Table Of ContentFESTKORPERPROBLEME XIX
ADVANCES NI SOLID STATE PHYSICS
REPROKTSEF
EMELBORP XlX
SECNAVDA NI
DILOS ETATS
Plenary Lectures of the Divisions
,HP SC,ISY "Semiconductor Physics"
"Metal Physics"
"Low Temperature Physics"
"Thermodynamics dna Statistical Physics"
"Magnetism"
"Surface Physics"
"Thin Films"
"'High Polymers"
of the German Physical Society
,retsn~iM March 12-17, 1979
Edited yb
.J Treusch, Dortmund
With 248 figures
V i gewe
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Festk6rper-Probleme = Advances in solid state
physics. - Braunschweig: Vieweg.
Frtiher auch im Verl. Pergamon, Oxford, Elmsford
(New York), Toronto (Canada), Sydney.
ISSN 0430-3393
NE: PT
19. Plenary lectures of the Divisions Semiconductor
Physics, Metal Physics, Low Temperature Physics,
Thermodynamics and Statistical Physics, Magnetism,
Surface Physics, Thin Films, High Polymers of the
German Physical Society, Miinster, March 12-17,
1979. - 1979.
ISBN 3-528-08025-6
NE: Deutsche Physikalische Gesellschaft/Fachaus-
schug Halbleiterphysik; Deutsche Physikalische
GeseUschaft/Fachausschug Metallphysik
ISSN 0430-3393
All rights reserved
(cid:14)9 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig, 1979
No part of this publication may be reproduced, stored in a retrieval system or transmitted,
mechanical, photocopying or otherwise, without prior permission of the copyright holder.
Set by Vieweg, Braunschweig
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Cover design: Barbara Seebohm, Braunschweig
Printed in Germany-West
ISBN 3-528-08025-6
Foreword
Volume XIX of "Festk6rperprobleme/Advances in Solid State Physics"
comprises a selection from the invited papers delivered at the Spring
meeting of the Solid State Divisions of the German Physical Society in
Mthnster, 1979. It is opened by H. M~iller-Krumbhaar, who was awarded
the Walter-Schottky prize for his outstanding contributions to the theory
of crystal growth. Five predominantly theoretical papers are concerned
with the electronic and excitonic properties of semiconductors as connect-
ed to their bonding properties and optical spectra, including the boun-
dary problem of spatially dispersive media. The four following contribu-
tions show that photoemission and LEED are still in the focus of experi-
mental and theoretical interest, yielding not only information on elec-
tronic states but also on surface defects, spin polarization, and many-
body effects. The recent revival of deep centers research is demonstrated
under the aspects of their electronic structure and of photoconductivity.
Sir N. Mott presented a plenary talk on hopping conduction. The Low
Temperature Division contributes with a paper by .M Tinkham on Non-
equilibrium Superconductivity. Picosecond spectroscopy, tunable lasers,
and new means for an effective use of solar energy are the topics closing
this volume, which reflects the stimulating atmosphere of activity and
excitement that characterized this springmeeting.
Thanks are due to all authors and to the Vieweg Verlag for their combined
efforts.
Joachim Treusch
V
Contents
H. Miiller-Krumbhaar
Surface-Dynamics of Growing Crystals
.11 lVendel, M. Martin
Electronic Properties, Chemical Bonding, and Lattice Dynamics 12
.W Hanke
The Role of Electron-Hole Interaction in the Optical Spectra of
Semiconductors and Insulators 43
.U ROssler
Fine Structure, Lineshape, and Dispersion of Wannier Excitons 77
E. .O GObel, .G Mahler
Extended Phase Diagrams of Excited Semiconductors 105
A. Stahl, .hC Uihlein
Optical Boundary Value Problem in Spatially evisrepsiD Media 951
M. Henzler
Defects in Surface Structure: Informations with LEED 391
A. Liebsch
Theoretical Aspects of Photoemission 209
.T Grandke
devloseR-elgnA Photoemission 225
E. Kisker M. Campagna, .W Gudat, E. Kuhlmann
Spin Polarization of Photoelectrons Emitted from Ferromagnets 259
H.-J. Hoffmann, F. StOckmann
Imperfections and Photoconductivity 172
IV
.G A. Baraff. M. Schliiter
Electronic Struture of Localized Defects in Covalent Semiconductors 303
N. F. Mort
Hopping Conduction and the Coulomb Gap; Applications to Fe304, Ti407
and Impurity Conduction in :iS P 133
M. Tinkham
Non-Equilibrium Superconductivity 363
D. .v d. Linde
Picosecond Spectroscopy: Methods and Applications 783
K Welling, D. FrMich
Progress in Tunable Lasers 403
A. Goetzberger, .V Wittwer
Fluorescent Planar Collector-Concentrators for Solar Energy Conversion 724
IIV
Festk6rperprobleme XlX (1979)
scimanyD-ecafruS of gniworG slatsyrC
Heiner M~ller-Krumbhaar
Institut ruf ehcsiteroehT ,kisyhP Universit~t ,revonnaH ,revonnaH ynamreG
Summary: The dynamic processes at surfaces of crystals during growth are described using a
variety of mathematical formalisms, depending on the characteristic lenght scales and times of
the processes. For surfaces without dislocations a master-equation formalism allows one to
calculate surface structures and growth rates to a very good quantitative precision. Surface
spirals originating from screw dislocations are described by a time-dependend Ginzburg-Landau
equation. The resulting anisotropic spiral structures are in agreement with Monte-Carlo simula-
tions and allow us to explain recent experiments. At temperatures above a predicted roughening
transition the growth rate is proportional to the difference of chemical potentials across the
crystal surface. Crystals growing from a super-saturated liquid in this regime develop an in-
stability of the interface, producing dendritic protrusions. The most popular example is the
snowflake. A dynamic stability analysis of these dendrites is in excellent quantitative agreement
with recent experiments.
I Introduction
Solids usually appear in crystalline form. The atoms or molecules as basic units then
are arranged in regular lattices. They oscillate with small amplitudes around their
average positions, while the lattice structure and the mechanical stability are con-
served usually up to the melting temperature. The mobility of the atoms in the
crystal is much lower than in the non-crystalline (liquid, gas) phase. If the crystalliza-
tion occurs not from a pure phase but from a mixed phase (solution, melt or vapor
with impurities), the composition and short-range structure near the phase-boundary
are frozen into the solid for high enough solidification rates. The solid, therefore, is
usually not in a state corresponding to thermodynamic equilibrium but in a long-
lived metastabel state. The structure of the crystal and its physico-chemical prop-
erties thus are determined to a large degree by the time-dependend processes at the
phase-boundary.
Even single-component crystals of monatomic substances exhibit a number of ex-
perimental phenomena which have been quantitatively explained only recently 1.
The usually observable phase transitions of such a single-component material are
sketched in the p vs. T diagram of Fig. .1 The transitions across the coexistence
lines between solid and liquid, liquid and gas and between solid and gas are of f'trst
order. Along the liquid-gas line beyond the critical temperature T e of second order.
The details of the latter transition have been examined theoretically in great detail
2 and most aspects are well understood. The presently more interesting problem
of first-order transitions has not made much progress over the concepts of mean-
field or nucleation theory 3, 4. In particular, the theory of the liquid to solid
transition is in a rather poor shape 5. One generally believes, that a pure hard-core
repulsion should be a sufficient ingredient for a system to show a liquid-solid transi-
tion 6. But analytic theories sofar have failed to predict ordering of the hard-core
fluid above a certain density, partly since one does not know what to take as an
order parameter 7. In order to formulate interface models for growing crystals the
liquid-solid transition, therefore, is not a good point to start.
The situation is somewhat better in the case of the vapor-solid transition. In addition
to the ordering phenomenon one has a drastic change in density during the phase
change such that one might take the density as an auxiliary order parameter, building
the possibility of ordering into the models right from the beginning. With this assump-
tion one can formulate lattice models 1 for the solid-gas transition, where an essen-
tially filled lattice corresponds to the solid phase and an essentially empty lattice
corresponds to the gas phase. This simplification of course restricts the applicability
of the models to temperatures below the triple point.
These lattice models now cannot only be used to stuffy static properties of the solid-
gas interface but one may also treat the dynamics of crystal growth. Assuming that
the atoms in the crystal are essentially localized at their equilibrium positions (hence
neglecting elastic deformations) dynamics are introduced as a stochastic process,
where atoms from the gas phase are assumed to impinge statistically on the surface.
There they may be incorporated immediately, diffuse around or re-evaporate. With
Fig. {
Phase diagram rof a single-component mona-
tomic material (schematic). The coexistence
diuqiL lines are accompanied by regions of meta-
.~Criticol stable states (with respect to nucleating one
Solid
phase within the other). In addition the so-
lid-gas interface (if free of dislocations) is
metastable (hatched area) against growth be-
~~ low the roughening temperature T R. If T R
J Triple soG is not below the triple temperature this inter-
.T I cT ='-T face-metastability can also exist along the
T t solid-liquid coexistence line.
these models one may describe most phenomena as long as no structures appear,
which have characteristic dimensions very much larger than the lattice constants, such
as dislocations or problems of dynamics instability. For these cases one introduces
coarsegrained phenomenological models, where the system parameters then again
are calculated from the microscopic models. In all of these cases electronic properties
of the systems do not enter explicitly but are incorporated in the definition of inter-
action energies, activation barriers etc.
Starting from equilibrium properties of a crystal-vapor interface, the implications of
the recently developed theories for the roughening transition upon growth rates is
described in section II. Section III describes the results obtained by a master-equation
formalism for dislocation-free surfaces, including adsorption-desorption and surface-
diffusion processes. Section IV summarizes the predictions of a phenomenological
kinetic equation for anisotropic surface spirals and gives a comparison with experi-
ments and computer simulations. The f'mal section V describes the mechanism of
dendrite formation and compares the theory with recent experiments.
II Roughening Transition
At very low temperatures the interface between a crystal and its vapor in thermo-
dynamic equilibrium is essentially fiat with only a few cluster of atoms adsorbed
onto the surface and holes within the surface. In order to grow the crystal, i.e. to
advance the interface by one layer, the clusters of adatoms have to increase in
number and diameter, until this next layer is f'dled. In terms of nucleation theory 3
the increase in area of the clusters is hindered by the free energy connected with the
surface step surrounding a cluster, unless by thermal fluctuations the cluster has ex-
ceeded a "critical" size. In other words, the interface has to overcome a nucleation
barrier before it proceeds by one layer. Growth, therefore, is extremely slow in the
low temperature range and the crystal surface is in metastable states for a long time.
This is the hatched region in Fig. 1 for temperatures O < T < TR. Some thirty years
ago already it was argued 8, that cooperative effects in the surface structure lead
to qualitative changes of the equilibrium properties and of the dynamic behavior,
as the temperature exceeds a "roughening" temperature TR. The surface structure
should change from "fiat" to "rough", roughness being measured in terms of the
internal energy U s of the surface. The roughening temperature T R was defined as
the temperature of fastest increase of U s with T. As it turned out only recently 9,
these early considerations, though at least in the right spirit, were not quite con-
clusive, since they did not allow for the type of long-wavelength fluctuations which
by now are considered to be of fundamental importance.
The introduction of a new model for an interface in a lattice system, the "discrete
gaussian" (DG) model 9-12, brought a breaktrough to our understanding of the
roughening phenomenon. The model is two-dimensional, corresponding to a two-
dimensional interface in a threedimensional lattice, the Hamiltonian being given as
H=J 2 Ihi-hilP, p = 2 (1)
<i,j)
oo < hi < + %o hi = integer,
-
the sum going over nearest neighbor pairs, i h here denotes the local height of the
interface above some reference plane parallel to the average interface. Below Tg the
interface accordingly is flat and localized (Fig. 2), while above TR long-wavelength
fluctuations appear, leading to a divergence of the moments >n2h< ~ (TR - T)- nO
with critical exponents n O [9-11] as TR is approached from below. Above TR these
moments are infinite. The free energy F of the surface is expected to have a very
weak ("essential") singularity such that one should not be able to detect the rough-
ening point by measuring thermal properties like the specific heat of the interface.
(But see ref. 13.) This predicted behavior is in contrast to ordinary critical phenom-
ena, where one has a divergence or at least a cusp in the specific heat at c . T The long-
wavelength fluctuations persisting above Tg suggest the picture of a line of critical
points for T i> Tg.
~:: {:{*~t~ .... " ~:<.~ x~'-':::..~.'~.~!
~ ~ ' '~}i~g~'gg~ ~ -{~ .~ii~"
Fig. 2 Crystal surface (schematic) in a lattice model of the type eq. (1). At low temperatures
(a) the surface is localized, only fluctuations of finite amplitude and extent exist. Above the
roughening transition (b), long-wavelength fluctuations are predicted to exist which lead to
a divergence of the fluctuation-amplitudes as the system size goes to infinity.
4
