Solid State Physics Advances in Research Volume 20

SOLID STATE PHYSICS VOLUME 20 Contributors to This Volume R. Brill Robert W. Keyes Philip C. K. Kwok Takeo Nagamiya M...

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SOLID STATE PHYSICS VOLUME 20

Contributors to This Volume R. Brill Robert W. Keyes Philip C. K. Kwok Takeo Nagamiya M. D. Sturge

SOLID STATE PHYSICS Advances in Research and Applications Editors

FREDERICK SEITZ Department of Physics, University of Illinois? Urbana, Illinois

DAVID TURNBULL Division of Engineering and Applied Physics, Harvard University Cambridge,Massachusetts

HENRY EHRENREICH Division of Engineering and Applied Physics, Harvard [Jniversity Cambridge,Massachusetts

VOLUME 20 1967

@

ACADEMIC PRESS 0 NEW YORK AND LONDON

COPYRIGHT @ 1967,

BY

ACADEMICPRESSINC.

ALL RIQHTS RESERVED NO PART O F THIS BOOK MAY B E REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM T E E PUBLISHERS.

ACADEMIC PRESS INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l

LIBRARY OF

CONGRESS CATALOG CARD

NUMBER:55-12200

PRINTED I N THE UNITED STATES OF AMERICA

Contributors to Volume

20

R. BRILL,Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin-Dahlem, Germany

ROBERTW. KEYES,IBM Thomas J. Watson Research Center, Yorktown Heights, New York

PHILIP C . K. KWOK,IBM Thomas J . Watson Research Center, Yorktown Heights, New York TAKEO NAGAMIYA,* Department of Material Physics, Faculty of Engineering Science, Osaka University, Toyonaka, Japan M. D. STURGE, Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey

* Present Address : Physics Department, University of Arizona, Tucson, Arizona. V

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Preface

In the first article of this volume, Brill surveys concisely X-ray methods of determining electron spatial distributions in crystals and their application to the various solid types. I n Volume 11 of “Solid State Physics,” Keyes discussed the effect of strain on the band structure of semiconductors. In the present volume, he shows how electronic effects may modify the elastic properties of semiconductors. Although the Jahn-Teller effect was predicted thirty years ago, it is only recently that its manifestation in solids has been extensively explored. Sturge summarizes critically the results of this exploration from both the experimental and theoretical viewpoints. The article by Kwok presents for the first time in this serial publication a systematic exposition of the application of the Green’s function technique to solids. It is hoped that the present application to lattice vibrations and phonon-photon interactions will be followed by articles applying this technique to other phenomena. In the final contribution to this volume Nagamiya presents a molecular field treatment of the theory of spin ordering, with special emphasis on helical arrangements, and its application to the interpretation of experiment. In a future contribution Nagamiya plans to discuss the role of conduction electrons in the exchange interaction.

FREDERICK SEITZ DAVIDTURNBULL HENRYEHRENREICH

October, 1967

Vii

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Contents CONTRIBUTORS TO VOLUME20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PREFACE ................................................................. CONTENTS OF PREVIOUS VOLUMES .......................................... SUPPLEMENTARY MONOGRAPHS ............................................. ARTICLESTO APPEAR SHORTLY .............................................

v vii xi xv xvi

Determination of Electron Distribution in Crystals by Means of X Rays

R . BRILL I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 I1. Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I11. Determination of the Degree of Ionization by Different Methods . . . . . . . . 10 IV. Difference Fourier Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Electronic Effects in the Elastic Properties of Semiconductors

.

ROBERTW KEYES

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Phenomenological Description of Elastic Constants . . . . . . . . . . . . . . . . . . . . I11. Effect of Free Electrons on Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . IV. Electronic Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Electrons on Donors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V I. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 39

40 67 78 89

The Jahn-Teller Effect in Solids

M . D . STURGE

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. The Jab-Teller Effect in Doubly Degenerate Electronic States . . . . . . . . I11. The Jahn-Teller Effect in Triply Degenerate States . . . . . . . . . . . . . . . . . . . IV. Optical Transitions Involving Jahn-Teller Distorted States . . . . . . . . . . . . . ix

92 115 151 178

CONTENTS

X

Green’s Function Method in Lattice Dynamics

PHILIPC . K . KWOK Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Dynamics of Lattice Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Phonon Correlation Function and Experimental Observables. . . . . . . . . . . . I11. Phonon Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Explicit Calculation of Phonon Green’s Function and Applications. . . . . . V. Phonon Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V I . Coupled Phonon-Photon System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helical Spin Ordering-1

214 214 220 240 262 279 290

Theory of Helical Spin Configurations

TAKEO NAGAMIYA Introduction . . . . . .

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

306

I . Elementary Theory elical Spin Ordering. . . . . . . . . . . . . . . . . . . . . . . . . . 307 . . . . . . . . . . . . . . . . 312 I1. Spin Waves in the Screw Structure . . . . I11. Effect of Anisotropy Energy on Spin C . . . . . . . . . . . . . . . . 316 IV . Effect of External Field on Spin Configurations. . . . . . . . . . . . . . . . . . . . . . . 330

V . Spin Waves in Various V I . Complex Spin Configura VII . Spin Configurations in S

in an Applied Field . . . . . . . . . . . . . . ................................. e ................... ........ VIII NQl Temperature and Spin Ordering for C ............ IX. Neutron Diffraction: Theory and Examples Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

348 362 376 395 404

AUTHORINDEX ...........................................................

413

SUBJECT INDEX ...........................................................

422

Contents of Previous Volumes

Volume 1, 1955 Methods Solids

the

of

One-Electron

Order-Disorder Phenomena in Metols

LESTERGUTTMAN Theory

of Phase Changes

DAVIDTURNBULL

JOHN R. REITZ Qualitative Metals

Analysis

EUGENE P. WIGNER FREDERICK SEITZ

of

the

Cohesion

in

Relotions between the Concentrations Imperfections in Crystalline Solids

of

F. A. KROGERAND H. J. VINK

AND

Ferromagnetic Domain Theory

C. KITTELAND J. K. GALT

The Quantum Defect Method

FRANKS. HAM The Theory in Alloys

of

Order-Disorder

TOSHINOSUKE MUTOAND YUTAKA TAKAGI Volence Semiconductors, Silicon

Volume 4, 1957

Transitions

Ferroelectrics and Antiferroelectrics

WERNERKANZIG

Germanium,

and

Theory of Mobility of Electrons in Solids

FRANK J. BLATT

H. Y. FAN The Orthogonalized Plane-Wave Method Electron Interaction in Metols

TRUMAN 0.WOODRUFF

DAVIDPINES

Bibliography of Atomic Wave Functions

ROBERTS. KNOX

Volume 2, 1956

Techniques of Growing

Nuclear Magnetic Resonance

G. E. PAKE

Zone

Melting

W. G. PFANN Nucleor

Electron Paramagnetism and Mognetic Resonance in Metols

Volume 5, 1957

W. D. KNIGHT Applications of Neutron Diffroction to Solid Stote Problems

Golvonomognetic Effects in Metals

C. G. SHULLA N D E. 0. WOLLAN

J.-P. JAN

The Theory of

~~~i~~~~~~~~ inSolids

Speciflc Heots and

lattice

of

Atoms

Thermomagnetic

JAMES

during

Groups and Their Representations

Irradiation

FREDERICK SEITZAND J. S. KOEHLER

G. K.

KosTER

Shallow Impurity Germanium

Volume 3, 1956 Group Ill-Group

and

CLIFFORDC. KLICKA N D H. SCHULMAN

Vibrations JULES DE LAUNAY Displacement

and Crystal

States

in

Silicon

ond

W. KOHN

V Compounds

H. WELKERA N D H. WEISS

Quadrupole Effects in Nu'cleor Resonance Studies in Solids

The Continuum Theory of Lattice Defects

M. H. COHENAND F. REIF

J. D. ESHELBY xi

Magnetic

CONTENTS OF PREVIOUS VOLUMES

Xii

Volume 6 , 1958 Compression of Solids by Strong Shock Waves M. H. RICE, R. G. MCQUEEN,AND

J. M. WALSH Changes of Stote of Simple Solid and liquid Metals

G. BORELIUS Electroluminescence

Photoconductivity in Germanium

R. NEWMAN AND W. W. TYLER Interactionof Thermal Neutrons with Solids

L. S. KOTHARIAND K. S. SINGWI Electronic Processes in Zinc Oxide

G. HEILAND,E. MOLLWO,AND F. ST~CKMANN The Structure and Properties of Grain Boundaries

S. AMELINCKX A N D W. DEKEYSER

W. W. PWERAND F. E. WILLIAMS Macroscopic Symmetry and Properties of Crystals

CHARLESS. SMITH Secondary Electron Emission

A. J. DEKKER Optical Properties of Metals

M. PARKER GIVENS Theory of the Optical Properties of Imperfections in Nonmetals

D. L. DEXTER

Volume 7, 1958 Thermal Conductivity and lattice Vibrational Modes

Volume 9, 1959 The Electronic Spectra of Aromatic Molecular Crystals

H. C. WOLF Polar Semiconductors

W. W. SCANLON Static Electrification of Solids

D. J. MONTGOMERY The Interdependence of Solid State Physics and Angular Distribution of Nuclear Radiations

ERNSTHEERAND THEODORE B. NOVEY

P. G. KLEMENS

Oscillatory Behavior of Magnetic Susceptibility and Electronic Conductivity

Electron Energy Bands in Solids

A. H. KAHNAND H. P. R. FREDERIKSE

JOSEPH CALLAWAY

Heterogeneities in Solid Solutions

The Elastic Constants of Crystals

ANDREGUINIER

H. B. HUNTINGTON Wave Packets and Transport of Electrons in Metals

Electronic Spectra of Molecules and Ions in Crystals Part II. Spectra of Ions in Crystals

H. W. LEWIS

DONALD S. MCCLURE

Study of Surfaces by Using New Tools

J. A. BECKER The Structures of Crystals

A. F. WELLS

Volume 8, 1959 Electronic Spectra of Molecules and Ions in Crystals Part I. Molecular Crystals

DONALD S. MCCLURE

Volume 10, 1960 Positron Annihilation in Solids and liquids

PHILIPR. WALLACE Diffusion in Metals

DAVIDLAZARUS Wave Functionsfor Electron-Excess Color Centers in Alkali Halide Crystals

BARRYS. GOURARY AND FRANK J. ADRIAN

CONTENTS OF PREVIOUS VOLUMES

...

Xlll

The Continuum Theory of Stationary Dislocations

Electron Spin Resonance in Semiconductors

ROLAND DE WIT

G. W. LUDWIGA N D H. H. WOODBURY

Theoretical Aspects of Superconductivity

Formalisms of Bond Theory

M. R. SCHAFROTH

E. I. BLOUNT

Volume 11,1960

Chemical Bonding Inferred from Visible and Ultraviolet Absorption Spectra

CHR. KLIXBULL JORGENSEN

Semiconducting Properties of Gray Tin

G. A. BUSCHAND R. KERN Physics at High Pressure

C. A. SWENSON The Effects of Elastic Deformation on the Electrical Conductivity of Semiconductors

ROBERTW. KEYES Imperfection Ionization Energies in CdS-Type Materials by Photoelectronic Techniques

RICHARD H. BUBE

Volume 14, 1963 g Factors and Spin-lattice Relaxation of Conduction Electrons

Y. YAFET Theory of Magnetic Exchange Interactionr: Exchange in Insulators and Semiconductors

PHILIPW. ANDERSON Electron Spin Resonance Spectroscopy in Molecular Solids

H. S. JARRETT BENJAMIN LAXAND JOHN G. MAVROIDESMolecular Motion in Solid State Polymers N. SAITO, K. OKANO, S. IWAYANAGI, AND T. HIDESHIMA Cyclotron Resononce

Volume 12,1961

Volume 15,1963

Group Theory and Crystol Field Theory

CHARLES M. HERZFELD AND PAULH. E. MEIJER Electrical Conductivity of Organic Semiconductors AND HIDEOAKAMATU HIROOINOKUCHI

Hydrothermal Crystal Growth

R. A. LAUDISEA N D J. W. NIELSEN The Thermal Conductivity of Metals at Low Temperatures

The Changes in Energy Content, Volume, and Resistivity with Temperature in Simple Solids and liquids

G. BORELIUS The Dynamical Theory of X-Ray Diffraction

R. W. JAMES The Electron-Phonon Interaction

K. MENDELSSOHN A N D H. M. ROSENBERG

L. J. SHAMAND J. M. ZIMAN

Theory of Anharmonic Effects in Crystals

Elementary Theory of the Optical Properties of Solids

G. LEIBFRIEDA N D W. LUDWIG

FRANK STERN

Volume 13, 1962 Vibration Spectra of Solids

SHASHANKA S. MITRA

Spin Temperature and Nuclear Reloxotion in Solids

L. C. HEBEL,JR.

Volume 16, 1964

Behavior of Metals at High Temperatures and Pressures

Cohesion of Ionic Solids in the Born Model

F. P. BUNDYA N D H. M. STRONG

MARIOP. TOSI

Dislocations in lithium Fluoride Crystals

F-Aggregate Centers in Alkali Halide Crystals

AND HERBERT RABIN J. J. GILMANA N D W. G. JOHNSTON W. DALECOMPTON

xiv

CONTENTS OF PREVIOUS VOLUMES

Point-Charge Calculations of Energy Levels of Magnetic Ions in Crystalline Electric Fields

M. T. HUTCHINGS Physical Properties and Interrelationships of Metallic and Semimetaliic Elements

KARLA. GSCHNEIDNER, JR.

The Fundamental Optical Spectra of Solids

J. C. PHILLIPS Crystal Symmetry, Group Theory, and Band Structure Calculations

ALLENNUSSBAUM Theoretical and Experimental Aspects of the Effects of Point Defects and Disorder on the Vibrations of Crystal-1

Volume 17, 1965 The Effects of High Pressure on the Electronic Structure of Solids

H. G . DRICKAMER

A. A. MARADUDIN

Volume 19, 1966

Electron Spin Resonance of Magnetic Ions in Complex Oxides. Review of ESR Results in Rutile, Perovskites, Spinel, and Garnet Structures

Theoretical and Experimental Aspects of the Effects of Point Defects and Disorder of the Vibrations of Crystals-2

W. Low

A. A. MARADUDIN

AND

E. L. OFFENBACHER

Ultrasonic Effects in Semiconductors

X-Ray Diffroction Studies of the Lattice Parameters of Solids under Very High Pressure

NORMAN G . EINSPRUCH Quantum Theory of Galvanomagnetic Effect at Extremely Strong Magnetic Fields

RYOGOKUBO, SATORUJ. MIYAKE,A N D NATSUKIHASHITSUME

H. G. DRICKAMER, R. W. LYNCH, R . L. CLENDENEN, AND E. A. PEREZ-ALBUERNE Shock Effects in Solids

DONALD G. DORANAND RONALD K. LINDE

Volume 18, 1966 Energy Loss and Range of Energetic Neutral Atoms in Solids

Interaction of Acoustic Waves and Conduction Electrons

D. K. NICHOLS AND V. A. J.

HAROLD N. SPECTOR

VAN

LINT

Supplementary Monographs

Supplement 1: T. P. DASAND 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, AND G. H. WEISS Theory of Lattice Dynamics in the Harmonic Approximation, 1963 Supplement 4 : ALBERTC. 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: ESYHERM. CONWELL High Field Transport in Semiconductors, 1967

xv

Articles to Appear Shortly

Transition Metal Oxides Polarons Production and Detection of Nuclear Orientation Band Electrons in External Fields: A E. BROWN Group Theoretical Approach H. CALLEN-R. TAHIR-KHELI Collective Properties of Pure and Impure Magnetic Insulators Optical Properties of Solids: ModulaM. CARDONA tion Techniques Magnetic Properties of the Rare Earth B. R. COOPER Metals Quantum Theory of Electronic States S. G. DAVISON at Crystal Surfaces J. R. DRABBLE-C. S. COUSINS The Third Order Elastic Constants Interactions between Defects in C. ELBAUM Crystals Plasmas in Solids M. GLICKSMAN The Semiconductor-Semimetal B. I. HALPERIN-T. M. RICE Transition The Lattice Dynamics and Statics of J. R. HARDY-A. M. KARO Alkali Halide Crystals Effect of Electron-Electron and ElecL. HEDIN-S. LUNDQUIST tron-Phonon Interactions on the Electron Band Structure of Solids S. K. JOSHI-A. K. RAJAGOPAL Lattice Dynamics of Metals Indirect Exchange Interactions in C. KITTEL Metals Dilute Magnetic Alloys J. KONDO Crystal Growth Mechanisms R. L. PARKER Diffusion N. L. PETERSON Noble Gas Crystals R. 0. SIMMONS W. ZAWADZKI Transport Properties of Semiconductors with Nonstandard Energy Bands D. ADLER J. APPEL A. BARKER WILLIAM

xvi

SOLID STATE PHYSICS VOLUME 20

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Determination of Electron Distribution in Crystals by Means of X Rays

R. BRILL Fritz-Haber-lnstitut der Max-Planck-Gesellschaft, Berlin-Dahlem, Germany

I. Introduction. .. . . . . . . . . . . . . . ........................ Theoretical Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Representation of Electron Distribution by Fourier Series . . . . . . . . . . . . . . 2. The Phase Problem.. . . . . . . . ........................... 3. Series Termination Effects. ...................... 4. Results Results on NaCl. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................. 111. Determination of the Degree o Ionization by Different Methods. . . . . . . . . . . D 5. Atomic Scattering Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration of Fourie Fourier Series .................... 6. Direct Integration IV. Difference Fourier Synthesis. . . 7. Diamond.. . . . . . . . . . . . . . . . ..................... 8. Other Compounds.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 2 4

5 99

10 10 16 20

1. Introduction

Diffraction of X rays is a process in which, exclusively, electrons participate. I n the electromagnetic field of X rays, electrons oscillate to become centers that emit radiation of the frequency of the irradiated wavelength. An incident beam of X rays generates coherently scattered X rays from all oscillating electrons. Hereby interference takes place, and the interference pattern contains information about the electronic arrangement. Since electrons are concentrated around nuclei of atoms, the patterns obtained by X-ray diffraction were used mainly to determine atomic arrangements, especially in crystals. For this purpose, it is sufficient to assume a centrosymmetric spherical electron cloud around the nucleus. The electronic distances in atoms are of the order of magnitude of the wavelength of X rays 1

2

R. BRILL

used for diffraction experiments. Consequently, interference between the radiation scattered by the different electrons of an atom occurs, and, the scattering amplitude depends upon the scattering angle as well as the wavelength. It depends, in fact, only upon the ratio (sins)/A. After the Fourier analysis was introduced as a tool to determine crystal structures by W. H. and W. L. Bragg, the first experiments to obtain information about the influence of the chemical bond on the electron distribution were performed by the author of this paper, together with Grirmn, Hermann, and Peters.’ Hereby at least the binding effect in diamond in comparison to an ionic lattice became evident. I n the following, a survey is given about the methods of evaluation of X-ray diffraction experiments with respect to electronic arrangements in connection with the chemical bond and about some salient results. II. Theoretical Background

1. REPRESENTATION OF ELECTRON DISTRIBUTION BY FOURIER SERIES

The intensity of an X-ray diffraction of the order hkl is proportional to the square of the absolute value of the structure amplitude I FhkZ l2 with F h k ~=

+ +k)).

C f n ( h k l ) exp ( 2 ~ i ( h ~ nkyn n

(1.1)

Here the summation is extended over all the n atoms of the unit cell with coordinates x , , y, , and z, , measured in units of the length of the corresponding axes of the unit cell. The f, are the atomic scattering factors of the n atoms in units of the scattering power of a single electron. Exp (ia) = cos a i sin a represents a vector of unit length in the Gaussian plane. The length of each vector in Eq. (1.1) is given by the magnitude of fn . Adding all the vectors in the sum results in the vector F h k l with the phase angle cp. Since f depends on the angle of scattering, it depends upon the order hkl of the reflection considered. The problem of any structure determination consists in determining the atomic coordinates x, , y, , z , . These coordinates can be found by calculating a sufficient number of I FhkZ I for a number of different plausible arrangements of atoms in the elementary cell and seeing which one gives a satisfactory agreement between calculated and observed 1 F h k z I values. That one, then, most probably contains the right x, , y, , z, values and, thus, gives the right structure. This procedure is the well-known method of trial and error. It works in this simple form only for simple structures,

+

R. Brill, H. G. Grimm, C. Hermann, and C. Peters, Ann. Physik [ 5 ] 34, 393 (1939).

ELECTRON DISTRIBUTION IN CRYSTALS

3

but, in principle, it must be used always in spite of the fact that methods exist to simplify the procedure so that even complicated structures could be resolved. The reason for the difficulties at structure determinations lies in the fact that, in principle, the experimental data are insufficient. Generally, F h k l is complex [Eq. ( l . l ) ] .However, from the intensities of X-ray reflections, only its absolute value can be determined. Hence, only the length of the vector F in the Gaussian plane is known, but not its phase angle (p, which defines the magnitudes of the real and the imaginary part of this vector. Furthermore, the procedure requires the knowledge of the atomic scattering factors for all atoms in the cell. This presumption is always fulfilled with good approximation, at least for spherical atoms, i.e., neglecting influences of the chemical bond. The arrangement of atoms in crystals is strictly periodic in the three dimensions of space. I n the diffraction of X rays, only the electrons play a role, and, therefore, we may consider a crystal lattice to be a periodic arrangement of electrons. Hence, the electron density at a point 2, y, z of the elementary cell can be described by a three-dimensional Fourier series : p ( z , y, z ) =

Khkz

exp { -2ai(hz

+ ky + lz) 1.

h,k.1

It can easily be shownZthat the Fourier coefficients K h k l are in close relation to the structure amplitude Eq. ( l . l ) ,viz., K h k l = ( l / v ) * F h k l . Therefore, p(z, y, z ) = v-l

Fhkl

exp { -2ai(hz

+ ky + lz) } .

(1.2)

h&,C

Here v is the volume of the unit cell. The equation connects Eq. ( 1 . 1 ) with the crystal structure. At any point where p has a maximum, an atom is located in the lattice, and the higher the maximum the heavier the atom. According to Eq. ( l . l ) ,F h k l represents a number of electrons, and, therefore p(z, y, x ) means the number of ele.ctrons per unit of volume and is commonly given in electrons per cubic angstroms. The usefuhess of Eq. (1.2) for structure determinations is evident. The phase angle can be introduced in ( 1 . 2 ) as follows. To each F h k l another one belongs with negative indices, i.e., FHii.’ Since F h k z = I F h k l 1 exp ( i ’ P h k l ) and Fski = I F h k l I exp ( - i P h k Z ) , p is real. Each such pair generates two terms

* R. W. James,

“The Optical Principles of the Diffraction of X-Rays”, p. 345. Bell & Sons, London, 1950. 3The same holds for any reflection with mixed indices, e.g., F h k i andFhkl are a pair with opposite signs of the indices.

4

R. BRILL

in the sum of Eq. (1.2), and, hence, Eq. (1.2) can be written in the form p(Z,

y , 2) = V - ’ [ F o o o

+2 21

Fhkz

I COS {2?r(hZ+ k y + l z ) - P h k z } ] .

h,B, 2

(1.3)

c’

The sum has to be taken over all combinations of h, k, and 1 excluding h = k = 1 = 0 and combinations the sign of which are opposite to any combination already used. Fooois the scattered amplitude of the zero order, i.e., of the radiation scattered in the direction of the primary beam. In this case, in Eq. (1.1) f n equals the number of electrons in the nth atom because no phase shift exists in this direction between the radiation scattered by the different electrons of an atom. It is, therefore, FoOo = 2, equal to the number of electrons in the cell. It is evident from Eq. (1.3) that a calculation of p is possible only if, besides the experimentally measured 1 F h k z I values, also the corresponding phase angles VhkZ are known. Unfortunately, no method exists yet to determine unambiguousIy and generally the phase angles directly from experimental data. Hence, a direct calculation of p in the general case is not possible. Equation (1.3) can be simplified considerably for lattices with a center of symmetry. If, in this case, the origin of the coordinate system of the unit cell is at the center of symmetry, to each atom at z, y , z another identical one belongs at 3,g, 2. Hence, the sum in Eq. (1.1) can be written as Fhkz =

+ + + exp { -2?ri(hzn + kyn + Zzn) 13

c’.fn[eXP {2?ri(h% kyn n

=

2

ZZn)

c’fn cos {2x(hz, + kyn + I & ) } . n

Therefore, in this case F h k l is always real. The phase angle cp [Eq. (1.3)] can only be either zero or ?r. In spite of the remaining ambiguity as to the phase angle, this case can be treated much easier than the general one. 2 . THEPHASE PROBLEM

Equation (1.2) requires the knowledge of the phase angle that is not available experimentally. Therefore, the most reliable determinations of electron distributions were carried out with substances where the atomic parameters z, y , z are uniquely given by the symmetry elements of the crystal lattice, i.e., where all the atoms are located at positions without a degree of freedom. In this case, the phase angles are calculated by means of Eq. (1.1) , provided that the atomic scattering factors of all atoms are

ELECTRON DISTRIBUTION I N CRYSTALS

5

also known well enough. As will be shown later, it is sufficient to start with the scattering factors of spherical atoms which are, indeed, known with sufficient precision in almost all cases. For centrosymmetric crystals, the determination of the sign of F is not difficult a t not too complicated structures. For substances containing per molecule, e.g., one heavy atom besides a number of light ones, the phase determination might even be unique. Having the signs, the structure is resolved by means of Eq. (1.2). 3. SERIESTERMINATION EFFECTS

But even if all phase angles are known with sufficient accuracy, another difficulty arises: Eq. (1.3) contains an infinite series. But our experimentally available values will always be limited. The termination of the series can have two reasons: (1) as to Bragg’s law, nX = 2d sin 6, for a given wavelength the order n of a reflection is restricted by the fact that sin 6 cannot be larger than 1; (2) this difficulty can be overcome, however, to a great extent by choosing the wavelength as short as possible. Even then, the intensities become weaker with increasing order, setting an experimental limit, which, however, can be extended by working at low temperatures. It is evident from Eq. (1.3) that each term of the Fourier series represents a plane density wave parallel to the lattice-plane hkl. Hence, missing Fourier terms may as well generate false maxima a t points where they contribute with negative signs or minima where their contribution is positive. This will be demonstrated by a two-dimensional example. The twodimensional projection is represented by a Fourier series in which the terms corresponding to the third dimension are missing. Equation (1.3) then degenerates to p ( Z , !/> =

(l/U>[Fooo

f 2

c’I

Fhkz

I COS

(2r(hz f ICY) - p h k o } ] .

(3.1)

Here a is the area of the unit cell onto which the projection is performed. Consequently, p ( z , y) is an electron density per unit area. Figure 1 shows the lattice of rocksalt looked upon in the direction of the projection of the lattice on the plane (110). A projection of the electron density of atoms (ABCD in Fig. 1) is given in Fig. 2. Sixty-one F values were used.4 One sees that the background is not smooth and that only the centers of the Na+ and C1- ions are spherical. The outer parts, especially a t Na+, show a remarkable deformation. Such a deformation is rather unbelievable and this irregularity is, indeed, not real but due to the seriestermination effect. This is especially made evident by the fact that “termination waves” surround, e.g., the heavy C1-. The maximum of such a C. Hermann, 2.Elektrochem. 46, 425 (1940).

6

R. BRILL

FIG. 1. Structure of NaCl, looked upon in the direction of the two-dimensional Fourier projection of the electron density of one atomic layer onto 110.

wave passes just through the center of the neighboring Na+, affecting the elongation of the Na+ in the (001) direction, and the next minimum in the direction toward the C1- affects the dents in the electron cloud of Na+. Of course, the irregularities in the background between ions are also caused by series termination. Hence, in spite of the fact that as many

FIG.2. Two-dimensional projection of the electron density of NaCl onto 110 without correction for series termination.

ELECTRON DISTRIBUTION I N CRYSTALS

7

X-ray intensities as possible were measured (the ratio of the strongest to the weakest one is 4500:1), the termination effect forges the electron distribution, i.e., diminishes the resolution of electron density in an unbearable manner. There are several possibilities to reduce the termination effect to such an extent that the details of electron distribution become reliable. Heat movement of the atoms in a lattice causes a decrease of the F values which

FIG.3. Same projection as in Fig. 2 with correction for series termination by means of an artificial temperature factor corresponding to 100°C.

is larger as the order of a reflection is higher. This means the waves corresponding to the terms of the Fourier-series are diminished the more the shorter they are. Consequently, the errors caused by the missing high orders are decreased and, therefore, also the irregular fluctuations of the electron density. The application of an artificial temperature factor5 helps to suppress the termination effect. Of course, details get lost this way, Cf. p. 13 for details on the temperature factor.

8

R. BRILL

and the lattice is looked upon as if at higher temperature. It is important not to exaggerate the “artificial” temperature. A criterion for the right amount of correction gives the observation that the background should just be smooth and close to zero at points where no electron accumulations are to be expected. Figure 3 shows the electron density projected in the same way as in Fig. 2 but with an artificial temperature factor applied’ which corresponds to 100°C. One sees that all the atoms have spherical symmetry now and that, on the lines parallel to the plane of projection where no overlapping of atoms takes place (cf. Fig. 1), the electron density between the ions decreases to zero practically, indicating no bonding electrons between C1 and Na, i.e., purely ionic bonding. (The small bridge of electron density slightly higher than 0.5 between two C1- will be discussed later.) Another method to get rid of the termination effect consists in a process of extrapolation which can be applied chiefly for simple structures consisting of simple chemical compounds and with exactly known parameters of atomic positions. In such cases, the F values, plotted with respect to (sin 8 ) / X = s, lie on smooth curves. These curves can be extrapolated16 eventually using theoretical F values and Eq. (1.1) to calculate F h k l values of higher orders.’ If the extrapolation is carried far enough, the influence of series termination is also suppressed. The advantage of this process is that no resolution is lost as if an artificial temperature factor is applied. But, on the other hand, it should be taken into account that the extrapolation introduces information that was not obtained experimentally and, hence, may be true to a certain extent only. Witte and WolfeP have performed very careful intensity measurements on rocksalt. They used for their Fourier synthesis 75 measured, 31 interpolated, and 450 extrapolated values. This means that most of the terms of the Fourier summation are not based on experimental data. This sounds rather strange, but one has to take into account that these 450 extrapolated values are not of extreme importance. They contribute more to the center of the atoms than to their exterior. The latter one is mostly of interest to problems of chemical influences on electron densities, and the corresponding information is contained just in the Fourier coefficients that correspond to the lower orders, i.e., small values of s. An example will be given later. Witte and Wolfel used their experimental results for the calculation Interpolation of experimental points by a smooth curve is a process that has been used, also. But this is a dangerous operation if applied a t small s values, as will be shown later in discussing the measurements on diamond. L. L. van Reijen, in “Selected Topics in X-ray Crystallography” (J. Bouman, ed.), p. 32. North-Holland Publ., Amsterdam, 1951. H. Witte and E. Wolfel, 2. Physik. Chern. (Frankfurt) [N.S.] 3, 296 (1955).

ELECTRON DISTRIBUTION I N CRYSTALS

9

of a three-dimensional series according to Eq. (1.3) and so obtained the electron density in three-dimensional space. Figure 4 shows the electron distribution in the plane (100). It is interesting to note that, in spite of using the process of extrapolation to get rid of the series termination effect, there is still a considerable deviation present of the density lines from circles a t the exterior of the ions. These inflections are not real, or, to be exact, their reality is not proved by the experiment. Probably they are caused by the omission of a number of index combinations, some of which are missing already among the 75 measured values, and maybe also by very slight errors of measurements and not by series termination. Errors of this kind disappear at projections because of the effect of overlapping.

4. RESULTS ON NACL It is interesting to note that both investigations on as simple a substance as NaCl (the one by Witte and Wolfel and the former one by Brill, Grimm, Hermann, and Peters) were carried out independently within a time interval of 16 years and using different methods of evaluation. Furthermore, all the F values of the older measurements were too low because of a slightly wrong correction for extinction which, however, has no serious influence on the final result. The limit of error of the older measurements amounts to about &5% in F , whereas the more recent values are claimed to be accurate within &1.5%. In spite of their differences and in spite of the fact that the much more elaborate three-dimensional analysis was

FIG.4. Three-dimensional Fourier synthesis of NaCl in the plane zy0.

10

R. BRILL

performed by Witte and Wolfel, the results of both investigations are identical. There is no evidence of a bond type other than the purely ionic one. The number of electrons in Na+ and C1-, determined by integration of p over a reasonable area or volume, found in both papers agree excellently (10.08 and 10.05 for Na+, 17.84 and 17.70 for C1-). There exists even an agreement in a finer detail. Figure 3 shows a very small electron bridge with a minimum of 0.53 electron/A2 between two chlorine ions. The same detail is suggested by the extended curvature in the direction of the diagonal in Fig. 4 of the lines corresponding to a density of 0.2 and 0.3 electron. It is to be expected that this effect is more pronounced in a projection. It was said previously that the deviations of the contour lines from circles in Fig. 4 probably are not real. One has to take into account that errors in the experimental values often accumulate in points of high symmetry, and the point t, 0 is one of this kind. Therefore, the agreement in both features may be not accidental but irrelevant. In any event, the discussion shows that in this simple case the main features of electron density distribution can easily be found even if the measurements are not performed with extreme accuracy, but that the evaluation of finer details is difficult, and that for this purpose extreme accuracy is required to the experimental procedure as well as to the evaluation.

a,

111. Determination of the Degree of Ionization by Different Methods

5. ATOMIC SCATTERING CURVES

a. Ionic Crystals

If information about the degree of ionization is wanted only, methods are known which do not require any correction for series termination. One of these methods uses f curves instead of Fourier projections. This method is applicable only for simple structures, where the atomic scattering factors can be determined directly from intensity measurements. We use again the best investigated substance, NaCl, as an example. Application of Eq. (1.1) to this structure shows that F h k l differs from zero only if all the three indices are simultaneously either even or uneven. For these reflections, it is Fhkz

=

4(fci f. h a ) ,

where the positive sign holds for the even, and the negative one for the uneven reflections. Consequently, plotting t F h k l with respect to s results in two curves

ELECTRON DISTRIBUTION I N CRYSTALS

11

corresponding to the sum and the difference of the atomic scattering factors of chlorine and sodium. Formation of half the sum of these two scattering curves gives fcl , whereas half the difference leads to fNa (cf. Fig. 5 ) . The atomic scattering factor a t s = 0 equals the number of electrons in the atom, and, consequently, extrapolation of the experimental curves could be used, in principle, to determine the degree of ionization. However, the extrapolation is difficult because the difference of scattering power between atoms of different charge takes effect only a t very small angles. According to Bragg's law, it is s = (sin 8 ) / A = 1/2d, where d is the distance of the reflecting plane. Now, for simple structures (and only for those can f values be found experimentally with sufficient accuracy) , the largest d values are of the order of magnitude of the distances between atoms, i.e., a few angstroms. This means that s in these cases cannot be smaller than about 0.15 to 0.2 A-I, and in this range the differences become very small. It is obvious from the atomic scattering curves of Fig. 5 that an extrapolation to s = 0 is ambiguous. 19

.'

' 0

. 4

\

3t

2l

51I

I

I

0.1

0.2

03

0.4

0.5

sin 13 -c-

x

FIG. 5. Measured structure amplitudes NaCl(+)

= fcl

+ fNa and NaCl(-)

=

and atomic scattering factors for Na+ and C1- [experimental values marked fci by 0 , values calculated according to Eq. (5.1) marked by 01.

12

R. BRILL

0.1

0.2

0.3

06 sin 8/X

FIG.6. Theoretical atomic scattering factors for Mn+, Mna+, and Mn3+ in comparison to the experimental values in MnO.9

In spite of these difficulties, the described method was used in several cases to determine the degree of ionization in compounds as, e.g., Mn0,9 CuzO,loMn4N,11and CaFz .12 As an example, Fig. 6 shows the measurements on MnO. The experimental points are compared with the theoretical scattering curves for Mn atoms carrying different charges. It is to be seen that the accuracy of absolute intensity measurements must be very high to discriminate between Mn+ and Mn++. The diffraction at Mn also requires corrections for the effect of anomalous dispersion. Hence, different X-ray wavelengths were used in performing the measurements. However, comparison of the coefficient of anomalous dispersion f’ = -3.91 of Mn, determined experimentally for Fe radiation with the very exactly calculated valuef’ = -3.587 by Guttmann and Wagenfeld,13shows that the deviation between these figures is at least as large as the difference between the scattering power of the neutral and the doubly charged Mn ion. The conclusions drawn from Fig. 6 imply, of course, also that the theoretically calculated scattering factors are correct. This is, indeed, so for small values M. Kuriyama and S. Hosoya, J . Phys. SOC.Japan 17, 1022 (1962); 18, 1315 (1963). Suzuki, J. Phys. SOC.Japan 16, 501 (1966). l1 M. Kuriyama, S . Hosoya, and T. Suzuki, Phys. Rev. 130, 898 (1963). la S. Togawa, J . Phys. SOC.Japan 19, 1696 (1964). l 3 A. J. Guttmann and H. Wagenfeld, Phys. Rev. (in press).

lo T.

ELECTRON DISTRIBUTION I N CRYSTALS

13

of s. In comparison with the theoretical scattering factors, also the Debye temperature factor has to be taken into account, since the scattering factors are theoretically calculated for motionless atoms, but measurements are usually taken at room temperature, and zero-point energy motion cannot be suppressed anyway. The temperature factor is applied separately to each atomic scattering factor in Eq. (1.1). In simple cases, the Debye factor (D)has the form

,

D

=

exp ( -Ms2)

M

=

(6h2/mW[+ (z) /z

+ 41,

Here h and k are Planck’s and Boltzmann’s constants, m the mass of the atom, and 0 the characteristic temperature. The influence of the temperature factor at small s values is small. Hence, it is determined by adjusting the theoreticalf values to the measured ones at higher s values, The measurements quoted here were carried out with crystal powders. For the determination of absolute values at single crystals, the influence of extinction14has to be evaluated also. A very elegant method by using the distance of Pendellosung fringes in measuring intensities was recently described by Hattori et aL15 to eliminate these difficulties. However, this method is restricted to ideal crystals.

b. Metals

A number of investigations were also performed on metals, as, for instance, Al, Fell6 Cr,l7 and Cu,16JS and alloys CoAIi9 and NiAl.l9 The atomic scattering factors of the metals were found to be too low at small s values. In the case of alloys, however, agreement was observed between experimental and theoretical f values calculated by the Hartree-Fock method. Since for free atoms (e.g., rare gases) the experiments gave an excellent agreement between observation and theory, the deviations observed at metals are supposed t o be real. They indicate that the electrons R. W. James, “The Optical Principles of the Diffraction of X-rays.” Bell & Sons, London, 1950. H. Hattori, H. Kuriyama, T. Katagawa, and N. Kato, J. Phys. SOC.Japan 20, 988 (1965). I6B. Batterman, D. Chipman, and J. De Marco, Phys. Rev. 122, 68 (1961). M. J. Cooper, Phil. Mug. [S] 7, 2059 (1962). 18 L. D. Jennings, D. R. Chipman, and J. De Marco, Phys. Rev. 136, A1612 (1964). ISM. J. Cooper, Phil. Mug. [S] 8, 811 (1963).

14

R. BRILL

in metals are more spread than in free atoms. The very rough picture of the electron gas in metals would suggest an effect of this kind. Using this model, one might be tempted to determine the number of electrons belonging to the core of the atom separately. As was said already, the extrapolation of f curves to s = 0 is too uncertain for this purpose. Now, by Hosemann and Bagchi20it was shown that atomic scattering factors for spherical atoms can be represented as a sum of Gauss-termsZ1:

ni and ai are constants.

From Eq. (5.1), the electron density as a function of the distance from the center of the atom is given as p(r) =

1 exp { - ( r / a i ) 2 } .

C{ni/(?r3’2ai3) i

Equation (5.1) may be used for an extrapolation to s approximate value for the total number of electrons:

n

=

=

0 to obtain an

Eni. i

This way the relation between f values at larger values of s is used to obtain an information about f at small angles. This method has the advantage that no comparison with theoretical f curves is required and, consequently, also no determination of a temperature factor that might be contained in cyi . To get an idea about the accuracy of this procedure, it may be demonThe f curves are represtrated on NaC1. The data in Fig. 5 were used.8f22 sented by

+ 3.09 exp [-0.463(2~)~], fcl = 7.00 exp [ - 8 . 3 7 8 ( 2 ~ ) ~ ]+ 3.33 exp [-2.025(2~)~] + 7.80 exp [-0.426(2~)~].

fNa

= 6.41 exp [ - 1 . 7 8 9 ( 2 ~ ) ~ ]

The agreement between calculated and observed values is within the limit of error. This gives 9.5 electrons for Na+ and 18.13 for C1-. Addition of a third term 0.5 exp [-161 ( s ) ~ to ] the expression forfNa would result in 2o

21 a2

R. Hosemann and S.N. Bagchi, Nature 171, 785 (1953); cf. also R. Hosemann and G. Schoknecht, 2. Nuturforsch. 12a, 932 (1957); R. Hosemann and G. Voigtliinder Tetzner, 2. Elektrochem. 63, 902 (1959). Cf. also V. Vand, P. F. Eiland, and R. Pepinsky, Acta Cryst. 10, 303 (1957). M. Renninger, A d a Cryst. 6, 711 (1952).

ELECTRON DISTRIBUTION I N CRYSTALS

15

10.0at s = 0, andfNa = 9.7and 9.10at s = 0.05and 0.1,respectively. Within the region of the measurements, this term would in no way influence the calculated values. However, a third term of this form would not be intelligible, because its Fourier transform leads to an electron distribution p ( r ) = 1.36 X exp [-(r/2.02)2], which has, at the center of the neighboring C1 ion, still 25% of its value at r = 0. Consequently, the missing half of an electron is due t o the lack of information at low values of s. This is obvious from the ambiguity of interpolation between 111 and 311 in Fig. 5.The smooth line was used for the calculation here, whereas, e.g., the dotted one would be permissible, too. In addition, the accuracy of measurements at reflections with uneven indices is less because of their low intensities. These inaccuracies appear more significant for the differences of observed F values than for their sum, and, thus, for C1- the extrapolated value agrees well with the expectation. There are other data available to check the usefulness of Eq. (5.1). The electron distribution in A1 was determined by means of a Fourier synthesis independently by two different teams of researcher^.^^ Both teams obtained identical results:

fNa =

(1) The electron density does nowhere in the lattice reach the value of zero. (2) The background density is, according to Brill et al., 0.18electronA-3 and, according to Bensch et al., 0.17 electron A-3. (3) This density corresponds to about’3 electrons per A1 atom uniformly distributed throughout the lattice. In agreement herewith, Bensch et al. determine by integrating over the spherical atoms a total number of 10.2 electrons/Al atom. In a later publi~ation,2~ the same authors derive from their measurements for the atomic scattering factor of A1 in metallic aluminum the formula

+

f~~ = 3.413exp [-0.294(2~)~] 6.300exp [-1.11(2~)~], and this gives a total number of 9.71 electrons for the A1 atom. This is close enough to 10 electrons for the AP+ Similar results were obR. Brill, C. Hermann, and C. Peters, Naturwissenschajten 32, 33 (1944); H. Bensch, H. Witte, and E. Wolfel, Z . Physik. Chem. (Frankfurt) [N.S.] 1, 256 (1954). 24 H. Bensch, H. Witte, and E. Wolfel, Z . Physik. Chem. (Frankju~t)[N.S.] 4, 65 23

(1955). The presence of three conductivity electrons in A1 is also indicated by measurements of the energy loss of electrons in A1 caused by the plasma.26 26 H. Raether, in “Springer Tracts in Modern Physics” (G. Hohler, ed.), Vol. 38, p. 84. Springer, Berlin, 1965; H. Ehrenreich, H. R. Philipp, and B. Segall, Phys. Rev. 132, 1918, (1963).

26

16

R. BRILL

tained a t Mg where two conductivity electrons were A measurement of the f curve of Mg carried out at 5", 9", and 296'K gave for the measurements at 296°K fMg

= 5.37 exp ( -2.5279)

f 5.00 exp ( -9.82s2),

leading to a total number of 10.4 electrons/Mg atom.28 Evaluation of the experimental values, given by Batterman et aZ.16 and by Cooper,17 on Fe and Cr lead to the following expressions for fpc and fcr : fFe

=

+

13.57 exp [-0.2595(2~)~] 9.40 exp [-2.440(2~)2],

fcr = 8.42 exp [-0.1009(2~)~]-I-11.7 exp [-1.736(2~)~],

i.e., about 23 electrons for Fe and 20 for Cr. It cannot be claimed that this method of evaluation is exact. But it is realistic, because the scattering power at large s values is given by the arrangements of the electrons in the inner part of the atoms. Vice versa, the scattering power at small values of s represents the arrangement in the outer shell. Thus, addition of a third term in the equations for the experimental f values, similar to what was just exemplified for Na in NaC1, could bring the number of electrons up to the theoretical value of the uncharged atoms but would correspond to the plasma of conductivity electrons. The reliability of this method, even if not very precise, depends very much upon the accuracy of the absolute intensity of X-ray reflections. The results of Cooper1gon the alloys NiAl and CoAl show different features. This author finds no charge a t the atoms and also finds regions in the lattice where the electron density is zero. He also reports a transfer of charge to the short bond positions. This suggests a bond type slightly deviating from the one found in pure metals. Further investigations have to show whether this difference between a metallic element and these alloys is significant. 6. DIRECTINTEGRATION OF FOURIER SERIES

The number of electrons belonging to an atom can also be determined by integrating p(z, y, z ) over the volume of the atom. It was already mentioned that for this integration the Fourier projection can be used. For this purpose, p, as a function of the distance from the center of the atom, is derived from the projection. Sometimes a numerical integration is 27

H. G. Grimm, R. Brill, C. Hermann, and C. Peters, Nutunuissenschaften 26, 479 (1938); R. Brill, C. Hermann, and C. Peters, Ann. Physik [5] 41, 37 (1942). R. Brill and K. L. Chopra, 2.Krist. 117, 321 (1962).

ELECTRON DISTRIBUTION I N CRYSTALS

17

required if p cannot be expressed in form of a treatable function. Furthermore, the array over which the integration should be performed is not defined unambiguously. The number n of electrons contained in a volume with the edges r - r’, s - s’, t - t’ (parallel to the edges of the rectangular elementary cell) is given by n =

2,

J,: J.: [:

AX,

y, 2 ) d x dy dz.

Inserting for p its value according to Eq. (1.2) gives, if r, s, t, and y, z are measured in units of the corresponding cell lengths,

n = FOw(r- r’) (s

+

?r-3

- s’)

(t

- t’)

C Fhktsin [?rh(r - 7-71sin [?rk(s - s’)]

sin [ d ( t - ~’)]WCZ)-~

h.k.[

X exp { -i?r[h(r

+ r’) + k ( s + s’) + Z(t + t ‘ ) ] ) .

(6.1)

This equation is simplified appreciably if the atom to be considered is located at the zero point of the coordinate system and the integration is performed from -r to +r, etc. In this case, the exponentials in (6.1) disappear. The term for h = k = I = 0 in the sum of (6.1) is the mean electron density of the crystal per volume of integration. The remaining part of the sum represents a positive or negative correction to the mean value. The result is only slightly influenced by the termination effect, and neither the atomic scattering factor nor the temperature factor has to be known. The method is especially suited for cubic lattices of the NaCl type. In this case, the choice r = s = t recommends itself. For an objective choice of the right values for r, the fact is used that between two neighboring atoms the electron density somewhere reaches a minimum at which no essential overlapping takes place. Hence, if the summation is carried out over a range of r values, a plot af n versus r, s, t should show a point of inflection indicating with good approximation the right values of r . As an example, Table I contains the values of n for Na in NaCl as a function of r. The minimum of the difference An is at r = 0.184, and hence n = 10.02. Hereby 107 independent F values29(up to 10, 10,O) were used. Taking into account only 77 values (termination a t 777) gives r = 0.173 and n = 9.98. The influence of the termination effect on n is of the order of a few per mill only, whereas its influence on r is much larger. 89

The data were taken from Table V of H. Witte and E. Wolfel, Z. Physik. Chem. (Frankfurt) [N.S.] 3, 296 (1955).

18

R. BRILL

TABLE I. DETERMINATION OF THE NUMBER OF ELECTRONS OF NA IN NACLBY MEANS OF EQ. (6.1) (b) 77 independent F values (highest order 777)

(a) 107 independent F values (highest order 10, 10, 0) r

n

0.180

10.0100

~ n104-

r

n

0.169

9.9774

0.170

9.9794

0.171

9.9811

0.172

9.9828

0.173

9.9843

0.174

9.9859

0.175

9.9875

0.176

9.9892

0.177

9.9910

0.178

9.9930

20

30 0.181

10.0130

0.182

10.0159

0.183

10.0187

0.184

10.0214

0.185

10.0241

0.186

10.0269

0.187

10.0298

0.188

10.0327

0.189

10.0358

~ n104.

17

29

17

28

15

27

16

27 28

16 17

29 29

18

20

31

The independent determination of n for C1- is more erroneous because in this case r > 0.25, and, therefore, overlapping takes place between the C1- at 000 and the C1- at 3, 3, 3. In this case, no inflection point occurs in the curve n = f ( r ) . At r = 0.5-0.184, it is n = 18.3 which is, in spite of the overlapping, still a good approximation. The radius of Na+ can be calculated from r = 0.184 to be 1.04 A, in excellent agreement with the value of 0.98 given by Goldschmidt30 as early as 1929. Corresponding calculations were carried out on MgO and LiF. For MgO, the measurements of Brill et aL31 were used applying a scaling factor to the low orders according to the measurements of T0gawa.3~The result is 10.28 electrons for Mg and, independently, 9.80 electrons for 0. These figures may be significant. For LiF,33the values are Li+ 2.12, F- 9.88. Because of the large V. M. Goldschmidt, Trans. Faraday SOC.26, 253 (1929). R. Brill, C. Hermann, and C. Peters, 2. Anorg. Allgem. Chem. 261, 151 (1948). S. Togawa, J . Phys. SOC.Japan 20, 742 (1965). 33Measurements of J. Krug, H. Witte, and E. Wolfel, 2.Physik. Chem. (Frankfurt)

*O

B.S.] 4, 36 (1955).

19

ELECTRON DISTRIBUTION I N CRYSTALS

dimensions of F-, an independent determination for this ion was not possible. The same procedure has been used in a very thorough paper of Calder et al.34The paper is concerned with the degree of ionization in LiH, where Li+ and H- are supposed to be present. Several methods were used, e.g., the forementioned comparison between theoretical and observed f values. It is shown that agreement between theory and experiment is obtained only if complete ionization is present and the overlap of electron density is taken into account. Especially, H u r ~ t ’ atomic s ~ ~ scattering factors for the hydride ion, based on the (‘open configuration” function, gives an excellent agreement. Finally, Eq. (6.1) is used to calculate the number of electrons in a cube with r = and hydrogen in the origin. The number of electrons in this volume is experimentally 1.52 f 0.07, whereas according to Hurst’s calculation a value between 1.51 and 1.52 is to be expected. This means that the electron shell around H- differs from the one in the free ion insofar as the charge cloud is contracted by the crystal field. Thus, Calder et al. come to the conclusion that the electron transfer in LiH is in the range 0.8-1 .O.

a

IV. Difference Fourier Synthesis

One of the most elegant methods for the localization of the binding electrons or the determination of deviations from the sphericity of the electron cloud of atoms is the difference Fourier synthesis. Hereby, series termination effects drop out automatically. However, temperature factors are required, especially in cases where the vibration of atoms is asymmetric. By Eqs. (1.2) and (1.3), p ( z , y, z ) may be calculated either with experimental or with theoretical F values [according to Eq. (1.1)]. The difference AP = Pobs

- Pcalc

=

(2/V)

AFhkl

=

I F h k l lobs - I F h k l

HX

=

hx

C’ I A F h k l I COS (2rHX -

lcalc

(PH),

(IV.1)

9

+ ky + lz

gives the deviation of the electron density from the one for spherical centrosymmetric atoms. The experimentally determined temperature factors have to be applied to the f s in Eq. (1.1). They should be determined R. S. Calder, W. Cochran, D. Griffiths, and R. D . Lowde, J . Phys. Chem. Solids 23, 621 (1962). s5 R. P. Hurst, Phys. Rev. 114, 746 (1959). 34

20

R. BRILL

with extreme care.36Furthermore, all the atomic coordinates are supposed to be known with high accuracy. The method may be demonstrated on the basis of diamond as a salient example.

7. DIAMOND a. The Asymmetry of the Carbon Atom

This substance is especially well suited for the study of electron distribution because its characteristic temperature is very high, i.e., the heat movement of the atoms is rather small at ordinary temperature. The consequence is that the electron density of the atoms falls rather steeply from a maximum to a low value. In such a case, the termination effect is rather large, so that either many extrapolated values have to be used or an artificial temperature factor has to be applied. Figure 7 gives the electron density in the plane 2x2 of the diamond The termination effect was eliminated by extrapolating the f values by means of Eq. (5.1). The difference between the NaCl and diamond is obvious. The electron density between two C atoms does not decrease to practically zero, as in NaCl, but the minimum between the atoms is still 1.67 electrons/A3. There are neither regions of negative electron density, which would indicate

FIG.7. Three-dimensional Fourier synthesis of electron density in the plane xxz of diamond. In the center of the atoms A , B, C, D, E , the electron density is 174 electrons-A-3. H. Lipson and W. Cochran, “The Determination of Crystal Structures,” pp. 278 and 302. Bell & Sons, London, 1957. 37 S. Gottlicher and E. Wolfel, 2. Ekktrochem. 63, 891 (1959).

36

ELECTRON DISTRIBUTION I N CRYSTALS

21

FIG.8. Three-dimensional difference Fourier synthesis of diamond in the plane xxz. The centers of C atoms are located at A , B, C, D,E .

errors of measurements or errors in phase angles, nor indications for the presence of series termination effects, except very slight undulations of electron densities in the background. The question whether the height of the electron density at the middle between two atoms is caused by overlapping of spherical atoms only or whether this is due to an electron accumulation by covalent bonding is answered easily by a difference analysis according to Eq. (IV.1). The result is shown in Fig. 8. The figure, also from the paper of Gottlicher and Wo1fel13’ shows that the electron distribution around the carbons (located a t A , B , C , D , E in Fig. 8) is not spherical. Between the atoms an electron accumulation is present with a maximum of 0.51 electron/A3 in the center. This should be due to the covalent bond in diamond.

b. Influence of Asphericity on the Atomic Scattering Factor As has been mentioned already, the contribution of the outer parts of an atom to the atomic scattering factor is reflected a t small values of s. On the other hand, deviations from spherical symmetry should effect the observed f values to deviate from a smooth curve, since the magnitude of the scattering factor depends in such a case on the direction a t which the radiation is scattered. Indeed, deviations of this kind are observed a t diamond. Figure 9 shows the theoretical scattering curve for the spherical atom. The observed values scatter around this curve by amounts appreciably larger than the limit of error. I n first approximation, the deviations can be represented as contributions of a separate small electronic charge

22

R. BRILL

4-

3-

i'T 7-

oc

'

'

'

'

'

a5

'

.

'

sin @/A

'

'

'

7

'

+

FIQ.9. Theoretical atomic scattering factor of carbon (solid line) and experimental values of diamond (dots). Systematic deviation from the theoretical curve at small (sin 8 ) / X shows the influence of the asphericity of the carbon atom.

with a scattering factor jclocated in the middle between two carbon atoms.38 Calculating the F values for this arrangement, one obtains @'iii

=

iF222

+

(f~/2~") =

2fc,

fc

,

3'220

~ F ~=o fc o

= fc

- 2ft

BF.122

,

,

= fc

@'3ii BF3.31

=

( f ~ / 2 ~ ") fc

= (fc/2l")

+fe

,

, (7.1)

.

It is evident that now FZz2# 0, whereas it would be zero for spherical atoms. To calculate the total amount of charge affecting the deviations from the smooth curve, the values of jcmay be used. These can be calculated in the following way. Equations (7.1) show that the contributions of fc to the reflections 220 and 422 are zero. The reflection 222 is due to fc only. Now, the values of s for 222 and 311 are not much different. Consequently, for both reflections jeshould be almost equal. By means of these conditions, three points at the f curve for the residual atom are fixed. To adjust an f curve 38

R. Brill, 2.Ekktrochem. 63, 1088 (1959); Acta C y s t . 13, 275 (1960).

23

ELECTRON DISTRIBUTION IN CRYSTALS

to these points, one may start with the scattering factor for the ordinary spherical carbon atom. Since the smallest value of s for diamond is' larger than 0.75, and since differences in ionic charges at those values disappear, such a procedure seems adequate. The adjustment can be performed by applying a temperature factor. A trial showed that more consistent results are obtained if the atomic scattering curve for C in the valence state is chosen. The evaluation is given in Table 11. The values measured by Gottlicher and WolfeP' are used. The exponent M chosen is 0.210. It is evident that this way the differences between observed and theoretical f values for 220 and 422 disappear. Furthermore, the sign of the deviations corresponds to that requested by Eq. (7.1). For the reflection 222, the directly measured value of Renni11gel3~was used. It is, indeed, very close to thef, of 311. fi can be represented by an equation f6

=

0.315 exp (-6.279).

This would mean about 0.3 electron is contained in the accumulation between C atoms. This figure depends, of course, very much on the choice of the theoretical f curve and the chosen temperature factor. Former measurements' gave, in a similar t r e a t ~ e n t , 3 ~

ff = 0.45 exp ( - - 8 . 1 ~ ~ ) . This corresponds to a higher number of electrons (0.45) and a flatter slope of the electron density. A former wave mechanical calculation by Ewald and Hon140 resulted in 0.535 accumulated electron. A similar, but TABLE11. DETERMINATION OF THE SCATTERING FACTOR fs OF THE ELECTRON ACCUMULATION BETWEEN CARBON ATOMS IN DIAMOND

111

3.025

2.988

3.282 f 7

+0.294

0.208

0.218

220

1.980

1.913

1.911 f 9

-0.002

-

-

311

1.779

1.700

1.592 f 7

-0.108

0.076

0.081

222

-

-

-

-

0.069

0.071

400

1.618

1.515

1.389 f 9

-0.126

0.063

0.045

331

1.545

1.428

1.428, f 5

+0.037

0.026

0.025

422

1.453

1.314

1.313

-0.001

-

-

3gM.Renninger, 2. Krist. 97, 107 (1937). P. P. Ewald and H. Honl, Ann. Physik [ 5 ] 26, 281 (1936).

40

24

R. BRILL

TABLE 111. VALENCE CONTRIBUTION OF CARBON IN DIAMOND AFTER KLEINMAN AND PHIL LIPS^^ Valence contribution Indices

BGHPa

GWb

Theor. KPc

111

0.99

0.98

0.88

220

0.07

0.18

0.01

311

-0.11

f O . 15d

222

400

-0.04

-0.15

-0.14

-0.14 -0.15 -0.13

BGHP = Brill, et aZ.1 GW = Gottlicher and Wolfe1.37 c K P = Kleinman and Phillips.41 Renninger.39

slightly indistinctive, consideration of Gottlicher and Wolfel led to 0.42 electron, whereas the same authors by integrating over reasonable areas of Fig. 8 obtain a value of 0.2 electron. Hattori et a1.,15 using the elegant method of determining the intensities from the distances of Pendellosung fringes at Si, found f = 0.45 exp ( -13s2), i.e., the same accumulation as Brill, but a wider spread of the electron density. Kleinman and Phillips41 have performed an analysis by subtracting the contribution of the 1s electrons from the observed X-ray intensities. These differences are compared with the theoretical contributions of the binding electrons by means of orthogonalized crystal plane wave functions. The agreement is shown in Table 111. The agreement between theory and experiment is fair and a little better for the older measurements of Brill et aZ.1 Measurements on Si and Ge gave results similar to those on diamond.15*37*42 These considerations show that the described method for the determination of the total amount of electrons shifted into the bond direction is not a very exact one. It is based on small differences of large values and, certainly, represents a rough approximation only. It is more adequate to obtain a representation of the electron distribution of the whole atom by using more sophisticated methods if warranted by the accuracy of measurements. 41 42

L. Kleinman and J. C. Phillips, Phys. Rev. 126, 819 (1962). J. J. De Marco and R. J. Weiss, Phys. Rev. 137, A1896 (1965).

ELECTRON DISTRIBUTION I N CRYSTALS

25

c. Expansion of p in Spherical Harmonics

Generally, any arbitrary nonspherical electron wave function or density can be represented by an expansion in spherical harmonics (atomic orbitals.) In this expansion, only functions with the appropriate symmetry are to be taken into account according to the cubic symmetry of the crystal (Kubic Harmonics of Bethe and von der Lage43). The number of the harmonics involved is supposed to be small, as the deviation from spherical symmetry cannot be large. The coefficients of the expansion can be adjusted to the observed values of the atomic scattering factors for the bonded atom. Starting with an expansion of the atomic wave function, Weiss4 assumed an electron density distribution according to p ( r ) = A (1

fc ~ x ~ z / T ~ ) ~ R ~ ,

(7.2)

where p is the electron density, A a normalization factor, 01 an adjustable coefficient of the Kubic Harmonics expansion, r the radial distance from the center of the atom, and R the Hartree-Fock radial wave function for the centrosymmetric C atom. Equation (7.2) gives a rather complicated expression for f. Comparison between calculated and observed intensities gave an excellent agreement. A very thorough analysis of the experimental results on diamond was performed by D a w ~ o n He . ~ ~modifies Eq. (1.1) by subdividing F M in components arising from centrosymmetric and antisymmetric charge densities. This is achieved by separating the atomic scattering factor into spherical, nonspherical centrosymmetric, and antisymmetric parts due to the corresponding contributions of the electron distribution. The diamond lattice has a center of symmetry in the middle between two C atoms. The bonded atoms themselves are not centrosymmetric, but possess the symmetry T d . The Kubic Harmonics expansion yields expressions for the charge distribution due to its spherical, nonspherical centrosymmetric, and its antisymmetric parts. To each of these parts the corresponding scattering factor is calculated. By means of Eq. (1.1), it is shown then that hi = 4n only centrosymmetric components are in all reflections with h; = 4n f 1 contain contributions involved, whereas reflections with of both centrosymmetric and antisymmetric components. Exclusively antisymmetric parts of the electron density distribution contribute to reflections hi = 4n 2. These reflections split into two groups. I n the first with group, a t least one index is zero (e.g., 200, 420), and the reflections are

c

c

+

43

45

H. A. Bethe, and C. von der Lage, Phys. Rev. 71,612 (1947). R. J. Weiss, Phys. Letters 12, 293 (1964). B. Dawson, Proc. Roy. SOC.Ser. 4 298, 255; 264 (1967). I thank the author for putting the manuscripts of the papers a t my disposal.

26

R. BRILL

genuinely forbidden. In the second group, reflections like 222 and 622 appear, which are forbidden for spherically symmetric atoms. Consequently, informations about the antisymmetric arrangement should be obtainable primarily from these reflections and, next, also from reflections with odd hi = 4n f 1 ) . indices To apply the derived equations to the measurements of Gottlicher and Wolfel (which are supposed to be more precise then the earlier ones of Brill et al.) , the differences are considered between these measurements and the intensities calculated for a spherical atom with a Hartree-Fock scattering factor to which the appropriate temperature factor is applied. I n agreement with the earlier results, the differences are restricted to reflections with low indices, the largest deviations observed a t 111 and 222. Furthermore, the deviations are restricted chiefly to reflections of the type 4n f 1. These are the ones influenced by antisymmetric components in the scattering factors. This indicates that the electron density, according to the analysis of Dawson, can be written approximately as

(c

p(r> = pC(r)

+ F ~ ( T{ z) y z l r 3 } .

(7.3)

Here pc is the centrosymmetric spherical Hartree-Fock charge distribution and F ~ ( T a) radial function, which, by adjustment to the experimental results, can be written in the simple form

F ~ ( T= ) 7.5~2exp ( - 2 . 2 ~ ~ ) .

(7.3a)

Equations (7.2) and (7.3) are different insofar as the distinction between F ~ ( T and ) p C ( r ) ,as made in (7.3), is not contained in (7.2). The antisymmetric parts of both equations are almost equal, and the square of the second term in the parentheses of (7.2) presumably is negligible. In accordance with the smallness of the deviations from centrosymmetric sphericity, it is not surprising to have a rather good agreement between observed and calculated intensities for both equations by a proper adjustment of the constants. The values of the atomic scattering amplitudes for the low-index reflections of diamond calculated by means of Eqs. (7.3) and (7.3a) agree fairly well with the experiment (see Table IV) . But there are still deviations between observed and calculated intensities, especially for 400. To get rid of this and other differences, another term has to be added. 400 is a reflection of the type 4n and, consequently, must be influenced by the centrosymmetric component. The next higher Kubic Harmonic with centrosymmetric symmetry is, therefore, added to Eq. (7.3) : ape =

G ~ ( T{ (s4 )

+ y4 + z4)>r4- $},

(7.4)

27

ELECTRON DISTRIBUTION I N CRYSTALS

and it is assumed that G4is a function similar to (7.3a) :

G4(r)

=

(7.4a)

const. r2 exp (-2.29).

Adding (7.4) to (7.3) and adjusting the constant in (7.4a) to the experimental results shows that the best agreement is reached if for the constant the value -2.0 is chosen. Table IV shows the final results of Dawson's calculation in comparison with wave-mechanical calculations by Clark46 and by Bennemann." In Clark's calculations, Slater's functions are taken to represent the atomic orbitals. The bond charge caused by the overlap between tetrahedral hybrid orbitals is replaced by a suitably chosen Gauss function. Bennemann performs a rather extended wave-mechanical calculation resuIting in approximate wave functions that give the density of the valence electrons besides their energy and the self-consistent crystal potential. Table IV contains also the reliability factors R for the different calcuTABLEIV. COMPARISON OF DAWSON'S EVALUATION WITH OBSERVED A N D THEORETICAL F VALUESOF DIAMOND hkl

F,.,GWa

FIIb

FIIIc

FIVc 17.947 14.807 8.581 0.992 11.344

F..,BGHPd 18.66 14.44

111 220 311 222 400 331 422

18.696 15.392 9.068 1.160 11.192 8.344 10.576

18.487 15.224 9.158 1.160 11.920 8.157 10.480

18.730 15.464 9.028 1.160 11.352 8.270 10.544

18.530 15.011 9.175 0.487 113 2 3 8.104 -

511 333 440 531 620 533

7.263

7.088 6.969 9.368 6.341 8.416 5.742

6.998 7.060 9.400 6.347 8.376 5.764

-

-

6.80 6.68 8.96 -

-

5.72

2.00

1.11

3.44

3.86(3.06)

R, %

-

9.144 6.302 8.408 5.736 ~~~~

a

41

FIb

~~~~

GW = Gottlicher and Wolfe1.37 FI = Dawson, Eq. (7.3); FII = Dawson, Eq. (7.4). FIII = Clark,46 FIV = B e n n e ~ n a n n . ~ ~ BGHP = Brill et al.1

H. Clark, Phys. Letters 11,41 (1964). K. H. Bennemann, Phys. Rev. 133, A1045 (1964).

-

-

8.48

11.05 7.76 9.33

-

28

R. BRILL

lations. It is evident that Dawson’s values in column FII have the lowest B index because his calculations are based on adjusting the parameters as well as possible to the experiment. Comparison between FI and FII clearly elucidates the effect of taking into account also the Kubic Harmonic with index 4. It is, furthermore, remarkable that the caIcuIation of Bennemann, reflecting a basically wave-mechanical treatment, leads to such an excellent

3 frl

t 2.0

10

FIG.10. The functions p c ( r ) and F ~ ( T )the , latter one as superimposed contributions of two carbon atoms, 1.54 A apart.

agreement with the observation. The R index becomes even smaller (3.06) if the earlier results of Brill ei! al., as given in the last column of Table IV, are used for the calculation of R. But the largest discrepancy between the experiment and Bennemann’s calculation at the value of Flll does not disappear. Here the figures of Gottlicher and Wolfel and Brill et aE. agree within the limit of error given by Gottlicher and Wolfel. Equations (7.3) and (7.4)probably give a better representation of the true electron distribution in diamond than the wave-mechanical calculation, where approxi-

ELECTRON DISTRIBUTION IN CRYSTALS

29

FIG.11. The function xyz/r3 in the plane containing the bond direction a t a constant distance r = 0.77 A = (a/8)31/2from the center of the atom (a = length of the edge of the unit cell of diamond).

I

II

I

I

I I I

\

'T

I

P

FIG. 12. Atomic arrangement in cyanuric acid (HNC0)a (Verschoor6l).

30

R. BRILL

[toil

T FIQ. 13. Three-dimensional difference synthesis in the plane of the molecule of cyanuric acid according t o Verschoor.61 Contour intervals 0.1 electron-A-3; broken lines: negative electron densities; chain dotted lines: 0 electron-A-3.

mations have to be used.4sThe paper of Dawson shows that very carefully performed experiments, indeed, allow a very accurate determination of the electron distribution, at least in solids of simple structure. To obtain similar accuracy in other substances, measurements at low temperatures and also exact determinations of the location of the atoms by means of neutron diffraction might be required. Diamond is, as mentioned already, one of the best suited substances, because its characteristic temperature is very high and because the atoms are located in positions without a degree of freedom. The electron distribution in the bonded C atom, as derived by Dawson and given by Eqs. (7.3) and (7.4) , is thoroughly described in the author’s It might, however, be interesting to note that the agreement between Fexp BGHP and Bennemann’s calculation, as well as the results of Kleinman and Phillips in Table 111, is best. Furthermore, some values of BGHP agree excellently with the ones of GW, as 111, 400, 440,and 533. This could, of course, be accidental. But, this agreement could also indicate that the earlier measurements are not as bad as commonly supposed. If this were the case, it could indicate that the well-known differences between different diamond crystals, on which the measurements of BGHP were performed, have also a n i d u e n c e on some of the X-ray intensities. Therefore, it might be interesting to repeat measurements on single crystals of different kind and origin. The larger deviation between Bennemann and BGHP at 111 might be because this intensity was also measured on a powder specimen of the same origin as that of GW.

ELECTRON DISTRIBUTION I N CRYSTALS

31

Y :

FIG. 14. Three-dimensional difference synthesis in the plane of the molecule of symmetric trinitrotriaminobenzene according t o Cady and Larson.52 Contour intervals 0.067 electron-A-3. Heavy lines represent positive values, light lines, negative values, and broken lines, the zero contour.

paper.45Here a brief description of the most important part is given as contained in Eq. (7.3). Figure 10 shows, along a line connecting two carbon atoms, the functions p c ( r ) and F 3 ( r ) , the latter one with ordinates five times enlarged. Since F3(r) describes a rather extended spherical shell around each atom with a maximum a t r = 0.674, and since the distance of two carbon atoms in diamond is 1.54 A, both shells are superimposed, resulting in a maximum a t the middle between the two atoms. I n the figure, the superimposed value is drawn. The function xyz/r3 by which F 3 ( r ) is multiplied is represented in Fig. 11, where its magnitude is drawn in angular coordinates at constant r = 0.77 A in the plane containing two carbon-carbon bonds forming an angle of 109.5'. According to the antisymmetry of the function, each lobe has opposite to it a negative one, so that the integration over the function is zero. The lobes have the effect that F3(r) (xyz/r$} has positive values only in the bond direction. Hence, the second term in Eq. (7.3) represents the bonding orbitals. Similarly, Dawson treated measurements on Si and Ge.45

8. OTHERCOMPOUNDS Since effects of binding electrons appear rather clearly in diamond, similar effects should be observable in organic compounds where the bond

32

R. BRILL

FIG.15. Three-dimensional synthesis of the cyclopropane ring according to Hartman and Hirshfeld.53 Contour interval 0.01, negative densities dotted lines, zero line broken.

type is the same. Peaks in the center of C-C bonds were found, e.g., by Cochran49 in a difference map of salicylic acid, by Masonso at acridine, and by VerschooF at cyanuric acid. Figure 12 shows the atomic arrangement and Fig. 13 the differenceFourier-map of cyanuric acid. Rather strong electron accumulations of height 0.2-0.5 ele~tron/A-~ are visible in the middle between bonded atoms. There is also a peak present near atom O2 which may indicate the lonely electron pair of an oxygen atom which forms a hydrogen bond to a neighboring NH group. An excellent example in this connection is also a difference map of symmetric trinitrotriaminobenzene obtained by Cady and Larsons2 reproduced in Fig. 14. Electron peaks of appreciable height appear between the bonded atoms. Furthermore, Hartman and Hirshfelds3found similar peaks at a cyclopropane compound ( I , 2,3 tricyanocyclopropane) . The peaks of the C-C bonds within the cyclopropane ring are, in agreement with the theory, located off the line connecting two carbon atoms in the ring. Figure 15 represents the result. Since in this figure there are more and W. Cochran, Acta Cryst. 6, 260 (1953); 9, 924 (1956). R. Mason, Proc. Roy. SOC.A268, 302 (1960); see also R. Mason and G. B. Robertson, Advan. Struct. Res. Diffraction Methods 2, 35 (1966). G. C. Verschoor, Nature 202, 1206 (1964). s2 H. H. Cady and A. C. Larson, Acta Cryst. 18, 485 (1965). 63 A. Hartman and F. L. Hirshfeld, Acta Cryst. 20, 80 (1966).

49

60

33

ELECTRON DISTRIBUTION I N CRYSTALS

deeper negative regions than positive ones, it might well be that the scale factor is in slight error, so that the positive peaks might be slightly higher. Another peak, not shown in the figure, occurs between the C atoms of the ring and the C atoms of the side chains. The peaks between the ring atoms and the ones between side chain and ring atoms are almost a power of 10 smaller than at the other organic .compounds (cf. Figs. 13, 15, 17, etc.). Since the R value for the structure of the propane-derivative suggest a high accuracy of the structure determination, the smallness of the peak heights may be caused by rather large thermal vibrations or by the refining process (see below). A very high accuracy of measurements is, of course, required to obtain details of bonding as clearly as in the examples given here. The authors discuss Even finer details are reported by O’Connell et 131.~~ very thorough experimental results on some benzene derivatives and compare the obtained difference maps with theoretical calculations. To obtain a higher accuracy, identical regions of symmetric molecules are averaged. Figure 16 shows such an averaged region of the symmetric trinitrotriaminobenzene based on the forementioned data of Cady and Larson. The additional features appearing this way are as follows. There is a slight difference in the electron density between the centers of the nitrogen atom of the NH, group and the NO2 group. Considering the resonance structure

0

H

FIG. 16. Averaged difference synthesis of symmetric trinitrotriaminobenzene according to O’Connell et ~ 1 . 5 4Contour interval 0.05 A--3, negative contours broken, zero contour chain dotted. A. M. O’Connell, A. J. M. Rae, and E. N. Maslen, A d a Cryst. 21, 208 (1966).

34

R. BRILL

C

H

FIG.17. Left: Theoretical difference synthesis for benzene in the molecular plane. Right: The same after refinement. Figure from O'Connell et a1.54 Contour interval 0.025 electron-A-3. Negative contours are broken and t,he zero contour is chain dotted.

(NO2)C- C(NH2)

FIG.18. Difference electron distribution between two carbon atoms perpendicular t o the molecular plane of s-triaminotrinitrobenzene.54 Contours are as in Fig. 17.

FIG. 19. Theoretical difference electron distribution between two carbon atoms perpendicular t o the molecular plane of bensene.64 Contours are as in Fig. 17.

ELECTRON DISTRIBUTION I N CRYSTALS

35

given by Cady and Larson, it is to be seen that the nitrogen of the NO:!

group is positively charged in each of the three structures, whereas only one of these structures carries a positive charge at the amino-nitrogen. This could explain the difference in electron density at these nitrogens. Whether or not this kind of evaluation is justified seems to be questionable, especially since no surplus charge is to be seen a t the location of the oxygen atoms which should carry negative charges according to the resonance structures. O’Connell et al. suggest that this may be due to the process of ~ ~ atoms with a noncentrosymrefinement. It was shown by D a w ~ o nthat metric arrangement of the electron cloud, as caused by bonding orbitals, may be misplaced by refinement procedures where scattering factors of centrosymmetric spherical atoms are used for the calculation of theoretical F values. This way, misplacements of light atoms by about 0.02 A and quite large temperature factor errors may be generated. This is demonstrated in Fig. 17, taken from the paper of O’Connell et al. Consequently, for demonstration of bonding effects between light atoms by a difference synthesis, the determination of the positions of the nuclei by means of neutron diffraction seems recommendable. Neutron diffraction often gives parameters that do deviate from the ones determined by means of X rays as far as atomic positions, as well as temperature factors, are concerned. It is also recommended to perform the measurements a t low temperatures. Another interesting feature, demonstrated by O’Connell et al., is the electron distribution in the plane perpendicular to the benzene ring of, e.g., s-triaminotrinitrobenzene as shown in Fig. 18. The drawing was obtained by the averaging process described previously. For comparison, the corresponding theoretical result for benzene according to the same authors is also shown (Fig. 19). The same effect was also observed by Mason.5o Thus, it may be stated that, by increasing experimental accuracy and by correcting for or elimination of errors, the X-ray method may become of increasing value to elucidate problems of chemical bond and to check theoretical predictions. 65

B. Dawson, A d a Cryst. 17,990 (1964).

This Page Intentionally Left Blank

Electronic Effects in the Elastic Properties of Semiconductors

ROBERT W. KEYES I B M Thomas J . Watson Research Center, Yorktown Heights, New York

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Phenomenological Description of Elastic Constants. . . . . 111. Effect of Free Electrons on Elastic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Ordinary Elastic Constants of Multiband Semicond 2. Crystal Stability.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Electronic Relaxation and Attenuation of Elastic Waves.. . . . . . . . . . . . . . . 4. Elastic Constants of Other Semiconductors. . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Thermal Propertie ......................................... 6. Third-Order Elast nts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Electronic Strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Photostriction. . . . . . . . . . . . . . ............................. 8. Doping by Electron Irradiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Electronic Magnetostriction in Bismuth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . agnetostriction and Magnetoelasticity in Germanium and ........................................................ ors . . . . . . . . . . . . . . . . . . . . . . . 11. Effect of Donors on Ordinary Elastic Constants. . . . . . . . . . . . . . . . . . . . . . . 12. Effect of Donors on Thermal Resistance.. . VI. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 39 40 40 52 53 56 59 63 67 68 68 70 78 74 83 78

89

1. Introduction

The electronic energy levels of a semiconductor depend on the state of strain of the semiconductor crystal. If some of the levels are occupied, the electronic contribution to the free energy of the crystal depends on the state of strain. The electronic contribution to the dependence of the energy on strain can be significant in moderately and heavily doped semiconductors and gives rise to easily observable dependences of elastic properties on doping. This work describes the electronic contribution to the elastic properties of certain types of semiconductors. The models of semiconductor band structures and the way in which they are affected by strain which are used here have been explained in 37

38

ROBERT W. KEYES

some detail previously.’ Briefly, the models are deformation potential models of the type originally introduced by Bardeen and Shockley2 and by Herring.3 The deformation potential model states that strain shifts a band in energy without changing the shape or other parameters of the band. Since this investigation is concerned with energetics, a band is completely specified by its density of states in energy for most purposes. The effect of strain, then, is to shift the density of states function by some then the density amount. If a strain shifts a band i by an amount Wi), of states function N ( ” ( E ) must be replaced by N ( i ) ( E - W ( 0 )in integrals that yield energetic quantities. The effects of small amounts of electrically active impurity on the elastic constants of a semiconductor can be surprisingly large. The reason is that the shifts in energy of the bands must always be compared with a Fermi energy or a thermal energy, which have values of only a few hundredths of a volt. Since the rate of change of a band energy with respect to strain is described by a deformation potential constant with a value of many volts, the strain is always multiplied by a factor of the order of lo2 or lo3 in expansions of the electronic energy in terms of the strain. Thus, even though the electronic energy itself is small, the coefficients that describe its strain dependence are quite large. Electronic energy is responsible for the binding and the elastic properties of many solids. Electronic models have frequently been used in calculations of the elastic constants of metal^.^-^ The characteristic feature of the present study is the use of semiconductor models that are useful in describing many other properties of semiconductors and whose parameters can be determined in a variety of ways. The study of electronic effects in the elastic properties of semiconductors has value for a number of reasons that are worth describing. Firstly, semiconductors are extremely important technologically. It is important to know their elastic properties in order that they may be properly used. Changes in moduli, thermal expansion, or magnetostriction with doping may have significant practical impact. The electronic effects in semiconductors furnish perhaps the only examples in which important mechanical properties can be accurately derived from physical models. At present, silicon and germanium are the only solids in which large elastic effects can be quantitatively explained R. W. Keyes, Solid State Phys. 11, 149 (1960). Bardeen, Phys. Rev. 76, 1777 (1949); W.Shockley and J. Bardeen, ibid. 77, 407 (1950);J. Bardeen and W. Shockley, ibid. 80, 72 (1950). C. Herring, Bell System Tech. J . 34, 237 (1955). K. Fuchs, Proc. Roy. SOC.A161, 585 (1935); A163, 622 (1936). H. Jones, Phil. Mag. [7] 41, 663 (1950); R.S. Leigh, ibid. 42, 139 (1951). For a review, see N. F. Mott, Progr. Metal Phys. 3, 76 (1952).

* J.

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

39

in terms of models of electronic band structure. It will also be seen that some elastic properties of semiconductors are completely dominated by electronic effects. Interpretation of these properties requires an electronic theory. Electronic effects in elastic properties provide a new method for investigating the electronic properties of semiconductors. Since they are derived from the same basic models, the electronic effects in the elastic properties of semiconductors can be correlated with other phenomena, such as transport phenomena, and can be used as an additional tool in the formulation of useful semiconductor models. Apart from its utility, the elastic method of studying the band structure of semiconductors is conceptually satisfying in that it depends only on the application of thermodynamics and statistical mechanics to models. It avoids the intervention of transport theory in the determination of such parameters as carrier density, valley symmetry, deformation potential constants, and density of states mass. As in practically all branches of semiconductor science, most experimental information concerning electronic effects in elastic properties relates to germanium and silicon. Thus, most of the experimental examples presented will refer to these semiconductors. This is, of course, a result of the availability of crystals of germanium and silicon with specifiable impurity content, exceptional freedom from crystalline imperfections, and large size. II. Phenomenological Description of Elastic Constants

An elastic property is here understood to mean any property that arises from the dependence of the free energy of a crystal on the strain. Attainable strains in solids are quite small, so th at the free energy may be expressed as a series in the strain components:

F

=

F,

+ F ~ : +E c : :

(EC)

+ C:::

(EEE)

+ ....

F1 is a tensor of rank 2, c is a tensor of rank 4, and C is a tensor of rank 6. Further terms in the expansion are usually not experimentally accessible. Consider F1 first. F1 must vanish in a crystal in equilibrium because if it differed from zero the crystal could lower its free energy by adopting some strain. There may, however, be electronic contributions to F which are conveniently regarded as contributions to F1. This term thus represents an effect of electrons on the form or size of a crystal. F1 can have only one independent component in a cubic crystal; it must be a scalar multiplied by the unit tensor. Its effect is, therefore, the same as the effect

40

ROBERT W. KEYES

of a hydrostatic pressure; it can only uniformly dilate or contract the crystal. On the other hand, F1 may have two components in a uniaxial crystal. Its effect then will, in general, be to change both the c/a ratio and the volume of the crystal. The coefficients c and C are known as second- and third-order elastic constants. They have forms that are consistent with the symmetry of the crystal in question, a requirement that will not be discussed in any detail here but that is adequately covered in many references.7s8The nature of electronic contributions to elastic properties is restricted by the fact that electronic contributions to c and C must also have the same forms that are prescribed by crystal symmetry. The electronic states that participate in electronic processes in semiconductors are extremal states; they are close to the maximum or minimum energy of their band. Extrema usually occur at points with special symmetry properties, so that energy bands in semiconductors are almost always not of the most general form that might exist in the crystal, but have special symmetries. Consequently, the electronic contributions to elastic constants usually do not have the most general form permitted by the crystal symmetries, but have special properties that arise from the special symmetries of the energy bands. Other properties related to the elastic constants, such as thermal expansion, propagation of elastic and thermal waves, and magnetostriction, will also be considered. The phenomenological description of these other properties is straightforward but not necessary for the purposes of this paper. 111. Effect of Free Electrons on Elastic Properties

The energy levels of a semiconductor change when the semiconductor is ~ t r a i n e d . The ~ . ~ electrons can redistribute themselves among the levels in such a way as to minimize their free energy in the strained crystal. Thus, some of the work needed to strain the crystal is recovered, and the effective elastic constant for the strain is .de~reased.~J" 1. ORDINARY ELASTIC CONSTANTS OF MULTIBAND SEMICONDUCTORS

The nature of the electronic contribution to the elastic constants of a semiconductor is found by expanding the free energy in powers of the 7

W. Voigt, "Lehrbuch der Kristallphysik." Teubner, Leipzig, 1910 (2nd ed., 1928).

* C. S. Smith, Solid State Phys. 6 , 175 (1958).

L. J. Bruner and R. W. Keyes, Phys. Rev. Letters 7 , 55 (1961). R. W. Keyes, IBM J. Res. Develop. 6 , 266 (1961).

lo

41

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

strain. Consider a semiconductor that contains several conduction bands (i).lO* The electronic free energy of such a semiconductor may be written a8

J

-m

Here nci) is the number of electrons in band (i), m

N ( < ) ( E ) f ( lE), dE,

n(i’ =

(1.2)

f(E, [) is the probability that a state of energy E is occupied, f(E,

r> = tl + exp C(E - l)/kT1lrl,

(1.3)

N ( S ( E ) is the density of states in energy of band (i), and l is the Fermi level of the electrons in the semiconductor. is used to indicate a sum over the bands (i). Let the semiconductor be strained, and call the shift in energy of band (i) caused by the strain Wi). It is then necessary to replace L V ( ~ ) ( by E) W ( E - Wi)) in Eqs. (1.1) and (1.2). The Fermi level also changes by an amount w, so that j- = lo w, introducing a subscript zero to denote values of quantities in the unstrained state. F,1 is to be expanded in powers of the parameters that depend on strain, the W ( ; )and w . Reference to the specific form of f ( E , I ) given in Eq. (1.3) reveals that the expansion of the integral in Eq. (1.1) can be expressed in terms of powers of ( Wci) - w ) and derivatives of f ( E , 1) with respect to E. Thus, F,, becomes

+

=

+ + kT

+co

Nci)(E) log [l

(n(i)(lo w )

+ (W(i)- w ) The coefficient of

1

N ( O ( E ) f ( Elo) , dE

(“(0 - w )

Wi)

=

- f ( E ,T o ) ]

dE

-m

-

+(Wi) - ~ ) ~ @ ( 9 (1.4) ] .

is n(i),Eq. (1.2). CPci) is the function

-1

N ( i )( E )[af(E,lo)/ a E ] dE.

(1.5)

df(E, TO)/aE is nearly a negative delta function in degenerate cases, which

are the ones of most interest. Therefore,

is essentially the density of

conduction band is a band in which the density of states vanishes below some energy. Usually the density of states is a monotonic increasing function of energy in the energy range of occupied states. The multiband and multivalley semiconductor models are of particular interest and importance because they describe the conduction bands of germanium, silicon, and the 111-V compound semiconductors.

l0.A

42

ROBERT W. KEYES

states at the Fermi surface. Ordinarily, w is determined by the condition that the total number of electrons in the bands (i) must be a constant. is constant. Thus, That is, N =

En(<)

Substituting for Eq. (1.4), gives

w

from Eq. (1.7) in the expression for the free ehergy,

Equation (1.8) illustrates two points. Firstly, the linear term in the strain (the WCi))is very simple: this change in free energy is equal to the average change in energy of the electrons. Secondly, the quadratic term in the free energy, which represents a contribution to the elastic constant tensor of the semiconductor, essentially samples the densities of states of the various bands at the Fermi level.11 Appreciable effects will be found only when there are at least two bands that have large densities of states at the Fermi energy. Furthermore, the form of the dependence of the W(i)on strain is limited by the symmetries of the crystal and the band structure. Only elastic constants corresponding to strains that shift one band edge with respect to another will be changed by the effect of Eq. (1.8). The same kind of criteria control the existence of large piezoresistance effects, and the origin of large electronic contributions to elastic constants may be recognized as easily as the origin of piezoresistance eff ects.’JJ2 It is also worth noting that aj(E, l o ) / a E = (kT)-lf(E, lo)in the nondegenerate approximation. The quadratic term in Fel then becomes

The most interesting case to consider, since it is the easiest to study experimentally and contains the examples in which experiments have been done, is the multivalley semiconductor. The minimum energy of the conl1

l2

P. J. Price, IBM Research Note NW-3 (1962) (unpublished). C. S. Smith, Phys. Rev. 94, 42 (1954).

43

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

duction band of a multivalley semiconductor occurs at some point other than the center of the Brillouin zone.3 Other energetically equivalent minima are generated by the symmetry operations of the crystal. The states in the vicinity of the various minima are called valleys, and the index (i) may then be used to enumerate the valleys. Scalar properties, such as the density of states function, N(O(E), are identical for all valleys. Tensor properties, such as the change in energy of a valley with some particular shear strain, are, in general, different for different vaIIeys but can be transformed from one valley to another by rotation of the coordinates. The integrals over the densities of states which appear in Eqs. (1.4) and (1.5) reduce to the well-known and tabulated Fermi integralsi3 for parabolic bands, that is, bands in which N(o(E) is proportional to Specifically, if the density of states function Nci)(E) is Nci)(E)

=

(47r/h3)(2m*)3’2(E- E,)l12,

=

-

no‘i’

(1.10)

=

(2/7r”2) N,F,/2 (7) ,

=

( 2 ” ~ ”(Nc/kT)F,/2’ ~) ( 7 ) , (1.11b)

(1.1la)

where N , = 2 (27rm*kT/h2)3/2, and 7 = (TO - E,) / k T has been introduced to represent the position of the Fermi energy. The valleys of all of the common cubic multivalley semiconductors, such as n-type silicon and germanium, and certain 111-V compounds, lie on axes of three- or fourfold ~ y r n m e t r y . ~Second-rank J~ tensors pertaining to such valleys have rotational symmetry about the axis involved. For example, the most general relation between the shift of energy of a valley due to strain and the strain tensor, which will frequently be used in analyzing the electronic effects in the elastic properties of a multivalley semiconductor, is c. (1.12) W‘i’ = s(i): Since aci) has rotational symmetry in the cases under consideration, it can be written in the form3 B C O =

gal + &a(i)a(i)

(1.13)

where a(i) is a unit vector along the axis of rotation. a(” is of the type 3-lI2 (111) for germanium and of the type (001) for silicon. The appearance of I3

l4

J. McDougall and E. C. Stoner, Phil. Trans. Roy. SOC. London A237, 67 (1938). H. Ehrenreich, J . A p p l . Phys. 32, 2155 (1961).

44

ROBERT W. KEYES

TABLE I. SUMSOVER VALLEYS OF CONTRACTIONS OF SECOND-RANK TENSORS t, u, v WITH THE VALLEY VECTORS a(i)

(111 ) valleys (germanium)

a I n the case of silicon, the sums can be expressed in the simple but not very useful forms ~,u:a(~)a(i)v:a(i)a(i) = BY(u,,v,, u,,,,vu,, + u.v,) and a(i)a(i)v: a(i)a(i)t:

+

cu:

a(i) in Eq. (1.13) and other tensors that characterize properties of the valleys of silicon and germanium leads to occurrence of the contraction of a(i) repetitively with various second-rank tensors in expressions for the electronic contributions to physical properties of these and similar semiconductors. The forms of the sums over valleys of such contractions are given in Table I as functions J(u, v) and K ( t , u, v) whose arguments are second-rank tensors. Results in the following sections will frequently be presented in terms of the functions of Table I. The form of the sum over

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

45

valleys of such contractions depends on the type of a(”), and a subscript to indicate a particular a(i)type will be added to J and K when appropriate. When Eqs. (1.11) -(1.13) are substituted into Eq. (1.8), it is found that the electronic free energy is

F,I

= -

(./a) ( 2 , ’ ~ ’ ’NoF112’ ~) ( 7 ) L 2 J( E, . E)

(1.14)

Here v is the number of valleys (see Table 11).The elastic energy of a cubic crystal to terms of second order in the strain is

F, = i[B(tr

+

e)2

4c44(ez,Z

+ +c’(ezz2 + ey1/2 + czaf -

+ + Ey2

ezzeyy - euueZz

- ezZeZZ) (1.15)

Ezz2)1.

Here B is the bulk modulus and c’ is the shear elastic constant 3 (c11 - ~ 1 2 ) . Comparison of Eq. (1.14) and Table I with the elastic energy shows that the electronic effect contributes to c44 in germanium and to c’ in silicon. These contributions are A ~ 4 4=

Ad

-$ ( 2 / ~ *NcF1/2’ / ~ ) ( 7 )Xu2

= - ( 2 / ~ ’ /NCF1/2’ ~) (7)Zu2

(Gel,

(1.16a)

(Si).

(1.16b)

The electronic effect changes the elastic constant for the shear that splits the symmetry degeneracy of the valleys. The change in the shear elastic constants is negative because part of the free energy needed to strain the TABLE 11. PROPERTIES OF GERMANIUM A N D SILICON, FROM VARIOUS SOURCES Property conduction band

a V

m*/mo au M

fft

ff1

gs g II

Germanium

Silicon

(111 )/31’2 4 0.22 16 eV 12.2 0.61 1.92 0.87

(001 ) 6 0.324 9 eV 5.2 1.09 1.9983 1 .9995

Elastic constantsb

B Cf c44

Q

0.750 0.401 0.668

0.977 0.509 0.796

m* is the density of states effective mass of one valley. Adiabatic elastic moduli at 300°K in 1012dyne/cm*.

46

ROBERT W. KEYES 1.2

0.4 -

0.2-

9

OO

!

2

I

T/TD

FIG.1. The functions L2(T/TD) and L 3 ( T / T ~that ) describe the temperature dependence of the electronic contributions to second- and third-order elastic constants. At high temperatures, L ~ ( T / T D )= $(TD/T) and LI(T/TD) = +(TD/T)' (after Keyedo).

crystal is recovered by the transfer of electrons from valleys that are raised in energy by the strain to valleys that are lowered in energy by the strain. A more useful form of Eqs. (1.16) is obtained by introducing the total number of electrons from Eqs. (1.6) and (1.11) and normalizing the temperature to the degeneracy temperature,

T D = (3N/87rv)2'3 (h2/2m*k).

(1.17)

The significance of T D is that the Fermi energy, { - E, , becomes equal to kTD at zero temperature. The electronic contribution to the elastic constants can be written15as Ac44 = -+(47r/3)213(m*~u2//h2))113Lz(T/T~) (Ge) , A d = -2 (27r)2'3 (m*Zu2/h2)N1/3L2( T / T D ) (Si).

(1.18a) (1.18b)

Here the temperature dependence of the electronic contribution is contained in the factor Lz( T / T D ), which is defined by the equations ( T/TD)

=

[3F1/2(40)/21-213,

Lz(T/TD) = $(TD/T)C~i/z'(~o)/Fiiz(~o)I. 16

(1.19) ( 1.20)

A formula for the [OOl] case which is similar to Eq. (1.18b) but is in error by a factor (2/3)2/3 was given by N. Einspruch and P. Csavinszky, A p p l . Phys. Letters 2 , l (1963).

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

47

The function L,(T/TD) is plotted in Fig. 1. Note that L ~ ( T / T D=) 1 at T = 0. Therefore, the other factors in Eqs. (1.18) are the electronic effects at the absolute zero of temperature. It is not possible to measure the electronic contribution to the elastic constants of a semiconductor by simply adding electrons, because any large body must be essentially electrically neutral. The only effective way to change the electron concentration in a semiconductor crystal is by doping with an electrically active impurity. There is, then, no unique way to separate the effects of electrons or holes on elastic constants from the effects of donors or acceptor atoms. I n practice, however, it appears reasonable to ascribe almost all of the effects of doping with an electrically active impurity to electronic effects because the electronic effects are much larger than the direct effects that impurities usually produce on the elastic constants of solids. Direct effects of impurities, such as are encountered in metals and ionic crystals, lead to relative changes in elastic constants which are only of the same magnitude as the fractional impurity concentration.16J7The electronic effects in degenerate semiconductors may be easily recognized by other features besides their large size, as will be seen below. They are sensitive to the symmetries of the band structure and thus affect some elastic constants strongly and leave others unchanged. Their effect is insensitive to the species of the impurity except insofar as it changes

,

6.3 0

I00

200

T

300

(OK)

FIG.2. c44 of pure and heavily n-type germanium as a function of temperature.g The lower line refers to a specimen containing 2.8 X 10'9 cm-3 arsenic donors. The concentration was determined by measuring the Hall effect. H. B. Huntington, Solid State Phys. 7 , 213 (1958). C. S. Smith and J. W. Burns, J . Appl. Phys. 24, 15 (1953);J. R. Neighbours and C. S. Smith, Acta Met. 2, 591 (1954);T. R.Long and C. S. Smith, ibid. 6,200 (1957); R.Bacon and C. S. Smith, ibid. 4, 337 (1956). 17s J. J. Hall, Phys. Rev. 137A,960 (1965).

16 17

48

ROBERT W. KEYES

the electron concentration. Finally, the temperature dependence of the change in elastic constant due to electrons is closely related to the degeneracy temperature of the electron gas. Figure 2 shows the elastic constant c44of a pure specimen of germanium and one containing 2.8 x 1019 arsenic donors as a function of temperature. A marked difference is apparent; the c44 of the heavily doped sample is reduced by the electronic effect. T 0

3.5

"5u

E

3

(OK)

100

200

300

-

:---\ ..

U

I I

I I

P

n

0

I

(b) Ge :1.5x10'eS b

"0

100

200

T

300

( O K )

FIG.3. Electronic effect$ in the c44 of germanium. (a) Obtained by subtracting the values obtained by Hal117*J8for the 1244 of the pure and heavily doped specimens shown in Fig. 2. (b) Values of A c 4 4 reported by Mason and Bateman's* for a specimen contain- ~ donors. The dotted lines are calculated from Eq. (1.18a) ing 1.5 X 10'8 ~ m antimony with m* = 0.22 and Eu = 16 eV.

Note the large size of the reduction; 0.06% arsenic impurity in germanium produces a 5.5% change in C M . The relative change in elastic constant is almost 100 times the atom fraction of impurity. The difference between the c44 of the doped and the pure sample is plotted in Fig. 3a. The prediction of the electronic theory, Eqs. (1.16)-(1.20), is shown by the dashed line. The electronic theory is also compared with a similar experment on germanium doped with antimony to a smaller concentration in Fig. 3b. The parameters of germanium which were used to calculate the theoretical curves are listed in Table 11. It is seen that the changes in the

ELECTRONIC EFFECTS IN ELASTIC CONSTANTS

49

TABLE 111. THEELASTIC CONSTANTS OF THE PUREAND HEAVILY DOPEDSPECIMENS OF GERMANIUM USEDTO OBTAINTHE DATAOF FIG.2 AT 300°K IN 10l2 DYNE/CM*

C44

C’

B

Pure

Doped

Change

0.6679 0.4014 0.7502

0.6409 0.3997 0.7482

-0.0270 -0.0017 -0.0020

elastic constants are in reasonable agreement with the predictions of the electronic theory, in regard to both magnitude and temperature dependence. The interpretation of the effect on c44 of doping germanium with arsenic as an electronic effect is supported by, in addition to the large size of the effect on c44, the small size of the effect on the other elastic constant^.^ Table I11 compares the complete set of elastic constants of the pure and doped crystals and shows that the major effect is a change of c44 .la The fact that the effect is produced by doping with either arsenic or antimony also shows that it is an electronic effect. Experiments similar to those summarized in Figs. 2 and 3 have also been performed with silicon, in which c’ rather than c44is changed by doping with don~r s . ~gJ~ The a difference arises because a shear strain with a (100) axis destroys the degeneracy of the valleys of silicon, whereas a shear strain with a (111) axis does not. Exactly the opposite is true for the conduction band of germanium. The measured changes in c’ are given in Table IV.

FIG.4. Values of Ac’ in n-type silicon taken from Table IV and plotted in the reduced form suggested by Eqa. (1.17) and (1.18). The line is fitted to the points by using the effective mass of Table I1 and choosing Eu = 9.3 eV.

J. J. Hall, to be published. W. P. Mason and T. B. Bateman, Phys. Rev. 134, A1387 (1964). 18 N. G. Einspruch and P. Csavinszky, A p p l . Phys. Letters 2, 1 (1963). 198 J. J. Hall, Phys. Rev. 161, 756 (1967).

18

186

50

ROBERT W. KEYES

TABLE IV. THECHANGE IN c'

N (cm-3)

Ac'

1 . 7 X 10'8 1.1 X 10'95 6.5 X 2 . 0 x 1019b (I

WITH

DONOR CONTENT IN SILICON

(T = 300°K)

-0.24 X 1O1Odyne/cma -0.75 -2.5 -1.29

Ad

(T = 78°K)

-0.50 X l O l o dyne/cmZ -0.3 -4.0 -2.47

* Einspruch and Csa.vinsky.'g

* ~a11.19.

Figure 4 is designed to fit the facts of Table IV to the electronic interpretation. As suggested by Eqs. (1.17) and (1.18), Ac'/N1I3 is plotted against T/N2I3.The theoretical line has been fitted to the points by choosing EU = 9.6 eV. This value of Euis in the range of values determined by other methods (see Table 11). Hall's study of the elastic constants of pure and doped silicon not only provided the points of Table IV and Fig. 4, but also included a complete measurement of the electronic effect as a function of temperature from 4' to 30O0K.lgaThe temperature dependence of Ac' follows the form of Eq. (1.20) and allows TD to be determined by fitting Ac' to Lz(T / T D ) . Combining the Hall value of N with independent knowledge of the effective mass also allows TD to be calculated. This calculated T D is 20% higher than that obtained by directly fitting the temperature dependence of A d , suggesting that the density of states effective mass may be slightly different in the impure material of the experiment from that obtained by cyclotron resonance measurements.lga The electronic effect decreases with increasing temperature in the way described by the function L2(T/TD)shown in Fig. 1. The decrease of the (negative) electronic contribution opposes the normal decrease of elastic constants with increasing temperature. The elastic constant may increase with temperature at low temperatures. Doping of semiconductors with electrically active impurities provides a method of controlling the teniperature dependence of elastic constants. The electronic contribution to the temperature dependence of the shear elastic constant of a multivalley semiconductor is found by multiplying the contribution to the elastic constant at 0°K by the derivative of the function L,(T/TD) defined by Eqs. (1.19) and (1.20). It is found that (1.21)

ELECTRONIC EFFECTS IN ELASTIC CONSTANTS

51

Here Q ( q o ) is defined by (1.22) . is dLZ(T/TD)/dT has a maximum value of (1.3/TD) a t T = 0 . 4 2 T ~It possible to use the electronic effect t o influence the temperature dependence of an ela.stic constant. Then it is appropriate to examine the electronic contribution to dc/dT (writing c for either shear constant) as a function of doping a t a particular temperature by writing it in the form d Ac/dT

=

{ 2 ~ - ” ~ b ~ N ~ & ~ /[2( gk) T 2 ’ 3~( )F ~ / z ( ~ o ) ) ” 3 Q ( ~ o ) ] .(1.23)

Here all the concentration dependence appears in the square bracketed factor. The constant b has the value t for Si ( Ac’) and & for Ge (Ac44). The bracketed factor has a broad maximum of 0.6 around N = 1 . 4 . ( 2 ~ ) - ~ ~ ~ v N , . It will be seen from Table 111 that the c‘ and B of germanium are changed by small but measurable amounts by doping. It has been suggested that these changes arise from excitation of electrons into the (001) conduction band, which is about 0.2 eV above the (111) As described, the (111) conduction band, which contains most of the electrons, is responsible for the large electronic contribution to ~4~ . Electrons excited into the (001) conduction band would, however, contribute to c‘, just as in the case of silicon. Furthermore, a uniform dilatation will shift the (001) band with respect to the (111) band and allow the energy of the crystal to be decreased by transfer of electrons to the band that has been lowered in energy. Thus there will be a decrease of the bulk modulus. Although further work will be required to identify the source of the observed effects in c’ and B, they have the correct sign (a decrease of the elastic constant with doping) and order of magnitude to be accounted for by excitation of electrons to the (001) conduction band. An alternative interpretation of the changes in c‘ and B has been investigated by Keyes and Price.21a The large concentration of donors in heavily doped germanium causes a certain amount of the (001) band wave functions to be mixed into the wave functions of the electrons, which are mostly the wave functions of the (111) band. A shear strain with (001) axis changes the energy of the (001) valleys and thus affects the amount of (001) in the wave function and the energy of the electrons. The dependence of the energy of the electrons on a (001) strain constitutes a contribution to c’. The relative motion of the bands with dilatation leads P. Csavinzsky, J . A p p l . Phys. 37, 1967 (1966). J. J. Hall, Bull. Am. Phys. SOC.[ 2 ] 10, 43 (1965). 218 R. W. Keyes and P. J. Price, unpublished (1966). 20

21

52

ROBERT W. KEYES

to a contribution to B. The quantitative theory of the effect involves a parameter, a matrix element of the impurity potential between the (001) and (111) wave functions, whose value is not known. If the value of the matrix element is chosen to be about the same as that between two (111) valleys, the effects calculated from this model are considerably smaller than the observed ones.21b A similar case, in which a large elastic effect is produced by a change in the mixture of various valley wave functions in a ground state wave function, will be encountered in Section 11, where the contribution of electrons trapped on donors to the elastic constants will be discussed. I n spite of the fact that a satisfactory quantitative interpretation of the dependence of B on doping has not been formulated, it seems clear that the effect has its origin in the proximity of the (001) or, perhaps, the (000) band to the (111) band; since a similar effect is not present in silicon, in which there are no other bands near the (001) extremum. Hall discovered a new kind of effect in c44 in his studies of electronic effects in the elastic constants of silicon. c44 is decreased by 0.32 X 1O1O dyne/cm2 at 25°C by doping with 2 x 10'9 cmP3 donors.lgaComparison with Table IV shows that this effect in c44 is about one-fourth as large as the effect in c'. The change in c44 cannot be explained by the deformation potential models considered so far. It is a result of the splitting of the degeneracy of the conduction band at the [OOl] Brillouin zone face (the XI point) by strain, which also appears in the effect of strain on spin resonance and in piezo-resistance.21w21e The dependence of energy on wave vector is changed in such a way that the energy of the electrons is lowered and c44 is decreased. A detailed quantitative theory of the effect has been worked out by Hall and Price.21e 2. CRYSTAL STABILITY

One condition that a crystal be stable is that its shear elastic constants must be positive. This condition will fail if the electronic contribution to the shear elastic constant, which is negative, becomes larger than the corresponding elastic constant of the pure crystal. The donor concentration at which such instability occurs can be readily calculated from the models described previously. It turns out to be about 1.5 x 1023cm-2 for both silicon and germanium, a concentration that is three orders of magnitude P. J. Price and R. L. Hartman, J. Phys. Chem. Solids 26, 567 (1964). J. C. Hensel, H. Hasegawa, and M. Nakayama, Phys. Rev. 138, 225 (1965). Z1d E. A. Makarov, Fiz. Tverd. Tela 8, 3636 (1966); see Sou. Phys.-Solid State (English Transl.) 8, 2902 (1967). 21e J. J. Hall and P. J. Price, to be published (1967). 21b

ELECTRONIC EFFECTS IN ELASTIC CONSTANTS

53

greater than the solubility of donors and is even greater than the concentration of host atoms. The electronic contribution to the shear elastic constants does not have an important effect on the stability of even the most heavily doped silicon and germanium crystals.

3. ELECTRONIC RELAXATION AND ATTENUATION OF ELASTIC WAVES The theory of the electronic effect on elastic constants presented previously is a static theory that assumes that the electrons are a t all times in equilibrium in the strained crystal. All of the experimental determinations of elastic constants referred to were made by measuring the velocities of sound a t frequencies less than lo8cps. There is little doubt that the distribution of electrons among valleys can follow the strain a t such frequencies, since, for example, the lowest intervalley scattering rate found by Weinreich and collaborators22from acoustoelectric experiments in germanium was 7 X lo8 sec-’ in a very pure specimen and was greater than 1010 sec-’ in specimens moderately doped with arsenic. Many of the interesting applications of elastic constants involve dynamic effects at high frequencies. The frequency dependence of the electronic effect in the elastic constants of a multivalley semiconductor and the application of the theory to the propagation of elastic waves have been considered in some detail by Adler.23Many of the questions involved were also discussed by Weinreich et a1.22As already noted, intervalley scattering brings the electrons into equilibrium a t very low phonon frequencies. Diffusive relaxation, the tendency of electrons to diffuse from places where their energy is high to places where their energy is low, brings the electrons into equilibrium a t high frequencies where the wavelength is small. The total relaxation time, T R , may be expressed in terms of the intervalley scattering rate, f~, the electron diffusion constant, D , the velocity of sound, s, and the frequency, oo, as22,23

has a maximum value of (s2/4DfI)”2.The electronic contribution to the elastic constants will be present throughout the frequency range if (W07R)mm is small compared to unity. This condition can be expressed in a simple form by substituting for D its value as given by elementary electronic transport theory, D = +v2r, where v is the Fermi velocity in a degenerate electron gas and T is the momentum relaxation time of the elec~ approximately the simple trons. Then the condition ( w ~ T R <)<~ 1~ takes

WOTR

** G. Weinreich, T. M. Sanders, Jr., and H. G. White, 28

E. Adler, I B M J. Res. Develop. 8, 430 (1964).

Phys. Rev. 114, 33 (1959).

54

ROBERT W. KEYES

form

>> (S/V) '. Since (s/v) is ordinarily 10-l or lo-' in semiconductors, it may be expected fiT

that condition (3.2) will be satisfied in a great many cases. The derivation of (3.2) assumed that macroscopic transport theory can be applied to the motion of electrons under the influence of the acoustic wave, that is, that the mean free path of an electron is small compared to a wavelength. Adler showed, however, that a condition like (3.2) can also Thus, be obtained when the mean free path is longer than a ~avelength.'~ it appears that the electronic effect will be present in the velocity of elastic waves except in unusual circumstances. The point of view adopted here also assumes that an electron can be localized in the field of the elastic wave, an approximation that cannot be correct if the wavelength of the electron is long compared to that of the elastic wave. An investigation of the dependence of the elastic constants on the wave vector of the elastic wave for high-frequency waves shows that the electronic contribution goes away for wave vectors larger than twice the wave vector of the electronic Fermi surface in a degenerate semiconductor in a manner similar to that described for metals by K ~ h n . ~ ~ ~ * The lattice frequencies involved are far above those at which ultrasonic experiments are carried out, so that such experiments should indeed reveal the electronic effects, but do represent phonons with energies of a few degrees, and thus may affect the interpretation of the electronic effect in the specific heat, which will be discussed in Section 5 . Experimentally, the fact that the electrons require a finite time to come into equilibrium in the presence of a strain shows up as an attenuation of acoustic waves. The absorption coefficient for a wave of circular frequency coo and velocity s is a! = ( o o ~ T / S ) (ACC,/C,). (3.3) Here Ca = ps2, the elastic constant of the wave, Ac, is the change in ca caused by the electronic effect, and T is the relaxation time for the redistribution of electrons among the valleys. It has been shown that there is a strong electronic absorption associated with the electronic effect on c44 in gerrnani~rn.'~ Acoustic waves of frequency 0.9 X 1O1O cps could not be propagated an appreciable distance in heavily doped germanium if their elastic constant contained a term in c44, because the strain associated with such waves destroys the degeneracy of the valleys and allows the 23a

W. Kohn, Phys. Rev. Letters 2 , 393 (1959); E. J. Woll and W. Kohn, Phys. Rev. 126,

1693 (1962).

P. J. Price and E. Adler, unpublished (1963). M. Pomerantz, R. W. Keyes, and P. E. Seiden, Phys. Rev. Letters 9, 521 (1962).

23b l4

ELECTRONIC EFFECTS IN ELASTIC CONSTANTS

55

electrons to absorb energy from the wave rapidly. Waves whose elastic constant does not contain c44 can be propagated in heavily doped germanium. The electronic absorption of waves whose elastic constant contains c44 decreases with decreasing frequency, as indicated in Eq. (3.3). The attenuation has been quantitatively measured at frequencies around lo8 see-’ and used to determine the electronic relaxation time.25The relaxation times calculated by combining the intervalley scattering times found from the low-frequency attenuation measurements with calculated diffusive relaxation times are shown in Fig. 5. It is seen that the relaxation is sufficiently fast to allow the electronic effect to appear at all frequencies, although only barely so for Sb-doped germanium at frequencies between lo9 and 10’0 see-1. The attenuation of acoustic waves by electrons in semiconductors and semimetals is a large field with many applications to problems in solidstate physics and to useful devices.26Most of these are only remotely re-

FIG.5 . Intervalley relaxation times,

TR,

in heavily doped n-type germanium (I?

>

10’8 cm-3) as a function of elastic wave frequency. The relaxation time is compounded

from a n intervalley scattering time and a diffusive relaxation time. The intervalley scattering time is greatly different for antimony and arsenic donors,18*so that separate curves are shown. The curve w ~ = ~ 1 and T a~ scale relating the wave frequency to temperature are also shown. z6

W. P. Mason and T. B. Bateman, Phys. Rev. Letters 10, 151 (1963). N. G. Einspruch, Solid State Phys. 17,217 (1965); W. P. Mason, “Physical Acoustics,” p. 237. Academic Press, New York, 1965; A. R. Hutson and D. L. White, J. Appl. Phys. 33, 40 (1962); A. R. Hutson, J. H. McFee and D. L. White, Phys. Rev. Letters 7,237 (1961); M. Pomerantz, ibid. 13, 308 (1964); H. N. Spector, Solid State Phys. 19, 291 (1966).

56

ROEZRT W. KEYES lo-6 P"'

I

1

3x10" P( ~ r n - ~

FIG.6. Acr4 of p-type silicon as a function of acceptor level at low temperature. The dotted line illustrates the proportionality of Act4 to PI13 at high doping levels. The experiments were performed by Mason and Bateman.18a

lated to the main theme of this study, however, and a survey of the field is not included here. 4. ELASTIC CONSTANTS OF OTHERSEMICONDUCTORS

It is apparent that there will be an electronic contribution to the elastic constants of many semiconductors, not only those that have the simple multivalley band structure. For example, the valence bands of all of the diamond lattice semiconductors are of the degenerate type.n Strain warps these bands in a complicated way, lowering the energy in some places and raising it in 0thers.~8*29 An electronic contribution to the elastic energy arises from the transfer of holes from those parts of the bands which are raised in energy to those parts which are lowered.IOTheories of the effect have been given which show that the order of magnitude of the change in elastic constants produced by doping is the same as in n-type semicond u ~ t o r s . ~ ~The J ' " ~theory ~ can be accurately formulated for germanium, in which the spin-orbit splitting is large enough to remove a part of the valence band from consideration." However, no experimental results are available for germanium or other semiconductors of high atomic number W. Shockley, Phys. Rev. 18, 173 (1950). E. N. Adams, Phys. Rev. 96,803 (1954). z9 G. E. Pikus and G. L. Bir, Fiz. Tverd. Tela 1, 1642 (1959); see Soviet Phys.-Solid State (English Transl.) 1, 1502 (1960). 30 G. L. Bir and A. Tursanov, Fiz. Tverd. Tela 4, 2625 (1962); see Soviet Phys.-Solid State (English Transl.) 4, 1925 (1963). 31 P. Csavinsky and N. G. Einspruch, Phys. Rev. 132,2434 (1963). 27 28

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

57

to which the calculation of Bir and Tursanov might be applicable. More approximate calculations have been made for silicon and used to interpret experiment^.^^ However, it is difficult to assess the adequacy of the approximations in the absence of independent knowledge of the deformation potential constants. The effects of holes in the valence band on the elastic constants of silicon are shown in Figs. 6 and 7. Figure 6 shows the change in c44 with doping at 1.5"K. The electronic contribution to c44 decreases rapidly with temperature, so rapidly that in some cases a region of anomalous temperature dependence occurs. That is, c44 increases with increasing temperature at low temperat~res.~5 The temperature dependence is complicated by the fact that the more lightly doped samples ( P 5 2.5 x 10l8cm-3) studied by Mason and Bateman2j are not statistically degenerate. The concentrations of carriers in acceptor states and in the valence band varies with temperature, so that Ac44 is not proportional to the function L2 of Fig. 1. It may be approximately expressed (for reasons that are not clear) as Ac44 = 0.013P2/3T-1'2, where Ac4, is in dynes per square centimeter, P is in centimeters+, and T is in degrees Kelvin. The values of Ac44 at 78°K are plotted in Fig. 7, which also shows changes in Ac' measured at higher doping levels.31 Large electronic effects have been observed in the elastic constants of SnTe, a semiconductor that can be doped with very high concentrations of acceptors.32Figure 8 shows the elastic constants of SnTe as a function of hole concentration. It is not possible to give a satisfying interpretation

FIG.7. Changes in the shear elastic constants of silicon at 78°K caused by doping with acceptors. The values of Ac44 are taken from Mason and Bateman's* and the values of Ac' from Csavinsky and E i n ~ p r u c h . ~ ~ aa

B. B. Houston, Jr. and R. E. Strakna, Bull. Am. Phys. Soc. [Z] 9, 646 (1964)

58

ROBERT W. KEYES

P"3

(cm" )

FIG.8. The elastic constants of SnTe as a function of doping level (data of Houston and Strakna3a).

of Fig. 8, since the band structure of SnTe is not known, although several experiments show that it is more complicated than those of n-type silicon and germanium. However, the most prominent features of Fig. 8, the linear dependence of the shear elastic constants on the one-third power of the carrier concentration and the relative lack of dependence of the bulk modulus on concentration, are consistent with a model in which the valence band is a single multivalley band, but the valley minima are a t general points of the k space rather than a t the points of high symmetry which they occupy in silicon and germanium.32A multivalley band with minima a t general points of the k space will contribute to both c' and c 4 4 , since any shear strain will destroy the degeneracy of the minima. The proportionality of the shear effects to W 3suggests that the holes are all going into one band. The lack of an electronic effect in the bulk modulus also suggests that only one band is involved, since, in general, dilatation will cause one band to move with respect to another and allow electrons to lower the bulk modulus as discussed from the case of germanium near the end of Section 1. However, two band models are usually invoked to explain

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

59

the transport properties of SnTe.33Further work will be required to clarify the situation. The possibility that the deformation potential constant for relative motion of two bands is accidentally near zero cannot be neglected. 5. THERMAL PROPERTIES The thermal excitation of elastic waves is one of the most important phenomena involved in the thermal properties of a crystal. When the elastic constants of a semiconductor are changed by doping, it may be expected that the thermal properties of the semiconductor lattice will also be changed. a. Debye Temperature

The Debye temperature, O D , is related to the elastic constants of a crystal in a straightforward way.a4Therefore, the change in shear elastic constants caused by doping of semiconductors with donors whould lead to a change in 8 D which can be calculated easily.35Bryant and Keesoma6 measured the 8 D of several heavily doped n-type samples of germanium and found an electronic effect. Their values of 8 D are plotted against in Fig. 9. The electronic nature of the effect is shown by its large size and by the fact that doping with silicon, which is a neutral impurity in germanium, produced no comparable effect.36Note also that samples doped with both arsenic and antimony donors are represented in the thermal measurements. Values of 8 D which have been calculated by using delaunay’s tables” from measurements of the elastic constants at very low t e m p e r a t ~ r e s ~ * ~ ~ and from the expected electronic effect on 1244 , Eq. (1.18a) , are also shown in Fig. 9. The elastic and thermal values of 8 D agree for pure germanium, but the thermal values me consistently less than the elastic ones for the heavily doped samples. The diagreeement, however, may not be larger than the combined error in the two kinds of measurements. I n any case, it appears that the electronic effect in the thermal 8 D is at least of the sign and order of magnitude to be expected theoretically. R. S. Allgaier and P. 0. Scheie, Bull. Am. Phys. SOC.[2] 6, 436 (1961); A. Sagar and R. C. Miller, Proc. Intern. Conf. Phys. Semicond., Exeter, Engl., 1962, p. 653. Inst. Phys. & Phys. SOC., London; R.F. Brebick and A. J.Strauss, Phys. Rev. 131,104 (1963); J. A. Kafalas, R. F. Brebick, and A. J. Strauss, Appl. Phys. Letters 4, 93 (1964); B. A. Efimova, V. I. Kaidanov, B. Ya. Moizhes, and I. A. Chernik, Fiz. Tverd. Telu 7, 2524 (1965); see Soviet Phys.-Solid State (English Trunsl.) 7, 2032 (1900). 34 J. de Launay, Solid State Phys. 2, 219 (1956). 3& R. W. Keyes, Bull. Am. Phys. SOC. [2] 7,237 (1962). 36 C. Bryant and P. H. Keesom, Phys. Rev. 124,698 (1961).

33

60

ROBERT W. KEYES

THERMAL

-

-99--

Y

--- -- - --0-

I

FIG.9. The effect of donors on the Debye temperature of germanium. The values of 8 D determined by specific heat measurements are taken from Bryant and K e e s ~ m . ~ ~ The errors given by those authors are also shown. The squares represent values of 9~ calculated from the elastic constants measured at very low temperatures by Bruner and and ) by Mason and BatemanlsS Keyese (the points at N = 0 and N = 2.8 X 1019 ~ m - ~ (the point a t N = 1.5 x 1018 0111-3). Since Mason and Bateman reported only a value for Acp4, the elastic constants used to calculate the point derived from their data were obtained by subtracting their AcP4from the elastic constant tensor of pure germanium measured by Bruner and Keyes. The dashed line shows the value of 8 D which is calculated by modifying the c44 of pure germanium by the theoretically predicted electronic effect, Eq. (1.18).

As discussed in Section 3, the electronic decrease in elastic constants is not present in phonons whose wave vector is smaller than twice the electronic wave vector at the Fermi surface. The electronic effect should be present in phonons with energies less than about 2°K at a concentration of 1 0 1 8 donors/cm3 and less than about 5°K at a concentration of l O I 9 donors/cm3 in germanium. Since the measurements of Bryant and Keesom extend to 0.3"K, it is to be expected that they should reflect the electronic contribution to the elastic constants. I n any case, the absence of the electronic effect in phonons of high frequency will raise the Debye temperature and, thus, will not account for the discrepancy between theory and experiment shown in Fig. 9. It is frequently sufficiently accurate and very convenient to use isotropic approximations in considering scalar thermodynamic properties of crystals. Isotropic approximations to the elastic properties of crystals are well knownnJ8 and have also been applied to the electrical properties 37

38

J. C. Slater, "Introduction to Chemical Physics," pp. 222-240. McGraw-Hill, New York, 1939. R. F. S. Hearmon, Phil. Mag. Suppl. 6, 323 (1956).

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

61

of multivalley semicondu~tors.~~ An isotropic approximation to the electronic effect on Debye temperature requires first a knowledge of the rate of change of OD with shear elastic constant c' ( = c44 for an isotropic crystal) , which is Bo-'(dOD/dc') = 7/2c'. (5.1) Here y is an elastic property that has a value close to unity and is exactly defined by

When the electronic free energy function obtained by substituting Eqs. (1.12) and (1.13) into Eq. (1.8) is averaged over all directions of the valley vector a(", the result may be described by writing J i S 0 ( U ) v)

= :Jool(u, v)

+ #J111(u, v) .

(5.3)

It is thus found that the electronic contribution to c' is C' =

-~(3n2y2)1/3(m*XU2/h2)N113L2( T/T)D.

(5.4)

The change in OD is given by (AOD/%)

=

- (27/15) (3a2y2)1/3(m*Zu2/h2c')N113. (5.5)

Equation (5.5) is an approximate formula that relates A e D to the density of states effective mass (#3m*) , deformation potential constant, and carrier concentration of a semiconductor but does not require any knowledge of the detailed, anisotropic features of the band containing the carriers.

b. Thermal Expansion Kontorova pointed out that the change in specific heat produced by the electronic effect on the elastic constants should cause an increase in thermal expansion.40The effect was sought for and found in n- and p-type germaniurn.4l An example, the fractional change in p, the thermal expansion coefficient, of an n-type sample of germanium, is shown in Fig. 10. The thermal expansion is increased. A simple analysis based on a Gruneisen model of the thermal expansion shows that4l Ap/p = (d log C / d e D ) AOD .

(5.6)

R. W. Keyes, Phys. Rev. 109, 43 (1958). T. A. Kontorova, Fiz. Tverd. Telu 4, 3328 (1962); see Soviet Phys.-Solid State (English Transl.) 4, 2435 (1963). 41 V. V. Zhdanova, Fiz. Tverd. Telu 6, 3341 (1963) ; see Soviet Phys.-Solid State (English Traml.) 6, 2450 (1964); V. V. Zhdanova and J. Kontorova, Fiz. Tverd. Tela 7, 333 (1965); see Soviet Phys.-Solid State (English Transl.) 7, 2685 (1966).

4o

62

ROBERT W. KEYES

Here C is the specific heat. The experimental values of (A/3//3) are two or three times the value predicted by Eq. (5.6) if A% is taken from the measurements of Bryant and K e e ~ o m(see ~ ~ Fig. 10). The change in /3 with doping is also anomalous in that it rapidly disappears and, perhaps, even reverses sign below 9O0K.4l A small change in the thermal expansion coefficient of doped silicon has also been c. Summary

It appears that the doping of germanium with electrically active impurities has a significant effect on lattice thermal properties. The effects are so large than they cannot arise from the normal effects of dilute solutes on a lattice; they must be electronic effects. Their sign is that to be expected from electronic effects; the lowering of Debye temperature and thc increase of thermal expansion correspond to a reduction of the elastic stiffness of the lattice. The thermal effects are not quantitatively accounted for by theory, however. The effects are generally somewhat larger than expected from straightforward application of the theory of electronic contributions to elastic constants. exp (Kontorow) colc (from A8,,)

---

- 0.04 P,

0 T

(OK)

FIQ.10. The effect of donors on the thermal expansion of germanium (after Zhdanova and Kontorova41). The difference between the thermal expansion of a pure and a heavily doped specimen of germanium divided by the thermal expansion at high temperatures is shown as a function of temperature. The value calculated from Eq. (5.6) is shown by the dotted line, using for A ~ the D value obtained by extrapolating the measurements of Bryant and Keesom36 to the required donor concentration (cf. Fig. 9 ) . 4a

S. I. Novikova, Fiz. Tverd. Tela 6, 333 (1964); see Soviet Phys.-Solid State (English Transl.) 6, 269 (1965).

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

63

6. THIRD-ORDER ELASTIC CONSTANTS When the expansion of the free energy in powers of the strain, as in Eq. (1.4), is continued to terms of third order in the strain, the resulting coefficients are known as third-order elastic constants. The electronic contribution to third-order elastic constants can be very large, as may be seen by considering the nature of the series expansion of the electronic free energy in powers of strain. When the W i ) ’ s of Eq. (1.8) are expressed in terms of the strain through Eqs. (1.12) and (1.13) and it is recognized is approximately 3N/2p in a dethat the density of states function W i ) generate semiconductor, it is seen that the series has the form

The a’s are dimensionless coefficients of order-of-magnitude unity. The Since the deformaseries, Eq. (6.1), is an expansion in powers of (&e/C). tion potential constant, Zu, is about 10 V, whereas the Fermi energy, C, is 0.1 V or less in semiconductors, the strain is multiplied by a large factor in the expansion. Thus, the coefficient of the third power of the strain is of the order of 100 times greater than the coefficient of the second power of the strain. As just seen, the term in e2 may represent a 5% contribution to an elastic constant. The values of the lattice third-order elastic constants are, however, of the same order of magnitude as the second-order elastic constants. Thus, the electronic contribution may completely dominate a third-order elastic constant. The calculation of the electronic contribution to third-order elastic constants is completely analogous to the calculation of the second-order constants.1° The term that must be added to Eq. (1.8) is

Again the essential role of the density of states at the Fermi level is apparent. Equation (6.2) simplifies to an equation analogous to (1.9) in .the nondegenerate approximation. Specializing Eq. (6.2) to the multivalley semiconductor with Eqs. (1.10) to (1.13), Fel

= ( ~ / 6 (2/~”~)N,F1/2”(7) ) E’”3(kT)-2K( E,

E, E ) .

(6.3)

64

ROBERT W. KEYES

The third-order terms in the free energy of a cubic crystal are

F,

+ ~,,3 + €22) + + - - -) + + + Cm%2+ +

= $$111(~~~~

+ + + +

$Cl12 ( E Z 2 E U V

eyy2ezz

2C144(€zz€y2

eyyEz82

’’’

2C166(EzzEz2

Cl23Ezz~yyEZZ

‘?zzEzP)

(6.4)

8C456EzyEyzEzz

in the convenient notation of Thurston and B r~ g g e r .4The ~ Cijt are thirdorder elastic constants. Comparison of the forms of K ( e , E, E ) given in Table I with Eqs. (5.4) and (6.3) shows that the electronic contribution to the third-order elastic constants is as follows. For germanium, AC456= ~((2/a112)N,Z,3(ICT)-2F11~”(11),

(6.5a)

and for silicon, AC111

=

ACm = -2AC11z

=

~(2/a11z)N,Zu3(ICT)-2F1/2”(11). (6.5b)

It is useful here also to normalize T to T D, Eq. (1.17), and to introduce the total number of electrons. Then AC456= +E (32a/3) 4/3Eu3 (2m*/h2)2N-1/3L3( T/TD) ACiil

[Gel,

=

ACn3

=

,1,(48a/3)4/3Xu3(2m*/h2)2N-*13L3( T / T D ) [Si].

(6.6a)

= -2AC112

(6.6b)

T/TD) is defined by Eq. (1.19) plus Here L3( LI(T/TD)

=

g(TD/T>2[F1/2”(110)/F1/2(110) 1.

(6.7)

The function &( T/TD) was plotted in Fig. 1. Hall44and Drabble and F e n d l e ~ have ~ ~ measured the electronic contribution to the c 4 5 6 of germanium. As in the case of the second-order constants, the C456of pure germanium and of germanium doped with arsenic or antimony was measured and the difference attributed to the electronic effect. The electronic effect completely dominates c45.5,in accord with the electronic theory. The results are presented in Table V, which shows that 43

44

45

R. N. Thurston arid K. Brugger, Phys. Rev. 133, A1604 (1964); K. Brugger, ibid. p. A1611. J. J. Hall, Phys. Rev. 137, A960 (1965). J. R. Drabble and J. Fendley, Solid State Conzmun. 3, 269 (1965); J . Phys. Chem. Solids 28, 669 (1967).

ELECTRONIC E F F E C T S I N ELASTIC CONSTANTS

65

TABLE V. THEELECTRONIC CONTRIBUTION TO THE THIRD ELASTIC CONSTANT C456 OF GERMANIUM AT 300°K. THEELASTIC CONSTANTS ARE GIVENI N 10" DYNES/Cht2. COMPARISON OF A C WITH ~ ~THEORY ~ A N D DETERMINATION OF Z, WITHOUT REFERENCE TO N ARE ALSO SHOWN

N (cm-3)~ T("W ACW (exp.) ACm (CalC.)d Ac44 (exp.Ib - (Ac44) (AC456)* a. (eV).

2.4 X 1019 300 1 .567b 1.64 -0.0252 0.0395 15.7

2.8 X 300 1.4956 1.57 -0.0270 0.0404 15.7

2.3 X 1018 293 0.78c 0.88 -

2.3 X 1018 77 3.65c 3.22 -

As determined by transport measurements. Ha11.44 c Drabble and Fendley.45 d Calculated from Eq. (6.6a) with the parameters of Table 11. See Eq. (6.8). 0

b

they are in excellent quantitative agreement with the calculated electronic effect. Hall and Drabble and Fendley also found, in agreement with the electronic interpretation, that changes in the other third-order elastic constants were much smaller than the change in C 4 5 6 . The other changes, however, were large enough to be measured, and it was suggested that they may represent electronic effects due to the electrons that occupy the (000) and (001) type minima in g e r m a n i ~ m . ~ ~ ~ ~ ~ The fact that the electronic contribution dominates certain third-order elastic constants makes the measurement of such constants a useful method of investigating semiconductors. As an example, in the case of germanium, consider the product of Eqs. (1.18a) and (6.6a) a t zero temperature: - ( AC456) ( Ac44) = ( 2 l ' ~ ~ / 3(W~ L) * ~ E ! , ~ ./ ~ ~ )

(6.8)

Note that dependence on N , the carrier concentration, has been eliminated. If one accepts cyclotron resonance results for m*, then a sensitive measurement of the deformation potential constant X, , which is otherwise difficult to determine, is made possible without establishing the value of N . It can be seen by multiplying out Lz(T / T D )and L3(T / T D )that the temperaturedependent factor that multiplies the right-hand side of Eq. (6.8) when T differs from zero is quite close to unity up to T = T D / ~The . function [ L 2 ( T / T ~ ) L 3 ( T / T ~is) ]shown 1 / 5 in Fig. 11. Thus, Eq. (6.8) can be applied to heavily doped samples at quite high temperatures. The calculation of %, from the data of Hall is also illustrated in Table V. The values ob, Table 11). tained are in good agreement with other determinations of ,"(see As another example, %, could be eliminated from AC.Mand Ac456 by forming the combination ( Ac44)3/ ( AC4j6)2. The carrier concentration N could

66

ROBERT W. KEYES

T/T,

FIG. 11. The function (Lz(T/TD)L3(T/T~))1/6, which expresses the temperature dependence of the value of suobtained from Eq. (6.8),as a function of T / T n . The function differs from unity by less than 4% if T < 0.5 TD .

then be determined without knowledge of %, . The temperature dependence of the function thus formed is quite strong, however, and it would be most useful a t low temperatures or as a function whose temperature dependence might be fitted to determine T D .The temperature dependence of Ac44 and Ac456 can, of course, be taken into account in analyzing data for values of Z, and N , and Hall has done so in analyzing his data.44Drabble and Fendle~ used ~ ~ the temperature dependence of the hC456 of their sample to determine the degeneracy temperature and, thus, the electron concentration. The value of %, could then be determined from the magnitude of . The values of Z, found by Hall and by Drabble and Fendley range from 15.8 to 16.5 eV. The carrier concentrations that they deduce from elastic measurements are in reasonable agreement with those determined by measuring Hall coefficients. Precise agreement cannot be expected, as the lack of a detailed theory of scattering and transport properties of heavily doped semiconductors creates doubt as to the quantitative interpretation of the Hall coefficient. Hall also measured complete sets of third-order elastic constants for pure silicon and the heavily doped silicon sample of Table IV with N = 2.0 X 1019cm-3.19aThe electronic effect predicted by Eq. (6.6b) was found. A convenient way of expressing the change in third-order constants caused by doping is19a A(C111

+

2C123

- 3Cnz)

=

6.05(f0.1) X

loL2 dyne/cm2.

Fitting the third-order elastic constant and the functional dependence of the second-order elastic constant A d on temperature (cf. Section 1) to the electronic theory, allowed the electronic concentration, deformation PO-

ELECTRONIC EFFECTS IN ELASTIC CONSTANTS

67

tential constant, and density of states effective mass to be determined. The electron concentration found was in good agreement with that given by the Hall effect, and the deformation potential constant has the value Eu = 8.6(&0.4) eV. A change in the third-order elastic constant ( c 1 4 4 - c 1 6 6 ) was also foundlga: A(C144

-

c166)

=

0.76(f0.15) X 10l2 dyne/cm2.

This effect is another result of the splitting of the degeneracy of the conduction band at the [OOl] Brillouin zone face by strain, which was mentioned in Section 1 as the source of the electronic effect in the c 4 4 of silicon. The change in third-order elastic constants with doping can be regarded in the following way. The energy of the electrons in, for example, the [OOl] or z valley depends on the xy component of strain (the electronic effect in the c44 of silicon). The number of electrons in the [OOl] valley is, however, changed by shear strain with an [OOl] axis. Thus the electronic contribution to c44is changed by [OOl] strain. Terms proportional to ezzezy2 and e2z~Zy2in the free energy are changed and a contribution to the thirdIf it is assumed that the order elastic constants c 1 4 4 a.nd C166 exists.19a~21e electronic contribution to c 4 4 is proportional to the electron concentration, N , then the change in the third-order elastic constants can be calculated from AcU and the electron transfer due to strain with an [OOl] axis. A(ci4a

- Ci66)/AC44

=

(~%/~~TD)L~(T/TD).

The experimentally determined value of the left-hand side of this expression is 240, whereas the calculated value for the right-hand side is 260, an agreement which amply confirms the postulated mechanism of the effect. IV. Electronic Strain

If the energy of a band depends on the strain, as in the deformation potential model, then the equilibrium form of the crystal depends on the number of electrons in the band. The crystal will strain itself so as to lower the energy of the bands that contain electrons. The amount of lowering of the free energy of the crystal which can be attained in this way is proportional to the number of electrons involved. The decrease of electronic free energy produced by the strain is counterbalanced by an increase in elastic free energy, a quadratic function of the strain. The equilibrium form of the crystal is determined by the minimization of the free energy function, which is the sum of a negative electronic contribution linear in the strain and a positive quadratic elastic contribution (see Section 11).

68

ROBERT W. KEYES

The simplest case in which electronic strain can be derived from known parameters is the excitation of electrons across the gap of a semiconductor. There is a direct comparison with the effect of pressure on the gap. An attempt to make this comparison is described in Section 7. Another case, which requires a knowledge of the absolute motion of a n energy band with dilatation rather than the motion relative to another band, is the effect of doping.1° This effect is, however, obscured by direct, nonelectronic effects of doping. Nevertheless, Section 8 describes a case in which the electronic effect has been rather clearly detected, the doping being accomplished by electron irradiation. Electronic strain can also be produced by application of a magnetic field to a semiconductor. Cases of this kind are described in Sections 9 and 10, together with associated magnetoelastic effects.

7. PHOTOSTRICTION One attempt to find the predicted electronic strain was carried out by Figielski, who created electron-hole pairs in germanium with light of photon energy larger than the band gap and measured the resulting change in length, or phot o~ trictio nThe .~ ~ fractional change in length caused by production of N electron hole pairs is (ALIL) = ( -E1/3B)N, where El is the deformation potential constant for change of the energy gap with dilatation ( - 4 eV for Ge) and B is the bulk modulus, Eq. (1.15). Figielski found an effect of the correct sign and order of magnitude, but it was two times larger than the prediction of the electronic theory. The factor of 2 is believed to be within the accuracy of calibration of the a ~ p a r a t u s . ~ ~

8. DOPING BY ELECTRON IRRADIATION North and B~schert~7.48 measured the change in length of electronirradiated n- and p-type germanium. Electron irradiation produces atomic defects that are electrically active. The concentration of charge carriers is changed by irradiation, and an electronic change in dimensions of a crystal is to be expected. The interpretation of the dilatations observed by North and Buschert as an electronic effect is complicated by several factors. There are atomic displacements that also directly influence the length. Electrons removed from the conduction band by irradiation may go to some kind of trap, where their energy still depends on strain. Results obtained by North and Buschert on a specimen of degenerate 46

47

48

T. Figielski, Phys. Status Solidi 1, 306 (1961). J. C. North and R. C. Buschert, Bull. Am. Phys. SOC.[a] 10, 123 (1965); Phys. Rev. Letters 13, 609 (1964). J. C . North and R. C. Buschert, Phys. Rev. 143, 609 (1966).

ELECTRONIC EFFECTS IN ELASTIC CONSTANTS

69

n-type germanium ( N O= 1.3 X 10l8 cm-3) are shown in Fig. 12. Here the dosage has been measured by its effect on the electrical conductivity of the germanium and converted into a change in electron concentration by multiplication of ( Au/u) by N o . The increase in length caused by electrons in the conduction band islo

+ Eu)/gB. & and %, were defined in Eq. (1.13) ; 3& + Z, X -6 aL/L

=

-m(3%d

(8.1)

eV. Thus, aL/L = 1.3 x AN. This calculated increase in length is shown by the dotted line in Fig. 12. It is seen that the electronic effect is of roughly the correct magnitude but does not provide an accurate explanation of the dilatation. North and Buschert analyzed their data more quantitatively, taking into account that the change in conductivity is not an exact measure of the change in carrier concentration, and concluded that the major part of the dilatation is electronic but that atomic effects are also significant. Smaller effects were observed in p-type germanium because electron irradiation has only a small effect on the concentration of holes. 0 annealing ---electronic theory

,I I

-

50 -

I

-I

\

-I

a P

0 25-

7-

FIG.12. The effect of electron irradiation on the length of a specimen of degenerate n-type germanium with a donor concentration (in the unirradinted condition) of No = 1.3 x 1 0 1 8 cm-3 (after North and Buschert48). The change in carrier concentration is measured by the change in conductivity in this figure; the maximum irradiation is about 3 X 10'6 cm-2 at 4°K and 7 X 10l6om-2 at 80°K. The points represent two annealing plateaus (4Oo-55"K and 65O-75"K) following the 4°K irradiation. The simple electronic (ALIL),which is calculated from Eq. (8.1) by assuming that the conductivity is proportional to the number of electrons in the conduction band and that t,hese are the electrons that produce the electronic strain, is shown by the dashed line.

70

ROBERT W. KEYES

9. ELECTRONIC MAGNETOSTRICTION IN BISMUTH Another example of electronic strain occurs in a semiconductor containing many valleys or bands in a magnetic field. A magnetic field will, in general, alter the densities of states of bands or valleys with different cyclotron and spin mass components in different ways and change their equilibrium populations. The electronically induced strain will then appear and constitute a magneto~triction.4~ The same magnetostriction can be regarded from a converse point of view.5oSome valleys or bands will have higher magnetic energies than others because of their differing susceptibilities. Strain that causes transfer of electrons from the valleys of high magnetic energy to those of low magnetic energy will appear. In fact, magnetostriction and stress dependence of susceptibility are manifestations of the same physical phenomena. Physical effects that affect the magnetic susceptibility will also normally affect its stress dependence and will, therefore, appear in magnetostricti~n.*~~ The magnetostriction can be calculated by minimizing the sum of the elastic free energy and the term in the electronic free energy which depends linearly on strain and quadratically on the magnetic field. Consider first the case of bismuth. Large magnetostriction effects in bismuth were reported by Kapitza in 1932.53Bismuth is a semimetal with trigonal crystal structure. The magnetostriction can be understood, at least partially, as an electronic effect arising from the changes in the number of electrons and holes which result from the changes in the densities of states in the conduction and valence bands in a magnetic field. Magnetic fields increase the concentrations of holes and electrons in b i s m ~ t h . 4Charge ~ , ~ ~ neutrality requires, of course, that an increase in the concentration of electrons be accompanied by an equal increase in the concentration of holes. Let the concentrations be increased by an amount M. The free energy of the crystal then contains a term ( M ) W , where W is the energy of the bottom of the conduction band minus the energy of the top of the valence band (a negative quantity in a semimetal such as bismuth). In a trigonal crystal, W depends upon strain as follows:

W

=

Wo - Ei(e,,

+

egg)

- EzezS .

(9.1)

J. J. Hall and R. W. Keyes, Bull. Am. Phys. SOC.[a] 9, 96 (1964). H. Jones, Proc. Roy. SOC.A147, 396 (1934). 61 A. Wolf and A. Goetz, Phys. Rev. 46, 1095 (1934). 52 B. S. Chandrasekhar, Phys. Letters 6, 27 (1963). 53 P. L. Kapitza, Proc. Roy. SOC. A136, 568 (1932). 54 G. E. Smith, G. A. Baraff, and J. M. Rowell, Phys. Rev. 136, A1118 (1964); G. A. Williams and G. E. Smith, IBM J . Res. Develop. 8, 276 (1964). 49

ELECTRONIC EFFECTS IN ELASTIC CONSTANTS

71

Here z denotes the trigonal axis of the crystal. The elastic free energy, including only those strains that can affect the overlap of the bands, is F g

=

*[c33€zz2

+ cll(ezz2 +

egg2)

+ 2C12~22Eyy+ 2C13(~zzEzz+

eggezz)].

(9.2)

Equations (9.1) and (9.2) only include the effects of strains which change the energy of the top of the valence band with respect to the average energy of the bottoms of the conduction band valleys. The deformation potential constants for these effects are approximately E2 = +3 eV and El = -3 eV.55~5~ More complicated effects can occur in bismuth because the conduction band is made up of three valleys. Strains and magnetic fields that destroy the degeneracy of the three valleys can also lead to magnetostriction. Since the deformation potential constants for the intervalley conduction band effects are not known, they have not been included in the present treatment. When the free energy, Eq. (9.2) plus the contribution from Eq. (9.1), is minimized with respect to the components of the strain, it is found that

+ 2si3Ei], AN[si3E2 + (811 + siz)Ei].

ezz

= AN[s33Ez

(9.3a)

€22

=

(9.3b)

eyy

=

Here the sij are elastic compliances. Substitution of the elastic constants of Eckstein et al." and the deformation potential constants into these equations gives e Z z = (3.1 X cm3)AN, (9.4a) e,

= egg =

- (1.4 X

om3) AN.

(9.4b)

Values of AN are not easily calculated in high magnetic fields because the energy bands are not parabolic and the g tensors are not completely and accurately known. It appears, however, that the number of carriers increases ( A N is positive) with magnetic field for all directions of the field.% Equations (9.4a) and (9.4b) thus predict that a magnetic field will cause the sample to lengthen along the trigonal axis and contract in the perpendicular direction. This result is in agreement with the findings of K a p i t ~ a . ~ ~ Smith and collaborators have determined values for AN with a magnetic field along a twofold axis by analyzing experiments on propagation of Alfv6n waves at microwave freq~encies.~~ The longitudinal magnetostriction calculated by substituting these values of AN in Eqs. (9.4) is plotted in Fig. 13 and compared with the measurements of Kapitza for a specimen 65

67

R. W. Keyes, Phys. Rev. 104, 665 (1956). A. L. Jain and R. L. Jaggi, IBM Res. Develop. 8 , 233 (1964); Phys. Rev. 136, A708 (1964). Y. Eckstein, A. W. Lawson, and D. H. Reneker, J . App2. Phys. 31, 1534 (1960).

72

ROBERT

W.

KEYES

20

50

100

20 10

5

2

I 0.5 250

H (KILOGAUSS)

FIG. 13. Longitudinal magnetostriction of bismuth along the binary axis. The solid curve shows the measurements of Kapitza53 a t 80"K, and the dashed curve is calculated from Eq. (9.4b) with the values of An as given by Smith and collaborators5* for 1.4"K.

with length direction near the binary axis. The electronic theory gives the correct magnitude for the effect at intermediate fields. Agreement is not quantitative, however, the experimental magnetostriction being twice the calculated electronic one at high fields. Reasons for the quantitative discrepancy may be found in several possible defects of the model: The deformation potential model may not be adequate in the case considered because the magnetic quantum is comparable in magnitude to the energy band gaps in bismuth. Kapitza's experiment was performed at 87"K, whereas the values of AN were determined at 1.2OK. And, finally, impurities and other bands may play a role. One other result is cIearIy implied by Eqs. (9.3) : the longitudinal magnetostriction will vanish at an angle with the c-axis defined by

if this equation has a solution. eo is calculated to be 56" for the elastic constants and deformation potentials quoted for bismuth. The dependence of longitudinal magnetostriction on angle as measured by Kapitza is shown in Fig. 14. It is seen that the magnetostriction passes through zero at 8 = 50", in reasonable agreement with the theory. The result expressed by Eq. (9.5) is independent of how depends on field and, in fact, on the direction of field, I9 referring to the direction of observation of the length change.

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

73

0

0

-E"

'0

-15

-30 90"

30"

60"

0

8

FIG.14. Dependence of the longitudinal magnetostriction coefficient of bismuth on the angle between the direction of measurement and the c-axis of the crystal. The solid curve shows the values of m obtained by Kapitza.53 Note the change in sign of m a t 50°, as predicted by Eqs. (9.3)-(9.5).

The magnetic susceptibility of bismuth exhibits well-known oscillations as a function of magnetic field. Because of the close connection of susceptibility with magnetostriction, oscillations are also to be expected in rnagnet~striction.~~ Such oscillations have been observed by Green and Cha ndr a ~e kha r.~ They ~ also observed that the nonoscillatory part of the magnetostriction was near zero a t 58" to the c-axis, in good agreement with the prediction of Eq. (9.5). All of the measurements and calculations described so far referred to temperatures of 87°K or lower. Magnetostriction has also been measured a t around 300°K. Although it is probably not appropriate to apply the simple electronic theory described a t 300°K because of the proximity of other band^,^^*^ the measurements illustrate two points. Wolf and Goetzsl found that the magnetostriction is very sensitive to small concentrations of impurities, another indication of its electronic nature. SchoenbergG1 measured the complete magnetostriction tensor of bismuth and found that it is necessary to include azimuthal variation around the c-axis to describe the effect for general orientations. The azimuthal variation probably reflects the multivalley nature of the conduction band or, perhaps, of some other band, which was not taken into account in the model used here. Magnetic fields change the density of states in a n electronic band. As shown in Eq. (1.9), the electronic effect in elastic constants is a measure of the density of states of the Fermi level. I n particular, if the Fermi level were near the peak of the density of states in a Landau level in a high B. A. Green, Jr. and B. S. Chandrasekhar, Phys. Rev. Letters 11, 331 (1963). M. H. Cohen, L. M. Falicov, and S. Golin, IBM J . Res. Develop. 8, 215 (1964). 6o L. Esaki and P. J. Stiles, Phys. Rev. Letters 14,902 (1965). 6 l D. Schoenberg, Proc. Roy. Soc. A160, 619 (1935).

58

59

74

ROBERT W. KEYES

magnetic field, the electronic effect should be increased.l' Thus, an oscillatory dependence of elastic constant on magnetic field is to be expected when there is an electronic contribution to the elastic constants. Such an effect has been observed.62

10. THEORY OF MAGNETOSTRICTION AND MAGNETOELASTICITY IN GERMANIUM AND SILICON The effects described in the preceding section for bismuth generally referred to phenomena at high magnetic fields, where the magnetic quantum might be comparable to or larger than lcT or the Fermi energy. Similar effects of smaller magnitude are to be expected at low magnetic fields. The low-field effects are most appropriately analyzed by expansions in terms of powers of the magnetic field. The nature of the low-field effects to be expected will be illustrated in this section by presenting the theory for cubic multivalley semiconductors.

a. Magnetostriction The free energy of an axially symmetrical valley, such as occurs in silicon or germanium, in a magnetic field, expanded to terms of first order in the strain and the square of the magnetic field according to the methods of Stoner,63is ~ , l ( i )=

nti)po- $(2/7r1/2)Nck~F3/2(7) - (2/"2)NcX'i''2pB2H2F1/2'(7)/kT

+

(Wt)

- u)(2/7r''2)NCF1/2(11)

+ (w(;) - u)(2/7r'l2)

(10.1)

( p B H / k T )2F1/2"(7) .

Also

n(i)= (2/7r'/2) N,F1/2(11)

+ (2/7r1/2)Ncx(~)'( p B H / k T )2F1/2"

- (W(i' - u)(2/7r'/2)NCF1/2'(7)/kT.

(7)

(10.2)

Here p B is the Bohr magneton and Xci)depends on the magnetic parameters of the valley and on the angle between the magnetic field and the valley axis as follows: X(02 = H . n ( i ) . H / p . 62

63

(10.3)

J. G. Mavroides, B. Lax, K. J. Button, and Y. Shapira,Phys. Rev.Letters 9,451 (1962). E. C. Stoner, Proc. Roy. SOC.A162, 672 (1935).

75

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

A(i)is the tensor A(i) = A d l

+ A,a(i)a(i)

+ (grI2- g ~ ~ ) a ( ~ ) a ( ~ ) ] - + [ a d + at(at - ~ r ~ ) a ( ~ ) a ( ~ ) ] .

= g[g121

(10.4)

The first part of A(i)represents the interaction of the magnetic field with the spin, and the second part is the orbital, cyclotron, or Landau term. The a’s here are reciprocal effective masses. Values of the a’s and g’s for silicon and germanium are given in Table 11. The orbital contribution to A(i)is considerably greater than the spin contribution in these semiconductors. Summing Eq. (10.1) over the valleys of a niultivalley semiconductor = const to Eq. (10.2) gives, for the part and applying the condition of Felwhich depends on H ,

+

Fe1 = - (N/lcT)llg2H2[Fi/2’(17)/F1/2(0)](nd $nu>

- N ~ , ~ , ( ~ c L B H / ~ T ) ~ [ F ~ / ~ ” (H~H)/ /HF2 )~. / ~(10.5) (~~)Y(E, The total number of electrons, N , has been introduced from Eq. (1.6), W ihas ) been expressed in terms of the strain by Eqs. (1.12) and (1.13), and reference has been made to Table I . The first term of Eq. (10.5) gives the magnetic susceptibility, and the second term gives rise to the magnetostriction. The magnetically induced strain is calculated by minimizing the sum of Fel , Eq. (10.5), and F , , Eq. (1.15). The results are

(P #

7)

€By =

0

(P

= 7)

€Or =

0

(P

#

[Gel

(10.6a)

[Si]

(10.6b)

r>

It is found from Table I1 and Eq. (10.4) that Au is negative in both silicon and germanium. Thus, the longitudinal magnetostriction consists of a n expansion in both cases. The magnitude of the coefficient of H 2 in Eq. (10.64 is shown in Fig. 15. The model is not applicable to germanium with donor concentrations less

76

ROBERT W. KEYES

ln

I

\OOK

I

I

3 N (ern+)

10’8

I 1019

FIG.15. The magnetostriction of n-type germanium as calculated from Eq. (10.6b). M is the factor that multiplies HpH, in Eq. (10.6a).

than about 2 X 10’’ cm-3. The temperature dependence of the magnetostriction is given by the function L3(T / T D ) , Fig. 1 and Eq. (6.7).

b. Magnetoelastic Effect The expansion of Eq. (10.1) can be continued to give terms proportional to the square of the strain and field. Such terms represent an effect of magnetic field on the elastic constants. The effect is present because a magnetic field changes the densities of states of the various valleys and, therefore, changes the electronic contribution to the elastic constants. It is only present when there is an electronic contribution to the elastic constants. The magnetoelastic energy of a multivalley semiconductor can be easily calculated by substituting for in Eq. (1.8) its value in a magnetic field :

The magnetoelastic energy is found to be

+ AUK(&, HWI. e,

(10.8)

The first part of Fmel(the first term in the brackets) represents a change in the electronic contribution to the elastic constants given by Eq. (1.14) due to the alteration of the density of states in a magnetic field. The second

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

77

term has a different tensor character. It is allowed to be present because it is not strictly necessary for the elastic constant tensor to have cubic symmetry in the presence of a magnetic field. The calculated change in the c44 of doped germanium in a magnetic T ) ~the electronic contribution field is of order of magnitude ( ~ B H / ~times to c44. It may amount to about one part in lo5 in a field of 30,000 G. c. Electron Transfer Magnetoresistance

The transfer of electrons between valleys caused by a magnetic field has been described. Incidental to this transfer there will be a magnetoresistance, because the contribution of an electron to the conductivity tensor depends on which valley it is in. Let the mobility tensor of valley (i) be p(" and the total conductivity be d. Then the change of conductivity due to transfer of electrons between valleys is Ad

(10.9)

AnCi) p("q.

=

In an axially symmetric valley, pci) has the form

+ ( p l - pt)a(i)a(i).

p = ptl

(10.10)

Dependence of p on magnetic field is ignored in order to demonstrate the contribution of electron transfer to the magnetoconductivity. The An(0 due to a magnetic field are obtained from Eq. (10.2). It is then found that the change of conductivity in a direction e is e.Ad.e

=

:;: (3

N-

-

q(pl

- p t ) A , J ( e e , HH).

(10.11)

The significance of this result is best appreciated by comparing it with the ordinary magnetoresistance in an example, say, germanium. The spin contribution to Au is small in germanium and will be dropped in the comparison. The quadratic magnetoconductivity of cubic semiconductors is usually described by the three coefficients introduced by Seitz,'j4 which are defined by HZ2 0 0 Au/u

=

PH21

+ 7HH + 6

0

H2

0

0

0

H2

(10.12)

It is sufficient for purposes of comparison to treat the conductivity and ordinary magnetoresistance in the simple energy-independent relaxation time approximation, in which p has the form p = (q7/mo)a. M

F. Seitz, Phys. Rev. 79, 372 (1950).

(10.13)

78

ROBERT W. KEYES

Comparing the form of J( ee, HH) with the magnetoresistance form, Eq. (10.12), and introducing Au from Eq. (10.4) shows that the electron transfer magnetoresistance can be described by the coefficients

The ordinary magnetoresistance of germanium is described by3,65@ (10.15) y =

-o=(D‘,(%+t).

at2a1

(10.16)

+

Here p = tr I. Equations (10.14) and (10.15) show that the change in the coefficient 6 may be described quite simply. The electron transfer contribution to 6 is of the magnitude that would be produced by a mobility p* =

p~~/3kT.

(10.17)

T must be approximately replaced by TD at low temperatures in a degenerate semiconductor. p* turns out to be in the neighborhood of 10 cm2/V-sec in heavily doped germanium. As p is in the range 30CL1000 cm2/V-sec in the same materials, the electron transfer magnetoresistance is not a large part of the total effect. It may be more important in low-mobility semiconductors. It should also be noted, however, that the electron transfer contribution destroys the condition y = -p, which is frequently used as a , ~ ~electron ’ transfer magnetotest of the (111) multivalley m ~ d e l . ~The resistance may play a relatively important role in this kind of experimental test in low-mobility semiconductors. V. Electrons on Donors

11. EFFECT OF DONORS ON ORDINARY ELASTIC CONSTANTS

Donors in silicon and germanium are centers of positive charge. Hydrogen-like electronic states are associated with each d ~ n o r . ~ The ~ - ~ states l B. Abeles and S. Meiboom, Phys. Rev. 96, 31 (1954). M. Shibuya, Phys. Rev. 96, 1385 (1954). 67 C. Goldberg and R. E. Davis, Phys. Rev. 102, 1254 (1956). 68 C. Kittel and A. H. Mitchell, Phys. Rev. 96, 1488 (1954). 69 M. A. Lampert, Phys. Rev. 97, 352 (1955). 7 0 W. Kohn and J . Luttinger, Phys. Rev. 97, 1721 (1955); 98, 915 (1955). 71 W. Kohn, Solid State Phys. 6, 258 (1957). 65

66

ELECTRONIC EFFECTS IN ELASTIC CONSTANTS

79

differ from the true hydrogen atom in several respects. The coulombic potential is reduced by the dielectric constant. An electron moves in the potential with its effective mass, which is, in general, anisotropic, so that the states are not simply scaled hydrogenic states. There are several kinds of electrons which differ only in the orientation of their effective mass tensors, introducing additional degeneracies of the bound states. Finally, the effective mass model of a donor is only an approximation. Deviations appear on account of the strong short-range part of the donor potential which differs from one species of donor atom to another. Since the wave functions of the bound states derive from the wave functions of the valleys, a change in the energies of the valleys by strain will produce a change in the energies of the bound state^.^^-?^ Shear strains also destroy the degeneracy of the valleys, producing large changes in the wave functions of the bound states which are reflected in large and striking changes in impurity c0nduction.7~The lowest state of a donor tends to be derived from the valleys that are lowest in energy. This tendency of an electron in the lowest state of a donor to follow the valleys that are lowest in energy under the influence of strain produces a n effect similar to the repopulation of the valleys described in Section 1. It also constitutes a negative contribution to the elastic constant.'O The quantitative model for the energy levels of a donor in a multivalley semiconductor has been described in the papers The theory of the effect of strain on the energy levels has been given by Price.?5 All of the effects that are relevant to the elastic constants in the case of germanium are included in the Hamiltonian

H =

in a representation based on the individual single-valley hydrogenic wave functions of lowest energy, which correspond to the 1s state of the hydrogen atom. Here W(i)is the change in energy of valley with strain as given by H. Fritzsche, Phys. Rev. 116, 336 (1959); 119, 1899 (1960). G. Feher, D. K. Wilson, and E. Gere, Phys. Rev. Letters 3, 25 (1959); D. K. Wilson and G. Feher, Bull. Am. Phys. SOC. [a] 6, 60 (1960). 74 G. Weinreich and H. G. White, Bull. Am. Phys. SOC. [2] 6, 60 (1960). P. J. Price, Phys. Rev. 104, 1223 (1956). 76 H. Fritzsche, Phys. Rev. 126, 1552 (1962); ibid. p. 1560 (1962). 73

80

ROBERT W. KEYES

Eqs. (1.12) and (1.13). The energy of the hydrogenic 1s state is the zero of energy. The presence of A and A in the Hamiltonian results from the fact that the effective mass approximation is not a completely adequate model of the donor states. Experiment shows that A and A are of the same order of magnitude and are different for different d o n o r ~ ? ~ JThe ~ - ~quantity ~ 4A is known as the “chemical splitting.” The eigenfunctions of the unperturbed (Wi) = 0) Hamiltonian of Eq. (11.1) divide into a singlet at -A - 3A, whose wave function is a simple normalized sum of the wave functions of A. It has been found that A > 0 the four valleys, and a triplet at -A for donors in g e r m a n i ~ m ? ~Hence, - ~ ~ - ~the ~ singlet state is the lowest in energy and is occupied by the electrons at the lowest temperatures. The change in the energy of the singlet quadratic in the strain is easily found by second-order perturbation t h e ~ r y . ’ It ~ , is ~~

+

E

=

- (64A)-l[4C W(O2-

( CW ( i ) ) 2 ] .

(11.2)

It is thus found, by using Eqs. (1.12) and (1.13), that the electronic contribution to c44 is Ac44

=

-NZU2/18A

[Gel

(11.3)

The effects are somewhat different in detail in silicon. There are six valleys, and the deviations from the effective mass approximation, the off-diagonal elements in the Hamiltonian corresponding to Eq. (11.1) , split the sixfold ground state into a singlet, a doublet, and a triplet, of which the singlet is lowest in energy. The singlet is perturbed in second order in the strain only by the doublet, from which it is separated in energy by 6A. Here A is the off-diagonal element that connects two valleys on different (001) axes. It turns out that the electronic contribution to the shear elastic constant c’ is Ac‘ = -NEU2/18A [Si] (11.4) The electronic contribution to a shear elastic constant given by Eqs. (11.3) and (11.4) refers to electrons in the singlet state of a donor. Since the singlet has been shown by experiment to be the lowest state of donors in silicon and germanium, it will be occupied at the lowest temperatures. With increasing temperatures, however, electrons will be excited into the triplet and, in silicon, also into the doublet states. When electrons occupy all of these states, the electronic effect, which results from the tendency H. Fritzsche, Phys. Rev. 120, 1120 (1960). G. Weinreich and H. G. White, Bull. Am. Phys. SOC.[2] 6, 60 (1960). 79 D. K. Wilson and G. Feher, Bull. Am. Phys. SOC. [a] 6, 60 (1960). 80 S. Koenig and J. J. Hall, unpublished data (1961). 80s R. E. Pontinen and T. M. Sanders, Jr., Phys. Rev. 162, 850 (1966). 77

78

81

ELECTRONIC EFFECTS IN ELASTIC CONSTANTS

of the singlet to follow the valley of lowest energy, will be greatly weakened. The activation energy for the excitation involved is the difference in energy between the singlet and the higher states, 4A in the case of germanium and about 6A in the case of silicon. It can be seen from Eqs. (11.3) and (11.4) that a shear elastic constant will vanish and the crystal will become unstable a t low temperatures when N = 18cA/Eu2 (letting c represent a shear constant). I n fact, this instability does not occur; the donor wave functions overlap a t high concentrations, and the interaction energy invalidates the simple model of the energy levels. The strength of the interaction between donor states is measured by K , the exchange integral between wave functions on different donor atoms. The isolated donor model certainly must be abandoned when K is equal to the energy difference between the singlet and triplet states, 4A in germanium. The corresponding concentration of donors may be estimated as the reciprocal of the cube of the distance at which K = 4A. The concentrations a t which K = 4A and the electronic contributions to an elastic constant at these concentrations are given in Table VI for some cases of interest. The elastic effects of Table VI are, roughly, the maximum TABLE VI. Crystal Ge Ge Ge Si Si Si

RELATING TO THE CONTRIBUTION OF DONORS I N SILICON AND GERMANIUM TO THE SHEAR ELASTIC CONSTANTS

PARAMETERS

Donor

A (lop3eV)

P

0.75 1.0 0.10 2.5 3.7 2.2

As Sb P As Sb

ND

(1017 c ~ - ~ ) OAc (lolo dyne/cm*)* 2.1 3.1 0.5 120 260 96

0.64 0.68 0.80 3.8 5.5 3.5

Xc

0.046 0.034 0.34 0.098 0.066 0.112

ND is an estimate of the concentration a t which overlap of donor wave functions destroys the contribution of the electrons trapped on donors to the elastic constants. This estimate is derived as follows. A value of (interatomic distance/Bohr radius) for which the overlap integral in the hydrogen molecule ion problemsob is equal to the splitting of the donor states (411 for Ge and 6A for Si) is obtained. This value is then multiplied by the b parameter of the wave function68-70 to obtain a n interatomic separation, R. ND is set equal to (6/7r)R-3. Note that in many cases ND is about equal to the degeneracy concentration, which suggests that as a function of concentration the donor contribution to elastic constants connects smoothly with the degenerate band contribution described in Section 1. Calcu1,ated by substituting ND into Eqs. (11.3) and (11.4). X = EU2/144cACZB~(see text). c = c p 4 in Ge and c = c’ in si.

L. Pauling and E. B. Wilson, “Introduction to Quantum Mechanics,” pp. 327-340. McGraw-Hill, New York, 1935.

ROBERT W. KEYES

FIG.16. Electronic effect in the elastic constant c44 of germanium a t very low temperatures. The solid line a t low concentrations represents the theoretical result, Eq. (11.3), for antimony donors with 4A=3.9X10-4 eV. The solid line a t high concentrations shows the theoretical effect of electrons in a degenerate conduction band, Eq. (1.18a) with L2(T/T,) = 1. The dashed portion of the line is an interpolation through the region of concentration in which no satisfactory theory is available. The two experimental points at high concentrations are obtained from Fig. 3. The experimental points a t low concentrations were measured by Hall on germanium samples doped with antimony.*Oc Theory shows (Eq. (11.3) and Table VI) that the effect on c44 a t low concentrations will be much smaller if arsenic donors are involved.

effects that can be produced by doping according to the donor model. A similar calculation can be made for silicon. Larger effects may, of course, occur a t much higher concentrations where the degenerate band model applies (Section 1 ) . The effect of low concentrations of donors has been observed by Hall in germanium.8ocHall’s measurements of Acu at very low temperatures in two specimens lightly doped with antimony are shown in Fig. 16. Doping with only 3.5 X 1015 C M - ~ Sb donors, an atomic fraction less than lo-’, changes cq4 by more than one part in a thousand. A line which shows the value of Ac44 calculated from Eq. (11.3) is also shown. (There is an uncertainty of about 30% in the position of this line because of the large spread in the values reported for the A of antimony donors.80a)Aca4 is less than proportional to donor concentration for the two samples studied. It is believed that the decrease in A c per ~ ~ electron is caused by the interactions between donors just described. Ac44 decreases with increasing temperature, as suggested by theory.1° J. J. Hall, unpublished data (1967).

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

83

Two points taken from Fig. 3 are also included in Fig. 16, thus summarizing the electronic effect in c44over four orders of magnitude in carrier concentration. One other factor that affects the contribution of donors to elastic constants must be considered. The energy of a donor center is affected strongly by strains near the donor, where the electronic wave function is large, but must be insensitive to strain at large distances from the donor, where the wave function is small. There is, therefore, an inhomogeneity of the electronic contribution to a shear elastic constant; the contribution is large within the wave function and small outside of it. Consequently, it is not strictly correct to calculate the elastic constant by assuming a homogeneous strain. The seriousness of the error that is introduced by assuming homogeneous strain depends on the magnitude of the inhomogeneity of the elastic constant; if it is small, the bulk elastic constant is not very sensitive to the exact method of averaging over microscopic regions.38To obtain a rough idea of the magnitude of the elastic inhomogeneity, assume that the energy of the donor pertains to a region with a diameter of about two ~ . elastic energy is then effective Bohr radii, or a volume of ( 2 a ~ ) The ( + ) c ~ ~ " ( a a= ~ )4~c a ~ ~Since, r ~ . according to Eq. (11.3), the electronic energy associated with a strained donor is Zu2d/36A, the fractional change in elastic constant in the vicinity of a donor is X = Eu2/144cAau3. The parameter X is listed in Table VI for various donor systems. The largest value of X is 0.25. The error introduced probably could be detected in such a case by very precise measurements, but is likely to be less than other uncertainties of measurement and calculation. Of course, if X > 1, the crystal can lower its energy by deforming in the neighborhood of the donor. A spontaneous local strain will appear, a sort of Jahn-Teller effect. Even though the numerical factor in the calculation has only been estimated very crudely, it appears that such deformations are not important in known semiconductors. 12. EFFECTOF DONORS ON THERMAL RESISTANCE

It has been found that the addition of donors to germanium produces an anomalously large thermal resistance at low temperatures.s1-83 The thermal resistance of n-type germanium is shown in Fig. 17 as a function of excess donor c~ncentration.~~ It is seen that very small atom fractions of impurity substantially increase the thermal resistance. (The thermal resistance of pure germanium samples of the size in question is almost E. Fagen, J. Goff, and N. Pearlman, Phys. Rev. 94, 1415 (1954). 82

J. F. Goff and N. Pearlman, Proc. 7th Intern. Conf. Low Temp. Phys., Toronto, Ont.,

88

1960, p p . 284-288. Univ. of Toronto Press, Toronto, 1961. J. F. Goff and N. Pearlman, Phys. Rev. 140, A2151 (1965).

84

ROBERT W. KEYES

t FIG.17. The effect of donors on the thermal resistivity of germanium at 2°K (after Goff and Pearlman83). The points represent different kinds of samples as follows: 0,antimony donors; 0 , arsenic donors; antimony donors, mostly compensated with gallium, the excess donor concentration plotted. A few points measured by Keyes and Sladeks' are also shown: 0, antimony donors; A, arsenic donors. Griffin and Carantimony donors; X, arsenic donors. ruthers86 calculated the following points:

v,

+,

entirely determined by boundary scattering a t 2°K.83--84) This property of donors can be understood as an electronic effect.%,%The inhomogeneity of the elastic constant which was described in the preceding section causes a scattering of elastic waves by donors. It has been seen that there is a localized energy of order (EU6)2/36A associated with a strained donor. A perturbation of the energy of a lattice by a term proportional to the square of the strain at a point couples phonons to other phonons and produces phonon scattering of the point-defect ty~e.~~-89 Thus, donors scatter phonons. It is easy to see that donor scattering is very strong compared to the most common type of point-defect scattering, isotopic, or mass-defect, scattering. To compare the perturbation of the lattice energy by a donor with that caused by the substitution of an atom of different mass in a host lattice, the replacement

(AM) v2e2--+ (Eu2/36A)e2

(12.1)

must be made.% ( ~ ) v x2 10 eV for an antimony atom in germanium, whereas EU2/36Ax 2 X lo4 eV. The electronic scattering is several orders of magnitude larger than the mass-defect scattering. 84

85 86

81 88

89

J. A. Carruthers, T. H. Geballe, H. M. Rosenberg, and J. M. Ziman, Proc. Roy. SOC. A238, 502 (1957). R. W. Keyes, Phys. Rev. 122, 1171 (1961). A. Griffin and P. Carruthers, Phys. Rev. 131, 1976 (1963). I. Pomeranchuk, J . Phys. USSR 6, 237 (1942). P. G. Klemens, Proe. Phys. SOC.(London) A68, 1113 (1955). P. G. Klemens, Solid State Phys. 7, 1 (1960).

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

85

The exhaustive study of the thermal resistance of germanium samples containing donors carried out by Goff and Pearlman showed several unusual effects that show that the phonon scattering or thermal resistance introduced by donors has an electronic rigi in.^^,^^ Firstly, the added thermal resistance is much larger than can be accounted for by ordinary impurity scattering, as discussed in connection with Eq. (12.1). Secondly, the thermal resistance introduced by doping was proved by a study of compensated samples to depend primarily on the concentration of electrons added to the sample rather than on the total number of impurity atoms added. Finally, the thermal resistance introduced by arsenic and antimony donors was quite different in the low-concentration range, where these impurities have different electronic properties, but is the same at high concentrations, where they produce a degenerate electron gas in the conduction band and are electronically Similar. These features are illustrated by Fig. 17. The characteristics of donor scattering are explained by an electronic interpretation of the thermal resistance as follows.85The large strength of the scattering has already been discussed and is caused by the high sensitivity of the energy of the donor state to strain. The high sensitivity to strain is a result of the small energy denominators that appear in a perturbation theory of the effect of strain on the energy. The perturbing states are only separated from the ground state by an energy 4 A . The difference in the strength of the scattering produced by arsenic and antimony donors is a consequence of the fact that the A of antimony is about seven times smaller than the A of arsenic (Table VI) . Since A appears in the denominator of the interaction energy, Eq. (12.1), and the strength of the scattering is proportional to the square of this energy, the scattering by antimony donors is 50 times that by arsenic donors. Goff and Pearlman also showed quite clearly by another experiment that the phonon scattering is electronic in origin. They compared compensated samples containing 10 or 100 times as many impurity atoms as electrons with singly doped samples, in which electron and donor atom concentrations are equal, and found that the thermal resistance depended only on the number of electrons. Only occupied donor states scatter. It develops that one other anomalous feature of the thermal conductivity of doped germanium is accounted for by the electronic interpretation. Goff and Pearlman found that the thermal conductivity is proportional to a power of T greater than three in the low-temperature range. Since lattice thermal conductivity may be written in the form Kth =

+C,Slm

,

(12.2)

where C, is the specific heat per unit volume, s is the phonon velocity, and 14 is the phonon mean free path, I+ must be increasing with increasing tem-

86

ROBERT W. KEYES

perature, a variation that results, in turn, from a decrease in l9 with increasing phonon frequency. The mean free path for scattering by donors decreases as the phonon frequency increases in a certain frequency range because, although phonons with wavelengths long compared to the Bohr radius of the bound electronic wave function are scattered in the way described, when the phonon wavelength is small its strain varies throughout the wave function and the averaging of the strain over the wave function reduces the interaction energy.ssgOThe effect of averaging the strain of a wave with wave vector k over an isotropic hydrogenic wave function with Bohr radius a* reduces the effective deformation potential constant by a factor (1 &~*~k2)-2.90 One more demonstration of the electronic nature of the donor thermal resistance in germanium was provided by the piezothermal conductivity measurements of Keyes and S l a d ~ kThese . ~ ~ experiments showed that the thermal conductivity of samples that have a large donor-induced thermal resistance is quite sensitive to elastic strains that affect the electronic structure of the ground state of the donor in the way described in Section 11. On the other hand, the thermal conductivity is insensitive to strains that do not destroy the degeneracy of the (1 11) valleys and do not affect the electronic structure of the donor (strains with a (001) axis). The thermal conductivity of samples that do not show strong phonon scattering by donors is also insensitive to strain. Several complicating factors tend to obscure a quantitative comparison of the theory of phonon scattering by an electron on a donor with experiment. As in all cases of point-defect scattering, some other kind of scattering must be introduced to limit a divergence of the thermal current integral at low phonon frequencie~.~~ Boundary scattering will always serve this purpose but is itself difficult to treat exactly. There is another divergence in the present case because certain elastic waves, e.g., a longitudinal wave in the (001) direction, does not shift the valleys with respect to one another, does not interact with a donor in the manner described, and is not scattered. The resulting divergence in the integral of the thermal current can again be removed by introducing boundary or phonon-photon scattering, but additional assumptions are required, and the uncertainty of the calculation is increased. Small residual strains in the lattice appreciably change the properties of some of the donors and make the simple model inapplicable.s0a*g2 Finally, it is difficult to determine the doping level of samples acc~rately.~~*~~

+

90 91

92

H. Hasegawa, Phys. Rev. 118, 1523 (1960). R. W. Keyes and R. J. Sladek, Phys. Rev. 126,478 (1962). R. E. Pontinen and T. M. Sanders, Jr., Phys. Rev.Letters 6,311 (1960); R. W. Keyes and P. J. Price, ibid. p. 473.

ELECTRONIC EFFECTS I N ELASTIC CONSTANTS

87

Griffin and Carruthers carried out a quite thorough theoretical analysis of the thermal conductivity of germanium containing antimony and arsenic donors for comparison with the results of Goff and Pearlman.83 They included several effects in addition to those discussed previously, briefly as follows. It is necessary to include boundary scattering for the reasons mentioned. Isotope scattering is not negligible, since, even though it is much weaker than donor scattering for many phonons, for the special directions mentioned and for high-frequency phonons for which the donor scattering is small, it may be important. There is a very strong resonance scattering for phonons that have just enough energy to excite an electron from the singlet to the triplet state of a d0nor.9~ By carefully taking into account all of the relevant scattering mechanisms, Griffin and Carruthers were able to obtain calculated values of the thermal conductivity which were in excellent agreement with the measurements of Goff and Pearlman. Two of the points calculated by Griffin and Carruthers are shown in Fig. 17. I n addition to calculating correctly the magnitude of the thermal resistance, they accurately reproduced its temperature dependence in the temperature range from 1.5' to 4°K. They provided a complete quantitative explanation of the thermal resistance caused by noninteracting donors in this temperature range. The explanation is entirely straightforward; it is based on independently known properties of germanium, and no fitting of any parameter is required. As described in Section 11, the donor states cannot be considered to be isolated a t high donor concentrations and the model of thermal resistance just described will not be appropriate. Nevertheless, Goff and Pearlman found an anomalously strong temperature dependence (proportionality of the thermal conductivity to T 3 9 through the degenerate doping range up to their most heavily doped sample, which had a donor concentration As seen in Section 111, 1, the electronic contribution to of 2.5 X 10l8 the c44 a t such a doping level is exactly that to be expected from a degenerate electron gas model. It is more appropriate to compare the thermal resistance in the degenerate doping range with the ordinary theory of the lattice thermal resistance of metals.g4This theory predicts a proportionality of the thermal conductivity to T2 .The observed temperature dependence is much stronger. Ziman has proposed an explanation of the stronger temperature dependence based on the fact that the wave vector of an electron at the Fermi surface of the degenerate electron gas of a moderately doped semiconductor is less than the wave vector of many of the phonons which contribute to the thermal conductivity, in contrast to the usual situation in metals. Ziman shows that it is difficult to satisfy the conditions of conCarruthers, Rev. Mod. Phys. 33, 92 (1961). J. M. Ziman, Phil. Mag. [S] 1, 191 (1956); ibid. 2, 292 (1957).

93 €'. g4

88

ROBERT W. KEYES

servation energy and of momentum simultaneously in this case and that the scattering of phonons by electrons is reduced. As the temperature and the average energy of the phonons increases, a region exists in which the thermal conductivity mean free path increases with temperature and an anomalously strong temperature dependence is pr0duced.9~It is difficult to apply the theory quantitatively because of the anisotropy of the conduction band of germanium. However, the values of the coupling constant of the theory needed to reproduce the magnitude of the experimentally observed thermal resistance are only a few volts, which seems rather small in comparison with the shear deformation potential constant, Xu = 16 eV. A t the lowest temperatures the normal metallic proportionality of thermal conductivity to T2should be found according to Ziman’s model. I n addition, the thermal resistance is independent of carrier concentration in the metallic model. These features are roughly in accord with observations of the thermal conductivity in the presence of a degenerate hole gas to very low temperature^.^^ The thermal conductivity was found to be proportional to the 1.5 power of temperature and the thermal conductivity of two samples whose concentrations differed by a factor of 5 were within a factor of 1.5 of one another. Doping of germanium with acceptors also introduces a large thermal resistance that is quite similar in qualitative aspect to that found in n-type g e r m a n i ~ m .An ~ - ~explanation ~ of the large scattering power of acceptors similar to that given for donors is probably applicable, although it has not been quantitatively worked o~t.85~8~ However, it has been demonstrated that elastic strain has a large effect on the valence band and on the properties of a ~ c e p t o r s .Also, ~ ~ *the ~ ~effective ~ ~ ~ Bohr radius of an acceptor wave function is about the same as that of a donor wave function. There is undoubtedly a quite strong scattering of phonons by acceptors at low temperatures which disappears, as in donors, when the phonon wavelength becomes small compared to the Bohr radius. The electronic theory successfully accounts for a great marry features of the low-temperature thermal resistance of doped germanium. The anomalous temperature dependence of thermal conductivity (Kth proportional to a power of T greater than three) is still not completely explained, however. The anomalous temperature dependence comes about in the following way in the model employed by Keyesa5 and by Griffin and Carruthers.86An electronic impurity appears to be a strongly scattering point defect for very long phonon wavelengths, that is, at very low temperatures. As the temperature is increased, a range is encountered in which the wavelength of the most important phonons becomes comparable to 95 g6

J. A. Carruthers, J. F. Cochran, and K. Mendelsohn, Cryogenics 2, 160 (1962). J. J. Hall, Phys. Rev. 128, 68 (1962); P. J. Price, ibid. 124, 713 (1961).

ELECTRONIC EFFECTS IN ELASTIC CONSTANTS

89

and then smaller than the Bohr radius of the impurity wave function. The strong scattering by the impurity goes away as temperature increases through this range, causing the thermal conductivity to increase more rapidly than T3. Thus, the thermal conductivity must vary as T3 at sufficiently low temperatures. The anomalous variation should only be present in a restricted temperature range. However, the experiments of Carruthers and collaborators on p-type T3.I from 0.3" to 4"K, germanium disclosed a specimen in which Kth more than an order of magnitude in temperat~re.9~ It seems doubtful that the anomalous temperature dependence introduced by the finite size of the wave functions could persist through such a wide range. Another model suggested by Ziman to account for the strong temperature dependence in a restricted temperature range also fails to explain the persistence of the anomalous dependence to the very low temperatures investigated by Carruthers and collaborators.94~95

-

VI. Conclusion

The recognition of the electronic energy as part of the energy of a semiconductor crystal has proved useful in a variety of ways. The most quantitatively successful application of the theory of the electronic contribution to the energy has been an accounting for large changes in second- and third-order elastic constants of germanium and silicon which are produced by doping with donors. The concept also makes possible an understanding of the large magnetostriction of bismuth and the great scattering power of donors for elastic and thermal waves in germanium. It provides a semiquantitative interpretation of significant changes in the specific heat and thermal expansion of germanium with doping. It furnishes a way of investigating semiconductors which has been used to determine new values of the deformation potential constant of silicon and germanium. The use of elastic methods to study the band structure of semiconductors has potential that has not yet been exploited. For example, it could be deduced from the fact that doping with donors changes c44 in germanium but does not change c' that the conduction band is a (111) multivalley band. The sensitivity of c' rather than c44to doping in silicon shows the (100) nature of the conduction band in that case. Similar conclusions could be reached from the effects of doping on the third-order elastic constants and from the crystallographic character of the electronic magnetostriction. Values of the electronic contributions to second- and third-order elastic constants can be combined to determine the donor concentration and the deformation potential constant. Study of the temperature dependence of

90

ROBERT W. KEYES

the second- or third-order elastic constants or a combination of the two yields a value for the degeneracy temperature. The density of states effective mass can be found by combining TD with the carrier concentration. The possibility of determining even more about the electronic structure of multivalley semiconductors by analysis of niagnetostriction and magnetoelastic effects will be apparent from an inspection of the results in Section 10. The applicability of the investigation of semiconductors by means of their elastic properties is somewhat limited, however, by the fact that most of the experimental methods involved require large, homogeneously doped specimens. The advanced technology needed to prepare such specimens exists for only a few materials. ACKNOWLEDGMENT

It is a pleasure to acknowledge the contributions of J. J. Hall and P. J. Price to this article. Explanations and suggestions from these colleagues clarified many points. J. J. Hall made available a large amount of unpublished experimental data, and P. J. Price provided a number of unpublished notes and memoranda. Finally, they must be thanked for their careful and helpful reviews of the manuscript.

The Jahn-Teller Effect in Solids” M. D. STURGE Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . 92 1. Crystal Defects and Transition Metal Ions in Insulating Crystals. . . . . . . 93 2. Collective Coordinates and the “Quasi-Molecular” Model 3. The Jahn-Teller Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The Static and Dynamic Jahn-Teller Effect: Some Orders of Magni 11. The Jahn-Teller Effect in Doubly Degenerate Electronic States 5. The Stat,ic Problem in a n Octahedral Complex. . . . . . . . . . 122 6. The Dynamic Problem: Vibronic Energy Levels.. . . . . . . . . . . . . . . . . . . . . . 7. The Transition from Stat,ic t o Dynamic Jahn-Teller Effect: Motional .............................. 126 8. Tunneling and Spin Resonance a t Low Temperatures. . . . . . . . . . . . . . 134 9. Acoustic Consequences of the Jahn-Teller Effect. . . . . . . . . . . . . . . . . . . . . . 10. E Terms in Tetrahedrally Coordinated Ions. . . . . . . . . . 11. Centers with Less Than Cubic Symmetry. . . . . . . . . . . . 12. Square Planar Comple .................... 143 13. Cooperative Jahn-Tell .... . . . . . . 146 III. The Jahn-Teller Effect in .................... 151 14. An Octahedral Complex in a ‘ T State, Coupling Only to el Modes.. . . . . . 152 15. Angular Momentum and Weak Spin-Orbit Coupling: The Ham Effect. . . . 154 16. Higher-Order Terms in the Ham Effect .......................... 159 17. Jahn-Teller Coupling to 7 g g Distortions ctahedral Complexes. . . . . . . . 162 18. Tetrahedral Complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 19. Vacancies and Other Defects in the Diamond Structure. . . . IV. Optical Transitions Involving Jahn-Teller Distorted States.. . . . 20. The Configuration-Coordinate Mode ............................ 178 es of Vibration. . . . . . . . . . . . . . . 182 21. Excitation of Nontotally Symmetric 186 22. Transitions from Doubly Degenerate States. . . . . . . . . . . . . . . . . . . . . . . . . . 23. Broadband Transitions to Doubly Degenerate States . . . . . . . . . . . . 188 s . . . . . . . . . . . 192 24. No-Phonon Transitions Involving Doubly Degenerat 25. Optical Transitions Involving Triply Degenerate States. . . . . . . . . . . . . . . . 193 . 194 26. The Case of Strong Spin-Orbit Coupling: F-Bands in Cesium Hal 27. Band Splittings Due to T~ Vibrations: KCl: T1+ and Isoelectronic C . 198 28. Fine Structure and the Ham Effect in Optical Spectra of Transition Metal 201 Ions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 29. Conclusions.. . . . . . . . . . . . . . . . . . . . .

*Part of this work was done while the author was a summer visitor a t the Department of Physics, Stanford University, with support from the Advanced Research Projects Agency through the Stanford Center for Materials Research. 91

92

M. D. STURGE

1. Introduction

The Jahn-Teller effect is the intrinsic instability of an electronically degenerate complex against distortions that remove the degeneracy. It was first predicted as a very general phenomenon in 1937l but had to wait until 1952 for the first unambiguous evidence for its existence.2Even then it was found in only a very limited class of systems and proved to be extraordinarily elusive when searched for elsewhere. As in the case of the dog in the night,3 the curious thing in most cases was that there apparently was no Jahn-Teller effect. As a result, until recently most of the effort in this field has been theoretical, and there has been more emphasis on mathematical sophistication than on the prediction of observable effects. Reviews of the field have reflected this mathematical emphasis and have made little contact with the needs of the experimentalist. In the past year or so, this situation has changed. Theoretical predictions have been made which can be tested rather directly against experiment. Experimental results have been forthcoming which show definite evidence for many of the predicted effects. This being so, there is a need for a review of the present experimental and theoretical position in the field, pitched at a level that the average experimentalist (such as the author) can understand. This review is intended to fill this need. In it, I have preferred to appeal to physical intuition rather than to mathematical rigor, and have concentrated on applications to experiment rather than on theoretical fundamentals. I have been able to do this with a clearer conscience in view of the existence of an authoritative study of fundamentals by Longuet-Higgin~,~ and of the detailed topological behavior of a vast number of molecules (actual and conceivable) by Liehr.5 I shall not attempt to compete with these theoretical treatments. The general plan of this review is as follows. Part I gives a general account of the basic theory and its possible applications. Part I1 applies this theory to the particular problem of a doubly degenerate electronic state interacting with a doubly degenerate vibrational mode. Part I11 extends the treatment to triply degenerate states, and Part IV deals with the problem of electronic transitions involving degenerate states. In each section, as the theory is worked out and applied to particular cases, exH. A. Jahn and E. Teller, Proc. Roy. Soc. A161, 220 (1937).

* B. Bleaney and K. D. Bowers, Proc. Phys. Soc. (London) A66, 667 (1952). a “ ‘Let me draw your attention . . . to the curious incident of the dog in the nighttime. ‘The dog did nothing in the nighttime.’ ‘That was the curious incident.’ ” A. Conan Doyle, “Silver Blaze,” in “The Complete Sherlock Holmes,” Vol. 1, p. 335, Doubleday, New York, 1953. ‘ H. C. Longuet-Higgins, Advun. Spectry. 2, 429 (1961). A. D. Liehr, J . Phys. Chem. 67, 389 and 471 (1963).

THE JAHN-TELLER

93

EFFECT I N SOLIDS

perimental results are quoted to illustrate how well (or how badly) the theory works out in practice. By this means, I have attempted to draw attention to gaps both in the experimental data and in our theoretical understanding. Since my intention is to stimulate experimental interest, I have not hesitated t o include results whose interpretation is still doubtful. Although the emphasis of the review undoubtedly reflects my own interest in the optical spectra of transition metal ions, I have tried to cover all important experimental methods of studying the Jahn-Teller effect in solids. Since the concern of this publication is with solids, I shall not consider specific applications to molecules, where the Jahn-Teller effect has complicated consequences in infrared and Raman spectra,6-13 and on the vibrational structure of electronic transitions.14 On the other hand, I shall make considerable use of the configuration coordinate aptppr~ach,~~-~~ which enables one to apply much of the theory developed for molecules to the case of an isolated impurity or defect in a crystal. Nearly every section contains a discussion of both theory and experiment. The reader who wishes to find out only about a particular system (for instance the F center, or the Cu2+ ion) should consult the first paragraph of Section 1, and Table I, where section references to all the systems mentioned in the article are given.

1. CRYSTALDEFECTS AND TRANSITION METAL IONS CRYSTALS

IN

INSULATING

In order to show a Jahn-Teller effect, a system of electrons and nuclei must have degenerate (or nearly degenerate) electronic energy levels that are reasonably sensitive to distortions of the nuclear framework. A system with extended wave functions can, in principle, show a Jahn-Teller effect W. Thorson, J . Chem. Phys. 29, 938 (1958).

B.Weinstock and G. L. Goodman, Advan. Chem. Phys. 9, 169 (1965). S. Child and H. C. Longuet-Higgins, Phil. Trans. Roy. SOC. London A264, 259 (1961).

M. S.’Child, MoZ. Phys. 6,391 (1962). lo M. S. Child, J . MoZ. Spectry. 10, 357 (1963). l1 M. S. Child and H. L. Strauss, J . Chem. Phys. 42, 2283 (1965). l* J. Herranz and G. Thyagarajan, J . MoZ. Spectry. 19,247 (1966). la J. Herranz, J. Morcillo, and A. Gomez, J . MoZ. Spectry. 19,266 (1966). l4 G . Hereberg, Discussions Faruday Soc. 36, 7 (1963) ; “Electronic Spectra and Electronic Structure of Polyatomic Molecules.” Van Nostrand, Princeton, New Jersey, 1966. A. von Hippel, Z. Physik 101,680 (1936). F. Seite, Trans. Faraday SOC. 36, 74 (1939). l7 C. C. Klick and J. H. Schulman, Solid State Phys. 6, 97 (1957).

l6

CD I+

TABLE I. CONFIGURATIONS OF TRANSITION A N D POST-TRANSITION METALIONS Ground term

Free ion config.

Crystal field

Oct. coord.

Tet. coord.0

d1

Jahn-Teller effect in ground term Oct. Tet.

W

W 0

d2

-

W

d3

W W

0

W

S

S

d4

d4 d5 d5 d6 d6 dl d? ds d9 d'OS2

d10s2p

S

0 W

W

W 0 W W 0 W

S

S

-

0 S 0

S S

W

S W

S

5E(t23e3)

o/

0

Examples and section numbers

Sc2+(8), Ti3+(15, 17, 23), v4+(lO, 15), Y2+(7,22),

Ce3+(6)c V3+(l, 16, 21, 28) V2+(18,21, 28), Cr3+(4,21, 28) F Cr2+(5; 9, 13, 18), Mn3+(5, 9, 13, 22) None known P Mn2+(4), Fe3+(15) u1 C04+, Ru3+ Cr0(15), Mn+(15), Fe2+(13, 15, 23, 28), C 0 ~ + ( 2 3 ) ~ CO~+~ M Mn0(15), Fe+(15), co2+(15,21, 28) Ni3+(5, 7, 9, 13, 22), Pt3+(5, 7 ) Ni2+(13, 15, 16, 18, 21, 28), Pt2+(12) Ni+(5, 7), Cu2+(5,7 , 8, 13, 18, 22, 24), Ag2+(22) T1+(27), Sn2+(27) TP(23)

Or cubal (eightfold) coordination. 0 = zero, W = weak, S = strong. c Excited configuration of Ce3+. Weak field in octahedral halogen coordination. Usually strong field in octahedral oxygen coordination (for exceptions, see G. B l a s ~ e ~ 5 ~ ) . f Spin-orbit coupling included. a

:

THE JAHN-TELLER

EFFECT I N SOLIDS

95

(indeed, superconductivity has been visualized in these terms1*J9).Although we shall discuss such systems briefly, we shall for the most part confine our attention to localized electronic states, since practically all the experimental evidence concerns these. In particular, we shall be primarily interested in various point defects, especially transition metal ion impurities, in insulating crystals. The transition metal ions appear a t scattered points throughout the chapter and have played a crucial role in the development of the theory. We shall therefore give a brief account of their relevant properties later in this section (referring the reader to the many excellent reviews and texts on the subject for details and greater depth). The point defects, on the other hand, are most conveniently introduced as we come to them; here we merely list them. We shall find evidence for Jahri-Teller effects in the F center (Section 26) and the R center in alkali halides (Sections 11 and 21), and in vacancies and related defects in diamond and silicon (Section 19). I n Section 11, we shall also deal with some molecular radicals: the Nz- radical in NaN3, and the CbH5 ring radical. We shall discuss heavy metal impurity ions in alkali halides (of which KC1:Tlf is the exemplar) in Section 27. There are many good texts and reviews on the properties of transition metal ions.z0-z3The object of the present introductory remarks is more to establish a point of view than to provide an introduction to the subject. The basic assumption of the “crystal field” or “ligand field” theory is that in an insulating crystal the d orbitals of a transition metal ion retain their identity, even though they are greatly affected by their surroundings. The simplest picture of this effectz4is that the ions in the lattice produce at the central (impurity) ion an electrostatic potential distribution (the “crystal field”) of a certain symmetry. The crystal field splits the energy levels of the central ion in a manner determined very largely by this symmetry. This picture can be j ~ s t i f i e in d ~terms ~ ~ ~of~ a more realistic molecular orbital model in which the d orbitals are mixed with the u and T orbitals of the nearest-neighbor (“ligand”) ions. In most cases, we shall be dealing specifically with ions of the 3d series. R. K. Nesbet, Phys. Rev. 126, 2014 (1962). R. Englman, Phys. Rev. 129, 551 (1963). zo B. Bleaney and K. W. H. Stevens, Rept. Progr. Phys. 16, 108 (1953). 21 D. S. McClure, Solid State Phys. 9, 399 (1959). 21 J. S. Griffith, “The Theory of Transition Metal Ions.” Cambridge Univ. Press, l8

19

London and New York, 1961. J. Ballhausen, “Introduction to Ligand Field Theory.” McGraw-Hill, New York, 1962. q4 H. A. Bethe, Ann.-Physik [5] 3, 133 (1929). 26 J. H. Van Vleck, J . Chem. Phys. 3, 803 and 807 (1935). G. Blasse, J. Appl. Phys. 36, 879 (1965). ra C.

96

M. D. STURGE

In such ions, the one-electron d orbitals of the free ion are subject to four main perturbations. (Mixing of free ion configurations is almost always, justifiably or not, neglected.) Two of these are strong (-10,000 cm-l) and of the same order of magnitude: the Coulomb (and exchange) interaction between d electrons, and the “cubic field.” The former is present in the free ion but vanishes for d’ and d9 ions. The cubic field is that part of the crystal field invariant under the operations of a cubic group: O h in octahedral (sixfold) or cubal (eightfold) coordination, and Td in tetrahedral (fourfold) coordination. The combined effect of the cubic field and interelectronic repulsion has been calculated “exactly” for all dn ions by Tanabe and Sugano.26 Two weaker effects (0-1000 cm-l) can usually be treated as perturbations on Tanabe and Sugano’s states. These are the effect of deviation from cubic symmetry (“axial” and “biaxial” crystal fields) , and spin-orbit coupling. Although it has been found necessary to include these effects exactly in order to account quantitatively for fine details of optical and microwave spe~tra~fi-29 we shall find the perturbation approach adequate for this article. A d orbital splits under crystal fields of successively lower symmetry, as shown in Fig. 1. (Spin-orbit coupling is neglected.) The cubic field

FREE ION

CUBIC (OCT.)

TRIQONAL TETRAOONAL,

‘

BIAXIAL

AXIAL

FIG.1. Splitting of a d orbital under crystal fields of successively lower symmetry (spin-orbit coupling neglected). The numbers in parentheses indicate the orbital degeneracy of each state. 26

27 2*

29

Y. Tanabe and S. Sugano, J . Phys. SOC.Japan 9, 753 and 766 (1954). S. Sugano arid M. Peter, Phys. Rev. 122, 381 (1961). R. M. Macfarlane, J . Chenz. Phys. 39, 3118 (1963). R. M. Macfarlane, J . Chem. Phys. 40, 373 (1964).

T H E JAHN-TELLER

(0)

FREE ION

(b) Oh

EFFECT I N SOLIDS

(C)

c3

97

(d)

c,+L.s

FIG.2. The energy levels of a d2 ion (e.g., V3+): (a) free ion (without spill-orbit coupling) ; (b) in a cubic (octahedral) crystal field; (c) in a trigonal field as in A1203; (d) with spin-orbit coupling included.

splitting shown has the correct sign for a d electron in octahedral coordination; it has the opposite sign in tetrahedral or cubal coordination. The signs and relative magnitudes of the other splittings depend on the nature of the distortion. The effect of successive perturbations on the energy levels of an ion with the d2 configuration is shown in Fig. 2. (The calculation of the levels of this system is given as a textbook example by McClure.21) The parameters are chosen29 to give the best fit to the optical absorption spectrum of V3+ in Ah03 ?0,31 The notation we shall use is illustrated here. We use upper-case Mulliken notation (A1 , A2 ,E , 1'2 , T2), with (where necessary) 30 81

M. H. L. Pryce and W. A, Ruriciman, Discussions Faraday SOC.26, 34 (1958). D. S. McClure, J. Chern. Phys. 36, 2757 (1962).

98

M. D. STURGE

+

a spin superscript (2s 1) and a parity subscript u or g for orbital states (“terms”) 32 transforming as irreducible representations (I.R.’s) of the cubic groups o h or Td . Orbital states of lower symmetry groups are careted. Combined spin-orbital states (whether of the single group in even-electron systems or the double group in odd-electron systems) are labeled in Bethe notation, e.g., rl through Tls for o h or Td . We shall also use r as a general . symbol for a state or an irreducible representation (abbreviated to I.R.) One-electron orbitals are in lower-case Mulliken notation. Thus, in Fig. 2, 3Tl(t22)means the spin triplet with orbital wave functions transforming in o h as T I , , and primarily derived from the configuration with two electrons in t z orbitals. I n a trigonal field, it splits into an orbital singlet 3A and doublet 3fi. These are further split by spin-orbit coupling (operating in second order in the case of ”). A list of ions of the various dn configurations is given in Table I, with their ground terms in octahedral and tetrahedral (or cubal) coordination. (This table also provides an index for finding the section where a particular ion is discussed.) For d4 through 8,one must’distinguish between the “weak field” ground term (the Hund’s rule ground term of maximum spin) and the ‘(strong field” ground term. The latter is the ground state obtained when the cubic field splitting is large enough to overcome interelectronic repulsion, so that the orbitals of Fig. 1 are filled without regard for Hund’s rule. (In practice, strong field ground terms of transition metal ions are only found in octahedral coordination, and this fact has been taken into account in constructing Table I.) Table I also gives an indication of the expected strength of electron-lattice interaction leading to a Jahn-Teller effect in the ground state. This classification is arrived a t as First, spin-orbit coupling is neglected, so that orbital singlets have zero Jahn-Teller interaction, and there is no stabilization of a T term by spin-orbit splitting. Secondly, we use the fact that in octahedral coordination only e electrons and in tetrahedral only t 2 electrons can hy. ~ ~ ~ ~of~the larger overlap, bridize with the u orbitals of the l i g a n d ~Because electron-lattice coupling in these orbitals is usually much stronger than in those which hybridize only with T orbitals, unless the orbital happens to be half (or completely) filled, in which case all first-order effects vanish. Ions having “strong” interactions by these criteria we shall call “JahnTeller ions,” and with occasional exceptions these are the ones that produce pronounced Jahn-Teller effects in the ground state. Practically any transition metal ion can show Jahn-Teller effects in one or more excited states. We shall make considerable use of the “effective Hamiltonian” for32

We use the word “term” t.0 mean “cubic field term” when discussing transition metal ions. J. D. Dunitz and L. E. Orgel, Phys. Chem. Solids 3 , 2 0 (1957).

THE JAHN-TELLER

99

EFFECT I N SOLIDS

malisma that enables us to isolate effects of interest from irrelevant complications. It is a logical extension of the well-known spin-Hamiltonian formalism of spin resonance.zoWe divide the electronic Hamiltonian into a “strong” part X(O) and a “weak” part X’, and we shall suppose that X(O) is simple enough to solve exactly. The Tanabe-Sugano HamiltoniaqZ6 including the cubic field and interelectronic repulsion only, is the usual choice for X(O) in an ion of the 3d group. Then X’ includes everything else (spin-orbit coupling, lower symmetry fields, vibronic interaction, external fields). We consider one degenerate eigenvalue of X@),labeled zs+lI?. The eigenfunctions can be written I i) = I SrM,e), where 0 is a component of r.35The first-order effective Hamiltonian X(l) for zs+iJ? has matrix elements xi$’) = (xrM,e I X’ I xrM,’e’), (1.1) where lj) =

1 xriw,’e’).

Similarly, the second-order effective Hamiltonian X(z)has matrix elements

(1.2)

where E ( F“) is another eigenvalue of X(O).This procedure can be extended to any accuracy required. If X(O)has nearly degenerate eigenvalues, we can lump them together and allow for their separation by including zerothorder terms in the effective Hamiltonian. Finally, we can calculate the matrix elements of x’by means of the Wigner-Eckart theorem.36 The matrix element of an operator can be written as the product of an “angular” factor (or factors) and a “reduced matrix element.” I n the notation of Griffith,37

(srM,e

I O Z F M ~I x’r’M,’e’)

=

-(“

-211,

S‘

M,‘

‘)(P

M

r’ :)(xr 11 -0

Bgr

11 s‘r!).

(1.3)

8’

Here O is a double operator with spin part transforming as I S M ) , and orbital part transforming as the lth component of the I.R.f ; the quantities a4 35

37

Y. Tanabe and H. Kamimura, J . Phys. Sor. Japan 13,394 (1958). By t,he “components” of an I.R.I-, we mean the linearly independent functions that form a basis for r. E. P. Wigner, “Gruppentheorie.” Vieweg, Braunschweig, 1931; C. Eckart, Rev. Mod. Phys. 2, 305 (1930). J. S. Griffith, “The Irreducible Tensor Method for Molecular Symmetry Groups.” Prentice-Hall, Englewood Cliffs, New Jersey, 1962.

100

M. D. STURGE

in angular brackets are the 3j symbols38J9 and their point group analogs the V coefficients.”~40(These are symmetrized forms of the Clebsch-Gordan coefficients.) The sign in (1.3) is to be determined by the rules given in Griffith.” (Sr 1 1 Ogr 1 1 fW) is the reduced matrix element, independent of M , , Ms’, 0, and Or, which can either be regarded as an experimental parameter or calculated in terms of more fundamental quantities by standard Although the effective Hamiltonian method is extremely powerful, it has its limitations. Its most important assumption, as far as we are concerned, is that the perturbations (in particular the Jahn-Teller interaction) We shall find cases must be small relative to the initial splittings due to x@). where this is not SO (Section 19). I n the case of transition metal ions, however, the assumption is usually well justified. 2. COLLECTIVE COORDINATES AND

THE

“QUASI-MOLECULAR” MODEL

The instantaneous potential seen by an electron in some nuclear framework (crystalline or molecular) can be divided into two parts. There is a static part V o ( q ) which , is a function only of the mean nuclear positions and was the basis of discussion in the previous section. There is also a dynamic part V ( q ,Q) , which depends on the displacements Q of the nuclei from their mean positions. We shall assume that V ( q , Q ) can be expanded in powers of Q: V ( q ,Q)

=

vo(~) + C (aVlaQia)Qia+ +E (a2Vv/aQiaaQi~)Qi,Qi8..., i,j,a,8

%.,a

(2.1)

where i, j label the nuclei and a, ,8 their Cartesian directions of displacement. This description of V is not very convenient in practice, since the Cartesian nuclear coordinates Q i a do not transform in a simple way under the operations of the symmetry group of the system. It is possible to choose linear combinations of the Qia which do transform according to the irreducible representations of the group. We call these combinations the collective coordinates Qk . Then

V ( Q )= Vo

+ C (aVlaQk)Qk + 4 E k

k.Z

(a2J‘/aQdQz)QkQz

+ *..,

(2.2)

38 E. P. Wigner, “Group Theory.” Academic Press, New York, 1959. 39M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr., “The 3j and 6 j symbols.” M.I.T. Press, Cambridge, Massachusetts, 1959. 4o E. Mauza and J. Batarunas, Lietuvos T S R Mokslu Akad. Darbai B3, No. 26,27(1961). 41 Y. Tanabe, Progr. Theoret. Phys. (Kyoto) Sdppl. 14, 17 (1960). 42 C. W. Nielson and G. F. Koster, “Spectroscopic Coefficients for the p n , d” and f” configurations.” M.I.T. Press, Cambridge, Massachusetts, 1963. 43 J. S. Griffith, MoZ. Phys. 6, 503 (1963).

THE JAHN-TELLER

EFFECT I N SOLIDS

101

23

t

FIG.3. (a) The octahedral XI’S complex. (b) The tetrahedral X Y , complex. The collective coordinates & k are linear combinations of the Cartesian coordinates shown; see Tables I1 and 111.

where the dependence on q is understood. Since V is invariant, av/a&k transforms in the same way as &k . Methods for finding the &k in terms of the &im are given in the For most purposes, we shall assume as a first approximation that only motions of the impurity or defect and its immediate neighbors contribute appreciably to V ( Q ) .This is usually quite a good approximation for a transition metal ion and not too bad a one even for the F center. We shall be interested primarily in two particular arrangements of ions: the octahedral xY6 complex ( O h symmetry) and the tetrahedral X Y I complex (Td symmetry). These complexes are illustrated in Fig. 3, and their collective coordinates &k are listed in Tables I1 and 111. The nuclear motions associated with those &k which are important for the Jahn-Teller effect 44 45

(6

V. Heine, “Group Theory in Quantum Mechanics.” Pergamon Press, Oxford, 1960. G. Herzberg, “Infra-red and Raman Spectra of Polyatomic Molecules.” Van Nostrand, Princeton, New Jersey, 1945. E. B. Wilson, J. C. Decius, and P. C. Cross, “Molecular Vibrations.” McGraw-Hill, New York, 1955.

102

M. D. STURGE

TABLE 11. COLLECTIVE COORDINATES FOR THE OCTAHEDRAL X Y 6 COMPLEX U

b

C

Qi

Qz Q3

Q4

Q5

Q6 Q7 Q8

Qs Qio Qii

Qiz

Q 13 Q 14 Q 15 ~~

We use Van Vleck’s notation.47 following Ballhausen, we use Greek letters for the symmetry labels of normal coordinates. c In odd parity modes, we hold the central ion, rather than the center of mass, fixed. (We also do this in Table 111.) a

are illustrated in Figs. 4 and 5 (notation of Van Vleck47) . I n any particular case, Tables I1 and I11 can easily be extended to include motions of further neighbors if these turn out to be important (see, for instance, Table I of Jahn and Teller‘). The Q 3 coordinate represents tetragonal (even parity) distortion of the octahedron or tetrahedron, lowering the symmetry from o h to Dh or from Td to D2d . Similarly, the combinations (Q4 Q5 Q 6 ) or ( Q 7 Q8 Q 9 ) represent trigonal distortion lowering the symmetry to D 3 d or Csv . There are other combinations of the Q’s which represent such axial distortions (see, for instance, Table IV) ; but a general linear combination represents a biaxial distortion. The splitting of a d orbital produced by these various distortions is illustrated in Fig. 1. I n most cases, static deviations from cubic symmetry are small enough that a zeroth order description in terms of o h or Td is adequate. These two complexes then have the convenient property of having only one totally symmetric coordinate Q1 , i.e., only one distortion (the radial or “breathing” mode) , which does not reduce the symmetry. This is not true for more elaborate complexes, or for lower symmetry groups.

+

47

J. H. Van Vleck, J . Chem. Phys. 7, 72 (1939).

+ +

+

THE JAHN-TELLER

EFFECT IN SOLIDS

103

FIG.4. Collective coordinates for the X Y S complex.

If we were dealing with a real isolated X Y s or X U 4 molecule, the Qk would be true normal coordinates, and the nuclear Hamiltonian could be written (in the harmonic approximation) as

where P k is the momentum operator conjugate to Qk , P k is the effective mass, and W k is the frequency of the kth normal mode. If there is more than one set of Q k transforming according to a particular I.R., we must choose the correct linear combinations of them for (2.3) to hold. (These combinations depend on the details of the system and are not necessarily the combinations chosen in Tables 11 and 111.) I n a real crystal, the &k are not normal coordinates. This does not destroy their usefulness for describing the electron-lattice interaction, since (2.2) still holds. Only when we use (2.3) for the nuclear Hamiltonian, and ask what is the frequency or effective mass associated with Qk , do we run into trouble. So long as our interest is concentrated on the electronic terms in the Hamiltonian, rather than the purely nuclear terms (X,,,), we shall find the “quasi-molecular” approximation, in which the Qk are iegarded as normal coordinates, to be quite adequate.

104

M. D. STURGE

FIG.5. Collective coordinates for the XY4 complex.

COORDINATES FOR THE TETRAHEDRAL XY4 COMPLEX TABLE111. COLLECTIVE

THE JAHN-TELLER

EFFECT I N SOLIDS

105

However, there will be occasions when we would like to know the precise form of X,,, . I n a perfect crystal, the normal coordinates are plane waves, and each one makes only an infinitesimal contribution to the displacement a t any particular lattice site. I n such a crystal, we can write Qk

=

(3N)-’

C [S~(KT)U(K~) + S,*(KT)U+(K~)]

(2.4)

Kr

and X,”, =

c

fiw ( K T ) [u+u

+ 31.

(2.5)

Kr

Here u + ( K ~ ) , u ( w ) are the creation and destruction operators for the phonon from branch r, with wave vector K and frequency w ( u r ) . N is the is)the projection of this phonon onto number of ions in the crystal. S ~ ( K T Q k , that is, a measure of the contribution of that particular phonon to the nuclear displacements described by Q k . Thus, Qk has associated with it not a single frequency wk but the whole range of the phonon spectrum, weighted by 1 X k 2 I (and, of course, by the phonon density of states). Since we are interested in impurities and defects rather than in the perfect crystal, we should in principle allow for the effect on the lattice vibrational spectrum of these local perturbations, which destroy translational symmetry. Formally, we replace ( K T ) by the generalized mode index s.48 Certain modes may have exceptionally large amplitudes near the defect: these are local modes if w ( ~ is ) outside the allowed frequency band for the perfect crystal, and quasi-local (resonance) modes if within it. All other modes are correspondingly reduced in amplitude at the defect. Even the quasi-local modes can often be quite sharp,48which means that &(s) is peaked over a small range of s and that w k w (s) . I n such a case, even though the frequency spectrum of Q k is continuous rather than discrete, (2.3) may be a better approximation to the effective nuclear Hamiltonian than (2.5). We have so far treated the electronic and nuclear Hamiltonians as if they were separate entities. When we come to consider the Jahn-Teller effect, we shall find that they are, in fact, inseparable. The eigenfunctions of the combined electron-nuclear Hamiltonian we call the “vibronic” wave function Q. I n the absence of the Jahn-Teller effect, the Born-Oppenheimer49approximation for 9 is valid. I n this approximation, 9 can be written as a product (2.6) 9 = #n(Q, Q ) P ( ~ Q, ) ,

+

where q and Q represent electron and nuclear coordinates respectively. Here #n is a solution of the Schrodinger equation for the electrons with *8 49

A. A. Maradudin, Solid State Phys. 18, 273 (1966). M. Born and J. R. Oppenheimer, Ann. Physik [4] 84, 457 (1927).

106

M. D. STURGE

the nuclei fixed at their positions Q, whereas p(n, Q) is a solution of the nuclear Schrodinger equation in which the electronic energy En(&)(i.e., the nth eigenvalue of the electronic Hamiltonian) is added to the “ordinary” internuclear potential V ( Q ) .[It is usually convenient to lump all electronic interactions except those of immediate interest into V ( Q ).] The wave function #p we call a Born-Oppenheimer product. Although the conditions for the validity of (2.6) break down when there is a Jahn-Teller effect (see Section 3), we shall still be able to write our overall “vibronic” (i.e., combined electron-vibrational) wave functions as linear combinations of a (usually small) number of Born-Oppenheimer products. We shall also, in most cases, still be able to define an energy surface V ( Q ) En(&) on which the nuclei move.

+

3. THEJAHN-TELLER THEOREM The Jahn-Teller theorem1 states that any complex occupying an energy level with electronic degeneracy is unstable against a distortion that removes that degeneracy in first order. Only if no such distortion is possible can the degenerate level be stable. This is obviously the case for Kramersmv51 degeneracy, and it happens that there are certain levels of a linear molecule whose degeneracy no distortion can remove in first order. (The latter case is obviously unlikely to be of interest in solids.) I n all other cases, we shall see that there exists a distortion that removes degeneracy. The instability arises because a linear splitting of a level necessarily leads to a state with lower energy than that of the unsplit level, the center of gravity of the level being unshifted to first order. Ultimately the distortion is limited to a finite value by the quasi-elastic forces [covalent and electrostatic : the V ( Q ) of the previous section] that resist it; but, because the original symmetry state would have been a position of equilibrium in the absence of electronic degeneracy, the quasi-elastic contribution to the energy contains no terms linear in the distortion. Thus, a new position of equilibrium is reached in which the local symmetry is lower than the point symmetry of the crystal; in general, this new symmetry will be low enough to remove all (except Kramers) electronic degeneracy. Now a distortion that removes electronic degeneracy must itself be degenerate,52 so there will be more than one position of equilibrium with equal energy. Thus, we have replaced the original purely electronic degeneracy with a more complicated, vibronic, degeneracy. 60 51

62

H. A. Kramers, Koninkl. Ned. Akad. Witenschap., Proc. 33, 959 (1930). E. P. Wigner, Nachr. Akad. Wiss. Goettingen, Math.-Physik. Kl., I l a . Math.-Physik. Chem. Abt., p. 546 (1932). Although this is not true for the tetragonal groups, it turns out that in their case positive and negative distortions are necessarily equivalent (see Section 12).

THE JAHN-TELLER

EFFECT IN SOLIDS

107

We shall now put the foregoing qualitative argument on a more precise basis. Jahn and Teller used perturbation theory to prove their theorem. Although this approach is certainly the best way to treat most real systems, it does obscure the full generality of the theorem, which is not dependent on the implicit assumptions made by Jahn and Teller. Furthermore, the alternative approach from the Hellmann-Feynman has a certain intuitive appeal.54We shall, therefore, use the latter approach to prove the Jahn-Teller theorem, and then show how it relates to the perturbation approach, which we shall use in working out the practical applications of the theorem. The basic assumption of the Hellmann-Fe~nman~~-~? theorem is that it is possible to write down the total potential energy of a system of electrons and nuclei as an explicit function of the coordinates, V ( q , Q), and that the first derivative with respect to each of the nuclear coordinates exists.58The Hellmann-Feynman theorem states that the generalized force F k acting on the nuclei in the sense to increase the coordinate Qk is59 F k

I

= - (i dV/dQ,

I i).

(3.1)

Here I i) is the electronic state. Equation (3.1) is the analog of the classical = -dV/dQk. Its proof and use contain certain pitfalls that have been discussed by many authors.60m61 If F k is nonzero for a certain nuclear configuration, the configuration is unstable and will spontaneously distort until F k is zero.62The Jahn-Teller theorem states that, if [ i) is a degenerate state when Qk = 0, the matrix

Fk

6 63

W. L. Clinton and B. Rice, J . Chem. Phys. 30, 542 (1959). A more formal proof, not explicitly using the Hellmann-Feynman theorem, is given by E. Ruch and A. Schoiihofer, Theoret. Chim. Acta 3, 291 (1965). H. Hellmann, “Quantenrhemie,” p. 285. Deuticke, Leipzig, 1937. 66 R. P. Feynman, Phys. Rev. 66, 340 (1939). 67 J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” p. 932. Wiley, New York, 1954. Clinton and Rice63 call this assumption the Born-Oppenheimer approximation. I t is not, however, the more restrictive assumption, more usually called the Born-Oppenheimer approximation, that the vibronic wave function can be written as the simple product (2.6). We are neglecting the dependence of the electronic energy on the nuclear momenta. To speak more pictorially, we assume we can “fix” the nuclei and solve the resulting electronic Hamiltonian. This is obviously valid if the nuclear mass is infinite, and it can be shown (Longuet-Higgins4) that it is accurate to order m / M . If there is no Jahn-Teller effect, the two approximations are equivalent. s9 Here & A is measured from some assumed “undistorted” configuration, which is not necessarily cubic. 6o T. Berlin, J. Chena. Phys. 19, 208 (1951). 61 M. L. Benston and B. Kirtman, J . Chem. Phys. 44, 119, 126 (1966). 62 For a formal treatment of stability criteria in molecules, see A. D. Liehr and H. L. Frisch, J . Chem. Phys. 28, 1116 (1958).

108

M. D. STURGE

element in (3.1) exists for some Qk , k # 1 (i.e., for some nontotally symmetric coordinate). This can be proved by elementary group theory. Let Qh (and hence a v / a Q k ) transform as some component of the irreducible representation (I.R.) f , and I i) transform as a component of r. If the matrix element (3.1) is not to vanish identically, r X f X r must contain rl,that is, f must be contained in the symmetric square [rz].44v63 If r is nondegenerate, [rz] = rl . The spontaneous distortion that then occurs is totally symmetric and does not lower the symmetry of the system. I n equilibrium, therefore, Qk = 0 unless f = rl .59 (If I i) is the ground electronic state, this proviso can be eliminated by a trivial change of origin for the totally symmetric coordinate Q 1 , and we can, without loss of generality, say Q k = 0, all k.) If r is degenerate, there will be one or more nonsymmetric I.R. contained in [rz].If the complex has a normal coordinate Q k transforming as one of these I.R.’s, there is no reason from symmetry why F k should be zero. I n general, it will not he, and spontaneous distortion will occur until equilibrium is reached a t some finite value of Q k . An important consequence of the requirement that [rz] contain T’ is that in a centrosymmetric system T must be of even parity. This greatly reduces the number of Q k with which we shall have to deal. Jahn and Teller’ enumerate all the normal coordinates (or rather, all their I.R.’s) of all possible molecules. If we ignore translational symmetry (as we have to in the case of a defect or impurity in a crystal), this includes all possible crystals. They show that, with one exception (the linear molecule), there is always at least one nontotally symmetric coordinate &k whose I.R. T’ is contained in the symmetric square of any degenerate I.R., [PI. Hence, with that one exception, there is always a distortion that will split the state r in first order. The question of how big this splitting might be in a real case we shall leave to the next section. As remarked earlier, this approach using the Hellmann-Feynman theorem, though physically appealing, is not very useful in dealing with a practical problem. This is because Eq. (3.1) is valid only for exact elec63

V , and hence a V / a & k , must be real in the absence of a magnetic field. We assume here that r is real; the extension to complex r is trivial so long as r* is linearly independent of r. The only relevant I.R. for which this is not the case is rs of the double groups 0*,O h * , and !I’d*. Among the crystallographic point groups, all other I.R.’s of this type (Wigner’s type c ; see G. F. Koster, J. 0. Dimmock, R. G. Wheeler, and H. Stat,z, “Properties of the 32 point groups.” M.I.T. Press, Cambridge, Massachusetts, 1963) are simply Kramers doublets. For such I.R.’s, must be contained in the antisymmetric square (J?) to have a first-order matrix element. However, in O h (for example) (r2) = rlo rSg rse= [r?], etc., and the physical situation is not altered by this purely formal complication.

r

+

+

THE JAHN-TELLER

EFFECT IN SOLIDS

109

tronic wave functions,61which we hardly ever have in practice. Furthermore, in order to be able to define the force F k in a degenerate electronic state, we must choose our functions I i) in a particular way. They must be those particular combinations of the basis states for which the offdiagonal matrix elements (i I a l / ’ / a & k I j ) , i # j , vanish. For a given Qk , such combinations can always be found, but they will not (in general) be the correct combinations for any other &k , even those &k transforming as other components of the same I.R. This makes it difficult to deal with all the relevant &k on an equal basis. Perturbation theory enables us to get over this difficulty by setting up an effective Hamiltonian for the system (in the state r) which describes it to any required accuracy. (Usually we shall not want to go beyond the first order of perturbation theory.) We shall set up a model that represents the essential features of real systems and will be the basis for our treatment of such systems in subsequent sections of this article. We start by dividing the electrons in our complex into two groups. One group consists of those few electrons that occupy degenerate orbitals and actively participate in the Jahn-Teller effect. We shall call these the “active” electrons. The remaining “passive” electrons form closed shells, covalent bonds, etc., and the possibility of exciting them to higher states will be neglected. I n the case of a color center, the trapped electron is “active”; all others are “passive”; in a transition metal ion, the d electrons (or some portion of them) might be “active,” whereas all electrons in s and p orbitals and the bonding ligand electrons are “passive.” Now suppose all “active” electrons have been removed. The Hamiltonian is now, as in (2.3), xnuc = 3 [pk2/Pk PkWk2&k2] V’(Q), (3.2)

c k

+

+

where V ’ ( Q ) has been added to represent anharmonic effects. The difficulties associated with the justification of this “quasi-molecular” nuclear Hamiltonian in a crystal have been discussed in Section 2. (We shall ra.ise them again, for the last time, at the end of this section.) Neglecting V’, the eigenvalues of (3.2) are E,(’?%l.’’nk...)

=

c k

&Wk(‘?%k

+ i),

(3.3)

where n k = 0, 1, . This spectrum is (in general) completely alteredoby the Jahn-Teller interaction. Now let us include the “active” electrons. These move in a potential V ( q , Q ) which is a function of the “active” electron coordinates q and the nuclear coordinates Q . We shall assume that V ( q , Q ) can be expanded in a Taylor series about the point &k = 0, all Q k . The Hamiltonian for the

110

M. D. STURGE

“active” electrons is then Xe = X,

+C

(av/aQk)Qk

k

+ ...,

(3.4)

since the electron kinetic energy operator does not contain Q k .‘j4 Here X, is the Hamiltonian for the “active” electrons in the static situation, i.e., when Q k = 0. Let us suppose that X, has a certain g-fold degenerate eigenvalue E, with eigenfunctions transforming as r (which we shall use as a label for the level). For simplicity, let us assume that it is well separated from other levels, so that we may use first-order perturbation theory. The first-order effective electronic Hamiltonian for r is a g X g matrix. Its matrix elements, to first order in Q k , are

c

X ..(O = ‘6ij

+C

hij(k)Qk

(3.5)

k

where h i j ( k ) = (i I d V / d Q k I j ) and I i) and I j ) are orthogonal substates of r. If Q k and d V / a Q k transform as the lth component of F, and F is contained in [r2],we have from the Wigner-Eckart theorem [see Eq. (1.3)]

where AF( = (I’ I I a V / a Q k I I r >) is a measure of the strength of the electronlattice coupling. If F = rl , h i j ( l ) = A & , so that there is an overall shift but no splitting. If r # I’I, Tr h i j ( k )

=

0

(3.7)

[Eq. (2.21) of Griffith%]. Thus, the centroid of r is not shifted to first order in Q k . It follows that at least one of the eigenvalues of (3.5) decreases, initially linearly, with increasing Q k . If we combine (3.2) and (3.5), we have the first-order Hamiltonian %(I). To first order, we can drop the for the whole system XI = X,,, anharmonic term V’, so that X,,, contains no cross-terms between the Qk . Then the matrix elements of XI are

+

Xij’ = E8ij

+C k

[!i(Pk’/1.1k

-I- I.1.W/c2Qk2)8ij

+

hij(k)Qk].

(3.8)

Zeroth-order splittings can be included by replacing l3 by Ei. If only one (degenerate) mode is important, we can drop the suffices on 1.1 and w and take the sum only over the components of this mode. Equation (3.8) is the basic first-order Hamiltonian for the Jahn-Teller effect, although in one or two special cases we shall need to consider 64

This is not to say that the electron kinetic energy does not depend on Q; it does, in general, through the dependence of the electronic wave function on Q (see BerlinGo).

THE JAHN-TELLER

EFFECT I N SOLIDS

111

higher-order terms, and, in at least one important case, the whole perturbation approach fails (see Section 19). The last two terms in (3.8) define a potential surface in & r ~ space. The presence of the terms linear in Qk ensures that the minima of this potential are not at the origin; this is merely another way of stating the Jahn-Teller theorem. Unless h i j ( k ) is diagonal for all k, the last term in (3.8) causes the nuclear motion to mix electronic states. It follows that simple BornOppenheimer products (2.6) are not, in general, eigenfunctions of (3.8). We shall see by considering specific examples in later sections that (3.8) has solutions with the form qrn

=

C amn#n(~t &) ‘~(n, &>

7

(3.9)

where the a,, form a square matrix (which can sometimes be diagonal). The “electronic” and “nuclear” wave functions, # and ‘p, are defined by Eq. (2.6). Note that there will be an infinity of possible cp for a given +, and that the cp’s associated with different #’s are not necessarily inde. ~as ~ accurate as the Bornpendent. Equation (3.9) can be s h o ~ nto~ be Oppenheimer approximation in the nondegenerate case, i.e., to order m / M , the ratio of electron to nuclear mass. It is important to note that 9, is as many dimensioned as #, . Thus, the overall degeneracy of a level (i.e., that degeneracy associated with the multiplicity of a given I.R.) cannot be reduced by the Jahn-Teller effect, since the overall Hamiltonian retains its symmetry. (We shall find in many cases that there are additional (‘accidental” degeneracies due to simplified Hamiltonians.) However, the original electronic degeneracy, in which an operation of the group changes the electronic and vibrational wave functions independently, is replaced by vibronic degeneracy, in which transformation of the electronic wave function is inextricably accompanied by transformation of the vibrational wave function. One of the advantages of the perturbation approach is that the electronic basis states I i) and I j ) need not be eigenfunctions of the static Hamiltonian X, We can use eigenfunctions of a simpler Hamiltonian, X(O), in which case we shall find that matrix elements (i I X, I j ) , i # j , exist. These “static terms” must be included in our first-order Hamiltonian (3.8). They may, by removing degeneracy, stabilize some eigenstates of X(0) against Jahn-Teller distortio+ (see Section 15). We can also include higher-order terms, such as (1.2), giving

.

(3.10) where 6s e6

I r’r)is some other eigenstate of

A. D. McLachlan, Mol. Phys. 4, 417 (1961). J. H. Van Vleck, Physica 26, 544 (1960).

Xo and E ( r ’ ) is its energy. Note

112

M. D. STURGE

that the second-order terms analogous to (3.10) which arise from the Jahn-Teller interaction are quadratic in the Qk . If these are included, we should also include first-order matrix elements of d2v/a&k6&?z. Another important term in the Hamiltonian which can, in practice, be of the same order is the anharmonic term V’. It is obvious that the situation gets very complicated once we go beyond the first-order Hamiltonian (3.8). There will be occasions (e.g., in Section 19) when the perturbation expansion implicit in (3.8) and (3.10) fails to converge. We have to be careful not to apply conclusions based on the effective Hamiltonian approach to such cases. We are reduced to using simple qualitative models, which, however, often prove singularly successful. Finally, before going on to consider the application to real systems, we would like to know how good our “quasi-molecular” approximation to the nuclear Hamiltonian is likely to be. Formally, we may replace the nuclear Hamiltonian (2.3) by the Hamiltonian (2.5) (generalized if necessary to take into account local and quasi-local modes). We replace (3.8) by [see Eq. (2.4)] 3c..(1) Y =

ESij

+ c $ f i w ( s ) [ a + ( s ) a ( s ) + $-JSij 8

+ (1/N)

hij(k)[Xk(S)a(s) k .a

+ fik*(s)a+(s)]* (3.11)

Hama7has shown that, if h i j j k ) is diagonal, the spectrum and eigenfunctions of (3.11) bear the same relation to their uncoupled values (ie., those obtained with h i j = 0) as those of (3.8) do to their uncoupled values. However, this is a very special case, since the vibronic spectrum is only modified in a rather trivial way by the interaction (see Section 14). As far as the optical consequences of the Jahn-Teller effect are concerned, the “quasi-molecular” approximation is as good as the semiclassicalFranckC o n d ~ n ~ approximation *~~ (see Section 20). This has been shown by rather general argumentb; its truth for the particular case of a doubly degenerate state has been demonstrated by O’Brien.S1 Slon~ewski~~ has considered the case of a doubly degenerate electronic level interacting with a finite number of doubly degenerate vibrational modes. He finds that the qualitative behavior of such a system is indeed similar to that of a single pair of degenerate modes (considered in Section 5 ) . However, Sloncewski runs into divergencies when the mode frequencies are continuous rather than discrete. 47

F.S . Ham, Phys. Rev. 138, A1727 (1965).

m J. Franck, Trans. Faraday SOC.21, 536 (1925). 6o

70 71 72

E. U. Condon, Phys. Rev. 32, 858 (1928). M . Lax, J . Chem. Phys. 20, 1752 (1952). M. C. M. O’Brien, Proc. Phys. SOC. (London) 86,847 (1965). J. C. Sloncewski, Phys. Rev. 131, 1596 (1963).

THE JAHN-TELLER

EFFECT I N SOLIDS

113

The general theorem that the consequences for the Jahn-Teller effect of using the nuclear Hamiltonian (2.3) are qualitatively the same as those of using the “correct” Hamiltonian (2.5) (suitably modified to allow for local and quasi-local modes) has still to be proved. We shall take it on faith for the purposes of this chapter.

4. THE STATICAND DYNAMIC JAHN-TELLER EFFECT: SOMEORDERSOF MAGNITUDE The last section set up the formalism for the Jahn-Teller effect, and showed that any complex (other than a linear molecule) in a state with electronic (non-Kramers) degeneracy is in principle unstable against a distortion that can remove that degeneracy. A first-order perturbation treatment leads to the basic Hamiltonian (3.8), which is a sum of electronic, lattice, and electron-lattice interaction terms. The basis states for this equation are Born-Oppenheimer products (2.6). The first two terms in the Hamiltonian are diagonal, but the electron-lattice interaction term can have off-diagonal elements, leading to a breakdown of the BornOppenheimer approximation. The electron-lattice interaction can be expressed in terms of a small number of parameters by means of the Wigner-Eckart theorem, as expressed in (3.6). The coupling of a state I r i )to a mode transforming as the lth component of the irreducible representation f is a product of a tabulated angular factor

and a “reduced matrix element” Ar , which is independent of i and 1. Furthermore, Ar , and hence the first-order coupling, vanishes unless the symmetric square [rz] contains f.63 In second-order perturbation theory, coupling to other modes is possible. The problem becomes much more complex, and it is best not to attempt a general treatment but to consider each case on its merits. We now turn to the question of when and how the Jahn-Teller effect is likely to be observable. Although the Jahil-Teller theorem is quite general, we can limit its effective range of application in practice by consideration of a few orders of magnitude. The magnitude of the Jahn-Teller distortion is determined by a balance between the distorting term h i j Q k and the “elastic” restoring term &14.#Jk2&k2 ( p k w k 2 is the effective force constant). The value of &k at which equilibrium is reached, &ko, is proportional to A r / p k w k 2 , and the corresponding stabilization energy (the “Jahn-Teller energy”) is proportional to A r / p k u k 2 . [The coupling constant AT is defined in Eq. (3.6) .]

114

M. D. STURGE

A level with only spin degeneracy, well separated from other levels (e.g., the 6S ground term of Mnz+), is extremely insensitive to distortion of its surroundings. A? is only nonzero when very high order spin-orbit perturbations are taken into account. The Jahn-Teller energy is correspondingly small, perhaps 0.01 to 1 cm-1, compared with the 100-10,000 cm-' that is obtained for an orbitally degenerate level of a transition metal Similarly, the energy levels of an ion in an f" configuration (i.e., all trivalent rare earths, most divalent ones in or near the ground state, and most highly ionized actinides) are very insensitive to the nuclear configuration, and so again the Jahn-Teller energy is very small. A v v a k ~ m o vhas ~ ~argued that even a very small Jahn-Teller energy can be significant in spin-resonance. His argument appears to confuse the Jahn-Teller energy, which is the reduction in the potential due to the Jahn-Teller effect, with the splitting of actual energy levels. Because of the kinetic energy term in the Hamiltonian, energy levels are not very sensitive to small changes in the potential. The general view,"J3 that Jahn-Teller energies very much less than the zero-point vibrational energy (typically 100 cm-1) can have no observable consequences, is borne out by c a l c ~ l a t i o nand ~ ~ by most experiment^?^ It is, therefore, reasonable to confine our attention to the orbitally degenerate terms of transition metal ions and point defects. Even amongst these, there are many levels that are so weakly coupled to the lattice that the Jahn-Teller effect may be ignored. We may cite as examples terms arising from configurations with half-filled shells, such as the 2E(tz3)term of octahedral Cr3+,16and the very extended states of a hydrogen-like impurity center in a semiconductor, which can only couple to very long wavelength phonons, of which there are too few to produce an appreciable Jahn-Teller effect." Assuming there is an appreciable Jahn-Teller effect, how are we going H. A. Jahn, Proc. Roy. SOC.A164, 117 (1938). V. I. Avvakumov, Opt. i Spektvoskopiya 13, 588 (1962); see Opt. Spectry. ( U S S R ) (English Transl.) 13, 332 (1962). 76 This is not to say that vibronic interactions cannot in principle contribute t o the cubic splitting of (say) a 6Sterm in a cubic environment. However, detailed calculation [T. J. Menne, private communication (1966)] shows that such a contribution will be exceedingly small. 'Ib A. L. Schawlow, A. H. Piksis, and S. Sugano, Phys. Rev. 122, 1469 (1961). 77 That the Jahn-Teller effect will be weak in such extended centers can also be seen as follows. The change in electronic energy per unit (uniform) strain for such a center is more or less independent of the center's radial extent. On the other hand, the larger the center, the larger the volume of crystal which has to be strained in order to produce this change. The elastic energy term opposing the Jahn-Teller distortion is therefore that much larger, and the Jahn-Teller energy should go down roughly as the inverse cube of the radius of the center.

73 74

THE JAHN-TELLER

EFFECT I N SOLIDS

115

to observe it? The answer to this question will become clearer as we proceed to consider special cases. There are three general ways in which the JahnTeller effect might be expected to have observable consequences. Firstly, we might see direct consequences of the lowering of site symmetry. For instance, this could show up in crystal structure, in spin resonance, in nuclear magnetic resonance (NMR), or in the Mossbauer effect. To be detectable in this way, the mean distortion must be finite when averaged over a certain period of time. We call such a "permanent" lowering of the site symmetry the static Jahn-Teller effect. If, on the other hand, the distortion averages out to zero in the characteristic time of the experiment, we have a dynamic Jahn-Teller effect. The same system can, in principle, show a static effect in an experiment with a short characteristic time, such atj spin resonance, and yet show no distortion (or a dynamic effect) in an experiment with a relatively long characteristic time, such as NMR. Another effect we might expect to see is the difference between the vibronic energy levels and the vibrational levels the system would have if there were no Jahn-Teller effect. This aspect has attracted a lot of attention in the case of rnolecule~."~ For an impurity or a defect in a solid, we only have a very rough idea of what the vibrational levels should be, and we can only expect to be able to identify such effects on the vibronic spectra if they are gross. For instance, splittings of broad optical absorption bands, in which the detailed vibrational structure is already washed out, can be characteristic of the Jahn-Teller effect (see Part IV) . Finally, we might expect to see the effect on electronic operators of replacing electronic degeneracy by vibronic degeneracy. Matrix elements of perturbing operators have to be calculated using the vibronic wave functions P! rather than the electronic wave functions t) [see Eq. (3.9)]." This can drastically affect the values of the matrix elements. It turns out that in many cases this is the most powerful method for detecting the Jahn-Teller effect in solids (see Sections 15 and 28). II. The Jahn-Teller Effect in Doubly Degenerate Electronic States

5. THESTATIC PROBLEM IN AN OCTAHEDRAL COMPLEX Let us first consider the simplest case of an octahedrally coordinated transition metal ion with an E (doubly degenerate) ground term.'&@ Examples are Cu2+ and Ni+ (d9 configuration, i.e., a single hole in the d shell) ; Mn3+ and Cr2+ (weak field d4, i.e., t23e) ; Ni3+ and Pt3+(strong U. Opik and M. H. L. Pryce, Proc. Roy. SOC.A238,425 (1957). W. Moffitt and W. Thorson, Phys. Rev. 108, 1251 (1957). 8o A. D.Liehr and C. J. Ballhausen, Ann. Phys. ( N . Y . ) 3 , 304 (1958).

7O

116

M. D. STURGE

field G?, i.e., tz6e). Because e orbitals can form u bonds in octahedral coordination, they are particularly sensitive to the nuclear positions, and strong Jahn-Teller effects may be e ~ p e c t e d . ~ 3On J ~the other hand, orbital momentum is quenched in an E term, and the complicating effects of spin-orbit coupling can be neglected for most purposes. As discussed in Section 2, the distortions of an octahedron can be analyzed into 15 normal modes, described as m g, e g , nu, 7lUa,nub , ~2~ , according to their symmetry. Of course, in the solid they are no longer normal modes, since the octahedron interacts strongly with its surroundings. The analysis into these modes can still be made, however, and, insofar as nearest-neighbor interactions predominate, it is a useful one. Since [Ez] = A1 El T~~ distortion does not split an E term in first order. Remembering that the totally symmetric distortion alg can shift but not split the term, we are ‘left with only the doubly degenerate e g mode to worry about. The electronic term has two orbital states lLu and +v , transforming as (2.9 - 9 - y2) and 31‘2(52- y2) , respectively. The normal coordinates Q3 and Q2 transform in the same way (see Table 11).The Hamiltonian for the electron-lattice coupling is, to first order in the Q’s [see Eq. (3.6)],

+

~ ( 1 )=

-A

[

-Q3

Q2

Qz Q3

1.

(5.1)

(Note that there is only one independent parameter, because of the Wignerthe electronic wave Eckart theorem.) This acts on the state vector b+y . 2A is the splitting of the E term by unit function being # = a$u distortion. The nuclei move in a potential V(Q2 , Q3) that is the sum of the ordinary “quasi-elastic” potential (due to bonding electrons, closed-shell repulsion, long-range Coulomb interactions, etc.) and the d electron energy 39). This statement, which is the Jahn-Teller analog of the Born-Oppenheimer theorem, is only approximately true but is adequate for our purposes (see Section 3). Adding the nuclear kinetic energy T , = (Pz2 P32)/2p to V , we arrive a t the first-order vibronic Hamiltonian:

[;I,

+

+

[ ]+ -Q3

XI = - A

Qz

Q2

3pwf2(Qz2

+

Q39

+ T,.

(5.2)

Q3

(Pk is the momentum conjugate to Q k .) Here we have assumed a harmonic quasi-elastic potential, in which the eu vibrations have a frequency w. and an effective mass p . (For an isolated octahedron, p is the ligand mass.)

T H E JAHN-TELLER EFFECT I N SOLIDS

117

V

t

FIG.6. The potential surface V ( Q 2 , Q3) for a doubly degenerate state in the lowestorder theory [Eq. (5.4)].

A2/2pfiw.3 is a dimensionless measure of the strength of the Jahn-Teller

coupling, relative to the quasi-elastic forces. It is convenient to use polar coordinates Writing Q3 = p cos 0, Qz = p sin 0, we have XI = -Ap

[

sin0

sin0

cost9

If the nuclei are regarded as fixed58(i.e., V , where eigenvalues E = T,

+

=

AAp

0) in the (Q&)

]+

-cost9

V

(p,

p

+ T,.

plane.

(5.3)

and 0 are given), XI has the

+ &.02pz.

(5.4)

V(p, 0) is the (double-valued) potential surface on which the nuclei move. Note that in the present approximation V is independent of 0. V has a minimum value of -6E = -A2/2pw,2 (the “Jahn-Teller energy”) at a radius po = I A l/pw,2. The surface V(p, 0 ) , commonly known as the “Mexican hat,” is illustrated in Fig. 6.s1 It is simply the surface generated by rotating the parabola (5.4) about the p = 0 axis. The electronic wave function with the nuclei “fixed,” on the lower branch of V(p, O ) , is (for A > 0) $- = $u sin $0

+

$v

cos 40,

(5.5a)

W. Moffitt and W. Thorson, in “Calcul des fonctions d’onde moleculaire” (R. Daudel, ed.), p. 141. Rec. Mem. Centre Nat. Rech. Sci., Paris, 1958.

118

M. D. STURGE

and, on the upper branch,

J.+

=

J.,, cos $0

- J.v

sin $0.

(5.5b)

For A < 0, these two equations should be interchanged. A is positive for a single e hole (as in Cu2+) and negative for an e electron. The vibronic problem, where the nuclei are not regarded as fixed, is taken up in the next section. Note the dependence of J. on 0/2 rather than 0. The physical significance of this dependence can be seen by considering a vertical section through Fig. 6, for instance, the section containing 0 = 0. This section shows two branches, one corresponding to J. = k ,the other to J. = J.u . It is possible to go continuously from one branch to the other by rotating through 180" about p = 0. Thus, the coeficients of fiU and J.u go from 0 to 1 (or vice versa) as 0 goes from 0" to 180";they must, therefore, be functions of 0/2. It follows that J. is double-valued, changing sign on a 360" rotation; the nuclear function must therefore also be double-valued (since the overall wave function must be single-valued) . As in the theory of electron spin,S2 this double-valuedness leads to half integral values of the quantum number j , which is associated with rotational (0) motion in ( p , 0) spaces3(see the next section). The physical reason for the cylindrical symmetry of the V ( p , 0) surface is as follows. The line 0 = 0 corresponds to Q2 = 0, Q 3 = p, that is, a tetragonal distortion along the x axis, transforming as (22 - x2 - y2). The line 0 = 120", for which &3 = - $ p , Q2 = (3l/2/2)p, can with a little algebra be seen to correspond to a tetragonal distortion along the x axis, transSimilarly, 0 = - 120" corresponds to tetragonal forming as (2z2 - y2 - 9). distortion along the y axis. Since the Hamiltonian has overall cubic symmetry, these three distortions must be physically equivalent : V ( p , f120') = V ( p , 0). This argument can easily be extended to show that the V ( p , 0) surface must have threefold symmetrys (to be precise, C3usymmetry with the threefold axis along p = 0 ) . Now let us make the further assumption, embodied in Eq. (5.2), that the system is harmonic and the Jahn-Teller interaction linear. Then V ( - p , 0) = V ( p , O ) , because the slopes of the two branches at p = 0 are equal and opposite. Thus, V ( p , 0) has at least sixfold symmetry about p = 0 ; but, since no terms higher than cosz 0 or sin2 0 appear in the Hamiltonian, such symmetry is only possible if all terms in 0 vanish. It is now obvious that anharmonic terms, by making V ( - p , 0) # V ( p ,O), destroy the cylindrical symmetry of V , leaving only the threefold 82

83

Chapter 6 of Griffith22. H. C. Longuet-Higgins, U. Opik, M. H. L. Pryce, and H. Sack, Proc. Roy. Soc. A244, 1 (1958).

THE JAHN-TELLER

EFFECT IN SOLIDS

119

symmetry required by the cubic symmetry of the Hamiltonian. Equations (5.5) for the electronic wave functions still hold, to a good approximation. A schematic contour map of V(p, 0) when anharmonic terms are included is shown in Fig. 7. There are three potential wells, corresponding to distortion along the x, y, and x directions, separated by saddles. The height of these saddles, depending as it does on anharmonic effects, is very variable but typically may be 200 cm-1.?8Although this is small compared with the Jahn-Teller energy 6E of (say) 3000 cm-', it is considerably greater than kT at low temperatures. Thus, we may well find that the system can be frozen into one of the valleys, leading to a permanent tetragonal distortion of the octahedron along one of the cube directions. This is called the static Jahn-Teller effect. If the Jahn-Teller active ions are far enough apart not to interact, the direction of the distortion will vary at random from ion to ion. (Concentrated crystals, in which cooperative Jahn-Teller distortions can occur, are discussed in Section 13.) At high temperatures,

T"

FIG.7. Schematic contour map of V ( Q 2 , Q 8 ) when anharmonic terms are included (after Herzberg").

120

M. D. STURGE

9

R (INTERNUCLEAR SEPARATION)

FIG.8. A typical internuclear potential curve.

thermal activation over the potential barrier that separates the valleys can occur, and the time-averaged symmetry of the surroundings of any particular ion will be cubic. Such averaging, by whatever means achieved, we call the dynamic Jahn-Teller effect. Clearly, the vibronic energy levels of such a system will be quite complicated (see the next section) ; fortunately, in solids we can usually use physical approximations that are adequate for our purposes. The sign of the tetragonal distortion to be expected is of some interest. It clearly depends on the sign of the anharmonic term in the internuclear potential. It is easy to see that if central forces are predominant the sign of this term will be such as to make Q 3 > 0 preferred, as illustrated in Fig. 4c. This is because a typical internuclear potential curve looks like Fig. 8; for finite displacements from the equilibrium position, the potential is greater when the internuclear separation is decreased than when it is increased. Thus, the Q3 > 0 distortion, in which two ligands move out and the other four move in a smaller distance (the mean internuclear separation remaining unchanged), requires less energy than the Q 3 < 0 distortion. In a tetrahedral complex, it is the ligand-ligand distances that change in an e distortion (see Fig. 5c). I n this case, anharmonic forces make Q3 < 0 the stable distortionla but their effect should be small. It has been pointed O U that ~ the ~ preceding ~ ~ argumenP ~ neglects second-order Jahn-Teller coupling; that is, it assumes the electronic energy to be linear in the nuclear displacements. Quadratic coupling terms in QZ and Q3 have the same qualitative effects as lattice anharmonicity in destroying the cylindrical symmetry of V ( p , O ) , since they destroy the 84

J. B. Goodenough, J . Phys. Chem. Solids 26, 151 (1964).

THE JAHN-TELLER

EFFECT I N SOLIDS

121

equivalence of V (+ p ) and V ( - p ) and introduce a dependence on cos 30. On a point-charge calculation, quadratic terms make Q3 < 0 (contraction) the stable distortion.m This is because electrostatic interaction falls off faster than linearly with internuclear separation. Thus, Q 3 > 0 produces a smaller Jahn-Teller splitting than Q 3 < 0, and the latter distortion is more stable. This argument holds only for the a2V/aQ2term (see Section 3 ) . Secondorder terms due to Jahn-Teller coupling with other electronic states can give either sign of Q3 . There are no terms within the d‘, d4, or d9 configurations which are coupled to the ground E term by E distortion. On the other hand, when covalency is taken into account, ligand orbitals are mixed into the d orbitals. Second-order terms can now have either sign.% In the case of Cu2+, valence bond arguments86 favor a square planar configuration, i.e., Q3 > 0. This is supported by a crude molecular orbital calculation on the CuFe4- c ~ m p l e x ~ which ~ . ~ a predicts a n elongation of This calculation also predicts appreciable coupling between cg and alg distortion, leading to a radial expansion of about 5%. All compounds containing octahedrally coordinated ions with E ground terms, in which there is a cooperative Jahn-Teller effect (see Section 13), show elongation rather than contraction along the tetragonal axis of the octahedron. The same is true for most isolated transition metal ions showing the static Jahn-Teller effect. (The sign of the distortion can be determined in spin resonance from the sign of 911 - g 1 , which depends on whether the electronic wave function is predominantly $u or $v .89,90) A possible exception to this generalization is Ni+ in LiF and NaF.gl Among the many spin resonance spectra of this ion, the one that is believed (from its annealing behavior) to arise from a “cubic” site unassociated with any defect has a predominantly $u ground state. For a hole, this implies Q3 < 0, i.e., contraction. M. H. L. Pryce, K. P. Sinha, and Y. Tanabe, Mol. Phys. 9, 33 (1965). L. Pauling, “The Nature of the Chemical Bond,” 3rd ed., pp. 153-161. Cornell Univ. Press, Ithaca, New York, 1960. 87 C. J. Ballhausen and H. Johansen, Mol. Phys. 10, 183 (1966). L. L. Lohr (Znorg. Chem., to be published) has made a more elaborate MO calculation on the CuCla4- complex. No additional “quasi-elastic” forces had to be assumed, these arising automatically from the interaction between the valence orbitals. H e finds that the principal term stabilizing the elongated configuration is the second-order interaction of the 3d(3z2-r2) orbital with the 4s orbital of Cu*+. The statement in Ballhausen and Johansen87 that this agrees with the observed 6% distortion of KZCuFa is incorrect, since in this case the octahedron is compressed [K. Knox, J. Chem. Phys. 30, 991 (1959)l. The distortion here is probably not due to the Jahn-Teller effect (see Section 13). 8 9 A. Abragam and M. H. L. Pryce, Proc. Phys. Sac. (London)A63, 409 (1950). A. Abragam and M. H. L. Pryce, Proc. Roy. Sac. A206, 135 (1951). 91 W. Hayes and J. Wilkens, Proc. Roy. Sac. A281, 340 (1964). 86

122

M. D. STURGE

6. THEDYNAMIC PROBLEM: VIBRONIC ENERGY LEVELS The general problem of calculating the vibronic energy levels of a doubly degenerate electronic state interacting with a doubly degenerate These calcuvibrational mode has engaged a number of authors.65J2~81,83,92 lations have also been extended to include higher-order terms that can lead to a static Jahn-Teller e f f e ~ t . ~ ~ ~ ~ Only a general outline of the calculation will be given here. We start from Schrodinger’s equation:

~ , ~Eq. ~ and V ( p ) is given by Eq. (5.4). We take as a trial f u n c t i ~ n[see (3.9)1 = *+cp+ *-cp1 (6.2)

*

+

and #+ and t,b- are given by (5.5). Note that, although the 9’s are orthonormal, the cp’s are not. Because V is independent of 8 in the present approximation, we may write cp5 = f,eiie, where j must be half-odd integral Substituting in Eq. (6.1) and integrating in order to make P! single-~alued.~~ out the electronic wave functions, we find that the f’s obey the coupled equationsg2

(6.3) The last term on the left couples f+ and f- , thus connecting the two sheets of the potential surface. If we neglect it, nuclear motion is confined to one sheet only, and the vibronic wave function is simply a Born-Oppenheimer product, #+cp+ or J / - c p - . This is a good approximation when the sheets are well separated. Under such circumstances, we usually find that the a / a p term is not very important, and, dropping it, we obtain the onedimensional Schrodinger’s equationV2:

- (fi2/2P) (aY-/aP2>

+Vh)f

=

Ef,

(6.4)

C. W. Struck and F. Herefeld, J . Chem. Phys. 44, 464 (1966). M. C. M. O’Brien, Proc. Roy. SOC.A281, 323 (1964). 94The quantum number j is only good in the approximation that V has cylindrical symmetry. It is variously called m,1, or )6 in the literature.

92

93

123

THE JAHN-TELLER EFFECT IN SOLIDS

where V j = AAp

+

Qpo2p2

+ V ( j 2+ t ) / 2 p p 2 .

(6.5)

The term in jz represents the nuclear kinetic energy due to rotation in (&2&3) space. Equation (6.4) is a good approximation for low-lying states (+L) in a strongly coupled situation,93since in such circumstances p po >> (fi/po)112, the rms amplitude of zero point motion. The term in j 2 is small in this case, and the solutions of Eq. (6.4) are harmonic oscillator wave functions centered about p = PO , depressed in energy by 6E = A2/2p02, and split into “rotational” levels by the j z term. Sl0ncewski7~shows that Eq. ( 6 . 4 ) is also a good approximation for highly excited states ((o+). A typical V j ( p ) curve (for j = 4) is sketched in Fig. 9. Besides the minimum in the lower branch a t p = po , where the “Jahn-Teller force” A is balanced against the ‘‘elastic force” -po2p, there is also a minimum in the upper branch at p = pm , where - A is balanced against “centrifugal force.” Usually pm << po , in which case we have po = A / p 2 (as before) and pm (h2/4pA)’I3. Because of the rapidly varying curvature in the vicinity of pm , vibrational levels in this region are widely and irregularly spaced. Sloncewski finds that, even in a solid, states near the upper branch minimum should be rather stable, decaying relatively slowly to the lower branch. It has been found possibleg5to fit the spin-resonance linewidths 6H of a number of octahedrally coordinated ions with 2E ground terms to an

-

+

FIG.9. The effective potential Vi(pj, f o r j

=

4 [Eq. (6.5)].

U. T. Hochli and K. A. Miiller, Phys. Rev. Letters 12, 730 (1964).

124

M. D. STURGE 0

-

“Orbach relaxation’196type of formula: 6H cc exp ( - W / k T ) , with W 1000 em-’. Since W is of the expected order of magnitude of the JahnTeller energy 6E, it has been suggestedg7that it represents the energy above the ground state of Sloncewski’s lowest centrifugally stabilized state. There is no experimental evidence other than this for the existence of these states or for their metastability.. For most practical purposes, we can ignore the centrifugal term in V j ( p ) (but see Section 23). I n the region where the two sheets of the potential surface approach each other, nuclear motion cannot be confined to one of them. The vibronic wave function now has the form (6.2). The eigenvalues of the coupled Schrodinger equation have been obtained numerically and t a b ~ l a t e d ~ l ~ ~ ~ ~ for a wide range of values of the coupling parameter A2/pliu3 (called 2 0 or k2 in the literature). When higher-order terms (anharmonicity and quadratic Jahn-Teller coupling) are included in the Hamiltonian, j ceases t o be a good quantum number. Because V ( p , 0) still has threefold symmetry, j’ = j modulo 3 remains a good quantum number.98Because of this, only levels with j = &$ are split in first order (see, for instance, Fig. 10). The effect on the energy levels of these higher-order terms has been c a l ~ u l a t e d we ~ ~ reproduce ~~; in Fig. 10 O’Brien’s results for the lower levels of a strongly coupled system [in the region where Eq. (6.4) holds]. When V ois large, the lowest E and A states coalesce to form a triplet, the three resulting states corresponding to distortion parallel to the 2, y, and z directions. There is a large excitation energy (of order V o )to the first “rotational” state. These “exact” calculations not only make precise predictions of the vibronic energy levels of simple molecules, but are also very helpful in understanding the qualitative consequences of the dynamic Jahn-Teller effect in solids. However, they cannot be expected to provide a quantitative prediction of the energy levels in a real solid, since the vibrational spectrum “ seen” by a n impurity ion is in general quite complicated, even in the absence of the Jahn-Teller effect. It may happen, of course, that coupling to a single local or quasi-local mode of the required symmetry is dominant in certain cases. The only reported observation of the individual vibronic levels arising R. Orbach, PTOC.Roy. SOC.A264, 458 (1961); C. B. P. Finn, R. Orbach, and W. P. Wolf, PTOC. Phys. SOC.(London) 77, 261 (1961). Note that the Orbach theory refers to spin-lattice relaxation rates, not to line widths, and the two are not necessarily related. 97 U. T. Hochli, K. A. Muller, and P . Wysling, Phys. Letters 16, 1 (1965). 98 The quantum number j’ = j modulo 3 is exactly analogous to the “crystal quantum number” in a system with a threefold axis of symmetry. See, for instance, E. Fick and G. Joos, in “Handbuch der Physik” (S. Flugge, ed.), Vol. 28, p. 205. Springer, Berlin, 1957 [especially their Eq. (22.5)].

96

THE JAHN-TELLER EFFECT IN SOLIDS

O V 0

I

10

I 20

125

I

30

"o/~o

FIG.10. Lower vibronic energy levels of an E state as a function of the anharmonic term (from O'Brieng3). The unit of energy is eo = K / 2 p p 0 2 = (hu)2/46E and is typically of order 10 cm-l. V Ois the height of the potential barrier. The levels on the left (counting respectively. Note how only t h e l j I = % from the bottom) have I j I = 8, $, $, 4, level is split in first order by the perhrbation.

y,

from the dynamic Jahn-Teller effect in a solid is that of Struck and Hersfeld.92They report a complicated absorption band associated with the 4f + 5d optical transition of Ce3+in CaFz . The excited state is a 2E state and should be subject to a Jahn-Teller effect in a cubic site. The observed energy levels can be fitted quite well to the eigenvalues of (6.3), although the relative intensities of the various transitions cannot be accounted for. The interpretation in terms of a Jahn-Teller effect has been criticizedsgon the grounds that there is good evidence that the spectrum arises from Ce3+ at a site tetragonally distorted by local charge compensation. I n such a site, the degeneracy of 2E is lifted, and the Jahn-Teller effect cannot occur. It is therefore argued that the vibrational structure is that of the crystal (modified by the presence of the impurity) and has nothing to do with the D. Sturge and M. H. Crozier, J. Chem. Phys. (1967)46, 4551. See also A. A. Kaplyanskii and V. N. Medvedev, Opt. i Spektroskopiya 18, 803 (1965); see Opt. Spectry ( U S S R ) (English Trartsl.) 18,451 (1965).

99M.

126

M. D. STURGE

Jahn-Teller effect. This view is supported by the observation of similar structure in emission.@ ' '

7. THE TRANSITION FROM STATICTO DYNAMIC JAHN-TELLER EFFECT: NARROWING MOTIONAL Now let us consider the situation at very low temperatures, where to a first approximation we can regard the complex as frozen into one of the three equivalent valleys. For definiteness, let us consider an e hole ( A > 0) frozen into the e = 0 valley (corresponding to distortion along the z direction). Then the electronic wave function is given by (5.5a) with 0 0. If the system remains strictly at 0 = 0, J. = J.. ; but zero point motion makes sin2 812 > 0, and so there is, on average, a nonzero proportion of #u present as well. This admixture of J.u manifests itself most clearly in the superhyperfine structure (SHFS) observed in spin resonance. SHFS arises from interaction with the ligand nuclei and provides a measure of the overlap of the central ion wave function onto the ligands.'O' Ni+ in LiF can occupy a site where the ground state (as determined from the g factor) is predominantly J.. .91 Since J.v transforms as (x2 - y2), it can only overlap the four ligands lying in the (2,y) plane, perpendicular to the local axis of tetragonal distortion. However, in fact there is a substantial SHFS from the axial ligands. This can only arise from admixture , transforming as (2z2 - x2 - y2). (It is perhaps a moot point whether of this can really be called a Jahn-Teller effect, since in this particular site Ni+ is apparently associated with a defect that presumably raises the degeneracy of 2E.)At another site, admixture of J.v into J.u causes the g value to deviate from its calculated value. It is not possible to estimate sin 20/2 from the data (but it certainly seems to be larger than O'Brien'sg3 theoretical estimate of 0.01). Similar effects of vibrational mixing are observed in the temperature dependence of the g values of Cu2+in tetragonally distorted s i t e P ; as the temperature is increased, more tLU is mixed into J.. , and the g values change in consequence.'O3 The effect of thermal activation over the barrier, producing a transition from the static to the dynamic Jahn-Teller effect as the temperature is

-

A. A. Kaplyanskii, V. N. Medvedev, and P. P. Feofilov, Opt. i Spektroskopiyu 14, 664 (1963); see Opt. Spectry. (USSR) (English Trunsl.) 14, 351 (1963). See also J. V. Nicholas, Phys. Rev. 166, 151 (1967). lol J. H. E. Griffiths, J. Owen, and I. M. Ward, Proc. Roy. SOC. A219, 526 (1953) ; M. Tinkham, ibid. A236,535 and 549 (1956). lo2 T. Rrtmasubba Reddy and R. Srinavasan, Phys. Letters 22, 143 (1966). 103 The anisotropy of the spin resonance spectrum of a ZE term in a distorted octahedron arises from the admixture of orbital momentum from excited terms by the spin-orbit coupling. This causes g 11 and g 1 to deviate from 2, in general by different amounts, and differently for $, and $v ) . a 9 3 3 loo

THE JAHN-TELLER

rAw u

127

EFFECT I N SOLIDS

I

rAw‘

FIG.11. Sketch of the way an anisotropic spectrum collapses int,o an isotropic one as the reorientation time is reduced.

raised, has often been observed in the spin resonance of isolated impurities and defects. It is perhaps the most convincing evidence for the occurrence of a Jahn-Teller effect when other causes, such as phase transitions of the host crystal, are ruled out. The transition with temperature can be visualized as the “motional narrowing” of an initially anisotropic spectrum when transitions between different directions of distortion become rapid relative to the microwave ~ p 1 ittin g .l~ ~ For the moment, we shall assume that thermal activation is the only way the system can reorient. Then the reorientation time T is (at least in order of magnitude) the same as the phase memory time, and conventional motional narrowing theory can be u ~ e d . ’ ~ ” (See ’ ~ ~ Englman and Hornlo’ for a more thorough discussion.) For simplicity, consider the case where there are two possible distortions, in which the g factors (for a given direction of magnetic field) are g1 and g2 . Then, if TAW >> 1 [Aw = (gl - g2)PH/z1],two g2)/2. lines will be seen; if TAW << 1, only one line will be seen at (gl The way the spectrum should vary with T is qualitatively illustrated in Fig. 11. Using the quantitative theory, one can determine T from such spectra. For thermal activation over a barrier 7 follows the Arrhenius equationlo8: T = T~ exp (Vo/k!P) (7.1)

+

lo4 lo5 lo6

lo*

H. S. Gutowsky, D. W. McCall, and C. P. Slichter, J . Chem. Phys. 21, 279 (1953). H. M. McConnell, J . Chem. Phys. 28, 430 (1958). A. Abragam, “The Principles of Nuclear Magnetism,” Chapter 10, Oxford Univ. Press, London and New York, 1961. R. Englman and D. Horn, in “Paramagnetic Resonance” (W. Low, ed.), Vol. 1, p. 329. Academic Press, New York, 1963. In this theory, phase memory is not completely lost on reorientation, with the result that the isotropic and anisotropic spectra can coexist over a range of temperature (see the next section). There appears to be some doubt as to whether this should not be T cc T-I exp (Vo/kT) [R. H. Doremus, J. Chem. Phys. 34, 2186 (196l)l; the difference is rarely significant [but see G. D. Watkins and J. W. Corbett, Phys. Rev. 134,A1359 (1964)l.

128

M. D. STURGE

where V Ocan be identified as the height of the potential barrier separating different directions of distortion (see Fig. 7 ) . The transition to the static Jahn-Teller effect a t low temperatures was first observed in ZnSiF6-6HzO:Cu2+by Bleaney and Bowers2; previous to this observation, Abragam and Prycesg had pointed out that the value of the isotropic g value found for Cu2+ a t room temperature in a variety of octahedral systems could only be explained in terms of a dynamic JahnTeller effect. The effect was also found in other crystals10gin which, as in ZnSiF6.6Hz0, the sites occupied by Cuz+ have trigonal symmetry, so that the 2E ground term is not appreciably split by the crystal field. In all these crystals, where the ligands are water molecules, the transition from the static to the dynamic effect occurs in the range 1O0-5O"K. Subsequent investigations have revealed the corresponding transition in A1203:Ni3+, CIA$+, and Pt3+,lloin YAG:Ni3+,I11and in CaF,:Y2+.112 In all these papers, the t,ransition with temperature is described only qualitatively. Quantitative measurements, from which reorientation activation energies Vo in the range 0.01 to 0.06 eV were deduced, have been made on a number of defects in ~ i l i c o n l ~ ~(see - l l ~Section 19). Quantitative measurements have also been made on CaO :Cu2+ and CaO :Ni+,l18and on AlZO3:Ni3+.l19In the latter case, although agreement of the spectrum with motional narrowing theory is not very good, the values of T ( T ) deduced from the temperatures a t which the spectrum collapses (TAW 1) for different values of AW are in quite good agreement with measurements of T by acoustic loss in the same system (see Section 9). More complicated effects due to tunneling, which will be discussed in the next section, have been seen in MgO :Cu2+,120,121 SrFz:Sc2+,and CaF,: Sc2+.lZ3 AgCl: Cu2+,122

+

B. Bleariey, K. D. Bowers, and It. S. Trenam, Proc. Roy. Soc. A228, 157 (1955). S. Geschwind and J. P. Remeika, J. A p p l . Phys. 33, 370 (1962). F. R.. Merrit,t, private communication (1965). YAG is short for yttrium aluminum garnet,. 112 J. R. O'Connor and J. H. Chen, A p p l . Phys. Letters 6, 100 (1964). 113 G. 1). Watkins and J. W. Corbett, Phys. Rev. 138, A543 (1965). 114 G. D. Watkins and J. W. Corbett, Phys. Rev. 134, A1359 (1964). 115 G. 1). Watkins and J. W. Corbett, Phys. Rev. 121, 1001 (1961). J. W. Corbet,t, G. D. W:tt,kins, R . M. Chreiiko, and R. S. Mcllonald, Phys. Rev. 121, 1015 (1961). 1' G. W. Ludwig and 11. H. Woodbury, Solid State Phys. 13, 223 (1962). 118 W. Low arid J. T . Suss, Phys. Letters 7, 310 (1963). n9 F. R. Merritt arid M. 1). Sturge, Bull. Am. Phys. SOC.[a] 11, 202 (1966). lZo J. W. Orton, P. Auxins, J. H. E. Griffiths, and J. E. Wert,x, Pmc. Phys. Soc. (1,072d o n ) 78, 554 (1961). lZ1 R. E. Coffman, Phys. Letters 19,475 (1965); 21, 381 (1966); J . Chem. Phys., in press. lz2 D. C. Buriiham, Bull. Ant. Phys. Soc. [ a ] 11, 186 (1966). Iz3 U. T. Hochli and T. L. Estle, Phys. Rev. Letters 18, 128 (1967) ; U. T . Hochli, t,o be published. log

THE JAHN-TELLER

EFFECT IN SOLIDS

129

In principle, similar information on the static Jahn-Teller effect to that provided by spin resonance can be obtained from the quadrupolar splitting in the Mossbauer e f f e ~ t . 1(This ~ ~ splitting must vanish in cubic symmetry.) Such splittings have not been seen for an octahedrally coordinated ion in an E state. They have been seen in tetrahedral Fe2+(5E)125 (see Section 13) and in octahedral Fe2+(5Tz)126 (see Section 1Ti).

8. TUNNELING AND SPINRESONANCE AT LOWTEMPERATURES Thermal activation over the barrier is not the only way the anisotropic spectrum can be averaged to make an isotropic one. It is also possible for the complex to go over from one potential minimum to another, even a t zero temperature, by quantum-mechanical tunneling. This is because the wave functions describing the system as localized in one or another valley are not orthogonal and are therefore not true eigenfunctions. Let us write the lowest vibronic states in each valley as Born-Oppenheimer products, good zeroth order wave functions when the motion is well localized in one va11ey12~J28J28a : \ki = cpil+bi

(i = 1, 2 , 3 ) .

(8.1)

Here \kl is the vibronic wave function for the valley corresponding to distortion in the x direction, and \kz and \k3 correspond to distortions in the y and x directions, respectively. The electronic wave functions l+bi are given by (5.5a), in which we approximate 0 by e, = 2 ~ / 3 4n/3, , 0 (i = 1, 2, 3). The vibrational wave function p i is the ground state wave function of a two-dimensional oscillator centered on the point ( P O , 0,) in (Q2&3) space. If we treat this as a harmonic oscillator, we have cpi

=

exp - $ [ a 2 ( q i -

(aa’/?r)1’2

po)2

+

a12q;’2].

(8.2)

Here pi, qi’ are the mutually orthogonal coordinates defined in terms of and &3 in Table IV. The corresponding frequencies are w , w ’ ; a+ = (fi/pw)l’Z, al-1 = ( f i / ~ w ’ ) represent ~/~ the rms deviation in the radial and tangential directions of Q2Q3 space. Although w w e , the frequeiicy of the

&z

+

G. K. Wertheim, “Mossbauer Effect,.” Academic Press, New York, 1964. M. Tanaka, T. Tokoro, and Y. Aiyama, J . Phys. SOC.Japan 21, 262 (1966). lZ6 D. N. Pipkorn and H. R. Leider, Bull. Am. Phys. Soc. [a] 11, 49 (1966). lZ7 V. I. Avvakumov, Zh. Eksperini. i Teor. Fiz. 37, 1017 (1959) ; see Soviet Phys. J E T P (English Transl.) 10, 723 (1960). lZ8 I. B. Bersuker, Zh. Eksperim. i Teor. Fiz. 43, 1315 (1962) ; see Soviet Phys. J E T P (English Transl.) 16, 933 (1963). lZ4

lZ5

Equation (8.1) is only correct if zero point motion can be neglected. A more accurate zeroth-order function would be: q1=pl[$l cos +(e-Oo)+(l/d3)($q-+~) sin+(e-eo)l, etc.

130

M. D. STURGE

TABLE IV. NONORTHOGONAL SYMMETRY COORDINATES FOR e DISTORTIONS q 1

=

q 2

=

-+

-t

Q 3

+ *(3)’/’&z

Q 3 - )(3)1/2Qz

4 3 = Q3 q l l = -1 Q z 4(3)1/2Qa q ’ 2 = -4 QZ - ) ( 3 ) 1 / z Q 3 4 3 = Qz

+

transforms as 2x2 - yz - 22 transforms as 2yz - 9 - 2 2 transforms as 222 - 5 2 - y 2 transforms as 3 1 / 2 ( y Z - 9 ) transforms as 3 1 / 2 ( 2 z- 2 2 ’ ) transforms as 3 1 / 2 ( 2 2 - y2)

mode in the absence of Jahn-Teller distortion, is approximately given by

e,

w’/w =

w’,

will be much less and

$ (Vo/GE) 112,

(8.3)

where Vo is the height of the barrier above the minimum, and 6E is the Jahn-Teller energy. The rate of tunneling is most easily calculated by the WKB method. As in the inversion of the NH3 the tunneling rate in the lowest vibrational state is given approximately by

w

=

w‘ I r ~ / f i= 2ir

[ 2 p ( V ( ~)

where the integration is along the path of minimum V. This path is close to the curve p = po , along which (to a first approximation) we may put

v = gvo(i - ~ 0 ~ 3 0 ) .

(8.5)

Then we havelZ7

where cos 300 ated :

=

1 - fiw‘/Vo. When

w

=

jlw’

<< V o, the integral can be evalu-

(w’/2ir) exp ( -2V0/fiw‘).

(8.7)

This formula is approximately correct even for low barrier height if we replace Vo by ( Vo - +jlw’). A more accurate calculationg3treats the problem as one of hindered rotation in Q2Q3 space. The energy levels so obtained are illustrated in Fig. 10. The splitting of the lowest (triply degenerate) level into a doublet and a singlet as the barrier height is reduced is the “tunneling splitting.” 159

C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy,” p. 302. McGrawHill, New York, 1955. For a general account of the WKB method, see L. I. Schiff, “Quantum Mechanics,” 2nd ed., p. 184. McGraw-Hill, New York, 1955.

THE JAHN-TELLER

EFFECT IN SOLIDS

131

However the tunneling splitting is calculated, the lowest (threefold degenerate) state is split into a orbital singlet at -2r, with the wave function

I A)

= (*I

+ + q-2

*3)/31/2,

r, with wave functions I Eu)= (\kl - \kz)/21/2, I Eb) = (2q3 - \kl - \k2)/6‘/’. Here the Ws are given by Eq. (8.1). Note that r

(8.8a)

and a doublet at

(8.8b)

is proportional to the overlap between the Ws. One of these overlaps is negative, the electronic I 1LZ) being -$. Hence r is negative, putting the “antisymintegral metric” E states below the “symmetric” A state (see Fig. 10). (This can also be regarded as a direct consequence of the fact that the nuclear wave function is double-valued, changing sign as e is increased by 2n. The antisymmetric combinations therefore have fewer nodes than the symmetric combination.) The spin resonance spectrum to be expected a t 0°K in a Jahn-Teller distorted 2E state has been calculated by a number of authors.93J21J23J30 It is necessary to take into account spin-orbit interaction with an excited T state (the 2T2state in a d1 or d9 system) in order to get any deviation from the isotropic spin-only spectrum. In the case where the tunneling splitting 3r is large compared with the Zeeman splitting gPH, we can deal with the E state and A state independently. The A state has an isotropic g factor: (8.9) g(A) = 2 +4u, whereas the E state shows cubic anisotropy :

g(E)

=

2

+ 4u f 2u{l - 3(t?2nz2+ m2n2+ n2P))l/z.

(8.10)

Here u = -A/A and is a measure of the spin-orbit mixing into the 2E term of the 2Tzterm an energy A above it; I, m, n are the direction cosines of the magnetic field vector. Equations (8.9) and (8.10) are to be compared with the cubic field results for a Kramers doublet: (8.11) g ( r 8 ) = 2 + 4 u f 4 u { l -3(Z2m2+m2n2+n2Z2)).

(8.12)

Note the change in coefficient of the anisotropic term. For a tetragonally distorted site, we would have89*90 911

=

2

+ 4u f 4u,

g1

=

2

+ 4u =I= 2u,

(8.13)

132

M. D. STURGE

xfz B 2u

Vo (ARBITRARY UNITS)

0

FIG.12. g factors for H 11 [l00] in a 2E state: (a) for the lowest vibronic level; (b) for a highly excited level. At the midpoint of the abscissa in (a), the tunneling splitting is roughly equal to the Zeeman splitting. The thickness of the lines indicates the transition probabilities (from O’Brien93).

where the sign is determined by the sign of the distortion. (The upper sign applies to Cu2+ in most instances.) Equations (8.13) are also obtainedg3J30in the limit r << 4pLpH and correspond to a static Jahn-Teller effect. The general problem of calculating the spin-resonance spectrum arising from one of the states of Fig. 10 has been treated by O’Brieng3and Bersuker.130O’Brien finds that to lowest order in u,for H 11 (loo), 9

A

=

2

+ 4 ~ ( +i (COSe ) ) , + (g - 2) - (3+ 6u/7) (cos e)].

= P[-K

(8.14) (8.15)

Here A is the hyperfine interaction parameter, P and K are constants, (cos 0 ) is the “spatial” average typically 300 G and 0.28, re~pective1y.l~~ (over &z and &3) for the states of interest. For instance, in the static limit, 0 takes the values 0, 2 ~ / 3 ,- 2 ~ / 3 or T, ~ / 3 ,- ~ / 3 , giving (cos 0 ) = f l , 7 4 , 74, and the g values of (8.13). On the other hand, if tunneling is rapid, (cos 6 ) = f +in the lowest E state and zero in the A state, leading to (8.9) and (8.10). The value of (cos 0), and hence the g factor for H ( 1 (Ool), varies with barrier height and from one vibrational level to another. The g factor for H 11(111), on the other hand, stays constant at 2 4u. Typical variationg3of the (001) g factors with barrier height is shown in Fig. 12. In (a)

+

I3O

131

I. B. Bersuker, Zh. Eksperim. i Teor. Fiz. 44, 1239 (1963); see Soviet Phys. JETP (English Trunsl.) 17, 836 (1963); I. B . Bersuker, S. S. Budnikov, B. G. Vekhter, and B. I. Chinik, Fiz. Tverd. Telu 6 , 2583 (1964); see Soviet Phys.-Solid State (English Trunsl.) 6 , 2059 (1965). B. Bleaney, K. D. Bowers, and M. H. L. Pryce, Proc. Roy. SOC.A228, 166 (1955).

T H E JAHN-TELLER

EFFECT I N SOLIDS

133

are the g factors for the lowest state. This shows a static Jahii-Teller effect (8.13) for a high barrier, and a dynamic effect with the three g factors given by (8.9) and (8.10) for a low barrier. In (b) are the g factors for a highly excited state. For a low barrier, this has the isotropic g factor, 2 4u, obtained by averaging over all possible directions in &2&3 space (i.e., (cos e) = 0 ) . The variation of the g factors with temperature can be found by taking a thermal average over the states.lo7To be useful, such a calculation should take the actual phonon spectrum into account. I n the spin-resonance of AgCl: Cut+, the anisotropic “low-temperature” spectrum and the isotropic “high-temperature” spectrum coexist over the range 90°-3000K.122This is to be expected from the theory (see Fig. 12 and Englman and Hornlo’). The variation of the individual g factors with temperature is found to be qualitatively what is expected from Eq. (8.14), [ (cos O ) [ falling to zero as the temperature is raised. Although the temperature range in which these effects are observed is a good deal higher than in other Cut+ systems, charge compensation and defect migration are not thought to be important in this case.132j133 The interpretation in terms of a dynamic Jahn-Teller effect is supported by the fact that the temperature dependence of the hyperfine constant obeys (8.15) quite well, if (cos 0 ) is deduced from the observed g values using (8.14). So far, we have only considered cases where there is a static JahnTeller distortion a t sufficiently low temperatures. In MgO :Cut+, there is anisotropy in the 4OK region which can only be accounted for by a dynamic effect. Coffman121 has studied the spin-resonance a t 10 kMc/sec. At 77”K, there is a single isotropic line with g = 2.192, whence u = 0.048. At 4°K and below, this is replaced by an anisotropic doublet spectrum described approximately by (8.10). It definitely does not follow (8.12). Th’is seems to be definite evidence for a dynamic Jahn-Teller effect, with the E state lowest, and with 3I’ >> gPH = 0.3 cm-l. The isotropic resonance (8.9) is not seen; presumably, the singlet state is depopulated, implying 3r 2 5 cm-l. This interpretation is supported by observation of the Cu HFS. CaF2:Sc2+, and SrF2:Sc2+.I23 Similar results are obtained for CaO: The latter two are cubally coordinated d1 systems, formally identical, except for the sign of u,to octahedral Cu2+. They show remarkably sharp resonances, and i t is possible to detect the isotropic resonance (8.9). From

+

la2 133

134

R. F. Tucker, Phys. Rev.112,725 (1958). Although the C d + ions are isolated a t room temperature, it has yet to be proved that they remain so as the temperature is lowered. (Vacancies are mobile in AgCl down to about 200’K.) Until this point is settled, the interpretation given here must be accepted with reserve [M. U. Palma, private communication (1966)l. R. E. Coffman, private communication (1966).

134

M. D. STURGE

the temperature dependence of its intensity, H O ~ h l is i ~able ~ ~ to deduce r = 8-10 cm-1.1348 These results are encouraging in that the role of random strain in the crystal does not seem to be important. Strain mixes the E and A states and tends to localize the system in one or the other direction of distortion. We shall see in the next section that random strains are of great importance when the tunneling splitting r is small. Presumably, in the systems discussed, I’ is sufficiently large to suppress the strain effect. A study of the effect of externally applied stress on the spectrum would be of interest. When strain broadening is large relative to the tunneling splitting, the system is effectively localized in one potential well (static Jahn-Teller effect). Tunneling is then no longer relevant to resonance experiments, but it can still show up in the relaxation type of experiment which will be discussed in the next section.

9. ACOUSTICCONSEQUENCES OF THE JAHN-TELLER EFFECT Because the ground state energy of a Jahn-Teller ion depends on the local crystal strain, the ion can interact with an acoustic wave, producing dispersion and attenuation. There are two ways in which this can occur: by a resonant transition between two (in general broadened) vibronic levels, or by relaxation between the different possible directions of JahnTeller distortion. The resonant process is considered by B e r ~ u k e r . ’He ~ ~ considers the transition between the E and A states defined in Eq. (8.8). The transition probability for the absorption of a long wavelength phonon with polari1348

135

A more satisfying explanation of Hochli’s (and incidentally of Coffman’s) results has been given by F. S. Ham (Phys. Rev. in press). Ham assumes that the Jahn-Teller effect in these systems is relatively weak, and that anharmonic effects can be neglected. He shows that the dynamic Jahn-Teller effect reduces the coefficient of the anisotropic contribution to g(r8) (424 in Eq. (8.12)) by a factor between 1 and 0 . 5 . (This is an example of the partial quenching of an orbital operator, to be discussed in Section 15.) The limiting value 2u (Eq. (8.10))is reached for quite weak Jahn-Teller effect (within 1% for 6E=hw). While this limit is reached for Cu2+ (Refs. 121 and 134), the reduction factor is only about 0.7 for SC*+(Ref. 123), corresponding to 6E-0.2 hw, which is a very weak Jahn-Teller effect. The isotropic resonance found by Hochli above 6°K cannot be assigned to the r6excited state, which is roughly hw above the ground state when the Jahn-Teller effect is weak. Ham interprets this resonance as the motionally narrowed rs spectrum. This interpretation is supported by the behavior of the hfs; the smaller the hyperfine splitting the lower the temperature a t which narrowing occurs. I. B. Bersuker, Zh. Eksperim. i Teor. Fiz. 44, 1577 (1963); see Soviet Phys. J E T P (English Transl.) 17, 1060 (1963) (in which Bersuker’s table is translated incorrectly).

THE JAHN-TELLER EFFECT IN SOLIDS

135

zation o,propagating in the direction 2, is

w

(9.1)

= J(47rA2(Q,,2)/ds3fi2),

where J is the energy density of the acoustic wave, s the velocity of sound, d the density of the crystal, A is defined in Eq. (5.1), and

+

(QaCZ)

= $ ( ~ . Q 2 * 2 ) ~$ ( ~ * Q 3 - i ? ) ' .

For an octahedron with its axes parallel to the crystallographic axes, the matrices Qz and Q3 are136

. .

-1 a -

_ . on=

where a is the interatomic spacing. Values of (Q,?) are given in Table V. The attenuation coefficient a for sound of frequency w is given by the difference between absorption and induced emission. (Spontaneous emission can be neglected at these low frequencies, and so we can use MaxwellBoltzmann statistics.) (w/J)Nfiwg(w)tanh (fiwl2kT)

(9.3)

= (2?r~~2(~,,2)/ds3k~)w2g(w),

(9.4)

a(w) =

when fiw << kT. Here N is the number of Jahn-Teller ions per cubic centimeter, a is in nepers per centimeter, and g ( w ) is the shape function for the TABLE V. DEPENDENCE OF ULTRASONIC ABSORPTION O N POLARIZATION A N D PROPAGATION DIRECTION, RELATIVE TO THE AXESO F A N OCTAHEDRON Propagation direction (001 ) (001 ) (111) (111) (110) (110) (110) la6

Polarization

6 (Qac2)/aZ

Long. Trans. Long. Trans. Long. (001 )

-1 0

(iio)

f

1 0 0

i

Their general form in different situations is discussed by A. S. Nowick and W. R. Heller, Advan. Phys. 14, 101 (1965).

136

M. D. STURGE

resonance [Jg(w) dw = 11. If the only source of broadening is the upperstate lifetime 7 , we have

where fiwo = 3 I I’ I is the energy separation of the E and A states. Zero field resonant absorption of this type by Jahn-Teller ions has never been observed. Acoustic absorption is observed, and it is very strong, but the frequency dependence is that of a relaxation rather than a resonant process. The reason for this is probably that real crystals are inevitably strained, so that the local symmetry a t any ion is lower than cubic. Consider a Jahn-Teller ion with well-defined potential minima, and suppose that at one particular site local strain lowers one of the minima by an amount uo relative to the other two. If uo 2 3I’, the system will (at 0°K) be localized in one or the other valley, and no resonant transition will be possible. At temperatures such that kT 2 uo , there will be a thermal distribution among the minima, but the resonance will be too broad to see. Now we ask the following question: if the system is perturbed by an additional (small) acoustic strain, how rapidly will thermal equilibrium be restored? If, instantaneously, we have a contribution to the appropriate elastic constant137( c = ds2), AC = a2F/dc2 =

where Z

=

2 exp ( - & / k T )

NkT(d2In z/ac2) s=o ,

(9.6)

+ exp [ ( u o + 2 p ~ ) / k T ]

and

p2 = 2A2(&,,2). Hence

AC = - (2Np2/kT)f (uo/kT)

(9.7)

Here c is the strain, 3p the splitting of the electronic state per unit strain, N the number of Jahn-Teller ions, F their contribution to the free energy, and 2 their partition function. The smooth but complicated function f (x) satisfies f (0) = 1, f (f00 ) = 0. A typical value for p is lo4 cm-’, and so we see that 1 ppm of a Jahn-Teller impurity ion at 4°K can produce a change in an elastic constant of the order of 1 in 1000, which is easily 137

Ac is calculated isothermally, since the impurity (by hypothesis) remains in thermal

equilibrium with its environment, which can be assumed to be an ideal solid with no difference between its adiabatic and isothermal elastic constants. The adiabatic value of Ac (corresponding to slow relaxation) is zero, as can be seen by substituting in Eq. (3.41) of W. P. Mason, “Piezo-electric Crystals and their Application to Ultrasonics.’’ Van Nostrand, Princeton, New Jersey, 1950.

THE JAHN-TELLER

0

137

EFFECT I N SOLIDS

I

I

0.05

0.10

I

0.15 (OK)-’

I

0.20

0.! 5

1 T

FIG. 13. Resonant frequencies of single crystal spheres of pure containing approximately 10 ppm Ni3+ (from Gyorgy el ~ 1 . l ~ ~ ) .

A1203,

and A1,03

measurable by a resonance technique.138The result of such an experiment139J40 (in AI2O3:Ni3+)is shown in Fig. 13. If relaxation is not instantaneous, but takes a finite time r , there is loss and dispersion. The attenuation coefficient is141 w AC

(.(a) = - -

wr

~

s c 1

+ w2?

- 2Noz -

uo cl.s3kTf(E)-’

w2r

(9.8)

This is identical to (9.4) if aor << 1 and uo << kT. If T varies rapidly with temperature, the attenuation coefficient has a maximum when wr(T) = 1. Typical behavior142of a (for Ni3+ in A1203) is shown in Fig. 14. From such data, we can deduce as a function of T as in Fig. 15. The data fit the solid curve quite well: 7-l

139

140 141

142

=

2v0 exp (-Vo/kT)

+ BTg(uo/kT),

(9.9)

D. B. Fraser, E. M. Gyorgy, R. C. LeCraw, J. P. Remeika, F. J. Schnettler, and L. G. van Uitert, J . A p p l . Phys. 36, 1016 (1965). E. M. Gyorgy, M. I>. Sturge, D. B. Fraser, and K. C. LeCraw, Phys. Rev. Letters 16, 19 (1965). E. M. Gyorgy, R . C. LeCraw, and M. D. Sturge, J. A p p l . Phys. 37, 1303 (1966). W. P. Mason and T. B. Bateman, J . Acoust. SOC.Ant. 36,644 (1964) ; M. Pomerante, Proc. IEEE 63, 1438 (1965). M. D. Sturge, J. T. Krause, E. M. Gyorgy, R. C. LeCraw, and F. R. Merritt, Phys. Rev. 155, 218 (1967).

138

M. D. STURGE

I

0.06 0

I

I

I

20 TEMPERATURE

I

I

40 (OK)

FIG. 14. Attenuation of 260-Mc/sec longitudinal acoustic waves in A1203:Ni3+; direction of propagation: (a) parallel to [OOOl]; (b) parallel to [llao] (from Sturge et ~ 1 . l ~ ~ ) .

where g(s) = s(2 =1

+ e-z)/3(1 - e-.) for small s,

-

-

and Y O , Vo, B, and uo are constants. The first term in (9.9) represents thermal activation over the barrier; we find typically vo 1013cps, VO 100 em-’. [Remember that Vo is not the Jahn-Teller energy, but the height of the barrier between potential minima corresponding to different directions of distortion, and that this is zero in the lowest-order approximation of Eq. (5.3).] The second term in (9.9) represents tunneling accompanied by phonon emission and is analogous to the “direct process” in spin-lattice relaxation. It has been discussed in a slightly different connection by Sussman and o t h e r ~ ’ ~in ~ Jthe ~ ;present case, we have

B

=

18r2p2k/~dfi4s5.

(9.10)

There is also a two-phonon term analogous to the “Raman’) process in 143 14*

J. A. Sussman, Physik Kondensierten Materie 2 , 146 (1964). R. Pirc, J. Zeks, and P. Gosar, J . Phys. Chem. Solids 27, 1219 (1966).

139

THE JAHN-TELLER EFFECT I N SOLIDS

spin-lattice relaxation, giving a contribution TB-~

=

(9.11)

9r2p4(kT) 3/~d2fi7~'0.

Although this process is negligible in A1203 , it might be important in materials with a lower Debye temperature. 0.2 em-', in reasonable agreethat r For Ni3+in A1203,it is ment with (8.7), whereas u0 2 cm-' (implying local strains of order a very reasonable value). Thus, it is not surprising that no resonance is seen. Resonant absorption might be seen in more weakly coupled systems, for which r is larger and uosmaller. Ultrasonic frequencies in the microwave

-

-

lo-'

0

0

260 MC

(1150)

+

ESR

iooMc 24kMC

lo-" 0

0.1

0.2 I / T ( DEC-1)

0.3

FIG.15. Relaxation time T for Ni3+ ions in AlzOa, deduced from acoustic loss at 260 Mc/sec and from spin-resonance measurements; solid curve is from Eq. (9.9).

140

M. D. STURGE

region would be needed. For larger tunneling splittings still, there should be effects on the thermal conductivity. Resonant scattering of phonons by impurities is a well-known phenomenon145; we might expect dips in the hwolk. thermal conductivity in the region T Because of their strong electron-lattice coupling, Jahn-Teller ions are likely to be good subjects for the detection of paramagnetic resonance by acoustic 1 0 ~ s . Such ~ ~ ~effects ~ ' ~ at ~ 10 kMc/sec have been reported in MgO:Cr2+ (5E ground state), a system in which no resonance can be detected by conventional means.148No analysis of the data has been given, but they do not appear to be explicable without invoking the dynamic Jahn-Teller effect. Since the spin-orbit splitting of the ground state is not quenched in this ion, even by a static Jahn-Teller effect, analysis is likely to be complicated. At a site that is not a center of symmetry, there can be an electric dipole moment associated with the transition between vibronic levels, and consequent paraeIectric resonance absorption. For instance, the tunneling splittings of the OH- ion in KC1 (which have nothing to do with the Jahn-Teller effect, of course) have been seen by this m e t h ~ d . ' ~This ~J~~ technique is not limited to such low frequencies as are ultrasonics, and the resonance can be tuned by application of an electric field, which is a great practical advantage. The electrical properties of a trapped electron at a tetrahedral site, which is probably the most promising case for experimental study, have been discussed theoretically by Sussmanlsl (see also Section 19). The mechanism that leads to acoustic loss also contributes strongly to the ferrimagnetic resonance losses in yttrium iron garnet doped with Mn3+.138J40J52 The theory of this effect has not yet been worked out. Because spin-orbit coupling and magnetostriction have to be taken into account, such a calculation is likely to be quite complicated.

-

10. E TERMS IN TETRANEDRALLY COORDINATED IONS Although we have only considered ions in octahedral coordination, the situation for tetrahedral coordination is formally the same. There is still R. 0. Pohl, Phys. Rev. Letters 8,481 (1962) ;M. V. Klein, Phys. Rev. 131, 1500 (1963) ; G. A. Slack, ibid. 134, A1268 (1964) ;M. Wagner, ibid. 136, B562 (1964) ; C. K. Chau, M. V. Klein, and B. Wedding, Phys. Rev. Letters 17, 521 (1966). 146 E. B. Tucker, Phys. Rev. Letters 6, 183 (1961). 14' I. S. Ciccarelo, R. Arzt, and K. Dransfeld, Phys. Rev. 138, A934 (1965). 14* J. R. Fletcher, F. G. Marshall, V. W. Rampton, P. M. Rowell, and K. W. H. Stevens, Proc. Phys. Soc. (London) 88, 127 (1966). 149 W. E. Bron and R. W. Dreyfus, Phys. Rev. Letters 16, 165 (1966). lSo G. Feher, I. W. Shepherd, and H. B . Shore, Phys. Rev. Letters 16, 500 (1966). 151 J. A. Sussman, Proc. Phys. SOC. (London) 79, 758 (1962). E. M. Gyorgy and R. C. LeCraw, A p p l . Phys. Letters 6, 32 (1965). 146

THE JAHN-TELLER

141

EFFECT I N SOLIDS

a single pair of E modes, which are the only modes that can split an E term. We would expect Jahn-Teller splittings to be much smaller than in octahedral coordination, because e orbitals do not form u bonds in tetrahedra. The static Jahn-Teller effect has not been observed in a dilute system but may occur in spinels containing tetrahedral Fe2+ (see Section 13). There is also some evidence for a dynamic effect (see Section 24). The expected weakness of the Jahn-Teller effect in such systems has been confirmed by molecular orbital calculations on the tetrahedral molecule VCl, .153,154 These show that 6E 5 liw, and that Vo 1 cm-’, so that no static Jahn-Teller effect is to be expected.

-

11. CENTERSWITH LESSTHANCUBICSYMMETRY

If the symmetry of the zeroth order Hamiltonian X(O) is axial rather than cubic, its eigenvalues are a t most doubly degenerate. (The same goes for vibrational degeneracy.) The treatment of Section 5 should apply to all degenerate states interacting with doubly degenerate vibrations; but we have to remember that spin-orbit coupling, which vanishes to first order in an E state of a cubic system, may be important in the lower symmetry case. We shall discuss three such systems; it so happens that in all three cases the electronic wave functions are molecular orbitals and cannot in any approximation be treated as “atomid’ states. An important example of an intrinsically axial center is the R center, which has CSvsymmetry. This center consists of three F centers (negative ion vacancies neutralized by a trapped electron) arranged in the form of a triangle in a (111) plane of an alkali halide crystal (see Fig. 16).155Some properties of this center are described in the review by Compton and Rabin.156 It contains three electrons, whose wave functions are not well known; but calculation157 suggests that the ground state is 2$. Spin-orbit splitting is apparently small in the crystals studied.I5* The 2i?state is subject to a dynamic Jahn-Teller effect. Second-order (“anharmonic”) effects are apparently not very strong; the vibronic ground state is E , with the lowest A1 state (see Fig. 10) at least 20 cm-’ higher. The E state is analogous t o the doublet state of MgO:Cu2f studied by CoffmaniZ1 (see Section 8 ) . 153

164 165 166

158

L. L. Lohr, Jr. and W. N. Lipscomb, 1norg. Chem. 2, 911 (1963). C. J. Ballhauseii and J. de Heer, J . Chem. Phys. 43, 4304 (1965). C. Z. Van Doorn, Philips Res. Rept. 12, 309 (1957). W. D. Compton and H. Rabin, Solid State Phys. 16, 121 (1964). J. 0. Hirschfelder, J . Chem. Phys. 6, 795 (1938). These do not iiiclude CsF, in which the spin-orbit coupling of the F center is relatively large (see Section 26).

142

M. D. STURGE

+ Ve 0 - ve

@

ION

.-. :../;

ION

F CENTER

FIG.16. The R center in the rock salt structure.

The R center has been studied (mostly in KCI) by optical a b s o r p t i ~ n ~ ~ ~ J m and spin resonance.161(The optical data will be discussed in Section 22.) In both cases, it is the splitting of the ground state under uniaxial stress that provides the clue as to its nature. The vibronic ground state is, in general, split by random strains in the crystal; this broadens the resonance beyond detection. If stress is applied in the (111) plane, splitting2fi by an amount large compared with kT,spin resonance is seen with (1 11 ) axial symmetry. In principle, this should be a powerful method for making spin-resonance detectable in orbitally degenerate states.162 The spin resonance of the N2- radical, which replaces the N3- molecular ion in a Dad site of NaN3, has been studied by Gelerinter and f3il~bee.I~~ In this ion, there is a single unpaired electron in a T* (antibonding) orbital, which in axial symmetry is doubly degenerate ("). The spin Hamiltonian is found not to be axial, but to have a complex angular dependence that can be accounted for if the axis of the N2- molecule is assumed to be tilted 4.5" from the C3 axis of the crystal. This tilt is to be expected from a static Jahn-Teller effect in Du symmetry, as can be seen from the sketches of Is9

D. B. Fitchen, R. H. Silsbee, T. A. Fulton, and E. L. Wolf, Phys. Rev. Letters 11, 275 (1963).

R. H. Silsbee, Phys. Rev. 138, A180 (1965).

D.C. Krupka and R. H. Silsbee, Phys. Rev. Letters 12, 193 (1964) ; Phys. Rev. 162, 816 (1966). 162

163

G. Feher, J. C. Hensel, and E. A. Gere, Phys. Rev. Letters 6 , 309 (1960). E. Gelerinter and R. H. Silsbee, J . Chem. Phys. 46, 1703 (1966).

THE JAHN-TELLER

EFFECT IN SOLIDS

143

the e B modes of the Dad x2Y6 molecule in Fig. 49 of Herzberg's book." The size of the g shift (which comes primarily from spin-orbit interaction with the upper Jahn-Teller branch) indicates that the Jahn-Teller splitting (36E) is only about 500 cm-'. It is surprising that such a weak Jahn-Teller effect is sufficient to produce a static distortion; it is suggested163that large random strains in the crystals (for which there is independent evidence) stabilize the static distortion. McConnell and others have studied t h e ~ r e t i c a l l y ' ~and ~ J ~ ~experimentally166J67 the spin-resonance of a number of pure crystals of molecular ring radicals having the formula C,H,. Like Nz-, these radicals have orbitally degenerate ground states due to an unpaired ?r* electron. Distortion of the ring shows up as anisotropy in the proton hyperfine structure, and by making different protons inequivalent. Whereas in C6H6-, C7H7, and other such radicals16*the hyperfine structure can be accounted for in terms of rhombic crystal field s p l i t t i n g ~ , ' ~the ~ . C5H5 ~ ~ ~ radical , ~ ~ ~ rotates rapidly in its own plane even at 25°K' and the rhombic crystal field is averaged Nevertheless, the proton hyperfine structure is anisotropic below 70"K, showing that the molecule is distorted. This is an €21 distortion ~), the symmetry from CSv to C Z v ,as (see Fig. 38 of H e r ~ b e r g ~lowering predicted by t h e ~ r y . ' ~Liehr5 ~,'~~ has enumerated the possible Jahn-Teller distortions of such planar molecules. There is some evidence for a Jahn-Teller effect in the spin-resonance of orbitally degenerate, metastable spin triplet states of aromatic molecules in glassy s01vents.l~~ The fact that in such solvents there is a continuous range of environment prevents the drawing of definite conclusions, however. 12.

SQUARE PLANAR COMPLEXES

The Jahn-Teller effect in a complex with tetragonal symmetry (of which the X Y 4 square planar molecular of Dlh symmetry is a simple exemplar) has some interesting features from a theoretical point of H. M. McConnell and A. D. McLachlan, J . Chem. Phys. 34, 1 (1961). H. M. McConnell, J . Chem. Phys. 34, 13 (1961). H. J. Silverstone, D. E. Wood, and H. M. McConnell, J. Chem. Phys. 41,2311 (1964). 167 G. R. Liebling and H. M. McConnell, J. Chem. Phys. 42, 3931 (1965). 168 M. G. Townsend and S. I. Weissman, J . Chem. Phys. 32,309 (1960). R. G. Lawler, J. R. Bolton, G. K. Fraenkel, and T. H. Brown [J Am. Chem. SOC. 86, 520 (1964)l have found evidence for a Jahn-Teller effect in the ground state of C6H6- in solution; see also A. Carrington, H. C. Longuet-Higgins, R. E. Moss, and P. F. Todd, MoZ. Phys. 9, 187 (1965). 170 L. C. Snyder, J . Chem. Phys. 33, 619 (1960). A. D. Liehr, 2. Physik. Chem. (Frunkjurt) [ns.] 9, 338 (1956). 172 M. S. de Groot, I. A. M. Hesselmann, and J . H. van der Waals, Mol. Phys. 10, 241 (1966). 164 165

144

++ M. D. STURGE

QL (819)

Q, (820)

FIG. 17. Even parity, nontotally symmetric normal coordinates for the square planar X Y 4 complex.

view.173-176The tetragonal point groups differ from all other crystallographic groups in that the symmetric square of a doubly degenerate irreducible representation (I.R.) contains nontotally symmetric but onedimensional 1.R.b. For instance, in D a ,

[Ez]

+ B1, + Bz, .

= .A10

(12.1)

Thus an E state is split in first order by the pl, and pz, distortions, QZand Q 3 , illustrated in Fig. 17. Note that QZand Q3 belong to different I.R.'s and are not degenerate. The first-order Hamiltonian is

(

-AQz

XI =

)+

BQ3

+3

+

(P3'/~3

+

~ ( P z ~ / LL ~~Z WzZ ~ Q ~ ~ )

BQ3 AQz

(12.2)

~ c 3 ~ 3 ~ Q 3 ~ ) ,

+

operating on the state vector (f). The electronic state is # = a+= l&, where $=and +u can be regarded (for simplicity) as transforming like (2,y) . The potential surface on which the nuclei move is then

~ ( Q 2z Q 3 )

+

+

= ~ L ~ ~ W Z ~ Q Z3 ~ ~ ~ 3 ~fQ (AzQz2 3 ~

B2Q3')l''.

(12.3)

This surface has twofold symmetry about the V axis. There are minima at QZ = &Qz0 = =tA/fi.&, and saddle points176 at Q 3 = &Q30 = f B / p 3 ~ 3 ~ , if Qz0 > &so, mutatti mutandis if &So > QzO. If by chance Qzo = Q3O, the potential surface happens to have cylindrical symmetry; but inclusion of n8 M. S. Child, Mol. Phys. 3, 601 (1960.) 174

lw

J. T. Hougen, J . Mol. Spectry. 13, 149 (1964). C. J. Ballhausen, Theoret. Chim. Acta 3 , 368 (1965). Not cusps, as indicated in Fig. 3d of Liehr.6

THE JAHN-TELLER

145

EFFECT IN SOLIDS

higher-order terms in the Q'S will restore twofold symmetry. So the static problem is quite different from that of Section 5. B a l l h a ~ s e n 'includes ~~ a spin-orbit coupling term XL-S in the Hamiltonian; he shows that there is no Jahn-Teller effect (in a spin-doublet) if I I > 46E. If I A I < 46E, the spin-orbit splitting of individual vibronic ; levels is reduced, especially in the lower levels (see Fig. 3 of Ballha~sen"~) this is a two-dimensional example of the Ham which we shall discuss in detail in Section 15. If coupling to a single mode (say Q2) is dominant, we can regard the V(Q2Q3)diagram as plane, as in Fig. 18. The electronic states corresponding to each parabola (in the absence of spin-orbit coupling) are simply & and # u . Since these states are orthogonal, no tunneling can occur between them. The vibronic levels are those of a simple harmonic oscillator, unchanged from the uncoupled case (except for a downward displacement of 6 E ) . Each is doubly degenerate, transforming as E of Dlh . Even if we include coupling to Q 3 , this degeneracy cannot be raised, since the overall Hamiltonian still has D4h symmetry. Thus, interaction with Q3 produces no further splittings. If interaction is strong, so that B 2 / p 3 ~ 3differs 2 from A 2 / p 2 w z 2by less than hw2 or h w 3 , we have a situation resembling the cylindrical potential of Section 5. The ground state is still doubly degenerate, but there are low-lying excited states corresponding to rotational excitation in Q2Q3 space (similar to those in Fig. 10). The square planar complex, because of its relative simplicity, should be a promising "model system" for studying theoretically the effects of such refinements as nonlinear interactions between modes. Unfortunately, there are apparently no experimental data on Jahn-Teller effects in such complexes. The optical spectrum of K2PtC14, in which the PtCL2- complex

Q2

FIG.18. V(Q2)for an E state in the square planar complex. In the absence of spinorbit coupling, the electronic states associated with each parabola are mutually orthogonal, and each vibronic state is doubly degenerate.

146

M. D. STURGE

is square planar,lT7 has been studied,lT8but only broad bands are seen without any definite evidence for a Jahn-Teller effect in the l8 excited A practical state. The same is true of other square planar ds comple~es.~7~ difficulty in such complexes is their tendency to distort out of the plane, leading to complicated second-order interactions with odd parity modes.1m 13. COOPERATIVE JAHN-TELLER EFFECTS This section is concerned with manifestations of the Jahn-Teller effect in concentrated systems, in which individual Jahn-Teller ions cannot be regarded as independent. It differs from other sections in that no attempt is made to include all the relevant experimental data. This is because there already exist two excellent and up-to-date reviews of the voluminous data on the structure of compounds containing Jahn-Teller ions.1s1Js2 The information that can be gained from these data concerning the JahnTeller effect is limited, the experiments having far outrun the available theory. On the other hand, there are aspects of the cooperative Jahn-Teller effect which have received relatively little attention, and we shall concentrate on these. We begin with some general considerations. A pure crystal can be in an electronically degenerate state, just as a single ion can. The electronic state transforms as an irreducible representation (I.R.) r of the space group of the crystal, and the condition for the possibility of a Jahn-Teller distortion of the whole crystal (“cooperative Jahn-Teller distortion”) is analogous to that for an isolated complex; the symmetric square [PI must contain the I.R. of some possible distortion. In the absence of perturbations such as impurities or excitons, this limits the possible distortions to K = 0 optical modes1@;the I.R. of the distortion is simply an I.R. of the parent point group. (Birmanls4 has considered the possibility of 177

The very large tetragonal field in such a complex splits the ‘Eg(t2e2)term of d8 by more than the ‘ E , - 3A2gseparation; the ground state is therefore l61,of h, not ’-429.

178

D. S. Martin and C. A. Lenhardt, Inorg. Chem. 3, 1368 (1964).

For a review of the data on such complexes, and a comparison with molecular orbital theory, see J. Perumareddi, A. D. Liehr, and A. W. Adamson, J . A m . Chem. SOC. 86, 249 (1963) ; H. B. Gray and C. J. Ballhausen, ibid. p. 260. 1@ C. J. Ballhausen, private communication (1966). J. B. Goodenough, “Magnetism and the Chemical Bond.” Wiley (Interscience), New York, 1963. la2 R. W. G. Wyckoff, “Crystal Structures,” 2nd ed. Wiley (Interscience), New York, 1963-1 965. l a 3 N. N. Kristofel, Fiz. Tverd. Tela 6 , 3266 (1964) ;see Soviet Phys.-Solid State (English Transl.) 6 , 2613 (1965). l E 4J. L. Birman, Phys. Rev. 126, 1959 (1962). 179

THE JAHN-TELLER

EFFECT I N SOLIDS

147

zone boundary distortions coupling to indirect excitons, for instance in diamond-type crystals.la) It turns out that in many structures there is no distortion satisfying this requirement. For instance, a structure in which every ion is at a center of symmetry (for instance, the NaCl or the cubic perovskite structure) has no even parity K = 0 modes and cannot have a first-order Jahn-Teller coupling. Kristofe11*3has pointed out that the selection rule K = 0 only holds if the Jahn-Teller energy is less than the electronic bandwidth that arises from interaction between ions. If, for instance, an electron is excited into a degenerate level in a band whose width is less than the Jahn-Teller energy, it will pay the lattice to distort locally, “self-trapping” the excitation at a particular site. Although the excitation can still move through the crystal, its probability of doing so (and therefore its bandwidth) is greatly reduced, since it has to “drag” its lattice distortion with it. The best known example of this phenomenon is the “self-trapped” hole ( V , center) in halide cry~tals.18~~~0 The selection rules for the Jahn-Teller interaction are now those derived from the point goup, not from the space group, and the problem can be treated (in principle) in terms of the quasi-molecular model discussed in previous sections. Most transition metal compounds are wide-gap insulators (we shall not of indiscuss the exceptions to this ~ t a t e m e n t ~ ~ ’ Jand ~ ~ the J ~ ~d )orbitals , dividual ions are reasonably well localized, so that we may use the tight binding approximation. Any crystal made up of ions with orbitally degenerate ground states will itself (in this approximation) have an orbitally degenerate ground state. We would, therefore, expect it to distort if it can. While translational symmetry is retained, zone boundary distortions can occur, increasing the size of the unit cell (for instance, in perovskites, from one molecule to eight per unit cell). See also J. C. Phillips, Phys. Rev. 139, A1291 (1965). T.G. Castner and W. Kanzig, Phys. Chem. Solids 3, 178 (1957). C. J. Delbecq, B. Smaller, and P. H. Yuster, Phys. Rev. 111, 1235 (1958). 188 W. Hayes and J. W. Twiddell, Proc. Phys. Soc. (London) 79, 1296 (1962). ISe J. Ramamurti and K. Teegarden, Phys. Rev. 146, 698 (1966). Igo This center is best understood as an isolated molecular ion with a singlet ground state and will not be discussed further in this review. Ig1 F. J. Morin, Bell System Tech. J . 37, 1047 (1958) ; Phys. Rev. Letters 3, 34 (1959). Ig2 The Jahn-Teller effect may possibly be involved in some of the metal-to-semiconductor transitions that occur in certain transition metal oxides as the temperature is reduced. For instance, in V z 0 8 ,the transition occurs at about 150°K and is accompanied by a reduction in symmetry from rhombohedra1 to monoclinic. See J. Feinleib and W. Paul, Proc. Intern. Conf. Phys. Solids High Pressures, Tucson, 1965 p. 571. Academic Press, New York, 1965, and Phys. Rev. 166, 826 (1967); D. Adler and J. Feinleib, Phys. Rev. Letters 12, 700 (1964); D. Adler, J. Feinleib, H. Brooks, and W. Paul, Phys. Rev. 166, 851 (1967).

148

M. D. STURGE

30 T (OK)

FIG.19. (a) Axial ratio at room temperature of the mixed spinel MgMn2,A12-bOa, as a function of Mn3+ content (Mn3+ here is octahedrally coordinated and has the

Jahn-Teller sensitive (from Irani et al.193).

6E

ground term). (b) Axial ratio versus T for pure MgMnzOl

The spinel structure has an cs (K = 0) mode and can distort tetragonally. A number of crystals with the spinel or related structure, which are cubic in the absence of Jahn-Teller ions, become tetragonal when enough JahnTeller ions are added. An example is given in Fig. 19a, where the c / a ratio is given as a function of Mn3+ (at room temperature) of MgMn2,A12-zz04 content.lg8 In general, as the temperature is raised from O'K, a transition to the more symmetric situation will occur, as in Fig. 19b. However, a good deal of local tetragonal distortion persists above the transition temperature, as evidenced by the quadrupolar splitting in the Mossbauer effect125 in ferrites, and by X-ray broadening.194 It has been shownlg5that 193

lo4 lo6

K. S.Irani, A. P. B. Sinha, and A. B. Biswas, Phys. Chem. Solids 17, 101 (1960) ; 23, 711 (1962). L. Cervinka, S. Krupicka, and V. Synacek, Phys. Chem. Solids 20, 167 (1961). R. J. Wojtowicz, Phys. Rev. 116, 32 (1959) ; G. G. Robbrecht and E. F. de Clerck, Physica 31, 1033 and 1575 (1965).

THE JAHN-TELLER EFFECT IN SOLIDS

149

in spinels this must be a first-order transition (although as much as 50% "disorder" may appear below the transition temperature). It is possible that in other systems the transition might be of second order. The necessary conditions for such a transition have been discussed by Landau and others,196and applied to the present situation by Haas.1s7Jss Data such as those in Fig. 19 are of great technical importance in magnetism and have been thoroughly reviewed by Goodenough.ls' The knowledge that an ion has a Jahn-Teller active ground state enables one to predict that in sufficient concentration it will not form cubic ~ r y s t a l s , a 3 J S ~ ~ ~ so long as there is a crystal distortion that can split the state. (Because of dynamic effects, even this weak statement is not always true; see below.) On the other hand, the converse cannot be argued safely; the existence of a compound with (say) tetragonal distortion is not in itself evidence for a Jahn-Teller effect, even if other sources of distortion, such as magnetostriction, can be ruled out. The reason is that the Jahn-Teller energy (at most 0.5 eV) is rather small compared with typical binding energies of crystals (2-3 eV/ion) and will only be important if other effects are rather finely balanced. For instance, in KzCuF4 the CuFs4- octahedron is tetragonal, with c/a < 1. This has been cited as a case of Jahn-Teller distortion with compression of the octahedron.%' However, it turns out that K2NiF4,with an orbitally nondegenerate ground term, has a similar tetragonal distortion,mZ as have many other compounds with this structure.'" It is probable that the distortion in these compounds is due to packing considerations rather than to the Jahn-Teller effect. After all, not all closed-shell ions form cubic crystals! The converse situation arises in spinels containing octahedral Cu2+. These are sometimes tetragonal and sometimes cubiclmaa t any rate down to 80°K.204The distortion is found empirically to be determined more by the concentration of the other, non-Jahn-Teller, ion than by the concenL. D. Landau, Physik. 2. Sowjetunion 11, 26 (1937); S. Strlissler and C. Kittel, Phys. Rev. 139, A758 (1965). 197 C. Haas, J. Phys. Chem. Solids 26, 1225 (1965). 198 An interesting example of cooperative ordering occurs in N ~ C ~ ~ . ~ N H . L ,This D , . is not actually a Jahn-Teller case but is closely analogous to it, the degeneracy being configurational rather than electronic. The ammonia molecules rotate freely at room temperature but become cooperatively frozen into a particular configuration below a well-defined temperature (8Oo-90"K) that is a function of z. The transition is detected by its pronounced effect on the spin resonance of Ni2+ [G. Aiello, M. U. Palma, and F. Persico, Phys. Letters 11, 117 (1964)l. L. E. Orgel, J. Chem. SOC.p. 4756 (1952). D. S. McClure, Phys. Chem. Solids 3, 311 (1957). 201 K. Knox, J. Chem. Phys. 30, 991 (1959). 202 D. Balz: and K. Plieth, 2.Elektrochem. 69, 545 (1955). 203 C. Delorme, Bull. SOC.Franc. Mineral. Crist. 81, 79 (1958). 204 M. Robbins and L. Darcy, J. Phys. Chem. Solids 27, 741 (1966).

196

150

M. D. STURGE

tration of Cuz+.A possible explanation of this anomalous behavior is that in the cubic crystals the Jahn-Teller effect is dynamic, and that the anharmonic effects that lead to a static distortion (see Section 5 ) depend critically on all the ions in the crystal. A dynamic Jahn-Teller effect, particularly if it is in any sense cooperative, should show up in X-ray diffraction. The large vibrational amplitude in the e mode should produce anomalies in thermal diffuse scattering and in the anisotropy of the Debye-Waller factor. Careful studies of such effects have yet to be made. A promising system for such an X-ray study of the transition from a dynamic to a static cooperative Jahn-Teller effect is NaNi02, in which Ni3+ has the strong field 2E ground term. Above 5OO0K, this has a very simple (distorted rock salt) structure. At this temperature, there is a transition (which does not occur in isomorphous crystals not containing Jahn-Teller ions) to a tetragonally distorted s t r ~ c t u r e . ~ ~ ~ ~ ~ ~ ~ I n spite of the qualified prohibition against a first-order Jahn-Teller effect in the perovskite structure, many perovskites containing Jahn-Teller ions (e.g., KCrF3 and LaMnOa) do show pronounced tetragonal distortions.181J8z~207 However, the majority of perovskites (particularly the oxides) containing no Jahn-Teller ions also show such distortions.*82It is not clear at present whether the distortion in the former case is due to packing requirements or to Jahn-Teller interactions. It is quite possible that even the distortions observed in perovskites with nondegenerate ground states are due to second-order Jahn-Teller effects, which are not prohibited by crystal symmetry. For instance, ferroelectricity, in which a spontaneous, K = 0, odd parity distortion of the whole crystal produces a macroscopic dipole moment, is common in perovskites. (Instability against zone boundary distortion leads to antiferroelectricity.) Such instability can arise from second-order Jahn-Teller interaction, in which a nondegenerate ground state is coupled to low-lying excited states by the odd parity distortion.208~209The matrix element for this interaction is just that which enters the polarizability, making the connection with the more usual semiphenomenological theories of ferroelectricity, in which 205 206

207

208

209

L. D. Dyer, B. S. Borie, and G. P. Smith, J . Am. Chem. SOC. 76, 1499 (1954). P. F. Bongers, Ph.D. Thesis, Leiden University (1957); P. F. Bongers and U. Enz, Solid State Commun. 4, 153 (1966). Some do not; for instance, LaNiOt is apparently rhombohedra1 down t o 4.2”K C. Koehler and E. 0. Wollan, Phys. Chem. Solids 2, 100 (1957)l. K. P. Sinha and A. P. B. Sinha, Indian J . Pure A p p l . Phys. 2, 91 (1964); I. B. Bersuker, Phys. Letters 20, 589 (1966). R. Englman, Solid State and Molecular Theory Group, Quarterly Progress Rept., No. 47, p. 74. M.I.T., 1963; “Microscopic Theory of Ionic Dielectrics,” A.E.C. Accession No. 14129, Rept. IA994 (1964) (Defence Documentation Center No. AD 612613).

w.

THE JAHN-TELLER

EFFECT IN SOLIDS

151

the polarizability is regarded as a parameter. Although it is not yet clear whether one gains anything in any practical case by this approach, it does hold out the attractive possibility of a unified treatment of the various distortions that the perovskites undergo. Koonce210 has considered the possibility that a crystal containing centers with (individually) very weak Jahn-Teller coupling might distort as a whole when the concentration of the centers is high enough. He considers in particular the shallow acceptor in silicon, which has a rs ground state, but only has a very weak Jahn-Teller coupling because of its diffuseness (see Section 4).At high concentrations, the strain due to the JahnTeller effect at one acceptor can enhance that at another, leading to a Jahn-Teller energy per acceptor of ND2/6c44 , where D is the splitting of rs by unit (111) strain and N is the acceptor concentration. The trouble is that at high concentrations these diffuse centers also interact electronically to form impurity bands, and the Jahn-Teller energy (roughly 2 cm-l/ion at N = 1 0 2 0 ~ m - ~has ) to be larger than the bandwidth for distortion to be energetically favorable. Koonce suggests that In (which, while being very soluble in Si, is electronically more localized than the other acceptors) might have a chance. 111. The John-Teller Effect in Triply Degenerate States

The Jahn-Teller effect in a state with threefold orbital degeneracy (e.g., a TI or T2 term) is, in principle, much more complicated than in a E, doubly degenerate state. For instance, in O h , TI2 = Tz2 = A1, TI, Tz, , so that a T term couples in first order not only to LYI, and 6, distortions but to r2, distortions as well. Moreover, since orbital angular momentum L transforms as TI, , it is not quenched as it is in an E term, and a T term is, in general, split in first order by spin-orbit coupling. There are, however, some simplifying factors that enable us to treat the complications as perturbations in many cases. In the 3d group of ions, spin-orbit splittings are of the same order as the size of vibrational quanta; thus, if the Jahn-Teller effect is strong enough to produce important effects, it is probably stronger than the spin-orbit coupling. (This is not true for heavier ions.) Furthermore, in octahedrally coordinated transition metal ions, q, distortions couple much more strongly to the electronic states than do rz0 distortions. This is because only E, distortions couple to the unpaired e electrons, which are primarily responsible for the Jahn-Teller effect. No such simplification is possible for tetrahedral coordination, in which it is the t2 rather than the e orbitals which form CT bonds and are

+

*lo

C. S. Koonce, Phys. Rev. 134, A1625 (1964).

+ +

152

M. D. STURGE

most strongly coupled to the nuclear motions. We shall discuss tetrahedral complexes in Section 18. 14. AN OCTAHEDRALCOMPLEX IN

A

‘T STATE,COUPLINGONLY

TO eg

MODES distortions, and anharmonicity, If we neglect spin-orbit coupling, the problem becomes remarkably simple. The lowest-order Hamiltonian is [from Eq. (3.8)]

[

;+ 0

-q1

XI = A

0

:q]

T,

+

4110,2(Q2~

+

&3?1

(14.1)

operating on the state vector

where the electronic wave function is

*

= u1*t

+

a2J.7

+ a3*r .

(14.2)

(Nonorthogonal coordinates q1 , q2 , 43 and q:, q2/, 43‘ are defined in terms of Q2 and Q3 in Table IV.) T , is the nuclear kinetic energy due to motion in &2&3 space; w, is the frequency of vibrations in this space, and p the effective mass. We have used a real basis (4, q , {) for the T term, transforming under proper rotations as (z, y, z ) or as (yz, zz, zy) . I n this representation, the Hamiltonian is diagonal; that is, the electronic states are not mixed by the vibronic interaction. The potential surface in (Q2, Q 3 ) space is therefore quite different from that for an E term. It consists, in fact, of three separate (“di~joint”~) paraboloids, one for each orbital electronic state: ‘v.I - -i p ~ P [ ( q i- ~

0

+ qi”] )

~

i

- 6E,

+

=

1, 2, 3.

(14.3)

PO is the value of the radial coordinate (Q2z Q,2)1’2 at equilibrium; A / p . 2 . The Jahn-Teller energy 6E = A2/2pw>, as for an E term. Vi is the potential surface on which the nuclei would have moved in the absence of the Jahn-Teller effect, but displaced along the p i axis by P O ,

Here po =

THE JAHN-TELLER

EFFECT IN SOLIDS

153

FIG.20. Potential surface V ( Q 2 ,Q 3 ) for a triply degenerate state interacting with zp distortion [Eq. (14.3) 1. The electronic states associated with each paraboloid are

mutually orthogonal (in a cubic system with zero spin-orbit coupling). The three arrows in the basal plane correspond to the linear combinations of Qz and Q 3 given in Table IV, i.e., to tetragonal distortion parallel to 2, y, and z directions, respectively (adapted from Liehrs, p. 437).

and downwards by 6E.The threefold surface (14.3)is illustrated in Fig. 20, and a vertical section along the q3 axis in Fig. 21. Note that the upper branch is doubly degenerate on this axis, being the line of intersection of the paraboloids with i = 1 and i = 2. The physical reason for the failure of es distortion to mix the electronic states, and for the consequent disjointedness of the three paraboloids, is as follows. Both e,, distortions, Q2and Q 3 , preserve the C2 (180' rotation) V

t

FIG.21. Section through Fig. 20 along the 43 axis.

154

M. D. STURGE

axes along the x, y, and z directions. Under each CZoperation, two of the basis functions 5, q, change sign, but the other one does not (for instance, a 180" rotation about [OOl] changes #Eand #,, but not # r ) . Thus, all three states remain mutually orthogonal, even in the presence,of eg distortion.211 It follows that the lowest vibronic level is threefold degenerate, with the same symmetry (T1 or T z ) as the term from which it derives. The three component states have wave functions *i

= Vi#i

(14.4)

1

where #i = #t ,#,, , #r ,for i = 1, 2, 3. The vibrational parts are of the form (8.2) with a = a' (in the harmonic approximation). Higher vibronic states have the same # i but more highly excited cpi . The implication of this result is that, in the absence of spin-orbit effects and coupling to rzOdistortion, a TI or T z term should always suffer a static tetragonal Jahn-Teller distortion. This is contrary to observation, and it will be shown in the next section that even quite weak spin-orbit coupling will drastically modify this conclusion. 15. ANGULAR MOMENTUM AND WEAKSPIN-ORBITCOUPLING : THE HAM EFFECT Consider the angular momentum in a T term undergoing the JahnTeller effect described in Section 14. The nonzero matrix elements of L within a T term are ( E I L, I S) = (S I L, 17) = (q I L , 1 E ) = iZA, where 1 is a number of order one, which depends on the particular term. Since L is a purely electronic operator (the conditions under which the momentum of the nuclei can be neglected are discussed by Longuet-Higgins4), the matrix elements between the Born-Oppenheimer product states *i have the same form: (*1

I L, I *3)

=

(*3

I L z I *z)

= (*z

I Lz I *l>

= zlAy, .

(15.1)

where Y = (cpl

I cpz),

since the vibrational overlap integral y is the same for any pair of unequal indices. Thus, the orbital momentum is reduced by the factor y, which in the harmonic approximation (8.2) is Y = (cpl

I cpz)

=

exp ( -3a2p02/4)

(The harmonic approximation for 211

cp

=

exp (-336E/2Aw).

(15.2)

may be expected to be a much better

M. H. L. Pryce, private communication (1964). The situation is analogous to that discussed in Section 12.

THE JAHN-TELLER

155

EFFECT IN SOLIDS

approximation here than it was for an E term, because the potential minima are separated by high cusps rather than by low saddles.) Exactly the same considerations apply to the spin-orbit coupling operator XS L. Because S does not operate on 9,the spin-orbit matrix elements are reduced in the same proportion as those of L. The same is true of any operator whose orbital part is exclusively off -diagonal in the real representation forced on the electronic term by the Jahn-Teller effect (i.e., the representation in which the Jahn-Teller interaction is diagonal). For instance, the potential due to a trigonal distortion (i.e., a trigonal field) and all its matrix elements are off-diagonal in the has T 2 (El 7, {) basis. Their values are therefore reduced by the same factor y. The T state is still split in first order into B and d states of the trigonal group (CSv, D P d , etc.) , and the direction of quantization is the trigonal axis, not one of the tetragonal axes of Jahn-Teller distortion; but the splitting is reduced by the factor 7. This general reduction in off-diagonal elements was first recognized by Ham" (although special cases of it had , and we call it the Ham effect. been implicitly noted earlier165*212-214) I n the limit y + 0, the matrix elements of off-diagonal operators go to zero. This is just what we would expect for the static Jahn-Teller effect; freezing the complex into a potential minimum quenches the orbital angular momentum; and a trigonal field (say, due to a [lll] stress) cannot distinguish between the three equivalent [100]-type distortions, so that there can be no trigonal splitting. Diagonal operators, on the other hand, such as a tetragonal field, are not quenched; again we can easily see that a stress parallel to a cube axis distinguishes between distortions parallel to and perpendicular to that axis, and so produces a splitting. The opposite limit, spin-orbit coupling strong compared with the Jahn-Teller effect, has been considered by Van Vleck.66 (The conditions under which sufficiently strong spin-orbit coupling will stabilize a complex against Jahn-Teller distortion have also been discussed by Opik and Pryce.'*) This limit is likely to be reached in ions of the 3d group only when the Jahn-Teller effect is in any case too weak to produce important h). effects (because in such ions the spin-orbit splitting parameter X It is a condition likely to apply in the T terms of 5d and other heavy ions, however, where spin-orbit splittings are large (see Sections 26 and 27). I n the rare earth ions, the condition certainly holds, but Jahn-Teller

-

-

212

213

214

I. B. Bersuker and B. G . Vekhter, Fiz. Tverd. Tela 6, 2432 (1963) ; see Soviet Phys.Solid State (English Transl.) 6, 1772 (1964). D. E. McCumber, unpublished work (1964); quoted in M. D. Sturge, Phys. Rev. 140, A880 (1965). S. Washimiya, private communication (1966) ; paper given at the Autumn Meeting Phys. Soc. Japan, 1963 (nnpublished).

156

M. D. STURGE

coupling is, anyway, too weak to matter in the 4f”configuration. If we apply spin-orbit coupling to a term as yet unaffected by the Jahn-Teller effect, we will, in general, split it into some levels that are, and some that are not, degenerate (over and above Kramers degeneracy). If they are not, distortion will produce no further splitting, and the Jahn-Teller effect cannot operate in first order. Even if they do retain some degeneracy, it is often the case that this is “primarily” spin degeneracy, in which case the matrix elements of strain may be sufficiently reduced that the Jahn-Teller effect is too weak to be of importance. According to Van Vleck,66this is the case for the lowest (r,) level of Fe2+ (6T2ground term) in MgO. However, this argument presupposes that the Jahn-Teller effect in the term as a whole (not just within a single spin-orbit level) is weaker than the spin-orbit coupling; otherwise, distortion can mix the spin-orbit levels, and the Jahn-Teller effect may occur.215For instance, calculations by Baltzer216 on the 3T1 term of Ni2+show that a moderate tetragonal distortion has a bigger effect on the rl level than on the I’5 , contrary to what one would expect from simple considerations of degeneracy (see Fig. 22). Thus, Van Vleck’s argument must be used with caution, particularly for 3d” ions. For 4d ions and a fortiori for 5d ions, the spin-orbit coupling is so large that even quite a strong Jahn-Teller effect may not mix the spin-orbit levels. Then each degenerate level (excluding Kramers doublets, of course) may show its own Jahn-Teller effect, a r3 level behaving like the doubly orbitally degenerate E term of Section 5 , and so on. The Jahn-Teller effect and by Child.217 in a rs leveP3has been discussed by Moffitt and Thor~on’~ Its behavior under distortion is essentially the same as r3(see also Section 26). I n this case of very strong spin-orbit coupling, not only the orbital angular momentum but the total momentum, spin included, of a r4or r5 level is quenched by the Ham effect.67 The main consequence of the Ham effect is that, even when spin-orbit coupling is not strong enough to prevent Jahn-Teller distortion from occurring, the qualitative appearance of the energy levels is the same as in the absence of the Jahn-Teller effect, but the first-order splittings are reduced. (Second-order splittings are discussed in the next section.) For instance, a 4T1term is split by first-order spin-orbit coupling into r7, rs, and rsr6levels, rs and r6 being degenerate in first order. The same order of levels, with the same selection rules, will be seen in the presence of the Jahn-Teller effect, but the splittings will be reduced by the Ham effect. *I6

*I6 *I7

This effect, the reduction in energy of a nondegenerate level by distortion of the complex, is sometimes called the “pseudo Jahn-Teller effect.” It seems to the present writer to be just as genuine as any other manifestation of the Jahn-Teller effec,t. P. K. Balteer, J . Phys. SOC. Japan 17, Suppl. B-1, 192 (1961). M. S. Child, Phil. Trans. Roy. SOC.London A266, 31 (1962).

THE JAHN-TELLER EFFECT I N SOLIDS

157

Similarly, the g factors will be isotropic (or have at least the site symmetry) , but the orbital contribution to the g factors will be proportionately reduced. The Ham effect has some remarkable consequences in optical spectra, which will be discussed in Section 28. Here we consider the effect on the spin-resonance spectra of the ground level. A number of ions with T1 or Tz ground terms in octahedral environments have been studied by spin resonance, and no static Jahn-Teller distortion has ever been seen. The consequences of the Ham effect are seen most clearly in the g factors of d6 and fl ions at interstitial sites in Si67 These sites have cubic symmetry, and ions in them behave as if they were in a weak octahedral field, as far &s ordering of terms is concerned.l17e218A d6 ion has a 6Tzground term, of which d'I is the lowest levelzlg; the spin contribution to the g factor is 3.0, and the orbital contribution (in the absence of orbital reduction) is 0.5. Whereas in MgO: Fez+ g is 3.43219and in CaO :Fez+it is 3.3OlZ2O in Si the isoelectronic Mn+ and CrO have isotropic g factors of 3.01 and 2.97, respecti~e1y.l~' Such strong quenching of the orbital contribution obviously cannot

I

0.9

I

I.o

I

1.1

,

I

12

c/o

FIQ.22. Effect of tetragonal distortion on the 3T1 term of Ni2+. Interaction with all other terms of da is included in the calculation. The numerical values for c/a shown only apply for a point charge model of the tetrahedral site in spinel, but the same qualitative picture holds for any Ni*+ complex (from Baltzer216). 118

*I9

H. H. Woodbury and G. W. Ludwig, Phys. Rev. Letters 6 , 98 (1960). W. Low and M. Weger, Phys. Rev. 118, 1119 and 1130 (1960). A. J. Shuskus, J. Chem. Phys. 40, 1602 (1964).

158

M. D. STURGE

/r Oh

c3

L.S

"EXACT"

(1st ORDER)

FIG.23. Splitting of a 2 T term ~ (single d electron), under a trigonal field and spin~ neglected. In this approximation, the ground orbit coupling. Mixing of 2E into 2 T is -1 ) for all positive v and h and Kramers doublet is purely I TZ + 1 ), I 2T2 has the g values 911 = 2 - 2k, g 1 = 0, where k = - ( t z + l I I , I t 2 + l ) .

++

-+

be explained by the traditional mechanism of covalency.zz1~2zz Even in CaO the Ham effect is probably operative. For d7 ions, the spin contribution is 10/3, and the calculated orbital contribution to the g factor is 1.0. The expected orbital contribution is ob~ other ~ ~ hand, ~ ~ ~ ~ served for NaF: Fe+,223 and it is 0.95 for MgO :C O ~ +On. the it is only 0.19 in Si:Fe+ and 0.03 in Si:Mn0.117Again, the last two figures are only explicable in terms of the Ham effect. There appears to be a moderate Ham effect operative in the zTzground term of Al2O3:Ti3+,a 3d1 system. The splitting of the ground term under the trigonal field v and spin-orbit coupling A, and the ground state g are given in Fig. 23. The observed g valueszn are gll = 1.067, gl < 0.1, from which we conclude that the order of the levels is as predicted, and that the orbital reduction factor k = 0.48. The lowest excited electronic levels have been determined by far-infrared spectroscopyzz8to be 37.8 and 107.5 cm-I above the ground state. Fitting these splittings to the lowest-order theory gives an effective value of A of about 50 cm-l, and of v of about 75 cm-l. These numbers are to be compared with the free ion 221

J. Owen, Proc. Roy. Soc. A227, 183 (1955).

The anyway implausible interpretation in terms of complete delocalization of the spin is ruled out for Mn+ by the observation of a large isotropic hyperfine structure arising from interaction of the electron spin with the nuclear moment of Mn. 223 B. Bleaney and W. Hayes, Proc. Phys. Soc. (London) B70, 626 (1957). 2z4 W. Low, Phys. Rev. 109, 256 (1958). 226 W. Hayes and J. W. Orton, unpublished work (quoted in Bleaneyz23). 226 D. K. Ray (D. K. M i ) , Piz. Tverd. Tela 3 , 2525 (1961); see Soviet Phys.-Solid State (English Transl.) 3, 1838 (1962). L. S. Kornienko and A. M. Prokhorov, Zh. Eksperim. i Teor. Fiz. 38, 1651 (1960) ; see Soviet Phys. J E T P (English Transl.) 11, 1189 (1960). 228 E. D. Nelson, J. Y. Wong, and A. L. Schawlow, Phys. Rev. 166, 298 (1967); E. D. Nelson, Ph.D. Thesis, Stanford University (1966).

222

THE JAHN-TELLER

EFFECT I N SOLIDS

159

values k = 1, A = 150 crn-l, and the value of v for other trivalent 3dn ions in A1203, which is about 800 cm-l. I n agreement with Ham’s theory, A and k are reduced by approximately the same factor, but v appears to be quenched more strongly. A more accurate calculation, including the effect of 2Tr2E mixing, is necessary to determine whether this discrepancy is significant. As we shall see in the next section, “second-order” effects, of which 2T2-2E mixing is only one, can be very important when the Ham effect is pronounced.228a A rather similar far infrared spectrum to that of Ti3+ is found in vanadium doped A1203229; this is attributed to V4+,which is isoelectronic with Ti3+. The case of MgO :Fez+ is puzzling. As mentioned previously, it has an isotropic g factor in spin resonance a t 24 kMc/sec, corresponding to a very weak or nonexistent Jahn-Teller effect (which is apparently quenched by spin-orbit coupling66).Nor is there any evidence for a Jahn-Teller effect in the acoustic paramagnetic absorption at 3 kMc/sec.’” On the other hand, the Mossbauer effect in dilute crystals of MgO: Fez+126 shows a small quadrupolar splitting below 14”K, which is evidence for axial distortion. Since the characteristic time for motional narrowing of the Mossbauer line (-10-7 sec) is much longer than that for spin resonance (fi/AgBH10-lO sec), a static distortion in the Mossbauer effect should show up in spin resonance. A possible reason for the discrepancy is that the Fez+ ions seen in the two experiments are a t different sites. This seems unlikely,2” although it is apparently the cause of a superficially similar discrepancy in NiAs: Fe3+.231 A more plausible explanation has been given by Ham,232 who shows that the quadrupolar splitting can be accounted for in terms of random local strain, without invocation of the Jahn-Teller effect. However, there is more work, both experimental and theoretical, to be done on this problem.

16. HIGHER-ORDER TERMS IN

THE

HAMEFFECT

So far, we have only considered matrix elements within the lowest vibronic level of a Jahn-Teller distorted T term. In a perturbation exA full calculation of the first- and second-order effects, including 2T2-ZEmixing, accounts quantitatively for the infrared and spin resonance data [R. M. Macfarlane, J. Y. Wong, and M. D. Sturge, (submitted to Phys. Rev.)]. *19 J. Y. Wong, M. J. Berggren, and A. L. Schawlow, in “Optical Properties of Iohs i n Crystals.” Wiley, New York (to be published). em D. N. Pipkorn, private communication (1966). ral G. Bemski and J. C. Fernandes, Phys. Letters 6, 10 (1963); T. L. Estle, Phys. Rev. 136, A1702 (1964). F. S. Ham, Phys. Rev. 160, 328 (1967). m a

160

M. D. STURGE

pansion, these would be called first-order effects. We now turn to higherorder effects, which involve matrix elements taking us out of the level of interest; we lump them together under the heading “second-order effects.” However small such matrix elements may be in the absence of the JahnTeller effect, if they are not reduced proportionately to the first-order matrix elements by the Ham effect they will ultimately become dominant. It will be shown that this limit corresponds to the static Jahn-Teller effect. There are two classes of second-order effect to be examined. The first class arises from matrix elements connecting different terms and would be there even if there were no Jahn-Teller effect. The second class arises from matrix elements between vibronic levels of the term and would not be there if there were no Jahn-Teller effect. In both cases, we shall see that only second-order matrix elements diagonal in the basis states (14.4) are usually of importance. The first class, if diagonal, is more or less unaffected by Jahn-Teller distortion, so long as the terms involved are well separated relative to the Jahn-Teller energy. (This is the condition for the validity of a perturbation approach, of course.) This can be seen as follows: the second-order matrix elements of a perturbing operator x’within the vibronic states (14.4) are given by [Eqs. (1.2) and (3.10)]

where Eo is the energy of the vibronic states of interest, and E ( k , n) is the energy of a vibronic state belonging to a different electronic term. If we assume that x’is a purely electronic operator, and that the separation between terms is large compared to the vibrational quanta or to the JahnTeller energy, we may write

The second sum is simply (cpi I cpj) by closure. The second-order matrix element is thus reduced by the same factor (cpi I cpj) as the corresponding first-order element (if the latter exists), i.e., by the factor y [see Eq. (15.2)] if it is off-diagonal, but not at all if it is diagonal. Thus, although we can often neglect off -diagonal second-order matrix elements, when the Ham effect is pronounced diagonal ones may become of paramount importance. Another way of putting this is simply to point out that, whatever the nuclear configuration associated with the electronic state 1 i), there exists somewhere a state I k) associated with the same configuration. The

THE JAHN-TELLER EFFECT IN SOLIDS

161

energy of I k ) will be changed, but only by something of the order of the Jahn-Teller energy 6E, which is ex hypothesi small compared to the term separation. When two terms are so close that they are appreciably mixed by the Jahn-Teller effect, closure can no longer be used, the distinction between first- and second-order effects breaks down, and each specific case has to be considered on its merits. The general formulas for the second class of second-order effects are complicated and are given by Ham.67 However, in the limit of a strong Jahn-Teller effect (6E >> nu), they simplify greatly; this is the range in which one would expect them to be most important. Consider Fig. 21. The minimum of the lower curve corresponds to the electronic state I S) associated with a certain nuclear configuration. The point on the upper curve, vertically above this minimum, corresponds to the degenerate states I q ) , I E ) in the s u m nuclear configuration. Thus, however strong the JahnTeller effect is, matrix elements connecting the two branches remain finite, giving contributions to X$) of the form I([ I X' I q)I2/36E (36E is the separation of the upper and lower curves a t q3 = P O ) . Because 6E, although large relative to nu, is usually small relative to term separations, the contributions of this type of second-order effect can be quite substantial. The matrix elements (and the energy denominators) that appear in (16.2) are evaluated in a given, Jahn-Teller distorted, nuclear configuration and, in general, will reflect the symmetry of this configuration. Thus, when the Ham effect is so strong that second-order effects are larger than first-order ones, the properties of the lowest vibronic level reflect the Jahn-Teller distortion; that is, we have a static Jahn-Teller effect. As a simple example of this changeover from the dynamic to the static JahnTeller effect, consider the 3T2excited term of V3f or Niz+in a cubic environment. If first-order effects are completely quenched, the lowest level associated with [OOl] distortion is the spin triplet I 3TzM,{), where M , = 0, f l . Interaction with other terms (in V3f, principally the 'TZ term) splits this degeneracy; we find a contribution to the splittingz33

E(M,

=

f l ) - E(M,

=

=

0)

3x2/[E(3Tz)- E ( ' T z ) ] ,

(16.3)

where AS-L represents the spin-orbit coupling operator. Another contribution to the splitting is from the upper Jahn-Teller branch of the 3T2 213

W. C. Scott and M. D. Sturge, Phvs. Reu. 146,262 (1966).

162

M.

D. STURGE

term; this is31,z33 h2C Af

f

c

f l ,{

1

I=[.?

S - L I 3TzMsflZ)Iz - 1(3T20,{ -3 6E

I

=

S - L I 3TzMafZ)(z -hx2/12 6E.

(16.4)

Turning to the g factors, we note that (3Tz{I L , I 3TzZ)= 0 for all Z, whereas (3T2{I L, 1 3T2v)# 0, so that 911 # g 1 . Thus, we arrive at the usual form of spin Hamiltonian for an X = 1 ion in a tetragonal fieldz0: X s

= gllPSzHz

+ g l P ( S z H z + SuH,) + D[S2

-

gS(S

+ I)],

(16.5)

where D is given by a sum of terms like (16.3) and (16.4). This spin Hamiltonian is appropriate for the 3 of the ions which have static distortions along the [OOl] axis; the others are described by permuting z, y, and 2. On the other hand, if the Ham effect is sufficiently weak that first-order spin-orbit coupling is still dominant, the 3Tz state is split into rZ, r5 , and r3r4levels, of which r3and r4are split apart by second-order effects. The r5level will have an approximately isotropic g factor, but that of the r4may be expected to become progressively more anisotropic as the contributions to the r3r4splitting of off -diagonal second-order matrix elements (containing factors like (qlI qZ)) are reduced relative to diagonal ones (containing (ql1 cpl)). 17. JAHN-TELLER COUPLING TO rZgDISTORTIONS IN OCTAHEDRAL COMPLEXES

If we neglect coupling to t odistortion, and concentrate on the rZgmodes, the Hamiltonian (3.8) becomes

XI

=

B[

0

&6

&6

0

&]

Q5

&4

0

&5

+ TT+

+ +

i@J~'(&4~

&5'

&6').

(17.1)

Here T, is the nuclear kinetic energy due to motion in Q 4 , &6 , &6 space, and w , is the frequency of rzOvibrations; as before, the Hamiltonian operates on the state vector

THE JAHN-TELLER

163

EFFECT IN SOLIDS

+

+

+

where the electronic wave function is = a& a2+? As before, we neglect spin-orbit coupling and anharmonicity. The fact that the first term cannot be diagonalized simultaneously with the other two makes the calculation for this case much more difficult than for coupling to E, distortion. We solve the static problem as in Section 5, by "fixing" the nucleiss and diagonalizing the electronic term in (17.1) alone. The secular equation for the electronic energy E is

+ +

E3 - B2E(Q42

Q5'

Q6')

- 2B3&4Q5Q6=

0.

(17.2a)

The potential in which the nuclei move is V(Q4, Q5 , Q6)

=

+ +

E 4-i p ~ r " ( Q 4 ~ Q5'

(17.2b)

Q6').

The exact form of V is complicated and not very important to us. We see from (17.2) that it has three sheets, and that its equipotentials are the surfaces in Q4Q5Q6 space defined by79 Q4'

+ + Q52

&62

= fi

,

Q4Q5Q6

= fz

,

where f1 and fi are complicated functions of V . The potential hypersurface (17.2b) has been imaginatively depicted by Liehr.5 The minima in V can be by differentiating (17.2). There are four equivalent minima corresponding to trigonal distortion in (111 )-type directions (see Fig. 4e); the potential at these minima is -6E = -2B2/3pw,Z. I n the "static" limit, 6E >> nu,, the lowest vibronic level is fourfold degenerate, corresponding to the fact that the distortion can be in one of four equivalent directions. In consequence, this lowest level T1 or A , TZ , and can transforms as a reducible representation, A2 be split by tunneling. The electronic wave function in the ith minimum, corresponding to [hkl] distortion ([hkl] = [lll] for i = 1, [Till for i = 2, [lTT] for i = 3, [iii] for i = 4 ) , is

+

*;

=

3-1'2(h*€

+

+ k*? + I*().

(17.3)

The zeroth-order vibronic wave function is \k = $ ~ i x i, where xi is the ground state vibrational wave function for nuclei moving on the potential surface near the ith minimum. As in the case of the static E,, distortion, there are two frequencies of vibration, W I I = W , for motion parallel to the axis of distortion, and WI = ( $ ) 1 ' 2 w 7 for doubly degenerate motion perpendicular to it. If the vibrational wave functions x i overlap, tunneling between potential minima occurs, and the A state, whose wave function \kz \k3 \k4}, is raised above the T states, is to zeroth order whose wave functions are

+

$[*I

- *z - \ k 3

+ \k41j

$[*I

+

+

- \k2

+

\k3

-941,

$[*I

+q z -

*3

- *4]. (17.4)

164

M. D. STURGE

k

FIG.24. Vibronic energy levels of a T 2 electronic state interacting with 7 2 distortion. The unit in the abscissa is B/(pLA0,3)1/2, and in the ordinate LAO, . For a TI electronic state, interchange all subscripts (from Caner and Englman234).

The vibronic energy levels of (17.1) have been calculated numerically by Caner and Englman.z34Their results for the lower levels are shown in Fig. 24. Here k2( = B2/pfiwT3)is a measure of the strength of the JahnTeller coupling. The energy unit is f i w r , and the symmetries are correct for a Tz electronic state. (For a Tl state, interchange all subscripts.) On the left are the levels for zero coupling, with n ( n = 0 to 3) 7Zg vibrations excited. The splitting for weak coupling is analogous to the spin-orbit splitting of a zS+lT state, where S can be identified with the angular momentum of the nuclei in r2 coordinate space.7sIn the strong coupling limit (extreme right), each energy level is at least fourfold degenerate, since there are four possible directions of distortion. For each direction, the levels are those of a three-dimensional harmonic oscillator with axial ( (111) type) symmetry, as discussed previously. The tunneling splitting of the lowest level (when small) i P 4

E(A1) - E(Tz)= 0.88fiwk2exp ( -0.827k2). 234

M. Caner and R. Englman, J . Chem. Phys. 44, 4054 (1966).

(17.5)

THE JAHN-TELLER

EFFECT IN SOLIDS

165

This has the same form but the opposite sign and roughly half of the magnitude of that given previou~ly.~3~ These calculations show that (as predicted by Ham6’) the qualitative consequences of the Ham effect obtain in this case. First-order matrix elements of off -diagonal operators, which now include not only orbital angular momentum and spin-orbit coupling, but also tetragonal fields, tend exponentially to zero in the strong coupling limit. Matrix elements of Tz (diagonal) operators, such as the trigonal field, remain finite, but at Q of their initial values. This reduction is because in the strong interaction limit the Tz states cannot be quantized along any particular (111) axis, as can easily be seen by writing out the wave functions in full, using (17.3) and (17.4), and rotating to the (111) axis. The exact dependence of the quenching factor on the coupling strength is given by Caner and Englman.2” We now turn to the much more difficult (but probably fairly common) situation where coupling both to eg and rZg modes is strong. (When the coupling to both modes is weak, they act independently, and nothing very new ernerge~.’~) When coupling is strong, the problem is mathematically intractable. opik and Pryce78 have treated the general static problem [determination of the potential surface V(Qz- .Q6)] in the linear approximation. They show that, at the potential minima, either E, or 72, distortion is nonzero, but never both a t once. There is a set of energy extrema corresponding to [llO]-type (orthorhombic) distortion, in which both e, and 7 Z g distortions are present, but these are saddle points, not minima. Although this statement is true with regard to the positions of the potential minima, i t does not necessarily extend to the dynamic problem. Certainly at finite temperatures one would expect rZg vibrations to cause transitions between the different potential minima in Qz , Q3 space, as discussed by Bers~ker.~~~ Bersuker and Vekhter212~235,236 treat the problem of weak T~~ coupling in the presence of strong e a coupling. They claim to find a small but finite static r2 distortion on top of the large tetragonal e distortion. Such a distortion mixes the electronic wave functions in different potential minima and produces a splitting of 3y2B2/pw?, where y is the overlap between the e ground state vibrational wave functions, (cpl 1 pZ).The trouble with this argument is that it predicts a splitting of a T vibronic state. [The ground state of the 7 2 vibrations, x ( Q 4 , Q5 , Q6), is nondegenerate, like that of any harmonic oscillator.] This is impossible, since the Hamiltonian still has cubic symmetry. The fallacy apparently lies in the substitution that leads which does not preserve the cubic symmetry of the to their Eq. (6),236

-

235

236

I. B. Bersuker, Zh. Eksperim. i Teor. Fiz. 43, 1315 (1962); see Soviet Phys. J E T P (English Transl.) 16, 933 (1963). I. B. Bersuker and B. G. Vekhter, Phys. Status Solidi 16, 63 (1966).

166

M. D. STURGE

problem. If one does the perturbation calculation c o r r e ~ t l y ,one ~ ~ simply ~.~~~ reproduces Moffitt and T h o r ~ o n ’ result s ~ ~ for weak 7 2 coupling. (The strong E interaction merely reduces the effective 7 2 coupling constant from B to rB, an example of the Ham effect.) As can be seen from Fig. 24, there is a shift but no splitting of the lowest (T) state. There is splitting in states containing r2 excitation, leading to the thermal effects mentioned previously. If coupling to 7 2 distortion were strong enough, one might imagine that the lowest state would become fourfold degenerate (A1 T 2 ) ,as in the case of pure 7 2 interaction. However, this would require y2B2 2 pfiw:. Since y ,- exp (-3A2/4pfiu,3), this is not consistent with the initial assumption that E coupling is stronger than r2 coupling (i.e., that A 2 / p w ? > B 2 / p w T 2 )unless w T << w. . This conclusion might possibly be upset by strong nonlinear interaction between E and r2 distortions. All treatments of the interaction with r2 distortion so far have used the quasi-molecular model. In particular, the foregoing argument disposing of Bersuker and Vekhter’s “inversion splitting” of a T state depends on there being a finite excitation energy for 7 2 vibrations. It may be that the quasi-molecular model is not so good an approximation as it was in the E case. Since there are effects that apparently involve 7 2 interactions and have not received any satisfactory explanation in terms of current theory (see, in particular, the end of Section 28), a theoretical treatment of interaction with a continuum of 7 2 vibrations would be of great interest.238 No satisfactory explanation of the anomalous temperature dependence239

+

F. S. Ham, private communication (1966). Such a calculation was attempted by K. W. H. Stevens and F. Persico [Nuovo Cimento [lo] 41B, 37 (1966)l. These authors considered an S = I ion (r4 or r6 ground state) interacting with a Debye spectrum of acoustic phonons. They predicted a splitting between the two allowed AM, = 1 transitions, which are coincident in the static cubic case. However, subsequent work by H. A. M. van Eekelen and K. W. H. Stevens [Proc. Phys. Soc. (London) 90, 199 (1967)] has shown that Stevens and Persico’s calculat,ion was incorrect. There is no splitting of a triplet state by interaction with the phonons, a t any rate within the random phase approximatlion. As for experiment, there are anomalies in the spin resonance line shapes of S = 1 ions, particularly Ni2+ in MgO and CaO [Orton et aL’20; W. Low and R. S. Rubins, in “Paramagnetic Resonance” (W. Low, ed.), Vol. 1, p. 79. Academic Press, New York, 19631. However, where detailed analysis has been attempted, as in t,he case of MgO:FeZ+, it has been found possible to account for the line shapes quantitatively in terms of random local strain [D. S. McMahon, Phys. Rev. 134, A128 (1964)l. 239 D. Bijl, Proc. Phys. Soc. (London) A63, 405 (1950); B. Bleaney, ibid. p. 407; B. Bleaney, G. S. Bogle, A. H. Cooke, R. J. Duffus, M. C. M. O’Brien, and K. W. H. Stevens, ibid. A68, 57 (1955); S. K. Dutta-Roy, A. S. Chakravarty, and A. Bose, Indian J. Phys. 33, 483 (1959); A. Bose, A. S. Chakravarty, and R. Chatterjee, Proc. Roy. SOC.A266, 145 (1960).

237 238

THE JAHN-TELLER

EFFECT IN SOLIDS

167

of the g values of Ti3f in CsTi alum has yet been given. Although interaction with rZadistortions may well be important, Bersuker and Vekhter’s treatment212appears to be inapplicable for the reasons just given. 18. TETRAHEDRAL COMPLEXES For coupling to 6 distortions, the tetrahedral problem is, in principle, identical to that of E,, distortions of the octahedron (Section 14). For r2 distortions, which might be expected to be a t least equally important in the tetrahedron,240the already complicated problem is made worse by the fact that there are two r2 modes in a tetrahedral complex. In consequence, In “Jahn-Teller movements in r2 space are inseparably six-dimen~ional.”~ view of this, we shall refrain even from writing down the Hamiltonian corresponding t o (17.1) for the tetrahedron. (The interested reader with a strong stomach for formulas is referred to Section 3.2 of Liehr.5) The cubal X U , complex is formally identical with tetrahedral X U ( , having an e a and two rZamodes. Although the tetrahedral problem might seem hopelessly complex, some simplification is possible in practice. We see from Fig. 5 that the ligand motion in the 72a mode ( Q 4 , Q5 , Q6) is radial, whereas that in the nb mode ( Q 7 , Q 8 , Q 9 ) , as in the E mode, is tangential. I n a molecular orbital treatment, rZacouples to u bonds, r Z b only to T bonds. We would therefore expect that in a covalent complex the coupling to the r Z a mode will be much stronger than to the rZb mode. Even if coupling to rZbis not weak, it is quite likely in any particular case that we can find linear combinations of rZaand rZbsuch that coupling to one combination is weak enough t o be neglected. For instance, consider the linear combinations Q6’ = ($) ‘I2& - ( f )112&9, Q; = (i)‘12&6 (4) ‘12&g (Fig. 25) . Q 6 ’ resembles the rza mode Q6 of the octahedron, whereas Q9’ resembles the rlUmode Q12, which cannot couple to the electrons in first order. I n a point charge calculation, we would expect to find that Q9’ couples much more weakly than Q6‘. If we can find some excuse for neglecting coupling to one set of 7 2 modes, the Hamiltonian for the tetrahedron reduces to (17.1). Furthermore, the important property of (17.1), that its potential minima correspond to trigonal distortions along (111) axes (as illustrated in Fig. 4e), is carried over to the tetrahedral case.5 The main difference is that there

+

240

This expectation is (perhaps fortunately) riot borne oat by the meager experimental data available on transition metal ions. The cooperative Jahn-Teller distortions observed in spinels containing large concentrations of tetrahedrally coordinated Cu2+ or Nit+ are tetragonal [R. J. Amott, A. Wold, and I). B. Rogers, J . Phys. Cheni. Solids 26, 161 (1964)l.The indications are that in Wurtzite type crystals CuZ+ and Cr2+ suffer tetragonal distortion in their ground states (see below). With two known exceptions, the only r2 distortions are in excited states.

168

M . D. STURGE

Qb FIG.25. Linear combinations of the T~ normal coordinates of the tetrahedral X Y , complex, chosen to illustrate their approximate “odd” and “even” character.

are now two possible (111)-type distortions, illustrated in Figs. 5f and 5g; the actual minimum will correspond to some linear combination of these two, as determined by the relative coupling to the rZa and rzbmodes. The qualitative consequences of Jahn-Teller coupling to rz distortions are essentially the same as for the octahedron. Since orbital angular momentum transforms as T I in T d , it is quenched by the Ham effect just as in On ; the same goes for spin-orbit coupling and any other off-diagonal operator. The Ham effect may be operative in the 2Tzground term of Cuz+ in Zn0,241~242 although a detailed analysis taking the Jahn-Teller effect into account has not been made. Remarkably small values of the orbital reduction factor k and effective spin-orbit coupling parameter A are found, although the energy level scheme is qualitatively what one would expect for a weak trigonal field. Although the low value of k could be explained in terms of very strong covalency, A is too small to be explained in this way. (This difficulty led the authors and his co11eagues,24zincorrectly in his present opinion, to a rather forced interpretation of the data in terms of a value of A closer to the free ion value.) It is probable that the Ham effect, rather than covalency, is primarily responsible for the large reductions seen. Support for the interpretation in terms of the Ham effect comes from the observation that A’, the off-diagonal spin-orbit parameter (A’ = -21’2(t2 +$, +1 I S - L I e +$, + I ) ) , is not reduced. That the Jahn-Teller distortion is tetragonal is suggested by the fact that the apparent value of V , the trigonal field parameter within the t z orbitals ( V = - 3 ( t z +1 I Vtrig 1 t z + l ) ) , is very much less than the off-diagonal trigonal parameter V’ (v’ = (tz +1 I VtrigI e +1)). The Ham effect is probably present in excited terms of Co2+and Niz+ in tetrahedral coordination; these are discussed in Section 28. 241

242

M. de Wit and T. L. Estle, Bull. Am. Phys. SOC.[ a ] 8, 24 (1963). R. E. Dieta, H. Kamimura, M. D. Sturge, and A. Yariv, Phys. Rev. 132,1559 (1963) ; see also R. E. Dietz, H. Kamimura, and M. D. Stnrge, Bull. Am. Phys. SOC.“.2] 8, 215 (1963).

THE JAHN-TELLER

EFFECT IN SOLIDS

169

Static distortions attributed to the Jahn-Teller effect have been observed243in the spin resonance of CdS: Cr2f. Similar results are obtained for ZnSe:Cr2+.244 Transitions are seen between the M , = &2 states of the 6T2 ground term of Cr2+. Six inequivalent sites are observed, with principal axes (in CdS) 58” from the hexagonal c axis in (1150) planes. These axes lie very close to the (001)-type directions of the tetrahedron (in a “cubic” coordinate system in which the c axis of the crystal is (111)). The spectrum broadens to vanishing at 20”K, but no isotropic spectrum appears. If the distortion is due to the Jahn-Teller effect, and not to some associated defect, the data can be explained in terms of an E distortion ( Q 3 of Fig. 5) lowering the symmetry to D 2 d . (All 12 nearest-neighbor cations are apparently equivalent in the SHFS,244ain agreement with this symmetry.) M0rigaki~~3 postulates a mty p e distortion (Q9’ in Fig. 25), lowering t,he symmetry to Czu . In the linear approximation (and neglecting the trigonal distortion of the tetrahedron in CdS, which is extremely small), such a distortion is not table.^ The nonlinear effects that stabilize the CZu distortion in the case of the negatively charged silicon vacancy (see Section 19) should not occur here. Not only are bonding requirements not critical (as evidenced by the fact that Cr2+has the high-spin, Hund’s rule, ground term), but in a four-electron system such as this they would, anyway, stabilize the tetragonal distortion. Static trigonal distortion along ( I l l )-type directions is observed in a spin-resonance spectrum attributed to V2+ in CaF2 .245 Here the d3 ion is cubally coordinated and has the 4Tl(t2e2)ground term. Although the distortion is probably due to the Jahn-Teller effect, it could be due to some associated defect, since no motional narrowing is observable as the temperature is raised. (The same is true of tetrahedral Cr2+ discussed in the previous paragraph.) If this is indeed a Jahn-Teller case, it is the only reported case where coupling to T~ modes is dominant in the ground state of a transition metal ion, so that the distortion is trigonal. A case where the Jahn-Teller effect may be important is that of Ni2+ in the tetrahedral site of the spinel structure. This site has cubic symmetry, and spin-orbit coupling brings the nonmagnetic rl state of 3T1 lowest (see Fig. 22). The contribution to the magnetic anisotropy of a ferrimagnetic spinel should therefore be small. It has been pointed out that this contribution could be large.216If there were a tetragonal Jahn-Teller K. Morigaki, J . Phys. SOC.Japan 18,733 (1963); 19, 187 (1964) ; T. L. Estle, G. K. Walters, and M. de Wit, in “Paramagnetic Resonance” (W. Low, ed.), Vol. 1, p. 144. Academic Press, New York, 1963. 244 T. L. Estle and W. C. Holton, Phys. Rev. 160, 159 (1966). 244a Superhyperfine structure. 245 U. T. Hochli, Bull. Am. Phys. SOC. [ a ] 11, 203 (1966). z43

170

M. D. STURGE

distortion splitting the 3T1 term is as shown in Fig. 22. An interesting point is that the large spin-orbit coupling in Ni2+ensures that the doublet (3E of S,) is just about as stable as the singlet. It would be necessary to assume that the doublet is lower in order to account for the sign of the observed anisotropy.216However, there are other explanations of the anisotropy data, and spin resonance and optical studies of Ni2+ in diamagnetic spinels would be of great interest. The r2 mode is infrared active in Td symmetry, and it should be possible to study the low-lying 7 2 vibronic levels of tetrahedrally coordinated ions with triply degenerate ground terms by far-infrared spectroscopy. No such work in solids has yet been reported. 19. VACANCIES AND OTHERDEFECTS IN THE DIAMOND STRUCTURE Radiation damage centers in diamond and silicon differ from the other centers in which we have been interested, in that Jahn-Teller coupling is strong, whereas other interactions, in particular interelectronic repulsion, are apparently relatively weak. For this reason, the effective Hamiltonian approach breaks down; there is no set of zeroth order states we can usefully choose which is not completely mixed up by the Jahn-Teller interaction.246 Fortunately, these centers can be understood in terms of a remarkably simple one-electron molecular orbital picture, in which covalent forces are all-important, electrostatic interactions play a minor role, and spin-orbit coupling can be neglected altogether. The model depends on the fact that in the perfect diamond lattice all the valence electrons form u bonds (the spa orbitals of organic chemistry). When an atom is removed to make a vacancy, the broken bonds “dangle”247from the neighboring atoms, as illustrated in Fig. 2K2& Unbroken bonds are assumed to be more or less unaffected. The occupation of the dangling bond orbitals depends on the charge state of the defect, which can be varied over quite a range in a semiconductor such as silicon by appropriate doping. This difficulty has led some authors (see, for instance, Liehrs) to doubt whether the distortions we discuss in this section can be attributed to “Jahn-Teller effects.” This seems to me to be a pointless distinction. The Jahn-Teller theorem is anexistence theorem; it states that orbital degeneracy leads t o distortion. The nature of the distortion is not specified (except that it must split the degeneracy). We shall find that the type of distortion that occurs is determined by bonding requirements and is sometimes inconsistent with the predictions of a linearized Hamiltonian such as (3.8). This merely shows that the Hamiltonian is inadequate in this case. The important fact is that, when there is electronic degeneracy, there is spontaneous distortion, and when there is not, there is no distortion. 247 W. Shockley, Phys. Rev. 91, 228 (1953). 2 4 8 See V. Heine [Phys. Rev. 138, A1689 (1965); 146, 593 (1966); 146, 568 (1966)l for critical discussions of this concept in terms of band theory.

*46

T H E JAHN-TELLER

EFFECT I N SOLIDS

171

Consider the hypothetical triply positive vacancy that we denote 03+. It has Td symmetry and has one electron outside closed shells. This electron can be in any one of the four bonding orbitals a, b, C, d (see Fig. 26). Overlap between the orbitals will split the fourfold degenerate state as indicated in Fig. 27a. This gives the basic one-electron scheme of the defeckZ49Let us neglect (for the moment) electrostatic and exchange interactions between the electrons. Then, as we add successive electrons to the defect, we shall fill up the levels of Fig. 27a. When a degenerate orbital is partially occupied, we have a Jahn-Teller sensitive state, and the defect will (in general) distort to remove the degeneracy. The presence of nearby impurities, or external fields, can of course also remove degeneracy. This model has been put on a more quantitative basis for diamond by Coulson and K e a r ~ l e yand ~~~ Y a m a g u ~ h i , ~who 5 ~ ~include ~ ~ ~ electrostatic interactions between electrons. They find that, besides the low-lying state of minimum spin predicted by the one-electron theory, there are other low-lying states (in particular, the “Hund’s rule ground state” of maximum spin). The relative positions of these various states depend rather critically on the details of the calculation. However, it is found experimentally that the ground state is always that predicted by the simple model, although in one or two cases an excited state of higher spin has also been found. In particular, whenever orbital degeneracy is predicted by the model, JahnTeller distortion is seen in spin-resonance.

FIG.26. “Dangling bonds” a t a vacancy in the diamond structure. G. D. Watkins, review an “Effets des rayoririements sur les semiconducteurs” (P. Baruch, ed.), p. 97. Dunod, Paris, 1965. 260 C. A. Coulson and M. J. Kearsley, Proc. Roy. SOC. A241, 433 (1957). See also J. Friedel, M. Lannoo and G. Leman Phys. Rev.,(1967) in press. 261 T. Yamaguchi, J . Phys. SOC. Japan 17, 1359 (1962). 252 T. Yamaguchi, in “Effets des rayonnements sur les semiconducteurs” (P. Baruch, ed.), p. 323. Dunod, Paris, 1965.

249

172

M. D. STURGE

o+

(0)

(a+ b -c-d) b c+d) (a-b+c-d)

+

(a+b+c+d)

*

(a

s4

Td

\

b2

- -

0-

c 2"

-

(a b-c +d)

*,

FIG.27. (a) Splitting of the bonding orbitals in cubic symmetry by overlap. The t 2 - a l splitting is thought to be about 0.5 eV. On the right is an indication of how these

orbitals are filled in the positively charged vacancy, o+.(b) Splitting of the bonding orbitals as the symmetry is reduced from T d , first to D2d (tetragonal distortion), and then to C z . (tetragonal plus trigonal distortion).

Watkins and his collaborators11"'16~249~252a~253 have studied a large number of defects in silicon by spin-resonance and ENDOR.253aThese techniques give not only the symmetry of the center, but also a measure of the unpaired spin density at the various nuclei surrounding the defect, through the hyperfine structure (HFS). The defects studied include the negatively charged, neutral, and positively charged vacancies, either isolated or associated with impurities, or with each other. The simplest case is the positively charged vacancy El+, which has three electrons. These fill the orbitals as shown on the right of Fig. 27a, giving a 2T2state. Such a state is unstable against trigonal ( ~ 2 ) or tetragonal ( E ) distortion. Spin-resonanceZ63shows that, in fact, there is a static tetragonal distortion of the Q 3 type (see Fig. 5). The HFS shows that the unpaired electron is shared equally among all four silicon atoms. G. I). Watkins, Phys. Rev. 166, 802 (1967). G. D. Watkins, J. Phys. SOC.Japan 18,Suppl. 11, 22 (1962). 26% Electron-nuclear double resonance.

252a

263

T H E JAHN-TELLER

EFFECT I N SOLIDS

173

The one-electron energy levels in the distorted situation are indicated at the left in Fig. 27b.254 As the temperature is raised, motional narrowing is observed, with an activation energy of 0.01-0.02 eV. With this activation energy, all motion should be frozen out at 2°K. In fact, however, reorientation of the direction of distortion occurs at a rate greater than 10 sec-I, even at this tempera t ~ r eThis . ~ ~unexpectedly ~ rapid reorientation must arise from some sort of tunneling process. As we have seen (Section 14), in the absence of spin-orbit coupling, tunneling in the usual sense is not possible in a tetragonally distorted T state. However, spin-orbit coupling destroys the orthogonality of the orbital wave functions and makes tunneling possible. The tunneling rate should be given by some such expression as (8.4), with u’ replaced by A/h. In a sense, this is a limiting case of the Ham effect. The neutral vacancy has a ‘E ground state (electronic configuration a12tzz)in the one-electron model (this is supported by detailed calculation), and no spin-resonance is observed. However, in diamond, a resonance is observed in an S = 1, tetragonally distorted state, 300 cm-l above the ground state.256This is presumably the Hund’s rule 3T1state of tZ2,JahnTeller distorted. An analogous spectrum is seen in a photoexcited state of the ( 0 A1)- center in This center also has four electrons (A1 being an acceptor). In this case, the orbital degeneracy of 3T1is already removed by the crystal field of the A1 atom, which reduces the symmetry to Car. The negatively charged vacancy 17- has three electrons in the t2 orbitals, and its Hund’s rule ground state is 4Az. This is the ground state predicted by Yamaguchi’s calculations.z52In fact, spin-resonance shows that the ground state is a spin doublet with strong Jahn-Teller distortion to Czu (biaxial) symmetry24g(see Fig. 28). It has been suggested=’ that the Jahn-Teller splitting of 2E, the lowest doublet of t 2 3 , is sufficient to bring one component down below 4 A z . This would only require tetragonal dis-

+

266

166

267

A resonance attributed to n+has been observed in diamond [J. A. Baldwin, Phys. Rev. Letters 10, 220 (1963)l. The g factor is isotropic. The anisotropic line width is attributed to a tetragonal Jahn-Teller effect but could have other causes. This reorientation is observed by applying a stress suddenly along (say) a (100) direction, thus raising the energy of a (100) distortion relative to that of an (010) or (001) distortion, and watching the subsequent changes in population by spin resonance. E. A. Harris, J. Owen, and C. Windsor, Bull. Am. Phys. SOC.[ a ] 8, 252 (1963). I t is not known for certain whether this is the neutral vacancy. It could be interstitial carbon, which has the same electronic configuration, or some other defect. A. B. Lidiard, in “Effets des rayonnements sur les semiconducteurs” (P. Baruch, ed.), p. 335. Dunod, Paris, 1965.

174

M. D. STURGE

(a3+ Qe)

FIG.28. The CZ. distortion of the negatively charged vacancy zero).

(&a

and

&B

non-

t0rtion.~~8 The lower symmetry can be explained in terms of the oneelectron model. The five electrons of 0- fill up the lower (nondegenerate) orbitals of Fig. 27b and partially occupy the e orbital ( e of D 2 d ) .To split this degeneracy, the center must undergo an additional 7 2 distortion (apparently of the Q6 type; see Fig. 5 ) . The symmetry drops to Go,producing the splitting indicated at the right of Fig. 27b. Note that only two orbitals, a and d, now appear in the zeroth-order wave function; this implies localization on two silicon atoms, as observed in the HFS. Confirmation of this picture of two independent distortions is provided by relaxation measurements. These show that the center “inverts” (i.e., it transfers between the two equivalent Cz. positions, averaging out the 7 2 distortion but retaining the E distortion) more easily than it “reorients” (averaging out the tetragonal e distortion). Similar spectra have been o b s e r ~ e d ” for ~ ~substitutional ~ ~ ~ . ~ ~ Ni- in Ge and for Pd- and Pt- in Si. These centers have 15 electrons, of which probably 10 fill the d she1P. The remaining five give a configuration just like the 0- center and produce a qualitatively similar distortion (illustrated schematicaIly in Fig. 2 8 ) . The g shifts are larger, presumably because the *5*

259

2m *61

The situation is rather analogous to that of Nizf in strongly covalent complexes, which tends to adopt a square planar configuration (see Section 12, footnote 177). Here also the 1E(lA 1) state is pulled down below the Hund’s rule 3Az state by what amounts to a strong Jahn-Teller interaction. G. W. Ludwig and H. H. Woodbury, Phys. Rev. 113, 1014 (1959). H. H. Woodbury and G. W. Ludwig, Phys. Rev. 126,466 (1962). Ludwig and Woodburyz69 say eight, but this seems unlikely, since no effects of the unfilled d shell are observed. Furthermore, the remaining seven electrons, going into bonding orbitals, would leave one hole whose degeneracy could be removed by a simple tetragonal distortion opposite in sign t o that observed in the O f center. The interpretation given here was first suggested by A. D. Liehr [private communication to G. D. Watkins (1960)l.

THE JAHN-TELLER EFFECT IN SOLIDS

175

valence orbitals hybridize with the metal d orbitals, where the spin-orbit coupling is relatively large. These centers appear to be the only known cases where the static JahnTeller effect causes the symmetry to decline from cubic to biaxial. Liehld has pointed out that this result is inconsistent with a Hamiltonian such as (3.8), in which only linear interactions with t and 7 2 distortions are included. The potential surfaces derived from such a Hamiltonian have minima corresponding to pure tetragonal distortion ( D M )or to pure trigonal distortion (C3v; two such distortions are illustrated in Fig. 5 ) ; but the Czv distortion of Fig. 28 corresponds to a saddle point, not a minimum. This illustrates the impossibility of describing the Jahn-Teller effect in these centers in terms of a linearized effective Hamiltonian. The molecular orbital model has, built into it, the nonlinear coupling necessary to stabilize the low-symmetry situation. If one of the nearest neighbors of the vacancy is a substitutional impurity such as P, the symmetry of the center in the absence of a JahnTeller effect is lowered to Clv.l14 The ground state of the five-electron P)O is still degenerate, however, as can be seen from Fig. 29. center ( 0 A second trigonal distortion occurs, placing the unpaired electron on a single silicon atom and lowering the symmetry to C, . This center is interesting because it is possible to follow the reorientation rate F* over 13 orders of magnitude, by motional narrowing at relatively high temperatures and by relaxation under uniaxial stress at low temperatures. Over the entire range, from 1010 to 10-3 sec-', Eq. (7.1) is obeyed, with an activation energy of 0.06 eV.114 Any tunneling must be slower than 10-3 sec-'. This low rate (relative to, say, A1203:Ni3+)142implies a relatively high potential barrier. The barrier might be expected to be high, since second-order

+

Td

CS

(0+ PI0

FIG.29. Effect of a trigonal field on the orbitals of Fig. 27, and of a subsequent Jahn-Teller distortion to C, symmetry.

176

M. D. STURGE

Jahn-Teller interactions with low-lying excited states, and also anharmonic effects, are likely to be strong. Similar spin resonance spectra are seen from more complex defects, for instance, the oxygen-vacancy pair (Si-A center) and the divacancy.113,115,116,262 If instead of a vacancy we have an isolated substitutional impurity with excess electrons (i.e., a donor), we begin to fill antibonding orbitals. Usually the wave function of such a center is too diffuse for the Jab-Teller effect to be appreciable (see Sections 4 and 13) ; this is apparently the case However, there are cases even for so compact a center as S+ in of donors that are reported to show a Jahn-Teller effect. The best-authenticated of these is the substitutional nitrogen donor in d i a m ~ n dThe . ~ ~ ~ ~ ~ ~ energy levels of Fig. 27 are inverted for antibonding orbitals, and the single antibonding electron is in a triply degenerate state. The CI3and NI4 HFS in the spin-resonance of this center show that the electron is localized primarily in one C-N “antibond,” so that the spectrum has [lll] symmetry. The corresponding distortion is that illustrated in Fig. 5f. There is no sign of reorientation even at room temperature265a (although studies by the stress technique2s to measure very low reorientation rates have not been made). With the exception of CaF2:V2+,245 this appears t o be the only well-established example of a static trigom1 Jab-Teller distortion. Even here, it is not absolutely proved that the distortion is due to the Jahn-Teller effect and not to some defect associated with the nitrogen. There is some slight which needs to be followed up, that the spin-resonance spectrum may be associated with mechanical imperfection of the crystals. Although isolated nitrogen donors in diamond have been tentatively identified optically,267it has not been proved that the optical and microwave spectra arise from the same center.268 There is some evidence for a dynamic Jahn-Teller effect, involving 262

263 3s4

G. Bemski, J . A p p l . Phys. 30, 1195 (1959). G. W. Ludwig, Phys. Rev, 137, A1520 (1965). W. V. Smith, P. P. Sorokin, I. L. Gelles, and G. J. Lasher, Phys. Rev. 116, 1546 (1959).

J. H. N. Loubser and L. du Preez, Brit. J . A p p l . Phys. 16,457 (1965). 2e6* L. A. Shul’man, I. M. Zaritskii and G. A. Podzyarei [Fiz. Tverd. Tela 8, 2307 (1966);

266 267 268

see Soviet Phys.-Solid State (English Transl.) 8, 1842 (1967)l have found that the nitrogen center reorient4 above room temperature, withan activation energy of about0.7 eV. Since this is much too low an activation energy for the diffusion of an impurity or defect in diamond, there can now be little doubt that the trigonal distortion is indeed a consequence of the Jahn-Teller effect. E. A. Faulkner, P. W. Whippey, and R. C. Newman, Phil. Mag. [S] 12,413 (1965). P. J. Dean, E. C. Lightowlers, and D. R. Wight, to be published. A. B. Lidiard and A. M. Stoneham, review presented at Intern. Ind. Diamond Conf., Ozford, 1966.

THE JAHN-TELLER

EFFECT I N SOLIDS

177

trigonal distortion, in the HFS of the F- donor in Be0.269More work on single crystals is needed to confirm this. The tetrahedral systems discussed in this section can have a substantial electric dipole moment and should be promising candidates for paraelectric resonance experiments. These would be particularly useful in cases where no magnetic resonance can be obtained, for instance, in a spin singlet ground state. Although the theory of such effects has been worked no experiments have yet been reported. In spite of the astonishing success of the one-electron molecular orbital model in accounting for the various types of Jahn-Teller distortion seen, particularly in silicon, it is even more difficult than usual to set up a quantitative theory.268 The nature of the electronic sta,tes, even in the absence of Jahn-Teller distortion, is not yet well understood, and it is not a t all clear what basis states to choose in setting up an effective Hamiltonian for the system. Furthermore, we have seen that a linearized Hamiltonian is, anyway, not going to be much use to us, whereas a nonlinear one will probably be too complex to handle. A possible approach has been indicated by Birman.270H e takes the u bonds as a whole. (The procedure can easily be extended to ?r bonds.) He asks what modes are contained in the symmetric square of the reducible representation spanned by the bonding orbitals. For instance, in the Oh XYO complex, Jahn-Teller interaction with T~~ modes becomes possible, since [ u X u] = [(A1, E, contains T1,.nlAlthough this idea has not yet been followed up, the approach would seem to be an appropriate one when Jahn-Teller coupling is strong compared to the exchange splitting of the bonding orbitals. One conclusion we can draw from the results on defects in silicon is that qualitative observations of the Jahn-Teller effect provide a powerful method for checking whether the ground state of a center is that predicted by theory. As so often happens, molecular orbital theory turns out to be more successful in a na'ive version that in its more sophisticated versions. This may only be because the calculations are for diamond, whereas most of the data are for silicon. The success of the simple model, in which the Coulomb and exchange interactions between valence electrons are neglected, should not be taken as evidence that these interactions are, in fact, weak. They are probably quite strong but because of strong configuration interaction tend to favor a low-spin rather than a high-spin ground state.250

+ +

26g

270

*'l

0. F. Schirmer, K. A. Muller, and J. Schneider, Physik Kondensierten Materie 3, 323 (1965). J. L. Birman, J. Chem. Phys. 39, 3147 (1963). A 7 I u distortion can give a permanent dipole moment. Such distortion might be observable by paraelectric resonance (see Sussmanl51). In concentrated crystals, it could lead to ferroelectricity (see Section 13).

178

M. D. STURGE

IV. Optical Transitions Involving John-Teller Distorted States

20. THE CONFIGURATION-COORDINATE MODEL

One of the most useful conceptual aids t o the understanding of electronvibrational interactions at impurity centers in crystals is the configurationcoordinate diagram.lsJ6 Many examples of the usefulness of this device as an aid to the understanding of optical processes in luminescent solids are given in the review by Klick and Schu1man.l’ We shall find it extremely helpful in visualizing the consequences of the Jahn-Teller effect in optical spectra. We shall adopt the “quasi-molecular” viewpoint set out in Section 2. That is, we shall make the (usually well justified) assumption that the electrons move in a potential V(Q) that is a function of only a few nuclear Q k (usually those of the nearest neighbors of the center). We shall (when necessary) treat the Q k as if they were normal coordinates, associated with more or less well-defined frequencies wk . As far as optical transitions are concerned, this can be justified when the semiclassical Franck-Condon approximation’O is valid (see below).n2 If coupling to local or quasi-local modes is dominant, the approximation is also justified. In the final analysis, we use it because it enables us to visualize a situation too complicated to treat any other way. This is its real justification. Let us for the present confine ourselves to nondegenerate electronic states, and consider the effect of nuclear motion on the energies of these states. Consider an ion with two such states, between which optical transitions can occur. I n general, the relative energies of the states will depend on the distance of the ion from its neighboring ions. Let us assume that all such distances can be specified by one parameter, for instance, the mean nearest-neighbor separation R. This is equivalent to confining our attention to the totally symmetric (“breathing”) mode of the system. In the language of Table 11, we have Q1 # 0, Qi = 0 (i # 1) ; Q1 is a measure of the deviation of R from its equilibrium value. This assumption is not so unreasonable as it looks, since only a totally symmetric distortion can produce a first-order shift in a nondegenerate level (Section 3). Besides the energy of the electrons of immediate interest, the “elastic energy” (ie., the energy of all other electrons) depends on QI ;since Q1 = 0 is the equilibrium position in the ground electronic state, the total energy must go initially as Q12. This is indicated by the lower curve in Fig. 30. The upper electronic state has a similar energy curve associated with it; but, because the separation between states contains a term linear in Q I , its 272

Lax70finds that not worry us.

wk

and

pk

must be regarded as temperature dependent. This need

THE JAHN-TELLER EFFECT IN SOLIDS

179

I\

FIQ. 30. Configuration-coordinate diagram for allowed transitions between nondegenerate states. X is the no-phonon line at energy E D .Such a transition is possible because the tails of the vibrational wave functions extend well outside the “classical” limits represented by the potential curves. A, B are “Stokes” lines in absorption; A’, B’ are “anti-Stokes” lines in absorption, “Stokes” lines in emission. E is the centroid of the absorption band. If the force constant pa2 is the same in both electronic states (i.e., there is zero quadratic coupling), the potential curves are given by Vi(Q1) =

tct~~Qi~,

minimum will be at finite Q1,&*. &f is a measure of the linear coupling. If there is no quadratic or higher-order coupling (i.e., no nonlinear dependence of energy separation on Q1),the curves will be the same shape. The relations given in the caption then hold ( A is the slope of the upper curve at Q1 = 0, and pw2 is the curvature). Possible transitions between the two electronic states are represented by vertical lines on the configuration-coordinate diagram. This can be regarded as a consequence of the Franck-Condon principle,6s70 according to which electronic transitions occur rapidly relative to nuclear motion, 80 that the nuclei do not move during the transition. A more precise way of looking at this is to write down the matrix element of (say) the electric dipole operator P between two Born-Oppenheimer product states:

PlP

= ( h 1

I P I +22(p2)

=

We)(2(p1

I cpz),

(20.1)

180

M. D. STURGE

where plz(e)=

($1

I p I $2)

if P does not operate on the nuclear wave function cp. I n writing (20.1), we have assumed that P is independent of the nuclear coordinates. (We shall not normally need to consider the many important cases where this assumption is not justified, since we shall be primarily interested in even parity distortions; but the configuration coordinate model with only vertical transitions does not, in fact, depend on this assumption for its usefulness.) Because of the overlap factor (cpl I cpz), Eq. (20.1) represents an integral over all possible vertical transitions, given an initial and final vibrational state.213 Now consider the vibrational energy levels of this system. If we really had an isolated molecule with only one degree of freedom (represented by Ql), the vibrational levels would be those indicated in Fig. 30; they are equally spaced if the potential curves are parabolic. I n fact, for a real system, such as a point defect in a crystal, phonons of all energies can contribute (in varying proportions) to Q1, and the vibrational spectrum is continuous rather than discrete. This fact does not affect the usefulness of the model (see Section 2). In particular, it does not alter the fact that the lowest vibrational energy level has some zero point motion, which spreads the wave function out along the Q1 axis in the Gaussian fashion indicated in Fig. 30. Because of this spread, even at 0°K possible transitions to the upper electronic state have a distribution of energies. If the equilibrium displacement in the upper state, Q+’, is large relative to the rms zero point displacement, we get an approximately Gaussian absorption band involving the excitation of a large number of totally symmetric ( a l g )vibrations. If, on the other hand, the coupling is sufficiently weak that the wave functions of the lowest vibrational levels of the upper and lower curves overlap, there is a “no-phonon” or a “0-0” transition between them, indicated by X in Fig. 30. This line (analogous to the Mossbauer line in y-ray emission) can be very sharp at low temperatures, even in solids, since its position is not dependent on any particular vibrational quantum. Its intrinsic width at 0°K is, in theory, determined by the lifetime of the upper state.214 In practice, the experimental width is usually determined by A no-phonon line has no random strain and impurity in the ~rystal.27~ Stokes shift; i.e., it is at the same energy in emission as in absorption, and 273

274

275

See D. L. Dexter [Solid State Phys. 6, 353 (1958)l for a generd discussion of optical absorption by point defects and impurities in solids. E. 0. Kane, Phys. Rev. 119, 40 (1960). A. L. Schawlow, in “Advances in Quantum Electronics 11” (J. R. Singer, ed.), p. 50. Columbia Univ. Press, New York, 1961.

THE JAHN-TELLER

EFFECT IN SOLIDS

181

the Einstein relations apply s t r i ~ t l yBesides . ~ ~ ~the ~ ~no-phonon ~ line, there is usually further structure in the absorption band ( A , B . . .) reflecting the vibrational levels of the excited electronic state; corresponding structure in emission (A', B'. - ) will reflect those of the ground state. If nonlinear coupling is weak (equal curvature in both potential curves), there will be mirror symmetry (about the no-phonon line) between the absorption and emission The probability for the mth vibronic transition (for an allowed electronic transition with linear coupling to a single vibrational mode) is found by calculating the overlap ( c p l j cp2) between displaced harmonic oscillator states. It turns out to be given by the Poisson distributionng

-

I P12(v)(m)12

=

I P12(412 e-sSm/m!,

(20.2)

where S = +a2(Q10) = ( E - Eo)/fiw (see Fig. 30) . S is a measure of the strength of the coupling. The factor e-S, the fractional intensity of the nophonon line, is the optical analog of the Debye-Waller factor.280 Equation (20.1) is quite general but is usually difficult to evaluate in practical cases. I n the case of strong coupling, we can make a useful approximation, which turns out to be of great heuristic value even when the coupling is not strong. This is known as the semiclassical Franck-Condon It depends on the fact that, when coupling is strong, E - Eo >> nu, so that most transitions end in rather highly excited vibrational states ( w is some effective mean frequency for the vibrations). In such a state, the vibrational wave function cp2 oscillates rapidly except near the classical potential curve V,(Q,). Hence, only this region of Q1 contributes appreciably to the integral, and we may write (cpi

Icp2)

+ c~i(Qi)G[hv- V~(QI)I,

where hv is the photon energy. Since we are treating the vibrations in the excited state essentially classically, their energy levels are to be regarded as continuous. Thus we may write for the absorption cross section per unit energy interval

I Pl2") 12 d(hv) = chv I f'12(e) 1' I cpi(Q1) l2 G[hv - V2(Q1)] dcpi ,

a(hv) d ( h v ) = chv

(20.3)

where c is a constant.53 If the initial state is a harmonic oscillator ground 276 277

278

280

D. E. McCumber, Phys. Rev. 136, A954 (1964). D. F. Nelson and M. D. Sturge, Phys. Rev. 137, A1117 (1965). R. E. Dietz, D. G . Thomas, and J. J. Hopfield, Phys. Rev. Letters 8, 391 (1962). M. Wagner, J. Chem. Phys. 41, 3939 (1964). See D. E. McCumber [Phys. Rev. 136, A1676 (1964)l for a more thorough treatment of no-phonon lines and vibronic spectra in the weak coupling limit.

182

M. D. S T U R G E

state, we have cpl(Q1) = a1/z(47r)-1/4 exp ( -a2QiZ/2),

(20.4)

a = (pw/zl)”Z,

and

hv

=

E - AQi

+ 3pdQ1 -

(20.5)

Here A is the electron-lattice coupling constant [see Eq. (3.6)]. Since, by hypothesis, aQi0>> 1, the last terms on the right in (20.5) can be neglected, and we have

ahv a(hv) = c [ Piz(’)1’ 2,,.1/2A

-

a2(hv - E)’ A2

,

(20.6)

which is the familiar “Gaussian” bandn3s2*lwith its maximum at hv = I3 and a width of 2A (In 2/a)’l2.We obtain the same result by letting S become large (and zlw small) in (20.2) and taking the envelope of the band. If we are only interested in the position of a band, not in its width, it is often sufficient to put cp1 = S(Ql),the “classical” value. We then obtain a line spectrum, absorption being represented by a vertical line at QI = 0, and emission by one at &I = Q10. However, this “classical FranckCondon approximation” can lead to qualitatively incorrect results when a Jahn-Teller effect is present, and we shall not use it in this chapter. 21. EXCITATION OF NONTOTALLY SYMMETRIC MODESOF VIBRATION

The selection rules for vibronic transitions in molecules have been discussed by Sponer and Teller.282An electronically allowed transition between nondegenerate electronic states can only be accompanied by the excitation of totally symmetric (a1) modes of the complex (if the force constants do not change in the transition). An electronically forbidden transition can be accompanied by one, and only one, vibration of a symmetry which makes the transition allowed. If one of the electronic states is degenerate and suffers a Jahn-Teller effect, large numbers of nontotally symmetric vibrations can be excited in a transition, as illustrated in Fig. 31 (compare Fig. 30). These modes are totally symmetric in the lower, Jahn-Teller distorted symmetry. The putative observation of progressions of such vibrations in crystals has occasionally been cited as evidence for a Jahn-Teller e f f e ~ t . ~ , ~Indeed, ‘-~*~ one author goes so far as to say “the discovery of uniquantal progressions C. K. Jorgensen, “Absorption Spectra and Chemical Bonding in Complexes,” Chapter 6. Pergamon Press, Oxford, 1962. 28) H. Sponer and E. Teller, Rev. Mod. Phys. 13, 75 (1941). 283 R. A. Ford and 0. F. Hill, Spectrochim. Acta 16, 493 (1960).

THE JAHN-TELLER

EFFECT IN SOLIDS

183

of asymmetric vibrations in electronic spectra. . . is an unambiguous indication of Jahn-Teller funny business.”5 Although this statement is true in principle, for impurity ions in solids we have no means of determining from other data what the symmetries of the modes involved are. I n the cited references, symmetries were determined from the observed frequencies by comparison with infrared and Raman spectra (of the host crystal). This procedure, usually justified in the case of molecules,7J4~2~ is fallacious when applied to crystals. We have no reason to suppose that the frequencies of the vibrational modes “seen” by the impurity ion are related to those of K = 0 optical modes of the host crystal, and symmetry assignments simply cannot be made by this method. In a pure crystal, assignment can

FIG. 31. Configuration-coordinate diagrams for transitions (a) from a Jahn-Teller distorted doubly degenerate state to a nondegenerate state; and (b) from a nondegenerate state to a doubly degenerate state. These are, in fact, sections through a three-dimensional diagram (see Fig. 6) that has cylindrical symmetry in the harmonic approximation. The dotted lines indicate the extent of the vibrational wave function in the initial state (see Sections 22 and 23). 18‘

R. S. Mulliken and E. Teller, Phys. Rev. 61, 283 (1942).

184

M. D. STURGE

(c) AL,O,

:v3+

CM-'

-I

(d)

MgO:Ni2+

FIG.32. Some vibrational progressions observed in the absorption spectra of transition metal ions in octahedral oxygen coordination. (a) ' A 2 + 4T2of MgO: V2+ (see Sturgezaa). (b) 4A,+ 4 T 2of A120a:Cr3+ (opolarization) (see McClure3l). (c) 3T1(3iZ)--* 3T2 of A1203: V3+ (n polarization) (see McClure31). (d) 3A2 + 3T1 of MgO:Ni2+ ) . known for certain if there is a Jahn-Teller effect in (see Pappalardo et ~ 1 . ~ It~ is~ not this case.

be more certain, but even there the very existence of a localized electronic excitation can substantially modify the vibrational spectrum of the lattice in its immediate vicinity. On the other hand, the argument could well be used in reverse. If we find that progressions of a certain frequency are always associated with transitions involving the Jahn-Teller effect, and not with other transitions, we can with some confidence assign this frequency to a local or quasi-local mode of the required symmetry. For instance, in the spectra of three ions in A1203(V3+,Cr3+, Vz+), there appear progressions with an interval of roughly 200 cm-1,"*3132859286 associated with transitions to the 3T2and 4T2 terms, which are known to suffer a tetragonal Jahn-Teller d i s t o r t i ~ n . ~ ~ , ~ ~ ~ I n association with transitions to Jahn-Teller distorted states of V2+,CoZ+, and Ni2+ in MgO, a similar progression with a 220-235 em-' interval is *s6 z86

B. N. Grechushnikov and P. P. Feofilov, Zh. Eksperim. i Teor. Fiz. 29, 384 (1955) ; see Soviet Phys. J E T P (English Transl.) 2, 330 (1956). M. D. Sturge, Phys. Rev. 130, 639 (1963).

T H E JAHN-TELLER

EFFECT I N SOLIDS

185

Some of these progressions are illustrated in Fig. 32. I n these systems, such progressions are not seen in conjunction with transitions not involving the Jahn-Teller effect. I n neither crystal is there any known feature of the phonon spectrum which might be correlated with the observed frequency interva1.290-2% We are probably justified in attributing this progression to a quasi-local (resonance) mode of cg symmetry, although it could, in principle, be cxlo . A crude calculation293of a quasi-local mode frequency for MgO containing transition metal ions supports this assignment. I n any case, whether or not this particular example is correct, this is, in principle, a powerful method for determining the symmetries as well as the frequencies of resonance modes. The prohibition of multiple excitation of nontotally symmetric modes is relaxed if the force constants for such vibrations differ between the ground and excited states. Such a difference provides a quadratic electron-vibrational coupling that permits the excitation of nontotally symmetric vibrations in pairs. Normally the intensity will be small; for instance, a change in force constant by a factor of 4 only transfers 5% of the total intensity to transitions involving such excitation.282One way in which such large changes in force constant can occur is when there is “accidental” degeneracy. For instance, consider the case where two curves on a configuration coordinate diagram cross1294as in Fig. 33. The accidental degeneracy a t the point of crossing will be lifted by interactions between the electronic states. (If the states are of different symmetry, such interaction can only be from nontotally symmetric vibrations.) When the states interact, the potential curves follow the solid lines, with greatly increased curvature (i.e., force constant) in the region of the crossover. This problem has been dealt with in more detail by Bron and WagnerlZg5 4f65d spectrum of Eu2+ who find evidence for such an effect in the 4 7 in alkali halides. This ion is associated with a vacancy, forming a complex ~een.2869*~6a-~89

---f

2mt M. D. Sturge, Phys. Rev. 140, A880 (1965). g87

e*g

R. Pappalardo, D. L. Wood, and R. C . Linares, Jr., J . Chem. Phys. 36, 1460 (1961). R. Pappalardo, D. L. Wood, and R. C. Linares, Jr., J . Chem. Phys. 36,2041 (1961). F. A. Kroger, If. J. Vink, and J. van den Boomgaard, Physica 18,77 (1952).

zw G. Peckham, Proc. Intern. Conf. Lattice Dynamics, Copenhagen,1964, p. 49. Pergamon Press, Oxford, 1965; Proc. Phys. Soc. (London)90, 657 (1967). *91 R. S. Krishnan, Proc. Indian ilcad. Sci. A26, 450 (1947) (the infra-red data quoted in this reference contain errors; see Barker292) ; S. P. S. Porto and R. S. Krishnan, J . Chem. Phys. 47, 1009 (1967). ‘92 A. S. Barker, Phys. Rev. 132, 1474 (1963). M. D. Sturge, Institute of Physics and Physical Society Conf. Phonons, Edinburgh, 1966 (unpublished).

M. H. L. Pryce, G. Agnetta, T. Garofano, M. B. Palma-Vittorelli, and M. U. Palma, Phil. Mag. [S] 10, 477 (1964). *= W. E. Bron and M. Wagner, Phys. Rev. 146, 689 (1966).

z94

186

M. D. STURGE

FIG.33. “Accidental” crossing of states in the configuration coordinate diagram; strong interaction (after Pryce et c d . * 9 4 ) .

_ _ _ zero interaction between states,

~

of CZV symmetry; two totally symmetric quasi-local modes of this complex (with frequencies of roughly 200 and 45 cm-l) have been identified in other spectra.296Eu2+shows progressions of these frequencies, in both emission and absorption, but in absorption an additional progression is observed, with an interval of about 800 cm-’. Such a frequency, more than twice the highest phonon frequency of the host crystal, would be difficult to account for in terms of a single quantum excitation, nor is there any explanation other than Bron and Wagner’s for its absence in emission. 22. TRANSITIONS FROM DOUBLY DEGENERATE STATES

Consider an ion such as Cu2+.in octahedral coordination; it has an E term lowest, undergoing a strong Jahn-Teller effect. An E term interacts with three modes of the octahedron, Q1,Q2, and Q3, so that the configuration coordinate diagram should be four-dimensional. However, the effect of QIis only to shift, not to split, the levels. If we neglect high-order effectss0 that couple Q1 to Q2 and Q3 , the effect of Q1 can be treated separately and added in (primarily as a broadening mechanism) after we have considered Jahn-Teller coupling to Q2and Q3. In the harmonic approximation, the configuration-coordinate diagram for an E term interacting only with Qzand Q3is simply Fig. 6. A vertical section through the potential surface of Fig. 6, taken along the Q3 axis, is shown in Fig. 31a (lower curve). The upper curve is the potential for a hypothetical nondegenerate excited term. The behavior of the 2Tzterm of Cu2+,which is relatively weakly coupled to the Jahn-Teller distortion, can be approximated by this curve. The probability distribution of the vibrational coordinate p, when the complex is in its lowest vibrational level, is represented by the dotted line in Fig. 31a. The 8 distribution is sensitive to anharmonicity but does not 21

M. Wagner and W. E. Bron, Phys. Rev. 139, A223 (1965).

THE JAHN-TELLER

187

EFFECT IN SOLIDS

affect the optical spectrum in any qualitatively important way. We can therefore use the harmonic approximation and regard the true configuration-coordinate diagram as the result of rotating Fig. 31a about the vertical axis p = 0. In the Franck-Condon approximation, only vertical transitions are possible in Fig. 31a. Thus, there will be two major transitions at low temperatures: one a t 4 SE, and one a t about A 2 6E.Both will be broad bands, not split by the Jahn-Teller effect, and will be roughly Gaussian if 6E >> liw. Unfortunately, Cu2+ in oxide or water coordination has A 26E, so that the two bands more or less coincide. However, three overlapping bands in the near infrared can be identified in the spectra of Cu2+ in a variety of “octahedral” environments297; two arise from transitions to the 2T2term and one from the upper branch of the 2E term. When the static Jahn-Teller effect is as strong as this, the complex is best regarded as a plane square rather than as a distorted octahedron.21 The transition between branches is not observed in the visible or nearinfrared spectrum of Mn3+ or Ni3+ in A1203,31*298 although these ions certainly have Jahn-Teller distorted ground states.138-140 Possibly 6E is substantially smaller than in Cu2+, so that the transition occurs far out in the infrared. The splitting of the 5E+ 5T2band of Mn3+ appears (from the polarization) to be due to the trigonal field of A1203rather than to tetragonal distortion carried over from the ground state. This is confirmation that the Jahn-Teller distortion is rather small. In KC1:Ag2+ (octahedral d9) , spin resonance shows that the ground state suffers a static tetragonal distortion (elongation) .299 The Ag2+ion has a strong absorption band in the blue. Pumping in this band with polarized light weakens the resonance associated with one direction of distortion and enhances the others. This is because light is preferentially absorbed by an ion with its local axis parallel to the electric vector; in the excited state, memory of the original orientation is lost, and the ion reverts to any one of the three possible ground states with equal probability. This is good evidence that the Jahn-Teller effect, rather than local charge compensation, is responsible for the distortion. O’Connor and Chen112have studied the optical absorption of CaF2:Y2+, a dl ion in a cubal environment. The 2E ground state shows a tetragonal distortion in spin resonance, attributed to a static Jahn-Teller effect. The transition to the 2T2 excited state is split into three bands; this splitting

+

-

0. G. Holmes and D. S. McClure, J. Chem. Phys. 26, 1686 (1957) ; R. L. Belford’ M. Calvin, and G. Belford, ibid. p. 1165. *** R. Miiller and H. H. Gunthard, J . Chem. Phys. 44, 365 (1966). C. J. Delbecq, W. Hayes, M. C. M. O’Brien, and P. H. Yuster, Proc. Roy. SOC.

M71, 243 (1963).

188

M. D. STURGE

is attributed to “carryover” of the tetragonal distortion into the excited state (classical Franck-Condon principle) . The splittings are remarkably large (-10,000 cm-l) . The picture is confused by the possible presence of interstitial F- ions, which can also produce tetragonal distortions. (These are supposed to have been eliminated by annealing.300)Furthermore, there is no independent check on the assignments of the optical transitions. The interpretation of these data in terms of a Jahn-Teller effect is still doubtful. The vibronic ground state in a center showing the dynamic Jahn-Teller effect is degenerate and, in general, can be split by uniaxial stress. If this splitting is greater than kT,dichroism can be induced in the absorption spectrum, even in a cubic crystal. Such an effect has been observed by SilsbeelGOin the spectrum of the R center in KCl. It can easily be seen that, in an axial center such as this, fi -+ A transitions should show pronounced temperature-dependent dichroism, whereas fi -+ fi transitions should not. This is because, if stress splits the fi ground state into f i z and fi, , of which only fiz is populated, only x-polarized light can induce transitions to an state, whereas any polarization will induce transitions to an fi state. (The dichroism is therefore zero in the lowest approximation, but inclusion of nonlinear Jahn-Teller interaction allows some dichroism to occur.) Silsbee is able to detect and identify several optical transitions of the R center by their dichroism, even though they are obscured in absorption by strong bands from other centers. TRANSITIONS TO DOUBLY DEGENERATE STATES 23. BROADBAND Now consider the case where the singlet is lowest and the doubly degenerate state highest (Fig. 31b). A t first glance, it might appear that the most probable Franck-Condon transition (band maximum) is to an unsplit upper state, and that it cannot tell us anything about the Jahn-Teller effect. However, this is incorrect. It is necessary to remember that there are two distortion coordinates, Q2 and Q3 , and that the curves of Fig. 31 are sections of surfaces of approximately cylindrical symmetry. The ground state vibrational wave function in the singlet is cpo = ffr-112

c

exp -ff2p2/21,

(23.1)

and the probability distribution over the p coordinate is p ( p ) d p = 2a2pexp [- a 2 p 2 ] d p .

This has its maximum, not at 3w

p =

(23.2)

0 (where there is no phase-space), but

Electric dipole transitions are allowed for a Yz+ ion associated with a defect (such as a n F- ion), whereas they are forbidden by parity for the isolated ion. Hence, the defect-associated ions might well dominate the optical spectrum, even if they form only a small minority of the total and do not show up in spin resonance.

THE JAHN-TELLER EFFECT IN SOLIDS

189

at p = (2~1)-l/~. Thus, in the semiclassical Franck-Condon approximation the most probable transition is not at p = 0 but at (2~x)-l/~,as indicated by the vertical lines in Fig. 31b. The band profile is approximately given by

u(hv)

0:

hv I E I exp ( -a2E2/A2) ,

(23.3)

where E = E - hv. The band thus has a “vibronic” splitting of 21l2A/cr = 2(liW6E)”2, which will be resolved 3 coupling to the totally symmetric mode is not too strong. O’Brien?’ has extended this treatment to the case of a real crystal with continuous (phonon or quasi-local) modes. She finds that (23.3) holds (at 0°K) if we replace (A/a)2 by x ( A , J a , ) 2 ,the sum being over all the vibrational modes of the crystal which contribute to eg distortion a t the Jahn-Teller ion. As the temperature is raised, the mean value of p in the ground state increases. According to the general treatment of Lax170we can simply replace ( A / a ) 2by

c (A8/ff8)2[2n(W8) -k 11 c =

(&/ff8)’

coth (hwg/2kT),

where n (w,) is the occupation number of mode s. This problem has been treated rigorouslya for a single-mode system at O”K, without use of the Franck-Condon approximation. Although the physical model is an unrealistically simple one, the Hamiltonian for the doublet is (5.2), and the calculation is directly applicable to the lowestorder problem of an E electronic term interacting with the eg modes of an octahedron. Some results for a few typical values of 6E/liw are plotted in Fig. 34. I n the approximation used, only transitions to vibronic states with I j I = 3 (see Section 6) are allowed. I n practice, the discrete vibrational structure will be smoothed out, except in the vicinity of the no-phonon line, and we need only consider the envelopes of the intensity distributions. The doublet-to-singlet (E -+ A ) transition is a single band, Gaussian if 6E >> nu, essentially the same as in the absence of the Jahn-Teller effect. On the other hand, the A -+ E transition is a split band. The separation of the band maxima varies as predicted by the semiclassical argument, even for quite small values of GElliw. The semiclassical argument based on the potential of Fig. 6 does not predict the difference between the widths of the two bands, as illustrated in Fig. 34. This can be corrected?’ by using the more accurate potential of Fig. 9, in which Slonce~ski’s~~ “pseudo-centrifugal” term (see Section 6) has been included in order to take account of the nuclear kinetic energy.” Comparing Fig. 9 with Fig. 6, we see that the slope of the upper branch is reduced, whereas that of the lower branch is increased (ap,,, < 1 in most important cases). Thus, the upper peak should be sharper than the lower one, both maxima are shifted to higher energy, and the splitting is slightly larger

190

M. D. STURGE

I

IIIIII, . . .

IIIIIIIIIIII

..

FIG.34. Vibrational structure of allowed transitions between a n electronic singlet and a Jahn-Teller distorted doublet. The parameter 2 GE/fiw is a measure of the strength of the Jahn-Teller coupling. The spacing of the individual transitions is approximately kw, but in solids only the envelope of the band is significant. The no-phonon line is indicated by the arrow (from Longuet-Higgins et ~ 1 . ~ ~ ) .

than the simple model predicts. Experimentally, the two bands are usually of roughly equal width. This may indicate that Sloncewski’s centrifugally stabilized states are too broad to observe in the cases studied. Vekhte91 denies that this gross splitting of an A + E band can occur. His argument is based on the assumption chis Eq. (2)] that the vibronic energy levels associated with the degenerate state are those of a harmonic oscillator. We have seen that, in some instances, such as a T state interacting only with an E vibration, this assumption is justified; then, aa Vekhter says, there can be no splitting of the band. In the case of an E state, however, the assumption is not correct (see Section 6 ) . The most dramatic examples of the splitting just predicted occur in the 6T2 + 5E transition of octahedrally coordinated high-spin d6 i ~ n s , ~ ~ ~ Fez+ and C03+. I n the earlier work”2 on complex halides, these absorption spectra were deduced from the diffuse reflectivity of powdered material and are not quantitative, but the splitting of the 5E(t23e3)band is unmistakable. The Jahn-Teller effect in the 5T2(t24e2)ground term is small B. G. Vekhter, Opt. i Spektroskopiya 20, 258 (1966); see Opt. Spectry. (USSR) (English Transl.) 20, 139 (1966). F. A. Cotton and M. D. Meyers, J . Am. Chem. SOC.82, 5023 (1960). ao3 G. D. Jones, Phys. Reu. 155, 259 (1967).

THE JAHN-TELLER

191

EFFECT I N SOLIDS

enough (since t2 orbitals do not form u bonds) to be completely suppressed by spin-orbit coupling. (This has been checked in many cases by spin resonance.) In FeSiF6.6H20, a barely resolved splitting in the 5Tz-5E band of about 1600 cm-1 is seen, whereas in K2NaCoFs the splitting is 2500 cm-1. The site symmetry in bot8h crystals is trigonal, and crystal field splitting of 5E can be neglected. If we take 6E 3000 cm-I (a reason200 cm-', we find 2(6Eh0)'/~ 1700 able value for an e hole) and ho cm-I, in reasonable agreement with the observed splittings. Jones303has observed the corresponding splitting in the absorption spectrum of Fez+in cubic crystals. The most clear-cut example is in KMgF3: Fez+ (see Fig. 35). The splitting increases with temperature, as predicted. A similar splitting, 1800 cm-I, is observed in the 2E(e) excited term of Ti3+ in single-crystal A1203.31 Again, the site symmetry is trigonal, and the splitting must be due to the Jahn-Teller effect. Some remarkable splittings of the type considered here occur in the re excited state of the F center in cesium halides. This state derives from a 2T1 term and will be discussed in detail in Section 26. When spin-orbit coupling is strong, so that rs is well separated from other levels, a re level behaves exactly like an E term in its interaction with e vibrations (but can, unlike an E term, also interact in first order with 7 2 vibrations). ball ha user^'^^ discusses the A-E transition in a square planar complex. The tetragonal point groups have the unique property that an E state couples to the nondegenerate PI, and p2, distortions (see Section 12). Whether one gets a splitting in the band depends on the relative magnitudes of the coupling to the & and P2, distortions. If the coupling to both happens to be the same, the problem is formally identical to that of coupling to e vibrations already considered, and we get a splitting. If coupling is to one mode only, the configuration coordinate diagram is Fig. 18. Each

-

-

-

3 300° K

roo0

10,000

13.000

cm-1

FIG. 35. -+ 5E absorption spectrum of KMgFs:Fe2+ (from Jones"3). Note that the band splitting increases with temperature.

192

M. D. STURGE

parabola is independent, and, just as in the case of coupling to Q1 (Fig. 30) , we get a Gaussian, unsplit band. Knoxm has pointed out that a qualitatively similar splitting can occur in symmetry-forbidden transitions, which cannot be attributed to the Jahn-Teller effect as ordinarily defined. A forbidden transition will, in general, be made allowed by some distortion Q. Assuming for simplicity that the matrix element is proportional to Q, and that the mode is nondegenerate, the transition probability is proportional to Qz I q ( & ) l2 Qz exp ( -a2Q2).If Q is an even parity distortion, it can split the final state by (say) 2AQ, and the band profile is given by

-

~ ( h v a) hvE2 exp ( -azE2/A2)

(23.4)

e

(where E = - hv). The splitting is even more pronounced than in the Jahn-Teller case [Eq. (23.3)]. If the transition is parity forbidden (as it is, for instance, in a d .--) d transition in a centrosymmetric site), Q is of odd parity, and the splitting of the final state is quadratic, say 2CQ2. The band profile is now ~ ( h va ) hv I E I exp

(-a2

IE

I/C2)

(23.5)

and still shows a splitting. One might have expected that C (being a secondorder term) will be rather small, and the splitting will not be resolved. However, a well-resolved splitting of about 1000 cm-' at 4"K, increasing + 2P3/~(s2p) transition of to 1700 cm-l a t 153"K, is observed in the 2P~/2 to the neutral TIoin KCl.N5 This is attributed to304 strong coupling of 2P3/~ T1,mode. (The triple degeneracy of this mode makes no qualitative difference but doubles the expected splitting.)

24. NO-PHONON TRANSITIONS INVOLVING DOUBLY DEGENERATE STATES

So far, we have only considered the splitting of the band maximum, from which only a small amount of information can be obtained. I n principle, it would be much more useful to study the no-phonon transition, in which transitions occur between the lowest vibronic levels of two electronic terms. If there is a splitthg or distortion in the initial or final state, it will be reflected in the no-phonon line. In particular, one might expect to see splittings due to tunneling, anisotropic Zeeman effect, etc. I n most of the cases we have considered up until now, coupling to the lattice is extremely strong, and the consequent reduction in overlap between the initial and final vibrational states makes the no-phonon transition too weak to be observed. The no-phonon line is only likely to be vis304

R.S. Knox, Phys. Rev. 154, 799 (1967).

305

C. J. Delbecq, A. K. Ghosh, and P. H. Yu'qter, Phys. Rev. 164, 797 (1967).

THE JAHN-TELLER

EFFECT I N SOLIDS

193

ible if the Jahn-Teller energy 6E is not more than a few vibrational quanta. This is the case for many of the triply degenerate terms of transition metal ions to be considered in later sections. The most likely candidate for a moderate Jahn-Teller effect in an E term is a singly occupied e orbital in tetrahedral coordination (where an e orbital cannot form c bonds). Cu2f enters the tetrahedral ( A ) site of ZnA1201 (spinel), which has Td symmetry, and the 2E excited term should not be split bycrystal fields or spin-orbit coupling. I n fact, a splitting of some 17 cm-l is seen in the no-phonon line,242 which is quite intense in this case. The relative intensities of the split components are consistent with a splitting due to tunneling. Tunneling sec) , because the Jahn-Teller effect may be rapid ( 10-l2 sec instead of is relatively weak. However, this interpretation is not certain, because the local cubic symmetry can be destroyed by inversion (i.e., the presence of Zn in A1 sites, and vice versa). Although this does not occur in the pure spine1,m6it is not known how extensive it is in the copper doped crystal. 25. OPTICALTRANSITIONS INVOLVING TRIPLY DEGENERATE STATES

Spin resonance is the ideal method for detecting the anisotropy associated with the static Jahri-Teller effect. Unfortunately, its virtual limitation to the ground state prevents it being a very useful tool for the study of dynamic effects, in particular the Ham effect, where the energy level scheme is of primary interest. Furthermore, only a limited number of Jahn-Teller sensitive ground terms are available, whereas there is a great variety of excited terms. It is not surprising, therefore, that much of the evidence for the Ham effect comes from optical spectroscopy. The splitting of the band maximum which occurs in transitions to E states (Section 23) is not usually seen in transitions to T states. We shall see that coupling to E distortion can only produce a splitting in the latter case if there is strong spin-orbit coupling (1>> 6E 2 liw) .nn This is unlikely to occur in 3d ions, although it might well obtain in 4d or 5d ions. In the F center of cesium halides, however, X is large relative to liw, and band splittings are observed when the Jahn-Teller coupling is in the right range (Section 26). Splittings are also seen in certain transitions where coupling to 7 2 distortions is strong (see Section 27). I n 3d ions, the relative weakness of electron-lattice coupling in many excited T terms enables us to see the no-phonon transition. This line or group of lines, connecting as it does the lowest vibronic levels derived from the ground and excited electronic terms, manifests most directly the effects of Jahn-Teller distortion. I n particular, if the ground term is orbitally m6

F. C. Romeijn, Philips Res. Rept. 8, 321 (1953).

307

The statement to the contrary in Scott and Sturge233 is incorrect.

194

1

M. D. STURGE

I

I

I

(4000

15000

1

16000

h v (cm-I)

FIG.36. The main absorption band of the F center in CsF. 0 , experimental points (from Hughes and Rabin315). , calculatio; of Moran.310 ~

nondegenerate, such effects can be unambiguously attributed to the excited term. There is a substantial body of data on optical absorption into excited T terms of first-row transition metal (3dn) ions in cubic or nearly cubic (weakly trigonal) environments. Many of these spectra show detailed fine structure on the low-frequency side of one or more absorption bands (particularly in the no-phonon transition). I n many cases, this structure shows qualitative evidence for the Ham effect and other consequences of the dynamic Jahn-Teller effect. We shall discuss these data in Section 28. 26. THE CASE OF STRONGSPIN-ORBITCOUPLING: F-BANDSIN CESIUM HALIDES

If spin-orbit coupling is small and interaction with r2 vibrations negligible, there can be no vibronic splitting of the band maximum by 6 distortion, such as occurs for a transition to an E term. This can be seen from Fig. 20, which shows the potential surfaces for a T term under these circumstances. Each paraboloid is independent and identical, so that the total transition probability for unpolarized light is simply three times the probability for a transition to any one of them. I n a transition between

THE JAHN-TELLER

195

EFFECT IN SOLIDS

two paraboloids, the fact that the axis of one is displaced relative to the other cannot possibly lead to a splitting. It can easily be seen by direct integration that the band is a distorted Gaussian, with its maximum close to the classical Franck-Condon maximum a t Qz = Q3 = Spin-orbit coupling causes the paraboloids to interact (that is, the electronic states associated with each paraboloid are mixed), and so they can no longer be treated independently, and the foregoing argument fails. The case of strong spin-orbit coupling (relative to the Jahn-Teller interaction) has been treated by M0ran.~l0Although his analysis (as it stands) applies only to spin and parity allowed transitions from a 2A1, to a 2Tlu term, it could without much difficulty be extended to the case of any other triplet term with strong spin-orbit coupling. Moran is interested in the main absorption band of the F center in cesium halides. The F center consists of a negative ion vacancy that has neutralized itself by trapping an electron.311Although the wave functions of this center are still a subject at least the ground state can be approximated to by an of active study1312 s-like (2AIg)linear combination of atomic orbitals on the six nearest neighbors of the vacancy. The F band arises from transitions to a p-like excited state (2T1,). In the halides of the lighter alkalis, the F band is a good Gaussian,313 as expected for weak spin-orbit coupling and strong coupling to alp and E, modes. In the cesium halides, however, the F band shows most pronounced in the case of CsF (see Fig. 36).315Spin-orbit coupling and 2P312 of a free splits 2Tluinto two levels, rs and I’8 (analogous to 2P1/z ion), and cannot by itself account for the existence of three peaks. Moran’s contribution310is to show that, just as in the case of an E term (Section 23), the r8 band can be vibronically split by the Jahn-Teller coupling. The first-order electronic Hamiltonian for Tluin the spin-orbital representation is 0.w89309

+

I

AQz

-2llzAQ2

AQz

X - AQ3

-2112A&3

-2ll2AQ2

-2”’A&3

X

AQ3

-2X

.

(26.1)

M. C. M. O’Brien, unpublished work (1965). Y. Toyoaawa and M. Inoue, J . Phys. SOC.Japan 20, 1289 (1965) ; 21, 1663 (1966). 310 P. R. Moran, Phys. Rev. 137, A1016 (1965). 311 F. Seitz, Rev. Mod. Phys. 26, 7 (1954). 312 B. S. Gourary and F. J. Adrian, Solid State Phys. 10, 127 (1960). C. C. Klick, D. A. Patterson, and R. S. Knox, Phys. Rev. 133, A1717 (1964). 314 H. %bin and J. H. Schulman, Phys. Rev. 126, 1584 (1962). 315 F. Hughes and H. Rabin, J . Phys. Chem. Solids 24, 586 (1963).

196

M. D. STURGE

This acts on the basis states

[1: I] [

$1.

rs - +

or

r6

-$

r 6

rs +

(26.2)

+3

(Kramers degeneracy is not, of course, lifted by 3C(I).) Here X is the spinorbit coupling parameter, and A is the splitting of rs by unit tetragonal distortion. [ A is the same as in Eq. (14.1) .] We have neglected interaction with 7 2 vibrations and have temporarily dropped the constant term representing interaction with the totally symmetric a1 mode Q1.When X

>> a-'A

(a2= p u . / h ) I

we can neglect the matrix elements connecting rSand rs states. Then the Hamiltonian for I's is exactly analogous to (5.2), the Hamiltonian for an E term, and has the same solutions. Thus, the 'A1+ rsband should split into two just as an A + E band does, and the zA1 + 'T1 band should have three peaks. If rs - r6 mixing is neglected, the electronic states of rs are given [as in Eq. (5.5)] by

1 f 2) - cos (40) ?rlrsF 3), (26.3) $z* = cos ($0) I 'T1rs f $ ) + sin ($0) I 2Tlrs ?= 3), and the ground state is +o* = I 'Airs f +). (As in Section 5, we have put fil*

&z = p sin 0, are simply

= sin ($0)

Q3

= p

cos 0.) I n this approximation, the electronic energies

+

E $A - $Ap, Ez = E + +A + +Ap, Ea = l3 - X, (26.4) where E is the mean energy of the 'Tlu term. In the semiclassical FranckEl

=

Condon approximation (see Section 20), we can write down the transition probabilities between the various electronic states for a given Qz and Q3 and average over these variables afterwards. For circularly polarized light, the transition probabilities are proportional to such terms a P 6

1 (+I* I P+ I $o*) l2 = sin2 ($0) I (2Ti+ 1 I P+ I ' A i ) \', I (+I* I P- [ #o*) l2 = 5 COS' (30) I ('TI - 1 I P- I 'A1) 12, etc. The ground state vibrational wave function is 316

exp ( -a2p2/2),

We use the fact that, the spin-orbital states are given by (Griffith,Zz Table A20) :

I *T1r8 +) )

=

I 2T1+I, + t ), I 2Tlrs- + ) =

(+)l/z

1 zT10, - 4

)

+

(+)'I2

I zT1 -1, + f) , etc.

THE JAHN-TELLER

EFFECT I N SOLIDS

197

and so the transition probabilities are to be simply averaged over 0 and weighted by the ground state distribution over p , i.e., by ( 2 3 . 2 ) . For instance, the probability of an RH polarized transition from $o* at energy hv is proportional to ~ J o m 2 p & - a 2 ~ [ 6 ( h-v E - +A - - A p )

+ 6(hv - E - $1 + + A p ) ] dp,

which has maxima at hv

=

E

+ +A f &A.

(26.5)

In this approximation, the transition to I's is at E3 = E - A. Moran310goes on to include I'c - mixing as a second-order perturbation. The electronic energies of $1 and I / J ~ now depend on 0, but, after averaging, the results are qualitatively the same as before. (The rs band is, of course, depressed by the interaction.) He also includes interaction with the totally symmetric mode Q1 ,which broadens the band without producing any further splitting. In fitting the data, there are three important parameters: the spin-orbit coupling A, the e interaction strength Wt ( = a - l A ) , and the a1 interaction strength W1. (In addition, the mean energy l? and the overall absorption strength are adjustable.) With these adjustable parameters, Moran is able to fit the F bands of all the cesium halides. However, he finds that W3 decreases with increasing anion mass, whereas WI increases, so that the splitting becomes less and less resolved; only in the case of CsF (and possibly CsC1) are the three components well enough resolved for the fit to be really significant. The fit to CsF is illustrated in Fig. 36. It is particularly encouraging that Moran's calculation reproduces the pronounced asymmetry of the I'6 band. The fit could perhaps be improved by full diagonalization of (26.1), since the criterion for the validity of the perturbation approach, A >> W 3 ,certainly does not hold. Besides the overall band shape, in which transitions from $o+ and $0are averaged, one can also measure the individual circularly polarized components of the band. This is done by applying a magnetic field and lowering the temperature so that kT << gPH. Moran finds qualitative agreement with the integrated circular dichroism in CsCl and CsBr.317,318 He 317

318

R. Romestain and J. Margerie, Cornpt. Rend. 268, 2525 (1964). Agreement is by no means perfect. The spin-orbit coupling parameter X can be deduced directly from the first moment of the circular dichroism. This is a more precise procediire than an attempt to divide the F band into r6and rs components, and can be used when the components cannot be resolved a t all. C. H. Henry, S. E. Schnatterly, and C. P. Slichter [Phys. Rev. 137,A583 (1965)l find X = -220 cm-1 for CsCl and CsBr by the moment method, compared with Moran's fitted valne of -280 cm-1. This discrepancy is in the direction expected for a perturbation approach (which underestimates the splitting for a given value of X).

198

M. D. STURGE

FIG.37. Band profile for a singlet to triplet transition, interacting only with 7 2 vibrations, in the semiclassical Franck-Condon approximation (from Toyozawa and Inoue,30Qp. 1668).

also finds good agreement with the peak value of Faraday rotation (which is essentially the dispersion of the circular dichroism) in CsBr. Recent work319 on the spectral dependence of circular dichroism in CsF confirms Moran's calculation qualitatively; again, a more exact calculation is needed for a quantitative check. So far, we have neglected interaction with r2 vibrations. If the interaction is weak enough to be treated as a perturbation to Moran's analysis, it merely adds to the effective value of W3and is indistinguishable in the It should, however, affect the absorption spectrum from c intera~tion.~'~ spectral dependence of the circular dichroism, since it can mix states otherwise orthogonal, such as I rs 4) and rs 3). The case where T Z interaction is strong is discussed in the next section.

+

I

+

27. BANDSPLITTINGS DUETO r2VIBRATIONS :KCl : T1+ AND ISOELECTRONIC CENTERS

The case of an A + T transition, coupling to a r2mode, has been treated in the semiclassical Franck-Condon appro~imation.~0sJO~ The band is found 319

T. A. Fulton and D. B. Fitchen, Bull. Am. Phys. SOC.[Z] 11, 245 (1966).

THE JAHN-TELLER

199

EFFECT I N SOLIDS

to have the curious shape depicted in Fig. 37. The splitting between the outer components is approximately [66Efiw,/coth (fiwJ2kT) Ill2, where 6E = 2B2/3pwrZ. The optical absorption of heavy metal ions with the s2 configuration (such as T1+, Pb2+, etc.) in alkali halides has been widely s t ~ d i e d . ~ ~ ~ ~ There are three main bands arising from the s2 + sp transition, 'So+ 3P1, 3 P ,~lP1 ('So + 3P0is forbidden in cubic symmetry) The spin-allowed lP1(I'4)band is split into a triplet in a manner very similar t o that predicted for 720coupling (Fig. 38). As expected, the splitting e0 coupling, increases with temperature. Toyozawa and I n o ~ ewho , ~ neglect ~ fit the data with coupling constants B ( f i / p ~ , ) ~in/ ~the region of lo4 cm-l, which vary from system to system in the way predicted by a point charge model.326 The 3Ppl(I'4)band is a doublet, not a triplet. Toyozawa and Inoue account for this splitting by the same no mechanism without introducing any new parameters; two of the three components are found to coalesce because of second-order Jahn-Teller interaction with 3 P .~ Na CP: Pb2+

--------48

50

52 (103

5

cm-')

FIG.38. Sketch of the 'SO-+ 'PI transition of NaC1:Pb2+ a t different temperatures (from Fukuda et ~ 1 . 3 2 3 ) . Charge compensation is not thought to be responsible for this structure, since a similar (though less well resolved) structure is found for monovalent ions. The tails of adjacent absorption bands have been substracted out.

R. Hilsch, 2. Physik 44, 860 (1927). F. Seits, J . Chem. Phys. 6, 150 (1938). 322 P. H. Yuster and C. J. Delbecq, J . Chern. Phys. 21, 892 (1953). 323 A. Fukuda, K. Inohara, arid R. Onaka, J . Phys. SOC. Japan 19, 1274 (1964). 324 R. Onaka, A. Fukuda, and T. Mabuchi, J . Phys. SOC.Japan 20, 466 (1965). Sz6 A. Fukuda, Sn'. Light (Tokyo) 13, 64 (1964). 326 We shall see that there is evidence that coupling t o e modes is in fact very strong. Although c modes can produce no splitting in the band (see Section 25), they may affect the band profile, both through quadratic 6 - 7 2 coupling and through the Ham effect. The latter will tend to reduce the band splitting although not, perhaps, by a very large factor. 32O

321

200

M. D. STURGE E

FIG.39. Configuration-coordiriate diagram for the 3 P arid ~ 3P0states of KCl:Tl+, assuming interaction only with eg distortion (adapted from Kamimura and Sugano330).

Although interaction with eg distortions is not important in understanding the absorption spectrum of these phosphors, it appears to play a crucial role in emission. For instance, KCl:Tl+, which has its SO -+3P1 absorption at 40,200 cm-l, emits at 32,800 and 21,000 cm-'. At 4°K (but not at 77"K), the upper emission band retains about 20% of the polarization of the exciting light, when this polarization is parallel to a cube axis, but is unpolarized when the exciting light is polarized parallel to (say) (lll).327The lower band is not polarized. Similar effects are seen in the s2 + s p transitions of other ~ystems.328,~~~ This result has been e ~ p l a i n e d ~ ~ J ~ l in terms of the configuration-coordinate diagram in Fig. 39, which is a section through the V ( Q 2 ,Q3) diagram in the Q Z = 0 plane. If there were no vibronic interaction between 3P1and 3P0, the potential curves would follow the dotted lines. Interaction between 3P0and the P , component of 3P1forces the P , curve up. Pumping with [OOlI-polarized light populates the P , state, and, if T~~ coupling can be neglected, emission will occur from the same electronic state, with the same polarization. On the other hand, pumping with [111]-polarized light populates all the P states equally, and no polarization is observed. This argument requires that the 32,800C. C. Klick and W. D. Compton, Phys. Chem. Solids 7, 170 (1958). K. Edgerton, Phys. Rev. 138, A85 (1965). 329 A. Fuknda, S. Makashima, T. Mabuchi, and R. Onaka, Proc. Intern. Conf. Luminescence, Budapest, 1966 (to be published). 330 H. Kamimura and S. Sugano, J. Phys. Soc:Jupan 14, 1612 (1959). 331 N. N. Kristofel, Opt. i Spektroskopiya 18,798 (1965) ;see Opt. Spectry. ( U S S R ) (English Transl.) 18, 448 (1965). 327

T H E JAHN-TELLER

EFFECT I N SOLIDS

201

cm-1 emission be from the P , minimum, and that eg coupling in 3P1 be assign the unpolarized stronger than r Z gcoupling. Kamimura and Sugan0~~0 21,000-cm-1 emission to the P,P, minimum. However, as can be seen from Fig. 20, the PzP, “minimum” is actually only a cusp. A more likely assignment332for the 21,0OO-~m-~emission is 3 P + ~ ‘80(made allowed by ng vibrations), as indicated in Fig. 39. 28. FINESTRUCTURE AND THE HAMEFFECT IN OPTICALSPECTRA OF TRANSITION METALIONS Many transition metal ions in crystals, with orbital singlet ground terms (see Table I ) ,have optical transitions to orbital triplet excited terms. I n some such spectra, the no-phonon transitions have a fine structure (either intrinsic, or as a result of the application of an external perturbation) that cannot be accounted for by ordinary crystal field theory. I n most cases, the Ham effect (Section 15) provides a qualitative explanation for the structure, but a quantitative interpretation has only been possible in a few instances. We shall consider first the lowest spin-allowed optical absorption of Al2O3:V3+.The energy levels of this ion are shown schematically in Fig. 2. Although the ground state in a cubic field is 3 T 1 ,the trigonal field of A1203 is sufficiently large (-800 em-1) to stabilize the ground state against the relatively weak Jahn-Teller effect expected for the tZ2configuration. Spinresonance measurements on the 3A2ground level show the expected splitThe no-phonon transition to the 3T2(t2e)term, ting and g va1ues.29,333,334 on the other hand, shows strong evidence for a tetragonal Jahri-Teller distortion in the excited state. McClure’s treatment31 in terms of a static Jahn-Teller effect, although fundamentally correct, was based on insufficient experimental data, the no-phonon lines not having been clearly resolved a t that time. Recent w0 rk 2 ~has ~ clarified the experimental situation and shown that it provides a very clear example of the operation of the Ham effect. The energy level scheme for the lowest vibronic level of the 3T2term, as deduced from the absorption spectrum, is shown in Fig. 40a. Assignments to representations of C3 and values of M , are deduced from the observed selection rules for transitions from the M , = 1 and M , = 0 components of the 3A ground level. I n Fig. 40b, we show the first-order splitting of 3T2 under the trigonal field v and spin-orbit coupling p. Comparison of R. Edgerton and K. Teegarten, Phys. Rev. 129, 169 (1963). Note that, although the assignment of electronic transitions in this reference is probably correc’t, the treatment in terms of a single (totally symmetric) configuration coordinate i5 inadequate. 333 G. M. Zverev arid A. M. Prokhorov, Zh. Eksperim. i Teor. Fiz. 38, 449 (1960) ; see Soviet Phys. J E T P (English Transl.) 11, 330 (1960). 334 S. Foner arid W. Low, Phys. Rev. 120, 1585 (1960).

332

202

M. D. STURGE

II

15 885.2

15 880.4

15 876.1

I

s.3

0

FIG.40. (a) The no-phonon transitions 32'1(~A)+ 3Tzof A1203:V3+. Heavy lines indicate the strongest transitions. (b) First-order trigonal and spin-orbit splittings of the initial and final states. The ground state is not subject to a Jahn-Teller effect (see text); y is the Ham reduction factor (vibrational overlap) in the excited electronic (lowest vibronic) state. (c) Same as (b), but including second-order spin-orbit splittings in the excited state. Note that M , represents quantization along the trigonal axis of the crystal, whereas M,' represents quantization along the tetragonal axis of JahnTeller distortion. There are further small splittings that have been neglected (from Scott and Stwge233).

Fig. 40a with Fig. 40b shows that v is reduced by roughly a factor of 0.022 from its ground term value of 800 em-'. This is rather direct evidence for the Ham effect, operating this time on the trigonal field rather than on the spin-orbit coupling. Spin-orbit coupling is weak in V3+ (( = 200 cm-1 in the free ion) , and a proportionate reduction in { would make the firstorder splitting less than the linewidth (1-2 em-'). I n fact, the secondorder spin-orbit splittings are dominant : two contributions to the splitting of 32were calculated in Section 16 [see Eqs. (16.3) and (16.4)]. The 32 and 3~ levels are split by the sum of these and higher-order effects, as

THE JAHN-TELLER

EFFECT I N SOLIDS

203

-u-

shown in Fig. 40c. Note that in the ground level spin is quantized along the trigonal axis, whereas in the excited states second-order effects tend to quantize it along the tetragonal axis of Jahn-Teller distortion. The model we have used here is essentially that for a perfect (“cubic”) octahedron, upon which a trigonal field is imposed from “outside.” It is remarkably successful in accounting for the data discussed previously. However, the nearest-neighbor octahedron in Al203 is in fact greatly distorted, and the elastic restoring forces (not to mention the nuclear kinetic energy and the whole question of choice of normal modes) must reflect this distortion. It follows that the directions in which Jahn-Teller distortion occurs need not be a t all those predicted by the model. That this is in fact of the effect of stress the case has been demonstrated by on the spectrum of Fig. 40a. The results are too complicated to go into here; suffice it to say that the qualitative conclusions of our model (in regard to the Ham effect and second-order splittings) are confirmed. On the other hand, a much more elaborate calculation would be needed to obtain a quantitative value for the Ham reduction factor y, or even to understand why the complex distorts in just the way it does. 336

M. D. Sturge, unpublished work (1965).

204

M. D. STURGE

STRESS (kG/mm2)

FIG.41. Splitting of the r3arid r4 substates of 3T2(Mg0:Ni2+)under [loo] and [111] stress. The lines show the splittings calculated from the ground state splittings of Watkiris and Feher.33’

These difficulties should not apply in a cubic site. The 3Az-+ 3Tztransition of Ni2+in MgO shows two sharp no-phonon lines a t 8015 and 8181 cm-l.*87 These arise from magnetic dipole transitions to the r3and r4levels of 3T2which are only split apart by second-order spin-orbit interaction (and whose separation, by the arguments of Section 16, should not be greatly altered by the Ham effect). These levels can be split by uniaxial and the matrix elements for this stress splitting can be calculated by straightforward crystal field theory from the observed stress splitting of the 3A2state.337The calculated and observed splittings are shown in Fig. 41. We see that, although the tetragonal splitting is in quite good agreement with theory, the trigonal splitting is reduced by a factor of about 0.1. This is presumably due to the Ham effect. Similar results are obtained in the 3T2term of KMgF3:Ni2+,338which apparently contradict a previous conclusion, based on the spin-orbit splitting, that the Jahn-Teller effect is not important in this term.339 I n octahedrally coordinated ions, the Jahn-Teller distortion has always 336

33’ 338

M. D. Sturge and K. A. Ingersoll, Bull. Am. Phys. Sac. [2] 12, 60 (1967). G. D. Watkins and E. Feher, Bull. Am. Phys. Sac. [2] 7, 29 (1962). R. E. Ilietz, A. Misetich, and F. R.. Merritt., unpublished work (1966). J. Ferguson, H. J. Guggenheim, and D. L. Wood, J . Chem. Phys. 40, 822 (196).

205

THE JAHN-T E L L E R E F FE CT I N SOLIDS

turned out to be tetragonal when its symmetry is known at all. This is not surprising, since in transition metal ions the principal contribution to electron-lattice coupling comes from the e electrons (see Section 1). I n tetrahedral coordination, on the other hand, either trigonal or tetragonal distortion might occur in a T state. We have seen (Section 18) that, in the ground states studied, the distortion is usually tetragonal; but, in excited states, we find some evidence (as yet unconfirmed) that trigonal distortion occurs. So far, we have discussed only the quenching of a trigonal field by a tetragonal Jahn-Teller effect. I n a cubic crystal, we should in principle be able to see the quenching of spin-orbit coupling, uncomplicated by the presence of lower-symmetry crystal fields. Two cases where the no-phonon line shows spin-orbit splitting, drastically reduced but still visible, are 5T2transition of shown in Figs. 42 and 43. In Fig. 42, we see the 5E(TI) Fez+in the tetrahedral site of MgA1204 .340 Transitions are only allowed to the two r5levels of 5T2.The first-order splitting between these two levels is apparently reduced by a factor of about 0.13 (which is uncertain, because ---f

r3

r4

cm-1

FIG.42. The 5E(I'l) -+ T2no-phonon lines of MgA1204:Fe2+. Only rl + rs is allowed; if there were no Ham effect, the two r6 substates of ST,would be separated in first order by 5X (X 100 cm-l in Fe2+). The observed splitting is 65 cm-l, but we do not know the second-order contribution to this (from Slack et ~ 1 . 3 ~ ~ ) . N

340

G. A. Slack, F. S. Ham, and R. M. Chrenko, Phys. Rev. 162, 376 (1966).

206

M. D. STURGE

L 10941.1

I

10902.4

I

10929.6

10917.4

r6

re

re

rr FIG.43. The 4A2 -+ 4T2no-phonon lines of KMgF3:V2+. The calculated spin-orbit splitting of 4T2 (in the absence of Jahn-Teller effects) is shown (from Van der Ziel and St~rge3~‘).

we do not know the exact second-order contribution to this splitting). This is, within a factor of 2, the reduction factor deduced from (15.2), using the values of GE/hw obtained from the fractional intensity of the no-phonon line (20.2) , neglecting contributions of modes other than the Jahn-Teller active one. Similar spectra are obtained for Fez+ in ZnS and CdTe. I n the latter case, three lines are seen, indicating an extra “vibronic” rs state 27 cm-l above the lowest one. This is evidence that r2 distortion is dominant, since this would bring a state of the required symmetry down, as shown in Fig. 24. I n Fig. 43, we see the 4A2+ 4T2transition of V2+ in Kh’IgF3 .341 All four components of 4Tzare seen, the reduction factor for the spin-orbit splitting being about 0.4. The picture is slightly complicated by the proximity of the 2E state, which is mixed into 4T2by spin-orbit coupling. Let us now consider Ni2+ and Co2+ in tetrahedral coordination; the absorption spectra of these ions in ZnO, ZnS, and CdS have been studied by a number of ~ o r k e r and s a~ wealth ~ ~ of~ detail ~ ~uncovered. ~ ~ ~ Al~ ~ ~ though most of these spectra show qualitative evidence for the Ham effect, in that the apparent spin-orbit coupling parameter is substantially reduced in one or more excited terms, a detailed analysis of a particular case has J. Van der Ziel and M. D. Sturge, unpublished work (1966). R. Pappalardo and R. E. Dietz, Phys. Rev. 123, 1188 (1961). 3 4 3 H. A. Weakliem, J. Chem. Phys. 36, 2117 (1962). a41

342

207

EFFECT I N SOLIDS

T H E JAHN-TELLER

yet to be made. For instance, a typical spectrum, arising from the 4&(t23e4)

-+

'T'l(tz4e3)

transition of CdS:Co2+ near 8000 cm-', is shown in Fig. 44. The coarse structure consists of four bands (of which the third is rather ~ e a k ) 3with ~~, separations of 280, 285, and 375 cm-'. These bands were originally assigned342to the spin-orbit split levels of 4T1, but this interpretation fails to account for the fine structure on each band (since the transitions to the spin-orbital levels can split into at most two lines, of different polarization, under the trigonal field, the ground state splitting being too small to be resolved). An interpretation in terms of the Ham effect would assign these bands to successive vibronic levels,343each of which is split by spin-orbit coupling. The band separations are rather close to the 350-cm-1 interval that is associated with a number of transitions in CdS. I n Fig. 45, we illustrate a possible interpretation of the fine structure. We compare a crude calculation of the spin-orbit splitting (including Ham effect) with the fine structure observed. On the extreme left are the spin-orbit splittings of 4Tl calculated by first-order perturbation theory. Next are shown the splittings calculated by diagonalizing the d3 matrix in Td symmetry; the parameters used are those of Pappalardo and diet^.^^^ CdS: C o Z t

4.2'K

z

0

t a

w

m a

5000

Boo0

7000

8000

hv(cm-1) FIG.44. 4 A z + 4Tlaabsorption spectrum of CdS:Co2+ (unpolarized) (from Pappalardo and diet^^^^). 344

The anomalous weakness of the third band may possibly be a manifestation of the splitting of the band maximum discussed in Section 27.

208

M. D. STURGE

Y * 0.16

7. I

CdS :Co2+ (oBs.1

FIRST “EXACT” ORDER

4A, d 4 T I 0-0

fa)

(b)

(c)

(d)

0-3

(0)

FIG.45. Spin-orbit splitting of the 4Tla term of Co2+ in tetrahedral coordination. (a) First-order splitting. The value of the splitting parameter X depends on the amount of mixing of the 4Tl(t22e) and ‘Tl(t2e2)terms. For CdS, X is calculated by Pappalardo and Dietz to be 0.48I . (b) Effect of diagonalizing the entire 8 matrix on the splittings of 4T1a; has its free ion value of 550 cm-l. (c) Result of reducing first-order splittings by a factor of 0.15 (Ham effect), leaving higher-order effects unchanged. (d) Observed splittings in the no-phonon band. (e) Observed splittings in the third vibronic band.

The changes are, of course, the consequence of second-order effects (in the sense of Section 16) and should not be substantially altered by the JahnTeller effect. The third column of levels shows the result of reducing. first-order splittings by a factor of 0.15, leaving the second-order effects unaltered. There is quite good agreement with the fine structure on the first (“no-phonon”) band; and, as might be expected, the reduction due to the Ham effect gets progressively less as we go to higher vibronic levels. Qualitatively similar effects are seen in the other systems mentioned previously. This interpretation of the spectrum is undoubtedly too simple and is only meant to be illustrative of the method. Stress and Zeeman effect measurementsa6 do not support an interpretation in terms of a tetragonal Jahn-Teller distortion. The effect of stress perpendicular to the c axis (predominantly tetragonal) is sma11, whereas stress parallel to the c axis (trigonal) has a big effect; furthermore, the trigonal crystal field, which should anyway be small in CdS, is apparently not quenched. A full analysis of the data will probably require coupling to both e and 7 2 distortions to a46

W. M. Yen and M. D. Sturge, unpublished work (1966).

THE JAHN-TELLER

EFFECT IN SOLIDS

209

be taken into account and has not been made, but the main Jahn-Teller distortion is probably trigonal. The necessity for going beyond the first-order theory of the Jahn-Teller effect when the interaction is strong is illustrated by the case of the 4A2-+ *Tz(h2e)transition of ruby. The theory of the 4Tzterm is essentially analogous to the 3T2term of V3+, except that all orbital matrix elements have the opposite sign.31The fine structure in the no-phonon lineN6is even more complex than in V3+ and is quite inexplicable if there is no Jahn-Teller effect; the same can probably be said of the stress splitting.347However, whereas in A1203:V3+ the no-phonon transition appears equally strongly in both polarizations, in A1203:Cr3+it is purely u polarized. This fact was explained by McClure3’ in terms of static tetragonal distortion, in fact extension, which in V3+ puts the orbital singlet lowest with the consequences discussed previously. For Cr3+, tetragonal distortion of the same sign (extension) would put the orbital doublet lowest (the matrix element of tetragonal field having the opposite sign). McClure showed that the transition to the lowest component of the doublet is forbidden in x polarization and thus explained the data. This argument is open to criticism, since there is no reason to suppose that the sign of the distortion will be the same for the two ions; rather, one would expect from Fig. 20 that the sign of the Jahn-Teller distortion would be such as to bring the singlet lowest in both cases, i.e., extension for V3+ but compression for Cr3+. This would lead, in Cr3+as in V3+, to a more or less unpolarized no-phonon line. The answer could lie in anharmonic effects in the crystal, which might make extension greatly preferred to compression. A more likely explanation348is that the trigonal field splitting (with some assistance from spin-orbit coupling) stabilizes the “tetragonal” doublet. The trigonal splitting is not only not quenched in the doublet but is actually enhanced (roughly doubled) by the mixing. (In V3+, this mixing has the opposite Jahn-Teller induced 4T2-4T1 effect, and partially accounts for the absence of trigonal splitting3‘ in the 3Tzband maximum.) Finally we come to the 4A2-+ 4T2transition in MgO:V2+ (isoelectronic with Cr3+),which is of considerable interest, since it may shed some light on what may happen when both e modes and rZ modes are Spin-orbit coupling is small, and its second-order effects are negligible. Measurements of the effect of stress on the no-phonon line show that there is a tetragonal Jahn-Teller effect, which quenches both spin-orbit coupling and the effect of a trigonal [111] strain. If this were the whole story, there would be a single transition to a threefold degenerate level, J. Brossel and J. Margerie, in “Paramagnetic Resonance” (W. Low, ed.), Vol. 2, p. 535. Academic Press, New York, 1963. 347 A. A. Kaplyanskii arid A. K. Przhevuskii, Phys. Status Solidi, 11, 629 (1965). 348 M. D. Sturge, Bd1. Am. Phys. SOC.[a] 11, 886 (1966). 346

210

M. D. STURGE

which could be split by tetragonal stress. I n fact, even in the absence of a stress there is a splitting (too large to be due to spin-orbit coupling), which behaves as if it were due to a not completely quenched trigonal field. This “trigonal field” is not present when the ion is in its ground state, as can be demonstrated by spin resonance. It splits the Tz vibronic level into a doublet and a singlet, the doublet lying some 40 cm-l below the singlet. To maintain cubic symmetry (and to account for the observed polarizations) , we must assume that the trigonal field may be oriented at random along any of the four (111)-type directions. The “singlet” is and the “doublet” eightfold then in fact fourfold degenerate (A1 Tz), ( E Ti TZ).~’ This picture is sheer speculation at present, since there is no theoretical basis for the “trigonal field.” We have seen (Section 17) that a perturbation treating the T Z interaction as much weaker than the e interaction, can give no such effect. Both interactions will have to be treated on an equal basis, and almost certainly nonlinear effects will have to be taken into account, before a satisfactory interpretation can be given on these lines. Ham3%has made an alternative approach that predicts the same observable results as the foregoing model. H e takes the lowest two levels of Fig. 24 in the zero coupling limit and assumes that nonlinear e - T Z interaction somehow pushes the lowest Tistate down until it is close to the lowest T2state. The next T2state is not only assumed to be pushed down to 40 cm-1 above the lowest state, but also mixed with it, so that there are zero-phonon transitions from the Az ground state to both Tz states. A tetragonal stress mixes the TI and T2 states, and Ham is able to reproduce the observed effects of stress with only two parameters, the separation between the T2states and their mixing. The most curious feature of this problem is the contrast between the extreme simplicity of the experimental data and the complexity of the theoretical interpretations so far attempted. It may be that a completely different theoretical approach will yield a simple and satisfactory explanation.

+ +

+

29. CONCLUSIONS We have seen that most of the predicted consequences of the JahnTeller effect have been qualitatively confirmed in a wide range of experi34g

350

The alternative hypothesis, tentatively put forward a t the end of Ref. 286a, that the no-phonon lines are transitions to E and A 1 vibronic states, is not tenable, since it cannot account for the observed intensities. F. S. Ham, in “Optical Properties of Ions in Crystals.” Wiley, New York (to be published).

THE JAHN-TELLER EFFECT I N SOLIDS

211

mental situations in solids. On the other hand, quantitative agreement between experiment and a “first principles” calculation has never been achieved. This is partly because of inadequacies in the experimental data, but it is primarily because the complexity of the theoretical problem has forced the adoption of oversimplified models. It is to be hoped that the recent spate of experimental results on relatively simple systems will stimulate theorists to make calculations based on more realistic models. On the experimental side, many techniques for studying the Jahn-Teller effect have only just begun to be exploited. Two such techniques are the Mossbauer effect and the detailed study of vibrational effects in X-ray diffraction. These two techniques are likely to be of most use in the study of cooperative Jahn-Teller effects, particularly in the understanding of dynamic effects near a transition, a subject barely touched on so far. Meanwhile, in the study of isolated centers, there is much still to be done with the “traditional” method of spin-resonance, in particular, by careful studies of the temperature dependence of hyperfine structure and spin-lattice relaxation in simple Jahn-Teller systems. The new technique of paraelectric resonance is a promising (although as yet untried) method for studying the Jahn-Teller effect in noncentrosymmetric systems. With the availability of better crystals and higher magnetic fields, we may expect to see a good deal more information come out of the stress effect and Zeeman effect in no-phonon optical transitions. T o these one can add the Raman eff ect,351two-photon absorption,3s2and (in noncentrosymmetric crystals) the Stark effect, all as yet virtually unexploited as tools for studying the Jahn-Teller effect. Finally, the techniques of far-infrared spectroscopy have been greatly improved in recent years, and far-infrared studies of the low-lying vibronic levels of systems with degenerate ground states, interacting with infrared active modes, would be of great interest. ACKNOWLEDGMENTS I am grateful to the following for communicating their work prior to publication and discussing it with me: D. C. Burnham, R. E. Coffman, P. J. Dean, R. E. IXetz, F. S. Ham, U. T. Hochli, G. D. Jones, R. S. Knox, D. E. McCumber, T. J. Menne, F. R. Merritt, M. C. M. O’Brien, J. Van der Ziel, G. I). Watkins, and W. M. Yen. Besides the foregoing, I have benefited from discussions (or correspondence) with I. Bersuker, C. J. Ballhausen, P. F. Bongers, H. L. Frisch, C. G. B. Garrett, S. Geschwind, E. M. Gyorgy, C. H. Henry, G. F. Imbusch, R. C. LeCraw, M. U. Palma, D. N. Pipkorn, M. H. L. Pryce, S. Strassler, Y. Toyozawa, and J. Y. Wong. I owe a special debt of gratitude to F. S. Ham, who has patiently enlightened my theoretical darkness on a number of occasions. Finally, I would like to express my gratitude to Professor A. L. Schawlow and the Physics Department of Stanford University, for hospitality during the summer of 1965, and for the encouragement without which this work would never have been begun. 35l 352

C. H. Henry, J. J. Hopfield, and L. C. Luther, Phys. Rev. Letters 17, 1178 (1966). J. J. Hopfield and J. M. Worlock, Phys. Rev. 137, A1455 (1965).

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Green’s Function Method in Lattice Dynamics

PHILIP C. K. KWOK I B M Thomas J . Watson Research Center, Yorktown Heights, New York

Introduction. . . . . . . . . . . . . . . . . . . .

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

I. Dynamics of Lattice Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. The Hamiltonian of Lattice Vibrations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

214

214 ............................ 215 2. Expansion in Nuclear Displacemen 3. Normal Modes and Normal Coord ............................ 216 11. Phonon Correlation Function and Experimental Observables . . . . . . . . . . . . . . . 220 4. General Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Definition of Phonon Correlation Function 6. Physical Properties of the Phonon Correlation Function 7. Spectral Representation of the Phonon Correlation Function, . . . . . . . . . . . 224 8. Relation of the Phonon Correlation Function to Experiments. . . . . . . . . . . 227 9. Ultrasonic Attenuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Inelastic Neutron Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. ODtical Measurements. . . . . . . . . . . . . . . . . . . 111. Phonon Green’s Function ..................... 12. Definition of Phonon en’s Function. . . . . . . . . 13. Functional Derivative Te 14. Green’s Function Equation. . . . . . . . . . . . . ...................... 245 15. Formal Procedure of Calc 16. Phonon Self-Energy Function ............................ 253 17. Dispersion and Damping 18. Perturbation Calculation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 IV. Explicit Calculation of Phonon Green’s Function and Applications. . . . . . . . . . 262 262 19. Calculation of the Phonon Self-Energy Function. . . . . . . . . . . . . . . . . . . . . . 20. Ultrasonic Attenuation: General Formulation. . . . . . . . . . . . . . . . . . . . . . . . . 267 21. Attenuation of Transverse Acoustic Phonons. . . . . . . . . . . . . . . . . . . . . . 22. Attenuation of Longitudinal Acoustic Phonons. . . . . . . . . . . . . . . . . . . . V. Phonon Boltzmann Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 23. Lattice Transport Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 24. Derivation of the Phonori Boltzmann Equat,ion.. . . . . . . . . . . . . . . . . . . . . . 281 VI. Coupled Phonon-Photon System, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 25. Interaction of Latt,ice Vibrations with the Macroscopic Elect.romagnetic Fields 26. Dispersion of the Optical Phonons. . . . . . . . . . . . . . . . . . . . . . . . . . . 27. Polariton Green’s Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 28. Dielectric Susceptibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 213

214

PHILIP C. K. KWOK

Introduction

This article is devoted to the application of the thermodynamic Green’s function method to the study of problems in lattice dynamics. It is intended to serve two basic purposes. The first one is to discuss the motivation of the introduction of the concept of Green’s function or correlation function from the physical point of view, and the second one is to present the computational technique by which significant physical quantities are calculated. The reason for choosing the topic of lattice dynamics is that it is one of the most fundamental and familiar problems in solid state physics. The Green’s function method has become an invaluable tool in the study of complicated systems of interacting particles in statistical physics. The method provides a rigorous and systematic approach in extracting essential physical information. There exist a large number of publications dealing with various aspects of the Green’s function method. For an overall view of the subject, the readers may find the work by the following authors helpful: Martin and Schwinger,l Bonch-Bruevich and Tyablikov? Abrikosov et u Z . , ~ and Kadanoff and Baym.4 For certain specific applications of Green’s function, one may refer to the book by Nozieress on interacting Fermi systems and that by SchrieffeP on superconductivity. An article by Cowley’ contains detailed calculations of anharmonic effects using the diagrammatic Green’s function technique. 1. Dynamics of Lattice Vibrations

1. THE HAMILTONIAN OF LATTICE VIBRATIONS

The fundamental starting point of obtaining a theory to describe the dynamics of lattice vibrations is, of course, the total Hamiltonian of all the nuclei and electrons in the solid. This formidable approach is greatly simplified if one makes use of the adiabatic or Born-Oppenheimer approximation.8 I n this approximation, the electrons are considered to be able to adjust t o P. C. Martin and J. Schwinger, Phys. Rev. 116, 1342 (1959). V. L. Bonch-Bruevich and S. V. Tyablikov, “The Green’s Function Method in Statistical Mechanics,” North-Holland Pnbl., Amsterdam, 1962. 3 A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, “Methods of Quantum Field Theory in Statistical Physics,” Prentice-Hall, Englewood Cliffs, New Jersey, 1963. L. P. Kadanoff and G. Baym, “Quantum Statistical Mechanics,” Benjamin, New York, 1962. P. Nozieres, “Interacting Fermi Systems,” Benjamin, New York, 1964. 6 J. R. Schrieffer, “Superconductivity,” Benjamin, New York, 1965. R. A. Cowley, Advun. Phys. (Phil. Mug.Suppl.) 12, 421 (1963). 8 M. Born and R. Oppenheimer, Ann. Physik [4] 84, 457 (1927). 2

GREEN'S

FUNCTION

METHOD IN LATTICE DYNAMICS

215

the instantaneous positions of the nuclei. The electronic states are thus unique functions of the nuclear co-ordinates. Furthermore, if the electrons are assumed always to remain in the ground state for any given set of nuclear coordinates, one can then obtain an effective Hamiltonian describing only the motions of the nuclei with the electrons eliminated from the problem. The correction to the latter approximation involves electronic transitions and represents the interaction between electrons and the lattice vibrations. For a review of the work of many authors on this topic, the readers are referred to the article by C h e ~ t e r . ~ The effective Hamiltonian for the nuclei may be written as

H

=

C $mi(axi/at)Z + a ( .. . , xi ,. . . ) ,

(1.1)

i

where mi and xi are the mass and co-ordinate of the ith nuclei. The first term is the kinetic energy, and the second term is the potential energy described by the function of the co-ordinates xi. I n this Hamiltonian, we have neglected the coupling of the nuclear motion to the transverse electromagnetic radiation field. This coupling has a n important effect on the long wavelength infrared active optical vibrational modes resulting in a hybridization of the phonon and photon spectra. Such a retardation effect was first considered in a classical manner by Huang.10 A quantummechanical treatment was later given by Fano" and Hopfield.12 The coupled optical phonon and photon modes have come to be known as the polaritons,12 and their dispersion has recently been observed by Raman scattering.13 2. EXPANSION IN NUCLEAR DISPLACEMENTS

in Eq. (1.1) has a unique We assume that the potential function absolute minimum corresponding to the nuclear positions xi". These positions then represent the equilibrium lattice sites of the crystalline s01id.I~ Adopting the notation in Born and Huang,I5 we rewrite the xi" as xo(;), where 1 denotes the lth primitive unit cell and k the kth nucleus in a cell. G. V. Chester, Advan. Phys. (Phil. Mag. Suppl.) 10,357 (1961). K. Huang, Nature 167, 779 (1951); Proc. Roy. SOC.A208, 352 (1951). l1 U. Fano, Phys. Rev. 103, 1202 (1956). J. J. Hopfield, Phys. Rev. 112, 1555 (1958). C. H. Henry and J. J. Hopfield, Phys. Rev. Letters 16, 964 (1965). 14 We assume that the positions also correspond to the absolute minimum of the free energy and thus exclude the possibility of a displacive phase transition. This assumption is, of course, invalid in the case of, for example, ferroelectrics. For a discussion, see P. C. Kwok and P. B. Miller, Phys. Rev. 161, 387 (1966). 15 M. Born and K. Huang, "Dynamical Theory of Crystal Lattice," Oxford Univ. Press, London and New York, 1956.

216

PHILIP C. K. KWOK

The nuclear displacements u(L) are defined as the deviations of nuclear co-ordinates xi or x(:) from their equilibrium values, i.e., U(L)

= X(k)

- XO(L).

The Hamiltonian (1.1) may now be expanded as a power series in the displacement u. The result is customarily expressed in the following form:

(2.2) where the a’s denote the Cartesian indices. There is no term in Eq. (2.2) linear in the displacements, because we are expanding the potential function about one of its extrema. The coefficient @@) in the term quadratic in the displacements is the harmonic coefficient that determines the lattice normal modes or harmonic phonons. The other coefficients for n 2 3 are the anharmonic parameters that couple the normal modes and give rise to phonon-phonon interactions. These coefficients satisfy various symmetry relations as a consequence of crystal symmetry. A summary of such symmetry properties may be found in the review article by Liebfried and Ludwig.16 A power series expansion for the lattice Hamiltonian such as Eq. (2.2) actually is only valid if the root-mean-square displacement due to quantum (zero-point) and thermal fluctuation is small compared to internuclear distances. The failure of a power series expansion for the nuclear potential energy is demonstrated in the case of solid helium, in which the zero-point motions of the nuclei are large.” One must then resort to other methods to calculate the lattice properties in order to obtain meaningful results.’*

3. NORMAL MODESAND NORMAL COORDINATES The dynamical equations of motion of the nuclear displacements describing the lattice vibrations can be readily deduced from the Hamiltonian (2.2) by using the following canonical commutation relations :

16

G. Liebfried and W. Ludwig, Solid State Phys. 12,275 (1961). L. H. Nosanow and N. R. Werthemer, Phys. Rev. Letters 16, 618 (1965). D. R. Fredkin arid N. R. Werthemer, Phys. Rev. 138, A1527 (1965).

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

217

where the operators are in the Heisenberg representation. One finds that

The frequencies of the normal modes of lattice vibrations or the harmonic phonons are determined by the harmonic part of Eq. (3.2), i.e.,

The solutions to this equation can be expressed in a simple manner if we neglect effects due to finite crystal size and use the periodic boundary ~0ndition.l~ Size effects on the phonon spectra in the long wavelength limit, in particular with regard to the validity of the Lyddane-SachsTellerIg relation between the longitudinal and transverse optical modes, was first investigated by Rosenstock.20Since then, several author^^^-^^ have carried out calculations of the normal vibrational mode spectra for finite and semi-infinite crystals. The conclusions one can draw from these works are that surface modes that do not extend throughout the crystal appear and that the spectra of the ordinary modes in the long wavelength limit depend on the geometry of the crystal. Neglecting size effects (and using the periodic boundary conditionZ4),one can look for oscillatory solutions to Eq. (3.3) of the form ua(L;t ) = A a ( k ) exp (ip-x? - i d ) described by a wave vector p. A a ( k ) is the amplitude that is independent of 1. One then obtains the normal modes characterized by the wave vector p and a branch index j . The square of the frequency wjpo of the (jp) mode is the eigenvalue of the dynamical matrixz5

(which is independent of 1 because of translational symmetry) belonging R. H. Lyddane, R. G. Sachs, and E. Teller, Phys. Rev. 69,673 (1941). H. B. Rosenstock, Phys. Rev. 121, 416 (1961). 21 T. H. K. Barron, Phys. Rev. 123, 1995 (1961). 22 A. A. Maradudin and G. H. Weiss, Phys. Rev. 123, 1968 (1961). 23 R. Fuchs and K. L. Kliewer, Phys. Rev. 140, A2076 (1965). 24 See, for example, Born and Huang,15 Appendix IV, for a discussion on the periodic boundary condition. 26 There are various models for the internuclear forces to calculate the dynamical matrix and the anharmonic coupling coefficients. The best known are the rigid ion model and the shell model. For a review of this topic, see B. G. Dick, in “Lattice Dynamics” (F. R. Wallis, ed.), p. 159. Pergamon Press, Oxford, 1965. lS 2o

218

PHILIP C. K. KWOK

to the eigenvector ekDl

( jp) .

(3.5)

The eigenvectors satisfy the following orthonormality and completeness relations : (3.6a) C e k a ( jp) e k a ( j'p) * = 6 j j r , ka

c

ekm

( jp) e k f a ' ( jp) * =

6oru"6kkf

.

(3.6b)

i

The normal mode coordinates or simply normal coordinates are defined by the expansion of uu(:;t ) in terms of these normal vibrations:

where N is the number of primitive unit cells in the crystal. The factor (Nmk)-1/2in Eq. (3.7) is chosen such that the normal coordinates Qjp(t) have the simple commutation relations CQjp

( t )7 Q j t p ~( t )1

=

(fi/46jj4,-pf

.

(3.8)

(3.8) follows directly from relation (3.1). Since the nuclear displacement e k U ( jp) = e k a ( j - p) *,15 the Qjp(t) also satisfy Qip+ ( t ) = Qj-p ( t )* (3.9)

u is real (or hermitian) and

The dynamical equations satisfied by these operators in the harmonic approximation are readily obtained from Eq. (3.3) to be Qip(t)

(3.10)

= -(~jpo)2Qjp(t)*

The harmonic Hamiltonian given by Eq. (2.2) may be expressed in terms of the normal coordinates as IT(')

=

C

+

+[QjpQj-p

(wjpo)2Qip&j-p1.

(3.11)

ip

This expression illustrates the independent character of the normal modes. One can also define the phonon creation and annihilation operators ujp+ and uip 26 as

+

Qjp

=

( fi/2wjpo)

Qj,

=

i-'(fiwjpo/2)'yujp -

'I2(

~ j p

ai-p+)

7

(3.12a) (3.12b)

From Eq. (3.8), it is readily verified that the operators ujp+and ujp satisfy 26

See, for example, J. M. Ziman, "Electrons and Phonons," Chapter I. Oxford Univ. Press, London and New York, 1960.

219

GREEN'S FUNCTION METHOD IN LATTICE DYNAMICS

the usual commutation relations [Uj,

, Uj?,!] = [UjP+,

UjfPI+]

=

(3.13a)

0,

(3.13b)

= 6jjdppl .

[UjP+, UjIp']

The harmonic Hamiltonian So) (3.11) is just a sum of independent harmonic oscillator Hamiltonians

H(O) =

C fiuipO(ajp+ajp+ $1.

(3.14)

jp

The study of various properties of lattice vibrations is much more convenient if one works directly with the normal coordinates and their interactions due to anharmonicity. Using definition (3.7) , one can rewrite the total Hamiltonian (2.2) as

H

=

c

4CQjPQj-P

+

(wjPo)2&jP&j-Pl

jp

+ C C (lln!) a,

n=3

U~~i.jzp ...., z jnpn&jipiQj2Pz.

* *&jnpn

9

(3.15)

lip)

where the coefficients U(") are obtained from the anharmonic coefficients W according t o

(3.16) These coefficients also satisfy various symmetry relations16 and, in particular, the lattice translational symmetry condition that the sum of their pn must be zero or a reciprical lattice vector. wave vectors p l p 2 The equations of motion (3.2) become

+ + -- +

QjP

(t) =

- (%PO>

(t)

220

PHILIP C. K. KWOK

II. Phonon Correlation Function and Experimental Observables

4. GENERAL DISCUSSION The experimental study of various properties of lattice vibrations is invariably carried out by applying appropriate external probes. The desired information is contained in the response of the crystal to these disturbances. This point of view is illustrated by the following discussion of several standard experiments. I n ultrasonic attenuation experiments measuring the lifetime of acoustic phonons due to anharmonic interactions, a sound wave is generated by a transducer bonded to the crystaLn The decrease in amplitude of the sound wave is measured after it has transversed the length of the crystal. Thus, one is observing the response of the crystal to a forced vibration of some nuclear displacements. Inelastic neutron scattering is one of the best available techniques used to measure the phonon dispersion c u r v e ~ . A ~ ~beam J ~ of neutrons of known energy is sent into the crystal, and the energy distribution of the inelastically scattered neutrons is measured at different angles. The various peaks in the distribution correspond to the frequencies of the phonons created and destroyed by the neutrons. I n optical measurements, the external probe used is electromagnetic radiation. Reflectivity and transmission experiments are commonly employed to determine the infrared absorption spectra of the solid.30One shines infrared light of different frequencies on the crystal and measures the intensity of the reflected or transmitted radiation. From these responses, one can calculate the frequency-dependent dielectric susceptibility and the frequencies and widths of the long wavelength infrared active optical phonons. I n Raman and Brillouin scattering130phonons with frequencies in the optical range, yet low enough not to cause real electronic transitions, are inelastically scattered, creating or absorbing optical and acoustic phonons, respectively. From the frequency distribution of the scattered light, one can determine the phonon dispersion curves as in inelastic neutron scattering. E. H. Jacobsen, in “Qiiantum Electronics” (C. H. Townes, ed.), p. 468. Columbia Univ. Press, New York, 1960. ** For a thorough review of the experimental as well as theoretical aspects of inelastic neutron scattering, see B. N. Brockhouse, in “Phonon and Phonon Interactions” (T. A. Bak, ed.), p. 221. Benjamin, New York, 1964. 28 Experimental data of the phonon dispersion curves of various materials may be found in “Lattice 1)ynamics” (R. F. Wallis, ed.), Sect. A. Pergamon Press, OXford, 1965. 30 See E. Burstein, in “Lattice Dynamics” (R. F. Wallis, ed.), Sect. C, p. 315. Pergamon Press, Oxford, 1965. 27

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

221

5. DEFINITION OF PHONON CORRELATION FUNCTION

It was shown in the previous section that the various experiments may be interpreted as measuring the responses of the phonon system to different external disturbances. The most natural question to ask, then, is whether these different types of experiment may be described in a unified way. We shall show presently that most of them may be described by a single phonon correlation function. The phonon correlation function determines the one-phonon transition probability involving the creation or absorption of a phonon, and also the linear response of the lattice vibrations to an external force. The crystal lattice is assumed to be in thermal equilibrium at a temperature T.The thermal expectation value of any operator 0 is defined in the usual manner by

(0)= Tr Iexp (-PH)*OJ/Tr (exp ( - P H I ) ,

(5.1)

where B is ( k B T ) - l , kg is the Boltzmann constant, and H is the total Hamiltonian of the lattice vibrations (2.2) or (3.15).We define the phonon correlation function as the thermal expectation value of two normal coordinates in the Heisenberg representation a t different times, narnelyl3l D>ip.jrp* (tt’)

=

(i/h>(Qip ( t )

(0>,

(5.2)

Qjp(t)

=

exp [ i ( H / h ) t ] * Q j p ( O )aexp [ - i ( H / h ) t ] .

(5.3)

Qipp‘

The factor i / h in the definition (5.2) is chosen for convenience. As the phonon system is in thermal equilibrium, one has crystal translational symmetry and translational invariance in time. Then it follows that the is only a function of the time difference correlation function D>ip,j.pr(tt’) t - t‘ and that it vanishes unless p’ is equal to -p,32 i.e.,

D>jp.yp.(tt’)= 8P,-p‘D>jp,jLp(t - t’). The abbreviation

D>jp,jt-p(t- t’)

=

D’jy,,(t

- t‘)

(5.4) (5.5)

will be frequently used in the following discussion. It is also useful to define the Fourier transform of the function (5.5) :

81

3)

This function is related to the displacement correlation function introduced by L. van Hove [Phys. Rev. 96, 249 and 1374 (1954)l by a simple transformation. Strictly speaking, p‘ must be in the first Brillouin zone and satisfied p reciprocal lattice vector K.

+ p’ = 0 or a

222

PHILIP C. K. KWOK

or m

D>jjt,,(w) = i-l

d t D > j j f , P (t t’) exp [ i w ( t

- t’)],

(5.7)

where the factor i is again chosen for convenience. 6. PHYSICAL PROPERTIES OF THE PHONON CORRELATION FUNCTION Let us first study the phonon correlation function in the harmonic approximation. The normal coordinate Q j p ( t ) = exp [ i ( H / f i )t].Qj,(O) exp [- i ( H / f i ) t ] is readily transformed into the following representation:

-

Qj,(t)

=

(fi/2~0jp’)’’~[~jp(O) exp (-iwjpOt) +aj-,+(O) exp (iWjpOt) J (6.1)

upon using the definition (3.12a) and the expression (3.14) for the harmonic Hamiltonian and the commutation relations (3.13). From (6.1), one can immediately write down the harmonic phonon correlation function, which will be denoted by the superscript (0) , as

DZij),,,(tt’)

=

[i/2(wj,”wj~,P)1~2][(ajp(0)ai~p~(O) ) exp (-iW,pOt - iwjtpjOt’)

+ (aj,(0)aj~-pf+(O)) exp ( -iwjpOt + iWjJp,Ot’) + (aj-,+(0)ai~,r(O) ) exp (iwjpOt - iwjtptot’) + ) exp (iwj:t + (~j-,+(o)~j*-,~+(O)

(6.2)

iwjtptot’)].

The thermal expectation values in (6.2) vanish unless j is equal to j’, because there is no coupling between the harmonic phonons. Furthermore, using the fact that the phonon occupation number is a good quantum (O)) number, one finds that the only nonzero terms are ( u~ p( 0 ) u~ ~ - p~ +and (aj,+(0)aj~,~(O) ), with j = j’ and p’ = -p. Thus, condition (5.4) holds, and we have

~ & ! ” (t t’)

=

6jjt(i/2wjp~) ([I

+

exp [-iojpO(t

~ ~ ( w j , ” ) ]

-

t’)]

+No(wjpO) Sexp [iwjpO(t - t ’ ) ] ) ,

(6.3)

where No(wj,”) is the thermal equilibrium phonon occupation number (aj,+(0)a~p(O) ), and No is the equilibrium distribution function

NO(o)= [exp(pho)

-

11-l.

(6.4)

The Fourier transform of the phonon correlation function defined by (5.7) may be immediately calculated from (6.3) to be

D;$:~
(W~,O)~)

(1

+ NO(0)).

(6.5)

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

223

The physical meaning of the phonon correlation function is clear if one studies expression (6.3). Let us take t to be greater than 1’. The first term in (6.3), which corresponds originally to the expectation value ( i / h )(uj, ( t )ujp+(t’) ), is proportional to the average probability amplitude of creating a phonon at time t‘ and finding the system in the same state at a later time t. Therefore, it describes the propagation of the phonons of the mode ( jp) in the medium. In the harmonic approximation, this amplitude is an oscillatory function of the time difference t - t’, and the corresponding probability is thus independent of time. When anharmonicity is present, this probability amplitude will decrease with time because of phonon interactions. One expects to find the time-dependent factor to be a product of an oscillating part and a decaying part, i.e., of the form exp [-iujp(t- t’) - yjp(t - t ’ ) ] . The frequency of oscillation wjp clearly represents the frequency of the real phonon in the anharmonic crystal or the renormalized phonon frequency, and the decay constant yip represents its inverse lifetime. The second term in (6.3) corresponds to the propagation of a phonon “hole,” and its time-dependent factor will become exp [ioj,( t - t ’ ) - yjp( t - t’) 3 when anharmonic interactions are included. For t < t’, the terms in the correlation function are simply the complex conjugate of the probability amplitudes discussed previously. The renornialized frequency wjp and widths yj, that describe the propagation of physical phonons in the anharmonic solid or, equivalently, the function D>jjtP( t - t’) are, of course, closely related to the frequency dependence of the Fourier transform of the correlation function. It is obvious from definition (5.6) that wjp and yip are just the magnitudes of real and imaginary part of the complex poles of D>jif,,(w).I n the harmonic approximation, D>jit , (0)is diagonal in j and j’, and its frequency dependence is 2, from (6.5) . Analytically, essentially described by 2a (w/1(J I) 6 (u2 - (ojpo) this function may be written as

showing that Dz!:)(a)has complex poles lying infinitesimally close to the real axis at f w j , o , the harmonic phonon frequency. The 6 function or (6.6) represents the frequency distribution of the normal modes of lattice vibrations of a given wave vector p. These excitations are undamped in the harmonic approximation and are just the harmonic phonons with frequencies q p 0 . When anharmonic interactions are taken into account, the frequency distribution of the vibrational modes or real phonons will no longer be

224

PHILIP C. K. KWOK

infinitely sharp. The correlation function becomes

D>jjt,p(@)= Aijt,p(w) (1

+No(w)),

(6.7)

which is often referred to as the spectral representation of the correlation functions and will be derived in the next section. The frequency-dependent function Ajj,,,(w) is known as the spectral function. In the harmonic approximation, A,jjl reduces to

.,

A;~!,,(U)= 6jjt2r(w/l w I)8(u2- ( ~ j , ~ ) ~ ) , (6.8) as can be verified by comparing (6.5) and (6.7). The physical meaning of the spectral function, however, remains unchanged. It represents the frequency distribution of possible excitations of the interacting phonon system for a given wave vector p. The nondiagonal part of Ajj.,,(w) which is zero in the harmonic approximation may be interpreted as the spectral distribution of excitations that are mixtures of phonons from two different branches. The diagonal part ( j = j’),on the other hand, essentially describes the spectrum of the physical phonon belonging to the branch j. When the anharmonic effects are sui5ciently small, Ajy,(w) will consist mainly of two well-defined peaks located at fwj, ,the renormalized phonon frequency, with widths given by yj, . It may also contain other well-defined but smaller peaks that representing collective excitations of the phonons, for example, the second sound mode, which describes phonon heat conduction in the low-frequency or collision-dominated limit.%.%Finally, there is also a continuum background representing multiphonon excitations.a6 These discussions will be put on a more quantitative basis when we actually calculate the correlation function and the spectral function.

7. SPECTRAL REPRESENTATION OF THE PHONON CORRELATION FUNCTION The phonon correlation function D>jj,,,(t - t’), according to (5.2), (5.4), and (5.5) and the definition (5.1) of the thermal expectation value, may be written as

D>jj..,( t

- t’)

( - p H ) X exp ( i ( H / f i ) t ).Qj,(0) Oexp ( - i ( H / f i ) t ) X exp (i(H/fi)t’)*Qjfl-p(0)-exp (-i(H/fi)t’)}/Tr (exp ( - p H ) ) , (7.1)

= Tr (exp

aaSee G. Baym, Ann. Phys. ( N . Y . ) 14, 1 (1961). I4 For example, the second sound mode, which has been shown by P. C. Kwok and P. C. Martin [Phys. Rev. 142, 495 (1966)l to appear explicitly in the spectral function for the acoustic branch in the low-frequency limit. I6 Actually, all well-defined peaks in the phonon spectral functions should be treated as physical phonons. However, for convenience, we shall call only those excitations with renormalized frequency w i D close to wjDO physical phonons. a6 For further discussion of the structure and physical meaning of spectral functions in other interacting systems, see Noaieres6 and Schrieffer.6

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

225

where we have exhibited explicitly the time dependence of the Heisenberg operators. Using the property that a trace is invariant under a cyclic permutation of the operators in it, the right-hand side of Eq. (7.1) is readily shown to be equal to Tr (exp (-pH)X exp [i(H/fi)t’].Qjf-,(0)-exp [-i(H/fi)t’]

+ iPfi)l.Qjp(O) X exp [ - i ( H / f i ) ( t + i@fi)])/Tr (exp ( - P H ) ). X exp I i ( H / f i )( t

(7.2)

From (7.1) and (7.2), one can immediately obtain the following identity for the phonon correlation functionn : D’jjf,P(t -

- t - ipfi).

t’) = D>jIj,-,(t’

(7.3)

The corresponding identity for the Fourier transform D>jj,, ( w ) is

D > j j r . p (= ~ )Pjgj,-,( - w ) -exp (pfiw).

(7.4)

I n the harmonic approximation, D>jjt,,(w) is given by 6jjf27r(w/l w I) X 6 ( w 2 - ( W ~ , O ) ~ ) (1 N o ( w ) ) and satisfies the foregoing relation because of the identities wjpo = wj-, 0 (7.5a) and 1 + N o ( w ) = -exp ( P f i w ) . ( l + N o ( - @ ) ) . (7.5b)

+

One is thus motivated to write the exact correlation function in a form similar to that of the harmonic correlation function, i.e., with the factor 1 No separated out:

+

D’jjf,p(w) The function A j j t , , ( w )

SO

=

(1

Ajjt,p(~)

+ No(w))*

(7.6)

defined is called the spectral function and satisfies

Ajj*,,(w) =

-Aj<j,-,(

-u)

(7.7)

on account of Eq. (7.4). Ajj?,, is given by Eq. (6.8) in the harmonic approximation, and its physical interpretation was discussed in the previous section. The spectral function A j p , , ( w ) , besides satisfying (7.7), also satisfies a sum rule as a consequence of the commutation relations (3.8) of the normal coordinates. Taking the thermal expectation value of (3.8) , one gets lim (Qjp (2) Qp-, (t’) (t’) Q j , ( t ) ) = (fi/i) 6jjr . (7.8) Qjt-,

tl-

37

t

D>jj,.p(t) is analytic in the portion of the complex t plane above t = -if@, as may be verified from (7.1). See G. Baym and N. I). Mermin [ J . Math. Phys. 2,232 (196111 for more det.ailed discussion.

226

PHILIP C. K. KWOK

Equation (7.8) may be expressed in terms of the phonon correlation function as lim ( a / ~ 3 t ) [ D ’ j j , , ~-( tt’) - B ’ j ’ j , - p ( t ’ - t ) ] = 6 j j f , (7.9) t --f

which, upon using definition (5.6), becomes

J-: :

- o [ D ’ ~ ~ ~ , ~-( D L >Jj)t j , - p (- u ) ] =

6jji

.

(7.10)

From (7.4) and (7.6), one finally obtains the desired sum rule:

As we have discussed in the previous section, the phonon spectral function describes the frequency distribution of the lattice excitations. We shall show presently that it is also related to the phonon density of states. Let us form the function v j given by Y ~ ( w >=

C

w

N(w/r)Ajj,p(U),

> 0.

(7.12)

P

According to (7.11), this function satisfies the simple sum rule

I,

rm

dWVj(W)

=

N.

(7.13)

One can readily interpret v i ( w ) as the number of phonon states in branch j lying in the frequency interval o and w do. Qualitatively, this interpretation follows from the physical meaning of Ajj,p(o).To see this conclusion on a more quantitative basis, one may work in the harmonic approximation. Upon using expression (6.3) for the spectral function, one obtains

+

where the summation over p has been replaced by an integral. Carrying out the integration, (7.14) becomes Vj(O)(w)

=

J

(N/8n2)

dSp/

1 VpwjpO 1,

(7.15)

0 j,o=w

where S , is the constant energy surface at wjpO expression for the phonon density of states.38

=

W.

(7.15) is the familiar

There exist critical points in the density of state function arising from the vanishing of the gradient I vDwjPO 1. See L. van Hove [Phys. Rev. 89, 1189 (1953)l for an introductory discussion.

227

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

8. RELATION OF

THE

PHONON CORRELATION FUNCTION TO EXPERIMENTS

Having defined the phonon correlation function in a rather abstract way, an obvious question to ask is, “How does this correlation function related to experimental observations?” I n this section, we shall consider two simple types of experiments to illustrate the desired relationship. In the first case, an external system coupled linearly to the nuclear displacements is made to transfer energy and momentum to the lattice while it undergoes a quantum transition. In the second case, the crystal is driven by an external force that is also coupled linearly to the nuclear displacements. These two types of experiments are actually closely related. I n fact, they may be described by a similar interaction term in the phonon Hamiltonian, namely, H I = -C Mjp(t)Qj-p(t). (8.1) iP

However, the quantity M that describes the external disturbance has slightly different meanings for the two cases. We first consider an experiment of the first type. M j p ( t ) is then readily shown to be proportional to the transition matrix element of the external system going from the initial state 1 to a final state 2. The momentum transfer to the lattice is fip = fi(pl - pz). The time dependence of M is described by M j p ( t ) = Mip(0) exp [- i ( E / f i ) t ] , where E is the transfer of energy El - E2. From (8.1), one can immediately obtain the total transition probability as

W12 = ( 2 r / f i )

C exp ( - 0 C i )

I(i I HI 1 f)lz 6(E

i,f

+

Ci

- tf),

(8.2)

where one sums over the final states ( f ) and averages over the initial states (i) of the lattice vibrations. The energy of the lattice vibrations has been denoted by C. We may rewrite the transition probability (8.2) in a slightly different way by using the following identity for the energy conservation 6 function:

6(E

+

m

Ci

- Cf)

=

[ J

( d t / 2 ~ f iexp ) [i(E

-m

+

ei

- er)t/h].

(8.3)

Substituting (8.3) into (8.2) and using expression (8.1) for H I we obtain )

Wlz

=

Irn

dt exp [ i ( E / f i ) t ]C M j p ( O ) M j t p * ( O )

(l/fi2)

-m

i,f

ji‘

228

PHILIP C. K. KWOK

where H is the total lattice Hamiltonian, and Q+jt,,(O) has been replaced by Q j J P ( O ) . The summation over the final states (f) drops out, and the average over the initial states (i) just gives the phonon correlation function defined by (5.2). Using the property (5.4) of the phonon correlation function and the definition of its Fourier transform, Eq. (8.4) may be written as

This result shows that the phonon correlation function enters explicitly into the transition probability. Equation (8.5) can be simplified by neglecting the terms in which j # j ’ . This is justified, as we shall see, because the nondiagonal part of the correlation function is usually small compared to the diagonal part. (In the. harmonic approximation, the correlation function is diagonal in j and j’.) Then Wlz becomes

This expression is easily recognized to represent the total transition probability of absorbing or emitting a single real phonon belonging to the different branches by the external system. Next we study the second case and show that the phonon correlation function is directly related to the linear response of the lattice vibrations to an external force. I n the absence of any outside forces, the thermal equilibrium average values of the nuclear displacements (ua(L;t ) ) are zero.= Let us apply a time-dependent force to the nuclei. This gives rise to an additional term in the Hamiltonian of the form (8.1),or, in coordinate representation,

M a ( : ; t ) is the force acting on the nucleus labeled by I and k. The averages (ua(:; t ) ) will now be nonzero and can be determined from time-dependent perturbation calculation.@The result is particularly simple 39

4O

We consider the crystal to be clamped so that there is no distortion of the crystal due to thermal strains. We also assume that the crystal undergoes no displacive transition in which the equilibrinm positions xo change (see Kwok and Miller14). See, for example, L. P. Kadanoff and P. C. Martin, Awn. Phys. ( N . Y . )24, Appendix A, 419 (1963).

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

229

if only the terms linear in the external force are retained, namely,

(8.8a) or

(8.8b) where ~ ( -t t’) is the step function V(X) =

=

1,

x >0

0,

x < 0.

The symbol M on the expectation value of the nuclear displacements indicate the presence of the external force. For theoretical discussion, it is more convenient to re-express Eq. (8.8a) or (8.8b) in terms of the normal coordinates, obtaining

(8.10a)

230

PHILIP C. K . KWOK

or

Mjp(t) in Eq. (8.10b) is the force acting on the vibrational mode ( j p ) given by

The function

is known as the linear response function.4l This is an obvious definition, as Eq. (8.10b) may be written as m

(Qjp(t))M

dt’

= jfpl

DRjp,j’p’(tt’)Mj~p‘(t’).

(8.13)

-m

From Eq. (5.2), one can immediately express DR in terms of the phonon correlation function D> 42: DRjp,jJpf(tt’) = ~ ( -t t’)[D’jp,jrpf(tt’)- D>jtpi,jp(t’t)]. (8.14) Then it follows from Eq. (5.3) that DRjp,j#pt(tt‘) is only a function of t - t’ and that it vanishes unless p’ = -p, i.e., DRjp,jtpr(tt’)= 6p,-ptDRjj,p(t - t’). 41

42

(8.15)

See Bonch-Bruevich and Tyblikov,2 Chapter 3. For a discussion of the response function in the hydrodynamic frequency regime and its relation to transport coefficients, refer to Kadanoff and Martin.40 I n conjunction with the definition of the phonon correlation function D’, one usually defines the function D<:

Thus one has the more familiar relation

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

231

Consequently, Eq. (8.13) can be written as m

dt’DRjjt,,(t- t’)Mjr-p(t’).

( Q j p ( t ) ) ~=

(8.16)

--m

j’

We define the Fourier transform of the response function as m

D R j j r , p ( w )=

/_mdtexp [ i w ( t - t ’ ) ] D R j j t . , ( t- t’).

(8.17)

[Note the absence of the factor i in contrast to definition (5.6) for the phonon correlation function.] ,The function DRjjf,p expressed in terms of the phonon correlation function according to (8.14) has the form DRjj.,,(w)

dw’

=

- D’j.j,,( - ( w i€)

D’jj!,p(w’)

w’

-0’)

.

+

(8.18)

The factor ie, where E is a positive infinitesimal, comes from the integral representation of the step function s ( t - t’) in the derivation

~ (-t t’)

=

i

J ;:

- exp

[-iw(t - t’)](w

+ ie)-l.

(8.19)

One may also express D R j j t , p ( ~ in) terms of the phonon spectral function defined by (7.6). Using (7.4), one finds that

Ajjt,p(w)

DRjj.,p(W)

=

Jm -m

do’

-

27r

Ajjt*p(w’)

w’ - ( w

+ i€)

(8.20) *

Thus we see from (8.14) or (8.18) that the phonon correlation function also determines the linear response of the phonon system to any external force. Our general discussion in this section will be illustrated by the specified examples mentioned in Section 4.

9. ULTRASONIC ATTENUATION In Section 4, it was mentioned that, in ultrasonic attenuation experiments, one is measuring the response of the lattice to a forced vibration of the nuclear displacements at one end of the crystal. What one measures t) is the displacement (ua(i;t ) ) M in the presence of a driving force Ma(;; as a function of distance (and therefore time) from the source of the disturbance. The time dependence of M is usually of the simple oscillatory form

M,

(i; M a(i) t)

=

exp ( - - i o t ) .

232

PHILIP C. K . KWOK

From Eq. (8.10a), we find that

where DR is the linear response function defined by (8.12). We have used the property (8.15) of the response function and have neglected for simplicity the off-diagonal part ( j # j ' ) . From (9.1), it follows that (ua(i; t ) ) M = (u.(:) ) M exp (-id). The spatial dependent amplitude (ua(:)) M is given by

DRjj,,(u) is the Fourier transform of the response function [see Eqs. (8.13)-(8.19)]. I n actual experiments, it is never possible to determine the force M a ( : ) that is being applied to the individual nucleus. But usually one can apply such an external force so that only one particular acoustic phonon branch (polarization) is e~cited.~' We shall therefore limit the summation over j in (9.3) to one acoustic branch. Since the external force is always localized and macroscopic in nature, we shall assume that the force M a (:) is nonzero only when the cells 1 lie on a plane and that it is identical in every cell. Taking this plane to be perpendicular to the z direction, (9.3) may be reduced qualitatively to

where X ( : ) is the perpendicular distance from the nucleus (I, k) to the plane on which the external force acts. The summation over p , and pa disappears because the sum over I' on a yz plane essentially gives rise to the 6 functions 6ppy,oand 6pp,,o. mja is an amplitude that depends linearly on M and on the eigenvectors of the branch j averaged over p , . In a later part, it will be shown that (9.4) is proportional to exp [ i ( w / c j ) X ( : ) CX.~(W)X(:)], where cj is the sound velocity of the branch j , and aj(u) is the attenuation coefficient inversely proportional to the lifetime of the

233

GREEN'S FUNCTION METHOD IN LATTICE DYNAMICS

phonon ( j , p , = w/cj) . The coefficient q(w) in various frequency ranges will be calculated using Green's function techniques.

10. INELASTIC NEUTRON SCATTERING The interaction between a neutron (mass nucleus at x(:) is given by

HI

= (27rV/rnN)U(k)63

[

rN

m N

-x

, coordinate

(31,

rN)

and a

(10.1)

where a ( k ) is the scattering length of the kth nucleus in the unit cell. This expression is obtained by replacing the finite range potential between the particles by a point interaction using the pseudo-potential meth0d.4~From (10.1), one gets the effective interaction Hamiltonian for the phonon when the neutron scatters from an initial plane wave state of wave vector ql to a final state q z :

We have separatedx(:) = x"(:) q1 - q z

.

+ u(:)

and used q to denote the difference

If one is interested only in the transition in which one phonon is created or destroyed, the exponential operator exp (iq-u) may be replaced by exp [ - + ( ( q . ~ ) ~ ) ]X iq-u.

(10.3)

This reduction is carried out and discussed in several place^.^"^^ To provide an adequate understanding of this formula, we shall derive it in an approximate manner. The transition matrix element of emitting one phonon (by a single nucleus) is proportional to

C e-Pfn(n n

+ 1 I exp (iqau)I n ) / C exp ( -W ,

(10.4)

n

where I n ) denotes a phonon state and I n f 1 ) the state with one more phonon or one less phonon. We expand the exponential in (10.4). Only terms with odd powers of the displacement contribute, since u is linear in the phonon creation and annihilation operators. Then (n

+ 1 I exp (iq-u)I n ) = (n + 1 I iq-u + (1/3!) ( i q ~ u+) ~... In). (10.5)

E. Fermi, Rie. Sci. [l] 7, 13 (1936) ; J. M. Cassels, Progr. NucL. Phys. 1, 185 (1950). R. J. Glauber, Phys. Rev. 98, 1092 (1955). 46 G. Baym, Phys. Rev. 121, 741 (1961). 46 H. J. Lipkin, Ann. Phys. 9, 332 (1960).

43 44

234

PHILIP C. K. KWOK

For simplicity, we assume that the phonons are noninteracting or harmonic. 1) is simply the direct product state of n ) and I n this case, the state 1 n the one phonon state I 1 ) = a+ lo), where I 0) represents the vacuum. As a result, one has for a general term in (10.5)

+

I

+ 1 I(iq.u)2m+lI n ) = (2m + 1)(1 I iq-u 10). (n I(iq.u)ZmI n). (10.6) The numerical coefficient (2m + 1) comes from the fact that each of the 2m + 1 displacements u contributes once to the matrix element between (n

I 1) and 10). Going back to (10.5) and keeping only terms up to third order in q (or u) , we obtain (n

+ 1 I exp (iq-u)In )

N

(1 I iq-u I O).(n I[1 - $ ( q - ~ ) ~ n] l) . (10.7)

Then (10.4) becomes approximately equal to (1 I iq-u I o m - 8 ( ( a 4 ” 1 , where angular brackets are used to denote the thermal average ((q.u)2)=

(10.8)

C exp (+en>(n 1(q.u)2I n)/Cexp (-/M. (10.9) n

n

Finally, consistent with our present approximation of keeping only terms quadratic in u, we may exponentialize (10.8), getting (1 I iq*u 10) exP c-3((s-u)z)l.

(10.10)

Hence, the effective interaction Hamiltonian for one phonon transition arising from the operator exp (iq-u) is approximately given by (iq-u) X exp ( -$((q.u)2)). The thermal expectation value ( (q u (i)) 2 ) is independent of 1 and gives rise to the Debye-Waller factor in the intensity of the scattered neutrons.28The interaction Hamiltonian is now linear in the nuclear displacement

or

in terms of the normal coordinates. We have denoted ([q.u(:)l2) by (pk(q).

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

235

Equations (10.11b) is identical to the interaction Hamiltonian (8.1) considered earlier if p is replaced by q and the matrix element M,, by the braced term in (10.11b). From the discussion in Section 8 [Eq. ( 8 . 5 ) ] , the inelastic neutron scattering probability is readily obtained to be

where hw is the energy transfer by the neutron hW

=

( h2qI2/2mN) - ( h2q22/2mN).

(10.13)

We have again neglected the terms in (10.12) which depend on the offdiagonal part of the correlation function ( j # j ’ ) . The transition probability W is proportional to the inelastic scattering cross section. Therefore, a measurement of the intensity of the inelastically scattered neutron as a function of w and q determines the phonon correlation and the dispersion curves of the phonons. 11. OPTICALMEASUREMENTS

In optical experiments, one studies the interaction of the electromagnetic field or photons with the lattice. However, the photons cannot be treated in principle as external radiation that can propagate unaffected through the medium. The reason is that a strong coupling exists between the transverse electromagnetic field and the long wavelength infrared active optical vibrations. This is the retardation effect,”Jwhich we have neglected. The correct theoretical treatment of the interaction between the electromagnetic radiation and the lattice is to consider the coupled photon-phonon system. All optical properties may then be calculated from the photon correlation function, which describes the propagation of electromagnetic wave in the crystal, for example, the transverse dielectric susceptibility. The mixed photon-phonon excitations, known as polaritons*2that exist in the long wavelength limit, will also show up naturally in the photon (and phonon) correlation function. However, such a rigorous approach is beyond the scope of the discussion at present. We shall ignore the retardation effect and regard the transverse electromagnetic field as an external disturbance. It turns out for the specific example discussed below that our incorrect treatment still gives the correct answer. I n a later part, the rigorous approach will be presented. The optical experiments we shall discuss are reflectivity and trans-

236

PHILIP C. K. KWOK

mission experiments in which one measures the infrared absorption spectrum of the crystal. This spectrum is determined by the frequency-dependent susceptibility function and represents the frequencies of the infrared action optical modes in the long wavelength limit. Let us suppose that a time-dependent transverse electric field Ea(x, t ) is applied to the crystal. Assuming the simplest case of each nucleus carrying an effective charge z k , the electric field then exerts a force Z k E a [ X ( L ) , t ] on the nuclei. This will give rise to a perturbation term in the Hamiltonian of the form (11.1)

For convenience, the spatial dependence of the electric field is neglected. Then (11.1) may be written as (11.2)

The term in the brackets is the total polarization of the crystal and is related to the polarization per unit volume Pa ( t ) by (11.3)

where V is the total crystal volume. One may express (11.3) in terms of the normal coordinates as

Pa ( t )

=

v-' c'( c j

( N / m k ) 'I2Zkeka

( j , 0 )) &jO ( t ).

(11.4)

k

I n (11.4), the summation on j is only over the infrared active modes for which c k z k e k a ( j , 0 )/(mk)'12does not vanish. The dielectric susceptibility function is determined from the linear response of the polarization to the electric field.47*48 We shall take the electric field to have the simple time dependence

E a ( t ) = E a ( o )exp ( - i w t ) .

(11.5)

Then, from (11.4) and (8.16) and the fact that the external force exerted on the normal mode Mi,( t ) defined by (8.11) is

c

( N / m k ) l12zkeka

( j , 0 )Ea ( t ) ,

ka

47 48

See, for example, R. F. Wallis and A. A. Maradudin, Phys. Rev. 126, 1277 (1962). For a classical discussion of the dielectric response of solids, see F. Stern, Solid State Phys. 16, 299 (1963).

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

fja

(N/mk)”zzkeka ( j , 0 ) .

=

237

(11.7)

k

The off-diagonal part of the response function has been neglected. The dielectric susceptibility tensor x a a r ( w ) is defined by

(Pa(a) ) E

c

=

ha’

(w)

(w)

,

(11.8)

at

where the E,(”t) represents the total electric field. Since we have ignored the retardation electric field that is coupled to the nuclear vibrations, the ) just total field is simply the external field E, . Thus, x a a ~ ( w is xaa’

(w) =

C

(11.9)

EjaEiatDRjj,O( w ) *

j

It is instructive to examine qualitatively the character of the susceptibility function when retardation effects are taken into account. Equation (11.6b) is still correct except that the phonon response function D R j j . o ( w ) must be calculated for the coupled photon-phonon system. However, the total electric field in (11.8) is no longer equal to the externally applied field E, . It contains the nonzero expectation value of the retardation or ~ by the external field. Its value may be internal field ( E a ’ ( w ) ) induced calculated in a straightforward manner by using time-dependent ,perturbation theory. The result, linear in the external field, will have the general form (11.10) @,‘(a) ) E = C Iad (w)Ea*( w ) a’

where Iaat is readily shown to be proportional to the mixed photon-phonon (Ea’(o)) E is then correlation functi0n.4~The total electric field Ea(w) given by (11.11) (w) = [Sad Iad ( w ) IEd (0)*

+

c

+

(I

The dielectric susceptibility according to (11.6b), (11.8), and (11.11) is readily obtained to be Xarr’((J)

=

c [el

EjafjarrD~ij,O(w)](l

a11

49

+ I(w))-la%# ,

(11.12)

j

Fourier transform of the function ( i / h ) (&j,(t)qcp(t’)), where qSDis the normal coordinate for the mode u of the quantized electromagnetic field.

238

PHILIP C. K. KWOK

where the tilde on the phonon response function indicates that retardation I ( w ) ] - l denotes the inverse of the effect has been included, and [l matrix Iaat ( w ) . The most interesting fact about the two expressions (11.9) and (11.12) for the susceptibility is that they turn out to be the same. This may be directly verified using a semiclassical description of the coupled photonphonon system.M Thus, for the dielectric susceptibility, the result based on a treatment that is not rigorous is the same as that obtained from a rigorous theory. The foregoing conclusion is, in general, not true. Let us consider firstorder Raman scattering, in which a lattice excitation is created or destroyed by an inelastically scattered high-energy photon. As the frequency of the incident photon is much higher than all the vibrational frequencies of the crystal (and lower than the electronic transition frequencies), it can propagate freely in the medium and therefore may be treated simply as external radiation. The mechanism for inelastic scattering of such photon has been discussed in several and, more mecently, by Loudon?3 We shall describe briefly Loudon’s calculation and show that the lattice contribution appears in a rather simple way. The total inelastic scattering probability is seen to be proportional to the phonon correlation function D>. In semiconductors, the dominant mechanism for first-order Raman scattering as calculated by Loudon is an indirect process involving the electron-phonon and electron-photon interaction. It consists of the following three transitions: (1) the initial photon with frequency w is absorbed; (2) a photon with frequency w‘ is emitted; and (3) a phonon with frequency wp = w - wt I is either created or destroyed. Each step is accompanied by a virtual electronic transition, and the three steps may occur in any time order. Denoting the interaction Hamiltonian between the electrons and the phonons by HgL and that between the electrons and photons by H E R , the transition probability of the first-order Raman scattering is readily deduced to be

+

+

I

x 6 ( w ’ f up - w ) . I n the preceding matrix element, HI represents the sum H E L

(11.13)

+ H E R, and

50See,for example, the treatment of the retardation effect in C. Kittel, “Quantum Theory of Solids,” Chapter 3. Wiley, New York, 1963. 51 M. Born and M. Bradburn, Proc. Roy. SOC. A188, 161 (1947). 62 H. M. J. Smith, Phil. Trans. Roy. SOC.(London) A241, 105 (1948). 5 3 R. Loudon, Proc. Roy. SOC. A276, 218 (1963).

239

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

m,n represent the intermediate state with energies E m and E n . An initial state 1 i) is the product state of a single photon with frequency w and a lattice vibrational state. The corresponding final state I f ) has one photon with frequency w’ and the same lattice state with one phonon added or the electrons are taken to be in their subtracted. I n both I i) and ground states. It is obvious that HEL appears only in one of the matrix elements, whereas for the remaining two matrix elements H’ is to be replaced by H E R .This is the case because H E L and HER are linear in the phonon and photon creation and annihilation operators, respectively. To show that (11.13) is proportional to the phonon correlation function, we make explicit use of the form of HEL , the electron-lattice interaction. TOlowest order in the nuclear displacements, i t may be written as54

If),

(11.14) jp

where the operator pjp contains only electronic variables. Substituting (11.14) into (11.3), one readily observes that the Raman scattering matrix element is simply the matrix element between appropriate lattice states of the following effective phonon interaction Hamiltonian: (11.15) where

Mi, =

C C (f’ I P ~ PI m’)(m’ I H E RI n’>(n’I H E RI i’)/(fiw mn

- Em)(fiw - En)

(f’I HER I m’>(m’I Pip I n’)(n’I HERI i’)/(fiw - Em)(fiw - En) -I- (f’ I H E RI m’)(m’ 1 H E RIn’)(n’I Pip I i ’ ) / ( f i m - Em)(fiw - En)) . -I-

(11.16) The primes in (11.16) denote that the states contain only electrons and photons. The transition probability is then given by TV

C

i(phonon states)

exp ( - D e i )

I (f I C

Qi-pMjp

1 i)lz6 ( w ’

f wip - w ) .

ip

(11.17) This, according to the calculation in Section 8, yields the desired result that W is proportional to the phonon correlation function Dij,, ( w - w’) . Thus, a measurement of the frequency spectrum of the scattered photon for different momentum transfer will enable one to determine the phonon correlation function and hence the dispersion curves of the excitations in s4

See, for example, L. J. Sham arid J. M. Ziman, Solid State Phys. 16, 221 (1963).

240

PHLIIP C. K. KWOK

the crystal. I n the experiment carried out by Henry and Hopfield,lathey studied the scattering a t very small momentum transfer and obtained the dispersion curve of the polariton. Therefore, to explain this experiment, it is necessary to calculate the phonon correlation function with retardation effects taken into account.

111. Phonon Green’s Function

12. DEFINITION OF PHONON GREEN’SFUNCTION We have seen in the previous part that a variety of properties of the (tt’) and phonons is described by the phonon correlation function D>jp,j~pI (tt’) . These functions can be the related linear response function DRjp,jrpt measured directly in a number of experiments. The task that must be taken up now is the calculation of these functions, starting from the anharmonic Hamiltonian (3.15) that describes the dynamics of the lattice vibrations. It turns out that it is more convenient to determine them from a related function, the phonon Green’s function. The reason is that well-developed analytic methods have been devised to calculate the latter function. is defined as The phonon Green’s function DjP,jtp#(tt’)

Djp,jfpt(tt’)= (i/fi) (C&jp(t)&j’P‘(t‘)l+>,

(12.1)

where the plus symbol denotes the positive time-ordering operator such that D j P . j ’ P # ( t t ’ ) = (i/fi> (&jP(t)&j’P’(t’) >, t > t’ =

( i / f i(&j’P‘(t’)&jP(t) )

>,

t

< t’.

(12.2)

Using definition (5.2) for the phonon correlation function, one may rewrite (12.2) as65 Djp,jfPr(tt’) = D>jp,jtpf(tt’), t > t’ = D>j*,t,jp(t’t),

t

< t’.

(12.3)

Then it follows from (5.3) that the phonon Green’s function satisfies

Djp,jtPt(tt’)= &,,-p#Djp,jt-p(t - 2’)

(12.44

Gp,-plDjjr,p(t - t ’ ) ,

(12.4b)

=

55 Refer to footnote4* for the definition of the function D<. Then Eq. (12.3) may be written in the more convenient form

241

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

so that Eq. (12.3) may be written as D..t 33 IP ( t - t’)

> 1’ t < t’.

- t’),

=

D>jji,P(t

=

D’.,. 3 3 , -p ( t ’ - 1)’

t

(12.5)

Let us express (12.5) in a slightly different way by using the identity (7.3) of the phonon correlation function. One finds that Djjr,p(t

- 2’)

- t’), - D’jjn,P(t - 2’ - ihp), =

D>jjt,p(t

t-t’>O t

- t’ < 0.

(12.6)

This relation suggests that a suitable way to study the phonon Green’s function is to use the concept of imaginary time. Following this suggestion, we take t and t‘ to be pure imaginary.56The time ordering of operators is defined by the convention that t - t‘ is considered positive time if it lies on the negative imaginary axis and negative time if it lies on the positive imaginary axis. Let us apply this definition to (12.6)’ choosing t and t’ such that t - t’ and t - t’ - if@ lie above and below the real axis; we obtain the following identities :

t’)

=

D’jjr,,(t

( t - t’ - ihp)

=

Djjf,,

D j j t , P ( t-

D>jjf,p

- t’ - if@),

( t - t’ - if@).

(12.7) (12.8)

Combining these two equations, one arrives at the very important periodicity condition of the imaginary time phonon Green’s function5’:

- E’)

=

Djjf,P(t

Djjr,P(t

- t’

-

dip).

(12.9)

The period T is given by -if@. If we always restrict the relative time variable t - t’ to be in the interval f 0, T ) , the phonon Green’s function may be represented by a Fourier series, namely,

- t’)

I ~ j j t , ~ ( t

= (1/r)

C exp [-iwn(t

-t

(12.10)

’ ) ] - ~ j j f , ~ ( w ~ ) ,

n

where wn is the imaginary frequency wn =

2 m / ~= 2m~/-ilip

(12.11)

and n is an integer. The Fourier coefficient of the phonon Green’s function is, according to (12.10), given by

Djjg,p(wn)

W 67

All times in the subsequent discussion will be pure imaginary unless stated otherwise. A thorough mathematical discussion of the analytic properties of Green’s functions and related functions may be found in Baym and Mermh3?

242

PHILIP C. K. KWOK

We shall now derive several useful relations between the phonon Green’s function and the phonon correlation function and response function. The intention of this is to show how such functions with direct physical significance may be obtained from the more abstract imaginary time phonon Green’s function. To begin, let us study the phonon Green’s function with t - t’ lying between 0 and r . Then Djjt,p(t - t’) = ( i / h )(C?jp(t)C?jr-p(t’)

(12.13)

>*

The right-hand side is just the phonon correlation function D > j j r , p ( t - t’) with its real time argument analytically continued on to the imaginary axis.67Expressing (12.13) in terms of the Fourier transform D > j j , , , ( w ) of the phonon correlation function defined by (5.6) we obtain

DjjP ., (t)

=

i

/

m

(dw/2a) exp ( -iwt)

~

j

(12.14)

j , t(0)

-m

where t’ has been set equal to zero for convenience. From (12.12) and (12.14) one can derive a relation between the Fourier coefficient D j p , , ( w n > of the phonon Green’s function and D>ijt,p ( w ) . The desired result is

I-, 03

~

~

~

~= ,

~

(dw’/2~)0>~~.,,(w’>[(l ( w ~ ) - exp

(-~fiw’))/w’

- con] (12.15a)

or Bjjt,,(Wn)

=

J

co

(dw’/2?r)D’jj3,,(o’) [(l

+ NO(o’))-l/w’

- Wn],

-m

(12.15b) where N o ( w ) is the equilibrium distribution function [exp ( p h w ) - 13-l. This relation takes on a more elegant form if the spectral representation (7.6) is used, namely, (12.16) One may immediately recognize that (12.16) is identical to the spectral representation (8.20) of the linear response function D R j j T , , ( w ) if wn is ie. Therefore, we have the following relation between the replaced by w Green’s function and the response function:

+

DRjjt,,(W)

=

Djj,,p(wn+ w

+ 2.4.

Finally, let us observe that the spectral function

Ajjt,,(w)

(12.17) that completely

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

243

determines the phonon correlation function and linear response function may be directly obtained from D j j , , ,(wn) according to (12.16) or (12.17) as Ajjf,, ( w ) = lim 2 Ini D j j r , p(wrL-+ w ie) . (12.18)

+

c-0

+

13. FUNCTIONAL DERIVATIVE TECHNIQUE

It was shown in the last section that, once the imaginary time phonon Green’s function is obtained, the phonon correlation function and linear response function can be immediately determined, and consequently so can many important physical properties of the lattice vibrations. In the remainder of this part, we shall devote ourselves to the task of calculating the phonon Green’s function. The calculation is based on an integrodifferential equation satisfied by the Green’s function derived from the dynamical equation (3.17) of the normal coordinates. The derivation of this equation is carried out with the help of the functional derivative technique. There are, of course, other methods to calculate the phonon Green’s function. One of the well-known methods is the diagrammatic technique discussed in detail by C ~ w l e y . ~ Before we go into the mathematical details of the derivation of the Green’s function equation, it is illuminating to describe briefly the structure of the equation. It will be shown it has the integrodifferential form [(a2/at2)

+(

WjPO) 2 ] 0 j p

,ifp!( t

- t’)

(13.1)

Equation (13.1) is also known as the Dyson’s equation for the phonon Green’s function. The function II is called the self-energy function.s8 In the harmonic approximation, the self-energy function II is zero, and we have the simple harmonic Green’s function equation

[(az/at2)+ (~j,”)~]D:.:)jt~t(t- t’)

= 6jp6p,-pt 6 ( t - t ’ ) .

(13.2)

n contains all the anharmonic effects, and the calculation of this function is equivalent to the calculation of the phonon Green’s function. We shall now discuss the basic concept of taking functional derivatives and discuss how it may be applied to derive the Green’s function equation. Let us consider the expectation value of the normal coordinate ( Q j , ( t ) ) J defined by (Qip(t) 58

>J =

Tr ( ~ X P(-PH)CfQjp(t)l+I/Tr

Sometimes it is called the polarization function.

I ~ X P(-PH)SJ’

(13.3)

244

PHILIP C. K. KWOK

where S is the time-ordered function

(13.3) is actually the expectation value of the normal coordinate in the presence of a fictitious time-dependent external disturbance defined in the imaginary time interval (0, 7 = --ih/3]. The disturbance is represented by an extra term

HI=

-

C

Jjp(t)Qjp(t)

ip

in the Hamiltonian, where J plays the role of an external force. (Q)J is called a functional of J, as J is a function of ( j p ) and time rather than a single parameter. According to (13.3) and (13.4), it is obvious that, if the fictitious force is set equal to zero, the expectation value (Q)J is just the thermal equilibrium average of the normal coordinate and therefore vanishes,59i.e., (13.5) lim ( Q j p ( t > >J = ( Q j p ( t ) = 0.

>

J-0

Let us introduce a small deviation to the force Jip( t ), i.e., replacing it 6Jjp(t).Expanding ( & ~ , ( ~ ) ) J + S Jin powers of SJ and only by Jjp(t) keeping up to terms linear in the deviation, we obtain

+

(Qjp

(t))J+~J=

(Qjp

( t ) >J

+ Irdt’ c DJjp,jrpf(tt’) 0

(t’)

ujtpr

+ .. * *

(13.6)

jlpl

The function D J j p , i t p t ( t t ’is) given by DJjppjtpt(tt’)

=

(-i/h) C ( C Q j p ( t > Q i ‘ p ~ ( t ’ ) I + ) ~ -

(Qjp(t)

)J(Qjfp*(t’)

)JI, (13.7)

where KQjP(t)Qi~p*(t’)

=

I+)J

T r (exp (-PH)[~Qjp(t>Qj~p~(t’)l+)/Tr Iexp (--PH)XI.

(13.8)

DJ is known as the functional derivative of ( Q ) J with respect to J and is most conveniently denoted by D J j p , j ~ p ~ ( ~ t ’= ) C8/6JJj’pf(t’)l(Qjp(t)

>J

.

(13.9a)

Because the normal coordinates commute, under the time-ordering sign D Jj.P ..j r P ~ ( 2 is t ’ )symmetric with respect to the interchange of ( jp; t ) and 69

See Kwok and Miller.14-39

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

245

( j’p’; t’) . Thus, we also have

DJjp,jrp< (tt’)

=

Cs/SJip ( t ) 1(Qifpt

(0>J .

(13.9b)

One can immediately recognize from (13.5), (13.7), and (13.8) that in the limit of the fictitious force J going to zero the functional derivative DJ is just the phonon Green’s function defined and discussed in the previous sections. Thus, (tt’) = Djp,jtpt (tt’) lim DJjp,jrpt J-0

= 6,,,~Djy,,(t

- t’).

(13.10)

One may now consider DJ as a functional of J and calculate its functional derivative, which is defined in a manner in similar to that for (Q)J (13.6), namely, as the coefficient of the term linear in SJ in the expansion of DJ+8J.One finds

Higher functional derivatives may be calculated in the same manner but will not be carried out explicitly here.

14. GREEN’SFUNCTION EQUATION The first step in deriving the equation for the function DJ defined by (13.7) or (13.9), which will also be called the phonon Green’s function, is to obtain the equation satisfied by the expectation value ( Q j , ( t ) ) ~ . We

246

PHILIP C. K. KWOK

begin by writing (Qjp(t) ) J in a more suitable form:

X

]+} / Tr [exp

Qjp( t )

( -PH)

(14.la)

taking the time ordering in the numerator into account explicitly. I n the denominator, definition (14.4) of S has been inserted for convenience. For further simplification, we define a function

so that S is just S(T,0). Then (14.lb) becomes (QiP(t)

>J

=

Tr { ~ X P(-PH).X(7,t > Q j p ( t > S ( t , 0)l/Tr { ~ X P(--PH)Sl. (14.3)

One may now derive the desired equation for (Q>J by differentiating Eq. (14.3) with respect to the time t . Taking the first derivative, we get

(a/W

(Qj,(t>

>J

x s(t,O)l/Tr

{exp (-PH)W).

(14.4)

The second term comes from the differentiation of the S functions, namely, ( a / a t ) S ( ~t ),

=

S ( T ,t ) [ - ( i / f i ) CJj~pf(t)Qj~p~(t)], (14.5a) jW

(14.5b) (a/at)S(t,0) = [(i/fi) CJjfp.(t)Qj~p~(t)]S(t, 0). ifpt

247

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

However, this second term drops out because the normal coordinates at equal times commute, and therefore (slat) ( & i P ( t )

>J =

=

Tr (exp ( - P W S ( 7 , t > Q i P ( t ) S ( t , O ) )/Tr (exp (-m)Sl (QiP(t)

(14.6)

>J

Taking the second time derivative, we obtain

upon using the commutation relation (3.8) and the fact that = 8,is readily calculated to be

S ( T ,t ) S ( t , 0) = S(T,0 )

(WW(Qh( t )

>J

We now substitute the dynamical equation (3.17) of the normal coordinate (which is also valid for imaginary time) into the right-hand side of (14.8) ; we finally get

+ C C [ ( n - 1) m

,....

! ] - l U ~ ~ ~ . j i p ii n - i p n - i ( Q i i P i ( t )

*Qin-ia.-i(t)

>J

n=3 lip1

= Jj-p(t).

(14.9)

In the following deviation, only the cubic and quartic anharmonic effects are considered. All higher-order anharmonic coupling parameters are set equal to zero. Then (14.9) simplifies to (a2/~t2> (Qip(t)

>J

+

( ~ i p O ) ~ ( Q i p ( t ) >J

+ (1/2!) C + ( 1/3 !>C uiCp

U ~ ~ p , j i p i . i z P 2 ( Q j i P i ( t Qiwz(t) ) >J

PI

,ilp,jzp2 , j a P 3 (QilP

1

(t>QiZP 2 (0QiSP 1( t ) >J

lip1

= Jj-p(t). (14.10) Mjp(t) in Eq. (8.10b) is the force acting on the vibrational mode ( j p ) The equation for the phonon Green’s function D J j p , i ~ p(tt’) ~ according to given by

248

PHILIP C. K . KWOK

the definition (13.9a) may now be immediately derived from (14.10) by taking the functional derivatives with respect to JjIpt(t‘) . We obtain (a2/at2)

DJjp, (tt’) jfPP

= 6jjt 6,,-,f

6(t

+ (Wjp”2DJjp,

jtpt

(tt’)

- t’).

(14.11)

The function 6 ( t - t’) that appears as a result of differentiating Jj,(t’) with respect to JjtPt(t‘)is defined by

l ‘ d t ’ 6 ( t - t’)f(t’) = f ( t ) ,

(14.12)

where f is an arbitrary periodic function defined in the imaginary time interval (0, T ) and satisfies the periodicity condition 6(t

- t’

+

T)

= 6(t

- t’).

(14.13)

From (14.12) and (14.13), one can easily show that it has the following Fourier representation : 6(t

- t’)

= (1/T)

C exp ( - i w n ( t

- t’)),

(14.14)

n

where wn and n have the same meaning as in Eq. (12.11). 15. FORMAL PROCEDURE OF CALCULATING THE PHONON GREEN’S FUNCTION We shall now develop Eq. (14.11) to an extent that it can be used to calculate the phonon Green’s function self-consistently. I n order to simplify the notations, we shall replace ( jp) by a single index p, and ( j - p) by - p . Furthermore, the convention of summation over repeated indices will be adopted. Then Eq. (14.11) becomes (@/at2)WP,* (tt’)

+ (w,”)‘DJUPr (tt’)

+ (1/2!) u~~,i,zCs/sJ,f(t’)l(QPi(t)QPz(t) + (1/3 (t’) 1 ( t ) ( t )Qr, ( t ) >J

U%irzraC6/&JPf

- 6P,-Pt 6 ( t

- t’).

(Qpi

Qrz

>J

(15.1)

Using identities (13.7) and (13.11) to replace the expectations values of

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

249

the normal coordinates in (15.1), one obtains a2 -

at2

+

DJPPt(tt’) ( ~ , ” ) ~ D ~ ~ , , ’ ( t t ’ )

where the fact that the anharmonic coefficients are symmetric in the indices ( p ) has been used. Upon taking the indicated functional derivatives, one gets a2

DJppl(tt’) at2

+

- 6#,+ S ( t

(tt’)

(~p”)2DJpp~

- t’).

+

UfLpIpZ(Qpt

(t) ) ~ D ~ p l p t ( t t ’ )

(15.3)

Before proceeding any further, let us recognize from Eq. (15.3) the very important fact that the phonon Green’s function is a functional of J only because of its dependence on the expectation values (Q)J . Therefore, DJ may be simply regarded as a functional of (Q>J . The reference to the fictitious force J can be eliminated from the Green’s function equation with the help of the following chain rule of functional differentiation:

(15.4)

An important consequence of (15.4) is the existence of an inverse phonon

250

PHILIP C. K. KWOK

Green’s function ( D J );;? (tt’) which satisfies

LT

dt’DJ,,t (tt’) (DJ)&! (t’t”)

=

IT

dt’(DJ);;’t(tt’)DJ,.,.J(t’t”)

0

=

6,,/. 6 ( t - t ” ) .

(15.5)

Equation (15.5) may be readily verified by applying (15.4) to Jptr(t”). One sees that the inverse Green’s function is just the functional derivative of J , considered as a functional of (Q)J because of Eq. (14.10)’ with respect to (Q)J . Henceforth, we shall from now on suppress the subscript J on the expectation values (Q).T and change the superscript J on the phonon Green’s function to Q, i.e., DJ = DQ.The limit of J going to zero is replaced by letting (Q) go to zero, and the desired imaginary time phonon Green’s function is obtained from lim (Q )+&Q. For convenience, we shall extend the repeated index summatmionconvention to include time integrations. Specifically, two or more identical time variables are to be integrated over the range { O , T I , with the exception of variables that are boldface. Taking these discussions into account, Eq. (15.3) may be written as

X

DQp5p~(t5t’)

- 6p,-pt 6 ( t - t’) .

(15.6)

This equation for the phonon Green’s function is now in a form that in principle can be used to determine the Green’s function exactly in a selfconsistent manner. This will become more obvious if one examines the equation for the inverse Green’s function. We multiply Eq. (15.6) by (DQ)-l,and then sum and integrate over appropriate indices. Then we obtain the following equation for the inverse

25 1

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

Green’s function :

(DQ);,*(tt’)= (D(”)$(tt’)

+ Ufp)lp~(Qpl(t)) 6(t - t’)

+ +uL”?,,z,*(Q,i(t>)(Qrz(t))s(tt’) + ~ut”?l,2,~(n/i)~Q,l,z(tt) 6(tt’> + 8UL”:,,*,, X ( A l i ) IC6/6(Q,r(t’)

x

>IDQP1r2(tt) I (Q,,(t>)

+

QuL”:P2P3

6 (li/i)2

6 (Q,J(t’) ) (DP,,,*(tt4) 6 (Q, 4 (t4) )

( tt)

DQPW2

)’ (15.7)

where (D(o))-l is the inverse phonon Green’s function in the harmonic approximation :

(D(’));i1(tt’)= [ ( # / a t 2 )

+

(~,,”)~]*6,,-,r

6(t

- t’).

(15.8)

The next step is to evaluate the functional derivatives in (15.7). Let us go back to (15.5)’ which in our present simplified notation may be written as DQ,i,z(tlt2)* (DQ);~N3(t2t3) = 6,1, a(tl - t 3 ) . (15.9) Taking the functional derivative with respect to (QC4(t4)), we get

(15.10) or

(15.11) Now it is not difficult to see that rule (15.11), together with Eq. (15.7)’ enables us to eliminate successively all the functional derivative and generate an infinite series for the inverse phonon Green’s function. Let us calculate one term in (15.7) to illustrate the procedure. We shall pick the term linear in U(3)and the functional derivative of D: +uL”?pz(fili> C(S/S(Q,#(0)oQpiwz(tt> 1.

(15.12)

252

PHILIP C. K. KWOK

FIG.1. Leading terms in the expansion of the phonon self-energy function in powers of the anharmonic coupling parameters and the exact phonon Green’s function.

Upon using (15.11), it becomes

The next step is to substitute (15.7) for the inverse phonon Green’s function, with the result

-4 uL”?w(fil4

{DQplpa

+

ul”,?WI6(Qp,(t3))

(C6/6 (Qp. ( t ’ ) )lC(DcO)) Glp5 ( t d 5 ) 6(t3

- tS)

+

‘*‘])oQp6pZ(t6t)}‘

(15.14)

The result of taking the indicated derivative is

- t’)

- 3 ~ k ” ? ~ 2 ( n / ~ > ~ Q p ~ p ~ ( t ~ 3 ) ~ ~ ? 6 / r 6 6 p r ’ p ~ 66(( tt 3

=

-3uk”,’#2

(n/i>DQP1P3

(tt’)DQp2p

6(

tt’)

ut”,?

6p‘ 7

- t5)

DQp6p2(t6t)

+

’’*

(15.15)

where only the leading term is retained. When all the functional derivatives are eliminated in the foregoing fashion and the expectation values (Q) 8et equal to zero, the resulting infinite series contains only the anharmonic coupling’parameters and the exact phonon Green’s function. It has the

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

253

form of =

D-1

+[u(3);

(7‘4);

(D(O))-1

D].

(15.16)

The brackets represent the infinite series that may then be used to calculate the phonon Green’s function exactly. However, it must be obvious that it is not possible to obtain an exact solution, because one can never find the infinite series. The only sensible thing to do is to terminate the series to include only those anharmonic effects one wishes to consider. For our purpose, the series will be truncated when all the terms proportional ( U ( 4 ) ) 2 , and lower powers of the anexplicitly to ( U ( 3 ) ) 4 , (U(3))2.U(4), harmonic coefficients are included. It will be shown later, when we discuss the perturbation calculation of the phonon Green’s function, that these terms contain the leading three (and four) phonon anharmonic effects. Then we have

D&

(tt’) =

(D(o’);:# (tt’)

+ +(fL/i) UfJPlPZDPlP2(tt) 6(tt’)

-

3(n/i)

x

UfL1P2P3DPlP4(

up~1P2DP1P3(tt’)DP2P4(tt’>

tt’>D P 2 P 6 ( t t ’ )

DP8P6(tt‘)

- ~(fL/i)’U~~1PZDP1C3(tt3)DP2P4(tt4)

x

DP6P,

(kt’)DP6P7

4(

2UfP)lP2P30p1P

6P7DP7P8

DP2P4(tt4)

+ 4 n/i)

(t4t’)

u$?;8Pf

tt’) DP2P

2U??1P2D~1P3

(

(t3t‘)

W 6P’

uiti6P6

‘$~;O:1OPf

(

6 t t 4 ) DP3P

6

(

f 3( f L / i2)u p i l P $ P 1 P 3

Ut”,’,6P6DP,P,(t4t’)DP6Pg(t4t’)

x DPW7 ( t 3 t ’ ) D P W S

‘it;

( h t 4 ) UL3)P7P8DP8W10(t4t’)

+ 3 ( fL/i) x uE; x

- 6(n/i)2

UL:)PWf

DP2P

@ w d ~

4 (tt3)

*

(tt‘)

‘t%7P8Pt

U$\)# 4s 6P6 (15.17)

The anharmonic terms in this equation are schematically represented by the diagrams in Fig. 1 to provide a glimpse of their physical meanings. As a final comment, let us note that expression (15.17) is not the perturbation series for D-1, because it still contains the exact phonon Green’s function D . The bonafide perturbation series is an expansion of t-he anharmonic coupling parameters and the harmonic Green’s function. Such an expansion will be derived in a later section. 16. PHONON SELF-ENERGY FUNCTION The inverse phonon Green’s function can be in the following exact form : (16.la) D;;’I(tt’) = (D(’);:t(tt’) - I I P P t ( t t ’ )

254

PHILIP C. K. KWOK

or =

D-;d.,jr,,(tt’)

(D(’))~i,j,,?(tt’)

- IIj,,,j*,,/(tt’).

(16.lb)

A slightly modified form of this result was quoted in (13.1). The function ll is known as the phonon self-energy function,60and it contains all the anharmonic corrections. In the last section, we calculated the first few terms of II as an expansion in the anharmonic coupling parameters (and the phonon Green’s function). From (15.17) and (16.la), we find that IIPPt(tt’) = - $ ( h / i ) Ut”,‘,P,PzDPIPz(tt) 6 ( t - t’)

x x x x x

DP1P3(tt’)DPZP4(tt’)

DPIP

4

+ (1/3!)

ui”,4P‘

(h/i)2uh4?lP2P3

( t’>DPZP 6 ( tt’>DP3P 6 ( t’) uit?6P 6P’

DP1P3 ( t t 3 ) D P 2 P 4 (

(3) UPPIP,

tt4)

ug?

Ux?7P8DP8P10(t4t’)

DP6P7 DPIP

+ $ ( h / i )U,$ilP2

4(

tt‘)DP2P

6 (tt4)

- $ ( h/i>2u,$21PZDP1P3 DPW7 (t4t’>

-

t(Yi)

x

DP6P7

DP3P

uht?

6 (tt4)

6P7DP7P8

4 ( tt4)

ul”,?

4(

uit?

(ht’) u$?8P’

6P 6

us”,?7P8P’

2Ui?1P,DP1P3(tt3)

DPZP

( t 3 t ’ ) ugL8Pt

( t 3 t ’ ) DP6P8

(t3t’)

- $(h/i>2ut”,’lPZP3

ut”,?lOPr

( tt’) DPZP

D#6P8(t4t’)

6C6DP6Pg

+ $ ( h/i)

4P 6P 6

(16.2)

.

Using lattice translational symmetry and time translational invariance, Eq. (16.2) can be written as

D~;t,,,(t- t’)

(D(O))$,,,(t - 2’) - I I j j r , , , ( t - t ’ ) .

=

(16.3)

The inverse harmonic phonon Green’s function (D(O))if.,p ( t - t’) is given by

(D‘’’)~;,,,,(t - t’)

=

6jjr[(az/dt2)

+

6(t

( W ~ ~ O ) ~ ]

- t’).

(16.4)

According to (15.8), all the functions in (16.3) are periodic in their imaginary time arguments, with period T = -if@. This is, of course, true for the inverse harmonic Green’s function on account of the periodicity property of the 6 function (14.13). The phonon self-energy function is periodic because its time dependence is determined by the phonon Green’s function. This fact may be directly verified for the approximate expression (16.2). Thus, we can define the Fourier series expansions for these functions similar to (12.10) and (12.12) for the phonon Green’s function, namely,

DT;,,,,(~- t ’ )

=

exp ( - i w n ( t - t ’ ) ) - D ? ~ r , p ( ~ (16.5a) n),

(1/7) n

ILjjr,,(t 60

See footnot,e58.

- t‘)

=

(1/7)

exp ( -ion(t - t ’ ) ) H j j ? , , , ( u n ) . n

(16.5b)

255

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

The Fourier coefficient for (D(O))-‘ is immediately obtained from (16.4) and (14.14) : (D(o))T;!,,(wn)= 6jjl( -wn2 (WjPO)2). (16.6)

+

Then (16.3) may be expressed as

DY;!, p ( ~ n )

=

(16.74

(D(O))Tf* , p ( ~ n ) - IIjj*,,(wn)

or

+

(16.7b) D T ~ ’ ! , ~= ( w6jjt( ~ ) -wn2 ( ~ j , ’ ) ) ” )- IIjj#,,(wn). The Fourier coefficient Djjl ,p (w,) of the phonon Green’s function which is the ultimate goal of our calculation is related to (D-’)jjr(wn) by Eq. (15.9) (with ( Q ) set equal to zero) or

C

(D-’)

Djji,p(ufi)

jij*,-p(mn)

= 6jj.

.

(16.8)

ji

Equation (16.8) implies that Djjr,p(w,), regarded as a matrix in the indices j and j ’ , is equal to the inverse of the matrix (D-l)jj.,-P(wn) or 6jj,[-wn2 (~~p”)~ ] IIjjf,-p(wn). More explicitly, upon substituting (16.7) into (16.3)’ we find that D is determined by

+

+

[-wn2

+

(~jpO)~]-’

C IIjji,-p(wn)Djii~,,(wn)

(16.9)

h

I n the harmonic approximation in which we neglect the phonon self-energy function, the Green’s function is first equal to D;o$,p(wn)= 6jjt/-wn2

+

(16.10)

(WjPO)2.

For later use, we shall find it convenient to define Djp@’(On) = 1/-wn2

+

(16.11)

(Ojp”)2.

Then Eq. (16.9) becomes

Djjt ,p (on) = 8jj’Djp‘O)(an)

+ C Dip‘’)

( w n )I I j j l , - p ( w n ) Dj1jf.p (on)

.

(16.12)

ji

One can solve this set of linear equations exactly and determine the phonon Green’s function in terms of the harmonic Green’s function and the phonon self-energy function. However, we shall only seek an approximate solution, which is sufficient for our purpose of illustrating the general structure of the Green’s function. We shall first determine the off-diagonal elements of the matrix Green’s function Djjt,p(~n), j # j‘. They are zero in the harmonic approximation and thus may be regarded as small compared to the diagonal elements. Then in the summation over j1 in (16.12), only the term with j , = j ’ , giving rise to the diagonal Green’s function D j T j ~ , ~ ( wis~ retained. ), Then

256

PHILIP C. K . KWOK

one obtains Djjt,,(wn)

=

(j f j')

Djp(O)(Wn)IIjj,,-p(On)Dj'j',p(wn)

(16.13a)

or D j j f , p(wn)

= [-wn2

+

(j

,p ( w n ) D j J j f , p(an)

(~jp~)']-'*njjt

f

j')

(16.13b) upon using the definition (16.11). For the diagonal part of the Green's function ( j = j ' ) , we have, according to (16.12), Djj,p(wn)

=

Djp'O'(wn)

+C

+

Djp(o)(~n)njj.-p(~n)Djj.p(wn)

Djp(O)(~n)~jjl,-p(~n)Dilj,p(~n).

(16.14)

ii#j

The approximate expressions (16.13) of the off-diagonal matrix elements of the phonon Green's function may now be substituted in (16.14), getting

C

Djjtp(wn>= ~ j , ( o ) ( w , ) / l - D j p ( o ) ( ~ n ) ~ j j , - p ( w n )-

Djp(O)njjl,-p(wn)

jifj

x D$;i(wn)

njij.-p

(16.15a)

(wn)

or D j j , p (wn)

(16.15b) For convenience, we shall define the function

so that (16.15b) may be written simply as Djj,p(wn)

=

{ (-an2 -

(OjpO)')

- g j p ( w n ) 17'-

(16.17)

The function fi jp ( w n ) is thus the effective phonon self-energy function for the phonons belonging to branch j (and wave vector p). It not only depends on the phonon self-energy lIjj,-pof branch j but also on the selfenergy function of other branches. Finally, putting (16.15) back into (16.13), the off-diagonal elements of the Green's function become explicit as Djjr.p(wn)

=

njjt,-p(Un)/[-Un2

+

- njj.,-p(Wn)nj'j,-p(wn)

(~jpO)~][-an2

+

( j Z j'),

(~~j*p)']

(16.18)

257

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

where all the anharmonic correlations in the denominator are neglected except the term that couples the branches j and j‘ directly, namely, JIjj‘ .-p (wn) II j.-p (an) * It is seen from definition (16.1) or, more explicitly, from (16.2) that the phonon self-energy function has the same analytic properties as the phonon Green’s function. Therefore, one may also define a spectral representation for the Fourier coefficient IIjj<,,(wn) similar to (12.16),6l i.e., jt

(16.19)

AT is the spectral function of the self-energy given analytically by A ~ j j,p, (0’)

=

lim 2 Im IIjjglP(wn+ w

+ ie).

(16.20)

r-o+

+ ie) as = lim Re IIw ( w + ie) , (a) = lim I m IIjjf ( w + ie) ,

Defining the real and imaginary part of IIjjt,,(o II’jj’ ,p (0)

(16.21)

,p

.+0+

,p

(16.22)

,p

t+Of

one obtains from (16.19) the following relations: (16.23)

IIjjr,p(w

+ ie)

L L OD

= 6

II’jjt,p(w) = 6

O3

dw’ II”jjJ,p ( w ’ ) -

7r

w’-w

+ in”jjj

,p ( w )

dw’ II”jjJ,p (a’)

(16.25)

-

n-

w ‘ - 0

(16.24)

*

The symbol 6 in (16.24) and (16.25) denotes the principle value. Such defined in relations also exist for the effective energy function fij,p(~n) (16.16). I n the next section, we shall find that the functions II’ and II” determine the dispersion and damping of the phonon excitations in the lattice. 17. DISPERSION AND DAMPING OF EXCITATIONS IN CRYSTALS In Section 6, the physical meaning of the phonon correlation function and spectral function was discussed in a qualitative way. We are now in See Kadarioff arid Baym,4 Chapter 4.

258

PHILIP C. K. KWOK

a position to give a more quantitative discussion, making use of the results obtained in Section 16. First we shall study the diagonal part of the phonon Green’s function D j j , p (on) given by (16.17). Let us analytically continue the pure imaginary frequency wn onto the real axis from the upper half complex frequency plane, i.e., wn +w ie, where e is a positive infinitesimal. Then Eq. (16.17) becomes

+

+ (wjPo)>”- 2iwe - f i j P ( w + &)I-’. The effective self-energy function fijp(o+ ie) may be written as IIjp(w +i€) + +

Djj,p(~ ie)

= [-a2

=

fiIjp(W)

211”.IP ( w ) 1

(17.1)

(17.2)

where f i I j p ( w ) and fi”jP(w) are the real and imaginary parts, respectively. Then we have Djj,p(o

+ ie)

=

[-d

+

(wjp’))”

- fi’jp(U)

- ifi”jp(~)]-l.

(17.3)

The imaginary factor -2iwe has been included in fi”jP(w). Recalling (12.17), one immediately sees that (17.3) is equal to the diagonal linear ; and, according to (12.18), one finds that the response function DRjj.p(~) spectral function A j j , p ( w ) is equal to twice its imaginary part. In the harmonic approximation, Djj,p(w i e ) has poles a t fwjpO - i e , whose real parts are just the harmonic phonon frequency. When anharmonic interactions are taken into account, the poles are generally complex, with a finite imaginary part. The position of the complex poles are determined by

+

w2

-

(Oj,0)2

+

filjP(O)

+

ifilIjp(W)

=

0.

(17.4)

We shall separately represent the real and imaginary parts of the solutions as (17.5) wp = up1- iw,”. The real parts uprgive the frequencies or dispersion of the excitations and the imaginary parts wP1‘ their damping. In order that they correspond to as well-defined excitations,62one must have wpl >> up”. I n that case, we have the following approximate equations determining w p t and uprt,respectively: (wp’)2 - (WjPO)2 f i ~ j P ( O P ~= ) 0, (17.6a)

+

wpll

=

(2wp’) - l f i l l j p (up’).

(17.6b)

When the anharmonic corrections are sufficiently small, there will be a solution to (17.6a) for the real part up1which is slightly shifted from the harmonic frequency wjpO. (We shall, for simplicity, just consider the positive 62

All well-defined excitations in the crystal are physical phonons.

259

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

solutions.) This is known as the renormalized phonon frequency and is it is approximately equal to63 denoted by wjP. According to (17.6~~)’

=

0‘ JP - wjpo

- (2wjp0)-1fi’jp(0jp0).

(17.7)

The damping or the inverse lifetime of the physical phonon is readily found from (17.6b) to be

-

up”-+ yj, = (2~j~)-~r”’j~(wj,)(2~j~~)-lfi”j,(~jp0).(17.8)

These expressions are only meaningful if the harmonic frequency shift Awj, = wj,

- 0jp0

= -(

2 ~ j ~ ~ ) - ~ f i ’ j ~ ( ~ j (17.9) ~~)

and the damping yj, are small compared to wjp or wjpo. There may be other solutions to (17.6a) besides oj, . They represent the frequencies of collective phonon excitations (in which the phonons from branch j take part in) which appear as a result of phonon interactions. Usually, they remain relatively undamped only for a limited frequency range. An example is the hydrodynamic mode of the acoustic phonons known as second sound which exists at low frequency and describes heat propagation.” The damping of these excitations is given by (17.6b). One can also determine the dispersion and damping of the excitations in the crystal by studying the frequency dependence of the phonon spectral function Ajj,,(o). According to (12.18), Ajj,,(o)

=

lim 2 ImDjj.,(w

+ ie)

e-0+

=

+

+

fi”jp(w)/[(w2 - (wjp’))” f i ’ j p ( ~ ) ] ~

(fi”jp(~))~.

(17.10)

By comparing this expression and (17.6) one immediately sees that the solutions w’ that correspond to the excitation frequencies are the positions of the peaks of the spectral function and their damping w” the half-widths of these peaks. The condition that the excitations are well defined, i.e., w‘ >> w”, is just that peaks are reasonably narrow. For the off-diagonal part of the phonon Green’s function, one can carry out a similar analysis starting from expression (16.18). It is obvious from (16.18) that the poles of the Green’s function Djjtp(w ie) ( j # j ’ ) represent excitations that both the phonons from branches j and j’ participate in. There is one particular point about the off-diagonal phonon Green’s function worth discussing. It concerns the case when the frequencies ojp0 and wjlp0 are degenerate. If the coupling does not vanish,

+

63

According to the second term in expression (17.16) for the self-energy function if the mode (jp) is not coupled to another mode (jlp) that is degenerate with it. There is a further discussion on this point later.

fiiD(ua),this approximate formula is only valid

260

PHILIP C. K. KWOK

the degeneracy will be removed. The renormalized frequencies are then of ) ~Ajp2]1’z. These solutions must also appear in the the form [ ( W ~ ~ Of diagonal Green’s function for the branches j and j ’ , implying that formula (17.6) is no longer valid. One must treat the term in fijp (16.16) which couples to the branch f with special care to determine the renormalized frequency wjp correctly, as in degenerate perturbation calculation. 18. PERTURBATION CALCULATION We have seen that the dispersion and damping of physical phonons in the anharmonic crystal are determined by the phonon Green’s function or, equivalently, by the phonon self-energy functional derivative technique (16.2). Even though (16.2) is a power series in the anharmonic coupling parameters, it is strictly not a perturbation series. The reason, as mentioned before, is that it contains the exact phonon Green’s function D, which, according to (16.1) or (16.3), still depends on II. The true perturbation series for the self-energy function is a power series in the anharmonic coescients and the harmonic phonon Green’s function D(O).I n obtaining this series, one must first expand the phonon Green’s function in terms of

88 (I)

(ii)

( 0 )H(’)

-0(b) 7d3)

FIG.2. Perturbation expansion of the phorion self-energy function.

GREEN'S FUNCTION METHOD IN LATTICE DYNAMICS

261

Do). From (16.1a), one finds D,pj(tt’)

=

DE?*(tt’)

+ D~O,?(tt1)II,,,,(t1tz)D,,,~(t2t’), (18.1)

which may be iterated, resulting in the following infinite series:

D,,* (tt’)

=

DL2r (tt’)

+

+ DL21(ttl)

&1,2

x DjPa,.(trt’) +

(ttl) II,,,, (tltz)DE,?.(tzt’) (tlt2) Dfz?a (tZt3) HI,,, 4 (td4) a * . .

(18.2)

Substituting (18.2) into (16.2) and iterating with respect to 11, one obtains the desired perturbation series. To order ( V3)) 4, ( U @ )2 -) U4), and ( U(4))2 it is readily calculated to be

These terms are represented by the diagrams in Fig. 2, which are identical to those in Fig. 1 except for the three diagrams that correspond to the second term in W)and the first and last term in P 4 ) which arise from the expansion of the phonon Green’s function. The self-energy function describes what should appropriately be called the one-phonon anharmonic effect because it arises from a single-

262

PHILIP C. K. KWOK

phonon Green’s function before the perturbation expansion. Since its time dependence is given simply by the 6 function, the Fourier coefficient II(l)( w , ) is independent of the frequency an . This implies that W)(o, + w ie) is real and therefore, according to the discussion in the previous section, only contributes to the anharmonic frequency shifts and not to the phonon lifetimes. It is analogous to the Hartree energy in an interacting particle system.64The self-energy 11(3)contains lowest-order three-phonon anharmonic effects due to cubic anharmonicity. This term gives rise to both frequency shifts and lifetimes, the latter as a result of real three-phonon processes. The function 11(4)contains higher-order three-phonon processes that occur via intermediate states. However, it essentially describes fourphonon anharmonic effects. The first two terms in contain four-phonon anharmonic corrections arising from the cubic anharmonic coupling parameter U(3) calculated in the second Born approximation.‘15The third term describes the lowest-order four-phonon effects due to the quartic anharmonicity alone. The last four terms contained the lowest-order fourphonon effects due to the presence of both W3)and V4). We shall conclude this section by finding the perturbation series for the effective phonon self-energy fijp(wn) for the diagonal part of the phonon Green’s function defined by (16.16). To the same order in (18.4), we obtain

+

-

IIjP(U,) = II:$Lp(un)

=

+ II%Lp(un)+ II$!-p(on)

IIE,-* (w,) nf:;Lp(wn) -wn2

+ ji#j

+

(WjPO)2

+ +

n,c:~,-p(wn>II,J~:-p(wn) ~~~~,-p(w,)II~~~,-p(wn) (1)

+c

-wn2

ji#j

(wipo)2

(18.5) where II(I), U3),and defined in (18.4).

II(4)

are the Fourier coefficients of the functions

IV. Explicit Calculation of Phonon Green’s Function and Applications

19. CALCULATION OF THE PHONON SELF-ENERGY FUNCTION In this part, we shall calculate the phonon Green’s function or the phonon self-energy function. The result is then applied to determine the G4 65

See, for example, Kadanoff and B a ~ r nChapter ,~ 4. P. C. Kwok and P. B. Miller, Phys. Rev. 146, 592 (1966).

263

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

ultrasonic attenuation coefficient. For simplicity, the quartic anharmonic coefficient V4)will be neglected. However, the analysis below can be readily extended to include the quartic and higher anharmonicity. Let us recall Eq. (15.7) for the inverse phonon Green’s function:

+

(DQ)s:t(tt’) = (D(O))i;/(tt’) Ut~)~,l(Q,l(t)) 6(t - t’) - $(Ti/i)UiYlp2 X DQ,,,,(tt3)DQPz~4(tt4) C(6/6(Q,t(t’)

)) (DQ)Lifi,(t3t4)]-

(19.1)

Here we have put V4)= 0 and carried out the functional derivative using the chain rule (6.11). To make contact with the diagrammatic technique7 of solving the phonon Green’s function, one defines the three-phonon vertex function r(3)as rrirzr3(tlt2t3) (3) =

lim C(8/6(Qm(t3)

)) (DQ)&z(~ddI.

(19.2)

Q-0

Then Eq. (19.1) becomes

0;:. (tt’)

=

(Do’) (tt’) ;;J

-

$ (Ti/i)U ~ ~ l p 2 D(tt3) p 1 D,,, p 3 (tt4)

wt

(t3t4t’)

,

(19.3)

which is represented by Fig. 3. This is the exact equation for the Green’s function in the presence of cubic anharmonicity alone. The vertex function as a power series in the anharmonic coefficients and the phonon Green’s function is readily determined by taking the functional derivative in (19.2). To first order in U3), one finds explicitly

rt;),,,,(tlt2t3) = u,$),~,~ 6(tl - t2) -6(tl - t3).

(19.4)

The remaining terms in the expansion of are known as vertex correpresents rections. I n Eq. (15.17)’ the term depending explicitly on ( U(3))4 the lowest-order vertex correction. When the quartic anharmonic coefficient U4)is included, one can similarly define a four-phonon vertex function

which is U;%z...,46(ti - tz) 6(tl - t3) 6(t1 - t 4 ) in the lowest approximation.

FIG.3. Exact phonon self-energy function (with only cubic anharmonicity)

264

PHILIP C. K. KWOK

One can now express D-1 exactly in terms of F4), and D as in (19.3). In the following calculation, we shall neglect all vertex corrections, so that

D%*(tt’) = (D(o))%p(lt‘)- 3 ( l i / i )Ut”:,,D,,,,(tt’)D,,,,(tt’) U$$4Bl. (19.5) The phonon self-energy according to (16.1) is

KI,#(tt’)

=

3(fi/4 ~t~”:,,D,l,,(tt’)~,,,,(tt’)

(19.6)

[which is just the second term in (16.2)]. The perturbation series is obtained as before by making use of (18.2). Then, including only terms up to ( U(3))4,one finds

rr,,.(U’) = 3 ( l i / i )u ~ ~ l ~ ~ ~ ” 3 ( t t ’ ) Dug)p4,,? ~~~,(tt’) -I- 3(n/i)zUt”?,,Dt“~(tt5)Ut3618,~DtOB?*(t~t8)DtO??p(t6t8)

x

u%9~1Pkt3 (t8t’)Dh%

,(tt’>

w’

-

(19.7)

The first term in (19.7) is just 11(3)defined by (18.4b), which describes the lowest-order three-phonon anharmonic effects and the second term is

(a) poles of No(z)

poles at

0 1

z-plane

(b) contour C

FIG.4. Integration on the complex frequency plane.

265

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

the first term in IF4) (18.44 containing higher three-phonon anharmonic corrections and the lowest-order four-phonon processes due to cubic anharmonicity. If vertex corrections had been taken into account, there would be an extra term in (19.7) proportional to ( U(3))4corresponding to the second term in (18.4~). We shall now calculate the self-energy function (19.6). Expressing the p’s in terms of the indices ( jp) and using translational symmetries, we have nji*,p(t

- t’> = i(fi/i) C I jP1

U3!~),jipi,jzpzDjija.pi(t -

tr>

The Fourier coefficient IIji, ,p (w,) is readily obtained upon using the definitions (12.10) , (12.12), and (16.5b) as njjr,p(wn)

=

i (fi/i)C

u1.3d.jipi,jzpzu~~~pi.j4-pzlj’~p

lip)

x

{ (1/7)

C

~jiia,pl.(wm)~j,ir,pz(~n

-

1,

(19.9)

m

where om= 27rm/r, with m being an integer. The frequency sum n (1/7) C ~ i i i a . p i ( w m ) ~ i z i r , P z ( ~-

(19.10)

m

is most conveniently carried out by expressing the phonon Green’s functions in terms of the spectral functions [Eq. (2.16)]. Then (19.10) becomes

(19.11) The simplest technique of evaluating sums of the form 7-1

C(wm-

Q1)-l(wm

- Qz)-1.

- (wm -

(19.12)

m

where 521 up to Qr are distinct complex numbers with finite real parts, is by contour integrati~n.~ Let us note that the phonon distribution function

NO(z)

=

(exp (fipz) - l)-l

(19.13)

of the complex variable z has simple poles at z = om= 27rm/-ifip dong Therefore, one can represent (19.12) the imaginary axis with residues l/fi@. by the following contour integral:

where C is a contour that wraps around the imaginary axis in an anticlockwise direction, as shown in Fig. 4. Deforming the contour C to enclose

266

PHILIP C. K. KWOK

all the simple poles in the integrand, one obtains i-’(NO(521)

(31

- 522)-’(31

- 33)-1.**

X

(32

x

(52, - 522)-1*

(522

- 523)-’.

’ (31

- Or)-‘

+

- (52, -

- 0,)-1

+ NO(52,)

+ NO(Qn,)(9, -

(522

-5 2 p

Ql)-’

(19.15)

Q74)-1].

The frequency summation in (19.11) is equal to

+ urn)-] = i(1 N’(w’)+ N 0 ( w ” ) )(w’ + w” (19.16) where the identities N o ( w + = NO(w) for any real w and NO( - w ) = -(1 + N O ( w ) ) have been used. Then, substituting (19.11) into (19.9), C(w’- wm)-’(w”

7-1

- wn

m

0,)

one obtains

X

Aiij,,pi(~’”zjr.pz(w’’)

x

(w’

+

w”

(1

+ no(^') + N o ( w ” ) )

- 6Jn)-1.

(19.17)

The real and imaginary parts of this self-energy function, when analytically iq are readily calculated to be continued to w

+

II’jjf,p(w)

= Re I I j j * , p ( ~ n+ w

X Ajzj,,p2(w”)[1

+ ie)

+ N 0 ( w ’ ) + N o ( w ” ) ] S(w’ +

- w),

LO’’

(19.19)

The symbol 6 is used to denote the principle value. It is obvious from (19.19) that the imaginary part of the self-energy n which determines the phonon lifetimes describes the interactions between three phonons. However, the phonons represented by the spectral functions (or w‘ and w ” ) are not harmonic phonons but are physical phonons containing anharmonic

267

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

+

corrections to all orders. The perturbation series for ITjj.,,(w i ~ )may be obtained by taking the Fourier transforni of (19.7) or, equivalently, by expanding the spectral functions in (19.18) and (19.19). To lowest order the spectral functions in in the cubic anharmonic coupling parameter U3), (19.18) and (19.19) are replaced by the harmonic spectral functions A(O) (6.8). I n particular, for the imaginary part, we have

x

{ (1

+ N o ( w o j i p i ) + No(wojzpz))

6(wojipi

+

+ 2(No(wojipi) - No(wojmz>)6(wojzpz - (1 + No(wojipi) + No(wojzpz)) +

wojzpz

wojipi

6(wojipi

- w)

- w)

wojzpz

+

1. (19.20)

The nature of the three-phonon processes this expression contains is obvious from the energy-conservation 6 functions. Taking w to be positive corresponding t o a real phonon, only the first two terms in the braces contribute. The first term is the contribution to the damping by the process in which the phonon with frequency w decays into two phonons wojlpl and wojzp, . The second term arises from the collision process in which the phonon w combines with a thermal phonon woj lp to produce the phonon woj zp . 20. ULTRASONIC ATTENUATION : GENERAL FORMULATION According t o Eq. (9.4), the amplitude of the sound wave generated at one end of the crystal is given by U(X) a

C exp (ipzX)DRjjtp.(w) PX

or

where X is the distance along the crystal, w is the frequency of the sound wave, and j is the acoustic phonon branch that is excited (representing the polarization of the sound wave). From Eq. (18.3) , we obtain U(X)

0~

C exp ( i p Z x ) / { -w2

+

(w0jp,)2

- fi’jpz(w) - zW’jp,(w) 1-1.

PX

(20.2)

The effective self-energy function

fi is defined by (16.16). Replacing the

268

PHILIP C. K. KWOK

summation over p , by an integral, (20.2) becomes

J

--m

(20.3)

This integral can be performed by taking a contour integration in the upper-half complex p , plane (since X is positive), getting

u( X )

Z j ( ‘ )( w )

a

exp (ikj(‘)(0) X - aj(l)( w ) X ) ,

(20.4)

r

where T sums over the poles of the integral, and c ~ j ( ~ ) ( are w ) the real and imaginary parts of the poles, and Z j ( r ) ( w ) is their residues. According to (20.4), we only need to consider those poles whose real parts are large compared to the imaginary parts. I n Section 17, it was shown that i ~ )has complex poles at the Green’s function D j j , , . ( w

+

0

= wlp,

- iw“,,

(20.5)

(0’ > > w ” ) , which represent the frequencies and damping of the excitations in the solid with wave vector p , . From (20.5), we find that k j ( r ) ( w ) and a+?)( w ) are determined approximately by

W’p.=ki(‘)(o)

(20.6)

= w,

a p ( w ) = (aW’,,/ap,)

praw”p.=hi

(7)

(w)

.

(20.7)

We shall for simplicity assume that the only excitations are the physical phonons with renormalized frequencies wjP . Then w’,, is equal to w j p , , given by (17.6), and is equal to yj,. (17.7), the inverse lifetime of the phonon. Thus, wjp.=ki(w) = w, (20.8) and a j ( w > = ( d w j p . / a p z ) p,=O?/jp.=ki * (20.9) (0)

The superscript T has been dropped for convenience. The factor (awjP,/apz)pz=o is just the renormalized sound velocity in the 2 direction and will be denoted by (20.10)

Uj(2) = (a~jp,/apz)p.=~

Therefore, the ultrasonic attenuation coefficient a t frequency w is simply equal to the inverse lifetime divided by the sound velocity of the physical phonon with wave vector p , ( = k j ( w ) ) and frequency qP,= w. Using (17.7) for y i p . , we can rewrite (20.9) more explicitly as cuj (0)= [ U j

(2)]-I( 2WjP,)-16”j p . ( w j p . )

(wip.

= w)*

(20.11)

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

269

If we neglect the dispersion of the acoustic phonon spectrum, the wave vector kj(w) describing the propagation of the sound wave is just kj(w)

=

(20.12)

~/uj(P).

However, we shall see below that dispersion plays a very important role for the attenuation of the longitudinal phonon and must therefore be taken into account. The longitudinal phonon has the higher energy of the acoustic branches. As a result, dispersiou together with the energy and momentum conservation laws forbid certain three-phonon processes that contribute to its attenuation to take place. On the other hand, dispersion is rather unimportant in the damping of the lower-energy transverse phonons.

21. ATTENUATION OF TRANSVERSE ACOUSTICPHONONS I n this section, we shall calculate the width or inverse lifetime Ytp of a transverse acoustic phonon from branch j = t with wave vector p. According to (17.7), -ytp is given by the imaginary part of the effective phonon self-energy fit, defined in (16.16). For simplicity, we shall neglect the second term in G;thus, 7t.p

=

(2Wtp)-ln”tt,-P(Wtp) ;

(21.1)

utp is the renormalized frequency of the phonon under consideration and

is determined by (Wt,IZ

-

+

(u0tp)2 n’tt,-p(wtp)

=

0.

(21.2)

The self-energy functions n’ and 11” are explicitly expressed in Eqs. (19.18) and (19.19). We shall consider the usual case that utp is small compared to the thermal frequency, i.e., Wtp

<< kBT/fi,

(21.3)

and the temperature T is small compared to the Debye temperature TD@:

T<
(21.4)

It follows from (21.1) and (21.4) that one may neglect all the Umklapp processes and the interactions between this low-frequency acoustic phonon and the optical phonons.6’ Let us first study the phonon damping or attenuation in the lowest approximation. The damping coeficient is then obtained from (21.1) by substituting the harmonic frequency uotpfor wlP and using the expression 66

67

Some average Debye temperature. Their contributions are smaller by factors of exp ( - m / k ~ T )where , s2 is a frequency of the order of the Debye frequency.

270

PHILIP C. K. KWOK

(19.20) for II” . The result is Ytp

cI

= (fir/4w0tp)

12(4~oj1P1Poj2P-P1)-1

~~%!p,jlPl.~2P2

(id X {

1 6(woi1p1+~oj,p-p1-uott,)

(1+N0(~Oj1p1> +No(aoj,p-p,)

+ 2(N0(~Oj1p1)-

No(wojzp-pi))

6(woj,p-pi

-~Oiipi wotp) 1,

(21.5)

where the fact that, for normal processes, the sum of the wave vectors in U@)must be zero was used. The first term in (21.5) represents the attenuation due to the decay of the transverse phonon into two acoustic phonons j , and j , . The second term represents the process in which the phonon is absorbed by a thermally excited phonon jlproducing another phonon j 2 . The selection rules for these three-phonon processes are discussed in a number of places.68~69As is well known, the dominant process for the damping of the transverse phonon in the frequency and temperature range C(21.3) and (21.4)] is the Landau-Rumer process,7owhich belongs to the latter class, with j~ = j 2 = 1, the longitudinal acoustic branch 1. Symbolically, it may be represented by t 1 1. The corresponding damping coefficient is

+

Ytp(LR)

=

~

c I u;!~),,~,1 2 ( 4 ~ 0 ~ ,

(fi?r/2w0tp)

l,zp-p

lwozp-pl)-~

P1

x

{NO(wOz,l) - NO(wOzp-pl) 1 6(w0zp-p1 - wozpl - WOtP). (21.6)

Contributions to the damping coefficient by other processes are smaller by powers of ( hwotp/kBT).For this particular three-phonon process, the dispersion of the phonon frequencies does not play any significant role and may be neglected. This is further justified by the fact that the frequency of the transverse phonon dtpand the frequencies of the longitudinal phonons, which are roughly equal to the thermal frequency k ~ T / hon account of the phonon distribution functions, are much smaller than the Debye frequency. Furthermore, for our purpose of presenting a general discussion, we shall assume that the phonon spectra are isotropic. The frequencies wotp and wolp may then be written as

ctop

(21.7)

w0zp = clop,

(21.8)

motp =

and R. E. Peierls, “Quantum Theory of Solids,” Chapter 1. Oxford Univ. Press, London and New York, 1956. 69 C. Herring, Phys. Rev. 96, 954 (1954). 70 L. Landau and G. Rumer, Physik. 2.Sowjetunion 11, 13 (1937). 68

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

27 1

where cto and clo represent the harmonic transverse and longitudinal sound velocities and p the magnitude of the wave vector p. To evaluate the summation in (21.6), one needs to obtain a more explicit form for the In the elastic continuum model discussed cubic anharmonic parameter U3). by Ziman,26one has

WtA i , j w

2,j.m

=

- (i/ ( P I 3’2)&lp

i.jw 2sj3ps~l~z~3

(21.9)

in the long wavelength limit in which p is the mass density of the solid is a function independent of the magnitude of the wave vectors. and Substituting (21.7)-(21.9) into (21.6) and changing the summation over p1 to an integration, one readily obtains

A

OlD

Ytp =

(

dm w4 -

aNo(w)

7) (21.10a)

or

(fiwzo

>> ~ B T ) ,(21.10b)

where WD is the Debye frequency for the longitudinal phonon, and [AZt2l] represents the appropriate angular average of Azt-, ,tP 2p-p over the angle between p and p l . We have neglected in (22.11) correction terms that are smaller by factors of ( fiwotp/kBT).The attenuation coefficient is just equal to (22.11b) divided by ct0, so that

This is the famous Landau-Rumer result for the attenuation of a transverse phonon or sound wave in an isotropic solid. It depends linearly on the frequency w and on the temperature T to the fourth power. We have just evaluated the width or inverse lifetime of a transverse phonon (tp) to lowest order. A natural question one may raise at this point is, (‘Under what circumstances is our calculation valid?” We recall that the present result is obtained by approximating the phonon Green’s functions in the self-energy function (19.6) by the harmonic Green’s functions, corresponding to the first term in the perturbation expansion (19.7). I n other words, we have neglected all the anharmonic interactions of the thermal phonons. Borrowing a field theoretical expression, the thermal phonons are treated as ‘(bare” particles that have not been “dressed” by the interactions with other phonons in the solid. Obviously, such a description is valid only if one encounters the thermal phonons

272

PHILIP C. K. KWOK

for a length of time short compared to the average duration between their successive collisions, which is simply their mean collision time or lifetime. To put these qualitative discussions on a more concrete basis, one may study the next and higher-order corrections to our results (21.10). They come from processes that include anharmonic interactions for the longitudinal phonons and correspond essentially to four-phonon and a higher number of phonon processes. One finds that the ratio of the successive terms in the perturbation calculation is given by powers of l / w ~ Z , where rZ is the average lifetime of the longitudinal phonons at thermal frequencies?’ It will be shown that T Z is proportional to T-5 for T << T D , the Debye temperature. Thus, if the period 2 ~ / wof our transverse phonon or sound wave is small compared to the mean collision time, so that O T ~>> 1, perturbation calculation and hence the result obtained previously are valid. This is often referred to as the high-frequency region. On the other hand, at lower frequency or higher temperature such that or1 << 1, the successive terms in the perturbation series for the damping coefficient will become comparable and eventually larger in magnitude. Then perturbation calculation ceases to be meaningful. One must go back to expression (19.19) for the phonon self-energy which contains the exact phonon Green’s functions or spectral functions to calculate the damping coefficient. We shall derive below, using some reasonable approximations, a formula for Ytp which is qualitatively correct for all frequencies and, in particular, extrapolates to expression (21.11) in the high-frequency limit. The damping coefficient ytp of a transverse phonon with frequency wtp due to interactions with two physical longitudinal phonons that contain anharmonic corrections to all orders is readily obtained from (21.1) and (19.19) to be hlr Ytp

=

~

4wtp

x

cI

u%zPl.zP-Pl

Pl

(1

l4

+ NO(0’) + N O ( w ” ) ) 6(w’ + w”

-

Wt,).

(21.12)

The spectral function for the longitudinal phonon according to (18.9) is Azz,~(w =)~ ~ ” Z ~ ( W ) / ( U ~(

+

W ~ Z ~ ) f ~i

+

’ ~ ~ ( w ) ) ~( f i ” z P ( ~ ) ) ’ ,

(21.13)

where f i l p is the effective longitudinal phonon self-energy function. If we replace A in (21.12) by the harmonic. spectral function, we obtain the ’l

P. C. Kwok, Thesis, Harvard University, 1965 (unpublished).

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

273

lowest-order attenuation coefficient studied previously. Obviously, it is impossible to evaluate (21 .la) without making appropriate approximations to simplify the spectral function (21.13). A reasonable approximation is to assume that the spectral function All,,(o) is sharply peaked a t the refi’z,,(w) = 0), and normalized frequency (determined by w2 - (wolp)2 consequently replace the frequency-dependent imaginary part of the selfenergy G ” l p ( w ) by g ” t p ( w ~ p ) .We then get

+

Ail,,(w)

= (w/I w

1)

*2wip~zp/(w2 - w2tp)’

+ (2wip~ip)~,(21.14)

where we have used (17.7) to express f i ” l P ( w t p ) in terms of the width ylP of the longitudinal phonon. Expression (21.14) is known as the Lorentzian approximation for the spectral function. This approximation is valid if fi”lp(w) does not vary too rapidly with frequency near w = alpand if the damping yzpis small compared to the frequency wlP .72 Using (21.14) in (21.12) and carrying out all the integrations, one finds that73J4

X r-’{tan-’[+(r

- l)UtpTl(O’)]

+ tan-l[+(r + ~ ) w ~ , T ~ ( w ’ ) ] ] . (21.15)

For simplicity, the phonon frequencies are assumed to be isotropic and dispersionless. ct and c z are the renormalized sound velocities, r is the ratio cl/ct ( > l ) , and ~ ~ ( w is ’ ) the lifetime of longitudinal phonon (yzP)-l a t wlp = w’. We see that (21.15) is very similar to (21.10a) except for the arc tangent factors. The frequency integral may be approximately carried out by making use of the fact that the integrand is rather sharply peaked at the thermal frequency w = k~T/h?’ Replacing the frequency-dependent lifetime ~ l ( w ’ )by its value a t the thermal frequency T Z = n ( w ’ ‘v k ~ T / h ) , we obtain the interpolation formula:

(hwm >> ~ B T ) .(21.16) 72

73

74 75

A. J. Leggett, and I). ter Haar, Phys. Rev. 139, AT79 (1965). P. B. Miller, Phys. Rev. 137, A1937 (1965). P. C. Kwok, P. C. Martin, and P. B. Miller, Solid State Commun. 3, 181 (1965). Unless there are processes other than anharmonic interactions, for example, point defect scattering, which dominates, so that it has strong frequency dependence at some particular frequency. For a detailed discussion, see Miller.73

274

PHILIP C. K. KWOK

The attenuation coefficient ( Y , ( ~ ~ ) ( wis) given by

X

(W/T){tan-'[+(r

- 1)wrJ

+ tan-l[+(r + 1)WTll). (21.17)

This is the desired result that is qualitatively correct for all frequencies. I n the high-frequency region such that W T Z >> 1 the arc tangents are exfor x > 1. The panded according to tan-' x = (7r/2) - (l/x) leading term of (21.17) is identical to our previous lowest-order perturbation result (21.11) except that all the sound velocities are renormalized velocities. The remaining terms are smaller by powers of l / w ~ Z . In the low-frequency limit, W T ~<< 1, which is often referred to as the hydrodynamic limit, the arc tangents are expanded as tan-' x = x - +x3 for x < 1. One then obtains

+ ...

+ ---

x W2Tt[1

+

O(W'TZ')

+

*"].

(21.18)

This well-known result can be obtained using the phonon Boltzmann equation to describe the attenuation of a sound wave in the hydrodynamic or collision-dominated limitj.76s77It is proportional to the square of the frequency and inversely proportional to T as 71 a T-5. At high temperature such that T is greater than the Debye temperature T D ( = ~ O Z D / ~,Bthe ) phonon lifetime 7 1 is inversely proportional to T , and the factor T4 in (21.18) which comes from the integral

lzD I dwI dw w4

aNO(w)

becomes linear in T , so that (Y*(w) becomes independent of temperature." 22. ATTENUATION OF LONGITUDINAL ACOUSTIC PHONONS

I n this section, we study the attenuation of longitudinal phonons with frequencies small compared to kBT/R and at temperature low compared to T D . For simplicity, we shall continue to use the elastic continuum model and assume that the phonon spectra are isotropic. To the lowest order in the cubic anharmonic coefficient U@),the damping coefficient of 76

77

A. Akhiezer, J . Phys. (USSR) 1, 277 (1939). T. 0. Woodruff and H. Ehrenreich, Phys. Rev. 123, 1553 (1961).

275

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

the longitudinal phonon is obtained from (17.7) and (19.20) to be

x

I(1

+

No(wojlPl)

+ 2(N0(w0iipi) -

+ NO(wOiz,-,l)) No(~oizp-pi))

6(U0j1Pl

6(uojzp-pi

+

woi2P-Pl

- woiipi

-

- O0ZP)

1.

~OZP)

(22.1) The allowed decay processes represented by the first term in (22.1) are 1$ t t and 1$ t L 6 8 Their leading contribution to y is readily calculated to be proportional to w4T, with numerical coefficients similar to the expression (21.10b). For the remaining processes represented by the second term in (22.1) , the only allowed ones are 1 t S I because of energy and momentum conservations and that in the elastic continuum model the transverse acoustic branches are degenerate. They also give rise to a damping that is proportional to w4T.’l I n real solids, the transverse phonon frequencies are not degenerate, and the phonon spectra are not isotropic. t t can occur. The corresponding damping Then the processes 1 coefficients have been calculated by Herring69 and are found to be pro- ~ , v is usually an integer lying between 2 and 4. portional to w ~ T ~where The process involving three longitudinal phonons 1 1 I (and 1 5 I 1) can also occur; however, its contribution is negligible compared to the foregoing processes. The most interesting fact one may immediately observe is that the attenuation of longitudinal phonons is much smaller than the attenuation of transverse phonons at comparable frequency and temper) ~ isotropic elastic conature. The factor of reduction is ( f i ~ / k ~inT the tinuum model and ( f i u / k ~ T”--l ) in real solids. However, experimental measurements of the attenuation coefficient of longitudinal sound wave a t ultrasonic frequency (-lo9 cps) and at intermediate temperature (-30°K) show that they are of the same order of mag n it~ d e .’~ The explanation proposed was the occurring of the “colinear” three-longitudinal process 1 1 1, not because of the anisotropy of the frequency spectrum, which is of little consequence, but because of the finite lifetime widths of the thermal phonons. This is large enough to relax the condition of energy c o n s e r ~ a t i o n . 7We ~ ~shall ~ devote the remaining part of the section to the development of this idea. The damping due to the process 1 1 S 1 in the lowest approximation

+

+

+

+

+

+

+

~

+

R. Nava, R. Arzt, I. S. Ciccarello, and K. Drausfeld, Phys. Rev. 134, A581 (1964); I. S. Ciccarello and K. Dransfeld, ibid. p. A1517. S. Simons, Proc. Phys. SOC.(London) 82, 401 (1963). *OH. J. Maris, Phil. Mag. 9, 901 (1964).

78

276

PHILIP C. K. KWOK

is, according to (22. ) , given by yzpW= __ li=

c

2w0zp P,

x

(NO(wOzp,) - NO(wOzp+p,)) 6(w0zp

+

w0zp1

-

wOzp+pl).

(22.2)

This vanishes when the phonon spectrum is isotropic and has a normal dispersion, i.e., wolp bends downward as the magnitude of increases because energy and momentum cannot be conserved simultaneously. The function A(P, PI)

=

w0lp

+

wolpl

- wolp+pt

(22.3)

1

which enters in the energy conservation 6 function, is never zero. This may be readily verified if we represent the phonon frequency by wozp = clop( 1

- Pp2),

(22.4)

where XO is the dispersion parameter and p is the magnitude of the wave vector p. Equation (22.4) provides a n adequate description if p is far from the Brillouin zone boundary. Then (22.3) becomes A

clop(1

-x

+ 3XX0pi2),

(22.5)

where x is the cosine of the angle between p1 and p. The frequency discrepancy is smallest when the phonons are colinear, x = 1, and is given by Amin E ~Z"p(3XOpl2)

o O ~ ~ ( ~ X O ~ I ~ ) .

(22.6)

Since the phonons w0lpl and wo~p+plthat appear in (22.2) are essentially a t thermal frequency, it is the value of Amin at such frequency which determines the amount of energy not being conserved. If this frequency difference is overcome, the foregoing process can occur and give rise to an attenuation similar to the Landau-Rumer result for the transverse phonons. We shall find below that such is possible when the finite widths of the thermal phonons are taken into account. The (partial) width of a physical longitudinal phonon at frequency wlp due to the interaction with two other physical longitudinal phonons is given by

dw' dw'

X [1

+ N 0 ( w ' ) + NO(W")]6(~' +

w"

- wzP),

(22.7)

similar to (21.12). Using the Lorentzian approximation (22.14) for the

GREEN'S FUNCTION METHOD IN LATTICE DYNAMICS

277

phonon spectral function and carrying out the frequency integrations, one obtains the damping coefficient h YzP(c)

=

~

2WlP

x

cI

w L - b , l - P l . z P + P 1 12(4~lPl~ZP+Pl)-~

PI

(NO(WlP1)

X (WP

+

+

- N0(WP+P1)) YZPl

-

WlPl

+

WlP+Pl)2

YZP+Pl

+

(YlPl

+

(22.8) YlP+PJ2

due to the process 1* I* l*; the asterisk is used to denote a physical phonon. I n Eq. (22.8), we have neglected all the terms that are smaller by factors of (Tzp,/wPl) and ( ~ ~ p + p l / w p + where p l ) , yzPland wp+plare the total longitudinal phonon widths. If we let the widths of the phonons go to zero, the foregoing expression reduces to a form identical to (22.2), except that all the harmonic phonon frequencies are replaced by the renormalized frequencies. The resulting damping coefficient will still be zero, because the renormalized frequencies, which are in fact the only measurable quantities, are assumed to have a normal dispersion. We shall represent the dispersion of the renormalized frequency by WZp

=

CZP(1 - W) 1

(22.9)

similar to (22.4) for the harmonic frequency. Then, using (21.10) for the cubic anharmonic parameter, one readily finds that the leading term of the damping coefficient for the colinear process is

X ?r-'[tan-'

(w

(

~ W~)

) T- ~tan-1 (o a f z(W ) T Z(a)) 3,

(22.10)

where T ~ ( w is ) the total lifetime l/yzplat wZpl = w and & ( w ) is the dispersion factor h ( W ) = %X(W/CJ2. (22.11) We have simplified the derivation by neglecting the dispersion in all other phonon frequencies appearing in the formula. Following the approximation procedure described in the previous section for evaluating such integrals, we replace the frequency-dependent functions & ( w ) and T ~ ( w )by their thermal values EZ and 7 1 . I n particular, we have %qA(k~T/h~z)~,

(22.12)

which is much smaller than unity when T << T D, because X is of the order

278

PHILIP C. K. KWOK

of a2, the square of the interatomic distance.al I n fact, one may express (22.12)a2qualitatively as fz b ( T/TD) ', where b is a numerical factor of order unity. Then from (22.10) we obtain the desired attenuation coeficient for the longitudinal phonon due to the colinear three physical phonon processes71J4 :

>>

B T ) .(22.13) There are three distinct frequency regions of interest contained in this expression. The first one is the high-frequency region, where both wrl and &WSZ are larger than unity. The attenuation is X w[tan-'(wn)

(fio~~

- tan-l(wtm)]

(W7Z

> 1; tcw.rz > 11,

(22.14)

which is independent of frequency. If we approximate total longitudinal phonon lifetime by lifetime due to the allowed lowest-order three-phonon processes, which are proportional to for some integer Y, as was discussed earlier, the thermal average rl will be proportional to TP5.This is a result quoted frequently before. The attenuation is then proportional to F/X.This high-frequency region is also the region in which one expects perturbation calculations to be valid. Indeed, if one calculates the four1& 1 t, which arises from the expansion of the phonon process 1 longitudinal phonon spectral function in (22.7), one finds that the attenuation behaves the same way." However, our result (22.14) for high frequency is only qualitatively correct because of the Lorentzian approximation used for the spectral function. For example, it cannot describe the attenuation due to the four-phonon process 1 1 1 I, which has been calculated to be proportional to wT6/X.a3 Next, we have the intermediatefrequency region, in which w.rl > 1 and f l w l < 1. This is possible because E l is much smaller than unity. Then a r & c ) ( o ) becomes

+

+

+

+

> 1; ~

( w ~ l

<

~ O T I1).

(22.15)

a linear chain with nearest-neighbor forces, the phonon frequency is proportional to sin (+up) E +up (1 - Au'p2). See Peierls.68 The Debye frequency w i D is of the order of ( c i l a ) . L. Landau and I. Khalatnikov, Zh. Eksperim. i Teoret. Fiz. 19, 1709 (1949). 111

GREEN’S FUNCTION METHOD IN LATTICE

DYNAMICS

279

The leading term is readily recognized to be identical to the Landau-Rumer result (21.12) for the transverse phonons except for numerical factors. It is proportional to the frequency and temperature to the fourth power. In this intermediate-frequency region, the colinear three- (physical) phonon process takes place because the widths of the phonons are large enough to compensate for the energy deficiency due to dispersion, i.e., { Z W T ~ < 1. Finally, we have the low-frequency or hydrodynamic region, where both W T Z and & W T ~ are smaller than 1. Then d c ) ( w ) reduces to the Akiezer form, similar to (21.18) for the transverse phonons:

Let us note in conclusion that, for both the intermediate and low-frequency regions, perturbation calculation for the particular interaction (22.7) is not valid. V. Phonon Boltzrnann Equation

23. LATTICE TRANSPORT PROPERTIES

The study of transport and hydrodynamic properties of the lattice vibrations is another important topic in lattice dynamics. It was Peierlss4 who first gave a detailed treatment of the problem of calculating the lattice thermal conductivity in an anharmonic crystal. He pointed out that in a pure crystal it is the Umklapp processes that are responsible for a finite thermal resistance. He based his calculation on the phonon Boltzmann equation, which describes the dynamical behavior of the phonon distribution function when the system is slightly perturbed from thermal equilibrium. Later on, others extended the calculation of thermal conductivity to include phonon scattering by impurities, defects, and boundaries.= For a review of this work, the readers are referred to the articles by Klemenss6 and car rut her^.^' Another hydrodynamic property of the phonon systems which is intimately related to heat transport is the existence of a collective or hydrodynamic mode of the acoustic phonons, known as 84

86

R. Peierls, Ann. Physik [5] 3, 1055 (1929). H. B. G. Casimir, Physica 6, 495 (1938). P. G. Klemens, Solid State Phys. 7, 1 (1958). P. Carruthers, Rev. Mod. Phys. 33, 92 (1961).

280

PHILIP C. K. KWOK

second sound.88Ward and Wilks89 demonstrated that the existence of such a mode follows from the macroscopic conservation laws for the phonon momentum and energy density. Their analysis was carried out for a single acoustic phonon branch. Sussman and Thellunggo subsequently considered second sound for three dispersionless acoustic branches. We shall discuss briefly the general principle underlying the calculation of second sound. For convenience, only the simplest case of a single isotropic and dispersionless branch is considered. The starting point is the Boltzmann equation describing the dynamical behavior of the phonon distribution function N(p; rt) . It has the form of

( a / a t ) N ( p ;rt)

+ cp.VrN(p; rt)

=

- (aN/at)collision,

(23.1)

where c is the sound velocity. The term on the right-hand side represents the rate of change of N due to phonon interaction and other scattering mechanisms, e.g., boundary and impurity scattering. From (23.1), one obtains first of all the equation describing the conservation of the phonon energy or heat density p( rt) : (a/at)p( rt)

+ V - j , ( rt) = 0.

(23.2)

The vector quantity j, is the energy or heat current. In arriving at (23.2), we have assumed that all the phonon scattering mechanisms are elastic. Another conservation equation that follows from (23.1) is the conservation of the phonon momentum density P( rt) . It appears as

(a/at)P,(rt)

+ VjTij( rt) = - ( ~ / T , ) Prt).~ (

(23.3)

T i j is the momentum stress tensor, and T, is the total effective collision time of all the scattering processes that do not conserve phonon momentum. These processes include Umklapp processes and boundary and impurity scattering. The simple form (23.3) in which T, appears is a result of the frequently used relaxation time approximation. Equations (23.2) and (23.3) can be combined by using the simple relation between the energy current and momentum density of the dispersionless phonons :

j,

The result is (d2/dt2)q

=

c2P.

- ViVjTij + ( 1 / ~ , )( d / a t ) p = 0.

(23.4) (23.5)

If the deviation from thermal equilibrium of the phonon system is small, one has the establishment of almost local equilibrium. In this case, all It was first discovered by Tisza [Compt. Rend. 207, 1035 (1928)l in his two-fluid model for super He4. 89 J. C. Ward and J. Wilks, Phil. Mug. [7] 43,48 (1952). J. A. Sussman and A. Thellung, Proc. Phys. Soc. (London) 81, 1122 (1963).

GREEN'S FUNCTION METHOD IN LATTICE DYNAMICS

281

thermodynamic quantities may be expressed entirely in terms of the conserved quantities. Thus, for the present isotropic system, we may write Tij

6ijT GS 6ij[(aT/aq)...q

+ D(ap/at) + ...I.

(23.6)

b

is a transport coefficient contributed by the normal process which conserves phonon momentum. Then (23.5) becomes ( d 2 / a t 2 ) q- U I I ~ V '~ d(a/at)V'q

+ ( 1 , ' ~ ~(a/'at)q ) = 0,

(23.7)

where we have denoted (aT/aq),, by U I I ~ Equation . (23.7) describes the propagation of heat in the phonon system. The dispersion relation is readily derived to be w2

- UII2k2

+ iw[bk2 +

(l/Tu)]

=

0.

(23.8)

and k are the frequency and wave vector of the propagation. At low frequencies such that W T ~<< 1, Eq. (23.8) may be approximated by -uI12k2 ( i w / ~ %= ) 0. This is the usual relation between the frequency and wave vector for heat diffusion. I n the frequency region w b / c z << 1 and w u>> 1, (23.8) corresponds to the dispersion of a slightly damped wave with velocity u I I . This wave is known as second sound, the collective mode describing the propagation of heat in an interacting phonon system. At higher frequencies, the damping is so large that second sound ceases to be a well-defined mode. The connection between thermal conductivity, second sound, and other hydrodynamic properties is discussed by Sussman and Thellunggo and more recently by Guyer and Kr~mhansl.9~ Finally, there is the question of the nature of the second sound mode. Besides being a collective excitation of the phonons due to anharmonic interactions, it has been shown by Kwok and Martin34that second sound must be treated as a n elementary excitation of the system, since it also appears in the phonon correlation function and phonon spectral function in the appropriate frequency range. w

+

24. DERIVATION OF THE PHONON BOLTZMANN EQUATION

We have seen that the phonon Boltzmann equation that describes the dynamical development of the phonon distribution function when disturbed from thermal equilibrium is the starting point of the investigation of a variety of problems in the hydrodynamic regime. It is of interest to know whether this extremely useful semiphenominological equation may be derived in a rigorous manner from the microscopic theory of lattice dynamics. The derivation was first carried out by Horie and Krumhand92 and later by Kwok and Martin" using a slightly different approach. 91 R.A. Guyer arid J. A. Krumhansl, Phys. Rev. 148, 766 and 778 (1966). 92

C. Horie and J. A. Krumhansl, Phys. Rev. 136, A1397 (1964).

282

PHILIP C. K. KWOK

However, the basic formulation of the two derivations is similar and is discussed by Kadanoff and Baym.4 We shall follow the procedure of the latter derivation. We define a function D l k a , l t k l a t ( t t ’ ) according to

The factor (??z.kmk’) l I 2 is chosen such that D is actually the phonon Green’s function in coordinate representation. All times are real unless specified otherwise. If the nuclear displacements are expanded in terms of the normal coordinates (3.7)’ one obtains Dzka,pkta*(tt’)

=

N-’

eka(jp)ekea’(j’P‘)

I ipl

eXp

+

iP‘.sZ.’)Djp,j’p’(tt’),

(24.2) where D j p , j t P , ( t t ’ ) is the phonon Green’s function in Fourier space defined by (12.1). When the system is in complete thermal equilibrium, the Green’s function D l k a , l t k f a f (tt’) depends only on the differences I - I ’ ( x Z 0 - xztO) and t - t’ on account of translational symmetries, i.e., Dzka,lrk’or’

(tt’) = D k a , k t o r . ’ ( I - l‘, t - t’).

(24.3)

This condition is equivalent to Eq. (12.4) that D j p , j p p r ( t t ’ ) = 6 p , - p t X Djit,P(t- t’). The actual derivation of the Boltzmann equation is carried out from the equations for two functions that are related to D , namely,

I n terms of the normal coordinates, Eqs. (24.4)and (24.5) become D>lka,ltva#(tt’)=

N-’

C ek*(jp)ek**’(j’p’)exp (ip-x? + ip’.xlto) I

ill1

(24.6)

X

D>j‘p‘,jp(t‘t)

7

(24.7)

where D > j p , j , p p (tt’) is the phonon correlation function (in Fourier space) defined by (5.2). I n thermal equilibrium, these functions also depend only

283

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

on 1 - 1‘ and t - t’, which for convenience will be denoted by 1,

=

t, =

1 - 1’

=

(Xi?

x10 - X l S O ) ,

(24.8)

t - t’.

Then Eqs. (24.6) and (24.7) may be written as D’lka,lrk’a’

(tt’)

=

D’ka,klal

(1,

=

N-’

eka( j p ) e k J a ’ (

7

tr)

j ’ p ) * eXp

(iP.Xl,O)D>jj’,p(tr),

ji’ I P

(24.9) D
=

D
=

N-l

7

tr)

eka( j p ) e k r a ’ (

j ’ p ) * exp

-t,).

( i p - ~ l , ~ ) D > j ~ j , - ~ (

ijl,p

(24.10)

Finally, using the definition D j j t , , ( t , ) ( 5 . 5 ) and the definition of its spectral representation (6.7), we put these expressions in the desired form: D’ika,l’k’al

(tt’)

=

D>ka,k*a’(Zr

=

~

-

C 1

, tr)

eka( jp)ekta’(

j ’ p ) * exp ( i p - x l , ~ )

ii’,p

In arriving at the final form of (24.12), we have used the identities Ajtj,-p( - u ) = -Ajjt,p(w) (7.7) and [l No( - w ) ] = - N 0 ( w ) . When the system has been perturbed from thermal equilibri~m?~ the functions E k $ l t k J a t ( t t ’ ) and D l k a , l t k r a , (tt’) are no longer just dependent on the relative coordinates 1, and 2,. They are now functions of 1, 1’ and t, t’. It is convenient to retain the variable 1, and t, and introduce the “center-

+

93 We

are studying how the system relaxes after it has been perturbed slightly from equilibrium. The relaxation is described by the Boltzmann equation. In Kwok and Martin,34 the Boltzmann equation is derived in the presence of a driving force.

284

PHILIP C. K. KWOK

of-mass” coordinates

to describe the additional spatial and time dependences. Thus we can still use expressions (24.12) for the D functions; however, the functions A j i . , , ( w ) and N o ( u ) must be replaced by functions that depend on I, and t, : (24.14)

For generality, we allow the function N to depend on the wave vector also. Then Eqs. (24.11) and (24.12) become

D>ika,zJkta’(tt’) = D’ku.k’a~(&;&) =

N-’

C e k a ( jp)ekf“’(j‘p)* eXp (ip*ZZ,O)i ijr,p

XiIm --m

do exp ( - i u t , ) A ~ ~ f , p ( w ; Z , t c ) N ( w p ; Z , t c )(24.16) . 2a

It is obvious that A ~ ~ J , Zctc) ~ ( uand ; N ( u p ; Zctc) represent the phonon spectral function and phonon distribution function in the perturbed system. From the relation between D> and D<, on account of definitions (24.4) and (24.5), one can readily show that they satisfy

similar to the equalities satisfied by their equilibrium counterparts. The derivation of the Boltzmann equation consists of obtaining the equations of motion for the functions D>< and hence N . The first step is to obtain the equation of the phonon Green’s function D l k a , ~ ~(tt’). k ~ aI~n imaginary time, this equation can be immediately obtained by transforming (25.2), (17.lb), and (17.4) into coordinate representation :

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

285

(24.18)

For later use, the equivalent equation

-

J dtl

C

D ~ ~ ~ , z l ~ i ~ l ( t ~ l ) ~ z i ~ i a i , ~ ~ ~ ~ ~ ~ ( ~

likiai

= 6ZZ# 6 k k r 6aar 6 ( t

- t’)

(24.19)

is also derived. Wz) is the harmonic coefficient in the expansion of the potential energy function (2.2), and n is the self-energy function in the coordinate representation :

C

~ ~ ~ ~ ~ =, ~ ~ - ~1 ~e k *~( jp)ek
x

rI+p,jt-p* (tt’) .

+ ip-xlfo) (24.20)

In previous analysis, the time integration has always been taken over the pure imaginary time interval (0, -if@], but at present we shall use the equivalent interval { T o , 70 - i h p ] , (24.21) where T~ is an arbitrary real number. The next step is to take the difference of the two equations (24.18) and (24.19), getting

- Dlka,zlklol(ttl) ~

(id’) 1.

z l k l ~ l , ~ w a ~

(24.22)

The equations for the functions D>< may now be derived by analytically continuing the complex times to real times. We shall only derive the equation for D< and simply quote the result for D>. Choosing t < t‘ in the interval (24.21), the Green’s function D l k a , z t k t a f ( t t ’ ) is equal to ( i / h ) X

286

PHILIP C. K. KWOK

(mkmkr)1/2(~u,(:; t’)u.(:; t ) ), which is simply the function D
- D < Z k U . 1 l k 1u 1(th)II>zlk 1u 1,

ZtktUt

(tit')1,

(24.23)

where the time integration was divided into appropriate intervals, and the We shall functions n><are defined from II similar to the functions D><. now continue t and t’ into real times (with no specific chronological relation) and let TO go t o - Q, . The result is

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

287

Here all times are real. Next we express all the positions I and time dependences of the function in (24.24) in terms of their respective relative and “center-of-mass” coordinates. Then, upon choosing convenient summation variables, Eq. (24.24) may be written as

288

PHILIP C. K. KWOK

(24.26) has been used explicitly. Equation (24.25) for the function D < is exact because no approximation has yet been made in the derivation. However, to obtain the Boltzmann equation, one makes the assumption that the deviation of the system from thermal equilibrium is both small and slowly varying in space and time. Then the functions in (24.25) are only weakly dependent on their %enter-of-mass” coordinates. We expand any function AZ, t, At) in powers of AZ (i.e., xol,+al - x1.O) and of the form F(1t; I, At, which is valid as long as AZ and At are small compared to the characteristic distance and time interval over which the deviation from equilibrium changes appre~iably.~ Carrying out the expansion and using representations (24.16) and (24.17) for the B functions and similar ones for the self-energy functions n<>,we obtain

+

[w(a/atc)

+

+ wojp~ojp.V , I A ~ ~ . ~ z( Wc t c; ) ~ ( w pictc);

= -$[n’jj.-p(u;

Z c t c ) A j j . p ( ~ Zctc)N(wp; ; Zctc)

- n<.. ~ ~ , - P ( C JU; c ) A j j , p ( ~ LtC) ; (1

+ N(wp; Lk)),

(24.27)

where u0jpis the phase velocity vpwOjp, and V, is the gradient operator with respect to $1, or simply 1, . Correction terms that are proportional to higher power of the derivatives a/atc and V, have been neglected. We have also neglected the off-diagonal parts of the spectral functions. A similar equation for D > j j , p ( w ;Zct,) or Ajj,p(w; ZC&) [l N ( w p ; Zctc)] may be derived in the same manner, starting from (24.22), but taking t > t’ instead. The result is

+

[o(a/atc> =

+ wojp~ojp. v

+

~ ~ A ~zctc) ~ (1. ~ N( (~~ ;Pzctc)) ;

-$[n’jj,-p (w ;Z c t c ) Ajj,p(W ; l c t c ) N (UP; Z c t c ) - n<jj,-p(~; Zctc)Ajj,p(a;

Zctc)

(1

+N(wp;

Zctc))].

(24.28)

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

289

Since the right-hand side of (24.28) is identical to that of (24.27), we find, by subtracting,

+ ~OjpUOjp.V]Ajj,p(W;rt)

[w(a/at)

=

0,

(24.29)

where we have put t, = t and replaced xO1, by the continuous variable r. The simplest solution to (24.29) is the harmonic spectral function = 2?~(w/Iw I) 6 ( w 2 - ( ~ 0 ~ ~, )which ~ ) is independent of r and t. Then (24.27) or (24.28) may be written as 2a(w/l w =

I)

6(w2

-

+

( ~ ~ j ~ ) ~ ) [ w ( a / a ~Ojp~Ojp.V]N(wp; t) rt)

--32r(w/l w

I)

6[02

- (~o~~)~][TI>jj,-~(w; rt)

X N(wp; rt) - II<jj,-p(u; rt) ( 1

+ N(wp; rt))].

(24.30)

Now, integrating this expression over w from 0 to co , we obtain [(a/at) =

+ uojp*V]Nj(p;

rt)

- (4wOjp)-1[n>jj,-p(wOjp ; rt)Nj(p; rt) - n<.. u.-p(wojp ; rt) (1

+ Nj(P; rt) )I1

(24.31)

where

Nj(p; rt)

=

N(w

=

wojpp;rt).

(24.32)

Equation (24.31) is the Boltzmann equation for the distribution function of the phonons from branch j. To complete the description, one must specify the self-energy function in the “collision” term on the right-hand side. Consistent with our approximation for the spectral function, we shall only calculate IT>< to the lowest order in the cubic anharmonic coupling Then, from (19.8) , it is readily found that constant U3).

(24.33)

X A$:i.z,p2(~ - wl)N(wlpl ; rt)N(w

- o1p2 ; rt).

(24.34)

Substituting these expressions into (24.31) and carrying out the frequency

290

PHILIP C. K. KWOK

integrations, we obtain the final result :

[(a/at>

+

v]Ni(p; rt>

uojp*

C I Uj!b,j1p1,jzpz

=

12(2~0jp.2~0j1pl.2~0jzp2>--1

Iipl

X

(6(Wojp

- Wojip,

- ~'j2p2>[(1+ N i i ( ~ 1; rt>>

+ Njz(Pz ; rt>)Nj(P;rt> - Nji(p1 ; rt>Njz(Pz; rt) X (1 + Nj(P; rt))] + 26(Wojp + WOjzpz) X (1

oojipi

X C(1

+

Nji(~1 ;

- Nji(P1 ; rt>(1

rt>>Niz(~2 ; rt>Nj(P;rt>

+

Njz(~2 ; rt>>(1

+ Nj(P; r t > ) l ) .

(24.35)

The only difference between this equation and the frequently used form of the Boltzmann equation68is that the phonon velocities uOjp and frequencies w0jp are the harmonic rather than the physical or renormalized ones. To obtain corrections to our present expression, one has to retain more terms in the expansion of (24.25) than we did in deriving Eq. (24.27). A discussion of this more complicated procedure may be found in Kadanoff and Baym.4

VI. Coupled Phonon-Photon System

25. INTERACTION OF LATTICE VIBRATIONS WITH ELECTROMAGNETIC FIELDS

THE

MACROSCOPIC

It was discussed in Section 11 that, to determine correctly the various optical properties of the dielectric solid, one has to consider the intrinsic coupling of the electromagnetic fields to the lattice vibrations. We shall study this rigorous approach in the present part. At the same time, the discussion below will serve as an introduction to the Green's function treatment of coupled quantum systems: in our case, the phonon and photon fields. We begin by determining the interaction between the phonons and the macroscopic electromagnetic fields?* Assuming that the nuclei are point ions with charges z k ) the equation of motion for the nuclear displacements in the harmonic approximation is

g4

See Fano" and Born and Huang.16

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

29 1

where is the harmonic coefficient that includes only the microscopic field effectss5, and E[x(:)] is the microscopic electric field a t x(:), the position of the (Zk) nucleus. In studying long wavelength phenomena in which the wavelength is much larger than the internuclear distances, one may make use of the following approximation: (25.2)

This approximation enables us to transform (26.1) in terms of the normal coordinates Qjp of the lattice vibrations. We obtain

The summation x k Zkeka ( jp)/ (Nmk)1‘2 is related to the polarization of the branch j, and it vanishes when j represents an acoustic mode in crystals having inversion symmetry a t every nucleus. We shall restrict ourselves to this simpler case and neglect the acoustic branch. Furthermore, in order to avoid unnecessary complications, we shall assume that only one optical branch is coupled to the electric field. The index j will be replaced by u to denote the polarization of this branch. However, the following analysis can readily be extended to include all phonon branches. The electric field E is determined as usual from the Maxwell’s equat i o n ~ It . ~is~ convenient to separate E into a longitudinal and transverse part. The longitudinal or irrotational part Ell satisfies the equation

V-E’l(x)

=

-4rV-P’I (x),

(25.4)

where P is the polarization per unit volume. On the other hand, the transverse or divergenceless electric field E A satisfies

EA(x) - c2V2E~(x)= -47rP

A

(x ) .

(25.5)

E l is often referred to as the retardation field. Equations (25.3)-(25.5) will completely describe the interaction between the phonons and the electromagnetic field, provided one expresses P in terms of the nuclear displacements or the normal coordinates. Consistent with the long wavelength approximation (25.2), we shall replace the continuous variable x by the discrete set of lattice vectors x t and define the polarization per unit volume a t x = x? as (25.6) 95

96

For a detailed discussion, see Born and Huang,15 Chapter V. J. D. Jackson, “Classical Electrodynamics,” Wiley, New York, 1962.

292

PHILIP C. K. KWOK

where vo is the volume of a primitive unit cell. I n terms of the normal coordinates, (25.6) becomes

Pa(XrO) = (l/Vo)

c c zk[ek.(up)/(~mk)’12]&up exp

(25.7)

k

UP

Only one optical branch enters in the summation over the phonon branches because of our previous assumption. The analysis below will be greatly simplified if another assumption is made, namely, that the optical phonons are purely longitudinal or transverse. I n other words, the eigenvectors eka (up) are either parallel or perpendicular to the wave vector p. We shall use u = 1 to denote the longitudinal phonons and u = t = tl or t 2 to denote the transverse phonons. Such separation is possible, in particular for a cubic crystal and for wavelength long compared to the internuclear distances. Because of symmetry, the harmonic frequencies woupare degenerate, all equal to wpO. The eigenvectors eka (up) have equal magnitudes :

eka(Up)

=

vk(p)da(‘Jp),

(25.8)

where 8(up) denotes the unit vectors parallel or perpendicular to p. Then (25.7) may be explicitly written as

P,(xrO) = ( 1 / W z ) ~ ( z ( p ) / v 0 ) 2 ~ ( u p ) exp Q . ~(~P-xzO), (25.9) UP

with

z(p)

zLvk(p) /mk’”.

=

(25.10)

k

One can similarly decompose the electric field E into a purely longitudinal and a purely transverse component:

E,(x)

=

(I/WZ) C & ~ ( a p > exp ~ , , (ip-x).

(25.1 1)

UP

The quantity Eupis related to the “normal coordinate” of the electromagnetic or photon field. Then, using (25.10) and (25.11), we can rewrite Eqs. (25.3)-(25.5) as Qzp = Qtp

and

=

- (wpo)zQzp+ z(p)Ezp , - (~pO))”Qtp + z ( ~ ) E t p, EzP

Etp

= - 4 d ~(PI

+ czpzEtp

=

/~oIQzP

(25.12a) (25.1213) 7

-4~[~(p)/voIQtp

(25.13) (25.14)

293

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS OF THE OPTICALPHONONS 26. DISPERSION

I n this section, we study the change of the dispersion curves of the optical phonons due to the microscopic electric field. First we consider the effect of the microscopic field on the longitudinal phonon. From (25.12a) and (25.13), one immediately obtains the following equation for the normal coordinate: QzP

+

+

[(upo)

vp2) I Q i p

=

0,

(26.1)

where

vp2 = (~*/vo) ( 4 ~ 12. )

(26.2)

Thus, the frequency of the longitudinal optical phonon is changed to =

W2lp

(opo)2

+

(26.3)

vp2.

If we neglect the coupling of the transverse electric field ELto the transverse optical vibration, that is the retardation effect, the transverse optical frequency will remain equal to upo (25.12b). The degeneracy of the frequencies of the longitudinal and transverse phonons is removed with the ratio w2lp/wZtp given by =1

&P/&P

+

(26.4)

(vP2/(wPo)2)-

This equality (usually evaluated at p = 0) is known as the LyddaneSachs-Teller re1ati0n.l~One may rewrite (26.4) in a more familiar form by expressing the right-hand side in terms of the transverse dielectric function E L , which is defined, as usual, as 1 4 ~ x 1 ,where X.L is the transverse susceptibility P L / E L .From (25.9) and (25.12b) we obtain

+

EL(WP)

=

1

+

(vp2/(wp0)2

- w”.

(26.5)

Hence the ratio (26.4) may be alternately written as W2Zp/W2tp

=

€qoP)/Eqw

= EL(OP),

+ 00,

PI ,

(26.6a) (26.6b)

since e ~ ( w+ 00, p) = 1. However, the Lyddandachs-Teller relation is only valid for p > lo3 cm-I. We shall see that, when the wave vector is smaller than lo3 cm-’, the retardation effect becomes important. At still smaller p , when the wavelength is comparable to the size of the crystal, size effect also has to be taken into account. I n fact, it has been shown that the ratio w2zp/wZtp in the limit of p -+0 depends on the geometry of the ~rystal.~~*~~ We shall now consider the dispersion of the transverse optical phonons due to retardation effect. The desired relation is obtained from the coupled

294

PHILIP C. K. KWOK

W

-

-ku

(u

F

+

‘u, oa

3

I+r-----y‘ /

w=w;

P

FIG.5. Dispersion of the polaritons.

equations (25.12b) and (25.14) by substituting the following oscillatory solutions : Qt, = Qt, ( 0 ) exp ( - i 4 ,

E,, = Etp(0) exp ( -iut) . The result is [WZ

- (u,O)>”](d- c”2)

- u2r],2

(26.7) =

0.

(26.8)

Equation (26.8) gives rise to the two hybridized phonon-photon excitations known as polaritons12with frequencies u21p

=

(UpO)2

+ + c”p” + [( Tp2

(up0)2

+ + rlp2

c2p2)2

- 4(u,0cp)2]’~2/2, (26.9)

w22,

= (up0)2

+ + c”p” r],2

- [( (up0)2

+ + Tp2

c2p2) 2

- 4 (w,Ocp) 2]1/2/2. (26.10)

I n the limit of small p , the frequency of the upper polariton branch 01, approaches the longitudinal optical phonon frequency wlp = ( (upo) qP2) and the frequency of the lower branch wW approaches cp/er(Op). At large p , 01, approaches the vacuum photon frequency cp, and u2, the noninteracting harmonic frequency upo.The retardation effect is strongest in the wave vector region in which upoN cp, i.e., p lo3 cm-’. These results are illustrated in Fig. 5.

+

27. POLARITON GREEN’SFUNCTION We shall now construct the Green’s function description of the coupled transverse phonon and photon or polariton system. First let us review briefly the quantum mechanics of the free photon field. The Hamiltonian

295

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

of the electromagnetic field in vacuum expressed in terms of the transverse vector potential A(x) is9’ Hphoton

+ ( ~ T ) - ~ x( VA ( X ) ) ~ ] ,

d3x[2ac2M2(x)

= V

(27.1)

where V is the volume of t,he crystal:

V

=

Nv~.

(27.2)

The vector operator M is also purely transverse, and it is the canonical conjugate variable of A. The nonvanishing commutation relation between M and A is [Ma (XI, AB(x’)1 = ( f i / i6’,p ) (X - x’). (27.3) Here 6 t , ~is the transverse singular function 6 t , ~ (-~ x’) = ( l / V ) c(6,~ - ( p a p , d p 2 ) exp (ip. (x

- x’) 1. (27.4)

P

As usual, it is more convenient to use the wave-vector representation. Accordingly, we define the operators Mtp and At, (t = tl or tz) :

M,(x)

=

( N V ~ ) - ’~/ ~( 4 ? r ~ ~ ) - ~ / ~ i ? , (exp t p ) (ipsx), M~,

(27.5)

tP

=

( N V ~ ) - ”C(4?rc2)1/2i?p(tp)~~, ~ exp (ip-x),

(27.6)

tP

Here i?(tp) are the transverse unit vectors perpendicular to p. The factors in (27.5) and (27.6) are so chosen so that the commutation relation between M,, and Atp has the simple form

EM,,

&P’I

=

(fi/i)8p,-p#

(27.7)

6ttJ *

I n deducing (27.7), we have used the identity

c

SOl(tp>~B(tP)= C6aS - (P=Ps/P2)1.

(27.8)

t=t1.t2

On account of the hermiticity of M(x) and A(x), we have

M+tp

=

Mt-p

,

A+$,

=

At-,

.

(27.9)

Now, substituting (27.5) and (27.6) into (27.1), we obtain Hphoton

=

3 C(MtpMt-p

+

C2p2AtpAt-p).

(27.10)

tP 97

W. Heitler, “The Quantum Theory of Radiation,” Oxford Univ. Press, London and New York, 1954. The more familiar expression is HDhoton= (I/%) J d3x(E2 B2), where E and B are the electric and magnetic fields related to M and A by E = - l ( l / ~ ) A = -47rcM, B = v X A.

+

296

PHILIP C. K. KWOK

For the free photon Hamiltonian (27.10), Mtp is equal to the time derivative of A,, as can be readily verified by calculating

This relation, however, is not true when the coupling of the photon to the transverse optical phonons is included. The quantity At, is readily identified as the normal coordinate of the transverse electromagnetic field.98 Let us now study the coupled phonon-photon system. We shall deduce the total Hamiltonian from the coupled equations (25.1213) and (25.14). First we express the transverse electric field variable Etpdefined by (25.11) in terms of Atp . Upon using the equality

- (i/c) (aiat)A,(xt),

E,(x~) = one finds that

Etp =

- (4?r/V0)’/~At~.

(27.12) (27.13)

Then (25.12b) and (25.14) become Qtp

=

Atp

= -CzpzAtp

-(~pO)~&tp

- VpAtp,

+

(27.14) (27.15)

VpQtp

(qP = ( 4 7 r / ~ ~ ) ~ ~ ~ Eq. z ( p(26.2)). ), The Hamiltonian H that gives rise to these equations of motion is easily determined to begg

H = 3 C(MtpMt-p tP

+3

c(QtpQt-p

+ c2PzAtpAt-p) +( + (upo)2

Vp2)&tp&t-p)

tP

(27.16) Once the coupling between the electromagnetic field and the phonons is taken into account, Mtp is no longer equal to At, . By straightforward 98

One can define the photon creation and annihilation operators bt, and bt,+ according to At, = (fi/2cp)”2(btp b f - D ) j

+

Mt,

= At, =

( l / i ) + ( f i ~ p ) ” ~ ( b-r ,bt-,+),

similar to Eq. (3.12) for the phonons. Then the Hamiltonian (27.11) becomes the familiar expression fiCp(bt,+bt,

HDhoton = tD

OQ

Compare expression with that in Kittel.60

+ 1)

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

computation, one finds Atp

=

M t p

showing that M t p

=

At,

+ -

VpQtp

297

1

(27.17)

VpQtp.

This relation originates in the so-called “velocity” coupling between the particle and electromagnetic field and is entirely equivalent to the more familiar identity that the kinetic particle momentum mv(A) is not equal to the canonical momentum p(M) but p - (e/c)A. We shall eliminate this inconvenient coupling and put the photon and phonon on the same footing by transforming the electromagnetic field variables according to MtP

At,

,

(27.18)

= (cp)-’at,.

(27.19)

=

--cp*qtp

Now qtp becomes the normal coordinate and atpthe conjugate momentum as they satisfy Cat, , P t ‘ P ’ 1 = ( J L / i ) 8 p , - p ’ 6 t t J . (27.20) The Hamiltonian (27.16) then becomes

H

=

+

3 C(atpat-p c 2 p 2 q t p q t - p ) tP

+ 3 C(6tpQt,+

((Wp”>”

+

Vp2)&tpQt-P)

tP

-

C

.

cp*VpqtpQt-p

(27.21)

tP

As the coupling is between normal coordinates, one has the more convenient relation a t p = Qtp * (27.22) Then finally, using (27.22), we obtain

H

=

3 C(atPat-p

+

c2p2qtpqt-p)

tP

+3 -

CCOtpQt-p tP

c

CPVPqtPQt-P

+( .

+

( ~ p ’ ) ~

Vp2)QtpQt-P)

(27.23)

tP

I n the following discussion, we shall only consider one transverse branch, because both branches are equivalent, and we shall drop the subscript t. The Hamiltonian (27.23) can be diagonaliaed to eliminate the coupling to exhibit the two polariton modes. After some simple algebra, one finds that the correct choice of the polariton normal coordinates (pip and (p2p is related

298

PHILIP C. K. KWOK

(27.24a)

Here ap is the coeficient = (1

ffp

+

C2P2rlP2

CW2lP

-

(27.25a)

- 4P21

(uPo)2

(27.25b) wlp and wZp are the two polariton frequencies (26.9) and (26.10) determined

previously. They satisfy the identities w21pw22p W2lP

+

w22p

(

=

c 2 p 2 wp0) 2)

=

(wp0)2

+ + qp2

(27.26)

c”2.

The resulting diagonalized Hamiltonian H is

H

=

3 X(iIPi1-P

+

~21P(P1P(P1-P)

P

+4

C(iZPi2-P P

+

w22P(PzP(Pz-P)?

(27.27)

which clearly illustrates the independent character of the polaritons. The Green’s functions for the polaritons are defined similar to the phonon Green’s function Eq. (12.1) , namely, I

Drp,rtp*(tt’)

where r

=

= (i/fi> (((Prp(t)(Prlp‘(t’)

I+),

(27.28)

1 or 2. By translational symmetry, we have I

-

Drp,r~p~(tt’) = 6p,-p*Drr~,p(t - t’).

(27.29)

Furthermore, since the Hamiltonian (27.27) is “harmonic,” i.e., the two polariton branches are not coupled, the Green’s function is also diagonal in r and r‘, i.e., Drp,rrp‘(ttr) = 6rrf6p.-p’Drp(t - t’). (27.30)

-

I

I n frequency space, the harmonic polariton Green’s functions d,, (w,) have the simple forms (27.31) d, (a,) = 1/ -w,2 wZrp .

+

When we include the anharmonic interactions of the phonons, the polaritons will no longer be independent. Let us insert in (27.27) the

GREEN’S FUNCTION METHOD IN LATTICE DYNAMICS

299

cubic anharmonic term coupling the transverse phonons (say for simplicity only from the branch we are considering)

(U39 C

~plpzp3QplQpzQp3

-

(27.32)

(PI

Then, by using (27.24), this term becomes a cubic anharmonic term for the polaritons ( I D ! ) C D r i p i . r z p 2 , r . w 3 ( ~ r i Pi q r m z ( ~ r 3 p 37 (27.33) (+PI

where o r i p i n p z ,r 3p 3

- a r ip i a r m z a r w 3 u p i p zp 3 7

(27.34)

arlpl= -(1 - a2p1)1/2, r1 = 1

-

r1

ffPIJ

= 2.

(27.35)

This gives rise to the anharmonic polariton Hamiltonian

H =3

+

z(+rp+r-p

uzrpVrp+r-p)

rp

+ (1/3!) C

B r i p i , r z p z , r , P , ( ~ , i p i ( ~ r 2 p z ( ~ r i P*a

(27.36)

lrpl

One may now proceed to calculate the polariton Green’s function exactly as for the anharmonic phonon Green’s function. But, for the remaining part of this chapter, we shall only discuss the harmonic polariton system. We conclude this section by calculating the phonon, mixed phononphoton and photon Green’s functions, which are defined as DP(t

- t’)

=

(i/fi)( ( Q P ( t ) Q - P ( t ’ ) ) + ) ,

(27.37)

FP(t

- t’)

=

(i/fi> ((QP(t)A-P(t’))+),

(27.38)

GP(t

- 2’)

= (i/fi> ((AP(t)A-P(t’))+),

(27.39)

from the known harmonic polariton Green’s function (27.31). The first step is to express Qp and A, in terms of prP. From (27.19) , (27.22) , (27.24), and (27.25), one obtains DP(t

- t’)

=

(1

+ FP(t

- t’)

=

- a P z ) ( i / h )( ( ( P l p ( t ) ( P l - P ( t ’ ) + ) .P“i/fi)

(l/CP)

(((PZP

( t )(Pz-P

(alat’>L--aP(l

+aP(1 -

(.P”l”“i/fi>

(t’)

I+),

- .Pz)l’“i/fi>

(27.40) (((PlP(Q(Pl-P(t’))+)

(((PZP(t)(Pz-P(t’))+)I,

(27.41)

300 GP(t

PHILIP C. K. KWOK

- t')

=

C l / ( c P > 2 1 C ~ P 2 ( i / f i > ((+lP(t)+l-P(t')

+ (1 -

aP2)

I+>

(i/fi) ((+2P(t)+2-P(t1))+)I

I+) + (1 - a p 2 ) (i/fi>( ( ~ 2 p ( t > ( ~ 2 - p ( t ' ))+)I 6 ( t - t ' ) 1.

= cl/(cP)211(a/at>(a/et'>Ca,"Vfi>

(((Plp(t)a-P(t')

+

(27.42)

The 6 function in (27.42) comes from the differences between ([+lp(t) b P ( t ' ) I+) and (slat) (a/at') ( ( a p ( t ) cpl,(t'))+) because of the equal time commutation relation [cpl, ( t ) , + I - ~(t) ] = in. Now, taking the frequency transform and using expression (27.31) for the polariton Green's functions, one obtains

Dp(on>=

1- ap2 -wn2

+

W2lP

+

aP2 -wn2

+

W2ZP

(27.43)

(27.45) For later use, we rewrite these functions in a slightly different way: (27.46)

(27.48)

301

GREEN'S FUNCTION METHOD I N LATTICE DYNAMICS

SUSCEPTIBILITY 28. DIELECTRIC We now calculate the frequency and wave vector dependent dielectric susceptibility function of the lattice in the presence of the retardation effect. Denoting the externally applied electric field by Eao(xt), the extra interaction term in the Hamiltonian isloo (28.1) We assume that the electric field Eois described by a wave vector p and frequency w, i.e.,

Ea0(x?,t )

=

E P o ( w ) 8 , ( t p )exp (ip-x? - i w t ) ,

(28.2)

where 8 is a unit transverse polarization vector. Then, in terms of the phonon normal coordinates, H I can be written as

HI

=

-W2z(p) Q,

(2)

EPo (0)exp ( - i d )

.

(28.3)

Q belongs to the branch t. As was discussed in Section 11, one now wishes to calculate the polarization per unit volume P, in the presence of Eo. From (25.9), one has

P m (~10,t )

=

(1/N1'2)C[z (p') /~01&a(t'p') Qt*pr ( t ) tfp'

X exp ( i p - x t ) .

Only the p'

=

p and t'

= t

(28.4)

terms contribute, and their magnitude is

Pp(t) = (l/N1")C z ( p ) / ~ ~ I Q t p ( t ) *

(28.5)

According to the derivation in Section 11, the expectation value of P,(t) in the presence of the external field also has same frequency dependence as Eo: (P, (t) ) E O = (P, ( w ) ) E exp ( - i w t ) , (28.6) with the amplitude (P,(w) ) E given by

(P,(w) ) E O = C . 2 ~ P > / ~ o l D P R ( ~ ) ~ P o ( ~ ) ~

(28.7)

D,"(w) is the phonon response function determined from the phonon Green's function by analytic continuation of the pure imaginary frequency, w, + w ie. The ratio P ( w p )/Eo(wp) is not the susceptibility, because the susceptibility is defined with the total electric field. The total (macroscopic) electric field consists of the external field and the induced internal field. Thus we must first obtain the retardation field. From (27.12), (27.19), and

+

lo0

See Section 11.

302

PHILIP C. K. KWOK

(27.22) , we obtain

( E p ( t ))EO

=

.

(4?rC2/Nffo)”2((a/dt)Ap(t) ) E O

-c-1

(28.8)

Its Fourier coefficient (Ep(u))EO is given by

(Ep(u))E”

= ‘b(4?r/ffo) ‘l2X (p) FpE ( -u)Epo(W )

(28.9a)

( E P ( ~)Eo)

=

- u > E ~ o ( w )7

(28.9b)

or iW?pFpR(

where FR is the photon-phonon response function (27.39) .I01 From (28.9), one immediately obtains the total electric field Ep”t(u) as Eptot(w) = (1

+ iOvpFpR( - w ) )

(28.10)

*Ep0(u).

Hence, the dielectric susceptibility is

x(uP) =

=

( P p ( ( J ) )Eo/Epht(u)

(1

+

(~‘((p)/~0)DpR(w)

hTpFpR(-u))-’.

(28.11) This expression can be explicitly evaluated by using the formulas derived for the various Green’s functions in the previous section. First we have, from (27.43),

DpR(u)= Dp(un4u = -u2

+ &)

+ c”p2/( -w2 +

+ c2p2) - u 2 ~ p 2(28.12) ,

( -u2

where the infinitesimal imaginary part of the denominator has been neglected. Then from (27.44) we get

(1

+ iOvpFpR(- u ) )

=

(1

+ iwvpFpR( -w,

Therefore,

4

+

X(WP) = c ~ “ P ) / ~ o l ( - ~ 2

--w

- i~))

(28.14)

(wPo)2)-1.

This expression is identical to the susceptibility function derived in Section 11, neglecting retardation effect. An alternate way of obtaining the susceptibility or the dielectric function

4WP) = 1

+ 4?rX(OP)

(28.15)

*

is the following. One works exclusively with the photon correlation or Green’s function Gp(a,).In vacuum, it is equal to Gp ( a n ) *01

=

( -an2

+~ ~ p ~ ) - ’ ~

F R ( w ) is the function FR(on)analytically continued from an to o

+ is.

(28.16)

303

GREEN'S FUNCTION METHOD IN LATTICE DYNAMICS

which may be derived from the Hamiltonian (27.10) of the free electromagnetic field or by putting qp = 0 in (27.48). When the coupling of the electromagnetic field or photon with matter is included, Gp(w,) can be represented as GP(wn)= [ - w n 2 c2p2 - ZP(wn)], (28.17)

+

where 2 , (a,) is the photon self-energy function or polarization function. From the structure of the Maxwell's equations for microscopic electromagnetic fields, one readily finds that the dielectric functions (in imaginary frequency) E(W,P) is related to the photon self-energy function by

+

E(W,P)

=

1

x (GP)

=

(1/4rwn2)Z, (u,).

or

(28.18)

(l/wn")p(wn)

(28.19)

I n other words, the photon Green's function may be represented by

Gp(w,)

=

+

[-E (W ~ P (J,' ) c2p2]-'.

(28.20)

A discussion of the derivation of this formula may be found in Abrikosou, Gorkov, and Dzyal~shinski.~ The complex dielectric function E (up) is obtained from E(w,~) by analytically continuing w, to w i ~ .By comparing (28.17) or (28.20) with (27.48), one immediately obtains the same result (28.14) for €(up) or ~(wp).

+

ACKNOWLEDGMENT The author wishes to express his deep appreciation to Professor Henry Ehrenreich and Dr. Peter Miller and Dr. Paul Marcus for their critical reading of the manuscript and many valuable suggestions.

This Page Intentionally Left Blank

Helical Spin Ordering-1 Theory of Helical Spin Configurations

TAKEO NAGAMIYA Department of Material Physics, Faculty of Engineering Science, Osaka University Toyonaka, J a p a n

Introduct,iori. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Elementary Theory of Helical Spin Ordering., . . . . . . . . . . . . . . . . . . . . . . . . . .

306 307 1. Simple Helical Spin Ordering at Absolute Zero.. . . . . . . . . . . . . . . . . . . . . 308 2. Molecular Field Theory for Finite Temperature.. . . . . . . . . . . . . . . . . . . . 310 .......................... 312 11. Spin Waves in the Screw Structure 111. Effect of Anisotropy Energy on Spin Configuration. . . . . . . . . . . . . . . . . . . . . . 316 3. Uniaxial Anisotropy Energy with an Easy Ax tudinal Component. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Uniaxial Anisotropy Energy with a n Easy Axis; Osc verse Component. . . . . . . . . . . . . . . . . . . . . . . . 5. Anisotropy Energy of Twofold, Fourfold, and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Easy Cone . . . . . . . . . . . . . IV. Effect of External Field on Spin Configurations. . . . . . . . . . . . . . . . . . . . . . . . . 330 7. Field Applied Perpendicnlar to the Plane of Spin Rotation.. . . . . . . . . . 331 332 8. Field Applied in the Plane of Spin Rotation.. . . . . . . . . . . . . . . . . . . . . . . . 9. Structure Changes in t8hePlane (No In Temperature) . . . . . . . . . . . . . . . . . . . . . . . . 10. Structure Changes with Anisotropy in the 11. Structure Changes of a Conical Arrangement 12. Experimental Observations. . . . . . . . . . . . . V. Spin Waves in Various Configurations in an Applied Field. . . . . . . . . . . . . . . . 348 349 13. Conical Arrangement; Field Parallel to z . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 14. The Fan with a Field Parallel to 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. Spin Waves in a Helix Subjected to a Weak Field VI. Complex Spin Configurations. . . . . . . . . . . . . . . . . . . . . . 16. Case 1: There I s a Single q That Is Equivalent to - q . . . . . . . . . . . . . . . . 363 17. Case 2: There Is a Single q That Is Not Equivalent to - q . . . . . . . . . . . 364 367 18. Case 3: There Are Two Wave Vectors, q and q’. . . . . . . . . . . . . . . . . . . . . 19. Lyons-Kaplan Theory. . . . . . . . . . . . . . . . . . . . 37 1 VII. Spin Configurations in Spinel Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 20. Crystalline and Magnetic Structur ............................ 377 ns . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 21. The NCel and Yafet-Kittel Configl 390 22. Multiple Cone Structure.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. NCel Temperature and Spin Ordering for Complex Lattices IX. Neutron Diffraction: Theory and Examples. . . . . . . . . . . . . 23. General Theory of Elastic Neut,ron Scattering. . . . . . . . . . . . . . . . . . . . . . . 396 . . 398 24. Examples of Helical Spin Configuration. . . . . . . . . . . . . . . . . . . . . 305

306

TAKE0 NAGAMIYA

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Al. Susceptibility of the Fan Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A2. Parallel Susceptibility of the Helical S t a t e . . . . . . . . . . . . . . . . . . . . . . . . . .

402 402 407

Introduction

The development of the study of helical spin ordering is a rather recent event. The first report on this subject appeared in 1958 at the International Congress on Magnetism in Grenoble, France, in a talk given by the present author. This report contained the prediction of a helical spin arrangement in MnOz , the theory having been worked out by A. Yoshimori in collaboration with the writer. Earlier observations on Mn02with neutron diffraction made by R. A. Erickson, which had given puzzling results, gave support to this prediction. It appeared that the report received little attention at that time. Since then, however, observations with neutron diffraction of satellite lines in chromium metal, made at Brookhaven, and findings at Oak Ridge of such lines in heavy rare earth metals that correspond to helical and modified-helical spin arrangements, as well as a study of MnAu2 under external magnetic field at Saclay, gave a good deal of impetus to a further development of the subject both theoretically and experimentally. In the meantime, the spin arrangement in chromium, which is sinusoidal, has come to be regarded as a spin density wave of the conduction electrons as proposed by A. W. Overhauser. In the present article, the writer has made an effort to give a complete description of the theory of helical and modified-helical spin ordering, confining himself, however, to molecular field treatments (except for a note in Section VIII) and spin-wave calculations. Modifications of a helical spin order arise from anisotropy energies and an external magnetic field. Also, a description of the theory of complex helical spin configurations in complex crystalline lattices is given. A basic assumption made is that there exist isotropic exchange interactions between atomic spin moments of further neighbors as well as between neighboring moments. The coefficients of these exchange interactions are assumed as given constants. In this sense, the spin system dealt with may be called the Heisenberg magnet. Experimental observations relevant to the theory are referred to as far as they are known to the writer and are useful to elucidate the theory. Part 2 of this article, in preparation, will deal with the role of conduction electrons in the exchange interaction, will review and discuss the observed magnetic and other properties of heavy and light rare-earth metals, and finally will discuss the spin density wave in chromium. Mention should be made that the original plan of writing this article was laid in 1963 while the writer was enjoying a stay in the Department of Physics of the University of California at Berkeley as the result of an

THEORY O F HELICAL S P I N CONFIGURATIONS

307

invitation extended by Professors C. Kittle and A. M. Portis. The writer would like to take this opportunity to express his warm thanks to these individuals. The text presented here deviates almost completely from the writer's lecture notes at Berkeley. Part 2 however, will cover the latter to a large extent. 1. Elementary Theory of Helical Spin Ordering

I n a ferromagnet that consists of atomic spins coupled with each other by exchange forces, the spin vectors are aligned parallel to each other, and there are spin-wave excitations. A spin wave is a motion such that a spin at a position R, precesses about the direction of the alignment with a phase angle q.R, - wt, m-here q is the wave vector of the spin wave and w its frequency, so that the spin vectors describe a t each instant a helix in space. In order that the ferromagnetic state be stable, w2 must be a positive function of q. If, however, w vanishes for a certain q, one may imagine that the spin wave with this q would stand still with an arbitrary amplitude. One may further ask what would happen if w2 were negative in a certain range of q. The ferromagnetic state would then be unstable, and a helical ordering of the spins with the maximum possible amplitude would be realized as the stable state. Yoshimori' looked into this problem in detail and predicted the existence of a helical ordering of spins in rutile-type crystals having magnetic ions as cations and nonmagnetic ions as anions. He analyzed the neutron diffraction lines of MnOz observed by EricksonZRand found that its spin structure is helical; the spins in the same c-plane of this crystal point in the same direction perpendicular to the c-axis, and this direction turns from one plane to the next by an angle of 129" ( = 5?r/7).It had already been pointed outzbthat in MnF2 , which is also of rutile structure and has a collinear antiferromagnetic spin arrangement, the spin-wave frequencies become negative in a region of q-space for H certain range of exchange constants. Soon after Yoshimori published his paper, Kaplan3 proposed a simple theory of helical ordering to interpret the neutron diffraction lines from chromium, and Villain4 also predicted helical ordering by a treatment of molecular field. Although the three papers have common features, Y oshimori's paper has been worked out most extensively, including discussions of anisotropy energy, susceptibility, the spin-wave spectrum, and neutron diffraction. Yoshimori confined himself to rutile-type crystals, A. Yoshimori, J . Phys. Sor. Japan 14, 807 (1959); T.Nagamiya, Proc. Colloq. Intern. MagnStisme, Grenoble, 1968, p. 15 [or see J . Phys. Radium 20, 70 (1959)l. R. A. Erickson, private communication (1957) (cited in Yoshimori'). zb T. Nakamura and 0. Nagai, private communication (1957). T. A. Kaplan, Phys. Rev. 116, 888 (1959). J. Villain, Phys. Chem. Solids 11, 303 (1959).

308

T A K E 0 NAGAMIYA

but his theory is applicable to all lattices with one magnetic atom per unit cell. 1. SIMPLEHELICALSPINORDERINGAT ABSOLUTEZERO

Consider a lattice of magnetic atoms, such as that of manganese in MnF2and Mn02, in which the unit cell can be so chosen that it contains only one magnetic atom. We put aside anisotropy energy for a moment and consider only exchange forces. On each magnetic atom we assume a classical spin. Between spins S,,,and S n at positions R, and Rn , an exchange potential of the form

(Rmn = Rm - Rn) (1.1) will be assumed, where J ( -Ln)= J ( h n ) .It is essential that this exchange potential is not confined to nearest neighbors in order that we obtain a helical ordering, as we shall soon see. We make Fourier transformations of the exchange coefficients and spins : -W(%n)Sm.Sn

J ( q ) = C J ( R n ) exp ( - i q * R n ) ,

(1.2)

n

S, = N-'12

C Sn exp ( - i q . R , )

(S-, = S,*),

(1.3)

n

where we denote by N the total number of atoms and we assume that one atom is at the origin and J ( R , ) = 0 for R, = 0. It is easy to see that J ( -q ) = J ( q ). The total exchange energy

We look for the lowest minimum of (1.4) or (1.5) subject to the condition Sn2 = const = S2 for all n. Instead of this condition, we shall first impose a milder condition ~ , 2= const, (1.6)

Cn

which can be written with Fourier components as

C s,*s-,

= const.

9

Under this milder condition, the minimum of (1.5) is attained simply by taking only that q for which J ( q ) is the highest maximum. Denoting this q by Q ( q = -Q being equally allowed), we have the minimum value

THEORY OF HELICAL SPIN CONFIGURATIONS

of (1.5) as4= -J(Q) (SQ-S-Q

309

+ S-Q-SQ).

(1.8)

+ S-Q exp (-iQ.R,)].

(1.9)

Also, we obtain from (1.3)

S,

=

N-1’2[S~exp (iQ-R,)

This equation can be written in components as

Snz

=

A cos (Q*Rn

+ a),

+P), Snz = C cos (Q.Rn + 71,

Sng

=

(1.10)

B cos (Q.Rn

where A , B, and a, p, y are arbitrary constants. Equations (1.10) represent a general elliptic helical ordering of spins with wave vector Q: that is, the spin vector S, rotates and describes an ellipse as the position R, advances in the direction of Q. Now, to satisfy the conditions Sn2 = const, this ellipse must be a circle. Taking the z-axis perpendicular to the plane of the circle, we have, instead of ( l . l O ) ,

S,,

=

+ a), Ssin (Q-R, + a),

s,,

=

0.

S,, = SCOS (Q*Rn

(1.11)

The corresponding energy, (1.8), is calculated, with the use of ( 1.3), to be

-NX2J(Q) .

(1.12)

Yoshimori called the spin ordering represented by (1.11) the screw structure of spins, since screw means a combination of rotation and translation. Generally, (1.11) represents an oblique screw structure, as the direction of Q is not necessarily perpendicular to the rotation plane. The relative orientation of the screw axis and the rotation plane may be fixed by an anisotropy energy; for instance, when there is an easy plane of magnetization in the crystal, the spins will rotate in this plane, whereas the direction of Q may be determined by an anisotropy in J(R,,). The structure in which the rotation plane is perpendicular to Q may be called the proper screw structure and the structure in which they are parallel the cycloidal structure (after Yoshimori) . A characteristic feature of the screw structure, or the helical arrangement of spins, as it is more commonly called, is that the period of the arrange4a

When Q = 0 or Q is a vector a t such a special point on the Brillouin zone boundary that Q and - Q are equivalent vectors, there is an additional factor on the right-hand side of (1.8) and of (1.9). I n this case, too, (1.12) is valid.

+

310

T A K E 0 NAGAMIYA

ment is, in general, incommensurable with the lattice period, since the magnitude of Q is determined solely by the exchange coefficients. Example: Consider a layer crystal with interlayer spacing c. The direction of Q will be assumed to be perpendicular to the layers. We sum J (Rmn)over all sites m in the layer that contains the site n and denote this sum by JO ; the sum over a neighboring layer we denote by J1 ; for a next-neighboring layer, we define J z , and so on. Then, for a q that is perpendicular to the layers, we can write

c J , exp co

J(q) =

( -iucq)

v=-m

=

Jo

+ W Icos (Cq) + 2J2 cos ( 2 ~ q +)

If we retain only Jo , J 1 , and J2 , we have from J ( q ) equation with q = Q :

[JI

=

*

..

(1.13)

max the following

+ 4 J 2 cos ( c Q ) ] sin (cQ) = 0.

This gives solutions Q = 0 (ferromagnetic spin arrangement), Q = a/c (antiferromagnetic spin arrangement) , and a helical arrangement having Q given by when I J1 I < I 4J2 I. (1.14) cos (cQ) = -J1/4J2 = Jo - (J?/4J2) - W Zcal2J1 ZJZ and culated from (1.14) is greater than both J ( 0 ) = J o J ( a / c ) = Jo - 2J1 2J2,provided that J z is negative. Thus, when Jz < 0 and [ J 1 I < I 4Jz I, the helical state has the lowest energy. When J 2 is positive, the ferromagnetic or antiferromagnetic state is more stable according as J 1is positive or negative; this is because spins on nextrneighboring layers wish to be parallel or antiparallel, depending on the sign of J 1 . On the other hand, for sufficiently large negative J 2 , there must be a compromise between the forces acting between next-neighboring layers to make spins antiparallel and the forces acting between neighboring layers to make spins either parallel or antiparallel, and as a result one has a helical arrangement.

It can be seen easily that the value of J ( Q )

+

+

+

2. MOLECULAR FIELD THEORY FOR FINITE TEMPERATURE The effect of a finite temperature, T,may be considered with the approximation of the Weiss molecular field. Denoting the thermal average of S, by (Sm),we can write the exchange magnetic field acting on S, as Hex,,

= -(~PB)-'

2 c J(Rmn) ( s m ) .

(2.1)

m

Here g is the spectroscopic splitting factor and

pB

= efi/2mc

the Bohr

311

THEORY O F HELICAL SPIN CONFIGURATIONS

magneton. When there is, further, an external field H, the effective field acting on S, is given by H Hex,, . The thermal average of S, must point in the direction of this field, and its magnitude is given by

+

I (S,) I

=

Sun

=

2 m exp (wBm I H + f exp (gpBm ] H +

Hex,,

I/kT)

m=--s

Hex,,

1

(2.2)

I/kT)

m=-S

namely,

Xn =

gpBS I H

+ Hex,, l/kT,

(2.4)

where Bs(x) is the Brillouin function. When H = 0, we may anticipate a helical ordering as a solution of Eqs. (2.3) and (2.4), and we may put un = const = u. Then, (2.3) and (2.4) become

x = gPBS 1 Hex (/kT, (2.5) where, from (2.1) and (S,) = XU exp [i(Q.Rm a)], (S,) being expressed as a vector in the complex plane, one has u =

Bs(x),

+

I Hex I

=

( g p ~ ) - l 2I CJ(Rmn)Su exp [i(Q*Rm m

=

(gpB)-l2

+ a)] I

I C J(Rmn)Sa exp ( i Q * L ).exp [i(Q.Rn -I- a)] I m

=

(gpB)-l 2J(Q) Xu.

(2.6)

Hence, from (2.5) follows u =

+

Bs[2J(Q)X2u/kT].

(2.7)

Since Bs(x) = [( 8 1) /3X]x - 0(x3) for small x, this gives immediately the NBel temperature by

TN = [S(X

+ 1)/3k]2J(Q).

(2.8) Here Q has to be such that J ( Q ) is maximum (and positive), since then this equation gives the highest, and thus the real, N6el temperature. Below this NBel temperature, u is given as the solution of (2.7). It varies with T like the saturation magnetization of a ferromagnet. Above the NBel temperature, the thermal average of each spin vanishes for no external field. For a finite field, the thermal averages of the spin vectors should be equal and parallel to the field direction; hence we put Q = 0 in (2.6) and obtain from (2.3) and (2.4) =

Bs([gpBSH

+ 2J(O) S2u]/kT),

(2.9)

312

TAKE0 NAGAMIYA

which determines u as a function of H and T. For small HIT, we have u =

[(S

+ 1)/3SkT][gp~SH 4-2J(O)S%],

so that

x

= gp,Su/H

c = (gpB)'S(S

=

C/(T - 0),

+ 1)/3k,

,

(2.10)

ep

=

2[S(S

+ 1)/3ky(O)i

(2.11)

where x is the susceptibility. The effect of an external field below the NBel temperature will be discussed in Section IV. II. Spin Waves in the Screw Structure

We shall discuss here the modes and frequencies of spin waves that can be excited in the proper screw structure.'*5Jj We shall assume the screw axis to be in the z-direction so that the spins rotate in the zy-plane as the position advances in the z-direction, as expressed by (1.11). We can introduce a n anisotropy energy for each spin of the form DS,2 with positive D , without disturbing the assumed proper screw structure. This anisotropy energy gives an effective field of -2DS, in the z-direction for each spin. For convenience of calculation, we shall introduce a local coordinate system [, 7, { in such a way that the {-axis coincides with the equilibrium spin direction at each lattice point, the [-axis is perpendicular to this direction in the zy-plane, and the 7-axis is parallel to the z-axis. Then the relative orientation of the [, &axes at R, and those at R, is such that the former is rotated from the latter by an angle of Q-Rmn. We have, therefore, the coordinate transformation {n

=

lm cos (Q. R,,) - $,. sin (Q-R,,) ,

.$n

=

lm sin (Q * R,,)

+ tm cos (Q .Rmn).

(11.1)

For small spin oscillations, the local {-component of each spin can be regarded as a constant, S, and the equation of motion (torque equation) can be written as

(11.2) where we have omitted a factor -gpB in the right-hand side. {, f stand for {, , f , so that S,r = S, and H n ~H,, HnIare the three components of the effective field acting on the nth spin. This effective field consists of 6

K. Yosida and H. Miwa, J. A p p l . Phys. 32,SS (1961). T.A. Kaplan, Phys. Rev. 124, 329 (1961).

THEORY OF HELICAL SPIN CONFIGURATIONS

313

the exchange field given by (2.1), the thermal average sign being dropped, and the anisotropy field -2DS,,, , namely,

Hnr

C zJ(Rmn)[Smrcos (Q-Rmn) - S m f sin (Q.Rmn)], = C 2J(Rmn)[Xmrsin (Q-Rmn) + Smc cos (Q.Rm,)], =

m

Hnc

(11.3)

m

H,,

=

C .2J(Rmn)Sm,,- 2DSn, . m

A factor by

was omitted in the right-hand side, and we understand to H , and Smthe respective local coordinates, i.e., f, , E m , respectively. Replacing Snr and Smr by S in (11.3) and

- (gpg)-l suffixes f and 5

5, and f m , writing

Snc

+ is,,,

S,C-

(2S)112a,

=

)

is,,,=

(2S)1/2an*, (11.4)

we have from (11.2) and (11.3) the following equation, after neglecting second-order terms in a and a*: fiu, = -2iSa,[

C J ( R m , )cos (QsR,,)] m

+is

C J ( R m n )( a m m

+

am*)

cos (Q-Rmn)

+is C J ( R m , )(am - am*) - iSD(an - a,*).

(11.5)

m

A similar equation in which a and a* are interchanged and i is replaced by -i is also obtained. These equations are further simplified by making the Fourier transformation: aq =

N-1/2

a, exp ( - i q . R , ) , n

a*,

=

N-1'2C a,* exp ( - i q . R,) .

(11.6)

n

Observing that exp ( - i q . R , ) = exp(iq-R,,) .exp ( - i q . R , ) ferring to (1.2), we obtain from (11.5)

and re-

Aciq = -2iXaqJ(Q)

+ 4iS(aq + a%)CJ(Q + q> + J ( Q - d l

+ iS(aq nu?,

=

-

a*,)[J(q) - 01,

+2iSa?,J(Q)

-3is(aq

+ &)CJ(Q + q ) + J ( Q - q > l

+iS(aq - d , ) [ J ( q ) - D ] .

(11.7)

314

TAKE0 NAGAMIYA

Putting a, = -iw,a, and a?, = -iw,u?, and solving the resulting homogeneous equations for a, and a?,, we finally obtain the frequency (11.8) and the amplitude ratio (a,

+ a*,)/(% =

- a?,) W(Q) - J(q)

S,E/.L‘Sq, Dl”’//W(Q> - V(Q

=

+

+

(I>- + J ( Q

- dY2. (11.9)

I n (11.9) , S,E and S,, are the Fourier transforms of SnE and S,, , respectively. I n this way, we have obtained formulas for the spin waves to be excited in the proper screw structure. oq given by (11.8) vanishes for q = 0, in which case S,, also vanishes, as can be seen from (11.9), and the mode of oscillation is such that the whole spin system rotates as a rigid body about the screw axis. Another special case of interest is q = =tQ when D = 0, in which case, too, the frequency vanishes and S,E also vanishes. I n this case, the spins oscillate with phase angles q.R, in the direction parallel to the screw axis; this implies a small canting of the plane in which the spin vectors rotate. If D is small but not zero, the spin waves with q = k Q have a small but finite frequency, and the spin vectors oscillate elliptically with a small amplitude perpendicular to the screw axis and a large amplitude parallel to it. I n this case, there is a n oscillating component of the total spin perpendicular to the screw axis, so that the mode should be active to a n imposed oscillating electromagnetic field. This can be seen in the following way. Since the x-component of the oscillating part of the a ) , the total z-component can be calculated nth spin is -SnE sin ( Q R, to be

+

n

n

- N-1/2Sqg exp

=

+ a)

(iq-R,) sin (Q-R,L

n

=

~ ( i / 2 ) N ~ 4 exp S , ~(=&a)

for q

=

k Q . (11.10)

Thus, the amplitude of the total z-component is +N1/2I S,E 1, which is nonvanishing for D # 0. I n the Holstein-Primakoff formalism, the spin operators are written with annihilation and creation Bose-operators, a, and an*,as

Snt

+ is,, = (2S)”2(1 - ~ n * ~ n / 2 S ) ”,~ ~ n

S,c - is,, = (2S)l/2an*(l - u , * u , / ~ S ) ~ ~ ~ , Snr = S - an*an

,

(11.11) (11.12)

315

THEORY OF HELICAL SPIN CONFIGURATIONS

where

[a,, a,*]

=

1.

(11.13)

This way of writing spin operators in terms of a, and a,* is exact so long as all the states with the number of Bose particles greater than 2 s are disregarded. If the factor (1 - a,*~,/2S)'/~is approximated by 1, Eqs. (11.11) reduce to Eqs. (11.4). From (11.6) and (11.13) follows also

[a,, a,*]

=

1.

(11.14)

The procedure of solving Eqs. (11.7) is equivalent to transforming a, and a,* to a new set of variables (the normal coordinates), a, and a,*:

+ a', sinh e, , a,* = a,* cosh e, + sinh 0, , coth20, = ( A + B ) / ( A - B ) , a,

where

A B

= a,

= J(Q) =

cosh e,

+ D,

- J(q)

J(Q) - +J(Q

+ q) - + J ( Q - q ) .

(11.15)

(11.16) (11.17)

T o see the foregoing, one may notice first that [a, 7

%*I

=

1,

(11.18)

which follows from cosh2 0, - sinh2e, = 1. Second, from (11.9), the transformation coefficients cosh 0, and sinh 0, have to be in the ratio cosh 0,/sinh 0,

= (All2

+ B1/2)/(A1/2- B1l2),

+

from which follows coth 28, = ( A B)/(A - B ) . The thermal average length of each spin in the helical ordering can be calculated in the following way. By (11.12) and (11.6),

(S,r)

=

S - (a,*a,)

=

S - N-l

(aq*aqr)exp [i(q' -. q).Rn], q,ql

which can be written with a, and a,* as

S - N-l

((aq*cosh 0,

+ a-, sinh 0,) (aqscosh Oqt + a?,* sinh eqt))

,,,I

exp [i(q' - q)-Rn]. Since there is no phase relation between spin waves with different q's, terms in this expression other than those with q = q' vanish; in other words, given a set of boson occupation Gumbers, { n,] , where n, = aq*aq, the traces of aq*aqtand a-,aT,, vanish for q # q', and the traces of a,*a_TqI

316 and

TAKE0 NAGAMIYA

a-,aqPall

vanish. Thus, we are left with

S - N-l

[ ( n , ) cosh28,

+ ( (n,) + 1) sinh2O,],

9

or S - N-I

[( ( n , )

+ 3) cosh 28, - 31,

(11.19)

P

where

(n,) = [exp (&w,/kT) - 11-I.

(11.20)

At low temperatures, only those spin waves that have small wq are important. Confining ourselves to this case and assuming that D is nonvanishing, we may consider only small values of q. If q is small, coth 28, can be approximated by 1 2 ( B / A ), since A and B can be approximated as

+

A

=

J(Q) - J(0)

+ D,

B

=

-3 C [a 2J(q)/aqiaqjl IQ i,i

X qiqi ,

where i,j = 2, y , z, so that B is small (positive). cosh 28, is then approxi~ . we obtain finally for the thermal average mately equal to ( 4 B / ~ l ) - "So length of each spin

S - (16dN)-'

=

///

(cosh 28, - 1) d3q - (8a3N)-'

j-1

(n,) cosh 28, d3q

const - (87r3N)-'

(11.21)

The constant term represents the spin length contracted by zero-point motion, and the second term is proportional to T2, since fiw, = 2S(AB)l'2 and B1'2 is proportional to I q

I.

111. Effect of Anisotropy Energy on Spin Configuration

I n this part, we shall be concerned with the alteration of a helical spin configuration by anisotropy energy. First, we consider an anisotropy energy with uniaxial symmetry giving rise to an easy axis of spin orientation, second, an anisotropy energy of twofold, fourfold, or sixfold symmetry giving rise to a sort of easy plane, and third, a more complex anisotropy energy giving rise to an easy cone. We assume that all these anisotropy energies are of one-atom type originating in the crystalline electric field acting on individual atoms and perturbing their spin states. Other anisotropy energies, such as magnetic dipolar,' electric q~adrupolar,~-* and 8

R. J . Elliott, Phys. Rev. 124, 346 (1961). H. Miwa and K. Yosida, Progr. Theoret. Phys. (Kyoto) 26, 693 (1961).

317

THEORY OF HELICAL SPIN CONFIGURATIONS

anisotropic exchangeg interactions will not be considered. The theory of the effect of anisotropy energy on the helical spin configuration has been developed by Kaplan,6 Elliott,’ and Miwa and Yosida.8 They explained successfully the various spin configurations and phase changes observed in heavy rare-earth metals. The description given here is somewhat different from theirs and somewhat more general, although essentially the same. 3. UNIAXIAL ANISOTROPY ENERGY WITH THE LONGITUDINAL COMPONENT

AN

EASYAXIS; OSCILLATION IN

The easy axis will be denoted as z. The simplest form of this type of anisotropy energy may be written as

w(S,) = D[S2 - + S ( S

+ l)],

(3.1)

with negative D, which vanishes for S = 3, however. The general form of the anisotropy energy to be considered can be written as a n even polynomial of S , of degree at most equal to 2S.9aWith this general function, w (S,), we shall proceed to the approximation of the Weiss molecular field. The exchange field acting on the nth spin is given by (2.1). For brevity, we shall omit the factor -(gpB)-l from it (and consequently the factor -gpB from the magnetic moment) and also eliminate the suffix “ex.” Then, it is given in components as

Hni

=

2

CJ ( R m n ) S U m i ,

Sum: = (S,i>,

i

(3.2)

= 2, 2, y.

m

The equations to determine uni (i

=

z, 2, y) are

where 6 = l/kT and “tr” means “trace.” To find the NBel temperature and the behavior of the spin arrangement below but close to it, we assume the exchange field to be small, and we expand the exponential function in (3.3) in powers of it. If we set H,, = Hnu= 0, a simple calculation yields the following result for the z-component : Xunz =

+ + -

+

PH,, tr S,2 exp [-Pw(Sz)] QP3H,2t r Sz4 exp [-awl t r exp [-Pw(S,)] $P2HnZ2 t r S,2 exp [-Pw]

+

*

(3.4)

If we retain only linear terms, we have simply Sun. = PHnz tr Sz2exp [-Pw(SZ)]/tr exp C -Pw(SZ)], T. A. Kaplan and D. H. Lyons, Phys. Rev. 129, 2072 (1963). Any function of S, takes 2S 1 eigenvalues and thus is expressible as a linear combination of 25 1 bases, for which we may take 1, S, , SZ2, , 529.

95

+

+

---

318

TAKE0 NAGAMIYA

and substituting (3.2) into this we obtain a system of linear homogeneous equations for u,,'~. So we put anz

= ~z

cos (q.Rn

+ a)

1

and utilize the definition of J( q) , Eq. (1.2). With the relation R, = Rmn

we then have

+ Rn

exp [-Pw]. Eliminating uz from both sides, we are left with the equation to determine the NBel temperature for the z-component of the spins. Since the last factor (the quotient of traces) is a decreasing function of T, being equal to S2 for T = 0 and to $ S ( S 1) for T = co , the highest NBel temperature, i.e., the real NBel temperature, is obtained for the maximum of J(q).So we put q = Q and have uz = 2PJ(q)uZtr Sz2exp [-pw]/tr

+

~ T = N 2J(Q) ( t r S 2 e x p [-w(S,)/kT~]/trexp [-w(s,)/kT~]). (3.5) If one assumes w ( S - 1) - w (S) >> ~ T ,Nthis equation reduces to

~ T = N W(Q)S2,

(3.6)

and in the opposite case to

For w = DSz2,or (3.1), we have for (3.7) the following:

(3.8) As we shall see later, a similar calculation for the z or y component leads to another NBel temperature, which is, however, lower than that just calculated. Therefore, below TN only the z-component, oscillates, down to another critical temperature. I n this temperature range, the oscillation of the z-component can be determined from Eq. (3.4) by putting H,, = H n , = 0. Immediately below TN we can put unz

=

uz

cos (Q-Rn

+ a),

+

(3.9)

H,, = 2J(Q)Su, cos (Q*Rn a). (3.10) More precisely, because of the presence of H:= and higher-order terms in (3.4), we have to include the third and higher odd harmonics in both unr and H,, . However, since with terms up to H:, we can determine only the amplitude of the fundamental, we neglect terms beyond H i , in (3.4), substitute (3.9) and (3.10) therein, and pick up only terms of the funda-

THEORY O F HELICAL SPIN CONFIGURATIONS

mental. Then we solve the equation to determine be written up to H:z as

uz

. Equation

319 (3.4) can

PHnzpl ( T ) - QP3HiZp3( T ),

(3.11)

t r S2 exp [-pw]/tr exp [-Pw], 3(tr 52 exp [ - p ~ ] ) ~ / l ( t rexp [-PW])~

(3.12)

- t r S,4 exp [-Pw]/tr exp [-awl.

(3.13)

Scnz =

where

T) p3(T) pl(

= =

Proceeding as mentioned, we obtain, using (3.5),

[

‘2:))]

(

-(kT)3 -~ * ( T N- T ) . (3.14) [J(Q)SIZ p3(T)dT T=TN Thus, uz varies as (TN - T )112, as might be expected. It can be shown that the amplitude of the third harmonic varies as ( TN - T )3/2, that of the fifth harmonic as (TN - T)6’2,and so on. Eventually, if the period of the oscillation is equal to seven layers of atomic plane, as in rare-earth metals Er and Tm, the fifth harmonic is equivalent to the second harmonic and the seventh one equivalent to the zeroth one; the zeroth harmonic is the ferromagnetic component of the whole system, and, it increases with decreasing temperature as ( TN - T)’I2. =

4. UNIAXIAL ANISOTROPY ENERGY WITH THE TRANSVERSE COMPONENT

AN

EASYAXIS; OSCILLATION IN

We now ask about oscillations in the x or y component. For this purpose, we return to Eq. (3.3) and consider the equation to determine unz . To first order in Hn, and Hn, , this equation can be written asgb

Sum = t r (8% exp P[HnzSz - w(S,)]

/

B

exp ( -A> CHnzSz - W ]

0

.(HnzSz + HnUSy) exp X[HnzSz - W ] dA)/tr exp P[HnzSz - w]. 9b

For noncommutative HOand H1, we have to first order in H I expC-S(Ho

+ HI)]

=

exp(-pHd - exp(-pHo)

+

exp(XH0)Hl exp(-XHo) dX. jo8 exp( -pHo)G(p), differentiating both

This follows by putting exp[ - p ( H o H1)] = the sides of this equation with respect to p, thus obtaining -Hl exp(-pH~)G(p) = exp( -oHo)G’(p), i.e., -exp(pHo)H1 exp( -pHO)G(p) = G‘(p), and then integrating G’(p) with the initial condition G(0) = 1. Thus, we have G(p) = 1 = 1 -

exp(XHo)Hlexp(-XHo)G(X) dX jo8 /oBexp(XHo)~lexp( -AHo) dX

+ o(H12).

320

TAKE0 NAGAMIYA

It can be shown that the term proportional to H,, vanishes. So we have, with (3.2) , linear homogeneous equations for unz. The preceding equation can be written, after a series of elementary calculations, in the following form : Sun, = +Hnz

5 { (S - S , +

1) ( S

+ 8,) expP[HnzSz - w(S2)]

S,S+l

+

1 - exp (-P)[Hnz w ( S 2 - 1) - w(S,)] Hnz w(Sz - 1) - w ( S z )

+

(4.1)

+

If we assume H,, w(S, - 1) - ~ ( 8 ,<< ) IcT for all the S , values, the factor in the second line reduces to 0. Then, in the approximation of neglecting the third and higher orders of P[H,,S, w(X, - 1) - w(S,)], Eq. (4.1) is written as

+

Sun, = +PHnz(trexp P [ H n z S z - w(SZ)]}-'

+ 1) - Sz21exp P C H n z S z - w ( X A 1 - (P/4)[w(S, + 1) + w(S2 - 1) - 2w(S,)1 + (P2/12) - ~ ( 8 2+) w ( S z - 1)12+ - w(8.z + 1) + w(SA12)1 - (P/4) 8, exp P C H n z S z - w(S,)] ( 2 H n z - w(Sz + 1) + w(Sz 1) - (P/3) -~(8,) + w(Xz - 1)12- - w(Sz + 1) + (4.2) X tr (CX(S

(CHnz

[Hns

-

[Hnz

([Hnz

~ ( s z ) ] ~ ) } ) .

It can be seen easily that (4.1) and (4.2) are even functions of H,, . H,, is a periodic function of R, with wave vector Q according to the preceding discussion. Hence either (4.1) or (4.2) may be expanded in a Fourier series with 2Q as the fundamental wave vector. I n these equations, H,, can be written by (3.2) as

and, in order to solve the equations, urn,and u,, may also be expanded in Fourier series with Q as the fundamental wave vector. If we retain the first two terms in the right-hand side of (4.2) and expand it in powers of H,, , the zeroth-order term will become $pH,, times tr [ S ( S

+ 1) - S2] exp [-Pw(S,)I/tr

exp [-Pw(Sz)l,

(4.4)

321

THEORY OF HELICAL SPIN CONFIGURATIONS

and the second-order term (@HnZ)( $bzHnz2)times

+ 1) + [ S ( S + 1) - SISzz- S21 exp (-Pw> x [tr exp ( -PW)]-~- t r { S ( S + 1) - S21 exp (-Pw) 1

tr(+S(S

x

[tr 82 exp ( -Pw> I/[tr exp ( -Pw> 7.

Abbreviating [tr Sz2exp (-pw)]/[tr [tr X i exp ( -pw)]/[tr

(4.5)

exp (-Pw)]

=

(S,2),

exp ( -Pw)l

=

(S2b ,

and

we can write (4.4) and (4.5) as

+ 1) - (Sz2)w fS(S+ 1) - S(SZ2)W + ( ( S 2 ) w ) 2 - ( S 2 ) w . S(S

and

(4.4a) (4.5a)

It is remarked here that (4.4a) is smaller than 2(S,2)w, i.e., 3(S2)w and that (Si),

> S ( S + 1)

> ((Sz2)w)2,

so that (4.5a) is a negative quantity. The first inequality is obvious, since w (8,) is a decreasing function of S 2 (the z-axis is the easy axis), and the second inequality follows from

[tr 8 2 exp (-pw)][tr =

exp (-Pw)]

3 C C ( ~ +2~

- [tr

exp ( -PW

~ ' 4 )

S2 exp ( -Pw)I2

- PW')

- C C S,2~~'2

8 s SZ'

sz

SZ'

X exp (-Pw - Pw') =

3 C C ( S z 2- X z ' z ) 2 exp ( -PW

- pw')

> 0,

8,

where w = w(S,) and w' = w(S,'). Using (4.3), (4.4a), and (4.5a), we can write (4.2) as unz

=

+ 1) - ( 8 , ~ ) w I + 3P2Hnz2[+S(S + 1) - g ( i 3 2 ) w + ( ( S 2 ) w ) 2 - (S14)w]).

P CJ(Rmn)umz([S(S m

(4.6)

To solve (4.6) approximately, we substitute (3.10) and expand u,, in a Fourier series : m

uy exp (ivQ-R,)

u,, =

where

u-,, = uv*.

(4.7)

-03

Then we obtain a set of linear relations between the Fourier coefficients as

322

TAKE0 NAGAMIYA

follows:

+

p - l ~= y J ( V Q ) U ~ U+~J [ ( v - 2 ) Q ] ~ , - ~ a ~ e x(2ia) p

+ V [ ( V+ 2) Q]u,+~azexp ( -2ia) , S ( S + 1) + uz = P[J(Q)SU~]'[C~S(S + 1) - s(82% + ( ( S , 2 ) W l 2

(4.8)

where a0

=

(SZ2)W

(4.9)

a2,

-

(sz4)W]. (4.10)

The infinite determinant of the homogeneous equations (4.8) should vanish, giving eigenvalues of f1. Since it was assumed that J(q) takes its largest value for q = .tQ, the largest eigenvalue of pl(= kT), which would give the second NBel temperature, is obtained mainly from the coefficients of and a l . Neglecting other coefficients, we have from (4.8) the following two equations : kTa-l = J(Q)a-lao kTal = J ( Q ) mao

+ $J(Q)u1a2exp ( - 2 i a ) ,

+ &J(Q)

u-1u2

(4.11) exp (2ia).

Thus, =

0,

Since a2 is a negative quantity as mentioned before, we have to take the lower sign, and this determines the second NBel temperature, TN', where the x-component of the spins starts to oscillate. Thus, the equation to determine TN' is (4.12) ~ T N=' J(Q) (a0 - +ad, where a. and a2 are defined by (4.9) and (4.10) , the temperature at which these are evaluated being TN'. The main term in the right-hand side of (4.12) is J ( Q ) [ X ( S 1) - (S,2)W],and when we compare this with the right-hand side of Eq. (3.5), which is W ( Q )(Sz2)W, we see that TN' is lower than TN . Furthermore, from (4.11) we have the ratio of u1 and a-1 to be -exp (ia) to exp ( - i c y ) , which means that a,, is proportional to sin (Q-R, a). I n other words, ordering in the x-component of the spins sets in below TN' in regions where the z-component is disordered, as the latter is given by (3.9) and is proportional to cos (Q R, a).

+

+

. +

323

THEORY O F HELICAL SPIN CONFIGURATIONS

If we neglect

a2

completely, we have for the second N6el temperature

~ T N=’ J(Q){ $ S ( S In the case of w

=

+ 1) + tr[S,2

- $S(S

+ l)]w(S2)/(2S + l ) k T ~ ’ } . (4.13)

+ l)], this becomes

D [S 2 - $X(X

+

~ T N=‘ ZJ(Q)$S(S 1)

D 4 X ( S + 1) - 3

}.

30

(4.14)

Comparing (4.14) with (3.8), we see that TN’ is lower than T N, since D is negative. The neglected terms will further lower TN’,and in the case of a large anisotropy energy the second NBel temperature might not exist. Examples of the theoretical results of Sections 3 and 4 are found in rare-earth metals Er and Tm. Neutron diffraction experiments have shown that in Er below 85°K and Tm below 56°K a longitudinal sinusoidal spin ordering sets in along the hexagonal axis (the crystal being hcp) with a period of seven hexagonal layers. In Er, the perpendicular component begins to oscillate a t 53°K and, furthermore, below 20°K the spin structure becomes conical. I n Tm, the perpendicular component never starts to oscillate, presumably because of a large anisotropy energy, and with decreasing - - -) , temperature the spin structure tends to a repetition of (+ four layers of up-spins and three layers of down-spins. Thus, in Tm the phase constant a,which was arbitrary in the foregoing arguments, becomes fixed in such a way that nodal planes do not coincide with atomic planes. According to experiment, a ferromagnetic component is perceptible below 40°K. The integral period of seven layers is evidently favorable for such to occur, and conversely the spin system seems to be stabilized by assuming an integral period and adjusting a so as to lower the energy.1° More discussions of rare-earth metals will be given in the forthcoming Part 2.

+++

5. ANISOTROPY ENERGY OF TWOFOLD, FOURFOLD, AND SIXFOLDSYMMETRY WITH AN EASY PLANE

We consider an anisotropy energy of p-fold symmetry ( p = 2, 4, 6) about the z axis. This anisotropy energy is assumed to make the xy plane the plane of easy magnetization. The simplest form of such an anisotropy Hamiltonian may be written as

+

~ ( 8 , )const.[(S,

+

i ~ , > p

+ (8%- iS,>pI,

(5.1)

where w(S,) is assumed to be the smallest for the smallest value of S,2. The constant in the second term should be small. This second term vanishes lo

T. Nishikubo and T.Nagamiya, J . Phys. SOC.Japan 20, 808 (1965).

324

TAKE0 NAGAMIYA

for S < p / 2 . I n the case of p = 2 , this term reduces to const. (SZ2- SY j l and in this case we can expect results similar to those that we have found in the preceding two sections, namely, below a certain NBel temperature, TN, only the y component oscillates (if the const is positive), and below another NBel temperature, TN', the x-component comes into oscillation, the resulting elliptic oscillation being confined to the xy plane. Similar calculations should be possible for this case. Thus, we shall henceforth be concerned only with the cases of p = 4 and p = 6 . I n the case of p = 4 or 6, a helical ordering in the zy plane appears below the NBel temperature if the second term of (5.1).is small. This can be seen by expanding the exponential factor in (3.3), w(S,) being replaced by (5.1), in powers of exchange field components and retaining the constant and linear terms, and then expanding these terms in powers of ( S , f is,). and retaining the constant and linear terms; since ( S , f is,)p are operators to change S, to S, f p , and since S, and S, appear a t most twice after the symbol tr, no contribution appears from the latter linear term when p > 2. We are therefore led to Eq. (4.1) or ( 4 . 2 ) , with H,, = 0. Thus, the NBel temperature is determined by the procedure that followed these equations. Assuming

it would seem that either the x or y component, or any one component in the xy plane, comes into oscillation below TN , but higher-order terms in the exH,,S, will couple the x and y components to pansion in powers of H,,S, give a helical ordering, i.e., if the x component has started to oscillate as cos (Q-R, a), then the y component must a t the same time have started to oscillate as f sin ( Q -R, a) to ensure the minimum free energy. Thus, below TN we have helical ordering in which the thermal average spin vectors rotate in the xy plane. With decreasing temperature, however, the anisotropy energy of fourfold or sixfold symmetry comes into play. If the turn angle of the helix between successive atomic xy layers, which is Q times the interlayer distance, is close to 60" in the case of p = 6, we may expect trapping of the spin vectors in the successive six potential valleys. If the turn angle is close to 30°, trapping of each pair of spin vectors in the successive six valleys may occur, as actually observed in the case of Ho below 20°K (although the structure is conical). For smaller turn angles, trapping of all spin vectors in a single valley is a possibility; this is the transition from the helical state to the ferromagnetic state observed in T b and Dy. We shall discuss this ferromagnetic transition in some detail in the following.

+

+

+

325

THEORY OF HELICAL SPIN CONFIGURATIONS

We shall first consider the situation at absolute zero or a t very low temperatures. I n the helical state, the anisotropy energy within the easy plane may modify the spin arrangement in such a way that the ( p - 1)th and (p 1)th harmonics are mixed to the fundamental, but this will only slightly change the energy, so that we shall simply take the energy to be -SzJ(Q) per atom [see (1.12) ; 2J(Q)X is the magnitude of the exchange field, and the energy per atom is minus half the product of this with the spin length S]. I n the ferromagnetic state, the energy due to the exchange interaction is -S2J(0), which is higher than -SzJ(Q), but we have an additional negative energy due to the minimum of the anisotropy energy, which we shall denote by wmin < 0. Therefore, the ferromagnetic state is more stable if S2[J:J(Q) - J(O)] < I Wmin 1. (5.3)

+

We now ask if the ferromagnetic state is really stable under this condition; i.e., we ask if all the ferromagnetic spin-wave frequencies are positive. For this purpose, we shall assume an out-of-plane anisotropy energy of the form of DXz2 with positive D, confining ourselves to small spin deviations from the xy plane, and an in-plane anisotropy energy of the form of GSy2 with positive G, considering only small spin deviations from the equilibrium x direction. These energies would give anisotropy fields -2DS, in the z direction and -2GSu in the y direction. If one assumes the in-plane anisotropy energy to be a classical function of azimuth Q and to be simply proportional to cos ( p ~ )namely, , W ( Q ) = Wmin

(5.4)

cos (PQ),

then

G

=

(p2/2S2) I Wmin

1,

(5.5)

since XQ = X u . Now, the effective field acting on the nth spin can be written in components as

Hnz

=

2J(O)S

+ H,

Hnu

=

2

C J(Rmn)Sm,

-

2GSnU 9

m

Hnz

=

2

C J(Rmn) S m z - 2DSnz

(5.6) 2

m

where we have included an external field H in the x-component. The equations of motion for the spins are fisnu =

SnzHnz - SnzHnz 7

fiSnz

= SnzHnU

- SnuHnz -

Substituting here (5.6) and putting En, = S , Sn, = b exp (iq-Rn - iwt), Sn, = c exp (iq.Rn - id), where b and c are constants, we obtain im-

326

TAKE0 NAGAMIYA

mediately the frequency formula:

(fiu/2S)'

=

+ (H/2X) - J ( q ) + 0 ] [ J ( 0 )+ ( H / 2 8 - J ( q ) + GI.

[J(O)

(5.7) We assume that D is so large that the first factor in the right-hand side of (5.7) is positive for any q. Then, the condition for stability is

J(0)

+ (H/2S)

- J(q)

+G > 0

for all q.

With the use of (5.5) ,and remembering that J( Q) is the maximum of J( q) , this turns out to be

S'[J(Q)

- J ( O ) ] < 3 ~ I 'Wmin I

+ 3SH-

(5.8)

This inequality is surely satisfied when (5.3) holds and p > 2. We are thus assured of the stability of the ferromagnetic state. If we now consider wminto be a function of temperature, we see that a transition occurs from the helical state to the ferromagnetic state a t the temperature at which the left and right sides of (5.3) become equal. An external field of course enhances this transition, in which case we have an extra positive term SH - 3xhH2in the right-hand side of (5.3) , where Xh is the susceptibility per atom of the helical state to be discussed in Part IV, Section 7. However, the inequality with this additional term in the right-hand side does not always assure the inequality (5.8), so that the ferromagnetic state may not be stable. I n fact, as we shall discuss in the next part, there is the interesting possibility of a fun structure in the presence of a magnetic field. In Dy, the NBel temperature is 179"K, whereas the temperature of transition to the ferromagnetic state is 85"K, where the turn angle of the helix is 26.5". The foregoing consideration may be applicable to this case. In Tb, the NBel temperature is 228"K, and only 7" below this the transition to the ferromagnetic state occurs; at the latter temperature, the turn angle of the helix is 18". For such a case, we have to make a high-temperature approximation, as we shall do below. In both Dy and Tb, the lowering of the (free) energy of the ferromagnetic state due to a magnetostrictive effect may play a considerable role, as was pointed out by Enz." We shall briefly discuss the ferromagnetic transition a t a high temperature. For this purpose, we compare the free energies of the two states. I n the approximation of the Weiss field, with a negligible anisotropy energy, the free energy of the helical state can be expressed as

Fh l1

=

--kTIntreXp (PHhS.)

U. Enz, Physicu 26, 698 (1960).

+ 3HhSUh ,

(5.9)

THEORY OF HELICAL SPIN CONFIGURATIONS

327

with

Suh

[ d / d ( B H h ) ] In t r exp (BHhS,) ,

=

(5.10)

where H h means the magnitude of the internal field in this state. The free energy, F f , of the ferromagnetic state can be expressed with H t and ut in place of Hh and U h , with an additional anisotropy energy term wmin of the given temperature. This w,in is the minimum value of the p-fold anisotropy Hamiltonian averaged over the Boltzmann distribution of the spins in the ferromagnetic Weiss field corresponding to an arbitrary direction of magnetization. We expand these free energy expressions in powers of Hh or Ht . To the lowest power, we have

Fh

-pl In ( 2 s

=

+ 1 ) - -&p3Hh4S(s + 1 ) (8' + + 4)

(5.11)

and a similar expression for Ff with an additional term w m i n . On the other hand, assuming a Brillouin function, we have XU = @ H S ( S for u

=

Uh

+ 1 ) - &((pH)3S(S+ 1 ) ( S 2+ S + 3) +

and H

Hh

=

=

Hh and also for u =

W(Q)suh,

~ T =N $ J ( Q ) S ( S

+l),

ut

Hr

and H =

=

(5.12)

Hr . Now,

W ( 0 ) S u t,

(5.13)

+ 1).

kTc = $ J ( O ) S ( S

(5.14)

Hence, using (5.12),we can express Hh as a function of T and T N and Hf as a function of T and TC . A calculation gives the results:

+ s + +)-' Hf2 = 15(kT)'(l - T / T c ) ( S 2+ S + +)-'

Hh2 = 15(kT)'(l - T / T N ) ( S 2 Thus, in order that Fh and T < T N :

< TN)r (T < Tc). (T

> Ft , the following inequality must hold for T < TC

(5.15) Since TC is lower than TN according to (5.14), the left-hand side of (5.15) is positive for T < TC and has a maximum at

T

=

+

~TNTc/~(TN Tc).

This maximum is small when Tc is close to TN . On the other hand, 1 Wmin I vanishes a t T N and increases with decreasing temperature. If this increase is rapid, the inquality (5.15) may become satisfied, or the ferromagnetic state may become more stable, at temperatures that are below the critical

328

TAKE0 NAGAMIYA

temperature determined by the equality of the two sides of (5.15). I t is emphasized that Tc must be close to T N in order that the ferromagnetic transition of this kind is predicted. I n other words, J ( 0 ) must be close to J ( Q ) , according to (5.14). Magnetostrictive energy in the ferromagnetic state may be included in wmin (cf. Note added in proof on page 403). 6. ANISOTROPY ENERGY HAVING AN EASY CONE

Neutron diffraction experiments have shown that in the rare-earth metals Ho and Er the spin structures below 20°K are such that the spins in each hexagonal basal plane are ferromagnetically ordered, and they precess on a circular cone in going from one plane to the next. The cone axis is the c-axis, and the vertex angle of the cone is abow 75" for Ho and 25" for Er. The turn angle in the projection of the spins on the c-plane is 30" and 44", respectively. In Ho, speaking in more detail, the turn angle deviates from 30" alternatingly by 26 and -26, with 6 = 9" a t 4.2"K, because of the anisotropy energy of sixfold symmetry.12 (For references, see the forthcoming Part 2.) We shall discuss briefly in this section the transition to such a conical structure due to a gradual change in the anisotropy energy with lowering temperature. Confining ourselves to low temperatures, we may assume that each atomic plane perpendicular to the vector Q (the hexagonal basal plane in the case of rare-earth metals) has a ferromagnetic saturation moment equal to that at absolute zero, and the anisotropy energy is a classical function of the direction of this moment and of temperature. Such an anisotropy energy for each atom may be written as w (cos28, sinp 8 cos pcp; T ),

(6.1)

where e and cp are the polar and azimuthal angles of the atomic moment under consideration, and p = 6 in the case of rare-earth metals. If w is a small quantity, we may have a helical arrangement whose rotation plane is determined by the minimum of w averaged over the orientations of the moments in that arrangement. Let us consider the simplest case that w is independent of cp and has a minimum a t e = 00 (so that it has also the same minimum a t e = a - eo) . The polar angle of the normal to the rotation plane of the helix will be denoted by 8 ;the azimuth should be indeterminate in this case. Since cose for the moment of the atom at R, is sin 8 cos (Q.Rn a) and the angle Q.R, a is distributed uniformly in the angular interval of 2a, provided that the period of the helm is not a simple rational multiple of the lattice spacing along Q , we have for

+

la

+

W. C. Koehler, J. W. Cable, M. K. Wilkinson, and E. 0. Wollan, Phys. Rev.161,414 (1966).

329

THEORY OF HELICAL SPIN CONFIGURATIONS

the average of the anisotropy energy

T ) d'p.

(6.2)

The minimum of this integral, with respect to 8, determines 8. It is easy to see that the rotation plane of the helix determined in this way cuts the easy cone determined by w(cos20 ) = min, with e = eo , the plane being closer to the polar axis. Eventually, if eo = 0, the plane will contain the polar axis, and, if Oo = r / 2 , it will be perpendicular to the polar axis. On the other hand, when the minimum of w at eo is deep enough, the structure should be a conical one. The vertex angle of this cone must be somewhat greater than eo because of the exchange interaction. This vertex angle will be denoted by el . Assuming again the independence (or a weak dependence) of w on 'p, we can determine el by minimizing the exchange plus anisotropy energy, which can be written as

-S2sin2OJ( Q) - S2cos2M ( O )

+ w (cos2el ; T),

(6.3)

where S s i n & is the rotating spin component and Scosel the parallel, or ferromagnetic, component. The minimization of this with respect to 61 gives S2[J(Q) - J ( 0 )] [d/d(cos2 el) ]W (COSZ el ; T ) = 0. (6.4)

+

The first term is positive, so that the second term must be negative, which means that ~ 0 ~ 2 must t h be smaller than cos2eo. We cannot go further without knowing the functional form of w. The structural change in Ho from a proper helix to a cone may be due to a deepening of the minimum of w at eo with lowering temperature. 00 should be smaller than 75" ( =el). Since the observed structure change is an abrupt one12 and since the rotation plane of the helical structure above 20°K remains perpendicular to the c-axis, there being no helix with an oblique normal as that predicted with (6.2), one must imagine that w has another well-defined minimum at e = ~ / 2and that the minimum at 00 gets deeper than this as the temperature is lowered. Furthermore, as mentioned before, the turn angle in the conical structure is alternately smaller and greater than 30", whereas the turn angle just above 20°K is a little greater than 30". The fitting of each pair of moments successively into potential valleys of the anisotropy energy must give a further stabilization of the conical structure below 20°K. I n Er, the structure above 20°K is such that components both parallel and perpendicular to the c axis oscillate with a common wave number. This wave number joins smoothly to that of the conical structure below

330

TAKE0 NAGAMIYA

20°K. A longitudinal sinusoidal structure a t high temperatures first changes, with decreasing temperature, to an elliptic structure at TN' discussed in Section 4, and then the plane of the ellipse gradually deviates from the c axis because the minimum of w moves from e = 0 to a nonzero 0 value, and, finally, as the minimum deepens, there occurs an abrupt change into a conical structure. This theoretical interpretation is consistent with all observations on Er. We shall mention the principle of calculating the energy of the structure in Ho below 20°K. The relative azimuthal angle between the moment of the atom a t R, and the moment of the atom a t the origin can be written as Q30*Rn- 6 6 cos (6Q30.Rn), where Q30 is the wave vector representing the helical structure with a turn angle of 30". To calculate the exchange energy for the rotating components, we have to take the cosine of this angle, multiply it by -52 sin2BJ(R,), and sum it over R, . The cosine of the angle can be expanded into a Fourier series with coefficients expressed in terms of Bessel functions. The Fourier components consist of the funda, harmonics, and, in general, mental having 0 3 0 , the fifth, seventh, (6 X integer f 1)th harmonics. Correspondingtly, the exchange energy can be expressed by J(Q30) ,J(5Q30),J(7Q30), etc., and by Bessel functions of the zeroth, first, etc., degrees. The result should replace the first term of (6.3). I n the anisotropy energy, given by (6.1), e is replaced by 01 and cos 6p by sin 66. However, actual calculations are not yet meaningful, because we do not know the exact form of w for Ho. On the other hand, the mentioned expansion of the angle into a Fourier series is useful in analyzing neutron diffraction data.

+

..-

IV. Effect of External Field on Spin Configurations

A weak external field applied to a helically ordered spin system induces a magnetization that is proportional to the field strength. At high fields, however, structure changes appear, some of which have no analogs in ferromagnetism and antiferromagnetism. Experiments to show these Structure changes are still few, and in this section we shall be concerned mostly with the theory.11J3-16 1s

l4

A. Herpin, P. Mdriel, and J. Villain, Compt. Rend. 249, 1334 (1959); A. Herpin and P. Mdriel, ibid. 260, 1450 (1960); J . Phys. Radium 22, 337 (1961). T. Nagamiya, K. Nagata, and Y. Kitano, Progr. Theoret. Phys. (Kyoto) 27, 1253 (1962).

Y. Kit.ano and T. Nagamiya, Progr. Theoret. Phys. (Kyoto) 31, 1 (1964). 16 H. Thomas and P. Wolf, PTOC. Intern. Conf. Magnetism, Nottingham, 1964,p. 731. Inst. Phys. Phys. SOC., London, 1965.

l6

THEORY OF HELICAL SPIN CONFIGURATIONS

331

7. FIELDAPPLIEDPERPENDICULAR TO THE PLANE OF SPINROTATION We shall first neglect the anisotropy energy and consider a field applied perpendicular to the plane in which the spin vectors rotate. The exchange field acting on each spin in the absence of external magnetic field has a magnitude of W ( Q )I (S)1, the factor -(gpB )-l being here again omitted for convenience. If a perpendicular external field of magnitude H is superposed on this, one will have a conical structure with the thermal average of any one spin having certain common components (S)ll and (S)Lparallel and perpendicular to the field direction. The effective field on each spin will have the components W ( 0 )(S)ll

+H

parallel to the field, perpendicular to the field.

2 J ( Q )( S > i Then, one should have

Thus, the effective field is W ( Q ) ( S ) ,where (S)is the thermal average of each spin vector. The magnitude of (S)is determined by the equation

I (S)I = SBs(x), x = W ( Q >I (S) I S h T . (7.2) One sees that I (S) I is unaffected by the field. This is, however, valid only when the field is so weak that the spins are not parallel to the field direction. At the critical field, H o , at which the spins come to point all parallel to the field direction, one has (S),l = I (S) I. From (7.1) follows

H

=

~(S)IICJ(Q)- J(0)l.

(7.3)

Thus,

I

Ho = 2 (S)I CJ(Q> - J(0)l. (7.4) (The right-hand sides of these equations is divided by gpB in the original notation.) Furthermore, from (7.3), one obtains the susceptibility perpendicular to the plane of spin rotation, which we shall denote by x z , namely,

xz = ( S ) I I / H= + [ J ( Q ) - J(O)]-'

(per atom).

(7.5)

[In the original notation, the right-hand side of (7.5) acquires a factor (gPB)2-1

The foregoing has a close analog in the two-sublattice antiferromagnetism, xz corresponding to the perpendicular susceptibility, XI , of the latter. I n actuality, the presence of an anisotropy energy modifies expression (7.5) to some extent.

332

TAKE0 NAGAMIYA

8. FIELDAPPLIEDIN

THE

PLANEOF SPINROTATION

When the external field is in the plane in which the spin vectors rotate, we encounter a more complicated situation. The theoretical treatment of this case was first made by Yoshimori,' then by Herpin and M6riel,l3who discovered the fan structure to be discussed below, by Enz," and by ~ ~ 4later most generally by Kitano and Nagamiya.15 Nagamiya et ~ 1 . and The plane of spin rotation will be taken as the xy plane, and the field is supposed to be in the x direction. At no field, the spin at R, has components

+ a), Sasin (Q-R, + a).

(SnZ) = Sunz = Sa cos (Q.Rn (S,,)

= Xu,

=

(8.1)

I n the presence of a field, H,these will change to16a Sanz =

S[az~

+ az1cos (Q.Rn + + azzc0~2(Q.Rn a)

+ ...I,

+a)

+ a) + a,2sin2(Q.Rn + a) + ...I, (8.2) being an even periodic function of the angle Q - R, + a and anyan odd

Sun, = S[a,lsin (Q-R, a ,

periodic function. Correspondingly, the effective field acting on the nth spin will become Hnz

=

+ W ( 0 )S U ~+OW(Q) Sazl cos (Q*Rn + a) + W ( ~ Q ) S Ucos, ~2(Q*Rn+ a) + ... , (8.3) W(Q)Sa,l sin (Q.Rn + a) + 2J(2Q)SaV2sin 2(Q.Rn + a) + - - .

H

H,, = If we write

(4% +

~ % y ) ' '= ~

an

7

(Hiz

+ H&,)1/2= H

(8.4)

n i

then a, will be given by the Brillouin function of SH,/LT, and, conversely, H , will be given by the inverse Brillouin function of a, , multiplied by k T / S . Even in the case where we have an anisotropy energy, the magnitude of the effective field (exchange plus external field) , H , , is given by a certain function of un and T similar to that for the case of no anisotropy energy. We shall write it as H*(an , T ). Then we have, in components, the following equations to determine unZand any: Hnz

= (anz/un)H*(an

7

T),

Hn, =

(an,/an)H*(an

7

T),

(8.5)

where unzand anyare given by (8.2) and by (8.3). If the external field H is small, a, will not be much different from a ; a t low temperatures, a,,will be practically equal to a for any strength of the external field; a t high temperatures, both a, and a will be small, so There will be a small change in the Q value when there is a field, which we shall neglect.

333

THEORY OF HELICAL SPIN CONFIGURATIONS

far as H is not extremely high. In all these cases, we can expand H* (a, , T) in powers of a, - a and can confine ourselves to terms of lower powers. Also, it will be shown later that a, coincides with a a t a critical field, Ho , where transition occurs from the f a n structure to the ferromagnetic structure, so that in the neighborhood of this critical field we can make the same treatment. Since by reversing the direction of the effective field the magnetization also reverses its sign, H*(an , T ) / a , is an even function of a, , or a function of an2.Thus, we can make the following expansion :

H*(u, , T) - H*(a, T) Qn

C T

a H*(a, T) + a-[(a2) ] . ( d-

a2)

+

0 . .

.

(8.6)

The first term of this expansion is equal to the exchange field at no external field, divided by a, so that it can be identified with W ( Q )S. With two terms of (8.6), we can therefore write Eqs. (8.5) as

where i = x, y. We substitute (8.2) and (8.3) into (8.7) and compare the Fourier coefficients of both sides. Then we obtain equations to determine uzo , uzl , q,1 ,etc. Leaving the details of the calculation to Kitano and Nagarniya,l5Jsa we mention here and in the next section the results only. I n the Appendix, we shall develop another mathematical method and derive susceptibility formulas. For weak field, we obtain the susceptibility formula for the helical state for a field in the plane: 2P2

sazo Xh=---=

H

2P2

+

Y

+ (1 +

xz

1

(8.8)

a2>r

where

(8.10)

In the present case, where we have no anisotropy energy, H * ( a , T) is (kT/S)Bs-'(a),as mentioned before. I n this case, y can be expressed, near the NBel temperature, as (8.11) Xh

and

xz thus coincide with each other a t TN . Below TN , X h is smaller

J ( 9) and q o in Kitano and Nagamiya16correspond to our 25V(q) and Q, respectively.

334

TAKE0 NAGAMIYA

than xz according to (8.8) and (8.11). For T 4 0, y tends to infinity, and ad. one has X h = x d ( 1 Because X h is smaller than x z below T N, a field applied in the plane of the spin rotation will make the spins flip in such a way that the rotation plane become perpendicular to the field direction, provided there is no anisotropy energy to keep the spins in the original plane. If this anisotropy energy is not zero but small, the flip will occur at. a critical field, H , ,which one obtains by equating a ( x z - Xh)H2 to the difference between the anisotropy energy values for the two configurations. This is analogous to the well-known spin-flip phenomenon in antiferromagnetism. When we are at temperatures sufficiently close to TN , the series (8.6) is well convergent, since both an2and u2are small, and the coefficients of the expansion are finite. It can be shown also that the Fourier expansions (8.2) and (8.3) converge well in this case. The calculation of the Fourier coefficients can be carried out completely after neglecting the second and higher harmonics, and the result is the following.15 At zero field, the hodograph described by the spin vectors is a circle of radius u. A weak field applied in this plane changes this circle into an ellipse whose minor axis along the field direction has a semiaxial length ' ' ~ being abbreviated as a,,),according to the calof uZl = (uz - 4 ~ 0 ~ ) (uzo culation, and whose major axis perpendicular to the field direction has the constant semiaxial length U . u0 increases with increasing field strength, first linearly, corresponding to the susceptibility X h , and then more rapidly. When a0 reaches +IS, uzl vanishes, and the magnetization curve then follows another branch. I n the latter region, we have a fan structure, in which only

+

FIG.1. Magnetization curve of a helical spin system for a field in the plane that contains the spin vectors. (a) At a temperature close to the NBel temperature (y < 8/11); helical below H , , fan between Ht and Ho , ferromagnetic above Ho . (b) At a low temperature (y > 8/11);helical below Ht', fan between H,' and HO, ferromagnetic above Ho . Ho is proportional to the thermal average spin length for zero field.

THEORY O F HELICAL SPIN CONFIGURATIONS

335

the y component of the spin vectors oscillates, whereas the x component is a constant, so that the hodograph described by the spin vectors is a line. The x component still increases with increasing field strength, whereas ” ~ , uo reaches u. the y component diminishes as u,l = [+(u - u ~ ~ ) ] until Above this field, which is equal to HO given by (7.4), the spins all align parallel to the field, and their magnitude increases with increasing field as in an ordinary ferromagnet. The transition from the helical structure to the fan structure a t uo = $a occurs a t a field H , given by (8.12) The magnetization curve looks like Fig. l a or b, according to whether y is smaller or larger than 8/11. I n the latter case, the actual transition from helix to fan should take place at a field H,’ at which the vertical line in Fig. l b divides the triple-valued part of the magnetization curve into equal enclosed areas. The transition is, therefore, of the first kind. We shall discuss these situations in more detail with different mathematics in the next section. ANISOTROPY, 9. STRUCTURE CHANGESIN THE PLANE(No IN-PLANE ARBITRARY TEMPERATURE) We would like to recall Eq. (5.7) of the preceding part for the spin-wave frequency of the ferromagnetic state in which the moments are all aligned parallel in one of the easy directions in the easy plane. I n that equation, D and G were so defined that DX,2 is the out-of-plane anisotropy energy and GSU2the in-plane anisotropy energy, both for small deviations from the easy direction, x. When D is large enough, positive frequencies are obtained only when the field strength exceeds a value H o defined by

Ho = 2X[J(Q) - J ( 0 ) -

GI.

(9.1)

We see that here D does not enter into H o . When G = 0, (9.1) is nothing but Eq. (7.4) for absolute zero. Below Ho , defined by (9.1), the ferromagnetic state is unstable, and we may have a fan structure. We might also infer that for an arbitrary temperature and no anisotropy within the plane we have ferromagnetic structure above Ho , defined by (7.4), and fan structure below Ho . In fact, this inference can be verified in the following way in the approximation of the molecular field. We once again consider Eq. (3.3). There w (S,) is the anisotropy energy that makes the xy plane the easy plane. H,, , H,, , H,, are the components

336

TAKE0 NAGAMIYA

of the effective field (exchange plus external field in the present case) acting on the nth spin. They are given by

Ha, = 2

C J ( R m n ) Sum, m

Hnz = 2

+H,

CJ ( L n ) S u m z m

We consider that urn,, as a function of the position R, , fluctuates about a constant value uo that satisfies the equation Suo =

t r S, exp P[H*S, - w(S,)] t r exp P[H*S, - w (AS,)]

where

H*

=

2

’

(9.3)

C J(Rm,)Sao + H m

+ H.

= 2J(0)S~o

(9.4)

We consider also that umuand umzfluctuate about zero. We shall be able to show that all these fluctuations vanish above Ho defined by (7.4) and that below Ho oscillation begins in am, with the wave vector Q and with an amplitude proportional to ( H o - H)’12. The oscillation in am, and a secondorder variation in urn,below Ho form a fan structure. umzvanishes above Ho and in a certain range below Ho . Putting unz = uo unZfand Hn,‘ = 2 CmJ( Rmn)Sum,’, we can write the energy of the spin a t R, as

+

+ Hnz’Sz + HnUS, + HnzSz - w(Sz)].

-[H*Sz

We substitute this into (3.3) and expand the exponential function in powers of H,,’, Hn, , H,, , using the mathematical method described in footnote 9b, page 319. Picking up only linear terms, we obtain equations of the following form: Sunzl = Hn,’A(P, H*),

Sun, = Hn,B(P, H*),

Sunz = HnzC(P, H*),

(9.5) where

A (p, H*) = t r S,

J

B

exp [ ( P - A> (H*s, - w)]~,

0

X exp [X(H*S, - w)] dX/tr exp [P(H*S, - w)] -

S2u02,

THEORY OF HELICAL SPIN CONFIGURATIONS

337

rS

B(P, H * ) = t r S,

I, exp [ ( P - A) (H*S, - w)]S,

X exp [A(H*S, - w)3dh/tr exp [D(H*S, C ( P , H*) = t r S ,

- w)],

B

j

exp [ ( p - A) (H*s, - w)]~, 0

X exp [X(H*S, - w)]dX/tr exp [P(H*S, - w)].

These are certain susceptibilities and are decreasing functions of H*. Namely, with increasing H*, the thermal average spin increases in the x direction, so that the squared spin deviations in the y and z directions decrease, and hence B and C decrease. It may also be imagined that the spin deviation in the x direction from its average value, Sao , decreases with increasing H*, and hence A decreases. A proof of these will be given in Section A1 of the Appendix. One should aIso be able to show that B is the largest, C the medium, and A the smallest ( B > C > A ) ;this follows from the fact that the susceptibility in the y direction is larger than that in the z direction (since there is no anisotropy energy to prevent the rotation of the spin in the xy plane, whereas there is such an anisotropy energy in the xz plane) and that the susceptibility in the x direction (which is the parallel susceptibility in the ferromagnetic state) is expected to be very small (see also the Appendix for the proof of B > A ) . It can be shown further that B(0, H*) = Sao/H*. (9.7) This can be proved in the following way. Consider t r S, exp PCH*S, - w(S,)],

(9.8)

and transform each of the operators after the symbol tr by exp (ieS,), which means a rotation by an angle of 8 about the z axis. We know that exp (ieS,) S, exp ( -ieS,)

=

S, cos 8 - S, sin 8,

exp (ieS,)S, exp ( -ieS,)

=

S, sin e

+ S, cos 8.

Then, expression (9.8) can be written, for small 8, as t r (OS,

+ S,) expP[H*S,

-

H*OS, - w(S,)].

(9.9)

Expanding the exponential in powers of e and taking the linear terms, we obtain tr OX, exp [D(H*S, - w)] - t r H*OS,

/,”

exp [ ( D - X) (H*S, - w)]S,

X exp [A(H*S, - w)]dX.

(9.10)

338

TAKE0 NAGAMIYA

Now, we note that (9.8) vanishes, which can be seen by a similar transformation with e = T about the x axis, by which S, and S , change sign. Hence, (9.10) also vanishes. Therefore, by the definition of Suo, (9.3), and the definition of B, (9.6),we obtain the result (9.7). It might be added that we can show in a similar way, starting with t r S , exp p[H*S, - w(S,)], that

loexp [ ( p r8

tr S,

A) (H*S, - w)]Syexp [A(H*S2 - w)]dA = 0. (9.11)

S , and S, can be interchanged in this expression. Now, to solve Eqs. (9.5), we put a,,’, any, a,, all proportional to exp (iq. Ri) , with amplitudes a / , ay , az , respectively, and substitute into / (9.5). Then Eqs. (9.2) for Hnyand H,, and H,: = 2 ~ m J ( R m , ) S u m we have =

W ( q ) A ( P H*)a,’, ,

all =

W(q)B(P, H*)% 9

=

2J(q)C(P, H*)a, *

a,’

These equations give vanishing a,’, a,, and 1

=

W(q)A,

1

=

a,

(9.12)

unless

W(q)B, and 1 = W(q)C,

respectively. However, in actuality, owing to the existence of nonlinear terms in H,,‘, H,, , H,, which we have neglected, we have nonvanishing a=’, etc., when, and only when, 1 5 W(Q)A, etc. This situation is similar to that we encountered in Chapters I and I11 in determining the NBel temperature. Furthermore, to determine the highest field H*, i.e., the real critical field, above which the fluctuations vanish, we have to take the maximum of J ( q ) , since A , etc., are decreasing functions of H*. So we put q = Q. Since B is the largest, there is a field a t which 1 = W(Q)B, whereas 1 > W (Q) A and 1 > W (Q) C. Below this field, anyshould become oscillating, whereas a,’ and unZremain still vanishing as far as first-order changes are concerned. By virtue of (9.4) and (9.7), the condition 1 = 2J(Q)B can be rewritten as

H* or

+

W(O)SUO H

=

W(Q)Sao,

H = 2[J(Q) - J(O)]Sao. I n this equation, Xuo can be put equal to Sa, where Sa is the thermal average magnitude of spins in the helical state in no external field. This is because the effective field in the present case is H* = W(Q)Sao, and the exchange field in the helical state is W(Q)Sa, and hence the equations to

THEORY OF HELICAL SPIN CONFIGURATIONS

339

determine u0 and u are the same. Writing Ho for H, we have, therefore,

Ho

=

2 [ J ( Q ) - J(O)]Sa.

(9.13)

This is nothing but Eq. (7.4). We can proceed to determining the amplitude of u, , as well as unzt, below Ho by taking account of higher powers of H,, and Hnz'in the molecular-field self-consistency equations. It turns out that Hn2 is of the order of H,,2 below H o . It can be shown also that an, and H,, vanish throughout. These calculations are described in the Appendix. We shall mention here the results only. Below Ho , the y component, un, , oscillates with the wave vector Q and an amplitude proportional to (Ho - H)'12. The 2 component, un2, consists of a constant part, uz0', and an oscillating part with wave vector ZQ,both parts being proportional to Ho - H. From uzot,the susceptibility of the fan structure can be calculated, and it is given by

(9.14) where Pz and y are defined by (8.9) and (&lo),y being zero at T N and infinite a t T = 0. We can now discuss the transition between the helical state and the fan state. The free energy of the helical state a t a field H, relative to that at H = 0, is -$XhHz. At H = Ho , we have seen that the effective field has the same magnitude as that at H = 0, so that the value of the entropy at H = HOand that a t H = 0 are equal. The energy a t H = Hois -3SuHo . Thus, the energy of the fan state a t a field H can be written approximately as -$SUHO SU(HO- H) - $Xran(Ho - H)'. (9.15)

+

The field of transition, Ht', which has the same meaning as that illustrated in Fig. lb, can be obtained by equating (9.15) with -$xhHz. We have

H,'

=

[(xz

- xfan) (x, - ~ h ) ] " ~Xfan

- Xh

(xz

- Xfan) Ho.

(9.16)

This falls between 0.5Ho and 0.414Ho. If the out-of-plane anisotropy energy is not large enough to keep $he spin vectors in the plane, there will be a field below Ho where the z-component starts to oscillate, giving rise to an elliptic oscillation. Upon further lowering the field, this elliptic oscillation will discontinuously transform to the helical ordering. These transitions were discussed by Nagamiya et al.14for T = 0, but we shall not go into this subject here.

340

T A K E 0 NAGAMIYA

10. STRUCTURE CHANGES WITH ANISOTROPY IN (Low TEMPERATURES)

THE

PLANE

We shall briefly consider magnetization processes in a system in which there is initially a helical ordering and there is an anisotropy in the plane of the spin rotation. Also, in the next section, we shall briefly discuss the case of a conical structure. Since these are of interest mainly at low temperatures, the entropy term in the free energy may be neglected, and the consideration may be confined to the problem of minimum energy. An external magnetic field is supposed to be perpendicular to the screw axis. Consider the case of a helical structure with an in-plane anisotropy energy. For a strong field applied in one of the easy directions in the plane, the spins may be aligned all parallel to the field direction. The spin-wave frequencies in this case were calculated in Section 5, and it was mentioned in Section 9 that some of the spin waves become unstable below H o given by (9.1). We may then have a fan structure. We shall here consider this problem in some detail. We have an anisotropy energy GS,z for small deviations of a spin in the y-direction from the easy direction x. For a deviation of an angle cp, this energy can be written as GS2q2; the corresponding restoring torque is -2GS2cp. An equivalent torque is obtained when a field of magnitude 2GS is applied in the x-direction. From this consideration follows the term -2GS in H o , Eq. (9.1). Now we assume that cp oscillates sinusoidally as cp, = q

(10.1)

sin (Q-R,).

Then, the mean value of GS2cpn2over n is

GS2(cpn2), = iGS2q2.

(10.2)

The mean value of the exchange energy per atom up to 0 ( q 4 ) is

-S2(CJ(Rmn) cos

(vm

- c ~ n )n)

m

=

-S2J(0)

+ 4Sz(CJ(Rmn)

(vm

- qnl2>n

m

-As2(CJ(Rnn)(cpm - q n I 4 > n m

=

- S 2 J ( 0 ) - +S2qz[J(Q)- J ( O ) ]

+ 3$Szq4[4J(Q) - U(0)- J(2Q)I.

(10.3)

The mean interaction energy of the spins with the external field, also up to 0 (q4) , is +HSq2 - &HSq4. (10.4) -HS(COSpn), = -HS

+

34 1

THEORY OF HELICAL SPIN CONFIGURATIONS

The sum of these three energies, (10.2), (10.3) , and ( l O . i ) , may be minimized with respect to 7. Then we have the following equation: 8(Ho - H )

" = 2S[4J(Q) - 3J(O) - J(2Q)I

-

H'

(10.5)

where (9.1) was used for H o . Assuming that H is close to H O, we may replace H in the denominator by H o . Then, q2 =

8(Ho - H ) 2 S [ 3 J ( Q ) - 2J(O) - J ( 2 Q )

+ G]

(10.6) '

From this, we can derive the differential susceptibility of the fan struc~ ) is~ ture. Namely, the mean x component of the spins is S(cos c ~ , which S ( 1 - s2/4) to order +', so that Xfan =

[3J(Q) - 2 J ( O ) - J ( 2 Q )

+ GI-'

(10.7)

[multiplied by (gpg)' in the original notation]. Equations (10.7) and (9.14) become identical in the limit of T = 0 (y = a ) and G = 0. Since Ho becomes lower and Xfan smaller by the introduction of G, the field of transition, H t f , between the fan and the helix, relative to Ho , becomes higher. This prediction is of course based on the assumption that the amplitude of the fan at the transition field is still small and the in-plane anisotropy energy can be written as GS2cp2. If, however, the anisotropy energy has a cp4-term, the situation becomes a little different. Assume, for instance, a n anisotropy energy of the form TlgGS2[1 - cos 6cpl.

(10.8)

+ --.

Expansion of this with respect to cp gives GS2cp2 - 3GS2cp4 . Correspondingly, we have, by a calculation similar to that we have made the following susceptibility formula: Xfan =

[ 3 J ( Q ) - 2J(O) - J ( 2 Q ) - 35G]-'.

(10.9)

Notice that the G-term in the denominator changed from +G to -35G. It would then be quite possible that the denominator becomes negative for somewhat large G. In such a case, the magnetization curve would look like Fig. 2. We may then have a transition of the first kind between the fan state and the ferromagnetic state at a field Ho' which is higher than HO. For high values of G, we might even have a direct transition between the helical state and the ferromagnetic state without a n intermediate fan state. I n fact, these transitions have been predicted by Kitano and Nagamiya15 by calculations in which the anisotropy energy (10.8) was exactly taken into account, but the fan state was approximated by a

342

TAKE0 NAGAMIYA

FIG.2. Magnetization curve of a helical spin system for a field in the easy plane, in the case where there is a large anisotropy energy within the plane; the field is applied along one of the easy axes; the temperature is absolute zero.

purely sinusoidal oscillation, and for the helical state the in-plane anisotropy energy was neglected. Actually, however, there should be certain modulations of the fan and helical configurations because of the anisotropy energy of a multifold symmetry. A more exact mathematical treatment is thus desirable. In the case where the external field is along one of the hard axes in the easy plane, discontinuous transitions helix-fan-ferro are predicted, as long as the anisotropy energy in the plane is not too large. On the other hand, if the anisotropy energy is large, one obtains a parallel alignment not in the field direction but near that potential minimum which is closest to the field direction. This parallel alignment appears between the helix and the fan. These results are shown by the examples in Figs. 3a and 3b. 11. STRUCTURE CHANGES OF

A

CONICAL ARRANGEMENT

Magnetization processes of a conical structure for a field applied perpendicular to the cone axis involve even more complex structure changes. For an axially symmetric anisotropy energy, it was predictedls that, depending on the functional form of this anisotropy energy, some or all of the following intermediate structures appear with increasing field (see Fig. 4) : (1) Conical structure, a little distorted and inclined toward the field direction. (2) Fan on a conical surface whose vertex angle, 8, is smaller than the angle of the original cone, el ,but tends to O1 for vanishing width of the fan. (3) Ferromagnetic alignment oblique to the field direction. (4) Fan in the meridian plane, obliquely disposed to the field direction.

343

THEORY OF HELICAL SPIN CONFIGURATIONS

Y

I I

Q252I

1

I

I

I

2 -

I -

36

36

1

36

(b)

FIG.3. Examples of structure changes due to a field applied along one of the easy axes in the easy plane (after Kitano and Nagamiya). (a) PZ = [J(Q) - J(2Q)]/[J(Q) J ( O ) ] = 8; (b) 8 2 = 3. The abscissa measures the strength of the anisotropy energy of sixfold symmet.ry within the easy plane: X = Ve/BS[J(Q) - J(O)], where V6 is the coefficient of the one-ion anisotropy energy of the form -VB cos 64. The ordinate measures the field strength: Y = H/2S[J(Q) - J(O)]. When H is increased, a firstorder transition from helix t o fan occurs a t H:, and then a second-order transition (left of the broken vertical line) or a first-order transition (right of the same) occurs from fan to ferromagnetic alignment. For high values of X , a direct first-order transition occurs from helix to ferromagnetic alignment. A broken, nearly horizontal line is a n approximation to H,' and to the field of direct transition helix-ferromagnetism. Lines in the fan region are contours of constant amplitude of the fan, the attached numerical a). values indicating E defined by sin(&/2) = 5 sin(Q.R.

+

344

T A K E 0 NAGAMIYA

4

t

FIG.4. Structure changes from a conical spin arrangement due to a n applied field. T h e anisotropy energy is assumed to be a function of the polar angle 0 only, to have a single minimum at a n angle between e = 0 and 0 = ~ / 2 and , to vary smoothly without showing pronounced fluctuations. Arrows indicate the path of structure changes with increasing field. The direct path going to the right is taken when the minimum of the anisotropy energy is shallow, and the downward round path is taken when the minimum is deep.

(5) Fan in the meridian plane, symmetrically disposed with respect to the field direction. (6) Ferromagnetic alignment in the field direction. The fifth structure may appear when the anisotropy energy has a maximum at the horizontal plane (0 = 7r/2). If it has a minimum (the second minimum) at 0 = 7r/2, a fan in the horizontal plane, rather than in the meridian plane, may appear. A brief discussion of the appearance of these structures will be given below. Denoting by en and cpn the polar and azimuthal angles of the nth spin, we can write the total energy of the system as

E

=

c c J(Rmn)[cos em cos + sin -SH c sin cos + w (cos2 -S2

8,

m

0, sin 8, cos (pm - pn)]

n

On

(11.1)

0,).

cpn

n

n

To derive a fan structure, we put 8, = 0 (on

=

+

~ ~ C (Rn.Q O S

%sin (Rn-Q

+ a),

+P),

(11.2)

THEORY OF HELICAL SPIN CONFIGURATIONS

345

and substitute these into (11.1). Then, up to the second powers of l and 4, we have

E/N

=

-S2J(0)

-

+ w(cos20) - J ( 0 ) } sin28 + SH sin O]t2 - J ( 0 )1 + SH sin 0 + (d2~/d02)][2. (11.3)

SH sin8

+[-2S2(J(Q) +[-2S2(J(Q)

If 5 2 and [2 are neglected, the value of 0 is determined from the minimum of the first line of (11.3) , namely, from -SH cos e - 2w’(cos20 ) cos 0 sin e

=

0,

or

SH

=

-2w’(cos28) sine.

(11.4)

The coefficient of t2in (11.3) can, therefore, be written as -[2S2(J(Q)

- J ( 0 )]

+ 2w’ (cos2O ) ] sin20.

(11.5)

By (6.4), this vanishes a t e = el , where el is the angle of the cone at zero field. Since w has a minimum near this angle, dw/dO is an increasing function of e near el , namely, w’(cos20 ) is a decreasing function of 0 near 81 . Thus, (11.5) is negative for e < el and positive fore > el . This would mean that t is nonvanishing for 8 < el and vanishing for e > el . The field, H I , corresponding to el can be calculated from (11.4) to be

HI = 2S[J(Q) - J ( O ) ]sin81 ,

(11.6)

where -2w’(cos2&) was replaced by 2S2[J(Q) - J ( O ) ] , since (11.5) vanishes for e = el. Below this field, t would be nonvanishing, and one might have structure 2, whereas above this field structure 3 might be realized. However, before concluding this, one has to make a few more considerations. The coefficient of l2in (11.3) must be positive in the neighborhood of el in order to have structures 2 and 3, as otherwise [ would be nonvanishing, and an oscillation in the &direction would take place simultaneously. Furthermore, the energy (11.3) a t 01 with $. = p = 0 must be lower than the energy of structure 1 extrapolated to H = H I . It can be shown after some calculations that the former energy is actually lower than the latter energy when the coefficient of l2is positive at 01 . Thus, with a t 81 , one obtains the assumption of the positive sign of the coefficient of I2 structures 2 and 3. The amplitude 5, of the cp-oscillation, can be calculated when the t4-term is included in the energy expression. It is more convenient to define t by sin ( d 2 )

=

(sin (Rn-Q

+ P)

(11.7)

than by (11.2). With this definition of t, the t4-term can be calculated

346

T A K E 0 NAGAMIYA

rather easily, and it can be shown that this term is expressed as

+ X 2 [ 3 J ( Q ) - 2J(O) - J ( 2 Q ) I sin2O a t 4 ,

(11.8)

which is positive. Thus, we can calculate l2below H I as a function of H or 0 by minimizing the energy with respect to t. When the coefficient of {2 in (11.3) a t 01 is negative, it can be shown that the energy (11.3) at el with 6 = { = 0 is higher than the energy of structure 1 extrapolated to H = H I . Although no detailed calculations have been made in this case, it is very likely that structure 1 persists beyond H1 and transforms discontinuously at a higher field to structure 4 or 5, or even 6. We can write the coefficient of l2in (11.3) , using (1 1.4) , as -282[J(Q) - J ( O ) ] - ~ w ’ ( c o s cos2 ~ ~ e)

+ 4w”(cos2e) sin2e cos28.

This is equal to -2S2[J(Q) - J ( O ) ] a t 0 = n/2, which is negative. For 0 close to 7r/2, it must also be negative. Thus, one has an oscillation in the .%directiona t a field that makes 0 close to or equal to s / 2 . Structures 4 and 5 result in this way. At a field higher than that which makes 0 equal to 7r/2, (11.4) is no longer valid, but the coefficient of t2in (11.3) will still remain negative until the field reaches a value, H o , given by Ho

=

2 X [ J ( Q ) - J ( O ) ] - S-l(dZW/de2)

10=r/z.

(11.9)

At this field, structure 6, namely, the ferromagnetic alignment in the field direction, will set in, and it will persist beyond Ho . These predictions are, however, valid only when the anisotropy energy, w, has a maximum at 0 = r / 2 , in which case the last term of (11.9) is positive. If w has a mini, term is negative and the coefficient of E2 in (11.3) mum at e = ~ / 2 this for 0 = n / 2 is negative a t H o , given by (1 1.9) , and will remain negative up to a higher field equal to 2 S [ J ( Q ) - J ( O ) ] . Thus, in this case, one should have a fan in the horizontal plane, rather than in the meridian plane, up to H = 2 X [ J ( Q ) - J(O)]. 12. EXPERIMENTAL OBSERVATIONS

The first example in which magnetization processes were observed by neutron diffraction is an ordered alloy MnAu2 . This was studied earlier by magnetic measurements by Meyer and Taglang”: there was a rather steep rise in the magnetization curve at about 10 kOe, followed by a gradual increase going to saturation (polycrystalline material). This behavior of the magnetization curve is called metamagnetic. By neutron diffraction experiment, Herpin and MBrieP studied structure changes in this alloy with l7 A. J. P. Meyer and P. Taglang, J. Phys. Radium 17, 457 (1956).

THEORY OF HELICAL SPIN CONFIGURATIONS

347

the application of a magnetic field, and they discovered the fan structure as the intermediate phase. They were also the first to propose the theory of the transitions from helix to fan and then to ferromagnetic alignment. The crystal of MnAuz consists of a tetragonal body-centered lattice of Mn having two (001) layers of Au between adjacent (001) layers of Mn ( a = 3.37 A, c / a = 2.60 a t room temperature). The NBel temperature TN is 365°K. The spin structure is helical, rotating in the c-plane and propagating in the c-direction. The turn angle measured by neutron diffraction is 51" at 300"K, which corresponds to a period of seven layers of Rfn, but decreases to 46" at 125°K and then increases to 47" at 87°K. The moment value of each spin extrapolated to T = 0 is 3.5 c(B . With an applied field, the helical structure changes abruptly to the fan structure at 10 kOe, the angle of the fan diminishing with increasing field and finally vanishing a t about 15 kOe. Although their samples were polycrystalline, they could select, by applying the field parallel or perpendicular to the scattering vector, reflections only from those particles for which the field was within the c-plane. However, there must have been particles for which the field was along the easy, hard, and intermediate directions in the c-plane. For rare-earth metals Dy and Ho, structure changes with applied magnetic field have been observed in detail with single crystals. These will be described in the forthcoming Part 2. A brief account will be given here, however. In Dy, in the range of temperature where the spin structure is helical a t zero field, there is evidence that simple helix-fan-ferro transitions occur when the field is applied in the hexagonal basal plane (the plane of the spin rotation). I n Ho, in the range between 133OK ( TN)and 80"K, in which there is no appreciable anisotropy energy within the hexagonal basal plane, also simple helix-fan-ferro transitions have been observed, but below 80"K, where there is an appreciable anisotropy energy in the plane, two intermediate fan structures have been observed. The original helical structure and these two fan structures all have different periods in space. In our theory, the function J ( q ) was assumed to be independent of the applied field, and so was the value of Q which makes J ( q ) maximum. The variation of Q with changes in the spin structure must be related to the state of the conduction electrons, but this is not yet fully understood. Also, our theory does not predict the appearance of two intermediate fan structures. I n Ho, in a small temperature range above 20°K, there is a single transition from helix to ferromagnetic alignment. This can be understood theoretically (see Figs. 3a and 3b in a high-anisotropy range). Below 20"K, we have a conical spin structure in zero field, and this transforms to a ferromagnetic alignment oblique to the field (structure 3) and then to the ferromagnetic alignment parallel to the field (structure 6).

348

TAKE0 NAGAMIYA

V. Spin Waves in Various Configurations in an Applied Field

I n this Chapter, we shall study spin waves in the following three spin configurations : (1) Conical arrangement arising from applied field or anisotropy energy or both, (2) Fan arrangement due to an applied field, (3) Helical arrangement in a weak field.

We shall particularly study those modes and frequencies that are resonant to an imposed oscillating magnetic field, for which we have to take account of the oscillating demagnetizing field. Spin waves in these structures have been studied mostly by Cooper, Elliott, and ~ o - w o r k e r s . ~ ~ J ~ J ~ ~ Taking a particular spin, S, , we denote its equilibrium direction by p, the direction perpendicular to this and the z-axis (cone axis in the first case, normal t o the plane containing the spin vectors in the second and third cases) by t , and the direction perpendicular t o { and by 7; the positive &direction will be assumed to coincide with the direction of right-hand rotation about the z-axis. Then, referring to Chapter 11, we can write

S,r

=

S

SnE

=

(3S>’”(an*

S,,

=

i(4S)1’2(a,*- a,).

- an*an,

+

an),

If we denote by e and (P, the polar and azimuthal angles of the equilibrium direction of S, (the polar axis being the z-axis), we can write

S,,

=

- S n ~sin (P, - S,, cos e cos (P,

S,,

=

+S,E cos (P, - S,, cos e sin (P,

S,,

=

S,, sine

+ Snr sin e cos

+ Snrsin e sin

(P, (P,

, ,

(V.2)

+ Snrcos 8.

With these expressions, we write the exchange energy, the anisotropy energy, the Zeeman energy, and the demagnetizing energy in terms of an* and a,. I n calculating the spin-wave frequences and modes, we take up only those terms which are quadratic in a,* and a, , and transform these variables to a,* and a, , defined by (11.6). Then we diagonalize the resulting Hamiltonian with another transformation. The condition for equilibB. R. Cooper, R. J. Elliott, S. J. Nettel, and H. Suhl, Phys. Rev. 127,57 (1962); B. R. Cooper, Proc. Phys. Soc. (London) 80, 1225 (1962). l9 B. R. Cooper and R. J. Elliott, Phys. Rev. 131, 1043 (1963). l Q O Most of their results had been obtained by A. Watabe, H. Miwa, and K. Yosida [reported i n a now discontinued mimeograph journal, Russeiron Kenkyu (in Japanese), April 19611.

349

THEORY OF HELICAL S P I N CONFIGURATIONS

rium follows either by putting the linear terms of the Hamiltonian equal to zero or by minimizing the constant term. We take the Hamiltonian to be X = -CJ(Rmn)Sm.Sn C [W(Sn) - H*Sn]

+

n

m,n

+ demagnetizing energy,

(V.3) where w (S,) is a one-atom type anisotropy energy, and we have written H for gpBH (more exactly, -gpBH, but then we may reverse the direction of H) . We shall consider the oscillating demagnetizing field only for uniform modes. Assuming an ellipsoidal shape of the sample, whose principal axes are along 2, y, z, and the volume of the sample to be 1, we can write the demagnetizing energy as ;[Nz(gpB

C X n z ) 2 + Ny(gpB C #nu)’ n

+ Nz(gI*B C

Snz)2],

n

n

+ + N,

where N, , Nu , N, are the demagnetizing factors (N, Nu However, t o simplify the notation, we shall write this as (1/2N)CNz( C n

Snz)2

+ Nu( C Snu)’ + Nz( C n

=

4a).

(V.4)

Snz)2],

n

where N is the number of atoms in unit volume; N, stands for N(gpB)ZNz, etc. 13. CONICAL ARRANGEMENT; FIELDPARALLEL TO z

If we have initially a proper screw structure, a field applied normal to the screw plane, namely, parallel to z, will change it to a conical one. If we have initially a conical arrangement, such a field will decrease the cone angle. I n either case, the cone angle, 8, is given as a function of H, and it depends also on the functional form of w ( S n ) .We assume that w(S,) is axially symmetric so that it is a function of S,, only (an even function of Sn,). The equilibrium direction of S, , or the value of 8, can be determined by considering S, as a classical vector and minimizing the total energy, (V.3), with respect to 8, where qn is put equal to QsR,. To determine 8, we put Xnr = X, Snc = S,, = 0, qn = Q-R , in the expression of the total energy and have -N[S2sin28J(Q)

+ S2cos2eJ(0)] + N ~ ( S c o s 8 )- N H S c o s e

+ +NN,S2cos28.

(13.1)

We differentiate this expression with respect to 8 and put it equal to zero: - ~ S C O S ~ [ J ( Q-) J ( O ) ] - w’(Scos8)

+ H - N,Scos8

=

0.

(13.2)

350

TAKE0 NAGAMIYA

Furthermore, out of the total energy we pick up

C [W (Snz)

- HSnz]

+ (1/2N) Nz ( C Snz)

(13.3)

n

n

and expand this around Snr = S , S,, = 0 in powers of S - Snr and S,, . Since S - Snr = an*anis a second-order quantity in the spin-deviation operators, a,* and a, , and S,, is a first-order quantity, we keep only those terms which are linear in S - Snr and Then we have for (13.3)

C { --w’

x,. ( S cos e) cos e ( s - Snr) + +w”(S cos e) sin2 ex:,

n

+ H cos e ( S - Snl) - N,S

C O S ~e ( S

+ ( 1 / 2 ~ ) sin2 ~ , e( C Snv12,

- Snr) }

n

or, with the use of (13.2),

C (2Sc0s2e[J(Q) n

- Snr)

- J(O)](S

+ ( 1 / 2 ~N) , sin2e ( C

+ DS:,,)

’,

(13.4)

snq)

n

where

D

=

+w”(S cos 0) sin28.

(13.5)

It is noted that H has disappeared in (13.4), although e is a function of H determined by (13.2). The total energy is now the sum of the exchange energy, the energy (13.4) , and the z and y parts of the demagnetizing energy. Those terms of the total energy which are quadratic in a,* and a, can be calculated to be the following:

C J ( Rmn)S ( [cos2 0 + sin20 cos Q m,n

R,,]

(a,*a,

+ a,*a,)

+ (1 + cos20) cos Q*Rmn](~m*~n+ an*am) + + sin2e[1 - cos Q - R,,] (am*an*+ a,a,) + i cos 0 sin Q - R,, (&*an - an*a,) } - +[sin2 0

+ C 2 s cos20[J(Q) - J(O)]an*an - C $DS(an* - an)’ + (1/4N)NzS[~(a,* + a,) sin Q-R, + i(an* - a,) cos 0 sin Q.Rn12 + (1/4N)N,S[~(an* + a,) cos Q-R, - i(a,* - a,) cos 0 sin Q.RRI2 n

n

n

n

- ( ~ / ~ N ) s N ,sin2e[C (a,* - an)]2. n

(13.6)

351

THEORY OF HELICAL SPIN CONFIGURATIONS

We make here the Fourier transformations: a, =

N-112

c

a, exp (iq. R,)

,

a,*

=

c

Nd1I2

9

a,* exp ( -iq.R,,).

9

Then, (13.6) becomes

C2Sa,*a,ICJ(Q) -

,

- 3J(Q

3J(Q -

+ q)l

+

- sin2e[J(q) - $ J ( Q - q) - + J ( Q

+ 30 + 3 cos CJ(Q + c S(a,*ai, + a&-,)

-

q) - J ( Q

+ s)I)

9

- +J(Q -

X {3sin2f(J(q>

+ q)]

- 3J(Q

+ q > l - 301

+ cos e)12 + a-a) (1 + cos e>12

- & ~ N , [ ( u Q *- uQ)(1 - cos e) - (u*Q - u-Q) (1

+ -&SNY[(a~*+ uQ)(1 - cos e) +

(uZQ

- + S N , sin28(ao* Excepting the terms with q

(13.7) =

0 and q

=

.tQ, we write this as

c 2S[Aqaq*aq+ ~B,(a,*a*, + a,a-,)].

(13.7a)

9

It is noted that A , # A_, due to the existence of a cos 8 term in the coefficient of aq*aqin (13.7), but B, = B-, . To diagonalize (13.7a), we make the following transformation (see Chapter 11): a, = a, cosh e, CUT,sinh 0, , (13.8) as* = a,* cosh e, a_, sinh 0, , with (13.9) coth 28, = ( A , A - , ) / ( -2B,).

+ +

+

Then (13.7a) becomes

2 S [ A , cosh28,

+ A _ , sinh28, + 2B, cosh 0,

sinh ~J,]cx,*cx,,

Q

or

S [ ( A , - A-,)

+ ( A , + A-,)

cosh 28,

+ 2B, sinh 2e,]a,*a,.

9

(13.10)

352

TAKE0 NAGAMIYA

Since cosh 28,

=

(1 - t a d 228q)-1/2 =

sinh 28,

=

(cosh2 28, - 1)--1/2 =

A,

+ A-,

[(A,

+ A_,)’

[(A,

+ A-,)2

- 4Bq2]1/21

-2Bq

- 4Bq2)1’2’

+

provided that A , A_, > O,*gb the coefficient of aq*aqin (13.10) , which is h times the frequency, is calculated to be the following: fiw, =

S(Aq - A-,)

+ S[(A, +

-

4Bq2]1/21 (13.11)

or

50,

=

S cos 8[J(Q - q)

+ 2SCJ(Q)

- J(Q

+ q)]

+

- +J(Q - q) - +J(Q q)]1’2 X {J(Q) - sin2OJ(q) - 3 cos2O[J(Q - q) J(Q

+

+ a)] + DJ1l2. (13.12)

In order that the assumed conical configuration be stable, all the frequencies must be positive. The first term of (13.11) or (13.12) changes sign when q and -q are interchanged, so that the absolute value of this term must be smaller than the second term. For small values of q, the first term is of order of q3, since J ( q ) is maximum at q = Q, whereas the second term is of order of q, so that the first term is surely smaller than the second term for small q. Also, near q = Q and q = -Q, it is easy to see that this condition is satisfied. The frequency near q = . t Q is small when e is close to 7r/2 and D is small. We are particularly interested in the cases of q = 0 and q = =tQ. In the case of q = 0, we have the last term of (13.7), but despite this, A . Bo vanishes when the Hamiltonian is written in the form of (13.74 ;hence the frequency, of the form of (13.11), vanishes. This is to be expected, since the mode of oscillation for q = 0 is such that the cone rotates as a rigid body about its axis (the z-axis). If we had an axial anisotropy (of hexagonal symmetry, for instance), the azimuthal, as well as the polar, distribution of the spin vectors at equilibrium would not be uniform; nevertheless, a rotation of the system about the z axis, such that the nonuniform distribution of the spin vectors on the undulatory cone surface remains unchanged in the vector space, would produce just a shift of the spin pattern along the z axis, so that the frequency of such a spin-wave mode would

+

19*

+

If A , A _ , is negative, we have a minus sign before the square root quantity; in this case, we have negative frequencies, so that the system is unstable.

353

THEORY OF HELICAL SPIN CONFIGURATIONS

be zero.’SCHowever, this is possible only when the period of the spin pattern is irrational in units of the lattice spacing, in which case the spin vectors cover the cone surface uniformly. If the period is rational, a finite number of spin vectors will fit in each potential valley, and the rotational oscillation will be coupled with a pulsation of the cone angle; hence the frequency will be nonvanishing. Evidently, such a motion would be active to oscillating magnetic field parallel to the z-axis. For q = &Q, we have extra terms in (13.7) due to the demagnetizing fields in the x and y directions. Thus, the total expression (13.7) has the form

+

+ + C(UQ*U-Q+ u?QuQ) + + uQuQ) + +D-(u?Qu?Q +

+

A+uQ*uQ A-U?QU-Q B(UQ*U?Q

UQU-Q)

+D+(uQ*uQ*

U-QU-Q)

. (13.13)

In the case of a sample that is axially symmetric about the z-axis, C , D+, and D- all vanish ( N , = N , ) , so that the frequency is expressed in the same form as (13.11) with q = Q or q = - Q . However, in this case, A,Q and BQ contain demagnetizing terms :

A*Q

=

+ c0s2e ) [ J ( Q ) - V ( 0 ) - V ( 2 Q ) + + s cos e [ J ( o ) J ( 2 Q ) - t(NZ+ N u ) ] ,

S { (1

$ ( N Z

+ Nu)] + D } (13.14)

-

BQ = S(sin28[J(Q) - *J(O) - 4J(2Q) - + ( N ,

+ N,)]

- D}. (13.15)

Corresponding to WQ and W-Q there are two modes. For WQ , one puts (Y-Q = CY?Q = 0 in (13.8) and obtains UQ/U?Q = cosh OQ/sinh8Q , or (UQ

+

U*Q)/(~Q -

=

~ Z Q )= SQE/~SQV (cash 6Q + Sinh 8,) /(cash 8Q - Sinh 8,)

=

cash 268

=

[AQ

+ Sinh 2 8 ~

+ A-Q - ~ B Q ] ” ~ / [ +A QA-Q + ~ B Q ] ’ ” .

(13.16)

For W-Q ( C ~ Q = C ~ Q * = o), cosh 6Q and sinh 8Q are interchanged, and one Since , , Sntand S,, for the has the negative of the ratio (13.16) for S Q ~ / ~ S.Q +Q mode vary as I S Q I~cos ( Q * R n- w Q ~ )and I SQ,,I sin ( Q . R n - W Q ~ ,) The writer is indebted for this explanation to Dr. Elliott, who has pointed out (privately) that there is always one mode with zero frequency. This is confirmed in Section 14 by an independent direct calculation for a special case. Cooper and Elliott,19 by an erroneous calculation, originally obtained a nonzero frequency for this case. See also the footnote 19g on page 361.

354

TAKE0 NAGAMIYA

respectively,lgd each spin vector describes an ellipse in the right-handed sense when one sees it opposite to its direction. The whole cone performs a right-handed rocking rotational motion about the z axis; in space, the wave propagates with wave vector Q. This mode is, therefore, active to a right-handed rotating electromagnetic field. For w-Q , the motion is just reverse in time, and it is active to a left-handed rotating electromagnetic field. 14. THEFANWITH

A

FIELDPARALLEL TO 2

In the fan, one has (P, oscillating with position R, , with wave vector Q;

e is ~ / 2 We . may put, for conveniencell* sin ((~,/2) = 26 sin (Q-R,) ,

(14.1)

where 6 is assumed to be small; it varies with field as ( H o - H)1’2,where HOis the fan-ferro transition field. In the following, we shall keep terms up to O(62). For the exchange energy, the following quadratic form in spin-wave amplitudes is obtained:

2s

C (CCJ(0) - J(dl + 4 W J ( Q )

- W(0)

9

+ c(q)llaq*aq

+ 262c(q)(%*a?q + aqa-9) - 2a2CW(Q) - J ( 0 ) - J(2Q) - C ( d l ( a ~ + Q % - Q &Q%+Q)

+ a2C(q)
(14.2) where

C(q)

=

J ( q > - +J(Q - q) - +J(Q

+ d.

(14.3)

We consider an anisotropy energy of p-fold symmetry about the z axis 19d

The classical picture is presented here. It follows from (13.8) that, for WQ,

ag = UQ cosh Bg

,

a-Q = U Q * sinh B Q ,

a?Q = U Q sinh 0 0 ,

aQ* = UQ* cosh BQ

so that

+ an)

Snc = (S/2)”*(an*

+

+ ( a - g + ag*) exp ( -iQ.Rn) 1 exp(iQ-R,) + exp(-iQ.R,)]

=

(S/2N)1/2[ ( a ~ a?a) exp (iQ-Rn)

=

(S/2N)l/*(cosh BQ

LT

cos(Q.R, - wgt

+ sinh + const).

eQ)[UQ

UQ*

+

Similarly, S ,, varies as sin (Q.Rn - w ~ t const).

355

THEORY OF HELICAL SPIN CONFIGURATIONS

( p = 2, 4, 6) and assume it to have the simplest formlgB

C {DX,,'- p-2S-p+2G[(Sn, +

iXn,)p

+ (Sn, - iSn,)*]).

(14.4)

n

This can be written in an* and a, as

C {-$DS(an* n

- an)2

+ GS[(2/p)an*an + [ ( p - 1)/2p](an* + an)']

.[1 - 4p26'(1 - cos 2Q.Rn)]),

(14.5)

and in spin-wave amplitudes, as* and a s , as

DS

C (aq*as - $ U ~ * C L ? ~ $u,cL,) 9

+ GS C { [(l + p-')aq*aq + &(1- ~-')(u,*u?, + u ~ u - ~ ) ] -( ~4 ~ ~ 6 ' ) + 2p(p + 1)6'(az+Qas-Q+ a:-Qas+Q) 9

+ p ( p - 1)6'(at+Qa?q+Q+ a:-Q&-Q + aq+Qa-q+Q+ aq-Qa-q-Q) 1.

(14.6) From the second term of (14.4) arises an additional anisotropy energy that prevents the spin from deviating from the xy plane. To see it, let us assume that the spin points almost in the z direction. Dropping the suffix n for the sake of brevity, we can write this term as

198

-x+

p-ZS-p+zG[S - ( S - S,) f iSy]P.

(a)

Expanding this in powers of (S - 8,) and S , , and writing

+ 1) - S,2 + (l/2S) (S,* + 52)

S - S, = S - [S(S

- S2]"2 (b)

under the assumption of large S, we can calculate (a), up to the second order of S , and S, , to be GSy2 (1lp)GSz'. (C)

+

+

(l/p)G in the z Thus, the total anisotropy energy has an effective coefficient D direction and G in the y direction. This situation is also manifested in (14.5). Namely, the second term in the first line of (14.5), which has a factor in the second line arising from the fan distribution, can be written as

GS[t(a,*

+ a,)'

- (1/2p) (a,,* -

(d)

and by (V.1) this can be expressed as

+ (1/p)GSn,2,

GSntZ

(el

where is the same as 2 in the present case. I n Section 5, after Eq. (5.3), we assumed an GS,Z and calculated spin-wave frequencies for anisotropy energy of the form DS,2 the ferromagnetic alignment in the 2-direction. In the present section, D (l/p)G appears in place of D.

+

+

356

TAKE0 NAGAMIYA

Furthermore, we have the Zeeman energy :

-H

c

snz

=

H

n

c { (1 - 46')aq*Uq + 26'(a:+Qaq-Q +

d-Qaq+Q)

}

P

(14.7) where we understand by H the field that includes the static demagnetizing field. The dynamical part of the demagnetizing energy can be written for an ellipsoidal sample, whose principal axes coincide with the coordinate axes, as follows:

-8Nz a2(aQ*- UQ

- U?Q

+ SN, 6'(ao* + -*SN.(ue*

+

+ )sNv(l

U-Q)~

- 86')

(Uo*

+ + a*zQ + a-zQ)

%) (az*~

+

Uo)'

~ Z Q

- uo)'.

(14.8)

As before, this part has been considered only for uniform modes. In (14.8), there is a coupling term between mode 0 and modes &2Q, but we shall disregard it, since this off-diagonal term will contribute to the frequency of mode 0 a correction of the order of 64. The mathematical problem we have to solve is to diagonalize the sum of (14.2), (14.6), (14.7), and (14.8). We neglect all terms aq*uq~, u,*a?,~, and a,u-,# with q # q', since these off-diagonal terms will also contribute a correction of 0 (a4) to the spin-wave frequencies. Then, except for q = 0 and q = &Q, the total Hamiltonian can be written in the following form : 2s CA,a,*a, 3B, (a,*a?, a,a-q> 3, (14.9)

+

c

+

9

where

+ 46'CW(Q> - W ( 0 ) + C(q) - b b + 11'5'1 (14.10) + +Lo+ (l/p)G] + 3G + (H/2S) (1 - 46'1, (14.11) Bq = -+[D + (l/p)G] + $6 + 46'[C(q) - $ p ( p - l)G]. A,

= J(0)

- J(q)

Here A , = A_, and B, written as

A,

= J(Q>

=

B-,

. Using Ho defined by

(9.1), A, can be

+ 3Co + (l/p)G] - 36 - J ( 0 ) + C(q) - $ ( p - l)(p + 2)GI - [(Ho - H)/281.

- J(q)

+ 46'CJ(Q)

(14.12) The second line of this quantity is proportional to Ho - H, since 62 is proportional to Ho - H.

THEORY OF HELICAL SPIN CONFIGURATIONS

357

As before, we have nu, = 2S[(Aq

+

( A , - Bq)-J'/2.

Bq)

(14.13)

This formula is not applicable to the cases of q = 0 and q = =tQ for which we have to take account of demagnetizing effect. The additional terms due to this effect can be obtained from (14.8). For q = 0, then, we have nu0 = 2 S [ ( A o Bo) (A0 - Bo)1'12, (14.14) with

+

Ao

+ Bo

=

J(Q) - J ( 0 ) - 4S2[J(Q)

- (2S)-'(Ho - H )

+ ;N,(l

and

Ao - Bo

=

- J(0)

J(Q) - J ( 0 )

-

+ ( p 2 - 1)GI

8P)

(14.15)

+ [ D + (l/p)GI - G

+ 4a2[J(Q) - J ( 0 ) - (P - 1)GI -

(2S)-'(Ho - H )

+ +Nz.

(14.16)

It can be seen from the functional forms of (14.14)-(14.16) that w o decreases with increasing Ho - H . The mode of this oscillation is such that the spin vectors describe, in unison, ellipses of the same size in space whose principal axes are along 5 and z, as one may expect from the demagnetizing coefficients, N , and N, , contained in (14.15) and (14.16). This does not mean that the fan oscillates as a rigid body, in which case the ellipses described by the spin vectors on both edges of the fan will have a smaller semiaxial length in the z-direction than the semiaxial length of the ellipse described by the central spin vectors. Thus, the actual motion of the fan includes a bending oscillation of the fan plane. For this reason, N, appears fully in (14.16), not with a factor (cos pn), averaged over n. If one had just one moment vector whose magnitude is equal to the average moment of the fan in the field direction, then one would have had a factor (cos cpn> to both N, in (14.16) and N, in (14.15). I n actuality, N, appears with a factor (cos2qn), which is equal to 1 - 8P. This is because the fan has a finite angular width, and it oscillates in the direction of cp as if it were a rigid body. In the case of q = f Q,we have the spin-wave Hamiltonian:

+

&S[AQ(UQ*UQ u?Qu-Q)

+ BQ(uQ*u?Q+ u ~ u - Q )

+ C Q ( ~ Q * ~+- Q~ E Q ~ Q )

358

TAKE0 NAGAMIYA

with AQ = 6[D

+ (l/p)G]

- 3G

+ 46'[2J(Q)

- W ( 0 ) - 3J(2Q) - + ( p

- (28)-'(Ho

- H)

BQ = -3[D

- 1)( p

+ N,6',

+ 2)GI (14.18)

+ (l/p)G] + 3G

+ 46'[J(Q)

-

i J ( 0 ) - 3J(2Q) - $p(p - 1)G]

- J(0)

CQ = -2S'[W(Q)

-

J(2Q) - 3 ( p

+ N,P,

- 1)( p + 2)G]

(14.19)

- N#, (14.20)

DQ = -2S'[J(Q)

- J ( 0 ) - $p(p

-

1)G] - N , P .

(14.21)

To diagonalize (14.17), we put, with Cooper and Elliott,19J9fas follows: p + = ;(aQ

+ aEQ+ a-Q + aQ*)

=

+ 8-Q.E),

(2~)-"'(8Q.E

p- = $(aQ +,a?Q - a-Q - a&*) = (28)-"'(8Q,E - 8-Q,E), -'@+ =

in-

+ 8-Q,z),

3(aQ- a*Q + a-Q - aQ*) = -i(28)-"'(8Q,z

= ;(aQ - a?Q - a-Q

+ aQ*) = +i(28)-"'(8Q,z

-

(14.22)

8-Q.2).

These variables satisfy the commutation rules : Cq+ 7 P+l = L-q-

, P-I

=

i, (14.23)

CP+ P-1 9

= [q+

, q-I

=

[P+ , q-I

= CP-

7

q+I

=

0,

so that p+ , q+ and p- , q- are pairs of canonically conjugate variables. Then, the Hamiltonian can be written as

+ BQ+ CQ f DQ)p+' + $(AQ - BQ+ CQ - DQ)q+' + $(AQ + BQ CQ DQ)p-' + 3(AQ BQ - CQ + DQ)q-'],

28[h(AQ

-

-

-

(14.24) and, correspondingly, the frequencies are found to be

+ BQ + CQ + DQ)(AQ - BQ + CQ - DQ)]112, fro- = 28[(AQ + BQ - CQ - DQ)(AQ - BQ - CQ + DQ)]"'.

fiU+

19f

=

28[(AQ

(14.25) (14.26)

Cooper and Elliott obtained incorrectly CQ/DQ= -4/2 for G = 0 and N , Their results (see Cooper and Elliottlg) are thus different from ours.

=

0.

THEORY OF HELICAL SPIN CONFIGURATIONS

359

Here we have

AQ

+ BQ + CQ + DQ =

26'[3J(Q) - W ( 0 ) - J(2Q) - ( p z - 1)GI - (2S)-'(Ho - H),

which vanishes identically for HO> H (thus giving explicitly as a function of Ho - H ) . The mathematical proof of this can be made with a calculation similar to that leading to (10.9) for p = 6. However, the vanishing of w+ [see (14.25)] can be visualized by looking a t the corresponding = S-QEand SQ,= S-Q, , since p - = q- = 0, mode. For w+ , we have SQE and furthermore SQ, = S - Q ~= 0, if the coefficient of p+2 in (14.24) vanishes, and hence q+ must vanish; in the analogy of a classical harmonic oscillator, the vanishing of the coefficient of p+z means an infinite mass, and hence vanishing of the displacement q+ . Thus, the mode of oscillation is such that the central vectors in the fan structure oscillate in the zy plane with a maximum amplitude, and the vectors a t the edges are at rest. This is illustrated in Fig. 5a. This mode may, thus, be called the cosine mode. When one sees the spin vectors opposite to the x direction, their tops will be on a sinusoidal curve running along the x direction, as shown by a full curve in Fig. 5b, when there is no oscillation. When the oscillation occurs, this curve will shift to a broken curve shown in the same figure. As may be seen, the oscillation merely shifts the full curve up and down in the x-direc-

p'3

-1

FIG.5

FIG.6

FIG.5. Cosine oscillation of the fan vectors. (a) Projection of the vectors on the easy plane. (b) Profile of the mode seen against the field direction, the easy plane being horizontal. The full curve is the'locus of the tips of the vectors at rest. This curve moves up and down when the vecto:.s oscillate as shown in (a). The frequency of oscillation is zero. FIG.6. Sine oscillation of the fan vectors. (a) and (b) have meanings similar to those of Fig. 5. The full curve in (b) oscillates in amplitude, as shown. The frequency is finite.

360

TAKE0 NAGAMIYA

tion. No energy change should be associated with this motion, and evidently the frequency must be zero. For w- ,we have a sine mode, shown in Figs. 6a and 6b. I n this case, we have Sgt = -S-Qt and Soz = -S-Q, , since p+ = q+ = 0. Correspondingly, the vectors move in the xy plane as shown in Fig. 6a and, with a phase , in the z direction. They describe ellipses in space. The advance of ~ / 2also ratio between the diameters of the ellipses in the 2: and z directions is

( A Q - BQ - CQ

+ DQ)"'/(AQ+ BQ - CQ - DQ)'/'

[the square root ratio between the coefficient of q 2 and that of pP2 in (14.24)].The profile of the oscillation is as shown in Fig. 6b. The frequency, w- , varies as 6 , or as (Ho - H)*12. I n the ferromagnetic range for H > HO, we have just to put 6 = 0, leaving H - Ho as it stands. Then, for =tQl we have

h+ = fiw- = { ( H - Ho)[2X(D This vanishes at H

=

15. SPINWAVESIN

A

+ (l/p)G - G) + ( H - Ho)]]"'. (14.27)

Ho . HELIXSUBJECTED TO

A

WEAKFIELD

This case presents a situation so complicated that one becomes almost uninterested in the analysis. No calculation has been reported which includes both the weak applied field and a small anisotropy energy in the plane, although the effect of an applied field only has been studied rather . shall not enter in detail (Cooper and Elliottlg and Watabe et ~ 1 . ' ~ )We into detailed calculations and shall discuss only a few points of interest. If we assume an anisotropy energy of the form (14.4), or

C{ DSnz - p-'S-"+'G[ n

(Snz

+

+ (Snz -

isnu)

isnu)

"I}

1

where G is small and D is positive (not necessarily small), and assume a weak H in the x-direction, then the azimuthal angle, qn of the nth spin in equilibrium will be given in the form qn =

Q-R,

+ el sin Q - R , +

f p

sin p Q - R n.

(15.1)

is proportional to H , and ep is proportional to G. To second order in H and G, the classical total energy (exchange, Zeeman, and anisotropy energies) per atom can easily be calculated to be the following: el

S'{ - J ( Q ) -J((p

+ tel"[w(Q) - J ( 0 ) - J ( 2 Q ) I + &p'[w(Q) - 1 ) Q ) - J ( ( p + 1 ) Q ) ] + (2S)-'He1 + p-'Gep).

(15.2)

THEORY OF HELICAL SPIN CONFIGURATIONS

361

Minimizing this expression with respect to el and ep , we obtain €1

=

-H/(XPJ(Q)

- J(0) - J(2Q)ll,

ep

=

-2G/ipPJ(Q)

- J((P - 1)Q) - J ( ( P

(15.3)

+ 1)Q)I)-

(15.4)

The spin-wave Hamiltonian to second order in H and G can be obtained without difficulty; the Zeeman and anisotropy terms are rather simple, but the exchange terms are fairly complicated. This Hamiltonian involves not only momentum-conserving terms, aq*aq, a,*a*, , U - ~ U , , but also ~ with momentum-nonconserving terms such as aq*a,t , U ~ * U -,~a-,a,~ Q' = Q, f 2 4 , q f (P - 1>Q,q f (P l > Qql f pQl q f ~ P Q This means that Bragg reflections occur at q = f $ Q , q = f Q , etc., because of the modulation in the helix, of the form of (15.1), caused by the external and anisotropy fields. The spin-wave frequencies are drastically modified at and near these Bragg points in the q-space. Not only near the Bragg points but also at points far apart from them, the frequency formula cannot be obtained in a simple way, because, although the coefficients of the momentum-conserving terms depend only linearly on H2 and G2, those of the nonconserving terms contain linear (as well as quadratic) forms of H and G, so that one has to include the nonconserving terms in the secondorder perturbation. Thus, the spin-wave frequency generally depends quadratically on H and G in a very complicated way. Like the q = 0 mode in the case of a conical arrangement and the cosine mode with q = .tQ in a fan arrangement, here we have also a mode with zero frequency, if the period of the spin pattern is irrational in units of the lattice spacing along the helix. The reason is similar to that mentioned in Chapter V, Section 13 in connection with the q = 0 mode and in Chapter V, Section 14 in connection with the cosine mode. Another way of reasoning the existence of a zero frequency mode in any case of an irrational period is the f~llowing.~g~ The spin pattern in the existence of an applied field and an anisotropy energy can be pictured by specifying the direction of the spin vector at R, as a function of Q-R, with a period of 27r. Since the period of the spin pattern was assumed to be irrational, Q.R, covers uniformly the whole angular range of 0 to 2~ when R, runs over all the atoms in the infinite crystal (which should have a uniform, infinite cross section perpendicular to the direction of Q) . I n such a case, the choice of the origin of R,'s, whether on a net plane or between net planes, is immaterial for the spin pattern; the same function of Q-R, with different choices of the origin gives spin patterns that are translated along Q with

+

+

Elliott and Langelgh give a mathematical proof of the existence of a zero frequency and the continuity of the frequency spectrum near zero. l g h R. J. Elliott and R . V. Lange, Phys. Rev. 162, 235 (1966).

l9g

362

T A K E 0 NAGAMIYA

respect to each other. I n other words, we can add an arbitrary phase constant, a,to Q-R, . A change in a just shifts the spin pattern in space, not always a small shift even if this change is small. However, no energy change should accompany it. The spin wave mode corresponding to a small change in a has, therefore, zero frequency. It may be noted that this spin-wave mode does not always correspond to a single q-value, namely, 0 or Q, but it consists generally of many harmonics. VI. Complex Spin Configurations

So far, we have confined ourselves to a simple helical spin arrangement and its modifications caused by anisotropy energy and an external magnetic field. It may be recalled that we have assumed lattices in which there is only one magnetic atom per unit cell. If we now have lattices whose unit cell contains several magnetic atoms, we may expect more complex spin arrangements. Yafet and Kittelm predicted for the spinel structure a triangular spin arrangement, which was subsequently observed by a neutron diffraction experiment in CuCrzOl .21 Kaplan et ~ l . ~ ~ a pre--f dicted, in 1961, for the same crystalline structure a ferrimagnetic multiple cone spin arrangement, which was then observed in MnCrzOl .23 We shall study such problems in this and the next parts. Our approach will be somewhat different from that followed by the authors mentioned, although it will depend upon their treatment. Throughout (except in Part VIII), we shall confine ourselves to absolute Let S,, be the spin vector of the vth atom in the nth unit cell and R, its position, where we understand by Rn a translational R,, = Rn lattice vector. The exchange energy will be written as

+

(VI.1) Y. Yafet and C. Kittel, Phys. Rev. 87, 290 (1952). E. Prince, Actu Cryst. 10, 554 (1957); R. Nathans, S. J. Pickart, and A. Miller, Bull. Am. Phys. SOC.[a] 6, 54 (1961). *la T. A. Kaplan, Phys. Rev. 119, 1460 (1960). 2zb D. H. Lyons and T. A. Kaplan, Phys. Rev. 120, 1580 (1960). 22cT.A. Kaplan, K. Dwight, D. H. Lyons, and N. Menyuk, J . Appl. Phys. 32, 13s zo

21

(1961).

zsdD.H. Lyons, T. A. Kaplan, K. Dwight, and N. Menyuk, Phys. Rev. 126, 540 (1962). BsN. Menyuk, K. Dwight, D. Lyons, and T. A. Ka.plan, Phys. Rev. 127, 1983 (1962). 2zfT. A. Kaplan, H. E. Stanley, K. Dwight, and N. Menyuk, J . Appl. Phys. 36, 1129 (1965). 2s

J. M. Hastings and L. M. Corliss, Phys. Rev. 126, 556 (1962). A concise presentation of Parts V I and VII, together with a few other materials, will be published by T. Nagamiya in J . Appl. Phys. S u p p l . (1968) (Proc. Intern. Congr. Magnetism, Boston, 1967).

363

THEORY O F HELICAL SPIN CONFIGURATIONS

We make Fourier transformations of the spins: S,, = S,

C d,,

(d-,” = d;,, v = 1, 2,

exp (iq-R,)

...,k).

(VI.2)

P

Here k is the number of atoms per unit cell. Then (VI.1) can be written

E

=

-C C C q

SpSvJmp,nv

exp (iq*Rmn)dqp.div,

(VI.3)

m . p n.v

where Rm, = Rm - Rn . This can be written also as EIN

c c c APY(q)d,p-d:Y ,

=

9

P

(VI .4)

V

where N is the number of unit cells and

Apv(q) = -SpSv

C

Jmp,nv

exp (iq*Rnn)

m

=

AL(-q)

= AvJ-q).

We want to minimize (VI.4) under the condition that St, which can be written as

[Ed,, exp (iq.R,)I2

=

1

(VI.5) =

S,2 for all n,

for all n and v.

(VI.6)

9

We shall study separately a number of cases.

16. CASE1 : THERE Is

A

SINGLEq THATIs EQUIVALENT TO -q

Such a q is either zero or equal to half a reciprocal lattice vector; in other words, q is at the origin of the reciprocal lattice space or at a symmetry point on the Brillouin zone boundary. If such a q minimizes the energy, then the problem is simplified to

EIN

=

C C Apv(q)dqp*dqv = min, c

(16.1)

v

d& = 1

for all v.

(16.2)

Here all quantities are real. If there are two atoms per unit cell, one will have EIN = All(@ A d q ) 2A12(q)dql.dqz = min,

+

+

from which will follow that dql and dqz are parallel or antiparallel according as A12(q) is negative or positive. The q-vector must be such that it correAzz(q) - 2 I Alz(q)). If q = 0, sponds to the lowest value of &(a) then S,, = S v d q v (v = 1, 2), so that one has a ferromagnetic or ferrimagnetic (antiferromagnetic when S1 = Sz) spin arrangement according

+

364

TAKE0 NAGAMIYA

as Alz(q) is negative or positive. When q is half a reciprocal lattice vector, one has exp (iq.R,) = +1 or -1 according as q-R, is an even multiple of T or an odd multiple of T , so that both the two sublattices are antiferromagnetic. When there are three atoms per unit cell, we can show that the three vectors d q l , d q z , dq3 must be coplanar for the minimum of the energy. If, in particular, the three sublattices are symmetrically related in such a way that AZ3(q)= &(q) = Alz(q), the three vectors are determined from &(q) (dqz.dq3

+

dq3.dqi

+

dqi-dqz) =

min,

which is equivalent to A23(q) ( d , l + d,z

+

d,d2 = min.

Thus, when Az3(q) is negative the three vectors are parallel, and when A23(q) is positive, they add up to zero. I n the case of q = 0, one has a ferromagnetic or a triangular spin arrangement corresponding to the negative or positive sign of A23(q). In the case of q equal to half a basic reciprocal lattice vector, one has three antiferromagnetic sublattices or an antiferromagnetic triangular arrangement. 17. CASE2: THEREIs

A

SINGLEq THATIs NOTEQUIVALENT TO - q

I n this case, we have the following problem:

E/N

=

c I

r

~[A,,,(q)d,,,.d& v

[d,,

exp (iq-R,)

+ complex conjugate]

+ d:,

exp ( -iq.R,)I2

=

=

min,

1.

(17.1) (17.2)

The complex vector d,, may be written as a combination of two real vectors, u,, and v,, : (17.3) d,, = +(u,, - iv,,,). Then we can write the condition (17.2) as

[u,, cos q-R,

+ v,,

sin q.RnI2

- 1~ u2, , ( l +cosZq-R,) + + v i v ( l - cos2q-R,) +uq,.vq,sin2q.R,

=

1.

From this equation follows that u,, and v,, are orthogonal unit vectors, except when Zq-R, is an integral multiple of T , namely, q is a quarter of a basic reciprocal lattice vector (q = K/4). In this exceptional case, u,, and v,, need not be orthogonal, since sin 2q.R, = 0, but they have to be unit vectors. Since S,, = Xy(uqycos q-R, v,, sin q.R,) , (17.4)

+

each sublattice has a helical spin arrangement of wave vector q. In the

365

THEORY OF HELICAL SPIN CONFIGURATIONS

case of q = K/4, it can be seen from this equation that each sublattice splits into two uncorrelated antiferromagnetic sublattices having u,, and v,, as magnetization axes. The energy (17.1) can be written as

E/N

=

C C t[Ap(q) P

(uq,*uq.

+

vqp.vqv

- iVpp*Uqv

+

iu,,.vpY)

V

+ complex conjugate] = c A””(q) + c CCRe A,dq)

(U,,*%V

C<”

Y

+ v,,-v,v)

+ Im A d q ) (vqP-uqv- ~ , , - v q v ) 1 .

(17.5)

It is not easy to solve generally the problem of minimizing this energy. However, in the case of two atoms per unit cell, it can be shown easily that the two sets of orthogonal unit vectors, uql, v,1 and uqZ, V,Z , lie in the same plane, and the sense of rotation in going from h1to vql is the same as that from uqZto vqZ, that is, the helical rotation has the same sense in the two sublattices. Actually, as we shall see in Section 19, these facts are valid in a general lattice. The relative phase angle, C#Jl2 , between the two helices in the case of two sublattices is determined from Re A l ~ ( p )cos C#Jl2

+ Im A ~ z ( qsin )

412

=

min,

namely, as 412

=a

+

where Alz(q) = I Alz(q) I

T,

(i.1.

~ X P

(17.6)

The corresponding energy is

E/N

=

Aii(q)

+ Azz(q) - 2 I Aiz(q) I.

(17.7)

The value of q corresponds to the lowest value of this energy. The value of q in case 1 also corresponded to the lowest value of the same expression. When q is a quarter of a reciprocal lattice vector (q = K/4), the foregoing argument fails partly. I n this case, each of the two sublattices splits into two uncorrelated antiferromagnetic sublattices, as mentioned before. u,1 and vql point arbitrarily, but u,2 and vqz are related to u,1 and vql . If 1 Re Alz(q) I > I Im A,,(q) 1, then uqZand vqZare parallel and antiparallel to uql and vql, respectively, according as Re A12(q) is negative or positive; if I Re Alz(q) I < 1 Im Alz(q) 1, uqZis parallel to vqland v,z antiparallel to uqlwhen Im A12(q) is negative, whereas antiparallel and parallel are reversed when Im Alz(q) is positive. I n the case where there are three sublattices symmetrically related to each other, so that A23(q) = &(q) = Alz(q) and Aij(q) = Aji(q) hold,

366

TAKE0 NAGAMIYA

we have a problem:

+ dq3'dil + dql.dlz) + c. c. A23(q)I d q l + + l2 min.

A23(q) (dqZ'dl3 or

dq2

dq3

=

min,

=

It follows that, if AZ3(q)is negative, d q l , d q 2 , dq3 point parallel to each other, so that there is a triple helical spin arrangement. If &(q) is positive,

+

+

dql dq2 dq3 must vanish, so that these three vectors form a triangle; the corresponding spin arrangement is such that this triangle rotates along the direction of q. An example of such a triangular helix has been observed in MnaSn at low temperatures by neutron diffracti~n.~~ This crystal is hexagonal, and the q vector points along the hexagonal axis. At temperatures above 270"K, the spin arrangement is triangular with q = 0, the transition from a finite q to q = 0 being abrupt. I n an isomorphous crystal, Mn3Ge, and a cubic crystal, MmRh, a triangular spin arrangement with q = 0 has been observed in the temperature range of spin ordering (below 77°K in Mn3Ge and below 600°K in Mn3Rh). JS

0

0

@

0

0

0

@

0

h

@

i

0

Q

8

Mn

0

Sn or Ge

z.$

0

, ,43

0.

FIG.7. Basal plane projection of the crystal structure of MmSn and MnaGe. Magnetic sublattices are labeled A , B, C. Moments on them form an equilateral triangle whose plane is perpendicular to the plane of the figure. In Mn3Sn, this triangle rotates as the position advances in the direction perpendicular to the plane of the figure (after Kouvel and KasperZ4). 24

J. S. Kouvel and J. S. Kasper, PTOC. Intern. Conf. Magnetism, Nottingham, 1964,p. 169. Inst. Phys. Phys. SOC.,London, 1965.

367

THEORY OF HELICAL SPIN CONFIGURATIONS

The value of q corresponds to the lowest value of the energy, which is

E/N

=

An(q)

+ Azz(q) + A33(q) + 6A23(d

when A23(q)

< 0,

18. CASE3: THEREARE Two WAVEVECTORS, q AND q’

If we assume two wave vectors, q and q’, to coexist in the spin arrangement, condition (VI.6) becomes [dqY exp

(iq-R,)

+ dqtuexp (iq’.R,) + c. c.12 = 1.

Writing d,, and d,., in the form of (17.3) , we can rewrite this equation as [us, cos q-R, v,, sin q.R, u,’, cos q‘.R, v,~, sin q‘.Rnl2

+ + + = +u:,(1 + cos 2q.R,) + +vq2,(1 - cos 2q.R,) + u,,.v,, + +uq2pu(1+ cos 2q’.R,) + $Vq2fy(l - cos 2q’*R,) + u q ~ , ~ v qsint , 2q’-R, + uqy.uqJy[cos(q- q’) .R,

+ (q + q’) *RJ + vq,*vq~,[cos(q - a’) *R, + u,,-v,~,[-sin (q - q’) -R,+ sin (q + 9’) -Rnl

sin 2q.R,

C O (q ~

+ v,,.uqf,[sin

(q - 9’) .R,

+ q‘)

+ sin (q + q’) .Rn] = 1.

(18.1)

I n order that this equation be satisfied for all R, when all the sines and cosines are different, the two real vectors u,, and v,, must be orthogonal and have the same length; also, uqtvand vqtvmust be orthogonal and have the same length, and, as can be seen from the last four terms of the equation, u,, and v,, must be orthogonal to both u,., and vqtv, which is impossible. We are thus led to a restriction to be imposed on q and q’. This restriction is such that there must be equivalent vectors among the nine vectors 0, &2q, f 2 q ‘ , & ( q - q’), and &(q q’). If we assume q to be a general vector, we have the possibilities in the accompanying table.

+

0 (a) q’

=0

(b) 9’ E -q’( (c) 2q’ s

fK) -2q’(q‘ = tK)

(d) 2q’

-2q(q’ = -q

(e) q‘

E

E 3q

=

or q’

3

-3q

+ fK)

2q

2q’

9 - 9’

0

2q

0

9

0

2q

0

0

2q

fK

0

2q

-2q

fK q - fK 2q - fK

0

2q

f6q

-2qor 4q

q -

q

+ q’ 9

q q

- fK

+ hK fK

4qor -2q

368

TAKE0 NAGAMIYA

Here K is a basic reciprocal lattice vector; in the table, on the right, are q‘. listed the corresponding five vectors 0, 2q, 2q‘) q - q’, and q In cases a and b, dstv is a real vector; it will be written as u,’ in the following. By (18.1), it must be orthogonal to both u, and v, (the subscript q being omitted for brevity), whereas u, and v, are orthogonal to each other and have the same length. It follows further from the same equation that u,2 u;2 = 1. (18.2)

+

+

If u, and u,’ are both nonvanishing, the spin arrangement in the vth sublattice is such that the q-component and q‘-component of the spin at R,, are, respectively, expressed by X,u,(i cos q.R,

+ j sin q.R,)

S,u,’k exp (iq‘.R,)

,

,

(18.3) (18.4)

where u, and u,’ are now scalars and i, j, and k are unit orthogonal vectors. Hence, in case a, where q’ = 0, the spin vectors rotate on a circular cone with a wave vector q, and in case b, where q’ = K/2, they rotate alternately on an up-cone and a down-cone. The spin arrangement in the whole lattice in such cases may be called a multiple cone structure. Kaplan’s ferrimagnetic cone structure (ferrimagnetic spiral, as he calls it) is an example of this. I n case c, u,’ and v,‘ need not be orthogonal to each other, since sin 2q‘.R, = 0 [see (18.1)], but they must have the same magnitude and be orthogonal to both u, and v, so that they are parallel to each other. Hence we may put )

d,,, =

+(1 - i)u,’k

or

+(1

+ i)u,’k,

(18.5)

where u,’is a real number. Equation (18.2) must hold in this case, too. The q‘-component of the spin vector at R,, is

S,[+(l F i)u,’ exp (iq’.R,) =

+ c. c.]

S,u,’(cos q’.R, f sin q’.R,),

(18.6)

which varies as (++- -) for the upper sign and (+- - +) for the lower sign for q’-R, = 0, n/2, n, 3 ~ / 2(mod 27r). Thus, the spin vectors rotate on two opposite cones, alternating in every two steps. In case d, we have a double helix, one helix running over those sites for which K-R, is an even multiple of 2n and the other over those of odd multiples. They have a common q but are uncorrelated. Case e corresponds to a mixing of the third harmonic. We shall not be concerned with these cases. Now we consider the interaction between different sublattices. TO simplify the problem, we shall henceforth be concerned only with the

369

THEORY OF HELICAL SPIN CONFIGURATIONS

case of two sublattices. The total energy is the sum of (17.5) for q = q and q = q‘. The q-part can be treated as in case 2, and we know that the rotation in sublattice 1 has the same axis and the same sense as in sublattice 2. As for the q’-part, we can see that, in cases a and b, u11.uzl appearing in (17.5) is replaced by 2u1’-uzl1since v1’ and v2) do not appear and Alz(q’) is real; in case c, v1’ and vzl are either parallel or antiparallel to ul’and uz’, respectively, as can be seen from (18.5). Using scalars u1, uz and u11, uzl defined by (18.3)-( 18.5) , we can write the energy as

E / N = A1l(q)ul2

+ Azz(q)u22 - 2 I A12(q) I

+ Azz(q’)uP - 2 I Re

UlUZ

+ A11(q’)u:2

or Im Alz(q’) I u11uzl.

(18.7)

Here u1 , uzand ul’,uz’ satisfy u12

+

u:2

= 1,

uzz

+ up = 1.

(18.8)

The choice of Re or Im in the last term of (18.7) is made as follows: Re in cases a and b, where Alz(q’) itself is real; Re in case c if the first (or second) choice in (18.5) is made for both sublattices; Im if opposite choices are made for the two sublattices. The spin configuration that results from the minimization of the energy (18.7) is a double cone configuration. By (18.8), we can put

u1 = cos 41,

u11 = sin 41 ,

uz = cos 42

,

UZ’ =

sin 42.

(18.9)

Then, abbreviating An(q) - All(q’) = a l , Azz(q) - Azz(q’) = U Z , I Alz(q)I = b, and I Re or Im Alz(q’)I = b’, we can write the equations to determine +1and 42 as a1

sin 241 - 2b sin 41 cos $2

a2

sin 242 - 2b cos c $ ~ sin 42

+ 2b‘ cos 41 sin 42 = 0, + 2b’ sin 41 cos 42 = 0.

Denoting cos tpz/cos qjl we have a1

- bc

= c,

+ b’s = 0,

(18.10)

sin 4z/sin +1 = s, uz - b/c

+ b‘/s = 0,

so that c

=

(bal

+ b’azcs)/(b2 - b”),

s = (b’ai

+ bazcs)/(b2 - b”).

(18.11)

Multiplying these equations side by side, we obtain a quadratic equation for cs. Solving it and substituting the result into (18.11) , we obtain

c

=

s

=

+ alaz f [( ( b + b’) - aiaz) ( ( b - b’) (1/2~&’){ b2 - 6” f [( ( b + b’)’ - aiaz) ( ( b - b’)’ (1/2~zb){ b2 - bt2

1,

UIUZ)]~’~

UIU~

UIU~)]”~}.

(18.12)

370

TAKE0 NAGAMIYA

From (18.10), we have c2 cos 41

from which we can determine equations: c2x2

+ s2 sin2

41 =

1,

(18.13)

This equation is equivalent to two

41.

+ sty2 = 1,

22

+ y2 = 1.

The crosspoint of these ellipse and circle gives the value of $ 1 . We may assume that both 41 and 42 occur in the first quadrant. Then we have

c>l>s>O

or

s>l>c>O.

(18.14)

This inequality imposes certain restrictions on the exchange constants, which we shall discuss briefly a t the end of this section. It can be shown that there are cases where the inequality (18.14) is satisfied. If 41and 42 turn out to be zero or ~ / 2 only , q or q‘ will survive. It can be shown that the minimum energy is expressible in terms of c and s as

E/N

= =

+ Azz(q) - 2 I A12(q) I - 1 Ai2(q) I (c”’ An(q’) + A22(q’) - 2 I Re or Im A d q ’ ) I Aii(q)

- c-”~)~

- I Re or Im A12(q’)I (s1lZ - s - ’ / ~ ) ~ .

(18.15)

Comparing (18.15) with (18.7) ,we see that the energy of the configuration in which q and q‘ coexist is lower than the energy of the configuration in which q or q‘ alone exists (i.e., u1 = uz = 1 or ul’ = u21 = 1 ) . Our mathematical problem will then be to minimize (18.15) with respect to q, but in this process, one of the components, that having q or q’, could vanish. Either of the inequalities (8.14) imposes certain restrictions on a1 , az , b, and b’. To study these restrictions, we assume that u2 > 0; if otherwise, the primed and unprimed A’s may be interchanged [and, a t the same time, c and s must be interchanged because of the definitions (18.9) and (18.10)]. Some elaborate analysis is necessary to find the conditions under which c and s are ensured to be real positive and either of the inequalities (18.14) is ensured to hold. Here the results only will be written. For simplicity, we write b/b’ = P,

al/b’

=

,

az/b‘ = ( ~ 2 .

The condition s > 1 > c > 0 requires the upper sign in (18.12). It requires further that one of the two sets of inequalities, (A) and (B) given below, must be satisfied:

(A)

az>P-l>c~i,

(B)

a2

> P - 1 > a1 ,

I > (fi-ai>(p-a2),

> ( P - all ( P - a z ) , P2 > (a1 + 1) (a2 + 1). 1

P2-1>~2(a1+2). a2(a1+

2)

> P2 - 1,

THEORY OF HELICAL SPIN CONFIGURATIONS

37 1

I n order that c > 1 > s > 0 be satisfied, one must take the lower sign in (18.12), and the following set of inequalities must hold:

(C)

- 1 > az> 0, P2 - 1 > az(2P - ad, a1>

1>

P

P2 >

(P

-

a d ( P - az),

+ 1) + 1). (a2

(a1

One can verify that within each of the groups, (A), (B) , and (C) , none of the inequalities is superfluous, nor is it conflicting with other inequalities. Hence, there may be cases where one of these groups of inequalities holds. If a1 is positive in (A) or (B), one can prove that ( p - l ) z > a1aZ,so that c and s given by (18.12) are real and positive. I n particular, if a1 = a2, it follows that P = 1, and hence al = az = 0; in other words, if the two sublattices are equivalent to each other and the Fourier transform of the exchange constants within each sublattice is nonvanishing, coexistence of two wave vectors is impossible. 19. LYONS-KAPLAN THEORY

We have studied spin configurations under the assumption that there is a single pair of wave vectors (q, -q) or there are two pairs (q, -q) and (q', -q') in the whole crystalline structure. We are, however, not sure whether or not the spin configuration of the lowest energy to be obtained under such a n assumption is the configuration of the lowest energy of all possible configurations. A mathematical theory developed by Kaplan and his co-workers22b-Jhelps us to elucidate this problem. We shall describe it below in a little different form. We have the condition (VI.6) imposed on each spin: s,2, = X,2. It can be written in another form:

c(,

dqv.dG-,~tv)

exp (iq"-R,)

= 1.

q"

Note that d,, = dr,, . I n order that this equation be satisfied for all R, the following equations must hold:

c

c,

d,,-d& = 1,

,

(19.1)

9

dqv'd;-,",

=

0

(q" # 0).

For a single pair of (q, -q), these equations reduce 2d,,*d;,

= 1,

di, = d;: = 0.

(19.2)

t 0 2 ~ ~

(19.3) (19.4)

When q and - q are equivalent vectors, the factor 2 in (19.3) drops; (19.5) is then replaced by dpv = uQpi.

24a

372

TAKE0 NAGAMIYA

(19.4) can be satisfied if we assume for d,, the form d,, = $uqy(i- ij),

(19.5)

where i and j are orthogonal unit vectors (that may depend on v) and u,, is a complex number. To satisfy (19.3), we must take I u,,[ = 1. For two pairs, (9, -q) and (q’, -q’), where q’ may be zero or K/2, or even K/4, according to the preceding section, we may also assume (19.5) and have to take 6,’” = dqrY =

i ( 1 - i)u,t,k

u,l$ or

when $(l

q‘

=

+ i)uqf,k

0 or K/2, when q’

(19.6) = K/4,

(19.7)

where up’,is a real number and k a unit vector perpendicular to both i and j. Equation (19.2) is then satisfied. Equation (19.1) is written as

I U,”

12

+I

Uq’v 12

=

(19.8)

1.

We see by these examples that (19.2) can be satisfied by appropriate forms of d,, , whereas (19.1) puts relations between the magnitudes of these vectors. Assuming that (19.2) can somehow be satisfied, let us confine ourselves to condition (19.1) and look for the lowest value of the exchange energy (VI.4). We may introduce a set of Lagrange multipliers {A,) ,v = 1,2, * ,k, and consider the problem

.

C C ~ A P Y ( q ) d q c . d G-Y EX,C dqY.dGY= min. ,

P

V

Y

(19.9)

9

Since this expression is diagonal with respect to q, we may first confine ourselves to a single pair of q and -q. Then our minimum problem will be reduced to solving the equations

CA,v(q)d,,

= XVd,”,

v = 1,2,

‘‘elk.

(19.10)

P

The negative of the left-hand side of this equation is the exchange field acting on the vth sublattice when there is a helical spin arrangement of wave vector q.

C A,,(q) d q c * d ~+, complex conjugate c

is the energy associated with the vth sublattice in this configuration. (The complex conjugate arises from -q.) It consists of the energy of interaction within the vth sublattice and half the energy of interaction with other sublattices, since when summed over v it gives the total energy (per unit cell). Hence, twice this energy is the energy of interaction between

373

THEORY OF HELICAL SPIN CONFIGURATIONS

each spin and the Weiss molecular field in the vth sublattice. It must be negative when a pair of q and -q gives the stable spin configuration; if it were positive that would mean that the spins are aligned opposite to the Weiss field, and hence the configuration is not stable. Multiplying Eq. (19.10) by d," and adding the complex conjugate, we obtain on the lefthand side the energy considered and on the right-hand side the quantity 2XvdqY-d~v. Hence, A, must be negative. Thus, we put Xv

=

X/P?,

(19.11)

where X is negative real and 0, real. Then (19.10) can be written as

c APv(q)dqP

v = 1,2,

= (X/P?)dqY,

.-.,k.

(19.12)

P

Rewriting this equation in the form

c

PPP"APY(Q)

(d,P/PP)

(19.12a)

= x(d,v/Pv),

P

one will see that the problem here concerned is an eigenvalue problem. The eigenvalues are obtained from or

--

For each eigenvalue,we can determine the eigenvector ( dql , dq2, * , dqk) from Eq. (19.12) or (19.12a). Since each d,, is a vector (complex vector) and Eq. (19.12) or (19.12a) implies that each of its three Cartesian components satisfies the same equation, we obtain the same ratio of u q l i , uq2i , .*., uqki for i = 2, y, z. This means that we can assume one of the forms (19.5)-(19.7) for the eigenvector, with i, j, and k independent of v. Then, uqvis determined from equations similar to (19.12) :

c

ApY(q)uqP = (X/P?)uqv,

v = I, 2,

a * * ,

k.

(19.14)

P

The eigenvalues are functions of q. We vary q and look for the lowest eigenvalue. This lowest eigenvalue will be denoted by Xo and the corresponding value of q by Q. The associated eigenvector will be denoted by (d& , ddz , -, d&) or (u& , ut2, ., u&) . Q , Xo , and the eigenvector depend still on parameters B y . To determine these parameters and simultaneously the value of Q, we try to satisfy condition (19.3) by assuming (19.5) when Q is a general vector. When Q is zero or K/2, we assume (19.6) and try to satisfy I d& ( 2 = 1. For Q = K/4, we may assume either (19.5) or (19.7) and consider (19.3). In all these cases, we are led to the condition 1 u& l2 = 1. If we are able to satisfy this condition by taking

--

--

374

TAKE0 NAGAMIYA

appropriate values of pv and to determine simultaneously the value of Q, and if the corresponding Xo is ensured to be the lowest nondegenerate eigenvalue among all the eigenvalues including those for different q values, then the spin configuration represented by (u& , u&, ., u&) is the configuration of the lowest energy, as we shall prove below explicitly. In this comparison of the eigenvalues, p i s are fixed at the adopted values. From (19.12) follows that the energy of the spin configuration under consideration is given by

Eo/N

=

cc c P

APv(Q)d&-d$

+ complex conjugate

V

By2 2 1 d&

= A0

12

=

Xo

V

c

(19.15)

py2.

V

(In the case of Q = 0 or Q = K/2, the term “complex conjugate” and the factor 2 before I d& l2 do not appear.) Now consider any other real or unreal spin configuration represented by a set of Fourier coefficients { d,,) that satisfy Eq. (19.1). These coefficients can be expressed as a linear combination of eigenvectors { d:“) in the form uqvi =

c

a:i

(i = Z,y, 2 1 ,

~ : v i

a

where i denotes Cartesian component and a specifies different eigenvectors (a = 1, 2, . . ., k). The replacement of q by -q always means the change of quantities into their complex conjugates. For convenience, u&i may here be normalized according to

c

u:vi(u:vi)

*/By”

=

V

cl/PY”. Y

We denote by X , ( q ) [ = X,( - q ) ] the eigenvalue corresponding to { d & ) . Then using (19.12), we can calculate the energy of the spin configuration { d q v ] as follows:

E/N

=

cc c cc cc c cc

A,v(q)d,,.d:v

P

= =

l

r

V

~ ~ P y ( q ) ~ ( e ~ : i f f ~ P i ) ( C ( ~ : l i ) * ( ~ ~ v i ) * )

q

v

i

a:i(a&) *[X,(q) /py”]u:vi(u$i) *.

a a ‘

I n the last line, the sum over v vanishes except when a eigenvectors are orthogonal: c ( u : v i / P Y )(u:vi/P”)* = V

0

= a’,since different

(a # a’).

THEORY OF HELICAL SPIN CONFIGURATIONS

Hence, utilizing the normalization condition for

(Y

375

= a‘, we obtain

(19.16) q

i

u

V

On the other hand, from (19.1) we have q

v

V

which can be transformed by a similar calculation into

so that (19.17) q

i

a

Thus, the energy (19.16) is a weighted mean of eigenvalues multiplied by C y1//3y2. Evidently, it is higher than the energy (19.15). There are k parameters PY to be determined from the k conditions I u& I = 1. However, one of these parameters can be put equal to 1, since , u& is determined from the eigenvalue only the ratio of u& , 2442 , equation. Although this determination of the parameters may be possible in some cases, there may be also other cases where real values of the parameters cannot be obtained. I n the latter case, we have to look for other spin configuration^.^^^ We may then examine spin configurations in which two wave vectors

--

a helical configuration having a wave vector Q is the configuration of the lowest energy, it must result from the eigenvalue equation (19.12), and thus the determination of 8. for this Q must be possible. However, i t will not immedfately follow that the eigenvalue Xo(Q) is the lowest of all eigenvalues for this set of 8“. That this configuration has the lowest energy would mean that Xo(Q) C,1/8.2 [see (19.15)] is the lowest of (19.16), i.e., by virtue of (19.17), that

24b If

CCC I Gi 12CXa(q) - ~o(Q)l.Cl/8.2 B 0. a i m If we were able to choose a:i arbitrarily, except that they are subject to condition (19.17), then the absolute minimum of Xo(Q) would follow from the preceding inequality. I n actuality, however, a:i are subject to further conditions that would result from (19.2), although no explicit use of such conditions was made in deriving (19.16) and (19.17). When the helical configuration having Q as its wave vector is the lowest in energy compared with other helical configurations, what we can say is only that XO(Q,8.)- Z l / @ p z 8. , being determined for Q, is the lowest of all X d q , aP).C,1/8.2, 0. being determined for q (not Q). It will then not necessarily follow that aXo(q, b,)/ a q = 0 for q = Q, i.e., that X,(Q, 8.) is the lowest of all Xo(q, By) when 8;s are determined for Q. I n other words, even if we fail in the procedure mentioned in the text, we cannot exclude the existence of a helical configuration as the stable configuration. A similar situation arises in the case of a configuration having two or more wave vectors.

376

TAKE0 NAGAMIYA

coexist. One of these two can be a general wave vector, which we shall denote by Q. The other must be a special wave vector, such as zero or K/2. In this case, the spin configuration is conical, since we must assume (19.5) and (19.6) for the two wave vectors, with i, j, and k perpendicular to each other, as we have shown in Section 18. Also, (19.8) must be satisfied for q = Q and q’ = 0 or q’ = K/2. In place of (19.9), we now have

c CCA (Q) + PV

P

%Pu;”

A,v(q’) u,%G

Y I

”

- C ~,(u,,u;, + u,+u;fy) = min. Y

Thus, we are led to the same equation as (19.10) for both u,, and uqrv, and the negativeness of A, can be seen in the same way as before for the stable spin configuration. Putting X, as in (19.11), we have Eq. (19.13) or (19.13a) to determine the same value of X for q = Q and q’ = 0 or K/Z. The eigenvectors are determined from (19.14) for q = Q and q = 0 or K/2. There are again k parameters p, and, in this case, two amplitudes of the 2 constants can be put equal to 1, and the eigenvectors. One of these k 1 are to be determined from the k equations (19.8) and remaining k the condition that the eigenvalues for the two wave vectors take the same value. The value of Q is determined simultaneously by the equation gradQA(Q) = 0. The equality of the two eigenvalues has been called “forced degeneracy” by Lyons and Kaplan, since the parameters are so chosen as to make the two eigenvalues degenerate. After the determinations just mentioned have been made, we have to check if the degenerate eigenvalues are the lowest of all eigenvalues for the adopted values of the pvfs and if there is no other degeneracy. When we succeed in all these procedures, we are sure that we have found the stable spin configuration. That this configuration has the lowest energy can be seen from (19.16) and (19.17). When we fail, we may proceed to investigating spin configurations having three or more wave vectors, but the calculations will become more complicated and might be unsuccessful.

+

+

VII. Spin Configurations in Spinel lattice

As an application of the preceding section, we shall here study possible spin configurations in the spinel-type lattice. A complete study of this problem has not yet been made, although detailed calculations have been carried out by Kaplan and his co-workers. We shall here limit our discussion to certain simple features of the problem. A brief account of the crystalline and magnetic structures of the spinel lattice will be given first.

THEORY O F HELICAL SPIN CONFIGURATIONS

377

20. CRYSTALLINE AND MAGNETIC STRUCTURES

In the spinel lattice, there are two kinds of cation sites, usually called

A and B sites, which are surrounded tetrahedrally and octahedrally, respectively, by anions. The structure is shown in Fig. 8, which can be derived from the NaC1-type lattice by removing every other cation along each line parallel to each of the principal axes and returning half of the removed cations to fill one-eighth of the vacant tetrahedral sites. I n MnCrz04, the Mn2+ occupy the A sites and Cr3+ the B sites; this type of cation arrangement is called normal. In FeaO4, as is well known, half of the Fe3+ occupy A , and the other half Fe3+ and Fez+ occupy B ; this type is called inverse. Chroniites are usually normal, but ferrites are often inverse. Some compounds, such as CuCrz04and FeCrzO4, are tetragonally distorted below a certain critical temperature by a cooperative Jahn-Teller effect, a theory of which has been worked out by K a n a m ~ r i . ~ ~ I n NBel's theory of ferrimagnetism,26the spins of the B sites are assumed to align parallel to each other and antiparallel to the spins of the A sites. The A-B exchange interaction is assumed to be dominant and antiferromagnetic. Yafet and Kittel2O have shown, however, that the NBel structure is no longer stable when the B-B (or A - A ) exchange interaction becomes I

-It

FIG.8. Structure of spinel lattice. Four (001) layers of atoms projected on the (001) plane are shown. The first layer a t zero height consists of cations 0 on B sites and The second layer a t a height of c / 8 ( = a/8 in cubic case) consists exclusively anions 0. of cations on A sites. The third layer is a t a height of c/4 and consists of cations 0 on B sites and anions (dashed circles). The fourth layer is a t 3c/8 and consists of cations 0 on A sites. The whole structure is generated by a vertical translation of c/4 combined with a rotation by 90"about the vertical line which passes the point C in the figure. I n actual crystals, the four anions surrounding each A site are displaced toward the latter. 25

26

J. Kanamori, J . Appl. Phgs. 31, 14s (1960). L. NCel, Ann. Phys. (Paris) [12] 3, 137 (1948).

378

TAKE0 NAGAMIYA

large and antiferromagnetic. They have shown that a triangular spin arrangement is more stable in this case; this is an arrangement in which the B (or A ) lattice is split into two equal sublattices having moments a t an angle less than 180" and the resultant of these moments is antiparallel to the A (or B ) moment. They have also shown that, when both the B-B and A-A interactions are large and antiferromagnetic, the B and A lattices both have antiferromagnetic arrangement (uncorrelated to each other). As mentioned before, copper chromite, CuCrzOl, has a triangular arrangement.Z1 Here the B lattice is split into two in such a way that each (001) net plane of the B atoms (Cr) is ferromagnetic, and its moment direction alternates from plane to plane. Jacobs2' made magnetic measurements at high fields on CuCrzOa and two other tetragonal spinels, Mn304 and FeCrz04 , as well as on several ferrite-chromite solid solutions, and observed a linear field-dependence of the moment, from which he expected that triangular spin arrangements exist in these substances. Mn304possesses a spin configuration characterized by a wave vector along [110],2s and FeCrz04has a multiple cone configuration at low temperatures, as will be described below. The magnetic structure of a cubic spinel MnCrzO4, observed by Hastings and C o r l i s ~ is , ~ the ~ first example of the conical structure predicted by Kaplan et uZ.22c-e(ferrimugnetic spiral, as they call it). The NBel temperature of this substance was observed to be about 43"K, and the neutron magnetic reflection lines down to 18°K corresponded to a NBel-type collinear spin arrangement (there was, however, an additional diffuse peak in the region where satellite lines develop below 18"K), but below 18°K additional sharp lines (satellites) appeared, which could be interpreted as being due to a transition to a conical spin arrangement. The Mn-spins (on sites 5 and 6 in our Fig. 8 ) , the Crl-spins (on sites 1 and 3), and the Cr2-spins (on sites 2 and 4) rotate on their respective cones under translation along [ l l O ] to equivalent sites in other unit cells. The rotation corresponds to a wave number of 0.98 A-1 (at 4.2"K, the lattice constant a t room temperature being 8.437 A). The cone axes are common and parallel to [liO], and the cone angles (measured from [lTO]) are 24", 152.5" ( = 180" - 27.5"), 104" ( = 180" - 76") for Mn, C n , Crz , so that the cones for Cr are opposite to the cone for Mn. These results are in good agreement with the theory. However, there are unexplained features, particularly concerning the magnitudes of the atomic magnetic moments. Also, the line intensities giving the magnitudes of the axial components of the moments did not change in going through the transition a t 18°K; together with a diffuse peak in the region of the satellites, this fact would suggest a disorder or large fluctuations in the transverse components 27 I. S. Jacobs, Phys. Chena. Solids 16, 54 (1960). 28 J. S. Kasper, Bull. Ant. Phys. SOC. [ 2 ] 4, 178 (1959).

THEORY O F HELICAL S P I N CONFIGURATIONS

379

above 18°K. Furthermore, some aspects of the theory have not yet been fully clarified, as will be mentioned later.28a Attempts to measure the cone angles by NMR have been The idea is that by applying a magnetic field the cone axis would become parallel to the field and the internal magnetic field acting on the nucleus of each magnetic atom, whose direction is parallel to the spin moment of the atom, would change proportionally to the strength of the applied field and the cosine of the cone angle. Under this assumption and from the observed linear field-dependence of the resonance frequency, the angles were determined to be 68",94",97" for Mn, Crl , Crz , which do not agree with the neutron diffraction results. ~ , ~crystal ~ I n FeCrz04 , a conical structure was also o b ~ e r v e d . ~This becomes tetragonal below 135°K. (The c-axis is only 3% smaller than the a axis; c = 8.21 A, and a = 8.46 A.) It has a NBel temperature of about 80"K, and below about 40°K it gives satellites that indicate a conical structure. The propagation vector is parallel to [llO] and has a value of 0.063 A-l (see Shirane and or 0.037 A-l (see Bacchella and P i n ~ t, ~ ~ ) or parallel to [OOl] and has a value of 0.026A-I (see Bacchella and P i n ~ t, ~ ~ ) depending on different indexing of the lines. I n the range between 80" and 40"K,the spin component perpendicular to the cone axis seems to be disordered or fluctuating. A cubic spinel CoCr204at low temperatures also produces neutron lines that can be interpreted by a conical structure.33The propagation vector is again parallel to [llO] and has a value of 0.62 x 2lI2/a (a = 8.332 A at room temperature). The cone axis is parallel to [OOl]. The cone angles have been determined to be 32", go", 150" for Co, Crl , Crz , and the magnetic moments of these ions to be 3 Bohr magnetons (the spin only value). The NBel temperature is 97°K. The temperature-dependence of magnetic properties reported by Menyuk, Dwight, and is extremely interesting. They performed a detailed calculation of the conical spin configuration over the whole temperature range from 0" to 97°K under the assumption of the molecular field approximation. Taking a parameter value u = 2.03 ( U = ~ J B B ~ B / ~ ,Jwhich A B ~will A appear in the next section), they found a very good agreement between the calculated and observed curves of the saturation magnetization versus temperature, except below 27°K. With increasing temperature, the calculated curve increases gradually up to about 20"K,then rises linearly with temperature, and attains a high broad 2Sn

29

30 31

33

The u parameter, which we shall discuss in Part VII, Section 22, is 1.6 in this example, which is too large to ensure the stability of the conical structure. T. W. Houston and A. J. Heeger, Phys. Letters 10, 29 (1964). H. Nagasawa and T. Tsushima, Phys. Letters 16,205 (1965). G. Shirane and D. E. Cox, J . A p p l . Phys. 36, 954 (1964). G. L. Bacchella and M. Pinot, J . Phys. 26, 537 (1964). N. Menynk, K. Dwight, and A. Wold, J . Phys. 26, 528 (1964).

380

T A K E 0 NAGAMIYA

maximum at 77"K, which is then followed by a steep decrease going to zero at 97°K. On the other hand, the observed curve decreases gradually up to 27"K, shows a break in slope a t this temperature, and then follows closely the calculated curve. The calculation showed that a transition occurs a t 86°K from the conical configuration to the NBel colinear arrangement. The calculation was very sensitive to the choice of u (f2.5% changes in the u value resulted in conspicuous changes in the magnetization curve). Despite the discrepancy below 27"K, the neutron line positions and intensities a t 4.2"K can be accounted for very well by the theoretical model. But, despite the agreement above 27"K, the satellite lines corresponding to the rotating component of the spins do not appear as sharp lines but appear only as diffuse peaks, whereas the magnetic contributions to the intensities of the fundamental lines, corresponding to the axially aligned magnetic moments, are in excellent agreement with those predicted theoretically. The authors note that the Lyons-Kaplan-Dwight-Menyuk theoryZzdof the conical spin configuration a t absolute zero predicts local instability (against small spin deviations) for u > 1.3 and that this may have a relationship with the low-temperature discrepancy. They note further that the absence of sharp satellites for the rotating component above 27°K may be due to a failure of the molecular field theory, which does not adequately takes account of correlations between the thermal fluctuations of neighboring spins. In contrast to the complicated spin structures mentioned previously, a very simple helical spin arrangement was observed in ZnCrzSe4 ,34 and it was interpreted theoretically.35 I n this crystal, there are magnetic atoms only on B sites. The spins in each lattice plane (001) are parallel and lie in the plane, and the helix propagates in the (001) direction with a turn angle of 42" (at 4.2"K) for adjacent (001) planes. Of course, the second (and further) neighbor interactions are necessary to account for the helical

21. THENAELAND YAFET-KITTELCONFIGURATIONS

It is not our intention to study completely the spin configurations in the spinel lattice. Our main purpose is only to show the main idea of how to find these configurations and how to discuss their stability. I n the cubic or tetragonal unit cell shown in Fig. 8, there are sixteen B sites and eight A sites, but if we take a regular or distorted rhombohedra1 unit cell defined by basic vectors (0, 3, 3) , (3, 0, 3) , and (3, 3, 0) , there R. Plumier, Compt. Rend. 260,3348 (1965). F. K. Lotgering, Solid State Commun. 3, 34T (1965). 36a ZnCrzOcbecomes tetragonal below 20°K ( = TN), with a decrease of about 0.05% in c / a . Precise measurements of the temperature variation of c / a were made by R. Kleinberger and R. de Kouchkovsky, Compt. Rend. 262,628 (1966).

34

s5

THEORY O F HELICAL, S P I N CONFIGURATIONS

381

are four B sites and two A sites. Taking the origin at one of these four B sites, we can write the regular or distorted rhombohedra1 coordinates of the six sites as 1 2 3 4 B : ( O , O , O > , (310, O ) , ( O , O , 31, (0,3,0);

A ..

5

6

(3 3 3)

(5 5 5)

8, 8, 8

8, 8, 8

1

-

In the original cubic or tetragonal coordinate system, these can be written as 1 2 B : ( O , O , 0 ), (0, a, 4) 1

A:

5

3 ($7

5

$70) 1 ($10, $1 ;

6

( 3 3 3) 8 , 81 8

(5 5 5). i

8 , 87 8

These sites change as 1 -+ 2 -+ 3 -+4 and 5 -+ 6 -+ 5 -+ 6 by successive operations of rotation by 90" about the vertical line passing through the point C in Fig. 8 and translation by $ parallel to this axis.3sbThis property we shall is useful in calculating the exchange matrix. As in Kaplan et take nearest-neighbor exchange interactions only. There are six B sites surrounding each B site. Two of these six are in the same (001) plane as the central site, and we shall denote the exchange constant between each of these two and the central site by J B B . The remaining four are not in the same (001) plane as the central site, two being below and two above, and we shall denote the corresponding exchange constant by JAB. There are also six A sites surrounding each B site, two at a level higher by 4 than the central site, two at a level -4, and one each at levels and -2. We denote the exchange constants corresponding to these by JAB , J A B , and JLB. (Conversely, each A site is surrounded by 12 B sites.) The A lattice is of the diamond type, so that we have only one kind of nearest-neighbor exchange constant, J A A. For convenience, we ~ r i t e ~ 5 ~

(21.1) SI + st

JkB = 21,

WAB

+ JLB

=

w.

3SbOur1, 2, 3, 4, 5, 6 correspond to 5, 4, 6, 3, 1, 2 of Kaplan et aLZ2 3 6 ~t, u, v, w are the same as those appearing in Menyuk et aLzZ* Here we shall restrict ourselves to t = 0 and 0 5 w 5 a.

382

TAKE0 NAGAMIYA

+

We shall assume JAB JAB < 0. As the exchange matrix, it is convenient to consider C,,(q) defined by

C,,(q) = APY(q)exp [in* (R, - RY)]/(~JAB =

-

c

SpSJ+np,n,

exp [iq.

+ J L B 1 SASB)

- Rnv)]/() JAB

(%p

+

I

J ~ BSAXB).

n

(21.2)

The summation with respect to n is taken over those points on the vth sublattice which are neighbors of a point on the pth sublattice. Using this C,,(q) and writing d,, for the previous d,, multiplied by exp (--iq.R,), the energy can be expressed as e =

E / ( N I JAB + J A B

I SASB)=

c c C C,,(q)d,,.d~,. 9

,

(21.3)

V

Furthermore, using u,, defined by (19.5) for a general q and (19.6) or (19.7) for the special vectors written there, we can write a single wave vector term of E (q or -q term for a general q) asasd (21.4)

Our mathematical problem is to fiIid the lowest eigenvalue of this expression by the procedure described in Part VI, Section 19 (the Lyons-Kaplan theory). For this purpose, we must calculate C,,(q) by (21.2) using the exchange constants appearing in (21.1) and the coordinates of the B and A sites. I n this calculation, it is convenient to use cubic or tetragonal coordinates. The results are as follows, where k, = aq,/4, k, = aq,/4, and k, = cq,/4:

ClZ = s’cos (k,

+ k,),

scos (k,

c 1 3

=

c23

= 8’ cos

+ k,),

( k , - kz) 9

(k, +k,),

c 1 4

= s’cos

c 2 4

= s cos ( k z

c 3 4

=

- k,)

9

s’ cos (k, - k,),

+ +kJ] + + w exp [(i/2)(kZ 4- k, - 3k2)]

CIS= 4 ( 1 - w){exp [(i/2) (-3k, k, exp [(i/2)(k, - 3k, k,)])

+

= c;6, c 2 6

=

=

+ k,) 1 + exp [(i/2) (3k, + k, + k,)]) + w exp C(i/2) ( --kz + k,

4 (1 - w) { exp [(i/2) ( -k, - 3k,

- 3kJl

Gs,

This is half of a single wave vector term for q equal to zero or half a reciprocal lattice vector, and represents half of the energy assiciated with q and - q for any q.

383

THEORY O F HELICAL S P I N CONFIGURATIONS

C35

=

3(1 - w) {exp [(i/2) (3kx - k,

+ k,)]

+exp[(i/2)(--kx+3k,+k,)])

c 4 6

=

c 6 :7

=

3(1 - w) { exp [(i/2) (k,

+ exp [(i/2) (-3k, c56

- k,

+wexp[(i/2)(--kx

+ 3k, + k 2 ) l - k, + k2)]1 + w exp C(i/2) (kz - k, - 3-k2)l

=

Ci5,

=

2t[cos (k, - k,) exp (ik,)

+ cos (k, + k,) exp ( -ik2)].

(21.5)

We shall first investigate two cases completely, i.e., the case that k is parallel to [OOl] and the case that k is parallel to [loo], the lattice being assumed to be tetragonal. Then we shall consider the case of k 11 [llo]. For simplicity, we shall put t = 0 and assume 0 < w < 4.

k

(1) ek =

=

(0,0, k) . I n this case, we can write the energy (21.4) as

3 { 8' cos k(uluz* + u1u4* + u3u2* + u3u4*) + s(ulu3* + uZu4*)

+ [(1 - w) exp (ik/2) + w exp ( -3ik/2)]

+ u4u6*) + [(I - w) exp (-ik/2) + w exp (3ik/2)](ulu6* + u3u6* + u2u5* + u4u5*) 1 x

(ulu5*

+ u3u5* +

+ complex conjugate.

(21.6)

Under a unitary transformation,

+ + u3 +

51

= 3(ul

23

=

25

= 2-ll2(U5

u2

- uh),

3(ul - u2

+

7-44),

u6),

+

- u3 - u4),

22

=

$(ul

24

=

*(ul - u2

56

=

2-'12(U5 - us),

24'2

- u3 + u4),

(21.7)

(21.6) becomes Ek =

+ Xp3* ( 1 - w)cos (k/2) + w cos (3k/2) ](x1x5* + x521*)

s' cos k(Xlxl* - x$3*) f ~ S ( ~ l xl *xfl2*

f 2'/'[

+ i2l/2[ (1 - w) sin (k/2) - w sin (3k/2)]

x&4*)

(x3x6*

- X P ~ * ) . (21.8)

.

24 Thus, the variables are separated into four sets: (21 , x5), 2 2 , ( 2 3 , 4, However, the three variables x3 ,x4,2 6 are redundant, since, by replacing k by k T,u2 and u4 change sign, and u5 and u6 change into iu5 and 4% , respectively, so that x1 , x2 ,x5 change into x3 , x4 , ix6 ,and by these changes the terms of (21.8) quadratic in x1 , 2 2 , 2 5 go over into those quadratic in x3, 2 4 , x6. Thus, we may put x3 = x4 = 2 6 = 0, or u1 = 242, u3 = 244,

+

384

TAKE0 NAGAMIYA

u5 = 2 4 6 . Hence, 21

= ~1

+

243,

22

= Ui

- U3 ,

25 =

2112U~.

(21.9)

From (21.8), we have a matrix for 2 1 and 2 5 :

s' cos k

+ as 0 (21.10)

and a one-dimensional matrix for 2 2 : (21.11)

-3s. Solving the equation s' cos k

i2+1

+ +s - x

w)

6)+ (31I w cos

COS

-va2

- w ) cos(;)+wcos(~)]

I =

0, (21.12)

we obtain two eigenvalues, of which the lower one is

Xl(k, p)

= +is' cos k

+ 3s - [(s' cos k + +s)2

+4p2(1 + c o s k ) ( l - ~ w + ~ w c o s ~ ) ~ ] " (21.13) ~]. From (21.11) we obtain another eigenvalue A2

(21.14)

= -3s.

If we put k = 0 and determine p in such a way that u1 becomes equal to -us , so that 21 = -21/2x5, then we obtain the NBel configuration as represented by the eigenvector x l , x5 (z2 = 0 ) . Namely, in the first row of (21.12) we put k = 0 and set the ratio of the two elements to be 1 to 2ll2. Then we have (21.15) p = +(1 - 4s - s') = pN2 (1 > 3s sl)

+

and, correspondingly, from (21.13) pN) =

-1

+ + $8

8 '

EZ AN

.

(21.16)

385

THEORY OF HELICAL SPIN CONFIGURATIONS

The energy of this configuration is calculated from e =

( CK 2 ) X = (4 + 2f2)X

to be EN =

+ 2s + 4s'.

-8

(21.17) (21.18)

The condition for the stability of the N&l configuration, within the restriction k = (0,0, k), is that XN is the lowest of h(k, PN) and lower than A t . That XN is the lowest of Xl(k, PN) is equivalent to the requirement that the determinant appearing in (21.12) is positive for X = AN when L # 0 and P = @N . This gives a condition s' < 3, provided 0 < w < 4, as one will see by a simple calculation. From XN < Xz ,followsthat s 9' < 1, which is stronger than the inequality written in parentheses in (21.15). Hence, the stability region of the NBel configuration becomes as shown in Fig. 9. There are two Yafet-Kittel configurations (for t = 0). In one of them, spins on sites 1 and 3 are parallel, and spins on 2 and 4 are also parallel, but the former spins make an angle with the latter spins, and their resultant is antiparallel to the spins on sites 5 and 6. This configuration, which we shall denote as YK1, has been observed in CuCrz04. The other configuration, YK2, is obtained from YK1 by the interchange of 3 and 2; in this configuration, spins within each (100) net plane of the B lattice are parallel to each other, but their direction alternates from plane to plane. If 3 and 4 are interchanged, this alternation will take place for (010) net planes, but

+

5'

4 '.

YK2

FIG.9. Stability regions for the three spin configurations, NBel, Yafet-Kittel 1, and Yafet-Kittel2, obtained from the study of the cases q (1 [OOl] and q 11 [lOO].

386

TAKE0 NAGAMIYA

FIG.10. Sublattice magnetizations in the Yafet-Kittel 1 configuration; tan e [(2s‘)9

=

- 111’2.

this configuration is equivalent to that for (100) net planes in tetragonal crystals. I n cubic crystals, YK1 and YK2 are equivalent. To obtain YK1, we put Xl(0, P ) = Xl(?r,

a>

(21.19)

*

Since each spin vector is given by S,, = X,u&, exp (iq-R,,) for a single q that is zero or a reciprocal lattice vector, where k, is an arbitrary unit vector that appeared in (19.6),36ethe relations u 1 = u2 , u3 = u4 , u 5 = u6 mean that for zero wave vector the spins on 1 and 2‘ are parallel, the spins on 3 and 4 are parallel, and the spins on 5 and 6 are also parallel, whereas for k = (0,0,T),or q = (0,0, 4?r/c), each of these pairs consists of antiparallel spins (spins on 1 are antiparallel to spins on 2, etc.). We note further that the eigenvectors associated with X1(O, p ) and h ( n , p ) have vanishing 2 2 ( = u1 - u3),so that the spins on 1 and 3 are parallel in the configuration represented by each of these eigenvectors. Hence, a superand k(o,o,,,at right angles, will position of these eigenvectors, with k(o,o.o, yield a YK1 configuration. Now, (‘21.19) implies

p2 = 3s’(28’ - 8)

(pyKl)2

if 2s’ > s,

8’

> 0,

(21.20)

and 82 = 0 if 2s‘ < s. We are not interested in the latter, since in this case the eigenvalue vanishes. For the former, we have a negative eigenvalue : (21.21) Xl(0, P Y K l ) = Xl ( ? r , P Y K l ) = $8 - 8’ X Y K l The corresponding energy is €yKl

=

2s - 4s’ - 2/s’.

(21.22)

The requirement that Xygl is the lowest eigenvalue gives an inequality s’ > s. From the eigenvalue equations, one obtains U l / u 5 ( = 2 1 / 2 ~ ’ ~ ~ 5= ) - 1/2s’ for k = 0, whereas for k = T,u1is arbitrary and us vanishes; thus, the two eigenvectors must be superposed with an amplitude ratio of 1 to [l - ( 1 / 2 ~ ’ ) ~ ] ~so ’ ~that , the spins on 1 have the same magnitude as the IsIn

the present case, q is zero or a reciprocal lattice vector, not half a reciprocal lattice vector; a reciprocal lattice vector appears, since we disregarded za , 24, $6 .

THEORY OF HELICAL SPIN CONFIGURATIONS

387

spins on 5 (Fig. 10). The stability region for the YK1 configuration is therefore characterized by s‘ > s and 2s’ > 1, as shown in Fig. 9. The YK2 configuration (see Fig. 11) can be obtained from Xl(0, P ) = Xr , with k = 0 for Xz . This equation gives

8’

=

$s(s

+

8’)

(PYKZ)21

XYK2

=

-S/2.

(21.23)

The corresponding energy is calculated to be EyK2

=

-2s - 4/(s

+ s’).

(21.24)

A similar consideration to that made for YK1 yields a stability region for YK2 as characterized by s’ < s and s’ s > 1 and shown also in Fig. 9. Since the three regions cover the whole s, s‘ plane, no other spin configurations are conceivable as long as k is considered parallel to [Ool]. (2) k = (k, 0 , O ) .Similar calculations can be carried out for this case. Under the same transformation (21.7) , the variables separate into (XI ,z b ) , x4, (xz , x6), x3 , and we can put xz = 2 6 = x3 = 0, since the quadratic form of the first three variables in the energy expression Ek goes over to that of the last three variables when k is replaced by k T . The lower

+

eigenvalue for

(XI

Xl(k, p)

+

, x5) is calculated to be

=

+ + s’) cos k - [(s’ + + s’) cos k)’ + 16P2(1 + cos k) + (1 - cos k)2]1’2} (21.25)

:{st

(S

(S

(W

W)

(S

- s’) cos k].

and the eigenvalue for 2 4 to be

Xz(k)

= -+[S‘

+

(21.26)

These eigenvalues are connected with the previous h ( k , P ) and XZ , respectively, at k = 0. It can be shown that XN , X Y K l , and XYKZ are the lowest eigenvalue in the respective regions of Fig. 9 when compared with (21.25) and (21.26) ,p being put equal to PN ,P Y K l , and PYKZ, respectively. The study of the present case thus adds no new results.

(3) k

=

(h, h, 0 ) , h

=

This case cannot be treated easily.

FIG.11. Sublattice magnetizations in the Yafet-Kittel 2 configuration; tan 0 = [(s

+ s’)* -

111’2.

388

TAKE0 NAGAMIYA

Here we will mention only a simple conclusion. Then in Section 22, we shall discuss the results obtained by Kaplan and his co-workers by extensive calculations. The fact to be mentioned is that the stability regions shown in Fig. 9 for the NBel and Yafet-Kittel spin configurations become narrowed by some energetically favorable states having propagation vectors included within the present case. I n the case at hand, the variables xi (i = 1, 2, ., 6) separate into two sets, ( 2 1 , 23 , z5) and ( 2 2 , x4 , s6). A further transformation,

--

91 =

(Xi

+

=

(Ui

93 =

(21

- Z3)/2112=

(U2

Z3)/2112

+ /2112, + U4)/2112, U3)

yz

=

(22

+ 24) /2112

94

=

(22

- 24)/2112 =

= ( U i - U3) /2112, (Uz

- U4)/21/2, (21.27)

separates the ( x 2 , x4 , x6)-space into a two-dimensional space and a onedimensional space; also, it simplifies the energy expression for (x1, 2 3 , 26). The calculated energy matrices are as follows. For (y1 , y3 , 2 5 ) one has

[

s' cos h

cos h

I

s' cos h

3s cos 2h

w

4s

w

+ (1 - W ) c 0 ~ 2 h ;

+ (1 - W ) cos 2h

0 cash

for y4 , ( -3s) ; and for ( y ,~x6) ,

[

-4s cos 2h

i ( 1 - 2w) sin h

-i(l - 2w) sin h

0

1.

(21.28)

(21.29)

From the one-dimensional matrix, we obtain a n eigenvalue XZ = -is, which is identical with (21.14). For the two-dimensional matrix, we may introduce P2 as in (21.12), and we have -4s cos 2h

-X

i ( 1 - 2w) sin h ~

-i(l

X3 = a{ -s cos 2h

- 2w) sin h

-

[s2

cos22h

=o,

(21.30)

-Alp2

+ 16p2(l - 2

~sin2 ) h]"2}. ~

(21.31)

This X3 becomes equal to X2 at h = 0 when s is positive. It can be shown again that for h # 0 this eigenvalue is higher than the eigenvalues for the NBel, YK1, and YK2 configurations in the respective regions of Fig. 9, provided P takes its respective values. We have thus to study the three-dimensional matrix (21.28). If we add -A, -A, and -X//?Z to the diagonal elements of (21.28)-since Pi's for 22s (i = 1, 2, 6) have been assumed hitherto to be 1, 1, 1, 1, P, Pa,

THEORY OF HELICAL S P I N CONFIGURATIONS

389

and calculate the determinant by putting X and /3 equal to , / 3 or ~ XYKl , or X Y K Z , &KZ, and see if the determinant is positive or not, then we shall be able to see whether the stability regions of Fig. 9 become narrowed or not. For simplicity, we confine ourselves to h = 7r/4. The point (7/4, 7r/4, 0 ) is the projection of the center of a hexagon of the Brillouin zone boundary on the (001) plane. It turns out that the determinant is no longer positive in the shaded region of Fig. 12, which is bounded by a hyperb o1a @yKl

ssr

+

sr2

- (1 - iW2)S - (3

-w

- w2)s‘

+ (4 - WZ) = 0

and two straight lines (1

+ w ) s - 2ws’ = 0,

(1

+ w ) s - 2s’ = 0.

We may therefore have to look for some other structure in this shaded region. Menyuk et a1.22e investigated in detail the stability regions for the N6e1, YK1, and YK2 configurations by examining many more k vectors, in particular those parallel to the symmetry directions (OOl), (loo), (110), and (101), and a special k vector at the edge of the Brillouin zone in the (201) direction. Those k vectors which are perpendicular to the c axis and are in the neighborhood of the origin were also examined. Furthermore, for certain ranges of the parameters, all k vectors in the first Brillouin zone were examined. These authors also investigated the “destabilizing wave vector” on the boundary of the stability region, namely, the wave vector for which the determinant vanishes and which is thus suggestive of a spin configuration outside that region. Much computer work was

FIG.12. Stability regions for the Nkel, YK1, and YK2 configurations, narrowed by the consideration of a wave vector (?r/4, ~ / 4 ,0). In the shaded region, one may have some other spin configurations. The figure was drawn for w = j.

390

TAKE0 NAGAMIYA

necessary for this study, and the results are not so simple as to be described here briefly. The reader is referred to the original paper.

22. MULTIPLE CONESTRUCTURE We have seen that in the shaded region of Fig. 12 (and possibly in a wider region) the stability of the three spin configurations studied is not ensured. Along the cubic line s = s‘, a multiple cone structure possessing a zero wave vector and a nonzero wave vector Q parallel to (110) seems to exist as the stable configuration, as discussed by Kaplan and co-workers.22e They showed that the NBel structure is stable for u (=2s) 2 9 , and a multiple cone structure is more stable than the previous three spin configurations and in fact more stable than any other spin configurations that possess “equal relative angles,” i.e., where S,,. S,, is invariant under lattice translations. Also, a multiple cone structure was shown to be locally stable (stable against small spin deviations) for Q < u < u” = 1.298 and locally unstable for u > u”.Extension of the calculation beyond u“ showed that the structure was coplanar for u > 3.817. Kaplan and coworkers report, however, that for all u > Q there are some wave vectors q in the Brillouin zone fo? which the inequality Xo(Q) < X,(q) is not satisfied. This would mean that the Lyons-Kaplan stability criterion cannot be applied to the multiple cone structure and helical structure studied, so that one does not know whether these structures are stable or not. In this study, nearest-neighbor interaction and the A-A interaction were neglected. The multiple cone structure can be expressed by

S,,

=

S,lCi cos (QnR,,

+ + j sin (Q-R,, + Y~)]sin 4, + k cos 4,, 7,)

(22.1)

where i, j, k are orthogonal unit vectors. Corresponding to (22.1), we have (22.2) uo, = cos 4, . UQv = sin +,.exp (iy,), (Our numbering of 1, 2, . . . , 6 is different from that of Kaplan and coworkers; see footnote 35b on page 381.) The three-dimensional matrix of (21.28) can be written in the present cubic case as 3s cos 2h

s cos h

s cos h

42

[

cos h

Q cos 2h

Qcos2h

+4

cash 0

I

+i

.

The corresponding secular equation, obtained by adding -A,

(22.3)

--h/ar2,

and

THEORY OF HELICAL SPIN CONFIGURATIONS

391

-Alp2 along the diagonal, will yield an eigenvalue Xl(h, a,p) . The condition for a minimum, &(h, a,p ) /ah = 0, the degeneracy condition, Xl(0, a, = ~ ~ ( a, h p, ) , and three more relations, I U h v l2 I uovl2 = 1 ( v = 1, 2, 5) will determine h, a, p, and the amplitudes of the two eigenvectors, and, thus, 4”and y v . [It is noted that this procedure is equivalent to minimizing the quadratic form of the energy, constructed from the preceding matrix with variables yl , y3, and 32.5 and summed over h = h and h = 0, under the conditions (19.8) in which q and q’ are replaced by h and 0, respectively.] If we restrict yV)s within (0,T ) , we can see quite easily from the energy expression that yz - y1 = y5 - y1 = T , provided s (or u = 2s) is positive, so that the helical component in the multiple cone structure rotates by an angle of ?r - h in going from one (110) atomic plane to the next. To visualize this situation, one may refer to Fig. 8. According to the computation carried out by Kaplan and co-workers, h (their p ) stays nearly at a constant value of about 0.92 for 8 < u < 3.817 and then increases slightly for u > 3.817. They also give +v and y y as funcThe axial components of the three subtions of u (Fig. 2 in Menyuk et uLZze). lattices, specified by (1, 3 ) , (2, 4 ) , and (5, 6 ) , are arranged more or less as in the NBel configuration. An interesting point reported is that a t u = 2 the eigenvalue Xz = -s/2, given before (21.29) [see also (21.14)1,becomes degenerate with XI (0,a,P ) and Xl(h, a,p ) , and for u > 2 no real values of a and are found. I n order to solve this situation, Kaplan and co-workers chose a and /3 so as to retain the threefold degeneracy Xl(0, a,0) = Xl(h, a,P ) = Xz for u 2 2, together with the condition dXl(h, a,p ) /ah = 0. A superposition of the three corresponding eigenvectors was made to construct the spin configuration. At u = 3.817, however, the coefficient of the eigenvector associated with h ( 0 , a,p) vanished, so that only h ( h , a,P ) = X P and aXl(h, a,@/ah = 0 were retained for u 2 3.817.35f

+

S6f

a)

The mixing of the eigenvector associated with XZ would result in the splitting of the (2, 4) sublattice into the (2) sublattice and the (4) sublattice. This is because the eigenvector associated with Xz consists only of y4 which means that the only nonvanishing u.’s are uz andua, related to each other by uz = - U P , where as the eigenvector associated with XI@, a, @), with either h = 0 or h # 0, consists of y1 , y3, and y b giving nonvanishing u1 = u3, up = u1, and U S = u6 . If the wave number associated with XZ is assumed to be zero, the superposition would give a difference in the axial component of the spins on the (2) sublattice and that on the (4)sublattices, but then the geometrical condition stating that the spin Iength should be the same on the two sublattices would be violated. If the wave number associated with XZ is assumed to be h, and if uzof XI@, a,@)and uzof XZ are assumed to have a phase difference of r / 2 , then the spins on the two sublattices would describe similar cones having the same half-cone angle, and the geometrical condition mentioned would be satisfied. For u > 3.817, in particular, one would have similar helices (the half-cone angle being r / 2 ) , since the eigenvector associated with X1(O, a, a) is lacking.

392

T A K E 0 NAGAMIYA

VIII. N i e l Temperature and Spin Ordering for Complex Lattices

The NBel temperature, TN , and the mode of spin ordering immediately below TN have been investigated by Kaplan and co-workersZzdJfor a general complex lattice and, in particular, for normal cubic spinels. Let Hn, be the exchange field acting on spin S,, . H,, may be assumed to be small immediately below T N. Then for the thermal average of Sn, one should have (in the molecular field approximation)

(snv)

= [sv(Sv

-I- 1)/3kT~]Hnv.

(VIII.1)

On the other hand, one has

Hnv

=

C

2

(VIII.2)

Jmp.nv(Smp).

m,P

By Fourier transformations,

(snv)

=

8,

C(dqY)

exp (iq*Rn),

(VIII.3)

9

Hnv

=

C Hq, exp (iq.Rn) ,

(VIII.4)

9

Eq. (VIII.2) changes into

H9v = - @ / X u >

cAPAd

(VIII.5)

(dcl,),

P

where A,,(q) is defined by (VI.5). Hence, (VIII.l) becomes (dqv)

=

-[2sv(sv

+ 1)/3kTNS?]c

Apv(q)

(dqc),

(VIII.6)

c

or, written in a symmetrical form,

Thus, A

=

- ($) ~ T must N be an eigenvalue of the matrix (VIII.7)

and in fact the lowest eigenvalue determines TN and simultaneously the value of q. Equation (VIII.6) will then determine the mode of spin ordering immediately below TN . In the case of normal cubic spinels, Kaplan and co-workersZzd find that for u 5 2.177 the eigenvalue giving rise to the NBel-type spin configuration is the absolute minimum over all a! (which specifies different eigenvalues) and q. For u > 2.177, the minimum eigenvalue of the matrix (V111.7)

393

THEORY OF HELICAL SPIN CONFIGURATIONS

yields a nonzero wave vector along CllO], the corresponding spin configuration being a C l l O ] helix, and this eigenvalue is the absolute minimum over all a and q. Since the NBel configuration is stable at T = 0 only for u < and a multiple cone structure has a lower energy for < u < 1.298 and is probably the stable configuration at T = 0 as mentioned in Section 22, it appears that for the latter range of u there will be a phase transition between T N and T = 0. For 1.298 < u < 3.817, the multiple cone structure has no local stability and hence is unstable at T = 0, and a certain deviated structure should be the stable configuration. Since neither the NBel structure nor a helix is stable at T = 0 in this range of u, there will also be a phase transition between T N and T = 0. Equation (VIII.l) is based on the molecular field approximation. A rigorous equation to determine the NBel temperature and the mode of spin ordering immediately below it may be derived in the following way. Let H,( r) be a fictitious magnetic field acting only on the vth sublattice and varying with position r, and H,, exp (iq. r) be a Fourier component of it. For convenience, one may assume that all H,, (v = 1, 2, ..-,k) point in the same direction, since one is dealing with a spin system having isotropic exchange interactions. The interaction between the set of Fourier components H,, exp (iq. r) (v = 1, 2, .-.,k) and the spin system will be

+

+

-C C Snv’Hqv exp (iq-R,,) u

=

n

-N

C S:,-Hq,

X’,

(VIII.8)

Y

where S,, is the Fourier transform of S,, and N the number of unit cells. The thermal average of the component of S,, in the direction of H,, will be

where Xo is the exchange energy. In the temperature range of vanishing spin order, one has t r S,, exp [-PXO] = 0, so that to the first order in X’, B

(S,,.H,,/H,,)

=

-tr exp (-Px~)/

exp (XX0)x’ 0

X exp ( - ~ X O(S,,~H,,IH,,) ) dXI[tr exp (-PXO)].

If m and n are eigenstates of Xowhich are connected by X’ and Em and En are eigenvalues of XO, and if one assumes I Em - En << p-’ = kT,then the preceding expression will become

I

394

TAKE0 NAGAMIYA

or where the thermal average is taken over the canonical ensemble for xo . Thus, f N ( S & * S , , ) is the q-dependent susceptibility matrix and may be written as X,,,(q), and the left-hand side of Eq. (VIII.S), i.e., the induced moment, may be written as

M,,

=

c X,,(q)H,,,

where xpy(q)= f N @ ( S : , . S q V ) . (VIII.10)

P

If we denote by xp,(q)-l the reciprocal susceptibility matrix, (VIII.10) can be written also as (VIII.11) M,”Xvr(q)-l = H,, *

c V

The NBel temperature is the temperature at which infinitesimalmagnetization sets in for vanishing field. Hence it is determined from det I x,,,(q)-l

I =0

or

det I xpy(q)I=

00.

(VIII.12)

This mode of spin ordering setting in a t TN is determined from (VIII.10) with H,, = 0. (VIII.ll) means that the largest eigenvalue of the matrix x,.(q) becomes infinite at TN , and of course the value of q which gives the highest TN should be adopted. This value of q will be the wave vector of the spin ordering setting in a t TN . If there are degenerate eigenvalues associated with different values of q, one will have a conical or some other spin ordering. Since S,, = N-’ S,, exp (-iq.Rnv)

C n

+

(Rn, = R, R, is used here instead of R, for the sake of convenience), the expression for the susceptibility given in (VIII.9) can be rewritten as xrv(q)

=

(/3/3N>

c C(~rn,.S?a”) exp (iq.Rrn,,nJ m

n

395

THEORY OF HELICAL SPIN CONFIGURATIONS

The first term represents the Curie law. The second term arises from correlations between different spins, and a prime attached to the summation symbol means that m = n is excluded when p = v, i.e., the correlation in the same atom has been separated as the first term. The second term may be calculated approximately by an expansion in powers of 8. The calculation to first order in with the use of (VT.1) (which is X,) and (VI.5) yields the following result:

where

C,v(q)

A,v(q) exp Ciq.(R, - R d I .

=

(VIII.15)

The determinant of x,.(q) does not become infinite. The usual mathematical trick in such a case is to calculate the reciprocal susceptibility to first order in 8 and to derive the NBel temperature from the vanishing of it. (VIII.14) can be written in a more symmetrical form:

xpv(q)

=

{CS,(S,

+ l ) X u ( S u + 1)1'''/3kT}

- (2/3kT)D,v(q)

{6vv

1,

where

Hence the reciprocal susceptibility is obtained to be

x,v(q)-'

=

{3kT/CS,(Sp

+ 1)Sv(Sv + 1)11''} + (2/3kT)D,v(q) 1. {6pv

(VIII.17) The NBel temperature is obtained from det I 6,,

+ (2/3kT)DPv(q)I

=

0.

(VIII.18)

Since D,,(q) is a matrix equivalent to BpY(q)defined by (VIIT.7), Eq. (VIII.18) is the same as that we derived earlier in the molecular field approximation. Kaplan et aL2'f carried out a calculation up to P3 and obtained results that suggest that one can expect to have the NBel ordering for values of u much greater than that predicted by the molecular-field approximation.360 IX. Neutron Diffraction: Theory and Examples

Neutron scattering experiments provide the most powerful means of determining an ordered spin arrangement in a crystal, as well as of oba58

Details have not yet been published.

396

TAKE0 NAGAMIYA

serving the spin-wave spectrum and correlations between neighboring spins. In the present part, we wish to give a brief discussion of diffraction lines from an ordered spin arrangement, in particular a helical order, and to mention a few interesting observations reported. 23. GENERAL THEORY OF ELASTIC NEUTRON SCATTERING

According to the standard theory,36,37 the amplitude of the neutron wave elastically scattered from an atom at the origin by the interaction between the magnetic moment of the neutron and that of the atom is given, apart from a universal constant, by (l/r) exp

(ih’r)Pspin

+

- e(Pspin-e) (i/K)e x Porbl’SX,

with

e Pspin

=

(+*,

= K/K,

C

sj

K

=

k - k’,

1kI

=

I k’I

=

(23.1)

k‘,

exp (iu- r&),

j

Porb= (1/2i) C(+*,

C exp ( i ~ . v,+> - (+, C exp (iu. rj>vj+*)I, rj>

i

i

where s is the spin vector of the neutron, x the spin function of the incident neutron, sj and rj the spin and position vectors of the jth electron in the atom, and t j the electronic wave function of the atom. From (23.1), it is seen that only the components of Papin and of Porb perpendicular t o K are effective in the scattering, K being the difference between the wave vector k of the incident neutron and the wave vector k’ of the scattered neutron. When the scattering atom is at position R,, , a factor exp (iu-R,,) will separate from Pspinand Po&, so that one will have, for the amplitude of the scattered wave from a crystal, (23.2)

+

where P,, is the value of Pspin- e(Pspin-e) (i/K)e x Porbassociated with the atom at R,, , the origin of the electron positions being taken at the nucleus of that atom. This P,, is in essence (apart from the magnetic form factor) the projection of the magnetic moment of the atom at R,, on the plane perpendicular to the scattering vector K. Now, consider that P,, varies sinusoidally with position R,, as

Pn,

=

[pyzcos (Q’Rnv

+ a,),

Pvusin (Q-Rnv

where the x and y axes are taken perpendicular to exp (iK~Rn,)[PYx~x cos (Q-R,, n,v

as 0. Halpern and M. H. Johnson, Phys.

K.

+

ay)],

(23.3)

Then (23.2) becomes

+ a,) + PWsusin (Q-Rn, + ay)lx,

Rev. 66,898 (1939).

G. T. Trammell, Phys. Rev. 92, 1387 (1953).

THEORY OF HELICAL SPIN CONFIGURATIONS

or

3

c c exp [ i ( ~+ Q) *Rnv+ iav](Pv=Sz- ~P,S,)X + 3 c c exp [ i ( ~- Q) -R,, - iavl(PVzsz+ iP,,s,)x. v

397

n

v

(23.4)

n

The first term of (IX.4) gives an interference condition

K+Q=K

k'=k+Q-K

or

(23.5)

and the second term another interference condition

K-Q=K

k'=k-Q-K,

or

(23.6)

where K is a reciprocal lattice vector. These conditions were first derived by Yoshimori.' Thus, for Q # 0, magnetic reflections will appear a t positions different from those of nuclear reflections, and in fact the former appear as satellites of the latter. If there are a number of wave vectors coexisting in the spin arrangement, one should of course have as many satellites as there are wave vectors. Taking the quantization axis of the neutron spin to be parallel to the scattering vector K, we write the spin function of the incident neutron as

x

= aa

with a = cos (0/2) exp ( -2$/2),

+ bP, b

=

(23.7) sin (0/2) exp (i4/2), (23.8)

where 0 and 4 are the polar and azimuthal angles of the incident polarization. Then (23.4) becomes, by virtue of (23.5) or (23.6), either

+ + Pv,(aP - ball exp (zK.Rv+ ia,)

$N c [ P y z ( a P ba)

(23.9)

Y

or

+

$N c [ P y z ( a P ba) - P,,(cCp

- ba)]

exp ( X - R , - ia,). (23.10)

V

R , is the position of the vth atom in the zeroth unit cell. The squares of the absolute values of these quantities, summed over the spin variable, give the intensities of the satellites. For unpolarized beam, 1 a l2 and I b l2 are equal on the average, so that (23.9) and (23.10) give intensities that are proportional to

I

c P,, exp ( X - R ,f Y

l2

ia,)

+I

P , exp ( z K - R ,f ia,)12.

(23.11)

V

When av is independent of v, the two satellites corresponding to the plus and minus signs of (23.11) have the same intensity, aside from a minute difference in the form factor due to the different values of K.

398

T A K E 0 NAGAMIYA

If the incident beam is polarized parallel to the x-axis, the x-component P,, will not alter the polarization, but the y-component P,, will reverse the polarization direction, as can be seen directly from (23.4). If the incident polarization is parallel to the scattering vector, i.e., a = 1 and b = 0, the polarization will be reversed, as can be seen from (23.9) and (23.10) , and one will have from (23.9) and (23.10) the intensities proportional to

I c ( P v zf P,,)

exp (zK.R, f ia,)12,

(23.12)

V

respectively. If, in particular, P,, = P,, or P,, = -Pvu , the vth sublattice will not contribute to the intensity of k’ = k - Q - K or that of k’ = k Q - I(,which means that one can, in principle, observe the sense of the spin rotation in a helical configuration. Theories similar to that just described have been published in a number of papers. The readers will find references in a paper by B l ~ m e . ~ *

+

24. EXAMPLES OF HELICAL SPINCONFIGURATION 1. Chromium is known to have a magnetically ordered phase below 310°K which is interpreted as being a “spin density wave” state.38aThe spin density varies sinusoidally with atomic positions, and this persists down to absolute zero. The wave vector is parallel to one of the cubic axes (the crystalline structure being bcc) and has a magnitude of 2a(l - & ) / a ( a : the lattice constant) a t 20°C, for instance. Between 310’ and 121”K, the spin vectors are perpendicular to the wave vector, whereas below 121°K they are parallel to the wave vector. With an unpolarized neutron beam, it is not possible to decide whether the spin configuration in the high9 a temperature phase is helical or linear-sinusoidal. Brown et ~ 1 . ~used neutron beam polarized along one of the cubic axes and observed depolarization in the diffracted lines. When, for example, the incident polarization was parallel to [OlO] and the line (0, 0, 1 - &) was observed, the ratio R of the intensities of the line with unchanged and reversed polarizations was 1.20 f 0.03, which ruled out the simple helical model and suggested the existence of domains with linear-sinusoidal spins parallel to [OlO] and [loo], in the ratio of 1.2 to 1. 2. Examples of helical or related spin order observed up to the present time (1966) are not many. Mn02 was the first example observed or, rather, interpreted, as mentioned in Section 1. MnAuz , referred to in M. Blume, Phys. Rev. 130, 1670 (1963). A discussion on the magnetic properties of Cr and the spin density wave will be given the forthcoming Part 2. w . P. J. Brown, C. Wilkinson, J. B. Forsyth, and R. Nathans, Proc. Phys. SOC. (London) 86, 1185 (1965). 38a

THEORY OF HELICAL SPIN CONFIGURATIONS

399

Section 12, and a number of rare-earth metals, referred to in many of the preceding sections, will be discussed in some detail in the forthcoming Part 2. These systems provide most interesting examples. Helical and more complicated spin configurations in spinel 1aOtices have been discussed in Chapters VII and VIII. MnI, , FeCl3 , and solid solutions of Crz03 and Fez03 also present helical spin arrangements, which will be described below. 3. MnIz crystallizes in a hexagonal layer structure of the CdI, type, i.e., the iodine ions form a hexagonal close-packed ABAB lattice, and the manganese ions penetrate into every other interlayer space to form hexagonal layers of the C type. Cable et aZ.40 observed by neutron diffraction the magnetic order in this crystal. The NBel temperature is 3.40"K1 and the magnetic order at 1.3"K is helical with a propagation vector pointing along the (307) direction, referred to the hexagonal unit cell, and a turn angle of 2a/16. When the plane containing the moments was assumed + assumed, normal to the helical axis, and a moment of 4 . 6 p ~ / M n ~was the calculated intensities of the Bragg lines were in good agreement with those observed. An interesting observation was that there was an apparent threefold symmetry about the c-axis in the reflections, which could be interpreted as being due to magnetic domains, which, in the absence of a magnetic field, grew with equal probability along three equivalent (307 ) axes. Application of a magnetic field favored that domain which had moments most nearly perpendicular to the field direction, and this domain grew at the expense of the other two until at saturation the entire crystal was transformed into a single domain. An attempt to interpret the observed magnetic order in terms of exchange interactions and anisotropy energies has been made by Moriya and the present writer,4l but it is not easy to attain an unambiguous conclusion. 4. FeCL is another compound that shows a helical order at low temperatures. It crystallizes in a hexagonal layer structure of the Bi13 type. The chlorine ions form a hexagonal close-packed ABAB lattice, as iodine ions do in MnI, , and the ferric ions penetrate into every other interlayer space to form, in this case, honeycomb layers. The unit cell has a c axis that corresponds to three such honeycomb layers. The same also made neutron diffraction observations of the magnetic order. The NBel temperature is at 15 f 2°K. The moment ordering is helical with a propagation vector pointing along (140) and with a turn angle of 2 ~ / 1 5 .The moment vectors 4O

J. W. Cable, M. K. Wilkinson, E. 0. Wollan, and W. C. Koehler, Phys. Rev. 126,

41

K.Moriya and T. Nagamiya, J. Phys. SOC.Japan (1968) (to be published).

42

J. W. Cable, M. K. Wilkinson, E. 0. Wollan, and W. C. Koehler, Phys. Rev. 127,

1860 (1962).

714 (1962).

400

TAKE0 NAGAMIYA

are normal to the propagation vector and have a magnitude of 4.3pB/Fe3+. The formation of magnetic domains similar to that in MnIz was observed. 5. Cr2O3and a-Fez03 possess isomorphous crystalline structures of the rhombohedral corundum type. The oxygen ions form a deviated hexagonal close-packed lattice and the metallic ions, imbedded in it, form deviated honeycomb layers. Each metallic ion is surrounded by six oxygen ions, of which three in the same c plane form a smaller equilateral triangle than that in the regular hexagonal close-packed lattice, and the remaining three, in the opposite c plane, form a larger triangle. The metallic ion at the center is shifted along the c axis toward the center of the larger triangle. Since the larger triangle and smaller triangle are alternately arranged in the same c plane, each honeycomb layer is uneven. I n the rhombohedral unit cell, there are four cations situated on the threefold axis, alternately apart and close, as A1 A& 2 3 2 . The ordered magnetic structure in Cr203 below TN = 310°K is such that these four cations have spins (+ -), the spin axis being parallel to the threefold axis. I n this spin arrangement, each honeycomb layer is antiferromagnetic. I n a!-Fe203,the ordered spin arrangement below TNE 950°K is of the type (+ - -) , which means that each honeycomb layer is ferromagnetic, and adjacent layers have opposite spins. The spin axis in Fe203is perpendicular to the threefold axis down to about -15°C but becomes parallel to this axis below - 15°C. As is known well, a weak ferromagnetism, due to the Dzyaloshinsky-Moriya interaction, is observed in the temperature range above -15°C. Crz03and a-Fe203 crystallize as solid solutions over the entire range of composition. Cox et ~xE.4~prepared samples of (1 - x)Cr2O3.xFeaO3for several values of x and made magnetic measurements, crystal parameter determination by X-rays, and spin structure determination by means of neutron diffraction. Starting from Cr203,the addition of Fez03 creates a helically ordered spin component superposed on the Cr203-type antiferromagnetically ordered component. Corresponding to the division of the Crz03-type antiferromagnetic order into two ferromagnetically aligned sublattices, which are coupled antiferromagnetically with each other, the helical component is divided into two sets, which have opposite phases in the helical rotation (i.e., the phase constant a! that appeared in Section 1 differs by T ) . The propagation vector is parallel to the c axis, and the helical moment vectors are perpendicular to the c axis. Hence, there is a conical spin arrangement on each of the two sublattices, the cones on one sublattice pointing oppositely to those on the other. The period in the helices decreases with increasing x (starting virtually from infinite period a t x = 0 ) , the magnitude of the helical component relative to that of the

+

+

‘8

D. E. Cox, W. J. Takei, and G. Shirane, J . Phys. Chem. Solids 24,405 (1963).

THEORY O F HELICAL SPIN CONFIGURATIONS

401

antiferromagnetic component increases with x, and the NCel temperature decreases with x to about half of the initial value at x = 0.2. When x exceeds about 0.2, the cone axis becomes perpendicular to the c axis, and at the same time the axial, antiferromagnetic component takes the Fez03-type order. The propagation vector for the helical component remains parallel to the c axis, so that this component constitutes a cycloidal spin arrangement. With increasing x, up to x = 0.35 or 0.4, the period remains nearly constant, but the cone angle decreases, until finally, when this becomes zero, the Fez03-type antiferromagnetic order is attained. All these are, however, the behavior at sufficiently low temperatures. The NBel temperature increases with increasing x. There appears the second critical temperature in the range 0.2 5 x 5 0.4, below which the helical component develops, but above which there is a simple Fez03-type spin arrangement. This critical temperature is also indicated by magnetic measurements. It appears to continue from the NCel temperature of the Crz03-richsamples but decreases more slowly with increasing x. For x > 0.4, no helical component is detected. Weak ferromagnetism is observed over the whole range of x > 0.25 but appears to vanish at a particular temperature, which is the second critical temperature. For x 5 0.2, no weak ferromagnetism is observed. These interesting observations challenged theoreticians for interpretation. On the assumption of a uniform distribution of Cr3+ and Fe3f and the consideration of four exchange constants, corresponding to apparently the most important superexchange paths, the the0ry~~s45,~h predicts only a pure helical configuration, other possibilities being the Crz03-type and Fez03-type antiferromagnetic configurations and the ferromagnetic alignment. [It may be mentioned in passing that recent measurements of the four exchange constants between Cr3+ions in ruby (AlZO3 Cr impurities) by M ~ l l e n a u e rwith ~ ~ the prizospectroscopic method give values such that predict that Crz03 would have the Crz03-type antiferromagnetic spin arrangement.] The theory is based on the Lyons-Kaplan method introduced in Section 19. The reason why a conical configuration cannot be obtained but only a pure helical configuration is predicted is as follows. When one considers cations and exchange interactions only and neglects oxygen ions, one can take a smaller rhombohedra1 unit cell containing two cations, say A1 and A 2 (or B1 and B,) . Since there is a point of in-

+

E. F. Bertaut, Proc. Intern. Conf. Magnetism, Nottzngham, 1964,p. 516. Inst. Phys. Phys. SOC.,London, 1965. * 5 N. Menyuk and K. Dwight, J. Phys. Chem. Solids 26, 1031 (1964). 46'1 The present author has made an independent calculation : T. Nagamiya, unpublished report (1965). 4 6 L. F. Mollenauer, Thesis, Stanford University (1965).

44

402

TAKE0 NAGAMIYA

version symmetry midway between A1 and Az , the exchange interactions within the A1 sublattice and those within the A2 sublattice are essentially the same; in particular, the Fourier transforms of the exchange constants defined in Section 16, All(q) for the A1 lattice and Azz(q) for the Az lattice, are the same. In such a case, a conical spin configuration is impossible, as mentioned at the end of Section 18. If there is a cation order in the solid solutions that destroys the inversion symmetry, so that the two sublattices are no longer equivalent, then one has the possibility of having a conical spin configuration. It is not possible to observe a cation order by X-rays. Nor was it detected by neutrons, since the calculated intensities are barely greater than the observable limit. Despite these facts, Cox et al. report that considerable order could exist. Theoretical consideration in such a case has not yet been given. 6. MnP is another interesting substance whose magnetic properties have become understood rather recently. It is ferromagnetic below 291’K and transforms to a metamagnetic phase at 50”K, as found by Huber and Ridgley” from magnetic measurements. By “metamagnetic” is meant the characteristic that the magnetization increases steeply in a certain range of applied magnetic field, approaching a near-saturation (as in FeClz and MnAuz). A theory for the electronic state in MnP and the like was proposed by G o o d e n o ~ g hand ~ ~he predicted a helical spin ordering in the metamagnetic phase of MnP, which is somewhat different from that observed by neutron diffraction experiment to be mentioned below. Hirahara and co-w~rkers~~ also imagined a helical spin configuration from the peculiar results of their torque measurements. The crystal structure of MnP is orthorhombic ( a > b > c for convention). The atomic arrangement can be derived from that of NiAs by displacements of the atoms; the b and c axes in MnP correspond to the hexagonal c and a axes in NiAs, respectively, and the a axis in MnP corresponds to the direction perpendicular to both the c and a axes in NiAs; the manganese atoms, lined up along the b axis in MnP, are displaced parallel to the a axis in a zigzag way (by f a / 2 0 , and each line is actually displaced parallel to the b axis by f b / 2 0 0 ) , and the phosphor atoms are displaced up and down parallel to the b axis (by f0.06b) ; the orthorhombic unit cell contains four manganese atoms, and the manganese lattice can be subdivided into two similar, approximately body-centered, orthorhombic lattices (exactly body-centered if the displacements by fb/200 were zero) ; these sublattices are displaced relative to each other by (a/lO, b / 2 , 0 ) . 47

E. E. Huber and D. H. Ridgley, Phys. Rev. 135, A1033 (1964). J. B. Goodenough, J . A p p l . Phys. Suppl. 35, 1083 (1964); J. B. Goodenough, M I T Lincoln Lab. Tech. Rep. 345 (1964). T. Komatsubara, K. Kinoshita, and E. Hirahara, J.Phys. Soc. Japan 20, 2036 (1965).

THEORY OF HELICAL SPIN CONFIGURATIONS

403

Neutron diffraction e x p e r i ~ n e n t srevealed ~ ~ * ~ ~ that the spin arrangement at 4.2"K is helical, the spin vectors rotating in the bc plane with a propagation vector parallel to the a axis. The period of rotation is about 9a and the phase difference in the two body-centered sublattices is such that the spin at a corner site (a/20, 0, 0 ) of one sublattice is parallel to the spin a t a body-center site (9a/20, b/2, c/2) of the other sublattice. We shall not mention here other observations so far made, but we would like to refer to the following fact (after Hirahara). When a magnetic field is applied in thg ac plane at 42°K and its component in the c direction is less than 2 kOe, a magnetic moment proportional to the field strength is induced in the field direction (the susceptibility in the a direction being eventually equal to that in the c direction), so that no torque is observed in this case. When the c component of the field exceeds 2 kOe (for total field values not exceeding 20 kOe) , a ferromagnetic alignment occurs in the c direction, with a little tilt toward the field direction. Thus, 2 kOe is the value of the critical field for the transition from the helix in the bc plane to the ferromagnetism in the c direction. The anisotropy energy to hold the spin vectors in the bc plane appears to be very large and the c axis seems t o be the easiest axis of magnetization. I n the torque curves in the ac plane there appears a range of vanishing torque in the neighborhood of the a axis, since there the c component of the field does not exceed 2 kOe. In fact, detailed features of the torque curves, as well as essential features of observed magnetizations curves, in the ac plane can be accounted for very well by the picture described above. When the field is in the bc plane, the change of the spin structure seems to be more complicated, possibly including the formation of a fan structure for a field along the b axis that exceeds 5.4 kOe. NOTEADDEDI N PROOF. The magnetostrictive energy is the main driving force for the helix to ferromagnetic transition in D y and Tb, according to B. R . Cooper (Phys. Rev. Letters (1967), to appear). The magnetostrictive energy of cylindrical symmetry about the hexagonal axis in the ferromagnetic state has a magnitude that exceeds by far the magnitude of the anisotropy energy of sixfold symmetry. Our formal analysis for high temperatures should still be valid. I n our low temperature analysis, wmin in the right-hand side of (5.3) must include this magnetostrictive energy, whereas the same quantity appearing in Eqs. (5.4) and (5.5) must not. Cooper shows that in Tb the magnetostriction is not frozen in the lattice for the q = 0 spin wave but it follows the motion of the magnetization, so that it is not effective for the frequency of this spin wave. It may follow, therefore, that wminappearing in (5.8) is the same as that appearing in (5.4) and (5.5), and hence is smaller than that in (5.3). The inequality (5.3) then might not ensure the inequality (5.8) for H = 0. However, in rare-earth metals, where p = 6, J. B. Forsyth, S. J. Pickart, and P. J. Brown, J. A p p l . Phys. 37, 1053 (1966); Proc. (London) 88,333 (1966). Phys. SOC. 51 G. P. Felcher, J. A p p l . Phys. 37, 1056 (1966).

404

T A K E 0 NAGAMIYA

it seems that (5.8) still holds, since a direct. helix to ferromagnetic transition has been observed. ACKNOWLEDGMENT The author would like to express his sincere gratitude to Dr. F. Seitz, for his continuing encouragement throughout the writing of this article. He also would like to thank Dr. Earl Callen and Dr. H. Miwa for reading the manuscript and making helpful suggestions.

Appendix

Al. SUSCEPTIBILITY OF THE FAN STRUCTURE We shall supplement here the mathematical proof of some facts mentioned in Section 9 and derive the amplitudes of the oscillating x and y spin components in the fan structure, as functions of the applied field (applied in the x-direction) . Also the formula for the susceptibility Xfan will be derived. We shall confine ourselves to field values in the vicinity of HO, given by (9.13) , where a transition occurs between the fan and the ferromagnetic structure. First we shall consider the quantity (Al.l) where H* is the effective field in the ferromagnetic state, which is given by

H* a.

=

+H,

(A1.2)

U(0)Sao

being defined by

Sao

=

tr S, exp P[H*S, - w(S,)] t r exp P[H*S, - w (S,)]

’

(A1.3)

As in Section 9, HA, and H , in (Al.1) are the fluctuating x and y components of the exchange field acting on the nth spin. We expand ( A l . l ) in powers of HAz and H,, . Since S, and Su do not commute with 8,, we have to make use of the expansion formula given in the first footnote of Section 4, up to any desired power. Then, there appears, for example, such a term as HA, times dX e@-X)KSz eXK,

where K = H*S, - w(S,).

(A1.4)

Also, a term H i u times (A1.5)

THEORY OF HELICAL SPIN CONFIGURATIONS

405

and so on appear. For the sake of brevity, we shall write, for instance, the integrals (A1.4) and (A1.5) as (1,

(1,

fL)l

s,, SU),

respectively, and S, times these as

(Sz XZ), 1

(SZ 1

xu

1

SU),

respectively. Then, we can show that the following relations hold: t r (1, S,)

p t r S, ePK,

M t r ( S , S,), (1, S, , S,) = +B t r ( S , , S,),

t r (1, f l u tr

=

1

flu) =

1

(A1.6)

H* tr ( S , , S,)

=

tr S, ePK,

H*2 tr ( S , , S, , S,)

=

H*3 t r ( S , , S, , S, , S,)

=

$H* t r ( S , , S,) - Str S, ePK.

The first three relations can be derived by elementary calculations of changing the order of integrations and with a theorem that, in tr, cyclic permutations of the factors are allowed. The rest follow with the method mentioned in Section 9, (9.8)- (9.11) ;this method is applied to t r S, exp pK and t r S, exp PK, and the expansion in powers of e is taken up to e3. By (Al.G), we can express with t r S, exp pK and tr ( S , , S,) all the other traces. Now, t r S, exp OK

=

Suo t r exp pK,

(A1.7)

by (A1.3), and t r ( S , , 8,) can be written in terms of the susceptibility of the ferromagn,tic state, Xf , by the following calculation. When we vary H* by 6H*, we have (XUCJH*+6H*

=

+

tr S, exp p[(H* 6H*) S, - w] trexpp[(H*+6H*)S, -w] '

The right-hand side can be written as

+ 6H* t r (X, , Sz) + 6H* t r (1, S,)

tr S, exp p[H*S, - w] t r exp p[H*S, - w]

'

Using relations (A1.6) , we can write this in terms of Su0 and t r ( 8 , , S,) evaluated at H*. Namely, we have

406

TAKE0 NAGAMIYA

Thus,

( S U O ) H * +~ H( *S U O ) H* -d(Sa0) = 6H*

dH*

-

t r ( S , , S,) - ~ ( S U O ) ’ .(A1.8) trexppK

On the other hand,

+ H . Writing

since H* = 2J(0)Sao

d(Sao)/dH

= Xr

,

(A1.lO)

which is the ferromagnetic susceptibility, we have from (A1.9) the following formula : d (Sao) -

dH*

Xf

+

(A1.11)

2xrJ(O) 1 . ( A l . l l ) , combined with (A1.8), gives t r ( S , , S,) in terms of xr and

uo .

We are now ready to calculate the quantities (A1.12a) and (Al. 12b) The results of the calculation are the following: SaL ( = Xunz - Sao) (A1 .13a)

(A1.13b)

From these follows also a relation (A1.14) To solve these equations, we put

(A1.15)

THEORY OF HELICAL SPIN CONFIGURATIONS

407

so that

+ a) + 2J(3Q)Su,3sin3(Q.Rn + a ) + + 2J(2Q)S& cos2(Q.Rn + a) + (A1.16)

H,,

=

2J(Q)SuUlsin(Q.Rn

HLz

=

2J(O)S&

a * * .

Substituting (A1.15) and (A1.16) into Eqs. (A1.13a) and (A1.14), we obtain equations to determine the Fourier amplitudes. Confining ourselves to the three amplitudes u Y l , a&, a:2 , we obtain the following equations: d o =

a:2

=

d (Sgo) 7 2J(O)& dH

+

(A1.17)

d (Sue) dH*

J(Q)%il,

~

(A1.18)

(A1 .19)

Near the field Ho( = 2[J(Q) - J ( O ) ] S u } ,we can write

H* - 2J(Q)Su0

[where H*

+H

=

2J(0)Su0

-HO(UO - U ) / U

+ ( H - Ho)

=

-H~uo/u

=

[1

=

(1 - XJ(Q) - J(O)lxr}( H - Ho),

=

+H]

- (Ho/c)(duo/dN)](H - Ho)

where X I , given by (Al.lO), is evaluated at Ho . The quantity calculated above appears in the right-hand side of (Al. 19). In other terms of equations (A1.17), (Al.l8), and (A1.19) near N o , Xuo/H* can be replaced by $ J < Q ) . Furthermore, we can utilize (Al.11) for d(Sao)/dH*. Then, from (A1.17)(A1.19) we obtain

d o

= -{ 1 -

2[J(Q)

- J ( O ) ] x r )( & / 4 ~ ) ,

(A1.20) (A1.21)

408

TAKE0 NAGAMIYA

Hence we obtain, for instance, doas su:o =

(1 - 2CJ(Q) - J ( O ) l ~ r } { 1- 2CJ(2Q> - J(O)lxr](H - Ho) 3J(Q) - z J ( 0 ) - J(2Q) - 4CJ(Q) - J(O)ICJ(2Q) - J(0)lxr' (A1.23)

The susceptibility of the fan structure immediately below Ho is given by

-

1

+ CJ(Q> + W ( 0 )- 3J(2Q)lxr

3J(Q) - 2J(O) - J(2Q) - 4CJ(Q) - J(O)ICJ(2Q)

- J(O>lxf.

(A1.24)

It follows from (A1.23), (A1.20) , and (A1.21) that dois proportional to H - Ho , uDl is proportional to (Ho - H)ll2,and u:2 is proportional to Ho - H . Other Fourier amplitudes that we have neglected are of higher orders. The susceptibility of the fan structure just derived and given by (A1.24) can be expressed in another way:

where p2 and y are defined by (8.9) and (8.10). To derive this, we notice that in the present case u0 stands for u of (8.10). So we have

(A1.26) [by (Al.ll)], from which follows

+

xr = { 2 ( 7 1)CJ(Q) - J(O)l]-'. (A12 7 ) Using this and (8.9) , we can write (A1.24) as (A1.25). The formula for X f a n was derived in a simpler way by Kitano and Nagamiya15by the method described in Section 8. At absolute zero, where y is infinity, Xf vanishes, and from either (A1.24) or (A1.25) we have Xfan =

[3J(Q)

- 2J(O) - J(2Q)]-',

which was derived earlier by Nagamiya et aZ.l4

(A128)

409

THEORY OF HELICAL SPIN CONFIGURATIONS

The coefficients A and B used in Section 9 ( C being disregarded in the following) are the coefficients of HAz and H,, , respectively, in (A1.13a) and (A1.13b), namely,

A

=

d(SUo)/dH*,

B

=

SUO/H*.

(A1.29)

By the definition of u 0 , (A1.3), one will be able to imagine easily that a0 is an increasing function of H*, going to saturation, a 0 4 1, for H* + w . The curve of uo versus H* must be a monotonically increasing, upward convex curve. Thus, both A and B are decreasing functions of H*, and B for nonzero finite H*. we see from (A1.29) that A d;

A2. PARALLEL SUSCEPTIBILITY OF THE HELICAL STATE With the method developed in Section Al, we can also derive the susceptibility of the helical state for weak field applied parallel to the plane of the spin rotation. We introduce a local coordinate system {, E, z such that at R, the { axis coincides with the direction of the thermally averaged nth spin in no external field, the z axis perpendicular to the plane of the spin rotation, and the 2: axis perpendicular to both. Then we have Eqs. (A1.12a) and (A1.12b), with { and E replacing x and y. H* is here related to u by

H*

2J(Q)Su,

(A2.1)

tr Sr exp P[H*Sr - w ( S , ) ] . t r exp P[H*Sr - w ( S , ) ]

(A2.2)

=

and, in place of (A1.3), we have

Sa

=

In (A1.13a) and (A1.13b), u0 is now replaced by u, and z and y by { and

E . We shall retain only terms linear in HLr and HnE. Then, putting (A2.3) we have simple equations:

SUL~= [ d ( s ~ ) / d H * ] [ H cos (Q'Rn) Sung = [ S a / H * ] [ - H sin (Q-R,)

+ €€;:I,

+ HA:].

(A2.4)

Here HA; and HAt are the components of an exchange field acting on the nth spin such that arises from a small change in the spin configuration due to the applied field H . In the original coordinate system, this small change in the spin configuration can be specified by Sa:, and Sa:,; these are related to Sa&

410

TAKE0 NAGAMIYA

and S U , ~ by Sd,

=

SaAr cos (Q .Rn) - Xunrsin (Q -Rn)

SU& = Xu& sin (Q*Rn)

+ S a n cos ~ (Q 'Rn).

The corresponding exchange-field components are

C J (Rmn) SUL, HAv = 2 C J ( L n ) XU&,

HL

=

2

m

(A2.5)

m

which are related to HA! and HAt by similar equations. Transforming (A2.4) back to the original coordinate system, we have

(A2.6a)

(A2.6b) I n order to solve (A2.6a) and (A2.6b), together with (A2.5), we expand u;, and uLy in Fourier components: uA,=u:~+u~cos~(Q*R + u~L) c o s ~ ( Q * R , ) aAu = ai2 sin 2(Q*Rn)

+

sin 4(Q.Rn)

+

+ -...

'

(A2.7)

(Odd harmonics appear only when we include nonlinear terms.) Substituting these into (A2.5) , (A2.6a), and (A2.6b), we obtain the following equations :

SU:O= +u[H

+ 2J(O)S&] + +b J(2 Q >S (d z+ &),

where

a

=

d(Sa)/dH*

+ (Su/H*),

b

=

d ( S a ) / d H * - ( S u / H * ) . (A2.9)

41 1

THEORY OF HELICAL SPIN CONFIGURATIONS

From these equations, we can easily obtain S&. With the use of (A1.11), (A2.1), and (A2.9), we can express the result as

Sdo H

Xh = __

-

+

1 ZCJ(Q> 2{2J(Q) -J(2Q) - J ( O )

+J(0) - 2J(2Q)l~r

+ CJ(Q)

-J(O)ICJ(O)

-J(2Q)lxr}

'

(A2.10)

where we understand by Xr the ferromagnetic susceptibility at H Using (A1.27), we can also express (A2.10) as

=

Ho .

(A2.11)

For T + 0, (A2.10) or (A2.11) reduces to Xh =

(2[2J(Q) -J(2Q)

-J(o)]}-'.

(A2.12)

These equations for Xh were derived by Kitano and NagamiyaI6 and earlier, for a special case, by Y0shimori.l

This Page Intentionally Left Blank

Author Index The numbers in parentheses are footnote numbers and are inserted to enable the reader to locate a cross reference when the author’s name does not appear at the point of reference in the text. A

Bennemann, K. I f . , 27 Bensch, H., 15 Benston, M. L., 107, 109(61) Berggren, M. J., 159 Berlin, T., 107, 110 Bersuker, I. B., 129, 131, 132, 134, 150, 165, 167, 210(236) Bertaut, E. F., 401 Bethe, H. A., 25, 95 Bijl, D., 166 Bir, G. L., 56, 88(29) Bird, R. B., 107 Birman, J. L., 146, 177 Bivins, R., 100 Biswas, A. B., 148 Blasse, G., 94 Bleaney, B., 92, 95, 99(20), 128, 132, 158, 162(20), 166 Blume, M. 398 B Bogle, G. S., 166 Bacchella, G. L., 379 Bolton, J. R., 143 Bacon, R., 47 Bonch-Bruevich, V. L., 214, 230 Bagchi, S. N., 14 Bongers, P. F., 150 Baldwin, J. A. 173 Borie, B. S., 150 Ballhausen, C. J., 95, 98(23), 115, 116(23), Born, M., 105, 214, 215, 217, 218(15), 238 120(80), 121(80), 141, 144, 145, 146 290, 291 Baltzer, P. K., 156, 157, 169(216) Bose, A., 166 Balz, D., 149 Bowers, K. D., 92, 128, 132 Baraff, G. A. 70, 71(54), 72(54) Bradburn, 238 Barker, A. S., 185 Brebick, R. F., 59 Bardeen, J., 38 Brill, R., 2, 8(1), 15, 16, 18, 22, 23(38), Barron, T. H. K., 217 24, 27 Batarunas, J., 100 Brdckhouse, B. N., 220, 234(28) Bateman, T. B., 48, 55-57, 59(25), 60, 137 Bron, W. E., 140, 185, 186 Batterman, B., 13, 16 Brooks, H., 147 Baym, G., 214, 225, 233, 241, 242(57), 257 Brossel, J., 209 262, 265(4), 282, 288(4) Brown, P. J., 398, 403 Belford, R. L., 187 Brown, T. H., 143 Bemski, G., 159, 176 Brugger, K., 64 413 Abragam, A., 121, 126(89), 127, 131(89) Abeles, B., 78 Abrikosov, A. A., 214, 303 Adams, E. N., 56, 88(28) Adamson, A. W., 146 Adler, D., 147 Adrian, F. J., 195 Agnetta, G., 185 Aiello, G., 149 Aiyama, Y., 129, 148(125) Akhiezer, A,, 274 Alder, E., 53, 54(23) Allgaier, R. S., 59 Arnott, R. J., 167 Arzt, It., 140, 159(147), 275 Auzins, P., 128, 166(120) Avvakumov, V. I., 114, 129

414

AUTHOR INDEX

Bruner, L. J., 40, 47(9), 49(9), 59(9), 60 Bryant, C., 59, 60, 62 Budnikov, S. S., 132 Burnham, D. C., 128, 133(122) Burns, J. W., 47 Burnstein, E., 220 Button, K. J., 74 Buschert, R. C., 68, 69 C

Cable, J. W., 328, 399 Cady, H. H., 31, 32 Calder, R. S., 19 Calvin, M., 187 Caner, M., 164, 165 Carrington, A,, 143 Carruthers, J. A., 84, 88, 89(95) Carruthers, P. 87, 279 Casimir, H. B. G., 279 Castner, T. G., 147 Cervinka, L., 148 Chakravarty, A. S., 166 Chandrasekhar, B. S., 70, 73 Chaterjee, R., 166 Chau, C. K., 140 Chen, J. H., 128, 187 Chernik, I. A., 59 Chester, G. V., 215 Chipman, D. R., 13 Chipman, D., 13, 16(16) Child, M. S., 93, 115(8), 144, 156 Chinik, B. I., 132 Chopra, K. L., 16 Chrenko, R. M., 128, 172(116), 176(116), 205 Ciccarello, I. S., 275 Ciccarelo, I. S., 140, 159(147) Clark, H., 27 Clinton, W. L., 107, 117(58), 163(58) Cochran, J. F., 88, 89(95) Cochran, W., 19, 20, 32 Coffman, R. E., 128, 131(121), 133, 141 Cohen, M. H., 73 Compton, W. D., 141, 200 Conan Doyle, A., 92 Condon, E. U., 112, 179 Cooke, A. H., 166 Cooper, B. R., 348, 358(19), 360 Cooper, M. J., 13, 16

Corbett, J. W., 127, 128, 172(113), 175 (114), 176(113) Corliss, L. M., 362 Cotton, F. A., 190 Coulson, C. A,, 171, 177(250) Cowley, R. A,, 214, 243, 263(7) Cox, D. E., 379, 400 Cross, P. C., 101 Crozier, M. H., 125 Csavinszky, P., 46, 49, 50, 51, 57 Curtiss, C. F., 107 D Darcy, L., 149 Davis, R. E., 78 Dawson, B., 25, 31, 35 Dean, P. J., 176 Decius, J. C., 101 de Clerck, E. F., 148 de Groot, M. S., 143 de Heer, J., 141 de Launay, J., 59 Delbecq, C. J., 147, 187, 192, 199 Delorme, C., 149 De Marco, J., 13, 16(16), 24 de Wit, M., 168, 169 Dexter, D. L., 180, 181(273), 182(273) Dick, B. G., 217 Dietz, R. E., 168, 193(242), 204, 206, 207 Dimmock, J. O., 108, 113(63), 156(63) Dinot, M., 379 Doremus, R. H., 127 Drabble, J. R., 64, 65, 66 Dransfeld, K., 140, 159(147), 275 Dreyfus, R. W., 140 Duffus, R. J., 166 Dunitz, J. D., 98, 116(33), 149(33) du Preez, L., 176 Dutta-Roy, S. K., 166 Dwight, K., 362, 371(22c), 379, 381(22e), 392(22d), 395(22f), 401 Dyer, L. D., 150 Dzyaloshinski, I. E., 214, 303

E Eckart, C., 99 Eckstein, Y., 71 Edgerton, R., 200

415

AUTHOR INDEX

Efimova, B. A., 59 Ehrenreich, H., 15, 43, 274 Eiland, P. F., 14 Einspruch, N. G., 46, 49, 50, 55, 57 Elliott, R. J., 316, 317, 348, 359(19), 360, 361 Englman, R., 95, 127, 133(107), 150, 164, 165 Enz, U., 150,326 Erikson, R.. A., 307 Esaki, L., 73 Estle, T. L., 128, 131(123), 133(123), 134 (123), 168, 169 Ewald, P. P., 23

F Fagen, E., 83 Falicov, L. M., 73 Fano, U., 215, 290 Faulkner, E. A., 176 Feher, E., 204 Feher, G., 79, 80, 140, 142 Feinleib, J., 147 Felcher, G. P. 403 Fendley, J., 64, 65, 66 Feofilov, P. P., 126, 184 Ferguson, J., 204 Fermi, E., 233 Fernandes, J. C., 159, 184(231) Feynman, R. P., 107 Fick, E., 124 Figielski, T., 68 Finn, C. B. P., 124 Fitchen, D. B., 142, 198 Fletcher, J. R., 140 Foner, S., 201 Ford, R. A., 182 Forsyth, J. B., 398, 403 Fraenkel, G. K., 143 Franck, J., 112, 179(68) Fraser, D. G., 137, 140(138), 137(138) Friedel, J., 171, 177(250) Fritzsche, H., 79, 80(72) Fredkin, D. H., 216 Frisch, H. L., 107 Fuchs, K., 38 Fukuda, A., 199, 200 Fulton, T. A., 142, 198

G

Garofano, T., 185 Gaym, G., 224 Geballe, T. H., 84, 88(84) Gelerinter, E., 142, 143 Gelles, I. C., 176 Gere, E. A., 79, 142 Gerschwind, S., 128 Ghosh, A. K., 192 Glauber, R. J., 233 Goetz, A., 70, 73 Goff, J., 83, 84, 85(82), 86(83), 87 Goldberg, C., 78 Goldschmidt, V. M., 18 Golin, S., 73 Gomez, A., 93, 115(13) Goodenough, J.B., 120,146,147(181), 149, 150(181), 402 Goodman, G. L., 93, 115(7), 183(7) Gorkov, L. P., 214, 303 Gosar, P., 138 GottJicher, S., 20, 21, 23, 24, 27 Gourary, B. S., 195 Gray, H. B., 146 Grechushwikov, B. N., 184 Green, Jr., B. A., 73 Griffin, A., 84, 88 Griffith, J. S., 95, 99, 100, 110, 118, 196 Griffiths, D., 19 Griffiths, J. H. E., 126, 128, 166(120) Grimm, H. G., 2, 8(1), 16, 24(1), 27(1) Gunthard, H. H., 187 Guggenheim, H. J., 204 Gutowsky, H. S., 127 Guttmann, A. J., 12 Guyer, R. P., 281 Gyorgy, E. M., 137, 138(142), 139(142), 140, 175(142), 187(138)

H Haas, C., 149 Ham, F. S., 112, 114(67), 115(67), 134, 145, 155, 156, 157(67), 159, 161, 165, 166, 205, 210 Hall, J. J., 48, 49, 50(19a), 51, 52(19a), 64, 65, 66(44), 70,80,82(80a), 86(80a), 88 Halpern, O., 396 Harris, E. A., 173

416

AUTHOR INDEX

Hartman, A., 32 Hartman, R. L., 52 Hasegawa, H., 52, 86 Hastings, J. M., 362 Hattori, H., 13, 24 Hayes, W., 121, 147, 158, 187 Hearmon, R. F. S., 60, 83(38) Heeger, A. J., 379 Heine, V., 101, 108(44), 170 Heitler, W., 295 Heller, W. R., 135 Hellman, H., 107 Henry, C. H., 197, 211, 215, 240 Hensel, J. C., 52, 142 Hermann, C., 2, 5, 8(1), 15, 16, 18, 2401, 2 7 ~ ) Herpin, A., 330, 346 Herranz, J., 93, 115(13) Herring, C., 38, 42(3), 43(3), 78(3), 270, 275(69) Herzfeld, F., 122, 124(92), 125(92) Herzberg, G., 93, 101, 115(14), 119, 143, 183(14) Hesselmann, I. A. M., 143 Hill, 0. F., 182 Hilsch, R., 199 Hirahara, E., 402 Hirschfeld, F. L., 32 Hirschfelder, J. O., 107, 141 Hochli, U. T., 123, 124, 128, 131(123), 133(123), 134, 169 Holmes, 0. G., 187 Holton, W. C., 169 Hod, H., 23 Hopfield, J. J., 211, 215, 240, 294(12) Herie, C., 281 Horn, D., 127, 133(107) Hosemann, R., 14 Hosoya, S., 12 Hougen, J. T., 143 Houston, B. B., Jr., 57, 58 Houston, T. W., 379 Huang, K., 215, 217,218(15), 235(10), 290, 291 Huber, E. E., 402 Hughes, F., 194, 195 Huntington, H. B., 47 Hurst, R. P., 19 Hutson, A. R., 55

I Ingersoll, K. A., 204 Inohara, K., 199 Inoue, M., 195, 198,.199 Irani, K. S., 148

J Jackson, J. D., 291 Jaeobs, I. S., 378 Jacobsen, E. H., 220 Jaggi, R. L., 71 Jahn, H. A., 92, 106, 114 Jain, A. L., 71 James, R. W., 3, 13 Johansen, H., 121 Johnson, M. H., 396 Jones, G. D., 190, 191 Jones, H., 38, 70 Joos, G., 124 Jorgensen, C. K., 182

K Kadanoff, L. P., 214, 228, 230, 257, 262, 265(4), 282, 288(4) Kafalas, J. A., 59 Kaidanov, V. I., 59 Kamimura, H., 99, 155(34), 168, 193(242), 200, 201 Kanzig, W., 147 Kanamori, J., 377 Kane, E. O., 180 Kapitza, P. L., 70, 71, 72, 73 Kaplan, T. A., 307, 312, 317, 363, 371 (22h), 381(22e), 391(22e), 392(22d), 395(22f) Kaplyanskii, A. A., 125, 126, 209 Kasper, J. S., 366, 378 Katagawa, T., 13, 24(15) Kato, N., 13, 24(15) Kearsley, M. J., 171, 177(250) Keesom, P. H., 59, 60, 62 Keyes, R. W., 38, 40, 42(1), 46, 47(a), 49 (9), 51, 54, 56(10), 59(9), 60, 61, 63 (lo), 68(10), 69(10), 70, 71, 79(10), 80 (lo), 82(10), 84, 85(85), 88 Khalatnikov, I., 278 Kinoshita, K., 409 Kirtman, B., 107, 109(61) Kitano, V., 330, 331, 333, 334, 339(14), 341, 342(15), 354(14), 406(14), 409

417

AUTHOR INDEX

Kittel, C., 78, 79(68), 81(68), 238, 296, Leigh, R. S., 38 Leman, G., 171, 177(250) 362, 377 Lenhardt, C. A., 146 Klein, M. V., 140 Lidiard, A. B., 173, 176, 177(268) Kleinman, L., 24 Liebfried, G., 216 Klemens, P. G., 84, 86(89), 279 Liebling, G. R., 143 Klick, C. C., 93, 178, 195, 200 Liehr, A. D., 92, 93(5), 107, 115, 120(80), Kliewer, K. L., 217, 293(23) 121(80), 143, 144, 146, 153, 163, 167 Knox, K., 121, 149 (5), 169(5), 174, 175, 182(5), 183(5), Knox, R. S., 192, 195 186(80) Koehler, W. C., 150, 328, 399 Lightowlers, E. C., 176 Koenig, S., 80 Linares, R. C., 184(287), 185, 206(287) Kohn, W., 54, 78, 79(70), 81(70) Lipkin, H. J., 233 Kontorovia, J. A., 61, 62 Lipscomb, W. W., 141 Koonce, C. S., 151 Lipson, H., 20 Kornienko, L. S., 158 Koster, G. F., 100, 108, 113(63), 156(63) Long, T. R., 47 Longuet-Higgins, H. C., 92, 93, 107, 115 Kouvel, J. S., 366 (B), 111(4), 118, 122(8), 124(83), 143, Kramers, H. A., 106 154, 163(58), 189(83), 190 Krause, J. T., 137, 138(142), 139(142), 175 Lohr, L. L., Jr., 121, 141 (142) Lotgering, F. K., 380 Krishnan, R. S., 185 Loubser, J. H. N., 176 Kristofel, N. N., 146, 200 Loudon, 238 Kroger, F. A., 185 Low, W., 128, 157, 158, 166, 201 Krug, J., 18 Lowde, R. D., 19 Krunahansl, J. A., 281 Ludwig, G. W., 128, 157(117), 174(117), Krupicka, S., 148 176, 216, 219(16) Krupka, D. C., 142 Luther, L. C., 211 Kuriyama, M., 12 Luttinger, J., 78, 79(70), 81(70) Kuriyama, H., 13, 24(15) Kwok, P. C., 215, 224, 228, 244, 259(34), Lyddane, P. H., 217, 293(19) 262, 272, 273, 275(71), 278(71), 281, Lyons, D. H., 317, 362, 371(22b), 381 (22e), 391 (22e), 392(22d), 395(22f) 283

M Landau, L., 270, 278 Mabuchi, T., 199,200 Landau, L. D., 149 McCa11, D. W., 127 Lange, R. V., 361 McClure, D. S., 97, 149, 162(31), 182(31), Lannoo, M., 171, 177(250) 184, 187(31), 191(31), 201 Larson, A. C., 31, 32 McConnell, H. M., 127, 143, 155(165) Lampert, M. A., 78, 79(69), 81(69) McCumber, D. E., 155, 166(213) Lasher, G. L., 176 McDonald, R. S., 128, 172(116), 176(116) Lawler, M. G., 143 McDougall, J., 43 Lawson, A. W., 71 Macfarlane, R. M., 96, 97(29), 159, 201 Lax, B., 74 Lax, M., 112, 178(70), 179, 181(70), 189 (29) LeCraw, R. C., 137, 138(142), 140(138) McFee, J. H., 55 McLachlan, A. D., 111, 122(65), 143 139(142), 175(142), 187(138) McLure, J. S., 95, 97, 187(21) Leggett, A. J., 273 McMahon, D. S., 166 Leider, H. R., 129, 159(126)

418

AUTHOR INDEX

Markarov, E. A., 52 Makashima, S., 200 Maradudin, A. A., 105, 217, 236, 293(22) Margerie, J., 197, 209 Maris, H. J., 275 Marshall, F. G., 140 Martin, D. S., 146 Martin, P. C., 214, 224, 228, 230, 259(34), 273, 278(74), 281, 283 Maslen, E. N., 33, 34(54) Mason, R., 32, 35 Mason, W. P., 48, 55(18a), 56, 57, 59(25), 60, 136, 137 Mauza, E., 100 Mauroides, J. G., 74 Medvedev, V. N., 125, 126 Meiboom, S., 78 Mendelsohn, K., 88,89(95) Menne, T. J., 114 Menyuk, N., 362, 371 (22e), 379, 381 (22e), 391, 392(22d), 395(22f), 401 MBriel, P., 330, 346 Mermin, N. D., 225, 241, 242(57) Merritt, F. R., 128, 137, 138(142), 139 (142), 175(142), 204 Metropolis, N., 100 Meyer, A. J. P., 346 Meyers, M. D., 190 Miller, A., 362 Miller, P. B., 215, 228, 244, 262, 273, 278 (74) Misetich, A., 204 Mitchell, A. H., 78, 79(68), 81(68) Miwa, H., 312, 316, 317, 348 Moffitt, W., 115, 117, 120(79), 122(81), 124(81), 163(79), 164(79), 165(79), 166 Moiehes, B. Ya., 59 Mollenauer, L. F., 401 Moran, P. R., 194, 195, 197, 198(310) Morcillo, J., 93, 115(13) Morigaki, K., 169 Morin, F. J., 147 Moriya, K., 399 Moss, R. E., 143 Mott, N. F., 38 Miiller, K. A., 123, 124, 177 Muller, R., 187 Mulliken, R. S., 183

N Nagai, O., 307 Nagasawa, H., 379 Nagamiya, T., 323,330,331,333,334, 339 (14), 341, 342(15), 354(14), 399, 401, 406,409 Nagata, K., 330, 339(14), 354(14), 406(14) Nakamura, T., 307 Nakayama, M., 52 Nathans, R., 363, 398 Nava, R., 275 NCel, L., 377 Nesbet, R. K., 95 Nettel, S. J., 348 Neighbours, J. R., 47 Nelson, E. D., 158 Newman, R. C., 176 Nicholas, J. V., 126 Nielson, C. W., 100 Nishikubo, T., 323 North, J. C., 68, 69 Nosanow, L. H., 216 Novikova, S. I., 62 Nowick, A. S., 135 Nozieres, P. 214, 224

0 O’Brien, M. C. M., 112, 122, 123(93), 124 (93), 125(93), 126, 130(93), 131(93), 132, 166, 187, 189, 195, 198(308) O’Connell, A. M., 33, 34 O’Connor, J. R., 128, 187 Onaka, R., 199,200 apik, U., 115, 118, 119(78), 120(78), 122 (83), 124(83), 155, 163(78), 165, 189 (83), 190(83) Oppenheimer, R., 214 Oppenheimer, J. R., 105 Orbach, R., 124 Orgel, L. E., 98, 116(33), 149(33) Orton, J. W., 158, 166 Owen, J., 126, 158, 173

P Palma, M. U., 133, 149, 185 Palma-Vittorelli, M. B., 185 Pappalardo, R., 184, 185, 206(287), 207 Patterson, D. A., 195 Paul, W., 147

419

AUTHOR INDEX

Pauling, L., 81, 121 Pearlman, N., 83, 84, 85(82), 86(83), 87 Peckham, G., 185 Peierls, R. E., 270, 275(68), 278, 279 Pepinsky, R., 14 Permareddi, J., 146 Persico, F., 149, 166 Peter, M., 96 Peters, C., 2, 8(1), 15, 16, 18, 24(1), 27(1) Philipp, H. It., 15 Phillips, J. C., 24, 147 Pickart, S. J., 362, 403 Piksis, A. H., 114 Pikus, G. E., 56, 88(29) Pipkorn, D. N., 129, 159 Pirc, R., 138 Plieth, K., 149 Plumier, R., 380 Podzyarei, G. A., 176 Pohl, R. O., 140 Pomeranchuk, I., 84 Pomeranta, M., 54, 55 Pontinen, R. E., 80, 82 (80a), 86 Porto, S. P. S., 185 Price, P. J., 42, 51, 52, 52, 67(21e), 54, 74(11) 79, 80(75), 88 Prince, E., 362 Prokhorov, A. M., 158, 201 Pryce, M. H. L., 97, 115, 118, 119(78), 120 (78), 121, 122(83), 124(83), 126(89), 131(89), 132, 154, 155, 163(78), 165, 184(30), 185, 190(83) Przhevuskii, A. K., 209

R Rabin, H., 141, 194, 195 Rae, A. J. M., 33, 34(54) Raether, H., 15 Ramamurti, J., 147 Ramasubba Reddy, T., 126 Rampton, V. W., 140 Ray (R6i), D. K., 158 Remeika, J. P., 128, 137, 140(138), 187 (138) Reneker, D. H., 71 Renninger, M., 14, 23, 24 Rice, B., 107, 117(58), 163(58) Ridgley, D. H., 402 Robbins, M., 149

Robbrecht, G. G., 148 Robertson, G. B., 32 Rogers, D. B., 167 Romeijn, F. C., 193 Romestain, R., 197 Rosenberg, H. M., 84, 88(84) Rosenstock, H. B., 217 Rotenberg, M., 100 Rowell, J. M., 70, 71(54), 72(54) Rowell, P. M., 140 Rubins, R. S., 166 Ruch, E., 107 Rumer, G., 270 Runciman, W. A., 97, 184(30) S

Sachs, R. G., 217, 293 Sack, H., 118, 122(83), 124(83j, 189(83), 190(83) Sagar, A., 59 Sanders, T. M., Jr., 53, 80, 82(80a), 86 Schawlow, A. L., 114, 130, 158, 159, 180 Scheie, P. O., 59 Schiff, L. I., 130 Schirmer, 0. F., 177 Schoenberg, D., 73 Schonhofer, A., 107 Schoknecht, G., 14 Schnatterly, S. E., 197 Schneider, J., 177 Schnettler, F. J., 137, 140(138), 187(138) Schrieffer, J. R., 214, 224 Schulman, J. H., 93, 178, 195 Schwinger, J., 214 Scott, W. C., 161, 162(233), 184(233), 193 (233), 201(233), 202 Segall, B., 15 Seiden, P. E., 54 Seitz, F., 77, 93, 178(16), 195, 199 Sham, L. J., 239 Shapira, Y., 74 Shepherd, I. W., 140 Shibuya, M., 78 Shirane, G., 379, 400 Shockley, W., 38, 57, 170 Shore, H. B., 140 Shul’man, L. A., 176 Shuskus, A. J., 157

420

AUTHOR INDEX

Silsbee, R. H., 142, 143 Silverstone, H. M., 143 Simons, S., 275 Sinha, A. P. B., 148, 150 Sinha, K. P., 121, 150 Slack, G. A., 140, 205 Sladek, R. J., 84 Slater, J. C., 60 Slichter, C. P., 127, 197 Sloncewski, J. C., 112, 122(72), 123, 189 Smaller, B., 147 Smith, C. S., 40, 42, 47 Smith, G. E., 70, 71(54), 72 Smith, G. P., 150 Smith, H. M. J., 238 Smith, W. V., 176 Snyder, L. C., 143 Sorokin, P. P., 176 Spector, H. N., 55 Sponer, H., 182 Srinavasan, R., 126 Stanley, H. E., 362, 371(22f), 395(22ej Statz, H., 108, 113(63), 156(63) Stern, F., 236 Stevens, K. W. H., 95, 99(20), 140, 162 (20), 166 Stiles, P. J., 73 Stoneham, A. M., 176, 177(268) Stoner, E. C., 43, 74 Struck, C. W., 122, 124(92), 125(92) Strakna, R. E., 57, 58 Strauss, A. J., 59 Strauss, H. L., 93, 115(11) Sturge, M. D., 125, 128, 137, 138, 139 (142), 140(140), 159, 161, 162(233), 168, 175(142),184(233),185(286), 193 (233), 201(233), 202, 203, 206, 208, 209 Sugano, S., 96, 99(26), 114, 200, 201 Suhl, H., 348 Suss, J. T., 128 Sussman, J. A., 138, 140, 177(151), 280 Suzuki, T., 12 Synacek, V., 148

T Taglang, P., 346 Takei, W. S., 400

Tanabe, Y., 96, 99(26), 100, 121, 155(34) Tanaka, M., 129, 148(125) Teegarten, K., 147, 201 (332) Teller, E., 92, 106, 182, 183, 217, 293 ter Ham, D., 273 Thellung, A., 280 Thomas, H., 330 Thorson, W., 93, 115, 117, 120(79), 124 (81), 163(79), 164(79), 165(79), 166 Thurston, R. W., 64 Thyagarajan, G., 93, ll5(12) Tinkham, M., 126 Tisza, 280 Todd, P. F., 143 Togawa, S., 12, 18 Tomatsubara, T., 409 Townes, C. H., 130 Townsend, M. G., 143 Toyozawa, Y., 195, 198, 199 Trammell, G. T., 396 Trenam, R. S., 128 Tsushima, 379 Tucker, E. B., 140 Tucker, R. F., 133 Tursanov, A., 56 Twiddell, J. W., 147 Tyablikov, S. L., 214, 230

V Vand, V., 14 van den Boomgaard, J., 185 van der Waals, J. H., 143 Van der Ziel, J., 206 Van Doorn, C. Z., 141 van Eekelen, H. A. M., 166 van Hove, L., 221, 226 van Reijen, L. L., 8 van Uitert, L. G., 137, 140(138), 187(138) Van Vleck, J. H., 95, 98(25), 102, 111, 114 (47), 155, 159(66) Vekhter, B. G., 132, 155, 165, 167, 190, 210(236) Verschoor, G. C., 29, 30, 32 Vink, H. J., 185 Villain, J., 307, 330 Voigt, W., 40 Voigtlander Tetzner, G., 14 von der Lage, C., 25 von Hippel, A., 93, 178(15)

421

AUTHOR INDEX

W Wagenfeld, I%.,12 Wagner, M., 140, 185, 186 Wallis, R. F., 236 Ward, I. M., 126 Ward, J. C., 280 Walters, G. K., 169 Washimiya, S., 155 Watabe, A., 348 Watkins, G. D., 127, 128, 172(249), 173 (249), 175(114), 176(113), 204 Weakliem, H. A,, 206, 207(343) Wedding, B., 140 Weger, M., 157 Weiss, G. H., 217, 293(23) Weiss, R. J., 24, 25 Weinreich, G., 53, 79, 80 Weinstock, B., 93, 115(7), 183(7) Weissman, S. I., 143 Wertheim, G. K., 129 Werthemer, N. R., 216 Werta, J. E., 128, 166(120) Wheeler, It. G., 108, 113(63), 156(63) White, D. L., 55 White, H. G., 53, 79, 80 Whippey, P. W., 176 Wight, D. R., 176 Wigner, E. P., 99, 100, 106 Wilkens, J., 121 Wilkinson, C., 398 Wilkinson, M. K., 328, 399 Wilks, J., 280 Williams, G. A., 70, 71(54), 72(54) Wilson, D. K., 79, 80 Wilson, E. B., 101 Windsor, C., 173 Witte, H., 8, 14(8), 15, 17, 18

Wolfel, E., 8, 14(8), 15, 17, 18, 20, 21, 23, 24, 27 Wojtowicz, 148 Wold, A., 167, 379 Wolf, A., 70, 73 Wolf, E. L., 328, Wolf, P., 330 Wolf, W. P., 124 Woll, E. J., 54 Wollan, E. O., 150,328, 399 Wong, J. Y., 158, 159 Wood, D. E., 143 Wood, D. L., 184(287), 185, 204, 206(287) Woodbury, H. H., 128, 157(117), 174(117) Woodruff, T. O., 274 Wooten, J. K., Jr., 100 Worlock, J. M., 211 Wyckoff, 146, 150(182) Wysling, P., 124

Y Yafet, 362, 377 Yamaguchi, T., 171, 173(252) Yariv, A,, 168, 193(242) Yen, W. M., 208 Yoshimori, A., 307, 312(1), 316(1), 330 (l), 332, 409 Yosida, K., 312, 316, 317, 348 Yuster, P. H., 147, 187, 192, 199 Z

Zaritskii, I. M., 176 Zeks, J., 138 Zhdanova, V. V., 61, 62 Ziman, J. M., 84, 87, 88(94), 89(94), 218, 239, 271 Zverev, G. M., 201

Subject Index

A Acoustic waves Jahn-Teller effects, 134ff. semiconductors, electronic effects, 53ff. Aluminum, electron distribution, 15 Aluminum oxide, acoustic attenuation, nickel ion effect, 137ff. Anharmonic effects, 247, 258, 263 Anisotropy energy fan structure, 354 helical spin effect, 360 helical spin system, magnetization effect, 340ff. spin configuration effect, 316ff. Antiferromagnetic, triangular, spin ordering, 364 Atomic scattering factors, ions and metals, l0ff

Color centers, R, 141 Configuration-coordinate model, optical spectra, 178ff Conical structure manganese chromite, 378 spin system, 343ff spin waves, 349ff Copper chromite, spin structure, 378 Correlation functions, 214 Crystal field, d orbital splitting, 96 Crystal field theory, assumptions, 95 Cupric ion Jahn-Teller effect, 128 optical spectra, 186 spin resonance, 133 Curie law, 395 Cyanuric acid, electron distribution, 296 Cyclopropane, electron distribution, 32ff

B Band structure, semiconductors, strain effects, 38, 67ff Benzene, electron distribution, 34 Bismuth magnetic susceptibility, 73 magnetostriction, 70ff Boltzmann equation phonon, 279-290 derivation, 281-290 second sound, 280 Born-Oppenheimer approximation, 107, 214 Bose-operators 314

D

Debye temperature elastic constants, 59ff semiconductors, electronic effects, 59ff Debye-Waller factor, 234 Deformation potential model, 38 Diamond electron distribution, 20ff structure factor, 27. Diamond structure, defects, Jab-Teller distortion, 170ff Dichroism cesium halides, 197 C optical spectra, 188 Cesium halides, dichroism, 197 Dielectric susceptibility, 236ff, 301ff Chromites, crystal structure, 377 retardation effects, 237 Chromium Donors electron distribution, 16 elastic constants effects, 78ff spin waves, 398 thermal conductivity effects, 83ff Chromium oxide, spin structure, 4 0 f f d orbital splitting, crystal field, 96 Clebsch-Gordon coefficients, 100 Dysprosium Cobaltous chromite, spin structure, 379 magnetic transition, 326 Collective coordinates, complex ions, 100ff spin structure, 347 422

423

SUBJECT INDEX

E Elastic constants Debye temperature, 59ff donor effects, 78ff Jahn-Teller effect, 136 phenomenological description, 39ff third order, electronic effects, 63ff Elastic properties free electron effects, 40ff germanium, electronic effects, 43ff semiconductors, doping effects, 47ff electronic effects, 37-90 multivalley, 43ff silicon electronic effects, 43ff Elastic waves, electronic effect.s, 53ff Electron distribution diamond, 20ff Fourier synthesis, 19ff metals, 15ff organic compounds, 29ff sodium chloride, 6ff x-rays, 1-35 Electron irradiation, semiconductors, strain effects, 68ff Electron-lattice interaction, evaluation, 113

Erbium, spin structure, 328ff Exchange energy, fan structure, 354 Excitations in crystals damping, 257-260 dispersion, 257-260

F Fan structure helical state transition, 339 spin waves, 354ff spins, 333ff conditions for, 344ff susceptibility, 341, 402ff Faraday rotation, CsBr, 198 F-Bands, CsX, spin-orbit coupling, 194ff F Centers CsF, optical spectra, 194 CsX, optical spectra, 191 Ferric chloride, spin structure, 399 Ferrimagnetic cone, spin structure, 368 Ferrous chromite, spin structure, 378ff Ferric oxide, spin structure, 400ff Ferrimagnetism, NBel theory, 377

Ferrites, crystal structure, 377 Ferroelectricity, perovskites, 150 Ferrous ions, optical spectra, 190ff Fourier coefficients, structure factor, 3 Fourier series, electron distribution representation, 2ff Fourier synthesis, electron distribution, 19ff Franck-Condon principle, 179 Free electron effects, elastic properties, 40ff Functional derivative technique, 243-245 G

Germanium Debye temperature, donor effect, 60 elastic constants donor effects, 78ff third order, electronic effects, 64ff elastic properties doping effects, 47ff electronic effects, 43ff electron distribution, 24 electron irradiation, strain-effects, 68ff Jahn-Teller distortion, impurities, 174ff magnetoresistance, 77 magnetostriction, 74ff thermal conductivity acceptor effects, 88 donor effects, 83ff thermal expansion, electronic effect, 61ff Green’s function method, 214, 233 anharmonic effects, 214 lattice dynamics, 213-303

H Ham effect optical spectra, 193, 201ff spin resonance, 156ff Hartree-Fock scattering factor, 25 Helical spin ordering, see also Spin ordering examples of, 398ff ferromagnetic state transition, 326 magnetic field effect, 330fT neutron diffraction, 396ff rutile structure, 307 symmetry effects, 323ff theory, 305-403 Helical spin structure, susceptibility, 409ff

424

SUBJECT INDEX

Helical spin system anisotropy energy, magnetization effect, 340ff fan structure transition, 335, 339 magnetization curve, 334, 342 Hellmann-Feynman theorem, 107ff Holmium, spin structure, 328ff, 347 Hund’s rule, ground state, 171

I Imaginary time, 241 Inelastic neutron scattering, 233-235 Infrared absorption spectrum, 236 Ionization degree, determination of, 10ff Iron, electron distribution, 16

J Jahn-Teller distortion crystal structure effect, 149 octahedral complexes, 204 silicon defects, 172 spinel structure, 377 tetrahedral complexes, 167ff trigonal, 176 Jahn-Teller effect acoustic consequences, 134ff concentrated systems, 146ff conditions for, 93 definition, 92 diamond structure, defects, 170ff dynamic and static, 115, 120 elastic constants, 136 heavy metal ions, 199 motional narrowing, 126ff observation of, 113ff octahedral complex ions, 115ff, 152 optical spectra, 178ff perovskites, 150 solids, 91-211 spin-orbit coupling, 155 spinel structure, 148, 169 static, 119ff, 160 tetragonal complexes, 143ff transition metal ions, 201ff triply degenerate states, 151ff Jahn-Teller energy, 113ff, 117 Jahn-Teller theorem, 106ff

K Kramers degeneracy, 106 Kubic Harmonics expansion, 25

Lattice dynamics, Green’s function method, 213-303 Lattice transport properties, 279 Lattice vibrations eigenvectors, 218 Hamiltonian, 214 harmonic approximation, 218 inelastic neutron scattering, 220 normal coordinates, 218 normal modes, 216ff optical measurements, 220, 235ff ultrasonic attenuation experiments, 220 Lead ions, NaCl, optical spectra, 199 Ligand field, see Crystal field Ligands, definition, 95 Lithium fluoride, ionization degree, 18 Lithium hydride, ionization degree, 19 Lyons-Kaplan theory, spin structure, 371ff 382

M Magnesium, electron distribution, 16 Magnesium oxide, ionization degree, 18 Magnetic susceptibility, bismuth, 73 Magnetic transition, helical-ferromagnetic, 326ff Magnetoelastic effect, semiconductors, 76 Magnetoresistance, semiconductors, 77 Magnetostriction bismuth, 70ff electronic, 70 semiconductors, theory, 74ff Manganese chromite, spin structure, 362, 378 Manganese-gold phase, spin structure, 346ff Manganese-intermetallic phases, spin structure, 366 Manganese iodide, spin structure, 399 Manganese oxide helical spin ordering, 307 ionization degree, 12 spin structure, 378 Manganese phosphide, spin structure, 402 Metals electron distribution, 15ff scattering factors, 13ff Metamagnetic behavior, 346, 402 Molecular field theory, spin ordering, 310ff

425

SUBJECT INDEX

Mossbauer effect, quadrupolar splitting, 129 Motional narrowing, Jahn-Teller effect, 126ff Multiple cone, spin structure, 368, 390ff Multivalley semiconductors, elastic properties, 43ff

N NCel temperature, 311 calculation, 317ff complex structures, 392ff definition, 394 N&l theory ferrimagnetism, 377 spin structure, 384ff, 390 Neutron diffraction, spin structure determination, 395ff Neutron-phonon interaction, 233, 234 Nickel ions A1203 acoustic attenuation effects, 137ff spin resonance, 157 NMR, spin structure measurement, 379 Nuclear displacements effective Hamiltonian, 215 expansion, 215 0 Octahedral complexes collective coordinates, l O l f f Jahn-Teller distortion, 116, 162ff, 204 Jahn-Teller effect, 115ff, 152 Optical phonons, dispersion, 293-294 Optical spectra broadband transitions, 188ff dichroism, 188 doubly degenerate states, 186 Ham effect, 201ff Jahn-Teller effect, 178ff transition metal ions, 201ff triply degenerate states, 193ff Organic compounds, electron distribution, 29ff P Paraelectric resonance, 177 Pendellosung fringes, scattering factor measurement, 13 Perovskites, Jahn-Teller effect, 150 Phase problem, structure determination, 4

Phonons, 214 Phonon correlation functions, 220, 239ff definition, 221 influence of anharmonic interactions, 223ff linear response, 228 physical properties, 222 relation t o experiments, 227-231 to transition probabilit.y, 228 spectral representation, 224ff Phonon density of states, 226 Phonon Green’s function, 242, 247, 299 calculation, 248 definition, 240 equation, 243, 245ff inverse, 250 perturbation calculation, 26&262 poles, 258-259 Phonon-photon Green’s function, 299 Phonon-photon system, interaction, 290293 Phonon response function, 230ff, 242 Fourier transform, 231 relation t o correlation function, 231 Phonon self-energy function, 253-257, 303 analytic continuation, 258 explicit calculation, 262-267 perturbation calculation, 262 Phonon spectral function, 259 Photon Green’s function, 299 Photon-phonon interaction, 236-238 Photostriction, 68 Piezoresistance, electronic contributions, 42 Polariton(s), 235, 240 Polariton Green’s function, 294300 definition, 298 Polariton Hamiltonian, 299

Q Quantum-mechanical tunneling, JahnTeller effect, 129 Quasi-molecular model, 178ff complex ions, lOOff validity, 112

R Raman process, spin-lattice relaxation, 138 Raman scattering, 238, 239

426

SUBJECT INDEX

Rare earth metals spin ordering, 323 spin structure, 328ff, 347 R Center, Jahn-Teller effect, 141 Rutile structure, helical spin ordering, 307 S

Scandium ion, spin resonance, 133 Scattering factors asphericity effect, 21 ions, 10ff metals, 13ff Screw structure, spins, 309 Second sound, 280 Semiconductors acoustic waves, electronic effects, 53ff band structure, strain effects, 38 Debye temperature, electronic effects, 59ff elastic properties, 37-90 doping effects, 47ff electron irradiation, strain effects, 68ff magnetoelastic effect, 76 magnetoresistance, 77 magnetostriction, 7 M multivalley, elastic properties, 43ff thermal conductivity, donor effects, 83ff thermal expansion, electronic effect, 61ff Series termination effects, electron distribution studies, 5ff Shear stability, semiconductor crystals, 52 Silicon defects, 172ff elastic constants donor effects, 78ff doping effect, 57 third order, electronic effects, 64ff elastic properties doping effect, 49ff electronic effects, 43ff electron distribution, 24 Jahn-Teller distortion, impurities, 17M magnetostriction, 74ff Sodium azide, spin resonance, 142 Sodium chloride electron distribution, 6ff ionization degree, 17ff structure, 6

Solids, Jahn-Teller effect, 92-21 1 Spectral function, 224-225 sum rule, 225-226 Spectral representation, 242, 257 Spin-lattice relaxation direct process, 138 Raman process, 138 Spin-orbit coupling F-bands, CsX, 194ff Jahn-Teller effect, 155 transition ion spectra, 97ff Spin-orbit splitting tetrahedral complexes, 206ff Spin ordering, see also helical spin ordering anisotropy energy effect, 316ff antiferromagnetic triangular, 364 complex structures, 392ff crystal structure effect, 377ff helical theory, 305-403 molecular field theory, 310ff NBel theory, 38M neutron diffraction measurement, 395ff rare earth metals, 323 spinels, 376ff, 392 Yafet-Kittel theory, 385ff Spin resonance Ham effect, 156ff Jah-Teller distorted state, 131 Jahn-Teller effect, 114 R center, 142 silicon defects, 172ff Spin structure Lyons-Kaplan theory, 371ff, 382 magnetic field effects, 343ff multiple cone, 368, 390ff N6el theory, 390 rare earth metals, 347 Spin waves chromium, 398 conical structure, 349ff fan structure, 354ff ferromagnetic, frequencies, 325, 335 helical structure, field effect, 360ff magnetic field effects, 348ff nature, 307 screw structure, 312ff Spinel structure Jahn-Teller effect, 148, 169 triangular spin structure, 362

427

SUBJECT INDEX

Spinels crystal structure, 377 spin ordering, 376ff, 392 Spins fan structure, 333ff screw structure, 309 thermal average length, 316 Stokes shift, 180 Structure factor, diffraction, 2ff Superhyperfine structure, spin resonance, 126 Susceptibility fan structure, 403ff helical spin structure, 408ff

Trinitroaminobenzene, electron distribution, 31ff Tunneling, acoustic attenuation effect, 138ff Tunneling splitting, 131ff

U Ultrasonic absorption, octahedral, 135 Ultrasonic attenuation, 231-233, 263 general formulation, 267-269 hydrodynamic limit, 274 Landau-Rumer limit, 2 i 9 Landau-Rumer process, 270 longitudinal acoustic phonons, 274-279 transverse acoustic phonons, 26S274

T

V

Tanabe-Sugano Hamiltonian, 99 Terbium, magnetic transition, 326 Tetrahedral complexes collective coordinates, l O l f f Jahn-Teller distortion, 167ff Jahn-Teller splittings, 140 spin-orbit splitting, 206ff Thallous ions, KC1, optical spectra, 200 Thermal conductivity semiconductors donor effects, 83ff isotope effects, 87 strain effects, 86 Thermal expansion, semiconductors, electronic effect, 61ff Tin telluride, elastic constants, doping effect, 57ff Transition metal ions configurations, 94 optical spectra, 184, 201ff

Vacancies, diamond structure, 170ff Vibronic energy levels, 122ff Vibronic transitions, selection rules, 182 “Vibronic” wave function, 105

W Wigner-Eckart theorem, 99 WKB method, tunneling rate calculation, 130

X X-rays coherent scattering, 1 electron distribution in crystals, 1-35 X-ray diffraction, intensity, 2

Y Yafet-Kittel theory, spin structure, 385ff Z

ZnCnSer, spin structures, 380

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