham potential and the von Barth-Hedin potential" in which the same computer program was used (so that differences due to different methods of calculation were eliminated). Use of the exchange-correlation potential lo*

D. G. Laurent, J . Callaway, J . L. Fry, and N. E. Brenner, Ph.ys. Rei?. B. 23, 4977 (1981).

208

J. CALLAWAY A N D N. H . MARCH

lowers the bands with respect to the exchange-only case by an amount which is roughly constant ( -0.15 Ry), almost regardless of symmetry. Changes in the relative position of levels were small; generally about 0.01 Ry. The more strongly attractive von Barth-Hedin potential produced a compression of the bands by about 0.008 Ry. The reason for the small effect on the band shape appears to be the following. The correlation part of the exchange-correlation potential is a slowly varying function of position, even in an atom where the electron density may vary rather rapidly. In particular, for large densities, the correlation energy per particle according to the free electron gas formula44 varies as In Y, (where r, = (3/4np)’l3,whereas the exchange energy per particle is proportional to r , The slow variation of the correlation part of the exchange-correlation potential means that its principal effect will be a lowering of the energy of electronic states rather than a relative shift. In contrast, if we try to simulate the effect of correlation by multiplying the exchange potential by a variable parameter (01 in the Xa method53), there is a fair degree of sensitivity to this quantity, which multiplies P ’ / ~ . An important and generally highly successful application of density functional theory in the local density approximation has been to the calculation of cohesive properties of metals. Moruzzi et ~ 1 . ” ~calculated the cohesive energy, equilibrium lattice constant, and bulk modulus for 32 metallic elements (including metallic hydrogen) up to indium ( Z = 49). In the course of this work they obtained energy bands, electronic charge densities, the density of states, and the susceptibility enhancement [Eq. (18.15)]. They employed an exchange-correlation potential of the von Barth-Hedin form weigh slightly different parameters. Spin-polarized calculations were made for iron, cobalt, and nickel and are discussed in Section 21. The calculations were made by using the Korringa-Kohn-Rostoker method of band calculation (see Callaway’’’ for a discussion of methods of calculation). They made a “muffin-tin” approximation to the electron density, in which the electron density is assumed to be spherically symmetric inside each of a set of nonoverlapping spheres centered on nuclear sites and constant outside. Only body-centered cubic (bcc) and face-centered cubic (fcc)lattice structures were considered-hexagonal close-packed (hcp) structures were replaced by fcc structures. We do not intend to review here the conclusions of Moruzzi et al. with regard to the trends of physical properties among the metallic elements.

’.

V. L. Moruzzi. J . F. Janak, and A. R. Williams, “Calculated Electronic Properties of Metals.” Pergamon, Oxford, 1978. J. Callaway. “Quantum Theory of the Solid State.” Academic Press, New York, 1974.

”’


209

What is of concern to us is the general level of accuracy achieved, since this indicates the degree of adequacy of present applications of density functional theory. In general, Moruzzi et al. found good agreement between theory and experiment for all metals that do not have a partly filled 3d shell. The calculated atomic volume for the alkali metals are about 10% smaller than the experimental values. Errors of 5% or less are found for Be, Mg, Al, Ni, Cu, Zn, and all of the metals with an open 4d shell. The systems without 3d electrons giving the worst results are the alkaline earth metals Sr and Ca (errors of about 14%),and the alkali metals Li, Na, K, and Rb (errors of about 10%). The calculated bulk moduli are again satisfactory except for magnetically ordered 3d metals. Determination of the cohesive energy requires comparison of calculated total energies for both the solid and the free atom. In fact, the same pattern of general agreement is obtained as for the lattice constant and the bulk modulus; serious errors occur for materials with open 3d shells. Evidently, there is much cancellation of common errors; however, there is reason to believe that the most important difficulties are in the calculation of the total energy of the atom. On theoretical grounds, this is expected because the charge density in an atom resembles that of a uniform free electron gas less than in a solid (the atomic charge goes to zero at large distance from the nucleus; a solid is periodic), and the conserved quantum numbers are different, etc. Practically, it seems that in the case of atoms with an unfilled 3d shell, application of straightforward local density theory frequently leads to a ground-state configuration with too few s electrons. Consider vanadium, for example. Local density theory predicts that the ground atomic state should have the configuration (3d)4 (4s)', whereas )~ is observed. Similar errors occur experimentally the (3d)3( 4 ~ configuration for Ti and Ni, whereas fractional 'occupancies of the 3d and 4s shells are predicted in Fe and Co (a phenomenon which is possible when there is a degeneracy at the Fermi energy). The result of possibly more severe errors in the atomic calculation than in the solid is an overestimate of the cohesive energy. Moruzzi et al. point out that it is not simply neglect of atomic multiplet structure which leads to difficulties because the same sort of errors are obtained in the case of Mn where the ground 6S state can be represented by a single determinant in the Hartree-Fock approximation, as in the case of other open-3d-shell elements. It is worth considering in more detail one example which illustrates the present state of calculations of this type. We select lithium, and the calculations are those of Callaway, Zou, and Bagayoko."' Numerical values are 'I'

J . Callaway, X. Zou, and D. Bagayoko, Phys. Reu. B 27, 631 (1983)

210


TABLEI. TOTALENERGY ET, EQUILIBRIUM LATTICECONSTANT a, BULK MODULUS B, AND COHESIVE ENERGY E, FOR LITHIUM Parameter

Calculated

Experimental

ET (Ry/atom)

- 14.914 6.52 0.138 0.125

- 15.072 6.60

a (a.u.)

B (Mbar) E, “Vatom)

0.123 0.122

given in Table I. A small contribution from the zero-point energy of the lattice vibrations has been included. Figure 2 shows the total energy as a function of lattice constant. It is seen that in the neighborhood of a = 11, a ferromagnetic state lies lower than the nonmagnetic state. There is a small region of unsaturated ferromagnetism near a = 11 (a ferromagnetic metal), followed by a saturated ferromagnetic region, with the occupied 2s band becoming isolated from higher bands. In this region, a ferromagnetic insulator is obtained. It is likely, however, that an antiferromagnetic insulating state, which was not explicitly calculated, would lie lower than the ferromagnetic state. However, it is to be noted that the curves of the total energy versus lattice constant for both the ferromagnetic and nonmagnetic states smoothly approach the atomic limits. (There is a small discrepancy of 0.004 Ry between the apparent limit of the solidstate total energy calculation and the atomic total energy calculation. This is presumably a basis-set effect. If the solid limit is used, the cohesive energy quoted in Table I would be reduced to 0.121 Ry/atom). In a Hartree-Fock calculation using a determinant of Bloch functions, one would find tbat the total energy of the nonmagnetic state approached too high a value due to the presence of spurious ionic components in the wave function. The essential point to be made here is that the local density calculation approaches a reasonable limit as the atomic separation is increased. The results can depend to some extent on the exchange-correlation potential employed. The work of Vosko and WilkL12indicates that the total energy of lithium might be changed by 0.02 Ry and the cohesive energy by 0.01 Ry if a different V,, were used. They also found effects of the same order of magnitude (0.01 Ry) for the cohesive energies of the other alkali metals due to differences of V,, . Finally, let us note that the occupied bandwidth at the equilibrium lattice I ”

S. H. Vosko and L. Wilk, Phys. Reo. B . 22, 3812 (1980).

DENSITY FUNCTIONAL METHODS : THEORY AND APPLICATIONS I

-14.950'

I

I

6.0

l

I

7.0

,

I

I

8.0

I

I

I

I

9.0

100 0 (0 u.)

I

I

I

I

110

120

I

I

2 11

I

I

13.0 1 4 0 A t o m

-

FIG.2. Total energy of lithium as a function of lattice constant. Crosses indicate calculated energies. A ferromagnetic state has lower energy beyond a 11, The inset shows the neighborhood of the equilibrium lattice constant.

parameter is 3.55 eV. The experimental value is about 4.0 eV. On the ~ 9.87 eV for other hand, a Hartree-Fock calculation of Pack et ~ 1 . " gave this quantity. This exaggeration of bandwidths is a characteristic defect of Hartree-Fock calculations in metals. We now consider energy bands and Fermi surfaces. In nonferromagnetic metals, a large amount of evidence has accumulated that shows that current calculations are capable of yielding Fermi surfaces which are at least qualitively in agreement with experiment. Moreover, comparison with angularly resolved photoemission experiments shows that the basic features of the band structure appear to be correct. No contention is made that agreement between theory and experiment is perfect, but we are not aware of major discrepancies. In ferromagnetic metals, covalently bonded semiconductors, and insulators, the situation is much less agreeable. Space limitations do not permit an exhaustive review of this topic. Rather, we shall be content with illustrating these remarks in the thoroughly studied case of copper, for which the Fermi surface has been carefully measured and in which photoemission has lead to the determination of much of the band structure. There are several calculations of the band structure, Fermi surfaces, and related properties for copper that are based on density functional 'I3

J. D. Pack, H. J . Monkhorst, and D. L. Freeman, Solid State Cornmun. 29,723 (1979).

212

J . CALLAWAY AND N. H . MARCH

theory.’ 1 4 - I ” If the band structure is calculated self-consistently with an exchange-correlation potential (it does not seem to matter precisely which one is used) and the Fermi surface is constructed, it is found that although the area of the (I1 1) “bel1y”orbit isin satisfactory agreement with experiment, the area of the(ll1) “neck” where the Fermi surface intersects the zone face is almost 20% too small. The neck area can be brought into quantitative agreement with experiment without damaging the agreement with regard to the belly orbit if the band structure is calculated with an X a exchange-only potential with CI = 0.77. The belly orbit is not sensitive to the potential because its size is essentially determined by the required volume of the Fermi sphere. However, the situation becomes less clear when the optical conductivity is considered. Janak et ul. found it desirable to “expand” the bands calculated with a = 0.77 by an empirical factor [Eq. (16.18)], with A = 0.08, in order to bring the position of the maxima in the calculated optical conductivity into better agreement with experiment. However, Bagayoko et al. observed that although the energy at which interband transitions begin is slightly too low in comparison with experiment (by about 0.3 eV; the experimental value is 2.15 eV) when an exchange-correlation potential is used, the positions of major additional structure in the optical conductivity curve are given reasonably correctly without any expansion of the band structure. That the general behavior of the band structure is reasonably correct can be seen immediately from Fig. 3, where the calculated band structure of Bagayoko et al. is compared with results of photoemission experiments of Thiry e f al.’ l 8 On the other hand, the d bands may be slightly too close to the Fermi energy (0.3 eV as mentioned before), according to the optical conductivity results stated earlier. (It is not clear whether the optical conductivity and photoemission measurements are consistent with each other to this accuracy). Finally, it is worth noting that the calculated Compton profile of Bagayoko rt al., which measures the electron momentum distribution, is in reasonable agreement with experiment, except possibly with regard to the directional anisotropy. The conclusion to be drawn from this is that theory and experiment match reasonably well, but that there are some specific aspects where improvement would be highly desirable, particularly with regard to the Fermi surface.

‘I4 ‘Is I“’ ‘I7

J. F. Janak. A. R. Williams. and V. L. Moruzzi. Ph,vs. Rev. B. 6, 4367 (1972). See Ref. 87 and Ref. 109. D. Bagayoko, D. G. Laurent, S. P. Singal. and J . Callaway. P/IJ~.c. Lctr. A 76A, 187 (1980). 0. Jepsen. D. Glotzel. and A. R. Mackintosh, Ph1.s. Rev. B. 23, 2684 (1981). P. Thiry. D. Chandesris, J . Lecante, C . Guillot. R. Pinchaux, and Y . Petroff. Ph1.s. Rev. Lrfr. 43, 82 ( I 979).


-v

2 13

-8

-9

B

X

FIG. 3. Calculated energy bands in copper along the T - K - X direction.''6 Experimental points are from the measurement of Thiry el a / . ; " 8 E = 0 indicates the position of the Fermi energy.

Wang and Rasolt"' have argued that there must be significant nonlocality to the exchange-correlation potential that is ignored by the conventional approximations. Their calculation suggests that a nonlocal modification of the exchange-correlation potential could lead to greatly improved agreement with experiment. We shall now consider a semiconductor. Silicon has been carefully studied and is the best example for analysis. In general, the conclusions are rather similar to the metallic case: the cohesive properties can be calculated quite accurately, but there are some problems with the band structure, possibly more severe than for most (non-3d) metals. The cohesive properties of silicon have been studied by Yin and Cohen.'20 These authors combined a pseudopotential treatment of the valence electron-core interaction with a density functional treatment of the valence 'I9

J . S. Y . Wang and M. Rasolt. Phys. Rer. B . 15, 3714 (1977). M . T. Yin and M. L. Cohen, Phys. Reo. Lett. 45, 1004 (1980).

214

J. CALLAWAY AND N. H. MARCH TABLE 11. TOTAL ENERGYET FOR SILICON (COREELECTRONS EXCLUDED), EQUILIBRIUM a, BULKMODULUS E, LATTICE CONSTANT AND COHESIVE ENERGYE, Parameter

Calculated

Experimental

ET (Ry/atom)

-7.909 5.451 0.98 4.67

-7.919 5.429 0.99

a

(4

B (Mbar) E, (eV/atom)

4.63

electrons. The amounts to a frozen-core approximation. The analysis for The pseudopotential employed was this approach was given by Ihm et that of Hamann et a1.122The correlation contribution to the exchangecorrelation potential was determined from W i g n e r ' ~expression ~~ for the correlation energy of a free-electron gas. There are two possible methodological difficulties here. Since the pseudopotential (treated as an external potential) is nonlocal, the ground-state energy is formally not a functional of the density but of the density matrix. In addition, Wigner's expression for the correlation energy of a free-electron gas is not correct in the high-density limit. However, since core electrons (which do have high densities) were excluded from the calculation, this error may not have much practical consequence. The results are extremely impressive, and they are summarized in Table 11. Yin and Cohen were able to investigate the total energies for various lattice structures. They were able to show that the observed structure (diamond lattice) has the lowest energy. A transition to the /?-tin structure occurs in silicon under high pressure, and Yin and Cohen were able to calc'ulate correctly the pressure at which this occurs. It is unfortunate that after such success with regard to cohesion, the energies of the excited (i.e., conduction band) states were not given correctly with respect to the valence band. The calculated band gap was 0.5 eV, rather than the observed 1.17 eV. It is characteristic of current calculations of bands in the local density approximation that the energies of excited states in semiconductors and insulators are too low with respect to the valence band. In addition to this example, it is useful to cite the work of Wang and K1eir1.l~~ They made all J. Ihm, A. Zunger, and M. L. Cohen. J . Phys. C 12,4409 (1979). D. R. Hamann, M. Schluter. and C . Chiang, Phys. Rru. Left. 43, 1494 (1979). ' 2 3 C. S. Wang and B. M. Klein, Phys. Rea. B ; 24, 3393 (1981).

12'

122


2 15

electron (no pseudopotential) calculations of energy bands in six semiconductors, including Si, by using an exchange-correlation potential (correlation from Wigner). The occupied bandwidth and x-ray form factors compare well with experiment, but the fundamental band gap was only 0.65 eV. A similar problem was discussed by Khan and C a l l a ~ a y with '~~ regard to the band structure of solid neon and argon. There, the calculated band gaps were found to be about one-half the experimental when calculated with an exchange-correlation potential. To obtain the correct gap, it was necessary to use an Xa potential with a 1.2. It is sometimes argued that agreement with regard to excited states would not be expected because Koopman's theorem does not apply in density functional theory. We have indeed seen that ti = dE,/dn,, where ni is the occupation number of state i. In finite systems, the observation that ti - tj is not the difference in energies between a situation in which state j is occupied and i is empty and one where i is occupied and j is empty is correct and important. The transition-state approach must be employed. On the other hand, in extended (infinite) systems, we ought to be able to relate eigenvalue differences directly to energy differences on the ground that the difference in potentials when one adopts the transition-state picture ought to vanish when the volume of the system becomes very large. In fact, sometimes one does obtain fairly good excitation energies in metals. In addition to the case of copper, a calculation of the optical conductivity of aluminum by Callaway and L a ~ r e n t showed '~~ that the calculated main peak (e.g., principal excitation energy) was located not more than 0.05 eV from the experimental peak near 1.6 eV in a calculation using the Kohn-Sham potential. In spite of the great success of Yin and Cohen's calculation of cohesive properties, we are inclined to believe that the current exchangecorrelation functional derived from the free electron gas may be to some extent inadequate in covalently bonded systems and in systems with large band gaps. One awaits with interest the introduction of nonlocal potentials and gradient corrections.' 2 5 a

-

"'M. A. Khan and J . Callaway, Phys. Left. A 76A. 441 (1980). 12*

J . Callaway and D. G . Laurent. Phys. Lett. A 84A. 499 (1981). The band gap in a semiconductor can be related to the differences in ground state energies in systems with N + I , N . and N - I electrons by: E, = (E(N + 1) - E(N) - (E(M E(N - I)). Problems in calculating Es for semiconductors may be related to the existence o f derivative discontinuities of the exact exchange correlation energy functional which are not present in the local density approximation that is actually employed in the calculations. See J . P. Perdew and M. Levy, Phys. Rei>.Lett. 51, 1884 (1983) and L. J . Sham and M. Schluter. Phys. Rev. Leu. 51, 1888 (1983).

216


2 1. MAGNETICALLY ORDERED METALS The most serious discrepancies between band and cohesive calculations based on density functional theory and experiment occur for magnetically ordered 3d metals. The calculation of cohesive properties by Moruzzi et ~ 1 . ” ~ revealed serious discrepancies with regard to the cohesive energy (too large), lattice constant (too small), and bulk modulus (too large). The problems with the lattice constant and the bulk modulus in their earlier calculations, in which spin polarization was not included, were attributed to the existence of magnetization in the actual materials (a “giant internal magnetic pressure”) or to volume magnetostriction, and their subsequent spin-polarized calculations yielded markedly improved values for these quantities in the ferromagnetic metals iron, cobalt, and nickel. The basic concept here is that a ferromagnet can reduce the cost in kinetic energy of magnetic ordering by undergoing a lattice expansion. However, the cohesive energy remains a problem, and there still are serious difficulties with the lattice constant and the bulk modulus for the antiferromagnetically ordered materials to which the proposed explanation for ferromagnets would not seem to apply. On the other hand, the calculations of Moruzzi et al. and of Callaway and Wang’26 for Fe and of Wang and Callaway’27 for Ni do produce ferromagnetic states with the correct magnetic moment per atom. The general charge, spin, and momentum densities appear to be in satisfactory agreement with experiment. Specifically, there is reasonable agreement with regard to x-ray form factors, neutron-scattering form factors, and Compton-profile measurements. In addition, the calculation of hyperfine fields at the nuclear site which are proportional to the value of the net spin density there yields results of the correct sign (negative minority spins dominate) and of a magnitude which, although quantitatively too small, suggest that the major omission is that of relativistic effects. The calculated Fermi surfaces follow the pattern discussed previously in Section 20, which is that of general agreement without all quantitative details being accurately reproduced by the theory. This is, in fact, a point of great significance: the agreement between theoretical and experimental Fermi surfaces indicates that the 3d electrons are itinerant and not localized. The first difficulty which bears directly on the question of the adequacy of spin-polarized exchange-correlation potentials is that of the exchange splitting of the d bands. Curiously, the disagreement between theory and experiment here exists for nickel but not for iron. Cobalt may also be a

IZh

J. Callaway and C . S . Wang, Phys. Rev. B : Solid State [3] 16, 2095 (1977).

”’ C . S. Wang and J. Callaway, Phys. Rer. Bc Solid Stare [ 3 ] 15, 298 ( 1977).


2 17

case where there is disagreement, but there are not yet any reliable selfconsistent, spin-polarized local spin-density calculation for this element in its actual (hcp) crystal structure, so we shall restrict our attention here to nickel and iron. Calculations for nickel based on the Kohn-Sham exchange potential give an exchange splitting BE = 0.88 eV. Use of the von BarthHedin potential gave AE = 0.63 eV.lZ7The experimental value is probably AE = 0.31 eV.lZ8Note that this is one of the few features where the von Barth-Hedin potential gives results which are significantly different from those using the Kohn-Sham potential. The implication is that the von Batth-Hedin potential unduly favors ferromagnetic alignment. However, the absence of disagreement for iron, where the exchange splitting is larger, is puzzling. More serious in our opinion are difficulties with the bandwidth and band shape in nickel. Reasonable agreement is obtained for iron. It is not possible to discuss this complicated situation in detail here: for more information In summary, the total d bandwidth see Callaway'2y and Himpsel et in nickel seems to be too small by an amount in the range of 12-25%, depending on the method of measurement. Certain levels at symmetry points (most notably L;) are in unexpected positions. Moreover, there is some indication that the exchange splitting of bands based on d functions of e, symmetry is smaller by a factor of 2-3 than that of bands based on t,, symmetry (to which the figure given in the previous paragraph pertained). Attempts to resolve this discrepancy have so far focused on calculations of the self-energy in second order with a Hubbard model Hamiltonian. We shall not discuss this topic here, but the investigation of this problem offers great possibilities to understand the possible corrections to band calculations in the local density approximation. In view of these complicated problems with the nickel band structure, it is noteworthy that the calculation of the spin-wave stiffness D according to Eq. ( 19.24)13'gives a value close to experiment. Specifically, the calculated D = 0.148 Ry a:, whereas the experimental value is D = 0.146 Ry Although this close agreement might be to some extent fortuitous in view of the questions raised earlier about the band structure, it does appear that density functional theory is capable of giving a reasonable account of magnetic excitations near T = 0. I t still remains to calculate the complete

D. Eastman, F. J . Himpsel, and K. A. Knapp. Ph1.s. Rec. Lett. 40, 1514 (1978). J . Callaway. Con/: Ser.-Inst. Phys. 55, 1 (1980). 130 F. J . Himpsel, P. Heimann. and D. E. Eastman. . I Appl. . Phys. 52, 1658 (1981). 1 3 ' J . Callaway. C. S. Wang, and D. G . Laurent. Phys. Rev. B : Condens. Mutter [3] 24, 6491 (1981). 1 3 2 H. A. Mook, J. W. Lynn, and R. M. Nicklow, Phys. Rev. Lett. 30, 556 (1973). lZ8

lZ9

218


susceptibility from the theory of Section 19, but some progress has been made along these VII. Discussion

Density functional theory provides an alternative approach to the quantum mechanics of many-body systems-alternative to the usual methods in that it does not make any initial assumption about the wave function of the system. At the highest level of generality, the Hamiltonian and therefore all properties of a system are determined if the ground-state (four-current) density is specified, subject only to the proviso that the “external” potentials involved are local. Of course, it is difficult to extract much information from so general a statement, but practical approximations have been developed over a period of years that have enabled rather accurate calculations. The theory has become widely adopted in solid state physics where the difficulties of the orthodox procedure of starting from the Hartree-Fock equations are severe. Hartree-Fock theory for solids is pathological as well as difficult to apply to realistic systems. Let us recall the basic pathologies: (1) The density of states vanishes at the Fermi energy. (2) A Slater determinant of Bloch functions may, in circumstances dependent on the spin state, describe at large atomic separations a state containing ions as well as neutral atoms. The energy of such a state is higher than that of a system of neutral atoms. We say that the dissociation limit is incorrectly described. (3) Bandwidths and band gaps tend to be too large. (4) The Hartree-Fock ground state of a free electron gas contains a spin density wave below a certain density.

All of these difficulties can be cured by going beyond Hartree-Fock; i.e., by considering explicitly electron correlation. However, it is a complicated and laborious process to apply the techniques of many-body perturbation theory quantitatively to realistic models of solids. It would be desirable to have a better starting approximation. Density functional theory in the local density approximation provides this better first approximation. This is the basis of its current acceptance and widespread utilization. In fact, the theory has proved to be surprisingly powerful both in a wide variety of cohesive property and band-structure calculations and in the formal development of linear response theory. 133

J . Callaway, S. P. Singhal, and L. Adamowicz, J . Appl. Phys. 53,2027 (1982).


2 19

Formulas can be obtained for response functions that are exact within the limitations of the local density approximation, have the formal structure of expressions from many-body theory, and which can be evaluated in a straightforward if possibly tedious fashion from the results of self-consistent calculations. Finally, the theory has been developed to the extentent that calculations for excited states are possible as well as for ground states, although perhaps with some loss of rigor. Growing experience with applications has, however, uncovered some vexing problems. From the point of view of results, the most serious difficulties seem at this time to be the following: (1) Erroneous prediction of the ground-state configuration from some atoms with an open 3d shell. (2) Failure to yield correct band gaps (and excited-state energies) in semiconductors and insulators. This problem does not seem to be so serious in metals without 3d electrons. (3) Apparently serious disagreements between calculations and experiment with regard to the band structure of nickel and (quite probably) cobalt.

These problems observed in applications may be related to one or more of the following basic problems. First, our knowledge of the basic exchange correlation functional (which includes some kinetic energy contribution) is derived from formulas for the ground-state energy of the free-electron gas, which is a highly idealized system. We need to undetstand matters such as the effect of large density gradients and the dependence of the functional on the conserved quantum numbers relevant to finite systems. Moreover, we do not understand how the functionals which arise in Theophilou’s discussion of excited states differ from those for the ground state. We need expressions for the energy-dependenk electron self-energy so that excited-state calculations in solids can be based on the appropriate form of the Dyson equation. Second, we lack understanding of the significance of the one-particle functions introduced by Kohn and Sham beyond their use for the construction of densities (if indeed, there is any such significance). In particular, under what circumstances, if any, is it legitimate to treat these objects as oneparticle wave functions for the calculation of transition probabilities, and what errors does one commit in so doing? Perhaps further developments in linear response theory are possible here. Having just raised these rather general issues, we shall conclude with a brief discussion of three specific cases to illustrate explicit difficulties which have arisen in applying density functional methods to solid state problems. The first is concerned with covalently bonded semiconductors (see also

220


Section 20). The second deals with d-electron bonds between transitionmetal atoms, whereas the third is an example of a metal-insulator transition. Kane'34 has shown that in silicon, any local potential will leave a discrepancy of about 15-20% between band gaps and effective masses. It is not possible to fit both to experiment, and the reason is that in the diamondlike structure we are dealing with highly directional and inhomogeneous electron density. Using the many-electron formalism of HedinI3' (see also Hedin and L ~ n d q v i s t ' ~Kane'37 ~), calculated the variation of the electron self-energy through the valence and conduction bands. Bennett and I n k ~ o n ' ~have ' extended Kane's calculation by using a dynamic screened interaction. They demonstrate that any static approximation is inappropriate for screened exchange energies in the conduction bands of silicon. This example cautions that the considerations of Section 16 of the present article are essential here. As a further example of directional bonding, this time involving d-electrons, we wish to refer briefly to ~ t ~ d i e on ~ the ~ ~nature ~ -of ~Mo-Mo ~ ~ ~and, ~ , ~ Cr-Cr multiple bonds. In the work of Goodgame and G ~ d d a r d , full '~~ potential energy curves for Mo, and Cr, have been examined by using correlated wave functions. They exhibit a competition between 5s-5s and 4d-4d bonding, and these authors claimed that a serious disagreement exists with the result of local density calculations (see references listed in Goodgame and Goddard' 39 ). However, other authors have demonstrated that more complete local spin density calculations which allow for asymmetric spin densities yield quite satisfactory results.' 40a*b*c In fact, the energies obtained from the local spin density calculations are superior to those obtained from the calculation using correlated functions. The success of local spin density theory in describing bonding in transition metal dimers is a major achievement, and is consistent with the results obtained for transition metal crystals. 41. '4 2 Our third and final example concerns a metal-insulator transition.

E. 0. Kane. Phys. Rer. B : SolidState [3] 4, 1910 (1971). L. Hedin. P h j s . Reu. 139, A796 (1965). 1 3 6 L. Hedin and S . Lundqvist. Solid State Phys. 23, 1 (1969). 13' E. 0. Kane, Phys. Re(>.B : Solid State [3] 5, 1493 (1972). 1 3 * M. Bennett and J . C. Inkson, J . Phys. C 10,987 (1977). 1 3 9 M. M. Goodgame and W. A. Goddard, Phys. Reu. Lett. 48, 135 (1982). I 4 O a B. Delley, A. J. Freeman, and D . E. Ellis, Phys. Reo. Lett. 50, 488 (1983). 140b J. Bernholc and N. A. W. Holzwarth, Phy.?. Reu. Lett. 50, 1451 (1983). I 4 O c B. D . Dunlap, Phys. Rev. 27, 5526 (1983). 1 4 1 D. G . Pettifor, J . Phys. F 8, 219 (1978). V . L. Moruzzi, A. R. Williams, and J . F. Janak, Phys. Rev. B. 15, 2854 (1977). 135


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Rose et a/.,'43have applied density functional theory to the phase transition of electrons in a lattice of fixed-point charges. Their earlier work on the Wigner transition in the low-density interacting-electron ensemble'44 has now been superceded by the computer calculations of Ceperley and Alder4' on this system with uniform positive background charge. It should be cautioned that the version of density functional theory used by Shore et was not quantitatively in accord with the reliable predictions of Ceperley and Alder. Therefore, in treating essentially the problem of hydrogen atoms l ~ ~well be making similar drastic approximations on a lattice, Rose et ~ 1 . may (see also Rose et Prior to this work, Ghazali and L e r o ~ x - H u g o n ' ~ ~ used density functional theory to estimate the critical density of the metalinsulator transition. Their work implicitly assumed that there was only a single transition and did not permit a determination of the order of the phase transition. Subject to the reservations made above, Rose et al. find a first-order metalinsulator transition at a critical density p c given by a,p,'I3 = 0.22, where a, is the Bohr radius. This transition is first order. However, Rose et al. also find a second-order transition associated with the onset of spin ordering (see, however, Rose et ~ 1 . ' ~ ~ ) . These three examples show both the very considerable success obtained in this field and something of the nature of current important problems. ACKNOWLEDGMENTS This review was begun at the International Center for Theoretical Physics in Trieste, Italy. We thank that organization for its support. A substantial portion was completed while one of us (JC) was a visitor in the Department of Mathematics, Royal Holloway College. He wishes to thank Professor M . R. C. McDowell for hospitality and support. Finally, the work of one of us (JC) has been supported in part by the Division of Materials Research of the U.S. National Science Foundation.

J . H. Rose, H. B. Shore, and L. M. Sander, Phys. Rev. B. 21, 3037 (1980). H. B. Shore, E. Zarembra, J . H. Rose, and L. M. Sander, Phys. Rev. B . 18, 6506 (1978). 1 4 5 J. H. Rose, H. B. Shore, and L. M. Sander, Phys. Rev. B. 24, 4879 (1981). 146 A. Ghazali and P. Leroux Hugon, Phys. Reu. Lett. 41, 1569 (1978). 143

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SOLID STATE PHYSICS VOLUME

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Surface- Enhanced Electromagnetic Processes ALEXANDER WOKAUN

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ETH Zentrutn Physical Chemistrv Lahoratorv

.

Zurich Swirzerlund

1. Introduction and Survey ............................................... I1. Surface-Enhanced Raman Scattering: Experimental Observations . . . . . . . . . . . . 1 . The Importance of Surface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Experiments on Metal-Island Films and in Ultrahigh Vacuum . . . . . . . . . . 3 . Inelastic Background Signals in SERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The Wavelength Dependence of SERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Surface-Enhanced Raman Scattering: Theoretical Models ..... ........... ........... 5 . Electromagnetic M 6 . Electron-Hole Pair ................... . . ........... 7 . The Adatom Model .................... ........... 8 . Surface Complexes ........... Induced by Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... 9 . Modulated Reflectance . . . . . ........... 10. Image Dipole Mechanism ..................... 1 I . The Electromagnetic Particle Plasmon Model: ........... Mathematical Description . . . . . . . . . . . . . . . . . . . . ........... IV . Evidence for the Eiectromdgnetic Model . . . . . . . . . . . . . . . . ........... 12. The Distance Dependence of SERS . . . . . . . . . . . . . . . . . . . . ........... 13. Correlation with Optical Properties . . . . . . . . . . . . . 14. The Frequency Dependence of SERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ........... V . Extensions of the Electromagnetic Model . . . . . . . . . . . . . .. ........... 15. The Lightning-Rod Effect ............................ 16. Radiation Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... .. ........... I7 . The Size Effect . . . . . . . . . . . . ............ ............. VI . Particle Dipolar Interactions . . . . ............. I8 . The Local Field . . . . . . . . . . . 19. Shift and Broadening of the Plasmon Resonance . . . . . . . . . . . . . . . . . . . . 20 . Retarded Dipolar Interactions in Regular Particle Lattices . . . . . . . . . . . ........... 21 . Directional Reradiation ..................... 22 . Simultaneous Resonating of Excitation and Raman Frequencies . . . . . . . . 23. Determination of Particle Dipole Moments from Reflectivity Measurements .................................... ~

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Copyright c.. 1984 by Academic Press lnc. All rights of reproduction in m y form reserved. ISBN 0-12-607738-X

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VII.

Enhanced-Surface Second-Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 24. Enhancement of S H G by Particle Plasmon Resonances . . . . . . . . . . 25. Experiments on Ag Island Films. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26. Determination of Local Field Factors from Measurements of Transmission and Reflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27. SHG from Regular Ag Particle Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Enhanced Absorption by Adsorbed Dyes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28. Enhanced One-Photon Absorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29. Enhanced Two-Photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Contributions to the Enhancement by Other Mechanism ........... 30. The Distance Dependence of SERS and the Role of Adatoms.. . . . . . . . . . 31. The Role of Extended Surface Plasmons., . . . . . . . . . . . . . . . . . . . . . . . . . . . 32. SERS from Unroughened Metal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. EfKects of Chemical Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Applications of Surface-Enhanced Phenomena ; Summary . . . . . . . . . . . . . . . . . . 34. Surface Andlytics Using Enhanced Raman Scattering, . . . . . . . . . . . . . . . . . 35. Surface Energy Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................ 36. Summary and Outlook . . .

269 270 27 1 273 274 275 276 219 283 284 285 286 286 287 288 290 293

I. Introduction and Survey

Relatively few mechanisms are known in the physical and chemical sciences that can increase spectroscopic observables by factors as large as lo6. Therefore, when such an amplification was discovered in the phenomenon of surface-enhanced Raman scattering (SERS), intense interest and extensive experimentation were focused on elucidating the origin of the enhancement. The motivating force behind this investigation was, of course, the hope that the underlying mechanism would be somewhat more general in nature and that it could be used for a variety of spectroscopic and other applications. The aim of this study is to present evidence that an electromagnetic mechanism involving localized surface plasmons provides a very important contribution to the enhancement. Starting from the predictions of the model, we demonstrate that other electromagnetic processes at or near the suitably prepared metal surfaces can be enhanced as well. Experiments on surface second harmonic generation, as well as on enhanced one- and twophoton absorption by adsorbed dyes, are described. They emphasize the generality of the phenomenon and provide further strong support for the electromagnetic model. In Part 11, characteristic observations in SERS experiments are briefly reviewed. When large Raman signals from pyridine adsorbed onto electrochemically roughened silver electrodes were first observed by Fleischmann et al.' in 1974, the effect was mainly attributed to an increased surface area.

' M. Fleischmann. P

J . Hendrd, and A. J . McQuillan, Climi. P/i.v.s. Lcrt 26, 163 (1974)

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Careful quantitative investigations by Jeanmaire and van Duyne’ in 1977 showed that the Raman cross section per molecule was enhanced by lo6. It was established that preparation of a microscopically rough surface was a necessary prerequisite for observing SERS. The metals silver and gold proved to be most efficient in providing the Raman enhancement. Along with the experimental investigations, a variety of theoretical models were proposed to explain the effect. They are surveyed in Part 111. The electromagnetic model, which forms the basis for the subsequent discussion, is described in some more detail, with the main emphasis on the physical origin of the effect. Strong experimental evidence in support of the electromagnetic mechanism is presented in Part IV, which includes (1) experiments in ultrahigh vacuum (UHV), testing in particular the distance and coverage dependence of SERS; (2) experiments on evaporated metal-island films, where SERS is correlated with optical properties (transmission, reflection); ( 3 ) experiments on regular arrays of uniformly sized and shaped silver particles, supported by lithographically prepared microstructures. In this case it is clearly demonstrated that the enhancement is due to localized surface plasmon resonances. The dependence of the enhancement on particle shape and on the medium surrounding the particles is in good agreement with the predictions of the model.

In Part V, the theoretical expression for the enhancement is analyzed and represented as a product of a purely geometrical “lightning-rod’’ factor and an electromagnetic factor representing the local field enhancement due to particle plasmon resonances. The model is then extended to include two effects depending on the volume df the metal particles: ( 1 ) The “size effect” represents additional losses of the metal conduction electrons due to collisions with the particle surface; it becomes important for small volumes. (2) “Radiation damping” accounts for the radiative losses of the oscillating electric particle dipole. It is a first-order correction to the electrostatic approximation and becomes important when the particle dimensions d increase and violate the condition d 4000 cm-’ below the excitation energy. Studying SERS from ~ 2 0 to from CN- adsorbed on roughened Ag electrodes, Otto” reported this strong inelastic background and a structure of two broad peaks around 1300and 1600cm- ‘ . I 6 Bergman et al.’found that theincreasein background signal with roughening was approximately proportional to that of the Raman signal. Quite generally, the continuum scattering is found to vary roughly in accordance with Raman signals; however, it is present even without adsorbates. A surface picosecond Raman gain experiment’ clearly showed that the continuum corresponded to zero Raman gain; it was interpreted as ’ ~ the backluminescence from the rough Ag surface. Chen et ~ 1 . model ground as being due to recombination of photoexcited electron-hole pairs (see later text).

’T. H . Wood and M . V. Klein, J . Vuc. Sci. Techno/. 16,459 (1979).

’(’T. H. Wood and M. V. Klein, Solid Srute Commun. 35,263 ( 1980).

T. H. Wood, M. V . Klein, and D . A. Zwemer. Sur/: Sci. 107,625 (1981). M. Moskovits and D. P. Dilella, Chem. Phys. Ltrr. 73, 500 (1980). l 3 P. N . Sanda, J . M . Warlaumont. J . E. Demuth, J . C. Tsang. K. Cristmann. and J . A. Bradley. PhJs. Re[>.Lerr. 45, 1519 (1980). l 4 J . E. Rowe, C. V. Shank, D . A . Zwemer, and C. A. Murray. Phys. Rer. Lerr. 44,1770(1980). A. Otto, Surj. Sci. 75, L392 (1978). l 6 This double structure. known as the “cathedral peaks,” is ubiquitous in SERS work. It has been interpreted as carbonate or graphite species generated by decomposition of carboncontaining adsorbates. 17 J . P. Heritage, J . G . Bergman. A. Pinczuk. and J . M . Worlock, Chem. Phys. Lerr. 67, 229 (1979). I ’ C. Y . Chen. E. Burstein. and S. Lundquist, Solid Srore Commm. 32, 63 (1979). I’


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4. THEWAVELENGTH DEPENDENCE OF SERS Creighton et al.' measured the wavelength dependence of the enhancement for pyridine in the electrochemical environment. When normalized to an internal standard (a Raman band from the electrolyte), the pyridine Raman signals were found to increase strongly with decreasing incident photon . ~ ~ enhanced Raman energy. Subsequently, Creighton et ~ 2 discovered scattering from pyridine adsorbed on silver or gold sol particles. When pyridine was added to a freshly prepared Ag sol, some coagulation was found to occur. The SERS excitation profile exhibited a maximum in the red spectral region and was correlated with a long-wavelength optical absorption band due to coagulated metal species. Wetzel and Gerischer2' were able to avoid sol coagulation with a different preparation technique. In their experiments, the Raman excitation was observed to increase toward the blue spectral region, where the sol absorption peaks near 400 nm, corresponding to the surface plasmon frequency of small Ag spheres. A similar wavelength dependence has been found for C O adsorbed on z 100-A-diameter colloidal Ag particles which were cocondensed with C O onto a cold window.22 A further characteristic feature of SERS experiments is that the usual Raman selection rules are due to interaction of the molecules with the surface. Vibrational modes that are Raman inactive in the free molecule may appear in the SERS spectrum. The relative magnitude of the enhancement of different vibrational modes depends on the orientation of the molecule relative to the ~ u r f a c e . ~ After this brief exposition of experimental observations, a survey on the various theoretical models for SERS shall be given.

Ill. Surface-Enhanced Raman Scattering : Theoretica I ModeIs

Several explanations have been proposed for the origin of SERS that differ substantially in the physical mechanism invoked. We shall not attempt to reproduce the various theories but only characterize them briefly and refer the reader to the original literature. The electromagnetic model, which " J . A. Creighton. M . G. Albrecht. R. E. Hester, and J . A. D. Matthew, Chem. Pkys. Lett. 65, 55 (1978). 2 o J. A. Creighton. C. G. Blatchford, and M. G. Albrecht, J . Chenz. Soc., Furuduy Trans. 2 75, 790 ( 1979). H. Wetzel and H. Gerischer, C/ienz. Phjs. Letr. 76,460 (1980). H . Abe, K . Manzel. W. Schulze, M. Moskovits. and D. P. Dilella, J . CAem. Phgs. 74, 792 (1981).

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forms our framework to interpret the experimental evidence presented later, is described in some more detail. A thorough review of theoretical models, with emphasis on the underlying physical phenomena, has been written by Furtak and Reyes,,; it includes experimental evidence up to mid-1979.

5. ELECTROMAGNETIC MODELS The enhancement is considered to be due to strong local electromagnetic fields at the surface, excited by the incident light. The model predicts that the enhancement should not be molecule specific, should not require direct contact of the adsorbate with the surface, and should extend as far as the amplified local fields. Two alternative mechanisms have been considered : the excitation of extended, delocalized surface plasmons on continuous surfaces, and the excitation of localized particle plasmons on metal-particle surfaces. a. Extended Surface Plasmons

The electric field amplification in extended surface plasmons was calculated, e.g., by Chen, Chen, and B ~ r s t e i nand ~ ~ by Chen, Ritchie, and B ~ r s t e i nThe . ~ ~coupling of light into extended surface plasmons by means of surface roughness was treated by Jha et ~ 1 . ~ ~ T o test the role of extended surface plasmons explicitly, a smooth evaporated Ag film was used.27Light was coupled into the plasmon by attenuated total reflection. Without roughening, no Raman signals from adsorbed pyridine were observed. After roughening, the SERS signals were increased by an additional factor of 4 when the extended surface plasmon was excited. Another set of experiments aimed at elucidating the role of extended surface plasmons was performed by Tsang e t ~ 1 . Al-AI,O, ~ ~ : films were deposited (1) onto a substrate covered with a rough CaF, film, or (2) onto a diffraction grating. 4-Pyridine-carboxaldehyde was chemisorbed onto the Al,O, and subsequently covered by Ag. In the first experiment both SERS and light emission from inelastic tunneling were found to increase with CaF, thickness, i.e., roughness. On the diffraction grating (experiment 2), a twenty-

T. E. Furtak and J. Reyes, Surf: Sci. 93,351 (1980). Y . J. Chen, W. P. Chen, and E. Burstein. Phys. Rec. Let/. 36, 1207 (1976). 2 5 W. P. Chen, G . Ritchie, and E. Burstein, Phjs. Rev. Lett. 37, 993 (1976). *'S. S. Jha, J . R. Kirtley, and J. C . Tsdng, Phys. Rec. Bc Condens. Matter [3] 22, 3973 (1980). 2 7 B. Pettinger, A. Tadjeddine, and D. M. Kolb, Chem. Phys. Ler/. 66,544 (1979). J . C. Tsang, J. R. Kirtley, and J. A . Bradley, Ph,ys. Rep. Lett. 43, 772 (1979). 23

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fold increase in the SERS intensity was observed at an incidence angle suitable to excite the extended surface plasmon.' 3 7 2 8

b. Localized Particle Plasmons The development of this mode129-35 was motivated by a microscopic examination of the surfaces that provide Raman enhancement. Scanning electron microscope (SEM) pictures36 of electrochemically roughened silver electrodes show well-separated globules and lumps of silver of various shapes protruding from the bulk of the silver bar. Typical dimensions are 50-500 A. Similarly, silver- and gold-island films in the thickness range which is optimum for SERS, i.e., around 50 A, consist of separate metal islands of approximately spheroidal shape and ~ 2 0 0A diameter. These observations suggested considering the electromagnetic problem of one isolated metal spheroid interacting with the incident laser light. This model is discussed in detail later. To anticipate the results, light of suitable frequency oocan excite surface plasma oscillations of the conduction electrons, which lead to large local fields close to the surface. An adsorbed molecule, the Raman scatterer, is polarized by this local field. The molecular Raman dipole in turn polarizes the metal particle, which acts as an antenna to amplify the Raman radiation.

6 . ELECTRON-HOLE PAIRMODELS Electron-hole (EH) pairs have been invoked by Gersten et al.j7 and by Burstein et aL3' The creation rate of EH pairs is increased by surface roughness. For the interaction with the adsorbed molecule, four alternative pictures have been drawn by Bursiein et aL3' : (1) The energy of the pair is transferred to electronic excitation of the molecule via Coulomb interaction. Subsequently, the molecule returns to a

M. Moskovits, J . Chem. Phys. 69,4159 (1978). E. Burstein, Y . J . Chen, and S.Lundquist, Bull. A m . Phys. Soc. [2]23, 130 (1978). 3 1 S. L. McCall. P. M. Platrman, and P. A. Wolff, Phys. Lett. A 77A, 381 (1980). 32 M . Kerker. D. S. Wang, and H. Chew, Appl. Opt. 19,4159 (1980). 3 3 J . I. Gertsen and A. Nitzan, J . Chem. Ph1.s. 73, 3023 (1980). 3 4 F. J. Adrian, Chem. Phys. Letr. 78,45 (1981). 35 D.-S. Wang and M. Kerker, Phys. Rer. B : Condens. Matter [3] 24, 1777 (1981). 3 6 See, e.g.. Phys. Today 33 (4), 18 (1980). 3 7 J. I . Gersten, R. L. Birke, and J . R. Lombardi, Phys. Reo. Lett. 43, 147 (1979). 3 8 E. Burstein. Y. J. Chen, C. Y . Chen, S. Lundqvist, and E. Tosatti, Solid Stare Commun. 29, 567 (1979). 29

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vibrationally excited electronic ground state, whereas the remaining energy hw, is transferred back to an EH pair. (2) The electron and hole scatter inelastically from the molecule, losing energy equal t o one vibrational quantum. (3) An electron (or hole) is excited to a virtual bound state of the molecule. Upon deexcitation, the molecule remains in a vibrationally excited state. (4) An initially excited polaron, formed by the interaction of the EH pair and the molecule, evolves into a vibrating molecule and EH pair at the energy ho,. The last step in all cases is the radiative recombination of the EH pair with emission of frequency oR.Only a weak dependence on incident photon energy, reflecting the excitation efficiency of the EH pair, is expected. Mechanisms (2)-(4) require close contact of the molecule with the surface, whereas a wider range is predicted for mechanism (1). The continuum background always present in SERS experiments (Section 3) is interpreted as radiative recombination of EH pairs after a series of inelastic collisions in the metal.” 7. THEADATOM MODEL

This model was developed by A. Otto and his research group, and is summarized in review^.^^,^^,^^ The central concept is that electron-photon couplings are increased by microscopic surface roughness features, i.e., adatoms. Once an EH pair is created, it interacts strongly with the molecules adsorbed at this site via one of the mechanisms proposed by Burstein et (Section 6). If, e.g., a resonant negative ion state is formed by transfer of the electron to the molecule, the Raman scattering cross section may be greatly enhanced, offsetting the small number of molecules adsorbed at active sites. In electrochemical systems, the anodic dissolution of the Ag electrode must be followed by redeposition of Ag from the solution in order t o observe SERS. This step was interpreted ascausing the formation of ad atom^.^' The adatom model was successfully used to interpret annealing experiments39341on thick Ag films evaporated in UHV systems at 120 K. When the substrate temperature is raised, the SERS intensity rises and reaches a maximum around 200 K. This observation is interpreted as being due to restructuring of the surface, whereby adatoms with higher numbers of nearest neighbor are formed; these are thought to be the active sites for surface enhancement. A. Otto. I. Pockrand, J . Billmann. and C. Pettenkofer. in “Surface Enhanced Raman Scattering” (R.K . Chang and T. E. Furtak, eds.), p. 147. Plenum, New York, 1982. 40 J . Billmann, G. Kovacs, and A. Otto, Surj: Sci. 92, 153 (1980). 4 1 I . Pockrand and A. Otto, Solid Stute Commun. 35,861 (1980); 37, 109 (1980). 39


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When the temperature is raised to 280 K, the surface is further annealed, and the SERS signals disappear irreversibly. In the adatom model, the inelastic background signal is due to electronelectron couplings at the surface defects. The variation of the background intensity with voltage42 in electrochemical systems and its time depend e n ~ e were , ~ ~ correlated with a model for the concentration of adatoms on the surface of the Ag electrode during voltage cycling.39

8. SURFACE COMPLEXES. RESONANT RAMANSCATTERING BY ADSORPTION INDUCED P h i l p ~ t and t ~ ~Efrima and me ti^^',^^ have proposed that the role of the metal surface is to shift (to lower energies) and to broaden the energy levels of an adsorbed molecule such that a transition coincides with the excitinglight frequency, and resonant Raman scattering is induced. Although surface roughness may enhance the coupling between substrate and molecule, it is not required; in fact, enhancement by Hg should be comparable to Ag (cf. Section 32). Large changes in the scattering cross section are predicted when the energy of the molecular transition is close to the energy of the extended surface plasmon on the metal plane.47 Direct contact of the adsorbate with the metal is again a necessary requirement for enhancement in this model. It is of interest to note that absorption and luminescence from dye molecules adsorbed onto Ag-island films have been m e a ~ u r e d . ~The ' excitation and luminescence spectra were similar to those of the dye on glass; thus, no large shifts of the electronic excitation energies due to the Ag were observed for the molecules investigated. Pettinger, Wenning, and K ~ l have b ~ formulated ~ a model where Raman enhancement originates in the formation of a specific surface complex. It is formed by individual metal atoms (adatoms) in the top layer of the metal, the adsorbed molecule, ions from the electrolyte, and solvent molecules. For certain geometries, an electronic transition of the complex can coincide with the incident photon energy such that conventional resonant Raman scattering may occur. The enhancement is restricted to the first adsorbate layer. A . Otto, J . Timper, J . Billmann, and I . Pockrand, Phys. Rev. Lett. 45,46 (1980). J . Timper, J . Billmann. A . Otto. and I . Pockrand. Surf Sci. 101,348 (1980). 44 M . R. Philpott. J . Chem. Phys. 62, 1812 (1975). 4s S. Efrima and H . Metiu, J. Chem. Phps. 70, 1939 (1979). 46 S. Efrima and H . Metiu, Surf. Sci. 92,433 (1980). 4 7 R. M. Hexter and M . G . Albrecht, Spectrochrm. Acta, Part A 35, 233 (1979). 4 8 A . M. Glass, P. F. Liao, J . G. Bergman. and D. H . Olson, Opt. Lett. 5, 368 (1980). 49 B. Pettinger U . Wenning, and D. M. Kolb, Ber. Bunsenyes. Phys. Chenz. 82, 1326 (1978) 42

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ALEXANDER W O K A U N

Only that fraction of the adsorbed molecules situated in a complex of suitable geometry will participate in the enhancement. experiment^^^.^ were performed by roughening single-crystalline Ag electrodes in a solution containing pyridine and halide ions. Resonance Raman scattering in an Ag-pyridine-halide ion surface complex was proposed as the enhancement mechanism. From electroreflectance spectra the authors concluded that roughness-induced excitation of extended surface plasmons was not involved in the enhancement. A puzzling feature of SERS experiments in electrochemical systems is that enhanced signals from water are often not observed. Typically, electrolyte concentrations of 0.1 M have been used. Strong enhancement of H, 0 or OH vibrations was only observed when the concentration of the electrolyte (NaCl, KC1, NaBr) was increased to 1 M 5 ’ or 10 M . 5 2 These ideas have been combined with the electromagnetic particle plasmon model. indicate that both large-scale surface roughness (i.e., particle plasmon enhancement) and adatom-adsorbate complexes are necessary to obtain intense surface Raman signals.

9. MODULATED REFLECTANCE This model has been proposed by Otto55 and McCall and P l a t ~ m a n . ~ ~ The charge density on the metal surface is modulated by the electrons of a vibrating, adsorbed molecule. Thereby the charge displacement is propagated to the surface either through the metal-molecule bond56 (in the case of chemisorption), or by Coulomb interaction^^^ in the case of physisorption. For enhancement through chemisorption, an extremely short range and an independence of incident photon energy are predicted. The experimental observation that Ag is enhancing Raman scattering more effectively in the green-wavelength range than Au or Cu is not reproduced by the theory. Experiments interpreted in terms of the modulated electroreflectance model include observations of CN- Raman from Ag electrodes by Otto” and F~rtak.~’ B. Pettinger and U. Wenning, Chem. Phys. Left. 56,253 (1978). M. Fleischmann, P. J . Hendra, I . R. Hill, and M. E. Pemble, J . Electroanal. Chem. 117,243 (1981). 5 2 B. Pettinger, M . R. Philpott, and J . G . Gordon, J. Chem. Phys. 74,934 (1981). 5 3 H. Wetzel, H . Gerischer, and B. Pettinger. Chem. Phys. Lrrr. 79,392 (1981); 80, 159 (1981). 5 4 B. Pettinger and H . Wetzel, Ber. Bunsenges. Phvs. Chem. 85,473 (1981). 5 5 A. Otto, Proc. I n f . Conf: Vibrations in Adsorbed Layers, p. 162. Jiilich, 1978. 5 6 S. L. McCall and P. M. Platzman, Bull. Am. Phys. SOC. [2] 24,340 (1979). 5 7 T. E. Furtak, Solid Sfare Cornmun. 28,903 (1978). 5’


235

10. IMAGEDIPOLE MECHANISM The behavior of a system consisting of an adsorbed molecular dipole and its image dipole below a metal surface has been considered by King et a1.,58 Efrima and me ti^,'^'^^ and Eesley and Smith.61In the point dipole approximation, the effective polarizability of the combined system diverges at a well-defined critical dipole-metal separation which depends on the dielectric function of the metal. Molecular dipoles at the critical distance are considered to be the source of the enhanced Raman signals. A very restricted range is predicted by the model. Surface roughness is not required for this mechanism, and Hg should be comparable to Ag in enhancing the Raman scattering. Considerably smaller enhancements are calculated when the point dipole is replaced by the extended charge distribution of a molecule. 1 1. THEELECTROMAGNETIC PARTICLE PLASMON MODEL :

MATHEMATICAL DESCRIPTION As mentioned earlier, the examination of effective Raman enhancing surfaces prompted several theoretical g r o ~ p s to ~ ~calculate - ~ ~ the Raman scattering from molecules adsorbed onto an isolated metal spheroid. The effects of a supporting conducting plane may easily be included.62 Here we shall concentrate on the fundamental problem of an isolated spheroid. The discussion of dipolar interactions between the particles on a surface is deferred to Part VI. The electromagnetic field distribution around a particle of arbitrary size can be calculated by using the formalism of Mie theory.63 Kerker32 has applied this approach to compute the Raman scattering from molecules adsorbed onto metal spheres. The solution is already quite involved for spheres,32and becomes even more complicated for spheroids. However, the problem can be considerably simplified if all particle dimensions d a r e small F. W . King, R. P. van Duyne, and G . C. Schatz, J . Chem. Phys. 69, 4472 (1978); G. C. Schatz and R. P . van Duyne, Surf. Sci. 101,425 (1980). 5 9 S. Efrima and H . Metiu, Chem. Phys. Lett. 60,59 (1978). 6 o S. Efrima and H. Metiu. J . Chem. Phys. 70, 1602, 2297 (1979). 6 1 G . L. Eesley and J . R. Smith. Solid Srate Commun. 31,815 (1979). 6 2 The main effect of the conducting plane can be visualized by placing image dipoles, both for the metal particles and the molecules, at mirror image positions below the metal surface.33 The Raman scattering cross section is thereby increased by a factor of - 4 relative to the isolated spheroid. Furthermore, in a multipole expansion33 of the electric field around the spheroid in terms of Legendre polynomials of the second kind, the even order terms are suppressed by the presence of the conducting plane. 6 3 G . Mie, Ann. Phys. (Leipzig) 25, 377 (1908). 58

236

ALEXANDER WOKAUN

compared to a wavelength; i.e., in the Rayleigh limit kd 1.0; thus, is real (in phase), parallel to, and smaller in magnitude than On the C. J. F. Bottcher, “Theory of Electric Polarization,” Vol. I , p. 79. Am. Elsevier, New York, 1973. h 5 This integral has been tabulated by E. C. Stoner [Philos. Mag. [7] 36, 803 (1945)l. Some examples will be given for spheroids of revolution, i.e.. ellipsoids. We choose to label the axis of revolution by a for both prolate and oblate cases. Thus A , = A,; furthermore Z, A, = 1 always holds. For a sphere, A , = 1/3. = a,b,c. For a prolate spheroid with a : b = 3: I , A,, = 0.1087; A , = 0.4456. For the limit of an extremely needle-like spheroid ( a :b co) A, + O ; A,+0.5. An oblate spheroid with a : b = I : 3 is characterized by Aa = 0.6354, A , = 0.1823.and thelimitingcasea:b+1:co b y A , + I , A , + O .

64

-+


237

other hand, for metals and semiconductors with carriers that are either free or excited by light, cl(w) can be < 1.0 and even negative, and c(w)is in general complex. Here, Ei,s,pcan be either parallel or antiparallel to and may be larger in magnitude than Eo,p.The maximum of the internal field occurs at a where the real part of the denominator in Eq. (11.1)vanishes; frequency asp, i.e., for (11.3)

tl(wsp) = 1 - l/Ao.

This condition corresponds to the excitation of the lowest order particle plasmon or localized surface plasmon.66 It is represented by a dipolar distribution of surface charges oscillating at frequency asp. It is interesting to make the connection with volume plasma oscillations. In the bulk medium the plasma frequency is given by the condition c 1 (cop)= 0. If the medium is limited in space, then boundary conditions must be fulfilled. For a half-space of carriers limited by a plane, the condition becomes tl(wp)= -1; for a sphere cl(wSp)= -2 ( A = 1/3); and for a spheroid the appropriate boundary conditions lead to Eq. (11.3). b. Polarization of’ an Adsorbed Molecule

We would like to calculate the Raman scattering from a molecule adsorbed at the tip of the long axis p along which the exciting field is applied. The appropriate average over adsorption positions shall be considered later. It is known64 that for large distances from the metal particle, the total field distribution equals the applied field plus the field of the equivalent particle dipole P,, located at its center. As a first step, we shall use the point dipole model to calculate the field at the site of the adsorbed molecule. Later we shall correct for the concentration of the electric field near the particle surface, which we shall term the “lightning-rod’’ effect. The dipole moment of the spheroid is directly related64 to the field inside the particle [Eq. (11.1)] and is given by Pe,p(w01 = V~,,p&“o )Eo,fl(wo

)9

(11.4)

where V is the particle volume, and the diagonal element of the particle susceptibility tensor a,(wo) is given by (11.5)

66

H . Raether. “Excitation of Plasmons and Interband Transitions by Electrons.” SpringerVerlag. Berlin and New York, 1980.

238

ALEXANDER WOKAUN

The equivalent point dipole, located at the particle center, produces a field Edip,p(OO)

(11.6)

= 2pe,p(00)/1i’

at the adsorbed molecule. This field polarizes the molecule and induces a The molecular polarizability amolis dipole moment pmol= amolEdip(wO). modulated by the vibrational motion of the nuclei, and thus the dipole moment contains a component oscillating at the Raman frequency oR: Pp(OR) =

(11.7)

)’

‘R,ppEdip,p(%

The details of the Raman scattering process have been cast into the form of a phenomenological Raman polarizability aR. c. Reradiation of the Raman Frequency

In addition to direct radiation, the vibrating molecule also polarizes the metal particle and uses it as an antenna to amplify its Raman r a d i a t i ~ n . ~ ~ This process is completely symmetric to the excitation process. The field produced by the molecule at the center of the spheroid is =

2p/J(WR

(11.8)

;

it induces a dipole Pp(yR)= V E ~ , ~ ~ ( W ~in) the E ~particle. ( W ~ ) Inserting Eqs. (11.4)-(11.7), one obtains6’ the particle dipole moment

(11.9)

aR,/JpEO,p(wO)‘

Relative to the Raman dipole of an isolated molecule subject to the same external field, the particle dipole is enhanced by the product f p d ( ~ R ) , f p i ( ~ O ) of local field enhancement factors at the Raman and incident frequencies: V

t(W) -

1

fp&4= __ 2711; 1 + [((a)- 1]A,’

(11.10)

where the subscript pd indicates that the point dipole approximation has been used. The radiated power integrated over the full solid angle is finally given by68 ‘(OR

)

=

(Wi/3c3 ) I pfl(WR ) 1 ’ = I ,fpd(WO I ’I ,fpd(WR) 1 ’

wmol(WR

)> (l ’” l)

where k%‘,,,ol(~R) is the power scattered by an isolated molecule. b7

P. F. Liao, J . G . Bergman, D. S. Chemla, A. Wokaun, J. Melngailis, A. M. Hawryluk, and N. P. Economou. Chem. Phys. Left.82,355 (1981). J. D. Jackson, “Classical Electrodynamics,” 2nd ed. Wiley, New York, 1975.


239

These equations for the local field enhancement shall be used to interpret the experimental observations presented in Part IV. Subsequently, the simplified point dipole model shall be corrected in Section 15 by multiplication of the enhancement with a lightning-rod factor. Furthermore, the model is extended to include two size-dependent effects (Sections 16 and 17).

IV. Evidence for the Electromagnetic Model

12. THEDISTANCE DEPENDENCE OF SERS

Of the theories mentioned, only the electromagnetic model and Otto's modulated reflectance theory5 predict long-range enhancement, whereas all

,

PYRlDlNE ON Ag

/ o

/ I

___

10

/ /

EXPERIMENTAL CALCULATION FOR 250 RADIUS Ag SPHERE

a

f

Y

I

I MONOLAYER

3

x

d4MOLECULESI~~'

I I I I

I

5

10

MONOLAYERS PYRlDlNE

FIG. 1. Distance dependence of the Raman enhancement (from Ref. 69). The intensity of the 991-cm- ring breathing mode of pyridine is observed on an Ag surface at 120 K as a function of pyridine coverage. The theoretical curve is calculated by using the results of McCall et aL3*and normalired to the intensity at 10 monolayers: (0)experimental; (---) calculation for 250-A radius Ag sphere; 1 monolayer = 3 x lOI4 molecules/cmz.

'

240

ALEXANDER WOKAUN

other models require close contact of the adsorbate with the surface. The explicit form of the distance dependence in the particle plasmon model is given by Eq. (11.9) where we now replace 1, by the distance R between spheroid center and molecule. Using Eq. (1 l.ll), we see that for a sphere of radius a, the Raman intensity enhancement is predicted to fall off with distance as ( a / R ) ” . The range of SERS was tested in a UHV experiment by Rowe et a1.,I4 where successive monolayers of pyridine were deposited onto an I,-etched Ag(100) surface. In agreement with electrochemical work,4 peaks corresponding to “chemisorbed” and “physisorbed” pyridine species were found. Although the intensity for the chemically bound species saturated at a coverage between one and two monolayers, the signals from the physisorbed species kept increasing beyond 20 monolayers coverage, indicating longrange e n h a n ~ e m e n t . ‘Data ~ are shown in Fig. 1; it is seen that they are well fit by an (a/R)” law, assuming a radius a = 250 A.69 Roughening Ag in various ways, it was verified that surfaces with smaller scale roughness yield shorter range enhan~ement.~’ Long range was also verified in a series of spacer experiment^.^' The scatterer, p-nitrobenzoic acid, was separated from a rough Ag film by a polymer layer of varying thickness. Enhancement was found to persist up to spacer thicknesses of 100 A.

13. CORRELATION WITH OPTICAL PROPERTIES Particle plasmon resonances have been known since the beginning of the century. They have been studied by the use of optical absorption, Rayleigh scattering, and Mie scattering from suspensions of small particles in gases or liquids. T o corroborate the electromagnetic model of Raman enhancement by localized surface plasmons, it was therefore essential to establish a correlation with optical properties. Creighton et a1.” first demonstrated that Ag sols, the color of which is due to excitation of surface plasmons on Ag spheres, did enhance Raman scattering from adsorbates. The SERS excitation spectrum did not match the absorption, apparently due to coagulation. In experiments by Wetzel and Gerischer’ ’ , where coagulation was avoided, the Raman enhancement increased toward the blue spectral range, where the absorption is maximum.

D. A. Zwemer, C. V . Shank, and J . E. Rowe. Chem. Phys. Lett. 73,201 (1980). T. H. Wood, D. A. Zwemer, C. V . Shank, and J . E. Rowe, Chem. Plzys. Letr. 82, 5 (1981). ” C. A. Murray, D. L. Allara, and M . Rhinewine, Phys. Reu. Leii. 46, 57 (1981).

6y

’O


24 1

Bergman et aL7, investigated Raman scattering from AgCN, formed by exposing Ag-island films in air to small amounts of HCN. Scanning electron microscope pictures show that these films consist of approximately spheroidal metal particles. With increasing film thickness, the absorption maximum shifts toward the red, as shall be discussed in Part VI (Fig. 9). The shift is due to changes in particle shape, size, and packing density, which determine the particle plasmon frequency. Alternatively, one can probe the absorption at a jixed frequency as a function of film thickness. Maximum absorption is observed at the thickness where the plasmon frequency of the film equals the excitation frequency. . ~ a~similar maximum for the Raman enhancement Bergman et ~ 1 observed versus film thickness that could be well correlated with the optical properties (transmission, reflection). The analytical form of this correlation is discussed in Section 26. 14. THEFREQUENCY DEPENDENCE OF SERS Since the excitation of localized surface plasmons is a resonant phenomenon, this mechanism can be tested most sensitively by investigating the frequency dependence of the Raman enhancement. The resonance frequency depends on particle shape according to Eq. (11.3). On a surface roughened by a statistical process (electrochemical, etching, evaporation), a broad distribution of shapes is obtained, and one does not expect to observe a clear SERS excitation resonance. Therefore a surface consisting of uniformly sized and shaped metal particles was prepared in the following manner.67 On a silicon substrate, a square array of SiO, posts, with a periodicity of 3000 A, was produced by using lithographic t e c h n i q ~ e s The . ~ ~ posts were slightly conical, ~ 5 0 0 0p\ tall and % 1300 A in diameter. Silver was obliquely evaporated onto this structure in a nonchanneling direction. In this manner one Ag particle is deposited onto the top of each post, whereas the area between the posts remains free of Ag. 72

73

J. G. Bergman. D. S. Chemla. P. F. Liao, A. M. Glass, A. Pinczuk. R. M. Hart, and D. H. Olson, Opr. Left.6,33 (1981). A silicon wafer was patterned by Liao et as follows. A 500 A thick layer of 50, was thermally grown. and 300 A of Cr followed by 1000 A of photoresist were deposited on top of the oxide. The photoresist was exposed twice to a holographic interference pattern. i.e., an intensity grating of3000 A period. By rotating the wafer between the exposures. a crossed grating was obtained. After developing the photoresist, the pattern was transferred into the chrome by ion milling, and tinally into the SiO, by reactive ion etching (cf. Ref. 67).

242

ALEXANDER WOKAUN

FIG.2. Silver particles on a lithographically produced array of SiO, posts6' (SEM photograph; length of calibration bar, 1 pm). 1200 8, of Ag were evaporated along the diagonal of the array at an angle of 68" to the surface normal.

An SEM picture of such a surface is shown in Fig. 2 in which 1200 A of Ag have been evaporated onto the substrate at an angle of 68". The Ag particles appear approximately ellipsoidal in shape with an aspect ratio of x 3 :1. The surface was exposed to HCN vapor, and the Raman intensity of the 2145 cm-' CN stretch mode was normalized to eliminate the trivial o4 dependence [Eq. (11.1I)], as well as instrumental sensitivity. The resulting enhancement is plotted against incident photon energy in Fig. 3 (open circle^'^). The Raman process clearly exhibits resonant behavior, with a maximum at A,, x 5300 A and a FWHM of x 1000 A. The curve can be analyzed by using Eqs. (11.9)-( 11.11) and the dielectric function data for Ag given by Johnson and Christy." An immediate conclusion is that the experimental enhancement curve is considerably wider 74

The solid line in Fig. 3 is merely a guideline to the eye.

'' P. B. Johnson and R. W. Christy, P h p . Rev. B: Solid State [3] 6,4370 (1972).


243

INCIDENT WAVELENGTH (nm) I

500

600

700 ’

I

I

I

I

60

W

f

50

4W

[r

40

t-

W z

I

8

3.0

z

a

I

E

20

z a z

2

1.0 0

I

I

I

I

1.8

2.0

2.2

2.4

2.6

2.8

INCIDENT PHOTON ENERGY (eV)

FIG.3 . Dependence of the Raman enhancement on particle shape. The normalized intensity of the C N - stretch mode (2145 cm-’) is shown as a function of incident photon energy.67 Ellipsoids of various aspect ratios were produced by varying the angle of evaporation : (a) particles with Y 3 : 1 aspect ratio, shown in Fig. 2 ; (b) particles with -2: 1 aspect ratio. Data were collected in a nitrogen atmosphere.

than predicted by the simple theory of Section 6. The theoretical enhancement spectrum, calculated by using Eq. (1l.ll), is double peaked. It maximizes when either the excitation frequency wo or the Raman frequency wR = wo w, coincides with the particle plasmon frequency wSp,corresponding to . [cf. Eq. (ll.lO)]. The width of large values of f ( w o )and f ( w R ) respectively each individual resonance is determined by the imaginary part tZ(w) of the dielectric function, which is small for Ag. The origin of the experimentally observed broadening of the resonance is discussed in Section 16. For the broadened curve, the resonance condition is approximately given by wSp= oo- w,/2; i.e., when plotted against w o , the enhancement maximum is shifted from the surface plasmon frequency toward higher energies by half a vibrational quantum. Using the function t l (0) from Johnson and C h r i ~ t y , ’an ~ aspect ratio of 3.9 : 1 is required to fit the observed resonance position ,lo zz 5300 A, using Eq. (11.3). This is in fair agreement with the shape seen on the SEM pictures, considering the crudeness of the approximations involved (modeling of the particles as ellipsoids and neglect of the supporting substrate). More important than the absolute value of the aspect ratio is the fact that the resonance varies with shape as predicted by theory. Evaporating Ag at a

244

ALEXANDER WOKAUN

more glancing angle (82") produces less eccentric particles, as seen from SEM photographs. Using Eq. (1 1.3), we predict a less negative value of c 1 for the plasmon resonance, i.e., a blue shift.7s This is in fact observed, as shown by the solid circles (dashed curve) in Fig. 3. Here the maximum can be fit with an aspect ratio of 3.2: 1. A second test is performed by varying the dielectric constant co of the surrounding medium. The electrostatic problem of a spheroid embedded in a dielectric is immediately obtained64 from the solution for the spheroid in vacuum by replacing c(w) of the spheroid by e(co)/cO. Making this substitution in Eq. (11.3), it is seen that c1(wSp)= (1 - l / A B F o becomes more negative with increasing t o ; i.e., the plasmon resonance is red shifted. The corresponding experiment is shown in Fig 4. A silver-particle surface, prepared as in Fig. 2, was immersed in water (to = 1.77) and cyclohexane (to = 2.04). In water the excitation maximum is predicted to shift to 1.9 eV and in cyclohexane to 1.5 eV. Although the maxima were not fully traced out due to instrumental limitations, the wings of the resonance curves are clearly INCIDENT WAVELENGTH (nm)

700 I

6.0

600

500

I

I

' \

1.

z W

5.0

t

W J

5

4.0

I-

w z I

Yz

3.0

a r

z

W

2.0

a

I

a (L

1.0

0 1.8

2.0

2.2

2.4

2.6

INCIDENT PHOTON ENERGY (eV)

FIG.4. Dependence of the Raman enhancement on the surrounding medium." The normaliLed intensity of the CN - stretch mode (2145 c m - ' ) on - 3 : 1 aspect-ratio particles is plotted versus excitation energy. Raman scattering was observed with the sample (a) surrounded by a nitrogen atmosphere (dielectric constant co = 1 ) ; (b) immersed in H,O ( f o = 1.77); and (c) immersed in cyclohexane (co = 2.04).


245

seen. Both the direction and the relative magnitude of the resonance shift agree with the theoretical prediction. The experiments mentioned in Sections 12-14 have verified the predictions of the electromagnetic particle plasmon model in the following aspects : distance dependence; correlation with optical properties; resonant character of the enhancement and its dependence on particle shape and surrounding medium. However, the simple point dipole version presented in Section 11 does not explain the magnitude as well as the width of the enhancement resonance. In the Section 15, the model is refined to include the lightning-rod effect, which results in a larger enhancement and size-dependent effects which broaden the resonance.

V. Extensions of the Electromagnetic Model

15. THELIGHTNING-ROD EFFECT In Section 11, a simple model for the Raman enhancement process was described in which the polarized metal particle was replaced by the equivalent point dipole located at the particle center. This model grossly underestimates the maximum enhancement, since the concentration of the electric field near the tip of a prolate spheroid or near the equator of an oblate spheroid is neglected. This effect has been called the lightning-rod effe~t.’~,’’ The maximum increase in enhancement due to the lightning-rod effect is easily calculated. Assuming that the exciting laser field E, is applied along the long axis a of the spheroid, the field at the tip just outside the metal is obtained from Eq. (11.1) by using boundary conditions (15.1)

This field is used to calculate the component of the molecular dipole at the Raman frequency, as in Eq. (11.7).The inverse problem, that of the polarization of the metal particle by the molecular dipole adsorbed at the tip, has been solved by Gersten and N i t ~ a n . ~As ~ .expected, ’~ excitation and reradiation

76 77

’*

P. F. Liao and A. Wokaun. J . Cheni. Phis. 76, 751 (1982). J. I . Gersten, J . Cheni. Phis. 72,5779 (1980). The depolarization factor A,, can be expressed in terms of the Legendre polynomial 01 the ~ ,~=: W / ( W - l ) , where W = Q,(tO)/ second kind Q , ( S ) used by Gersten and N i t ~ a n A [ B = x,y,z. (18.3) Here, E,oc(r,,)has taken over the role of the external field in Eq. (16.3), and


257

aeff( w ) contains all the properties characterizing an individual particle, including radiation damping and size effect, as discussed in Sections 16 and 17. Inserting the dipolar fields into Eq. (18.1) yields

with the self-consistent solution for the local field,

It is seen that Eq. (18.4a) has the form of a feedback equation, with the gain coefficient 9 =

"%ff,pp i

1.

1D,,(rn)eik"" n

(18.4~)

The situation Re{ g) < 0 corresponds to negative feedback, where I ElocI < lEol. For 0 < Re{g} < 1, positive feedback occurs, resulting in an amplified local field. If Re{y} > 1 , the system reaches a stationary state where Elocis antiparallel to E, such that the feedback counteracts the applied field. For Re{g) >> 1, the local field tends to zero. The dipole moment of the reference particle is obtained from Eq. (18.3) by using the solution for the local field and the effective susceptibility a,, given by Eq. (16.9). The result can be written in the form

Here the effective depolarization factor A,, includes the effects of dipolar interactions and of radiation damping, (18.6) The implications for the position and width of the plasmon resonance are discussed next in Section 19. OF 19. SHIFTAND BROADENING

THE

PLASMONRESONANCE

The particle dipole moment found in Eq. (18.5) corresponds68 to a local field enhancement factor

f ( 4= 4 4 / { 1 + C4w) -

1lA,ff,,(k!l?

(19.1)

258

ALEXANDER WOKAUN

for molecules adsorbed at the tip of the spheroid axis /3. By comparison with Eq. (16.4) we see that the effect of the dipolar interactions is to introduce an additional term in the effective depolarization factor A,, besides the radiation damping correction given in Eq. (16.4). To maximize the intensity enhancement I f ( o )I with respect to frequency, the simple relationship Eq. (11.3) is no longer sufficient. First, the effective depolarization factor A , , , is complex; second, the numerator in Eq. (19.1) increasing toward the red. The is itself a function of frequency with maximum of is obtained for

I~(u))~

Cl(Wres)

+ [2(R - B)]-’{- 1 2tJ + B - C ” 2 [ C + 4t,I + ~ E : B ] ’ ’ ’ } ,

= 1

-

(1 9.2)

+

where R = Re{Aeff,s),I = Im(Aeff,s),B = IAeR,p12, and C = 1 B - 2 R . Thus, the resonance position is influenced by both real and imaginary parts varies only slowly of the effective depolarization factor. For cases where &(a) across the plasmon resonance bandwidth, Eq. (19.2)can be approximated by (19.3) 1 -(Re{ Aefi,) I / I A,,,, I 1Under the same conditions, the frequencies oljZ, where )f(o)I’drops to half its peak value, are given by zz

61(~1/2)

- 6j(mres) x -1-

[62

-(Im(A,,,B}IJAeff,BJ2)].

(19.4)

Equipped with these formulas, we shall now interpret the results of an explicit calculation to see how is influenced by the dipolar interactions.

20. RETARDED DIPOLAR INTERACTIONS IN REGULAR PARTICLE LATTICES

To calculate the effective depolarization factors, the following expression6’ for the retarded dipolar field generated by the dipole P, was used: EDn(rn)= [3d,(d, *P,) - P,](r;)-’(l

+ (ii,

+ ikr,)eikrn

x P,) x ii,[(kr,)2/r~]eikrn,

(20.1)

where ii, = r,,/r,,. Besides the static dipolar interaction characterized by an F 3dependence, two additional terms appear which decrease as r P 2 and

’, respectively; the latter becomes dominant in the radiation zone. Effective depolarization factors A,, [Eq. (18.6)] were evaluated by computer summation over a square particle lattice.85The coordinate system was defined by the plane of the surface xy and the plane of incidence yz. The incidence angle is do, and the angle between the x axis and the particle rows is denoted by 2. r-


259

To represent the results by functions that are independent of the particular sample used, it is convenient to write A,, in the form A,,,

=

A,

+ C,(V/d31,

B

= X,

Y, Z,

(20.2a)

with (20.2b) from Eq. (18.6). Here, I/ is the particle volume and d is the lattice constant of the array. The dimensionless quantities C,, which we shall call dipolar interaction constants, depend on the ratio d/A and the angles 8, and z but are independent of particle shape and volume. According to our notation, C, corresponds to an s-polarized exciting field, whereas C , and C , are needed to calculate the effects of a p-polarized field; the z component is perpendicular to the wafer plane. We have assumed that the principal axes of the particles coincide with x, y, z . As an example, real and imaginary parts of C, are shown in Figs. 10 and 11 as a function of a/,? for an incidence angle 0, = 60"and particle rows aligned with the x axis (z = 0'). The following points can be seen from Figs. 10 and 11: (1) The real part of C,, which is negative for d/A+O, changes sign near d/A = 0.2. It can considerably exceed unity for higher values of d / l , corresponding to large resonance shifts [Eq. (19.3)]. (2) The imaginary part is negative for all values of d / A and thus always effects a broadening according to Eq. (19.4). There is an overall cubic

d/X

FIG. 10. Real part of the dipolar interaction constant C, defined in Eqs. (20.2a) and (20.2b). for a square particle array of lattice constant d. Results are shown as a function of the ratio d/A. An angle of incidence 0, = 60" and a tilt angle T = 0" (particle lines parallel to the x axis) were

assumed.

260

ALEXANDER WOKAUN t

0

0.5

1.0

1.5

d/X

FIG. 1 1 . Imaginary part of the dipolar interaction constant C, as a function of the lattice constant to wavelength ratio din. Incidence angle Oo = 60” and tilt angle T = 0” were chosen as in Fig. 10. The overall cubic dependence on d/A is due to the radiation damping term in Eq. (20.2b).

dependence on d/;l due to the radiation damping term in Eq. (20.2b), as well as three pronounced cusps at d / A = 0.54, 1.08, and 1.33. These features correspond to increased broadening. (3) Three corresponding cusps, where Re(C, } becomes strongly negative, appear at d / A = 0.53, 1.06, and 1.30. These cusps or extrema in the dipolar interaction constants are due to the assumed regularity of the particle array. They arise when the dipolar fields from a large number of particles add up in phase at the site of the reference particle. This phenomenon is similar to Bragg scattering; however, the mathematical conditions are different, since the total phase 4 of the dipolar field is composed of the phase k-r,, of the incident plane wave exciting a particle n, and the phase retardation I k 1 I r, I along the distance r,, back to the origin. Particles producing fields with the same total phase are located on confocal ellipse^.'^*^^ This is due to the fact that we are evaluating the phase at a reference point in the surface. By contrast, for Bragg scattering a diffracted plane wave is registered by a distant observer. The elliptical shape of the loci of constant phase seems at first incompatible with a cooperative effect from a square lattice. It is only the long range of the retarded dipolar interactions that makes the occurrence of the interaction extrema (“cusps”)possible. A set of parallel particle lines is selected, making an arbitrary angle a with the basis vectors of the lattice. (It might be called a set of “lattice lines” by analogy with the lattice planes in the three-dimensional case.) For certain selected values of the lattice constant d , these lattice lines are tangential to successive ellipses of total phase Qj = j2n ( j is an integer). In


261

the far field, the ellipses have small curvature and are well approximated by their tangents over large distances. Over this range all the particles on the tangents produce dipolar fields which add up in phase at the origin. This gives rise to the cusps in the dipolar interaction constants seen in Figs. 10 and 11. The analytical condition for the occurrence of such “resonances” has been given by Liao et a/.,” and is fully verified by the numerical calculations. The strong variation of the dipolar interaction constants offers an attractive possibility to tailor the plasmon resonance frequency of a particle array. An approximate discussion is possible with the aid of Eq. (19.3). A positive value of Re{ C, f increases Re{AefiaB 1: c,(wres) becomes less negative, corresponding to a blue shift of the resonance.” Negative values of Re(Cs} result in a red shift. As seen from Fig. 10, Re(Cs) varies between positive and strongly negative values close to a cusp; thus, resonance shifts in both directions are possible. Care must be taken of the concomitant broadening, as expressed in the imaginary part of C,; e.g., at d / l = 0.53, Re{Cx>= -5.15 and Im{C,} = -6.50, so the red shift is comparable to the broadening. For d / A = 0.54, Im{ C,) = - 11.45, and the broadening prevails. The second parameter at the experimenter’s disposal is the particle volume. Shifts and broadening are determined by the product of C , and V / d 3 , according to Eq. (20.2a). Scaling the dimensions of a particle array corresponds to varying d, hence d/A, while V / d is kept constant. It is seen from Fig. 11 that Im{ C , } becomes strongly negative for increasing a/)., because the larger volume particles suffer higher radiative losses. Thus, broadening is always dominant for d/A > 1. To make use of a cusp for shifting the plasmon resonance, an array with 0.5 < d / A < 1.0 is desirable. For a given array, the position of the resonance can then be influenced by the incidence angle and by the alignment of the particle rows in the plane of the ~urface.’~ We would like t o discuss an experiment where a single array can resonate both the excitation frequency ooand the Raman frequency w,. For this purpose, we shall first consider the role of dipolar interactions in the reradiation step. 2 1. DIRECTIONAL RERADIATION

Spontaneous Raman scattering is an incoherent process; i.e., the phases of emitting molecules located on different particles are random. At first glance it might appear that this randomness will lead t o zero average dipolar field at the Raman frequency and that the dipolar interactions will cause only broadening of the plasmon resonance amplifying the reradiation. However, a distant observer selectively detects radiation emitted into a well-defined

262

ALEXANDER WOKAUN

direction k’. From a surface covered with randomly phased emitters, the observer will only register one particular spatial Fourier component, which is suitable for emitting a plane wave into the direction k’. The components of the molecular dipoles which contribute to this particular Fourier component are in a well-defined phase relationship. The average dipolar interaction between these dipole components is nonzero and does give rise to a plasmon resonance shift. In fact, the problem of radiation into direction k’ is completely analogous to that of excitation from direction k. To verify this statement, let us assume that a molecular dipole ,u(oR) with random phase (DYis adsorbed on the tip of each metal particle q. The molecule induces a dipole moment (21.1) in the particle, where we have corrected Eq. (1 1.8) for the lightning-rod effect (Section 15). The total dipole P, in the particle y will also contain contributions from the neighboring dipoles: PY,JuR)

= Rq,.(wR)

+ Vaefi,,.(wR) C Dffn(rn- rq)Pn,a(UR).

(21.2)

n

The intensity radiated by the array into direction k’ is proportional to68 (21.3)

x exp( - ik’r,,)

c Daa(rn

-

r4)exp(ik’(r, - rq).

(21.4)

4

We recognize that the summation over q on the right-hand side of Eq. (21.4) exp(ik‘.r,). is independent of the subscript n and can be rewritten as CsDffff(rs) Thus Eq. (21.4) is solved for the desired sum, yielding pq,a(wR)

exp(- ik’rq)

Y

=

1

Rq,a(mR)

exp(-ik’rq)

Y

{l -

vaeK,au(oR)~

Daa(rs) S

I-’

exp(ik’’rs)

.

(21.5)

It is seen that this result is completely analogous to Eqs. (18.4a) and (18.4b),


263

and thus the effects of the dipolar interactions are the same as in the excitation step, with k replaced by k'. Using the explicit from of M,,,,,(w~),we obtain the final result:

with Aeff(k')given by Eq. (18.6). Here, N is the total number of molecular scatterers. As a consequence of the random phases 4qof the Raman emitters, the intensity is proportional to N , not to N2.92The double enhancement of SERS is visible in Eq. (21.6): The molecular Raman dipole Ip(wR)I2is amplified by the excitation intensity enhancement I .f(w,)l [Eq. (19.1)], whereas the amplification of reradiation by the antenna array is shown explicitly in Eq. (21.6). The new feature compared to the isolated metalparticle case is that the two steps are characterized by potentially different effective depolarization factors of the array A,, (k) and A,,(k'), respectively. This offers the possibility of simultaneously resonating both w , and wR by proper choice of excitation and detection angles, is discussed next in Section 22. 22. SIMULTANEOUS RESONATING OF EXCITATION AND

RAMAN FREQUENCIES

We would like to maximize the Raman enhancement in a given experimental situation, e.g., a surface analytical application. The excitation laser wavelength A o , the frequency of the vibrational mode to be observed, and are given. We shape our particles such that hence the Raman wavelength iR the isolated particle resonance lies between A. and A R . Dipolar interactions in the array are then adjusted such .that for the incident beam, the plasmon resonance is blue shifted into coincidence with A o , whereas for the direction of detection, the plasmon resonance is red shifted to coincide with ,IR. As an example, we choose the 3500 c m p l mode of H,O adsorbed onto Ag; ;lo= 4579 A and A, = 5450 A. The Raman shift is large such that conventional surfaces used in SERS studies cannot amplify both wavelengths simultaneously. Since we require a blue shift for the excitation, we choose d / A , = 0.80 from Figs. 10 and 11, where Re{C,) is large and positive, whereas I Im{ C,) I is moderate. This yields d = 3660 A and d/;lR = 0.67. For both relevant values of d / A , the dependence of the dipolar interaction constants on angle is calculated. The imaginary part of C, is plotted in Fig. 12. We choose an incidence angle Bo = 40", where IIm{C,} I is minimum [Re{C,) = +4.58]. For detection, an angle OR = 25" is selected just 92

I

When X,,Rq, wp, mL % wp were compared. (3) Diphenylanthrancene, where in linear excitation o1>> wp, {oL% up, whereas for two-photon ( 2 0 , ) excitation, w l , wL % w,. a. One- and Two-Photon Excitation of Rhodamine B Luminescence at 590 nm was observed from a monolayer of rhodamine B on an Ag-island film excited at 0.53 pm (linear absorption) or at 1.06 pm 120

S . Garoff, D. A. Weitz, T. J. Gramila, and C. D. Hanson. Opr. Let(. 6 , 245 (1981). C. F. Eagen, A p p / . Opt. 20,3035 (1981). A . M . Glass, A . Wokaun, J. P. Heritage, J . G. Bergman, P. F. Liao, and D. H Olson. Phys. Rev. B : Condens. Matter [3] 24,4906 (1981). A. Nitzan and L. E. Brus. J . Chem. Ph,y.~.75,2205 (1981).

280

ALEXANDER WOKAUN

(two-photon absorption). Results are shown in Fig. 18.' 2 2 Again, luminescence intensities are reported relative t o the same number of molecules in solution. For linear absorption, the results discussed earlier in Section 28 are reproduced: Even after a peak enhancement of z 5 relative to the dye on sapphire, the luminescence intensity is only 4% of the solution value, a consequence of low radiative quantum yield due to energy transfer to the particle. For two-photon excitation, the luminescence peaks a t a considerably higher mass thickness (d, x 100 A). Here a true enhancement by a factor of 150 relative to the solution value is observed.

\

a 6

150

'SAPPHIRE

s K 2

a

U z

2

t-I

2

u)

K c

-

e >

5

0.06

z

0.05

E c

k

e 100

5 + E

w 0

W

0

5:

0.04

-

W

z

-

u)

0.03 0.02

5

0.01

0

5 3

4

K 4

z W v)

50

z

2

W

4

2

0

0

0

40

80

120

Ag MASS THICKNESS

160

(i)

FIG. 18. Enhanced one- and two-photon absorption of rhodamine B on Ag island films.'22 Fluorescence as a function of film thickness is observed at I , = 0.59 pm following (A) onephoton excitation at Li = 0.53pm or (B) two-photon excitation at li= 1.06 pm. Luminescence intensities are given relative t o the same number of dye molecules in solution. In the lower diagram. the local field factorsf2 (0.53 pin) a n d f 4 (1.06 pm) obtained from transmission and reflection measurements are plotted against film thickness.


28 1

To test whether this enhancement is consistent with the electromagnetic model, we determine the local intensity enhancement factors f ( w )I from transmission and reflection measurements, as described in Section 26. The factors lf’(0.53 ,urn))’ and lf(1.06 pm)I4 as calculated from Eq. (26.3) are plotted in the lower half of Fig. 18. It is seen that the luminescence intensity closely follows the excitation rate R [Eq. (29.1)] predicted by the electromagnetic model. The local intensity enhancement at 0.53 pm is relatively broad, with a maximum at d, x 45 A. The local field at the IR wavelength 1.06 pm peaks at a higher mass thickness, as expected. Since the squared intensity enhancement { lf(1.06 pm)I ’} is plotted, the resonance is considerably sharpened. The shift, the narrowing, and the higher value of 1 f l4 are all reflected in the two-photon luminescence curve. The fact that the luminescence mimics the excitation rate so closely implies that the product Qf’(w,) in Eq. (29.2) can vary only slowly in the thickness range of interest. This point is investigated further with the studies of luminescence from nile blue.

I

’

b. Luminescence of Nile Blue following UV and Visible Excitation

Nile blue has two (one-photon) absorption bands at ~ 6 3 and 0 ~ 3 6 nm. 0 Although the first wavelength falls within the Ag plasmon resonance bandwidth, the Ag film is transparent at the second. For both excitations, luminescence is emitted at 686 nm. The dependence of the luminescence intensity from a nile blue monolayer on the thickness of the supporting Ag-island film is shown in Fig. 19. For excitation at 633 nm, the characteristic particle plasmon peak is observed. However, for excitation at 356 nm no enhancement is observed; the luminescence even shows a slight decrease in the particle region. An amplification of the incident UV light was not expected in this case because the wavelength lies outside the plasmon resonance bandwidth. The total lack of an enhancement peak in the luminescence demonstrates that the reradiated wavelength is not enhanced either, i.e., Qf’(w,) must be w l [Eq. (29.2)]. In this respect the luminescence results are different from SERS or surface SHG, where we have seen that amplification of the reradiated light does significantly contribute to the enhancement. The difference may be connected with the finite lifetime of the luminescing state compared to the “instantaneous” processes of Raman scattering and SHG. Since the energy transfer from the dye molecule to the metal particle is most efficient when the emitted frequency equals the plasmon frequency [cf. Section 281, any local field enhancement f’(w,) may be offset by a lower radiative quantum yield Q. The participation of higher order multipole modes’23 has been discussed as an alternative explanation.

282

ALEXANDER WOKAUN

200

150

100

Ag MASS THICKNESS

50

0

(fi)

FIG. 19. Luminescence from nile blue adsorbed onto Ag island films as a function of film thickness.”’ Intensities are given relative to the same number of molecules in solution. Following one-photon excitation at either ii = 356 or 633 nm, luminescence is observed at AL = 686 nm.

c. One- and Two-Photon Excitation of Diphenylunthracene

Diphenylanthracene has an excited state at 260 nm and luminesces at 430 nm. In a one-photon excitation experiment, a monolayer of the dye o n a n Ag-island film was excited at li= 268 nm. Again, the excitation wavelength lies outside the plasmon resonance bandwidth, whereas 1, = 430 nm could be resonated by a thin (d, x 10 A) Ag film. Results are plotted in Fig. 20; the lack of observed enhancement is consistent with the conclusions drawn from the nile blue experiment. However, we did expect enhancement for excitation with two photons of li= 514.5 nm, where I f’(0.51 pm) l4 is known to be large. Quite to the contrary, the luminescence shows a slight decrease in the particle plasmon region (Fig. 20). This may be due to the fact that the excited-state energy of diphenylanthracene considerably exceeds the Ag plasmon energy and lies within the spectrum of interband transitions in silver.66The lack of enhancement is likely to be caused by rapid energy transfer from the molecule to Ag interband excitations.lZ2 To summarize, the experiments reported here in Section 29 have lead to the following conclusions:


THICK Ag

I

'

I

I

I

Ag WEDGE

-4

I

28 3

1 SAPPHIRE

ln t

3 K >

6 -

d

LUMINESCENCE

t m m 5 >

t ln

4 -

5 + z

Y

V

W z

x w

z

2 -

A 3

0

I

1

I

I

I

I

172

150

120

90

60

30

I 0

FIG. 20. Luminescence from a monolayer of diphenylanthracene on Ag-island films following (A) two-photon excitation at 514.5 nm or (B) one-photon excitation at 268 nm. In both cases. luminescence was observed at 430 nm. (From Ref. 122.)

(1) Absorption by adsorbed dye molecules can be enhanced, and the enhancement follows the predictions of the particle plasmon model. (2) Enhancement is particularly efficient for nonlinear processes (here, two-photon absorption), which depend on higher powers of the local intensity enhancement. (3) For excited states with a finite lifetime, as in luminescence, energy transfer to the metal particle becomes important. This is the subject of further study outlined in Section X.

IX. Contributions to the Enhancement by Other Mechanisms

The experiments presented in the preceding sections provide strong evidence for the local field enhancement by particle plasmons. Results from a variety of processes (SERS, SHG, optical properties, luminescence following one- and two-photon absorption) have been consistently interpreted in terms

284

ALEXANDER WOKAUN

of the electromagnetic model. When radiation damping is taken into account and the lightning rod effect is averaged over adsorption sites on a spheroid surface, the calculated enhancements typically lie in the order of % lo4. This suggests that other mechanisms might be contributing a factor of % 10' to yield the experimentally observed enhancement of z lo6. In Part IX we would like to discuss experiments which have been aimed at establishing other contributions to the enhancement. We shall address the distance dependence of SERS, the role of extended surface plasmons, observations of SERS from smooth metal surfaces, and chemical bonding effects.

30. THEDISTANCE DEPENDENCE OF SERS A N D

THE

ROLEOF ADATOMS

The coverage dependence of the Raman intensity from pyridine adsorbed on a Ag( 11 1) surface with a slight modulation of 1 pm periodicity has been investigated in ultrahigh vacuum by Sanda et d.' The enhancement was reported to decrease rapidly with distance from the surface' '. Enhancement for the second monolayer was % lo2 times smaller than for the first adsorbed layer. This result has been contrasted with the experiments of Rowe et ~ 1 . ' ~ and Zwemer et ~ l . where , ~ ~a comparatively long range of the enhancement was found to support the particle plasmon model (cf. Section 12). Billmann and Otto'24 have investigated SERS from an electrochemical cell with both C N - and pyridine present in the solution. By changing the voltage applied to the Ag electrode, Raman spectra from either exclusively pyridine or cyanide were observed. Although a voltage-dependent exchange between a charged species (CN-) and the neutral molecule pyridine is plausible in the first adsorbed layer, the authors state'24 that both species would always be present in the succeeding layers of the liquid. From the disappearance of signals for the nonadsorbed species, it was concluded that enhancement was restricted to the closest adsorbed layer.' 24 The authors also point to the absence of enhanced signals from water, which would be abundant in the second layer. A possible explanation for these differing results on distance dependence may be differences in surface preparation." The particle plasmon model predicts an ( u / R ) I 2dependence, where n is the radius of curvature and R the particleemolecule distance, as mentioned in Section 12. Even moderate differences in particle dimensions due to preparation conditions will drastically alter the range because of the indicated twelfth-power dependence. In J. Billmann and A. Otto. Appl. Surf: Sci, 6 , 356 (1980).


285

fact, a series of experiments in UHV demonstrates that smaller scale roughness corresponds to a shorter range.70 The observations on the distance dependence of SERS have been reviewed by C. A. Murray.12’ Intensive research has been devoted to determine the role of adatoms or atomic scale roughness, as reviewed by Otto6a339and briefly mentioned in Section 5. In an attempt to estimate directly the adatom contribution to the enhancement, Wood’ 2 6 performed the following experiment: Small additional quantities of silver were evaporated onto a thick, annealed Ag film at 180 K in UHV. No SERS signals from pyridine were observed until an additional layer of 150 A of Ag had been deposited. It was concluded’26that the adatom contribution to the enhancement was below lo3, as given by the detection sensitivity ofthe apparatus. However, it has been questioned39 whether the evaporation procedure was suitable for generating adatoms.

3 1. THE ROLEOF EXTENDED SURFACE

PLASMONS

The excitation of extended surface plasmons on statistically rough surfaces has bees discussed by Jha et aLz6 They consider how momentum conservation conditions for surface plasmon excitation can be fulfilled, as the incident light selects a suitable spatial Fourier component of the random surface roughness. There is agreement that SERS intensities are increased by approximately one order of magnitude when an extended surface plasmon is deliberately excited. In the relevant experiments the surfaces d o already exhibit enhanced Raman signals when the conditions for exciting extended surface plasmons are not fulfilled. When coupling into the plasmon is achieved, either in the Kretschmann configuration with light entering through a or by * ~ ~ signals * ~ ~ were found to increase by use of a grating s t r u c t ~ r e , ’ ~SERS factors of 4-20. Since the coupling into extended surface plasmons by random surface roughness is at least an order of magnitude less efficient than either of the two methods mentioned, it must be concluded that the contribution of extended surface plasmons to the enhancement on statistically roughened surfaces is limited.”*

125

IZh

’*’

C . A . Murray, in “Surface Enhanced Raman Scattering” ( R . K. Chang and T. E. Furtak, eds.), p. 203. Plenum, New York, 1982. T. H. Wood, Phy.~.Rer. B : Condens. Mutfer [3] 24, 2289 (1981). R . Dornhaus, R. E. Benner, R. K. Chang, and I . Chabay, Surf. Sci. 101,367 (1980). H. Tom, C. K. Chen, Y . R . Shen, and A. R. B. de Castro, Bu//. Am. Phys. Soc. [2] 26,338 ( 198 1 ).

286

ALEXANDER WOKAUN

32. SERS FROM UNROUGHENED METALSURFACES An early report of Raman signals from iodine adsorbed onto an unroughened Pt electrode'29 is thought to be due to resonant Raman scattering, caused by the coincidence of the Ar+ 514.5-nm line used for excitation with an absorption band of I,. More recently, SERS has been observed from Hg surfaces in two geometries. Naaman et ~ 1 . ' ~ 'reported enhanced Raman signals from pyridine, benzene, and cyclohexane adsorbed onto a hanging mercury drop. Two other groups unsuccessfully tried to reproduce these results.I3'*' 3 2 Sanchez et a/.' 3 3 employed a Pt electrode dipped into mercury. On this electrode, Raman signals were observed from adsorbed pyridine approximately 10 times smaller than from roughened Ag electrodes. The signals were identified as originating at the electrode surface by their voltage dependence. If it proves true that these signals d o indeed arise from an atomically smooth, liquid-like part of the surface, the enhancement will have to be explained by one of the other mechanisms mentioned in Part 111. Enhanced Raman spectra from molecular oxygen adsorbed onto polydiacetylene single crystals have been observed.' 34 They were interpreted in terms of a resonant charge transfer from the hydrocarbon to the oxygen molecule.

33. EFFECTS OF CHEMICAL BONDING Surveying the list of molecules for which room temperature SERS spectra have been r e p ~ r t e d ~one - ~ ~finds , a prevailing of ionic, strongly dipolar or highly polarizable species. This is not surprising since these properties promote strong adsorption onto the surface. On the other hand, work'in UHV suggests that SERS can be observed from any molecule that sticks to a cold Ag (or Au) surface. Comparing the enhancement from these two sets of experiments, it is seen that the modification of the Raman scattering cross section due to chemical bonding does not exceed one order of magnitude (barring cases where the free molecule exhibits resonant Raman scattering).

12"

I3O

R. P. Cooney, P. J . Hendra, and M . Fleischmann, J . Raman Specrrosc. 6, 264 (1977). R. Naaman, S. J . Buelow, 0. Cheshnovsky, and D . R. Hershbach, J . Phys. Chem. 84,2692 (1980).

J . G . Bergman, P. F . Liao, and R . M . Hart, private communication. 1 3 * R. P . van Duync, private communication. 1 3 3 L. A . Sanchez. R. L. Birke. and J. R. Lombardi. Chern. Phys. Lett. 79,219 (1981). D . N . Batchelder, N . J. Poole, and D. Bloor, Chern. Phvs. Lett. 81, 560 (1981). I3l


287

An example where chemical differences between adsorbates appeared not to affect the enhancement, is a cyclic voltammetry study by Benner et The complexes Ag (CN)E-', n = 1-4, have all been observed through their enhanced Raman signals without an indication for largely varied enhancements. Interesting experiments aimed at elucidating the role of chemical bonding and ~ "by Furtak et al.' 3 s b effects have been performed by Loo and F ~ r t a k ' ~ A roughened gold electrode was illuminated with 514.5 nm light. N o enhanced Raman signals from adsorbed pyridine or cyanide could be observed at this wavelength. However, when a submonolayer of Ag was deposited onto the Au electrode, enhanced Raman signals from surface pyridine and cyanide were detected with a relatively weak intensity. The authors conclude that the Ag layer establishes a particular chemical bond between the substrate and the molecule.'35a They state, however, that an unambiguous distinction from the generation of microroughness by the Ag-deposition process is not possible.'35b At the present moment, it seems premature to decide between the various possibilities of contributions to Raman enhancement by other mechanisms. These additional contributions may be system specific and different in origin, depending on surface preparation and absorbate. Since many investigations have been aimed at proving the exclusive prevailance of one particular mechanism, more work is needed to establish the relative size of factors which may be simultaneously contributing to the enhancement. Reliable theoretical estimates of the particle plasmon contribution such as those provided by the model presented in Part V will be helpful toward achieving this goal.

X. Applications of Surface-Enhanced Phenomena ; Summary

The applications of surface-enhanced phenomena can be divided into the two broad classes of surface analytics and surface energy conversion. We shall mention selected examples of each type and conclude with a summary and outlook based on the results of the present study.

R . E. Benner. R. Dornhaus. R . K . Chany. and B. L. Laubc. Surf Sci. 101,341 (1980). B. H . Loo and T. E. Furtak. Chem. Phys. Lett. 71,68 (1980). '35b T. E. Furtak, G . Trott, and B. H . Loo, Surf Sci. 101,374 (1980). 135

135a

288

ALEXANDER WOKAUN

34. SURFACE ANALYTICS USINGENHANCED RAMAN SCATTERING An attractive feature of SERS, which in fact prompted the discovery of the phenomenon,' is the possibility of studying directly the surface composition of electrodes in the electrochemical environment. As an example of the detailed information that can be obtained, we shall mention some studies of cyanide adsorption onto group Ib metals. Using an optical multichannel analyzer, the time development of SERS signals from C N - and heterocyclic organic molecules was followed during the oxidation-reduction cycle of an Ag electrode.' 36 Spectra were identified with the complexes Ag(CN):-', n = 1-4, and correlated with the corresponding oxidation-reduction peaks in the cyclic voltammograms.' The formation of CuCN on roughened copper electrode^,'^^ and SERS from Au(CN); adsorbed onto gold colloids'38 have been studied. No enhanced Raman signals for C N - were observed for voltage-cycled Pt electrodes,' 3 9 because the metal is not efficiently roughened, and the high value of reduces the local intensity enhancement [Eq. (1 7.1)]. Of special analytical value are bands associated with stretching of metalligand bonds. These vibrations carry direct information on the binding state of the adsorbate. The observation is made difficult by the low frequency (typically, 100-400 cm-') of these modes, which are superimposed on a strong scattering background from the rough electrode. For the case of the Ag-pyridine system, a band is observed which shifts between 210 and 243 cm- ' with applied electrode voltage. References to experimental work on this vibration have been given by Dornhaus and ChangI4'. From their observations, the authors conclude that the band is not an Ag-N stretch but rather that of an X--pyridine complex adsorbed onto Ag, where X- is a halogen ion from the ele~trolyte.'~' Leaving the field of electrochemical systems, we turn to the possibility of applying SERS as a diagnostic tool in heterogeneous catalysis. Here, SERS complements other methods that are used for surface analysis such as electron-energy-loss spectroscopy,' 4 2 low-energy electron diffraction (LEED),'43 ultraviolet photoemission s p e c t r o ~ c o p y , 'x-ray-induced ~~ photoelectron spectroscopy (ESCA),'45 Auger electron s p e c t r o ~ c o p y , 'secondary-ion ~~ mass s p e c t r o s ~ , o p y ,and ' ~ ~ EXAFS.14' A detection sensitivity corresponding 136

13'

13*

'41

14'

R. Dornhaus, M. B. Long, R. E. Benner, and R. K . Chang, Surf: Sci. 93,240 (1980). R. E. Benner, K . U . von Raben. R. Dornhaus. R . K . Chang. B. L. Laube. and F. A. Otter, Surf: Sci. 102, 7 (1981). K. U. yon Raben. R. K. Chang, and B. L. Laube, Chem. Phys. Lett. 79,465 (1981). R. K. Chang. R. E. Benner, R. Dornhaus, and K . U. von Raben, in "Lasers and Applications" (W . 0. N. Guimaraes, C. T. Lin, and A. Mooradian, eds.), p. 55. Springer-Verlag, Berlin and New York, 1981. A. Y.-C. Yu, W. E. Spicer, and G. Hass, Ph4.s. Rea. 171, 834 (1968). R. Dornhaus and R. K. Chang, Solid State Commun. 34, 811 (1980).


289

to 1% of a monolayer has been demonstrated for SERS," which equals the best sensitivities achieved with other methods such as LEED. A necessary requirement for the observation of SERS is the use of a metal surface suitable for surface enhancement. This condition being fulfilled, any adsorbed atom or molecule can be studied because observable vibrations either within the molecule or between metal and adsorbate always d o exist. The Raman selection rules, which are somewhat restrictive in isotropic phases, are partly lifted by the presence of the surface. First, we shall survey the metals suitable for SERS studies. Strongest signals have been obtained from the coinage metals, Ag, Au, and Cu. Enhancement has been reported from finely dispersed Ni,""" from Pd,'4yh pt,149c ~ ~ , 1 4 Y d Cd,I5' and Hg.130.133 Several of these metals are highly important for catalytic applications. For example, silver powder is used to catalyze the oxidation of ethylene.15' SERS from oxygen on evaporated Ag films has already been observed by Wood et a/." The fact that the coinage metals exhibit particularly strong SERS signals is understood from the requirements on the dielectric function, i.e., negative c,(w) and a small value of c2(oi) [cf. Eqs. (11.3) and (17.1)]. Surveying the optical properties of metals,152it is seen that only a few metals satisfy this condition in the visible wavelength range. Work is presently in progress to demonstrate SERS for selected other metals by using UV and near-IR laser excitation. If t 2 ( w )is large, some residual nonresonant enhancement due to the lightning-rod effect may occur (A1).'53The local intensity enhancement at

'"

D. C. Joy a n d D. M . Mahcr. J . PhJ,.s.E 13, 260 (1980); J . Microsc. (O.\-/or.d)124, I (1981). M . A . Hove and G . A . Somorjai. Surf. %i. 114, 171 (1982). I" M . W . Roberts. A d r . Cuttrl 29, 55 (1980). 'I5 J . Hcdman and K . Siegbahn. eds.. "Elcctron Spectroscopy." Phys. Scr.. Vol 10. p 169. R . S w r d . Acad. Sci.. Stockholm. 1977. '41 C. C. Chang. in "Characterization of Solid Surfaces" (P. F. Kane and G. B. Larrabee. eds.). p. 509. Plenum. New York, 1974. I" K. Wittmaack, in "Inelastic too-Surface Collisions" (J. C. Tully a n d C. W. White. eds.). p. 153. Academic Press, New York. 1977. 14' B.-K. T e o a n d D. C. Joy, eds.. "Extended X-Ray Absorption Fine Structure ( E X A F S ) Spectroscopies-Applications to Materials Science.'' Plenum. New York. 1981. W . Krasser. C. R.-Conf' SpccIro.s(~.Inr. Conf Rar?icin, 7rh. 1980 p. 420 (19x0). M . Fleischmann. P. R . Graves. 1. R . Hill, a n d J . Robinson, Chvrii. P/7j,.s.Lerr. 95,322 (1983). IJ9' W. Krasser a n d A . J . Renouprez, SolidSrnrc. Cor?iri7un.41,231 (1982): R . E. Benner. K . U. von Raben, K. C . Lee, J . F. Owen, R. K . Chang. a n d B. L. Laubc. Chcwi. Phi,.s. L r l t . 96, 65 ( I 983). P. A . Lund, R. R . Smardzewski, a n d D. E. Tevault. Clirrn. P/71x Lett. 89, 508 (1982). B. H. Loo. J . Ck0rii. P / i ~ , . s75, . 5955 (1981). I s ' X. E. Verykios. F. P. Stein, a n d R. W. Coughlin. Corn/. R w - S c ~ i . Enq. 22, 197 (19801. I s ' G . H a s a n d L. Hadley. in "American Institute of Physics Handbook" ( D . E. Gray. ed.). 3rd ed.. p. 6. McGraw-Hill. New York. 1972. I s ' P. F . Liao a n d M . B. Stern. Opr. Lrtr. 7, 483 (1982).

290

ALEXANDER WOKAUN

the tip becomes ( f I 2 x A P 2 [Eq. (17.1)], and the SERS enhancement averaged over the surface is W = q,4-4Wm0, [Eq. (16.5)]. For a 3 :1 prolate spheroid, this would correspond to an enhancement of 1.6 x lo2; i.e., 100 times smaller than for a resonant Ag spheroid of the same geometry. From this consideration only a weak enhancement can be expected for Pt, due to its relatively high loss t2(0).1403149c The limitations imposed on the application of SERS by the metals that can be used are partly offset by the specificity of the information obtained. The SERS signals arise from a well-defined adsorption region close to the surface. Since the observed quantities are molecular vibrations, the detection of elements with low atomic number represents no problem. Studying band positions, band contours, and metal-ligand vibrations in particular, one can obtain detailed information about the binding of adsorbates to the substrate, changes in bond strengths in the adsorbed molecule, and adsorptiondesorption dynamics. The information content of SERS with regard to these questions is just being recognized, and important developments are likely to emerge. In contrast to electron or high-energy photon-bombardment methods, SERS is generally a nondestructive probe, as long as low light intensities are used. Thus the information on elementary composition obtained from LEED and the chemical and structural information of SERS complement each other. It is hoped that an increased understanding of the molecular processes occurring on catalyst surfaces will aid in the successful design of efficient catalysts for industrial processes.

35. SURFACE ENERGY CONVERSION Experiments demonstrating enhanced one- and two-photon absorption by dye molecules on surfaces have been described in Part VIII. Enhanced luminescence from dyes interacting with the microstructural resonances of dielectric cylinders (glass fibers) has been reported.' For many of the dye molecules used, radiative quantum yields are quite high in solution. Studying these dyes adsorbed onto rough metal surfaces obviously does not aim at enhancing the absorption cross section for the dye molecules themselves; rather, they serve as convenient model systems to investigate energy-conversion processes on the surface with high sensitivity. As discussed in Part VIII, two processes are competing on the surface: enhanced absorption due to the amplified incident field, and loss of the molecular excitation energy by transfer to the metal particle followed by radiationless decay. This transfer is manifested directly in the shortening of fluorescence lifetimes. Lifetime studies of emitters close to smooth metal surfaces have been described by Lukosz and Meier,' where references to

'


29 1

earlier work are given. For the same layer of fluorescing centers [Eu(III) benzoyl trifluoroacetone chelate], the lifetime was seen to change by a factor of 5 when a smooth Ag layer was mechanically approached from large distances up to direct contact with the fluorescing film. A drastic shortening of fluorescence lifetimes by three orders of magnitude was observed in experiments by Weitz et ul.,' 5 4 where Eu(II1) complexes were deposited directly onto (rough) Ag-island films. The balance between enhancement and damping can be influenced by varying the distanced between the particle surface and the adsorbed molecule, as pointed out by Nitzan and Brus.ls5 The coupling between the particle dipole and the molecular transition dipole, which is responsible for the enhancement, decreases as ( u + d)-3, where a is the particle radius of curvature. The damping, on the other hand, decreases with distance from the 100 A or larger dimensions, d can be consurface as d - '. For particles of n veniently varied to optimize the overall luminescence yield by separating the molecule from the particle through a spacer layer.' 5 5 These predictions have recently been verified by experiments.' s 6 Strong variations in the intensity and angular distribution of the Eu(II1) fluorescence have been observed by Lukosz and Meier,' l 5 when an Ag film was mechanically approached to the emitters in the experiments mentioned earlier. The use of particle plasmon resonances for enhanced photochemistry has 5 They distinguish two situations. been proposed by Nitzan and B r ~ s . ' ~5 ~ ,' First, if the chemical process following absorption, e.g., bond breaking, is very fast, the photochemical yield will be proprotional to the enhanced absorption, which decreases monotonically with distance. As an example, dissociation of a model I, molecule adsorbed on an Ag sphere is shown in . Th e absorption of the free I, molecule occurs near 4500 A, Fig. 21123,'55 and the plasmon resonance of the'sphere near 3540 A. For the combined system, the photodissociation yield shows two peaks as a function of excitation energy. There is enhanced absorption at 4500 8, due to the amplified local field and a second dissociation maximum due to energy transfer from the particle plasmon. Both peaks are seen to decrease with molecule-surface distance. Second, if the dissociation process requires accumulation of energy in the molecule, such as in multiphoton dissociation, energy transfer back to the metal competes with the enhanced absorption. This is shown in Fig. 2212' for a model SF, molecule adsorbed on an InSb sphere. Both the SF, absorption ' S. Garoff, C. D. Hanson, T. J . Gramila, and J . 1. Gersten, J . Lumin. 24/25,83 . A . Weitz. (1981); Opt. Lett. 7, 89 (1982). A. Nitzan and L. E. Brus, J . Chem. Phjs. 74,5321 (1981). A. Wokaun, H.-P. Lutz, A. P. King, U. P. Wild, and R. R. Ernst, J . Chem. Phys. 79, 509 (1 983).

154D

'55

292

ALEXANDER WOKAUN

PHOTON ENERGY

(lo3c m - ' ~

FIG.21. Enhanced absorption of an iodine n~oleculenear a silver sphere (from Ref. 123). The molecule is oriented perpendicular to the surface. Calculations were performed for separations of d = 5 and 50 A from a sphere of radius a = 500 A. The intrinsic absorption of iodine is shown as the bottom curve.

and the sphere plasmon resonance of suitably doped InSb coincide with the 10.4-pm CO, laser wavelength. The photodissociation yield is seen first to rise with distance, reach a maximum at d = l0A, and then to decrease toward the free-molecule value. This example demonstrates the existence of an optimum spacing for a particular energy-conversion process. It is not evident that carrying out a photochemical reaction on a suitable enhancing surface will actually result in increased production yields. Photochemical reactions are usually carried out in the volume of a gas or solution. The number of molecules in 1 cm3 of gas at atmospheric pressure is typically lo5 times larger than in a monolayer of area 1 cm'. Thus a high rate of starting material adsorption followed by product desorption is necessary to generate significant quantities of product. However, there are certainly cases where the presence of a solid catalyst is necessary; e.g., when selection rules preclude a photochemical transformation in the homogeneous, isotropic medium. In such cases the utilization of surface enhancement will result in a strong increase in product-formation rate.


293

w (crn-1)

FIG.22. Enhanced absorption of an SF, molecule near an InSb sphere of radius ( I = 500 A (from Ref. 123). Calculations are shown for 5, 10. 150 8, and infinite surface-molecule separation. Note that the distance dependence is nonmonotonic and exhibits a maximum at d = 10 A.

36. SUMMARY AND OUTLOOK Over the past four years, intensive research efforts have been devoted to elucidating the origins of surface-enhanced Raman scattering. Claims were often made to explain the phenomenon as a whole in terms of just one mechanism, leading to the oversight of observations supporting another. In the present study, we have compiled recent evidence for the electromagnetic particle plasmon model. For surfaces exhibiting discrete metal particles or protrusions (low-temperature evaporated films, island films, lithographic structures), the following facts have been reported: (1) A variety of electromagnetic processes is enhanced at or near the surface. The enhancement is particularly large for nonlinear processes, consistent with the concept of a local field enhancement. (2) For SERS, the distance dependence follows the predictions of the particle plasmon model. (3) An excitation resonance has been observed which varies with particle shape and surrounding medium as expected for a localized surface plasmon. The width of this resonance and its variation with particle volume are understood in terms of radiation damping. (4) The calculated peak enhancement in the scattering cross section per molecule, when averaged over adsorption sites, is on the order of lo4, depending on material, surrounding medium, and shape and volume of the particles.

294

ALEXANDER WOKAUN

( 5 ) Local field enhancement factors can be determined independently from transmission and reflection measurements. The Raman enhancement correlates well with the local field factors. (6) Enhanced SH generation was demonstrated for island films and successfully correlated with optical measurements of the local field enhancement. (7) Dye molecules adsorbed on island films exhibit enhanced one- and two-photon absorption. The dependence on film thickness for various dyes, in linear and nonlinear absorption, is consistent with the predictions of the particle plasmon model.

This combined evidence strongly suggests that amplified local electromagnetic fields due to particle plasmon excitation are indeed present on the surfaces studied. This conclusion opens up the way to further experiments in the following directions:

(1) A systematic search for other materials exhibiting surface enhancement by using the guidelines of the electromagnetic model. (2) Quantitative determination of additional contributions to the enhancement, e. g. due to surface chemical effects or adatoms. (3) Application of SERS to problems of surface analytics, in particular to studies of reaction mechanisms on heterogeneous metal catalysts. (4) Investigations of energy-transfer kinetics between adsorbate and metal using time-resolved luminescence. In particular, the prediction of an optimum distance for energy accumulation on the molecule has been tested by spacer experiments.' 5 6 As this concept is now verified, surface-enhanced photochemistry may become feasible. (5) Extensions of the concept of surface-enhanced absorption with a view to efficient conversion and storage of the incident light energy. Enhancements of the order of lo6, as observed in SERS, are certainly not frequent in the natural sciences. It is hoped that the mechanistic studies devoted to the origin of this enhancement will contribute t o the realization of some of the fascinating applications that are presently emerging. ACKNOWLEDGMENT It is a pleasure to thank Paul F. Liao for stimulating discussions

SOI.ID STA r E PHYSICS. VOLUME

38

The Dynamics of Excitons JAI SINGH Rcseurch School o f Cheniisrrx. Ausrrulicin Mutioiiul Unirersity, (iinherra. Ausrruliu'

I.

11.

Ill. IV.

V.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ............ I . Excitons.. . . . . , . , . , . . . . . 2. Theoretical Methods for Cr Theory of Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Hamiltonian in Second Quantization ......................... 4. Wannier Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The Wannier Exciton Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Frenkel Excitons . . . . . . . . . .... ............... 7. The Frenkel Exciton Hamilt 8. Motion of Frenkel Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Critical Observations on the Frenkel Exciton Theory. . . . . . . . . . . . . . . . . . . . 10. Comparison of the Wannier and Frenkel Excitons . . . . . . . . . . . . . . . . . . . . . . Exciton-Phonon Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I I . The Wannier Exciton-Phonon Interaction Operator . ... .... 12. The Frenkel Exciton-Phonon Interaction Operator. . . . . . . . . . . . . . . . . . . . . Composite Exciton-Phonon States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. The Energy Eigenvalue and Eigenvector of the Composite Exciton-Phonon States (0-0 Phonon Transition). . . . . . . . . . . . . . . . . . . . . . . Exciton Reactions.. ..... .......................... .... 14. The Reaction R a t e . , . . . . . . . , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. Binding and Decay of Excitons in Pure Crystals . . . . . . . . . . . . . . . . . . . . . . . . 16. Exciton- Exciton Collisions .............. 17. Fission and Fusion of Excitons , , . . . . . . . , . . , . . . . . . , . . . . . . . . . . . . . . . . . . .. .. ... .. . ... . . . 18. Exciton-Charge Carrier Interactions . . . . . . . . . . . . 19. Rates of the Nonradiative Decay of Excitons _........_..

295 296 291 30 I 30 I 304 314 315 32 1 321 324 326 328 329 332 336 337 34 I 342 342 352 358 363 368

I. Introduction

The subject of exciton dynamics is prominent in almost every study of the spectroscopy and conduction mechanisms of semiconductors and insulators. The theory of Wannier excitons and Frenkel excitons has been known for

'

Present address : Department of Physics, National University of Singapore, Kent Ridge, Singapore. 295 Copyright t i . . 1984 by Acndemic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-17-60773X-X

296

JAI SINGH

about five decades; the former is mostly studied by physicists working on inorganic crystals and the latter by chemists working on molecular or organic crystals, with there being little attempt to bridge the disciplines. Many nonradiative exciton decay processes are known, usually as intermediate channels in luminescence and the generation of charge carriers in nonmetallic crystals, but they have rarely been reviewed in a single article. Here the theory of Wannier and Frenkel excitons is presented first, applying a unified approach, and their results are compared. Second, the various known radiationless processes of excitons in inorganic as well as organic crystals are described. The theory of excitons is given in Part 11. The exciton-phonon interaction is discussed in Part 111, and the nonradiative states formed between the exciton and phonon, as composite exciton-phonon states, are considered in Part IV. In Part V the theory of a few nonradiative processes is described, and the rates of several known processes are given. It is not the aim of this article to compare these rates in different materials; instead, the rates and reaction mechanisms are given in forms suitable to applications. The references quoted are not exhaustive. It is almost impossible to refer to all papers in this very large subject. Often a recent work is cited so that the reader can find earlier references if needed. 1. EXCITONS

Excited but nonconducting electronic energy states produced usually by the absorption of photons in a crystal are called exciton energy states. Such excited states are possible in nonmetallic crystals that have a large energy gap between their conduction and valence bands so that the number of free charge carriers is small. The absorption of a photon or photons, in such crystals, can excite an electron from the valence to the conduction band, creating a positive charge vacancy (a hole) in the valence band. The attractive Coulomb interaction between the excited electron and the hole thus created binds the two together to form a neutral compound system of two charge carriers. Such a system of charge carriers is called an exciton and carries a crystal momentum, equivalent to the vector sum of the individual momenta of the electron and hole, that enables it to propagate through the crystal. Because of the attractive Coulomb interaction between the electron and hole in an exciton, the exciton states lie below the conduction band by an energy equivalent to the binding energy of the exciton in the crystal. In a perfect crystal the motion of an exciton is uniform. This can, however, be hindered by the presence of crystal imperfections such as physical defects, impurities, and structural disorders. Most of these imperfections tend to act

THE DYNAMICS OF EXCITONS

297

as traps for excitons, resulting in either the dissociation of excitons-radiatively or nonradiatively-or localization of the exciton in the traps. The lattice vibrations (phonons) also play a very important role in the processes of dissociation of excitons. In what follows we will study some of the dynamical behavior of excitons in crystals previously mentioned. First of all we shall work out the theory of excitons by applying a manybody approach. A theory of excitons was first formulated by Frenkel’ and subsequently developed by Peierls’ and Wannier and M ~ t t Two . ~ approaches are possible for deriving the energy eigenvalues and eigenfunctions of excitons: one is for the tightly bound, or Frenkel, excitons and the other is for weakly bound excitons (also called large-radii orbital excitons), or Wannier excitons. The background philosophy of the two approaches is different; however, as we will see later on, they produce similar results for the energy and wave functions of excitons. There are excellent books and reviews available on the theory of e x c i t o n ~ ~ and - ~ the reader is referred to them for alternative treatments.

2. THEORETICAL METHODS FOR CRYSTALLINE SOLIDS In practice the experiment is carried out with crystals that usually have lattice defects and impurities, whereas the theory begins with ideal crystals and the impurities are introduced later. This choice, or lack ofit, is due mainly to the availability of simple systems for experiment and theory. We consider an ideal crystal as a model for finding the eigenfunctions and eigenvalues of an exciton. An ideal crystal is a collection of a large number of identical atoms, ions, or molecules (about per cubic centimeter)

’ J. Frenkel. Ph-vs. Rro. 37, 17 and 1276 (1931). R. E. Peierls, Ann. Phy.s. (Lripzig) [ 5 ] 13,905 (1932). G. H. Wannier, Phys. Reit. 52, 191 (1937); N. F. Mott, Proc. R. SOC.London. Ser. A 167,384 (1938). R . S. Knox, Solid Stare Phys. 5, Suppl., 000 1963; D. L. Dexter and R.S. Knox, “Excitons.” Wiley, New York, 1965. R.J . Elliott, in “Polarons and Excitons” (K. G . Kuper and G . D. Whitfield, eds.), p. 269. Oliver & Boyd. Edinburgh and London, 1962. J . 0. Dimmock, in “Semiconductors and Semimetals” (R.K. Williardson and A. C. Beer, eds.), Vol. 3, Chapter 7. Academic Press, New York, 1967; S. A. Rice and J. Jortner. in “Physics and Chemistry of the Organic Solid State’‘ (D. Fox, M . M. Labes, and A. Weissberger, eds.), Vol. 3, p. 199. Wiley (Interscience), New York, 1967. ’ D. P. Craig and S. H. Walmsley. “Excitons in Molecular Crystals.” Benjamin, New York, 1968. A. S. Davydov, “Theory of Molecular Excitons.” Plenum, New York, 1971. H. Haken, “Quantum Field Theory of Solids.” North-Holland Pub].. Amsterdam. 1976.

298

JAI SINGH

arranged on a Bravais lattice whose unit cells are constructed by a set of basic vectors a, b, and c. The Hamiltonian, without the spin interactions, for our ideal system is written as

where fi is a sum of the kinetic-energy operators of all nuclei and electrons and all Coulomb interaction terms; ri denotes the electronic coordinates and X, the nuclear coordinates; and Vi and V, represent the gradients with respect to ri and X,, respectively, and 2, and Z , are the atomic numbers. We intend to solve the following Schrodinger equation for our model system : RY(r,, r z , . . .; x , ,xi,. . .) = EY(rl, r z , . . .; x,, x,, . . .),

(2.2)

where Y(r,, r,, . . .; X , , X,, . . .) represents a crystal wave function of the Hamiltonian [Eq. (2.l)] with the corresponding eigenvalue E . We solve Eq. (2.2) usually within the Born-Oppenheimer approximation, which enables us to write Y as a product of the electronic and nuclear wave functions,

Y(rl, r z , . . .; X , , X,, . . .)

=

@(rl, r Z , . , . ) x ( X , , X,, . . .).

(2.3)

Assuming that Y is as given in Eq. (2.3), we can solve the Schrodinger equation (2.2) first with the electronic part of the Hamiltonian in Eq. (2.1), ignoring the kinetic-energy operator of the nuclei, for a particular nuclear configuration X = (X, ,X,, . ..) of atoms (molecules) in the crystal. The energy eigenvalue thus obtained, as a function of the nuclear configuration, is then employed as the potential-energy term to solve the Schrodinger equation of nuclear motion. In this section, however, we shall not solve the problem of nuclear motion and shall concentrate only on solving the Schrodinger equation for electrons for any particular nuclear configuration as a parameter. For convenience and without loss of generality, we can choose the nuclear configuration of atoms (or molecules) as that at equilibrium. The third term of Eq. (2.1), representing the nucleus-nucleus interaction, then introduces just a constant energy term in the electronic part of the Hamiltonian, and therefore it can be taken into account together with the electron-nucleus interaction term, the fourth term of Eq. (2.1). Our problem thus reduces to solving the Schrodinger equation Ae@(rl,r 2 , . . .) = Ee@(rl,r 2 , . . .),

(2.4)

299


where

It is not explicitly indicated but to be understood that the eigenvalue E , depends on the nuclear configuration as a parameter. We have thus simplified our problem to a great extent; nevertheless, to solve the Schrodinger equation (2.4) for a crystal with about loz3 atoms (molecules) per cubic centimeter is an impossible task. For further simplification, therefore, we assume that the electrons move independently of one another so that @(rl,r2, . . .) is separable into antisymmetrized products of orthonormal one-electron wave functions: @(r,

9

r2'

. . .)

=

A$1(r1)$2(r2). . .,

(2.6)

where A is the antisymmetrizing operator. There are two standard approximations for one-electron wave functions in a crystal: (1) Bloch-type wave functions $r,k(r),where t denotes the electronic energy bands and k is a reciprocal lattice vector, and (2)wave functions $f,n, obtained through the tight-binding approximation as atomic or molecular functions, where f denotes a set of quantum numbers associated with the particular electronic state of a molecule at the lattice site n. The former is of the form of a propagating plane wave and is therefore a delocalized representation of the one-electron wave function, whereas the latter is a localized representation of the one-electron wave function in real space. a. Bloch-Type Functions

The ideal situation considered for describing the electronic ground state of an insulator is that in which the valence band is fully occupied by electrons. This enables us to express, in the independent-particle approximation, the electronic wave function of an unexcited crystal as @O(rl

9

r2'

' '

')

= A$o,k,,,(rl)$o,k,,,(r2)

".

$ O , k , , ~ ( ~ i )'

'

. 3

(2.7)

where 0 represents the valence band and M and p are the electron-spin quantum numbers and respectively. The one-electron wave functions $O,k,u used in Eq. (2.7) are usually expressed as product wave functions:

4

$O,k,a(r)

= $O,k(r)qa=

-4, 112

and

$0,k,/3(r)

= $O,k(r)qp=

- 1/29

(2.8a)

where $t,k(r) = V i 1 I 2 exp(ik*r)u,,,(r),

t = 0, 1,

(2.8b)

t is a band index, and u,,k(r)a periodic function that has the same period as that of the crystal lattice; Vois the total volume of the crystal; q l i 2 and q-l,2

300

JAI SINGH

represent eigenfunctions of the spin projection operator S, corresponding to eigenvalues fh/2. The normalized spin functions are written as column vectors, q l j 2= and q - l / 2 = ( y ) , so that

(A)

An electronic excitation in an insulator/semiconductor is created by exciting an electron from the valence band to the conduction band, which is well separated by an energy gap from the valence band. The energy gap between the valence and conduction bands is a characteristic of the individual crystal. Let us assume that the excited electron in the conduction band has wave vector k and spin CT. As in Eq. (2.7), we can then write the electronic wave function of the excited state of a crystal as @1

= A$0,k,,n(rl)~0,k1,8(r2)

' ' '

$l,k,~(~i).

' ' 9

(2.10)

where t = 1 denotes the conduction band. Such representations of the ground and excited states are commonly used for crystals of inorganic materials where the overlap of the electronic wave functions is large, making it difficult to localize electrons on individual lattice sites. h. Functions in the Tight-Binding Approximation For crystals in which the electronic interaction among the atoms or molecules is rather weak, the electrons, although assumed to be independent of one another, can still be considered as localized at lattice sites because of their interactions with the nuclei at individual lattice sites. If we now denote by Y g nthe ground-state wave function of an atom/molecule at the lattice site n,the ground-state wave function of the crystal can be written as @o =

n

ygn.

(2.1 1)

n

Likewise, the excited state of a crystal corresponds to having one of its sites in the excited state. Assuming that the mth lattice site is excited to its fth state, the wave function of the excited state of a crystal becomes (2.12) The crystal, however, possesses translational symmetry, allowing every lattice site equal opportunity of being excited, hence Eq. (2.12) does not correspond to a proper eigenstate of the crystal. Applying Bloch's theorem however, a linear combination of Ws [Eq. (2.12)] that propagates through the crystal like a wave offers the satisfactory solution (2.13)


30 1

where K is a reciprocal lattice vector and N is the number of unit cells in the crystal. Equation (2.13) is written for crystals with one molecule per unit cell, and K has N possible values in the first Brillouin zone. The descriptions of the ground and excited states as given in Eqs. (2.1 1) and (2.13), respectively, are usually applied to molecular crystals. A comparative observation of the two descriptions of the crystal wave functions, Bloch and tight binding, shows that while in the former every electron is free to move through the whole crystal [see Eqs. (2.8) and (2.10)], in the latter only the excitation of a molecule/atom is free to do so [see Eq. (2.13)]. In Eq. (2.7) the individual electron is important, while in Eq. (2.11) the individual atom/molecule is important. A general analytical form of the Bloch functions ut,,(r) [Eq. (2.8)] is not known; however, these depend more on the characteristics of a crystal and not of the individual atoms or molecules in Eq. (2.11) is regarded to be constituting the crystal. On the other hand Ygn known for individual atoms or molecules of the crystal and does not depend on the crystal structure of the solid. Hence, in terms of Bloch single-electron functions, the individuality of the crystal constituents is lost, whereas it is retained in terms of electron functions from the tight-binding approximation. The former are normally employed for studying the theory of Wannier excitons, and the latter for Frenkel or molecular excitons.

II. Theory of Excitons

For working out the theory of excitons we will adopt a microscopic approach by treating the crystal as a many-particle system. One of the elegant ways of attempting a many-body problem is to use the method of second quantization,lO-’ which we shall use here.

’

3. THEHAMILTONIAN I N SECOND QUANTIZATION In inorganic crystals, in which the overlap of the atomic electronic wave functions is large, we use the delocalized form of single-particle functions described as Bloch waves [Eq. (2.8a)l. The picture of a Wannier exciton is framed essentially within this property of a crystal. We have the electronic

lo

P. L. Taylor. “A Quantum Approach to the Solid State.” Prentice-Hall. Enylewood Cliffs,

I

New Jersey, 1970. A . L. Fetter and J . D. Walecka, “Quantum Theory of Many-Particle Systems.” McGrdwHill, New York. 1971.

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JAI SINGH

Hamiltonian H e given in Eq. (2.5) written in second quantization (see Taylor' O) as He

=

Hc=

H,

+ H,,

(3.1)

c c c c

( ~ l , k l , o ~ ; r 2 , k 2 , o 2 l U ( l-r r 2l l ) l

(3.2b)

i i , k i , o i i r , k z . ~ zt3,ks,u3 t4.k4.64

'3

1

k3

O3; '4,

k 4 3

04)u~l,kl(ol)u~~,k~(02)ut~,k~(03)~t~,k~(04)'

It, k, G) are eigenvectors of one-electron Bloch states, as given in Eqs. (2.8a) and (2.8b), of a crystal in its energy bands t (= 0 for valence and = 1,2, . . . for conduction), with wave vector k and spin o.In analogy with Eq. (2.8), these eigenvectors are a product of the form

(3.3a)

It, k, 0) = It, k)lo),

where It, k ) = l$t,k(r)) and lo) are orthonormal :

= q*,,2,

(olo') =

and, as in Eq. (2.9), the spin kets (3.3b)

'6,u,.

The fermion operators u:~(cJ)and q k ( o )are, respectively, the creation and annihilation operators of an electron with spin o and wave vector k in the tth energy band. These operators obey the anticommutation relations of Fermi operators : ['ti,ki(ol)>

U12rk2(02)l+

= 'tI,tz'ki,kZ'uj,uZ)

['il,kl(ol)r

ai,,k,(ol)l+

= [ ' ~ , , k l ( o l ) ~ u12,k2(02)l+

(3.4) = O'

According to Eqs. (2.8) and (3.3),the matrix elements ofEq. (3.2)are written as

303


where U((r, - r 2 ( )= e2/lr, - r21.

In a translationally invariant lattice only the diagonal form k , = k, of the matrix element Eq. (3.5) is nonzero, and it represents the energy of an independent particle (k) in the tth electronic band. Equation (3.6) is the twoparticle interaction energy matrix element. In order to have a conceptual grasp of the Hamiltonian in Eq. (3.1)microscopically, it is essential that we have a physical interpretation of both the matrix elements, Eqs. (3.5) and (3.6), in terms of the scattering of particles in the crystal. Equation (3.5) represents the scattering of an electron from an initial state ( t , k,, c)to a final state (t,k, , g) by the crystal periodic potential V(r). Because of the translational symmetry of the crystal, the periodic potential cannot scatter a particle from one energy band to another. The matrix element [Eq. (3.6)] represents the scattering of two electrons initially in states ( t ,,k, , c3)and (r4, k,, 04)by the Coulomb potential U ( ( r , - r2 I) to their respective final states ( t Z ,k,, c2) and ( t , , k , , 0,).We shall adopt a convention here: In the two-particle interaction matrix element [Eq. (3.6)] the two central electronic states within the angle brackets, nearest to the scatterer U ( l r , - r21), represent the initial (right) and final (left) states of particle 2 (coordinate r 2 ) , and the two outer electronic states are that of particle 1 (coordinate rl). This is clearly expressed in the integral form of Eq. (3.6), and illustrated in Fig. 1, but is recognized as a shortcoming of the second quantization presentation, where the convention cannot be expressed so clearly. For understanding the theory of excitons on a microscopic basis we will find this convention useful. The electronic Hamiltonian in Eq. (3.1) is the most general one, as it accounts for all the possible energy operators of the charge carriers in all energy bands, excluding spin interactions. two particles involved

--final

states-scatterer-initial

states-

FIG. 1. Interpretation of the two-particle interaction matrix element [Eq. (3.6)] in terms of particle scattering by the Coulomb potential U(lr, - r21).

304

JAI SINGH

4. WANNIEREXCITONS

An electronic energy state with all the electron states fully occupied in the valence band and completely empty in all the conduction bands is defined as the vacuum or ground state of a semiconductor/insulator crystal. Let us denote such a state by the ket 10). The minimum energy gap between the valence and the first conduction band is denoted by E , (see Fig. 2). Now if a photon of frequency hv 2 E, is incident on the crystal, one of the electrons from the valence band can be excited to the first conduction band by absorbing the photon and leaving a positive charge vacancy (hole) behind in the valence band (see Fig. 2). Let us denote the valence band by t = 0 and the first conduction band by t = 1. We assume that no other conduction bands are either excited or able to interact in the excitation. The fermion operators operating on the vacuum eigenvector obey, then, the following relations, which, in another way, define the vacuum state of the crystal as well. a&,(o)lO) = 0

and

a,,,(a)lO)

=

0.

(4.1)

The pair of charge carriers thus generated can remain free, producing the photoconductivity in the crystal, or because of their attractive Coulomb interaction, the electron and hole can form a bound energy state just below the conduction band. (This does not mean that the photons create the charge carriers first, which then form bound states. These bound states are

E hv

E,

FIG.2. Electronic energy bands and the generation of an electron ( - ) and hole (+) in nonmetallic crystals.


305

created directly as well.) Our aim here is t o calculate the energy and eigenfunction of this bound state called an exciton. It would, however, be convenient if, like for electrons, we defined the creation (dt) and annihilation ( d ) operators for holes as well: -k(

uA,k(‘)

-

aO,k(o)= di,-k(-o)

do,-k(-o)lo) = 0.

and

(4.2)

The hole operators also obey the anticommutation rules of fermion operators. a . The Wave Function and Energy Eigenvalue of a Wannier Exciton

Let us assume that an exciton with wave vector K and spin S is created in the crystal by exciting an electron with wave vector p and spin 0, in the conduction band and a hole with wave vector K - p and spin oh in the valence band, so that the total crystal momentum is conserved as

P+K-P=K electron

hole

(4.3) exciton

+ oh = S = 0 ce + oh = S = 1 ce

(singlet exciton), (triplet exciton).

(4.4)

The eigenvector of the exciton state can now be written as a linear combination of all the product eigenvectors of such free charge carriers created in the crystal: (K, V ; S ) =

C8 C,@, K - B)(B, K - P; S ) ,

(4.5)

where v is a quantum number of the exciton’s internal energy states, which will be described later on, and p sums over the first Brillouin zone. Cf(P, k - b) is the expansion coefficient and will be determined at a later stage. The product eigenvectors lp, K - 6; S) are given as

(4.6)

for singlet excitons, and

(4.7)

for triplet excitons. Spin angular momentum is expressed in units of h.

306

JAI SINGH

To evaluate the exciton eigenvalue W,(K, S) we solve

Re1K, V ; S)

=

WJK, S)IK, V ; S).

(4.8)

The standard procedure for solving Eq. (4.8) is to operate on it by any term of the product eigenvector in Eq. (4.9, say, Ik’, k”; S’) from the left-hand side. Thus we get

(k‘, k”; S’lfIelK,V; S) = W,(K, Sj(k’, k”; S’IK, V ; S),

(4.9)

where (k’, k”; S’1 = (lk’, k ; S ’ ) ) ? . For calculating the left- and right-hand sides of Eq. (4.9) the following procedures may be quite useful : 1. Choose any term of Ik’, k”, S’) from Eq. (4.6) or (4.7), for example, (4.10)

lk’, k”, o’, o”) = a~,k.(o’)d~,k,,(o”)lO).

2. Similarly, choose any term of the eigenvector in Eq. (4.5), for example,

Ic;

p 3

-

p; O e ? Oh)

=

c,(p>

- P)a:.P(a,)d~,K-P(o,,)Io).

(4.11)

3. Use Eqs. (3.1), (4.10), and (4.11) in Eq. (4.9) to determine the matrix elements

(k’, k”, o’, o”lfIelC; p, K - p; oe,o,,) and (k’, k”, o f ,o”lC; p, K

-

p; o e ,oh)

by converting the electron and hole operators into one form through Eq. (4.2) and then using the anticommutation relations of Fermi operators [Eq. (3.4)]. According to Eqs. (4.1) and (4.2), many undesired terms will thus vanish. 4. Calculate Eq. (4.9), eventually using Eqs. (4.6) and (4.7), accounting for all possible spins of the electrons and holes. The first result thus obtained from Eq. (4.9) is a well-established one,

[(k‘, k”; S‘IA,IK, V ; S )

=

Wv(K, S)(k’,k’’;S’IK,v;S)]6,,,.,

(4.12)

that there is no nonzero matrix element between the singlet and triplet exciton states of the Hamiltonian in Eq. (3.1). The final form of Eq. (4.12) is then obtained as [W,

+ E,(k’) - E,( - k”) - W,(K, S)]C,(k’, k”) = C CJp, K - P)[(1, k’; 0, p KIUIO, -k”; -

1, p)

P -

(1 - S)( 1, k ; 0, fl - KI UI 1,

0, - k”)],

(4.13)

noting that the second term on the right-hand side of Eq. (4.13) vanishes for triplet excitons (S = 1). Also note that any quantity or eigenvector with a


307

negative wave vector and/or spin represents a hole following the definition of Eq. (4.2). The quantities in Eq. (4.13) are as follows: W,

=

2

1 Eo(k) + 1 (2(0, k l ; 0, kZlUI0, kz; 0, k l ) k

ki,Lz

- ( o , k , ; 0 , k 2 p p ’ k 1 ; O,k,)),

(4.14)

where U = U(lr, - rz/), Eo(k) = (0, kl - hzV2/2m, + V(r)(O, k) is the energy of a free (noninteracting) electron and W , represents the total electronic energy of the crystal before it is excited. E,(k’) = El(k‘)

+ 1 (2(1, k’; 0, kl UIO, k; 1, k‘) k

-

(1, k’; 0, kl UI 1, k’; 0, k))

(4.15)

is the total energy of an excited electron in the conduction band (t = l), with wave vector k’, including its interaction energy with the rest of the electrons in the valence band [note that no wave vector is negative in Eq. (4.15)].

E,(-k”)= E,(-k”)+C(2(0,

-kf‘;O,klU1O,k;O, -k”)

k -

(0, k”; 0, kl UIO,

-

k”; 0, k))

(4.16)

is the total energy of an excited hole in the valence band, with wave vector - k”, including its interaction energy with the rest of the electrons in the valence band. Neither E,(k’) nor E,( - k ) contains the interaction energy between the excited pair of the electron and hole in the crystal. This is given by the two terms on the right-hand side of Eq. (4.13). All contributions from the Coulomb interaction operator E?, [Eq. (3.2b)l to the energies W , [Eq. (4.14)], E,(k‘) [Eq. (4.15)], E,( -k”) [Eq. (4.16)], and the interaction energy of the excited electron and hole given by the righthand side of Eq. (4.13) have two terms. As is customary, the first term is called the Coulomb interaction, and second the exchange interaction, and the main difference is that compared to the Coulomb interaction term in the exchange interaction, either the initial or final states of the two particles get exchanged, leaving the ordering of the particles as given in Fig. 1 unchanged. The two interaction energy terms have opposite signs, and except for the particles of the same energy states, the exchange interaction is usually considered smaller than the Coulomb one. b. Singlet and Triplet Excitori States In the energy equation (4.13) the spin dependence is seen only through the exchange interaction energy term with 1 - S [the second term of the right

308

JAI SINGH

side of Eq. (4.13)]. The spin dependence of the coefficients is assumed to be negligible. It is therefore only the exchange interaction, which vanishes for triplet excitons, between the electron and hole that essentially distinguishes the two spin states of an exciton. Consequently, as is obvious from Eq. (4.13), the interaction energy between an electron and a hole in their triplet state is larger than in singlet states. It is this interaction energy that is responsible for the formation of an exciton (bound pair of an electron and a hole) state. c. Solution of the Energy Equation (4.13)

Although Eq. (4.13) is simplified, we cannot solve it for W,(K, S ) and C,(B, k - B) exactly. In fact, it is very difficult to solve it even approximately for the case of singlet excitons with nonzero exchange interaction. The first simplifying assumption to be made therefore, is, that the exchange interaction between the electron and the hole is negligible. Thus the second term on the right-hand side of Eq. (4.13) can be neglected. The justification in neglecting the exchange interaction is based on the fact that such an interaction is of short range.5 If the separation between the excited electron and hole in the crystal is large, the exchange interaction between them will be relatively small and can therefore be neglected. As a result of this approximation, the Wannier excitons are called also the large-radii orbital excitons, a name given

FIG.3. Illustration of the effective-mass approximation


309

and derived from the theory and suitable only for singlets. A solution of Eq. (4.13) including the exchange interaction has not been attempted yet. Neglect of the exchange interaction, however, leads to the following equation, which can be solved only approximately:

[Wo + E,(k') =

-

Ev(- k")

1 C,,(p,K

-

-

W,,(K, S)]C,,(k', k")

p)(1, k'; 0,p

-

KIUIO, -k"; 1,p).

(4.17)

B

In general it is convenient to transform Eq. (4.17) into real space for a solution. In doing so we consider that k' and k" are small and measured, respectively, from the minimum of the conduction band, assumed to be at k = K O , and from the maximum of the valence band, assumed to be at k = 0, as illustrated in Fig. 3. With the help of the effective-mass a p p r o ~ i m a t i o n ~ ~ ' ~ applied to electrons and holes, we can then write E,(k')

N

+ h2kI2 2m,* '

Ec(Ko)

~

(4.18a) (4.18b)

where m,* and mz are the effective masses of an electron in the conduction band and a hole in the valence band. In Eqs. (4.18) we replace k' by - iV, and k" by -iV,, where V , and V, are the gradient operators with respect to r l and r 2 , respectively, the position coordinates of the electron and the hole. We now use Eqs. (4.18) in Eq. (4.17) and multiply Eq. (4.17) thus obtained from the right-hand side by exp(ik'-r, ik"*r,) and then sum over all the k' and k" within the first Brillouin zone to find

+

where E,(Ko) = Ec(Ko) - EV(0),the energy gap between the minimum of the first conduction band and the maximum of the valence band, G(r,, r,)

=

1

C,(k', k")eik"rleik"'rz,

(4.20)

k',k"

(4.21)

3 10

JAI SINGH

Equation (4.19) can only be solved provided that we know the matrix element (1, k; 0, fi - K J U ( 0 ,-k”; 1, fi), obtained through Eqs. (2.8b) and (3.6) as (1, k‘; 0, J3 - KIU10, -k”; 1, J3)

uCJ, - kt’(r2)u1,b(r1)e

i(p- k’).rlei(K

~

p- k”).rz

d3r, d3r,.

(4.22)

Evaluation of the integral in Eq. (4.22)is not possible because of the unknown analytical form of the Bloch functions. To evaluate the integral analytically, therefore, one has to find a way to get rid of the Bloch functions. As the product of Bloch functio~sis also periodic, it may be expanded in a Fourier series: (4.23a)

where L, and L2 are reciprocal lattice vectors and I , and I, are the Fourier coefficients given by (4.24) Substituting Eqs. (4.23) in Eq. (4.22), we obtain

‘LI+@-k‘

where

6L 1 + ” + K , k ’ + k ’ ’

(4.25)

r

U L I + p - k= l V;’

J

U(lrl)ei(L1+P-k‘)’r

d 3r,

r

=

rl

-

r2.

(4.26)

Following Landsberg,’ we assume that the main contribution to I comes from L, and L2 being nearly zero. We can then write (4.27)

’* P. T. Landsberg, ed., “Solid State Theory: Methods and Applications.” Wiley, New York. 1969.

31 1


Equation (4.27) enables us to write Eq. (4.19) in the form of a Schrodinger equation, using Eqs. (4.21) and (4.25):

=

(4.28)

Y ( K ) G @ ' , r2),

where the spin index S from W,,(K,S ) is dropped, as the operators on the left-hand side of Eq. (4.28) d o not depend on the spin. Thus G ( r l , rz) becomes the eigenfunction of the exciton Schrodinger equation (4.28). d . The Problem in a Dielectric Medium

We have thus far reduced the many-body Schrodinger equation [Eq. (4.8)] to that of an electron and a hole interacting through their attractive Coulomb potential U(lr, - r2)), similar to that of a hydrogen atom or positronium. In the case of a hydrogen atom, however, there is only one electron and one proton, and their attractive Coulomb interaction is, therefore, written for a vacuum (the dielectric constant 6 = 1). Equation (4.28)is derived in a crystal in the presence of a many-particle field and therefore the Coulomb potential between the electron and the hole cannot simply be assumed to be the same as that in a vacuum. The most crucial approximation that we have made in deriving Eq. (4.28) is Eq. (4.27), which effectively makes the Bloch functions orthonormal : ~ r ~ ( r ) u ~ ,=~ d, (, ,r, .) Consequently, the Coulomb potential U(lrl - r21) of Eq. (4.28) remains the same as in Eq. (3.6), the two charges interacting in a vacuum ( 6 = 1) [Eq. (4.28) is equivalent to considering the charge carriers as plane waves]. However, U(lr, -'r21) in Eq. (3.6) is correct, because each charge carrier individually is interacting with others in the vacuum. The lack of an analytical form for the Bloch functions thus prevents the determination of the correct dielectric function and interaction potential between the electron and hole. Haken' has suggested the use of the static dielectric constant of the crystal. This approximation is essentially as follows. As we have assumed k' and k" to be small, and from Eqs. (4.25) and (4.27), K = k' k" = p (K - p). We may assume that k' and k" K - p, and then to the zeroth order we may write that

+

-

+

uT,k,(r1 )u1, @('I)

ry

-

uT,k,(r

l,k'(r1)

-

u:,@-K ( r 2 ) U 0 ,

u,*,~k..(r,)u,,-k,,(r2) =

t- 'P,

'' H . Haken. J . Phys. Chem. Solids 8, 166 (1959); in "Polarons

-

-

k"(r2)

(4.29)

and Excitons" (K. G. Kuper and G. D. Whitfield, eds.), p. 295. Oliver & Boyd, Edinburgh and London, 1962.

312

JAI SINGH

where E is the static dielectric constant of the crystal and is assumed here to be equal to the square of the normalization constant of the Bloch functions. The Schrodinger equation (4.28) then becomes

=

(4.30)

W,,(K)G(r,, I-').

As Eq. (4.30) has been solved by several workers in the theory of excitons, we should provide only the main results here. However, there are two different methods generally used for solving Eq. (4.30)and they yield slightly different results, particularly for the wave function of the exciton. We will describe the two approaches here briefly. e. The Perturbation Approach

Apply the transformation r

=

rl

- r2,

R = $r,

+ rz),

K

=

+

k'

+k ,

(4.31)

+

so that the gradient operators become V , = V, i V , and V, = - V, $VRr, where V, and V, are with respect to r and R. Using Eq. (4.31) in Eq. (4.30) and changing the variables accordingly in G(r, , r 2 ) to G(r, R), we get G(r, R) = V;1/2eiK'R 4,m

(4.32)

where q5,(r) is an eigenfunction of

(4.33)

If we consider the K * p (p = ih V,) term of Eq. (4.33) as a perturbation, the eigenvalue of a crystal with one exciton is obtained as (the second-order correction to the energy van is he^'^)

where M* is the exciton effective mass and E , the energy eigenvalue of

(l4

Vl

-

"> cr

4 3 r ) = Evq5:(r).

See, e.g , G . Dresselhaus. J . Ph,ls. Chem. Solids 1, 14 (1955).

(4.35)

313


E , = -pe4/2h2~’v2 represents the binding energy of an exciton in the hydrogen atomlike energy states with principal quantum number v = 1,2,3, ... (s, p, d, . . .). The first-order eigenfunction cPl,(r)of Eq. (4.32) is obtained as

(4.36) f . Center of Mass Coordinate Transformation

Instead of using Eq. (4.31), if we transform the electron and hole coordinates as Rc

=

m:r,

+ m*r ’,

M*

r = rl

-

(4.37)

r2,

we get the same energy eigenvalue W,(K) from Eq. (4.30) as obtained in Eq. (4.34). However, then G(r, R,)

=

(4.38)

V;’i2eiK’Rc4:(r)

becomes an exact eigenfunction of the Schrodinger equation (4.30), unlike Eq. (4.32), where 4 t ( r )is only the unperturbed part of the eigenfunction. Both the transformations [Eqs. (4.31) and (4.37)] are identical if m,*= mt. The energy eigenvalue [Eq. (4.34)] and eigenfunction [Eq. (4.32) or (4.38)] clearly exhibit the nature of an exciton state in the crystal. The exciton has rn; to move the kinetic energy that enables the total mass M* = m: throughout the crystal, and the binding energy E , with which the electron and hole form bound hydrogen atom-type energy states. The value of E , is negative and therefore the discrete exciton states lie below the edge of the conduction band, as shown in Fig. 4. The equation v = 1 represents the ground state of a Wannier exciton and usually E , = , is of the order of 10 meV in inorganic solids. In terms of exciton states one may redefine the conduction band as the continuum of exciton states. The wave vector K is called the exciton wave vector and K 0 in an optical excitation. The kinetic energy (h2K2)/2M*of the exciton is usually of the order of k,T ( k , = the Boltzmann constant) at room temperature.

+

-

-

g. Determination of the Coeficients C,($, K - p)

We first defined the exciton eigenvector as given in Eq. ( 4 3 , with the unknown coefficients C,(p, K - p). Later on we derived two other forms of the exciton wave function, as given in Eqs. (4.20) and (4.32) or (4.38). One should note that the form in Eq. (4.5) is the most accurate, followed by that in Eq. (4.20), and then the rest. Here, however, we determine C,(p, K - p) by

314

JAI SINGH

E I

I I I

-

I

1

-

I

.,k -

0

FIG.4. Wannier exciton's internal energy states ( v duction band.

=

wo

1.2.3.. . ; s,p.d.. . .) below the con-

considering Eqs. (4.20) and (4.38) as equivalent, and then multiplying both sides of the equations by V;' exp{i[p-r, + ( K - p)-r2]}. By integrating over rl and rz, we eventually obtain

For the internal ground state ( v

=

1) of the exciton we have

47(r) = 71- l 1 2 ~ 3 1 2exp( -ar)

and

a = p/M*a,c,

and then from Eq. (4.39) we get

In actual problems involving excitons we use the exciton eigenvector [Eq. (4.5)] with coefficients as given in Eq. (4.39) or (4.40) for the ground state.

5. THEWANNIER EXCITONHAMILTONIAN In the previous section we showed that the electronic Hamiltonian A, [Eq. (3.1)] of the crystal does have exciton states with eigenvalues WJK) [Eq. (4.34)] and eigenvectors (K, v ; S ) . We may now redefine the exciton eigenvector [Eq. (4.5)] to write it in terms of exciton operators:

IK, V ; S)

=

1 C,(&K - p)Ifl, K - p; S ) b

3

Bi(S, ~)(0),

(5.1)

315


where BL(S, v) represents the creation operator of a Wannier exciton with wavevector K, spin S, and internal energy quantum number v. The expression for BA(S, v) in terms of the electron and hole operators is obvious from Eqs. (4.5)-(4.7). Assuming the completeness relation of exciton states as

multiplying by one such operator on the left and another on the right of the Hamiltonian [Eq. (3.1)] and then using Eq. (4.8),the Wannier exciton Hamiltonian is obtained as

I?,":

Wv(K)BL(S,v)BK(S,v).

=

(5.3)

K.v

A sum over S = 0 and 1 in Eqs. (5.2) and (5.3) is implied. The exciton operators are assumed to obey the commutation relations of Bose operators. In fact, assuming that the number of excited electron-hole pairs in the crystal is small, it can be shown from Eqs. (4.5)-(4.7) that the exciton operators do behave like Bose operators: [B,(S? v),

Bi,(S', v')]-

[BK(S,v), BK,(S',v')]-

(5.4a)

= BK,K'BS,S'B~,~',

=

[BA(S, v), BA.(S', v')]-

= 0.

(5.4b)

It is easily verified that the Hamiltonian [Eq. (5.3)] does satisfy the Schrodinger equation [Eq. (4.8)]. In deriving Eq. (5.3) using Eqs. (5.1) and (5.2) in Eq. (4.8), we actually get the operators on the right-hand side of Eq. (5.3), with the vacuum-state vector as

BLP,

v)]o)(o(B,(s, v).

(5.5)

This is typical of an operator in the second quantized form obtained from the single free-particle eigenstates." The properties of the Hamiltonian [Eq. (5.3)] are not influenced by the operator (O)(Ol.

6. FRENKEL EXCITONS

In organic crystals or molecular crystals where the molecular overlap of the wave functions is small, the one-electron wave functions are best represented by the electronic molecular wave functions of the individual molecules, as described in Section I. It is therefore more appropriate when studying these crystals to express the electronic Hamiltonian [Eq. (3.1)] in

316

JAI SlNGH

the real crystal space as He

=

Ho

+ Hc,

(6.1) (6.2a)

where

Fermion operators ~ J J , ~ (and G) ~ ~ , ~ are( ethe) creation and annihilation operators of an electron with spin (T in the electronic state j of the molecule at 1. The Hamiltonian equation (6.1) is obtained by replacing the ts by j s and the ks by Is in Eq. (3.1), and therefore each term of Eq. (6.1) can be understood in real crystal space in exactly the same way as those of Eq. (3.1) in k space. The single-electron eigenvectors lj, I, c) are product ket vectors as in Eq. (3.3a), that is, Ij, I, c) = ( j ,1)1c), where ( j ,I) = l$j,l(r - I)) is the oneelectron molecular eigenvector of the electronic state j of the molecule at I. Here we do not need to know the form of tjj,&r - 1) for deriving the formal theory of Frenkel excitons; we only assume that these are localized at the individual molecular sites of the crystal. At the end of the section, however, we will specify what is commonly used in actual calculations. a . Eiyenvectors of’ Frenkel Excitons Suppose that an incident photon of appropriate frequency excites one of the molecules at I in the crystal such that an electron is created in its f th excited state, as well as a hole in the ground state g. Denoting the creation I; S ) of operator of the excitation by B;,,(S), we can write the eigenvector such excitations as

If,

If, where

1;

s) = Bj,l(s)lo),

s=0

or

1,

(6.4)


317

Equations (6.5) and (6.6) are similar to Eqs. (4.6) and (4.7) in k space for Wannier excitons. The definition of the hole operators follows from Eq. (4.2), except that we do not change the sign of the lattice sites in converting from the electron to the hole operators, that is,

dg,l(- a) and ag,,(a)= d;,,( - a). (6.7) The ket 10) in Eq. (6.4) defines the electronic ground state of the crystal, that is, all molecules are in their ground state, which is completely occupied (closed shell), so that a:,,(a)

E

d,,,(o)IO)

=

a f , , ( o ) ( o )= 0.

(6.8)

The translational symmetry of the crystal, however, provides no way of distinguishing a particular molecule excitation by an incident photon; any one of the molecules may be excited. Thus we must write the eigenvector of a crystal with a single excitation, and with one molecule per unit cell, as I,f,K;S)

=

N-’12 ~ e i K ” B J , , ( S ) I O ) ,

(6.9)

I

where K is the reciprocal lattice vector and N is the number of unit cells in the crystal. The left-hand side of Eq. (6.9) can be defined in terms of another operator B J , K ( S )like , Eq. (6.4), as B f ,K ( S ) = N-’12 ~ e i K ” B ~ , , ( S ) .

(6.10)

I

Although Bj,K(S)is only a Fourier transform of Bj,l(S),the former is usually called the creation operator of a Frenkel exciton with wave vector K, and latter the creation operator of a localized excitation on a particular molecule. Many workers’,’ like to distinguish between the exciton and excitation with this point of view, that an exciton’moves throughout the crystal, while the excitation is localized and does not move. However, as far as the definition of Eq. (6.10) is concerned, the operator Bj,K(S)is localized at K in the reciprocal lattice space but delocalized in real space, whereas Bj,l(S)is localized at 1 but delocalized in the k space, which is obvious from the inverse transform of Eq. (6.10). I t is important t o understand that the delocalization of Bj,K(S)in Eq. (6.10) does not mean that there is motion of the excitation from one lattice site 1 to another, as is often misunderstood from the sum over 1. In other words, Bj,K(S) does not refer t o the motion of an excitation through the crystal, but it only signijies the indistinguishability of one molecule from another in a pure crystal lattice. I t is therefore only a matter of tradition and of no logical consequence to call Bj,K(S)the operator of an exciton that moves and Bj,I(S)that of an excitation that is localized. The motion of a Frenkel exciton or excitation depends on the intermolecular interactions, which we shall deal with in Section 11,6,a.

318

JAI SINGH

Equation (6.9) represents the eigenvector of a crystal with one Frenkel exciton at K and with spin S created by an incident photon that can excite to the f t h excited state of individual molecules in the crystal. As the electronic properties of the individual molecules are thus retained to a large extent in the crystalline form in organic solids, the Frenkel exciton is also called the molecular exciton. Equation (6.9) is the same as Eq. (2.13), but in a different representation (second quantization).

6. Energy of Frenkel Excitons We shall solve the Schrodinger equation f i , ( f , K; S) = W,(K S ) ( f ,K; S ) ,

(6.11)

where W,(K, S) is the energy eigenvalue of a Frenkel exciton with wave vector K and spin S. We operate on Eq. (6.11)by the complex conjugate eigenvector ( f , K; S ( = ( I f , K; S ) ) + and then use Eqs. (6.1) and (6.5)-(6.10) to obtain the energy eigenvalue as Wf(K, S ) = Wo

+ Ef

- E,

+ Eb(S) + L,-(K, S),

(6.12)

where 12

(6.13)

-

E,N

=

( j , ~ ; g , ~ z ~ U ~ j , ~ ; g , j~=z f) ) ~ or ,

g,

(6.14)

N - l C [ ( f , 1; g, l p l g , 1; f , 1) I

- S ) ( f , 1; g, IJUlf, 1; g, I)] exp[iK.(m - I)]M[,,, Lf(K, S) = N - '

(6.15)

- (1

(6.16a)

I,m # I

M[,,,

=

-

[(g, 1; f , ml ~

- (1 - S)(g,l;

l f 1;, g, m) f , mlUlg, m; f,l)],

S=0

or

1. (6.16b)

In deriving Eq. (6.12), the matrix elements with two excited electrons and holes are omitted. The values Wo, E,, and E , correspond, respectively, to

319


the energies W,, E,(k'), and E,( -k") in Eq. (4.13) of the Wannier exciton. Unlike Eqs. (4.14)-(4.16), where the hole was designated by a wave vector with negative sign, in Eqs. (6.13)-(6.16) the hole states are not easy to recognize. What we have followed here, however, is that 1, and 1, represent unexcited molecular sites, so lg, I , ) and lg, 12) represent the state of valence or ground-state electrons, whereas 1 and m represent the excited molecules and thus lg, I) or 18, m) represent the states of the excited hole, and I f , I) or m) the states of the excited electron (note that this assignment is only a matter of convenience, as I , , l,, 1, and m are dummy variables). Accordingly, W , is the total electronic energy, including the interaction energy of all the electrons in the unexcited crystal. Ef is the energy of an excited electron in the f t h excited state and E , is that of a hole in theground stateg of a molecule at 1, including their interaction energy with the rest of the valence electrons of other unexcited molecules in the crystal; Eb(S) is the interaction energy between the excited electron and hole on the same molecule I, and L,(K, S) represents the same between the excited electron and hole lying on different molecules. In a perfect crystal E,, E,, Eb(S), and LJK, S) do not depend on the location of the excited site I, thus the sum over I in these quantities means summing N equal terms. The total energy Eb(S) + L,(K, S) is the counter part of the term on the right-hand side of Eq. (4.13) in k space. The energy Ef - E , is equivalent t o the band gap energy as obtained in the case of the Wannier exciton. The interaction energy Eb(S)is responsible for holding the excited pair of electron and hole (exciton) on one molecular site, while L,(K, S ) is responsible for transferring them (electron and hole) to other molecular sites of the crystal. The matrix element MI,,, [Eq. (6.16b)l is known as the resonance transfer matrix element of the Frenkel exciton. The form of the Frenkel exciton energy [Eq. (6.12)] is very convenient for a comparative study of Frenkel and Wannier excitons, as discussed earlier. W,(K, S) is, however, usually written in the literature in a slightly different form that can readily be obtained by rearranging the terms of Eqs. (6.13)(6.15) as

If,

W f ( K ,S ) = W,, + AE,(S)

+ D, + L,(K, S ) ,

S = 0 or

1. (6.17)

A E f ( S )consists of all the diagonal terms 1, = I of energy E, - E , from Eq. (6.14) plus the binding energy E,, and it represents the difference between the excited state (singlet or triplet) and the ground-state energy of an isolated molecule. 0,represents the difference between the energy of interaction of an excited electron and a hole at a molecular site, with all ground-state electrons on neighboring molecular sites. Df is given by (6.18)

320

JAI SINGH

where D{l* = [2((f, 1; g, 1,Iulg, -

4;f , 1)

((.f,1; g, 1,Iulf, 1; g, 1,)

- (g,l; g, I,IUlg, 1,; g, 1)) - (g, 1; g, 121 ulg, 1; g, I,))].

(6.19)

D f d a is independent of the location of the excited lattice site 1 in a pure is known as the dispersive interaction matrix element.8 Dl,12of crystal. D1,12 Eq. (6.19) includes the exchange interaction matrix elements (the last two terms), and the factor 2 associated with the Coulomb interaction matrix elements is obtained from spin considerations. c. The Efectiue Mass of a Frenkel Exciton

T o write the energy of a Frenkel exciton in a form analogous to that of a Wannier exciton [Eq. (4.34)] we expand L,(K, S) in terms of a Taylor’s series about K 0; K is known to be small for optical excitations. The first term of the series L,(O, S), although indicating the transfer of the excitation from one molecule to another in the lattice through the matrix element M;,, [Eq. (6.16b)], represents a part of the binding energy [M;, is negative; Eq. (6.16b)l of the exciton when the electron and hole are located at different molecular sites (owing to the carrier transfer matrix element) or when the electron and hole are in the process of being transferred to different molecules (owing to the energy transfer matrix element). L,(O, S) arises from the translational symmetry of the crystal used in writing the eigenvector [Eq. (6.9)] at K = 0. The second term of the Taylor series vanishes and the third term contributes to the kinetic energy of the Frenkel exciton as h2K2 (6.20) L,(K, S) = L,(O, S) 2M* ’

-

+

~

where M * is the effective mass of the Frenkel exciton given by (6.21) As L,(K, S ) is independent of the excited molecular site I, it is chosen to be at I = 0 in Eq. (6.21). M* is obviously different for singlets and triplets [see Eq. (6.16b)l. Substituting Eq. (6.20) into Eq. (6.12), we get the energy of the Frenkel exciton analogous to that of the Wannier exciton,

14a

Usually D, has been used in the literature8 as the matrix element of the intermolecular interaction operator. which includes the electron-nuclei interaction operator as well. However, as is obvious from Eq. (6.19), the electrons-nuclei interaction operator does not contribute to this interaction matrix element.

32 1


Equation (6.20) may also be used in Eq. (6.17) to write the expression for the energy of the Frenkel exciton. 7. THE FRENKEL EXCITON HAMILTONIAN The Frenkel exciton Hamiltonian can easily be derived in a way analogous to that used to derive the Wannier exciton Hamiltonian in Section 5. Using the completeness relation of the Frenkel exciton eigenvectors [Eq. (6.9)],

c IK,~;S)(K,~;S[

=

1,

S =O

or

1,

(7.1)

K .I

and the electronic Hamiltonian equation (6.1), we get the Frenkel exciton Hamiltonian HIx through Eq. (6.11):

HLx =

c Wf(K, S)B;,,(S)Bf,,(S),

S

=

0 or

1.

(7.2)

f.K

Like in case of the Wannier exciton [Eq. (5.5)] in Eq. (7.2), we also get the vacuum-state or ground-state operator 10) (01, which has been omitted here. The Hamiltonian fiTx [Eq. (7.2)] is written in the reciprocal lattice vector space in which it is diagonal. However, the Frenkel exciton Hamiltonian is often useful in real space as well. Applying the inverse transformation of Eq. (6.10) to the exciton operators in Eq. (7.2), and using the properties of the translational symmetry of the crystal, we obtain H:x in real space as

firx =

c [W, + AEf(S) + c D;,lB:,,(S)Bf,,(S)

f,I

mZI

The Hamiltonian in Eq. (7.3) is not diagonal in real space. Here again the difference between firx of Eqs. (7.2) or (7.3) and H e of Eq. (6.1) is that H e consists of more terms that do not satisfy the Schrodinger equation (6.11). No terms of Eq. (7.3) depend on the location of the excited molecule in a perfect crystal.

8. MOTIONOF FRENKEL EXCITONS The difference between the excited state of an isolated molecule and a Frenkel exciton is that the latter is the excited stale of a crystal composed of the molecules. The essential difference is therefore the introduction of the intermolecular interactions in the crystalline state. It is obvious from the Frenkel exciton energy eigenvalue in the form of Eq. (6.17) that D, and L,(K, S) are thus the additional energy terms incurred in going from isolated

322

JAI SINGH

molecules to crystals. We need not worry about W , here, which is only a constant. D, is an energy associated with the excited state of the crystal such that one of the molecules of the crystal, say, at I, is excited, causing a change equal to Of in the ground-state intermolecular interaction energy of the crystal, and because of this energy the excitation remains in a stationary state at 1 in the crystal. This, however, does not mean that we are fixing the excitation on any one particular site of a pure crystal and violating the law of translational invariance. This excited site could be any one in the crystal, but whereuer it is, the excitation will remain there because of the electrostatic interaction energy Ds. The effect of Df on the Frenkel exciton is illustrated in Fig. 5, taking the example of a linear lattice and using Eq. (6.18). On the other hand, the energy LJK, S) is due to the resonance transfer interaction energy matrix elements M,,,, [Eq. (6.16b)], which consist of the Coulomb and exchange interaction matrix elements. The Coulomb interaction matrix element (g,l; f ,m(UI f , 1; g, m), keeping in mind Fig. 1, transfers a hole from its initial site m to a final site 1, and at the same time an excited electron from the site 1 to m, owing to the potential U(lr, - r2)),as shown in Fig. 6. As the molecular overlap of the electronic wave function is small in organic crystals, the contribution of the Coulomb interaction to M,,,, is very small.Theexchangeinteractionmatrixelement(1- S)(g,I; f , mlUIg,m; f , l ) vanishes for the triplet exciton ( S = 1). However, for singlets it provides what is known as the hopping motion of the Frenkel exciton; that is, initially an excited electron at I recombines finally with the hole at I and the energy thus liberated is transferred to create another excited electron and hole on another

323

THE DYNAMICS OF EXCITONS e

0

0

-1

m Molecular s i t e s

FIG.6 . Motion of a Frenkel exciton due to the intermolecular electronic interaction. The arrows show the direction of transfer of the charge carriers (e = electron. h = hole).

molecule at m,as illustrated in Fig. 7. The contribution to the motion of a Frenkel exciton from the Coulomb interaction between the excited electron and the hole involves the actual transfer of individual charge carriers (electron and hole) from one molecular site to another (Fig. 6 ) in the crystal. The exchange interaction, however, transfers the energy of excitation from one molecule to another (Fig. 7). Therefore, while both mechanisms (carrier and energy transfer), the latter being dominant, are responsible for the motion of a singlet Frenkel exciton, only the former takes care of the transfer of a triplet exciton in the crystal. Appropriately, we may refer to the Coulomb interaction matrix element as the carrier transfer, and to the exchange interaction as the energy transfer matrix element. As the magnitude of the e

q

e

z1

a? I c

W W

,,

Wf

+

Molecular sites FIG.7. Illustration of the hopping motion of a singlet Frenkel exciton due to the exchange interaction between the excited electron (e) and the hole (h) in the molecular crystal (single arrows indicate the direction of recombination of charge carriers, and the double arrow the direction of the energy transfer).

324

JAI SINGH

carrier transfer matrix elements are smaller than that of the energy transfer matrix elements, the kinetic energy of a triplet Frenkel exciton is less than that of a [see Eqs. (6.20) and (6.21)]. D, and L,(K, S) thus act as competitive components of the motion of a Frenkel exciton: While D, tries to hold the excitation on one site, L,(K, S) tends to move it away to other sites. Typically, for a singlet D, is of the order of 1000 cm-’ and L,(K, S) I 100 cm-’, resulting in an exciton staying sec on a molecule before moving to another molecule in a time for 10-12 Sec.l6.l7

--

-

9. CRITICAL OBSERVATIONS ON THE FIZENKEL EXCITON THEORY

1. Consider the expressions for W,, in Eq. (6.13) and Ef and E , in Eq. (6.14). In the first term E,,l,,l, of W,, we summed over all the lattice sites (sum over 11) and this is supposed to take into account the single-particle energy of all the electrons in the ground state g. The question then arises as to how many electrons per molecule we are considering in the ground state of the molecule? Have we assumed only one electron per molecule in the electronic ground state of the crystal? These questions, however, d o not arise for the first term (E,,l,l) of E, and E, I I of E , [Eq. (6.14)], which represent clearly, and respectively, the energy of one excited electron in the electronic state f and of one excited hole in the ground state g. But again, in the twoparticle interaction terms of W,,, E,, and E , , the second summation over 1, takes into account only the molecular sites and it is not clear as to how the interactions between one excited carrier and all the other ground-state electrons (whatever their number may be) of each molecule in the crystal are actually evaluated. 2. We have not specified so far the form of the one-electron molecular functions $j,l(r - I). When we write, for instance, $,,,(r - 1) as the oneelectron wave function of the ground state of the molecule at 1, what is the form of this wave function? As long as we present the theory only formally, these questions, even if they are relevant, d o not create any problem. In actual calculations, however, the importance of these questions cannot be ignored and we ought to define g in some other way. The only way in which this problem can probably be

J . Singh, in “Radiationless Processes” (B. DiBartolo and V. Goldberg. eds.), Vol. 62. p. 465. Plenum, New York, 1980. F. Gutmann and L. E. Lyons, “Organic Semiconductors.” Wiley. New York, 1967. ” J . B. Birks, “Photophysics of Aromatic Molecules.” Wiley (Interscience). New York, 1970.

’’


325

solved is by using the molecular orbital method, which will be considered in what follows. The Molecular Orbital Approach

Most of the organic molecular crystals that have extensively been studied for the excitonic processes consist of large closed-shell molecules, for example, naphthalene, anthracene, and tetracene. For these molecules the only electronic functions known are in the semianalytical form of the molecular orbitals (MOs). The calculation of MOs is, however, beyond the scope of this text. The electronic ground state of a closed-shell system is a singlet. Let us denote by g ithe fully occupied MOs up to the highest occupied one, which we shall denote by g. Thus in the ground state we have the MOs gl, g 2 , g 3 , . . ., g; g1 is at the bottom of the energy scale, and g at the top (the number of g is 4 2 , n being the total number of electrons in the ground state of a molecule). Let us now identify the one-electron molecular wave function of the M O g iof a molecule at I, with the one electron eigenvector Isi,I ; 0) with electronic spin cs as in Eq. (6.3). Then the ground-state electronic wave function of the molecule at 1 can be written as the antisymmetrized product of these functions”: (9.1)

IN)

where = ICS= i) and Ip) = ICS= - r2 ). The ground-state eigenvector of a crystal, denoted by as a product of all lg, I) in the crystal, as in Eq. (2.11):

lo), can be expressed (9.2)

The excited state of a molecule is defined as a promotion of one electron from the highest occupied MO g to the lowest empty MO f of a groundstate molecule. The excited-state eigenvector I f , 1, S ) of a molecule at 1 is “ S o m e quantum chemists might express some concern over the structure ot the molecular = Iyi, I)) are calculated as the eigenvector [Eq. (9.1)], because usually the MOs $,,.,(l$q,,l) linear combination of atomic orbitals, and therein the electron spin functions are associated with every atomic orbital. Here, however. from Eq. (9.1) we construct the ground-state function [Eq. (9.2)]of a crystal and assume that the MOs are already calculated. Thus the $y,,l do not involve the electron spin functions, but only the calculated coefficients and the corresponding atomic orbitals. The structure of Eq. (9.1) produces no errors at all. neither in the formal theory nor in any quantitative calculations.

326

JAI SINGH

then written as

If71; S)

=

c

J+I,fh

- 1))l%)I+l3&r2

-

I))l%)

bc*bh

x A

Y-i

-~ ) ) l ~ ) l ~ I , & Z

I+l,&1

- I)>IS>

(9.3)

91

or

If,

I; S )

= Bj.,(S))O),

S

=

0 or

1,

which is identical to the eigenvector in Eq. (6.4), except that in Eq. (6.4) f and g denote the electronic states of the molecules, whereas in Eq. (9.3) they refer to the MOs. The definition of a Frenkel exciton eigenvector [Eq. (6.9)] and the corresponding creation operator [Eq. (6.10)] is analogous. The exciton energy expressions (6.12) and (6.17) remain unchanged. However, the energies Wo [Eq. (6.13)], Ef,and E , [Eq. (6.14)] are written correctly as s

+ 1 1 [2(gi,

wo = 1 I1

( 2 ~ q , , 1 ~ , 1 ~

Y,

11; g l ,

12~~(r -1rZ)Igf,12;

gi, 1 1 )

12 4 d J :

(9.4)

E~ = ~

-

1 c [w, 1; gi,1 , l ~ r ~r2)lgi,1,;

F1( E ~ , + ~,~

g

-

j , I>

12 Y i

- ( j , 1;

si,1 2 1 ~ ( r 1 r 2 ) l j ,I ; gi,I,)]),

j = .f

or g.

(9.5)

We now have summations over all the occupied MOs gi in Eqs. (9.4) and (9.9, taking into account all interactions of electrons in the ground state. 10. COMPARISON OF THE WANNERAND FRENKEL EXCITONS

We have developed the theory of Wannier and Frenkel excitons from the same electronic Hamiltonian H e [Eq. (2.5)] of a crystal. However, for Wannier excitons we used H e in reciprocal lattice space [Eq. (3.1)], and for Frenkel excitons H e was real crystal space [Eq. (6.1)]. Many of the further steps in obtaining the ultimate results of the eigenvalues and eigenfunctions of the two excitons are also based on similar grounds. EgriI8” has presented a unified approach with a one-dimensional model for the theory of Wannier and Frenkel excitons considering a parametrical dependence on the binding I . Egri. J . Phys. C 12, 1843 (1979)


321

and localization in the exciton bands. Nevertheless, there are the following marked differences between the two excitons. 1. The Wannier exciton eigenvector [Eq. (4.5)] is written as a linear combination of the eigenvectors of various possible pairs of charge carriers ofdifferent wave vectors. The probability amplitude coefficients C,,(P, K - P) are different, and hence the probability IC,,(P, K - P)I2 that a pair of charge carriers binds into a Wannier exciton is different, for different wave vectors P. In the Frenkel exciton case, however, the probability of any molecule being excited is the same, and hence the probability amplitude coefficients are simple phase factors [Eq. (6.9)]. 2. Neglect of the exchange interaction from the energy equation [Eq. (4.13)] of the Wannier exciton produces its binding energy E , of Eq. (4.359, equivalent to that of hydrogenic states formed by the electron and the hole, and E , is obtained independent of the exciton’s spin. Consequently the energy WJK) [Eq. (4.34)] of the Wannier exciton is also spin independent. In Frenkel excitons, on the other hand, the exchange interaction between the excited electron and the hole cannot be ignored, as both occupy the same molecule. Therefore the binding energy of the Frenkel exciton [Eb(S)] in Eq. (6.15) is spin dependent. The Coulomb interaction between the excited electron and the hole [first term of Eq. (6.15)] in the binding energy Eb(S)is not negligible here, as it would be if the charge carriers resided on different molecules. In fact, the Coulomb and exchange interactions in Eb(S) CEq. (6.15)] are expected to be of comparable magnitude. This leads one to conclude that the magnitude of the binding energy of a singlet Frenkel exciton &,(S = 0 ) is less than that of a triplet exciton; the exchange interaction vanishes for triplet excitons in Eq. (6.15), that is 0

IE,(S

=

o)( < ]Eb(S= 1)1-

(10.1)

Usually for organic crystals Eq. (10.1) is satisfied. Accordingly, from Eq. (6.12) it is obvious that since Eb(S)is negative, the singlet exciton energy is higher than that of a triplet exciton in molecular crystals. This is a wellestablished experimental result for molecular crystal^'*^^'^; it is in fact a consequence of the well-known results17 for isolated molecules. As can be seen from Eq. (6.17), AEJ(S = 0), the singlet transition energy of an isolated molecule, is larger than its triplet transition energy AEf(S = 1). 3. The motion of a Wannier exciton is due to the individual kinetic energies of the electron and the hole and does not depend on the interactions between the two. We must, however, keep in mind the approximations used in deriving Eq. (4.25), which enabled us to obtain the potential of interaction between the electron and the hole as wave vector independent. The kinetic energy of a Wannier exciton, therefore, does not depend on its spin state; in

328

JAI SINGH

other words, it is not possible to draw the conclusion that the signlet Wannier exciton moves faster or slower than the triplet Wannier exciton. The case of a Frenkel exciton, however, is slightly different, because a Frenkel exciton’s motion from one molecule to another depends on the intermolecular interactions. As discussed in Section 9, the kinetic energy of a singlet Frenkel exciton [Eq. (6.20)] is greater than that of a triplet Frenkel exciton; in other words, the crystal volume traversed by a singlet Frenkel exciton in a particular time interval is larger than that traversed by a triplet Frenkel exciton in the same time interval. 4. We have renamed the Coulomb and exchange interaction matrix elements of the electron-hole pairs as the carrier and energy transfer matrix elements in molecular crystals. This is, however, applicable to inorganic crystals as well; the Coulomb interaction matrix element [Eq. (4.22)] represents the electron (p) getting transferred to k’ and the hole ( - k to f! - K) remaining in their respective conduction and valence bands. This is a carrier transfer process, and the exchange interaction represents the energy transfer process. In the theory of Wannier excitons we have found the carrier transfer matrix element to be more important, although for the sake of convenience we did not incorporate it in the transfer of excitons from one region of the crystal to another, as described in (3). In actual excitonic reactions, as will be shown in later sections, the carrier transfer matrix elements (or carrier transfer processes) are usually dominant in Wannier excitonic reactions, and energy transfer is usually dominant in Frenkel excitonic reactions. This distinction is due to the difference in the characteristics of inorganic and organic solids. 5. The Wannier exciton Hamiltonian :?I [Eq. (5.3)] and the Frenkel exciton Hamiltonian fiFx [Eq. (7.2)] are of similar form in reciprocal lattice space. Except in some particular problems, the indices v [Eq. (5.3)] and’ f [Eq. (7.2)] are not usually used. Therefore, dropping v and f from the exciton operators in Eqs. (5.3) and (7.2), we can write the Wannier and Frenkel exciton Hamiltonian in a unified form:

EJ(K)BL(S)B,(S), J

H:x =

=W

or F,

(10.2)

k

where Ew(K) = WJK) [Eq. (4.34)] for the Wannier exciton and EF(K)= W f ( K ,S ) [Eq. (6.12),(6.17), or (6.20)] for the Frenkel exciton.

111. Exciton-Phonon Interactions

In the previous sections we discussed the theory of excitons, assuming the crystal lattice to be at equilibrium. Now we introduce lattice vibrations.

;3-9


There are two reasons for studying phonons in solids. First, in practice they are an intrinsic part of a solid, and, second, their incorporation does assist in arriving at the desired results in theory. For instance, the reaction of formation of an exciton state from a free electron-hole pair or vice versa is not possible in a crystal without the participation of phonons required for the conservation of energy and momenta. And for the same reason phonons play an active role in several other excitonic reactions in solids. As there are several books’ 0*19 and articles20-21treating phonons, we will present only a brief comparative study of the Wannier exciton and phonon and Frenkel exciton and phonon interaction operators in a form suitable for our use in the theory of excitonic reactions. 1 1. THEWANNIEREXCITON-PHONON INTERACTIONOPERATOR

The Wannier exciton-phonon interaction is derived from the electronphonon interactions in inorganic solids. There are two forms of electronphonon interaction operators used in inorganic solids: (1) Frohlich’s Hamiltonian and (2) the deformation potential interaction operator. In the former the electrons are assumed to be scattered by the lattice vibrations or ionic motions of the nuclear lattice, whereas in the latter they are carried with the ionic motions creating strains in the lattice. Frohlich’s Hamiltonian is derived from the Taylor expansion of the matrix element of the ionic potential about the lattice equilibrium.’ O J ’ The first-order term of the expansion thus obtained gives the electron-phonon interactions that are linear in the lattice displacement vectors. However, up to the first order the form of Frohlichs Hamiltonian and that of the deformation potential operator are similar.2‘ Here we will describe these operaiors very briefly and then derive the Wannier exciton-phonon interaction operator linear in the lattice displacement vectors. The ionic potential V(r) =

1 V(r - I), I

.

where

V’(r - I) =

Z,e2 Ir - 11’

--

as in Eq. (2.1), contributes to the single-particle energy matrix element in Eq. (3.2); I is used in place of XI [Eq. (2.1)] to denote the position vector of the lattice site 1 at equilibrium. Owing to the lattice vibrations, we assume that 1 is displaced to I + R,,R,being the vector of lattice displacement; and J . M. Ziman. “Electrons and Phonons.“ Oxford Univ. Press, London and New York, 1960. J. Bardeen and D. Pines, Phys. Rea. 99, I140 (1955). 2 1 L. J . Sham and J . M . Ziman, SolidStuie P h j ~5, . 223 (1963). l9

330

JAI SINGH

introducing this in the matrix element of Eq. (3.2), we obtain through Eqs. (2.8), ( 3 3 , (4.23), and (4.27) the matrix element of V(r): (t, kllV(r)lt, k2)

=

xexpi[(k, - k 2 ) . ( l + R l ) ] V i l - k 2 ?

(11.1)

I

where

(11.2) Assuming that the lattice displacement vectors are small, such that the translational symmetry of the crystal is still preserved, we expand Eq. (11.2) in a Taylor series about the equilibrium 1 and retain only terms linear in R,, and express R,in terms of lattice waves:

where s and q denote the branch and wave vector of a phonon with frequency o,(q), I is the ionic mass coefficient and the e,(q) are unit polarization vectors. ba,s represents the creation operator of a phonon with wave vector q and in branch s. Substituting then, Eq. (11.1) thus obtained in Eq. (3.2), the first term of the Taylor's series gives the usual single-particle energy matrix elements diagonal in k, and the second term the first-order electron-phonon interaction operator I -- N-1i2

cc

M:(Q)a:k+q(o)a,,k(a)(btq,s

+ bq,s),

(11.4)

t,k,o q,s

where N is the number of atoms in the crystal, with one atom per unit cell. The interaction operator in Eq. (11.4) has no contributions from the twoparticle interaction operator (A,) of Eq. (3.2b). In inorganic crystals, since the interatomic overlap of the electronic wave functions is large, it is not possible to localize the charge carriers at individual atoms of the crystal. Mj(q) is given by (11.5) It is obvious from Eq. (11.5) that for transverse phonons M:(q) = 0 [q-e,(q) = 01 within the first Brillouin zone, that is excluding the Umklapp processes. M:(q) is also zero for phonons with q = 0.


33 1

As considered in the theory of Wannier excitons (Section 4), we assume the involvement of only two electronic energy bands, t = 0 (valence band) and t = 1 (conduction band), in Eq. (11.4). Converting a operators into d operators according to Eq. (4.2) for holes in the valence band, we get from Eq. (1 1.4) the Wannier exciton-phonon interaction operator

I??

=

c1

[',r(q)af,k+q(a)a,,k(a)

- M,O(cl)d~,-k(-a)dO,-k-q(-a)l

k,a q,s

x

W , , S

+ bq,A

(1 1.6)

where the superscript W stands for the Wannier exciton. The exciton-phonon interaction operator @' [Eq. (11.6)] is derived within the rigid ion approximation. However, if we use Mj(q) and M;(q) in Eq. (11.6) as (11.7) we find the form of the interaction operator obtained from the deformation potential method.22*24*25 The variables C , and Co are known as the deformation potential parameters and are of the order of atomic energies. The Wannier exciton-phonon interaction operator E?pv [Eq. (1 1.6)] is in a very suitable form for actual calculations, because the exciton eigenvector [Eqs. (4.5)-(4.7)] is usually expressed in terms of the electron and hole operators. However, Eq. (11.6) can be written in terms of Wannier exciton operators [Eqs. (5.3) and (5.4)] as well. Repla~ing'.~a l , k + q ( a ) a , , k ( a ) and d&,-k(--a)do,k-q( -a) by B$+,(S)B,(S) in Eq. (11.6), we obtain the Wannier *' The deformation potential23 of the exciton-phonon interaction in an inorganic semiconductor is

where r , and r2 represent the electron and hole coordinates. Using the Bloch wave functions [Eq. (3.3a)I for electrons and holes in the deformed lattice as a first-order approximation for their deformed Bloch states, and applying the principles of translational symmetry, we obtain H , kom H , in the second quantization, with M:(q) as given in Eq. (1 1.7). The equivalence between the rigid ion approximation and the deformation potential is presented in detail by Sham and Ziman." 23 J. Bardeen and W. Shockley, Phys. Reu. SO, 102 (1950). 24 A. A. Lipnik, Sou. Phys.-So/icf State (Enyl. Trans/)I , 661 (1960); 2, 1835 (1961); 3, 1683 (1962). 25 J . Barrau. M .Heckmann, J . Collet, and 1w.Brousseau, J . Phjs. C h m . So/ids34,1567 (1973).

332

JAI SINGH

exciton-phonon operator

12. THEFRENKEL EXCITON-PHONON INTERACTIONOPERATOR In organic solids the expression for Frenkel exciton-phonon interaction is usually d e r i ~ e d ~ *from ~ ~ the * ~Hamiltonian ’ [Eq. (7.3)] by expanding it in a Taylor series about the lattice equilibrium. It is done by assuming that the displacement of molecules from equilibrium, due to lattice vibrations, is small. The main reasons for t h s assumption are (i) the translational symmetry of the crystal can still be assumed to be preserved so that the transformation [Eq. (6.10)] of Bj,,(S)into BJ,k(S)and vice versa are applicable in a vibrating lattice as well, and (ii) the Taylor series can then be terminated at the first-order term linear in displacement vectors; the higher-order terms can be neglected. Therefore, in a vibrating lattice the Frenkel exciton Hamiltonian retains the same form as Eq. (7.3), however the indices 1 and m which denote molecular positions at equilibrium must be considered to represent the instantaneous positions of molecules. In order to present this even more clearly, we may replace 1 and m in Eq. (7.3)by I’ and m’for a vibrating lattice, and without the loss of any generality we can consider I‘ = 1 + R, and m’= m R,; R, and R, are the displacement vectors from the equilibrium position of molecules at I and m. Thus in a vibrating lattice the Frenkel exciton Hamiltonian can be written from Eq. (7.3) as

+

+

1

M{,m,

(R)B),,,(S)Bf,m. (S);

(12.1)

f,l’.m‘ +I‘

1’

=

I

+ R,

and

m’= m

+ R,,

where E?Fx(R)denotes the Frenkel exciton Hamiltonian in a vibrating lattice at a displaced configuration R in the configuration space of molecules such that at R = 0, fiFx(0) [Eq. (12.1)] is identical with the exciton Hamiltonian [Eq. (7.3)] in the equilibrium configuration [W, is set to zero]. Expanding Eq. (12.1) about the equilibrium configuration in a Taylor series up to the first order we get

r?Fx(R)= E?rx(0)+ fir, ” R. M . ” D.

Hochstrasser and P.N. Prdsad, Excited States 1, 79 (1974) P. Craig and L. A. Dissado. Chem. Phys. Left. 44,419 (1976).

(12.2)


333

where H r is the first-order Frenkel exciton-phonon interaction and written as

A:

=

Tl(R)

+ T2(R) + T3(R),

(12.3)

(12.4b)

( i2 . 4 ~ ) 1’= I

+ R,

and

m’= m

+ R,

In T,(R) [Eq. (12.4a)l AEf(S)does not depend on the displacement vector R; however, according to Eq. (6.10), the excitation operators Bft,l.(S)do depend on 1’ or on the displacement vector R because 1’ = 1 + R,. It may be emphasized here that the excitation operator BJ,l.(S) is the same as the wave function of an excited molecule I’ [see the analogy between Eqs. (2.13) and (6.9)]. According to the tight-binding principle, then the excitation operators must remain localized on the excited molecule and move with it during the lattice vibrations. In view of this we can write Eq. (12.3) as

fir = Tl(R) + T’JR)

+ T\(R) + T!(R) + TY(R)

(12.5)

where

(12.6a)

(12.6b)

334

JAI SINGH

In order to evaluate Eqs. (12.4a), (12.6c), and (12.6d) we have to evaluate the derivatives like

From the inverse transformation of Eq. (6.10) we can write

Bf,,,(S)= N-'I2

c exp(ik.I)Bf,k(S),

(12.8)

k

and Eq. (12.8) gives

[";('I

= R,=O

c ik exp(ik.I)Bf,,(S) + N - 1 / 2c exp(ik-I)

NP'j2

k

(12.9)

k

In the second term of Eq. (12.9) it may be assumed that dBf,,(S)/dR, = 0. As one can see that the right-hand side of Eq. (6.10) sums over all lattice sites (1) Bf,k(S) is independent of any position coordinates of molecules in the lattice. Likewise, Bf,l(S)is independent of k. Thus we can write2' (12.10) Using Eq. (12.10) in Eq. (12.9) we get = R,=O

N-1/2

c (ik) exp(ik-I)Bf,,(S).

(12.11)

k

Using Eqs. (1 1.3), (12.8), and (12.1l), and the translational symmetry properties of the crystal in all the terms of Eq. (12.6), we get the first-order

'' J. Singh, Chem. Phys. 75,371

(1983)

335


Frenkel exciton-phonon interaction operator fif as2'

fir = N-112

c c (F,(k, q) + XA) + f

q)

+ J,@.

q))

k, w

+

x B:,Lfq(S)B/.~(S)(btq,, bq,J7

(12.12)

where the coupling functions F,(k, q), x,(q), Is(k, q), and J,s(k,q) are:

(12.13)

(12.15) (12.16) The coupling functions F,(k, q) and xs(q) are obtained by combining Ti@) [Eq. (12.6a)l and T'JR) [Eq. (12.6b)], ZJk, q) is obtained from T,(R) [Eq. (12.4a) and TY(R) [Eq. (12.6c)], and J,(k, q) is obtained from T:'(R) [Eq. (12.6d)I. exciton-phonon interaction In earlier c a l c ~ l a t i o n s ~of, ~the ~ ,Frenkel ~~ operator contributions of the coupling functions Z,(k,q) and J,Q,q) are completely ignored without mention of any reasons. It is to be noted that the origin of the coupling functions Zs(k, q) and J,(k, q) is due to aBf,,(S)/aR, [Eq. (12.9) being nonzero]. On the contrary, however, the previous res u l t ~ ~are, derived ~ ~ * ~ by ~considering aBJ,,(S)/i3R, = 0. This implies that in the previous c a l c ~ l a t i o n s ~it*might ~ ~ * ~have ~ been assumed that although the molecules move away from their positions of equilibrium due to the lattice vibrations, the molecular excitation wave-functions remain still localized at the equilibrium and hence do not move with the molecules. As mentioned earlier, this contradicts directly the principle of the tight-binding theory, which is the key principle of achieving the Frenkel exciton formalism, and hence it is not acceptable. The correct Frenkel exciton-phonon interaction operator [Eq. (12.12)] has been derived.28 The magnitudes of the four coupling functions F,(k,q), Xs(q), l,(k, q) and J,(k, q) depend on the particular crystals. In general however xs(q) plays apparently the dominant role in case of transverse phonons and phonons with q = 0, otherwise it is IJk, q) that dominates. J,(k, q) is zero when k or

336

JAI SINGH

k.e,(q) vanishes. The quantitative estimates of the coupling functions I&@, q) and Js(k, q) are given for anthracene by the author.28 A comparison between Eq. (12.15) and Eq. (11.5) suggests that the derivation of the coupling function Z,(k, q) in Frenkel exciton-phonon interaction is analogous to that of M:(q) in the Wannier exciton-phonon interaction. A part of the energy W,@, s ) in Eq. (12.15) involving no intermolecular interactions [see Eq. (6.12)] contains explicitly the difference between the interaction energy of an excited electron with nuclei exactly as M,'(q)- M:(q) in Eq. (11.8). Therefore Z,(k, q) is the only term that contains contributions common to the Frenkel exciton-phonon interaction and the Wannier exciton-phonon interaction." Reference 28 may be seen for details. However, MunnZsa has recently argued, but not derived, that the past results are essentially correct and the new coupling functions derived by SinghZ8 are not applicable for the Frenkel exciton-phonon interaction. Munn has also argued that it is proper to consider the excitation operators Bj,,(S) as independent of the lattice displacement vectors. In view of Eq. (12.11), and Eqs. (12.6), (12.15), and (12.16), it is obvious that Munn's arguments are incorrect, and consequently his results for the Frenkel excitonphonon interaction are also incorrect. It has been discussed in detail by Singh. The exciton-phonon interaction Hamiltonians of Eqs. (1 1.8) and (12.12) can be written in an unified form as

fi;

GJBL+,(S)BL(S)(bLq,,+ bq,s);

= N-1'2

J

=

W

or F , (12.17)

,.k,q

where by substituting GW = Mb(q) - M;(q) we get the Wannier excitonphonon interaction and GF = F,(k, q) + xs(q) + Z,(k, q) + J,(k, q), the Frenkel excito1:-phonon interaction operators. In writing Eq. (12.17)the index f for the excited state of the Frenkel exciton and v for the internal energy state of the Wannier exciton are dropped out.

IV. Composite Exciton-Phonon States

The total Hamiltonian of the exciton and phonon with the linear excitonphonon interaction operator in a crystal can be diagonalized further. The diagonalized Hamiltonian thus obtained is an operator whose eigenstates represent the composite exciton-phonon states in a crystal. These states '".I

2Bb

R. W. Munn, Chem. Phps. 84,413 (1984). J. Singh. Chem. Phys. To be published.

337


give rise to the tail structure that is more evident in emissionz9 than in the absorption spectra of crystalline solids at very low temperatures. A photon incident on a crystalline solid can create an exciton with a certain wave vector, as we have studied in the foregoing sections. Because of the exciton-phonon interaction, however, the possibility that the photon can also excite an exciton and some phonons conserving the total momentum and energy cannot be excluded. Under such circumstances what can be the structure of these excited states in a crystal? 13.

ENERGY EIGENVALUE AND EIGENVECTOR OF THE COMPOSITE EXCITON-PHONON STATES(0-0 PHONONTRANSITION)

THE

The Hamiltonian (flip) of the exciton, phonon and exciton-phonon interaction can be written from Eqs. (10.1) and (12.9) as (13.1)

where Hph

=

1

hws(q)(bA,sbq,s

-k

3).

(1 3.2)

s*q

The phonon Hamiltonian is not derived in this article; however, its form is familiar. We assume that initially (before the excitation) the phonon population in the crystal is given by the ket In),

).1

=

Inl, nz, . .., n,, . ..),

(13.3)

with n,, being the occupation number with wave vector q. We now excite an exciton, assuming its wave vector to be K, without changing the initial phonon population. The eigenvectoi of such an exciton state, in interaction with the phonons, can be written as the linear combination of the eigenvectors of all excited states with one exciton and all possible creations and/or annihilations of phonons conserving the total crystal momentum. However, there can be an infinite number of terms in such a linear combination, and the resulting secular equation cannot be solved. An approximate solution is possible. As the exciton-phonon interaction Hamiltonian is only of the first order, involving the interaction of the exciton with a created or annihilated phonon, we can in a consistent way limit the linear combination to eigenvectors of one exciton plus or minus one phonon state. Within this approximation, however, we must choose a parent state-the one that is being excited by the photon, for example, a pure exciton state (0-0 phonon transition) or an exciton plus a phonon state (0-1 phonon transition)-and z9

K.P. Meletov. E. I . Rashba, and E. F. Sheka, JETP Lett. (Engl. Trans/.)29,

165 (1979)

338

JAI SINGH

the corresponding eigenvector can be called the parent eigenvector or ket. In what follows we will describe the energy eigenvalues and eigenvectors for a 0-0 phonon transition in a crystal. Similar procedures can be applied for 0-1 and 1-0 phonon transitions. Craig and Singh3' and Singh3' have calculated the energy eigenvalues in some organic crystals for 0-0 and 0-1 phonon transitions. However, as it is obvious from the Hamiltonian [Eq. (13.1)], the method and results are applicable to both inorganic ( J = W) and organic ( J = F) solids. The eigenvector of a 0-0 phonon exciton state can be written, accordingly, as

,

+ C,(k, K

-

k ; n - l)B,t$k-K]}) 10; n ) ,

(13.4)

where we have omitted the spin index S from the exciton operators and the branch index s from the phonon operators to avoid complications. The term (K, S ; n ) represents the eigenvector of an excited state (singlet or triplet) of a crystal expressed as the linear combination of (1) the parent eigenvector [first term of Eq. (13.4)] of a pure exciton (K), with no change in the initial phonon population, (2) the eigenvector of a state with an exciton (k) accompanied by the creation of a phonon (K - k), and (3) the eigenvector of a state with an exciton (k) accompanied by the annihilation of a phonon (k - K). In other words, the eigenvector of the composite exciton-phonon state is written as a linear combination of the eigenvectors of the parent state and of states that differ by f 1 in the number of phonons from the parent state. The probability amplitude coefficients C, and C, apply to the creation of an exciton without any change in the initial phonon population, and to the creation of an exciton together with the creation or annihilation of one phonon with conservation of the total wave vector. 10; n ) is the initial state of the crystal before it is excited, a product of the initial electronic state 10) (as the valence band or ground state) and the phonon state In). We can determine the eigenvalue AJ ( J = W or F) of the Hamiltonian E?:, [Eq. (13.1)] associated with the eigenvector IK, S ; n ) by solving

I?ipIK, S ; n )

=

AJIK, S ; n ) ,

(13.5)

which provides through Eqs. (lO.l), (12.9), (13.1), and (13.2) the following

3"

D. P. Craig and J. Singh, Chem. Phys. Leii. 82,405 (1981) .I.Singh, to be published.

339


three equations:

[WJ(K) - AJ]C,(K, 0; n)

+ C GJICl(k, K - k; IZ + 1) k

X

(1 f

5K-k)

f C,(k, k - K;

-

0,

(13.6a)

+ 1) = 0,

(13.6b)

=

l)fik-K]

G*JCo(K, 0; n)

+ [WJ(k) + hw(K - k) - IZJ]Cl(k, K

-

k; n

-

K; II

G*JCo(K, 0; n)

+ [WJ(k) - ho(k - K) - AJ]Cl(k, k

-

1) = 0,

(13.6~)

where

WJ(k) = EJ(k)

+ 1ho(q)(fi, + i).

(13.7)

q

+

1) from Eq. (13.6b) and C,(k, k Substituting C,(k, K - k; n from Eq. (13.6~)into Eq. (13.6a), we get the secular equation WJ(K) - AJ

=

C (GJ)’ k

(

1

+

-

K; n - 1)

fi,-k

WJ(k) + ho(K

-

k) - IZJ

-

+ WJ(k) - hw(k - K) - AJ “k-K

(13.8)

with fik = [exp(hcu(k)/k,T) - 1 1 - l .

Normalization of the eigenvector [Eq. (13.4)] gives

{

Co(K, 0; n ) = N - ’

[’

+

(GJjZ

(

1+ + ho(K - k) + AJI2 fiK-k

[WJ(k)

-

nk-K

+ [WJ(k) - ho(k - K)

-

AJI2

j]}-1’2.

(13.9)

The secular equation (13.8) gives the energy eigenvalues AJ for the composite exciton-phonon states in a crystal. However, to solve the secular equation (13.8) is difficult. In k space the number of eigenvalues depends on the number of phonon modes and branches, acoustic or optical. In inorganic crystals with N atoms the number of lattice modes is 3N - 3 and the degree of the secular equation becomes 6N - 5 . In organic crystals with N molecules and n atoms in each molecule, we have a total of 3nN - 3 modes: (3n - 6 ) N intramolecular modes and 6 N - 3 lattice modes. It is not always possible to separate the intramolecular from the lattice modes in molecular crystals, as there is considerable interaction between the two. It is therefore

340

JAI SINGH

attractive to solve the secular equation (13.8) in energy space by converting the sum over k into an integration and then using the density of states to convert the integral from k space into energy space, for example, (13.10) where p ( E ) is the density of states of the particles involved and f(k) is an integrand. For a known p ( E ) it is often easy to evaluate the left-hand side of Eq. (13.10). For organic crystals, with Eq. (13.10) the secular equation (13.8) has been solved in the low-temperature limit by Sir~gh.~' The Hamiltonian l?ipcan now be written in a diagonal form. Assuming the completeness relation of the eigenvectors [Eq. (13.4)] for every In) [as was done for the exciton states; see Eqs. (5.2) and (7.1)] and using Eq. (13.5), we can write the Hamiltonian A:, in the following form:

fi;,

=

1 ,lfAk(i)AK(i),

J

=

W or F,

(13.11)

K.i

where the operators Ak(i) and AK(i)are defined as A&(i)= C,(K, 0; n)Bk

+ Cl(k,

+

[C,(k, K - k; n

+ l)BIbL-k

k

- k;

+ l)Bibk-K]lo;

(13.12)

n)(oEpl,

and AK(i) is the complex conjugate of Ak(i) [Eq. (13.12)]. The index i refers to the ith eigenvalue of the secular equation (13.8); accordingly, every eigenvalue will have its own set of probability amplitude coefficients. loEp) is the vacuum state of the composite exciton-phonon particles. Note that the vacuum-state operator (O)(Ol does not appear in Eq. (13.11) as it did in Eqs. (5.3) and (7.2), because we have incorporated it by introducing the new vacuum state loEP).It is obvious from Eqs. (13.1)-(13.8), but not stated explicitly in order to avoid complicated notation, that the functional form of the operators A&(i)is different for inorganic and organic crystals. The operator Ak(i) is analogous to a single-particle creation operator.' Assuming the composite exciton-phonon to be a boson, we can define the operators in terms of their occupation numbers. Writing N i , K for the occupation number in the ith eigenstate with K, we can define A&(i)and AK(i)as the symmetrized product of the eigenvectors":

Ak(i) = 1f l i , K I N l , K ?

N2,K,

. . . ?N i , K ) ( N l , K ?

N2,K,

Ni,K

-

1,. . - 1 ,

(13.13)

Wt AK(I')=

zfii,K(Nl,K, {Nl

N2,K,-..,Ni,K

-

l,..*) (13.14)


34 1

where { N ) means to sum over all terms of the symmetrized products. The operators of Eqs. (13.13) and (13.14)satisfy the boson operator commutation relations. V. Exciton Reactions

This section is devoted to the processes of radiationless decay of excitons in inorganic and organic crystals. A number of such processes have been observed and studied theoretically; however, to the author’s knowledge they have not yet been compiled in any single review or book. In the previous sections we dealt with the stationary states of exciton and composite excitonphonon states having a certain lifetime, after which they may either dissociate nonradiatively into pairs of charge carriers or recombine radiatively. There are many other forms of stationary-state excitons, for example, biexcitons, polyexcitons, or their Bose condensations, which are possible due to high exciton concentrations in crystalline solids and which have drawn a great deal of interest in crystals like Si, Ge, CuCl, and CuBr. The formation of biexcitons is analogous to that of a H2 molecule, provided that the excitonic concentration in the crystal is sufficiently high. Under such circumstances polyexcitonicmolecules can also be formed in semiconductors with multivalley conduction bands, like Si and Ge. The subject of biexciton and polyexciton molecules, particularly in inorganic crystals, has been extensively reviewed by some of its pioneer^,^' and therefore it will not be discussed here. However, experimental results on biexcitons and polyexcitons are available only for inorganic crystals, where the results of different experiments are at variance and hence have been interpreted differently. Theoretical attempts have .been made to investigate biexcitons33-34a and triexcitons3’ in molecular crystals as well; however, the author is not aware of any experimental results in molecular crystals, although the exciton bands are known to have multivalleys, like cosine and sine functions. There may be several reasons for this: (1)The rates of radiative emission may be faster than the process of formation of excitonic molecules, although in the case of the triplet Frenkel exciton this may not always be

’’ V. S. Bagaev, J. Biellmann, A. Bivas, J . Goll, M. Grosman, J. B. Grun, H. Haken. E. Hanamura, R. Levy, H . Mahr, S. Nikitine, B. V. Novikov, E. I. Rashba, T. M . Rice, A. A. Ragachev, A. Schenzle, and K. L. Shaklee, Springer Tracts Mod. Phys. 73,000 (1975). 3 3 M. R. Philpott. J . Chew. Phys. 48,5631 (1968); K. Patzer, Phys. Status Solidi32,l I (1969). 3 4 C. Mavroyannis, Chew. Phys. Lett. 14,497 (1972). 34a D. Fox. Chem. P h p . 61,477 (1981). 35 M. Trlifaj, J . Lumin. 18/19, 215 (1979).

342

JAI SINGH

true; (2) the Frenkel exciton picture is not analogous to that of a free hydrogen atom, but localized at individual molecular sites; and (3) this area of investigation in molecular crystals may not be considered of potential importance for experimental work. In what follows we will discuss the rates and mechanisms of some of the nonradiative excitonic reactions. It is not possible to describe the theory of all reactions in detail in this article; we will therefore present the details of a few more interesting and important reactions and give only the rates of other reactions in Table IV in Section 19. RATE 14. THEREACTION The reaction or transition rate in crystals is usually calculated by assuming that the crystal goes through a quantum transition from an initial state to a final state. The quantum transition arises owing to interactions among the particles present in the two states, and the rate of the reaction (transition) R depends on the magnitude of the transition matrix element (flfii,,li) between the initial li) and final I f ) states of the crystal. R is given by”,36 (14.1) where the operator flintrepresents the interaction among the particles in the initial and final states. Eiand E, are the energy eigenvalues of the initial and final states and S(E, - EJ) conserves the energy of the crystal before and after the transition. The sum over f takes into account all the possible final states in the continuum. If the matrix element (ffii,,,li) is the same for eyery If), Eq. (14.1) can be replaced by ( 14.2)

where p ( E f )is the density of continuum states. For the calculation of reaction rates in crystals either of the preceding two forms can be used. 15. BINDING AND DECAY OF EXCITONS IN PURECRYSTALS

Having considered the theory of the formation of exciton states in crystals in the previous sections, two simple questions come to mind: (1) Can an exciton be formed between any free electron-hole pair present in a pure A. S. Davydov, “Quantum Mechanics.” Pergamon, Oxford. 1976


343

crystal? and (2) can an exciton dissociate spontaneously into a free electronhole pair? The importance of the first question is associated with decreasing the conductivity or photoconductivity of a crystal, and the importance of the second question is associated with enhancing it. We consider here rates of the reactions of binding and dissociation of excitons in pure nonmetallic crystals. It should, however, be obvious that the binding and decay of excitons is greatly influenced by the presence of impurities, lattice defects, other excitons, and excess charge carriers in crystals. This will be considered later on. The rate of binding and decay of excitons is measured directly in SiZ5and nonmetallic crystals. calculated t h e o r e t i ~ a l l y f~or~ several ~ ~ ~ - ~inorganic ~ Indirectly, the studies of binding and dissociation are of use in studying the electroluminescent properties of crystal^.^' In organic materials, however, the spontaneous binding and dissociation of an exciton in pure crystals is overlooked and considered to be insignificant. Instead, the motion of Frenkel e x ~ i t o n s ~ and l - ~ ~the transport of energy have drawn a great deal of attention in organic crystals. In a recent review of excitons in rare-gas crystals, F u g 0 1 ~has ~ described exciton binding. As both the charge carriers (electron and hole) in the Frenkel exciton are excited (created) simultaneously on the same molecule and the energy required to excite an organic molecule, even to its lowest excited state, is rather large, it is not possible to have thermally generated free charge carriers to spontaneously form exciton states in pure crystals. Only optical excitation can generate these charge carriers in molecular crystals, and then the number of excitons

’’ J . Barrau, M. Heckmann, and M. Brousseau, J . Phys. Chem. Solids 34, 381 (1973). M. Lax, Phys. Rel;. 119, 1502 (1960). A. A. Lipnik, Sou. Phys.-Tech. Phys. (Engl. Trunsl.) 2, 2575 (1957). 40 See. e.g., P. J . Dean, R. A. Faulkner, S. Kimura, and M . Ilegems, Phys. Rev. B : Solid State [3] 4, 1926 (1971); and M. Welkowsky and R . Braunstein, ibid. 5,497 (1972), and references therein. 41 See, e.g., H. C. Wolf, Adv. A t . Mol. Phys. 3, 319 (1967); P. Avakian and R. E. Merrifield, Mol. Cryst. Liq. Cryst. 5, 37 (1968); G. Durocher and D. F. Williams, J . Chem. Phys. 51, 1675 (1969); M. Grover and R. Silbey, ihid. 54,4843 (1971). 4 2 R. W. Munn, J . Chem. Phys. 58,3230 (1973). 4 3 See the papers by R . Voltz and P. Kottis; V. Ern and M. Schott; P. Reineker and H. Haken and R . Volts, in “Localization and Delocalization in Quantum Chemistry” (0.Chalvet, R. Daudel, S. Diner, and J. P. Malrieu, eds.), Vol. 1 I , Part 111, p. 187. Reidel Publ., Dordrecht, Netherlands, 1976. 4 4 W. L. Greer, J . Chem. Phys. 60,744 (1974); 65, 3510 (1976). 4 5 For a review. see R. Silbey. Annu. Rev. Phys. Chem. 27,203 (1976). 46 I . Y. Fugol, Ado.. Phys. 27, 1 (1978). 47 0. H. LeBlanc, Jr., J . Chem. Phys. 35, 1275 (1961); J . I. Katz. S. A. Rice, S. Choi, and J. Jortner, ihid. 39, 1683 (1963); L. Friedman, Phys. Rev. 133, A1668 (1964); R . Silbey, J. Jortner, and S. A. Rice, J . Chem. Phys. 42, 15 15 (1965). 38

39

344

JAI SINGH

generated can be expected to be proportional to the number of incident photons (excluding nonlinear processes). We can calculate theoretically the binding and dissociation of Wannier as well as Frenkel excitons due to phonons in pure crystals. Having obtained a unified exciton Hamiltonian [Eq. (lO.l)] and the exciton-phonon interaction operator [Eq. (12.9)] for Wannier and Frenkel excitons, a unified method can be applied to the calculation of their binding and dissociation probabilities. However, the details of the final results would be different for Wannier and Frenkel excitons, and hence a very complicated notation would be required. It is therefore more convenient to consider the two cases separately. The rates of binding and dissociation are identical, provided that the energy and momentum conservation laws are satisfied. Thus the number of excitons converted from the thermally generated pairs of charge carriers cannot increase or decrease the exciton population in crystals at a particular temperature. However, in case the charge carriers are generated through optical excitations at very low temperatures, their relaxation into excitonic states of the crystal would be irreversible, as then the conservation of energy cannot be achieved. This is how the high concentration of excitons is usually created in a crystal. In the following we will calculate the rate of conversion of a pair of free charge carriers (electron and hole) into a free Wannier and Frenkel exciton.

a. Rate of Conversion of Charge Carriers into the Wannier Exciton

We consider that initially a crystal has a free electron in the conduction band and a hole in the valence band and lattice vibrations (phonons). These particles interact in such a way that finally the electron and hole form a bound state of the Wannier exciton below the edge of the conduction band, and the excess energy is conserved by one phonon. We require only a lowenergy phonon, because the binding energy of the exciton is 10 meV. The acoustic phonons, therefore, can be expected to be the major participants in the transition. Even if the energy of the free charge carriers is much higher than at the bottom of the conduction band, the actual reaction is likely to take place only after the charge carriers have relaxed so that their energies are close to the minimum of the conduction band and the maximum of the valence band. The probability of direct transition from very-high-energy pairs of charge carriers into excitons is very small. We can clearly see the two possibilities of the transition from the initial state to the final state of the crystal through the participation of a phonon: (I) A phonon is created in the final state or (11) a phonon is annihilated in the initial state. These two

-

345


possibilities of the reaction can be described as (15.1)

+

e

(11)

h

(k)

hoa,(q)

-

4-

(K-k+q)

i,

(15.2)

- ' d h one acoustic phononq

(K)

e

where e, h, and

I represent, respectively, a free electron, a free hole, and a

h

free exciton, and hoac(q) is the frequency of an acoustic phonon. yb and yd represent the rates of binding and decay of an exciton. The wave vectors under each particle in parentheses represent the conservation of momentum as would be considered in writing the eigenfunctions of the initial and final states given in Eqs. (15.3) and (15.4): li>l

c

=

.>

If), = BLq(S7v)q,,IO; for possibility (I), and for (11) li)il

=

n>,

u ~ , k ( ~ ~ ) ~ ~ , K - k ( ~ ~ ) l o ;

01.02

1

(15.3a) ( 1 5.3b)

n>,

(15.4a)

u ~ , ~ ( ~ ~ ) ~ ~ , K - k + ~ ( ~ ~ ) ~ q , ~ l o ;

01.02

lf>iI = BUS, v)p; n),

(1 5.4b)

where the charge carrier and Wannier exciton operators are written according to Eqs. (4.2), (4.6), (4.7), (5.1), and (5.3). 10; n ) represents the product of the eigenvectors of the electronic ground state 10) and phonon state [Eq. (13.3)] (i.e.. 10; n ) = l0)ln)). To calculate the energies of the initial and final states we have to consider the total electronic Hamiltonian 8, [Eq. (3.1)] plus the phonon Hamiltonian A p h [Eq. (13.2)] and the exciton-phonon interaction operator as given in Eq. ( 1 1.6). Thus the total Hamiltonian of the system becomes

8 = A,

+ A,, + By,

where 8, is as in Eq. (3.1), but only with t are then obtained from

Alj),

=

E?lj),,

=

(15.5)

0 and 1. The energy eigenvalues

j = i and f, and P = I and 11.

(15.6)

We assume no interaction between the electron and the hole in the initial state; however, the interaction is nonzero in the final state where the exciton

346

JAI SlNGH

state exists due mainly to this interaction. It is easy to see that fip does not contribute to the initial- and final-state energies Eiand Ef calculated from Eq. (15.6) for both possibilities. We obtain from Eq. (15.6)

E:

=

h2k2 h2(K - k + pq)2 w, + E, + __ + - pLhws(q) + Eph 2m,* 2m,*

=

Wo

(15.7)

and

E,P

+ E, + E , +

where p = 0 in P = I and p

- ('

-

2M*

=

p)q12 + (1 - p)hws(q)+ E,,,

(15.8)

1 in P = 11.

represents the total phonon energy that remains unchanged during the transition, and if desired it can be set to zero. The transition matrix element (J1fiint/i) can be calculated using Eqs. (15.3)-( 15.5). As state earlier, the interaction operator represents the interactions among all the particles involved: electron-hole, electron-phonon, and hole-phonon. It is easy to see that the electron-hole interaction operator [a part of fi, given in Eq. (3.2v)ldoes not contribute to the transition matrix element. It is only the electron-phonon and hole-phonon interaction (or exciton-phonon interaction) operator that causes the transition from l i ) to I f ) in the crystal. The matrix element (flfiz,li) calculated for singlet and triplet excitons for both of possibilities I and I1 are obtained as

- C,*(k,K

-

C:(k, K

-k

-

k

+ q)M:(q)](l + fi-q)

+ q)M:(q)]ii-q

(singlet), (15.9a)

(triplet),

( 15.1Ob)


347

where ii, represents the Boltzmann value for phonons [Eq. (13.8)]. The coefficients C,(k, K - k) are the probability amplitude coefficients of the exciton binding due to the electron-hole interaction as given in Eqs. (4.5) and (4.39). From Eqs. (15.9) and (15.10) we obtain (15.11) for both possibilities. As the square of the transition matrix element for the singlet exciton is larger than that for the triplet exciton, the rate of binding of the singlet exciton would be higher, provided that the corresponding densities of states are equal. At very low temperatures, T -+0, the crystal can be assumed to be in the 0. The second possibility (11) in Eq. (15.10) zero phonon energy state, ii, will therefore have zero transition matrix element and the reaction cannot take place through this channel; a phonon cannot be annihilated from the zero phonon energy state. Only possibility I [Eq. (15.9)] will support the binding of an exciton at low temperatures. At higher temperatures, however, where 6, >> 1 both possibilities will have nearly equal transition matrix elements and hence contribute equally to the binding of Wannier excitons. We consider here the probability per second of binding R , of the singlet Wannier exciton in its internal ground state (v = 1) through process I taking place at T = 0 (nq 0). For a triplet exciton this can easily be estimated using Eq. (15.11); however, in determining R, at higher temperatures (nq # 0), one has to determine the total rate as the sum of the rates RL and Rf of processes I and 11, respectively, which can be derived following treatments similar to those used for calculating the rate at low temperatures. Using Eqs. (14.1), (15.9a), (15.7), ahd (15.8), we get the RY for the singlet Wannier exciton:

-

-

-

C;e,(k, K - k

+ q)M,O(q)]I26(Ef- E;),

(15.12)

where Ni = N , = N , are the number of free electrons or holes present in the crystal. The binding coefficients C,= l(k, K - k) for the internal ground state of the exciton are given in Eq. (4.40). The coupling functions M,’(q) and M:(q) are from Eq. ( l l S ) , with V t = -4nZe2/Voq2 = - V:, Z being the atomic number of the atoms in the crystal. 6(Ef - E;) can be simplified using Eqs. (15.7) and (15.8):

6(Ef - Ej) = 6

(h2p’ 2p ~

2K2 +--h2M*

h2(K - 9)’ 2M*

348

JAI SINGH

where

P=

m:k

-

m,*(K - k)

M"

a2EH (15.14) M* P and EH is the ground state energy of a hydrogen atom. L i ~ n i k ' ~and -~~ have calculated the binding probability of a Wannier exciton, Barrau et neglecting the acoustic phonon energy hw,(q) in Eq. (15.13). 6(Ej - E i ) then becomes independent of q, thus simplifying the evaluation of the matrix element. However, the magnitudes of the energies within the large parentheses of Eq. (15.13) are of about the same order. The kinetic energy of the particles, for instance, is - k g T at room temperature (k,T = 0.025 eV) and E,=, 10 MeV. The acoustic phonon energy hw,(q) = hu(q(,v being the velocity of sound, and Aw,(q) can therefore be very small for q+O. It is, however, clear from Eqs. (15.3a) and (15.3b), that q cannot be restricted to any one value. The neglect of ko,(q), therefore, cannot be justified in Eq. (15.13); instead, we may assume that the reaction takes place as the following: The total excess energy (excluding the kinetic energy) of the electron and hole in the initial state is given to a single acoustic phonon ho,(q) in the final state, that is, hw,(q) = - E l . As stated before, we assume that the free charge carriers in the initial state have to first relax to energies close to the minimum of the conduction band and the maximum of the valence band before the reaction takes place. This enables us to write Eq. (15.13) as 9

Ev=l

= --

9

-

m S[P - (m/M*)ql + S[P + (m/M*)sl, (15.15) 191 nh2 where m = (rn,*m,*)"'. Assuming that m = m,*= m: and Eqs. (15.12), we get the total probability of binding per second per unit volume (sec-' cmP3)as -

where n, = n, = nh represents the concentration of free electrons and holes in the crystal, p is the mass density, and Q, N A; volume of a unit cell of lattice constant A , . Converting

in Eq. (15.16), we find that the integral can easily be evaluated by contour integration and we obtain pl;N = yl;Nn',

(15.17)

349


where

Y?

=

1'

1

2m(Ze2)' (5a2 - 24) npRgvcr5hZ a

a

=

M* 2m

-.

(15.18)

In an intrinsic semiconductor the statistical estimate of the excited free charge carrier population at temperature T is (15.19) and ni increases as the temperature and effective mass increase, but decreases as the energy gap increases. The total probability Pp represents the concentration of excitons generated per second due to the participation of acoustic phonons. yp [Eq. (15.19)] is evaluated in the low-temperature limit, taking i i . = 0, and therefore it is temperature independent. At higher temperatures with ii. # 0, however, yp will be temperature dependent. Barrau et aL2' have measured the temperature dependence of yp and their theoretical result3' is yp K T p ' / 2in , close agreement with their measurement. The order of magnitude of yr, calculated at T = 0 from Eq. (15.18) in Si, is l o p 4cm3 sec-'. This is in agreement with the experimental and theoretical results of Barrau et a1.25.37The binding probability :7 of a Wannier exciton is tabulated in Table I for several inorganic crystals. TABLE 1 THEBINDING RATECONSTANT A N D THE RATEOF TRANSITION (R,> ) FROM ONEINTERNALSTATE(v) OF WANNIER EXCITON TO ANOTHFR in ORGANIC CRYSTALS ;Ih

( ~ 8 ' )

Crystal

Si

Nature of binding

e

Rate R v v , (sec- I)

e

+ h-1

h I I" K 300' K

Ge CU2O e Si

"

I&

Rate constant yh (cm-l sec- I )

jlbl

=

;*h2

=

:lb3

=

ybl =

yb2 =

1.14G"(25Jh 2.41G (25) 3.4G (25) 0.6 x lO-'(24) 6 x 10-6(24)

e

I

-h

h

V

V'

R,, = 1.02

x

1 0 ' ' (25)

G = 1 0 - 5 T - ' / 2 ;yb,, = rate constant of binding into the vth internal state of the exciton The numbers in parentheses are references.

350

JAI SINGH

L a n d ~ b e r ghas ~ ~given the statistical estimate of the exciton concentration (nex)at temperature T as nex = 4 (M*kT)3"e-EeJk~T[erf,/& ___ 2nh2

-

2(E)1'2e-a],

(15.20)

where a is a parameter as a measure of the kinetic energy ( = ak,T) of the exciton; at room temperature a 1. If a+ co, the quantity within the square bracket becomes unity, and then nex gives the effective density of free exciton states. Adding nexto Eq. (15.17), we obtain the total concentration of excitons at temperature T :

-

N,

=

nex

+ PYzex,

(15.21)

where t,, is the exciton lifetime. N , is not the net concentration of excitons and to obtain that we shall have to subtract from N , the rate of recombination ( P y ) of excitons, as described in what follows.

b. Decay of Wannier Excitons Mediated b y Phonons The decay probability of a Wannier exciton mediated by acoustic phonons can be calculated by following the same approach as that used for binding. The initial and final states described in the previous section for binding become the final and initial states for decay, respectively, as can be seen from the reaction equation (15.1) and (15.2). The magnitude of the transition matrix element is the same, and therefore the decay probability y y of a Wannier exciton is also the same as y Y . The net generation of excitons in a pure inorganic crystal can then be given by Nne, =

nex + PTzex - P y z e - h ,

(15.22)

where P y and T , - ~ are, respectively, the total probability of the decay of excitons per second per unit volume and the lifetime of a free electron-hole pair in the crystal. The lifetime is inversely proportional to the magnitude of the transition matrix element, which is the same for the binding and the decay of an exciton. Thus P r = P y , and therefore the conversion by phonons of a free electron-hole pair into an exciton does not contribute to the net exciton population in a pure crystal at constant temperature. One must emphasize that nex is the thermally generated population of excitons, and not that generated initially by thermally creating electron-hole pairs that are eventually converted into excitons. 48

P. T. Landsberg, in "Handbook on Semiconductors'' (T. S. Moss, ed.). Vol. 1 . NorthHolland Publ.. Amsterdam (1978).

35 1


c . Binding of Frenkel Excitons

It is not possible to have a thermally generated population of Frenkel excitons in molecular crystals, as the threshold energy is too large [which is obvious from Eq. (15.20)]. The free charge carriers or excitons can be created only by optical excitation. In general, the binding energy E , of a Frenkel exciton [Eq. (6.191 is about an order of magnitude higher than the ground-state binding energy E , , , [Eq. (4.35)] of a Wannier exciton. The conservation of energy, therefore, during the binding or dissociation reaction of a Frenkel exciton requires a phonon of relatively high energy; otherwise the mechanism of reaction for Frenkel and Wannier excitons are identical. Both of possibilities I [Eq. (15.3)] and I1 [Eq. (15.4)] are applicable here; however, we consider only briefly the reaction through the former process. The initial- and final-state eigenvectors can be taken from Eqs. (15.3a) and (15.3b), with the Wannier exciton operator replaced by the Frenkel exciton operator [Eq. (6. lo)], and the creation operator of the free charge carriers replaced by ei".laJ,,(o),

atJ 9 k(a)= N P ' I 2

j =f

or g.

(15.23)

I

Note the distinction between the index f used in Eq. (15.23) for the excited state of the molecular crystal and I f ) for the ket vector of the final state being used in this section. Using Eqs. (15.3a) and (15.3b) and the Frenkel exciton-phonon interaction operator of Eq. (12.12), we find the transition matrix element for a singlet Frenkel exciton : CflflPli)

where

=

S(K,

-w + C , ) / J Z ,

W ?9) = l s ( k Q) + J s ( k 9) + F , ( k 9) + xs(9).

(15.24) (15.25)

The rate of transition R: [Eq. (14.1)] is then obtained as

where Ni = N , = N , is the number of free charge carriers generated by the optical excitation and N is the number of unit cells, with one molecule per unit cell in the crystal; F denotes a Frenkel exciton. The equation of conservation of energy in the case of a Frenkel exciton can also be written as in Eq. (15.13), using Eq. (6.20) for the energy of the Frenkel excition. The energy of the free electron and hole is their kinetic energies as in Eq. (15.7). Using E,, = - E , and Eq. (15.15), the rate of

352

JAI SINGH

transition R, can be evaluated. The analytical evaluation of Eq. (15.26) is quite cumbersome, as S(k, q) is a complicated function of q. A simplifying assumption can be made. In the coupling function S(K, -9) [Eq. (15.25)], it is Zs(K, -q) that plays the dominant role for longitudinal phonons with nonzero qs. We can therefore write S(k, -q) Zs(K, -q). The total probability P: of binding for the Frenkel exciton at low temperatures (fip 0) is obtained using Eqs. (12.7) and (15.15) in Eq. (15.26):

-

-

(15.27) where rn, a, p, and V, are, as given in Eq. (15,16), the corresponding quantities in molecular crystals. W,(K, S = 0) is the singlet-state energy of the Frenkel exciton. R, is used to write 6 ( k ) = !2A’36(QAi3k). Using Eq. (15.27) for an anthracene crystal, with W,(K = 0, S = 0) 2: 26,700 cm-’ and m being the electronic rest mass, we find :y cm3 sec-’. Comparing for Si [calculated from Eq. (15.18)] with y r in anthracene, we find y r ‘v lO-’y:. P r in Eq. (15.27) is proportional to the square of the exciton energy. A large W,(K, S ) means that less excess energy is to be conserved during the transition. [W, (K, S ) usually cannot be greater than the total energy of the free electron and hole in the initial state of the crystal.]

-

yr

16. EXCITON-EXCITON COLLISIONS The process of exciton-exciton collision in nonmetallic crystals is useful in two ways: (1) The exciton-exciton collision can lead to the generation of charge carriers in the crystal and therefore (2) it can compete with the formation of biexciton or polyexciton molecules. The reaction rate has been calculated for inorganic”-” and organic5 crystals. The experimental evidence for the reaction is obtained from the observation of nonlinear photoconduction in a n t h r a ~ e n eand ~ ~ in benzophenone crystal^.'^ (For a review on molecular crystals see LeBlanc.”) Exciton-exciton collisions can take place between excitons that are localized or trapped or when one is free and the other is trapped or when both are free. A pair of charge carriers, F. Culik, Czech. J . Phys. B16, 194 (1966). P. I . Khadzhi, Sor. Phg.y.-Sernicond. (Enyl. Trans/.)2, 190 (1968). ” S. Choi and S. A. Rice, J . Chrrn. Phys. 38,366 (1963). 5 2 J . Singh. .I. Phys. C 13, 3639 (1980). 53 D. C. Northrop and P. Simpson, Proc. R. SOC.London, Sur. A 244, 377 (1958); M . Silver, D. Olness. M . Swicord, and R . C. Jarnagin, Phgs. Rec. Lei/. 10, 12 (1963). 5 4 J . B. Webb and D. F. Williams. J . P h p . C 11,3245 ( 1978). s 5 0. H . LeBlanc, Jr., in “Physics and Chemistry 01 the Organic Solid State” (D. Fox, M . M . Labes. and A. Wessberger. eds.). Vol. 3, Chapter 3. Wiley (Interscience).New York. 1967. 49


353

usually with higher energy, is generated after the reaction. There are two probable channels through which the reaction can proceed: (A) through the carrier transfer process and (B) through the energy transfer process: (16.1) ( 16.2)

where the symbols are defined in Eqs. (15.1) and (15.2). The arrows indicate the direction of the recombination between the charge carriers. There are two possibilities for each channel: Any of the pairs of charge carriers in Eqs. (16.1)and (16.2)can recombine to generate a pair of free charge carriers. One would usually observe electron as well as hole mobilities in the crystal. However, if only one type of charge carrier mobility (electron or hole) is observed, as in the experiment of Webb and Williams,54 it would mean that one of the charge carriers is finally trapped in the crystal. In general, the photoconductivity due to exciton-exciton collisions in a crystal can be observed only if the reaction takes place at the surface of the crystal. In the carrier transfer process [Eq. (16.1)] the two excitons in the initial state of the crystal interact in such a way that the electron of one exciton recombines with the hole of the other, transferring the excess energy to the remaining charge carriers and making them free in the final state. The energy transfer process [Eq. (16.2)], however, proceeds by the recombination of an electron and a hole of the same exciton, transferring the excess energy to the other exciton to dissociate it into an electron-hole pair. As the molecular overalp of the electronic wave functions is small in organic crystals, the carrier transfer process is usually less probable, and without the spin-orbit interaction the energy transfer process [Eq. (16.2)] is not allowed for triplet excitons; at least one of the excitons has to be a singlet.

a. Frenkel Exciton-Exciton Collisions The transition rate is calculated here only for the interaction between a free Frenkel exciton and a bound or trapped exciton generating a free and a trapped charge carrier in a molecular crystal. As far as the calculation is concerned, this reaction is at an intermediate level; the calculation of the rate of the reaction between two trapped excitons is simpler, and that between two free excitons is more complicated. Culik" has calculated the reaction rate between two free Wannier excitons. In the calculation of the rate of exciton-exciton collisions the etfect of phonons is usually i g n ~ r e d . ~As ~ . the ' ~ experiments are performed at low

354

JAI SINGH

temperatures, it is assumed in theory that phonons are not expected to take an active part in the reaction, even though phonons can possibly be emitted during the reaction even at very low temperatures. As the electron-phonon or exciton-phonon interaction operators present the interaction within the single-phonon approximation (see Sections 11 and 12), the multiphonon interaction operators are extremely complicated, and it is not possible to calculate the rate of a reaction in which more than one phonon is emitted. We assume that both the excitons are of the same spin state. Under such constraints the eigenvectors of the initial and final states are written as

or If>I1 =

a:,,k-q(al)dgt,p'(62)b~,,10;

n),

(1 6.4b)

9.01,02

where the creation operators are as in Eqs. (6.5)-(6.7), (6.10), and (15.23). The terms f , , f , , and ,f,represent the excited states of the free exciton, the exciton trapped at site p, and the free electron generated in the final state, respectively. Two types of final states are possible: A free electron and a trapped hole at p' can be generated (I) without emitting or (11) emitting a phonon. The two processes are independent. The interaction operator is a part of E?,, say, [Eq. (6.2b)], which is responsible for the transition from l i ) to If), of Eq. (16.4a), and the interof action operator HL E?: [Eq. (12.4)] causes transition from li) t o Eq. (16.4b). The transition matrix element is then obtained for a crystal with one molecule per unit cell as

+

5b

The transition does not occur due to the total Coulomb interaction operator Hc; it is due only t o a part of Hc that represents the interaction 01 the particles in the initial and final states; and one can sort it out from the operator H e . However. it is usually easier for the calculation of the transition matrix element to calculate( fll?cli), and to obtain from this the matrix elements responsible for the transition.

355


Equation (16.5) is obtained through the conservation of momentum =k G , G being zero in the first Brillouin zone. The first matrix element of each square bracket of Eq. (16.5) represents the energy transfer matrix element, which vanishes for triplet excitons (S = 1). (In the first square bracket the mechanism of energy transfer occurs due to the recombination of the free exciton, which transfers its energy to the trapped exciton at p, letting its electron move freely to 1’. In the second square bracket this mechanism takes place due to the recombination of the trapped exciton, which gives its energy to the free exciton, letting its electron move freely to 1’. Both of the energy transfer matrix elements, however, create a free electron in the crystal and can be considered equivalent.”) Likewise, the second terms of each square bracket are equal and represent the corresponding carrier transfer matrix elements that are nonzero for both singlet and triplet excitons. Within the single-phonon approximation the transition matrix element emitting a single phonon in the final state vanishes [Eq. (16.6)]. Therefore only the Coulomb interaction is responsible for the transition at low temperatures. At higher temperatures, where the same number of phonons can be considered t o be emitted or absorbed in the initial and final states of the crystal, a nonzero transition matrix element due to the operator I?: of Eq. (16.6) will be obtained. The transition matrix element [Eq. (16.5)] is thus obtained in the lowtemperature region as

K

+

,(f)i?;)i) = 2 J Z N - I

1exp[ik.(I

- l‘)]

1.1‘

qqf,,1; f z . P) - w,, 1’; g, 11 qf,, p; f , 0 3 . x [(I

-

S)(f,, 1’; g,

3

(16.7)

The second term of Eq. (16.7) can be neglected for singlet excitons, since the major contribution comes from the first term representing the reaction through the energy transfer process. The initial- and final-state energies are obtained according to Eqs. (6.17) and (6.20) as (16.8) (16.9)

+

+

where E;: = AEJ.,(S) D f , Lf(O,S),the energy of excitation of a molecule in the crystal, and Ei;2 is that of a molecule at p; E;: = Eft - E , , as given in Eq. (6.14).

3 56

JAI SINGH

Using the rate equation (14.2), we get the total transition probability P," of singlet-singlet Frenkel exciton collisions in molecular crystals with one molecule per unit cell :

P," = yznex cm-3 sec-',

(1 6.10)

where (16.11) n p and n,, are, respectively, the concentrations of the trapped and free excitons in the crystal. By neglecting the kinetic energy of the exciton, lkl can be calculated from the conservation of energy equation: (16.12) =

exp[ik.(l - l')](ft, 1'; g, 1IUlf,, 1; fz, P>.

(16.13)

1.1'

Equation (16.10) is obtained by assuming that the free electron density of states in the crystal is p ( k ) = m,Vok/n2h2. In Eq. (16.13),Z can be calculated only numerically in molecular crystals.52 However, one can expand it, as Z involves a transition from the excited state to the ground state of the molecule at I, in terms of the dipole and quadrupole transition moments. Retaining only the terms of the dipole moments, 2 can be evaluated approximately. Such methods have usually been used for evaluating matrix elements like Z in k space for inorganic crystal^.^' Equations (16.10) and (16.11) have been derived for more than one molecule per unit cell by Singh.52

b. Wannier Exciton-Exciton Collisions Culik4' has calculated the rate of the reaction of exciton-exciton collisions for free Wannier excitons. The approach followed before for the Frenkel exciton case can easily be adjusted for Wannier excitons; one has only to work in reciprocal lattice space. Culik found the probability of collisions for Wannier excitons (Py)through the energy transfer process and within the dipole transition approximation: w- w pc - Y c nexr 4"n' ke21r,o(z x3 k2hZ (16.14) y y =h (1 + x2)6' = __ pe2 ' 9 where r,o represents the dipole transition moment and p is the reduced


357

TABLE11. RATECONSTANTS (p, IN cu3 SEC-I)OF EXCITON-EXCITON COLLISIONS FOR NONMETALLIC CRYSTALS AT Low TEMPERATURES Crystal

Calculated y,

Experimental y,

Anthracene Benzophenone Fluoranthene TCNB-durene CdS

cu,o

GaP GaSb InP ~

6.4 x l o - ' ' (52) 1.4 x 10-I3(56a) 2.5 x 10-l3 (56a) 10- (49) 9.2 x (50) 3.2 x 1 O - I 2 ( 5 O ) 1.3 x l o - " (50) 2.5 x lo-'* (50)

~~~

The numbers in parentheses are references

exciton mass. In crystals with rI0 = 0 one has to consider the higher-order transition moments in evaluating the transition matrix element. The rate constant of a transition through the carrier transfer process 7: is obtained as49 4'01r7

7:

ke21r,012 1 - x-' arctan x x(l + x2)4 *

=3A

(16.15 )

In deriving Eqs. (16.14)and (16.15), it is assumed that both the excitons are in their ground state (v = 1). Culik4' has also studied the reaction for excitons in their first excited state (v = 2). In CdS crystals, according to Culik, the orders of magnitude of y," and y z are the same, whereas in molecular crystals y,"t is usually very small. KhadzhiS0has proposed a continuum model for calculating the probability of exciton-exciton collisions in which the interaction potential between the excitons is assumed to be a sum of Morse and Van der Waals potentials. The transition rate thus obtained has two unknown parameters, and the model does not distinguish between the carrier and energy transfer processes. The orders of magnitude of the rate coefficients y, of two excitons, both free, both trapped, or one trapped and one free, are usually the same. This is true for Wannier as well as Frenkel excitons. Some results are given in Table 11.

56b

T . Kobayashi and S. Nagakura, Mol. Phys. 24,695 (1972). A. Bergman, M. Levine, and J. Jortner, P h p . Reu. L e t t . 18,593 (1967)

358

JAI SINGH

17. FISSION AND FUSION OF EXCITONS In the 1960s the reaction of fission and fusion of excitons was one of the most interesting areas in the spectroscopy of organic crystals. Kepler et aLs7 and Hall et al.58 were first to observe the delayed fluorescence in anthracene due to the annihilation of two triplet excitons. This is another aspect of exciton-exciton collision in which two triplet excitations at different molecular sites combine, owing to their mutual interaction, to form a singlet, and this is usually called exciton fusion. The reaction is only possible in molecular crystals where twice the energy of the triplet excited state is less than that of the singlet state. The reverse reaction is also possible: A singlet exciton dissociates into a pair of triplets, and this is called the exciton fission. Swenberg and Stacys9 proposed a hypothetical reaction of exciton fission that was first observed by Geacintov et aL6' and Merrifield et ~ 1 . ~in' tetracene crystals. Subsequently there has been tremendous interest in the reaction of fission and fusion, e ~ p e r i m e n t a l l y ~ as ~ . ~well as theoretiBoth processes are thermally assisted for the conservation of energy; however, at very low temperatures68.68athey can be induced optically, as has been suggested by Vaubel and B a e ~ s l e r . ~ ~ The probabilities of exciton fission and exciton fusion should be identical, as one process is the reverse of the other, provided that the energy and momentum in the crystal are conserved. Both reactions have been observed6' in tetracene, where the difference AE between the singlet exciton energy and twice the triplet exciton energy is 2WT - W, = 1520 cm-1.63 In other aromatic crystals the gap is larger. The activation energy AE can be provided by lattice vibrations, intramolecular vibrations, or infrared photons. It is not possible to distinguish on the basis of frequency between phonons and intramolecular vibrations in molecular crystals. However, as fission and R. G . Kepler. J . Caris, P. Avakian, and E. Abramson, Phys. Reu. Lett. 10,400 (1963). J. L. Hall, D. A. Jennings, and R. M. McClintock, Phys. Rer. Lett. 10,400 (1963). 5 y C. E. Swenberg and W. T. Stacy, Chem. Phys. Lett. 2,327 (1968). 6o N. Geacintov, M. Pope, and F. Vogel, Phys. Rev. Lett. 22,593 (1969). 6 1 R. E. Merrifield, P. Avakian, and R. P. Groff, Chem. Phys. Letr. 3, 155 (1969). 6 2 R. P. Groff, P. Avakian, and R. E. Merrifield, Phys. Rev. 8:SulidSfate [3] I, 815 (1970). 6 3 Y . Tomkiewicz, R . P. Groff, and P. Avakian, J . Chem. Phys. 54,4504 (1971). 6 4 J. Jortner, S. A. Rice, J . L. Katz. and S. Choi, J. Chem. Phys. 42, 309 (1965). 6 5 C. E. Swenberg, J . Chem. Phys. 51, 1753 (1969). 6 6 M . Trlifaj. Czech. J. Phys. B22, 832 (1972). " J. Singh, J . Phys. Chem. Solids 30, 1207 (1978). G . Klein, R. Voltz, and M. Schott, Chem. Phys. Left. 19, 391 (1973). 6 8 a W. M . Moller and M. Pope, J . Chem. Phys. 59,2760 (1973); C. E. Swenberg, M . A. Ratner, and N. E. Geacintov, J . Chem. Phys. 60,2152 (1974). " G. Vaubel and H. Baessler. Mol. Cryst. 15, 15 (1971). 57

58

359


fusion are caused by bimolecular triplet excitons, it is expected that the localized intramolecular vibrations would be more actively involved in the

reaction^.^^ We assume that in the initial state the crystal has one singlet exciton, localized intramolecular vibrations, and lattice vibrations. In the final state there are two triplet excitons and intramolecular and lattice vibrations. The initial- and final-state eigenvectors can be written as li)

=

N-’/’

1eiK’PB:s,p(S= 0)lO; vf,(p);n), C e i ( k ~ . ~ ~ + k zf t. .~P 1z ()S~ =t 1)

(17.1)

P

If)

= N-1

P I +P2

x 10; vf,(PI))Bftt,p2(S= 1)10; vft(P,))ln’),

(17.2)

where f , and f , represent the MOs corresponding to the singlet and triplet excited states. The exciton operators are written as in Eqs. (6.5), (6.6), and (6.10),and K, k, ,and k, are the reciprocal lattice vectors such that k , + k, = K + G ; G = 0 in the first Brillouin zone. The ket 10; vfs(p); n ) E 10) lvfs(p)) In) represents the product of the kets of the electronic ground state and the corresponding intramolecular vibrational state “0) = 10, vg(1)),]. The ket of the intramolecular vibration associated with the singlet excited state f , of the molecule at p is given by Ivf,(p)) = lvf,,,@)), where u is the vibrational quantum number and the kets of the lattice vibrations In) and In’) are as given in Eq. (13.3).In Eq. (17.2) the product is split into two factors as 10; u,,@,)) In’). The ket B;,,,(S)lO;uf,(p)) represents a vibronic excitation as localized at p that is similar to a purely electronic excitation Bftm:p(S)lO), in Eq. (6.4). The product of two creation operators of triplet excitation has nine terms [see Eq. (6.6)]; however, onlythree of them, with their projection spins (Sz, Ss) as (1, - l), (- 1, l), and (0, 0), will contribute6’ in order to conserve spin. The interaction operator consists of a part of A, [Eq. (6.26)], say, and the exciton-phonon interaction operator [Eq. (12.12)]. However, the excition-phonon interaction operator does not contribute to the transition matrix element; the situation is similar to that of the transition matrix element of Eq. (16.6). The spin-orbit interaction, known to be small for hydrocarbons, is neglected. The transition matrix element thus obtained is

n,

nu

(flfict19=

*c 1

-

~

Cexp[ik, .(PI

-

PI1

PI +P

+ exp[i(K x

Qft,

-

kJP,

-

P)I}(f,? P I ;

f , ; P , ) F ( f , ?g; PI,

. f t 3

P l q g , P1; f , ; P) (17.3)

360

JAI SINGH

in

t] c in

W

Molecular sites

FIG.8. Microscopic illustration of the reactions of exciton fission and fusion in molecular crystals.

where F(f,,f , ; p,) and F ( f , , g; p) are the Franck-Condon overlap factors between the triplet (f,) and singlet (f,) states of the molecule at p l , and between the triplet and ground (g) states at p, respectively, as given by

Q f , , f,; Pl) =

rI (V,~,"(P~)lV,~,"(Pl)),

(17.4a)

v

(1 7.4b) The transition matrix element represents the mechanism of the reaction as shown in Fig. 8. In deriving Eq. (17.3), it is assumed that the phonon population remains unchanged during the transition. It is interesting to note that except for the exponential and FranckCondon factors, the carrier exchange interaction matrix element of Eq. (17.3) is the same as that of the corresponding term [the second term of each square bracket of Eq. (16.5)] of the transition matrix element in the reaction of exciton-exciton collision. The difference between the reaction of excitonexciton collision and exciton fusion is that in the former the free charge carriers are created in the crystal and the excess energy is given to them; intramolecular and lattice vibrations play essentially no roles. In the latter, however, a bound state (singlet exciton) is created in the crystal, and the excess energy has to be conserved either by lattice or intramolecular vibrations. The exciton fusion is therefore another possible channel for excitonexciton annihilation, where intramolecular vibrations take active part. The initial and final state energies of the crystal are obtained as (1 7.5a)


36 1

where M,* and M,* represent, respectively, the effective masses of singlet and triplet excitons of(u) is the intramolecular vibrational frequency associated with the electronic state f and vibrational mode u, and E,, is the phonon energy that can be set to zero. In writing Eq. (17.5b), it is assumed that the energies of the intramolecular vibrations of the two molecules with triplet excitons in the final state are equal. The kinetic energy terms in Eqs. (17.5a) and (17.5b) can safely be neglected. The transition rate ( R )can now be calculated67using Eq. (14.1). However, we have to calculate the Franck-Condon vibrational overlap factors [Eq. (17.4)]. There are two limits in which the calculations can be done approximately7’: (1)the weak coupling and ( 2 )the strong coupling between electrons and intramolecular vibrations. In the strong-coupling limit the vibrational frequency o f ( u )is approximated by the mean frequency (0) = N ; C, of(u), Nu being the number of normal modes of vibrations, whereas in the weak coupling limit o f ( u ) is approximated by the maximum frequency w,. By triplet assuming that the frequencies of vibrations in the singlet [wfS(u)], [o,,(u)], and ground [o,(u)] states are equal [ o f s ( u ) = ofI(u) = o g ( u ) = ~ ( u ) ] and that the corresponding normal modes of vibrations are also the same, we obtain the rate of transition in the strong-coupling limit at low temperatures as67

where lcfN (flficfli) is calculated, including only the nearest-neighbor interaction. The values of K and K - k , are assumed to be small, being associated with the optical excitation, so that the exponential factors in Eq. (17.3) can be approximated by 1. Eml and Em2are known as the molecular deformation energies, corresponding, respectively, to the excitation or relaxation of the singlet and triplet and the triplet and ground states of a molecule in the crystal. A E = IE;; - 2E;:I is the difference between the energy of a singlet state and twice that of the triplet state [see Eq. (17.5)]. In the weak-coupling limit the rate of transition is obtained as

where emland em*are the molecular deformation energies, like Eml and Em* for modes with maximum frequency om, and g, is the number of such modes. It is obvious from Eqs. (17.6)and (17.7)that the rate of transition decreases with increasing energy gap. In the strong-coupling limit the decrease is R. Englman and J. Jortner, Mol. Phys. 18, 145 (1970).

362

JAI SINGH

TABLE111. RATEOF EXCITON FISSION (R) AND RATECONSTANT (yF) IN MOLECULAR CRYSTALS

OF

EXC~TON FUSION

a. Fission Molecular crystal

A E (cm- ')

Anthracene" Neutral CT Naphthalene Tetracene

4,000 (61) 1.690 (71, 72) 10,400 (17) 1,520 (63)

~

10" (71) -

l o R (62) 1013 (73)

(optically induced)

Calculated R (sec-l)

Experimental R (sec-')

lo9 (68a)

107-109 (67)

(67) (67) 10'o-10'2(59) 108-1010 (66) 10'-10" (67) 109-1011

b. Fusion Molecular crystals Anthracene Biphenyl Naphthalene Tetracene

Experimental yF (cm3 sec- I ) l o - " (57,74) (2.8-6.7) x lo-" (58, 75-78) 3 x 10-12(79) -

9 x lo-" (62)

Calculated yF (cm3sec-') 1 0 - " - 1 0 ~ ' * (64) 4 x l o - " (65) (65) IO-"(65)

" CT stands for the charge transfer state of anthracene, and neutral means the first singlet neutral exciton state. The numbers in parenthescs are references.

faster than exponential, whereas it is not so steep in the weak-coupling limit. It is known'O that for unimolecular processes in large aromatic molecules the strong-coupling limit is inapplicable; however, for bimolecular. processes in the crystalline state the strong-coupling limit is apparently more appr~priate.~' The rate of exciton fusion, is in principle identical to the rate of fission, since the transition matrix elements are identical. In actual calculations. G. Klein, R. Voltz. and M. Schott, Chem. Phys. Lett. 16, 340 (1972). 72

M. Pope and J . Burgos, Mol. Cryst. 1, 395 (1966).

73

R. R. Alfano. S. L. Shapiro, and M. Pope, Opr. Commun. 9,388 (1973).

P. Avakian and R. E. Merrifield, Mol. Cryst. 5 , 37 (1968). S. Singh. W. J . Jones, W. Siebrand, B. P. Stoichefl, and W. G. Schneider, J. Chem. Phjs. 42, 330 (1965). 76 T. A. King and H. G. Siefert, in "The Triplet State" (A. B. Zahlan, ed.), p. 329. Cambridge Univ. Press, London and New York, 1967. 71 P. Avakian, E. Abramson, R. G. Kepler, and J. C. Caris, J. Chrm. Phys. 39, 1127 ( 1 9 6 3 ) . W. Helfrich and W. G. Schneider, J . Chem. Phys. 44,2902 (1966). N. Hirota and C. A. Hutchinson. Jr., J . Chem. Phys. 42,2869 (1965). 74

75

'' ''

363


however, the probability of fusion is usually calculated in terms of rate coefficients yF, as in the reaction of exciton-exciton collision in the previous section. The total probability of fusion P , ( ~ m sec-') - ~ is thus F '

=

(17.8)

YF"exY 2

where nex is the concentration of triplet excitons in the crystal. In tetracene yF, determined from experiment,62 is cm3 sec-l. Taking a typical valueofn,, = 10'' cm-3, weget the total probability P , z 10" cm-3 sec-', which is very close to the rate of exciton fission ( R 109-10" sec-') in tetracene, as expected. The rate of exciton fission ( R ) and the rate constant of exciton fusion (yF) are listed for several molecular crystals in Table 111. Exciton fission has been observed in tetracene, and exciton fusion in both anthracene and tetracene. However, exciton fission is also possible from the charge transfer singlet state in anthracene, where the energy gap AE is almost the same as in tetracene (see Table 111). Energetically, arrangements of singlets and triplets are not the same in inorganic crystals as in molecular crystals. Therefore exciton fission and fusion have not been observed nor calculated for Wannier excitons.

-

-

18. EXCITON-CHARGE CARRIER INTERACTIONS

The interaction between an exciton and an excess charge carrier is observed in molecular crystals in terms of the decrease" in prompt81,82and delayed fluorescence, detra~ping,'~ and external p h o t o e m i ~ s i o nof~ ~electrons. There are two possible ways in .which the interaction between an exciton and excess charge carrier may take place: (a) a hot charge carrier (electron or hole) is created because the exciton dissociates and transfers its energy to the excess charge carrier either by promoting it to higher conducting states or increasing its kinetic energy, or both, and (b) a complex charge carrier is generated in the crystal as the interaction between the exciton and the charge carrier is often found to be attractive, thus enabling the formation of a bound state between the three charge carriers. Schlotter and Baesslers5 have directly observed the effect of hot electrons as evidence

'' M. Pope, .I.Burgos. and N. Wotherspoon, Chem. Phys. Lett. 12, 140 (1971). '' N . Wakayama and D. F. Williams, J . Chem. Phys. 57,1770 (1972). '' L. Peter and G. Vaubel, Phy.7. Stutus Solidi B 48,587 (1971). x3 84

*'

A. Many, J . Levingson. and 1. Teuchner, Mol. Cryst. Liq. Crjsr. 5, 121 (1968). D. Haarer and G . Castro, Chem. Phys. Lerr. 12, 277 (1971). P. Schlotter and H. Baessler. Chem. Phys. Left.24,450 (1974); Chem. P h p . 19,353 (1977).

364

JAI SINGH

of possibility a. Possibility b has not been observed, but calculations’ 5,86-89 strongly support the formation of complex charge carriers in organic crystals. The author is not aware of any work on mechanism (b) in inorganic crystals. Process (a), however, is similar to the Auger processes in inorganic semiconductors. The mechanisms of both processes are shown here: e

(a)

6)

l+e+e h

,

A

hot electron

e

(ii)

I +h

+

h

h ,

(18.1)

6 hot hole

e

(b)

(i)

I + e+e-h-e,

h e

(ii)

I + h+

h-e-h,

(18.2)

h

where e-h-e and h-e-h represent complex charge carriers formed by two electrons and one hole and two holes and one electron, respectively. A quantitative account of both processes a and b of the reaction between an exciton and an excess electron in molecular crystals has been published.’ The results of the reaction between an exciton and an excess hole [processes a(ii) and b(ii)] are expected to be similar. 5388389

a . Generation of a Hot Electron

The rate constant yhe for the generation of a hot electron due to the interaction between a free Frenkel exciton and a free excess electron in molecular crystals is calculated assuming that in the initial state the crystal has an exciton and an electron, and in the final state a free electron in some higher energy state. The transition operator consists of part56 of A,, denoted by plus the exciton-phonon interaction operator fir [Eq. (12.12)]. However, the exciton-phonon interaction operator does not contribute to the transition matrix element, which is similar to Eq. (16.6). In Ref. 89, however, it has been assumed that phonons do not take active part in the reaction. The participation of phonons does contribute to the transition matrix

R,

H6

V. M. Agranovich and A. A. Zakhidov, Chem. Phys. Lett. 68,86 (1979). 1. Dalidchik and V. Z. Slonim, JETP Lett. (Enyl. Trans/.) 31, 112 (1980). J . Singh, Phys. Status Solid; B 103,423 (1981). J . Singh. J . Chem. Phys. 75,4603 (1981).

” F.


365

element of the interaction operator & by a factor of the phonon population, provided that one assumes that an equal number of phonons are created or destroyed in the initial and final states. At very low temperatures this will have little influence on the transition matrix element. It can therefore be confidently concluded that phonons play no active role in the reaction. The rate R,, and rate constant yhe are calculated, using Eq. (14.2) for crystals with two molecules per unit cell, ass9 Rhe

=

?he

=

(1 8.3)

?henex,

64m,*(2m,*E,,,)”2fl, I Z,Iz 7Ch4

(1 8.4)

where

z,= ( 2 N ) - ” 2 c ( f e ,

1’; g, I p l . f o 31; f o , 1’).

(18.5)

1,l‘

The sum over 1 and 1’ corresponds to the individual molecular sites and not to the unit cells of the crystal; f , represents the molecular orbital corresponding to the initial state of the crystal occupied by the exciton and excess electron, and f , that of the final state carrying the hot electron; Ehotis the excess energy given to the free electron and is obtained from the conservation of energy. It is found that the rate constant for a singlet exciton is at least one and a half times largera9than that for a triplet. For anthracene, from Eqs. (18.3)(18.5), yhe 7 x cm3 sec-’, in agreement with the experimental value determined by Schlotter and B a e ~ s l e r ,who ~ ~ used the concentration of excitons n,, 10l6 ~ m - ~ .

-

-

b. Formation of a Complex Charge Carrier We assume that a complex charge carrier (CCC) (exciton and electron) with wave vector K is formed by the interaction between a free Frenkel exciton with wave vector k and a free electron with wave vector K - k. The exciton-phonon interaction apparently has little influence on the potential of interaction between the exciton and the electron, as was demonstrated by Hochstrassergo; it can therefore be neglected. The eigenvector of the electronic state of a CCC in the crystal can thus be written as

IK, f , S ; f ’ )

=

C Cff,(k, K

-

k;

v)B:,k(s)a~,,,-k(a)lo),

(18.6)

k.0

where ,f and f ’ label the MOs corresponding to the excited electronic states of the exciton and electron; Css.(k,K - k; v) represents the probability R. M . Hochstrasser, Acc. Chrm. Res. 6,263 (1973).

366

JAI SINGH

amplitude of the formation of a CCC due to the interaction between an exciton (k) and an electron (K - k); v is a quantum number associated with the internal energy state of the CCC, as in the case of a Wannier exciton [Eq. (4.5)]. Using Eqs. (6,1), (6.5), (6.6), (6.10), and (15.23), and the procedure of Wannier exciton theory, we solve the Schrodinger equation, AJK, .f, S; f ‘ )

=

wfff(K)IK, f , S; f ‘ ) ,

(18.7)

and obtain the energy eigenvalue equations9: [Ey(k‘)

+ E;,(k”) + Wo

+

C,,,(k, K

-

Wff,(K)]Cfff(k’, k”; V )

-

k; v) exp{i[(k - k’).I + (K - k

-

k”).l’])

t

x [U,,,,(ll

- 1’))

-

Uf.,,(ll - l’l)]

=

0,

K

=

+ k”,

k‘

(18.8)

where U f , , j ( l-~

1‘ 1 )

=

2( j , I; f ’ , 1’1 ~ -

( j , I;

l f ’ , 1’; j , I)

f ’ , I’lUlj, I; f ’ , l’),

j

=

f’

or g. (18.9)

E;”(k’) = AEJS) + Of + LJk, S), from Eqs. (6.17) and (6.20), represents the energy of a free exciton, and E;(k”) is the corresponding energy of a free electron. Equation (18.8) is obtained for crystals with one molecule per unit cell, and by neglecting the three and four molecular center integrals. The terms after the summation sign in Eq. (18.8) give the interaction matrix element between the exciton and the electron instantaneously being on a pair of molecules at 1 and l’, respectively. There are obviously N 2 such pairs, N of which will be such that the exciton and electron occupy the same molecule. In case the exciton and electron occupy different molecular sites (1 # 1’); the potential of interaction U ( R ) = U f , , f ( R ) U f , , , ( R ) ; R = 11 - 1’1, is obtained” : 2eR U ( R )= - 7 { [ p f ( 0 ) - pg(0)] R

+ ( a f f ’- &’).

E(R)},

(18.10)

where E(R) = eR/tR3, and the energy of a CCC is Wff,(K) = Wo + E;’(O)

+ E;(O) + E , + -.h2K2

(18.11)

M,

The internal binding energy E, of a CCC is given by

(- 5V i +

U(R)

4.Cf‘(R)= E,4Gf’(R).

(18.12)

367


In these equations p J ( 0 )( j = f or g), is the permanent dipole moment of a molecule in its electronic state corresponding to the MO j , and cdf' is a second-rank tensor of the static polarizability of the molecule; E(R) represents the electric field induced on the molecule at a distance R by the excess electron, and E is the static dielectric constant of the crystal; MI = M * + ml and pI-' = (M*)-' + (m:)-'. For centrosymmetric crystals, such as naphthalene and anthracene, U ( R ) [Eq. (18.10)] between nearest neighbors on the b axis is found to be U ( R ) -0.08 and -0.05 eV,89and for noncentrosymmetric crystals of phtahlazine mixed in naphthalene U ( R ) is repulsive and is equal to 0.5 eV (although in noncentrosymmetric crystals the sign of U ( R )[Eq. (18.lO)l is expected to depend on the sign of R). The probability amplitude coefficients are found to be

-

C,,,(k, K - k; v)

=

N-'

1 exp[ - i

(k -

RfO

which determine the eigenvector [Eq. (18.6)] of the CCC. In case the exciton and electron occupy the same molecular site, the situation is a little different. The potential energy of interaction, denoted by U , ,is then obtained from Eqs. (18.8) and (18.9) and by assuming that f = f ' : U, =

(f, 1; f,qqf,1; f ,1) - 2(g, 1; f ,I1 Ulf,

1; g, 0,

(18.14)

and the energy of CCC is A2K2 + 2E7(0) + U , + __ ,

EF(0) = E;(O). (18.15) 24 The probability amplitude coefficients C,,(k, K - k; v) are now equal to unity and the CCC behaves more like a free wave of wave vector K. The potential energy U , [Eq. (i8.14)] can be calculated numerically; for naphthalene, anthracene, and phthalazine mixed in naphthalene U , is, respectively, - 1.16, -0.95, and -2.97 eV.89 The formation and motion of a CCC in molecular crystals can be described as follows. The attractive potential of interaction is found to be strongest when the exciton and electron both occupy the same molecule. Therefore, the most favorable condition for the formation of a CCC arises when both are at the same molecule or, in other words, close together. This situation is, however, uncommon for large-radii orbital excitons or Wannier excitons, and therefore the CCC would be improbable there. The motion of a CCC in molecular crystals can be described as the hopping type; that is, when the exciton and electron are apart, there may not be a CCC and process a of a hot electron may dominate. However, as soon as the exciton and electron get close together, such a formation becomes possible. In the mixed crystal Wrr(K) = W,

368

JAI SINGH

of naphthalene-phthalazine the depth of U , at a phthalazine molecule, as discussed above, is even deeper, which implies that it would act as a shallow trap for the CCC, allowing it to reside there for longer than on naphthalene molecules. 19. RATES OF THE NONRADIATIVE DECAY OF EXCITONS

In addition to the excitonic reactions of the nonradiative decay of excitons described in the previous sections of this article, there are many other possible reactions that can lead to the dissociation of excitons in crystals. For instance, the influence of crystal defects and impurities on the nonradiative decay of excitons has not yet been mentioned. In Table IV the rate ( R ) or the rate constant (y) of several known reactions, caused by the presence of impurities such as acceptors or donors, are listed. A brief description of the mechanism of each reaction is also presented in the second column of Table IV. TABLE Iv. THERATE(R) AND/OR RATECONSTANT (7) OF THE NONRADIATIVE OF EXCITONS IN NONMETALLIC CRYSTALS" PROCESSES

Crystal

Reaction mechanism e

Cds

I +i

Rate R (sec-')

Rate constant y (cm" sec- ')

+ nd

4.5 x 10-'(91)

+ i d A h + n d

4.0 x lO-'(92)

+ i d A h + n d

2.0 x 10-5 (93)

+ i d d h + n d

4.8 x

d A h

h e GaAs

I

h

e Ge

I

h e

Si

1

(93)

h

e

CdS

I +n

d A e +id

2.9 x

10-13

(94)

h e cu,o

1 +i

h

a A e

+ na

2.6 x 10-'(91)

.

369


TABLE IV. Continued Crystal

Reaction mechanism e

CdS

1 +n

a L h

+ ia

a A h

+ ia

Rate constant y (cm' sec-')

Rate R (sec-')

4.4 x lo-" (94)

h

cu,o

e

1 +n

l o i o (95)

;tz

= 2.6 x

(91)

h

1.8 x 1013(96)

NaCl e

Si and Ge

I

h

+ e A e + e + h (impact ionization)

10-7

(97)

e

Anthracene

I + metal interface

h

'- e + (h in metal) '' h + (e in metal)

10" (98) 10" (98)

'' * h + (e in metal) e

Anthracene

1

autoioRnization +

h Hopping rate ( R ) Hopping rate at 77" K Hopping rate at 77" K

Anthracene Fluoranthene TCNB-durene e Carbazole (CA) substituent in I h polydiacetylene crystal (DCHD)t_hcI at nCA

',

5.0 x 101z(lO1) 2.6 x 108(56a) 2.9 x lo8 (56a)

e

' I

1.6 x

1013

(102)

h

zzGii7 chain

e

I

h

at CTCA

',

e +

1

3 x 10" (102)

h

z chain za

id. ionized donor; nd, neutral donor; ia, ionized acceptor; na, neutral acceptor; y , , rate constant corresponding to the internal ground state ( v = I ) of the Wannier exciton; y,, rate constant corresponding to the internal first excited state ( v = 2) of the Wannier exciton; nCA, neutral singlet exciton state of carbazole; and CTCA, charge transfer singlet state of carbazole.

370

JAI SINGH

In crystals interacting strongly with the electromagnetic field of radiation there arises another noiiradiative stationary state due to the interaction between the exciton and photon, called the p o l a r i t ~ n ' ~or~ optical ~'~~ exciton state. The development in this area of spectroscopy of solids has been vast in the past years and has been r e v i e ~ e d ~ . ~ at ~ ,different ' ~ ~ , ' stages. ~~ As it can only be summarized in a relatively large volume, polaritons cannot be incorporated in this article. ACKNOWLEDGMENTS

I am indebted to Professor D. P. Craig, F.R.S., for many valuable discussions at the onset of this work, and to Dr. R. W. Munn for suggestions and many useful comments. Thanks are due to Dr. B. R. Markey and Dr. R. N. Lindsay for critically reading the manuscript. 1 am very grateful to Anne Dowling and Irene Page for their patience and their accurate typing of the manuscript.

M . Trlifaj, Czech J . Phys. B15, 780 (1965). Richard and M. Dugue, Phys. Status Solid B 50,263 (1972). 93 J . Singh and P. T. Landsberg. J . Phys. C 9,3627 (1976). 94 M . Trlifaj, Czech. J . Phys. B14, 227 (1964). " Z . Khas. Czech. J . Phys. B15, 568 (1965). 9h R. Fuchs. Phj,.r.Rev. 111, 387 (1958). '' T. Harada and K. Morigaki, J . Phys. SOC.Jpn. 32, 172 (1972). ' ( l i J . Singh and H. Baessler. Phj,.s. Stutus Solidi B 62, 147 (1974). " H. Killesreiter and H . Baessler. Phys. Sraius Solid B 51, 657 (1972); Chrn?.Ph.vs. Lert. 11, 411 (1971). ''I S. Choi. Phys. Rev. Lett. 19, 358 (1967). "" J. Jortner. Phys. Reo. Lett. 20, 244 (1968). I o Z K . Lochner, H . Baessler, L. Sebastian, G . Weiser, G . Wegner. and V . Enkelmann. Chen?. Phy,s. L e f t . 78, 366 (1981). J. J . Hopfield. Phy.s. Rru. 112, 1555 (1958). S. I. Pekar, Sou. Phyx-JETP (Enyl. Trans/.)34,813 (1958). l o 5 B. Fischer and J . Lagois, Top. Curr. Phys. 14, 183 (1979). I o 6 E. Burstein and F. DeMartini, eds.. "Polaritons." Pergamon, Oxford. 1974. " C.

Author Index Numbers in parentheses are reference numbers and indicate that an author's work is referred to although his name is not cited in the text.

A

Aas, E. J., 82 Abe, H.. 229 Abe, R.,I18 Abey, A. E.,33, 34. 35 Abramson, E., 358, 362 Adamowicz, L., 218 Adrian, F. J.. 231. 235(34) Agranovich, V. M.. 364 Albrecht, M. G.. 229.233. 240(20) Alder, B. J., 167, 220 Alexopoulos, K., 15. I6( 14, 15), I9,25( 14, 15). 26, 27. 61 Alfano, R. R..362 Allcock, G. R., 118 Allara, D. L.,240 Allen, P. C . , 65, 66. 67(64). 68(64), 70 Allen, S. J., Jr.. 71 Alterovitz, S., 62 Andeen. C . G.. 48. 52(d, k). 5 5 , 56(53a), 57 (53a), 58, 59(53a). 60(53a). 61(53a. 54). 62(53a, 54) Anlage, S., 124 Arbman, G., 194 Arends, J., 29 Arisawa, K.,118, 119(67) Avakian, P.,343,358. 362, 363

B Baessler, H., 358, 363, 365, 369(98, 99, 102) Bagaev, V. S.,341, 348(32) Bagayoko, D., 209, 212,213(116) Baraff, G. A., 178, 181 Barber, P. W., 276, 290(116) Bardeen, J., 141, 329, 331 Barends, E. J., 178 Barker, J. R., 90(45) 37 I

Barr, L. W.. 2, 11(2), 12(2), 13(2), 16(2), 19(2), 23(2), 28(2), 32(2). 33(2), 44(2), 48, 54(2). 55(2). 78. 79(2) Barrau. J.. 331, 343, 348, 349 Barsis, E., 54 Bassett, W.A.. 65 Batchelder. D. N., 286 Baym. G., 151 Beattie, A. M., 187, 188 Beleznay. F., 82. lOO(13). lOl(13). 102(13), IZh(13). 131(13) Benner, R. E., 285, 287. 288 Bennett, M.. 219 Bergman, A.. 357 Bergman. J. G., 227, 228. 233, 238, 241, 242 (67), 243(67), 244(67). 251(67). 252(67). 253(67). 256(67), 270.271( 105). 272(105), 276(48). 277(48), 279,280(122), 282( 122), 283(122). 286 Bethe, H. A.. 141 Beyeler. M.. 23. 24(23). 27(22). 34, 35 Biellmann, J.. 341, 348(32) Billman. J.. 232. 233, 284. 285(39) Birke. R. L.. 231,286, 289(133) Birks. J. B.. 324. 362(17) Bivas, A. 341. 348(32) Blatchford, C. G., 229. 240(20) Bloembergen. N.. 269,271(97. 102). 272( 102) Bloor. D.. 286 Blotekjaer, K.,82 Boardman, A. D.. 82.91(20). lOS(20) Bogolubov, N. N., 101 Bogolubov. N. N.. Jr.. 101 Born. M., 248,265 Boswara, I. M.,13. 43(10) Bottcher, C. J. F., 236, 244(64). 266(64) Boyer. L. L..47 Bradley, J. A., 228. 230, 23 I ( 1 3, 28). 284( 13). 285( 13. 28) Braunstein. R.,343

372

AUTHOR INDEX

Brenner, N. R., 207 Brosens, F., 82, 123, 126, 127 Brousseau, M., 331, 343, 348(25, 37), 349(25, 37) Brown, F. C., 83, 104(39), 269, 272(96) Brueckner, K. A., 166, 208(44) Brun, J. B., 341, 348(32) Brus, L. E., 279, 281(123), 291, 292(123), 293 (123) Buelow, S. J., 286, 289(130) Burgos, J., 362, 363 Burley, G., 65 Burstein, E., 226(5), 227, 228, 230, 231, 235 (30), 286(5), 370

C Callaway, J., 148, 194, 198, 199,203,204, 206. 207. 208, 209, 212, 213(116), 215. 216, 217,218 Caris, J. C., 358, 362(57) Castro, G., 363 Catlow, C. R. A., 47, 77, 79(78, 79) Cava, J., 73, 74(75) Ceperley, D. M., 167, 220 Chabay. I., 285 Chadwick, A. V., 52(d, j, k), 55, 56(53a), 57 (53a), 59(53a), 60(53a), 61(53a), 62(53a) Chandesris, D., 212, 213(118) Chang, C. C., 289 Chang, R. K.,269,270,271(97,102),272(102), 273(107), 276, 285, 287,288, 290(116) Chatterjee, A. K., 199, 203 Chell, G. G., 199 Chemla, D. S., 238,241,242(67), 243(67), 244 (67), 251(67), 252(67), 253(67), 256(67) Chen, C. K., 270,271,273(106), 274,285 Chen, C. Y., 226(5), 227, 228, 231, 286(5) Chen, W. P., 230 Chen, Y. J.. 230, 231, 235(30) 286, 289(130) Chesknovsky, 0.. Chester, G. V., 126(32) Chew, H., 231, 235(32), 248(32) Chiang, C., 214 Choi, S., 343, 352, 357(51), 358, 362(64), 369 (100) Christmann, K., 228,231(13), 284(13), 285(13) Christy, R. W., 242, 244(75), 250(75), 278(75) Claesson, A,, 199 Clark, S. P., Jr., 79

Cohen, M. L., 214 Collet, J., 331, 343(25), 348(25), 349(25) Conwell, E. M., 93 Cooney, R. P., 286 Corish, J., 77, 78(79) Coughlin, R. W., 289 Craig, D. P., 297, 317(7), 324(7), 327(7), 332, 335(27), 338, 348(27, 30) Craighead, H. G., 276 Creighton, J. A., 229, 240(20) Culik, F., 352, 353, 356, 357

D Dalidchik, F. I., 364 Davoli, I., 227 Davydov, A. S., 297, 317(8), 320(8), 327(8), 332(8), 342, 348(36) Dean, P. J., 343 de Castro, A. R. B., 270,271(104), 274,285 De Martini, F., 370 Demuth, J. E., 228, 231(13), 284(13), 285(13) Dernier, P. D., 71 De Sitter. J., 114, 115(56), 116 Devreese, J., 82, 83, 87, 91, 100(13), 101(13), 102(13, 19), 103, 104, lOS(12, 19), 106 (12, 22), 111(19), 114, 115(56), 116, 118 (35), 119(35), 123, 124(17), 125. 126, 127, 13l(13) Dexter, D. L., 297 Dilella, D. P., 228, 229, 240(12) Dimmock, J. O., 297 Dirac, P. A. M., 136, 138, 140 Dissado, L. A., 332, 335(27), 348(27) Dorain, P. B., 270, 273(107), 276, 290(116) Dornhaus, R., 285,286,288 Dresden, M., 83, 126, 131(82), 132(82) Dresselhans, G., 312 Ducuing, J., 269 Duminski, A. N., 269 Durocher, G., 343 E Eagen, C. F., 277,279 Eastman, D. E., 217 Economou, N. P., 238, 231(67), 242(67), 243 (67), 244(67), 251(67), 252(67), 253(67), 256(67)

AUTHOR INDEX

Edwards, D. M., 206 Eesley, G. L., 235 Efrima, S., 233,235 Egri, I., Eliott, R. J., 297,309(5) Engineer, M., 118 Englman, R., 361,362(70) Enkelmann, V., 369(102) Ern, V., 343 Evrard, R.,82,87,91,105(12), 106(12), 114,

115(56), 124(17)

373

Garrett, K. W., 64 Geacintov, N.E., 358 Geldart, D. J. W., 188 Geller, S., 69 Cell-Mann, M., 166,208(44) Gerischer, H., 229,234,240 Gerlich, D., 62 Gertsen, J. I . , 231,235(33), 236,245,291 Ghazali, A,, 221 Gilbert, T.L., 147 Gillan, M.J., 12,13,15, 27,32,43(9), 62,77,

79(9) Glass, A. M., 233. 241, 270, 271(105), 272

F

(105), 276, 277(48), 279, 280(122), 282

(122), 283(122)

Glotzel, D., 212 Clyde, H. R.,19 Goddard, W. A., 220 Gombas, P.,136,189(3), 190(3), 209(3) Goodenough, J. B., 73 Goodgame, M. M., 220 Goovaerts, M., 114,115(56) Gordon, J. G., 234 118,121(29), 123(29,30,33),125,127,128 Gordon, J . P.,248,250,258(85), 260(85), 261 (85) (29), 130,132,141 Gordon, R. G., 167 Fischer, B., 370 Goss, J., 341,348(32) Fleischmann, M., 224,227,234,286. 288(1), Gramila, T.J.,279,291 289 Greer, W.L., 343 Flores, F., 197 Groff, R.P.,358,362(61, 62,63), 363(62) Flygare, W. H., 70 Gross, E. P.,118 Flynn, C. P.,18, 32(17), 40(17), 41(17) Grossman, M., 341,348(32) Fontanella, J . J., 48,52(d, k), 55, 56(53a), 5.7, Grover, M., 343 58, 59,60,61,62(53a, 54,5 5 ) Guillot, C., 212,213(118) Fox, D., 341,348(34a) Gunnarsson, O . , 164,167,174,183 Freeman, D. L., 21 1 Gupta, U., 168 Friedman, L., 343 Gu, Shih-Wei, 118 Frenkel, J., 297 Gutmann, F., 324,327(16) Friauf, R.J., 37,41(40) Frohlich, H., 85 Fry, J. L., 207 Fuchs, R.,369(96) H Fugol, I. Y., 343 Fulton, T., 118 Haarer, D., 363 Furtak, T. E., 230,234,287 Hadley, L., 289 Haken, H., 297,31I , 342,343,348(32),370(9) Hall, J. L., 358,362(58) G Hamann, D. R., 214 Hanamura, E., 341,348(32) Garcia-Moliner, F., 82 Hanson, C. D., 279,291 Garoff, S., 279,291 Hara, M., 65,67(67) Faulkner, R. A,, 343 Faux, I. D., 11, 12(8), 13,27(8), 43(8) Fawcett, W., 82,91(20), 105(20) Fedyanin, V. K., 123 Fermi, E., 136,137,209(2) Fetter, A. L., 166,301,342(11) Feynman, R.P.,82,83,84,85, 86,90(30), 92 (29), 93(29), 99, lOl(30). 103, 104, 109,

374

AUTHOR INDEX

Harada, T., 369(97) Hardy, J. R., 11. 12(7) Harriman, J. E., 162 Hart, R. M., 241,247,252, 286 Harvey, P. J., 29 Hass, G., 288, 289, 290(140) Hawryluk, A. M., 238, 241(67), 242(67), 243 (67), 244(67), 251(67), 252(67), 253(67), 256(67) Hayden, L. M.. 58, 61(54), 62(54) Hayes, W., 44, 45(44) Heckmann, M., 331, 343, 348(25, 37), 349(25, 37) Hedin, L., 148, 169, 192, 207, 220 Hedman, J., 289 Heffner, H., 269 Heinmann, P., 217 Heinz, T. F.. 270, 273(106) Helfrich, W., 362 Hellwarth, R. W., 82, 83(29), 84, 92(29), 93 (29), 109(29), 121(29), 123(29), 125, !27, 128(29), 130, 132 Henderson, G . A,, 186 Hendra, P. J., 224, 227(1), 234, 286, 288(1) Heritage, J. P., 227, 228, 270, 271(105), 272 (105), 279, 280(122), 282(122), 283(122) Herman, F., 183, 191 Herring, C., 202(99) Herrmann, P., 37 Hershbach, D. R., 286, 289(130) Hester, R. E., 229 Hexter, R. M., 233 Hill, I. R., 234 Himpsel, F. J., 217 Hinze, E., 65, 66, 67(65) Hirota, N., 362 Hochstrasser, R . M., 332, 335(26), 348(26) Hodges, C. H., 186 Hohenberg, P., 143, 187 Hong, H. Y.-P., 73 Hoodless, I. M., 29 Hopfield, J. J., 370 Hoshino, H., 65, 66, 67(66), 74, 75(77) Hove, M. A,, 289 Howarth, D. J., 82, 123(3) Hubbard, J., 189 Huberman, M., 126(32) Huggins, R. A., 70 Hultsch, R. A., 28 Humphrey, L. M., 267

Hutchinson, C. A., Jr., 362 Huybrechts, W., 116

I Iddings, C. K., 82, 83(29), 84, 92(29), 93(29), 109(29), 121(29), 123(29), 125, 127, 128 (29), 130, 132 Ihm, J., 214 Ilegems. M., 343 Inkson. J. C., 219 Ishiguro, M., 65, 67(67) Itoh, K., 72 Izuyama, T., 205

J

Jackson, J. D., 238,248(84), 257(68), 258(68), 262(68), 266 Jackson, J. M., I19 Jacoboni, R., 91 Jacobs, P. W. M., 77, 78(79) Janak, J. F., 169, 194,202(100),208(109),212, 216(109), 220 Jarnagin, R. C., 352 Jeanmaire, D. L., 225, 227(2), 250(2) Jennings, D. A,, 358, 362(58) Jepson, O., 212 Jha, S. S., 230, 269, 271(102), 272(102), 285 Johnson, P. B., 242, 244(75), 250(75), 278(75) 164 Jones, R. 0.. Jones, W., 186, 196, 197, 199, 362 Jortner, J., 297, 343, 357, 358, 361, 362(64, 70), 369(101) Joy, D. C., 289

K Kadanoff. L. P., 82, 84, 87, 88, 89, 94(7), 106 (7), 123(1, 4), 125(1, 4), 127 Kafalas, J. A,, 73, 74(75) Kahn, A. H., 32, 79 Kajita, K., 83, 104(40) Kane, E. O., 220 Kartheuser, E., 82, 87, 104, 105(15), 114, 115 (56), 119(15), 124(15, 17) Katriel, J., 177

AUTHOR INDEX

Katz, J. I., 343, 358, 362(64) Kellersreiter, H., 369(99) Kenkre, W. M.,126, 131(82), 132(82) Kepler, R. G., 358, 362 Kerker, M.,231,235(32, 35),236,248(32) Keyes, R. W., 3, 6(6), 17(6), 25 Khadzhi, P. I., 352, 357 Khan, M. A., 215 Khas, Z., 369(95) Kim, D., 205 Kim, Y. S., 167 Kimura, S., 343 Kinbara, A,, 256, 260(91) King, F. W., 235 King, T. A,, 362 Kirsnits, D. A,, 186 Kirtley, J. R.,230,231(28), 285(26,28) Kittel, C., 2, *I), 5(1), 23(1), 32(1) Klein, B. M.,214 Klein, G., 358, 362 Klein, M. V., 228, 289(11) Klinger, M. I., 124 Knapp, K. A,, 217 Know, R. S., 297 Knox, R. S., 297 Kobayashi, T., 357, 369(56a) Kohn, W., 137, 143, 153, 156, 159, 175, 186, 187, 192, 194, 199 Kolb, D. M., 230, 233, 234(49), 285(27) Komiyama, S., 83, 104(40) Kondo, K., 72 Kost, R. E., 28 Kottis, P., 343 Kovacs, G., 232 Krasser, W., 289 Kreibig, U., 254 Kubo, R., 89,90,205 Kummer, J. T., 71, 72 Kunz, R. E., 276 Kurnick, S. W., 34, 35

L Lagois, J., 370 Lallemand, M., 52(g), 55, 58 Lam, L., 158 Landau, L. D., 248(83) Landsberg, P. T., 309(12), 310,350, 368(93) Lang, N. D., 137

375

Langreth, D. C., 82, 123(1, 24), 125(1), 164, 189, 190 Lannoo, M.,178, 181 Lansiart, S., 34, 35 Larsen, D. M.,118, 119 Latter, R., 183 Laube, B. L., 270, 273(107), 287, 288 Laurent, D. G., 194, 206, 207, 212, 213(116), 215, 217 Lax, M., 91, 92(49), 343, 348(38) Lazarus, D., 2,3,7(4), 17(4,1 I), 23,24,25(1 I), 26,27,44,45(47), 47(47), 48,52(c, f, g, m), 54, 63, 64, 65, 66, 67(64), 68(64), 70 Le Blanc, 0. H., Jr., 343, 352 Lecante, J., 212, 213(118) Lee, C. H., 269,271(97, 102), 272(102) Lemmens, L. F., 123 Leroux Hugon, P., 220 Levine, M.,357 Levingson, J., 363 Levinson, T. B., 119 Levy, M., 146, 179 Levy, R., 341, 348(32) Liao, P. F., 233, 238, 241, 242(67), 243(67), 244(67), 245. 248, 250, 251(67), 252(67), 253, 256(67), 258(85), 260(85), 261, 267, 270,271(105), 272(105), 276(48), 277(48), 279, 280(122), 282(122), 283(122), 286, 289 Lidiard,A. B.,2,4(1), 5(1), 11, 12(2,7,8), 13, 16(2), 19(2), 23(1, 2), 27, 28(2), 32, 33(2), 43(8, 13), 44(2), 48, 54, 55, 77, 78, 79(2) Lieb, E. H., 143 Lifshitz, E. M.,248(83) Lipnik, A. A., 331, 343, 348, 349(24) Liu, K. L., 206 Lochner, K., 369(102) Loje, K. F., 35, 36 Lombardi, J. R., 231, 286, 289(133) Long, M. B., 288 Loo, B. H., 287, 289 Low, F. E., 82, 118(23) Lowdin, P. O., 162 Lowndes, R. P., 35, 36, 52(i), 62 Ludwig, W., 15, 16(15), 25(15), 27 Lukosz, W., 276,290,291 Lundqvist, B. I., 164, 167, 174, 183, 192 Lundqvist, S., 137, 220, 226(5), 228. 231, 235 (30), 2W5) Lynch, D. W., 29

376

AUTHOR INDEX

Lynn, J. W., 217 Lyons, L. E., 324, 327(16)

M McCall, S. L., 231, 234,235(31), 236(31), 238 (31) McClintock, R. M., 358, 362(58) MacDonald, A. H., 169 McFee, J. H., 227, 228(7) Mackintosh, A. R., 212 Mackrodt, W. C., 77, 79(78) McQuillan, A. J., 224, 227(1), 288(1) McWhan, D. B., 71 Magistris, A,, 39 Mahan, G . D., 82, 123(2), 125(2) Maher, D. M., 289 Mahr, H., 341, 348(32) Many, A., 363 Manzel, K., 229 March,N. H., 136.137,140,141.143,146(11), 148, 187, 188, 195, 196, 197. 199, 209(4) Marien, M., 104 Marshak, R. E., 141 Martin, G . , 24 Martin, P. C., 1 I7 Masumi, T., 83, 104(40) Matthew, J . A. D., 229 Mavroyannis, C., 341, 348(34) Mehl, M. J., 189, 190 Meier, M., 276, 290, 291 Meletov, K. P., 337, 348(29) Melngailis, J., 238, 241(67), 242(67), 243(67), 244(67), 251(67), 252(67), 253(67), 256 (67) Melo, F. E. A,, 64 Mendes Filho, J., 64 Mermin, N. D., 152, 153, 169 Merrifield, R. E., 343, 358, 362, 363(62) Metiu, H., 233, 235 Metropolis, N., 141 Mie, G . , 235, 248(63) Miglio, L., 141 Mitchell, D. E., 274 Mitchell, J. L., 24 Moller, W. M., 358 Monkhorst, H. J., 21 1 Mook, H. A,, 217 Moore, I. D., 141, 146(11) Moreira, J. E., 64

Mori, T., 65, 67(67) Morigdki, K., 369(97) Morlin. Z., 39 Moruzzi, V. L., 194, 208, 212, 216, 220 Moskovits, M., 228, 229, 231, 235(29), 236, 240(12) Munch, W., 147 Munn, R. W . , 336, 343, 348(28a) Murin, A. N., 34, 35 Murin, I. V., 34, 35 Murphy, D. R., 186 Murphy, D. V., 270, 273(107) Murray, A. M., 187, 195 Murray, C. A,, 228,240,284(14), 285

N Naarnan, R., 286,289(130) Nachtrieb, N. H., 2, 3, 7(4), 17(4) Nagakura, S., 357, 369(56a) Nettel, S.. 124 Nicklow, R. M., 217 Nijboer, H., 29 Nikitine, S., 341, 348(32) Niklasson, G . , 188 Nitzan, A., 231, 235(33), 236, 279, 281(123), 291, 292(123), 293(123) Northrup, D. A., 36, 352 Novikov, B. V., 341, 348(32) Nozieres, P., 166, 189 Nusair, M., 167

0 Oberschrnidt, J., 44, 45(47). 47(47), 48, 52 (c. f, g, m), 54, 63, 64, 65 Ohrnurd, Y . , 197 Okarnoto, K., 82, 118, 119(11), 123(11) Olness, D., 352 Olson, D. H., 233, 241, 270, 271(105), 272 (105), 276(48), 277(48), 279, 280(122), 282(122), 283(122) Ortenburger, 1. B., 188, 191 Osaka, Y., 82, 83, 84(28), 86(34), 94, 118(8, 34), 119(8), 123(10, 28). 125(28), 127 Otter, F. A., 288 Otto, A., 228,232, 233,234, 239, 284, 285(39) Owen, J . F., 276, 290(116)

377

AUTHOR INDEX

P Pack, J. D., 21 1 Painter, G. S., 167 Pant, M. M., 148 Parks, R. E., 269,272(96) Parr, R. G., 186 Part, M. M., 199 Pavlidou, C. M. E., 185 Peeters, F. M.,82,83,87(19), 91, 101(19), 102 (19), 103, 104(19), 105(19), 106(22), 11 1 (19), 118(35), 119(35), 124(19), 125, 126 (22, 80) Peierls, R., 89 Peierls, R. E., 297, 335(2) Pekar, S. I., 370 Pelzer, H., 85 Pemble, M. E., 234 Perdew, J. P., 164, 182, 183, 189, 190, 199,202 Pershan, P. S., 269 Persson, B. N. J., 277 Peter, L., 363 Petroff, Y.,212, 213(118) Pettenkofer, C., 232,233(39), 285(39) Pettifor, D. G., 220 Pettinger, B., 230, 234, 285(27) Pettinger, P., 233, 234(49) Peuckert, V., 189 Pezzati, E., 39 Philpott, M. R., 233,234,341,348(33), 370(33) Pierce, C. B., 23 Pinchaux, R., 212,213(118) Pinczuk, A., 227, 228, 241 Pines, D., 82, 118(23), 166, 189, 329 Platzman, P. M., 82,83(29), 84,92(29),93(29), 109(29), 119, 121(29), 123(29), 125, 127, 128(29), 130, 132, 158,231,234, 235(31), 236 238(31) Pockrand, I., 232, 233, 285(39) Pope, M.,358,362,363 Poole, N. J., 286 Porsch, M.,118 Prasad, P. N., 332, 335(26), 348(26) Preuss, H., 185 Puff, R. D., 118, 119

R Radzilowski, R. H., 72 Raether, H., 237,282(66)

Ragachev, A. A., 341, 348(32) Rahman, M. A., 206 Rajagopa1,A. K., 137,148,155,167,168,169, 203 Ramana, M. V., 169 Rashba, E. I., 119, 337, 341, 348(29, 32) Rasolt, M., 188,213 Ratner, M. A,, 358 Rauk, A., 178 Reggiani, L., 91 Reichl, L. E., 153 Reineker, P., 343 Reiss, J., 147 Remeika, J. P., 71 Revzen, M., 82,88,89,94(7), 106(7) Reyes, J., 230 Rhinewine, M., 240 Ricard, D., 270,273(106) Rice, S. A., 18,297,343,352,357(51), 358,362 (64) Ritchie, G., 227, 230 Roberts, M. W., 289 Rodriguez, C., Rose, J. H., 221 Rowe, J. E., 228, 240, 284, 285(70) Rudnick, J., 269

S Saghafian, R., 52(d, k), 55, 56(53a), 57(53a), 59(53a), 60(53a), 61(53a), 62(53a) Saito, S.,72 Saitoh, M.,83, 118, 119, 122, 123 Samara, G. A,, 3, 13,26,27(12), 29,30(29), 31 (29), 32(29), 36,38(39), 39(39), 40(39), 41 (39), 43(12,29, 39), 44,45(45), 46(45), 47 (49, 48, 49, 50, 51, 52(b, I), 53(45), 54 (45, 46), 57(45), 61(45), 62, 77(12) Sampanther, S., 140 Sanchez, L. A., 286,289(133) Sanda, P. N., 228,231(13), 284,285(13) Sander, L. M., 221(145) Saunders, V. R., 77, 79(78) Sawaoka, A,, 72 Schatz, G. C., 235 Schenzle, A., 341, 348(32) Schimoji, M.,74, 75(77) Schiraldi, A., 39 Schlotter, P., 363, 365 Schluter, M., 178, 181, 214

378

AUTHOR INDEX

Schneider, W. G., 362 Schrock, R. N., 65, 66, 67(65) Schoonman, J., 47 Schott, M., 343, 358, 362 Schuele, D. E., 35. 36, 61, 62(56) Schultz, T. D., 82, 118(27). I19(6), 123(27) Schulze, W.. 229 Schweber, S. S., 149, 150(27) Schwinger, J., 130 Scott, G. D.. 256 Sebastian, L., 369(102) Seitz, F.. 185 Sennett, R. S.. 256 Shaklee, K. L., 341, 348(32) Sham, L. J., 156, 186, 187, 192, 194, 199. 329, 331 Shank. C. V., 228,240,284(14.69,70), 285(70) Shapiro, S. L., 362 Sheka, E. F., 337, 348(29) Shen, Y. R., 270,271(104),273(106), 274,285 Shimoji, M., 65, 66, 67(66) Shockley. W., 32. 79, 106, 331 Shore. H. B., 221 Siebrand, W., 362 Siefert, H. G., 362 Siegbahn, K., 289 Silbey, R.,343 Silver, M., 352 Simon, H. J.. 274 Simpson, P. 352 Singal. S. P.. 212, 213(1 16). 218 Singh, J., 324,334,335(28), 336,338,340,348 (28, 28b, 31), 352, 353(52), 355(52), 356, 357(52), 358, 359(67), 361(67). 362, 364, 365(89), 366(89), 367(89), 368(93), 369 (98) Singhal, S. P., 198 Singwi, K. S., 166, 186, 188, 198 Sivkov, V. P., 34, 35 Sjolander, A,, 188 Skillman, S., 183 Sleeper, A. M., 269, 272(96) Slater, J. C., 136, 140, 169, 171, 172, 182, 186, 208(53) Slonim, V. Z., 364 Smith, J. R., 235 Somorjai, G. A., 289 Sondheimer, E. H., 82, 123(3) Sonnenberg, H., 269 Spicer, W. E., 288,290(140)

Stacy, W. T., 358, 362(59) Stein, F. P., 289 Stern, E. A,, 269 Stern, M. B.. 267 Stoddart, J. C., 148, 187, 188, 197 Stoicheff, B. P.. 362 Stoll, H., 185 Stoneham, A. M., 197 Stoner, E. C., 236 Stoney. P., 188 Stott, M. J., 197 Swain, S., 82,91(20), 105(20) Swenberg, C. E., 358, 359(69), 362(59,65) Swicord, M., 352

T Tadjeddine, A., 230, 285(27) Takahashi. T., 65 Takeda, S., 82, 118(11), 119(11). 123(11) Taylor, A., 54 Taylor, P. L., 301, 315(10), 329(10), 340(10) Teller, E..141 TCO,B.-K., 289 Tessman, J. R., 32, 79 Teuchner, I., 363 Theophilou, A. K., 175, 181, 191 Thiry, P., 212,213 Thomas, L. H., 136. 137 Thornber, K. K., 82, 83, 84, 90(30), 99, 101 (30), 103, 104, 109, 123(30). 127, 130, 132 Timper, J., 233 Tom, H., 285 Tomizuka, C. T., 33.34.35 Tomkiewicz, Y.,358. 362(63) Tong, B. Y., 199 Tosatti, E., 231 Tosi, M. P., 141, 166, 186, 198 Trlifaj, M., 341, 348(35), 358, 362(66), 368 (94), 369(94) Tsang, J. C., 228, 230, 231(13, 28), 284(13), 285( 13, 26, 28) Trott, G., 287

v van Duyne, R. P., 225, 226, 227(2, 4), 229(4), 235, 240(4), 242(4), 250(2), 286

379

AUTHOR INDEX

van Dyke, J. P., 191 van Hove, L., 126, 127, 132 Van Puymbroeck, W., 82. 106(22), 126(22) Van Royen, J., 82, 100(13), 101(13), 102(13), 104, 123, 126(13), 131(13) Van Vechten, J. A., 18, 36(18) Van Vliet, K. M., 126. 127(83), 132(83) Varotsos, P., 15, 16, 19. 25, 26. 27, 61 Vashista, P., 137, 153, 159, 175 Vaubel, G., 358, 363 Verykios, X . E., 289 Vogel, F., 358 Volts, R., 343 Voltz, R.. 343, 358, 362 von Barth, U . , 148,169,178,180, I8 I , 194,207 von Fragstein, C., 254 von Raben, K. U., 270,273(107), 288 von Weizsacker, C. F., 186 Vosko, S . H., 167, 169, 199, 202, 206. 210

W Wakayamd, N., 363 Walecka, J . D., 166, 301, 342(11) Walmsley, S. H., 297, 317(7), 324(7), 327(7) Walters, L. C.. 36 Wang, C. C., 269 Wang,C. S., 194,203,204,206,207(107),214, 216,217 Wang, D. S., 231, 235(35), 236, 248(32) Wang, J. S. Y., 213 Wang, W. P., 186 Wdnnier, G . H., 297 Warke, C. S., 269 Warlaumont, J. M., 228,231(13), 284(13), 285 (13) Watson, J. G . ,274 Webb, J. B., 352, 353 Weber, W. H., 277, 279 Wegner, G . , 369(102) Weiser, G . , 369(102) Weitz, D. A,, 279, 291 Welkowsky, M., 343 Wenning, U., 233, 234 Wert, C., 2, 6(3), 17(3), 20(3) Wetzel, H., 229, 234, 240 Weyland, A,, 82, 93 White, R. J., 143 Whitfield, G. D., 118, 119

Wigner, E. P., 166, 185,214 Wilk, L., 167, 210 Williams, A. R., 194, 208(109), 212, 216(109), 220 Williams. D. F., 343, 352, 353, 363 Wintergill, M. C., 48, 52(d, j, k), 55, 56(53a), 57(53a), 58,59(53a), 60(53a). 61(53a, S S ) , 62(53a, 55) Wittmaack, K., 289 Wokaun, A., 226(6), 238,241(67), 242(67), 243 (67), 244(67), 245, 247, 248,250,251(67), 252, 253(67), 256(67), 258(85), 260(85), 261(85), 267, 270, 271(105), 272(105), 279,280(122), 282(122), 283(122) Wolf, E., 248, 265 Wolf, H. C., 343 Wolff, P.A,, 231, 235(31), 236, 238(31) Wolter, H., 273 Wong, C., 61, 62(56) Wood, T. H., 228. 240, 284(70), 285, 289 Worlock, J. W., 227,228 Wotherspoon, N., 363

Y Yamaguchi, T., 256, 260(91) Yanagiya, H., 74, 75(77) Yin, M. T., 213 Yokota, T., 118 Yoon, D. M., 13, 17(11), 23(11), 24(11), 25 (1 l), 26, 27 Yoshida, S., 256, 260(91) Yound, W. H., 140 Young, W. H., 186 Yu, A. Y.-C., 288, 290(140) Yu Yao, Y. F., 71, 72(71)

2

Zakhidov, A. A,, 364 Zarembra, E., 220 Zener, C., 2, 6(3), 17(3), 20(3) Ziegler, T., 178 Zienau, S., 85 Ziman, J. M . , 329, 331 Zou, X., 209, 212(111) Zunger, A,, 182, 183,214 Zwemer, D. A,, 228,240,284,285(70), 289(11)

Subject Index

A

Absolute reaction-rate theory, 2 Absorption enhanced one-photon, 276-279 bands, nile blue, 281 luminescence intensity, 276-277 metal-island film interaction with adsorbed dye, 277 two-photon, 279-283 nile blue, 28 1-282 spectra, Ag-island films, 255-256 Activation energy, 2 rubidium silver iodide, 68-69 Activation volume, 8-10 alkaline earth fluoride, 52, 55-56, 59 comparison of calculated and experiment, 76 formation, 10 alkaline earth fluorides, 57-58 alkali halides. 25-27 cesium chloride, 30-31 lead fluoride, 51 silver chloride and bromide, 34-35 thallium halides, 43 hard-sphere model, 10 intrinsic regime, 10 lead chloride and bromide, 64 migration, 25-26,34-35 motion of vacancy, 18, 20 motional, 10 alkaline earth fluorides, 58 cesium chloride, 31 lead fluoride, 49 thallium halides, 43 NaCl and isomorphs, 28 pressure dependence, 8-10 rubidium silver iodide, 68-69 RX,-type compounds, 52 temperature dependence lead fluoride, 49, 51 silver iodide, 66-67

silver sulfobromide and sulfoiodide, 7475 thallium halides, 42 thallium halides, 26, 42-43 Adatom, role and distance dependence of SERS,284-285 Adatom model, 232-233 Ag, deposited on Au electrode, 287 Alkali halides, ionic conductivity cesium chloride, 28-32 migration enthalpies, 24 NaCl and isomorphs, see NaCl and isomorphs silver halides, 32-36 Alkaline earth fluorides, ionic conductivity, 54-62 activation enthalpy, 59 activation volume, 52, 55, 59 temperature dependence, 56-57 bulk values, 6 1 dielectric relaxation measurements, 58-59, 61 formation volume, 56-58 interstitial jump process, 60 mode Griineisen parameters, 59-60 models, 61 -62 motional activation volume, 58 pressure effects, 55 vacancy and interstitial mobility, 55, 60 8-Aluminas, ionic conductivity, 70-73 ionic size relative to channel size, 71 resistivity, pressure dependence, 72-73 Atomic multiplet problem, 177-182 determinantal wave functions, 179 energies of state, 178-179 Hamiltonian, expectation value, I78 Hartree-Fock approximation, I80 local density values, 180- 181 wave function 2's state, 179 23S state, 179-180 mixed state, 180

380

38 1

SUBJECT INDEX

B Binding rate constant, 349 Bloch-type functions, 299-300 Boltzmann equation, 85-91, see also Feynman polaron model Hamiltonian; Frolich polaron Hamiltonian arbitrary electron-phonon coupling strength, 87-90 arbitrary operator, 90-91 assumptions, 87 collision term, 88 distribution at zero electric field, 92 Feynman polaron model Hamiltonian, 8687 first moment, 90-91 Frohlich polaron Hamiltonian, 85-86 linear operator, 92 normalization constant, 92 scattering processes included in, 88-89 second moment, 106 validity, 89

C Cadmium fluoride, ionic conductivity, 62-63 Carrier transfer, see Coulomb interaction, matrix element Cesium chloride, ionic conductivity, 28-32 activation volume, 30-31 ionic transport, 29 structure, 23, 28 temperature dependence, 29-30 CO, wavelength dependence of Raman signals, 229 Complex charge carrier eigenvector, 363 energy, 366 formation and motion, 367-368 internal binding energy, 366 Compressibility, 16, 20 Copper, energy bands, 212-213 Coulomb interaction matrix element, 303, 322-323, 328 operator, 302, 307 Crystal Bloch-type functions, 299-300 holes, creation and annihilation, 305

ideal defined, 297-298 Hamiltonian, 298 orthonormal one-electron wave functions, 299 Schrodinger equation, 298-299 tight-binding approximation, functions, 300-301 total electronic energy, 307 unexcited, electronic wave function, 299 vacuum state, 304 wave function excited state, 300 ground-state, 300 orthonormal one-electron, 299 unexcited state, 299 Current density, 149 D Density functional methods, 135-221 application to phase transition of electrons, 220 atomic multiplet problem, 177-182 Dirac-Slater local density treatment of exchange, 139-141 exchangecorrelation functional, see Exchange-correlation functional excited states, 174-177 Hamiltonian, 175-176 density operator, 175 inhomogeneity correction, single-particle kinetic energy, 186-187 length scales and local density approximation, 188-191 linear response theory, see Linear response theory local density approximation, see Local density approximation magnetically ordered metals, 2 16-2 18 nonmagnetically ordered solids, 207-215 agreement between theory and experiment, 214, 209-210 band structures, 207-208 cohesive properties of metals, 208-209 current calculations, 214-215 energy bands and Fermi surfaces, 21 1212 ground-state configuration, 209

382

SUBJECT INDEX

Density functional methods (continued) one-electron eigenvalues, 169-171 one-particle Green’s function and selfenergy, 191-195 partial summations of gradient series, 187188 self-interaction correction, 182- 185 single-particle equations, see Single-particle equations Thomas-Fermi theory, 137-139 finite temperature, 141-143 transition state, 171-174 internal energy, 172 total energy differences between two states, 172-173 Density matrix functional, 153- I55 one-body reduced, 161-162 operator, 153 Depolarization factor, 236, 257-259 Dielectric constant effective, and film absorption, 273 size-dependent, 254 in terms of dipoles, 265 Dielectric continuum model, thallium halides, 40-4 1 Dielectric function, and suitability for SERS, 289-290 Dielectric matrix, periodic system, 198 Dielectric relaxation measurements, 58-59,61 Diffraction grating equation, 274 Diffusion coefficient, 6 dynamical theory, 10 Diphenylanthracene, one- and two-photon excitation, 282-283 Dipolar field, retarded, 258 Dipolar interactions, 255-269 constants, 259 angular dependence, 264 cusps, 260 imaginary part, 259-260 real part, 259 and variation of plasmon resonance frequency, 261 directional reradiation, 261-263 gain coefficient, 257 intensity enhancement maximization, 258 local field, 256-257

particle dipole moments, determination from reflectivity measurements. 264269 p-polarized incident field, 267-268 retarded, regular particle lattices, 258-261 role, 225-226 shift and broadening of plasmon resonance, 257-258 simultaneous resonating of excitation and Raman frequencies, 263-264 s-polarized incident field, 266 Dipole point, 235, 238 size, 249 Dipole moment, 257 particle, 238, 246, 248,264-269 spheroid, 237 Dirac 6 function, integral representation, 90 Dirac-Slater local density treatment of exchange, 139-141 Directional bonding, 220 Displacement field, elastic part, 12 Drifted Maxwellian approach, 97- I06 electron density-density correlation function, 99 electron velocity distribution function, 100, 105

polaron distribution function, 98, 100 polaron momentum translation operator, 99 polaron velocity as function of electric field, 101-102 scattering processes included, 102 Thornber-Feynman result, 97- 100 validity, 100-106 velocity-electric field characteristic, 101102, 103, 106 no stable state exists, 104 no stationary state exists, 103, 105 Dyes, absorbed, 275-283 one-photon, 276-279 two-photon, 279-283 Dynamical model, 18-19 alkaline earth fluorides, 61 -62 macroscopic Griineisen approximation, 53 microscopic, silver halides, 36 NaCl and isomorphs, 25 thallium halides, 43 Dyson equation, 192

SUBJECT INDEX

E Eigenvalue, one-electron, 169-171 Eigenvector, 314, 338 complex charge carrier, 365 composite exciton-phonon states, 338-339 excited-state, 305, 325-326 ground-state, 325 initial and final states, fission and fusion of excitons, 359 Elastic volume relaxation, 3 Electromagnetic models, 230-23 1 absorption as function offilm thickness, 241 correlation with optical properties, 240-241 dependence o n dielectric constant of surrounding medium, 244 electron collision frequency, 254 enhancement at spheroid tip, 252 extended surface plasmons, 230-23 1 frequency dependence, 241 -245 lightning-rod effect, see Lightning-rod effect localized particle plasmons, 23 1 particle shape dependence, 243 polymer replicas, 252-253 power radiated at tip. 247 radiation damping, 225, 248-254 resonance and shape, 243-244 resonance condition, 243 SERS distance dependence, 239-240 size effect, 225, 54-255 surface of uniformly sized and shaped particles, 241-242 Electromagnetic particle plasmon model, see Particle plasmon model Electromagnetic processes, surface-enhanced adsorbed dyes, see Dyes, adsorbed enhancement-damping balance, 29 1 future research, 294 second-harmonic generation, see Secondharmonic generation summary and outlook, 293-294 Electron fastest, energy equation, 137-140 collision, frequency, 254 mobility, 83 Electron gas density change, 195-196 inhomogeneous density, 137 kinetic energy density, 138

383

number of electrons, 137 response function, 196 total ground-state energy, 138 total kinetic energy, 138 Electron velocity distribution function, 110111 Electron-electron density correlation function, 108 Electron-hole pair models, 231 -232 Electron-phonon coupling, polaron Boltzmann equation, 87-90 Emitters, lifetime studies, 290-291 Empirical model, 15-16, 19-20 activation volume, 76-77 NaCl and isomorphs, 25-26 Energy activation, 2, 7 conversion, optimum spacing, 292 defect-formation, 12- 13 eigenvalue complex charge carrier formation, 366 crystal, 312-313 electrostatic interaction, 32 1-322 Gibbs formation, 15 Gibbs free energy, .see Gibbs free energy interaction, 318 internal binding energy, complex charge carrier, 366 lattice statics model, 12 surface conversion, 290-293 total electronic, 3 18 transfer matrix element, see Exchange interaction Energy functional, 151 correlation-only, 185 ground-state, 152 kinetic, 157-158, 162 Enthalpy, activation, 59 Evolution operator, 107 Exchange correlation energy, 189, 191 self-interaction corrected, 184 Exchange-correlation functional, 159, 163169 correlation potential, 168 exchange-correlation hole, 165 Hamiltonian, 164 pair correlation function, 165 Exchange-correlation hole, 165 Exchange-correlation potential, 157, 159

384

SUBJECT INDEX

Exchange-correlation potential (continued) and band shape, 207-208 change in, 204 local density approximation, 166-169 local spin density approximation, 166- 167 parameterized, 167 self-interaction corrected, 184 spin-polarized, 216-217 temperature-dependent, 160, 168- 169 Exchange energy, 187 density, 139 per electron, 139 total, 140 Exchange interaction Frenkel exciton, 322-323 matrix elements, 328 Exchange potential, 187 Dirac-Slater. 140 Excitation rate, molecule, 279 Exciton, 295-370 biexciton, 341 defined, 296 energy states, defined, 296 Hamiltonian, second quantization, 301303 large-radii orbital, see Wannier exciton optical state, 370 polyexciton, 341 stationary-state, 341 theoretical methods for crystalline solids, see Crystal traps for, 296-297 two-particle interaction matrix element, 303 Wannier, see Wannier exciton Exciton reactions, 341 -370 binding and decay, 342-352, see also Wannier exciton, charge carrier conversion exciton-exciton collisions, see Exciton-exciton collisions fission and fusion, 358-363 eigenvectors, initial and final states, 359 energy, initial and final states, 360 Franck-Condon overlap factors, 350 microscopic illustration, 361 probability, 363 rate, 361-362 transition matrix element, 359 Frenkel exciton binding, 351 -352 nonradiative decay rates. 368-370 rate, 342

Wannier exciton decay, 350 Exciton-charge carrier interactions, 363-368 complex charge carrier formation, 365-368 hot electron generation, 364-365 rate, 364-365 Exciton-exciton collisions, 352-357 carrier transfer process, 353 energy transfer process, 353 Frenkel. 353-356 rate constants, 357 Wannier, 356-357 Exciton-phonon interactions, 328-336 deformation potential, 331 Frenkel exciton-phonon interaction operator, 332-336 Wannier exciton-phonon interaction operator, 329-332 Exciton-phonon states, composite, 337-341 eigenvector, 338-339 Hamiltonian, 337, 340 Expectation value of function, 90

F Fast ion conductors, 65-75 fi-aluminas, 70-73 models, 78 NASICON, 73-74 rubidium silver iodide, 67-70 silver iodide, 65-67 silver sulfobromide, 74-75 silver sulfoiodide, 74-75 Fermi energy, 202 Fermi hole function, 140 Feynman polaron model, 83 Hamiltonian, 84, 86-87, 98 Feynman’s polaron theory, extending to finite temperature, 118-1 19 Feynman- Hellwarth-Iddings-Platzman result absorption of polarons, 1 12- 1 13 absorption spectrum peaks, 115-116 electron velocity distribution function, 110Ill memory function, 114- 125 rederivation, 107-109 validity, 109-1 16 Fluorites, see RX,-type halides, ionic conductivity

385

SUBJECT INDEX

Four-current density, 149, 155 Franck-Condon overlap factors, 360 Frenkel defects activation volumes, 34-35 concentration, 4 formation, 10 formation volume, 14, 51, 57 thermodynamic relationship, 14 Frenkel exciton, 315-321 binding, 35 1-352 comparison with Wannier exciton, 326-328 Coulomb interaction matrix element, 322323 critical observations, 324-326 delocalization of creation operator, 317 differentiation from excited state of isolated molecule, 32 I effective mass, 320-321 eigenvector, 316-318 excited-state, 325-326 ground-state, 325 electrostatic interaction energy, 322 energy, 318-320, 326 exchange interaction, 322-323, 327 Hamiltonian, 315-316, 321, 328, 332 interaction energy, 318 molecular orbital approach, 325-326 motion, 321-324, 327-328 intermolecular electronic interaction, 322-323 rate of transition, 351 Schrodinger equation, 31 8 total electronic energy, 318 transition matrix element, 351 Frenkel exciton-exciton collisions, 353-356 eigenvectors, initial and final states, 354 energy, initial and final states, 355 transition matrix element, 354-355 transition probability, 356 Frenkel exciton-phonon interaction Hamiltonian. 336 operator, 332-336 Frolich polaron Hamiltonian, 85-86

G Gibbs formation energy, I5 Gibbs free energy for motion, 18-19 system of noninteracting fermions, 160

Grand potential, 153, 155, 170 change in, 171 decomposition, 160 noninteracting particle system, 160 Green’s function, 197 one-particle, 191-192 Griineisen parameters choice of, 76 macroscopic, 9, 5 1, 53 microscopic, silver halides, 36 mode, 9, 18 alkaline earth fluorides. 59-60

H Hamiltonian acting on Kohn-Sham functions, 203 composite exciton-phonon states, 337, 340 effective, 205 exchange correlation functional, 164 excited states, 175-176 expectation value, 178 Feynman polaron model, 86-87 Frenkel exciton, 321, 328, 332 Frenkel exciton-phonon interaction, 336 Frohlich polaron, 85-86 ground-state energy, 150 Hartree-Fock, 182 Hohenberg-Kohn theorem, 145. 148 ideal crystal, 298 interaction, 204 potential part, 164 second quantization, 30 1-303 and spin densities, 200 total, 345 Wannier exciton, 314-315, 328 Hard-sphere model, motional activation volume, 50 Hartree-Fock approximation, I80 Hartree-Fock Hamiltonian, 182 Hartree-Fock theory, pathologies, 21 8 Heisenberg equations of motion, 87 Helmholtz free-energy, 161 Hohenberg-Kohn theorem, 143- 155 finite-temperature theory, 152-1 55 functional, 146-147 ground-state energy, 145-146, 149-151 functional of density matrix, 147 Hamiltonian, 145 particle density function, 144

386

SUBJECT INDEX

Hohenberg-Kohn theorem (continued) physical content, 151-152 proof, 144-147 spin inclusion, 148-149 summary, 146 Hydrostatic pressure, effects on ionic conductivity, 2-3

I Image dipole mechanism, 235 Impedance function, 117 Intensity, 270 local enhancement factor, 273, 281 Intrinsic defects, Tee also Activation volume variance of parameters with temperature and pressure, 78 Iodine, absorption enhanced, 291-292 on Pt electrode, 286 Ionic conductivity in solids, 1-80 absolute reaction-rate theory, 2 activation energy, 2, 7 activation volume, see Activation volume alkali halides, see Alkali halides, ionic conductivity association regime. 7 conduction regimes, 7-8 elastic volume relaxation, 3 extrinsic versus intrinsic regime, 7 hydrostatic pressure effects, 2-3 intrinsic regime, 6 lattice defects, see Lattice defects log rrTversus T - ' , 20-22 superionic conductors, 2 temperature dependence, 5-8

K Kadanoff-Boltzmann equation, 84

L Latter correction, 183-184 Lattice defects, see also Frenkel defects; Schottky pairs intrinsic defect concentration, 5

and ionic conductivity, 4-5 model calculation of properties, 11-20 dynamical model, 18-19 empirical model, 15-16, 19-20 formation energies and volumes, 11-16 general procedure, 11 lattice statics models, 11- 15 motional energies and volumes, 16-20 strain-energy model, 17-18 rate traverses barrier, 6 vacancy-formation volume, 16 vacancy migration enthalpy, 18 vibrational period of butting, 18-19 Lattice statics models, 11-15 activation volume, 76-77 defect-formation energies, 12-13 displacement field, elastic part, 12 energy, 12 formation volume of defects, 14-15 macroscopic polarization, I 1 volume change o n creating defect, 13-14 Lead bromide, ionic conductivity, 63-64 Lead chloride, ionic conductivity, 63-64 Lead fluoride, ionic conductivity, 45-53 cubic-orthorhomic phase transition, 45, 47 cubic phase, 46-47 motional activation volume, 49 formation volume, Frenkel pairs, 51 orthorhomic phase, 47-53 enthalpy for defect formation and anion motion, 47-48 motional activation volume, 49 pressure dependence, 47, 49 temperature dependence, 47, 49 sublattice melting, 46-47 temperature dependence, 45-47 log pressure derivative, 49-50 Lead iodide, ionic conductivity, 64-65 Lightning-rod effect, 237,245-247 defined, 245 factor, 246-247 reduction, 246-247, 252 Linear response theory, 195-207 density change for displacements, 199 dielectric function and Kohn anomaly, 195199 spin susceptibility ferromagnets, 202-207 paramagnets, 199-202 Lithium, 210-21 1

387

SUBJECT INDEX

Local density approximation calculations of bands, 214-215 difficulties, 219 error decomposition, 190 exchange-correlation potential, 166- 169 and length scales, 188-191 Local field enhancement factor. 257 Luminescence diphenylanthracene, 282-283 intensity, 279, 281 dependence on film thickness, 28 1-282 rhoddmine B,279-280

M Magnetization, 201,204 Mass operator, 192-193 Memory function, 109, 114-1 15 Metal-ligand bonds, stretching, 288 Migration enthalpy, alkali halides, 24 Modulated reflectance, 234 Momentum density function, 158 N

NaCl and isomorphs, ionic conductivity, 2328 activation volume, 28 bound cation vacancy motion, pressure dependence, 28 dynamical model, 25 empirical model. 25-26 formation activation volumes, 26-27 migration activation volumes, 25-26 pressure dependence, 25 pressure-induced transition of RbCI, 27-28 strain-energy model. 25 structure, 23 NASlCON and related materials, ionic conductivity, 73-74 Nernst-Einstein relation, 6 Nickel, bandwidth and band shape. 217 Nile blue, luminescence following UV and visible excitation, 281 -282 0 One-particle equation, in presence of magnetic field, 199

P Pair correlation function, 165 Particle density, 152 function, 144 Particle plasmon model, 235-239, 251-252 adsorbed molecule polarization, 237-238 dipole moment of spheroid, 237 and distance dependence of SERS, 284 equivalent point dipole, 238 field at center of spheroid, 238 field distribution to calculate Raman scattering, 235-236 field inside particle, 236-237 metal particle, role, 236 particle dipole moment, 238 particle plasmon resonance, 236-237 radiated power, 238 reradiation, 238-239 Particle plasmon resonance, enhancement of second-harmonic generation, 270 Phonon, 297, see al.yo Exciton-phonon interactions acoustic energy, 348 Photodissociation yield, 291 -292 Plasmon extended surface. 230-231 role. 285 localized particle, 231 particle resonance, 236-237 resonance frequency, and dipolar interaction constants, 261 Polariton, 370 Polarization, macroscopic, I 1 Polaron absorption, 112-1 13 defined, 85-86 distribution function, 98, 100 impedance function, see Feynman-Hellwarth-Iddings-Platzman result mass, 117 temperature dependence, 118-123 temperature limits, 119-120, 122 momentum operator, 98-99 momentum translation operator, 98-99 nonparabolicity of energy spectrum, 105 physical picture of, 117-1 18 relaxed excited states, 114-1 15 states, transition rate between, 89 velocity, function of electric field, 101-102

388

SUBJECT INDEX

Polaron mobility, 8 I 133, see trlvo Drifted Maxwellian approach S(kT/hwLO)problem, 123-127 electron-phonon coupling, 126 function of inverse temperature, 124, 126 impedance function, see Feynman-Hellwarth-Iddings-Platzman result interrelation between theories, 128 linear dc conductivity, 127 nonlinear dc conductivity, 127-128 relaxation-time approximation. .see Relaxation-time approximation Potential deformation, Wannier exciton-phonon interaction, 331 difference, 203 effective, 160 exchange-correlation, 183 external effective, 157 external, I54 grand, see Grand potential interaction, complex charge carrier formation, 366 ionic, Wannier exciton-phonon interaction operator, 329 local energy-dependent, 193-194 one-body, 197- 199 Pseudopotential, 214 Pyridine, wavelength dependence of Raman signals, 229 -

R Radiation damping, 225, 248-254 inhomogeneous broadening model, 252 local field enhancement factor, 249 local intensity enhancement dependence on particle volume, 250 radiated power, 249 wavelength dependence of SERS fit, 251 Radiation reaction field, 248-249 Raman scattering, surface-enhanced, 226-229 AgCN, 241 Ag sols, 240 distance dependence, 239 inelastic background signal, 228 metal surface, role, 233 metal-island films and in ultrahigh vacuum, 227-228

surface analytics, 288-290 surface roughness, 227 theoretical models, 229-239 adatom, 232-233 electromagnetic models, see Electromagnetic models electromagnetic particle plasmon, see Particle plasmon model electron-hole pair, 231-232 image dipole, 235 induced by adsorption, 233-234 modulated reflectance, 234 surface complexes, 233-234 wavelength dependence, 229 RbCI, pressure-induced transition, 27-28 Reflectivity, 266 Ag-particle array, 267-268 coefficient, 265 Relaxation-time approximation, 9 1-97 absorption of polaron, 97 average electron velocity, 93 impedance function, 91-95,97 scattering rate, 94-95 and impedance function, 94-97 and LO phonon emission, 95 Reradiation, directional, 261-263 Rhodamine B interaction of metal-island film and dye monolayer, 276-277 luminescence, 279-280 dye monolayers, 277-278 one and two-photon excitation, 279-281 Rubidium silver iodide, ionic conductivity, 67-70 RX,-type halides, ionic conductivity, see also Alkaline earth fluorides; Lead fluoride activation volumes, 52 cadmium fluoride, 62-63 enthalpies, 47-48 lead bromide, 63-64 lead chloride, 63-64 lead iodide, 64-65 pressure effects, 44-45 strontium chloride, 54 structure, 23, 44

s Schottky pairs activation volumes, 25-26

389

SUBJECT INDEX

concentration, 4 formation enthalpy, 78 formation volume, 14- I5 Schrodinger equation electrons for any nuclear configuration, 298-299 formation of complex charge carrier, 366 Frenkel exciton energy, 318 ideal crystal, 298 quantum electrodynamics in Fock space, 148 Second-harmonic generation, enhanced surface, 269-275 diffraction grating equation, 274 enhanced Ag-island film experiments, 172-173 by particle plasmon resonances, 270 roughened Ag electrodes, 270 film absorption and imaginary part of effective dielectric constant, 273 function of mass thickness, 272-273 geometry, 275 intensity, 270 local enhancement factor, 273 thickness dependence, 271 local field factors, 273-274 output order condition, 274 regular Ag-particle arrays, 274-275 sapphire substrates, 271 surface, 272-273 weak, 269-270 Self-energy, 193-195 Self-interaction correction, 182-185 correlation-only energy functional, 185 exchange correlation energy, 184 exchange-correlation potential, 183 Hartree-Fock Hamiltonian, 182 Latter correction, 183-184 SERS distance dependence, 284-285 effects of chemical bonding, 286-287 enhanced Raman scattering, 288-290 extended surface plasmon role, 285 information content, 290 metals suitable for, 289-290 time development of signals, 288 unroughened metal surfaces, 286 SF,, enhanced absorption, 293 Silicon cohesive properties, 2 13-21 5

energy values, calculated versus experimental, 214 local potential, 220 Single-particle equations, 156-163 effective potential, 160 exchange-correlation functional, see Exchange-correlation functional exchange-correlation potential, see Exchange-correlation potential external effective potential, 157 Gibbs free energy, 160 grand potential decomposition, 160 Helmholtz free-energy, 161 kinetic energy functional, 157-158,162-163 momentum density function, 158 occupation probabilities, 161 one-body reduced-density matrix, 161-162 separating functional, 156 total energy, 158 variation of density with orbitals, 157 Silver halides, ionic conductivity, 32-36 Silver iodide, ionic conductivity, 65-67 Silver sulfobromide, ionic conductivity, 74-75 Silver sulfoiodide, ionic conductivity, 74-75 Spin density, 159 Spin density vector, 152 Spin susceptibility ferromagnets, 202-207 non-self-consistent, 201 paramagnets, 199-202 self-consistent, 201, 206 static, uniform, 202 Spin-wave energies, 206 Spin-wave spectrum, 205 Spin-wave stiffness, 206-207 Strain-energy model, 17- 18 activation volume, 51, 53 alkaline earth fluorides, 61 NaCl and isomorphs, 25 silver halides, 36 thallium halides, 43 Strontium chloride, conductivity, 54

T Thallium halides, ionic conductivity, 36-44 Thermal expansion coefficient, 16 Thomas-Fermi theory, 137-139 finite-temperature, 14-143

390

SUBJECT INDEX

Thomas- Fermi theory (continued) kinetic energy, electron density. 143 minimum-energy principle, 138-1 39 Thomas-Fermi-Dirac theory Euler equation, 140 kinetic energy, 143 Thornber-Feynman result. see Drifted Maxwellian approach Tight-binding approximation, 300-301

V Volume derivative method, formation volume. 62

W Wannier exciton. 304-315 center of mass coordinate transformation, 313 charge carrier conversion, 344-350 acoustic phonon energy, 348 binding rate constant and rate of transition, 349 exciton concentration estimate, 350 Hamiltonian, 345 low temperatures, 347 transition matrix element, 346

coefficient determination, 31 3-314 comparison with Frenkel exciton, 326-328 Coulomb interaction matrix elements, 328 decay, 350 dielectric medium, 31 1-312 effective-mass approximation, 308 eigenvector, 326-327 energy bands and electron and hole generation, 304 energy eigenvalue, 305-307 energy equation solution, 308-31 1 energy gap between minimum of first conduction and maximum of valence band, 309 Hamiltonian, 314-315, 328 internal energy states, 314 motion, 327 perturbation approach, 312-313 reciprocal lattice vectors, 310 singlet and triplet states, 307-308 wave function, 305-307 Wannier exciton-exciton collisions, 356-357 Wanner exciton-phonon interaction operator, 329-332 ionic potential, 329 potential, deformation, 331 vector of lattice displacement, 329-330 Wave function, see nlro Crystal, wave function excited state, 300 one-electron, 299 unexcited crystal, 299

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