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. 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.

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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