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 Ryo...

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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.

This Page Intentionally Left Blank

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


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


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 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


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


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.


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


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-'.


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 .


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).


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.


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


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.


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.


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

I

I

I

I

I

I

I

I

I

I\ 0

7 \

-

I\\ I

\-I

-1000-

\

\

]

-

CUCl

do=25.080 CM-'

\\

5 -2oooI

\ \

3 " 4

.,.

-3000-

-

CWI

-4000

-5000

1 ,

-

o

I

20

I

40

I

60

I

ao

I

100

I

o.

120

.

'23,550 CM-'

-

--. I

140

I

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


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


,

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).


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


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).


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.


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.


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.

I4>300

0

20

40

60

80

100

0

P I KILOBARS

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

"0

I00

200

300

400

5

P,KILOBARS

FIG.69. Resistance versus pressure for powdered pentacene.


67

T, OK

I

I50

700

"

125

80

I00

Y

1

I

I P-208 KB

105

2 P.213 KB 3 P.221 KB 4 P=377 K B

i

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,

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- * ; $ I10 TRANSFORMATION INITIATES

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b

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.

-

200 -

-

d

TEMP, OK

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).


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.


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'

0

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


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


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).


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.


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.


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.


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).


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


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

I

100

200

300

400

P, KILOBARS

I

5M)

FIG.92. Resistance versus pressure for sodium.

i

I

1 600


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).


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).


r

I

I

I

97

I

28-

1.21

0

I

I00

I

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).


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


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

I

I

I

I

I

0

I

I

I

I

I

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

I

I00

I

200

I

I

300 400 P, K I LOBARS

I

500

FIG.102. Phase diagram for barium.

I

600

1

103


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---+

I

296'K

A

/

h

\

o

ISOBARS

\ 0 21

0

I

I 00

I

200

I

I

3nr) 400 P, KILO ?A Fc')

I

.LOP

600

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).


105

1.0-

w V

z

W [L

0.8-0=0-0--0-,,

0 7. 0

I

I

I00

200

I 400 3 00 P .K ILO B A RS

5 00

600

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

,

-- -

ISOTHERMS ISOBARS 7 7 ° K ISOTHERM BASED 011 ISOBARS

06t

I

0

1

I00

200

390

L 4CO 500 600

P, KILOBAF(5

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

I

1

I

I

o

0

I00

200

_ _ II ' 300 400 P . K I L O B A R5

500

I

ISOTHERMS ISOBARS

600

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


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

I

1

I

I

12-

o

A

1 ~

z

Q

~

;

2

-

ISOBARS

0

V lo' w 8

I

ISOTHERMS

6

-

'

-

K

-

G 6-

Ya

-

4-

0 0

-

77'K

2-

I

I

I

I

I

I

I

I00

200

300

400

500

600

700

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.

I.o W

U

z

2

9 0.8 VI

W

[L

0.6

0.4L 0

I

100

I

I

200

300

I

400 P , K I LOBARS

I

500

I

600

I

700

I

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

I

I

1

I

I

300

400

500

600

700

P,KILOBARS

FIG.112. Resistance versus pressure for erbium.

110

H. G . DRICKAMER

21

I

I

0

I00

200

I

I

300 400 P,KI LO B A RS

I

I

500

GOO

i

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


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).


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


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).


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.


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


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


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.


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


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.


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


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


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


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 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).


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


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


h

Pd1+

X

77

2.1698


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. 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 > a; (1.2a) and case (B) long-range force (slowly varying potential),

1 > 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


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


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

> 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-


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


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-’

(ls= 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).


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 > 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 > 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 >> 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)


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 > fiw/l

and

> k T >> fiw/l is

8mw2(> 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: > fiw/l). It is obvious that the conditions E , >> fiw/l

312

RYOGO KUBO, SATOHU J. MIYAKE, AND NATSUKI HASHITSUME

and 2mv,w > fiw/l and k T >> 8mw2, respectively. (B) When k T > k T and 2mv,w > 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> p

- fiQ/2

Q

>>

kT

=

>>

eH/m,c) :

(mcw2fiQ)112,

=

(8.33)

315


>> 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

{ - 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).


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


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) a ) ,the solution ( G N ~ . (f does not depend appreciably upon p,, so long as I p , I . -

( 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)


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 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


€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.


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


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


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.


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 {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

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


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).


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 , [ > 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)

> 1, b > 0,c

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