SOLID STATE PHYSICS VOLUME 11
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SOLID STATE PHYSICS Advances in Research and Applications Editors FREDERICK SEITZ
DAVID TURNBULL
Department of Physics University of Illinois Urbana, Illinois
General Electric Research Laboratory Schenectady, New York
VOLUME 11
I960 AN A C A D E M I C PRESS REPLICA REPRINT
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Contributors to Volume i t
RICHARD H . BUBE,RCA Laboratories, Princeton, New Jersey
G.A. BUSCH, Laboratoriumfur Festkhperphyaik, Eidgenbs8ische Technische Hochschule, Zurich, Switzerland
R. KERN,Lmborcrton’um fur Festkhperphysik, Eidgenbssi8che Technische Hoehchule, Zurich, Switzerland
ROBERTw. KEYEB, Research hboratin’ies, Westinghouse Electric COTporation, Pittsburgh, Pennqlvania
BENJAMIN LNC,Lincoln Laboratmy, Massachusetts InatitUte of Technology, Lexington, Massachusetts JOHNG . MAVROIDEB, Lincoln Laboratory, Massachwetk Inditute of Technology, Lezingkm, Maamchwtk
c. A. SWENBON, Institute for Atom& Research and Department of Physice, Iowa Slale University, Ames, Iowa
V
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Preface
Two of the articles in the present volume carry forward the coverage in this series of materials that have been of especial interest in solid state studies. One, by Busch and Kern, treats the preparation and properties of gray tin, which is of interest as an elemental semiconductor. In the other, by Bube, important aspects of the behavior of imperfections in cadmium sulfide type materials are discussed. The coverage of semiconductor behavior is continued in the article by Keyes in which the effect of elmtic deformation on electrical properties of semiconductors is reviewed. Two fields that have become especially prominent recently in solid state studies are reviewed in the other two articles of this volume. One is “Physics at High Pressure, ” reviewed by Swenson, and the other is “Cyclotron Resonance,” reviewed by Lax and Mavroides. Also included in this volume is a cumulative index of the first ten volumes of the series. This index is not a detailed one but is rather a topical index constructed from the outlines of the articles.
FREDERICK SEITZ DAVIDTURNBULL June, 1960
Vii
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Contents Contributors to Volume 11.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Contents of Previous Volumes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Articles Planned for Future Volumes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
Semiconducting Properties of Gray Tin G. A. BUSCHA N D R. KERN
I. 11. 111. IV.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The P t)9, Transition of Tin.. ..................... ......... 2 Preparation of Gray Tin Specimens.. . . . . . . . . . . . . . . . General Physical Properties.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 V. Semiconducting Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 VI. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . ............................. 39
Physics at High Pressure C. A. SWENSON
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. ReslRts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41 44
73
The Effects of Elastic Deformation on the Electrical Conductivity of Semiconductors ROBERTW. KEYJLS
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Phenomenological Description of Resistance and Pieeoresistance.. . . . . . . . . 111. Messurement of Piezoresistance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Properties of Semiconductors. . . . . . . . . . . . . . . V. The Effects of Hydrostatic Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. The Effects of Shear Strain.. . . . . . . . . . . . . . . . VII. Related Phenomena.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
149 150 156 169 213
Imperfection ionization Energies in CdSType Materials by Photoelectronic Techniques RICHARDH. BUBE
I, Imperfections in Insulators.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11. Photoelectronic Terhniques Applied to CdS-Type Materials. . . . . . . . . . . . . 230 ix
CONTENTS
X
I11 . Trends in the Ionization Energies in CdS-Type Materials. . . . IV. Indications of Complex Ionization Processes . . . . . . . . . . . . . . . . . . . . . . . . . . .
252
Cyclotron Resonance BENJAMIN LAX AND JOHNC . MAYUOIDIDS I. I1 111 IV
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V Summary and Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
261 264 276 370 388
AUTHORINDEX ..........................................................
401
SUBJECT INDEX .........................................................
412
CUMULATNE TOPICAL INDEX
421
. Cyclotron Resonance of Free Charged Particles. . . . . . . . . . . . . . . . . . . . . . . . . Cyclotron Resonance of Carriers in Solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
FOR VOLUME0
1
TO
10 . . . . . . . . . . . . . . . . . . . . . . . . .
Contents of Previous Volumes VOIUIIM
Volume 3
1
Methods of the One-Eloctron h r y of Solids
Group Ill-Group V Compounds
JOHN R. REITZ
The Continuum Theory of Lattice Defects
Qualitative Analysis of the Cohesion in Metols
J. D. EBHEL~Y
EUGBNBP. WIONERAND F ~ D E R I CSEITZ K
LESTSR GU~TMAN
H. WELXERAND H. Wcrss
Order-Disorder Phenomena in Metals
Phase Changes
Tho Quantum Defect Method
FRANK8. HAM The Theory of Order-Disorder Transitions in Alloys
TOBHINOSUKE M m YUTAXA TAKAGI
AND
Valence Semiconductors.
Germanium, and
DAVIDTURNBULL Relations between the Concentrations of Imperfections in Crystalline Solids
F. A. K R ~ G EARN D H. J. VINIC Ferromagnetic Domain Theory
C. KITPELA N D J. K. GALT Volum.
silkon
4
Ferroelectrics and Antiferroelectrics
H. Y. FAN
WERNERKANZIG
Electron Interaction in Metals
DAVIDPINES
Theory of Mobility of Electrons in Solids
FRANK J. BLATT
Volume 2
The Orthogonalired Plane-Wave Method
Nuclear Mognetic Resonance
TRUMAN 0.WOODRUFV
G . E. PAXE
Bibliography of Atomic Wave Functions
Electron Paromognetism and Nuclear Magnetic Resonance in Metals
ROBERTS. KNOX Techniques of h e Melting and Crystal Growing
W. D. KNIGHT
W. G . PFANN
Applkations ot Neutron Diffraction to Solid Stote Problems
Volume 5
C. G. SHULLA N D E. 0.WOLLAN
Galvanomagnetic and Thermomagnetic E6cts in Metols
The Theory of Speciflc Heah and Lattko Vibratiam
JULES
DE
J.-P. JAN
LAUNAY
luminescence in Solids
CLIFFORDC. KLICKA N D JAMES H. SCHULYAN
Displacement of Atoms during Irradiation
FREDERICK SEITL AND J. 8. KWHWB
xi
xii
CONTENTS OF PREVIOUS VOLUMES
Space Groups and Their Representations
Study of Surfaces by Using New Tools
G . F. KOBTER
J. A. BECEER
Shollow impurity Stotes in Silicon and Germanium
The Structures of Crystals
A. F. WHLW
W. KOHN
Volume 8
Quadrupole EtTects in Nuclear Magnetic Resonance Studies in Solids
M. H. COHENAND F. REIF
Electronic Spectra of Molecules and Ions in Crystals Part 1. Molecular Crystals
DONALD S. MCCLURE
Volume 6 Compression of Solids by Strong Shock Waves
Photoconductivity in Germanium
R. NEWMAN A N D W. W. TYLER
M. H, RICE, R. G . MCQUEEN,A N D J. M. WAL~H
Interaction of Thermal Neutrons with Salidr
Chonges of Stote of Simple Solid and Liquid Metals
Electronic Processes in Zinc Oxide
G.
I,. S. KOTHARIA N D K. S. SINCWI
RORELIUS
Electroluminescence
W. W. PIPERA N D F. E. WILLIAMS Macroscopic Symmetry and Properties of Crystols CHARLES
s.
S. AMELINCKXA N D
W. DEKEYBER
Volume 9 lor Crystals
H. C. WOLF
A. J. DEKKER
Polar Semiconductors
Optical Properties of Metals
W. W. SCANLQN
M. PAEKZR GIVENS Theory of the Optical Properties of Imperfections in Nonmetals
Static Electrification of Solids
D. J. MONTGOMERY The Interdependence of Solid State Physics and Angular Distribution of Nucleor Rodiations
DEXTER
Volume 7 Thermal Conductivity and lattice Vibrational Modes
The Structure and Properties of Grain Boundories
The Electronic Spectra of Aromatic M ~ S C V -
SMITH
Secondary Electron Emission
1). L.
G. H E I L A N D , E. MOLLWO,A N D F. S T ~ C K X A N N
'
1'. G. KLEMENS
ICRNST HEER A N V THEODORE B. NOVEY Oscillatory Behavior of Magnetic hscepti. bility and Electronic Conductivity
Electron Energy Bands in Solids
A. H . K A H NA N D H . P. R. FREDERXKSE
J O ~ E P CALLAWAY H
Heterogeneities in Solid Solutions
The Elastic Constants of Crystals
ANDRE GUINIER
H. B. HUNTINGTON Wave Packets and Transport of Electrons in Metals
Electronic Spectra of Molecules and Ions in Crystals Part 11. Spectra of Ions in Crystals
H. w.
DONALD S. M C C L U ~
llEWIS
CONTENTS OF PREVIOUS VOLUMES
...
Xlll
Volume 10 Positron Annihilation in Solids and Liquids
PHILIPR. WALLACE Diffusion in Metals
DAVIDLAZAXUB Wove Functions for Electron-Excess Color Centers in Alkali Halide Crystals
BARRYS. COWRARY AND FRANKJ . ADRIAN
The Continuum Theory of Stationary Dislocations
ROLANDDE WIT Theoretical Aspects of Superconductivity
M. R. SCHAFROTH
This Page Intentionally Left Blank
Articles Planned for Future Volumes The Direct Observation of Dislocations
5. AMELINCKX
PHILIPW.ANDERSON-CONYERSFerromagnetic and Antiferromagnetic Exchange Interactions HERRING Galvanomagnetic Effects
ALBERT C. BEER WERNERBRAN-Y.
H. PAO
Physics of High Polymers
B. N. BROCKHOUSE
Determination of the Normal Modes of Lattices by Neutron SpectrosCOPY
P. BUNDYFRANCIS M. STRONG HERBERT
Behavior of Metals a t High Temperatures and Pressures
ELIASBURSTEIN-G. PICUS
Infrared Spectra Arising from Foreign Atoms in Semiconductors
ELIASBURSTEIN-MELVIN LAX
Infrared and Related Properties of Ionic Crystals
NICOLASCABRERA
Theory of Crystal Growth
R. G. CHAMBERS
Determination of the Fermi Surface in Metals
ALAN H. COTTRELL
Work Hardening
J. FRIEDEL
Theory of Solid State Solutioris
FAUSTO FUMI
Theory of Ionic Crystals
JOHN J. GILUN
Dislocation Generation and Propagation in Lithium Fluoride
ROLFEGLOVER
The Properties of Thin Films
A. V. GBANATO-KURT LWKE
Internal Friction Phenomena Due to Dislocations xv
xvi
ARTICLES P L A N N E D FOR FUTURE VOLUMES
PAULHANDLER
Properties of Semiconductor Surfaces
CHARLES M. HERZFELDP. H. E. MEIJER
Group Theory and Crystal Field Theory
VAIN^ HOVI
Thermodynamic and Physical Properties of Ionic Solid Solutions
H. INOKUCHI-H.AKAMATU
Electrical Conductivity of Organic Semiconductors
C. K. JCARGENSEN
Chemical Bonding Inferred from Visible and Ultraviolet Absorption Spectra
C. KITTEL
Cyclotron Resonance
JANKORRINGA
Group Theoretical Discussions of the Spin Hamiltonian for Magnetic States in Crystals
G. LEIBFRIED
Theory of Anharmonic Phenomena in Crystals
R. J. MAURER
Transport Phenomena Crystals
ELLIOTT W.
Theory of Lattice Vibrations and Specific Heats
~~OSTROLL
in
Ionic
H. M. ROSENBERGK. MENDELSSOHN
Thermal Conductivity of Metals and Semiconductors at LOW Temperatures
G. W. SEARS-S. S. BRENNER
Growth and Properties of Whiskers
H. SUHI,
Magnetic Resonance in Ferromagnetic Materials
G. H. WANNIEH
Dynamics of Bloch Electrons
H. WITTE--ERICHWOLFEL
X-Ray Determination of Electron Density in Crystals
YAKOYAFET
Spin-Orbit Coupling in Solids
Semiconducting Properties of Gray Tin G. A. B u s c ~A N D R. KERN Loborcrtorium f i r FeeLk(irperphynik, Eidgnrbs8irchc Tcehnischc Hochrchule, Zurich, Switzerland
I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. The P * @ Transition of Tiri. . . . . . . . 1. General Rcmarks. .......................................... (I Transformation. . . . . . . . . . . . 2. The @ 3. The Q -+ @ Transformation . . . . . . . . . . . . 4. Dependence of the Transition Temperatu 5. Stabilization of the P Phase.. . . . . . . . . . . . 6. The a-8 Element.. . . . . . . . . . . . . . . . . . . . . 111. Preparation of Gray Tin Specimens.. . . . . . . 7. Gray Tin Powder., . . . . . . . . . . . . . . . . . . . . 8. Coherent Samples... . . . . . . . . . . . . . . . . . 9. Single Crystals... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. General Physical Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Structure, Density. . . . . . . ........... 11. Specific Heat. . . . . . . . . . . . . . . V. Semiconducting Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. General Rcmarks.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Experimental Results.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. Determination of Characteristic Data. . . . . . . . . . . . . . . . . . . . . 15. Effect of Impurities.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
2
+
14 17
1. Introduction
Of the group-four semiconductors, silicon, germanium, and gray tin, all of which have the diamond structure, gray tin is obtained the most easily. It forms spontaneously if metallic tin is kept at temperatures below 13°C. In spite of this fact, systematic investigations of the mechanism of e1ect)ricalconduction in gray tin were started comparatively late, after the semiconductor properties of silicon and germanium had already been studied ext,ensively. This situation is explained by the fact that, unless special precautions are taken, gray tin is always obtained in the form of a powder on which electrical measurements are difficult to perform. Considerable progress has been made in the prepahtion of samples 1
2
0. A. BUBCH AND R.
KERN
since the discovery of the semiconducting properties of gray tin in 1950. Methods have been developed for producing compact pieces in the form of filaments and films which enable more reliable determinations of electrical data. Recently Ewald and co-workers succeeded in growing single crystals by crystallization from a mercury solution. Final values for the semiconductor parameters can be expected only from measurements on such single crystals. Electrical measurements are now being made by Ewald and co-workers. Their conductivity values seem to differ only slightly (personal communication) from those obtained earlier' on powdered samples so that no significant corrections of the values reported here are to be expected.'" II. The a ++
0 Transition of Tin
1. GENERALREMARKS
The existence of a low-temperature modification of tin was first reported in the literature by Erdmann (1851).* The phase transformation which leads to this nonmetallic gray tin or a phase has been studied extensively by Cohen and his co-workers'-" and by a number of other authors. The CY phase, which is stable below 13.20C16forms spontaneously if metallic or @-tinis kept below this temperature for a sufficiently long time (days to years). The transformation is accompanied by a large increase in volume (27%). If a compact polycrystalline piece of white tin G. Buech, J. Wieland, and H. Zoller, Helv. Phys. A d a 24, 49 (1951). Note added in proof. Results of electrical measurements on single cryetsls have recently been published [A. W. Ewald and 0. N. Tufte, Phys. and Chem. Solids 8, 523 (1959)l.In the intrinsic range the values of conductivity, Hall coefficient, and magnetoresistance coincide with those from filaments obtained by the phase transformation. In the extrinsic range the single crystals show a larger conductivity and magnitoresistance, indicating a higher mobility of the charge carriers. * 0. L. Erdmann, J . Prakf. Chem. 62, 428 (1851). a E. Cohen and C. van Eijk, Z . physik. Chem. 30, 601 (1899);93,57 (1900). E. Cohen, Z . physik. Chem. 36, 588 (1900). E. Cohen and K. D. Deltker, 2.physik. Chem. 127, 178 (1927). E. Cohen and A. K. W. A. van Lieshout, 2.physik. C k m . 173, 1, 32, 67 (1036). E. Cohen, W.A. T. Cohende Meester, and A. K. W.A. van Lieshout, 2.physik. Chem. 173, 169 (1935). * E. a h e n and A. K. W. A. van Lieshout, 2.physik. Chem. 177,331 (1936);178,221 (1936). * E. a h e n 8nd A. K. W.A. van Lieshout, Proc. Acad. Sn'. Amsterdam 39, 352, 596 ( 1936). E. Cohen, W.A. T. Cohende Meester, and J. Landsman, Proc. A d . Sci. Amstetdam 40, 746 (1937);2. physik. Chcm. 181, 124 (1937). I1 E. Cohen and W. A. T. Cohende Meester, Proc. A d . Sci. Amsterdam 41,462,860 (1938);2. physik. Chem. 182, 103 (1938);183, 190 (1939). 1
Is
SEMICOKDUCTISG PROPERTIES OF BRAY T I N
3
is transformed, a powdery ma69 possessing cracks will result. After a certain incubation period, the transformation starts at one or several points on or near the surface and spreads out spherically a t a more or less constant rate until the whole piece is transformed. Density measurements reveal, however, that the process is never completed, i.e. a small percentage of the original phase always remains untransformed. I* This is also true for the reverse transfomiation. Gray tin converts back to white tin in a comparatively short time as soon as the temperature is raised above the equilibrium temperature. This transformation has been studied less extensively as a result of the lack of availability of compact pieces of gray tin as a starting material. A correct description of the -+ a transformation requires that two subsequent processes be distinguished : during the incubation period nuclei of the a phase are formed. This nucleation time 1, is followed by spherical growth of each nucleus with a linear rate of growth L.Both the time of nucleation and linear rate of growth are temperature-dependent and may be influenced by a number of factors: impurities, previous treatment (cold-working, annealing, number of previous transformations), shape and size of sample, presence of other substances acting as seed crystals, and the surrounding medium. The atomic mechanism underlying the B 4 a transformation has not yet been e~tablished.~* On the other hand a number of observations seem to indicate that the a -+ B transformation is rather of the diffusionless or martensitic Two experimental methods have been applied successfully to obtain quantitative information about the transformation : (1) determination of the fraction transformed as a function of time by means of a liquid dilatometer which makes u8e of the large change in volume; and (2) direct observation of the motion of the phase boundary. The former method is more suited if a formal description of the reaction in the sense of chemical kinetics is required. The latter enables separate determination of the nucleation time t, and the linear rate of transformation L, which are interpreted more easily in terms of the mechanism of the fundamental processes. 2. THEB-,
(Y
TRANSFORMATION
a. Crystdbgtaphic Relations
If the 9, + a transition were of the diffusionless type, a simple relation between the crystallographic orientation of the growing a phase and the disappearing B phase should be expected. X-ray investigations on transI* W. 0.Burgers and L. J. Groen, Digcwrsias Faraday Soe. 13, 183 (1957). "A. W. Ewdd and 0. N. Tufte, J . A p p l . Phys. 49, 1007 (1958).
4
G . A. BUSCH A N D R. KERN
forming single crystals of 8-tin containing 0.1% of mercury by Burgers and Groen1*-14revealed no such relation, a t least not on a macroscopic scale. The shape of the phase boundary does not indicate a dependence of the linear growth rate on the relative orientation of the two lattices. On the other hand, the fact that the transformation in either direction is never completed might be explained by an unfavorable orientation of the residual untransformed regions with respect to the growing lattice.
b. Nucleus Formation According to Burgers and Groeri, l 2 nucleation seems to take place only at the beginning of the transformation. In white tin which has undergone no previous transformation, a small number of nuclei is formed after a long incubation period and the gray phase spreads out spherically around each nucleus. In white tin which has already been subjected to one or more transformations, in every particle a limited number of nuclei is formed. These are supposed to originate in small regions of the gray phase which did not, transform back previously. At higher temperatures (above - 30°C) the nucleation time seems to increase with increasing temperature, 15 but no marked difference in nucleation time was found at -30°C and -78°C. The time of nucleation is reduced considerably by inoculating white tin with gray tin or with some substances isoniorphous with gray tin (InSb and CdTe,l8 Gel'). Further i t is reduced by cold work1.8.18-20and neutron irradiation.21 In contrast, annealingl*8v20and slow growth of the initial white tin crystal22 have the opposite effect. These facts may be explained by assuming that gray tin nuclei are formed preferably a t lattice distortions. The nucleation of gray tin also is influenced by surface conditions. Oxidation of the surface has a retarding effect'.8.20while certain electrolyte^^.^^**^ accelerate the nucleation. Groen2a has shown that only electrolytes which are capable of dissolving tin or tin oxide show the effect. This seems to indicate that mechanical obstruction by an oxide layer plays an important role. A number of impurities such as AI, Zn, Mg, Te, Co, and I v h reduce the nucleation time of Banka tin subjected to cold work considerably.l.8 K. Kuo and W. G . Burgers, Proc. Koninkl. Ned. Akad. Wefcnschap.B69,288 (1956). J. H. Becker, J . Appl. Phys. 28, 1 110 (1958). ION. A. Goryunova, Doklady Akad. Nauk S S S R 76, 51 (1950). R. R. Rogers, J. F. Fydell, J . Electrochem. SOC.100, 161 (1953). I'E. S. Hedges and J. Y. Higgs, Nature 169, 621 (1952). le H. Ishikawa, J . Phys. SOC.Japan 6, 531 (1951). so G. Tammann and K. L. Dreyer, Z . anorg. Chem. 198, 97 (1931). J. Fleeman and G. J. Dienes, J. Appl. Phys. 26, 652 (1955). C. W.Mason and W. D. Forgeng, Melds & Alloys 6, 87 (1935). *a L. J. Groen, Thesis, Delft (1956). 16
SEMICONDUCTING PROPERTIES O F GRAY T I N
5
No effect was observed with Fe, Xi, and Cu whereas Bi, Sb, Pb, Au, Cd, and Ag increase the nucleation time. 7*8.17.20
c. Kinetics of the Transformation DilatomeQic measurements a t constant temperatures on white tin powder which was subjected to several transformations previously yield S-shaped curves if the fraction f(t) transformed is plotted against the logarithm of time t . According to Groen2' part of these curves may be described by Avrami'sZ4equation f(l) = I
- e-Al'
(1)
in which k takes values between 2.65 and 3.55 and A = acL3is a constant which is determined by a shape factor a, the number c of nuclei formed at the beginning of the process per unit volume and the linear rate of growth L. A value 12 = 3 should be expected if three-dimensional growth of the nuclei takes place. Other experiments by the same author gave values of k between 1.45 and 2.4 which are explained by the small number of nuclei and the small particle size. Dunkerley and Mudge16 describe their results with a similar equation assuming k 1. Thus they show some formal resemblance to a homogeneous chemical reaction of first order. L could be determined from the experimental value of A if the values of a and c were known. By comparison with the experimental value of L for one temperature, Groen has determined L values from his isothermal transformation curves for a number of temperatures. The results are in satisfactory agreement with values obtained from direct observation of the motion of the phaae boundary. The influence of different liquid media (xylene, ethyl alcohol, alcoholic pink salt solution, and distilled water) on the rate of transformation has been studied by Smith and Raynor.26The highest rate was observed with alcoholic pink salt solution. This is explained by the reducing nature of this medium.
-
d. Temperature Dependence of the Rate of Growth A number of authors have investigated the temperature dependence of the linear rate of growth. In pure tin a maximum was found by Cohena.4 at -50°C (volume rate of growth), by Tammann and Dreyer,*OKomar and LazarewZQ and by Becker" between -30" and -35°C and by *I
M. Avrami, J . Clhem. Phye. 7 , 1103 (1939);8, 212 (1940). F. J. Dunkerley and W. L. Mudge, Jr., Tech. Rept. Univereity of Pennsylvania (1950).
''R. W. Smith and C . V. Fhynor, Proc. Phys. Soc. B70, 1135 (1957).
*-
A. Komar and B. Lazarew, Phyuik. 2.Sowjetunion 7 , 468 (1935).
6
G . A. BUBCH AND
R. KERN
Groenza between -35" and -50°C. Figure 1 shows the linear rate of growth as a function of temperature. Becker has given a theoretical treatment of this temperature dependence based on the following assump tions: the final state is reached by passing through an activated state separated from the initial state by the energy of activation
AG* = A H * - TAS*. After nucleation, the reaction takes place only a t the white-gray interface. Among the atoms a t these sites, only a fraction A are a t reactive sites, namely those a t highly stressed regions where a smaller energy is
35
4D
4 5 m 5 0
bd
T
FIG. 1. Linear rate of transformation 88 a function of temperature [after J. H. Becker, Thesis, Cornell University (1957)j. X , experimental data obtained by Becker (high-purity tin); 0 , experimental data obtained by Tammann and Dreyer; A,B, curves calculated by Bucker; C,D, curves calculated by Cagle and Eyring (corresponding to the experimental values of Tammann and Dreyer).
required for an atom to jump. If the atoms pass one by one from the white to the gray phase the linear rate of growth is given by
L
=
kT A&+AS'/ke-AH8fkT(1
- e-AO&kT)
(2) h where k, h, T have the usual meaning, AS* is the entropy of reaction, AH* is the heat of activation, AGwgis the difference in free energy between white and gray tin and the asterisk indicates that one degree of freedom haa been removed from the function. The average distance moved by the phase boundary for a single atom jump is D.Since the difference in free energy is known from other experimentslS7the heat of activation J. N. Broenrtsd, 2. phya'k. Chcm. 88, 479 (1914).
SEMICONDUCTING PROPERTIES O F GRAY TIN
7
may be determined graphically from the experimental L values by means of this formula if the temperature dependence of A is neglected. Becker obtains a value
AH
=
(8.7 It 1.5)lO' cal/mole a t 0°K.
The calculated L curves show the qualitative features of the experimental data (Fig. 1); however, the temperature of maximum rate of growth is shifted to higher values. This shift probably is related to the assumption of a constant value of A. Similar calculations have been made by Cagle and EyringZs who find a heat of activation of 2600 cal/mole. The same authors obtain a heat of activation of 5000 cal/mole with a different calculation baaed on the activation energy for self-diffusion in white tin determined by Fensham.
e. Injlzcence of Impurities and Mechanical Treatment on the Linear Rale of Growth Mechanical treatment seems to have little influence on the linear rate of transformation.11 This is explained by the assumption that the number of reactive sites created by internal stress associated with the volume change during the transformation is much larger than the number created by mechanical treatment. According to Becker, the rate of transformation in a thin foil of white tin is reduced if it is cemented to a substrate. This behavior is explained by the dependence of the rate of transformation on pressure, which has been investigated by Komar and Ivanov.'o The rate of growth seems for free samples to be independent of sample thickness in the range from 0.01 to 1 mm; however, the rate decreases rapidly with thickneea for very thin filrns.I6 The influence of impurities on the linear rate of growth has been investigated by a number of authors. 1-16~17.20.2cl An accelerating effect has been observed with A1 and Ga if present in small concentrations." In, As, Sb16 and Pb, BiZaseem to have a retarding effect. According to Croen,*' Hg also accelerates the transformation in the presence of impurities such as I3i which normally have a retarding effect. Compact pieces of the gray phase often occur if tin containing 0.1 weight per cent of Hg is trsnsformed.1' Small amounts of Al increase the rate of transformation, whereas larger amounts decrease it again. According to Buach, Wieland, and Zoller' the so-called Heyn-Wetzel effect" cannot **F.W. Cagle, Jr., and H. Eyring, J . P h p . Chem.67, 942 (1953). ** P. J. Fenahsm, Australian J . Sci. Research AS, 91 (1950). JgA. Komar and K.Ivanov, J . Expll. Thcoret. Phys. (UGSR) 6,256 (1936). E. Heyn and E. Wetzel, Z.MelalUc. 14,335 (1922).
8
G . A. BUSCH AND R. KERN
be the only reason for the accelerating influence of A1 since it also is observed in the complete absence of water. 3. THEQ -+ j3 TRANSFORMATION a. Phenomenological Observations
The transformation back to the white phase starts as soon as the temperature is raised a few degrees centigrade above the equilibrium temperature. Microscopic examination of gray tin powder which is transforming back to the white phase shows that the phase boundary does not move continuously within the particles but that it moves s t e p wise.2* Small domains of 0.04-0.2mm length are formed within a few seconds. The formation of a domain is followed by an interval of several minutes after which time a new domain is formed and so on. Similar observations have been made with single crystals of gray tin.1a This behavior is a characteristic feature of a diffusionless transformation, i.e., of a process in which domains of the new phase are formed by coordinate movement of a large number of atoms, even though the time for domain formation for a true martensitic process has been estimated to be of the order of lo-' see. Another observation supporting this view has been reported by Ewald and Tufte:" when a droplet of mercury is placed on the surface of a single crystal of gray tin kept above 1"C, a volume of the crystal comparable to that of the droplet is converted to the white phase. The boundaries of the transformed region coincide with principal planes of the gray tin lattice and thus indicate a diffusionless process. Dilatometric measurements by Smith and Raynor*' have shown that the rate of transformation depends on the surrounding medium. The rate was greater in water or ethyl alcohol than in xylene for mat.erial which had undergone one previous 0 -) a transformation. It is suggested that xylene behaves as an inert liquid whereas the effects observed with other liquids are the results of the presence of a hydroxyl group. GroenZahas measured the linear rate of growth of the white domains as a function of temperature by microscopic observation. An activation energy for growth of 50 kcal/mole was obtained from these values. Addition of Bi and Pb seems to have a retarding effect on the transformation comparable to that, in the reverse direction. In contrast, the addition of 0.1% of Hg has only little effect contrary to its influence on the B -+ a transformation." b. Kinetics of the Transformation
Groenxaconcludes from the observations already described that the isothermal a -t B transformation curves obtained from dilatometric
SEMICONDUCTING PROPERTIES OF GRAY TIN
9
measurements require an interpretation which is different from that for the /3 -P u transition. The fraction f transformed is not determined by the rate of growth but by the number of nuclei formed per unit time per unit volume which grow to the final volume of the domains in a comparatively short time. Assuming a timeindependent rate of nucleation N and an equal volume u for all white domains, the reaction is described by the expression f(t) = 1
- e-N.r
(3)
which has a formal resemblance to the Avrami formula with k = 1. Indeed k values close to unity have been found experimentally in many cases; however, values up to 2.4 have been obtained for some specimen8 of high-purity tin. These deviations are explained by induced nucleation as a result of lattice deformation during the transformation. Putting
N =c
+ af(t)
where c denotes the constant rate of spontaneous nucleation and af(t) represents the induced nucleation, a different equation, namely
+
log [ p f ( t ) 11 p = a/c,
- log (1 - f ( t ) l m = (a
=
+ c)/2.3
m~t
(4)
is obtained. Values of p and m may be determined graphically from this. c. Temperature Dependence of the Rate of Nucleation According to Groen, p increases with decreasing temperature, that is, induced nucleation becomes more important at lower temperatures. This behavior is explained by the decreasing efficiency of recovery so that the disppearance of distorted regions is retarded. Groen has calculated c values a t various temperatures from corresponding values of p and m, assuming v = lo-' mm'. The rate of spontaneous nucleation increases with increasing temperature. An activation energy for spontaneous nucleation of 120 kcal/mole was calculated from these values.
4. DEPENDENCE OF THE TRANSITION TEMPERATURE ON IMPURITIE~ AND PRESSURE Careful dilatometric measurements by Raynor and Smith" revealed that the transition temperature of tin from different sources is not the same. Vulcnn tin (99.997%) containing Fe as the main impurity showed a transition temperature in the range from 9.9 to 10.8OC,whereaa Pass&tin (99.997%) with Pb as the main impurity had a transition temperature in the range from 13.0 to 13.6OC.The authors attribute this to the '*0. V. Raynor and R. W. Smith, Proc. Roy. Soc. -44, 101 (ISM).
10
G. A. BUSCH AND R. KERN
different condition of the material, i.e., the amount of lattice strain stored in the white modification during the previous transformation, which depends in turn on the type of impurities present. Pb, Bi, and Sb were found to increase the transition temperature whereas Zn and A1 showed no influence and Fe lowered the transition temperature. As the free energy of the white modification is increased by lattice strain, the lower transition temperature observed with the less active material is supposed to be nearer to the true transition temperature. The transition temperature is lowered by hydrostatic p r e s ~ u r e . ~ An estimate of the order of magnitude of the effect may be made by applying the equation of Clausius-Clapeyron to the transition a c)8. Inserting the heat of transformation Q = 537 cal/mole, one obtains the value
at T = 286.4"K. This is more than twice the experimental value" (dT,,)/(dp)= - 2.0 X 10-*"K/atmos. 5. STABILIZATION OF THE a PHASE
The conversion of gray tin to the metallic modification above 13.2OC is inhibited by the addition of small amounts of germanium. This discovery was made in 1954 by E ~ a l d . Gray ~' tin containing 0.75 weight per cent of germanium transforms back to the white phase at an appreciable rate only at temperatures above 60°C. Measurements of the electrical conductivity up to this temperature do not indicate any phase change. If the conductivity is plotted in a logarithmic scale as a function of 1/T, the intrinsic conductivity line is extended 50" above the normal transition temperature. The possibility of producing gray tin which is stable at still higher temperatures is limited by the solubility of germanium in metallic tin. Ewald's observations have been confirmed by Raynor and Smith." Careful dilatometric measurements by these authors showed however that the true transition temperature actually is lowered by the addition of germanium. In any case the formation of white tin at a very small rate was observed at temperatures down to 7°C. 6. THEa-8 ELEMENT
Measurements of the emf of the galvanic chain a-Sn/elecitrolyte containing Sn++ or Sn++++ions/&tin have been performed by a number aa
H. Enz, Diplomarbeit, Eidgenhiache Technieche Hochachule, Zilrich (1961), unpublished. A. W. Ewald, J . A p p l . P h p . 26, 1436 (1954).
11
BEMICONDUCTINQ PROPERTIES OF GRAY TIN
of authors.'J7.a6 Since the difference in free energy, and therefore the emf, is zero if the two modifications are in equilibrium, these meaaurements provide a means of determining the equilibrium temperature T,. Measurements by Cohen and van Eyck' yielded T, = 20°C, whereas recent measurements by Antenena6(unpublished) gave T, = 13.0 f 0.2"C1 in close agreement with the value obtained from rate data (13.2 f 0.1OC) by Cohen and van Lieshout.6 The latter is accepted generally as the true value. According to Antenen, the emf of the galvanic chain &n
I AN solution of pink salt, (NHASnCl,, in ethanol or propanol I &Sn
does not depend on temperature in the range from -120°C to 0°C nor does it depend on the electrolyte concentration. A difference in free energy of 1340 cal/mole is calculated from the constant value of 14.5 mv in this temperature range. The large discrepancy between this value and that obtained from calorimetric measurements by Broensteds7 probably is the result of surface effects.
+
111. Preparation of Gray Tin Specimens
7. GRAY TINPOWDER Gray tin powder always is obtained when compact pieces of metallic tin are transformed without special precautions. The time of transformation may be reduced either by factors favoring nucleus formation or by factors increasing the linear rate of growth of the new phase. Inoculation with gray tin powder, cold work,and exposure to a temperature of about -30°C are used most frequently to accelerate the transformation. Oxidation of the forming gray tin powder, which is quite severe if the sample is kept in air, may be prevented by a protecting atmosphere or by high vacuum. Electrical measurements on gray tin powder suffer from the wellknown diaadvantages. Thus special methods have been developed to determine absolute values of the electrical resistivity.1 Since the transition temperature is lowered with increasing pressure and since traces of the metallic modification may cause severe deviations from the true value, compreseion of the samples has to be avoided. 8.
COHERENTSAMPLES
Small lumps of gray tin a few millimeters in length were found by Kendall" among large quantities of powder when he transformed 1 kg of spectroscopically pure tin. "A. Schertel, J . prakt. Chem. 19, 322 (1879). K. Antenen, Diplomarbeit, Eidgenwische Techniache Hochschule, Zurich (1952), unpublkhed. "J. T. Kendall, Proe. Phys. Soc. B83, 821 (1950).
12
G. A. BUSCH
AND R. KERN
According to Groen and Burgers,2a.a*compact pieces of gray tin also are obtained by converting metallic tin containing at least 0.1 weight per cent Hg. The solubility of mercury in white and gray tin, determined by Groen with x-ray and metallographic technique, was found to be 0.1 weight per cent and 0.02 weight per cent respectively. The excess mercury forms a second phase (HgSnlp) which is presumed to act as a binding substrate. Similar experiments have been performed by Hall, who offers a somewhat different explanation. If at least one dimension of the white tin specimen to be transformed is reduced considerably, coherent pieces of gray tin may be obtained. Becker" has studied carefully the factors influencing the quality of the gray tin formed, the quality being defined in terms of the number of cracks per square centimeter of surface and the grain size. His results may be summarized as follows. The quality decreases with increasing 'sample thickness. The best samples are produced by transformation close to the transition temperature. For a given thickness, purity, and mechanical treatment, the quality decreases with decreasing transformation temperature. Mechanical working of the white tin decreases the quality of the forming gray tin. The quality is also influenced by a number of impurities. Gal As, and A1 seem to improve the quality, whereas Sb has a negative influence. If thin sheets or films are transformed, the manner in which they are bonded to a substrate has an influence. Filaments of gray tin have been prepared by Ewald and K ~ h n k e . ~ O . ~ * Filaments of metallic tin are produced first by drawing out glass capillaries filled with tin to a core diameter of about 0.1 mm. The glass is then removed with hydrofluoric acid and the filaments are transformed by storing them in gray tin powder a t a temperature of -3OOC. The time required for complete transformation is of the order of one day for the samples of the highest purity. Contacts are soldered to the samples with solder of low melting point. X-ray Laue patterns show well-defined spots which indicate that the filaments probably are single crystals. @ '
9. SINGLE CRYSTALS
Of the numerous attempts to produce compact pieces of gray tin by processes different from the white-to-gray phase transition, only one has been successful. In 1958 Ewald and Tuftel* succeeded in growing single crystals of gray tiin from a liquid amalgam. Earlier attempts by Busch "L. J. Groen and W. G. Burgers, Ptoc. Koninkl. Ned. Aka. Welenschap. B67, 79 (1954).
E. 0. Hall, Nature 176, 165 (1955). 40 A. W.Ewald, Phys. Rev. 81, 244 (1953). 41 A. W. Ewald and E. E. Kohnke, Phys. Rev. 87, 607 (1955). as
SEMICONDUCTING PROPERTIES OF GRAY T I N
13
al.,a3,42 Kendall," and Becker" to deposit tin atoms directly into the gray tin lattice by electrodeposition, vapor condensation, or by chemical reactions below the transition temperature all yielded negative results, even though the conditions had been varied over a wide range. The method developed by Ewald and Tufte consists of continuous crystallieation from a saturated solution in mercury a t a temperature of about -30°C. The concentration of tin is maintained by continuoua dissolution of the metal in the warmer part of the apparatus. Single crystals up to more than 1 cm in length have been grown thus far. They are reported to show well-developed crystal faces and to be fractured readily to expose surfaces of high luster having the general appearance of broken germanium. According to the authors, the mercury content of these crystals-if properly cut to eliminate occluded mercury in small pockets-is remarkably low. Attributing the entire residual resistance after transformation back to the metallic phase to the presence of mercury, a content of 0.001 atomic per cent was calculated.
et
IV. General Physical Properties
10. STRUCTURE, DENSITY
Gray tin has the diamond structure with a lattice constant of 8 atoms. The
a = 6.489 A at 25OC." The elementary cell contaim density is 5.765 g cm-' a t 13°C.'
The corresponding data for white tin are given for comparison. White tin has a tetragonal lattice with a = 5.831 A and c = 3.182 A at 26°C." The density is 7.285 g em-' at 18OC.' 11. SPECIFICHEAT Hroenstedz' has measured the specific heat of both the white and gray modification in the temperature range from 80°K to 290°K and the heat of transformation a t 0°C which is 532 cal/mole. He calculated the heat of transformation aa a function of temperature and the free energy from his data. Lange" has extended the measurements down to 13°K. His reaults differ markedly from those of Broensted. According to Lange neither the specific heat of a-Sn nor that of &tin can be described by a 41
J. Wieland, Diplomarbeit, Eidgenkieche Technische Hochschuie, Ziirich (1949),
unpubliehed. J. T. Kendall, Phil. Mag. [7] 46, 141 (1954). "H. E. Swaneon and R. Fuyat, NaU. Bur. StandarL (U.S.) Circ. 689, Vol. 11, 12 4a
(1953).
H. E. Swaneon and E. Tatge, Natl. Bur. Sfandardr (U.S.)Circ. 689, Vol. I, 25 (1953). ('F. h n g e , 2.physik. Chem. 110, 343 (1924).
14
G. A. BUSCH AND R. KERN
Debye function. The relation
Q = heat of transformation T , = temperature of transformation c = difference in specific heat of the two modifications which follows from the second law of thermodynamics and the theorem of Nernst is confirmed within the limit of error. Hill and Parkin~on’~ have measured the specific heat of gray tin in the temperature range from 2°K to 110°K. Their results agree very well with Lange’s values in the region where these overlap. Values of the thermodynamic functions were calculated in the temperature range from 7 to 100°K. The dependence of the Debye temperature of gray tin upon temperature is similar to that of diamond, silicon, and germanium, possessing a minimum a t low temperatures. The specific heat of the white modification has been investigated by Keesom and van den Ende4sand by Keesom and E;ok4* in the temperature range 1.3 to 21°K. A rapid change of the specific heat was detected between 3.70 and 3.72”K, coinciding with the transition from the superconductive to the normal state. V. Semiconducting Properties
12. GENERALREMARKS
There is still some doubt about the true values of such semiconductor parameters of gray tin as the energy gap, lattice mobilities, and effective masses, This is partly a result of experimental difficulties. As remarked earlier, pure gray tin was available only in the form of powder, filaments, and films until recently, so that an accurate determination of the absolute values of the conduct,ivity and Hall coefficient met with difficulties. This is illustrated in Fig. 2 which shows the conductivity versus 1/T curves of “pure” gray tin measured by different authors. The discrepancies between individual curves in Fig. 2 probably cannot be explained by systematic experimental errors. Even small concentrations of impurities may alter the “intrinsic ” properties considerably, as the conductivity measurements in Fig. 4 show clearly. Representative values are to be expected only from the purest material available. On the other hand, the simple band model containing two types of charge carriers may not be 47
R. W. Hi11 and D. H. Parkinson, Phil Mag. [7] 43, 309 (1952). W. H. Keesom and J. N. van den Ende, Proc. Acud. Sci. Amsterdam 36, 143 (1932). W. H. Keeeom and J. A. Kok,Proc. A d . Sci. Amsterdam 86, 743 (1932).
SEMICONDUCl’INO PROPERTIES OF GRAY TIN
15
adequate for gray tin. However, since there are no measurements from which the existence of a third type of charge carriers (slow electrons or holes) can be deduced conclusively, most evaluations of characteristic data are based on the simple two-carrier model. A determination of characteristictic data which is free of any arbitrary assumptions does not exist. The interpretation of electrical measurements
FIQ.2. Electrical conductivity of *‘pure” gray tin as a function of temperatun memured by different authors. The Roman numerale refer to Table I. u[O-r cm-I],
“[OK].
is complicated by the fact that intrinsic behavior is approached only a t temperatures near the transition point. Impurity scattering seems to be important up to these temperatures, although intrinsic charge carrier concentrations are reached at much lower temperatures as a result of the mall energy gap. We start from the expressions for Hall coefficient R and conductivity u, namely
16
Q.
A. BUBCH AND R. KERN
in which
P/? = scattering factor, n, p = concentration of electrons and holes respectively,
mobilities, and e = electronic charge. A question arises concerning the value of the scattering factor in (6) and the manner in which the mobilities pn and p, depend on temperature. Usually 7/f2is put equal to 3*/8, corresponding to thermal scattering, and a power law of the form pa, p,, =
An,,,T-Tw (8) is assumed for the mobilities in the range of lattice scattering. Since the mobility ratio is of the order of unity. evaluation of the activation energy from Hall data is not possible without knowledge of the mobilities. The energy gap is usually determined from conductivity messurementa assuming a T-l law for the mobilities. Additional information about carrier mobilities, effective masses, and scattering mechanism has been gained from an analysis of magnetoresistance, field dependence of the Hall coefficient, and thermoelectric power. The value of the intrinsic energy gap obtained from electrical measurements has been confirmed by magnetic and optical measurements. However, the temperature dependence and the absolute values of the lattice mobilities, determined by different authors, as well as the values of the effective masses, disagree. The first attempt to measure the electrical conductivity of gray tin was made by de Haas, Sizoo, and Voogd60in order to decide whether or not it is a superconductor at low temperatures. No superconductivity wai detected. Later Sharvin'l showed by nieans of magnetic susceptibility measurements that gray tin did not become a superconductor a t least at temperatures above 1.2"K. Normal metallic conductivity of gray tin was reported by Moesveld (1937)! The semiconducting properties of gray tin have been established in 1950 by Busch el UZ.,'*~~ Kendall," and Blum and G o r y ~ n o v a A . ~ number ~ of authors have contributed to our knowledge of the mechanism of conduction of a-tin since then. Messurementa of conductivity, Hall effect, and magnetoresistance have been reported by Busch and Wieland,'L Kendall," Ewald et uZ.,41-6' and '0 W. J. de Haas, G. J. Sixoo, and J. Voogd, Communs. Phys. Lab. Unw.Lcidcn lala, pn.,,=
41 (1927). G. Sharvin, J . Phys.
(USSR)0, 350 (1945). A. L. Th. Moesveld, Z. phyuik. Chcm. A178, 455 (1937). '' G. Busch, J. Wieland, and H. Zoller, Hclu. Phys. A c h 13,528 (1950). I4 A. Blum and N. A. Cmryunova, Doklady Akad. Nauk SSSR 76,367 (1950). ** G. Buach and J. Wieland, Hclv. Phys. A& 36, 697 (1953). *' E. E. Kohnke and A. W. Ewald, Phyr. Rcv. 104, 1481 (1956). b*
SEMICONDUCTINO PROPERTIES OF GRAY TIN
17
Becker." The thermoelectric power haa been measured by Blum and Goryunova,**Kamadzhiev," and by Goland and Ewald.'@ Buwh and Mooserdo-'* have investigated the magnetic properties and have shown that the contribution of the charge carriers to the susceptibility may be separated from that of the lattice by virtue of the dependence on temperature. Optical measurements have been performed by B e ~ k e r . ~No ' infrared transmission WBB obtained; however, the optical energy gap could be determined from photoconductivity measurements.
13.
EXPERIMENTAL
RESULTS
a. Electrical Conductivity
Figure 2 shows the electrical conductivity of "pure" gray tin measured by different authors as a function of temperature. The experimental conditions, the origin of the sample, the absolute value of the conductivity at 0°C and a t 13.2"C respectively and the activation energy determined from the slope of the curves in the high-temperature range are presented in Table I. The somewhat low value of the activation energy obtained by Kendall" and the high value of the conductivity probably is a result of the presence of the metallic modification in his samples, which had been prepared by compressing gray tin powder a t liquid nitrogen temperatures. The shift of the curve of Busch, Wieland, and Zoller' to higher values is explained by the lack of accuracy in determining absolute values by means of the method of high-frequency loss. The rest of t h e curves yield values of u a t 0°C lying in the range (2.1 f 0.5)10* i2-l cm-I; the slopes correspond to activation energies in the range 0.09 It 0.01 ev. Hitherto n-type material has always been obtained by the B + a transformation of "pure" white tin. The shape of the curves a t low temperature seems to be determined largely by the kind and amount of the residual impurities. The influence of the addition of known amounts of impurities on the conductivity has been investigated by a number of authors. Busch el a2.l have used All which acts as an acceptor, as a doping agent. Later the effect of Mg,Zn, Af, Gal In, Pb, As, Sb, and Bi has been investigated. Typical curves are shown in Fig. 3 for Sb and in Fig. 4 for In. These I'J,
H. Becker, Thesis, Cornell University (1957).
aP. R. Kamedrhiev, Czcclroslov. J . Phys. 6, 60 (1955). '*A. N. Coland and A. W. Ewald, Phys. Rev. 104, 148 (1956). Buach and E. Mooeer, Hclv. Phys. Acfu 14,329 (1951). IIG. Busch and E. Mooser, 2. physik. C k m . 198, 23 (1951). I r G . Bunch and E. Mooller, Hclo. Phys. A& 16,611 (1953). lo G.
TABLEI. MEASUREMENTS OF ELECTRICAL CONDUCTIVITY Reference
No. (Fig. 2) Author I Kendsll.
I1
Busch et aLb
Origin of tin, main impurities [per cent) "Spectroscopically pure ',
Shape of sample preparation Lumps
Johnson-Matthey Powder Pb 0.002,Sb 0.001
Experimental method dc potentiometer High-frequency loes
0
(OT)
AE
[n-l cm-11
Ievl
2.5 X 10'
0.098
g5
x
10'
so.1
111t.t
Busch and Wielandc Johnson-Matthey Powder Pb 0.002, Sb 0.001
X 10' (13.2"C) iz } potentiometer 2.7(talc.)
IV
Kendalld
Johnson-Matthey Cylindrical, Pb 0.0012,Sb 0.001 compressed from powder
dc potentiometer
4.1 X 10'
Ewald and Kohnke.
Johnson-Matthey Filaments (single Pb 0.001,Sb 0.Wl crystals or Johnson-Matthey slightly Fe 0.001 polycrystalline)
dc potentiometer (no potential leads)
2.25 X 10'
0.088
2.09 x 10'
0.082
Vulcan Detinning Film or rather Company thin plate Fe 0.003,Sb O.OOO8
dc potentiometer
(1.8k 0.4)x 10'
0.087 to 0 . 1
Vt Vl V1
Beckerf
Johnson-Matthey Pb 0.002,Sb 0.001 Fe 0.0045 J. T. Kendall, Proc. Phya. Soc. B63,821 (1950). * G . B m h , J. Wielsnd, and H.Zoller, H e b . Phys. A d a 24, 49 (1951). G . Busch and J. Wieland, Hdu. Phys. A d a 26,697 (1953). J. T.Kendall, Phil. Mag. (71 46, 141 (1954). A. W. Ewald and E. E. Kohnke, Phys. Rw. S7.607 (1955). J. H.Becker. Thesb, Cornell University (1957).
0.08+ 0.008
0 ?
0.064
E
w X
SEMICONDUCTING PROPERTIES O F GRAY TIN 0
I '!
.OOL
,004
-loo I
.Ow
- 200. c
-150
I'
.008
!
.010
10
1
* * * q a
.OIZ
.Ol4
t
FIG.3. Electrical conductivity of gray tin doped with Sb [after A. W. Ewald and E. E. Kohnke, Phys. Rev. 97, 607 (1955)J.The broken line refers to pure tin.
Fro. 4. Electrical conductivity of gray tin doped with In [after A. W. Ewald and E. E. Kohnke, Phvs. Rev. 97,607 (1055)). The arrow^ indicate the approximate tempentursr at which tho oonduotivity chargee type.
20
G . A. BUSCH A N D R. KERN
O
20
40
60
80
-P-
FIQ.5. Electrical conductivity at low temperatures for one p-type and two n-typc specimens [after J. H. Becker, Thesis, Cornell University (1957)l.
Id.r= 1.6 I
0.6 0
-0.6 -I
-
l.6
Q
" 7 FIG.6. Hall coefficient of pure gray tin and of gray tin doped with As and In [after E.E. Kohnkc and A. VC'. Ewald, Phye. Rcv. 102, 1481 (1956)l.
R [cma ~oulornb-~],T["K].
SEMIC0NDUC"INQ PROPERTIES OF GRAY TIN
21
curves, obtained by Ewald and Kohnke," exhibit the following features. The slope of the curves in the "intrinsic" range is increased by increaeing amounts of any impurity, whereas the absolute value of the conductivity is lowered. The impurity concentration of the samples underlying Figs. 3 and 4 are calculated from the mass of impurity added to the melt prior to transformation. Samples doped with an n-type impurity show a negative temperature coefficient of conductivity at low temperatures, whereas the curves for p t y p e samples tend to become flat for higher impurity
198p Fro. 7. Hall coefficient at low temperaturn for one ptype and two n-typs speuimen8 [after J. H. Becker, Thesis, Cornell University (1957)l. R [cmJcoulomb-I], T("K1.
concentrations. The effect of compensation is shown clearly in Fig. 4. The conductivity of the originally n-type material is first decreased by increasing amounts of a p t y p e impurity and is then increased again. Very low values of the conductivity are reached at low temperatures by this means. Figure 5 shows two curves of such partially compensated material in the temperature range 100' - 5°K obtained by Becker."
b. Hall Efect Typical curves obtained from measurements on filaments by Kohnke and Ewaldl@are shown in Fig. 6 for pure material and for gray tin doped
22
a. A.
BUSCH AND R. KERN
with arsenic and indium (1.2 X 1 0 ' 8 atoms per ema), respectively. The field dependence and the behavior a t low temperatures are illustrated in Fig. 7 for one p-type and two n-type specimens.b' Samples of p-type with low impurity concentrations show a change of sign of the Hall coefficient, whereas specimens with high acceptor concentrations show a positive Hall coefficient up to the highest temperatures. The curves for the ptype samples are typical for a semiconductor with a mobility ratio subst,antially larger than unity.
FIQ. 8. Transverse magnetoresistance at high fields. A p / p o versus H for vari0Ue temperatures [after G . Busch, J. Wieland, and H. Zoller, Helu. PAYS. A d a 2 4 40 (1961)1.
Magnetoresistance The transverse magnetoresistance has been investigated by several authors. The results obtained by Busch et al.1 for spectroscopicdly pure a-tin powder are shown in Figs. 8 and 9. The change of resistivity A p l p o is plotted as a function of H for high fields (Fig. 8) and as a function of H a for low fields (Fig. 9) for various temperatures. As theory predicts, A p l p , which is equal to A p / p o for small fractional changes, obeys a H 2law for low fields whereas A p / p ~varies approximately linearly with H for high fields. The coefficient of magnetoresistance B(T)in the expression c.
APlPO =
W)(/JoW2
(9)
SEMICONDUCTING PROPERTIES OF QRAY TIN
23
(weak fields) is increased for pure material by lowering the temperature. Its temperature dependence is illustrated in Fig. 10 for pure gray tin and for gray tin doped with various amounts of Al." The angular dependence of magnetoresistance has been investigated by Be~ker.~'Figure 11 shows the dependence on the position of the magnetic field with respect to the primary current for both magnetoresistance and for the voltage between the Hall probes. The sample was I*I% *
0
* Po
.&
4
s
I
0
0
2
4
6
0
K)
19
[~Rdz]
FIG. 9. Transverse magnetoresistance at low fields. A p / p , versus H afor various temperatures [after G . Busch. J. Wieland, and H. Zoller, Helv. Phyu. A d a 84, 49
(lQSl)].
parallel to the magnetic field for 0 = 109" and 289". The transverse magnetoresistance and the normal Hall voltage are measured at 0 = 19". The magnetoresistance follows a cos2 ( 0 19") law approximately. The residual value at 8 = 109" and 289" may be a result of slight misorientation of the sample or of slight deviations from spherical symmetry of the energy surfaces.
-
d. Thermoelectric Power Curves obtained with gray tin filaments by Goland and Ewald" me shown in Figs. 12 and 13 for pure material and for a number of doped
24
G . A. BUSCH AND R. KERN
FIQ.10. Coefficient of transverse magnetoresistance aa a function of temperature for gray tin doped with Al [after G . Bumh and J. Wieland, H d v . Phye. A& 36, 697 (195311.
FIG.11. Dependence of magnetoresistance and “ f i l l probe voltage” on the orientation of the primary current with respect to the magnetic field [after J. H. Backer, Thesis, Cornell Univemity (1957)l. H 6080 gauw, Jp 60 ma, T = 78”K, e = rumple holder angular poeition.
-
-
SEYICONDUCMNQ PROPERTIES OF GRAY TIN
100
25
T p r l as0
FIO.12. Thermoelectric power of pure gray tin and of gray tin Joped with Sb [dbrA. N. &land and A. W. Ewald, Phys. Rm. 104, 148 (1966)].
Fxo. 13. Tbermoehtric power of gray tin containing ptype impuritk [after A. N. Goland and A. W. h d d , Phyr. Rw. 104, 148 (1966)l.
26
G. A. BUSCH AND R. KERN
samples. Pure samples show negative values of the differential thermoelectric power Q over the whole temperature range investigated. The quantity Q reaches a maximum of approximately -100 w/”K near 100°K. The maximum is higher and is shifted to higher temperatures for n-type samples with higher impurity content. All p t y p e samples show a crossover of the thermoelectric power from plus to minus which is shifted to higher temperatures with increasing impurity content. e. Magnetic Susceptibility
Measurements of the magnetic susceptibility as a function of temperature yield additional information about the band structure. Since
-
FIQ.14. Magnetic susceptibility of pure gray tin [after G . Busch and E. Mooeer, Helv. Phys. A c h 26, 611 (1953)]. Ch = Chempur, JM Johnson-Matthey (mscaptibility per 0).
this quantity is not affected by the geometry of the sample, the method is particularly suited for gray tin, which has been available only in powder form. Busch and MooseF have shown that the experimental curvea (Fig. 14) can be interpreted by assuming a temperature-independent lattice contribution over which the susceptivility X L of the charge carriers is superposed. For a nondegenerate intrinsic semiconductor they obtain XL
= ATte-ABI2kT.
(10)
Here A is a constant which may be calculated by means of electron theory.
SEMICONDUCTING PROPERTIES OF GRSY TIN
27
The constant lattice susceptibility may be evaluated by extrapolating the experimental curves to 'I' -+ 0. Gray tin is diamagnetic. The temperature dependence of two samples of high purity from different sources is shown in Fig. 14. Curves for gray tin contaiiiing various amounts of p-type and n-type impurities are shown in Figs. 15 and 16 respectively. f. Photoconductivity
Measurements of photoconductivity have been reported by Becker.5' His results at 5'K for an n-type sample (n = 2 X l O l o cm-a) of thickness
FIQ. 15. Magnetic susceptibility of gray tin doped with A1 (after G . Busch and E. Mooaer, Helv. Phya. A d a 26, 611 (1953)j. 7.10-a cm are shown in Fig. 17. Photoconductive response is virtually independent of wavelength in the range from 5 t o 12 microns. I t reaches a maximum at 16 microns and decreases again for larger wavelengths. The same author has made attempts t o determine the optical absorption coefficient in the infrared region. However, no transmission was found in the range from 2 t o 35 microns at either liquid nitrogen or a t liquid helium temperatures. The samples were thin polycrystalline films with thicknesses of 15 X lo-' cm and 30 X lo-' cm and donor concentrations of lo1*cm-a and 5 x 1 0 1 6 c i r a respect,ively. The transmission lo-', lo-', and lo-? at wavelengths sensitivity of the apparatus was leb, of 3, 12, 20, and 30 microns, respectively.
28
G . A. BUSCH A N D R. KEHN
FIG. 16. Magnetic susceptibility of gray tin doped with Sb [after C. B w h and E. Mooeer, Helu. Phys. Acla 26, 611 (1953)j.
A [microns]
FIQ. 17. Spectral sensitivity of photoconduction for a high-purity n-type sunpie at 5°K [after J. H. Becker, Thesis, Cornell University (195711.
14. DETERMINATION OF CHARACTERISTIC DATA
a. Activation Energy Most evaluations of characteristic data start from the determination of the intrinsic energy gap from conductivity measurements. If log u is plotted as a function of 1 / T , the slope of the resulting curve in the high-
SEMICONDUCTING PROPERTIES O F GRAY TIN
29
temperature range is given by the activation energy. This method is based on the aseumption that there is no degeneracy and that the mobilities of the charge carriers vary as T-1. Results obtained by different authors are summarized in Table I. These values have to be corrected if the temperature dependence of the mobilities differs from a 2’-1 behavior. The temperature dependence of the mobilities of a number of aemiconductors may be described by (8)) y being larger than 1.5 and different forelectrons and holes. Values of y from 1.5 up to 3.5 have been estimated for gray tin, so that the values in Table I are rather a lower limit. Assuming the same temperature dependence for both electrons and holes the value of A E in Table I must be increased by 20 and 40% for y values of 2.0 and 2.5, respectively. Degeneracy reduces the apparent energy gap further. The activation energy may be calculated from Hall data if the mobility ratio b = pJpp is known. This is done by calculating the carrier concentrations A and p from (6) which may be written in the form
and from the Hall coefficient in the exhaustion range, namely
The relation
n. = N D - N A = n
-p
ia wed. A straight line the slope of which is determined by Ah’ should reault if log ( n p / T * )is plotted as a function of 1/T. Unfortunately this method is not reliable in the case of gray tin for which the mobility ratio ie close to unity. In this case the Hall coefficient is very sensitive to the value of b and its temperature dependence. Aseuming a constant mobility ratio b = 1.28, evaluated from other measurements, Becker” found an energy gap of A E = 0.22 ev for an n-type sample of high purity. The large discrepancy between this value end that obtained from conductivity measurements indicates that the aseumptions made are not justified. A different treatment has been given by Kohnke and Ewald.s6 It is based on the aasumption that the temperature dependence of the electron mobility ia the same for gray tin and germanium (y = 1.65). Assuming that a power law is also valid for the holes, the lattice mobilities consistent with Hall and conductivity data were determined (Section c). The carrier concentrations may be calculated from the resulting mobility ratio and
30
G . A. BUSCH AND R. KERN
the Hall coefficient. A value of AE = 0.094 ev (Fig. 18) is obtained for the energy gap a t O'K, in satisfactory agreement with the values determined from conductivity data. The determination of the energy gap from magnetic susceptibility measurements is based on the assumption that the lattice susceptibility xo (electrons on inner shells plus valence electrons) does not depend on temperature. This assumption has not been confirmed for the other group IV semiconductors silicon and germanium. The close agreement
FIG.18. Determination of AE from Hall data. n p / T * as a function of 1/T [after E. E. Kohnke and A . W. Ewald, P h y s . Rev. 102, 1481 (1956)J.
between A R values obtained from conductivity data and from magnetic susceptibility in the ease of gray tin is, however, an indication that the assumption made above is a good approximation. This view is supported by the fact that the lattice susceptibility of the isoelectronic 111-V compound InSb seems to be practically independent of temperature. The charge carrier susceptibility X L is given by X L ==
Xtot
- xr:
in which x ~ , is, ~the measured total susceptibility and xo is the lattice contribution determined by the extrapolation indicated in Fig. 14. If now log ( x ~ T - 1 )is plotted as a function of 1 / T , one should obtain a straight line, the slope of which is determined by the activation energy
31
SEMICONDUCTING PROPERTIES OF GRAY TIS
AX. Results of Busch and Mooser are shown in Fig. 19. The deviations from a straight line at high temperatures are attributed to degeneracy. An activation energy of Ali" = 0.082 ev was obtained. Assuming that the long wave limit of photoconductivity is determined by the intrinic energy gap, these irieasureinents yicld an additional means of determining AE. The curve measured by Beckerb7does not fall off sharply for long wavelengths and a question arises concerning the long wave limit. Photon energies corresponding to the peak of the response curve and to the wavelength a t which the response h a fallen
F
FIG. 19. Determination of AE from magnetic susceptibility data. log (xLT-') as a function of 1/T [after G. Busch and E. Mooser, Helu. Phys. A d a 26, 611 (1953)l.
to one-half of its maximum value linve been calculated. An average for the two methods and for two different samples yields AEopt = 0.075 ev.
Measurements of as a function of temperature have not yet been performed. Thus the problem of whether or not the activation energy depends on temperature is stilt open to question. If it is assumed that the temperature coefficient B of the activation energy in A E = A& BT is reduced by the same factor as the activation energy itself in going from one group I V semiconductor to another, a P value of the order of 5 X lo-' ev/deg is obtained for gray tin from the values of j3 for Ge and Si, which are of the order of 5 X ev/deg. This value has been used by Kohnke and Ewald66 with Hall effect and conductivity data, to evaluate the product of the effective masses.
-
32
0. A. BUBCH AND
R. KERN
b. Concenttdion of Churge Cam'ers The product of the carrier concentrations may be crrlculated from the activation energy A E by means of the law of mass action
provided the product of the effective maaaes m,, and mp is known. The difference n, = n - p may be evaluated from the Hall coefficient at low temperatures so that values of n and p may be calculated separately,
FIG.20. Temperature dependence of the carrier concentration8 A and p and of tht Fermi level r [after G . Busch and J. Wieland, Helu. Phyr. A d o 46, 697 (lQM)].
Kendall" has adopted this procedure. He assumes arbitrarily that m.,, = mp = m so that the calculated values of n and p cannot claim a high degree of probability. A similar method which takes into account degeneracy and valuee of the effective masses different from the free electron mass waa used by Busch and Wieiand." Inserting the effective mwses m,, = 0.67 m and m,,= 3 m, which are consistent with magnetic susceptibility data, the neutrality condition n(l>- P ( l ) = N D [ ~ f@o, f>J (15) 1
J(ED1
= 1 + e(BD-r)/kT
BEYICONDUCI'ING PROPERTIE8 OF GRAY TIN
33
ie solved graphically and the Fermi level { is obtained as a function of temperature. If we take No = 7.4 X 10" cm-' and the value AED = 0.005 ev, derived from Hall data obtained at low temperature, the temperature dependence cjf the quantities {, n, and p obtained is shown in Fig. 20. The figure shows that the Fermi level lies within the range of kT from the bottom of the conduction band for temperatures higher than about 200°K.Thus deviations from classical statistics are to be expected. Similar curve8 for c, n, and p have also been calculated by Kohnke and Ewald,'O who used the values of the effective masa m, = mp = 0.68 m. These were obtained from high-temperature Hall effect and conductivity data. Aseuming equal effective maases for electrons and holes, the Fermi level approaches the middle of the forbidden energy gap with increasing temperature. Thus deviations from classical statistics should occur at higher temperatures than for the case m, < mp. c. Mobilities
An estimate of the mobilities corresponding to different scattering mechaniems-lattice scattering, scattering by neutral impurities, and scattering by ionized impurities-has been made by Busch and Wieland. 66 It is based on an energy gap of 0.08 ev, an impurity activation energy of 0.008 ev, and a dielectric constant of 47. The results for two different impurity concentrations (lo1' and lo**crn-', respectively) are shown in Fig. 21. Since the resulting mobility is given approxiniately by
'=C' F
Fk
k
puro lattice scattering may be expected only at the highest temperatures even for pure samples. Once the charge carrier concentrations are known as a function of temperature, the mobility ratio b = (pr/pp) may be calculated from the Hall coefficient by means of (11). This has been done by Busch and Wieland. In the high-temperature range, where thermal scattering is to be expected, b values of 1.26 and 1.20 were calculated for temperatures of 250' and 300"K,respectively. Values of p, and p,, may be calculated eeparately from the electrical conductivity, the mobility ratio, and the carrier concentrations. The result is shown in Fig. 22. Above 250°K the electron mobility may be described approximately by p,,[cm*/vsec] = 6.3 X lo6 2'-t. An electron mobility larger by a factor of about 2 has been determined by Kohnke and Ewald'e at 250OIC Assuming a temperature dependence
34
0 . A. BUSCH AND R. KERN 20 10
I8
I4
2
I
.
1000 1
FIG.21. Temperature dependence of the mobility for different scattering mechenisms [after G . Busch and J. Wieland, Helv. Phys. Acta 26, 697 (1953)]. pi, scattering by ionized impurities; p-, scattering by neutral impurities; p , , scattering by lattice vibrations. Impurity concentrations: 1017 crn-s, -; 1018 cm-3
FIG.22. Temperature dependence of the mobilities [after G. Busch and J. Wieland, Helv. Phys. Acta 26, 697 (1953)l.
35
SEMICONDUCTING PROPERTIES OF GRAY TIN
of the form (8), they obtain a working equation e(ND
- N A ) A , A , = U T ~ ~-AuT’PA. , - 3lr 8
RUZT(I*+TP)
(16)
by combining ((i), (7), (8), and (13).The unknown values of N D - N A , A,,, A,, yn,and yp may be determined from this in principle by inserting a number of corresponding values of R, U , and T. Considering only n-type samples and assuming in addition that
y,, = 1.65, the
authors obtain
pn[cm2/vsec] = 3.02 X 10’ pp[cni2/vsec] = 2.18 X lo* T-**O. This implies that the mobility ratio is smaller than unity in the whole range of temperature considered and reaches unity slightly above the transition point. Such a mobility ratio is however in contradiction to the measurements on p-type samples which show a crossover of the Hall coefficient from minus to plus with decreasing temperature. This indicates a mobility ratio larger than unity. From the Hall effect maximum R,,. above the crossover and the Hall coefficients R.. in the exhaustion range, the mobility ratio may be estimated by means of the relation
which is due to Breckenridge.aa A value of b = 10 was calculated by Becker” for a p-type sample with an effective impllrity concentration of 3 X lo1’cm-J. If the carrier concentrations for the temperature a t which the Hall coefficient changes sign is known, the mobility ratio may be calculated from the relation
This method gives reasonable values of b only for samples in which the Hall coefficient changes sign at comparatively high temperatures. Values slightly larger than unity were obtained by Becker” near the transition point. Similar calculations have been performed by Goland and Ewald60on p-type samples. Values of the effective masses are determined first by combining thermoelectric power and Hall effect data. Approximate values of n, p, and b, which are improved by an iteration process, are then obtained. Separate values of pn and p, are calculated subsequently by means of conductivity data. In this way b was found to be 1.03 and 1.21, “R. G. Breekenridge, R. F. Blunt, W. R. HosIer, H. P. R. Frederikse, J. H. Becker, and W.Oshinsky, Phys. Reu. 96, 571 (1954).
36
AND R. KERN
0. A. BUSCH
respectively at 150°K. Both samples reached the same value b = 1.05 at 270°K. Both mobilities follow a T-1 law approximately in the temperature range extending 100" below the transition temperature. The mobilities are approximately 3000 cm2/vsec a t the transition temperature. The mobilities increase more rapidly than T-1 a t lower temperatures. In the extrinsic range, the mobilities are given by the quantiLy Ra, apart from a scattering factor which is of the order of unity. Ra values as high as 70,000 cm2/vsec at 80°K and as high as 55,000 cm'/vsec at 22°K were obtained by Beckers' for an n-type sample. The values for p-type samples were 2000 cm2/vsec and 1200 cm2/vsec in the ranges from 5" to 20°K and from 60" to 120"K, respectively. Measurements of the transverse magnetoresistance have been used by Busch el aZ.,lsssKendallJ4*and by Ewald and Kohnke" to evaluate carrier mobilities as a function of temperature. It is difficult, however, to determine electron and hole mobilities separately. Instead a type of mean value is calculated on the assumption that only one type of carrier is present. In this case the magneto-resistance is given by
for pure lattice scattering and low fields. A temperature dependence of the mobility of the form (8) with y between 3 and 3.5 WBB obtained by Busch and Wieland." This result has been confirmed by Ewald and Kohnke. d . Eflective Masses
By analyzing magnetic susceptibility data, Busch and Mooser" have evaluated the effective masses. The constant A in (10) is given by
where p B is the Bohr magneton, p is the density, and fn.9 are the numbers of freedom which are defined by the relation fn,,, = m/mn.,. is a quantity which is related to the curvature of the energy surfaces E ( k ) = const and is given by
p=LL7z fi..
=
4r2ma 2 E ( k )
h' d k d k , .
f,
8
2, y, 2.
-
The set of values fn
= 1.5
F,2
= 41
fp
?F,
= 0.33 = 23
BEMICONDUCTING PROPERTIEB OF GRAY T I N
37
was found to be consistent with susceptibility data from pure and doped specimens. The relation
Fn.p)
= .fn,p2
should hold for spherical energy surfaces. The discrepancy between the values of f and F indicates that the surfaces of constant energy deviate appreciably from spheres. This conclusion is supported by the result of theoretical considerations by Herman and Callaway.64 Values for m,, and m,, of the same order have been obtained by Ewald and co-workers6'.6* from measurements of the Hall effect, conductivity and thermoelectric power. The effective mass product mnmpmay be calculated from the carrier concentrations if the activation energy and its temperature dependence are known. Using the value (14a) hE[ev] = 0.094 5 X 10-'T, the effective mass product (m,m,)' = 0 . 6 8 ~ ~
-
is obtained. If the Fermi level obtained from thermoelectric power data is taken into account, the effective masses are mn = 1.02(0.97)m
mp = 0.45(0.48)m.
Becker" concludes from the small ratio of the longitudinal magnetoresistance to the transverse magnetoresistance that the energy surfaces in gray tin are spherical or are warped very little. This stands in contradiction to the results of Busch and Mooser" but agrees with Herrnm's't predictions. According to Becker, a n effective mass of the order of 0.01m, predicted by Herman, mould account for the observed optical absorption at long wavelengths. ?
15.
EFFECTOF IMPURITIEB
The addition of small amounts of impurities influences both the "intrinsic " and "extrinsic" properties of gray tin. Numerous doping elements have been investigated. In accordance with their position in the periodic system, the group three elements -41, Ga, and In act as acceptors and the group five elements As, Sb, and Ri act as donors.'~4'~4".'L~b6.'g~6~ Their extrinsic activation energy is very small. It was estimated by Busch and Wieland" to be of the order of 0.005 ev from low-temperature Hall data. Donor and acceptor activation energies of 0.004 and 0.005 ev, respectively, were found to be consistent with magnetic susceptibility data in eamples doped with Sb and Al." These values are in satisfactory agreement with those calculated by means of the hydrogen model which yields F. Herman and J. Callsway, Phys. Rev. 89, 518 (1953).
F. Herman, J . Ebdronies
1, 103 (1955).
38
G . A. BUSCH AND R . KERN
Here IH = 13.5 ev is the ionization energy of the hydrogen atom, fn,, are the numbers of freedom and t is the dielectric constant. With the refractive index n = 6.5,0° a value c = 42 is obtained. If we use the Moss relation" AE[ev] X e2 = 174, which holds for group four semiconductors, a value c = 47 is obtained. An impurity activation energy of 0.005 ev is calculated if we takef = 1 and c = 50. In many cases, the impurity concentration determined from lowtemperature Hall data is considerably smaller than that calculated from the mass of material added to the metallic phase prior to the transformation. This may be explained by postulating that there is precipitation of part of the impurity atoms a t grain boundaries or elsewhere or by assuming that some is located at interstitial positions. In both cases the atoms which are not in substitutional places will not give rise to additional charge carriers. On the other hand, usually one can only designate a lower and an upper limit of the charge carrier concentration, differing by a factor of about two, from Hall data because the exact scattering factor is not known. The number of charge carriers per added impurity atom is roughly 1 for Al, 0.2 for Bi,660.5 for Sb, 0.5 for Ga, and 0.06 for Pb.4aIn contrast to these results Ewald and Kohnke" have found that A1 is 40 times less effective than In in bringing n-type material to the neutral condition. A thorough discussion of the behavior of different types of impurities, i.e., H-like, He-like, and Li-like, when one is concerned with magnetic properties has been given by Busch and M o o ~ e r . ~Their * experimental results may be summarized as follows: acceptors: Al, Ga, In Zn, Cd donors:
(H-like) (probably He-like) Au (probably Li-like) As (partly), Sb, Bi (partly), (H-like)
neither acceptors nor donors: cu Si, Ge Se, Te V, Co, Ni Ti, Fe, Mn
(probably substitutionally on lattice sites forming spa hybrids) (probably precipitated as small crystallites)
Early magnetic measurements6a seemed to indicate that Mn is substitutionally situated on lattice sites and forms sp* hybrids. Hall effect T.S. Mom, Proc. Phy8. Soc. A64, 590 (1951).
SEMICONDUCTING PROPERTIES OF GRAY TIS
39
measurements on Mn-doped samples lead to the same c o n ~ l u s i o n . ~ ~ However, a reinvestigation of the magnetic propertiesds showed that Mn probably does not enter into the gray tin lattice and that the paramagnetic susceptibility observed is probably associated with a Mn-Sn compound which forms a second phase. Lattice defects which act as donors also may be generated thermally. By heating pure gray tin to 30°C for 10 minutes and then quenching, Busch and Wielandbbobserved an increase of the low-temperature carrier concentration by a factor of 1.6. A question arises as to whether the carrier concentration of the order of 10’7 cm-8 normally found in spectroscopically pure gray tin is mainly a result of an incomplete transformation @ + a. The influence of impurites on the “intrinsic” properties is not yet understood. Ewald and Kohnke4l observed an increase in slope of the o-T curves in the “intrinsic” temperature range whereas the absolute value of the conductivity in the same temperature range was reduced by certain impurities. The apparent increase in activation energy and the consequent reduction in carrier concentration cannot alone account for the reduction in conductivity. Both the activation energy and the carrier mobility are influenced according to the authors. VI. Conclusions
Gray tin is a semiconductor characterized by a very low energy gap (AE = 0.09 f 0.01 ev). The carrier mobilities lie in the range from lo00 to 3000 cm2/vsec a t the transition point. The mobility ratio is near unity at this temperature and increases with decreasing temperature. The temperature dependence of the mobilities may be described by power laws in a temperature range extending about 100 centigrade8 below the transition temperature. Exponents of and more have been reported for both electrons and holes. The effective masses of the charge carriers seem to be of the order of the normal electron mass. Magnetic measurements indicate that the energy surfaces are not spherical. The comparatively large discrepancies in semiconductor parameters reported by different authors are only explained in part by different techniques of measurements. In the main they appear to be a result of applying t o experimental results different methods of analysis which are based on different assumptions. So far no set of characteristic parameters has been given which is consistent with all experimental facts. Table I1 contains a comparison of the values of parameters which are thought to be the most reliable ones with those of the other group four semiconductors and of the isoelectronic compound TnSb.
-+
”G.
Buaeh and K. A. Muller, Helv. Phys. Acta 28, 319 (1955).
T. Fiecher and K. A. Muller, Helv. Phys. A d a SO, 223
(1957).
TABLE11. COMPARISON OF CHARACTERISTIC DATAOF GRAYTIN,SILICON; GERMANIUM,. A N D INDIUM ANTIMONIDE
x 10at 0°C b a t [ev/deg] [cm*/vsec] [cm*/vaec] 0°C fin
x 10-
pp
6 X lo4 at 0°C
a
AEo
A
lev]
Si
5.43086
1.19
4 .O
1.8
0.6
3.0
2.6
2.3
0.97 0.19
0.16
Ge
5.65748
0.77
4.4
4.4
2.2
2.0
1.66
2.33
1.58 0.082'
0.042 0.34'
a-Sn
6.489
0.085
-
1.4
1.2
1.2
1.5
1.5
0.67* 1 .w
3 .od 0.46'
InSb
6.475
0.26
2.9
80
-1
-80
Yn
1.66
1.
>1.66
m.lm
0.013
m./m
0.52~
0.18
Y.Fan, &lid StoLc P h p . 1,283 (1955). With Si and Ge the effective masses are anisotropic. The figures refer to longitudinal and transversal m869ecs determined from cyclotron resonance. c With Si and Ge two aorta of holea with slightly anisotropic but different masees are present. The figures refer to light and heavy holes respectively. Valuea consistent with magnetic susceptibility data IG. Busch and E. Mooeer, Helv. Phyu. A& 46,611 (1953)]. V d u a coneistent with Hall effect, conductivity, and thermoelectric powerdata [A. N. Goland and A. W. Ewald, P h p . Reu. 104, 148 (1956)l. See H.
b
? a
9 0 X
> Z
U
s
E
Physics at High Pressure C . A. SWENSON Institute for Atomic Reaeareh and Department of Physics, Iowa SMc Univcraity, A m , Iowa
I. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................ .. 1. The Preasure Scale.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Techniques-Mainly Room Temperature.. . . . . . . . . ............ 3. Low-Temperature Techniques.. . . . . . . . . . . . . . . . . . . 111. Results ................................ 4. PVT Data for Solids.. . . . . . . . ............ ..................... 5. Electrical and Magnetic Meas 6. Optical Measurements.. . . . . . . . . . . . . . . . . . . . ................. 7. Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Miecellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Fluids: P V T and Transport Data. ......................
11. Experimental Methods. . . . . . . . . . . . .
41 44 44 47 73 130 137 142
1. Introduction
The application of high-pressure techniques to problems in solid state physics has expanded greatly in the past few yearn. Before World War 11,high-pressure work was to be a great extent exploratory in nature, snd waa hindered by a need for the development of techniques. Since 1946, industrial interest in high-pressure chemical processes haa made commercial equipment available a t reasonable cost, and, if advertisements in trade journals are to be believed, it is now possible to purchase complete facilities for performing almost any type of measurement at pressures from 10,OOO to even 150,000 atmos. This is a relatively new development, however, and most of the high-pressure work which will be discussed in the following has involved custom built (“homemade”) equipment, although in many cases commercial valves and hydraulic pumps have been used. The development of reliable materials and techniques has considerably expanded current types of high-pressure research over those which are derscribed, for instance, in Bridgman’s book, “The Physics of High Pressures.”l The pressure range has been expanded to static pressures of ’P.W. Bridgman, “The Physics of High Preamres.” G . Bell, London, 1949 (with supplement). 41
42
C. A. SWENSON
over 300,000 stmos, with working temperatures that may be as low a8 0.05”Kor as high as 5000°C: for limited parts of the pressure region. Measurements of the pressure dependence of electron spin resonance at microwave frequencies near 20°K required t i rather unique combination A surwy o f the litcrature reveals that most of extreme c~~nditioris. branches of physics hiive profited froin t lit* use of high-pressure techniques. With this great cxpansiori of the use of high pressures as a tool in research, it seems a bit presumptuous to assume that such a field 88 “high-pressure physics” exists. The results of the high-pressure measure ments in many cases can be described best in their relationship with other, more detailed, measurements a t normal pressures, and should be included in review articles restricted to specialized fields. This is particularly true of high-pressure measurements on semiconductors, for instance, where considerable background is necessary to appreciate the significance of the high-pressure results. Thus, the purpose of a chapter bearing the title, “Physics at High Pressure,” cannot be to discuss all (or even a number of) high-pressure measurements and their theoretical implications in detail; to do this would lead to an encyclopedia of solid (and fluid) state physics. There are many problems of technique, however, which are common to several different types of work, and this chapter is intended as an introduction to the types of measurement which are being made at high pressures, with little regard in many cases, for the fundamental significance of the measurements. Various recent developments in high-pressure techniques will be described first, with particular emphasis on those techniques which are applicable to the relatively new low-temperature region. The following sections will form, essentially, a slightly expanded bibliography of many of the more recent experiments in which these techniques have been used. Certain boundary conditions have been established for the discussion. Very little of the work which was covered by Bridgman in his 1946 review article2 will be mentioned, making the starting point of the references, roughly, June, 1945, or immediately post-war. Anyone desiring a background in high-pressure physics certainly should read thia very comprehensive article of Bridgman’s. Secondly, the term high pressure” usually will be reserved for pressures greater than loo0 atmos, or pressures requiring other than readily available commercial equipment of the type that automobile body repair shops use. This latter restriction allows pressures of a few hundred atmospheres a t liquid helium temperatures to be considered as “high pressures.” The pressure region above 25,000 atmos requires special techniques in general, and will receive a more complete coverage in a fieparate article in this series.
* P. W. Bridgman, Reus. Modem Phya. 18,1 (1946).
43
PHYSTCS A T HIGH PRESSURE
The widespread interest in high-pressure work has resulted in the publication of several books and review articles in recent years. Standard references by Bridgman’ and NewittJ have been supplemented with books by Comings‘ and Hamannb which arc of particular interest to the chemical engineer and physical chemist and also serve as excellent introductions to the art. Many of the gcophysical aspects of high-pressurc work have been discussed by while more recently Hall and Kistlers have surveyed new high-pressure developments with particular emphasis on the ultra-high pressure and temperature region. Lawsong has reviewed the theoretical and experimental aspects of the effects of pressure on electrical resistivity, although since the publication of this article several pertinent papers have appeared which are concerned with low-temperature effects. Hridgnian has also summarized his work on mechanical properties a t high pressures in a book, “Study in Large Plastic Flow and Fracture.””J Finally, attention is called to one of the Fsraday Society Discussions,” “ The Physical Chemistry of Processes at High Pressures,” and to two issues of the periodical, “Industrial and Engineering Chemistry,”l* which are devoted to high-pressure work of commercial interest. Not all of the papers contained in these last three references will be included in the following discussion, the last two are of particular interest for their information on the type and availability of commercial equipment. Relatively short general discussions of the field of high pressures can be found in Bridgman’s Nobel Lecture,” and in two more recent papers by him,14-16 as well as in an article by Newitt.” Finally, a statement must be made about the units which will be used. There are three practical choices; the atmosphere, the kg/cm2, and the bar. These are all of roughly the same order of magnitude, and in practice the choice of one or the other seems to depend on personal
ID. M. Newitt, “High Pressure Plant and Fluids at High Pressure.” Oxford Univ. Preee, London and New York, 1940.
E. W. Comings, “High Pressure Technology.” McGraw-Hill, New York,
1956.
‘5. D. Hamann, “The Physico-Chemical Effects of Pressure.” Academic Press, New York, 1957. IF. Birch, J . Geophys. Research 67, 227 (1052). ‘F.Birch, Trans. Am. Geophys. Union 36, 79 (1954). ‘8.T.Hall and S. S. Kistler, Ann. Rev. Phys. Chem. 8,395 (1958). ‘A. W. Lawson, Prop. in MeM Phys. 6 , 1 (1956). I$€’.W. Bridgman, “Studies in I a g e Plastic Flow and Fracture.” McGraw-Hill, New York, 1952. llDiscuseions Faraday SOC.22 (1956). ”Ind. Eny. Chem. 48, 826 (19561, 49, 1945 (1957). W. Bridgman, J . Wash. Acad. Sci. 38, 145 (1948). I4P. W. Bridgman, Endeavor 10, 63 (1951). 18P. W. Bridgman, Mech. Eng. 76, 111 (1953). I’D. M. Newitt, Chartered Mcch. Eng. 3, 14 (1956).
44
C. A. BWENBON
prejudice, In standard geophysical practice the bar (0.98692 atmos) is used, while Bridgman and many others prefer to use the kg/cm2 (0.96784 atmos). I n the following discussion all results (where the correction has been made when accuracy to more than a few per cent is required) will be expressed in terms of the standard atmosphere (1 atmos = 1.01325 X lo6 dynes/cm*)
or, for certain elastic constants, dynes/cm*. This unit is used because of the author’s greater familiarity with it, and the lack of any more compelling reason for using either of the other two. II. Experimental Methods 1. T H E
PRESSURE SCALE
The interpretatiori of experimental high-pressure data in terms of theoretically calculated quantities is only possible if the experimental pressure scale is identical to the thermodynamic pressure scale. To this end, standards for the calibration of secondary pressure measuring devices have been set u p in terms of the fundamental definition of pressure; namely, force per unit area. Bridgman” and, more recently, EbertI8 and Newitt,lO have discussed in considerable detail the setting up of such a scale for the various pressure regions. For pressures greater than a few hundred atmospheres the scale is determined experimentally by mean8 of the free piston gauge (sometimes called a pressure balance or deadweight gauge) which is sketched in Fig. l a . In this instrument, the weight on the piston is supported a t equilibrium by the hydraulic pressure as it acts upward on the base of the piston which is enclosed in a closely fitting cylinder. The absolute pressure is given by the quotient of the supported weight and some average of the piston and cylinder areas. T h e details of a practical free piston gauge may vary considerably from those of the elementary sketch of Fig. la, and various designs are described by Comings. The major difficulty encountered in the use of a free piston gauge at high pressures (greater than 1000 atmos) is due to the elastic distortion of the cylinder with internal pressure, and a change in the effective area which cannot be calculated. Bridgman suggested that this effect could be minimized through the use of the controlled clearance principle, and Newhall’s application of this is Ahown in Fig. lb.10 The gap between the P. W. Bridgman, “The Physics of High Pressures,” p. 406. G . Bell, London, 1948, H.Ebert, 2. angew. Phys. 1, 331 (1949). E. W. Comings, “High Pressure Technology,” p. 82. McCraw-Hill, New York, 1956, D. P.Johnson and D. H. Newhall, Trans. A S M E 76,301 (1953).
PHYSICS AT HIGH PRESSURE
45
piston and cylinder (of the order of a few micro-inches) is kept constant by the use of the second source of pressure to balance the expansion due to the inner (unknown) pressure. The effective size of the gap can be monitored by observing either the leak rate of the oil past the gap or the rate of fall of the piston at constant load. The general problem of elestic distortion in the use of a free piston gauge has been discussed by Johnson et uZ.,~' who compare the design of a conventional type gauge with that of the controlled clearance type.
UrnNOWW
PRESSUllE (a)
PRESSURE ( bl
FIG.1. Basic principle8 of two free piston gauges. (a) Conventional type. (b) Conhlled clearance type [D. P. Johnson and D. H. Newhall, Trans. A S M E 7 4 301 (1953)J.Not shown are the center of gravity of the weight (which is ueually well below the piaton) and a piston rotating mechanism which is wed to reduce friction between the piston and cylinder.
Other approaches for evaluating or minimizing this elastic distortion have been tried. Basset22,28 has reduced the effect of the expansion through the use of sintered tungsten carbide pistons and cylinders; the use of this material effects an improvement since its elastic modulus (90 X 10' psi) is three times that of steel. Dadson*' has developed a method for determining the variation of the cylinder area with pressure through the use of two dimensionally identical piston and cylinder combinations which are made from different materials. Capacitance measurements**have also been used to determine the gap width as a function of D.P. Johneon, J. L. Cross, J. D. Hill, and H. A. Bowman, Ind. Eng. Chem. 49,2048 (1957). James Baaset, Chimie & industn'e 63,303 (1945). Jam= Baeeet and Jacques Basset, J . phy8. radium lS, 57a (1954). R. 8. Dadson, Ndure 176, 188 (1956). J. Bultemann and M. Schuster, 2. angtw. Phy8. 9, 29 (1957).
46
C. A. SWENSON
pressure. Bett, Hayes, and Newitt*' describe the use of a primary mercury standard to calibrate with considerable precision, a free piston gauge to 3000 atmos, and they give many references to older work of a similar nature. The pressure scale below 1000 atmos is probably known to an accuracy of much better than 0.1%. However, the uncertainty grows at higher pressures and it is quite difficult to estimate the accuracy with which absolute pressures can be measured at 10,000 atrnos, although Basset claims an accuracy of 1 :2500.23The free piston gauge is a bulky piece of equipment, and this leads to problems when it is used as a standard a t high pressures. The intercomparison of gauges from various laboratories is difficult. A reliable secondary pressure measuring device can be made which utilizes the pressure dependence of the resistance of manganin wire. Bridgman used a free piston gauge to show that the resistance-pressure relation for manganin is almost linear to 13,000 atmos. In order to aid in the calibration of such secondary manometers, the freezing point of mercury a t 0°C (7640 kg/cm2 or 7394 atmos) was established as a fixed point. Bridgman discusses other possible choices for fixed points, but concludes that the mercury point probably is the most convenient standard.17 Initially the pressure range was extended to 20,000 atmos by B linear extrapolation of this calibration, but, later, Bridgman established a second fixed point through the use of the very sharp BiI-BiII transition at 25,420 kg/cm2 (24,600 atmos) a t 30°C.27These two fixed points are easily detected through the use of an electrical resistance method de scribed by Bridgman. 28 The deviation from linearity of the manganin gauge is only of the order of a few per cent up to 25,000 atmos. This makes the gauges rela. tively ideal as secondary standards and they have been widely used 88 such. RridgmanZ7and othersZ0sa0 have described seasoning cycles which are designed to insure stability of the zero pressure resistance. The rela. tive change in resistivity per atmosphere depends on the source of the wire, but is of the order of 2.5 X lo-" (atmos)-', with possible sensitivitiea of the order of a few atmospheres being reported for gauges of 300 ohms normal resistance. Warschauer and Paula1 describe a simple bridge for use with such a gauge when maximum accuracy is not required. For some applications, when the temperature of the manganin gauge is.appreciably different from room temperature, slight variations in the a'K. Bett, P. F. Hayes, and D. M. Newitt, Phil. Trans. Roy. Soc. A247, 59 (1954). P. W. Bridgman, Proc. Am. Acad. Art8 Sci. 74, 1 (1940). P. W.Bridgman, Rev. Sn'. Znslr. 14, 400 (1953). *O D.Lazaru8, Phys. Rev. 76, 545 (1949). a0 H. E. Darling and D. H. Newhall, Trane. A S M E 78, 311 (1963). *I D. M. Wamchauer and W. Paul, Rev. Sci. Znstr. 39,678 (1958).
l7 2*
PHYSICS AT HIOH PRESSURE
47
temperature can have a serious effect. Darling and Newhallaohave reported a gold-2.1 % chrome alloy which, although it has only one-third the pressure sensitivity is much easier to season than manganin, and has almost zero temperature coefficient of resistivity up to 100°C. Howea* discusses this in an article on high-pressure illstrumentation and control. Ebert and Gielessenasalso have investigated the pressure and temperature coefficients of resistivity of many alloys to roughly 5000 atmos near room temperature. From this information their suitability for use in pressure gauges can be evaluated. The mercury and bismuth fixed points, as determined by Bridgman, essentially furnish the definition of the manganin pressure scale to 25,000 atmos. Johnson and NewhallZoused the controlled-clearance free piston gauge to verify Bridgman's value for the mercury point to within the expected accuracy of their experiments, about one per cent. There seems to be little other information in the literature, outside of Bridgman's work, which would aid in a judgment of the accuracy of this pressure scale. This deficiency is not serious, however, since it is possible to reproduce the fixed points to 0.1 %, making the correction of current data to a new scale quite simple. It is very important in a given precision experiment to specify the definition of the pressure scale which is used but this is not always done. Ideally, a thermodynamic investigation of the pressure scale could be made on a basis similar to that used to investigate a vapor pressure temperature scale from purely thermal measurements, but this would be difficult. Ultrasonic techniques (with which the adiabatic compressibility can be measured as a function of both temperature and pressure) furnish another possibility. Smith and Lawsonabhave concluded from a survey of the existing data for water (both ultrasonic and volumetric as functions of temperature and pressure) that the manganin pressure scale below 10,OOO atoms is possibly uncertain to about 0.5%. This is to be compared with Bridgman's estimate that his free piston gauge pressures were accurate to 0.1% at 13,000atmos. However, serious discrepancies seem to exist between ultrasonic and volumetric data for solids in this same pressure range, so further work seems to be necessary. A further refinement of high-pressure techniques will undoubtedly lead to a need for further investigation of the accuracy of the pressure scale.
2. TECHNIQUES-MAINLY ROOM TEMPERATURE The complications which arise in the design and construction of highpressure equipment depend greatly on the pressures which are to be I r W . H.Howe, ZSA Journal 2, 77, 101)(1955). ' I H.Ebert and J . Gielesen, Ann. Physik 161 6, 1, 229 (1947). "A. H. Smith and A. W.Lawson, J . Clem. Phys. 22, 351 (1954).
48
C. A. SWENSON
used, and on the type of work which is to be done. An examination of published work indicates very little difficulty in the use of pressures up to 5000 atmos, with several successful designs for pressure seals, electrical lead-ins, and even windows for visual observation being readily available. The extension of working pressures to 10,000 atmos offers somewhat greater, but by no means excessive, difficulties. Complications increase rapidly for higher pressures, however, with the result th a t very few experimentalists have worked a t pressures between 20,OOO and 50,000 atmos, while the higher (100,000 atmos and above) pressure regions have, indeed, been investigated oiily by a dedicated few. Bridgman,' Bssset," Comings,' and Hamann6 have discussed in some detail the various problems which are encountered and their solutions. The purpose of this section is to review briefly the techniques necessary for work in the various high-pressure regions, and to point out a few of the problems which are encountered. Many of the techniques such as the design of high-pressure optical windows, are specialized to certain experiments, and a discussion of these will be left for the sections in which the results of the experiments are given. The specific difficulties which arise in high-pressure experiments at temperatures below 80°K have resulted in the development of specialized techniques, and these will be discussed a t some length in ti separate section.
a. Pressure Generation The general use of hydraulic methods in industrial equipment ha led to the development of inexpensive hand-operated or mechanized oil pumps which produce pressures from 10,000 psi to 40,000 psi (nearly 3000 atoms). Flexible hydraulic fluid tubing and quickly demountable couplings, good to 10,000 psi, are also available commercially, and may be used with a wide selection of hydraulic rams. The beginning of any highpressure work, which inevitably involves these low pressures and the generation of large forces through the use of them, is facilitated by the existence of this commercial equipment. Some indication of the type of equipment available can be found by scanning through the issues of " Industrial and Engineering Chemistry "l* which have been referred to previously. Warschauer and PaulJ1 also have suggested some uses for such equipment as tools in a high-pressure laboratory. While a simple single stage pump is adequate for generating pressun up to a few thousand atmospheres, higher pressures require some form of a pressure multiplying device, which may be called a press or an intensifier. A simple press, which has been used for optical work, it sketched in Fig. 2.a6The area ratio is chosen so th a t fluid at a pressun
** E. Fishman and H. G. Drickamer, Anal.
Chem. 28,804 (1956).
PHYSICS A T HIGH PRESSURE
49
near 10,OOO psi is used to create a force which in turn will generate a maximum pressure of 12,aoO atmos in the lower cylinder. It is often convenient to pre-compress the fluid in the high-pressure chamber to the lower pressure through the direct use of a pump, and a second opening on the aide of the high-pressure chamber is used for this purpose. The
FIG.2. A high-pressure presa which has been used for optical measurements to [E.Fishmsn and H. G . Dricksmer, A d . Chem. 48,804 (1956)).
12,000 atmoe
high-pressure end may be used directly as an experimental chamber as shown in Fig. 2, or, by a suitable valve arrangement, it may be used in a cyclic manner as a pump to deliver the high-pressure fluid to a separate pressure vessel. These features are not present in the relatively small press in Fig. 2, which is depicted only to show a simple example of a piece of high-pressure equipment. Other examples of equipment for use to 10,OOO atmos are given by Bridgman' and Comings,' while there are
50
C. A . SWENSON
also several papers which describe in detail the construction and performance of apparatus for use in this pressure range.*6J8 Commercial intensifiers, while identical in principle with that sketched in Fig. 2, can be quite large pieces of equipment capable of delivering quantitiee of fluid a t pressures up to 13,000 atmos. A description of some of these has been given by NewhalLa9 The production of high pressures in gases is a bit more difficult, and decidedly more dangerous. The high compressibility of gases means that considerable quantities of fluid must be handled, and usually the gas must be separated from the oil by a mercury U tube separator in order to avoid contamination. Since mercury has been known to attack steel a t pressures above 5000 atmos,'O the use of a direct means of compreesion is desirable.a'JI n many applications, such as in the study of spectra, the presence of impurities (as for instance from lubricants in pistons) can have serious effects. Because of these difficulties, thermal compression has been used by several w o r k e r ~ . ~ l -High ~ * pressures may be obtained, for instance, by sealing off at 63'K a high-pressure bomb which was filled with solid nitrogen, and then allowing it to warm to room temperature at constant volume. Cyclic thermal compressions using this principle have been suggested, and used in some cases, although the techniques do not seem to have general applicability. The ideal situation for high-pressure work occurs when a nonviscous fluid is used to apply a truly hydrostatic pressure to the sample. A t room temperature, however, the upper limit of the range of purely hydrostatic pressures is 25,000 to 30,000 atmos since, under these conditions most fluids either have solidified or have become very viscous. Out of many substances which were tested, Bridgman found a mixture of pentane and isopentane to be the most useful," since it neither solidified nor became excessively viscous at 30,000 atmos at room temperature. Gases are sometimes used (and, indeed, must be used for work at very high temperatures), but even nitrogen freezes at room temperature near 30,000 atmos, and hydrogen, a potential fluid with a higher freezing point, cannot be used since it is known to cause pressure vessel failure at pressures C. E. Weir, J . Research Natl. Bur. Standards 46, 468 (1950). D. S. Hughes and W. W. Robertson, J . Opt. SOC.Am. 46, 557 (1956). ** D. W. Robinson, Proc. Rov. Soc. A446, 393 (1954). So D. H. Newhall, Znd. Eng. Chem. I @ ,1949 (1957). d o P. W. Bridgman, "The Phyaica of High Pressures," p. 94. G . Bell, London, 1949. 4 ' Ya. S. Ksn, 3. Tech. Phys. (USSR) 18, 1156 (1948). 42 J. Robin and B. Vodar, J . phys. radium 17, 500 (1956). 4'5. S. Boksha, Krystallograjiya 4, 198 (1957), Soviet Phys., Crystallography 2, 191
*'
( 1958).
P. W. Bridgman, Proc. Am. Acad. Ails Sci. 77, 115 (1949).
I4
51
PHYSICS A T HIGH PREStiURE
of the order of a few thousand atmosphere^.'^ The use of argon and nitrogen to 30,000 atmos will be discussed further along in this section in conjunction with high-temperature work. In the pressure regions where fluids do not exist, methods must be used in which the applied pressures are only approximately hydrostatic. These were developed initially by Bridgman for work to 50,000 atmos and were later applied for work to 100,OOO a t m o ~ . ~The ~ . ~method ’ which is most applicable to the det,ermination of P-V isotherms is shown in 1
SAMPLE
MOLDER
(b)
IC)
Fro. 3. Methods which have been used to apply approximately hydrostatic pre% iurea to solids a and c [P. W. Bridgman, “The Physics of High Prmures,” pp. 397404. G. Bell, London, 1949; Proc.Am. ..lead. . 4 d s Sri. 81, 165 (1952)j and to liquids (b) 18. D. Hamann, “The Physico-Chemical Effects of Pressure,” -4cademic Press, New York, 1957; S. D. Hamann and D. R. Teplitzky, Discussion8 Faraday Soc. 38, 114 (1956)). The press which applies force to the pistons is not shown in any of the sketches. The size of the extrusion rings in a, and the gap hetween the piston and cylinder in both a and b are exaggernted, ao is the sample thickness in c.
Fig. 3a. The sample is placed in a cylinder which is closed at each end by a closely fitting piston, and force is applied to the pistons, the pressure being just the force divided by the area of the pistons. If the sample material is soft, the pressure will be roughly hydrostatic. If it is hard or brittle, then the sample must be enclosed in a thin sheath of indium or some other soft material which will act like a pseudo-liquid to create an approximately hydrostatic pressure. In order to investigate melting phenomena, Rridgman sealed substances which are normally liquid at ‘‘E. W.Comings, “High Pressure Technology,” p. 69. McGraw-Hill, New York, 1956. “P. W. Rridgmen, “The Physics of High Pressures,” pp. 397-404. G. Bell, London, 1949.
“P. W.Bridgman, Proc. Am. A d . A& Sci. 81, 165 (1952).
52
C. A . SWENSON
room temperature in a lead capsule.48Similar work has been done using a polytetrafluoroethylene (Teflon) or polyethylene capsule with a tapered stopper so as to form a self-sealing joint (see Fig. 3b).*a40These methods are not restricted to the very high-pressure region, but may be adapted with simple apparatus for relatively crude (a few per cent) P-V work at almost any temperature and for pressures up to 20,000 atmos. A discussion of the use of this method is given in the section on low-temperature experimental techniques. I n a second method, which has been used for developing pressure having an approximately hydrostatic nature, a thin, cylindrical sample is placed between the truncated conical ends of press pistons (Fig. 3c). Due to the shape of the piston tips, pressures much greater than the yield stress of the piston material can be generated, and the friction forces at the top and bottom of the thin disk are great enough to prevent lateral extrusion. With a proper radial and vertical stacking of various materials, Bridgman has used this technique to measure resistance changes to 100,OOO atmos,” while Griggs and Kennedy (and others) have used it t,o explore phase changes in geological samples to high temperatures and pressures.6o Bridgman also has determined, from the torque necessary to rotate one of the pistons, the shear strength of various substances to 100,000 atmos; many of the phenomena that he observed in these investigations are as yet unexplained.” In apparatus, built by Drickamer and his collaborators (described in the optical prop erties section), a modification of this principle is used to obtain absorption spectra in various samples to pressures as high as 250,000 atmos. I n principle, any of the foregoing terhniques can be used above room temperature by placing the experimental vessel in a bath or a n external furnace. The softening of ordinary tool steels at temperatures of the order of 300°C seriously limits both the pressure and temperature range; however, the range can be extended to some extent through the use of high-speed steels. Kurnick has used conventional fluid transmission techniques t o 400°C and 8000 atmos pressure using DC200 silicone oil and an external furnace.62The Griggs and Kennedy apparatus (known as the “Simple Squeezer”) has been used to 20,000 atmos at 1000°C using this type of heating. The temperatures a t which truly hydrostatic pressures can be applied can be extended almost indefinitely upward through the use of a 1’. W. Bridgmen, Proc A m . Acad. Arfs Sci. 77, 129 (1949). ‘@S.I). Hamann and I). K.Teplitzky, I>iscussaons Faraday Soc. 12, 114 (1956). n. T. Griggs and G. C.Kennedy, Am. J . Sci. 264, 722 (1956). r 1 P. W. Bridgman, “The Physics of High Pressures,” p. 409. G. Bell, London, 1948. *=S.W. Kurnick, J . Chem. Phya. 10, 218 (1952).
PHYSICS A T HIGH PRESSURE
53
furnace which is immerwd in the pressure transmitting fluid and which surrounds the sample. The fluid must of necessity be inert, and, hence, a gas (either argon or nitrogen) must be used. While the temperature of the furnace (and the sample) may be lO0O”C or higher, the walls of the pressure vessel remain relatively cool and do not soften with time. Yoder has described such an apparatus,6s and recently this technique of “internal heating” has been used widely.&& Birch et aLS4have described an apparatus of this type in which 27,000 atmos and 1400’C is obtained. The upper pressure limit for this type of apparatus is given by the freezing point near room temperature of the gas which is used. The techniques can be extended to much higher pressures and temperatures through the use of approximate methods, and Hall66 has given a good introduction to some of the design considerations for this type of work. One major difficulty which is encountered in internal heating work is to know the temperature of the sample. Bridgman has investigated the effects of pressures to 12,000 atmos on the thermoelectric properties of various pure metals,’ while Birch has studied the effect of 4000 atmos on both chromel-alumel and Pt-Pt, 10% Rh thermocouples at temperatures up to 580°C.66With the extension of internal heating techniques to much higher temperatures and pressures, the need for more extensive data has become apparent, and at least one laboratory has plans for carrying out these experiment^.^' ( 1) Limitations. The design of high-pressure equipment becomes more difficult as the working pressures are increased. For instance, a steel ve8sel which can be used indefinitely a t 12,OOO atmos has only a limited lifetime at 20,000 atmos and may not be useable at all for higher pressures. The pressure limit for a simple, single-walled, carefully heat-treated steel vessel is probably somewhere near 10,OOO atmos. The limiting working pressure is reached when the stress on the inner surface of the vessel H. 6.Yoder, Trans. Am. Geophye. Union 31, 827 (1950). t a s P r o f ~F. r Birch (private communication, 1959) has written the author as follows with reapect to the history of internal heating techniquea: “The internal furnace goes back at least as far as 1911, when J. Johnston and L. H. Adams (Am. J . Sci. 181, 501-17) measured the effect of preeaure on the melting points of Sn, Bi, Cd and Pb to 2000 bars. Further developments were deacribed by F. H. Smyth and L. H. Adams, J . A m . Chem. Soc. 46,1172 (1923); R. E. Gibeon, J . Phys. &vt. 82, 1199 (1928); R. W. Goranson, A m . J . Sei. 422, 481 (1931), F. Birch, P h p . Rev. 41,641 (1932); ale0 Am. J . Sci. 288,192 (1940), and BuU. Qeol. SOC.Am. 64, 263 (1943).”
I’F. Birch, E. C. Robertson, and S. P. Clark, Jr., Znd. Eng. Chem. 49, 1966 (1957). “H.T.Hall, Rev. Sci. Znsfr. 29, 267 (1958); 11,125 (1960). I’F. Birch, Rev. Sci.Znsfr. 10, 137 (1939). I’F. P. Bundy and H. M. Strong, private communication (1959).
54
C. A. SWENSON
exceeds the yield strength of the vessel material. While it is possible to increase this limiting pressure by increasing the wall thickness, after a certain point a further increase has no useful effect. A detailed discussion of design criteria is given by Comings, together with comments on various construction niaterials.‘ The maximum internal pressure which a vessel can withstand can be increased, however, hy placing the material a t the inner circumference under a state of compressive stress. This can be accomplished either by a proper strain hardening and stress relaxation cycle (autofrettage) or by using one or more bands which have been shrunk on the outside of the vessel.68 Both of these involve permanent changes in the stress at the inside of the vessel so th at the application of a n internal pressure first reduces the compressive stress to zero, and then causes a tensile stress to be built up. Bridgman has developed a modification of this technique which is more satisfactory for extending the working pressure This is illustrated in Fig. 4 which is a sketch of the apparatus described by Birch et ~ 2 1 The . ~ ~inner vessel has a tapered outer surface, and a series of conical rings which have been ground to fit this taper are forced onto the inner vessel by a hydraulic ram simultaneously with the increase in the internal pressure. By a judicious choice of ram pressures and taper, it n i possible to maintain the inner diameter of the pressure vessel roughly constant as the internal presaure is inrrease to 27,000 atmos. Bridgman has used this design with a sample holder of the type illustrated in Fig. 3a to insure that the area of the cylinder did not increase more rapidly than the diameter of the piston at pressures up to 40,000atmos. Rridgman obtained higher working pressures than Birch by using Carboloy (sintered tungsten carbide) pistons and cylinders of smaller diameter. For much higher pressures (up to 100,OOO atmos), Bridgman used a two-stage apparatus in which the highest pressures are created in a sample holder (Fig. 3a) which is totally immersed in a fluid at 30,000 atmos.60 The need for external support depends to some extent on the inside volume of the pressure vessel. When a sample holder of the type shown in Fig. 3a is used with a short sample (ratio of sample length, diameter, and cylinder length about as shown in Fig. 3a) the support of the part of the cylinder wall which is not directly stressed is such as to increase the effective internal pressure which the cylinder will stand. We have D. M. Newitt, “High Pressure Plant and Fluids at High Pressure,” Chapter 111. Oxford Univ. Press, London and New York, 1940. so P. W. Bridgman, PTOC. Am. Acad. A r k Sci. 76, 9 (1945). 1’. W. Bridgman, Proc. Am. A d . Art8 Sci. 76, 55 (1948). 6n
PHYBICS A T HIGH PRESSURE
55
used a type 85B Carboloy cylinder, 0.250 in. i.d. and 0.850 in. o.d., for over one hundred cycles to 21,000 atmos a t room temperature before failure, with no external support of any kind. Jacketing with a shrink-fit hardened beryllium copper sleeve did not extend the life appreciably, but
FIQ.4. An illustration of Bridgman's method of external support as used by Birch [F. Birch, E. C. Robertson, and S. P. Clark, Jr. 2nd. Eng. Chem. 49, 1965 (1957)l. (a) betion through apparatus used for measuring chemical equilibria. Tapered cylinder E in forced into supporting rings D by ram B, 8 inches in diameter with 500-ton thrust. Five-inch ram C generates preeaure in E. (b) Tapered cylinder and supporting ring. Gae at 2000 bars is introduced at top. Preaaure is measured with Manganin coil a t B. Cylinder bore is inch. (c) Bottom closure and insulated electrical lead. Horimmtally ruled material is lava, diagonally ruled material is Teflon, and remaining parta are metal. Samples placed in the furnace F can be heated to temperatures of the order of 1400"C, while the walls of the pressure chamber [A, Fig. (b)] remain relatively cool. Argon or nitrogen gas is used to transmit the pressure to the sample.
did prevent a catastrophic rupture when the Carboloy finally failed with a radial crack. The pistons (also Carboloy) gave some trouble, and actually set the upper pressure limit since they showed a tendency to grow slightly in diameter upon successive cycles with a maximum pres-
56
C. A. GWENSON
sure of 25,000 atmos. A proper choice of Carboloy type and adding support to the head of the piston should eliminate this trouble since Bridgman has used unsupported pistons of this type to 40,000 atmos, although some growth in diameter was always observed. The success of the apparatus sketched in Fig. 3c depends on the support of the relatively small and highly stressed area of the piston tips by the larger area of the base of the cones. This interesting and useful design illustrates what Bridgman has called the “principle of massive support.” It is because of this “massive support” that the cone tips can support a stress which is much greater than their yield stress without deforming. As another example, if a hardened Carboloy piston is pressed against a plane steel surface of much greater area, the surface will be deformed appreciably only after a stress is applied by the piston which is much greater than the yield strength of the steel. Thus, tungsten carbide tool blanks can be “backed-up” by much softer, but larger, hardened steel blocks with little deformation of the blocks. The essentially ferromagnetic nature of ordinary steels and sintered tungsten carbides causes difficulties when high-pressure work on magnetic properties is attempted. While the choice of steels and the carbidee is quite large, the choice of high tensile strength nonmagnetic materials is quite limited. Austenitic stainless &eels (the 300 series) will workharden quite easily, are almost diamagnetic, and can be used to roughly 5000 atmos maximum pressure^.^ They have, in addition, favorable low temperature properties in that their thermal conductivity is low and they remain ductile to absolute zero. The most useful nonmagnetic material, however, is a 12 at % (2.1 wt %) alloy of beryllium in copper which can be hardened by precipitation a t 325°C to give a Rockwell C hardness of about 40, and a yield strength in excess of 10,OOO atmos. This alloy also remains ductile to very low temperatures, in contrast with the brittle behavior of ordinary steels and carbides. Hardened beryllium copper was first used by Benedek and Purcell, and has made many heretofore unexplored areas of the high-pressure field accessible to experiment. There are however, two disadvantages of this material. First, it ha8 a low elastic modulus (about 15 X 10’ pRi) and, a thermal expansion about twice that of steels, so care must be taken in making connections where temperature changes are involved. Secondly, the great toxicity of beryllium in any form makes it necessary to exerciee great care in the fabrication of any high-pressure parts from beryllium copper. While machining under oil should involve no hazard, grinding should be done only under carefully controlled conditions.o1 From II 8’
N. I. Sax, “Dangerous Properties of Industrial Materials,” p. 357. Reinhold, New York, 1957.
PHYSICS A T HIOH PRESSURE
57
purely technical standpoint, when the above limitations are noted, it is a material with which machinists enjoy working.61*
b. High-pressure Seals In the previous discussion the technical difficulties which are encountered in the design and construction of high-pressure equipment, other than those which are involved in the choice of construction materials was not considered. The manufacture of satisfactory high-pressure seals (both moving, as in Fig. 2, and static, as for electrical leads) can be accomplished in several different ways. These have been surveyed by Bridgman,’ Comings,’ and Hamann,s and will not be summarized in detail here. NiemieP also has considered various types of seals from both a theoretical and experimental viewpoint. The particular technique which will be used in a given situation will depend to some extent on the type of work which is being done (and the pressure range). The recent accelerated interest in high-pressure work has resulted in the utilization of new materials and methods which make the job of assembling high-pressure equipment much easier. Some of these will be considered in the following. Warschauer and Paula1have described several innovations and design modifications. Probably the most useful suggestion made by them is that harddrawn, small diameter (0.125 in. 0.d. X 0.024 in. i.d.) type 316 stainless steel tubing can be used to contain pressures in excess of 15,000 atmos, the size being in contrast with traditional high-pressure tubing which has had a large outside diameter for high-strength. By using this small diameter tubing the experimental vessel may be isolated mechanically from the source of high pressure; also its use allows coneiderable flexibility in the planning of an experiment. The connection of this tubing to high-pressure apparatus is attended by some problems, but practical solutions to these are also given. The construction of seals for pressures of the order of thousands of atmospheres is, to a great extent, routine. The major principle in any technique must involve either high sealing pressures at contact points, or a design in which the sealing efficiency increases as the pressure to be contained increases. The classic example of this latter type of seal was introduced by Bridgman, and makes use of the “unsupported area” principle.” This is illustrated in Fig. 5a. The nut is tightened sufficiently A rather complete diecussion of experiences with nonmagnetic pressure v& bss been given by W. Paul, G . B. Benedek, and D. M. Warschauer, &. Sei. Z&r. SO, 874 (1959). (‘€3. A. Niemier, Trans. ASMB 76, 389 (1953). u P. W. Bridgman, “The Phpice of High Preasurea,” p. 32. C. Bell, London, 1949. (1.
58
C. A. SWEKSON
a t zero pressure to produce an initial seal. As the internal pressure is increased, the total force on the base of the seal, P X A , must equal the total force on the gasket, P’ X A ’ , wherc A’ < A (conventionally, .1 = 1.25.4’). Sinre the gasket pressure P’ is always greater than the iiit,crnal pressure P , leakage is impossible in principle. Typical gasket mtit,erials are Neoprene or Teflon for room temperature use and lead for higher temperatures; potassium niet:tl has been used at low temperatures. The area ratio in this type of seal cannot be increased indefinitely, however, since too high a gasket pressure can cause the stem of the plug to be “pinched off ” resulting in the ejection of the top of the stem with a dangerously high velocity. Hence, the yield strength of the stem limits SELF-EXTRACT0R INTENSIFIER PISTON
PRESSURE
Fro. 5. High-pressure seals. a, Bridgman unsupported area [P. W. Bridgman, “The Physics of High Pressures,” p. 32. G.Bell, London, 19491. b, 0 ring intensifier [from W. B. Daniels and A. A. Hruschka, Rev. Sci. Znslr. 28, 1058 (1957)l.
the area ratio for a given pressure. This limitation can be modified somewhat if the gasket thickness is decreased and use is made of the massive support principle. Experiments by Paul and Warschauer“ have shown that area ratios of the order of 4 can be used with gaskets that are & in. thick. A limitation of most high-pressure gasketing is that, for sufficiently high pressuresI the gasket material can extrude through the gaps between the retainer plate and the vessel wall. To prevent this (and also to reduce the need for precision machining), small triangular ertrusion ring8 (Figs. 3 and 5) can he used so that the internal gasket pressure effectively maintains zero gap.06These extrusion rings are extraordinarily effective for almost all high-pressure applications and should be more widely 64
8)
D. M. Warschnuer and W. Paul, Rev. Sci. In&. 48, 62 (1957). P. W.Bridgman, “The Physics of High Pressures,” p. 34. G. Bell, London, 1949.
uw:d than ttwy urc. An an cximplt~,w~ line one of thrse rings (0.030in. X 0.030 in., hardened heryllium copper) 0 1 1 each end of compression samples which are placed in mmple holders of the type shown in Fig. 3a. After several compressions of a soft sample (indium, for instance) to 20,OOO atmos, the pistons can he removed ciisily t)y hand, with no extrusion of the sample material into the gap between the piston and cylinder walls. I n most applications, it is not important th at the piston fit closely t o the cylinder initially, since the increase in the diameter of the cylinder upon the application of pressure is much greater than the increase of the diameter of the piston due to the compressive stress. A poor initial piston fit (-0.002 in.) can be compensated for quite effectively by the use of extrusion rings. Howman et nl., descrilw t h r use of these rings with a MoS2 impregiirrted Teflon Rridgmari pnc.king to 10,OOO atrnos6' A second and more recent type of uiisupported area packing is given by the use of 0 rings. I ~ w s o nfirst ~ ~dcscrihed the use of 0 rings to seal the piston of an intensifier at pressures u p to 10,000atmos. Later, Gugan6* reported discussed their use for static seals and Daniels and Hruschka6@ on an extension of these techniques which made use of extrusion rings (Fig.5b). When the 0 ring cannot extrude, it appears that its range of tlpplicahility should be as great as that of the ordinary, Neoprene, Rridgman-type packing. Jlaniels and FIruschka have reported using the Oring packing to 16,000 atmos, and they stress th a t it has the additional advantage that no I' pinch off" can occur as in the Rridgman seal. Whalley and Lavergne have rc-ported also on their experiences with 0 rings to 10,OOO atmos both for valves and for fixed and moving seals.'& Even at low pressures (10,000 psi), we have found it useful to replace the leather packings in commercial hydraulic rams with simple 0 ring packings eimilar to those shown in Fig. 5b; this resulted in a decrease in friction and an increase in reliability. In moving seals the friction often is large; it can be minimized by the right choice of packing material (such as the MoS2 impregnated Teflon), or by making the gasket very thin. Another possibility for low-pressure friction seals involves the use of the controlled clearance principle, and Newhall has discussed several applications of this principle in seals in addition t o its use in free piston gauges.?O H. A. Bowman, J. I,. Cross, I). P. Johnson, J. D. Hill, and J. 6. Ives, Reu. Sei. Instt. 17,550 (1956).
"A. W.Lawson, Rev. Sei. Instr. 26, 1136 (1954). ''1). (hRan, J . Sri. Insfs. 33, 160 (1056). " W . R. 1)anirlx and A. h . Hrusrhka, Rw. Sci. In&. 28, 1058 (1957). 'In E. Whalley a d A. I,avorgne, J . Sci. In&. 38, 46, 47 (1959). lo[). H. Newhall, Ind. Rny. C h m . 49, 1993 (19571.
GO
C. A. 6 W E N S 0 N
The problem of introducing electrical leads into a pressure v e w l sccms to be difficult to solve in an easy manner. Bridgman used insulating conical pipestone sleeves, the design of which is based on the unsupported area principle, for pressures up to 30,000 atmos.60.010 Several variations of these are described by Birch et al." (Fig. 4), and their use at low temperature also has been reported. * I However, these seals require precise machining to be useful, and various other types have been used, mostly for lower pressures. Gugane8 describes a high resistivity (10" ohm) Araldite seal which is good to 5000 atmos. Bowman'' also has described an easily demountable high-resistivity seal which has a commercial sapphire or quartz instrument bearing and an 0 ring. A seal described by Simon72 contains commercial fittings; Gibbs and Jarman': have suggested one, good to 4500 atmos, which appears difficult to make, but which should be inherently satisfactory. Vallauri and Forsbergh" have described a wide band seal good from 0 to 3000 Mc at 1500 atmos. The subject of valves also has been discussed occasionally in the literature; see for example referen~es.~~-'' One of the difficult problems in high-pressure work is the detection of small leaks. In one effective technique for leak detection7* a minute amount of a halogen-containing compound (chloroform, methyl chloride, etc.) is included in the hydraulic fluid. The seepage of the fluid then ie detected by means of a commercial leak detector which is particularly senrsitive to the vapors of these halogen compounds. It is claimed that by this technique leaks are detected easily which would not measurably affect a sensitive pressure gauge for several hours. 3. LOW-TEMPERATURE TECHNIQUES
The application of high pressures a t very low temperatures requires an approach which is different from that which can be used for similar pressures a t room temperature and above. The available techniques depend to a large extent on the type of measurement which is being considered; many of the difficulties which arise resemble those which are encountered in the pressure region above 30,000 atmos, where there also are no transmitters of truly hydrostatic pressure. The difficulties are much more serious a t low temperatures, however, since even helium, the H. A. Bowman, Nall. Bur. Sfandards ( U . S . ) Tech. Newa Bull. 39, 71 (1Y55). I. Simon, Rev. Sci. Znstr. 28, 963 (1957). 7 a I ) . F. G i l h and M. Jarman, J . Sci. Inslr 36, 472 (1958). kl. (;. Vallauri and 1'. W. Forsbergh, Jr., Hru. Sci. In&. 28, 198 (1957). '6 E. W. Comings anti I f . G. lfric.kamcsr, I&v. Srz. fnsfr. 24, 1028 (1!)51). 7 8 1). W. Itobirimon, J . Sri. lnstr. 80, 483 (i!)Sd). 77 It. 1,. Mills, Kcu. Sci. fnstr. 27, 332 (19.56). 7" W. Paul and I). M. Warschauer, Rcv. Scz. Inatr. 26, 731 (1955). 71
61
PHYSICS AT HIGH PRESSURE
fluid with the lowest melting temperatures, solidifies a t 25 atmos at absolute zero, and at 140 atmos and 1800 atmos, respectively, at the boiling points of liquid helium (4.2’K) and liquid hydrogen (20.4’K). Various ingenious techniques have been devised to overcome this limitation, although approximate methods of obtaining very high presa r e s (i.e., using plastic solidR as transmitters of pressure) also have been used successfully. Most of the stimulus for high-pressure work at low temperature is due to the early work by Laearew and K a r ~ , who ’ ~ showed that pressures of a few thousand atmospheres produced a measurable effect on the transition temperatures of superconductors. Thus, the discussion of both techniques and results will draw heavily on the work on superconductors. a. Fluid Transmitters
Kushida, Benedek, and Bloembergenuo have reported that a 50-50 mixture of n-pentane and 2-methyl-butane can be used at 10,000 atmos and 200°K without appreciable loss of fluidity. This temperature probably corresponds to the practical lower limit for the use of normal fluids at pressures of the order of 10,OOO atmos. Bridgmansl found it necessary to use helium gas to measure the effects of pressure on electrical resistivity at liquid nitrogen temperatures (77’K). The maximum pressure which he used (about 7500 atmos) was limited by the materials which were available at that time, and the danger of a failure in the high-pressure system. From a cryogenic point of view, improvements in this type of apparatus should be possible through the use of materials which have since become available. The apparatus of Kushida et al.,*O which consisted of a beryllium copper bomb connected to a source of pressure by small diameter, hard drawn, stainless steel tubing, should need little basic modification for use with a gas a t much lower temperatures. The beryllium copper and stainless steel alloy^ have roughly the same thermal expansion coefficient,02 and both have excellent low-temperature mechanical properties. The design of an efficient low-temperature high-pressure system should be relatively easy because of the low thermal conductance of the stainless steel tubing. The upper limit for hydrostatic pressures in such an apparatus would correspond at liquid nitrogen temperatures to the yield strength of the beryllium copper (about 10,OOO atmos), while at lower temperatures it would correspond to the solidification pressure of the helium (see Sec”B.Laaarew and L. S. Kan, J . Phys. (USSR) 8, 193 (1944). ‘OT.Kushida, G. B. Benedek, and N. Bloembergen, Phys. Rev. 104,
1364 (1956).
’’ P. W. Bridgman, “The Phyeica of High Preesuree,” p. 427. G. Bell, London, 1949. ‘‘J. J. M. Beenakker and C. A. Swenmn, Rev. Sci. Instr. 26, 1204 (1955).
62
C . A. SWENSON
tion 4b). Some extension of the fluid region could be obtained through the use of helium of isotopic mass 3, but the small difference in the melting pressures arid the danger of losing ail investment of thousands of dollars in the event of u leak would niuke this procedure unprofitable. Some results at temperatures near ahsolute zero have been obtained at pressures below the freezing pressure of liquid helium, and here the advantage to be gained by using helium 3 would be relatively greater, since the difference in melting pressures of the isotopes is of the same order of magnitude as the applied pressures. b. Ingenious Devices IJsiny S o l d Transmitters ( 1 ) The icc-bomb kchniques. The initial work on the production of high pressures at liquid helium temperatures was done by Lazarew and k'anl9 who measured the effects of pressure on superconducting transitions. Their so-called " ice-bomb " technique makes use of the anomalous expansion of water on freezing a t constant pressure, and the related increase in pressure which accompanies freezing a t constant volume. A sketch of their equipment is shown in Fig. 6. Lazarew and Kun found that with a slow cooling rate the ice froze homogeneously with a fairly uniform pressure distribution, atb: was indicated by the sharpness of the superconducting transitions which were obtained. The maximum pressure generated in the bomb is that of the ice I-ice III-water triple point. From this pressure, and the measured strain in the walls of the bomb as a function of temperature, Lazarew and Kan estimated the pressure a t liquid helium temperatures to be about 1700 atmos in the case of tin and indium. Recent measurements in other laboratories, using different methods, indicate that the observed temperature shift (0.0'37")x3 probably should he associated with a pressure of between 1900 and 2000 a t r n ~ s .The ~ ' difference could be due to a n increase with decreasing temperature nf the elastic modulus of the beryllium bronze from which the bomb was made, a correction which was not mentioned in their estimates. Similarly, if the sample fills the bomb fairly well, differences in thermal expansion from one sample to another could cause significantly different final pressures, depending, of course, on the relative compressibilities of the ice and the .sample. I n their earlier work, Lazarew and Kan introduced electrical leads into the bomb t)y means of (aapillm-ies which contained both the wire arid a drop of water (Fig. ti). When the water froze, reliable high pressurr seal was formed. The nonm:qpctic nature of the bomb material made possible the use of a ballistic or alternating current method and 8a
I3 Lazarew and L. Kan, J . Phys. (ITSSR) 8, 361 (1944). 1,. D. Jcnnings and C. A. Swenson, Phya. Rev. 112, 31 (1958).
PHYSICS AT HIGH PRESSURE
63
Fra. 6. The “ice bomb” of B. Lazarcw and L. Kan [ J . Phya. (USSR)8, 361 (1944)). The bomb A waa sealed a t each end by high pressure plugs B, D and suspended by a ring G. EIectrical connections were made to one end of the sample S through a copper wire H to the bomb body La,L,, and to the other end by means of wires L1, Lt which passed through a channel K in the pipe T. Thia channel and the bomb were filled with water W, and sealed a t room temperature and zero pressure by a plug C and rubber sleeve R. Water in thechannel WBB froren by means of liquid nitrogen held at the level N-N in order to seal the leads against high pressure as the main body of the bomb was cooled slowly to 77°K.
external coils in later superconducting experiments. Thus high-pressure electrical lead-ins were no longer needed.86 The small size and rtmncc of high-pressure conaeetions from outside the cryostat make a technique of the ice-bomb type particularly attractive. Its flexibility was demonstrated by its use to obtain the first highpressure results below 1°K (the critical field curve to 0.06”K for the superconductor, cadmium, a t 1550 atmos).8e It has also been used for the study of galvanomagnetic effects in nonsuperconductors at low temperatures, as will be discussed in a later section. IrL. S. Kan, B. G. Lazarew, and A. I. Sudovtsov, J . E x p t l . Theorel. Phys. (USSR) 18,825 (1948). (8
N. E. Alekseevskii and Yu. P. Caidukov, J. Exptl. Theorel. Phya. (USSR) 29, 8% (1955), Soviet P h p JETP 2, 762 (1956).
64
C. A. SWENSON
The major drawbacks of the ice-bomb type technique are the limited number (one or two) of pressures which are available, since water is the only practical working substance, and the need for warming a sample to room temperature in order to change the pressure. Within these limitations, it is a satisfactory and convenient method for attaining 2090 atmos a t low temperatures without conventional high-pressure equipment. (2) The $zed clamp method. Chester and JonesB7have devised a method of extending the pressure range to 50,000 atmos in a n apparatus which has many of the advantages (and disadvantages) of that used in the ice-bomb technique. This apparatus is a variation of the “simple squeezer’’ of Fig. 3c, in which a thin disk of a superconductor is placed inside a nonmetallic retaining ring between the faces of the conea. With the apparatus a t room temperature, a force which was applied to the pistons by a press was locked in by a clamp, so that the stress on the sample was retained when the assembly was removed from the press and cooled to low temperatures. The final pressure was estimated from the applied force and the area of the faces, with suitable corrections for thermal expansions, change in elastic moduli with temperature, etc. This method is applicable only to superconductors whose transition temperatures can be obtained from susceptibility measurements with external coils. From comparisons with other results the effects of sample deformation and lack of pressure homogeneity, which are unknown, do not seem to have been important in these experiments. The highest pressures used in experiments a t liquid helium temperature were those, roughly 50,000atmos, reached in the work on superconducting transitions. (3) Solid helium. The foregoing methods suffer from two drawbacks. First, the pressure cannot be varied except a t room temperature, and, second, deformation and annealing effects are of indeterminate importance. There is, also, a fundamental uncertainty in the actual pressure exerted on the sample, since the corrections are quite large. This means that even if the first two difficulties are not important, precision measurements are still not possible. A variation of these techniques has been used by Dugdale and €Iulberts8 to measure the change in the resistivity with pressure of various metals at liquid helium temperatures. The actual method was suggested by the results of Dugdale and Simonsgon the thermodynamic properties of fluid and solid helium a t high pressures. This work showed that solid helium is quite compressible, so that if fluid helium, initially P. F. Chester and G . 0. Jones, Phil. Mag. 171 44, 1281 (1953). J. S. Dugdale and J. A. IIulhert, Can. J . Phy3. 16, 7N (1957). J. S. Dugdale and F. E. Simon, Proc. Roy. SOC.A118, 291 (1953).
PHYSIC8 AT HIGH PRESSURE
65
at 3000 atmos pressure, is cooled under pressure at constant volume through the solidification temperature, the pressure drop in the container due to the density change on solidification is quite small. The thermal expansion of the solid under pressure is also small, so that once the helium has solidified practically isobaric conditions are maintained as the temperature is varied. The pressure drop upon solidification under these conditions is very much a function of the initial pressure and temperature since the compressibility decreases rapidly with pressure; thus there is a practical limit on the pressures attainable. This limitation can be overcome by a modification of the procedure in which the helium is solidified slowly from the bottom at constant pressure."O Although this should extend the available pressure range considerably, it introduces the problem of possible deformation if the freezing occurs initially at conatant pressure and then at constant volume due to the blocking of the filling capillary during solidification. A sketch of the Dugdale and Hulbert apparatus is given in Fig. 7." The beryllium copper bomb was used over the temperature range from liquid helium temperatures to room temperature with pressures up to about 3000 atmos. Above the solidification temperature, fluid helium was used as a pressure transmitting medium, and pressures were measured with a Bourdon gauge. A t lower temperatures, the helium was solidified at constant volume, and pressures were calculated from the equation of state of solid helium. One disadvantage of this method is that, again, pressures may not be varied a t will a t low temperatures. For many purposes, however, the solidification temperatures are so low (28.3"K at 3000 atmos, for instance) that annealing effects do not occur. The experience of Dugdale and his c o - w ~ r k e r sis~very ~ ~ ~encouraging ~ with respect to the homogeneity of the pressures, and the lack of deformation of the samples. For more precise work, it is possible to determine the pressure in the solid helium below the solidification temperature through the use of the elastic properties of the bomb itself. A capacitance proximity gauge or a resistance strain gauge which measures the strain in the bomb wall can be calibrated at low temperature against the fluid helium pressure up to the solidification pressure. In this temperature range the calibration should not be a function of temperature, and, thus, it should be possible to follow directly the change of pressure in the solid with temperature. This method was first suggested and used by Mapother for relatively low (loo0 atmos) pressure work on superconductors.** P. F. Chester, private communication (1957). J. S. Dugdale and D. Gugan, Proc. Roy. SOC.A241, 397 (1957). D. E. Mapother, private communication (1959).
I@ b1
66
C. A. SWENSON
TO WGH PRfSSIRf GAS f n T f Y
n
-A
SPECIMEN
FIG.7. The high-pressure resistivity apparatus of J. S. Dugdale and J. A. Hulbert [Can. J . Phye. SS, 720 (1957)l. High-pressure helium gas was fed into the high-prmure bomb A by means of a thick-walled stainless steel tube B, these high-pressure components being enclosed in a vacuum space C. The bomb was sealed by a lens ring F. The temperature of the bomb was measured by the thermometers D and E, while electrical leads to the specimen G were brought out through a seal produced by freezing silicone oil in liquid nitrogen in the side tube H.
c. Direct Use of Solids
The previous methods are most useful for electrical or magnetic measurements in which only electrical coniiections are required. Heat leak from surrounding regions a t room temperature and the resultant evaporation of low-temperature liquids (helium or hydrogen) are not
PHYSICS AT HIGH P R E S S P R E
67
serious problems. However, the study of the equation of state at low temperature requires essentially volume measurements and the use of fundamentally different techniques which may, in turn, be modified for electrical and magnetic experinleiits. The approach which has been most useful in this work is identical with that of the piston displacemelit method which was first used by Bridgman a t pressures up to 40,000 atmos.40 I n the simplest version, the sample holder, which is identical with that shown in Fig. 3a, is placed between the platens of a press which has long stainless steel compression and tension members extending to regions at room temperature (see Fig. 8). The forces are quite large (roughly eight tons for 20,000 atmos on a 0.25041. dism sample), but fortunately stainless steel has the desirable combination of high yield strength and low thermal conductivity. In the proper design of a press for work a t low temperatures allowance is made for maximum contact between the evaporating helium vapor and the press support members, since the refrigeration contained in helium vapor a t 4"K, for instance, is much greater than the latent heat of vaporization of liquid helium. A ten ton press currently in use a t Iowa State University (Fig. 8) uses about 120 cc of liquid hydrogen per hour,@awhile a smaller, nonmagnetic, press (4 ton) uses 2; liters of liquid helium for an 8-hour experiment a t 3"K.*' The nominal sample pressure is calculated from the force which is applied (as measured by the hydraulic pressure in the ram at room temperature) and the cross-sectional area of the sample. In the apparatus of Fig. 8, the change in length of the sample is measured by means of a commercial dial indicator (0.4-in. trave!, IO-'-in. divisions) a t room temperature, the body and pin of which are connected by means of quartz rods to the top and bottom pistons of the sample holder. Friction effects in the ram and in the sample holder itself w e compensated for by taking dial gauge readings a t both increasing and decreasing pressure (Fig. 9), with the average pressure for a given displacement being taken a8 the true pressure corresponding to the dispIacement.sBThis procedure has meaning only if the pressure changes arc monotonic, and this is ensured by the direct use of ti free piston gauge as the source of pressure. The friction and the observed chsnges in length are both directly dependent on sample length, 90 that an empirical compromise must be made in any given case. The friction generally becomes more important at low temperatures, and in mme cases the low-temperature data cannot be precise. The major corrections which must be applied to the data are due to R. I. Eeecroft and C. A. Swenaon, "The experimental equation of state of sodium," to be publhhed.
68
C. A . SWENRON
the compression of the pistons and the expansion of the cylinder walls around the sample, both of which are proportional to the applied force. These two correction^ are determined simultaneously by measuring a sample for which the compression versus pressure relationship is known (indium), and using the difference between the measured ALILO and the
HYDRAULIC RAM
T-O ,
FREE PISTON GAWL
FIG.8. A ten-ton hydraulic press used for low-temperature compression measurements to 20,000 atmoe (R. I. Beecroft and C. A. Swenuon, “ The experimental equation of state of sodium,” to be published]. The Carboloy sample holder is identical with Fig. 3a. A metal Dewar vessel, refrigeration coil and transfer tube, and heater, as well as a housing for the top of the press, are not shown. In operation, the space around the press can be evacuated, and the dial gauge viewed through a window.
expected AV/Vo for each pressure as the press correction. This correction is sensitive to sample length (through the change in sample area) and must be determined for a standard sample of the same length as the unknown. A typical correction curve is shown in Fig. 9 for a Carboloy piston and cylinder combination. The use of Carboloy (or any sintered tungsten carbide) has the advantage that both the corrections and their temperature dependence are much smaller than for steel or beryllium
PHYSICS AT HIGH YRXSSUHE;
I
I
1
I
I
I
I
I
I
I
I
I
n w
u
E
h 0
z B
l l l l r 9 3 4
w
a
80ODIU
u
SAUPLL LEMGTM
9W-U
g
#
I 1
I
4
I
6
0.159 INCHES
I 8
I
I0
I
I2
I
I
I
14
I6
I0
1
m
I Lo
PRESSURE , ATYOSPHERES a I Q 8
FIO. 9. An experimental length versus nominal preesure curve as obtained for mdium a t mom temperature [R.I. Bekroft and C. A. Sweneon, “The experimental equation of state of sodium,” to be published]. The dashed curves are the mean preu8urc.a for a given dirplacement, while the arrowe:indicate either the increase or decrease of premure. The anomalous curvature at each end of the curve irP due to the reveresf of the sign of the frictional forces, and is ignored in the calculation of the displacement. The correction curve which must be subtracted from ‘the smoothed data in order to give the “true” change in sample length is a h shown.
70
C . A . SWENYON
copper.98 This is presumably due to the high elastic modulus and extreme hardness of Carboloy. The major approximation in this type of work is due to the assumption that the sample is relatively plastic: that is, the shear stress set u p in the sample is small when compared with the total applied p r e s s ~ r e . ’ ~ Stewartg4.96 and others96 have used an extrusion criterion for investigating the magnitude of the shear yield stress for various metals and nonmetals as a function of trmperature. In general, the extrusion pressures for metals seem to increase by a t least an order of magnitude a s the temperature is decreased from 300°K to 4°K. The simpler crystalline solids (neon, argon, oxygen, for instance) are quite plastic below their triple points, but may, as is the case for argon and oxygen, become brittle a t liquid helium temp erat~ res.9 ~ I n the investigation of polymorphic transitions in substances which show large friction effects great care must be taken if the transition volume changes are small since these may be obscured or a n apparent transition indicated by irregular piston motion. If one is interested only in detecting polymorphic transition pressures and measuring the volume change at the transition, the piston contraction and cylinder expansion corrections (although not the friction) are relatively unimportant. When the sample to be measured is brittle at low temperatures it may be enclosed in a plastic sheath similar to th a t used by Bridgman for very high-pressure work.46 Indium is again most suitable for this at low temperatures, and Stevensons7was able to make a marked improvement in his transition data for solidified gases hy using a thin indium sheath to reduce the friction between the sample and the cylinder wall. Similarly, Boweng8 has encased cylindrical samples of superconductors in silver chloride, and placed them in a piston and cylinder arrangement like that shown in Fig. 3a. Instead of using a press with long support members to apply the pressure a t low temperature, the force was applied to the pistons at room temperature, and then retained upon cooling to low temperatures by means of a clamp of a type similar to that used by Chester and Jones. The corrections are quite large, but the data are in good agreement with those of other investigations. Experiments on the compression of solidified hydrogen,Q6.ggled to the ’4 J. W. Stewart, “Some Measurements at High Premures and Low Temperatures,” Doctorate thesis, Harvard University (1954), unpublished.
O‘J. W. Stewart, Phye. Rev. 97,578 (1965). OSC.A. Swenson, Phy8. Rev. 100, 1607 (1955). O’R. Stevenson, J . Chem. Phye. 17, 1656 (1957). D. H. Bowen, Proc. 6th Intern. Conf. on Low Temperature Phys. and Chem., M a d ~ m , W18., 1967,p. 337 (1958). J. W. Stewart and C. A. Swenson, Phye. Rev. 94, 1096 (1954).
71
PHYSICS AT HIGH PREBSURE
suggestion that it might suffice as a transmitter of approximately hydrostatic pressure for electrical measurements at liquid helium temperatures. It was so used first by Hatton,loOwho measured the effects of pressures up to 5000 atmos on the electrical resistivity of small wires which were imbedded in solid hydrogen. Essentially the same technique was used later for high-pressure nuclear magnetic resonance experiments
n
’ER
-
0 SCALE
-INCHES
Fro. 10. A sample holder in which solid hydrogen was used as a “bath” to apply approximately hydrostatic pressure to various superconductors [L. D. Jennings and C. A. Swenson, Phys. Rev. 112, 31 (1958)l.The top part of the press (which waa at room temperature) and the liquid helium Dcwar vessel are not shown.
on solid orthohydrogen by Fairbank and %lcCormick,lol and for measurements of the pressure effect in superconductors by Jennings and Swenson. The sample holder which was used by the latter workers is shown in Fig. 10, since it illustrates some of the problems encountered in work with solid hydrogen or solid helium a t these pressures. The cylinder bore is closed a t the bottom to simplify sealing problems, and, initially, the top is also sealed by a thin disk. The capillary is used to evacuate the Phys. Rev. 100, 681 (1955). W. M. Fairbank and W. D. McCmrmick, Bull. Am. Phye. Soc. 121 3, 166 (1958).
lo0J. Hatton,
Iol
72
C. A. SWENSON
cylinder and then to condense into the cylinder a known amount of the gas which is to be studied or to be used as a pressure transmitter. This is donedby a suitable manipulation of the temperature,l02 after which the liquid is frozen (if it is not liquid helium) by decreasing the temperature of the sample holder. The seal is then broken by applying force to the piston, and the sample compacted. The initial experimentson solid hydrogen showed that near 2000 atmos the solid hydrogen extruded violently from the sample holder, presumably through the small gap between the piston and cylinder walls, the magnitude of which is a function of pressure. Later experiments on solid hydrogen, successful to 10,OOO atmosg@and then 20,000 atmos,lOawere made possible by the use of a potassium gasket and extrusion rings in a Bridgman seal (Fig. 10) which effectively closed the gap and prevented the extrusion. The extrusion problem is even more serious with solid helium, since liquid helium can only be solidified under a pressure of 140 atmos at 4.2%. Thus, as the force on the piston increases there is a high probability that the liquid helium will extrude before it is solidified unlesa the gasket seal is perfect. Stewart has used the potassium gasket succew fully in work with solidified helium to 20,OOO atmos,lO*but reports about fifty per cent of his attempts were failures due to leakage of the liquid. Even with solid hydrogen, the initial pressure must be applied to the sample quite rapidly so that the gasket will seat and form a seal before extrusion takes place. The use of hardened beryllium copper in the sample holder shown in Fig. 10 made it possible to use an ac mutual inductance technique at 33 cps to detect the superconducting transition. The coils shown in the figure were used for this measurement. The pressure gradients in the solid hydrogen along the sample were estimated from the transition data. This led to the surprising conclusion that, although gradients of several hundred atmos seemed to exist after an appreciable change in pressure (and, indeed, caused some deformation of the softer samples), these gradients "annealed out" and became very small within a half hour or 5 0 . ~ ' Similar annealing effects in solid hydrogen were observed earlier by Fiske and his coworkers.10' d. Remarks
The need for reliable high-pressure data at low temperatures is quite great, and the necessary techniques now exist for this work. The direct A cryostat and some general suggestions for this type of work have been given by C. A. Swenson and R. H. Stahl, Rev. Sn'. Znstt. 26, 608 (1954). 101 J. W. Stewart, Phye. ond C h m . Solids 1, 146 (1956). M. D. Fiske, private communication (1957).
101
PHYSICS AT HIGH PRESSURE
73
use of the piston displacement method seems to offer the only possibility for the study of the equation of state near absolute zero, although it is restricted to the more plastic materials and the accuracy is only a few per cent. Various alternatives exist for the electrical and magnetic measurements with perhaps the most promising involving the use of fluid helium to the solidification temperature, and then solidified helium below this temperature. The experience of Dugdale and his co-workers shows that it is possible to apply pressures of several thousand atmos at low temperatures in such a manner that the sample properties are not altered. The use of a liquid helium 3 cryostat or a magnetic refrigerator should make it possible to extend these measurements below 1°K in much the same manner as the ice bomb technique has been used to measure the effects of pressure on the critical field curve of cadmium. The safety aspect must not be ignored, however, and the danger involved in working with high-pressure gas means that adequate safety precautions must be taken. 111. Results
4.
PVT DATAFOR
SOLIDS
a. Equation of State
The equation of state of a solid ( V ( P , T ) )and its thermal properties can be derived from the Helmholtz free energy (F(V,T) = U - TS) through the use of the relationships,
P
=
-(aF/aV)T,
and
S = -(aF/aT)v.
(4.1)
Conversely, measurements of the temperature variation of CV (the specific heat at constant volume) at one volume and the equation of state can be used to derive F( V ,T), since
(4.2b) (4.2~) (4.2d) where @ = V-'(aV/aT)p is the volume coefficient of thermal expansion, and kr = - V-l(aV/aP)r is the isothermal compressibility.10c V o ( V )is the internal energy of the d i d at absolute zero, and, in general, contains contributions from the lattice energy (purely static in nature) and the lo(
J. C. Slater, "Introduction to Chemical Physics," Chapter 11. McGraw-Hill, New York, 1939.
74
C . A . SWENSON
zero point vibrations (zero point or residual energy) of the atoms. These may or may not be separable for any given substance. The expression for the pressure [Eq. (4.1)) can be written a s the aum of two terms:
P(V,T) = Po(V)
+ P * ( V , T ) = -dC‘o/dV
(aU*/av)* + T(a S / d V ), = Po(V + ( B / ~ Td)T . (4.3) -
loT
Here Po is the pressure which would be required to obtain the volume V a t absolute zero, and P* is an additional (‘thermal’’ pressure due to the lattice vibrations. P * may be considered qualitatively a s being responsible for the thermal expansion a t constant pressure. I n general, the thermal properties and, hence, P* are difficult to calculate from first principles, since the calculation requires not only a knowledge of the vibrational spectrum of the crystalline lattice, but also its variation with temperature and volume. The simple assumption that all vibrational frequencies are changed in the same manner by a change in volume leads to the Mie-Grueneisen equation of state,
P
=
+ rP*(V,T)/V,
Po(V)
or
P*
=
rC*/V
(4.4)
where r = BV/CvkT, the Grueneisen constant, is a number of the order of two for most substances and is relatively independent of temperature; however, i t may be a function of the volume.Io6 I;*may be determined explicitly if a model (Einstein or Dehye) is assumed, but Eq. (4.4) is believed t o be fairly general and to be a good approximation when the substance is isotropic and anharmonic vibrations are not involved. Rice et a1.,’O7 as well as Gilvarry,’O* have discussed various temperature dependent equations of state in some detail in terms of this postulate, as well as the problem of the consistency of the various possible definitions of r. More recently Sternloghas given a theoretical discussion of the effect of anharmonic terms in the equation of state. While U*(V,T)is difficult to determine theoretically, Uo(V) is often more amenable to calculation. This is particularly true of the simpler solids such as the solidified rare gases and the alkali metals, and some success is being attained for other substances. Essentially Vo(V) is usually calculated with the pressure-volume curve obtained from Eq. (4.1) and with F = U o ( V ) ; the compressibilities are determined from J. C. Slater, “Introduction to Chemical Physics,” pp. 215-220, 238-240. McGrawHill, New York, 1939. lo’ M. H. Rice, R. G. McQueen, and J. M. Walsh, Solid State Phys. 8, 1 (1958). I o 8 J. J. Gilvarry, J . A p p l . Phys. 28, 1253 (1957). log E. A. Stern, Phys. Rev. 111, 786 (1958). loo
PHYSICS A T IIIOII PRESSURE
75
kr = ( V d*lTo,/dV2)-l..4n initial check for the validity of a theoretical model involves the comparison of theoretical and experimental pressures for a given volume, although a more rigorous test is afforded by the comparison with theory of the variation of the compressibility with pressure (or volume). First we will discuss low-temperature work, where the Cro(V) term in the free energy predominates. This will he followed by a discussion of experiments which give information about the temperature-dependent part of the equation of state. The experiments which lead t o U , ( V ) involve conipression measurements (see Section 3c) which ideally are made a t absolute zero. Since such measuremerits are both practically and theoretically impossible, the question arises as to the highest temperature which can be used for ti given pressure to obtain a volume which isapproximately the same as would t)e obtained in a measurement at absolute zero. The condition to be satisfied is that P , be much greater than P * , or, stated in another W R Y , the thermal expansion (which in general decreases with pressure i n c r a w ) must be effectively zero at PO. Typical values of P * a t zero pressure are 13,000 atrnos for copper at room temperature, and 1700 atmos for argon a t its triple point (84°K). The variation of P* with pressure (or volume) is difhcult to determine a priori, since I' and U * are both functions of volume. The shock-wave high-pressure experimentslo7can be interpreted to give r as a function of volume, with the result that I' should be the dominant volunie-dependent factor in Eq. (4.4) a t high temperatures. Experimentally, P* appears to decrease with increasing pressure at low temperatures for both argonioa and the alkali metals,"0 but no general conclusions can be drawn. In general, except for helium, 10,OOO atmos applied a t 20°K should produce a volume which is identical with that produced by the same pressure a t absolute zero. This is convenient experiment ally because the latent heat of vaporization of liquid hydrogen is ten times that of liquid helium. Experimental data for the pressure-volume isotherms for the simpler solids a t relatively high pressures and low temperatures have been available only in the past few years. The experiments of Dugdale and Simon,8B in which calorimetric measurements a t constant volume were used, gave quite precise data, but this method would seem to be limited to helium. The methods which have been used by Stewart and others to obtain data to 20,000 atmos a t liquid helium temper:ktures are relatively crude (see Section 3c), but die magnitudes of the compressions are quite large, so that accuracies of a few per cent in the total volume change are possible. It is fortunate that the solids of greatest interest are also quite mft and, hence, lend themselves readily to the use of these techniques.
$lac.A. Sweniwn, Phys. Rev. 99, 423 (1955).
76
C. A. SWENSON
Stewart's data for hydrogen, deuterium, heiium, neon, and argonlo' are given in Fig. 11. The zero pressure molar volumes and initial compressibilities of the various solids reflect the influence of the zero point energy. This leads to expansion of the lattices of the lighter elements to volumes which are much larger than purely classical considerations would predict. As the pressure is increased, however, the effect of the zero point
SOLlDlFEO PERMANENT OASES
3
10
IS
20
PRESSURE, THOUSANDS O f ATMOS
F ~ Q11. . Pressurevolume isotherms for the solidified permanent gases [J. Stewart, Phy8. and Chem. Solids 1, 146 (1956))
energy becomes smaller relative to the increased lattice energy, and above a few thousand atmos the relative volumes are in the order which would be expected from gas viscosity determinations of the molecular diameters. The difference between neon and helium a t low pressures is striking, although the data for the hydrogen isotopes show that the effect of the zero point energy does not disappear even at 20,000 atmos. The data which are plotted for argon are for 65°K and 77°K; at lower temperatures argon is brittle and because of this the corresponding
PHYSICS AT HIGH PRESSURE
77
P-V data are unreliable. The relatively small initial value of P* (1700 atmos a t 84'K) and its apparent decrease with pressure make it reawnable to conclude that the molar volumes which are observed for pressures above 10,OOO atmos are virtually independent of temperature."' The slight eeparation between the isotherms which is shown is in the direction of a negative thermal expansion, but it is also well within the expected experimental accuracy. Recently Bernardesll' has calculated theoretical f-V relationships for solid Ne, A, Kr, and X e at absolute zero. The results of these calculations are in good agreement with the neon data, but in doubtful agreement with the argon data. It is surprising that the deviation is greatest for argon, since the relatively uncertain quantum mechanical corrections which must be made for the zero point energy are greatest for the lighter atom. Similar data for the alkali metals are of interest because of calculations by Brooks and his c o - ~ o r k e r s . ~ ~The a * ~pressure-volume ~' isotherms for all five of these metals (Li, Na, K, Rb, and Cs) have been measured to 10,OOO atmos at both 4'K and 77'K.I1O The agreement with theory is not especially good, in part perhaps because the initial comparison has been with an assumed form of U Owhich was used by Bardeen:"'
The constants were obtained from theoretical data for three values of r, (the atomic radius) near the equilibrium value. It had already been shown that the experimental P-V isotherms at liquid helium temperatures could not be represented accurately by such an expression if experimental values of the lattice constant, cohesive energy, and compressibility at cero pressure were used to calculate the constants.IlO More recent data on eodium to 20,000 atmas aver 8 wide temperature range are in essential agreement with these earlier data.O' The problem of obtaining an analytic expression for experimental F V isotherms is important both for interpolation and extrapolation, and for the determination of compressibilities by differentiation. A semiempirical relationship which was derived by Birch from Murnaghan's theory of finite strain' has proven to be quite useful for these purposes. According to Birch, the pressure P a t a given volume V is given by
P = (3/280)[~7 - yb][l - €(Y*
- 1)l.
(4.6)
Recent unpublished nieasurements at 22°K by J. W. Stewart (Univ. of Virginia) show that this i N essentially true above 2000 ntmos (private communication, 1959). n'N. Bernardes, phys. Rev. 112, 1534 (1959). I18F. S. Ham, Solid Stale Phy8. 1, 127 (1955). '"H. Brooks and F.S. Ham, Phy8. Rev. 112, 344 (1958). Ik1J. Bardeen, J . Chern. Phyu. 6, 364, 372 (1938).
Here, 9 = ( V O / V ) arid ~ , V Oand P o are, respectively, the volume and compressibility a t zero pressure, while t is an adjustable parameter. In most cases, the data can be fitted with t = 0, a condition which may he defined as “normal” behavior. A negative t , then, indicates a “hard” substance (its compressibility decreasing more rapidly with pressure than “normal”), while a positive E indicates a “soft” substance (its compressibility decreasing too slowly with pressure). A t low temperatures, Na, K, and R b are essentially “normal,” lithium is “hard,” and cesium is “soft.” As would be expected, the condensed gases are also “ hard ” in this sense, since the lattices are expanded by the zero point energy. I t must be emphasized, however, that Eq. (4.6)cannot be interpreted in terms of atomic theories, and is of use primarily for interpolation and extrapolation. The previous discussion has been restricted to the temperature region where P*, the thermal pressure, is negligible. The higher temperature region is also of interest because an experimental determination of the equation of state over a wide range of temperatures and pressures can be combined with thermal data at atmospheric pressure to calculate both P* and U*, and thus, to investigate their relationship. These experiments can be performed best with the simpler, more compressible, substances (such as the solidified rare gases and the alkali metals), where the magnitude of the effects are sufficient to he measured readily by the relatively crude experimental techniques which are available. The earliest data were those on solid helium from 4°K to 25°K and to 2500 atmos obtained by Dugdale and Simon,**who used a constant volume calorimetric method. Dugdale has combined these data with Stewart’s 4°K results to obtain the equation of state of helium to 20,000 atmos and the melting curve.116More recently, apparatue of the type discussed in Section 3c has been used to obtain P-V data over a wide range of temperatures for indium and thallium to 10,000 atmos,g‘ for mercury to 13,000 atmos,l17 and for Teflonl1*and sodiums’ to 20,000 atmos. These data all show an expected decrease in thermal expansion and compressibility with pressure, although, except for sodium, the data, for various reasons, are not sufficiently accurate for a quantitative calculation of the variatiori of the thermodynamic functions with pressure. Data have also been obtained for polythene to 2000 atmos over the range from 25°C to I(i0”C and the thermodynamic properties calculated.11” Figure 1% gives four experimental isobars for sodium over the temJ S. Ihydnle, Nuovo cantrnlo 9, Suppl., 27 (1958). C . A. Swenson, P h y s Rev. 111, 82 (1958). l L U R. I . Beecroft and C. A. Swenson, J . A p p l . Phys. 80, 1793 (1959). W. Parks and R. B. Richards, Trans. Fataday Sm.M, 203 (1949). 11(
11’
79
PHYSICS A T HIGH PRESSURE
perature range from absolute zero to 360.K (melting point 3 f 1 ° K ) . g S These isobars and the accompanying plot of the compressibility as a function of temperature for the same pressures (Fig. lab) represent the type of behavior which is to be expected for a normal substance. The temperature dependence of the volume decreases very markedly with increased pressure, as does the temperature dependence of the compressibility. An analysis of these data showed the compressibility to be a function of volume only and independent of the temperature. The I
I
I P.0
I
c.
1
1 FIG. 12. The molar volumes of sodium as a function of temperature for four pres- V-’(dV/JP)r) as a function of temperature for the same pressures [R. I. Beecroft and C. A. Swenson, “The experimental equation of atate of sodium,” to be published]. awes, and the cornpressibilities (kT =
significance of this will be considered later in this section when ultrasonic measurements on Cu, Ag, and Au are discussed. In spite of the emphasis which has been placed on the low-temperature work, most of the available P-V data for solids has been obtained near room temperature. Bridgman has used a lever piezometer to measure the linear compressibilities of many polycrystalline cubic metals and single crystals as a function of hydrostatic pressures to 30,000 atmos.11gJ20 These experiments are quite precise (changes in length of 2 X 10-6 cm could be detected in samples roughly 1.8 cm long), and show clearly that length hysteresis with increasing and decreasing pressure is a natural phenomenon in most samples, even if the pressure is truly hydrostatic. “‘P. W. Bridgman, €‘roc. Am. Acad. Arts Sci. 78, 89 (1948). IroP.w.Bridgman, Proc. Am. Acad. Arts Sn’. 77, 187 (1949).
80
C. A. SWENSON
The behavior of zinc and tellurium was particularly unusual:la0 zinc having linear compressibilities that differed by a factor of eight in the two crystallographic directions, while tellurium showed a small negative linear compressibility along the trigonal axis. The total compressibility must always be positive, however. Bridgman was interested in the variation of compressibility with pressure, and usually found that kr decreased with pressure. In hie earlier work, he used the expression :'*I AV/Vo = -aP
+ bP2
(4.7)
to represent his data, while in his more recent work (119, 120 for example) he prefers to quote actual values of the volume (or length for anisotropic crystals) at intervals of 5000 kg/cm2. Although most of this new data could be represented by Eq. (4.7), the value of b increases with pressure for some of the substances (silver, for instance). These linear compression measurements are not absolute, but are relative to iron as a standard. Hence, the accuracy of the AV/VO data which are derived from them depends to a great extent (especially for the least compressible substances) on the accuracy with which the expression AL/Lo = - a 3 brP2 (4.8)
+
is known for iron. This was determined by Bridgman in separate experiments; initially to 12,000 atmos,'*' and later to 30,000 atmos.lZzIn these later experiments the linear compression of an iron sample which was 8.4 cm long was measured. The compression was determined by measuring both the relative motion of the sample and the interior of the bomb and the absolute extension of the interior of the bomb as a function of pressure to 30,000 atmos. The linear term was essentially the same in the two determinations [aL,,." = 1.942 X lk7 (kg/cm2)-', aL.1d = 1.953 X lo--7 (kg/~m*)-~]. There are considerable experimental difficulties in measuring the second quantity and an uncertainty of 20 or 30% in the second degree coefficient (bL) would be possible in view of the scatter which Bridgman observed. The contribution of this term is about 3% at 30,000 atmos, and the maximum deviation from a straight line through the end points is about 0.9%. Indeed, as will be mentioned later, ultrasonic experiments on the variation of the compresaibility of Cu, Au, and A1 with pressure to 10,OOO atmos would be consistent with a br. value of 0.4 X 1@1* (kg/ 1'1
P. W. Bridgman, "The Physics of High Pressures," Chapter VI. G . Bell, London,
I**
P. W. Bridgman, Proc. A n . Accul. Arfa Sci. 74, 11 (1940).
1949.
PHYBICS A T HIQH PREBlJURE
81
and cm*)-s, as compared with Bridgman’s “new ” value of 0.23 X While this “ultnrsonic” value is in good his ‘‘old” value of 0.80 X agreement with an earlier determination by Ebert to 5000 atmos,1~2Js* it would seem to be outaide the range of probable error of Bridgman’s data to 30,000atmos. The possibility exists that a simple second d e w equation is not adequate for expressing the compreaaions over the whole range to 30,000 atmos, and that different effective values of br. must be used for different pressure ranges. Bridgman comments on this, and notes that it would be presumptuous to attempt to analyze any of the data for deviations from the second degree relationship of Q. (4.8). Newer experimental techniques (such as those used by Reitzel ei uZ.lZ4) might be adapted for a redetermination of this absolute compression with less uncertain corrections. Bridgman’s older resulta, calculated on the basis of his initial value for the linear compression of iron, have been quoted. These older data [which are given in terms of the a and b of Eq. (4.7)J must be corrected 80 ~II to be consistent with the new data for iron. Slaterlabhas done this, and his results disagree with those which would be obtained using relations given by Bridgman.I2’ Slater’s corrections were obtained ueing the formula for the older values of the volume compression of iron that are given by Bridgman.lZk The second-order term in this equation (0.21 X differs from that which is quoted earlier1**(0.23 X lo-‘), and which follows from the “old” linear compression data given. The general relationship between the “new ” and the “old a and b values are, essentially IM given by Bridgman, as follows (in kg/cms);1*7
..-
,,..,
where AaL = (as - aL d d ) and AbL = (bL - bL &) for the linear expansion of iron in Eq. (4.8). Bridgman also comments on the incorrect reduction of some of his linear compression data to volume changes in earlier work, 80 caution must be observed on this account in the recalculation of the old data to the new basis. It must be emphasized that the
H.Ebert, Phyaik. 2.86, 386 (1935). J. Reitsel, I. Simon, J. A. Wdker, Rev. Sn’. Znstr. 28, 828 (1957). J. C. Slatar, Phys. Rev. 67, 744 (1940). 1“P. W. Bridgman, “The Physics of High Preesureu,” p. 417. G. Bell, London, 1949; Ram. Modern Phys. 18,23 (1946). P.W. Bridgman, “The Physics of High Prwures,” p. 154. G. Bell, London, 1949. lU “4
”‘The aeriounnw of this diecrepancy was firat pointed out to the author in correspondence with Dr. C. 8.Smith.
82
C. A. SWENSON
relative data are much more precise than the absolute data, so the absolute accuracy can be improved as better data becomes available for the standard substance, iron. The zero pressure compressibility (the constant a in Eq. (4.7) or the initial slope of the curves in Fig. 11) can be determined also from ultrasonic measurements of elastic constants a t zero pressure. These constants give directly the adiabatic compressibility, while the bulk measurements TABLE I. A COMPARISON OF ULTRASONIC AND BULKCOMPRE~SION DETEBYINATIONB OF THE CONSTANTS a A N D b I N EQ. (7) The ultrasonic adiabatic compressibility is given also for comparison. The lineer compression data for silver show anomalous behavior, and cannot be expressed a d e quately in terms of Eq. (47).The variation of the bulk modulus (BT = l/h) with pressure is given for sodium in place of the constant b. a = kT x (dynes/cm')
b X 10" (dynes/cml)-*
-1
k, x 101' Solid
Ultrasonic
A (65°K) 620 cu Ag Au
A1
Na
6
7.51 9.85 6.02 13 75 161
Bulk (dynes/cm*)-'
630 7.26 9.75 5.76 13 85
490 7.30 9.65 5 79 13 09
160
151
Ultrasonic -
1.9 3.4 1.4 5.6
dB* -3.0 dP
,
Linear
References
-
0
1.5 2.0 ( 1 ) 0.8 4.9
d BT -3.30 dP
L.C
bd b+
c,d
*
E. R. Dobbs and G. 0. Jones, R e p f s Progr. in Phya. 10, 516 (1957). W. B. Daniels and C. S. Smith, Phya. Reo. 111, 713 (1958);J. R. Neighbours and G . A. Alers, ibid. p. 707. P.W.Bridgman, Proc. A m . Acad. Arts Sci. 77, 187 (1949). R. E. Schmunk and C. S. Smith, Phya. and Chem. solid8 9, 100 (1959). W. B. Daniels, Bull. Am. Phys. Soc. 121 4, 131 (1959).
always give the isothermal compressibility. These two are connected by the thermodynamic relationship : kT =
I C , C ~ / C=, k,(l
+ P * T V / ~ , C=~ k.(l ) +B m ) .
(4.10)
This correction usually is not important (a few per cent), but it isespecially significant in the case of argon for which there was a large apparent discrepancy between the static data of Stewartgs and the zero pressure ultrasonic data of Dobbs and Jones. lZ8 A redetermination of the volume expansion removed the discrepancy, and the actual agreement (well within experimental error) is shown in Table I. The ultrasonic kT and k, 128
E. R. Dobbs and G. 0. Jones, Repts. Progr. in Phys. 20, 516 (1957).
PHYSICS AT HIGH PRESSURE
83
values which are given in this table illustrate the magnitude of the correction which must be applied in various cases. A rather complete summary of the work which has been done on elastic constants (mostly ultrasonic) has been given in a recent article by Huntington.'** Ultrasonic techniques also furnish an elegant method for determining the effects of pressure on elastic constants (and, hence, on the compressibility). The distinction between adiahatic and isothermal constants is ignored in this work because i t is found to be relatively small. This method is inherently more accurate than the bulk methods because the quantity of interest, ( a k T / a P ) T , is obtained essentially by drawing a line through the actual data points, while the results of bulk measurements must be differentiated twice to obtain the same information. The first high-pressure measurements using ultrasonic techniques were made by Lazarus29 on typical solids with cubic structures and at pressures up to 10,000atmos. The ultrasonic method is potentially more accurate than the bulk method and in addition has the advantage (if measurements are taken for a sufficient number of directions with both shear and longitudinal waves) of giving all the elastic constants of the crystal and their variation with pressure. Lazarus chose for his studies copper and aluminum, beta-brass, and the salts KC1 and NaCl as being representative of the face-centered cubic, body-centered cubic, and simple cubic structures, respectively. Other ultrasonic work at high pressures in which single crystals were used has been done by McSkimin on germanium,l*D Daniels and Smith on copper, silver, and gold,13' Schmunk and Smithla2 on aluminum and magnesium, and Daniels on sodium.1a*~13'Hughes1*6q1*6has worked with polycrystalline specimens of metals and geological specimens at pressures up to 10,OOO atmos. His method differs from the other high-preasure methods in that a transit time technique applicable only for polycrystalline samples is used; this contrasts with the pulse-echo techniques of the other workers. One of the major difficulties in high-pressure work is to bond the transducer t o the sample, and both Lazarus, and Daniels and Gmith, report that their quartz crystals were shattered after a run to
'"H. B. Huntington, Solid Stale Phys. 7 , 273 (1958). IlOH.J. McSkimin, J . Acousl. SOC.Am. 80, 314 (1958). Ia1W.B. Daniels and C. 6. Smith, Phy8. Rev. 111, 713 (1958). latR. E. 8chmunk and C. S. Smith, Phy8. and Chem. Solide 9, 100 (1959). Ia8W.B. Daniels, Bull. Am. Phy8. Soc. [2] 4, 131 (1959). "'The author is indebted to Dr. C. S.Smith and Dr. W. B. Daniels tor supplying him,
in advance of publication, with preprints of the papers on aluminum, magnesium, and eodium. labD.S. Hughes and J. H. Cross, Geophysics 18, 577 (1951). la*D.8.Hughes and C. Msurette, J . Appl. Phys. 37, 1184 (1956).
84
C. A. SWENSON
10,OOOatmos. McSkimin, whose maximum pressure was only 3500 atmos, did not mention this effect. Table I gives a comparison of ultrasonic and bulk (mostly linear compression) determinations of the constants a and b in Eq. (4.7). The isothermal compressibilities disagree by 3 to 4%, which appears to be well outside experimental error, for copper and gold. If the linear b values are corrected by the systematic addition of 0.5 X lo-*', the agreement is good. This suggests that the iron data are possibly in error, &a has been discussed ear1ier.la*There seems to be little doubt that the ultrasonic data for b are to be preferred. Bridgman's data for the cubic metals were all obtained with polycrystalline samples, but it is doubtful if the discrepancies are due to polycrystallinity. The anomalous behavior which Bridgman found for silver (definite deviation from a second degree expression) was not observed by Daniels and Smith. Lazarus, and later Neighbours and A l e r ~ , considered ~*~ the problem of the temperature dependence of the isothermal compressibility at constant volume. A particularly simple assumption is that kT is a function of the volume only, and that its isobaric temperature dependence is due solely to the change in volume with thermal expansion. To investigate this, one can write (after Lazarus'g) ; (d In k ~ / d T ) p=
(a In
kT/aT)v
+ (d In kr/aV)T(aV/aT)p.
(4.11)
The term on the left is obtained from zero pressure measurements, while the second term on the right can be evaluated from thermal expansion data and Eq. (4.7). The calculation can be made for all of the elastic constants, but the k T data lend themselves particularly to physical interpretation. The beat available data for argon, copper, silver, and gold are summarized in Table I1 in terms of these derivatives. Except for the pressure derivative of the compressibility for argon, all of the data are from ultrasonic measurements, with the tabulation for the metals essentially as given by Neighbours and Alers.Ia7 The experiments on sodium cited before (Fig. l2)*' showed directly that, within a few per cent, k T is independent of the temperature a t constant volume. Except for solid argon and sodium, for which the assumption that k T has no intrinsic temperature dependence seems to be valid within the experimental uncertainties, there is a definite decrease in compressibility as the temperature is increased at constant volume. This behavior for the metals can be attributed qualitatively to an increase of the thermal pressure with temperature. The implications of this behavior can be investigated through the use of Eq. (4.4), if the assumption is made that the variation of r with l S r J . R. Neighbourn and G. A. A l m , Phys. Rev. 111,707 (1958).
a5
PHYSICS AT kIQH PRESSURE
temperature is small. The fourth column can then be written;
(a In k r / a T ) v = B((a In ( C . / V ) / a In V ) , + (d In I'/d In V)].
(4.12)
The first term in the brackets can be estimated from the Debye model for 8 solid, and for copper it varies from - 1 for temperatures greater than Or, to a maximum of +0.6 at 80'K. These figures should be of roughly the same order of magnitude for silver and gold. In order for the left-hand side of Eq. (4.12) to be - 1.1 X lo-' for copper at room temperature (B = 0.5 X lo-'), (d In I'/d In V) must be of the order of - 1, and certainly not positive. This is in direct contrast with the results of Rice et d.,lo7 who find that this quantity is +6 for copper and silver, and TAEM 11. A SUMMARY OF THE TERMS WHICH ENTERINTO A DETERMINATION OF THE INTRINEIC TEMPERATURE DEPENDENCE OF THE ISOTHERMAL COMPaEBSIBILITY FOR VARIOUB &LIDS-
d In kr
Solid
A (86°K) cu
4 Au
ain kr
a l n kr
("K-1)
(OK-1)
133 1.60 1.88 1.66
145 2.66 3.42 2 60
(OK-')
- 12 - 1.06 - 1.64 - 0.94
The argon data are taken from E. R. Dobbe and G . 0. Jones (Repts. Progr. in 516 (1957)], while the data for the metala are M given by J. R. Neighboure rpd 0. A. Alm [Phys. Rbv. ill, 707 (1958)], based in part on work by W.B. Danieln and C. 8. Bmith [ibid. p. 7131. 6
&6.40,
considerably higher for gold. Evidently more data on the volume dependence of 'I are needed in order to resolve this problem. In particular, an extension of the high-pressure ultrasonic measurements to a range of temperatures would be of great value and possibly worth the extra experimental difficulties which would arise. In addition to providing the accurate data on linear compreseibilities mentioned above, the static method has been used for relatively rough determinations of compreasions for many other substances. Weir has used a piston-displacement technique with a fluid as a pressure transmitter" to measure the compressions with respect to the fluid of many different Substances to 10,OOO atmos near room temperature. These have included organic compounds, as well as the glassy and crystalline modifications of both selenium and glucose (see Section 9 ~ ) . ~ * * - ~ ' ~ c. E.weir, J . Research N d . Bur. Standards 46,207 (1951). 18*C.E. Weir, J . Rceearch NOU. Bur. Standards 60, 95 (1953). l"C. E. Weir, J . Rcearch NaU. Bur. Standards 68,245 (1964). lt1C. E. Weir, J . l h a u c h Natl. Bur. Stan&r& 66, 187 (1956).
86
C. A. SWENSON
PRESSURE (THOUSANDS OF kg/cm*)
Fro. 13. Prewure-volume iwtherms to 100,000 atmos for various solids at morn temperature [from P. W. Bridgman, Endeaaor 10, 63 (1951)l.
Brldgman also has used the piston-displacement technique to determine the approximate compressions of many substances to pressures as high as 50,000 atmos, with the samples themselves (or an indium sheath) serving as the pressure tran~rnitter.'~~~'J~~~*-~~~ In these experiments he was concerned principally with anisotropic substances for which single crystals were not available (such as indium, thallium, and many comu2P. W. Bridgman, Proc. Am. Acad. Artr Sci. 74, 21 (1940). P. W. Bridgman, Proc. Am. Acad. Arta Sci. 76, 55 (1948).
PHYSICS A T HIGH PRESSURE
87
pounds and alloys). In addition, he has obtained pressure-volume data to 100,OOO atmos for many of these same s u b ~ t a n c e s Typical . ~ ~ ~ ~ ~ ~ ~ ~ data for a variety of substances are given in Fig. 13. The total compresaon at the maximum pressure varies from 63% of the initial volume for cesium, to 1.8% for diamond. No details of these experiments will be given here, although some of the results will be discussed in the section on polymorphic transitions. Data at much higher pressures have been obtained through the use of shock wave techniques. These experiments have been discussed elsewhere by Rice et aZ.lo7
b. Polymorphic Transitions and Melting Two modifications of a pure substance can be in thermodynamic equilibrium only if the molar Gibbs free energies G of the modifications are equal. An equilibrium line P ( T ) is defined by this requirement, although thermodynamics cannot be used to predict whether it is possible for the transition to occur a t a finite rate on or near this line. Transitions are usually classified as I'first " or "second " order, depending on whether the first (and higher) derivatives of the Gibbs function (8,V , and their derivatives) are discontinuous, or whether the first derivatives are continuous and the second derivatives (kT, 8, C V )are discontinuous. The distinction often is difficult to make if the actual transformation is not ideal and the entropy change A S and the volume change AV are small and/or are spread over ft range of pressure and temperature. Nevertheless, unambiguous examples of first-order transitions do occur (melting and vaporization phenomena in pure substances, and many crystallographic transformations), and some truly second-order transitions are believed to exist (e.g., order-disorder transitions). Experimentally, it may be difficult to decide whether a given transition is actually second order, or of a still higher order. These problems have been discussed elsewhere, and will not be considered further here."# In this the discussion will be devoted primarily to '(first-order " transitions in which an entropy change (or latent heat) and a volume change are found. The relationship between A S , AT', and the equation of the transition line P ( T ) , is given by the Clausius-Clapeyron equation,
d P / d T = A S / A V = L/TAV
(4.13)
where L is the latent heat of transition. For a pure one-component system, 14(P. W.Bridgman, Ptw. Am. Acad. Arts Sn'. 76, 1 (1945). 14'P. W.Bridgman, Proc. Am. Acad. Arta Sci. 84, 112 (1955). See for example, A. B. Pippard, " Classical Thermodynamics," Chaptera 8 and 9. Cambridge Univ. Press, London and New York, 1957.
88
C. A. BWENSON
two phases may coexist over a range of either pressure or temperature (defined by the transition line), but three phases can be in equilibrium only at a “triple point” on the P-T diagram. This triple point may be between three states (solid, liquid, vapor), or between two solid phases and the liquid (or fluid), or between three solid phases. The equilibrium diagram for nitrogen, a substance which has all three types of triple ~
SOLID NITROGEN
0’
I
x
I-
0
10
20
30 40 TEMPERATURE
..
5 0 6 0
10
K FIQ. 14. The phase diagram for nitrogen at low temperatures. The shaded area on the lower right of the figure representa the liquid region [C.A. Swenaon, J . C h . P h w 28, 1963 (1955)l.
points,147is shown in Fig. 14; the low-temperature vapor pressure line is not shown, but the “zero pressure’’ solid-solid and solid-liquid transitions actually occur in contact with the vapor. These triple points for nitrogen (in particular, the one a t high pressure) can serve as an illuetration of a requirement for thermodynamic consistency. The slopes of the various transition lines and the volume changes are not all independent, but must satisfy the relationships; 2 AV = 0 ,
2 AS = L: A V d P l d T = 0
(4.14)
where the summations are taken about a closed path which encloses the triple point.148The thermal data (which are known for the a-@-vapor triple point), also must be consistent with Eq. (4.13). 14’
c. A. swemon, J . Chem. Phy8. 21, 1963 (1955). For a diacueeion of thermodynamic parameters along transition lines, see P. W. Bridgman,” The Physics of High Preseurea,” Chapters 7 and 8. G. Bell, London, 1949.
PHYSICS AT HIQH PRESSURE
89
The existence of a first-order transition may be detected experimentally either through the release of the latent heat L or by the change in volume AV. At low pressures, both of these indications are commonly used, and if quantitative measurements are possible, the two measurements and the slope of the transition line must again be consistent with Eq. (4.13). Transitions at high pressures have been studied in the peat almost exclusively by means of techniques in which AV is measured mechanically.148 Recently, however, the change in sample temperature caused by the latent heat of the transition (as measured by a differential thermocouple) has been used as an indication of the transition in highpressure work. This latter “thermal arrest” or “differential thermal analysis” method has the advantage that friction in the piston packing, for instance, has no effect. However, rapid transition rates may cause an uncertainty in the measured transition temperatures. This technique was used a t high pressures by Yoder6*in order to investigate the high-low quartz inversion to 10,OOO atmos and 140OOC. Yoder (and subsequent workers) used gas as a pressure transmitter and a manganin gauge to measure the pressure directly. The temperature of the sample waa varied by means of an internal heater. If the heat which is evolved can be calibrated by means of a second, known, transition which occurs a t a temperature and pressure near those a t which the first transition occurs, the method of “thermographic analysis” can be used to measure latent heats directly. As will be discussed in some detail later in this section, both methods (AV and thermographic analysis) have been used for the investigation of the cerium transition under pressure, and the results are in close agreement. The Clausius-Clapeyron equation places certain thermodynamic requirements on the temperature dependence of a transition. In addition the empirical generalization that equilibrium lines, with the exception of the liquid-vapor curve, terminate only a t triple points is consistent with all the known facts. Transitions often become “sluggish” a t low temperatures or high pressures, and, indeed, do not take place except with definite subcooling. Nevertheless, it is possible by proper manipulation to obtain either one or the other form a t a given temperature and preaaure along the extrapolated equilibrium line. The existence of a critical point would be an idealized termination of the equilibrium line along which the corresponding properties of the two forms would approach each other gradually and become identical a t a critical temperature, in direct analogy with the liquid-vapor critical point. Finally, the third law of thermodynamics requires that entropy differencesbetween equilibrium states must approach zero a t absolufe rero. Thue, dP/dT in Eq. (4.13) must go to zero for reversible transitions at low temperatures. This ia the case for the melting curves of the two
90
C. A . SWENSON
isotopes of helium, both of which can be solidified only under pressure The unique phenomenon of a minimum near absolute zero (Fig. 15).149*150 in the melting curve of helium 3 was predicted theoretically on the basis that the nuclear spin entropy of the liquid would be less than that of the solid a t some temperature close to absolute zero. 161 Recent experiments
TEMPERATURE ,'K
FIQ.15.The melting curves near absolute zero for the helium isotopes [B. Weinstock, B. M. Abraham, and D. W. Osborne, Phys. Rev. 86, 158 (1952);F. E. Simon and C. A. Swenson, Nature 166, 829 (1950)l.The three theoretical curves represent the predictions for three different magnitudes of the nuclear spin-spin interaction in solid He 3. The center curve is believed to be the most reliable [N. Bernardes and H. Primakoff, Phys. Rev. LeUers 2, 290 (1959);8, 144 (1959)l.
by Baum et a1.Ib2have confirmed these predictions, quantitatively, and give results which resemble the middle theoretical curve of Fig. 15. The only crystallographic transformation which has been studied at various temperatures down to the boiling point of helium is the a-7 140
B. Weinstock, B. M. Abraham, and I). W. Osborne, Phys. Rev. 86, 158 (1952).
F. E.Simon and C. A. Swenuon, Nature 166, 829 (1950). ls1 N. Bernardes and H. Primakoff, Phys. Rev. Letters 2, 290 (1959);3, 144 (1959). 11* J. L. Baum, I). F. Brewer, J. G . Daunt, and 1). 0. Edwards, Phya. Rev. Letfcta 3, 160
127 (1959).
PHYSICS AT HIGH PRESSURE
91
trannition in solid nitrogen (Fig. 14).14’Again, d P / d T is seen to approach zero a t absolute zero. (1) Melting curves. The pressure dependence of the melting temperature has been the subject of considerable discussion because of potential similarities between the behaviors of the melting line and the vapor pressure curve. I n particular, the possibility that a crystal-fluid critical point exists has been studied in some detail both experimentally and theoretically. Many of the theoretical aspects of this problem have been summarized by Domb in a recent paper,16awith conclusions which are in essential agreement with the qualitative discussion which follows. The experimental evidence tends to support the conclusion that the melting line continues indefinitely with increasing temperature and pressure. Simon154observed that a plot of In P versus In l’ resulted in a series of straight lines for different substances at high pressure, with all the lines having roughly the same slope. This led him to postulate the following form for the equation of the melting curve: (a) P + P o = bTc;
or
(b) P/Po
+ 1 = ( T / T O ) ~(4.15)
which is in very good agreement with the experimental data. Here, Yo is an internal pressure (which presuma1)ly must be overcome to melt the solid a t absolute zero), and b and c are constants, with c being of the order of 2 for most nonmetals. This equation fits the experimental data well and has received some theoretical support in recent years from Salter’s work relating c to the Gruencisen constant I?. 155 However, Eq. (4.15) implies nothing about the existence of a critical point. The form (h) of Eq. (4.15) was used by Simon as a reduced melting pressure equation, with To (of the order of the triple point temperature) and P o being a characteristic temperature and pressure, respectively. Simon16‘ argued that, in analogy with the reduced representation of the liquid-vapor equilibrium in terms of the critical constants of 11 liquid, one should be able to represent all melting phenomeiia approximately by Eq. (4.15b). Thus, in order to investigate the possible existence of a solid-fluid critical point, work should be done with the substance with the highest possible reduced pressures (P/Po) and temperatures ( T / T o ) which is helium. The liquid-solid transition in ordinary helium (He 4) is not normal at low temperatures due to the onset of superfluidity and the extraordintcrily rapid decrease of the entropy of the liquid along ’UC. Ilomb, N~ioaoctntenlo 9, Suppl , p. 9 (1958). 164 F. E. Simon, in “I,. Farkas Mrmorial Voliirnr,” Itescrtrrh C h n r i l of Israel BpmiRI Pub. No. 1, p. 37, Jerxualem, 1952. ”6L. Salter, Phil. Mag. 171 46, 369 (1954).
92
C. A . S W E N 8 0 N
the melting line below 13°K (see Fig. 15).160 Nevertheless, above 4°K the data for He 4 can be represented by Eq. (4.15b) with an experimental P o = 17.30 atmos and T o = 0.90”K. These are to be compared with a P o = 271 atmos and TO= 14°K for hydrogen. Thus, helium could be expected to behave qualitatively in the same manner at 1000 atmos and 14°K as would hydrogen at 15,000 atmos and 196°K. According to this reasoning, if a liquid-solid critical point exists, it should occur a t a lower pressure for helium than for any other solid. Simon and his collaborators have done considerable work on the melting curve of helium (and other gases) at high pressures, with Holland, Huggill, and Jones”“ showing that the transition occurs at 7300 atmos at 50°K. This approach is rather uncertain, however, since if the behavior of the melting line is similar to the behavior of the liquidvapor curve, measurement)s of the transition pressure will give no warning of the imminence of the critical point. Dugdale and Simon**undertook to measure both the entropy and volume changes for the fluid-solid transition in helium 4 by a calorimetric method. This was done, since for the liquid-vapor case, the onset of a critical point is definitely indicated by a rapid decrease in both A S and A V as the temperature approaches T,. While Dugdale and Simon only worked to pressures of 2500 atmos at 25”K,their data show quite conclusively that. both A S and AV approach relatively constant values as pressure and temperature increases, with no indication of unusual behavior. The results for solid He 4 have been summarized in a review article by Domb and D~gda1e.l~’ More recently, Mills and Grilly16*have determined the volume change on melting for both He 3 and He 4 to 3500 atmos, with results for He 4 that agree (both for AV and A S ) with those of Dugdale and Simon. Recently, Ebert169 also considered this problem and analyzed the data on much more complex substances. He shows that for a number of substances measured values of A S and A V extrapolate to zero at the same temperature, but i t is difficult to rely on an extrapolation of data to pressures an order of magnitude greater than the maximum pressure attained experimentally. A similar extrapolation by Crilly and Mills*8o on A V data for nitrogen gives 18,500 at,mos and 256°K as the point at which A V would extrapolate t o zero. As they remark, this result cannot I6*F. A. Holland, J. A . W. Huggill, and G. 0. Jones, Proc. Roy. SOC.A107, 268 (1951). C. Domb and J. S. Dugdale, in “Progrew in Low Temperature Physics” (C. J. Gorter, ed.), Vol. 11, p. 388. Interscience, New York, 1957. 1‘8 R. I,. Mills and E. R. Grilly, Proc. 6th Intern. Conf. on Low Temperature Phys. and Chem., Madison, Wis., 1.967, p. 106 (1958); Ann. Phys. ( N . Y . ) 8, 1 (1959). *6OL. Ebert, &&r. Cheniker-Zlg. 66, 1 (1954). loo E. R. Grilly and R. L. Mills, Phys. Reu. 106, 1140 (1957). 11’
93
PHYSICS AT HIGH P R E S S U R E
he taken seriously since the pressure is five times aa great as the maximum experimental pressure. The melting curves of the simpler substances are of considerable intrinsic interest, and a number of melting curve determinations have been published in recent years. The most interesting of these are for the isotopes of helium (He 3 and He 4) and hydrogen (HI, Ds, and TI) obtained by Mills and Grilly.i81,i6*They also have worked with 02,N,, and Ne. Their data, all to 3500 atmos, are represented well by the Simon I
I
I
I
I
I
I
1
TEMPERATURE, *K
FIU.16. The melting pressure curves for the iaotopea of helium and hydrogen [after E.R. Grilly and R. L. Mills, Phys. Rm. 89, 480 (1955); 101, 1246 (1956)l. melting equation. The results of the isotope experiments are given in Fig. 16. It is interesting to note that He 4 apparently would solidify quite normally if it were not for the X transformation (Fig. 16); however, the negative value of P o for He 3 seems to indicate that it is basically a substance which would remain liquid to absolute zero under normal pressures. The differences in the melting pressures for both the heliums and the hydrogens are due to the zero point energy which blows up the liquida and solids, and results in a lower solidification temperature for E. R. Grilly and R. L. Mills, Phye. Rm. 09, 480 (1955). ll*E. R. Grilly and R. L. Mills, Phys. Rev. 101, 1246 (1956).
94
C. A . SWENSON
the lighter isotope. At high temperatmes, the effects of the zero point energy should disappear, and this seems to be indicated by each set of data. Earlier data on H z and D2 by Chester and Dugdale163indicated a constant separation in pressure betwecn the curves, but the latest data show this separation begins to disappear at, higher pressures. Domb in discussing the effects of isot,opic niitss on the melting line''j4 remarked that, although this high-pressure behavior would be expected for the helium isotopes, there is no reason for exprcting it for hydrogen since the hydrogen molecules are slight,ly altered by the change in nuclear mass. Most of the above data were obtained using the blocked capillary technique. In this method, the center o f n. fine capillary U tube is placed in a temperature bn.t,h which niaintains a cold spot on the tube. One end of the capillary is connected to a pressure generator and the other to a pressure sensing device. With the cold spot temperature held constant, the pressure is slowly increased until the sensing device shows that the capillary is blocked. A decrease in pressure will show the onset of melting in a return of continuity. Since a solid never subcools on melting, this melting pressure is the more reliable indication of the true transition pressure. Two problems concerning the experimental method should be considered. First, Bridgman raised the question as to how the melting point would be affected by pressure gradientasin the block.ls6 However, some rather lowpressure (but highly accurate) work on helium has shown that these effects are entirely negligible,16sas Grilly and Mills have also concluded. Second,'the method becomes inapplicable a t high pressures because of the increased viscosity of the fluid, a difficulty which is not easily overcome. Robinson38in his work on the melting curves of nitrogen and argon used the magnetically induced mot,ion of an iron weight in the fluid it,self to detect the presence of the solid. Unfortunat'ely, his data on nitrogen do not agree with those of Grilly and Mills, probably because his sample temperatures (due to the latent heat of transition or normal heat leak down the sample) were higher than his external thermometer indicated. In the course of st,udies on liquids Bridgman'* had difficulty determining the melting point,s of various silicones by the piston-displacement technique, but finally concluded that this behavior probably was due to the high viscosity of the liquids rather t,han to the existence of a critical point.167 Recently addit,ional melting pressure data a t relatively high pressures P. F. Chester and J. S. Dugdale, Phlys. Rev. 96, 278 (1954). C . Domb, Pror. Phya. Soc. 70B, 150 (1957). lo 1'. W. Bridgnian, IZevs. M o d e r n l'hyx. 18, 27 (1946). l o ) C . A. Swenson, I'hys. Rev. 89, 538 (1953). In' P. W. Bridgnian, J . ('hem. I'hys. 19, 205 (1951). IEa
PnYsics AT
nwn
PRESSURE
95
and temperatures for many metals, including Bi, Sn, T1, Cd, Pb, Zn, Sb, Cu, Al, P, and In, have become available. The results are not unusual, except that the melting point of antimony continues to decrease with increasing pressure. A triple point above 550°C and 40,000 atmos was postulated to account for this abnormal behavior.16* These experiments were done by Butuzov and his collaborators who worked to 1500°C and 35,000 atmos. A description of the apparatus which they used (in principle, much like that described by Birch6‘) and an excellent bibliography to their work are given in a paper by Butuzov.16@Similar work on eight alkali halides also has been reported recently. lboo Data on transitions at much higher temperatures (to 2400°C) and pressures (to 180,000 atmos) have been obtained from measurements of electrical resistivity changes at the transitions and with the use of solid pressure transmitters. Hall has measured the melting curve of germanium to 180,000 a t m o ~ , ’ ?while ~ Bundy has studied the phase diagrams and melting curves of bismuth171and r u b i d i ~ m ”to~ pressures of the order of 150,000 atmos. Bundy and Strong’?’ have also reported data on the melting curves in this pressure range of nickel, platinum, rhodium, and iron at temperatures up to 2400°C. Strong1’* has commented on the geophysical significance of the iron data. The rubidium melting curve is unique in that dF’/dT, which is initially positive, becomes infinite and then negative as the pressure is increased. This is the only example of this type of behavior, and is not u n d e r ~ t o o d . 1 ~ ~ (2) Phase changes in solids. The existence of more than one modification of a solid is far from a rare occurrence, as is evidenced by Bridgman’s statement that out of several hundred substances which he has investigated, roughly one-third have shown polymorphic transitions.‘4 In general, moqt solids will tend to transform at high pressures to a more tightly packed structure; however, if the molecules are asymmetrical this structure may not be close-packed. The various modifications of some of the more common elements, along with their relative volumes, are indicated in Fig. 13. The data which are shown are for room temperaItaV. P. Butuzov, E. G . Ponyatovskii, and C. P. Shakhovskoi, Doklady Akad. Nauk SSSR 109, 519 (1956). IScV. P. Butuzov, Kriatallografiya 2, 536 (1957), Soviet Phya., Crystallography 2, 533 (1957). 141.S. P. Clark, Jr., . I . Chem. Phys. 11, 1520 (1959). no H. T. Hall, J . Phya. Chem. 69, 1144 (1955). F. P. Bundy, I’fiys. Rev. 110, 314 (1958). Ir1F. P. Bundy, Phys. Rev. 116, 274 (1959). F. P. Bundy and 11. M. Strong, Phya. Rev. 116, 278 (1959). 1’‘ H. M. Strong, Nature 183, 1381 (1959). fnO
The recent data on the melting of metals at very high pressures has been discuesed in Home detail by 11. M. Strong, American Scientist 48, 58 (1960).
96
C. A. SWENSON
ture and extend to 100,OOO atmos. Bridgman has collected a great deal of additional information for many substances over the smaller pressure range to 50,000 atmos.48~bs~ss~14*-14b Recent high-pressure work at both low temperatures (on N2,147 Op,171 COS,”’ CO,’TL CH4,97.176 CD4,170H2S,lT7CF4,171and Hg,l17) and
WC)
FIQ.17. The phase diagram for bismuth [after F. P. Bundy, PAyu. Rev. 110, 314 (lQS8)l.
high temperatures (Bi, P, Ce) either has confirmed low-pressure measurements, or established the existence of new phases. Probably the most surprising result is that solid He 4s9 exhibits a phase transition with a triple pcint along the melting line a t 15°K. Presumably the transition is R. Stevenson, J . Chem. Phy8. 27, 673 (1957). J. W. Stewart, Phys. and Chem. Solids 12, 122 (1960). The data obtained for agree with Stevenson,1” but the CH, data do not.97 R. StevenRon, J . Chem. Phys. 27, 147 (1957). 17* J. W. Stewart and It. I. LaItock, J . Chern. Phy8. 28, 425 (1958).
01
PHYSICS A T HIGH PRESSURE
97
between the normal hexagonal close-packed s t r u ~ t u r e " and ~ a facecentered cubic structure. A transition has also been reported in solid He 3 below 4'K,l68 but this has not been studied in detail. Usually, very little can be said about the significance of high pressure transitions, and for a cataloging of those which exist, reference is made to Bridgman's reviewal.* and to more recent papers which are mentioned here. 1700 The phase diagram for ice is of a complex type, since seven different modifications of ice have been found.ls0 Also the phwe diagram of elemental bismuth (Figs. 13, 17),17' which exhibits eight solid forms along with seven triple points is very complex. The high-temperature forms were found by Bundy using the discontinuity in the electrical resistivity at the transition, with results that are in substantial agreement with the work of Bridgman and Rutuzov, who used piston displacement and thermal analysis techniques, respectively. The 111-IV, IV-V, and V-VI transitions were found by Bridgman only in his piston displacement work, and presumably because of very small differences in the resistivities of the two phases, these were not apparent from either Bridgman's or Bundy's resistivity measurements. The breaks in Bundy's curves would eaem to indicate triple poir ts corresponding to these forms. The exact nature of the structure change in a high-pressure transition is difficult to determine except under special conditions. Often a transition which will take place at high temperatures and high pressures will not revert back to its low-pressure form upon rapid cooling at high pressure, 80 that it is possible to obtain samples of the new phase for x-ray study at normal pressures and temperatures. The most notable example of this is the transition from graphite too diamond. In this system the diamond form is stable only at high pressures, and even at 100,ooO atmos diamonds can be produced a t an appreciable rate only a t very high temperatures (2500°K).8*1a1 Yet, upon cooling to room temperature, the diamonds 80 formed are completely stable to all practical purposes, although they are not in thermodynamic equilibrium.18* The quenching technique has been used by Coes188to obtain new forms of certain minerals, and Wentorfls4 has used it to produce a cubic form of boron *I'D. G. Hemhaw, Phyu. Rev. 109,328 (1958). IT*A discussion of memurementa on the plutonium phase transitions has been given in an article by P. W. Bridgman, J . A p p l . Phys. 80, 214 (1959). IroFor a diagram, see J. C. Slater, "Introduction to Chemical Physics," p. 168. McGraw-Hill, New York, 1939. '"F. Y. Bundy, H. T. Hall, H. M.Strong, and R. H. Wentorf, Jr., Nature 176, 51 (1955).
I8'R. Berman and F. E. Simon, Z. Elektrochem. 69, 333 (1955). Ir8L.Gea, J . Am. C'eram. Soc. 88, 298 (1955); Science 118, 131 (1953). OcR.H. Wentorf, Jr., J . C h m . Phye. 46, 956 (1957).
98
C. A. SWENSON
nitride which is as hard a s diamond and considerably more stable. In general, this method constitutes a powerful tool for use by the geophysicists in their study of minerals. The “simple squeezer”60and the apparatus of Birch et al. (Fig. 4)64 were developed for work of this type, where rapid quenching of the samples is of importance. Typical applications have been described by Birch and his collaborators, MacDonald, and Ringwood. 1*&18Q New solid forms of CS, and phosphorus (black phosphorus) which are stable at room temperature and zero pressure have been produced by BridgmanIQOunder certain specialized conditions. The irreversible formation of black phosphorus a t temperatures up to 200°C and pressures up to 50,000 atmos has been investigated by Piitz,lgl who shows definitely that this is a true allotropic modification of phosphorus. Keyesl82 ha investigated the electrical resistivity of this substance, and finds that it is a semiconductor with an energy gap that should decrease to zero near 20,000 atmos. Bridgnian has found a sluggish reversible transition in black phosphorus that occurs a t about 50,000 atmos at room temperature, but which reverses itself only a t pressures less than 25,000 atmos.’O It has heen established as definitely reversible, however. Butuzov and his collaborators also have looked at the phase diagram of phosphorus in some The existence of a new phase of mercury which is stable at low temperatures and pressures was originally predicted by Bridgman on the basis of data obtained near 200”K.191 Experiments a t normal pressures gave no indication of its existence, hut recent high pressure work a t low temperatures has shown that :b p phase can be produced which is the thermodynamically stable phase at zero pressure below 79°K.”’ The new phase anneals out irreversihly a t temperatures near 90°K at atmospheric pressure, so the study of its properties requir,es continuous refrigeration of the metal. Nevertheless, work has been done on the compression, 117 superconducting properties,IQ4and x-ray structure of fl-mer~ury.’~~ Some effort has been made to do x-ray work a t high pressures, with 1.; C. Robertson, F. Hirch, and <;. J. F . MarIkmald, Am. J . Sci. 266, 115 (1957). E. C;. Flobrrtson, and F. Birch, Am. J. Sci. 266, (328 (1957). (;. J. F. MacIhnaid, .4m. Mineralogist 41, 744 (1956). I * * 6. P. Clark, Jr., .4m. Minerdogis! 42, 564 (1957). In9 A. E. Ringwood, Hzill. Gpol. Soc. A m . 69, 129 (1958). 1’. W. Bridgman. “The Physics of High Pressures,” p. 424 (CS,), p. 383 (P). G. Bell, randon, 1949. I g 1 K. l’atz, Z . anol-g. ‘11. allgenz. (,’hem. 286, 29 (1956). R. W. Keyes, Phys. Rev. 92, 580 (1953). 183 1’. W. Bridgman, Phiis. Rev. 48, 893 (1935). I g 4 J. E. Schirber and (1. A. Swcnson, Phys. ICev I.Pfters 2, 296 (1959). l g 5 M. Atoji, J . E. Schirbcr, and C. A. Swenson, J. Chem. Phys. 31, 1628 (1959).
Inas.P. Clark, Jr.,
PHYSICS A T HIGH PREBSURE
99
moderate success. X-ray windows of sufficient strength are difficult to design, and both polycrystalline and single crystal beryllium and diamond have been used. Lawson and his collnbortitors have used both types of windows in their study of the phase transit ion3 in cerium i ~ n dAgI.186.197 Guegant and \‘odar1B8have studied KNOr and Ant, while J s m i e ~ o r i ~ @ ~ also has studied Kh’Oa using Lawson’s tli:~niondt)oml). Vereshvhikgin and his co-workersLoO.*O1 have described n beryllium window bomb for x-ray work to 30,000 atnios, and have investigated the bismuth transitions (Figs. 13, 17) as we11 as the transitions in lib1 and RbC1. Lithium was used as a transmitter in these experiments. Jamiesonzozhas also described a new single crystal diamond bomb with which KI, Cd, and the calcite phase diagram t o 24,000 atmos were studied. This paper is quite complete and contains an excellent description of the methods used and the problems which are eiicountered.202a Probably the most striking result to come from the x-ray work is the discovery that in the room temperature phase transition in cerium, which occurs a t about 7000 atmos, there is no change i n the crystal btructure (both phases being face-centered cubic), while there is a 14% change in the volume.203This volume change is enormous relative to other systems (Fig. 13 contains several examples) in which the larger changes are of the order of a few per cent. It has been postulated that this new “condensed cubic” phase is due to the movement of an electron in the ion from a 4f to a 5d orbital, a possibility which is quite plausible in view of the occurrence of either trivalent or quadrivalerit cerium in chemical compounds.20aThe abrupt decrease in the resistivity of the metal by a factor of two a t the transition also lends credence to this postulate. Bridgman has discussed his previous work in some detail in describing the resistivity result^.^^-*^^ The cerium transition seems to be quite sensitive to impurities in the metal, and discrepancies between various workers as to transition pressures and volume changes can perhaps be traced to this. The transition has been studied us a function of temperature by both lD8A.W.
Lawson and N. A. Riley, Rev. Sci. In&. 20, 763 (1049). I*’A. W. Lawson and T. Y. Tang, Rev. Sci. Instr. 21, 815 (1950). IorL.Guegant and B. Vodar, Compl. rend. acad. sci. 239, 431 (1954). 1°J. C. Jamieson, 2. KnSf. 107, 5 (1956). *OoL. F. Vereshchagin and I. V. Brandt, Doklady Akad. Nauk SSSR 108, 423 (1956).
rolL. F. Vereshchagin and S. S. Kabalkina, Doklady Akad. Nauk SSSR 113, 797 (1957).
rorJ.C. Jamieson, J . Geol. 66, 334 (1957). roroThe“single queezer” has been modified for X-ray work by J. C. Jamieson, A. W. Lawson and N. H. Nachtrieh, Rev. Sci. In&. SO, 1017 (1959). A. W. Lawson and T. Y. Tang, Phys. Rev. 76,301 (1949). ro4P.W. Bridgman, Proc. A m . Acad. Arts Sci. 79, 149 (1951).
100
C. A. SWENBON
the thermographic*06.2o6and piston-displa~ement~~~~~O~ methods, with substantial agreement. The piston-displacement data given by Herman and Swenson disagree slightly at^ to transition pressures and greatly 88 to volume changes with similar work by Likhter et at. These measurements have been checked in more recent experiments,209and the new
a-
I-
6‘-
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/
/
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\ \
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. 1
FIQ. 18. The transition pressures and relative volume changes for the cubiccondensed cubic transition in cerium aa functions of temperature (R. I. Beecroft C. A. Swenson, “On the existence of a critical point for the phase transition in cerium,” to be published in P h p . and Chem. Solidej. The open circles at low temperature repreeent zero pressure data.
room temperature value of the volume change (13.5%) and the slope of the transition line (dP/dT = 43 atmos/deg) (see Fig. 18) give a calculated value for the latent heat at room temperature (910 cal/mole) which zo6
M. G. Gonikberg, G. P. Shakhovsko, and V. P. Rutuaov, J. Phys. Chem. (USSR) 51, 350 (1957).
‘OrE. G. Ponyatovskii, Doklody Akad. Nauk SSSR 120, 1021 (1958); Soviet P h y ~ . , Doklady 9, 498 (19581. z07 A. I. Likhter, Yu. N. Ryabinin, L. F. Vereshchagin, J. ExpU. Theorel. Phya. (USSR) 55, 610 (1957); Soviet Ph2/8., JETP 6, 469 (1958). *08 R. Herman and C. A. Sweneon, J . Chem. Phys. 29, 398 (1958). *09 R. I. Beecroft and C. A. Swenson, “On the existence of a critical point for the p h w transition in cerium,” to be published in Phys. and Chem. Solids.
P H Y S I C S A T HIGH PHEYYUHE
101
is in excellent agreement with the thermographic analysis value of 880 cal/mole given by Gonikberg et a1.206 The slope of the transition line in these new experiments also is in excellent agreement with the earlier Russian work.*07 The magnitudes of the volume changes are not in agreement, however, but do show quantitatively the decrease with increasing temperature which Likhter rt al., commented 011. The most recent piston-displacement work on the cerium transitioriPoD wag stimulated by the results of Ponyatovskii*06which indicate that the latent heat of the cerium transition approaches zero somewhere above 280°C and 18,000 atmos pressure, and that the phases should become indistinguishable above this point on the phase diagram. The pistondisplacement data were obtained by placing the lower end of the press, which is sketched in Fig. 8, inside a cylindrical furnace, and by using directly a Carboloy sample holder of the type shown in Fig. 3a. The problems encountered at low temperature and high temperature have much in commnn, although the annealing of the steel parts of the press at high temperatures set a practical upper limit of about 300°C to the temperatures which could be used. The resulting data for both the transition pressure and volume change are plotted as functions of kmperature in Fig. 18. A critical point near 630°K (358°C) and 20,000 atmos would seem to be indicated if the extrapolation can he relied upon. The situation is not as simple as this, however, since data taken at 575"KZo9 indicate that the transition, although definitely reversible, is considerably spread out in preseure at this temperature, and it would seem impossible to consider it as first order. In view of the large change in the resistivity of the cerium at the transition, electrical resistivity measurements as a function of both temperature and pressure would be useful both to check on the existence of the critical point, and to aid in deciding the electronic configuration above the critical point. This transition in cerium would seem to be one of the few transitions in which a critical point is possible, since the structures of the two phases are identical and differ only in their lattice constants. A similar transition has been postulated for cesium at about 45,000 atmos at room temperature. Cesium normally has a body-centered cubic structure, with a small transition at 23,000 atmos [AI'/V = 0.5% (Fig. 13)] which is believed to involve a change to the more compact face-centered cubic structure. However, it is unlikely that the transition at 45,000 atmos8O (AV/V = 10%) is due to a crystal structure change, and Sternheimer has proposed that a change in electronic configuration occurs, with an electron moving from a 6s to a 5d orbital.*10The resistivity shows a large anomaly in the transition region, but not a discontinuous change." The low-temperature com'lo R. L. Sternheimer, Phys. Reu. 78, 235 (1950).
102
C. A. SWENSON
pression data show a n interesting effect which may be due t o this transition.Il0 Cesium was found to behave quite differently from the other alkali metals in th at it was too “soft” rtt 4% [see Eq. (4.G)and the following discussion]. When the low-temperature pressure versus volume curve was extrapolated from 10,000 to 50,000 atmos, it appeared to join more smoothly with that for the new condensed phase than with that for the room temperature, low-pressure, form. More low-temperature work at higher pressures must be done to verify that this is actually the case. Recent work by Rundy on the phase diagram of rubidiuml72 has shown that this alkali metal does not have a transition of the cesium type above room temperature and a t pressures to over 100,OOO atmos. One other unusual type of trsrisition should be mentioned. The plastic polytetrafiuoroethylene (Teflon) was found by Bridgman14sto he unique in that it exhibited a phase transition a t high pressures and, unlike other plastics, behaved very much like a crystalline solid. The phase diagrarr and equation of state of Teflon have been studied over a range of temperatures by Wier to 10,000 atmos18g.140 and Beecroft and Swenson to 20,000 atmos.118X-ray studies indicate that this plastic is partially crystalline, and indeed the crystallites “melt ” in the solid at temperatures which depend on the pressure in just the same manner as normal melting phenomena.*l‘ Bridgman has reported a similar transition in a plastic called F ’ h o r ~ t h e n e . ~ ~ ~ Bridgman has predicted that in the next decade of pressures (up to 106 atmos) an increasing number of the electronic transitions will be found. The theoretical aspects of some of these have been discussed by Behringer,*I2 with particular emphasis on a possible transition to the metallic state in LiH. It has been reported that this transition does occur in shock-wave experiments. The extension of static techniques into the next high pressure region will be difficult, and there seems to be some doubt that the shock-wave experiments will give transition data which are easily compared with static data. For instance, a possible transition in iron a t 120,000 atmos waa inferred recently from the results of experiments in which dynamic techniques were used, 21a However, Bridgman’s electrical resistance measurements at room temperature did not confirm this finding although a pressure of 180,000 atmos was attained before the apparatus failed “ c a t a ~ t r o p h i c a l l y . ”The ~ ~ ~comparison is not quite valid, however, since some transitions (as in bismuth) do not show u p in resistivity work, and the temperatures were not comparable. Similar *llP.L. McGeer and H. C. Duus, J . C‘hem I’hys. 20, 1813 (1952). 2 1 1 R. E. Behringer, Phys. Rev. 113, 787 (1959). m* D. Bancroft, E. L. Peterson, and S. Minshall, J . A p p l . Phys. 27, 291 (1956). 314 P. W. Bridgman, J . A p p l . Phys. 27, 659 (1956).
PHYSICS A T HIGH P R E S S U R E
103
result8 have been reported (as in Li.Z1H4, for instance) in which resistivity measurements by the static and dynamic approaches failed to give the same r e s ~ l t s The . ~ experimental ~ ~ ~ ~ ~ ~ situation in the region of overlap of the two techniques is not very clear.
5.
ELECTRICAL A N D
MAGNETIC h4EASUHEMENTS
a. Resistivities of Normal nlelals and Semiconductors ( 1 ) Nortnul metals. The theoretical and experimental aspects of the effects of pressure on the electrical resistivity of metals have been dis, ~ hence, these cussed at some length in a review article by L a ~ s o n and, topics will not be dealt with in detail here. Nevertheless, since the time of his article, various fundamental high-pressure experiments have been done at both high and low temperatures. One addition'4bcan be made to Lawson's rather complete bibliography of Bridgman's work. This publication146 includes resistivity data at high pressures for yttrium, rhenium, technetium, and strontium. The strontium data are of particular interest since strontium previously had been found to have an anomalous dependence of resistivity on pressure and temperature. However the new experiments, in which the data were taken over a much more limited pressure range (7000 atmos rather than 30,000 atmos) but over a wider temperature region (to 200°C rather than 75"C),showed no anomalous beh a v i ~ r . ~The ' ~ possibility of an impurity effect was not ruled out, however. The principle new high-temperature experiments involve the use of electrical resistance measurements to iiivestigate phase changes, as in the work by Hall, Bundy, and Strong already mentioned (Section 4b). Their method is applicable to the measurement of the changes in resistance which are indicative of phase transformations but not to the measurement of absolute resistivities. The fact that changes in structure occur without significant changes in electronic properties limits the usefulness of this method. Data obtained by more conventional techniques are more often in a form which allows a direct comparison with theoretical calculations. Lawsone concludes that (with notable exceptions, such as the alkali metals, alkaline earths, bismuth, and antimony) the effect of pressure on the temperature-dependent part of the electrical resistivity is well understood. This is borne out by recent work at low temperatures (see the
*I'D. T. Griggs, W. C. McMillan, E. D. Michael, and C. P. Nash, Phys. Reu. 109, 1858 (1958).
B J. Alder and R. H. Christian, Discussions Faraday SOC.29,44 (1956); Phys. Rev. 1% 660 (1956).
104
C. A. BWENSON
following), although the same work indicates that this is not true for effects of pressure on the resistivity a t very low temperatures. In order to discuss the low-temperature results, it is convenient to write the electrical resistivity of a metal in two parts, as suggested by Matthiessen :217 P = PO
+ PdT)
A PO
+ P.(B/T).
(5.1)
Here, po is a temperature independent contribution (called the residual resistivity), the magnitude of which is a qualitative measure of the purity and state of strain of the metal, and pi (the ideal resistivity) is a temperature-dependent contribution due to the interaction of the electrons with the lattice vibrations. The ideal resistivity (p,) goes to zero at absolute zero, and becomes approximately linear in the absolute temperature a t high temperatures. This temperature dependent part of the resistivity can be approximated quite closely by a semiempirical relationship d- to r weneisen, from which a dependence of the resistivity on 6 / T follows, 6 being a characteristic temperature of the order of the Debye t e m p e r a t ~ r e . ~ ” The pressure dependence of the ideal resistivity can be interpreted with considerable success in terms of the variation of 6 with pressure, Since 0 usually increases with pressure (and [ d p / d P ] is ~ always positive for a pure metal), a decrease in resistivity with pressure would be expected for temperatures a t which p , predominates. This is always observed for metals which behave normally in other respects, and the work of Dugdale and Gugangl on copper is in good agreement with the theory for both the temperature and pressure dependence down to 20°K and for pressures up to 3000 atmos. Similar behavior has been found a t low temperatures by Kan and Lazarewalsin their work on single crystals of zinc, tin, and gold to 2000 atmos, using the ice-bomb technique. The effects of pressure on the residual resistivity are, for reasons which are not understood, of the opposite sign, PO increasing with preseure. This is illustrated in Fig. 19 where Dugdale and Gugan’s data for copperB1 have been plotted as a function of temperature. The results which were obtained when the resistivity was analyzed into its residual and ideal components are also given. Kan and Lazarew have observed similar behavior for zinc and gold ( a p / a P being zero a t about % O K ) , and for tin, where the change in sign occurs a t about 10°K. According to the latter workers, the pressure coeficient of the resistivity never changes sign for bismuth, but remains anomalously positive over the whole temperature **‘C. Kittel, “Introduction to Solid State Physics,” 2nd ed. pp. 304, 306. Wiley, New York, 1956. ZlaL. S. Kan and B. G. Laearew, J . Ezptl. Theme;. Phye. (USSR) 84, 258 (1968); Sot& Phys., JETP 7, 180 (1958).
105
PHYSICS A T HIGH PRESSURE
range, in agreement with Bridgman's room temperature results.47 Dugdale and Gugan were looking specifically for the influence of pressure on the resistivity minimum in copper,g1and found little effect. In another set of very difficult experiments, Hattori has used solid hydrogen as a pressure trsnsniitter to measure the effect of 5000 atmos pressure at 4°K on the resistances of small wires of Cu, Ag, Au, Pt, Sn, In, T1, Ta, Bi, and single crystals of antimony and a r ~ e n i c . ~ ~As '~*~*
+9
0 ACTUAL RESISTANCE
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Fxo. IS. The preasure dependence of the electrical resistivity of copper [after J. 8. Dugdele and D. Gugsn, Proc. Roy. SOC.A l l l , 397 (1957)l.
Dugdale and Gugan*l have pointed out, the hysteresis and lack of reproducibility from cycle to cycle which Hatton observed may result from a deformation of the samples by the solid hydrogen on the increase and decrease of pressure; however, the behavior of some of the CUNW of resistivity versus pressure cannot be accounted for this simply. Hatton also notes a correlation between the positive sign for ap/aP for nonsuperconductors and a negative sign for metals which become superconducting at lower temperatures.z1B He suggests that this distinction may be '1)
J Hatfnn Phva RPU inn- 1167 (iQ5aI
106
C . A . SWENSON
significant. His data for tin, however, are in direct contradiction with the single crystal ice-bomh data, and Kan and Lazarew find “normal” behavior for zinc which also becomes superconducting.**SI t is possible that the metals for which Hatton noted this anomalous sign (tin, thallium, and indium) were so pure that he was actiinlly observing the pressure dependence of the ideal resistivity. (‘ertainly, these points should be investigated in more detail and by other methods. Dugdale arid his co-workers also have determined the effect of ZOO0 atmos pressure. on the low-temperature resistivities of rubidium,8* sodium, and lithiuiri.220While the results of these experiments are in qualitative agreemcnt with gcrieralizst ions discussed above, the deviation of the temperature dependenc~of the zero pressure resistivity of rubidium from the simple Grueneisen relationship and the existence of martensitic transformations in lithium and sodium make any quantitative discussion impossible. The electrical and magnetic properties of bismuth have proven t o be interesting subjects for investigation a t high pressure because this metal exhibits anomalous behavior in many resperts. The phase diagram of bismuth, with its many modifications, has already been mentioned (Fig. 17). .4lekseevskii and Rrandt221have measured both the Hall and magnetoresistive effects in bismuth a t pressures up to 2000 atmos and at temperatures down to 4°K. I m g r impurity effects were found which n ere decreased t)y increasing pressure. Their results were qualitatively explained by the variation in electron concentration with both pressure and impurities. Abnormal behavior in the temperature variation of the pressure coefficient of elcctrical resistivity was also found by Kan and Lazarew.2’8Likhter and Vereshchagin222 have measured the Hall effect in bismuth to 30,000 atmos a t room temperature, and find a decrease in the Hall coefficient hy three orders of magnitude in the course of the 1-11 and 11-111 phase transitions. They describe normal bismuth as a “semimetal,” arid bismuth 111 as a “real” metal. This agrees with the results of Chester and Jcnes,67 which showed that, when bismuth is cooled froin room temperature t o 4°K a t about 25,000 atmos, it exhibits superconducting behavior, presumably because the I1 or I11 phase is frozen in. Finally, many of the electrical and magnetic properties of bismuth D. Gugan and J. S. Dugtiale, Proc. Filh IrLlern Conf. on Low Temperature Phys. and Chem., Madison, Wis., 1967, p. 376 (1958). 221 i X. E. Alekseevskii and N. €3. Brandt, 1. E x p t f . Theoret. Phys. (USSR) 28, 379 (1955); Soviet Phys., JETP 1, 384 (1955). D2z A. I. Likhter and L. F. Vereshchagin, J . Ezptl. Theoret. Phys. ( U S S R ) 32, 618 (1957); Soviet Phys., J E T P 6, 511 (1957). 220
PHYSICS A T HIGH PRESSURE
107
strongly exhibit the de Hans-van Alphen effect. Overton and Berlincourt221 have measured the influence of 100 atrnos pressure at 4°K on the de I-Iaas-van Alphen parameters for t,he eIcct.rical resistance of bismuth. Verkin, Ihnitrcnko, and l,:izur(w,?2i :LS well as Hrandt. a1111 Veiitsiil’,22b have measured the effect of 1500 atnios prcasiire, a g : h at. licluid helium temperatures, on b0t.h the nscillat ion amplitude and frequency of the susceptibility variations by measuring the force couple on an ice bomb in which a bismuth single crystal was imbedded. Little effect was found on the frequency, while a reduction in the amplitude was observed. Dmitrenko, Verkin, and LLtzarew226also have investigated the de Haas-van Alphen effect in zinc a t 1700 atnios pressure and a t temperatures down to l.G°K. Both the mnplitude and frequency of t.he oscillations were strongly pressure-depeiident. l’hcse pressure effects were quite revcrsible upon release aritl renpplication of pressure. These data give perhaps the best evidenw for the homogeiieity of the pressures generated in the ice-bomh tech~iiciuc. (2) Semicondi~ctor~. A &:tailed discussion of the effects of pressure on the properties of semiconductors, which would fit best into a general discussion of this subject, will not he presented here. A considerable amount of high pressure data on semiconductors has accumulated, however, and this section will be devoted to a listing of somc of this work. Bridgman has surveyed the effects of temperature and presswe to 50,000 atmos on the resistivities of ten semiconductors, including the elements germanium and Earlier work on the effect of pressure on the resistivity of germanium had been done by Taylor,229 and by Hall, Bnrdeen, and l ’ e a r ~ o n ,the ~ ~ latter ~ workers concerning themselves with p-n junctions. Bridgman’s eciuipnient n-ss used by Paul and to determine the resistaiice of germanium to 30,OOO atmos at 25°C and 76”C, and to 7000 atmos over the range from 77°K to 35OOC. Benedek, Paul and Brooks232subsequently measured the conductivity, *lS br.
C. Overton, Jr., and T. G . Rerlincourt, Phys. R e v . 99, 1105 (195.5).
**‘ B. I. Verkin, I. hl. Dmitrenko, and B. G . Lazarew, J . Ezptl. Theoret. I’hys. (USSR) 31, 538 (1956); Soviet Phys., J R T P 4, 432 (1957). N. B. Brandt aud V. A. Ventsd’, J . E x p t l . Theoret. Phys. ( U S S R ) 36,1083 (1958); Soviel Phys., JETE-‘ 8, 757 (1958).
lX6
rlsI. M. Dmitrenko, B. I . Verkiri, and B. C. Laaarew, J . Ezptl. Theoret. Phys. ( U S S R ) 33, 287 (1957). Soviel I’hys., J E T P 6 , 223 (1957); J . Exptl. Thcotet. Phys. (USSR) 96, 328 (1958); Soviet Phys., J E T P 8, 2% (1958). xpl P. W. Bridgman, l’roc. A m . Acad. Arts Sn’. 79, 125 (1951). ‘ M P. W. Bridgman, Proc. A m . Acad. Arts Sci. 82, 71 (1953). J. H. Taylor, Phys. Rev. 80, 919 (1950). *,OH. H. Hall, J. Bardccn, and C;. 1 4 . Peareon, I’hys. Rev. 84, 129 (1951). **I W. Paul and H . Brooks, I’hys. Heu. 84, 1128 (1954). B. Benedck, W. h u l , and €I. Brooks, I’hys. ICev. 100, 1129 (1955).
108
C. A. BWEN60N
Hall effect, and magnetoresistance for germanium to 10,OOO atmos, while Landwehrza3hns studied the effect of pressure on the drift mobility of holes in gcrmnnium over a similar pressure range. More recently, Paul and Wanhauer have investigated the effect of 10,000 atmos pressure on the absorption edge of germanium,la4 silicon,2a6and germaniumsilicon alloys;za6the optical and resistivity measurements were in good agreement. These " low-pressure" hydrostatic data on the optical prop erties of germanium and silicon agree with results t o 100,OOO atmm obtained by Slykhouse and DrickamerZs7 who used an approximate technique for producing their pressures. The absorption edge for germanium was found to show a blue shift to about 35,000 atmos, a t which pressure the shift changes sign and returns towards the red direction. Them results agree with predictions from other work. Silicon showed no such spectacular behavior, but only a monotonic red shift as the preasum was increased. Also the pressure dependence of the resistivity of silicon has been investigated recently to 350°C and 7000 atmos by Paul and PearaonZa8and hy Ryabininzaoet al., to 35,000 atmos. The only semiconducting compound which has been investigated in some detail is indium antimonide. LongZ4Ohas measured the electrical resistivity and Hall effect in InSb to 2000 atmos. K e ~ e s * has ( ~ made resistivity measurements on InSb to much higher pressures (12,000 atmos) and over a much wider range of temperature (- 78°C to 300°C) ; also in his paper he gives a rather complete discussion of the results. Gielegaen and von Klittingz" have studied both the Hall effect and electrical resistivity of indium antimonide a t 20°C as a function of both magnetic field (14,000 gauss) and pressure (7000 atmos). Long,*'* in a later public* tion, has given Hall effect and resistivity data to 2000 atmos on several other semiconducting compounds (InAs, GaSb, and Mg&) as well 88 on Gel InSb, and tellurium. Two semiconducting elements, tellurium and the black modification of phosphorus, which was mentioned previously, are of interest becauee there is evidence that in both the energy gap should disappear at presG . Landwehr, 2. Nalurforseh. lla, 257 (1956). W. Paul and D. M. Warachauer, Phys. and Chem. Solids 6,89 (1908). W. Paul and D. M. Warschauer, Phys. and Chem. So1id.a 6, 102 (1958). labW. Paul and D. M. Warschauer, Phys. and Chem. Solids 6, 6 (1958). aa7 T.E. Slykhouse and H. G . Drickamer, Phys. and Chem. Solids 7 , 210 (1958). W. Paul and G. L. Pearson, Phys. Rev. 98, 1755 (1955). Yu. N. Ryabinin, L. D. Livachic, and L. F. Vereshchagin, J . Tech. Phy8. (USSR) 28, 1382 (1958); Sovied Phys., Tech. Phys. 8, 1284 (1968). **O D. Long, Phys. Rev. B9, 388 (1955). R. W . Keyee, Phys. Rev. 99,490 (1955). J. Gieleaeen and K. H. von Klitring, 2.Phy& 146, 151 (1956). D.Long, Phys. Rev. 101, 1256 (1956).
la) la'
PHYSICS AT HIGH PRESSURE
109
surw sufficiently high that they would become metallic in nature. The case of tellurium waa originally discussed by Bardeen"' on the basis of data dve to Bridgman. Further high-pressure work on tellurium, in which the ice-bomb technique was used, has been done at low temperatures by Alekseevakii, Brandt, and l i o ~ t i n a However, .~~~ this work did not shed any light on the problem of the metallic state of tellurium. More recently, Long"' has made measurements to 2000 atmos on both the resistivity and Hall effect in tellurium at various temperatures; also he reconsidered the problem of the variation of the energy gap with pressure. Nussbaum, Myers, and Long246 have extended these measurements in their investigation of the anomalous Hall coefficient reversal in single crystals of tellurium. The resistivity of black phosphorus has been investigated by Keyes"2 to 8000 atmos at various temperatures to 350°C. He concluded that the new form of phosphorus found by Bridgman'goat 25,000 atmos is probably metallic in nature, since the energy gap of black phosphorus definitely extrapolates to zero by this pressure. Butuzov'" alrso has studied these transitions in his investigation of the phase diagram of phosphorus. The resistivity of another semiconducting element, selenium, has been investigated by Kozyrev and N a s l e d o ~ ~to' ~30,000 atmos over the temperature range from 20°C to 125°C. Two additional experiments with ionic conductors are worth noting. Kurnick'g has studied the effect of 8500 atmos pressure on the ionic conductivity of AgBr at temperatures up to 400°C. The results indicated that more than one mechanism of ionic conductivity are operative in this compound. Secondly, Hughes2'* has measured the effect of pressure on the electrical conductivity of peridot, a gem stone variety of olivine. Hie data extend to pressures of 8500 atmos and temperatures of 1300°C. The results show that peridot is an ionic semiconductor, with a decrease in conductivity due to pressure superimposed on an increase with temperature. These results are in agreement with optical absorption edge studies by R u n c ~ r n , ~ who ' ~ used a modified "simple squeezer."
b. Superconductors The first indication that mechanical stress could produce a reversible change in the properties of a superconductor was reported in 1925 by J. Bardeen, Phys. Rev. 76, 1777 (1949). 14'N. E. Alekseevskii, N. B. Brandt, and T. I. Kostina, J . Ezpfl. Theorct. Phys. (USSR)81,043 (1956); Soviet Phys., JETP 4, 813 (1957). $40 A. Nussbaum, J. Myers, and D. Long, Phys. Rev. Letters 2, 6 (1959). "'D. N. Nseledov and P. T. Kozyrev, J . Tech. Phyo. (USSR) 24, 2124 (1954); Doklady Akad. Nauk SSSR 110,207 (1956). H. Hughes, J . Ueophys. Research 60, 187 (1955). r4c8.K. Runcorn, J . A p p l . Phyr. 27, 598 (1956).
14'
110
C. A. SWENSON
Sizoo and Kamerlingh Onnes,260who observed a decrease in the superconducting transition temperature of tin and indium under pressures generated with liquid helium. About twenty years later, Lazarew and Kan7g.8’developed the ice-bomb technique and with it obtained results on tin and indium. Since that time, they and other workers have used this technique to measure both the shift in transition temperature with pressure at zero field ( d T J d P ) , and the shift in critical field with pressure for temperatures down to 0.06°K, for various substances. The magnitudes of the effects are small; for most superconductors, lo00 atmos will produce a shift in the critical field curve of a few gauss, or it change in the zero field transition temperature of a few hundredths of a degree. Various other methods also have been used. The very small shifts in critical field which can be obtained by using liquid helium directly as B transmitter (pressures from 50 to 200 atmos, depending on the tempersture) have been measured precisely by numerous workers; perhaps the most extensive and reliable data of this type are due to Fiske.261Chester and JoneslS7work with a fixed clamp has extended the pressure region to 40,000 atmos, and it showed that bismuth becomes a superconductor at pressures above 20,000 atmos. Finally, approximately hydrostatic methods, with the samples imbedded in solid hydrogen, have been used by Hatton*’* and by Jennings and Swenson (referred to as J and S in the following).*‘ These measurements all give, directly or indirectly, ( d T e / d P ) ,and, in addition, some of them give (aH,/aP)T as a function of temperature. The usual thermodynamic treatmentzb2shows that if the critical field is a function of pressure and temperature, a volume change must also exist a t the transition which is given by:
AT.’/V = (l’,, - v , ) / V ,
=
(aHC2/c3P)T/&.
(5.2)
This change in volume is of the order of 5 X for tin. The corresponding change in length a t the transition in a magnetic field, has been measured by Lazarew and Sud0vstov2~~ (for tin), by Olsen and his collaborators,z54and by Cody.2bbTheir results are in qualitative agree‘*O
G. J. Sicoo and H. Kamerlingh Onnes, Communa. Phys. Lub. Univ. Leiden No. 18Oc (1926).
M. D. Fiake, P h y ~and . Chem. Solids 2, 191 (1957). ”‘For a discussion of tho thermodynamics of euperconductors, see A. B. Pippard “Classical Thermodynamics,” pp. 18!J-145. Cambridge Univ. Press, London and New York, 1957; also, I). Shoenherg, “Superconductivity,” 2nd ed., pp. 73-77. Cambridge Univ. Preas, London and New York, 1952. 16*€3. G. Lszarew and A. I . Sudovstov, Doklady Akad. Nauk SSSR 00, 345 (1949). *m J. L. Olsen and H. Rohrer, Helv. Phys. Acta SO, 49 (1957). r r c G .D. Cody, Phy8. Reu. 111, 1078 (1957).
111
PHYSICS A T HIGH PRESSURE
ment with the pressure results, and show, in addition, that the ALIL values are quite anisotropic and depend strongly on the various crystallographic directions. Equation (5.2) must be modified to express this;
(Ltn - Le,J/Le., = ( a H 0 ' / * - 9 ) ~ / 8 ~
(5.3)
where 7 g is the stress in the direction 8. Grenier et a1.,2b6 previous to the work of Olsen et al., actually had observed this anisotropy in the variation
Fra. 'LO. Critical field curve for c*adniiumunder zero pressure (right) and 1550 J . Exptl. Theorel. Phys.
atiiios (left) [after N. 15. Alekseevskii and Yu. 1'. Gaidukov, ( U S S K ) 2@, 898 (1!)55); Sovirt f'hys. JK'1'p 2, 762 (l956)J.
of H , when tin and mercury single crystals were stressed elastically in various crystallographic directions. One of the striking results which has come out of the length change work, and which will be discussed later, is that ( d H , / & e ) , in thallium has opposite signs for directions parallel to and perpendicular to the C axis. The results obtained with the various techniques are in essential agreement, although there are disagreements on details. A typical curve **'C. Grenier, R. Spondlin, and C. F. Squire, Physica 19, 833 (1953); C. Grenier, Compt. rend. acad. sci. 288, 2300 (1954);240, 2302 (1955); 241, 1275 (1955).
112
C. A. SWENSON
representing the variation of the critical field with pressure is given in Fig. 20. This result is for cadmium and was obtained by Alekseevskii and Gaidukov,8eusing the ice-bomb technique. The temperature range is not Of the ela typical but these are the only results available below l0K.*&& ments for which data exist, only thallium (and possibly vanadium, and lanthanum) show a behavior which is different from this. Both T,(the transition temperature in zero field), and H O(the critical field a t absolute zero), BB well as the whole H,(T) curve, usually decrease with pressure. As will be explained further on in this section, data of high accuracy, similar to those represented in Fig. 20, are essential if one hopes for a comparison with theory. Also it will be pointed out that data of accuracy sufficient for this comparison do not exist a t present. (1) Theoretical. The theory of Bardeen, Cooper, and Schrieffer"' contains, in principle, the means for calculating the variation of the superconducting transition temperature with volume or pressure. Luthi and Rohrer,268Cody,*" and also Morel,26Dhave pointed out how a comparison can be made with theory. .4ccording to the theory,"' the transition temperature is given by
LT. = 1.14ho exp ( - l / N ( O ) A )
(5.4)
where w is a characteristic phonon frequency and is proportional to the Debye 8, N ( 0 ) is the density of states a t the Fermi level, and A is an average matrix element for the electron-electron interaction.26g"A contains two parts ; the first involving the electron-phonon interaction which is believed to be responsible for superconductivity, and the second involving ordinary screened Coulomb interactions. This latter term can, presumably, be calculated, and its volume dependence estimated. The density of states a t the Fermi level, however, cannot be calculated, and must be derived from experiment. To calculate tho variation of T,with volume, one writes: d In T, - d In BE) 1 d i n N(0) Id In A dinv d i n v +N(o) d i n v + A d l n v *
(5.5)
The first term is given by the Grueneisen constant, the other two must be obtained either from experiment or theory. It is possible to derive the 1-
Data on Al to 0.4"K are now available aleo: J. L. Oleen, Helv. Phya. A d o 89,310 (1959).
Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Reu. 108, 1175 (1957). B. Luthi and H. Rohrer, Hclv. Phys. A c h S1,294 (1958). 1~ P. Morel, Chcm. and Phyr. Solids 10, 277 (1959). ~m A is used here instead of the V which was used in the original publication in order to avoid confusion with the symbol for volume.
1"J. N*
PHYSICS AT HIGH PRESSURE
113
second term on the right from measurements of the critical field curve of superconductors as a function of temperature and pressure. Therefore the theory might be checked by calculating A and its volume dependence, and comparing them with the experimental values derived from Eq. (5.4) and (5.5). Unfortunately, the various terms are of the same order of magnitude and of opposite signs, YO that accurate data are necessary for this comparison. N ( 0 ) is related directly to the electronic specific heat of the normal metal by the r e l a t i ~ n s h i p : ~ ~ '
Cn/T =
7 =
2r2k2N(0)/3.
(5.6)
Since both y and d In r/d In V follow directly from critical field measurements as a function of temperature and pressure, N ( 0 ) and d In N(O)/d In V can be obtained from measurements on superconductors. To show this, we begin with the usual expression for the difference in theelectronic heat capacities of the superconducting and normal states : z b z
H,(T,V) is conveniently expressed in terms of a power series, n
n-2
where t = T / T , ( V ) . The expansion may be in terms of all powers of t greater than unity, or only in terms of even powers of 1, depending on the data. Usually 0 2 is negative, and of the order of unity, with the further requirement on the an that H,(T,) = 0. A t small values of t , C,. approaches zero as exp ( - l / T ) , while the normal electronic specific heat is proportional to T . Thus, in the limit as T approaches zero,
The so-called "similarity l 1 principle requires that all critical field curves have the same shape, and differ only by a constant factor which multiplies both Ho and T,. That is, the an are independent of the parameter which is being varied (which may be the average isotopic mass of the ions, for instance, or the volume), and the ratio of (Ho/T,) remains constant. This implies that y is dependent only on volume, and in a linear fashion, for the variations being considered. It has been shown that the similarity principle holds quite strictly
114
C. A. SWENSON
when changes in the average isotopic mass of tin are considered,26oand, hence, it can be concluded that the electronic specific heat per unit volume is unchanged when the average mass of the ions is varied. The postulate has been made that this superconducting similarity principle should hold also for changes in volume. It does not seem plausible, however, that the electronic heat capacity (or density of states) should be as simple a function of volume as this postulate would imply. In order to use critical field data to investigate the volume dependence of the electronic heat capacity, Eq. (5.9) can be differentiated with respect to volume to give: d 1 n L 1 + 2 - -d- In - - H2 O d In I/
dT-V
d In T ,
d In ( - a z )
dlnV+
dlnV
*
(5.10)
Measurements of ( ~ H , / ~ Pas ) Ta function of temperature, as well aa H , ( T ) at zero pressure, can be used to evaluate the right-hand side of this equation. To show this, the volume dependence of the critical field curve [Eq. (5.8)] can be written as: n
( a ~ e / a ~ )=T d
+1
~ o / (1 d ~
jntn)
(5.11a)
n-2
where the same values of n are used as for the critical field curve. For pressures below about 2000 atmos, the volume of most substances is linearly related to the pressure, and
Thus, it should be possible to describe the experimental data for (aH,/aP)r by a power series in t as in the foregoing. In order to evaluate (5.10), it is noted that the only term in (5.11) which contains (d In a z / d In 1.') is jz.A straight-forward substitution gives (5.12)
where (d In Hold In V ) = -(dHo/dP)/koHo, and lzo is the compressibility a t absolute zero. If the similarity principle holds, a2 = -f2, and d In r / d In V = 1. The foregoing discussion may be summarized as follows. Measure ments of the critical field curve as a function of both temperature and J. M. Lock, A. B. Pippard, and D. Shoenberg, Proc. Cambridge Phil. Soc. 47,811 (1951).
PHYSICS A T HIGH PRESSURE
115
pressure may be used to evaluate both the coefficient of the absolute temperature in the electronic specific heat (7)and its pressure (or volume) dependence.2ah If the critical field curves and their pressure dependence are expressed in the form of power series ((5.8)and (5.1l)],Eqs. (5.9) and (5.12) can be used to calculate both these quantities. The only important restriction is that the coefficients uz and fs must be evaluated near absolute zero where the effects of higher order terms are negligible. There is no a ptioti reason for believing that a simple quadratic dependence of (BH,/~~P)T on temperature can be used over the whole temperature range, although this type of dependence seems to fit the available data within experimental accuracy. A rough calculation of the magnitude of the effect of the third-order terms in the pressure dependence [using (5.11b) and assuming that the coefficients, u2 and u3 in the expression for the critical field curve all have roughly the same volume dependence] shows that neglect of the higher order terms may not be justified for lead, for instance, even for temperatures of the order of T J 2 . (2) Experimental results. Measurements of (aH,/aP)T have been published for ten superconductors. The data were obtained either from direct critical field measurements or from the change in length of the sample at the transition in a magnetic field. In the discussion that follows, data will be described as low pressure when liquid helium was used as a hydrostatic pressure transmitting medium; high pressure when the ice bomb, clamp, or solid hydrogen techniques were used; and AL/L when volume change methods were used. The specific high-presaure techniques have been described previously, and will not be discussed here. Where thermodynamic relationships are used in the calculation of (BH,/I~P)T (such as in the A L / L o or zero field measurements), care should be taken that the critical field curve which is assumed is that which corresponds to the sample which was used. The problem is not serious for “soft” superconductors (tin, indium, mercury, etc.), but may be critical for the impurity-sensitive “hard ” superconductors such as tantalum, vanadium, and niobium, which normally give broad transiThese quantities may be used to calculate directly the thermal expansion due to the electrons in the normal and superconducting metals. To show this, note that S,’ YT,and by a Maxwell relationship (Eq. 4.26),
-
Standard thermodynamic formulas for @,,a - B,*) can then be used to calculate B.’.llll Thb calculation has been made for tantalum [C. H. Hinrichs, “Effecta of Pressure on the Superconducting Propertiea of Tantalum,” M.S. Thesis, Iowa State University (1960), unpublished]. This application of the high-pressure data on superconductors was suggested independently by P. G. Klemens [Bull. Am. Phys. Soe. [2] 6, 164 (19SO)l.
116
C. A . SWENSON
tions. The lack of well-defined and reproducible sample states may be a partial reason for the wide spread in the data which is found for many superconductors. These effects seem to be more serious at large values of the magnetic field, as will be indicated in the following discussion. The AL/L data for anisotropic metals indicate a general behavior and are of qualitative interest; however, they cannot be used to calculate the values of ( ~ H , / ~ Punless ) T the orientation of the crystal is known, and measurements are made for different orientations. The anisotropic effects in hL/L which have been observed were often in polycrystalline samples in which the crystals had a preferred orientation. It is difficult to interpret the data on such samples quantitatively. Nevertheless, in the case of thallium in particular, the information derived from the data help to explain the pressure measurements which determine only a bulk volume effect.
Tin. This metal is the easiest with which to work, since it can be obtained in excellent purity, has a relatively high transition temperature (3.73"K), and is, technically, an ideal soft superconductor. Unfortunately, as the measurements of Grenier,266Olsen and R ~ h r e r , *and ~ ' Cody"' show, the pressure effect in tin is highly anisotropic, so that the comparison of the pressure and volume change measurements is difficult. The data obtained by various methods seem to agree within experimental error. It appears that the most accurate measurements are those of Garber and Mapother*@'and FiskeZs1on long single crystals at low pressures which give ( d H C / d P ) ~and , those of J and SB4(Fig. 21) which give T, as a function of pressure. There was some sample deformation (about 2%) in the latter investigation, and polycrystalline samples were used. The disagreement (about 8 %) between the low-pressure and highpressure data seems to be outside experimental error, and the reason for the discrepancy is not clear. In order to investigate this difference, the zero pressure critical field curve for tin was investigated as a function of sample shape, crystal size, and deformation at liquid helium temperatures.*@* The effects were small, and most likely did not contribute to the disagreement between the high- and low-pressure data. The data of J and S were of sufficient sensitivity so that curvature could be detected in the T , versus pressure curve for pressures above 2000 atmos. The curvature essentially disappeared when the T , data to 10,000 atmos were plotted against the volume estimated by taking into account its nonlinear pressure dependence. An extrapolation of these *el
M. Garber and D. E. Mapother, Phys. Rev. 91, 1065 (1954). W. Heaterman "Effecta of Deformation on Superconducting Metals," M. S. Thesis, Iowa State College (1958), unpublished.
*** V.
117
PFiYSICB AT HIQH PREBSURE
data pass through the transition temperature obtained by Chester and Jones*’ at 17,500 atmos. Three sets of data for (aH,/aP)T as a function of temperature exist. These include the low-pressure data of Fiske,26*and the ice-bomb dnta of Kan et aZ.,*Land Muench.*6aAll show scatter, and there is some uncertainty as to the nbsolute pressures in the ice-bomb measurements.
I
I
0
too0
1
1
I
1
I
f W
a
c a
ijl W
I-
4000
6000
8000
I0000
1OOOO
PRESSURE I ATMOS
FIG.21. The variation of the zero field superconducting transition temperature for tin aa a function of pressure [after L. n. Jennings and C. A. Swenson, Phys. Rev. 112, 31 (195S)l. Nevertheless, the agreement is close enough so that one can choose ( ~ H , / ~ P ) T ,= T ,-7.2( f0.4) and ( d H o / d P ) = -3.8( f0.4) (both in units of lo-* gausslatmos), with a quadratic temperature dependence of ( ~ N , / ~ P[Eq. ) T (5.1la)l.
Indium. This metal is also a “soft” superconductor with
T, = 3.40”K.It has many of the advantages (high purity, convenient ‘“N.L. Muench, Phys. Rev. 99, 1814 (1955).
118
C. A. SWENSON
T,)and disadvantages (anisotropic) of tin. Single crystal data obtained by CodyZLLsnd RohrerZo4are in essential agreement with the highpressure critical fie!d data of MuenchZs3and the zero field data of Hatton210and J and S . 8 4 The effect of uniaxial stress (Eq. (5.3)] is very anisotropic, being zero perpendicular to the tetragonal axis at all temperatures. E h l y results given by Karl ct u L . , * ~ are perhaps not reliable, since they lead to an excessive value for ( a H , / a T ) p a t T,, and exhibit thermodynamic inconsistencies. If these results are excepted, indium is perhaps the only metal for which there are no serious disagreements between the various published results. Lead. This metal is difficult to work with because of its high zero field transition temperature (T, = 7.18"K). It is, however, face-centered ) T be calculat,ed as a funccubic in crystal structure, so that ( ~ H , / ~ Pcan tion of temperature directly from single crystal or polycrystalline U / L data. There is excellent agreement between the low pressure, directly gauss/atmos) as given measured, value of (aH,/aP)T, (-9.95 X by Hake and hlapother,*66and the value obtained by both Olsen and RohrerlzS4 and Cody266 from an extrapolation to T,. IJnfortunately, the ratio of this quantity a t absolute zero to its value at T , as given by Olsen and Rohrer is 0.57 a s compared with the value of 0.83 given by Cody. The reason for this discrepancy is not known; however, more recent work on the critical field curve of lead by Mapother and his co-workersz'6suggests that below 4"K, unless great care is taken in sample preparation, nonideal transitions are the rule for lead , rather than the exception. Thus, these A L / L data may be characteristic of the particular samples used, and not lead in general. Mercury. Mercury is also a soft superconductor, with a transition temperature of 4.15"K. Its normally rhombohedra1 structure leads to anisotropic effects.26oThe various measurements are somewhat in agreement at T,, with the low-pressure data of Fiskez6*giving a substantially ~ the high-pressure work of J and S.84*26" higher value for ( i l H , / a P ) than Fiske's work extended over a range of temperature, but showed considerable scatter. The high-pressure work is complicated by the agpearance of a new crystalline modification"' (Section 4b), the transition in these superconducting experiments being only partially complete at m H.
Rohrer, Phil. Mag. 4, 1207 (1959).
*@'R.R. Hake and D. E. Mapother, Phys. and Chem. Solick 1, 199 (1956). *" D. L. Decker, D. E. Mapother, and R. W. Shsw, Phys. Rev. 112, 1888 (1958). Recent work on eingle crystals of mercury gives good agreement with J and S. [J. L. Olsen, Communication to the IUYAP Superconductivity Conference, Cambridge, England (July, 1959), unpubliahed.]
PHYSIC8 A T HIGH PREf38URE
119
10,OOO atmos. The new phase has been shown to have a zero field transition temperature of 3.94OK,'@4and a body-centered tetragonal structure.*QbThe difference in the transition temperatures of the two modifications can be explained quantitatively in terms of the known difference in density between the two modifications and the pressure dependence of the zero field transition temperature of the normal m~dification.'~~ It would be of interest to see if the critical fields of the two modifications possess the same pressure dependence.
L
Tantalum. The data on this metal, although quite extensive, are perhaps the most confusing of any that exist. The transition temperature in zero field is about 4.4% and the crystal structure is body-centered cubic, both properties being ideal for experimental work. Unfortunately, tantalum belongs to the class of "hard" superconductors, and its properties, especially in a magnetic field, are very sensitive to purity, with dissolved gases being of unknown (but possibly of primary) importance. The AL/L data of Olsen and Rohrer,*S4and Cody,2ss as well as the lowpressure ( I ~ H , / ~ PdataIza1 )T were obtained from magnetic transitions a t constant temperature, with samples which differed widely in annealing and degassing treatment. The results are essentially in quantitative ~'~ and J and S,8' were all taken agreement. The data of H a t t ~ n ,Bowen,Q6 in zero (or the earth's) magnetic field over a wider range of pressures, with results which are also in substantial agreement, but a value of (BHC/t3P)~ is obtained which is roughly a factor of four less than that which is given by the magnetic data. In order to investigate this discrepancy further, the original zero field data have been supplemented by other high-pressure data (obtained with the apparatus and method of J and S) using various tantalum These additional samples included a portion of one of Fiske's samples (a. = 4.369"K), aome tantalum dendrites (T,= 4.451°K), and a wire of highly purified tantalum (T,= 4.481"K) which exhibited ideal magnetic transitions in a separate experiment. The results were independent of the value of To or the widths of the zero field transitions (up to 0.02" waa observed for some of the samples), and the data could be represented in each case by a straight line of slope d. Tc = -2.65( k0.l) X 10-a deg/atmos. dP ~-
From this, a value of ( ~ H , / ~ P=) T-0.90( kO.04) X gauss/atmos can be calculated, using an initial value of ( a H , / a T ) p = 340 gauss/deg which was measured for the high-purity sample.
*"
C. A. Swenaon, unpublished results (1959); also Communication to the IUPAP Superconductivity Conference, Cambridge, England (July 1959), uupublished.
120
C. A. SWENSON
The remarkable insensitivity of (dT,/dP) to sample purity and to T, (the initial J and S sample gave T , = 4.30"K) suggests that the cause of the discrepancy between the zero field measurements and the magnetic measurements lies in the nonideal behavior of the transitions which was commented on in all the reports on the magnetic experiments. I t is difficult to understand how the sign reversal of the pressure effect for tantalum which was observed by Olsen and RohrerZb4at 3.5"K can be explained in terms of sample inhomogeneity or impurity. The use of samples which exhibit ideal transition behavior would assist in clearing up this problem. Recent experiments by Olsen and Rohrer on highpurity samples give results which are much closer to the high-pressure results, but the agreement is still not good.2e7a Thallium. Early data on thallium were quite contradictory, in that the low-pressure work showed an anomalous increase of transition temperature with pressure,261,*b8 and the high-pressure data showed a normal decrea~e.~' Hatton21B*26g was the first to obtain data which actually showed that the sign of (d!l',/dP) reverses at a pressure of a few thousand atmospheres. Later data have confirmed this result; a curve given by J and S8'is shown in Fig. 22. The accuracy in these data is limited by the uncertainties in the relatively small temperature shifts which were observed, but the data on two separate samples are in quantitative agreement. These samples were deformed by about six per cent in the course of the measurements, and it is not known whether or not this had a serious effect. Most of the deformation occurred at the initial compression, however, so that the state of the samples probably remained relatively constant while the data which are shown were taken. The data in Fig. 22 extrapolate to the value of transition temperature shift (- .W0) which was obtained by Chester and Jonesn7for 13,400atmos. In this case, the AL/L data have offered a clue to a qualitative understanding of the T o - P behavior. Both (%en and RohrerlZs4and CodylZrs find that the change in length associated with the transition has different signs for the directions parallel to and perpendicular to the hexagonal c axis. Thus, since the pressure effects must depend on an averaging over all crystallographic directions, the observed sign of ( ~ H J ~ P )can T *('a
J. L. Olsen, Private communication (1059). Recent experiments at Iowa State
(see reference to Hinrichs in footnote 260a) which used solid helium at 2000 etmos to determine (aH,/aP)T to l.l°K for very pure tantalum are in agreement with the J and S zero field results quoted above. They show ( a H , / a P ) T roughly independent of temperature. aa'L. S. Kan, B. C . Lazarew, and A. I. Sudovtsov, Doklady Akad. Nauk SSSR 69, 173 (1949). *(O J. Hatton, Phye. Rev. 100, 1784 (1955).
121
PHYSICS A T HIGH PHENSCRE
depend on the relative magnitudes of the ( d H , / d r e ) ~values. If at low pressure, ( ~ ~ H J ~ Twhich ~ ) T ,is positive, is the dominant term, then ( ~ H , / ~ Pwill ) T be positive. But, if an increase of pressure should cause ( a H , / a r , , ) ~which , is negative, to increase more rapidly than ( ~ H , / ~ T ~ ) T , then the slope of the T, versus P curve can change sign and become negative, also. This is undoubtedly the case for thallium. It is unfortunate, that, while the low pressure and AL/L data at T, agree quite well, there is considerable disagreement in the AL/L data at I
2.4
XK)"
I
I
OEO /ATNOS
1
&%o
Y
I
Cb
I
\,
eO00
4000 PRESSURE, ATYOS
FIQ.22. The pressure dependence of the zero field transition temperature of thallium [after L. D. Jennings and C. A. Swenson, Phys. Rev. 112, 31 (195S)l.
lower temperatures. Olsen and Rohrer give dHo/dP = 0, while Cody's data indicate that it should be roughly equal to its value at T,. The r w o n for the discrepancy is not clear. However, the length changes are quite small and they cannot be measured conveniently over a wide range of 1 = T/T,because of the relatively low transition temperature (2.4'K).
Aluminum. For aluminum MuenchZescalculates d- Te = dP
-2.0( It 0.2) X lowbdeg/atmos
from magnetic ice-bomb data. The transition temperature for aluminum is 80 low (1.2'K) that adiabatic demagnetization techniques or a liquid
122
C. A . SWENSON
helium 3 cryostat are necessary to obtain the temperature dependence of the pressure effect. Recently Olsen has used the ice-bomb technique to obtain data on aluminum down to 0.4"K which are in essential agreement with Muench's data nenr Tc.2Kot
Vanadium. Miiller arid I ~ o l i r c ~hnve ~ ~ " nieanured A L / I , for this cubic superconductor, and find that it is of opposite sign from that of all other metals, except for thallium perpendicular to the c axis, and lanthanum.26d"Thus, (aH,/aP)T is also positive. No pressure effect data exist, probably because of the high transition temperature (5.4'K). Vanadium, like tantalum, is a "hard " superconductor; thus it is difficult to obtain well-defined magnetic transitions. Certainly, the results are intriguing and vanadium should be investigated with other techniques under well-controlled sample conditions. Cadmium. These data have been mentioned previously, and are plotted in Fig. 20. Outside of the difficulty of determining absolute temperatures accurately in the temperature region below 0.5'K, the data are ideal in that they extend over a range of t from 0.1 to 1. The H,(T,P) curves at 0 and 1550 atmos are parabolic in t, so the derived (aH,/aP)T curve is also strictly parabolic in temperature. Bismuth. This metal is not normally a superconductor, but many of its compounds are. However, Chester and Jonesa7have found that the element becomes superconducting at pressures from 20,000 to 41,000 atmos with T, = 7°K. Since these pressures were applied at room temperature, it was suggested that one of the Bi phases which exists at room temperature below 30,000 atmos (Fig. 17) was frozen in upon cooling to liquid helium temperatures. Hall effect datazz2show that Bi I11 is more metallic in nature than the lower pressure Bi I and Bi I1 phases. However, more data are needed to establish the true source of the superconductivity. Compounds and alloys. The ice-bomb technique has been used to investigate various compounds for the influence of pressure on superconducting transition temperatures. In particular, it has been found that BiBNiand Bi4RhZ7'as well as Bi2K,27z show an anomalous increase of T, with pressure, while AuzBi (which has the same structure and lattice constants aa BizK), PbT1271and BiLi*" exhibit normal behavior. z70
z7l *7*
J. Miiller and H. Rohrer, Helu. Phys. Acto S1, 289 (1958). N. E. Alekseevskii, J . Ezpll. Theorel. Phys. (USSR) 19,358 (1949). N. E. Alekseevskii and N. B. Brandt, J . Expll. Thoref. Phye. (USSR) PP, #w) (1962).
*'IN. E. Alekseevskii, N. B. Brandt, and T. I. Kostina, Zzvert. Akud. Nouk SSSR, Ser.Fir. 18, 233 (1952).
PHYSICS AT HIGH P R E S S U R E
123
It has been concluded that those compounds tend to show anomalous behavior because of an excess of bismuth. In addition, the pressure effect in the two modifications of BisPdz7‘ has been studied. The a phase is monoclinic, with T , = 1.70°K and (dT,/dP) = -2.5 X deg/atmos. The 0 phase is tetragonal with T, = 425°K and (dT,/dP) = -5.6 X lo-& deg/atmos. This latter figure seems to be dependent on deformation to some extent, and values as low as -3.6 X lob6have been observed. The dependence of the pressure effect a t zero field on impurity concentration was determined in tin-indium alloys of low indium concentration (up to 3%).276Very little impurity effect was found, the results for the alloys essentially being the same as those for pure tin to within experimental accuracy (about one per cent). (3) Summary. The foregoing survey of high-pressure data on superconductors indicates that the order of magnitude of the effects is known, as well as their complexity. The disagreement on details emphasizes the need for single crystal U / L O data of the type which exist for indium, and the need for data on well-defined and carefully prepared samples of the cubic “hard ” superconductors. Olsen and his coworker^^^^-^^^ have published summaries of their data (weighted by that obtained by other methods) in which they calculate the volume dependence of both y and the BCS interaction term. The range in the numbers which they obtain is quite large. Because the reliability of the data which was used is unknown at present, no attempt will be made to summarize these calculations here. Nevertheless, it is important to note that these pressure (and volume change) experiments on superconductors can give data on the variation with volume of the electronic density of states which cannot be obtained in any other way.
c. Magnetic Properties (Including Resonance Work) A study of the variation with pressure of the magnetic properties of highly magnetic materials presents various difficulties. It would be of interest to know the variation of both the Curie temperature and the saturation moment with pressure. Since the Curie temperatures of most substances are quite high it is necessary to use special techniques. The measurement of saturation moments can be done at normal temperatures (although low-temperature measurements would be useful) , but since these measurements usually require external magnetic fields it is necessary to use nonmagnetic metals in the pressure vessels. Austenitic stainless steels and hardened beryllium copper are acceptable materials, but lrcN.
E. Alekseevskii and I. I. Lifvanov, J . E x p l l . Theoret. Phyu. (USSR)80, 405
(1956); Soviet Phys., J E T P 8, 294 (1956).
C. H. Hinricha and C. A. Swemn, to be published.
124
C. A. SWENSON
the pressure region in which they can be used is limited to a maximum of about 10,000 atmos. The major systematic study of the effects of pressure on the Curie He used an alternating current temperature was made by method with the metal specimen forming part of a transformer which was immersed in the high-pressure fluid. This method, although of doubtful validity for accurate Curie point determinations, was considered satisfactory for the measurement of changes in Curie temperatures. At low temperatures (below 400°C) a liquid system was used with external heating, while at high temperatures (to 1 lOO"C), internal heating w a ~ used with argon as the pressure transmitter. The pressure range in each case was up to 8000 atmos. Patrick's low temperature data are reproduced in Fig. 23, which shows the variation with pressure of the Curie temperature for various substances in the range from 10°C (Gd) to 360°C (Xi), The high-temperature data (on Fe, Co, and two Fe-Si and a Xi-Fe alloys) showed considerable scatter and are not reproduced here. Even the agreement with qualitative theoretical predictions does not seem to be good, although S m o l u ~ h o w s k istates ~ ~ ~ that the change for nickel is about what would be expected. S t a ~ e has y ~ used ~ ~ a technique which gives only approximate results to measure the variation of magnetic saturation with pressure. Thin samples in the form of wafers were placed between truncated magnetic pole tips of a magnet and approximately hydrostatic pressures of up to 10,OOO atmos were obtained by squeezing the sample and using the magnet as an integral part of the press (Fig. 3c). Stacey's results contradict previous data, and more recent work have shown them to be incorrect. Both von Klitzing and G i e l e s ~ e n , and * ~ ~ Gugan,*80have made measurements using truly hydrostatic pressures, and show that in contrast with Staceys results, which gave a 12% change in saturation moment, pressure has essentially no effect on the saturation moment for nickel. The reason for the error in Stacey's measurements is not known, but it may be in the pressure generation. This is indicated since Gugan used the same magnetic techniques to measure magnetic moments and the reliability of his measurements on nickel, various copper nickel alloys, and cobalt is confirmed by the excellent thermodynamic agreement with other low-pressure data on forced magnetostriction. Gugan and Row1"L. Patrick, Phys. Rev. 93,384 (1954). a7' R. Smoluchowski, Phys. Rev. 93, 392 (1954). 17aG. 0. Jones and F. D. Stacey, Proc. Phys. SOC.B66, 266 (1953); F. D. S b e y , Can. J . Phys. 34, 304 (1956). 179K. H. von Klitaing and J. Gielessen, 2.Physik 146, 59 (1956). *m D. Gugan, Proc. Phys. Soe. 72, 1013 (1958).
PHYSICS A T HIGH PREBSURE
125
lands2*1have used a semiempirical analysis of some low field data which were obtained in these experiments to give the pressure dependence of the ferromagnetic anisotropy for theBe same substances.
FIQ.23. The variation of the Curie temperatures of various metals with pressure [afterL. Patrick, Phys. Rev. 93, 384 (1954)l.
The foregoing measurements were all made at room temperature and above. At lower temperatures, Gal'perin, Larin, and Shishkov"* have measured the effect of 2000 atmos pressure on the saturation of iron at *" D.Gugan and G . Rowlands, Proc. Phye. SOC.72, 207 (1958). la*F.M. Gal'perin, S. Larin, and A. Shishkov, Doklady Acad. Nauk SSSR 89, 419 (1 953).
126
C.
A. SWENSON
liquid nitrogen temperatures. Kondorskij and S e d ~ v * have ~ * measured both the saturation magnetization and resistivity of Fe-Ni alloys at high pressures and in strong fields near absolute zero. The variation of eaturation moment with pressure is in disagreement with theory, and the resistivity data presumably give some clue that electronic changes are taking place. The study of the effects of pressure on nuclear and electronic paramagnetism represents a fairly new extension of high-pressure work which has been made possible by the existence of the completely nonmagnetic beryllium copper alloys. Benedek and P ~ r c e l l , *who ~ ' did the first work of this kind, used the spin-echo technique to measure the effect of 10,OOO atmos pressure on the proton relaxation time TI in water and five organic liquids. T Iis closely related to the viscosity of the fluid, and the pressure was used as a tool for increasing the viscosity at constant temperature. Although simple theory leads to the result that Tlq (where rl is the vie cosity) should depend only on the absolute temperature, this is not confirmed by the experiments. According to Benedek and Purcell, the result observed presumably indicates that the decrease in free volume of the liquid hinders the migration of a molecule ( q ) far more than its ability Where nonequivalent nuclei do not exist, the spin+cho to rotate (T,). technique can also be used to determine self-diffusion constants BS a function of pressure, and this was done for water and methyl iodide. High-pressure nuclear resonance techniques also can be used to study diffusion in solids, and the work of Smith and Squirez8' on solid hydrogen will be discussed briefly at the end of this section. Barnes, Engardt, and Hultschzs6have determined the activation volume for self-diffusion in lithium at room temperature from the pressure dependence to 3000 atmoe of the spin-spin relaxation time, T;, of the Li, nucleus in dispersed lithium. The resonance technique would seem to be the only one for obtaining these data since no radioactive traces for lithium exists. The pure quadrupole resonance of C136in paradichlorobenzene was measured as a function of pressure by Dautreppe and Dreyfus*87 and as a function of temperature and pressure to 10,000 atmos by Kushida, Benedek, and Bloembergen.80Kushida et al. have made similar investigations for C136in KCIOI, as well as for the C U ' ~nucleus in CuzO. They analyzed their data, and deduced the volume dependence of the electric 18*
E. I. Kondorskij and V. L. Sedov, J . Ezpll. Theoret. Phys. ( U S S R ) 36,845 (1958); Soviet Phys., JETP 8, 586 (1958). G. B. Benedek and E. M. Purcell, J . Chem. Phys. 22,2003 (1954). G . W. Smith and C. F. Squire, Phys. Rev. 111, 188 (1958). R. G. Barnes, ft. D. Engardt, and R. A. Hultsch, Phys. Rev. Lellets 2, 202 (1959). D. Dautreppe and B. DreyfuB, Compt. rend. acad. sci. 241, 795 (1955).
127
PHYSICS AT HIGH PRESSURE
field gradient a t the nucleus with the help of a simplified equation of state. They have also used the resonance to detect the various phase changes in paradichlorobenzene and have discovered a new modification of this substance. Kushida and Benedek"8 also have measured the pressure dependence of the pure quadrupole resonance in metallic gallium to 8000 atmos at I
Vp.0
* 7,388,542
I
1
I
I
I
-
CP8
-
-
-
.D
-c
-40
-
-
-60W 0
= -70-
B K
-80-
3 W
-
-90
(3
-100
0
-110
\ L-
I
0
I
2000
I
I
4000
1
I
6000
I
1
8OOO
I
J
lop00
FIG. 24. The pressure dependence of the Na**resonance frequency at 24.4OC (after G . B. Benedek and T. Kushida. Phvr. and Chem. Solida 6,241 (195S)l.
- 75°C. While ( a v / a P ) decreases ~ with temperature, no detailed analysis was given because of difficulties with the anisotropy of the metal. Probably the most accurate resonance experiments which have been made a t high pressures are those of Benedek and KushidaZa9in which the pressure dependence of the Knight shift ww measured for the alkali metals, Li, Na, Rb, and Cs. The data for sodium are given in Fig. 24. OOC, -29.8OC and
ye
Kushida and G. B. Benedek, Bull. Am. Phys. Soc. 121 3, 167 (1958). 0. B. Benedek and T. Kushida, Phys. and Chcm. Solids 6,241 (1958).
128
C. A . 8 W E N 8 0 N
Thege data were combined with theoretical work by Pines to obtain information on to the electronic structure of the metals. There is fairly good agreement between the results of the analysis and of the direct calculations. From independent zero pressure data, they also concluded that the Knight shift contains an explicit temperature dependence ~8 well as a volume dependence; their calculation resembles that for the compressibility [Eq. (4.11)]. The realm of high-pressure resonance work also has been extended into the microwave region by Walsh and Bloembergen,29o who measured the effect of 10,000 atmos a t room temperature on tshe electron spin resonance of nickel fluosilicate. Their results, which indicated a change in sign of the zero field splitting with increase of pressure, were in qualitative agreement with zero pressure results in the conclusion that the splitting is strorigly volume-dependent. This is presumably due to the anisotropy of the crystalline fields. Kushida and BenedekZQ1have reported nuclear resonance work on antiferromagnetic MnF2. They determined the variation of the NBel point with pressure using pressures up to lo00 atmos a t 35.7"K. Kaminov et aLZg2have reported on ferrimagnetic resonance experiments with yttrium iron garnet to 10,OOOatmos at room temperature. These represent extensions of high-pressure resonance techniques to new fields. Interest in high-pressure nuclear resonance experiments on hydrogen at low temperature was stimulated by a suggestion by F. LondonZegs that it should be possible, because of their asymmetrical shape, to align the ortho molecules in solid normal hydrogen by application of pressure. The alignment could be studied by means of the nuclear susceptibility of the protons in these molecules. While this phenomenon has not been observed as yet, two separate nuclear resonance experiments have been done with solid hydrogen under pressure. At relatively high temperatures (10°K) the line width of the resonance signal narrows with increltsing temperature due to the increasing import,ance of self-diffusion in decreasing the relaxation time. Smith and Squirezss have studied this line-width transition as a function of pressure to roughly 230 atmos, and have observed a shift of the line-narrowing region from 10°K to about 13'K even though the pressure range was not large. They estimate from their data the activation energy for self-diffusion as a function of pres190
W. M. Walsh, Jr., and N. Bloernbergen, Phys. Rev. 107, 904 (1957); see ~ B W. O M. Waleh, Jr., Phye. Rev. 114, 1473, 1485 (1959). T. Kushids and G. B. Benedek, Bull. Am. Phys. Soc. 121 4, 183 (1959); Phys. Rcv. 118, 46 (1960).
I. P. Kaminov, W. Paul, and R. V. Jonea, Bull. Am. Phys. Soe. (21 4, 177 (1959). F. London, Phye. Rev. 102, 168 (1956).
lo*
PHYSICS AT HIGH PRESSURE
129
sure, and obtain values varying from 350 cal/mole at zero pressure to 560 cal/mole at 230 atmos. These data for the activation energy could,
presumably, have been used also to calculate the activation volume for diffusion. At much lower temperatures (about 1.6"K) there is a second change in the nuclear resonance signal due to the removal of the threefold rotational degeneracy which normally exists in the ground state of the orthohydrogen molecule. Smith and Squire could observe no shift in the temperature of this transition with the pressures available to them, but Fairbank and McCormicklol have reported a shift of the transition to higher temperatures with the application of pressures of the order of 4000 atmos. Although this transition has been studied calorimetrically at zero pressure, the nuclear resonance work is particularly valuable since it offers the only possibility for studying how the beginning of the degeneracy disappearance depends on volume.
d. Dielectrics and Ferroelectrics The study of the variation of the dielectric constants of solids with pressure has not received as much attention as has similar work with gases. Mayburgzs4has measured the change in the dielectric constant at lo00 cps with 8OOO atmos applied at room temperature for a representative group of simple dielectrics, which included MgO, LiF, NaCl, KCl, and KBr. The pressure range in this work was limited by the failure at higher pressures of the special, low capacitance electrode which was used. In general, the dielectric constant decreased with pressure more rapidly than would be expected from the volume changes, implying a reduction in the internal fields with increasing pressure. A comparison of the high-pressure data with zero pressure data at various temperatures showed that the dielectric constant cannot be considered to be a function only of volume, but must contain an explicit temperature dependence. R a o ' ~ objection *~~ to Mayburg's interpretation of the data was shown by MayburgZQ6 to arise from conflicting definitions of the compressibility. ReitzelZQ7 has determined the effect of pressure on the dielectric constant of vitreous silica to 4000 atmos, and also finds a decrease in polarizability (as defined by the Clausius-Mosotti equation) with pressure. Lynch and ParsonsZs8have given similar data for polythene to 600 atmos. Barium titanate has been the most studied of the ferroelectrics. Vul S. Mayburg, Phys. Rev. 79, 375 (1!f50). A. S. N. Itno, Phys. Rev. 82, 118 (1!151). **( S. Mayburg, Phye. Rev. 85, 1072 (1951). J. Reitzel, Nature 178, 940 (1956). A' C. Lynch and P. 1,. Parsons, Nalurc. 179, 686 (1957) 294
a*s 1). A.
130
C. A. SWENSON
and VereshchaginZ9¶have measured the variation with 2500 atmos pressure of the capacitance of a condenser containing barium titanate as B dielectric. Merz*O0has published data on the variation of the dielectric constant and Curie temperature of this substance with pressure. The Curio temperature was decreased at pressures of thc order of 2500 atmos. Forsberghaolsubsequently showed that ti two-dimensional pressure had the opposite effect, although there are theoreticala0*and experimentalaoa reasons for believing that his results were due to an alignment of the domains by stress. Kozlobaevso4 and Shirane and Satoaos also have studied effects of pressure on barium titanate, and the latter workers also obtained data on polycrystalline barium strontium titanate. Merzao6 has measured the effect of hydrostatic pressure on the hysteresis loop in guanidine aluminum sulfate hexahydrate to 5OOO atmos at room temperature. Also there has been some interest in the effects of pressure on the piezoelectric properties of quartz crystals. Michels and Perezao7measured the frequency changes due to loo0 atmos pressure for A T and BT cut crystals, while Perez and Johannin*O*worked to 5000 atmos with Y cut crystals. In both cases the frequency changes which were observed were of the order of a few parts in 10' per atmos, and were presumably due to changes in the elastic moduli. The suggestion is made that since resonances remain sharp under pressure, the frequency shift should otrer a means for measuring pressure. Susse has analyzed changes in the frequencies of quartz crystals of various cuts, and has determined the changes in the elastic moduli of crystal quartz with loo0 atmos pressure. Susse8Ogfinds that the variations in these moduli (+ for c44, for c66 and c14) are of the same order of magnitude as the variations in the frequencies.
-
6. OPTICALMEASIJREMENTY Techniques for the study of the effects of pressure on the optical properties of both fluids and solids have been developed in several laboratories. Usually the objective of this work has been to study molecuB. M. Vul and I,. F. Vereshchagin, Compl. rend. acad. sci. URSS 48, 634 (1945). W. J. Merz, Phys. Rev. 78, 52 (1950). aol P. W. Fombergh, Jr., Phys. Rev. 98, 686 (1954). lo' E. Sawaguchi, Busseiron Kenkyu 74, 27 (1954). H. Jaffe, D. Berlincourt, and J. M. McKee, Phys. Rev. 106, 57 (1957). )04 1. P. Kozlobaev, Doklady Akad. Nauk SSSR 104,387 (1955). )06 G. Shirane and K. Sato, J . Phys. Soc. Japan 6, 20 (1951). W. J. Merz, Phys. Rev. 103, 565 (1056). A. Michels and J. 1'. Perez, I'hysica 17, 563 (1951). *08 J. P. Perez and 1'. Johannin, J . phys. radium lS, 428 (1952). C. Susse, J . phys. radium 16, 348 (1955).
aoo
PHYAICR AT I f I G H PRESSURE
131
lar interactions through the u5e of absorption spectra. Also, as WBB mentioned previously, there is considerable interest in the effects of pressure on the absorption edge of semiconductors because of the need for information about the energy gap and its pressure dependence. Most workers have used the Poulter seal (Fig. 2)s10in conjunction with s a p phire windows for moderate pressures (below 10,OOO atmos), although Hughes and Robertsona7have used an 0 ring seal with such windows up to 6OOO atmos pressure. Fishman and Drickamer3&have described in some detail their techniques for the construction of windows, and Parsons and Drickamer3" have discussed attempts to use XaC1 and CaFz windows in a similar manner. The optical work which has been done at high pressures falls rather naturally into three categories according to the type of substance investigated: gases, organic fluids, and solids (mostly alkali halides and semiconductors). Robin and Vodar have described an apparatus for studying the spectra of gases to 6000 atrnos in which, in order to reduce contamination,(*only thermal compression is used. The optical cell is similar to that described by Robin and for use with internal heating techniques at temperatures to 1100°C and pressures to 1500 atmos. Vodar, Robin, and their collaborators have published the results of several studies on the effects of gas (argon or nitrogen) pressure on reaonance spectra (potassium and mercury),31athe effects of 5000 atmos preasure on two of the bands in oxygen gas,314and the effects of pressure on the spectra in gaseous mixtures. a16.a10 Hare and Welsh3" have investigated the pressure-induced infrared absorption of pure hydrogen and mixtures of other gases with hydrogen at pressures up to 5000 atmos. This paper is quite useful for its technical information, and it also contains numerous references to earlier work. May, Stryland, and Welsh also have investigated the Raman spectra of Hz and CH, as a function of pressure to densities approaching the liquid density."'" Drickamer and his collaborators have studied the effects of pressure on the absorption spectra of various orgacic liquids, all to 10,OOOatmos C. Poulter, Phys. Rev. 86, 297 (1930).
nlR.W. Pamom and H. G. Drickamer, J . Opt. Soe. A m . 46,464 (1956). J. Robin and S. Robin, J . phy8. radium 17, 499 (1956).
J. Robin and R. Vodar, Compt. rend. acad. an'. 242, 2330 (1956). #I4 J. Robin, M. Desmaret, and F. Ubelmann, Compt. rend. mad. sci. 043,1750 (1956). (I'R. Coulon, L. Galatry, J. Robin, and B. Vodar Diaczcssias Faraday Soc. 44, 22
(1956). II'J.
Robin, R. Bergeon, L. Galatry, and B. Vodar Discussions Faraday SOC.PP, 30
(1956).
W.F.J. Hare and H. L. Welsh, Can. J . Phys. 96,88 (1958). rlaA. D. May, J. C. Stryland, and H. L. Welsh, J . Chem. Phys. 80, 1099 (1959).
)"
132
C. A. SWENSON
near room temperature. al+sal These experiments used techniques which were discussed in the previously mentioned paper. Robertson, Babb, and their co-workers have used the apparatus of Hughes and Robertson” to investigate the near ultrnviolet absorption of pure benzene and benzene diluted by various gases to GO00 a t r n ~ s . ~ ? ~ All of the work on the optical properties of solids (except for that done on semiconductors below 10,000 ntmos) has involved the alkali (or silver) halides in one form or another. These compounds have been preferred because the theoretical interpretation of the results is simple and they are quite plastic under pressure. Jacobsazasurveyed the effect of 8000 atmos pressure on F-center absorption in seven alkali halidea (NaCl, NaBr, KCI, KBr, KI, RbC1, CsCl), and found fair agreement between theory and experiment. Reiffe1s24studied the decay with time of the fluorescent and phosphorescent emission from Nal(T1) crystals which had been irradiated with y rays. He concluded that the rather large pressure effects observed, with pressures up to 2400 atmos, were due to significant changes in trap depths. The more conventional hydrostatic techniques do not work for pressures above 10,OOO atrnos, and methods which give only approximate results must be used. R u n c ~ r has n ~ used ~ ~ transparent (sapphire) pistons in a “simple squeezer” to study the ultraviolet absorption edge of a thin slice of olivine t o 30,000 atmos; his results are consistent with those of Hughesz4*on the pressure and temperature dependence of the conductivity of this ionic semiconductor. The method, although simple, seems to have been somewhat difficult to apply, and the results of only one run are Fitch, Slykhouse, and Drickameraz6have described two more complicated sets of apparatus by which optical studies may be made to either 50,060 or 200,000 atmos. In both of these an alkali halide (usuJly NaCl) is used as a pressure transmitter. The alkali halides are especially E. Fishman and H. G. Drickamcr, J . Chcm. Phys. 24, 548 (1956). A. M. Benson and H. C. Drickamer, Discussions Faraday SOC.22, 39 (1956). 8’’ R. R. Wiederkehr and H. G . Drkkamer, J . Chem. Phys. 18, 311 (1958). W. W. Robertson, S. E. Babb, Jr., and F. A. Matsen, J . Chem. Phys. 36,367 (1957); W. W. Robertson and S. E. Babb, Jr., ibid. 26,953 (1958); S. E. Babb, Jr., J. M. Robinson, and W. Robertson, ibid. 30, 427 (1959). aaaI.S. Jacobs, Phys. tlev. 93, 994 (1954). aZ4L.Reiffel, Phys. Rev. S4, 856 (1954). s*b A deacription of infrared work with a diamond press has recently been publihed by C. E. Weir, E. R. Lippincott, A. vsn Valkenberg, and E. W. Bunting, J. Research Natl. Bur. Standards 63A, 55 (1959). a*s R. A. Fitch, T. E. Slykhouse, and H. G . Drickamer, J . Opt. Soc. Am. 47, 1015 *lo
(1957).
strtidactory in this application sirtcc: the solid is an excellent transmitting fluid and is transparent at these pressures; it is also very viscous and will not extrude (after appropriate seasoning) through reasonably sized openings. A sketch of the 50,000 atmos apparatus is shown in Fig. 25. The samples are mounted as thin sections normal to the centerline of the
G C
Flu. 25. An apparatus whirh has been uvcd IR. A. Fitch, T. E. Slykhouse, and Am. 47, 1015 (1957)) for optical measurements to 50,OOO atmos. The dark regions S represent salt windows which are held in place by the retaining nuts P, and which support the high internal pressure by the high viscosity of the salt. The sample to be studied L surrounded by or dissolved in the same or another salt R which forms the experimental chamber (%in. diameter). The cell is placed in a press where force is applied to the steel jacketed Carboloy pistons C . The Carboloy cylinder A is a h supported by a shrunk-on steel jacket B, and the retaining nuts G .
H. G. Drickamer, J. O p f . Soc.
windows. With these techniques, Drickamer and his collaborators have studied the effect of pressure on the spectra of various ions dissolved in alkali halide lattices (CII;-,sz6 TI+**’) as well as on color centers in the alkali halides. *** Slykhouse and Drickamer also have determined the preasure dependence to 200,000atmos of the absorption edge in silicon as well as in the silver halides, AgCl, AgBr, and ‘*IT. E. Slykhouse and H. G. Drickamer, J. Chem. Phyu. 27,
1226 (1957).
’:’ R. A. Eppler and H. G. Ikickamer, Phys. and Chem. Solids 6, 180 (1958). W. G. Maisch and H. G. Drickamer, Phys. and Chem. S o l i L 6, 328 (1958). ::*T.E. Slykhouse and H. G. Drickamer, phy8. and Chem. Solids 7, 275 (19.58).
I
I
c l o
-
’
-100
I
TRANSITION AY1s870
I
3 t i 4
I
I
I 0
I
1
100 PRESSURE, ATMOS (a)
I
Po0
K
o3
I
50
I
I
100
I50
PRESSURE ,ATMOS X
lo3
(b)
FIG.26. Rermlts of T. E. Slykhouse and H. G. Drickamer on the effect of pressure on the absorption edge of AgCl (a) and AgI (b) [Phys. and Chern. Solids 7,207 (1958)l.Note the different scales. The AgCl transition at 87,000 atmos is presumably from the fcc structure to the simple cubic (CsCI) structure. The AgI transition at 2900 atmos is from the normal wurtzite I3tNCtUI-e to the fcc, but the transition at 112,OOO atmos is anomalous in that it shows a blue shift, opposite to the case for the presumably similar AgCl transition.
PHYSICS AT HIGH P R E S S U R E
135
and AgI."O A plot of some of their results on the silver halides is given in Fig. 26. Finally, Parsons and D r i ~ k a r n e rhave ~ ~ ~looked a t the spectra of Ni(I1) and Cr(1II) complexes which have HzO and S H 3 as ligands. Their data show the increase in the crystal field strength with 130,000 atmos pressure, and also show a transition in (Si(H?O),)SO, at 65,000 atmos. Other investigations have been made on a pressure-induced ligand peak in KzReCls,a 3 z and the near-ultraviolet spectra of some fused-ring aromatic crystals. 3 3 3 These experiments graphically illustrate the use of ingenuity in extending physical measurements into a very difficult pressure range. 7.
DIFFU6ION
High-pressure techniques were first applied to the study of diffusion in solids by Nachtrieb et al.,s34who studied self-diffusion in sodium to 12,000 atmos a t temperatures up to 13O"C, using conventional sampling methods. In this paper the concept of the activation volume for selfdiffusion, which can be obtained directly from the pressure dependence of the diffusion constant, was introduced. This laid the groundwork for future discussions of the subject. The relative magnitude of the activation volume with respect to the molar volume is believed to furnish a sensitive test of the diffusion mechanism. The work on sodium, as well aa later work on white phosphorus,336lead,3a6lithium,286and a fcc silver-zinc alloy,8*' shows that the activation volume for diffusion for the solid metals is of the order of one-half the molar volume of the metal. On the other hand, the activation volume for diffusion in liquid mercury and liquid gallium338 is less than this by an order of magnitude. Bosman el al.,'3@ have used the time rate of decrease of the permeability of iron to determine the activation volume for interstitial diffusion of Nt in iron. Their activation volume is of the same magnitude as those for liquids. Nachtrieb has been concerned with how the finding that the activation volume for diffusion is less than the molar volume for fcc metals can be interpreted on the basis of the activated state theory. An alter I'aT. E. Slykhouse and H. G . Drickamer, Phys. and Chem. Solids 7 , 207 (1958). l i l R . W. Parsons and H. G . Drickamer, J . Chem. Phys. 29, 930 (1958). Ir*D.R. Stephens and H. G. Drickamer, J . Chem. Phys. SO, 1364 (1959). 10S. Weiderhorn and H. G. Drickamer, Phys. and Chem. Solids 9, 330 (1959). H. Nachtrieb, J. Weil, E. Catalano, and A. W. Laweon, J . Chem. Phys. 20, 1189 (1952).
H. Nachtrieb and A. W. Lawson, J . Chem. Phys. 23, 1103 (1955). IrcN.H. Nachtrieb, 11. A. b i n g , and S. A. Rice, J . Chem. Phys. 81, 135 (1959). I1'G. W. Tichelaar and D. Lazarus, Phys. Rev. 113, 438 (1959). ''cN. H. Nachtrieb and J. Petit, J . Chem. Phys. 24,746, 1027 (1956). #*'A. J. Bosman, P. E. Brommer, and G. W. Rathenau, Physica 28, 1001 (1957). 1:".
136
C. A. 8WENSON
native dynamical formulation of the diffusion problem developed by Ricea40has been applied to the pressure dependence of the diffusion constant (and, hence, the activation volume) 3 4 1 with some success. Tichelaar and Lazarusass have discussed this problem of activation volume in connection with their indirect diffusion measurements in the Ag-Zn alloys. Also E i e y e ~ ~has ' ~ calculated the activation volume for diffusion on the basis of various models. The temperature variation of the activation volume in a single phase can be used to give useful information. The rapid decrease of the activation volume with an increase of temperature in white phosphorusaa6has been attributed to premelting phenomena. A similar measurement on ~ulfura'~ has been explained from the supposed decrease of the shortrange order in the liquid with increasing temperature. The use of nuclear resonance techniques to study diffusion in lithium"' and solid hydrogenzs5 has been mentioned in a previous section; this method seems to offer special promise for those elements for which no radioactive tracer is available. There is some doubt about the applicability of the nuclear resonance method to elements with high atomic weights, however, due to the relatively small contribution of the diffusion process to the relaxation time. Various correlations have been found for diffusion phenomena. Nachtrieb et al.aa4-aJs established a relationship between the pressure dependence of the activation energy for self-diff usion and the presaure dependence of the melting temperature. In fact, for all of the metals which they studied, covering both the liquid and the solid states, AH', the activation energy for self-diffusion, was proportional at each pressure to the melting temperature T , at that pressure. This was true even for liquid gallium,a**where an anomalous decrease in T , with pressure is observed. Some basis for these correlations is provided by the theory of Rice and Nachtrieb."' Lawsona" also has been able to use rough thermodynamic arguments to correlate the activation volume for self-diffusion (AV*) with diffusion measurements at constant pressure. This correlation permits a rough estimate of A V * when data on pressure effects are lacking. Keyesa4*also has related the activation volume, activation energy, and the compressibility for various models. All the measurements discussed above were made on cubic crystals or on liquids. Liu and D r i ~ k a r n e r 'determined ~~ the effects of stress on rtQS. A. Rice, Phys. Rev. 112, 804 (1958). 8 4 1 5. A. Rice and N. H. Nachtrieb, J . Chem. Phys. 31, 139 (1959). u * R . W. Keyes, J . Chem. Phys. 29, 467 (1958). 8'8 D. R. Cova and H. C. Drickamer, J. Chem. Phys. 21, 1364 (1953). 844 A. W. Lawson, Phys. and Chem. Solids 3, 250 (1957). 84' T.Liu and H. G . Drickamer, J. Chem. Phys. 32,312 (1954).
PHYSICS A T HIGH PRESSURE
137
eelf-diffusion in single crystal and polycrystalline zinc, with various stress (both uniaxial and hydrostatic) and diffusion-direction combinations. In general, the diffusion constant waa found always to decrease with the application of compressive stress. The largest diffusion constant, which is for a direction parallel to the C axis was decreased the most. Lawsona4*has analyzed these data in terms of tensor relationships, and has shown that they are inconsistent to several orders of magnitude. A suggestion is made aa to the probable, experimental source of the error. Drickamer and his co-workers have used an apparatus to investigate diffusion in liquids to 10,OOO atmos in which radioactive tracers and a scintillation counter are situated in the transmitting liquid. Koeller and Drickamer"' have described the method in some detail, and give results for selfdiffusion in CSI. Other papers have elaborated on various aspects of the t e c h n i q ~ e . J 4 ~Other - ~ ~ ~results at high preasures (obtained using this technique or more conventional techniques) are for various liquids (CSrorganic mixtures, a61 water and sulfate solutions,'62 alcoholic solutions,"* CC14-Sn14, and polymer solutions8s6).Rutherford and Drickameraso have outlined the theory of thermal diffusion in liquids. The results of high-pressure experiments have been used to investigate the predictions of this theory. In contrast with the ordinary diffusion results, the thermal diffusion measurements seem to agree quite well with the theoretical prediction^.'^' 8. MISCELLANEOUS a.
The Thermal Conductivity or Solid Helium
The use of helium as a "model" substance to investigate the behavior of the melting pressure curve at high reduced pressures and temperatures already has been discussed (Section 4b). Solid helium, because of its
[email protected].
Lawson, J . Chem. Phya. 22, 1948 (1954). R. C. Koeller and H. G. Drickamer, J . Chem. Phys. 21, 267 (1953). l"K. D. Timmerhaus, E. B. Giller, L. H. Tung, R. B. Duffield, and H. G. Drickamer, Rev. Sn'. Zmtr. 21, 261 (1950). A. J. Reinsch and H. G. Drickamer, J . A p p l . Phyr. 23, 152 (1952). (Io H. G. Drickamer, K. D. Timmerhaua, and L. H. Tung, Chcm. Eng. P r w . 49,603 847
(1953). #I1 R.
C. Koeller and H. G . Drickamer, J . Chem. Phys. 21, 575 (1953).
caR. B. Cuddeback, R. C. Koeller, and H. G . Drickamer, J . Chem. Phys. 31,
589
(1953).
R. B. Cuddeback and H. G. Drickamer, J . Chem. Phys. 21, 697 (1953). la4E.P. Doane and H. G. Drickamer, J . Chcm. Phys. 21, 1359 (1953). #'&H.Emery, Jr., L. H. Tung, and H. G . Drickarner, J . C h .PAys. 22,961 (1954). a''W. M. Rutherford and H. G . Drickamer, J . Chcm. Phys. 22, 1157 1% (1954). Ir7W. M. Rutherford, E. L. Dougherty, and H. G . Drickrmer, J. C h . Phyr. 33, 1289 (1954).
138
C. A. SWENSON
high compressibility, also has been used for the determination of the effect of density changes on the thermal conductivity of dielectric crystals.868 Wilks and his collaborators found that the thermal conductivity increases by a factor of roughly 300 when the pressure is increased from the melting line to 2000 atmos (a density change of about two).J68J'0 These results are in essential agreement with the concept that the major factor governing thermal conductivity in dielectric crystals is the Debye temperature, 8 0 . Furthermore a remarkable resemblance is found between the behavior of solid helium (where 80 varies from 25" to 90') and results obtained with sapphire (eo 'II980") and diamond ( O D 1:1840') when the differing values of 80 are taken into account. Webb and Wilks found some inconsistencies in the results of their highest density experiments, presumably because the helium did not solidify as a single crystal. b . The Rare Earths
The elements in this group exhibit similar chemical behavior, and the free atoms differ only in the number of electrons in the 4f shell. However, some of the physical properties of the metals are quite dissimilar as is shown by the high-pressure measurements on twelve of them (La, Ce, Pr, h'd, Sm, Gd, Dy, Ho, Er, T m , Yb, Lu) made by Bridgman."' The data summarized in this paper include the volume compressions to 40,000 atmos, and the resistance change to 100,000 atmos, both a t room temperature. Bridgman also investigated the shear strength of the metals to, roughly, 100,000 atmos. Anomalies (sometimes minor in nature), which are different for each of the metals, are the rule rather than the exception. In a later paper, Bridgman*45reports similar measurements on yttrium which show no anomalous behavior. Yttrium, while strictly not a rare earth, resembles the rate earths in many of its properties and chemical behavior. Two relatively major anomalies which were found in cerium and ytterbium are worth mentioning. The unique phase transition in cerium in which a 4f electron presumably moves into a 5d orbital has already been discussed in some detail in Section 4b. Ytterbiumaa1also shows unusual behavior in that its volume compression from zero to 40,OOO atmos is greater by a factor of two than for the other rare earths. A h its resistivity increases by a factor of 16 as the pressure is increased to 50,000 atmos, a t which point the resistivity suddenly drops to roughly its initial value. The abnormal behavior of ytterbium is possibly due to It. Berman, F. E. Simon, and J. Wilks, Nature 168, 277 (1951). aseF.J. Webb, K. R. Wilkinson, and J. Wilks, Proc. Roy. SOC.AP14, 546 (1952). lE0 F. J. Webb and J. Wilks, Phil. Mag. 171 44,663 (1953). **I P. W. Bridgman, Ptoc. Am. dead. Arts Sci. 83, 1 (1954).
PHYSICS AT HIGH PRESSURE
139
the divalent nature of the ions in the metal, in contrast with the normal trivalent state of the ions of the other rareearth metals. Just as the abnormal behavior of cerium can be interpreted in terms of the existence of both a trivalent and a quadrivalent state, the abnormal behavior of ytterbium possibly can be interpreted in terms of the existence of both a divalent and a trivalent state. Bridgman made no measurements on either europium or terbium, both of which were unavailable in pure metallic form at the time of his measurements. Europium is similar to ytterbium in that it is normally divalent in the metallic state, and measurements to 13,000 atmos a t both %OK and room temperature confirm that its compression behavior is more similar to that of ytterbium than to that of the other rare earths.862 No anomalous behavior was found. Similar measurements to 20,000 atmos on terbium at 78"K, room temperaturc, and 100"C36ashow no obviously anomalous behavior and a small initial compressibility. This would be expected from the trivalent nature of the ions in the metal. Unfortunately, no other high pressure measurements have been made on these two metals, so further comparison with Bridgman's work is not possible. c. Glasses
The structure of glasses, being intermediate between that of crystalline solids and liquids, results in unique properties in some cases. The open structure of the amorphous solid makes the compressibility of the glass larger than that for the corresponding crystalline solid. The effect of this is especially striking for quartz glass, where an abnormal increase of compressibility with pressure is found (Fig. 27) until 35,000 atmos, where the behavior abruptly becomes normal.'4S One interpretation of this change is that voids which are contributing to the excess compressibility are totally collapsed a t this pressure. The compressibility of the glass above this pressure is still much greater than for the crystal, however.14a Weir also has determined the difference in compressibility between the crystalline and glassy states for selenium and glucose.s64 Bridgman has measured the compressions of various glasses to 40,000 atmos at room t e m p e r a t ~ r e , "while ~ Weir and S h a r t s i ~ ~ have ' ~ made a systematic study of the compressions to 10,000 atmos of six selected glass systems, using various compositions within each system. The existence of an expanded structure for glass as compared with rtrF.H. Spedding, J. J. Hanak, and A. Daane, Trans. A I M E 212, 379 (1958). "* R. I. Beecroft and C. A. Swenson, unpublished work (1959). rt4C.E. Weir, J . Research Natl. Bur. Standards 62, 247 (1954). C. E. Weir and L. Shartsis, J . A m . Ceram. Soc. 98, 299 (1955).
140
C. A. SWENBON
that of the crystal makes it possible for permanent increases of density to occur during the application of very high pressures. Bridgman and Simon"' first studied this up to 150°C at approximately hydrostatic pressures of up to 200,000 atmos applied in D "squeezer" type of apparatus. The density of quartz glass, for instance, was found to be close to that of the crystal after such processing, although x-rays showed that the structure was still amorphous, and the density increase could be made to disappear on annealing. In order to check on the effects of the shear which inevitably occurs in these " squeezer " experiments, a BIO,
0
lop00
20DOO
3WOO
4 0 m
PRESSURE hg/sm*
FIG.27. The deviations from linearity of the compresaion of quarts glass. The behavior below 36,000kg/cm* represents curvature of the P-V relationship in the anomalous direction (the compressibility increasing with preasure), while that above this pressure represents normal behavior.
glass was subjected to 40,000 atmos hydrostatic pressure, a corresponding change in density*66 was observed. In a later experiment, Bridgman exposed soda glass to a more nearly hydrostatic pressure of 100,OOO am08 (transmitted by a lead jacket) applied in his resistivity equipment."' No change in the thickness of the specimen was observed after this treatment. This is contrary to what was expected from previous measure ments in which, however, some shearing occurred. Other workers have investigated these effect^*^^-^^^ and distinguish between a '' densifica*04
P. W. Bridgman and I . Simon, J . A p p l . Phys. 14, 405 (1953).
r470. L. Anderson, J . A p p l . Phyu. 27, 943 (1956). a'8 C. E. Weir, 9. Spinner, I. Malitson, and W. Rodney, J . Research Natl. Bur. Stand. arde 68, 189 (1957).
PHYSICS AT HIGH P R W U R E
141
tion,” which disappears with time, and a true “compaction” which is permanent.
d. Mechanical Properties of Metals and Alloy8 The mechanical behavior of many substances at stresses beyond their yield stress depends strongly on the magnitude of an applied hydrostatic pressure. Bridgman was interested in this field for many years, principally because of the need for information which would assist in the construction of satisfactory high pressure equipment. Many of his experiments and ideas have been published in the volume entitled, “Studies in Large Plastic Flow and Fracture,”1o with some supplementary data on the rarer metals (Ni, Ta, Nb, Mo, W, Sb, gamma-brass, Ge, and Cr) and other materials (B20, glass and Melmac 404 plastic) appearing in later p a p ~ r s . * ~In~ general J~~ the strength and ductility of most substances increases as the pressure increases. One of the most striking examples of unusual behavior was furnished by a single crystal of sapphire which, when immersed in a fluid a t 23,000 atmos, failed by twinning a t a compressional load of 50,000 atmo~.’~O Bridgman’s experiments also showed that the strength and mechanical qualities of metals a t normal pressure is increased more by work hardening at high hydrostatic pressures than by a similar treatment a t atmospheric pressure. The difficulties in this type of procedure are such as to almost preclude its use on a commercial scale. There is some indication that the properties of hard-drawn wires are influenced to a great extent by the relatively high pressures which must exist in the die in the drawing process.1o Most of the fundamental work which has been discussed in previous wtions has been done with elements or chemical compounds of fixed composition. Bridgman’s most recent work has been the study of the effects of pressure on the properties of various alloy systems; most of the initial work was on systems containing bismuth because of the intereating phase diagram of pure bismuth. The results of these measurements, which include compression to 40,000atmos, resistivities up to 100,OOO atmos, and shearing strength under pressure, for various binary alloy systems are presented in six lengthy papers. Some of the components in the 26 low melting point alloy systems were Be, Ca, Si, Pb, Zn, and Tl’7’-a78 Also some relatively high melting point alloys of the noble and W. Bridgman, J . Appl. Phya. 24, 560 (1953). W. Bridgman, “Studies in Large Plastic Flow and Fracture,” p. 120. McCrawHill, New York, 1952. I7’P. W. Bridgman, Proc. A m . Acad. Arts Sci. 82, 101 (1953). lap. W. Bridgman, Proc. A m . Acad. Arts Sci. 83, 149 (1954). 8sP. W. Bridgman, Proc.Am. Acad. A r b Sci. 84, 1, 43 (1955). Il1P.
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C. A. SWENSON
other metalsa7' were investigated. It would be difficult to discuss, in a general way the effects of alloy cornposition on the results, and for the present purpose, it is sufficient merely to call attention to the existence of this work. Bridgman found that a hitherto unknown intermetallic compound, BiSnls7lformed a t pressures greater than 20,000atmos. This discovery bolsters his thesis that at sufficiently high pressures many hitherto unknown forms of common materials should be produced which are stable upon the release of pressure. Previous examples of this were given for the CSZand phosphorus systems (Section 4b). Finally two papers by Johannin and Hai Vu3" discuss the deformation of polycrystalline Zn, Cd, Al, Cu, and several brasses under truly hydrostatic pressures of up to 9OOO atmos. Deformation was found for Zn, Cd, and the brasses, but not for the cubic metals Cu and Al. 9. FLUIDS: PVT
AND
TRANSPORT DATA
The previous discussion was devoted principally to the behavior of solids at high pressures, although some mention of the absorption spectra of gases and diffusion in liquids was made. This section, although strictly speaking it does not deal with solid state physics, is included for completeness, since it describes an active field of high-pressure work. The general factors which govern the changes in the mechanical properties of gases with pressure are well understood in a qualitative, if not quantitative, sense. The high-pressure work on gases has had as its principal objective a better understanding of the forces between mole cules through a study at high densities of P I'T relationships, viscosity, thermal conductivity, and diffusion. For an ideal hard sphere gas with no interatomic forces, both the viscosity and the thermal conductivity are independent of density and have a temperature dependence which can be predicted. Actually molecules do not behave as hard spheres and, even at moderate pressures, deviations are observed from the elementary theories. Enskog and Chapman have developed basic theories for the volume dependence of both the viscosity and thermal conductivity, and the determinations of these quantities at high pressure have been carried out with the objective testing the theoretical predictions. Michels has summarized the work of his group, which has been probably the moat active in work on gases, in a recent paper.376 I n the past fifteen years, Michels and his co-workers have continued to use their tested techniques to obtain precision pressure-volume iso-
*"
P. W. Bridgman, Proc. A m . Arad. Arts Sri. 84, 131, 179 (1955). P. Johannin and Hsi Vu, Compl. rend. a d . sci. 241,566 (1955); 242,2579 (1956). A. Michels, Nuouo cimento 9, Suppl., p. 152 (1958).
PEYBICB AT HIQH PRESSURE
143
therms from 0°C to 150°C a t pressures up to 2800 atmos for argon,lT7 p r ~ p e n e ,xenon,a7@ ~~~ air,a8o hydrogen and deuterium.a80" They also dwribe a new low-temperature device for obtaining isotherms to -180°C,a8' and give isotherms down to this temperature for air,au* argon,a8aand hydrogen and deuteriumSB0"to 1000 atmos. Thermodynamic properties have been calculated for these substances as well as for nitrogen."' Other work to about 3000 atmos, in which these same techniques were used, has been reported for nitrogen and ethylene.a86 The upper temperature limit of the experiments discussed above is limited by the use of external heating and the vapor pressure of the mercury which is used in the pycnometer, while the pressure range is limited by the direct use of a deadweight gauge for pressure measurements. Both of these limitations are tolerated because of the high precision which is desired. TsiklisaB6has extended the pressure range to 10,OOO atmos for both KH3 and nitrogen for temperatures up to 100°C. KennedyaB7 has worked with both water and carbon dioxide a t moderate pressures (1400 atmos), but a t temperatures up to 10oO"C. Basset and Bameta88have described methods for obtaining PVT data at temperatures and pressures up to 1200°C and 10,OOO atmos. SaureizSohas described a constant density apparatus in which internal heating is used to overcome the difficulties due to the vapor pressure of mercury, and J'rA. Michels, Hub. Wijker, and Hk. Wijker, Phymcu 16, 627 (1949);A. Michele, R.J. Lunbeck, and G. J. Wolkers, ibid. 16,689 (1949);A p p l . Sci. Research Ap, 345 (1951). 'll A. Michels, T. Wassenaar, P. Louwerse, R. J. Lunbeck, and G . J. Wolkem, Physica 19,287 (1953). JTBA. Michels, T. Wassenaar, and P. Louwerse, Physicu 20, 99 (1954);A. Michels, T. Wassenaar, G. J. Wolkers, and J. Dawson, ibzd. 22, 17 (1956). 'toA. Michele, T.Wwenaar, W. van Seventer, Appl. Sei. Research A4, 52 (1953). J1o. A. Michels, W. de Graaf,T. Wassenaar, J. M. €1. Levelt, and P. Louwerse, PhySica 26,25 (1959). I8*A. Michels, T. Wasaenaar, and T. N. Zweitering, Physica 18, 67 (1952). J(' A. Michels, T. Wwenaar, J. M. Levelt, and W. de Graaf, Appl. Sei. Research A4, 381 (1954);A. Micheh, T.Wassenaar, and G. J. Wolkers, ibid. A6, 121 (1955). 'IJA. Michels, J. M. Levelt, and W. de Graaff, Phym'cu 24, 659 (1958);A Michele, J. M. Levelt, and G. J. Wolkers, ibid. 24, 769 (1958). 884A.Michele, R. J. Lunbeck, and G. J. Wolkers, Phyaiccr 17, 801 (1951);A p p l . Sei. Rerearch AS, 197 (1952). J81 W . P. Hagenback and E. W. Comings, Znd. Eng. Chem.46,606 (1953). llaD. S . Tsiklis, Doktady Akad. Nauk S S S R 79, 289 (1951);91,589 (1953). G. C. Kennedy, Am. J . Sei. 248, 540 (1950);212, 225 (1954). "'James Basset and Jacques Basset, J . phys. radium 11, Suppl. to No. 1, 47A (1954) I t o J . Saurel, R. Bergeon, P. Johannin, J. Dapoigny, J. Kieffer, and B. Vodar, Discussions Faruday Soc. 22, 64 (19.56);J. Saurel, J . recherche8 centre d .recherche 8 6 . 11, 21 (1958).
144
C. A. SWENSON
data are given for nitrogen to 1OOO"C and lo00 atmos pressure. Finally, in order to obtain much higher temperatures and pressures, RyabiniP has used adiabatic compressions, and states that argon under these conditions acts like a quasi-ideal gas from 2100 to 7000 a t m ~ sAlso . ~ ~ he ~has investigated the electrical conductivityap2and optical propertiesagaof argon and mixtures under these conditions. Except for the work on air by Michels, the only new high-pressure PVT data for gases a t low temperature would seem to be for hydrogen. Kaaarnovskii and Sidorova04give data for hydrogen to 1800 atmos at 0°C and -85"C, while Zlunitsyn and Rudenkoss6 have worked with hydrogen to lo00 atmos a t 65"K, 77.7"K, and 90.6"K. Johnston and Whiteag6give a tabulation of the PVT relationships obtained at Ohio State for gaseous normal hydrogen from its boiling point to room temperature a t pressures up to lo00 atmos; however, few experimental details are given. Bennett and Dodge"' have measured the compressibility of mixturea of hydrogen and nitrogen to 3OOO atmos in order to determine the general type of behavior. Their results are not in accord with Dalton's law of partial pressures, but are in correspondence with Amagat's law of additive volumes, as would be expected from the approximate theory. To check the PVT measurements, Lacarnag*measured the velocity of sound a t frequencies from 2 to 5 Mc/sec for argon, nitrogen, methane, and propane to 1200 atmos. No evidence of relaxation phenomena were found, and calculations were made of the adiabatic compressibility and specific heat ratios. The results of the PVT measurements on argon led Michels et a1 to conclude that there should be a change in the polarizability with pregsure, so measurements were made of the change in the dielectric constant of argon with pressure to 2700 atmos from 25" to 125°C. Some change was foundIaDgand the Clausius-Mosotti function showed a small presYu. N. Ryabinin, J . Ezptl. Theoret. Phye. (USSR) 23, 461 (1952). Yu. N. Ryabinin, A. M. Markevich, and I. I. Tamm, J . Ezpll. Themet. Phua. (USSR) 24, 107 (1953). A. 5. Karpenko, A. M. Markevich, and Yu. N. Ryabinin, J . ExpU. Theoref.Phyc. (USSR) 23,468 (1952). a@J Yu. N. Ryabinin, N. N. Sobolev, A. M. Markevich, and I. I. Tamm, J . Ez& Theorel. Phys. ( U S S R ) 23, 564 (1952). *e'Ya. 8.Kazarnovskii and I. P. Sidorov, J . Phys. Chem. (USSR)21, 1363 (1947). a** A. Zlunitsyn and N. 5. Rudenko, J . Expll. Theorct. Phye. (USSR)18, 776 (1946). ao@H. L. Johnston and D. White, Trans. Am. SOC.Mcch. Engts. 72, 785 (1960). * @ I C. 0. Bennett and B. F. Dodge, Znd. Eng. Chem. Ul180 (1952). A. Lacam, J . reckchee centre nall. recherche sci. Wl 25 (1956). *O*A. Michels, C. A. ten Seldam, and S. D. J. Overdijk, Physica 17, 781 (1951). a@l
*Q*
PHYSICS A T HIGH P B W U B E
145
Bure dependence but little temperature dependence. Vereshchagin and Duginam ale0 found that the Clausius-Mosotti function for ethylene changed but little up to 2150 atmos. Recently, Phillips,'O' and Vallauri end Forsbergh" have investigated relaxation phenomena by measurements of the dielectric constant which extended into the microwave region. The PVT data can be combined with meaaurementa of viscosity and thermal conductivity to obtain the desired check on the besic theories for these quantities. Various types of viscosity meaaurement are poseible: the viscous damping of an oscillating body, the rate of fall of a ball (or other body) in a fluid, or the flow of a fluid through a capillary due to a given pressure difference. The first of these is ruled out becauee of the complication due to the effect of pressure on the elastic constanta of the euspension (unless an empirical calibration is used).'O* The second is most useful for liquids of moderate to high viscosity (both r ~ l l i n g ~ ~and *.~~' falling40sspheres have been used, while Bridgman used a falling vane44) end the third seems to be most popular for work with gases since the theory for it is well understood. Michels and his co-workers again have been most active in this field. They have worked at pressures up to 2000 atmos and temperatures between 25OC and 125OC. The capillary flow method which they requires a precise knowledge of the equation of state, but a method developed by Kuss4O' does not suffer from this requirement. Lasarre and V0dar4O8 also give details of a capillary flow method useful to 3000 atmos. The gases which have been studied are h y d r ~ g e n , ' ~deuterium,409 ~.~~~ methane,407."0 argon,"' carbon dioxide,''* and nitrogen.'08 In all these methods mercury is used in the measurement of the pressure difference; thie limits the maximum possible temperature and sample gaa to some extent. Ross and Brown'lo have developed a new type of apparatus in which a bellows is substituted for the mercury, and data for nitrogen, F. Vereehchagin and N. S. Dugina, Doklady Akad. Nauk SSSR 68, 41 (1947). C. S,E. Phillips, J . Chem. Phyu. 98, 2388 (1955). 'w M. Krotech, Ezptl. Tech. Phyaik 6 , 116 (1957). 'uL. Heyne, EzpU. Tech. Phyaik 6, 261 (1957). 4'4E. M. Griat, W. Webb, and R. W. Schieaeler, J . Chcm. Phyr. 88, 711 (1968). 'oE. KUBB, 2. angew. Phys. 7,372 (1955). 'WA. Michele and R. 0. Gibson, Proc. Roy. Soc. AlM, 288 (1931). '"E. Kum, 2.angew. Phys. 4, 203 (1952). (@IF.Lasarre and B. Vodar, Compl. rend. mad. sci. 948, 487 (1956). 'O'A. Michele, A. C. J. Schipper, and W. H. Rintoul, Phy,sico 19, 1011 (1953). 'IOJ. F. Rose and G. M. Brown, Znd. Enp. Chem. 49,2026 (1957). 4llA. Michele, A. Botzen, and W. Schurrman, Physics. 80, 1141 (1954). 41'A. Michele, A. Botzen, and W. Schurrman, Phytieo W , 95 (1967). 'ML.
481
146
C. A. SWENSON
helium, and methane are given to 10,000 psi. I n general, the accuracy claimed by all workers is about one per cent, and where comparison with theory can be made, the agreement is unsatisfactory. The volume and temperature dependence of the thermal conductivity of gases is also of interest. Michels and Botzen41' have described a parallel plate conductivity cell with which they have measured the thermal conductivity of nitrogen"' and argon4'&from 0°C to 75°C up to 2500 atmoa pressure. J ~ h a n n i hss n~~ used ~ a coaxial cylinder apparatus with internal heating to obtain data on nitrogen to 1600 atmos and 1000°C. Again, the accuracy in both cases is about one per cent, and Johannin states that the agreement with theory is satisfactory. Diffusion experiments with gases are necessarily difficult, and the current status of this work is given in a paper by Mifflin and Bennett"' on selfdiffusion in argon. The pressure range is only 300 atmos, and definite agreement with theory was not established. At pressures greater than lo00 atmos it is very difficult to distinguish between liquids and gases: indeed, above the critical temperature and pressure the distinction disappears completely. The only very highpressure work on liquids has been done by Bridgman who measured compressions to 40,000 a t m o and ~ ~ ~viscosities to 30,000 a t m ~ s . T ~he' compression work, mostly on organic liquids and silicones, shows that a t these pressures (at normal temperatures) all substances have either become crystalline solids, or are so viscous that they resemble glsasy solids. The viscosity measurements, done with the falling vane viscometer, bear this out, and indicate which substances are suitable for use aa tram mitters of truly hydrostatic pressure. Bridgman has remarked that the range of variation of the measured values of viscosity (a factor of 5 X 10') is greater than the measured range for any other experimental parameter. Other work has been done on various hydrocarbons a t somewhat lower pressures. Webb and his co-workers have studied the compreasions418 and viscosities4o4of high molecular weight hydrocarbons to l0,OOO atmos. K U S S ~has ~ * determined the viscosities of many allcohola to 2000 atmos. Cornish and Sirnon4lgalso have described a new method for measuring the specific volume of liquids at a few thousand atmoa '1'
~4
A. Michela and A. Boteen, Physica 18, 605 (1952). A. Michela and A. Boteen, Physica 19, 585 (1953).
A. Michels, A. Boteen, A. S. Friedman, and J. V. Sengers, Phyeicu 14,121 (1966), P. Johannin and B. Vodar, Znd. Eng. Chcm. 4Q, 2041 (1957); P. bJohannin,J. recherche8 centre natl. recherche a&. 43, 116 (1958). 417 T. R. MifRin and C. 0. Bennett, J . Chem. P h p . 29,975 (1958). 418 W. G. Cutler, R. H. McMickle, W. Webb, and R. W. Schiessler, J . C h m . Phya. 49, 418
727 (1958). *IB
R. M. Cornish and I. Simon, Reu. Sn'. Znstr. SO, 565 (1959).
PHYSIC8 A T HIGH PREEISURE
147
preaure and over a temperature range to 300°C. This method seems to have distinct advantages over the more conventional methods. Finally, the pressure dependence of ultrasonic propagation has been studied in several liquids. H ~ l t o n ‘ *and ~ Smith and Lawsona4nieaaured the velocity of sound in water at various temperatures and pressures. Their results tend to disagree on certain details. Litovitz and Camevale‘** also have worked with water and various electrolytes. They concentrated on absorption phenomena, and not on the velocity of propagation. uoG.Holton, J . A p p l . Phys. 22, 1407 (1951). “IT.A. Litovitz and E. H. Camevale, J . A p p l . Phys. 26, 816 (1955); J . Acouat. Soc. Am. 90, 610 (1958).
This Page Intentionally Left Blank
The Effects of Elastic Deformation on the Electrical Conductivity of Semiconductors ROBERT W . KEYES Research Laboratories. Weetinghousc Eleetric Corporation. Pithburgh. Pennsylvania
I. Introduction ....................................................... 149 I1. Phenomenological Description of Resistance and Piesoresistance .......... 150
. .
1 Notation ........................................................ 2 The Requirements of Crystal Symmetry ............................ 3. Piesogalvanomagnetic Effects.....................................
I11. Measurement of Pieeores
.................................
.
4 Effecta of Hydrostatic ................................ 5. Effects of Shear Streas ...... .................................. 6. Dimensional Corrections .......................................... 7 Adiabatic and Isothermal Constants ................................ 8 Transformations of the Piesoresistance T e m r .................
.
151 152 153 156 156 157 160 161
. 9. Energy Level Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10. Deformation Potential Model of the Effect of Strain . . . . . . . . . . . . . . . . . 167 ...... ..................... 168 11. Transport Properties V. The Effects of Hydms ............................... 169 170 12. Effects of Pressure on the Band Structure ...........................
IV. Properties of Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
179 13. The Thermodynamic Interpretation ................................ . . . . . . . . . . . . . . . . . . 184 L4. The Bonding Interpretation . . . VI . The Effects of Shear Strain . . . . . . . . . . . . . . . . . . . . . . . 187 . . . . . . . . . . . . . . . . . . 187 15. The Multivalley Semiconducto 210 16. semimetals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............................. 211 17. Degenerate Bands . . . . . VII. Related Phenomena ................................................ 213 IS. The “Minor Effects” ............................................. 213 19. Pieco-optical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 20. Dependence of Energy Gap on Chemical Composition ................ 218
.
1 Introduction
In general, the physical properties of a solid depend on ita state of strain. In particular, the electrical resistivity is a function of the state of etrain. This phenomenon has been known for many years, and haa 149
150
ROBERT W. KEYE8
found application in various resistance strain and pressure gauges.‘4 The change of the resistivity of a solid as a result of the introduction of an elastic strain into the solid is known as the elastoresistance effect. Alternatively, it is called the piezoresistance effect if the change in resistivity is regarded as being caused by application of a stress. Since the elastic constant tensor defines the relationship between the stress and the strain, a knowledge of one effect implies a knowledge of the other in a material with known elastic constants. Usually discussions of the interpretation of the effects in terms of fundamental parameters of the material are more conveniently formulated with respect to the elastoresistance effect, whereas problems of measurement are most easily discuased from the viewpoint of piezoresistance. It has been found in recent years that the pieaoresistance effect is quite large in many semiconductors.10 These large effects can be interpreted in terms of parameters of the electronic wave functions of the semiconductor, and, in fact, constitute a useful tool for the investigation of various features of the electronic structure. This aspect of the subject of piezoresistance will be treated in the present article. It. Phenomenological Description of Resistance and Piezoreristance
Since the elastic shear strain that can be introduced into a crystalline solid usually is quite small, only the linear theory of the relationship between the resistivity and the strain is of general interest. The resistivity of a material, unstrained or in any state of homogeneous strain, can be described by a symmetric second rank tensor, containing a set of six W. B. Dobie and P. C. G. Isaac, “Electric b k t a n c e Strain Gauges.” English Universities Press, London, 1948. * W. M. Murray and P. K. Stein, “Strain Gage Techniques.” Mass. Inst. Technol., Cambridge, Mass., 1956. * C. C. Perry and H. R. Lissner, “The Strain Gage Primer.” McGraw-Hill, New 1
York, 1955. J. J. Koch, R. G. Boiten, A. L. Biermasz, G. P. Roezbach and G. W. van Santen, “Strain Gauges.” Philips Technical Library, Eindhoven, Netherlands, 1952.
J. Yarnell, “Resistance Strain Gauges.” Electronic Engineering Soc., London, 1951. 6 W. P. Mason and R. N. Thurston, J . Acowf. Six. Am. 29, 1096 (1967). F. P. Burns, J . Acoust. Soc. Am. 29, 1096 (1957). eb C. Landwehr and K.-F. Zobel, Z. Inatrumdenk. 66, 220 (1957). k W. P. Mason, BeU Labs. Record 87, 7 (1959). ’ P. W. Bridgman, “The Physics of High Pressure,” G . Bell, London, 1952. 8 A . Michels, Proc. A d . Sci. Amst. 82, 1379 (1329); A. Michels and M. Lensen, J . Sci. Inalr. 11, 345 (1934). ’H. E. Darling and D. H. Newhall, Trans. Am. Soc. Mech. Engrs. 76,311 (1953).
6
10
C. 8. Smith, Phys.
Rev.94, 42 (1954).
PIEZORERIBTANCE O F SEMICONDUCTORS
151
independent coefficients. The stak of etrain of the material can also be described by a symmetric second rank tensor, or another set of six coeficients. Thus the most general linear relationship between the resistivity and the strain can be expressed by six linear equations, involving thirtysix coefficients. 1. NOTATION
It ie desirable to introduce a compact notation to describe the relationships between various physical concepts, each of which is defined by a number of parameters. In this chapter quantities having a number of components, i.e., vectors and tensors, will be represented by boldface letters. Relationships between these quantities are expressed by the dot product operation. For example, the electric field vector F in a crystal is related to the current density j by
where p k l are the components of the resistivity tensor g. In vector notation Eq. (1.1) is written
F
= e-j.
(1.2)
The piezoresistance equations relate Gp,,, the change in pil that is caused by the application of a stress, to the stress tensor K k l . The most general linear dependence of 6e on u is
where If is the piezoresistance tensor. I n analogy with Eq. (1.2), (1.3) will be written 6p = II:u (1.4) where the double dot (:) signifies the dot product over two indices. Since most of the second-rank tensors that will occur are symmetric and contain only six independent components, the fourth rank tensors that we will use can be displayed in the form of six by six matrices as is customary in the theory of elastic properties. In using this notation the symmetric tensors are regarded as six-component vectors in which, for example, pt+ = P I , pVV = p2, pal = ~ 3 ,pv. = p4, pl2 = P,, pZv = P6. The components of a fourth rank tensor then are characterized similarly by two subscripts. In this notation Eq. (1.3) is written tl
152
ROBERT W. KEY=
This representation of the fourth rank tensor n requires that HI, = llszzz, etc. A factor two must be inserted, however, if the second index of IT,,,, is 4, 5 , or 6, e.g., II,, = 2II.-,, for the offdiagonal terms in u appear twice in the double contraction operation.
I I l t ,= ,II
2. THEREQUIREMENTB
OF
CRYSTALSYMMETRY
Tensors, such as 8 and n, which represent properties of a crystal, must be invariant under the symmetry operations of the crystallographic point group. This condition leads to certain relations between the different tensor components for each crystal class which, in most cases, reduce TAELEI. THEF o w
SYMMETRIC SECOND RANKTENSORB FOR TW CRYSTAL IN THIS PAPER C L A ~ ~TREATED ES All of the offdiagonal componenb vanish for these classes. OF
Trigonal Tetragonal Hexagonal Cubic
I
P I I = PIZ f 0, pat f PI1
0
= PZZ = P I 1 # 0
TABLE 11. THEMOST GENERAL PIEZORESISTANCE TENSORS M R CRYSTALS HAVINO AXES OF 3-, 4, A N D &FOLDSYMMETRY A N D FOR CUBICCRYSTALS. IN MANYCRYSTALS ADDITIONAL SYMMETRY ELEMENTSRJKJUIRE THAT SOMEOF THE COEFFICIENTS SHOWN HERE VANISH.
PIEZORESISTANCE OF SEMICONDUCTORS
153
the number of independent, nonvanishing components to considerably less than the total number. The influence of point group symmetry on tensor properties of crystals haa been discussed in many places. Particularly valuable are the book of Voigtll and the recent review of Smith.” A thorough treatment of the phenomenological theory of the piezoresistance effect is given in the latter work and will not be repeated here. The crystals of greatest interest in semiconductor physics have a t least one axis of threefold or higher symmetry. Specific discussions and examples in this and the succeeding sections are limited to such cases. Problems pertaining to crystals of lower symmetry are worked out easily along the same lines with the aid of the paper by Smith.’* For reference, the forms of second and fourth rank tensors for the crystals under coneideration are given in Tables I and 11. The second rank tensors of greatest interest here are the electrical conductivity and the resistivity. The pertinent fourth rank tensors are those which relate the resistivity or the conductivity to the stress or the strain. The elastic constant tensor will also be of interest.
3. PIEZOOALVANOMAGNETIC EFFECTS Meaaurements of the effect of elastic strain on the galvanomagnetic coefficients of semiconductors sometimes have proved useful in elucidating certain features of models of the band structure and scattering processes of semiconductors. 18--17 Therefore a discussion of important aepects of the phenomenological theory of these effects is also included here. The phenomenological theory of galvanomagnetic effects is baaed on an expansion of the resistivity tensor in powers of the components of the magnetic field vector H.To terms of second order in H the expansion hae the form pij(H) = P i j ( 0 )
+2 k
Psjk(‘)Hk
+
z
psjk1(*)HkHt.
(34
The sets of coefficients p,Jktl) and plJktr(2) are tensor properties of the crystals, and, therefore, are subject to the restrictions imposed by the crystal W. Voigt, “Lshrbuch der Kristellphyaik.” Teubner, Leipsig, 1928. I*C. 8.Smith, Solid Slds Phys. 6, 175 (1958). I’R. W. Keyea, Phyr. Rw. 99, 1655(A), 1633(E)(1955). “R. W. Keyeu, Phyr. RGV.108, 1240 (1956). 1 I R . W. Keyea, in “Semiconductors and Phoephore,” (M. &h6n and H. Welker, eds.) pp. 236-246. Inbrecience, New York, 1958. I I R . W. Keyes, Phys. and Chetn. Solidu t , 102 (1957). C. Maynard, Theeia, Harvsrd University (1957), unpublished. 11
154
ROBERT W. KEYES
symmetry. I n addition, they satisfy the Onsager conditions,18 =
PJ8(-H)*
Here the third rank tensor PtJk") will be designated the Hall effect tensor, and the fourth rank tensor p,jk1(2) will be designated the magnetoresistance tensor. This usage accords with that of Voigt," although different terminology has been used by some authors. With respect to the first two terms of Eq. (3.1), the Onsager relationshipla implies both that pll(0) = pJl(0), as discussed by Smith, and th a t p,,k(') = - Pjsk'". This means that p,Jk'" has only nine nonvanishing components. Hence it can be expressed in the form 19e20
where 6,j1 vanishes if any two of its subscripts are the same, is 1 for even premutations of the subscripts, and is -1 for odd permutations of the subscripts. Equation (3.2) shows that the Hall effect is also a second rank tensor property, here denoted by This second rank Hall effect tensor is not symmetric in general, as is the resistivity tensor. I t can be decomposed, however, into a symmetric and a n antisymmetric part :zk p ~ ' = ~ 'Ra
(Ku = RLI,s l k
slk,
=
-SAI)-
(3.3)
The antisymmetric part may be written further as sik
=
8rkdLt.
(3.4)
Here b is now a vector property of the crystal. Summing u p this notation, one may use conventional vector notation t o express that part of the electric field arising from the Hall part of the resistivity tensor, and obtain2" F(1) = j X ( R - H ) j X (H X b) (3.5)
+
where R is a symmetric second rank tensor. The effect of elastic stress on the Hall effect is now clear. The dependence of the symmetric tensor R. on the stress is given by a tensor of exactly the same form as th at which describes the piezoresistance effect, exhibited in Table 11. It is known th at a vector property b can characterFor references to the application of Oneager's relations to electrical transport phenomena,see J.-P. Jan, Solid Slate phy8. 6,8 (1957). le C. Goldberg and R. E. Davq Phys. Rar. 94, 1121 (1955). loL. P. Kao and E. Katr, Phys. and Chem. Solids 6,2B(1958). P. J. Price, ZBM J . Rcaearch Deuelsp. 1, 239 (1957). I*
PIEZORESILITASCE OF SEMICOSDUCTORS
155
ize only a limited class of crystals, iianiely, those which can have a pyroelectric effect. The effect of strain on such a vector property is exemplified by the piezoelectric effect. The appearance and the symmetry of the pyroelectric and piezoelectric effects in crystals have been discussed in detail by Smith;I2 this discussion applies also to the Hall vector b and its dependence on strain. Since most of the common semiconductor crystals are found to fall in only a few crystal classes, specific mention of the appearance of the symmetric part of the Hall effect in these classes is worthwhile. In the cubic class m3m (diamond, NaCI) neither the pyroelectric nor the piezoelectric effect can occur, so that the Hall tcnsor e(H)has the same form as the resistivity tensor in zero field, e(O), and the effect of stress on e(H) is given by a tensor IVH)having the same form as the piezoresistance
TABLE111. THEPIEZOQALVANOMAONETICEFFECT IN
THJE CUBICCBYSTAU CLASSES432, 8 2 , AND &ma The change in resistance arising from the application of a magnetic field is defined in terms of the tensor P , ~ U ( * ) ,Eq. (3.1), to second-order terms in the magnetic field: pi,(Yg) &,~p,~u(*)B&. The coefficients Qw.h of this table deacribe the change in the coefficients p,>u(a) as B result of the stress. OF THE
-
f
b,i*s(*)
=
QIIIG,
dP,,j,(')
=
QI~IC~S
bPijt,(*)
i .
Q4tZ(tlt
bPp,s)kff)
QLa4e)k
= Q41&j b**,(f) = QlS,ft,
6P.j&t(')
+
f
C!IIZ(*~J+ f t k ) Q1z2cjJ (3))
f
+
Q~tatkk
Qa'ltkt
6P*jt,(*) QPiis~(')
Qll66>
==Qtsfi@r,t
(1957), unpublished; R. W. Keyen, Weatinghouee Hesearch Report 8-1038-R1I (1956), unpublished.
.C. Maynard, Thesis, Harvard IJniversity
tensor n. In the ZnS (zincblende) structure e(") is still symmetric, but an antisymmetric part b can be induced by stress. In the ZnO (wurzite) structure, b can be different from zero even in the unstrained state and can vary with stress. The dependence of b on the stress is given by a third rank tensor of the piezoelectric type. The piezo-Hall effect in cubic crystals has been described previously.laJ7.21The decomposition of the Hall effect represented by Eq. (3.2) is given by Voigt." Voigt, however, considers only the case in which the tensor p ( H ) is symmetric. The author knows of no theoretical or experimental evidence to support the view that this case is of general validity. On the other hand, he is not aware of any specific model of electronic structure and transport processes which exhibits the effect described by the vector b. R. W. Keyea, Weetinghouse Reeearch Report 8-1038-Rll (1966), unpublished.
156
ROBERT W. KEY=
Studies of the form of the tensor which describes the dependence of the aecond-order magnetoresistance coefficients on stress for the class m3m have been carried out by the author21and by Maynard." The t e w r involved is of sixth rank and has 12 independent coefficients. These are displayed in Table 111. In most of the problems to be discussed in this article the magnetic field will be assumed to be zero. In general, this fact will not be indicated explicitly; thus, for example, p should be understood to mean g(O). 111. Measurement of Piezoretistance
The determination of a piezoresistance coefficient essentially involve8 measurement of the change in a component of the resistivity of a crystal caused by the application of a small, known stress. There are only two simple types of stress which can be applied to a solid conveniently: (1) hydrostatic pressure, and (2) uniaxial tension or compression. Techniques permitting the measurement of resistivity under each of these types of stress have been described in the l i t e r a t ~ r e . ~ Only ~ ~ ~ the ~~~ principles -'~ involved will be discussed here, and descriptions of apparatus will not be given. 4. EFFECTS OF HYDROSTATIC PRESSURE When a specimen is subjected to a hydrostatic pressure P, the effects produced on the electrical conductivity are comparatively simple. Hydrostatic pressure cannot change the crystal symmetry; hence, any component of the resistivity which vanishes for reasons of symmetry in the unstrained crystal vanishes also in a crystal subjected to hydrostatic pressure. (These remarks exclude, of course, those cases in which the application of high pressures causes a transformation to a different crystal structure.) Thus, the pressure effects can be described by giving the rate at which the nonvanishing components of the resistivity vary with pressure. Since only crystals with at least one axis of threefold or higher symmetry are considered here, this means that the effects of hydrostatic pressure on resistivity can be described by at most two coefficients.12In the case of a cubic or isotropic substance only one constant is required. For the purpose of determining a complete piezoresistance tensor, 19
1) '4
1' 9'
M. Allen, Phya. Rev. 12, 848 (1932).
R. F. Potter and W. J. McKean, J . Research N d l . Bur. Standards 69, 427 (1957). A. J. Tuzeolino, Phys. Rev. 108, 1980 (1958). M. Pollak, Rev. Sci. Znslt. 29, 639 (1958). F.J. Morin, T. H. Geballe, and C. Herring, Phyr. Rev. 108, 625 (1957).
157
PIEZORESIBTANCE O F SEMICONDUCTOR8
however, it is useful to know the relations of the pressure coefficients These relations are worked to the ordinary piezoresistance coefficientsnuA. out easily with the use of Eq.(1.4), since the stress tensor for hydrostatic pressure is simply Y = P1, where 1 is the unit tensor and P is the pressure. The results relating the pressure derivatives of the nonvanishing resistivity components to the n,, are given in Table IV. TABLEIV. THE RELATIONSHIP OF PRESSURE DERIVATIVE OF RESJSTIVITY TO PIEZORESISTANCE COEFFICIENTS COMPONENTS Crystal systems Trigonal
PI1
Hexagonal or tetragonal
ma
Cubic 23, m3
dP
n1r
~
PI1
+ nir + nia 2na1 + naa rill + + + 2n1,
nii
Pll ~
Cubic 432,33m, m3m
- dPii _
lteoiotivity component
nla
nll
5. EFFECTSOF SHEAR STRESS The effect of shear stresses on the resistivity also must be studied to determine the pieaoresistance tensor completely. The geometrical arrangements that can be used to investigate these effects are limited by eeperimental considerations. It is necessary that the stress be homogeneous over a considerable portion of the sample. The practical way of achieving this is to apply a tensile or compressive stress along the axis of a long cylindrical specimen. If the length of the specimen is large compared to the transverse dimensions, there will be a region in which the stress field is not perturbed by the grips used a t the ends to apply the stress. For convenience in preparation and measurement the sample ordinarilv has n rectangular cross section. The resistivity measurements that can be made with relative ease on such specimens may be divided into three types. These are illustrated in Fig. 1. The experiment which is performed with the arrangements shown in Fig. 1 is as follows: the current which passes through the specimen, j, is drawn from a constant current source. The voltage V is measured when X,the tensile force per unit area, is equal to zero. A finite stress then is applied, and the voltage change bY is measured. By varying the stress increments it is verified that 6V is proportional to X. Since the applied force is the directly meamred mechanical quantity, a knowledge
158
ROBERT W. KEYES
of the cross-sectional area also is required for the determination of X. It should be noted, however, that the important quantity required for the interpretation of the results is the ratio of the change in a component of resistivity to a resistivity. In configurations (a) and (b) this ratio is given by 6 V / V , except for a wn2tll correction associated with changes of dimensions which will be described Inter. Thus accurate measurement of the current and of the dimensions of the electrode configurations is not required. Certain specific comments about the arrangements of Fig. I will aid in evaluating their relative usefulness. (a) This type of arrangement allows one to measure a diagonal component of the resistivity tensor. That is, the component of the electric
FIG.1. Three convenient arrangements for measuring the effect of stress on components of the resistivity tensor. Arrangements (a) and (b) measure diagonal componenta and arrangement (c) measures an off-diagonal component. X is the tensile force per unit arca applied to the specimen.
field along the direction of the current is measured. It is the geometry customarily used for resistivity measurements and gives the best determination of the piezoresistmce constants. However, as will be seen in the following, a complete set of piezoresistance constants cannot be determined by measuremeiits of this type. Thus measurements by one of the other methods must be made. One may also make use of measurements under hydrostatic pressure. (b) This type also measures a diagonal component of the resistivity tensor. The geometry is far from the optimum for a resistivity measurement, however. The area over which the current is distributed is large and the distance over which the voltage is measured is small, so that a comparatively large current is rcquired to produce an easily measurable voltage. In addition, it is difficult to plare electrodes in such a way as to
PIEZOREBISTANCE OF SEMICONDUCTORS
159
use the four terminal method for the resistivity measurement. Hence the problem of applying large area, low resistance electrode surfaces to the material must be solved. Moreover, the electrodes are not extended to the ends of the specimens, since it is desirable to avoid including contributions to the conductivity from regions of the sample near the grips. Thus the current spreads out of the region defined by the electrodes. This introduces a need for a correction to the measured piezoresistance, which has been discussed by Smithtofor cubic crystals. (c) This configuration measures an off-diagonal component of the resistivity. It avoids most of the difficulties of method (b), but has other aharacteristic disadvantages. These arise from the fact that, in contrast to arrangements (a) and (b), the electrodes that are used to measure the bV generally are not suitable for the measurement of an ordinary resistive voltage. The latter must be found from a separate set of electrodes. Thus the calculation of the ratio of a piezoresistivity to a resistivity requires an accurate knowledge of the dimensions of the electrode configuration. In addition, the piezoresistance and the resistance are measured on different regions of the material. Since each of these quantities, when determined separately, is much more sensitive to impurity content than their ratio, inhomogeneity of the sample may have a serious effect on the measured ratio. For a particular orientation of a sample, each of the schemes described in the foregoing allows the measurement of a certain linear combination of the ~T,,A. The complete determination of a piesoresistance tensor involves the measurement by these methods of as many linearly independent combinations as there are independent coefficients in the tensor. As a check, a larger number of combinations may be determined. Moreover, it sometimes is possible to investigate particular features of a mode1 of:the energy band structure from a knowledge of fewer coefficients. To illustrate the determination of a piezoresistance tensor by the methods just described, certain cubic and hexagonal crystals will be discussed in more detail. In the arrangements shown in Fig. 1, let a unit vector parallel to the axis of the specimen, and therefore, parallel also to the applied force, be designated by e. The stress tensor then is Y = Xee. (This notation means K,j = Xe,e,.) The change in resistivity produced by this stress is 6p = X U : (ee). In the “longitudinal” arrangement, Fig. la, the ee component of the resistivity is measured, i.e., 6V/V
= e
6p
*
e/e . p . e.
Cubic crystals of the classes 432, 332, mid m3m, for which there are only three independent piesoresistance constants, will be discussed first. This cme has been considered by Smithlo and by Potter and McKean.2’
160
ROBERT W. KEY=
Evaluation of the formulas of the preceding paragraph shows that
)JV/v = X[(ITn
+ITd +
(IT11
- ITIS - n4,)(es4 + ey4 + eS4)l/p
(5.1)
for the longitudinal arrangement. Here the quantities (e., e, e,) are the components of e when referred to the system of the cubic axes of the crystal. Equation (5.1) shows that only two linear combinations of the IT, can be determined from the longitudinal measurement, and that another type of measurement is necessary for the complete determination of n. One suitable type is the effect of hydrostatic pressure, aa can be seen from Table IV. Various transverse measurements can also be used for the determination of the third coefficient, and examples of this method are described by Smith.lo There are eight independent piezoresistance coefficients for the moet general hexagonal crystal. The 6V measured in the longitudinal arrangement is*'
6v/v
=i
x[nll(l
- eat)* 4- (nn 4- nai -t 2l%+)e~*(1- er') + nasea4]/ + (pa; - pll)e~*I (5.2) b11
where e.3 denotes the component of e along the hexagonal axis. Note that the hexagonal fourth rank tensor actually has complete rotational symmetry about the hexagonal axis, so that only the orientation of the axis of the specimen with respect to the hexagonal axis is significant. Measurements of the longitudinal type allow the determination of only three combinations of coefficients in this case. It is a general result for all crystals that a longitudinal measurement permits the determination of only the sum of coefficients which are symmetrically placed in the tensor, e.g., (IT81 IIIs)in this case. The pressure effect allows the determination of two more combinetions, 80 that three measurements of a transverse type are required. These are selected easily. Additional symmetry elements require that IT,' and lT,, vanish in the more common hexagonal crystals; thus the problem is somewhat simplified. I n the tetragonal and trigonal crystals, the orientation of the specimens with respect to the crystal axes perpendicular to the c axis is also important.**The formulas are slightly longer than those for the simpler examples discussed, but the selection of a set of arrangements to make the required measurements is equally straightforward.
+
6.
DIMENSIONAL CORRECTIONS
The treatment presented thus far, particularly that leading to equations such as (5.1), is based on the assumption that the only effect of I*
M. Allen, Phys. Rev. 48, 248 (1936).
PIEZOREBIBTANCE OF SEMICONDUCTORS
161
stress is to change the reaistivity tensor. I n fact, the stress also changes the dimensions of the specimen, so that a correction for the effect of dimensional change is necessary to obtain the true piezoresistance constante from measured values of bV. Often the correction is only a negligible part of the piezoresistance coefficient, but it is sometimes significant. The value of the correction term is obtained directly from elasticity theory, and will not be described here. The principle involved and formulas for certain caaes in cubic crystals have been given by Smithlo and Potter and McKean."
7. ADIABATIC AND ISOTHERMAL CONSTANTS The theory of the piezoresistance effects is formulated most conveniently with the assumption that the stress is applied isothermally. On the other hand, the measurement of the effects is frequently performed under adiabatic conditions; i.e., the measurement is completed in a short time compared to the time required for thermal equilibrium between the twmple and its environment to be established after the stress is applied. A correction then must be applied to convert the measured constant to an isothermal constant. The correction often turns out to be quite negligible. It can be expressed by the equationz8 (7.1)
where the tensors e and K have been defined previously, is the thermal expansion coefficient tensor, C is the specific heat per unit volume, and the subscripts T and S mean isothermal and isentropic (adiabatic) derivatives, respectively. 8. TRANSFORMATIONS OF THE PIEZORESISTANCE TENSOR
As discussed in the preceding section, the relationship that is measured directly in the conventional piezoresistance experiments is that between the resistivity and the stress. For convenience in making comparisons with theoretical models, a knowledge of the elastoconductivity tensor, the dependence of the electrical conductivity on the strain, is more useful. The transformation of the tensor which describes the former relationship to that which describes the latter will be explained in this section. The relationship between the conductivity and the resistivity is given simply by d =
e-'.
(8.1)
If 8 is altered, by the application of a stress, for example, by a small ** M.Pollsk, Thesis, University of Pittsburgh (1958), unpublished.
162
ROBERT W. KEYEB
amount 6e, it is verified easily from (8.1) that the resulting change in d is (8.2)
6d = - d - b p - d .
An equivalent way of writing (8.2) is to regard the relation between 6d and 6p as defined by a fourth rank tensor, S; thus
(83)
6d = -S:bp.
Here the components of S are S11 = Srz = SSS= uu2, Sir = uw', S, = SSS= U i p a a , and a11 other components vanish. The relationship between the stress and the strain is also well known, being r = C:e. (8.4) Here e is used to designate what is called the tensor strain by Smith, and has the meaning of the quantities ZAof Srnith.l2 Since the elsstic constants are usually referred to the conventional strain, the elastic constants cuadiffer from the conventional elastic constants by a factor two when X = 4, 5, or 6: &A = 2C,a when X = 4, 5, 6 c u A = C,a otherwise.
tA
are equivalent to the yUi in Smith's notation. The Now, substituting (1.4) and (8.4) in (8.3) gives 6d
=
-s:n:C:e.
(8.5)
The elmtoconductivity tensor M is defined by the relation
M:e and is, therefore, the coefficient of e in Eq. (8.5). Thus 6d
M
=
(8.6)
-S:n:Z5. (8.7) Tensors of the fourth rank have been defined here in such a way that the double dot product operation corresponds to ordinary matrix multiplication of the six by six matrices which represent the tensors in accordance with Section 1.2- Thus the evaluation of (8.6) is quite straightforward if the elastic constant tensor is known. As an example, the relation of M t o n for the cubic classes 432, 232, and m3m is recorded here: =
+ +
Mll = ul2(nllC1l 2n12c121 M12 = ulz(n12cll n11c1.r = 2t7i2n4rC44.
+ nnclz)
The present definition of the elastoconductivity tensor is intended to preserve the matrix multiplication form of equations such as (8.6) and (8.7), and diffep from the convention of other authors (Smith,1o Herring".") who have been concerned with cubic crystals by a factor two in m,,.
PIEZORESISTANCE OF SEYICOSDUCTORG
163
IV. Properties of Semiconductors 9. ENERGY LEVELSTRUCTURE
The elastoresistance effects observed in a semiconductor arise primarily from the fact that the electronic energy levels of a crystal depend on the state of strain of the crystal. The usual theory of the energy levels of a crystal is a oneelectron theory in which the electronic state is specified by designating those levels which are occupied by electrons. Under equilibrium conditions, the probability that a level is occupied is determined by Fermi statistics. The theory of energy bands will not be discussed here, but certain results of importance will be summarized. The reader is referred to recent reviews of the for more details and references. In a perfect crystal, each state can be characterized by a value of the crystal momentum p, and the states can be grouped into bands. The values of p represented in each band span the same region of p space, a region known as the Brillouin zone. Thus a state can be specified by giving the band in which it lies and its value of p. All of the values of the energy E corresponding to states of a particular band lie within a finite range. The levels are closely spaced within a band; various parameters of the states can be regarded as continuous functions of a continuous variable p. An important example of the last property is the dependence of E on p. In problems in semiconductor physics only states having energies near the lowest energy of the band are of interest. The dependence of E on p for such states often is quadratic to an adequate approximation, and can be written in the form
E
=
-
p . a p/2m.
(9.1)
Here a is a dimensionless reciprocal effective mass tensor and m is the electron mass. In Eq. (9.1) the zero of energy has been taken to be the lowest energy of the band. Transport theory shows that neither a full band (all states of the band occupied) nor an empty band can carry an electric current. Moreover, only states with energy relatively close to the Fermi level participate in the conduction process in a partially filled band. I n a semiconductor, the Fermi level is so situated that all bands are either nearly empty or neaTly full. Thus the states which are important in the transport, problem have energies either near the maximum or the minimum energy
” c .Kittel, “Introduction to Solid State Physics,” pp. 234-311. 1953.
F. Herman, Rwu. Modern Phys. 80, 102 (1958). J. Callaway, Solid rstds Phys. 7,99 (1958).
Wiley, New York,
164
ROBERT W. KEYE8
of their band. The term semiconductor is used here in a rather broad sense in order to include the so-called semimetals and degenerate semiconductors. Some examples of band structures which are encompassed by the present definition are illustrated in Fig. 2, and the terminology by which they are customarily described is shown. The important prop erty of a semiconductor that is required here is that the energy can be
“i“ I
n 1 (0)
Eitrinric 5
-
( d ) Degenerate Voirnce Bond Type
( b ) Intrinsic
---
( ( 1 Two bond Conduction Bond
FIG.2. Density of states 1x3 energy for certain types of semiconductors. N(&)d# is the number of energy states between E and E dE. The position of the Fermi level is marked t. The Fermi function, the probability that a state is occupied by electron, is shown by the dotted line.
+
represented accurately &B a function of p by a simple expansion, for example, Eq. (9.1). Since the terms introduced by Fig. 2 will recur throughout the text, some brief introductory and explanatory remarks about them will be made here. (a) Extrinsic n-type semiconductor. Electrons in only one band are important, and the number of electrons in the band is constant, in particular, not dependent on temperature or strain. Usually the term “extrinsic semiconductor” implies the nondegenerate case, that is, it is
PIEZORESISTANCE O F SEMICONDUCTORS
165
assumed that (IFc) - {) is a t least several times kT. The number of electrons per unit volume in the band is related to E(.) and I by n = N , exp [-(I?(,) - t ) / k T ]
(9.2)
where N. = 2(2m,*kT/h2)’ is the density of thermally available statee and me* is an appropriate density of states effective mass. In the case of the valley of Eq. (9.1), m* = m/(det a)’. A corresponding case in which the Fermi level is just above the highest energy state of the relevant band can occur. In this case there is a low concentration of empty states near the edge of the band. The empty states are known as “holes,” and have properties which are essentially those of electrons with positive charge. (b) Intrinsic semiconductor. The Fermi level assumes a position such that the concentration of electrons n is equal to the concentration of holes p, namely, p = n = (NcN.)+exp ( - E 0 / 2 k T ) . (9.3) Here N, has the same significance for the valence band as N, for the conduction band. There is a continuum of cases intermediate between case (2) and case ( l ) , but these are easily treated if the two limiting cases are understood, and deserve no further attention here. (c) Degenerate extrinsic semiconductors. This situation is essentially the same as that in case (1) ; however, the fact that Fermi statistics must be used greatly complicates the treatment of many problems. Thus many results which are derived easily for case (1) are not applicable here. The equation for the number of carriers per unit volume in the band is 2
n = N . -F,[(E(’)- {)/kT]
4;
(9.4)
where F({) is the Fermi integraLJ2 (“Degenerate” here has a statistical eignificance, in contrast to its use in the next example.) (d) Eztrin8ic p-type semzconductor with degenerate bands. This type of valence band occurs, so far as is known, in all of the semiconductors with the diamond and zincblende structures. The principal feature of interest is that the quadratic terms in the expansion of the energy is terms of the momentum of the hole do not have the simple form given in Eq. (9.l).J* (e) Semimetal. This is an intermediate case between the semiconductors and metals. Degenerate statistics always must be used for at least one of the bands. Although these materials often are classed with *I J. McDougalf and E. C.Stoner, Phil. Trans. Roy. SOC.M 7 , 67 (1938). W.Shockley, Phyr. Reu. 78, 173 (196).
166
ROBERT W. KEYE8
the metals, they satisfy the condition required of the materials to be considered here, namely, all of the states of interest are close to a band edge. (f) Two band model. Several cases are known in which it appears that there is-almost an accidental degeneracy between extrama a t two different points of p space. The second band reveals itself by producing anomalies in the temperature and pressure (Section 9) dependence of transport properties and complicated magneto- and piezoresistance effects.abaa (9) There is another type of degeneracy, of great importance in the theory of the piezoresistance effects, which is not illustrated in Fig. 2. This originates in the fact that the energy function in p space must have point group symmetry that corresponds to that of the crystal. Thus if an energy extremum of a band occurs at a point PO in p space, there must also be extrema with the same energy a t all of the points to which pa can be transformed by the symmetry transformations. The group of states near one of these extrema is known as a valley. Since the valleys can be transformed into one another, they are identical except for orientation. In particular, a scalar property must have the same value for all valleys. A band with valleys that can be transformed into one another by the transformations of crystal symmetry is known as a multivalley band.” A cross section of a constant energy surface of a multivalley band will be illustrated in Section 16. Certain properties of a valley, e.g., the contribution to the conduotivity of the electrons in a valley, can be characterized by tensors. These tensors can have general form; however, certain points of p space have the property that, under some of the symmetry transformations of the crystal, they transform into themselves. Such points are referred to aa special points. Associated with each special point po there is a point groups* G(po) which transforms po into itself. Any tensor that describes a property of a valley a t PO must be invariant under the operations of G(p0). Special points appear frequently as band extrema, because the special symmetry usually requires that some of the derivatives of the energy vanish a t these points. The density of states in a multivalley band is just the density of states 14
*b
M. Glicksman, Phys. Rev. 100, 1146 (1955); lOa, 1496 (1955); M. Glicksman and S. M. Christian, Phys. Rev. 104, 1278 (1956). A. &gar, Thesis, University of Pittsburgh (1959), unpublished; Phyr. Rw. lip,
1533 (1958). A. &gar, Phys. Rev. 117, 93 (1960). 1 7 C. Herring, Bell System Tech. J . 34, 237 (1955). *‘L.P. Bouokaert, R. Smoluchoweki, and E. Wigner, Phye. Rat. 50, 58 (1936). 16
PIEZOREBISTANCE OF SEMICONDUCTORS
167
of a single valley multiplied by the number of valleys. Thus any of the bands depicted in Fig. 2 might he a multivalley band. 10. DEFORMATION POTENTIAL MODELOF
THE
EFFECT OF STRAIN
In many cases, the effects of elastic strain on the conductivity of a semiconductor can be interpreted in terms of a very simple model which describes the effect of strain on the energy band structure. It is assumed that the strain shifts the energy of all of the states associated with a given band extremum by the same amount, which is proportional to the strain. The states are otherwise unchanged. In other words, it is assumed that a valley moves along the energy scale as a unit, and that the various parameters such as effective masses and matrix elements which are associated with it are unchanged by the strain. In this model, the property of a band or valley which must be known in order to develop a theory of the effect of strain on the conductivity is the dependence of the extremal energy on strain. The most general linear form which this dependence can take is
gE(0 =
g(0:e
(10.1)
in which 8")is a symmetrical second rank tensor. The superscript ( i ) denotes the valley or band referred to in a complex model; E(') is the extremum energy of valley ( i ) and bE(') is the change in E(')caused by are known as deformation the strain L. The constants of the tensor S(') potential constants. For that reason the model described in this section will be referred to hereafter as the deformation potential model, even though the phrase "deformation potential " has additional implications which will not be of interest here.*g Bardeen'O and Herring" apparently are responsible for the introduction of this model into the literature. When the deformation potential model is correct, it is obvious that there will be no effect of strain on the electrical conductivity if all the electrom belong to a single extremum. The consequences of the shifting of the extremal energies on the electrical properties in more complex models will be discussed in succeeding sections. In some caaes it is necessary also to allow other band parameters, such aa the effective masses, to depend on the strain. Moreover, other, uoually small, effects, such as the change in the phonon spectrum caused by the strain, may not be entirely negligible. Such effects will be mentioned at appropriate places in later sections.
'' W.Shockley and J. Bsrdeen, Phys. Rev. 77,407 (1950); J. Bsrdeen and W. Shockley, ibid. 80,72 (1950).
toJ. Berdeen, Phye. Rev. 71, 1777 (1949).
168
ROBERT W. KEYES
11. TRANSPORT PROPERTIES
Many expositions of the theory of transport processes in semiconductors are available. A selection is given in the references.*0,17*41.4* Only statements of some important results which will be used later will be given here. A current is carried by the particles in each of the valleys when an electric field is applied to a semiconductor. The current in valley (i) is related t o the electric field by the conductivity tensor of the valley, jCi) = d(i) . F.
A basic assumption made in most of the succeeding sections is that the current in a valley is determined solely by the electric field and the prop erties of the valley in question, and is independent of the current in other valleys. Thus the total conductivity of the crystal is (11.1)
An exception occurs in the case in which electron-electron scattering h significant.4a Important properties of the single valley conductivity are: (a) d(’) is invariant under the operations of C(p&’)),where PO(*) is the point in p space which locates the energy extremum ( 2 ) . (b) An electron mobility can be defined by the equation = d(l)/n(l)e,
~ ( 1 )
In the case of nondegenerate statistics d(’) is proportional to n“) and y“’ is independent of n(”.I n the degenerate case p(’)may depend on (E(’) l), the difference between the extremum energy and the Fermi level. (c) The mobility 1) is determined by scattering processes which cause transitions from one electronic state to another. These transitiona take place as a result of deviations from perfect periodicity in the lattice. Such deviations originate either in the lattice vibrations or in static defects. I n a simple version of transport theory, the effect of the scattering is desdribed in terms of a relaxation time T , which represents the rate a t which the electron distribution function would return to equilibrium if the electric field were switched off suddenly. Usually T can be regarded
-
C. Herring and E. Vogt, Phye. Reu. 101, 944 (1956). W. Shockley, “Electrons and Holes in Semiconductors.” Van Nostrand, New York, 1950.
R. W. Key-, Phya. and Chem. Solids 8, 1 (1958); Bull. Am. Phyr. Soc. [2] P, 313 (1957).
PIEZORE818TANCE OF SEMZCONDUCMRS
169
88 a function of E, the electron energy measured from the extremal energy of the valley. The relation between the conductivity and the relaxation time ia
(11.2)
In the nondegenerate caae,fo = exp [({ to d =.
- E ) / k T j , and Eq. (11.2)reduces
2nega(Er)/3mkT
(11.3)
in which the angular brackets designate the Boltzmann average of a function of the energy: F(E)E+e-g'krd E (11.4) (F(E))§ /om Eie--E'kT dE
lo"
In the special case in which it is aasumed that I is independent of energy, Eq. (1I.2) becomes d = nesw/m. (11.5) Again a mobility independent of n can be determined. (d) In the case of certain scattering mechanisms, in particular, for scattering by lattice vibrations, the only energydependent factor in T is the density of the final states into which the electron may be scattered. If it is further aasumed that the energy is a quadratic function of p, and that the electron energy is conserved in the scattering process (negligible phonon energy), when this type of scattering occurs, 7 is proportional to E-i. A very important manifestation of the influence of the density of final states is found in the effect of band-band scattering on piezoresistance. This effect will be discussed in Part VI. (e) A common approximation, in which the integrals of transport theory have relatively simple form, is that 7 is proportional to some power of the energy, for example, = PE'. In t h b approximation Eq. (11.2) takes the form (11.6)
V. The Efiects of Hydrostatic Pressure
The UM of the derivatives of the components of resistivity with respect to pressure in determining the piezoresistance tensor haa been described in Section 4. It is important to note that the values of the pressure under which resistivity measurements can be performed are ordera of magnitude larger than the shear stresses which can be wed.
170
ROBERT W. KEYES
Thus, in contrast to the case of shear stress, important nonlinear effects, not describable by the first-order piezoresistance tensor II, often can be observed. Therefore, the interpretations of the effects of hydrostatic pressure in terms of electronic energy band structure will be referred directly to the pressure dependence of resistivity, rather than to the piezoresistance coefficients n,~. 12. EFFECTS OF PRESSURE ON
THE
BANDSTRUCTURE
The effects of hydrostatic pressure on the electronic properties of a crystal are simpler than the effects of shear stress, in the sense that the
Pressure (10) hg/crnz) FIQ.3. The effect of preseure on the electrical conductivity of a specimen Of indium antimonide [after R. W. Keyes, Phys. Rev. 09, 490 (1955)J.
application of hydrostatic pressure does not destroy any of the symmetry elements of the crystal. The symmetry degeneracies in the energy band structure are not removed. Thus, according to the deformation potential model of the piezoresistance phenomena, the parameters and conduction properties of a single band model are not affected by pressure. The important effect which the application of hydrostatic pressure can have on the energy band structure is to shift, with respect to one another, band edges which are not required by symmetry to be degenerate. If there is more than one band edge sufficiently cloee to the F e d
PIEZOBE8ISTANCE OF SEMICONDUCTORS
171
level to participate in the conduction process, the shifting of the band energies causes electrons to be transferred from one band t o another. This ahift can produce a pressure dependence of the conductivity. Among the band structure of Fig. 2, an effect of this type can occur in the intrinsic semiconductor, the semimetal, and the two-band models. The case of principal interest here centers about the effect of pressure on the intrinsic semiconductor. An illustration of the effect of pressure on the conductivity of a typical semiconductor is provided by Fig. 3,44 which presents data obtained from a sample of p-type indium antimonide at several temperatures and for pressures extending to 12,000 kg/cm2. These data can be understood readily in terms of the deformation potential model. A t the higher temperatures and atmospheric pressure the sample is intrinsic, i.e., most of the conductivity is associated with electrons that are excited across the gap from the valence to the conduction band. The energy gap increases as pressure is increased. Thus the number of electrons excited across the gap decreases in accord with Eq. (9.3), and the conductivity decreases rapidly. When the number of intrinsic charge carriers becomes sufficiently small compared to the number of acceptor impurities, the sample becomes extrinsic and the effect of pressure on the conductivity disappears. At 0°C the concentration of intrinsic carriers is completely negligible at pressures above 6000 kg/cm2, and the constancy of the conductivity from 6OOO to 12,000 kg/cm* is in good agreement with the deformation potential model. At still lower temperatures, at which the sample is extrinsic at atmospheric pressure, there is practically no effect of pressure on the conductivity. The interpretation of the large effect of pressure on the conductivity of an intrinsic semiconductor in terms of the change of energy gap between the valence and conduction bands was given originally by Bardeen.‘O The theory of the effect in terms of the deformation potential model is worked out easily. The conductivity is simply =
+
nebu,
(12.1)
pp)
where n is the intrinsic carrier concentration given in Eq. (9.3) and pI and p P are the mobilities of the electrons and holes, respectively, which
are assumed here to be independent of crystallographic direction. The dependence of u on pressure arises entirely from the pressure dependence of EQ in the model under discussion and therefore, from Eq. (9.3)) d-logu dP 14R.
dlogndP
W.Keyes, Phys. Rev. QQ, 490
(1955).
(- _2kT_) ddp’ Eo 1
(12.2)
172
ROBERT W. KEY=
TABLE V. PARAMETERS ASBOCIATEDW I T H THE EXCITATIONOF A N ELECTRON A C ~ O S S T H E GAP IN SEMICONDUCTORS FOR WHICHTHE PRESSURE EFFECT HAS BEEN MEASURED, INCLUDINQ REFERENCES TO THE HIGH PREB~URE EXPERIMENTS
Semiconductor Method' Tellurium Ge (111) Ge (000) Ge (100) Silicon Phosphorus InSb Inh MgSn GaSb
se
CdS
Ti02
el. opt. el. opt. opt. opt. opt. el. opt. el. el. el. opt. el. opt. opt. opt. opt.
dEo/dp (ev/106 kg ern-¶)
Activation enthalpy h(ev) 0.37
-
0.78
1.21 -
0.38 0.27 0.47
-
Activation free energy** V (ev) (cma/mola) 0.37 0.33 0.66 0.662
-
0.803
-
1.09 1.111 0.46 0.17 0.36 0.33
0.33
0.20
-
0.70 1.8 2.4 2.95
-
-
3.03
- 16
- 18 4.9 7.4 4 11 --I -2 -2 24 15 5 8 5 16 13 4 1.1
-
-
* Optical measurements and their interpretation will be discussed in Section 19. ** A t 300°K. *** Derived from thermoelectric power rneasurementa or cyclotron resonance, where available. Otherwise estimated from mobility according to R.W. Keyes, J . Appl. Phys. SO, 454 (1959). *P. W.Bridgman, Proc. Am. Acad. A d s . Sn'. 68, 94 (1933); 74, 195 (1938). * J. Bardeen, Phya. Rm. 76, 1777 (1949). D. Long, Phys. Rev. 101, 1256 (1956). L. J. Neuringer, Bull. A m . Phya. Soc. [2] 2, 134 (1957); Phys. Rev. 113, 1495 (1959). 'P. H. Miller, Jr., and J. Taylor, Phya. Rev. 76, 179 (1949); J. Taylor, ibid. 80, 919 (1950).
H. H. Hall, J. Bardeen, and G. L. Pearson, Phys. Rm. 64, 129 (1951).
W. Paul, Phys. Rm. 90,336 (1953). W.Paul and H. Brooks, Phys. Rev. 94, 1128 (1954). i A. Michels, J. van Eck, 6.Machlup, and C. A. ten Seldam, Phys. and Chem. Solidc
0
10, 12 (1959).
173
PIEZOREBISTANCE OF SEMICONDUCTORS
TABLE V (Continued) -
~~
Activation Effective massea*** entro>y El X-' conductor m.*/m. m.*/m. ev/lO' OK (106 kg em-*) (ev) Tellurium
0.2
0.25
Ge (111)
0.55
0.3
Phosphorua InSb InAs
Mg& GaSb s0
as
TiOt
0 0 4.4 4
0.196
-
0.76
-
4
2.3
0.5
1 0.037 0.064
0.7 0.18 0.33
4.1 4 -2.6 3.0 3.7 3.5 4.2 4.1 9 6.5 10.3
0.98 0.34 0.43 0.6
-
0.6 0.56 0.25 0.55 2.1
3.1 3.5 -3.8 5.7 -3.0 -8.0 0.8 2 2 8.2 -6.5 -3.3 -5.1 -3 -9 3.2 -2.4 -2.4
Explicit entropy 1080 s (k unite) (OK)-1
51
-
18
-
-3X
1.9 2.1 4.3 3.5 4.1 3.0
-
100 16 16
-
20 21 130 20 33
5.0 4.6 6.5 2.3 3.7 3.1 4.2 2.6 15
7 11
€ Y.I. Fan, M. L. Shepherd, and W. Spiteer, in "Photoconductivity Conference," p. 184. Wiley, New York, 1956. D. M. Wamhauer, W.Paul, and H. Brooke, Phys. Rev. 88, 1193 (1955); W. Paul and D. M . Warachauer, Phyr. and C h . Solids 6, 89 (1958). IT. E.Slykhouse and H. G. Drickamer, J . Phys. Chcm. Solids 7 , 210 (1958). li W.Paul and G. L. P e a m n , Phys. Rev. 89, 1755 (1955). M. I. Nathan and W. Paul, Bull. Am. Phys. Soc. [2] 2, 134 (1957). * W.Paul and D. Wamhauer, Phys. and Chcm. Solids 6, 102 (1958). 'R. W . Keyes, Phys. Rev. 92,580 (1955). * D. Long and P. H. Miller, Jr., Phys. Rev.98, 1192 (1955); D. Long, ibid. 99, 388
*
(1955).
'R. W: Keyes, Phys. Rev.99, 490 (1955). * J. Taylor, P h p . Rev. 100, 1593 (1955). 'J. Taylor, Bull. Am. Phys. Soc. 121 3, 121 (1958). "H.L. Suchan, 6.Wiederhorn, and H. G. Drickamer, J . Chem. Phys. 31,355 (1959). *For amorphous Se, aee R. 8.Caldwell and H. Y. Fan, Phys. Rev. 114,664 (1959). E. Guteche, Naturwiasensca~tm46, 486 (1958). 'H.L.Suchan, A. S. Balchan, and H. C.Drickamer, Phys. and Chcm. Solids 10,343 (1959).
174
ROBERT W. KEYES
The rate of change of the gap is usually referred to the volume and expressed in terms of a parameter E l : (12.3)
where x is the compressibility. The quantity EI haa the dimensions of an energy and has the magnitude of the shift of an energy band per unit of strain. The value of El has been determined for many semiconductors by studies of the pressure dependence of the intrinsic conductivity. A compilation of results of this type is presented in Table V. It is seen that the order of magnitude of E l is several volts. The dimensionless quantity d log u/d log V is of order E l l k T , or lo2.
Pressure
kg/cm*
FIG.4. The effect of preasure on the electrical resistivity of n-type G a h and InP in the extrinsic regime. The increasing elope of the G s h curve at high preee~n,b believed to be a result of the presence of another conduction band minimum [Ifter A. Sagar, Thesis, UniverEity of Pittsburgh (1959), unpublished; Phys. RCV.119,1633 (1958); 117, 101 (1960)l.
In Home cases an appreciable effect of hydrostatic pressure on the electrical conductivity is found even in an extrinsic single band semi~onductor.*~,a6.44-4' Examples are given in Fig. 4. I n these cases the effect cannot be interpreted in terms of the deformation potential model. An explanation must be sought in terms of a variation of mobility with pressure. The source of such a pressure dependence of mobility can be understood by considering the pressure dependence of the effective maea of an e l e ~ t r o n .This ~ ~ . dependence ~~ is apparentl6 from the f-sum rule:" D. Long, Phy8. Rm. 99, 388 (1955); 101, 1256 (1956). A. %gar, Phy8. Ra,.117, 101 (1960). " F. hits, "The Modern Theory of Solids," pp. 849-662. McGraw-Hill, New York, " 'O
PIEZORERISTANCE OF SEMICONDUCTORS
175 (12.4)
Here v is a band index and Y = 0 designates the band whose properties are being studied. By changing the interband energy differences EO- E,, hydrostatic pressure can produce changes in the effective masses and, consequently, in the mobilities. In certain semiconductors, namely, those in which the minimum of the conduction band occurs a t p = (OOO), this effect has been found t o be particularly simple. I n these materials the right-hand side of Eq. (12.4)is dominated by a single term, namely, that which represents the interaction of the conduction band with the valence band. The effective mass then is approximately proportional to the energy denominator of the dominant term, namely, the energy gap EQ.Using this proportionality, an approximation in which it is supposed that the mobility is inversely proportional to some power of the effective ma~s,a@,~a p (m*)-., for example, and Eq. (12.3), it is found that
-
d log P dP
-
d log EQ = XrEJE,. x r dlog V
(12.5)
The parameters of Eq. (12.5) are tabulated in Table VI for those semiconductors which have an (O00) conduction band minimum. The value of Eax-ld(log p)/dP is also given there. According to Eq. (12.5) this quantity is equal to rEl. It is seen that rE1 is approximately constant for the semiconductors considered, although d(log p)/dP varies by almost an order of magnitude. By comparison with the values of El, it is also seen that r sz 1. This value of r is in reasonable agreement with another estimate.4aIt is also seen from Table VI that the dimensionless elastoresistance coefficient x-1 d(1og p ) / d P can be as large as 25. An analysis of the experiments of Long“ suggests that the situation in tellurium may be analogous to that in the materials listed in Table VI. Long found that the mobility of holes in tellurium increases with increasing pressure a t a rate d log p/dP = 4-11.5 X lo-* (kg/cm2)-l. If this figure and the data of Table V are used in Eq. (12.5),it is found that r = 2.5. This value, although somewhat larger than those derived from Table VI, is not Thus the data are consistent with the view that the energy gap in tellurium is the denominator of the dominant term of Eq. (12.4),and that it is, therefore, a direct gap. As was seen in Fig. 3, the pressure dependence of electron mobility 1940; A. H. Wilson, “Theory of Metals,” pp. 46-47. Cambridge Univ. Prese, 19%. The content of Eq. (12.4) which ie important for the present argument can be &ply derived from perturbation theory (see Kittel,*@ pp. 683-586). “R. W. Keyes, J . A p p l . Phy8. SO, 454 (1959).
176
ROBERT W. KEYES
which is deduced from Eqs. (12.4) and (12.5) is not found in the hole mobility of the semiconductors of Table VI. This might seem surprising, since the principal interaction in Eq. (12.4) which was invoked to explain the mobility effect in n-type semiconductors was that between the conduction band and the valence band. The explanation is to be found in the fact that the p-type materials have the degenerate band structure illustrated in Fig. 2d. Most of the holes are found in the band with the large& density of states, the heavy maas band. The matrix elements in Eq. (12.4) for interaction of this band with the conduction band vanish, however. TABLEVI. ANALYSISOF THE VARIATION OF MOBILITY WITH PRESSURE FOB ELECTFLONS IN CONDUCTION BANDEXTILEMA at p = (000)
Material (kg/cm*)-l InSb
InAs GaAe InP GaSb
Ge
(ev)
60"J 354
0 27 0.47
9.6'
1.53 1.34
81
-
-
0.8 0 8
7 9 11
8
-
6.5 4 -
__ 9 9
.
At 300°K. D. Long and P. H. Miller, Jr., Phys. Reu. 98, 1192 (1955); D. Long, ibid. 99, 388 (1955).
* R. W.Key-,
Phys. Rev. 99, 490 (1955). D. Long, Phys. Rcv. 101, 1256 (1956). J. Taylor, Phys. Rev. 100, 1593 (1955). ' A. Sagar, The&, University of Pitteburgh (1959), unpublished; Phys. Rw. 119, 1533 (1958).
A. Sagar, Phys. Rw. 117, 101 (1960).
Thus the mass of this band is determined by interaction with bands much farther away in energy, and the effect described by Eq. (12.5) is negligibly small. The light mass band does interact with the conduction band and the mobility of the light holes should vary in accord with Eq. (12.5); however, because of the low density of light hole states the fraction of the holes in the light band and their influence on the conductivity are quite small. Thus, the extrinsic part of Fig. 3 represents a measurement of the mobility of the heavy holes. The effects of hydrostatic pressure on the properties of germanium have been intensively studied by workers at Harvard University and others. A review of some of this work has recently been given by Paul.48 W. Paul, Phys. and Chem. Solids 8, 196 (1959).
PIEZORESISTANCE O F SEMICONDUCTORB
177
Considerable information in addition to that given in Table V hss been derived from this work, and a summary of the way in which certain energy levels are found to vary with pressure is presented in Fig. 5. Some comments concerning important features of these results follow. (a) The gap increases at a rate dEa/dP = 5 X lo-' ev/(kg/cm*).*I-'O The pressure experiment does not allow the rate of change of the energy gap to be separated into a part due to the valence band and a part due to the conduction band. However, the rate of change of the conduction band
FIG. 5. The effect of pressure on the band structure of germanium. The left-hand aide shows the band structure of germanium [after H. Brooks, Advances in E l e ~ f t a k 8 and Eloelton Phys. 7 , 117 (1955)]and the right-hand aide shows the variation with Preaeure of the important band extrema [after W.Paul, J . phys. Chem. Solids & 198 (1959)l. The pressure dependence of levele esmciated with gold impurity are shown.
minimum has been deduced by Herring and ~ o - w o r k e r s by ~ ~ .the ~ ~U B ~ of a theory of the mobility which will be briefly discussed in Section 15h, and found to be dE("/dP = +1.3 X 10-8 ev/kg cm-2. The increase of gap with pressure can also be confirmed by optical However, there is some variation in the exact optical value of dEa/dP reported by different workers, which appears to be due to the use of
'Ow.Paul and H. Brooks, Phys. Rw. Q4, 1128 (1954).
c. Herring, T. H. Ceballe, and J. E. Kunder, Bell S y e h Tech. J . 88, 657 (1959). *'D. M.Warechauer, W.Paul, and H. Brooks, Phys. Rw. 98,1193 (1955); W.Pad and D. M . Warschauer, Phyr. and Chem. Solids 6, 89 (1958). "H.Y.Fan, M. L. Shepherd, and W. Spitzer, in "Photoconductivity COnference" (R.G . Breckenridge, B. R. Russell, and E. E. Hahn, eds.), p. 184. Wiley, New York, 1956. "T. Slykhouse and H. G. Drickamer, Phys. and Chem. Solids 7, 210 (1958). ''L. J. Neuringer, Bull. Am. Phys. Soc. [2]4, 134 (1957);P h p . Rcv. 118,1495 (1959). I*
178
ROBERT W. KEY=
different methods of analyzing the data.40.6*This topic will be diacuesed in more detail in Section 19. (b) The r2- minimum of the conduction band is too high in energy to have an effect on the electrical conductivity. It affects the optical absorption a t high absorption levels, however. An analysis of the preseure dependence of the a b s o r p t i ~ nshows ~ ~ ~that ~ ~ the energy gap between raI?¶'+, that is, the direct gap a t p = (OOO), increases with increasing pree sure at a rate 11 X lo-" ev/kg cm-'. (c) :The fenergy difference between the (100) minima and the top of the valence band decreases slowly with pressure. More important, however, is the fact that the energy of the (100) minima decreases with respect to the energy of the (111) minima. Therefore, even though at minima are a t least 0.15 ev above the (111) zero pressure the (W) minima and contain negligibly few electrons, above 10,OOO kg/cm* enough electrons occupy the (100) minima to affect the electrical resistivity."J' The resistivity of an extrinsic specimen below 10,OOO kg/cm* has only a very weak dependence on p r e s s ~ r e . b Above ~ - ~ ~ 10,OOO kg/cm* the resistivity increases sharply with increasing pressure for two reasons: (1) the electrons in the (100) minima have lower mobility than those in the (111) minima; (2) the (100) minima provide additional final states for scattering, so that the mobility of electrons in the (111) minima is d e c r e a ~ e d . The ~~-~ increase ~ continues to 50,OOO kg/cmz, and then the resistivity begins to decrease as the (111) minima becomes sufficiently high in energy that their influence on the conductivity starts to disappear." This interpretation of the phenomena is confirmed by measurements of the effect of pressure on the optical absorption spectrum in Ge64 and in Ge-Si alloysa0and by measurements of the drift mobility of the electrons." Other semiconductors in which two band models may be necessary to account for complicated pressure effects are tellurium,6¶*6aGBAS,*~," and GaSb.'" (d) The pressure dependence of energy levels sssociated with gold '6
6'
F. Herman, Phys. Rev. 96, 847 (1954). P. W. Bridgman, Proc. Am. A d . Arb Sci. 79, 127 (1951); S18 165 (1952); 8&71 (1953).
11. Brooke and W. Paul, BuU. Am. Phys. Soc. [2] 1,48 (1956). 69 M. I. Nathan, Thesis, Harvard University (1958), unpublished. ( 0 W. Paql and D. M. Warschauer, Bull. Am. Phys. Soc. [2] 1, 226 (1956). a* A. C. Smith, Bull. Am. phys. Soc. 121 9, 14 (1958); Thesis, Harvard Univemity (1958), unpublished. 6' H. Callen, J . Chcm. Phye. 31, 518 (1954). a* A. Nussbaum, J. Myers, and D. Long, Phy8. Rw. Leftere 3, 6 (1959). 6' W. Howard and W. Paul, private communication (1959). '8
PIEZORESISTANCE O F SEMICONDUCTORS
179
impurity in germanium is also shown in Fig. 5.64aMore complete studies of the pressure dependence of impurity levels in silicon are available.‘@ An important general feature of the results is that the impurity levels have a pressure dependence close to that of the top of the valence band and shift with respect to the conduction band.‘@KO interpretation of the effects is available. (e) Mertsurements of the galvanomagnetic effects61 and pieaoresistance tensor66 under high pressure suggest that, the pressure dependence of the parameters of the (1 11) conduction band extrema up to 10,000 kg/ cm* is weak. However, the effect of scattering into the (100) band on the electron mobility is measurable even below 10,OOO kg/cm2. The resistivity of relatively pure extrinsic n-type specimens increases by 8 % when The resistivity of the pressure is increased from 0 to 10,OOO specimens in which the mobility is limited by impurity scattering increases at a somewhat more rapid rate with pressure, h o ~ e v e r . ~ ’ . ~ ” For example, Bridgman found that the resistivity of a specimen with an electron mobility of about 600 cm2/V sec (0.004 ohm em) increases by 17% when the pressure is raised to 10,OOO k g / ~ m ~ This . ~ ’ difference between the pure and heavily doped materials is consistent with the band-band scattering model. When coulombic impurity scattering is the dominant scattering mechanism the contribution of high energy electrons to the current is larger than it is in the case of lattice scattering. The band-band scattering only affects the high energy electrons a t low pressures, because the (100) minimum is several kT above the ( 1 1 1 ) minimum. Thus, a t low pressures, band-band scattering, which produces the pres8ure effect, is relatively more important in samples in which coulombic impurity scattering limits the mobility. (f) Measurements of hole mobilityo1and piezoresistance‘6 also show that the parameters of the valence band have a very weak pressure dependence. 13. THETHERMODYNAMIC INTERPRETATION
The concepts used up to this point to discuss the effects of pressure on the electronic properties of semiconductors are those most useful in the theory of transport processes and optical phenomena. However, models of solids which are based on other concepts are useful for many purposes and provide insight into the physical factors which determine ‘4.
M. G . Holland and W. Paul, Bull. A m . Phy.9. Soc. [2] 4, 145 (1959).
‘* G. B. Benedek, W. Paul, and H. Brooks,Phy.9. Rev. 100,1129 (1955); M. I. Nathan, (4
W. Paul, and H. Brooks, Bull. Am. Phy.9. Soc. “4 t, 14 (1958). M. Pollek and R. W. Keyea, Bull. Am. Phy.9. Soc. [ Z ] 4, 185 (1959). M. Poll& and R. W. Keym, unpublished meauurements.
180
ROBERT W. KEYES
the sign and order of magnitude of the constant El of the deformation potential model. In order to relate the effects of preseure on the energy band structure to the parameters of other type of models the thermodynamic interpretati~n~’.’~ of the pressure effects will be described in this section. Let us regard a semiconductor as an open phase, that is, a phase in which the content of holes and electrons may be varied. The state of the system is defined by thermodynamic variables of the usual type, and by N, and N I , the numbers of electrons in the conduction band and of holes in the valence band, respectively. The Gibbs free energy of the semiconductor then is (13.1) h
Here Nk is the number of atoms of type k in the crystal and the P’B are the corresponding chemical potentials. Only the intrinsic caw, N. = N A = Ni will be considered, and pclc p. will be set equal to k.Ob Thus (13.2) G = Z N h p k N#.
+
+
The partial entropy, volume, and enthalpy associated with the formation of an electron-hole pair are (13.3) u =
(g)
(13.4)
T
h = p - T(g)
(13.5)
P
where B is the volume thermal expansion coefficient and x is the compressibility. To illustrate the physical origin of these thermodynamic excitation parameters, it is instructive to consider a simple model which accounta for the moat obvious properties of solids. Let the energy levels of the crystals be expressed in the form R N
(13.6) H. Jamee, in “Photoconductivity Conference” (R.G. Breckenridge, B. R. Ruasell, and E. E. Hahn, eds.), pp. 204-214. Wiley, New York, 1956. (* H. Brooks, Advancea in Elcetronica and Elecfron Phya. 7, 117 (1955). 6. This p should not be confused with the mobility. 6’
181
PIEZORESIBTANCE O F SEMICONDUCTORS
This model is a common one, discussed, for example, by Slater.bgHere, however, a dependence on the electronic state of the crystal, defined by the set of numbers [ l , j ] ,has been introduced. The electronic state can be specified by giving the states of the conduction band which are occupied by electrons and the states of the valence band occupied by holes; the sets of numbers [Z] and Lj] designate these states, respectively. Each takes on Ni different values, where Ni is the total number of intrinsic carriers. U(V,[l,j])is the lowest energy the crystal can have in the electronic state [l,J and depends on the volume. The second term on the right-hand side of (13.6) is the energy of excited lattice modes. The modes are numbered by the parameter a ;J a is the excitation number of mode a,and Ya(V,[l;il) is the frequency of mode a. The frequency also depends on V and on the electronic state. It is assumed that N , << N , the total number of atoms in the crystal, 80 that U and va can be expanded, and that, except for the kinetic energies of the electrons and holes, the dependence of U and vo on the electronic state occurs only through N i . Thus N.
N.
(13.8) The last two terms in the expansion of U represent, respectively, the kinetic energies of electrons and holes with respect to their band edges. In each case the summations extend over the Ni occupied states. The effective maases of the conduction and valence bands are m, and m.. The function w ( V ) and the parameters ha have been introduced as the coefficients of Ni in an expansion in powers of Ni. When Ni is given a particular value, the sum over states is readily carried through by standard methods for a crystal whose energy levels are given by (13.6), (13.7), and (13.8). I n the approximations of small Ni (classical statistics) and high temperature (classical excitation of the lattice oscillators), the free energy is found to be 8N
+ 2 kT log (hv,(V)/kT) + 2NikT (log N<- 1) - NikT log (NJV.Vz) + N,w(V) + 3NikTX (13.9)
F(V,TjNi) = U O ( V )
a-1
where N,and N. are defined in Section 9, and i; is an average value of A:
''J. c.Slater, "Introduction to Chemical Physics," p. 216. McGraw-Hill, New York, 1939.
182
ROBERT W. KEY=
X=
2
hJ3N.
(13.10)
a-1
It is found from standard thermodynamic relationships that ~1 =
kT log (Ni2/NcN,V’)
The equilibrium value of Ni is N, = (N,N,)iV exp
+ w + 3kTX.
[ -[” ;y].
(13.11)
(13.12)
The partial thermodynamic functions of Eqs. (13.3)-(13.5) are s = -k log (Ni2/N,N.V2)
+ 3k - 3kX + 2kTb - B(dW/d log V)
v = 2kTx - x(dw/d log V) h = w 28kTZ 3kT @T(dw/dlog V).
+
+
-
(13.13) (13.14) (13.15)
The important points to be made concerning these relationships are the following. (a) The expression (w 3kTX), which appears in Eqs. (13.11) and (13.12), usually is known as the “activation energy.” As is shown here, and more generally by James67 and Brooks,e* it actually resembles an “activation free energy.” If the effective m w e s , and therefore No and N,, are known, this quantity is determined through Eq. (13.12) by a measurement of Ni. A comparison of Eq. (13.12) with Eq. (9.3) shows that (w 3kTX) is the same as the Eo of the previous discussion. fb) The quantity fw - BT(dw/d log V ) ] is the slope of the plot of log (N,2/N,?.N,VS) versus T-1. This quantity is also sometimes known 88 the “activation energy” and sometimes as the “activation energy at absolute zero.” It is the quantity tabulated aa “activation energy” in Table V. As may be seen from Eq. (13.15) this quantity is essentially the activation enthalpy, for the terms 28kTa and 3kT are negligible. Even the term BT(dw/d log V) is usually small, so that h = w. (c) The entropy, given by Eq. (13.13), consists of a configurational part (the first two terms), a part which arises from the thermal expansion of the lattice (the last two terms), and an “explicit” part or activation entropy a t constant volume, -3kX. It is apparent from the definition of X that the explicit part is a result of the effect of electronic excitation on the vibrational frequencies of the lattice. The configurational entropy can be calculated from a knowledge of N,and N.. The contribution from thermal expansion is equal to ( B / x ) u . Thus the explicit entropy, -3kX, can be calculated from s if the effective masses and u are known. (d) The activation volume u is determined by measurement of the
+
+
PIEZORFXISTANCE OF SEMICONDCCTORS
183
pressure dependence of N,:v = -2kT(a log Ni/dP)T. Aside from a small term in kT, u = --El. The values of the activation thermodynamic parameters are given in Table V for crystals for which the pressure data are available. The sign of the effects is worth noting. The activation volume is positive for most of the crystals and the significance of this will be discussed in the next section. It is also Been that the entropy of activation at constant volume is positive for all of the crystals, i.e., the excitation of electrons lowers the vibrational frequencies. The sign and magnitude of the effect is in accord with a theory developed by Fan.7O In theories of electronphonon interaction, the strength of the interaction is measured by a deformation potential constant of the type introduced in Section 10. Fan found that the activation entropy is proportional to the square of this constant. The same measure of electron-phonon interaction determines the lattice mobility in the theory of Bardeen and Shockley,’* however, and by using their formula to eliminate the deformation potential constant from the formula of Fan’O it is found that the temperature dependence of the energy of a band extremum is (13.16) where Q is the volume of a unit cell and P L is the lattice mobility. Substituting a n average value of fit = 3 X 10-8 cm and the empirical )t volt-aec into Eq. (13.16) result‘8 that a t 300°K ~ ~ ( m * / n=~200/cm* gives dEci)/dT = - k . Thus a gap may be expected to decrease with increasing temperature at a rate of about 2k. This is seen from Table V to be the observed magnitude. I n most of the cases this “Fan effect” is the dominant part of the temperature variation of the gap. The lattice vibrations were treated classically in the foregoing for reaons of simplicity and clarity. If the lattice oscillators are treated quantum-mechanicafly, instead, it is found that the term 3kTX in Eq. (13.11) is replaced by 3N
(1/N)
1h
X [exp (hv,/kT)
Ya a
- 11-l.
(13.17)
a-1
Differentiation of this expression with respect to T shows that in Eq. (13.13) the electron-vibration contribution to 8, - 3kX, is replaced by aN ..
8ev
=
2 (2)
-3 N!a-1
lcH. Y.Fan, Phys. RGV.a,900 (1961).
exp (hva/kT) [exp (hv,/kT) 112’
-
(13.18)
184
ROBERT W. KEY=
The form of this contribution to s thus resembles that of the vibrational specific heat C.. I n fact, if A, = X, independent of a,then (13.18) becomes 8.. = -XC./N. The contribution of the thermal expansion of the lattice to s also haa a temperature dependence similar to that of C,.6sOptical studies of energy gaps in semiconductors, which will be discussed in detail in Section 19, confirms that the temperature dependence of the gap ie qualitatively similar to that of C.. At. temperatures sufficiently high that the lattice vibrations are classically excited and C. has the Dulong and Petit value, s is a constant, and the treatment summarized in Eqs. (13.9)-(13.15) is applicable. At low temperatures C. becomes very small and, correspondingly, the temperature dependence of the gap is very weak. The quantum-mechanical treatment of the lattice vibrations produces a minor modification of Eq. (13.11) even in the high-temperature limit. Because of the temperature dependence of s, a linear extrapolation of the high temperature activation energy to T = 0 does not give w, or the gap at T = 0. Thus a more appropriate way of writing p at high temperatures is p =
+ w + 3kX(T - To).
kT log ( N , 2 / N , N , V * )
(13.19)
The effect described by T o is represented by a constant “8” in the paper of James.67An estimate of T o can be obtained by expanding (13.17) in the high-temperature limit, setting X, = X, and representing the distribution of vibrational frequencies by a Debye model. This calculation gives TO= (9) 00, where eo is the Debye temperature. The data for germanium and silicon (see Section 19) suggests TO= 0.2 8 0 . 14. THEBONDINCI INTERPRETATION Some understanding of the sign and magnitude of the values of v can be gained by recognizing that the bonding in semiconductors is primarily electronic. Since the electron states which are studied in experiments on the pressure dependence of transport properties participate in the bonding a rough idea of certain of the characteristics of these states can be obtained16 from established ideas concerning the nature of the covalent bond. The bonding of a covalent solid can be understood in the following way: the behavior of a certain set of electronic energy levels is followed through the process of bringing isolated atoms together to form the solid. It is found that some of the energy levels in the solid are below the initial atomic levels and others are above the initial atomic levels. The binding energy of the solid originates in the occupation by electrons of the depressed levels. These states, which are the source of binding, form the
PIEZORESISTANCE O F SEMICONDUCTORS
185
valence band in homopolar semiconductors. The situation is illustrated in Fig. 6. Those energy levels which are raised above the atomic levels have an antibonding character, since to the extent that these levels are occupied, the energy of the system is lowered in going from the solid to the separated atoms. As is shown in Fig. 6, these antibonding states form the conduction band in a simple semiconductor. This accounts for the fact, noted in the preceding section, that the equilibrium volume of a semiconductor usually is increased when electrons are excited into the conduction band.1b*64 The effects are analogous to those produced by the electronic excitation of m o l e ~ u l e s . 7 ~ ~ ~ ~
P Atomic
Stator
I
1
Equilibrium Valuo
L
Lattice Constant
ho. 6. An illustration of the relative poeitions of energy lev& of mmmon samioonductoraarieing from varioua atomic s t a h . Differentmaterials differ in the relative height of the d band, shown by the dotted line. It ie much higher than shown here in mrny cryetala, and has no influence on the properties of the d i d .
The expected order of magnitude of v is also apparent from this viewpoint: the excitation of one electron per atom would essentially destroy the bonding of the solid and produce a change of volume of the same order of magnitude as the volume itmlf. Therefore u must be of the order of magnitude of the molar volume of the crystaI. Thia is seen to be the c & ~in e Table V. However, the quantity u is large and negative in tellurium and in phosphorus, in contradiction to the prediction of the simple bonding I' R. W. Key-, J . C h . Phyu. 49,623 (1958); Ndure la* 1071 (1958). '1. R. W. Krvss, BUU. Am. Phyu. Soc. I 21 4, 145 (1959).
186
ROBERT W. KEYE6
picture. A suggestion for the explanation of this anomaly in the case of tellurium may be found in the work of Callen.O* He proposed that the conduction band and the valence band in tellurium originate in different atomic states. More specifically he suggested that the valence band is derived from atomic p function, in conformity with the observed 90' bond angles, and that the bonding d functions are lower in energy than the antibonding p functions a t the equilibrium lattice parameter. Thus the bonding d functions form the conduction band. The situation is illustrated by the dotted line in Fig. 6, which shows that the d conduction band can have a bonding character which is stronger than that of the valence band. The excitation of an electron across the gap into the conduction band therefore decreases the equilibrium volume of the lattice and provides an explanation of the negative value of v. A similar situation may occur in phosphorus. The foregoing description of electronic states in terms of bonding concepts cannot be expected to give a quantitative description of the effects of strain. Only one-electron states that are very close to the band edge are observed in experiments on semiconductors. The behavior of these states may not be typical of the behavior of the band as a whole. In fact, the experiments on germanium, cited earlier, show that the energy variexj with pressure a t considerably different rates in different parts of the conduction band. Also, even in relatively simple cases, e.g., germanium, the mixture of atomic states other than those which form the valence band into the conduction band wave functions may be significant. Nevertheless the qualitative features of the effect of pressure are accounted for by the bonding interpretation. The connection between the energy gap and electronic bonding is also supported by a study of semiconductor phases. In a caae in which bonding d functions are not much higher in energy than the bonding p functions, the energy gap may vanish if the pressure is sufficiently high. I n this case electrons will occupy the bonding d function even in the unexcited state. It might be expected that other crystal structures which take better advantage of the overlap of the d functions would then become more stable. Phase transitions actually have been found in tellurium" and in black phosphorusTsat about the pressures a t which the energy gaps vanish. It is worth noting that this suggested participation by the d band in the electronic phenomena is found in those cases in which the bond angles are 90" (tellurium and phosphorus). This indicates the presence of a valence band and bonding energy arising from atomic p functions. The P. w.Bridgman, Phya. Rev. 48,896 (1935); Proc. Am. Acad. Arts Sci. 81,165 (1952). 7J
P. W. Bridgman, Proc. Am. A d . Arla Sci. 76.55
(1948).
PIEZORESISTANCE OF SEMICONDUCTOR8
187
participation of the d band is not found in crystals having tetrahedral bonding, in which the bonds are hybrids of 8 and p functions. The latter type of bond allows a much greater lowering of the energy than the former, and removes the valence band farther from the d levels. VI. The Effects of Shear Strain
An active interest in the effects of elastic shear strain on the electrical conductivity of semiconductors was stimulated by the experiments of Smith.lo They showed that the shear piezoresistance effect is sensitive to certain details of the energy band structure. The most fruitful application of the technique has been to the multivalley semiconductor; a large portion of this section is devoted to that subject. As mentioned earlier, the extent to which a solid can be deformed in elastic shear is quite small, strains less than lo-* ordinarily being employed in experiments. Thus the effects of principal interest are linear in the strain, and are described by the phenomenological coefficients introduced in Section 1. However, significant nonlinear effects have been observed in germanium at low temperature.” 15. THE MULTIVALLEY SEMICONDUCTOR
a. The Electron Transfer EIffect The interpretation of the large piezoresistance effects in semiconductors in terms of the multivalley model was developed by Smith,’O Herring,” and Adarns.” The discussion of this section is baaed on the work of these authors. Only relatively simple cases which illustrate the principles involved will be discussed in detail. The multivalley band has been described in Section 9. The basis for a theory of the effect of shear strain on the conductivity properties of such a band was given in Sections I0 and 11. The origin of the effect is illustrated qualitatively in Fig. 7, which shows a constant energy surface which could occur in a two-dimensional square lattice. There are two types of symmetry operation in this p space which allow any one of the valleys to be transformed into any other, namely, Ud, reflection in the (11,) lines (a) and (b), and C,, which is a rotation by r / 2 about the origin. Thus the minima are degenerate in the unstrained state, and are occupied by the same number of electrons. Addition of the conductivies associated l4€ I.
Fritcsche, Phya. and Chem. Solids 8, 257 (1959); Bull. Am. Phyr. Soc. [2] 4, 185
(1959).
‘6E. Nl Adams, Chicago Midway Laboratories Technical Report, CMGTN-PS (1964), unpublished.
188
ROBERT W. K E Y W
with each of the valleys shows that the total conductivity is isotropic in the plane. If now the crystal is strained by a tension in the x direction, the x axis no longer is equivalent to the y axis, and the aymmetry elements Ud and Cc are destroyed. The degeneracy of the minima is thereby removed. Suppose, for example, that the energy of those on the y axis is lowered with respect to those on the x axis. Electrons then will be transferred from the x valleys to the y valleya. If the energy surfacea have the shape shown, the conductivity of the y minima will be higher in the
FIG.7. A constant energy surface of a multivalley semiconductor with an u b of fourfold aymmetry.
x direction than in the y direction. Hence the conductivity will become anisotropic in the plane, being larger in the z direction. Also it can be seen that tensile forces applied to the square lattice in a (11) direction will not destroy the symmetry element 01. Therefore, the symmetry degeneracy of the valleys will not be removed, and no pieco-
resistance effect will be produced. Thus, the symmetry propertiea of the valleys can be derived from a study of the dependence of the piesoresistance effect on crystallographic orientation. In fact, the results of Smithlowere among the first to give indications of the multivalley nature of germanium and silicon. A quantitative discussion of the effect will be limited here to the caae of a single, nondegenerate, multivalley band. Intervalley scattering will be neglected at first. Thestrains ordinarily used in experiments involving elastic shear are quite small; the effects produced can be regarded as
189
PIEZOREBISTANCE OF SEMICONDUCTOR6
differentials. Thus the change of a quantity which a d from the introduction of a strain into B crystal will be denoted by placing a prefix 6 before the quantity. The conductivity of a valley, in the notation of Part IV,is
(13.1)
d(l) = n(*)evt*)
and the change in the conductivity produced by the strain quence of the electron transfer mechanism is ad(*) =
88
a conse-
(15.2)
~ ~ ( I ) Q ( I ) ~
It can be seen from Eq. (9.2)that the bn(" are related to the shifts in the energy extrema by the equation 6n(" = -nb(E(*'
- {)/kT
(15.3)
in which n is the value of n(*)in the unstrained state. Since a single band is under discussion, the total number of electrons must remain constant:
Zbn(" = 0 = n(-l/kT)[Z6E(*)-
vat].
(15.4)
Here v is the number of valleys in the band and Z denotes summation over all of the valleys. If (15.4) and (10.1) are substituted into (15.3), it is found that = n( - l / k T ) [ S ( * ) v-'ZS(')] :e. (15.5) We obtain, using (15.5),(15.2),and the fact that the change of the total conductivity is given by 6d = Z6d(*), 6d = ne(- I/kT)Zy(')[II'*)- v-lZS(*)]:r
(15.6)
which is the desired result. The factor multiplying the strain tensor e is the fourth rank elastoconductivity tensor M,which waa defined in Section 8. The relationship between M and the piezoresistance tensor ll also was described in Section 8. In the following paragraphs some comments will be made concerning Eq. (15.6),its application to special c a w s , and its modifications in certain situations. (a) The trace of d is not affected by the strain; i.e., tr 6d = 0. The reamn for this is clear from Eq. (15.2). The trace of the mobility tensor is independent of (i). Hence t r 6d = e(tr y)Zbn(') = 0. (b) I n the case of a pure dilation 6d = 0. This result can be understood in the following way: a pure dilation implies e = d. However, W : l = t r W. Since the valleys can be transformed into one another by rotations and the trace of a tensor is invariant under rotation,
190
ROBERT W. KEYES
tr SCi) = t r S, independent of (i). Thus the contraction of 1 and the bracketed factor in Eq. (15.6)vanishes. If we multiply by the elastic compliance tensor C-l, it is seen that the result 6d = 0 also holds for the strain produced by hydrostatic pressure in a single band model. The reason 8d vanishes for these strains is that they do not remove the symmetry degeneracy of the E(0 and hence do not produce electron transfer. (c) The tensors v(') can be transformed into one another by the crystal symmetry operations. The same is true of the S(a).Thus the summation in Eq. (15.6)can be evaluated from a knowledge of the tensor for a single valley plus a knowledge of the crystal symmetry." Therefore formulas relating Zv(i)a(i) to the components of a single and a single S can be given for each crystal class. The most important operations are rotation axes, and the formulas for the sum Zv(i)I(i) are given in Table VII for valleys which can be transformed into one another by three-, four-, and sixfold axes. The effect of additional symmetry elements, such as a reflection plane or a twofold rotation axis, usually is to make some of the terms of the summations vanish in a simple way." The formulas for the important case of crystals of the cubic classes 432, 33m and m3m are also included in Table VII. (d) In cubic crystals the conductivity is a scalar, d = a1 = (1/3)neu tr p. It is customary to define the elastoconductivity tensor as m = u-I M, a dimensionless quantity, in this case.28oA component of m is of order of magnitude E,/kT, which may be lo2or 10'. (e) If the anisotropy of v is independent of temperature, Eq. (15.0) shows that m is proportional to T-*. To the extent that the elastic constants are independent of the temperature, this is also true of cr-ln, the piesoresistance tensor divided by the conductivity. (f) In the well-studied cubic semiconductors germanium and silicon, the valleys lie on the axes of three- and fourfold rotational symmetry, respectively. This implies that the second rank tensors associated with the valleys have rotational symmetry If a(i)is a unit vector lying along the axis of rotation of valley (i), g has the form gCi) =
Zdl + Eug(s)a(*)
(15.7)
in the notation established by Herring and V ~ g t . Then ~ ' v(') is customarily expressed in a form such as
(15.8) Equation (15.8) means that the conductivity perpendicular to the axis of rotation is K times the conductivity parallel to the axis. The use of '6
R. W. Keyes, J . El~ctronics2, 279 (1956).
PIEZORESISTANCE OF SEMICONDUCTORS
191
TABLEVII. THECOMPONENTS OF THE FOURTH RANKTENSOR V-~~(~)W) [SEE &J. (15.6)]I N TERMB OF THE COMPONENTS OF THE y A N D 8 OF A SINQLEVALLEYFOB CBYSTAMWITH CERTAIN SIMPLESTYMETRIES Cmmponenta
Value Threefold axis and sixfold axiabVc
11
12 13 31 33 1 .Id 41d 15d
51d 16
44 15
Components not listed vanish. Compare with Table 11. See R. W. Keyea, J . Elsctronica 2,279 (1956). Axis 3 is the rotation axis. For a hexagonal axis the 14, 15, 41, and 51 components vanish. The axe8 1, 2, 3 are the cube axes.
192
ROBERT W. KEY=
(15.7), (15.8))and Table VII giveszb = mlt = 0
(15.9)
for Ge(a = (lll)/&), wall=
z,, K - 1 3 -kT 2K + 1 ~
z,, K - 1
m1'=*kT2K
+1
(15.10)
rn4( = 0
for &(a = (001)). A transformation which introduces the pieroresistance tensor shows that the corresponding coefficients T,,A also vanish in the same way as the m , ~ .
9
Fro. 8. The effect of degeneracy on electron transfer. The ordinate is the factor by which the &a(') of Eq. (15.3) are reduced by the effect of degeneracy. In the relaxation time approximationwith the scattering exponent u = 0 this is also the factor by which the elaetoconductivity, %. (15.6), is reduced. (I f / k T .
-
(g) In the case of the semiconductors of paragraph (e) with deformation potential tensors of the form of Eq. (15.7) the rate of change of the band edge with dilation, # t r E,is (a, # Ea). (h) Statistical degeneracy of the electron gas reduces the piezoresietance and weakens its temperature dependence. The modifications of the reasoning of Eqs. (15.1) to (15.6), which are required to take account of the effects of degeneracy, are as follows." (1) I n the degenerate c w , Y ( ~ )in Eq. (15.1) depends on (E") - {) in general; therefore, Eq. (15.2) must be modified by the addition of a term n(')e6p"). (2) Equations (15.3)
+
193
PIEZORESIBTANCE OF BEYICONDUCTORB
muat be replaced by one baaed on Eq. (9.4). It is found that the quantities h(Q appearing in Eq. (15.3) are reduced by a factor F+({/kT)/2F&/kT), which ie plotted in Fig. 8. If 8p(i) = 0, the result for the elastoconductivity, Eq.(15.6), is reduced by this same factor. In any case dyci)/d(Eci)- 1) is subject to the same symmetry restrictions aa is y("; therefore, the ' ) {) to Eqs. (15.2)-(15.6) addition of a term proportional to d ~ ( ' ) / d ( E (does not change the symmetry of the elastoconductivity tensor. In the relaxation time approximation of Eq. (11.6)) 7 is proportional to E' and 8 ~ " can ) be calculated easily. The calculation shows that the contribution of 6y(n can be significant and that the result given by Eq. (15.6) is reduced by a factor (8
+ +>F.-,(T/kT)/F.+t(l/kT).
If
8 = 0, by(') = 0, and the elastoconductivity arises entirely from the 8n") term. For nondegenerate statistics the degeneracy factor reduces, of course, to unity. In the extreme degenerate case it amounts to replacing the factor (kT)-l by the temperature-independent factor (8 #/{. It is because of this degeneracy effect that metals do not exhibit the large piezoresistance effects which are common in semiconductors.
+
b. Intervalley Scattering Another important contribution to the piezoresistance of the multivalley eemiconductors, which may be of the eame order of magnitude as the contribution from electron transfer, arises from intervalley scattering. In this scattering process the initial and final electron states are in different valleys. There is a contribution to the piezoresistance aa a result of intervalley scattering because the density of final states depends on the relative position of the valleys in the density When an electron is scattered from valley ( i )to valley (j), of final states is proportional to the square root of the energy difference between the final state and the extremum energy of valley ( j ) . Thus the in which E is the density is proportional to [E (E(n E(')) f ha]+, energy measured in valley ( i )and liw is the phonon energy. The plus sign corresponds to phonon absorption and the minus to phonon emission. In the unstrained state E(i) - E(') = 0 for valleys of the aame band. If, for example, EO is increaeed relative to E(i),the density of final states for the scattering process (i)+ ( j ) is decreaaed. Thus the scattering probability for an electron in valley (i) is decreased, and the conductivity arising from an electron in valley (i) is increased. Since the same energy difference a180 causes a transfer of electrons from valley 0') to valley (i), both effects cause the conductivity in ( i ) to increase, and reinforce one another.
-
-
194
ROBERT W. KEY=
A quantitative evaluation of the effect of intervalley scattering on elastoresistance requires detailed theories of both the intravalley and intervrdley scattering. Qualitatively, it is to be expected that the stronger the intervalley scattering is, the larger will be its effect on the elastoresistance. Some insight into the relationship between the strength of the intervalley scattering and the enhancement of the elastoresistance can be obtained by applying a detailed balancing condition to a relatively simple model of the intervalley scattering processes.14The assumptions of the model, which was first introduced by Herring," are &s follows: (1) all of the scattering processes involved can be described by relaxation times; (2) in the unstrained state, the probability that an electron in valley (i) will be scattered to some state in valley 0') is independent of (21 and ( j ) . Under these assumptions the relaxation time for an electron in valley (i) can be written
(15.11)
-
Here T O is the relaxation time for intravalley scattering, is the relaxation time for (i) (j)scattering and Z' means summation with omission of the term j = i. In the unstrained state all of the scalar functions of 8 multivalley band are equal. According to assumption (2), then, all of the .Ci ) have the same value, say T r . In the presence of a strain, the differential of (15.11)is
(15.12) since, according to the deformation potential model, T O is independent of strain. The relaxation time ~ ( ' 1 ) depends only on the energy difference E") - E(i)and the dependence can be written
(15.13) All of the scalar functions such as f and the T'S are functions of the electron energy. Now consider the principle of detailed balancing. The condition that the number of electrons scattered from valley (i) to valley (j)be equal to the number scattered in the reverse direction is
(15.14) The averages, have been defined in Eq. (1 1.4). Taking the differential of
195
PIEZORESIBTANCE OF SEMICONDUCTORB
Eq. (15.14),applying Eqs. (15.13)and (15.3),and recalling that T ( V )
=
TZ
to zeroth order, it is found that
'L)
=
2($).
(15.15)
\TI
The relaxation time approximation to the conductivity is given by Eq. (11.3). From this
(15.16)
It is found from (15.12)and (15.3)that (15.17) If (15.17)is substituted into (15.16), (15.18) Thus it is seen that the effect of electron transfer is enhanced by the bracketed factor. Using (15.15), the bracketed factor in Eq. (15.18) can be rewritten
where K = (l/zI)(Ez2)/(Er). The physical interpretation of the intervalley scattering factor written in the form of Eq. (15.19) is clear. The quantity (Efr*/zZ*)/Cf/~z*)(~z*) is a numerical factor of order unity, whose exact value is a function of the form of the energy dependence off, z, and TI. If for example, f, 7, and zz are assumed to be energy-independent, the value of this factor is exactly one. The factor K is of order TITI, the fraction of the total scattering which arises from a single (i) --$ (j) type of scattering. Its exact value also depends, of course, on the form of the energy dependence of z and 71. However, the enhancement of the piezoresistance is roughly proportional to the strength of the intervalley scattering, as predicted by physical intuition. It is apparent that the intervalley scattering component of the piezoresistance can be of the same order of magnitude as the electron transfer component. By assuming particular mechanisms for the intravalley and intervalley scattering, it is possible to evaluate the factors of Eq. (15.19) quantitatively. This has been done for phonon-induced intervalley scattering in germanium and sili~on.*6**~ The T-' dependence of the piezoresistance on temperature, a~ given,
196
ROBERT W. KEY=
for example, by Eq. (15.9), is distorted by intervalley scattering." The excitation of the intervalley phonon is controlled by a Planck function. For temperatures comparable to L / k , the excitation is a rapidly increasing function of temperature. Thus, the intervalley scattering is very weak a t low temperatures, since there are few excited phonons which can be absorbed and few electrons with sufficient energy to emit a phonon. Ae the temperature is increased and intervalley scattering appears, the resulting increase of the K in Eq. (15.19) weakens the 2'-l temperature dependence of M.At temperatures sufficiently high that the phonon ie excited classically M becomes proportional to T-' again. The influence
I
1
I
I
V4
1/2
I
I
I
2 4 Temperature in Units h w / k
I 8
-
FIQ.9. The effect of intervalley scattering on the temperature dependence of elmtoresistance. The dmhed lineg ehow the T-1 dependence characteristic of the highand low-temperature limits.
of intervalley scattering on the temperature dependence of M is illustrated in Fig. 9. The value of the characteristic temperature Aw/k ie expected to be comparable to the Debye temperature. Galvanomagnetic effects in semiconductors are particularly sensitive to the relaxation times, and provide well-known methods for the measurement of mobilities. It might be expected, therefore, that the effect of strain on the scattering times as described in this section could be m e w ured by determining the effect of strain on the galvanomagnetic coefficients. This is in fact the case; a general theory of the elastogalvanomagnetic effects, including the effect of intervalley scattering, can be worked out along the lines of this section and Section 15s. It shows that the elastogalvanomagnetic effects are sensitive to the strength of the intervalley scattering. 1 * ~ 4
PIEZOREBISTANCE OF BEMICONDUCTORS
197
Their Interpretafion Since the initial work of Smithlo on silicon and germanium, piezoresistance studies have been carried out on many other materials. Large effects have been found in several of these, that is, a component of the dimensionless tensor m which ie of order of magnitude 100. It cannot always be concluded that these substances are multivalley semiconductors, for large effects can arise in another way, a~ will be seen in c. Ezperimenlrrl Studies and
\
\ Germonium
I
Silicon
FIQ.10. The room temperature piemresistance effect meesured in the longitudinal arrangement (a) of Fig. 1 in germanium and silicon for sample axea normal to (110). Valuee of II ( - a p / p X ) are in units of lo-'* cm*/dyne. [Based on data of C. 6.Smith, Phy6. Rav. 94, 42 (1954).] a eucceeding section. Theory'O and other types of experiment'' suggest, however, that in the n-type cubic semiconductors the multivalley model ie the appropriate one to adopt to explain a large piezoresistance effect. An analysis of the dependence of the effect on cryetallographic orientation gives some information about the symmetry properties of the valleys and their location in p space in those semiconductors in which a large effect ie found. AE an example the resulta of Smith10 are preeented in Fig. 10. This figure ie a polar plot of the longitudinal piezoreeiatance I7 B. Lax, Rwa. Modmr Phy6. 80,!1!22 (1958).
198
ROBERT W. KEYES
effect [arrangement (a), Fig. 11 in germanium and silicon as a function of the direction of the specimen axis. The anisotropy is very large. A comparison of the results for germanium with Eq. (15.9) shows that the criterion for (1 11) valleys is satisfied very accurately. For silicon, Eqs. (15.10) also give a very good representation of the data, although the deviation from the deformation potential model is more apparent than in germanium; for example, the pieeoresistance in the (1 11) orientation is almost 10% of that in the (001) orientation. The examples of Fig. 10 illustrate two especially simple piezoresistance tensors which allow the models on which they are based to be recognized readily. The analysis of more complicated tensors may not be equally clear, although a model sometimes can be formulated with the assistance of the results of other experiments.abJ6The important qualitative feature to be kept in mind is that strain may destroy some of the symmetry elements of the crystal. A large elastoresistance effect is produced if the elements destroyed are those which were the source of the degeneracy of the valleys. For example, the valleys in silicon are transformed into one another by rotation about any of the threefold axes. Extension along one of the threefold axes does not destroy the “threefoldness” of that axis and therefore does not remove the degeneracy of the valleys; consequently, according to the deformation pcitential model, there would be no elastoresistance effect. On the other hand, if a silicon crystal is extended along, say, the (001) axis, the degeneracy of the (001) valley with the (100) and (010) valleys is removed and there is a large effect. This is seen from the results presented in Fig. 10, which show that the piezoresistance for a silicon sample having its axis in a (111) direction is an order of magnitude smaller than that for a sample in which the axis is in a (001) direction. The principal parameters that enter into the theory of the pieaoresistance effect are the shear deformation potential constants, of which there is only one, Z,, in the model associated with Eq. (15.7). The determination of this parameter from equations such as (15.9) and (15.10) is complicated by two circumstances: (1) the value of K is not known; and (2) these equations neglect intervalley scattering. This difficulty haa been resolved for germanium by the calculations of Herring and Vogt41 and D ~ m k e , ’which ~ will be described in a succeeding section. However, there ia much uncertainty in the determination of Z, for less intensively investigated materials and only estimates can be obtained. One fact which should be noted in Eqs. (15.9) and (15.10) is that the m , ~are fairly insensitive to K if K is much different from one, as is usually the caae. ‘8 W.Dumke, P h p . Rw. 101, 631 (1966).
PIEZORESISTANCE OF SEMICONDUCTORS
199
Only “small effects” are found in other materials, i.e., all components of m are less than about 10. While such a result could arise in a multivalley model if Z, happened to be small, another, and more probable, explanation is that the extremum of the conduction band is a t the point p = (0oO) in p space. In this caae there ia only one such extremum and TABLEVIII. RESULTSOF PIEZORESIBTANCE MEASUEEMENTS ON S E M I C O N D ~ C ~ R S , INCLUDKNG THE LOCATION OF THE BAND EXTREMA AS DEDUCED PROM THESE MLABUREMENTS, AND ESTIMATES OF THE SHEAR DEFO~MATXON POTENTIAL CONSTANTS ~~~~
~
Material
~
Crystal structure
Ge Si InSb InA8 GaAa GaSb
Cubic Cubic Cubic Cubic Cubic Cubic
InP Bi Te
Cubic Trigonal Hexagonal
a
~~
Inforniation about band structure (111) valleys. (conduction band) (100) valleysa (conduction band)
S, (ev)
17‘-’ 76
(OOO) min’*p (conduction band) (OOO) m i d (conduction band)
(000) min’ (conduction band) (OOO) min and (conduction band) (111) valleysi,1 (OOO) minB-*(conduction band) Two bands, one multivalleyi~a Consistent with pressure results”
2Oi.i
C. S. Smith, Phys. Rev. 04, 42 (1954).
* C. Herring and E. Vogt, P h p . Reu. 101,944 (1956).
c C. Herring, T. H. Gebde, and J. E. Kuntler, Bell S y s h Tech. J . 58,657 (1959). ’H. Fritwche, phy8. Rev. 116, 336 (1959). ‘ G . Weinreich and H. G. White, Phya. Rev. 106, 1104 (1957); G. Weinreich, T. M. Sanders, Jr., and H. G. White, ibid. 1 1 4 3 3 (1959). R. I?. Potter, Phya. Rw. 108, 652 (1957); A. J. Tutmlino, ibid. 100, 1980 (1958). F. P. Burns and A. A. Fleischer, Phys. Rev. 107, 1281 (1957). @
J. ”urzolino, Phgs. &. 112, 30 (1958). ‘ A . %gar, Thesis, University of Pithburgh (1959), unpublished; Phya. Rev. 119,
a A.
1533 (1958). A. &gar, Phya. Rev. 117,93 (1960). A. Sagar, Phya. Rw. 117, 101 (19600). M. Allen, Phyr. Rev. 42, 848 (1932); 48, 248 (1936). R. W. Key-, phy8. Reu. 104, 065 (1956). a D. b n g and R. Pu,Bull. Am. Phys. SOC.[2] 8, 15 (1958).
no electron transfer effect is possible. Although one might question the appropriateness of calling this a “multivalley ” case, the theory presented doea apply with v = 1 and %, = 0. A summary of information about the band structure of several materials which can be obtained from their piezoresistance coefficients is given in Table VIII.
200
BOBERT W. KEYElJ
Still another line of investigation was stimulated by the work of Smith, namely, the use of piezoresistance aa a tool for obtaining more detailed knowledge of the band structure and acattering proceasea in germanium and for establishing the validity of the model on which the theory of the piezoresistance effect is baaed. A brief summary of this work will be given as an illustration of the types of information which can be derived from piezoresistance studies. As I-malready been described, one of the most important facts shown by Smith's experiments was that the coefficients mll and mll vanish in germanium, in agreement with the model which places the energy minima on the (1 11) axes in p space. This model is also established by other evidence." Measurements of the piezoresistance under high hydrostatic preaeUrela8show, however, that the coefficients do not satisfy Eqs. (7.9) above about 15,000 kg/cm2. This result is explained by the model of the effects of pressure on the band structure of germanium which WBB described in Section 12. Proportionality of the large piezoresistance coefficient of germanium to T-1 has been wtablished over a wide temperature range.26JP.80This proportionality provides a good verification of the electron transfer mechanism, and also indicates that in the temperature range in question (up to 380°K) intervalley scattering is not strong. Accurate values for the intervalley scattering rates in the temperature range 20°K 160'K were obtained by Weinreich et aL8I from their measurements of the acoustoelectric effect. The deactivation of the intervalley phonon a t low temperatures is apparent from the acoustoelectric results, and the rate of phonon-induced intervalley scattering can be written in the form l/rr = (loll sec-l) exp (-315'K/T). The figure 315'K representa the activation temperature of the intervalley scattering phonon. The phonon becomes excited classically a t high temperatures, and the intervalley scattering rate then varies as the 8 power of the temperature. Cruder estimates of the strength of intervalley scattering have been derived from several sources: (1) small deviations of the piezoresistance fromithe T - l law;" (2) deviation of the mobility from a T-1 law;*'.** (3) pieeogalvanomagnetic effects. 1 8 ~ 1 4These estimates suggest that intervalley scattering is roughly 10%--15% of the total scattering a t 3000K, which ia somewhat more than is found by extrapolating the results of
-
"R. W.Keym, PAYS.Rar. 100, 1104 (1955). a0 H. Fritsache,
PAYS.Rw. 116, 338 (1959). The author indebted to Dr. hitmobe for a preprint of this paper. G. Weinreioh and H. G. White, Phyr. Rm. 106, 1104 (1957); G. Weinreiob, T.M. Sanders, Jr., and H. G . White, &id. ll&33 (1959). *a F. J. Morin, PAp. Rev. Or, 82 (1954).
PIEZORESIUTANCE OF SEMICONDUCTORS
20 1
the acoustoelectric effect to 300°K.81In any case it appears that intervalley scattering is very weak in germanium. Weinreich et uZ.”~ found for the ratio of the coupling constantsa7for intervalley and intravalley scattering w2/wI = 0.02. Intervalley scattering is probably much stronger in siliconI2’and the weakness of the effect in germanium can be understood the~retically.~’ It has also been shown that various types of foreign atoms in the germanium lattice can induce intervalley scattering. luu A very interesting application of Eq. (15.9) has been made by Paige.6* This application is baaed on the observation that the T of Eq. (15.9) is the electron temperature, and that, consequently, if the electrons are in equilibrium among themselves but not in thermal equilibrium with the lattice, a measurement of m14measures the elecfron temperature. In many semiconductors electron-lattice equilibrium in high electric fields is not attained, and the resulting nonohmic conduction is known as the “hot electron” effect. Values of m4( for “hot electrons” in germanium were measured by Paige. These measurements can be interpreted in terms of an effective electron temperature, although some question might be raised as to how closely the electronic energy distribution iq the “hot electron ” caw approximates a thermal distribution.sao An effect related to elastoresistance, the effect of strain on the Seebeck coefficient of n-type germanium, was observed by Drabble and grove^^.^^ They were able to deduce a value for the anisotropy of the phonon-drag part of the Seebeck tensor for a single valley from their results. Drabble has also given a theoretical calculation of the elasto-Seebeck tensor according to the deformation potential model.”’ Most of the piezoresistance experiments mentioned, and also other types of experiments which can elucidate the band structure of semiconductors, have been performed on relatively pure materials. It has been shownls6 however, that the piezoresistance of n-type germanium also can be interpreted on the basis of the established model of the band structure for highly doped material which contains up to 3 X donors per cm‘. The results for heavily doped germanium are shown in Fig. 11. As described earlier it is necessary to use degenerate statistics in the theory in interpreting these results; thus a determination of the degeneracy temperature as a function of carrier concentration is possible. Since the degeneracy temperature depends on the density of electronic etates, about which some question had existed, the latter quantity was E. G . S. Paige, Prm. Phys. Soc. 72, 921 (1968). 88‘S. H. Koenig, Proc. Phys. Soc. 78, 959 (1959). *‘J. R. Drabble and R. D. Groves, Phyu. Rsv. Lctrerr 2,451 (1959). J. R. Drabble, J . Electroniw and Control 6, 362 (1958). M. Pollak, Phyu. Reu. 111, 798 (1958).
202
ROBERT W. KEYE8
also found. As shown by Fig. 11 the data were fitted by the theory involving degeneracy with 60") = 0, which corresponds to 8 = 0 in the relaxation time approximation. This value of s agreed with that deduced from a study of the temperature dependence of the mobility.86 A rather direct confirmation of the deformation potential model hse been provided by the results of Rose-Innes.*' He measured the cyclotron
7s
100
IS0
200
Tom pe r at ure
300 (OK
1
FIG. 11. The effect of statistical degeneracy on the piemreaistsnce of %-type germanium. The curves for samples, B, C, and D were calculated by multiplying the nondegenerate piezoresietance, obtained by fitting a T-1 curve to ample A , by the degeneracy factor given by Fig. 8. [After M. Pollak, Phys. Rw. 111, 798 (1958).1
resonance in elastically strained n-type germanium and silicon, and found that the shape of the valleys was unchanged by the strain, but that the intensities of the resonance lines changed in a way which could be understood on the basis of the electron transfer effect.
d. Comparison of Piezoresistance and Magnetoresistance Another technique which has proved valuable in the identification and study of the models discussed in the present section centers about the measurement of magnetoresistance. As pointed out in Section 3 [see Eq. (3.1)], the magnetoresistance effect can also be described by a fourth rank tensor e(*). The interpretation of g(*) for germanium was developed practically simultaneously with that of the piesoreabtanae. 8'
A. C. RoePInnes, Proc. Phys. Soc. 74, 921 (1968).
PIEZORESISTANCE O F SEMICONDUCTORS
203
tensor.88.8QA comparison of the relative usefulness of these techniques in the study of multivalley models is in order because of the overlap in their ranges of applicability. An important advantage of the piezoresistance technique hinges on the insensitivity of n to the scattering mechanism. As seen in Eq. (15.6), the piezoresistance coefficients depend only on the form of v(”, whereas interpretations of the magnetoresistance coefficients are based on relaxation time theories of ~ ( 4 ) . Since the latter type of theory is not adequate in the impurity scattering regime, the criteria used for the identification of multivalley models from their magnetoresistance tensors are not applicable to impure materials.00 On the other hand, the effect of impurity scattering appears in the piezoresistance tensor only through its influence on K. The work of Pollak,86 quoted previously, demonstrates that the identification of the (111) valley model from the piezoresistance tensor in germanium is unambiguous for material containing up to 3 X 1Olg donors per cmS. Impurity scattering reduces the mobility to less than 0.1 times the lattice mobility at 300°K in this heavily doped material. GaSbs6 is another semiconductor for which piezoresistance measurements on impure, statistically degenerate, material have proved valuable. The fact that the mobility is reduced by impurity scattering also makes the measurement of the magnetoresistance tensor in impure materials difficult. The magnitude of the magnetoresistance effect is proportional to the square of the mobility for a given magnetic field. Thus, in the example of the germanium cited in the foregoing, the size of the magnetoresistance effect is reduced by u factor greater than 100. The piezoresistance effect in the impure material is reduced by the effect of degeneracy. However, this reduction is only by a factor of about two. The mobility may be so low as to make the measurement of the magnetoresistance tensor difficult even in pure materials. A complicating factor in the piezoresistunce tensor is its dependence on the shear deformation potential constants E,,.In the absence of a well-developed theory ior these coilstants, the possibility that an accidental smallness of one of them may obscure important features of a model must be considered. Moreover, different relative values for the various can lead to quite different shapes for the tensor n in complicated models. I n contrast, the form of e(*) is essentially determined only by the 0%. Thus it appears that the piezoresistance tensor probably provides a relatively more useful tool for the determination of band structure in &#.
*O
B. Abeles and S. Meiboom, Phys. Rev. 96, 31 (1954). M. Shibuya, Phys. Rev. 96, 1385 (1954). C. Goldberg and W. E. Howard, Phys. Elm. 110, 1035 (1958).
204
ROBERT W. KEYES
materials having low mobility or in those which are difficult to purify. In relatively pure materials having high mobility, the magnetoresistance may offer certain advantages and may also permit a determination of K or of the shape of the @). The optimum utilization of the effects occurs of course when they are used as complements to one another in the manner exemplified by the work of Herring and Vogt." e. Low Temperatures and Impurity Stales Recently Frit~sche~4.80has extended measurements of the piesoresistance coefficients of germanium to very low temperatures. He finds that the variation of the coefficients as T-I persists down to 7°K.The dimensionless elastoresistance constant m44 has a value of 4000 at tbis temperature. In such a case a strain of 5 X lo-', which is easily attained experimentally, corresponds to a change in resistance greater than the initial resistance. As is to be expected in such a case, important nonlinear effects are observed. The germanium is not extrinsic in the sense discussed in Section 9 at the low temperatures at which the nonlinear effects are observed. Some of the electrons are trapped at localized levels associated with impurity atoms. The number of such electrons is, of course, an exponential function of reciprocal temperature. However, shear strains do not change the number of electrons in the conduction band in first order, and the linear piesoresistance effect is still given by the calculation of Sections 15a. On the other hand, the nonlinear effects are influenced by changes of the effective activation energy with strain, and are, therefore, sensitive to details of the impurity states. I n the effective mass approximation the electronic states associated with a donor impurity atom are given by a theory very similar to that of the hydrogen atom, the differences being that the electron mass must be replaced by the effective maas tensor and the dielectric constant must be taken into account in the potential.9' The solution for the ground state in germanium actually is fourfold degenerate, since one solution can be constructed from the wave function of each of the four valleys."-o' The deviation from the effective mass approximation must be taken into account, however, to explain the low temperature nonlinear piezoresistance in germanium. This deviation can be regarded as a perturbation H. C . Torrey and C. A. Witmer, "Crystal Rectifiers," p. 66. McGrpw-Hill, New York, 1948. C . Kittel and A. H. Mitchell, Phys. Rev. 96, 1488 (1954). @*M. A. Lsmpert, Phys. Rev. 97, 352 (1955). O4 w.Kohn and J. Luttinger, Phy.9. Rev. 97, 1721 (1955); 96,915 (1955). O' W. Kohn, Solid Skrts Phyu. 6, 258 (1957).
P1EU)REBIBTANCE OF SEMICONDUCTORS
205
with identical matrix elements between any two of the degenerate ground-state wave functions. Thus the impurity energy levels are determined by solving the perturbation problem of the Hamiltonian
(15.20)
in which the E(0 are the extremal energies of the valleys, which were discussed in previous sections. The matrix element A. is known as the chemical shift. In the unstrained state the E(') are identical, and the lowest impurity levels are split into a threefold level and a singlet level. The effect of strain is due to the dependence of the E(I)on the strain [Eq. (10.1)]. The problem of Eq. (15.20) for the strained lattice has been solved by Pricee6 for certain special strains, and a theory of the elastoconductivity based on these solutions has been given by Fritzsche.80The details are complicated and will not be reproduced here; however, the following points should be noted. (a) The c u e s of Sb and As are quite different. A, = 0 is a good approximation in the former, whereas Ae is large in the latter. (b) The effects are sensitive t o the value of A,. (c) The strains employed by Fritz~che'~.'~ produced splittings of the E(" of the same magnitude as the A,, so that the alteration of the solutions of Eq. (7.12) was quite significant. The results conformed to the proposed model in all details, assuming that the parameters have the values %, = 19 ev and A, = 0.0010 ev, with the singlet state being the ground state of the impurity level. In the temperature range covered by the experiments of Fritzsche, the electrons in the conduction band determined the conductivity, and the change of the effective activation energy with strain was the significant impurity effect. At lower temperatures the number of electrons in the conduction band becomes negligible and impurity band conduction is the dominant conductivity mechanism; therefore quite different effects are to be expected. f. Thermodynamics Some additional aspects of the deformation potential model are most conveniently discussed from a thermodynamic viewpoint. This viewpoint will be presented here by writing down the formula for the Gibbs free @@P. J. Price, Phya. Rev. 101, 1223 (1956).
206
ROBERT W. KEYES
energy G of an elastically strained multivalley semiconductor. The important contributions to C are as follows. (a) The elastic energy, ~:C-':KV. (b) The change of the electronic energy due to the stress, 2di)W: C-':uV. (c) The configurational entropy of the electrons, Zn(')k[log (N,/n(')) l]V. Here N , is the thermally available density of states for a single valley, and is defined in the text following Eq. (9.2). N , is, of course, the same for all valleys in the single band multivalley model. (d) The free energy of the lattice vibrations. This depends on the n(", since the vibrational frequencies depend on the n"). For the purpoaee of this section the dependence of u, on the n(" and on the stress will be expressed in the form
+
where N arid V have the same significance as in Section 13. If, as in Section 13, a classical theory is used to describe the lattice vibrations, G is found to be
G
+ C,(*)g(i):C-l:u
= { ~ : C - I : ~
+ z n ( i ) k T [ l o g(n("/N,) - I]} V
+ kT
3N
2 [log (hu,(O)/kT)+ 1Aa(%(')/(N/V)
a-1
Equation (15.22) has two interesting consequences: (1) The chemical potential of an electron in valley (i) is6& p,Ci) =
+
S('):€W~:K kT log n(')/(N/V)
+ kT
3N
3N
1Aa/N + 1kT[a"':wN. (I-
1
a-1
It is seen that the dependence of I ( ' ) on stress contains a term which arises from the electron-vibration-stress interaction in addition to the term considered in Sections 15a-c. The effective deformation potential constant is %N
a-1
This temperature dependence of H was neglected in Sections 16a-q
PIEZORESISTANCE OF SEYICONDUJCTOBS
207
because it is small and does not change the essential features of the piezoresistance effects which were described there. It has, however, been observed in germanium by Fritzsche.80 He finds that The inclusion of higher order terms in the expansion of the free energy in the thermodynamic description of the pressure effects, Section 13, leads in a similar way to a temperature dependence of El. (2) The equilibrium state of strain of the lattice is c =
c-1: K
+ Zn(’)8(’):c-1.
(15.23)
Equation (15.23) implies that the equilibrium “shape” or configuration of the lattice depends on the valley occupation numbers n(;).The physical significance of this fact is that if the concentration of electrons in a particular valley is increased, then the energy of the crystal can be lowered by straining it in such a way as to lower the energy of the valley with the increased electron concentration. Thus the equilibrium lattice configuration of a crystal is sensitive to its electronic state as specified by the concentration of electrons in each valley. Shear strains associated with this effect have not been observed in cubic semiconductors because there is no simple way of adding electrons to a single valley of a symmetrically degenerate set, although one might hope to do so by removing the degeneracy with a magnetic field, for example. In noncubic crystals, however, the addition of electrons can produce changes in the shape of the unit cell. JonesD1has, in fact, suggested that such an effect can account for the effect of impurities on the c / a ratio of magnesium. The model used by Jones*’ for the band structure of magnesium is essentially that illustrated in Fig. 2e, each of the bands being a multivalley band. In the model elements of valency greater than two act aa donors when dissolved in magnesium and increase the number of electrons in the conduction band and decrease the number of holes in the valence band. Monovalent solutes have the opposite effect. Since the energies of the bands depend on the strain, in particular on the c/a ratio, the value of c/a which minimizes the energy is a function of the solute concentration. Unfortunately, no experimental values for the deformation potential constants based on piezoresistance measurements and mobility theory are available for magnesium. As illustrated in Table VIII, accurate values have been derived only for germanium and silicon. Jones therefore estimated the deformation potential constants from a “nearly empty lattice” theory, which will be described in the next section. Jones’theory H.Jones, Phil. Mag. I71 41, 663 (1950).
208
ROBERT W. KEY=
agreed quantitatively with experimental data on the e/u ratio of many solutions of metals in magnesium. g. The “ N e a d ~Empty ~ LaUice” Model
In the nearly empty lattice model of energy bands*O the crystal potential is treated as a weak perturbation, and most of the electron energy is attributed to the free electron kinetic energy. The energy gaps a t the zone boundaries are caused by the perturbing effect of the crystal potential. Because of the fairly close correspondence between the nearly empty lattice band structures and those determined experimentally, to which Hermanaohas drawn attention, and because of Jones’ success in interpreting the c/a ratios of magnesium alloysg7on the nearly empty lattice model, it might be hoped that this model would provide a useful means for calculating the deformation potential constanta of semiconductors. If it is assumed that the perturbations of the free electron energy by the crystal potential are small, then the change in the energy of a band edge caused by a strain is equal to the change in its free electron kinetic energy. The energy extrema of the bands occur at points of high symmetry in the perturbed free electron model. The value of the momentum at an extremum, P, satisfies an equation P * t = n, in which n is a constant and t is a primitive translation vector of the lattice. In the presence of a strain e, t is changed by an amount 6t = t L. Thus it can be seen that 6P = -P . e and that 6E = -P * e . P/m. The expression for this result in the deformation potential notation is 8 = -P P/m. It would be interesting to check the “nearly empty lattice” approximation by comparison with the known deformation potential constants of germanium, in which the band extrema satisfy an equation P t = constant. This procedure does not appear to be possible for the shear constants. The conduction band minima are a t points of the type h(3,1,1)/2u in p space in the “nearly empty lattice” model. There are 24 such points, and several combinations of them which have the proper symmetry (Lt) for a particular ( 1 11) valley can be formed. The proper weight of the various combinations in the actual wave function depends on the details of the potential, and is not known. The appropriate value to take for a particular valley is not determined, since the tensors P(f)P(i) am different for the various combinations of plane wavea. The situation in regard to the dilatational effect is different. The energies of all the (3,l,1)/2a points vary in the same way with dilatation. The rate of change can be calculated by the foregoing method to be
dECe)/dlog V =
- 1 lh*/l2mul =
-9 ev.
PIEZORESISTANCE OF BEMICONDUCTORS
209
The calculations of Herring and ~oliaborators,*~*‘~ however, give for dE(c)/d log V ( = E d &&) a value of about - 1 ev. The discrepancy is more striking in the case in which the gaps between valence and conand InSb). duction band extrema lie at the point p = (OOO) (Ge, I&, Both extrema arise from the same points of the “nearly empty” p epace in this case. The gap is, therefore, independent of dilation according to the theory. It is seen from Table VI, however, that El rangee from -4 ev to -9 ev for these cases. Thus, although the “nearly empty lattice ” approximation accounts for the c/a ratios of magnesium alloys, it does not give the correct results for the dependence of the energy gaps of the common cubic semiconducting crystals on dilatation. In germanium, for which the value of the shear deformation potential constant is very well established, a comparison with the “nearly empty lattice” approximation is not possible.
+
8. Mobility Themy The use of the concept of the deformation potential to calculate the scattering of electrons by lattice waves was introduced by Shockley and Bardeen. A very thorough discussion has been given by Shockley.@* The theory is based on the effect described by Eq. (5.4). In the presence of a lattice wave both the strain c and the extremum energy are functions of position. Shockley and Bardeen’O showed that the bE given by Eq. (10.1) can be regarded as a potential which scatters the electrons. They calculated the matrix elements for scattering by this potential, and thus arrived a t a formula for the mobility of electrons in a spherical band in terms of E d , the deformation potential constant for this case. The calculation has been carried through by Dumke” and by Herring and Vogt4‘ for the important case of a valley with an axis of rotational symmetry. [See Eqs. (15.7) and (15.84 The details are much more complicated than in the spherical case. Certain features are illustrated by the formula for the probability for a transition from a state p to a state p’: a@
Here CL and CT are the elastic constants for longitudinal and transveree waves, respectively, (elastic isotropy is assumed) and 0 is the angle between a, the valley axis, and q = (p - p’), the wave vector of the
” W. Shockley, “Electrons and Holes in Bemiconductom,” pp. 481-543. trmd, New York, 1950.
Van Nos-
210
ROBERT W. KEYE8
scattering phonon. It is seen that, depending on the values of the 3’8, the scattering can be highly anisotropic because of the dependence of W on 8. In particular, the simple relaxation time approximation is not an adequate description of the scattering. The solution of the Boltzmann equation with the Scattering term given by Eq. (15.24) entails considerable numerical complication, but approximate solutions have been obtained. 41*78 The principal significance of the theory of the mobility of electrons in an anisotropic valley for the problems of this paper is that it aids in the determination of E, and E d . The problem arises because the determination of $, from the elastoresistance by means of Eq. (15.9) (for germanium) requires a knowledge of K, the anisotropy of the conductivity. However, K depends on E, and Zdthrough Eq. (10.1) and the twociated solution of the transport equation. The deformation potentials therefore must be determined by obtaining a solution of Eq. (15.9) which is consistent with the measured mobility and piezoresistance coefficients, and with other information relating to K, such as the measurements of magnetoresistance. This program has been carried through for germanium, and the results, which have been mentioned previously in this paper, %, = 17 ev, Z d = -6.9, have been f o ~ n d . ~ ~ . ~ ’ 16. SEMIMETALS
The band structure of a semimetal has been illustrated in Fig. 2e. If one of the bands involved is a multivalley band, piezoresistance effects of the type described for the multivalley semiconductor also will be found in a semimetal. The effect will he complicated by the presence of two bands, however. Their magnitude will be reduced by the degeneracy effect discussed in Section 15. Semimetals were the subject of the early studies of piezoresistence by Allen.s*~*7~QQJOo Detailed analysis of Allen’s results in terms of semiconductor concepts is not possible for a number of reasons: (1) some components of the piezoresistance tensor were not determined by this work; (2) effects which do not arise in the deformation potential model, namely, the so-called “minor effects” which will be discussed in a sub sequent section, are relatively more important because of the reduced magnitude of the deformation potential effects; (3) the complications arising from the presence of two bands. The order of magnitude of the effects appears to be reasonable, however, and the transition from the metallic to the semiconducting case is evident. The relationship of the elastoresistance of the semimetals to that of a m M.Allen, Phys. Rev. Is, 569 (1933). loo
M. Men, Phys. Rw. 62, 1246 (1937).
21 1
PIEZORESISTAKCE OF SEMICONDUCTORS
typical extrinsic multivalley semiconductor is illustrated in Fig. 12, in which an important elastoresistance coefficient is plotted against resistivity for a multivalley semiconductor (germanium) and for the semimetals meaaured by Allen. The resistivity wrves as a crude measure of the charge carrier concentration, and, hence, of the degree of degeneracy, of the materials. The complete elastoresistance tensor for the semimetals cannot be determined from the data of Allen; the dimensionless constant plotted is *oa/paCaa with the elastic constant Caa taken from the measurements of Bridnrnan.'*' While it is to be exDected that manv details of the models
-g
I
I
100 -
I
1
-
7-
c
0
50-
V
?i!
In
-eIs
0 0 0
0
ii
/ O
0
0
'
-
300%
Germanium
/ 10--O
5L
-
/*
0
.t 20-
0
/o
1
I
semimetals
-
1
FIG. 12. Dimensionless elastoresistance constante for the semimetals compared with values for a multivalley semiconductor. For the semimetala the quantity plotted k the magnitude of n,,/p,C,, without regard to sign. The data is that of Allen and the points from left to right repreaent Znl, Cd'J, Snloo, Sb", and Bill. The d i d line represents mo/2 as measured by M. Pollak [Phys. Reu. 111, 798 (1958)l on n-type germanium. All data refer to 300°K.
will influence the values of the elastoresistance constants, it is seen nevertheless that the values for the semimetals are, in order of magnitude, a natural extrapolation of those for germanium. Zn, Cd, and Sn are statistically degenerate and essentially metallic, having resistivities of about ohm cm, and show small elastoresistance effects. The typical semimetals, Sb and Bi, have higher resistivities and have elastoresistance constants intermediate between those of the highly metallic elementa and germanium.
17. DEGENERATE BANDS
The preceding interpretations of the effects of shear stress on the atructure of a multivalley band are applicable to the common n-type lo1 P. W. BriQmm, PWC.N d l . A d . Sci. U.8. 10, 411 (10%).
212
ROBERT W. KEYES
semiconductors. Experiments of Lawrence'Ol and Smithto showed that the hole mobility is also strongly dependent on strain. A variety of experimental and theoretical evidence has established, however, that the valence band extrema in silicon and germanium are not of the multivalley type, but rather of the degenerate type illustrated by Fig. 2d. Thus, the origin of the large piezoresistance effect in the p t y p e materials requirea another explanation, which was originally given by Adarns.'6,1@' Brookslo4has presented an isotropic version of the theory of the effect of strain on the energy levels of a degenerate band which illustrates the important features of the effect. According to Brookslo4the energy shift of a state p in band ( j ) ( j = 1 and j = 2 denote the two bands of Fig. 2e) has the form (17.1)
for values of p not too close to the origin. In (17.1) Ed' and k' are constants similar to the deformation potential constants previously introduced; Ed' represents the effect of a pure dilatation and E,' represents the effect of a pure shear. The meaning of (17.1) is that the change in energy of a state p depends on the angle between p and the axis of shear. If e is a unit vector defining an axis of pure shear, i.e., c = c(cc - +l), then 6E(i)(p) = fZ,'t(cos* cp - +), where cp is the angle between c and p. Thus the surfaces of constant energy are distorted by the shear, and anisotropy of the velocity on a surface of constant energy is produced. This effect is the main source of the pieeoresistance in the degenerate bands.'6St0' It leads to dimensionless elastoresistance coefficients of order of magnitude E'lkT, which are of the same order of magnitude as those found in the multivalley semiconductors. The piezoresistance effects in germanium and silicon are in fact highly anisotropic."J The advantage of the isotropic approximation is that it allows a theory of the mobility and the elastoresistance to be formulated*@' in terms of the constants Ed' and Eg',and the measured mobility to be compared with the elastoresistance along the lines of the BardeenShockleyag theory. Brooks finds from the piezoresistance the value 3.' = 1.66 ev for germanium. The value of Ed' is difficult to determine from the mobility theory, as the anisotropy is probably important. However, combination of the previously quoted result that the dilatational eflect (Ed +Eu)= - 1 ev for the conduction bandk1 with the value EI = -4.5 ev gives Ed' = -3.5 ev. The sign of Ed' indicates that dila-
+
lo*
R. Lawrence, Phys. Rev. 89, 1295 (1953). E. N. Adarne, Phys. Rw. 96,803 (1954).
H.Brooks, Advances an Elsetronics and Electron Phys. 7,
151 (1956).
PIEZOREBISTANCE OF SEMICONDUCTOR8
213
tation shifb the valence band extremum upward, as indicated in Fig. 5. If the value ZL = -3.5 ev is used in Brooks’ formula for the mobility, it is found that either P,’ = 7.0 ev or Z.’ = - 3.8 ev is required t o match the mobility of p-type germanium at 77’K. The sign of the elastoresistance effect suggests that the former of these values for E,’, +7 ev, is to be preferred, although the predicted magnitude of the elastoresistance is much too high. However, a value 3,’= 4.5 ev has been deduced recently from optical experiments.lo6This value is also much greater than that required to explain the elastoresistance effect. The essential point here is that the value of E,’ suggested by the piezoresistance data is considerably smaller than the values derived from the optical experiment and from the mobility theory with Ed’ taken from other experiments as described in the foregoing. The probable explanation is that the effect of anisotropy is very important in the piezoresistance effect. Other features of the results of piezoresistance experiments on p-type semiconductors are also in qualitative accord with the degenerate band model. By making some simplifying assumptions, it can be argued that XI should be proportional to P I , in rough agreement with experiment. 103436 In contrast to the situation for n-type germanium and silicon, the values of XI for the p t y p e materials are very sensitive to impurity content. This fact also is qualitatively consistent with the model, for the light holes are scattered strongly by impurities, and their contribution to XI is expected to be dependent on the impurity concentration. The large degenerate band piezoresistance effect has also been observed in p t y p e InSb.*4Jo’ VII. Related Phenomena
18. THE “MINOR EFFECTB” The phenomena which can be interpreted within the framework of the deformation potential model were discussed in Parts V and VI. It was seen that such phenomena are characterized by fractional changes in resistance of the order of magnitude of 100 times greater than the strain which causes them. In some cases, uaually for reasons of symmetry, the deformation potential model predicts that a strain should produce no change in resistance. In such cases it is found that the elastoresistance effect in fact does not vanish exactly. Instead it is one or two orders of magnitude less than the deformation potential effects. Some of the sources of these “minor effects” will be described briefly in this section. One of the minor effects, which, like the deformation potential effects, ID* W. H. Kleiner and L. M.Roth, Phys. Rw. Letters 4,334 (1959). lo(
R. F. Potter, Phys. Rw. 108,652 (1957).
214
ROBERT W. KEYEB
has its origin in the changes in the band structure arising from the strain, was discussed in Section 12; namely, the alteration of the effective mass and the change of mobility arising from a change of band-band interaction terms. In Section 12 it was shown that this effect can give rise to a dimensionless elastoresistance coefficient of order of magnitude 10. Band structures in which a similar effect could he produced by shear strain can be imagined easily. The mechanism depends, however, on the existence of a strong interaction between the band under observation and a band that is nearby in energy. Thus it is only likely to be important in bands possessing a rather small effective mass. Another source of such small effects is the dependence of the lattice frequencies on the state of strain of the crystal. This dependence can change the scattering of the electrons. The magnitude of the change in vibrational frequencies can be estimated by using a simple model of 8. solid. It is found to be approximately two times the Griineisen constant times the strain. Thus the change in mobility can be of the order of 2 to 5 times the strain. This effect is often the predominant one in metals, and has been discussed in considerable detail by writers on the effect of prw sure on the electrical conductivity of metals. *07 Other sources of minor effects have been described by Herring.” The minor effects can be quite small in spite of the many possible contributions, however. For example, the deformation potential model predicts that the shear elastoresistance coefficient (+)(mil - mlz) should vanish in n-type germanium. The value measured by Smithlo at 300°K is only 0.5, or 1/200 times the value of m44.On the other hand m14should vanish in n-type silicon. However, its value actually is as large as (+)(ml1 - mu). The importance of the minor effects diminishes with decreasing temperatweso because of the proportionality of the principal effects to T-l.
+
19. PIEZO-OPTICAL EXPERIMENTB
The foregoing sections have dealt primarily with the use of the effecta of stress on electrical transport properties as a method for the investigation of the band structure and scattering processes in semiconductors. Another important technique for the investigation of semiconductor band structures is that of measurement of the optical absorption spectrum. An extensive discuasion of the optical properties of semiconductors and their interpretation has recently been given by Moss.lo8 Optical methods have also been applied to the study of the effects of stress on the energy level structure of semiconductors. The most 101
For a recent review see A. W. Lawson, Prop. in M e t d Phys. 6, 1 (1956).
‘08
T.8.MOW,“Optical Propertieg of Semiconductors.” Academic Press, New York, 1959.
PIEZORESISTANCE O F SEMICONDUCTORS
215
common form of such experiments has been the measurement of the effect of pressure on intrinsic optical absorption. (The intrinsic optical absorption in a semiconductor is the process by which absorption of a photon raises an electron from a state in the valence band to a state in the conduction band.) The significance of the intrinsic optical edge in terms of the statistical-thermodynamical parameters of Section 13 has been discussed by Jamesa7and by Brooks.o8They have argued that the optical gap is the free energy of activation to a good approximation, and that the entropy and volume of activation can be deduced from the temperature and pressure dependence, respectively, of the optical gap. This result can be seen in the semiconductor model of Section 13 by using the energy levels defined by Eqs. (13.6), (13.7), and (13.8) in Brooks'(* formula (4.24). The energy required to optically excite an electron from a state j' in the valence band to a state I' in the conduction band is found to be =
w(V)
+ 3XkT + p112/2mo+ p,lS/2m..
(19.1)
This result shows that the minimum energy of intrinsic optical absorption is w ( V ) 3XkT, the activation free energy. In some caaes the optical absorption edge is determined by a transition in which absorption of a phonon also takes place.lOO~l10 In these cases the optical gap is smaller than the activation free energy by an amount equal to the phonon energy. This is a quite small amount. Since the phonon energy is not strongly dependent on pressure or temperature, the derived values of v and s are not affected by the phonon participation. Some results of measurements of the dependence of optical gaps on pressure and temperature have been used in the discussions of Sections 12-14 and a compilation of optical data was given in Table V. References to the high-pressure optical experiments were given in connection with this table. Table V shows that in most cases in which the activation parameters have been determined by both the optical and the electrical methods there is substantial agreement between the results of the two methods. The most notable exception occurs in the caae of the pressure dependence of the gap between the valence band and the (111) conduction band extremum in germanium. Two different types of results have been obtained in this There appears to be general agreement that, for absorption constants a > 20 cm-I, the curves of a v 8 hv shift to
+
*ogG.G. Macfarlane and V. Roberts, Phys. Rev. 97, 1714 (1955); G. G. Macfarlane, T. P. McLean, J. E. Quarrington, and V. Roberts, ibid. 108, 1377 (1957). *1oG. G. Macfarlane and V. Roberta, Phyu. Rev. 98, 1865 (19%); G. G. Macfulane, T. P. McLean, J. E. Quarrington, and V. Roberta, Phyu. Rtv. 111, 1245 (1958).
216
ROBERT W. KEY=
higher energy with increasing pressure a t a rate of about 7.5 X lo-’ ev/kg cm-2. Fan et d.“ and NeuringeF find that in fact the entire absorption curve down to values a = 1 cm-’ shifts a t the same rate. Therefore, they conclude that 7.5 X ev/kg cm-’ is the rate of change of the gap. On the other hand, Paul and Warachauerbxfind that the ahape of the a versus hv curve for a < 20 cm-l changes with pressure. Such a change is to be expected on the basis of the theory of optical transition8 involving interaction with a phonon, which waa given by Bardeen, Blatt, and Hall.ll1 According to this theory the dependence of a on photon energy can be written in the form hv = Aai
+ E(C).
(19.2)
Here A contains a number of factors; it is pressure dependent through the fact that
(19.3) is the energy of the conduction band edge In Eqs. (19.2)and (19.3), and E(Oo0)is the energy of the conduction band extremum a t p = (OOO), both measured from the top of the valence band. The linear relationship between a+ and hv can be accurately verified in the range 4 cm-l < a < 25 crn-l.lo0 A phonon energy is neglected in (19.2)and (19.3).If Eq. (19.2)is differentiated for fixed n, a = ao,say, (19.4)
and from Eq. (19.3)
Equations (19.4)and (19.5)can be solved self-consistently for dA/dP and dE(e)dPby using measured values of d(hv)/dP, dE(Oo0)/dPand A. For example, using the values d(hv)/dP = 7.5 X lo-’ ev/kg cm-’ for a0 = 20 cm-1, A-1 = 65 ev cm-4, dE”ooo)/dPfrom Table V and E(Oo0) = 0.15 evlOgJ1*gives dEcC)/dP= 5.4 X lo-’ ev/kg cm-? This value is in agreement with that obtained by the electrical methods (see Table V) and is close to that found by Paul and Warschauer’x by direct extrapolation of their curves in the region of a < 20 cm-1 to a = 0. There is some uncertainty as to the best values to use for the various
-
J. Bardeen, F. J. Blatt, and L. H. Hall, in “Photoconductivity Conference” (R.0. Breckenridge, B. R. Russell, and E. E. Hahn, eds.), p. 146. Wdey, New York, 1966. lil 5. Zwerdling, B. Lax, and L. Roth, Phys. RGV.100, 1402 (1967).
PIEWREBIBTANCE OF 8EMICONDUCN)RB
217
mameters such 88 d(hv)/dP and dE(Oo0)/dP.However, the important eature which emerges is that in an interpretation which is consistent Kith the indirect transition theory"I the preasure effect on hv at 20 cm-I will be appreciably different from dE(*)/dP,the preasure shift of the bsnd edge.)' A few observations of the effects of elastic shear strain on optical transitions in semiconductors have also been carried out. One of t h e e ie the experiment of Macfarhne et a L 1 1 * on the absorption due to the direct transition exciton in germanium. The specimen was cemented to glaea in this experiment, and the strain was produced by the differential thermal
t/8, ho. 13. The dependence of the optical energy gape of germanium and dioon on the temperature when the latter is reduced by the Debye temperature. The ordinata are normalized in such a way that the s l o p are the -me at high tempemtureo.
contraction of the glaas and the germanium. It was, therefore, not known quantitatively. The results, however, have been andyaed by Kleiner and Rothto' and an estimate of the shear deformation potential constant of the valence band which waa used in Section 17 waa obtained. Another important streae-optical experiment ie that of Weinreich et d.l14These authors memured the effect of shear etrain on oertain tranaitione among the electronic energy levels saeociated with d c impurity atoms in germanium. The intent of this experiment waa to cl*G. Q. Mlcfarlane, T.P. M c h n , J. E. Qlumngbn, and V. Roberta, P h p . Rso. Lsusrr I,262 (1969). 114 G. Weinreich, W. 8. Boyle, H. G. White, and K. F. Rdgem, Phvr. h. Lahrr I, 97 (1969); 8, UI (1959).
218
ROBERT W. K E Y E S
decide whether the triplet or singlet ls state is lower. The results indicated that the singlet is lower, in agreement with the conclusions drawn by Fritzeche14*80from piezoresistance measurements, and by Feher, Wilson, and GereliLfrom electron spin resonance experiments. The optical method of measuring energy gaps permits the determination of the gaps t o be extended to very low temperatures. The temperature dependence of the activation energy has been discussed in Section 13, and very accurate data on germanium"J9 and silicon11oare available for comparison with this discussion. The data are presented in an appropriate reduced form in Fig. 13. The temperature is reduced by dividing it by eo, the low-temperature Debye temperature. The change of the energy gap between 0°K and temperature T is reduced by dividing it by a constant which makes the slopes of the curves - 1 at high temperatures. This reduction removes the effect of differences in the average electronphonon interaction constant X. The figure illustrates that the form of the functional dependence of the gap on (T/&)is the same for germanium and silicon. It shows that the value of To [Eq. (13.19)] is 0.2160 for the= materials. Less detailed data suggest that To is different in other CBBBB.
COMPOSITION 20. DEPENDENCE OF ENERGY GAP ON CHEMICAL The fact that both energy gaps and lattice parameters vary in systematic ways among the semiconducting elements and compound8 is well known. As an example, the lattice parameters increase and the energy gaps decrease with increasing atomic number in the fourth group of the periodic table. These two observations may be interpreted aa a relation between energy gap and lattice parameter. A question which naturally arises then is: to what extent is the dependence of energy gap on lattice parameter which is observed in high pressure experiment8 and which was discussed in Part V the same as the dependence of gaps on lattice constants among different materials of similar crystal structure? This question will be examined in the present section. The most extensive data concerning the dependence of energy gap on lattice parameter with respect to both the effect of pressure and the effect of composition are those derived from observations on cubic materials with the diamond and xincblende crystal structures. The pertinent data for these cubic crystals are presented in Fig. 14. The available evidence indicates that in all of these materials the maximum energy of the valence band occurs a t the point p = (0oO). Data relating to gaps between the valence hand and three different types of conduction band minima are available, however. The three cmes are plotted wpa. rately in Fig. 14 and discussed individually here. One general statement l' G. Feher, D. K. Wilson, and E. A. &re, Phycl. Rar. Ldlcrs 1,25 (1959).
219
PIPZOREYISTANCE O F SEMICONDUCTORS
may be made, however: the assumption that the energy gaps are a universal function of lattice parameter is not quantitatively correct. The (0) minima: Here the universality of the relation between energy gap and lattice parameter may be described as semiquantitative. The dashed line, which roughly represents the variation of the gap from one material to another, corresponds to an average value of
In sb 0 52
1
I 5.6
1
1
6.0
I
1
64
Lotticr Conrtanl (A)
FIQ.14. The relation between energy gap and lattice parameter for semiconductora with the diamond and zincblende crystal structures. The effwts of the application of hydroetatic pressure and of alloying (H. Weies, Z. Nakrforclch. lla, 430 (1956); R. Braunstein, A. R. Moore, and F. Herman, Phys. Rev. 109, 695 (1958)l are ehown where available. References to the high-pressure data are given in Table V. Thc effect of prewure on the (100) and (111) extrema in germanium nnd silicon is taken from T. Slykhoum and H. G. Dricknmer [ P h p . ond Chern. Solids 7, 210 (1958)).
El( = &a dEo/da) of - 3 ev. The values of E l derived from the effects of hydrostatic pressure and from the gaps of the alloys of the type InAsPl,116 are somewhat larger than this. They lie in the range - 4 ev to - 10 ev. The (111) minima: The values of R1 for germanium derived from the effect of hydrostatic pressure and from measurements on germaniumsilicon alloys117have the same sign and differ in magnitude by a factor *La H.We&, Z. Noturforach. 111, 430 (1956). ''' R. Braunetein, A. R. Moore, and F. Herman, Phys. Keu. 109,695 (1968).
220
ROBERT W. KEYES
two. A comparison of germanium with CaSb gives a quite different value for El, however. The (la))minima: There is no suggestion of a unique relationship between energy gap and lattice parameter for the (001) minima. In fact, the effects of alloying and of pressure give values of oppoeite sign for 61. The more striking failure of a universal relationship in the caoe of the (001) minima than in the others is probably msociated with the fact that the location of the (OOO) and (111) type minima in p space is completely determined by symmetry, whereas the position of the (001) minima contains one general coordinate which may vary with composition and with pressure. There are two other cases in which the relationships between energy gaps and lattice parameters aa revealed by the effect of hydrostatic pressure and by the effect of changing composition can be compared. One of these cases is that of tellurium. The value of El which is derived from the effect of dissolved selenium on the gap of tellurium118J1’is -4 ev. The value derived from the effects of hydrostatic pressure has the same magnitude, but the opposite sign. Thus it is apparent that the effect of added selenium on the energy gap is not a result of its effect on the lattice parameter. The other case for which the desired comparison can be made is that of M g d n and the analogous compounds, MgtGe and Mgai. The relationship between the energy gaps and the lattice parameters of these three compounds corresponds to a value El = -2.5 ev.lZ0 The value derived from the effect of pressure on the energy gap of MgzSn is quite close to this. The significance of this result aa compared to that illustrated by Fig. 14 is uncertain, however, as no information concerning the location of the band extrema of the magnesium compounds in p space ia available. The present result and the fact that the carrier mobilitiea do not vary greatly from one compound to another may be interpreted as evidence that the positions of the extrema are the game in all of the compounds. It should also be noted that extrapolation of the relation between energy gap and lattice parameter to Mg2Pb, which has the same crystal structure as the other compounds, fails. The extrapolation predicts an energy gap of about 0.25 ev for Mg2Pb; however, the compound is, in fact, metallic. 111
190
J. Lofemki, Phys. Rw. Bb, 707 (1954). T. 8. Mom,“Optical Propertias of Semiconductors,” pp. 169-170. Academic Ram, New York, 1959. A compilation of the properties of group 11-IV compounds with referencss to the original work ia given by J. M. Whelan in “Semiconductors” (N. B. Hannay, ed.), pp. 424-426. Reinhold, New York, 1969.
PIEZOBEBIBTANCE OF BEMICONDUCTORB
!22l
In summary of this section, it may be said that the variation of energy gap among materials of similar crystal atructure, but with differing chemical compmition, uaually corresponda to a decrem of energy g.p with inmewing lattice parameter. ThL effect is qualitatively and even aemiquantitatively in agreement with obeervations of the effects of hydroetatic preesure on the gaps of semiconductors in most cases. The striking exceptions to the similarity of the effects of application of preaaure and of variation of chemical composition are found in those materiala whose properties were found in Section 14 not to be in agreement with the simple bonding interpretation.
This Page Intentionally Left Blank
Imperfection Ionization Energies in CdS-Type Materials by Photoelectronic Techniques RICHARD H. BUBE RCA Laboratories, Princeton, New Jersey
. . . . . . . . . . 223 I. Imperfections in Insulators.. . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................ 1. Introduction. . . 2. Imperfections and Crystal Binding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Effecta on Photoconductivity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 4. Photoconductivity Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 5. Imperfection Sensitization of Photoconductivity.. , . . 11. Photoelectronic Techniques Applied to CdSType Mate 6. Conductivity and Hall Effect versus Temperature. . . ...................
Emission of Luminescence. . .............................. Variation of Photoconductivity with Light Intensity. . . . . . . . . . . . . . . . . Thermal Quenching of Luminescence and Photoconductivity.. . . . Optical Quenching of Luminescence or Photoconductivity. . . . .. Thermally Stimulated Luminescence or Photoconductivity.. . . . . . Space-Charge-Limited Currents ........................ 111. Trende in the Ionization Energies in CdS-Type Materials. . . . . . . . . 15. Summary of Meaaured Energiee.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16. Approximate Correlation with Cation and Anion Constituents of the Material.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Indications of Complex Ionization Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. Variation of Ionization Energy with Imperfection Concentration.. . . . . . 18. Double Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. Multiple Levels.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. 10. 11. 12. 13. 14.
235 235 237 240 243 247 249 249 249 252 252 253 259
1. Imperfections in Insulators
1. INTRODUCTION Many of the most significant photoelectronic properties of insulators such aa CdS are associated with the presence of crystal imperfections. These imperfections may consist either of foreign impurities or of crystal defects such as vacancies. On the basis of a phenomenological understanding of the role of such imperfections in photoelectronic processes, 223
224
RICHARD H. B U B E
it is possible to utilize photoelectronic techniques for the investigation of these imperfections themselves. It is the purpose of this article to summarize some of the recent results of such investigations, emphasizing particularly those techniques useful in determining the location of energy levels which cannot be located by standard semiconduetor techniques.'
2. IMPERFECTIONS AND CRYSTAL BINDING Let us consider imperfections present in a hypothetical compound MX, where M and X are divalent elements The nature of the imperfections depends upon whether the binding is ionic or covalent. Actually in most materials, including the CdS-type materials, the binding ie of some intermediate type between purely ionic and purely covalent. But, aa will be seen, our qualitative conclusions concerning the behavior of a given type of imperfection are not dependent upon the model chosen for TABLEI. PossmLt EFFECTS OF IMPURITY INCOE~POXUTION ~
Impurity
R+' 2 R+a
R+' 2
R+1
A-1 2 A-1 A2 A-8
Substitutes for
M +' 3 M+' M+' 2 M+'
X-'
x-' X-' 3 x-'
2
Effect
Terminology
Free electron at high T M vacancy Free hole at high T X vacancy Free electron at high T M vacancy Free hole at high T X vacancy
Donor Compensated acceptor Acceptor Compensated donor Donor Compensated acceptor Acceptor Compensated donor
the binding. For pure ionic bonding, each M atom gives up two electrons completely to become an M+'ion. Each X atom takes on two electrons completely to become an X-' ion. The transferred electrons are localized on the X-2 ion and there is negligible charge density between the ions. For pure covalent bonding, on the other hand, each M and each X form four bonds, each bond being composed effectively of electron from the M and l+ electrons from the X. The charge density is concentrated in the interatomic spaces. Consider the effects of incorporating a trivalent R atom in place of an M atom. According to the ionic picture, the removal of an M atom leaves behind two positive chargea. The R will take up these two positive charges, but requires a third positive charge to fulfill its valence requirements; therefore it gives up an electron which becomes weakly bound
+
* This article is based to a large extent on material discussed by the author in more detail in "Photoconductivity of Solids." Wiley, New York, 1960.
IONIZATION ENERGIES IN CD8-171PE MATERIALS
225
to the R ion in the crystal. However, this electron may be freed to contribute to the conductivity when the temperature is sufficiently high. According to the covalent picture, the removal of an M atom breaks four bonds and leaves a deficiency of two electrons. The R atom contributes two of its three outer shell electrons to meet these bond requirements, and then ie free to donate the third electron at sufficiently high temperatures. On the basis of either picture, therefore, the R impurity acts like a donor. There is another possible substitution which permits the incorporation of R impurity into the MX compound. If two trivalent R atoms replace three M atoms, the crystal requirements are satisfied by the formation of an M vacancy. Because of the absence of the M ion, this vacancy will, in efTect, have two negative charges with respect to the rest of the crystal; it is an ionized or compensated acceptor center. A number of such possible substitutions with their effects are summarized in Table I. The queation of whether R will substitute for M to give donor behavior or to form compensated acceptors is largely determined by the local conditions of growth when the incorporation takes place; a high M pressure will favor donor behavior, wb3ress a high X pressure will favor formation of compenaated acceptors..
3. EFFECTS ON PHOTOCONDUCTIVITY Impurities incorporated in CdS-like photoconductors have an obvious effect on the conductivity and, in addition, can have three basic effects on the photoconductivity. (1) They may change the photosensitivity. (Throughout this article the term photosensitivity will be used to mean photoconductance per unit excitation intensity. The specific sensitivity is measured in units of cm'/ohm watt, obtained by multiplying the photoconductance by the aquare of the electrode spacing, and dividing by the excitation power abaorbed. The specific sensitivity is a property of the material and is independent of geometry, applied field, or excitation intensity, if the photoconductivity varies linearly with field and intensity.) Imperfections which act as efficient reconibination centers decrease the photosensitivity by decreasing the free carrier lifetime. On the other hand, imperfections (like doubly negative centers) which have a large cross section for capturing photoexcited holes, but a small cross section for capturing photoexcited electrons after hole capture, may increase the sensitivity (we Section 5 ) . (2) Imperfections may change the speed of response. Those imperfections which act as trapping centers, by localizing free carriers until thermally freed, decrease the speed of response. Likewise those imperfec-
226
RICHARD H. BUBE
tions which act to increase photosensitivity decrease speed of response by increasing the carrier lifetime. On the other hand, imperfections which decreaae uensitivity increase the speed of response. (3) Imperfections may extend the spectral response of photoconductivity to the long-wavelength side ‘of the abeorption edge. Since direct excitation from an imperfection center with its level lying in the forbidden gap requires less energy than excitation across the band gap, the apectral response is extended to longer wavelengths. 4.
PHOTOCONDUCTIVITY PROCESSES
The discussion of several of the photoelectronic techniques described in the following section is based upon a phenomenological treatment*’ of the photoconductivity process in CdS-like materials. I n order to prepare for this discussion, therefore, it is appropriate here to consider some of the main points of this treatment. All imperfection centers which can exist in an insulator are considered to be either trapping centers or recombination centers. Trapping centers are those for which the probability of thermal freeing of the trapped carrier is greater than the probability of recombination with a carrier of opposite type. Recombination centers are those for which the probability of thermal freeing of the captured carrier is less than the probability of recombination with a carrier of opposite type. Trapping centers affect speed of response; recombination centers affect lifetime and photosensitivity. It is convenient to define the demarcation level as the energy level at which the probability of electronfree hole (or hole-free electron) recombination equals the probability of thermal ejection of the electron (or hole) into the conduction (or valence) band. The occupation of a level lying above the electron demarcation level is determined by the conditions of thermal equilibrium between the levels and the conduction band; similarly the occupation of a level lying below the hole demarcation level is determined by the conditions of thermal equilibrium between the levels and the valence band. The occupation of levels lying between the electron demarcation level and the hole demarcation level is determined by the recombination kinetics of the material. Figure 1 shows schematically the relationship between the demarcation levels and the steady-state Fermi levels for an insulator.
* A. Rose, RCA Rev. 12, 362 (1951). * A. Rose, i n “Photoconductivity Conference,” p. 3. Wiley, New York, 1950. A. Rose, Proc. I R E IS, 1850 (1955). A. Rase, Phys. Rev. 97, 322 (1956). A. Rose, i n “ Progress in Semiconductors,’’ Vol. 2, p. 109. Wiley, New York, 1957. R. H. Bube, Proc. IRE 43, 1836 (1955). * R. H. Bube, Phys. & Chem. Solids 1, 234 (1957).
4
’
IOSIZATIOX ENERGIES IN CDS-TYPE MATERIAL8
227
The steady-state electron Fermi level is defined as:
El, = kT In INJn]
(4.1)
where. J?,* in the energy difference between the steady-state Fenni level arid t h ~ N J ~ V J Y I : ~4 the ~:mdm\vmh a d , .Ye is the eflwtire density d states in the conduction band, and ri i s the density of free electrons. The steady-state hole Fermi level is similarly defined. Under conditions of pure thermal equilibrium, Ern Efp = EQ, where EQ is the band gap,
+
/////, CONDUCTION
-.-.
BAND /
.--.- .--.
-ELECTRQH FERYLEVEL ELECTRON OLUllRILnON LEVEL
--------------- -----__- --__-_________ 1-
"t,T,
(-3
-.-.-
-
HOLE FERYI- LEVEL
X
-HOLE
DEUARUTIQ1 LEVEL
Fra. 1. Steady-state Fermi level and demarcation level relationships for an insulator like CdS.
but in the presence of excitation, the steady-state Fermi levels have the following relationships :
+
Ern Efp= EQ - kT In (np/ni2) (4.2) Ern = Ern kT In [(m./mr)'(p/n)l (4.3) where ni ia the density of free carriers in an intrinsic material, and m, and
+
are the effective electron and hole masses respectively. The relationship between the electron Fermi level and the electron demarcation level for a certain type of center may be derived in the following way. When the electron demarcation level is located a t the level corresponding to these centers, it follows by definition : ??8h
228
RICHARD H. BWBE
nrS,.N& exp (-Ed,/kT) = nIpvS,
(4.4)
where nl is the density of occupied levels, S. is the capture c r w section for free electrons, v is the free carrier thermal velocity, p is the density of free holes, and S, is the capture cross section for free holes. The lefthand side of Eq. (4.4) is the rate a t which electrons are thermally excited out of the centers into the conduction band; the right-hand side of the equation is the rate of recombination of free holes with the captured electrons. If N , is replaced using Eq. (4.1) we obtain, Efn
=
Edn
4-kT In [S,p/S,n].
(4.5)
Since in the steady state, np,uSn = pn,vS,, where p , is the density of recombination centers for electrons (i.e., the density of recombination centers which have captured a hole) and n, is the density of recombination centers for holes, it follows: =
Edn
4-kT In [pr/n,l.
(4.6)
A similar relationship exists between the hole Fermi level and the hole demarcation level for these centers. Since a demarcation level ie determined by the values of S , and S , of the centers involved, each different type of center, as characterized by different values of S , and S,, has ita characteristic demarcation levels. An important part of our analysis is concerned with the changeover of compensated acceptor centers from hole trapping centers to recombination centers; this will be described more fully in the next section. Since we are interested in the location of the hole demarcation level, but measure experimentally only the free electron density, it is necessary to derive a relationship between the electron Fermi level and the hole demarcation level. This is a natural thing to do since both depend on the free electron density. When the hole demarcation level is at the level of the centers under consideration, pmS.v = prSJ?,v exp (- Ed,/kT>
(4.7) where pr is the density of captured holes, and N. is the effective density of valence band states. The right-hand side of Eq. (4.7) is the rate of thermal freeing of holes from the levels into the valence band; the lefthand side is the rate of recombination of free electron8 with captured holes. If n is replaced using Eq. (4. l ) , we obtain:
Edp = Ej, i- kT In (S,/S,) (4.8) except for a small term which goes to zero if m, = mr.A similar relationship exists between the hole Fermi level and the electron demamation level.
IONIZATION E N E R G I E S I N CDS-TYPE MATERIALS
229
6. IMPERFECTION SENSITIZATION OF PHOTOCONDUCTIVITY It is found experimentally that an increase in photosensitivity accompanies the incorporation of certain imperfections in CdSlike photoconductors; them imperfectionr, have in common the characteristic that they are comperlrwbd wceptorn, hence they have an effective negative charge with respect to the rest of the crystal. An energy level representation of the sensitization process is given in Fig. 2. In a pure un-sensitized material (Fig. 2a), there are present recombination centers I of such a
YECTROW rEml LEVEL
--
--
---MOLE
'
J
'
OEUARCATION
LEVEL
----
TI-+
(a)
f
(b)
.*
.
FIQ.2. Schematic representationof imper-hion sensitizationof photoconc xtivity in 8 material like CdS. (a) Material with only large crose-section recombination cantere; (b) incorporation of small cross-section centers and their behavior M hole trap; (c) sensitizing effects reeulting from behavior of small cross-aection centers 98 recombination centem; (d) optical quenching of photoconductivity.
type as to produce a small free lifetime for photoexcited carriers. These centers have always been present in the purest crystals yet grown and their identity is still virtually unknown. They behave like neutral imperfections; thus, once one type of carrier were captured, there would be a strong Coulomb attraction for recombination with the other type of carrier. Consider the effect of adding compensated acceptor centers 11; these centers will have a large cross section for hole capture, and then a emall cross section for electron capture. For low light intensities and/or high temperatures, the hole demarcation level for the I1 centers will lie above the levels and the added imperfections will function only as hole traps with effectively no change in the sensitivity (Fig. 2b). But for high light levela and/or low temperatures, the hole demarcation level will
230
RICHARD H. BUBE
lie below the levels (Fig. 2c). Then a sensitizing effect comes about in the following way. (1) Holes captured by the I1 centers have a longer life there before recombination than holes captured by I centers, because of the small cross section of I1 centers containing holes for capturing free electrons. (2) Therefore, the I1 centers become occupied principally by holes and, if the concentration of I and I1 centers is much larger than the density of free carriers, the electrons initially in I1 centers will be effectively transferred to I centers. (3) The lifetime of a free electron ia increased because now it will encounter mainly centers with a small capture cross section. Three photoconductivity characteristics are associated with thie basic mechanism for sensitivity. (1) When the hole demarcation level is lowered through the I1 level by increasing light intensity at fixed temperature, the photosensitivity increases with increasing light intensity; therefore, the photocurrent varies with a power of light intensity greater than unity (sometimes called superlinearity or supralinearity). (2) When the hole demarcation level is raised through the I1 levels by increasing temperature at fixed light level, the photosensitivity decreases with increasing temperature, i.e., thermal quenching of photoconductivity occurs. (3) If electrons are optically excited from the valence band to I1 levels occupied by holes (Fig. Zd), the holes are freed and may be captured by I centers, thus reversing the sensitizing process, i.e. optical quenching of photoconductivity occurs. II. Photoelectronic Techniques Applied to CdS-type Materials VERSUS TEMPERATURE 6. CONDUCTIVITY A N D HALLEFFECT Measurement of the variation of conductivity and/or Hall constant with temperature is one of the standard techniques of determining imperfection ionization energies in semiconductors. In a material in which it is possible to have conductivity governed by carriers of one type, the carrier deneity varies exponentially with the ionization energy of the controlling imperfection. Applying Eq. (4.1), to an n-type material with N D donors, partially compensated by N A acceptors,@we obtain:
and the following expression for the free electron density:
@
FL 18. Bube, J . C h .Phyr. 08, 18 (19SS).
IONIZATION ENERGIES IN CDS-TYPE MATERIALS
231
where E is the ionization energy in question. Since the Hall constant is proportional to l/n, E can be determined directly from Eq. (6.2) with measurements of the temperature dependence of the Hall constant. If only the conductivity can be meamred, SB is sometimes the case in high-rmrirrtivrty mntt.ria1, slcimr: rrretrrumption must he m d e about the trwuwraturc clqwrtdernw of t tit: carrier mobikty. In the birnplest cam?, the temperature dependence of the density of states in the band ( aT')
VT
, i '16'
FIG.3. Dark conductivity as a function of temperature for bromine donors in a ZnSe crystal. Indicated points are only illuatrative since curve was continuously recorded.
just cancels the variation of mobility due to lattice scattering ( a V). The corrections to E required by assuming that this cancellation obtains generally are not likely to amount to more than f 10%. ThiR technique can be used to study imperfections in insulators, which become semiconducting with increasing temperature, but it becomes increasingly difficult to apply for ionization energies greater than about 0.3 ev. Many of the levels in CdS-like materials, particularly those associated with acceptor centers, do have ionization energies greater than 0.3 ev.
232
RICHARD H. BUBE
One example of a successful application of this technique for bromine donors in ZnSe is shown in Fig. 3.'O The log of the conductivity variee exponentially with 1/T over seven decades of conductivity, from to (ohm cm>-l. The points of Fig. 3 actually are taken from a continuously recorded curve. The ionization energy for bromine donors in ZnSe ia seen to be 0.21 ev. The ionization energies of donors in CdS also have been determined from the dependence of conductivity on temperature; these have a value of about 0.03 ev.@JIJ*Acceptor energies in ZnTe," and both donor and acceptor energies in CdTel4-ld have been determined from conductivity versus temperature data.
7.
ABBORPTIOY
Measurement of optical absorption is another technique which has been widely used with semiconductors, but we shall comment on it only briefly here. Direct absorption by imperfections can be detected, and the lowest energy required to give both absorption and photoconductivity gives a measure of the optical ionization energy of the imperfections. It should be noted that some of the techniques give thermal ionization energies and some give optical ionziation energies; because of the FranckCondon principle, these two energies need not be the same. Frequently, for 11-VI compounds the two energies do seem to be approximahly equal. However, measurements on other materials such as GaSe indicate that the two energies" can differ by as much as a factor of two. Measurement of absorption becomes difficult when the imperfection concentration is low. In this case it becomes more convenient to meaaure excitation spectra and draw conclusions about ionization energies from their wavelength dependence. Measurements of reflectivity can be substituted for those of transmission to determine the absorption. Diffuse reflectivity has been used to determine the ionization energies of silver and copper impurities in microcrystalline ZnS power. 10
11
R. H. Bube and E. L. Lind, Phys. Rw. 110, 1040 (1958). F. A. Kroeger, H. J. Vink, and J. Volger, Philip8 Reaearch Rep&. 10, 39 (1955).
I* 1'
R. H. Bube and S. M. Thornsen, J . Chem. Phya. 48, 16 (1956). R. H. Bube and E. L. Lind, Phys. Rev. 106, 1711 (1957).
F. A. Kroeger and D. DeNobel, J . Elecfronicu 1, 190 (19M). D. A. Jenny and R. H. Bube, Phys. Rev. 06, 1190 (1954). 1.E. L. find and R.H. Bube, unpublished data (1958). 1' R. H. Bube and E. L. Lind, Phys. Rev. 116, 1169 (1969). ISR. H. Bube, Phyu. Rev. 90.70 (1963). 8'
1'
IONIZATION ENERQIES IN CDS-TYPE MATERIAL8
8.
EXCITATION OF
233
LUMINESCENCE OR PHOTOCONDUCTIVITY
When the absorption of light raulta in either luminescence emission
or photoconductivity, messurementa of the latter quantities constitutes a eenaitive means of detecting absorption. Such mecrsurements have been widely made for 11-VI aulfidee and selenides. The minimum photon
WAVELENGTH, A
FIQ.4. Photoconductivity spectral response associated with Cu impurity in ZnSe crystals, showing both the intrinsic response with maximum at 4700 A and the Cu reaponse with long-wavelength limit at about goo0 A.
energy required to produce photoconductivity, or to produce luminescence if accompanied by photoconductivity, in the imperfection range, is interpreted aa the ionization energy of the impurity. As an example, consider the caae of copper impurity in ZnSe;lL excitation spectra for photoconductivity are given in Fig. 4.10 The intrinaic peak for ZnSe Whenever we speak of the excitation of an electron from an acceptor center Like copper, we mean that we have initially a compensated center pnwrent in the cryatal. In most of the materials here discussed, this has been brought about by the incorporation of donor impuritiea. The excitation of a copper center therelore mcum Cu++ Cu+* 6. The exoitation of a cation vaouicy may mean either Vc+-r
+
+
V C + ~ e. or YC-r VC+
+
6.
234
RICHARD H. BUBE
occurs at about 4700 A at room temperature; the long-wavelength response out to 6000 A corresponds to direct cxcitation of electrons from copper centers to the conduction band. The long-wavelength limit of W A corresponds to an optical tranliition of about 2.1 ev. Therefore, the level from which the transition originatea must lie 0.6 ev above the top of the valence band, since the band gap of ZnSe is 2.7 ev. The falling off of photocurrent a t the long-wavelength limit seems gradual in the linear plot of Fig. 4 but its sharpness is emphasized by a logarithmic
FIG.5. Photoconductivity spectral response associated with Ag impurity crystale, showing long-wavelength limit at about 6ooo A.
plot. Figure 5 shows the photoconductivity spectral responae associated with silver impurity in ZnSe.10 At room temperature the photoconductivity response decreases by three orders of magnitude between 5600 A and 6ooo A. The long-wavelength limit indicates that the silver level also lies about 0.6 ev above the top of the valence band. As other examples, we may cite the long-wavelength limits for excitation of photoconductivity of copper in CdS and CdSe. I n CdS the longwavelength limit is 9OOO A; since the band gap is 2.4 ev this corresponds to a copper level lying 1.0 ev above the top of the valence band. In CdSe, which haa a band gap of 1.7 ev, the long-wavelength limit is 11,OOO A,
IONIZATION ENERQIES IN CDB-TYPE MATERIALS
235
corresponding to a copper level lying 0.6 ev above the top of the valence band. 9. EMISSIONOF LUMINESCENCE
In establishing the higher energy terminus of a transition from measurements of absorption and excitation of luminescence, we are guided by the concurrence of photoconductivity. Thus we attribute the absorption and excitation bands of Mn impurity in ZnS, which are not accompanied by photoconductivity, entirely to inner shell transitions of the Mn ion. The use of luminescence emission spectra to determine imperfection ionization energies is somewhat more difficult than the aforementioned techniques, for in general the higher energy origin of the emission transition is not known. However, it is found in some cases, ZnS phosphors for example, that the high-energy limits of the emission spectra for copper and silver agree well with the Eow-energy limits for excitation of luminescence or It is possible therefore to use emission spectra as additional information, to confirm excitation data.
10. VARIATIONOF PHOTOCONDUCTIVITY WITH LIGHTINTENSITY In Part I of this article we discussed the effects of lowering the demarcation level through the region of the levels associated with compensated acceptors on the variation of photoconductivity with light intensity. Figure 6 shows the variation of photocurrent of a CdSe crystal‘* 88 a function of light intensity for eleven different temperatures between -182°C and 99°C. A slope equal to or less than unity, Sl, is found a t low temperatures and a t intermediate temperatures for high excitation intensity. A slope greater than unity, Sz, is found at intermediate ternperatures for low excitation intensity and at high temperatures for high excitation intensity. A slope equal to or less than unity, S1, is again found at’high temperatures and low excitation intensity. If a sufficiently large range of light intensities were available at an intermediate temperature, all three slopes would be found in measurements of photocurrent versus light intensity at fixed temperature. The data illustrate the opposing effects of temperature and light intensity; the changeover from SI to Si, and from St to S, sets in at a higher light intensity the higher the temperature. The location of the electron Fermi level corresponding to the break from S1to S r , and to the break from S2 to Sslwas calculated from the conductivity and the temperature of the break, according to Eq. (4.1). From this it was found that the region where the slope is Szis bounded by locations of the Fermi level between 0.3 and 0.6 ev below thc bottom of la R. H. Bube, in “Photoconductivity Conference,” p. 575. Kiley, New York, 1956.
236
RICHARD H. BUBE
the conduction band in CdSe. To a firat approximation, the same value of the Fermi level is calculated for the break from SIto S,, as measured at different temperatures; similarly, to a first approximation, the same value of the Fermi level is calculated for the break from SIto S , aa meaaured at different temperatures. The Fermi levels calculated at different
L O I EXCITATION
INTtNIlW
h a . 6. Photocurrent ra a function of light intensity at different temperatures for 8 crystel of a&..
temperatures are the same only to a firat approximation; the break from SIto S1, for example, occurs when the hole demarcation level for the sensitizing centers is at the level corresponding to these centers, and there is a temperature variation of the location of the hole demarcation level with respect to the electron Fermi level, as given in Eq. (4.8). We shall consider how to determine the hole ionization energy for the sensitic-
IONIZATION ENERGIES IN CDS-TYPE MATERIALS
237
ing centers and their capture cross-section ratio S,/S, in the following section.
11. THERMAL QUENCHING OF LUMINESCENCE AND PHOTOCONDUCTIVITY The effect on the photoeensitivity is the w e , whether the demarcation level is moved relative to the eemitiaing levels by variations in light intensity at fixed temperature, or by variations in temperature at fixed light intensity. Data such aa those presented in Fig. 6 can therefore be interpreted in the same way as data on the thermal quenching of photoconductivity. If the levels through which the demarcation level moves
Fxa. 7. Photocurrent as for crystal of Cd&.
B
function of temperature at diffetent light intensitits
correspond to activator centers for luminescence, then there is an in-
creased probability for thermal hole freeing relative to radiative recombination. This leads to thermal quenching of luminescence. Thus to a first approximation thermal quenching of luminerrcence and photoconductivity are quite similar. Figure 7 shows the variation of photoconductivity with temperature for daerent intensitiea of excitation for a CdSe crystal.' The same data are plotted in Fig. 8 as the variation of photoconductivity with light intensity for different temperatures ; experimental points have been omitted from Fig. 8. Straight lines were drawn through those points lying in the range where the photocurrent varies as a power of hght
238
RICHARD H. BUBE
intensity of greater than unity, to intersect with the limiting curves at very high and very low temperatures. The temperature breakpoint from high to decreasing sensitivity in Fig. 7, and the light-intensity breakpoint from high to decreasing sensitivity in Fig. 8, occur when the hole demarcetion level is located at the sensitizing levels. According to the definition
Fro. 8. Same data as for Fig. 7, replotted to show photocurrent as a function of light intemity at different temperatures. The straight lines drawn here give the outline of the variation to show how points of intersection for use in analysis are obtained. The actual experimental ranges where the photocurrent varies as a power of light intensity greater than unity are considerably smaller than tha t indicated by the straight lines, rounding offthe curves occurring at both upper and lower ends.
of the demarcation level in Eq. (4.7), the following condition holds at this breakpoint : (11.1) In nb = In (N,S,/S,) - E*/kTb where nb is the density of free electrons a t the breakpoint, E* is the hole ionization energy for the sensitizing centers, and Tb is the temperature of the breakpoint. If In nb is plotted against l / T b Ja straight line is obtained with slope E*/k and intercept a t 1/Tb = 0 of In (N,S,/S,). A simple analysis* also suggests that the following condition attends the end of thermal quenching or decreasing photosensitivity with decreasing light intensity at fixed temperature:
-
IONIZATION ENERGIES IN C D W N P E MATERIALS
In n. = ln (N.NI/N,r)
- E'/kT,
239 (11.2)
where n. is the density of electrons at the end of quenching, Nr is the density of centers with a large cross section for electrons, NIIis the density of centers with a small cross section for electrons, and T. is the temperature at the end of quenching. The particular simple form of Eq. (11.2) results from the assumption that the electron cross section of the I centers is equal to the hole cross section of the I1 centers; it is therefore not a
FIG.9. Plot of photocurrent at the breakpoint from high sensitivity to decreaahg sensitivity, and from decreasing sensitivity to low sensitivity, plotted as a function of the temperature at the breakpoint, according to Eqs. (11.1) and (11.2).
general equation like Eq. (lLl), and in practice other conditions may mark the end of quenching. The results of plotting the data of Figs. 7 and 8 in accordance with Eqs. (11.1) and (11.2)are shown in Fig. 9. The slopes of the two lines are equal and give a value of E* = 0.64 ev. The intercepts of the two lines, sssurning N. = lo**cm-', give SJS, = 8 X 10' and N I / N ~= ~2, when the values of photocurrent are translated into values of n. An alternative way of analyzing the data of Figs. 7 and 8 is to plot the calculated electron Fermi level for the breakpoint from high sensitivity aa a function of the temperature at which this breakpoint occurs,
240
RICHARD H. BUBE
i.e., E,,,, versus Ts.Following Eq. (4.8) such a plot will show a straight line with slope k In (SJS,) and intercept at Te = 0 of E l .
-
12.
OPRCAL QUENCHING OF LUMINESCENCE OR PHOTOCONDUCTIvITY
As mentioned in Part I, optical quenching of luminescence or photoconductivity can be caused by the optical freeing of holes from activator or sensitizing centers; this occurs in the temperature range below that at 1
I
I
I
I
I
III
I 0
I
ori
1
I
I
I
I
I
I0
ao
SO
40
80
*O
TIME,
SEC
Fxo. 10. Dynamic quenching curvea for a crystal of CdS for eecondary radiation by wavelength of (1) 6500 A; (2) 6750 A; (3) 7000 A; (4) 7250 A; (5) 7500 A; a d (6) 80oO A.
which temperature quenching of photoconductivity sets in because of thermal freeing of these holes. An example of the type of transient behavior usually encountered in such meaaurements of optical quenching is shown in Fig. 10 for a CdS crystal.20 A background or bias photocurrent ie produced by light near the absorption edge of the material, in this case 5350 A light. When a second source of radiation is allowed to illumia*R. € Bube, I. P l y . Rw. 89, 1106 (1955).
IONIZATION ENEBOIEB IN C D m P E MATERIALS
241
nate the crystal, the current versus time variations are as ahown in Fig. 10, depending on the wavelength of the secondary radiation. Light with wavelength between 6500 A and 7250 A produces both excitation of photoconductivity, due to excitation from imperfection levels to the conduction band, and quenching of photoconductivity, reeulting from excitation from the valence band to the sensitizing centers. For wave lengths greater than 7500 A, only quenching ia found, the energy of the
PHOTON ENERGY, aw.
Fro. 11. Infrared quenching epectra at room temperature for four typical crystals of CdS. Curve 3 ia for an inaenaitive crystal, curvtyl1 and 4 for crystals of intemndiate wnaitivity, and curve 2 for a 8eMitiVe crystal.
light now being too small to produce excitation. The shape of the curves ia determined by the different rates at which excitation and quenching by the same wavelength light establish themselves; in general the excitation is more rapid, producing an initial maximum when the eecondary radiation is turned on, and a minimum when the eecondary radiation ie turned off. Typical optical quenching curves for CdS are shown in Fig. ll.M Per cent quenching is defined as the ratio of the Merence between the
242
BUBE
RICHARD H.
l t Y C C R 4 l U R L , .C
100
0
I
-
u
-
Y I
S80
-
c
-
I Y 0
e
Y
a
-
-
-
0. -Loo
I
I
-100
I
I
0
I
I IOQ
d
TEYCLRATUIE. *C
FIQ.12. (a) Temperature variation of per cent quenching for a CdS crystal for photons of (1) 1.65 ev; (2) 1.35 ev; and (3) 0.89 ev. Curve (4) shows the temperature dependence of the relative magnitude of initial stimulation’caused by 1.65-e~redhtion; the behavior probably ia due to emptying of traps. Curve ( 5 ) in the inset gives the temperature dependence of the photocurrent at constant primary radbtbn intensity. (b) Temperature variation of per cent quenching in CdSe crystal for photonr of (1) 1.20 ev; (2) 1.05 ev; and (3) 0.79 ev. The curve in the h e t gives the temperotUre dependence of the photocurrent a t constant primary radiation intensity.
maximum current and tbe long-time steady current with the secondary radiation on, to the maximum current. The measured quenching spectrum consists of two main “bands”; a narrow band exists with maximum at about 0.9 ev, separated somewhat from a much broader “band.” Investigation of the phenomenon has shown that the broad “band” h really not a band a t all. The decrease in quenching at higher energies ia only apparent; it is due to the radiation in this wavelength range cauaing
XONIZATION ENEROIEB IN CD&TYPE MATERIAL8
243
both excitation and quenching. In special cases quenching can be measured down to energies within 0.1 ev of the absorption edge, with a lowenergy cutoff a t about 1.1 ev. The highenergy cutoff of the quenching spectrum is determined therefore by the long-wavelength photoconductivity response spectrum; the further out the long-wavelength response extends, the lower the highenergy limit of the quenching spectrum. It can be seen from curve 3 in Fig. 11 that the 0.9ev quenching peak does not occur in insensitive crystals. Measurements of quenching as a function of temperature gives additional information on the properties of this 0.9 ev quenching band. Figure 12a shows the temperature variation of quenching by 0.89, 1.35, and 1.65 ev photons in CdS. The inset shows the temperature variation of the corresponding photocurrent, measured a t constant primary light intensity. This intensity was chosen to give the aame room temperature conductivity as that used for measurements of infrared quenching. All optical quenching starts to disappear at about the same temperature that thermal quenching sets in, the holes beipg freed from sensitizing centers thermally rather than optically. In addition, however, quenching by 0.89 ev also disappears at low temperatures, indicating the existence of a thermal step in this process. We are led to propose the existence of two levels for the sensitizing centers: one lying 1.1 ev above the top of the valence band, and another lying about 0.2 ev above the top of the valence band. Excitation from the lower to the upper level, followed by thermal freeing of the hole from the 0.2 ev level, produces the 0.9 ev quenching band. Direct excitation from the valence band to the 1.1 ev level produces the main quenching. This is our first example of multiple levels, which we shall discuss further in Part IV of this paper. Figure 12b shows similar temperature dependence of the quenching data for a CdSe crystal; the low-energy limit for quenching in this material is a t about 0.6 ev, and thermal freeing of holes from these levels occurs below room temperature. 13. THERMALLY STIMULATED LUMINESCENCE OR CONDUCTIVITY
Thus far our discussion has been concerned almost exclusively with recombination centera. I n this section we consider a technique useful for determining the location and density of trapping centers. Suppose that a crystal is heated in the dark after being excited at a low temperature so as to fill trapping centers. Then the excess of the measured thermally stimulated current, contributed by carriers freed from traps, over the normal dark current provides a measure of the density and energy distribution of trapping centers. Similarly if the thermally freed carriers recombine to produce luminescence emission, the emiaeion may be detected rather than the stimulated current. For
244
RICHARD H. BUBE
thermally stimulated current, the basic equation is:
An = - ( d n t / d t ) ~
(13.1)
where An is the density of free electrons resulting from the thermal emptying of electron traps as the temperature is raised, dnt/dt is the rate of trap emptying, and r is the lifetime of an electron freed from a trap. For thermally stimulated luminescence, the basic equation is: AI =
- (dnt/dt)&
(13.2)
where A1 is the emission intensity resulting from the recombination of electrons freed from traps with holes a t activator centers, &is the efficiency of the radiative recombination process, and it is assumed that the rate of trap emptying governs the kinetics. We shall confine our discussion to thermally stimulated conductivity in the following. I n order to determine the proper form for dnt/dt, it is necessary to comider the occurrence of retrapping during the thermally stimulated proceas. There are two simple alternatives: (a) aasume negligible retrapping; or (b) assume strong retrapping. In the ease of negligible retrapping, dnt/dt = -nP,which integrates to give:
dnddl
=
-ntP exp [-J'(P/B)dT1
(13.4)
where nt, is the initial density of trapped electrons; P is the probability for thermal freeing, P = NJLu exp (- Et/kT);and B is the linear heating rate, B dt = dT. The temperature T,, corresponding to the maximum value of An for a given trap depth E,, can be calculated by setting d(ln An)/dT = 0. If the temperature dependence of 7, Nc, S., u, and the electron mobility can all he neglected in the region of the thermally stimulated maximum, the following simple relationship reaults: (13.5)
I n order to obtain values of El and S , from experimental data, it is necessary to make several measurements for different valuea of the heating rate 8. Then if Q. (13.5) is rewritten: In (Tn*/B)= Et/kTm
- In (N,,S,,uk/Et)
(13.6)
it is evident that a plot of In (T,*/B) as a function of l/Tm will give a straight line with slope Et/k. Knowing Et, the intercept when l/Tm = 0 can be used to evaluate S,. I n the cam of strong retrapping, we may aaaume that the traps are effectively in equilibrium with the conduction band. Then the trap depth
IONIZATION ENERGIES IN CDS-TYPE MATERIALS
245
E, can be calculated from curves of thermally stimulated current by calculating the location of the Fermi level corresponding to the conductivity and temperature of the maximum thermally stimulated current, according to Eq. (4.1). Figure 13 shows a simple curve, with only one predominant maximum, for thermally stimulated current in a CdS crystal. The results are shown for two heating rates, which differ by a factor of seven. If the Fermi level 2.01
I
TEYPERATURE,Y
FXO.13. Thermally stimukted current CUNBB for B Cd8 crystal, for two different heating ratea.
corresponding to the maximum is calculated, it is found that the shift in the maximum temperature due to the lower heating rate is just sufficient to counterbalance the lower maximum conductivity. The calculated Fermi level is the same for both curvea, about 0.35 ev. However, the trap depth calculated from Eq. (13.6), assuming negligible retrapping, is 0.26 ev. Excluding special traps with a very small capture cross section, it aeems likely that the Fermi-level evaluation is the more reliable. Simple thermally stimulated current curves like that of Fig. 13 appear to be the exception rather than the rule. Typical thermally stimulated
246
RICHARD EL BUBE
current. curves for four CdS crystals are shown in Fig. 14.*O These show evidence for at least 7 different trapping centers with depths between 0.2 and 0.8 ev. Data accumulated by several investigators over the past few years indicate that about 7 trap depths are reproducible and charecteristic of CdS.*@-” With this number of trapping centera to account for, it is no easy matter to assign certain imperfections to specific trapping centers, as has been done in the case of recombination centers. This is still an active field of research; present indications are that vacancies and vacancy pairs are almost certainly involved.
I
I
I
I
TEM P E R ATU R E
.
I
*C
FIO.14. Illustrative thermally stimulated current c u n w for four Cd8 cryetalr, ohowing indicetione of about Beven reproducible trap depth.
One of the best identifications of trapping centers with particular impurities is that which has been made for ZnS phosphors using the results of studies on thermally stimulated luminescence.** It has been possible to identify specific emission peaks with each of the impuritiee: aluminum, scandium, gallium, and indium. Since we have emphasiaed the correlation between compensated acceptors and sensitizing centera in past sections, it is fitting that we should here emphasize the correlation between compensated donors and trapping centers. Each of the afore11
J. Woods, J . El&ronia and Control 6, 417 (lQS8). R. H.Bube and L. A. Barton, RCA Rcu. 20, 564 (1959). W. Hoogenetraaten, J . E~!&ru&m. Soe. 100,368 (1963).
IONIZATION ENERGIES I N CDS-TYPE MATERIALS
247
mentioned trivalent catiom is present aa a compensated donor in ZnS. Likewise anion vacancies, which certainly play a role in trapping proceases, are also present aa compensated donors. The density of traps corresponding to a particular thermally stimulated current peak can be calculated from the area under the curve. In order to do this, the lifetime of the thermally freed electrons must be known. This lifetime can be approximated from steady-state photoconductivity data, measurements being made at such a low light level as to give a photocurrent equal to the thermally stimulated current at the same temperature. 14.
SPACE-CHARQE-LIMITED
CURRENTS
When a sufficiently large field is applied to an insulator with ohmic contacts, electrons will be injected into the bulk of the material to form a current which is limited by space-charge effects. The magnitude of this current in a trap-free material can be quite large, but in practice it is reduced by trapping effects. Space-charge limited currents in a solid are about lo-' those in a vacuum diode, mainly because the average velocity of carriers in a vacuum (v = l O Y ) is much larger than that in a solid (u = pV/Z). The expression for the space-charge limited current density in a solid is: j = lW1aV2p~/Z1 amp/cm2. (14.1) For a voltage of 10 volts applied across cm of a trap-free material with p of about 100 cm2/volt sec and c = 10, j comes out t o be about 10 amp/cm*. When trapping centers are present, they capture many of the injected carriers, thus reducing the density of free carriers. The form of the current versus voltage curve in the region of space-charge-injected currents can therefore be used to obtain information about trapping cer~ters.~~-~' Figure 15 shows some experimental results with CdS which illustrate this technique.28 The straight line a t the top of the graph represents the current expected in the absence of traps. The measured current exhibits three distinct types of behavior, namely:. (a) at low voltages, the dependence of current on voltage obeys Ohms law; (b) at intermediate voltages the current varies as the square of the voltage; and (c) at higher voltages, the current rises very rapidly with voltage toward the curve characteristic
*'R. W. Smith and A. Row, Phys. Rev. 97, 1531 (1955). "A. Rose, Phya. Rev. 97, 1538 (1955). I'M. A. Lampert, Phys. Rev. 103, 1648 (1956). a' M. A. Lampert, A. Row, and R. W.Smith, Phya. & Chem. Solids, in prees. 1.R. W.Smith, RCA Rar. 20,W (1969).
248
BICHARD H. BUBE
of trapfree behavior. The Ohms law behavior is associated with the preinjection condition in which the bulk conductivity is dominant; the conductivity corresponds to a location of the Fermi level about 0.95 ev below the bottom of the conduction band. The region in which current varies as the square of the voltage can be interpreted as resulting from a discrete level of density N ,lying B: below
Fro. 15. Space-charge-limited current aa e function of applied voltage on e crystal of CdS. (After R W. Smith.)
the bottom of the conduction band. As long as the Fermi level, which rises when current injection occurs, is further from the bottom of the
conduction band than E:, and moves in a region of the forbidden gap where the level density ia much less than N t , the ratio of free electrons n to trapped electrons n1is constant and independent of applied voltage. In thie region, the current density ie given by: j = lO-*V~ptr/Z~ amp/cm*
(14.2)
IONIZATION ENERGIES I N CDS-TYPE MATERIALS
249
where I is given by: 1
= n/nt =
5 Nt
e-Bilkr.
(14.3)
The extremely steep part of the curve of Fig. 15 corresponds to the condition in which the traps have been essentially filled. The density of traps which have been filled can be calculated from the voltage VV, corresponding to the condition of filled traps:
(14.4) where C is the capacity of the crystal, and A is the cross-sectional area. A value of N t = 3 X 10" cm-* is computed. When r is evaluated by comparison with the curve for trap-free behavior, [Eqs. (14.1)and (14.2)] and N c = 3 X 10" cm-' is inserted in Eq. (14.3),a value of Ec = 0.8 ev is obtained. Another way of calculating the location of the trap level is to make use of the fact that the Fermi level is located a t the trap level when V = 2Vtj/3; calculating the Fermi level from the current corresponding to this voltage again gives a value of Et = 0.8 ev. 111. Trends in the Ionization Energies in CdS-type Materials
15. SUMMARY OF MEASURED ENERGIES The ionization energies of imperfections in CdS-type materials, measured by the various techniques described in this article, are summarized in Table 11. The method by which the energy was found is also indicated. Some of the assignments of observed ionization energies with specific imperfections is difficult. In the case of Ag in CdS, for example, spectral response in powders and layers indicatea that the Ag center definitely has a smaller ionization energy than the Cu, whereas thermal quenching data on single crystals with relatively small Ag concentrations indicate the same hole ionization energies for the two impurities. It ie quite likely in the latter case that the actual sensitizing centers are cadmium vacanciee rather than the silver centers. 16. APPROXIMATE CORREIATIONWITH CATIONAND ANION OF MATERIAL CONSTITOENTB The data of Table I1 suggest an approximate correlation between the observed ionization energies and the individual cation and anion constituents of the material. To a first approximation the electron ionization energiea are dependent primarily on the cation present in the material,
250
BICHARD H. BUBE
Material
zna
(Eu= 3.7 ev)
Imperfection
a
Electron ionisation energy, ev
Ga
0.25 0.25 0.25 0.35 0.42
In
0.6
Br Al sc
cu
Hole ionisation energy. ev
Technique used
Ts Ts Ts Ts Ts Ts 1 .O
Ab,
Ex, TQ, c
OQ
-
ZnSe
(Eu 2.7 ev)
0.65
Ab,Fh,TQ
Con
d
cu
0.6
1
Sb
0.6 0.7
Ex, TQ, OQ Ex, TQ, OQ
A8
0.7
Br
0.21
&
-
CdS (EQ 2.4 ev)
a,Br, I All Gal In cu
ZnTe
-
*
Ex,TQ,OQ
’
Ex,%
d
Con
0.03 0.03
*J
Con Exl TQ
b
Ex,TQ Ex,TQ,OQ
4
I.I
Vod
1.0 Il.0 1.0
cu
0,ll
Con
I
Con Ex,TQ,OQ Con, TS Ex,TQ
I
0.64
&
(EQ= 2.0 ev)
Sample referend
Cd&
(EQ 1.7 ev)
CdTe (Eo = 1.60~)
Cl, Br, I
cu
0.03
VEh VOd
0.14
I Li
0.01
0.6
Sb P
0.27 0.36 0.38
Ns
0.29
Vod
0.3
scs nolcsfor
hb16
on opporifc poqs
Con Con Con Con
Con Con
b
b
m II)
I ,
. I
a
IONIZATION ENERGIES IN CDS-TYPE MATERIAL5
251
Fro. 16. General summary of characteristic donor and acceptor energies in CdS type materiale.
with only second order differences existing for different imperfections. Similarly, to a first approximation the hole ionization energies are dependent primarily on the anion preaent in the material. The electron ionization energies are of the order of a few hundredths of an electron volt in cadmium compounds, and of the order of several tenths of a volt in zinc compounds. The hole ionization energies are of the order of a volt in sulfides, about 0.6 of a volt in selenideg, and a few t e n t h of a volt in telluridm. Firlire 1 R slimmarims thin r e n ~ r a lc?nrrelat.ion.
N d for Table I I Key: Ab, Optical absorption. Con, Conductivity VBIUUB T. Ex, Excitation spectrum of luminescence or photoconductivity. "8, Thermally stimulated luminescence or conductivity. TQ,Thermal quenching of luminescence or photoconductivity. Og, Optical quenching of lumineecence or photoconductivity. F. A. Kroeger, Phyiku 22, 637 (1956). * H.A. Kleeens, J. Electrochem. Soc. LOO, 72 (1953). a R. H.Bube, Phys. Rw. 90,70 (1953); G . F.J. Garlick and A. F. Gibeon, J . ON.Soc. Am. 89, 935 (1949). 'R. H.Bube and E . L. Lind, Phyu. Rw. 110, 1040 (1958). * R.H.Bube, J . Chcm. Phys. 28, I8 (1955). 'F. A. Kroeger, H. J. Vink, and J. Volger, Philip8 Ruearch Zbpfa. 10, 39 (1955). R. H.Bube and 8. M.Thornsen, J . Chem. Phyu. 28, 15 (1955). * R. H . Bube, Phvs. & Chem.Sdida 1, 234 (1957). 'R. H.Bube, Phys. Rev. 99, 1105 (1955). f R.H.Bube and E. L. End, Phys. Rev. 1 06, 1711 (1957). R. H. Bube and L.A. Barton, J . C k m . Phyu. 29, 128 (1958). IF.A. Kroeger and D. DeNobel, J. ElGdronics 1, 190 (1955). D.A. Jenny and R. H.Bube, Phys. Rw. 98, 1190 (1954). mE.L.Lind and R. H.Bube, unpubliahed data (1966).
252
RICHARD H. BUBE
That donor energies should be determined primarily by the cation of the compound involved is consistent with the simple picture of the bound electron of an un-ionized donor center being shared by neighboring cations; similarly, the bound hole of an un-ionized acceptor center ie considered to be shared by neighboring anions. Investigations of the trapping centers associated with compensated aluminum, scandium, gallium, and indium donor centers in ZnS, by means of thermally stimulated luminescence emission, showed that the trap depths do not depend on the particular activator used.lg In addition, it was found that no new traps are introduced by making solid solutions of ZnS-ZnSe, but new traps are introduced by making solid solutions of ZnS-CdS. Similarly, no new luminescence centers are introduced by making solid solutions of ZnS-CdS, but new centers are introduced by making solid solutions of ZnS-ZnSe. The magnitudes of the ionization energies given in Table I1 agree well with the available informatian on the types of conductivity cornmonly found in the various materials. The preparation of high n-type conductivity material is easy for cadmium compounds but difficult for zinc compounds. The extreme difficulty of this preparation for ZnS indicates that defect compensation is the favored form of imperfection incorporation. High p t y p e conductivity is easy to achieve in the tellurides, but is much more difficult in the selenides and sulfides. If the ionization energy can be taken as a measure of the effective mass of the corresponding free carrier, the results summarized in Fig. 10 show that the effective masses of both electrons and holes decream 88 the atomic number of the constituent atoms increases. IV. Indications of Complex Ionization Processes
17. VARIATIONOF IONIZATION ENERGY WITH IMPERFECTION
CONCENTRATION It is well known that the ionization energy in semiconductors decreaeea with increasing imperfection concentration. There are indications that the same type of phenomena may also occur in insulators in which imperfections have much larger values of ionization energy. Measurements on certain single crystals of CdS and CdSe, prepared or annealed under conditions which would favor a high concentration of sensitizing centers, have shown the existence of sensitizing centers with hole ionization energies in the 0.1 to 0.3 ev range, instead of the normel 1.0 ev and 0.6 ev for CdS and CdSe, respectively.2*2b These low-ioniseH.A. Klaaens, J . El&rochnn. Suc. 100, 72 (1953). R. B.Bube, J . C h .P h p . 80, 206 (1959).
253
IONIZATION ENERGIES IN CD%TYPE MATERIALS
tion-energy centers seem generally to be aesociated with surface regions of the crystal. In mme casea reversible photothermal processes are observed; this suggests that the agglomeration of the centers results in low ionization energies, and that their dispersal results in normal ionization energies. A more thorough investigation of thie phenomenon has been carried out with CdS photoconducting powders, containing Ga and Cu impurities.'O The Cu concentration waa varied from 4 X lo*' to 2 X 10*O cm-', but always with a constant ratio of Cu concentration to Ga concentration of 1.05. The hole ionization energy, as determined from the thermal quenching of photoconductivity decreases from about 1.0 ev for low Cu TABLE111. E ~ M P L E SOF DOUBLE LEVEU Ratio of First Second second to h t Ionization ionization, ionization, Technique hole ioniaaof ev ev ueed tion energies Material Imperfection CdS cdh ZnSe cdse
Vm, (Cu) Holes VCd
Holes
1.Oev 0.6
(Sb)
Holes
07
vs
Electrons
0.14
l.6ev
SCLC
1 .o 1.3 0.6
TQ,OQ Ex,OQ Con, "8
cdS
cdse ZnSe
1.6 1.7 1.9
Key: Con, Conductivity versua T. Ex, Excitation of photoconductivity. TS, Thermally etimuldad conductivity. TQ, Thermal quenching of photoconductivity. OQ, Optical quenching of photoconductivity. SCLC,Space-charge-limited current injection.
concentrations to about 0.3 ev for 2 X 1O'O Cu cm-'. An invariance of the long-wavelength limit for photoconductivity excitation with Cu concentration indicates that the initial level does not move toward the valence band, but rather that interaction results in an effective decrease of the energy required to excite the hole to the valence band. This interaction probably concerns low-lying levels which are part of the Cu center, aa will be discussed in the final section of this paper. 18.
DOUBLE LEVEU
Imperfections with multiple levels are well known in semiconductors, as for example gold in germanium, which has four measured levels. We R. H. Bube and A. B. Dreeben, Phye. Reu. 116, 1578 (1959).
264
BICEARD R. BUBE
might expect at least double levels for such imperfections as cation and anion vacancies in CdS-type compounds. Some evidence haa been accumulating that such levels do in fact exist. Data on these levels are summarized in Table 111, together with an indication of the technique UBBd to locate the second level. The observation, by means of the space-charge-limited current injection technique, of the level in CdS lying 1.6 ev above the top of the
WAVEWOT)(,
Y I C M
ho. 17. Photoconductivity spectral reeponee for a Zn8e:Br:Sb crystal.
valence band was discussed previously.** Concurrent photoconductivity data indicated that this center is not a simple trapping center, but a recombination center though not the principal one. Its properties seemed to be best explained by assuming it to be a doubly negative center of the same type as the singly negative normal recombination center. This means that the level lying 1.0 ev above the top of the valence band should be identified with a V$ center, and the level lying 1.6 ev above the valenae band with a V Ocenter. ~
IONIZATION ENERGIES IN CDB-TYPE MATERIAL8
255
The double levels in ZnSe with incorporated Sb and Br impurities can be detected in both the spectral response and in the optical quenching.10 The spectral responee curves of Fig. 17 show a main peak near the absorption edge of ZnSe, a shoulder breaking at about 7000 A, and a long-wavelength tail with threshold wavelength of about 1.1 microns. The two long-wavelength shouldera indicate transitions of about 1.8 ev and 1.1 ev, respectively. The long-wavelength response is much less at -177°C than at 25°C principally because of the strong simultaneoue
I
'Ot
u
60
r'
9
2
c
cI
50
-
40
-
n
N
+
1'
a
so-
10
*- l
0
0
0.S
LO
I .s
ZD
PHOTCn ENLROI,r
FIG.18. Infrared quenching spectra for the ZnSe:Br:Sb crystal of Fig. 17, both st -177OC and at 25°C. Curvos for two different wavelength primary rediathM are shown at 25'C.
quenching caused by light of such long wavelengths. Subtraction of the above energies from the band gap value of 2.7 ev for ZnSe indicates levels lying roughly 0.9 and 1.6 ev above the top of the valence band. An identification of these energy differences, which may be more accurate, can be obtained from the curves of optical quenching of photoconductivity. These are given in Fig. 18,for quenching both at - 177OC and at 25OC. Optical quenching at the low temperature is associated with transitions from the valence band to a level about 0.7 ev above the top of the valence band, and to higher levels. Optical quenching at room
256
RICHARD H. BUBE
temperature is associated with transitions to levels lying about 1.2 to 1.3 ev above the top of the valence band. The absence of the lowenergy optical quenching at room temperature is caused by the thermal release of holes from theee oentere. The evidence for double levels in CdSe haa been found with crystals which were initially insulating and insensitive, but were made more conducting and sensitive by annealing in vacuum.11 A definite difference L found between these crystals and sensitive crystals made by incorporating Cu and I impurities. Although both types of crystale have
FIQ. 19. (a) Photocurrent M a function of light intemity at mom temperature. (b) Photocurrent aa a function of temperature for a light inteneity of 0.6 f t c .
approximately equivalent photosensitivities a t high light intensitiea and/or low temperatures, and although both types show a temperature quenching phenomenon occurring over the same temperature range, the annealing-sensitized crystala are 10' to 10' times more sensitive at low light intensities and/or high temperatures. This is illustrated by the experimental curves shown in Fig. 19. Figure 19a shows the variation of photocurrent with light intensity at room temperature, and Fig. 19b shows the variation of photocurrent with temperature at a constant low
** R. H. Bube and L. A. Barton,J . Chem. Phys. 28,
128 (1958).
IONIUTION ENEROIEB IN CD&TYPE MATEIUALB
257
light intensity. The photosensitivity of the CdSe:I:Cu crystal drops to that of an untreated cryatal in one atep; this decrease can be as great aa a factor of 10' in temperature quenching. However, the photoaemitivity of the annealing-sensitised CdSe decresaes only by a factor of about 10%to 10' in temperature quenching; it remains, over the entire temperature range of the measurements, about 10' to 10' t i m a greater than that of the insensitive, untreated cryatale. If this residual sensitivity corresponds to holes captured at higher lying sensitiring centers which are not
0
01
0.s
I.o PHOTON EMRQY.a
1.t
ha 20. Infrared quenching spectrum for an annealing mnritbed cryatid of Cdk at mom temperature.
thermally emptied at room temperature, then it ahould be possible to observe infrared quenching at room temperature. Figure 12b show that in CdSe: I :Cu no such quenching can be observed at room temperature becaum all holea have been thermally freed. Infrared quenching at room temperature is obaerved in the annealing sensitized crystals at room temperature, however, and the quenching spectrum is given in Fig. 20. This curve indicates that the higher lying level ia located about 0.9 to 1.0 ev above the top of the valence band. Evidence for the existence of the electron ionisation energiea was found in the =me experiments with annealing aensitiwd CdSe
ws
RICHARD H. BUBE
The presence of a level 0.14 ev below the conduction band was indicated (a) by the occurrence of a new trapping center with depth of 0.14 ev in annealed crystals which were still insulating after annealing, (b) by the slope of the log conductivity versus 1/T plot in annealed crystala which were conducting after annealing, giving an activation energy of 0.14 ev and (c) by the exponential variation of photocurrent with T in annealed conducting crystal, which could be attributed to 0.14 ev centers acting as efficient recombination centers when occupied by electrons. The presence of a level 0.6 ev below the conduction band WBB indicated (a) by the occurrence of a new trapping center with depth of 0.6 ev in annealed crystals which were still insulating after annealing; and (b) by TABLE IV. TRANIITIONS INVOLVED IN LUXINIDBCENCE, sv ~
Abeorption edge Cut emission Cu, emission Ag, ernkion A g r emiegion Infrared excitation
Infrared emieeion
~
~
~
_
~
~
_
ZnS
CdS
cdse
3.7 2.3 2.7 2.8 3.3 1.6 0.91
2.4 1.2 1.6 1.6 2.1 1.6 0.82
1.7 1 .o 1.36
0.M
0.76
0.70 0.09
0.60
0.67
the slope of the log conductivity versus 1/T plot in annealed crystah which showed only a slight increase in dark conductivity afterannealig. Another kind of double level ssaociated with certain impuritiea in CdS-type materials haa been detected through measurements of luminescence emission. It haa been shown that there are two emission bands associated with Cu and Ag impurities in ZnS, Cd8, and CdSe, one band being predominant when the activator (acceptor) concentration is equal to or less than the coactivator (donor) concentration, and the other band becoming evident when the activator concentration exceeds the coactivator concentration.as A summary of these transitions***8 is given in Table IV, where the subscript 1 indicates the emiesion found for activator
w.van Cool, Philip8 Research Repf. 13, 157 (1958). P. F. Browne, J . Eledronico 2, 154 (1956). s4 E. Gnllot and P. Guintini, Compt. r m d . mod. sci. 388,802 (1963). a' E. Grillot and P. Guintini, Compt. r m d . a c d . en'. 38S, 418 (1964). G . F. J. Garlick and M. J. Dumbleton, Proc. Phyr. Soc. B67, 442 (1964). IT P. F.fBrowne, J . EZecironico 2, 1, 96 (1966). G. Meijer,Phyr. & C h . Sdidr 7,163 (19s).
IONIZATION ENEBGIEB IN CDB-TYPE MATERIAL8
259
concentration equal to or lem than coactivator concentration, and the subscript 2 indicates the emission found for activator concentration greater than that of the coactivator. In each case the “2” emission is of higher energy then the “1” emission. A variety of models have been proposed to explain these phenomena, but no one hss definitely been established. Measurements on CdS:Ag provide an indication that the “1” emission may be the reault of a transition from the conduction band to a level lying above the valence band, whereas the “2” emission may be the m u l t of a transition from a level lying below the conduction band to the valence band.*+’* It is possible in thm case that the l o c a l i d levels involved in the “2” transition are anion vacancy levels formed to compensate the Ag in e x c w of the coactivator concentration. 19.
MULTIPLE LEVELB
The most complex level structure about which any definite information ia available ie probably that sssociated with Cu impurity in 2x8 or VISIBLE EMIssloN
1.2 au L6rv
A B
OBeW
c
047w OROw 0.mw
F D
INFRARED EMlSSKm
E
4 AL
Y IONIZED L/9QBLLLyEL [C””] -O
[cq‘
FIG.21. Illustrative energy level ffiheme for a possible eet of energy levels a m ciated with Cu impurity in Cd8.
CdS. In our previous discussion of infrared quenching of photoconductivity in CdS, in which the imperfections probably are cation vacancies, it waa shown how two levels were needed to explain the two quenching bands and their temperature dependence. The Cu impurity center peeme to have thia same level structure, and in addition there is evidence, from measurements of excitation and infrared luminescence emiwion from “J. h b e md C.C. Klick, Phv8. Rev. B8,909 (1955). “J. h b e , Phy8. Rw. S6, 985 (1956). “ J. b b s , Pky8. Rsrr. 100, 1S86 (1965). “ J. h b s , p.Z h - la, 1715 (1956).
260
RICHARD H. BUBE
thew materials, for several other levels. The traneition energies are summarized in Table IV. There are two principal excitation tramitions which correspond very cloeely to the two infrared quenching bands previously described. There are, however, three infrared emiseion bands. A hypothetical energy level scheme for Cu impurity in CdS ia indicated in Fig. 21. It ia very likely that future m a r c h into the ionization procesees and excitation proceeses in insubtore will continue to demonstrate that the concept of single energy levels associated with a given imperfection is a gram approximation.
Cyclotron Resonance* BENJAMIN LAX
AND
JOHNG. MAVF~OIDFS
L i d n Ldordny, Mauachuutla Z d i r J s oj T d W , Inrington, Xamochwsl(r
I. Introduction.. .....................................................
261
.............................
284 273 276
.................................
311
11. Cyclotron Rsronrncs of Bee Charged Puticlsl,. ........................
.................
2. I0m . . . ......................................................... 111. Cyclotron Resonance of Carriers in Solids.............................. 4. Phenomena in Metals.. .
7. Millimeter Cyclotron Resonance.. .................................. 8. Croas-Modulation... . . . . ........................... 9. Cyclotron Remonmce in Strained Germanium and Sicon.. ............ 10. Magnetoacoustic Resonance.. ..................................... 11. Cyclotron Reronance Generaton m d Amplifiers.. .................... v. Summary and Future Prorpecta....................................... Acknowledgmenta. ..................................................
264
370 374 377 379 383
388 808
I. Introduction
The phenomenon of cyclotron resonance has been known for some time and its manifeatatione in i o n i d gmea have been invmtigatd extenaively. Studies of the phenomenon in solids were begun only quite recently but dramatic results already have been obtained. Both from a historical and academic viewpoint, it, is desirable to discuss cyclotron resonance of free electron6 and ions as it may occur in ionized gases mnce the effect is fundamentally simpler in gases and many of the fundamental considerations apply ale0 to mlida. The basic idea nec88BLIly for understSndin8 thie phenomenon ie that an electron in a dc magnetic field traces out a helical path with the axis of the helix along the direction of the magnetic field and with the well-known cyclotron frequency eH 0.
=
mc
* Thin article w u prepued et Lincoln Laboratory, a center for remaroh opsrrted by Manwhuoetta Inntitutu of Teohnobgy with the joint mpport of the U. 8. Army, Navy, md Air Fbm. 261
262
BENJAMIN LAX AND JOHN 0. MAVROIDES
where e is the electron charge, m the electron mass, H the magnetic field, and c the speed of light. If now an alternating electric field of frequency w is impressed on the system transverse to H , then in addition to ita rotational motion a t the cyclotron frequency we, the charged particle will oscillate simultaneously a t the frequency w as well. Furthermore, if w = we, the particle will gain energy resonantly from the alternating electric field and increase its radius of orbit in an ever-increasing spiral. This will continue until finally the charged particle will collide with B neutral atom. In order that there be a significant effect of the magnetic field on the interaction between the moving particle and the rf field, the time between collisions T for a particle must be sufficiently long that it travels at least +T of a revolution, i.e., O,T 2 1. Cyclotron resonance in solids differs from that in ionized gasea in two main respecta, both of which are a consequence of the crystalline properties of the solid. In the first place, as the electron moves in the periodic field of the crystal, it does not behave as a particle with a simple mass. For the simplest situation, the dependence of energy on wave vector k is given by & = (A2k2/2m*).The scalar masa m*, known as the effective mass, differs from that of a free electron and is usually smaller than the free electron mass for most of the aemiconductm and metals which are treated in this article. By use of quantum mechanics, it can be demonstrated that, in a nearly full band, the electron behavee aa if it had a negative masa. By convention, the mass is assumed positive and the charge of the electron, which is normally negative, is therefore assumed positive; this carrier is known as the hole. Although it is an electron that is actually moving in the solid, it leaves behind an equivalent positive charge in the vacancy, and as another electron movea into the vacancy, the positive vacancy also move8, but in the opposite direction to the electron. For a nearly full band it is this positive charge and its effective properties which are of interest to us. This may be demonstrated in another way. By definition from quantum mechanics l/m* = (l/h2)(a*&/ak*), i.e., the effective mass is a measure of the curvature of the energy surface in energy-k space. The simplest quantummechanical model that illustrates this phenomenon is treated in standard texts;' in this model an electron moving in a crystal of periodic structure is represented by a plane wave, with the wave function $, which haa the characteristic periodicity of the lattice. When the wave vector of the electron approaches * * / a , where a is the lattice spacing, the electron undergoes a Bragg reflection. If the Schriidinger equation is solved by means of the usual perturbation methods, it can be shown that the energy C. Kittel, “Intmduction to Solid State Physics.” Wiley, London, 1967; F. &its, “The Modem Theory of Solids.” McGraw-Hill, New York, 1940.
CYCLOTRON RESONANCE
263
versus k curve has a discontinuity at the points k = **/a, the curvature of the upper curve being positive and that of the lower curve being negative at this point as shown in Fig. 1. Furthermore, to firat order the effectivemsse is given by -m- = l + - 4 b m* A&
where G. is the energy at the gap and A& is the separation in energy at = &*/a. When the splitting is small and the energy at the top of the band is large, the effective maas can be fairly small. In an actual crystal, the situation is not really so simple and the dependence of energy on wave vector can become quite complex. The effective maas must then be represented by a tensor in the form l/mj* = (l/A*)(a2&/dk,dk,) in which the individual term8 themselves can still be functions of the wave vector. This we will find indeed to be the case in semiconductors and metals. If the energy wave vector relationship is quadratic, i.e., an
k
t" FIG. 1. The variation of the energy E ( k ) with wave vector k for electrons that are nearly free in a periodio potential. -Wh
0
=h
k,
ellipsoid, the effective mass ie a simple tensor which ie diagonalired in the principal coordinate system of the ellipsoid. Thus one of the principal objectives of cyclotron resonance studies is to determine the components of the effective mass tensor, or curvature of the energy surfaces, at the extreme of the conduction and vdenae bands, or at the Fermi surface. This ie done by observing mwnance abeorption peaks in an experiment and by using the cyclotron equation
-
0,
=
eH m*c
(3)
where at m n a n c e oc w , the rf frequency; since the magnetic field H and the frequency are known, m* is determined from Eq. (3). The other aspect which distinguishes cyclotron resonance in solids from that in a gas is the scattering mechanism. Whereas in a gas the scattering time usually is determined by collisions with neutral atom or molecules, in the solid collision or scattering arises from the inter-
264
BENJAMIN LAX AND JOHN 0. YAVROIDEB
action of the carriers with lattice vibrations, impurities, and imperfections in the cry~tal.If the dominant procees that limits the mean free time is the interaction with the lattice vibrations, which of course is temperaturedependent, the mean free time can be increased by placing it in a liquid nitrogen or liquid helium bath and thus decreasing the temperature of the crystal. On the other hand, in order that the bath be effective, it is necessary that the concentration of impurities and imperfections be reduced eufficiently that the scattering by them centers is lesa than that due to the thermal vibrations even at these low temperatures. Such a situation has been achieved in the case of germanium and silicon where the impurity concentrations have been reduced to the order of 10IJ/cmJ, but not in most other materials. In this respect, cyclotron resonance in a gas presents a somewhat simpler problem since the scattering time can be made sufficiently long by merely reducing the preeaure of the gas. Thus for helium, the scattering time r = (4 X 10-'o)/p where the preasure p is in millimeters. At l-mm pressure and a wavelength of 10 cm, (JT = 8, which is easily sufficient for resolution. In the catw of germanium, the initial experiments with a comparable value of wr were carried out a t 3 cm or 9OOO Mc/sec, the corresponding ecattering time being equal to about lo-'* second in very pure crystals. Although this value ia cloee to the best that has been achieved in a solid, it can easily be exceeded in a gas by using a lower preeaure. 11. Cyclotron Resonance of Free Charged Particles
1. ELECTRONS a. Ionospheric Propagation
The finst to consider the electron cyclotron resonance were the ionospherio physiciets who were concerned with electromagnetic propegation in the presence of the earth's magnetic field.l'~*~JThey used the simple free electron concepts just diecussed to explain s u c c d u l l y the eelective absorption of radiofrequency waves in the ionosphere. The minimum range of propagation shown in Fig. 2 occurs a t a wavelength of about 200 metera or a frequency of approximately 1500 kc/sec corresponding to an electron reeonance in a magnetic field of about 0.5 gauss. This is in agreement with the average value of earth's magnetic field which varim from about 0.33 gauss at the magnetic equator to about 0.62 g a w at the poles. l.E. V. Appleton, Proc. phg8. Soc. 8'7, 16D (1926).
* H.W. Nichob and J. C. bholling, Bell SvsrCm Tuh. J . 4,215 * A. H. Taybr and E. 0. Hdburt, P h p . Rw. W, 189 (1828).
(1925).
26s
CYCLOTRON REBONANCE
1 0
I
I
1
I
1
1
1
1
1
200 400 600 800 (ooot2ooHooldoo1#K)zooo WAVELENGTH, X ( moton
FIG.2. The propagation ranges for radio waves, under full daylight conditi0r.w averaged throughout a year, for uniform transmitting conditions, M a function of wavelength. The minimum is due to cyclotron remnance absorption by the ionosphere [after A. H. Taylor and E. 0. Hullburt, Phys. Rw. 27, 189 (1926)].
b. Breakdown in Gaaes It was not until sometime later that analogousexperimentawerecarried out with magnetic fields produced in the laboratory. E. W. B. Gill' measured the high-frequency electric field required for discharge in air as a function of applied magnetic field at low pressures. A characteristic breakdown minimum was found a t the magnetic field corresponding to cyclotron resonance. More complete experiments on air were performed by Townsend and Gill' and later on pure nitrogen and helium by A. E. Brown' with the same apparatus. A more quantitative study of a helium discharge at microwave frequencies was subsequently conducted by the group at Massachusetts Institute of Technology.' The helium eyetem used is relatively simple since, by virtue of the experimental fact that the collision frequency is essentially independent of the electron energy, the analysis is more amenable to calculation. Using the single-electron analysis, we represent the equation of motion of the electron aa dv
m-
dt
= -e(E
+ v X H/e) - mv
E. W. B. Gill, Nature 140, 1061 (1937). 'J. S. Townsend and E. W. B. Gill, Phil. Mag. [7] 26,290 (1938). a A. E. Brown, PhiZ. Mag. [7]49, 302 (1940). 'B. Lax, w.P.A . h , and 8. C. Brown, J . Appl. Phvr. 31, 1297 (1950).
266
BENJAMIN LAX AND JOHN 0. MAVROIDES
where the first term on the right is the force due to the electric field E and magnetic field H and the second term is a frictional term due to the collision of electrons with neutral gas atoms. The foregoing result can be obtained rigorously from the Boltzmann transport theory by assuming z independent of energy and integrating the distribution over momentum space so that the velocity in Eq. (1.1) is the average over all eleCtrOM. Since the electric field E = Eoebt, the velocity v = vo vlebt; for our purposes only the oscillatory component of the velocity is of interest. t. The corresponding rf current density J = nevl = u E, where is the tensor conductivity and R is the electron density. If Eq. (1.1) is solved with H in the z direction, one obtains the following results for the conductivity tensor in an orthogonal coordinate system:
+
-
where
. ' U
=u =
go ~
1 +am
and
uo =
x. ?&?4
The power P absorbed by the electrons from the electromagnetic field in an infinite medium is P = +ReJ.E*. (1.3) For purposes of analysis, it is simpler to decompose a linearly POW wave into two circular counter-rotating components and to consider the effect of each independently. Then the resultant is the sum of the two components, i.e., P = (P+ P-). It can be shown from Eqs. (1.2) and (1.3) that
+
or
P
-
+
1
+ + (W'
We')T2
- [l (we* - @*)?*I* + 40"* where Po is the total power for 8 linear wave. Figure 3 ie a plot of the power absorbed for a linearly polarized wave, as given by Eq. (1.5). This result holds for an infinite medium, i.e., the ionosphere in which the electron density is constant and uneffected by the electromagnetic wave.
i%
267
CYCLOTRON REBONANCE
It is seen that an absorption peak is clearly resolvable when WT 2 1. The pesk of the absorption corresponds t o the resonance condition w = we. The physical meaning of WT 2 1 is that the electron has rotated a t least through one radian before colliding with a neutral atom. In the actual experiments of Lax, Allis, and Brown,' the resonance was observed in a microwave cavity containing either air or helium gas at low pressure and a t a frequency of 3000 Mc/sec. The source of microwave energy was a tunable continuous wave (cw) magnetron. The power into the cavity was increased until the gas broke down and became ionized and the level of the threshold of breakdown was observed with a sensitive microammeter on the output side of the cavity. The power level
, 1
4-
n
-
-
FTQ.3. The microwave power absorption P as a function of magnetic field, or of d o ,where we is the cyclotron frequency. POia the abeorption when
H
[after B.Lax, H.J. Zeiger, and R. N. Dexter, Phyaiccl90,818 (1954)).
0 and o
0
at breakdown was measured as a function of magnetic field for a range of preeeures between 1 and 30 mm. I n analyzing this experiment, we shall not consider the more complicated cases, but shall limit ourselves to the simple one-electron theory for one particular situation in a flat cylindrical cavity. This corresponds to studying the resonance phenomenon exhibited by the breakdown of the gas in an equivalent one-dimensional case. The problem in its simplest form is as follows: in a small volume of ionized gas at breakdown, a steadystate condition exists; i.e., the number of electrons created by ionization is balanced by the number diffusing out of this volume. This can be expressed by the equation: c. an - = n v , - V - r =nvi+V.(D.Vn) - 0 at
268
BENJAMIN LAX AND JOHN
a. MAVROIDES
..
where vi is the rate of ion production per electron in the presence of the electromagnetic field, C is the flow vector, and D the diffusion tensor in a magnetic field. For the magnetic field in the z direction, the diffusion tensor becomes D,, D , 0 D, 0 (1.7) 0 D,. where
1
.. "-12
D , = D,, = D ,
D,. = - D ,
=
=
D
1
+ WC%2
DWJ
1
+
wc272
and
Thus the diffusion equation becomes:
The second condition is that the rate of energy gain of the electrons in an elementary volume from the electromagnetic field is equal to the rate of energy loss due to the diffusion of electrons out of this region, i.e.,
aE2 = nv,ii (1.9) where ii is the average energy gain of a single electron between collisions and nvi is given by the solution of Eq. (1.8). The solution of this equation in any geometry can be written as: (1.10)
where At and A, are constants determined by the geometry of the microwave cavity containing the ionized gas. For the cam of parallel plates of separation L, Al = L/rr and A,+ a. We shall also m u m e that
ti
is unaffected by the magnetic field 80 that
UnEn' uE2 nti Re -= Re - = - = constant (1.11) D, D A2 or by use of the tensor components given in expressions (1.2)and (1.7)
CYCLOTRON REBONANCE
269
The same results have also been obtained by the use of the Boltzmann transport theory after neglecting two compensating higher order effects due to finite geometry and a nonuniform electric field. However, these two effects can be taken into account and the resulta are shown in Fig. 4. It is seen that the agreement between theory and experiment is excellent. The two principal effects which are included in the single-electron theory and brought out by the curve are the cyclotron resonance phenomenon corresponding to the deep minimum, and the diffusion which is superimposed on the resonance. The diffusion has the over-all effect of reducing the electric field necessary for breakdown as the magnetic field is increased.
0
W O 2000 Moo 8 , MAGNETIC FIELD (gauss)
-
FIG.4. Breakdown of helium at I-mm pressure in a cylindrical cavity (diameter 7.60 cm, height = 0.318 cm). &lid line is obtained from the Boltsmsnn theory and points are experimental (after B. L a x , W. P. Allie, and S. C. Brawn,J . Appl. phy8. 21, 1297 (1950)].
Another type of experiment which has been carried out on ionized gases is one in which cyclotron resonance of electrons occurs in the afterglow.' I n this instance, the gas, located in a cylindrical microwave cavity oscillating in the TEIIImode, is ionized by pulsing a magnetron. An rf signal of fixed frequency is fed from a klystron into the cavity during the decay period and the variation of the reflected signal with time is observed by means of an oscilloscope for a given value of magnetic field. When the decaying plasma sweeps a resonance of the cavity through the reference frequency, a resonance pip appears. With a magnetic field, two such reaonances appear at different plasma densities corresponding to the two counter-rotating modes. The experimental observation of 'B. Lax, Phyr. Rw. 64, 1074 (1651).
270
BENJAMIN LAX AND JOHN
a.
MAVROIDES
electron resonance in a cavity with or without a magnetic field is given in Fig. 5. From the perturbation theory for a degenerate system, it can readily be shown that the complex frequency shift Au* of the two normal modes can be evaluated from' i
Aw*
-
2to
Iv E* IvEo* - E* c*Eo* .
dV (1.13)
dV
where Eo*is the complex conjugate of the unperturbed electric field, E* is the perturbed electric field corresponding to either of the two
TIME (mlcrorrsondr)
FIG.5. Resonance of a T E o l lcavity containing ioniced pleama in an afterglow;0, no magnetic field; b, splitting of degenerate modes with magnetic field [after B. and A. D. Berk, IRE NaU. Conu. Record 1, Pt. 10, p. 70 (1953)j.
modes, and V is the volume of the cavity. The distribution of electron density is calculated from the diffusion equation, the solution of which for a completely filled cavity is a Bessel function. From this type of measurement one can calculate in principle the complex conductivity of such a medium from the frequency shift of the resonance and from the change in Q. Since the electron density is not known from independent measurements, the theory may be checked by comparing the frequency shift of the two counter-rotating circularly polarized modes in the cavity as a function of magnetic field. The results are shown in Fig. 6, which shows the ratio of frequency shift AMJAW+ as a function of magnetic *B.Lax and A. D. Berk, ZRE Natl. Conu. Record 1, Pt. 10, p. 70 (19S3).
271
CYCLOTBON RESONANCE
field. Using Eq. (1.13), the calculated theoretical expression is -Aw= Aw+
w w
- 0.94~~
+ 0.94~c
(1.14)
where the factor 0.94 comes from integrating the electron density over the entire cavity; this factor would have been unity if the ionized gae had been located in a small container of cylindrical symmetry at the center of the cavity in which case the cross-over point of Fig. 6 would occur at resonance. This type of experiment also permits the evaluation
z++ 0.8
9
J' a
0.6
I-
0.4
5
HELlUY AT 0 . 6 1Hq ~ AXIAL MAGNETIC FIELD
TrnORY
5 0.2
Au, = Awe
t-.94wc/w t + J 4 oc /W
> V
gz o
-0.2
8-0.4
5 -0.6 0
-0.8 4 .o
FIG.6. Resonance studies of helium afterglow at 3000 Mc/sec. The interseetion of the curvea determinea cyclotron resonance [after B. Lax and A. D. Berk, IRE Nd. Cow. Rscord 1, Pt. 10, p. 70 (1953)j.
of the variation of the diffusion coefficient D , with magnetic field, since the electron density decays exponentially with time. Combining Eqs. (1.6) and (1.10) we have an= at
or
n = no exp
-.[%+&I
[ - (s+ $)I
(1.15)
t.
(1.16)
In another method for studying cyclotron resonance paramagnetic resonance techniques are used. Ingram and Tapley ' 0 observed absorption lines and measured the line widths in a gas discharge at low pressures. They attributed the absorption mainly to electron cyclotron resonance. D.J. E. Ingram and J. G . Tapley, Phys. Rw. 97,238 (1955).
272
BENJAMIN LAX A N D JOHN 0. MAVROIDEB
Jones and co-workers11 studied simultaneously the cyclotron resonance of electrons and the spin resonance of any paramagnetic species present in nitrogen and oxygen afterglows; the steady-state nature of these systems made possible more direct measurements of electron parameters than are possible in the breakdown type of experiment. The afterglows were excited by microwaves and pumped through a quartz t u b e at the center of a TEoll cavity. The power transmitted through the cavity was observed by either a bolometer or crystal diode as the magnetic field wm slowly swept through resonance. A typical curve of the derivative of the absorption line for low powers is shown in Fig. 7. These results were obtained in an experiment in which the magnetic field was modulated at 80 cps. Such meaaurements were made a t 9500 Mc/sec as a function of
I
2.3rnm PRESSURE
13 MICROWATTS
CAVITY POWER
hI \
ELECTRONS
FIQ.7. Derivative of a sharp cyclotron rwnance sbeorption line in a nitrogen afterglow at a frequency of 9OOO Mc/sec (after R. V. Jones, Ph.D. Thesis, Univenrity of California, submitted in January, 1956).
power for different pressures, and line widths between 10 and 100 gauss were found. From these results the low-energy electron collision crow section waa determined" as a function of effective electron temperature. In addition, from the simultaneous measurement of cyclotron and spin resonances, the relative and absolute abundance of electrons and the various paramagnetic species present could be determined by comparing signal heights. It haa been observed that cyclotron resonance of electrons also occurs in acetylene-oxygen flames'P at 24,300 Mc/sec. At atmospheric pressures there is no indication of cyclotron resonance; at 100 mm Hg a broad resonance appears which sharpens considerably as the pressure ia decreased. At a preasure of 7.5 mm Hg a cyclotron resonance line with I1
R. V. Jones, W. Dobrowolsky, W. B. Kunkel, and C. D. Jeffries, Bull. Am. Phyc. SOC.SO, No. 3, 46 (1955); R. V. Jon-, Ph.D. Thesis, University of California, eubmitted in January 1956. J. Schneider and F. W. Holmann, Phye. Rw. Letters 1, 408 (1958).
CYCU)TBON RESONANCE
273
= 25 waa obtained. The microwave signal from a klystron WM fed from a wave-guide horn to a low-pressure veseel containing the burner and positioned in an electromagnet. After the signal was transmitted through the flame it was picked up with another wave-guide horn and fed to the detector. The concentration of free electrons was varied over a wide range by introducing into the flame fine sprays of alkali salt solutions from an atomizer. Using the expression for the conductivity, i.e., om of Eq. (1.2), both the crvrier concentration and the collision time were determined. The most recent experiment on cyclotron resonance in a gas discharge has been carried out by Fukuda and co-workersIa. They meaaured the dispersion of cyclotron resonance at 24,000 Mc as a function of magnetic field. This experiment is analogous to that of Laxe in that the frequency shift increases with magnetic field close to resonance and changea sign at fields greater than those required for resonance. The difference b that Fukuda et al.'sls measurements were made on a steady-state diacharge, while those summarized in Fig. 6 were for a decaying afterglow.
w
2.
ION8
Cyclotron resonance of ions also has been under investigation, and the problems associated with its observation are closely related to those encountered with observation of the phenomenon in metals, semimetale, and degenerate semiconductors a t low temperatures. In an ionized plasma consisting of hydrogen atoms, the resonant frequency of the protons ie given by W H = ( e H / M c ) which equals 4.258 kc/sec per oersted. This means that, even with fields as large as 5000 to 10,OOO gauss, which might be readily available a t present, the cyclotron frequency would be only of the order of 30 Mc/sec. The penetration of the electromagnetic wave into the plasma will be limited either by the electron cloud or by the ions themselves. It can easily be shown from Maxwell's equations that for the simplest case of propagation along the magnetic field
I"*! = where I' = a
--w'c,,p0
+ iqw*
(2.1)
+ iB is the coniplex propagation constant
the first term in the conductivity expression referring to the electrone and the second, to the ions. The propagation corresponds to counter-rotating 18 K. Fukuda, H. Matumota, Y. Uehida, and H.Yorhimura, J . Phyr. Soc. Japcm 14, 543 (1959).
274
BENJAMIN LAX AND JOHN 0. MAVROLDES
circularly polarized waves with the propagation vector along the magnetic field. The difficulty of observing cyclotron resonance of the ions can readily be deduced from the above equation in the following manner. For the moment assume that we are a t a sufficiently low pressure so that the collision times are extremely long both for the ions and the electrons and that we are near the cyclotron resonance frequency of the ions. Under these conditions
Now suppose that we could ignore the ions. Since the electron plasma itself, independent of the ions, is a medium below cutoff, then the penetration depth 6 equals about 100 cm for a typical density of 10" per cm' and a frequency of 10 Mc/sec. Now considering the ions we can show that the collision time is still larger than that of the electrons, and hence obtain a penetration of about 1.5 cm. This conclusion holds even if we assume operation with a plasma whose ion temperature is close to that of room temperature where the collision time T M , as determined by Coulomb scattering," is about 2 X 10-7 second. Actually the ion temperatures may be much higher than room temperature since in Stix's experiments,'b which are discussed in the following, the ions were heated ohmically and also by the rf field. If we assume temperatures of the order of lo4to 106 O K as being reasonable, then T M is between 0.5 X lo-' and lo-* second and 6 lies between lo-' and 2 X lo-' cm. Thus under the foregoing conditions the skin depth is determined primarily by the ions and as suggested by Kaner," it is only possible to observe the occurrence of resonance in regions within the skin depth. The existence of this particular problem has been demonstrated etrikingly in studies on semimetals and will be discussed later. Another possible configuration is that in which the magnetic field is perpendicular to the direction of propagation of the electromagnetic wave. I n this case, even for an infinite medium, complex magnetoplrtsma effects are present. For a dense plasma in which the displacement current can be neglected, the cyclotron resonance singularity for equal densities of electrons and ions corresponds to a mass which is the geometrical mean between that of electrons and the ion involved. We shall treat this problem in detail in connection with magnetoplasma effects associated with metals, semimetals, and degenerate semiconductors. 1( L. Spitrer, Jr., "Physics of FuUy Ionized Gwea," p. 65. Interscience, New York, 1956.
T. H. Stix and R. W. Pslladino, Phys. Fluida 1,446 (1958); Proc. 8nd U . N.I n l a . emf.on Pecrccful Use8 of Alomu Energy, Gencva 81, p. 282 (1958). E. A. Kaner, Sovict Phys. Jh'TP 6, 425 (1958).
275
CYCLOTRON RESONANCE
Recently Stix and Paladino16 have reported successful ion cyclotron resonance experiments at a frequency of 10 Mc/aec with a plaama of about 10" carriers per cma confined by a magnetic field in the B-65 stellarator. These cyclotron resonances were observed with an external coil which was coaxial with the plasma and energized for several values
m o a J a K
g5 Q O a a I,
a
0.5
-
-3.0 1.5
5.0
-
PLASMA CURRENT
-
(3.09mc
10.0
-
100.0
-
= 1.0
PRESSURE 1
1
1
1
1
1
l
microns HELIUM 1
1
1
1
1
1
1
1
1
1
FIG. 8. Ion cyclotron resonance plasma loading vs confining field for various frequencies. Vertical lines are drawn for cyclotron field values for H+ and He+* ions. Insert drawing shows plasma current versus time, and indicates time for which loadinn pointa were taken [afterT. H. Stix and R. W. Palladino. Phus. Fluids 1.446 (195811.
of power ranging from milliwatts to kilowatts. By measuring the power absorbed by the plasma as the confining magnetic field waa varied, resonances corresponding to ion cyclotron motion of H+, He+, and He++ ions were observed. The appearance of a large H+ absorption peak was attributed to hydrogen released from the walls of the container. Typical curves showing the I{+ and He++ resonances for various rf frequencies are given in Fig. 8. The absorptipn and resonance width messurements
276
BENJAMIN LAX AND JOHN G . MAVROIDES
were compared with theoryi7 and the agreement was found to be semiquantitative. Another possible interpretation of the major resonance in Fig. 8 occurring a t the lower magnetic field is that it is due to the He+ and He++ ions. According to Kaner’s theory,“ resonances can occur when a charged particle rotates in and out of the penetration depth of the rf field at the surface of the plasma while the electromagnetic field oscillates through a multiple number of periods for one rotation of the charged particles. Thus at a frequency of 11.50 Mc/sec, which corresponds to the middle diagram of Fig. 8, the primary resonance for He+ ions should occur at about 30,000 gauss with harmonics at 15,000, IO,OOO, 7500, etc., and for He++ ions at 15,000 gauss with harmonics at 7500, 5000, etc. Perhaps experiments at higher frequencies and magnetic fields may be more favorable for the observation of the anomalous “skin” resonances of ions in such a discharge. In addition to the expected cyclotron resonance, further loading was observed at fields above the resonance field, and these were presumed to be “ion cyclotron waves,” the short-wavelength, lowdensity limit for the extraordinary hydromagnetic waves discussed by Alfven and also by h , r o m . l s 111. Cyclotron Resonance of Carriers in Solids
The problem of cyclotron resonance in solids was first considered in 1951 by DorfmanIg and also by Dingle.*O Dorfman in a short note suggested the possibility of doing cyclotron resonance experiments on metals while Dingle worked out the quantum theory for the behavior of B system of free charged particles in a magnetic field. The first crystallized proposal for carrying out an experiment was made by Shockley“ who suggested that such resonances might be found in weakly doped germanium at approximately lOoK and at microwave frequencies. The actual experiments were carried out on germanium a t liquid helium temperature independently by Dresselhaus, Kip, and Kittel (DKK)**and the group at Lincoln L a b ~ r a t o r y . * *The ~ * ~ initial results which showed IT
T. H. Stix, Phys. Fluids 1, 308 (1958). H. Alfven, Arkiu Mat. Astron. Fysik 2SB, No. 2 (1942); E. Ikltrom, Arkiu Fyaik 3, 443 (1951).
J. Dorfman, Doklady Akad. Nauk S.S.S.R. 81, 765 (1951). lo R. B. Dingle, Ph.D. Theais, Cambridge University (1951) unpublished; Proc. Intern. C a f . on Very Loco-Temperatures, Ozford p. 165 (1951) Proc. Roy. Soc. lo
A212, 38 (1952). W. Shockley, Phya. Reu. 90, 491 (1953). 12 G. Dreaselhaus, A. F. Kip, and C. Kittel, Phys. Reu. SS, 827 (1953). ** B. Lax, H. J. Zeiger, R. N. Dexter, and E. S. Roaenblum, Phya. Rcv. 03, 1418 (1964). R. N. Dexter, H. J. Zeiger, and B. Lax, P h p . Rm. 06, 667 (1954). l1
CYCLOTRON RESONANCE
277
resonance for both holes and electrons were reported by the Berkeley group (DKK). The more complete data of the Lincoln group included results on the anisotropy which gave quantitative data on the effective masses of holes and electrons in germanium; these results defined the energy surfaces at the top of the valence band and bottom of the conduction band. The parameters for electrons in silicon were reported jointly by the two groupsz6and the results for holes in silicon were published later by Dexter and Lax.Z*The Berkeley group also reported cyclotron resonance in germanium-silicon alloysz7 but in these experiments their resolution was much poorer. Difficulties were encountered in getting reproducible results for indium antimonide ;** nevertheless DKK did obtain reliable data for electrons in this material.*9 Luttinger and Kohnao in their comprehensive treatment of the quantum theory of cyclotron resonance in semiconductors laid the foundation for the detailed study of holes and predicted the existence of quantum effects for theae carriers. Fletcher and co-workersal subsequently found these effects in very pure germanium at 4.2'K and 1.3°K.a1a Plasma effects in indium antimonide were studied both theoretically and experimentally by DKK.a*They predicted nonresonant absorption in metals in a magnetic field at low temperatures and microwave frequencies. Such effects in bismuth were observed by Galt and co-workera" and also by Dexter and Lax;" they were treated theoretically by P. W. Anderson." Dexter and Lax demonstrated the feasibility of determining cyclotron resonance experimentally from the inflection point of the absorption curve. These early results were then analyzed theoretically by Tinkham,a6and in greater detail by the Lincoln group." The most R. N. Dexter, B. Lax, A. F. Kip, and G . Dresselhaus, Phyr. Rw. W, 222 (1954). R. N. Dexter and B. Lax, Phy6. R w . 96,223 (1954). *I G. Dreeselhaw, A. F. Kip, Han-Ying Ku, G . Wagoner, and S. M. Christian,Phys. Rev. 100, 1218 (1955). I* R. N. Dexter and B. Lax, Phys. Rw. 90,635 (1955A). 1' G . Dresselhaus, A. F. Kip, C. Kittel, and G . Wagoner, Phys. Rw. 98, 666 (1955). J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955). R. C. Fletcher, W. A. Yager, and F. R. Merritt, Phys. Rev. 100, 747 (1955). Recently microwave cyclotron remnance hati been obeerved in CdS by Dexter [ J . Phys. Chcm. Solids 8,494 (1959)] and in C d A s r by M. J. Stevenwon [Phys. Rw. h f f e t 8 8, 464 (195911. IrG. Drceselhaua, A. F. Kip, and C. Kittel, Phy6. Rw. 100,618 (1955). J. K. Galt, W. A. Yager, F. R. Merritt, B. B. Cetlin, and H. W. Dail, Jr., P h p . Rw. 100, 748 (1955). "R. N. Dexter and B. Lax, Phys. Rw. 100, 1216 (1955). w.Andereon, Phy6. Rw. 100, 749 (1955). M. Tinkham, Phys. Rw. 101, 902 (1956). "B. Lax, K. J. Button, H. J. Zeiger, and L. M. Roth, Phy6. Rut. 109, 716 (1966). 1' 1'
278
BENJAMIN LAX AND JOHN G. U V R O I D E S
successful and significant results using the inflection technique were obtained experimentally for graphite by Galt and co-workers.a8 The results of these experiments were subsequently analyzed by Lax and ZeigersQin terms of harmonics, and more completely by Noziere~.'~ Using similar techniques, Datars and Dexter" made a quantitative study of antimony at microwave frequencies. After these developments infrared techniques for extending cyclotron resonance studies to other materials were developed by the groups at the Naval Research Laboratory and Lincoln Laboratory. Using static magnetic fields of the order of 60,OOO gauss and a wavelength of 41 microns, ~ cyclotron resonance of electrons in indium Burstein et ~ 1 . 4reported antimonide. Keyes and c0-worker~,4a using pulsed magnetic fields up to 300,000 gauss and a wavelength of 12.7 microns, observed resonance of electrons in samples of indium antimonide and also, for the first time, a resonant absorption in a metal, namely bismuth. These techniques were extended to wavelengths of approximately 100 microns in the far infrared by the groups at the Bell Telephone Laboratories (BTL) and Lincoln Laboratory. This permitted detailed investigation of plasma effects in bismuth by Boyle, Brailsford, and G a l P (BTL) and in indium antimonide by Lipson, Lax, and Z ~ e r d l i n g .The ~ ~ pulsed cyclotron resonance results have been treated theoretically by Wallis.46A detailed quantitative analysis of cyclotron resonance in indium antimonide and bismuth has been developed by Lax, Mavroides, Zeiger, and Keyes." Another experimental approach has been the extension of the measurements to wavelengths in the millimeter range. One of the first experiments reported was that of Bagguley and ~o-workers'~in which cyclotron resonance of holes was observed in gold-doped germanium at 8.8 rnm and 77°K. Heller and c o - w o r k e r ~have ~ ~ observed the resonance of electrons in germanium at liquid nitrogen and 2.65 mm. Cyclotron absorp 18
a* 'I
'8
J. K. Galt, W. A. Yager, and H. W. Dail, Jr., Phya. Rev. 108, 1586 (1956). B. Lax and H. J. Zeiger, Phys. Rev. 106, 1466 (1957). P. Nozieres, Phys. Rev. 109, 1510 (1958). W. R. Datars and R. N. Dexter, Bull. Am. Phys. SOC.[2] 2, 345 (1957). E. Buratein, G . S. Picus, and H. A. Gebbie, Phya. Rev. 103, 825 (1956). R. J. Keyes, 8. Zwerdling, 8. Foner, H. H. Kolm, and B. Lax, Phya. Rev. 101,1804
(1956). W. S. Boyle, A. D. Brailsford, and J. K. Galt, Phys. Rev. 109, 1396 (1958). 4L H. G. Lipson, S. Zwerdling, and B. Lax, Bull. Am. Phys. Soc. [2] 3, 218 (1958). R. F. Wallis, J . Phys. Chem. Solids 4, 101 (1958). *' B. Lax, J. G. Mavroides, H. J. Zeiger, and R. J. Keyes, to be published. D. M. S. Bagguley, J. A. Powell, and D. J. Taylor, Proc. Phya. Soc. A70, 759 (1957). 40 J. J. Stickler, J. B. Thaxter, G. S. Heller, and C. J. Rauch, unpublished. "
CYCLOTRON RESONANCE
279
tion in bismuthlKO graphite," and zincb2a t 4 mm has been studied by the BTL group using improved resolution. Silicon has been investigated by Rauch and co-workersbaa t 2-mm wavelength from 1.7"K to 50°K. In a series of theoretical papers, Azbel and Kaner6' proposed a new kind of cyclotron resonance in metals in which the anomalous skin depth region in the magnetic field plays a role analogous to that of the dees of a cyclotron. This resonance only occurs with the magnetic field parallel to the surface of the metal. Nonresonant absorption under anomalous skin conditions with the magnetic field perpendicular to the surface has been treated by Chambers.bb The former configuration has also been proposed by Kaner for studies of cyclotron resonance of electrons in ionized gases." This phenomenon was first observed in tin and copper by Fawcett.b6 More quantitative results on bismuth were obtained subsequently by Aubrey and chamber^.^' Kip and co-workers*8 also published their research on tin in which the effect waa demonstrated dramatically by the appearance of many subharmonics. The Azbel-Kaner effect has been observed in antimony by Datars and Dexter" and again in tin and also lead by Bezuglyi and Galkin.bsMore recently, the anomalous cyclotron resonance has been observed in zinc by Galt and coworkers," in aluminum by Langenberg and Mooreaoand also Fawcett," and in copper by Langenberg and Moore.6z The theoretical resulta of Aabel and Kaner have been derived classically by Heine" and treated by the Boltzmann theory by Mattis and D r e s ~ e I h a u 6and ~ ~ also by Rodriguez.O b K. Galt, W. A. Yager, F. R. Merritt, B. €3. Cetlin, and A. D. BraLford, Phys. Reo. 114, 1396 (1959). '1 J. K. Galt, W. A. Yager, and F. R. Merritt, Proc. 3rd Conf. on Carbon, 1967 p. 193 8OJ.
(1959). 1s
J. K. Calt, F. R. Merritt, U'. A. Yager, and H. W. Dail, Jr., Phys. Rw. h.fter8 9, 292 (1959).
'* C. J. Rauch, J. J. Stickler, G. S. Heller, and H. J. Zeiger, Phys. Rev. Lcllets 4, 64 (1960).
M. I. AEbel and E. A. Kaner, Soviet Phya. J E T P 3, 772 (1956); 6, 730 (1957). a6 R. G. Chambers, Phil. Mag. [8] 1, 459 (1956). 6 8 E. Fawcett, Phys. Rev. 103, 1582 (1956). "J. E. Aubrey and R. G. Chambere, J . Phys. C h .Solida S, 128 (1957). A. F. Kip, D. N. Langenberg, B. Rosenblum, and C. Wagoner, Phys. Rev. lob, 494 b4
(1957).
P. A. Bezuglyi and A. A. Galkin, Soviet Phys. J E T P 6, 831 (1958); 7, 163 (1958). so D. N. Langenberg and T. W. Moore, Phys. Rev. Letters 3, 137 (1959). (1 E. Fawcett, Phya. Rev. Leltera 1, 139 (1959). 6' D. N. Langenberg and T. W. Moore, Phys. Rev. Letters S, 328 (1959). O V . Heine, Phya. Rev. 107, 431 (1957). 64 D. C. Mattis and G. Dresselhaus, Phys. Rev. 111, 403 (1958). 'Is. Rodriguez, Phys. Rat. 119, 1616 (1958). 6@
280
BENJAMIN LAX AND JOHN (3. MAVROIDES
A variety of phenomena aesociated with cyclotron resonance have been considered. K r o m e P has proposed the use of negative maeses for microwave amplification. The negative maae aaeociated with heavy holm in germanium has been observed by cyclotron resonance by Dousmanis and ~ o - w o r k e r s .The ~ ~ possibility of using the harmonics associated with the warped holes for generation of millimeter wavelengths was experimentally examined by Dexter, Zeiger, and Lax (DZL)6* but without success. This idea also has been suggested by Tager and G1adun;O' they also considered the possibility of using harmonics of the heavy holes for cyclotron resonance masers. Maiman'" has examined, both theoretically and experimentally, the use of magnetic field modulation of cyclotron resonance for generating harmonics. Tager and Gladunaoalso considered this and in addition suggested that such a phenomenon be used for parametric devices. Lax71 has made a quantitative study of cyclotron resonance masers involving optical excitation and magnetoabsorption phenomena. Another potentially useful phenomenon associated with cyclotron resonance is that of modulation of the dc conductivity of a semiconductor by microwave cyclotron reeonance. Zeiger and co-workeraTa have studied this effect in several semiconductors. 3. MICROWAVE CYCLOTRON RESONANCE IN SEMICONDUCTORS a. Experimental Techniques
The first cyclotron reeonance experiments were carried out at microwave frequencies using techniques similar to those in paramagnetic reeonance, except that in cyclotron resonance the dc magnetic field was perpendicular to the rf electric field rather than to the rf magnetic field. Basic spectrometers have been built for frequencies of 9O00 Mc/aec, 24,000 Mc/sec, and some for wavelengths in the millimeter region. Although the original experiments were performed at 9OOO Mc/sec, it was found desirable t o go to higher frequencies in order t o improve the resolution and 24,000 Mc/eec became standard since components became readily available. A block diagram of a cyclotron resonance spectrometer"
0'
H. K r h e r , Phys. Rev. 109, 1856 (1958). G. C. Dousmanis, Phys. Rcv. LcUers 1.55 (1958); G. C . Dousmanis, R. C. Duncan, J. J. Thomas, and R. C. Williams, ibid. p. 404. R. N. Dexter, H. J. Zeiger, and B. L a x , Phys. Rev. 101,637 (1956). A. S. Tager and A. D. Gladun, . I Ezptl. . Thcotet. Phys. (U.S.S.R.) 86, 808 (1958).
To
P. H. Maiman, Solid State Millimeter Generation Study, Hughes Aircraft Company
94
e7 6*
(1956-1957).
B. Lax, in "Quantum Electronics" (R. Tilley, ed.), p. 428. Columbia Univ. Presr, New York, 1960. I* H. J. Zeiger, C . J. Rauch, and M. E. Behrndt, Phye. Rev. Lclfcrs 1, 59 (1958).
CYCLOTRON RESONANCE
281
is given in Fig. 9. Microwave power is fed from a klystron to the sample cavity via a hybrid junction. The klystron is stabilized to the sample cavity frequency by a reflected signal through a directional coupler. The power reflected from the sample cavity splita up, some going into a matched load in one arm of the tee and some going into a crystal detector in the remaining arm which is connected to a narrow band amplifier. A change in the loaded Q of the cavity resulting from cyclotron absorption in the sample changes the amount of reflected power reaching the crystal detector and for small signals this change in power is proportional to the absorption. SLIOIIIQ STUO TUNCR
FIQ.9. Block diagram of an experimental arrangement for microwave cyclotron rmnance [after R. N. Dexter, H. J. Zeiger, and B. Lax, Phys. Rw. 101,637 (1966)l.
Since at very low temperatures semiconductors like germanium and silicon are ementially insulators, in which the carriers are froaen into impurity levels or into bands and hence are not free to contribute to conduction processes or resonance, some means of producing free carriers is necessary. The three methods used for germanium are breakdown of the impurity levels by the microwave field, breakdown of the impurity levels by an applied dc electric field, and excitation by light. In the electrical breakdown method it ie poeeible to give the few carriers available in the bands enough energy so that they ionize a neutral impurity upon collision with it; thus the number of carriers in the conducting state ie inorecrsed. Furthermore, only one type of carrier is excited by this method. Although the ionization method provides fairly reliable information on the location of the peaks in the absorption curve, it has several didvantages. In
282
BENJAMIN LAX AND JOHN 0. MAVROIDES
the first place, the rf level must be kept fairly low, otherwise the effective temperature of the free carriers will be raised well above that of the lattice. In addition, for carriers with complicated energy surfaces, the carrier energy may be distributed over a wide energy range at high rf fields. This distribution in energy may obscure the carrier aniaotropy since the effective mas8 will depend on energy. Another disadvantage of the avalanche technique is that the observed line width does vary with rf power level. Furthermore, the carrier concentration, particularly a t resonance, will be a function of magnetic field; this is so since the conductivity per carrier on resonance u, = e%/m will be large due to the small electron masses, and hence for a given electric field the energy gain per electron will be larger than for heavier electrons. The intensities of low field lines will be enhanced, making the interpretation of line shapes difficult. A method of excitation first used by DZL6*in which the carriers are excited by light has proved to be much more satisfactory. By euitably filtering the light, only the impurity levels are excited and thus carriere of predominantly one type are produced. In addition, in this method the number of carriers is independent of the magnetic field and rf field, which is a t a very low level, so that the interpretation of the line widths is a meaningful procedure. The low rf level allows the electron temperature to approach closer to that of the lattice than is the case when higher levels are used. In materials such as silicon, with high impurity ionization energies (-0.05 ev) and deep traps, the available rf microwave power was not sufficient to produce ionization and thus photoexcitation had to be used. In the optical excitation method shown in Fig. 9, light chopped at 90 cps totally modulated the number of carriers and hence the power absorbed. The modulated absorption signal was sent through a narrow band amplifier and then a lock-in detector. The reference signal for the lock-in detector was obtained from an auxiliary light source which W M modulated by the same chopper. The detector output, which wm a rectified signal proportional to the absorption waa recorded aa a function of the magnetic field; this was produced by a magnet with an electronic control circuit for sweeping the field. Magnetic field measurements were obtained with a rotating flip coil, which had been calibrated with a proton resonance magnetometer, and markers were placed manually on the recorder. Thus a plot of the absorption versus magnetic field wan obtained. In the original DKK22experiments on germanium, it was necessary to verify the sign of the charged carriers involved. This was accomplished by using circularly polarized waves. Absorption was then observed only
CYCLOTRON RESONANCE
283
when the direction of the circular polarization corresponded to the direction of rotation of the carrier. Circularly polarized waves were produced by the microwave equivalent of an optical quarter-wave plate. The plane polarized wave from a rectangular wave guide was fed through a gradual transition into a circular wave guide. Then it passed through 8 section of guide which was made elliptical and so adjusted that the two components, each at 45' to the original plane of polarization, were equal in amplitude and, after traversing the ellipsoidal section, ninety degrees out of phase. The ellipticity and orientation of the elliptical section was determined experimentally by analyzing the resulting microwaves with a rectangular wave guide and crystal detector. The Lincoln identified the carriers by a different technique. They obtained relatively pure samples on which they carried out Hall memurements at 77°K; from these results the excess carrier concentration and its sign were determined. Consequently, by the use of the rf breakdown or long wavelength optical excitation methods, the identification of the electrons and holes was straightforward.
b. Experiments on Germanium and Silicon We have already stated that the effective mass in a solid could have a complicated form which may or may not be a simple tensor, but which would reflect the properties of the particular crystal involved. Theoretical predictions by Herman" indicated that the energy-momentum relationship for electrons in germanium can be represented by an ellipsoidal surface. S h o ~ k l e y ~ ' * lalso 4 * ~ deduced ~ the possibility of reentrant or warped energy surfaces from the experimental results of Pearson and Suhl76 for the magnetoresistance of holes in germanium. He suggested, therefore, that the effective mass obtained from cyclotron resonance studies on germanium, as interpreted by Eq. (3), would show structure and also anisotropy. In order t o observe such effects, the Lincoln group****' oriented single crystals of germanium so that the crystal was rotated with the magnetic field in the (TlO) plane through 90' from the [Ool]axis to the [110] axis. This was done both for holes and electrons in germanium and silicon. The type of data obtained for germanium is shown in Figs. 10 and 11. Before discussing the quantitative data, it is appr0priat.e to consider 7aF. Herman, Phys. Rev. 88, 1210 (1952); F. Herman and J. Callaway, ibid. 89, 518 (1953); F. Herman, ibid. 93, 1214 (1954); 96, 847 (1954); Phyuiea 40, 801 (1954); Proc. I . R . E . 43, 1703 (1955). l4 W. Shockley, Phys. Rev. 78, 173 (1950). l 6 W. Shockley, Phys. Rev. 79, 191 (1950). 7'G. 1,. Pearson and H. SuhI, Phys. Rev. 89,768 (1951).
BENJAMIN LAX AND JOHN 0. MAVHOIDEY
c
THEORY --- EXPERIMENTAL
wC/W
@C/W
FIG.10. Microwave absorption in intrinsic n-type germanium at 4.2% for four different directions of B in the (710) plane as a function of the magnetic field. Ptfrepresenta the absorption with B at 35" to the [Ool]axis [after B. Lax, H.J. Zeiger, and R N. Dexter, Physicu 20,818 (1954)j.
0
300
600
900
1200 4500 1-
Fro. 11. Resonance absorption of holea in p t y p e Ge at 8900 Mc/aec with infrared excitation. a, b, and c represent B along [Ool],[ I l l ] , and [I101direction% rapectively. htermediate peak is weak resonance due to electrons [after B. Lax, H.J. %@, end R N. Dexter, Phyaica 20,818 (1954)l.
the classical analysis of cyclotron resonance for these two types of carriers. ( 1 ) Electtone. For electrons, one can write an equation of motion analogous to Eq. (1.1) d -
;ii( m .v) = -e(E
+ v X H/c)
., - m7. v *-
(3.1)
285
CYCLOTRON RESONANCE
Hince the applied electric field E(1) = EeG' we can write the velocities v(t) = vo ve'"'. We are interested only in the time-varying component of velocity; thus Eq. (3.1) becomes
+
( + -3-m v = -e(E + v X H/c). iw
This vector equation can be solved for v in terms of component equations" or in vector fom.7' From this solution one can obtain the tensor u conductivity since J = nev = u - E,where J is given by
v /H. X . H\ ,
I
Iml
I
(3.3)
where 1711.1 = mlmzm~,is the determinant of the maas tensor. The power absorbed by the electrons is given by the relation P = Re J * E*.The components of the conductivity tensor are obtained for J the current density and can be found in explicit form in Lax et a1.l6 For distinct resolution of cyclotron resonance WT >> 1 ; consequently the denominator of the above equation gives a resonance when
+
where a, 8, and y are the direction cosines of the magnetic field relative to the principal axes of the ellipsoid. Using Eq. (3.4) it is a simple procedure to interpret data such as that given in Fig. 10. Since germanium has cubic symmetry, it is necessary that the family of ellipsoids also have cubic symmetry. Thus, if we consider ellipsoids along the cube edges, we must consider either three or six depending on whether the minima of the energy surfaces are located a t the edge or inside the fbt Brillouin zone. Under similar conditions, if the ellipsoids are along the cube diagonals, there must be either four or eight. Cornistent with the above cubic symmetry, the individual ellipeoide are really spheroids with ml = m, = ml and m, = ml, therefore
M. Shibuya, Phye. Rev. @5,1385 (1954). 7.B. Lax, H.J. Zeiger, and R. N. Dexter, Phyuica 90, 818 (1964). 1'
286
BENJAMIN LAX AND JOHN 0. MAVROIDEB
where we = (eH/rn&) and e is the angle the magnetic field makes with with the principal axis of the ellipsoid. We see from Fig. 10 that there is only one peak with the magnetic field along tho (0011 direct,ion. This is therefore the high symmetry direction, and the ellipsoids must t,hen be located dong the [ 11 11 directions. With this in mind, one readily obtains that one of the mass values
0.04
t& 0
20
40
60
ANGLE (d.grwr in 110 pione from
80
100
[mi] axis)
FIQ.12. Effective maas of e'iectronsin germanium at 4'K for magnetic field dimtions in a (110) plane [after G . Dresselhaus, A. F. Kip, and C. Kittel, Phys. Rw. S8, 368 (1955)].
in the [ l l l ] direction is mIl1*= mt and one, in the [llO] direction is nzllo* = d G . Choosing the low value in the I1111 direction for m111' and the larger value in the Ill01 direction for mllo*, the following values are obtained:'B mJm = 0.0819 & 0.0003;mr/m = 1.64 _+ 0.03, giving a mass ratio ml/mr = 20.0 f 0.4. Using these values, the complete anisotropy was studied with the results shown in Fig. 12. For the electrons in silicon, only a single resonance was observed with the magnetic field in the [ l l l ] direction (see Fig. 13), thus the ellipsoids must be located along the cube edges. From two suitable points of Fig. 14, it was simple
CYCLOTRON RESONANCE
Y*OIILTIC FIELD
287
(0.1
Fxo. 13. Cyclotron resonance trace in silicon near 4'K and 23,000Mc/sec; external magnetic field watt nearly parallel to the [111] axis [after R. N. Dexter, H.J. Zeiger, and B. J k , Phyu. Rm. 104, 637 (1956)).
I
1
I
288
BENJAMIN LAX AND JOHN 0. MAVROIDES
to obtain the following longitudinal and transverse values:68 mr/m = 0.98 k 0.04.'&
mt/m = 0.19 f 0.01;
(a) Holee. The situation for holes is more complex than for electrons. The interpretation of the cyclotron resonance results in terms of band structure was given by DKK7g*80 who first observed the light and heavy holes. Their development, which we shall discuss subsequently, was derived from the earlier work of S h ~ c k l e y 'and ~ Herman7' and leads to the following energy wave vector relationships for the three bands &(k) =
-" ( A k 2 +_ 2m
&(k)
--x - AkZ 2m
=
+ C*(k,2k,2+ ky2k,* + k,*k,*)]i)
(B2k4
(3.6)
(3.7)
where the plus sign is associated with the holes of the small effective mass, and the minus sign with holes of the larger effective mass. Here the k coordinate system is coincident with the cubic ax-, X is the spinorbit splitting energy, and A, B, and C are constante to be determined experimentally. In view of this model of the valence band, the interpretation of the cyclotron resonance results requires writing the force equation in terms of the crystal momentum p in the following manner:
2
= qv
x
H/c
where v = V&. For our purposes we shall use cylindrical coordinatea p, 4, and p ~with , the p~ axis taken along the direction of the magnetic field H.Thus or (3.10)
since o0 = (qH/m*c) = 2 r / T where T is the period.76Using Eqs. (3.6) and (3.10) one can obtain for the case of p~ = 0 and H parallel to the (110) plane the following appropriate expressions0for the effective mass:
-
Very recent experimente at millimeter wavelengths by Rauch and co-workera~~hwa given the following results for electron mmwa in silicon: mr/m 0.192 f 0.001; ndm 0.90 f 0.02. 7.G. Dreseelhaw, A. F. Kip, and C. Kittel, Phyu. R e . 96, 568 (1954); C. Kittel, PhysiCo 40, 829 (1954). G. Dreaeelhaus, A. F. Kip, and C. Kittel, Phys. Reu. 98, 368 (1955). 7-
-
209
CYCLOTRON RESONANCE
1
m+ m
--a
A k [B*
+ (C/2)*]+
- 3 ey + *] ( * M [ B * + ( Ccyi / 2 ) p l ) ( Af [B* + (C/2)*1'} COB'
*
*
(3.11)
where e is the angle the magnetic field makes with the [lOOJ direction. From this equation and three suitable experimental values of effective
v) v)
W LL
w
.I
cool1 0
ClIllJ I
I
30
0
[I
1 1
1
I
io]
1
90
60
8 \6egTtt.b)-
FIG.15. Effective mass of holm in silicon; magnetic field in the (110) plane. C u r ~ e ~ are theoretical and points experimental [after R. N. Dexter, Doctoral Thesis, University of Wisconsin (1955)].
mms, one can determine A, B, and C. For example, in the case of silicon we find, uaing Eq. (3.11) and Fig. 15, that for the [ 1111 direction
A
+dB'+
A
-
m
-
1
C2/4 = - -
m = d B 2 + C2/4 = m~'l1
0.005
0.157
VZL'''
1
0.57
* 0.01
and for the [lo01 direction A
- d B z + C'/4
+ 16 d B 2cz+ Cz/4
=
--m mHlo0
=
1 0.46 f 0.01
290
BENJAMIN LAX AND JOHN 0. MAVROIDES
>
W
a W
2 W
K
-
(000) a
PURE GERMANIUM
L2' L3 W
Lt
a W 2 W
L3'
K
= (100)
FIG.16. Schematic diagrams of the energy band contours in germanium and silicon along the [ I l l ] and [loo] axes in the Brillouin mne. The darkened portions designate the maxima and the minima of the valence and conduction bands. The spiit-off valence band below the degenerate bands is produced by spinorbit intaraction [after F. Herman, Phys. Rev. 96, 847 (195411.
291
CYCLOTRON RESONANCE
where mL is the light hole mass and mH the heavy hole m a . Solving these three equations simultaneously we find for silicon, and in an analogous manner for germanium, the following values:68*81 Ge : Si :
A 13.1 _+ 0.4 4.0 f 0 . 1
B 8.3 & 0.6 1.1 f 0 . 4
C 12.5 k 0 . 5 4 . 1 f 0.4.
Knowing these parameters, it is possible using the results of the perturbation treatment of the valence bands, Eq. (3.7),to determine the effective mass of the split-off J = 4 bands for both germanium and silicon. It is found that for germanium and silicon respectively m,+* = 0.074mand 0.25m.In an analogous manner DKKB0and Dumke8* have estimated the effective mass of the rs- conduction band in germanium which a t 4'K is 0.896 ev8J34above the valence band. The value of effective maas so obtained is 0.034m and it has been verified experimentally by the magnetoabsorption measurements of Zwerdling and Lax.84 From cyclotron resonance, infrared, and other experiment^^^-^^-^^ and his theoretical work, Herman" has drawn a picture of the band structure for germanium and silicon (Fig. 16).It turns out that there are four [Ill] conduction band minima for germanium, each an ellipsoid of revolution located at the edge of the zone. There are six [lo01 conduction band minima for silicon, also prolate spheroids, but located between the center and the edge of the Brillouin zone. The energy surfaces for the valence band in germanium and silicon consist of two concentric warped spheres, which for the heavy holes protrude in the [Ill]directions. For the light holes, the surfaces are depressed along the [ 11 11 and protrude along the [ 1001 directions. c. Theory for Warped Spheres
( I ) k * p Perturbation treatment. The form of the energy surfaces at the edge of the valence bands was derived by DKK79Joand followed from the work of Shockley7' and Herman.7*Shockley suggested that, if the extremum of the energy band lies at the center of the Brillouin zone, the wave functions would be triply degenerate with p-type symmetry of the form 9,= Z U z ( Z f , y * , Z * ) = zu*(y*,z*,z*) = ZU.(Z*,Z*,a/'). (3.12) *I R. N. Dexter, Doctoral Thesis, University of Wisconsin (1955). W. Dumke, Phys. Rev. 106, 139 (1957). W. C. Dash and R. Newman, Phys. Rev. 99, 1151 (1955). S. Zwerdling and B. Lax, Phys. Rev. 106, 51 (1957). G. G . Macfarlane and V. Roberta, Phys. Rev. 97, 1714 (1955); 98, 1865 (1955). ** J. H. Crawford, Jr., H. C. Schweinler, and D. K. Stevens, Phys. Reu.99,1330 (1955). *' B. Lax and J. G . Mavroides, Phys. Reu. 100, 1650 (1955). la
l4
292
BENJAMIN L A X AND JOHN 0. MAVROIDES
Taking the spin into account, the states become sixfold degenerate. The observation in cyclotron resonance studies of two rather than three holm suggested that the spin-orbit coupling had lifted the degeneracy, splitting the valence band edge into two levels, the upper level being fourfold degenerate and the lower band twofold degenerate. If we expand the wave function in Bloch functions, i.e., enk(r) = U.,k+' where n denotes the band and k the wave vector, the Schrodinger equation becomes, neglecting spin-orbit coupling,
where V ( r ) is the periodic potential. This equation can be solved at k = 0, and then for points near the origin the wave functions may be expanded as a perturbation in term of the wave functions of the various bands at k = 0 by use of the k * p method. Thus the periodic functions are linear combinations of (3.14) "m
where U I Orepresents the wave functions other than those of the degenerate states. Using standard perturbation methods, we can write the perturbetion matrix as
U"*
where the matrix D because of cubic symmetry is of the form
(3.16)
and L,M ,N are three constants defined by
L
=
111 m'
2
En0
- Em0
u-0
(3.17)
CYCLOTRON RESONANCE
293
The introduction of spinsrbit coupling aa a perturbation,
where d is the Pauli spin matrix vector of the potential energy of the electron, leads to an extra term in the Hamiltonian and splits some of the degeneracy so that the fourfold J = 8 band lies above the twofold J = band. If the spin-orbit splitting X is sufficiently large, there is no appreciable mixing between the two sets of degenerate bands, and we can consider the two sets independently, writing two independent matrices, a 4 X 4 and a 2 X 2. Since in cyclotron resonance the energies involved are always much less than A, it is the former matrix which is of particular interest to us. From this matrix which is discussed in detail by Luttinger and KohnaOand DKKaO we can write the energy wave vector relationships of the bands in the forma7"already quoted, namely
+
&(k)
=
-hZ (Ak' 2m
* [B'k' + CZ(kz*ky2+ k,nk,2+ kzZk,*)]t) (3.6)
where - ( h 2 / 2 m ) A = +(L
+ 2 M ) + h2/2m
(E)
B = *(L - M )
From the 2 X 2 matrix, the solution of the secular equation yields
&(k)
= -A
-'' Ak' 2m
(3.7)
where aa we have already mentioned, X is the spin-orbit splitting, which equals about 0.04 ev for silicona*and 0.3 ev for germanium.aeQ2 (8) Classical Boltzmunn treatment. In obtaining the expression, Eq. (3.11))for the effective mass of holes we used the semiclassical treatment in terms of the crystal momentum and obtained a solution which Note that with these definitions, A , B, and C are dimensionless and differ from those of DKK by a factor V / 2 m . @IS. Zwerdling, K. J. Button, B. Lax, and L. M. Roth, Phya. Rev. Lc&s 4, 173 (1960) have determined the spin-orbit splitting in silicon, with the result X 0.044 f 0.001 ev from experiments on internal impurity levels. (* A. Kahn, Phya. Rev. 97, 1647 (1955). *o H. B. Briggs and R. C. Fletcher, Phys. Rw. 91, 1342 (1953). W.K a i i r , R. J. Collins, and H. Y. Fan, Phya. Rcu. 91, 1380 (1953). **R.B. Dingle, Phyu. Rcu. 99, 1901 (1955). 0.'
-
294
BENJAMIN LAX A N D JOHN
c).
MAVROIDEl
applies in the presence of the magnetic field alone. We neglected, however, the electrical field and did not solve for the conductivity in a manner analogous to that used for the electrons. This problem ia rather complex and haa been treated in terms of the Roltamann transport theory by Luttinger and Goodman,g*McClure,g' and Zeiger, Lax, and Dexter.g' One of the shortcomings of the classical treatment is that it takes into account only the contributions of the holes which rotate about the magnetic field on the energy surfaces at the equatorial plane, i.e., p~ = 0.
MAGNETIC FIELO
( 0 11
FIG.17. Cyclotron resonance trace in germanium near 4'K and at 23,000 Mc/sM; external magnetic field was 10' out of (110) plane and 30" from [lo01 direction. Orientation was selected to show the eight resonances observed in germanium [after R. N. Dexter, H. J. Zeiger, and B. Lax, Phya. Rm. 104, 637 (1956)l.
The Boltamann theory, however, includes the contributions from holes with all valuee of p ~ In . addition, the treatment accounts for the harmonics found both in germanium and silicon and which are indicated in Fig. 17 for germanium. Zeiger, Lax, and Dexter write the Boltzmann equation in the form e[E
+ v X H/c]
*
VJ
+ v - V,f+
~;di. af =
-- lo 7
where the distribution function f (p,r,t,H,E) is normalized so that
If dP = n J. M. Luttinger and R. R. Goodman, Phys. Reu. 100, 673 (1955). J. w.McClure, Phys. Rev. 101, 1642 (1956). OL H.J. Zeiger, B. Lax, and R. N. Dexter, Phya. Rev. 106, 496 (1957). 93
O4
(3.18)
295
CYCLOTRON RESONANCE
V, and V, are gradients in momentum and coordinate space respectively, v = V,,&(p), f o is the distribution function in the absence of E and H, and the collision time 7 may in general be a function of E . They assume a
uniform distribution in coordinate space so that V,j = 0, a distribution function of the form j = f a - CP afo/a&, a constant H, and that E and CP have an e" dependence. Substituting into Eq. (3.18) and neglecting terms of the order of E' the following equation is obtained:
- Te(v X H/c) . V#
7eE * v
- (1
+ im)@= 0.
(3.19)
The current is thus
J
=
eJfvdp = -eJ(afo/as)@vdp.
(3.20)
For spherical and ellipsoidal energy surfaces, the Boltzmann theory gives results essentially equivalent to the classical theory; it does not add new information except possibly for the case in which the collision time is energydependent or for resonance under conditions of high electron energies where the curvature changes or conductivity mo d ~l at i o n l ~ conditions are important. Thus we shall restrict ourselves to the application of the Boltzmann theory to slightly warped surfaces where the corrections and new information on harmonics are of importance. For warped spheres the energy wave vector relationship which is given in Eq. (3.6) can be written in the form
(3.21)
* +
where p = hk, the average effective mass iit* = m[A (B4 +C*)i]-l, and g ( p / p ) is a small, nonspherical term which is a function of angle only and is given by
d P / P ) = .[b./P>'
+ (PY/P>' + (P./P>'
- +I.
(3.22)
Here =
f C' 4(B2 +C*)+[Af (B'
+
+ +C1)+]'
Introducing spherical coordinates ( p , 8 , 4 ) in momentum space with the z direction chosen along the direction of the magnetic field, aseuming CP = x(e,+) per/**, where T is energy-independent, and substituting
this into Eq. (3.19) the following approximate solution is obtained by expanding x in a Fourier series in 4
x
=
Zxr(e) cos I4
+ x-,(e) sin Z+.
(3.23)
296
BENJAMIN LAX A N D JOHN 0. MAVROIDES
This leads to a set of linear equations in which, to first order in g, only
xl and x - ~are coupled. Inserting this expression for x into Eq. (3.20), the current density J is found and then the power absorbed per unit volume, from P = Re (J E*).
+
*
+ i~r)(S~u~ + S-zu-t) sin 8 d e } . E
(1 (1 1
+ iw)*+ [IW,(l + Ro)lZT1
(3.24)
where n is the number of carriers per cmx,
(1
+
+
+ g)+
sin 8 d e d 4
+
I-'
,
- 2a2 3a4) 7(1 10a2- 15a4) c0s4 el, and u = cos 8, e being the angle between the magnetic field direction and the [Ool) axis. Ro = & ~ [ 3 ( - $
FIG.18. Theoretical curves of cyclotron resonance of heavy holes for H dong: 8, [OOl]and b, [ill]. Curves show power absorption (arbitrary scale) versu magnetic field, in unita of & / w = eH/rfi*w. Parameters chosen were o+ = 5.5, K 0.854. Curvea of the center contour Lorentz lines are shown dashed for cornparinon [after H. J. Zeiger, B. Lax, and R. N. Dexter, Phys. Rev. 106, 495 (195711.
-
In Eq. (3.23) the subscripts (I) and (-1) refer, respectively, to the cos 14 and sin 14 harmonic components and 1 = 0 refers to a constant term.Thus 1 = 2, 3 . . . correspond to resonant absorption of energy at
CYCLOTRON RESONANCE
297
harmonics of the fundamental cyclotron frequency. The problem is then to evaluate Eq. (3.24) using the experimental values for A , B, and C. For the light holes the anisotropy is very slight and therefore the harmonic cyclotron resonance is much too small to observe experimentally. Zeiger and co-workers consider therefore only the cyclotron resonance of the heavy holes and obtain expressions for the power absorbed in transverse cyclotron resonance at the fundamental and first two harmonics for E along the [ l i O ] direction and H in the plane normal to E. Corresponding expressions also were worked out for longitudinal cyclotron resonance absorption, Using these results, Zeiger et at. have calculated line shapes for the warped surfaces of the heavy holes with the magnetic field along the [Ool], [ 1111, and [ 1101 directions. For comparison they also calculated the
MAGNETIC FELD (0.1
FIQ.19. Experimental tracert of cyclotron resonance of holes in germanium for H along: a, \loo] and b, 1111). Dashed lines are mirror images and indicate the Mymmetry [after R. N. Dexter, H. J. Zeiger, and €3. Lax, Phys. Rcv. 101, 637 (1956)l.
line shape for the center contour ( p H = 0) using a Lorentzian line. These curves are plotted together in Fig. 18; in this figure the solid lines represent the experimental situation in which the actual line is an envelope of the resonance for the contours of all values of pH. Including contributions from holes with all values of p H has two effects. First it makes the lines unsymmetrical and secondly, it shifts the peaks slightly from &/a = 1 in such a way that the experimental lines reflect a variation in effective masses which are not as anisotropic as those for the center contour alone. Hence the uncorrected parameters A , B, and C have values which in germanium differ by about 5 % from the true values. This correction was taken into account by Zeiger and co-workers both for silicon and germanium in obtaining the values of these parameters given in Section 3b. The theoretically predicted rwymmetry is quite consistent with the line shapes obtained experimentally and shown in Fig. 19. It is
298
B E N J A M I N LAX A N D J O H N 0. MAVROIDES
seen that the line in the [ill] is asymmetrical in the direction of higher fields. Starting with the Boltzmann transport equation, Luttinger and Goodman,gahave obtained results similar to those of Zeiger et al. by using the following general expression for the current density J in terms of the harmonics 1 e2v'v-' . E dfo - dS d e dp.
J=F/+
a&
(1
iw)
- ME,p.)r
(3.25)
where the magnetic field is taken in the z direction and the angular variable dS = { d p / [ ( e / c ) vX HI). The solution of the Boltzmann equation for the warped spheres indicates the existence of the harmonics. From the solution of the equivalent semiclassical equation of motion as obtained from Eq. (3.8) it can be seen that the physical origin of the harmonics arises as follows: The equation is nonlinear in the momentum parameter so that an electron traversing around the warped contour excites a motion in which higher order oscillations of multiples of the cyclotron frequency of the fundamental motion are also excited. This gives rise t o resonance of harmonics a t two, three, four times, etc., the frequency. Conversely, for a given frequency resonances occur at $, &, f of the applied magnetic field which corresponds to the fundamental resonance. Indeed the existence of such harmonics is observed (see Fig. 17). The existence of any particular harmonic is a function of the symmetry of the particular contours involved, the direction of the magnetic field relative to the crystal axes, and the polarization of the electric field relative to the magnetic field. The intensities of these lines have been examined theoretically as well as experimentally and it is found that the correlation between the two is not very However in these experiments depolarizing effects due to the sample geometry would distort the relative intensities parallel and perpendicular to the dc magnetic field and thus account for some of this discrepancy. More careful experiments are required to settle this point. d. Quantum Theory of Cyclotron Resonance
(1) Free carriers. The problem of the energy levels of electrons in a magnetic field was first considered by Landau who solved it for the c m of free electrons. In connection with cyclotron resonance, this problem also has been considered by Dingleso who solved the following equation 0'
J. M. Luttinger, Phyu. Reu. 102, 1030 (1956).
299
CYCLOTRON RESONANCE
in cylindrical coordinates: (3.26) where the magnetic field is located along the z direction. The solution ie = e-il+yll2e-~l2Ln+1I(Y)
EL
=
(n
+ +)hisc
and
(3.27) &I = -*h2
2m
kU 2
where EL and &I refer to the motion transverse and parallel to the magnetic field respectively and y = (eHr2/2hc); Ln+i'(y)
and M ( - n , 1
= (-1)l
(((nn!l!I ) ! ) * ) M(-n,I + +
1, y>
+
1, y) are the confluent hypergeometric functions. Considering the interaction between the electric field E of the incident radiation and the electric dipole moments of the electrons the transition probability from (n,I ) G (n’,I’) is, for the magnetic field along the z axis, woDortiona1 to E,re*I+ (3.28) [eJJ#ni$n*v* E,re*.'+ t dr &$I2. Ezt
Substituting Eq. (3.27) into this expression and evaluating the integral it is found that all electric dipole moments vanish except for the following:
where Nn.1 =
A"
'
y'e-"{ Ln+i(y)I dy = ( (n
+ I ) !1 S / n !
The second expression for electric dipole moments, given in Eq. (3.29), leads to transitions between states of the same energy and thus does not contribute to the absorption. The first, i.e., D(n, 1; n 1, I - 1) leade to transitions between states differing in energy by
+
(3.30) which is in agreement with the classical result for free electrons in the
300
BENJAMIN LAX AND J O H N 0 . MAVROIDES
magnetic field. Dingle also considers the magnetic dipole moment and in this case he finds that there are no transitions between different states. (6) Carriers with an ellipsoidal energy surface. We can consider the same problem for an ellipsoidal energy surface in which the magnetic field is arbitrarily oriented relative to the principal axes of the ellipsoid. Luttinger and Kohnaohave justified the representation of the Schrodinger equation in the effective mass approximation. Therefore the Hamiltonian can be written as
where in the second form the coordinates have been transformed such that Ai’ = piA,
and
2,’ = p,-’x,.
(3.32)
Here p. = d m / m i ; choosing A = +H X r, then H,’ = ~ , - ~ p ~ p ~ p , In H,. the primed coordinate system the energy surfaces are spherical so that the motion of the electrons in a plane is perpendicular to H‘.Hence the vector normal to the plane of motion makes an angle y with H,the magnetic field in free space, such that
Hence the a,are the directional cosines of the magnetic field H relative to the principal axis of the ellipsoid. We can now rotate into a new coordinate system in which the z” axis coincides with H’. Then
a
and since w c = (eH”/mc)
- 1_ m*
--
=
(pH/m*c),
1
dm.m,m,
d?m,
-+
aZ2rnu
-I-aa2m.
(3.33)
so that the energy levels for an ellipsoidal energy surface in this coordinate system are dependent on the angle the magnetic field makes with the principal axes of the ellipsoid and on the mass parameters which determine the energy-momentum relationship. Hence the Landau levels are given by the expression of Eq. (3.30) for the spherical case except that the isotropic mass is replaced by the effective mass given in Eq. (3.33).
301
CYCLOTRON RESONANCE
An interesting aspect of cyclotron resonance for the caae of anisotropic surfaces is longitudinal cyclotron resonance. This can be demonstrated for the situation in which the energy surfaces are ellipsoidal. If the electric field is parallel to the magnetic field it can be shown that for spherical energy surfaces the matrix for the electric dipole transitions vanishes. I n the ellipsoidal case, however, with the electric field in some arbitrary direction relative to the principal axes we transform the Hamiltonian into the prime coordinate system which now includes the timedependent potential for the electric field, namely E * r. The electric field E' in the new coordinate system will now have a component perpendicular as well as parallel t o H',since E transforms differently from R.
0
0.08
0.16
024
0.32
FIG. 20. Theoretical curve8 for transverse and longitudinal reeOnance in germanium with B in the [I101direction and E in the 6101 and [I101directions respectively [after B. Lax, H. J. Zeiger, and R. N. Dexter, Phyaica 20, 818 (1954)l.
The component of magnetic field perpendicular to E' results in nonvanishing matrix elements, giving rise t o resonance but not, of course, 80 intense as that for a transverse magnetic field. Indeed such experiments have been carried out by LZD7*and some of the results are given in Fig. 20. The figure shows only one peak with the magnetic field along the (1101 direction for germanium. This peak corresponds to the longitudinal resonance for those ellipsoids along the [lll] and the flli] axes which all lie in the plane containing these three directions. The other two sets of ellipsoids lie in the [ill] and [lil] directions; thus they have surfaces perpendicular t o the magnetic field so that the field is along one of the principal axes. This results in a vanishing matrix for longitudinally induced transitions. (3) Carriers with warped spherical energy surfaces. It has been shown by Luttinger and Kohnaoand more recently by Luttinger" by a modified procedure that the quantum theory of cyclotron resonance for warped spherea can be deduced from the matrix repreaentation of the energy
302
BENJAMIN LAX A N D J O H N 0 . MAVROIDES
surfaces as derived by DKK.*O In the presence of a magnetic field represented by the vector potential A,the energy levels are obtained by solving the coupled equations
where the repeated indices a,B are summed over x,y, 2 and the summation over j' is over the number of degenerate states for the band edge of interest; this number would be three for germanium or silicon. The p . are the momentum operators ( - i a / d x , ) and the wave function 9 is given by
(3.35) where the 4, are the Rloch functions at the edges of the degenerate valence bands and Fl(r) are the envelope functions obtained from Eq. (3.34). The coefficients Djjla@,are given by
(3.36) Here the primed summation is over all those states of other bands at the center of the zone not belonging to the degenerate set j , EO is the energy of the degenerate set, and the PI,= are the momentum matrix elements between the different bands evaluated a t k = 0. For the valence band of germatiium or silicon, the p-like valence band edge is split by spin-orbit interaction into a fourfold ( p , ) and a twofold ( p , ) degenerate edge. Cyclotron resonance indicates that the fourfold degenerate level lies above that of the twofold one so that the valence bands can be described by a 4 X 4 effective mass Hnmiltonian for energies near the band edge. In writing down the effective mass equations, Luttinger has demonstrated that additional parameters not contained in the expression for the energy surfaces must be included if there is an external magnetic field. The way these parameters enter can best be eeen by writing the effective mass Hamiltonian D,j i - h2Za@Dija~k.k~, where hk = p (eA/c), in terms of the 4 X 4 angular momentum matrices J , , J,, J , . These matrices and their products can be used to represent any 4 X 4 matrix and their transformation properties under rotation are known. Luttinger finds that the most general form of the effective mass Hamiltonian for the spin orbit case with cubic symmetry is
+
303
CYCLOTRON REBONANCE
D
h'
= m ((TI
+ F)
- y2(kz2Jz2+ ku2JUZ+ k12Jg2)
- 2 ~ r [ l k z k , I( J J , )+ I k A J (JJsI+ {U*I{JJzII e + e uJ. H + h4 q(JraHz+ J U J H ,+ J I a H I ) ]
(3.37)
+
where {kzk,] = +(k.k, k,k,). The first three terms give the Hamilare defined in terms of tonian of DKK; the three parameters yl, y t , those for the energy surfaces given in Eq. (3.6) and those of DKK:
The two additional terms arise from the noncommutivity of the operators given in the parentheses of Eq. (3.34), and the more important constant u is given in terms of the matrix element sums of Eq. (3.36) without spinorbit interaction by 1
-
m
K
(3u
+ 1) = - K
D,yru
(3.39)
-Dx~Y~.
The second constant q is introduced by the spin orbit interaction and is estimated to be very small. To obtain an explicit form for D one introduces a particular representation of the J ' s :
0
J,
0
0
=
(3.40)
0
o
id8
0
I 0
0
0
-8
I
304
BENJAMIN LAX AND JOHN G. MAVROIDES
The resulting effective mass equation (for q = 0) is shown in (3.41), opposite. The characteristic values of the general Hamiltonian of Eq. (3.37) or (3.41) give the energy levels of the system, but the general solution is extremely complicated and has not yet been obtained; however, Luttinger has found an exact solution for H 11 [ 1111 and k H = 0. Furthermore, for the special case yz = 7 3 = f (or L - N = N) and q = 0, Luttinger finds that with kH = 0 the solutions can be obtained readily. When 7: = 7: the effective mass equation has spherical rather than cubic symmetry, so that in this case anisotropy in the energy surfaces is neglected; this condition is fairly well satisfied in germanium but not in silicon. We can take the magnetic field to be in the z direction and introduce creation and destruction operators of at and a,
which have the following well-known properties when operating on the harmonic oscillator functions Un (3.43)
the latter holding for properly chosen phases of the u.. Expression (3.37) or (3.41) then reduce to
+ 2i(a* - ~ ~ ~ ) ( J+J ~ I I(3.44) KJ*)
or, in matrix form, to the expression (3.45), shown on the facing page. This breaks u p into two 2 X 2 blocks, corresponding to mJ = +#, -; and mJ = +, - Q. If the first of these is denoted by “ 1 with eigenvalues GI and eigenfunction rL1, with similar notation for the second block “2,” then the eigenfunctions are of the following form involving harmonic oscillator functions (3.46)
Letting n = 0, 1, 2, 3, . . . the eigenvalues then can be found as the solutions of two by two secular determinants, except for n = 0, 1, in which c w we take al = 0 and bl = 0. There are four sets of levels, which are described more fully in the following.
where
D = -k H ?nc
H
306
BENJAMIN LAX AND JOHN 0. MAVBOIDEB
Experimentally the levels are not the same in all directions even in germanium, and the anisotropy is taken into account by applying perturbation theory to the difference between the original Hamiltonian of Eq. (3.37) and the approximate one of Eq. (3.44). Luttinger restricts himself to the case which is of pertinence to the experimental investigation of cyclotron resonance, namely that in which the magnetic field is in the (110) plane. Calling 8 the angle between the field and the t axis he chooses a (1, 2, 3) coordinate system so that
k,
=
-ski
+
CkS
where 8 = sin 8 and c = cos 8. Transforming also the J’s so that the J a , J,,, J , are related to J1,Jz, Jj in the manner of Eq. (3.47) and setting k, = 0, Luttinger finds that
D = Do+Di+Dz
(3.48)
-
where D Ogives the energy to first order in [ (L- M ) N]and D 1and Dt are higher order terms. DO,which reduces to the 4 X 4 matrix of Eq. (3.45) for yz = y3, can now be written similarly to Eq. (3.44) as
where
+ %(a* - at’) { JlJz)]+ K J I ) (3.49) 7’ = tI(3C2 - 1)2y* + 3sy3c2 + l ) y 4 y“ = 6[(3 - 2c2 + 3C’)yz + (5 + 2c’ - 3c4)yJ].
The Hamiltonian D Ois soluble in the same manner as that of Eq. (3.44) and the following two sets of solutions are found.
*
([Y’n
f (Ti
-K
- 47”)]’+ 3Yr‘%(7Z- 1) 1’1
(3.50) where for the plus sign n = 0,1,2, . . . ,and for the minus sign n = 2,3,4. These energy levels can be represented schematically as shown in Fig. 21. Crudely these levels can be considered to be associated with
307
CYCLOTRON REBONANCE
light and heavy holes, since for higher quantum numbers the spacings are nearly equal; however, this interpretation is not as meaningful for the lower quantum numbers since, for these, the spacings of the levels are not equal. The solutions obtained have several significant consequences, some of which have been termed by Luttinger and Kohn aa quantum effects. This description appliea particularly to those features which have t o do with the lowest levels, their unequal spacings, and possible transitions between these levels. Evidently in order to see such fine structure in the cyclotron resonance spectrum it is necessary to use: very pure samples, very low temperatures, and an extremely sensitive
H IU
[loo]
1
ntd Y
0
n=3
50
n=2 -
n= 2
n . ( 10
n=o 0
I
b*
a b-
FIQ. 21. Schematic representation of the energy levele for holm in germanium with H in [loo].Here refers to light holes and -, to heavy holea (after L. M. Roth, unpublished).
+
microwave spectrometer. Under these conditions the energy levels corresponding to hole transitions are individually resolved. Experiments in which the above requirements were fulfilled have been performed by Fletcher and co-workers.*l Their results, which have been analyzed by Goodman," did prove the existence of these effecta from the h e structure of cyclotron resonance. The complex nature of the energy level structure has been demonstrated in a rather direct fashion and correlated quantitatively with the results of the magneto-optical experiments a t infrared frequen~ies.~8.9~ Another consequence of this quantum theory
'' R. R. Goodman, Doctoral Thais, University of Michigan (1958). S. Zwerdling, B. Lax, L. M. Roth, and K. J. Button, P h p . Rw. 114, 80 (1959). **L.M.Roth, B. Lax, and 8. Zwerdling, Phy8. Rm. 114,90 (1959). 9*
308
BENJAMIN LAX AND JOHN
a.
MAVROIDES
of cyclotron resonance, i.e., the existence of harmonics for the case of warped energy surfaces, follows from the solution of the moat general case in which the higher order terms are considered. In this case it ie necessary that harmonic eigenfunctions in addition to Un and Un-2 be considered. This means that any single eneqy level is an admixture of several states; hence in calculating the matrix elements or transition probabilities between states, transitions corresponding to IAnl > 1 would be allowed, with decreasing probabilities for the higher orders. These correspond to the harmonics, which have been observed. The quantum-mechanical solutions obtained above for the valence bands are only valid for relatively small energies where it can be assumed that the energy momentum relation is essentially a quadratic relation in momentum with the anisotropy in the angular coordinate expressed as in Eq. (3.21). However, it is conceivable that with proper optical excitation cyclotron resonance may be possible at energies which lie deeper in the bands. In such an event the observed curvature of the bands would be considerably different from the curvature at the bottom of the bands and the quantum-mechanical solution would have to take into account higher order terms. This is particularly significant for silicon where the spin-orbit coupling is relatively small, i.e., 0.044 ev. Hence the interaction of the bands is large. The changing curvature resulting from this interaction has been worked out by Kanelo0and must be taken into account. This would be relatively difficult to do for germanium and silicon. However, as we shall show, such a treatment for the conduction band of indium antimonide is relatively straightforward.
e. Experiment8 012 Compound Semiconductors Cyclotron resonance experiments a t microwave frequencies have been carried out on materials other than pure germanium and silicon, but not with as great success as with the experiments on these two elements. One of the first of such experiments was made by Dexter and Lax*8on relatively pure indium antimonide. The absorption was rather broad and with linear polarization no resonance peak was observed. However by analyzing the absorption line in detail they were able to obtain a n estimate of the electron effective mass m ' = 0.02m, with an WT = +. Using p-type indium antimonide with an acceptor impurity concentration of less than 5 X lO1'/cm*, Dresselhaus et al.¶¶observed resonance at 2.2'K and 24,000 Mc/sec as shown in Fig. 22. The low field line, which showed no snisotropy, is associated with an electron having a mass value (0.013 f 0.001)m.From the line width a relaxation time T = 1 X 10-1' sec waa estimated. They also reported two broad resonance lines associated with E. 0. Kme, J . Phyr. C h . Sdadr 1, 82 (1958).
309
CYCLOTRON RESONANCE
m w e s m* = 0.18m and m* > 1.2m with some apparent anieotropy. The former value is associated with holes and is in reasonable agreement with the masses mk* = 0.2m estimated by BursteinlO1 and also by Motd01 from their experimental results. Independent attempts to confirm the existence of the higher mass have failed. The Berkeley group also observed cyclotron resonance at microwave and millimeter frequencies in germanium-silicon alloys27 containing from a fraction of an atomic per cent up to 5.4% silicon. Although resonance was observed the resolution was not sufficient to distinguish the small deviations of the electron and hole parameter from that of pure
I
'
1
I
I
0
500
too0
1500
2000
STATIC MAGNETIC FIELD
2500
3Ooo
(0~8td8)
FIG.22. Power absorption (relative scale) versus static magnetic field intensity, indium antimonide at a frequency of 23,975 Mc/sec [after G. Dresselhaus, A. F. Kip, C. Kittel, and G . Wagoner, Phyu. Rcu. OB, 556 (1955)j.
germanium and that due to possible uncertainties in crystal orientation. It appears that more fruitful results would be obtained from experiments with higher frequencies in the millimeter range. One of the most recent successes of cyclotron resonance studiee on a compound semiconductor has been achieved by Dexter'O' who observed resolvable lines from a pure single crystal of cadmium sulfide. Both electron and hole resonances were obtained a t 4.2'K and a t 1.3'K. Using visible light an isotropic mass m* = 0.35m was attributed tentstively to the electron. Using infrared illumination camera with a msse in the range of 0.07m with definite anisotropy were identified with holes. A trace obtained with the highest resolution is given in Fig. 23. E. Burstein, G. Picus, and N. Sclar, Proe. Phoiocductivity Conf., AUcrdie City, 1964 p. 353 (1956); T. S. Moss,ibid. p. 427. lo* R. N. Dexter, J . Phys. Chem. Solids 8, 494 (1959).
310
BENJAMIN LAX AND JOHN 0. MAVROIDES
Considerable effort has been expended both by the Berkeley and Lincoln groups in an attempt to observe cyclotron resonance in many other materials a t microwave frequencies. lo% Intermetallic compounds such as indium arsenide, gallium arsenide, lead sulfide, and lead selenide with impurities of the order of 10*8/cmJhave shown some magnetoconductivity a t microwave frequencies but no resonances were resolved. All attempts a t these frequencies and low temperatures to measure resonances in silicon carbide, diamond, and other high gap materiala have failed.
H (kilogourr)
FIQ.23. Cyclotron resonance in cadmium sulfide at 1.3'K and 23,500 MC/W with preferential excitation of the 0.36 macarrier and the carriers on ellipida. is nm1Y bfw perpendicular to the c axk, w = 10 corresponding to T = 6.7 X R. N. Dexter, J . Phys. C h . Solids 8,494 (1959)j.
Perhaps the best results obtained from compound semiconductors are those reported by Stevensonlo* on cadmium arsenide (CdAsr) single crystals grown from spectrographically pure cadmium and arsenic. The data were taken at frequencies between 20,000 Mc/sec and 24,000 Mc/sec at temperatures below 2°K. The lines were well resolved with 07 E 3. The results were unique for a number of reasons. Experimental identification of holes and electrons by Hall effect, impact ionization, and excitation with filtered infrared radiation indicated that the electron ma8888 were greater than those of the holes. Furthermore the energy surfaceq which Private communications.
108
M.J. Stevenson, Phys. Rm. Leuera a, 464 (1959).
311
CYCLOTRON RESONANCE
0. C AXIS
4s-
lo'
45.
so'
rs.
ANGLE (drgrma In C - A ~ I Od MH
so* A AXIS
from A onis)
Re. 24. Effective maw of holm and electrons in cadmium d d e in the fquaCY range between 26,000 Mc/sec and 24,000 Mc/eec. The magnetic field is in the C-A plane at an angle e from the C axie [after M. J. Btevenaon, Phys. Rcv. LcfrCrs 8,464 (lQ59)l.
were presumably located at the center of the Brillouin zone, were oblate spheroids for both the conduction and valence bands. The ma& p m eters were as follows:
for the electrons, ml/m = 0.16 and mt/m = 0.57 for the holee, ml/m = 0.11 and ml/m = 0.32. The anisotropy traces with the magnetic field making an angle with the c axis are shown in Fig. 24. No anisotropy was observed when the magnetic field was rotated in the plane perpendicular to the c axis.
312
BENJAMIN LAX AND JOHN G. MAVROIDES
4. PHENOMENA IN METALS
a. Nonresonant A bsorplion ( 1 ) Introduction. Although Dorfman'O and Dingle20 suggested that cyclotron resonance should be first investigated in metals, the direct observation of a resonance peak at microwave frequencies and low temperatures was really not possible. This was pointed out by DKK'2 in their paper on plasma effects in indium antimonide. In experiments on microwave absorption in a magnetic field, the metal to be studied is usually made a portion of the wall of the cavity. The electromagnetic wave therefore is absorbed in a small skin depth in the surface of the metal. It can be shown that, in order to calculate this absorption, it ia merely necessary to evaluate the Poynting vector of a plane wave normally incident upon the surface of a metal. Let us consider the classical problem in which the propagation constant 'I = a i8. Choosing the E coordinate perpendicular to the surface with the boundary at E = 0, and a wave in the medium varying as e-ra, we obtain from the boundary conditions for E and H in the t,y plane at z = 0 that
+
Eo + E' = E
r ro
Eo-E'=-E
where the Eo and E' respectively refer to the incident and reflected wave8 in free space. From the real part of Poynting's vector, i.e., P=fRe[EXH*] and the relations between the vectors in free space and in the medium we find that ratios of P the power absorbed in the medium to Po, the incident power, is _ P -4B80 (4.2) Po a2 (a B O P '
+ +
For a metal,
u2
+ 8' > Bo* + 2880,and Eq. (4.2) reducea to
The dispersion relation between the propagation vector r, the frequency and the magnetic field is found from the first two of Maxwell's equations by assuming E(r) = E exp [ - (r r id)] and eliminating the rf magnetic field. We thus obtain w,
-
V X V XE
=
W'E&
- iwpof - E
(4.4)
313
CYCLOTRON RESONANCE
and
- E - E) = w*t& - iwfiof
E
r2(yy
(4.5)
where y is the unit vector in the direction of propagation, t the dielectric constant of the medium, and f the conductivity tensor. Equation (4.5) may be written as three homogeneous linear equations in terms of components of the electric vector E. By equating the resulting determinant to zero a biquartic equation for r is found, whose solution is in general rather involved. However, the problem can be simplified considerably by the experimental procedure of orienting the magnetic field along a symmetry direction of the crystal, so that in many cases the conductivity tensor can be of the form'ob
In this particular case the magnetic field is along the z direction. Also, for the most convenient configuration r is either in the direction of the magnetic field or perpendicular to it. Under these conditions
+ iwr,,u.,,
I'*
(4.7)
= -w2~p0
where u,ft = u, - iu, consists of combinations of components of the conductivity tensor appropriate to the particular configuration being considered. For r in the direction of magnetic field
and for r perpendicular to the magnetic field U . N
=
Qeif
-
or
(4.9)
u81
- wc(Yz*UzZ
+ r,*u,,) +
of
t(.,.,z
+
- i(n*uZZ
-
Uxpyy)*
(4.10)
Y S ~ O ~ )
When the propagation is longitudinal, as in the case to which Eq. (4.8) appfies, E is in general elliptically polarized normal to the static magnetic field H and becomes circularly polarized when u, = uU3.When the propagation is transverse there are two cases, expression (4.9) corresponding to linear polarization with E 11 H and (4.10) corresponding to elliptical polarization with E I H.In a metal the conduction current I0a0
When the magnetic field is in an arbitrary direction, or when the crystal is not cubic (for example, in bismuth), it is possible that all nine components of the conductivity tenaor may be needed.
3 14
BENJAMIN LAX AND JOHN 0. MAVROIDES
usually is much greater than the displacement current. Then neglecting factors which do not depend upon the magnetic field, the power absorbed is : (4.11)
Let us now consider the simplest case, namely that of a single carrier in an isotropic medium with the magnetic field perpendicular to the
-
I
-. m 4
[
40
I
I
I
I
-8
-6
1
I
I
I
I
I
I
I
I
I
2 4 H o t x 10 -3 (oersteds 1
6
8
10
I
I
I
-4
-2
I
0
I
FIQ. 25. Abeorption coefficient versus dc magnetic field for circularly polarired radiation incident on the (00.1) plane of bismuth at 4.2”K. These data were taken at 24,000 Mc/eec. The vertical ecale ie only approximately linear. The magnetic field L normal to the (00.1)plane. Zero absorption is eomewhat below the axia of a b e e b [after J. K. Galt, W. A. Yager, F. R. Merritt, B. B. Cetlin, and H. W. Dail, Jr., Phys. Rev. 100, 748 (1955)].
surface and circular polarization. In this case we find from Eq. (4.8) that the conductivity components are given by (4.12)
so that Q O b
QO
ur =
1
+ (w f w,)42’
Qi
= 1
+
(0
f web f
we)42
and the power absorbed is
This is equivalent to Anderson’s expressiona6 and indeed dsscribes nonresonant absorption. This type of absorption was observed in bismuth
3 15
CYCLOTRON IWSONANCE
by Galt and co-workersJJ and also Dexter and Lax;" it is illustrated by Fig. 25. Dexter and Lax obtained some interesting results by differentiating Eq. (4.13) twice and equating the result to zero. They obtained for the inflection point we =
0
+ 1/.
d3
(4.14)
with a similar result for linear polarization. This suggests that one can estimate the approximate cyclotron resonance value from the nonresonant absorption by use of Eq. (4.14). This theoretical result was the
1
I
1
0
I
1
2000
I000
0
200 H
I
1
I
3000
400 600 lorrrtrbr)+ (b)
I 4000
800
FXO.26. The derivative of the abeorption versus magnetic field in bismuth. H parallel to trijpnal axis and perpendicular to surface of metal (after R. N. Dexter and B. Lax, Phys. Reu. 100, 1216 (1955)J.
basis upon which Dexter and Lax based their experiments for detecting the cyclotron resonance condition in a metal. By these experiments they demonstrated that the slope of the absorption would show a peak at the inflection point aa in Fig. 26. This was accomplished by using a coil to modulate the magnetic field which then automatically gave them a derivative curve. The same technique was subsequently exploited quite successfully by the group at BTL*h60in their work on graphite and bismuth.
316
BENJAMIN LAX A N D JOHN G. MAVROIDES
For a single carrier in an isotropic medium with the magnetic field parallel t o the surface and with linear polarization, it can be seen from Eq. (4.9) that ueII = uzz.This result indicates no dependence of conductivity on magnetic field and consequently no resonance when the collision time is independent of energy. The situation is the exact analog of that for magnetoresistance in a metal slab with the magnetic field perpendicular to the applied electric field and isotropic carriers. However, for an energy-dependent relaxation time there will be some magnetoresistance. Analogously for a n energy-dependent relaxation time one would expect a resonance in a metal; but the effect would be of second order and its detection would probably be difficult. The possibilities in a two-carrier system are more interesting. In this case it is not difficult to show that for the metallic case the conductivity takes the form
+
where b = ( ( e H r ) / [ c ( l i w r ) ] } and the subscripts 1 and 2 refer to the two carriers. The plus or minus sign signifies carriers of the same or opposite signs. One of the most important consequences of the above expression is that, for transverse propagation in a two-carrier system, even for a n energy-independent relaxation time, there is a dependence upon the magnetic field. The startling result, however, is that there is only one resonance point corresponding to a n effective mass (4.16)
where R = n 1 / n 2 . If R = 1, then m* = 4 G ;R << 1, m * = m l ; R >> 1, m* = mz.Thus we see that the resonance effective mass is a function of the relative carrier concentration as well ns the individual effective masses. k'or intrinsic material the resonance effective mass is the geometric mean. Furthermore, when the carrier concentrations are considerably unbalanced as in extrinsic samples, the resonance gives the mass of the minority carrier. Analysis of the cyclotron absorption of such a two-carrier system yields the interesting result that, if the minority carriers have a mass smaller than t hat of the majority carriers, they will give rim to a sharp peak with enhanced intensity in the derivative of the absorption aa shown in Fig. 27. Of course the situation is more complex for metals with multiple energy surfaces such as bismuth. It has been shown by Lax and co-workersa7that, in this case, for the magnetic field perpendicular t o the surface, the effective conductivity for the combined holes and
317
CYCLOTROS RESONANCE
electrons is:
This expression is based upon the JonesShoenberg model which is discussed subsequently. Also it has been aasumed that the scattering 1.8
1.2
0
-06
-1
0
1
2
3
-p
Fla. 27. Derivative of the power absorption, as a function of magnetic field in
-
normalized units. wC is the cyclotron frequency of the majority carrier. The mass ratio of minority carrier to majority is 1 :3. The concentration ratio is 1:5. WI 3 for both carriers. Note the sharpnees and enhanced intensity of the minority carrier resonance, und the small shift in peak position [after B. Lax and H. J. Zeiger, Phys. Rew. 106,1466 (195711.
time t is isotropic and the same for both holes and electrons. This sssumption may not be justified, but it is used as a first approximation. In Eq. (4.17), U A = nhef/mh, where n h is the hole concentration and mi is the effectivemaas of holes in the trigonal plane; W . = (eH/nnc) ;R = n./m is the ratio of electron to hole concentration; r l , rz, t i , and r4 are the ratios of the electron mass tensor components to mh and
is the ratio of the resonance mass for electrons, with
H along the trigonal
3 18
BENJAMIN LAX AND JOHN 0. MAVROIDFS
axis, to that of holes. Using Eqs. (4.11) and (4.17), the dependence of power absorption upon wcr may be calculated and typical curves are given in Fig. 28 for several values of R, with WT = 2 and mh = 0.035m.10" For intrinsic bismuth (R = 1 ) the absorption curve shows a typical increase as a function of magnetic field, the absorption for poeitive wcr being due primarily to holes and for negative wer, to electrons. An unbalance in 16
-400
-80
-60 - 4 0
-20
0
20
40
60
80
FIG.28. Absorption versus W ~ Tfor w = 2' with B perpendicular to the surface of the sample. The isotropic hole mass waa choaen to be 0.035~1 and the m a e m of the electrons are those of Shoenberg. The solid curve shows the result for an intrineic sample and the others represent an excees of holes to electrons in the ratios indicated [after B. Lax, K. J. Button, H. J. Zeiger, and L. M. Roth, Phys. Rev. 102,715 (1956jl.
concentration, e.g., R = +, gives a curve similar to that of Fig. 25 obtained by Galt and co-workers, with a peak for negative values of magnetic field corresponding to a minority concentration of electrons. Figure 29 shows the absorption for different values of w r and a fixed value for R. Curves of this type were also obtained using the Abeles and Meiboom model, which will be discussed later, for the magnetic field parallel aa well as perpendicular to the surface. 'En the former case, i.e., with the lo*
-
Recent measurements by Galt and co-workers'@show that the appropriate value for rnr 0.068m.
319
CYCLOTRON RESONANCE
magnetic field parallel to the surface, the analysis ie quite complicated. Thus, from a theoretical point of view, the beat configuration for determining the effective masses of the carriers is that in which the magnetic field is perpendicular to the surface and with a circularly polarid system. The crystal is cut along the principal axes such that two of the principal axes are contained on the polished surface. Then it is placed perpendicular to the magnetic field in a circularly polarized system such as that developed by the group a t the Bell Telephone Laboratories and also by Dexter for his work on antimony.
I I24 I
-
0I
-0
I
I
-40
I
I
0
I
I
40
I
I
80
*r FIG.29. Theoretical curves of absorption versus O.T for di5erent vdua of WT and a relative concentration of holes to electrons of 4 to 1. The other parameters are the same BB in Fig. 28 [after B. Lax, K. J. Button, H. J. Zaiger, and L. M. Rath, P h p . Reu. 101, 715 (195S)l.
(9) Ezpen’mental techniques. A typical circularly polarized system for cyclotron absorption in metals is the one developed by Calt and ~ ~ - ~ ~ r kand e m is shown * ~ ~in ~Fig. * 30. ~ ~In ~this arrangement, a degenerate cylindrical cavity is excited on a T E I 1 ,mode a t 24,000 Mc/sec, or a TEw mode at 72,000 Mc/sec, via a coupling hole at one end of the cavity. This hole is placed on the broad side but off center of the wave guide so as to produce a circularly polarized field near the axis of the cavity. The wave guide is bent into a U shape in order to fit into the
320
BENJAMIN LAX AND JOHN 0. MAVROIDES
helium Dewar. The microwave signal travels down one arm of the guide, and is incident upon the cavity and also the sample which covers a hole on the end of the cavity opposite the coupling hole; thus the sample forms the wall of the cavity in an area where circularly polarized radiation is incident upon it. The resultant reflected signal is observed with a detector on the other arm of the U. The degree of circular polarization waa checked a t 24,000 Mc/sec by observing the ratio of intensities of paramagnetic resonance in a sample of calcium copper acetate hexahydrate, located near the bismuth sample, for the dc magnetic field in FROM
FIQ.30. Diagram of experimental geometry for cyclotron reaonanee experimenb in bismuth [after J. K. Galt, W. A. Yager, F. R. Merritt, B. B. Cetlin, and A. D. Brailsford, Phys, Rev. 114, 1396 (195911.
opposite directions. This ratio, which was at least 10 in all the experiments, was between 20 to 30 in most experiments. Because of difficulty in finding a suit,able paramagnetic material which had a resonance at sufficiently low magnetic fields, this type of check for circularity waa not made at the higher frequency. However, the fact that data at 72,000 Mcl sec fitted the same theory as that for the 24,000 Mc/sec data waa taken as a confirmation of the circularity. In achieving circularly polarized radiation, it was necessary to make the cavity perfectly degenerate. This required extremely careful machining tolerances and also adjustments in the form of small cylindrical studs which were moved in and out of the cavity on screw threads.
CYCLOTRON RESONANCE
321
For improvement of the sensitivity at 24,000 Mc/sec a somewhat novel microwave bridge was used. For bismuth, for which the magnitude of the signals was large in most cases, this bridge was a convenience rather than a necessity. By placing a current actuated ferrite isolator in the wave guide after the cavity and square wave modulating its current at 27 cps, the microwave was also square-wave modulated. A dummy wave-guide arm was modulated with another isolator in opposite phase to that of the signal arm and both signals were fed to the detector. The dummy arm was adjusted so that at zero magnetic field the two signals were equal and thus no 27-cps signal was present at the detector. As the magnetic field was varied the signal passing through the cavity changed, causing an unbalance in the bridge and a 27-cps signal to appear at the detector. Very low signal levels can be observed with this system by means of a lock-in amplifier and phase detector. (3)Results. Bismuth. It is not surprising that the first metal investigated for cyclotron resonance was bismuth. Considerable was known about its electronic properties from the g a l v a n o m a g n e t i ~and ~ ~ de ~ . Ham-van ~~~ Alphen'O' measurements. Also it is available in relatively pure form and large crystals are easily grown from it. Bismuth is referred to as a semimetal because it has overlapping bands. In order to explain the galvanomagnetic measurements, Joneslo4proposed a model for the band structure which consisted of two ellipsoids, one for the conduction band and one for the valence band. This model, however, was not too successful. On the basis of Blackman's theoretical work'O' and his de Haas-van Alphen measurements, which gave information only about the electrons, Shoenberglooproposed a model for the conduction band consisting of a set of three ellipsoids, each tilted out of the trigonal plane by 6' and with rotational symmetry about the trigonal axis. One of the inclined ellipsoidal surfaces was represented in k space by the relation where the subscripts 1,2, and 3 refer to the binary, bisectrix, and trigonal axes, respectively and the a ' s are dimensionless constants. Two more such ellipsoids are generated from this one by 120' rotations about the trigonal axis. The combination of a single ellipsoid for the holes and three inclined ellipsoids for the electrons will be referred to as the JonesShoenberg model. Subsequently, Abeles and Meiboom lo6 found that the lo'
H.Jones, Proc. Roy. Soc. Al66,653 (1936). B. Abeles and S. Meiboom, Phys. R w . 101, 544 (1956). D. Shoenberg, Proc. Roy. SOC.A170, 341 (1939);Phil. Tram. Roy. Soe. A246, 1
10'
M.Blackman, Proc. Roy. Soe. A166, 1 (1938).
10'
(1952).
322
BENJAMIN LAX AND JOHN 0 . MAVROIDES
simplest model which could explain their galvanomagnetic measurementa was that of an ellipsoid of revolution for the holes with major axis along the trigonal axis and a set of three ellipsoids in the trigonal plane for the electrons. One electron ellipsoid has a principal axis along a binary axis and another principal axis along the trigonal axis; the other two ellipsoids were generated from this one by 120' rotation about the trigonal axis. This model is essentially a combination of the models proposed by Jones and Blackman. Although this model explained their measurements the Jones-Shoenberg model was still not ruled out since it would predict longitudinal magnetoresistance in the trigonal direction and such measurements were not reported. Using these two models Lax and c~-workers*~ derived expressions for the effective conductivities for different proportions of hole and electron densities. The mass tensors of the tilted ellipsoids, normalized to the free electron mass, were represented by (4.19)
where axes 1, 2, and 3 are again the binary, bisectrix, and trigonal axes, respectively. The Abeles-Meiboom model can be represented by setting m4 = 0. The other two ellipsoids have mass tensor m / m and mJm derived from (4.19) by the two transformation matrices -1
s&c= f
f
l
4 3 o
+d3 0 -1
0
O2
/.
(4.20)
From these expressions the effective conductivity and power absorbed were calculated for various ratios of excess hole to electron densities, values of wr and of effective mass for the holes in the trigonal plane. The results of some of these calculations have already been indicated in Figs. 28 and 29. A similar analysis was carried out in great detail by Galt and coworkers.6oTheir analysis included a term for the displacement current which at microwave frequencies is small and can be neglected. At infrared frequencies, of course, it leads to dielectric anomalies or magnetoplasma effects where the effective dielectric constant may go to zero. We shall treat these effects later. Galt el al.5Qcarried out their experiments at 24,000 and 72,000 Mc/eec, with the apparatus which we have already described, on pure bismuth and bismuth alloyed with tin and tellurium. In pure bismuth the number of holes equaled the number of electrons. Alloying with tin increased the
323
CYCLOTROS RESOXANCE
relative number of holes, while alloying with tellerium increaeed the relative number of electrons. They carried out experimental measurements with the crystals cut and etched in such a way that the three principal crystal directions, i.e., the trigonal, binary, and bisectrix axes, were normal to the surface. Magnetic field modulation techniques were used to determine the derivative of the signal as a function of magnetic field. However, because the effects observed resulted in broad lines, this 8 , X lo' IN WEOERS/ML ti IN OERSTEDS (72,000 Mc/ccc -18 -42 -6 0 6 12 18 XfO'
110-a
c
z
w
0 LLLL
1 "10-2
$ ;3.48 2
o F; a 2.90
'
g3 a m 4
2.32
S 1.74
(L
1.16
g
0.58 0.00
-14 -12 -10
-8 -6 - 4 -2 0 2 4 6 ti IN OERSTEM (24,000 MCIHC) 0 , X f04IN WEBERS/M'
8
10
12
14 XIO'
FIG.31. Power absorption in bismuth with H normal to the sample surface and along a twofold axis. Vertical arrows indicate cyclotron resonance fields. Zero abmrp tivity for experimental curves is offset vertically from that of the theoretical curve& Dashed line shows how the 24,000 Mc/sec experimental curve would look in the absence of saturation which occurs in these experiments at this frequency. Another dashed line shows approximately how the 72,000 Mc/sec data would look in the absence of the reflected wave of the wrong circularity [after J. K. Galt, W.A. Ysger, F. R. Merritt, B. B. Cetlin, and A. D. Brailsford, Phys. Rcv. 114, 1396 (1959)j.
method gave less information than the microwave bridge detector technique which provided a signal proportional to the power absorption. Figure 31 gives both the theoretical and experimental variation of the power absorption coefficient with magnetic field in pure bismuth with the magnetic field normal to the sample surface and along a twofold axis. When the magnetic field was placed normal to the sample surface but along the trigonal axis the absorption coefficient varied as shown in Fig. 32. The effect of adding tin to the bismuth, which increases the hole density to a value five times that of the electrons, is given in Fig. 33 for
324
BENJAMIN LAX A N D JOHN 0 . MAVROIDES
the magnetic field parallel to a twofold axis and normal to the sample surface. The cyclotron resonance masses were estimated from the absorption curves on the basis of the following premise: since WT >> 1 in the pure crystals at high frequencies, the extrapolation of the curve at high 8, a 10‘ IN WEBERSIM~ H IN OERSTEDS 172.000 k/sac.) -18 -12 -6 0 6 12 l6xtO’
x10-2
523 4.65
2
4.06
3
0
-
I
a
-H 3
c
0
rd 4- r4.65
5
3.48
;
; 2
290 4.06 346
2.32
k
$! 290
1.74
s
-5 2.32
1.16
&
5 1.74
0.58
2
*
0.00
416
5 0.58 z
2 0.00 44
-12 -10 -6
-6 -4 -2 0 2 4 6 8 H IN OERSTEDS (24.000 Mc/src) 8, a lo4 IN WEBERSIM~
10
12
0
a
z*!::
14a10’
FIG.32. Power absorption in bismuth with H normal to sample surface and along a threefold axis. Vertical arrows indicate cyclotron resonance fields. Zero abeorptivity for experimental curves is offset vertically from that of the theoretical curves. Dashed line shows how the 24,000 Mc/sec experimental curve would look in the a b e n c e of ssturation which occurs in these experimenta at this frequency [after J. K. G d t , W.A. Yager, F. R. Merritt, B. B. Cetlin, and A. D. Brailsford, Phys. Rcv. 114, 1386 (1959)j.
fields will intersect the axis of zero absorption at a magnetic field corresponding to cyclotron resonance as shown in Fig. 34. Based on this concept and detailed analysis of the data the results obtained for bismuth gave values of effective masses valid to within 10%. For the electrons from the ellipsoidal model for
325
CYCLOTRON RESONANCE
M IN OERSTEDS (24000 Y c h K ) B, x 10' IN WEBERS / Ma
FIG.33. Power absorption i n alloy of bismuth and tin with H normal to sample surface and along a twofold axis. Amount of tin present is such that there are five times aa many holes as electrons. Zero absorptivity for experimental curvea is offset vertically from that of the theoretical curves. Dashed line shows how the 24,000 Mc/ sec experimental curve would look in the absence of saturation which occurs in these experiments at this frequency [after J. K. Galt. W. A. Yager, F. R. Merritt, B. B. Cetlin, and A. D. Braigford-Phys. Rev. 1UI 1396 (1959)j.
I w I .
8 c.3
m
a X
d n FIG. 34. Theoretical plot of power abeorption coefficient versua H for cyclotron absorption in a metal with one group of isotropic carriers and the magnetic field normal to the sample surface [after J. K. Galt, W. A. Yager, F. R. Merritt, B. B. Cetlin, and A. D. Brailsford, Phyr. Rev. 114, 1396 (1959)j.
326
BENJAMIN LAX AND JOHN
a.
MAVROIDES
ma* = m[(mzma- m42>ml/mz]t= 0.0091m - a''*
L
3ml
+
mz
~
]
-
"I."
and similarly for the holes the masses are for
B 11 axis 3 B 11 axis 1 B 11 axis 2
m* m*
=
=
m(m,m2)i = mml
=
0.068m
m(m?m,)i = m(mlm3)+ = 0.25m
m* = m(mlmr)( = m ( m z m J )=~ 0.25m.
From these results the values of the effective m-8 in the ellipsoidal coordinate system may be calculated and these are given in Table I along with the results of Shoenberg as obtained from the de Haas-van Alphen effect. TABLEI. ELLIPSOIDAL MASSPARAMETERS FOR BISMUTH [After Galt el al:]
For elwtrons
Galt et al.0 Shoenbergb
0.0088 0.0024
1.80 2.50
0.023 0.05
k0.16 +0.25
For holes
Galt el al.0
0.068
0.068
0.92
J. K. Galt, W. A. Yager, F. R. Merritt, B. B. Cetlin, and A. D. Brailsford, Phy6. Rev. 114, 1396 (1959). * D. Shoenberg, Proc. Roy. Soc. A170,341 (1939); Phil. Tram. Roy. Soc. A246,l (1952).
Antimony. Datars and Dexter'] have conducted similar experiments on antimony. They deduced their results from the peaks of the derivative curves which corresponded to the inflection points of the absorption curves. It was found that the energy bands are very similar to those of bismuth, that is, the electrons move on tilted ellipsoids and the holes on an ellipsoid of revolution with major axis along the trigonal direction. The results from cyclotron resonance for electrons agree quite well with those from the de Haas-van Alphen effect,lo8 namely ml = 0.05m, lo' D.Shoenberg, Physiccr 19,791 (1953).
327
CYCLOTRON RESONANCE
m 2 = 1.00m, m3 = 0.52m,and mr = -0.65m. The mass components of the ellipsoid for the holes are nl = m2 = 0.021m and ms = 0.032m. Graphite. Perhaps the most dramatic demonstration of the cyclotron absorption technique in metals were the results obtained on graphite by Galt and co-workers.a8.61 The curves shown in Fig. 35 indicate a b r p tion with nearly equal concentrations of holes and electrons, and are consistent with the theoretical analysis of this type of absorption in pure b i ~ m u t h . * The ~ * ~ curves ~ bear a striking resemblance to the claasical analysis as represented by Fig. 27 in which was plotted the resulta for a majority and a minority carrier of the same sign. The analysis indicated that the intensity of the derivative peak is higher and the line is apparently narrower for the minority carriers than for the majority carriers
FIG. 35. a, Derivative of the power absorption versus H for circularly polarired radiation at 24,000 Mc/sec at normal incidence on (00.1)plane of graphite at I.1"K; b, Expanded plot of the low-field data of (a)[aft.erJ. K. Galt, W. A. Yager, and H. W. Dail, Jr., Phyd. Rm. 108, 1586 (1956)).
even though the concentration of the former is less. This interpretation was given by Lax and Zeiger (L-Z)aBwho showed, from the de Haas-van Alphen data on the masses and the energies, that the mean free path and skin depth were of the same order of magnitude. This indicated that the classical theory is essentially applicable to this problem and that phenomena associated with the anomalous skin effect are not important. The argument goes as follows: since the surface of the crystal which was exposed was the hexagonal plane, it is necessary to estimate the mean free path 1 in the hexagonal direction and compare this with the penetration depth 6. The de Haas-van Alphen data for graphite'O8 indicate two masses in the hexagonal plane, which are 3.6 X 10-'m and 7 X lo-%, and a mass of -10%~ along the hexagonal direction; also Fermi energies of ev, and carrier densities of -10" - 1O1a/cma are indicated. Consequently a t the Fermi surface one calculates the velocities ut 3 X lo' cm/sec and 111- 5 X loKcm/sec, where t and 1 refer to the hexagonal
-
328
BENJAMIN LAX AND JOHN 0. MAVROIDES
directions, respectively. The penetration depth d (which for wr >> 1 equals c/wp, where the plasma frequency wp = \/ne¶/m*co, c o being the dielectric
-
constant) is found to be -lo-* cm for the smallest mass value of the majority carriers and with n 1018/cma. From cyclotron resonance at 24,000 Mc/sec, a value for LJT of 2 to 3 is indicated so that r t , the relaxasec. tion time for the transverse direction, is approximately 2 X If we assume that the collision time is nearly isotropic, i.e., T I = T ~ the , mean free path I = o m along the hexagonal axis would be about lo-' cm. TABLE11. EFFECTIVE MAEBEEI N GRAPHITE [After Soule:] Measurement
m.*/m
mk*/m
Reference
de Haegvan Alphen Cyclotron resonance
0.036 0.031
4.07
Oscillatory galvenomagnetic effects
0.030
Shoenberg' Gelt ef d.8 Norierd Soule'
0.086 0.060
D. E. &ule, Phys. R w . 114, 708 (1958). D. Shoenberg, Physica 19, 791 (1953). J. K. Galt, W. A. Yager, and H. W. Dail, Jr., Phys. Rcv. 103, 1586 (1956). P. Nosiereu, Phys. R w . 108, 1510 (1958).
On the other hand if we assume, as in bismuth, where the energy surfaces are anisotropic, that T is anisotropic and follow the approximate relation which was found for that material, namely that T I / T ~ (ml/mr)i,then TI T,xp(ml/mt)+ 1Wpsec. This gives for the mean free path a posaible upper limit, probably too high, of -5 X lo-' cm which is essentially the same order of magnitude as the skin depth. Consequently, graphite appears to be a material to which the classical analysis of cyclotron absorption can be applied with reliable results. With this in mind Lax and Zeiger deduced that the charged carriers had the following m m values : m,*/m m*/m majority carriers 0.05 0.07 minority carriers 0.015 0.028
-
-
-
The existence of minority carriers also has been indicated by the work of SoulelO@who carried out extensive measurements of the oscillations of the Hall effect and also of magnetoresistance. His summary of the effective masses of the majority carriers as determined by different methods are reproduced in Table 11. Soule associates the minority carriers with the warping of the energy m D. E. &ule, Phyu. R w . 112,708 (1958).
CYCLOTRON RESONANCE
329
surfaces which, according to the theoretical analysis of McClurello and Nozieres,'O are essentially elongated ellipsoids. The Brillouin zone of graphite consists of a doubly spaced hexagonal pill box in which the height of the zone is essentially equivalent to the reciprocal of two lattice spacings. This is due to the two nonequivalent layera in which the atoms
Fro. 36. The Fermi surface for pure graphite. The central surface containa holm and the outer surfaces contain electrons. The length-to-width ratio of each surface ia about 13. The trigonal anieotropy ir exaggerated for clarity [after J. W. McClure, Phya. Rev. 108, 612
(1957)l.
in the hexagonal lattice are alternately staggered so that the bonding directions of atoms in one layer make a 60"angle with those of the other layer in the hexagonal lattice. The band structure of a two-dimensional lattice has been worked out by a number of authors including Sloncreweki and Weh1'' who also considered the three-dimensional aspect of the 11. 11'
J. W. McClure, Phyu. Rw. 108, 612 (1957). J. C. Sloncww8ki, Ph.D. The&, Rutgem University (lW),unpubllhed; J. C. Slonclewski and P. R. W&, Phyr. Rev. 100,272 (1958).
330
BENJAMIN LAX AND JOHN 0. MAVROIDES
problem. These authors showed that the Fermi level lies a t the corners of the two dimensional zone in a I or antisymmetric band, on a doubly degenerate level. Because of the large interlayer distance, the interlayer interaction is small and the energy change in going to a three-dimensional crystal can be treated by considering wave functions of the tight-binding type in the k, direction. For changes in the k, and k, directions in the neighborhood of the zone edge the problem is treated to first order by using the k . p perturbation method. The results are rather complicated; 4
-4
-
FIG.37. Theoretical curve of the derivative of the power absorption dP/d€f v e m € inI graphite for WT 2.5. The two arrows mark the aingularitiee due to the rnbnty carriera [after P. Nozierea, Phys. Rcv. 108, 1510 (1958)l.
however the pertinent information can be illustrated by the threedimensional diagram of the Fermi surface, Fig. 36, as obtained by McClure and by Nozieres. The L - P interpretation differs from that of Nozieres in the identification of the resonance lines. The former indicated the existence of majority carriers which are associated with energy surfaces of the large dimensions but with different cross sections for the holes and electrone. In addition to the fundamental resonance lines for minority camera, they identified pronounced peaks in the Galt, Yager, and Dail data appearing a t 8 and 4 the field for the fundamental as eecond and third harmonica. However, it can easily be shown from the harmonic selection rules that 8 single contiguous surface with trigonal symmetry would not show a third harmonic. Consequently they deduced on a phenomenological b& that the intereection of the Fenni level with the energy surfaces muat ocm
331
CYCLOTRON REBONANCE
away from the edge of the zone and should be associated with anisotropy and warping as shown in Fig. 36. Nozieres carried out a detailed analysh from which he concluded that the effective masses for the holes range from 0.066 to 0.05m and those for electrons from 0.054m to 0. With the appropriate repreaentation of the energy momentum relation as a function of k, Nozieres evaluated the absorption, and also derivative curves such as shown in Fig. 37. In obtaining this figure the warping of the energy surfaces which are associated with the minority carriers was neglected. However, this effect was analyzed independently and it was TABLE111. CALCULATED A N D OBSERVED FIELDSFOR HARMONICS OF MINORITY CARRIER RESONANCES I N GRAPHITE [After Nosieres:] Harmonic Electrons
Calculated H(oe) Observed H(oe)
+1 +4 +7 10 -2 -6 -8
+ Holea
-460 -115 -66 -46
+230
-2
-65 -45
+a
+240 +91
+115
+460 120
-230
-230
+57
+1 $4
- 120
+a
+
*P. Norieres, Phye. Rw. lOg, 1510 (1958).
found that the warping does not shift or broaden the minority carrier absorption lines but rather gives rise t o harmonics. Because of the trigond symmetry of the warped portion of the energy bands, for circular polarization harmonics should exist only a t a frequency wc(l 3n), where n is any integer positive or negative. On this basis Nozieres analyzed the data of Galt and co-workers taking the fundamental a t Ho = 460 oersteds with the results given in Table 111.
+
b. Anomalous Skin Effect (Atbel-Kaner Efect) ( 1 ) Theory. In resonance experiments on a metal a t low temperatures, where the condition that w7 >> 1 is satisfied, the usual type of cyclotron resonance as observed in semiconductors, is not possible.”i* This is due to the small distance of penetration of the electromagnetic field into the metal; this distance is smaller than the mean free path for the electrons at these temperatures. Consequently the motion of the electron between For a theoretical treatment of the expected results see R. G . Chsmbera.”
332
BENJAMIN LAX AND JOHN 0. MAVROIDES
collisions is not localized in a region in which the electric field can be assumed constant over the mean free path. Hence it is not possible to relate the current density J and electric field E in the sample by a point u relation J = u * E. Let us consider the fundamental quantities involved. Since the first criterion for cyclotron resonance is satisfied, the medium is a dispersive one and we can on a semiquantitative basis treat the electrons as a semistatic electron gas. For a metal such as copper for which we shall assume that m+ = m and n *v 10aa/cm8we can show tfiat the penetration depth 6 equals about lo-' cm. In this same metal, which has a Fermi level of -7 ev, the Fermi velocity V F equals 1.6 X 108cm/sec, and for a pure sample at 4°K the relaxation time is between 10-oand lo-" sec; consequently, the mean free path 1 lies between lea to 10-I cm, i.e., we are in the anomalous region. In bismuth where the electron density is between lo1' and 101*/cm*and the effective mass is as low as 10-2m, the skin depth 8 lo-' cm. With the Fermi level a t about 0.018 ev, U P looand the mean free path I lies between lo-* and 10-l cm. Thus for bismuth the conditions correspond to the region of the extreme anomalous skin effect where 1 >> 6. The other condition with which we are concerned is the radius of the electron orbit relative to the skin depth. In copper the cyclotron radius r = ( U F / W ~ ) lo-' cm for a frequency of 23,000 Mc/sec; for bismuth, because of its small effective m w , the cyclotron radius, for the same frequency, is -6 X lo-' cm. In either case 6 < < r < 1. The chief consequence of the condition in which the radius of electron orbit is larger than the skin depth is that, for the magnetic field H parallel to the surface of the metal, the electron which executes helical motion around H circles in and out of the anomalous skin region, but does not drift out of it. If the rf electric field is then placed either parallel or perpendicular to HI some electrons can come into the proper phase to gain energy in a manner analogous to that in which energy is gained across the dee's of a cyclotron. Furthermore, the possibility exists that as the electron rotates in its magnetic orbit the constructive phase relation between the electron and the electric field may be such that the latter may have gone through 1, 2, 3, or . . . n cycles during a single rotation of the electron and yet impart energy resonantly to the electron. This means that there would be peaks in the absorption at subharmonic frequencies such that o = nuc. In describing this situation we have considered essentially electrons which are moving with the Fermi velocity in a plane perpendicular to the magnetic field. Actually the distribution of velocities is over all angles; consequently it is the component of velocity in the perpendicular direction which is of significance. Those electrons which have velocities very nearly parallel t o the magnetic
-
-
-
333
CYCLOTRON RE60 NANCE
field give no contribution but remain essentially within the skin depth. The electrons which contribute most to the absorption are those with the largest possible orbits or zero velocity component along the magnetic field. J. C. Phillips”* has considered the relation of the absorption to the properties of the energy surfaces; he has shown more rigorously that the electrons for centrosymmetric surfaces with k H = 0 contribute most significantly to the absorption. Furthermore he has demonstrated that such electrons will produce resonant absorption even when the magnetic field is tilted relative to the surfaces of the specimen. The anomalous skin resonance effect was predicted by Azbel and Kaner (A-K)64 who solved the Boltzmtrnn transport equation; further by the use of Maxwell’s equations they obtained an expression for the impedance of the metal surface and predicted the existence of harmonics. The same results were later obtained by Heine63 using the “ineffectiveness concept’’ of Pippard’l’ and more recently by Mattis and Dresselhaus (M-D)64 and also Rodriguez.a6 We shall follow the treatments of the latter two since they are somewhat simpler than that of Azbel and Kaner. For simplicity they assumed as the boundary condition specular reflection instead of diffuse scattering for electrons a t the surface, even though it has been demonstrated that the latter condition is closer to reality. The justification for this approach is not only that it simplifies the mathematics but that the essential fentures of the phenomenon are satisfactorily described. The Boltzmann transport equation applies in this case and can be written in the following form: (4.21)
where f o is the Fermi distribution function, Ho is the dc magnetic field, v is the velocity of the electron, and j is the distribution function given by fo f l ;f l is the time-dependent component of the distribution function and is assumed to have the same time dependence as the electric field, namely ewt. The distribution function is substituted into the Boltsmann equation which is then linearized and expressed in spherical coordinates v, 8, and 4 in velocity space with the z coordinate or polar axis coinciding with the magnetic field Ho;in this case HOis taken to be parallel to the surface of the metal. The Boltzmann equation can then be rewritten as follows:
+
(1 112 1~
+ iut)fl +
OJ
+
1 af vr sin 8 sin Q af -1 =
a+
aY
-df ezE 0
v.
(4.22)
dE
J. C. Philiips, Phys. Reu. Letter8 3, 327 (1959). A. B. Pippsrd, Repl. 10th S o h y Coqf. on Solid Slate Phys. p. 123 (1954) ; Proc. Roy. SOC.A424, 273 (1954).
334
BENJAMIN LAX AND JOHN
a.
MAVROIDES
Next the quantities j(s), &(s), and @pl(s,B,+), the Fourier transforms of f l respectively are introduced, the linear partial differential equation in a1 equivalent to Eq. (4.22) is solved,l1W and the current density J is evaluated from
J, E, and
or the Fourier transform equivalent (4.24)
where
is defined by (4.25)
The principal assumption which is involved in integrating over the distribution function in k space is that, a t these low temperatures, which are much lower than the Fermi temperature, -dfo/d& = b(& - &), where & P is the Fermi energy and b the Dirac delta function. From Eq. (4.24) one obtains the conductivity tensor which relates j(s) to &(s), or J to E. The components of this tensor are expressed in series form, the h t terms of which are given as follows:"
.. L
-I
u.,, = - u y z =
+
.1
+
* .
'I
(4.26)
+
where y = (1 ~ C N ) / W ~ T a, = sl/(l iwr), I = U F T , and u is the bulk electrical conductivity of the sample. In order to find the impedance 2 of the surface, it is necessary to use Maxwell's equations within the medium and, thus, the following result is obtained:
solution is simplified by assuming specular scattering. In general one must write the Boltzmanrl transport equation in two parts, one for electrons with velocity components in the positive direction and one somewhat altered for electrons with negative velocity components. For specular reflection, however, one can assume a mirror image of the metal for negative valuea of y and the same Boltcmann transport equation holds throughout all apecea with the boundary condition t h a t at E = 0 the electric field is continuous and hae a cusp, i.e., E'(+0) = -E'( -0).
Ilao This
335
CYCLQTRON REBONANCE
where the argument ( + O ) indicates the limit is to be evaluated for positive values of y. Using Eq. (4.24) and Eq. (4.26) it is then possible to eliminate j(s) in Eq. (4.27) and thus evaluate &(s) and then E(y) by taking the inverse transform. Also from Maxwell's equations we can obtain the component of the rf magnetic field H and hence evaluate the surface impedance both for longitudinal and transverse cyclotron resonance respectively as follows: (4.28)
4x E,(O) 22 -- ----=---. c HJO)
4riO E,(+O) cz Ed( +O)
(4.29)
Considering first the longitudinal case and taking the electric field E parallel to the z axis, it is found that J. = uSaEa;solving Eq. (4.27) making use of Eq. (4.28) and the first term in the ua, expansion of Eq. (4.26) 2.
=
8iw/c2
lo" + ( 3 d i w / c 2 l ) coth ( r y ) ] - 5 ds.
(4.30)
[s*
Integration of Eq. (4.30) yields
Z.
=
Q
(qy
(1
+ i 4 3 ) tanhi (ry).
(4.31)
For the transverse case, we can show by writing out the two components of Maxwell's equations that the relation between the current density J, and electric field E, is given by J, = u,E.; for the case at hand ul = u,, (uw2/uuu)since the displacement current can be neglected. Rodriguez has evaluated this effective conductivity and has shown that uz, >> (uw*/uw). Thus, if only first-order terms are retained, the imped,dce again is given by Eq. (4.31). The real part of this surface impedance, R, which gives the power absorbed and corresponds to the quantity usually measured experimentally, has been evaluated by M-D" as
+
R = - lhrw cos
+ r)]
[+(a
349 (3r2ow/c*vlrz) coshz (?T/w=T) sin2 ( r w / w , ) cos2 ( r w / w c ) ) * x " [cos2 -smh2( r(wr //wwcC7)>cosh2 (r/wC.) sin2 ( r w / w , ) sinh* ( r / w c t ) ] *
+ +
1.
(4.32)
This expression which holds for parabolic energy bands is plotted in Fig. 38. Essentially this same result was obtained by A-KJ4who depicted this phenomenon as indicated in Fig. 39. The plot of M-D is more nearly representative of the experimental results, since these are usually carried out at a fixed frequency with the magnetic field as a variable. Figure 39, on the other hand, which is essentially a plot of the components of surface impedance as a function of the inverse of magnetic field, shows
336
BENJAMIN LAX AXD JOHN (3. MAVROIDES
0
.5
1.0
1.5
2.0
2.5
3w
FIG.38. Theoretical plot of R' = RfI&u/3tc1)-'(3*Coo/c4~)~,where R k the resistive component of the surface impedanre, versus W,/W for * 1 and 10. The (JT
first five harmonics are indicated by arrows. The fundamental and first harmonic are appreciably shifted toward lower magnetic fields. Thls shift remains even for longer relaxation times [after D. C. Mattis and G. Dresselhaus, Phys. Rev. 111, 403 (1958)]. 2
R(H)( R(O)
0
'
M
2
3
2
3
WC
2
x(x) x (0)
0
t
W
WC
b
FIQ.39. Theoretical plot of: a, the resistive component; and b, the reactive component of surface impedance as a function of (u/uc)for various values of UT [attar M. I. Aabel and E. A. Kaner, Soviet Phys. JETP I,730 (1957)l.
337
CYCLOTRON RESONANCE
a periodic variation for this phenomenon. It is convenient to analyze the data in this way. It is interesting to note that, for ellipsoidal masses and high w7, the minima corresponding to the resonance are shifted by approximately 10% to lower magnetic fields. Also it may be seen that the peaks have a shape asymmetry which decreases with increasing order of the harmonic. Asymmetrical lines of this sort have been reported for copper by Langenberg and Moore." Azbel and Kaner"' have extended their original surface impedance calculations. They have studied the magnetic field and temperature dependence of the surface impedance tensor, assuming an arbitrary energy-momentum relationship and collision probability, and have shown that, in principle, the shape of the Fermi surface and the electron velocities on this surface can be determined completely from the experiments. A set of rules is given for making qualitative deductions about the surface from experimental results without extended analysis. Some of the important conclusions are as follows. For an ellipsoidal surface, the fundamental resonance should be observed for an arbitrary direction of magnetic field H relative to the crystal axes, but parallel to the surface, and an arbitrary polarization of the rf field a t a frequency corresponding to the expression of Eq. (3.10). The relative depth of the resonance is given by
R"/R(O)
-
(w)-i;x.-/X(O)
-
(wr)f.
For surfaces which are not ellipsoidal, resonance occurs for extremal values of the effective mass, m* = (1/21r)(dS/d&).,~, with respect to the projection of the momentum on H, i.e., p~ = p H/IHI. Resonance at frequencies corresponding to the elliptical limiting points of a surface, i.e., portions of a closed surface for which the V,& is along H, occurs only when the rf field is parallel to H. Resonance corresponding to a central section of the Fermi surface occurs only when the rf field polarization, the direction of motion of the electrons involved and the surface of the metal are all parallel. Resonance due to an extremal mass neither at the limiting points nor the central section will be observed with any direction of rf polarization. The relative depth of the resonance differs for maximum and minimum values of the effective m=. For a maximum effective mass
-
R"/R(O) '14
-
(w)-';
X"/X(O)
-
(wr)-+
M. I. Aebel and E. A. Kaner, J . Phys. Chem Solids 6, 113 (1958); E. A. Kaner, and M. I. Acbel, J . Ezptl. Theorel. Phys. (U.S.S.R.) 88, 1126 (1958); E. A. Kaner, ibid. 85, 962 (1958); M. I. h b e l , J . Phys. Chcm. Solids 7, 105 (1958).
338
BENJAMIN LAX A N D JOHN Q. YAVROIDEB
and for a minimum effective mass
R"'/R(O)
-
(w)-t;x"./X(O)
-
(or)+.
Away from resonance, the impedance depends only weakly on the form of the energy-momentum relationship. For magnetic fields just above the resonant field with H strictly parallel to the surface, and for 8 quadratic energy-momentum relationship 2 = Z(O)[1 - exp ( - 2 r i w / o ,
- 2r/o,r))b.
(4.33)
For strong magnetic fields [oc>> ( 2 r / r ) ,2rw] the impedance is independent of the shape of the energy surface and is given by 2
-
+ o V ) *exp i + arc tan w ) / 3 .
H-'d(l
(T
For high frequencies (w >> 1) and intermediate magnetic fields [2rw << << (r/2)u+],the impedance components are
wc
R
-
w2H-1; X
-
wH-1.
There is practically no variation of the impedance with magnetic field if the field is not exactly parallel to the surface of the metal. For fields much lower than the resonant field the impedance changes by a few per cent at most. For very weak fields
Z/Z(O)
-1
-
H2.
In another paper, Kaner116has discussed the possibility of cyclotron resonance in a metal with the magnetic field inclined to the surface. He expands the surface impedance in powers of (~5.,~/r)rwhere the effective (Pi)'and the remaining variables are skin depth, Cen = I(c2/4w)Z(0)l as previously defined. Kaner finds that the impedance is independent of the magnetic field to zeroth power in (cL/r)+, nonresonant to first and second powers and weakly resonant (for certain polarizations) in still higher terms. The magnitude of this resonance is small because it is produced by the few electrons which return many times into the akin depth. Kaner's results are valid for the angle between the magnetic field and metal surface not close to zero or to r / 2 , i.e., sin 241>> i L / r , Also considered in this paper is the surface impedance Z(H)for an arbitrary reflection law of electrons according to which the fraction p is reflected specularly while the remainder are distributed after reflection in the equilibrium distribution fo(E). Detailed calculations show that, for all values of p # 1, the surface impedance near resonance or in the highfield region is independent of p and the earlier impedance results" are valid. For p = 1, i.e., specular reflection, the impedance haa a non-
-
11)
E. A. Kaner, J . Ezptl. Theoret. P h y . (V.S.S.R.) 88, 1135 (19s).
339
CYCLOTRON RESONANCE
-
resonant dependence on field to zero order in (&,/r)t, the resonance appearing only in higher order terms. For strong fields, Z ( H ) H-i for p = 1 and Z(H) H-1 for p # 1. Kaner also has considered the theory for resonance in thin films.116 (2) Experimental observations. The first evidence of the effects predicted by A-K was presented by Fawcetts6 who carried out experiment8 on single crystals of tin and copper a t 24,000 Mc/sec and 4.2”K. He measured the absorption in the sample located at the end of a wave guide through which microwave radiation of two principal modes could be propagated so that the electric field could be made perpendicular or parallel to the surface. His detection of absorption was by means of a sensitive carbon composition thermometer attached to the back of the specimen a t the end of the wave guide. Fawcett interpreted his data from the first maximum and suggested that in tin the mass varies from 0.23 to 0.43m. In copper, the lines were so broad that no quantitative estimate of the mass could be made. Similar effects in bismuth were independently in experiments in which the magnetic field observed by Foner el d.J117 was parallel t o the surface. They used the derivative technique which can be shown theoreticallly to accentuate the oscillations a t low fields and minimize those a t high fields. Their results, however, were not analyzed quantitatively. Quantitative results on bismuth for both the absorption and dispersion were presented by Aubrey and Chambers (A-C)“ and compared with the somewhat simplified theoretical expression [Eq. (4.33)] for the surface impedance due to Azbel and Kaner. This comparison is shown in Fig. 40.These results were obtained at 9OOO Mc/ sec with single crystal rods of bismuth forming the center of a coaxial cavity similar to that used by Pippard in his studies of the anomalous skin effect. They interpreted their results in terms of the ellipsoidal model of Shoenberg as deduced from the de Haw-van Alphen data. The results were in essential agreement; however the numerical values differed somewhat, in particular Aubrey and Chambers found m4 = O.llm in contrast to the Shoenberg value of 0.25m. A-Cb7suggest that this discrepancy could be due to a slight misorientation of the Shoenberg ~pecimen.1~” The effective masses obtained by them gave the results
-
ml = 0.006m,mt = 1.0m,m, = O.O2m, m4 = 0.llm.
Kip and co-workerss8 also carried out experiments on tin with the 11‘
E. A. Kaner, Soviet Phya. Dokludy 1,314 (1958). Foner, H.J. Zeiger, R.L. Powell, W. M. Wslsh, and B. Lax, Bull. Am. Phya.
117s.
lI1.
Soe. [2] 1, 117 (1956). For a more detailed comparison with values obtained using other techniques, nee Table I.
340
BENJAMIN LAX A N D JOHN 0. MAVROIDES
magnetic field parallel to the surface of the sample. They used coin-shaped single crystals of very pure tin and rotated the sample relative to the magnetic field so as to observe the effects of anisotropy. Using the derivative technique they obtained along certain crystallographic directions as many as 15 subharmonics. l h m the results, they determined effective masses ranging froni 0.2m to 0.3m, and in several orientation8 the patterns indicated the superposition of harmonics due to two or more masses. More recently, Galt and colleaguesb2 reported cyclotron resonance effects in zinc with the magnetic field both parallel and perpendicular to 0.3 7
1
n ( OOU88 FIQ.40. Variation of surface resistance and surface reactance of bismuth with magnetic field, for (31, 21) orientation. Experimental pointa (lefthand ordinate scale): 0 AR/R(O); 0 A X / R ( O ) . Theoretical curves (right-hand ordinate scale): -AR/R(O); - - -AX/R(O). Plotted from Eq. (4.33), assuming z(0)= e"", w 3, and m * / m = 0.11 [after J. E. Aubrey and R. G . Chambers, J . Phya. C h . Solids 1, 128 (1957)].
-
-
9
the sample at about 1.3"K and at both 24,000 Mc/sec and 72,000 Mc/sec. I n the parallel case with the magnetic field in the hexagonal plane and along a sixfold axis they observed harmonics as shown in Fig. 41. They interpreted their results in terms of the A-K theory and obtained a ma88 of 0.55m f 5%, 02 20, and a phase of e-ir'a in Z(0). Also from Fig. 41 there are indications of absorption at low fields associated with the small effective mass of 0.015m which was found from measurements of the de Haas-van Alphen effect. With the magnetic field along the twofold axis they obtained a mass of 0.43m k 10%. Bezuglyi and Galkinsocarried out cyclotron resonance measurementa in the 10,OOO Mc/sec region on single crystal wires of tin and lead. The
-
CYCLOTRON RESONANCE
341
specimen was placed along the axis of a coaxial copper resonator in the manner of Fawcett. Measurements of the wire surface resistance were taken a t 4.2"K and a t 2°K and the effective masses were determined from the positions of the minima. For tin the effective mass m* of the electrons turned out to be m ' = m in agreement with the estimate of Borovik118from galvanomagnetic phenomena. A second much weaker minimum suggested possibly another set of electrons with mass m* = 0.25m. In lead at 4.2"K they found that the resistance dropped off monotonically with the magnetic field. Decreasing the temperature to 2°K increased the relaxation time so that a deep resonance minimum * followed by a maximum was found. From the resonance minimum they obtained an effective mass of 0.8m.
H IN OERSTEDS (72poO MCISEC) 112 x d IN WEBERS/Y'
FIG.41. Cyclotron resonance at 1.3"K in zinc with the magnetic field in the sample plane and along a sixfold axis (after J. K . Galt, F. R. Merritt, W. A. Yager, and H. W. Dail, Jr., Phys. Rev. Letlere 1, 292 (1959)).
The experimental results for copper,6O and have been compared with theory by Azbel and Kaner"' and in all case6 it is found that there is qualitative agreement between experiment and theory. Cyclotron resonance in aluminum has been observed by Langenberg and Moore (L-M)O0 and also by Fawcett8' under anomalous skin conditions a t 24,000 Mc/sec and 36,000 Mc/sec respectively. Decided anisotropy was observed by both groups and a t 4.2"K the lines indicated an WT 10. L-M found an oscillation corresponding to a mass of 1.5* with no noticeable anisotropy. In addition they found a small mass of 0.18m with anisotropy of the order of 50%. According to their interpretation the masses are associated with holes located a t the corners of the first Brillouin zone. This interpretation is in accordance with the theory of
-
118E. S. Borovik, Doctoral Thesis, Inat. Tech. Phys. Aced. Sci. Ukrainian S.S.R. (1954).
342
BENJAMIN LAX AND JOHN G. YAVROIDEB
band structure by Heine119and the observations of Gunnereenl*ofrom the de Hass-van Alphen effect. GM6O associated the high mass resollsnce with the major part of the Fermi surface which is located in the second zone. Fawcett found effective masses ranging from 0 . l m t o 0.4m. Thew. results are qualitatively consistent with those obtained by L-M. The
0
4
8
12
H (hllo- o w rtrdrl
FIG.42. Cyclotron reeonance in copper a t 24,470 Mc/sec. The upper panel, a, 8 plot of reciprocal magnetic fields a t the absorption derivative maxima against integers; m* = c/[ocA(l/H)], where A(l/H) is the slopeof the 1/Hplot. Note the large “phase” discrepancy between the two nearly equal massea. The lower panel, b, shows a recorder trace of absorption derivative for H in a (110) plane 10” from a [loo] axis. J,, in 8 (110) plane and 45 degrees from E,55 degrees from [loo]axis [afterD. N. Langenben and T. W. Moore, Phys. Rev. Letters 8,328 (1959)j.
-
former compares an effective mass m* 0.125m for holes with the value 0.15m found by Gunnersen for the same orientation. The early results6*J*lof cydotron resonance studies on copper were not definitive. Recently, Langenberg and Moore6* using very pun synthetic crystals of copper and improved techniques have obtained the well-resolved results shown in Fig. 42. As many as twelve subharmonies 11* V. Hehe, Proc. Roy. Soc. 1w0,340 (1957). IroE.M. Gunnersen, Phil. Trans. Roy. Soc. A949, 1 (1955). lrl D. N. Lungenberg, A. F. Kip, and B. Rmenblurn, Bull. Am. Phyu. Soc. [2] 8,418 (1958).
CYCLOTRON RESONANCE
343
were observed along certain orientations. Anisotropy studies were made with the magnetic field in the (1 10) plane. The data were taken with and without magnetic modulation but somewhat better data were obtained by the former method which is more sensitive. The mass varied from l.lm to 1.4m for most orientations. Values for the mass along the principal axes are given in Table IV. The value of 1.3m in the 11111 direction is in good agreement with that obtained by Shoenberg"2 from the de Haas-van Alphen effect. There were other mssses between the extreme limits of 0.5m to 5m observed along certain directions, but no detailed analyses of these were made. It is significant that these results are consistent with the model of the Fermi surface of copper which was proposed by Pippard**' on the basis of his study of the anomalous skin effect. In TABLEIV. CYCLOTRON MASSESFOR COPPERWITH H IN (110) PLANE TO PRINCIPAL CRYSTAL Axm PAEALLEL [After Langenberg and Moore:]
M W
Behavior of mess with reepect to orientation
(1.32f 0.02)m (1.30 f 0.03)m (1.12f 0.02)m
Maximum Increasing as H rotated toward [llO]axis Minimum
~~
"NI [I111
UlOl
D. N. Langenberg and T. W. Moore, Phys. Rev. LeuCrs 8,328 (1959).
general the masses are well behaved since they are associated with m n a n c e s of the nearly spherical part of the Fermi surface. Along certain directions, however, where the energy surface makes contact with the edge of the Brillouin zone the masses should have singularities. Indeed one such singularity was found in the (110) plane with the magnetic field 18" 1" from a [llO] axis. Another interesting result is that shown in Fig. 42a in which the "phase" of the oscillation is not consistent with that predicted by theory. The phase shift, which is easily determined from the intercept of the 1/H plot, was found to vary with magnetic field orientation. Discrepancies of the phase have also been observed for bismuth by Aubrey and Chambersb7and for zinc by Galt and co-workers.'*
*
5. MAONETOPLASMA PHENOMENA Theory Magnetoplasma phenomena in met:.!!: and eemimetale can have a very pronounced effect on the resonance. of carriers in such media. In a.
D. Slmenberg, N d u r e 188, 171 (1959). '**A. B. Pippard, Phil. Tram. Roy. SOC.A%@, 325 (19573. 1)'
344
BENJAMIN LAX AND JOHN 0. MAVROIDES
connection with metals we briefly indicated the behavior of the absorption under conditions in which the electron concentration was BO large that one could neglect the displacement current a t microwave frequenciea. However, in semiconductors even a t these frequencies and in semimetals a t millimeter and far infrared frequencies one must include the displacement current also. A number of phenomena appear under such conditions. These related phenomena have been described by a variety of terms, such as plasma resonance,J2 depolarising eff ect~'~-l24 and dielectric an~malies.'~.~' The first of these terms appears less appropriate than the latter two although each of them has been applied to different geometrical situations. The broad term of magnetoplasma phenomena seems to provide a more suitable description for these effwts which are a consequence of the behavior of an electromagnetic wave in infinite, semiinfinite, and finite bounded media. Let us first consider the situation for propagation in an infinite medium with a single carrier. If we neglect losses due to scattering, then a = 0 and the propagation constant 'I = iB where B* for longitudinal propagation or circular polarization is given by (5.1)
and for transverse propagation by
where 8-2 = w%po, and up2= ne9/mc. Equation (5.2) holds when the electric field E is perpendicular to the dc magnetic field, and Eq. (5.3) when E is parallel to the dc magnetic field. In terms of the propagation there is a cutoff frequency a t which B = 0, often called the plasma frequency, and corresponding to the solution which obtains either from setting & or B1 = 0:
Expressions (5.1) and (5.2) also show singularities with 8 3 a.Thi occurs for positive circular polarization when w = wc and for B1 when w2 = cooz wpz. For the case of the semi-infinite medium that applies to , ~ ~ the experiments are the infrared cyclotron resonance W O T ~ , in~ ~which
+
Lax, in "Solid State Physics in Electronics and Telecommunicstiona" (M. Desirant and J. L. Michele, eds.), Vol. 3: Magnetic and Optical Properties, p. 508. Academic Press, New York, 1960.
I14B.
345
CYCLOTRON REBONANCE
done by reflection, one obtains the total reflection coefficient R, from the transmission coefficient T as given in Eq. (4.2), a.9
R where
= 1
- T = (-) B-Po'
+
(5.5)
B Bo is the free space phase constant and is given by Bo' = w*two.
-5
-2
-3
-4
0
-1
2
t
-
3
s
4
WC
W
t.0 ----7 -
0.8
1
-
( b)
I
I
I
0.6
a
-
0.4
-
I
-
I
I
I
0
1
2
I
3
1
4
.
5
0
1
2
I
I
I
3
4
s
WC W
Fxo. 43. Magnetoplasma effects. Variation of reflection coefficient R versus a, circularly polarized longitudinal propagation; b, linearly polarized longitudinal propagation; and c, transverse propagation (EI&) with no Ionsea and (o0/u) for :
e
= 16.
From Eq. (5.5) it ia seen that total reflection occurs for @ equal either to 0 or a.Since away from the singularitiea [@I is usually >>,30, the other condition of interest, R = 0 or B = pol falls close to the condition for cutoff and is hardly distinguishable from it. If we neglect losses it ie a relatively simple matter to study the reflection coefficient as a function of magnetic field using Eqs. (5.1) and (5.2). The resulting relation,
346
BENJAMIN LAX AND JOHN G . MAVROIDEB
for the magnetic field perpendicular to a semi-infinite plane and for two senses of circular polarization, is represented in Fig. 43a. The plot for negative and positive fields corresponds to left-handed and right-handed circular polarization, respectively; the latter corresponds to the sense of polarization for cyclotron resonance. Using this sense of polarization we find that for electron concentrations such that up < w , the reflection coefficient a t zero magnetic field is reduced below that corresponding to 'the dielectric medium without carriers. Then as the magnetic field is increased the reflection coefficient goes to zero, corresponding to the condition of unit index of refraction. This value of magnetic field, as found from Eq. (5.1), is given by wc/w =
1-
__- (w,/w)*.
(c:
1)
The reflection coefficient then increases abruptly to total reflection, corresponding to cutoff with zero index of refraction. From Eq. (5.1), it follows that this occurs for wc/o =
1
- (w,/w)'
(5.7)
which for the usual semiconductors or semimetals, where c 2 10, lies just above the zero reflection. Cutoff or total reflection persists until wc = w at cyclotron resonance where the index of refraction changes abruptly to infinity. Above this field, R decreases with increasing field and approaches the asymptotic value for the pure dielectric. Hence at very large fields a t which the frequency is well above the cyclotron fre guency it is possible to determine the dielectric constant c from measurements of R. In the region for left-handed polarization, where the magnetic field is reversed, the reflection coefficient rises again to the asymptotic value at large magnetic fields as shown in Fig. 43a. When the frequency is lower than the plasma frequency, i.e., o < w,, the medium is totally reflecting a t zero field, since the index of refraction is imaginary until the magnetic field corresponding to cyclotron resonance is reached; then the refractive index becomes real and again R drops and approaches the asymptotic value. A unique feature of the circular polarization configuration is that for wp > w the sharp resonance edge occurs for negative values of the magnetic field, or for the opposite seme of polarization thsn for wp < o as shown in Fig. 43a. For linear polarization with the magnetic field perpendicular to the surface one obtains 8 resultant for the reflection coefficient R which equals the sum of the two counter-rotating circularly polarized modes &s represented in Fig. 43b. The situation for the magnetic field parallel to the surface can be analyzed in a similar manner; we shall consider only the case in which the electric
347
CYCLOTRON REBONANCE
field is perpendicular to the static magnetic field. As shown in Fig. 43c, the curve looks similar to that for the previous cases with the conditions for zero R being given by an expression analogous to Eq. (5.6) but obtained from Eq. (5.2). The condition for total reflection, when the index of refraction equals zero, is again given by Eq. (5.7) and holds until the magnetic field is sufficiently high to give an infinite refractive index. For (w,/w)~ >> 1, the refractive index is imaginary until the magnetic field is sufficiently large so that the index of refraction is zero, i.e., ( u J u ) ~= [(w,/w)~ 11'. Curves for the reflection coefficient when losses
-
I
0
I 2
I
I
I
4
I 6
I
I 8
l
l (0
%" FIQ.44. Magnetoplasma effects. Plot of the reflection coefficient R as a function for circular polarization and various values of the parameter (wp/w)*. Here e = 9 and ur 10 [after H. J. Zeiger and S. Hilsenrath, Lincoln Laboratory QPR, ( h u p 35, p. 54 (February 1957)].
of
wc/o
-
are included have also been calculated126 and are shown for the case of circular polarization ( + H ) in Fig. 44. They demonstrate the essential features discussed when losses are neglected, namely, that the sharp rise near cyclotron resonance shifts to lower magnetic fields with increasing up/@as indicated. These particular curves have been worked out for the rather large value of WT = 10. In some of the experiments which have been carried out WT has not been as large and the maxima and minima have not been as pronounced. l r r H . J. Zeiger and S. Hilsenrath, Lincoln Laboratory QPR, Group 35, p. 5L (1 February 1957).
348
B E N J A M I N LAX A N D J O H N G. MAVROIDES
At microwave and millinieter frequencies experiments are usually done with resonant cavities which essentially fix the frequency at which the spectrometer must operate. I t is therefore necessary to sweep the magnetic field through resollance or the appropriate singularities which then determine the resonance condition. However, at infrared frequencies where gratings and prisms are used it is advantageous to vary the wavelength or frequency and fix the magnetic field. Under conditions such that the frequency is much higher than that of the plasma frequency no unusual effects are encountered if the magnetic field is sufficiently high to obtain resonance. However, when the plasma frequency approaches or becomes greater than that of the electromagnetic field, it is possible to 1
FIQ.45. Theoretical curves for the magnetoplasma effect of isotropic carriers for
>> 1 and wc < u p . ( w C / w p= 0.2.) The energy splitting of
the plasma edge for both longitudinal and transverse propagation equals ha, [after B. Lax and G . B. Wright, Phys. Rev. Lefters 4, 16 (196O)l. wr
use the technique of varying the frequency to good advantage. Let us first consider the limit in which the magnetic field is too small t o obtain cyclotron resonance and wC << w p . Then Eq. (5.4) holds and it can be shown that Wf
z up f
w
-: 2
w2 + 82. wp
The significance of this result is that, a t a frequency slightly above the value it gives, when the index of refraction goes to unity, the reflection coefficient of either circular component for longitudinal propagation and of the linear polarization for transverse propagation goes to zero. Furthermore two reflection minima have appeared whose separation is given by the cyclotron frequency as shown in Fig. 45. Indeed this effect has been observed in indium antimonide and mercury selenide by Wright”’ and l*n
B. L a x and G. B. Wright, Phye. Rev. Leffers 4, 16 (1960).
349
CYCLOTRON REBOXANCE
it has been used to evaluate the effective mass of the electrons in these materials. Lax and Wright*2ohave carried out the analysis for semiconductors with more complicated energy surfaces and have shown how the technique can be utilized for energies in the linear and quadratic region to determine the mass parameters of such surfaces. The other limit that we wish to analyze is that in which wc > wp and where it is possible, as with frequencies in the far infrared, to go through cyclotron resonance. Again it is instructive to consider the longitudinal and transverse cases separately. For the former it can be seen from Eq. (5.1) that the reflection coefficient has a dominant minimum or zero I 100
90
eo
I#.
50 i-
2
pw a
40
30 20 10
0
45
-
Fro. 46. Theoretical curves for the rnagnetoplasma effect of isotropic carriers for OT >> 1 and wI > u p (w,/w, 5.) The curve on the right shows the magnetoplasma shift of the cyclotron resonance reflection edge corresponding to the value given by the curve at the left which is the magnetoplasma resonance wP1/oe [after 8. Lax and Ci. B. Wright, Phye. REV.Lefteta 4, 16 (1960)l.
+
reflection at the frequencies corr.esponding respectively to o 2: we (wp*/o,) and o = ( w P z / w , ) . This is shown in Fig. 46. The separation is again given by the cyclotron frequency. In addition, since one can now fix the plasma frequency from the position of the first minimum, and the dielectric constant from the reflection coefficient at high frequencies, it is possible to determine the electron density. Similarly the situation for transverse propagation cnn be analyzed and once more we obtain two minima with some structure. As in the previous case one can determine the mass parameters. Of course, the analysis becomes more complicated for anisotropic surfnces. However, in the limit of o,<< wp the resonant singularities corresponding to the plasma edges as well as the other minima will show anisotropic effects
350
BENJAMIN LAX AND JOHN G. YAVROIDES
which provide information on the energy bands of the semiconductor or semimetal under study. The intermediate case in which the cyclotron and plasma frequencies are of the same order of magnitude is not so attractive either for experimental or theoretical studies as the two limiting cases. Thus, by design, the experimentalist can select either limiting case as the suitable one for a given situation. However, discussion of pertinent experimental results will be deferred until the infrared work is considered. Next will be considered those magnetoplasma effects which have been encountered experimentally with semiconductors such as indium antimonide,aP+28 a t microwave frequencies.lZ7These effects are a result of depolarizing charges built up by the free carriers in the surfaces of samples whose dimensions are small compared to a wavelength or the effectiveskin depth, whichever is smaller. The conductivity tensor in an isotropic plasma is obtained from the equation of motionso which yields for the rf component of the velocity: m*(v
+i w ) ~
=
e(Ei
+ v x H/c)
(5.9)
where v is the collision frequency, i.e., l / r , and E,the internal field in the (+
medium is related to the external field E by the depolarizing tensor L as follows: (5.10)
where xo is the dielectric susceptibility. When the equations are solved for the velocity components, one can obtain the conductivity tensor whose components are (v
+ iw) -
1 1 (5.11)
where A =
(v
+
iw)2
- i(v + iw) LZL,
- _wPo4 _ w2 I*’
(1
+ L,xo)(l + Z,Z)
B. L a x and I,. M. Fbth, Phys. Rev. 98, 548 (1955).
+
35 1
CYCLOTRON RESONANCE
and
The geometries which are of interest are disks, thin flat slabs, or long cylinders. In each case the magnetic field may be transverse or parallel to the plane or axis of the sample respectively. In only one of these situations is the magnetoplasma effect eliminated. If the magnetic field is perpendicular to the face of a thin slab and the rf field is in the plane then L, = L, = 0, L. = 1 and the conductivities are the usual ones for the infinite medium. This results in an absorption which is independent
1
.8
0;
-/c*
.6
.4
.2
0
1
2
3
4
5
6
w‘/W
-
FIG.47. Magnetoplasma effect for a circular cylinder with the magnetic field along the axis. Here w+ = 1, W’ o - (u,,’)*/w hae the values shown. The absorption for linear polarization is proportional to u+ u- (after B. Lax and L. M. Roth, unpublished).
+
of the depolarizing factors. Of course, the other situations give rise to depolarizing or magnetoplasma effects. A situation which yields very interesting results for analyzing these geometrical effects is that in which a cylindrical sample with the magnetic field along its axis is used 88 shown in Fig. 47. In this case L, = 0, L, = L, = +, and
352
BENJAMIN LAX AND JOHN G. MAVROIDES
where up‘ =
@PO ____-
d2+0
and
A =
(V
4- i w
-
+ wC2.
~W,,’~/W)~
To analyze the phenomenon it is again convenient to consider the absorption of the positive and negative circularly polarized components. The results are shown in graphical form in Fig. 47 for different values of the plasma frequency or electron density as a function of the magnetic field. The plot represents the following analytical expression -1
(5.14)
where uo = (ne%/m*) = w ~ ~ % ~The T . startling result shown is that the negatively circularly polarized field reverses role with the positively circularly polarized field as the density goes above a critical value given by wpl = w. This is the phenomenon which DKKa2designated as magnetoplasma resonance. When wp‘ << w we obtain the usual cyclotron resonance curve shown in the diagram, which shows a peak for case of the positive circularly polarized field. Experimentally it is more convenient to use linear polarization for which the absorption curve is the sum of u+ and u-. In this case for w7 = 1 no resonance peak occurs; thus to observe cyclotron absorption it is necessary that WT >> 1 . When up’= w, the critical value for the plasma frequency, no resonance peak is observed, independent of the value of W T . When wpl > w the peak can be shifted above the value corresponding to normal resonance, i.e., wc = w. A further interesting effect, which occurs for linear polarization even with (JZ < 1, is that B I6 resonant” peak appears when wp) is very large. Such a peak of course does not occur for low plasma densities. b. Experimental Results ( 1 ) Magnetoplasma resonance. The foregoing phenomena have been observed both at microwave and infrared frequencies. A t this point we shall discuss the results of DKK*z and of Dexter2*for n-type indium sntimonide at microwave frequencies. DKK used an irregular disk 1.4 X 1014/cma. sample roughly 0.4 mm X 0.4 mm X 0.1 mm with n The experimental curve obtained with the magnetic field both parallel and perpendicular to the large face of the sample is shown in Fig. 48. These results confirm semiquantitatively the depolarizing effects of the plasma. Thus for the samples used, Ll = 0.15 and L , = 0.70 and the plasma frequency wPo = 5.6 X lo1*,which is considerably higher than the microwave frequency. Consequently the lower order terms in wpo in Eq. (5.12) can be neglected and the coordinates of the maxima, or the
-
353
CYCLOTRON RESONANCE
derivatives at these points as experimentally observed by DKK, correspond to the '' magnetoplasma resonance " absorption. Hence in this case with H I to the sample (5.15)
and for H
11 to the sample (5.16)
Using the foregoing values and assuming c = 16 we find from Eq. (5.15) that at 24,000 Mc/sec the magnetic field required for resonance is about 7000 oersteds as compared to the experimental value of 2500 oersteds.
0
1
2
3
4
5 6 7 8 H (kilo owrledr)
-
9
40
11
FIG.48. Experimental plasma resonance absorption signals obtained with carrier modulation in a thin disk of n-type indium antimonide at 77'K, at 9OOO MC/W and 24,000 Mc/sec as indicated. The static magnetic field is directed normal to or parallel to the plane of the disk in separate runs. The broken linen connect the curvea below goo0 oersteds with single terminal points determined at higher fields. The reaOnance condition is determined approximately by the crossover points [after G. Dreaselhauit, A. F. Kip, C. Kittel, and 0. Wagoner, Phys. Rev. 96,556 (1955)l.
The discrepancy between theory and experiment is attributed by DKK to a probable error in estimating the depolarizing factors. The important fact to note both from the above equation and the experimental data is that in this region of high plasma densities, the magnetic field for the resonance edge shifts to higher values as the frequency is lowered from 24,000 to 9OOO Mc/sec; essentially the field varies inversely with the frequency.
354
BENJAMIN LAX A N D JOHN 0. MAVROIDES
( 2 ) Dimensional egects. When the dimensions of samples approach a wavelength or the skin depth, whichever is smaller, then the boundary value problem cannot be treated by the depolarizing factors, since in this treatment a uniform field within the medium is tacitly assumed. In this case one should solve the boundary value problem exactly, which is very difficult. However, the solution is fairly straightforward for the cme of normal incidence of the electroinagnetic wave on slabs which are of infinite dimension in the transverse but finite in the longitudinal direction. This configuration has been treated for isotropic media.128The
02
InSb n-TYPE
f
. 0
2 3 , 0 0 0 MCPS
Ea
P 0
0 1
0
P
0
2
4
6
0
1 0 1 2
0
FIQ.49. a, Experimentaltrace of power absorption for a slab of indium antimonide against the wall of a cavity. b, Theoretical curve for the absorptionof a plane polarized wave in n slab of finite thickneas backed by a perfect conductor, with HI to the slab. Here d o c 10, e 9, and the thickness equals + / ( w / c ) or about skin depth on resonance (after B. Lax and L. M. Roth, unpublished).
- -
solution is equally simple for the anisotropic case since, when an effective conductivity is used, it is redly equivalent to that for the isotropic case. The basic information desired can be obtained from the interpretation of experiments on the effects of the dimensions and walls of the specimens on the microwave absorption line or on the infrared reflection or transmission coefficients. From cyclotron resonance experiments on n-type indium antimonide a t 4’K, it became apparent that a t this temperature the free electrons could not be frozen into the impurity levels. With carrier densities of the order of to 1016/cm*and even dimensions as small as of the order of 0.1 mm, phenomena associated with the skin depth were observed. If one takes a semiconducting slab of this dimension and pIaces it against J. A. Stratton, “Electromagnetic Theory,” p. 511. McGraw-Hill, New York, 1941.
CYCLOTROB RESONANCE
355
a conducting wall of a resonant cavity and studies the absorption as a function of temperature, one observes a peak in the absorption.26 I n the material in which this occurred, no true cyclotron resonance for a linear polarized field was expected, since WT < 1. This apparent resonance can be explained as follows; if the thickness of the sample is small compared to a skin depth, the absorption should decrease with magnetic field in proportion to u [l W ~ ? T * ] - - I , whereas if the skin depth is smaller than the sample thickness, the absorption should increase as in a metal in proportion to a-t. In this case the skin depth, initially smaller than the sample thickness, increased with magnetic field. The absorption, therefore, increased as in a metal until the skin depth became of the order of the sample thickness, after which the absorption decreased as in a semiconductor. An experimental trace of such an absorption curve is shown in Fig. 49a. A more rigorous calculation of the absorption curve has been carried out by evaluating the real part of the Poynting vector for a plane wave incident on a semiconducting slab backed by a perfect conductor with the magnetic field perpendicular to the face of the slab. The result is shown in Fig. 49b where the absorbed power normalized by the incident power is plotted against the magnetic field for m = 0.5. The graph shows qualitatively the features discussed in the foregoing. The analysis also shows that the peak moves toward higher values of magnetic field with increasing slab thickness and electron density.
-
+
6. INFRARED CYCLOTRON RESONANCE
I n order to extend cyclotron resonance studies to semiconductors other than germanium and silicon, it was logical to consider experiments at infrared frequencies. Since pure samples of intermetdlic, polar, and alloy semiconductors are not available impurity scattering in these compounds is large so that wz < 1, even at low temperatures. Even in n-type indium antimonide and in semimetals where the scattering can be reduced sufficiently, resonance at microwave frequencies could not be observed directly because of the high electron densities which give rise to depolarizing or magnetoplasma effects, i.e., w < wI. Hence cyclotron resonance at infrared frequencies offered the possibility of overcoming the difficulties due to both scattering and plasma effects. Even at room temperature, where the scattering time T = lo-” second on resonance at 100 microns, w 2 1; thus for a semimetal such bismuth having a typical carrier concentration of -lO1*/cma, the plasma wavelength A, = 100 microns. Consequently experiments in which are used wavelengths from a few microns to about 100 microns, and temperatures as high as room temperature appeared quite feasible. However, there was the problem that, even with small effective masses of the order of 0.01m,
356
BENJAMIN LAX A N D JOHN 0 . MAVROIDES
frequencies of the order of 3 X 10la would require magnetic fields of the order of 100,OOO gauss for resonance. Two approaches were available for obtaining these high fields. In the first, which was taken by Burstein, Picus, and Gebbie,'* a water-cooled Bitter coil, with an inner diameter of 4", capable of reaching fields of 60,000gauss was used. In the second approach taken by the Lincoln group (Keyes et al)", a pulse magnet capable of reaching fields of three quarter million gauss was utilized. This magnet waa developed for these studies by Foner and K ~ l m . l * ~ a. dc Magnetic Fields The resonance experiments of the NRL group" were carried out at a wavelength of 41 microns on p-type indium antimonide; at room temperature this material had an intrinsic carrier concentration of approximately 1016/cma.They carried out a transmission experiment, in which the magnetic field was parallel to the sample surface, on a sample 0.02 mm thick, and a reflection experiment, in which the magnetic field was perpendicular to the surface, on a sample 0.5 mm thick. From the curves, shown in Fig. 50, a value of m* = 0.015m was found. Boyle and Brailsford (B-B)la0have carried out experiments on indium antimonide at liquid helium temperature at wavelengths in the far infrared between 70 and 120 microns. They used a grating spectrometer and magnetic fields of the order of 12,000 gauss to 20,000 gauss. Their samples were relatively pure with n-type impurity concentrations of approximately 2 X loL4and 8 X 10". At this temperature and with these values of magnetic field the transitions involved were between the Landau levels n = 0 and n = 1. However as B-B pointed out the possibility existed that the Landau levels involved were perturbed by the Coulomb field. In concept their experiment is similar to the Zeeman experiments at infrared frequencies,1a1.1a2,1)a except that in the high-field limit it is no longer appropriate to consider that the transitions are simply between a 1s ground state and an excited 2 p state; this is so since, with a small effective mass, a high dielectric constant, and magnetic fields in e x c w of 10,000 gauss, the Coulomb term in the Hamiltonian becomes small compared to the diamagnetic or quadratic term in magnetic field. ev and the energy correspondThus the Coulomb energy is -1.5 X ing to the lowest Landau levels is -7 X lo-' ev. The Coulombic energy is even more insignificant in comparison with the energies of the excited S. Foner and H. H. Kolm, Rev. Sci. Znstr. 48, 799 (1957). W.9. Boyle and A. D. Brailsford, Phys. Rcv. 107, 803 (1957). 8. Zwerdling, K. J. Button, and B. Lax, Bull. Am. Phya. Soc. 121 4, 145 (1959). H.Y.Fan and P. Fisher, J . Phys. Chem.Solids 8, 270 (1959). ma W. 8 Boyle, J . Phyr. Chem. Solids 8, 321 (1959).
I**
lBo
357
CYCLOTRON RESONANCE
states. Consequent.ly one can treat the electrons as quasi-free with a Coulombic perturbation in which the trial functions are of the form used by Dingle,*O#“t, multiplied by a term of the form exp ( - z 2 / 4 a l ) . By a suitable variational solution the perturbed Landau levels were found. The experimental results of B-Bla0are shown in Fig. 51 ; to interpret these they supposed the transitions to be between the states n = 0, n = 1, A1 = 0 but for two values of 1, namely 1 = 0 and 1 = 1. The theoretical splitting of these two lines was equal to 0.05hw, in good agreement with
0.9
.-.
t
\
0.8
1: L ; . : J : :0.7
60
f,
50-
I
i
a6 0.5
40
0
0.4
I n S b NO.126
o.3
W
0.2
-
Ql
-
~
30-
WAVELENGTH * 41.1 MICRONS TEMR=293*K H # R U L € L TO SAMPLE
I
I
l
I”.
---_ .-J
---
EXP.
----wLG.
-
W U
a
at 2 0 -
10-
I 0
-
l
1
m m w 4 o s o c o m
o
InSb No126 WAVELENGTH 4t.l MICRONS TEMP= P93.K
-
H PERPENDICULAR TO SAMPLE
to
bo
i o i o
20
A i ; o
FIG.50. Infrared cyclotron resonance in intrinsic indium antimonide a t 41.1 microns and room temperature: a, relative transmission versus magnetic field for a sample 0.02 mm thick; b, reflectivity versus magnetic field for a sample 0.5 mm thick. The reflectivity is that of one surface only since the sample was strongly absorbing. The calculated curve is bawd on m* = 0.015m and wcr = 16 [after E. Burstein, G. 8. Picus, and H.A. Gebbie, Phga. R w . 101, 825 (1956)].
the experimental value of 0.055hw,. From their data, B-B obtained an effective mass m* = 0.0146m.This is consistent with the low-temperature value of the microwave results and the band structure ss indicated in Fig. 54. Experiments have also been carried out on n-type indium antimonide by Lipson et at.'= a t the far infrared wavelengths of 63,83, and 94 microns obtained by using reflections from four restrahlen plates of KCl, KBr, and KI, respectively. The radiation from a hot globar waa selectively reflected from the surfaces of these restrahlen plates giving a resultant
358
BENJAMIN LAX AND JOHN 0. MAVROIDES CONDUCTION BAND
IMPURITY LEVELS
2
3
FIG.51. Energy level diagram for bound and free electrons in indium antimonide at a magnetic field of 18,000 gauss as calculated from an extension of the results of Y. Yafet, R. W. Keyes, and E.N. Adams, J. Phys. Chem. Solids 1, 139 (1956) lafter W. S. Boyle and A. D. Brsilsford, Phys. Rev. 107, 903 (1957)l.
K I
0
02
04
1.94~
0.6
0.8
10
$2
7.4
1.6
W C/W
FIG. 52. Far infrared cyclotron resonance in indium antimonide showing the magnetoplasma effect which shifts reflection minimum and croas-over to lower fields [after H. G. Lipson, S. Zwerdling, and B. Lax, Bull. Am. Phya. Soc. [2) 3,218 (1958))
radiation with a band pass of about 10% of the peak wavelength. The radiation from this restrahlen monochromator then was reflected from a polished surface of indium antimonide which was placed between the pole pieces of an electromagnet so that the magnetic field was parallel to the sample surface. The experiment was carried out a t room temperature on a variety of samples whose carrier concentration varied from 10" to lO"/cm'. The object was to study the magnetoplasma effects a m -
CYCLOTRON ItESONANCE
359
ciated with high carrier concentrations. Indeed these effects were observed since for the higher concentrations of approximately lO1'/crna the plasma frequency was greater than that of the incident radiation. The results of particular interest are those for the purest sample and are shown in Fig. 52. From these metlsureoients, in conjunction with the absolute reflectivity at high magnetic fields, it was possible to obtain the dielectric constant, which came out to about 15. The plasma frequency or wavelength was found from the reflection at zero field to be approximately 130 microns. This corresponds to a carrier concentration of about 1O1&/cma which is consistent with the intrinsic carrier concentration for indium antimonide. Using the data at the three wavelengths shown in Fig. 52 an effective mass quite consistent with that from other infrared data was obtained. b. Pulsed Magnetic Fields
The technique used by the Lincoln group43 inyolved the generation of large magnetic fields by discharging a 2000 pf bank of condensers through a beryllium-copper coil. This coil was in series with a spark gap, the discharge of which was triggered by a spark coil. A damped oscillatory magnetic field with a half-period of approximately 150 psec was produced in this manner. The resonance line was observed by taking a photograph of the oscilloscope trace during the first half-cycle. Because of the very short duration of the pulse field; it became necessary to develop a rapidly responding infrared detector. This was achieved by means of a 1-mm cube of zinc-doped germanium placed at the bottom of a hollow stainless steel tube immersed in a helium bath. The tube, which was filled with dry helium, contained a suitable infrared window at the top, and a parabolic mirror at the bottom for focusing the infrared onto the detector. The infrared radiation also was focused o:i the sample which was located at the center of an +in. diameter coil. The signal, either transmitted through or reflected from the sample, was then passed through a prism monochromator to the detector which was placed about six feet away from the coil in order to minimize electromagnetic pickup. The output of the detector was fed to a specially designed circuit which reduced the time constant and over-all response time of the detector t o 2 psec. The amplified signal was then recorded by photographing the oscilloscope trace, triggered by light from the spark gap switch of the pulse magnet. I n d i u m antimonide. The initial transmission experiments were carried out at a wavelength of 12.7 microns with fields up to 300,OOO gauss on samples of indium antimonide and indium arsenide which were -200 microns thick. The traces showed broad absorption primarily because of dimensional broadening and perhaps also because of the apparent
360
BENJAMIN LAX A N D J O H N G . MAVROIDES
increase of the effective mass a t higher magnetic fields. Since the samples were rather thick, the skin depth became less than the thickness of the sample as the magnetic field increased toward resonance. Thus the signal w a s almost completely absorbed well below resonance, because the skin depth on resonance was of the order of 10 microns. Above resonance the skin depth remained less than the sample thickness until very high fields were reached; this effect contributed further to the broadening of the resonance absorption. In order t o eliminate the effect, reflection experiments were performed. More recently R. J. Keyes"' has carried out transmission experiments with pulsed fields on thin samples a few microns I
1
I
I
1
I
1
1
1
1
TRANSMISSION
B
x
-
W
2:
*
I-
.50! ~ 60 I
I
70
I
I
I
1
1
00 90 100 ((0 120 MAGNETIC FIELD (hilogaurr)
1
(30
I
140
150
FIQ. 53. Reflection and transmimion tracea of pulsed field cyclotron rwnance photographs in indium antimonide at 19.4 microns. The tranamiwion minimum correlates with cross-over of reflection curve (after K.J. Keyes, private communication).
thick, placed on a suitable backing of germanium. He obtained resolvable resonance absorption which checks quite well with the reflection data. From these resonance traces, the resonant lines shown in Fig. 53 were sketched. Both the reflection and transmission for indium antimonide show a sharp trace at 19.4 microns. The transmission minimum at 110,OOO gauss gives an effective mass of 0.0194m and corresponds to the center of the reflection line. Using such data for different values of magnetic field and corresponding wavelengths, the dependence of the apparent effective mass on magnetic field was evaluated from the ueual cyclotron resonance expression of Eq. (3). Figure 54 shows the values of the effective maas in indium antimonide as a function of magnetic field. This experiment was carried out a t wavelengths between 10 and 22 microns, which are sufficiently removed from the plasma frequency 80 that I*'
R. J. Keyes, private communication (1957).
CYCLOTRON RESONANCE
361
plasma effects were negligible. I t is of interest that the effective mass increased with magnetic field. However, this is not altogether surprising, since the perturbation theory1a*does show that higher order terms must be included in the energy-wave vector relations. These relations can be written in the approximate form,4a*lS'
&(k) Then.
=
h2kz - - ah4k4. 2m*
heH h2kZz + . . . ; L = (n + +) 7 + -*-2m0 mo c
6, = L - 4arnj2L2
(6.2)
If we msume that cyclotron resonance at high magnetic fields corresponds
MAGNETIC FIELD (kllOgOuSB)
FIG.54. Variation of apparent effective mass with magnetic field in indium antimonide. The theoretical curves are obtained from E q . (6.11) (after B. Lax, J. G. Mavroides, H. J. Zeiger, and R. J. Keyes, to be published).
to the transition between the quantum states n = 0 and n = 1, then, taking k, = 0, we obtain
heH m +c
AE1 = - = hw,
or m* 'Ib
=
mo*(l
- 8am0*~(Aw,)~
+ 8am0*~hw,]
(6.3) (6.4)
E. 0. Kane, J . Phys. Chem. Solids 1, 249 (1957). 9. Zwerdling, R. J. Key-, S. Foner, H. H. Kolm, and B. Lax, Phys. Rat. 104,1805 (1966).
362
BENJAMIN LAX AND JOHN 0. MAVROIDES
where a may be found from the slope of Fig. 54, and ma* is the mass at the bottom of the band. Wallis46has worked out a similar expression also on the assumption that resonance is between levels n = 0 and n = 1;however, he takes into account the population of the levels which he aasumes are given by the Maxwell-Boltzmann distribution law. This results in an additional term so that Eq. (6.4) becomes
Under the experimental conditions, namely very high magnetic fields, this correction is very small. No comparable analysis has been carried out for a degenerate semiconductor to which Fermi-Dirac statistics apply as would be the case at room temperature for materials such as highly doped indium antimonide, mercury selenide, etc. The foregoing theory, which gives a linear variation of mass with magnetic field in indium antimonide, is only applicable at moderate fields up to perhaps 60,000 gauss. To be applicable to the pulsed field experiments of Keyes et al. in which the fields were as high as 330,000 gauss it is necessary to reformulate the theory. It is known from the theoretical work of KanelSbthat the energy of the conduction band in indium antimonide can be expressed as & = - h2k2 $-
dEg2 -+- 8P2k2/3- &,
2m
2
where P = - i ( h / m ) ( S ~ p J Z ) S , being an unperturbed wave function of the conduction band and Z one of the unperturbed wave functions of the valence band. The above expression is obtained from a k * p perturbation in which the interaction between the conduction band and the valence bands, including the spin-orbit splitting, are taken into account. We can rewrite the energy-momentum relation by expressing the matrix P in terms of the effective mass at the bottom of the band. This is done by neglecting the first term in Eq. (6.6) and writing for small values of k
Then,
4-+ &a2
E =
where p
=
hk.
4ELl
(2mo*
p 2 )
2
- &,
363
CYCLOTRON REBONANCE
In order to solve this equation for the energy levels in the presence of the magnetic field the momentum operator p is replaced by the generalized momentum operator II = p (e/c)A, where A is the magnetic vector potential. Then the Hamiltonian is rewritten so that the square root in Eq. (0.8) is eliminated, i.e.,
+
(6.9)
This expression is essentially equivalent to the Schriidinger equation for a quasi-free electron in a magnetic field. If it is solved as before, the eigenvalue becomes
This equation can be rewritten to include spin as follows:1sh
where 19 = eh/2mc and the effective g factor has been obtained by Roth" and is given by geff = 2 ( 1 [l - (m/mo*)][A/(3&, 2A)]). This has a value at 4°K of gefr = -52 and a t room temperature of getf = -74. The correction for the spin splitting of the conduction electrons has been ignored since it is small when the differences between levels are taken for the cyclotron resonance transition, i.e., An = + 1 , Am = 0. In the pulse field experiments it is reasonable to assume that in a relatively pure sample the electrons occupy the lowest Landau level even a t room temperature. Hence the transitions we are concerned with are those between the levels n = 0 and n = 1. In interpreting the data, Keyes et ~ 1 defined . ~ an~ effective mass from the resonance equation which we shall call an apparent egkctiue mass m*. In terms of our theory this becomes
+
+
(6.11)
where mo* is the mass at the bottom of the band, w, = eH/mo*c is the Since the writing of this article, R. Bowers and Y. Yafet [Phys. Rev. 116, 1165 (1959)l have published their results on the magnetic susceptibility of InSb. They obtained an expression for the energy by solving a cubic equation; under the same approximations made by us, their result reduces to the above equation (6.10b). Although Bowers and Yafet set up the eigenvalue equation including the magnetic potential as an 8 X 8 matrix, their result is equivalent to ours since the energy surfaces in this case are spherical although nonparebolic.
1aeo
364
BENJAMIN LAX AND JOHN G. MAVROIDES
corresponding cyclotron frequency, and it has been assumed that k, = 0. By selecting the appropriate value of the mass mo*and using the value of the energy gap &, = 0.18 ev obtained from magnetoabsorption data, we can draw a theoretical curve from Eq. (6.11) for the apparent effective mass as a function of magnet.ic field. The appropriate value of mo* at room temperature was deduced from the cyclotron resonance data at microwave frequency carried out at 4°K which gives m0*(40K) = O.O13m, the measured value of the energy gap &, = O.24"7 at 4OK, and the theory of Kane which gives for small values of k2: (6.12)
Using an estimated value of 0.9 ev'35 for the spin-orbit splitting A, we calculate mo*(3OO0K)= 0.010m from the above equation. With this value of mO*we obtain the theoretical curve shown in Fig. 54, which is consistent with the experimental data obtained at room temperature by the pulse cyclotron resonance expression and also the value found by the NRL group42 a t lower fields.137"Thus we see that all the cyclotron resonance data reported is quite self-consistent and in accordance with the model of the conduction band calculated by Kane. Indium arsenide. Similar experiments (both reflection and transmission) were carried out on n-type indium arsenide samples (Fig. 55), one 2 x 1016/cma and t,he other with n 2.3 X 10". The purest with n sample showed a resonance corresponding to a n effective mass m* = 0.031m; however, the effective mass in the sample with the higher carrier concentration varied from 0 . 0 2 5 ~to ~ 0.030m between 100,OOO and 230,000 gauss and it would extrapolate to a value mo* = 0.015m at the bottom of the band. This value is consistent with the mass determinations of Sladek"8 and Frederikse and HoslerlaBfrom oscillatory effects in indium arsenide. For low electron concentrations (n lo1'), they obtained a n effective mass m* = 0.018m a t liquid helium temperatures. If again we employ the results of KaneIJbfor the mass a t the bottom of the band as given by Eq. (6.12), and taking &,(4'K) = 0.45 ev,l'O
-
-
-
V. Roberts and J. E. Quarrington, J . Blccltonirs 1, 152 (1955). Using Eq. (6.10h) we'' have made ralculations subsequently to the foregoing, taking into account the electron spin with gel, = -74 and assuming that the transitions take place between n = 0 and n = 1 for m, (eince gem is negstive, the positive m. gives the lower energy). By choosing a value of effective ms86 mo*(300"K) .L 0.011m, somewhat better agreement with the experimental data is found than that shown in Fig. 54. I*' R. J. Sladek, Phys. Rev. 110, 817 (1958). lag H. P. R. Frederikse and W. R. Hosler, Phys. Rev. 110, 880 (1958). ld0F. Oewald, Z.Naturforsch. 14a, 374 (1959).
la'
IwO
- ++
365
CYCLOTRON RESONANCE
&,(300"K)= 0.36 ev,'4' and X = 0.46 ev,'** we calculate an effective mass mo* = 0.015m. This value agrees quite well with the mass obtained by extrapolation of the experimental curve. The resulta for the somewhat purer sample are at present unexplained. Although the net impurity content as determined from Hall measurements may be low, the impurities may have been compensated in this sample; such compensation, if it existed, might account for the apparent anomaly. At the time of these ex&rirnents only polycrystalline samples of relatively unknown impurity .035r
1
I
I
I nAs
0
- 2. i x ~ O ~ ~ C A R R I E R S
I , I
0200
100
200
300
4
-0
content were available. These experiments should be repeated preferably with dc magnetic fields since single crystals of greater purity are now available. Bismuth. Experiments in which pulse techniques were used also have been carried out on bismuth. In these experiments reflection was from well-polished surfaces of two sets of crystals with faces chosen so that resonance could be observed in the spectral region of 10 to 20 microns and a t fields of several hundred thousand gauss. One set had faces which contained the trigonal axis and a bisectrix axis or which were in the binary plane (1120) with the magnetic field perpendicular to them. The faces of the other set were in the bkectrix plane (1070) and also were positioned perpendicular to the magnetic field. I n accordance with the results of Table I, the expected resonances were indeed observed by Keyes el al. and again with the result that the apparent effective mass varied with magnetic field. S. Zwerdling, B. L a x , and L. M. Roth, Phys. Rw. 108, 1402 (1957). F. Stern and R. M. Talley, Phyr. Rar. 108, 158 (1957).
366
BENJAMIN LAX AND JOHN 0. MAVROIDES
On the basis of Fig. 44, Zeiger14*has pointed out that, in order to see resonance at all at the lower magnetic fields and longer wavelengths, (oJw)* must be approximately 0.7 or lower. This would give a negligible plasma correction at the short wavelength end of Fig. 56. The net effect of the plasma shift is to raise the values of effective maas given in Fig. 56 at the low magnetic fields. However, even in this case the mass valuee would still vary with magnetic field and be different from those obtained
0
0
A
0
SAMPLE SAMPLE SAMPLE SAMPLE
50
1. (1070) '2 (1OTO) .1 ( i l f 0 ) *2 (1120)
100 150 MAGNETIC FIELD (hilogours)
200
FIG.56. Variation of apparent effective mass m*/mo with magnetic field in bismuth (after R. J. Keyes, private communication).
at low temperatures using the de Haas-van Alphen effect. This contention is further supported by the experimental traces obtained at resonance. These have the appearance of the typical reflection curves shown in Fig. 53; therefore the experimental frequencies must be above the plasma frequency. The most unsatisfactory situation occurs for the resonance of the lowest masses. Even in this case it can be shown that the correction to the apparent effective mass is not excessively large at low magnetic fields and decreases essentially to zero at high fields. The situation for the larger effective masses is not at all serious even though resonance occurs at longer wavelengths. This is due to the lighter carriers whose presence makes the plasma correction quite small a t these longer wavelengths; since these carriers have already passed through resonance, H. J. Zeiger, private communication (1959).
CYCLOTRON RERONANCE
367
they have the effect of increasing the positive value of the effective dielectric constant. The plasma correction is not sufficient to explain the variation of the effective masses with changing magnetic field; thus, as in indium antimonide, the curvature of energy bands in bismuth must increase with energy. I n analyzing the foregoing data we shall assume that at the high magnetic fields used the transitions involved are those between the Landau levels14u n = 0 and n = 1. This assumption can be justified on the following basis. Even a t the lowest value of magnetic field, i.e., 30,000 gauss, for an effective mass m* = 0.006m the lowest Landau level lies approximately 0.02 ev above the bottom of the band and the second Landau level a t approximately 0.06 ev. At room temperature the latter value is greater than 2kT. Furthermore since the Fermi level lies about 0.018 ev above the bottom of the band Boltzmann statistics apply; thus the assumption of transitions only between the first two levels is justified. For the two larger masses which are of the order of 0.015m at 75,000 gauss, the values for the fi st two levels a t this field are equivalent to those for 0 , W gauss which we have just discussed. From these the lower mass a t 3 arguments, as well as from the apparent linear variation of the effective masses with magnetic field, it might be deduced that the energy-momentum relation for the bands can be represented in terms of the principal ellipsoidal coordinates as (6.13)
where C = 1/&,and mi are the masses a t the bottom of the band. Equation (6.13) leads (6.14) As shown by &hen and Blount (C-B),144the Landau levels are split by spin-orbit effects, with an anisotropic g value as high as 200; however, as in indium antimonide, we shall, BB a firsborder approximation, neglect this effect since Am 0. 1'4M. Cohen and E. I. Blount, The g-factor and de Haas-van Alphen Effect of Electrons in Bismuth, Westinghouse Scientific Paper 6-94760-2-Pl9 (Dec. 8, 1958). 1440 If we were t o include the s p in a b it effect and the anomalous values of the g factor then in accordance with the results of C-BI4( 1480
-
(6.15)
368
BENJAMIN LAX AND JOHN 0 . MAVROIDES
where we is the cyclotron frequency for each ellipsoid which of course depends on the orientation of the ellipsoid relative to the magnetic field. We have as usual neglected the momentum along the direction of the magnetic field. Hence if we linearly extrapolate our curve to zero magnetic field, we obtain the apparent cyclotron resonance mass values at the bottom of the band of approximately 0.004m10.006m, and 0.008m. These masses are approximately one-half those obtained a t low temperatures at the Fermi surface. From the theory of cyclotron resonance” it is found that the low mass values in the two orientations are nearly equal. The high mass value along the bisectrix axis should be approximately twice that of the low mass; this is apparently the case. However, in addition there was obtained for the (1120) samples a center line giving a mass m* = 0.006m which is not accounted for by the ellipsoidal model of the electrons. One of the factors which has been neglected up to now has been the spin splitting of these levels. It has been shown by C-B”‘ that this splitting can be rather large along certain directions, i.e., for the magnetic field along axis 2 of the ellipsoids (refer to Section 4 s for a definition of the axes). This therefore affects the position in the band and relative population of these levels so that it is conceivable that, in addition to the n = 0 to n = 1 transition, the n = 1 to n = 2 transition with a larger apparent effective mass would also be observed. Such an effect may account for the presence of the extra line. Another possible explanation for this additional line lies in the fact that at room temperature the energy gap and the thermal energy are comparable. Thus the additional line could arise from the cyclotron resonance of the holes in the light mass band at the edge of the Brillouin zone. The experimental evidence of the high-field cyclotron resonance, which strongly suggests the existence of nonparabolic bands, can be supported from independent theoretical considerations. An analysis similar to that of KaneIa6using the k p perturbation method for two bands separated by a small gap at the edge of the Brillouin zone would immediately lead to this conclusion. Since the spin-orbit splitting is large, of the order of 1.86 ev, and also the symmetry of the bands is low in this region, only the two bands need be considered. One can then derive a Hamiltonian for the two bands which includes spin and takes the form 9
(6.16)
where d is the spin vector, bottom of the band, and
is the inverse effective mass tensor at the is the effective spect.roscopic splitting fac-
369
CYCLOTRON RESONANCE
The latter two tensors have been derived from second-order perturbation theory by GB. l4' The solution of this cyclotron resonance Hamiltonian in the presence of a magnetic field is then obtained as
where the effective mass m, has been given by C-B"" as ~ ( C X - ~ ) This ,,. equation defines the magnetic levels for both the conduction and valence bands at the edge of the Brillouin zone. It can readily be shown that, for the conduction bands, where G, < &, the approximate solution of Eq. (6.17) is similar to that of Eq. (6.14); however, it is anisotropic and predicts the linear variation of the apparent effective mass with magnetic field a t low fields. However, at much larger fields, when the energy becomes comparable to &, the solutions of the quadratic equation have to be applied in a manner analogous to their application to indium antimonide. At these higher fields the curve of effective mass as a function of magnetic field should begin to bend downward. The mass values appear to be the right order of magnitude and the general relation for the two orientations also corresponds to that of the de Haas-van Alphen results. The effective masses evaluated a t the Fermi level (-0.018 ev) from the pulsed cyclotron resonance experiments are definitely smaller than those of the de Haas-van Alphen and the microwave experiments probably because of the temperature difference and magnetoplasma e f f e ~ t e . 'This ~ ~ ~suggests that the energy gap has decreased with temperature and it is logical to assume that the valence band, which is located a t the center of the zone and overlaps with the conduction band, probably has also shifted with temperature. To further confirm these hypotheses the pulse experiments, which are exploratory in nature, should be repeated with greater spectral resolution; or preferably when comparable static magnetic fields are available, the experi144b
I(*
+
+
For the effective g factor C-B obtain g.rr = 4 g 1 ~ X ~ *g t * X t y gayXty, where the & are the directional cosines of the magnetic field relative to the principal ellipsoidal axes which are essentially identical for the two bands. Also
Recently, a more careful analysis" of the cyclotron reaonance results of Fig. 56 has been carried out, using Eq. (6.17) and inciuding plasma corrections. The calculations confirm quantitatively that the effective massea are much amaller at the bottom of the band and that the energy gap at room temperature is considerably maller than at liquid helium temperatureg.
370
BENJAMIN LAX AND JOHN cf. MAVROIDES
ments can be carried out with static fields over a temperature range from liquid helium to room temperatures. With large dc magnetic fields and relatively short wavelengths it will be possible to explore the curvature of the bands in pure crystals with greater accuracy. IV. New Developments
7. MILLIMETER CYCLOTRON RESONANCE Although in the course of reviewing the resonance work at microwave frequencies we have mentioned the use of millimeter wavelengths in extending some of the measurements, it is appropriate to discuss this aspect of cyclotron resonance by itself for a number of reasons. First of all, by going into the millimeter range the stringent requirement that or be greater than one is more easily fulfilled. Furthermore, in semiconductors such as germanium and silicon in which cyclotron resonance already has been observed at microwave frequencies, observations at temperatures higher than 4'K are possible. The reason for this may be apparent by considering the properties of the purest germanium or silicon which is available where, at 77"K, the relaxation time T and at 2 mm wz 1. With this pure material the carrier concentration is about 101*/cm3 so that the plasma frequency is below that of the applied electromagnetic wave; hence the electromagnetic field easily penetrates the semiconducting sample which makes it possible to observe cyclotron resonance by the usual techniques. Another possibility is that at liquid helium temperatures and with wavelengths of millimeters, the impurity concentrations below which cyclotron resonance can be observed may be increased perhaps by a factor of 100; also from the line width one can study the properties of the scattering, i.e., whether it is from neutral or ionized impurities. Also one can study the scattering properties of any imperfections and vacancies which have been introduced. The use of millimeter wavelengths allows the observations of cyclotron resonance in materials in which the scattering is high not only because of impurities but also because of other constituents. This was the case for the germanium-silicon alloys studied by the Berkeley group" at 6 mm. However, considering the uncertainties in crystal orientation, the resolution obtained was really not good enough to examine differences in maas parameters between the alloys and the pure crystals. The quantum effects in p-type material, barely detected by Fletcher and co-workers,*I could be examined quantitatively a t 2-mm wavelengths. The conditions for this experiment would be more favorable at these higher frequencies since in essence we are requiring that transitions between individual states be observable and also resolvable. Considering
-
-
CYCLOTRON RESONANCE
371
-
this point we see that the energy corresponding to 2-mm quanta is -5 X lo-' ev and that at 1.5"K, k'l' 1X ev; consequently under quasi-equilibrium conditions, it should be possible to see transitions between the lowest and next lowest lying levels and to clearly resolve them. In order to see the quantum effects for the two different transitions for light holes between the levels n = 0 and n = 1, shown in Fig. 21, we need at 2-mm wavelength fields of 2400 and 4800 gauss. At these magnetic fields the separation for the heavy holes between levels n = 2 and n = 3 would be about 1 X ev to 2 X lo-' ev; thus at 1.5"K only the very lowest levels would be occupied due to the favorable Boltzmann factor; this makes possible the observation of these two quantum transit i o n which ~ ~ ~apparently ~ were not detected by Fletcher et aLalI n addition the quantum effects associated with the heavy holes would be better resolved. Although it is apparent that it is advantageous for a number of reasons to go to millimeter wavelengths the two factors which have deferred this have been the difficulty in obtaining components for instrumentation and the necessity of high magnetic fields to explore properly materials with relatively high effective masses. To obtain 2-mm radiation, klystrons operating approximately at either 24,000 Mc/sec, 35,000 Mc/ sec, and 75,000 Mc/sec with a suitable harmonic generator of either the fourth, third, or second harmonic respectively are required. The harmonic generators are usually hand-made and of the whisker-type silicon diode which is specially fabricated by the individual investigator. The detecton at these frequencies are also similarly custom-made. The cavity techniques which are required are quite difficult and delicate. This situation applies to instrumentation and techniques in general. However, the recent development of more stable and higher power klystrons which operate at 4-mm wavelength and in the tens of milliwatte power range, plus the development of nonreciprocal and other critical components at these frequencies promise brighter prospects for the future. In addition to the germanium-silicon alloy work of the Berkeley group, various experimental results a t millimeter wavelengths already have been reported. Bagguley and c o - ~ o r k e r s ~reported ~ J ~ ~ cyclotron resonance in gold-doped germanium at 8.8 mm and at temperatures up to 90°K. The anisotropy of the heavy hole which is observed with pure samples was apparently absent. Both holes were well resolved with relaxation times of 3 X lo-'* to 6 X lo-'* sec at 90°K and approximately Quantum effects in the vicinity of the value of magnetic field required for the light bole resonance have recently been observed by Rauch and Zeiger.14'6 ''t.. C. J. Rauch and H. J. Zeiger, private communication (1959). u6 D. M. 6. Bagguley and J. Owen, Progr. in Phys. 20, 304 (1957).
lo
372
BENJAMIN LAX AND JOHN G . MAVROIDES
5 X lo-'* sec from 4" to 65°K. Galt and co-workers have built a 4-mm spectrometer, including a provision for circular polarization, which has been used to study cyclotron resonance in semimetals, such as bismuth,'* graphite,61and zinc.62 In the former two they observed a nonresonanttype absorption and also the derivative curve of the absorption. These results are essentially the same as their observations at 24,000 Mc/sec, but as expected the resolution was greater. I n zinc resonance of the A-K type under the anomalous skin conditions was much better defined at this higher frequency; as a result there was obtained for zinc one of the most dramatic and definitive observations of this type ever seen in any metal. Stickler et aZ.'¶ carried out successful experiments on n-type germanium at 2 mm and at a temperature as low aa 1.5"K. The lines were extremely well defined with an WT 50 and with an arbitrary orientation four resonance peaks were observed corresponding to the four ellipsoidal energy surfaces at the edge of the Brillouin zone. Also they have observed resonance in indium antimonide;"' their result gives an effective electron mass value in agreement with that of DKK and Wagoner. 29 The work of the Lincoln group68 has been extended by Rauch and co-workersbawho have made extensive studies of cyclotron resonance in high-purity silicon and of the resonance line width a t 136,000 Mc/sec and from 1.2"K to about 50°K. From the line width they deduced a scattering time as a function of temperature such as shown in Fig. 57. Essentially this represents a measurement of the mobility with great accuracy; at these temperatures it is difficult to obtain the mobility by Hall effect or other means. In addition Rauch et al. developed a scheme for determining the effective mass parameters for electrons even though the magnetic field was not necessarily parallel to the (110) plane. The consequence of this is that all three resonance lines are obtained aa the crystal is rotated relative to the magnetic field and the mass parameters are found as follows:
-
where mz,my,and m. are the effective maases associated with the three different ellipsoidal surfaces along the cube edges, m, and ml are the transverse and longitudinal components of the effective maas ellipsoid and mill* is the effective mass in the [ l l l ] direction. The effective mass m10* l''
is found from the maximum of the sum of
+ [& +1m'u
as a
J. J. Stickler, G . S. Heller, J. B. Thaxter, and C. J. Rauch, private communication (1959).
373
CYCLOTRON RESONANCE
TEMPERATURE (OK)
FIG.57. The variation of w as a function of temperature for cyclotron resonance absorption in high-purity n-type silicon at 135,950 Mc/sec. The obaervations were made on the low-field electron resonance with H along the [I101axis. Also shown is a curve of m*ofi/c where is the high-temperature conductivity mobility for the electrons [after C. J. Rauch, J. J. Stickler, H. J. Zeiger, and G. 8. Heller, Phyu. RGU. Laltstu 4 6 4 (lW)].
function of orientation, i.e., 1
+
[is
1
1
+
=
mllo*' - +
41
1
[;;.
+
(7.2)
An additional relation is obtained from the intersection of a plot of the experimental values of +[(l/m.*) (l/mvy)] and l/msz as a function of orientation. The intersection of the two curves occurs when
+
Using thew relations they were able to evaluate the effective mtw parameters and obtain the following values: mJm = 0.192 f 0.001
and
ml/m = 0.90 f 0.02.
The longitudinal maas value appears to be somewhat smaller than that obtained from the microwave experirnents."**O
374
B E S J A M I N L A X A S D J O H N 0. MAVHOIDES
8. CHOSS-MODULATION
The phenomenon of cross-modulation is one which is well known to ionospheric physicists. Perhaps the best known manifestation of the phenomenon is the Luxembourg effect148in which a modulated powerful radio transmitter tuned to 1.5 hIc/sec, the cyclotron frequency of the electrons in the earth's magnetic field, was beamed up at the ionosphere. Another signal at a different frequency was transmitted through the region in the ionosphere radiated by the Luxembourg transmitter and the Luxembourg program was detected by receivers tuned to the second frequency. To explain the phenomenon Bailey and Martyn postulated that at the cyclotron frequency the electrons are heated up; thus their energy is increased and their mean free time is decreased so that the conductivity of the ionosphere is changed. The modulation of the conductivity correspondingly modulated the electromagnetic wave at the second frequency. This resulted in a mixing or cross-modulation. Fletcher et dJal and the Berkeley2* and Lincoln68 groups observed that the cyclotron resonance line width was effected by the rf power level. The former" were able to narrow the line in very pure germanium by cooling down to 1.5'K and using extremely low rf power levels thereby obtaining an WT = 80. The Lincoln group observed the phenomenon in their early investigations in which the power level was increased to milliwatts; this resulted in broadening of the line prior to the onset of avalanche breakdown. Zeiger72~1~s~x60 et at. decided to make use of this phenomenon to increase the sensitivity of experiments for detecting cyclotron resonance in semiconductors. They conceived the idea of modulating the dc conductivity of a sample as the static magnetic field was slowly passed through cyclotron resonance for particular carriers in the usual manner. The resonance peaks were indicated by changes in the dc conductivity of the sample. With this technique the usual crystal microwave detector, together with its inherent conversion loss, was eliminated; thus the sensitivity of the system was increased by at least one order of magnitude. In observing cross-modulation cyclotron resonance it was found, 89 shown in Fig. 58 for germanium, that some of the resonances occurred with an increase of signal and others with a decresse; in particular the relative amplitudes of the second and third harmonics of the heavy hole were amplified. Since the signal direction could be reversed by a small change in orientation or light intensity, the effect was only partially V. A. Bailey and D. F. Martyn, Phil. Mag. (7118, 369 (1934); 8. H. Zhevakin and V. M. Fain, Soviet Phys. JETP 3, 417 (1956). 140 H. J. Zeiger, C. J. Rauch, and M. E. Behmdt, J . Phyu. Chem. Solida 8,496 (1959). H. J. 7 ~ i g e rLincoln , Laboratory QPR, Division 8, p. 37 (1 August 1958.)
375
CYCLOTRON. R E S O N A S C E
correlated with the sign of tho carriers. To indicate a possible origin of this effect Zeiger made a simplified analysis of the problem assuming a semiconductor with a number of carriers of different spherical energy surfaces, each characterized with an effective mass ma, relaxation time T,, and carrier concentratio11 ni. Choosing the dc magnetic field in the z direction, he calculated the conductivity due to all the carriers. Then applying a microwave signal on resonance for the j t h carriers the conductivity change due to the energy change (and therefore mean relaxation time change 6rj) was calculated in terms of 67,. Zeiger applied this
605
1110
1700
2160
3220
2700
H ( gouts I
FIG.58. Resonance trace of the cross-modulation effect in germanium showing the enhancement of harmonice [after H. J. Zeiger, C. J. Rauch, and M. E. Behmdt, J . Phys. Chem. Solids 8, 496 (1959)l.
calculation to two hypothetical circuit configurations, one for observing cyclotron resonance modulation of the transverse magnetoresistance and Hall effect and the other for measuring the longitiidirial magnetoresistance. The circuit consisted of a sample in series with a resistance R‘, much less than the sample resistance, and a battery. For the first case the changes in voltage across R’, 6E{,and Hall voltage, ~ E Hare ,
i
376
B E N J A M I K LAX A N D JOHN G. M A V R O I D E S
1
(2 + i
1
and for the second, the longitudinal case,
Here 60, = (n,e*/m,)T,, wm = ( e H / m , c ) , and J is the current density through the sample. Examination of the first case (transverse magnetoresistance) indicates that 6E1‘ can change sign when OJ for the majority carrier is greater than one. For a system with carriers of both signs, ~ E may H also change with increasing field; 6Et’ does not change sign. In conclusion it should be borne in mind that this analysis was carried out for the spherical energy surfaces. For ellipsoidal or warped surfaces these results would not hold in detail although reversals could still occur for these cases. Another cross-modulation phenomenon in cyclotron resonance waa observed by the Lincoln group14” in experiments in which two rf frequencies, one in the microwave and the other of which was chosen in the millimeter range, were used simultaneously. The sample was located in a cavity, which was resonant to both of these frequencies, in a position where the electric fieId for both modes did not vanish. Radiation at medium power levels was fed into the cavity at the lower frequency thereby raising the energy of either the electrons or the holes within the band. The magnetic field was then varied as in the usual cyclotron resonance experiment and at the same time a low level signal at the millimeter wavelength was sent into the cavity. At this wavelength the usual resonance bridge was used. The cyclotron resonance absorption curves were observed in exact detail at both frequencies, although the magnetic field went through resonance for holes and electrons at the microwave frequency. The tentative interpretation of this experiment is that the electron energy is modulated by the microwave frequency with energy peaks occurring when the magnetic field sweeps through cyclotron resonance. Under conditions corresponding to the peaks the
CYCLOTRON RESONANCE
377
recombination time of the electrons in the band and thus the carrier density, changes. This density change alters the carrier conductivity at the millimeter wavelength in the same proportion. If this situation were reversed and the energizing signal were at the millimeter wavelength, undoubtedly the resonance would be observed similarly at the lower frequency. This type of cross-modulation is nonlinear, and by appropriately utilizing the interaction of the semiconducting plasma it is conceivable that parametric devices, similar to those using diodes and ferromagnetic materials, could be built. 9. CYCLOTRON RESONANCE I N STRAINED GERMANIUM AND
SILICON
The effect of strain on the relative motion of the energy bands has been demonstrated quite strikingly by the cyclotron resonance experiments of Rose-1nneslb1on germanium and silicon specimens which were put under tension and contraction. These experiments were performed a t 3.2 cm wavelength on samples which were mounted on silica rods with tap grease which acted as a cement at low temperatures. For thin wafers mounted on one side, the differential coefficient of expansion between silica and the semiconductor w a s such that the sample was put under tension, since the thermal expansion of the silica was negligible; i.e., 10-4 aa compared to - lo-' for germanium. Contraction was obtained by making a sandwich consisting of the semiconductor plate between two silica surfaces, one of which was free mounted. The effects obtained were quite consistent with those found by C. S. SmithI5*in his piezoresistance experiments. For n-type material, for which as we have already seen the energy band minima consist of sets of ellipsoids, tension applied to the crystal along a certain crystallographic direction raised the energy of those minima which lay along the corresponding direction in momentum space whereas compression lowered their energy. The relative change in population of the different ellipsoids was reflected in the intensity of the cyclotron resonance lines as indicated in Fig. 59 for silicon which was effectively contracted in an [001] direction. Here the magnetic field waa in 8 (100)plane at about 35' from an [OOl] direction so that the electron resonances for the ellipsoids lying along the [OOI]and (0101 directions were clearly resolved into the peaks labeled B and A , respectively. A t low temperatures where the sample wm contracted, the energy of the minimum in the (Ool]direction was lowered and this resulted in a relative increase in the line intensity aa shown in Fig. 59b. Similar experiments were also performed on germanium with similar results. For holes, whose energy minima are warped sphem
+
ls*
A. C . Rose-Inn-, Proc. Phg8. Soc. 79, 514 (1968). C. 9. Smith, Phy8. Rw. 94, 42 (1959).
378
BENJAMIN LAX AND JOHN G. MAVROIDES
degenerate at k = 0, strain decreased the intensity of the resonances. However the intensity for the light hole decreased even more. This result is consistent with the interpretation of the piesoresistance data for p ty p e germanium and silicon16*~164 that strain lifts the degeneracy, lowering the light hole band below that of the heavy hole.
HEAVY HOLE HEAVY HOLE
0
to00 2000 H (wrrtrdl
3ooo
0
f 0 0 0 2000 xK)o H (omtodl
FIG.59. The effect of strain on silicon: a, cyclotron rmnance spectrum for unstrained crystal with H in a (100) plane at about 35O from an [001] direction; b, spectrum of strained crystal with H in name direction aa (a), but with effective contraction in an [Ool] direction [after A. C. Rose-Innee, Proe. Phys. SOC.74, 614 (1958)).
For germanium a quantitative analysis of the data was made and information on the intervalley scattering parameter wae deduced. It can be shown that the relative population of two valleys, displaced in energy by an amount A&, is given by
where n1and n2 are the equilibrium number of electrons in the lower and higher valleys respectively, rr and il2,rZ1are the relaxation times for recombination aad intervalley scattering, and TIZ/TJI = exp (AE/kT). Theoretical curves, calculated from Eq. (9.1), of log n l / n 2 versus AE/kT were plotted. Data obtained from the experiment were then fitted to the theoretical curves as shown in Fig. 60 by adjusting the value of A&/kT for the 4.2'K point so that the points lay on one of the calculated curves. r 0.16 and A& = 7 X lo-' ev. From this it was deduced that r 2 1 / ~= Using the value of the strain component along the (1111direction and A&, a shift of the conduction band edge of approximately +2 ev per unit strain was estimated. This result resolved the sign ambiguity of this shift . The experiments of Rose-Innes should be repeated at millimeter wavelengths since the higher resolution of the resonances would permit 168 H. Brooke, Advanece in Electronics and Electron Phys. 7,85 (1955). u4E. N. Adame, Phyu. Rcv. 96,803 (1954); F.J. Motin, T. H. Gebdle, and C. H & ibid. 106, 525 (1957).
CYCLOTRON RESONANCE
379
AE/h T
FIG.60. Variation with temperature of the relative populations of two conduction band minima when due to strain one valley is at a higher energy than the other [after A. C. Rose-Innes, Prm. Phy8. Soc. 72, 514 (1958)l.
greater accuracy in determining the relative intensities of the electron resonances and, hence, the relative population of electrons in the conduction band minima as a funct.ion of strain. Furthermore, the experiments could be extended to higher temperatures, giving a wider range of A&/kT and thus permitting a more reliable measurement of the parameters represented in Fig. 60. 10. MAGNETOACOUSTIC RESONANCE Bommel16' discovered a variation of the attenuation of high-frequency ultrasonic waves a t liquid helium temperatures in very pure tin with magnetic field with attendant resonances. Pippard's' proposed a phenomenological explanation of these results based on the concept of a magnetoacoustic resonance which has a resemblance to cyclotron resonance. Similar effects in and tin*6Q have since been observed H. E. Bommel, Phrys. Rev. 100, 758 (1955). A. B. Pippard, Phil. Mag. [8] 2, 1147 (1957). ~7 R. W.Morse, H. V. Bohm, and J. D. Gavenda, Phys. Rev. 109, 1394 (1958). 1'' R.' W. Morse and J. D. Gavenda, Phryu. Rev. Letter8 2 , 2 5 0 (1959). l''R.vW.'Morse, H. V. Bohm, and J. D. Gavenda, Bull. Am. Phy8. SOC.[2] 1, 44 (1958); T. Olaon and R. W. Morse, ibid. 1, 187 (1959). 1'6
16'
380
BENJAMIN LAX A N D JOHN 0. MAVROIDEB
by Morse and co-workers and also in bismuth160.161by Reneker using both longitudinal and transverse sound waves. Typical results of such experiment,a 3re given in Fig. 61. Subsequent theoretical work by Mikoshiba, la* Rodriguez, la* Harrison, l d 4 Kjeldaas, labKjeldaas and Holstein, 186
4-
-
P-
fl
E
4
0-
0
m
-
1
c
*
-I
% -4
-
-6
-
-10t.5
I
2
1
I
I
I I111
I
3 4 5 6 7 8 t O 20 X H I gourr em
-
1J (I)
FIG.61. Acoustical attenuation aa a function of AH between '1 and 4.2OK for a 75 MC/EW[OOl]longitudinal wave in a copper crystal. Curve a is for H in the [lo01 direction, and b is for H in the 11 101direction. Attenuation is memured rrlative to that for H = 0, with curve b arbitrarily displaced downward by 6 dh/cm (after R. Morse and J. D. Gavenda, phys. Rev. Leflerv 1, 250 (1059)l.
w.
and Cohen and co-workers'67 has clarified the nature of the phenomena. It appears that there are a number of phenomena which can be observed in pure metals, depending on the wavelength of the sound, the magnitude D. H. Reneker, PhhyS. Rev. LeUers 1, 440 (1958). D. H. Reneker, Phys. Rev. 116, 303 (1959). N. Mikoshiba, J. Phys. Soc. Japan 18, 759 (1958). I.* 8. Rodriguez, Phys. lieu. 111, 80 (1958). M.J. Harrison, Phya. Rcv. Leffers 1, 442 (1858). IebT.Kjeldaaa, Jr., Ph.ys. Rev. 113, 1473 (1959). 1.4 T. Kjeldaaa, Jr., and T. Holstein, Phys. Rev. Lellers 1,340 (1959). 14' M.H.Cohen, M. J. Harrison, and W.A. Harrison, Phyu. Rw. 111, 937 (1960). I*'
CYCLOTRON RESONANCE
381
of the magnetic field, and the mean free path of the electrons. Theee phenomena might be designated as temporal cyclotron resonance, spatial cyclotron resonance, and the de Haas-van Alphen oscillations. The theoretical treatment using the Boltzmann equation is not unlike that presented previously for the anomalous skin resonance. Here the spatial derivative of the distribution function introduces the properties of the acoustical wave which then appear in the results. An additional term involving the local velocity of the ions is also included. The conductivity tensors for this problem have been obtained by R,odriguezleafor specified directions of the wave vector q of the acoustic wave, the particle velocity u, and the external magnetic field H. In calculating the attenuation using these results, Rodriguez predicted oscillations for transverse waves with the magnetic field along the direction of propagation. Because he assumed a zero Hall field along the wave vector q, he did not explain the observed oscillatory behavior for transverse waves with the magnetic field perpendicular to the direction of propagation. This was pointed out by Kjeldaas and Holstein,166who then showed that indeed the magnetoacoustic phenomena would show oscillations for the latter configuration also. Kjeldaas and Holstein further predicted oscillations in the sound absorption for longitudinal waves and a transverse magnetic field. Without going into the detailed mathematics the observed phenomena can be differentiated on a physical basis as suggested by Cohen, Harrison, and Harrison. 167 For low magnetic fields the electron orbit size will be large compared to A,, the wavelength of sound, and one may encounter the phenomenon of temporal cyclotron resonance where the phonon frequency is of the order of the cyclotron frequency.’62 We have previously discussed this case in connection with electromagnetic waves. The simple classical analysis holds with the usual Bohr frequency condition w = nwc and it is required that WT 2 1 for an observable resonance. Since even in pure metals this condition is only satisfied at low temperatures and in the microwave region, it is evident that the meaning in this instance of low magnetic fields merely categorizes the relative size of the magnetic orbit to the acoustical wavelength. The actual value of the magnetic field can be quite high. This is seen by translating the WT requirement in terms of the magnetic field; we find that for the fundamental resonance (n = 1) vpm*c el
H > -
(10.1)
where as before 1 is the mean free path of the electron and U P is the Fermi velocity. The distinction between this and the usual cyclotron resonance in the bulk is that in the acoustical case resonance can occur at sub-
382
BENJAMIN LAX AND JOHN
a.
MAVROIDEB
harmonic values of the magnetic field. Physically this means that when the electron is moving in the direction of the acoustically induced electric field it is in phase with the field a t one instant and again at one period later in time. The electric field can of course go through several cycles during the electron orbit and this is what gives rise to the subharmonics. For obtaining information on semiconductors this phenomenon does not have any advantage over the usual cyclotron resonance measurement since electromagnetic waves penetrate into the bulk of semiconductors with relative ease. The method is most advantageous in the case of metals because with it magnetoplasma and associated anomalous skin effects are avoided; these effects do not occur since the electric field is a locally induced field rather than an electromagnetic wave. At higher magnetic fields spatial cyclotron resonance can take place essentially as indicated by Pippard.1ba In this case the electron orbit size and A, are of the same order of magnitude. As the electron rotates in the magnetic field it comes in and out of phase with the ultrasonically induced spatially alternating electric field to produce resonance. The requirement for this spatial resonance as given by Pippard is that the radius of the orbit r be given by1(7a
r = (271
+ 1)Xg/4
(10.2)
where n = 0, 1, 2, etc. The condition for observing this resonance is that the mean free path of the electron, 1 2 XJ2r or that ql 2 1. This requirement is easily met for many of the pure metals since ql = (wT)vr/v,, where v, is the velocity of sound, and for many metals v p / v . lO*.’”* This is a far less stringent condition than the condition for temporal cyclotron resonance. From Eq. (10.2) one can write
-
n
(10.3)
This result indicates that the magnetic field required for resonance is increased by the ratio of the Fermi velocity to the sound velocity. Hence for the mass of the free electron, the magnetic field for resonance at 30 Mc/sec is about lo00 gauss. Thus even for low effective masses the method is capable of giving a well-resolved resonance. It may be seen from Eq. (10.2) that spatial resonance occurs for subharmonics also; i.e., for magnetic fields which are submultiples of the fundamental in which case the electron traverses 3, 5, 7, etc. half-wavelengths of the ultrasound. This type of resonance essentially provides a measure of the la’*
A more detailed theory (Cohen1.7) gives as the condition t = nX,/2. Thia implies that w 2 lo-*; thus resonances can be observed 88 low aa 10 M c / w .
CYCLOTRON RESONANCE
383
product of the Fermi velocity and the effective m w , i.e., the momentum and hence unlike temporal cyclotron resonance doee not yield a meaaure of the effective mass directly. However, from the dependence of the attenuation on orientation it is possible for certain crystal configurations to measure the Fermi velocity and hence determine the effective masses.181 Although for a given acoustical frequency a higher magnetic field is required for spatial than for temporal cyclotron resonance, the magnetic fields a t which each of these resonances can be observed are of the same order of magnitude because of the resolution requirement. Rewriting the condition ql 2 1 in terms of the required magnetic field we have (10.4)
Comparing this spatial cyclotron resonance resolution condition, Eq. (10.4), with that for the temporal case, Eq. (lO.l), we see that actually the required magnetic fields are approximately equal. At still higher fields the possibility exists for de Haas-van Alphen oscillationslaO in which the attenuation is influenced by the relative change in population of the electrons in the magnetic field just above and below the Fermi level; at low temperatures this separation is large compared to kT and the radius of the cyclotron orbit is small compared t o the wavelength of the ultrasound. As in magnetoresistance and magnetic susceptibility measurements, maxima and minima will appear in the ultrasonic attenuation with the peaks corresponding to the condition &F = (n 3 6)hw,, where d is a phase constant. For the approximation n = 0, a proper exploration of this phenomenon requires magnetic fields of the order of 10,000 to 20,000 gauss in the case of the lowest mass in bismuth and still higher magnetic fields for materials with larger effective masses. For some metals, fields as high as 100,OOO gauss are needed.10e*108
+ +
11. CYCLOTRON RESONANCE GENERATORS AND AMPLIFIERS
a. Masers The possibility of utilizing cyclotron resonance for generating millimeter and infrared radiation has been considered by a number of people. Perhaps the first attempt to achieve such utilization was that of DZLa8 who had observed and identified the harmonic absorption of the heavy hole resonance. I n view of this observation it was not surprising that they looked for emission a t 6 mm in germanium when it was strongly irradiated with 1.2-cm waves in a cavity. For reasons that are now well understood no such signal was found at 6 mm. An analogous scheme was once again
384
BENJAMIN LAX AND JOHN 0 . MAVROIDES
considered by Tager and Gladun (T-G)eSwho proposed using the anharmonic properties of the heavy hole resonance for generation and also for amplification. Their thoughts were bwed on the observation of the harmonics which indicated that transitions between Landau levels occurred for the selection rules An = k 1, +2, & 3, etc.; these transitions were observed since, due to the warping of the energy surfaces. the normal selection rules were violated and higher quantum jumps became allowed. They proposed that energy with a pumping frequency wp = nu, be supplied at such power levels that a large number of carriers would be excited to the higher states and induced downward at a signal frequency w, = lo, where I < n or I > n. The latter condition corresponds to the scheme of the Lincoln groupss discussed in the foregoing and the former is essentially a proposal for a cyclotron resonance maser. However, the idea aa suggested by T-G is not feasible for at least two reasons. In the first place in a system with equally spaced quantum levels a population inversion cannot be created by pumping power in at the resonant frequency. Since transitions up or down are equally probable, under steady-state conditions an increase in the average energy of the electrons results in a distribution of energy such that the largest number of electrons lie in the lowest states. Secondly even with a n inverted population or one with a population peak for states of high quantum numbers the system would still not be emissive with equally spaced levels, since transitions up or down with equal probability will merely spread the distribution. The net effect will be one of absorption. In addition T-G did not consider the transition probabilities of the harmonics which are considerably lower than that of the fundamental. This means that the power level required to create the necessary population inversion in an appropriate quantum system would be rather large, and deleterious effects such as breakdown and line broadening would probably result. Previously we have shown that there are semiconductors with unequally spaced Landau levels. This is quite evident for the valence band in germanium and silicon in which the lowest levels shown in Fig. 21 with quantum numbers n = 0 and n = 1 have quite different spacings than the others. Materials in which the curvature of the bands varies with energy will also have levels which are unequally spaced. The conduction band of indium antimonide and the valence bands in germanium and silicon at high magnetic fields also would show this property. E. 0. Kane*oo*la* has shown that the curvature of these bands for energies of the order of 0.1 ev or more exhibits marked changes with energy; consequently the spacing of the Landau levels also would vary. Typical values of the magnetic field at which these effects become marked range from approximately 50,000 gauss and up. The masses of interest may vary
385
CYCLOTRON RESONANCE
from 0.02m to approximately 0. lm,corresponding to cyclotron reaonance in the far infrared from about 40 to 200 microns. I n order to excite such systems it would be necessary to excite to a given level selectively in the near infrared or at optical wavelengths. This is possible as shown by the magnetoabsorption studies of the Lincoln g r o ~ p For . ~ this ~ ~excitation ~ ~ ~ germanium would be the most suitable substance since the direct transition which is involved occurs in the infrared near 1 micron a t which wavelength greater energy is available. However Lax71has shown that such a system in which germanium
f
P
FIG. 62.Illustration of possible negative resistance for cyclotron reswance in a nonparobolic energy band indicated by the horizontal dotted lines. The dotted line for the distribution aaeumea a “eink” for carriers at the bottom of the band [after B. Lax, in “Quantum Electronice” (R. Tilley, ed.), p. 428. Columbia Univ. Press, New York, 19601.
waa used would not quite be satisfactory. Nevertheless with some possible improvements the method might be feasible with germanium or a aimilar material with higher energy gap in a more favorable region of the optical spectrum. Zeigerl“ has suggested a somewhat different but related system for using cyclotron resonance of bands with varying curvature which might be useful in the microwave, millimeter, or infrared regions. Again the direct transitions permit selective excitation of carriers deep into the band, i.e., 0.1 ev below the bottom of valence band in germanium. This could create a population inversion in which the slope of the distribution curve would be poeitive as in Fig. 62. If the microwave cavity or millimeter or infrared interferometer is tuned to resonance at energies corresponding to the levels in the inverted regions the system would be emissive since the net induced transition would be downward. The system la*
H. J. Zeiger, Lincoln Laboratory QPR, Division 8, p. 43 (16 October 1959).
386
BENJAMIN LAX AND JOHN 0. MAVROIDES
is emissive provided the effective mass above and below the region of the dotted line differs sufficiently to give a selective resonance for this region. According to the results of Kane,'OO such a region apparently exists for the light hole valence bands of germanium and silicon. Again preliminary estimates71 indicate that for the direct transition in germanium such a system may have possibilities when used with a strong optical source of one watt or more.
b. Harmonic Generators (Parametric Amplijiers) A scherr e which embodies the principles of parametric amplifiers has been considered by Maiman'O for the generation of millimeter waves using cyclotron resonance. He considered an rf oscillating magnetic field H ( t ) = HlesULparallel to and modulating the dc magnetic field H o a t the applied frequency w. The rf electric field which was perpendicular to the dc magnetic field also had a frequency w . The equations of motion were solved as a power series in the ratio wl/w, where w l = e H l / m c is the cyclotron frequency corresponding to the rf magnetic field. It can be shown that the velocity components contain denominators with second, third, and higher harmonics in ascending powers of w l / w , thus indicating resonance for w, = nw when w, = eHo/m*c. It was shown theoretically that the ratio of the net power flowing out of the sample a t the second harmonic to that absorbed by the sample at the driving frequency is
(11.1)
where the theoretical limit of efficiency is 25%. Maiman actually considered two modes of operation, one in which his modulating frequency w,,, = we, and the harmonic generation or signal frequency o, = 20,. The other mode required that the modulating frequency be one-half the cyclotron frequency urn= 4 2 and that generation occur a t we. A gas discharge system was built to operate according to the latter scheme in which 9 watts of input power a t lo00 Mc/sec was converted to 1.3 milliwatts at 2000 Mc/sec. Still another version of this technique has been suggested by (T-G)Ogfor parametric amplification and generation of highfrequency oscillations. They proposed that the pump frequency be the modulating magnetic field wm = 2wJn where n is an integer and that the system, which is then unstable a t we, be used to amplify a t this frequency in a polarization perpendicular to the dc magnetic field. Another possible system for a parametric device is one in which the pumping field is at the
387
CYCLOTRON REBONANCE
cyclotron frequency, the idler or modulating field w, ia lower, and hence the signal frequency w, is the difference (i.e., w, = we urn).
-
c . Negative Mass Effects The concept of a negative mass amplifier or generator originated from the study by Kromer’e of the properties of the heavy mass holes in germanium and silicon. This work led Dousmanis and co-worker@ to look for the presence of negative masses in germanium by means of &
&
&
FIG.63.Unbalanced nonequilibrium distribution for positive and negative c a r r i e ~ ~ . In parabolic bands u I -[(;zrerrn*4~)/(3~ll)jl)jl&ta~~/d&d& is shown to be positive for positive m w carriera; i.e., absorptive, and can be negative for negative IIIME Wriers; i.e., emisuiue [after B. Lax, in “Quantum Electronics” (R. Tilley, ed.), p. 428. Columbia Univ. Prees, New York, 19601.
cyclotron resonance studies. Subsequently the theory for the properties of negative mass systems was considered by Kittel,laDKBUS,~’OMattie ~ * of the interesting and Stevenson, 171 Zeiger,16*and D o ~ s m a n i s . ~One features of a negative mass system which differentiates it from a positive mass system is shown in Fig. 63. This figure indicates two parabolic bands in which the Landau levels are equally spaced. The distribution in each is shown with a peak well inside the band so that the negative mase band is emissive but the positive mass band is not. According to the C. Kittel, Proc. NaU. Acad. Sci. U.S.46, 744
1@@
(1959).
P. G u s , Phy6. Rcv. h t t e r s 8, 20 (1959). 171 D. C. Mattis and M. J. Steveneon, Phys. Rw. Letters 8, 18 (1959). 1’) G. C. Dounmanis, in “Quantum Electronics” (R. Tilley, ed.), p. 468. Columbia Univ. h, New York, 1960.
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BENJAMIN LAX AND JOHN 0 . MAVROIDES
explanation as provided by Mattis and Stevenson and also by Zeiger, not only the distribution of the carriers must be considered, but also the density of states. Since for a parabolic band conductivity is proportional to u
- - 18
n(&) d&
(11.2)
for the positive mass the density of states increases with energy and the integral as shown in Fig. 63 is negative, thus making the conductivity positive. This means that there is a transfer of energy to the electrons which are absorptive. For the negative mass the integral has a positive value and the conductivity is negative. Thus the negative mass system is emissive. The experiments of Dousmanis and co-workers involved the negative mass region of germanium as shown in Fig. (54. The transverse effective mass mT along a [lo01direction is given byas
The condition for a negative transverse mass is C* > 2 B ( A - B ) . Using the parameters given in Section 3b, we find mT/m = -0.22 for germanium and mT/m = -0.43 for silicon. This value of the mass remains negative and increases in magnitude for a cone of about 17" as shown in Fig. 64 whereupon it becomes positive. Similar results are also obtained for silicon. Using circular polarization and excitation by light the resonance spectrum shown in Fig. 65 WBQ obtained. A dip below the baseline on the electron aide of the resonance for mass values of approximately 0.22m and larger was interpreted as evidence for emission by the holes of negative mass. According to the theoretical analysis, such emission can only occur under nonequilibrium conditions and only if the distribution for the negative mass is a t least partially inverted. Dousmanis has argued that in germanium, in which approximately 3.5 % of the holes have negative masses at equilibrium, holes were optically excited to an inverted population thus giving rise to emission. This result seems to demonstrate that cyclotron resonance can be utilized to generate microwave energy. V. Summary and Future Prospects
It is evident from the material that has been presented that the cyclotron resonance technique from its modest beginnings in studies of ionized gases has proved to be a most sophisticated and valuable tool
389
CYCLOTRON REBONANCE
NEGATIVE MASSES: 4.5% IN AT EOUlLlBRlUU
Go. 2 . 4 %
IN
si
*4
FIG.64. Energy contours for the heavy holes in germanium and silicon (left panel), and the two branches of the effective maas of the heavy holes in germanium (right panel). The mass is evaluated in the directions perpendicular to, and radially outwards from, the [lOO] axis [after G. C. Dousmanis, an “Quantum Electronics” (R. Tilley, ed.), p. 458. Columbia Univ. Press,New York, 19601. t
-
6
-
c .C
I0 OHM-CM P TYPE GO GO FILTERS USED
*
t 4
FREQUENCY-I70 KMC
f
H 1(100)AXIS
.B
NEGATIVE MASS HEAVY MOLES
!2
g m 4
0
2800
17%
600
0
-600
--n
-ti OERSTEDS
-
-47%
-2800
FIG. 65. Cyclotron resonance spectrum of germanium taken with circularly polarized microwaves under optical excitation. The new resonance on tbe -H side is aaaigned to carriers with negative effective maaaea (the smaller branch of the effective maas curve of Fig. 64) [after G. C. Dounmanis,” in “Quantum Electronics" (R.Tilley, ed.), p. 458. Columbia Univ. Press, New York, 19601.
Q
TABLEV. E P ~ C T I VMASSES E OF ELECTEONS AND HOLESI N SEMICONDUCTORS AND METALS Effective matsea
-
Semiconductors
Ge
Holes
ml/m
mdm
Comments ms/m
4O
4.3 4.3 x lo-'
Electrons
Si
Holes
InSb
InAs
4O
8.2 X lo-* 8.2 X lo-* 1.64 4.1 X lo-* 3.6 X lo-* 4 . 5
4O 4O 300" 4"
4.17
4O
0.21 0.19 0.19 0.19 0.19 Holes 4.18 1.3 X 10-9 Electrons 1 .o x 10Electrons 1 . 8 X lo-* 1.5 x 10-2 -3 x 10-3 x lo-' Electrons
Temp.
4O 0.98 f O . 0 4 4 O 0.90f0.02 <50° 4O 4" 300" 4" 300" 300"
("K)
Energy surfaces
Warped spherical surfaces with about 30% anisotropy Warped spherical surfaces with dight anisotropy [11 11 ellipsoids k 0 band
Warped spherical surfaces with about 20% anisotropy Warped spherical surfwea with small anisotropy Spherical split-off band [ 1001 ellipsoids Some anisotropy Spherical nonparabolic Spherical
Technique' hferencer, mcr
I
m
2
*
CI
mcr ma
t
mcr
4
3
B
E* z
U
ira mcr mmcr rncr
6
i; X
6 7
P
8
K
ircr
9
0
w
10 9 11
ircr ies ma
3
z
%
s
CdS
Holes Electrona C d h , Holes Electrons HgSe Electrons GaSb Electrona
0.07 0.35 0.32 0.57 4.5 x 10-3 4.7 x 10-3
Gah CdSb ZnSb ZnO
7 . 8 x 10-3 0.15 0.15 6 x 10-3
Electrons Electrons
1.3" 0.32 0.57
0.11 0.16
to 4" <2"
Definite anisotropy Isotropic Ellipsoid at k = 0 Ellipsoid at k = 0 Spherical nonparabolic Spherical
300" 1.5" and 4.2" 300" Sign of carrier not 1.5" yet determined 1.5" 300"
mcr
19
mcr
13
mP ma
14 16
ires mcr
16 17
ire8
18
n
*0
s2
6 z X
B V
z
w
z
w
(0
TABLEV. EFFECTIVE MASSES OF ELECTRONS A N D HOLES I N SEMICONDUCTORS A N D METALS(Continued) ~
Effective masaesb
mdm
Metals Cubic A1 (8)
(b)
cu
m h
ni,/m
0.15 0 . 1 to 0 . 4 0 . 1 8 an d 1 . 5
(8)
(b)
Temp.
(OK)
4"
4O
-1.2" -2O -2"
1.3
Pb
m./m
4"
0.71 0.9tol.l
AU
Comments
-1.2"
0.08
-1.3
Ag
N
~~
~~
1.1 0.75
-1.2"
0.8
2"
Energy surfaces
Techniqueo
References
Mass a t Fermi level
dHvA
19
No detailed interpretation of the surfaces; two distinct carriers Anisotropic spherical surface with "humps" at zone edge Mass at Fermi level Mass a t Fermi level; no detailed interpretation of the surfaces Msss a t Fermi level
A-K A-K
$0
A-K
e.9
El
2: t'L
*x *2
dHvA dHvA
CS 24
dHvA
19
2 X
2
P
P
4
A-K
M
dHvA A-K A-K
19 l?6
M
A-K
¶?6
dHvA
19
s 0,
U
0.10 0 . 2 3 to 0 . 4 3 0 . 2 to 0 . 3 0 . 2 5 and 1 .O
2
-1 .2"
M ~ e as t Fermi level
4 O
Patterns indicated two or more masses 2"
and 4"
In
5 5
d
Tetragonal
Sn
m m z
0.3
-1.2"
Mese at Fermi level
M
m
TABLE V. EPWCTNE MASSES OF ELECTRONS A N D HOLESI N SEMICONDUCTORS A N D METALS (Continued) Effective massesb
Metals Orthorhombic Ga
(a) (b)
(4
mt/m
>o
2 0.1 0.2
mt/m
mdm
>0.15
C0.05
0.3
0.03
0.02
0.4
Commenta m,/m
Energy surfaces
Technique.
References
-1.2"
Mass at Fermi level
dHvA
19
-1 .'La
Mess at Fermi level
dHvA
ID
Warped ellipsoide
nra
68
og
29
nra nra nra og nra nra dHvA
so
Temp. (OK)
Hexagonal
c
).(
3 . 6 X 10-1
(b) Holes
7
Electrons
(a)
(b)
200 25 to 700
x
lo-' 6 . 6 x 10-1 6 . 0 X 10-1 7 . 0 X lo-* 2 . 8 X 10-1 3.1 X 10-t 3.0X 10-1 5.0 X lo-* 1.5 x lo-'
4"
4"
5 . 3 x lo-' 1 . 6 X 10-1
2 . 3 X lo-'
1.7
0.42
1.5 x 10-1
0.2
-1.2"
Majority carriem Minority carriers Warped ellipsoids Majority camera Minority carriers Mass at Ferrni level
30 28
69
so so 19
See 31 for
2.5 1 . 1 x lo-'
1 .3O
N o detailed interpreta-
A-K
detailed list 36
tion
0.43 0.55
cd
0.4
TI
0.35
-1.2"
Maas st Ferrni level
dHvA
19
TABLE V. EFFECTWE MASSESOF ELECTRONS A N D HOLES I N SEMICONDUCTORS AND METALS(Continued) Effective massesb MetaIs
m,/m
Be
3
Rhombohedra1 Bi Holes
Electrons -4
6.8 X 10-1 6 X 10-8 2.4 X 10-J
6.8 X 10-1 1 .o 2.5
8.8 X 10-8
1.8
x
10-8
4.9 -1.0 1 .oo
0.92 2 x 10-1 5 X 10-1
m4/m
Temp.
( O K )
Energy surfaces
0.11 -0.25c
-1.3" -1.3O -1.2"
Ellipsoid at k = 0
Msae at Fermi level (4.G18ev) 2.3 X 10-1 + O . 16 -1.3" Ellipsoids at zone edge; surfaces nonquadratic -1 X 10-1 -8 X 10-1 300" Mssr, a t bottom of conduction band -1.2 X 10-2 -8 X 10-9 1.2" and 4.2' 0.52 -0.65 -1.3" Ellipsoids at mne edge
Holes
2.1 X 10-2 0.15
2.1 X 10-2
3.2 X 10-1
(a)
3 . 1 X 10-1
3 . 1 X 10-1
(b)
0.193
1.07
0.23 1.78
m4 As
mt/m
Technique"
References
nra A-K dHvA
33
nra
33
ircr
36
x 10-2
4 x lo-' 5 x 10-
Sb
ml/m
Comments
Ellipsoid at k
-1.2"
Technique by which values were obtained: mcr microwave cyclotron resonance mmcr millimeter cyclotron resonance ircr infrared cyclotron m n a n c e nrn non-reeonant absorption A-K Ahl-KBner effect mp magnetoplasma
mar
-1.33
magnetoacoustic resonance oscillatory galvanomagnetic measurement dHvA de Ham-van Alphen effect ma magneto-absorption ira infrared absorption ires infrared electrical clusceptibility og
-
0
34 19
z; i-
z
U
4
mar dHvA
36 19
DrB 57 nra 37 dHvA 19 dHvA See 32 for detailed list
0
3: 2.
?
*In the cams of bismuth, tin, and arsenic, the m a w s m,,m2,mr,and m, are the elements of the mass tensor ml 0 m = 10 mt 0 m,
*.
in the coordinate system corresponding to the expressions for t.he energy surfaces in Eq. (4.18). m 4should be positive (see 36). I. G . Dresselhaus, A. F. Kip, and C. Kittel, Phys. Rev. 92, 827 (1953); R. N. Dexter, H. J. Zeiger, and B. Lax, ibid. 96, 557 (1954). 2. B. Lax, H. J. Zeiger, R. S . Uexter, and E. S. Rosenblum, Phys. Rev.. SS, 1418 (1954); R. N. Dexter, H. J. Zeiger, and B. Lax, ibid. 104, 637 (1956). 3. S. Zwerdiing, B. Lax, and L. ?ti.Roth, Phys. Ra.. 108, 1402 (1955). 4. R. N. Dexter and B. Lax, Phys. Rev. 96, 223 (1954). 6. S. Zwerdling, K. J. Button, B. I-, and L. M. Roth, Phys. Rev. Letters 4, 173 (1960). 6. R. h'. Dexter, €3. Lax, A. F. Kip,and G . L)resselhaus, Phys. Rert. 96, 222 (1954). 7. C. J. Rauch, J. J. Stickler, H. J. Zeiger, and G. S. Heller, Phys. Rev. Letters 4, 64 (1960). 8. G. Ihsselhaus, A. F. Kip, H.-Y. Ku, G . IVagoner, and S. 31. Christian, P h y s . Rev. 100, 1118 (1955). 9. B. Lax, J. G . Mavroides, H. J. Zeiger, and R. J. Keyes, to be published. 10. R. J. Sladek, Phys. Rev. 110, 817 (1958); H. P. H. Frederikse and W. R. Hoslcr, ibid. p. 880. 11. W. G . Spitzer and H. Y. Fan, Phys. Rm. 106, 882 (1957). IC. R. K.Dexter, J . Phys. Chem. Solids 8, 494 (1959). 23. &I. J. Stevenson, Phys. Rev. Letters S, 164 (1959). 2 4 . B. Lax and G . B. Wright, Phys. Rev. Letters 4, 16 (1960). 16. S. Zwerdling, B. Lax, K. J. Button, and L. ht. Roth, J. Phys. Chcm. SoZids 9, 320 (1959). 16. W.G . Sphzer and J. M. Whelan, Phys. Rev. 114, 59 (1959). 27. M. J. Stevenson, to be published. 18. R. J. Collins and D. A. Kleinman, J. Phys. C M . Solids 11, 190 (1959). 19. D. Shoenberg, Physak 19, 791 (1Y53). 80. E. Fawcett, Phys. Rev. Lelters 3, 139 (1959). d f . D. N. Langenberg and T. W . Moore, Phys. Rev. Letters 3, 137 (1959). 99. D. N. Langenberg and T. W.Moore, Phys. Rev. Leffcrs 8, 328 (1959). 23. D. Shoenberg, Nature la, 171 (1959). E4. D. Shoenberg, to be published.
P. A. Betuglyi and A. A. Galkin, Sooiel Phys. JETP 6 , 831 (1958); 7, 163 (1958). t6. E. Fawcett, Phys. Rcv. 103, 1582 (1956). 97. A. F. Kip, D. N. Langenberg, B. Hosenblum, and G. Wagoner, Phys. Rev. 108, 494 (1957). 88. J. K. Galt, W. A. Yager, and H. W. Drril, Jr., Phys. Reu. 109, 1586 (1956); J. K. Galt, W.A. Yager, and F. H. Memitt, Prx. 3rd Con/. on Carbon, 1957 p. 193 (1959); P. Nosieres, Phys. Rcv. 109, 1510 (1958). d9. L). E. Soule, Phys. Rev. 112, 708 (1958). 30. J . K. Galt, R. A. Yager, and H. W.Dail, Jr., Phys. Rev. 109, 1586 (1956); B. Lax and H. J. Zeiger, ibid. 106, 1466 (1957). S1. B. L a x , Revs. Modern Phys. 80, 121 (1958). SZ. J . K. Galt, F. R. Merritt, W. A. Yager, and H. W. Dail, Jr., Phys. Rev. Letters 0, 292 (1959). 33. J. K. Galt, W. A. Yager, F. R. Merritt, B. B. Cetlin, and A. D. Brailsford, Phys. Rnl. 114, 1396 (1959). 34. J. E. Aubrey and R. G. Chambers, J . Phys. C k m . Solid 3, 128 (1957). 35. R. J. Keyes, private communicatiou (1957). 36. 1). H. Reneker, Phys. Rev. 116, 303 (1959). 37. W.R. Datam and R. K . Dexter, Bull. .4m. Phys. Soc. 121 2, 345 (1957).
95.
Q U
m z >
4
Ez
CYCLOTRON RESONANCE
397
for studying the basic electronic properties of charged carriers in solids. The experimental knowledge and theoretical understanding have advanced considerably since the initial efforts in the early 1950’s. Already experiments with different degrees of success have been cnrried out on eight semiconductors, namely germanium, silicon, indium antimonide, indium aruenide, cadmium sulfide, cadmium antimonide, zinc antimonide, and cadmium arsenide, and eight metals including bismuth, tin, copper, antimony, lead, zinc, aluminum, and copper, using the variety of techniques which have been described. The results for these materials aa well as others obtained by other techniques are summarized in Table V. A t first the principal objective of cyclotron resonance studies was to measure the effective m a w s of carriers and thereby provide information about the energy band structure either a t the extrema or a t the band edges ad in semiconductors or at the Fermi surface as in metals; however, it appears now that the techniques which have been developed permit us to go beyond these initial objectives. It is now possible to study the properties of these bands a t energies above and below the Fernii surface or the band edges. In addition some of the related magnetoplasma phenomena will supplement the cyclotron resonance experiments on semiconductors and metals over a wide spectra range, possibly into the ultraviolet. The galvanomagnetic effects which have been fruitful for elucidating the band structure in materials, particularly semiconductors and to a lesser extent metals, still remain a useful and complementary tool in this area. However the elegance and the directness of the cyclotron resonance and related experiments will overshadow this classical method not only in the quality of the detailed structure and information which these provide but also in the number of new materials to which the advancing techniques will be successfully applied. The de Haas-van Alphen effect, which has been so skillfully employed by Shoenberg*D6JD* and others, stands out as perhaps the most effective of the classical methods for metals. Nevertheless it is slowly but surely being challenged by the cyclotron resonance techniques. The results of the de Haas-van Alphen measurements which have been instrumental in determining the band structure of bismuth have been now confirmed and extended. The numerical values of the effective masses a t the Fermi surface can be potentially more accurately determined from the cyclotron resonance results. Furthermore the data for the holes, which were not obtained by the de Haas-van Alphen technique, has finally been determined. Lastly, the cyclotron resonance data for bismuth at high magnetic fields and at room temperature give m=s above and below the Fermi surface. The rwults, of cyclotron resonance and de Haas-van Alphen experimentR are
398
B E N J A M I S LAX A N D J O H N 0 . MAVROIDES
in reasonable agreement for graphite ; however, the theoretical elucidation of the complex band structure of this substance was advanced greatly by the detection of minority carriers with their harmonics in cyclotron resonance studies. It appears that we are facing similar experiences with other metals in the foreseeable future. Cyclotron resonance studies have the further advantage over de Haas-van Alphen experiments of providing the prospect of measuring effective masses at higher temperatures in metals. Microwave resonance and other techniques have not been particularly fruitfuI for studying the alloys, because the lines are broadened or the oscillations obscured by impurity scattering; however, these difficulties might be circumvented by resonance studies in the far infrared with the accompanying high magnetic fields. At low temperatures the effect of alloying should be t o displace the Fermi level as well as perturb the band structure. Resonance results under these circumstances would be effective in extending our understanding of some of the more complicated systems. The problem of applying microwave and even millimeter techniques to semiconductors is that the purity of most of the materials has not been sufficient to permit the detection of resonance at these frequencies. Nevertheless the encouraging advances in the purification of crystals, such aa cadmium sulfide and cadmium arsenide, indicate that such progress will be repeated in other materials so that microwave techniques will still remain quite useful. However for the bulk of the elementary and compound semiconductors for which purification still remains a problem the infrared experiments at high magnetic fields again offer a possible means of obtaining useful information on electronic structure. These experiments of course will require infrared spectrometers with large optics, stronger sources, and high magnetic fields. At present pulse techniques permit the creation of high magnetic fields up to approximately 500,000 gauss in small volumes for periods of the order of one millisecond, which are somewhat larger periods than those used in the early experiments ; nevertheless the observation of resonance by the lowtemperature sensitive detectors on a transient basis beyond 40 microns seems unlikely. At best this combination would allow the measurement of effective masses up to perhaps one-sixth that of the electron mass. In arriving at this estimate the fact that the field has to exceed that of resonance for suitable observation was taken into account. The most promising technique for the future involves the use of steady-state dc magnetic fields of the order of 200,000 gauss or more with accessible optical apertures; these apertures will permit experiments in the far infrared where phase detection and integration over long periods of time will allow greater sensitivity in the energy starved region of the electro-
CYCLOTRON RESONANCE
399
magnetic spectrum. It is hoped that coherent generators in the submillimeter region, which will overlap the far infrared, will provide monochromatic sources of somewhat greater energy in this region of the spectrum. Thus at 500 microns and 200,000 gauss, masses of the order of unity will be measurable. Finally, it is only appropriate to mention some of the practical applications of cyclotron resonance that may be realized in the future. Already devices such as harmonic generators, parametric amplifiers, negative mass amplifiers, infrared and millimrtcr masers have been considered. However, the glaring weakness in some of the proposab is the lack of information about many of the basic processes and the lack of quantitative knowledge of the band structure. Sew semiconductors must be found which can be made sufficiently pure for cyclotron resonance studies. For masers the semiconductors should have energy bands whose curvature varies strongly with energy and whose energy gap is in a suitable region of the optical spectrum. Cyclotron resonance will have to be studied in these materials under optical excitation, deeper into the bands than heretofore. The processes of scattering and relaxation due to interband transitions under these conditions of optical excitation will have t o be quantitatively examined. In brief the scope of the cyclotron resonance techniques and tools will have t o be extended for studying the lifetime and the details of the transition processes under a variety of conditions. Acknowledgments
It is a pleasure t o acknowledge the contributions and suggestions of many of our colleagues in the preparation of this article. In particular we have benefited from many discussions with H. J. Zeiger in connection with resonance in metals, cross-modulation effects in semiconductors, and problems related to cyclotron resonance masers. We are greateful to L. M. Roth for her suggestions and criticisms of the presentation of the quantum theory for warped spheres. The authors have made good use of unpublished notes and calculations by K. J. Button on electromagnetic propagation in metals and semiconductors. We would also like to thank R. J. Keyes for use of his experimental data on infrared pulsed cyclotron resonance, G. S. Heller, C. J. Rauch, J. J. Stickler, and J. B. Thaxter for their millimeter results, and S. Zwerdling and K. J. Button for their infrared data on silicon prior to publication. Recent theoretical and experimental work on magnetoplasma effects in semiconductors which were carried out by G. B. Wright were very helpful. Prepririts on the g factor and de Haas-van Alphen effect of electrons in bismuth, and the magnetic field dependence of ultrasonic attenuation in metals by Pro-
400
BENJAMIN LAX AND JOHN
a. MAVROIDES
fessor M. If. Cohen and co-workers were very timely. We also wish to thank M. J. Stevenson for a preprint of his cyclotron resonance work in CdAs, and G. C. Dousmanis for a preprint of his work on negative effective mass effects. Finally we would like to thank Mrs. Nancy Peterson and hfrs. Patricia Dougherty for their patience in typing and re-typing this manuscript.
Author Index The numbers in parentheses are footnote numbers and are ineerted to enable the reader to locate a crow reference when the author’s name does not appear at the point of reference in the text. Basset, James, 45, 46(23), 48, 143 Baum, J. L.,90 Abeles, B., 203,321 Becker, J. H., 4,5,6,7(15), 12, 13,17,18, 20, 21, 22(57),23, 24, 27, 28,31, 35, Abraham, B. M.,90 Adams, E. N., 187,212, 358,378 36 53 Adams, L. H., Beecroft, R. I., 67,68,69,70(93), 77(93), Alder, B. J., 103 78, 79, 84(93), 100, 101(209), 102 112, (118),139 Alekseevskii, N. E.,63,106,109,111, 122, 123 Beenakker, J. J. M., 61 Behringer, R.E.,102 Alers, G.A., 82,84,85 Behrndt, M.E.,280, 295(72),374,375 Alfven, H.,276 Benedek, G. B., 57, 60(61a), 61, 107, Allen, M.,156, 160, 199,210 126, 127, 128, 179 Allis, W.P.,265,267(7),269 Anderson, 0.L.,140 Bennett, C. O., 144, 146 Renson, A. hl., 132 Anderson, P.W.,277, 314(35) Antenen, K.,11, 13(36) Bergeon, R.,143 Appleton, E.V.,264 Berk, A. D., 270,271 k t r o m , E.,276 Berlincourt, D.,130 Atoji, M.,98, 119(195) Berlincourt, T.G., 107 Aubrey, J. E.,279,339(57),340,341(57), Berman, R.,97, 138 Bernardes, N.,77, 90 343, 394(34),396 Bett, K.,46 Avrarni, M., 5 Azbel, M.I., 279, 333(54),335(51), 336, Bezuglyi, P. A., 279, 340(59), 341(59), 392(M),396 337, 338(54),341(114) Biermasz, A. L.,150 B Birch, F.,43, 53, 55, 60,77(6),95, 98 Blackman, M.,321 Babb, S. E., Jr., 132 Blatt, F.J., 216, 217(111) Bagguley, D. hf. S., 278,371 Bloembergen, N.,61, 126, 128 Bailey, V. A., 374 Blount, E.I., 367, 369(144) Balchan, A. S.,173 Blum, A., 16, 17(54) Bancroft, D.,102 Blunt, R. F.,35 Bardeen, J., 77, 107, 109, 112, 113(257), Bommel, H.E.,37!) 167, 171, 172, 175(39),209, 212(39), Bohm, H.V., 379 216, 217(111) Boiten, R. G.,150 Barnes, R. G., 126, 135(286), 136(286) Boksha, 8. S.,50 Barton, L. A., 246, 251, 252(22), 256, Borovik, E. S.,341 257(31) Bosman, A. J., 135 Basset, Jacques, 45, 46(23),48, 143 Botcen, A., 145, 146 A
401
402
AUTHOR INDE X
Bouckaert, L. P., 166 Bowen, D. H., 70, 119 Bowers, R., 363 Bowman, H. A., 45, 59, 60, 96(66) S., 217, 278, 344(44), 356, 357 Boyle, (130), 358 Brailsford, A. D., 278, 279, 315(50), 318 (50 see 103b), 319(50), 320, 322(50), 323, 324, 325, 326, 327(50), 344(44), 356, 357(130), 358, 372(50), 394(33), 396 Brandt, 1. V., 99 Brandt, N. B., 106, 107, 109, 122 Braunstein, R., 219 Breckenridge, R. G., 35 Brewer, D. F., 90 Bridgman, P. W., 41, 42, 43, 44, 46, 48, 49, 50, 51, 52, 53(1), 54, 57, 58, 60 (60),61, 67(46, 59), 70(46, 59), 79, 80,81,82,86,88,89(148),94,95(14), 96(48, 59, 142-145), 97, 98, 99, 101 (47, W), 102, 103(145), 105(47), 107, 109, 138, 139(143), 140, 141, 142, 145, 146(44, 48), 150, 156(7), 172, 178, 179(57), 186, 211 Briggs, H. B., 293 Broensted, J. N., 6, 11(27), 13 Brommer, P. E., 135 Brooks, H., 77, 107, 172, 173, 177, 178, 179, 180, 182(68), 210, 215(62, 68), 217(52), 378 Brown, A. E., 265 Brown, G. M., 145 Brown, S. C., 265, 267(7), 269 Browne, P. F., 258 Bube, R. H., 224(1), 226, 230, 232, 233 (lo), 234(10), 235, 237(8), 238(8), 240, 241(20), 246, 251, 252, 253, 255 (lo), 256, 257(31) Bultemann, H. J., 45 Bundy, F. P., 53, 95, 96, 97, 102(172) Bunting, E. W., 132 Burgers, W. G., 3, 4, 12 Burns, F. P., 150, 199 Burstein, E., 278, 309, 344(42), 356, 357, 364(42) Busch, G., 2, 7(1), 11(1), 16, 17, 18, 22, 23(65), 24, 26, 27, 28, 31, 32, 33, 34, 36, 37, 38(55, 62), 39, 40
w.
Button, K. J., 277,293,307,316(37), 318, 319, 322(37), 327(37), 356, 368(37), 385(98), 390(6), 391(16), 395 Butuzov, V. P., 95, 98(169), 100, 101 (2051, 109 C
Cagle, F. W., Jr., 7 Caldwell, R. S., 173 Callaway, J., 37, 163, 283 Callen, H., 178, 186(62) Camevale, E. H., 147 Catalano, E., 135, 136(334) Cetlin, B. B., 277, 279, 314, 315(33, M)), 318(50 8ee 103b), 319(33, SO), 320, 322(50), 323, 324, 325, 326, 327(MJ), 372(50), 394(33), 396 Chambers, R. G., 279,33l(see l l l e ) , 339 (57), 340, 341(57), 343, 394(34), 386 Chester, P. F., 64, 65, 94, 106, 110(87), 117, 120, 122 Christian, R. H., 103 Christian, S. M., 166, 277, 370(27), 390 (81, 395 Clark, S. P., Jr., 53, 55, 60(54), 95(54), 98(54) Cody, G. D., 110, 112,116,118(255), 119, 120 Coes, L.,97 Cohen, E., 2, 4(3, 7, 8 ) , 5(3, 4, 7, S), 10 (91, 11(3, 61, 13(5) Cohen, M., 367, 369(144) Cohen, M. H., 380, 381(167), 382(167n) Cohen de Meester, W. A. T., 2,4(7), 5(7) 395 Collins, R. J., 293, 391(28), Comings, E. W., 43, 44, 48, 49, 51, 54(4), 56(4), 57, 60, 143 Cooper, L. N., 112, 113(257) Cornieh, R. M., 146 Coulon, R., 131 Cova, D. R., 136 Crawford, J. H., Jr., 291 Cross, J. H., 83 Cross, J. L., 45, 59, 96(66) Cuddeback, R. B., 137 Cutler, W. G., 146
D Daane, A., 139 Dadson, R. S., 45
403
AUTHOR INDEX
Dail, H. W., Jr., 277, 278, 279, 314, 315
Drickamer, H. G.,48, 49, 80, 108, 131,
(33,38), 319(33, 38), 327, 328, 340 (52),341, 343(52), 372(52), 393(28, SO, Sg), 396 Daniels, W. B., 58, 59, 82, 83,85 Dapoiqny, J., 143 Darling, H.E.,46, 47, 150 Dash, W.C.,291 Datars, W.R.,278,379(41),3%,394(37), 396 Daunt, J. G., 90 Dautreppe, I)., 126 Davis, R. E., 154 Dawson, J., 143 Decker, D. L.,118
132,133,134,135,136,137,173,177, 178(54), 185(54),219 Duffield, R. B., 137 Dugdale, J. S.,64, 65,66, 75,78, 92,96 (89),104, 105, 106 Dugina, N. S.,145 Dumbleton, hl. J., 258 Dumke, W.,198, 209, 210(78),291 Duncan, R.C.,280 Dunkerley, P. J., 5
de Graaf, W.,143 de Haas, W.J., 16 Dekker, K.D.,2, 13(5) DeNobel, D., 232,251 Declmaret, M.,131 Dexter, R.N., 267,276,277,278,279(41),
Ebert, H., 44,47,81 Ebert, I'., 92 Edwards, 1). O.,90 Emery, H., Jr., 137 Engardt, R. D.,126, 135(286),136(286) Enz, H.,10 Eppler, R. A., 133 Erdmann, 0.L., 2 Ewald, A. W.,2, 3,8(13), 10,12, 16, 17, 18, 19,20, 21, 23, 25, 29,30, 31, 33,
280, 281, 282(68), 283(23, 24), 284, 285, 286(68),287, 288(68), 289, 291, 294, 296,297,301(78), 309,310,315, 326, 341 (34),350(28), 352(28), 355 (26),372(68), 373(68), 374(68), 383 (68), 384(68),390(I,f!, 4, 6, IO), 391 ( I f ! ) , 394(37), 395, 396 Dienes, G. J., 4 Dingle, R. B.,276, 293, 312, 357(20) Dmitrenko, I. M.,107 Doane, E. P.,137 Dobbs, E. R.,82, 85 Dobie, W.B., 150 Dobrowoleky, W.,272 Dodge, B. F., 144 Domb, C., 91,92,94 Dorfman, J., 276, 312 Dougherty, E.L.,137 Dousmanis, G. C.,280, 387, 389 Drabble, J. R.,201 Dreeben, A. B.,253 Dresselhaus, G., 276, 277, 279, 282(22), 286, 288, 291(79, 801, 293(80), 302 (80),308(29),309,333(64),335(64), 336, 344(32), 350(32, 80),352(32), 353, 370(27), 372(29), 373(80), 374 (22), 390(1,6, 8), 395 Dreyer, K. L., 4, 5(20), 7(20) Dreyfus, B.,126
Duus, H.C.,102
B
35,36, 37,38, 39, 40 Eyring, H., 7
F Fain, V. M., 374 Fairbank, W.M.,71, 105(101),129 Fan, H. Y.,40, 173, 177, 178(53), 183,
215(53), 216, 293, 356, 390(11), 395 Fawcett, E., 279, 339(56), 341(56, 61), 342(56),392(%'0, I%), 395, 396 Feher, G.,218 Fensham, P.J., 7 Fiwher, T.,39 Fisher, P.,356 Fishman, E., 48, 49, 131, 132 Fiske, At. D.,72, 110, 116, 117, 118,119
(251),120(251) Fitch, R. A., 132, 133 Fleeman, J., 4 Fleischer, A. A., 199 Fletcher, R. C.,277, 293, 307, 370(31),
371, 374(31) Foner, S.,278,339,344(43),356,359(43),
361,363(43)
404
AUTHOR INDEX
Griggs, D. T., 52,98(50), 103 Forgeng, W. D., 4 Grillot, E., 258 Forsbergh, P. W., Jr., 60,130, 145 Frederikse, H. P. R., 35, 364, 390(1~3), Grilly, E. R., 92, 93, 94 Groen, L. J., 3, 4,5, 7(23), 8(23), 12 395 Groves, R. D., 201 Friedman, A. S., 146 Guegant, L., 99 Fritzsche, H., 187, 199,200,204,205,207, Gugan, D., 59, 80,65, 104, 105, 106, 124, 214(80), 218 125 Fukuda, K., 273 Guintini, P., 258 Fuyst, R., 13 Gunnersen, E. hl., 342 Fydell, J. F., 4, 5(17), 7(17) Gutlrche, E., 173 G
H Gaidukov, Yu. P., 63, 111, 112 Galatry, L., 131 Calkin, A. A , , 209, 340(59), 341(59), 392 @6), 396
Gsl’perin, F. M., 125 Galt, J. K., 277, 278, 279, 314, 315(33, 38, SO), 318(50), 319(33, 38, 50), 320, 322(50), 323, 324, 325, 326, 327, 328, 340(52), 341, 343(62), 344(44), 372 (50,51, 52), 393(Z8, 90, SZ),394(3S), 396 Garber, hl., 116 Garlick, G. F. J., 251, 258 Gavenda, J. D., 379, 380 Geballe, T. H., 156, 177, 195(26), 199, 209(51), 210(51), 213(26), 378 Gebbie, H. A., 278, 344(42), 356, 357, 364 (42) Gere, E. A., 218
Gibbs, D. F., 60 Gibson, A. F., 251 Gibson, R. E., 53 Gibson, R. O., 145 Gielessen, J., 47, 108, 124 Gill, E. W. B., 265 Giller, E. B., 137 Gilvsrry, J. J., 74 Gladun, A. D., 280, 3&4(69), 386(69) Glickaman, M., 166 Goland, A. N., 17,23,25,35,37(59), 40 Goldberg, C., 154, 203 Gonikberg, M. G., 100, 101 Goodman, R. R., 294, 298(93), 307 Goraneon, R. W., 53 Goryunova, K. A., 4, 16, 17(54) Grenier, C., 111, 116, 118(256) Grieat, E. M., 145, 146(404)
Hagenback, W. P., 143 Hai Vu, 142 Hake, R. R.,118 Hall, E. O.,12 Hall, H. H., 107, 172 Hall, H. T., 43, 53, 95, 97 Hall, L. H., 216, 217(111) Ham, F. S., 77 Hamann, S. D., 43, 48, 51, 52, 57 Hanak, J. J., 139 Hare, W. F. J., 131 Harrison, M. J., 380, 381(167), 382(167 see 167s)
Harrison, W. A., 380, 381(167), 382(167 8ec
167a)
Hatton, J., 71, 105, 110(219), 118(219), 119(219), 120
Hayes, P. F., 46 Hedges, E. S., 4 Heine, V., 279, 333(63), 342 Heller, G. S., 278, 279, 288(63 8ee 7b), 372, 373, 390(7), 395
Henshaw, D. G., 97 Herman, F., 37, 163, 178, 197(30), 208, 219, 283, 288(73), 290, 291
Herman, R.,100 Herring, C., 156, 162(8ee 284, 166, 167, 168, 177, 187, 190, 192(28a), 1Q3(37), 194, 195(26, 371, 196(37), 198, 199, 200(26), 201(26, 37, 411, 204, 209, 210(41, 51), 213(26), 214, 378 Heaterman, V. W., 116
Heyn, E., 7 Heyne, L., 145 Higgs, J. Y., 4 Hill, J. D., 45, 59, 96(66)
405
AUTHOR INDEX
Hill, R. W., 14 Hilsenrath, S.,347 Hinrichs, C. H.,115, 12O(o(See2itip), l’t3 Holland, F. A., 92 Holland, M.G.,170 Holmmn, F. W.,272 Holstein, T.,380, 381(166) Holton, G.,147 Hoogenatraaten, w., 246 Hosler, W.R.,35, 364, 390(10),395 Howard, W.E.,178,203 Howe, W.H., 47 Hruschka, A. A.,58, 59 Huggill, J. A. W., ‘32 Hughes, D. S., 50,83, 131, 132(37) Hughes, H.,loY(248) Hulbert, J. A., 64,65(88),66 Hullburt, E.O.,264,265 Hultsch, R. A,, 126, 135(286), 136(286) Huntington, H.B.,83
Jones, H., 207, 208(97), 321 Jones, R. V., 128, 272
K
Kebalkina, 8.S., 99 Kahn, A., 293 Kaiser, W.,293 Kamadzhiev, P. R.,17 Kamerlingh Onnes, H.,110 Kaminov, 1. P.,128 Kan, L. S., 61, 62, 63, 104, 106(218), llO(79,83), 120 Kan, Ya. S.,50 Kane, E.O.,308,361,362(135),368(135), 384(100, 135),386(100) Kaner, E. A., 274, 276, 279, 333(54),335 (54),336, 337, 338, 339, 341(114) Kao, L. P.,154 Karpenko, A. S., 144 Katz, E.,154 I Kaus, P.,387 Kacarnovakii, Ye. S.,144 Ingram, D. J. E., 271 Keesom, W.H.,14 Isaac, P.c.G.,150 Kendall, J. T.,11, 13, 16, 17, 18, 36, 38 Ishikewa, H.,4 (43) Ivsnov, K., 7 Kennedy, G.C., 52,98(M)), 143 Ivea, J. S.,59, 96(66) Keyes, R. J., 278, 344(43), 356, 359(43), 360, 361, 363(48), 384(47 8ee 137s), J 365,366,390(9), 394(36),395,396 Keyee, R.W.,98,108,109,136, 153, 154 (13),155,156(21),168,170,171,172, Jacobs, I. S., 132 173, 174(15,44), 175, 176, 179, 185, Jaffe, H.,130 190, 191, 192(76), 194(14), 196(13, James, H., 180, 182(67), 184(67), 215 14), 199,200,201(16),358 (67) Kieffer, J., 143 Jamieson, J. C.,99 Kip, A. F., 276, 277, 279, 282(22), 286, Jan, J.-P., 154 288, 291(79, 80), 293(80), 302(80), Jarman, M.,60 308(29), 309, 339(58), 342, 344(32), Jeffries, C. D.,272 350(32, 80), 352(32), 353, 370(27), Jenninge, L. D.,62, 67(84),71, 72(84), 110(84), 116(84), 117, 118(84, 117), 372(29), 373(80), 374(22), 390(1, 6, 8), 392(lV), 395, 390 119(84), 120(84), 121 Jenny, D. A., 232, 251 Kistler, 8. S., 43, 97(8) Johannin, P., 130, 142, 143, 146 Kittel, C., 104,163,168,174(see47),204, Johnson, D.P.,44, 45,47, 59, 96(66) 262, 276,277, 282(22),286, 288,291 Johnston, H.L.,144 (79,801, 293(80), WWO), 308(29), Johnston, J., 53 309, 344(32), 350(32, eO), 352(32), Jones, G.O.,64,82.85, 92,106, 110(87), 353, 372(29), 373(80), 374(22), 387, 117, 120, 122, 124 3W(1 1, 395
406
AUTHOR INDEX
Kjeldaas, T., Jr., 380, 381(166) Klasens, H.A., 251, 252 Kleiner, W.H.,213, 217 Kleinman, D.A.,391(18),395 Klemens, P.G . , 115 Klick, C. C.,259 Koch, J. J., 150 Koeller, R.C.,137 Koenig, 6 . H.,201 Kohn, W.,204, 205(95), 277, 203(30),
Lax, B., 197,200(77),216, 265, 269,270,
271, 273(83, 276, 277, 278,280,281, 282(68), 283(23, 24), 284, 285, 288 (68),287,288(68),291,293,294,298, 297, 301(78), 307,315,316(37),317, 318, 319, 322(37), 327(37, 39), 339, 341(34),344,348,349(120),350,3SI, 352(28), 354, 355(26), 356, 357(45), 358,359(43), 361,363(43),364(47 ace 137a), 365, 368(37), 372(68), 373 300(30),301(30) (68),374(68), 383(68),384(68),385, Kohnke, E. E.,12, 16, 18, 19,20,21, 29, 30,31, 33, 36, 37(56), 38,39 386(71), 3871390(1, 6, 3,4,4 8 , g), Kok, J. A , , 14 391(14, 16), 393(30, Sf), 394(31), Kolm, H. H.,278, 344(43),356, 359(43), 395, 396 361, 363(43) Lazarew, B. G.,5, 61, 62, 63, 104, 106 Komar, A., 5,7 (218), 107, 110, 117(85), 118(85), Kondorskij, E. I., 126 120 Koatina, T.I., 109, 122 Lazarre, F., 145 Kozlobsev, I. P.,130 Lazarus, D.,46,83, 84, 135, 136(337) Kozyrev, P.T.,109 Lensen, M., 150 Kroeger, F. A., 232, 251 Levelt, J. M. H., 143 Krtimer, H., 280, 387(68),388(66) Lifvanov, I. I., 123 Krotsch, M.,145 Likhter, A. I., 100, 101(207), 106, 122 Ku, H.-Y., 277, 370(27),390(8), 395 (222) Kunkel, W. B., 272 Lind, E. L., 232, 233(10), 234(10), 251 Kunzler, J. E.,177, 199, 209(51), 210 Lippincott, E. R.,132 (511 Lipaon, H. G., 278, 344(45), 357(46), Kuo, K., 4 358 Kurnick, S. W., 52, 109 Lissner, H. R., 150 Kuahida, T.,61, 126, 127, 128 Litovitz, T.A., 147 Kuss, E.,145, 146 Liu, T.,136 Livschia, L. D.,108 L Lock, J. M.,114 Loferaki, J., 220 Lacam, A,, 144 London, F.,128 Lambe, J., 259 Long, D.,108, 109, 172, 173, 174, 175, Lampert, M. A., 204, 247 176, 177(45), 178, 199 Landsman, J., 2 Louwerse, P., 143 Landwehr, G.,108, 150 Luthi, B.,112 Lange, F., 13 Lunbeck, R.J., 143 Lsngenberg, D. N., 279, 337(62), 339 Luttinger, J. M.,204,277, 293(30), 284, (58), 341(60), 342, 343, 392(61, %d, 298, 300(30), 301(30, 96) %7),395, 396 Lynch, A. C.,129 Lasin, S.,125 Idtock, R. I., 96 M Lavergne, A., 59 Lawrence, R.,212 McClure, J. W., 294, 329 Lawson, A. W.,43, 47, 59, 99, 103, 135, McCormick, W.D.,71, 105(101), 129 MacDonald, G.J. F., 98 136, 137, 147,214
407
AUTHOR INDEX
MeDougall, J., 165 Macfarlane, G. G.,215, 216(109), 217, 218(109, 110),291 McGeer, P. L., 102 Machlup, S., 172 McKean, W.J., 156, 159(23), 161 McKee, J. M., 130 McLean, T. P., 215, 216(109), 217, 218 (109,110) McMickle, R. H.,146 McMillan, W.C.,103 McQueen, R. G.,74, 75(107), 85(107), 87(107) McSkimin, H. J., 83 Maiman, P. H., 280, 386(70) Maisch, W.G.,133 Maliteon, I., 140 Mapother, D.E.,65, 116, 118 Markevich, A. M., 144 Martyn, D. F.,374 Mason, C. W.,4 Mason, W.P., 150 Mataen, F. A., 132 Mattis, D.C.,279,333(64),335(64),336, 387 Matumoto, H., 273 Maurette, C.,83 Mavroides, J. G., 278, 291, 361, 364(47 .we 137a), 390(9), 395 May, A. D., 131 Mayburg, S.,129 Maynard, C.,153, 155, 156(17) Meiboom, S.,203, 321 Meijer, G.,258 Merritt, F. R., 277, 279, 307(31), 314, 315(33, 50), 318(50 8ee 103b), 319 (33,38, 50), 320, 322(50),323, 324, 325, 326, 327(50, 51), 340(52), 341, 343(52), 370(31), 371 (31), 372(50, 51, 52), 374(31), 393(f?8, %), 394 (331,396 Mers, W. J., 130 Michael, E. D.,103 Michels, A., 130, 142, 143, 144, 145, 146, 150, 172 Mifflin, T.R., 146 Mikoshiba, N., 380, 381(162) Miller, P. H.,Jr., 172, 173, 176 Mills, R.L.,60, 92,93,94 Minshall, S.,102
Mitchell, A. H., 204 Moesveld, A. L. T., 16 Moore, A. R.,219 Moore, T.W., 279,337(62),341(60),342, 395 343,3'33(t1,B), Morin, F. J., 156, 195(26),200, 213(26), 378 Morse, R. W., 379, 380 Mooser, E., 17,26,27,28,31, 37,38, 40 Morel, P.,112 Moss, T.S.,38, 214, 220, 309 Mudge, W.L.,Jr., 5 Miiller, J., 122 Muller, K. A., 39 Muench, N. L., 117, 118, 121(283) Murray, W.M., 150 Myers, J., 109, 178
N Nachtrieb, N. H., 99, 135, 136 Nash, C. P., 103 Nasledov, D. N., 109 Nathan, M. I., 173, 178, 179 Neighbours, J. R.,82, 84, 85 Neuringer, L. J., 172, 177, 178(55), 215 (55),216 Newhall, D. H., 44,45,46,47,!50,59, 150 Newitt, D.M.,43, 44,46, 54 Newmen, R.,291 Nichols, H.W.,264 Niemier, B. A., 57 Noeieres, P.,278, 328, 329(40), 330, 331, 393(98),396 Nussbaum, A., 109, 178 0
Olsen, J. L., 110,112, 116,118, 119(254), 120, 122(256a,266a) Olson, T., 379 Osborne, D.W.,90 Oshinsky, W.,35 Osweld, F.,364 Overdijk, 6. D.J., 144 Overton, W.C., Jr., 107 Owen, J., 371
P Pate, K., 98 Paige, E.G.S., 201
408
AUTHOR INDEX
I’alladino, R. W., 874, 275 Parkinson, D. H., 14 Parks, W., 78 P8lWnS, P. L.,129 Pawns, R. W., 1 3 1 , 135 Patrick, L., 124, 125 Paul, W., 46, 48, 57, 58, 60,105, 108, 128, 172, 173, 176, 177, 178, 179, 215(49, 52), 216, 217(52) Pearson, G. L., 107, 108, 172, 173, 283 Perez, J. P., 130 Perry, C. C., 150 Peterson, E. L., 102 Petit, J., 135, 136(338) Phillips, C. S. E., 145 Phillips, J. C., 333 Picus, G. S., 278, 309,341(42), 356, 357, 364(42) Pippard, A. B., 87, 110, 113(252), 114, 115(252 see 2604, 333, 343, 379,382 (156) P O W , M., 156, 161, 179, ZOO(66), 201, 202,203, 211 Ponyatovekii, E. G., 95, 100, 101 Potter, R. F., 156, 159(23), 161, 199, 213 Poulter, T. C., 131 Powell, J. A., 278, 371(48) Powell, R. L., 339 Price, P. J., 154, 205 Primakoff, H., 90 Pu, R.,199 Purcell, E. M., 126
Q Quarrington, J. E., 215, 216(109), 217, 218(109, llo), 364
R
Rao, D.A.A.S.N., 129 Rathenau, G. W., 135 Rauch, C. J., 278,279, 280, 287,288, 295 (72), 371, 372, 373, 374, 375, 376 (145~1,390(7), 305 Raynor, G. V., 5, 8(26), 9, lO(32) Reiffel, L., 132 Rainsch, A. J., 137 Reitcel, J., 81, 129
Reneker, D. H., 380, 383(160, 1611, 394 (36), 395(56), 396 Resing, H. A., 135, 136(336) Rice, M. H., 74, 75(107), 85, 87 Rice, S. A., 135, 136 Richards, R. B., 78 Riley, N.A., 99 Ringwood, A. E., 98 Rintoul, W. H., 145 Roberta, V., 215, 216(109), 217, 218(lM, llO), 291, 364 Robertaon, E. C., 53, 55, 60(54),95(54), 98 Robertson, R. W., 50, 131, 132 Robin, J., 50, 131 Robin, S., 131 Robinson, D. W., 5 0 , 6 0 , 9 4 Robinson, J. M., 132 Rodgem, K. F., 217 Rodney, W., 140 Rodriguez, S., 279, 333(65), 334(65), 380, 381(163) Rogers, R. R.,4, 5(17), 7(17) Rohrer, H., 110, 112, 116, 118, 119(254), 120, 122 Rose, A,, 226, 247 Rose-Innes, A. C., 202, 377, 378, 379 Rosenblum, B., 279, 339(58), 342, 392 (67),396 Rosenblum, E. S., 276, 283(23), 39O(C), 395 Ross, J. F., 145 Roszbach, G. P., 150 Roth, L. M.,213, 216, 217,277,293,307, 316(37), 318, 319, 322(37), 327(37), 350, 351, 354, 363, 365, 368(37), 385(98, 141), 390(5, 6 ) , 391(16), 395 Rowlands, G.,125 Rudenko, N. S., 144 Runcorn, 5. K., 109, 132 Rutherford, W. M., 137 Ryabinin, Yu. N., 100,101(207), 108,144 5 Sagar, A., 166, 174, .176, 178(35, 36), 198
(35, 36), 199, 203(36) Salter, L., 91 Sandere, T. M., Jr., 199, 200,zOl(81) Sato, K., 130
AUTHOR INDEX
Saurel, J., 143 Sawaguchi, E.,130 Sax,
N.I., 56
Schelling, J. C.,264 Schertel, A., 1 1 Schiessler, R. W.,145, 146 Schipper, A. C.J., 145 Schirber, J. E.,98, 119(194, 195) Schmunk, R. E.,82,83,84(132) Schneider, J., 272 Schrieffer, J. R.,112, 113(257) Schurrman, 145 Schuster, M.,45 Schweinler, H.C.,291 Sclar, N.,309 Sedov, V. L.,126 Seitz, F.,174,262 Sengers, J. V.,146 Shakhovekoi, G.P.,95,100, IOl(205) Shartsis, L., 139 Sharvin, G.,16 Shew, R. W.,118 Shepherd, M. L., 173, 177, 178(53),215
w.,
(53),216(53) Shibuya, M.,203, 285 Shirane, G., 130 Shiehkov, A., 125 Shockley, W.,165,167, 168,175(39),209, 212(39),276, 283, 288(74, 751,291 Shoenberg, D.,110, 113(252), 114, 115
(252 bee 26Oa), 321, 326, 327(108), 328, 343, 383 (106,1081,392U9,$3, 841, 393(19), 394(19),395 Sidorov, I. P., 144 Simon, F. E.,64, 75, 78, 90,91,92, 96 (89),97, 138 Simon, I., 60,81, 140, 146 Sizoo, G.J., 16, 110 Sladek, R.J., 364, 390(10), 395 Slater, J. C.,73,74, 81,97, 181 Sloncceweki, J. C.,329 Slykhouse, T.E.,108, 132,133,134,135, 173, 177, 178(54), 185(54),219 Smith, A. C., 178, 179(61) Smith, A. H.,47, 147(34) Smith, C.S.,81,82,83,84(132), 85, 150, 152, 155(12), 156(10, 12), 159, 160, 162, 179(10), 187, lsS(lO), 193(10), 197, 199,212, 214(10),377
409
Smith, G. W., 126, 128, 136(285) Smith, R. W.,5, 8(26), 9, 10(32), 247,
248, 254(28) Srnoluchowdci, R.,124, 166 Smyth, F. H., 53 Sobolev, N. N., 144 Soule, D.E.,328,393(f?9),396 Spedding, F.H.,139 Spinner, S.,140 Spitcer, L.,Jr., 274 Spitcer, W. C.,173, 177, 178(53), 215
(53),216 (53), 390(11),391(16), 395 Spondlin, R.,111, 118(256) Squire, C. F., 111, 118(256),126,128,136
(285) Stacey, F. D.,124 Stahl, R. H.,72 Stein, P.K.,150 Stephens, D.R.,135 Stern, E.A., 74 Stern, F.,365 Sternheimer, R. L.,101 Stevens, D.K., 291 387,39 Stevenson, M.J., 277, 310, (is’f7),395 Stevenson, R.,70,96 Stewart, J. W., 70,72,75(103),76,77,82,
96 Stickler, J. J., 278, 279, 287, 28863 we
7&), 372,373,390(7),395 Stix, T.H., 274, 275, 276 Stoner, E. C., 165 Stratton, J. A., 354 Strong, H.M.,53, 95, 97 Strylsnd, J. C.,131 suchan, H.L.,173 Sudovteov, A. I., 63, 110, 117(85), 118 (85),120 Strong, H.M.,95 Suhl, H.,283 Sum, C.,130 Swaneon, H.E.,13 Swenwn, C.A., 61,62,67,68,09,70,71,
72, 75, 77(93, llO), 78, 79, 84(93), 88, 90, 91(147), 92(150), W, 96 (117, 147), 98, 100, lOl(#)9), 102 (110, ll8), 110(84), 116(84), 117, 118(84), 119, 120(84), 121, 123, 139
410
AUTHOR INDEX
T Tager, A. S.,280,384(69), 386(69) Talley, R. M.,366 Temm, I. I., 144 Tamman, G.,4,5(20), 7(20) Tang, T.Y.,99 Tapley, J. G., 271 Tatge, E.,13 Taylor, A. H., 264, 265 Taylor, D.J., 278, 371148) Taylor, J. H.,107, 172, 173, 176 ten Seldani, C.A., 144, 172 Teplitzky, D.R.,51,52 Thaxter, J. B.,278, 372 Thomas, J. J., 280 Thomsen, S. M.,232,251 Thurston, R. N.,150 Tichelam, G.W., 135, 136(337) Timmerhaua, K.D.,137 Tinkham, M.,277 Torrey, H.C.,204 Townaend, J. S.,265 Trih, 0.N.,2 Taildia, D.S.,143 Tufte, 0.N.,3, 8(13), 12(13) Tung, L. H.,137 Tuzaolino, A. J., 156, 199, 213(24)
Vink, H. J., 232,251 Vodar, B.,50,99, 131, 143, 145, 146 Vogt, E., 168, 190,198,199,'201(41), 204, 209 Voigt, W., 153, 154, 155(11) Volger, J., 232,251 von Klitzing, K.H., 108,124 Vul, B. M.,130
W
Wagoner, G., 277,279, 308(29), 309, 339 (58),353, 370(27), 372(29), 390(8), 392(6?), 395, 396 Walker, J. A., 81 Wallis, R. F., 278, 361(46), 362 Walsh, J. M.,74, 75(107), 85(101), 87 (107) Walsh, W. M., Jr., 128, 339 Warechauer, D. M., 46, 48, 57, 58, 60, 108, 173, 177, 178, 215(52), 216, 217 (52) Wsssenaar, T., 143 Webb, F. J., 138 Webb, W., 145, 146 Weiderhorn, S., 135 Weil, J., 135, 136(334) Weinreich, G.,199, 200, 201, 217 Weinstock, B.,w) Weir, C. E., 50, 85, 102(139, la), 132, U 139, 140 W e b , H., 219 Ubelmann, F., 131 W e b , P.R,329 Uchida, Y.,273 Welsh, H.L., 131 Wentarf, R. H., Jr., 97 V Wetzel, E., 7 Whalley, E.,69 Vallauri, M. G., SO, 145 Whelan, J. M.,220, 391(18), 395 van den Ende, J. N., 14 White, D.,144 van Ekk, J., 172 White, H. G.,199,200, 201(81), 217 van Eijk, C.,2, 4(3), €431, 11(3) Wiederhorn, S., 173 van Gool, W., 258 van Lieshout, A. K. W. A., 2, 4(7, 8), Wiederkehr, R. R.,132 Wieland, J., 2, 7(1), 11(1), 13, 16, 17, 18, 5(7, 81, 10(9), 11(6) 22, 23(55), 24, 32, 33, 34,36, 37, 38 van Santen, G.W., 150 van &venter, W., 143 (55),39 Wigner, E.,166 van Vslkenberg, A., 132 Wijker, Hk., 143 Venteal', V. A., 107 Vereahchagin, L. F.,99, 100, lOl(U)7), Wijker, Hubert, 143 Wilkinson, K. R., 138 106, 108, 122(222), 130, 145 Wilks, J., 138 Verkin, B. I., 107
411
AUTHOR INDEX
Williams, R. C., 280 Wilson, A. H.,174(47), 175 Wileon, D.K., 218 Witmer, C.A.,204 Wolkers, G.J., 143 Wooda, J., 246 Wright, G. B., 348, 349(126), 391(14), 395
Y Yafet, Y., 358,363 Ysger, W. A., 277,278,279,307(31),314, 315(33, 38, 50), 318(50 see 103b), 319(33, 38, 50), 320, 322(50), 323, 324,325,326,327,328,340(52), 341, 343(52), 370(31), 371(31), 372(50, 51, 52), 374(31),393(88,So, 38), 394 (W, 396 Yarnell, J., 150 Yoder, H.S.,53, 89 Yoshimurs, H., 273
z Zeiger, H . J., 267,276,278,279,280,281, 282(68), 283(23, 241, 284, 286, 286 (68), 287, ZsS(53 we 78s, 681, 291 (681, %295(72), 296, 2971301(78)1 316(37), 317, 318, 319, 322(37), 327 (37,39), 339, 347, 361, 304(47 we 1378),366,368(37),371,372(53,68), 373, 374, 375, 376(145a), 383(68), 384(68),385,387,390(1,8,7,B), 393 (SO),395, 396 Zhevakin, S. H., 374 Zlunitsyn, A,, 144 Zobel, K.-F., 150 Zoller, H., 2, 7(1), 11(1), 16, 17, 18,22, 36(1) Zweitcring, T.N.,143 Zwerdling, S., 216,278,291,293,307,344 (43,45), 356, 357(45), 358, 359(43), 361, 363(43), 365, 385(98, 141), 380 (3,b), 391(16), 395
Subject Index A
AbelsaMeiboom model, bismuth, 322 Acoustic attenuation, copper, 380 Activation volume, diffusion, 135 Alkali halides, optical spectra, pressure effect, 133 Alkali metals, Knight shift, premre effect, 127 P-V isotherms, 77ff resistivity, pressure effect, 106 Aluminum, compressibility, 82 cyclotron resonance, 3418 superconductivity, pressure effect, 121 Amplifier, negative mass, 387 Anomalous skin effect, 279, 331ff ace also Acbel-Kaner effect, Antimony, cyclotron resonance, 326 effective m w , charge carriere, 3’26 melting point, pressure effect, 95 P-V isotherm, 86 Araldite aeal, g0 Argon, solid, compreesibility, 85 P-V isotherm, 76 Arbel-Kaner effect, 279, 3318 bismuth, 339
B Band gap, germanium, optical, tempersture dependence, 217
semiconductors, 172 composition effect, 218ff silicon, optical, temperature dependence, 217 Band structure, bismuth, 3216 germanium, 290 germanium, pressure effect, 176ff strain effect, 378 magnesium, 207 neatly empty lattice model, semiconductors, I63 pressure effect, 170-179 semimetal, 165, 210 silicon, 290 strain effect, 378 Bardeen-Cooper-Schrieffer theory, 112 Barium, P-V isotherm, 86 Barium titanate, dielectric constant, pressure effect, 130 Beryllium copper, in high pressure devices, 56 Bismuth, band structure, 321 ff cyclotron resonance, 314ff, 320ff effective msss, charge carriers, 324ff, 324ff, 339 magnetic field effert, 365 phaee diagram, 96 P-v isotherm, 86 resistivity, premure effect, 106 superconductivity, presaure effect, 122 Boltcmann transport theory, 294ff Boron nitride, cubic form, 97 Bourdon gauge, 66 Bridgman, high-pressure seal, 58 Bridgman method, 412
SUBJECT INDEX
preaeure generation, 64 Brillouin zone, graphite, 329 C
Cadmium, superconductivity, preesure effect, 111, 122 Cadmium uaenide, cyclotron resonance, 310 Cadmium aelenide, infrared quenching spectrum, 257 photoconductivity, 2368, 256ff Cadmium sulfide, see CdS Carboloy, in high pressure devices, 55 CdS, copper impurity levels, 259 c u m n t-tem perat ure relat ion, 215ff cyclotron reaonance, 301) donore, ionization energy, 232 effective mass, 309 optical quenching, 241 photoconductivity, 240ff space-charge effect, 248 CdStype materials, conductivity, 230ff imperfections, 223-260 ionization energies, 2528 ionization energy table, 250 multiple levels, 2mff luminescence, 2338, 258 photoconductivity, 2336 light intensity effect, 2aff space-charge effects, 2478 trapping centers, 2436 Ckerium, d i d , phaee transition, 99ff, 101 Cesium, P-V isotherm, 86 Clauaius-Clapeyron equation, 87 ClauniukMosotti function, preaoure dependence, 144 Compound semiconductors, cyclotron resonance, 3088 effective mass, holes, 308ff Compressibility,
413
adiabtic, 82 bulk, 82 pressure variation, 80 temperatw dependence, 84 ultraeonic, 82 Copper, acoustic attenuation, 380 cyclotron resonance, 3416 effective mam, chuge carriem, 343 resistivity, preaaure effect, 105 Covalent binding, 184 Critical field, superconductivity, pressure effect, 1128 Critical point, liquid-eolid, 91 Cross-modulation, 374ff Crystal symmetry, piezoresistivity, 152 Cubic crystah, piesogalvanomagnetic effect, 155 piezo-Hall effect, 155 Curie temperature, pressure effect, 123ff Cyclotron frequency, 261 aluminum, 341ff amplifiers, 38311 Cyclotron reaonance, antimony, 326 bismuth, 314ff, 3208 cadmium m n i d e , 310 compound semiconductors, 308ff copper, 3416 crow modulation, 374 electrons, free, 264ff g-, 265 generators, 383 germanium, 2838, 294, 301, 389 holes, 297 germanium, gold doped, 371 germanium-silicon alloys, 371 graphite, 327ff holes, 288, 296 indium antimonide, 354 indium arsenide, 364 infrared studies, 278, 355 ions, 273ff lead, 340 magnetic field effect, 3676 mssers, 280
414
SUBJECT INDEX
metals, 312ff experimental techniques, 319ff microwave frequencies, experimental techniques, 2&08 millimeter wavelengths, 37M photons, 273 plasma, 275 quantum theory, 298ff semiconductors, 28oR silicon, 283ff, 372 solids, 276-400 spatial, 382 spectrometer, 281 temporal, 382 tin, 340 zinc, 34ofi
D Deformation potential model, elastoresistivity, 167ff de Haas-van Alphen effect, 397 antimony, 326 bismuth, 321ff graphite, 327 pressure effect, 107 Demarcation level, Fermi level, 227 insulators, 226 Deuterium, solid, P-Visotherm, 76 Diamond, synthesis, 97 Dielectric constant, solids, preseure effect on, 129 Differential thermal analysis, 89 Diffusion, activation volume, 135 gases, 146 liquids, preasure effect, 137 melting correlation, 136 pressure effect, 1358 stress effect, 137 Ductility, preasure effect, 141
E Effective mass, apparent, 363 Effective mass, charge carriers,
antimony, 326 biemuth, 324f€, 339 copper, 343 definition, 262 magnetic field effect, 361ff metals, 340ff determination, 316 table, 390-394 pressure dependence, 174 semiconductors, 173 table, 390-394 silicon, 373 Effective mass, electron, germanium, 286 silicon, 287, 289 Effective m w , holes, compound semiconductors, 308ff Elastic constants, pressure effect, 83 ultrasonic techniques, 83 Elastoconductivity, semiconductors, 187ff Elastoconductivity tensor, 162 Elastoresistivity, 150 deformation potential model, 167ff Bemiconductors, 167ff, 175 semimetals, 211 Electric breakdown, gases, 265ff Electron-hole pairs, semiconductors, activation energy, 182 activation volume, l82ff thermodynamic treatment, leoB Electron mobility, cryetals, 168 lattice vibration effecta, 209 semiconductors, pressure effect, 176 Electrons, uee also effective mass effective m w , 289 pressure dependence, 174 Electrons, free, cyclotron resonance, 264ff magnetic field levels, 298ff Electronic specific heat, 113 Energy gap, tin, gray, 28ft Energy levels,
415
SUBJECT INDEX
semiconductore, 185 Equation of state, alkali metals, 77 Mie-Crueneieen, 74 solid, 738 Extrinsic eemiconductor, band structure, 164
F Fermi level, demarcation level relation, 227 insulators, 227 Fermi surface, graphite, 329 G
Gallium arsenide, resistivity-pressure relation, 174 Galvanomagnetic effecb, semiconductors, 1536, 196 Gases, compression technique, 50 diffusion, 146 P-V-Trelations, 142 thermal conductivity, pressure effect, 146 Germanium, band gap, optical, temperature dependence, 217 band structure, 290 preasure effect, 176ff strain effect, 378 charge carriers, effective m w , 40, 286 mobility, 40 crow modulation effect, 375 cyclotron resonance, 283ff, 294, 301, 389 holes, 297 strain effect, 377 energy gap, 40 holes, energy levels, 307 holes, heavy, 387ff impurity levels, pressure effect, 179 magnetoresistance, 202 microwave absorption, 284
optical behavior, pressure effect, 108 pieeoresietance, 197, 212 temperature effect, '204 Seebeck coefficient, strain effect, 201 Germanium-silicon alloys, cyclotron resonance, 308 GlW, compressibility, 139 Graphite, Brillouin cone, 329 cyclotron resonance, 327ff effective mass, charge carriers, 327ff Fermi surface, 329 Gray tin, 8ee tin, gray Gruenekn constant, 74, 91
H Hall coefficient, 15 tin, gray, 21ff Hall effect, elmtic strew, 154 sem iconductom, temperature dependence, 23OfT Hall effect tensor, 154 Harmonic generators, 386ff Helium, solidification, 64 Helium ges, cyclotron resonance, 265ff , 269 Helium isotopes, melting curves, 9Off Helium, solid, phase transitions in, 96 P-V isotherm, 76 thermal conductivity, presmue effect, 137 Hexagonal crystal, pietoreaistance coefficients, 160 High-preseure, dielectric constant of solido, effect on, 129 diffusion effect, 135ff experimental techniques, 44ff fixed clamp method, 64 fluid transmittere, 6lff generation, 48
416
SUBJECT INDEX
ice-bomb technique, 62 Knight shift effect, 127 low-temperature techniques, 6Off magnetic property effects, 1238 materials for, 55, 57 metals, mechanical property effects, 141ff nuclear magnetic reeonance, 1268 optical measurements, 49 optical spectra effect, 131ff resistivity, effect on, 103ff resistivity measurement, 60,66 seals, 57ff semiconduction effect, 1078 solid transmitters, 62 superconduction effect, 109ff x-ray measuremente, 99 Holea, 289 see abo effective mass cyclotron resonance, 296 germanium, energy levels, 307 Hydraulic preas, low-temperature, 68 Hydrogen, as pressure transmitter, 71 nuclear magnetic resonance: pressure efiect, 128 Hydrogen gae, P-V-T relations, 144 Hydrogen isotopes, melting c w e e , 93 Hydrogen, solid, P-V isotherm, 76
I Ice, phrrse transitions, 97 Ice-bomb technique, 62 Imperfections, photoconductivity effect, 225ff Indium, superconductivity, pxwsure effect, 117 Indium antimonide, charge carriers, effective mass, 40 mobility, 40 cyclotron resonance, 364, 369
cyclotron resonance, infrared, Wff effective mesa, magnetic field effect, 361 energy gap, 40 magnetoplasma effect, 358 magnetoplasma reeonance, 352ff plasma effects, 277 resistivity-pressure relation, 170 Indium arsenide, cyclotron resonance, 359, 364 effective mass,charge carriers, magnetic field effect, 364 Indium phosphide, resistivity-pressure relation, 174 Intrinsic semiconductor, band structure, 165 Ions, cyclotron resonance, 2738 Iron, compressibility, 80 magnetic properties, pressure effect, 125
J Jones--Ghoenberg model, bismuth, 321
K Knight ehift, pressure dependence, 1278
Landau levels, 300 transitions, 384 Lattice vibrations, pressure contribution, 74 Lead, cyclotron resonance, 340 superconductivity, pressure effect, 118 Liquids, diffusion, pressure effect, 137 Lithium, P-V isotherm, 86 self-diffusion, preaeure dependence, 126
4 17
BUBJEC" INDEX
Lithium hydride, phase transition, 102 Lumineacence, CdStype material, 2338, 258 excitation, 233 optical quenching, 2thermal, 243ff thermal quenching, 237ff Luxembourg effect, 374
M Magnesium , band structure, 207 Magnetic fields, high, 356 pulsed, 359 Magnetic saturation, pressure effect, 124 Magnetic susceptibility, tin, gray, 26, 30 Magnetoacoustic resonance, 3798 Magnetoplasma effect, 274, 343-355 indium antimonide, 358 isotropic carriers, 349 Magnetoplasma resonance, 349, 3628 indium antimonide, 352ff Magnetomistance, germanium, 202 semiconductors, 202 tin, gray, 21ff Magnetoresistance tensor, 154 Manganin, pressure gauge use, 46 Masers, 280,383ff Matthiesen equation, 104 Mechanical strength, pressure effect, 141 Melting, zero point energy effect, 93 Melting temperature, pressure dependence, 91 Mercury, phase transition, 98 superconductivity, preseure effect, 118 Metale, cyclotron resonance, experimental techniques, 31917 effective maas, charge crrriem,
determination,-316 table, 390-394 mechanical properties, pnrraura effect, 141ff reaistivity , pressure effect, 103ff Mie-Grueneieen equation, 74 MultivaIley model, semiconductors, 187ff
N Ne81 point, preaaure dependence, 128 Neon, P-V isotherm, 76 Nitrogen, gee, cyclotron resonance, 272 Nitrogen, eolid, phase diagram, 88 Noble metals, compressibility, 82, 85 Nuclear magnetic reeonance, preseure effect, 126ff Nucleation, gray tin, 4 0
Optical behavior, measurement at high preesure, 49 semiconductors, impurity effecta, 232 pressure effect, 214ff Optical quenching, luminescence, 240ff photoconductivity, 2408 Optical spectra, preasure effect, 131ff
P Parametric amplihen, 3868 Phase transition, measurement of, 89 Phosphorus, black, formation of, 98 reaistivity, preasure effect, 108 Photoconductivity,
418
SUBJECT INDEX
cadmium selenide, 236ff a s , 240ff CdGtype material, 233ff light intensity effect, 235K imperfection effecta, 226ff imperfection sensitization, 229ff impurity effects, 225ff optical quenching, 240ff thermal quenching, 237ff tin, gray, 27 zinc selenide, 233ff Photosensitivity, definition, 225 Piezogalvanomagnetic effects, 153ff cubic crystals, 155 Pieeo-Hall effect, cubic crystals, 155 Pieao-opticnl effects, semiconductors, 214ff Piezoreshtance, 150 crystal symmetry, 152 germanium, 197, 212 temperature effect, 204 measurement, 156ff semiconductors, 187ff, 199 temperature dependence, 196 semimetals, 210 silicon, 197, 212 Piesoresistance coefficients, hexagonal crystal, 160 Piesoresistance tensor, transformations, 161 Plasma, cyclotron resonance, 275 Plasma frequency, 346ff Polymorphic transitions, 95 Poulter seal, 131 Poynting vector, 312 Pressure, see also high-pressure meesurement, standards, 46 melting point, effect on, 91 various materiala, 95 Pressure gauges, 45ff Pressure scale, 44ff Pressure transmitters, 6lff Pressure-volume isotherms, various solids, 86
PrOtons, cyclotron remnance, 273 Pyroelectric effect, 155
0 Quartz glees, compreasibility, 139
B Radio waves, range, 265 Rare earths, preesure effects on, 138 Rare g w s , solidified, P-V ieotherms, 76 Recombination centers, insulators, 226 Reaistivity, high-preasure measurement, 80,88 metals, preesure effect, 1036 p r e ~ ~ ueffect, re 166 shear streae effect, 157ff Rubidium, phaee diagram, 102 P-V isotherm, 86 S
Seebeck coefficient, germanium, strain effect, 201 Gelenium, P-V isotherm, 86 resistivity, pressure effect, 109 Semiconduc tion, pressure effect, 107ff Semiconductors, aee ale0 compound semiconductors, 308 band gap, 172 composition effect, 218ff band structure, 163 preseure effect, 170-179 binding in, 184ff cyclotron resonance, 280ff
419
SUBJECT INDEX
effective mass, charge carriers, 173 table, 390-394 elastoresistance coefficient, 175 elastoreaistivity, 167ff electron-hole equilibria, 18OfT electron-hole pairs, activation energy, 182 activation volume, 182ff electron mobility, pressure effect, 176 energy levels, 185 galvanomagnetic effects, 196 galvanomagnetic coefficients, 1538 Hall effect, temperature dependence, 230ff lattice parameter, composition effect, 218 magnetoresistance, 202 multivalley model, 187ff optical behavior, impurity effects, 232 presaure effect, 108 pieso-optical effects,214ff pieroreaistance, 187ff, 199 temperature dependence, 196 thermodynamic treatment, 206 transport properties, 168 two band model, 166 Semiconductors, valence, absorption edge, pressure effect, 133 Semimetals, band structure, 165, 210 pieroresistance, 210 Silicon, band gap, optical, temperature dependence, 217 band structure, 290 strain effect, 378 charge carriera, mobility, 40 cyclotron resonance, 2838, 372 strain effect, 377 effective mass, charge carriers, 40, 373 effective mass, electron, 287 effective mam, holes, 289 energy gap, 40 holes, heavy, 387ff
impurity levels, pressure effect, 179 optical behavior, pressure effect, 108 piesomistance, 197, 213 Silver halides, absorption edge, pressure effect, 133 “Similarity ” principle, 113 Simon equation, 91 Sodium, compressibility, 69 diffusion, pressure effect, 135 P-V isotherms, 77ff, 86 Solids, equation of state, 736 PVT data, 73, 86 Sound velocity, pressure effect, 147 Space-charge effectti, CdStype material, 247ff Superconductivity, critical field, p m u r e effect, 112ff p m u r e effect, 109ff Superconductors, “hard,” 115
T Tantalum, superconductivity, pressure effect, 119 Tetlon, phase transition, 102 Tellurium, resistivity, pressure effect, 109 Thalliu m, superconductivity, p m u r e effect, 120 Thermal conductivity, gases, pressure effect, 146 helium, solid, preasure effect, 137 Thermal pressure, 74 Thermal quenching, luminascence, 237ff photoconductivity, 237ff
420 Thermoelectric behavior, presaure effect, 63 Thermographic analysis, 81) Tin, cyclotron resonance, 340 superconductivity , preeaure effect, 116ff transition, a-8, 2ff crystallography, 3ff deformation effect, 7 free energy, 11 impurity effects on, 7 kinetics, 5, 8-9 nucleation, 4 temperature, 9-10 Tin, gray, charge carriers, concentration, 32 effective mass, 36, 40 mobility, 33ff, 40 electrical propertiea, 14ff impurity effects, 378 energy gap, 28ff, 40 formation of, Iff growth of, 5 Hall coefficient, 21ff magnetic susceptibility, 26, 30 magnetomistance, 22ff nucleation of, 4 photoconductivity, 27 physical propertiea, 13ff single crystal preparation, 12-13 specimen preparation, 1 Iff
SUBJECT INDEX
stabilization of, 10 thermoelectric power, 23, 25 Transitions, fmt order, 87 second order, 87 Transport properties, semiconductors, 168 Trapping centers, CdStype material, 2438 insulators, 226
U Ultrasonic, compreeeibility measurements, 82
v Vanadium, superconductivity, pressure effect, 122 Viscosity, fluids, pressure effect on, 145
z Zero point energy, melting curve effect, 93 Zinc, cyclotron resonance, 340ff diffusion, pressure effect, 137 Zinc selenide, photoconductivity, 2338, 2548
Cumulative Topical Index A
for Volumes 1 to 10
Aluminum-copper alloys, G. P. zonea in, 9, 336-345 Activity coeficienta, Alums, ordering alloys, 9, 2018 ferroelectric behavior, 4, 21 Adsorption, Ammonium halides, ion gauge studies, 7, 386ff neutron diffraction, 8, 154 Alkali halidea, Amorphous structures, color centers, 10, 128-247 neutron diffraction, b, 163-167 continuum models, 10, 167-168 Anelasticity, semi-continuum models, 10, 188ff elastic constanta, 7,330 crystal properties, 10, 128ff Angular correlation, dielectric constanta, 10, 146ff nuclear radiation, 9,2028 elastic conatante, 7,2798 Angular distribution, electron energy bands, 7, 202ff nuclear radiation, 9, 199-255 ion a i m , 10, 143ff experimental techniques, 9, 2478 optical spectra, 10, 134ff Anomalous skin effect, 6, 3298, 351ff Alkali metals, Antiferroelectrics, 4, 1-197 band calculations with QDM,1 , 184Crystal BtNCtUre, 4, 148-173 190 dielectric behavior, 4, 124ff cohesive energy, 1, 187 domain structure, 4, 147 compremibility, 1, 187 electric field effect, 4, 131ff core polarization, 7, 125 electromechanical behavior, 4, 135, 145 crystal potential, 7, 123ff b t o p e effects, 4, 174ff elastic constanta, 7,2878 optical properties, 4, l45ff electron energy bands, 7, 119-147 Perowkite-type, 4, 1288, 141, 146, equilibrium lattice constant, 1, 187 16lff, 167ff Fermi energy, 7, 1218 solid solutions, 4, 175ff ground state energy, 1 , 185 apontaneous strain, 4, 13Sff Knight shift, 7, 143ff thermodynamics, 4, 124-145 optical behavior, 6, 348 transition heat, 4, 139ff quantum defect data for, I , 174-184 Antiferrornagnetism, Allotropic modifications, neutron diffraction studiea, 8, 184-217 metals, 1, 100 Antimony, sone rething, 4, 467 Alloys, Hall effect, 6,46ff Aromatic crystab, Alloys, ordered, electronic spectra, 9, 1-81 irradiation behavior, 8, 425-433 Aromatic hydrocarbons, Alnico, electronic spectra, 8,26-38 magnetism, 3,613ff Atom displacement, Aluminum, during irradiation, 8, 305-448 elastic constants, 7, 2918 Atomic orbitale, orthogonalimd, I , 54 optical behavior, 6,350 Atomic wave functions, zone refining, 4, 465 bibliography, 4, 413-421 421
422
CUMULATIVE TOPICAL ISDEX FOR VOLUMES
Augmented plane wave method, 1, 86-88 Avalanche radiation, 6, 137
B Band gap, semiconductom, 11, 218ff Band structure, see also electron energy bands bismuth, 9, 277ff compound semiconductors, 111-V, S, 716 germanium, 8, 52ff polar semiconductors, 9, 109ff pressure effect, 11, 170-179 semiconductors, 11, 163ff Bardeen-Cooper-Schrieffer theory, superconductivity, 10, 465ff Barium titanate, crystal structure, 4, 162ff domain structure, 4, 101, 113 electromechanical behavior, 4, 47ff ferroelectric behavior, 4, 21ff hystereais, 4, 121 spontaneous polarization, 4, 56 transition heat, 4, 62 Barkhausen effect, 9,456ff Baroody theory, secondary emission, 6, 287 Bismuth, band structure, 7, 198ff, 9, 2778 De Haas-van Alphen effect, 9, 276287 zone refining, 4, 467 Bloch formulation, magnetic monance, 2, 19-25 Bloch wall, 3, 473-481 energy, 3, 476ff thickness, 3, 475ff Bloch waves, 1, 19-23 Body-centered cubic structure, 7, 4948 zone structure of, 1, 36-37 Body-centered structures, lattice vibrations, 8, 269-290 Bogoljubov theory, superconductivity, 10, 473ff Boltzmann equation, electron transport, 4, 214-287 metals, 4, 217-238 semiconductom, 4, 238-260
1 TO 10
Born-von K h h lattice, two-dimensional, 8,257-263 Born-von Kbrmh theory, d , 237-248 Bow gau model, superconductivity, 10, 422ff Bragg-William approximation, orderdisorder transition, I, 210 Bravak latticee, 6,194-21 1 Brillouin zones, 1, 26-38, 2, 252 Bubble model, grain boundaries, 8, 391 Burger’s formula, dislocations, 10, 26Uff C
Cadmium sulfide type materials, conductivity, 11, 230ff Hell elfect, 11, 230ff imperfections, 11, 223-260 ionization energies, 11, 249ff luminescence, 11, 233ff, 243ff optical behavior, 11, 232, 24Off photoconductivity, 11, 233-247 Carrier mobilities, germanium, 8,Mti Cellular method, 1, 61-73 empty lattice test, 1, 72 Cellular precipitation, 3,301 Central forces, d, 248.251 CRsium chloride structure, 7, 496ff grain boundark, 8, 339 Cesium, quantum defect data, 1, 181, 184 Charge carriers, recombination and trapping in valence semiconductors, 1, 354-305 Charge transfer spectra, 9,442ff, 602-511 Charge transport, luminescent materials, 6, 154ff Chemical shift, 8, 57ff, 108-110 Close-packing, 7, 4388 Coercive force, 3,452ff, 5196 order, effect of, 1, 267 small particlea, 9,607-519 Coheaion in metals, I , 97-126 table, 1, 108 Cohesive energy, alkali metals, 1, 187 metals, interpretation, 119-123
CUMULATIVE TOPICAL INDEX FOR VOLUMES
Collision theory, coulomb encounters, basic equations, I,311-328 energy relations, classical, E, 311-314 Color centers, alkali halides, 10, 128247 continuum models, 10, 167-188 semi-continuum models, 10, 188ff electron nuclear double resonance, 10, 141ff electron spin resonance, 10, 141ff hyperfine interactions, 10, 162ff, 218ff point-ion-lattice models, 10, 210-244 Composite lattices, application of cellular method to, 1, 73 Compound semiconductors, electroluminescence, 8, 164ff electron energy bands, 7, 176ff Compound semiconductors, 111-V, 3, 1-79 band structure, 3,716 binding properties, 3, llff conductivity, 3, 128 Hall effect, 3, 126 heat conductivity, S, 42ff infrared absorption, 3,49ff phsee diagrams, S, 6ff preparation, 3,8K reaistivity, magnetic field effect, 3, 33ff thermopotential, 3,42ff Compmibility, alkali metals, 1, 187 met&, 6, 798 Conduction bands, germanium, 6,273 silicon, 6,272 Conduction electrons, Bee electrons, conduction Constitutional supercooling, 4, 605 Copper, band structure, 7, 193 pile irradiation, E, 408 radiation damage recovery, I,411-424 radiation effects, 2, 398.41 1 Core polarization, correction for in multivalent atoms, 1, 165-169 Correlation effects in cohesion, 1, 103-106
1 T O 10
423
Correlation energy, of free electron gas, 1, 391-400 lattice effects on, 1, 405 Coulomb encounters, heavy particles, 2, 335-336 in relativistic range, 8, 328-336 Rutherford electron scattering, E, 329-334 screening, effect of, 8, 33&336 theory, 2, 311-328 Cross modulation, cyclotron reaonance, 11, 374E Cryolite structures, 7, 465 Crystal dynamics, neutron scattering relations, 8,161168 Crystal field theory, 9, 400-438 Crystal growth, 3, 2798, 289 gray tin, 11, 11-12 from melt, 4, 501ff polar semiconductors, 9, 948 techniquea, 4, 423-621 Crystal imperfections, ~ e imperfections e in crystals Cryatal orbital methods, 1, 84-88 Crystal orbitals, general properties, 1, 18-23 Crystal perfection, measurement, 4,5196 Crystal potential, 7, 1068 O y S t d S t N C t U R , 7, 426-503 elements, 7,474 nuclear quadruple resonanoe relation, 6,395-405 polar semiconductors, 9 , s Crystal symmetry, classification, 6, 186ff operators, 6, 182 piezoelectric effect, 11, 152 point groups, 6, 183ff property relations, 6, 175-249 Crystal-vapor equilibrium, 3,3298, 342, 347, 378, 418 Crystals, cleesification, 6, 186ff nucleation, 3,283ff point groups, 6, 183ff tensor properti-, 8, 196-249 Cubic crystals, nuclear quadrupole spectra, 6,368-383
424
CUMULATIVE TOPICAL INDEX FOR VOLUME8
Cubic lattices, space group repreeentation, 6,224229
Curie temperature, order, effect of, 1 , 268 Cyclotron resonance, 11, 261-400 amplifiers, 11, 383ff charge carriers, 11, 276-370 c m modulation, 11, 3748 free electrona, 11, 264-273 generators, 11, 383ff infrared, 11, 355-370 ions, 11, 273ff metale, 11, 312-343 millimeter wavelengths, 11, 370ff semiconductors, 11, 280-311 solids, 11, 276-400
D d bands,
structure and width of, 1, 107-116 Debye temperature computation, 8,257 Debye theory, specific heat, 9,225-237 “Defect, 1))’’ 1, 157-166 De Haas-van Alphen effect, 9, 257-291 bismuth, 9,276-287 metals, 9, 287ff semiconductors, 9,257-291 Deaorption, ion gauge studies, 7, 3Diamagnetic alloys, effect of order on, 1, 261-262 Diamagnetinm, orbital electron, electron interaction effect on, 1, 424 Diamond, band structure, 7, 1Mff localiz6d bond orbitals for, 1, 91 positron annihilation, 10, 45 Diamond structure, 7, 478 lattice vibrations, 8, 290-303 space group repreeentation, 6,231 mne structure of, 1, 38 Dielectric, surface charge distribution, 9, 195 Dielectric behavior, antiferroelectrics, 4, 124ff ferroelectrics, 4, 12-29 Dielectric. breakdown, 6, 102ff
1
TO
10
Dielectric conatantrr, alkali halides, 10, 146ff Dielectric loss, impurity effecte, 9, 392ff Dielectric susceptibility, crystal symmetry relation, 6, 217ff Diff usion, ace also selfdiffusion binary alloys, 10, 101-1W experimental techniques for, 10, 109114 grain boundaries, 8, 454-468,10, 1216 high pressure, 11, 135ff impurity effects, 3,3928 metals, lO,71-126 chemical, 10, 119ff grain boundaries, 10, 121ff impurities in, 10, 958 selfdiffusion, 10, 115ff streaaed, 10, 123ff surface, 10, 121ff polar semiconductors, 8, 98ff temperature dependence, 10, 80-89 theory, 10, 73-101 Dipolar broadening, nuclear magnetic resonance, d , 39-47 Dislocation models, grain boundaries, 8, 329-358,393-419 Dislocation rings, from displacement spikes, 9,369.370 Dislocations, Burger’s formula, 10, 260ff continuous distributions, 3, 136ff continuum theory, 10, 240-291 elastic constante effect, 7, 34211 energy, 10, 265, 277-280 helical, 10, 290 image effecta, 3, 127ff interaction energy, 3, 1238 Kroner’s theory, 10, 267-282 in melt grown crystals, 4, 513ff motion, 9, 130ff nuclear quadrupole effects, 6,411ff optical behavior effect, 6,4ooA partials, 8, 416 Peach-Koehler formula, 10, 2836 Somigliana, 9, 89 in valence semiconductors, 1, 319 Dislocations, edge, interaction between, 8, 393ff
CUMULATIVE TOPICAL INDEX FOR VOLUMES
Dislocations, screw, interaction between, 8, 408ff Disorder, antistructure, 3,347 from thermal spikes, 4, 360 Displaced atoms, in radiation damage, b, 390-391 Displacement spikes, a, 323-327,368-375 in electron bombardment, a, 430 Displacement theory, for radiation darnage, 8, 378-391 Distribution coefficient, solid-liquid, 4, 426-437 Domain structure, 8Ce also ferromagnetic domains antiferroelectrics, 4, 147 ferroelectrics, 4, 97-124 Drude theory, 6, 321 Dugdale-MacDonald relation, 6,44, 6Off Dulong-Petit law, b, 220-221
E Effective mass, in semiconductors, 1, 290-291 valence semiconductors, I, 298-303 Einstein theory, specific heat, 8, 221-225 Elastic constanta, 7, 213-351 alkali halides, 7, 279ff alkali metals, 7, 287ff alloying effect, 7, 332ff aluminum, 7, 291ff anelasticity effecta, 7, 330 atomistic theories, 7, 226-243 definition, 7, 2248 dislocations effect, 7, 3428 ferromagnetic materials, 7, 305ff hexagonal metals, 7, 2968 meaaurement of, 7, 256-272 noble met&, 7, 287ff order, effect of, 1, 272-276 phase transitions effect, 7, 338ff pieaoelectrice, 7, 3108 polycrystah, 7, 316ff pressure effect, 7, 3208 radiation damage effect, 7, 345 rare gas aolids, 7,303 from mund velocity, 4, 263-269 superconductivity effect, 7, 341
1
TO
10
425
temperature effect, 7, 320ff zinc blende structure, 7, 299ff Elastic deformation, semiconductivity effect, 11, 149-221 Elastic field, energy-momentum tensor, 3, 10lff Elasticity, crystal symmetry relation, 6,241ff theory, 3,84ff, 10, 252-257 Elasticity moduli, 7, 219ff semiconductors, f 1 , 149-221 Elastoresistance, wmimetals, 11, 210-211 Electrical resistivity, crystal symmetry relation, 6,208-217 high pressure, If, 103-123 metals, 4, 217ff order, effect of, 1, 253-261 theory, 4, 200-366 Electrodynamics, superconductivity, 10, 366-401 Electroluminescence, 6, 166ff, 6,95-173 compound semiconductors, 6, 104ff emission, 6, 128-137 energy transfer in, 6, 125-128 exritation, 6, 113-125 valence semiconductors, 6, 16Off zinc sulfide, 6, 137-160 Electromagnetic theory, 6, 316ff, 355ff Electromechanical behavior, antiferroelectrics, 4, 135, 145 ferroelectrics, 4, 33-51 Electronelectron, scattering, metals, I, 414-416 Electron emission, secondary, 6,251-311 angular distribution, 6, 305 elemental semiconductors, 6, 257 field dependent, 6,306ff imperfections effect, 6, 303 incidence angle dependence, 6, 2978 insulators, 6, 262, 300 metals, 6,2548 theory, 6,263ff, 286ff time lag, 6,305 velocity distribution, 6,311ff yield, 6,2548 Electron energy ban&, 7, 99-212 alkali halides, 7, 2028 alkali metals, 7, 119-147 compound eemiconductors, 7, 176ff
426
CUMULATIVE TOPICAL INDEX FOR VOLUMES
density of states, 7 , 113ff elemental semiconductors, 7, 158-176 noble metals, 7, 1938 transition metals, 7, 180-197 Electron excitation, energy loss from, le, 347-350 during irradiation, 8, 338-351 Electron gas, correlation energy of, I, 391 Electronic conductivity, impurity effects, 3,392ff oscillatory behavior, 9, 257-291 Electronic spectra, aromatic hydrocarbons, 8,26-38 ionic crystals, 9, 400-525 theory, 9,400-452 lanthanides, 9,453-476 molecular crystals, 8, 1-47, 9, 1-81 “oriented gas ” model, 9, 69 transition metal ions, 9, 476-502 Electron interactions, collective description, 1, 376-407, 446 metals, 1 , 367-450 lattice effects, 1, 400-407 relation to many-body problema, I, 449 Electron-lattice interaction, I, 441-449 Electron nuclear double reaonance, color centers, 10, 141ff Electron paramagnetism, in metals, 2,93-136 theory, 8, 98-101 for various elements, 8, 116 Electron-phonon interaction, 7, 748 Electron spin resonance, 8, 84 color centers, 10, 141ff Electron transport, Boltzmann equation, 4 , 214-287 metals, 7, 353-378 acattering in, 4, 287-366 theory, 4, 200-366 Electrons, correlated motions of, I, 14 exchange energy of, I, 92-95 mobility in solids, 4 , 200-366 Rutherford scattering of, 8, 329-334 Electrons, conduction, angular momentum, 7, 365-374 impurity scattering, 7, 365-374 phonon interactions, 7, 374
1 TO 10
scattering of, theory, g, 393ff Electrons, core, exchange interaction with valence electram, I, 169-174 Electrona, free, cyclotron rewnance, 11,264-273 statistical theory, 4, 203-214 Electrona, primary, back scattering, 6,2738 characteristic energy losses, 8,283ff reflection, 8, 279ff Electrons, valence, exchange interaction with core electrons, I, 169-174 Electro-optic effect, ferroelectrics, 4, 886 Elemental semiconductors, Bee also valence semiconductors electron emission, secondary, 8,257 electron energy bands, 7, 158-176 gray tin, 11, 1-40 Elementary particlea, decay, 9, 210, 245 Energy, atructural, 8,68ff Energy bands, see oleo electron energy bands Energy bands calculation, quantum defect method, I, 140-144 Energy bands in cryatsla, I, 38-44 Energy gap1 polar semiconductors, 9, 111-118 valence semiconductors, I, 303-309 Entropy, structural, 6,688 Equation of state, fluids, 11, 142ff Mie-Grueneisen, 8,41 solids, 22, 73-103 from shock wave techniques, 8, 40-60 Exchange energy, free electrona, I, 92 tightly bound electrona, 1, 94 Exchange energy, magnetic, 3,4571 Exchange interaction, one-electron approximation, 2,Exchange mechaniame, indirect, 8, 214-217
CUMULATIVE TOPICAL INDEX FOR VOLUMES
Excitone, molecular crystals, 8, 24
F Face-centered cubic structure, grain boundaries, 8, 3408, 356 zone structure of, 1, 34-36 Face-centered structures, lattice vibrations, 2, 269-290 F center, model, 10, 2 11 ff, 227ff Fermi energy, alkali metals, 7, 121ff non-stoichiometric compounds, 9, 360 Ferrielectrice, 4, 7 Ferrimagnetrc compounds, neutron diffraction, 8, 191-195 Ferroelectrics, 4, 1-197 crystal structure, 4, 148-173 dielectric behavior, 4, 12-29 domain structure, 4, 97-124 electromechanical behavior, 4, 33-51 electronic polarization, 4, 1808 electro-optic effect, 4, 88ff hyeteresis, 4, 118ff ionic polarization, 4, 1876 isotope effects, 4, 1738 neutron diffraction, 8, 158-162 optical behavior, 4, 88-97 orderdisorder treatment, 4, lQlff phase transitions, 4, 60-08 polarization, 4, 51-60 refractive index, 4, 96 solid SOlUtiOM, 4, 1758 strain, spontaneous, 4, 61-60 thermodynamics, 4, 68-88 transition heat, 4, 60-65 Ferromagnetic domains, boundaries, viscous damping, 3, 5486 boundary motion, 3, 534-567 cavitiee, 3, 601 energy, 3,457-473 experimental studies, 9,49Off flux-closure configurationa, 3, 483 grain boundaries, 9, 502 origin, 3, 4458 eim, 3,450 small particles, S, 502-519 structure, 9, 481-490
1 TO 10
427
theory, 3,437-564 wall maee, 3,546 wall stiffness, 3,645 Ferromagnetism, electron interactions, effect OD, 1, 427 order, effect of, 1, 262-272 Field emission microscope, experimental studies, 7, 394-420 grain-boundary studies, 8, 373 Field ion micmecope, 7 , 420ff Fine particles, magnetism, 9, 502-539 Fluids, equation of state, 11, 142ff Fluorescence, molecular crystals, 8, 41ff Fluorite structures, 7, 4 4 f f Fock equation, 1, 3-14 c*rystalorbitals in, f, 12 localized solutions, f, 12 Franek-Condon principle, 6, 387ff Frenkel defecta, 3,3338, 342 thermal spike generation of, 2,364-306 G
Galvanomagnetic behavior, high frequency, 6,506 liquid metals, 6, 52ff metals, 6, 1-96 General precipitation, 3,299 Germanium, acceptor states, 6, 297-306 field effecta, 6, 309 magnetic susceptibility, 6, 312 band structure, 7, 1728, 8, 52ff carrier mobilities, 8, 66ff conduction bands, 6, 273 conductivity tensor, 4 , 277ff crystal perfection, 8, 65 crystal preparation, 8, 64-78 crystal purity, 8, 65 displacement threehold, d , 435 donor states, field effects, 6, 308-31 1 magnetic susceptibility, 6, 312 electronic structure, 8, 52-56 impurity states, 8, 54ff impurities, 6, 258-320 diffusion, 8, 758
428
CUMULATIVE TOPICAL INDEX FOR VOLUME8
double resonance, 6, 2668 electron spin resonance, 6, 266K encrgy levels, 5, 261ff, 8, 708 lattice vibration interaction, 6, 316 magnetic susceptibility, 6, 27 1 optical absorption, 6, 317 solubility, 8, 75R infrared detection by, 8, 104ff irradiation of, d , 433-441 photoconductivity, 8, 49-107 valence bands, 5, 300 zone refining, 4 , 463 Glasses, thermal conductivity, 7, 67 Gold, pile irradiation, d , 408 quenching of defects in, 8, 3 9 6 3 9 8 Grain boundaries, see also sub-boundaries body-centered cubic structure, 8, 344ff bubble model, 8, 391 cesium chloride structure, 8, 339 diffusion, 8, 459-468, 10, 12117 dislocation models, 8,329-358,393-419 electrical properties, 8, 491ff energy, 8, 414-435 experimental studies, 8, 358-391 face-centercd cubic structure, 8, 340ff, 356 melting, 8, 4738 migration of, 8, 47G491 precipitation, 8, 468-473 properties, 8, 327-499 resistance, 8, 4928 segregation, 8, 468-473 semiconductors, 8, 492ff structure, 8, 327-499 tilt, 8, 350 twin, 8, 434 twist, 8, 4146 Grain growth, 8, 476-491 Graphite, band structure, 7, 1598 positron annihilation, 10, 43 Green's tensor function, 10, 254ff Grueneisen constant, volume dependence, 6, 438 Guinier-Preston zones, 9, 293-393 see also zones kinetics of formation, 9, 3038
1 TO 10
H Hagens-Ruben relation, 8, 319 Halides, structures, 7, 449-467 Hall coefficient, order, effect of, 1, 261 Hall effect, alloys, 6 , 46ff cadmium sulfide type materials, 11, 230ff compound semiconductors, 111-V, s, 12R metals, 5, 15-54 polar semiconductors, 9, 118 superconductors, 4 5 3 theory, 6, 38ff in valence semiconductors, 1, 326336 Hartree equation, 1 , 11 Hartrer-Fock equation, 1 , 1W139, 7, 104ff, 10, 152ff Heitler-London method, I , 88-91 Helium, liquid, positron annihilation, 10, 53ff Helium, liquid 11, excitations, 8, 162 Hexagonal lattice, space group representation, 6, 234 zone structure of, 1 , 37 High pressure, diffusion, 11, 1358 electrical behavior, 11, 103-123, 129 experimental techniques, 11, 4478 magnetic behavior, 1 I , 1236 optical behavior, 11, 1308 High pressure physics, 1I , 41-147 Hole energy calculations, f , 123-126 Hooke's law, 2,263 Hydrides, neutron diffraction by, 8, 150 Hydrogen bonded structures, 7, 476, 482 Hydrogen, metal, band structure, 7, 145 Hydrogenic wave equation, solutions, 1, 190-192 Hyperfine coupling constant, 8, 119-131 Hyperfine interactions, color centers, 10, 162ff, 2188
CUMULATIVE TOPICAL INDEX FOR VOLUMES
Hystensis, ferroelectrica, 4,118ff magnetic, 3, 4628 I
1% neutron diffraction, 2, 151 Imperfections in crystals, S, 307-436 OM ah0 lattice defects atomic, 3, 3106 cadmium sulfide type materials, 11,223-260 ionization energies, 11, 249ff electron emission, wcondary, 6, 308 electronic, 3, 316ff electronic conductivity, 9, 328ff equilibria, S, 3238 Frenkel, 3,333, 342 impurity atoma, 3,373-416 insulators, 11, 223ff ionic crystals, nuclear quadruple studies, 6,412-422 metals, 6, 363 migration, 3, 326tf nonmetals, optical behavior, 6, 355-411 nuclear quadruple effecb, 6, 352, 363, 410-429 photoconductivity, 11, 225ff Schottky-Wagner, 3, 344, 349 thermal conductivity effect, 7, 15ff, 69ff, 9Off x-ray diffraction effects, 9, 296-305 Impurities, control in crystal growing, 4 , 472-493 in valence semiconductors, 6, 268-320 Independent electron model, 1, 371-374 Infrared, cyclotron resonance, 11, 355-370 Infrared detection, germanium crystal method, 8, 104ff Insulators, imperfections, 11, 223ff Interfaces, in solids, 3, 269ff thermodynamics, 3,236ff Ionic conductivity, 3,326ff impurity effecta, 3,3928
1 TO 10
429
Ionic crystals, charge transfer rpectra, 9, 442ff, 502-51 1 crystal field theory, 9, 400-438 electronic spectra, 9, 400-525 imperfections, nuclear quadruple studiea, 6, 412-422 irradiation behavior, 8, 308-311 molecular orbital theory, 9, 438ff nuclear quadruple effect, 6, 346-358 positron annihilation, 10, 41ff radiation damage in, 8, 442-448 spin-orbit coupling, 9, 4248 Ions, cyclotron resonance, 11, 273ff Ion sizes, alkalr halides, 10, 143ff Iron, band structure, 7, 189ff zone refining, 4, 465 Ising model, orderdisorder transition, 1, 2378 theory, 9,204-210 Ising problem, approximate solutions, 3, 192ff Ivey's laws, 10, 1348
J Jonea mne, Brillouin mne, comparison with, 1,32
K Kerr effect, 3, 495ff Kersten theory, 3, 521ff Kinetica, orderdisorder transitions, 3,2lOff phase transitions, 3,21 1, 252-306 precipitation, 9,298-306 solid state transitions, 3, 293ff solidification, S, 281-293 Knight shift, 8, 54ff,93-134 alkali metals, 7, 143ff in alloys, 2, 129-133 magnetic field effects, 2, 126-129 measurement of, 2, 110-113 metals, 2, 121-123 temperature effect, C, 120-129
430
CUMULATIVE TOPICAL I N D E X FOR VOLUMES
Knocked-on atoms, behavior, S, 319-321 Kroner's theory, dislocations, 10, 267-282
L Landshoff-Lawdin method for ionic crystals, 1,88-91 Lanthanides, electronic spectra, 9, 453-476 properties, 9, 453 Lattice, normal modes, 7, 7 8 Lattice constant, alkali metals, I, 187 Lattice defects, see also imperfections continuum theory, 9, 79-144 point, in anisotropic media, 3, 122 crystal distortion effect, 3, 107ff solid solutions, 3, 115ff x-ray diffraction effects, 3, 113ff in semiconductors, 1,285-287 in valence semiconductors, 1, 315-321 Lattice dynamics, 8, 187 Lattice energy, polar semiconductors, 9, 86 Lattice vibrations, 2, 219-303, 7, 1-98 in body-centered structures, 8,269-290 in diamond structure, S, 290-303 in face-centered structures, 2, 269-290 frequency distribution, 8, 251-257 heat transport by, theory, 7, 7-45 effect on orderdisorder transitions, 1,246-250 Laves phases, 7, 472 Linear Combination Atomic Orbitals (LCAO) Method, 1, 5 S 6 1 Liquids, crystallization kinetics, 3,2818 neutron diffraction, 2, 163-167 positron annihilation in, 10, 1-69 Liquid-vapor equilibria, 3,370ff Lithium, band structure, 7, 1286 quantum defect data, 1, 178, 182 Low temperature, experimental techniques, 11, 60-73
1
TO
10
Luminescence, 6,97-172, 6, 105ff absorption processes, 6,99-119 cadmium sulfide type materials, 11, 233ff, 2438 continuous dielectric model, 6, 116ff cmission procetJse8, 6, 99-119 energy transfer in, 6,1198 impurity-sensitized, 6, 1228 phosphors, 6,146ff zinc oxide, 8, 223-236
M Magnetic anisotropy, order, effect of, 1, 266 Magnetic anisotropy energy, 9, 463ff Magnetic energy levels, periodic lattice, 9,274 Magnetic field effect, thermal conductivity, 6, 54ff thermoelectric power, 6,618 Magnetic form factors, 8,208-214 Magnetic Properties, electron interaction effects on, 1, 416-429 high pressure, 11, 1238 valence semiconductors, 1, 296-208 Magnetic resonance, 8, 2-25 aee obo nuclear magnetic reeonanca Bloch formulation, d, 19-25 rotating coordinate methoda desorip tion, t, 8 Magnetic susceptibility, metals, 2, 113-119 oscillatory behavior, 8, 257-291 Magnetism, neutron diffraction studies of, 2, 177-217 rocks, 3,513 Magnetoacoustic resonance, f 1, 379ff Magnetocrystalline anisotropy, 3, 507 Magnetoelmtic energy, 3,4678 Magnetoplasma phenomena, 11,343-355 Magnetoresistance, anomalous, 6,34ff metals, 6,15-54 theories, 6,24ff in valence semiconductors, I, 336-340 Magnetostatic energy, 3, 472ff Magnetostatice, 10, 268ff
CUMULATIVE TOPICAL INDEX FOR VOLUMES
Magnetostriction, 3, 471ff order, effect of, 1, 268 Manganeae, magnetic structures, 2, 191 Many-body problems, relation of electron interaction to, 1, 449 Many-electron problem, 7, l O l f f Martensite transitions, jl 2958 Mathieu equation, one dimensional, momentum eigenfunctions for, 1,41-44 Mathieu problem, I, 38-44 M center, 10, 234ff Mechanical properties, order, influenced by, I, 272-282 Meissner effect, superconductivity, 10, 31 1-366 Melting, 9, 241ff, 291 order-disorder treatment, 6, 766 Metals, band structure, 7, 119-158 compressibility, 6, 79ff cyclotron resonance, f l , 312-343 De Haas-van Alphen effect, 9, 287ff diffusion, 10, 71-126 impurities, 10, 958 electron-electron scattering, I, 414-416 electron emission, secondary, 6, 2548 electron interaction in, 1, 367-460 electron paramagnetism in, 9, 93-136 electron transport, 7, 353-378 theory, 4, 217-238 electrons, scattering of, 4,287-332 galvanomagnetic behavior, 6, 1-96 Hall effect, 6,15-54 imperfections, nuclear quadrupole studies, 6, 4116 irradiation behavior, 8, 308-311 magnetoresistance, 6, 15-54 nuclear magnetic resonance in, 1, 420, .9, 93-136 optical behavior, 6, 313-352 order-disorder transitions, 3, 140-223 p h w transitions, 6, 65-94 positron annihilation, 10, 36ff radiation damage, 9, 391-433 selfdiffusion, 10, 115ff specific heat, electron interaction effects, 1, 407-409
1
TO
10
431
thermal conductivity, 7, 7 0 thennomagnetic behavior, 6, 1-96 transport properties, 1, 412-414 x-ray emission spectra, 409-412 Mie-Grueneieen, equation of state, 6, 41 Mobility, polar semiconductors, 8, 1l9ff Moirb fringea, grain boundary studies, 8, 388ff Molecular crystals, electronic spectra, 8, 1-47, 8, 1-81 anisotropy effects, 8, 23 aromatic hydrocarbons, 8, 26-38 energy levels, 8, 12 solid solutions, 8, 38 symmetry considerations, 8, Sff transition probabilities, 8, 16fl excitons, 8, 24 fluorescence, 8, 418 vibrationelelectronic states, 8, 1W Molecular orbital theory, 9, 438ff, 10, 2036 Molecular solids, irradiation behavior, t , 308-31 1
N Neb1 theory, 3, 525ff Networka, periodic, 7, 429-434 Neutron collisions, t , 33G338 Neutron diffraction, amorphous structures, 8, 163-167 antiferromagnetic structures, 8, 184-217 from crystals, 8, 149-177 ferroelectric structures, 9, 158-162 by hydrogen-containing structures, t, 149-177 liquids, 2, 163-167 magnetic studies, t , 177-217 rare earth metals, !?,183 solid state problems, application, 8, 137-217 transition elements, 8, 177-217 Neutron scattering, croae-eection, 8, 116-131 crystal dynamics relations, 8, 151-188 inelmtic, 9,167-177 phonons, 8, 131ff
432
CUMULATIVE TOPICAL INDEX FOR VOLUME8 1 TO
theory, 8, 112-139 from various elements, d, 143-145 Neutrons, see obo thermal neutrons Neutrons, cold, scattering, graphite, 8, 144ff isotropic lattices, 8, 139ff lead, 8, 149 Nickel, band structure, 7, 192 Nickel arsenide structure, 7, 484ff Noble metals, elastic constants, 7, 287ff electron energy bands, 7, 1936 optical behavior, 6, 343ff Normal modes, lattice, 7, 7ff Nuclear magnet, 8, 28 Nuclear magnetic relaxation, 2, 12-19 Nuclear magnetic resonance, 8, 1-91 chemical shift, 8, 108-110 impurity effects, 8, 106-108 in liquids, d, 67ff measurement, 8,25-39 metals, 8, 93-136 electron interaction effect on, 1, 420 magnetic field effect, 2, 105-106 temperature effect, 8, 105-106 volume effect, 8, 105-106 quadrupole effects, 6, 321-438 spectrum characteristics, 8, 39-67 superconducting transition effect, 2, 133-134 Nuclear quadrupole effects, dislocations, 6, 4118 ionic crystals, 6, 346-358 in solids, 6, 321-438 Nuclear quadrupole Hamiltonian, 6,326ff Nuclear quadrupole interaction, 8, 5054 Nuclear quadrupole momenta, 6, 396 Nuclear quadrupole resonance, crystal structure effects, 6,395-405 experimental techniques, 6, 394 Nuclear quadrupole spectra, 6,333-345 imperfection effects, 6, 352, 363, 410-429 line broadening, 6, 365-383 orientation dependence, 6, 337
10
powders, 6,3388 single crystals, 6, 3408 Nuclear radiation, angular correlation, 9,2028 angular distribution, 9, 198-266 experimental techniquea, 8,2478 Nuclear relaxation in solids, t, 77-84 Nuclear spin equilibrium, 2, 7 Nuclear spin-lattice relaxation, 8, 67-91 quadrupole contribution, 6, 383-394 Nuclear spin-spin interactions, electron coupled, d, 58-67 Nucleation, in condensed systems, 3,2668 crystals, 3, 283ff heterogeneous, 3, 2758,286 homogeneous, 3,266,283 liquid in vapor, 3, 257ff orderdisorder transitions, 3, 213 of precipitates, 3,269ff theory, 3,256-279 transient, 3,268 Nuclei, oriented, 9, 206 0 Octahedral coordination, 7, 449-463 Olivine structure, 7, 468 One-electron approximation, 6, 362 exchange interaction in, I , 92-95 historical survey, 1, 15-18 One-electron potential, 1, 130-135 One-electron theory of solids, 1, 1-95 Optical behavior , alkali metals, 6, 348 aluminum, 6, 350 anisotropy effects, 6, 395ff antiferroelectrica, 4, 1Gff cadmium sulfide type materials, 11, 232, 240ff colloidal particles effect, 6,402 dislocations effect, 6, 4008 ferroelectrica, 4,88-97 high pressure, 11, 13OtT imperfections in nonmetab, 6, 356-411 metab, 6, 313-352 experimental methods, 6,3368 noble metab, 6,343ff polar semiconductare, 8, 109-135
CVYULATIVE TOPICAL INDEX ?OR VOLUYEB
eemiconductora, stream effects, 11, 2148 valenoe semiconductors, 1,294-296 rinc oxide, 8, 216-236 Optical spectra, 10, 134ff Order, influence on electrical properties, 1, 262-261 effect of irradiation on, I,4331 long-range, 3, 152ff, 201, 210 mechanical properties, influence on, I, 272-282 short-range, 3, 152ff, 201, 210 Orderdiaorder, in ferromagnetic alloys, 1, 208-272 magnetic properties, influence on, 1, 261-272 Orderdieorder transitions, 1, 203 SIIOYS,1, 193-282 theory, 1, 205-252 Bragg-Williama approximation, 1, 210 cooperative nature, S, 188ff Ising model, 1, 237ff kinetica of, 1, 250-252, S, 210ff metals, 3, 146-223 energy change, 3, 198ff nucleation, S, 213 quesi-chemical theory, 1, 227-237 statistical theory, S, 191-210 Takagi method, 1,227 temperature effect, 3, 178 thermodynamics, 3, 178-191 Order parameters, 1, 197 measurements, 3, 164ff, 2OOff Ordered alloys, activity coefficienta, 3, 201ff saturation moment of, 1, 262 structure, 5, 148-178 “Oriented gaa” model, electronic spectra, 9, 69 Orthogonal transformations, 6, 175-177 Orthogonaliced plane-wave method, I , 74-83, 4, 367-411 Chcillatory behavior, electronic conductivity, 9, 257-291 magnetic susceptibility, 9, 257-291 Overhauser polarization of nuclei, I,87 Oxides, structures, 7, 449-467
1 TO 10
433
P Pairdenaity functions, 3, 148, 166 Parmagnetic dloya, effect of order on, 1, 261-262 Param.gnetic scattering, neutrom, S, 190-208 Paramegnetiam, 8,236ff Paramagnetism, spin, electron interaction effecta on, 1, 416 p bands, structure and width of, 1, 107-116 Peach-Koehler formula, dislocations, 10, 263ff Periodic networks, 7, 42Q-434 Perovakiten, magnetic structure, 8, 191 Perovskite structure, 7, 464 Perovakite-type, antiferroelectrica, 4, 128ff, 141, 146, 1616, 167ff Perovskite-type compounds, neutron diffraction, d , 19tj-1W Phase stability, theory, S, 240-252 Phase transitions, 3, 226-306 elastic ~ ~ t r m t 7, a ,338ff ferroelectrica, 4,60-68 gray tin, 11, 2-11 kinetica, S, 211, 262-306 formd theory, 3, 262ff metals, 6,6!j-94 nuclear quadruple atudies, 6, Uff shock wave effects, 6, 12ff strains, 5, 272ff fhermodynamica, 9,226-240 Phonon drag, 4,367ff Phonons, electron interaction, 7, 74ff, 374 neutron scattering, 8, 131ff Phosphors, ZnStype, 6, 1468 Photoconductivity, cadmium sulfide type materials, 11,233-247 germanium, 8, 49-107 extrinsic, 8, 798 intrinsic, 8, 92ff quenching, 8, 101 euperlinearity, 8, 103
434
CUMULATIVE TOPICAL INDEX FOR VOLUMES
imperfection effecta, 11, 2258 recombination process, 8, 59ff theory, 8, 56-64 Photoelectric effect, internal, 6, 326ff Piesoelectricity, crystal symmetry relation, 6, 2248, 11, 152 thermodynamics, 4,68-81 Piezoelectrics, elastic constants, 7, 3106 fiemgalvanomagnetic effects, 11, 153ff Piesoresistance, 11, 150-162 crystal symmetry relation, 6, 232ff meaeurement, 11, 156ff preasure effect, 11, 166 shear stress effect, 11, 157ff Pile irradiation, copper and gold, I, 408 Plasma, degrees of freedom, 1, 382 Plasma oscillation, frequency of, 1,400 solids, excitation of, 1, 429-441 Plastic properties, order, effect of, 1, 276-280 Point defects, see lattice defects, imperfections Point groups, 6, 180-194 crystals, 6,1838 Point-ion-lattice models, color centers, 10, 210-244 Polarization, crystals, 6,1978 ferroelectrics, 4, 51-60 Polar semiconductors, band structure, 9, 109ff crystal growth, 9, 94ff cry8tal structure, 9, 85 diffusion, 9, 98ff electrical properties, 9, 109-135 energy gap, 9, 111-118 Uall effect, 9, 118 impurities in, 9, 106ff lattice energy, 9, 86 mobility, 9,119ff optical behavior, 9, 109-135 plaaticity, 9, 87ff recombination in, 9, 134 thermal conductivity, 9, 133 thermoelectric power, 9, 131
1 TO 10
trapping in, 9, 134 vacancies, 9, 92 Polygonization, 8, 442-459 Polyhedra, coordination, types, 7,470 open packings, 7, 437 space filling arrangementa, 7,435 Polymorphic transitions, 3,2461 Positron annihilation, 10, 1-69 diamond, 10, 45 gamma rays, angular correlation, 10, 22' -36 graphite, 10, 43 helium, liquid, 10, 53ff impurity effects, 10,64ff ionic crystals, 10, 416 lifetime, 10, 22-46 metals, 10, 36ff superconductors, 10, 44 Positronium, 10, 468 Positrons, mobility in diamonds, 10, 45 Potassium, band structure, 7, 136ff quantum defect data, 1 , 179-180, 183 Precipitation, continuous, 9, 324 discontinuous, 9,324 grain boundaries, 8, 468-473 kinetics, 3,298-306 Pre-precipitation, 9, 325ff Pressure, ace abo high pressure Pressure scale, 11, 44ff Proton exchange, I,63 Proton magnetic resonance, in 1,2dichloroethane, 8, 47-SO Pyroelectric effect, 6,200-207
Q Quantum defect method, 1, 66,127192 energy bands calculation, 1, 140-144 Kuhn and Van Vleck treatment, 1, 136-139 low energy extrapolation, 1, 167-11 WKB justification of, 1, 150-157 Quadrupole interaction, 9,217-232
CUMULATIVE TOPICAL INDEX FOR VOLUMES
Quadrupolar relaxation, 6,383-394 Quasi-adiabatic approximation, 10, 148ff Quasi-chemical theory, orderdisorder transition, 1, 227-237
B Radiation damage, 8, 306-448, 9, 2418 in copper, 8, 398-411 cross-eections in, 8, 321-327 displacement theory, I?, 37&391 elastic ronstants, 7, 345 electron excitation, 8, 338-351 ionic crystals, 8, 442-448 in metals, 2, 391-433 ordered alloys, 2, 425-433 recovery from, 2, 41 1 4 2 4 thermal conductivity, 7, 69 valence crystals, 2, 433-442 hnge-energy relations, 2, 350-351 Rare-earth metals, neutron diffraction, 2, 183 Recombination, luminescent materials, 6, 15M Recovery, from radiation damage, 3,411424 Recrystallisation, 8, 479 Refractive index, ferroelectrics, 4, 96 Resistivity, w e electrical resistivity Retrogression, 8e4 reveraion Reversion, tones, Q, 367ff Rochelle salt, crystal structure, 4, 169ff dielectric behavior, 4, 176 domain structure, 4, 100, 111 electromechanical behavior, 4, 42ff hysteresis, 4, 120 solid solutions, 4, 176 spontaneous polariation, 4, 55 transition heat, 4, 62 Rock salt structure, 7, 487ff Rubidium, quantum defect data, 1, 180-181, 183 Rutherford scattering, 8, 314-315 Rutile structure, 7,4668
1
TO
10
435
S
saturation moment, ordered alloys, 1, 262 Schottky-Wagner disorder, 3,3446, 349 Screening, Rutherford scattering effect, 2, 315-319 Secondary emission, ace electron emission Secular equation, 8, 261 Selfdiffusion, metals, 20, 1151 Bemiconductora, uee also compound semiconductom ace obo elemental semiconductors ace also polar semiconductors, see also aemiconductors, valence band gap, 11,218ff land structure, 11, 163ff pressure effect, 11, 170-179 cyclotron resonance, 11, 280-311 De Ham-van Alphen effect, 9,257-291 effective maas in, I , 290-291 elastoresistance, 11, 149-221 electron distribution in, 1, 287-288 electrons, scattering in, 4, 332-357 electron transport theory, 4, 238-260 grain boundaries, 8, 492ff lattice imperfections in, 1, 286-287 multivalley, 11, 187-210 optical behavior, stress effects, 11,214ff thermal conductivity, 7, 94ff transport properties, 11, 1688 valence, 1, 283-385 band structure, I , 298-303 conduction phenomena, 1, 321-364 crystal properties, 1, 291-321 dislocatione in, I , 319 energy gap, 1 , 303-309 Hall effect, 1, 326-336 high field conduction, 1, 345-349 impurities in, 1, 304-315 magnetic properties, 1, 295-298 magnetoresistance in, 1,336-340 optical propertieu, 1, 294-296 thermoelectric power, I , 340-345 trapping phenomena, 1, 360-366 Zener current, 1,348
436
CUMULATIVE TOPICAL INDEX FOR VOLUMES
zone refining, 4, 464 Semiconductom,extrinsic, 1, 288-290 Semiconductor theory, 1, 284-281 Semimetah, elaetozeaietance, 11, 210-21 1 Shock waves, conservation relations, 6, 7 measurement of, 6, 178 production of, 6, 15 solids, effects on, 6, 1-63 stability, 6, 9ff Short range order, solid solutions, 9, 308-317 Side bands, 9,3566 Silicon, acceptor states, 6, 297-306 field effects, 6, 309 band structure, 7, 168ff conductivity tensor, 4, 2778 conduction bands, 6,272 donor states, field effect, 6, 308-311 impurities, double resonance, 6,2668 electron spin resonance, 6, 2668 energy levels, 6, 261ff magnetic susceptibility, 6, 271 impurity states, 6, 268-320 OPW method, application, 4,38441 1 valence bands, 6, 297 zone refining, 4, 464 Single crystals, zinc oxide, 8, 1998 Skin effect, anomalous, 6, 3298, 351ff Sodium, band structure, 7, 133ff eigenvaluea, 1,188 quantum defect data, 1, 178-179, 182 Sodium chloride structure, 7, 487ff Solidification, kinetics, 9, 281-293 Solid-liquid equilibria, 9, 370ff, 428 Solids, cyclotron resonance, 11, 276-400 equation of state, 11, 73-103 from shock wave techniques, 6,4040 magnetic behavior, high preeaum, 11,1238 poritron annihilation in, 10, 1-69
1 TO 10
mltivity, high pressure, 11, 103-123 thermal conductivity, 7, 1-98 Solid eolutions, 3,416-431 heterogeneities in, 9, 293-398 precipitation, 9, 3248 ehort range order, 9, 30a317 theory, 9, 3218 lone., 9, 328-396 Solid-state transitions, 3,2458 kinetics, S,293ff Solutions, 3,248ff Somigliana dislocations, S, 89 Sound, propagation in crystals, I, 203-267 Sound velocity, elaetic constant determination from, I, 263-269 Spece group, 6,174-256 irreducible representations, 6, 211224 Specific heats, crystals, 2, 219-303 metal, 1, 407 Specific heats and lattice vibrations, 2,219-303 Spectra, electronic, molecular crystals, 9, 1-81 Spectrometers, nuclear magnetic mmnance, S,31 Spherical harmonics, wave function expansion in, I, 66-72 Spin, ace abo nuclear spin Spin echo methods, nuclear magnetic resonance, 1, 32-38 spin interactions, I, 12-19 Spin-lattice interactions, 8, l b 1 9 Spin-lattice relaxation, 2, 67-91 Spinorbit coupling, 9, 4248 Spinorbit interaction, 1, 106 Spin-spin interactions, 8, 12-16 Spinel structure, 7, 468 Stark effect, semiconductor impurity stcrtea, 6, 311 Static electrification, 9, 139-197 experimental studiea, 0, 144-174 theory, 9, 17M S t N C t U B Of Crystab, 7, 426-603
CUMULATIVE TOPICAL INDEX FOR VOLUMES 1 TO
Sub-boundaries, generation, 8, 436-458 structure, 8, 356 Superconductivity, 10, 296-498 Budeen-Cooper-Schrieffer theory, lo, mff Bogoljubov theory, 10, 473311 Boee gas model, 10,4228 elmtic constante effect, 7, 341 electrodynamics, 10, 366-401 energy gap, 10, 43Off Meissner effect, 10, 31 1-366 nuclear magnetic reaonance effect, 8, 133-134 pair correlations, 10, 45tbW single electron theories, 10, 430-451 surf~aeenergy, 10, 308ff thermodynamics, 10, 297-311 two-fluid models, 10, Wff Superconductors, Hall effect, 6, 53 positron annihilation, 10, c4 Supercooling, constitutional, 4 , Surface diffurion, 10, i2lff &Irfaces, aw i n t e r f m experimental studies of, 7, 379-424 Symmetry, scc crystal symmetry
T Takagi method, orderdisorder transition, 1, 227 Temperature spike, atomic rearrangemenb in, 8,366 disordering, t, 360 Frenkei defect generation, 8, 364-366 theory, t, 351-378 Tensor properties, 6, 196-249, 7,2438 Tetrahedral coordination, 7,4428 Thermal conductivity, boundary re8istance, 7,418 compound semiconductom, 1114, 3,42ff electronic, 7, 70ff formal theory, 7, 13 damen, 7, 67 grain boundary effects, 7,65ff imperfections effect, 7, lSff, 59ff, 8off
10
437
lattice component, 7, 786 magnetic field effsct, 6, Mff metah, 4,22t)t?, 7 , 7 W nonmetda, 7, 45-70 polar semiconductom, 9, 133 radiation damage effect, ?,a scattering proceums, 7, 16-29 semiconductors, 7, Wff size effect, 7, 566 solids, 7, 1-98 Thermal expamion, crystal rymmetry relation, 6,22ofi Thermal neutrons, gae. model, 8, 1 M sccrttarbg, 8, 171ff slowing, 8, 168-188 solids interaction, 8, 109-190 Thermal spikes, 8,350-378 see also temperature spikm Thermodynamica, interfaces, S, 236ff orderdlisorder transitions, 3, 178-191 phaae transitions, S, 226-240 superconductivity, 10, 297-311 Thermoelectric poner, magnetic field effect, 6, 6lff metda, 4,220fF polar semiconductors, 9, 131 rinc oxide, 8, 298lf Thermornagnetic behavior, metab, 41-90 Thermopotential, compound semiconductors, 111-V, 3,42ff Thin films, electrical conductivity, 4, 268ti Tight binding method, I , 46-58,6,W R Tilt boundark, 8, 350-378 Time reversal, 6, 249 Tin, band structure, 7, 176 Lone refining, 4,468 Tin, gray, 11, 1-40 phame transition, 11, 2-11 physical properties, 11, 13-14 preparation, 11, 11 semiconductivity, 11, l c l o single crystab, 11, 11-12 Tramition metal ions, electronic spectra, 8, 47tHO2
438
CUMULATIVE TOPICAL INDEX FOR VOLUMES
Transition metals, electron energy bands, 7, 180-197 neutron diffraction, 8, 177-217 Transmission micmmpy, grain-boundery studies, 8,387 Transport properties, metals, I, 412-414 Transport theory, electrons in solids, 4,200-366 Trapping phenomena, in valence semiconduetors, I, 360-305 Tungaten, crystal, hydrogen on, 7, 405ff, 415ff oxygen adsorption on, 7, 401ff surface migration on, 7, 397, 41ofI surface model, 7, 383ff work function, 7, 399 Twist boundary, energy, 8, 414ff Two-fluid models, superconductivity, lO,3OOff
U U center, 10,2 4 M Uncertainty principle, 7,368ff V
Vacanciers, polar semiconductors, 9,92 Valence crystals, irradiation behavior, 8, 308-31 1, 433-442
Valence electron interaction, correction for in multivalent atoms, I, 165-169 Valence semiconductors, I, 283-366 ace Olso elemental semiconductors electroluminescence,6,160ff impurity states, 6, 268-320 Variational principle for periodic lattices, I, 7b79 W
WSVe fUn&OM, @lor centem,
alkali halides, 10, 128247 Wave functions, atomic, bibliography, 4,413-422 Wigner-Soitr spproximation, I, 62-66
1 M 10
Work function, impurity adsorption effect, 7,399 Z
Zener current, in valence semiconductors, 1,348 Zinc blende structure, space group representation, 6,229 Zinc oxide, adsorption of gases by, 8, 237ff catalysis by, 8,239-246 conductivity, field effect, 8, 273 measurement, 8,246 oxygen effect, 8, 276 radiation effecta, 8, 278-296 single crystal, 8, 2556 surface effecta, 8,304323 volume effects, 8, 296-301 electron mobility, 8, 298ff electronic proeeaees, 8, 191-323 imperfections, di&ion, 8, !203-210 hydrogen diffusion in, 8, 209ti lattice &NCtUrB, 8, 193 luminescence, 8,223-238 optical behavior, 8,216-236 phyricd properties, 8, 193 preparation, 8, 196ff single cryatdo, 8, 1 M thermoelectric power, 8, 298ff zinc diffusion in, 8, 203A Zinc sulfide, band etructure, 7,208 eleotroluminmcence,6, 137-180 Zone leveling, 4,468-472, mff Zone melting, 4,423-521 solid phase, 4,4626 temperature gradient, 4, 403-4W Zone refining, continuous, 4,4628 practice, 4,460-468 principles, 4,437456 Zone rtructure for various latticem, I, 34-38
Zones, kinetics of formation, 9, 371-384 properties, 9,384-398 reversion, 9, 367ff solid solutions, 9, 328396
