Table Of ContentFESTKORPERPROBLEME XVI
ADVANCES IN SOLID STATE SCISYHP
R E P R O K T S E F
IVXEMEIBORP
HISECHAVDA
gHos ETATS
SCISYHP Plenary Lectures of the Divisions
"Semiconductor Physics"
"Metal Physics"
"Low Temperature Physics"
"Thermodynamics dna Statistical "'scisyhP
of the German Physical Society
Freudenstadt, April 5-9, 1976
Edited yb
.J Treusch, Dortmund
With 781 figures
geweiV
1976
All rights reserved
(cid:14)9 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig, 1976
No part of this publication may be reproduced, stored in a retrieval
system or transmitted, mechanical, photocopying, recording or
otherwise, without prior permission of the copyright holder.
Set by Vieweg, Braunschweig
Printed by E. Hunold, Braunschweig
Bookbinder: .W Langeliiddecke, Brauns~l~Ig-
Cover design: Barbara Seebohm, .~w~LsnuarB t~
Printed in Germany-West
ISBN 3 528 08022 1
Dedicated to the memory of
Walter Schottky
1886-1976
Foreword
This volume of "Festk6rperprobleme/Advances in Solid State Physics" is dedicated
to the memory of Walter Schottky, the first editor of this series, who died a few
weeks before the 1976 Freudenstadt meeting of the German Physical Society.
Acknowledging his pioneering work in Semiconductor Physics the selection of in-
vited papers presented in this volume once more favors work on semiconductors.
As may be easily recognized, scientific and technological interest is mainly directed
towards material engineering and applications. "New" materials, such as one- and
two dimensional crystals, III-VII semiconductors, mixed valence compounds,
certain glasses and polymers are investigated as to their basic electronic, thermal
and magnetic properties, whereas, in the case of the well known semiconductors,
such as Silicon, structural and compositional control is of main importance. In two
symposia on "Solid State Device Technology" and on "Microdefects and Micro-
inhomogeneities in Silicon", discussions were focussed on the latter point, simult-
aneously yielding an opportunity of closer contacts between academic and in-
dustrial research.
Solar cells were covered by two invited talks, the one of B6er being a public lecture
in the evening. Though there si necessarily some redundance, it seemed worthwhile
to publish both of them, so offering a sort of balance between the academic, the
technological and also the economical point of view.
The first paper is the prize-lecture given by Franz Wegner, the fourth recipient of
the Walter Schottky prize, which he won for his outstanding contributions to the
theory of phase transitions.
My compliments and thanks to the publisher and the authors whose cooperation
made it possible to meet the high standards set by my predecessors, Otfried Maclelung
and Hans-Joachim Queisser: this volume is published only ten weeks after the
meeting.
I hope the critical reader will decide, that also selection and contents of this volume
meet the standards set in the years before.
Joachim Treusch
Dortmund, June 1976
Contents
Franz J. rengeW
Phase Transitions and Critical Behaviour
rl Richter dna R o land Zeyher
Resonant Raman Scattering in Semiconductors 15
Philippe Schrnid
Phonons in Layer Structures 47
reteP-snaH hcireseG dna Lothar Pintschovius
Polymeric Sulfur Nitride, (SN)x - A New Type of One-Dimensional Metal? 65
Gernot Giintherodt
Configurations of 4f Electrons in Rare Earth Compounds 95
Koichi Kobayashi
Optical and Electronic Properties of m-VII Compounds - Thallous Halides 117
Alfred Seeger
Atomic Defects in Metals and Semiconductors 149
A. J. Rudolf de Kock
Characterization and Elimination of Defects in Silicon 971
Heinz Beneking
Material Engineering in Optoelectronics 591
Thomas Ricker
Electron-Beam Lithography - A Viable Solution? 217
HaraM Overhof
Hopping Conductivity in Disordered Solids 239
S. Hunklinger, H. Sussner and K. Dransfeld
weN Dynamic Aspects of Amorphous Dielectric Solids 267
Dieter Bonnet, Matthias Selders and Helmut Rabenhorst
Solar Cells and Their Terrestrial Applications 293
K. rI BOer
Large Scale Energy Utilization - The esU of Thin Film Solar Cells 315
FestkSrperprobleme XVI (1976)
esahP Transitions dna Critical ruoivaheB
Franz J. Wegner
Institut fiir Theoretische Physik, Universit~it Heidelberg, Heidelberg, ynamreG
:yrammuS nA introduction to eht theory of lacitirc anemonehp dna eht noitazilamroner group
sa promoted by nosliW si .nevig ehT main sisahpme si on eht idea of eht fixed point nainotlimah
(asymptotic ecnairavni of eht lacitirc nainotlimah under egnahc of the htgnel )elacs dna eht
gnitluser ytienegomoh .swal
The renormalization group procedure as a scale transformation on the effective
hamiltonian has fumed out to be a very fruitful theory to explain critical pheno-
mena. The study of the structure of this theory has confirmed most of the pheno-
menological assumptions and heuristic observations on critical systems, and has
reproduced the features of exact model solutions. Moreover, the theory yielded
new results and more information on critical phenomena. At the same time, the
theory gives a deep insight into the complicated nonanalytic behaviour at the
critical point.
In view of the numerous papers on this subject, most often only review articles
are cited.
1. Introduction to critical phenomena
A. Critical behaviour
Consider a fluid in equilibrium with its vapour. By increasing both temperature and
pressure along the coexistence curve, the density of the fluid will decrease whereas
the density of the vapour will increase. Frequently one can follow the coexistence
curve until both phases become equal in density and lla other properties at the
critical point. Above the critical temperature T e only one homogeneous phase
exists. The phenomena observed in the vicinity of this point are called critical
phenomena. Close to this critical point the system exhibits rather strong fluctua-
tions which is apparent in the critical opalescence. Due to these fluctuations in the
local density a beam of light passing through a fluid near a critical point is scattered
so strongly that the entire volume of fluid appears to glow.
This phenomenon of two (or more) phases becoming identical at a critical point is
not restricted to liquid-vapour systems. A ferromagnet, for example, consists of do-
mains of magnetic moments of different orientation, thus constituting several
phases in equilibrium. On approaching the Curie temperature T e from below the
magnetic moment of each domain vanishes continuously. Above T e the net ma-
gnetic moments are zero: the system is in the homogeneous paramagnetic phase.
Some other systems exhibiting critical behaviour are binary mixtures, alloys, anti-
ferromagnets, ferroelectrics, superfluid helium, superconductors, systems under-
going structural and elastic phase transitions.
It is common to all these systems that they can be described in terms of an order
parameter m. The magnetization serves for the ferromagnet and the density differ-
ence/9 - Pc (where Pc is the critical density) for the liquid-vapour system as the
order parameter. The unified description in terms of the order parameter allows
one to restrict to one class of systems to explain the main features of critical phen-
omena 1 . We will use the description for a ferromagnet here. Since critical phen-
omena appear on a macroscopic scale we will often assume the variable S(r) of the
local magnetization to be a continuous classical variable (c-number).
B. Critical exponents
In most cases the behaviour of the magnetization (order parameter) can be des-
cribed by a power law close to T c
m=A mlrl fl, r<0 (1.1)
with
r = (T - Te)/T~. (1.2)
Similarly the susceptibility (in case of the liquid-vapour system the compressibility)
diverges like
X = A x Irl -~. (1.3)
This divergence reflects the strong fluctuations near the critical point since
(cid:141) ~f < S(0) S(r) > ddr. (1.4)
The specific heat contains a nonanalytic contribution often described by
Csinz = A e Irl -~. (1.5)
The exponents a,/3, and 3' are called critical exponents.
The oldest theory which explained critical phenomena qualitatively (Van-der-Waals-
equation, molecular field theory by P. Weiss, Landau theory) 2 predicted
Og=0 ' ~=1
~, 7 = 1 (1.6)
where the amplitudes A x of the susceptibility and rA of the specific heat differ
above and below T e. Experiments yield exponents close to
a~0, #~, 1 7~. 4 (1.7)
The explanation of these exponents remained a puzzle for more than half a century.
Various models like the spherical model in three dimensions (a = - 1,/3 = (cid:1)89 = 2)
and the two-dimensional Ising-model (a = 0,/3 _-1~ , 7 = ~) could not resolve this
puzzle. Estimations of the critical exponents from various types of expansions
yielded numbers quite close to (1.7), but did not provide an understanding of
critical phenomena. They showed, however, convincingly that the exponents dep-
end on the dimensionality d of the system and its symmetry.
C. Homogeneity law
It was Widom 3 who made the very useful assumption (we give a simplified version)
that the free energy as a function of the magnetic field h and the temperature dif-
ference r consists of a regular and a singular part
F0", h) = Freg (r, h) + Fsing ('r, )11 (1.8)
where the singular part is responsible for the ciritical behaviour. He assumes that
the singular part is a homogeneous function of the variables "~ and h
Fsing (r, h) = r 21 -'~ +_f (#) (1.9)
where the + indicates that the function is different above and below Te. A is called
the gap exponent.
Let us discuss some consequences. One obtains the specific heat by differentiating F
twice with respect to r. This yields for vanishing field h
Csi.g ~ I rl -~ f+ (0) (I .10)
which was the reason for calling the exponent in equation (1.9) 2 - .~, The magneti-
zation and the susceptibility is obtained from F by differentiating with respect
to h 4
)11.1(
)21.1(
with
#=2-a-A )31.1(
7=a+2A-2. (1.14)
