Solid State Physics: Advances in Research and Applications, Vol. 17

SOLID STATE PHYSICS VOLUME 17

Contributors to this Volume H. G. Drickamer Norman G. Einspruch Natsuki Hashitsume

Ryogo Kubo

w. Low Satoru J. Miyake

E. L. Offenbacher

SOLID STATE PHYSICS Advances in Research and Applications Editors

FREDERICK SEITZ

DAVID TURNBULL

Department of Physics University of Illinois Urbana, Illinois

Division of Engineering and Applied Sciences Harvard University Cambridge, Massachusetts

VOLUiME 17

1965

ACADEMIC PRESS NEW YORK AND LONDON

COPYRIGHT @ 1965,

BY

ACADEMICPRESSINC.

ALL RIGHTS RESERVED

NO PART OF THIS BOOK MAY B E REPRODUCED IN ANY FORM BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. 111 FIFTHAVENUE

NEWYORK,N . Y. 10003

United Kingdom Edition

Published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London, W.1

Library of Congress Catalog Card Number 55-12200

PRINTED IN THE UNITED STATES OF AMERICA

Contributors to Volume 17 H. G. DRICKAMER, Department of Chemistry and Chemical Engineering and Materials Research Laboratory, University of Illinois, Urbana, Illinois NORMANG. EINSPRUCH, Physics Research Laboratory, Texas Instruments Incorporated, Dallas, Texas NATSUKIHASHITSUME, Department of Physics, Ochanomizu University, Tokyo, J a p a n

RYOGO KUBO,Department of Physics, The University of Tokyo, Tokyo, J a p a n W. Low,* Department of Physics and National Magnetic Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts SATORU J. R ~ I I Y A K E , Department ~ of Physzcs, The University of Tokyo, Tokyo, J a p a n E. L. OFFENBACHER,Department of Physics, Temple University, Philadelphia, Pennsylvania

* Permanent address: The Hebrew University,

Jerusalem, Israel.

t Present address: Department of Physirs, Tokyo Institute of Technology, Tokyo, Japan.

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Preface

The first article in this volume, by Drickamer, surveys the effect of high pressure upon the electronic structure of solids. This is the fourth article in this series on high pressiire physics, the preceding articles being by Rice, McQueen, and Walsh (Volume 6), Swenson (Volume ll), and Bundy and Strong (Volume 13). In the second article, which is by Low and Offenbacher, the results of electron spin resonance experiments on magnetic ions, in various oxide crystals are reviewed. Other aspects of ESR spectroscopy have been treated in earlier articles in this series by Low (Supplement a),Ludwig and Woodbury (Volume 13), and Jarrett (Volume 14). In the third article, Einspruch describes the application of ultrasonic techniques to studies of the properties of semiconductors. In the final article of this volume, Kubo, Miyake, and Hashitsume treat the theory of the galvanomagnetic effect at high magnetic fields. Earlier articles by Jan (Volume 5) and by Kahn and Frederikse (Volume 9) treated closely related topics.

May, 1965

FREDERICK SEITZ DAVIDTURNBULL

vii

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Contents

CONTRIBUTORSTO VOLUME 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PREFACE ................................................................. OF PREVIOUS VOLUMES.. ........................................ CONTENTS SUPPLEMENTARY MONOGRAPHS. ............................................. ARTICLESPLANNED FOR FUTURE VOLUMES.. ................................

v vii xi

xv xvi

The Effects of High Pressure on the Electronic Structure of Solids

H. G. DRICKAMER I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Nonmetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 89

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

W. Low AND E. L. OFFENBACHER I. Introduction., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... 11. Outline of ESR Spectra in Inorganic Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Single-Ion Contribution to Anisotropy Energy. . . . . . . . . . . . . . . . . . . . . . . . . . IV. The Spectra of Transition Elements in Simple Oxides (MgO, CaO, SrO, ZnO, andA1203). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Rutile: TiO,. . . . . . . . . . . . . . . . . . . . . ........................ VI. Perovskites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Spinels.. . . . . . ................................................... VIII. Garnets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

136 138 147 153 154 162 180 185 193

Ultrasonic Effects in Semiconductors

NORMAN G. EINSPRUCH

I. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Summary of Classical Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Measurements Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C:eneralReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

217 218 226 230 268

X

CONTENTS

Quantum Theory of Galvanomagnetic Effect a t Extremely Strong Magnetic Fields

RYOGO KUBO, SATORU J. MIYAKE,AND NATSUKIHASHITSUME I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Inelastic Collision with Phon ....................... IV. Collision Broadening. . . . . . . . ....................... V. Non-Born Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Apendix A. Collision Broadening Effect upon Oscillatory Behavior.. . . . . . . . Appendix B. Approximate Form of the Green Function in a Magnetic Field. .

270 274 336 356 362

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

365

SUBJECTINDEX ...........................................................

373

Contents of Previous Volumes Order-Disorder Phenomena in Metals

Volume 1

LESTERGUTTMAN

Methods of the One-Electron Theory of Solids

Phase Changes

JOHN R. REITZ

DAVIDTURNBULL

Qualitative Analysis of the Cohesion in Metals

Relations between the Concentrations of Imperfections in Crystalline Solids

EUGENEP. WIGNERA N D FREDERICK SEITZ

F. A. KROGERAND H. J. VINK

The Quantum Defect Method

C. KITTELA N D J. K. GALT

Ferromagnetic Domain Theory

FRANK S. HAM The Theory of Order-Disorder Transitions in Alloys

TOSHINOSUKE MUTOAND YUTAKA TAKACI

Volume 4 Ferroelectrics and Antiferroelectrics

WERNERKANZG Theory of Mobility of Electrons in Solids

Valence Semiconductors, Germanium, and Silicon

FRANK J. BLATT

H. Y. FAN

The Orthogonalized Plane-Wave Method

Electron Interaction in Metals

TRUMAN 0. WOODRUFF

DAVIDPINES

Bibliography of Atomic Wave Functions

ROBERTS. KNOX

Volume 2

Techniques of Zone Melting and Crystal Growing

Nuclear Magnetic Resonance

G. E. PAKE Electron Paramagnetism and Magnetic Resonance in Metals

W. C. PPANN

Nuclear

Volume 5

W. D. KNIGHT Applications of Neutron Solid State Problems

Galvanomagnetic and Effects in Metals

Diffraction to

J.-P. JAN

C. G. SHULLAND E. 0. WOLLAN The Theory of Specific Heats and Lattice Vibrations

JULES DE LAUNAY

Luminescence in Solids

CLIFFORDC. KLICKA N D JAMES H. SCHULMAN Space Groups and Their Representations

Displacement of Atoms during Irradiation

G. K. KOSTER

FREDERICK SEITZAND J. S. KOEHLER

Shallow Impurity States in Silicon and Germanium

Volume 3 Group Ill-Group

Thermomagnetic

W. KOHN

V Compounds

H. WELKERAND H. WEISS The Continuum Theory of Lattice Defects

Quadrupole Effects in Nuclear Magnetic Resonance Studies in Solids

J. D. ESHELBY

M. H. COHENA N D F. REIF xi

xii

CONTENTS O F PREVIOUS VOLUMES

Volume 6 Compression of Solids by Strong Shock Waves

Photoconductivity in Germanium

R. NEWMANAND W. W. TYLER Interaction of Thermal Neutrons with Solids

M. H. RICE, R. G. MCQUEEN,AND J. M. WALSH

L. S. KOTHARIAND K. S. SINGWI

Changes of State of Simple Solid and liquid Metals

G. HEILAND, E. MOLLWO,A N D F. STOCKMANN

G. BORELIUS

Electronic Processes in Zinc Oxide

Electroluminescence

The Structure and Properties of Boundaries

W. W. PIPER AND F. E. WILLIAMS

S. AMELINCBX A N D W. DEKEYSER

Macroscopic Symmetry and Properties of Crystals

CHARLES S. SMITH Secondary Electron Emission

A. J. DEKKER Optical Properties of Metals

M. PARKER GIVENS Theory of the Optical Properties of Imperfectionsin Nonmetals

D. L. DEXTER Volume 7 Thermal Conductivity and Lattice Vibrational Modes

P. G. KLEMENS Electron Energy Bands in Solids

JOSEPH CALLAWAY The Elastic Constants of Crystals

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

H. W. LEWIS Study of Surfaces by Using New Tools

Grain

Volume 9 The

Electronic Spectra Molecular Crystals

of

Aromatic

H. C. WOLF Polar Semiconductors

W. W. SCANLON Static Electrification of Solids

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

ERNST HEERAND THEODORE B. NOVEY Oscillatory Behavior of Magnetic Susceptibility and Electronic Conductivity

A. H. KAHNA N D H. P. R. FREDERIKSE Heterogeneitiesin Solid Solutions

ANDREGUINIER Electronic Spectra of Molecules and Ions in Crystals Part 11. Spectra of Ions in Crystals

DONALD S. MCCLURE Volume 10

J. A. BECKER Positron Annihilation in Solids and Liquids The Structures of Crystals

A. F. WELLS

PHILIPR. WALLACE Diffusion in Metals

Volume 8 Electronic Spectra of Molecules and Ions in Crystals Part 1. Molecular Crystals

DONALD S. MCCLURE

DAVIDLAZARUS Wave Functions for Electron-Excess Color Centers in Alkali Halide Crystals

BARRYS. GOURARY AND FRANK J. ADRIAN

CONTENTS O F PREVIOUS VOLUMES

The Continuum Dislocations

Theory

of

Stationary

...

Xlll

Dislocations in lithium Fluoride Crystals

J. J. GILMANAND W. G. JOHNSTON

ROLAND DE WIT Electron Spin Resonance in Semiconductors Theoretical Aspects of Superconductivity

M. R. SCHAFROTH

G. W. LUDWIGAND H. H. WOODBURY Formalisms of Band Theory

E. I. BLOUNT

Volume 11 Semiconducting Properties of Gray Tin

G. A. BUSCHAND R. KERN

Chemical Bonding Inferred from Visible and Ultraviolet Absorption Spectra

Physics at High Pressure

CAR. KLIXBULLJORGENSEN

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

ROBERTW. KEYES

Volume 14 g Factors and Spin-lattice Relaxation of

Conduction Electrons

Imperfection Ionization Energies in CdSType Materials by Photoelectronic Techniques

RICHARDH. BUBE

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

PHILIPW. ANDERSON BENJAMIN LAXAND JOHN G. MAVROIDES Cyclotron Resonance

Electron Spin Resonance Spectroscopy in Molecular Solids

Volume 12

H. S. JARRETT

Group Theory and Crystal Field Theory

CHARLESM. HERZFELD AND PAULH. E. MEIJER Electrical Conductivity Semiconductors

of

Molecular Motion in Solid State Polymers

N. SAITO,K. OKANO, S. IWAYANAGI AND T. HIDESHIMA Organic

HIROOINOHUCHI AND HIDEOAEAMATU Hydrothermal Crystal Growth

R. A. LAUDISEAND J. W. NIELSEN The Thermal Conductivity of Metals at low Temperatures

Volume 15 The Changes in Energy Content, Volume, and Resistivity with Temperature in Simple Solids and liquids

G . BORELIUS

The Dynamical Theory of X-Ray Diffraction K. MENDELSSOHN A N D H. M. ROSENBERG R. W. JAMES Theory of Anharmonic Effects in Crystals

G. LEIBFRIEDAND W. LUDWIG

The Electron-Phonon Interaction

L. J. SHAMA N D J. M. ZIMAN

Volume 13 Vibration Spectra of Solids

Elementary Theory of the Optical Properties of Solids

SHASHANKA S. MITRA

FRANK STERN

Behavior of Metals at High Temperatures and Pressures

Spin Temperature and Nuclear Relaxation in Solids

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

L. C. HEBEL,JR.

xiv

CONTENTS OF PREVIOUS VOLUMES

Volume 16 Cohesion of Ionic Solids in the Born Model MARIOP. TOSI F-Aggregate Centers in Alkali Halide Crystals W. DALECOMPTON AND HERBERT RABIN

Point-Charge Calculations of Energy Levels of Magnetic Ions in Crystalline Electric Fields M. T. HUTCRINGS Physical Properties and Interrelationships of Metallic and Semimetallic Elements KARLA. GSCHNEIDNER, JR.

Supplementary Monographs

Supplement 1: T. P. DASAND E. L. HAHN Nuclear Quadrupole Resonance Spectroscopy, 1958 Supplement 2 : WILLIAMLow Paramagnetic Resonance in Solids, 1960

E. W. MONTROLL, AND G. H. WEISS Supplement 3: A. A. MARADUDIN, Theory of Lattice Dynamics in the Harmonic Approximation, 1063 Supplement 4: ALBERTC. BEER Galvanomagnetic Effects in Semiconductors, 19G3 Supplement 5 : R. S. &ox Theory of Excitons, 19G3 Supplement 6: S. AMELINCKX The Direct Observation of Dislocations, 1961

In Preparation JORDAN J. MARKHAM F Centers in Alkali Halides J. W. CORBETT Electron Radiation Damage in Semiconductors and Metals

xv

Articles Planned for Future Volumes

F. G. ALLEN-EVAN 0. KANEG. W. GOBELI

Photoelectric Emission Semiconductors

WILLIAMA. BARKER

Production and Detection of rjuclear Orientation

B. N. BROCKHOUSE

Determination of the Normal Modes of Lattices by Neutron Spectroscopy

ELIASBURSTEIN-G. PICUS

Infrared Spectra Arising from Foreign Atoms in Semiconductors

ESTHER CONWELL

Transport Properties of Germanium in High Electric Fields

ELBAUM CHARLES

Interactions between Defects in Crystals

MAURICEGLICKMAN

Plasmas in Solids

ROLFEGLOVER

The Properties of Thin Films

A. V. GRANATO-KURTLUCKE

Internal Friction Phenomena due to Dislocations

H. D. HAGSTRUM

Interaction of Surfaces and Ions xvi

from

ARTICLES PLANNED FOR F U T U R E VOLUMES

xvii

VAINOHOVI

Thermodynamic and Physical Properties of Ionic Solid Solutions

MARSHALL I. NATHAN

Semiconductor Lasers

D. K. NICHOLS-V. iz. J. VAN LINT

Energy Loss and Range of Energetic Neutral Atoms in Solids

ALLENNUSSBAUM

Crystal Symmetry, Group Theory, and Band Spectra Calculations

J. C. PHILLIPS

Fundamental Optical Spectra of Solids

HARRY SUHL

Magnetic Resonance in Ferromagnetic Materials

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The Effect of High Pressure on the Electronic Structure of Solids*

H. G. DRICKAMER Department of Chemistry and Chemical Engineering and Materials Research Laboratory, University of Illinois, Urbana, Illinois

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . ................... 11. Nonmetals. ............................................ ..

2

ers in Ionic Crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Crystal Field Effects.. .......................... .......... 3. Band Structure and the Approach to the Metallic State.. . . . . . . . . . . . . . . . 4. Organic Crystals.. . . . . . . . . . ........................ 111. Metals.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Electronic and Metal-Nonmetal Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The Electronic Structure of Hexagonal Close-Packed Metals. . . . . . . . . . . . . 7. The Electronic Structure of Iron a t High Pressure.. ....................

19 34 57 89 91 112 127

1 2

1. Introduction

During the past decade there has been a considerable expansion in the amount and variety of high-pressure research. In part this has been caused by an increased interest in experimental geophysics. Much of the industrial high-pressure effort has been triggered by the successful synthesis of artificial diamonds at the General Electric Company. In large part, however, this research is a response to the increased awarness of the significance of interatomic distance as a prime parameter in physical science, and particularly in solid state research. In the pressure range to 12 kbar, experiments of a variety and sophistication comparable to that a t 1 atm are possible. In the range to 30 kbar, liquid media can still be used. Relatively accurate and direct measurements of pressure are possible to perhaps 70 kbar. In the range from 100 to 600 kbar static measurements become more difficult. There arc available two recent books surveying the field of high-pressure research in an extensive way.’.z In this paper we shall

* This work was supported in part by the United States Atomic Energy

Commission. “Solids Under Pressure” (W. Paul and D. M. Warschauer, eds.). McGraw-Hill, New York, 1963. “High Pressure Physics and Chemistry” (R. S. Bradley, ed.), Vols. 1 and 2. Academic Press, New York, 1963. 1

2

H. G . DRICKAMER

review the experimental activities in our laboratory on the electronic structure of solids a t pressures to perhaps 600 kbar. Experiments in the range beyond 100 kbar encounter limitations as to the variety of measurements that is possible, as to the accuracy with which the pressure is known, and as regards hydrostaticity of the pressure. There are, however, important compensations. The volume decreases are large (20-50% in metals and frequently more in nonmetals) ; and one can observe second- and third-order effects beyond the direction and magnitude of the first-order pressure shift of a variable. Many qualitatively new events are observed-continuous and discontinuous transitions from nonmetal to metal, electronic transitions, and changes from ferromagnetic to nonferromagnetic states. Included in this discussion are optical absorption measurements to 160 kbar, phosphor emission and decay to 50 kbar, electrical resistance measurements to about GOO kbar, X-ray diffraction studies to 500 kbar, and Mossbauer measurements to 250 kbar. The experimental methods are not included, but they can be found in the The plan of the discussion is as follows. In the first two sections optical absorption studies involving localized electronic energy levels are discussed. In the next two sections optical absorption phosphor emission and electrical resistance measurements involving insulators, semiconductors, and the transition to the metallic state are covered. The last sections are concerned with the effect of pressure on the structure of metals, including electronic transitions, the interaction of the Fermi surface with Brillouin zone boundaries in noncubic metals, and the electronic structure of transition metals a t high pressure. II. Nonmetals

1. IMPURITY CENTERS IN IONIC CRYSTALS

a. Color Centers in Alkali Halides The alkali halides are the most thoroughly studied of all ionic crystals. Perhaps the most interesting impurity phenomena in these crystals are the color centers produced by X-irradiation or addition of excess alkali metal. The simplest of these is the F center which is generally assumed to R. A. Fitch, T. E. Slykhouse, and H. G. Drickamer, J . Opt. Soc. Am. 47, 1015 (1957). D. W. Gregg and H. G. Drickamer, J . Appl. Phys. 31, 494 (1960). 6 A. S. Balchan and H. G. Drickamer, Rev. Sn’. Inst. 31, 511 (1960). 6 H. G. Drickamer and A. S. Balchan, in “Modern Very High Pressure Techniques” (R. H. Wentorf, ed.). Butterworth, London and Washington, D. C., 1962. 7E. A. Perez-Albuernc, I<. F. Forsgren, and H. G. Drickamer, Rev. Sci. Znstr. 36, 29 4

( 1964).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

3

be a halide ion vacancy with an electron trapped in it. The usual simple treatment approximates the center as a “particle in a box178which predicts the energy E as proportional to 1/1*. Using peak energies for the F center and lattice parameters for a number of alkali halides, Mollwogshowed that this relationship is roughly accurate. I ~ e y analysis ~ s ~ ~ of the data gives

EAU1.84 = 1.76 X 10-19 eV m2

(1.11

if E is in electron volts and Ao is in centimeters. Jacobs1] showed that Ivey’s relation is an approximation to a more general relationship. He obtained d l n E - 3ybT dlnAo Ee

+

nQfF

where y is the Gruneisen constant, 0 is the Debye temperature, E is the energy of the F center, T is the absolute temperature, ng is the change of F-center energy with local compressibility, and f~ is the ratio of local t40bulk compressibility. Jacobs showed that, a t low pressures, at least, the second term dominates. From elasticity theory he obtained an approximate relationship for f F : fF = 1 2(K/P) (1.3)

+

where K is the bulk modulus and P is the shear modulus. Figure 1 shows the fractional change in F-center frequency with density for NaC1, NaBr, and K C P J 3 using Bridgman’sI4 density data. The line marked Ao-2 shows the particle in the box approximation. At low pressuresfF = 2.1 for XaC1 and 2.3 for NaBr, which are close to Jacob’s values. At high pressure SF approaches 1.0-1.5. It is intuitively reasonable that the local compressibility should approach the bulk compressibility a t high pressure, as with increasing compression one would expect increased overlap among electron clouds of nearest-neighbor ions. One can show that Eq. (1.3) predicts this qualitatively. The bulk modulus of NaCl increases by 60 to 70% from 10 to 50 kbar. In the same pressure range Bridgman’s15 results indicate that the LLresistance to shear” increases by about a factor of 3.3. From these values one would estimate anfF of 1.6 a t 50 kbar. For KCl the change in bulk modulus is 2.7 and the *E. Bumtein and J. J. Oberly, Nut. BUT.Std. (U.S.), Circ. 6109, 285 (1952). * E. Mollwo, Nachr. Ges. Wiss. Gottingen. Math-Physik. KZ.,Fachgruppen I, 97 (1931). ‘OH. F. Ivey, Phys. Rev. 72, 341 (1947). ’’ I. S.Jacobs, Phys. Rev. 93, 993 (1954).

’’ w.G. Maisch and H. G . Drickamer, Phys. Chem. Solids 6, 328 (1958). ’*R.A. Eppler and H. G. Drickamer, J . Chem. Phys. 32, 1428 (1960). ’‘ p. W. Bridgman, PTOC.A m . Acad. Arts Sci. 76, 1 (1915). ’’p. W. Bridgman, PTOC.Am. Acad. Arts Sci. 71, 387 (1937).

4

H. G . DRICKAMER

FIG.1. Log(v/vo) versus l o g ( p / p o ) for the F band in alkali halides.

increase in resistance to shear is 5.2, giving f~ (50 kbar) = 1.6. These are qualitatively similar to the experimental values. An interesting, but as yet unexplained, phenomenon is the appearance of a new pressure-induced band on the high-energy side of the F band in certain alkali halides. Figure 2 exhibits this phenomenon in KCI. At 8

O

O

0

L

-

I

I

12

13

-1 I4

L I5

@Po

FIG.2. Log(v/vo) versus l o g ( p / p o ) for KCI.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

5

kbar the band is one-tenth as intense as the F band; at 13 kbar the new band (called K’) is ten times as intense as the F band. It shifts with pressure approximately as does the F band. Both bands can be bleached by light monochromated near either peak. A change of pressure in the appropriate region restores either band. It may be that the K‘ band is associated with an electron trapped at a lattice defect, but no treatment of the problem is yet available. If a crystal with F centers is irradiated with light near the energy of the F peak, the peak is bleached and new peaks labeled R1, Rz, M , and N appear on the low-energy side. These are associated by Seitz16 wit,h a union of the F center with (a) a vacant negative-ion site, (b) another F center, (c) a pair of vacancies of opposite sign, and (d) a unit containing three negative-ion vacancies. The interpretation of these bands has undergone considerable change since the first attempt at interpretation by Seitz. It now seems likely that the M center correspond^^^ to a pair of F centers and that the other bands are the result of more complex coagulation of F centers. The Rz and N bands can be observed in KC1 on the higher and the lower frequency sides of the M band,Is respectively, although they were very weak. The peak heights of the M band in the high-pressure phase, the CsCl (SC)structure, are greater by a factor of 2 in KCI, 3 in KBr, and 3.4 in KI than those in the face-centered-cubic (fcc) structure. In all cases, however, after returning to the fcc structure from the high-pressure phase the peak heights were less than the original heights in this structure. The peaks of the M bands in the four alkali halides and those of the Rz and N bands in KCI shift to higher frequencies with increasing pressure except during a phase transition. The change in the frequency of the peak of the M band with pressure is plotted in Fig. 3; vo is the frequency of the peak of the M band at atmospheric pressure. The results are also plotted as log ( v / v o ) versus log ( p l p o ) in Fig. 4, where p/po is the fractional change in density according to Bridgman.14The rates of change in the frequency of the M band with fractional density are almost identical in the three potassium halides. However, the discontinuities at the phase transitions are different. That is, the M band in KC1 shifts to a higher frequency, but the &f bands in KBr and KI shift to a lower frequency with increasing density; also K I shows a greater red shift than KBr. The rate of change in the frequency of the M , Rz, N , and F bands in KC1 w-ith fractional density is compared in Fig. 5. The Rz band cannot be resolved in the sc structure, F. Seitz, Rev. Mod. Phys. 26, 7 (1954). ” J . H. Shulman and W. D. Compton, “Color Centers in Solids.” Macmillan, New York, 1962. Minomura and H. G. Drickamer, J. Chem. Phys. 33, 290 (1960). I‘

’*s.

6

H. C . DRICKAMER

but the absorption coefficient in the frequency range where the R, should be located is much greater than that before the phase transition. The N band shifts to a lower frequency with increasing density at the phase transition; on the other hand, the M and F bands shift to higher frequencies. If for the M center one accepts as a zeroth approximation the model consisting of a particle in a box, log Em should be proportional to log Ao, where Emis the frequency of the peak of the M band a t atmospheric pressure and A , is the lattice constant. The data can be expressed by the relation E , A O ’ . ~=~ 26.0 (eV A2) (1.4) which is, except for a slight difference in the coefficient, the same as that derived by Ivey.’O If the compressibility in the neighborhood of the M center were the same as the bulk compressibility, the rate of change in the frequency with density,

x=

1

d log (./yo

a log W P O )

should have the value 1.56/3 = 0.520. However, x is 20 to (30% larger than this, as shown in Table I. The value of x for the M center is somewhat smaller than that for the F center.I3 That is, the compressibility of the M center is generally lower than that of the F center. However, the ratio of the value of x for the M center to x for the F center increases and approaches unity with increasing lattice constant. Table I1 shows the variation of X M / X F with lattice constant in five alkali halides. I

I

I

I

vo

I

I

NhCl 13740 KCI 12100 K B r 10825

KI

0

I

I

I

1

i

CM-’

KCI

9535

I

I

1

10

20

30

I

I

40

50

I

I

60

70

80

P, KBAR

FIG.3. AV versus pressure for the M band in NaCl, KCI, KBr, and KI.

HIGH PRESSURE AND ELECTHONIC STRUCTURE

FIG.4. I,og(u/vo) versus l o g ( p / p O ) for the M band in NaCI, KCI, KBr, and KI.

r

I

I

I

I

I

I

I

0 12-

/

FIG.5. L O ~ ( ~ /versus V , , ) log(plp0) for the F , M , N , and R$ bands in KCl.

7

8

H. G. DIlICXAMEIl

TABLEI. THE RATE O F CHANGE IN THE FREQUENCY O F THE h f BANDWITH DENSITYIN FOURALKALIHALIDES

x =

Crystals KCl KBr

BI NaCl

a log(v/vo) a log(PiPo)

fcc structure

At the phase transition

sc structure

0.813 0.850 0.850 0.633

0.077 -0.092 -0.665

0.543 0.543 0.543 0.633

-

The values of z at the phase transition are different for the three potassium halides, as given in Table I. The values are negative in KBr and K I and positive in KC1. The shift to lower frequency at the phase transition may be caused by the effect of the polarization of ions which contracts the shift to higher frequency caused by the volume contraction. The contraction in the fractional volume at the phase transition increases in the order of KCI (-0.110) > KBr (-0.105) > K I (-0.085). The electronic polarizability of ions increases in the order of I- (6.16) > Br- (4.13) > C1- (3.00) > K+ (0.97) in units of cm.19,20 The algebraic values of x for the three potassium halides could be accounted for by the difference of the above two effects, the volume contraction and the electronic polarizability . The values of z for the M , Rz, N , and F bands in KC1 are given in Table 111. The value of z increases in the order of F > N > M > Rz as long as no phase transition occurs. At the phase transition the F and M bands shift to higher frequency, but the N band shifts to lower frequency. If the F , M , and N centers are approximated by models of particles in a negative vacancy, and in two and three pairs of vacancies of opposite sign, respectively, then the numbers of the first neighboring ions to the TABLE11. THE VALUESOF

l9 2o

X M / X F AND

LATTICECONSTANTS

LiCl

NaCl

KC1

KBr

KI

XM/XF

0.20

0.49

0.58

0.77

0.81

A O(A)

5.14

5.62

6.28

6.58

7.06

E. Burstein, J. J. Oberly, and J. W. Davisson, Phys. Rev. 86, 729 (1952). C. J. F. Bottcher, Rec. Truv. Chim. 62, 325, 503 (1943).

9

HIGH PRESSURE AND ELECTRONIC STRUCTURE

TABLE 111.

THE

RATEOF CHANGE IN THE FREQUENCY F BANDSWITH DENSITY IN KC1 z =

f cc

M

0.813 0.633 1.102 1.208

F

AND

a lOg(PlP0)

structure

N

M , Rz, N ,

a l og(y/Yo) -

Bands

RP

OF

At the phase transition 0.077 -

-0.547 0.207

sc

structure 0.543 0.872 1.005

color centers are eight positive ions in the F center, eight positive and eight negative ions in the M center, and eleven positive and eleven negative in the N center. The electronic polarizability contributed by the nearestneighbor ions to the color centers increases in the order of N (43.67) > M (31.76) > F (5.28) in units of cm3. The algebraic values of 2 at the phase transition could thus be accounted for by the effect of the electronic polarizability which counteracts the effect of the volume contraction. The peak heights of the ill bands in the three potassium halides increase by a factor of 2 to 4 across the phase transition from the fcc structure to the sc structure. The M-center density in the sc structure should increase by a factor of about 1.25 over that in the fcc structure according to Bridgman’s compressibility data. If the oscillator strength times the Mcenter density is proportional to the peak height times the half-width for the M center, as derived for the F center by Smakula,21the oscillator strength of the M center in the high-pressure phase should be greater than that in the low-pressure phase by a factor of over 1.6. On the other hand, centers may be formed from F centers in passing through the transition. The Rz band cannot be resolved in the sc structure, but the absorption coefficient in the frequency range where the Rz band should be located is much greater than that in the fcc structure. It could be ascribed to the broadening of the Rz band caused by interaction with lattice imperfections, 01 to the enhancement of a tail on the high-frequency side of the M band associated with transitions to discrete levels above the first excited state, as well as to the increase of the Rz-center density. A check run returning to the fcc structure from high pressure showed that the M, R2, and N centers can be considerably bleached by going through the phase transition in both directions. a A. Smakula, 2.Physik 63, 762 (1930).

10

H. G. DRICKAMER

If alkali halide crystals containing small amounts (from 0.05 to 2%) of silver halide impurities are subjected to X-rays, five bands appear in addition to the F band. This was discovered independently by Katsz2 and by the group a t the Naval Research L a b o r a t ~ r y , ~who ” ~ ~named these bands A through E. Etzel and Schulman26made an extensive investigation of the properties of these bands. More recently, Maenhout-van der Vorst and DekeyseP and Ishiguro et aLZ7have made further studies on these bands. Although it is true that the nature of these centers is not yet certain, models have been proposed for them as discussed below. The A center is almost certainly a hole phenomenon. Etzel and SchulmanZ5propose that the center is a substitutional silver ion which has been trapped by a hole, while Maenhout-van der Vorst and Dekeyser26 propose that it is a substitutional silver adjoining a V center (hole trapped in a positive-ion vacancy). It is generally agreed25-28 that the B center is a substitutional silver ion adjoining an F center (electron trapped at a negative-ion vacancy). The fact that the strength of the C band is strongly dependent on concentration had led both Etzel and Schulman and Maenhout-van der Vorst and Dekeyser to postulate that there are two silver ions in the center. The lattice adds to this two F centers, while the former only one. The greatest amount of controversy concerns the D band; Kats attributed this band to the B center. Etzel and Schulman’s bleaching experiments led them to the conclusion that no unbound electrons were involved in the center, and they proposed a center consisting of a substitutional silver atom adjoining a V center. Maenhout-van der Vorst and Dekeyser, however, found the band in additively colored crystals where only electron centers are to be found. They attribute the band to a substitutional silver ion adjoining an M center (electron trapped at a negativeion vacancy plus a vacancy pair). Maenhout-van der Vorst and Dekeyser attributed the E band to an interstitial silver ion adjoining an F center. Ishigur0,~7however, has found that while bleaching in the F center at room temperature enhances the B band, bleaching at liquid nitrogen temperature enhances the E band. Moreover, the E center is thermally unstable at only slightly above room 22

23

z4

M. L. Kats, Dokl. Akad. Nauk. SSSR 86, 539 (1952). E. Burstein, J. J. Oberly, B. W. Henuis, and M. White, Phys.Rev. 86, 225 (1952). H. W. Etzel, J. H. Shulman, R. J. Gintler, and E. W. Claffy, Phys. Rev. 86, 1063

(1952). H. W. Etzel and J. H. Shulman, J . C h m . Phys.22, 1549 (1954). 26 W. Maenhout-van der Vorst and W. Dekeyser, Physica 23, 903 (1957). 27 M. Ishiguro, T. 0 . Kuno and W. Veda, Mem. Inst. Sci. I n d . Res., Osaka Univ. 13, 69 (1956). zB H. N. Hersh, J . Chem. Phys.30, 790 (1959). 25

HIGH PRESSURE AND ELECTRONIC STRUCTURE

11

t e m p e r a t ~ r e .From ~ ~ all this Ishiguro concludes that the E center is an electron trapped in the field of a substitutional silver ion, or, in other words, a B center which has lost its associated vacancy. High-pressure measurementsz9 have been made on XaC1 crystals containing 0.1 and 1.0% AgCl, on KCl crystals containing 1.0% AgC1, and an KBr crystals containing 0.1% AgC1. The original paper may be consulted for typical curves and experimental details. The conclusions are summarized below. The data on the A band are scanty and of very poor quality because of the band’s location far into the ultraviolet. The shift with increasing pressure appears to be slightly to lower energy. This is consistent with the generally held conclusion that this is a hole phenomenon (that is, a phenomenon associated with the absence of a normally present elertron), although the data give no basis to define the model further. The data on the B band are of excellent quality and provide strong confirmatory evidence to the proposal that this center is a substitutional silver ion adjoining an F center. The shift with pressure is to higher energy, in magnitude roughly one-half that of the F band. A rough Ivey-like relation for the B center shows about twice the shift predicted from the change in bulk density and the “particle in the box” model as was observed for the F center.12J3 However, the strongest evidence for this model of the center is the emergence of a B‘ band on the highenergy side of the B band, and a t the expense thereof. in potassium bromide. This occurrence is analogous to the emergence of the K‘ band in the same crystal (see above ) . A somewhat unusual phenomenon occurred with the C center. I n the rest of this color-center work (in Ag+-doped crystals), the intensities of the bands are relatively independent of pressure. In the case of the C center, however, the intensity of the band increases rapidly with increase of the pressure; often more than an order of magnitude in 50 kbar. The shift in frequency of the spectrum is to higher energy, although somewhat less so than the B band. This work is riot inconsistent with the consensus of opinion that this band involves the interaction of two silver ions and the electron or electrons adjoining them. The model of Maenhout-van der Vorst and Dekeyser involving the joining of two B centers to form the C center is perhaps favored. A great deal of controversy centers around the D band. Several authors have proposed models for the band, but no consensus exists. When presSure is applied, the band shifts to higher energy initially, but levels off mound 100 kbar. The shift is in magnitude similar to the B center. The ’a

R. A. Eppler and H. G . Drickamer, J. Chem. Phys. 32, 1734 (1960).

12

H. G . DRICKAMER

strong shift to higher energy seems inconsistent with a hole picture for the center as proposed by Etzel and Schulman. On the other hand, a t first glance, the other model proposed-that of a silver ion adjoining an M center-also appears to be inconsistent, because the M band in pure LiCl has a much smaller pressure shift than the F band,’3 while the D and B bands have comparable shifts. The data on the E band are confined almost exclusively to potassium chloride, where the band is quite strong. Little or no pressure dependence of v, is detected. This is consistent with Ishiguro’s model of an electron trapped in the field of a substitutional silver ion.

b. Color Centers in BaF2 and CaF,

It is possible to induce color centers by X-irradiation in a number of crystals besides the alkali halides. Smakula30has studied three absorption bands of the color centers produced in CaFz and BaF2 crystals by X-ray irradiation in the region from 220 to 1000 mp (580, 400, and 335 mp in CaF,; 670, 480, and 380 mp in BaF,). On the other hand, M o l l ~ o has ~ l found two absorption bands in natural CaF2 crystals by additive coloring (525 and 370 mp). According to Smakula, as compared with alkali halides the absorption spectra of colored CaF2 and BaF, show some similar and some different properties and the nature of the color centers may be generally the same as in alkali halides with electrons trapped in lattice defects. The effect of pressure has been measured on the absorption bands of the color centers produced in CaF2and BaF232crystals by X-ray irradiation up to 54 kbar, and two pressure-induced absorption bands were found in each crystal in addition to the three bands found by Smakula. Figure 6 shows typical results for CaF,. The bands in BaF, are similar but are shifted to lower energy and have greater overlap. The bands described by Smakula show very little shift and only a small change in intensity with pressure. They are also not very susceptible to bleaching. In contrast to this, the pressure-induced bands grow steadily in intensity with increasing pressure. This growth is accompanied by a shift to higher energy (see Fig. 7). Upon releasing pressure the (pressure-induced) bands return to their original energy, but with a marked further increase of intensity. The pressure-induced bands can be bleached by light of their own wavelength, but the band a t 367 mp is not bleached by 523-mp light, and vice versa. There is little bleaching of the Smakula bands. A second compression of the crystal after bleaching restores the bands. 80

az

A. Smakula, Phys. Rev. 77, 408 (1950). E. Mollwo, Machr. Ges. V’iss. Gottingen. Math.-Physik. K l . , Fachgruppen I, 79 (1934). S. Minomura and H. G. Drickamer, J . Chem. Phys. 34, 670 (1961).

WAVE NUMBER

FIG 6 . Spectra of X-irradiated CaFz at various pressures.

H. G . DRICKAMER

I

1

0

10

20 P,ATM

30

x

1

40

I

1

50

10-3

FIG.7. AV versus pressure for pressure-induced bands in CaF2 and BaF2.

In Fig. 8 are plotted In (v/vo) versus In ( p / p o ) for the pressure-induced bands. Bridgman’s compressibility data for CaFZ33 to 30 kbar and for BaF234 to 12 kbar were extrapolated to 54 kbar. Within the accuracy of the extrapolation one obtains a straight line of slope 0.444. This slope is somewhat smaller than that obtained for the F or 144 center in alkali chlorides or b r ~ m i d e s , ~but ~ J *the fluorides may well behave differently. It should be noted that there is apparently a phase transition in BaF2 near 30 kbar, as the light cuts off at that point and is restored at 35 kbar. There seemed to be no discontinuity in the shifts at these pressures. The properties of the pressure-induced bands (the shift to higher frequency with increasing pressure, the growth with pressure change, and the

FIG.8. Log(v/vo) versus log(p/pO) for pressure-induced bands in CaFz and BaF2. 38

a4

P. W. Bridgman, Proc. Am. Acad. Arts Sci. 77, 187 (1949). P. W. Bridgman, “The Physics of High Pressure.” Bell, London, 1949.

15

HIGH PRESSURE AND ELECTRONIC STRUCTURE

bleaching) all indicate that these centers are associated with electrons trapped a t lattice defects. From the bleaching characteristics there is no connection between the high- and low-energy centers such as that between the F and M centers in alkali halides. The bands consistent with Smakula’s observations seem to be associated with colloidal metal deposits or impurity ions in the lattice. The pressure-induced color centers are probably formed by dispersion of electrons from colloidal metal to pressure-induced lattice defects by processes such as the following: Ca

+ 2V+ + Ca++ + 2(V+ + e ) , +

where T’+ is a negative-ion vacancy, e is an electron, and (V+ e ) is a color center. The process is not reversible with pressure at room temperature, but the electrons are released from the color centers by light of the appropriate energy. c. Alkali Halide Phosphors

One of the most thoroughly studied phosphor types is the alkali halide l ~ ~S e i t proposed ~~~ some time crystal doped with TI+ ion. Von H i ~ p e and ago that a one-dimensional configurational coordinate system be used to treat simple phosphors. The usefulness and limitations of this treatment have been considered by Klick and Schulman3’ and by Kamimura and S u g a n ~The . ~ ~calculations of Williams and ~ o - w o r k e r using s ~ ~ ~this ~ ~ treatment for KCl: T1 yield reasonably quantitative results. The limitations in Williams’ treatment have been pointed out by Knox and Dexter41; nevertheless, his work can be used as a basis for a first-order discussion of many of the pressure effects. Williams’ treatment assumes that the observed absorption band corresponds to a transition from the ‘So to the 3P1state of the thallous ion (Seitz’s A peak) and that the energies can be represented effectively on a single configuration coordinate. Making use of the available empirical information, he is able to calculate energy versus configurational coordinate for the ground and first excited state of TI+ in KCl. These curves are shown as the solid lines in Fig. 9. (Presumably similar curves would be obtained for T1+ in other alkali halides having the face-centered-cubic structure. ) A. von Hippel, 2.Physik 101,680 (1936). F.Seits, Trans. Faraduy SOC.36, 79 (1939). 37 C . C. Klick and J. H. Schulman, Solid State Phys. 6, 97 (1957). 38 H. Kamimura and S. Sugano, J . Phys. SOC. Japan 14, 1612 (1959). s8 F. E. Williams and P. D. Johnson, J . Chem. Phys. 20, 124 (1952). 40 F. E. Williams, J. Chem. Phys. 19,457 (1951). 41 R. S. Knox and D. L. Dexter, Phys. Rev. 104, 1245 (1956).

16

1%. G . DRICKAMER

0 -04

0

t 0.4

t0.6

Aa,l\

FIG.9. Configuration coordinate diagram for alkali hailed systems.

P, KBAR

P, KBAR

Av

- 200 CM"

(0)

NhCl TI

-400

FIG. 10. Initial frequency shift versus pressure for ten alkali halides activated by TI+. (a) Crystals in the NaCI structure. (b) Crystals in the CsCl structure.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

17

- 800

--i 1

I

I

20

I

I

40

PRESSURE, KILOBARS

FIG.11. Apv versus pressure for KCl, KBr, and KI activated by TI+.

Johnson and Williams42have developed a simple theory of the effect of pressure on the absorption spectra of KC1:Tl. They assume no distortion of the energy levels of Fig. 9, and no displacement of one level with respect to the other. These assumptions should be reasonable at low pressures. Then the work done by pressure in causing a displacement along the configuration coordinate is equated to the change in potential energy due to the displacement. An expression for the change in transition energy is obtained as a function of pressure, an effective area, and the force constants of the ground and excited states. The essential feature as far as this discussion is concerned is that the change of AE with pressure is proportional in the slope of the excited-state energy curve at the origin (i e., at the minimum of the ground-state curve). For a system such as KC1:Tl where the 3P1curve minimum lies inside the 'So curve minimum, the slope is negative and a shift to lower energies (red shift) with increasing pressure is predicted. If the excited-state minimum were outside the groundstate minimum (dashed curve in Fig. 9 ) , a shift to higher energies Lvith increasing pressure would be predicted. (2

P. D. Johnson and F. E. Williams, Phys. Rev. 96, 69 (1954).

18

H. G , DRICKAMER

I 0

I I 50 100 PRESSURE, KILOBARS

I50

FIG.12. AV versus pressure at high pressure for Tltactivated alkali halides.

At higher pressures other effects could possibly become important. There could be a displacement of the minimum of one curve with respect to the other. One might expect that the curve with the minimum a t higher values of the configuration coordinate ( X ) would be displaced more than the other. For the first case above, the ground-state curve should move in with respect to the excited state. In the early stages a t least, this would predict a further red shift. In the second case (dashed curve) the excited state should be shifted in. This would result in a red shift a t higher pressure also. A further effect would be the increase in zero point energy. Because of the shapes of the curves, this effect would be larger for the ground state, resulting in a red shift for all systems at sufficiently high pressures. The effects of pressure have been presented in two articles.43 Figure 10 shows the initial shift with pressure for a number of systems. As predicted by the theory, KCI, and indeed all systems having the fcc structure, show a red shift with pressure. CsBr and CsI, which have the sc structure, show 4s

R. A. Eppler and H. G . Drickamer, Phys. Chem. Solids 6, 180 (1958); 16, 112 (1960).

HIGH PRESSURE AND ELECTRONIC S3TRUCTUR.E

19

an initial blue shift with pressure. One would then postulate that in the sc structure the excited-state energy curve has a minimum at larger values of the configurational coordinate than does the ground-state curve (dashed curve in Fig. 9). At 15-20 kbar the blue shift levels off and reverses. One can ascribe this to the displacement of the excited state with respect to the ground state or to the increase in zero point energy discussed above. The potassium halides have a phase transition a t about 20 kbar, going from the fcc (NaC1) to the sc (CsC1) structure. In view of the results discussed above for the two phases a t low pressure, one would predict a displacement of the excited state outward with respect to the ground state, and thus a blue shift at the transition. Figure 11 shows the results for KCI, KBr, and KI. The first two halides show the predicted blue shift. It is not easy to give a completely satisfactory explanation of the red shift of KI. Probably the large and polarizable iodide ion accentuates the difficulties in simple configuration coordinate theory pointed out by Knox and Dexter. The level portion of the K I curve just above the transition corresponds to the level section of the CsI curve a t about the same pressures. Figure 12 shows the shifts of the absorption bands a t higher pressures. All systems show a relatively large red shift, probably caused mainly by the increase of zero point energy in the ground state as discussed above. 2. CRYSTAL FIELDEFFECTS

Since there exist excellent detailed reviews and discussions of crystal field t h e ~ r y ,only ~ ~ a, ~brief ~ outline of those results which show direct dependence on interionic distance will be given here. A 3d electron on a free transition-metal ion exists in a fivefold degenerate ground state. A series of excited states are possible, and their separation from the ground state is due to repulsion between the electrons in the 3d shell. These separations can be calculated using the integrals of Condon and Shortley, but it is more convenient to express them in terms of the Racah parameters A , B, and C which are combinations of the CondonShortley integrals. The simplest version of crystal field theory pictures the central transition-metal ion surrounded by point charges (or point dipoles). These provide a n electric field of less than spherical symmetry which partially removes the degeneracy. For such point ligands there would be no change in the Racah parameters. The degree of splitting would depend only on the symmetry of the ligand arrangement and the ligand-ion distance. The ru D. S. McClure, 46

Solid State Phys. 9, 400 (1959). J. S. Griffith, “The Theory of Transition Metal Ions.” Cambridge Univ. Press, London and New York, 1961.

20

H. G. DRICKAMER

symmetries usually considered are tetrahedral, octahedral, and cubic corresponding to 4, 6, and 8 nearest neighbors. The potential can then be expanded in Legendre polynomials and the separation of each of the d levels from the free-ion state can be calculated. For, say, the d, and d&,2 levels in a field of octahedral symmetry this separation between these calculated levels is called lODq, where the symbols arise for historical reasons. From symmetry considerations it is clear that in this case one has one doubly degenerate and one triply degenerate level. For the other symmetries the splittings differ; e.g., for the same ion-ligand distance Dq (octahedron) = -$ Dq (tetrahedron) = -# Dq (cube). The first nonzero term in the expansion of the potential is V4. Calculation thus indicates that the field intensity (1ODq) should vary as RPS where II is the ion-ligand distance. This is a conclusion which can be directly tested by pressure measurements. Since, for these transitions, AL = 0, they are LaPorte forbidden. They occur with nonzero intensity because the excited-state wave function contains some admixture of, say, 4 p with the 3d wave function. This mixing is caused by the electrostatic potential mentioned above. It may come about either because the system at equilibrium lacks a center of symmetry or because thermal vibrations instantaneously remove the center of symmetry. The first nonzero term in the perturbation of the wave function varies as R-4, so the intensity should vary as R-8. I n reality, of course, the ligands are not point ions but consist of nuclei and electron clouds. Not only the net ionic charge but also the positively charged nucleus of the ligand interacts with the 3d electrons on the central ion. This tends to spread out the 3d electron cloud and reduce the interelectronic repulsion. An accurate picture of the situation would call for a molecular orbital calculation, but for our purposes it can be expressed in

I I0 Dq (CRYSTAL 1 FIELD STRENGTH)

FIG.13. Schematic diagram for energy levels of transition-metal ions.

HIGH PRESSURE AND ELECTItONIC STRUCTURE

21

terms of reduced values of the Racah parameters B and 6. (The A parameter cancels out identically.) Figure 13 shows a schematic picture of the events we have described. This interaction between 3d electron and ligand nucleus is frequently rather loosely referred to as “covalency.” One would expect the covalency to increase with increasing pressure and consequently that B and C would decrease. The situation involving a rare-earth ion is qualitatively very similar to that of the transition-metal ion except that electrons from the partially filled 4f shell are involved and the intensity depends on mixing with 5d states. Since the 4f electrons are shielded from the crystal field, much smaller splittings are involved, but interesting details concerning intensity changes and thermal population of states can be investigated and are discussed below. As can be seen from Fig. 13, transitions are possible which measure lODq directly, independent of the Racah parameters. There are also transitions which depend on both the crystal field strength and interelectronic repulsion, and finally there are those which depend on B and C only. In our discussion lODp, B, and C (where it occurs) are treated as empirical parameters to be measured as a function of pressure and codguration, whose change with interatomic distance can be used as both a qualitative and a quantitative test of theory. Before proceeding to discuss specific experimental results, it will be helpful to summarize the effects one would expect to observe with increasing pressure. ( 1 ) Absorption peaks which represent transitions that depend on lODq or lODq plus B and C would be expected to shift to higher energy with increasing pressure. (2) The change in 1ODq with pressure should in the zeroth order be independent of the transition used to calculate it for a given ion in a given crystal. I n actual crystals the Racah parameters would be expected to decrease with increasing pressure, and therefore there might be discrepancies in calculating lODq from different transitions, holding B constant. (3) Transitions which depend on B and C only would be expected to shift to lower energy with increasing pressure because of the abovementioned change in B and C. (4) One might, in the simplest case, expect the crystal field strength to vary as R-6, but more complex considerations might be important in actual cases. (5) One would expect the integrated intensity to increase with increasing pressure, roughly as R-*.

22

€1. G . DRICKAMER

These and a few more subtle considerations are discussed below, based on experimental work done in this laboratory.46-51

a. Peak Shifts I n Fig. 14 the shifts of two Ni2+peaks in MgO are shown as a function of pressure. The low-energy transition depends on lODq only, the highenergy transition on both lODq and B. Both shift to higher energy as pre-

MgO.NiZ'

0 0

yo

8,845 C M - '

= 24,500CM-'

P, KILOBARS

FIG.14. Frequency shift with pressure for MgO:Ni2+.

dicted by the theory. Figure 15 shows the change in lODq with pressure for four ions in MgO. In all cases the crystal field strength increases with pressure, in qualitative agreement with theory. Figure 16 shows the change in lOUq with pressure as calculated from two different transitions in Cr3+:A1203(ruby), using the 1-atm value of B. As predicted, the changes in lODq are qualitatively the same, but differ by an amount significantly beyond experimental error, which indicates that B changes with pressure. Figure 17 shows that the interelectronic repulsion parameter indeed does decrease with increasing pressure by about 4% in 120 kbar, indicating an increase in ion-ligand interaction. 46

47

R. W. Parsons and H. G. Drickarner, J. Chem. Phys. 29, 930 (1958). D . R. Stephens and H. G. Drickamer, J . Chem. Phys. 34, 937 (1961); 36, 424, 427, 429 (1961).

K. B. Keating and H. G. Drickamer, J . Chem. Phys. 34, 140, 143 (1961). 49 S. Minomura and H. G . Drickamer, J . Chem. Phys. 36, 903 (1961). 50 J. C. Zahner and H. G. Drickamer, J. Chem. Phys. 36, 1483 (1961). 61 R. E. Tischer and H. G. Drickamer, J . Chem. Phys. 37, 1554 (1962). 48

23

HIGH PRESSURE AND ELECTRONIC STRUCTURE

0

I00

50

I ‘0

P,KILOBARS

FIG.15. Change in 1ODq with pressure for four ions in MgO.

The Mn2+ ion exhibits two transitions whose energies depend only on B and C. I n Fig. 18 it is shown that these peaks indeed shift to the red (to lower energy) with increasing pressure, as predicted above. Figure 19 shows the calculated changes in B and C in MnCL and MnBrp as a function of pressure. I n the theoretical discussion above it was mentioned that simple theory would predict the crystal field strength to increase as R-5 (p513for isotropic I

I

- 4T,2(F)

I

50 P, KI LOBARS

100

FIG.16. Change in lODq with pressure for two transitions in &O3:Cr3’

H. G . D M C K A M E R

mo

: M

100

I50

P, KILOBARS

FIG.17. Change in B with pressure for A1203:Cr3+.

compression). Figure 20 exhibits a test of this theory for a series of ions in AlzOa, using Bridgman’@ compressibility data. The agreement is surprisingly good. The ions are present substitutionally in A13+ sites having essentially octahedral symmetry. The lattice is very rigid with high cohesive energy. The 02-ions have low polarizability. ’ 1 ure These latter two conditions are ideal for application of the theory. T’g 21 shows a similar test for ions in MgO. The sites again have octahedral symmetry with oxygen ligands, but the lattice has lower cohesive energy. I n all cases the change in 1ODq is larger than would be predicted from the compressibility. While a number of factors are operative, the major one probably involves local relaxation and higher local compressibility near the foreign ion. Figure 22 shows the same test for ions in ZnS. Again the shifts are greater than predicted, indicating an increased local compressibility. I n Fig. 23 we see the shift of the lODq peak for Ni2+ in Ni(NH3)&12 This crystal has a first-order phase transition a t about 60 to 65 kbar. The crystal field is supplied by the six NH3 ligands in octahedral array around the nickel ion. The transition rearranges the C1- with respect to the complex Ni(NH3)2f ions but supposedly does not disturb the symmetry of the complex ion. Nevertheless, it is interesting to note that there is a measurable change in crystal field energy at the transition, which indicates F.*

P. W. Bridgman, Proc. Am. Acad. Arts Sci. 77, 220 (1949).

HIGH PRESSURE A N D ELECTRONIC STRUCTURE

FIG.IS. Shift of two peaks not dependent on the crystal field, in MnC12.

P, KILOBARS

FIG.19. Effect of pressure on Racah parameters B and C in MnC12 and MnRr?.

26

26

H. G . DRICKAMER

-

7

-

r

1

1

-

x

a' 0

0 0

I

I

I

20

I

40

P , KILOBARS

FIG.21. Test of R-5 law for MgO.

P, K i LOBARS

FIG.22. Test of

R-6

law for ZnS.

60

HIGH PRESSUHE AND ELECTRONIC STRUCTURE

27

FIG.23. Pressure shift for Ni2+peak in Ni(NH3)&12.

that second-nearest neighbors significantly perturb t.he crystal field a t the Ni2+ ion. In the earlier discussion of ruby it was indicated that the Cr3+ion is in a situation of essentially octahedral symmetry. However, the Cr3+ ion is a little too large for the site, and there is a measurable trigonal distor-

>O""

t..-/6, 1 500 0

50

I00

I50

P, KILOBARS

FIG.24. Trigonal distortion versus pressure for A1201: Cr3+.

28

H. G. DRICKAMER 4

I

I

A1203 : T r 3 '

vo 11 =

17,870 CM-I

7)OL = 17,720 CM -' 3I

P X

7

2-

I v (> 7-

a

//

,-.

n d

: -

I-

0

I

I

50

0

I00

I50

tion. The trigonal component can be obtained from the difference in peak location for crystals oriented parallel and perpendicular to the c axis. Figure 24 shows the trigonal component as a function of pressure. Up to about 60 kbar the pressure effect is negligible and the compression is essentially isotropic. Above this pressure the trigonal component increases rapidly, indicating that further compression can only take place a t the expense of increased distortion. Figure 25 shows a similar situation for M2O3:Ti3+ except that here the Ti"+ion is smaller than Cr3+and both the initial distortion and absolute increase of distortion with pressure are less.

b. Intensity Effects I n the earlier discussion it was indicated that crystal field effects could also be observed for rare-earth ions. I n this case the crystal field splitting is a relatively small perturbation on the free-ion spectrum. Since the trivalent rare-earth ions are not located at a center of symmetry, the observation of intensity effects as a function of pressure provides a useful test of theory. Figures 26 and 27 show the change of intensity with pressure for peaks in PrC13and NdF3. The increase is by a factor of 1.4 to 1.45 2.0r

P,KILOEARS

FIG.26. Intensity ratio versus pressure for 3H4 431'0 transition in PrC13.

H I G H P R E S S U R E AND ELECTIZOXIC S T R U C T U R E

29

2.0r

FIG.27. Intensity ratio versus pressure for

- 2Gj/2

2Gs/2 transition

in NdF,.

in '00 kbar and about 1.65 to 1.75 in 170 kbar. This corresponds to a 15% compression in 100 kbar and 20-22% in 170 kbar. These are quite reasonable values; e g., silver chloride compresses lGYo in 100 kbar. Figure 28 shows the peak shift and half-width as a function of pressure for TmCI3. These results arc typical of observations of a large number of rare-earth ions in a variety of crystals. The shifts are small, but the shift in 100 kbar amounts to about 15 to 30Y0 of the observed atmospheric splitting (2EO-400 cm-I), which is comparable to the shifts noted for transition metal ions. In every case the shift for the rare-earth ion was accompanied by an increase in half-width of the peak. These results can be explained in terms of the diagram of Fig. 29. The peak observed is the sum of various transitions from the ground state (split by the crystal field) to the similarly split excited state weighted according to the occupation probability of the levels. The increase in pressure increases the crystal field effect and therefore permits transitions over

u

I 5,o P , A T M X 10-

200

I

Y Q

w-I00

a

FIG.28. Half-width ratio and peak shift versus pressure fur in TmCla.

3Hti

~

30

H. G . DRICKAMER I - - - - - - -

FREE ION

CRYSTAL FIELD

--

PRESSURE-INTENSIFlED CRYSTAL FIELD

FIG.29. Schematic representation of splitting of rare-earth ion levels in crystal field a t atmospheric pressure and high pressure.

A

P=l57,000 ATM (a X=1.070)

h

P:103,200 ATM ( a X=1.049)

P- 50.800 ATM (a X-0.548)

P=31,200 ATM (a X=0.522)

P- 16 800 ATM (a X.0.487)

I

17,000

I

I

I

I

I

17,500 WAVE NUMBER, CM-'

FIG.30. Peak shape as a function of pressure for Nd(C2HjSOJ3-9HzO peak at 17,250 em-'.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

31

a broader energy range; hence, the increase in half-width with increase in crystal field. Finally, it should be mentioned that in some cases it was possible to operate at a sufficiently narrow slit to permit observation of the crystal field components of the bands. Figure 30 illustrates one such case. It can be seen that the higher energy component shifts red and increases markedly in intensity relative to the lower energy component. This is probably due to an increase in population of the lower lying Stark level of the ground state. Since the splitting is of the order of 1 to 2 kT,a 20-30y0 increase in splitting would significantly affect the relative occupation of the levels. c. Local Symmetry and Compressibility i n Glass

The results on ions in A1203,MgO, and ZnS indicate that the change in crystal field energy might be a useful zeroth-order approximation to the local compressibility, especially in cases where no other techniques are applicable. One such case involves transition-metal ions in glass. It is found that when transition-metal ions are dissolved in glass they may enter into one or more of three different kinds of sites. One type of site has octahedral symmetry and, from the sharpness of the peaks, the symmetry is sub-

0.07

0.06

OCTAHEDRAL SYMMETRY

-

TETRAHEDRAL SYMMETRY OCTAHEDRAL SYMMETRY BULK VALUES AFTER BRIDGMAN

P, KILOBARS

FIG.31. Local compressibilities for various sites in silicate glass.

32

IT. G . DRICKAMER

stantially as regular as that of a good crystal. h second type of site has tetrahedral symmetry also of a high degree. A third type of site exhibits a rather “sloppy” octahedral symmetry with relatively broad peaks. Measurements of the change in lODq for ions in these sites give an approximation to the local compressibility. Figure 31 shows the calculated local compression for these sites in a silicate glass. The glass composition and details of sample preparation are given in the original paper.61It is observed that the highly symmetric octahedral sites show a compression of about 4% in 100 kbar, which is comparable to a garnet or sapphire. The tetrahedral sites exhibit a somewhat greater compression (about 6% in 100 kbar). Again this is the order of, say, spinel compressibility. Finally, the “sloppy” octahedral sites exhibit a much greater compressibility, of the order of the observed bulk compressibility of the glass.52 Apparently the first two types of sites are essentially crystalline in nature, while the third t$yperepresents the more nearly amorphous areas. The local compressibility depends on the thermal history of the glass. (All the above samples had, of course, identical heat treatment.) In general, the glasses were held at temperature for several days and quenched to preserve the structure stable at that temperature. Most glasses exhibit

Q

1

0 50 O b 0 L200 - -

L 600 I I I000 I STAB I L I ZAT I O N TEM P, ‘ C

1

1400 1

I:IG. 32. Locitl compressibilities in silicate glass, as a furiction of heat treatment.

H I G H PItESSUItE AND ELECTItONIC STRUCTURE I

I

1

I

I

L 100

120

33

1

0.6

0

0

.

20

2 40

0 60 80 P , KILOBARS

L

FIG.33. Intensity changes in absorption bands due to Ni2+in tetrahedral sites.

a transformation temperature involving rapid softening, which can be established by differential thermal analysis. For the silicate glass discussed here the transformation temperature was 600°C. All glasses were initially quenched from 1100°C before being stabilized. Figure 32 shows the local compressibility of various sites as a function of stabilization temperature. The compressibility is a minimum for glass stabilized at the transformation temperature. Glasses quenched from a higher temperature exhibit a more open structure and a higher local compressibility. It is more difficult to understand the high compressibility of the glasses stabilized below 600°C. Apparently even 144 hr at 400°C is not sufficient to remove completely the open structure quenched in from 1100°C. Finally, an unusual intensity effect is observed for tetrahedral sites. As discussed above, the normal behavior for crystal field spectra is an increase in intensity with the decrease in ion-ligand distance. Initially, this occurs for all the sites. Above about 20 to 30 kbar, the intensity of the tetrahedral sites decreases rapidly with increasing pressure. By 100 kbar the peaks are about 75y0gone. Figure 33 illustrates the intensity change. At the same time that the tetrahedral sites are diminishing in intensity, new peaks are appearing at locations consistent with octahedral symmetry. The phenomenon is entirely reversible. McC1u1-e’~~~ calculation of the “site preference energy” would indicate an increased preference for ocD. S. McClure, Prog. Inorg. Chem. 1, 23 (1959).

31

H. G. DRICKAMEH

tahedral sites a t high pressure, but the mechanism whereby the ions transfer to octahedral sites, or the sites change symmetry reversibly a t room temperature and high pressure, is hard to understand. 3. BANDSTRUCTURE

AND T H E

APPROACHTO T H E

METALLIC STATE

The most fruitful single idea concerning the electronic structure of solids and the relationship among insulators, semiconductors, and metals is that of energy bands. The general theory is developed in every standard text on solid state physics. The elucidation of the details of band structure is the subject of many sophisticated experiments and much advanced the0ry.5~Nevertheless, considerable information can be obtained from optical absorption and electrical resistance measurements a t high pressure. Those parts of theory directly relevant to explain the results of such measurements as a function of interatomic distance and crystal structure are reviewed briefly below. A n electron on a free atom or ion can exist in a series of discrete energy states. The lowest available state is the ground state. At energies above the series of bound excited states exists a continuum which corresponds to ionization. As an array of atoms is brought closer together so that their wave functions overlap significantly, the situation is modified as shown in Fig. 34. Since the Pauli principle permits only two electrons per stationary state, the n-fold degeneracy (n atoms each containing one electron of a given type) is removed and one has bands of closely spaced allowed levels, separated by gaps of “forbidden” energy. The width of the band is determined by t.he type of atom and the degree to which the electron is bound to it. For tightly bound electrons where the band width is of the order of the thermal energy of the electrons the usefulness of the band description becomes questionable. (See the discussion of Section 4 on organic crystals.) The spacing of the levels within the band is determined by the number of atoms in the crystal, but for any normal experiment this spacing is very small compared with thermal energies. If the highest filled state of the free atom contains only one electron as in monovalent metals, only the lower half of the states in the highest occupied band is filled. This situation is represented in Fig. 34a. There are then unoccupied states within easy reach and electrons can travel through the lattice under the impetus of an applied potential, so that one has an electrical conductor. The electrons are not truly free, as they still feel the periodic potential of the lattice, but an “almost free electron treatment” is frequently applicable. See Chapter by W. Paul and D. M. Warschauer in “Solids Under Pressure” (W. Paul and D. M. Warschauer, eds.). McCraw-Hill, New York, 1963, and also references contained therein; see also P a u P .

HIGH PRESSURE AND ELECTRONIC STRUCTURE

35

If the highest occupied state of the free atom or ion contains two electrons, all states of a band are filled, and there are no unoccupied states within reach, if the forbidden gap is large compared with the thermal energy. One then has an insulator or semiconductor. This is the situation in most ionic, molecular, or valence crystals, as represented in Fig. 34b. If, however, the highest filled band overlaps the nearest empty band, one can still get electrical conductivity such as one obtains for the divalent metals. The filling of the electron levels a t any temperature is given by FermiDirac statistics. The energy boundary between filled and empty levels (which is sharp a t sufficiently low temperatures) is known as the Fermi surface. For a n intrinsic semiconductor of simple band structure it lies half way between the top of the highest filled band (valence band) and the bottom of the lowest empty band (conduction band). An important role for optical absorption measurements in studying energy gaps is easy to see. If light of an appropriate wavelength impinges

1

I

I

ISOLATED

METAL

R (INTERATOMIC DISTANCE)

1

E ISOLATED

I

klETAL

'I

I INSULATOR

SEMICONDUCTOR R (INTERATOMIC DISTANCE)

(.bl

FIG,34. Schematic diagram-nergy

-

versus interatomic distance.

36

H. G . DRICKAMER

on the crystal, it will excite electrons from the top of the valence band to the bottom of the conduction band. Since this is generally a n allowed transition, very intense optical absorption is observed at this wavelength. The shift of this absorption edge with pressure measures the change of the gap with pressure. In simple cases one would expect it to decrease monotonically to zero. The role of the electrical resistivity is a little more complex. From elementary theory one can write p =

(3.1 1

(ripe)-'

where p is resistivity, n is the number of carriers, p is the carrier mobility, and e is the charge. Both n and depend on temperature (and pressure). The number of carriers is determined by the probability of exciting carriers from the Fermi energy to the bottom of the conduction band n

-

exp(-Eg/2kT)

(3.2)

where E, is the energy gap and the factor 2 arises from the location of the Fermi surface discussed above. In simple cases the mobility is limited by lattice scattering and is proportional to P I T . One would then expect some increase in mobility with pressure due to increase in Debye 8. For insulators and semiconductors the controlling factor in the resistance is t.he exponential. One then anticipates a continuous red shift of the absorption edge to zero with increasing pressure corresponding to the decrease in the energy

E

(a)DIRECT

TRANSITION

(bl

INDIRECT TRANSITION

E

CONDUCTION

INDIRECT TRANSITION

E

-

re, I M P U R I T Y

(C)

k--

If,MONOVALENT METAL

td)

EXCITON STATE

(hl

SEMIMETAL

E

-k-+

--k-

(g) DIVALENT METAL

FIG.35. Schematic picture of typical band types.

HIGH PlLESSUllE AND ELECTRONIC STRUCTURE

37

gap, and a corresponding exponential decrease in resistance. As will be seen in the discussion of experimental results, this fits some solids but is too crude for many of them. The electronic energy is a function of the propagation vector (k) of the wave function. One can plot this energy in a space generated by the components of this vector. The periodicity of the atoms in real space introduces a periodicity in k space. The ‘(unit cell” of this k space structure is known as the Brillouin zone. Figure 35 represents some of the possible types of electronic events which one might observe. The cross-hatched areas represent filled states. In Fig. 35a one sees a direct transition (Ah = 0 ) from the top of the valence band to the bottom of the conduction band. This is the normal allowed transition. At temperatures above absolute zero the atoms of the lattice are vibrating. The propagation vector of a phonon may add to or subtract from the propagation vector of the ground state and give indirect transitions where Ak # 0 as shown in Figs. 35b and 35c. There may exist bound excited states below the conduction band as shown in Fig. 35d. These can be detected at atmospheric pressure by lack of photoconductivity in a n optically excited crystal. Such experiments have not yet been performed a t very high pressure. Impurity atoms with energy levels above the top of the valence band may furnish electrons to the conduction band as seen in Fig. 35e. Figures 35f and 35g represent the band situation in simple monovalent and divalent metals where one would expect resistance to increase linearly with increasing temperature. Figure 35h shows a more complex type of metal whose resistivity may not be linear in temperature. For the purpose of discussion of experimental results in this part of the review, a semiconductor (or insulator) is d e h e d as a material whose resistance decreases exponentially with increasing temperature, while a metal is a material whose resistance increases with temperature, whether or not the increase is linear. Figure 35f-h still presents a greatly simplified picture. The top of the filled zone is shown as independent of k , which implies a spherical Fermi surface. This would be accurate for strictly free electrons, but for actual metals the Fermi surface is frequently very complex. Finally, the above discussion assumes that the atomic arrangement is independent of pressure. The cohesive energy involves differences of relatively large terms, and differences in cohesive energies between different crystalline phases are frequently very small. It is not surprising, then, that over the relatively large pressure range discussed here, firstorder phase changes may not infrequently occur. These may involve discontinuous changes in absorption edge and in resistance.

38

H . G . DRICKAMEH

a. Elements

Figure 36 shows the change in the optical absorption edge with relative density for four nonmetallic elements.55 The energy gap decreases by a very sizable fraction in the range of pressures covered. The shapes of the curves for these elements are surprisingly similar.

I

1.0

1

I

0.90

1

I

0.80

II

0.70

r,/P FIG.36. Shifts of absorption edges of some elements with density.

A much more complete study of the approach to the metallic state has been made on i0dine.56>~7 Iodine crystallizes in a base-centered orthorhombic structure with the I2molecules in the ac plane. It is quite practical to grow crystals of usable size from the vapor phase. The measurements include: ( I ) optical absorption measurements (location of the absorption edge) as a function of pressure to about 90 kbar, (2) measurements of electrical resistance both parallel and perpendicular to the molecular plane to over 400 kbar, (3) measurements of the temperature coefficient of resistance between 77°K and 296°K from 60 to 400 kbar. 66 66 67

H. L. Suchan, S. Wiederhorn, and H. G. Drickamer, J . Chem. Phys. 31, 355 (1959). A. S. Balchan and H. G . Drickamer, J . Chem. Phys. 34, 1948 (1961). B. M. Riggleman and H. G . Drickamer, J . Chem. Phys. 37, 446 (1962); 38, 2721 (1963).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

39

Figure 37 shows resistance versus pressure measured both perpendicular and parallel to the molecular plane. In our apparatus it is not possible to correct for contact resistance, so that the curves have been placed relative to each other by correcting for sample geometry only. Below 50 kbar the resistances are too large to be measured in our apparatus, but they must be decreasing by many orders of magnitude. For measurements made in the ac plane, the rapid drop continues to about 230 to 240 kbar, where there is a relatively sharp break. Beyond this pressure the resistance decreases at a rate which would be expected for a relatively compressible metal. The broken curve in Fig. 37 represents measurements made perpendicular to the ac plane. The curve is qualitatively like the one discussed above, but the break comes a t 160 kbar. In the high-pressure region the resistance perpendicular to the ac plane is apparently 5 to 7 times greater than in the other direction, although corrections for contact resistance could alter this number. Figure 38 shows the measured optical absorption edge as a function of pressure (black triangles). Compared with this is shown twice the ac7 1--

I

6 0

5-

I 0 \

PERPENDICULAR T O OC P L A N E IN

OC

PLANE

b 4-

n

3-

u

0,

2-

I-

I

I

I

I00

200

3 00

4

P. KILOBARS

FIG.37. Log resistance Venus pressure for iodine.

40

H. G. DRICKAMEIt

I

I

I

A 0

1

I

I

Eq O P T I C A L G A P 2 A E RESISTANCE TO OC P L A N E

DATA

I

P, KI LOSARS

FIG.35. Energy gap versus pressure for iodine.

4

3

r.3 [L

\ (L

2

240 KBAR 2

I TO

OC

PLANE

I

/ I

/

I

I00

I

I8O

1

T,OK

I

260

340

FIG.39. Resistance versus temperature for iodine.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

41

tivation energy for electrical conductivity measured in the ac plane (open circles) and perpendicular to the ac plane (black circles). In the pressure region where both optical and electrical measurements could be made, the agreement is excellent, confirming that the simple band picture is a reasonable description for iodine. The activation energy measured perpendicular to the ac plane vanishes at 160 kbar where the break in the resi-atancepressure curve occurs. As one would expect, below 160 kbar the activation energy is independent of direction. Above 160 kbar the activation energy in the ac plane tails off to zero by about 220 kbar. Figure 39 shows a resistance-temperature plot obtained at 240 kbar and measured perpendicular to the ac plane. It shows the linear increase of resistance with temperature which one would expect for a typical metal. Essentially identical curves were obtained from 170 to 400 kbar, indicating that the transition to the metallic state occurs in a very small pressure range. Above 240 kbar measurements in the ac plane also revealed typical metallic behavior. In the region between 1GO and 220 kbar the electronic

\

u."

P W B DATA

108-

106-

R'

-

lo4-

102-

I-

4

0

I

1

I

I

80

I60

240

32(

P. KILOBARS

FIG.40. Relative resistance versus pressure for selenium.

42

H. G . DRICKAMER -

16-

14-

12-

2

610-

W

0.8-

06-

040LI

'

20

40

' $0 ' P, KILOBARS

sb

'

'

IbO

lid

FIG.41. Energy gap versus pressure for selenium.

a

OA

oA 0

A

a 0

0

0

o a

I2

a

a

-10-

G

- 0 - A -

0

T I C I - 27,300CM-' T I Eir -

0 TI I

a

23.950 CM-'

I - 21,840 I

CM-'

I

I

I

HIGH PRESSURE AND ELECTRONIC STRUCTURE 0

I

w6 -

I

I

43

I

I

-

0

8 -

8

-2-

-I

8

I

U

8

2 8

VO 0

t 0

8

PbC12- 32,300 CM-'

A PbBl',-

26,400CM-'

2 n

8 * I

50

I

I

I

I00 P.Y\ILOBARS

i

1 0 1 I50

FIG.43. Shift of absorption edge versus pressure for Vh16 structures.

properties are very highly directional, in a general way analogous to the behavior of single-crystal graphite. ~~*~~ Figure 40 exhibits the resistancepressure curve for ~ e l e n i u m .The resistance drops continuously to 130 kbar and then shows a discontinuous change of 2 to 3 orders of magnitude. It is not yet clear whether this represents a first-order phase change or the discontinuous transition to the metallic state predicted by Mott5* and suggested also for selenium by H ~ m a n Figure . ~ ~ 41 compares the absorption edge with twice the activation energy for conduction. The agreement is again excellent. Above 130 kbar the resistance increases substantially linearly with increasing temperature in typically metallic style.

b. Simple Ionic and Molecular Crystals The conventional picture of a simple ionic compound or crystal is one in which the valence electron (s) have been completcly transferred from the cation to the anion. The valence band should then be made up entirely of anion wave functions while the conduction band should be constructed from cation wave functions. Since the conduction band represents an excited state, one would expect it to be more sensitive to pressure than the valence band. One would then expect the shift of the absorption edge to be relatively insensitive to the anion involved as long as the symmetry remains constant. Figure 42 shows the shift of the edge versus density for three thallous halides60 all of which crystallize in the CsCl (simple cubic) 69

N. F. Mott, Can.J . Phys. 34, Suppl. 12A, 1356 (1956). R. A . Hyman, Proc. Phys. SOC.B69, 743 (1956).

80

J. C. Zahner and H. G. Drickamer, Phys. Chem. Solids 11, 92 (1959)

68

44

H. G. DRICKAMER

-6000

I

0

CI;

-

sn14

- 19.500 CM-I

lF5,5OO:M-'

,

20 40 P.KILOBARS

,\

1 60

FIG.44. Shift of absorption edge versus pressure for molecular iodides.

structure. As can be seen, there is little difference in shift from C1- to Brto I-. A similar conclusion can be obtained from PbC12and PbBr,, as shown in Fig. 43. I n contrast t o ionic crystals, molecular crystals involve noncharged units held together by van der Waals' forces. In this case both valence and conduction bands will involve wave functions for both atomic species present. Figure 4461shows the shift in absorption edge with pressure for three molecular iodides, while Fig. 45 represents the same data for three mercurous halides of the same crystal structure. The shifts depend distinctly on both species present. In a qualitative way these data justify our simple picture of the difference between ionic and molecular crystals.

Silver Halides The silver halides offer a more complex problem in band structure. Since the chloride and bromide behave almost identically,'j2~'j3 results for the chloride and iodide only are shown here. Figure 46 shows the shift of the absorption edge of AgCl as a function of temperature and pressure. There are a number of significant features: ( I ) I n the low-pressure (KaC1) phase the shift of the edge with pressure is considerably smaller than it is in most ionic crystals. This can be interpreted in terms of Seitz'se4 suggestion that the tail on the absorption c.

61 62

63

a

H. L. Suchan a d H. G. Drickamer, Phys. Chem. Solids 11, 111 (1959). T. E. Slykhouse and H. G. Drickamer, Phys. Chem. Solids 7, 207 (1958). A. S . Balchan and H. G. Drickamer, Phys. Chem. Solids 19, 261 (1961). F. Seite, Rev. Mod. Phys. 23, 328 (1951).

45

HIGH PRESSURE AND ELECTRONIC STRUCTURE

20

40 P, KILOBARS

0

60

FIG.45. Shift of absorption edge versus pressure for mercurous halides.

edge of AgCl and AgBr is due t,o an indirect transition such as is shown in Fig. 35c. (2) At about 83 kbar there is a transition presumably to the simple cubic (CsC1) structure accompanied by a large red shift of the edge. (3) The high-pressure phase exhibits a small red shift which is accelerating a t the higher pressures. I

-

I II I

I V

-

25OoC

crn-

I

~,=25,200 CM-'

-

46

H. G . DHICKAMER

(4) A t atmospheric pressure the edge shows a large red shift with increasing temperaturc which is presumably associated with a high concentration of defects in the lattice a t high temperature, probably Frenkel defects. (5) At temperatures above lG5"C, the absorption edge of the lowpressure phase shifts blue with increasing pressure. This blue shift can be associated with the inhibiting effect of high pressure on the formation of Frenkel defects. (6) The absorption edge of the high-pressure phase shifts red with increasing pressure at all temperatures. Apparently in this closed-packed phase at high pressure the formation of Frenkel defects is inhibited even at the highest temperatures reached in this work. Figure 47 shows the absorption edge of AgI as a function of pressure and temperature. At 1 atm and room temperature AgI has the zincblende structure. It transforms to the NaCl structure at 5 kbar with a large red shift of the absorption edge. The face-centered cubic phase behaves quite differently from the similar phases of AgCl and AgBr. The edge shows a large red shift with increasing pressure and a small blue shift with increasing temperature. It seems likely that a direct transition may be involved in this case. At 97 kbar there is a first-order transition accompanied by a distinct blue shift of the edge in contrast to the corresponding transitions in AgCl and AgBr.

!

0 100 OC

A 165'C -2000

0

I\

Uoz 22400 CM-1

25OoC

,l0O0C

a

-8000

0

-

-

50

d

.

-

100

-

I

L

I50

-

P,K I LOBAR S

FIG.47. Shift of absorption edge of AgI a t high pressure and temperature.

HIGH PRESSURE A N D ELECTRONIC STRUCTURE

47

3t 0

80

I60 P,KILOBARS

240

FIG.48. Resistance versus pressure for AgI.

As one would expect from the large energy gap and small shift of the edge with pressure, AgCl and AgBr have negligible conductivity even a t 600 kbar. In contrast to this the conductivity of AgI can be measured throughout the pressure range.57 It can be seen from Fig. 48 that substantial conductivity was found even at low pressures where the optical gap is quite large. The resistance is distinctly nonohmic in this region. It seems probable that this represents ionic conductivity. Ionic condu~tivities6~ for pressed pellets of AgI, the source being Mallinckrodt powder, have been reported in the order of 1 X 10-4 a-1 cm-1 at room temperature. This was of course for the zincblende lattice, but one might still expect appreciable ionic conduction in the fcc phase. The resistance, however, decreases with pressure, indicating an increasing contribution from electronic processes, since the ionic conduction should be hindered by compressing the lattice. Shimizu.66for instance, observed a large decrease in the ionic conductivity of AgCl with hydrostatic pressure; thus one might expect similar behavior in AgI. At 70 kbar, the change of slope, which continues to the transformation, probably indicates that electronic conduction has become predominant. Howa

J. M. Mrguclich, J . Electrochem. SOC.107, 475 (1960). R.N. Shimizu, 12ev. Phys. Chem. Japan 30, 1 (1960).

48

H. G. DRICKAMER

ever, attempts to arrive a t E, through resistance-temperature measurements resulted in values of a few tenths of an electron volt, considerably less than those from the optical work. They might correspond to impurity ionization energies. The optical data show an energy change of 1160 cm-1 (0.135 eV, or 3.1 kcal) across the transition a t 97 kbar. It is interesting to observe that this corresponds closely to what is predicted from the resistance plot a t the transition. In the high-pressure phase both resistivity and optical gap vary quite slowly with pressure. Because of the long tail on the absorption edge in this phase, it is difficult to calculate accurately the true gap for which a = 0. It is estimated at 8000 to 10,000cm-I (1-1.2 eV). This is quite consistent with the electrical activation energy which is approximately 0.5 eV (10-12 kcal) in this region.

d. Silicon, Germanium, and Zincblende Compounds Because of both their theoretical and practical interest, silicon, germanium, and a few 111-V compounds with the zincblende structure have been very thoroughly studied. There is a vast literature and several definitive review papers,54 on the effect of pressure on the band structure as determined by a variety of different measurements. There are a number of additional features revealed by measurements in the higher pressure range which will be reviewed here.

- 3 d

20

do

$0 io P,KILOBARS

I60

li3

,Lo

FIG.49. Shift of absorption edge versus pressure for silicon and germanium.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

c

49

I

3000

FIG.50. Shift of absorption edge versus pressure for GaSb.

Figure 49 shows the shift of the absorption edge of both Si and Ge67 with pressure. Silicon exhibits an essentially linear red shift with a slope of about 2 x10-3 eV/kbar. A t low pressure the Ge absorption edge shifts blue a t an initial rate of about 7.5 x 10-3 eV/kbar. Near 35 kbar the direction of shift reverses and at high pressures the red shift is almost identical with that shown for silicon. Figure 50 shows the corresponding data for GaSb.Os Initially there is a blue shift of 12 x 10-3 eV/kbar. At about 18 kbar there is a distinct change of slope to about 7.3 x 10-3 eV/kbar, corresponding closely to the initial shift for germanium. Around 50 kbar the shift of the edge is changing direction. Apparently, a t high pressure it would shift red like the silicon absorption edge. These results can be explained in terms of Fig. 51, which exhibits the salient features of the band structure of these materials, although it is not identical to that of any of them. It is known from cyclotron resonance experiments that the transition observed at 1 atm in silicon is the indirect

'' T.E. Slykhouse and H. G. Drickamer, Phys. Chern. Solids 7, 210 (1958). '*A. L. Edwards and H. G. Drickamer, Phys. Rev. 122, 1149 (1961).

50

H. G. DRICKAMER

FIG.51. Generalized band structure; Si, Ge, and GaSb.

transition to the band minimum at A, in the 100 direction. In germanium, the initial transition is to the band minimum in the 111 direction. In GaSb the direct transition is initially observed. The 000 minimum shifts to higher energy relative to the valence band maximum at a relatively fast rate. The 111 minimum shifts to higher energy at a somewhat lower rate, while the 100 minimum shifts to lower energy. In GaSb at first the 000 minimum is lowest, but beyond 18 kbar I

I

I

I

I

I

I

FIG.52. Shift of absorption edge versus pressure for ZnS, ZnSe. and ZnTe.

51

HIGH PRESSURE AND ELECTRONIC STRUCTURE IOOOl

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I

I

I

I

I

I

I

I

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-

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do=25.080 CM-'

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.,.

-3000-

-

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-

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60

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ao

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100

I

o.

120

.

'23,550 CM-'

-

--. I

140

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160

I

180

the 111 minimum is lower in energy. At about 50 kbar the 100 minimum becomes lowest. In germanium initially the 111 minimum is lowest, but at about 35 kbar the 100 minimum takes over. Theoretical analysis has not proceeded as far for 11-VI compounds with the zincblende structure. These compounds combine a significant amount of covalent and ionic structure. Figure 52 shows the shift of the absorption edge for ZnS, ZnSe, and ZnTe. All three compounds exhibit an initial blue shift of the edge which persists throughout the available pressure range for ZnS. ZnSe exhibits a red shift beyond 120 kbar, and ZnTe shows the same effect beyond 45 kbar. CuC1, CuBr, and C U P crystallize in the zincblende structure at 1 atm in spite of the largely ionic character 'of the binding. Figure 53 shows the shift of the edge with pressure for CuCl and CuI. The interesting feature is the presence of two first-order phase transitions in CuCl and three in CuI. Apparently there are a number of arrangements differing only slightly in energy. No theoretical work is yet available. Experimental absorption edge shifts with pressure have been published for a number of other 111-1and 11-VI ~ompounds.~~.69 The diamond (or zincblende) lattice is a four-coordinated structure. Each atom is bound to its four nearest neighbors with covalent bonds at tetrahedral angles. The structure is stable because of the high electron 69

A. L. Edwards, T. E. Slykhouse, and H. G. Drickamer, Phys. Chern. Solids 11, 140 (1959).

52

H. G . DRICKAMER

I

Ce

N

I

I

I I I I

\

'.

--__

--1

I

I

I

I

I00

200

300

400

500

P, K I L O B A R S

FIG.54. Resistance versus pressure for silicon and germanium.

concentration possible along the bonds. It is, however, a relatively open structure and one would expect that a t high pressure other more closely packed phases would be more stable. We have seen that transitions occur in CuCl and CuI a t relatively low pressure. Figure 54 shows the resistance of silicon and germanium70 as a function of pressure. Silicon shows a gradual decrease to 190 kbar and then a sharp drop by orders of magnitude. At high pressure the resistance decreases slowly with increasing pressure. Germanium exhibits a slight maximum which can be explained in terms of the shift of the absorption edge and a very sharp decrease a t 115-120 kbar. In both cases7I it has been shown that the resistance in the high-pressure phase increases with increasing temperature in typically metallic fashion. X-ray studies72 on the high-pressure phases of both compounds show the tetrahedral white tin structure, so the transition is quite analogous to the white tin-gray tin transition. Figure 55 shows resistancepressure curves for a number of 111-V compounds. 6 . Minornura and H. G. Drickamer, Phys. Chem. Solids 23, 451 (1963). S. Minomura, G. A. Samara, and H. G. Drickamer, J. Appl. Phys. 33, 3196 (1962). 72 J. C. Jarnieson, Science 139, 762 (1963).

70 71

HIGH PRESSURE AND ELECTRONIC STRUCTURE

53

FIG.55. Resistance versus pressure for 111-V compounds.

The details of the band structure are not available in most cases, but all show a sharp drop in resistance at high pressure. In all cases the highpressure phase exhibits metallic conductivity. X-ray studies7”” on a few of these and related compounds indicate that the high-pressure structure is the diatomic analog of white tin, just as zincblende is the diatomic analog of diamond. Figure 56 contains resistance-pressure data for three 11-VI compounds having the zincblende struct~re.7~ In the zinblende phase the resistances of ZnS and ZnSe were greater than could be measured in our apparatus. A11 three compounds exhibit a very large drop in resistance at high pressure, and in each case the high pressure phase is metallic. Preliminary X-ray J. C. Jamieson, Science 139, 845 (1963). P. L. Smith and J. E. Martin, Nature 196, 762 (1962). 75 A. J. Darnel1 and W. F. Libby, Science 139, 1301 (1963). 76 S. Geller, D. B. McWhan, and G. Hull, Jr., Science 140, 62 (1963). 7 7 M .D. Banus, R. E. Hanneman, A. N. Mariano, E. P. Warekois, H. G. Gatos, and J. A . Kafalaa, Phys. Letters 2, 35 (1963). 78 G. A. Samara and H. G. Drickamer, Phys. Chern. Solids 23, 457 (1963). 73

74

54

H. G . DRICKAMER

c .__---

i

--_

-________

---7

I

5

ZnSe

4

Zn 5 J

I

-I

I

I

i

-31 0

’I

\

I

I

100

200 P.

300

3

KI LOBARS

FIG.56. Resistance versus pressure for ZnS, ZnSe, and ZnTe.

work indicates that the high-pressure phases probably have the sodium chloride structure. All of the above materials exhibit considerable metastability. In fact, InSb which transforms at 300°K at 23 kbar can be quenched in at 78°K and maintained in the high-pressure structure at 1 atm. The high-pressure phases of this and analogous materials have been shown to be super~onducting.7~7’ This metastability means that the transformation pressures obtained with increasing applied force on single-crystal materials may bear little relationship to the equilibrium pressure between phases. X-ray patterns on silicon powder taken as low as 105 kbar have shown definite evidence of the presence of the white tin structure. ZnS powder transforms at or below 200 kbar. has shown that it is Recent work a t the General Electric C0mpany7~-~~ possible to quench in high-pressure phases of silicon and germanium which 79

F. L. Bundy and J. S.Kasper, Science 139, 340 (1963). R. H. Wentorf, Jr., and J. S.Kasper, Science 139, 338 (1963).

55

HIGH PRESSURE A N D ELECTRONIC STRUCTURE

,

I

-4800

-

1

r

I

I

I

I

I I I -

- 5600I

I

P, KILOBARS

FIG.57. Shift of absorption edge versus pressure for CdS.

have rather complex structures neither that of diamond nor of white tin. The range of true stability (if any) of these structures has not yet been established.

e. T h e Wurtzite Xtructure-CdS At 1 atm, CdS crystallizes in the wurtzite structure. At about 23-25 kbar it transforms to a new phase which has been shown to have the fcc (NaC1) structure."' The transition is accompanied by a large red shift of the absorption edge as shown in Fig. 57.c9 The resistance behavior (shown in Figs. 58 and 59)y8is rather interesting. There is a very sharp drop a t the 24-kbar transition. The resistance then rises by several orders of magnitude and starts to level around 300 kbar. At 350 kbar a second rise initiates and there is a distinct maximum a t 4G5 kbar. This behavior is very reproducible and quite independent of any modest doping of the sample. The material remains a semiconductor a t all pressures. f . Olivine Olivine is a crystalline phase of potassium silicate which is present in rocks which have originated relatively far below the surface of the earth. I t has been postulated that much of the mantle of the earth consists of this phase. Figure 6OU2shows the shift of the absorption edge of olivine N. B. Owen, P. L. Smith, J. E. Martin, and A. J. Wright, Phys. Chem. Solids 24, 1519 82

(1963). A . S.Balchan and H. G. Drickamer, J. Appl. Phys. 30, 1446 (1959).

56

H. G . DRICKAMER

50Ll

40

I

200

300

I

400 P, KILOBARS

L 500

--I

_L

600

FIQ.59. Resistance versus pressure for CdS(high-pressure region).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

57

FIG,60. Shift of absorption edge versus pressure and temperature for olivine.

with pressure and temperature. The edge is originally in the near ultraviolet. A rather gross extrapolation indicates that a t 1000°C and 1000 kbar the gap between the conduction band and valence band will have disappeared and metallic conduction will result. Well before this point is reached, the absorption edge will have moved into the near infrared and markedly affect the redistribution of heat by radiation within the earth. 4. ORGANIC CRYSTALS

While organic semiconducting crystals have not been as thoroughly studied as inorganic crystals, they present some very interesting problems. The crystals usually have relatively low symmetry, but the molecules frequently have rather high symmetry. I n particular, the A electrons present on fused-ring aromatic compounds represent systems which are tractable to a surprising amount of theory. I n this section we discuss first some optical and electrical studies on fused-ring aromatic hydrocarbons, including the approach to the metallic state. Then a n irreversible phenomenon involving a novel high-pressure reaction is reviewed. I n this connection graphite is treated as the limiting case of a n aromatic hydrocarbon. Next we consider optical absorption, emission, and decay of two organic phosphors. Finally, a brief discussion of Davydoff splitting is introduced.

58

H. G . DRICKAMER

The structural formulas for the organic crystals discussed are shown in Figs. 6la and 61b.

a. Fused-Ring Aromatic Hydrocarbons A series of :studies has been made on the optical and electrical properties of fused-ring aromatic hydrocarbons including, especially, the three-, four-, five-, and six-ring polyacenes, violanthrene, and c0ronene.8~-~~ In addition to the reversible phenomena expected and obtained, irreversible

/

/

ANTHRACENE

TETRACENE

P E N T A C EN E

HEXACENE

AZULENE

CORONENE VIOLANTHRENE

(a) FIG.Gla. S. Wiederhorn and H. G. Drickamer, Phys. Chem. Solids 9, 330 (1959). G. A. Samara and H. G. Drickamer, J . Chem. Phys. 37, 474 (1962). 85 R. B. Aust, W. H. Bentley, and H. G. Drickamer, J . Chem. Phys. 41, 18.56 (1961). 83 84

HIGH PRESSURE AND ELECTRONIC STRUCTURE

59

behavior of particular interest was observed and is discussed later in this section. (i) Optical absorption. Figures 62-64 show the shift with pressure'-of the components of the first measurable absorption band in anthracene, tetracene, and pentacene. This band is labeled 'L, by Klevens and Platts6 and is assigned to a singlet-singlet transition. The excited states of aromatic

PHOSPHORS

FLUORESCEIN

DICHLOROFLUOROSCEIN

CYANINES

I-

FIG.G l . Structural formulas: (a) fused-ring hydrocarbons, (b) phosphors and cyanincs. 8E

H. B. Klevcns and J. R. Platt, J . Chem. Phys. 17, 470 (1949).

60

FI. G . DRICKAMER

J

30.020

20,0001 0

I

I

40

80 P.KILOBARS

I

I20

1

FIG.62. Shift of low-energy absorption peaks versus pressure for anthracene.

crystals have been widely treated. McCIure’s8freview contains most of the pertinent references. In general, theory predicts 5t red shift going from the free molecule to the crystal as a result of excitation exchange between molecules. In the dipole approximation the shift would be proportional to the inverse cube of the intermolecular distance (i.e., roughly proportional to the density). No compressibility measurements are available for these crystals. I n Fig. 65 the shifts of the low-energy peak are plotted versus a generalized relative density developed by Samara and DrickameF4 for aromatic hydrocarbons. The close agreement between tetracene and pentacene is probably coincidental. The important feature is that the shift is markedly more rapid than linear in the density. The most probable cause is an increased dipole moment for the excited state, although an increased importance of quadrupole interactions cannot be eliminated. Broadening of the peaks and difficulties with the high-pressure optical system for these compounds made accurate determination of area changes impossible.

*’ D. S . McClure, Solid State Phys. 8, 1 (1959).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

61

Figures 66 and 67 show the shift in the low-energy absorption peaks of azulene with pressure. It is very interesting to note that all components of the peak show a n initial blue shift which reverses a t about 60 kbar, and a t high pressure a strong red shift is observed. Azulene has a dipole moment in the ground state, so that a t low pressure the transition to the excited state is apparently accompanied by a decrease in moment. As the pressure increases, the dipole moment of the excited state increases until ultimately it is greater than the ground-state dipole moment. This tends to confirm the result inferred from the tetracene and pentacene data above. The increased dipole moment of the excited state with increasing pressure implies an increasing charge separation which is consistent with the decreased activation energy for electrical conductivity a t higher pressures discussed in the next section.

I

!O

FIG.63. Shift of low-energy absorption peaks versus pressure for tetracene.

62

H. G . DRICKAMER I

10,ooot,

Ib

io

io

4b

510

P. K I L O B A R S

FIG.64. Shift of low-energy absorption peaks versus pressure for pentacene.

(ii) Electrical conductivity. The electrical properties of organic semiconductors have been discussed in many papers and in several revieurs.RS-91 Only those features directly applicable to our results are mentioned here. The resistivity of a solid, as discussed earlier, can be represented by the equation p = l/Npe (4.1 1 where N is the number of carriers, p is the mobility of carriers, and e is the charge. For sufficiently high resistances it is usually assumed that the limiting process is carrier production and that this step is an activated C. G. B. Garrett, in “Semiconductors” (N. B. Hannay, ed.). Reinhold, New York, 1959. 89 H. Inokuchi and H. Akamatu, Solid State Phys. 12, 93 (1961). O0 D. R. Kearns, Advan. Chem. Phys. In press. 91 A. N. Terenin, H. Kallman, and M . Silver, eds., “Symposium on Electrical Conductivity in Organic Solids.” Wiley (Interscience), New York, 1961.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

63

process. One then writes p = poeE/kT

(4.2)

For simple intrinsic semiconductors E is one-half the band gap. The band description, however, is not adequate when the width becomes of the order of a few k T . Furthermore, it is difficult to eliminate impurities as a source of electrons in organic crystals, even for the multiply sublimed material used in this work. Therefore, the experimental results are expressed here in terms of the empirical activation energy E. For a simple metal the controlling factor in the resistance is the mobility; thus for lattice scattering, the resistance is proportional to the temperature. For the purposes of this discussion the term “metallic” is used to describe all systems in which the resistance increases (reversibly) with temperature. The most extensive work was done with pentacene, and these results are described first in some detail, then a few remarks are appended concerning the other substances.

FIG.65. Peak shift versus fractional volume change for polyacenes.

n.

64

G . DRICKAMER

I

17.000

2 15,000

I

0

20

I

I

40 60 P,KILOBARS

I

I

80

100

I

FIG.66. Shift of low-energy absorption peaks versus pressure for azulene.

Pentacene crystals grow with a well-developed 001 face (perpendicular to the c axis). Resistance measurements were taken along the c axis and in the two directions perpendicular to this axis which have been labeled a' and b'. These latter of course do not correspond to simple crystallographic directions in the material. Figure 68 shows typical isotherms parallel to the c axis. At room temperature the resistance drops by a factor of approximately 10'2 in the first 200 kbar (5 to 6 orders of magnitude were observable on the electrical apparatus) and then levels at a value between 10 and 100 Q . This level region extends to the limit of the pressure apparatus-between 500 and 600 kbar. There is a noticeable upward drift of the resistance beyond 250 kbar; when the pressure is increased the resistance drops, but then drifts up with a decreasing rate if the pressure is held constant. The 78'K isotherm drops rapidly for 250 kbar and then decreases much more slowly at higher pressures. Around 240 kbar the 78°K isotherm crosses the 296°K isotherm. At high pressures no upward drift of the resistance is noted. Isotherms obtained in the a' and b' directions did not differ significantly.

HIGH PRESSURE A N D ELECTRONIC STRUCTURE

65

Figure 69 shows isotherms for powder fused by pressure into thin platelets. While the shapes of the curves are qualitatively the same as the single-crystal results, the break at high pressure is significantly less sharp. Figures 70 and 71 show plots of log R versus 1/T for four different pressures on single-crystal pentacene. The portions of the curves for temperatures greater than 180°K will be considered under irreversible effects. The slopes of the curves decrease with increasing pressure and pass through zero, the point at which the sample becomes "metallic." Isobar 3 corresponding to a pressure of 211 kbar, illustrates an anomalous effect (a maximum in the resistance) observed in isobars very close to 220 kbar at temperatures between 140 and 180°K. The effect is more evident in Fig. 71. Possible explanations include the effect of temperature and pressure on the number and efficiency of trapping centers (lattice defects or impurities). The isobars are reversible below 180°K; that is, if a sample is taken to pressure at 78"K, heated to some temperature less than 180"K, cooled to 78°K and reheated, the second isobar duplicates the first.

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100

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FIG. 67. Expanded scale-shift

of low-energy peak versus pressure for azulene.

66

H. G . DRICKAMER

P,K I L O B A R S

FIG.^^^. Resistance versus pressure for pentacene (parallel to the c axis).

I 02t

-

1

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200

300

400

5

P,KILOBARS

FIG.69. Resistance versus pressure for powdered pentacene.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

67

T, OK

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Fro. 71. Resistance versus temperature for pentacene (parallel to the c axis).

68

H. G. DRICKAMER

0

30

I

I

200

II C A X I S

A

II

0

POWDERED S A M P L E

I

300

0'

I

P, KILOBARS

AXIS

I

400

I

500

FIG.72. (d log R / d T ) p versus pressure for pentacene.

Figure 72 is a plot of (dlog R ) / d T versus pressure showing the similarity between the results obtained in two directions in the single crystal, and the marked difference in the fused powder results. Figure 73 shows the calculated values of E, the activation energy. Limited data taken in the b' direction were consistent with results shown for the a' direction. The atmospheric pressure values of the resistivity and activation energy for conduction are 3 X 1013 f2 cm and 0.75 eV, r e s p e c t i ~ e l y . ~ ~ The decrease in the resistance with increasing pressure can be totally explained by the observation that the activation energy for conduction goes to zero. Thus, exp (0.75/kT) gives a factor of 1012, the amount by which the resistance was observed to drop. Initially, as the activation energy E decreases, the increased conductivity is certainly due to an increase in the number of charge carriers. However, when E becomes the same order as the thermal energy, it is difficult to consider it a classical activation energy. In this region band theory would predict a broadening of the bands, due to increased overlap, which corresponds to a higher mobility of the carriers, thus increasing the conductivity. From the standpoint of an activated mobility process the energy 92

D. C. Northrup and 0. Simpson, Proc. Roy. SOC.A234, 124 (1956).

HIGH PRESSURE A N D ELECTRONIC STRUCTURE

69

associated with the mobility decreases with increasing pressure, and the electron may go several lattice distances after being activated before being “trapped” on a pentacene molecule. The activation energy eventually goes to zero (250 kbar for the singlecrystal samples), and pentacene exhibits “metallic” character with a positive temperature coefficient of resistance. A linear extrapolation of the shift of the 670-mp optical peak (Fig. 64) predicts the transition energy going to zero around 200 kbar. If it is assumed that this energy, which corresponds to the ‘La transition, is directly associated with the production of charge carriers, then “metallic” behavior would be predicted above 200 kbar. Considering the nature of the extrapolation, the agreement between the observed and predicted pressures is good. The “metallic” pentacene is not a simple metal but probably should be considered a semimetal. The band structure is certainly complex with large anisotropies. The resistance-temperature plots showed various curvatures (some were linear) which could be explained by a varying number and efficiency of trapping and scattering centers. The trapping and scattering centers could be either lattice defects or impurity molecules.

0

II C A X I S

A

II a‘ A X I S

0

POWDERED SAMPLES

I5

FIG.73. Activation energy versus pressure for pentacene.

70

H. G . DRICKAMER

The data for powdered samples are on a different curve and show the sample becoming metallic at a significantly higher pressure (360 kbar). This illustrates the influence of the increased number of grain boundaries. The trapping or scattering efficiency of these boundaries varies with temperature and most likely masks the “metallic” behavior of the pentacene. Only a very limited supply of hexacene was available, and any substantial purification was impossible. Typical isotherms are shown in Fig. 74. Essentially the same type of reversible effects observed in pentacene were noted for hexacene, and most of the preceding discussion applies. The initial drop in resistance is probably explained by the decrease in the activation energy for condition. “Metallic” behavior was not noted in hexacene, although the apparent activation energy decreased at high pressure. The use of powdered samples and the presence of impurity molecules could mask possible metallic behavior as was described in the case of powdered pentacene. (iii) Irreversible eflects. An irreversible transformation was observed in pentacene. There is evidence of this both from the electrical resistance measurements under pressure and from optical absorption measurements on the transformed sample.

100

200

300

400

P. KILOBARS

FIG.74. Resistance versus pressure for hexacene.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

71

I n the first place, an upward drift of resistance with time was observed a t high pressure and 296°K. This drift was not noticeable at 78°K. The isobars used to measure the activation energy for conduction of the untransformed pentacene provide the most conclusive electrical evidence of the transformation. Between 180 and 200°K there is a sharp increase of the resistance with increasing temperature. At pressures where the pentacene behaves as a semiconductor the curve passes through a minimum and then increases with upward drift. At pressures where the pentacene is metallic there is an abrupt change in the slope of the resistance-temperature curve between 180 and 200°K. The transformed material is a semiconductor within the available pressure range. The temperature dependence of the resistance of the transformed material was obtained by running a 296°K isotherm to some portion of the level region and then cooling the sample to 78°K while maintaining the pressure. The ratio of the resistance at 78°K to the resistance at 296°K remained essentially constant (10 to 15) a t high pressures. It should be noted that a t pressures greater than 270 kbar single-crystal pentacene is "metallic." The irreversible behavior is shown in Fig. 75. A 78°K isotherm is run to 350 kbar and then the sample is heated to 296"K,

,

I

, 1 - _ __ _ HEAT TO 29G°K -_ QUENCH TO 78'K I

I

I

IC

w

I

i 0

a IC

lo'

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300 P,KILOEARS

FIG.75. Resistance versus pressure-irreversible

400

effects in pentacene.

72

n.

G. DRICKAMEH

showing metallic behavior. When cooled to 78'K, the transformed material behaves as a semiconductor. Its heat-up curve gives a linear relationship between log R and 1/T. Upon reheating to 296'K the resistance returns to the value previously obtained at 350 kbar and 296'K, indicating no additional transformation has occurred. The absorption spectra of pentacene and the transformed material were measured a t atmospheric pressure. The transformed material was obtained by recovering the sample from a 296'K isotherm with singlecrystal pentacene as the origirial sample. Figure 76 shows these spectra for equal weight concentrations of the two materials in sodium chloride pellets. The pentacene peaks in the visible region have essentially disappeared in the transformed material while the peak a t 280 mp has remained constant. The transformed material is black with a broad absorption band throughout the visible region. Material transformed to temperatures below 296'K exhibits spectra intermediate between those shown in Fig. 76. The density of the transformed material as established by sinkfloat technique was measurably greater than ordinary pentacene (1.32 as compared with 1.30). This latter value compares well with the results in the 1iteratu1-e.~~ The irreversible transformation may be explained in terms of crosslinking between neighboring pentacene molecules, much like a Diels-Alder product. Photodimerization of anthracene has been studied for many year^.^^,^^ Figure 77 shows the resulting dimer. If this same type of polymerization occurs in the pentacene transformation, then the dimers shown in Fig. 77 would be obtained, assuming the various rings have equal reactivity and noting that the centers of adjacent molecules are essentially opposite each other. It is possible that the cross-linking involves more than two pentacene molecules and becomes a high-order polymerization. It should be noted that the cross-linking disturbs the ?r electron distribution and decreases the portion of the original molecule over which the ?r electrons are mobile. The visible peaks in the absorption spectra of pentacene essentially disappeared in the polymerized material, while the 280-mp peak remained about the same intensity (see Fig. 76). The visible peaks in pentacene are due to the lowest energy transition of the ?r electrons. If the conjugation of the pentacene molecule were disturbed by cross-linking, this spectra should change to that of another conjugated system. I n general the absorption spectra of a mixture of conjugated systems may be obtained by adding the spectra of the components, weighting each inividual spectra R. B. Campbell, J. M. Robertson, and J. Trotter, Actu Cryst. 14, 205 (1961). J. Fritzsche, J. Prukt. Chem. 101, 333 (1867). 95 R. Luther and F. Weigert, 2.Physik. Chem. 61, 297 (1905). 93

73

HIGH PRESSURE AND ELECTRONIC STRUCTURE

FIG.76. Absorption spectra for pentacene and transformed pentacene.

a

b

C

FIG.77. Proposed structures for photodimeriaation (upper) and cross-linking (lower).

74

H. G . DRICKAMER I

I

I

(1:

t

a: m

200

FIG.78. Absorption spectra of violanthrene,ytransformed and untransformed.

by the amount of that component present.96 Thus, referring to Fig. 77, spectra similar to that of benzene, naphthalene, anthracene, and tetracene would be expected from the dimers shown. Tetracene has an absorption band beginning a t 5.50 ml.c and extending into the ultraviolet region. When its spectrum is superimposed on the spectrum of pentacene, the combination gives absorption over the entire visible region. The bulk of the polymerized material should have conjugated portions corresponding to benzene, naphthalene, and anthracene as illustrated in parts b and c of Fig. 77. These configurations do not contribute to absorption in the visible region; however, they have a peak around 280 mp which is characteristic of all aromatic compounds. Thus, the 280-mp peak in the polymerized material would be expected to remain at the same intensity as in pentacene, and this was observed. X-ray powder patterns were taken of both pentacene and the transformed material. The lines found for pentacene, which is triclinic, were found in the transformed material, indicating that the same lattice regularities are present. There may well be differences in the relative intensity of lines, but this is hard to establish definitely from powder patterns on such a complex structure. 86

H. H. Jaffe and M. Orchin, “Theory and Applications of Ultraviolet Spectroscopy.” Wiley, New York, 1962.

75

HIGH PRESSURE A N D ELECTRONIC STRUCTURE

Pentacene crystallizes in a triclinic structure with the long axes of the molecules parallel.93The distance of closest approach is about 3.6 A. In the range of several hundred kilobars the crystal compresses by an estimated 3040%. The distance of closest approach could readily be reduced to 2.7-2.8 A, which is roughly half the dimension obtained for the new lines which appear in transformed graphite which is discussed below. The observed transformation in hexacene is explained in the same manner as that in pentacene. Cross-linking occurs between neighboring hexacene molecules. The same temperature for the initiation of the transformation was found for both compounds, indicating that the transition is the same. High-pressure resistance measurements have also been made on tetracene, coronene, and violanthrene at 296°K. The resistances a t high pressure were too large to permit any work below room temperature. No significant irreversible effects were found in the spectra of tetracene or

I

I

I

I

I

1

lo0

100

200

300

P.

400

500

600

700

KI LOBARS

FIG.79. Resistance of graphite versus pressure measured perpendicular to the c axis (below transition).

H. G . DRICKAMER

PYROLYTIC

I .5

1

I

I

I

coronene, but violanthrene transformed very much like pentacene and hexacene. Spectra of the transformed and untransformed material are shown in Fig. 78. The results are quite analogous to those of pentacene and can be discussed in the same terms. Apparently, it is important to have a polyacene chain of considerable length if transformation is to take place. b. Graphite

A transformation has been observed in single-crystal graphiteS5sg7 which is in many ways analogous to the irreversible phenomenon discussed above for fused-ring aromatic compounds. It has been found in single-crystal materials obtained from a number of sources and purified by a variety of techniques. Evidence is also found for some transition in pyrolytic graphite. The most convincing evidence for the transformation is obtained from electrical resistance measurements. Graphite crystallizes in a planar O7

R. B. Aust and H. G. Drickamer, Science 140, 817 (1963).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

77

hexagonal lattice. The planes have atoms arranged in hexagons very closely similar to those in large aromatic molecules. The r electrons are in orbits above and below each plane as in the organic systems. The planes are spaced along the c axis at the relatively large distance of 6.7 A with an intermediate plane displaced horizontally between them. Pyrolytic graphite is vapor deposited. The layers are apparently rather well formed, but the stacking is relatively erratic. Resistance measurements have been made both parallel to and perpendicular to the c axis. Figure 79 is a typical isotherm measured perpendicular to the c axis at 296°K. The resistance decreases slowly with increasing pressure to about 296°K. At this point there is a very sharp rise in resistance accompanied by drifting upward with time. The total rise was at times by a factor of several hundred. Upon release of pressure the resistance increased as illustrated, so that the transformation is irreversible. The dotted curve represents an isotherm on pyrolytic graphite, where the effect is noticeable but much smaller. Figure 80 shows a 296°K isotherm measured along the c axis. The events are qualitatively similar, but the total rise is by a factor of only 7 to 8. The dotted curve represents pyrolytic graphite. In this direction no significant rise is observed in the latter material. At low pressures (-25

0

SINGLE-CRYSTAL

(1C - A X I S )

GRAPHITE

P-410

KBAR

PARTIALLY TRANSFORMED GRAPHITE

J

FIG.81. Resistance of graphite versus temperature measured perpendicular to the c axis.

78

H. G . DRICKAMER

TABLEIV. NEWX-RAYDIFFRACTION LINESIN TRANSFORMED GRAPHITE~ SINGLE-CRYSTAL

hkl

d

(111) (200) (210) (220) (221) (300) (222) (321)

3.208 2.770 2.467 1.961 1.844 1.600 1.485

Intensity W

m m W

W* W* W

These lines appear also in untransformed graphite, but their relative intensity is markedly higher in the transformed material; a = 5.545 A.

kbar ) single-crystal graphite behaved metallically both parallel to and perpendicular to the c axis, while pyrolytic graphite was semiconducting in both directions. At high pressures (i.e., above 300 kbar) and 296°K both single-crystal and pyrolytic graphite exhibit metallic behavior along the c axis, and semiconducting behavior perpendicular to it. A t 78°K the resistance decreases with increasing pressure to the highest pressures reached (-500 kbar). On heating samples a t 410 kbar the transition initiated a t 1FO to 190°K as can be seen in Fig. 81. When X-ray powder patterns were taken on the transformed material, in addition to graphite lines, seven new, or distinctly more intense, lines appeared as shown in Table IV. These can be indexed in terms of a cubic structure with lattice parameter 5.545 A. The partially transformed graphite has a density of 2.35 to 2.40. Initially this event was interpreted as a first-order phase transition to a cubic structure. With 24 atoms per unit cell a density of 2.803 would be predicted. This would correspond to a conversion of 20 to 25%, which is not unreasonable in view of the intensity of the new lines relative to the graphite pattern. The behavior of the graphite transition is in so many ways similar to the irreversible effects discussed earlier for fused-ring aromatic hydrocarbons that it would seem more probable that graphite is also cross-linking. The pressures a t which the transformations occur are very similar. In neither case does the reaction proceed at 78"K, and in both cases heating to 180-190°K initiates the transformation. Both involve a n irreversible increase in resistance and a t least a partial transition from metal to semiconductor. If cross-linking is occurring in graphite, it is clear why the single-crystal material reacts much more strongly than the pyrolytic graphite, as the planes are much better lined up in the former material.

HIGH PRESSURE A N D ELECTRONIC STRUCTURE

79

It is not entirely clear how to account for the new lines in transformed graphite if cross-linking is occurring. As pointed out earlier, however, it is not unreasonable that the distance of nearest approach of pentacene molecules is about 2.7 to 2.8 A at the pressure where they react and this is about half the 5.545 A regularity found in the graphite. It should be pointed out that Libbyg8has discussed the possibility that very high pressure may greatly enhance the reactivity of organic compounds, and he has predicted, in a general way, results such as were obtained here. c.

Organic Phosphor Decay

Another useful approach to the study of n-electron energy levels involves absorption and emission spectra and decay characteristics or organic phosphors99 dissolved in a glassy medium. Two energy diagrams which have application to many organic phosphors are presented in Fig. 82. The diagram in Fig. 82a was presented by Lewis100JOl for fluorescein, and the diagram in Fig. 82b is a simplified energy versus configurational coordinate diagram of Fig. 82a. It illustrates one additional necessary condition for phosphorescence in many organic phosphors, the crossing or close approach of the energy levels of the S1 and TI states (singlet and triplet states). Transitions between states of the same multiplicity, S, + S , or T, + T,, are spin allowed, but transitions between states of different multiplicity, S + T or T --f S, are spin forbidden. However, there are conditions under which spin-forbidden transitions can take place, but with considerably less probability than spin-allowed transitions. It is this sort of transition which is responsible for phosphorescence in organic compounds. The excitation process involves a So + S1 step followed by a rapid transition to the T I state, where the electron is trapped and released later as phosphorescence. The So + S1transition is spin allowed and appears as a strong peak in the absorption spectra; whereas the transition So--+ T I is spin forbidden and is usually not detected in the absorption spectra. Since the transition S1+ Sotakes place in the order of sec, it is necessary for the S1 + Tl transition to be highly allowed, or all the excited electrons will return to the ground state without being trapped. Since this transition is spin forbidden, a possible explanation for its probability is presented with the aid of Fig. 82. The transition probability between two W. F. Libby, Proc. Natl. Acad. Sci. U.S. 48, 1475 (1962). D. W. Gregg and H. G. Drickamer, J . Chem. Phys. 36, 1780 (1961). loo G . N. Lewis, D . Lipkin, and T. Magel, J . Am. Chem. Soe. 63, 3005 (1941). Io1 G. N. Lewis and M. Kaaha, J . Am. Chem. Soc. 66, 2100 (1944). 88

99

80

H. G. DRICKAMER

electronic states is inversely proportional to the square of the energy difference between them. If the S1 and T I states have an energy crossing or position of close approach as illustrated at y, the transition probability would be high at this point, even between states with different spins. After the electron is trapped in the TI state it can return to the ground state by several paths, two of which emit phosphorescence. The emitting paths are (1) the direct transition from the TI state to the Sostate, called beta emission, and (2) the thermal re-excitation of the electron through 1 state from which it then makes the radiative transition point y to the S to the So state, called alpha emission. The int,ensity of the alpha emission is quite temperature dependent, whereas that of the beta emission is not as much so. At room temperature, depending on the phosphor, one of these processes may be controlling, or they both may take place with nearly equal probability. A spin-forbidden transition is totally forbidden if the spin and orbital momentum of the electron are completely separate. However, under the proper conditions there may be a certain amount of coupling between them,

5

W

w z

I

X (CONFIGURATIONAL

COORDINATE)

( b)

FIG.82. Energy diagrams for organic phosphors.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

81

making the transition partially allowed. The coupling allows a state of spin a to mix with a state of spin b, and the degree of mixing may be found from second-order perturbation theory and is given by44

The mixing coefficient may be abbreviated to K{,l/AE, where K is of the is the coupling coefficient. I n terms of oscillator order of unity and strength one has f a b = fo(K{nl/AE)’ (4.4)

cnl

where f o is the oscillator strength of an allowed transition between states of the same multiplicity. Forster102 shows that the oscillator strength is related to the decay time as follows: 1 / r = Kf, where K is a coefficient containing several terms which are not important for this argument. One then obtains: r = K‘(AE)2

(4.5 1

The above discussion applies mainly to beta decay where the transition is from the Tlstate to the So state. If the model in Fig. 82 is valid and if the alpha, beta, and any monomolecular quenching decay processes are the only means by which the electron trapped in the triplet state returned to the ground state, one would expect a n exponential decay. The deviation from a n exponential decay is most likely due to different molecules having different interactions with the surroundings. This is discussed thoroughly by other who propose representing the decay as a summation of exponentials. The reasons for representing the decay as a summation of exponentials are not important for the interpretation of the pressure data and thus will not be considered here. In this work they are represented as such mainly because it is a convenient method of characterizing them. Fluorescein and dichlorofluorescein have similar structural formulas and similar characteristics under pressure. They are conveniently studied T. Forster, “Fluoreszeny Organischer Verleindungen.” Vandenhaek and Ruprecht, Gottingen, 1951. logA. Baczynski and M. Czajkowski, Bull. Acad. Polon. Sci., Ser. Sci. Chim. 6, 653 (1958).

R. Bauer and M. Baczynski, Bull. Acad. Polon. Sci., Ser. Sci. Chim. 7, 113 (1958). lo6A. Jablonski, Acta Phys. Polon. 16, 471 (1957). A. Jablonski, Bull. Acad. Polon. Sci., Ser. Sci. Chim. 6, 589 (1958). lo’ M. Frackowiak and J. Held, Acta Phys. Polon. 18, 93 (1959). lo8M. Frackowiak and H. Walerys, Acta Phys. Polon. 19, 199 (1960). loD H. Walerys, Bull. Acad. Polon. Sci., Ser. Sci. Chim. 7, 47 (1959). lo4

82

H. G . DRICKAMER

in a matrix of boric acid glass. The results are presented and discussed in three parts: (1) the effect of pressure on the absorption spectra and decay rates of the two compounds, (2) the effect of pressure on the emission spectra of fluorescein, and (3) an interpretation of the results. Figure 83 shows the shifts of the absorption maxima with pressure. These correspond to the So+ S , transition in Fig. 82b. They both exhibit a single peak which shifts red with pressure, dichlorofluorescein shifting slightly more than fluorescein. However, the most important characteristic that they both exhibit is the large red shift of their low-energy edges. These shifts, presented in Fig. 84, are much larger than the shift of their respective peak maxima. They are important because they represent the shift of the lower edge of the S1 state from whence the alpha emission takes place. This large shift of the red edge is not completely understood. It cannot be explained by a simple broadening of the peak, since there is no similar effect observed on its blue edge. It corresponds to a change of shape of the S, state as illustrated in Fig. 82b. Atmospheric decays were measured for both fluorescein and dichlorofluorescein each with concentrations ranging from to 10W gm/gm. In both cases the decay was found to be independent of concentration. A filter with a transmission peak at 22,720 cm-' was used on the exciting I

5 :

0 O\

0

A -IC

I

2

A

PRESSURE R U N

A A AFTER 5 4 KILOBARS

1

\

-

0

\

'9 o \

\o

V

b

I-

k

I v)

Y

4

g -20

9

F L U OR ESC EIN 10-5 GM/GM v 0 = 2 2 , 7 6 0 CM-I

8-

4.2

. \

x

-DICHLOROFLUORESCEIN 1.2 x 10-4 GM/GM u,=22,350 C M - '

0-

\"

\

0

-30 10

20

30

40

50

1

P, KILOBARS

FIG.83. Shift of absorption peaks versus pressure for fluorescein and dichlorofluorescein.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

83

light so as to excite only the first excited singlet state, and a constant shutter speed of 54.6 rpm was maintained for all the decay measurements. It was found that the decay of both fluorescein and dichlorofluorescein could be represented by a summation of two exponentials. The component decay times as a function of pressure are presented in Figs. 85 and 86 for fluorescein. The results for dichlorofluorescein were qualitatively similar, although both fast and slow decay rates were 4 to 5 times faster at each pressure. The effect of pressure on the decay of two concentrations of fluorescein was measured and found to be the same, so pressure effects were measured for only one concentration of dichlorofluorescein. I n all cases both components showed shorter decay times a t the higher pressures. At the same time the fraction of the initial intensity due to the rapid decay increased with pressure, from 30% a t 1 kbar to about 50% a t 54 kbar. The question of how each path of decay, alpha and beta, is being affected by pressure will be discussed below. Measurements indicated that the total initial intensity of emission was substantially independent of pressure for both compounds. The emission spectra of fluorescein at pressures from 0 to 54 kbar is presented in Fig. 87. The dotted portions of the curves in these and similar figures indicate regions where the film sensitivity changes rapidly so that the darkening density could not be established accurately. At atmospheric pressure there are two distinct peaks located at 17,560 and 20,480 cm-I, representing beta and alpha emission respectively. The red peak does not v,:

20 950 CM-I

- 0-DICHLOROFLUORESCEIN

1.2x 10-4 GM/GM

do -20210 cv-'

- I000

0 0

PRESSURE

A A

AFTER 54 KILOBARS

RUN

I

0

10

20

I

30 40 P, KILOBARS

1

50

FIG.84. Shift of low-energy edge of absorption peak versus pressure for fluorescein and dichlorofluorescein.

84

H. G . DRICKAMER

'

Oo0-

CONCENTRATION

PRESSURE

AFTER

RUN

5 4 KBAl

CGM/CM)

,,

- 002

44

0

10

x

10-6

20 30 40 KILOBARS

y,

FIG. 85. Fluorescein in boric acid-slow two concentrations.

A A

0

50

1

component decay time versus pressure for

J

W V

a a Q

K

50

0

i

10

20

30

40

50

60

P, KILOBARS

FIG.86. Fluorescein in boric acid-rapid component decay time versus pressure for two concentrations.

' I

HIGH PRESSURE AND ELECTRONIC,STRUCTURE

85

shift measurably with pressure; however, the blue peak shifts red roughly between 1200 and 1400 cm-l in 54 kbar. Both the location and the shift of this peak corresponds closely with that of the lorn-energy edge of the absorption peak. This evidence helps to justify the assignment of this peak to alpha emission, the S1 + So transition, because one would expect the emission peak to be located near the low-energy edge of the absorption peak for transitions between the same two states.'OOIt would thus shift with this edge. It is also noted that there does not appear to be a large change in the relative intensities of the two peaks. There may be a slight decrease in the relative intensity of the blue peak, but this is hard to verify with certainty.

WAVE NUMBER, CM-'

FIG.87. Emission spectra of fluorescein in boric acid.

The emission spectra of fluorescein in Fig. 87 shows how the relative positions of its Sl and Tl states are changing with pressure. This is important because the rate of beta decay depends on the amount of mixing that the TI state has with S states near it, or thus the amount of singlet character it assumes. This mixing is a function of the energy difference between T1and S states. Since the Sl state is much closer to the T I state than any other S state, it probably contributes eFectively all of the singlet character present in the T1state. As shown in Eq. (4.5) the part of the decay time associated with this process is proportional to the square of the

86

H. G . DRICKAMER

TABLE V. MEASURED AND CALCULATED RELAXIONTIMES ~~

71

(msec)

r 2 (msec)

Pressure

E (cm-1)

Measured

Calculated

Measured

Calculated

1 atm 54 kbar

2920 1600

900 285

-

271

194 70

58.3

-

energy difference between the T1 and S1 states. The emission spectra also show that there is no large change in peak heights between the alpha and beta emission with pressure, indicating that the alpha-emission process is also being enhanced with pressure. This means that the energy crossing, point y , is moving to lower energy along with the S1 states. A rough estimate can be made as to how much the change in the energy difference between the X 1 and T1 states affected the decay rate of the phosphor. Since this energy difference can be related to the beta decay only, the decay will be assumed to be completely of the beta type. From Eq. (4.5) the following relationship would hold:

A E ( P = 0)2 0) A E ( P = 54 kbar)2 54 kbar)

T(P =

T ( P=

(4.6)

Using the AE measured from the emission spectra and the atmospheric rn a t 54 kbar can be calculated and compared with the measured values. As is seen in Table V, if the decay was all of the beta type, the change in the energy difference between the S1 and T1 states with pressure would more than account for the decrease in decay times. However, there is a large portion of alpha decay, probably about 30 to 40% (estimated from emission spectra), so the change in the decay time cannot be entirely described in terms of beta decay. T,,, the

d. Davydof Splitting in Cyanine Spectra

The spectra of many crystalline organic molecules frequently show more structure than the spectra of similar molecules in solution. This phenomenon was first described by Davydoff,"" whose explanation has since been generalized. McClures7 has published a comprehensive review. Briefly, the explanation is that when a molecule is placed in a lattice site, the transitions allowed depend on the site symmetry, not on the molecular symmetry. The number of peaks depends on the site symmetry, but the magnitude of the splitting is established by the degree of interaction between neighboring molecules. In hydrocarbons the splitting is diKcult to observe 110

A . R. Davydoff, Zh. Eksperim. i Teor. Fiz. 18, 210 (1948).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

t -

87

’

v, C M - ’x I O - ~ FIG.88. Typical crystalline cyanine spectra for cyanine A (n = 1); and 54 kbar.

0 A

uo = 202 10 CM-I

0

U o = 17600

I

/

O

uo= 19100

A

I

/

I

CENTER OF G R A V I T Y

/

/

I

/

I

A’

/’

/

/

O

’

/

/ /

-400 I

0

10

20

30

40

50

P, K B A R

FIG.89. Davydoff splitting and shift of center of area versus pressure for cyanine A (n = I).

88

H. G . DRICKAMER

/

o

A

/

A/

/

A

d

/ /

/

,

-8001 0

10

20

30 P I KBAR

40

5.0

FIG.90. Davydoff splitting and shift of center of area versus pressure for cyanine B (n = 0).

with the slits necessary in the high-pressure apparatus. The cyanine dyes (see Fig. 61b) offer the opportunity to observe this phenomenon. Spectra have been taken on cyanines dissolved in cellulose acetate (effectively, solution spectra) and as crystallites.11l Figures 88-90 show typical spectra and the shifts of the component peaks for two dyes. There are two features to be noted. First, there is an increase of splitting with increasing pressure; the higher-energy peaks shift blue, the lower-energy peaks red, with increasing pressure. Second, there is a redistribution of intensity among the peaks; the higher-energy peaks lose intensity while the lower-energy peaks gain. As a result, the center of gravity of the total peak shifts red with pressure at a rate substantially equal to that observed for the ‘‘solution” spectra. The difference is probably due to differences in compressibility. The increased splitting is caused by the increased molecular interaction with decreasing interatomic distance. The redistribution of intensity can be explained in terms of the Boltzmann factor for transition probability; G. A. Samara, B. M. Riggleman, and H. G. Drickamer, J . Chem. Phys. 37, 1482 (1962).

HIGH PRESSURE AND ELECTHONIC STRUCTURE

89

while it is difficult to establish areas under the various peaks accurately as a function of pressure, the fractional changes are of the magnitude one would calculate from the change in splitting. 111. Metals

Pressure measurements above 50 kbar on the properties of metals date back to the pioneering work of Bridgman on pressure-volume measurements112 and electrical resistance.113 Bridgman’s results and other early ~ data have been reviewed and interpreted at length by L a ~ s 0 n . l ’More recently, Kennedy and his c ~ - w o r k e r s ~have ~ ~ -made ~ ~ ~ extensive and precise phase equilibria studies to 70 kbar over a range of temperatures. Kaufmanlls has made a thorough investigation of melting and solid-solid transitions in iron and its alloys, and Bundy and Strongllg have made similar studies. Jura and his colleagues120J21 have made p-v measurements on ytterbium, strontium, and dysprosium; Jayaraman et uZ.122J23have made X-ray and related studies on certain rare earths to 60 kbar; and Jamie~on72J3J2~ has made extensive X-ray measurements on metals having first-order phase transitions below about 150 kbar. In this discussion we review measurements of electrical resistance, studies of lattice parameters by X-ray diffraction, and the Mossbauer effect (energy of recoilless radiation) to several hundred kilobars at 300°K or below. Probably the most common measurements made on metals at high pressure is the electrical resistance. The theory of the effect of pressure on the resistivity of metals has been reviewed by Lawsonll4and by Pad.125 P. W. Bridgman, Proc. Am. Acad. Arts Sci. 74, 425 (1942). W. Bridgman, Proc. Am. Acad. Arts Sci. 81, 165 (1952). 114 A. W. Lawson, Progr. Metal Phys. 6, 1 (1956). G. C. Kennedy, A. Jayaraman, and R. C. Newton, Phys. Rev. 126, 1363 (1962). ‘16 W. Klement, Jr., A. Jayaraman, and G. C. Kennedy, Phys. Rev. 129, 1971; 131, 1, 112

ua P.

632 (1963). ‘lrA. Jayaraman, W. Klement, Jr., and G. C. Kennedy, Phys. Rev. 130, 540, 2277; 131, 644 (1963); Phys. Rev. Letters 10, 387 (1963); Phys. Chem. Solids 24, 7 (1963). L. Kaufman, in “Solids Under Pressure” (W. Paul and D. M. Warschauer, eds.). McGraw-Hill, New York, 1963. ‘19 F. P. Bundy and H. M. Strong, Solid State Phys. 13, 81 (1962). uoP. C. Souers and G. Jura, Science 140, 481 (1963). P. C. Souers and G. Jura, Science 146, 575 (1964). D.B. McWhan and A. Jayaraman, Phys. Letters 3, 129 (1963). ua A. Jayaraman, Private communication. la, J. C. Jamieson, Science 146, 572 (1964). W. Paul, in “High Pressure Physics and Chemistry” (R. S. Bradley, ed.), Vol. 1. Academic Press, New York, 1963.

90

H. G. DRICKAMER

Basically, the theory predicts that the resistance should decrease as the pressure increases because the Debye temperature increases and therefore the lattice scattering decreases. While resistance measurements are relatively straightforward to make, in principle at least, there are grave difficulties in interpreting the results in terms of theory. In the first place, it is not practical to insert more than two leads in the very high pressure electrical resistance apparatus. Thus it is not possible to calculate resistivities accurately. An even more serious problem is the fact that the resistance is quite sensitive to a wide variety of factors in addition to the interatomic distance. Impurities, the concentration and type of dislocation, grain boundaries, and lattice strains can all affect the measurements. Since the conditions in the high-pressure cell are not truly hydrostatic, it is clear that a quantitative comparison with theory is out of the question. Nevertheless, electrical resistance measurements at high pressure can reveal some very useful and interesting information. One can detect ordinary first-order phase transitions between crystalline phases, which are normally accompanied by a discontinuity in resistance. These, in fact, comprise the most widely used calibration points in high-pressure work. One can also detect melting in this manner. Furthermore, one can obtain leads as to the possibility of a more interesting type of transition, known as an “electronic transition.’’ In such a transition an electron is promoted from a partially filled shell to an empty shell, changing the electronic structure of the atom, but not necessarily the crystal structure. In terms of band theory an empty band of higher energy than the conduction band is lowered in energy until conduction electrons are scattered into it, and it becomes the conduction band. In principle, what occurs is not greatly different than the situation we have discussed for germanium or GaSb, except that there the conduction band is an excited state which normally contains no electrons, whereas in metals the conduction band contains electrons in their ground state and therefore such equilibrium properties as the atomic volume are affected. An electronic transition was first proposed126-‘28 to explain the large drop in resistance in cerium at 5 kbar, accompanied by a discontinuous change in volume but no change in structure. 111 this case the promotion of the 4f electron to a band arising from the 5d shell was postulated. SternheimerI29 proposed that a cusp in the resistance of cesium a t 41 kbar accompanied by a volume discontinuity be explained in terms of proA. w. Lawson and T. Y . Tang, Phys. Rev. 76, 301 (1949). I. Lihkter, N. Riabinin, and L. F. Vereschaguin, Soviet Phys. J E T P (English Tranel.) 6, 469 (1958). lZ8R. Herman and C. A. Swenson, J . Chem. Phys. 29, 398 (1958). R.Sternheimer, Phys. Rev. 78, 238 (1950). lz7

HIGH PRESSURE A N D ELECTRONIC STRUCTURE

91

motion of a 6s electron to the empty 5d band. His analysis is undoubtedly not quantitative, as he assumed a single, spherically symmetrical 5d band; but the idea is fruitful. Later in this section we discuss resistance data on alkali, alkaline-earth, and some rare-earth metals, and in a number of cases electronic transitions seem quite possible. X-ray diffraction measurements are helpful in the interpretation of many high-pressure measurements, as the prime variable is interatomic distance rather than pressure. Many noiicubic crystals compress in an anisotropic manner. The results of these measurements used in conjunction with electrical resistance or other studies can give an interesting picture of the relative position and movement of the Brillouin zone walls and the Fermi surface. Some examples are discussed below. The Mossbauer effect has proven to be a very sensitive tool in solid state research. One can measure the local magnetic field in ferromagnets, and the s-electron density at the nucleus in a number of materials. The final section of this review deals with some recent studies.

5. ELECTRONIC AND METAL-NONMETAL TRANSITIONS a. Alkali Metals

Bridgman’s work on the alkali metals showed that none of them exhibited the modest decrease in resistance expected from simple theory. The cusp he found in the resistance of cesium near 42 kbar has been the basis for much of the speculation concerning electronic transitions. His work as well as the theoretical studies of BardeenI3O and Frank131are reviewed by Lawson. More recently, Ham132has made extensive calculations on the alkali metals. It is difficult to apply these theories in any quantitative way to data at very high pressure, in part because of the existence of first-order phase transitions to phases of undetermined structure. The individual elements are discussed below.I33 (i) Lithium. Figure 91 shows the two resistance-pressure isotherms for lithium. The open circles are terminal points of isobars. A t 296°K the resistance rises to a maximum value at 70 kbar and then drops abruptly. Beyond this drop the resistance exhibits a minimum. At 77°K the drop in resistance was found to be smeared out and subsequent data were taken after first pressing to 100 kbar at 296°K and then cooling. As observed in the diagram, the resistance rises slowly but continuously at higher pressures. J. Bardeen, J . Chem. Phys. 6 , 367 (1938). N. H. Frank, Phys. Rev. 47, 282 (1935). F. S. Ham, Phys. Rev. 128, 2524 (1962). 138 R. A. Stager and H. G. Drickamer, Phys. Rev. 132, 124 (1963). 130

lal

92

H. G. DRICKAMER

---

-

77 OK

-mooo--

0

I

I

I

FIG.91. Resistance versus pressure for lithium.

Since at atmospheric pressure and room temperature lithium has a bcc structure, a transformation to a closer packed structure such as fcc or hcp is the most likely explanation for the discontinuity in resistance. The smearing out of the transition at 77°K suggests a first-order, diffusioncontrolled transformation. The slow rise in resistance with pressure at high pressures may be due to narrowing of the conduction band, as has been suggested by several authors. I

01

0

I

I

I

I

I

I

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100

200

300

400

P, KILOBARS

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5M)

FIG.92. Resistance versus pressure for sodium.

i

I

1 600

HIGH PRESSURE AND ELECTRONIC STRUCTURE

93

(ii) Sodium. A t 296°K a minimum in resistance is observed at 40 kbar, after which there is a continuous rise ending in a very broad shallow maximum a t about 360 kbar (Fig. 92). The 77°K curve, starting after the 50-kbar room temperature minimum, shows a similar, but less pronounced, rise, which never reaches a maximum even at 600 kbar. While it is not surprising that the rise in resistance with pressure a t 296°K becomes less and less with increasing pressure, the particular shape of the curve may indicate sufficient increase in the Debye temperature, e,, to bring sodium into the region of T / b = 0.15. Below 0.15 the resistance is less sensitive to changes in e,, and so should be less sensitive to pressure at higher pressures.

FIG.93. Resistance versus pressure for potassium.

(iii) Potassium. The resistance of potassium as a function of pressure is shown in Fig. 93 for isotherms obtained a t 296 and 77°K. The main feature of the 296°K isotherm is the very large continuous rise of resistance with pressure. The increase is by a factor of about 50 in 500 kbar and contrasts markedly with the modest rises in sodium and lithium. Probably some form of interband scattering is taking place here,

94

H. G. DRICKAMER

as there is no evidence from Ham’s calculations that there could be sufficient band narrowing to give this result. The 77°K isotherm has two unusual features in addition to the large rise exhibited by the 296°K isotherm. At about 280 kbar there is a distinct discontinuity in slope of the resistance-pressure curve. The size of the discontinuity varied from run to run, as would be expected from a sluggish phase transition. A t 320 kbar a series of isobars were obtained by alternately heating and cooling between 77°K and room temperature until the same terminal values were obtained for successive cycles. The new phase is metallic and is apparently stable at room temperature when established in this fashion. It is not clear why the transition does not occur during a 296°K isotherm. At 360 kbar and 77°K a second transition took place, with a very sharp increase in resistance in contrast to the one discussed above. The highpressure phase also showed a large increase in resistance with increasing pressure. An isotherm obtained at 197°K was very similar to that at 77°K. It is believed that this second transition can be explained with the aid of the isobars shown in Figs. 94 and 95.

FIQ.94. Resistance versus temperature for potassium (500 kbar).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

95

From Fig. 94 it is seen that the resistance drops with increasing temperature (points 1 to 2 ) to about 230"K, then increases to 270°K (2 to 3). The sharp drop a t 270°K (3 to 4) is the reverse transition. On cooling (4 to 5) the material remains metastably in the lower resistance phase, but transforms back immediately when pressure is applied. Evidently the slight shear accompanying pressure application is sufficient to initiate the transition. From Fig. 94, one could conclude that the high-pressure phase is a semimetal with an energy gap a t low temperatures and overlapping bands a t high temperatures. Figure 95, however, shows a cycle in which the heating is interrupted a t 160°K by recooling (2 to 3 ) to 77°K. The resistance-temperature curve (1 to 2) is not reversible. On reheating, the material returns to its former state a t point 2 (now state 5 ) . The cycle then continues. The 360-kbar transition is very likely martensitic on the basis of the following observations : (a) There is a temperature above which the transition does not run with pressure, which is between 197" and 296°K. (b) The transition is sharp a t temperatures a t which a diffusioncontrolled first-order transition is usually very metastable.

34-

"'"9,

2 7LpL--60 100

150TOK 200

250

FIG.95. Resistance versus temperature for potassium (490 kbar).

H . G. DRICKAMER I

I

I

FIG.9G. Resistance versus pressure for rubidium.

(c) Martensitic transitions have been found in lithium and sodium a t atmospheric pressure, but not in potassium (Barrett134).The behavior is qualitatively similar for these transitions. (d) Upon heating up, the reverse transition occurs at about 270"K, depending slightly on the pressure. This would be the Md (martensitic critical) temperature. The irreversible nature of the initial resistance drop in Figs. 94 and 95 could indicate that this drop is due to the removal of strain in the sample. The subsequent rise would then indicate that the high pressure phase is metallic. (iv) Rubidium. As seen in Fig. 9G,6J33 the resistance of rubidium rises with pressure a t the lowest pressures obtainable in this equipment. There is a distinct discontinuity in slope near 70 to 75 kbar, which is undoubtedly the transition (probably bcc to fcc) observed by B ~ n d y . Above ' ~ ~ thk point the resistance rises with increasing slope. Near 1CO kbar there is an abrupt rise accompanied by much drifting upward with time. At higher 134 1%

C. S. Barrett, J . Znst. Metals 84, 43 (1955). F. P. Bundy, Phys. Rev. 116, 274 (1959).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

r

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I

I

97

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0

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200

1

300

I

400

P. KI LOBARS

FIG.97. Resistance versus pressure for cesium.

pressures a downward drift initiates, and there is a broad maximum near 425 kbar. The higher-pressure features are similar but sharper at 77°K. The abrupt rise is at 210 kbar and the maximum at 510 kbar. Typical isotherms and terminal points of isobars are shown in Fig. 96. The sharp rise at 190 kbar and 296°K could be melting, judging by the extension of Bundy's melting curve which showed a negative slope at high pressures. In view of the fact that it occurs at only slightly higher pressure at 77°K' it would seem more likely that it is an electronic transition. (v) Cesium. Cesium was by far the most difficult of the alkali metals to handle, and it was hard to obtain reproducible pressures due to flowing of the sample when pressure was first a~plied.'~6 All of the electrical resistance features occurred on every run, but the 41-kbar cusp appeared at apparent pressures from 20 to 70 kbar. Figure 97 shows a composite average of twelve successful runs. The 22-kbar transition is smeared out by problems of making contact, etc. The cusp at la

R. A. Stager and H. G. Drickamer, Phys. Rev. Letters 12, 19 (1964).

98

H. G . DRICKAMER

41 kbar is of the same magnitude as found by Bridgman. The important new feature is the very sharp rise in resistance initiating at an apparent pressure of 175 kbar. The rise is accompanied by a strong upward drift with time. There is a definite maximum at a higher resistance than the first cusp. It should be emphasized that, while the pressures shown are nominal because of problems in handling cesium, the sharp rise and drifting occurred on every run. (There was no drift with time except in this region.) The maximum always occurred at the same pressure relative to the minimum, and always at a resistance higher than that of the first cusp. All of the features occurred on release of pressure as well as on application of pressure. A number of runs were also made at 77°K. The features occurred at about the same pressures, and were similar in character, except that the rise initiating at 175 kbar was somewhat more sluggish. To summarize the alkali-metal data: lithium and sodium show relatively small effects of pressure on resistance, potassium a very large rise in resistance with no maximum up to 600 kbar, rubidium a sharp rise at 190 kbar and a maximum beyond 400 kbar, and cesium a cusp at 41 kbar and a second maximum near 200 kbar. In a qualitative way these results are consistent with s-d interband scattering as no d bands are available for lithium and sodium, and one would expect the energy separation to be , least for 6s-5d. (The d bands are, of largest for 4s-3d, less for 5 . ~ 4and course, split in the crystal with different degrees of degeneracy at different symmetry points, but the separations should go qualitatively in this order. ) On the other hand, the increase in resistance in potassium and rubidium is by a factor of 50 or more. (Any correction for contact resistance would tend to increase the factor.) This seems large for interband scattering. Also, to explain the second maximum in cesium in terms of interband scattering it is necessary to assume scattering into successive bands created by the splitting of the 5d levels. It would appear that complete theoretical reanalysis of the problem is desirable.

b. Alkaline-Earth Metals

Resistance-pressure measurements have been made on four alkalineearth metals.137Since magnesium is discussed separately in Section 6a, this section contains information on calcium, strontium, and barium only. Since these atoms involve filled shells only, in the simplest model they would be expected to be insulators. As they are, in fact, metals, there must be holes in the highest “filled” zone and electron overlap into the next higher Brillouin zone. Indeed, as discussed in Section Ba, this has been shown to be true for magnesium. One could then imagine a modification of the struc187

R. A. Stager and H. G . Drickamer, Phys. Rev. 131, 2524 (1963).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

99

ture which would make a semimetal or semiconductor from these elements. Such modifications do exist for calcium and strontium. (i) Calcium. Earlier measurements of resistance versus pressure for calcium by BridgmanlI3 to 65 kbar showed a continuous rise in resistance to the highest pressure he obtained. Our measurements (Fig. 98) indicate a rather complex and interesting behavior which is difficult to resolve quantitatively because of the sluggishness of the transitions involved. Below about 70 kbar it is difficult to separate the effects of changing contact from the events characteristic of calcium. Thus, we could not identify the transition noted by Bridgman1I2a t 60 kbar. Above this pressure, at 296"K, there is a very small rise in resistance with pressure to about 140 kbar. At this point the resistance increases much more rapidly and drifts upward with time as is characteristic of a sluggish first-order phase transition. The drift dies out in a few moments but reinitiates when a further increment of pressure is applied. By 300 kbar the total rise above 140 kbar is about a factor of 5. At about 300 kbar a new phenomenon appears. On application of pressure the resistance rises initially but then starts to drift downward. At higher pressures the downward drift is accelerated, until there is a considerable net decrease in resistance with pressure. The 77°K isotherm had a qualitatively similar behavior except that the first rise was more sluggish initially, but greater eventually. At 390 kbar the resistance was some 20 times the 140-kbar resistance. Above 390 kbar, the resistance would, on some runs, drop precipitously, while on other runs it would drop off slowly.

FIQ.98. Resistance versu8 pressure for calcium.

100

H. G. DRICKAMER

I n order to establish the temperature coefficient of resistance accurately as a function of pressure, a series of isobars was obtained a t 100, 200, 255, 360, 390, and 430 kbar. At each pressure the cell was cooled to 77"K, heated to 296"K, recooled, and so on. The cycle was continued until the terminal values did not change from cycle to cycle. The isotherms shown in Fig. 98 were then constructed, combining isotherms and isobars. The points shown are from isobars. The results can be summarized as follows. At low pressures calcium has the fcc structure (phase A ) . At 140 kbar a sluggish transition to phase B initiates. At 300 kbar, before the A-B transition is complete, a second transition starts ( B - 6 ) and runs, also somewhat sluggishly. There exists also the possibility of the low-pressure transition mentioned by Bridgman.ll2 The calcium atom contains only completely filled atomic orbitals. The solid is metallic because there is an overlap between a filled shell and a neighboring empty shell. From the isobars it is evident that phase B is semiconducting (Rv/REw = 1.33 a t 390 kbar). The true energy gap may, of course, be larger as the calcium was not zone refined. The third phase, C, is clearly metallic. (ii) Strontium. Bridgman1I3observed a sharp maximum in the resistance of strontium a t about 40 kbar. This maximum was also observed in our studies. Just beyond the maximum there is a sharp drop in resistance ac1

FIG.99. Resistance versus pressure for strontium.

1

101

HIGH PRESSURE AND ELECTRONIC STRUCTURE

companied by a drift downward with time, which is typical of a first-order phase transition. There is a maximum also in the 77°K isotherms, which appears to be slightly displaced to higher pressures, although our equipment is not well adapted to studies in this pressure range. Results are shown in Fig. 99. As far as could be determined from our isobaric measurements, there was substantially zero temperature coefficient of resistance near the maximum. In this pressure range in our apparatus there are small pressure drifts which could have concealed a small gap a t the maximum. Such a gap was actually observed with ytterbium, as discussed later. After the phase transition, strontium is definitely metallic. At 296°K there is a minimum a t 100 kbar and a broad maximum at about 300 kbar. The 77°K isotherm has a slight minimum a t about 320 kbar. Like all the alkaline-earth elements, strontium has filled atomic shells. The solid is thus metallic only because of overlap between bands. Apparently the overlap decreases with pressure until, near the maximum, strontium is a semimetal, if not a semiconductor. Recently Jayaraman et u Z . , " ~ worked out a phase diagram for strontium using equipment more adaptable to this pressure range, and they found definite proof of a first-order phase transition at the resistance maximum. McWhan and Jayaraman'22 have shown that the high-pressure phase is bcc. (iii) Barium. Figure 100 shows the pressureresistance characteristics of barium a t 296, 197, and 77°K. After a low-pressure minimum the resistance rises linearly. There is a sharp rise a t 58.5 kbar, a second linear region, and another sharp rise at 144 kbar. A definite maximum follows. I

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0

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FIG.100. Resistance versus pressure for barium.

I

102

H. G. DRICKAMER

FIG.101. Resistance versus temperature for barium (440 kbar).

A t 197°K a similar curve is obtained, except that the second rise occurs at 190 kbar. To avoid dragging out the 58.5-kbar transition, for low-temperature isotherms, cooling was done after this transition. At 77"K, beyond 100 kbar a gradual rise in resistance with pressure is observed. At 240 kbar a marked break in the curve is observed, after which the rise is much steeper. In striking contrast to &hetwo higher-temperature

0

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200

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300 400 P, K I LOBARS

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500

FIG.102. Phase diagram for barium.

I

600

1

103

HIGH PRESSURE A N D ELECTRONIC STRUCTURE

isotherms, the high-pressure maximum in resistance appears abruptly, after which an essentially flat region is observed. The resistance breaks downward at 380 kbar to show a minimum at 540 kbar and then an accelerating rise in resistance. Figure 101 is a typical isobaric run at 440 kbar. The rise from 77 to 160°K is characteristic of a solid metal. The sharp rise at 160°K is quite analogous to the transition observed at 144 kbar and 296°K. The hightemperature phase shows a small positive temperature coefficient of resistance. From a series of isotherms and isobars the phase diagram shown in Fig. 102 has been constructed. In our original paper we speculated that the phase at high temperature and pressures above 140 kbar was possibly a liquid, since this was consistent with the maximum in the melting curve at 1040°K and 15 kbar found by Kennedy et al.,Il7 and with the relatively small temperature coefficient of resistance of this phase. Recent X-ray experiments in this laboratory show, however, that the high-pressure phase is crystalline, probably fcc. c. Rare-Earth Metals

The rare-earth metals present very interesting possibilities for unusual electronic structure, as the bands arising from the 4f and 5d shells must lie very close or overlap in some cases. While electrical resistance data are insufficient to resolve these problems and sufficient other data are avail-

a 0

ISOBARS ISOTHERMS

I

300 400 P,K ILOBAR S

I

I

500

600

FIG.103. Resistance versus pressure for cerium.

1

104

H. G. DItICKAMEK

able only in very few cases, it is hoped that these results138together with other information which may be forthcoming will lead to a satisfactory theoretical attack. (i) Cerium. Figure 103 shows two isotherms and terminal points of two isobars. The electronic transition at 5-7 kbar discussed in detail else~h e r e ' ~~- loccurs *7 below our effective range. There is a cusp a t 00-05 kbar a t 290°K which occurs at 85-95 kbar a t 77°K. At about 1GO kbar and 296°K a distinct drop in resistance with some drifting with time occurs. This behavior usually typifies a first-order transition. This transition occurs a t much higher pressure (-365 kbar) on the 77" isotherm. However, isobars taken a t 220 kbar by cooling from room temperature and reheating indicate that the low-temperature phase a t this point is the high-pressure high-temperature phase, so that the dotted curve probably more nearly represents equilibrium conditions.

m---+

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FIG.104. Resistance versus pressure for praeseodynium.

(ii) Praseodynium. Figure 104 represents the isotherms for praseodynium. The 77°K isotherm was located by terminal points of a series of isobars. All the features shown are from the twelve 77°K isotherms obtained, but varying contact resistance gave varying placements of the curves so that this represents an average isotherm. The maximum in resistance a t 40 kbar and 296°K was also found by Bridgman. At 77°K it apparently occurs above 100 kbar, but an 80-kbar isotherm indicates that, a t that point, the maximum should already have occurred (dotted curve), 138

R. A. Stager and H. G. Drickamer, Phys. Rev. 133, 830 (1964).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

105

1.0-

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5 00

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FIG.105. Resistance versus pressure for neodynium.

so that there is apparently metastability at low temperatures. There is a small but definite hump in the curve at 100 kbar-296"K, 150 kbar197"K, 190 kbar-77"K, probably representkg a change in electronic structure. There is a very broad maximum at 340-360 kbar and 296°K. At 197°K the maximum is sharper, and at 77°K it is very sharp. (iii) Neodynium. Three isotherms for neodynium are shown in Fig. 105. At 65 kbar and 29G'K either a point of inflection or a maximum was obtained, depending on the degree that contact had been stabilized. There

0

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ISOTHERMS ISOBARS 7 7 ° K ISOTHERM BASED 011 ISOBARS

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FIG.106. Resistance versus pressure for samarium.

106

H. G . DRICKAMER

is a sharp minimum at 120 kbar and a broad maximum beyond 200 kbar. At 197°K the first maximum is a t 100 to 110 kbar and the minimum a t 135 to 140 kbar. There is a broad maximum above 300 kbar. At 77°K the sharp maximum occurs at 1'70 to 175 kbar, the minimum is not observable, and the high-pressure maximum occurs as a change of slope above 350 kbar. (iv) Samarium. Samarium data are shown in Fig. 106. For the 296°K isotherm there is an inflection in the slope a t about 50 kbar, a distinct drop in resistance a t 160 to 170 kbar, and a broad maximum near 400 kbar. A 77°K isotherm is shown, as well as a dotted curve constructed from a series of isobars. The most important feature is the distinct rise in resistance above 200 kbar. The very small temperature coefficient of resistance especially above 300 kbar is a point worth noting. I

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ISOTHERMS ISOBARS

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700

FIG.107. Resistance versus pressure for europium.

(v) Europium. The isotherms for europium are shown in Fig. 107. At 296°K the resistance rises with pressure to about 150-160 kbar where there is a very sharp rise typical of a first-order transition. There is a small but distinct maximum a t about 175 to I80 kbar, beyond which the resistance falls very slowly with increasing pressure. The 197°K isotherm is quite similar except that there is a distinct rise in resistance a t the highest pressures. The 77°K isotherm has an entirely different appearance, and one which reproduced itself very precisely on nine different loadings. There is a small maximum a t 175 to 180 kbar, a sharp maximum at 210

107

HIGH PRESSURE AND ELECTRONIC STRUCTURE

ISOEARS

P. KI LOBARS

FIG.108. Resistance versus pressure for terbium.

kbar, a minimum a t 310 kbar, and a broad maximum near 500 kbar. A very large number of isobars were run. Apparently, over most of the pressure range the phases along an isobar differ a t 77°K and 296°K. I

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108

H. G. DltICKAMElZ

(vi) Terbium. Figure 108 shows the terbium isotherms. The 296°K isotherm shows a drop with increasing pressure, a small but distinct minimum near 150 kbar, and a broad maximum near 220 to 230 kbar. The 77°K isotherm has a minimum a t 60 to 70 kbar, a distinct maximum near 220 to 230 kbar, and a shallow minimum just above 300 kbar. Although the isotherms differ considerably in shape, no distinct evidence of a firstorder transition was found, either from the isotherms or isobars. (vii) Gadolinium. Figure 109 shows isotherms for gadolinium. At 296°K there is the possibility of an inflection a t low pressure, masked in our apparatus. There is a distinct change in slope above 200 kbar. The 77°K isotherm does not differ radically.

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600

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FIG.110. Resistance versus pressure for dysprosium.

(viii) Dysprosium. In Fig. 110 are shown three isotherms for dysprosium. At all three temperatures there is a distinct inflection a t 60 to 80 kbar. The 296°K isotherm has no other prominent features except an increased slope a t very high pressure. At 197°K there is a shallow minimum above 200 kbar and a broad maximum below 500 kbar. The drop-off a t high pressure is very noticeable here. At 77°K there is a distinct maximum at 200 to 210 kbar, a minimum near 350 kbar, and a broad maximum near 500 kbar. These features, which sharpen at low temperature, would seem to be associated with electronic transitions. (ix) Holmium. Isotherms for holmium are shown in Fig. 111. They are similar in general characteristics, with an inflection a t 60 to 80 kbar and

109

HIGH P R E S S U R E AND E L E C T R O N I C S T R U C T U R E

FIG.111. Resistance versus pressure for holmium.

o

0

I00

200

ISOTHERMS ISOBARS

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I

1

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I

300

400

500

600

700

P,KILOBARS

FIG.112. Resistance versus pressure for erbium.

110

H. G . DRICKAMER

21

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200

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300 400 P,KI LO B A RS

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FIG.113. Resistance versus pressure for thulium.

R' =RESISTANCE AT 77'K AND 100 KB

0

40

80

120 160 P , K I LOBARS

200

240

FIQ.114. Resistance versus pressure for ytterbium.

280

HIGH PRESSURE A N D ELECTRONIC STRUCTURE

111

296°K which occurs at increasing pressure at lower temperature, but no other important features. (x) Erbium. The curves for erbium are shown in Fig. 112. The characteristics are much like gadolinium, with the possibility of an inflection at low pressure and no other distinctive features. (xi) Thulium. The thulium isotherms are shown in Fig. 113. The 296°K isotherm shows a minimum near 60 to 80 kbar and a maximum a t 150 to 160 kbar. The 77°K isotherm has a minimum a t 190 to 200 kbar. (xii) Ytterbium. Figure 114 shows resistance versus pressure for ytterb i ~ mat' ~77~and 296°K. The most striking features are the sharp maximum in resistance at about 40 kbar and 296°K and the even larger maximum at the same pressure at 77°K. From about 20 kbar to just beyond the resistance maximum ytterbium is a semiconductor. More detailed results have been obtained by Hall and Merrill140 and by Souers and JuralZ0in apparatus better adapted to the low-pressure range. The maximum is accompanied by a first-order phase transition to a bcc structure. Since ytterbium has only closed shells, this transition would appear to be analogous to those in strontium and calcium discussed in the previous section.

d. Similarities in Electronic Behavior among Alkaline-Earth and Rare-Earth Metals

A really satisfactory analysis of the electronic structure of the alkalineearth and rare-earth metals cannot be made based on available measurements and theory. There are a number of similarities in electronic structure among these elements which will be helpful in ultimately unraveling the details of their structure. Certain of these analogies have previously been noted by Jayaraman et al.,l17 by Souers and Jura,120and by J a ~ a r a m a n . ' ~ ~ There is a distinct resemblance between the high-pressure behavior of calcium and of strontium. Each has a region in which it exhibits semiconducting or semimetallic behavior, at around 40 bar for strontium and at much higher pressure for calcium. The band structures must be rather similar, and mixing of bands from s, p , and d shells may be involved. The similarity between ytterbium and strontium is even more striking. The outer electronic structure of the atoms is, of course similar, the same crystalline phases are involved, and the band structures must be very much alike. AS noted above, this similarity has been discussed by a number of authors. Again, europium and barium show similar phase d i a g r a m ~ ~and ~~J~~ both have a maximum in the melting curve. The transition at 140 kbar and 296°K in barium is very simiiar to the 150-kbar transition in europium, with a sharp rise in resistance, a distinct maximum, and a very flat high-

''*R. A. Stager and H. G . Drickamer, Science 139, 1284 (1963). 140

H.T.Hall and L. Merrill, Inorg. Chem. 2, 618 (1963).

112

H.

G.

DRICKAMER

pressure curve. The half-filled 4.f shell in europium is a stable configuration and permits behavior much like that of a filled-shell element. With the present state of theoretical and experimental knowledge, it would not be useful to speculate a t length on the detailed interpretation of these similarities, but they may well provide the basis for a sound theoretical analysis of electronic structure in these metals. 6. THE ELECTRONIC STRUCTURE OF HEXAGONAL CLOSE-PACKED METALS X-ray diffraction measurements on hexagonal close-packed metals as a function of pressure permit the determination of the changes in both the c and a axes and therefore the change in both the size and shape of the Brillouin zone as the interatomic distance is changed. Especially when combined with electrical resistance measurements, significant information about the electronic structure can be obtained. To date, studies have been made on the elements magnesium, cadmium, and zinc. Before discussing the specific results, some general information concerning hcp structures will be reviewed.

L

FIG.115. Brillouin zone for hcp structure.

A hexagonal close-packed arrangement of rigid spheres would have an ideal c/a ratio of 1.633. Figure 115 shows the Brillouin zone for the hcp lattice with some of the important symmetry points indicated. Such a zone does not have energy discontinuties across all its zone boundaries, and the “Jones zone” shown in Fig. 116 is frequently used in discussion of electronic structure. Harrison141has shown that a reasonable approximation to the Fermi surface of a number of real metals can be obtained from the free-electron approach wherein the surface is represented by a sphere centered a t each 141

W. A. Harrison, Phys. Rev. 118, 1182, 1190 (1960).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

113

atom. When this is mapped into the zone of Fig. 115 for a n element with two valence electrons per atom, four important features are observed: (1) There is a complex-shaped hole in the second zone, known as the “monster.” It has twelve tentacles stretching to the points H and is connected by tubes passing through the points 2. The tentacles come down and inward to the ring so that there are electrons a t and near K in the second zone. (2) There is a pocket of electrons in the third zone in the form of an oblate spheroid known as the “pillow,” centered a t r. (3) There are long, relatively narrow, cigar-shaped pockets of electrons along the zone boundary of the third zone near the points K. (4) I n the third and fourth zones there are rather complex electron pockets near L known as stars.

FIG.116. Jones zone for hcp structure.

A simple compression of the lattice holding the axial ratio c / a constant should change the Brillouin zone and Fermi surface proportionally and would not affect these features. An increase in c/a a t constant volume Would result in a relative compression of the 002 axis a t the Brillouin zone and an increase in the pockets at r and L and a decrease in the pocket at K and possibly the holes a t H . A decrease in c/a would have the inverse effect. I n addition, there are, of course, important effects due to the lattice Potential. The size of the energy gaps can affect markedly the electron

114

H. G . DRICKAMER

distribution and therefore the size and shape of electron pockets and the holes. As will be pointed out below, the energy gaps and their change with pressure are significant in the interpretation of the experimental data for the individual compounds. J0nes~~ZJ~3 has developed a theory of the interaction between a pocket of electrons overlapping a Brillouin zone boundary with that boundary. He showed that the stress exerted by the electrons tends to inhibit the expansion of the corresponding Brillouin zone wall. As discussed in the section on magnesium, the details of Jones' picture for that substance are not correct; however, the principle seems sound. Go~denoughl~~ has expanded the theory to show that there is an attractive force exerted by the Fermi surface approaching a Brillouin zone boundary before it actually intersects. He further showed that a repulsive force is generated by the increase in energy, due to distortion, of an intersecting Fermi surface. When overlap takes place this repulsion is partially relieved. At sufficiently large overlap all interaction of Fermi surface and Brillouin zone boundary is nulli6ed. These theories have been largely applied to explain the existence and axial ratios of the hcp phases of alloys of the noble metals and to discuss the change of axial ratio of magnesium alloys with changing electron to atom ratio.

a. Magnesium F a l i ~ o v has ' ~ ~ made a detailed orthogonalized plane-wave (OPW ) calculation of band structure of magnesium incorporating available de Haasvan Alphen and magnetoacoustic measurements into his description of the Fermi surface. It is qualitatively similar to the free-electron picture, but the pockets at L are considerably smaller, the pocket at r is a little smaller, the tentacles-at H are smaller, while the pocket at K is larger. In Jones' original discussion of magnesium he assumed no overlap at r. However, the work of Smith and his colleag~es,146-1~9as well as the calculations of Falicov, and indeed the free-electron picture, show that this is incorrect. Electrical resistance measurements have been made on magnesium to 500 kbar137 and X-ray diffraction results have been obtained to 300 H. Jones, Proc. Roy. Soc. A147, 396 (1934). H. Jones, Phil. Mag. 171 41, 633 (1950). lU J. B. Goodenough, Phys. Rev. 89, 282 (1953). '16 L. M. Falicov, Phil. Trans. Roy. Soc. London A266, 55 (1962). 146 J. R. Reits and C. S. Smith, Phys. Rev. 104, 1253 (1956). 117 T. R. Long and C. S. Smith, Acfu Met. 6, 200 (1957). 148 R. E. Smunk and C. S. Smith, Phys. Chem. Solids 9, 100 (1959). 1*9 S.Eros and C. S. Smith, Acfu Met. 9, 14 (1961). 143

HIGH PRESSURE AND ELECTRONIC STRUCTURE

0

X-RAY

DATA

A BRIDCMAN DATA X SHOCK WAVE DATA

0.7-

I

0

I

I00 200 PI K ILOBARS

I

300

FIG.117. V/Vo versus pressure for magnesium.

115

116

H. G. DRICKAMER

FIG.119. (c/a)/(e/a)oand resistance versus V / V Ofor magnesium.

kbar.150Figure 117 shows V / V Oversus pressure. There is a small but measurable irregularity at about 140-150 kbar. The data of Bridgmanlsl to 100 kbar and the high-pressure shock-wave data162corrected to 23°C are also shown. In view of the large temperature correction to the shock data, the agreement is reasonable. Figure 118 shows c/co and a/ao as a function of V / V o .In Figure 119 ( c / a ) / ( ~ / ais) plotted ~ versus V / V o .As V / V odecreases at first the a axis is slightly more compressible than the c axis and c/a decreases with increasing pressure (decreasing V/Vo). In the region V / V o= 0.83-0.80 the c axis shows a marked decrease in compressibility and the a axis a compensating increase. The slight irregularity in V / V o versus pressure would indicate that the compensation is not perfect. There is thus a sharp rise R. L. Clendenen and H. G. Drickamer, Phys. Rev. 136, 1643 (1964).;also unpublished data of R. L. Clendenen. 161 P. W. Bridgman, Proc. Am. Acad. Arts Sei.76, 55 (1948). 162 M. H. Rice, R. W. McQueen, and J. M. Walsh, Solid State Phys. 6 , 1 (1958).

160

HIGH PRESSURE AND ELECTRONIC STHUCTURE

117

TABLEVI. CHANGEOF OVERLAPA N D HOLESWITH V/Va FOR MAGNESIUM CALCULATED FOR FREEELECTRONS”

v/v, = 1.0

(Falicov)

0.91

0.805

0.76

0.72

r pocket r+A r -+M

0.0852 0.342

(0.058) (0.255)

0.0811 0.338

0.9005 0.364

0.0939 0.374

0.1051 0.399

K +M K +H K +r

0.032 0.203 0.061

K pocket (0.044) 0.0361 (0.277) 0.229 (0.073) 0.0676

0.0367 0.229 0.0652

0.0346 0.231 0.0650

0.0332 0.229 0.0626

L+A L +H L +M

0.051 0.251 0.085

(0.050) (0.043) (0.050)

0.054 0.260 0.093

0.055 0.274 0.095

0.057 0.281 0.099

0.154 0.343

0.140 0.350

0.137 0.345

Contact of holes a t H (0.044) 0.112 0.114 (0.012) 0.094 0.101

0.119 0.105

0.123 0.107

1.6285

1.651

L pocket 0.053 0.260 0.090

Holes in second zone Thickness of “monster” r+M 0.135 r +K 0.321

5

H +K H +L

0.118 0.097

C b

1.523

(0.037) (0.201)

0.144 0.334

1.5995

1.6245

All dimensions are in atomic units.

in c/a in this region, to a value well above the initial 1.623. At higher densities the c axis again becomes significantly more compressible than the a axis, and c / a decreases with increasing density. Current work on a series of dilute magnesium alloys indicates that their behavior is qualitatively very similar to pure magnesium. Both figures show also the change of relative resistance with V/V,. In the low density region the resistance decreases with increasing density as one would expect. There is a minimum near V / V o = 0.82, and in the region to V / V o= 0.74, resistance increases with density. Beyond this point resistance drops normally to the highest pressures obtainable in the electrical apparatus. Preliminary measurements on a number of alloys153 indicate that the behavior is quite similar to pure magnesium over the density range V / V o = 1.0-0.70. 153

R. L. Clendenen, Private communication.

118

H. G . DRICKAMER

I

I

I

I

I

1

FIG.120. Resistance versus pressure for cadmium.

It seems clear that the rise in resistance and the distinct change in the relative compressibility of the c and a axes are associated and are the result of a modification of the electronic structure. While a complete interpretation is not yet feasible, some remarks can be made concerning the possibilities. It would seem possible to explain the changes in c/a in terms of changes in the stresses discussed by Jones and Goodenough, due to changes in size and shape of the electron pockets and holes. I n Table VIlS4are shown the critical dimensions of the pockets and holes calculated from the free electron approximations at V/Vo = 1.0, 0.91, 0.805, 0.76, and 0.72 for the measured values of c / a a t these densities. Falicov’s values for these parameters at V / V , = 1.00 are shown in parentheses. While there are changes in the dimensions with density they seem far too small to account for the relatively large changes in c/a. If the energy gaps were held constant presumably Falicov’s values for the parameters would change proportionally. It seems clear that a rigid band model is insufficient to explain the results and one must msume that the energy gaps change significantly with pressure. Cohen and Heine156have discussed the change in gap with alloying. I n the density region 0.824.76 the resistance increases with density. This can most logically be explained by assuming that one or more of the pockets in the third zone is collapsing into the holes in the second zone, E. A. Perez-Albuerne, Private communication. R. W. Lynch and H. G. Drickamer, Phys. Chem. Solids 26, 63 (1965).

119

HIGH PHESSUIlE AND ELECTRONIC STRUCTURE

I-

4

-

2 0.95I

I

I

I

reducing the amount of available Fermi surface and therefore the conductivity. This, of course, could only occur if there is significant change in the energy gaps, and confirms our discussion above. b. Cadmium and Zinc

Cadmium and zinc also crystallize in the hcp structure, but the c axis is considerably extended so that the c/a values (at 25°C and 1 atm) are 1.886 and 1.856, respectively. k Figures 120 and 121 show resistance versus pressure for cadmium and zinc.*56For cadmium the resistance drops in a “normal” manner to about 150 kbar, where a minimum occurs. There is then a gradual rise to a broad maximum above 200 kbar. A t higher pressures the resistance drops monotonically. The behavior of zinc is very similar except that the minimum is at a somewhat lower pressure (110-120 kbar). As discussed in the last section magnesium, which has the hcp structure but has a nearly ideal c/a, behaves very analogously. X-ray measurements have been made on cadmium to over 300 kbar which give rather accurate measurements of lattice parameters in this range. Zinc gives very poor diffraction patterns using the molybdenum radiation which is necessary in our apparatus. Data have been obtained, but over a more limited pressure range and with less accuracy than for cadmium. 156

R. W. Lynch and H. G. Drickamer, Phys. Cheriz. Solids 26, 63 (1965).

120

H. G . DRICKAMER

Figure 122 shows V/VO versus pressure for cadmium. The pressures were calculated from the shifts of lines for an MgO marker mixed with the sample. There is a distinct irregularity in the region of 90-130 kbar ( V / V o = 0.88435). Shock-wave densitiesIs2 are also indicated in the figure. At values of V/VO below 0.85 there is a measurable disagreement between shock-wave densities and our measurements, which is rather unexpected as for many other substances the agreement is excellent. Figure 123 shows c, a, and relative resistance as a function of V / V o for cadmium. Figure 124 exhibits c/a and relative resistance for the same substance. The a axis is at first very incompressible, but at about V/Vo = 0.96 its compressibility increases markedly; near V/Vo = 0394.88 it becomes more incompressible, while at higher densities the compressibility again increases. The behavior of the c axis tends to mirror that of the a axis, with an initial high compressibility which decreases with increasing density and then shows a marked increase near V/V0 = 0.894.88. This compensation is only partial as there is a distinct kink in the P versus

0.90-

VPJO

0.85-

-

0.80

L

0

X I

I

I

100

200

300

P , K I LOBARS

FIG.122. V / V , versus pressure for cadmium.

HIGH PHESSURE AND ELECTRONIC STRUCTURE

121

V / V Ocurve near I’/VO = 0.88-0.85. The axial ratio c / a (Fig. 124) shows a large drop in the region V / V o= 1.04.90, then it levels near V / V o= 0.90 and drops rather abruptly beyond V / V o = 0.88. Figures 126 and 126 show the results for zinc. The results are qualitatively very similar to those for cadmium in the same range of V/Vo.The a axis is at first quite incompressible (though more compressible than the a axis of cadmium). Its compressibility is larger in the region V / V o = 0.96-0.90. At higher densities it becomes smaller again. The c axis is at first relatively compressible, becomes quite incompressible in the region V / V O= 0.94-0.91, and then increases in compressibility at high density. The ratio c/a decreases from 1.856 at 1 atm to about 1.80 at V / V o = 0.95. It exhibits a small maximum near V / V , = 0.91-0.92, and then decreases to the highest pressures obtainable. Since it was impractical to use a marker with zinc, shock-wave densities were used plotting resistance as a function of V/VO.

FIG.123. Lattice parameter c and a and resistance versus V / V ofor cadmium.

122

H. G. DRICKAMER

FIG.124. Axial ratio c/a and resistance versus V/Vo for cadmium.

Table VII shows the sizes and shapes of the electron pockets for cadmium at various values of V / V oand the measured c/a, using free-electron theory. The most obvious effect of the greatly increased axial ratio is the disappearance of the pockets a t K a t 1 atm. There are measurable changes in the sizes of some of the other features. Harri~on’*~J~7 has made OPW calculations for zinc and cadmium. Gibbons and FalicovIss have used detailed magnetoacoustic and de Haasvan Alphen data to construct Fermi surfaces and to discuss electronic properties of cadmium and zinc. While there are a number of quantitative differences, the free-electron picture is qualitatively correct for zinc. For cadmium the most striking feature is that the “monster” is pinched off at the points 2 , so that it is not continuous throughout the zone. This is not predicted from freeelectron theory. 167 158

W. A. Harrison, Phys. Rev. 126,497 (1962). D. F. Gibbons and L. M. Fdicov, Phil. Mag. 181 86, 177 (1963).

HIGH PRESSURE AND ELECTRONIC STRUCTURE

123

It is desirable to consider to what extent the results can be described from a rigid-band model with all gaps independent of pressure. As mentioned in the section on magnesium, for this model the changes in size of the pockets for the Fermi surfaces calculated by Falicov and Gibbons, or by Harrison, should be roughly proportional to the changes calculated for the free-electron picture. The results for cadmium will be discussed first. As can be seen from Table VII the most important change with density involves the pockets at K which do not exist at 1 atm. Small pockets appear at V/VO = 0.95 which may or may not be present at V / V o = 0.90, but a t higher densities they increase in size rapidly. At 1 atm the Fermi surface must be very near but not touching the Brillouin zone boundary near the point K . As the density increases it must contact the boundary with little or no overlap but with a resulting distortion of the Fermi surface. At higher densities there is a n increasing degree of overlap. One might explain the initial very low compressibility of the a axis from the attractive interaction of a Fermi surface and a Brillouin zone wall

5;1\ .0

FIG.125. Lattice parameter

v/ v, c

1

I

0 90

0 85

and a and resistance versus V / V Ofor zinc.

124

H. G. DRICKAMER

it approaches without contact, as shown by Goodenough. As the Fermi surface intersects the Brillouin zone wall without overlap, it is distorted, raising the energy and decreasing the attraction. Hence the increased compressibility of the a axis in the region V / V , = 0.95-0.88. The reduced compressibility in the region V / V o = 0.88-0.84 could result from the “partial pressure” of electrons in the small pockets formed at these densities in the third zone (again using Goodenough’s theory). Finally when the overlap gets large enough, the interaction with the Brillouin zone boundary is negligible and the compressibility again increases. There are several objections to this explanation. The Jones-Goodenough theory applies to changes in c and a at constant volume not with changing volume. The overlap considered is with the faces of the Brillouin zone, overlap a t corners (like K ) gives only higher order terms. As seen from the tables, the overlap of the faces at I? is large and does not change radically with pressure. As discussed below, the ultimate explanation probably involves changes in the energy gaps with pressure. I

1.15

I

1.10

R

1.05

I2 I.oo

C/O

1.8

1.7

I

0.95

V/V,

0.90

0.85

FIG.126. Axial ratio c/a and resistance versus V / V Ofor zinc.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

125

TABLEVII. DIMENSIONS OF ELECTRON POCKETS A N D HOLESIN CADMIUM MODEL” BASEDON FREE-ELECTRON

v/vo = 1.00 ~

~

0.95

0.90

0.86

0.82

0.79

0.1252 0.4234

0.1159 0.4124

0.1111 0.4076

~~

r pocket r+A r+M

0.1489 0.4453

K +M K+H

-0.004

-

K-rr

-0.0068 ~~~~

0.1338 0.4285

+0.015 f0.159

0.1251 0.4197

K pocket -0.004 -

+0.015

$0.0277

+0.0154 +0.1544

$0.0227 $0.1865

+0.0300

f0.043

S0.0256 +0.2022 +0.0495

~

L pocket L-tA L+H L+M

0.040 0.214 0.069

0.407 0.233 0.078

0.0446 0.2455 0.0831

0.0458 0.2503 0.0849

0.0496 0.2620 0.0897

0.0517 0.2686 0.0944

0.1197

0.1283 0.3343

0.1344

Holes in the second zone Thickness of “monster”

r

a

-r M r+K

0.1026 0.3020

H-rK

H+L

0.2958 0.1579

c/a

1.886

0.1069 0.3097

0.1171 0.3188

0.3242

Contact of holes a t H 0.1197 0.2039 0.3210

0.3421

0.1399

0.1309

0.1312

0.1283 0.1230

0.1344 0.1189

1.806

1.759

1.752

1.708

1.686

All dimensions are in atomic units.

The behavior of zinc is qualitatively similar to that for cadmium, and a similar explanation can be offered. Table VIII shows the calculated dimensions of the electron pockets and holes a t various densities from freeelectron theory using the experimental values of c/a. The higher initial compressibility of the a axis may be associated with the presence of small electron pockets at K at 1 atm. The details of interpretation should not be pressed too far for zinc because of the limited precision of the data. There does not seem to be any phenomena directly associable with the pinching off of the “monster” a t Z in cadmium. I n both cadmium and zinc there is a minimum in resistance followed by a maximum at higher pressure. In both cases the minimum occurs at a

126

H. G . ,DRICKAMER

TOLE VIII. DIMENSIONS OF ELECTRON POCKETS A N D HOLESIN ZINC BASED ON FREE-ELECTRON MODELO vp-0 =

r --,A r + nr

1.00

0.95

0.92

0.87

0.1552 0.4919

0.1465 0.4842

0.0066 0.105G 0,0130

0.0137 0.1543 0.0269

0.0454 0.2a3 0.0873

0.0497 0.2764 0.0933

0.1231 0.3505

0.1310 0.3613

Contact of holes a t H 0.2301 0.2378 0.1561 0.1622 1 .SO7 1.818

0.2089 0.1532 1.774

r pocket 0.1503 0.4848

0.1603 0.4010

Ii pocket K K K

+ AT --+

H

j r

0.0007 0.0330 0.0002

0.008% 0.1175 0.0102

L pocket L +A L +H L + 'u

0.0412 0.2472 0.0800

0.0461 0.2629 0.0873

Holes in second zone Thickness of "monster" r -+M O.lli1 r +K 0.3399

H ---t K H -+L c/a a

0.3030 0.1686 1.856

0.1205 0.3451

All dimensions are in atomic units.

density distinctly higher than that a t which the hump in the c axis compressibility and in c/a occurs. This feature is also common to magnesium, as discussed in the previous section. As indicated there, the increase ill resistance can most logically be associated with the absorption of an electron pocket from the third or fourth zone into the holes in the second zone, reducing the number of free electrons and the available Fermi surface. This could only take place as the result of a relatively large change in energy gap. Since this process must compete with thc lowering of resistance due to stiffening of the lattice with increasing pressure, it may well initiate at the hump in the c and c/a curves, and become significant enough to cause an actual increase in resistance only a t higher density. This reinforces our argument that the unusual features in the curves of c and c / a versus T.'/Vo must be associated with distinct changes in the energy gaps.

HIGH PRESSURE AND ELECTRONIC STRUCTURE

127

7. THE ELECTRONIC STRUCTURE OF IRON AT HIGHPRESSURE The electronic structure of the transition metals, and particularly of iron, occupies an enormous literature which it would be impossible to review here. Problems concerning the phase diagram of iron as a function of temperature and pressure have been reviewed by K a ~ f m a n . At ’ ~ ~room temperature and atmospheric pressure iron has the bcc structure and is, of course, ferromagnetic. At high temperature, above the Curie point, it transforms to the fcc structure. For this transition d p / d T is negative. Shock-wave investigationslGO revealed a transition in iron at about 50°C and 130 kbar. This was confirmed by static electrical resistance measurements.lG1 Lawson and Jamieson1G2 first found definite indications that the high-pressure phase is hcp. Later work in this laboratory163confirmed their

‘“ 0.95-

0.90-

v/vo hcp ’

0.85-

“\

PHASE r \ O

X X

0.80

SHOCK W A V E DATA

100

200

300

400

F: KILOBARS FIG.127. V / V oversus pressure for iron.

analysis. Figure 127 shows the volume change as a function of pressure. An interesting feature is the strong tendency for metastability in the bcc phase, Although the transition pressure is 130 kbar, on one occasion traces of the phase persisted to 300 kbar, and it was usually present in measurable amounts to 200 kbar. Figure 128 shows the c/a ratio for the hcp phase as a function of pressure. The compressibility is decidedly anisotropic with the c axis some 2.6 times as compressible as the a axis. This point may be significant in the interpretation of the Mossbauer data. L. Kaufman, in “Solids Under Pressure” (W. Paul and D. M. Warschauer, eds.). McGraw-Hill, New York, 1963. D. Bancroft, E. L. Peterson, and S. Minshall, J . A p p l . Pkys. 27, 291 (1956). 181 A . S. Balchan and H. G . Drickamer, Rev. Sci. Instr. 32, 308 (1961). lB2 A. W. Lawson and J. C. Jamieson, J . A p p l . Pkys. 33, 776 (1962). It. L. Clendenen and H. G. Drickamer, Pkys. Ckem. Solids 26, 865 (1964). 160

128

H . G . DRICKAMER

I 551

I

100

0

200 300 P. KILOEARS

400

FIG.128. Axial ratio c/a versus pressure for hexagonal close-packed iron.

095-

.. .. . . . *

-.

0901-

I

:

:. ..... :. . . . . . . . . . .. .. - * . . *

.

. *

..

*.

*

FIG.129. Mossbauer spectrum for iron a t low pressure. H. Frauenfelder, “The Mossbauer Effect.” Benjamin, New York, 1962. R. L. Mossbauer, Ann. Rev. Nucl. Sci. 12, 123 (1962). ls6 A. F. J. Boyle and H. E. Hall, Rept. Progr. Phys. 26, 441 (1962). 167 R. S. Preston, S. S. Hanner, and J . Heberle, Phys. Rev. 128, 2207 (1962). 16s D. N. Pipkorn, C. K. Edge, P. Debrunner, G. De Pasquali, H. G. Drickamer, and H. Frauenfelder, Phys. Rev. 136, 1604 (1964). 164

1c5

HIGH PRESSURE AND ELECTRONIC STRUCTURE

I .. 0.95-

.. -. .

*C

..

*

..

.,.*

129

... .,. ..- ... '. .. .. ..... *

*

*

.-

..

0.90-

0.85I

FIG.130. Mossbauer spectrum for iron at the transition region.

The six lines arise from the splitting of the ground state and excited state by the local magnetic field, and the degree of splitting is a measure of the field strength. The location of the center of gravity measures the s-electron density a t the iiucleus relative to that of the absorber (the isomer shift). Figure 130 shows a spectrum a t about 145 kbar. In addition to the six-line VELOCI TY, CM/SEC

TOWARD

AWAY -010

I

FIG. 131. Mossbauer spectrum of the high-pressure phase for iron.

130

H. G . DRICKAMER

t 0.015

t 0.OlOC

8

I

I

I

I

1

I

-0.01 5

spectrum of the bcc phase, a seventh line near the center is appearing. This was first observed in the work of Nicol and J ~ r a . ' 6In ~ Fig. 131 is shown the spectrum of the high-pressure phase. There is only a single line. (Measurements have been extended over a considerable velocity range without finding any other structure. ) The high-pressure phase is apparently not ferromagnetic. If it is antiferromagnetic, the splitting is too small to be observed in our experiment. It is also quite possible that 300°K is above the Nee1 temperature. There is no observable quadrupole splitting in the high-pressure peak. Figure 132 shows the isomer shift as a function of pressure. Figure 133 shows the same data plotted against relative volume change. The center of gravity of the spectrum is determined by the second-order Doppler shift which is proportional to the mean-square velocity of the atoms and by the isomer shift which is proportional to the s-electron density at the nuclei. It can easily be shown that the eflect of pressure on the second-order Doppler shift is small.170 The main effect of pressure must be a change in the isomer shift as a result of a n increase in the electron density at the nucleus. An estimate of the contributions of the 4s and inner s electrons can be made from the lES 170

M. Nicol and G. Jura, Science 141, 1035 (1963). D. N. Pipkorn, Ph.D. thesis, University of Illinois, Urtmna, Illinois, 1964.

HIGH PRESSURE AND ELECTRONIC STItUCTUItE

131

$0015

1

hcp phase

-0.0I 5

-0.0201 0

0

I

I

0 04

0 08

ao%6%gD 0 12

-A”/Vo

FIG.133. Isomer shift versus -AV/Vo for iron.

work of Walker et aZ.171 Using free-ion Hartree-Fock wave functions and the FermiSegre-Goudsmit formula, they derive a plot of s-electron density at the nucleus versus number of 4s electrons per atom for various numbers of 3d electrons. They arrive at a calibration for the isomer shift in terms of electron density by associating the calculated electron density for the configuration 3d6 and 3d5with the measured shifts for the most ionic divalent and trivalent compounds respectively. If we assume that the only effect of changing the volume is to change the scale but not the shape of the 4s wave function, the volume dependence of the isomer shift can be found from the line 3d74s2.The shift produced by one 4s electron occupying a volume V ois -0.14 cm/sec. Decreasing the volume by the small fraction AV/V, is equivalent to adding that fraction of one 4s electron to the configuration 3d74s. The corresponding shift can be written &/a(Tr/Tro> = 0.14 cm/sec (7.1) Pound et aZ.,17* have made an accurate measurement of the isomer shift to 3 kbar using a hydrostatic medium. Their results were consistent with Eq. (7.1) to a few percent. 17*

L. R. Walker, G. B.Wertheim, and V . Jaccarino, Phys. Rev. Letters 6 , 98 (1961). R. V. Pound, G. B. Benedek, and R. Drever, Phys. Rev. Letters 7, 405 (1961).

132

H. G . DRICKAMEK

A straight line of the slope given by Eq. (7.1) is plotted in Fig. 133. Evidently the main effect of pressure is an increase in s-electron density in proportion to the decrease in volume. It is believed, however, that the deviation at the highest pressures is larger than experimental error, so that over a sufficiently large range of volume change, shielding cannot be neglected completely. The change in isomer shift across the transition is very large (4.017 cm/sec). This is about four times the amount predicted by density change alone. Some of this may be due to the change from ferromagnetic to para6 7 only a change of magnetic behavior, although Preston et ~ 1 . ~ ~found -0.001 cm/sec at the Curie point in the bcc phase. The largest part of the change in isomer shift must be attributed to change in the band structure. The usual picture of the band structure of the transition metals involves a broad 4s band overlapped by a rather narrow set of bands arising from the 3d shell. A change in structure from bcc to hcp could easily involve a change in shape of the 3d band or a shift in its center of gravity vis A vis the 4s band, which would result in a redistribution of electrons between 3d and 4s bands. Very roughly, an increase in s-electron character of the outer electrons of 6% would account for the anomalous shift. The change in isomer shift with volume change in the high-pressure phase is relatively small, less than the simple scaling on density would predict. As noted earlier, the compression in the hcp phase is quite aniI

I

I

\

\ \

\ \ \O

s

?

0

50

I00

0

P, KILOBARS

FIG. 134. Magnetic field intensity versus pressure for iron.

HIGH PRESSURE AND ELECTHONIC STRUCTURE

133

sotropic. This could favor a redistribution of electrons from 4s to 3d, partially counteracting the eKect of volume decrease. Mossbauer experiments’67 have shown that the magnetic field a t the nucleus is large and opposite in direction to the magnetization. iClar~hall’7~ ind Watson and Freemanl74 have shown that the largest contribution is due to core polarization. Figure 134 shows the fractional change in field strength with pressure lbtained in our experiments. Litster and Benedek175have measured the internal field in iron to GS kbar using nuclear magnetic resonance. Their :xperiments involved a somewhat more hydrostatic medium than ours. They obtained a substantially linear change of field 1Tith pressure, having the slope ,1.69 X

(kbar)-l

(7.2)

This line is superimposed in Fig. 134. The extension to 145 kbar shows a measurable but quite small deviation from linearity. The magnetic resolance signal arises mainly from nuclei in the domain ~ a 1 l s . The l ~ ~ Moss3auer effect does not distinguish between nuclei in the domain walls and ;hose on the interior. Since the two kinds of measurements are in good igreement both as to temperature coefficient and pressure coefficient of the held, one can infer that the field in the domain is quite uniform. Benedek”7 ?as given an extended discussion of the change of field strength in terms if changes in spin density. His discussion will not be repeated here except co note that the results can be accounted for in these terms. It is of interest to note that the pressure dependence of the isomer shift (Fig. 132) and that of the field strength (Fig. 134) are significantly lifferent; i.e., the field is much more linear in pressure than is the isomer shift. The fact that they are indeed different is not unreasonable, since the isomer shift involves primarily the’ 4s electrons, while the core polarization which is the main cause of the field is primarily controlled by 3d ?lectrons. ACKNOWLEDGMENT

It is a pleasure to acknowledge financial assistance from the United States Atomic Energy Commission for much of this work. A number of students, especially E. A. Perez-Albuerne, made helpful comments on the manuscript. W. Marshall, Phys. Rev. 110, 1280 (1958). E. Watson and A. J. Freeman, Phys. Rev. 123, 2027 (1961). ” J. D. Litster and G. B. Benedek, J . A p p l . Phys. 34, 688 (1963). 76 A. C. Gossard, A . hl. Portis, and W. J. Sandle, Phys. Chem. Solids 17, 341 (1961) ” G . B. Benedek, “blagnetic Resonance a t High Pressure” Wiley (Interscience), New York, 1963. ” R.

This Page Intentionally Left Blank

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

w-.Low* Department of Physics and National Magnet Laboratory,** Massachusetts Institute of Technology, Cambridge, Massachusetts AND

E. L. OFFENBACHER*** Department of Physics, Temple Universiiv, Philadelphia, Pennsylvania

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... 11. Outline of ESR Specta in Inorganic Crystals.. . 1. Energy Levels in Octahedral and Tetrahedral Symmetries.. ........... 2. Remarks on the Interpretation of Paramagnetic Resonance Results. . . . 3. Remarks on the ESR of Rare Earth Spectra.. . . . . . . . . . . . 111. Single-Ion Contribution to Anisotropy Energy. . . . . . . . . . . . . . . IV. The Spectra of Transition Elements in Simple Oxides (MgO, CaO, SI.0, ZnO,

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

136 138 138 141

154

. . . . . . . . . 156 . . . . . . . . . 158 . . . . . . . . . . . 161 VI. Perovskites. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . 164

........... VIII. Garnets.. . . . . . . . . . . . . . . . . . 14. ESR Spectra. .

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

* Permanent address: The Hebrew University, Jerusalem, Israel.

** Supported by the U.S. Air Force Office of Scientific Research. *** Supported in part by the National Science Foundation. 135

188

136

W. LOW AND E. L. OFFENBACHER

1. Introduction

During the last few years paramagnetic resonance has dealt, with some degree of success, with the magnetic properties of paramagnetic ions in complex multiple oxides. This review summarizes the main results for three classes of multiple oxides: perovskites, spinels, and garnets. It also discusses the magnetic properties of ions in rutile, a major constituent in some of these compounds. The complex multiple oxides in the concentrated magnetic forms have interesting properties. Some of these are ferromagnetic, others antiferromagnetic. The magnetic ions can occupy sites which have different point symmetry within each structure. Many of these compounds show one or more phase transitions which affect the magnetic properties. Some of the crystals, in particular those of the perovskite structure, have very large dielectric constants and are ferroelectric. Bulk magnetic phenomena, since they involve cooperative interactions, are difficult to calculate. The properties of isolated magnetic ions are much better understood. A knowledge of the behavior of the lowest energy levels in a n applied magnetic field of isolated ions in these complex compounds can contribute to some extent to the understanding of the behavior of concentrated magnetic complexes. We have selected the systems AB2, AB203,AB2O4,and A3B5012 for detailed discussion. The representative substances of these structures are rutile (TiOz), barium titanate (BaTiOs), lanthanum aluminate (LaA103), spinel (MgAI2O4),and garnet (Y3Fe6OI2).The choice of these crystal types was dictated mainly by their relative importance for different fields of physics; the titanates and aluminates for their cooperative ferroelectric behavior, the spinels and garnets for their cooperative magnetic properties. Moreover, there are now sufficient results for these compounds to permit a review. Many of their paramagnetic properties, in particular those of isolated S or effective X-state ions, have been studied. As often in science, these investigations solved some problems and created new ones. Spin resonance proved to be a very important structural tool. It often determined the site preference of a particular magnetic ion. I n some cases it permitted the determination of the point symmetry, in others the distortions from cubic symmetry, the axes of these distortions, and the number of inequivalent magnetic sites. It was found in many cases that some ions, even when present in small concentrations, are exposed to strong axial or lower symmetry crystal fields. Previously it had often been assumed that these fields arise because of cooperative effects. For some perovskite compounds, the spin resonance results established the nature of the phase transitions. They showed that in these highly polarizable substances there

ELECT RON S P I N RE S ONANCE OF MAGNETIC I O N S

137

is a correlation between the temperature dependence of the splitting of the energy levels, caused by changes in the crystal field, and the temperature dependence of the dielectric effects. Spinels are very complicated substances and often show a disordered structure. Resonance has given some indication of the degree of order in these substances. Finally, from the parameters of the spin Hamiltonian, the magnetic axes of the individual substructures, and the number of the inequivalent ions, it has been possible to calculate the “single ion” contribution to the magnetic anisotropy. We have tried to cover as much as possible of the published electron spin resonance (ESR) material. However, in these days of proliferation of journals it has become diEcult to retrieve all the significant information. We feel certain, however, that we have dealt with the major contributions in these particular fields.’ We have in some cases discussed critically some of the conclusions reached by various authors and arrived a t different conclusions. The optical data are mentioned briefly, whenever they have a bearing on the main theme of this review. Unfortunately, only very little reliable optical data are available a t present, with the possible exception of data on the garnets. This review is divided into nine parts. In Part I1 we briefly summarize the main features of the energy level schemes of transition elements, defining in Section 2 our notation so that the subsequent discussion, and in particular the parameters of the spin Hamiltonian tables, are consistent. Part I11 is a discussion of some of the theoretical considerations in the calculations of the single-ion contribution to the anisotropy energy. The main features of the resonance results of simple oxides are given in Part IV. We have not attempted to be exhaustive, since this would necessitate a monograph in itself; only those facts are mentioned which may serve as a comparison for the more complicated structures. In Part V the resonance results in rutile are discussed in detail. We felt that this highly polarizable substance is an important constituent of the more complicated titanates and deserves a more detailed evaluation. The results on perovskite structure materials, with particular emphasis on barium titanate and strontium titanate are given in Part VI. Finally, Parts VII and VIII deal with the ESR data of spinels and garnets. In the -4ppendix ESR parameters for the various structures are listed in a number of tables. A large fraction of the work reported here has been obtained in a few institutions. The main contributions are from Batelle Memorial Institute, Geneva (K. A. Muller) The Hebrew University, Jerusalem (W. Low) Naval Research Laboratory (V. J. Folen) Oxford University, England (W. P. Wolf) Raytheon Research Division (L. Rimai and G. A. deMars) RCA Laboratories, Princeton (H. J. Grrritsen)

W.

138

LOW AND E.

L.

OFFENBACHER

It is hoped that this review will induce a few more scientists to work on these complex oxides and will Iead to a better understanding of the behavior of relatively isolated magnetic ions as well as of dense magnetic materials. II. Outline of ESR Spectra in Inorganic Crystals

During the last few years a number of good textbooks have appeared discussing crystal field theory and the structure of the energy levels in sites of different point syrnmetrie~.~-~ The effect of these crystal fields on the Zeeman splitting of the low-lying energy levels, as measured by It suffices, paramagnetic resonance, has also been discussed in therefore, to give only a brief summary of those aspects of the theory which are of importance to this review. 1. ENERGY LEVELSIN OCTAHEDRAL AND TETRAHEDRAL SYMMETRIES

Let us consider first the iron group elements. For the configurations d1 to d5 the lowest atomic energy levels of these ions are 2D, 3F, 4F,5D, and 6S.The crystal field splits the ground states as well as the higher levels of the dn configurations into a number of Stark levels. (See Fig. 1.) This splitting depends on the symmetry of the crystal field. For octahedral symmetry, for example, it can readily be shown by group theory that the above orbital states split up as follows (neglecting spin-orbit coupling and admixtures from other energy levels of the same or other configurations). E(+6) D + Tz(-4)

F + T1(-6)

s

+

+ + 5“2(+2) + &(+I21

(1.1)

A,(O),

where we have used the Mulliken notation for the crystal field symmetry of the split states. It is customary to label the splitting of the energy levels of a single4 electron in an octahedral field [ & ( T z )- &(I?)] as 10 Dq. The numbers in L. E. Orgel, “Introduction to Transition Metal Chemistry.” Wiley, New York, 1960. C. J. Ballhausen, “Introduction to Ligand Field Theory.” McGraw-Hill, New York, 1962. 4 C. K. Jgrgensen, “Absorption Spectra and Chemical Bonding in Complexes.” AddisonWesley, Reading, Massachusetts, 1962. 6 J. S. Griffith, “The Theory of Transition-Metal Ions.” Cambridge Univ. Press, London and New York, 1961. 8 W. Low, “Paramagnetic Reaniianre in Solids.” Academic Press, New York, 1960. 5. A. Al’tshuler and B. M. Kozyrrv, “Elwtrnn Paramagnetic Resonanre.” Academic Press, New York, 1964. 2

3

ELECTRON S P I N RESONANCE O F MAGNETIC l O N S

S

A,

139

d 3 ( S = 3 / 2 ), d S ( S = I )

d 5( S = 5 / 2 )

FIG.1. Energy level splittings of the ground state for the d" configuration in weak or medium crystal fields. Perturbation arising from spin-orbit coupling or other higherorder effects arc neglected.

the parentheses in Eq. (1.1) are the energies of the states in units ofDq. In octahedral complexes the 10-Dq splitting for iron group elements is found to be about 8000-10,000 cm-l for many divalent ions and 15,00020,000 cm-1 for trivalent ions. The identification and order of the energy states of various symmetries is determined by the relative strengths of the crystal field energy V,, the spin+rbit coupling energy VLS and the interelectron Coulomb energy V,. The ground states shown in Fig. l ' a r e appropriate if V , > V , > VLS. However, if V, > V , > VLs, then one has the so-called strong field scheme. From the calculations of Tanabe and Suganos one can determine the ground states in these cases. New ground states are obtained only for the configuration d4 to # as follows: d4:6E + 3T1

d6:5T2+ 'A1 (1.2)

d5: 6A, + 2Tz d7: 4T1+ 2E.

Of course, in many of the crystal systems considered here the point symmetry is far from octahedral. It usually involves a distorted octahedron. *y.Tansbe and S. Sugano, J. Phys. SOC.Jupan McClure, Solid State Phys. 9, 419-425 (1959).

9, 753 and 766 (1959); or see I).

S.

140

W. LOW AND E. L. OFFENBACHER

However, the lower crystal fields, such as occur in axial or orthorhombic distortion, are a t least an order of magnitude weaker than the cubic field. I n some cases the ion is exposed to a crystal field of approximately tetrahedral symmetry. The Stark energy levels of a given ion when situated in this symmetry are in reverse order compared with its levels in octahedral symmetry. As the order also reverses when a d n ( n < 5 ) is compared with a dIo-. configuration, one can state that the levels of a d n ion in tetrahedral symmetry are similar to those of a d10-. ion in the octahedral case. A pointcharge calculation yields for the ratio of the crystal field strengths (Dp) tetrahedral/ (Dp) octahedral

=

-4/9.

Consequently one may expect a considerably smaller crystal field for tetrahedral coordination, and the negative sign shows the reversal of the order of the levels. I n other words, in going from octahedral to tetrahedral symmetry the ground states interchange as follows: TSi3 E and T I e A 2 . A smaller crystal field has indeed been found by experime~it.~*'~ For the divalent ions Dq is about 3500-5000 cm-I and for trivalent ions it is 60009000 cm-'. These very large changes in Dq, the reversal of sign, and the absence of a center of inversion for the tetrahedral sites, produce marked differences in the spectra. These differences enable one to identify which site the given ion occupies in a complex structure. They are briefly summarized.

a. Optical Spectra Differences in optical spectra are revealed by (1) the line width, (2) the line intensity, and (3) the number of lines in the optical range of the spectrum. (1) The line width of optical transitions in tetrahedral complexes is usually narrower. It often permits one to resolve the fine structure caused by spin-orbit interaction or by lower crystal field symmetries. (2) The relative intensity of the optical spectra of tetrahedrally coordinated compounds is stronger by a factor of 10-103. In general, optical spectra in crystals are weak since they arise from transitions between levels of the same parity. Electric dipole transitions are found, however, because either the ion is not quite in the center of symmetry, or because an odd vibration is coupled to the electronic state. In the case of tetrahedral symmetry, however, electric dipole transitions are permitted because the odd part of the crystal field potential, T'3, admixes configurations of opposite parity. For example, the V 3 term will give rise to matrix elements of the type (d I V3 I f ) and (d I V3 1 p ) from odd parity configuration dn-'p and dn-'f. This small admixture will not appreciably affect the position of the energy levels. However, it will influence the intensity of the transition since the 9

R. Pappalardo, D. L. Wood, and R. C. Linares, J . Chem. Phys. 36, 1460 (1961). R. Pappalardo, D. L. Wood, and R. C. Linares, J . Chem. Phys. 30, 2041 (1961).

lo

ELECTRON S P I N RESONANCE O F MAGNETIC IONS

141

intensity is a sensitive function of the admixture. (3) The smaller spacing of the Stark levels for tetrahedral sites produces more lines in the near infrared. b. Paramagnetic Resonance Spectra

As far as the paramagnetic resonance spectra is concerncd, tetrahedral symmetry (1) affects the g factor, ( 2 ) shortens the spin lattice relaxation time, and (3) permits a shift in the resonance line proportional to the electric field. ( I ) On account of the inversion of levels already mentioned, an ion with a configuration dtL(n < 5) in tetrahedral symmetry corrcsponds to a n ion with a dl0pnconfiguration in an octahedral site. However, as the sign of the spin-orbit coupling is opposite for these two configurations, the order of the levels will be reversed. I n the case of effective S-state ions, for which the orbital moment is nearly completely quenched, the contribution to the g factor from the spin-orbit coupling (gLs) will be of opposite sign for the two cases. Moreover since gLS is proportional to X/Dq, this contribution will be larger for tetrahedral symmetry. ( 2 ) Similarly, the lower Dq shortens the relaxation time because there is a greater admixture of the higher levels to the orbitally nondegenerate ground state. (3) Finally, a n applied electric field will give rise to a shift of the resonance line which is proportional to the applied electric field.11-13 The reason for this is essentially similar to the effect of the odd crystal field V 3 on the electric dipole intensity of optical spectra. One only has to substitute the odd operator eE-r for the electric dipole operator, where E is the applied electric field. The effect of this operator will be to shift the energy levels of both the fine and hyperfine structure. In the case of centrosymmetric complexes there will be much smaller shifts proportional to E2 (shifts of the order of 10-3 G per 1000 V/cm).14 2. REMARKS ON

THE

INTERPRETATION OF PARAMAGNETIC RESONANCE

RESULTS This review assumes familiarity with the theory and the techniques of electron spin resonance. We shall give here only the pertinent formulas which will be used in the discussion of the data and the various tables.I5 N. Bloemhergen, Science 133, 1363 (1961). G . W. Ludwig and H. H. Woodbury, Phys. Rev. Letters 7, 200 (1961). la G . W. Ludwig and F. S. Ham, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, Vol. 2, p. 626. Academic Press, New York, 1963. l4 M. Weger and G. Feher, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, l1

le

1962, Vol. 2, p. 628. Academic Press, New York, 1963. The notation here follows essentially that of M. T. Hutchins, Tech. Note 13, Contract A F 61(053)-125 R63. (See also M. T. Hutchins, Solid Stale Physics. 1701. 16. Academic Press, New York, 1964.)

142

W. LOW AND E. L. OFFENBACHER

The crystalline field potential a t a point r(r, 8,

cp)

caused by charges

qj a t points dj is given by

~ ’ ( r , 8‘PI , =

C

(qj/I

dj

i

- r I).

(2.11

This potential is usually expanded either in Cartesian or spherical polar coordinates. For cubic point symmetries the expansion in Cartesian coordinates is V ( X ,y, z )

=

C ~ Cx: C-

gr41

+ ce[c + ~

(xi2xj4c-

xi6

+$~6)],

(2.2)

with i $ j = 1 , 2, 3 and x1 = x. x2 polar coordinates it is

=

y, x3

+ De[ Yen-

= z;

or in terms of spherical

( Ye4

+ Yew4)].

(2.3)

One needs to retain only the fourth-order terms for states described by d-type wave functions, whereas for f electrons, it is necessary to consider also the sixth-order terms. We have omitted odd powers of xi or odd spherical harmonics, as found in the case of tetrahedral complexes, because these do not yield finite matrix elements between states belonging to the same configuration. The values of the constants Cn and D , for four-, six-, eight-, and twelve-fold coordinations of point charges are listed in Table I. It is usually most convenient to calculate the matrix elements of the crystal field by the use of the operator-equivalent method introduced by Stevens.16 I n this method the crystal field expansion in terms of x, y, z is rewritten so that x, y, z is replaced by J,, Jv, J,, respectively, allowance being made for the noncommutation of the components of J. The resulting expression then operates on wave functions specified by the angular momenta quantum numbers and the matrix elements within a given J or L manifold are computed. Reference 6 has several tables of these matrix elements for the operator equivalents corresponding to commonly occurring symmetries. For a cubic field, for example, one writes for the crystal field part of the fourth-order term of the Hamiltonian, X, =

c4C n

l6

(xi,:

(c4/20){ ( 3 5 ~ ~3orzz2

- $rn4)= n

8.W. H. Stevcns, l’roc. I’hys. SOC.(London) A66, 209 (1952).

+ 3r‘)

143

ELECTRON SPIN RESONANCE OF MAGNETIC IONS

TABLE I. EXPRESSIONS FOR CUBICCRYSTAL FIELDPOTENTIAL PARAWETER~

Cd in units

c 6

D4in

D6 in

B40 in

units

units

units

iinit,s

_-i 0

221 --

56 --

+-329

+-187

4-fold

_-85

112 9

_-28

I6 7 +s +-36

6-fold

-

&fold

9

9

9

35

_-8

27

14 6

_-21

70 8

12-fold

27

2

13

+ -4 x -

21 2

3

+2

7

13

1 9

--

+-181

_-14

_-3

+-327

+64 -x-

32

3

-4 x - 2

_-6

B6° in unit,s

64

3

18

4

The 4th- and 6th-order potentials are for the various coordinations in the ratio of -16:-8:18:-9 and -64:-32:-27:(27 X 13/4). (See recent review by M. T. Hutchins, Solid State Physics, Vol. 16, Academic Press, New York, 1964.)

where n is to be summed over all magnetic electrons. Inserting the operator equivalents one obtains Wc = (C4/20)@~(?)([35J,4- 3 0 ( J ) ( J -

25JZ2- 6(J)(J

+ 1)Jz2

+ 1 ) + 3J'(J + l)']

+ 5 [ 3 ( +~ ~ + ( J =- 4 ~ 4 1 1 1

(2.51

i ~ , ) 4

The multiplicative factor @ J is a numerical constant depending on n, L, S; ( r 4 ) is the expectation value of r4,indicating that in the last step the r integration has already been performed. The operator expressions in the square brackets are abbreviated by 0 4 O and 0 4 4 respect,ively and the coefficients are designated BkOand B44 or, in general,'? X, = C BnmOnm. (2.6) n.m

The coeEcient of p ( r 4 ) for example is often called AqO,so that Bnm = Anm(P)On,and On is the numerical constant a,p, y for n = 2, 4, 6,respectively. The products Anm(rn) are usually treated as phenomenological constants because the point-charge model is in general not valid and the radial dependence of the wave functions in a solid is not well known. I'J.

M. Baker, B. Rleaney, and W. Hayes, Proc. Roy. SOC.A247, 141 (1954).

114

W. LOW AND E. L. OFFENBACHER

I n general there can be many Bnmdepending on the point symmetry of the site. The restrictions are that n must be even and that n _< 21 where I is the quantum number of the electrons of the particular configuration, i.e., I = 2 for d electrons and I = 3 for f electrons. The explicit expressions for X, for the three most common point symmetries (cubic, tetragonal, and trigonal) are Xcubio =

Xtetragonai = Xtrigoonal =

B4'[04' -k 5O4*]

+ B4'04' B,'OZ' + B4'04' Bz'Oz'

+ B6'[06' - 2 1 0 ~ ~ 1 ; + B44044+ B6'06' + B64064; + B6'06' + B14'[04' + 5044] + B16'[06'

(2.7)

(2.81

- 210~41. (2.9)

I n the last equation the terms 0 4 0 and 06' are separated into two parts. The unprimed terms Bdfland Be0represent the distortion referred to the trigonal axis. The primed terms are referred to cubic axes. The relation between the various constants C, D, and B are given in Table I for the 4, 6, 8, and 12 cubic coordinated substances. Using a Hamiltonian of the type in Eq. (2.6) we express the behavior of the lowest energy levels in an external magnetic field by means of an effective spin Hamiltonian, such as

XS=

c

[PHqgqSq

+

SqDqSq

+

SqAqIq

-

PgnqHqIq

f IqpqIq].

(2.10)

'1

Here, y = z, y, x , arid gq and qnqare the electronic and nuclear spectroscopic splitting factors respectively. A , and P , are the components of the hyperfine and quadrupole tensors. Higher-order spin variables such as Sq3, Sp4, etc. are required for manifolds of S = $ and higher. For iron group elements in the presence of axial symmetry, the spin Hamiltonian is usually written as

+ HgS,)] + A S J , + BCSJ, + SJ,],

XB= P [ g l I H z X z

qJ-(HzSz

D[Sz2 - * S ( S

+ I)] (2.11)

where we have omitted the quadrupole interaction. It should be stressed that in the majority of the cases with 3dn configurations the crystal field is stronger than the spin-orbit coupling or the Zeeman splitting. The effective S is then based on the number of levels which fall within the range of measurement. I n Eq. (2.11) the D term arises from the crystal field. The coefficient D is related to the corresponding coefficient Bnmin Eq. (2.8) or Eq. (2.9), namely D = 3Bz0. I n Table I1 are listed the relations between the Brim's and the other symbols commonly used for the coefficients of the crystal field terms in the spin Hamiltonian.

145

ELEC T RON S P I N RESONANCE O F MAGNETIC I O N S

TABLE 11. RELATION OF VARIOUS CUBICFIELD CONSTANTS IN SPIN HAMILTONIAN~ B20

=

THE

fD = fb,O

B22 = E

=

Lb 2 3 2

a I n the literature pertaining to iron group elements the constants a , D, E, and F are usually used. For rare-earth ions the notation c, d, and b," is found.

Recently Koster and Statz,l8Hauser,lgand Ray20 have given a different formulation of the spin Hamiltonian taking into account the full group theoretical expression. These Hamiltonians may contain more constants than mentioned here. However, it has not been found necessary to use these extended Hamiltonians and in the subsequent discussion we shall only refer to the conventional expressions. 3. REMARKS ON THE ESR

OF

RARE-EARTH SPECTRA

The optical spectroscopic information on the f" configuration of free ions is not as complete as that of the dn configuration. The existing results all indicate that adjacent configurations are relatively close. This gives rise to configuration interaction, resulting sometimes in appreciable deviation from the Land6 interval rule. The 4f" ions, imbedded in an inorganic complex, are exposed to a relatively small crystal field. In general, this perturbation is weaker than the spin-orbit coupling. To a first approximation J is a reasonably good quantum number. The crystal field acting on each J level will split the de1 into a number of Stark components. In some cases the generacy of 2 J crystal field will admix levels of different J's.

+

I8

G. F. Koster and H. Statz, Phys. Rev. 113, 443 (1959); 116, 1568 (1959). W. Hauser, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, T'ol. 1, p. 297. Academic Press, New York, 1963. T. Ray, Proc. Roy. Sac. A277, 76 (1964).

146

M'. LOW AND E. L. OFFENBACHER

TABLE111. GROUND STATESOF RAREEARTHIONSFOR SIX-,EIGHT-, TWELVE-FOLD COORDINATION Ground state

Twelve-

Sixcoordinated

ra r5,r3 rS,rs r4,r3 rs r3,rs rs

r7 1'1

I's,

r 6

rl, r5 r7 rz,rl r7,rs r3,rl r7,rs rr,rl ra:

1'5

rl r6 + r7 + rs

r,, rs r,, r3,r6 rh,r7 rl r6 r7 rs

+ +

I n the case of a cubic field the crystal field is given by Eq. ( 2 . 7 ) . The order of the Stark levels will depend on the relative magnitude and sign of Bd and Be.21Table I11 lists the ground state of the various 4fn ions and the possible lowest Stark levels for the various coordinations. At present there are often not sufficient data to decide which will be the lowest Stark level, although all evidence points to B4 > BBin the six- and eight-coordinated compounds.22 The wave function of the ground state in the crystal can in general be written as

+

=

C I J, ak

Jk)

k

+C

I J',

(3.1 1

J'k),

k'

where b k f <
+ 2 s l I +)

--$

Si(+

I

J,

I +> =

(3.2)

gq.

I n general, for lower symmetries, one measures the g tensor components along three directions. One tries to derive the wave function of Eq. (3.1

+

21

22

K. R. Lea, M. J. M. Leask, and W. P. Wolf, Phys. Chem. Solids 23, 1381 (1962). E. L. Offenbacher and L. Shapiro, Phys. 1,etters 10, 16 (1964).

ELECTRON SPIN RESONANCE OF MAGNETIC IONS

147

which would yield the experimentally measured gnfactor. If # is determined. one next derives the crystal field potential, i.e., the magnitIde of Brim or Zlnm ( P ). However, there are always more crystal field parameters than measured g factors for a particular level. Hence, without any additional data such as the knowledge of g factors of some higher-lying Stark levels or the energy level separations of adjacent Stark levels originating from the same J level, one cannot determine uniquely the magnitude of these parameters. 111. Single-Ion Contribution to Anisotropy Energy

As mentioned before, some of the systems to be discussed in the later chapters, such as the garnets and spinels, are ferromagnetic or antiferromagnetic in their concentrated form. They show a large magnetic anisotropy. Part of this anisotropy arises from the contribution of the various anisotropic terms in the single-ion spin Hamiltonian, Eqs. (2.7)- (2.9). We briefly discuss these aspects which are related to the ESR data of many of the ions in a number of crystal structure^.^^ Let the magnetization vector M have direction cosines al, az,a3 with respect to the crystallographic axes. If no external magnetic field is applied, the magnetization will lie in the direction in which the free energy F is a minimum. This direction is called the easy axis of magnetization. With the application of an external field along a direction other than that of the easy axis, M will tend toward a new direction. This direction will be defined by the balance of the torques exerted by the external field and that of the anisotropy energy. If the anisotropy energy is small, M can be oriented in any given direction by applying a reasonably strong magnetic field along this direction. The anisotropy may be defined in terms of the free energy as a series expansion in the direction cosines a. For example, for a cubic crystal this is

where K1, K 2are the anisotropy constants. For crystals of lower symmetry, there are more constants to be taken into account. The contributions to the anisotropy energy are many. The main classes of interactions are the following: ( a ) Magnetic dipolar interaction. The interaction between two dipoles is a second-rank tensor. If the crystal is cubic and the y factor isotropic,

*'

J. Kanamori haa reviewed in detail the effecta of anisotropy and magnetostriction in ferro- and antiferromagnetic materials; see J. Kanamori, in "Magnetism" (C. T. . Rado and H. Suhl, eds.), Vol. 1, p. 127. Academic Press, New York, 1963.

148

W . LOW AND E. L. OFFENBACHER

the contribution of the dipolar energy is zero to first order, but there is a contribution to second order in the perturbation development. For noncubic symmetries, the dipolar terms contribute in the first order. ( b ) Anisotropic exchange energy. Van VleckZ4has shown that the interaction between two spins can be expressed in general as a second-rank tensor in spin variables when S = $ for the two spins. He calls this the "pseudodipolar" interaction. The pseudo-dipolar interaction will contribute only to second order, but higher multipole interactions which exist for S 2 1 produce first-order perturbation terms. (c) Crystal field effects on single ions. The crystal field Hamiltonian x, = CB,mOmrn contains the various crystal field parameters which are anisotropic. For those ions which have an orbital ground state, the crystal field induces anisotropy in the orbital motion. For those ions which have an effective singlet ground state the small admixture from higher orbital states via the spin-orbit coupling induces anisotropy; this is reflected in the various parameters such as BZ0 and Bso in the spin Hamiltonian. The first two mechanisms are caused by the interaction of two or more ions. The third mechanism is caused by the single ion. Assuming that the single-ion parameters in the spin Hamiltonian do not change significantly from the dilute to the concentrated sample, one can calculate this contribution to the magnetic anisotropy. Let us consider first a simple Hamiltonian with only a B20020term for the crystal field. The explicit energy levels as a function of the angle 8, the direction in which spins point on the average, measured from the z axis, will be given for an orbitally quenched ground state by Whf

=

MgpH,rf

+ Bzo(3

C O S ~8

-

1)020.

(I11.2)

One can write down the partition function

for a sublattice of N ions. The free energy obtained as usual is

Assuming that the crystal field parameter Bzois small compared with k T , one may expand the terms in the exponential containing the crystal field. 24

J. H. Van Vleck, Phys. Rev. 62, 1178 (1937); see also J . Phys. Radium 12, 262 (1951).

149

ELECTRON S P I N RESONANCE O F MAGNETIC I O N S

Retaining the leading terms one obtains F1

=

N[Fo(v)

+ Bz03

COS'

(111.4)

O(Oz')av],

where Fo(v) is that part of the free energy which is isotropic rl =

exp[(-gPHed/kTI

and (OzO)av is the therinal average of the operator (02')av

=

t ( S z 2- + S ( S

02':

+ 1))av.

(111.5)

Another example is the fourth-order axial field which can be written as X = gpHofr. S

+

B4'04'.

The energy levels for a d5 configuration of electrons can be written

+

Ef5/2 =

+%PHerr

E+3/2 =

&gypHerr - 3b4'

E+112 =

f$gpHeff

b4'

(1II.G)

+ 2b4'.

Computing the partition function and the free eiiergy in an analogous manner as before, we obtain

F

=

-[b4'/(60

X 8 )][3 5 ~ ~ 3 ~3003'

+ 3](04°)av,

(111.7)

where (O4')av

=

[35(Sz")av - 3OS(S -I- 1)(Sz')av -

GS(S

25(S,2)av

+ 3S2(S

+ l)].

1)'

(111.8)

For a cubic field one has to superimpose the three fourth-ordcr axial field components along the three cubic axes. One then finds that F1

=

+

- (b4"/12)[35~~- 30~~3' 3](04°)av.

(111.9)

has given explicit expressions for the (040)av and (020)av as a function of temperature. They can be expressed as

F

=

+

N { F o ( ~ )WCos*Op(v)

+ b4°+r(v) + [(b~o)2/ICT]cos40t( v ) , (111.10)

H. P. Wolf, Phys. Rev. 108, 1152 (1957).

150

W. LOW AND E. L. OFFENBACHER

where 77 =

p(7) = T(?)

exp [(-g/3Heff)/kT];

zo-'(5 -

= zo-l$(-i

t(77) =

+ = 12m2+ m2nz+ n212

+ 5771' + 377- 2+ - 2,,3 + 3774-

9

- 477' - 4q3 - q4

775)

c - n(7)/ln771 - 8 S h ) + 3CP(77)I2

+ 71 + 167' + 16q3+ v4 + 2 5 ~ ~ ) ZO-'~ (75 - 5777 - 3677' + 36q3 + 57q4 - 7 5 ~ ~ )

~ ( 7 = ) 20-'(25 n(7) =

and 2 0

=

1

+ + v2 + + v4 + 77

TI5.

In general, one has to sum over several ions per unit cell, taking into consideration the individual local magnetic axes and their relation to the axis of the Hew If, however, the over-all symmetry of the crystal is cubic, one can simplify the expression given in (111.10) to some extent. To preserve a total cubic symmetry, despite the fact that each ion has a Hamiltonian of the form X

=

+ Bd'(04' + 5044),

Bz"0z"

there must be a number of ions in the unit cell. Their crystal fields must be such that the sum over the axial contribution cancels out identically. The first-order contribution to the magnetic anisotropy can then be written as Ki

=

2hnr(77)

+ y[(b~')~/kT]t(~);

(111.11)

y is -2/3 if the axis of distortion is along the (100) directions and 6 if it is along the four (111 ) directions. An important example of these considerations in the case of Gd3f is garnet.26The structure of the garnets will be discussed in Section 13 in more detail. For the present it suffices to know that we can take two of the local axes of symmetry parallel to the [ l l O ] and the third parallel to the [loo] direction (z axis) of the crystal. There are six inequivalent ions in the unit cell. The spin Hamiltonian of Gd3f in the local symmetry axes is given by

+ Bz'Oz' + BZ20z2+ B4'04' + B42042+ B44044+ B6'06' (111.12) + B62062+ +

X = g/3H. S

~

26

~

4

0 ~~ ~ 46 0 ~ 6 .

J. Overmeyer, E. A. Giess, M. J. Freiser, and B. A. Calhoun, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, Vol. 1 , p. 224. Academic Press, New York, 1963.

151

ELECTRON SPIN RESONANCE OF MAGNETIC IONS

The exchange field in gadolinium iron garnet is of the order of 2.5 X lo6 G. The "easy" axis of magnetization is along the [lll] direction. The dominant term would then be ypHefrS along this direction and the various operations will have to be transformed for each particular ion into this direction. The transformation matrices have been given by Baker and Williamsz7. Sinre the total symmetry of the garnet is cubic, we shall be restricted immediately to those terms which depend on the fourth power of a where CY is measured with respect to the exchange field. At T = 0 we are only concerned with the lowest energy level of each of the six ions. The t,otal contribution to the anisotropy is given by the difference in the energy measured along the easy [lll] and the [loo] direction for all sites:

K , (per ion)

=

$C(wIl1- w1m).

(111.13)

One finds this

Kl (per ion)

+

=

-

Vh4),

(111.14)

and the second-order contribution

- (21/48gp&)[

(b,')'

+ 10bzobz2 - 7 (bz')'].

(111.15)

It is found that a large fraction of the anisotropy in gadolinium garnet is determined by the anisotropy of the parameters of the single ion as determined from the ESR spectra in the dilute sample. If the ground state is orbitally degenerate the calculations are more complicatcd. The same holds true for the rare-earth ions.28 It is clear from the above discussion that the determination of the parameters in the spin Hamiltonian are important in the determination of one and sometimes more than one of the dominant contributions to the magnetic anisotropy. In some simple cases one can predict that sublattices are canted with respect to one an0ther.2~In an iron garnet we have to consider the effect of the effective magnetic field of the iron lattice on the rare-earth sublattice. I et us take as a n example a rare-earth ion with only one Kramers doublet ( 8 = $) populated. I n the absence of the exchange field an ESR experiment would yield g,, g, gZ measured along the appropriate local axes. 27

J. M. Baker and F. I. B. Williams, Proc. Phys. Soe. (London) 78, 1340 (1961).

** K. A. Wickersheim and R. L. White, Phys. Rev. Letters 8, 492 (1962).

We follow here the outline by W. P. Wolf, M. T. Hutehins, M . J. M . bask, and A . F. C. Wyat,t, J . Phys. Soe. Japan 17, B-1 443 (1962).

152

W. LOW AND E. L. OFFENBACHER

With the exchange field He present we obtain an anisotropic exchange G-tensor, which can be evaluated from optical or far infrared measurem e n t ~The . ~ ~free energy then can be written X

2

He.G. S

+ PH0.g. S.

(111.16)

The lowest eigenvalue is

where x, y) z are the three principal directions of the g tensor. The component of the magnetic moment M i along one of these directions is given by aF/aHo,i provided that the exchange energy is not a function of Ho. This is usually true for small external fields and a t low temperatures.

At T

=

O’K, Mo,iis then given by (111.18)

for N number of ions.

If He lies in one of the principal planes of the g tensor, then the induced magnetization will also lie in this plane but will not be along the same direction. The spontaneous moment produced only by He can be found by putting Ho = 0. If 1L. is the angle between M and the direction i and 8 the angle between H e and the same direction, the two angles are related by (111.19) For Yb3+in an iron garnet, the magnetic g’s and the exchange G’s are very different and, therefore, 1L. and e are different. The amount that M will deviate from the easy direction (the [111] direction in the garnet) will depend on the anisotropy of both g and G, Wolf et al.29 have estimated that the canted sublattice in YbIG is about 20 deg between the Yb3+and the Fe3+ magnetic moments. 30

K. A. Wickersheim, Phys. Rev. 122, 1376 (1961); see also the review article by K. A. Wickersheim, in “Magnetism” (G. T. Rado and H. Suhl, eds.), Vol. 1, p. 269. Academic Press, New York, 1963.

ELECTRON SPIN RESONANCE OF MAGNETIC IONS

153

IV. The Spectra of Transition Elements in Simple Oxides (MgO, CaO, SrO, ZnO, and A1203)

It is not our purpose to discuss in detail the spectra of transition elements in simple oxides. We will mention the main facts, which have a bearing on the results of the multiple oxides. The resonance data on these simple oxides are a guide for the interpretation of the data for the more complex structures which are being dealt with in the following sections. These data are compiled in Tables AI-AV in the Appendix. They serve as a convenient reference in identifying the paramagnetic impurity in various coordinations in the more complicated structures of the multiple oxides. The following generalizations can be drawn for paramagnetic ions in systems of the A 0 type.

(I) Iron elements can be easily substituted in all oxides. Rare-earth elements can be incorporated in CaO and SrO, and as a minor impurities in MgO and ZnO. (2) Trivalent or monovalent ions of the iron group can also be substituted for the 0 ions. The dominant fraction of these incorporated ions is exposed to the octahedral field of the host lattice and therefore charge compensation does riot generally occur in the neighborhood of the ion. (3) A smaller fraction of trivalent iron group ions is at sites of lower symmetries. An example of this is Cr3+ in MgO which shows a predominantly cubic spectrum, but also a tetragonal and orthorhombic spectrum. The lower symmetries are believed to arise from charge compensation. (4) Trivalent rare-earth ions are usually a t a site of tetragonal symmetry. A small fraction is a t a site of cubic symmetry. I n the case of CaO and SrO this fraction, however, may in some cases be of the order of 50%. (5) Unusual valence states of the iron group elements can be stabilized Conversion in these crystal hosts. Examples are Pel+, Nil+, Co*+,and N3+. from one valence state to another can be induced by ultraviolet, X-ray, or y-ray, by bombardment with electrons or neutrons, by heat treatment in various atmospheres, or by partial compensation with another impurity ion of different valence state. (6) Ions in ZnO show a spectrum characteristic of tetrahedral coordination with a trigonal field superimposed. (7) The line width of many of these ions in the octahedral A 0 are angularly dependent. Very often the narrowest line width is found along the [lll] direction. It is thought that the contribution to the line width comes from small tetragonal components, probably arising from internal strains which distort the cubic crystal fields.

154

W. LOW AND E. L. OFFENBACHER

Iron group elements can be substituted for the aluminum ion in ALO3. The spin Hamiltonian given by Eq. (2.9) is characteristic of a distorted octahedron with a trigonal field. Geschwind and Remeika31 have shown that one can substitute gadolinium as a minor impurity in A1203. A detailed analysis of the spectra has indicated that not all the A13+ ions are magnetically equivalent. Experiments using external electric fields on ruby A1203:Cr3+show a linear Stark shift.32This is consistent with X-ray data indicating that the Cr3+ion is not a t a site of center of symmetry. V. Rutile:

Ti02

For the purpose of this review our interest in rutile stems from the fact that its Ti4+ion is surrounded by six octahedrally coordinated oxygen ions. This is the same as that of Ti4+in the titanates. Furthermore, the ferroelectric behavior of some of the titanates seems to be closely linked with the properties of Ti02.33Similar to other highly polarizable substances, it has a large dielectric constant and a relatively low loss factor both in the microwave and in the infrared regions.34Since this is the prototype of many of the subsequent compounds we shall discuss its ESR spectra in detail. Titanium dioxide crystals are usually grown by means of the flame fusion method. These crystals are often not of stoichiometric composition and have the formula Til+z02-z, indicating oxygen vacancies. Nonstoichiometric crystals may be semiconducting with a variable energy gap deThey have a metallic appearance. The “pure” TiOz pending upon 2.35s36 crystal is an insulator with an energy gap of 3.03 eV.37However, all transition metal ions shift the fundamental absorption from the ultraviolet into the visible region, thus impeding the investigation of optical absorption or emission spectra in these regions of the spectrum. 4. CRYSTAL STRUCTURE

The crystal structure of rutile is based on a tetragonal unit cell and can be described by the space group P42 I m2, I n2 I m or D4h14. A subunit is 31

33 34

36

37

S. Geschwind and J. P. Remeika, Phys. Rev. 122, 757 (1961). J. 0. Artman and J. C. Murphy, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, Vol. 2, p. 634. Academic Press, New York, 1963; E. B. Royce and N. Bloembergen, Bull. Am. Phys. Soc. [2] 7 , 200 (1962). A. von Hippel, Rev. Mod. Phys. 22, 221 (1950). P. Ehrlich, Z . Elektrochem. 46, 362 (1939). F. E. Grant, Rev. Mod. Phys. 31, 646 (1959). H. P. R. Frederikse, J . Appl. Phys. 32, Suppl., 2211 (19GO). D. C. Cronemeyer, Phys. Rev. 87, 876 (1952).

155

ELECTRON S P I N RESONANCE O F MAGNETIC IONS

n,.4+

.-

-(TI'+)*

b FIG.2. Structure of rutile-TiOz. Fig. (2a) shows the position of the oxygen ions around the octahedrally coordinated substitutional (S) ion and Fig. (2b) the position of the oxygen ions around the interstitial (I) ion.

the Ti06octahedron which is the basic unit in the resonance study. Crystallographic studies have been made by a number of investigators.3a41 Figure 2 shows the Ti0 6 octahedron using the data of Baur. The point symmetry a t the Ti site is orthorhombic. The distorted octahedron can be considered to arise from an octahedron of cubic symmetry by pushing in the four ions in the equatorial plane unequally, so that they form the forners of a rectangle rather than the corners of a square. The dimensions of the rectangle are 2.959 by 2.520 A with the four oxygen ions a t a distance of 1.944 A from the central Ti ion. The remaining two oxygen ions are 2.2% farther away a t 1.988 The'point symmetry a t the Ti site is D 2 h . There are two octahedral sites which are equivalent except for a 90" rotation around the tetragonal axis ( c axis) of the crystal. The two ESR spectra which correspond to these two inequivalent octahedra coincide when the magnetic field is directed along the c axis. A resonance spectrum for substitutional ions will in general show a n anisotropic g factor with the magnetic axes along the [llO], [TlO], and

A

A.

w.H. Baur, Actu Cryst. 9, 515 (1956). 'OS. Anderson, B. Collen, U. Kuglaustierno, and A. Magneili, Actu Chem. Scand. 11, 'O

'I

1641 (1951). D. T.Cromer and K. Herrington, J. Am. Chem. SOC.77,4708 (1955). C. Legrand and J. Delville, Compt. Rend. 236, 944 (1953).

156

W. LOW AND E. L . OFFENBACHER

[OOl] directions for one system and along the [IlO], [Oil], arid [OOl] directions for the second system. We label the [llO], [TlO], and [OOl] directions x,y, and x respectively. The tetragonal c axis of the crystal [OOl] coincides with the z direction, I n our discussion we will refer only to the first set of octahedra. The iiiagnetic axis, which is most nearly tetragonal, is for most ions along the x direction, although for some ions it turns out to be along the z direction. This will be explained later.

5.

I N T E R S T I T I A LPOSITIONS

Another way of looking a t the structure is illustrated in Fig. 3. I n this figure it is shown how the various octahedra are stacked in the rutile crystal. Certain adjacent stacks are separated by open spaces, or channels, which are parallel to the c axis. These open channels may be considered as stacks of oxygen octahedra where the central Ti# ion is missing. The volume of this oxygen octahedron (10.91 k 3 )is larger than the volume of the TiO, octahedron (9.89 k3). I n general, the tetragonal magnetic axis of the oxygen octahedron parallel to the line AB in Fig. 2 is oriented a t an angle a! with respect to the 110 crystal direction. For interstitials a! # 0. If the interstitials are located a t the centers of the rectangular faces of the unit cell, then a! = f12'38' (see Fig. 2b) whereas if they are located a t the center of the edges, a! = 90 =t12'38'. Therefore there are four inequivalent oxygen octahedra. It is possible that an interstitial impurity may be a t or near the center of these octahedra. Assuming that the paramagnetic impurity

I O k - 2 520 Basic Ti0, 0

8

Octahedron Oxygen Titonium

Stacking

of octahedro in

loftice

FIG.3. The stacking of various octahedra in rutile (TiO?).

o.,:r

ELECTRON S P I N RESONANCE O F MAGNETIC I O N S

157

Number of 0'ons pushed

0.24

-None

E 0.36 0.48

0.84

'six 1.08

FIG.4. Impurity ion radius and possible fit as, substitutional (S) and interstitial (I) sites in rutile (Tion).Based on a radius of 1.26 A for 02ions.

ions do not further distort the octahedra, one will observe in general four identical but inequivalent spectra. The ESR spectra will coincide along the c axis. Table AVI in the Appendix summarizes the ESR results. We have listed the ions which are substitutional as well as those which have been identified as interstitials. At present there is no theory for the site preference of any particular ion and valence state. The distribution among the sites may be determined in part by the nature of the crystal-growing process. To obtain a better picture of the geometrical difference for the interstitial and substitutional sites, we .graphically illustrate in Fig. 4 the relation between ion size and its ease of accommodation for these two positions. It is easily seen that an ion with a radius greater than 0.73 A may tend to prefer an interstitial position. The ESR experimental techniques are influenced by the large dielectric with a constant of TiO,. Thus, the crystal may act as its own multitude of modes of various rf configurations and different Q's. Some of these modes are such that the rf is mainly a t the surface of the crystal. The variety of modes permits the observation of many so-called forbidden transitions. We shall now comment briefly on the individual spectra. V. Volterra, M. Sc. Thesis, Jerusalem (1959). Okaya, Proc. Z.R.E. (Znst. Radio Engrs.) 48, 1921 (1960).

" A.

158

W. LOW AND I?. I,. OFFENBACHER

6. INDIVIDUAL SPECTRA

a. Sdl, Ti3+,and Chester44945 was the first one to observe Ti3+in reduced rutile crystals. Three separate spectra A , B, and C were obtained when the temperature of observation or state of oxidation was changed. Chester associates spectrum A with interstitial Ti3+.The identification of the interstitial ion rests on the observation of four inequivalent spectra with a = 19". According to Gerritsen and Sabisky46the fact that a is greater than 12" arises from the distortion induced when the large Ti3+ion pushes apart the two closest oxygen ions. The spectra B and C are not well understood as yet. Gerritsen47 has reported the simultaneous observation of interstitial a ~ observed ~ lines (which are and substitutional spectra of Ti3+. O k a ~ has attributed to O-H bonds) together with strong lines from substitutional Ti3+. These O-H bonds may play an important role in the thermal and electrical conductivity of reduced rutile. with the magnetic z axis The V4+ ion occupies a substitutional site49f50 along the [liO] direction. Recently Yamaka and Barness1 have observed a well-resolved h y p e r h e structure. Their interpretation is that this is caused by the interaction of the unpaired (de) electron with the Ti47 and Ti49nuclei which occupy the two nearest Ti ion sites along the c axis.

b. 3d3, Cr3+,and Mn4+ Another The main spectrum is that of Cr3+ at a substitutional ~ite.~2J3 weaker spectrum is so far unexplained. The relaxation processes have been studied in detail, in part because of the possibility of using this material for masers. The spin-lattice relaxation shows the usual complications of iron group elements, including cross-relaxation effects.54~55 P. F. Chester, J . Appl. Phys. 32, 866 (1961). P. F. Chester, J . Appl. Phys. 32, Suppl., 2233 (1961). 46 H. J. Gerritsen and E. S. Sabisky, Phys. Rev. 126, 1853 (1962). 47 H. J. Gerritsen, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1.962, Vol. 1, p. 3. Academic Press, New York, 1963. 48 A. Okaya, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, T'ol. 2, p. 687. Academic Press, New York, 1963. 49 G. M. Zverev and A. M. Prokhorov, Soviet Phys. JETP (English Transl.) 12, 160 44

45

(1960). EJI

D.K. Rei, Soviet Phys.-Solid Stab (English Transl.) 3, 1845 (1962).

E. Yamaka and R. G. Barnes, Phys. Rev. 136, A144 (1964). See for example, H. J. Gerritsen and S. E. Harrison, Phys. Rev. Letters 2, 153 (1969), and J . Appl. Phys. 31, 1566 (1960). 53 I. Sierro, K. A. Muller, and R. Lacroix, Arch. Sci. (Geneva) 12, 122 (1959). 5' J. H. Pace, D. F. Sampson, and J. S. Thorp, Proc. Phys. Soc. (London) 77, 251 (1961). 66 A. A. Manenkov, V. A. Milyaev, and A. M. Prokhorov, Soviet Phys.-Solid State 51

5%

(English Transl.) 4, 980 (1962).

ELECTRON S P I N RESONANCE O F MAGNETIC I O N S

159

The spectrum of Mn4+is relatively simple. The initial splitting is smaller The larger than in the case of Cr3+and the g factor is nearly isotropi~.~6*~7 y factor is indicative of a sizable crystal field with 10 Dq probably of the order of 25,000 cm-l. There are two important differences between the Cr3+ and Mn4+spectrum. (1) The Mn4+ion has a resolved hyperfine structure which indicates that the magnetic electrons are interacting with the nuclear spin of the Ti ions, whereas the Cr3+ does not show such a hyperfine structure. (2) The tetragonal magnetic axis for Mn4f lies along the c axis ( x axis), in contrast to the more common [ l l O ] axis (x) which is found for Cr3+. These differences are attributed to the strong tendency of manganese to form covalent bonds. This tendency for Mn to form covalent bonds is also apparent from the appearance of the superhyperfine c. 3d4, Mn3+

The additional spectrum shown by many crystals containing manganese a t and below 77°K has been identified as Mn3+.This is the first time that Mn3+ has been identified in paramagnetic resonance. For d4 in an octahedral field the 5D ground level of the ion gives a lowest state of 5 E . The crystal field splits this tenfold degenerate state into singlets. The lowest pair is specified by m, = &2 and it has a zero field splitting of 6B2O. The experimental results can be satisfactorily described by a Hamiltonian which neglects the off-diagonal terms B4202 and B44044:57*58 X

=

gpHiSi

+ AiSiIi + BZoOZo+ BZ202*+

B4'(04'

+ 5044)

(6.1)

where S = 2 . The quartic term which appears in the Hamiltonian for S = 2 is important in describing the angular dependence of the spectrum. The total zero field splitting is large, about 13.7 cm-'. d. 3 ~ Fe3+ ~ ,

The spectrum is characteristic of a substitutional S-state ion in a n orthorhombic geometry. It has been analyzed fully by Carter and O k a ~ a . ~ ~ Volterra42has included in his treatment all the quartic terms BqO,B2, and H. G. Andreson, Phys. Rev. 120, 1606 (1960). Stahl-Brade and W. Low, Unpublished data (1958). 67a These covalent bonds are formed with the four oxygens in the plane passing through 66

" R.

the c axis. This bonding has the effect of widening the angle between the oxygens along the y axis (which originally is 89.8" in TiOp). A simple crystal field calculation shows that this will shift the tetragonal axis from the x to the z direction. H. J. Gerritsen and E. S. Sabisky, Phys. Rev. 132, 1507 (1963). " L). Id. Carter and A. Okaya, Phys. Rev. 118, 1485 (1960).

160

W. LOW AND E. L. OFFENSACHER

B44.However, these terms represent only a small correction. Foner et aZ.60 have utilized the level scheme of Fe3+ to operate a maser at a frequency near 72 kMc. e. Sd7, Co*+

Co2+ has a ground state 4T, which is orbitally degenerate. Therefore, the values of g and the crystal field parameters are particularly sensitive to the orthorhombic symmetry of the crystal field. Yamaka and BarnesGl have interpreted the spectrum of Co2+a t a substitutional site, neglecting second-order corrections such as (15/2)X/lO Dq. Their evaluation of the spectrum yields an orbital reduction factor, k, of 0.69; this indicates an unusually large amount of charge transfer which is attributed to the high polarizibility of the crystal. The orbital reduction factor, however, is increased upon inclusion of the second-order correction in the calculation. Spin-lattice relaxation measurements were made by Zverev and P r o k h o r o ~ , 6who ~ ! ~ ~conclude that the next excited doublet is a t 107 = t 5 cm-l, assuming an Orbach mechanism for the relaxation. The spectrum of Co3+ has not been observed, presumably because the ground state is diamagnetic. f. S#+, Ni3+

There are in general two spectra.46The main spectrum is caused by a nickel ion in a n interstitial site with a = 9.1”. The g values indicate an + tetragonal orbitally quenched state and they are similar to those of C U ~in symmetry. I n the strong crystal field a 2 E level may be the lowest level, and the g factor would then be close to 2. The variation in the g factor, in particular that of gz > gz, is consistent with the fact that for an interstitial site two oxygen ions form a compressed octahedron in contrast to the slightly elongated octahedron for a substitutional site. A second spectrum is obtained upon irradiation of the crystal with ordinary light and quenching it below room temperature. The magnetic axes coincide within the experimental accuracy of 0.2” with the axes of the octahedron of the substitutional site. These spectra have been attributed to Ni3+, but this assignment is not definite since they could also arise from Nil+. (A similar situation exists in SrTiOe and is discussed there.) Foner, L. R. Momo, J. B. Thaxter, G. S. Heller, and R. M. White, “Quantum Electronics,” p. 555. Columbia Univ. Press, New York, 1961. 61 E . Yamaka and R. G. Barnes, Phys. Rev. 126, 1568 (1962). 62 A. M. Zverev and A. M. Prokhorov, Soviet Phys. J E T P (English Transl.) 16, 303 (1963). 88 A. M. Zverev and A. M. Prokhorov, Proc. 1st Intern. Cmf. Paramagnetic Resonance, Jerusalem, 1.969,1’01. 1, p. 13. Academic Press, New York, 1963. 60s.

161

ELECTRON SPIN RESONANCE OF MAGNETIC IONS

g. Sd8, Ni2+

Most of the nickel appears as a divaleiit ion in interstitial sites.46This is in contrast to other divalent ions such as Co2+and Cu2+.The reason for the appearance of these interstitial spectra is not clear. h. Sd9, Cu2+

The large ionic radius of Cu2+ (0.72 A) would make it fit better as a n interstitial ion. However, it is found in a substitutional position. (The splitting between doublet and triplet levels as observed from optical spectra is similar to that found with hydrated salts and this is generally also true for other paramagnetic ions in r ~ t i l e . ~ ~ )

i. 4d1:Nb4+,&lo5+, and 5d1Ta4+ The spectra of the ions of Nb4+,Ta4+,47and Mo5+64 indicate that these ions are also a t substitutional sites. j. 4f1:Ce3+

Chester45has detected a spectrum which can be interpreted to arise from a Ce3+ in a substitutional site. This was presumably produced from Ce4+ upon vacuum heat treatment of the originally colorless crystal. k.

4f7,

Gd3+

YamakaA5has observed a Gd3+ spectrum a t a substitutional site in a n oxidized Ti02 crystal. The seven lines were fitted to a spin Hamiltonian X, = @ H - S

+ B2'02' + B4'04' + B6'06' + BZ20z2+ B42042+

B44044,

(6.2) neglecting the Be"O6" operators. He observed a sharp positive temperature dependence of the axial parameter B2', which is a t present not understood.

1.

4fn, Er3+

Gerritsen and Sabisky have reported the detection of Er3+ spectrum presumably a t a substitutional ~ i t e . ~ 7

7. GENERAL CONCLUSIONS (1) The majority of the ions with d* configurations can be incorporated in the TiO, lattice. Most of these ions are to be found a t a substitutional site, and a few occur a t the interstitial site. The distribution between the two different sites is not yet completely understood.

R.T.Kyi, Phys. Rev. 128, 151, 1962; T. Chang, Bull. Am. Phys. Soc. E. Yamaka, J . Phys. SOC.Japan 18, 1557 (1963).

[218, 464, 1963.

162

W. LOW AND E. L. OFFENBACHER

(2) Rutile has a great tolerance for different and unusual valence states, probably more so than other ionic lattices. However, the axes of the orthorhombic symmetry are preserved even for misfits such as divalent ions a t the Ti4+ site. This shows that charge compensation is far away or at random. (3) The paramagnetic resonance spectra can all be explained with the usual spin Hamiltonian used for ionic lattices. The ESR spectra and the few optical spectra that have been observed follow a pattern similar to that found in other inorganic lattices. No particular effects of the large polarizability of the TiOz lattice have been observed on these spectra. (4) The majority of the substitutional ions have their tetragonal magnetic axes along the [ l l O ] direction. However, there are some that have it along the c axis. Although this may eventually be explained on the basis of a theory of covalent bonding, no such theory has as yet been developed in detail. ( 5 ) The rare-earth ions can be substituted for the Ti ion despite the is far larger than the space available. fact that the ionic radius (1.34 This is of importance in interpreting the more complex spectra of rare-earth ions in the titanates.

A)

VI. Perovskites

The perovskite structure has the composition A B 0 3 , of which BaTiOy is the best known example. Many of these compounds have very interesting properties. Some of these such as BaTi03 and PbTiOe arc ferroelectric a t room t e m p e r a t u r ~ . Others ~ ~ . ~ ~remain nonpolar or paralectric over nearly the whole temperature range. An example of this is SrTi03.6* In reduced form this crystal is a semiconductor and becomes superconducting a t very low temperature^.^^ Other systems such as LaMnOs are antiferromagnetic,70whereas mixed LaMn03 and SrMn03 are ferromagnetic.71 For a discussion of ferroelectric and antiferroelectric properties of perovskite materials see Jona and Shirane66and Kanzig.67 The majority of these compounds show phase transitions a t welldefined temperatures. Others, however, such as LaA103, show no phase transition but a deformation of the lattice which is a function of the temperature. I n addition, there are classes of compounds of the type KMgF, F. Jona and G. Shirane, “Ferroelectric Crystals.” Macmillan, New York, 1962. W. Kaneig, Solid State Phys. 4, 1 (1957). s8 H. Griincher, Helv. Phys. Acta 79, 210 (1956). $9 J. F. Schoolery, W. R. Hosler, and hl. L. Cohen, Phys. Rev. Letters 12, 474 (1964). 70E.0. Wollan and W. C. Koehler, Phys. Rev. 100, 545 (1955). 71 C. H. Jonker and J. H. Van Santen, Physicu 16, 337 (1950). 67

ELECTRON S P I N RESONANCE O F MAGNETIC I O N S

163

which preserve the ideal perovskite structure down to very low temperatures. These will not be discussed in detail. Such striking and varied behavior can be studied in part by substituting a paramagnetic iinpurity for a small fraction of the positive ions. Changes in the phase transition, the nature of these transitions and domain structures as reflected in changes of the parameters in the spin Hainiltoiiian can fruitfully be investigated by ESR spectra. The study of the many different compounds having the perovskite structure has hardly been exploited. The only system where a fairly systematic study has been undertaken is SrTi03. One of the reasons is the difficulty in obtaining reasonably good single crystals. Since many of these compounds have large dielectric constants which are temperature dependent, temperature stability has to be insured if consistent results are to be obtained. It is likely that considerably more effort will be spent on the study of these systems when new and better crystals become available. The ideal perovskite structure is shown in Fig. 5. The ion B4+ (usually Ti4+)is surrounded by a perfect octahedron of oxygen ions whose axes are along the cubic axes. The octahedron is similar to the Ti06 structure in rutile except that it is not distorted. The A ion has twelve nearest oxygen neighbors and eight Ti neighbors. Goldschmidt has introduced a tolerance factor t, defined such that RA Ro = t f l ( I Z B R o ) where the various R’s are the ionic radii of A , B, and 0 ions. This tolerance factor may vary between 1 and 0.77 and it permits a large variety of compounds. Within these limits one can substitute different ions A and B in the perovskite structure. Many of the iron group elements are mainly a t the B site. Other ions such as the rare-earth group will primarily go into the A site. I n some cases the valence state is trivalent for both the A and B ions; in others the valence state of A may be monovalent with B divalent while the surrounding anions are monovalent.

+

A

+

0 0

FIG.5. Structure of the ideal perovskite. The A ion (i.c., Ba2+) is surrounded by twelve oxygen and eight B (Ti4+)inns. The B inn is a t a center of a prrfwt octahedron of oxygen ions.

164

W. LOW AND

E.

L. OPFENBACHER

Very few of the crystals have the ideal structure. The majority show deformations of one kind or another. These will be discussed for the various particular systems. In this review we shall be concerned primarily with BaTi03 and SrTi03.

8. BARIUM TITANATE (BaTiO 3) This is the most extensively investigated ferroelectric crystal. Above 120°C the crystal is cubic and centrosymmetrical. Below 120°C the crystal becomes tetragonal with a point group 4 mm and the fourfold axis along the ‘(c” tetragonal axis. One edge becomes elongated and the other two edges compressed. There are six equivalent possibilities for the direction of the polar axis, resulting in two sets of three different domains. The two sets differ only by being parallel or antiparallel to each other. At temperatures between 5 and -90°C there is an orthorhombic phase, the polar axis lying along one of twelve [llO] directions. At temperatures below -90°C the symmetry becomes rhombohedral, the polar axis lying along the [lll] direction. At all these phase transitions there is a discontinuous change in the dielectric constant. The dielectric constant in the cubic phase obeys a Curie law of the type E

=

C/(T

-

To) with

To= 118°C.

(8.11

The spontaneous polarization shows a discontinuity near the Curie temperature, jumping from zero to a high value at 120°C which is indicative of a first-order phase transition. Other properties such as the birefringence, lattice constants and thermal expansion show discontinuities near the various phase transitions. The tetragonal phase can be described by selective movements of Ti and 0- - ions parallel and antiparallel to the c axis. These shifts are of the order of 0.1 to 0.05 A, about 2 to 5% of the bond distances. Before describing the ESR results, a few general remarks should be made concerning the experiments. (a) The large dielectric constant causes the crystal to act as its own cavity. The rf configuration is unpredictable, and the high Q mode will not always yield the best signal-to-noise ratio. This is similar to what has been found in rutile. (b) Because of the peculiar rf configurations and the possibility of two or more modes coexisting at a given frequency, so-called “forbidden transitions” corresponding to AM = 2, 3, 4, 5 can be seen for nearly all directions of the magnetic field. The shape of the spectral line may change as the

ELE CT RON S P I N RESONANCE O F MAGNETIC I O N S

165

magnetic field is moved from H I to , the c axis to a different angle corresponding to part absorption part dispersion or to full dispersion.?2 The most extensively studied spectra are the S-state ions, in particular Fe3+7 w 4 and Gd3+.75Most of these results relate to the tetragonal spectrum and in particular to “c” domains, i.e. domains which have their polarization perpendicular to the plates of the crystal. Both the Fe3+ and the Gd3+ spectra clearly show a tetragonal distortion. A typical spectrum of Fe3+ is shown in Fig. 6. I n this figure are indicated all the permitted AM = f l transitions, as well as the weaker AM = 2, 3, 4, 5 transitions. Both the Fe3+and Gd3+spectra show abrupt changes in the spectrum near the transition temperature. Below the transition temperature the tetragonal splitting is found to be temperature dependent. This is shown in Fig. 7. 10,000

6000

; ; 2000 v)

? l o

-m -2000 -6000

-10,000

I000

3000

5000

7000

~ ~ a u s s )

FIG.6. Observed ESR transitions of the Fe3+ spectrum in barium titanate (BaTi03).76 The results of these spectra are briefly summarized as follows. (a) The cubic-tetragonal phase transition has been clearly demonstrated to affect parameters in the spin Hamiltonian, at the correct phase transition temperature. (b) In the tetragonal case there are in general three sets of lines corresponding to three different domains. It is found that the “c” domains predominate. Low and ShaltieP have also studied lines of nearly pure “n” domains. D. Shaltiel, Ph.D. Thesis, Technion, Haifa (1959). The surface of the BaTiOj apparently polarizes the marker such as D.P.P.H. when placed in intimate contact. W. Low and D. Shaltiel, Phys. Rev. Letters 1, 51 (1958). ?* A. W. Hornig, R. C. Rempel, and H. E. Weaver, Phys. Chem. Solids 10, (1959); see also D. Shaltiel, Ph.D. Thesis, Technion (1960). 76 L. Rimai and G . A. deMars, Phys. Rev. 177, 702 (1962). 70 W. Low and D. Shaltiel, Unpublished data (1958). ?*

166

W . LOW AND E . L. OFFENBACHER

7400

7'300 6600 D

6200

2 5500 D -

=

5400

'

5000

-

4600 4200

3RO

I

20

30 40

50

60 70

80

90

I00 I10 120 130 140 150

FIG. 7. Temperature dcpendence of ESR spectrum of Gd3+ in barium titanate (BaTiO3).T5To be noticcd is the sudden collapse of the axial spectrum into the cubic spectrum, indicative of a first-order phase transition. L. Riniai and G. A. debfars, Phys. Rev. 127, 702 (1962).

(c) The transition from tetragonal to cubic is drastic with orily a small temperature hysteresis. A.s one approaches the cubic region from below, the tetragonal spectrum decreases in intensity and the lines broaden. Near the transition temperature it is very difficult to keep the system sufficiently stable. Above the transition temperature the lines are weak and increase in intensity with T. The lines, however, are broad, presumably because of relaxation effects.At still higher temperatures the lines decrease in intensity because of the influence of the Boltzmann factor. (d) The line widths of the individual transitions are strongly angularly dependent and are only relatively sharp when they are parallel to one of the cubic axes. This can possibly be taken as an indication that the various domains are not quite parallel to one an0ther.~~176 (e) Hornig et ~ 1and. Low ~ ~and Shaltie173have observed peculiar phase relations of individual lines, and a temperature dependence of the phases of the lines. Some lines have dispersion or emission character. It is believed that this arises from the high dielectric constant and the many nearly coincident modes with different coupling constants. The distribution of these modes is strongly temperature dependent because of the changes in the magnitude of the dielectric constant. (f ) Merz77 has shown that the piezoelectric strains are proportional to the square of the instantaneous polarization P2, which is indicative that the crystal is centrosymmetric. Rimai and d e M a r P have shown similarly that the axial variation of the field parameter bro is proportional to P'. i7

W. Merz, Phys. Rev. 91, 513 (1953).

ELECTRON SPIN RESONANCE O F MAGNETIC IONS

167

(g) Resonances were seen for Fe3+ a t the two lower phase transitions, but the nature of the spectra have not been studied. (Recently Sakudo and Unoki have reported the spectra of Fe3+in the rhombohedra1 phase.??b) (h) A comparison of the cubic field splitting parameters of Fe3+ in BaTiOa with other cubic crystals indicates that it is smaller by a factor of 2-3 compared with Fe3+in MgO, SrTiOs, or TiOz. It is about twice as large as that in CaO. Nevertheless, it is believed, mainly because of ionic radii considerations, that the iron substitutes for the titanium ion. The reduced magnitude of the initial splitting is a t present unexplained. (i) The Gd3+ ion is believed t o substitute for the Ba ion. The small magnitude of b40 (of the same order as in SrTiOa and SrO) seems to confirm this.

( j ) Application of the electric field by Hornig et al. has not resulted in an appreciable shift or splitting of the lines. On the basis of the polarization data one would have expected such a shift (see f ) . It is possible that experimental difficulties such as poor contact of the electrodes prevented the observation. Low and Shaltiel have applied ac fields and have obtained relative changes in the intensity with an ac modulation. A spectrum of Mn2+has also been observed a t room temperature. Only 3 3 -4 transitions could be disti1iguished.7~It is thought that the Mn2+ is substituted a t the Ba site. 9. STRONTIUM TITANATE (SrTi03) a. General Properties

There are a number of reasons for the present interest in this crystal system. (1) The crystal has the ideal perovskite structure a t room temperature. It undergoes a phase transition a t 110 f 2 . 5 " K as determined from ESR spectra; ESR spectra may shed some light regarding the nature of this transition. (2) Many iron and rare-earth group elements can be substituted at the Ti and Sr sites respectively. These ions are located a t octahedral and dodecahedra1 sites and this permits the investigation of the magnetic properties a t these two sites. ( 3 ) It is hoped that there is some correlation between the temperature dependence of the ESR spectra and the polarisability of the crystal. The room-temperature dielectric constant e of SrTi03 measured a t 1 kc/sec is about 370. Above 110°C e follows a Curie-Weiss law. At 1.4"K the dielectric constant increases t o about 18,000. No definitive evidence has been produced indicating that the crystal is ferroelectric. 78

T.Sakudo and H. Unoki, J . Phys. SOC. Japan 19, 2109 (1964) W. Low, Private communication (1900).

168

W. LOW AND E. L. OF F E N BA CH ER

I

Temperaturel'Ki

FIG.8. Temperature dependence of ESR spectrum of Gd3+ in strontium titanate (SrTiO8).7jThere is a gradual transition from the axial spectrum to the cubic spectrum near the transition temperature. L. Rimai and G. A. deMars, Phys. Rev. 127,702 (1962).

The ESR spectra show that there is a smooth transition from cubic symmetry to tetragonal symmetry. This is shown in Fig. 8 for the case of Gd3+ and will be discussed more fully later. Below the transition temperature down to 2°K the spectra can be interpreted to arise from three magnetically inequivalent ions. For impurity ions, substituting for the Ti ion, it can be shown that they are exposed to a dominant cubic field potential with some small tetragonal component along the cubic a ~ e s . 7These ~ three types of spectra, often called domains, have also been recently observed optically.80The smallness of the axial distortion (which is temperature dependent) has been confirmed in part from nuclear magnetic resonance 011 the Ti ion, which was studied near the phase transition temperature.81 No change in the Ti resonance was observed upon going through the phase transition. This suggests that the ionic displacements are small, and that only a small distortion of the titanium-oxygen octahedron occurs. All spin resonance results on iron group elements are consistent with this picture. However, the results on the rare-earth group elements, discussed in detail later, cannot at present be explained in a consistent manner. It is believed that the majority of the rare-earth ions substitute for the Sr2+ion. I n the absence of additional charge compensation this ion would be surrounded by twelve oxygen ions and eight titanium ions. Table I shows that these two crystal fields are of the opposite sign if one assumes point charges and absence of polarization effects. The contribution of the eight titanium ions overbalances that of the oxygen ions. From this picture one would, there79

K. A. Muller, Helv. Phys. Acta 31, 173 (1959). A. Kikuchi, and Y. Kodcra, J . Phys. SOC.Japan 18, 459 (1963). M. J. Weber and R. R. Allen, J . Chem. Phys. 38, 726 (1963).

80E. Sawaguchi,

ELECTRON SPIN RESONANCE OF MAGNETIC IONS

169

fore, expect only a small negative cubic crystal field.&*The tetragonal component could play the more important role in the interpretation of the resonance results. Nuclear magnetic resonance has revealed a detectable change in the Sr resonance a t the phase transition.81This may indicate a larger displacement of this ion. It is also consistent with a large fraction of the ESR data on rare-earth ions which indicate a stronger axial contribution for the Sr site than for the Ti site. It should be pointed out that measurements near the phase transition are not easy to make. The temperature has to be carefully controlled since the spectrum changes rapidly. Moreover, there seems to be a time lag while the various domains are being formed, and therefore one has to change the temperature very slowly if consistent results are to be obtained.

b. Spectra of Iron Group Elements 3d3, Cr3+,and Mn4+ The chromium spectrum in SrTiOs consists of a narrow line flanked by the four resolved hyperfine lines.83Both the g factor and the hyperfine structure constants are similar to those found in MgO. This is a rather general result for nearly all the iron group elements. The optical data indicate that the spectrum of Cr3+in SiTrOHis similar to that in AI2O3,and that it has a cubic field strength slightly larger than in Mg0.84f85This is a strong indication that the Cr3+substitutes at a site of octahedral coordination, i.e., a t the Ti site, without any noticeable charge compensation in the neighborhood. Weaker lines have been observed corresponding to Cr3+ion in a n axial site. The axial distortions are presumably due to charge compensation.86 Below the transition temperature the cubic line splits into a number of anisotropic lines. If the magnetic field is parallel to the [loo] direction, one observes in general five lines. Along the [lll] direction the quintet collapses into a single line. The quintet pattern can be explained to arise from a tetragonal field acting along the cubic axes. For the ions with the tetragonal component parallel to the [loo] direction, one observes a triplet for which the two outermost components are separated by 4bz0.For the ions with the tetragonal components along the [OlO] or [OOl] direction the separation of the spectrum components 2bz0, is only 4 X 10-4 cm-1 a t room temperature. This represents a very small tetragonal distortion. The value of b,O is somewhat temperature dependent. R. S. Rubins and W. LOW,Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1963, Vol. I, p. 59. Academic Press, New York, 1963. 83 K. A. Miiller, 7 2me Colloque AmpBe. Arch. Sci. (Geneva) 11, 150 (1958). 84 K. A, Miiller, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, Vol. 1, p. 51. Academic Press, NeF York, 1963. 85 L. Rimai, T. Deutsch, and B. D. Silvrrman, Phys. Rev. (to be published). 86 W. Low, Unpuhlished results (1963). 82

170

W. LOW AND E. L. OFFENBACHER

TABLE IV. SPINHAMILTONUN PARAMETERS FOR Mn4' A

Crystal host MgO A1203

Ti02 SrTiOt

b

9

in units of lo-' cm-l

1.992 1.994 1.993 1.994

71.4 f 0 . 5 69.4 f 1 71.8 f0 . 3 69.4 f 1

Reference n b c d

W. Low, Unpublished results (1962). S.Gcschwind, P. Kisliuk, M. P. Klein, J. P. Remeika, and D. L. Wood, Phys. Rev.

126, 1684 (1964). c H. J. Gerritsen and E. S. Sabisky, Phys. Rev. 132, 1507 (1963). d K. A. Muller, Phys. Rev. Letlers 2, 341 (1959).

Muller has identified a characteristic six-line pattern spectrum arising from a manganese ion, as due to Mn4+.*7The evidence of this valence state arises from ( I ) the smaller g factor, ( 2 ) the reduced hyperfine structure constant, and (3) the position of the forbidden transitions A M = f l , A m = ~ 1 The . last mentioned evidence is very convincing and will be reproduced here. The simple spin Hamiltoniari X = gPH.S

+ AS-I

-

gnPnH.I

(9.1)

has energy levels which are given to second order by

H(M

+ 1, m

--$

M , m)

=

H a - Am - ( A 2 / 2 H 0 )

X (I(I

+ 1) - m ( m + 1 ) + ( 2 M - l ) m }

(9.2)

and

H ( M + 1 , m - 1,M, m ) = Ho - Arrz -

M 2- m2

+ + S(S+ 1)

(A2/2Ho)(I(I 1 )

-

+ 4 M m - 3 M + 3 m - 2 } - (gnPn/g)Ho.

(9.3)

The latter transitions are forbidden and therefore are much weaker and are reduced in intensity. For H,f 11 H o this reduction is about

(I

+ m)(I - m + ~ ) ( A / H o ) ~ . +

The forbidden transition depend on S ( S 1 ). Herice au accurate measurement of the individual positions of these lines would determine S. Muller showed that he could fit the spectrum with S = $ but not with S = 5, which established that the spectrum is due to Mn". Table I V gives the values of the various parameters of Mn4+ in different octahedral hmts. 87

I(.A. Muller, Phys. Rev. Letters 2, 341 (1959).

171

ELECTRON SPIN R E S O N A N C E O F MAGNETIC IONS

TABLEV. INITIAL SPLITTINGOF Fea+

Host

MgO BaTiOa

SrTiOa KTa03

b

IN

Temperature (OK)

bro 10-4

295 425 300 300 77 4.2 4.2

102.5 51 .O 51 .O 98.8 110.4 112.8 172.5

PEROVSKITE TYPE-MATERIALS btO cm-'

Reference

a b 930 c

7.3 16.1

cl

e

W. Low, PTOC.Phys. SOC.(London) B69, 1169 (1956). A. W. Hornig, P. C. Rempel, and H. E. Weaver, Phys. Chein. Solids 10, 1 (1959). K. A. Miiller, Helv. Phys. Acta 31, 173 (1959). W. I. Dobrov, R. F. Vieth, and M. E. Browne, Phys. Rev. 116, '79 (1969). W.Wemple, Ph.D. Thesis, M.I.T. (1963).

Actually, the lines are not exposed to a pure cubic field even a t room temperature. A small splitting of a few gauss can be observed with H parallel to the [loo] direction. The nature of this splitting is a t present riot ~ n d e r s t o o d Cooling .~~ below the transition temperature induces additional splittings which are very similar to the case of Cr3+. 3d5, Fe3+. This spectrum has been studied intensively by Muller.79 A t room temperature the spectrum is again similar to Fe3+ in MgO and the parameters do not differ significantly. Again this can be taken as strong evidence that the Fe3+ion substitutes for the Ti4+ion. The lines are relatively sharp and indicate that, on the whole, long-range order must exist in this crystal. Below the transition temperature three sets of domains are found with only a small axial splitting. This is very different from the iron spectrum in BaTiOa in which the axial splitting predominates. (See Table V.) Dobrov et aLS8have shown that both the axial and the cubic part are temperature dependent. Rimai et aLS5have analyzed the temperature dependence and have been able to obtain a good fit with an equation cm-1, b1 = -6.2 x 10-6 cm-1, b, = 100 x where bo = 115 x cm-I, and To = 49°K. They have also found that the pressure dependence of b d O (and similarly the g shift for Cr3+) is about twice as large in SrTiO3 as for Fe3+in MgO. These results, with some additional inferences from the optical data, lead these authors to conclude that the local conipressibility is about twice as large as the bulk value. They feel that the polarizatiori plays only a minor role. 8q

\T. I . Dohrov, R. F. Vieth, and M. E. Browne, Phys. Rev. 116, 79 (1959).

172

W. LOW AND E. L. OFFENBACHEH

Recently iMuller et aLs9 have found an axial spectrum with 911 = 2.0054 f O . O O 1 , 91 = 5.991fO.001 (at 1 = 3.2cm), and g r = 5.961 fO.OO1 (at X = 1.85 cm). They infer from the frequency dependence of g 1 that Fe3+ is exposed to a tetragonal field with bzO = 1.42 f0.15 cm-'. Lowg0 has examined this spectrum using magnetic fields up to 60,000 G in the hope of inducing the Q 43 transitions which would occur a t magnetic fields corresponding to 2bk0f hv. No such transitions were observed and the initial splitting, therefore, must be considerably larger. The strong tetragonal field is attributed to local charge compensation a t a nearestneighbor site. d7, ds, dg; nickel. Rubins and Lowx2have studied the nickel spectrum in SrTi03. These spectra may be classified as follows: (1) A broad line of about 120 G width a t room temperature and of about 90 G a t 80°K.At higher microwave power and low temperatures a narrow line is found at g = 2.204. These lines are attributed to the normal and to the double quantum transitions r e s p e ~ t i v e l yFrom . ~ ~ the observed g factor one concludes that this spectrum is caused by Niz+(d8) a t a Ti4+site. S o additional splitting is observed below the transition temperature. This can be interpreted to mean that the random deviation from cubic symmetry (inferred from the large line width) must be larger than the small tetragonal splitting caused by the domain formation. (2) An isotropic spectrum a t 203°K with g = 2.180.The line narrows as the temperature decreases. Below the transition temperature one observes an axial spectrum corresponding to the three directions of the domain formation. The glI and g1 of this spectrum are temperature dependent, Ag = glI - 91 increasing as the temperature is lowered, but with the center of gravity of the g factor (gll 2g~)/3remaining approximately constant. The interpretation of this spectrum is not certain. Rubins and Low have indicated that the spectrum arises from Ni3+ in a strong octahedral field which would result in a 2E level being lowest. However, in the absence of a dynamic Jahn-Teller effect this would result in more than one line. A dynamic Jahn-Teller effect may under certain conditions explain the experimental results.92 We feel that the assignment Ni1+(d9)cannot be ruled out. The main argument presented against this assignment is that for the isoelectronic

+

E. S. Kirkpatrick, and R. S. Rubins, Phys. Rev. 136A, 86 (1964). W. Low, Unpublished results (1964). 91 J. W. Orton, P. Auzins, and J. E. Wertz, Phys. Rev. Letters 4, 128 (1960). 82 K. A. Muller and R. S. Rubins, To be published. We are grateful t o see a preprint of

s9 K. A. Muller, go

this paper. These authors give a lengthy argument for the ( b ) spectrum to arise from a d7 configuration.

ELECTRON S P I N RESONANCE O F MAGNETIC IONS

173

Cu2+(3d9)one always finds that yll > qr. However, Low and Sussg3recently have shown that Nil+ in CaO has 911 < g r and that also in this case Ay is strongly temperature dependent (see Table AII). It is not quite clear whether the tetragonal spectrum below the transition temperature is connected with the domain formation or with a JahnTeller effect or both. The temperature dependence of Ag probably is connected with an increase of the deformation, as can be inferred from Fig. 8. (3) An axial spectrum is observed above the transition temperature. Below the transition temperature each line splits into a quintet. The quintet arises from the domain formation. However the additional tetragonal component is smaller than the original tetragonal distortion. Rubins and Lows2assume that this spectrum arises from Nil+. The axial field arises from a charge compensation. Muller and Rubinsg2attribute this spectrum to Ni3+with a local charge compensation. In addition, Muller and Rubins find that a fourth unstable spectrum with 911 > gr can be induced with ultraviolet light. They attribute this spectrum to Nil+. From the above it is obvious that a t present there is no certain identification of the Ni spectrum, except for Ni2+. To some extent the same is found to be true in Ti02. Diverse spectra. A number of scientists have attempted to detect the spectrum of Go2+or Co3+.In most crystals cobalt apparently is in the trivalent state. In a strong octahedral field the ground state of Co3+(d6)is a singlet, and no resonance would be expected. A weak signal presumably caused by Go2+was observed, but it is likely that it was mainly caused by Go2+ a t the surface layer of the crystal.94 A spectrum due to Ta4+has been detected but needs still further analysis.94

c. Rare-Earth Spectra

As mentioned before, the main difficulty in the interpretation of the rare-earth spectra is the lack of knowledge on the nature of the crystal field: whether the ions are exposed to a strong cubic field and weaker axial field as found for the iron group elements or vice versa. It is usually assumed that the rare-earth ions are a t the Sr2+site, and that no charge compensation is present a t nearest-neighbor sites. There is no experimental evidence for either of these assumptions. The main argument for assigning the rare-earth ions to the strontium site is from consideration of the respective ionic radii. In this connection it should be remembered that the spectrum of Gd3+has been observed in Ti02 which proves that rare-earth ions may be substituted a t the smaller titanium site a t least in small con-

’’ w. Low and J. T. Suss, Phys. Letters 7, 310 (1963). w.Low, Unpublished results (1964). 94

17-1

\\'. LOW A N D E. L . OFFEKBACHER

centrations. Consideration of ionic radii when only small amounts of impurity ions are involved, must a t present be very circumspect. 4f', Ce3+. All rare-earth ions, with the exceptions of S-state ions and Yb3+ can be observed only a t liquid helium temperatures. This indicates that the spin-lattice relaxation is fast and that there are energy levels within 100-200 cm-l from the ground state. For Ce3+a line is observed a t 911 = 3.005,91 = 1.12. Within the J = 8 manifold the closest fit is given by the wave function8*

+* = 0.93 I J , =

>

A0.37 I J ,

=

TS),

(9.5)

with calculated g factors of gll = 3.36, g 1 = 1.32. The deviation can be accounted for by admixing a contribution from the nearby J = 4 level. The question then is what potential can account for such a wave function. In general the tetragonal potential can be written as

v = Vcubic+ Azovzo + A40V40,

(9.6)

where the last two terms are the axial contributions. The level J = 8 is split in a cubic field into a rs quartet and r7doublet. The quartet is expected to be the lowest level in a twelve-coordinated symmetry (b4 < 0 ) and the I'7 level for a sixfold coordination (see Table I ) . The center of gravity of the experimental g factor cannot be fitted to a r7level, and it seems to rule out a titanium site (assuming that the cubic field a t the Ti site dominates the spectrum). Awave function of the type (9.5) can be constructed if there exists a very strong tetragonal distortion which leaves the J , = 8 at the lowest level. Indeed this is found for Ce3+ in CaFz(b4 < 0 ) where the g values are similar to those in SrTi03 (gl I = 3.045, g1 = 1.384).A strong tetragonal distortion can arise from the relative displacement of the Sr2+ ions with respect to the Ti4+ions. An alternative description is to assume a dominant cubic field, leaving the I's lowest (b4 < 0), superimposed upon a small tetragonal distortion. f3, Rid3+.The J = 8 level is split into two quartets rsand one r6.According to Table I11 the r6level can be lowest for b4 > 0, but only for large values of &/&. For b4 < 0 a r8 quartet is the lowest for any reasonable b6 v a l ~ e . ~ ~ . 9 5 The spectrum can be fitted again either with a I'6 level (giving g = 8) or by one of the doublets of the quartet which is split by the tetragonal field. It is remarkable however, that the g factor shows only small anisotropy. One would be inclined to attribute this to a small tetragonal field and a dominant cubic field. These results are as yet not sufficient to determine whether the Nd3+ is a t the Sr2+or at the Ti4+site. 95

L. Rimai and G. A. &Mars, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalri:,, 1862, Vol. I , p. 51. Academic Press, New York, 1963.

ELECTRON SPIN RESONANCE OF MAGNETIC IONS

175

f13, Yb3+. Next to the Ce3+ this is the simplest spectrum. Similarly to the Nd3+ spectrum it is found that the anisotropy in the g factors is not large.82 The weighted average of the g factors is similar to that of a doublet, 5, and cannot be fitted to a r7doublet. It is seen from Table I11 that a dominant octahedral b40 parameter would result in a rs doublet being lowest. A dominant potential b4O from the twelve oxygen ions would result in a r7level, but never in a re doublet. Hence the g factor can only be fitted i f either the Yb3+ ion i s at a site of the Ti4+or i f the potential at the Sr2+i s determined mainly by the eight Ti4+ions. In this latter case the fourth-order potential term would have to be very small. Since the spectrum can be observed even a t liquid nitrogen temperature we take this as an indication of a slow spin-lattice relaxation caused by a fairly large splitting of the various doublet states. This would be achieved a t the Ti4+site because of the large value of the octahedral field. We feel that this is strong evidence for Yb3+being a t a Ti4+substitutional site f7, Gd3+ and Eu2+. This spectrum has been investigated in detail by Rimai and de R i f a r ~The . ~ ~gadolinium ~~~ spectrum is particularly instructive since it clearly shows the difference of the behavior of this ion in BaTi03and SrTi03. In Fig. 8 the temperature dependence of the individual seven lines is shown. The smooth variation across the transition temperature should be compared with the abrupt change as seen in Fig. 7. This clearly indicates that the phase transition is not of the first order, but probably of the second order. The structural changes must, therefore, be different from those found in BaTi03. In addition, unlike BaTiOa, there is no loss of intensity or drastic change in line width as one goes through the transition temperature. Hence, a t least a large fraction of the whole lattice must move coherently from the cubic to the tetragonal positions. It is, therefore, likely that the tetragonal phase has no permanent polarization and indeed there is no definitive evidence for ferroelectricity. As was found for the iron group elements the axial parameter bzo changes appreciably with the temperature below the phase transition. The isoelectronic ion Eu2+ shows some striking differences from Gd3+: (a) the cubic field component is much larger, (b) the axial component below the transition temperature is relatively small, (c) there is hardly any temperature dependence of b40, whereas there is such a dependence for Gd3+. There are no definite explanations for these differences. One possibility is to assume that in the case of Gd3+ there is considerable polarization. If the Gd3+is a t the Sr2+ site the extra charge would attract the surrounding 02-and cancel out the cubic field contribution from the Ti4+ ions. This would explain the small cubic field and the relatively large tetragonal component. It would also account for the temperature dependence of b4O 96

L. Rimai and G. A. deMars, Phys. Rw. 127, 702 (1962).

176

717. LOW AND E. L . OFFENBACHER

since the degree of cancellation of the two fields would depend on the lattice distances. I n the case of Eu2+the field of the eight Ti4+would predominate, and the relative displacement below the phase transition would not affect the various parameters very much. A simpler but ad hoc assumption is to consider the two ions as substituting a t different sites, E u ~ a+t Ti4+,and Gd3+ at Sr2+. For Gd3+ the cubic field is small, but below the phase transition there are relatively large tetragonal components, as shown from the NMR spectra. The Eu2+ ion, on the other hand, experiences a large cubic and small tetragonal field. The above discussion has shown that it is a t present impossible to give a unique fit with one set of potentials for all the rare-earth ions. It is not ruled out that some rare-earth ions substitute for the Sr and others for the Ti ions. The necessity of further work is clearly indicated. 10. ~IISCELLANEOUS PEROVSKITE STRUCTURES

a. Lead Titanate: Pb Ti03 Lead titanate is ferroelectric a t room temperature and has a Curie temperature of 490°C. The structure shows a tetragonal distortion from the ideal perovskite structure and is isomorphous with the tetragonal phase of BaTi03. The phase transition to the cubic structure is of the first order as determined from specific heat anomalies and from changes in the lattice parameters.66 Gainon$’ has measured the Fe3+ spectrum. The very large gr 5.97 f 0 . 0 2 is consistent with a large axial field as found from X-ray measurernents.g* His measurements were not carried beyond 300°C and therefore he did not determine the spectrum near the phase transition. One would expect the initial splitting and gr to be temperature dependent. Gairion also reports measurements down to -1120°C but he did not find any appreciable change in the spectrum. It has been reported that this crystal shows another phase transition a t approximately - 100°C.99The new phase reportedly is not ferroelectric. Gainon’s measurements do not indicate this phase transition. The large g1 factor shows that the individual doublets are split off considerably. Apparently the tetragonal distortion in PbTiO, is larger than in BaTi03.

-

b. Potassium Tantalate: KTa03

The crystal structure of KTa03 is the ideal perovskite structure even at law temperatures. No phase transition has been detected down to 1°K. D. A. Gainon, Phys. Rev. A134, 1300 (1964). J. Kobayashi and R. Ueda, Phys. Rev. 99, 1900 (1955). 99 J. Kobayashi, S. Oksmoto, and R. Ueda, Phys. Rev. 103, 830 (1956) 97

98

ELECTRON SPIN RESONANCE OF MAGNETIC IONS

177

The dielectric constant is very high. It varies from a few hundred to about 4500 a t liquid helium temperature. In mixed KTa03-KNb03 a second-order phase transition has been reported. All the evidence from the dielectric and polarization data point to a great similarity between SrTi03and KTa03 except that the transition temperature of the latter, if it exists, is very low. WemplelOO has studied the Fe3+spectra in the crystal at 4.2"K. He found the expected cubic symmetry. The initial splitting is about 50% larger than in SrTi03 a t the corresponding temperature. It is a t present not understood why this splitting is so large. The temperature dependence of the splitting has not been carefully studied. Wemple detected in addition an axial spectrum with g1I = 2.0, 91 = 6.0. This is attributed to charge compensation from a nearest-neighbor oxygen vacancy. Additional resonances have been found but their origin is uncertain. Wemple applied electric fields and found that the iron line shifts with the field. It should be noted that a field of 10 kV/cm requires a dielectric constant of about 1130 in order to obtain a polarization P of 1 pC/cm2. The axial field parameter bz0 will depend primarily on the linear strain E = da/a. If there is a center of symmetry then the strain is proportional to P2. Wemple finds experimentally that b20 = 3 X P2,where b20 is measured in units of cm-' and P is in pC/cm2. This is a strong indication that the ion remains in a center of symmetry. It is also the largest quadratic Stark effect so far observed. Extrapolation of these results to BaTi03 would indicate that a noticeable Stark effect should be observed there as well. c. Lanthanum Aluminate: LaA103

The crystal structure of Lah103 is a distorted perovskite structure which has been determined by Geller and Bala.'O1 It is rhombohedra1 and the space group is probably R3m. The primitive unit cell contains two molecules of LaA103. The distortion from cubic symmetry can be described as follows. The cubic cell is stretched along one of the body diagonals. In addition there are slight changes in the coordinates of the three ions, in particular those of the oxygen ions. The X-ray data'O'J02 indicate that (1 ) there is a continuous change of the cube as the temperature is lowered, (2) that there is no sudden change in the structure near the transition temperature, and (3) that a t 708 f 2 5 " K the structure becomes the ideal perovskite structure. Granicher and Miiller102find that the change in rhombohedra1 angle A@ = p - 90" shows

loow.Wemple, Ph.D. Thesis, M.I.T. (1963). lo's.Geller and Y. B. Bala, A d a Cryst. 9, 1019 (1956). lo*

H. Griinicher and K. A. Miiller, Nuovo Cimento Suppl. 6, p. 1216 (1956).

:aN:

178

W . LOW' AND E. L. OFFENBACHER

'5

% *0°

I00

0

200

400

600700

TEMPERATUREPK)

FIG.9. Temperature dependence of the axial crystal field parameter b P in lanthanum aluminate (LaA103).1°3The phase transition occurs a t 720 f l 5 " K .

a nearly linear dependence on the temperature over the temperature range 100-700 'K. The spectra then can be expected to behave similarly to those of SrTiOs, except that the distortion in this case is along the [111] direction rather than along the cubic axes. The domains therefore may possibly be aligned along four different directions. The The spectra of Gd3+ and Cr3+ have been studied in detai1.103J04 appropriate spin Hamiltonian for the Gd3+ion is X

=

gPH* 8

+ BzoOzo+

B4'04'

+ B6°060+

b'4 3 0 4 3

+ Bc3063+ Bs60a6. (10.1)

Low and Zusiiian103 have shown how to determine these parameters. They assume that the Gd3+ ion takes the position of the La3+. The temperature dependence of bzO is shown in Fig. 9. It clearly indicates that over a fairly large temperature region bzO varies linearly with the temperature. In general bzo can be expected to depend on the axial field V,, as follows bzo = AVax BV2,,. Since A is probably larger than B one may infer that the axial crystal field varies nearly linearly with the tem15°K in agreement with perature. The intercept for bzo = 0 is a t 720 the X-ray data. This is one of the strongest temperature variations of a crystal field parameter observed. It has been pointed out that this property could be used in thermometry. The fourth-order parameter b4O is also temperature dependent, but to a much smaller extent. The small value of b40 is similar to Gd3+in SrTi03and may confirm in part the conjecture that the Gd3+ion is located a t the Sr site. Kiro105 has measured the spectra of Cr3+ and Fe3+. The spectrum of Cr3+ seems to follow the conventional axial spin Hamiltonian. A strong

+

Io3 Io4

Io5

W. Low and A. Zusman, Phys. Rev. 130, 144 (1963). D. Kiro, W. Low, and A. Zusman, Proc. 1st Intern. Conf. Paranzagnetic Resonance, Jerusalem, 1962, Vol. 1, p. 44. Academic Press, New York, 1963. D. Kiro, M.Sc. Thesis, Jerusalem (1962).

ELECTRON S P I N RESONANCE O F MAGNETIC I O N S

179

temperature dependence of b20 is found similar to the Gd3+ spectrum. The Fe3+ spectrum is not completely understood. It behaves similarly t o the Cr3f spectrum. However, when measured along a direction other than [111] there seems to be a doubling of the lines. Probably not all A1 sites are equivalent. Granicher and Miiller84J02~'06 have also performed experiments on ceramics of LaA103 containing Gd3+and Pr3+.

d . Potassium Magnesium Fluoride KMgV3 There are a considerable number of optical and NMR measurements for KMnF3 and KNiF3.107J08 Much effort has been expended in the interpretation of these results, in particular as far as covalency in crystal field theory is concerned.105-111 We shall not concern ourselves with the theory of covalency in ionic crystals sinre this would lead us too far from the main object of this review.112 The diamagnetic lattice KMgF3 is of the ideal perovskite structure. For the Mg ion, situated a t an octahedral site, one can substitute a number of iron group elements. ESR spectra show the conventional spectra of octahedral complexes complicated by the additional hyperfine structure caused by the fluorine ions. The Hamiltonian, therefore, has the more general form

x = gBH. s + A I . s + B * 0 ( 0 4-~ 5044) + CXFy

(10.2)

where the last term describes the interaction of the magnetic electrons with the six fluorine nuclei. The more detailed expression for the XF depends on the model of the bonding assumed. In the molecular orbital approach the antibonding orbitals are usually defined by e,:

!be = N , (cpe

-

Xsx?s - X . , X ~ , , ~ (10.3)

tzo:

!bt =

Nt(cPt -

XsX2pa),

where the N ' s are the normalization constants, the p's are the 3d orbitals (t, or e , ) , and the X ' S are the appropriate linear combination of 2s, 2pu, and 2pn atomic orbitals associated with the six neighboring F- ions. H. Granicher, K. Hubner, and K. A. Mhller, Helv. Phys. Acta 30, 480 (195i). R. G. Schulman and K. Knox, Phys. Rev. Letters 4,603 (1960). 10sK. Knox, R. G. Shulman, and S. Sugano, Phys. -Rev. 130, 512 (1963). 109 R. G. Shulman and 8. Sugano, Phys. Rev. 130, 506 (1963). 110 S. Sugano and R. G. Shulman, Phys. Rev. 130, 517 (1963). 111 R. E. Watson and A. J. Freeman, Phys. Rev. 134, A1526 (1963). 112 See Watson and Freeman"' for the main referenres t o the theoretical considerations. 106

107

180

W. LOW AND E. L. OFFENBACHER

Similar wave functions cau be written for the bonding molecular orbitals which will also take into account covalent mixing. Assuming that the p function of the F- ion is directed along the cubic axes, one can write, irrespective of the role of the bonding or antibonding orbitals, X F= ~ A'SrIr"

+ B'(S,I," + S E I E ~ ) ,

(10.4)

where the { axis coincides with the direction of the pu orbital and 71, with the p~ orbital; A' and B' are the fluorine hyperfine parameters which are measured parallel and perpendicular to the a-bond axis. It is these parameters which can be determined experimentally. In practice it is useful to measure the hyperfine structure along the [ l l l ] direction because when the magnetic field is parallel to this direction all the six fluorines are equivalent and thus simplify the measurement. The interpretation of these results however is still somewhat controversial. Hall et aL113have carefully measured a number of iron group elements. They have interpreted their results using essentially the theory of Shulman and Sugano with slight modifications. This theory takes into account only the covalency of the antibonding electrons. The bonding electrons are passive partners in their calculations. Watson and Freeman"' have pointed out that the unpaired bonding electrons contribute heavily to the covalency and the transferred hyperfine structure and that the contribution of the antibonding electrons is irrelevant. The results of Hall et al. have not yet been interpreted taking into account the latest theory. VII. Spinels

11. CRYSTAL STRUCTURE

The mineral spinel has the composition ABz04 where A represents a divalent ion such as Mg, Zn, or Cd, and B represents A1 or other trivalent spinels ions. Other combinations are possible. For example in Lio.sAlz.504 the A site occupied by equal numbers of Li+ and Al3+ ions has only a n average valency of two. The smallest unit cell that has cubic symmetry contains eight AI?z04 molecules and, therefore, thirty-two anions and twenty-four cations. The unit cell has a total of sixty-four tetrahedral sites and thirty-two octahedral sites. In a stoichiometric crystal eight of the tetrahedral and sixteen of the octahedral cation sites are occupied. (They are called A and B sites respectively.) Two octants of the spinel structure are shown in Fig. 10. From the right-hand octant of this figure one sees that the A site is tetrahedrally surrounded by four oxygen ions. It is also surrounded by a perfect tetrahedron of metal ions. The occupied tetrahedral sites are the A ions in the center and four of the 113

T.P. P. Hall, W. Hayes, R. W. H. Stevenson, and J. Wilkers, J . Chem. Phys. 38, 1977 (1963).

ELECTRON SPIN RESONANCE O F MAGNETIC IONS

- -

-0-

0

0

~~~

Tetrahedrally coordlnoted ton Octohediolly Cuord noted ton l x y g e ” “n

181

I A s8te) I B site)

FIG.10. Crystal structure of spinel (AB204).A sites are the tetrahedrally coordinated ions and are indicated by a black sphere. Octahedrally coordinated B sites are shown as small open spheres and the oxygen ion as a large open sphere. A parameter u is defined as a measure of the displacement.

eight corners of one of the octants. The adjacent octant has no A ion in the center. The octahedral B sites are found in this adjacent octant only. The position of each B ion is on a body diagonal with a symmetrically placed oxygen ion on the same diagonal. The point symmetry of the A site in a perfect spinel is cubic since it is surrounded both by a perfect tetrahedron of oxygen and metal ions while this is not so for the B site. Each B site is surrounded by an octahedron of oxygen ions. However, the nearestneighbor B ions surrounding a given B site induce an axis of symmetry in the [lll] direction. Since any of the directions along the four-body diagonals occur equally as symmetry axes, the over-all symmetry remains cubic. A paramagnetic ion a t a B site would be exposed to a trigonal field, and there would be four magnetically inequivalent sites. The perfect spinel structure is seldom maintained. Often one finds deviation in the anion sublattice, in part because of considerations of relative size. The A sites have a volume too small for some of the metal ions which can be substituted a t this site and tend to force out the four oxygen ions along the body diagonals. The cubic symmetry ( T d ) is maintained for the A site. However, the oxygen octahedron becomes distorted and tends to reinforce the trigonal component at the B site. One defines a parameter u as a measure of the displacement (see Fig. 10). When u = Q the spinel is called perfect. For a hard sphere model

RA

+ Ro =

(U

-

;)UG RB + Ro

=

(5 - U ) U

(11.1)

where RA, RB, and Ro are the radii of A , B, and the anion. I n a perfect spinel there are eight divalent ions in the A sites and sixteen trivalent ions in the B site. Such a spinel, called normal spinel, is MgALO4. There are spinels in which the eight divalent and the eight trivalent ions are distributed a t random at the B sites, with the remaining

182

W. LO\\’ A N D E. L . O F F E N B A C H E R

eight trivalent ions a t the A sites. Such a spinel is called an inverse spinel. One can find in principle all kinds of different distributions and these are called intermediate spinels. Normal spinels are usually those in which the A site is occupied by ions with filled shells such as Zn2+or Mg2+.When the divalent ion is a transition metal ion, the spinel is often inverse. This is a strong indication that most 3dn transition elements prefer the octahedral to the tetrahedral site. A simple theory, using the crystal field approach, has been proposed to estimate this site preference en erg^."^-"^ In some inverse spinels the ions having different valencies can be “ordered.” Ordering has been found for octahedral B sites in Lio.5A12. 504.118 In general, “ordering” is accompanied by change in the space group, and also often by changes in specific heat and electrical conductivity. Electron spin resonance can contribute considerably to the understanding of the distribution of the magnetic ions over the A and B sites. Unfortunately, most crystals show some disorder to start with. The best crystals are natural MgA1,04 spinels. Synthetic spinels usually have considerable disorder and random distribution of the cations a t the A and B sites. We shall discuss the information derived from ESR data in detail. 12. ESR SPECTRA a. Cr3+ in MgAZ2O4and in .%&zO4

The chromic ion is found in the natural ruby spinel (MgA1204) and has been studied e x t e n s i ~ e l yThe . ~ ~spectrum ~ ~ ~ ~ ~can be described by a trigonal spin Hamiltonian with the trigonal axes for the four inequivalent magnetic ions along the four crystallographic body diagonals. The experimental g factor is very close to 2. The initial splitting has been determined directly by OvermeyerlZ1who found it to be -1.84 cm-I for MgA1204and -1.86 cm-I for ZnAlz04. This is almost twice as large as estimated by Stahl-Brade and The fact that the ground state is an effective S state with g close to 2 is strong evidence that the chromic ion primarily occupies the B site.11g If it were a t the A site there would be an additional orbital contribution to the g factor. The trigonal spin Hamiltonian with the very large initial splitting reveals the relatively large distortion of the oxygen octahedron. D. S. McClure, Phys. Chern. Solids 3, 318 (1957). J. D. Dunitz and L. E. Orgel, Phys. Chern. Solids 3, 30 and 319 (1917). 116 A. Miller, J . A p p l . Phys. 30, 245 (1959). 117 See also the discussion of J. B. Goodenough, in “Magnetism” (G. T . Rado and H. Suhl, eds.), Vol. 3, p. 1. Academic Press, New York, 1964. P. B. Braun, Nature 170, 1123 (1952). 119 R. Stahl-Brade and W. Low, Phys. Rev. 116, 561 (1959). l20 Ti. A. Atsarkin, Soviet Phys. JETP (English Transl.) 16, 353 (1963). 1*1 J. Overmeyer, Private communication (1964). 114

115

ELECTRON SPIN RESONANCE O F MAGNETIC IONS

183

This distortion must be larger than that found in aluminum oxide, and probably it is connected with the intrinsic imperfect octahedron. Brun et C L Z . ~ ~ *as well as Atsarkinl2’Jhave studied the line width in natural and synthetic spinels. In natural spinel the linewidth of the Q + -3 transition is quite narrow, and not very dependent on angle. The Q + Q transition, found at high magnetic field, is very much wider. This can be interpreted to arise from a misalignment of various domains, as well as from random distributions of the parameters in the spin Hamiltonian. Apparently even in natural spinel there is some disorder in the cation distribution. Heat treatment increases the line width considerably.lZ2 I n synthetic spinels the line width is wider than in natural spinel and is dependent on angle. Synthetic spinels usually do not have a stoichiometric ratio of Mg:A1 and probably do not have a normal distribution of the cations on the B sites; this may result in a larger distribution of the values of the various parameters in the spin Hamiltonian. The angular dependence of the line width may arise from additional terms in the Hamiltonian. Further study of the Q + 3 transition and a study of the line width as a function of the stoichiometric composition and heat treatment are indicated. b. Mn2+in MgA1204and ZnAlz04

The spectrum of manganese in ZnL&04 consists of the typical six-line The interpretation given hyperfine pattern with a g factor close to 2.119,*23 is that it results from divalent manganese with hardly any fine structure. Stahl-Brade and Lowiigconjecture that the Mn2+is a t the A site. The support for this is (1) the absence of the fine structure, indicating either a small trigonal field component or none at all, ( 2 ) the small cubic field, and (3) the magnetic equivalence of all the ions. This is consistent with a tetrahedral site a t which the cubic field is about one-half that a t a n octahedral site. The cubic field splitting depends on a high power of the cubic crystal field in the case of S-state ions; this would explain the absence or the small magnitude of the fine structure. Additional support for this interpretation comes from the small value of the hyperfine structure, which is similar to that found in other tetrahedral compounds (see Table AIV). This interpretation is consistent with neutron diffraction data on concentrated manganese spinel.121 The line width of the Q + Q transition is only of the order of 10 G. This gives a n upper limit of a few gauss for the cubic field or axial field splitting. W a l dne ~-has ’ ~~studied the “forbidden” transitions in MgA1204 correspond-

’*’E. Brun, s. Hafner, H. Loelinger, and F. Waldner, Helv. Phys. Actu 33, 966 (1960). F. Waldner, Helv. Phys. Acta 36, 756 (1962). J. M. Hastings and L. M. Corliss, Phys. Rev. 104, 328 (1956). lZ6 F. Waldner, Helv. Phys. Actu 36, 756 (1962). lZ3

lZ4

184

W. LOW AND E. L. OFFENBACHER

ing to Am = 2 1 , M = 3 -+ -3 transitions, and measured their relative positions and relative intensity compared with the Am = 0, M = 3 -+ lines. He assumes an axial distortion, the direction of which is distributed with equal probability over all solid angles, and this would result in a spectrum similar to a powder pattern. Analysis of these transitions is concm-'. sistent with a n axial spin Hamiltonian parameter 2b20 < 35 X This interpretation is somewhat doubtful since it is unlikely that the distortion is randomly distributed in space; it is more likely directional. Such a large value of bzo would also result in stronger angular dependence and line width broadening of the main transitions.

-+

c. Fe3+ in MgAZ204

This spectrum has been studied by many scientists. However, in most spinels, both natural and synthetic, only one very wide line is observed (more than 150 G ) ; this is the -+ -3 transition. Brun et aZ.lZ6have analyzed the low field AM = 4, 5 transitions. These lines are usually much narrower. Their position permits the approximate determination of the spin Hamiltonian parameters. The trigonal field component and the four inequivalent sites with the trigonal axes along the body diagonals indicate that the Fe3+ion is at a B site. This is opposite to what is apparently found for Mn2+ and is contrary to the results of theoretical analysis using the crystal field stabilization energy.114-116 It should be pointed out that the initial splitting as determined by Brun et ~ 1 . ~ 2is6 smaller than that for Cr3+, although the line width is much greater. The large line width for Fe3+ probably has a different origin than simply a random distortion of the diamagnetic lattice, which would result in comparable line widths for Cr3+ and Fe3+. I n the case of Al2O3the spin Hamiltonian parameters of Cr3+ and Fe3+ are of similar magnitude. The distortion of the octahedron in spinels is apparently sensitive to the particular impurity introduced. Whether the paramagnetic ion is a t a center of inversion has not yet been experimentally determined.

+

d. Fe3+ in L ~ O . ~ Aand / ~ .in~ Li0.5Ga2.604 O~ The lithium spinel is an inverted spinel and can be studied both in the ordered and disordered condition. Braun118 has determined from X-ray measurements that the transition from the order to disorder state changes the symmetries for both the A and B sites. The symmetry of the A site changes from 3 to 3 3m and that of the B site from 2 to 3m. The axis of distortion for this type of spinel, as inferred from the X-ray data, is for the A site along one of the four-body diagonals and for the B site along one of the six [ l l O ] directions. This is very different from the normal spinel. lea

E. B m , H. Loelinger, and F. Waldner, Arch. 813.(Geneva) 14, 167 (1961).

ELECTRON S P I N RESONANCE: OF MAGNETIC IONS

185

Folen has studied this system intensively.127-129 For the lithium aluminum spinels he finds that the Fe3+.is a t both A and B sites in disordered spinel, but only the B site has been analyzed. In ordered spinel the iron is primarily a t the A site. The assignment of the sites assumes that the X-ray determination of the point symmetry is correct and is the same for the dilute paramagnetic impurity. I n the gallium spinel Folen finds primarily the ordered structure. The spectrum of Fe3+ shows that a large fraction of the ions are at the A site, in agreement with the aluminum inverted spinel, but in disagreement with the data in the normal spinel. Folen calculates the single-ion anisotropy contribution from these parameters and finds it to be about 1.3 x 10-4 cm-' per ion for the ordered structure a t T = 126°K. This is about 65 times smaller than the measured anisotropy in lithium ferrites. The single-ion contribution, therefore, is not the major contribution to the magnetic anisotropy of these ferrites.130 It can be concluded that ESR can contribute significantly to the determination of the site distribution of different transition elements. Most crystals, even the natural spinels, seem to show crystal field inhomogeneities. It has not been possible to obtain synthetic crystals with a high degree of order. Further work on better crystals with different non-8-state ions is clearly indicated. The magnitude and distribution of these crystalline field inhomogeneities may be important to the ferromagnetic resonance line width, as Callen and Pittelli131 have pointed out. VIII. Garnets

The garnets are of considerable interest since they permit the study of ferrimagnetism in a family of cubic compounds. The structure is very stable; it has low microwave losses, and the compound YIG (yttrium iron garnet) has been utilized in microwave devices. If rare earths are substituted for the diamagnetic yttrium, it is found that the spontaneous magnetization c h a n g e P : the spontaneous magnetization is a function of the particular rare-earth ion and varies with temperature.133Different measurements, such as susceptibility, far-infrared, optical, and Mossbauer studies, have been reported on the garnets. We shall be concerned mainly with the V. J. Folen, J . A p p l . Phys. 33, 1084 (1962). V. J. Folen, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1966, V O ~1, . p. 68. Academic Press, New York, 1963. ''@ V. J. Folen, Private communication (1964). la' V. J. Folen, J . A p p l . Phys. 31, 166s (1960). lal H. B. Callen and E. Pittelli, Phys. Rev. 119, 1523 (1960). Ia2 R. Pauthenet, Compt. Rend. 242, 183 (1956); 243, 149 and 1737 (1956). W. P. Wolf, Rept. Progr. Phys. 24, 212 (1961).

"*

186

W. LOW AND E. L. OFFENBACHER

diamagnetic garnet in which the paramagnetic ion is only a small constituent. 13. CRYSTAL STRUCTURE

The crystal structure is quite complicated. It has been studied extensively for a number of corn pound^.^^*-^^^ The general formula is A3BSOI2, where B can be A13+, Ga3+, and trivalent iron group elements; A can be Y3+, Lu3+, or rare-earth and uranium group trivalent ions. The space group is Oh(10)-Ia3d and the over-all symmetry is cubic. There are eight formula units in the unit cell. The room-temperature lattice constant of YIG is 12.376 k and varies for different substituerits such as Y3+and Fe3f.136 The most important aspects of magnetism can be derived from the point symmetries of the cations. There are three types of cation sites, with different coordination of oxygen ions (shown in Fig. 11). The B ions can occupy either the sixteen octahedral a sites or the twenty-four tetrahedral d sites. The octahedral cations form a body-centered cubic structure. The A ions are a t the 24 dodecahedral c sites. The tetrahedral and dodecahedral cations are along a bisector of the faces of the cube. The octahedron is also shown in Figs. 11 and 12. These are not regular octahedra but are distorted and twisted. This can be best pictured as follows. The octahedron is distorted along one of the threefold axes; this trigonal axis coincides with

0

Octahedral [16(0)]

0 Tetrahedral [ 2 4 ( d ) ]

8

Dodecahedra1 [24(c)l

FIG.11. Crystal structure of garnet. 134 136 136

S. Geller and M. Gilleo, Phys. Chem. Solids 3, 30 (1957). S. Geller and M. Gilleo, Phys. Rev. 110, 73 (1958). F. Euler and J. Bruce, Proc. 20th Conf. Difraction, Pittsburgh, Pennsylvania, 1962.

ELECTItON SPIN RESONANCE O F MAGNETIC IONS

0

Go”Oi

187

Fe3‘

0 o2

FIG.12. The distorted octahedra in garnet. The trigonal axis is along the [lll]direction of the crystal. Each octahedron is rotated through opposite angles 01.1~’s. Geschwind, Phys. Rev. 121, 363 (Fig. 3 ) (1961).

the [lll] direction of the crystal. This would give a point symmetry C 3 h , and, therefore, four inequivalent magnetic spectra. I n addition, the octahedra are rotated about the [lll] direction through opposite angles hta! (about 28”). This then gives rise in general to eight magnetic spectra which can be described by the same spin Hamiltonian. The tetrahedra are similarly distorted. This can be visualized (see Fig. 13) by inscribing the tetrahedron in the cube whose axes coincide with the edges of the unit cell. The cube is distorted by pulling along a unit cell edge. The point symmetry is S4and could result in three inequivalent

0

G03‘Gr Fe3’

0 Y3+

(.&

00 2 FIG.13. The distorted tetrahedra in garnet. The tetrahedra are inscribed in a cubo whose axes coincide with the edges of the unit cell. Earh cube is rotated through opposite angles j3.1:7 S. Geschwind, Phys. Rev. 121, 363 (Fig. 2 ) (1961).

188

W. LOW AND E. L. OFFENBACHER 2

3

3

4

4

FIG.14. The orientation of the g tensor axes with respect t o the cubic unit cell axes. There are six inequivalent ions, but in the (110) plane there can be seen four inequivalent ions. H Ois the magnetic field.

spectra. Similar to the octahedral case, the cube is rotated through angles j $ about 15.6'. Therefore, one observes six inequivalent spectra. The rare-earth ions also have six inequivalent sites. They are situated in a distorted cube with point symmetry Dz.Each of the six sites can be described by the same g tensor. The axes of the tensor g,, gy, and gz do not coincide with the X, Y , Z axes of the cube. This is shown in Fig. 14 where it can be seen that if the magnetic field is kept in the [ l l O ] plane, there are only four inequivalent sites. 14. ESR SPECTRA

a. Iron Group

The spectrum of Fe3+in YGaG has been extensively studied by Geschwind.137 This spectrum is of course of particular interest since Fe3+ is the main constituent in YIG. Using the parameters deduced from the ESR spectra, one can calculate the single-ion anisotropy contribution as outlines in Part 111. This explains in part the observed anisotropy of these ferrimagnets The analysis of the spectra is complicated by the many inequivalent sites, since Fe3+ is situated both at the octahedral and tetrahedral sites. As in the spinels it is found that the Fe3+ ion prefers the octahedral coordination. The spectrum of the octahedral ion is given by the trigonal spin Hamiltonian of Eq. (2.9). It is convenient to take the z axis along the [lll] direction. The main AM = f l transitions are then given to second order 137

S. Geschwind, Phys. Rev. 121, 363 (1961).

ELECTRON SPIN RESONANCE O F MAGNETIC IONS

189

by j=$ +

f$

H

=

4bz0 f Q[2(b4')c -

Ho

34'1

- H[(b4'),2/(H

f 2b,O)I,

Ho F 2bz' T 8[2(b4')~ - 3h0] V [(b4°)cz/(H T 2bz0)], (14.1) H = Ho - #(b4')2[2H/(H2 - 4bz0)]. 3 -+ -4 These equations permit the evaluation of the main parameters. The angular variation of the spectrum determines the parameter (b40)c.137 It is found that the g factor and the (b40)care very similar to that found for Fe3+ in MgO. I n a similar manner the parameters were determined for the tetrahedrally coordinated Fe3+. The cubic field splitting parameter is found to be smaller, as expected for the smaller cubic field in the tetrahedral case.138--141 The convenient axis of quantization is along one of the cubic axes. Using the measured parameters Geschwind calculates the contribution of the various octahedral and tetrahedral cations to the single ion anisotropy. Care must be taken in averaging correctly the contribution of each distorted octahedron since the distortion along the [lll] direction in the cubic crystal does not coincide with the local body diagonals of the octahedron. The anisotropy then can be written as j=$ -+

xt+

H

=

+

KI

=

+

Ktrt(~) K J ~ ( v ) ,

(14.2)

where Kt and KO are the coefficients of the tetrahedral and octahedral contributions to Eq. (111.11). It is found that these results differ from the experimental results on YIG142 by about 50%. The disagreement is rather large for the contribution of the octahedral site where the single-ion contribution for YGaG would predict a much larger value than observed. It is a t present not clear whether the discrepancy arises from the deficiencies in the single-ion approximation or from neglect of the interactions among the spins. Possible sources for discrepancies can arise from different values of b4O, a, and 0 in YGaG compared with YIG or possible inclusion of impurities. The Cr3+ ion in YGaG and YAlG has also been s t ~ d i e d . ' ~ ~The J ~ 4ion is primarily situated a t an octahedral site, as found in other multiple oxides. The axial distortion parameter bzo, which is sensitive to small changes in 138

141

J. R. Gabriel, D. F. Johnston, and M. J. D. Powell, Proc. Roy. SOC.264, 503 (1961). M. J. D. Powell, J. R. Gabriel, and D. F. Johnston, Phys. Rev. Lett. 6, 145 (1960). W. Low and G. Rosegarten, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, Vol. 1, p. 314. Academic Press, New York, 1963. W. Low and G. Rosegarten, J . MoZ. Spectry. 12, 319 (1964). G. P. Rodrique, H. Meyer, and R. V. Jones, J . Appl. Phys. 31, 376s (1960). J. W. Carson and R. L. White, J . A p p l . Phys. 32, 1786 (1961). S. Geschwind and J. W. Nielsen, Bull. Am. Phys. SOC.[2] 6, 2521 (1960).

190

TV. LOW AND E. 1,. OFFENBACHER

the lattice parameters, is very large-about twice that found in A1203and of the opposite sign. It is difficult, therefore, to extrapolate these parameters to other garnet structures containing Cr3+. b. Rare-Earth Group

Most of the work on ESR and susceptibility measurements were made by Wolf and his associates in They measured the magnetic properties of the individual rare-earth ions in one diamagnetic garnet lattice in the hope that this set of measurements would be sufficient to explain the role of the rare-earth ions in a YIG lattice. However, on the basis of their studies on four diamagnetic systems LuGaG, YGaG, LuAlG, and TAlG, they found that there is no simple correspondence between the YIG lattice and diamagnetic lattices such as YGaG. Their results are contained in Table ,4IX. There are large changes in the g values on substituting aluminum for gallium which occupies the a and d sites. For the ions Dy3+ and Er3+there seems to be even a change in symmetry. It is difficult, therefore, to predict the g values in concentrated rare-earth paramagnetic lattices. The general crystal field of the rare earth garnet can be written as X,

=

BZ0O2'

+ B4'04' + B6'Otj' + Bz20Z2+

+

+

B42042

Bh40h4

+ 2h406* +

B62062

(14.3)

It contains, therefore, nine parameters. The resonance measurements were carried out mainly on odd-electron systems. The ground state in this low symmetry is a n isolated Kramers doublet. One can obtain a t most three g factors, which do not determine the nine Bnm.Wolf et a1.146J47 had hoped that by measuring the whole rare-earth series they could detect regularities which would establish the magnitude of the Brim. However, as mentioned above, the individual Bnmseem to be strongly influenced even by small changes in the lattice distances. At present it is not possible to calculate (from the g factor) the crystal field parameters uniquely or with any accuracy using only the ESR data. Some success has been achieved in the case of Yb3+in garnets. Here the 1,s splitting is very large, of the order of 10,000 cm-I. The observed g factors in YGaG are g, = 3.73, g, = 3.60, and gz = 2.85. The mean value g 8 ) / 3 is within a few percent of the g value of a I'7 level, of g = (gz g, which is 3.43. The crystal field splitting has been found to be large. One of

+ +

Tech. Rept. AFCRL-63-192, Clarendon Laboratory, Oxford, 1963. W. P. Wolf, M. T. Hutchins, M. J. M. Leask, and A. F. G. Wyatt, J . Phys. Soc. Japan 17, 487 and 943 (1962). 147 M. Ball, G. Garton, M. J. M. Leask, D. Ryan, and W. P. Wolf, J . Appl. Phys. 32, 146 lP6

2675 (1961).

ELECTRON SPIN RESONANCE OF MAGNETIC IONS

191

the excited levels has been located at 550 cm-1.148Susceptibility measurements are in agreement with this and suggest that the next level is a t 700 cm-1, presumably a ral e ~ e 1 . 1These ~ ~ experimental data in conjunction with the g factors are of help in determining the wave function

1 4 ) + 4 I -$>I + Pc(T%.)1'2I -3) + a[+ 1 4 ) - W / 2 ) I -$>I r[(A)l'zI -$)

l4 =

[(fl/2)

+

+ - &)l/z

I ++)I I +>I, (14.4)

where the coefficients p , q, and r are functions of the Bnm. The g factors are given by gz = gj[3

gg = gj[3

+ 4 flql + 6p - 2 flq]

(14.5)

gj = 8/7. 9. = gjC3 - 6 p - 2 Gq], It should be noted that the g factors do not involve r ; hence, a measurement of the g factors does not give the wave function in this case. However, one can find values (two sets) for p and q.

Further progress can be made by assuming a model. Hutchins and Wolf150 applied a point-charge calculation to estimate the field parameter, Anm(rn).However, these estimates are always in doubt, in part because of our lack of knowledge of (r"). They calculated, therefore, the five ratios A,m/A,m' which are independent of (r").This reduces the number of unknowns to four: Az0(r2), Az2(rz),and Aao(r6).Hence, the combination of the optical data and resonance results is more than sufficient to determine these parameters. For the crystal field parameters they found Az0(r2) = -86 cin-1, Apz(rz)= 297 c n r l , A40(r4)= -193 cm-I, A42(r4)= 159 cn~-I,A44(r4)= 535 cm-1, = 72 cm-l, As2(T6) = -229 cm-', A s 4 ( r 6 ) = 1315 cm-1, and A s 6 ( r 6 )= -233 cm-'. This gives rise to energy levels a t 0, 517, 697, and 796 cm-1, which is in good agreement with the optical data. The predicted g values of 0.43, 2.03, and 1.75 are also in fair agreement with the experiment. Further experimental data on the g values of the other levels could determine uniquely these various parameters. The ESR optical and susceptibility measurements indicate that for YbIG the exchange splitting is smaller than the crystal field interaction. One can estimate from these data the exchange anisotropy at T = 0°K. It is found that this estimate accounts for the bulk of the measured anisotropy. It should be pointed out that ytterbium ions should be canted with respect to the easy [111] axisI46 (see Part 111). Such a canting angle has been observed for HoIG.lS1 R. Pappalardo and D. L. Wood, J . Chem. Phys. 33, 1734 (1960). R. H. Brumege, C. C. Li,and J. Van Vleck, Phys. Rev. 132, 608 (1963). 150 M. T. Hutchins and W. P. Wolf, J . A p p l . Phys. 36, 1060 (1964). 151 A. Herpin, W. C . Koehler, and P. Merial, Compt. Rend. 261, 1359 (1960). 148 149

192

TV. LOW AND

E.

L. OFFENBACHEK

I n this connection it is interesting to mention the recent work by Wickersheim and White on YbIG.152In the molecular fieId approximation one can write the exchange Hamiltonian as X =

-p*Heff,

(14.6)

where p = pgS' is the total magnetic moment of the ion, g the paramagis netic resonance tensor, and S' = +. Now Hcff= -AM== where the net magnetization of the iron lattice and A is the anisotropic Weiss constant. The exchange field is related to the effective field as f0llows~~3: Hex =

[gj/z(gj

-

1)IHeff.

(14.7)

Wickersheim and White noticed that the exchange field acts only on the spin component, whereas the G S interaction will interact with the orbital component. The anisotropy in the orbital part will therefore be reflected through the strong L-S coupling in the exchange interaction. Applying the concept of an exchange potential they rewrite (14.6) as X = gp[l

+ aYZ0 + b(Y2' + Y2-2)]S1*XMFe

and X = gPH,rr{ +GzoOzo

+ Gz20z2]8,. So,

(14.8) (14.9)

where S1 is the Yb3+ ion spin and So is the unit vector in the direction of the magnetization. By fitting the wave function to the measured g factors of the J = $ and J = 4 state they calculate the three parameters GZo,Gz2, and Heffto fit the six experimentally determined exchange splittings. This simple theory, using the g data, gives the remarkable agreement (to within 10%) between experiment and theory. The S-state ions, in particular Gd3+, have been studied by Overmeyer et a1.26 and Rimai and deMars.IS4For an S ion it is expected that the magnetic behavior in the GdIG should be explained by the single-ion theory. Overmeyer et al., using the theory as outlined in Part 111, calculated the singlecm-' in remarkable ion anisotropy per ion between - 750 and - 800 X agreement with the measured value of -910 X cm-1.142J55 The calculated value of Rimai and deMars seems to be in error. ACKNOWLEDGEMENT We greatefully acknowledge the permission by Dr. Rimai to reproduce Figs. 7 and 8 from Rimai and deMars75 and Dr. S. Geschwind for Figs. 12 and 13. This review article was written while one of the authors (W. L.) held a Guggenheim fellowship.

K. A. Wickersheim and R. L. White, Phys. Rev. Letters 8, 483 (1962). W. P. Wolf and J. H. Van Vleck, P h p . Rev. 118, 1490 (1960). 154 L. Rimai and G. A. deMars, J . Appl. Phys. 33, 1254 (1962). 162

153

166

R. F. Pearson, J . A p p l . Phys. 33, 1236 (1962).

ELECTRON SPIN RESONANCE OF MAGNETIC IONS

193

Appendix

I n the following Tables A1 to AIX we list the values of the g factor, of the hyperfine structure constants A and B, and the crystal field parameters brim for transition group ions in the systems

I I1 I11 IV

v VI VII VIII IX

magnesium oxide calcium oxide strontium oxide zinc oxide aluminum oxide titanium oxide perovskite spinel garnet

The tables list in each crystal host the inividual ion, the frequency band, and the temperature a t which the parameters were measured. The frequency bands are called X band for frequencies near 9 Gc/sec, K,, about 16 Gc/sec, K about 24 Gc/sec and Q band about 35 Gc/sec. The temperatures are listed in degrees Kelvin. Room temperature has been uniformly designated as 290°K. The errors of measurement are indicated in parentheses; i.e., 2.01 f 0 . 0 2 is abbreviated as 2.01 (2) and g = 2.0017 f0.0012 as 2.0017(12). The g factor without subscript means that the g factor is isotropic within the limits of error. cm-', except where The values of A , B, and bnm are given in units of explicitely indicated otherwise. We have used the crystal field parameters bnm rather than the conventional D, E , a, and F. The conversion from the brim to the usual nomenclature is given in Table I1 in the text. A difficulty arises when b40 is used for two different types of measurements; for example for crystals which have cubic and trigonal symmetry. The conventional way is to designate the fourth-order cubic component as a and the fourth-order trigonal component as F . We have listed these as (b40)c and b40, where (b4°)c = a/2 and b40 = F/3, so that ( a - F) = 2(bd0), - 3b4O. We have not included measurements on powdered samples, except when these contain significant information riot available from measurements on single crystals. The references are listed separately for each individual table. Mr. L. Shapiro of Temple University helped in the compilation of these tables. We gratefully acknowledge this assistance.

TABLEAI. ESR DATA FOR TRANSITION GROUPIONSIN MAGNESIUM OXIDE (MgO)

Ion

V”+

Frequency band

x X

Cr3+

K,, X X

Temperature (OK)

290 290

290, 77 290

9

A

1.9803 (5) 1 .9800 (5)

74.24(2) -75 1(1)

1.9800(5) 1.9797 911 = gs = 1.9782

16.0(3) 16 .O

(b4’) o

bz’

Remarks

References

I

A third pattern arises a, d from ions with axes e, f 819.4

of symmetry in the face diagonal [ l l O ] type direction

P

3

m

? MnP+

K K X X

290 70 290 290

2 J016 (1) 2.0015(1) 2 .0014(5) 1.9942 (5)

-81.2(5) -S1.3(5) -S1.0(2) 70.S

9.33(15) 9.33(15) 9.33(15)

9

“Anomalous spectrum,” probably

h e

Mn4+

Fe3+

X

290

2.0010

K X X

77 290 290

2.0037 (7) 2.0037 (7) 2.0030

-81.1

10.1(2) 11 $4

9.50

102.5(5)

a,

h 101.9

e

4j

? *

t

G

? -3

e-

-

f

e 0

R

e-

-a-a-a---

ddd,

h l - k

ELECTRON SPIN RESONANCE OF MAGNETIC IONS

a-

7

R

9

8

f

S

-f,

&

-

e-

P-

r-

195

TABLEA1 (Continued) Frequency

Temperature

Refer-

Ion

Cu'+

X

57

RLI-'

X

Cr

il

2.1697

Identification uncertain

h

Rho

x

77

2.1708

Identification uncertain

h

Pd1+

X

77

2.1698

Identification uncertain

h

Er3f

x

20 20 20 20

Q

Q Q

2.190 (2)

4.62, gioo gioo = 11.84, 9110 gioo = 3.576, giio 9100 = 3.625, giio

~ i o o=

= = = =

19(1)

Line with anomaly h below 1.2'K, evidence for anisotropy

3.86, gill = 3.60 12.13 12.13, giii = 4.29 12.13, 9111 = 4.29

W. Low, Ann. N . Y . Acad. Sci.72, 69 (1958). W. Low, Phys. Rev. 101, 1827 (1956). J. S. Van Wieringen and J. G. Rensen, Proc. 1st Intern. Conj. Paramagnetic Resonance, Jerusalem, 1.962, Vol. 1, p. 105. Academic Press, New York, 1963. W. Low, Phys. Rev. 106, 801 (1957). W.M. Walsh, Jr., Phys. Rev. 122, 762 (1961). J J. F. Wertz and P. Auzins, Phys. Rev. 106, 484 (1957). W.Low, Phys. Rev. 106, 793 (1957). a

*

P. Auzins, J. W. Orton, and J. E. Wertz, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, Vol. 1, p. 90. Academic Press, New York, 1963. W. Low, Proc. Phys. SOC.(London) B69, 1169 (1956). i E. S. Rosenvasser and G. Feher, Bull. Am. Phys. SOC.[Z] 6, 116 and 117 (1961). W. Low and M. Wegcr, Phys. Rev. 118, 1130 (1960). J. W. Orton, P. Auzins, J. H. E. Griffiths, and J. E. Wcrtz, Proc. Phys. SOC.(London) 78, 554 (1961). W. Low, Phys. Rev. 109, 256 (1958). " B. Bleaney and W. Hayes, Proc. Phys. SOC.(London) B70, 626 (1957). I).J. I. Fry and P. M. Llewellyn, Proc. Roy. SOC.A266, 84 (1962). p J. W. Orton, P. Auzins, and J. E. Wertz, Phys. Rev. 119, 1691 (1960). q W. Low, Phys. Rev. 109, 247 (1958). W. Low, Bull. Am. Phys. SOC.121 1, 398 (1956). a J. W. Orton, P. Auzins, and J. E. Wertz, Phys. Rev. Letters 4, 128 (1960). D. Descamps and Y. Merle D'Aubigne, Phys. Letters 8, 5 (1964).

TABLEAIIa. ESR

Ion V2f

Frequency band

X

Temperature (OK)

Oieot

FOR

IRON GROUPELEMENTS IN CALCIUM OXIDE

A

Remarks

(b490

References

290 77 20

1.9683 (5) 1,9683(5) 1.9683(5)

76.04(5) 76.15(5) 76.22(5)

a, b

a, b

2

c

3

Cr3+

X

290, 77

1.9732(5)

17.0(1)

Mn2+

X

290 290 77 20 77 77

2.0009 (5) 2.0011 (5) 2.0011 (5) 2.0011 (5) 2.0015(5) 1.9931 (5)

80.8(2) 80.7(1) 81.6(1) -81.7 (1) 81.4 72.8

77 77 20

2.0052 (5) 2.0059(6) 2.0059 (6)

4

;" r

2.95(15)

A and a are of opposite signs

a, b

l-

m r

+3.0(2) Powder spectrum probably

d

Mn4+

0

r r m

z Fe3+

X

Fez+

X

4.2, 2 2

3.30 3 .298(3) 6.58

Fe1+

X

4.2, 2

4.1579(6)

10.5(5)

31.9(2) +32.2(2) +32.6(2)

C

a

Broad line, Narrow double quantum Symmetric AM = f 2 transition

33.9(2)

e

f

$

2m 0

co2+

X

20 20, 4.2

4.372(2) 4.3747 (2) 2.327 (1)

Nil+

X

20,4.4

Nil+

X

77

132.2(2) 131.5(1)

f

Double quantum transition

a, 9

g factor slightly temperature

i

M

cu2+

X

77 4.2

2.2814 (6) 911 = 2.0672(6), gs = 2.3828(6)

2.2201 (6) 2.2223(10)

Ao3 = 21.6(3) A63= 29.1(8)

Note all lines show angular anisotropy with the minimum along the [Ill] axis.

dependent; below 65°K 3 sets of lines of tetragonal symmetry Both g and A are strongly temperature dependent; below 1.2"K 3 sets of tetragonal lines with g I 1< g A ; in addition] there are many weak lines a t T < 2°K

F

M d

80 9 0

M

u, 0

2 3M

N

TABLEAIIb. ESR DATAFOR RARE-EARTH IONS IN CALCIUM OXIDE

Ion

Freq. band

Eu2+

X

Gd3+

X X

Dy3+

X

Er3+

X

Temp. 9

A

77

1.9941 (5)

290

1.9914(10)

77 4.2

1.9917 (10) 1.9918(10)

A161 = 30.1(2) = 13.4(2) = 29.03(10) = 13.05(20) A'61 = 30.09 (10) A161 = 30.16 (10) = 13.46(10)

290, 4.2 290 77 4.2

1.9922(5) 1.9913(5) 1.9908(5) 1,9925 ( 10)

(OK)

20 20, 4 . 2

x

20

b4O

6.60(5) 911 = 3.09(2), 01

-

Remarks

b 6'

24.0(5)

-1.6(5)

25.1(1) 25.7(5)

-2.1(5) -l5(5)

h

3

1.2(1) -1.15(10) -1.16(10) -1.19(1)

15 3 sets with tetragonal axes along cubic axes; in addition, a broad line 9100 < 1.5

glo0 = 4.84(1), gloo = 3.85(1), gill = 3.50(1)

2.585 (3)

A171

Refcrences

28.8

-12.2(1) -11.6(1) 1 2 . 1(1) 12.2(1)

911 = 4.730(5) Yb3+

8

gr = 7.86(1)

i

z m z i

= 698(6) ~

m

$

~~~

a W. Low and R. S. Rubins, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1969, Vol. 1, p. 79. Academic Press, New York, 1963. * W. Low and R. S. Rubins, Phys. Letters 1, 316 (1962). c A. J. Shuskus, Phys. Rev. 127, 1529 (1962). P. Auzins, J. W. Orton, and J. E. Wertz, Prac. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, Vol. 1, p. 90. Academic Press, New York, 1963. A. J. Shuskus, J . Chem. Phys. 40, 1602 (1964). W. Low and J. T. Suss, Bull. Am. Phys. Soc. [2] 9, 36 (1964). 0 W. Low and J. T. Suss, Phys. Letters 7, 310 (1963). A. J. Shuskus, Phys. Rev. 127, 2022 (1962). W. Low and R. S. Rubins, Phys. Rev. 131, 2527 (1963).

TABLE AIII. PARAMAGNETIC RESONANCE DATAFOR TRANSITION ELEMENTS IN STRONTIUM OXIDE Frcq.

Ion

band

Temp. ( O K )

9i

b4'

11

77 77 20 4.2

1 ,9520 (5) 1,9683 (6) 1.9683 (6) 1,9686 (5)

290 81 77 20 4.2

2.0012 (5) 2.0014(5) 2 .OOlO(G) 2.0012(G) 2.0010 (6) 2.0008 ( 5 )

Ni3+

4.2

911 = 4 . 3 6 f l ) , 91 = 4.647(5)

Eu*+

1.6-77

1.991(1)

1 A'61I I A153 I

1.991(1)

1A1511

Cr3+

Mn2+

Gd3+

290 70 4.2

Yb3+

4.2

1.991(1) 1.991 (1) 1.989(1)

Remarks

References

17.2(5) 17.3 (4) -7s .7(2) -80.2 (2) -80 .0 (2) -so .2 (2) -80.7(2) -80.9 (2)

I A153 I

= = = =

29.9 13.2 30.1 13.3

2.15(4)

<0.5 <0.5
Assignment doubtful

<0.5 5.9(3) 6.5(3) 5.8(5)

2.578(5) There exists line width anisotropy for all lincs. The narrowest line width is along the [111] direction and the widest along [lOO] direction

a P. Auzins, J. W. Ort,on, and J. E. Wertz, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, Vol. 1, p. 90. Academic * W. Low and J. T. Suss, Phys. Letters 11, 115 (1964). Press, New York, 1963. c L. V. Holroyd and J. L. Kolopus, Phys. Stat. Sol. 3, NO. 12, K456 (1963). W. Low, Unpublished results (1964). dB. A. Calhoun and J. Overmeyer, J . A p p l . Phys. 36, 989 (Atlantic City Conf.).

N

52

202

W. LOW AND E. L. OFFENBACHER

TABLEAIV. ESR DATAFOR TRANSITION

Ion v2+

Mn2+

Fe3+

Fre- Temperquency ature band (OK)

91I

9i

1.3

K X

77 300

2.0016(6) 2.0012 (2)

- 76 .0(4)

X

290

2 :0060 (5)

9.02(2)

1.977(1)

-2

x

1.3

2.243(1)

2.2791(2)

X

4.2 1.3

2.1426(5)

4.3179(1)

Cu2+

X, K

1.2

0.74(2) 0.7383(3) ~

b c

e

1.50(2) 1.5237(3)

46.7 174.1(5)

Co2+

4

B

x

Ni3+

~

A

BS

16.11(5)

219 (14) 198(3)

3.00(3)

235 (3) 224(1)

~~

T. L. Estle and M. deWit, Bull. Am. Phys. Sac. [2] 6, 445 (1961). P. B. Dorain, Phys. Rev. 112, 1058 (1958). J. Schneider and S. R. Sircar, 2. Naturforsch 17a, 570 (1962). W. M. Walsh, Jr. and L. W. Rupp, Jr., Phys. Rev. 126, 952 (1962). W. C. Holton, J. Schneider, and T. L. Estle, Phys. Rev. 133, A1638 (1964). H. Kamimura and A. Yariv, Bull. Ant. Phys. Sac. [2] 8, 23 (1963). R. E. Dietz, H. Kamimura, M. D. Sturge, and A. Yariv, Phys. Rev. 132, 1559

(1963). h M. deWit and T. L. Estle, Bull. Am. Phys. Sac. [2] 8, 24 (1963).

203

ELECTRON SPIN RESONANCE O F MAGNETIC IONS

GROUPIONSI N ZINC OXIDE(ZnO)

a

-216.9(22) 1236.2 (4)

-1. 0(. 25) 5. 23 (5)

Sign of D = sign of A = -sign of

4 f 5

Sign of D = sign of A = -sign of

b e

(b04)c

-593.7 (10)

19.95 (2. 5)

d

(b04)c

2.75 (15)

a e

204

W. LOW A N D E. L . OPFENBACHEIL

TABLEAV. ESR DATA FOR TRANSITION Ion

Freq. band

Temp.

("K)

9

91 I

X

4.2

1.067 (1)

300

1.97

C?+

X X

4.2

1.90 (2)

V2+

,Y

300

Cr3+

s

4.2 300

Ti3+ V4f

X

A

((0.1

I

1.97

11.32

1.991

1.991

-73.538 (8)

1.984 2.003 (6)

1.984 2.002

16.2(3)

1.9937(7)

1.9937(7)

-69.608 (8) 170.0 I (5)

2.0017(10)

2.0001 (2)

-79.6(5)

K

4.2 1.6-295

Mn2+

s

300

Fe3+

25-40 9

s

290 4.2 77 299

X

4.2

2.292 (1) 2 .808 (3)

4.947(3) 4.855 (5)

32.4(1) 20 .8 (5)

X

I .6

2.316(5)

4.98 (1)

33.8

Xi3+

K

290-50

Ni2+

x x

290-4.2 290

2.1957 (13) 2.196 (4)

2.1859 (13) 2. 187 (4)

cu3+

I<

1.4

2.07134(5)

2.0772(5)

Rh4+

17

91

X Co2+

1.993

2.003 (1) 2.003 (1) 2.003 ( I ) 2.003(1)

Q

2.146

=

-64.316 (13) Cu65 =

-68 3 9 3 (13)

Ru3+

K

20

Gd3+

K

290

K

79 4.2

Pt+(3

+ 6)

<0.06

2.430

1.9912(5)

2.011 (6)

2.220 (1) 2.328(4)

0 L. S. Bornienko and A . M. Prokhorov, Soviet Phys. J E T P (English Transl.) 11, 1189 (1960). b J. Lambe and C. Kikuchi, Phys. Rev. 118, 71 (1960). G. h4. Zverev and A. M . Prokhorov, Soviet Phys. J E T P (English Transl.) 7, 707 (1958); 11, 330 (1960). S . Foner and W. Low, Phys. Rev. 120, 1585 (1960). R. H. Hoskins and B. H. Soffer, Phys. Rev. 133, A490 (1964). J N . Laurance and J. Lambe, Phys. Rev. 132, 1029 (1963). 0 R. W. Terhune, J. Lambe, C. Kikuchi, and J. Baker, Phys. Rev. 123, 1265 (1961). J. E. Geusic, Phys. Rev. 102, 1252 (1956). S . Geschwind, P. Kisliuk, M. P. Klein, J. P. Remeika, and D. L. Wood, Phys. Rev. 126, 1684 (1962).

205

ELECTRON SPIN RESONANCE O F MAGNETIC I O N S

GROUPIONSI N CORUNDUM (A120a)

B

b02

Remarks

b04

References U

I 1.32 I

b

< 0.15

e

(in cm-1)

-74.267 (30) 16.2 -70.480 (33)

170.0 1 (5) -78.8 (8)

-1601.2(3)

f

- 1907.8 (10) 1930 (10)

9 h

- 1956 .O (3) - 1957 (1)

f 1

194.2(10) I1684(3) I 1719(1) 1716(1) 1679(1)

I131(10) 1 I112(2) I I llS(2) I I120.5(10)

1

2b04 - 3b04 = 334(2) f339 (2) +337 (2) +329 (2)

Two inequivalent sites ildditional splittings a t 4.2"K

97.2(5) 151.0(11) 97.4

j k 1 1 1

m, n m, n 0

P

-60.03 (2)

13760 a t 300°K 13287 a t T 0°K 13850(20)

P

-1883.8(5)

5

r

-64.305 (17)

5

P

1032.9(20)

26.0(10)

b06

= 1.0(5)

t

b2 = 18.3(10) be3 5 1. O baa = 5.0(5) P

W. Low and J. T. Suss, Phys. Rev. 119, 132 (1960). L. S. Kornienko and A. M. Prokhorov, Soviet Phys. J E T P (English Trunsl.) 6 , 620 (1958). G. S. Bogle and H. F. Symmons, Proc. Phys. SOC.(London) 73, 531 (1959). G. M. Zverev and A. M. Prokhorov, Soviet Phys. J E T P (Eng. Trl.) 9, 451 (1959). G. M. Zverev and A. M. Prokhorov, Soviet Phys. J E T P (Eng. 2'72.) 12,41 (1961). ' J. E. Geusic, Bull. Am. Phys. SOC.[2]4, 261 (1959). s. Geschwind and J. P. Remeika, J.A p p l . Phys. 33, 370 (1962). * S. A. Marshall, T. T. Kikuchi, and A. R. Reinberg, Phys. Rev. 126, 453 (1962). S. A. Marshall, and A. R. Reinberg, J . A p p l . Phys. 31, 3368 (1960). W. E. Blumberg, J. Eisinger, and S. Geschwind, Phys. Rev. 130, 900 (1963). S. Geschwind and J. P. Remeika, Phys. Rev. 122, 757 (1961).

206

W. L O W A N D E. L. O F F E N B A C H E R

TABLEAVI. ESR DATAFOR TRANSITION (Y

is the angle between one of the magnetic axes in the plane perpendicular to the c interaction with the Fro quency band

Ion

Ti'+

X

V4t

Temperature

a

(OK)

g;ro

4.2

19"

1.974

1.977

1.941

Not given 4.2

0"

1.975

1.978

1.953

X+ K

0"

1.9566 1.955(1)

1.912s 1.912(1)

1.97(1)

1.97(1)

A'

4.2and78 77

0"

1.915 1.913(.1)

X+K

4.2and350

0"

1.97(1)

Mn4+ X

4.2and300

0"

1.9909(5) 1.995(5)

MJla+ 2-70 kMc/sec

4.2 and 77

0"

Cra+

FC'+

Bc

gllo

x

CO?+ X

X

142 141.5(7)

l.99(1)

1.4 and 78

0"

Z.OOO(5)

2.000(5)

2.000(5)

0" 0"

5.88(2) 5.860(1)

2.19(05) 2.090(1)

3.75(1) 3.725(2)

0"

Nia+

x+ K

4.2 and 78

2.272(3)

2.050(3)

2.237(3)

Nia+

X+K

4.2 and 290 9.l(f0.3)

2.084 (3)

2.254 (3)

2.085(3)

Nig+

X + K

4.2

2.10(5)

2.20(5)

2.10 (5)

cue ?

4.2 and 77

0"

2.105

2.344

2.093

Nb4+ X

4.2 and 25

0"

1.973

1.981

1.948

X

4.2 a n d 77

0"

1.8117(10) 1.9125(10) 1.7884(10)

Mo6+

31s 30.9(3)

1.9898(5)

4.2 4.2

5.4"

n,o

..lllO

-19

-88

8.04

1.75

2.474(25)

65.85(15)

Ce:+

P

4.2

Od'+

X X

295 77 1.8

0"

Era+

Not given 4.2

0"

<0.1

<0.1

?

1.979

1.979

1.945

<2.5

<2.5

0"

1.5945

1.4731

1.4463

92.0

40.5

T8'+ X W'+

Not given

1.4and 10 63

4.394

2.06s

3.866 1.9930 1.9941 1.9986 15.1

495

P. F. Chester, J . Appl. Phys. 32, Suppl., 2233 (1961). H. J. Oerritsen. Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1062,Vol. 1, p. 3. Academic Press, New York, 1963. H. J. Oerritsen and II. R. Lewis, Phys. Rev. 119,1010 (1960). 0. M. Zverev and A. M. Prokhorov, Soviet Phys. JETP (English Transl.) l2,160 (1960). H . J. Gerritsen, S. E. Harrison, H. R. Lewis, and J. P. Wittke, Phys. Rev. Letters 2, 153 (1959); H. J. Oerritsen, S. E. Harrison, and H. R. Lewis, J. Appl. Phys. 31, 1566 (1960). f I. Sierro, K. A. Mfiller, and R. Lacroix, Arch. Sci. (Geneva)l2,lZ (1959). H. 0. Andresen, Phys. Reu. 120,1606 (1960); J . Chem. Phys. 31.1090 (1961). Ir II.J. Oerritsen and E . 9. Sabisky, Phyr. Reo. 132, 1507 (1963). a

207

ELECTRON S P I N R E S O N A N C E O F MAGNETIC I O N S

GROUPIONSIN TITANIUM DIOXIDE(TiOr) axis and the [110] direction. The Ti4?and T i 4 9 nuclei

AT'

are the hyperfine constants for the superhyperfine

References

Remarks

Other spectra have been observed but these are not understood

a b

43 441(3) 16.7 15. (2)

-m(50)

--6800(50)

72.7(2)

4000

$9900

6780(+0.5%)

2210(5%)

AmTi < 0.47; &oTi ACT'= 0.93

i83(*20%)

F

< 0.47;

= -170(fM)%) (10-4

cm-9

26(1)

8. 9

1

j k

25.0(3)

1

Interstitial, light generated -83CW(+2%)

1

1

1370(+5%)

-29

b, 1

2.1

b, m

30.5( 2 )

n m 143.2 120.9 107.8

+22.8

+20.5 +20.2 +21.5

b4Z = -2.4 h44 = -182.0 56' = -1.0

0

h4= -30.0

b

-2.7

562 = bs6 = +4.8

m

63.9

Superhyperfine lines also observed: ATi = 2.5 AiioTi = 3.0 A f i = 4.0

P

' D. L. Carter and A. Okaya, Phys. Reo.118,1485 (ISM). E.Yamaka and R. Q. Barnes, Phys. Reo. 196,1568 (1962). Q. M. Zverev and A . M . Prokhorov, Souiet Phys. JETP (English Trand.)16,303 (1963); Proc. Is2 Intern. Con/. Paramagnetic Resonance, Jerusalem, 1962, Vol. 1, p. 13. Academic Press, New York. 1963. H. J. Qerritsen and E. S. Sabisky, Phys. Reo. l26.1853 (1962). P. F. Chester, J . AppZ. Phys. 39,866 (1961). Ru-Tao Kyi, Phys. Rev. 188,151 (1962). O E. Yamaka, J.Phys. SOC. Japan 18,1557 (1963). T.Chang, BUZZ.Am. Phys. Soe. (219,568 (1964). E. Yamaka and R . 0. Barnes, Phys. Reo. 136, A144 (1964).

'

*

TABLE AVIIa. ESR DATA FOR PEROVSKITE-TYPE MATERIALS-DATA FOR BaTiOa

AND

h3 0

SrTiOa

00

Host material

Ion

Freq. band

BaTi03

Fe3+

X

SrTiOa

Temp. (OK) 425 300

X

Feat

300 300 80 77 4.2 1.9 295 295 295 295 80

TT

n

Mn4f Cr3+

Refer9

X KP X X

91

A

9r

bdO

mces

1 7 (1)~ 51.0 f (10)

a, b

96.14 98.8 (50)

c

7.7(3) 7.3(3) 110.(6) 16.1(7) 112.8(10) 17.9(10) 115(5)

c

bz"

2.003 (1) 930 2.004 (1) 2.004(1) 2.004 (1) 2.004 (1) 2.004 (1) 2 ,004 ( I ) 2.0054(7) 2.0054(7)

5.993 (1) 5.961(1) 7*5(1) 15.8(1)

1.994(1) 1.978(7)

d d

3 F

2 +

d e

5

f

7

9

0

m 4

r

M 2

m

Ion

Frequency band

Temp.

9

811

QI

A

B

Ref.

SrTiOs (rontinued)

NiZ+(A)

X

80

2.204( 1)

Ni3+(B)

X

203 80 20 4.2

2.180 (2)

Nil+(C)

X

203, 80, 20

h 2.172 (1) 2.136 (1) 2.110(2)

2.184(1) 2.202 (1) 2.213 (2)

h

2.029 (1)

2.352 (1)

h

$

X

Nil+(L))

77

2.375 (2)

2.084(2)

i h

Ce3f

X

4.2

3.005(5)

1.118(3)

Nd3+

X Kll

4.2 2 2

2.609(3) 2.61(1) 2.62(1)

2.472(3) 2.470(5) 2.740 (5)

50 2 77 65 60 2

2.18 (1) 2.11(1) 2.25(1)

2.720(5) 2.780 (5) 2.67(1) 2.70 (1)

2.170(5) 2.10(1)

2.720 (5) 2.785(5)

Q h', h',

Yb3+

Q

F

M

d

20

j

Z

u, A171

21

B171 720(5)

530(20)

Z

m

Bin 11(5)

M

u,

0

2 TABLEAVIIb. ESR

Host material

Ion

BaTiOI

Gd3+

firTiO3

Gd3+

EU2+

Frcyucwc.y Temp. harid ("I<)

3M

PEROVSKITE-TYPE MATERIALS--SSTATEIONS

0 4

F0 9

li, K,

425 300

K,

300 1.992(2) 77 1.992(2) 4 . 2 1.992(2) 200 1.990(1) 2 1.990(1)

K,

D A T A FOR

1.995(3) 1.995(3)

b,'

b4' 1v6

-293.6(1.0) -233.6(5) -3G2.5(5)

-10(4)

4.0(1.0)

bGO =O l.G(l.0)

-5.7(2) 0.5(3) -4.8(5) -0.25(5.0) -:%.24(50) 1.4(5) 105.9(2) 1.1(2) 106.6(2) 0.7(2)

bn4

bG4

b$

Itcf. k

-2.0(1.0)

r ~ 0

$

2 3

k -2.5(3) -4.2(5) 1 . 1 4x2 6.7 f2

0.1(5) -0.69(50) j h3

!2

TABLE AVIIb (Continued)

0

Frequency Temp. band (OK)

Host material

Ion

LaA1O3

Gd3+

X

689 575 505 415 358 293 273 203 80 20

9

1.991 1.991 1.9910 1.9910(5) 1.9915(5) 1.9908 (5) 1.9911 (5) 1.9904(5) 1.9911 (5) 1.9909 (5)

67 .O (10) 150.6 (10) 208.3 (10) 307 .O(10) 337.0(10) 371.2(10) 383.8 426.7 479.2 +490.7

be0

b4'

b2'

5.60(3) 5.80(3) 5.99(3) 6.13(3) 6.17(3) 6.23(3) 6.42(3) 6.42(3) +6.46(3)

b,4

Ref.

blj4

0.9(2) 1.0(2) 1.0(2) 1.0(2) 1.0(2) 0.9(2) 0.9(2) 0.9(2) +0.9(2)

9.9(12) 7.6(12) 9.4(12) 7.2 10.2

+8

1

.o

TABLEAVIIc. ESR DATAFOR PEROVSKITE-TYPE MATERIALS-DATAFOR PbTiO, KTa03, LaAIOa, KMgFs ~

~~

Temp. Ion

Host

Fe3+

PbTi03 KTa03

Cr3+

LaA103

(OK)

290 4.2 4.2 291 273

so 20

9

bzO

br0

A

A'

B'

References

911 = 2.009(5) 91 = 5.97(2)

172.5

1.99 f 0.01 911 = 1.99 gs = 6.0

1.9825(5) 1.9825(5) 1,9825(5)

n n

450(1) 475.5 (10) 587 (6) 600 (2)

0

KMgFs

77

1.9720 (2)

Cr3+

77

1.9733(2)

Cr'+

77

2.0005 (5)

2.25( 100)

300

2.0015(5)

3.25(25)

Fe3f

77

2.0031 (2)

Ni2f

20, 77

2.2797 (4)

Cd+

4.2

\T2+

Mn2+

86.2(2)

25.75 (3)

91.0(5)

7.1(2)

2.0(2)

9.4(2)

2.3(2)

23.0(5)

17.5(5)

23.9(5)

18.0(5)

36.0(5)

18.0(5)

p

M M

F 0

$ 1

4.28

A. W. Hornig, R. C. Rempel, and H. E. Weaver, J . Phys. Chem. Solids 10, 1 (1959). D. Shaltiel, Ph.D. Thesis, Haifa Inst. of Technology (1960). K. A. Muller, Helv. Phys. Acta B1, Suppl., 173 (1959). d W. I. Dobrov, R. F. Vieth, and M. E. Browne, Phys. Rev. 116, 79 (1959). E. S. Kirkpatrick, K. A. Muller, and R. S. Rubins, Phys. Rev. 136A, 86 (1964). f K. A. Muller, Phys. Rev. Letters 2 , 341 (1959). K.A. Mfiller, 78me Colloque Amp $re. Geneve, Arch. Sci. (Geneva) 11, 150 (1958). R. S. Rubins and W. Low, Proc. 1st Intern. Conf.Paramagnetic Resonance, Jerusalem, 1962, Vol. 1 , p. 69. Academic Press, New York, 1963. K. A. Miiller and R. S. Rubins, To be published. j L. Rimai and G. A. deMars, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, Vol. 1, p. 51. Academic Press, New York, 1963. L. Rimai and G. A. deMars, Phys. Rev. 127, 702 (1962). W. Low and A. Zusman, Phys. Rev. 130, 145 (1963). ln D. A. Gainon, Phys. Rev. A134, 1300 (1964). W. Wemple, Ph.D. Thesis, M. I. T. (1963). D. Kiro, W. Low, and A. Zusman, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, Vo1. 1, p. 44. Academic Press, New York, 1963. * T. P. P. Hall, W. Hayes, R. W. H. Stevenson, and J. Wilkens, J . Chem. Phys. 38, 1977 (1963).

rn

4

m

M C

u:

z9

2m 0

q

6

m

zz

rn

E c

W. LOW AND E. L. OFFENBACHER

TABLEAVIII. ESR DATAFOR TRANSITION

Ion

Host lattice

Sit,e B

Frequency band

X

Temperature ("K) 290

Lio.SAL.$ 0 4 Ordered Disordered

911

=

1.986(1)

g s = 1.989(2)

K, Q

Fe3+

9

B

X

290

A A

X X

290 290

2.0015(3) 2.0002 (1)

B

'X

290

2.001 (7)

A

K

290

2.006(2)

B

K

290

A

K

290

2.008(3)

B?

K

290

2.0023(1)

R. Stahl-Brade and W. Low, Phys. Rev. 116, 561 (1959). V. A. Atsarkin, Soviet Phys. J E T P (English Trunsl.) 16, 593 (1963). c J. Overmeyer, Private communication (1964). d E. Brun, S. Hofner, H. Loelinger, and F. Waldner, Helv. Phys. Acta 33, 966 (1960). 6 F. Waldner, Helv. Phys. Acta 36, 756 (1962). f E. Brun, H. Loelinger, and F. Waldner, Arch. Sci. (Geneva) 14, 167 (1961). 0 V. J. Folen, J . A p p l . Phys. 33, 1084 (1962). h V. J. Folen, Proc. 1st Intern. Conf. Paramagnetic Resonance, Jerusalem, 1962, Vol. 1, p. 68. Academic Press, New York, 1963. V. J. Folen, J . Appl. Phys. 31, 166s (1960). 7 R. H. Kelly, V. J. Folen, M. Hass, W. N. Schreiner, and W. G. Beard, Phys. Rev. 124, SO (1961). a

b

213

ELECTRON S P I N R E S O N A N C E O F MAGNETIC IONS

GROUPIONS

A

IN

SPINELS( A B 2 0 r )

bz'

I z(b4').

(b4').

- 3b4" I

-0.92

a,

-0.93

d d

75.4(6) 74.9(5)

e 10.247 I(1) 10.104

I

0.13

10.080

77.2 (1)

References

Remarks

I

0.024 I(2) 0.01

1

0.046 (2)

If (bP)),and bto. are of opposite sign

0.0166

A rhornbic cornponent is present as well

f

bJ

'1

TABLEAIXa. ESR DATAFOR IRONGROUP IONSIN GARNETS

FreIon

Host

quency band

Cr3+

YGaG

K

Q YAlG

Q

YGaG

K

Temperature

(OK)

9

290 300 77 300

gll 1.9767(2) 1.98 1.98 1.98 1.98

295 4.2 295 4.2

2.003(1) 2.003(1) 2.0047 (5) 2.0047 (5)

62’

(bra)a (d2)

3b4‘

(F)

Remarks

gs = 1.9757(2)

References a

b

3500 3490 2550 2620

3

* 3 m ? 0

Fe3f

-1295(3) -1320(4) -885 (5) -880 (6)

+93(2) +95(3) 31 (2)

26(4) 34(7) 37(4)

31(2)

38(5)

Octahedral(a) site Tetrshedral(d) site

c

9

9

i2m bm

TABLE AIXb. ESR DATAFOR RARE-EARTH IONSIN GARNETS All measurements a t X band and 4.2"K except Yb3+,which was measured a t 20°K.

M

r

8

Ion

Host

Nd3+

LuGaG YGaG LuAlG YAlG

2.083 (7) 2.027(8) 1.789(5) 1.733(2)

1.323(7) 1.251(8) 1.237(6) 1.179(2)

LuGaG YGaG LuAlG YAlG

13.45(10) 11.07(20) 2.29(3) 0.73(15)

0.57 (10) 1.07(14) 0.91 (5) 0.40 (20)

3.41 (3) 7.85 (10) 16.6(3) 18.2(4)

d d d

12.62 (10) 10.73(5) 8.43 (4) 7.35(8)

d d d d

92

9u

9 2

3.550(7) 3.667(8) 3.834(5) 3.915 (26)

References d d d d

3 u1

2

z

m

M

u,

DyS+

Er3+

Yb3+

LuGaG YGaG LuAlG YAlG

3 .183 (15) 4.69(3) 6.93(2) 7.75(9)

3.183(15) 4.03 (2) 4.12 (3) 3.71 (2)

LuGaG YGaG LuAlG YAlG

3.653(13) 3.73(2) 3.842(5) 3.87 (1)

3.559 (13) 3.60(2) 3.738(7) 3.78(1)

2.994 (11) 2.85(2) 2.594 (4) 2.47(1)

0

2 2M

e

0 %!

z

k 25 c?

d fl

d d

9

5z

v,

Host YGaG Temprrature ("10 300 9 b2' b22

b44 bs' bs2 b64 be6

KJon &/ion References

I

1.991 +440.7 $216.1 -43.2 +0.3 f36.1 +O .3 +O .04 -0.7 t3.9 -611 - 23 h

LuGaG

LuGaG

LuAlG

YGaG

YGaG

YAlG

300 1.99 +275 +228 -44.9

4.2 1.99 +279 $235 -47.3

300 1.989 +571.5 f115.4 -50.2

300 1.992(3) 441.3(5) -269.7 (5) - 42.2 ( 5 )

4.2 1.992(3) 448 ( 5 ) - 283.8 ( 5 ) -45.2 (5)

300 1.990(3) 777.7(5) -85.1 ( 5 ) -46.9 (5)

+39.2

+43.3

+40.4

22.1(5) +4.15(5)

23.2(5) +3.77(5)

21.1(5)

+O.G(5) 0.1(5)

1.28(5)

2.43 (5)

YAlG 4.2 1.990(3) 776 .4 ( 5 ) -96.9 (5) -48.6(5) 23.5(5) 4.62(5) 4.29(5)

4 F 0

*

z u

M

-658 -20 h

-715 -21 h

-699 -25 h

189.5 11.2

i

205.5 11.8

i

.

211.8 15.1

1

222.2 1 G .O 1

t. 0

r r

Ultrasonic Effects in Semiconductors NORMAN G. EINSPRUCH Physics Research Laboratory, Texas Instruments Incorporated, Dallas, Texas

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ........... 11. Summary of Classical Analysis ............................ 1. Hooke's Law, Elastic Cons ts, Wave Equation.. . . . 2. Scattering.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... 3. Diffraction.. . . . . . ........................................ 4. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Measurement Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Megacycle Techniques-Pulse Echo. . . . . . . . . . . . . . . . 6. Gigacycle Techniques-Microwave Ultrasonics.. . . . . . 7. Samples and Bonds.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Electronic Attenuation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Some Generalizations on the Elastic Moduli. . . . . . . . . . . . . . . . . . . . . . . . . . 10. Electronic Contribution to the Shear Moduli.. . . . . . . . . . . . . . . . . . . . . . . . . 11. Acoustoelectric Effect.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Piezoelectric Interaction.. . . ............................... ............. 13. Radiation Damage.. . . . . . . . . . . . . . . . . . . . . . . 14. Phonon-Phonon Interactions. . . . . . . . . . . General References. ........................ ......................

217

224 225

230 235 237 247 264 265 26s

1. Introduction

A useful technique for investigating the physical properties of materials in the solid state is to measure the characteristics of the propagation of mechanical waves in the solid. These characteristics are the velocities of propagation and the spatial or temporal rates of absorption of energy from the mechanical wave. Data on the attenuation of mechanical waves have been interpreted over a range from 70 Gc sec-I, in quartz at low temperatures,' to frequencies corresponding to the time required for geological processes2 to take place. The purpose of this article is to review exploitation of this technique in the elucidation of the physical properties of semiJ. B. Thaxter and P. E. Tannenwald, A p p l . Phys. Letters 6, 67 (1964). J. R. Macdonald, J . A p p l . Phys. 32, 2385 (1961). 217

218

NORMAN G. EINSPRUCH

conductors. Particular emphasis will be placed on the 1 to 200 Mc: sec-’ frequency range. The term uZtrasonic will be used to describe mechanical waves of frequency greater than 1 Mc sec-l; waves of frequency greater than 500 Mc sec-l have been described in the literature as hypersonic and microwave ultrasonic. The usefulness of the ultrasonic technique will be demonstrated in discussions of a wide variety of experiments in which ultrasonics is the research tool. The review will begin with a brief exposition of classical analysis, including elasticity theory, wave propagation, scattering, and diffraction. Measurement and laboratory techniques will then be discussed briefly. The bulk of the review is concerned with discussion and interpretation of experiments on a variety of semiconductors, both piezoelectric and nonpiezoelectric. 11. Summary of Classical Analysis

1. HOOKE’SLAW,ELASTIC CONSTANTS, WAVEEQUATION

Since the strains3s4 normally encountered in ultrasonic measurements to lo-’, calculations based upon classical linear are of the order of elasticity, as reviewed by Love6 and Sokolnikoff16are normally valid. Principally for the purposes of definition, a brief outline of classical wave propa. gation in solids is presented here. The strain tensor, eij, is defined as eij =

h[

+

(dui/dxj) (duj/dxi)], i, j

=

1, 2, 3,

where the xi are the components of the coordinate vector of the point at which the strain is defined in a rectangular Cartesian coordinate system; the ui are the components of the displacement vector representing the change in position of a material point produced by the deformation. As defined here, the strain is a symmetric tensor. The general linear relationship between stress and strain,

is a statement of Hooke’s law; repeated indices indicate summation over these indices. The C i j k 1 are the elastic constants of the material. The reF. G. West and N. G. Einspruch, J . Acoust. Soc. Am. 32, 1160 (1960). L. T. Claiborne and N. G. Einspruch, Phys. Letters 7, 301 (1963). 6 A. E. H. Love, “A Treatise on the Mathematical Theory of Elasticity,” 4th ed. IEeprinted by Dover, New York, 1944. 6 I. S. Sokolnikoff, “Mathematical Theory of Elasticity,” 2nd ed. McGraw-Hill, New York, 1956. 3 4

ULTRASONIC EFFECTS IN SEMICONDUCTORS

219

quirements on the symmetry of the tensor of elastic constants upon interchange of indices, and the postulation that there exists a strain-energy potential function which is quadratic in the strains, reduce the number of independent elastic constants from 81 to 21. Upon consideration of crystalline symmetry, the number of elastic constants may be further reduced; for example, a cubic material has three independent elastic constants, an isotropic material has two elastic constants. It is frequently convenient to use the following contracted notation: c,,

=

Cijkl,

r = i, s =

for

i

k, for k

=

j

=1

i # j # m

r=m+3,

for

s = p + 3 ,

for k # l # p ,

where r, s

=

1, 2,

..-,6.

It is these reduced constants that one finds compiled in the literature.7 Table I contains the arrays of reduced elastic constants for several crystal systems of high symmetry : isotropic, cubic, hexagonal, tetragonal, rhombohedral, and trigonal. By considering the rotational and translational equilibrium conditions for an element of volume in a material in the deformed state, it is readily shown that the stress tensor is symmetric. An objection to making the elastic energy independent of rotations was raised by Lavala; his proposal was to reformulate classical elasticity in terms of a stress tensor which is not symmetric; the reformulation is accompanied by an increase in the number of elastic constants required to describe the properties of a given crystal symmetry class. Measurements utilizing optical techniques have been made by LeCorreg and by Joel and Wooster'O on ammonium dihydrogen phosphate (ADP) ; these results lent support to Laval's hypothesis which, if valid, would have great impact on the tremendous body of literature dealing with the elastic constants of materials. Jaffe and Smith" have measured two of the modified Laval elastic constants of ADP by a piezoelectric resonance technique and by the ultrasonic pulse-echo technique, thereby obtaining an excellent internal check on their measurements. They indicate that the probable maximum difference between the Laval H. B. Huntington, Solid State Phys. 7, 214 (1958). '5. Laval, Cbmpt. Rend. 232, 1947 (1951); 238, 1773 (1954). ' y . LeCorre, Bull. SOC.Franc. Mineral Crist. 77, 1363 (1954); 78, 33 (1955). I' N. Joel and W. A. Wooster, Acta Cryst. 13, 516 (1960). H. Jaffe and C . S. Smith, Phys. Rev. 121, 1604 (1961).

220

NORMAN G . EINSPRUCH

TABLE I. THEFORM OF THE CRYSTALLINE ELASTIC CONSTANTS cij FOR ISOTROPIC, CUBIC,HEXAGONAL, TETRAGONAL, RHOMBOHEDRAL, A X D TRIGONAL SYMMETRY

x

0

0 0 0

P

Isotropic:

c44

0

c44

c 3I

0 0 0

0 0 0

c44

0 0 0 0

0 0

0

~

C3l

0 0 0

0 0 0

0 0

c13

c13

c13 r13

Trigonal :

0 0 0 0

c44

c13

Rhombohdral:

0 0 0

0

0

CI

Tetragonal:

P

0

CI I

0 0 0

Hexagonal :

0 0

0 0 0 0 0

c12

. CIZ

Cubic:

0 0 0 0

c44

c14

- 04

c44

0 0 0 0

0 0 0 0 0 f(n1 -

C44

0 0 0 0 0

0

r66

0 0 0 0

0 0 0 0

r.13

0

0 0 0

0

4 4

c14

0

c14

f (Cll

(‘44

(‘I3

c14

(‘13

-c14

cs:l

0 0 0

-c25 c25

c44

0 0

0

c44

c25

c14

0

-

c12)

c12)

0 0 0 c25

c14 3(Cll

-C

d

I I I I I

shear constants which they studied was no greater than O.l%, rather than 6% or greater as was previously reported. Consequently, the classical formulation will be employed throughout the present discussion.

ULTRASONIC EFFECTS I N SEMICONDUCTORS

221

J x3

FIG.1. Equilibrium of an infinitesimal volunie in a stressed solid.

Consider a small, rectangular parallelopiped of material in the deformed state, as in Fig. 1, of lateral dimensions of Axl, Axz, Ax3. In general, the stress will vary from point to point in the material; the net force acting in the x1 direction is

{[TI,

+

ax1

Axl] - TlI] Ax2 Ax3

+ [[TIZ+

Ax2] - Tlz} Axl Ax3 ax2

where p is the density of the material in the deformed state, and ji is the body force per unit mass. Upon rearranging terms, one obtains d~~~ ac3 + -+ - + dxz dx3 8x1

a~,,

-

P j l = PUl.

The same argument applied to the other two sets of surfaces yields d Tij + pfi dXj

=

PUi,

the equations of motion for an elastic material. Hooke's law for an isotropic material is

222

NORMAN G . EINSPRUCH

where p and are the Lam6 constants of the material and 6 i j is the Kronecker delta, 1 i=j 6..= for 0 i # j.

[

For an isotropic material, in the absence of body forces, the equations of motion reduce to the well-known wave equation: pv2u

+ + A)V(O~U) (p

= pu.

The wave equation yields as solutions one longitudinal wave traveling with 2 p ) / p ] 1 ’ 2 and one transverse wave traveling with velocity velocity [(A [ p / p ] 1 / 2 . Since the three components of the displacement vector completely specify the deformation of the material, the redundancy implied in the six “independent” components of the stress tensor is removed upon satisfaction of the six equations of compatibility of strain, which are given below :

+

a2e33 -+-=2ax32 ax2?

a2e23

ax2 ax3

, etc.

and d2e11 ax2 ax3 -=

a ae23 de31 ae12] -[-+ ax2 + axg , ax1 ax, -

-

etc.

Detailed analysis of wave propagation in anisotropic solids indicates that waves may be propagated in any direction in a n extended solid. For an arbitrary direction of propagation, three plane wave velocities of propagation are predicted by the wave equation. Under certain circumstances, the normal to the wave front coincides with the direction along which energy is propagated. This situation occurs for propagation along puremode directions for which two of the plane waves are identifiable as transverse waves and the third wave is identifiable as a longitudinal wave. Since the analysis of experiments utilizing pure modes of propagation is not hindered by the presence of waves other than the one under study, pure modes are studied whenever possible. The problem of selecting the pure-mode directions has been studied by SakadiI2 and, later, by Borgnis,13 for several crystal classes. Unfortunately, for crystal classes of symmetry less than cubic, measurement of a complete set of elastic constants requires at least one measurement in a direction which is not a crystallographic axis or in a symmetry plane. 12 13

Z. Sakadi, Proc. Phys. Math. SOC.Japan 23, 539 (1941). F. E. Borgnis, Phys. Rev. 98, 1000 (1955).

ULTRASONIC E F F E C T S I N SEMICONDUCTORS

223

FIG.2. Pure-mode directions and related elastic constants of a hexagonal crystal.

Rather than a presentation of a detailed discussion of the solution of the eigenvalue problem associated with wave propagation in anisotropic media, an illustration of the requirements of an experiment in two crystal systems will be discussed. For a hexagonal crystal, such as CdS, the pure modes14 of propagation are shown in Fig. 2. The longitudinal wave propagated along the hexagonal axis yields c33;the transverse wave, of arbitrary polarization, propagated along the hexagonal axis yields c44; the compressional wave propagated in the base plane yields ell; the transverse wave propagated in the base plane and polarized in the base plane yields +(ell - clz); the transverse wave propagated in the base plane and polarized perpendicular to the base plane yields c44. As a result of the degeneracy of the elastic properties in the base plane, the direction of propagation in the base plane is arbitrary. Four of the five independent constants, ~ ,thus obtained from pure-mode measurements. and a cross check on c ~ are I n their evaluation of the complete set of elastic constants of zinc, Alers and NeighboursI5 made measurements in the base plane and along the hexagonal axis, as indicated above; to obtain c13 measurements were made in a skew direction, 16.6 deg with the hexagonal axis. l4

l6

P. C. Waterman, J . A p p l . Phys. 29, 1190 (1958). G. A. Alers and J. R. Neighbours, Phys. Chem. Solids 7, 58 (1958).

224

NORMAN (3. E INS P RU C H

For a cubic crystal, such as germanium and silicon, it is possible to orient the sample such that all three elastic constants can be evaluated from measurement of the three independent velocities in one direction of propagation. For propagation in the [110]: v1

2.

= [(Cll

+ + 2C44)/2PI1~*, ClZ

u

II C l l O l

SCSTTERING

In an experiment in which the diminution of the intensity of a plane elastic wave is measured, there are three classes of processes which contribute to the attenuation. The first of these is elastic scattering which occurs when the incident wave impinges upon a region in the sample of elastic properties different from those of the matrix in which the region is located. I n every case other than normal incidence upon a n extended planar discontinuity, some of the elastic energy is deviated from the direction of the incident wave. To a sensor located to receive the plane wave signal from the source, such scattering appears as an attenuation; it is not, however, a loss or absorption of energy, since sensors stationed around the periphery of the elastic sample would receive the energy which is deviated from the beam. The literature on the scattering of acoustic and elastic waves begins with a detailed discussion by Rayleighle of scattering by elastic discontinuities of small size; he found that for isotropic acoustical materials, the scattering cross section in the ka << 1 (where a = radius of the spherical scatterer, k = 2r/X = wave number, X = wavelength) limit is proportional to k4a6.The problem of scattering of plane longitudinal waves by a spherical obstacle in a homogeneous, isotropic elastic solid has been studied in detail by Ying and TruellI7 and by Einspruch and Truell*susing an analysis similar to the one of Herzfeld,’g who studied scattering by an elastic sphere in a viscous fluid. The shear wave scattering problem was treated by Einspruch et aLZ0White2I performed a similar analysis for a cylindrical discontinuity. The results of these analyses indicate that in elastic materials in the Rayleigh limit, the scattering cross Lord Rayleigh, “The Theory of Sound,” 2nd ed., Vols. I and 11. Reprinted by Dover, New York, 1945. 17 C. F. Ying and R. Truell, J . Appl. Phys. 27, 1086 (1956). l a N. G. Einspruch and R. Truell, J . Acoust. SOC. Am. 32, 214 (1960). l9 K. F. Herzfeld, Phil. Mag. [7] 9, T41 (1930). 20 N. G. Einspruch, E. J. Witterholt, and R. Truell, J . A p p l . Phys. 31, 806 (1960). R. M. White, J . Acoust. SOC.Am. 30, 771 (1958). 16

ULTRASONIC EFFECTS I N SEMICONDUCTORS

225

section depends on k4a6 to first order. The problem of multiple scattering has been studied by Waterman and TruelP using field averaging concepts due to FoldyZ3;their results, based upon the previously reported single scatterer results, have been applied to a radiation damage problem.

3. DIFFRACTION The second type of apparent loss process appears in attenuation measurements a t low megacycle frequecies in samples of low attenuation. This loss is manifest experimentally as irregular variations of the apparent attenuation with the distance traversed by the sound wave. Seki et al.,24 following an analysis due to Huntington et al.25 and Williams,26have shown that the observed irregularities can be attributed to a diffraction effect due to the deviations from plane wave propagation which occur when the source of sound is too few sound wavelengths in lateral extent to be considered a plane wave source. In attenuation measurements in germanium2' at 5 Mc see-' using a ;&.-diameter source, almost 90% of the loss could be ascribed to diffraction. A rough, but easily remembered, estimate of the diffraction contribution to the attenuation is 1 dB per a2/X, where a is the radius of the source. McSkiminZ8has experimentally investigated the effect of diffraction on the velocity of propagation of ultrasonic waves; he finds that diffraction errors of less than O.OZyo are readily achieved by not extremely stringent restrictions on the experimental conditions. For a typical value of the sound velocity, 5 x 105 cm sec-I, a diffraction error of 0.0270 is incurred when a I-cm-diameter transducer operated a t 10 Mc sec-I is used with a total path length of 10 cm. 4. DEFINITIONS

The third contribution to the attenuation is the absorption and eventual dissipation of elastic energy as heat by the sample. It is this class of mechanisms which yields information on the microscopic interactions in solids. Henceforth, the term attenuation mill refer only to this class of losses. Consider a damped plane stress wave described by T ( z ) = Ae-CLtz exp [-i(Wt - k z ) ] where at is the attenuation coefficient and

w

is the angular frequency of

P. C. Waterman and R. Truell, J . Math. Phys. 2, 512 (1961). L. L. Foldy, Phys. Rev.67, 107 (1945). 24 H. Seki, A. Granato, and R. Truell, J . Acoust. Soc. Am. 28, 230 (1956). ?5 H. B. Huntington, A. G. Emslie, and V. W. Hughes, J . Franklin Inst. 246, 1 (1948). ?6 A. 0. Williams, Jr., J . Acoust. Soc. Am. 23, 1 (1951). A . Granato and R. Truell, J . A p p l . Phys. 27, 1219 (1956). 2 8 H. J. McSkimin, J . Acoust. SOC. Am. 32, 1401 (1960). ?*

23

226

NORMAN’ G . EINSPRUCH

+ Ax) corresponding to (t + Axk/w) the wave is

the wave. At a point (x described by

T(x

+ Ax) = A exp [-a’(x + Ax)] exp [ - i ( ~ t - kx)],

thus for small a’ a’ =

(Ax)-’ loge [ T ( z ) / T ( z

+ Ax)];

a’ therefore has the dimensions Np/length. A convenient unit of measure of energy ratios is the decibel, which is introduced as follows: a’ = (Ax)-’ 20 IOgm [T (x)/T (Z

+ Ax)];

a’ here has the dimensions dB/length. Obviously a’ [dB/length]

=

8.686~2’[Np/length],

a’ [dB/length]

= 20 (log,, N/loge N ) a ’ [Kp/length]

since where N is any number. The logarithmic decrement, 6, is defined as the ratio of half the energy dissipated per cycle to the total elastic energy stored per cycle.

It can be shown that 6 = a’X;

6 thus has the dimensions N p or dB. The energy lost per unit time is given

by a = a’v

where v is the velocity of propagation of the elastic wave. A convenient unit of time in ultrasonic experiments is the microsecond; thus a[dB/p sec]

=

a’[dB/cm]v[cm/p

sec].

Finally, the quality factor, Q, of the system is related to the decrement by Q6

= T.

Frequently, loss data are reported in terms of the reciprocal of the quality factor, &-I. 111. Measurement Techniques

5 . MEGACYCLE TECHNIQUES-PULSE ECHO The development during World War I1 of radio-frequency pulse techniques for radar applications led directly to the use of megacycle-persecond sound waves in studying the physical properties of solids. Hunt-

227

ULTRASONIC EFFEW'I'Y I N SEMICONDUCTOliS

, Arenberg ultrosonic laboratory PG 650 C pulsed osclllotor

Echoes

Tektronix 545 oscillostope

1

preamp

Diode limiting circuit

Quartz transducer

genera tor

multiplier

FIG.3. Pulse-echo apparatus (after Einspruch and Manning31).

ingtonZ9in 1947 presented a detailed discussion of the use of 10-Mc sec-I ultrasonic pulses in measuring the elastic constants of single crystals of some of the alkali halides. The pulse-echo technique has since been used in a number of laboratories in contributing to a detailed literature on the mechanical properties of solids. A discussion of the design and construction of a pulse-echo system has been given by Forgac~.~O A system similar in principle of operation has been constructed in this laboratory31 and will be described here; a block diagram is shown in Fig. 3. A radio-frequency pulsed oscillator produces a, typically, 10-Mc sec-' pulse of about 1.5-psec duration and also furnishes a trigger pulse for the oscilloscope. Several hundred pulses per second excite the transducer which is bonded to the sample. The mechanical pulse thus introduced into the sample is reflected back and forth between a pair of carefully prepared parallel faces. After each transit through the sample, a small portion of the energy in the mechanical pulse is reconverted to electrical energy by the transducer; this electrical signal is amplifled and displayed on an oscilloscope. The sawtooth circuit which controls the time base of the oscilloscope has been modified in order to allow greater precision in measurement. The diode limiting circuit prevents saturation of the amplifiers by the direct pulse from the transmitter; saturation can also be avoided by using two transducers-one for transmitting, one for receiving. The times between the first half cycle of successive unrectified echoes, which are displayed, are measured with the modified delay 29

30 31

H. B. Huntington, Phys. Rev. 72, 321 (1947). R. L. Forgacs, Proc. Natl. Electron. Conf. 14, 528 (1958). N. G. Einspruch and R. J. Manning, J . Acoust. SOC.Am. 36, 215 (1963).

228

NOHMAN G . EINSPHUCH

which is calibrated against a crystal-controlled time-mark generator. The transit time and thickness of the sample yield the velocity of propagation. Typical accuracies for this technique are ~ described a compact instrument for use in attenuaChick et 1 2 1 . ~have tion and velocity measurements in the 5 to 200 Mc sec-l range. The instrument contains a pulsed oscillator, superheterodyne receiver, video amplifier, and detector, as well as synchronization circuits, exponential wave-form generator, and a cathode ray tube with display circuits. An instrument of this type is in use in this laboratory and has been found extremely useful in making attenuation measurements and in making velocity measurements to accuracies of 10-2. This technique has been used a t Brown University a t frequencies as high as 1000 Mc sec-I. Waterman and T e u t ~ n i c ohave ~ ~ described an interference phenomenon which has been used by Stein et ~ 1 . ~ ~ 0in3 5measurements of the temperature dependence of the elastic constants of silicon and germanium. If the same rf transmitter signal is used to drive transducers on two identical samples at a given temperature, an exponential decay pattern will be observed for the combination. If the temperature of one sample changes slightly, the small difference in the travel time of a sound wave due to the temperature dependence of the elastic constants and to thermal expansion produces a n interference effect which can be used to obtain the temperature dependence of the moduli. McSkimin36has developed a pulse superposition technique which yields accuracies of 2 x 10-4 in velocity measurements. Instead of measuring the time delay between successive echoes, one controls with a continuous wave oscillator the pulse repetition rate such that it corresponds to the time delay between echoes. When the rate is properly adjusted, superposition of echoes results; the rate is then counted and the transit time evaluated. 6. GIGACYCLE TECHNIQUES-MICROWAVE ULTRASONICS

The common technique for achieving higher measurement frequencies for ultrasonic pulse-echo experiments is to operate a thin transducer at a high harmonic resonance mode. A frequency of around I Gc sec-I was thus achieved by Ringo et 121.~7 An alternative approach, due to B a r a n ~ k i i , ~ ~ B. B. Chick, G. P. Anderson, and R. Truell, J. Acoust. SOC.Am. 32, 1S6 (1960). P. C. Waterman and L. J. Teutonico, J . Appl. Phys. 28, 266 (1957). 34 F. Stein, N. G. Einspruch, and R. Truell, J. Appl. Phys. 30, 820 (1959). 36 F. Stein, N. G. Einspruch, and R. Truell, J . Appl. Phys. 30, 1756 (1959). 36 H. J. McSkimin, J. Acoust. SOC. Am. 33, 12 and 606 (1961); 34, 609 (1962). 37 G. R. Ringo, J. W. Fitzgerald, and B. G. Hurdle, Phys. Rev. 72, 87 (194’7). K. N. Baranskii, Soviet Phys. “Doklady” (English Transl.) 2, 237 (1957). 32

ULTRASONIC E F F E C T S I N SEMICONDUCTOKS

229

is to place a thick quartz plate into a high-frequency electric field. Elastic waves are excited a t the surfaces of the piezoelectric material and are propagated into the bulk. J a c ~ b s e nused ~ ~ a Green's function approach in solving the inhomogeneous wave equation which describes the wave motion in a piezoelectric medium; he demonstrated that the sound wave is generated a t the piezoelectric stress discontinuity a t the surface of the quartz rod. Moreover, he was able to show that the microwave and the radio-frequency techniques for piezoelectric sound generation are electrically and mechanically equivalent. Bommel and Dransfeld40 reported the generation of longitudinal ultrasonic waves a t 2.5 Gc sec-' in a n X-cut quartz rod, a portion of which was placed in a microwave cavity such that an end of the rod was in a region of high electric field; transverse waves were also generated in a Y-cut sample. The presence of the ultrasonic wave was detected by the diflraction of a light beam by the strained quartz samples. By placing the other end of the X-cut rod in an identical microwave cavity, which acted as a receiver, the propagation velocity of ultrasonic pulses was measured. Jacobsen41 extended the frequency range to 10 Gc sec-' in measurements on quartz by a similar technique a t liquid helium temperatures.

7 . SAMPLES AND BONDS In ultrasonic attenuation and velocity measurements made with the pulse-echo technique, errors can be introduced if the sound wave undergoes successive reflections between sample faces that are not strictly parallel. It has been found that tolerances of 50 p in./in. are readily obtained by ordinary sample lapping techniques and are acceptable for precision measurements. A second source of error in precision velocity measurements in anisotropic media is introduced as a result of crystal misorientation. The problem of obtaining the elastic constants of crystals from measurements made in arbitrary propagation directions has been treated by Arenberg.42 Neighbours and Neighbours et al. ,44 Fein and Smith,45and Neighbours.46 Relationships between the elastic constants of materials of sym39 40

E. H. Jacobsen, J . Acovst. SOC.A m . 32, 949 (1960). H. E. Bommel and K. Dransfield, Phys. Rev.Letters 1, 234 (1958); Phys. Rev. 117, 1245 (19GO).

*' E. H. Jacobsen, Phys. Rev.Letters 2, 249 (1959). 42

43

46

D. L. Arenberg, J . A p p l . Phys. 21, 941 (1950). J. R. Neighbours and C. S. Smith, J . A p p l . Phys. 21, 1338 (1950). J. R. Neighbours, F. W. Bratten, and C. S. Smith, J . Appl. Phys. 23, 389 (1952). A. E. Fein and C. S. Smith, J. A p p l . Phys. 23, 1512 (1952). J. R. Neighbours, J. Acoust. SOC.Am. 26, 865 (1954).

230

NORMAN G. EINSPRUCH

metry as low as orthorhombic and velocities obtained from propagation in skew crystallographic directions have been derived. Waterman47 has made a detailed study of the perturbed eigenvalue problem associated with wave propagation along nearly pure mode directions in various crystal systems. He finds that the error introduced in velocity measurements on cubic crystals depends on the square of the polar misorientation angle, 8 , and is maximum for zero azimuthal misorientation. For shear wave propagation in a [lo01 direction (Av/v),,,,,

=

E@.

E depends on the elastic constants of the material, and for silicon E -0.93. For 8 = 1 deg = 0.0174 rad, ( A ~ / u > , , = -2.8

x

=

10-4,

which obviously is significant in velocity measurements of 0.01% accuracy. The experimenter is thus required to maintain strict tolerances on crystal orientation, or to correct his measurements accordingly. One problem encountered in experimental work is preparation of the bond between the transducer and the sample. Phenyl salicylate (salol) forms a thin, uniform layer which melts at 43"C, convenient for roomtemperature work. I n low-temperature work, differential thermal expansion across the sample-bond-transducer sandwich often leads to fracture of the bond after cooling. Nonaq48 stopcock grease and the 200 Series Silicone F l ~ i d s , ~which g have rather low freezing points, have been used successfully in low-temperature work, as well as numerous other resins and plastics. The commercially available transducers commonly used are polished discs cut from natural crystals of quartz. For compressional waves, X-cut quartz is used; for shear waves, Y- or AC-cut quartz is used. IV. Semiconductors

8. ELECTRONIC ATTENUATION

I n 1957, Blattso suggested intervalley scattering as a mechanism for the attenuation of ultrasonic waves in n-type germanium and silicon. The stress wave raises the band edge of one group of the ellipsoidal energy surfaces and lowers the band edge of another group of ellipsoids. To reP. C. Waterman, Phys. Rev. 113, 1240 (1959). Fisher Scientific Company, Fairlawn, New Jersey. 4 9 Dow Corning Corporation, Midland, Michigan. 50 F. J. Blatt, Phys. Rev. 106, 1118 (1957). 47

4*

ULTRASONIC EFFECTS I N SEMICONDUCTORS

231

achieve an equilibrium distribution, electrons are transferred from the higher lying valleys to the lower lying valleys by the intervalley scattering mechanism. Intravelley scattering also takes place, but since the relaxation time associated with this process is very short, the former mechanism determines the over-all relaxation time. Weinreich51 subsequently pointed out that the mechanism proposed by Blatt is closely related to his explanation of the acoustoelectric effect, which will be discussed in Section 11. Although little data are available on the absorption and amplification of ultrasonic waves by electron-phonon interactions in semiconductors, a plethora of theoretical work exists and awaits experimental confirmation. Spector,52 M i k ~ s h i b a ,E~~~k s t e i n Chick~ashvili,~~ ,~~ and Pokatilov56 have performed detailed calculations applicable to semiconductors and semimetals, many of which are based upon the approach taken by Cohen et a1.57 to explain the frequency, magnetic field, and mean free path dependence of the ultrasonic attenuation in metals. Pomerantz et aE.58have investigated the propagation of 8.9-Gc sec-1 phonons in pure and doped germanium in order to study the electronic contribution to the attenuation at frequencies closer to the reciprocal of the intervalley relaxation time than is possible in the ultrasonic frequency range. The hypersonic phonons were generated and detected at 4 2 ° K by spin-wave phonon interactions in thin Ni-Fe films evaporated directly on the end of a germanium rod, as described by Pomerantz,59 and Bommel and DransfeId.60The germanium specimens were prepared from pure, less than 1014 donors and heavily doped, greater than 1019 arsenic donors cm-3, material. T h e results of the experiment are summarized in Table 11. [ l l O ] and [loo] propagation directions were selected allowing study of five types of waves. The striking result of this experiment is that all of the waves, regardIess of direction of polarization and direction of propagation, propagate in pure germanium, while only those waves whose elastic constant does not contain c44 can propagate in the heavily doped material. G. Weinreich, Phys. Rev. 107, 317 (1957). sz H. N. Spector, Phys. Rev. 120, 1261 (1960); 126, 1192 and 1880 (1962); 130, 910 (1963); 131, 2512 (1963); 132,522 (1963); 134, A507 (1964). 63 N. Mikoshiba, J . Phys. SOC. Japan 14, 22 (1959); 16, 982 and 1189 (1960); 16, 895 (1961). s4 S. G. Eckstein, Phys. Rev. 131, 1087 (1963). 5s Y. A. Chikvashvili, Soviet Phys.-Solid State (English Transl.) 6, 264 (1963). 66 E. P. Pokatilov, Soviet Phys. J E T P (English Transl.) 11, 835 (1960). 67 M. H. Cohen, M. J. Harrison, and W. A. Harrison, Phys. Rev. 117, 937 (1960). 58 M. Pomerantz, R. W. Keyes, and P. E. Seiden, Phys. Rev. Letters 9, 312 (1962). 69 M. Pomerants, Phys. Rev. Letters 7, 312 (1961). 6o H. Bommel and K. Dransfeld, Phys. Rev. Letters 3, 81 (1959). 6L

232

NORMAN G. EINSPI1UCI-I

TABLE11. PROPAGATION OF 8.9-Gc

SEC-’

PHONONS IN GERMANIUMO Propagation in Ge

Wave Propagat,ion direction

Polarization direction

Elast,ic constant

Piire

Heavily doped

a After M. Pomerantz, R. W. Keyes, and P. E. Seiden, Phys. Rev. Letters 9, 313 (1962).

The waves whose elastic constant contains cq4 destroy the equivalence of the valleys and are consequently attenuated by the intervalley relaxation mechanism, precisely as predicted by Keyes’Gl theory of electronic contribution to the shear moduli of many valley semiconductors. Mason and Batema@ have studied the effects of doping in n-type germanium and p-type silicon on the temperature dependence of the attenuation and elastic moduli. Measurements on arsenic-, antimony-, and phosphorous-doped germanium at frequencies as high as 500 Mc sec-l can be interpreted using the intervalley scattering mechanism; the relationship between the measurable quantities a’, the attenuation, and Ac/c, the fractional modulus change, is given by a’ = (w%/v) (Ac/c),

where w is the angular measurement frequency, T the relaxation time, and 2’ the velocity of propagation of the sound wave. The formula is valid for WT < < 1 with wl/v < 1, where 1 is the mean free path for electrons. Hence measurement of both the attenuation and fractional modulus change yields a direct measure of the relaxation time. For dopings of 1018 and 3 x 1019 and 2.3 x 10-13 sec, independent of arsenic atoms ~ m - ~T ,= 4 X temperature. Since the electronic contribution to the attenuation is small, the method becomes insensitive at donor concentrations less than 1018 cm-3; fortunately, the acoustoelectric technique, sensitive a t much lower concentrations, complements the absorption and velocity measurements well. Measurements on boron-doped silicon with 3 X 10’9 acceptors cm-3 yield a temperature-independent relaxation time of 10W3 sec for the repopulation of holes. 81 62

R. W. Keyes, I B M J. Res. Develop. 6, 266 (1961). W. P. Mason and T. B. Bateman, Phys. Rev. Letters 10, 151 (1963); Phys. Eev. 134, A1387 (1964).

ULTRASONIC E F F E C T S I N SEMlCONDUCTOltS

1

30

I

I

35

40

233

I 50

I

45

H(kG)

FIG.4. Longitudinal-wave magnetoacoustic resonance in p-type PbTe (after Shapira :tnd L a x 9 .

An oscillatory effect in the magnetic field dependence of the ultrasonic attenuation and wave velocity in the semimetal bismuth was reported by Mavroides et ale3 Alers and Swime4 observed oscillations in the sound velocity in gold in the presence of a suitably oriented intense magnetic field. Thse oscillations have been described as being of the de Haas-van Alphen type. The excellent agreement of the spacing with H of the Av/v minima of the acoustic and magnetic susceptibility measurements on gold by Sh0enbe1-g~~ substantiates the interpretation. Subsequently, Shapira Shear waves c

0 ._ c

0 3

W C

c

0

c .al m

0 C

5 20

30

40

50

60

70

80

H(kG)

FIG.5. Shear-wave magnetoacoustic resonance in p-type PbTe (after Shapira and Lax66). 63

J. G. Mavroides, B. Lax, K. J. Button, and Y. Shapka, P h y s . Rev. Letters 9, 451 (1962).

O4 65

G. A. Alers and R. T. Swim,P h y s . Rev. Letters 11, 7 2 (1963). D. Shoenberg, P h i l . Trans. R o y . SOC.London A266, 85 (1962).

234

NORMAN G . EINSPRUCH

TABLE111. REDUCEDELASTICCONSTANTS OF Crystal

co

THE

DIAMOND-TYPE CRYSTALS'

B*

C44*

CS*

C*

1.080 1.289 1.182 1.170 1.183 1.244 1.261 1.ll 1.072 1.140

1.405 1.050 1.055 0.925 0.906 0.873 0.815 0.56 (hex) 0.536

1.159 0.671 0.633 0.508 0.505 0.474 0.411 0.24 (hex) 0.224

1.30i 0.898 0.886 0.758 0.746 0.714 0.653 0.43 0.297 0.412

(10'2 dyn/cm2)

Cb

Si. Gecad GaAsc GaSbf AlSbg InSbf." ZnSi CdSj CdTek

4.09 0.759 0.636 0.646 0.477 0.476 0.371 0.753 0.572 0.372

After R. W. Keyes, J . A p p l . Phys. 33, 3371 (1962). H. J. McSkimin and W. L. Bond, Phys. Rev. 106, 116 (1957). c H. J. McSkimin, J. A p p l . Phys. 24, 988 (1953). d M. E. Fine, J . A p p l . Phys. 24, 338 (1953). C. W. Garland and K. C. Park, J . A p p l . Phys. 33, 759 (1962); T. B. Bateman, H. J. McSkimin, and J. M. Whelan, J . A p p l . Phys. 30, 544 (1959). f H. J. McSkimin, W. L. Bond, G. L. Pearson, and H. J. Hrostowski, Bull. Am. Phys. Soc. [2] 1, 111 (1956). g D. I. Bolef and M. Menes, J . A p p l . Phys. 31, 1426 (1960). h R. F. Potter, Phys. Rev. 103, 47 (1956); L.D. de Vaux and F. A. Pezzarello, Phys. Rev. 102, 85 (1956). i S. Bhagavantam and J. Bhimasenachar, PTOC. I n d . Acad. Sci. A20, 298 (1944). i D. I. Bolef, N. I. Melamed, and M. Menes, Bull. Am. Phys. Sac. [2] 6, 169 (1960). kH. J. McSkimin and D. G. Thomas, J . A p p l . Phys. 33, 56 (1962). (1

b

6

and I.ax66 reported the observation of de Haas-van Alphen type oscillations in the ultrasonic attenuation in PbTe; this is the first application of magnetoacoustic techniques, which have been extremely successful in elucidating the physical properties of metals, to a study of the properties of semiconductors. An oscillatory attenuation was observed a t 4.2"K for 57-Me see-' longitudinal ultrasonic waves with v 1 1 H I I [loo], as shown in Fig. 4; the material was p-type and had a carrier concentration of 4.8 x 10l8 a t 77°K. A single period with AH-l = (18.8 f 0.6) X 10-7 G-1 was found. Figure 5 shows the results for a 43-Mc sec-l shear wave a t 1.8"K; a single period with AH+ = (18.3 f 0.5) X lo-' G-' is observed for fields between 20 and 45 kG;between 60 and 90 kG the periodicity is close to that a t low frequency but the peaks show a not completely resolved splitting. The periodicity observed a t the lower fields is twice that 66

Y. Shapira and B. Lax, Phys. Letters 7, 133 (1963).

235

ULTRASONIC EFFECTS I N SEMICONDUCTORS

TABLEIV. APPROXIMATE VALUESOF Class

B*

c44*

Ge, Si

1.2 1.2 1.1

1.05 0.8-0.9 0.5-0.6

111-v 11-VI ~~

a

THE

~

~

REDUCED ELASTICCOWSTASTS~ c*

ca*

0.65 0.4-0.5 0.2

0.9 0.65-0.75 0.3-0.4

~ ~ _ _ _ _ _

After R. W. Keyes, J . Appl. Phys. 33, 3371 (1962).

expected from the interpretation of susceptibility data by Stiles et al.67 Although the discrepancy is not yet resolved and only preliminary data on n-type PbTe are available, Shapiro and Lax have demonstrated the feasibility of the ultrasonic technique in studying quantum effects in semiconductors. 9. SOMEGENERALIZATIONS ON THE ELASTIC MODULI

Keye@ has made some interesting generalizations concerning the elastic properties of semiconducting crystals of the diamond, zinc blende, and wurtzite structures. If a normalizing elastic constant co = q 2 k 4 is defined, where q is the electronic charge and b is the nearest-neighbor distance, and the ordinary elastic constants expressed in units of co are as follows:

B* c44*

(cii

+ 2Ci2)/3Co

= c44/co

cs* =

(c11

- C12)/2CO,

then these reduced elastic constants exhibit a much smaller variation from material to material than do the elastic constants themselves. In addition, the average shear modulus for a general crystal structure,‘jg c = h(C11

+ c2Z + c33 - c12 - c13 - c23) f B(c44 +

c55

f

C66)r

after reduction to c*

= c/co,

can be treated as well. A summary of the reduced elastic constants for several semiconductors is given in Table 111. One can make the following generalizations: The reduced bulk moduli all lie in the range 0.95 to 1.3; a rough average value is B* N” 1.2. The lowest values are those of diamond and the 11-VI compounds. The shear moduli decrease in the following ” P.

J. Stiles, E. Burstein, and D. N. Langenberg, Phys. Rev. Letters 6, 667 (1961). R. W. Keyes, J . A p p l . Phys. 33, 3371 (1962). 69 W. Voigt, “Lehrbuch der Kristallphysik,” Appendix 11. Teubner, Leipzig, 1928.

236

NOBMAN G . EINSPRUCH

TABLEV. REDUCED ANISOTROPY MODULIJS~ ~

~~

~~

r

Crystal

~-

GaAs GaSb AlSb InSb a

-0.834 -0.802 -0.798 -0.808

L. J. Touchard, J . A p p l . Phys. 34, 2694 (1963).

order: fourth-group elements, 111-V compounds, 11-VI compounds. The decrease of c,* with increasing separation of the component elements in columns of the periodic table is relatively faster than the decrease of CM*. Rough average values for the various crystals are given in Table IV. Touchard70 then made the additional observation that the reduced anisotropy modulus r = (ell - c12 - 2 C 4 4 ) / C u is essentially constant for the 111-V compounds as shown in Table V. Steigmeie? incorporated Keyed observations into the work of Marcus and Iien1iedy7~on the relation between the 0°K Debye temperature, OD ( 0 ) : and the elastic constants of a material-with the following result: OD

(0)

=

c4.19

x

10-8(a3~)1/21 (cll/co)l’2f(Tir

~ 2 )

where

a M f

=

= =

lattice constant mean atomic mass angular average in k-space of the reciprocal of the sound velocities

T1

= (c11

rz

=

-

ClZ)/C11

c44/c11.

Results of this analysis are presented in Table VI; the quantity G, which is included in the table, is defined by the relation

G

= (cii/co)”2f(Ti,

This scheme is useful for estimating data.

QD

~ 2 ) .

(0) in the absence of experimental

L. J. Touchard, J . A p p l . Phys. 34, 2694 (1963). E. F. Steigmeier, A p p l . Phys. Letters 3 , 6 (1963). 72 P. M. Marcus and A. J. Kennedy, Phys. Rev. 114,459 (1959). ’0

71

237

ULTRASONIC E F F E C T S I N SEMICONDUCTORS

TABLE J‘I. THEELASTIC CONSTANTS AND THE DEBYETEMPERATURES OF THE 111-V COMPOUNDS~

nf

a

co

(10’2

(A) InSb AlSb GaSb InAs InP GaAs AlAs AlP GaP BAS BP

BN a

118.37 6.478gb 74.45 6.13566 95.75 6.0955b 94.91 6.0585b 72.95 5.8688b 72.37 5.65356 50.95 5.62c 29.00 5.451d 50.57 5.45066 42.82 4.777e 20.87 4.53fP 12.42 3.615f

c11

f(rr, 7 2 )

G

OD(0)

(10”

dyn/cm2) dyn/rm*)

0.371 0.463 0.47i 0.487 0.553 0.646 0.656 0.743 0.i43 1.257 1.546 3.833

6.720 8.948 8.858 9.06h 10.28h 11.888 12.20h 13.82h 13.82h 23.35h 28.73h 71.20h

0.645 0.660 0.684 0.667i O.67li 0.690 0.695< 0.704< 0.704<

0.868 202.5 0.916 292.O 0.931 265.5 0.913i 262 0.929i 320.5 0.936 344 0.949i 417 0.9621 588 0.9621 446 1.01@ -625 1.03@ -985 1.116k -1900

After E. F. Steigmeier, Appl. Phys. Letters 3, 6 (1963).

* G. Giesecke and H. Pfister, Acta Cryst. 11, 369 (1958).

R.Wyckoff, “Crystal Structures,” Vol. I. Wiley (Interscience), New York, 1960. A. Addamiano, A d a Cryst. 13, 505 (1960). 6 J. A. Perri, S. LaPlaca, and B. Post, Aeta Cryst. 11, 310 (1958). f R.H. Wentorf, J . Chem. Phys. 26, 956 (1957). 0 R.W. Iceyes, J. Appl. Phys. 33, 3371 (1962). h From interpolation of cll/co. i From interpolation of f ( n , T I ) . i From interpolation of a plot of G as a function of a. k From extrapolation of a plot of G as a function of a. c

d

10. ELECTRONIC CONTRIBUTION TO THE SHEAR MODULI

In his study of the piezoresistance effect, Smithy3showed that the electronic properties of germanium were sensitive to the state of strain of the crystal. Keyes74 later did calculations on what is essentially the inverse interaction, the contribution of the electrons to the elastic properties of a multivalley semiconductor. Keyes predicted that the elastic constant c44 of degenerate n-type germanium would be depressed by increasing the donor concentration; a similar effect, but two orders of magnitude smaller, was predicted for degenerate p-type material. As a result of crystal symmetry and the orientation of the energy minima along [lll] d’irec~ the temperature tions, only c44is influenced. Brurier and K e y e ~ ’measured C. S. Smith, Phys. Rev. 94, 42 (1954). See Keyes.61 76 L. J. Bruner and R. W. Keyes, Phys. Rev. Letters 7, 55 (1961) 75

74

238

KORMAN G . EINSPRUCH

TABLEVII. ELASTIC CONSTANTS OF GE (in units of

1011

AT

4.2“Ka

dyn cm-2)

~

Pure Heavily doped a

6.80 6.42

4.06 4.04

7.5s 7.66

After L. J. Bruner and R. W. Keyes, Phys. Rev. Letters 7, 55 (1961).

dependence of the elastic constants of pure, less than 1 0 1 4 donors and heavily doped, 3.5 X l O I 9 arsenic donors ~ m - ~and , substantially confirmed the theoretical prediction, as shown in Table VII and Fig. 6. The

b>

3.5x IOl9 donors

66

100 TPK)

0

200

FIG.6. Shear-modulus depression by impurities in Ge (after Bruner and K e y e ~ ? ~ ) .

constant c44 was found to be depressed by about 5.5%; the other elastic constants were essentially unaltered. The quantitative theory predicts in the low-temperature limit

(s4)

= - (4)6’3(7?~3m*Eu2N1~3

c44

T=O

m*

N Z, h

)

h2c44

(for N

3.5 X

crn+);

=

-7.5%

= =

density of states effective mass donor concentration shear deformation potential constant for electrons Planck’s constant.

= =

=

10’9

,4dler75a has examined, using a Boltzmann equation approach, the theoretical problem studied by Keyes. 75a

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

ULTRASONIC EFFECTS IN SEMICONDUCTORS

0

0

X

5 9 I

3

0

X

5 9 I

a3

c m

22

-- dx

Lo

4-

SQ

239

240

NOllMAN G . EINSPHUCH

Einspruch and Csavinszky76 extended Keyes theory to the case of heavily doped n-type silicon, and predicted that only c’[ = (c11 - c12)/2] mill be influenced by the carrier concentration. The difference is due to the number and orientation of the equivalent valleys in k-space. The quantitative theory predicts in the low-temperature limit

(F) -4E)

213 7n*8u2N1/3

T=O

h2c‘

=

‘

which is very similar in form to the result for germanium. For finite temperatures (6C’/C’)T = (6c’/c’)T=OL(T), where L ( T ) = IC~F~,Z(~O)~~/~CF‘~,Z(~U)/F~/~(~U)I; 70 is related to the Fermi energy loby ?lo = Co/kT;F1I2 ( 7 0 ) is a Fermi function and is tabulated by McDougall and Stoner.77A summary of measurements a t 78°K on degenerate phosphorous-doped silicon is given in Table VIII. The experimental observations agree with the theoretical predictions within 10% over the entire doping range considered. The good agreement between the theory and experiment confirms that X u = 11eV is a reasonable value of the deformation potential constant, as suggested by Wilson and Feher.78 Csavinszky and Einspruch79 have also extended Keyes’ theoretical work on the dependence of the elastic constants on hole concentration in degenerate p-type Ge to the case of degenerate p-type Si and have studied the latter case experimentally. The analysis for p-type silicon assumes parabolic valence bands and spherical isoenergy surfaces, but includes consideration of the splitoff band, which lies only 0.044eV below the degenerate light and heavy hole bands a t k = 0. The analysis predicts for doping into the splitoff band

where m*,“=

(m*,13/2

+

+

m*y23/2

1n*,33/2)2/3,

m*,,= effective mass appropriate to ith sub-band X = spin-orbit splitting X,’ = shear deformation potential constant for holes. N. G. Einspruch and P. Csavinszky, A p p l . Phys. Letters 2, 1 (1963). J. McDougall and E. C. Stoner, Phil. Trans. Roy. Soc. London A237, 67 (1938). 78 D. K. Wilson and G. Feher, Phys. Rev. 124, 1068 (1961). 79 P. Csavinszky and N. G. Einspruch, Phys. Rev. 132, 2434 (1963).

-76

77

c

*

F

T-LE IX. VALUESOF

THE

HOLESHEAR DEFORMATION POTENTIAL CONSTANT Er' AS

A

:: FUNCTION OF DOPING CONCENTRATION pa

%

0

2CI

%' (eV)

P

21

(cm-3)

(106 cm sec-1)

Two-band approx. with masses a t k=O

Three-band approx. with masses a t k=O

Two-band approx. with masses a t large values of k

Three-band approx. with masses a t large values of k

Quadratic strain dependence

M 9

m9 CI

4

TI:

1.1

x

c (

1020

1.4 x 1019

3 . 2 X 10IS 7.0 X 1016*

0.4622 0.4691 0.4705 0.4711

Not applicable 8.5 .5 .x -

11.8 Not required Not required -

Not applicable i.1

4.9 -

9.94 Not required Not required -

3.46 3.28 3.28 -

2,

or m

5

CI

0

After P. Csavinszky and N. G. Einspruch, Phys. Rev. 132, 2434 (1963); and P. Csavinszky, to be published. * Reference sample.

3 2c3 0 2 TI:

242

NOHMAN G. EINSPRUCH

I

Energy (eV)

(

S p l i t - o f f band

‘-Light

hole band

H e a v y hole band

FIG.i . Structure of the valence band of Si (aftcr Csavinszky and E i n ~ p r u c h ~ ~ ) .

The valence band structure and the boron concentrations used in their analysis are shown in Fig. 7 and their results are shown in Table IX. They find a striking dependence of the shear deformation potential constant on hole concentration, ranging from 4.9 to 9.94 eV over the range of doping considered. Subsequently, CsavinszkyEOincluded in a theoretical calculation the assumption that the strain dependence of the valence sub-band shifts is not purely linear but contains terms which are quadratic in the strain, and succeeded in accounting for the variation in 8,’with doping, as shown in the last column of Table IX. HallEoahas measured the influence of doping on the third-order elastic constant c456 of n-type germanium. He found that -3 X 1019 arsenic atoms ~ m produced - ~ a reversal in sign of c456 a t room temperature. Interpreting these results and results of measurements of c44 in terms of Keyed analysis yields the value 17.0 f 0.2 eV for the shear deformation potential constant and a value for the carrier density in excellent agreement with transport measurements. P. Csavinszky, t o be published. J. J. Hall, Phys. Rev. 137, A960 (1965).

ULTRASONIC E F F E C T S I N SEMICONDUCTORS

243

11. ACOUSTOELECTRIC EFFECT

When a traveling elastic wave propagates through a semiconductor, a transfer of momentum from the wave to the carriers can occur. The net momentum is in the direction of propagation of the elastic wave, and is balanced by a n induced electromotive force, which may be detectable. If the ends of the sample are insulated, an electric field-the acoustoelectric field-is maintained. For the effect to occur, there must be a n interaction between the stress wave and the carriers, and a mechanism for momentum absorption and energy loss must exist. When the traveling elastic wave passes through the sample, there is a spatial variation of the potential energy associated with the wave. The carriers will attempt to achieve an equilibrium distribution which favors the regions of lowest potential energy. Since a finite relaxation time is required for achieving this distribution, and since the wave is a progressive wave, equilibrium is never achieved, and the carriers always lag the elastic wave; energy is thus transfered from the ultrasonic wave to the carriers. Since carriers of both signs are present in a semiconductor, it is possible to produce appreciable spatial bunching of carriers without producing space charge. If only one type of carrier is present, as in a metal, electrostatic repulsion prevents appreciable bunching and thus precludes a substantial effect. At low frequencies, the equilibrium between the carriers and the wave is almost exact; a t high frequencies the carrier distribution is almost unaffected by the ultrasonic wave. A related phenomenon is "phonon drag" which is the contribution to the thermoelectric power due to momentum transfer to electrons from thermal phonons streaming down a temperature gradient. The acoustoelectric effect has had considerable theoretical attention since i t was first proposed by Parmenter,81 who with MikoshibaE2predicted that a n effect should be observable in metals. Other contributions to the theory have been made by Van den B e ~ k e lHolstein,E4 ,~~ Parmenter,85 Blount,*6Takimoto,87 Mertsching,8s Sandomirskii and Kogaqsg as well as by Weinreich?o Greebegl has pointed out that in piezoelectric semiconR. H. Parmenter, Phys. Rev. 89, 990 (1953). N. Mikoshiba, J . Phys. SOC.Japan 14, 1691 (1959); see Mikoshiba.53 83 A. Van den Beukel, A p p l . Sci. Res. BB, 459 (1956). "

'*

T. Holstein, Westinghouse Research Memo 60-94698-3-M15 (1956). R. H. Parmenter, Phys. Rev. 113, 102 (1959). 86 E. I. Blount, Phys. Rev. 114,418 (1959). 87 N. Takimoto, Progr. Theoret. Phys. (Kyoto) 26, 659 (1961). " J . Mertsching, Phys. Stat. Sol. 2, 747 and 754 (1962). v. B. Sandomirskii and S. M. Kogan, Soviet Phys.-Solid State (English Transl.) 6, 1383 (1964). Q. Weinreich, Phys. Rev. 104, 321 (1956). c. A. A. J. Greebe, Phys. Letters 4, 45 (1963). 84

244

NORMAN G . EINSPRUCH

ductors, measurement of the frequency and temperature dependence of the acoustoelectric effect should yield information concerning impurity centers. Particularly, the evaluation of the cross section for capture of electrons by donors should be possible. Mikoshibag2 has carried out the theoretical analysis of the influence of an external magnetic field on the acoustoelectric effect. He predicts that the various magnetoacoustic resonances observable in ultrasonic attenuation measurements should also be observable in the acoustoelectric effect. Unfortunately, this body of theoretical work is accompanied by a dearth of experimental verifications. Detection of the acoustoelectric effect was first reported by Weinreich.93 Measurements reported a t about the same time by Sasaki and Yoshida94 9 ~ a spurious thermoelecwere subsequently attributed by Weinreich et ~ 1 . to tric effect. A detailed experimental study of n-type germanium was made by Weinreich et al. and will be discussed in some detail below. The acoustoelectric effect in n-type germanium is directly related to the intervalley scattering mechanism for electrons, and, as has been pointed out previously, is related to the electronic contribution to the ultrasonic attenuation and to the depression of the shear elastic constants. Since the valleys in the conduction band of germanium lie on [111] axes, the deformation potential energy of the a t h valley can be expressed as 1'") = [&&j EuKz(a)K3(a) 1e,j,

+

where the K , ( ~are ) the components of a unit vector directed from the center of the Brillouin zone to the a t h valley, and Ed and & are the deformation potential constants. If one considers the field strength, @, of the acoustical wave to be defined such that @2 is the energy density of the field, then the ith type of particle in the sound field experiences a local potential energy

u, = qa";

qz is called the acoustic charge. For an acoustic wave propagating in a [loo] direction and polarized in a [OlO] direction, the acoustic charge for valley a is given by qa(a) = f 8 , , / 3 (c44)"2,

where the sign is determined by whether the components K,(a) and Ky(a) have the same or opposite signs. I t can be shown that the acoustoelectric field E,, is given by =

(qa2F/fJ2qkB T ){ W

2 d [ 1

+

(WTR)']

1

N. Mikoshiba, J . A p p l . Phys. 34, 510 (1963). G. Weinreich, Phys. Rev. 107, 317 (1957). * W. Sasaki and E. Yoshida, J . Phys. SOC.Japan 12, 979 (1957). 95 G. Weinreich, T. M. Sanders, and H. G. White, Phys. Rev. 114, 33 (1959). 92

93

ULTRASONIC E F F E C T S I N SEMICONDUCTORS

245

where F is the acoustic power density, and T R is the relaxation time of the distribution. Relaxation of the electron distribution proceeds by two mechanisms-intervalley scattering and the spatial redistribution of electrons. The intervalley scattering time is related to T R by TR-l

=

4 --I

f k2D,

where kB is Boltzmann’s constant and D is the averaged diffusion constant. Results of measurements over the temperature range 20-160°K a t 20 and 60 Mc sec-1 on a series of arsenic-doped germanium samples, impurity , shown in Fig. 8, content ranging from 1014 to 10’6 impurities ~ m - ~are where 7-1 is plotted as a function of temperature. The data yield a value of 1 E , I = 16 eV, in good agreement with independent estimates. At higher

20

60

100 T in O K

140

T in OK

FIG.8. Temperature dependence of the intervalley scattering rate in n-type Ge (after Weinreich et aLg6).

246

NORMAN G. EINSPltUCH

temperatures, the curves are very similar, indicating the dominance of phonon-scattering processes; the low-temperature rates vary from sample to sample, indicating that an impurity scattering mechanism dominates. Wangg6reported the observation of an acoustoelectric effect in CdS a t 33 Mc sec-’ using an ultrasonic amplifier configuration. The acoustoelectric field was found to be about six orders of magnitude greater than that observed in n-type Ge; the stronger effect is due to the piezoelectric interaction in the polar material. It was observed that a finite negative acoustoelectric field was observed a t zero attenuation, contrary to Weinreich’s notions which require an energy absorption for the existence of a n acoustoelectric field. Eckstein97 has calculated the acoustoelectric effect considering the exchange of energy between the acoustic wave, the carriers, and the external electric field; substantially better agreement with Wang’s results were obtained than were obtained on the basis of Weinreich’s earlier calculations. E ~ k s t e i nin , ~a~later ~ paper, makes further comments on the range of validity of Weinreich’s analysis and treats in detail the semimetals problem. Beale and Pomerantzg8 have produced the acoustoelectric effect in GaAs using microwave phonons. One end of the semi-insulating (108 electrons ~ m - ~GaAs ) sample was placed in a re-entrant microwave cavity resonant a t 9 Gc sec-I. The piezoelectric properties characteristic of the sphalerite crystal structure are sufficiently strong to allow use of GaAs in the “self-transducer” mode. Contacts a t the opposite end of the bar were used to pick up the acoustoelectric current which was found to depend on the input microwave power, intensity of illumination, distance traveled by the acoustic wave, and temperature. The dependence of the peak-to-peak acoustoelectric current, I,,, on temperature is shown in Fig. 9. The fall in I,, in the low-temperature region is attributed to the decrease in concentration and mobility of the electrons. The constancy of I,, in the midrange is attributed to balance between a decreasing ~ q o / k T , which enters the theory for the magnitude of I,, (q0is related to the amplitude of the acoustic wave) and an increasing electron mobility with temperature. The decrease in I,, in the high-temperature range is attributed to an increase in the acoustic wave attenuation by thermal phonons. Beale98ahas studied the problem of the acoustoelectric effect in a piezoelectric semiconductor when the sound wave has a large amplitude and is of high frequency; these conditions obtain in the experiment on GaAs. W. C. Wang, Phys. Rev. Letters 9, 443 (1962). S. G. Eckstein, Bull. Am. Phys. SOC.[2] 8, 254 (1963). 978 S. G. Eckstein, J . Appl. Phgs. 36, 2702 (1964). 9s J. R. A. Beale and M. Pomerantz, Phys. Rev.Letters 13, 198 (1964). 98% J. R. A. Beale, Phys. Rev. 136, A1761 (1964). 96

97

ULTRASONIC E F F E C T S I N SEMICONDUCTORS

0

1

10

20 Temperature

30

247

40

(OK)

FIG.9. Temperature dependence of tlhe acoustoelectric current in semi-insulating GaAs (after Beale and Pomerantzg8).

It is rather clear that experiments on the acoustoelectric effect lag the theoretical work considerably. It appears that verification of the many facets of the predictions of the theoretical work, much of which is applicable to semimetals, would be a very fruitful field of research. 12. PIEZOELECTRIC INTERACTION a. Photo E$ect

Qualitative observations of the effects of illumination and temperature variation on the propagation of elastic waves in CdS were reported by Gobrecht and B a r t s ~ h a t who , ~ ~ observed that the mechanical properties of the crystals were strongly influenced by the concentration of activator impurities. Their measurements were made using the piezoelectric properties of CdS to construct a crystal-controlled oscillator in which the sample was the control element. Subsequently, Nineloomeasured the attenuation in CdS by the ultrasonic pulse-echo technique. He studied the effects of illumination by white light and monochromatic light in the visible region on the attenuation in the 10-200 Me sec-I range and on the electrical conductivity. He observed no effect for shear waves propagated along the c axis and a n anisotropy in behavior for compressional waves propagated along and normal to the c axis. Under illumination by white light both decreases and increases in attenuation as a function of light intensity were measured in not deliberately doped crystals prepared under similar conditions. Hutson*olsuggested that the observations could be explained in terms of a piezoelectric interaction between the elastic waves and the charge carriers. H. Gobrecht and A. Bartschat, 2.Physik 163, 529 (1959). H. D. Nine, Phys. Rev. Letters 4, 359 (1960). 101 A. R. Hutson, Phys. Rev. Letters 4, 505 (1960). 99

loo

248

NORMAN G . EINSPRUCH

Kine and Truelllo2 subsequently investigated the attenuation in illuminated CdS with the conventional pulse-echo technique and using the piezoelectric sample as its own transducer. The latter technique is particularly advantageous a t low temperatures since it obviates the transducerto-sample bond, which often fractures upon cooling. Once again two distinct types of behavior were observed in studies of the attenuation as a function of light intensity and temperature; for example, the change in the ultrasonic velocity at room temperature was depressed by 0.5% in one set of crystals and by 0.04% in the second type of crystal in going from dark to illuminated conditions. Measurements of the temperature dependence of the electrical conductance correlated particularly well with the absorption measurements in the former group of crystals. The relationship between the polarization and the stress in a piezoelectric material is given by

Pi

=

dijkTjk,

where the dijk are the piezoelectric coefficients. For CdS :lol d33 = 3.2 (lo-' statC/dyn), dsl = -1.1 and d16 = -4.3; d31 = d32, = d16 in the conventional reduced notation; all of the other components of the tensor vanish in the 6-mm crystal symmetry class. The measurements of the photosensitive ultrasonic attenuation for propagation along the hexagonal axis and normal to the hexagonal axis of both longitudinal and transverse waves (polarized along and normal to the hexagonal axis) correlate very well with the idea of a piezoelectric interaction. The ultrasonic waves corresponding to the components of the stress tensor which produce a polarization through the piezoelectric interaction, are precisely the waves which demonstrate the photosensitive effects. The authors also reported that under suitable conditions of excitation in the self-transducer technique, the decrease in attenuation upon going from dark to light conditions a t low-excitation pulse amplitudes in some of the crystals could be made to reverse, producing apparent gain, at higher pulse amplitudes. The experiments on the influence of illumination on the ultrasonic attenuation provided the impetus for the further experimental and theoretical study which led to the discovery of the ultrasonic amplification effect in piezoelectric semiconductors. b. Depletion Layer Transducer

Whitel03 has demonstrated a new electromechanical transducer which operates as a result of the piezoelectric properties of some semiconductors Io2

H. D. Nine and R.Truell, Phys. Rev. 123, 799 (1961). D. L.White, IRE (Inst. Radio Engrs.), Trans. Ultrasonics Eng. 9, 21 (1962).

lo3

ULl’ltABONlC EFFECTS I N SE:MICONDUCTOI1S

249

semiconductor

X

FIG.10. Metal-semiconductor contact depletion layer transducer (after White103).

possessing the sphalerite or the wurtzite structure. The theoretical range of operation of the transducer is 0.1 to 10 Gc sec-I. The active portion of this type of transducer is a high-resistance depletion layer which occurs at a p-n junction or a t a metal-semiconductor rectifying contact. This device has two prominent advantages over a conventional quartz piezoelectric transducer: (1) The depletion layer is thin; consequently the fundamental frequency-thereby the frequency for the maximum electromechanical conversion efficiency-is high. (2) The thickness of the depletion layer ‘depends upon the biasing voltage; hence the resonant frequency of the transducer can be altered by varying the dc bias level. In the p-n junction transducer, a narrow junction in low-resistivity material is reverse biased with a dc voltage. Since the resistance in series n-ith the junction is low, most of the voltage drop occurs across the junction, giving rise to an intense electric field. I n a piezoelectric semiconductor, the field produces a mechanical stress which, in turn, elastically strains the junction. If an ac signal is superimposed upon the dc biasing voltage, an alternating stress is produced at the junction, thereby producing a n elastic wave which propagates into the bulk. In the rectifying contact transducer, the depletion layer is formed by reverse biasing the junction between a metal electrode and bulk n-type material. The principles of operation of the two types of devices are identical. Although Strauss104 has recently presented a n elaborate theoretical analysis of the performance of the transducer, White’s earlier description of the behavior of the device, based on a one-dimensional model, will be summarized here. The thickness of the depletion layer-hence the fundamental frequency of operation-can be estimated by a straightforward application of Gauss’ law, neglecting the piezoelectric interaction. Consider the rectifying metal-semiconductor contact with the geometry shown in Fig. 10. I n the plane of the contact, x = 0, the biasing voltage produces a negative charge; in order to preserve charge neutrality, a distribution lo‘

w. Strauss, J . A p p l . Phys. 36, 2106 (1964).

250

NORMAN G . EINSPRUCH

of positive charge, the depletion layer, is induced in the n-type semiconductor, and extends a distance d into the bulk. Since the low-resistivity bulk material is grounded, V = 0 a t x = M . I n a semiconductor dD/dx = e ( a E / d x )

=

Nq,

D is the electric displacement, E is the electric field (= -aV/d;c), e is the dielectric constant, N is the density of ionized doiiors, and q is their charge. The differential equation thus derived is readily integrated to yield M = [-2Vo',~/Nq]'1~, where Vo combines the effects of the biasing voltage and the difference to of the Fermi levels. Transducer thicknesses in the range cm are readily obtained, corresponding to resonant frequencies of 100 Mc sec-' to 10 Gc sec-l, depending upon the biasing voltage and the concentration of impurities. In one experimental arrangement, shown in Fig. 11, a (111) face of a 0. l-ohm-cm gallium arsenide crystal suitably plated to produce a rectifying contact, was bonded to one end of an X-cut quartz rod. The other end of the rod was placed in an 830-Mc sec-l resonant microwave cavity and served as the source of ultrasonic waves. The waves generated a t the cavity were detected at the depletion layer transducer; conversely, waves generated by the transducer were detected at the cavity. Transducer action was thus demonstrated conclusively. Although the theoretical high efficiency of these transducers is yet to be realized, it appears that this device will be useful in studying ultrasonic effects in piezoelectric semiconductors, in which the sample and the transducer are integral; the device will also be useful, after development of improved circuitry, in ultrasonic studies in the UH F and microwave regions in a wide variety of materials.

FIG.11. Resonant cavity-depletion layer transduccr experiment (after Whitelo3).

ULTRASONIC E F F E C T S I N SEMICONDUCTORS

251

Metal

FIG.12. Diffusion layer transducer (after Fostei-106).

Foster105has described a diffusion layer transducer, which is an interesting variation on the depletion layer transducer. The structure of the device bonded to a delay line is shown in Fig. 12. The active region is a resistive layer formed by diffusing copper into the surface of a highly conductive single crystal of CdS; thc diffusion process introduces deep acceptor levels which lead to a marked decrease in conductivity. The principles of operation are similar to the White device. Fundamental frequencies of operation as high as 200 Mc sec-I have been observed. Integrated transducer-delay line structures have been fabricated in l-in.-long CdS crystals; this technique for ultrasonic wave generation should be very valuable in the practical realization of ultrasonic amplifiers. Thin films of insulating CdS epitaxially deposited on A1203and MgO have been used by de Klerk and Kellyloc to generate phonons in the 0.5 to 2 Gc sec-' range. Transducers deposited on Z-cut quartz and fundamentally resonant a t 3 Gc sec-l have been driven at 9 Gc sec-l and were found to have twice the electromechanical conversion eficiency of X-cut quartz at that frequency. De Klerk and Kelly speculate that it might be possible to deposit films which will resonate at 10l2 cps, near the upper limit of the phonon spectrum.

Ultrasonic Amplification Formal theories of wave propagation in piezoelectric materials have been derived by Laws0n,~07Kyame,losKoga et al.,"Jg Pailloux,llo and have been commented upon by Ragajopallll ; these theories incorporated the

c.

N. F. Foster, J . Appl. Phys. 34, 990 (1963); IEEE Trans. Ultrasonics Eng. 10, 39 (1963). l 0 6 J. de Klerk and E. F. Kelly, A p p l . Phys. Letters 6, 2 (1964). Io7 A. W. Lawson, Phys. Rev. 62, 71 (1942). lo* J. J. Kyame, J . Acoust. SOC. Am. 21, 159 (1949); 26, 990 (1954). lo9 I. Koga, M. Aruga, and Y. Yoshinaka, Phys. Rev. 109, 1467 (1958). H. Pailloux, J. Phys. Radium 19, 523 (1958). E.S. Ragajopal, Phys. Letters 1, 70 (1962). lo5

252

NORMAN G . EINSPRUCH

effects of internal electric fields on the elastic stiffness of the propagation medium. Kyame's analysis involved simultaneous solution of Maxwell's equations and the mechanical-piezoelectric equations of state. For an arbitrary direction of propagation, he found a secular determinant coupling the three acoustic plane waves ordinarily found in a crystal to two plane electromagnetic waves. For certain directions of propagation, the acoustic waves are accompanied by longitudinal electric fields which increase the elastic constants. Observation of strong p iezo electri~ ity~ ~ some semiconducting J13 ~in materials led Hutson and White114 to analyze the elastic wave propagation problem taking into account the currents and space charge which are produced by the longitudinal electric fields which accompany the elastic wave. They derived a linear one-dimensional theory including carrier drift, diffusion, and trapping for both extrinsic and intrinsic semiconductors.

rL $': '-1

A t t e nu o t or]

Drift field pulse

rf

output

4-1 Atienuotor

I ,Transducer Buffer

T\Tronsducer

FIG.13. Ultrasonic amplifier configuration (after Hutson et c ~ l . ~ l ~ ) .

I n a rather remarkable experiment, Hutson eB aZ.lI5 demonstrated that substantial amplification of ultrasonic waves could be produced in photoconductive CdS by applying a dc electric field in the direction of wave propagation. In a 7-mm-long crystal, 18 d B of gain was obtained a t 15 Mc sec-l, and 38 d B was obtained a t 45 Mc sec-1. The experimental arrangement is shown in Fig. 13; the Y-cut quartz transducer is excited by an rf pulse from the transmitter producing a shear wave which propagates in the CdS crystal which is oriented such that the ultrasonic wave propagates in a basal plane with particle displacement along the hexagonal axis. The dc drift pulse is applied to the semiconductor through diffused and evaporated indium contacts. The buffer rods serve as delay media and 112

113

114

115

See Hutson.lol H. Jaffe, D. Berlincourt, and L. Shiozawa, Proc. 14th Ann. S y m p . Freynency Control, U.S . Arwiy Res. Develop. Lab., Fort Monwlouth, New Jersey, 1960 p. 304; see also 11. Berlincourt, H. Jaffe, and L. Shiozawa, Phys. Rev. 129, 1009 (1963). A. R. Hutson and D. L. White, J . A p p l . Phys. 33, 40 (1962). A. R. Hutson, J. H. McFee, and D. L. White, Phys. Rev. Letters 7, 237 (1961).

253

ULTRASONIC E F F E C T S I N SEMICONDUCTORS

for electrical isolation. Measurement of the change in output signal with respect to the signal observed in the dark with no drift field yielded the following results: ( I ) With no illumination, the drift field pulse had no effect. (2) With no drift field, increasing illumination yielded increasing attenuation and increasing conductivity, as had been observed in the earlier experiments on the photoeffect. (3) With the sample illuminated by the 5770- and 5790-A mercury lines, the 5-psec-drift field pulse had no effect unless it overlapped the 3.5-psec transit time of the 1-psec ultrasonic pulse in the sample. At overlap, the output signal was either increased or decreased with the maximum effect corresponding to complete overlap. (4) With the sample illuminated and complete overlap, variation of the drift field strength produced changes as shown in Fig. 14.

-751

'

160014001200 I000800 600 400 200 Ed ( V/crn)

1-25 0 -200

FIG.14. Observation of ultrasonic amplification (after Hutson et

aZ.115).

Two striking results of the observation are the appearance of acoustic gain and the crossover from loss to gain a t 700 V cm-1; the drift field a t crossover corresponds to a drift velocity equal to a shear wave velocity for CdS for electrons of mobility 285 cm2 V-1 sec-1; the previously reported Hall mobility of about 300 cm2 V-l sec-1 is in excellent agreement with this determination. It was also observed that with no input signal, in the presence of the drift field acoustic oscillations were produced by amplification of noise in the crystal.

254

NORMAN G. EINSPItUCH

More recently, MidfordllSahas reported the observation of light-sensitive ultrasonic amplification in CdSe. I n measurements made at 45 Mc sec-l, the drift mobility evaluated a t the loss-to-gain crossover was found to be in good agreement with results of transport measurements. The amplification experiment stimulated a great deal of theoretical work on the details of the interaction of phonons with charge carriers in semiconducting materials. Contributions to the theory, in addition to those mentioned in Section 8, have been made by Spector,116 Kazarinov and Skobov,117Gurevich and Kogan,ll* Gurevich,llg Pippard,'20 Paranjape,121 Tsu, lZ2Eckstein, lZzaQuate,lZzbMucha, 122c Grinberg,122dand Ashley. lZze Whitel23 extended the earlier phenomenological calculations on wave propagation in piezoelectric semiconductors to include the effects of a drift field on the sound absorption process. Since his work was so successful in describing many of the details of the observations, a brief sketch of this theory is included. From the piezoelectric equations of state

D

=

dX

4- EE,

and the equations of motion for a n elastic material, a wave equation p(d2u/diZ) = dT/dx = c ( ~ ~ u / ~ ex ( ~d E) / d z )

can be derived; it should be noted that this is a one-dimensional analysis with the strain X defined as duldx; d is the piezoelectric constant, E the electric field, D the electric displacement, and E the dielectric permittivity. T. A. Midford, J . A p p l . Phys. 36, 3423 (1964). H. N. Spector, Phys. Rev. 127, 1064 (1962); Phys. Letfers 10, 163 (1964). 117 R. F. Kazarinov and V. G. Rkobov, Soviet Phys. J E T P (English Transl.) 16, 628 (1962); 16, 1057 (1962). 118 V. L. Gurevich and V. D. Kogan, Soviet Phys.-Solid State (English Transl.) 4, 1785 115a 116

(1963).

V. L. Gurevich, Soviet Phys.-Solid State (English Transl.) 6, 89% (1963). 120 A. B. Pippard, Phil. M a g . [8] 8, 161 (1963). 121 B. V. Paranjape, Phys. Letters 6, 32 (1963). 122 R. Tsu, J . A p p l . Phys. 36, 125 (1964). 122* S. G. Eckstein, Phys. Letters 13, 30 (1964). 122b C. F. Quate, J . Electron. Control 17, 33 (1964). 1 2 2 0 T. Musha, J . A p p l . Phys. 36, 3273 (1964). 122d A. A. Grinberg, Soviet P h y s -Solid State (English Transl.) 6, 701 (1964). m e J. C. Ashley, J . A p p l . Phys. 36, 528 (1965). 123 D. L. White, J . A p p l . Phys. 33, 2547 (1962). 119

ULTRASONIC EFFECTS I N SEMICONDUCTOItS

25 5

Applying Gauss' law, the equation of continuity and the expression for the current density J in an n-type material] J

=

qpneE

+ pDn(ane/ax),

the following equation can be obtained:

-a2D ax at

~-

a {(q n o - f z 1 - D n a3D p 1 ax ax3

where p is the electron mobility, n, the density of electrons in the conduction band, D, is the electron diffusion constant. The density of electrons in the conduction band may be written as n e = no

+ fnsl

where no is the equilibrium concentration of electrons] n, the space charge density, and f is the fraction of the space charge which contributes to the conductivity; in the absence of traps, f = 1. Under small signal assumptions] the electric field and material displacement are given by

+ El exp [i(kx

E(x, t )

=

Eo

u(x, t )

=

u exp [i(lcx

- wt)]

- wt)];

Eo is the field due to the applied dc voltage and El is the amplitude of the sinusoidal electric field which accompanies the ultrasonic wave. Combining the equations of state, the wave equation, and the third-order partial-differential equation describing D, a dispersion relation is derived. Since lc is complex, taking real and imaginary parts of the wave number yields for K 2 << 1 and a << w / v ,

where us is the phase velocity of the sound wave, K = d 2 / q we the dielectric relaxation frequency (= nOqp/c),wD the diffusion frequency (= v:/fDn), and y = 1 - ( f p E O / u . ) . The electric field dependence of the attenuation and velocity dispersion are plotted in Figs. 15 and 16; the good agreement of the theory with the observations is apparent. The analysis points out that a n ultrasonic wave in a piezoelectric material is accompanied by an alternating electric field. In a piezoelectric semiconductor, the alternating electric field gives rise to currents which produce space-charge bunching.

256

NORMAN G. EINSPRUCH

FIG. 15. Theoretical prediction of drift field dependence of ultrasonic attenuation and amplification (after WhitelZ3).

The currents cause the electric field and the stress to lag the strain; the currents also give rise to Joule heating, a source of ultrasonic attenuation. The space charge is manifest as a conductivity modulation; hence a direct current in the semiconductor is always accompanied by an alternating electric field. In a sufficiently large external electric field, the drift velocity of the mobile carriers exceeds the sound velocity, and the phase of the total alternating electric field is such that the stress leads the strain with the consequent transfer of energy to the ultrasonic wave. The mathematical condition for amplification is y < 0. The possible utility of a device based on the acoustic amplification effect in signal processing applications in which simultaneous amplification and delay are required has stimulated a considerable amount of research. Performance of acoustic amplifiers has been described by Hickernell and Saki~tis,'?~ and by Blotekjaer and Quate1z6;it appears that the most serious limitations to practical embodiments of the device are the substantial losses incurred in the electromechanical conversion processes, and the spontaneous generation of acoustic noise. Chubachi et al.125a have succeeded in integrating two diffusion layer transducers and an acoustic amplifier in a single bar of CdS. The transducers are separated from the amplifier portion of the device by diffused conducting layers which serve as the contacts for applying the drift field across the amplifying region and as contacts for the transducer layers. At 54 Mc sec-', application of a drift field produced 35 dB of amplification relative to the zero field measurement. Truell et aZ.lZ6 measured the temperature dependence of the attenuation of 30-Mc sec-' ultrasonic waves in CdS from room temperature to 380°C, 124

F. S. Hickernell and N. G. Sakiot's, PTOC. IEEE 62, 194 (1964).

126

K. Blotekjaer and C. F. Quate, Proc. IEEE 62, 360 (1964).

125s 126

N. Chubachi, M. Wada, and Y . Kikuchi, Japan J . Appl. Phys. 3, 777 (1964).

R.Truell, C. Elbaum, and A . Granato, J . Appl. Phys. 36, 1483 (1964).

ULTRASONIC EFFECTS IN SEMICONDUCTORS

257

FIG. 16. Theoretical prediction of drift field dependence of ultrasonic velocity dispersion (after Whitel23).

and of the resistance of the specimen from room temperature to 600"C, in order to verify the prediction of the Hutson and White127theory that the attenuation should increase with temperature with an activation energy equal to that for electrical conductivity. Their thermal measurement of the energy gap, E, = 2.30 eV, was in good agreement with the optical detenninationlZ8E, = 2.26 eV. Within experimental error, it was found that theory described the temperature-dependent part of the attenuation. E ~ a k i , ' *in~a study of the low-temperature magnetoresistive properties of bismuth at high electric fields, discovered a sharp break, leading to voltage saturation, in the current-voltage characteristics measured in a transverse magnetic field. He found that the velocity corresponding to the magnetic field H required to produce the kink at fixed electric field E , v = C E / H , was an appropriate shear wave velocity in the material; C is the velocity of light. This, of course suggested that a strong electronphonon interaction was responsible for the kink. Hopfield130 and Miyake and Kubo131 then analyzed models which qualitatively described the observed characteristics. Dumke and H a e r i ~ ~presented g'~~ a phenomenological theory for sound amplification by semimetals in crossed electric and magnetic fields. The condition for amplification is as in semiconductors: the drift velocity of the carriers must exceed the sound velocity. The only reported observation of acoustic amplification in a semimetal is contained in a preliminary account of studies on bismuth by Toxen and Tansa1,133 who observed gains as large as 14 dB cm-I a t 15 Mc sec-l for sound waves propagated in a bisectrix direction; buildup of acoustic oscillations with no input signal was also observed. See Hutson and White.114 M. Balkanski and R. D . Waldron, Phys. Rev. 112, 123 (1958). Iz9 L. Esaki, Phys. Rev. Letters 8, 4 (1962). 130 J. J. Hopfield, Phys. Rev. Letters 8, 311 (1962). S. J. Miyake and R. Kubo, Phys. Rev. Letters 9, 62 (1962). 132 W. P. Dumke and R. R. Haering, Phys. Rev. 126, 1974 (1962) A. M. Toxen and S. Tansal, Phys. Rev. Letters 10, 481 (1963). Iz8

258

NORMAN G . EINSPRUCH

20 D d

c 0 .-

$

10 0

C

2 -10 ; i

- 20 - 30 -40

I

2 50

1

1

.

1

I

I

I

I

750 1250 1750 Drift field (V/crn)

2 '50

FIG.17. Inhibition of ultrasonic amplification by acoustic flux buildup. Curve A-no flux; curves B and C correspond to increasing flux (after McFeel").

Pomerantzla3*extended earlier theoretical work on ultrasonic amplification via a deformation potential interaction to include the effects of an applied electric field. The calculation predicts that the attenuationamplification crossover occurs when the drift velocity of the carriers equals the sound velocity, as is the case in a piezoelectric semiconductor. In measurements a t 9 Gc sec-' on germanium crystals containing 10l6arsenic donors ~ m - ~amplification , was indeed observed when 20-A current pulses were applied to the samples, which measured 1 cm by -0.01 cm2. Hot electron effects were included in the theory by C0riwe11,~~~~ and a substantially improved estimate of the magnitude of the amplification was obtained.

d. Space Charge and Generation Eflects Deviations from ohmic behavior in measurements of the current-voltagc characteristics of semiconducting CdS have been reported by Smith.I3 Nonohmic behavior accompanied by ultrasonic flux buildup and reducec gain in ultrasonic amplifiers using CdS and ZnO as active elements ha: been reported by M C F ~ ~Behavior . ' ~ ~ similar to that reported by McFec has been observed by Wang and Pua136 in measurements of the acousto electric field in CdS. Smith found that the current saturated in unilluminated 10-ohm cn samples, having electron mobility of 300 cm*V-1 sec-l, at an applied electrii M. Pomerantz, Phys. Rev. Letters 13, 308 (1964). E. M. Conwell, Proc. IEEE 62, 964 (1964). R. W. Smith, Phys. Rev. Letters 9, 87 (1962).

1wb 134

135 136

J. H. McFee, J. A p p l . Phys. 34, 1548 (1963); 36, 465 (1964). W. C. Wang and J. Pua, Proc. IEEE 61, 1235 (1963).

259

ULTRASONIC E F F E C T S I N SEMICONDUCTORS

y

/

=-I /

0

In G,

3

0

k - A k h i e z e r loss /

/

/

I

10'

, IO ' O

I

I

I

I

10"

W

-

FIG.18. The plots of the frequency dependence of the ultrasonic amplification of shear waves in CdS. An Akhiezer type of loss is also shown. For oc = l O 1 0 sec-1, u 0.01 (ohm ern)-'; for oc = 10" sec-1, u 0.1 (ohm cm)-l (after Hutson137).

-

field of 1600 V cm-I. The carrier {drift velocity in $his field is -5 X lo5 em sec-', corresponding to a sound velocity in CdS. Although direct observations of ultrasonic wave propagation were not made, a piezoelectric interaction was suggested as being responsible for the electrical effect. At voltages greater than that required to produce the kink in the characteristic, rf oscillations corresponding in frequency to an acoustic oscillation of fundamental wavelength equal to the interelectrode spacing. Similar effects, not described in detail, were found in GaAs. The inhibition of the amplification process by the buildup of ultrasonic flux after application of the drift field to a CdS amplifier is shown dramatically in the data of McFee presented in Fig. 17. The amplification of a 45-Mc sec-I ultrasonic pulse timed to occur before buildup of flux is shown in curve A; in curves B and C, the signal pulse is retarded 15 and GO psec respectively and reduced gain is observed. Since the time required for saturation of the flux buildup was SO psec in the 7-mm-long crystal, there is a dependence of the gain reduction on the delay time of the signal pulse. H ut ~on' ~7 presented arguments based upon his earlier analysis of the amplification of ultrasonic waves in piezoelectric semiconductors to ex137

A. R. Hutson, Phys. Rev. Letters 9, 296 (1962).

260

NORMAN G . EINSPHUCH

plain the current saturation effect in semiconductors and the voltage saturation effect in bismuth. The expression for the frequency dependence of the amplification in the presence of an external drift field is plotted in l ~ ~ loss estiFig. 18, as well as a quadratic phonon-phonon h k h i e ~ e r type mated from the attenuation measurements on quartz by Bommel and D r a ~ i s f e l dThe . ~ ~ ~gain is maximum a t U* = U , W ~ and decreases a t higher frequencies due to carrier diffusion. There will be net gain up to a critical frequency determined by the intersection of the gain and loss curves. At yet higher frequencies there is net attenuation even in the presence of a n applied electric field. If the amplitudes of the acoustic waves are sufficiently great under amplification conditions, a nonlinear loss mechanism can offset the gain mechanism, thereby producing a steady state. A pair of waves of the group being amplified can interfere and produce a wave a t their sum frequency which is beyond the critical frequency. This high-frequency wave is then attenuated by the phonon-phonon scattering process and is eventually dissipated thermally. Solution of the nonlinear amplifier theory shows that there is a net dc acoustoelectric current which accompanies the steadystate acoustic flux. The analysis shows that in the acoustoelectric component of the current, the carriers flow in the direction of the elastic wave during attenuation, arid in the opposite direction during amplification. When the drift field exceeds the threshold required for amplification, the acoustoelectric current due to the ultrasonic wave subtracts from the ohmic current, and thereby produces the current-saturated kink in the volt-current characteristic. In a semimetal, as in a semiconductor, there is a critical frequency above which linear loss exceeds linear gain. Under conditions for amplification, the electrons and holes are driven against the acoustic flux; however, as a result of the interaction with the external magnetic field, there is a net additive component of both the electron and hole acoustoelectric current in the direction of the ohmic current, thereby producing voltage saturation. has succeeded in producing both current and voltage saturation in the same CdS crystal, in this way demonstrating the similar nature of the two effects, as suggested by Hutson. The requirements for observation of the Smith effect are that the drift velocity exceed the sound velocity and that the sample have sufficient conductivity and length to provide substantial amplification and flux buildup. The requirements for observation of the voltage saturation effect in bismuth were a high (PH>> 1) magnetic field and no Hall field set up to oppose the cross drift of the 1 3 8 AAkhiezer, . J . Phys. (t7.SS.R.) 1, 277 (1939). See Bommel and Dransfield.*o 140 A. R. Moore, Phys. Rev. Letters 12, 47 (1964).

I39

ULTltASONIC EFFECTS I N SEMICONDUCTORS

26 1

carriers. I n Moore's experiment pH M 2 was achieved with pulsed magnetic fields, and proved sufficiently large to see the effect. The Hall voltage was eliminated through the use of the Corbino disc geometry, in which the sample is in the shape of a washer, the applied electric field and current are radial, the external magnetic field is axial, and the cross-drift current is circumferential. With this experimental arrangement, the Smith effect in the radial current was seen a t H = 0, and the Esaki effect was seen in the circumferential current a t H > 0. Kroger et aZ.l4l have reported the observation of an anomalously slow ultrasonic wave propagating in CdS under amplification conditions. The wave has been described as a second-sound type of wave, analogous to the collective phonon wave known to propagate in liquid helium. Second sound is the transport of a thermal fluctuation by wave propagation rather than by thermal diffusion. This mechanism for heat transfer can be viewed as the propagation of an energy disturbance in the phonon gas; thus it proceeds with a velocity related to the sound velocity. Contributions to the theory of second-sound propagation in crystals have been made by Chester,142Prohofsky and K r ~ m h a n s l , ' Guyer ~~ and K r ~ m h a n s I , 'and ~~ P r ~ h o f s k y . 'Measurements ~~ were made a t room temperature in the 10 to GO Me see-1 range on shear waves propagating in the basal plane and polarized along the hexagonal axis in a standard amplifier configuration. The anomalous wave, which was not observed in the dark, appeared upon illumination in the presence of a dc drift field. The slow wave was identified as "almost completely" a shear wave of the same polarization as the injected wave and appeared only after an amplifying transit through the sample. The amplitude of the anomalous pulse was found to be temperature dependent, voltage dependent, light sensitive, and related nonlinearly to the amplitude of the main pulse. The anomalous pulse could be made to disappear by cooling the amplifier to 200°K; it reappeared upon cycling back to room temperature. The slowest room-temperature velocity observed was vJl.6; v, is the velocity of the ordinary sound velocity of the main wave, which is the only mode coupled to the gain mechanism for shear wave propagation in the basal plane. The theoretical analysis predicts that a collective oscillation in the phonon field can propagate a t low temperatures with velocity 2111 = V S / G 14' 14' 143 144

H. Kroger, E. W. Prohofsky, and R. W. Damon, Phys. Rev. Letters 11, 246 (1963). M. Chester, Phys. Rev. 131, 2013 (1963). E. W. Prohofsky and J. A. Krumhansl, Phys. Rev. 133, A1403 (1964). R. A. Guyer and J. A. Krumhansl, Phys. Rev. 133, A1411 (1964). E. W. Prohofsky, Phys. Rev. 134, A1302 (1964).

262

NORMAN G . EINSPRUCH

Amplitude A, of fundamental (dB)

FIG.19. Relative amplitude of the second harmonic (A2) as a function of the amplitude of the fundamental (A1). Curve A is the greatest slope observed; curve B is typical of the most common behavior (after Elbaum and Truell148).

in a spherically symmetric phonon distribution. The analysis also predicts that the slow wave can propagate a t higher temperatures if an appropriate gain mechanism, such as the ultrasonic amplification process, is operative. The small deviation from G in the ratio vJv11 was attributed to the deviation from sphericity of the phonon distribution under amplification conditions during which the distribution is peaked in the direction of carrier ~ ~ 6 also observed long period oscillations in the curflow. Kroger et ~ 1 . have rent flowing in CdS under amplification conditions. These oscillations, which are unlike those reported by Smith and McFee, are also interpreted as being due to the interaction between long-wavelength collective phonon waves and the mobile carriers. A survey of the second sound concepts and the experimental observations is given by Damon et Observation of sustained acoustoelectric current oscillations in illuminated CdS has been reported by Okada and mat in^.'^^^ Acoustoelectric current saturation effects in ZnS and CdS have been observed by Spear and Le Observation of harmonic generation, resulting from a nonlinear interaction between mobile electrons and the piezoelectric field which accompanies properly selected sound waves in CdS, was reported almost simulH. Kroger, E. W. Prohofsky, and H. R. Carleton, Phys. Rev. Letters 12, 555 (1964). R. W. Damon, H. Kroger, and E. W. Prohofsky, Proc. ZEEE 62, 912 (1964). 146b J. Okada and H. Matino, Japan J . A p p l . Phys. 3, 698 (1964).

146

146s

14*0

W. E. Spear and P. G. Le Comber, Phys. Rev. Letters 13, 434 (1964).

ULTRASONIC EFFECTS I N SEMICONDUCTORS

263

FIG.20. Amplitude of the second harmonic as a function of electron concentration (after K r ~ g e r ' ~ ~ ) .

taneously by K r o g ~ r ' ~and ' by Elbaum and T r ~ e 1 1 . Solution '~~ of the onedimensional nonlinear wave equation yields

A2

=

LAi2k2X,

where A1 and A2 are the amplitudes of the fundamental and second harmonic, respectively; L is a combination of second- and third-order elastic constants; and x is the distance traveled by the sound wave in the sample. The dependence of APon A, for 10-Mc sec-I fundamental waves in CdS is shown in Fig. 19. The greatest slope measured, that of curve A, is 1.9, in good agreement with the prediction of quadratic behavior. The slopes measured at different light intensities vary over the range 1.5 to 1.9, suggesting that L is a function of A1,which is also a prediction of the nonlinear theory. I n both experiments, the amplitude of the second harmonic was found to decrease a t high electron concentrations, a t which the internal piezoelectric fields are being shorted out by the mobile carriers. The results of measurements of the relative amplitude of the second harmonic of a 15-Mc sec-I fundamental as a function of carrier concentration are shown in Fig. 20; once again, good agreement with nonlinear theory was observed. The electron drift velocity also affects the harmonic generation process. Results of measurements of the dependence of the second harmonic amplitude on an externally applied electric field are shown in Fig. 21, as well as the theoretical prediction. T ~ l l ' ~has 8 ~also studied harmonic generation resulting from the nonlinear electron-lattice interactions responsible for

'" H. Kroger, A p p l . Phys. Letters 4, 190 (1964). C. Elbaum and R. Truell, A p p l . Phys. Letters 4, 212 (1964). B. Tell, Phys. Rev. 136, A1761 (1964).

264

-

NORMAN G. EINSPRUCH

Electric iield ( V / c m )

FIG.21. Dependence of the second harmonic amplitude on the applied drift field (after Kroger"7).

ultrasonic amplification; his measurements on CdS biased a t the crossover field also showed the second harmonic power to be proportional to the square of the fundamental power. 13. RADIATION DAMAGE

Truell et ~ 1 . have ' ~ ~ studied the effects of collimated neutron irradiation on the elastic properties of single-crystal silicon by the ultrasonic doublerefraction technique. The crystals were oriented such that each cubic sample had three equivalent { 100] faces. The neutron beam, incident normally to one of these faces, produced damaged regions which are elongated and aligned with the flux. Measurements made with transverse waves propagating in the direction of irradiation showed essentially no effect. Shear wave measurements made transverse to the direction of irradiation showed a strong interference effect when the direction polarization of the shear wave was between the two [loo] directions in the { 100) plane. The interference pattern, absent before irradiation, is ascribed to an anisotropic depression of 2% in the shear modulus measured in the direction of the neutron flux. I n this experiment, no change in attenuation accompanied the change in shear velocity. One of the few examples of the use of elastic wave scattering theory to explain the results of an experiment on a single crystal is given by T r u e l P using a multiple-scattering calculation by Waterman and Truell.I6' To explain observations on a neutron-irradiated silicon crystal, Truell R. Truell, L. J. Teutonico, and P. W. Levy, Phys. Rev. 106, 1723 (1957). R. Truell, Phys. Rev. 116, 890 (1959). 151 See Waterman and Truell.22 149 160

IJLTRASONIC E F F E C T S I N SEMICONDUCTORS

265

approximated each damaged region by a spherical cavity in a surrounding undamaged elastic medium. Scattering theory predicts a change in velocity which can be calculated in terms of the fractional volume associated with the damaged regions. Moreover, the attenuation can be estimated in terms of the density of scatterers and the elastic wave scattering cross section. From the change in a compressional wave velocity in silicon and the observation that there was no change in the attenuation, the size of a damaged region-described as a spherical cavity-was found to lie in the range 0.01 to 0.27 p. A better description of a damaged region undoubtedly would be a spheroid, although the single scatterer problem for spheroidal geometry has been the multiple-scattering problem has not yet been undertaken.

INTERACTIONS 14. PHONON-PHONON The theory due to Granato and L u ~ k e , which ' ~ ~ has been very successfull54 in describing mechanical loss and dispersion phenomena in metals and in the alkali halides on the basis of dislocation-phonon interactions, was used to interpret room-temperature attenuation measurements on germanium155J56in the 5 to 300 Mc sec-l range. However, as Mason and BatemanI57have pointed out in their detailed study of the phonon-phonon interaction in silicon and germanium, measur e me n t~ '~ 8ofJ ~the ~ mobility of a dislocation in a diamond-type lattice indicate a high Peierls barrier, which suggests that energy dissipation resulting from a dislocation vibration mechanism should occur only at elevated temperatures. Lamb et uZ.,l6O working at frequencies as high as 800 Mc sec-l, measured the frequency dependence of the ultrasonic attenuation in a series of silicon and germanium crystals. They examined n- and p-type conductivity, a range of resistivities, and a range of dislocation densities from a few hundred to 104 cm-2, and found little difference in the attenuation in going from sample to sample. Dobbs et aZ.'el measured the attenuation of compressional and shear waves a t frequencies as high as 650 Mc sec-I and a t temperatures as low N. G. Einspruch and C. A. Barlow, Jr., Quart. A p p l . Math. 19, 253 (1961). A. Granato and K. Lucke, J. A p p l . Phys. 27, 583 and 789 (1956). Is4 See, for example, Acta Met. 10 (April 1962) for papers presented a t the Conference on Internal Friction, Cornell University, July 1961. 155 See Granato and True1L27 156 L. J. Teutonico, A. Granato, and R. Truell, Phys. Rev. 103, 832 (1956). W. P. Mason and T . B. Bateman, J. Aeoust. SOC.Am. 36, 644 (1964). lS8 M. N. Kabler, Phys. Rev. 131, 54 (1963). 159 V. Celli, T. Ninomiya, and R. Thomson, Phys. Rev. 131, 58 (1963). 160 J. Lamb, M. Redwood, and Z . Shteinshleifer, Phys. Rev. Letters 3, 28 (1959). 161 E. R. Dohbs, B. B. Chick, and R. Truell, Phys. Rev Letters 3, 332 (1959) 153

266

NORMAN G . EINSPRUCH

as 1.5"K in an n-type germanium crystal with a net donor concentration of 10l2 ~ m - ~They . suggested that phonon-phonon scattering processes contribute to the temperature-dependent part of the measured attenuation. Further analysis of these data in terms of phonon-phonon processes was given by Dobbs162;Verma and J o ~ h i found l ~ ~ good agreement with theory using a temperature-independent Griineisen constant selected to fit the data a t 70°K. A phonon-phonon scattering mechanism proposed by Akhiezer164 and subsequently developed by Bommel and Dran~field,16~ and Woodruff and Ehrenreich16'j to explain the observed frequency and temperature dependence of the ultrasonic attenuation in quartz, has been applied to explain losses in pure semiconductors. In essence, the sound wave modulates the equilibrium thermal phonon distribution in the crystal; the modified distribution tends to relax back to equilibrium via phonon-phonon collisions. The relaxation process absorbs energy from the sound wave and is manifest as ultrasonic attenuation. The strength of the coupling of the thermal phonon modes to the ultrasonic phonon modes is given by the third-order elastic constants, which are a measure of the deviation of the lattice from harmonicity. Millerl67 has used the result of the Woodruff and Ehrenreich formulation to explain his measurements on germanium; the theory predicts for the phonon-phonon contribution to the attenuation alp =

8.68(y2)wTStan-l(2wr) P (v >5 2wr

)

where (y2) is an averaged squared Gruneisen constant, X is the thermal conductivity, (v) is an averaged sound velocity, the relaxation time r is defined as r = ~S/C,(Z~)~, where C, is the specific heat. In deriving the expression for alp, valid in the w r << 1 limit, simplifying assumptions on the detailed nature of the relaxation process were made, as well as the assumption that the lattice is in thermal equilibrium with the bath. Under the assumption that the theory is correct, an averaged Gruneisen constant was derived and was E. R. Dobbs, Proc. 7th Intern. Conf. Low Temp. Phys., Torunto, Ont., 1960 p. 291. Univ. of Toronto Press, Toronto, Canada, 1961. 163 G. S. Verma and S. K. Joshi, Phys. Rev. 121, 396 (1961).

162

164 165

lB6

16'

See Akhiezer.138 See Bommel and Dransfield.40 T. 0. Woodruff and H. Ehrenreich, Phys. Rev. 123, 1553 (1961). B. I. Miller, Phys. Rev. 132, 2477 (1963).

ULTRASONIC EFFECTS I N SEMICONDUCTOHS

267

found to have a temperature dependence similar to that observed in measurements of thermal properties. Miller was unable to reproduce the . ~ ~ at ~ 280°K in measattenuation peaks found by Blitz et ~ 1in germanium urements at 4 and 6 Mc sec-I and attributed by them to a n electron-phonon interaction. Miller also estimates, on the basis of Weinreich's calculations, that the relaxation times for electrons are much too short to allow appreciable interaction a t this temperature. Deviations from harmonic behavior give rise to a number of important thermal and mechanical properties of solids. Among these are thermal expansion, thermal resistance, and mechanical loss due to phonon-phonon interactions. The problem of relating observables, particularly the dependence on stress of the ultrasonic velocities, to the third-order constants of crystalline materials, has been treated by Birch,169Seeger and Buckl70 Toupin and Ber1istein,~7~ Einspruch and Manning,l72 Thruston a complete set of third-order and Brugger,lT3 and B r ~ g g e r . 'Obtaining ~~ constants requires far more experimental work than obtaining a complete set of second-order elastic constants of a given material at a fixed temperature. For example, a cubic crystal, which has three second-order constants, has either six or eight third-order constants, depending upon the details of the crystal symmetry. Most frequently, the third-order constants are evaluated by measuring the stress derivatives of the second-order constants. The six third-order elastic moduli of germanium have been ~ ~ ; and Andreatch have remeasured a t 25°C by Batemari et ~ 1 , ' McSkimin ported measurements on germanium176and silicon177 at 25°C and - 1953°C made over the pressure range 0 to 20,000 psi. I n a subsequent paper, McSkimin and Andreatch report the results of further measurements of the third-order elastic moduli of silicon and germanium using more accurate experimental techniques. Although these are only six independent moduli, fourteen experimental parameters were measured and were consistent to within three parts in lo5a t 10,000 psi. J. Blitz, D. M. Clunie, and C. A. Hogarth, Proc. Conf. Semicond. Phys., Prague. 1960 p. 641. Czechoslovak Academy of Sciences, Prague, 1961. F. Birch, Phys. Rev. 71, 809 (1947). 170 A. Seeger and 0. Buck, 2. Naturforsch. 16a, 1056 (1960). lil R. A. Toupin and B. Bernstein, J. Acoust. SOC. Am. 33, 216 (1961). li2N. G. Einspruch and R. J. Manning, J . A p p l . Phys. 36, 560 (1964). li3R. N. Thruston and K. Brugger, Phys. Rev. 133, A1604 (1964); 136, AB3 (1964). 174 K. Brugger, Phys. Rev. 133, A1611 (1964). 175 T. Bateman, W. P. Mason, and H. J. McSkimin, J. A p p l . Phys. 32, 928 (1961). l i 6 H. J. McSkimin and P. Andreatch, Jr., J . A p p l . Phys. 34, 651 (1963). lii H. J. McSkimin and P. Andreatch, Jr., J . A p p l . Phys. 36, 21G1 (1964). 17ia H. J. McSkimin and P. Andreatch, Jr., J . -4ppl. Phys. 36, 3312 (1964). 169

268

NORMAN G. EINSPRUCH

Mason and BatemanI78 have extended the Akhiezer model, incorporating the recently measured third-order elastic constants, to explain their measurements of the temperature-dependent ultrasonic attenuation in high-purity n-type germanium and p-type silicon containing fewer than lOI4 impurity atoms The anisotropic Gruneisen constants required in the analysis are calculated directly from the measured moduli. For both silicon and germanium, the extended theory predicts the large observed differences in the measured longitudinal and shear wave attenuation and predicts the magnitudes of the attenuations within a factor of two. teiiin o ssstt~ v ~ ~ ~ ACKNOWLEDGMENTS The author acknowledges with thanks a critical reading of the manuscript by Dr. R. Stratton and the able assistance given him in the preparation of this manuscript by Mrs. Kay Byers. GENERAL REFEREWES

H. B. Huntington, The elastic constants of crystals, in “Solid State Physics” (F. Seits and D. Turnbull, eds.), Vol. 5, pp. 214-349. Academic Press, New York, 1958. W. P. Mason, “Physical Acoustics and the Properties of Solids.” Van Nostrand, Princeton, New Jersey, 1958. W. P. Mason, ed., “Physical Acoustics: Principles and Methods,” Vol. 1, Parts A and B, 1964 (Vols. 2-4, to be published). Academic Press, New York. R. True11 and C. Elbaum, High frequency ultrasonic stress waves in solids, in “Handbuch der Physik” (S. Flugge, ed.), Vol. 11, Part 11. Springer, Berlin, 1962.

”*See Mason and Bateman157; the authors also cite measurements of the third-order elastic constants of germanium and silicon made by J. R. Drabble and M. Gluyas and presented a t the International Conference on Lattice Dynamics, Copenhagen, 1963. 179 R. Hooke, London, 1678.

Quantum Theory of Galvanomagnetic Effect at Extremely Strong Magnetic Fields

RYOGOKUBOAND SATORU J. MIYAKE* Department of Physics, The University of Tokyo, Tokyo, J a p a n AND

NATSUKI HASHITSUME Department of Physics, Ochanomizu University, Tokyo, Japan

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

270

11. Basic Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 274 1. Motion of Crystal Electrons in a Magnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 2. General Expression of the Conductivity Tensor. . 3. Formulas for the Case of Elastic Scattering.. . . . . . . . . . . . . . . . . . . . . . . . . . 286 4. Asymptotic Behavior of Conductivity at Extremely Strong Fields. . . . . . . 289 5. Logarithmic Divergence in the Perturbational Treatment. . . . . . . . . . . . . . . 295 111. Inelastic Collision with Phonons ........................... .... 6. Effect of Inelasticity on the 7. Formula for Scattering by Phonons.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Acoustic Phonons. . . . . . . . . . . . . . . . . . . ........................ 9. Optical Phonons and Piezoelectric Interactions, . . . . . . . . . . . . . . . . . . . . . . .

299 299 302 307 316

IV. Collision Broadening. . . . . . . . . . . . . . . . . . . . . . . . .................... 10. Effect of Collision Broadening on the Logarithmic Divergence. . . . . . . . . . ............................ 11. Damping Theoretical Formulation.. . ....... 12. Davydov-Pomeranchuk Theory. . . . . . . . . . . . . . . . . . . . . . 13. Scattering by Gaussian Potential: Short-Ra ................ 14. Collision Broadening versus Inelasticity. ... ................

317 317 320 322 326 333

V. Non-Born Scattering.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 15. Effect of Interference of Waves on the Logarithmic Divergence.. . . . . . . . 336 16. Conductivity Tensor Expressed in Terms of Scattering Operators.. . . . . . 337 17. Skobov-Bychkov Theory: Short-Range Fo . . . . . . . . . . . . . . . . . . 340 . . . . . . . . . . . . . . . . . . 343 18. Method of Partial Waves.. . . . . . . . . . . . . . . 19. Non-Born Scattering versus Collision Broa ............ . . . . . . 353 Appendix A. Collision Broadening Effect upon Oscillatory Behavior.. . . . . . . . 356 Appendix B. Approximate Form of the Green Function in a Magnetic Field.. 362

* Present address: Department of Physics, Tokyo Institute of Technology, Tokyo, Japan. 269

270

RYOGO KUBO, SATOIKJ J. MITAKE, AND NATSUKI

HASHITSUME

1. Introduction

Since the first observation by Shubnikov and de Haas' in 1930 of oscillations in the magnetoresistance of bismuth crystals, the quantum theory of galvanomagnetic eff ect-especially at strong magnetic fields-has long been an outstanding problem, in contrast to the de Haas-van Alphen effect. The work of Titeica2 in 1935 and that of Davydov and Pomeranchuk3 in 1940 marked important progress, but their interpretation of the electric conduction in terms of the center migration process was given a sound logical basis only a few years ago."-? On this basis many theoretical studies have recently been done. Effects of the, quantization of electronic motion on transport phenomena through changes in the density of states of electrons and the scattering mechanisms appear first in the Shubnikov-de Haas oscillation at strong magnetic fields and second in the monotonous change of resistivity a t extremely strong magnetic fields. I n this article we shall mainly review theories on the second effect; the first effect shall be treated only briefly (in Appendix A ) , since this has been reviewed by Kahn and Frederikse.8 Since there are not many experimental data available a t the present moment-although more will come out in the future-the emphasis is put here on the logical structure of the theory. In this respect the subject provides us with a very interesting example of transport phenomena for which the traditional method of using a kinetic equation becomes powerless or at least dubious, and therefore a more fundamental approach is seriously required. Thus we start in this review from the general formula of conductivity and discuss its applications to degenerate or nondegenerate electrons with various mechanisms of scattering. Critical reviews of previous theories will be given from this unified point of view together with some results of our unpublished work. The galvanomagnetic effect shows considerably different features depending on the difference in the strength of the magnetic field and in the scattering mechanism. We characterize the field strength H by comparing the cyclotron angular frequency fi

=

eH/nzc

L. Shubnikov and W. J. deHaas, Leiden Commun. 207a, c, d; 210a (1930). S. Titcica, Ann. Physik [5] 22, 129 (1935). B. Davydov and I. Pomeranchuk, J . Phys. (USSR) 2, 147 (1940). R. Kubo, J . Phys. SOC.Japan 12, 570 (1957). R. Kubo, H. Hasegawa, and N. Hashitsume, J. Phys. SOC. Japan 14, 56 (1959). E.N. Adams and T. D. Holstein, Phys. Chem. Solids 10, 254 (1959). 7 P. N. Argyres and L. M. Roth, Phys. Chem. Solids 12, 89 (1959). A. H. Kahn and H. P. R. Frederikse, Solid State Phys. 9, 257 (1959).

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

27 1

(rn being the effective mass of an electron) with its mean free time (or the relaxation time of electric current) rf and the representative energy of an electron I?, and distinguish the following cases:

<< E ; << E ;

(I.lb)

and hfi 2 E ;

(Llc)

5 1 and hfi

( a ) weak field,

if

(b) strong field,

if firf >> 1 and hfi

(c) extremely strong field, if

fir1

fir1

>> 1

(I.la)

As the energy l? we take the chemical potential { for degenerate electrons arid the thermal energy kT for nondegenerate electrons. At weak fields the problem may be treated semiclassically on the basis of the Boltzmann kinetic equation or its equivalent, e g. the pictorial kinetic method.9 The same approach may be applied also for the strong field case, as long as the effect of quantization upon the scattering process is ignored. The theory given by Lifshitz et aZ.’O is a typical one which contains a discussion of asymptotic forms of the resistivity tensor p in the limit of fir1 >> 1. They pointed out, in particular, that the magnetoresistance component corresponding to the direction along which an open orbit extends, increases proportionally to H2 without saturation. This prediction is very useful in studying the topology of the Fermi ~urface.1l-l~ The validity of the kinetic equation in case (b ) is investigated by several authors. Stinchcombe14 showed that when hfi, h/rf are small compared with the Fermi energy, and hfi is less than the thermal energy, the Boltzmann equation is derived for the k-space distribution function which is closely related to the Wigner phase space distribution function, and from which the current is calculated in the usual manner. For nondegenerate electrons, Gurevich and Firsov15 confirmed the results obtained from the kinetic equation on the basis of a new formalism, which will be used when we discuss the monotonous change of resistivity at extremely strong magnetic fields. At strong magnetic fields, the orbital quantization makes its appearance mainly in the Shubnikov-de Haas oscillation. Owing to the second condi’W. Shockley, Phys. Rev. 79, 191 (1950); R. G. Chambers, Proc. Phys. SOC.(London) 66, 458 (1952); R. B. Dingle, Physica 22, 671 (1956). I. M. Lifshitz, M. Ya. Azbel’, and M. I. Kaganov, Zh. Eksperim. i Teor. Fiz. 30, 220 (1956); 31, 63 (1956); see Soviet Phys. J E T P (English Transl.) 3, 143 (1956); 4, 41 (1957).

1. M. Lifshitz and V. G. Peschanskii, Zh. Eksperim. i Teor. Fiz. 36, 1251 (1958); 38, 188 (1960); see Soviet Phys. J E T P (English Transl.) 8 , 875 (1959); 11, 137 (1960). l2 J. M. Ziman, Phil. Mag. [S] 3, 1117 (1958). l 3 R. G. Chambers in “The Fermi Surface” (W. A. Harrison and M. B. Webb, eds.), P. 100. Wiley, New York, 1961. R. B. Stinchcombe, Proc. Phys. Soe. (London) 78, 275 (1961). 15 v. L. Gurevich and Yu. A. Firsov, Zh. Ekspen’m. i Teor. Fiz. 40, 199 (1961); 41, 512 (1961); see Soviet Phys. J E T P (English Transl. 13, 137 (1961); 14, 367 (1962).

272

RYOGO KUBO, SATORU J. MIYAKE,

AND NATSUKI

HASHITSUME

tion of Eq. (I.lb), the quantum numbers of Landau levels are still large, that we may use the kinetic equation for the density matrix. The theory developed by LifshitzI6 and supplemented by Kosevich and Andreev,I7 and the theories given by ArgyresI8and by Hajdu's follow this line. I n the following, we shall not be so much concerned with weak and strong magnetic field cases. We shall concentrate mainly on the extremely strong magnetic field case where the effect of orbital quantization becomes really dominant. Then, as mentioned before, we can no longer put much confidence in the use of kinetic equations because the duration time of collision may not be short compared with the time between collision. Thus we use as the basis of our treatment the general formula of conductivity which will be briefly described in Part 11. The lowest-order approximation to this rigorous expression, however, has the unpleasant drawback of divergence, which was seen in the old treatment of Davydov and P~ m eran ch u k .~ The origin of this divergence lies in the approximation which is essentially the lowest Born approximation and allows an electron repeatedly to interact with a given scatterer if it is running slowly along the direction of the magnetic field. The stronger the magnetic field is, the more slowly most electrons move in the direction of the field, which makes the difficulty greater. There are three different mechanisms which prevent the divergence. We shall discuss them separately, although in practice they may appear together. SO

(1) The first is the inelasticity of a collision, which we shall discuss in Part 111. This can lead to a finite conductivity, even in the Born approximation. This was done for the phonon-scattering case by Titeica12 by Gurevich and Firsov,15 and the present authors.20 (2) The second mechanism is the collision broadening of the Landau levels. The stronger the magnetic field is, the longer and thinner the wave packet of an electron becomes in the direction of the magnetic field (Section 1). This means more chance for an electron to be scattered by a second I. M. Lifshitz, Zh. Eksperim. i Teor. Fiz. 32, 1509 (1957); see Soviet Phys. J E T P (English Transl.) 6, 1227 (1957); Phys. Chem. Solids 4, 11 (1958); I. M. Lifshitz and A. M. Kosevich, Phys. Chem. Solids 4, 1 (1958). 17A. M. Kosevich and V. V. Andreev, Zh. Eksperim. i Teor. Fiz. 38, 882 (1960); see Soviet Phys. J E T P (English Transl.) 11, 637 (1960). l8 P. N. Argyres, Phys. Rev. 109,1115 (1958); 117,315 (1960);see also Argyres and Roth. l9 J. Hajdu, 2. Physik'l60, 47, 481 (1960); 163, 108 (1961); Can. J . Phys. 41, 533 (1963). 2o R. Kubo, H. Hasegawa, and N. Hashitsume, Busseiron-Kenkyu (mimeographed circular in Japanese) [2] 4, 170 (1958). This note contains errors in the estimation of integrals, which have been corrected by S. J. Miyake, Master Thesis, University of Tokyo, 1959. I n this paper the conductivity tensor is calculated for the ellipsoidal energy surface.

l6

QUANTL'M THEORY Oh' GALVANOMAGNETIC EFFECT

2'73

scatterer before returning to the first one. This type of mechaiiism was considered by Davydov and P ~ m e r a n c h u k We . ~ shall formulate this approach in Part IV. For phonon scattering, the inelasticity is effective a t low temperatures, whereas the collision broadening is effective a t high temperatures. (3) I n the case of impurity scattering, which may be assumed to be elastic, the third mechanism comes in before the second becomes effective if the concentration of scatterers is sufficiently low and the force range is short compared with the radius of the electron orbit. Namely, when the wave packet of an electron moving slowly in the direction of the field collides repeatedly with a scatterer, it produces many scattered waves, one for each collision, and these scattered waves do the same. All these waves are coherent and interfere with one another. Thus in this case we have to solve the scattering problem more exactly. This will be given in Part V. Kahn,zl Skobov,22and BychkovZ3have treated this problem by introducing the scattering amplitude f. Their treatment corresponds to the s-wave approximation of the scattering problem in the absence of magnetic field. We shall give the partial-wave method a t extremely strong magnetic fields. The relative importance of these different mechanisms mainly depends on the nature of the conduction electrons and of the scatterers. For simplicity we shall consider only point-defect scatterers (static potential of impurities) and acoustic phonons and refer only to some results for optical phonons and the piezoelectric interaction (Section 9). The calculation is generally difficult to be carried out to the end, so that we shall consider two extreme cases of impurity potential : case (A) short-range force (&functionlike potential), 1 >> a; (1.2a) and case (B) long-range force (slowly varying potential),

1 << a,

(1.2b)

where 1

=

(fLc/eH)1/2=

=

2.566 X

(fL/mQ)1/2

H-'/2 cm (H in Oe)

(1.3)

is the classical radius of the ground Landau orbit.

*'A. H. Kahn, Phys. Rev. 119, 1189 (1960). V. G. Skobov, Zh. Eksperim. i . Teor. Fiz. 37, 1467 (1959); 38, 1304 (1960); see Soviet Phys. J E T P (English Transl.) 10, 1039 (1960); 11, 941 (1960). 23 Yu. A. Bychkov, Zh. Eksperim. i . Teor. Fiz. 39, 689 (1960); see Soviet Phys. J E T P (English Trunsl.) 12, 483 (1961).

*I

274

I ~ Y O G OKUBO, S A T O I ~ UJ. MIYAKE, AND NATSUKI HASHITSUME

For the phonon scattering we shall consider only case (C) semiconductors or semimetals, A >> l / q m a x (1.4)

] q m a x the where A is the de Broglie wavelength [A = h / ( 2 ~ n E ) ” ~and maximum wave number of phonons. Metals will not be considered because extremely strong fields cannot be realized for metals. II. Basic Theory

1.

iklOTION OF CRYSTAL

ELECTRONS IN A

h’IAGNETIC FI E L D 5

Motion of electrons in crystals in magnetic fields is generally very complicated, but if we neglect the interband elements of electron operators, we can describe it approximately by an effective Hamiltonian, which is obtained in the following way. Let us consider electrons in a band with the energy-momentum relation Eo(p), where p is the crystal momentum hk, k being the Bloch wave vector, According to PeierlsZ4and L ~ t t i n g e r , ~ ~ the effective Hamiltonian in a magnetic field

H

=

rot A

is obtained by substituting for p the quasi-momentum operator o = p

+ eA/c,

(1.1)

which satisfies the commutation relation x x x =

(fie/ic)H.

(1.2)

Thus the effective Hamiltonian of an electron is where and U(r) is the perturbation potential due to applied electric fields or the scatterers, either impurities or phonons, its interband elements being omitted. The coordinate vector r stands for the discrete lattice points to which the Wannier functions are referred, but it will be treated as continuous as in the usual effective mass approximation. We shall thus neglect the Harper broadening of the Landau levels.26 It. E. Peierls, Z . Physik 80, 763 (1933). J. M. Luttinger, Phys. Reu. 84, 814 (1951). 26 P . G. Harper, Proc. Phys. Soc. (London) A68, 874, 879 (1955); G. E. Zilberman, Zh. Eksperim. i Teor. Fiz. 30, 1092 (1956); see Soviet Phys. J E T P (English Transl.) 3, 835 (1956).

24

25

275

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

I n the following the magnetic field is always assumed to be uniform and along the z-direction. Then r 2 = p , and the commutation relation (1.2) and other commutation relations are written explicitly as [rZ,ru] = (lie/ic)H,

La,, 1 ' = Y1 =

CrZ,

=

CaZJ

Y1 '1

[Pz,

XI =

[*U,

[r,, p,] = Cry, pz] = 0 ,

=

[PZ,

=

Cr,J

' = 1 '1 = Y 1 = 0.

(1.5a) (1.5b)

'/iJ

h U J

CPZ,

(1.5~)

Now let us define the relative coordinates of the cyclotron motion by

E

=

(c/eH)auJ

71 =

- (c/eH)a,

(1.6)

and its center coordinates by X=x-E,

(1.7)

Y=y-q,

for which the commutation relations are easily found to be

[EJ 771

[EJ [EJ

=

-iP,

[ X , Y]

pZ1 =

[VJ

p21 =

=

[VJ

=

Ex,

=

pZ1

[t, 1 '

iP,

=

=

J' [

[VJ

pZ1

"1

=

= [EJ

OJ

'1

(1.8) =

[?J

'1

=

O>

where I means the elementary length (1.3) which corresponds to the classical radius of the lowest Landau level of an electron with a spherical mass m. We see a t once from the commutation relation [ X , Y] = iZ2that the position of the center is quantized in such a way that there exists only one center in a region of area 2 ~ 1 2= hc/eH according to the uncertainty principle AX-AY = 2 ~ 1 ' . (1.9) When the magnetic field becomes extremely strong, the radius of orbit 1 diminishes to zero, and so we may regard the center coordinates (XJ Y ) and the relative coordinates ( E , 11) as commutable. Under the action of the potential U(r), an electron moves following the equations of motion, which are obtained easily from the assumed Hamiltonian (1.3 ) : ?i,

=

(i/fi)[x, r,]

=

eH aEo(.x)- dU(r) c aru ax ' (1.10)

?i, =

(i/fi,)[x, r,] =

eH d E o ( x ) dU(r) -___ - -, c

a*,

aY

276

RYOGO KUBO, SATORU J . MIYAKE, AND NATSUKI HASHITSUME

or

i s -aE0 --aa,

c ar; eHay

’ (1.11)

c ali eH dx

a~~ aa,

q=-+--.

.]

.]

Also we find easily from the commutation rules that k

= vz

jl

3

= (i/fi)[X,

V, =

(Z/fi)[~,y]

( i / h ) [ X e ,t ]

aEo/ar,,

=

(i/fi)[Xe,

=

( i / f i ) [ X e ,y] = ( i / f i ) [ X e ,73 = dEo/aa,.

=

=

(1.12)

Therefore Eq. (1.10) is the familiar equation of motion including the Lorentz force. By the definitions (1.7) or directly by the commutation rules we have

x

= ( i / f i ) [ X ,X

]

( i / f i ) [ U ,X]

=

c au , eH a y

= --

(1.13)

Y

= ( i / f i ) [ X , 1’1 =

( i / f i ) [ U ,Y ]

=

c ac ---. eH ax

These are the equations of motion for the center coordinates (X, Y ) which stay constant if there is no perturbation potential U . This observation justifies the identification of (X, Y ) with the center coordinates of cyclotron motion. In the x-direction, the velocity is given by

i

=

aEo/ap,,

(1.14)

and the motion follows the equation

p,

=

-

(aU/az).

(1.15)

The above equations hold either classically or quantum-mechanically. The function Eo (r,,r,, ps) must be well ordered in the quantum-mechanical case with respect to the noncommuting variables a, and a,. The unperturbed cyclotron states are characterized by three quantum numbers, N , pz’, and X’ (or Y’ depending on which of noncommutable coordinates X and Y is chosen to be diagonal). The explicit form of the eigenfunction for the state ( N , pz’, X ’ ) depends on the representation. For instance, if the variables (r,,r,), ( X , Y ) and (pz,z ) are chosen as the canonical variables, the (a,, Y , z ) representation of the eigenfurictiorl

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

277

will be of the form z) a L-’m(a,) exp [ (ipz’z/fi) - ( i X ’ Y / P ) ] , where the cyclotron eigenfunction PN (n,) is the solution of $ N . ~ , J , x(az, ~ y,

Eo[x,, - (fieH/ic) (d/da,), p = ’ l v(az) ~ = E N ( P ~ ’ ) ((az). PN

(1.16) (1.17)

Or, by the canonical transformation

exp (-ZqX/P)l exp ( i q X / P )

+X

=

exp ( - i q X / P ) Y exp ( i q X / P ) = Y

=

x,

+ q = y,

(1.18)

we may choose the variables ( x , q ) , ( X , y), and (p,, x ) and take the ( x , y, z ) representation, in which the eigenfunction takes the form $ N , ~ Q , x ~ (y, x ,z ) a

where

q~

L-’PN(~- X ’ ) exp [ ( i p B ’ z / h ) - (iX’y/P)J,

(1.19)

(z) is the solution of

(1.20)

This is equivalent to (1.17), but it in fact corresponds to choosing the vector potential as (0, H x , 0). Since this particular representation is convenient for practical calculations, it will be often used in the following. The wave functions (1.16) or (1.19) are normalized within a volume V = L3 (the domain of (z, Y ) or (2, y ) is L 2 ) .If any summation is made over possible values of p,‘ and X’, it will be replaced, in the limit of L + m , by the integrals over p,’ and X’ as (1.21)

The quantization of cyclotron motion is conveniently visualized in the momentum space.27 At weak field, where the quantum number N is generally large, the cyclotron motion

eH dEo T I =

c da,

,

.

eHdEo

a,=--

c

arz

is quantized by the semiclassical condition, (1.22)

*’L. Onsager, Phil. Mag. [7]43, 1006 (1952).

278

RYOGO KUHO, SATOKU J. MIYAKE, AND NATSUKI HASHITSUME

I I I I I

N=Nmax

I I I 7

I

I

I

b'

-\

\

TY

I I

TX

(y is a constant). This quantum condition defines a cylindrical surface in the (rz,rg,pz) space for a given N.28This is illustrated by the familiar figure (Fig. l ) , which shows this quantization for a free-electron case. The Fermi surface cuts the cylinders as shown in the figure to define the states occupied by electrons a t 0°K. I n this article, we shall not consider the so-called extended or open orbits, so that the Fermi surface will be assumed to be of some relatively simple closed form. As the magnetic field becomes stronger, the spatial degeneracy of each cyclotron state increases with H as

L2/2r12= L2eH/ch and each cylinder will grow in the diameter so that the outer cylinders will disappear successively a t the Fermi surface. When the field is extremely strong there remains only one cylindrical surface corresponding to the eigenvalue N = 0, i e. to the ground Landau level. The length of this last cylindrical surface becomes shorter as its cross-sectional area becomes larger. Thus we see that the number of electrons with small speed v, in the field direction becomes larger as the field increases. This causes the increase of electrical resistance by the following reason. If an electror 28

R. G . Chambers, Can.J. Phys. 34, 1395 (1956).

279

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

with v, nearly equal to zero once hits a scatterer, it will be scattered again atid again by the same scatterer, as if it were captured there. This means that it will contribute little to the conduction. A wave packet corresponding to the corpuscular concept of the electron can be made up of those wave functions that correspond to points on the cylindrical surface mentioned above. Since in an extremely strong magnetic field all these wave functions have small absolute values of momentum p,, and since they belong to the ground Landau level N = 0, the resultant wave packet will look like a cigar (Fig. 2), which is much elongated in the direction of magnetic field and has a small cross section ?r12 = H-l. We may estimate the extension of wave packet in the z-direction as follows. In the case of degenerate electrons, we put { =

3hB

+ A,

(1.23)

A being the maximum value of the kinetic energy E , in the direction of magnetic field. Then the allowed range for momentum p , is

I p, I

5 (2mA)1’2a A112,

(1.24)

and the required extension is of the order of 2h/(2mA)1’2and increases in proportion to H , if the number density of electrons is kept constant [cf. kT. Eq. (4.10)]. I n the case of nondegenerate electrons, we may put A When this wave packet which is moving slowly in the direction of magnetic field interacts with a scatterer, its center will be transferred within the plane perpendicular to the magnetic field by a distance of the

-

wave packet

FIG.2. Shape of a wave packet in an extremely strong magnetic field.

280

RYOGO KUBO, SATORU J. MIYAKE, A N D N A T S U K I HASHITSUME

order of 1 or 12/a depending on whether the range a of the scattering potential is short or long, respectively, as we shall see in Section 4. If the wave packet is very much elongated into a needle shape in a n extremely strong magnetic field, and if its length becomes of the order of the mean distance of the scatterers, it can be scattered simultaneously by two scattering centers. A kinetic equation of the familiar type will no longer be useful in this situation. 2. GENERALEXPRESSION OF THE CONDUCTIVITY TENSOR

According to the theory of irreversible processes developed by one of the a ~ t h o r s the , ~ electric conductivity tensor can be generally expressed by the formula

which gives the exact amplitude and phase of induced electric current in an applied electric field oscillating with frequency w . Here p is l / k T and ir is the current along the p-direction in the volume V ,which can be written as

by using the one-electron current operator 3,

=

-ev,

(2.3)

and the quantized wave functions 4*(r) and 4*+(r) normalized in the volume V . Many-electron operators will be hereafter expressed in Gothic letters. The Heisenberg operator i, ( t ) represents the natural motion of current in the absence of external electric field, namely

i P ( t ) = exp (iXt/A)i, exp ( - i X t / A ) ,

(2.4)

where X is the Hamiltonian of the system. The average denoted by brackets in Eq. (2.1) means the equilibrium average with the equilibrium density operator, i.e., ( A ) = trace (@*A), (2.5) where e* = Cexp {--P(X- ")I, (2.6)

C the normalization factor. This equilibrium average must be also taken over the s t a b of scatterers or the probability distribution of the scatterers.

( being the chemical potential, N the number operator, and

QUANTUM THEOHY O F GALVANOMAGNETIC EFFECT

281

The important point in Eq. (2.1) is that it expresses the conductivity in terms of spontaneous fluctuation of current in the equilibrium state represented by p*, in (2.6), where the average current naturally vanishes, i.e., (ill> = 0 ( P = 2, Y,2 ) (2.7) but the fluctuations do not. In fact it can be proved that B

dA(iv(-ifiA)i,,(0))

V-l!

=

e2n(me-1),,,

(2.8)

0

where the effective mass tensor me-' is defined by (me-l),,v= n-l trace { f ( E o )(d2Eo/dp,d p y ) )

n being the number density of electrons. The integrand of (2.1), B

d+(t)

=

T--I/

dA(iv(-ifiA)i,(t))

(2.9)

0

is a correlation function of fluctuating current density components which decays in time. The conductivity U , , ~( w ) is the one-sided Fourier transform of such a correlation function. The general formula (2.1) can be applied to the present problem in the following way.5 The components of the current carried by a n electron in the x-y plane (perpendicular to the magnetic field) can now be written as (2.10) j, = -e(+ Y) j, = -e(i XI,

+

+

corresponding to Eq. (1.7). We may conveniently use the complex representation, (2.11) v, = (l/v2)(vz f iv, v, = x* g*,

+

The current components are

i* = -e(%

+ (**I, -*(r)dr.

(2.12)

When this decomposition is made for the current components appearing in Eq. (2.1), i t is found that a remarkably simple result is obtained for the static case, i.e. w = 0, if the Fermi surface is closed. Then the cross terms between X and & components will drop out and we have our basic

282

RYOGO KUBO, SATOIlU J. MIYAKE, AND NATSUKI HASHITSUME

formulas

(E

++0) \

(2.13)

u Z z(0) =

g

/a

dt e-‘l

0

/

B

dX (v, (- ifiX)v, ( t ) ).

(2.16)

0

For the proof, let us first observe that

(t**) = 0

arid (X*) = 0, (2.17) if the Fermi surface is closed. Then every cyclotron orbit is closed and so the variables t and r] are bounded. Thus the first equation follows. The second equality is obtained from the first and Eq. (2.7). Since the correlation of dynamic variables which are bounded will be safely assumed to vanish a t an infinite time, we have

) ) (t*T)(t**)= 0. lim ( t * F ( - i h ~ ) t * * ( t = t-m

Therefore we can write ~ = d l ~ 9 ~ i h ( i * , ( - i s X ) t t ( l= ))= =

f

dX(t*,(-ifix>~**)

(l/ifi) trace

- (l/ifi) trace (p*[t*r,

= - (I/ifi) =

([e*, t*Fjt*+)

t*J)

Kt*r, %!**I)

fi( n c / e H ) V ,

(2.18)

283

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

where n denotes the number of electrons per unit volume. The transformation of the second expression into the third expression is made by the identity [exp (-OX), A]

=

\

0

dX p exp (XX)[A, X] exp (-AX)

0

=

ifi

B

(2.19)

dX p A (-&A),

and the last expression is obtained by the commutation rule

Similar calculations show that

Lrn

dt

dX (i**(--fix)

(** ( t ) )

=

(2.21)

0

and

= -(I/ifi)

trace

(e*[X,, c*T])

=

0, (2.22)

With the aid of these relations, Eqs. (2.13)-(2.16) are derived when Eqs. (2.12) are inserted into Eq. (2.1). I n the following we shall assume that the magnetic field is sufficiently strong so that we may neglect the second terms in Eqs. (2.14) compared with the first terms, i.e.,

.,(O)

= - (nec/H) =

-uuz(0).

(2.23)

Titeica2 also made this approximation, by which the Hall effect becomes normal : RH = - (nec.)-l, (2.24)

284

I ~ Y O G O KUBO, S ATO I NJ J . MIYAKE, AND NATSUKI HASHITSUME

and the transverse resistivity components are given by pZz(0 ) =

pYu(0

(H/necI2auu(0 ),

1=

(H/necI2azz(0 ),

(2.25)

since we may assume

<< I a z u ( 0 ) 1, ....

UZZ(0)

The inequality will be fulfilled a t strong and extremely strong fields. From the elementary expression of transverse components of conductivity tensor azz(0) =

ne2 m 1

-

+

rf (8rr)2

’

a,(O)

ne2 Qrf2 m 1 (0rr)2’

= --

.

+

(2.26)

we have in fact azz(O)/l

U,(O)

1

=

1/fhf

<< 1.

This situation will not be changed by the quantum expressions, although it should be justified by evaluating the second terms of Eqs. (2.14). In the approximation mentioned above we need only to calculate the components uzz(0) and uuY(0) in their lowest-order approximations with respect to (Orf)-l. The expressions of these components, i.e. the components of the symmetric part of conductivity tensor, can be rewritten into more convenient forms.5 Remembering the definition of the Heisenberg operators (2.4), we obtain

I”

dX (X (- ifiX)X (- t ) )

=

i,”

dX trace (exp [- (P - X)X

+ PlN]X exp (-AX)

exp ( - i t X / f i ) X

X exp (itiC/h)]/trace [exp (-PX =

[

dp trace {exp (-pX)X exp (-itiC/h) exp [ ( P

- P)X

+ P l N )]

+ P[N]X

X exp (itXP)}/trace [exp (-PX

+ PlN)] (2.27)

by putting

p =

/3 - A, and hence

(2.28)

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

285

The somewhat troublesome X integral can be eliminated as follows:

1"

B

dt/

--a,

dk(X(-ifiX)X(l)) = 0

/m

B 0

-m

=

=

dX(X(O)X(t

dl/

I"

+ ink))

dX /m+ifiA d s @ ( O ) X ( s ) ) --m+ihA

PJm

dt(X(O)X(t)).

(2.29)

--m

Here the path of integration was shifted. This will be justified because the integrand seems to have no singularity between the new and the old paths in the complex s-plane. By making use of Eq. (2.29), we may write Eq. (2.28) as (2.30)

Similarly we have

We have shown that the transverse conductivity in a magnetic field can be expressed in two ways, Eq. (2.1) or Eqs. (2.13)-(2.14). I n the first expression, the direction of current rotates with angular frequency Q, as the electron revolves on its orbit, and is changed a t scatterings. This means that the spectral density of velocity, i.e. the Fourier transform of the velocity correlation function, has a peak at the frequency Q. This corresponds to the cyclotron resonance line, which is broadened by the scattering process. I n the weak magnetic field case, where fl is small, we may use this direct way of calculating the expression (2.1) for the static conductivity ( 0 )is .usually done by making use of the kinetic equation. tensor ~ ~ ~ This The detailed calculation of the static conductivity is nothing else than the determination of the precise shape of the broadened cyclotron line in its

286

RYOGO KUBO, SATORU J. MIYAKE, AND NATSUKI HASHITSUME

tail a t w = 0. The usual theory assumes the line shape to be Lorentzian, and the condition under which the Lorentzian shape is valid is just the one under which the kinetic equation can be used. But the latter condition is not necessarily fulfilled at extremely strong magnetic fields, whence the line shape may be different from the Lorentzian shape. The second expressions, Eqs. (2.13)-(2.14), on the other hand, are useful for strong magnetic fields. If there is no scattering for the electrons, the center of cyclotron motion simply glides in the zy plane perpendicular to the electric field (see Eq. (1.13) with the electrostatic potential in place of V ) ,so that only the components uzy = -uyz = -nec/H remain. The components uzz and uyy become finite by the presence of scatterers. Thus the current fluctuation is now pictured as the migration of the center coordinates which make a certain zigzag motion. It can be shown that the correlation of successive steps in this zigzag motion persists very long in weak magnetic fields, but it is lost quickly in strong magnetic fields, where the condition c h i >> 1 is fulfilled. This corresponds to the expansion of the familiar expression of conductivity tensor (2.26), uzz(0) =

(ne2/mQ2)7t1( 1 - (Qrj)-2

u,,(O) = - (nec/H) [ 1

- (%j)-2

+ - - - ),

+ - - - 1.

(2.33)

In the limit we may ignore higher expansion terms. Correspondingly, in the first approximation, each displacement of the center may be regarded as independent, and the scattering process can be treated separately. In the language of perturbational calculation, 1/71 is proportional to the square of the perturbation matrix elements in the lowest approximation, and the higher-order terms in l/rf appear only in higher-order perturbations. 3. FORMULAS FOR

THE

CASEOF ELASTIC SCATTERING

In the following parts, except Part 111, we shall consider the case of elastic scattering, assuming the scattering potential U to be static. We shall here show that our basic formulas are greatly simplified in this case. We shall not explicitly consider the effect of Coulomb interaction between electrons,16 for it makes the calculation very complicated and also the effect is, to some extent, taken into account in the band electron picture. Then the total Hamiltonian of the system composed of conduction elec-

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

287

trons and scatterers can be written as

x=

4 * + ( r ) x t * ( r )dr

+ Xe,

(3.1)

where xsis the Hamiltonian of scatterers, which may be treated as a constant a n d thus neglected in the elastic case. But the average with respect to variable (e.g. positions) of scatterers contained in the interaction term U should not be forgotten when we evaluate expressions (2.30)- (2.32). With this Hamiltoniari we can reduce these many-electron expressions t o one-electron expressions in which the operators of second quantization disappear. It is easily shown that (X(O)X(~= ) ) (trace ( j ( ~ ) 2 ( 0 ) { 1 f ( x ) } X ( t ) ) ) i , (3.2)

where we have defined the Heisenberg operator

X(t)

=

exp ( i t ~ / f i ) exp X (-itx/n)

(3.31

-

and the trace in the one-electron space, the average ( - )s denoting the average with regard to the scatterers’ variables. By the use of this reduction formula, we can make the time integration in Eq. (2.30) as follows: m

=

V-’/” dt --m

Irn

dEf(E)(trace [ 6 ( E - X ) X ( 1 - f ( 3 C ) )

-a

x

exp ( i t ~ / )fXi exp (- itx/fi) 1)s

288

HYOGO KUBO, SATOHU J . MIYAKE, AND NATSUKI HASHITSUMK

Here f ( E ) denotes the Fermi distribution function corresponding to the chemical potential {, and 6(s) the delta function of Dirac. Noticing the relation f(E){1- S(E)J = - k T ( 4 f ( E ) / a E ) , (3.5) we obtain the required expression5 *fie2

/

uZZ(0)= -

V

dE

-m

(-”> (trace (6(E dE

-

X ) X F ( E - X)X)),, (3.6)

and in the same way 0,,(0)

=

0,,(0) =

(-$) (trace (F(E &e2 -/ d E (-”> (trace (F(E- X ) v , 6 ( E dE v

/m

dE

- X)Y6(E

.x)Y)),,

(3.7)

- X)v,)),.

(3.8)

-

--m

-m

These expressions are exact as long as the scattering is elastic and the interactions between electrons can be neglected. Comparing the formula for the energy-level density29 p ( E ) = V-l(trace

(F(E- X ) ) ) , ,

(3.9 1

we see that the two delta functions appearing in formulas (3.G)-(3.8) mean the level densities at the initial and the final states for transitions by collisions. Their energies E are equal with each other in accordance with the conservation law. Now one could try the simplest method for the evaluation of expressions (3.6)-(3.8), namely the perturbation expansion in the process of the scattering potential. As was mentioned in the previous section, this would be a good approximation a t strong magnetic fields. Although such a direct perturbation calculation suffers from the difficulty of divergence, which will be discussed in the following sections, we shall give here the result of the simplest perturbation and use this in the next section for a qualitative discussion of the asymptotic behavior of conductivity at extremely high magnetic fields. Since the operators X and Y are of the order of U by virtue of the equations of motion (1.13), the lowest-order expression of the conductivity component uzz(0)-when we treat the scattering potential U as a perturbation-is given by *fie2

1 dE

uZz(O) = -

V

-m

(-3)(trace dE

( 6 ( E - X , ) X S ( E - X , ) X ) ) * , (3.10)

where x,is the kinetic energy of conduction electron defined by Eq. (1.4). 29

C.Kittel, “The Elements of Statistical Physics.” Wiley, New York, 1958.

QUANTUM THEORY O F GALVANOMAGNETIC E F F E C T

Remembering that

x

=

( i / R ) [ U ,X I ,

289

(3.11)

we can write down the trace explicitly in terms of the matrix elements of the operators appearing in it. It is convenient to use the ( N , X , p z ) representation introduced in Section 1, in which the unperturbed Hamiltonian X, is diagonalized and has the eigenvalues EN(^,) defined by Eq. (1.17) or (1.20). Thus we have

*

(1

( N , X , pz

I u I N’, X’, pz’)

I’)g8(E - EN’(p.’)) ( X - X’)’. (3.12)

The factor 2 stands for the degeneracy due to electron spin, which will not be considered more explicitly than this factor. After integration with respect to E , we obtain the formula

’

(2r/fi)(I ( N , X , p z 1 u l N ’ , X ’ , p z ’ ) I’))ss(EN(pz)- EN’(pz’)), (3.13)

which is just the formula derived by Adams and Holstein6 by a perturbational treatment.

4. ASYMPTOTIC BEHAVIOR OF CONDUCTIVITY AT EXTREMELY STRONG FIELDS Adams and Holstein6 pointed out that the magnetic-field dependence of resistivity at extremely strong magnetic fields manifests rather remarkably the difference in the scattering mechanisms. We shall discuss this here in a somewhat more elementary way, assuming elastic scattering. As was stated in the Introduction, we consider only the two extreme cases of short- and long-range scatterers.

a. Short-Range Forces. Transverse Effect Let us first consider the case (A) or ( C ) of short-range force, in which the radius of orbit I in (1.3) is much larger than the force range a of a scatterer [inequality (1.2a)l. When an electron is in the lowest Landau level N = 0 (quantum limit), the component p , of its momentum in the direction of magnetic field either remains unchanged (Ap, = 0 ) or changes to the opposite direction (Ap, = -2p,) owing to the conservation of energy: E N ( p z ) = E”(pz’). Since the short-range potential has the

290

RYOGO KUBO, SATORU J. MIYAKE, AND NATSUKI HASHITSUME

Fourier components for almost all wave numbers lql Va (4.1 1 and since the allowed range for 1 p , I is a t most of the order of fill as shown in the inequality (1.24), both of these two processes can satisfy the momentum conservation Apz = f i q z (4.2) and can occur with almost equal probability. In the representation used in Eq. (1.19) we see that the center displacement by scattering is

AX = - (Ap,/mQ) = - (fiq,/mQ) = -12q,. This can be estimated roughly to be

(4.3)

1AXI-k because an electron will interact most strongly with the Fourier components of the scattering potential which is approximately equal in wave1/1) and such components are certainly length to itself (namely I q I available by condition (4.1). In other words, the transverse component of electron velocity can change its direction without any restriction, so that the center of orbit can jump isotropically within the plane perpendicular to the magnetic field. The collision rate, 1/7, can be estimated as follows. The cross section of one scatterer is 47rf2. Here we have introduced the scattering amplitude f for electrons with long wavelength under no action of magnetic field in order to treat the scattering by a short-range potential. As we have seen in Section 1, the electron encounters one scatterer many times while the wave packet passes by the scatterer in the direction of the magnetic field. The number of encounters for the duration time 7d of a collision is of the order of 7&/(21r), 27r/O being the period of cyclotron motion; Td will be of the order of h / E , , because the extension of wave in the direction of magnetic field is of the order of h/l p , I and the kinetic energy E , is of the order of p,v,. Thus the effective cross section is given by 4r$rdQ/2r = 47r$fiO/E,, and the required collision rate becomes

-

7-l

-

n h f 2 (fiQ/E,) 1 v,

I

=

nS47rf2(fiQ/I p , 1 ),

(4.4)

where n, stands for the number of scatterers per unit volume. This value should be averaged with respect to p,. Thus we obtain from Eq. (3.13) uzz(0)

-

(neffe2/kT)3Pns47rf2fiQ(lllP ,

I >,

(4.5)

where (1/1 p , I ) is a certain average of 1/1 p , 1 of incident electrons. Now we evaluate the H dependence of u z z ( 0 ) for the following cases: (I) If the chemical potential is kept constant with reference to the ground Landau leve1,Z the field dependence of u z z ( 0 ) arises from that of

QUANTUM THEOKY OF GALVANOMAGNETIC EFFECT

291

effective number neff(p,is kept constant), which is proportional to the degeneracy factor 1/ ( 2 ~ 1 in ~ )the level density, uzz(0) = H

(I

>> a,

{

- fi02/2

const)

=

(4.6)

for both the degenerate and the nondegenerate electrons. (2) n = constant; liondegenerate case. If the number n of conduction electrons is kept constant, the chemical potential [ changes with magnetic field ( p z is constant and is determined by the thermal energy k T ) . For riondegenerate electrons, neff n does not depend on H , and

-

u z z ( 0 ) a HO

(1

>> a,

n

nondegenerate,

=

const).

(4.7)

-

(3) n = constant; degenerate case. For degenerate electrons, neff kTp (I is) proportional to the level density a t the Fermi surface p ({). The general formula of the level density (3.9) in the lowest approximation is given in the case of spherical mass by p

(E)

=

V-I trace (6 ( E - X,) )

(N

+

;)fi0]-1’2,

for

E

> 63/2,

(see Fig. 3),

FIQ.3. Density of unperturbed states in a magnetic field.

(4.8)

292

ItYOGO KUBO, SATOICU J . MIYAKE, AND NATSUKI HASHITSUME

+

Nmax being the maximum value of N that makes E - ( N 1/2)fifl nonnegative. In the case of extremely strong magnetic fields N,,, = 0, and we have p ( { ) = (27rZ2)-1 ({ - fifl/2)-’/2 a H/A1/2. (4.9) On the other hand the number of electrons per unit volume is given approximately by

n

=

/

m

d E p ( E ) f ( E )=

J’

p(E)

dE

cc

(4.10)

HA112,

nn/ 2

--9

I)

-

and therefore All2 0~ H-’ and p ( { ) cc H2. The average (1/1 p , in Eq. (4.5) also depends on the magnetic field. We estimate it as (1/1 p , I ) 1/(mA)ll2 a H , and we arrive a t azz(0) a H 3

(1 >> a, degenerate,

n

=

const).

(4.11)

These results are summarized in Table I in terms of the field dependence of the resistance. Remembering Eq. (2.23) : Ptrnns a

(H/nI2 gzz(O),

(4.12)

we obtain the first row of Table I for the field dependence of transverse resistivity in the case of short-range potential. This field dependence is just the same as given by Adams and Holstein6 in the cases where scattering mechanisms are due to “point defect” and to “high-temperature acoustical phonons.” Their “low-temperature acoustical11mechanism will be discussed in Section 7. b . Short- Range Forces. Longitudinal E$ect

The field dependence of the longitudinal resistivity can be determined in a similar way. The effect of orbital quantization will appear in the relaxation time of longitudinal velocity, which is of the order of T determined TABLEI FIELDDEPENDENCE OF MAGNETORETISTANCE FOR THE CASE OF A SHORT-RANGE FORCE n Scattering mechanisms

I - +fin = const

~

Degenerate

=

const

~___~___ Nondegenerate

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

293

above (4.4), since the extension of the wave packet in the direction of magnetic field is much larger than the force range a. By making use of the usual expression for the longitudinal conductivity uzz(0)

we obtain uzz( 0 )

-

-

(ne2/m1 (uz27)/ (v,2>,

(4.13)

( n e 2 / m )( ( lpZl)/nS4?rf2fin).

(4.14)

( I ) When the chemical potential l is kept constant with reference to the ground Landau level, the number density n is proportional to 1/(27rP) a H , and thus

(I

a z Z ( 0 )0: HO

>> a, l

const).

- fiQ2/2 =

(4.15)

(2) When the number density n is kept constant and the electrons are degenerate, then (Ip,l) cc H-l and u z z ( 0 ) a H-2

(I

>> a,

degenerate, n

=

const).

(4.16)

(3) For the nondegenerate electrons we have a Z Z ( 0 )a H-l

(I

>> a,

nondegenerate,

n

=

const).

(4.17)

Thus the longitudinal resistivity Plong

=

[uzz

(0)l-l

(4.18)

is found to have the magnetic field dependence which is shown in the second row of Table I . This field dependence has also been obtained by Adams and Holstein.6

c. Long-Range Forces. Transverse Efect We shall next consider this case (B) of long-range force, in which the radius of orbit I is much shorter than the force range a (Eq. (1.2b)). I n this case, as was mentioned in Section 1, the center coordinates mayrlbe treated as classical. The cross section of a scatterer is of the order of aa2, and the collision rate is given by (4.19) 1/7 = n,?ra2 I v, I. When the wave packet encounters a scatterer, its center ( X , Y ) is moved by the potential of the scatterer according to the equation of motion

x. = -c au(x, Y , Z ), eH

aY

y=--

c eH

au(x,Y , Z ). ax

(4.20)

Here we have written X , Y in place of x, y on the right-hand sides of equations, because the 'relative coordinates may be neglected. Although U (r)

294

RYOGO KUBO, S A T O I ~ U J. M I Y A K E , AND NATSUKI HASHITSUME

comes from all of the scatterers, we assume here that the scatterers are separated from each other by a distance greater than the size of the wave packet. We see from Eq. (4.20) that the center moves in the direction perpendicular both to the magnetic field and to the force exerted by a scatterer which the electron is passing by. The displacement AX during one encounter may be estimated to be (4.21 )

where 7d stands for the duration of a collision and the double bar for the time average; T d is of the order of all v, I if the force range is still larger than the length of the wave packet in the direction of the magnetic field. Equation (3.13) may now be evaluated as

(4.22)

Thus we have the field dependence of transverse conductivity :

arr(0) a

H-’

(l<< a, p - fiQ2/2 = const);

Hp2

(I

<< a,

nondegenerate,

(I

<< a,

degenerate, n

,I

I.

n =

=

const);

(4.23)

const).

The field dependence of the transverse resistivity is given in the first row of Table 11. The scattering mechanism of “ionized impurity” in the paper by Adams and Holstein6 is considered to correspond to case (B). Our results coincide with those given by Adams and Holstein. TABLEI1

FIELDDEPENDEWE O F hfAGNETORESISTANCE FOR THE OF A SLOWLY VARYINGPOTENTIAL

CASE

n = const

Scattering mechanisms

b - +Pi2 = const Degenerate

Nondegenerate

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

295

d. Long-Range Force. Longitudinal Effect The relaxation time for the longitudinal component of the velocity is of the order of T estimated above, and therefore, from Eq. (4.13) we obtain

n

If p

- 3 K f l is kept constant, then u.,(O)

n

H and

a

(I << a, p - KQ/2

H

If n is kept constant, we estimate 1 v,

(4.24)

I

-

=

const).

( A / V L ) I ’ ~and

Ho

(I

<< a,

nondegenerate,

H

(1 <
degenerate, n

n

=

(4.25)

obtain

const);

(4.26)

(7.22(0)

=

const).

The field dependence of longitudinal resistivity is tabulated in Table 11. (Note: plong for degenerate electrons predicted by Adams and Holstein6 behaves similarly to ptrsns; only at this point does our simple estimation differ from theirs.)

5. LOGARITHMIC DIVERGENCE IN

THE

PERTURBATIONAL TREATMENT

I n the preceding section we made only an order-of-magnitude estiappearing in the expression of uzz(0) without any mation of (l/l p , explicit calculation. If, however, this factor is computed by a n honest perturbational calculation, a difficulty will immediately arise from the logarithmic divergence of the integral over a region containing p , = 0:

I)

In the argument of the logarithmic function k T is inserted because the energy E , appears in the calculation in the ratio to k T . The physical reason of this divergence is easily seen; namely, if the velocity component v, in the direction of magnetic field is vanishingly small, the electron encounters the same scatterer infinitely many times, so that the duration of collision 7 d becomes effectively infinite. This divergence was first pointed out by Davydov and P ~ m e r a n c h u k . ~ In order to look into the nature of this divergence more in detail, let us trace their theory assuming a single band, although Davydov and Pomeranchuk

296

RYOGO KUBO, SATORU J. MIYAKE, AND NATSUKI HASHITSUME

gave many-band formulas. Our starting point is Eq. (3.13).If the potential U is due to N , scatterers, i.e., Ns

V(r)

=

c u ( r - Rj),

(5.2)

j=1

the Fourier transform of U is o(q)

=

1

N s

exp ( - i q - r ) V ( r ) dr

=

V-I

V

exp (-iqRj)a(q).

(5.3)

j-1

The position vectors Rj of scatterers may be regarded as independent random variables, so that the average over the scatterers givcs ( m N * ( q ’ ) ) s = & L d ( n s / V ) I .ii(q)

I?; (5.4)

a ( q > = L e x p (-iq-r)u(r) dr,

where n, = N,/V is the number density of scatterers. In the ( N , X , p p ) representation the matrix elements of U are generally given by

( N , x, Pz

I u I N’, X’, Pz‘) =

ca ( q ) J N . N ’ ( x ,

42,

X’)~X+12*,,X’

b,-ii**,P,~,

(5.5)

4

where we have introduced after Argyres the notation

=

J * N ~ . N ( X-qS, ’, X ) .

(5.6)

The Kronecker symbols appearing in Eq. ( 5 . 5 ) mean the conservation of momentum in the y- and z-directions: AX = -12q,, Apz = fiq., which have been used in Section 4. We see in the case of long-range force, I << a, that the inequality (4.1) is more restrictive than the inequality AX 5 1 and thus AX 1 12/a.By making use of this value of A X , the reader will be able to derive the field dependence of resistivity obtained in Section 4.

I

-

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

297

According to Eq. (5.5), the square of the matrix elements becomes

(1

( N , x,

1'2

I I'

I N', X ' , p z ' ) 1%

( X , qz, X ' ) J * N , N ~ ( X , qz', X ' ) .

X

JN.N~

(5.7)

- -

If the force range is short, i.e. a << 1, the qz dependence of J N , N t is more 0 in the B's. Then rapid than that of 8,so that we may put qz 0, qz' we can use the relation1*

C J N , N( X~, qz, X ' )

=

L P N ( - X ) ' P N(~- X ' ) ,

(5.8)

92

and therefore we obtain

(1

( N , x', P

L

I l i I",

X', P z ' )

1')s

1,emembering that ( P N ( X is ) an even or odd function of X . Following Davydov arid Pomeranchuk, let US replace ns I ii l2 by an appropriate constant W , which can be expressed in terms of the scattering amplitude f :

W

=

n,[ (2?m2/m)fI2.

Then Eq. (3.13) can be written in the form

(5.10)

298

RYOGO KUBO, SATOIXJ J. MIYAKE, AND NATSUKI HASHITSUME

where El

cIx

=

-

max ( [ N

x'1' I

(PN

+ +]fin, [N' + 3]fisl), and we used the relation (x') 1'

(X>'f'A'#

X,X'

(5.12)

The integral in Eq. (5.11) converges when N # N', while it diverges when N = N'. I n the case of extremely strong magnetic fields we are left only with the term N = N' = 0, which has the logarithmically divergent integral

behaving just like the integral in (5.1). Obviously the origin of divergence is the overlap of two level densities, which in turn arise from the two delta functions in Eq. (3.10). One of the delta functions corresponds to the level density of initial states, while the other represents the energy conservation in elastic scattering. The latter appears in the transition rate, which is calculated in the Born approximation. Thus we see that there are three points which must be improved in order to avoid the logarithmic divergence: (1) the elastic scattering, (2) the use of the unperturbed level density, and (3) the Born approximation. In reality there must be corresponding physical mechanisms a t work. They may act simultaneously, but in many cases only one of them will play the dominant role. Thus we shall discuss them separately, and limit ourselves to obtain the nondivergent results in their lowest order. When we discuss, for example, the effect of inelasticity, we shall use the unperturbed level density and the Born approximation, and so on. It is worth noting that in order to cut off the divergence we have to analyse the physical process in detail, because the electrons with small longitudinal momentum I p , 1 becomes the majority as the magnetic field increases extremely, as was seen in Section 1; but the main dependence of the resistivities on the magnetic field is not altered by the detailed theories.

299

Q U A N T U M THEORY O F GALVANOMAGNETIC EFFECT

111. Inelastic Collision with Phonons

6. EFFECT OF INELASTICITY ON THE LOGARITHMIC DIVERGENCE

Let us first consider the effect of inelasticity, which brings the energy arguments in the two unperturbed level densities apart and avoids the divergence. One of important inelastic scattering mechanisms is the electron-phonon interaction, which will be discussed in this part mainly for the example of the Debye phonons. The case of optical phonons is discussed in the paper by Gurevich and Firsov.lS The piezoelectric interaction is important for some semiconductors6 and is treated in parallel to the acoustic phonon interaction. We shall only refer to the results for this case in Section 9. For inelastic scattering we cannot apply Eq. (3.6), and we must go back to Eq. (2.30), in which we make an approximation. In the lowestorder perturbational treatment, as was done in Section 3, we may replace x in Eq. (2.30) by

xo

=

xe + xs,

(6.1 )

where Se is the Hamiltonian of electrons: (6.2)

The Hamiltonian of scatterers commutes with Xe,and thus the average (2.5), which we shall denote by ( - )o, is decomposed into the succession of the grand canonical average with respect to electrons (. ) e , and the canonical average with respect to scatterers, which we shall denote by (. )s. Furthermore, the motions of electrons and of scatterers are independent of each other in our approximation. We denote the Heisenberg operators thus obtained by

-

--

-

XO(t )

=

exp ( i t ~ o / f )X i exp ( - i i t ~ o / f i ) ,

- - -.

(6.3)

Then we may use the formula (3.2) with respect to the average (. (X ( ~ ) X o ( t )o)

=

(trace ( f ( x e ) X (01 11- s(x,)1 x 0 ( t )

--

L,

)e:

(6.4)

where truce stands for the trace for the electron states and

20(t )

=

exp ( i t ~ , / f) iexp (itx,/fi ) X exp ( - i t ~ , / f)i exp (- itxS/fi ). (6.5)

The unitary operators exp (it=&) and exp (-itx,/fi)induce motion of the scatterers appearing in the interaction U in X = ( i / f i ) [ U ,X I , and thus the energy of scatterers enters into the energy conservation. If the

300

I1YOC;O KUBO, SATOItU J. MIYAKE, AND NATSUKI HASIIITSUME

t; has such a property that

interaction energy

exp ( i t ~ , / n ) Uexp (-itxc,/fi) =

c U,(r) exp (iw,t),

(6.6)

9

where g distinguishes the normal modes of scatterer system with eigenfrequencies w,, then we can carry out the time integration in Eq (2.30) as follows:

X

dt --m

x

cexp (it&/fi) in [ U , , X ] exp (iwqt) exp (-itE/fi) -

q

c6 ( E

- fiw,

-

X,)

9

;[rq.

c/ dEf(E)(l f(E

) I X s

m

=

27rfi

-

q

x

- nu,))

--m

(trace ( 6 ( E - x,) ( i / f i ) [ U X , ] 6 ( E - fiw, - x,) ( i / f i ) [ L r qXI)),. ,

(6.7) Making use of the identity

f ( E ) ( 1- S ( E -

fiwri)I = {

f(E - fiw,) - f ( m l ~ V , ,

(6.8)

where

N,

=

[exp (liw,/kT) -

(6.9)

we obtain the approximate expression of transverse conductivity

1 2a .-(trace ( 6 ( E - Ke)[X, /1]6(E 2fi

fiw, -

X e ) [ L r q , XI)),.

(6.10)

Comparing this with expression (3.10) for the case of elastic scattering, we see that the unperturbed energy E is certainly not conserved. In the

301

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

same way as was done for Eq. (3.10)) by making use of the ( N , p,, X ) representation, we can obtain an expression of the type of Einstein relation with a diffusion constant for the migration of center of orbit, i.e. an expression similar to Eq. (3.13). To see the effect of the shift in energy arguments, let us assume that the electrons are nondegenerate. Then Eq. (6.10) reduces to an equation similar to Eq. (3.10) : 2e2

XI)

)S.

(6.11)

If the interaction potential U is such that its Fourier transform changes 1/Z, by a similar procedure as what led us to Eq. (5.13) little as long as q we shall arrive a t the integral

2

( - a f / a E ) dE

($$),

exp [ (.( - h Q / 2 - fiwq/2)/kT] kT KO

-~

(6.12)

-

where &(z) is the modified Hankel function. If nu, is much smaller than the mean energy of an electron: Z? kT for modes interacting effectively with electrons, we obtain

L:Q+fi,, 2

( - a f / a E ) dE ( E - fiQ/2)’/2 ( E - nn/2 -

fiwq)”*

where In y = C is Euler’s constant, i.e. y = 1.781072. The exponential factor before the logarithmic function is absorbed into the number density (cf. Eq. 8.20’)) and the logarithmic factor is just the cutoff factor corresponding to Eq. (5.1). We can interprete the result (6.13) as follows. The logarithmic factor may be derived by cutting off the lower part of the integration range of Eq. (5.1) at a typical energy (hw,):

(6.14)

302

RYOGO KUBO, SATORU J. MIYAKE, AND NATSUKI HASHITSUME

-

We have done in Eq. (6.13) almost the same thing as this. This shows that h/Ez the duration of a collision T d cannot be given by the formula T d used in Section 4, when E , becomes smaller than the energy (fiw,). When E , is small, the electron velocity is almost zero in the direction of magnetic field, but in such a case the duration time T d is determined by the motion of the scatterers. This is most easily seen in the case of electron-acoustic phonon interaction. An acoustic phonon is moving with the velocity of sound w, so that the duration time T d cannot be longer than l/w, if we assume the extension of the wave packet of the electron being of the order of 1. Then the formula T d h / E , cannot be applied, when h / E , 2 l/w, i.e. E , 5 hw/l. On the other hand, .1/1 is the wave number of phonons, which interact most strongly with electrons, and thus fiw/l is the typical energy of an acoustic phonon:

-

-

(fiw,)

fiw/l.

(6.15)

We may also say that the energy of an electron is always fluctuating by emitting or absorbing phonons and therefore the fluctuation of energy is of the order of fiw/1, so that the energy of an electron cannot be smaller on the average than fiw/Z. In the general case of Eq. (6.14), the energy E , cannot be smaller on the average than the typical value of energy exchanged a t a collision (nu,),so that we must cut off the integration range a t ( n u g )as was done in Eq. (6.14). It will be worth noting that when the matrix elements of interaction U are not slowly varying, we may not always expect a logarithmic function as the cutoff factor. the form of which should be determined by the mechanism of scattering.

7. FORMULA FOR SCATTERING BY PHONONS Now let us examine the general results obtained in the preceding section in detail for the case of scattering by phonons. The Hamiltonian of acoustic phonons is X8 = Efiw,bqtb,, (7.1 1 9

where bqt and b, are the creation and the annihilation operators of a phonon with wave vector q. We neglect here interactions between phonons, but we simply assume the thermal equilibrium for phonons a t a temperature 5". The interaction between an electron and phonons is assumed to be of the usual form U(r)

=

c' (C(q)b, [exp (iq-r)/V'/*] 4

+ C*(q)bqt[exp ( - i q ~ r ) / V ~ / ~ ] ] , (7.2)

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

303

where the summation extends over q’s inside one half of the unit cell of the reciprocal lattice. The equations of motion (1.13) are now written as

);(

=

T’E ( “) - 9.

exp (iq-r) (C(y)b,

1‘112

- C*(q)b,

exp (-iq.r) vl/2

(7.31

and the operators (6.5) are given by

exp (-iq.r

- C*(!7)b,

+ iu,t)\

21‘J

1 exp (-itX,/fi)

(7.4)

Then, by making use of the same technique employed in the preceding section and the well-known formulas for the equilibrium average of phonori quantities (bqtbqrt)S= 0,

(bqbqOs= 0, (bqtbqJ>s= Nq6,,,J,

(bqbqtt)s= ( N ,

+

l ) & , q P ,

(7.51

we obtain the equation corresponding to Eq. (6.10) :

- 27r - [ ( N , + 1) trace (6(E- X,) exp (iq-r) n

- {I - f(xc,))6(E

- fiw,

- X,) exp (-iq-r))

+ N , trace ( 6 ( E - x,) exp ( - i q - r ) { 1 - f(Xe)) - 6 ( E + nw, - X,) exp (iq-r))]

(7.6 1

- 2-n7r [ ( f ( E - nu,) - f ( E ) )trace ( 6 ( E - X,) - exp (iq-r)G(E - nu, - X,) exp (-iq-r))

+ { f ( E ) - f(E +

nu,) 1

- exp (-iq.r)6(E +

trace ( 6 ( E - %)

fiw, -

x,) exp

(iq-r))].

(7.7)

304

RYOGO KUBO, S.4TOltU J . M I Y A I X , AXD NATSUKI HASHITSUME

The second term in the square brackets may be transformed into the same fiw,: expression as the first term by the substitution of E for E

+

- N , ( N , + 1) kT trace ( 6 ( E - 6 ( E - X,) exp ( - i q . r ) ] . fiW

- Xe)

exp (iq-r)

(7.81

fiw, -

Here we have removed the restriction on the q summation by virtue of the factor 2 thus obtained. Before proceeding further, let us note two extreme cases of high and low temperatures. (1) First let us consider scmiconductors a t high temperatures. Assuming noridegenerate electrons, we have

trace [ 6 ( E -

x,) exp

(iq.r)G(E -

fiw,

- Xe) exp (-iq-r)]. (7.9 1

-

If the temperature T is sufficiently high, and if the logarithmic divergence kT, is left out of consideration, we may neglect fiw, compared with E and have

*

trace [ 6 ( E -

x,) exp (iq-r)G(E - x,) exp (-iq-r)]. (7.10)

This shows that the interaction with a phonon of wave number q may be represented by the Fourier component C ( q ) of a static potential. This approximation can be used for acoustical phonons, if the magnetic field is weak. Adams and Holstein6 called this the “high-temperature acoustical” or the “high-temperature piezoelectric” scattering mechanism, depending on whether C ( q ) represents the deformation potential or the piezoelectric interaction. (2) Next let us consider the low-temperature limit. If we assumed the elastic scattering first and then made T -+ 0 in Eq. (7.6), we would obtain

305

CJUANI‘UM THEOI1Y OE’ GALVANOMAGNETIC EFFECT

only the term corresponding to the spontaneous emission of phonons:

- trace [ 6 ( E - X,) exp (iq-r)G(E - X,) exp (-iq-r)]. (7.11)

Adams and Holstein6 called this the “low-temperature acoustical” or “low-temperature piezoelectric” scattering. The correct result in the lowtemperature case, however, is given by taking the limit T -+ 0 in Eq. (7.8). I n contrast to Eq. (7.11), the scattering process involving a highfrequency phonon (fiw, >> k T ) is improbable owing to the presence of a factor N,(fiw,/kT). This factor was replaced by one for the emission and by zero for the absorption in deriving Eq. (7.11). As mentioned before, the contributions from the emission and the absorption are equal. At low temperatures, the absorption becomes infrequent because the number of phonons decreases, while the emission (including the spontaneous one) also becomes infrequent since the electrons rarely acquire enough energy to emit a high-frequency phonon. This situation is overlooked in deriving Eq. (7.11) on account of the approximation of elastic scattering, which is invalid in the low-temperature case. It gives the absurd result that electrons are scattered by lattice vibration even a t the absolute zero. Now let us return to Eq. ( 7 . 8 ) .In the ( N , p,, X ) representation, this equation is written as uzz(0)

2e2 1-2

= --

*

c, 2kT c c{

I

(z2qu)22?r - C ( q ) /2N,(N,

fi

fCEN(P2)

-

+ 1)

fiwsl - f C E N ( P z ) l l

N , X , P S N’ *

6 [ E l v ( p z )-

fiw, -

EN!( p z - f i ~ z ) ] I J N , N ~ qz, ( XX, - L2p,)

12,

(7.12)

where J N , . V t are defined by Eq. ( 5 . 6 ) . The matrix elements JN,.vf in Eq. (7.12) are calculated if the explicit forms of the wave functions, V N (Z - X ) , are known. I n the following we shall carry analytical calculations only for electrons with a spherical mass, for which ‘PN(5

- X)

=

exp - I J: - x j2/212) HN (2”! tF1)’/2

rq), (7.13)

306

RYOGO K U B O , SATOKU J. MIYAKE, AND N A T S U K I HASHITSUME

where

H N

J0,O

is the Nth Hermite polynominal. In particular we have

(X, q2, X

f l2qu = exp { - 1412 (qz2

+ qu2) + ihz(X

f

a h u )1 ,

therefore

I

J0,o

(X,qz, x

f Pqu)

12

=

exp { - $1'

+ 4u2) 1

(7.14)

*

I n an extremely strong magnetic field we may retain only the terms with N = N' = 0 in the sum (7.12); namely =

Uzz(O)

-2e2

V

*

1

I

-~ dq (z2q')2 2T - C ( q ) ( 2 N,

( 2 ~ 2kT ) ~ h

(N,

+ 1)

sCEo(pz) - fi% - E o b z - fiqz)I I JO,"(X, 42, x - Z2q,) 12, (7.15)

where sCEo(p2) - nu, - Eo(Pz - h ) ] = (dfi I q z 1)6CP=- 3 f i q z - ( ? W l / q z ) l (7.16) because we have E o ( p z ) = ifin (pz2/2m)

+

for electrons with a spherical mass. The result (7.15) or its generalization for weaker magnetic field could be derived, as was done by Gurevich and

Firsov,15with the aid of the analytical expression of the free-electron Green function in a magnetic field. This method uses the explicit expression of the density matrix or the propagator of nondegenerate electrons in a magnetic field as derived previously by Sondheimer and Wilson.30Since this is simply a matter of mathematical technique, we shall not enter into the details of such a calculation. Inserting these expressions into Eq. (7.15), and introducing the notation

K

=

f{+fiQ

+ (2m)-1[3fiqz + (mu,/q2)]2 - nu,) - f(3fiQ

+ (2m)-"3fiqz + (mu,/qz)l"1,

(7.17)

we can write Eq. (7.15) explicitly:

X exp { -312(q22 30

+ qu2)}R, (7.18)

E. H. Sondhejmcr and A . H. Wilson, Proc. Roy. SOC.A210, 173 (1951).

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

307

where the g integration should be taken over the unit cell of reciprocal lattice. If the unit cell may be replaced by a sphere q 5 qmax, we get

+ 1)R,

X exp (-Z2p12/2)Nq(Nq

where q l = (qz2 ing sections.

(7.10)

+ qy2)1/2.We shall carry out the integration in the follow-

8. ACOUSTIC PHONONS I n the case of acoustic phonons, we shall assume, as usual, that an electron interacts only with longitudinal phonons. The potential U (r) is then V(r) = D div u, (8.11

u being the displacement vector of a lattice point at r. We shall replace the unit cell of reciprocal lattice by a sphere q 5 qmax = k e / ( f i w ) of the Debye model. Then the Fourier component of the potential C(p) is given by C (a) = D ( f i / 2 ~ o ~ ) ’ / ~ i q , (8.21 and the dispersion law by oq =

(8.3 1

wq.

Here D is the coupling constant, p the density of crystal, and 8 the Debye temperature. Inserting these expressions ( 8 . 2 ) and (8.3) into Eq. (7.19), we obtain20

+

X exp (-Pql?/2)Nq(NqI ) R , (8.4)

and, from Eq. (7.17),

(8.5)

Before carrying out the integration of Eq. (8.4),we notice here that, if we put { = 5 Q / 2 in our expressions (8.4) and (8.5), we just have the equation (35) in Titeica’s paper2 and his K respectively. Titeica has assumed { fiQ/2 by saying that all the electrons are in the energy level N = 0. This assumption will be discussed later.

-

308

RYOGO KUBO, SATOltU J. MIYAKE, AND N A T S U K I HASHITSUME

For the purpose of making an approximation in the integral of Eq. (8.4), let us introduce new dimensionless variables

after Titeica,2 and use them in place of q~ and q p : q1 = q[ (t - l)/t]'/Z,

qz

=

(8.7)

q/t"2.

Then Eq. (8.4)is written as

where

'1c =

(

(ZmkT)--lJ? mwq q, -

;')

-

=

(""")'" (t 2k Tt

-

2mw2

'

(8.9)

The integral over E and t in Eq. (8.8) is complicated, but under certain conditions simplifying approximations can be used. Let us first examine the conditions imposed upon transitions by the conservation of energy and momentum. If the kinetic energy E , of the motion in the field direction is larger than the energy fiw, of a phonon involved in scattering, the change of E , is negligible and the scattering can be specified by whether the sign of the longitudinal velocity is changed or not. The momentum change in the longitudinal direction is about fiw,/u, in the case of forward scattering, while it is about, 2mu, in the case of backward scattering. If 2mv,w is larger than k T , the backward scattering scarcely occurs, since those phonons are few whose energy is larger than kT. On the other hand, if 2mv,w is much smaller than k T , the backward scattering occurs as often as the forward

QUANTUM THEOItY OF GALVANOMAGNETIC EFFECT

309

scattering. I n the last case (i.e. 2wiv,u1 << leT), the wave number vectors of most phonons involved in scattering are directed almost perpendicular to the z-direction. This is true because the perpendicular component of the phonons, q l , is of the order of kT/Rw or 1/1, depending on whether kT/fiw is smaller or larger than 1/1, since q l is restricted by Eq. (7.14) and by the condition q l 5 q < kT/fiw. If kT/Rw >> 1/1, q l (-l/Z) is larger than qz(-2mv,/R), because in the quantum limit Rs2 = R2/m12>> mv,2/2 is assumed. If 1/1 >> kT/Rw, ql(-kT/hw) is larger than qa(-2mvZ/R), because kT >> 2mv,w is assumed. Thus wheii E , >> nu, and 2mv,w << k T , t = q2/qz2>> 1 holds arid the factor ( t - l ) / t in Eq. (8.8) can be approximated by unity. When E , is comparable to Rwq, the momentum change in the z-direc/ ~ Rw, . >> mw2, Rql is larger than tion is of the order of ( 2 m f i ~ , ) l If Rq,[

-

(2mh~,)'/~]

and ' ( t - l ) / t can be approximated by unity. Making use of this approximation, and transforming the integration variable from t to u,we obtain for the integral in Eq. (8.8)

du

([exp ( u 2 - z )

+ 11-'

-

[exp (u2- z

+ () + 11-1) (8.10)

with u1 = (mw2/2kT)l/2- 1 ( k T ( / 2 m ~ ~ ) ' / ~ .

(A) When kT >> Rw/l (i.e. (kT)2/(mw2R0) >> 1) is assumed, most phonons have energy XU, 5 Rw/l (mw2fi0)"2. If the conditions E , >> Rw/l and 2mv,w << kT are satisfied, the approximation leading to Eq. (8.10) can be used and the lower limit of the u integration u1 can be replaced by - 0 0 . For, the contribution to the u integration comes mostly from the region u2 < E,/kT, while an estimation of u1 for the representative value of ( ( - R w / l k T ) gives u1 ( m ~ ~ / 2 k TL ) ' / (~R 0 / 4 m ~ 2 ) "which ~ , is smaller than - (E,/kT)'12by virtue of the assumed inequalities RQ >> kT >> mw2 and ( R C ~ / ~ W >~>) E,/k ~ ' ~ T . Since the representative value of [ (-fiw/Zk T )

-

-

+

310

RYOGO KUBO, S A T O I ~ UJ. MIYAKE, AND NATSUKI HASHITSUME

is smaller than one, the factors in Eq. (8.10) can be approximated by

- 5-2

[ (& - 1 ) (1 - e-E)]-' and [exp (u' - z )

+ I]-'

-

[exp (u' - z

+ 5 ) + 11-'

(8.11)

Thus the integral (8.10) is approximated by

de

[exp

(E

- z)

+

13-I

[l

+ exp ( z -

e)]-',

(8.12)

where the variable e = u2 corresponds to E,/lcT. (a) When electrons are degenerate (i.e. z >> l ) , the last factor in the integral (8.12) can be approximated as usual by 6 (e - z ) . Finally we obtain Uzz(O)

=

e2 D2m3(fiQ)3 kT 2 __ ma22 ( 2 n ) 3 f i 7 pT~ z- fifl/2

(8.13)

(8.13')

where we have introduced the number density of electrons

no = 2[ (2mTo)3/2/6n2h3] and the relaxation time

71

(8.14)

a t the Fermi level: (8.15)

both in the absence of magnetic field. The number density n in the presence of magnetic field is related to no by (8.16)

in the present approximation. If the number of electrons is kept constant, i.e. if n = no, Eq. (8.16) gives { -

ifin

=

{0(2r0/3fifi)~,

(8.17)

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

31 1

whence the expression (8.13’) becomes a,,(O)

=

-

(ne2/mQ2rr)

(fiQ/23-0)5.

(8.18)

This corresponds to the previous expression (4.11). In the present case, the conditions E , >> fiw/l and 2mv,w << k T become j- - fiQ/2 >> fiw/l

and

<< ( k T ) 2 . (8.19) fiQ/2 >> k T >> fiw/l is

8mw2(<- fiQ/2)

These conditions are satisfied, since fiQ >> l assumed. (b) When electrons are nondegenerate (i.e. - 2 >> l ) , the last factor in the integral (8.12) can be approximated by ez-f. Integrating over e, we obtain for (8.12)

where K O(z) is the modified Bessel function whose approximate behavior for x: << 1 is given by &(z)

-

(y = 1.781

In (2/yz)

- - .).

Using this approximation, finally we obtain e2 D2m3( f i Q ) 3 azz(0) = 2 exp ma22 (2~)~fi7pw2

4 (< -k ThQ/2) (-(2ey)I/2 ”‘T) fiw/l 111

4

-

kT

(8.20)

(8.20’)

where the electron number is given by

n

=

2[ (2m)3/2/4~3/?fi3]3fiQ(kT)1/2 exp [ (< - i f i Q ) / k T ]

=

no(fiQ/kT)exp [ (l - fiQ/2 - l o ) / k T ]

(8.21)

and the relaxation time of an electron with energy equal to k T in zero magnetic field is given by 7i-l

=

ID2(mkT)3/2/v2afi4pw2].

(8.22)

The expression (8.20’) corresponds to the previous one (4.7), and agrees with the result given by Gurevich and Firsov15 by the method of Green’s function. We see that the cutoff factor (6.13) or ((3.14) with cutoff energy (0.15) appears (the factor 4/(2e7)1/2 = 1.28539 may be omitted by virtue of the assumption k T >> fiw/l). It is obvious that the conditions E , >> fiw/l

312

RYOGO KUBO, SATOHU J. MIYAKE, AND NATSUKI HASHITSUME

and 2mv,w << k T are satisfied, since the conditions are equivalent to k T >> fiw/l and k T >> 8mw2, respectively. (B) When k T << fiw/l (i.e. ( k T ) z / ( m w z f i Q < )< 1) is assumed, those phonons are effective in scattering for which fiw, 5 k T . If the conditions E , >> k T and 2mv,w << k T are satisfied, the approximation leading to Eq. (8.10) can be used and u1 can be set to - co . An estimation of u1 for the representative value of f (-1) gives u1 (mw2/2kT)lI2- ( k T / 2 m ~ ~ ) ' / ~ , which is smaller than - (E,/kT)I/z since inequalities k T >> mw2 and lcT/8mw2 >> E,/kT are assumed. The condition E , >> k T holds when electrons are degenerate. (a) Degenerate case ({ - fin/2 >> k T ) . The integral over u in Eq. (8.10) becomes

-

for z

>> f . Putting this into Eq.

+

(8.10), we obtain

e2 D 2 m ( k T ) 5 fin mQ24 (2~)3fi7pw6{ - fiQ/2

azz(0) = 2 -

(8.23)

(8.23')

where we have introduced the functioii (8.24)

313

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

and the relaxation time T € which appears in the expression for the conductivity at low temperatures in zero magnetic field Tf-l

=

D 2 ( k T ) 5 $5[2w(2m{o)1/2/kT] 1 6 (2m)112fi4p~6 ~ 5.0~~~

(8.25)

1

no being given by Eq. (8.14) and C0 by Eq. (8.16).In the low-temperature case under consideration, e / T is usually much larger than one. The integrals $ , , ( e / T ) are approximated by = r(n l){(n). If the number of electrons is kept constant, we obtain

+

by making use of Eq. (8.17).The condition 2mvZw<< k T becomes in this case 8mwz(!: - hQ2/2)<< ( ~ c T ) ~ . (8.27) The condition holds only when f - RL?2/2<< RQ, i.e. in the extreme quantum limit. On the contrary, when { - fiL?/2 5 RQ, the condition 8mw2({ -- RQ/2) >> ( ~ C T ) ~

(8.28)

will be satisfied instead of the condition (8.27).I n this case, the backward scattering is very weak compared with forward scattering and correspondingly the lower limit of the u integration u1 should be set equal to zero. The value of the conductivity becomes one half of that given by Eq. (8.23) or (8.23'). (b) Nondegenerate case ({ - RO/2 << k T ) . When electrons are nondegenerate, the conditions E , Ru, and &aq>> 2mw2 hold for most phonons if k T >> mw2 is satisfied. The integral (8.10) can be approximated by

-

Since the important contribution to this integral comes from [ asymptotic expansion e"2Ko ( ( / 2 ) (T/()'/~

-

2

2, the

314

RYOGO KUBO, SATOIlU J. MIYAKE, AND NATSUKI HASHITSUME

may be used to evaluate the integral. Finally we obtain Uzz(O)

=

e2 D2m(kT)4AD p - AQ/2 2mil2 4 ( 2 ~ ) ~ f i ~ p w ~

(8.20)

rf-l =

[D2( m k T ) 3 / 2 / a ~ h 4 p ~ 2 ]

(8.30)

and the integral (8.31) When O / T

>> 1 , we can use the approximation g n ( 8 / T )E S n ( a ) = r ( n + 1){(n

+ I).

Titeica calculated the transverse conductivity under the assumptions [ - hQ/2

-

0

and

hQ/lcT >> IcT/mw2 >> 1 ,

which correspond nearly to those of the case (B.b). Though the approximation used by Titeica to evaluate the integral (8.10)seems questionable, the result obtained by him agrees qualitatively with Eq. (8.29). These calculations can be extended to electrons with an ellipsoidal mass. Here we give only the results for each of the cases treated above.20Let

E o ( x )= ( n . a . x / 2 m o ) be the electron energy for the quasi-momentum

(A') m,

x.

(8.32) Then we have

m,w2fiD<< ( k T ) * ( m , = nio/(azza,, -

(a) Degenerate case (KQnz,/m, mo(a-l).z)

>> p

- fiQ/2

Q

>>

kT

=

>>

eH/m,c) :

(mcw2fiQ)112,

=

(8.33)

315

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

>> k T >>

(b) Nondegenerate case [fiQm,/m,

(mcw2fiQ)11?]

f - fiQ/2 cyzze2 D2m,2m,(fiQ)3 moQ22 ( 2 ~ ) ~ f i ' p w ~ kT

(

uzz(0) = 2 -

(B')

) x In (- 4 (2ey

kT

). (8.34)

)l l 2 (mcw2fiQ )' I 2

mcw2fiQ>> ( k T ) 2 :

>> k T >> [rn,w2(f - fiQ/2)]'/21 (a) Degenerate case { (mcw2fiQ)1'2 (When (m,w2fiQ)112 >> k T and [m,w2 (5 - fiQ/2)]'/2>> k T , the result below should be divided by 2 ) :

(8.35) (b) Nondegenerate case [ (mcw2fiQ)1/2 >> k T

>> m,w2]

(8.36) Before concluding this section, we refer to the longitudinal component of the conductivity tensor azz( 0 ): we shall not give the derivations here. Corresponding t o the case ( A a ) , we have uzZ(O)=

-

(noe2n/m) 3C(f -

3fiQ)/fiQl,

(8.37)

where we have used the relaxation time (8.15) and the number density (8.14), or if n = no, u Z Z ( 0 )= (ne%r/rn)

- g(2{o/fiQ)3,

(8.38)

and in the case (A.b),

-

a z z ( 0 )= (4ne2~r/3t';m) 3 ( k T / f i Q ) .

(8.39)

where the number density is given by Eq. (8.21) and the relaxation time by Eq. (8.22).

316

RYOGO KUBO, SATORU J . MIYAKE, AND NATSUKI

HASHITSUME

9. OPTICALP H O N ~AND N S PIEZOELECTRIC INTERACTIONS Gurevich and Firsov16 discussed the case where optical phonons dominantly scatter electrons in a polar crystal. The Fourier component of the phonon-electron interaction will be then of the form

I C ( q ) l2

=

Q/q2,

(9.1)

and the dispersion law may be approximated by const

wq =

= wa.

Assuming the inequality fin >> k T and fiwo expression (for nondegenerate electrons) u z z ( 0 ) = (ne2/mD2n)

-

>> k T , they have derived the (fiD/2kT)

(9.2)

for the case Q

The relaxation time 7f-1

7f

=

>> wo.

is given by (Q/2?m2)( 2 m / f i ( ~ ~exp ) ” ~ (-ficc0/kT).

(9.3 1

The result (9.2) differs from that given by Adams and Holstein6 only by the logarithmic factor. For the case D

<< wo

they also pointed out the possibility of a new oscillatory phenomenon: the conductivity oscillates periodically in 1/H, and the peaks appear when D = w0/M, M = 1, 2, -. This phenomenon was discussed by Efros3I and Gurevich et ~ 1 . ~ ~ I n a crystal with low symmetry, the piezoelectric interaction may become important.33It is the interaction between acoustic phonons and electrons through the polarization induced by the piezoelectricity of the crystal. For this interaction, the interaction constant in Eq. (7.2) may be assumed, as the simplest model, as

--

1 C(q)

12

=

PZ(fi/2PW)

(9.4)

(where P is a coupling constant). Calculation can be carried out in parallel with the preceding calculations. Electrons are assumed to have a spherical energy surface. A . L. Efros, Fiz. Tverd. Tela 3, 2848 (1961); see Soviet Phys.-Solid State (English Transl.) 3, 2079 (1962). 32 V. L. Gurevich, P u. A. Firsov, and A. L. Efros, Fiz. Tverd. Tela 4, 1813 (1962); see Soviet Phys.-Solid State (English Transl.) 4, 1331 (1963). 33 J. G . Meijer and D. Polder, Physica 19, 255 (1953).

QtJANTUM THEOltY OF GALVANOMAGNETIC EFFECT

317

(A) Piezoelectric interaction, high temperature (fiw/Z << k T ) (a) Degenerate case (fiQ >> { - fiQ/2 a,,(O)

=

>> k T >> fiw/Z)

e2 P2m2(fiQ)2 k T 2mQ24 (2n)3fi5pW2 { - f i ~ / '2

(b) Koiidegenerate case (fiQ >> k T

(9.5)

>> fiw/Z)

(B) Piezoelectric interaction, low temperature (fiw/Z >> L T ) : (a) Degenerate case { ( n ~ w ~ f i Q )>> ' / ~k T >> [mw2({ - fiQ/2)]1/2) [When (mw2fiQ)1/2 >> k T and [mw2({ - f i Q / 2 ) ] * / 2>> k T , the result below should be divided by 2. ] U,.(O)

=

e2 P 2 m ( k T ) 3 fiQ 2mQ24 (2n)Vi5pw4{ - fiQ/2

(b) Nondegenerate case [ (mw2fiQ)112 >> k T az,(0)

=

e2 P2m ( l ~ T ) ~ f i Q 2mQ2 4 (2n)Vi5pw4

>> mw21

(9.7)

fiQ2/2 kT

{ -

)

IV. Collision Broadening

10. EFFECT OF COLLISION BROADENING ON THE LOGARITHMIC DIVERGENCE

As we have seen in Section 5 , the lowest-order perturbation fails to apply to elastically scattered electrons. Here we treat an effect of multiple scattering which in the presence of scatterers with higher concentration brings about the broadening of energy levels and gives finite answers to the conductivity problem. Electrons with small momenta in the z-direction (along the magnetic field) are represented by wave packets elongated in this direction which will interact with two or more scatterers simultaneously. We shall not, however, consider interference effects, which will not be important in the first approximation as far as the scatterer distribu-

318

HYOGO KUBO, SATOIZU J. MIYAKE, AND NATSUKI HASHITSUME

tiori is random. The effect to be considered may be explained in a classical picture: an electron with small momentum p , scattered by a scatterer is scattered by another scatterer before coming back to the first scatterer. This means, in the quantum-mechanical picture, that the electronic states have only a finite lifetime. If the time interval between successive collisions becomes smaller than the duration time T d of a collision, T d loses its meaning. I n other words the estimation of T d by Td

-

fi/Ez

(10.1)

does not hold and the scattering cross sections stay effectively finite. Roughly speaking, this is equivalent to cutting off the divergent integral at E , 5 n/T r which will yield

J',

-

kT dEz ln-. (Ez)1/2(Ez)1/2 r

(10.2)

More exactly, we have to replace the Unperturbed level density l / ( E z ) 1 / 2 by a broadened one instead of cutting off the lower part of the integration region as in Eq. (10.2). For example, in the expression of unperturbed level density (4.8) :

we replace the delta function by the Lorentzian function with a constant level-width r and a constant energy-shift A, and obtain

2 (m)1'2 -~ (2TZ)'fi

x

E( h'

-

(N

N=O

+ +)hQ- A + [ E - ( N + $)fin- A ) ' + {E- (N+;)~~Q-A)~+P (10.3)

For the ground Landau level N = 0, the unperturbed level density 1/(Ez)'" is thus replaced by the new density (cf. Fig. 4) {[E>

+ (E,Z +

r2)1/21/2

(E,'

+

rz))1/2.

319

QUANTUM THEOltY O F GALVANOMAGNETIC EFFECT

-

The new density coincides with the old unperturbed one for E , 2 r, does not diverge a t E , +0, and tails off in proportion to I E , 1- 3/2 as E , + - a. If this (10.3) is inserted into the integral (5.13) with the Fermi function j ( E ) to obtain

E

- fiQ/2 - A

+ [ ( E - fiQ/2 - A ) 2 + + r2)

21 ( E - fiQ/2 - A ) 2

where we have introduced the notations z = ({ - fist/2 - A ) / ( k T ) , E = ( E - fiQ/2 - A ) / ( k T ) . If the breadth I' is very small compared with k T , the integral (10.4) can be evaluated as

Irn --m

cosh2 {

I

dE (E

- ~)/2)

-

8e*ln

r5)

(10.5)

in accordaiice with Eq. (10.2). Davydov and Pomeranchuk3 have estimated the breadth r fi/r in an extremely strong magnetic field by making use of an ingenious argument, which we shall reformulate in a more precise form.34After Davydov and Pomeranchuk, let us consider the case of short-ranged scattering potential. I n this case the lifetime T has already been given in Eq. (4.4). Or, if we use the mean free time 71 in the absence of the magnetic field

-

7f-l

we obtain T-'

-

-

TL~~T~~(~{~/?TA)'/*,

(EL)-'/'.

[fiQ/~,

(10.6) (10.7)

According to the preceding discussions, T d given by Eq. (10.1) becomes of the order of T given above when E , approaches the order of r, so that

r/fi r

- [ns2/Tf ({,)l/q [( f i / T f )

(r)-1/2,

(fi~/{~1/2)]2/3.

(10.8) (10.9)

The power of f i / q with index Q is characteristic of the line breadth estimated in the Davydov-Pomeranchuk theory. We shall return to their theory in Section 12. I

S. J. Miyake, Ph.D. Thesis, University of Tokyo, 1962.

320

RYOGO KUBO, SATOltU J . MIYAKE, A N D NATSUKI HASHITSUME

11. DAMPING THEORETICAL FORMULATION' When the scattering is elastic, the diagonal components of the conductivity tensor can be expressed exactly in the form given in Eqs. (3.6), (3.7), and (3.8). It is convenient now to introduce the resolvent operator for the Hamiltonian X defined by

R ( s ) = (X - s)-', (11.1) where s is a complex variable. The delta function appearing in the conductivity formulas can be expressed as 6(E

- X)

=

rlla

lim v-fO

(X

- El2

+ t2

= '

lim

R(E

+ iq) - R ( E - iq) 27ri

7-i-0

(11.2) Thus we can evaluate the conductivity components by any useful approximation for the resolvent R ( s ) . The matrix element of R ( E f i0) in the r representation G*(r, r') = lirn (r I R ( E =F iq) I r') (11.3) ?++o

is the Green function for the Schrodinger equation. Now we make the damping theoretical expansion of the resolvent R (s) after van Hove3': (11.4) R(s) = D(S) - {D(s)UD(S)}nd where {

---

+ ...)

}nd

stands for the nondiagonal part and

+

(11.5) D ( s ) = [X, G(s) - s1-l is the diagonal part of R (s) in the representation, in which X, is diagonal. In Eq. (11.5), the operator G(s) is diagonal and is determined by solving the equation G(s)

=

{ v ) d - { uD(S)Cu -

--

G(s)l}d

+ (?m(S)cu- G ( s ) ] D ( s ) [ U -

G(S)]}d

-

*-*,

(11.6)

where { * } d stands for the diagonal part in the representation mentioned above. From Eqs. (11.3) and (11.5), we find that the diagonal part of the operator 6 ( E - X ) appearing in our expressions of conductivity tensor

S(E)

=

{ 6 ( E- X ) ) d

(11.7)

is determined in the form (11.8) 36

L. van Hove, Physica 21, 901 (1955); 22, 343 (1956); N. M. Hugenholts, Phyaica 23, 481 (1957).

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

321

where we have defined the diagonal operators by lim G ( E f iv)

=

A ( E ) =F i r ( E ) .

(11.9)

P+o

We can easily show that the eigenvalues of r ( E ) are all n~n-negative.~~ We see that the exact level density (3.9) can be written as

and, by comparing this expression with Eq. (10.3), that the eigenvalues of the operators I' ( E ) and A ( E ) would just give the line breadth and the energy shift of the eigenstate under consideration respectively, if they were constant. In reality they cannot be constant, especially for the true of the Hamiltonian ground state. The energy EG of the ground state \ k ~ X is determined as the lower bound of E that makes p

( E ) = 0,

or all the eigenvalues of

r ( E ) vanish,

(11.11)

because there are no eigenstates below EG. The energy dependence of the eigenvalues of r ( E ) and A ( E ) makes the shape of broadened line differ from the Lorentzian shape as assumed in Eq. (10.3), and thus will result generally in a cutoff factor different from the logarithmic function. It is in general very difficult to solve Eq. (11.6), and we have to be satisfied with finding the first or the second approximation. In the following we shall thus confine ourselves to the lowest approximation, in which we retain only the first two terms in the expansion (11.6) and only the first term in the expansion (11.4), viz. we approximate R ( s ) by D ( s ) , whence 6 ( E - X ) by its diagonal part S ( E ) given by Eq. (11.8). Eqs. (3.6), (3.7), and (3.8) are then approximated by

uyy(0) =

*/" v

dE(-g)

-/v

dE

(trace ( S ( E ) Y S ( E ) Y ) ) s .

(11.13)

(-z(trace ) (S(E)V,S(E)U,)),,

(11.14)

-m

u z z ( 0 )= d i e 2

O0

8.f

-a?

and the diagonal operator G (s) is determined by

322

RYOGO KUBO, SATORU J.

MIYAKE,AND

NATSUKI HASHITSUME

where we have put

{U),= 0

(11.16)

without loss of generality by shifting the origin of energy. If the scatterers are distributed a t random, the average represented by ( - - . ) . in Eqs. (11.12), (11.13), and (11.14) should be taken over the distribution of scatterers. We make further an approximation uzz(0)=

a,,(O) =

e/mJ’

dE -m

(-g)

(trace ((S(E)),X(S(E)),X)),, (11.17)

%Irn v “““Irn (-2) dE(-$)

(trace ( ( A ‘ S ( E ) ) ~ Y ( S ( E ) ) ~ Y(11.18) ))~,

dE

(trace ( ( S ( E ) ) , V , ( S ( E ) ) , ~ , ) ) , ,(11.19)

-m

u z z ( 0 )=

J’

-m

determining (S( E ))s by

In this approximation, as will be seen in the next section, the scattering of an electron by each scatterer is treated independently, and each scattering process in the lowest Born approximation. We have taken into account only the effect that the coherent amplitude of wave function of an electron decreases during the revolution of orbit on account of the interaction with scatterers.

12. DAVPDOV-POMERANCHUK THEORY Davydov and Pomeranchuk3 have discussed the conductivity of a bismuth single crystal at low temperatures in a strong magnetic field, considering three groups of electrons and three groups of holes with ellipsoidal energy surfaces. The scattering mechanism considered by them is due to the lattice distortion induced by randomly distributed atoms of admixtures, and the range of scattering potential is assumed to be short compared with the de Broglie wavelength of electron. Their theory is based on both the kinetic equation and the center-migration picture, and gives the logarithmically divergent conductivity described in Section 5. They cut off the divergence by the argument given in Section 10. In this section we shall reconstruct the Davydov-Pomeranchuk theory in view of the damping theory formulated in the preceding section. For

QUANTUM THEOHY OF GALVANOMAGNETIC EFFECT

323

the sake of simplicity, let us consider the case of single band with spherical mass. We assume the inequalities (I.lc), (1.2a), and (12.1)

and finally nS-’13

>> 1.

(12.2)

The last inequality meam that the diameter of a wave packet in a plane perpendicular to the magnetic field is much smaller than the mean distance between scatterers. This inequality will be satisfied when the magnetic field is sufficiently strong. The condition (12.1) is satisfied when the temperature is not very low: higher than about 1°K for bismuth. First let us solve Eq. (11.21), which can be written as

in the ( N , p,, X ) representation given in Section 1. I n order to solve this equation we have to determine the matrix elements of U more precisely than that given in Eqs. (5.9). For this sake, we shall approximate the potential from one scatterer u(r - Rj)in Eq. (5.2) by a delta function potential u (r) = (27rfiz/m) fS (r), (12.4) where f stands as before for the scattering amplitude for electrons with a long wavelength in the absence of magnetic field. Then taking the random distribution of scatterers into account, we obtain

(I ( N , x,pz I u I N ’ , X’, Pz’ ) I * =

“v L2

/m

)a

dX“ 1 ‘PN(X”)‘P”(X’’

+ x - X ’ ) P,

(12.5)

W being given in Eq. (5.10). The integration with respect to X”/L is just the averaging procedure over the position of scatterers. Since the matrix elements (12.5) depend only on X - X’, N , and N‘, Eq. (12.3) has a solution ( G N , x ,(s) ~ . )s that depends only on N . Carrying out the summations with respect to X’ and p,’ in Eq. (12.3), and taking the limit s 3 E f 20, we obtain (AN

( E ))a f i ( r N ( E ))a

(12.6)

:324

HYOGO KUBO, SATOHU J. MIYAKE, A N D NATSUKI HASHITSUME

where we should take the branch with the positive imaginary part for the root. It is still difficult to solve these simultaneous equations for ( A N ( E ) ) ,f i ( r N ( m ) s . In this section we shall thus be satisfied only with finding such a solution that gives the order of magnitude. More rigorous solutions will be determined in the next section for a more special type of potential, since in reality the delta-function potential cannot be used for the exact discussion be. we omit the summation and cause of the divergence of ( A N ( E ) ) s First retain only the term N‘ = N . This procedure will be justified for the lower Landau levels, when the magnetic field is sufficiently strong, so that the transitions between different widely separated Landau levels may be neg+)fiQ - E equal to zero on the right-hand side lected. Next we put ( N of Eq. (12.6). This procedure means that we are seeking for the values of ( A N( E ))s and (Jh( E ))B near the peak of level density corresponding to the N t h Landau level. In reality, we cannot neglect the energy depend~ ( r N ( E ) ) s a t the tails of level density. Then, by ence of ( A N ( E ) )and making use of the relaxation time given by

+

7f-1 =

w

(m3/2/7rfi4)

(2{0)’/2,

(12.7)

we obtain AN)^ f i

( h ) s =

- ( f i / T f ) [ ~ f i n / ( { ~ ) ” ~ ]AN)^ ( fi(rN)s)-’”,

(12.8)

which has the solution

We see that Eq. (12.8) corresponds to Eq. (10.8) and the solution (12.9) to Eq. (10.9). Now we shall evaluate the conductivity component a Z Z ( 0 )in the approximation (1 1.17). The matrix elements of X can easily be determined as

I ( N , x,pz I x I”, =

X’?

PZ’)

l2

( c / e H ) z I ( N , X , p , I au/ay I N’, X’, pZ’>l2

(12.10)

325

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

Inserting this element into Eq. (11.17) written in the ( N , p,, X ) representation, we obtain

[:

(SY W L (1E (-%) c c (El c ( E )

-

I

(2*13 1 2

{

NJJ'

(SN,,,

(SN!,Y,'

)sl{

PZ

PZ

'

)sl

(N

+ b + " + 3), (12.11)

where we have used Eq. (5.12). If we use the constant values (12.9) for ( A N ( E ) )and ~ (I'hi(E))s,the level densities appearing in Eq. (12.11) is just the one in Eq. (10.3) :

E - (N+$)fin- AN)^+[{ E - (N+a)fifi- AN)^}'+{ { E - (iv+;)fia- ( A ~ ) J Z + {( r N w

>.

( ~ N ) ~ I"' ~I"' (12.12)

Since there is no divergence trouble in the terms N # N', AN)^ and ( r N )may s be neglected there, then we have

[ g:[ E

- (Ar

(N + " + 1)nn + 4)filS2]'/' [ E - (N' + $)fiQ]1/2

326

RYOGO KUBO, SATORU J. MIYAKE, A N D NATSUKI HASHITSUME

where the prime on the first summation symbol stands for the sum to be taken over all pairs ( N , N ' ) such that N # N' and E 2 ( N +)fin, E 2 (N' +)fiQ. Equation (12.13) gives the corrected form of Eq. (5.11). I n the case of an extremely strong magnetic field, we may retain only the term with N = 0 in Eq. (12.13), and by making use of Eqs. (10.4) and (10.5) we obtain the final result for nondegenerate electrons

+

+

or

(12.15)

This result is the one that will be obtained by following the idea of Davydov and Pomeranchuk. A constant breadth and a constant shift such as given in Eq. (12.9) cause difficulties, although they are often assumed in simple theories.36 For example, if we would use Eq. (12.13) instead of Eq. (12.11) to analyse the de HaasShubnikov oscillation, the sum in Poisson's summation formula would diverge. The origin of this divergence is in that the level density (12.12) for each level N has the long tail in the lower energy side. In reality, the level density has to vanish at a certain energy higher than the ground level given by Eq. (1 1.11). An example will be given in the next section. 13. SCATTERING BY GAUSSIANPOTENTIAL: SHORT-RANGE FORCE

I n order to carry out explicit calculation of the damping theory, we assume now a Gaussian potential for the scatterers, which is given by u (r) = [2ah2f/m (27ra2)3 / 2 1 exp ( -r2/2a2),

(13.1)

where f denotes the true scattering amplitude for an electron with zero energy as before, and a gives a measure of force range. We have used the true scattering amplitude f, so that the results obtained by using this potential can have a real meaning in the case (A) (Part I) of shortrange force, because in case (A) the detailed shape of the potential is immaterial. 36

R.B. Dingle, Proc. Roy. SOC.A211, 517 (1952).

QUANTUM THEORY OF GALVANOMAGKETIC EFFECT

327

The matrix elements of the potential U (5.5) are now evaluated as follows. The Fourier transform (5.4) of our potential (13.1) is given as N q ) = (2rfi2f/m) exp

c-

(a2/2)q21,

(13.2)

and the function (5.6) becomes

- X' . jx -I 6 sgn (N' - N ) + i1 6 ( I 3.3) where N 1 = max ( N , N ' ) and N z = min ( N , N ' ) ; sgn (s) stands for the sign function, and L,") (2) for the associated Laguerre polynomials:

L,Cr)(x)

=

(e'z-+/n !) (dn/dxn) (e--zxn+r).

(13.4)

By making use of Eq. (1 3.3), and remembering the first equation of (5.4), we obtain for Eq. (11.21)

It can easily be seen that ( G N ~ , ~f ( Eiq) )& has a solution which is independent of X. When ( E - EG) satisfies the condition ( E - EG)<< fiQ << fi2/2ma2 ( EGbeing the lowest bound to the energy spectrum and the second E ill))% inequality coming from our assumption 1 >> a ) ,the solution ( G N ~ . (f does not depend appreciably upon p,, so long as I p , I << fi/a. Since such , f iq) ). can be values of p , are important, the p , dependence of ( G N ~( E ignored and (GNp,( E f iq) )s is simply written as ( G N ( Nf ir]) ).. Summing over q, we have

( G N ( Ef

)I

328

RYOGO KUBO, SATOHU J. MIYAKE, AND NATHUKI HASHITHUME

where

f(N, N ’ )

=

& // (:.

dq, dq, I

JN,”

j(0, N’)

=

and K”

=

[(2m)”2/fi]{ (N’

( X , Q, X 1

+ 12q,) l2 exp (-a2qL2),

+ 2(u/1)2)-(”+”

1,

(13.7)

+ +)fin + ( G N , ( Ef i71)>. -

( E f ill))1’2

(Re KN’

> 0).

(13.8)

The relaxation time rf in Eq. (13.6) is given by Eq. (12.7). If the force range a is set equal to zero, i.e. the potential is of the delta) (uK“) is equal to one function type, (2/tG)f(N, N’) exp ( U ? K ? N <erfc and the real part of the right-hand side of Eq. (13.6) diverges. When the N ~ )(uK”) force range a is finite, the factor (2/ts)f(N, N ’ ) exp ( u ~ K ~erfc plays the role of the cutoff factor as N‘ tends to infinity. The behavior of this factor is roughly given by (2/tk) f ( ~N,’ ) exp

(usK~“)

- (a/\&)

erfc ( u K ~ )

(CZK”)-~

X exp { - ~ ( U / Z ) ~ N ‘ ) (N’ >> 1?/2a2>> N ) .

(13.9)

It is interesting to note that Eq. (13.6) has only real roots when E is below a certain critical value EG,which gives the lowest bound to the energy spectrum. It is given by (13.10)

(13.11)

and C is determined by (13.12)

with

g(N’)

=

+ C)hn) X erfc { [(2ma2/ h2) (N’ + C)hQ]1/2].

(2/t&) f(0, N ’ ) exp { (2mu2/h2) (N’

(13.13)

When the magnetic field is strong, the right-hand side of Eq. (13.12) is much larger than unity. In this case, the approximate solution for C is obtained as c = 4-48 (eo/hQ), (13.14)

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

329

where we have put €0

{ (h/27f)

=

[nn/(ro) ""g(0)

(13.15)

)2'3.

When E is in the vicinity of EG, only the states with N = 0 are important. Hence we shall look for the solution (Go ( E f ir]) ) s . The contributions to Eq. (13.6) from terms with N' >> 1 in the sum are approximately constant so long as I E - EG I << TiQ. The contribution from the term with N' = 0 can be treated separately. I n this approximation, Eq. (13.6) is simplified by introducing the following notations :

E ( A o ( E ))s

=

+

A E ~ 21/3e~,

(13.16)

+A +

(13.17)

A

=

=

h0/2

+ ccx(e,)

A

and

EOE,,

>.

(rO(E)

= EOY(E,).

(13.18)

Thus we have for Eq. (13.6)

x(4

=F i Y ( E 2 ) = - [ x ( e z )

=F i Y ( 4

-

(13.19)

€z]-1/2.

The solution of this equation is given as

x (E,)

= - [2a2(€*)]-',

Y ( 4 = ([441-'

- [2a2(~z)1-2)1'2

+ ($[1 + (~,3/27)] + $[1 + (4e,3/27)]1/2)1'3} + {+[1 + ( ~ > / 2 7 ) ]- +[1 + (4ez3/27)]'~2}'~3

a(€,)= +ez

for eZ

> EG

=

-3/4'13

(i.e., E

> EG)

and X ( 4 = -[2P2(€z)I-' Y(E2)

=

I

+ (c2P2(41-2 - [P(41-1)1/2'I

0;

P ( e 2 ) = Qe,

(13.20)

+ {t[1 + (8~,3/27)+ (4e,6/729)]}"6

for E?

5

eG

(i.e. E 5 EG).

!

(13.21)

The solution for (Ao(E))s and (I?o(E))s can be expressed with the help of Eqs. (13.20) and (13.21) as =

and

A

+ eox[$(E -

fiQ -

A)/Eo]

(13.22)

330

RYOGO KUBO, SATORU J. MIYAKE, AND NATSUKI HASHITSUME

€2

FIG.4. Functions ~ ( e . ) and -y(eZ) defined in Eqs. (13.20) and (13.21). These functions together represent shifting and broadening of levels. -y(eZ) is proportional to the density of states in the lowest Landau level perturbed by scattering. Broken line represents l/(ez)1'2 which is proportional to the density of unperturbed states. Thin line represents the density of states if each level took Lorentzian shape with a constant width.

-

€2

FIG.5 . Broadened levels given by damping theory [Eqs. (11.20) and (13.18)]. For pz2/2m = 0, €0, 2a0, 3€0,4€o, 5e0, and 1 0 ~The . shape of levels with small p , differs considerably from Lorentaian shape.

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

33 1

The functions ~ ( e , ) and - y ( e Z ) are plotted in Fig. 4. The energy shift and broadening of levels can be calculated from Eqs. (13.22) and (13.23). The function (So,,,(E) is plotted in Fig. 5 for p,2/2m = 0, €0, 2e0, 3 ~ , 460, 5e0, and 10e0. The results (13.22) and (13.23) can be used to caIculate the density of states (ll.lO), the transverse and the longitudinal conductivities (11.17) and (11.19), when hQ >> { - EG for degenerate electrons or hQ >> kT for nondegenerate electrons, so that the contributions from low E values axe important. The results are )8

p( E ) =

[(2m) 3’2/2?r21i3][+hQ/(eo) 1/2]y(e,)

,

(13.24)

and

with

and

G Z z ( E )= (e2/m)p(E)C2(ro(E) >J-l 2e0 I a =

-

(noe2dm) ~(EO/{O) (1

+ ~ ( u / Z ) ~ } I a ( e , ) y ( e , ) 12,

no being given by Eq. (8.14). The functions are plotted in Figs. 6 and 7. I n the region ez >> 1, the approximations

I r ( 4 l2 and

Ia

(€2)

p( E ) =

-

Y(€2)

can be used and we get

I2

1 - y ( e Z ) l2

and

-

=

nge2n -3 m

{1 + 2

I a ( e z ) - y ( e z ) l2

(13.29)

ez

[( 2 m )3’2/4~2fi3] (ifin)( E’)-’I2,

and

(13.27)

(13.28)

€El,

[l

G,,(E)

l2

( e Z ) ~ ( d

(13.30)

+ 2(u/Z)2]--2,

(13.31)

c)’}

(13.32)

-

-

,

332

RYOGO K U B O , SA'l'OltU

J. MIYAKE, A N D N A T S U K I H.4SHll'SUMIC

I.o

0.9 0.8 0.7

0.6 0.5 04 0.3 0.2 0. I

9- 5 - 4 - 3 - 2

-I

0 I

2 3 4 5 6 7 8 9 10 I I 12 13 14 15 €2

FIG.6. Function ( y ( ~ ~which ) ) ~ is proportional to the contribution to ~ ~ ~from ( 0 ) electrons with energy eZ. Broken line represents l / e , which is the corresponding quantity if the broadening of levels is neglected. Thin line is obtained if each level took Lorentsian shape with a constant width.

where E' = E - A - li12/2 may be interpreted as the kinetic energy associated with the motion parallel to the magnetic field. These results agree with the results by the perturbational calculation. I n the region e, 5 1, p ( E ) , G,,(E), and G,,(E) behave differently from the results by the perturbational calculation. Instead of rising up to infinity, p ( E ) and G,,(E) tend to zero as E' goes down. As a result, the logarithmic divergence of u=, does not occur and it is cut off at energy E' tO.Performing the integrals in Eq (13.25), we obtain the following results:

-

9876-

-EZ

FIG.7. Function ( a ( e . ) y ( e z ) ) Z which is proportional to the contribution to UZZ(O) from electrons with energy eZ. Broken line represents e, which is the corresponding quantity if the broadening of levels is neglected.

333

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

13.34) no and 71 being given by Eqs. (8.14) and (12.7). (b) Nondegenerate case 1

rzz(0)

=

(1

2k T

+ ~ ( u / Z ) ~ In) ~ ( y r o ) ,

13.35)

~

(13.36) Here we have used the number density (8.21) ({ being replaced by { - A ) and the relaxation time rf-l

=

W ( m 3 ' z / ~ h(2k 4 ) T )lI2.

By making use of this relaxation time, the characteristic energy by Eq. (13.15) can be written as

(13.37) €0

defined

(13.38)

BROADENING VERSUS INELASTICITY 14. COLLISION I n the preceding sections we have seen that the logarithmic divergence described in Section 5 is cut off by the mechanism of collision broadening at an energy of order rOgiven by Eq. (13.15) or (13.38). On the other hand, the cutoff energy due to inelastic collision is of the order of hw/l for the electron-acoustic phonon interaction, as was shown in the preceding chapter. The mechanism with the largest cutoff energy by nature cuts off the logarithmic divergence first. Let us examine which of the two mechanisms works first in the case of electron-acoustic interaction. As was discussed in Section 10, the effect of collision broadening beconies appreciable when the concentration of scatterers, i.e. the number of phonons, is sufficiently large; the number of phonons increases as the

334

RYOGO KUBO, SATOKU J. M I Y A K E , A N D N A T S U K I H A S H I T S U M E

temperature increases. On the other hand, the cutoff energy due to inelastic scattering hw/l does not change appreciably with temperature, and is of the same order for any crystalline material. Thus we can infer that the mechanism of collision broadening becomes effective a t high temperature, while the mechanism of inelastic collision is more effective a t low temperature, provided that the sample is sufficiently pure and perfect, i.e., when the scattering by impurities and imperfections may be neglected. As an example, let us consider a very pure and perfect n-type germanium single crystal. The velocity of sound w has been estimated to be equal to ' inserting this value into the formula 5.4 X lo5 ~m / s ec.~Thus, hw/l

=

(mow22pBH)1/2 = 4.1093 X =

2.9772 X

X wH1l2 erg

lop8 X

wH"'

OK,

(14.1)

in which the magnetic field H should be measured in oersteds, we obtain

hw/l

=

X H1I2 "K

1.61 X

( = 5.1"K

for H

=

lo5Oe).

(14.2)

The relaxation time T f can be estimated from the observed value of niobility which has been given by the experimental formula38 p =

4.90 X lo7 X T-1.66 cm2 V-' sec-'.

(14.3)

This formula is valid in the interval 100°K < T 5 280"K, and the main reason why the index of power of T is different from the theoretical value -3/2 given in Eq. (8.22) is the contribution of optical phonons. Below 60°K the observed data seem to lie on the curve corresponding to the power index - 1.5. Thus we may estimate the relaxation time by making use of the equation (e/m)Tf

=

31';

X 4.9 X lo7 X

cni2 V-l sec-',

(14.4)

where the effective mass m should be the one appearing in the level density, i.e., m = (1.58 X 0.082 X 0.082)1/3X mo = 0.22 X mu, (14.5)

mo being the true mass of electron. From Eq. (14.4), we get =

8.3 X

X T-1.5 sec.

(14.6)

If the magnetic field is applied in the direction of the axis of a spheroidal energy surface, the effective mass for the cyclotron motion is equal to W. L. Bond, W. P. Mason, H. J. McSkimin, K. M. Olsen, and G. K. Teal, Phys. Rev. 78, 176 (1950). as F. J. Morin and J. P. Maita, Phys. Rev. 94, 1525 (1954). 37

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

335

0.082 mo,so that the energy of cyclotron motion becomes hQ/2

=

pBH/0.082

=

1.13 X 10-'9 X H

erg

=

0.82 X 10V X H

O K

(14.7)

and the cylotron frequency is given by D = 2.1 X lo8 X H

sec-1,

DTf =

.*.

1.73 X HT-'.'.

(14.8)

Inserting Eqs. (14.6) and (14.7) into Eq. (13.38), we obtain €0

= 0.72 X

x

( H T ) 2 / 3 erg

= 0.52

X lop4X ( H T ) 2 / 3 OK.

(14.9)

In Fig. 8 we have plotted Eqs. (14.2) and (14.9). The strength of magnetic field should be much larger or the temperature much lower than the value determined by hQ AT N

H, oersteds

FIG.8. Cutoff energy due to inelasticity of collision and to collision broadening for n-type Ge. H is parallel to the axis of a spheroidal energy surface. Cyclotron mass m = 0.082m0.Velocity of sound 20 = 5.4 X 106 cm/sec. Thick line represents cutoff energy from inelasticity as a function of H . Thin lines represent cutoff energy from collision broadening for various temperatures. At a given temperature and a given field, inelasticity is effective as long as the thin line corresponding to the temperature is below the thick one.

336

I ~ Y O G O KUBO, SATOHU J. MIYAKE,

AND NATSUKI

HASHITSUME

in order that the condition of quantum limit can be applied. From this figure we can ascertain that the inelasticity is more important for our sample a t an available magnetic field than the collision broadening. V. Non-Born Scattering34

15. EFFECTOF INTERFERENCE OF WAVESON THE LOGARITHMIC DIVERGENCE

We shall assume in this part the elastic scattering. Then, if the concentration of scatterers is sufficiently low, the collision broadening is not effective for cutting off the logaritbmic divergence, which will be, however, avoided by solving the scattering problem more exactly than the (lowest) Born approximation. An electron moving very slowly in the direction of the magnetic field encounters the same scatterer repeatedly and produces an infinite number of scattered waves, which are coherent and interfere with each other and with the incident wave. If we take all these coherent waves into account, we shall have an exact incident wave and exact scattered waves which are quite different from the Born approximation and have the exact transition amplitude in place of the matrix element of scattering potential U in Eq. (3.13), as will be seen from Eq. (16.14). I n this section we shall roughly estimate the effect of interference in terms of the cutoff energy. Let us denote the phase shift of the scattered wave relative to the phase of the incident wave by 6. As is well the wave function obtained in the Born approximation becomes very poor when the phase shift 6 is larger than about ~ / 4 I:6 ! > 0 (1). The phase shift 6 in the Born approximation can be estimated in the following way. If we estimate the force range with the scattering amplitude f in the case of shortranged potential, the duration of a collision is given as T d f/vz = f m / p z = f m / ( h k Z ) .During this time the phase of the incident wave proceeds by f/(Z2k,), while the phase of the scattered wave does not the angle QTd effectively change by the definition of scattering amplitude. Thus the phase shift will be (15.1) I 6 I (f/i2kJ.

-

-

-

Therefore the lowest Born approximation is inapplicable for those electrons which have I k , I 5 ( f/Z2), or E , 5 (h2f2/2mZ4). (15.2) The lowest Born approximation overestimates their contribution to the conduction. Thus we should cut off the integration range of Eq. (5.1) a t 39

N. F. Mott and H. S. W. Massey, “The Theory of Atomic Collisions.” Oxford Univ. Press, London and New York, 1949.

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

335

ri2j2/((2m14) to obtain

(15.3) for nondegenerate electrons. In the case of long-range force, the situation is more complicated, so that we shall not discuss it here. IN TERMSOF SCATTERING 16. CONDUCTIVITY TENSOREXPRESSED OPERATORS

Since we shall assume the scattering to be elastic, our starting point is Eqs. (3.6) and (3.7), which we shall transform into a form more suitable for the use of the exact solution of the scattering problem. Following the formal theory of ~ c a t t e r i n g ,we ~ ~ introduce the scattering operator deh e d by (16.1) T ( s ) = U - UR(s)U, where s stands for a complex variable, and U is the scattering potential. U is defined The resolvent operator R(s) for the Hamiltonian X = X, by Eq. (11.1) , and is easily found to satisfy the identities

+

R(s)

=

Ro (s)(l - U R ( s ) )

=

(1 - R ( s ) U } R o ( s ) ,

(16.2)

where Ro(s) is the resolvent operator for the unperturbed Hainiltonian X,. By making use of these identities, we can prove the relations

R(s)U

=

Ro(s) T ( s ) ,

UR(s)

=

T(s)Ro(s),

(16.3)

which give in turn R(s)

=

Ro(s) - &(a) T ( s )Ro(s)

(16.4)

and

T(s)

=

U ( l - R o ( s ) T ( s ) }= (1 - T ( s ) R o ( s ) ) U .

(16.5)

Equations (16.5) are just equations satisfied by the scattering operator, and if these equations are solved, we can determine the resolvent in terms of this solution by Eq. (16.4). I n order to express the delta function 6 ( E - X) appearing in our basic formulas (3.6) and (3.7) in terms of the scattering operator, let us introduce the operators

T ( * ) ( E )= lirn T ( E f is),

(16.6)

F-w

R(*)(E) = lim R ( E f i ~ ) , v+o

Ro(*)(E)= lim Ro(E f is).

(16.7)

V-w

4oB. A. Lippmann and J. Schwinger, Phys. Rev. 79, 469 (1950); J. M. Luttinger and W. Kohn, Phys. Rev. 109, 1S92 (1958).

338

ILYOOO KUBO, YATOILU J. M I Y A K E , AND NATSUKI HASHITSUME

Then the following relations can be proved :

6(E--X) =S(E-X,) -

Elo(+’ ( E )T(+)( E )Ro‘+’( E )- Elo‘-’ ( E )T(-)( E )Ro(-)( E ) 7 2ai

6(E--X) U -

no(+) ( E )T(+)( E )- Ro(-)( E )T(-)( E ) 1

2ai U6 ( E - x ) -

T(+’( E )Ro‘+’ ( E )- T(-)( E )Ro(-)( E ) 9 2ai

liS(E-X) U

T‘+’( E ) - T(-)( E ) -2ai

= TI*’ (

E )6 ( E -

x,)T(+)(E ) ,

U G ( E - X ) (l+URO‘*’(E) 1 -

T‘?’ ( E )6 ( E-X,)

,

(l+Ro‘*’(E) C’]G(E--X)

u

=6(I3-Xc) T ( F ) ( E ) .

(16.8) By making use of these relations, remcinberirig [Xe, X] transform Eq. (3.6) into the u,,(O)

=

?re2 fi

-

/ m

dE

-m

=

0, we can easily

(-g)

(trace ( 6 ( E -

x,)[X,

( E )IS( B -

~ ( 7 )

x,)[T(*’ ( E l , XI) )s, (16.9)

and in the saiiie way Eq. (3.7) into the form u,,(O) =

fi

/ m

--m

dE

(-$

These expressions are exact for elastic scatterings, and they are our basic expressions in this part.

QUANTUM THEORY OF GALVANOMAGNETIC E F F E C T

339

When the scattering potential is given as the sum of contributions from many scatterers, i.e. when U is given by Eq. (5.2)) we can expand the scattering operator T ( s ) in terms of the scattering operators t j ( s ) associated with scatterers, each of which is considered to be isolated from other scatterers :

u(r - R j ) { l - Ro(s)tj(s))

11 - t j ( s ) R o ( s ) } u (r Rj). (16.11) As is well known,a the expansion formula is tj(s) =

+

C

=

t i ( s ) R o ( s ) t j ( s ) R o ( s ) t ~( s )* * . .

(16.12)

i ,j,k(i#j#k)

Since we are interested in the case of low concentration of scatterers, we may retain only the lowest-order term: (16.13) and furthermore, in the expression obtained from Eq. (16.9) by inserting Eq. (16.13) we may neglect the cross terms including different scatterers. Thus we have in the ( N , p,, X ) representation

N'J',pz'

2a 9

-

n

Ns

(

C I ( N , X , p , I tj'*t'(E)I N', X ' , pz')

I2)s

6(E - E N ' ( p z ' ) ) ,

j=l

(16.14) where we have used the relation t (s) = ( t (s*) } t or t(+)( E ) = f t(-) ( E )} +. The longitudinal component of the conductivity tensor a,,(O) , Eq. (3.8)) cannot be transformed in the same way, because the coordinate x does not commute with the unperturbed Hamiltonian x,,although the velocity component v 2 = X commutes with x,. If the series (16.12) and (11.4) were summed up, and if we had convergent results, we would obtain the same result for the conductivity in either way. The evaluation of the higher terms is practically very difficult, so that we have to be satisfied by determining only the first few terms either by the damping theoretical method developed in the preceding part if the perturbation U is weak, or by the method developed above if the potentials u are local. These two ways of calculation thus correspond to the two different standpoint^.^

340

RYOGO KUBO, SATOIZU J. MIYAKE,

AND NATSUKI

HASHITSUME

17. SKOBOV-BYCHKOV THEORY: SHORT-RANGE FORCE

Approxiniate solutions of the scattering problem in a magnetic field have been obtained independently by Skobov,22by Kahn,21and by B y c h k o ~ . ~ ~ Kahn and Bychkov have assumed the scattering potential being of the delta-function type (12.4), and they have been led to certain divergence, which they have cut off. Skobov has treated such a scattering potential, that the force range is short compared with the de Broglie wavelength of electron and the mean distance between scatterers, in a more refined way. In this section we shall give the Skobov theory. If we introduce the wave function W * ) N , Xsatisfying ,~~ the integral equation

From this expression we see that the matrix elements required for the calculation of the conductivity (16.14) is determined by the value of the wave function \k(*)N,x,ps(r) within the force range of scattering potential u: (17.3)

Ir - R j [ 5 a.

In order to obtain the solution for Eq. (17.1) within this region, we may approximate the unperturbed Green function (r I Ro(*)(E)I r’) as follows (cf. Appendix B) :

(r I Ro(*)( E ) I r’ )

=

(m/2afi2) { (I r - r’ I)

-l

(1

f iK ( E )}

r - r’ I << I ) , (17.4)

where we have introduced the notations

K(E)

K’(E)

=

[(afi)”m]p(E),

=

K ’ ( E ) =F iK”(E);

K”(E)

=

1/ 1

[2NE

+ 1 - E/(fif2/2)]1’2’ (17.5)

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

341

p ( E ) being the unperturbed level density given by Eq. (4.8), N E the smallest integer that exceeds E/iX - 9. By making use of the approximate expression (17.4), we can easily solve the integral equation (17.1), and obtain

*(*)~,~.p,(r) = (+N.X,p.(Rj)/C1 f i f K ( E ) I ) P ( r ) ,

(17.6)

where @(r) is the solution of the integral equation in the absence of magnetic field: @(r)

=

1- -

(17.7)

and f stands for the scattering amplitude (17.8) both for an electron of zero energy. Inserting the solution (17.6) into Eq. (17.2), we obtain the approximate expression for the required matrix elenients

( N , x,pz I t i ’ * ) ( E N ‘ ( P Z ’ ) ) I ”, X’, pz’)

W being given by Eq. (5.10). Comparing this expression with the expression (12.5), we can see the effect of wave interference: if either K ’ [ E N . ( ~ , ’ ) ] or K”[EN, ( p z ’ ) ] diverges, i.e. if ps’ approaches zero, the present expression (17.10) vanishes, while the old one (12.5) does not, and the range of 1 p,’ I in which the expression (17.10) is much smaller than (12.5), may be estimated from the condition fK’ 2 1 or jK” 2 1 as I p,’ I 5 Tzf/P, i.e. just as the relation (15.2). Now let us evaluate the conductivity (16.14). By making use of Eq. (5.12) as was done to obtain Eq. (5.11), we can carry out the summation

342

HYOGO KUBO, S A T O I ~ UJ. MIYAKE, AND NATSUKI HASHITSUME

over X and the integration with respect to X", and get

5

e2m2W

(N

+N'+

+

dE" 3)hO]'/' [ E - (N'

1)hO

N.N'=O

*

1

[E - (N

+ 3)hO]1/2 (17.11)

Hereafter we shall retain as usual only the term with N = N'

=

0:

(17.12)

Since we take

K'(E)

=

N E =

1, the explicit f o r m of K ' ( E ) and K " ( E ) are

(m/2) lI2Q ( E - fiQ/2) ' I 2 '

K"(E) =

(m/2) 1/52 (3hQ/2 - E)'/**

(17.13)

I n the degenerate case we niay replace the negative derivative of the Fermi function by th2 delta function, and obtain

(17.14)

where no is given by Eq. (8.14), and r f by Eq. (12.7). If the number density of electrons n is kept constant, the chemical potential { varies with the magnetic field H according to Eq. (8.17). Then the assumption { - hQ/2 << hO means that hQ >> {o, or XO >> 2, where Xo = h/(2m{o)'l2is the de Broglie wavelength in the absence of magnetic field. On the other hand if we may assunie for the short-range potential that f/l

<< Z I X O ,

(17.15)

we find that the factor representing the wave interference niay be approximated by one, and we get uzz(0) =

(ne2/m02Tr)

(hO/2{0)s.

This is the result obtained by SkobovZ2for the degenerate case.

(17.16)

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

343

In the nondegenerate case we may replace the Fermi function by the Maxwell-Boltzmann factor:

’

lm + E,[1

exp ( - E , / k T ) dE , (17.17) (m/2)112!df/(fin - E,)1/2T m(f2f)2/2 ’

+

In the denominator of the integrand, we may replace the coefficient of E , by one. The integration with respect to E , can be carried out by making use of the exponential integral E i ( x ) , which may be approximated by the logarithmic function, if we assume that

f/l where XT

=

<< l / X T

(17.18)

h/(2mkT)1/2.Thus we obtain

{

’

e2m2W fin exp [({ - +fin)/kT]In leT (2n) 3h6k T rm(f?f)2/20

u,,(O) = 2 _ _ _ -

(17.19)

where y = 1.781072, or, by making use of the relaxation time (13.37) and the number density (8.21), (17.20)

Thus we have arrived at the cutoff factor (15.3). Skobov22has given the expression including the exponential integral Ei ( x ). 18. METHOD OF

PARTIAL

We shall give an exact method of solving the scattering problem in an extremely strong magnetic field in the case of such a scattering potential ~ ( r Rj) that is axially symmetric around the direction of magnetic field and symmetric on the reflection with respect to the plane perpendicular to the magnetic field and through the center of the scatterer Rj. Then it is more convenient to use a representation, in which the z component of angular momentum xp, - yp, = fiLs is diagonal. Let us first change the gauge from A = (0, Hx, 0 ) to A = ( - Hy/2, Hx/2, 0) : the wave function (1.19) is transformed into $N.X,P*

(r)

=

exp c i x Y / ( 2 ~ 2I +)N , X . P . (r).

(18.1)

344

RYOGO KUBO, SATORU J. MIYAKE, A N D N A T S U K I HASHITSUME

Next let us change our representation from the ( N , X , p z ) representation to the new (N, m , p z ) representation, in which

x2+ Y2 = (X, lio 212

=

212(N

-

g)

-

212 - ( 2 p z - ypJ li

+ + - L,)

(18.2)

is diagonalized instead of X . Here we have assumed a spherical mass tensor. Introducing the cylindrical coordinates r = ( p , ‘p, x ) , we obtain the eigenfunctions of X,:

where we have introduced the normalized Laguerre function

(18.4)

+

with n = N ( m - 1 m 1)/2. The associated Laguerre polynomials are Y 2 should be nondefined in Eq. (13.4). Since the eigenvalues of X 2 negative, the quantum number m characterizing the eigenvalues of - L, cannotbesniallerthan-N:m = -N, - N + 1 , - . . , 0 , 1 , 2 , --..Theorthop ) taken as normalization relations of ‘ p ~ , ~ ( are

+

(18.5) and thus the functions (18.3) make a complete orthonormal set of eigenfunctions in the cylindrical coordinate system. The eigenvalues are EN(^,) = ( N +)hQ p,2/2m as before. If we introduce the cylindrical coordinates by r - Ri = ( p , ‘p, 2) and diagonalize ( X ( Y - Yjl2 in the gauge A = ( -H [ y - Y j ] / 2 , H [ x - X j ] / 2 , 0 ), where we have put Rj = ( X j , Y i , Zj) , we obtain the same eigenfunctions (18.3) , in which we replace ( p , ‘p, z ) by ( p , ‘p, 2) ; z = z - Zj. If we do the same in the gauge A = ( -H y / 2 , H x / 2 , 0 ), we must change the gauge and obtain

+

(r)

+~.m,p,

+

= ~ X PCi(xYj =

C

+

- ~ X i ) / 2 1 ~ 1 6 ~ , (r m ,p , Rj)

( - l ) % v * , x , p s t ( r* ) 8 N , N ~ ~ p l . p . ~ ~ N+m X(i X )

N’sX,p,’

2a12

XjYj

+ i- X Y j 12

-

iq).

(18.6)

3 4ti

QUANTUM THEORY O F GALVANOMAGNETIC E F F E C T

Hereafter we shall use these wave functions. It must be noted that the angular momentum fiL, is now the one around the scattering center a t Rj. Since we are interested in the case of an extremely strong magnetic field, we shall consider only the lowest Landau level N = 0, whence we havem = 0 , 1 , 2 , ..., a n d n = 0 : ~ p ~ , ~= ( p [Z(m!)1/2]-’ )

exp

(-p2/412)

(p2/212)42.

(18.7)

Since the energy is conserved in the elastic scattering, the value of should be conserved, if we neglect the transitions between different Landau levels. Furthermore, by the assumption of axial symmetry of the scattering potential, the quantum number m should be conserved. Thus we may expect only the transitions from an initial state (0, in, p z ) to final states (0, m, = t p z ) , which correspond to the forward and the backward scatterings respectively. We should have as the asymptotic form of the solution a(+)of our scattering problem

I p, 1

@+)o.m.p.

(r)

-

exp [ i ( x Y j - yXj)/212]~o,m(~> [exp ( -imp)/(2r)1/21

+

( I 0, P2(exp ( i p z V f i ) F+,m,p,exp

*I

( i p Z 2 / f i ),

for 2 + +.o (18.8)

\ I v,

(exp ( i p z 2 / f i )

+ F-,m,pzexp ( - i p , Z / h ) ) , for Z--t

-00,

where we have changed the normalization: v, = pZ/m. On the other hand, by virtue of the assumption of reflection symmetry, the parity concerning this symmetry should be conserved. Therefore it is convenient to classify stationary solutions of our scattering problem into symmetric and antisymmetric solutions with respect to the inversion of 2. We can introduce the phase shift 6 for these symmetric or antisymmetric solutions: am8(I p , I) for the symmetric solution and 6,”( I p , I) for the antisymmetric solution. They are introduced by assuming that the asymptotic forms of stationary solutions for I 2 I + 00 are given by @P(8)o,m.p,(r)

-

exp [ i ( x Y j - y~j)/2Z~]po,,(p)[exp (-imcp)/(2r)1/2] *

and *(a)O,m,p,(r)

-

@/I

flz

1)1’2cos { ( I P Z Z Ilfi)

+

6m8(l

Pz

I)

(18.9a)

1 1

exp [ i ( ~ Y-j yXj)/2Z2]~o,m(~) [exp ( -imcp)/(2r)1/21 *

(2/1 8. 1)1’2~in { (1 pzz

I/n)

+

Sma(l

pz

I) 1 SP

2,

(18-9b)

34G

RYOGO KUBO, SATORU J. MIYAKE, AND NATSUKI HASHITSUME

respectively. Comparing these solutions with the previous one (18.8), we obtain

=

1 4 * 0 . ~ , ~ . ( r ) u ( rRj)@(+)o,m,p,~(r) dr, (18.12) -

where the normalization of the plane wave part of +O,rn,p,(r)should be changed in the same way as in Eq. (18.8), and the volume integration is to be taken over a region between two planes z = f constant sufficiently far from Z = 0. By making use of the Schrodinger equation, we can rewrite the right-hand side of Eq. (18.12) into the form

Inserting the asymptotic forms of c $ * ~ , ~and , ~ sion and remembering p,’ = fp,, we obtain

,@ J ( + ) ~ , ~ , into ~,,

(0, m, pz I 2i‘+’(Eo(p,’)) 10, m, P Z ’ )

1) ( W )

x

({exp [2isms(l p ,

=

(Pz/l pz

this expres-

I)] - I ) + sgri (plpz’) {expC2iSma(l p z 111 - 11). (18.14)

Since the matrix elements are related to the normalization such as used in Eq. (18.8), we have to multiply I v, I/L in order to return to the original

QUANTUM THEORY O F GALVANOMAGNETIC E F F E C T

347

normalization used in Eqs. (18.3) and (18.6). Thus we obtain

x

({exp [2i8M8(I p ,

I)]

- 1)

+ sgn (paz‘)Iexp [2i8M‘(1

p 2 I)] - 113,

(18.15) or, in the ( N , p,, X) representation,

Then the average of squared matrix elements appearing in the expression of conductivity (16.14) is given by

- ({exp [2isM8(lp , I)]

- I } +sgn (p2p,’>texp [2iSMa(l

P, 111- 11) .’1 (18.17)

Comparing this expression with the one obtained in the preceding section (17.10), we see that in the Skobov-Bychkov theoryZ2J3we have neglected all the partial waves except the one with zero angular niomentum in the direction of magnetic field ( m = 0), and put

for small I p , I. By this first relation the phase shift Sos(l p , I) does not diverge even when I p , I becomes zero, in contrast to the one determined by the lowest Born approximation (15.1).

348

RYOGO KUBO, SATORU J. MIYAKE, AND NATSUKI HASHITSUME

Now we can express the transverse conductivity (16.14) in terms of the phase shifts SmS(l p , I) and Smu(l p , 1) :

- [sin2

I) + sin2 {SUm+1(l p , 1) (SS,I(l

pz

-

Sm8(l

- Smu(l

I) 1 p , I) ) ] I ~ , I = [ z ~ ( B - - ~ L ~ / z ) I ~ / (18.19) ~.

Pz

This expression is exact for elastic scatterings, except that we have made use of the approximation (16.13) and retained contribution from the lowest Landau level only.

a. Short-Range Forces In this case we can apply the theory of effective range,41and express the phase shifts in terms of the scattering length and the effective range. We shall first consider the symmetric solution of the scattering problem in an extremely strong magnetic field, which satisfies the boundary condition (18.9a), or

for

I z I + +a,

(18.20)

where we have changed the normalization and put Rj = 0 for simplicity. By making use of the Schrodinger equation and of Green's theorem, we obtain the following identity

(18.21)

On the other hand, the function defined by

(18.22) 41

H. A. Bethe, Phys. Rev. 76, 38 (1949).

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

349

satisfies the Schrodinger equation with zero scattering potential, but has discontinuous derivatives with respect to z at z = 0. The identity corresponding to Eq. (18.21) becomes

(18.23)

- [the same expression as abovelf:?:

.

Remembering the boundary condition (18.20) and the definition (18.22), we thus have

Sm8(l p ,

1) - n

tan S m 8 ( l p,'

I)],

(18.24)

or, by taking first the limit I p,' tan

Sm8(l

Pz

I>

-

I + 0 and considering a small value of 1 p , I, (18.25a) CV(lP z Ifm('))l + (I Pz I l n ) a m ( s ) ,

where fin(') stands for the scattering length defined by (18.26a) and

the effective range defined by

{ I @(a)o,m,ps(r) 12 - I +(8)~,rn.p,(r) 12) dr.

(18.27a)

As is well known,4l the asymptotic expression (18.25a) may be used for I p , J such that 1 p , [ << f i / a or xo(E) >> a, (18.28)

XO(E)being the de Broglie wavelength and a the force range of the scatterh g potential. I n the same way we obtain for the antisymmetric solutions cot LQ(l pz

1)

- (n/I

p,

/fm(a))

-

(1

pz [/n)am(a), (18.25b)

350

ItYOGO K U B O , SATOItU J. MIYAKE, A N D N A T S U K I HASHITSUME

where we have defined the scattering length and the effective range respectively by (18.26b)

am(a)

=

lim lPzl-0

' 1( 1

2

+(a)o,m,p,(r)

12

-

I 4(a)o,m,p,(r) 121 dr,

(18.27b)

the wave functions + ( a ) O . m , p , (r) and +(a)o,m,p, (r) being defined in a similar way by remembering Eq. (18.9b). From the asymptotic expression (18.25a), we find that 6,"( I p , I) tends to a/2 nr ( n = 0, f l , f 2 , . - - ) as I p , I approaches zero, i.e. the phase shift of the symmetric solution behaves essentially in the same way as that of the solution corresponding to an unbound state in the onedimensional problem. In contrast to this situation, from the expression as (18.25b) we find that Sma([p , 1) tends to na ( n = 0, f l , &2, 1 p , I + 0, i.e. the phase shift of the antisymmetric solution behaves like the radial solutions corresponding to an unbound state in two- and three-dimensional problems. When the scattering potential is weak, the scattering length f m ( s ) becomes much larger than the force range a, and the phase shift I Sms(l p , I is always large ( Z r / 2 ) for a small value of I p , 1. Thus the exact symmetric wave function differs from that obtained in the lowest Born approximation for such values of I p , I that satisfy ,

+

. . a )

I)

I ~ " ( 1pZ I) I 2 r/4

Ip, I

or

5

li/fm(s)

<< h/a.

(18.29)

This means that the cutoff energy is given by lt2/2m(f m ( s ) ) 2 .When the scattering potential is strong, fmla) is of the order of a, and the lowest Born approximation is expected to be iuvalid for

I

Pz

I 5va;

(18.30)

i.e. up to fast electrons. I n the case of short-range force ( a << Z), in which case we may assume Eq. (18.28) and thus use Eqs. (18.25a) and (18.2513) (cf. Section 13), the amplitude of wave functions near the scatterer is very small except that of the symmetric wave function with m = 0, the scattering lengths are very large except f0("), and the effective ranges are very small except a0("), whence only the phase shift 6os is appreciable for a wide range of small p , We may neglect the contributions from phase shifts other than cYOs, just as was done in the Skobov-Bychkov theory, in which we have

I I.

fo(8)

-

1"f.

(18.31)

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

35 1

By the assumptions (17.15) [or (17.i8)] and (18.28), the scattering potential has to satisfy the relations

>> XO (or AT) >> a,

(18.32)

i.e. it must be weak, in order that the main result (17.16) or (17.20) of the Skobov-Bychkov theory can be applied. b. Long-Range Force

I n this case we can approximately reduce the scattering problem to a one-dimensional problem in the direction of magnetic field, and treat the components of motion perpendicular to the magnetic field classically. If we neglect the transitions between different Landau levels because of the large separation h0 between them, the stationary solution of the scattering problem is approximately given by (m

@(r) = cpo.,(p)[e-im~/(2?r)1’2]F(~)

=

0, 1 , 2 ,

. a * ) ,

(18.33)

where the function F ( z ) satisfies the one-dimensional equation

h2 d2 Here we have put Ri

=

(18.34)

0 and defined p, by p,2/2m

=

E

-

(h0/2).

In Eq. (18.34) the effective potential u,(z)

/d ub-1I

(18.35)

is defined by

m

urn(z)

=

cpO.m(P)

PPdP,

(18.36)

where we have assumed as before the scattering potential u ( r - Rj) due to the scatterer a t Ri having the axial and the reflection symmetry, i.e. u(r) depending only on p and [ x I. By the assumption of long-range potential ( a >> I), the potential changes very slowly compared with the function p I cpo,,(p) 12, which has a peak a t p (am 1 ) W [cf. Eq. (18.7)]. Thus we may use the approximation (18.37) um(z) = [u(r)I p = ( 2 m + ~ ) 1 / ~ 1

-

+

which may be used up to a sufficiently large value of m: m << (a/Z)4. We may regard p = (2m 1)W as a continuous variable, which behaves as the impact parameter. Then Eq. (18.37) means that the motion of center of orbit may be treated as classical and the phase difference - 6,

+

352

I ~ Y O G O KUBO, SATOHU J. MIYAKE, AND NATSUKI HASHITSUME

appearing in the expression of transverse conductivity (18.19) may be replaced by the derivative : b l ( l

Pz

I)

- 8 4 1 Pz

I)

-

I

(12/P)Ca6(P,

Pz

(18.38)

I>/aPI,

the function 6 being defined by 6(P,

I P z I)

=

MI P z I).

(18.39)

In general we have to determine the motion in the direction of the magnetic field quantum mechanically by solving Eq. (18.34), in which we may use Eq. (18.37). If the wavelength & ( E ) = h/l p , I in this direction is smaller than the force range, i.e. if

I P z 1 >> h/a,

(18.40)

we can make use of the WKB approximation. When the scattering potential is attractive, or when the kinetic energy p2/(2m) is larger than the repulsive berrier, the phase shifts in the WKB approximation are given by 68(P,

I P z I)

= P(P,

I Pz I)

-

1

J_," t (P," - 2mu(r>)1'2- I Pz I I

dz,

(18.21) where in the integrand we have to fix the value of Inserting this into Eq. (18.38), we obtain

p

in the potential u(r).

dz

, (18.42)

where we have defined (1,(p, 2) =

(18.43)

eH ap

and v,(p, x ) =

(18.44)

[(2/m) ( E - ~ A Q- u(r) )]1'2.

We find a t once that (1, is just the classical angular velocity of the rotation of orbit center around the axis of symmetry of the potential and v z the classical velocity in the direction of magnetic field. Then Eqs. (18.42) mean that the phase difference 6wl - 6, is equal to one half of the angle of rotation around the axis of potential. We can interpret the quantity

(I

AX

1'

+ I AY 12)1'2

=

2[

p s h ( 6 w 1 - 6,)

I

(18.45)

353

QUANTUM THEORY O F GALVANOMAGNETIC EFFECT

as the displacement of center during the collision, and rewrite Eq. (18.19) into the form

or, by introducing the unperturbed level density p ( E ) defined in Eq. (4.9) and the incident velocity v 2 = [ 2 ( E - hQ/2)/m]'I2, a,,(O)

=

e2

lrn(-$) lrn 5 (7 +) g I dE

I

n. I v, 2ap dp.

1 IAXI2

p(E)

2

fi0/2

(18.47) The quantity da = 2np dp is just the differential cross section corresponding to the impact parameter, and nz ] v z I du the transition rate: Eq. (18.47) corresponds to the classical picture of the diffusion of orbit center. When the scattering potential is strongly repulsive and the electron is recoiled, we obtain Eq. (18.47), although we have the angle of rotation given by 6smt1(1 Pz

I)

-

LS(l Pz I) =

aam+l(l

Pz

I)

-

Sm.(I

Pz

I)

where f z o stands for the classical turning points determined by vz(p,

20)

=

(18.49)

0.

The phase shifts for the symmetric and the antisymmetric waves are different in this case (18.50) 6'(P, I Pz I) W P , I P z I)

-

+

and the forward scattering does not occur, while in the previous case of Eq. (18.41) the backward scattering does not occur. 19. NON-BORN SCATTERING VERSUS COLLISION BROADENING In the case of the elastic scattering by impurities, the logarithmic divergence is cut off either by the mechanism of collision broadening or by the interference of electron waves occurring at a scattering. Let US examine which of the two mechanisms works first, in the case of shortrange force and for the nondegenerate electrons.

354

RYOGO K U B O , SATORU J. MIYAKE, AND N A T S U K I HASHITSUME

According to the result of the preceding chapter, the cutoff energy due to collision broadening is given by Eq. (13.38), or eo = (h2/2m) (n,47rf2/Z2)2 / 3 1

(19.1)

-

1. where we have used the definitions (13.37) and (5.10) and put g ( 0 ) On the other hand, the cutoff energy due to the interference mechanism is of the order of h2f2/(2mZ4),as was shown in Eq. (17.20). Thus we may expect that the cutoff energy due to the collision broadening is larger or smaller than that due to the interference mechanism, if the number density of impurities n, is higher or lower than .f/(47rZ4), respectively. Since we have assumed the relation (12.2) (n, << l/Z3) in order that each collision may be treated separately at least for the motion perpendicular to the magnetic field, and the relation (17.18) ( f << Z2/X~) , our criterion is not very sharp. However, we may say generally that the collision broadening is important when the density of impurities is high and the magnetic field is not extremely very strong, whereas the interference of waves cannot be neglected when the density is low and the field is extremely strong. Let us examine the case of n-type indium arsenide as an example. The effective mass is given by m = 0.02 X mo, mo being the true mass. If we regard the experimental value of mobility p = 16150 cni2 V-I sec-l at 1.25°K42as due to the impurity scattering, we obtain 7f

=

1.84 X

see.

(19.2)

If we may make use of Eqs. (12.7) and (5.10) ,we can estimate the scattering amplitude f : f

=

8.4 X 10/(n,)1’2 cm,

(19.3)

-

where n, should be measured in units of Since n, is of the order of 1017 cm-3, f becomes of the order of cm. If we take n, = n = 1016 cm. This value is in accord with 7.6 X 10l6~ m - we ~ , obtainf = 3.0 X cm estimated in the Born approximation by the value f = 1.5 X the relation f = 2me2a2/(h 2 ~ from ) the Thomas-Fermi screening length cni corresponding to n = 7.6 X 10l6 ~ m and - ~ from the a = 1.4 x dielectric constant of the crystal K = 12. On the other hand, the classical radius of cyclotron orbit is given by Eq. (1.3). Thus we get eo = 2.05 X

lopz9X

( n , H ) 2 / 3 erg

=

1.49 X

X ( n , H ) 2 / 3 OK

(19.4) 42

R. Sladek, Phys. Rev. 110, 817 (1958).

355

QUANTUM TIIEOHT O F GALVANOMAGNETIC EFFECT

and (h2/2m)( f / P ) Z

= 6.4

X H 2 erg = 4.6 X

X

X H2

OK.

(19.5) Since we obtain f/4d4

=

5.5

x

loio x H 2 c 1 r 3 ,

(19.6)

the criterion n, z f/(4?rZ4) can now be written in the form

n,/H2

5.5 X

loio

c n r 3 Oe-2 , or H/(n,)1’2 3 4.3 X 10+

Oe (19.7)

We can ascertain from Fig. 9 our previous expectations.

-

H. oersteds

FIG.9. Cutoff energy due to collision broadening and to wave interference. Effective mass = 0.02mo.Scattering amplitude f = 3.0 X 10-7 cm. Thick line represents cutoff energy from wave interference as a function of H . Thin lines represent cutoff energy from collision broadening for various density of scatterers. At a given density of scatterers and a given field, collision broadening is effective as far as the thick line is below the thin one.

356

RYOGO KUBO, SATORU J. MIYAKE, AND NATSUKI HASHITSUME

Appendix A. Collision Broadening Effect upon Oscillatory Behavior

At strong magnetic fields [cf. Eq. (I.lb)], there appears a singular behavior in the density of states and in the related quantities, which results in a periodic variation as a function of 1/H in various physical quantities such as diamagnetic susceptibility, specific heat, and transport coefficients, when electrons are degenerate. The singular behavior mentioned above is blurred on account of various reasons, among which thermal averaging and collision broadening are the most important. The theory of the collision broadening effect was first examined by Dingle.36He assumed that each Landau level is broadened and has a Lorentzian shape , he obtained an expression for diamagnetic suscep with a width ? i / 2 ~ ,and tibility of electron gas. In his expression, each harmonic component of oscillatory part is modified by an exponential factor which is commonly called the Dingle factor. The modifying factor is expected to appear in an expression for physical quantities other than diamagnetic susceptibility, and is widely used in analyzing experimental data. Although the importance of the Dingle factor is established experimentally, his introduction of the Lorentzian broadening assumption seems intuitive and open to a criticism. Moreover, the assumption leads to a bad divergence when the calculation is rigorously carried out. (See the last paragraph of Section 12.) In this appendix we shall show that when the scattering potential is of a short-range type the Dingle factor naturally follows from the damping theoretical treatment described in Part IV, and that the divergence d f i culty does not arise in this refined treatment. Let us start from Eq. (13.6). When the condition (2mE)'l2 << A/a is satisfied, the N dependence of ( G N ( Ef iq) may be neglected, since the exact value of (GN(E f iq) is required only for small N such that NMl 5 E << h2/2ma2,where (GN(E f iq) does little depend on N . If we neglect the N' dependence of ( G N r ( E f iq) in the right-hand side of Eq. (13.6), the summation over N' can be performed with the aid of Poisson's sum formula )8

5 f(" + 4) /=dxf(z) + 2 5 ( - 1 ) r /=dxf(z) cos (27rrn). =

Nf=0

0

i=l

0

(All For sufficiently large values of N and N' (but [ (2N Z/a) , we may use the approximation

f ( N , N')

=

exp {-2(a/02(N

+ 1)(2N' + 1)]'/2 <<

+ N' + 1)1,

(A21

357

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

which is derived from the expression

=

1/[(N

+ N' + 1 ) P q ~-~ ( N - N')'

- (Z2q~2/2)2]1/2(A3)

obtained by Titeica2 in the WKB approximation. We also use the following estimation of integrals which is valid when a >> 1, b > 0,c << 1, and 6% << 1:

id 2

2 -erfc

\$

+

erfc { c[x - ( a ib)]1'2) [ x - ( a ib)]'/2 dx

+

=

2 [x - ( a

=

(2/\$)

-

z[-

{c[-(a

(a

2 + i b ) I 1 /ta 2 7 e r f c{ c [ x - ( a + i b ) ] 1 / 2 )

c-'exp [c2(a

+ ib)]

+ ib)]'/2 ( z / G ) erfc { c [ -

+ ib)]"2] = 1 --2 C ( - 1 ) n n-0

+ib)];), {c[- ( a + n!(2n+ 1)

(a

(-44)

ib)-J'/2)2n+l 1

(AS) erfc {c[x - ( a

=

+ i b ) ] 1 / 2 )cos (2nrx) E

(2r)-1/2 exp { i(2ara 0

+ $a) - Barb)

cos (2nrx)

l

m

dx

[z

cos (2arx) ib)]'/2

- (a

+

358

RYOGO KUBO, SATORU J. MIYAKE, AND NATSUKI HASHITSUME

Thus we obtain to the lowest order in hQ/E and 2ma2E/h2,

(G(E

+i

h.1

2

{z

~ )= ) ~ 27f 2

- 2(-

E - (G(E

nn

+ i ~ ).)

and

+ (2)““ 2 T=l

In a similar fashion, we obtain the level density (11.10) and the transverse and longitudinal conductivities (11.17) and (11.19) as follows:

(2m) 4r2n3

( (PO)

+ (r( E ))>) lI2 + E’ 2lO

359

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

[

+

c

3 ha7 (-1Ir A , - 3- flQ _In 1 - exp ( - 4 ~ ( r ( E ) ) ~ ) ]4 2E' fla 2 2 E -1

and

W4) where we have put

Equations (A9) and (A10) can be solved by iteration or by some other method. If the oscillatory part is small compared with the nonoscillatory part in the right-hand side of Eq. (AlO), the first approxiination for (r( E )) s and (A ( E )) s is obtained as

and (A ( E )),(l) = - (h/27t) (h 2 / 2 ~ m ~ 2 5 . 0 )

(A171

Substituting Eqs. (A16) and (A17) into the right-hand side of Eqs. (A9) and (AlO), we get the second approximation for ( I ' ( E ))s and ( A ( E )),, and so on.

360

I ~ Y O G OKUBO, SATORU J. MIYAKE, AND NATSUKI HASHITSUME

On the other hand, if the oscillatory part happens to be comparable to or larger than the nonoscillatory part, it is convenient to start from Eq. (13.6). I n the case stated above, a dominant contribution to ( r ( E ))s comes from the term with N' = No, where N o is such that ( N o ;)fin is just above or below E - ( A ( E ) Retaining only one term with N' = N O in the right-hand side of the imaginary part of Eq. (13.6), we are led to an equation similar to that which was met when the quantum limit case was considered (cf. Eq. (13.19)). The solution for ( I ' ( E ))s and ( A ( E ))* is given as

+

)6.

210 *

In order that the oscillatory part is comparable to or larger than the nonoscillatory part, the condition

should be satisfied. The condition is equivalent to QTf

2

2{0/h0

and

[E

-

(No

+ $)fin - A 1

5 (fiQ/210)fiQ. (A221

The solutions for ( r ( E ) ) ,obtained above [Eqs. (A16)-(A20)] are es~ with the sentially the same as those obtained by B y ~ h k o vin~connection discussion of the collision broadening effect on the de Haas-van Alphen effect. When the E dependence of ( F ( E ))s and (A(E) ). can be neglected, the E integration in Eqs. (A13) and (A14) can be performed with the aid of a formula

La, a,

ds sech2

2 - so ( 7 sin (bs) + c ) = n.2a2. b cosech ( m b ) sin (bso + c ) .

(A231 Yu. A. Bychkov, Zh. Eksperim. i Teor. Fiz. 39, 1401 (1960); see Soviet Phys. JETP (English Truml.) 12, 977 (1961).

QUANTUM THEORY OF GALVANOMAGNETIC EFFECT

361

The results are

and

where we have defined

The last three terms in the curly brackets in Eq. (A24) come from those transitions for which the quantum number N does not change. If (I'(T) )s is set equal t o zero, the third and the fourth terms in the curly brackets in Eq. (A24) will diverge. The divergence corresponds to the divergence pointed out by Davydov and P o m e r a n c h ~ k .It ~ is cut off a t the energy (I'({) )s in our case. When the amplitude of oscillations is large and the Fermi level lies in the very vicinity of a certain Landau level, the cutoff energy (r(())s agrees qualitatively with that given by Davydov and ) s is approximated by l i / % f . Pomeranchuk. In other cases, (I?({)

362

RYOGO KUBO, S A T O I ~ UJ. MIYAKE, AND NATSUKI HASHITSUME

If the Dingle temperature T’ is defined by

T’ = (r(r)>slsk,

(A281

the exponential factor exp ( - 2 ~ r ( r ( { ) )&Q) in the coefficients of the oscillatory terms becomes exp ( - 27r2rkT’/hQ). I n a relatively low field region where the hyperbolic sine can be approximated by the exponential function, the effect of collision broadening amounts apparently to raising the temperature by T’ in the thermal broadening factor. There are two kinds of contributions to the oscillatory part of the transverse conductivity, the one from the transitions for which the quantum nuiiiber N changes [the second term in Eq. (A24)] and the other from those for which N does not change [the fourth and fifth ternis in Eq. (A24)]. The relative magnitude of the two contributions depends upon the magnitude of (r ({) )s, since the fourth term in Eq. (A24) contains a factor which increases as In { hQ/47r (r({) ) s ) when (r({) ) s becomes small. Although the fourth term in Eq. (A24) contains an extra factor (hQ/2{’)lI2 compared with the second term, the increase of the factor In { hQ/47r (r(i-) I may overcome this extra factor. This may occur when {/fin is not so large and (r({))s is much smaller than ha. In this case, the oscillation phase may differ from 7r/4. Appendix B. Approximate Form of the Green Function in a Magnetic Field

We shall consider the asymptotic form of the Green function in a magnetic field in the limit 1 r - r’ I + 0. Remembering Eqs. (1.19) and (5.6), we can easily derive the exact expression

QUANTUM THEORY O F CALVANOMAGNETIC EFFECT

363

where the integer N E is defined as the smallest integer which is not smaller than E/(liQ) - 4; X N ( E ) stands for the de Broglie wavelength in the z-direction, X“E)

=

li/[2m 1 E - ( N

+ +)finI y2.

032)

According to Eq. (13.3), the function J N , N ( x ,[Y - ~ ‘ ] / 1 ~x’) , is essentially the Laguerre polynoinial : JN,N

(

2,

- y’ , 12

.’)

By the assumption of short-range force ( a << I ) , we need to consider only values of coordinates such that

Ir

-

Rj I << I ,

:. I r respect to 1 r

I r’ - Rj1 << I ,

If we retain the lowest-order term with J N , N ( ~ [y ,

- y‘]/P,

x’)

-

- r‘

I << 1.

(B4)

- r’ ] / I , we obtain (B5)

1.

I n the first sum and the first term of the second sun1 of Eq. ( B l ) we niay use this approximation and neglect the exponential factors in Eq. ( B l ) to obtain

*

N B-1

n

N=O

[2m{E - ( N

+ +)liQ]-J1/2

+

li

[am( ( N E

+ +)fin-

E]]”2’

(B6) For the remaining terms of the second sum we cannot use the same approximation, which leads us to a divergent series. We make use of the asymptotic expression

364

RYOGO KUBO, SATORU J. MIYAKE, AND NATSUKI HASHITSUME

which is valid for N >> 1, and approximate the sum over N integration with respect to N . The result is

> NE by the

in which we may replace the Bessel and the exponential function by one. Adding Eqs. (B6) and (BS), we obtain

Author Index The numbers in parentheses are footnote numbers and are inserted to enable the reader t o locate a cross reference when the author’s name does not appear at the point of reference in the text.

A Adams, E. N. 270, 289, 292, 293, 294, 295, 299(6), 304,305, 316 Adler, E., 238 Akamato, H., 62 Akhiezer, A., 260, 266 Alers, G. A., 223, 233 Allen, R. R., 168, 169(81) Al’tshuler, S. A., 138 Anderson, G. P., 228 Anderson, S., 155 Andreatch, P., Jr., 267 Andreev, V. V., 272 Andresen, H. G., 159, 206 Arenberg, D. L., 229 Argyres, P. N., 270, 272, 297(18) Artman, J. O., 154 Aruga, M., 251 Ashley, J. C., 254 Atsarkin, V. A., 182 183, 212 Aust, R. B., 58, 76 Auzins, P., 172, 196, 197, 200, 201 Azbel’, M. Ya, 271

B Baczynski, M., 81 Baker, J. M. 143, 151, 204 Bala, Y. B., 177 Balchan, A. S., 238, 43(56), 44, 55, 96(6), 127 Balkanski, M., 257 Ball, M., 190 Ballhausen, C. J. 138 Bancroft, D., 127 Banus, M. D., 53, 54(77) Baranskii, K. N., 228 Bardeen, J., 91 Barlow, C. A., Jr., 265

Barnes, R. G., 158, 160, 207 Barrett, C. S., 96 Bartschat, A., 247 Bateman, T. B., 232, 265, 267, 268 Bauer, R., 81 Baur, W. H., 155 Beale, J. R. A., 246, 247 Beard, W. G., 212 Bell, M., 216 Benedek, G. B., 131, 133 Bentley, W. H., 58, 76(85) Berlincourt, D., 252 Bernstein, B., 267 Bethe, H. A., 348, 349(41) Birch, F., 267 Blatt, F. J., 230 Bleaney, B., 143, 197, 216 Blitz, J., 267 Bloembergen, N., 141 Blotekjaer, K., 256 Blount, E. I., 243 Blumberg, W. E., 205 Boakes, D., 216 Bommel, H., 231 Bommel, H. E., 229, 260, 266 Bogle, G. S., 205 Bond, W. L., 334 Borgnis, F. E., 222 Bottcher, C. J. F., 8 Boyle, A. F. J., 128 Bratten, F. W., 229 Braun, P. B., 182, 184 Bridgeman, P. W. 3, 14, 24, 32(52), 89, 99, 100, 116 Browne, M. E. 171, 211 Bruce, J., 186 Brugger, K., 267 Brumege, R. H., 191 Brun, E., 183, 184, 212

365

366

AUTHOR INDEX

Bruner, L. J., 237, 238 Buck, O . , 267 Bundy, F. L., 54 Bundy, F. P.. 89, 96, lll(119) Burst,ein, E., 3, 8, 10, 235 Button, K. T., 233 Bychkov, Yu. A., 273, 340, 347, 360 C

Calhoun, B. A,, 150, 192(26), 201, 216 Callen, H. B., 185 Campbell, R. B., 72, 75(93) Carleton, H. R., 262 Carson, J. W., 189, 216 Carter, D. L., 159, 207 Celli, V., 265 Chambers, R. G., 271, 278 Chang, T., 207 Chester, M., 261 Chester, P. F., 158, 161, 206, 207 Chick, B. B., 228, 265 Chikvashvili, Y. A,, 231 Chubachi, N., 256 Claffy, E. W., 10 Claiborne, L. T., 218 Clendenen, R. L., 116, 117, 127 Clunir, D. M., 267 Cohen, M. H., 231 Cohen, M. L., 162 Collen, B., 155 Compton, W. D., 5 Conwell, E. M., 258 Corliss, L. M., 183 Cromer, D. T., 155 Cronemeyer, D. C., 154 Csavinsxky, P., 240, 242 Czakowski, M., 81 D Damon, R. W., 261, 262 Darnell, A. J., 53, 54(75) IYAubigne, Y. Merle, 197 Davisson, J. W., 8 Davydoff, A. S., 86 Davydov, B., 270, 272, 273, 295, 319, 322, 361 Debrunner P.. 128 deHaas, W. J., 270

Dekeyser, W., 10 deKlerk, J., 251 Delville, J., 155 de Mars, G. A., 165, 166, 168, 174, 175, 192, 211, 216 DePasquali, G., 128 Descamps, D., 197 Deutsch, T., 169, 171(85) de Wit, M., 202 Dexter, D. L., 15 Dietz, R. L., 202 Dingle, R. B., 2i1, 326, 356 Dobbs, E. R., 265, 266 Dobrov, W. I., 171, 211 Dorain, P. B., 202 Dransfield, K., 229, 231, 260, 266 Drever, R., 131 Drickamer, H. G., 2, 3, 5, 6(13), 9(3), 11, 12, 14(13, IS), 22, 38, 43, 44, 47(57), 49, 51, 52, 53, 55, 58, 60, 64(12), 76, 79, 88, 91, 96(6, 133), 97, 98, 104, 111, 114(137), 116, 118, 119, 127, 128 Dumke, W. P., 257 Dunitz, J. D., 182, 184(115)

E Eckstein, S. G., 231, 246, 254 Edge, C. K., 128 Edwards, A. L., 49, 51, 55(69) Efros, A. L., 316 Ehrenreich, H., 266 Ehrlich, P., 154 Einspruch, N. G., 218, 224, 227, 228, 240, 242, 265, 267 Eisinger, J., 205 Elbaum, C., 256, 262, 263,268 Emslie, A. G., 225 Eppler, R. A., 3, 6(13), 11, 12(13), 14(13), 18 Eros, S., 114 Esaki, I d . , 257 Estle, T. L., 202 Etzel, H. W., 10 Euler, F., 186

F Falicov, L. M., 114, 122 Feher, G., 141, 197, 240

367

AUTHOR INDEX

Fein, A. E., 229 Firsov, Yu. A,, 271, 272, 299, 306, 311, 316 Fitch, R. A., 2, 9(3) Fitzgerald, J. W., 228 Foldy, L. L., 225 Folen, V. J., 185, 212 Foner, S., 160, 204 Forgacs, R. L., 227 Forsgren, K. F., 2, Forster, T., 81 Foster, N. F., 251 Frackowiak, M., 81 Frank, N. H., 91 Frauenfelder, H., 128 Frederikse, H. P. R., 154, 270 Freeman, A. J., 133, 179, 180 Freiser, M. J., 150, 192(26), 216 Fritzsche, J., 72 Fry, D. J. I., 197 G Gabriel, J. R., 189 Gainon, D. A., 176, 211 Garrett, C. G. B., 62 Garton, G., 190, 216 Gatos, H. G., 53, 54(77) Geller, S., 53, 54(76), 177, 186 Gerritsen, H. J., 158, 159, 160(46), 161, 170, 206, 207 Geschwind, S., 154, 170, 187, 188, 189, 204, 205, 216 Geusic, J. E., 204, 205 Gibbons, D. F., 122 Giess, E. A,, 216 Giess, M. J., 150, 192(26) Gilleo, M., 186 Gintler, R J., 10 Gobrecht, H., 247 Goodenough, J. B., 114, 182 Gossard, A. C., 133 Granicher, H., 162, 177, 179 Granato, A., 225, 256, 265 Grant, F. E., 154 Greebe, C . A. A. J., 243 Gregg, D. W., 2, 79 Griffith, J. S., 19, 138 Grfiths, J. H. E., 197 Grinberg, A. A., 254

Gurevich, V. L., 254, 271, 272, 299, 306, 311, 316 Guyer, R. A., 261

H Hawing, R. R., 257 Hafner, S., 183 Hajdu, J., 272 Hall, H. E., 128 Hall, H. T., 111 Hall, J. J., 242 Hall, T. P. P. 180, 211 Ham, F. S., 91, 141 Hanneman, R. E., 53, 54(77) Hanner, S. S., 128, 132(167), 133(167) Harper, P. G., 274 Harrison, M. J., 231 Harrison, S. E., 158, 206 Harrison, W. A., 112, 122, 231 Hasegawa, H., 270, 272, 274(5), 281(5), 284(5), 307(20), 314(20), 320(5), 339(5) Hashitsume, N., 270, 272, 274(5), 281(5), 284(5), 307(20), 314(20), 320(5), 339(5) Hass, M., 212 Hastings, J. M., 183 Hauser, W., 145 Hayes, W., 143, 180, 197, 211 Heberle, J., 128, 132(167), 133(167) Held, J., 81 Heller, G. S., 160 Henuis, B. W., 10 Herman, R., 90 Herpin, A., 191 Herrington, K., 155 Hersh, H. N., 10 Herzfeld, K. F., 224 Hickernell, F. F., 256 Hofner, S., 212 Hogarth, C . A., 267 Holroyd, L. V., 201 Holstein, T. D., 243, 270, 289, 292, 293, 294, 295, 299(6), 304, 305, 316 Holton, W. C., 202 Hooke, R., 268 Hopefield, J. J., 257 Hornig, A. W., 165, 166, 171, 211 Hoskins, R. H., 204

368

AUTHOR INDEX

Hosler, W. R., 162 Hubner, K., 179 Hugenholta, N. M., 320,321(35) Hughes, V.W., 225 Hull, G., Jr., 53,54(76) Huntington, H. B.,219,225,227,268 Hurdle, B.G., 228 Hutchins, M.T., 141,151, 152(29), 190, 191,216 Hutson, A. R., 247,252,257,259 Hyman, R.A., 43

I Inokuchi, H., 62 Ishiguro, M., 10,ll(27) Ivey, H.F.,3,6

J Jablonski, A., 81 Jaccarino, V.,131 Jacobs, I. S.,3 Jacobsen, E.H., 229 Jaffe, H., 219,252 Jaffe, H. H., 74 Jamieson, J. C., 52,53,89,127 Jayarman, A., 89,101,103(117), 111 Joel, N., 219 Johnson, P. D., 15,17 Johnston, D. F.,189 Jona, F.,162,176(66) Jones, H., 114 Jones, R. V., 189,192(142) Jonker, C. H., 162 J$rgensen, C. J., 138 Joshi, S. K., 266 Jura, G., 89,111, 130

K Kabler, M. N., 265 Kanzig, W., 162 Kafalas, J. A., 53,54(77) Kaganov, M.I., 271 Kahn, A. H., 270,273,340 Kallman, H., 62 Kamimura, H., 15,202 Kanamori, J., 147 Kasha, M., 79 Kasper, J. S.,54 Kats, M.L., 10 Kaufman, L., 89,127

Kazarinov, R. F, 254 Kearns, D. R.,62 Keating, K. B.,22 Kelly, E. F., 251 Kelly, R. H., 212 Kennedy, A. J., 236 Kennedy, G. C., 89, 101(117), 103, 111(117) Keyes, R.W., 231,232,235,237,238 Kikuchi, A., 168 Kikuchi, C., 204 Kikuchi, T.T., 205 Kikuchi, Y . , 256 Kirkpatrick, E.S.,172,211 Kiro, D., 178,179(105), 211 Kisliuk, P.,170,204 Kittel, C., 288 Klain, M.P.,170,204 Klement, W., Jr., 89,101(117), 103(117), lll(117) Iilevens, H. B.,59 Klick, C. C., 15 Knox, K., 179 Knox, R. S.,15 Kobayashi, J., 176 Kodera, 168 Koehler, W. C., 162,191 Koga, I., 251 Kogan, S.M., 243 Kogan, V.D., 254 Kohn, W., 337,339(40) Kolopus, J. L., 201 Kornienko, L. S.,204,205 Kosevich, A. M., 272 Koster, G. F.,145 Kozyrev, B.M., 138 Kroger, H., 261,262,263,264 Krumhansl, J. A., 261 Kubo, R.,257,270,272, 274(5), 280(4), 281(5), 284(5), 307(20), 314(20), 320(5), 339(5) Kuglaustierno, U.,155 Kuno, T.O., 10,ll(27) Kyame, J. J., 251 Kyi, R., 161,207

L Lacroix, R., 158,206 Lamb, J., 265

369

AUTHOR INDEX

Lambe, J., 204 Langenberg, D. N., 235 Laurance, N., 204 Laval, J., 219 Lawson, A. W., 89, 90, 104(126), 127, 251 Lax, B., 233, 234 Lea, K. R , 146 Leask, M. J. M., 146, 151, 152(29), 190, 191(146), 216 Le Comber, P. G., 262 Le Corre, Y., 219 Le Grand, C., 155 Levy, P. W., 264 Lewis, G. N., 79, 85(100) Lewis, H. R., 206 Libby, W. F., 53, 54(75), 79 Lifshite, I. M., 271, 272 Lihkter, I., 90, 104(127) Lin, C. C., 191 Lmares, R. C., 141 Lipkin, D., 79, 85(100) Lippmann, B. A., 337, 339(40) Litster, J. D., 133 Llewellyn, P. M., 197 Loelinger, H., 183, 184, 212 Long, T. R., 114 Love, A. E. H., 218 Low, W., 138, 159, 166, 167, 169, 170, 171, 172, 173, 174(82), 175(82), 178, 182, 183, 189, 196, 197, 200, 201, 204, 205, 211,212 Ludwig, G. W., 141 Lucke, K., 265 Luther, R., 72 Luttinger, J. M., 274, 337, 339(40) Lynch, R. W., 118, 119

M McClure, D. S., 19, 33, 60, 81(44), 86, 182, 184(114) MacDonald, J. R., 217 McDougall, J., 240 McFee, J. H., 252, 253(115), 258 McQueen, R. W., 116, 120(152) McSkimin, H. J., 225, 228, 267, 334 McWhan, D. B., 53, 54(76), 89, 101 Maenhoubvan der Vorst. W.. 10 Magel, T., 79, 85(100) Magneili, A., 155

Maisch, W. G., 3, 11(12), 64(12) Maita, J. P., 334 Manenkov, A. A., 158 Manning, R. J., 227, 267 Marcus, P. M., 236 Mariano, A. N., 53, 54(77) Marshall, S. A., 205 Marshall, W., 133 Martin, J. E., 53, 55 Mason, W. P., 232, 265, 267, 268, 334 Massey, H. S. W., 336 Matino, H., 262 Mavroides, J. G., 233 Meijer, J. G., 316 Merial, P., 191 Merrill, L., 111 Mertsching, J., 243 Merz, W., 166 Meyer, H., 189, 192(142) Midford, T. A., 254 Mikoshiba, N., 231, 243, 244 Miller, A., 182, 184(116) Miller, B. I., 266 Milyaev, V. A., 158 Minomura, S., 5, 12, 14(18), 22, 52 Minshall, S., 127 Miyake, S. J., 257, 272, 319, 336(34), 338(34), 343(34) Mossbauer, R. L., 128 Mollwo, E., 3, 12 Momo, L. R., 160 Moore, A. R., 260 Morin, F. J., 334 Mott, N. F., 43, 336 Mrguclich, J. M., 47 Muller, K. A., 158, 168, 169, 170, 171, 172, 173, 177, 179, 206, 211 Murphy, J. C., 154 Musha, T., 254

N Neighbours, J. R., 223, 229 Newton, R. C., 89 Nicol, M., 130 Nielsen, J. W., 189, 216 Nine, H. D., 247, 248 Ninomiya, T., 265 Northrup, D. C., 68

370

AUTHOR INDEX

0

R

Oberly, J. J., 3, 8, 10 Offenbacher, E. L., 146 Okada, J., 262 Okamoto, S., 176 Okaya, A., 157, 158, 159, 207 Olsen, K. M., 334 Onsager, L., 277 Orchin, M., 74 Orgel, L. E., 138, 182, 184(115) Orten, J. W., 172, 197, 200, 201 Overmeyer,J., 150, 182, 192, 201, 212, 216, Owen, N. B., 55

Ragajopal, E. S., 251 Ray, T., 145 Rayleigh, Lord, 224 Redwood, M., 265 Rei, D. K., 158 Reinberg, A. R., 205 Reitz, J. R., 114 Remeika, J. P., 154, 170, 204, 205 Rempel, R. C., 165, 166(74), 170, 211 Rensen, J. G., 196 Riabinin, N., 90, 104(127) Rice, M. H., 116, 120(152) Riggleman, B. M., 38, 43(57), 47 (57), 88 Rimai, L., 165, 166, 168, 169, 171, 174, 175, 192, 211, 216 Ringo, G. R., 228 Robertson, J. M., 72, 75(93) Rodrique, G. P., 189, 192(142) Rosegarten, G., 189 Rosenvasser, E. S., 197 Roth, L. M., 270 Rubins, R. S., 169, 172, 173, 174(82), 175(82), 200, 211 Rupp, L. W., Jr., 202 Ryan, D., 190, 216

P Pace, J. H., 158 Pailloux, H., 251 Pappalardo, R., 140, 191 Paranjape, B. V., 254 Parmenter, R. H., 243 Parsons, R. W., 22 Paul, W., 34, 48(54), 89, 104(125) Pauthenet, R., 185 Pearson, R. F., 192 Peierls, R. E., 274 Perez-Albuerne, E. A., 2, 118 Peschanskii, V. G., 271 Peterson, E. L., 127 Pipkorn, D. N., 128, 130 Pippard, A. B., 254 Pittelli, E., 185 Platt, J. R., 59 Pokatilov, E. P., 231 Polder, D., 316 Pomeranchuk, I., 270, 272, 273, 295, 319, 322, 361 Pomerantz, M., 231, 246, 247, 258 Portis, A. M., 133 Pound, R. V., 131 Powell, M. J. D., 189 Preston, R. S., 128, 132, 133(167) Prohofsky, E. W., 261, 262 Prokhorov, A. M., 158, 160, 204, 205, 206, 207 Pua, J., 258

Q Quate, C. F., 254, 256

S Sabisky, E. S., 158, 159, 160(46), 161, 170, 206, 207 Sakadi, Z., 222 Sakiotis, N. G., 256 Sakudo, T., 167 Samara, G. A., 52, 53, 55(78), 58, 60, 88 Sampson, D. F., 158 Sanders, T. M., 244, 245(95) Sandle, W. J., 133 Sandomirskii, V. B., 243 Sasaki, W., 244 Sawaguchi, E., 168 Schneider, J., 202 Schoolery, J. F., 162 Schreiner, W. N., 212 Schulman, J. H., 15 Schulman, R. G., 179 Schwinger, J., 337, 339(40) Seeger, A., 267 Seiden, P. E., 231 Seitz, F., 5, 15, 44

371

AUTHOR INDEX

Sugano, S.,15,139,179 Seki, H., 225 Suss, J. T., 173,200,201,205 Shaltiel, D., 165,166,211 Swenson, C. A., 90 Shapira, Y., 233,234 Swim, R. T., 233 Shapiro, L., 146 Syrnmons, H. F., 205 Shirnizu, R. N.,47 Shiozawa, L., 252 T Shirane, G., 162,176(66) Takirnoto, N., 243 Shockley, W., 271 Tanabe, Y., 139 Shoenberg, D., 233 Tang, T . Y.,90,104(126) Shteinshleifer, Z., 265 Tannenwald, P.E., 217 Shubnikov, L., 270 Tansal, S.,257 Shulman, J. H., 5,10 Teal, G. K., 334 Shuskus, A. J., 200 Tell, B., 263 Sierro, I.,158,206 Terenin, A. N., 62 Silver, M., 62 Terhune, R. W., 204 Silverman, B. D., 169,171(85) Tetonico, L. J., 228 Simpson, O., 68 Teutonico, L.J., 264,265 Sircar, S.R., 202 Skobov, V. G., 254,273,340,342,343,347 Thaxter, J. B., 160,217 Thomson, R.,265 Sladek, R.,354 Slykhouse, T. E.,2, 9(3), 44, 49, 51, Thorp, J. S.,158 Thruston, R. N.,267 55(69) Tischer, R. E.,22 Smakula, A., 9,12 Titeica, S.,270,272,283,290(2), 307,308, Smith, C. S.,114,219,229,237 357 Smith, P.L.,53,55 Trotter, J., 72,75(93) Smith, R. W., 258 Touchard, L.J., 236 Smunk, R.E., 114 Toupin, R. A., 267 Soffer, B.H., 204 Toxen, A. M., 257 Sokolnikoff, I. S.,218 Truell, R.,224, 225, 228, 248, 256, 262, Sondheimer, E.H., 306 263,264,265,268 Souers, P. C., 89,111 Tso, R., 254 Spear, W.E., 262 Spector, H.N., 231,254 U Stager, R.A., 91,96(133),97,98,104,111, 114(137) Ueda, R., 176 Stahl-Brade, R., 159,182,183,212 Unoki, H., 167 Statz, H., 145 V Steigmeier, E. F.,236 Van den Beukel, A., 243 Stein, F., 228 van Hove, L., 320,321(35) Stephens, D. R.,22 Van Santen, J. H., 162 Sternheimer, R.,90 Van Trleck, J. H., 148,191,192 Stevens, K. W.H., 142 Van Wieringen, J. S., 196 Stevenson, R. W. H., 180,211 Veda, W., 10,ll(27) Stiles, P. J., 235 Vereschaguin, L.F., 90,104(127) Stinchecombe, R.B., 271 Verma, G. S.,266 Stoner, E. C., 240 Vieth, R. F.,171,211 Strauss, W., 249 T'oigt, W., 235 Strong, H.M., 89,lll(119) Volterra, V., 157,159 Sturge, M.D., 202 von Himel. A.. 15.154 Suchan, H. L., 38,44 .A

I

I

I

372

AUTHOR INDEX

W

Williams, A. O., Jr., 225 Williams, F. E., 15, 17 Williams, F. I. B., 151 Wilson, A. H., 306 Wilson, D. K., 240 Witterholt, E. J., 224 Wittke, J. P., 206 Wolf, W. P., 146, 149, 151, 152, 185, 190, 191, 192, 216 Wollan, E. O., 162 Wood, D. L., 140, 170, 191, 204 Woodbury, H. H., 141 Woodruff, T. O., 266 Wooster, W. A., 219 Wright, A. J., 55 Wyatt, A. F. C., 151, 152(29), 190, 191(146), 216

Wada, M., 256 Waldner, F., 183, 184, 212 Waldron, R. D., 257 Walerys, H., 81 Walker, L. R., 131 Walsh, J. M., 116, 120(152) Walsh, W. M., Jr., 196, 202 Wang, W. C., 246, 258 Warekois, E. P., 53, 54(77) Warschauer, D. M., 34, 48(54) Waterman, P. C., 223, 225, 228, 230 Watson, R. E., 133, 179, 180 ' Weaver, H. E., 165, 166(74), 170, 211 Weber, M. J.; 168, 169(81) Weger, M., 141, 197 Weigert, F., 72 Weinreich, G., 231, 243, 244, 245 Y Wemple, W., 171, 177, 211 Wentorf, R. H., 54 Yamaka, E., 158, 160, 161, 207 Wertheim, G. K., 131 Yariv, A., 202 Wertz, J. E., 172, 196, 197, 200, 201 Ymg, C. F., 224 West, F. G., 218 Yoshida, E., 244 White, D. L., 248, 249, 250, 252, 253, 254, Yoshinaka, Y., 251 256, 257 Z White, H. G., 244, 245(95) White, M., 10 Zahner, J. C., 22, 43 White, R. L., 151, 189, 192, 216 Zilberman, G. E., 274 White, R. M., 160, 224 Ziman, J. M., 271 Wickersheim, K. A,, 151, 152, 192 Zusman, A., 178, 211 Wiederhorn, S., 38, 58 Zverev, G. M., 158, 160, 204, 205,206,207 Wilkens, J., 180, 211

Subject Index

A Absorption Edge Pressure Effect, Compounds, 43ff Pressure Effect, Elements, 38 Pressure effect, Semiconductors, 48ff A Center Alkali Halides, 10 Acoustic Amplifiers, 256 Acoustic Phonons, 307ff Acoustic Scattering, 224, 304ff Acoustoelectricity Cadmium Sulfide, 262 Semiconductors, 243ff Akhiezer Loss, 259 Alkali Halides Color Centers, 2 Phase Transition Effect, 5ff F Center Density Effect, 3ff Impurity Color Centers, 10 M Bands Density Effect, 7 Phase Transitions, 8 Thallium Activated Pressure Effect, 15ff Alkali Metals Phase Transitions, 91ff Resistivity-Pressure Effect, 9lff Alkaline-Earth Metals Phase Transitions, 98 Resistivity-Pressure Effects, 98ff Aluminum Oxide Transition Metal Ions ESR, 204 Anisotropy Energy Single-Ion Contribution, 147ff Anisotropy Modulus Semiconductor Compounds, 236 Anthracene Optical Absorption Pressure Effect, 60 Photodimerization, 72

Aromatic Hydrocarbons Optical Absorption Pressure Effect, 58ff Arsenic Band Gap Pressure Effect, 38 Azulene Optical Absorption Pressure Effect, 64

B Band Gap, 36 Pressure Effect Iodine, 40 Non-Metals, 38 Band Structure Hexagonal Metals, 1126 Pressure Effects, 34iT Semiconductors, 50 Silicon, 242 Types, 36 Barium Phase Diagram, 102 Resistivity-Pressure Effect, 101 Barium Fluoride Color Centers Pressure Effects, 12ff Barium Titanate Dielectric Constant, 164 ESR, 164ff, 208 Phase Transition, 164 B Center Alkali Halides, 10 Bismuth Magnetoresistance, 322 Born Approximation Magnetoresistance, 336 Brillouin Zone, 37 Hexagonal, 112

C Cadmium Fermi Surface, 122ff Resistivity-Pressure Effect, 119 373

374

SUBJECT INDEX

Cadmiun Selenide Ultrasonic Amplification, 254 Cadmium Sulfide Absorption Edge Pressure Effect, 55 Acoustoelectric Effect, 246, 262 Piezoelectric Interaction, 247 Ultrasonic Amplification in, 253 Ultrasonic Waves, 247, 259 Calcium Phase Transition, 100 Resistivity-Pressure Effect, 99 Calcium Fluoride Color Centers Pressure Effects, 12ff Calcium Oxide Rare-Earth Ions ESR, 200 Transition Metal Ions ESR, 198 C-Center Alkali Halides, 10 Cerium Resistivity-Pressure Effect, 104 Cesium Resistivity-Pressure Effect, 97 Color Centers Alkali Halides, 2 Phase Transition Effect, 5 6 Barium Fluoride Pressure Effects, 128 Calcium Fluoride Pressure Effects, 12ff Conductivity Magnetic Field Effect, 281ff Conductivity Tensor, 280 Scattering Operations, 337ff Corundum See Aluminum Oxide Crystal Field Octahedral Splitting, 138 Potential Calculation, 142 Tetrahedral Splitting, 140 Crystal Field Energy Compressibility Relation, 3 1 Crystal Field Splitting Pressure Effects, 21ff Rare Earth Ions, Pressure Effects, 28ff Crystal Field Theory, 19ff

Cuprous Halides Absorption Edge Pressure Effects, 51 Cyclotron Mass Electrons in Germanium, 334 Cyclotron Motion Electrons, 274ff

D Davydov-Pomeranchuk Theory, 319 Magnetoresistance, 322ff Davydov Splitting Cyanine Spectra, 86ff Debye Temperature Semiconductor Compounds, 237 D Center Alkali Halides, 10 deHaas-van Alphen Effect, 233, 270 Collision Broadening, 360 deHaas-Shubnikov Oscillation, 326 Density of States Magnetic Field, 291 Depletion Layer Transducer, 248 Diamond Crystals Elastic Constants, 234 Diffusion Layer Transducer, 251 Dysprosium Resistivity-Pressure Effect, 108

E E Center Alkali Halides, 10 Elastic Constants Crystal Classes, 220 Diamond Crystals, 234 Lava1 Theory, 219 Pulse-Echo Terhiques, 226 Semiconductor Compounds, 237 Elastic Waves, 218 Scattering, 224ff Electrical Conductivity Magnetic Field Effect, 281ff Tensor, 280 Electron-Electron Interaction, 317ff Electron-Phonon Interaction, 299ff Electronic Structure Pressure Effects, pp. 1-133 Electrons Elastic Scattering, 286ff Magnetic Field Motion, 274ff

375

SUBJECT INDEX

Electron Spin Resonance See ESR Energy Gap Iodine Pressure Effect, 140 Erbium Resistivity-Pressure Effect, 111 ESR Barium Titanate, 164ff Garnet, 185ff Inorganic Crystals, 138ff Ions, Data Tables, 194ff Oxides, Magnetic Ions in, pp. 135-215 Perovskites, 162ff Rare-Earth Ions, 145 Rutile, 156ff Strontium Titanate, 167 Europium Resistivity-Pressure Effect, 106

F F Center Frequency Density Effect, 2ff Fermi Energy, 36 Fermi Surface Hexagonal Metals, 112ff Magnetic Field Relation, 278 Semiconductors, 35 Ferrimagnetism Garnet, 185 Fluorescein Optical Properties, 79ff

G Gadolinium Resistivity-Pressure Effect, 108 Gadolinium Garnet Magnetic Anisotropy, 150 Gallium Antimonide Absorption Edge Pressure Effects, 49 Band structure, 50 Gallium Arsenide Acoustoelectric Effect, 246 Gdvanomagnetic Effect Theory, pp. 269-364

Chrn et Crystal Structure, 186 ESR, 185ff Ferrimagnetism, 185 Magnetic Anisotropy, 150 Rare-Earth Ions ESR, 190, 215 Transition Metal Ions ESR, 188, 214 (hmanium Absorption Edge Pressure Effects, 48ff Acoustoelectric Effect, 244 Band Structure, 50 Elastic Constants Electronic Effects, 238 Gruneisen Constant, 266 Intervalley Scattering, 245 Magnetoresistance, 334 Phonon, Propagation in Impuritv Effects, 231 Resistance Pressure Effect, 52 Shear Modulus Impurity Effects, 238 Sound Velocity, 334 Gigacycle Techniques Microwave Ultrasonics, 228 Glass Transition Metal Ions Crystal Field Splitting, 31 Graphite Phase Transition, 76ff Resistance Pressure Effect, 75ff Green Function Magnetic Field, 362 Gruneisen Constant Semiconductors, 266

H Hall Effect, 283 Hankel Function, 301 Hexacene Resistance Pressure Effect, 70 Hexagonal Crystal Wave Propagation, 223 Hexagonal Metals Electronic Structure, 1128

376

SUBJECT INDEX

Lithium Phase Transition, 92 Resistivity-Pressure Effect, 91 Lithium Spinels ESR, 184

Hexagonal Structure Brilloun Zone, 112 Jones Zone, 113 Holmiun Resistivity-Pressure Effect, 108 Hooke’s Law, 218 Hypersonic, 218

M

I

Indium Arsenide Magnetoresistance, 354 Intervalley Scattering Germanium, 245 Iodine Band Gap Pressure Effect, 40 Resistance Pressure Effect, 39 Temperature Effect, 40 Ionic Crystals Impurity Centers, 2ff Iron Electronic Structure, 127ff Hexagonal Phase, 127 Mossbauer Effect, 128 Iron Group Ions Crystal Field Effects, 138ff

J Jahn-Teller Effect Strontium Titanate, 172 Jones Zone Hexagonal Structure, 113

L Landau Levels, 274 Collision Broadening, 272 Land6 Rule, 145 Lam6 Constants, 222 Lanthanum Aluminate ESR, 177 Lava1 Theory Elastic Constants, 219 Lead Tantalate ESR, 176 Lead-Telluride Magnetoacoustic Resonance, 233 Ligands, 19

’

Magnesium Band Structure, 114 Resistivity-Pressure Effect, 114 Magnesium Aluminate ESR, 182 Magnesium Oxide Transition Metal Ions ESR, 194 Spectra, 23 Magnetic Anisotropy Single-Ion Contribution, 147ff Magnetic Field Green Function, 362 Magnetoacoustic Resonance Lead Telluride, 233 Magnetoresistance, 281ff Bismuth, 322 Born Approximation, 336 Collision Broadening, 353ff Davydov-Pomeranchuk Theory, 322ff Field Dependence, 292ff Germanium, 334 Indium Arsenide, 354 Partial-Waves Method, 343 Skobov-Bychkov Theory, 340 M Bands Alkali Halides Density Effect, 7 Pressure Effect, 5 M Center Structure, 5 Mercury Halides Absorption Edge Pressure Effect, 44tf Metallic Transition Pressure Effect, 34ff Metals Resistivity-Pressure Effect, 89ff Microwave Ultrasonic, 218 Mossbauer Effect, 91 Iron, 128

377

SUBJECT INDEX

N Neodynium Resistivity-Pressure Effect, 105

0 Olivine Absorption Edge Pressure Effect, 57 Optical Absorption Organic Compounds Pressure Effect, 57ff Optical Absorption Edge Pressure Effect, Compounds, 43ff Pressure Effect, Elements, 38 Pressure Effect, semiconductors, 48 Optical Phonons Piezoelectric, 316ff Optical Spectra Tetrahedral Ions, 140 Organic Crystals Electrical and Optical Properties, 57ff Organic Phosphors Optical Properties, 79ff Organic Semiconductors Pressure Effects, 626 P Paramagnetic Resonance Tetrahedral Ions, 141 Pentacene “Metallic” Transition, 69 Optical Absorption Pressure Effect, 62 Resistance Pressure Effect, 66ff Perovskites ESR, 162ff, 208 Structure, 163 Phase Transitions Alkali Metals, 9lff Alkaline-Earth Metals, 98 Phonons Acoustic, 307ff Electron Interactions, 299ff Optical Piezoelectric Interactions, 316

Phonon Drag Semiconductors, 243 Phonon-Phonon Interactions Semiconductors, 265 Phonon Propagation Semiconductors Impurity Effects, 231 Phosphorus Band Gap Pressure Effect, 38 Photodimerization Anthracene, 72 Piezoelectrics Equations of State, 254Polarization-Stress Relations, 248 Piezoelectric Interaction Semiconductors, 247 Piezoelectric Scattering, 304ff Potassium Phase Transition, 94 Resistivity-Pressure Effect, 93 Potassium Chloride Color Bands Pressure Effect, 4 Potassium Halides M Bands Pressure Effects, 6ff Phase Transition, 19 Thallium Activated Pressure Effects, 15ff Potassium Magnesium Fluoride ESR, 179 Praseodynium Resistivity-Pressure Effect, 104 Pressure Electronic Structure Effects, pp. 1-33 Pulse-Echo Technique Elastic Constants, 226 Pyrolytic Graphite Resistance Pressure Effect, 75ff

R Racah Parameters, 19 Manganese Salts Pressure Effects, 25

378

SUBJECT INDEX

Rare Earth Ions Calcium Oxide ESR, 200 Crystal Field Splitting Pressure Effects, 28ff Garnets ESR, 190, 215 Ground States, 146 Perovskites ESR, 176, 208 Resistivity-Pressure Effect, 103ff Rutile ESR, 161 Strontium Titanate ESR, 173 Rubidium Resistivity-Pressure Effect, 96 Ruby Spinel ESR, 182 Rutile Crystal Growth, 154 Crystal Structure, 154 ESR, 156ff Rare-Earth Ions ESR, 161 Transition Metal Ions ESR, 158, 206

S Samarium Resistivity-Pressure Effect, 106 Serond-Sound Propagation in Crystals, 261 Selenium Band Gap Pressure Effert, 38 Resistance Pressure Effect, 11 Semiconductors Absorption Edge Pressure Effect, 48ff Acoustoelectric Effect, 243ff Band Strurture, 50 Grimeisen Constant, 266 Phonon Drag, 243ff Phonon-Phonon Interactione, 265 Piezoelectric Interaction, 247 Ultrasonics, pp. 217-268 Ultrasonic wave attenuation, 2308

Semiconductor Compounds Anisotropy modulus, 236 Debye Temperature, 237 Elastic Constants, 237 Semiconductor Compounds, 111-V Resistance Pressure Effect, 53 Shubnikov-deHaas Oscillation, 270 Si1icon Absorption Edge Pressure Effects, 48ff Band Structure, 50, 242 Elastic Constants Electronic Effects, 239 Elastic Properties Radiation Effects, 265 Phonon Propagation Impurity Effects, 232 Resistance Pressure Effect, 52 Silver Halides Absorption Edge Pressure Effect, 44ff Silver Iodide Phase Transitions, 46 Skobov-Bychkov Theory Magnetoresistance, 340 Sodium Resistivity-Pressurc Effect, 92 Sodium Chloride M Bands Pressure Effects, 6ff Silver-dopcd Color Centers, 11 Sound Velocity Measurement, 228 Spinels Crystal Strurture, 180 ESR, 180ff Inverse, 182 Normal, 181 Transition Metal Ions ESR, 212 Stark Levels Tetrahedral Sites, 140 Strontium Phase Transition, 101 Resistivity-Pressure Effect, 100 Strontium Oxide Transition Metal Ions ESR, 201

379

SUBJECT INDEX

Strontium Titanate Dielectric Constant, 167 ESR, 167, 208 Transition Metal Ions ESR, 169 Sulfur Band Cap Pressure Effect, 38

T Terbium Resistivity-Pressure Effect, 108 Tetracene Optical Absorption Pressure Effect, 61 Thallium Halides Absorption Edge Pressure Effects, $2 Thulium Resistivity-Pressure Effrct, 111 Titanium Oxide See Rutile Transducers Depletion Layer, 248 Diffusion Layer, 251 Fabrication, 230 Transition Elements Oxides, Spectra, 153ff Transition Metal Ions, 22 Aluminum Oxide ESR, 204 Barium Titanate ESR, 165 Calcium Oxide ESR, 198 Crystal Field Effc cts, 138ff Crystal Field Splitting Pressure Effects, 20ff Garnets ESR, 188, 214 Glass Crystal Field Splitting, 31 Magnesium Oxide ESR, 194 Perovskites ESR, 176ff, 208 Rutile ESR, 158, 206 Spinel ESR, 182, 212

Transition Metal Ions Strontium Oxide ESR, 201 Strontium Titanate ESR, 169

U Ultrasonics Gigacycle Techniques, 228 Pulse-Eeho Technique, 227 Semiconductors, 217-268 Ultrasonic Amplification, 251ff Ultrasonic Double-Refraction Technique, 264 Ultrasonic Waves, Attenuation, 224 Diffraction EtTects, 225 Semiconductors Attenuation in, 230ff

V Violanthrene Absorption Spectra, 74 Y Ytterbium Resistivity-Pressure Effect, 111 Yttrium Garnets ESR, 188ff

z Zinc Fermi Surface, 125ff Resistivity-Pressure Effect, 119 Zinc Aluminate ESR, 182 Zincblende Structure, 51 Zinc Compounds, 11-VI Absorption Edge Pressure Effect, 50 Resistance Pressure Effect, 54 Zinc Oxide Transition Metal Ions ESR, 203 Zinc Sulfide Transition Metal Ions Spectra, 26

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