David Adler Massachusetts Institute of Technology Cambridge. Massachusetts
and
Brian B. Schwartz Institute for Amorphous Studies Bloomfield Hills. Michigan and Brooklyn College of the City University of New York Brooklyn. New York
DISORDERED SEMICONDUCTORS Edited by Marc A. Kastner. Gordon A. Thomas. and Stanford R. Ovshinsky
LOCALIZATION AND METAL-INSULATOR TRANSITIONS Edited by Hellmut Fritzsche and David Adler
PHYSICAL PROPERTIES OF AMORPHOUS MATERIALS Edited by David Adler. Brian B. Schwartz. and Martin C. Steele
PHYSICS OF DISORDERED MATERIALS Edited by David Adler. Hellmut Fritzsche. and Stanford R. Ovshinsky
TETRAHEDRALLY-BONDED AMORPHOUS SEMICONDUCTORS Edited by David Adler and Hellmut Fritzsche
A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further information please contact the publisher.
Disordered Semiconductors
Edited by Marc A. Kastner Massachusetts Institute of Technology Cambridge, Massachusetts
Gordon A. Thomas AT&T Bell Laboratories Murray Hill, New Jersey
and Stanford R. Ovshinsky Energy Conversion Devices, Inc. Troy, Michigan
Plenum Press · New York and London
Library of Congress Cataloging in Publication Data
Disordered semiconductors.
(Institute for Amorphous Studies series) Bibliography; p. Includes index. 1. Amorphous semiconductors. I. Kastner, Marc A. II. Thomas, Gordon A. III. Ov­
shinsky, Stanford R. IV. Series. TK7871.99.A45D57 1987 ISBN-13: 978-1-4612-9028-5 DOl: 10.1007/978-1-4613-1841-5
© 1987 Plenum Press, New York
537.6'22 e-ISBN-13: 978-1-4613-1841-5
Softcover reprint of the hardcover I st edition 1987 A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013
All rights reserved
86-30539
No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher
To HELLMUT FRITZSCHE
In honor of his outstanding career and service to physics on the occasion of his 60th birthday
HELLMUT FRITZSCHE
PREFACE
Twenty-four years ago, Hellmut Fritzsche came to our laboratory to evaluate our work in amorphous materials. He came many times, sometimes bringing his violin to play with our youngest son, to talk, to help, to discover, and to teach. The times with him were always exciting and rewarding. There was a camaraderie in the early years that has continued and a friendship that has deepened among Iris and me and Hellmut, Sybille and their children.
The vision that Hellmut Fritzsche shared with me, the many important contributions he made, the science that he helped so firmly to establish, the courage he showed in the time of our adversity, and the potential that he recognized put all of us in the amorphous field, not only his close friends and collaborators, in his debt. He helped make a science out of intuition, and played an important role not only in the experimental field but also in the basic theoretical aspects. It has been an honor to work with Hellmut through the years.
Stanford R. Ovshinsky Energy Conversion Devices Troy, Michigan
vii
CONTENTS
PART ONE: THE METAL-NONMETAL TRANSITION
Impurity Bands in Silicon and Germanium.......................... 3 N. Mott
Critical Phenomena Near the Metal-Insulator Transition........... 11 M. Kaveh
The Metal-Insulator Transition at Mi11ike1vin Temperatures....... 23 T.F. Rosenbaum
Magnetic Field Induced Transitions in Disordered Metals.......... 29 G.A. Thomas
Compensation Tuning Study of Metal Insulator Transition in Sl:P......................................... 37
W. Sasaki, Y. Nishio, and K. Kajita
The Metal-Insulator Transition in Compensated Silicon............ 45 M.J. Hirsch and D.F. Holcomb
The Spectroscopic Investigation of Negatively Charged Donor Ions (D- States)..................................... 57
R.A. Stradling
Magnetic Properties of Donors and Acceptors in Silicon: Similarities and Differences............................... 65
A. Roy, M. Levy, M. Turner, M.P. Sarachik, and L.L. Isaacs
The AC Conductivity in n-Type Silicon Below the Meta1- Insulator Transition....................................... 73
T.G. Castner and R.J. Deri
Low Frequency Conductivity Anomalies of Strongly Disordered Semiconductors.................................. 83
W. Gotze
Localization Effects in Quasi-One-Dimensional Lithium Quench-Condensed Microstructures.................. 97
D.J. Bishop, G.J. Dolan, and J.C. Licini
Potential Disorder in Granular Metals........................... 107 C.J. Adkins
X-Ray and Neutron Scattering Studies of Graphite Intercalated with Two-Dimensional K-NH3 Metal-Ammonia Solutions................................... 115
S.A. Solin
S. Minomura
The Superconductor-Semiconductor Transition in Cation­ Substituted Lithium Titanate, Li[MxTi2-x]04:
M = Li+, Al3+, and Cr3+................................. 135 P.M. Lambert, M.R. Harrison, D.E. Logan, and
P.P. Edwards
Photoinduced and Radiation-Induced Structural Transformations in Vitreous Arsenic Chalcogenides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.T. Kolomiets and V.M. Lyubin
New Aspects of Photoinduced Paramagnetic States in Chalcogenide Glasses................................... 155
S.G. Bishop, U. Strom, and J.A. Freitas, Jr.
On the Relationship Between ESR and Photodarkening in Glassy As 25 3. . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . 163
J. Hautala, J.Z. Liu, and P.C. Taylor
Photo-Induced Effects in Amorphous Ge-S......................... 173 T. Shimizu, G. Kawachi, and M. Kumeda
Polarization Memory of Photoluminescence and Photoinduced Optical Anisotropy in Chalcogenide Glasses...................................... 185
K. Murayama
Changes in the Photoelectronic Properties of Glassy Chalcogenides Induced by Chemical Doping, Irradiation, and Thermal History.................. 205
M.A. Abkowitz
A Model for the Electrical Doping of Chalcogenide Glasses by Bismuth........................... 219
S.R. Elliott
D.D. Gibson and M.A. Kastner
Optical Determination of the Fundamental Energy Gap of Amorphous MOS3..................................... 247
R.N. Bhattacharya, C.Y. Lee, F.H. Pollak, and D.M. Schleich
PART THREE: STRUCTURE AND BONDING IN AMORPHOUS SEMICONDUCTORS
Structure of Amorphous Semiconductors........................... 257 J.C. Phillips
The Sil1ium Model............................................... 261 D. Weaire and F. Wooten
High Brilliance X-Ray Sources and the Study of Amorphous Materials.................................... 269
A. Bienenstock
J.S. Lannin
K. Murase and K. Inoue
Bonding and Short Range Order in a-GexTel-x Alloys. . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
M.G. Fee and H.J. Trodahl
Refractive Index Dispersion and Structure of Ch.alcogenide Glasses...................................... 317
W. Burckhardt
K.L. Chopra and S. Kumar
PART FOUR: TRANSIENT, PHOTOEXCITED AND SPIN PHENOMENA IN TETRAHEDRAL AMORPHOUS SEMICONDUCTORS
Picosecond Photomodulation Studies of Carrier Trapping in a-Si: H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
Z. Vardeny and J. Tauc
Photoluminescence Studies of Band-Tail States in a-Si:H................................................. 349
B.A. Wilson
Recombination in a-Si:H Based Materials: Evidence for Two Slow Radiative Processes................. 357
T.M. Searle, M. Hopkinson, M. Edmeades, S. Kalem, I.G. Austin, and R.A. Gibson
Thermal and Optical Quenching of the Photo- conductivity in a-Si:H-Films.............................. 369
R. Carius, W. Fuhs, and K. Weber
Recombination and the Standard Model in Amorphous Hydrogenated Silicon (a-Si:H): An. Essay................................................... 379
E.A. Schiff
A. Rose
Determination of the Electronic Density of States of n-Type Hydrogenated Amorphous Silicon from Transient Sweep-Out Experiments.............. 401
M. Silver, D. Adler, and H.M. Branz
Electron Correlation Energies in Hydrogenated Amorphous Silicon......................................... 407
M. Stutzmann, W.B. Jackson, R.A. Street, and D.K. Biegelsen
Defects in a-Si:H............................................... 415 K. Morigaki, M. Yamaguchi, I. Hirabayashi,
and R. Hayashi
Optical Absorption Edge of Hydrogenated Amorphous Silicon......................................... 425
K. Tanaka and S. Yamasaki
Preparation of a-Si and Its Related Materials by Hydrogen Radical Enhanced CVD........................... 435
J. Hanna, N. Shibata, K. Fukuda, H. Ohtoshi, S. Oda, and I. Shimizu
Application of In Situ Ellipsometry to the Growth of Hydrogenated Amorphous Silicon.................. 447
R.W. Collins, J.M. Cavese, and A.H. Clark
Density of Gap States in Undoped and Doped Amorphous Hydrogenated Silicon Obtained by Optical Spectroscopy................................... 459
A. Triska, J. Kocka, and M. Vanecek
xii
Structural and Electronic Properties of Polymeric Amorphous Semiconductors Si1-xCx:H........................ 469
S. Nitta, S. Nonomura, R. Takagi, N. Kojima, and A. Yasuda
Structural Characterization of Amorphous Silicon and Germanium............................................. 479
R. Tsu
M.-L. Theye, A. Gheorghiu, T. Rappeneau, and D. Udron
DC Electrical Conductivity of Highly Disordered Elemental Semiconductors.................................. 499
A.n. Yoffe and R.T. Phi11ip.s
PART SIX: MULTI LAYERS AND INTERFACES
Resonant Tunneling Through Quantized States in a-81 :H. .. . . . . . .. . .• ..• . . . . . . . . .• . . . . • . . . . . . . . . . . . . . ... • • 511
M. Hirose, Y. Ihara, andS. Miyazaki
Persistent Photoconductivity in Amorphous Silicon Alloys............................................ 519
S.C. Agarwal and S. Guha
The Thermal Equilibration Model for Persistent Photoconductivity in Doping Modulated Amorphous Silicon......................................... 529
J. Kaka1ios and R.A. Street
Band Edge Alignment and Quantum Size Effects in Hydrogenated Amorphous Si1icon/ Germanium Super1attice Structures......................... 541
P.D. Persans, C. Wronksi, and B. Abeles
Carrier Recombination Kinetics in Amorphous Doping Super1attices...................................... 551
L. Ley and M. Hundhausen
Material DeSign by Structural Modulation of Amorphous Semiconductors.................................. 563
S. Oda, H. Shirai, A. Tanabe, J. Hanna, and I. Shimizu
Junction Capacitance Studies of Hydrogenated Amorphous Silicon Doping Superlattice Films............... 577
J.D. Cohen, C.E. Michelson, and J.P. Harbison
The Effects of Band Bending on the Optical, Electrical, and Photoe1ectronic Properties of a-Si:H Thin Films in Surface Cell Structures.......................... 587
G.N. Parsons, C. Kusano, and G. Lucovsky
Effect of Band Tails on the Electric Field of Amorphous Silicon Solar Gells.......................... 603
R.S. Crandall
Hydrogen Motion and the Staebler-Wronski Effect in Amorphous Silicon............................... 613
D.E. Carlson
J.D. Joannopoulos, D. Adler, and Y. Bar-Yam
Dangling Bonds and Metastability in Semiconductors.............. 625 J.1. Pankove
Recombination-Enhanced Defect Formation and Annealing in a-S1:H....................................... 635
D. Redfield
Composition and Thermal Stability of Glow- Discharge a-Si:C:H and a-Si:N:H Alloys.................... 641
W. Beyer and H. Me1l
Investigation of the Electronic Structure of Amorphous Silicon Based on Studies of the Amorphous-to-Crystalline Transition................... 659
M.A. Paes1er and L.E. Mosley
PART EIGHT: THEORIES AND MODELS FOR THE ELECTRONIC PROPERTIES OF AMORPHOUS SEMICONDUCTORS
Theory of Covalent Amorphous Semiconductors..................... 673 D. Adler
Tails in the Density of States.................................. 681 E.N. Economou, C.M. Soukoulis, M.H. Cohen,
and S. John
Localization Effects in Amorphous Semiconductors................ 697 U. Dersch and P. Thomas
Band-Edge Conduction in Amorphous Semiconductors................ 705 D. Monroe
The Statistical Shift of the Fermi Energy and the Prefactor of the DC Conductivity in a-Si:H................ 713
H. Overhof
Luminescence and Relaxation of Energy in Disordered Organic and Inorganic Materials........................... 723
B. Movaghar, M. GrUnewald, and B. Ries
Thermoelectric Power of Anderson-Mott Insulators with an Exponential Density of States Distribution. . .. . . . .. . .. . . . . . . . . . . . .. . . .. . . . . . . . . . . .. . . . . . 745
L. Friedman and M. Pollak
Dramatic Effects of Disorder on Small-Polaron Formation and Motion...................................... 751
D. Emin
xiv
INTRODUCTION
It is hard to imagine now, in the 1980's, how little interest there was in the 1950's in the electronic properties of disordered semiconductors (amorphous or crystalline with high impurity or defect density). Heavily-doped semiconductors were considered dirty systems, unworthy of careful study. Today, one can find in many textbooks a section on the metal-insulator transition which inevitably includes or refers to the famous figure from the research group of Fritzsche at the University of Chicago elaborating the properties of this trnsition in doped, compensated Ge. That work began an evolution which has made the problem of the metal-insulator transition in disordered materials one of the most sophisticated and elegant in solid-state physics.
History repeated itself in the 1960's. Turning to the problem of electronic conduction in amorphous semiconductors, He11mut Fritzsche found himself criticized again for studying ·'dirt." This time his ideas (the CFO model, the long-range potential model of the absorption edge, the valence-alternation model) helped to shape the field as much as the experiments of his research group. The experiments included crucial ones on cha1cogenide glasses (the observation that the glasses are diamagnetic which was the basis of Anderson's negative correlation energy and the defect models) as well as on amorphous hydrogenated silicon (the observation that hydrogen diffuses out of a-Si:H at elevated temperature, providing the first direct evidence for and measurement of hydrogen content). The interest in this field is now enormous as evinced by the number and size of conferences on the subject.
This volume has been assembled, with the enthusiastic support of so many authors, to honor He11mut Fritzsche on his 60th birthday for his leadership over the past 30 years. He has set a high standard as a colleague and teacher. He has always been able to cut through rhetoric with simple arguments which emphasize the essential physical problem. Because of this, his collaborators have often appeared stronger than they really were. They have benefited from the relative ease of solving a well-stated problem compared to the impossibility of solving the same one clouded in jargon. They have also benefited from the chance to identify the crucial experiment when the problem was reduced to simple terms.
We honor He11mut, too, for his intellectual honesty. He is as critical of his own models when they disagree with experiment as he is of those of the theorists. And we honor him for his service to the community: his leadership as organizer of many conferences, as Chairman of the condensed-matter section of the American Physical Society, and, most of all, as Chairman of the Physics Department of the University of Chicago.
He1lmut is very young at 60. This volume is not meant to celebrate the end of a great career, but rather as a gesture of thanks for what He1lmut has done up to now, with the expectation of much more to come.
This volume contains eight sections addressing all the current issues relevant to the electronic and structural properties of heavily-doped crystalline and amorphous semiconductors. The first section contains a summary by N.F. Mott of the current theory of the metal-insulator transition followed by theoretical and experimental discussions of all aspects of this forefront problem. Section II contains nine articles on the electronic and optical phenomena in cha1cogenide glasses. Here, there is special emphasis, as in the paper by Ko1omiets, on photostructura1 and photoelectronic effects. In Section III are two theoretical and several experimental discussions of structure and bonding in amorphous semiconductors. The fascinating results of transient, photoexcited, and spin resonance experiments on amorphous silicon and related materials are presented in section IV. Section V provides a view of the new preparation techniques being used for amorphous silicon as well as the measurements characterizing this and related materials. The exciting field of amorphous silicon mu1tilayers, heterojunctions, and interfaces is discussed in section VI. The stability of amorphous silicon is a crucial technological issue. This problem is reviewed and discussed in section VII. Finally, in section VIII are papers presenting diverse general models and theories for the electronic properties of amorphous semiconductors.
The editors of this book, as well as the general editors of the Institute for Amorphous Studies Series, would like to thank Ms. Ghaza1eh Koefod, the assistant director of the Institute. Ms,. Koefod has been responsible at every stage of the production of this book, from the announcement of the Festschrift to the editing of many of the papers to the compilation of the important aids such as indices. Without her dedicated effort this book would not have been possible. We all thank her.
2
Marc A. Kastner Massachusetts Institute of Technology Cambridge, Massachusetts
Gordon A. Thomas AT&T Bell Laboratories Murray Hill, New Jersey
IMPURITY BANDS IN SILICON AND GERMANIUM
ABSTRACT
Cavendish Laboratory Cambridge, U.K.
An account is given of some of the developments which have led to a better understanding of conduction in impurity bands, since the work of He11mut Fritzsche opened up the subject.
Professor He1lmut Fritzsche is known, among his many other achieve­ ments, for his work from 1959 onwards on impurity conduction in doped and compensated semiconductors 1,2,3. The idea was first put forward by Gudden and Schottky4 as early as 1935; these authors mentioned the possibility of a conduction process in which an electron tunnels from an occupied to an equivalent unoccupied centre in a semiconductor. The first evidence for this form of conduction comes from the work of Busch and Labhart 5 in SiC and of Hung and G1eissman6 in Ge. Ginsbarg7 pointed out that with large concentrations of impurity the overlap between the wave functions becomes so great that carriers are no longer localized around individual impurities; conduction can then proceed without compensating acceptors. Mott8 in 1949 wrote a paper suggesting that, as the concen­ tration of donors is increased, the transition to the metallic state should be accompanied by a discontinuous change in the concentration n of electrons. The concentration nc at which this occurs was predicted to be
1/3 nc ~ = 0.25
(aH is the hydrogen radius of the donor or acceptor), a result in fair agreement with much experimental data, even though the derivation given in 1949 is no longer acceptable. Miller and Abrahams9 in 1960 first gave a quantitative theory of thermally activated hopping conduction between occupied and empty sites, and in 1958 Anderson lO gave his theory of the localization of non-interacting electrons in a random lattice, a model directly applicable to impurity conduction in a compensated semi­ conductor. Fritzsche1 ,2,3 between 1955 and 1965 investigated impurity conduction in Ge in detail, introducing donors by slow neutron bombard­ ment which greatly improved our understanding of compensation. He also investigated the effect of stress, and in 1960 introduced11 , as well as the energy £1 needed to excite an electron into the conduction band and
3
the activation energy €3 for hopping. the energy €2 needed to promote an electron from a donor onto another occupied donor. where in uncompensated or lightly compensated material it is mobile and forms a band of levels. sometimes called the "upper Hubbard band".
We attempt here to describe the use made of this pioneering work in the last 30 years. It has become a major part of solid state physics. both for theory and experiment. As regards hoppin~ conduction Mott12 in 1968 extended the theory of Miller and Abrahams to take account of hopping over larger distances than those between nearest neighbours. and obtained the result that at low temperatures the conductivity a should vary with temperature as
\l a = A exp {- (T /T) } o v = 1/4. (1)
a form of charge transport known as variable-range hopping. The pheno­ menon has been extensively observed both in non-crystalline semiconductors and doped germanium. though to determine the index accurately is not easy and Fritzsche (ref. 3. p. 226) reported that for some measurements v = 1/2 or 1/3 gives a better fit. The value of the index is of particu­ lar interest since Efros and Shk1ovskii13 showed in 1975 that. when Coulomb interaction between electrons on different sites is taken into account. the index. at least in the limit of low temperatures, should be 1/2, both in three and two dimensions. The evidence on whether this is so is somewhat contradictory. Their work is based on their proof that the one-electron density of states vanishes at the Fermi energy, behaving like const (EF - E) 2, although the relaxed density of states, when all electrons are in equilibrium. is believed to be finite at the Fermi energy. Very recently Vigna1e et a1.1~ have maintained that a dynamic model, which takes into account the effect of the hopping process on the density of states, suppresses.the Coulomb gap in two dimensions (in agreement with experiment for inversion layers for which v = 1/3 is predicted and observed) and also for rapid hopping in three.
The hopping model is of course based on Anderson'slO paper of 1958. "Absence of diffusion in certain random lattices", as is much other work. Anderson started from a "tight binding" model, that of Kronig and Penny16. with a potential as illustrated in Fig. 1a. This is in three dimensions. and a band formed from s-like wave functions will have width
B = 2z1 (2)
I = f1/l H 1/1 d 3x n n+l (3)
H is the Hamiltonian, z the co-ordination number and 1/In an s-like wave function on atom n. The tight-binding (Bloch) wave functions are of the form
'II = L exp (ika ) 1/1 n n n
(4)
where k is a wave number and the an are lattice points. Anderson then added a random potential Vn to each well extending over a range Vo as shown in Fig. lb. For small values of Vo this will introduce a finite mean free path ~ where
(5)
4
( 0)
E
v
The potential energy of an electron in the Anderson model.
(a) For a crystalline lattice (b) With random potential
The density of states is also shown. (From Reference 18, fig. 2.1)
When Vo = B, it follows that t = a, and according to loffe and Regel17 this is the smallest mean free path possible; the present author (with E. A. Davis) has supposed1 8 that the wave function is then of the form
'l'o = Len exp(i ~n) 1/In ' (6)
the cn being random constants, and ~n random phases. Using for the con­ ductivity of a metal the equation
(7)
where SF is the Fermi surface area. For a half-full band, and with t a, we find
1 a="3 e /-lni,
the loffe-Regel value.
(8)
As Vo/B increases, a transition occurs to a non-conducting state in which states are localized when, according to recent calculations (Elyutin et al. 19 )
VB=1.7. o
(9)
5
It is supposed that the wave functions are then of the form
'I' = exp (-r/E.) 'I' o
(10)
where '1'0 is given by (6). E. is called the localization length; it is infinite at the transition and decreases with increasing Vo/B. For attempts to measure it, see Mott and Davis 18 , p. 138, and Shafarman and Castner20 •
If Vo/B is less than the critical value (9), the present author first pointed out 21 that "tails" of localized states must exist, with a critical energy Ec separating the localized from the non-localized range. This limiting energy was postulated independently by Cohen, Fritzsche and Ovshinsky22 and called by them a "mobility edge". Here too, at energies on the localized side of the edge, wave functions are of the form (10), with E. depending on energy E according to the equation
-\I E. = const (E - E) • c
Several authors propose that V = 1 for non-interacting electrons.
(H)
The conductivity at zero temperature of a partly filled band just on the metallic side of the transition has been a matter of controversy. The present author in a number of papers proposed that there existed a "minimum metallic conductivity", estimated to be
a i = 0.03 e2 /-ff a • m n (12)
Abrahams et a1. 23 in 1979 first used a scaling theory to show that, again in a theory of non-interacting electrons, no minimum metallic conductivity exists, the zero temperature conductivity tending to zero with E - Ec, probably linearly in (E - Ec), and, as we shaH see1 this prediction is in accord with experiment. Mott24 and Mott and Kaveh~5 have investigated the form of the wave function that would lead to this conclusion and suggest that the constants cn in equation (5) have long-range fluctua­ tions between zero and a maximum value, with wave-length A tending to infinity with (E - Ec)-l.
Another way of understanding why a(T = 0) should go to zero near the transition is to realize that the effect is a result of quantum inter­ ference between multiple scattering processes leading from one value of the wave number to another. This approach has been used by Bergmann26
for the case of two-dimensional systems. Kawabata27 first proposed an equation for the conductivity a in the case when these corrections are small. This is, as modified by Mott and Kaveh25
(13)
Here a B is the Boltzmann conductivity given by (7); g is the factor intro­ duced by the present author, given by
g = N(EF)/N(EF) t crys (14)
where N(EF)cryst is the density of states with no disorder (Vo = 0 in fig. 1); kF is the wave vector at the Fermi surface, i is the elastic mean free path, c is a cut-off number probably equal to unity and L
6
either (a) the size of the specimen, or (b) the inelastic diffusion length Li given by
(15)
D is here the diffusion coefficient and 'i the time between elastic collisions, or (c) in the presence of a magnetic field H, the cyclotron radius
~ = (c-H'/eH)1/2 • (16)
1/'i is expected to vary as 1/T2 if inelastic collisions with electrons are under consideration, or liT in the case of collisions with phonons above the Debye temperature.
Although (13) is deduced under conditions when the correcting term is small, it serves reasonably well for extrapolation to the Anderson transition where 0 vanishes. In the case (T = 0) where L is infinite, then if c = 1, since when! = a kF! ~ 3, the conductivity goes linearly to zero as g increases, and vanishes when g ~ 1/3, the value deduced from the calculations of E1yutin et a1.19. At the mobility edge when L is finite, we find
(17)
(18)
The value of the constant will be increased if the transition occurs in the conduction band of many-valley type (Bhatt and Ramakrishnan28 ), but not in an impurity band.
The equation (18) has been applied by the author29 to the conduction band of amorphous silicon to evaluate the constant 0 0 in the equation
(19)
representing the conductivity when charge transport is by electrons with energies at a mobility edge Ec. It can also be obtained from the drift mobility ~ by writing
~ = ~o exp(- ~E/~T) } (20) with
Here N(EA) is the density of states at the bottom of the conduction band, averaged over a range kBT.
In this problem it is supposed that 0 0 is given by
o o (21)
7
and that Li is caused by interaction with phonons. Interaction with phonons will of course broaden the mobility edge, but it is found that the broadening is not great, being given by
~E/E 0
Results are in fair agreement with experiment.
In this problem, interaction between electrons does not occur. The main applications of the concept of Anderson localization, however, have been to impurity bands; an impurity band in a compensated semiconductor corresponds closely to the Anderson model, except that the potential wells are at random positions in space, giving additional disorder. For compensated semiconductors the partly occupied donors give rise to an impurity band. At low concentrations all states are localized, for higher concentrations a mobility edge Ec will exist. If the Fermi energy at zero T lies below Ec ' conduction at low T will be by variable-range hopping, at higher T by excitation to the mobility edge with conductivity of the form (19) with crQ given by (21). As the concentration increases a metal-insulator trans1tion will occur as EF - Ec changes sign to a positive value. In the metallic region the zero-temperature conductivity should behave like
where ~ is the localization length for an equal energy below Ec. If ~ varies according to (11), we see that the conductivity will behave like
cr = const { (n - n )/n }v c c v = 1 (23)
where n is the number of electrons per unit volume and nc the value of n for which cr vanishes. Compensated semiconductors, and materials such as amorphous NbxSil-x • show a variation of the conductivity near the tran­ sition according to (23).
Both equations (13) and (18), if L is the inelastic diffusion length Li' show the surprising result that inelastic collisions increase the conductivity. This is to be understood because the correcting term in (13) is a consequence of multiple scattering, two multiple paths in k-space showing destructive interference. If not enough real space is available for this to occur, the conductivity increases (Bergmann26).
When the mean free path is short, electron-electron interaction can have a major effect on the conductivity, especially on its variation with temperature. As first shown by Altshuler and Aronov30 • a change in the density of states at the Fermi level leads to a conductivity of the form
cr A + BTl/2
an effect which has been observed in doped semiconductors and also in amorphous metals 31 •
8
We turn now to uncompensated semiconductors, typically Si:P. If, as we believe, conduction for concentrations near the metal-insulator transition is in an impurity band, the Hubbard U, namely the intra-donor interaction energy <e2/kr12>, will now playa role. For a crystalline array of one-electron centres the Hubbard U should lead to a discontinuous transition; as at zero T the number N of donors increases, the number n of free electrons should jump discontinuously from zero to a finite value (cf. Mott 32). No discontinuous change in n or a is however observed in doped silicon or germanium and this is believed to be a consequence of disorder. Thomas and co-workers 33 have shown that for uncompensated semi­ conductors investigated until now, near the transition and indeed over a wide range of concentrations,
a = const (n - n )v c v ~ 0.5 (24)
It has to be said that an explanation of the index v = 0.5 has not been given up until now. If the conductivity behaves like e2/~~, this will mean that
const/(n _ n)1/2 c
(25)
If - as for non-interacting electrons - the dielectric constant K behaves like 1/~2, then measurements of the dielectric constant3~ show that (25) must be valid. On the other hand, Mott35 has given arguments that, if
~ ~ l/(n - n) c (26)
v cannot be less than 2/3. Also measurements of v in (26) from variab1e­ range hopping v seem to show values of v slightly less than unity. We believe that the correct value in (26) is v = 1, and that another explana­ tion of (24), perhaps on the lines of that of Mott and Kaveh3~ must be given.
We see, then, that though much is now understood, there are still outstanding problems.
REFERENCES
1. H. 2. H. 3. H.
4. B. 5. G. 6. C. 7. A. 8. N. 9. A.
10. P. 11. H. 12. N. 13. A.
14. G.
Fritzsche, J. Phys. Chem. Solids 6:69 (1958). Fritzsche and M. Cuevas, Phys. Rev. 119:]238 (1960). Fritzsche in: "The Metal Non-metal Transition in Disordered Systems," L. R. Friedman and D. P. Tunstall, ed., SUSSP Publi­ cations, University of Edinburgh (1978) p. 193. Gudden and W. Schottky, Z. Tech. Phys. 16:323 (1935). Busch and H. Labhart, He1v. Phys. Acta 19:463 (1946). S. Hung and J. R. G1eissman, Phys. Rev. 79:726 (1950). S. Ginsbarg, PhD Thesis, Purdue University (1949). F. Mott, Proc. Phys. Soc. (London) 62:416 (1949). Miller and S. Abrahams, Phys. Rev. 120:745 (1960). W. Anderson, Phys. Rev. 109:1492 (1958). Fritzsche, Phys. Rev. 119:1899 (1960). F. Mott, J. Non-Cryst. Solids 1:1 (1968). L. Efros and B. I. Shkovskii, J. Phys. C: Solid State Phys. 8:L49 (1975). Vigna1e, Y. Shinozuka and W. Hanke (preprint).
9
15. G. Timp, A. B. Fowler, A. Hartstein and P. N. Butcher, Phys. Rev. B 23:3570 (1986).
16. R. Kronig and W. Penney, Proc. R. Soc. (London) 130:499 (1931). 17. A. F. Ioffe and A. R. Regel, Proc. Semicond. 4:327 (1960). 18. N. F. Mott and E. A. Davis, "Electronic Processes in Non-Crystalline
Materials," 2nd ed., Oxford (1979). 19. T. V. Elyutin, B. Hickey, G. J. Morgan and G. F. Weir, Phys. Stat.
Solid (b)124:279 «1984). 20. W. N. Shafarman and T. G. Castner, Phys. Rev. B 33:3570 (1986). 21. N. F. Mott, Adv. in Phys. 19:49 (1967). 22. M. H. Cohen, H. Fritzsche and S. R. Ovshinsky, Phys. Rev. Lett.
22:1065 (1969). 23. E. Abrahams, P. W. Anderson, D. C. Licciarde110 and T. V. Ramakirsh-
nan, Phys. Rev. Lett 42:673 (1979). 24. N. F. Mott, Phil. Mag. B 49:L75 (1984). 25. N. F. Mott and M. Kaveh, Phil. Mag. B 52:177 (1985). 26. G. Bergmann, Phys. Rev. B 28:2914 (1983). 27. A. Kawabata, Solid State Commun. 38:823 (1981). 28. R. N. Bhatt and T. V. Ramakrishnan, Phys. Rev. B 28:6091 (1983). 29. N. F. Mott, Phil. Mag. B 51:19 (1985). 30. B. L. Altshuler and A. G. Aronov, Zh. Eksp. Teor. Fiz. 77:2028 (1979). 31. M. A. Howson and D. Greig, Phys. Rev. B 30:4805 (1934). 32. N. F. Mott, "Metal-Insulator Transitions," Taylor and Francis,
London (1974). 33. G. A. Thomas, Physica B 117:81 (1983). 34. N. F. Mott and M. Kaveh, Adv. in Phys. 34:401 (1985). 35. N. F. Mott, Phil. Mag. B 44:265 (1981).
10
ABSTRACT
We review the recent developments regarding critical phenomena near
the metal-insulator transition. A discussion is presented of the experi­
mental evidence for the effect on the transition due to localization and
electron interaction. The critical phenomena fall into two classes. A
theoretical explanation of these phenomena and their classification is
presented and additional experiments are suggested.
1. Introduction
The subject of the metal-insulator transition has been studied for a
period of about 35 years. Mott (1949) was the first to point out that
electron-electron interactions can drive the metallic state into an insu­ lating phase. This is known as the Mott transition. In 1958, Anderson
discovered that increasing the disorder of the system results in the ab­
sence of diffusion for critical disorder. This is known as the Anderson transition. The experimental state-of-the-art before 1979 for the Anderson
transition in compensated doped semiconductors was given by Fritzsche (1978).
In 1979, a scaling theory was developed (Abrahams et al 1979) in which it
was proposed that the Anderson transition in a three-dimensional system is
continuous and can be described as a second-order phase transition with a
diverging length scale on both sides of the transition; the localization
length below the transition and a correlation length (Imry 1980) on the
metallic side. In that same year, 1979, it was also realized by Altshuler
and Aronov (1979) that disorder dramatically affects the electron-electron
interaction, resulting in a breakdown of Fermi-liquid theory and yielding a
reduction in the single-particle density of states (DOS).
11
!.: This leads to the well-established T:2 term in the temperature dependence
of the conductivity at low temperatures in the metallic phase. In recent
years, it has become clear that long-range electron-electron interactions
may induce a metal-insulator transition (MIT) under certain conditions.
The numerous experimental studies of the metal-insulator transition since
1979 have revealed a complex picture.
In this review, we present a possible explanation of critical phenomena
near the MIT. In particular, a material classification will be given of
the transport phenomena. The experimental observations will be reviewed
first and the theoretical approaches later.
2. Critical Behaviour of the Conductivity
The critical form of the conductivity is given by
(1)
where n is the electron density and nc is its value at the transition. The
data seem to fall into two classes. Materials for which v = 1 and materials
for which v =~. The first class of materials (v = 1) contains the amor­
phous metals and amorphous semiconductors, whereas the second class (v = ~) contains the uncompensated many-valley doped semiconductors (see Kaveh,
1985, table 1). We do not discuss here the magnetic semiconductors which
may form a third class (Mott and Kaveh 1985).
3. Classification of Phenomena The two classes of materials seem to have differing signs for do/dT
for electron densities n ~ 1.lnc (Thomas et a1 1983). Writing the conduc­
tivity as
(2)
yields m > 0 for materials for which v = 1 and m < 0 for materials for
which v =~. Interestingly, when n + nc ' m changes sign (for materials
with v = ~) and becomes positive (Thomas et al 1983, Long and Pepper 1984).
Another difference between the two classes of materials is the value
of the scaling conductivity 00 in (1). For materials with v = 1, the data
indicate (see in Kaveh and Mott 1986b, table 1) that 00 ~ (1-2)0 . , where mln o. - 0.03 e2/ha is the minimum metallic conductivity of Mott. For ma­mln terials with v = ~, the data (Thomas et al 1982) yield 00 ~ (13-15)0 . , mln which is about an order of magnitude larger than 00 for the v = 1 class.
We thus see that the two classes of materials can be characterized by three
numbers: v, m and 00 , The first class is characterized by v = I, m > 0,
12
o ~ 0 . , whereas the second class by v = ~, m < 0, 00 ~ 100 . . An in-o mln mln teresting question is whether a crossover is possible from one class to ano-
ther. Theoretically, a transformation from the v = ~ class into the v = 1 class is possible when a strong magnetic field is applied (Castellani et
al 1984, Mott and Kaveh 1985) but this has not yet been observed experimen­
tally. An important observation (Thomas et al 1982) is that doped semicon­
ductors are sensitive to compensation. Moreover, it was demonstrated that
Ge:Sb, which belongs to the v = ~ class, is transformed continuously to the
v = 1 class by compensation.
4. Effect of Electron Interactions on the DOS
According to the interaction picture of Altshuler and Aronov (1979),
when the exchange interactions dominate over the Hartree interactions, a
reduction in the density of states (DOS) is expected. Scaling theories for
this class (v = 1) suggest (MacMillan 1981, Imry and Gefen 1984) that the
DOS will decrease to zero as n + nc with the same critical exponent as the
conductivity; namely, N = No((n-nc)/nc)n=1 Such behaviour was observed in
tunnelling experiments in amorphous metals (Hartel et al 1983). For a re­
view, see Mott and Kaveh (1985). For the v = ~ class no tunneling experi­
ments have yet been performed. It has been suggested (Kaveh 1985) that for
this class, the DOS will not vanish at the transition.
Specific-heat measurements (Thomas et al 1981) as well as magnetic­
susceptibility data demonstrate an enhancement of the DOS near the transi­
tion for Si:P
5. Effect of Magnetic Field
The magnetic field dramatically affects the transport properties of a disordered metal. We now summarize the observed phenomena.
(i) Negative magnetoresistance This is caused by the supression of the quantum interference effects, thus enhancing the conductivity. The data are usually analyzed within
the formula of Kawabata (1981) and an inelastic scattering time is de­
duced. For reviews, see Kaveh and Wiser (1984), Lee and Ramakrishnan
(1985) and Mott and Kaveh (1985).
(ii) Positive magnetoresistance
This is due to the suppression of the Hartree terms when gVBH > kBT
(Altshuler et a1 1980) and has been widely observed (Thomas et al
1983, Long and Pepper 1984).
(iii) Strong magnetic fields
In this region, the Hartree terms as well as the localization effects are strongly suppressed (Finkelstein 1983, Altshuler and Aronov 1983,
13
Castellani et al 1984) and the metal-insulator transition is induced
solely by the exchange interactions. The critical phenomena are ex­
pected to belong to the v = 1 class with the conductivity having the
form
a = AT1/ 3 (3)
for n + nco This was recently observed by Newson and Pepper (1986)
and analyzed by Kaveh et al (1986).
(iv) Effect of Magnetic Field on the Insulating Phase
In the insulating phase, two important phenomena have been observed,
one in the weak magnetic field limit and the second for very strong
magnetic fields (~9T).
a) Castner's group (Shafarman et al 1986) measured in Si:As Mott
variable range hopping a = aoexp t(To/T)~] and deduced that the
critical exponent of the localization length is unity. This was
obtained by using the relation To «~-3 By applying a weak mag­
netic field, it was found that the critical exponent is changed
to ~, in agreement with the theory of Hikami (1981).
b) For strong magnetic fields, it is expected that the exchange in­
teractions will dominate, yielding a reduction in the DOS on the
metallic side and a Coulomb gap in the insulating phase. Thus,
for strong magnetic fields on the insulating side of the transi­
tion, one expects a crossover from Mott variable range hopping to 1
a = aoexp~(T1/T)~] behaviour due to the Efros-Shklovski Coulomb- gap theory. Such behaviour has recently been reported for InSb
by Biskupski et al (1986).
6. Dielectric Constant Accurate measurements (Thomas et al 1983) of the dielectric constant
of Si:P yield
This result establishes the continuous nature of the metal-insulator tran­
sition in Si:P and that the critical exponent of (1 is twice that of the
conductivity.
The Knight shift of Si:P was recently re-analyzed experimentally by
Jerome et al (1985). The data indicate a very large electron concentra­
tion near the phosphorous nuclei. Kaveh and Liebert (1986) calculated
the Knight shift at the Si nuclei and at the P nuclei using a tight-binding
14
model for the wavefunctions. They concluded that in order to obtain agree­
ment with the data, one must assume that the MIT occurs in an impurity
band. They also find agreement with the specific-heat data (Thomas et al
1981) and show that its n1/ 3 dependence may also result (accidentally) from
an impurity band.
We now review the recent theoretical developments of the metal-in­
sulator-transition.
8. Pure Disorder
The scaling theory of Abrahams et al (1979) implies a continuous tran­
sition. Following this suggestion, an ( expansion treatment (Wegner 1980)
suggests a critical exponent v = 1 for the conductivity.
A useful formula for the conductivity,which obeys the scaling theory
and agrees with numerical calculations (Elyutin et al 1984) that the DOS
due to disorder is reduced by a factor g = 1/3 at the transition, is given
by Kaveh and Mott (1982, 1986a)
This formula leads to
g (KFR.)
(5)
1/3n2
Near the MIT, a = Gce2jhLi where Li is the diffusion inelastic length Li =~. This leads (Imry 1981) to a = ATP/ 3 where P is the exponent of
. -1 P the inelastic scatter~ng rate Ti - T Recently, Kaveh, Newson, Ben-
Zimra and Pepper (1986) have shown that near the MIT, P = 1 due to the electron-electron scattering. Thus, near the MIT, a = AT1/3. It has also
been pointed out (Kaveh and Mott 1986a) that the T1/3 law holds also in
the case when electron interactions are taken into account, provided that
the exchange interactions dominate.
According to the scaling theory, there is a critical conductance,
Gc = aL in three dimensions which is independent of the length L and se­
parates the metallic phase from the insulating phase. This conductance
can be deduced either from the temperature dependence of a near the MIT
(The T1/3 law) or from the electron density dependence. It has been
pointed out (Kaveh and Mott 1986a) that Gc will be different in these
15
(6)
or
(7)
The value of Gc(Li) was calculated by Kaveh and M~tt (l986a) and was found
to vary between -0.03 and -1 depending on the value of L.. The smaller 1.
values are obtained for large values of Li . The values of Gc(~) were found
to vary between -0.06 and -1. The data seem to support these calculations
(for a detailed discussion, see Kaveh and Mott 1986a).
10. Effect of Magnetic Field on the Anderson Transition
Shapiro (1984) has proposed a phase diagram for the Anderson transi­
tion which applies to the pure disorder limit. According to Shapiro, the
mobility edge is first shifted to lower values and then is pushed back to
positive values. For a detailed discussion and applications of this phase
diagram, see the review by Mott and Kaveh (1985). This prediction has not
yet been observed experimentally. The reason may lie in the fact that
there is probably no material for which one finds a pure Anderson transi­
tion. The interactions will always affect the transition and may prevent
the initial shift of the mobility edge to lower values. This conclusion
may be supported by the experiment of Shafarman et al (1986). They found
~ - (n _n)-l at zero magnetic field and upon the application of a weak c 1 --
magnetic field, ~(H) - (nc(H)-n)-~, in agreement with the pure disorder theory (Hikami 1982). However, nc(H) was not found to be smaller than
nc(o). Another important fact is that the critical behaviour of the trans­
port properties is not universal and falls into two classes. This indi­
cates the importance of electron-electron interactions near the MIT.
11. Classification of Critical Transport Properties
It was recently suggested (Kaveh 1985) that the critical behaviour near the MIT can be classified according to the competition between the
exchange interaction and the Hartree interaction. The exchange interac­
tion dominates for materials which belong to the class for which v = I
(amorphous metals and amorphous semiconductors). On the other hand, the
Hartree terms dominate for materials which belong to the class for which
v = ~.
The conductivity due to exchange interactions combined with localiza­
tion effects is given by (Kaveh and Mott 1982, 1986b)
16
2 [ 1 R, 0= gOB 1- (1- -)- g2 (KFR.) 2 Li
C exc (1- L)]
(8)
where C ~ 1 and T_ = ( D/kT)l/s is the interaction length. exc -1'
Equation (8) leads to the following critical behaviour (Kaveh and
Mott 1986b)
0 0 - O.06e /ha. (See Eq. (1)).
Thus, eq. (8) accounts for all the critical phenomena of a near the MIT
for the first class of materials.
13. The Class v = ~ The conductivity due to Hartree interactions combined with localization
effects for n » nc (in the weak-disorder limit) leads to an equation si­
milar to (8), except that c~ replaces C . This is a fundamental change lHar exc because ~ar is negative. This yields the following temperature dependence
2 2 00 = (Siar/3~ )e /~Lr (9)
1
which leads to the mT~ term but with m < o. Thus, the class v = ~ may be
characterized by the condition I~arl > Cexc which is sufficient to make m negative. We now discuss why m changes and becomes positive near the MIT
(Thomas et al 1983, Long and Pepper 1984). Two reasons may be responsible
for this. First, Kaveh et al (1986) have shown both theoretically and ex­
perimentally that near the MIT, L. becomes smaller than T_. Moreover, for -1 11 -1'
n + n , they showed that L. «T~. Thus, for n + n one may always write ell c a = oo+mT~ with m > 0 due to the inelastic diffusion length which becomes the smallest length scale. In addition, it was pointed out (Kaveh and
Mott 1986b) that (8) does not hold near the transition when I CHar I > Cexc .
The effect of localization is to reduce the Hartree terms relative to the
exchange terms. Thus, near the MIT, one always expects to find a positive 1
mT~ term for both classes of materials. Only for n > nc is m negative for
the v = ~ class.
14. Origin of the Critical Exponent v = ~ A theory which takes into account localization effects and interaction
effects to all orders is still lacking. What we now have is a scaling
theory for two cases
A theory including localization, exchange and Hartree interactions is still
17
missing. The importance of the Hartree terms is demonstrated from the ri­
gorous treatment of Finkelstein (1983) and Castellani et al (1984). They
deduced that interactions alone can not drive the metallic phase into a
transition to an insulating phase. Only when a strong magnetic field is
applied are the Hartree terms reduced and a continuous transition induced
with a critical exponent v = 1. For such strong magnetic fields, the lo­
calization effects and Hartree terms are suppressed, leaving the exchange
interactions to dominate the nature of the transition.
What happens for zero magnetic field, when localization, exchange in­
teractions and Hartree interactions act simultaneously?
Kaveh (1985) suggested that when the exchange terms dominate one gets
a critical exponent v = 1 for 0, whereas v = ~ when the Hartree interac­
tions dominate. Indeed, all the materials which are in the class v = ~ 1
consistently show a negative T~ term (for n ~ l.lnc)' which is due to the
dominance of the Hartree interactions. The argument of Kaveh (1985) was
recently used (Kaveh and Mott 1986b) to explain why 0 0 in (1) is larger
by an order of magnitude for the class for which v = ~ than for the other
class (v = 1). The reason is that when the Hartree term dominates
(I~arl > Cexc)' eq. (8) cannot be extrapolated down to the transition.
In this case, the conductivity hear the MIT is given by
which leads to a value 00 - e2/ha in (1) which is about 100 .. m~n
15. Material Classification
We have pointed out that the class for which v = 1 is represented by
the amorphous metals and amorphous semiconductors. The class for which
v = ~ is represented by the many-valley doped semiconductors. Why then do
the exchange interactions dominate in the first class, whereas the Hartree
terms dominate in the second class?
This question was addressed recently by Kaveh and Mott (1986b). They
pointed out that the answer lies in the value of the Hartree factor F which
is given by (Fukuyama 1982, Lee and Ramakrishnan 1985)
F = <V(K» V(o) (10)
where <V(K» is the Fourier transform of the electron-electron interaction
averaged over the Fermi surface. In order that the Hartree term will domi­
nate over the exchange terms, one needs F > 0.95. This means an extremely
short-range potential implying a very large DOS near the transition. This
18
cannot be achieved. Thus, for most materials the exchange interaction
will dominate, yielding the class v = 1. However, for a many-valley semi­
conductor, the dominance of the Hartree term~ i~ easily ~~hiYveQ, for example, for Si:P with 6 valleys, one needs only F > 0.15, and for Ge:As
with 4 valleys, one needs F > 0.22. This explains why the class v = ~ is
represented by the many-valley doped semiconductors, since only then does
the Hartree interaction dominate. Moreover, when these materials are com­
pensated, strong intervalley scattering is induced and the many-valley
character is lost. In this case, a transformation into the v = 1 class is
expected. This is what was observed by Thomas et al (1982) for compensated
Ge:Sb. The critical exponent was found to change from v = ~ (uncompensated)
to v 1 (compensated).
16. Relationship Between (1 and a
2 2 For pure disorder (Anderson transition), a ~ e /h~ and (1 ~ ~ (Imry
et al 1982, Thomas et al 1983, Kawabata 1984). Thus, the critical expo­
nent of the dielectric constant is twice that of the conductivity,
n -n (_C_)-2v
nc ( ex: 1 (11)
Since v = 1 in this case, it is expected that the critical exponent of (1
will be 2. It was argued (see, for example, Lee and Ramakrishnan 1985)
that the relationship between the critical exponents of (1 and a holds
also in the presence of interactions. This would imply for Si:P or Ge:As _k
where v = ~ that ~ ~ ((nc·n)/nc) 2 This problem was recently studied ex-
perimentally for Ge:As by Shafarman et al (1986). They deduced that the
critical exponent of the localization length is different from the critical
exponent of the conductivity. The critical exponent of the conductivity is v ~ ~, whereas the critical exponent of ~ is unity. This result was re­
cently interpreted by Kaveh and Mott (1986b). They deduced that the
general relationship between (1 and a is
(12)
For pure disorder or when the exchange interactions dominate (class v = I),
a ~ e2/h~ and (1 ~ ~2 For materials in the class for which v = ~, 2 ~ a ~ e /(a~) and from (12) it follows that (1 ~~. This explains all the
data. The critical exponent of ~ is unity, the critical exponent of 1
a ~ ~-~ is v = ~ and the critical exponent of (1 N ~ is unity.
17. Summary We have reviewed the experimental and theoretical developments of the
metal-insulator transition. It is clear that the localization and electron-
19
interaction effects act together to induce the transition. The Hartree
terms are important for uncompensated many-valley doped semiconductors
yielding a critical exponent v =~. It would be interesting to measure
the critical exponent in strong magnetic fields or under pressure. In
both cases, the Hartree contribution is suppressed and a change to v = 1
is expected. The amorphous materials have a single-valley band (as well
as strong compensation in the many-valley doped semiconductors) and there­
fore the exchange terms will dominate in this case, leading to a critical
exponent v = 1. It would be interesting to search for a material in which
the exchange interaction is almost cancelled by the Hartree interaction.
In this case, the transition will be of the Anderson type. (See a detailed
discussion by Mott and Kaveh 1985). Then, for strong magnetic fields, a
discontinuous transition is expected (Schmeltzer and Kaveh 1986) with a
minimum metallic conductivity ~0.03e2/~a.
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21
Thomas F. Rosenbaum
The James Franck Institute and Department of Physics The University of Chicago Chicago, Illinois 60637
The physics of the metal-insulator transition has been explored for over thirty years now, but our understanding is still incomplete. As emphasized in the classic papers of Mottl and Anderson2 , electron cor­ relation and disorder are essential ingredients in the phenomenon of electron localization. However, their exact manifestation in real sys­ tems remains a subject of active study and dispute.
The plethora of recent experiments on a variety of systems which ex­ hibit metal-insulator transitions 3 rests heavily on the foundation laid by Fritzsche4 in his encompassing characterization of the electrical proper­ ties of doped germanium crystals. We reproduce in Fig. 1 his measurements of a series of Ge:Ga crystals spanning four orders of magnitude in accep­ tor concentration. The samples systematically move from a metallic to an insulating temperature dependence as the acceptor concentration is reduced, with a critical density nc~l x 1017cm- 3 • The £1, £2, and £3 denote dif­ ferent activation processes in the insulating state.
Measurements down to liquid helium temperature paint the broad picture of electron localization, 'but lower temperatures are required to reveal the fine details of the (zero temperature) critical behavior. We can understand why by considering the characteristic energies involved. For doped Ge and Si the Fermi temperature is of order 30K to lOOK. The energies of interest behave critically; if we assume a linear dependence on n-nc ' then within 1% of the transition the characteristic temperature is of order 1K. Hence, we need a dilution refrigerator in order to measure doped semiconductors at energies which approximate T~ when compared to a characteristic ~lK.
The contrast between liquid helium and mi11iKe1vin experiments is illustrated in Fig. 2. We plot the conductivity cr of Si:P at three dif-
ferent temperatures over a range of n-nc ~ ± 0.01. The intrinsic critical nc
form is thermally smeared beyond recognition at T = 4.2K, and only approa-
ches its inhomogeneity broadened limit at a few mi11iKe1vin. We note that the rapid variation of cr(T ~ 0) with n in Si:P requires a resolution which can be provided only with the use of uniaxial stress. This powerful tech­ nique, which allows fine adjustment in the overlap of donor wavefunctions, was pioneered by Cuevas and Fritzsche5 in 1965.
23
24
7
19
1.7 -16
6 .3 -\7 15 2 .6 -18
""=::!:;:-r:::-=-~-:!::-~.c:-~+?==_~_~216-.,Jl:----,J 5 .' -18 L.. 27 \.3-'9 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7
lIT lK-1)
Fig. 1: Resistivity measurements by Fritzsche4 on a series of Ge:Ga sam­ ples map out the metal-insulator transition.
2'.---.---~---~---r--~
S l klXIr l
Fig. 2: The necessity of milliKel­ vin temperatures for determining the true critical behavior is illustrated by these measurements6 on Si:P.
The Si:P data6 can be fit to the scaling form cr(T~)= cr (n-nc)v, with o --yr
v~1/2. The exponent Qf 1/2 presently remains unexplained. Amorph8us alloys
(e.g. a-Nb:Si7 , a-Si:Au8 , a-Ge:Au 9), compensated Ge:Sb 10 , and the magnetic semiconductor Gd3S~11 fit the same scaling form, but with v~l. Both the scaling theory of 10ca1ization12 and a theory considering electron inter­ actions alone in the weak disorder 1imit13 give v=l. It appears likely, however, that a theory incorporating both localization and interaction ef­ fects will be required to account for the full spectrum of electronic, magnetic, and dielectric critical behavior.
Recent measurements on the critical form of the Hall conductivityl~ in r~:Sb samples (grown by Fritzsche in the 1960's!) with v=l illustrate the complexity of the problem. The scaling theory of localization predicts a drop in the conductivity at the approach to the metal-insulator transition over and above the more gradual decrease expected from the decreasing number of carriers. In particular, Shapiro and Abrahams15 predict that the Hall coefficient Ru should remain constant as n~nc' We plot l/~ vs. n at
T=8mK in Fig. 3. The values of ~ = dPH/dH were determined from linear
fits to PH(H ~), shown for a sample with n = 3.95x1017cm- 3 in the figure
inset. The Hall coefficient data clearly go to zero as n~ n and can be c
fit to a critical form with an exponent vRH = 0.7±0.lS. Although this is
a disordered system with v=l, no present theory can account for the measured vRH•
A very different system in which to study the metal-insulator transi­ tion is the narrow gap semiconductor Hg1 Cd Te. Here too, nonetheless, -x x mi11iKe1vin temperatures are essential. The dominant energy scale is the Coulomb interaction with a characteristic temperature T ha ~ 4K for n ~101~cm-3. New physics emerges when T« T h' c r
c ar
We plot in Fig. 4 the variation of the longitudinal resistance with mag­ netic field as a function of temperature for a sample with n= 2.3x101~cm-3 and x = 0.225. The extreme quantum limit, where the electrons are confined to the lowest spin-polarized Landau level, is reached by a few kOe, followed
6 6
e4 E
0 ..... :c
a:: ......... .-
2
Fig. 3: Critical behavior of the inverse of the Hall coefficientl~ RH in Ge:Sb. Theory predicts no dependence on n (see text).
25
12
8
c::
;:: a::
4
°OL-~~~~~--L-~~~ 40 60 80 H (kOe)
Fig. 4: The longitudinal resistance as a function of magnetic field at different temperatures. The extreme quantum limit was reached by a few kOe.
by the transition into the insulating state at a critical field Hc. We show in Fig. 5 the linear dependence of H on T determined previ~~sllj6 from mea­ surements of the Hall resistivity onca sample with n=1.4xlO cm and x=0.24.
The linear relationship between Hc and T emerges naturally from the viewpoint of the melting temperature of an electron crystal. At low tem­ peratures the quantum mechanical fluctuations, whose scale is set by the magnetic field, dominate, giving Tc n H. At high temperatures only the thermal fluctuations are important and Tc should saturate. We explicitly show this saturation in Fig. 6, where we 17 have extended the measurements in Fig. 5 to higher temperature and field. It is clear from the phase dia-
26
10
9
0.2 0.4 0.6 0.8 T(K)
Fig. 5: The linear dependence of the cri­ tical field marking the transition from metal to insulator, Hc ' on temperature for a sample 16 of Hgl Cd Te with n=1.4xlOl~cm-3. -x x
100~---.---r--~-"-....,
4 5
Fig. 6: Phase diagram determined1 ' by extending the measurements of Fig. 5 to higher Hand T. The properties of the solid are only accessible for study in the mi11iKe1vin regime.
gram of Fig. 6 that measurements in the liquid helium range18 are sensitive to the properties of the correlated electron fluid, but that the physics of the electron solid can be studied only at mi11iKe1vin temperatures.
I am indebted to my co11eages at The University of Chicago and at AT&T Bell Laboratories, who were essential to these experiments. This work was supported by the National Science Foundation under Grant No. DMR85-17478.
REFERENCES
1. N.F. Mott, Proc. Camb. Phil. Soc. 32, 281 (1949); N.F. Mott, Proc. Phys. Soc. London A62, 416 (1956).
2. P.W. Anderson, Phys. Rev. 109, 1492 (1958).
3. See, for example, P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985); and N.F. Mott and E.A. Davis, "Electronic Processes in Non­ Crystalline Materials" (Oxford Univ. Press, 1979).
4. H. Fritzsche, Phys. Rev. 99, 406 (1955); H. Fritzsche in "The Metal Non­ Metal Transition in Disordered Systems", ed. by L.R. Friedman and D.P. Tunstall (SUSSP Pub1., Edinburgh, 1978) p. 193.
5. M. Cuevas and H. Fritzsche, Phys. Rev. 139, A1628 (1965); Phys. Rev. 137, A1847 (1965).
6. T.F. Rosenbaum, R.F. Milligan, M.A. Paa1anen, G.A. Thomas, R.N. Bhatt, and W. Lin, Phys. Rev. B~, 7509 (1983) and references therein.
7. G. Hertel, D.J. Bishop, E.G. Spencer, J.M. Rowell, and R.C. Dynes, Phys. Rev. Lett. 50, 743 (1983).
8. M. Yamaguchi, N. Nishida, T. Furubayashi, K. Morigaki, H. Ishimoto, and K. Ono, Physica Bl18, 694 (1983).
27
9. B.W. Dodson, W.L. McMillan, J.M. Mochel, and R.C. Dynes, Phys. Rev. Lett. 46, 46 (1981).
10. G.A. Thomas, Y. Ootuka, S. Katsumoto, S. Kobayashi, and W. Sasaki, Phys. Rev. B25, 4288 (1982).
11. S. von Molnar, A. Briggs, J. F1oquet, and G. Remenyi, Phys. Rev. Lett. 51, 706 (1983).
12. E. Abrahams, P.W. Anderson, D.C. Licciarde110, and T.V. Ramakrishnan, Phys. Rev. Lett. 42, 693 (1979).
13. C. Castellani, C. DiCastro, P.A. Lee, and M. Ma, Phys. Rev. B30, 527 (1984).
14. S.B. Field and T.F. Rosenbaum, Phys. Rev. Lett. 55, 522 (1985).
15. B. Shapiro and E. Abrahams, Phys. Rev. B24, 4025 (1981).
16. T.F. Rosenbaum, S.B. Field, D.A. Nelson, and P.B. Littlewood, Phys. Rev. Lett. 54, 241 (1985).
17. T.F. Rosenbaum, S.B. Field, D.H. Reich, D.A. Nelson, and P.B. Little­ wood, to be published.
18. See, for example, "Physics of Narrow Gap Semiconductors", ed. by E. Gor­ nik, H. Heinrich, and L. Pa1metschofer, Springer Lecture Notes in Physics, Vol. 152 (Springer, Berlin, 1982), and "Applications of High Magnetic Fields in Semiconductors", ed. by G. Landwehr, Springer Lec­ ture Notes in Physics, Vol. 177 (Springer, Berlin, 1983).
28
AT&T Bell Laboratories Murray Hill, NJ 07974
A disordered metal can be driven through a metal-insulator transition by the application of a sufficiently large magnetic field. A comparison of recent results in Ge:As to other materi­ als suggests that an instability may occur toward a spin or charge density wave state. This state, while related to Wigner crystallization, may only have short range order. The transport properties exhibit a well defined crossover field and suggest the possibility of a phase transition at zero temperature.
INTRODUCTION
Hellmut Fritzsche and coworkers pioneer1d the study of the metal insulator transition in semiconductors. In particular they utilized extensively the technique of driving a single sample throu~h the transition with the application of an extern­ al parameter. Such parameters have included uniaxial stress and magnetic field. In magnetic field the insulating state may have different translational symmetry from the original metal. Furthermore, the application of uniaxial stress produces an anisotropy which favors the formation of a new ground state in magnetic field.
Generally, the fields necessary to drive metals into an insulating state are easily accessible only for doped semicon­ ductors because of their relatively small characteristic ener­ gies. However, in these materials there is a random potential caused by the impurities which produce the metallic state.
Theoretical suggestions to describe this insulating state are qualitatively helpful but appear to be inadequat~ at pres­ ent. These include a simple wave function shrinkage (magnetic freeze-out) with a retention of translational symmetry, based on a consideration of isolated impuiities -- an approach successful in describing very low densities. In this case the transi-
29
tion is expected at point described by a field dependent Mott criterion.
(1)
where the critical density is nc ' and the impurity Bohr radius is aB along the field but is reduced perpendicular to the field to a(H) which is related to the magnetic length
LH=(hc /eH)1/2.
In a metal the electron energy in field, EH, is proportional to the cyclotron frequency
* EH=neH/(cm ).
For example, in Ge:As, Eij(meV) is about 1.4H(Tesla). The quan­ tum limit occurs when th1s energy equals the Fermi energy.
EH=EF•
The number density of conduction electrons above the point of magnetic freeze-out would be small at low temperatures.
where HO is a constant.
Alternate models take into consideration a breaking of the tsa9slational symmetry as a result of field-induced aniso­ tropy - and are formulated for metallic electrons, but in a uniform positive background. The state produced by the broken symmetry could involve charge deRsity waves, spin density waves and Wigner crystallization. In the case of spin densi­ ty waves, the wavelength would be given by
1=2eH!(nhc).
For example, r~e c~rrier density of the Ge:As sample discussed below of 6x10 cm- gives 1=120A at H=15T. However, the random potential with finite spin-orbit interactions may desgroy the long range order and the symmetry under spin rotation. In this case, there could be a charge or spin density wave formed with short range order and a well defined crossover magnetic field, Hc' at finite temperature.
AN ANISOTROPIC, ORDERED POTENTIAL
To emphasize the importance of anisotropy in favoring the formation of charge or spin density waves and as a background for the consideration of spin density waves in the disordered potential, we mention briefly the cases of highly anisotropic compounds whIOe1~harge or spin density waves form. These mater­ ials include - Nbse3' TaS3,(TaSe4)2I, and K 3Mo03' where condensation occurs in the aosence of magnetic· field and has been established by scattering experiments.
Of particular i~terest is the case of graphite studied by lye and Dresselhaus and illustrated in Figure 1. Although scattering experiments have not been carried out, a field induc-
30
ed state was observed by measuring the resistance in the a,b plane of the carbon sheets as a function of a perpendicular magnetic field. At the temperature of 455mK where the measure­ ments were made, the conduction is two dimensional with an ex­ tremely high resistance in the c direction. An important obser­ vation that is not illustrated is that the resistance along the field direction undergoes a smaller change. The anisotropy is thus reduced at the anomaly, suggesting that the condensation occurs primarily in the plane of the sheets.
T-455mK
200
lImA)
0097
(/ 0201
1. Transverse resistivity versus magnetic field in graphite (Ref. 14) •
The figure also illustrates that the resistance in the high field state drops as the applied current is increased, an effect which also argues in favor of sliding charge or spin density waves. (The resistance below 21T varies only slightly with current but is offset for clarity as indicated on the left axis. The inset shows other anomalies that occur at lower fields.)
AN ISOTROPIC, DISORDERED POTENTIAL
Figures 2 and 3 show magnetic1Sield induced anomalies in InSb:n observed by Shayegan i~ ale and in Hg 76Cd 24Te as reported by Rosenbaum et ale • In both cases the·resistivity in the Hall geometry is plotted as a function of magnetic field, so that the applied current, measured voltage, and field are in the x, y, and z directions respectively. The different curves show results from different temperatures as labelled.
31
u 5
B( Tl
2. Hall resistivity versus magnetic field in InSb:n (Ref. 15).
80 SGa :is
40 HlkOe)
0 d:500mK
H(kOe)
3. Hall resistivity versus magnetic field in H9. 76 Cd. 24Te (Ref. 16).
The main points that we wish to make about these results are the qualitative similarity, the well defined crossover fields, and the sharpening of the crossover as temperature is lowered. The similarity argues for a similar nature of the field induced insulating state in both cases. The significant rounding near Hc leads in part to our suggestion that charge density waves may form with only short range order. The possi­ ble analogy to the random field problem also suggests that three dimensions may be below the lower critical dimensionality for a phase transition. Therefore, although the data establish He as a crossover field, they do not appear conclusive in establishing a finite temperature phase transition with long range order. The sharpening of the crossover region seen in the figures suggests that an extrapolation to zero temperature indicates a transition at that point.
The crossover fields are shown in the inset of Figure 2 for InSb:n and in Figure 4 for Hg 76Cd 24Te. The linear behavior seen su~~orts the suggestion that charge density waves are formed. This type of condensation is also suggested by a decrease in the resistanr7 above Hc in the presence of relative­ ly small electric fields •
In the cases of both InSb:n and Hg 76Cd 24Te the band structure is isotropic since both have Q1rect energy gaps with the electron valley at the zone center. The magnetic field produces an anisotropy of a factor of about 2 with the higher resistance perpendicular to the field. Qualitatively this ani­ sotropy is in the direction of that expected from a simple
32
~ M T(K)
4. Crossover field versus tempera­ ture in H9. 76Cd. 24Te (Ref. 16).
shrinkage of the wave functions in the perpendicular plane. However the similarity in functional form of the resistance versus field curves iY6the field direction and perpendicular to it can be interpreted as indicating the formation of a slight­ ly anisotropic Wigner crystal.
AN ANISOTROPIC, DISORDERED POTENTIAL
The case of Ge:As differs from the direct gap semiconduc­ tors in that the donor electrons reside in valleys away from the zone center. The material then becomes asymmetric when the electrons are redistributed among these valleys, as by the ap­ plication of a [111] stress. In analogy with the ordered mater­ ials, the anisotropy should favor the formation of spin or charge density waves.
Figure 5 shows a plot of one aspect of the high field anom­ aly18 in stressed Ge:As, plotted so as to suggest some similar­ ity to the results discussed above. In this linear plot of the resistance transverse to the magnetic field, crossover fields can be defined as shown. The qualitative features of the cross­ over region are similar to those above and we would draw similar conclusions from them. However this data set is selected to show this point and most of the results show striking differen­ ces. For example, preliminary measurements of the Hall resis­ tance show only a slight increase in slope near H suggesting a nearly unchanged carrier concentration in the higfi field state. This small change in n differs from the isotropic semiconductors discussed above and differs strongly from the expectation based on simple wave function shrinkage (Eqn. 2). The anomaly also
33
occurs at a field lower than that PI7dic~ed from this model, since for the critical density 3x10 cm-, Hc should be about 25T according to Eqn.1.
34
H (Tesla)
6. Longitudinal to transverse resis­ tivity ratio versus magnetic field in Ge:As (Ref. 19).
The change in the transverse resistance shown in Figure 5 also turns out to be quite small compared to t~~ substantially increased resistance along the field direction • This aniso­ tropy is shown in Figure 6 at two different temperatures as labelled. The relatively large parallel resistance is particu­ larly remarkably compared to the expectation from a simple model of wave function shrinkage, as it is in the opposite direction. The behavior in Figure 6 can be understood as the formation of a spin density wave state primarily in the direction of magnetic field. A spin density wave is favored here over a charge densi­ ty wave because at these fields both spin states are occupied. This situation differs from the large spin splitting in H9. 76Cd. 24Te where the g-value is unusually large.
CONCLUSIONS
Results in a variety of systems suggest magnetic field induced charge or spin density waves with the instability favor­ ed by anisotropy. In the presence of a random potential a crossover appears to occur to this ground state with probable short range order and a possible critical point at zero tempera­ ture. Interpretation of the results remains uncertain pending improved theoretical treatment and further experiments, but existing theories of simple wave function shrinkage appear par­ ticularly inadequate.
I thank my collaborators in the work on Ge:As M. Burns, P. Hopkins, R. Westervelt, and B. Halperin. I also thank R. Bhatt, R. Birgeneau, H. Fukuyama, P. Lee, P. Littlewood, T. Rosenbaum, and M. Shayegan, for helpful discussions. Part of this work at Harvard University was supported by grant number DMR83-16969.
REFERENCES
1. See, for example, H. Fritzsche, Phys. Rev. 99, 406 (1955)~ and in "The Metal Non-Metal Transition in Disordered Systems", ed. by L.R. Friedman and D.P. Tunstall (SUSSP Publ., Edinburgh, 1978), p.193.
2. M. Cuevas and H. Fritzsche, Phys. Rev. 139, A1628 (1965)~ Phys. Rev. 137, A1847 (1965).
3. See, for example, G. DeVos and F. Herlach, in "Applications of High Magnetic Fields in Semiconductors", ed. by G. Landwehr, Springer Lecture Notes in Physics, Vol. 177 (Springer, Berlin, 1983), p.378.
4. N.F. Mott, "Metal-Insulator Transitions," (Taylor and Francis, London, 1974).
5. V. Celli and N.D. Mermin, Phys. Rev. 140, A839 (1965).
6. W.G. Kleppmann and R.J. Elliott, J. Phys. c~, 2729 (1975).
7. H. Fukuyama, Solid State Comm. 26, 783 (1978), D. Yoshioka and H. Fukuyama, J. Phys Soc. Japan 50,725 (1981).
35
8. E. Wigner, Phys Rev. 46, 1002 (1934).
9. B.I. Halperin, M.J. Burns, P.F. Hopkins, G.A. Thomas and R.M. Westervelt, Proceedings of the Argonne Conference, 1986, in press.
10. N.P. Ong and P Monceau, Phys. Rev. B16, 3443 (1977).
11. R.M. Fleming and C.C. Grimes, Phys. Rev. Lett. 42, 1423 (1970); R.M. Fleming, Phys. Rev. B22, 5606 (1980). --
12. A. Zettl, G. Gruner, and A.H. Thompson, Solid State Comm. 11, 899 (1981).
13. J. Duman, C. Schlenker, J. Marcus and R. Buder, Phys. Rev. Lett. 50, 757 (1983).
14. Y. lye and G. Dresselhaus, Phys. Rev. Lett. 54, 1182 (1985).
15. M. Shayegan, V. J. Goldman, H.D. Drew, D.A. Nelson and P.M Tedrow, Phys. Rev. B32, 6952 (1986).
16. T. F. Rosenbaum, S.B. Field, K.A. Nelson and P.B. Littlewood, Phys. Rev. Lett. ~, 241 (1985).
17. S.B. Field, D.H. Reich, B.S. Shivaram, T.F. Rosenbaum, D.A. Nelson and P.B. Littlewood, Phys. Rev. B33, 5082 (1986).
18. M.J. Burns, P.F. Hopkins, B.I. Halperin, G.A. Thomas and R.M. Westervelt, unpublished.
19. M.J. Burns, P.F. Hopkins, B.I. Halperin, G.A. Thomas and R.M. Westervelt, preprint 1986.
36
w. Sasaki, Y. Nishio, and K. Kajita
Toho University
INTRODUCTION
Low temperature transport properties in doped semiconductors have attracted interest of solid state physicists, and many important achievements have been obtained. The genuine expe,imental work in this field started with a contribution by H. Fritzsche •
The problems of impurity conduction are divided into three categories: the hopping conduction, the metallic conduction, and the metal-insulator transition. The very important concept of the loc~lization of electrons in random potential field was introduced by Anderson in the same year as Fritzsche's paper. Later development in this field has demonstrated that the Anderson localization plays a central role in all these three categories.
The idea of hopping conduct~on was established at an early stage of the impurity conduction research • The classical hopping picture was formulated on the basiZ of an electron localized on a donor. An idea of variable range hopping was developed by which hopping of electrons extended over a wider region, corresponding to the electrons localized in the sense of Anderson, is included. A relation among the size of the localized state, the density of states and the characteristic temperature was introduced.
The wave function is approximated by a plane wave modulated by an envelope function exp(-r/2a), where a is the localization length, which gives the measure of the extent of the localized wave function, and must be much larger compared to the average impurity distance when the state is situated just below the mobility edge. The present article is partly related with those states.
Anomalous tr~n~port properties, including negative magnetoresistance' and7tHe temperature variation of resistivity which is observed even below 30mK ' , were left unsolved for a long time9 teese problems were also settled as precursor eff,cts of localization' •
A scaling approach by Abrahams et ale provided a powerful device to handle the problem of impurity conduction from a unified view point. Their argument starts from a renomalization group equation, in which a non-dimensionalized conductance g appears as the independent variable. In 3-d system there exists a critical value of g, corresponding to the boundary between metal and insulator. Physical quantities should show a critical behavior near the transition. In case of doped semiconductor the electron concentration n is a candidate for this parameter. For metallic
37
side the critical behavior of canductivity is represented by 0=0 (n/n -1) (1)
where n is the criticil co&centration of the transition and s is the critical exponent. For insulator side the dielectric constant is given by ,
E:=1+X (1-n/n )-s • (2) A simple relation o c
s'=2s (3) is an important requirement for the case wh'2e both quantities are governed by the common randomness parameter •
Experiments on the critical behavior in doped semiconductors indicate a variety of s value depending on the material and compensation. A series of stress tuning expe;!m'fts of metal-insulator transition was carried out on uncompensated Si:P , • Stress tuning enabled detailed study of conductivity and dielectric constant over a region very close to the transition, and the relation (3) was confirmed to hold for this case. The value of s was estimated to be 0.55±0.1. For many 95 fge1,morphous materials s values of approximately 1 are reported ' , ,and for other materials, including compen~~t,~ semiconductors, the values of s are scattered betweem 18and 0.5 ' • For Ge:Sb the s value seems to differ by compensation ratio •
2r--------ro------~--.~ I a .-: po _
• o ",." • , ;; "' ... , I • • I ./
E ! ol 01 ~ 'l-~.-.-'" <.>0. 0 I /r/ _" E 0 ~ ". ,,',
.s:: • 0 j' / /" ~ . • o 0 I .-. Ci ~ .A~" /-:: Z en 0 _/ ? ~ ~,! 3· 0 _0 .1 0 7 /.,"" _.-" 0 0 0
o~..,......-................. 0 00 0 3.6 • ". -"' .. ~'" 0 0 0: ... ......... 6..
0 / •• ~ .JP i 0 ... Ao a aDo a 0 0 a
~• '8.f~"" 0' 0 0 3.8 ... "" ...... " 'I' ... .. ",. .... or ...........
• •
( T-" 4/ K -v4l
Fig. 1 The Mott Pt§t ~S resistivity versus temp'8at~e data for a Si:P crystal (N =5.1x10 cm ) irradiated with 1.5x10 cm fast neutrons. Fi~e~3inRicate the room temperature carrier concentration in unit of 10 cm • The plot "5.1" is the data before irradiation.
38
These facts tempted us to study the effect of compensation on the critical behavior in Si:P system. Chemical doping of both of the majority and minority element is not appropriate for systematic study of the transition. For the present purpose a physical processing, fast neutron irradiation followed by isochronal annealing, was adopted.It is w~6l known that the lattice defects produced by irradiation act as acceptors • The concentration of these defect can be decreased by annealing. By careful stepwise annealing, the acceptor concentration 2~ be varied almost continuously with the donor concentration fixed • By decreasing the acceptor concentration, the potential fluctuation is made smaller and the mobility edge is lowered. At the same time, the concentration of electrons responsible to conduction is increased and the Fermi level becomes higher. Thus the physical picture of the crossing over from insulator to the metallic state is very clear. This is another merit of the present method of tuning.
EXPERIMENTAL
Samples are single crystals of silicon doped with phosphorus. In the following, the value of donor concentration ND calculated from the room temperature Hall coefficient ~ by
N =1/eRu (4) is used. ND values ~hus ootained should be multiplied by a factor of approximately 1.3 to compare with those obtained fro'aIr!3ne curves. The samples were irradiated with fast neutrons of 1.5x10 cm • 10 minutes isochronal annealings were done with temperature interva