Table Of ContentFESTKORPERPROBLEME
ADVANCES IN SOLID STATE PHYSICS 29
FESTK(JRPER
PROBLEME
ADVANCES IN
SOLID STA'I~
PHYSICS 29
PlenaryLecturesof the Divisions
Semiconductor Physics
Thin Films
Dynamics and Statistical Physics
Magnetism
Metal Physics
Surface Physics
Low Temperature Physics
of the GermanPhysicalSociety (DPG),
M~Jnster, April 3 to 7, 1989
Edited by
Ulrich RSssler
With 204 Figures
Friedr.Vieweg & Sohn Braunschweig/Wiesbaden
Editor:
Prof.Dr.Ulrich R6ssler
Institut fdr TheoretischePhysik
Universit~it Regensburg
P.O. Box397
D-8400 Regensburg, FRG
Vieweg isa subsidiary company ofthe BertelsmannPublishing Group.
Allrightsreserved
© Friedr. Vieweg& SohnVerlagsgeseIlschaft mbH, Braunschweig 1989
No part of this publication may be reproduced, stored in a retrieval
system or transmitted, mechanical photocopyingor otherwise, without
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Printedby Lengericher Handelsdruckerei, Lengerich
Bound by Hunke+ Schr6der, Iserlohn
Coverdesign: Barbara Seebohm,Braunschweig
ISSN 0430-3393
ISBN 3-528-08035-3
Foreword
The annual spring meetingof the SolidState Physics Division (Arbeitskreis Fest-
k6rperphysik) of the German Physical Society took place in 1989 once more in
Manster. As in previous years the plenary and invited talks presented at this meeting
attracted and hold together the larger part of the attendents every day, which
otherwise were dispersed over the numerous parallel sessions. A selection of these
plenary and invited talks is the content of this volume of Advances in SolidState
Physics/ Festk6rperprobleme. They are representative for the widespread activities
in condensed matter physics and signalize particular achievements and concen-
tration of research activities on special topics.
The Walter-Schottky Prize 1989 was sbared by three scientists, U.Eckern,G.Sch6n
and W. Zwerger, who received the award for their theoretical contributions to the
understanding of quantum effects in superconducting tunnel-contacts and granular
films. Pattern formation in liquid crystals, localprobing ofatomic arrangements in
condensed matter by EXAFS, surface studies by scanning-tunnel microscopy, an-
isotropic propagation of heat pulses, lattice dynamics of surfaces includingrecon-
structions, and fast relaxation ofexcitons and free carriers in semiconductors are
topics of individual contributions. Two groups of papers, presented mainly in
symposia duringthe conference, cover the current investigations ofdeep centers in
semiconductors and the fabrication and physics oflow dimensional semiconductor
structures.
In order to lower the cost of production and likewise the market prize thisvolume
is producedfrom camera-ready manuscripts. On the otherhand, the intention was
to conserve the homogeneous appearance as a printed book. Thus, theauthorshad
to carry also the burden of preparing aTEX-nicaUy perfect manuscript. Not all of
them succeeded and their manuscripts had to be rewritten, at least partially. This
tremendous work has been done with great care by AngelaReisser. The TEX-
instructions have been prepared by Dr.FranzMalcher. Their help and effort was
essential for the completion ofthis volume. Finally, I acknowledge the eversmooth
cooperation with Bj6rn Gondesen (Vieweg-Verlag).
Regensburg, April 1989 Ulrich ROssler
V
Contents
Ulrich Eckern andGerd SchOn
Charge Transfer betweenWeakly CoupledNormalMetals and
Superconductors at Low Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . .
Wilhelm Zwerger
Quantum Effects and the Onset of Superconductivity in
GranularFilms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
IngoRehberg,BernhardL. Winkler, Manuel dela Tor,eJuarez,
Steffen Rasenat,and Wolfgang SehOpf
Pattern Formation inaLiquidCrystal . . . . . . . . . . . . . . . . . . . . . . . . . . 35
BrunoLengeler
X-Ray Absorption and Reflection in Materials Sciences . . . . . . . . . . . . . . . 53
JamesP. Wolfe
Propagation of Large-Wavevector Acoustic Phonons
New Perspective fromPhononImaging . . . . . . . . . . . . . . . . . . . . . . . . . . 75
WolfgangLudwig
Theory of Dynamical Surface States and Reconstructions
at Crystal Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Henning NeddermeyerandStephanTosch
Scanningtunnelingmicroscopy and spectroscopy onclean
and metal-covered Si surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Jiirgen Kuhl, Alfred Honold,Lothar Sehultheis, and Charles W. Tu
Optical Dephasing and Orientational Relaxationof Wannier-Excitons
and Free Carriers inGaAs and GaAs/AlxGat_xAs QuantumWells . . . . . . . . 157
Ulrich Kaufmann
The Spectroscopic Evidence forthe Identity ofEL2andthe
ASGaAntisite in As-Grown GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
VI
BrunoMeyer,KlausKrambock, Detlev Hofmann, andJohann-Martin Spaeth
On the ChargeStateofthe EL2Mid GapLevelin Semi-Insulating
GaAs from a Quantitative Analysis ofthe Compensation . . . . . . . . . . . . . . 201
Patricia M. Mooney
Deep DonorLevels (DX Centers) inIII-V Semiconductors:
RecentExperimentalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Matthkzs Scheffler
Chemical Binding, Stability and Metastability of Defects in Semiconductors . 231
Rolf E. Hummel
A New Lookat the Reliability ofThin Film Metallizations for
Microelectronic Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Mark A. Reed,John N. Randall, JamesH. Luscombe, WilliamR. Frensley,
RafJ. Aggarwal,RichardJ.Matyi, TomM. Moore,andAnnaE. Wetsel
QuantumDot Resonant Tunneling Spectroscopy . . . . . . . . . . . . . . . . . . . 267
Detlef Heitmann, Thorsten Demel,PeterGrambow,andKlausPloog
DCandFarInfrared Experiments onDeep MesaEtchedSingleand
Multi-Layered QuantumWires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
C. W.J. Beenakker, H.vanHouten,andB. J. van Wees
Coherent electron focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Peter Marquardt and Guenther Nimtz
The Size-Induced Metal-Insulator Transition andRelatedElectron
Interference PhenomenainModem Microelectronics . . . . . . . . . . . . . . . . . 317
VII
FestkSrperprobleme 29 (1989)
Charge Transfer between Weakly Coupled Normal Metals and
Superconductors at Low Temperatures
Ulrich Eckern
Institut fEr Theorie der Kondensierten Materie, Universit;;t Karlsruhe, P.O.Box 6980,
D-7500Karlsruhe, Federal Republic of Germany
Gerd SchSn
Department of Applied Physics, Delft University of Technology, Lorentzweg 1, NL-
2628 CJ Delft, The Netherlands
Summary: We give a review ofrecent advances in the theoretical description of
ultrasmall tunneljunctions,i.e. weaklycouplednormalmetals and superconductors,
at verylowtemperatures. Thedescription,whichisbasedonthemicroscopictheory,
accounts forthe quantum effects associatedwith Cooperpair tunneling,and forthe
dissipation due to single electron tunneling (SET) or normal current flow. The
former is describedbytile familiar periodic potentialwhich,inthe quantum regime,
leads to energy bands, and tocoherent Bloch oscillations inresponse to an external
currcnt. The stochastic SET, which may bc blockcd by the Coulomb intcraction,
yields voltage oscillations cvcn in normal junctions, and also rnodifics the Bloch
oscillationsin Joscphson junctions. Thetheoryisextended tonctworks and granular
superconducting films as well.
1 Introduction
During the past ten years, a great deal of theoretical as wellas experimental
efforts have been devoted to what is now commonly called Quantum Me-
chanics of Macroscopic Variables [1]. The experimental systemwhich has
been studied extensively is a Josephsonjunction, i.e. a system oftwo weakly
coupled superconductors. It is wellknown that the relevant variable for the
description ofthe dynamics of a Josephson junction is the difference ~ of
the phases of the superconducting orderparameters in the two electrodes.
If the capacitance ofthejunctionis large enough, the dynamics ofthephase
is determined by the following classical equations:
t~i~ = 2ev ; cfz + v/n + Z~si,,~ = z= (1)
where the firstis Josephson's equation, and the second expresses the balance
of currents; V, I=, C, R, andartdenotethe voltage, the externalcurrent, the
capacitance, the resistance, and the criticalcurrent,respectively. Obviously,
Eq. (1) describes the damped motion of a particle in a tilted periodic
potentialU(~), given by
u(~) = -E~cos~, - hz=. ~/2~ (2)
where the Josephson energy Ej = hIc/2e and the charging energy Ec = e2/2C
are the important scMes of energy. Generally, R-1 can be thought of as
a sum of two contributions, R-1 = R~1+ R71, where Rt is the (strongly
temperature dependent) resistance due to quasiparticle tunneling, and R,
denotes an external shunt (if present).
Of course, a description based on Eq. (1) is incomplete in several ways.
First of all, it is a general theorem of statistical mechanics that dissipation
and fluctuations are intimately related; for the present case, this means
that I~ in Eq. (1) has to be replaced by I~ +5i, where 6I denotes a random
contribution to the current. Considering the tunneling of electrons, it was
noted eighty yearsago by Schottky [2]that in situations, inwhichthe energy
gained by tunneling from one metal to another is much larger than the
thermal energy, i.e. eV >> kT, the resulting Schroteffekt or shot noisecan
considerably exceed the thermal noise. Shot noise is a direct consequence
of the charge being transported in elementary units; thus it proved to be
an independent (and at that time [2]quite accurate) possibility to measure
the charge of an electron. The classical stochastic description of a normal
tunneljunction, and in particular the influence of the Coulomb interaction
(Coulomb blockade) on the dynamics, is discussed in the next chapter.
Secondly, the classical description fails in the quantumregime of very low
temperatures, where the phase has to be considered an operator-valued
quantity, with the charge Q being its canonically conjugate variable [3]
according to the commutatorrelation[Q,~b]= -2e. i. In the non-dissipative
case it is obvious that a discussion ofquantum mechanicaleffects is to be
based on the tIamiltonian
= ~.~/2c+ tr(~). (3)
On the other hand, the general treatment [4], which starts from micro-
scopic thcory, proceeds most conveniently by working with path integral
methods in the Lagrangian formulation. In particular, an earlierapproach
[5] in which the coupling to the environment, i.e. the microscopic degrees
of freedom, is modeled in a phenomenological way, can be confirmed and
extended [4]. The microscopic theory of weakly coupled superconductors is
briefly reviewed in Ch. 3. In the finaltwo chapters, we describe some ofthe
consequences ofthe quantum description: macroscopic quantum tunneling
[6], energy levels [7], macroscopic quantum coherence, Sloch oscillations [8,
9], and further consequences ofsingle electron tunneling (SET), and give a
brief discussion ofgranular superconducting films and networks of Joseph-
sonjunctions [10]. A detailed reviewis in preparation [11].
2 Normal TunnelJunctions
We firstreview the classical description of stochastic tunnelingof electrons
in a normaljunction,which proceeds analogous to [2]. The tunneling cur-
rent is composed of statistically independent SET processes, forward and
backward across the barrier, at randomtimes {t+} and {t~'}. Sincethe time
for tunneling is negligible, the current is given by a sum of delta-functions
as follows:
z(t) = e Z S(~ - ~+,) - ~~_,S(~ - q) (4)
i j
The statistics governing these proccsses is Poissonian, characterized by the
rates v+ and v-. Thus the probability density for n particles to tunnel
forward at times 0 <t+,...,t+ < t! is given by
P,~+ (q+,.. .,t+) = e-"% •(~+)"/-! (5)
and similar for the backward tunneling. Detailed balance determines the
ratio v+/v - = exp(-e/kT), wheree is the energy differenceneeded for the
forward tunnelingprocess. The average current and the current fluctuations
are expressed by v+ and v- as follows:
< z(,) >= e(~+ - ~-) (o)
< sr(0sI(t') >=< t(l)z(t') > - < z >== e=S(~ - ~')-(~+ + ~-) . (7)
Let usassume that for a fixed voltage, which meansfixed energy difference
e+ = -)-eV, the tunneling current voltage characteristic It(V) is known (in
simple cases it is given by L(V) = V/R~). This provides a relation for the
average current,I,(V) = e(v+ - v-), and combined with the detailed balance
requirement it fixes the rates to
(s)
Thus the fluctuations of tile voltage biased junction are given by
< s±(~)~I(0) >= e6(0I,(V)¢oth(z-~.~) (9)
which obviously interpolates between Schottky's result and the standard
Nyquist formula forthermalnoise (eV >> kT and eV << kT, respectively).
After generalization to the quantum mechanical form, the power spectrum
is found to be given by [12]
e eV 4-hw
&(w) = ~ ~*,(V± ~ / e ) c o t h ( ~ ) . (10)
+
In more generalsituations, the energy is not fixed by the external voltage
but depends on the state of the system. In particular in junctions with
very small capacitance, the energy is dominated by the charging energy
E(Q) = Q2/2C, which changes in the tunneling processsince Q ~ Q=l=e. The
detailed analysis [13] shows that the tansition rates are still givenby Eq.
(8), however, the energy differences depend on the charge according to
e:i:(Q) = E(Q + e) - E(Q) . (11)
At zero temperature the rates reduce to
v+(Q,T = O) = e-1 • I~(le±l/e) for e± < 0 (12)
and vanish for e+ > 0. Thus tunneling occurs only if during the process
the energy decreases, i.e. only for charges exceeding IQI > e/2. This phe-
nomenonis called the Coulomb bloc'1"Imdv. At finite temperature, also tran-
sitions occur by thermal excitation which are unfavourable with regard to
energy.
We consider now a normal tunnel junction with E(Q) = Q~/2C which is
biased by an external current Iz(t). This current increases thecharge on the
junction electrodes in a deterministic, continuous fashion. But the charge
changes alsostochastically,in discreteunits ofe due to tunneling with rates
givenby Eq. (8). Thus we may write
dQIdt = I~(t)+ (~t,....z,.g • (13)
The statistical properties of the last term on the rhs have been discussed
above. Because of the discrete natureof the tunneling, the allowed values
of the charge Q at each moment can take values only from the discrete set
Q = Q~(t)+ q- e, where q is an integer, and
Oz(t) = dr' r~(t') . (14)
oo
The segmentation of the current in units of e and the related charging
and discharging of the junction leads to an oscillatory time dependence
of the voltage across the junction. On the average, it oscillates with the
flmdamental frequency [13]
/SET = I~/e. (15)
The strict discreteness of the tunneling, combined with the Coulomb inter-
action which favours transitions to states with small charge, leads also to
long-time correlations in <Q(t)Q(O)> which are noticable forsmallcurrents
and low temperatures [13, 11].
From the time averaged charge we obtain the dc current-voltage character-
istic which is shown in Fig. 1. In particular, the voltage forsmall currents
is
Vo ---- (~r-r~R,e/2C)'/2 for I, RtC << e, T = 0 (16)
whereas at large currents the I-V-curve becomes linear but shifted with
respect to the Ohmic line:
Vo = I~R, + sign/~•e/2C for l~RtC >> e . (17)
