Lecture Notes in
Physics
Edited by H. Araki, Kyoto, .I. Ehlers, MiJnchen, K. Hepp, Z~irich R. Kippenhahn, M(Jnchen, H. A. Weidenm~iller, Heidelberg and J. Zittartz, K61n
207 S.W. Koch
Dynamics of First-Order Phase Transitions in Equilibrium and Nonequilibrium Systems
Springer-Verlag Berlin Heidelberg New York Tokyo 1984
Author Stephan W. Koch Institut ffJr Theoretische Physik, Universit~t Frankfurt Robert-Mayer-Stra6e 8-10, D-6000 Frankfurt/Main 1
ISBN 3-540-13379-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-38?-13379-8 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright.All rights are reserved,whetherthe whole or partof the material is concerned,specificallythose of translati(~n,reprinting,re-useof illustrations,broadcasting, reproduction by photocopyingmachineor similar means,and storage in data banks. Under § 54 of the GermanCopyright Law where copies are madefor other than private use, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2153/3140-543210
Contents I)
II)
Introduction
Survey
I)
Basic
Features
2)
Phase
Transitions
3)
Scope
of
Equilibrium I)
2)
III)
and
this
Phase
Dimensional
Transitions Systems
.........
2 8
......................................
...............................
Separation
............................
Nucleation
b)
Phase
c)
Spinodal
d)
Spinodal Decomposition as G e n e r a l i z e d Nucleation Theory ...................................
Phase
Theory
Separation
................................... in B i n a r y
Decomposition
Transitions
in T h i n
Mixtures
Films
Phase
a)
The
Solid-Liquid
The
Commensurate-Incommensurate
Transitions
Systems
.....
.........................
b)
Phase
.................
in One-Component
Transition
................... Transition
..........
............................
The Plasma Phase Transition in Highly Excited Semiconductors ..........................................
b)
I
............
a)
a)
2)
in L o w
Transitions
of P h a s e
Nonequilibrium I)
First-Order
Book
Phase
Dynamics
of
.....................................
10
16 16 18 24 32 45 47 48 62
74
74
Electron-Hole Droplet Nucleation in I n d i r e c t - G a p Semiconductors ......................................
77
The Plasma Phase Transition in D i r e c t - G a p Semiconductors ......................................
86
Optical
Nonequilibrium
Systems
...........................
a)
First-Order Nonequilibrium Phase Transitions in L a s e r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
b)
O p t i c a l B i s t a b i l i t y o f T w o - L e v e l A t o m s in a Resonator ..........................................
c)
Deterministic
d)
Dispersive
e)
Optical
Chaos
in B i s t a b l e
Bistability
Bistability
due
Systems
in S e m i c o n d u c t o r to
Induced
94
95 98
.............
108
Etalons
.....
114
.......
129
Absorption
I: I n t r o d u c t i o n and Survey
The topic of this book + is the theoretical d e s c r i p t i o n of d y n a m i c a l aspects of d i s c o n t i n u o u s phase t r a n s i t i o n s
in p h y s i c a l
systems.
It in-
cludes effects like the d e v e l o p m e n t of phase s e p a r a t i o n - i.e. nucleation and spinodal d e c o m p o s i t i o n - , freezing and melting,
as well as
phase t r a n s i t i o n s in systems w h i c h are far from thermal equilibrium. Examples are the f o r m a t i o n of e l e c t r o n - h o l e d r o p l e t s in highly laser excited s e m i c o n d u c t o r s and transitions b e t w e e n d i f f e r e n t n o n e q u i l i b r i u m states in optical systems like optical bistability.
The usual m e t h o d s to d e s c r i b e these phase t r a n s i t i o n s is to treat only a few r e l e v a n t v a r i a b l e s explicitly.
In such a m a c r o s c o p i c theory these
r e l e v a n t v a r i a b l e s assume the role of order parameters. r e l e v a n t v a r i a b l e s lead to nonlinearities,
The other,
d i s s i p a t i v e effects,
noise c o n t r i b u t i o n s in the equations of the order parameters.
non-
and to
Hence,
it
is m o s t a p p r o p r i a t e to treat these p r o c e s s e s w i t h i n a p r o b a b i l i s t i c formalism.
In this way one is able to i n v e s t i g a t e the d y n a m i c a l evolu-
tion of a system after changing the e x t e r n a l control parameters.
If dis-
c o n t i n u o u s phase t r a n s i t i o n s take place the i n i t i a l l y stable states d e c a y through b u i l d - u p and growth of h e t e r o p h a s e fluctuations.
The pro-
babilistic f o r m a l i s m is well suited to d e s c r i b e these stochastic processes.
It is one of the intentions of this book to p o i n t out some common aspects of the seem±ngly quite d i f f e r e n t physical p r o b l e m s p r e s e n t e d in the v a r i o u s chapters.
Especially,
c o n n e c t i o n s will be e s t a b l i s h e d bet-
w e e n the d i s c u s s e d n o n e q u i l i b r i u m t r a n s i t i o n s and phase t r a n s i t i o n s in the v i c i n i t y of thermal equilibrium.
However,
one may not expect a uni-
fied theory on the level of the theory of critical p h e n o m e n a for continuous phase transitions,
This theory u t i l i z e s the o c c u r e n c e of fluc-
tuations on all length scales leading to u n i v e r s a l behaviour.
Such fluc-
tuations do not occur in c o n n e c t i o n w i t h d i s c o n t i n u o u s phase transitions. On the contrary,
a w e l l - d e f i n e d length scale is set by the spatial ex-
tension of the h e t e r o p h a s e f l u c t u a t i o n s driving the r e s p e c t i v e transition.
E s p e c i a l l y these h e t e r o p h a s e f l u c t u a t i o n s and their c o n s e q u e n c e s
are a c o n t i n u o u s theme of this book.
+This book is based on the h a b i l i t a t i o n thesis of the author, w h i c h has been a c c e p t e d in D e c e m b e r 1983 by the Physics D e p a r t m e n t of the U n i v e r sity Frankfurt, Fed. Rep. Germany.
I.I Basic F e a t u r e s of F i r s t - O r d e r Phase T r a n s i t i o n s
The b a s i c features of f i r s t - o r d e r e q u i l i b r i u m phase transitions are d i s c u s s e d m o s t easily in the framework of the Van der Waals theory for the liquid-gas transition.
Historically,
this theory is the first
successful d e s c r i p t i o n of a d i s c o n t i n u o u s phase transformation. dates back to the year 1873 when V a n der W a a l s in his thesis
It
[I] pro-
posed the famous state e q u a t i o n for a c l a s s i c a l liquid-gas system:
(p + ~ ) ( V - b ) = R T
Here, V is the m o l e c u l a r volume,
(I.I)
p is the pressure,
ture, and a and b are m a t e r i a l parameters.
T is the tempera-
The p r e s s u r e c o r r e c t i o n
term a/V 2 is a c o n s e q u e n c e of the attractive part of the i n t e r m o l e c u l a r i n t e r a c t i o n potential. part, Fig.1
The co-volume b takes into account the repulsive
shows e x a m p l e s for the isotherms r e s u l t i n g from the Van der
W a a l s equation.
IT! c
L , A B
Fig.
I :
I C
l O
I E
T Tc.~ V
Isotherms of a Van der Waals system
The function p(V) IT=const d e f i n e d by Eq.(I.1) ty. Therefore,
one has m u l t i p l e
(schematically)
shows a cubic n o n l i n e a r i -
solutions for T
rate at the so d e f i n e d critical t e m p e r a t u r e T c. become
simply hyperbole,
For T>T c the isotherms
w h i c h are also d e s c r i b e d by the ideal gas
state equation. For T~T c the isotherms m a y be s u b - d e v i d e d into parts c o r r e s p o n d i n g to d i f f e r e n t states of the system
(see Fig.l).
For V
the d e r i v a t i v e
~p/~vIT is
quite large and n e g a t i v e i n d i c a t i n g that a small volume
i
c o n t r a c t i o n leads to a large increase of the pressure. izes the h o m o g e n e o u s
fluid state.
Correspondingly,
This c h a r a c t e r -
for V>E the system
is in the h o m o g e n e o u s gas state. The actual values of A and E, i.e. of V f l u i d and Vgas , are d e t e r m i n e d w i t h the help of the s o - c a l l e d M a x w e l l construction.
This c o n s t r u c t i o n is a c o n s e q u e n c e of the condition,
that in h o m o g e n e o u s systems the
chemical
potentials
~ for c o e x i s t i n g
phases have to be equal ~fluid = Pgas The i n c r e a s i n g part of the i s o t h e r m
(region B-D in Fig.l)
describes
states of the h o m o g e n e o u s system that are m e c h a n i c a l l y u n s t a b l e
>0
.
T On t h e
other
hand,
the
states
belonging
m e c h a n i c a l l y stable. Nevertheless,
to
the
regions
A-B and
D-N a r e
they are no t h e r m o d y n a m i c equilibri-
um states because they do not have the lowest free energy. However, due to their m e c h a n i c a l
stability they have a finite lifetime and may
v e r y well be o b s e r v e d experimentally.
The states of regions A-B and
D-E are called o v e r - h e a t e d fluid phase and o v e r - s a t u r a t e d v a p o r phase, respectively. F r o m the p-V d i a g r a m
(Fig.l) one may c o n s t r u c t the binodal line in a
T-V d i a g r a m by connecting the points of liquid-gas e q u i l i b r i u m .
l
Tc
~pbin
vc Fig.
line
odol line inodQI
2 : Phase d i a g r a m of a Van der Waals
v system
(schematically).
The binodal line is also known as phase s e p a r a t i o n line,
since it se-
parates the h o m o g e n e o u s gas and liquid states. For p a r a m e t e r s w i t h i n
the binodal existence.
line the e q u i l i b r i u m system shows spatial two-phase coThe dashed line in this
c o e x i s t e n c e region
been c o n s t r u c t e d from the p-V d i a g r a m for the c o n d i t i o n
(see Fig.2)
~p/~V T=0. For
this curve, V a n der Waals i n t r o d u c e d the name "spinodal line" In the region b e t w e e n binodal and spinodal, An a m e t a s t a b l e
state. At this point,
has
[2] .
the h o m o g e n e o u s system is
it is important to mention,
that
such a strict d i s t i n c t i o n between m e t a s t a b i l i t y and i n s t a b i l i t y is a direct c o n s e q u e n c e of the e q u a t i o n of state y i e l d i n g isotherms of the form shown in Fig.1. Moreover,
a rigorous t r e a t m e n t of these effects
goes clearly beyond the framework of e q u i l i b r i u m thermodynamics.
The
c o r r e s p o n d i n g states are n o n e q u i l i b r i u m states w h i c h are c h a r a c t e r i z e d by a finite lifetime.
Physically,
this is a c o n s e q u e n c e of the charac-
t e r i s t i c fluctuations driving the system towards equilibrium.
The Van der Waals theory is a typical example of a m e a n - f i e l d theory. In Refs.
[3-6]
the r e s p e c t i v e authors show, that the state equation
(I.19 is an exact d e s c r i p t i o n for systems w i t h a p o t e n t i a l U acting on a m o l e c u l e at point r given by
[3,5]
~
r
=
(I.2a)
U(r) or in the limit K+0 by
otherwise
~ _]
r
[7]
U(r)= The e x p l i c i t
"IUI
e x p r e s s i o n for these p o t e n t i a l s
(I.2b)
(e.g. Eq. 1.2a) may be
d e r i v e d as effective s e l f c o n s i s t e n t - f i e l d a p p r o x i m a t i o n of the d e t a i l e d m a n y particle
interaction
[3]
Other examples of successful m e a n - f i e l d theories are, e.g., theory of f e r r o m a g n e t i s m conductivity
[9]
[8]
and the G i n z b u r g - L a n d a u theory of super-
. These theories m a y be u n i f i e d by introducing the
concept of an order parameter.
The order p a r a m e t e r can be any q u a n t i t y
r e p r e s e n t i n g an a p p r o p r i a t e m e a s u r e transition.
the Weiss
for the order arising at the phase
For the liquid-gas t r a n s i t i o n one u s u a l l y chooses the den-
sity d i f f e r e n c e P-Pc as an order parameter.
Here,
Pc is the density at
the critical point.
In m a n y cases,
it is p o s s i b l e to expand the free energy in terms of the
order p a r a m e t e r q:
F(q,t)=F(0,T)+qF'(0,T)+
~
F"(0,T)+ ~
F"' (0,T)+ ~
F ....(0,T)+ ...
:F(0,T)+~q + ~ q2 +~ q3+~ q4+ ... This expanded form of F(q,T)
(I.3)
is known as Ginzburg-Landau potential.
It is one of the basic ingredients of the classical Landau theory of phase transitions
[10]. In the framework of statistical mechanics
the most probable value qm of the order parameter is calculated by minimizing the free energy. Consequently,
the probability distribution
for q, -F/kBT f(q,T)=Ne (N is a normalization constant)
has its maximum at q=qm"
The various phase transitions are usually classified using the expanded form of F(q,T). Systems with a=0=¥ and B=b (T-T c ) exhibit continuous phase transitions.
These transitions are also
called phase transitions of second order. As shown in Fig. 3 minimum of F(q),
i.e. qm' shifts continuously from qm=0 for T>T c to
qm ~ 0 for T
•
Fig.
, the
F~
.T>T c
T: T¢
3 : Free energy F versus order parameter q for a continuous phase transition (schematically).
If however the q u a n t i t y T is not equal to zero, ~0
,
the free energy m a y have the structure shown in Fig. 4 . In this case, the phase t r a n s f o r m a t i o n appears discontinuously.
A c o m m o n feature of
these first-order phase transitions is the a p p e a r a n c e of two-phase coexistence
(two simultaneous m i n i m a of F(q) at q=0 and q~0), which gives
rise to the p h e n o m e n o n of hysteresis.
In the case of t e m p e r a t u r e
changes,
in a m e t a s t a b l e
the system may still remain
state (relative
m i n i m u m of F) a l t h o u g h this is not the state of lowest free energy. The p o t e n t i a l b a r r i e r can only be surmounted with the help of r a n d o m fluctuations
which, however,
are always present in a realistic system.
They give rise to the o b s e r v e d finite lifetime
of the m e t a s t a b l e
states.
% Fig.
4 : Free energy F versus order p a r a m e t e r q for a d i s c o n t i n u o u s phase t r a n s i t i o n (schematically). (TI>T2>...>T6,T~=T c)
To summarize,
the Van der W a a l s theory is a m e a n - f i e l d theory for a
first-order phase t r a n s i t i o n in an e q u i l i b r i u m system. The transition is an e q u i l i b r i u m phase transition,
since the state,
into w h i c h the
system evolves is a true e q u i l i b r i u m state in the sense of thermodynamics. Generally, (thermally)
e q u i l i b r i u m transitions occur in systems, w h i c h are
coupled to a r e s e r v o i r
(heatbath). S t a r t i n g from thermal
e q u l i l i b r i u m b e t w e e n bath and system,
the bath t e m p e r a t u r e may be
c h a n g e d to d e s t a b i l i z e the e x i s t i n g state initiating a t r a n s i t i o n to a new phase, w h i c h is again in e q u i l i b r i u m w i t h the bath at the new temperature,
One distinguishes
between these e q u i l i b r i u m phase transitions
and the so-called n o n e q u i l i b r i u m common features
(see e.g.
phase transitions one reservoir peratures
transitions.
the discussion
take place in systems,
[12,13]
preventing
are called
. These reservoirs have to be at different
between different "nonequilibrium
the e q u i l i b r i u m
case,
of the respective
potentials equation
they may be classified
[.11-74] e.g.
for the probability
-
_~
In this equation,
~
~qi
one can induce
These transitions
Completely
according
analogous
to
to the behaviour
the so-called
Explicit
"generalized
expressions
for these
in the framework of a F o k k e r - P l a n c k
distribution
Ki(~)f +I2
function
f(q,t):
~2
[ ZqiZqj Qijf
the drift term
of the order parameter dynamics.
(~)
(1.4)
contains
the d e t e r m i n i s t i c
The diffusion
the influence of the stochastic of eq.
states.
steady
The Landau theory may be generalized:
one discusses
potentials"
tem-
nonequilibrium
the bath parameters
nonequilibrium
order parameter.
are calculated,
~f ~t
At most,
phase transitions".
instead of the free energy, Ginzburg-Landau
11), n o n e q u i l i b r i u m
which are coupled to more than
thermal equilibrium.
states may be realized. By changing transitions
A l t h o u g h both show m a n y
in Ref.
fluctuations.
term
part
(~Q) accounts
The stationary
for
solution
(I.4)
fst(~)=f(~,t=~)
defines the Ginzburg-Landau f st (~)=Ne
Once ~ is known,
potential
~ ,
-¢(q)/kBT
the terminology
of e q u i l i b r i u m
theory may be applied
directly. In some special cases ted directly
(systems with detailed balance)
from the so-called potential
tionary solution of the F o k k e r - P l a n c k detailed balance condition In more general
# may be calculayielding
Consequences e.g.
numerical methods have to be applied. phenomena,
Approximation This becomes which arise,
phase transitions.
the sta-
of the
, in Ref.
the stationary F o k k e r - P l a n c k
usually cannot be solved analytically.
text of n o n e q u i l i b r i u m
equation.
are discussed by Graham,
cases however,
when dealing eith dynamical
conditions,
12.
equation
schemes and/or
even more necessary e.g.
, in the con-
At this point of r a n d o m sitions.
A central
transitions scales.
it is a p p r o p r i a t e
fluctuations
is
Hereby,
particular
In contrast,
pour
stable
is shifted
fluctuations
into w h i c h
II.1
fluctuations states.
[5,20]
phase
are well b o u n d e d
of c r i t i c a l
for more
are found
spatial
ex-
realizations A typical
in an o v e r s a t u r a t e d
The a p p e a r a n c e feature
by w h i c h
potential
in n e a r l y
local
for a
transitions.
at the transition.
droplets
is a c h a r a c t e r i s t i c
phase
fluctuations
phase
in their
represent
details).
tran-
specific
. Such
in d i s c o n t i n u o u s
fluctuations
Ginzburg-Landau
effects
from p r o p e r t i e s
properties
They are the m e c h a n i s m
(generalized)
importance
o c c u r o n all length
the s y s t e m evolves
is the f o r m a t i o n
for these
fluctuations
do not appear
heterophase
(see section
rophase
the
most
These
of the phase example
the e m p h a s i s
scales
the g e n e r a l
and n o n e q u i l i b r i u m
of the u n i f i e d theory of c o n t i n u o u s
critical
system to u n i V e r s a l
on all length
tension.
issue
that
to e m p h a s i z e
for e q u i l i b r i u m
va-
of these hete-
in the d e c a y of m e t a -
the p o t e n t i a l
may be crossed.
all topics
discussed
barrier
in
Examples in this
book.
1.2
Phase T r a n s i t i o n s
The d i s t i n c t i o n is o r i g i n a l l y a transition dynamic
in L o w D i m e n s i o n a l
between
continuous
due to E h r e n f e s t of order
potential
i-I d e r i v a t i v e s dentvariables,
and d i s c o n t i n u o u s
[171
• According
i, if the i-th d e r i v a t i v e
is d i s c o n t i n u o u s
are continuous.
are not equal
phase
transitions
to his classification, of the r e l e v a n t
at the phase
transition,
thermo-
whereas
all
For a s y s t e m with p and T as indepen-
the free e n t h a l p y
first d e r i v a t i v e s
Systems
is the r e l e v a n t in both phases
potential.
If the
A and B, i.e.
VA#VB and SA~S B (v v o l u m e
per particle,
is of first order. hand,
s entropy
per particle),
In a s e c o n d - o r d e r
phase
VA=V B
SA=S B
the phase
transition,
one has
but the r e s p e c t i v e
constant
and
pressure
Cp,A~Cp,B and the isothermal
compressibilities
,
specific
heats
transition
on the other
Generally,
phase transitions are c o n n e c t e d w i t h d i s c o n t i n u i t i e s of
t h e r m o d y n a m i c potentials.
Genuine d i s c o n t i n u i t i e s of these p o t e n t i a l s
can only occur for e q u i l i b r i u m systems in the t h e r m o d y n a m i c limit (N,V÷~,N/V=finite)
[18]. In finite systems, however,
the t h e r m o d y n a m i c
p o t e n t i a l s are always continuous functions of their i n d e p e n d e n t variables
[19]
systems.
. In this sense, there are no
phase
The changes of states in realistic
should therefore,
strictly speaking,
phase transitions.
Nevertheless,
t r a n s i t i o n s in finite
(i.e. finite)
systems
not be i d e n t i f i e d as t h e r m o d y n a m i c
for s u f f i c i e n t l y large systems this
may still be a very m e a n i n g f u l approximation.
As an example, Fig.5
shows isotherms of Van der Waals systems of d i f f e r e n t sizes.
0.4 A
0.3
._&
O.2
~,"
0.1
~~/
T constant
N -- I0 N --50
Fig.5: E q u i l i b r i u m portion of the isotherms in Van der Waals systems (T
10 The function p(v) varies continuously for any finite systems. With increasing system size it approaches quite rapidly the characteristic shape of the thermodynamic limit. Therefore, it is justified to neglect the differences between very large systems particles) However,
(of the order of N =1022
and systems in the thermodynamic
limit.
the situation is different for smaller systems.
In numerical
simulations one often works with less than 1.000 particles
(see section
II). Here, it is certainly important to investigate the influence of the finite size effects on the equilibrium properties of the system. This can be done in a pragmatic way by simply increasing the number of atoms until the results become sufficiently size independent.
This
limit is reached as soon as the linear extensions of the system are larger than its characteristic
length scale determined by the investi-
gated effects. Besides the finiteness,
also the dimensionality of a system may affect
the nature of phase transitions.
This is stated, e.g., in the universa-
lity hypothesis of modern theory of critical phenomena
[5,20]
.
Critical phenomena are mostly investigated in connection with continuous phase transitions.
In the vicinity of the critical point the only
relevant quantities are found to be the dimensionality d, the number of components of the order parameter and the symmetry and the range of the interactions.
Especially,
the possible dependence of the order of
phase transitions on the dimensionality has triggered investigations of low dimensional
systems. A recent example is the melting and freez-
ing transition in two-dimensional
systems
(see section II.a for more
details).
1.3
Scope of this Book
The two main chapter of this book deal with equilibrium systems ter II) and nonequilibrium systems As a starting point,
(chap-
(chapter III).
in chapter II.1 a discussion of the dynamics of
phase separation is presented.
A typical model
is an originally
spatially homogeneous system which has been quenched into the coexistence region of its phase diagram.
Depending on whether the system is
quenched into the metastable or the unstable region, phase separation takes place by nucleation or spinodal decomposition. both mechanisms are investigated separately
(sections
To begin with, (II.la-c).
In section II.la the classical Becker-D6ring theory is presented. This theory is very useful not only for the description of a supersaturated vapour but also as quite general approach for the understanding of the decay of metastable states in equilibrium and nonequilibrium systems. However, several assumptions are made in the classical nucleation theory which permit no appropriate treatment of phase separation in the critical region. Here,
field-theoretical methods are more
adequate. In section II.Ib an example of such a semi-phenomenological continuum theory is presented for a model of a binary system. This may be a mixture of two fluids or two metals, which precipitate into the pure phases below a critical temperature.
The field-theoretical approach allows e.g. the calculation of the timedependent structure factor of the phase separating system. Also, an expression for the distribution function of the relative concentrations of the pure phases may be obtained. This distribution function shows quite clearly the break-up of the originally homogeneous system into regions of high and low concentrations of the constituents. On the other hand, the structure factor contains the integrated information of the separation process. It may be measured directly in light-scattering experiments, allowing a comparison between theory and experiment. Field-theoretical techniques have been applied to the kinetic Ising model, because many results for this system are available from extensive Monte Carlo computer simulations
(see section II.Ib).
Following the treatment of binary systems, chapter II.Ic is devoted to the study of spinodal decomposition in one-component systems. Here, decomposition of the homogeneous state leads to spatial liquid-gas coexistence. A hydrodynamic fluctuation theory for this problem is presented. Applying the Enskog theory of dense fluids, the relevant transport coefficients can be expressed in terms of the state equation. An approximation for this state equation for a model of an atomic system with Lennard-Jones interaction is obtained in the framework of liquidstate perturbation theory. Thus, a closed approximation scheme is constructed, which is evaluated for the linear regime.
Molecular dynamics simulations of spinodal decomposition in one-component fluids are discussed at the end of section II.lc. The short-time results allow a comparison with the hydrodynamic fluctuation theory.
12
In a d d i t i o n
, the simulations yield d i s t i n c t growth laws for the liquid
clusters and universal
scaling of the p e r t i n e n t structure factor. The
long-time b e h a v i o u r is found to be in good a g r e e m e n t w i t h the pred i c t i o n s of an a s y m p t o t i c analysis of the k i n e t i c e q u a t i o n s
for the
growth of clusters in a s u p e r s a t u r a t e d vapour.
Section II.2a of this book is devoted to the still somewhat controv e r s i a l p r o b l e m of m e l t i n g in t w o - d i m e n s i o n a l
systems. A topological
theory of m e l t i n g p r e d i c t s the p o s s i b i l i t y of a s e c o n d - o r d e r phase transition.
This is in c o n t r a s t to the t h r e e - d i m e n s i o n a l case, where
u n d o u b t e d l y a f i r s t - o r d e r t r a n s i t i o n takes place. Unfortunately,
com-
puter simulations of d i f f e r e n t authors yield results w h i c h seem to contradict each other to some extend.
Moreover,
laboratory experiments
have not been able to u n a m b i g u o u s l y clarify the situation,
neither.
Some aspects of this p r o b l e m will be d i s c u s s e d for the example of n e a r l y t w o - d i m e n s i o n a l noble gas films on a graphite substrate. of c o m p u t e r simulations indicate,
Results
that m e l t i n g is of first order in
this case. The n u m e r i c a l l y g e n e r a t e d s n a p s h o t pictures of atomic trajectories are very illustrative transition.
as a v i s u a l i z a t i o n of the phase
The t r a n s f o r m a t i o n b e t w e e n liquid and solid states can be
seen d i r e c t l y , a s well as the h e t e r o p h a s e
fluctuations w h i c h precede
the transition.
The c o m m e n s u r a t e - i n c o m m e n s u r a t e
t r a n s i t i o n for k r y p t o n on graphite is
d i s c u s s e d as final example of an e q u i l i b r i u m phase t r a n s i t i o n
(section
II.2b). E x p e r i m e n t a l m e a s u r e m e n t s of the structure factor seem to indicate the e x i s t e n c e of a well c o r r e l a t e d liquid phase for intermediate c o v e r a g e s b e t w e e n the c r y s t a l l i n e c o m m e n s u r a t e and i n c o m m e n s u r a t e phases.
C o m p u t e r simulations reproduce
the e x p e r i m e n t a l
findings but
the " i n t e r m e d i a t e phase" turns out to be the c o e x i s t e n c e of c o m m e n s u r a t e regions separated by a network of i n c o m m e n s u r a t e domain walls. Many features of these walls suggest a t h e o r e t i c a l d e s c r i p t i o n as solitons. Therefore, veloped.
a g e n e r a l i z e d S i n e - G o r d o n t h e o r y for this p r o b l e m is de-
The p e r t i n e n t soliton s o l u t i o n s d e s c r i b e p e r p e n d i c u l a r cuts
through the domain walls. The c a l c u l a t e d t e m p e r a t u r e d e p e n d e n t wall w i d t h is in good a g r e e m e n t w i t h the results of the n u m e r i c a l simulations.
F o l l o w i n g the d i s c u s s i o n of e q u i l i b r i u m phase transitions,
the com-
plete chapter III of this book is d e v o t e d to the study of d y n a m i c a l
13
aspects of n o n e q u i l i b r i u m phase t r a n s i t i o n s the most important examples
in p h y s i c a l systems. One of
in this context is a s e m i c o n d u c t o r that
is strongly e x c i t e d by laser irradiation.
On this system v e r y m a n y
e x p e r i m e n t a l and t h e o r e t i c a l studies have been performed. reasons is that such s e m i c o n d u c t o r s have the nice feature,
One of the that ex-
ternal v a r i a t i o n s of the e x c i t a t i o n i n t e n s i t y induce large changes in the d e n s i t y of the g e n e r a t e d e l e c t r o n i c excitations. exceed several orders of magnitude.
E x a m p l e s of the e l e c t r o n i c excita-
tions are c o n d u c t i o n - b a n d electrons,
v a l e n c e - b a n d holes and excitons.
They have the c h a r a c t e r of q u a s i - p a r t i c l e s . e l e c t r o n - h o l e pair states
These changes may
In certain a p p r o x i m a t i o n s ,
(excitons) may be r e g a r d e d as analogues to
the h y d r o g e n atom. However,
the excitons e x h i b i t a finite lifetime
because of possible decay by radiative and n o n - r a d i a t i v e recombination. They may be g e n e r a t e d by a b s o r p t i o n of photons.
Thus,
a highly excited
s e m i c o n d u c t o r is truely a system far away from thermal equilibrium. Permanently,
energy is pumped into the system,
d i s t r i b u t e d to other degrees of freedom.
and this e n e r g y is being
The t r a n s i t i o n s b e t w e e n the
s t a t i o n a r y states of h i g h l y e x c i t e d s e m i c o n d u c t o r s have the c h a r a c t e r of n o n e q u i l i b r i u m phase transitions.
A first example is the plasma phase t r a n s i t i o n b e t w e e n the exciton gas phase and the e l e c t r o n - h o l e
liquid phase. The e x c i t o n s have a relative-
ly long lifetime in i n d i r e c t - g a p semiconductors,
i.e.
in m a t e r i a l s
with an indirect gap b e t w e e n the c o n d u c t i o n - b a n d m i n i m u m and the v a l e n c e - b a n d maximum. occurs.
In those m a t e r i a l s spatial phase c o e x i s t e n c e
E l e c t r o n - h o l e droplets are formed in a s u p e r s a t u r a t e d e x c i t o n
gas. The process of e l e c t r o n - h o l e d r o p l e t n u c l e a t i o n is d i s c u s s e d in section III.la as a n o n e q u i l i b r i u m analogy to c l a s s i c a l d r o p l e t nucleation. A s u i t a b l e g e n e r a l i z a t i o n of the B e c k e r - D ~ r i n g theory is p r e s e n t e d w h i c h allows a s a t i s f a c t o r y d e s c r i p t i o n of the f o r m a t i o n and decay of e l e c t r o n - h o l e d r o p l e t s in i n d i r e c t - g a p
On the other hand,
the lifetime of e x c i t o n s
semiconductors.
in d i r e c t - g a p s e m i c o n d u c -
tors is c o m p a r a t i v e l y short. This causes d i s t i n c t m o d i f i c a t i o n s of the plasma phase t r a n s i t i o n in these materials.
Up to now,
it has not been
possible to observe the effect of phase s e p a r a t i o n e x p e r i m e n t a l l y . This s i t u a t i o n is d i s c u s s e d in section III.Ib. A g e n e r a l i z e d hydrodynamic d e s c r i p t i o n is p r e s e n t e d and the results of a s i m u l a t i o n of the m a s t e r e q u a t i o n for the d y n a m i c s of cluster f o r m a t i o n are discussed. Both m e t h o d s show, that only e x t r e m e l y small e l e c t r o n - h o l e d r o p l e t s may be formed because of the short e x c i t o n i c lifetime.
The region of
14 two-phase c o e x i s t e n c e in the phase d i a g r a m of d i r e c t - g a p semiconductors is t h e r e f o r e strongly reduced.
The c h a r a c t e r of the n o n e q u i l i b r i u m
phase t r a n s i t i o n is m o d i f i e d to be only w e a k l y of first order.
In part 2 of chapter I I I a is presented.
d i s c u s s i o n of optical n o n e q u i l i b r i u m systems
Here, u n d o u b t e d l y the laser is the m o s t important example.
It shows complete
sequences of n o n e q u i l i b r i u m phase transitions.
A semi-
classical t h e o r e t i c a l t r e a t m e n t of the laser is given in section III.2a. In the framework of the optical M a x w e l l - B l o c h equations, transitions will be discussed, which take place, e.g., saturable absorber. transitions
However,
discontinuous
in lasers with a
the vast m a j o r i t y of n o n e q u i l i b r i u m phase
in lasers is of second order. Therefore,
they are not the
topic of this book.
D i s t i n c t i v e l y d i s c o n t i n u o u s b e h a v i o u r is found in o p t i c a l l y b i s t a b l e systems. Here,
the n o n e q u i l i b r i u m transitions take place b e t w e e n states
of high and low transmission.
The n e c e s s a r y n o n l i n e a r i t y is due to an
intensity d e p e n d e n c e of the d i e l e c t r i c function, c o e f f i c i e n t and the refractive
i.e. of the a b s o r p t i o n
index of the i n v e s t i g a t e d material.
In
section III.2b, a d i s c u s s i o n of optical b i s t a b i l i t y of two-level atoms in a F a b r y - P e r o t resonator is presented. are investigated.
Examples of dynamical effects
The t r a n s i t i o n p r o b a b i l i t y b e t w e e n states of high
and low t r a n s m i s s i o n is c a l c u l a t e d taking into account the relevant noise contributions.
An e i g e n v a l u e analysis of the F o k k e r - P l a n c k
e q u a t i o n is p e r f o r m e d and the smallest n o n - z e r o e i g e n v a l u e is calculated approximately.
Very recently,
t i m e - d e p e n d e n t p e r i o d i c and chaotic solutions have
been found in certain p a r a m e t e r regions of the optical M a x w e l l - B l o c h equations.
Details of this i n s t a b i l i t y are d i s c u s s e d in chapter III.2c
for the example of two-level atoms in a ring r e s o n a t o r with a short r o u n d - t r i p time. The route to chaos is shown to follow the F e i g e n b a u m szenarium.
In section III.2d an i n v e s t i g a t i o n of dispersive optical b i s t a b i l i t y in s e m i c o n d u c t o r etalons is presented.
The n o n l i n e a r p r o p e r t i e s of the
m e d i u m are c a l c u l a t e d within a q u a n t u m - m e c h a n i c a l m a n y - b o d y theory. U t i l i z i n g these results,
a set of e q u a t i o n s
for the n o n l i n e a r r e s o n a t o r
is derived.
This allows a d e s c r i p t i o n of dynamical aspects of optical
bistability
in these systems.
As an e x p l i c i t example,
a d i s c u s s i o n of
the b i e x c i t o n b i s t a b i l i t y in the s e m i c o n d u c t o r m a t e r i a l CuCI is pre-
15
sented. The analytical treatment is supplemented by numerical simulations of the full stochastic equations.
The dynamics of the
switching processes between the nonequilibrium steady states and the stability of metastable states is investigated. Finally,
section II!.2e contains a discussion of optical bistability
due to induced absorption.
Here, the exciting light beam changes the
absorption of the medium. Light of high intensity is absorbed more effectively
than light of low intensity.
to saturable absorption.
This is the opposite effect
It is shown to give rise to optical bista-
bility, even without resonator feedback. Additionally,
the theoretical
calculations predict intrinsically bistable spatial structures caused by the formation of regions with different excitation density.
II.
E q u i l i b r i u m Phase T r a n s i t i o n s
This chapter is d e v o t e d to the d i s c u s s i o n of phase transitions
in
e q u i l i b r i u m systems. These transitions are n o r m a l l y initiated by external changes of some control parameters.
The system reacts to these
changes since it is no longer in thermal equilibrium. ditions,
For certain con-
e v o l u t i o n to a new e q u i l i b r i u m is c o n n e c t e d with a phase
change in the system. Dynamical aspects of these e q u i l i b r i u m phase t r a n s i t i o n s are i n v e s t i g a t e d in the following sections.
II.1 Dynamics of Phase S e p a r a t i o n
The s u p e r s a t u r a t e d vapour of a Van der Waals system is a typical example of a m e t a s t a b l e fluctuations.
state.
It may decay because of localized density
These fluctuations must have a large amplitude and exceed
a certain spatial extension.
They are c o m m o n l y called "critical drop-
lets". The i n s t a b i l i t y of the h o m o g e n e o u s system against formation and growth of d e n s i t y fluctuations
is shown in Fig.6.
The spontaneous formation of critical droplets is a typical example of a heterophase
fluctuation.
L o c a l i z e d liquid regions appear w i t h i n the
s u p e r s a t u r a t e d v a p o u r phase.
These h e t e r o p h a s e f l u c t u a t i o n s are the
origin of the finite lifetime of m e t a s t a b l e
states in e q u i l i b r i u m and
n o n e q u i l i b r i u m systems.
On the other hand, even very small fluctuations are sufficient to initiate the decay of unstable states. A s p a t i a l l y h o m o g e n e o u s system w i t h i n the spinodal region of its phase d i a g r a m d e c o m p o s e s because of i n f i n i t e s i m a l l y small density fluctuations. "spinodal decomposition".
This is the process of
In Fig.6 it is shown s c h e m a t i c a l l y in com-
parison to nucleation.
Even though they m a y be of i n f i n i t e s i m a l l y small amplitude,
these
d e n s i t y f l u c t u a t i o n can initiate spinodal decomposition, p r o v i d e d they e x c e e d a critical w a v e l e n g t h
Ic" This c o n d i t i o n is c o n n e c t e d with the
i n c r e a s i n g surface e n e r g y by forming a spatially i n h o m o g e n e o u s distribution.
The surface energy has to be c o m p e n s a t e d by the energy gained by
a p p r o a c h i n g the new equilibrium.
The c r i t i c a l w a v e l e n g t h
I c gives rise
17
Ph~le Dip
•
\..oh,e_/ Regions
Metastable Re¢jion: PhaseTransition by Nucleation Uniform Fluid
Density Fluctuation
-
Growth -
R >Re =" IOA Unstable Recjion: PhaseTransition by Sloinodal Decomposition
P o ~ X > Xc ~ 50A
Fig. 6: Schematical representation of density fluctuations leading to spinodal decomposition or to nucleation. to the appearance of a characteristic precipitation pattern in the early stages of spinodal decomposition. Phase separation by nucleation or spinodal decomposition is well-known to occur also in binary systems. Here, the local concentration c(~,t) of one of the constituents of such a system is the relevant order parameter.
The dynamical evolution of this quantity is diffusively,
whereas the order parameter
(density)
in one-component systems varies
by hydrodynamic flow. From a more fundamental point of view, nucleation and spinodal decomposition are not entirely different processes.
In both cases, the dri-
ving mechanism is the formation and growth of density fluctuations.
One
would therefore expect a gradual transition between nucleation and spinodal decomposition.
Clearly,
in the vicinity of the binodal line,
phase separation occurs by nucleation.
On the other hand, near the
18
center
of the u n s t a b l e
m e c h a n i s m . These
region,
processes
spinodal
decomposition
are d i s c u s s e d
separately
chapters. T h e n a u n i f i c a t i o n is given by a t r e a t m e n t position
in the f r a m e w o r k
II.la N u c l e a t i o n
In the year
1935 B e c k e r
in m a n y
references
Fore more
saturated
theory vapour
typically
No direct
the co m p l e t e
change
Fig.7:
Schematical mechanism.
gn is the gain
rate,
i.e.
A
is indiscussion.
in a superto be a
. The index n
is assumed.
to the p r o c e s s e s
and c l u s t e r
sketched
Thus, of
shrinkage
The c o n c e n t r a t i o n
due to the r e a c t i o n - s c h e m e
transitions.
n in the r e s p e c t i v e
the c l u s t e r s
of m o n o m e r s
respectively.
representation
is c o n s i d e r e d
and clusters
is r e d u c e d
in con-
references.
condensation
are c o n d e n s e d
between
of the s y s t e m
of m o n o m e r s ,
A n may
Here,
interaction
presentation
to the given
(monomers) which
phase
for the s u b s e q u e n t
T. The v a p o u r
molecules
of m o l e c u l e s
dynamics
point
famous
problems
of it have been r e v i e w e d
. The f o l l o w i n g
deals w i t h d r o p l e t
growth by c o n d e n s a t i o n
evaporation
decom-
theory.
their
the basic
improvements
is r e f e r r e d
of t e m p e r a t u r e
of i n d i v i d u a l
cluster.
21-27]
the reader
the number
cluster
[e.g.
published
states in f i r s t - o r d e r
and some
only as a s t a r t i n g
details
Nucleation
labels
theory
[21]
It deals w i t h
the d e c a y o f m e t a s t a b l e
to serve
mixture
and D ~ r i n g
nucleation.
The B e c k e r - D 6 r i n g
tended
of spinodal
nucleation
Theory
theory of droplet text with
of a g e n e r a l i z e d
is the d o m i n a n t in the f o l l o w i n g
by
of clusters
in Fig.7.
of the c o n d e n s a t i o n - e v a p o r a t i o n
the rate of c o n d e n s a t i o n
onto a cluster A n . £n is the p e r t i n e n t
loss rate.
This
of m o n o m e r s
simplified
19
d y n a m i c s of droplet n u c l e a t i o n is known as c o n d e n s a t i o n - e v a p o r a t i o n mechanism.
It is the dominant dynamical effect in systems with a rela-
tively dilute cluster concentration,
w h e r e the m o b i l i t y of the clus-
ters is very small in c o m p a r i s o n to that of the monomers. A m a t h e m a t i c a l d e s c r i p t i o n of these p r o p e r t i e s may be given in terms of a m a s t e r equation for the p r o b a b i l i t y d i s t r i b u t i o n fn(t). This d i s t r i b u t i o n function m e a s u r e s the relative p r o b a b i l i t y of finding a cluster w i t h n m o l e c u l e s
at time t. The m a s t e r e q u a t i o n is:
~fn 3t = J n - 1 - J n
'
n_>2
(II. I )
where Jn=gnfn-£n+ifn+1
(II.2)
The q u a n t i t y Jn is the net p r o b a b i l i t y current b e t w e e n clusters A n and An+ I. The gain rate is given by the current of m o n o m e r s t h r o u g h the cluster surface gn=b(T)nl (t)n(d-1)/d
Here, nl(t)
is the c o n c e n t r a t i o n of m o n o m e r s
(II.3)
in the vapour,
n (d-1)/d
is p r o p o r t i o n a l to the surface of a cluster in a s y s t e m with d dimensions. The loss rate is given in terms of the t h e r m i o n i c e m i s s i o n current through this surface: £n=a (T) n (d-1)/dex p (o (T)/n I/3) ,
(II. 4)
Here, the term e x p ( o ( T ) / n 1/3)
takes into account the s u r f a c e - t e n s i o n c o r r e c t i o n of the m o l e c u l a r work function in a cluster.
(More details are given in [21-27]
.)
The m a t h e m a t i c a l t r e a t m e n t may be s i m p l i f i e d c o n s i d e r a b l y by n e g l e c t i n g the discrete nature of the v a r i a b l e n. This c o n t i n u u m a p p r o x i m a t i o n replaces the d i f f e r e n c e - t e r m s In this way,
in the m a s t e r e q u a t i o n by differentials.
the s o - c a l l e d F o k k e r - P l a n c k a p p r o x i m a t i o n is obtained.
F o r m a l l y more correct w o u l d be a K r a m e r s - M o y a l e x p a n s i o n the m a s t e r equation.
[28-30]
The individual c o n t r i b u t i o n s are e x p a n d e d
of
20
according to
£n+ I fn+ 1=m~0 [~.w ~n~l £nfn Hence, the right-hand side of equation
(II.1)
is expressed in terms of
fn" The Fokker-Planck equation is obtained by neglecting derivatives of higher than second order. Generally,
a Fokker-Planck equation need not be a good approximation
of a master equation.
The Pawula theorem
[30]
states,
that the
Kramers-Moyal expansion either stops for m<2, or infinitely many terms occur. The latter is true in the theory of droplet nucleation.
There-
fore, the Fokker-Planck equation is clearly only an approximation to the full master equation. The validity of this approximation has to be examined carefully. Without going too much into the details,
the Fokker-
Planck approximation is apparently not very good when relatively small droplets are important.
This happens
in the problem of electron-hole
droplet nucleation in direct-gap semiconductors For lagge clusters
however,
(see section III.Ib).
the continuum limit is justified. Hence~
the dynamics of these large clusters may
be
described in the framework
of the Fokker-Planck approximation to the master equation f(n,t)= - ~ J(n,t) ~n
~t
(II.1) (II.5)
Here, the continuum expression for the probability current is J(n,t)=Ig(n,t)-~ (n)- ~I ~
Q(n,t)if(n,t )
(II.6)
The diffusion coefficient in the Fokker-Planck equation can be written as Q (n,t)=g (n,t)+~ (n) This form is typical for particle fluctuations
(II.7) (shot noise). Here, the
second moment of the random fluctuations is always the sum of all gain and loss rates
[31]
Becker and D~ring
[21]
restrict their calculations to the situation
of a constant concentration of monomers nl(t)=n I . This assumption is justified in cloud-chamber experiments with a constant flow of gas into the experimental volume, while the larger clusters are removed. these conditions,
For
Becker and D6ring derive an expression for the nulea-
21
tion rate Jnuc" This rate measures the formation of sufficiently large clusters exceeding the critical size n c. Here,
the critical size is
determined from (nc) =g (nc ) Subcritical droplets
ing(n) I have the tendency to reevaporate,
whereas larger droplets will continue to grow. The Becker-D6ring theory yields _@cl (nc)/kBT Jnuc=J0e
(II.8)
Here, J0 is the rate prefactor and ~Cl(n c) is the energy required to form a critical droplet
(see Fig.8).
10cl n
Fig.8:
Energy difference ~cl between a cluster with n molecules and the supersaturated vapour (schematical representation).
For situations without conservation of the total number of particles, Eq. (II.8) yields a good description of the experimental results.
Diffi-
culties arise however in the respective microscopic calculations of the rate prefactor and the activation energy ~Cl(nc). For details see, e.g., [22-24] The approximation n~(t)=n1=const,
is not allowed when the long-time
behaviour of a closed system is calculated.
In this case, particle con-
servation yields an effective mutual restriction for the growth of large clusters.
The late-time regime is most easily described in terms
of the temporal evolution of the mean cluster radius R(t). To obtain an equation for this quantity,
one makes use of the Langevin equation
for the number n of molecules in a cluster
22
an dt - g (n, t)-£ (n) +F (n,t)
(II. 9)
This L a n g e v i n e q u a t i o n is s t o c h a s t i c a l l y e q u i v a l e n t to the FokkerPlanck e q u a t i o n
(II.5). The noise term in Eq. (II.9) d e s c r i b e s a Gauss-
M a r k o v process =0 and =Q(n,t)6(t-t')
Microscopically,
(II.10)
the origin of the stochastic f l u c t u a t i o n s is the coup-
ling of the system to a bath. The d y n a m i c a l b e h a v i o u r of the m o s t probable value of the system v a r i a b l e can be o b t a i n e d as s o l u t i o n of the b a t h - a v e r a g e d L a n g e v i n equation. Since the m e a n cluster radius and the m e a n number of atoms in the cluster are related by R~n I/d
,
the b a t h - a v e r a g e d e q u a t i o n
(II.9) m a y be t r a n s f o r m e d to
dR dt - aA (t)-be~/R The s u p e r s a t u r a t i o n
A(t)
is d e f i n e d as
A(t)=nl(t)/n s where n
(II.11)
,
(II.12)
is the thermal e q u i l i b r i u m density of the vapour,
a, b and
s
in Eq. (II.11) are t e m p e r a t u r e d e p e n d e n t quantities. is not relevant for the following discussion. note,
that Eq. (II.11)
T h e i r e x p l i c i t form
It is interesting to
does not d e p e n d on the d i m e n s i o n s of the system.
C l u s t e r s e x c e e d i n g the critical radius Rc~n c I/d continue to grow,
since
their loss rate is smaller than the gain rate. The growth rate for these s u p e r c r i t i c a l l y large clusters will be c a l c u l a t e d in the following. Here,
it is well j u s t i f i e d to e x p a n d the e x p o n e n t i a l in Eq.(II.11)
powers of I/R and to n e g l e c t the h i g h e r order terms. The result is
dt
c
In this e q u a t i o n Rc (t) ~I/A (t)
- ~)
(II.13)
in
23
is the critical radius, c is a (temperature dependent) sides Eq.(II.13)
constant.
Be-
one has to include the equation that guarantees the
conservation of the total number of particles in the system
Q0=A0+q0
=A(t)+q(t)
(II.14)
Here, Q0 is the total supersaturation
at time t=0
, A 0 is the super-
saturation in the vapour only, and q0 represents the molecules,
which
are already condensed in clusters at t=0 . q(t) may be expressed as moment of the probability distribution f(n,t) q(t)= ~
~dn nf(n,t)
(II.15)
s
Thus, Eqs.(II.5),
(II.14), and
(II.15) form a closed set of equations
for the long-time behaviour of droplet growth. A very similar set of equations has been solved by Lifschitz and Slyozov
[32]
with a method that is asymptotically correct. The result
is independent of the systems dimensions
[33]
. Lifschitz and Slyozov
investigate the growth of clusters in a binary mixture. As discussed in the next chapter of this book,
for binary systems
the rate equation
(II.9) has to be replaced by a diffusion equation. This can be transformed to
The prefactor D is proportional to the diffusion coefficient. (II.16),
(II.14), and (II.15), together with a Fokker-Planck
for the cluster probability distribution R(t)~R
c
function yield
(t)=I/A(t)
Eqs. equation
[32] (II.17)
and R~t I/3 .
(II.18)
These results are correct for large times. A Lifschitz/Slyozov-type
analysis of the equations for the condensa-
tion-evaporation mechanism has been presented in Ref.34. Relation (II.17) is also correct in this case, but the growth law is found to be
24
R~t I/2
(II. 19~
It is not necessary to go through the complete
asymptotic
in order to understand
and
the growth laws
(II.18)
analysis
(II.19). These laws
follow directly from the structure of the equation
for dR/dt,
if a
power
Eqs.(II.13)
and
law dependence
of R on t is assumed
[35]
.
(II.16) may be combined to
d--~dR ~ R-y (Rc__~t) ~I I
(II.20)
-
Here,
y=0 for Eq.(II.13)
and y=1 for Eq.(II.16).
If Eq.(II.20)
is to be
satisfied by a power law R~t 8 Eq.(II.17)
(II.21)
has to be correct,
i.e. Rc(t)~R(t) , and
~=(y+2) -I
This is the g e n e r a l i z a t i o n
II.Ib Phase Separation Alternatively
tion may be described ously varying
of Eqs.(II.18)
theory,
Often,
dynamical
aspects of phase separa-
the relevant variables
This may be the local density
systems or the local concentration mixture.
(II.19).
in the framework of semi-phenomenological
In such theories, fields.
and
in Binary Mixtures
to nucleation
nuum theories.
(II.22)
in one-component
of the constituents
it becomes necessary
conti-
are continuof a binary
to deal with sets of field vari-
ables. The dynamic equations
for the respective
using gradient expansions.
Low-order
reduces the region of applicability. the short-time
fields are commonly derived
truncation of these expansions Either,
regime of the dynamical evolution,
the control parameters
close to the critical
however experimentally
quite relevant.
tical region find much interest Historically,
one is restricted
or one has to choose
point.
Especially
to
Both regimes are
systems
in the cri-
in current literature.
the first successful
field theoretical
description has
25 been formulated by Cahn and Hilliard spinodal
decomposition
tension to a stochastic Langer and coworkers tical descriptions San Miguel,
theory.
[39-41]
can be found, e.g.,
and Sahni
binary metallic phase diagram.
quenched
in the review articles by Gunton, and Halperin
theory are sketched
[42]
in the following
for a simple model of a
into the m i s c i b i l i t y
a m o d i f i e d version of the Cahn-Cook
gap of its
the relevant
~t
c(r,t)=
The equation
is the local concentraof m o t i o n
V J(r,t)+F1(r,t)
This is a nonlinear Langevin equation. a Gauss-Markov
systems.
field variable
of one of the metals.
process,
that
theory may be used to describe
also in n o n e q u i l i b r i u m
In a binary alloy,
to describe
came from
of the field theore-
In a later chapter of this book it will be shown,
phase separation
tion c(r,t)
of
gave the ex-
contributions
has been d e v e l o p e d
alloy,
[38]
summaries
[26] and by H o h e n b e r g
The theory
for the process
Cook
Important
. Detailed
The basic ideas of the Cahn-Cook paragraphs.
[36,37]
in binary systems.
is (II.23)
The thermal noise is assumed
i.e.
=0 and
(II.24) = - 2 k B T M V 2 ~ ( r - r
The current
in Eq. (II.23)
can be expressed
as
~= - M ~ (~) Here, M is the mobility
(II. 25)
and ~(~)
p(~)_
)~(t-t')
is the local chemical
6F 6c(~)
potential
(II.26)
with K
-~
F{C}= ~ d3r (~(Vc) 2+fh (c)) as G i n z b u r g - L a n d a u of Eq.(II.27) The term fh(c) system.
free-energy.
(II.27)
The first term on the right-hand
side
is the surface energy caused by local inhomogeneities. stands
for the free energy density of the homogeneous
It is advantageous
to transform the equations
to the variable
26
u(~,t), which describes
the local deviation
from the mean concentration
7: u(r,t)=c(~,t)-7 The homogeneous
part of the chemical potential ~fh ~= ~c .....
is expanded
in terms of 7. Thus, one can derive a functional Fokker-
Planck equation
for the probability
functional
f({u},t)
~(J)
~2uJ) f (II.28)
22 +2 . ~u--~kBTMv f} Here,
the abbreviation ~(J)=~J~/~cJ
has been used. The functional chastically
equivalent
forming from c(~,t) and integrating equation
c=7
Fokker-Planck
equation
to the Langevin equation
to u(~,t).
Multiplying
over all configurations
for the density auto-correlation
(II.28)
(II.23)
is sto-
after trans-
Eq. (II.28) by u(r,t)u(0,t)
~ Du yields the dynamical function S(r,t)
S(r,t)= f D u u(r)u(0)f({u},t) (II.29) = fdu(r) In this equation f2(u(r),u(0~t)
~du(0)f 2 (u(r),u(0),t)u(r)u(0)
is the two-point probability
.
distribu-
tion, i.e., the joint probability for simultaneous realization of the local values u(r) and u(r=O) at time t. Suitably normalized, S(r,t) is also known in the literature correlation
fuction g(r,t).
as radial distribution The normalization
that g approaches unity for uncorrelated
function or pair
is commonly choosen so
atoms
lim g(r,t)=1
The probability
of finding two
atoms a distance
r apart is incresed
(decreased) in comparison to the uncorrelated situation if g(r) is larger (smaller) than unity. The Fourier-transform of the radial
27 distribution
function
This quantity
is the time dependent
structure
can be m e a s u r e d by light scattering
formation of Eq.
(II.28)
Trans-
yields
1 s=-2~2((Kk2+~(1))s+½ 7~2)s3+g
__ ~t
factor S(k,t).
experiments.
(3)
S 4+ •.. ) +2MkBTk2 (II.30)
where
Sn= ;d~ e i ~ ~du(;)du(O)u n-l(~)u(O)f2 [u(r) ,u(O),t) (II.31) Eq.
(II.3o)
possible
couples S(k,t)
hierarchy of equations archy yields Langer,
directly to the higher moments
extension of the calculations for all moments.
a certain approximation
Bar-on,
and Miller
ximation scheme. is sufficient
Realizing
to calculate
Truncation
to the
(LBM) published that knowledge all moments,
f2(u(r) ,u(o),t)=fl[u(r),t
S n. One
would be to establish
a
of the hier-
exact result.
In Ref.
41
a far more elegant approof the joint p r o b a b i l i t y
they made an ansatz
f2
for f2
Ill [u(O),t){1+¥(r)u(r)u(O)}
. (II.32)
Here,
fl(u,t)
is the single-point
probability
has the meaning of a concentration Eq.
(II.32)
is a simple factorization
product of individual expression
probabilities.
in brackets
in Eq.
which
For y=O,
of the joint p r o b a b i l i t y
(II.32)
as the b e g i n n i n g
Inserting Eq.(II.32)
relation
in terms of S and
moment of the single-point
function.
into a
For y~O one may regard the
series expansion. for y(r)
distribution,
distribution
probability
into
of a power-
(II.29) yields
. Here,
a recursive
is the second
distribution
fl . Generally,
the moments of fl are given by = fdu unfl(u,t)
To simplify
(II.33)
the further treatment,
which expresses
the distribution
LBM make an additional
function
ansatz
fl as sum of two d i s p l a c e d
Gaussians
f1(u,t)=
(b2(t)
{(27)
exp{
(bl(t)+b2(t))b3(t)} -I w 2 2 {u-b/ (t)} {u + b 2(t) } }+ b I (t) e x p { 2 } 1 2h~ 2(t) 2b 3 (t)
(II.34)
28
This ansatz contains three p a r a m e t e r s bl(t) , b2(t) , and b3(t) w h i c h can describe some e s s e n t i a l
features of the system under consideration.
If, e.g.,
fl shows only one m a x i m u m around u=0 no phase s e p a r a t i o n has
happened,
contrary to the situation of a double p e a k e d structure of fl
w i t h the m a x i m a at a p o s i t i v e and a n e g a t i v e value of u, respectively.
W i t h the h e l p of the full F o k k e r - P l a n c k e q u a t i o n
(II.28)
one can derive
e q u a t i o n s of m o t i o n for the p a r a m e t e r s bl, b2, b 3. These equations, t o g e t h e r with the e q u a t i o n s the moments
, n=2,3,4
for the structure factor
, e s t a b l i s h a closed s y s t e m of integro-differ-
ential equations, w h i c h can be solved numerically. t h e o r y for the kinetic Ising model. simulations
[43]
(II.30), and for
LBM evaluate their
Details of this model and n u m e r i c a l
are d i s c u s s e d below. An example for the results of
the LBM theory is p r e s e n t e d in Fig.9.
3¢
$
J .2
Fig.9:
.4
.6
.e
,.o
~.z
,A
kla 0
Structure factor S(k,~) for spinodal d e c o m p o s i t i o n in the kinetic Ising S y s t e m (from Ref. 41) . The insert shows the d i s t r i b u t i o n f u n c t i o n fl for two c h a r a c t e r i s t i c times after the quench of the Ising system into its phase c o e x i s t e n c e region, a 0 is the lattice constant.
This figure shows the structure factor and the d i s t r i b u t i o n function fl for d i f f e r e n t times after the Ising system has been q u e n c h e d into the coexistence region of its phase diagram. One clearly sees the d e v e l o p m e n t of a double p e a k e d structure of fl as d i s t i n c t sign for the occurence of phase separation. pattern m a n i f e s t s
The c o a r s e n i n g of the p e r t i n e n t p r e c i p i t a t i o n
itself in the shift of the peak p o s i t i o n kma x of the
structure factor to smaller k-values.
Here, kma x is p r o p o r t i o n a l to
I/R, with R b e i n g the m e a n radius of the formed clusters.
A d e t a i l e d analysis shows that the LBM theory yields n o n - s a t i s f a c t o r y results for the long-time regime and for p a r a m e t e r s outside the critical
29
region
[44,45]
. It does not c o r r e c t l y reproduce the R~t I/3 growth
law for large clusters. However,
up to now no c o n s i s t e n t n o n l i n e a r
theory of spinodal d e c o m p o s i t i o n has been c o m m o n l y accepted, w h i c h goes far b e y o n d the L B M treatment.
A g e n e r a l i z a t i o n of Eq.(II.23) sented in Ref.39. [39b,40]
to the N - d i m e n s i o n a l
In this framework,
case has been pre-
a n u c l e a t i o n theory is d e r i v e d
. The g e n e r a l i z e d F o k k e r - P l a n c k e q u a t i o n can be w r i t t e n as ~f _ %t
~Ji ~ ~i
'
(II.35)
where Ji = - [ Mijl ~F
~f
The sum over i implies summation of the r e l e v a n t p a r a m e t e r H i as well as i n t e g r a t i o n over r. M. • is the m o b i l i t y m a t r i x and F £s a func13 tional of the free energy. It has its m i n i m a at the p a r a m e t e r values of the pure phases.
These m i n i m a are s e p a r a t e d by a saddle point.
Similar to classical n u c l e a t i o n theory,
a s o l u t i o n of the s t a t i o n a r y
F o k k e r - P l a n c k e q u a t i o n is constructed, w h i c h c o r r e s p o n d s to a c o n s t a n t value of the p r o b a b i l i t y current.
The n u c l e a t i o n rate is o b t a i n e d
as the intensity of this p r o b a b i l i t y current i n t e g r a t e d over the saddle point region.
Formally,
cleation theory
the result agrees w i t h that of c l a s s i c a l nu-
(Eq. II.8).
As m e n t i o n e d above,
the LBM theory has been e v a l u a t e d for the example
of the kinetic Ising model. This model and c o m p u t e r s i m u l a t i o n s
for
the p e r t i n e n t dynamic features are d i s c u s s e d in the f o l l o w i n g paragraphs of this section.
The Ising model for a binary alloy of metals A and B is c h a r a c t e r i z e d by the H a m i l t o n i a n
[47]
E= - J
[n(~i)n(~ j)
The c o u p l i n g c o n s t a n t J is positive.
(II.37) The v a r i a b l e
n(~ i) assumes the
values +I or -I if the lattice point r i is o c c u p i e d by an atom respectively.
This Ising model
model of a m a g n e t i c system
A or B,
is isomorphic to the spin-I/2 Ising
[48]
and to the lattice gas
model of a
fluid system. To describe the m a g n e t i c ( l a t t i c e gas s y s t e m ) , o n e has to replace,
e.g.,
'molecule B' by
'molecule A' by 'spin down'
'spin up'
('vacancy').
('occupied lattice site')
and
30 The Ising model exhibits an o r d e r - d i s o r d e r phase transition for w h i c h the d e t a i l e d phase d i a g r a m is known t w o - p h a s e coexistence,
[48]. Since it has a region of
one may i n v e s t i g a t e the process of phase separa-
tion. This has been done, e.g., by means of Monte Carlo simulations
[47,35]
to d e t e r m i n e
properties
(MC) c o m p u t e r
. Such MC techniques are by now a standard m e t h o d of
systems in t h e r m o d y n a m i c equilibrium.
n u m e r i c a l algoritbam has been i n v e n t e d by M e t r o p o l i s et al. reviews see e.g. Refs. processes,
[49]
The , for
50 - 52 . In the context of phase separation
one is m a i n l y i n t e r e s t e d in applications of MC methods to
d e t e r m i n e d y n a m i c a l features. Therefore,
the stochastic nature of time ~
e v o l u t i o n in MC systems is d i s c u s s e d briefly.
In MC simulations successive c o n f i g u r a t i o n s x
of a m a n y particle sys-
tem are g e n e r a t e d by changing the positions of one or more atoms according to the p r o b a b i l i t y W(x ÷x
,). The p r o b a b i l i t y for such a move
depends on the p e r t i n e n t change of the energy A E = E ( x ,)-E(x
)
W is choosen such that time averages over s u c c e s s i v e c o n f i g u r a t i o n s converge to the correct t h e r m o d y n a m i c e q u i l i b r i u m value ever,
this r e q u i r e m e n t does not d e t e r m i n e W uniquely.
[51]
. How-
Two common
choices are W(x +x
,)~ e - S A E / ( l + e -SAE)
W ( x +x
~ e "SAE, AE>0 ,) [I, otherwise
and
(II.38)
The i n t r i n s i c a l l y stochastic dynamics of the MC m e t h o d is the reason why the sequence of g e n e r a t e d c o n f i g u r a t i o n s does not n e c e s s a r i l y correspond to the true physical time evolution.
One may h o w e v e r introduce
a MC analogue of time, w h i c h in the f o l l o w i n g Will be called T . It is measured, m e n t s of an atom,
e.g.,
'MC time'
in units of the number of a t t e m p t e d displace-
i.e., in units of M C S / A
(MC steps per atom). The re-
lation of the MC time • to the real p h y s i c a l time t is strongly dep e n d i n g on the p r o p e r t i e s of the s i m u l a t e d system. An almost linear conn e c t i o n between t and T should be e x p e c t e d only if the true physical d y n a m i c s is c o r r e c t l y
mimiked
K e e p i n g these r e s t r i c t i o n s
by
the M e t r o p o l i s dynamics.
in mind, one may discuss the MC dynamics of
phase s e p a r a t i o n in the kinetic Ising model. As a result,
these simula-
tions yield the atomic distributions on the available lattice sites.
31
Fig.lO presents some typical examples. been obtained for different times homogeneous equilibrium system
•
The shown distributions have after quenching an initially
(at T>Tc)
to T
i.e. to a point with-
in the phase coexistence region.
.~"~,'
*': 50MCSIA
~,,~,.,~.~_.~,~ . ~ f
Fig.10:
1000 MCSIA
50000MCSIA
,,~
Snapshots of the cluster growth pattern obtained from a MC simulation of spinodal decomposition in a lattice gas system. The occupied lattice sites are symbolized by dots. (From Ref. 35.)
Fig.tO shows clearly the formation of regions whith densities different from the mean value. At the early stages one recognizes the characteristic precipitation pattern. As mentioned earlier, sequence of the critical wavelength tiating spinodal decomposition.
this is a con-
lc of the density fluctuations ini-
A quantitative analysis of the data is
given in terms of the radial distribution
function g(r,T)
and its
Fourier-transform S(k,T). Numerically, g(r,T) is calculated by simply counting the number of atom pairs at distances between r and r+A r, and by correctly normalizing the computed function. the intervals factor S(k,T)
This yields g(r,T)
for
Ar and correspondingly the time dependent structure at discrete points in k-space.
An example of the results is shown in Fig.11. The full lines are obtained theoretically from the LBM analysis
[41]
. The agreement be-
tween simulation and theory is quite satisfactory.
More details of the
results of the MC simulations will be discussed at the end of the following chapter in comparison to the findings of molecular dynamics
32
•
,I
. . . . . .
;
" •
'. . . . .
. . . . . . . .4 .6 .0 1.0
.2
~.. :,.~
1.2
14
1.6
I
° .1
• !
•
;
•
i
•
!
• v •
!
• g •
i
-
20 / ~
S
T : ~7.?
/
~ o
(41 t : 970
k
6
S
•
i
o
T,II.3
o
0t f
.
,
.
.2
, .4
o .
,
.
,6
,
.
.8
,
(D) t = 190
o o
.
I.O
0
,
.
1.2
,
.
1.4
,
.
/
. . . . . . 0.~ 0.4 0,~
0.~
I.(~
1,2
o. 1.6
1.4
k
4G e
1.6
(e)
k
T:
97.3
: 1640
3C •
'
'
'
'e
"
;
'
'
"
*
"
r
(C)
!
;
"
"
S
: 26.1 : 440
S
2G
•
•2
e
A
.6
e
J
I0
•
1.2
1,4
k
1.6
IJI
.2
4
.l
.O
~0
L2
1.4
i~
k
Fig.t1: MC result for the structure factor S(k,T) (dots) in comparison to the findings of the LBM theory (full lines). The simulations have been done for an Ising system with 30×30×30 lattice sites, periodic boundary conditions, quenched to T=0.8 T c . k is in units of a~ I (see Fig.9). (From Ref. 41.) simulations for spinodal decomposition in one-component systems. Also the scaling relations
for the spinodal peak of the structure factor
and the different growth laws for the atomic clusters are presented at this point.
II.Ic Spinodal Decomposition in One-Comp0nent Systems In the area of spinodal decomposition most laboratory experiments have been done on binary mixtures.
Here, the characteristic time scales are
much longer compared to the conceptually simpler, one-component fluid system , which
separates into the liquid and vapour phases. The dynamic
33
effects in binary fluid systems occur w i t h i n
seconds
long as hours or even days in m e t a l l i c alloys systems on the other hand,
theory p r e d i c t s
tion on the time scale of picoseconds.
[46]
and
may last as
. For o n e - c o m p o n e n t
l i q u i d - v a p o u r phase separa-
No p r e s e n t - d a y l a b o r a t o r y ex-
p e r i m e n t s of spinodal d e c o m p o s i t i o n are able to probe this regime. Therefore,
one uses the m e t h o d of n u m e r i c a l s i m u l a t i o n s to o b t a i n in-
formations on the phase separation dynamics. d i s c u s s e d in the following paragraphs. lysis of spinodal d e c o m p o s i t i o n
Some of the results are
Before that, a t h e o r e t i c a l ana-
is presented.
Phase s e p a r a t i o n in o n e - c o m p o n e n t systems may be discussed, the level of f l u c t u a t i n g h y d r o d y n a m i c s
[53]
e.g., on
. One has the c o n s e r v a t i o n
laws for the local fluid density and for the m o m e n t u m density. be e x p r e s s e d in terms of the h y d r o d y n a m i c equations, d e r i v e d in the textbook by L a n d a u and L i f s c h i t z
[54]
They can
w h i c h are, e.g., . Methodically,
the h y d r o d y n a m i c d e s c r i p t i o n is a n o t h e r example for a field theory. The fundamental d i f f e r e n c e to the C a h n - C o o k d e s c r i p t i o n of phase separation in binary systems is the d i f f e r e n t number of r e l e v a n t cons e r v a t i o n laws. There is only one c o n s e r v a t i o n law centration)
(for the local con-
in t w o - c o m p o n e n t systems, w h i c h at the level of C a h n - C o o k
theory decouples
from the other variables.
The h y d r o d y n a m i c e q u a t i o n s for a o n e - c o m p o n e n t fluid s y s t e m may be f o r m u l a t e d for the mass density, respectively.
Stochastic
the m o m e n t u m and the energy densities,
fluctuations due to local strains and t h e r m a l
fluxes are i n c o r p o r a t e d into the e x p r e s s i o n s
for the stress tensor and
the heat flux, respectively.
in the f o l l o w i n g the hydro-
For simplicity,
d y n a m i c e q u a t i o n s are only d i s c u s s e d in linear a p p r o x i m a t i o n . A full n o n l i n e a r t r e a t m e n t is still to be done.
As v a r i a b l e s of linear h y d r o d y n a m i c s one m a y choose the d e n s i t y deviation ~p from the mean density P0, the d i v e r g e n c e of the v e l o c i t y field , and the d e v i a t i o n
~
from the m e a n t e m p e r a t u r e T O . The t r a n s v e r s e
part of the v e l o c i t y field c o m p l e t e l y d e c o u p l e s ties. The set of r e l e v a n t v a r i a b l e s
from the other q u a n t i -
[53]
(II.39)
satisfies the e q u a t i o n s
S4
a--~ ~o+P°~=0
oo
%-~+v
p-vv
P°Cv ~-t AT+T0
Here,
c
is the heat capacity v stress tensor is given by ~v.
~v k
(II.40)
'
(II.41)
: o =0 ~p ~AT
p ~+?q=0
at constant volume.
2
.
(II.42)
The viscous
3v (II.43)
and the heat flux is ~T
qi = - K ~-~. +gi l In Eqs. shear,
(II.43)
and
(II.44)
(II.44) K , n and { are the thermal
and bulk viscosities,
formulae,
e.g.,
explicit
expressions
respectively.
for the example of an argon liquid, for the dependence
conductivity,
For the evaluation
of the
one has to derive
of these transport
coeffi-
cients on the variables ~ . This can be done with the help of the Boltzmann equation.
From the Enskog theory of dense fluids
[56]
, one
obtains n=q0(1+O.8y + 0 . 7 6 1 y 2 ) / Y
,
~=q0(1.002y2)/Y K=<0(1+1.2y + 0 . 7 5 5 y 2 ) / Y where
q0, K0 are the low-density
thermal conductivity,
, values of the shear viscosity
respectively.
and
The quantity
Y=Y/(~ ~P0) is the factor by which the frequency in comparison
of binary collisions
to the low density limit.
This factor is linked to the
equation of state by the relation
y=
is increased
p01BT[T(~-~T ) p]-1
35
Following Landau and Lifschitz
[54] , a Gauss-Markov behaviour of the
stochastic hydrodynamic fluctuations is assumed. Physically this implies, that these fluctuations are uncorrelated on the time scale of the dynar~ical process. Thus, the first moments of
s÷ and g vanish
<~÷ > =0, =0 and the second moments are 6-correlated in space and time
<Sik(~1,tl)Sgm(~2,t2)> 2 =2T0 (n(6i£6km+6im~k£)+(¢ - ~ ~ )~ik~£m ~ x~(rl-r2)6(tl-t 2)
,
=2
< S
The equations (II.40)-(II.42) be solved by Fourier-Laplace
>=0
.
are an initial value problem, which can . The result is
transformation
~(~, z) =~ (~,t=0)+F (~, z)
(II.45)
Here, the transformed variables have the arguments (k,z). ~ is the vector of the noise terms and the matrix
~÷ contains the coefficients of
the equations. Inverse Laplace-transformation yields an expression for Ap(k,t), from which an analytical result for the time-dependent structure factor can be obtained S(k,t)=
[53]
3 { . (ai+a~)t (ai+aj)t } ~ PiPjS (k, t=0) e -QiQ~ (1-e ) i,j-1 (II.46)
In this equation e i(k), i=I,2,3 of the matrix
M , ~d
, are the complex eigenmodes
P, Q are combinations of the eigenmodes and the
36
m a t r i x elements Mij , respectively.
The e x p l i c i t
e x p r e s s i o n s are given
in Ref.53. E v a l u a t i o n of the modes ~i yields i n t e r e s t i n g informations on the h y d r o d y n a m i c i n s t a b i l i t i e s c o n n e c t e d with spinodal decompositions. Fig.12 shows the results for the example of an argon liquid.
04 •
0~(~ =iJ \ , \
w--,,,
2)
-
"~'~"~
-1.2
O.4
Fig.12:
0.8
1.2
1[6
k
2~0
D i s p e r s i o n of the h y d r o d y n a m i c modes el(k) w i t h i n the liquidv a p o u r coexistence region of a L e n n a r d - J o n e s system. (From Ref. 53.) The units for e. and k are ~-/To and 0 -I , respec1 . tively. Here, m is the atomlc mass and c, o are the parameters of the i n t e r a t o m i c p o t e n t i a l (Eq. II.48).
The real parts of the modes ~2, ~3 whereas the real part of ~i
(see Fig. J2). A c c o r d i n g to Eq. growing structure factor.
are found to be always negative,
is p o s i t i v e in a certain region of k - v a l u e s (II.46)
down by the n o n l i n e a r contributions, treatment.
Nevertheless,
this growth rate is slowed
w h i c h are n e g l e c t e d in the present
the result for
kc, for w h i c h ~l(k) has its maximum, critical fluctuations
this causes an e x p o n e n t i a l l y
In real systems,
~i is instructive.
The value
d e t e r m i n e s the w a v e l e n g t h of the
in spinodal d e c o m p o s i t i o n
. The m a g n i t u d e of
~l(kc ) is a m e a s u r e for the time scale on w h i c h the initial decomposition process occurs.
yields ~i I (kc)~1 ps
I n s e r t i n g the argon p a r a m e t e r s
into
.
It is usual to classify the h y d r o d y n a m i c modes a c c o r d i n g to their longw a v e l e n g t h properties.
E x p a n d i n g e i for small k - v a l u e s shows, that ~3is
the mode of heat d i f f u s i o n and ~ i, e2 are the sound modes, ly. In the phase coexistence region,
respective-
the c o m p r e s s i b i l i t y becomes nega-
37
tive causing a purely imaginary speed of sound. Therefore,
one of the
sound modes becomes undamped.
A comparison
with e x p e r i m e n t a l
the theoretical predictions.
findings w o u l d be a d e s i r a b l e test for
However,
as m e n t i o n e d above,
e x p e r i m e n t a l results available for spinodal d e c o m p o s i t i o n ponent sYstems.
Therefore,
there are no in o n e - c o m -
one is r e s t r i c t e d to n u m e r i c a l simulations.
Some results of m o l e c u l a r dynamics s i m u l a t i o n s L e n n a r d - J o n e s s y s t e m are given in Ref.
for a t h r e e - d i m e n s i o n a l
57. The s h o r t - t i m e v a r i a t i o n
of the spinodal peak of the atomic s t r u c t u r e factor is presented.
A
c o m p a r i s o n of the t h e o r e t i c a l l y c a l c u l a t e d u n d a m p e d mode ~ 1 w i t h the mode ~ e f f c o n s t r u c t e d from the s i m u l a t i o n s shows quite good a g r e e m e n t (see Fig.~3). 0.75 1.5 ps
045
, a.u ",~, "~%
•
o.15
"~,\,,,
-o. 5 -0.7!
0.2
0.6
1.0
1.4
\
1.8
k
Fig.13:
Critical
C o m p a r i s o n of the t h e o r e t i c a l growing mode el (full line) with the e f f e c t i v e a m p l i f i c a t i o n factor ~eff (dashed line) obtained from the results of m o l e c u l a r d y n a m i c s s i m u l a t i o n s of a t h r e e - d i m e n s i o n a l L e n n a r d - J o n e s fluid q u e n c h e d to the unstable region of the phase diagram. (From Ref. 53.)
i n t e r p r e t a t i o n of the comparison
shows however,
truely linear regime for spinodal d e c o m p o s i t i o n
that the
in o n e - c o m p o n e n t sys-
tems is found at most in the t i m e - i n t e r v a l up to 2 ps after the quench. Thus,
the region of a p p l i c a b i l i t y of a l i n e a r i z e d theory is quite
small. N e v e r t h e l e s s , instabilities.
it yields some valuable
insights into the basic
A more complete analysis of the h y d r o d y n a m i c e q u a t i o n s
is still an open problem.
For studies of n o n l i n e a r p h e n o m e n a of spinodal d e c o m p o s i t i o n c o m p o n e n t systems, one is r e s t r i c t e d to n u m e r i c a l f o l l o w i n g paragraphs,
in one-
simulations.
the results of recent m o l e c u l a r d y n a m i c s
tions for a t w o - d i m e n s i o n a l L e n n a r d - J o n e s
system are s u m m a r i z e d
In the simula[55].
Such a system is a quite r e a l i s t i c model of a noble gas film w h i c h is
38
p h y s i s o r b e d onto an e x t r e m e l y smooth surface. The study of phase transition in these films is a quite topical project. More examples are discussed in the next section of this book.
In m o l e c u l a r d y n a m i c s computer simulations one n u m e r i c a l l y solves N e w t o n ' s equations of m o t i o n for a given number of atoms in a cell. By imposing p e r i o d i c b o u n d a r y conditions u n w a n t e d surface effects are eliminated.
The conservative nature of N e w t o n ' s equations guarantees
c o n s e r v a t i o n of the total energy w i t h i n the n u m e r i c a l accuracy. On the other hand,
one may also simulate e.g.
conserve the total energy. ling to a heatbath.
isothermal systems, which do not
In this case, one has to introduce a coup-
A constant mean t e m p e r a t u r e may be o b t a i n e d by
simply r e s c a l i n g the v e l o c i t i e s of all atoms at time intervals ~T ' so that the m e a n kinetic energy corresponds to the desired t e m p e r a t u r e [58]
. P h y s i c a l l y m o r e satisfactory,
however,
is to e x p l i c i t l y incorpo-
rate the h e a t b a t h action into the dynamic equations. d i s s i p a t i v e terms and stochastic fluctuations. of
i n t e g r a t i n g these equations
is described,
This gives rise to
A numerical method e.g.,
in Ref.
59. It
has been applied r e c e n t l y
[60]
in o n e - c o m p o n e n t systems.
Since the results are still prelimi-
nary,
to simulate spinodal d e c o m p o s i t i o n
in the f o l l o w i n g only a summary of c o n s t a n t t e m p e r a t u r e simula-
tions is given, w h i c h have been p e r f o r m e d w i t h the v e l o c i t y r e s c a l i n g procedure
[55]
The m o l e c u l a r d y n a m i c s s i m u l a t i o n s have been done for a classical atomic system w i t h p a i r - w i s e L e n n a r d - J o n e s
(LJ) i n t e r a c t i o n
#(r)=4e{(~112-
In this equation, tial parameters.
(r16}
#(r)
.
(II.48)
r is the i n t e r a t o m i c d i s t a n c e and e, ~ are the potenThe LJ interaction law is a good a p p r o x i m a t i o n to the
i n t e r a t o m i c p o t e n t i a l of rare gas atoms
[61]
. The phase d i a g r a m
of
a strictly t w o - d i m e n s i o n a l LJ s y s t e m has been c a l c u l a t e d in the framework of l i q u i d - s t a t e p e r t u r b a t i o n theory
[61,62]
p o r t i o n of this phase d i a g r a m is shown in Fig.14.
. The l i q u i d - v a p o u r It resembles strongly
that of a classical Van der Waals system. As indicated in Fig.14, liquid state p e r t u r b a t i o n theory yields a spinodal line. This is due to the s e p a r a t i o n of the i n t e r a c t i o n p o t e n t i a l into hard core r e p u l s i o n and the remainder.
The hard core system acts as the r e f e r e n c e system,
the c o r r e c t i o n s are c a l c u l a t e d p e r t u r b a t i v e l y
[52]
. The m e a n - f i e l d
a p p r o x i m a t i o n to this m o d e l may be e x p r e s s e d by Eq.(I.2a),
for w h i c h
39 the Van der Waals equation of state is valid exactly.
1.0
I
I
I
I
0.9
~0.8 L_
0.7 ~o. E 0.6
Metastable region
0.5
0
0.2
0.4 Density
0.6
0.8
po 2
Fig.14: Phase diagram of a two-dimensional Lennard-Jones (From Ref. 55.) The computer
'experiments'
system.
of spinodal decomposition have been done by
quenching the system from an initially homogeneous configuration into the unstable region of the phase diagram
[55]
. As result of the mo-
lecular dynamics simulations one obtains the temporal changes of all atomic positions.
Examples for the atomic distributions of a 5~O41-
atoms system are shown in Fig. 15 for different times after the quench. The break-up of the originally homogeneous system into regions with different densities is clearly visible. The wavelike decomposition process starts immediately after the initialization.
It yields highly intercon-
nected regions, which are quite similar to the findings of the Monte Carlo simulations for the kinetic Ising model the Lennard-Jones
(see section II.Ib).
In
system, these regions separate into isolated clusters
after approximately 30-40 ps. The liquid density is quickly approached within the larger clusters.
They continue to grow by condensation of
monomers from the vapour phase. This causes the supersaturation to decrease and the smaller clusters to re-evaporate. realization of the condensation-evaporation
Thus, one has a good
mechanism
(see section
40
• ~.. k~-~.d~:,~.~ e~ ....; ~.-. ~-~:.'~.
-,- , ~ ~-,, ..~ 20 ps
•" - ' . , . ¢ . ~ - ~ . . ' ~ , : , ~ : . . "
Fig.15:
t
,Zo o.
/
Zo o.
S n a p s h o t p i c t u r e s of the atomic d i s t r i b u t i o n in a m o l e c u l a r d y n a m i c s s i m u l a t i o n of 5 0 4 1 L e n n a r d - J o n e s atoms. Each atom is s y m b o l i z e d by a dot. The time is given in ps after the quench. (From Ref.55.)
II.la). S t r i c t l y speaking,
the process of spinodal d e c o m p o s i t i o n
is
c o m p l e t e d by the time at w h i c h the liquid d e n s i t y is reached in the clusters.
A q u a l i t a t i v e analysis of the data requires c a l c u l a t i o n of the radial d i s t r i b u t i o n f u n c t i o n g(r,t).
The results for the different atomic con-
figurations of Fig.15 are p l o t t e d in Fig.16.
41
3,0~_ 1.51
5ps
3.0
10ps
6.0 ~ 2.0~
150ps
20ps
.~ 6.0
200ps
F
~4.5 "~
'
U.
6.0j+
lOOps
2.0~
t
I
g 2.o
~- 1 . 5 .9
e~
~ 5.0
i5
[~ ~ 1,5
~5 6.01
40ps ---[
~ 2.o
"
_
6.0 r
I
6.0
350ps
60ps
I 450ps
'
2 . 0 ~ 0
20
40
0
60 r/o
0
20
J
40
60
Fig.16: Sequence of radial distribution functions calculated from the atomic configurations shown in Fig.15. (From Ref. 55.) This figure shows that g(r,t)
contains two superimposed structures.
These are the short-range oscillations tion which has a growing amplitude.
for r<5o and a damped oscilla-
The small-wavelength
structure cha-
racterizes the atomic order in the high-density regions of the phaseseparating system. This short-range order becomes time invariant after the liquid density is reached inside the clusters. simulation,
In the discussed
this occurs around 30 ps after the quench.
42 The l o n g - w a v e l e n g t h oscillations, for all times,
on the other hand,
continue to grow
i n d i c a t i n g the i n c r e a s i n g order in the system due to
the e x p a n d i n g clusters.
It is w o r t h w i l e to isolate the l o n g - w a v e l e n g t h
o s c i l l a t i o n s from the full g(r,t). of the radial d i s t r i b u t i o n by a v e r a g i n g g(r,t)
This defines the cluster part G(r,t)
function. O p e r a t i o n a l l y ,
G(r,t)
is o b t a i n e d
over the w a v e l e n g t h of the short-range oscillations.
The average size R(t) of the g r o w i n g liquid d r o p l e t s can be e s t i m a t e d by the radius, distances
at w h i c h G(r,t)
(r
,
G(r,t)
firstly equals unity.
For smaller
is larger than unity, because inside the
clusters the p r o b a b i l i t y to find atomic pairs is increased. o t h e r hand, Thus,
On the
the g r o w i n g droplets are s u r r o u n d e d by a d e p l e t i o n zone.
G(r,t)<1
for
r
average cluster size
slightly larger than R(t)
R . In Fig.
17, the
is p l o t t e d as function of time on a double
l o g a r i t h m i c scale.
I
'
u ,i
I
'
'
'
I
'
'''I
'
'
'
'i -i
20tO.5
\
10
v'"
o/o"
O.O"i
l
q.e"
ooO ct"
0.0."
5 tO.2
-"
O/
O...O."
.-
2 I
5
,
,
,
, I
10
i
20
,
,
I
,
,
,
, l
50 1O0 time [ps]
,
,
200
Fig. 17: A v e r a g e cluster size R(t) c a l c u l a t e d from the coarse g r a i n e d radial d i s t r i b u t i o n function G(r,t) see text. (From Ref. 55.)
,
I
500
,
43
One
clearly recognizes two dynamic regions.
spinodal regime. It the cluster.
The e a r l i e r one is the
ends after the liquid density has been reached in
The second time regime is c h a r a c t e r i z e d by cluster growth I/2 law, w h i c h has been c a l c u l a t e d a s y m p t o t i c a l l y
a c c o r d i n g to the R=t
for the c o n d e n s a t i o n - e v a p o r a t i o n m e c h a n i s m
(see section II.la). This
t h e o r e t i c a l result is thus r e p r o d u c e d by the m o l e c u l a r dynamics computer simulations.
Two d i f f e r e n t dynamic regimes
of phase s e p a r a t i n g systems have also
been o b s e r v e d in Monte Carlo simulations of the k i n e t i c Ising model [35,47,63]
. Heermann et al. [63]
c h a r a c t e r i z e these regimes as
" c o m p a c t i f y i n g of the o r i g i n a l l y not compact clusters" "classical growth".
followed by
These findings are quite similar to the p r e s e n t e d
results for spinodal d e c o m p o s i t i o n in o n e - c o m p o n e n t systems. It is w o r t h w i l e to m e n t i o n at this point,
that the growth law in the
first dynamic regime is not yet well understood: the growth law R~t 0"2, i n d i c a t e d in Fig.17,
One should not take
too seriously.
It is not
clear, w h e t h e r there really is a simple p o w e r - l a w b e h a v i o u r at these short times. The numerical data are also c o m p a t i b l e w i t h other fits. A l s o the c r o s s o v e r time to the R~t I/2" growth regime has not yet been i n v e s t i g a t e d in detail. There are some indications
[60]
, that this
c r o s s o v e r shows no universal features.
On the other hand, the p h y s i c a l p r o p e r t i e s of the t I/2 regime are much b e t t e r understood.
A s s u m i n g that the density m o r p h o l o g y remains appro-
x i m a t e l y t i m e - i n v a r i a n t in this regime, one may p o s t u l a t e an u n i v e r s a l cluster radial d i s t r i b u t i o n function G(X)
G(X)=G(X,t)
T r u e l y u n i v e r s a l b e h a v i o u r of G(X)
,
X=r/R(t)
is realized,
(II.49)
if the complete dynam-
ics of the phase separating system is c o n t a i n e d in R(t). the results of the computer simulations accuracy,
a good scaling of G(X)
[55]
Fig. 18 shows
. W i t h i n the n u m e r i c a l
is found. Similar scaling b e h a v i o u r
has been o b t a i n e d in the Monte Carlo simulations of the k i n e t i c Ising system
[35,47]
. The common reason may be u n d e r s t o o d through a
simple model calculation
[55]
. In this model,
one c o n s i d e r s the
growth of one liquid droplet w i t h i n the v a p o u r phase.
The m o r p h o l o g y
of the total system is a p p r o x i m a t e d by a spherical droplet w h i c h is s u r r o u n d e d by a d e p l e t i o n zone of density zero. the d e p l e t i o n region, the mean density Pm
(See Fig.19.)
is assumed.
Outside
The d e n s i t y
44
1
I
I
2 X
3
1.6 1.4 ~1.2 1.0 O8
Fig.18:
-
0
1
4
Scaled radial d i s t r i b u t i o n function G(X) (Eq. II.49) for different times w i t h i n the R~tl/2 growth regime. The squares are the results of the m o l e c u l a r dynamics simulations and the full line is o b t a i n e d from a model c a l c u l a t i o n (see text). (From Ref.55.)
profile is given by
£ 0
p=
i~
RI
(II.50)
The two length scales R1 and R 2 are related because of particle conservation
Thus,
the complete dynamics of the system in c o n t a i n e d in R](t).
45
Fig.19:
Model for the c a l c u l a t i o n of the radial d i s t r i b u t i o n f u n c t i o n G(X). The inner circle of radius Rl(t) symbolizes the c l u s t e r of density p~ which is s u r r o u n d e d by a d e p l e t i o n zone of density 0 . Outside this zone the d e n s i t y approaches the m e a n value Pm"
By construction,
the r e s u l t i n g radial d i s t r i b u t i o n function is u n i v e r -
sal in the sense of Eq. (II.49).
It may be c a l c u l a t e d a n a l y t i c a l l y
[55].
The result is p l o t t e d in Fig.18 t o g e t h e r w i t h the scaled cluster dist r i b u t i o n function of the n u m e r i c a l simulations.
The m o d e l r e p r o d u c e s
the e s s e n t i a l features of the m o l e c u l a r d y n a m i c s results.
Still b e t t e r
a g r e e m e n t could be o b t a i n e d for a more r e a l i s t i c density profile.
II.Id Spinodal D e c o m p o s i t i o n as G e n e r a l i z e d N u c l e a t i o n T h e o r y
A f t e r the d i s c u s s i o n of spinodal d e c o m p o s i t i o n and n u c l e a t i o n as separate processes
in the previous chapters,
the f o l l o w i n g p a r a g r a p h s
are devoted to an attempt of a u n i f i e d d e s c r i p t i o n of phase s e p a r a t i o n dynamics. 64,65]
The t h e o r e t i c a l model is due to B i n d e r and coworkers
[33,44,
. It treats spinodal d e c o m p o s i t o n in the framework of a g e n e r a l -
ized n u c l e a t i o n theory. The basic assumptions may be s u m m a r i z e d as follows: diately inside the binodal line of the phase diagram, b a r r i e r to n u c l e a t i o n is r e l a t i v e l y high. Therefore,
In the area immethe e n e r g e t i c only few
48
unstable
fluctuations exist.
However,
these critical droplets may
g r o w almost u n r e s t r i c t e d l y t h r o u g h c o n d e n s a t i o n of m o n o m e r s out of the s u p e r s a t u r a t e d vapour phase. Towards the center of the phase coexistence region, decreasing,
on the other hand, the energy b a r r i e r is c o n t i n u o u s l y
causing an i n c r e a s i n g d e n s i t y of u n s t a b l e fluctuations.
The growth of these m a n y fluctuations
is r e s t r i c t e d because of the con-
servation of the total number of particles. A d e p l e t i o n zone develops around all the individual tiny clusters.
Under certain conditions this
may lead to the q u a s i p e r i o d i c pattern, w h i c h is c h a r a c t e r i s t i c
for spi-
nodal decomposition.
Binder and coworkers analyze this model w i t h i n a kinetic theory. The correct inclusion of the c o n s e r v a t i o n of p a r t i c l e s into the t h e o r e t i c a l c a l c u l a t i o n s is very essential. Otherwise,
the p e r t i n e n t m u t u a l growth
r e s t r i c t i o n of the clusters cannot be described.
The kinetic e q u a t i o n
for these processes is d d-~ An=
~ I n-1 ,[1Sn+n',n'An+n' -2n,_[_1Sn'n'An+ n = (II.51)
I n-1 + 2 n ' ~ 1 Cn-n''n'An'An-n'-n'~1Cn'n'An'An= Here, An(t)
is the c o n c e n t r a t i o n of clusters with n molecules.
first term on the r i g h t - h a n d side of Eq.
The
(II.51) d e s c r i b e s the increase
of clusters w i t h n atoms due to d i s s o c i a t i o n of large clusters (n+n') ÷ (n,n'). The second term is the p e r t i n e n t loss rate n').
(n) + (n-n',
The rate c o e f f i c i e n t s are d e n o t e d by S. The third and fourth
terms c h a r a c t e r i z e increase
(n',n-n') ÷ (n+n') and decrease
(n+n') of A n with the rate c o e f f i c i e n t s C, respectively.
(n,n') ÷
A s s u m p t i o n of
d e t a i l e d balance allows an e x p r e s s i o n of C in terms of S.
This model has been treated by B i n d e r and coworkers for m a n y p h y s i c a l situations.
They could show
[33,65]
, that with certain simplifica-
tions Eq. (II.51) may be reduced to the m a s t e r e q u a t i o n
of classical
n u c l e a t i o n theory or to the C a h n - C o o k theory of spinodal decomposition. For an a n a l y t i c a l t r e a t m e n t one t h e r e f o r e has to m a k e use of a d d i t i o n a l approximations.
A n u m e r i c a l solution of Eq. (II.51)
for spinodal decom-
p o s i t i o n of the k i n e t i c Ising model yields good a g r e e m e n t with the M o n t e Carlo simulations.
Furthermore,
it can be shown,
that this gene-
r a l i z e d n u c l e a t i o n theory is c o n s i s t e n t with the t I/3 growth law of the L i f s c h i t z - S l y o z o v analysis.
47
To summarize, ralization spinodal process
the s k e t c h e d
decomposition. of phase
croscopic
basis
It yields
independent
different
in Thin
presented
such that even
example
solid t r a n s i t i o n of s e c o n d - o r d e r established
Realistic
effects
melting
of nearly
films
the laws of atomic
tems
are easily m a n a g a b l e
sional
those phase systems,
no direct
incommensurate the lattice
Details
this book.
m a y be
from n u m e r i c a l These
are also k n o w n
transition
investiga-
phenomena. simulations
The
may
be
influ-
separately. of noble
gas
from t h r e e - d i m e n -
transitions
One example
solid phases
depending
substrate.
and the sys-
.
films may exhibit analogy.
films, sys-
the s i m p l e s t
simulations
situation.
and m e l t i n g
chapter
which
of s u b s t r a t e
of this
to the
These
on a g r a p h i t e
can be i n v e s t i g a t e d
for f r e e z i n g
between
. Among
of a d s o r p t i o n
system.
in the f o l l o w i n g
transition
liquid - i
m a y be thin
Such e x p e r i m e n t a l
two-dimensional
of f r e e d o m
physisorbed
[67]
of a v a r i e t y
transitions
by the
surfaces.
adsorbates
can be o b t a i n e d
results
surate or i n c o m m e n s u r a t e , meters.
of
a dimension
the p o s s i b i l i t y
are known quite well
of the p h y s i c a l
constants
not a feature
in c o n t r a s t
systems
on smooth
experimentally.
three-dimensional
systems,
in Dash's book
to study the strictly
ence of the third degree
Besides
to be
in three dimensions.
interaction
informations
are p r e s e n t e d
shown
transitions
predict
two-dimensional
tions have led to the d i s c o v e r y
films
studies
are rare gas
Here,
Some of the n u m e r i c a l
were
one expects
c o u l d be p r o v i d e d
or m o l e c u l e s
in detail
for r e a l i s t i c m o d e l s
is, however,
in t w o - d i m e n s i o n a l
of atoms
of such m o n o l a y e r
This
chapters
On the contrary,
. Theoretical
melting
tems are d i s c u s s e d
extended
of the
on a mi-
Films
the order of the phase
for these [66]
examples
Supplementary
theory
and
dimensions.
first-order
e.g. m o n o l a y e r s
for a gene-
of n u c l e a t i o n
understanding
a quantitative
in the p r e v i o u s
in general.
in d i f f e r e n t
A possible
qualitative
However,
of the d i m e n s i o n a l i t y .
transitions
dependence,
a
is an ansatz theories
is still missing.
of the results
phase
model
of the c l a s s i c a l
separation.
II.2 Phase T r a n s i t i o n s
Most
theoretical
and u n i f i c a t i o n
which
have
is the c o m m e n s u r a t e -
of the adsorbate.
and a d s o r b a t e
m a y be e i t h e r
on the t h e r m o d y n a m i c are d i s c u s s e d
commen-
control
in section
Here,
II.2b
paraof
48
II.2a The S o l i d - L i q u i d Phase T r a n s i t i o n
F r e e z i n g and m e l t i n g are physical effects which are w e l l - k n o w n daily life. A s t o n i s h i n g l y enough, these processes
from the
the theoretical u n d e r s t a n d i n g of
is by no means complete.
There is no g e n e r a l l y accep-
ted theory of m e l t i n g and even the t h e r m o d y n a m i c c l a s s i f i c a t i o n of the phase t r a n s i t i o n in t w o - d i m e n s i o n a l Undoubtedly, transition.
systems is somewhat controversal.
in three dimensions m e l t i n g takes place as a first-order However,
in c o m p a r i s o n to other f i r s t - o r d e r transitions it
exhibits only a relatively weak d i s c o n t i n u i t y of the order parameter. The m a g n i t u d e of the density d i f f e r e n c e between liquid and solid is m u c h smaller than that between liquid and gas. This weak d i s c o n t i n u i t y may be r e g a r d e d a s a hint, that p o s s i b l y the order of the m e l t i n g transition becomes continuous in lower dimensions.
A theory of
the s o l i d - l i q u i d t r a n s i t i o n in a t w o - d i m e n s i o n a l
has been p r o p o s e d within the s o - c a l l e d d i s l o c a t i o n model. emphasized
is
system
Particularly
the role of t o p o l o g i c a l defects for the m e l t i n g of two-
d i m e n s i o n a l solids. The basic ideas of this approach are due to Kosterlitz and Thouless
[68]
and Feynman
[69]
. The d e t a i l e d theory of
d e f e c t - m e d i a t e d m e l t i n g has been w o r k e d out m a i n l y by H a l p e r i n and Nelson
[.70,71]
and by Young
ory is the possibility,
[72]
. An important result of this the-
that in two d i m e n s i o n s the t r a n s i t i o n from
solid to liquid m a y take place by two s e c o n d - o r d e r phase transitions.
With i n c r e a s i n g t e m p e r a t u r e T, at a value T=T~
, d i s s o c i a t i o n of dis-
location pairs takes place and free d i s l o c a t i o n s appear in the crystal. Thus, new,
the resistance to shear is lost and a continuous t r a n s i t i o n to a liquid-like phase occurs. H a l p e r i n and Nelson
[71]
show, that
this phase is c h a r a c t e r i z e d by e x p o n e n t i a l l y d e c a y i n g spatial correlations but a l g e b r a i c decay of the o r i e n t a t i o n a l order. These features are typical for some liquid crystals. lattice with s i x - f o l d anisotropy,
In the case of a t r i a n g u l a r
the authors call this new phase the
"hexatic phase". With i n c r e a s i n g temperature,
this hexatic phase trans-
forms c o n t i n u o u s l y into an o r d i n a r y fluid due to the appearance of u n b o u n d disclinations.
H a l p e r i n and N e l s o n m e n t i o n in their paper
that these two stages of melting, possibility. tion,
solid÷hexatic÷liquid,
[71],
are only one
They cannot rule out, that a direct d i s c o n t i n u o u s transi-
solid÷liquid,
Occurs at a lower temperature.
49
Nevertheless, much
the p o s s i b i l i t y
attention
perimental, conference
theoretical,
those
laboratory have
exposed more
experiments
layers.
gas atoms
tion of a d s o r p t i o n therms
F o r the e x p e r i m e n t a l
layers
of the system
simulations
up of noble
increasing
can be seen as steps
[77]
. An example
in recent
melting
nearly
on a two-di-
surface
gas-pressure The
some of
theory.
gas atoms
a graphite
to the surface.
ex-
w i l l be discussed,
forming
studies
With
become b o u n d
important
paragraphs,
of the t o p o l o g i c a l
is made
c a u s e d very
of the
are s u m m a r i z e d
are p h y s i s o r b e d ,
to a rare gas atmosphere.
noble
studi~s
and c o m p u t e r
system
The atoms
of m a t t e r Many
. In the f o l l o w i n g
as tests
physical
substrate.
mensional
[75,76]
been p e r f o r m e d
investigated
graphi t e
community.
and n u m e r i c a l
proceedings
which
The
of a new phase
in the s c i e n t i f i c
more
successive
in the a d s o r p t i o n
of such an i s o t h e r m
is and
formaiso-
is shown
in
Fig.2©.
[(9
~C(3D)
plPo
{
3
,8
;/
/
i
©
2
O
I
1
'
~
O, 1
~
© (3)
0
~
~
121
0
Fig.2©:
S c h e m a t i c a l e x p l a n a t i o n of the steps in an a d s o r p t i o n isotherm. 8 is the c o v e r a g e in units of m o n o l a y e r s and P0 is a r e f e r e n c e pressure. (I) t w o - d i m e n s i o n a l gas, (2) t w o - d i m e n s i o n al c o n d e n s e d phase, (3) t h r e e - d i m e n s i o n a l gas. All these p h a s e s are in m u t u a l e q u i l i b r i u m , the t w o - d i m e n s i o n a l p h a s e s are phys i s o r b e d on a substrate. (From Ref.77.)
The very steep parts of the i s o t h e r m s tional n o b l e
gas layers.
The a l m o s t
indicate flat parts
the a d s o r p t i o n
of addi-
are due to a d s o r p t i o n
50
of individual atomic
transitions. Refs.
atoms
density
into an e x i s t i n g
layer.
in the layer increases,
Surveys
78 and 79.
of all the o b s e r v e d
In the p r e s e n t
book,
the t w o - d i m e n s i o n a l causing
transitions
transition
and the c o m m e n s u r a t e - i n c o m m e n s u r a t e
various
in
will be discuss-
for x e n o n
transition
phase
can be found
only two e x a m p l e s
ed in some detail. These are the m e l t i n g phite
Thus,
eventually
on gra-
for krypton
on
graphite.
In both
cases
is adsorbed. Xenon
one p r e p a r e s
the system such that e s s e n t i a l l y
This m o n o l a y e r
on g r a p h i t e
exhibits
is a r e l a t i v e l y
in two dimensions.
The c r y s t a l l i n e
a lattice
which
strate. the next
constant
Details chapter
of the g r a p h i t e (II.2b).
t h e phase
surface
i n t e r a c t i o n only gives the atoms
with
the i n t e r a c t i o n s
for the phase d i a g r a m
constructed
by a d e t a i l e d
of such
important
adsorbate.
to an e x t e r n a l It does
not
in
to note,
gas i n t e r a c t i o n
of a p h y s i s o r b e d
analysis
investigations
130~,
rise
sub-
in-
The
sub-
field, interfere
layer.
An e s t i m a t e
sults
it is only
in form of a layer.
in that
film has
will be d i s c u s s e d
of an i n c o m m e n s u r a t e
strate-adsorbate
xenon
to that of the g r a p h i t e
only the d i r e c t noble
transitions
features.
for a study of m e l t i n g
structure
At this point,
which
constrains
two-dimensional
state of the a d s o r b e d
is i n c o m m e n s u r a t e
that to a good a p p r o x i m a t i o n fluenc e s
almost
good c a n d i d a t e
one layer
are
I
xenon
of the a d s o r p t i o n
summarized
I
film m a y be
isotherms.
The re-
in Fig.21.
I
Xenon on Graphite 120
?
s
E I00
90
Ll®Ld
'loIsIl l.tl~i t 1.0 Coverage
Fig.21:
1.5
2.0
Phase d i a g r a m for xenon on graphite. The o r i g i n a l data are from e x p e r i m e n t s of T h o m y and Duval [79] . The results for 1.5 to 2 m o n o l a y e r s are extrapolated. (From Ref. 80.)
51
More
informations
on the nature
system m a y be o b t a i n e d [80-84] ture
. Here,
A narrow This
one m e a s u r e s
factor S ( k , t = ~ ) = S ( k ) .
formations
structure
liquids
the first B r a g g
characterizes
solid
state.
structure
(triangular
r e c t l y yields dynamic
control
Especially,
lattice
parameters
phase
o c c u r continuously,
tion of h i g h e r
order,
nuous
These
changes.
in c o n t r a s t
of kma x diof the thermo-
of the s t r u c t u r e
factor.
in a c h a r a t e r i s t i c
to a s i t u a t i o n
[85]. for
lattice
Changes
one o b s e r v e s
are c h a r a c t e r i s t i c
length
the v a l u e
density.
are r e f l e c t e d
in-
are found
w i t h known
gases),
atomic
struc-
yield
state of the adsorbate.
structures
adsorbate
cause v a r i a t i o n s
transitions
If these v a r i a t i o n s
peak of the static
a large c o h e r e n c e
for noble
the t w o - d i m e n s i o n a l
experiments
kma x and l i n e s h a p e
Broader
or gases . F o r a c r y s t a l l i n e
in this
diffraction
and the t h e r m o d y n a m i c
factor
for the
x-ray
Peak p o s i t i o n
on the d e n s i t y
is typical
of the phase t r a n s i t i o n s
from s y n c h r o t r o n
manner.
a phase t r a n s f o r m a -
with
almost
for a f i r s t - o r d e r
disconti-
transition,
respectively.
In Refs.
80 and 83 the authors
ing f i r s t - o r d e r more
recent
reported
coverages
A series this
findings
of c o m p u t e r
pological
d ynami c s
[74,86,88]
ition.
Nevertheless, w h i c h were
melting
to
two-dimensional
for the to-
systems
with differ-
. Molecular
model
of xenon
investigations is a f i r s t - o r d e r
for the xenon
some of the e x p e r i m e n t a l
come
on
to the
transition
film on the gra-
a discontinuous
interpreted
investigate
tests
[74,86-88]
for a r e a l i s t i c
ex-
for small
performed
As i n d e p e n d e n t
simulated
also the m o d e l
. These
transition
coverages.
in two d i m e n s i o n s
previously
[82,84]
are
solid-liquid
findings
trans-
could be repro-
as i n d i c a t i o n s
of c o n t i n u o u s
[89]
One of the b a s i c ably also
reasons
for this
for the d i f f e r e n t
lations
[87])
is found
tigated
system.
Working
perature
strictly
is found to exhibit
duced,
show-
However,
for h i g h e r
. M o s t of these n u m e r i c a l
. Moreover,
substrate
melting
has been
behaviour.
have been
that m e l t i n g
results
on graphite.
a first-order order
have been o b t a i n e d
[88,89]
conclusion,
to second
theory,
xenon
continuous
simulations
interaction
results
graphi t e
with
diffraction
c o v e r a g e s of about one m o n o l a y e r
seem to indicate
crossover
melting
ent laws of
phite
for h i g h e r
changing
interesting
x-ray
for s u b - m o m o l a y e r
experiments
to be c o n s i s t e n t
perimental xenon
melting
present
does not a l l o w
seemingly
interpretation
discrepancy
(and most
prob-
of some other
computer
simu-
in the t h e r m o d y n a m i c under
conditions
for t w o - p h a s e
constraints
of c o n s t a n t
coexistence,
of the
pressure
in c o n t r a s t
inves-
and temto the
52
situation of c o n s t a n t density and temperature.
This is important,
be-
cause a system with l i q u i d - s o l i d coexistence may show a p p a r e n t l y continuous changes of the atomic structure factor,
simply because changes
of the control p a r a m e t e r s cause d i f f e r e n t p e r c e n t a g e s of liquid to solid. Nevertheless,
In Ref.
the phase t r a n s i t i o n is of first order.
74, a very d e t a i l e d d i s c u s s i o n of this i n t e r p r e t a t i o n p r o b l e m
is p r e s e n t e d for the example of c o m p u t e r simulations for a two-dimensional L e n n a r d - J o n e s here.
(LJ) system. This d i s c u s s i o n will not be repeated
In the following paragraphs only some examples of the m o l e c u l a r
dynamics results are shown, w h i c h lead to the u n a m b i g u o u s conclusion, that m e l t i n g in a LJ system and in the m o d e l
for xenon on g r a p h i t e
is
indeed a f i r s t - o r d e r phase transition.
For reference,
in Fig.22 the complete phase d i a g r a m of a t w o - d i m e n s i o n -
al LJ system is r e p r o d u c e d
[62]
solid
~j
~J
10" tn
10.2
0.3
Fig.22:
0.4 0.5 06 Temperolure klle
0
0.5 Densily ~ 2
10
Phase d i a g r a m of a t w o - d i m e n s i o n a l L e n n a r d - J o n e s system in the p r e s s u r e / t e m p e r a t u r e plane and in the d e n s i t y / t e m p e r a t u r e plane, respectively. (From Ref. 62.)
In the Monte Carlo
(MC) or m o l e c u l a r dynamics
(MD) simulations of melt-
ing, the initial system is always p r e p a r e d in an e q u i l i b r a t e d homogeneous
state. Then a stepwise v a r i a t i o n of either pressure or tem-
p e r a t u r e is performed. At each step, the s y s t e m is e q u i l i b r a t e d again. Thus,
one a c c o m p l i s h e s various cuts through the p / T - p h a s e diagram.
The
o b s e r v e d changes of density and free e n t h a l p y clearly indicate the order of the phase transition. An example for the results of MC simula-
53
tions for a two-dimensional
LJ system is shown in Fig.23.
O.9~-"-"~"""-'---._
-4,0
[ Density-~ - ' ~
¢~o 0'8 t
t~. 1Melt -3.0 -1-
0.6
o
-2.o
0.4[ Pressure, po2/e = 0.05 0.2
0.3
0.4
\
0.5
Temperature, kT/e
Fig,
23: Equilibrium density and free enthalpy per atom yersus temperature, Results of the Monte Carlo simulations of a two-dimens~onal Lennard-Jones system w~th 256 atoms, periodic boundary conditions, and constant pressure. (From Ref. 74.9
One recognizes the characteristics of a first-order phase transition: discontinuous variation of the order parameter and hysteresis. These MC results have been reproduced by MD simulations tems in Refs. 74 and 88. Moreover, tuitive
illustrations
nerates
so-called
for much larger sys-
the MD calculations
of the dynamics
allow very in-
of the melting process.
the atomic trajectories
for certain time intervals.
state, the atoms vibrate around their respective yields the snapshot picture
In the crystalline
lattice sites. This
shown in Fig.24b.
.,,:..*..,-...:.,--,-,.-,..,, ....
....
• ........ .,....,
Fig.24:
One ge-
snapshot pictures of the system by simply plotting
b)
Trajectory pictures of a 576 atom system: a) for the liquid state, b) for the solid state.
54
The atoms of a liquid are not bound to lattice sites. On the contrary, they have a relatively high mobility which causes long trajectories (Fig.24a). The darker appearance of Fig.24a in comparison to Fig.24b does not imply a higher density but only a higher mobility of the atoms. The temporal evolution of the density during the liquid+solid transition is shown in Fig.25.
(N
0.88 0.84
f-
C~
0.80
0
10000 20000 Ti me Steps
30000
Fig.25: Temporal variation of the density in the liquid÷solid transition of a two-dimensional Lennard-Jones system. Result of molecular dynamics simulations of a 576 atom system with periodic boundary condition. One timestep corresponds to 0. O1 ps, po2/e=2.6 , kBT/e=0.7. The shown behaviour of the density is a typical example for the variation of the order parameter in a first-order phase transition. (From Ref. 88.)
Initially the density fluctuates around the value of the liquid phase. The system remains in this metastable state until a sufficiently large heterophase fluctuation occurs. In the present case, this heterophase fluctuation is the formation of a super-critical crystallite causing nucleation of the system into the solid phase. Quite similar behaviour can be observed during melting
(see Fig.26). Fig.26 shows a temporal
sequence of trajectory pictures for the melting of an over-heated crystal. At the early times, one recognizes the spontaneous build-up and decay of premelted regions. This causes temporal fluctuations of the density around the solid value. Only after approximately 8,000 time steps, a super-critical heterophase fluctuation occurs and the total system melts.
To summarize, it has been shown by the computer simulations that in the two-dimensional LJ system the solid+liquid transition is of first order. No indications for a 'hexatic phase' are found.
These investigations have been extended in Refs. 88 and 89. Here, the authors simulate the realistic system xenon on graphite. To construct a computer model, one needs an expression for the total potential
55
.C< Z << X'Y'F:.F:'FZ-;vFZ .'FD;<'};';'X+;';';,;;"7.','-FD
:::::::::::::::::::::::::::::::::: ================================
.:.....•.,.~"~:.......-........ 2000 .v v...::c., v.'::.v, v.v:
::':':';:"-"~"
6000
,
00
.'.'.'<.FD"; FFFFFFD:--.v.Z-D"
================:==,::::-:::::.. ====== :.:.:.:.:.:.. .:..:.:.:..' .:.:.:..:.:.:.:,;.:.:.:.:.: :.:.:.::::. - ::.:.:.:.:: -~
..::i:i$i:!:f:i:i:!~:~!"
:::::::::::::::::::::::::::::::::::::::::::3000
<..v'.~.:: ~.":v.~:-.,..w
..:";~;~..:-~i:~:" :::::::::::::::::::::::::: 7o0o ~ o
oo
•~
O0
..:.X.Y.Z':.';.:.:::..:-.:'k'.DF::
.-i:i:~:i:i:i~:~!$i:i:i::::" .:~,~"'~:~:::::.....
....,.:,'t.':;:; .......
, "..V.:v.v:
.::i!i:!::"!ili!.~i~i,=~::...::'::::::
.'~:<:::,:'.:v..:;::::~v.... 4000
0 z)~;-~';~.~
,': ::;:~:::-.',..::~, ~-.-:.....'-'-':::::::::::
/FD::':':F[.:<.'.''~.~.'.<.[.Dy
~
~
"-
~!i:.iii::::::" ~ii"
....:~" .::ii:::N~..' -. ..:.'..:.:. ~ : : i : : : ~00
~
0
0
,30 O0
Fig. 26: Trajectory pictures showing the melting of a twodimensional Lennard-Jones solid at constant temperature and constant pressure (po2/~ =0.05, kBT/~= o.45). The individual snapshots show the atomic configurations of a system with 576 atoms for different times after the onset of melting. The respective time is indicated at the lower right corner of each snapshot picture. It is given in units of time steps of the molecular dynamics simulation. One time step corresponds to o.oi ps. (From Ref. 74.)
56 energy E for a given configuration above the graphite infinite
surface.
of N noble gas atoms ri, i=1,2,...N,
The graphite
crystal of carbon atoms Rj,
is assumed to be pair-wise
9=1,2 . . . .
as semi-
The total interaction
additive.
E=Eng+E s where the potential
solid is represented
,
(II.52)
energy due to the mutual noble gas atom interaction
is
N Eng=.[.~ng(Iri-rjl) 1,3 i>j
and the energy contribution
(II.53)
due to noble gas - graphite
interaction
can be written as N
~
~
(II.54)
ES= [ [ #ng_c(Iri-Rji) i 9 {ng} { c } Here,
~ng and ~ng-c are the Lennard-Jones
potentials
(II.48) between
two noble gas atoms and between a noble gas atom and a carbon atom, respectively. At the investigated temperatures, one m a y assume a rigid graphite
solid.
graphite
lattice
has shown
This approximation is considerably
[90]
that the potential
as a Fourier-series the graphite
in the plane parallel
z i denotes
311 [E0(zi)+Ej(zi)f9(~li'~2i)l'=
(II.55)
the distance of a noble gas atom from the surface and of the position vector parallel
Due to the rapid convergence
of the Fourier
series,
order term, ~E 2 , is already two orders of m a g n i t u d e first-order
to
surface
~I, ~2 are the components face.
Steele
~ng-c can be expressed analytically
in the position variables
N Es=i=1 [ Here,
is justified by the fact that the
stiffer than the adsorbed film.
term.
Therefore,
one usually restricts
to this surthe second
smaller than the
the calculations
to
the leading order of the potential N
Es This truncation Ref.
[ [ E 0 (zi~-E l(Zi) f i(~1i, ~2i) I i=I
is not essential
in the numerical
(II.56) simulations.
88, the full potential has been taken into account,
ing notable differences
to the results using eq. (II.56).
E.g.
without
in
find-
57
In
Eq. (II.56)
fl({l,~2)= - 2 { c o s 2 ~ { l + c o s 2 n { 2 + c o s 2 ~ ( ~ l + ~ 2 ) ) The e x p l i c i t 2.34
expressions
for E0(z), El(Z)
(II.57)
are given in Ref. 90 in Eqs.
and 2.35, respectively. W i t h e m p i r i c a l l y d e t e r m i n e d p o t e n t i a l
parameters (II.56)
Eng , ~ng and Eng_c,and Ong_c , Eqs. (II.52),
(II.53), and
constitute the n u m e r i c a l m o d e l for noble gas films on a gra-
phite substrate.
In the f o l l o w i n g p a r a g r a p h s some of the s i m u l a t i o n results for x e n o n on g r a p h i t e will be d i s c u s s e d
[88,89]
. Fig.27 shows the g e o m e t r y of the
c o m p u t a t i o n a l cell.
X" Fig.27:
G e o m e t r y of the c o m p u t a t i o n a l box for the s i m u l a t i o n of noble gas films on graphite. The rare gas atoms are in the volume V=XL.YL.ZL. The surface of the s e m i - i n f i n i t e graphite substrate defines the x - y plane at z=0. P e r i o d i c b o u n d a r y conditions are assumed in x and y direction. (From Ref. 88.)
In the simulations,
the xenon atoms are free to move in a p a r a l l e l o g r a m
compatible not only w i t h the t r i a n g u l a r lattice of a t w o - d i m e n s i o n a l noble gas solid but also with the g r a p h i t e lattice.
Therefore,
the
linear d i m e n s i o n s XL, YL have to be choosen as YL= I / T XL
(II.58)
and XL=m ~ respectively.
aG
,
(II.59)
Here, m is an integer and a G is the g r a p h i t e lattice
58
constant.
For the almost two-dimensional p in the three-dimensional parallel
to the graphite
noble gas films,
the ordinary pressur e
systems has to be replaced by the pressure Pll
surface
[91,92]
. This quantity plays a role
similar to the surface tension in three-dimensional ternal surface A . Pll is calculated n u m e r i c a l l y virial theorem [93]
=NkBT
_ I
:
N
(xi-x~)2
Z
'
i
1,3>i
+(Yi-YJ
)2
(ri-rj)2
Here, < ..... > denotes configurational placed by temporal averaging.
systems with an in-
by applying the
a%(ri_rj )
>
"
(II. 60)
a(ri-rj ) averaging,
which may be re-
f A
0.2
%
~, > - .
/
/////
o.o 0.0
Fig.28:
0
z/o
2.0
Number density versus normal distance z from the graphite surface. Results of m o l e c u l a r dynamics simulations for xenon on graphite. With increasing parallel pressure, PH , successively a second adsorption layer is formed. The constant temperature in the simulations is kBT/e=0.7 . (From Ref. 88,)
In the numerical the formation
simulations
for xenon on graphite
of a first and a second adsorption
[88]
one observes
layer , depending
o n the value of the applied pressure Pll . The results are summarized
59
in Fig.28. The two peaks in the density distribution function define the two layers at z=z m and z=2Zm, respectively.
They are separated by
a broad region with almost zero density. Thus, one may unambiguously separate the properties of both layers. With increasing pressure, more and more atoms are promoted into the second layer. However,
for the
investigated values of Pll one always observes a low density fluid state in this layer
(see Fig~29).
Piio2/e = 2.0 ~"
-,"
"~,
Piio2/e = 2.6 v-
-,
-
_.~-
Piio2/e = 2.2
Piio'2/e = 2.8
Piio2/e = 2.4
Piio2/e = 3.0
Fig.29: Trajectory pictures of the atoms in the second adsorption layer for the conditions shown in Fig.28. (From Ref. 88.) The liquid to solid transition takes place in the first adsorption layer. The temporal evolution of this phase change is shown in Fig.30. Again, the heterophase
fluctuations preceeding the phase transition
are visible in form of small crystallites within the liquid. The pertinent temporal variation of the density is completely analogous to
60
'-'-'.-:-'-4 &~:.'--:' ~r+ ,
•
2000
~:~:5:ii:i:{:~:{:~:i':!"
,.jy
•
========================
~
t
~ff=4000 ========================== t= ,oooo •,:.:: ::. ,: DUUU
,,'.-.-:.:,:.-..'.:.:,.,.
• ..,.-..,.,,
.,,.. ....
,,
_
t -
12000
Fig.30: T r a j e c t o r y pictures for the atoms in the first a d s o r p t i o n layer. The i n d i c a t e d times are in units of time steps of the m o l e c u l a r d y n a m i c s simulations (1 time step ~O.O5 ps). At t=0 the p r e s s u r e has b e e n increased from Pl ~ 2 / E = 2 . 5 to 2.6. At the lower pressure, the system was in the state of a homogeneous liquid, kBT/e=O.7. (From Ref. 88.)
that of the t w o - d i m e n s i o n a l
system
(see Fig.25).
The results of c o n s t a n t p r e s s u r e simulations for v a r i o u s t e m p e r a t u r e s are s u m m a r i z e d in Fig.31. constant density situation
They are c o m p a r e d to MD findings for a [89b]
. The constant p r e s s u r e results
clearly show d i s c o n t i n u o u s v a r i a t i o n of the density, w h i c h is indicative for a f i r s t - o r d e r transition.
Under c o n s t a n t d e n s i t y conditions,
the m e a n d e n s i t y of the first layer varies continuously,
because one
crosses the liquid - solid c o e x i s t e n c e region.
To summarize,
the d i s c u s s e d c o m p u t e r simulations c l e a r l y show that the
m e l t i n g t r a n s i t i o n in the n u m e r i c a l m o d e l systems is of first order. Even t h o u g h some results of the l a b o r a t o r y e x p e r i m e n t s
[82] may be
61
reproduced quantitatively
[89b]
, there is still a c o n t r o v e r s y con-
cerning the i n t e r p r e t a t i o n of these findings. of some unknown reason, h e x a t i c phase
[93]
It could be, that b e c a u s e
the s i m u l a t i o n techniques cannot r e p r o d u c e the
. This point has to be i n v e s t i g a t e d in future
studies.
0.92
i
i
i
i
i
• Conslont Aree end Coverege • Conslont Pressure end Eoverage
, ~ 0.90
~' 0.88,
"\
u= 0.86
\•
-~ 0.84
i
'-" 0.82 I
0.62
I
I
I
I
0.64 0.66 Temperoture kT/~
0.68
Fig.31: V a r i a t i o n of the p r o j e c t e d t w o - d i m e n s i o n a l d e n s i t y in the first a d s o r p t i o n layer. Results of the m o l e c u l a r d y n a m i c s s i m u l a t i o n s for the n u m e r i c a l model of x e n o n on graphite. The dots are obtained for constant d e n s i t y s i t u a t i o n s and the t r i a n g l e s are results under c o n s t a n t p r e s s u r e conditions, respectively. (From Ref. 89b.)
A n o t h e r not c o m p l e t e l y u n d e r s t o o d feature is the role of the s u b s t r a t e in the e x p e r i m e n t a l findings. This has been studied, niques,
in Ref.
again by MD tech-
89a. Here, m e l t i n g of s u b m o n o l a y e r xenon, k r y p t o n and
argon has been simulated. W i t h o u t substrate, all three systems are c o m p l e t e l y identical, noble gas p a r a m e t e r s
the n u m e r i c a l m o d e l s of b e c a u s e the r e s p e c t i v e
~ng' ~ng m a y be e l i m i D a t e d by proper scaling of
the t h e r m o d y n a m i c variables. noble gases on graphite.
However,
this is no longer true for the
The a d d i t i o n a l p a r a m e t e r s
e x p l i c i t l y in the calculations.
~ng-c'
~ng-c appear
T h e y are the basic reason for the
quite d i f f e r e n t b e h a v i o u r of the three systems as soon as they are phys i s o r b e d onto the substrate.
A solid krypton film,
for example, may b e c o m e c o m m e n s u r a t e to the
lattice of the g r a p h i t e surface.
This gives rise to the c o m m e n s u r a t e -
incor~mensurate t r a n s i t i o n d i s c u s s e d in the f o l l o w i n g c h a p t e r of this book.
The n u m e r i c a l model of the argon film, on the other hand,
shows
d i s c o n t i n u o u s m e l t i n g w i t h o u t substrate, but an a p p a r e n t l y c o n t i n u o u s
82
t r a n s i t i o n after p h y s i s o r p t i o n onto graphite
[89a]
. The c o n f i g u r a -
tions of the argon adsorbate are found to be strongly influenced by the p r e s e n c e of the substrate. Therefore,
one has to conclude that m e l t i n g
argon on graphite is no proper example for truely t w o - d i m e n s i o n a l behaviour.
II.2b The C o m m e n s u r a t e - I n c o m m e n s u r a t e
Transition
Two length scales are c o m p e t i n g in the compound s y s t e m of a noble gas adsorbate on a graphite substrate. One of it is due to the atomic i n t e r a c t i o n between the rare gas atoms. This defines the e q u i l i b r i u m lattice constant of the t w o - d i m e n s i o n a l
crystal. The second length
scale is p r o v i d e d by the p e r i o d i c v a r i a t i o n of the substrate p o t e n t i a l p a r a l l e l to the g r a p h i t e surface. The amplitude of this v a r i a t i o n is c o n s i d e r a b l y smaller than that of the i n t e r a t o m i c rare gas p o t e n t i a l [90]
. If the two length scales are incommensurate,
the p r o p e r t i e s of
the adsorbate are d o m i n a n t l y d e t e r m i n e d by the direct noble gas interaction. An example for this s i t u a t i o n is the s y s t e m xenon on graphite, w h i c h has been d i s c u s s e d in the p r e v i o u s chapter.
The present chapter deals with a system, can become commensurate.
This is r e a l i z e d for k r y p t o n on graphite.
a certain range of coverages,
In
one observes the o c c u r a n c e of a complete-
ly r e g i s t e r e d k r y p t o n adsorbate, W i t h increasing coverage,
the s o - c a l l e d c o m m e n s u r a t e phase.
a t r a n s i t i o n to the i n c o m m e n s u r a t e situation
/....,../...,../
takes place.
js
/ 11
Pig.:32:
in w h i c h both length scales
/
t
i/
•
i /
'--I z " / / "-.,/ / /
Atomic structure of a graphite surface. Four unit ceils are plotted w i t h two carbon atoms per cell. The center of the d a s h e d hexagon is a possible a d s o r p t i o n site.
TO discuss details of this c o m m e n s u r a t e - i n c o m m e n s u r a t e
transition,
it
63
is important to consider the geometrical properties of the graphite surface lattice. Four graphite unit cells are shown schematically in Fig.32. Each graphite unit cell contains two carbon atoms. The lattice of these carbon atoms has hexagonal symmetry. This is indicated by the dashed line in Fig.32. The centers of the hexagons are the energetically most favourable positions
for a physisorbed atom. The graphite sur-
face may be regarded as lattice of possible adsorption sites. This is shown in Fig.33.
Fig.33: Lattice of the possible adsorption sites on a graphite surface (hexagons). Physisorbed krypton atoms are symbolized by circles. An incommensurate domain wall is indicated, which connects two commensurate regions A and B (see text). (From Ref. 94.) The equilibrium distance between two krypton atoms is roughly 4% smaller than /~ times the distance between two possible adsorption sites [90] . If the crystalline krypton film is rotated by 30 ° relative
to
the base line of the graphite unit cell, the system gains energy by slightly expanding the distance between the krypton atoms. At certain coyerages,
this allows all krypton atoms to be at adsorption
sites,
forming the so-called /~ x /3 R 30-commensurate phase. Here, one third
64
of all adsorption sites is occupied by krypton atoms,
leading to three
energetically degenerate ground states of the total system. The pertinent sublattices are denoted by A, B, and C, respectively (see Fig.33). By increasing the coverage above one commensurate monolayer,
it is no
longer possible that all krypton atoms are at commensurate positions. The system becomes more and more incommensurate,
until it eventually
forms a uniform solid which is almost entirely determined by the krypton-krypton interaction potential. Many details of this commensurateincommensurate transitions have been studied extensively in laboratory experiments
[78-80,95-99~
of computer simulations
I
13o A
1zo
11o
E
, theoretically
[~05-IO9]
I
"~'
[100-104,94]
and by means
.
I
]
I
]
Krypton on 5rophite
/ Flu,d
9O 80
OZ
Fig.34:
0.4
06 08 1.0 I.Z C . . . . . . . /commensurotel uv~,uy~ ~.monolayerS /
t4
I6
Phase diagram for krypton on graphite. Most of the experimental data are from Ref.99. The results for coverages above one monolayer are extrapolated. (From Ref. 80.)
A p h a s e d i a g r a m for krypton on graphite is reproduced in Fig.34. It has been constructed ments of Ref.99.
[80] utilizing the detailed specific-heat measure-
Some of the details in this phase diagram have been ob-
tained simply extrapolating existing data and may have to be modified in the future. In the following paragraphs,
recent investigations of the
transition region between the commensurate and the incommensurate phase will be summarized.
Very-high resolution synchrotron x-ray scattering experiments have been performed for coverages around one commensurate monolayer
[98]
. The
observed diffraction profiles show a somewhat unexpected behaviour (see Fig.35).
65
, , ,
0.05 E
i
I
i •
0.04
,
,
,
Experiment
E 13 S i m u l a t i o n
x
/H~\\
0.03 w--
"1-
O. 0 2
I
"o
.--
\\
I I
\
0.01
q-
/
cO
"1-
0
I
m,
\
,
I
1.71
i
I q 1.75
I
I
I
1.80
kmax [ ~ - 1 ]
Fig.35: W i d t h of the first B r a g g - p e a k of the atomic s t r u c t u r e factor for krypton on g r a p h i t e at coverages s l i g t h l y above one c o m m e n s u r a t e monolayer. (From Ref. 106 .)
The first B r a g g - p e a k has been found to b r o a d e n c o n s i d e r a b l y for coverages s l i g t h l y above one c o m m e n s u r a t e m o n o l a y e r
[98]
. This was regard-
ed as a hint for the p o s s i b l e e x i s t e n c e of a w e l l - c o r r e l a t e d
fluid
phase between the solid c o m m e n s u r a t e and i n c o m m e n s u r a t e phases. However, no
detailed
informations on the p e r t i n e n t m i c r o s t r u c t u r e of the kryp-
ton film could be o b t a i n e d from the laboratory experiments. has been closed by c o m p u t e r s i m u l a t i o n s not only to reproduce the e x p e r i m e n t a l
[106-109] findings
This gap
. It is now p o s s i b l e
[106]
, but also to
identify the atomic structure of the w e a k l y i n c o m m e n s u r a t e phase.
The n u m e r i c a l model is exactly identical to that of the p r e v i o u s chapter, only the xenon p a r a m e t e r s are r e p l a c e d by the r e s p e c t i v e krypton values. Ref.
The simulation results for the e x p e r i m e n t a l c o n d i t i o n s of
98 are
plotted
in Fig.35.
simulation and experiment.
Quite good a g r e e m e n t is o b t a i n e d between
The m i c r o s t r u c t u r e of the w e a k l y incommensu-
rate k r y p t o n layer in the n u m e r i c a l m o d e l
is
found to be a c o e x i s t e n c e
of c o m m e n s u r a t e regions s e p a r a t e d by a network of i n c o m m e n s u r a t e domain walls
(see Fig.36).
The concept of domain walls has been i n t r o d u c e d p r e v i o u s l y in theoretical m o d e l c a l c u l a t i o n s for the zero t e m p e r a t u r e state of the incomm e n s u r a t e phase
[1OO-104]
. For k r y p t o n on graphite,
assumed to consist of an array of h e x a g o n a l l y
this phase was
shaped c o m m e n s u r a t e
66
regions. This "honeycomb structure" n e e d not be a very regular one b e c a u s e of the p o s s i b i l i t y to reshape the honeycombs by "breathing" [100,101]
. The b r e a t h i n g has to occur w i t h o u t changing the total length
or the number of i n t e r s e c t i o n s of the i n c o m m e n s u r a t e domain walls.
22212 Krypton A
Fig.36:
.
S n a p s h o t picture of the i n c o m m e m s u r a t e krypton atoms (shown as dots) for a 2 2 , 2 1 2 - k r - a t o m s y s t e m on graphite. The effective coverage in the first layer is 1.O23, the t e m p e r a t u r e is 97.5 K. (From Ref. 106.)
In other t h e o r e t i c a l calculations, included
[1OO,101,94]
finite temperature effects have been
. This leads to t e m p e r a t u r e d e p e n d e n t renormali-
zations of the ground state p r o p e r t i e s of the w e a k l y i n c o m m e n s u r a t e phase, e.g., to thermal f l u c t u a t i o n s of the domain wall n e t w o r k and to a b r o a d e n i n g of the wall width.
Some details of these theoretical cal-
c u l a t i o n s will be d i s c u s s e d in the f o l l o w i n g paragraphs. the findings of the m o l e c u l a r dynamics simulations the c o m m e n s u r a t e - i n c o m m e n s u r a t e
Beforehand,
for the dynamics of
t r a n s i t i o n are summarized.
As an example for the n u m e r i c a l results,
the sequence of Figs.
37-40
shows the temporal e v o l u t i o n of the w e a k l y i n c o m m e n s u r a t e phase of k r y p t o n on g r a p h i t e at d i f f e r e n t temperatures. rature
kBT/e = O.1
(Fig. 37)
At the very low tempe-
one finds the h o n e y c o m b network. The
domain walls consist of s t r a i g h t segments w h i c h are aligned
parallel
to the three symmetry d i r e c t i o n s of the g r a p h i t e substrate. M i c r o s c o p ically,
one has to d i s t i n g u i s h b e t w e e n two c o n f i g u r a t i o n a l l y d i s t i n c t
types of walls. A c c o r d i n g to the c l a s s i f i c a t i o n of Ref. called
'light' and
'heavy' walls.
104, they are
The atomic density in a heavy wall
is h i g h e r than in a light one. Both types may be easily i d e n t i f i e d by
67
their o r i e n t a t i o n and by the c l a s s i f i c a t i o n of the comm e n s u r a t e ground states s e p a r a t e d by this wall. (See e.g. Fig.1 Ref.
104.)
in
In the p r e s e n t e d s n a p s h o t p i c t u r e s of Fig.37, only heavy
walls are found.
Krypton/Graphite kT/e = 0.1
0 = 1.028
Time 2
4
6
12
14
( Fig.37:
10
8-~
16
Time sequence of snapshot p i c t u r e s of the i n c o m m e n s u r a t e krypton atoms in the m o l e c u l a r d y n a m i c s s i m u l a t i o n s of a 20,736 atom s y s t e m w i t h p e r i o d i c b o u n d a r y conditions. The time is given in units of t h o u s a n d time steps. One time step corresponds to 0.05 ps. The atoms at c o m m e n s u r a t e p o s i t i o n s are not shown, only the o c c u p i e d s u b l a t t i c e s are d e n o t e d by A, B, and C, respectively. The c o v e r a g e is 8=1.O28 and the t e m p e r a t u r e kBT/~=O.I. (From Ref. 107.)
The continuous b r e a t h i n g of the h e x a g o n s may be seen by c o m p a r i n g the individual s n a p s h o t pictures at d i f f e r e n t times. The low t e m p e r a t u r e situation agrees well with V i l l a i n ' s t h e o r e t i c a l assumptions. d i c a t i o n s for a striped phase are found,
No in-
in w h i c h the c o m m e n s u r a t e re-
gions w o u l d exist in form of an array of p a r a l l e l stripes. W i t h inc r e a s i n g temperature, distorted
(see Figs.
ens considerably. unambiguously,
the domain wall n e t w o r k becomes more and m o r e 38-40). The walls b r o a d e n and their surface rough-
However,
those domain walls, w h i c h can be c l a s s i f i e d
are always found to be heavy w a l l s
[107]
. Around
k B T / e = O . 7 , the s y s t e m begins to melt. L i q u i d - s o l i d c o e x i s t e n c e occurs, because the total d e n s i t y of the s y s t e m is fixed. process,
During the h e a t i n g
the p e r c e n t a g e of c o m m e n s u r a t e to i n c o m m e n s u r a t e atoms de-
creases from 75 % c o m m e n s u r a t e atoms at kBT/e=O.1
to roughly 60 % at
68 kBT/e=O.6
[1o7]
The subsequent
decrease for higher temperatures.
onset of melting causes a much sharper As long as the system is in the
Krypton/Graphite
(~
kT/e = 0.3
0 = 1.028
Time 2
4
6
8
10
12
14
16
Fig. 38: Same as in Fig. 37, but for the temperature (From Ref. Io7.)
1l
kBT/e=o.3.
Krypton/Graphite kT/e = 0.5
8 = 1.028
Time 2
4
6
10
12
14
8 -~
16
Fig. 39: Same as in Fig. 37, but for the temperature (From Ref. 1o7.)
kBT/e=o.5.
69
solid state, the increasing percentage of incommensurate atoms is due to a broadening of the domain walls. The average wall width is plotted in Fig.41 for different temperatures. The simulation results allow to
Krypton/Graphite kT/e = 0.7
8 = 1.028
Time 2
4
/c" Ac C
10
6
Ac' ! 12
8--~
v ~: Ac 14
16
Fig.40: Same as in Fig.37, but for the temperature kBT/e=O.7. (From Ref. 107.) determine a width distribution, which is indicated by the bars in Fig.41. This distribution becomes considerably broader at higher temperatures, because of the increasing surface roughness of the walls. The presented results of the molecular dynamics simulations of a system with 20,000 krypton atoms have been confirmed recently by studies of systems with more than 100,O00 atoms
[109]
The solid line in Fig.41 is the result of a theoretical calculation [ 9 4 ] . This theory maps the system of monolayer krypton on graphite onto a Sine-Gordon model. The domain walls are the pertinent soliton solutions. To construct the Hamiltonian of the system, it is justified to restrict the Fourier expansion of the krypton-graphite potential to the leading order
(Eq.II.56). The resulting Hamiltonian is
H=E'I~ ~ (~t ÷S£ml 2+ 21 [ F£'m'i' £ m i S£m,iS£'m',i '+A ~ { 3-cos(g S£m,1) -cos(g S£m,2)-cos g(S£m,1+S£m,2))}l
(II.61)
70
40
I
I
I
'
I
'
/
I
I
~< _="
I
4-'
30
/
2O
I
I
I
0
I
0.2
I
I
0.4
0.6
I
0.8
Temperature, kT/e Fig.41:
Here,
W i d t h of The solid the bars dynamics
÷S£m
the i n c o m m e n s u r a t e domain walls versus temperature. line is a t h e o r e t i c a l r e s u l t [94] . The dots with show the w i d t h d i s t r i b u t i o n found in the m o l e c u l a r simulations. (From Ref. 107.)
denotes
of the n e a r e s t
the d e v i a t i o n
adsorption
r£m=£a 1+ma 2
The q u a n t i t i e s krypton The
E ~, A,
term is the k r y p t o n - k r y p t o n
£ m i
(II.62)
and g are c o m b i n a t i o n s
(the explicit
term in eq. (II.61)
F£'m'i ' ~
atom from the p o s i t i o n
(£,m=0,+I ,+2 .... ) .
on graphite
first
of a k r y p t o n
site
expressions
of the p a r a m e t e r s are given
for
in Ref.
represents
the kinetic
energy,
interaction
in h a r m o n i c
approximation
kr-kr
94).
the second
(II.63)
~r£m,i~r£,m,,i ,
I
and the third term is the k r y p t o n - g r a p h i t e state e n e r g y
has been
subtracted.
interaction.
The g r o u n d
71 Motivated by the computer
simulation
results,
the domain wall network
is assumed to consist of wall segments,
w h i c h are aligned parallel
the symmetry directions
surface
To calculate sufficient
of the graphite
the temperature
(see e.g. Figs°37-40).
dependent width of these walls,
to deal with a p e r p e n d i c u l a r
Following
Ref.
positions
is denoted by s(~). Here,
94, the deviation
to
it is
cut through one of the segments.
of the atoms from the commensurate £ labeles
the commensurate
posi-
tions along the cut. s(£)=0 indicates
a completely
A self-consistent, guration
, for all
commensurate
temperature
(II.64)
situation w i t h o u t
dependent
approximation
any wall. for the confi-
s(£) may be obtained with the help of the B o g o l u b o v
inequality
for the free energy F I
F= - ~ £n Z with
,
(II.65)
Z=tr e -SH
and 8=I/kBT.
The B o g o l u b o v
(II.66)
inequality
is w r i t t e n
F T
,
as
(II.67)
where F T is given by Eq. (II.65) with H T instead of H. The average <'''>T
is defined with respect to H T. The test H a m i l t o n i a n
be chosen to describe
the interesting properties
allow an exact evaluation
of the right-hand
In Ref. 94 the authors make an ansatz ÷ S£m , yielding
+½
side of Eq. (II.67).
for H T which is quadratic
< S£m>T=S (£) The resulting equation
H T has to
of the system and to
in
(II. 68)
for F 0 is
+ ~F T ,q!
+A2£{ {3- (71+72 )coS
+A 1 [ { 2 s ( Z ) 2 - s ( £ + l ) s ( £ ) - s ( ~ ) s ( £ - l ) } £,m
s(£))-72}
(II.69)
72
The explicit The quantity
expressions
sence of the wall. figuration
for A[~(J),q),'~ A1 and A 2 are given in Ref.94.
~(J) (q) in Eq.(II.69)
is the energy of phonons
It has to be determined
s(£) by m i n i m i z i n g
Minimizing
=
together with the wall con-
F 0 with respect to these quantities.
free energy of the test Hamiltonian FT
in the preThe
is
I ÷[ log(1_e-8~ q,3
(J)(q)
~
m (j )+÷[ . ) (3) 2 q,3
°
(II.70)
F 0 with respect to the wall configuration 6F0 (II.71)
=0
6s(~) yields s(Z+1)-2s(£)+s(£-1)-B Here, B is a combination has the structure
(II.72)
sinlg s(£)!=0
of the system parameters
of a discrete
Sine-Gordon
[94]. Eq. (II.72)
equation.
The depth of the
nonlinear potential
B sinlg s(~)) depends on the phonons. krypton-graphite
This leads to an effective weakening of the
potential
due to the thermal vibrations
of the kryp-
ton atoms. The incommensurate connecting
by the competition interaction. teraction
domain walls are the m i n i m u m energy configurations
two commensurate
regions.
Their width
between the krypton-krypton
For example,
strong krypton-graphite
would lead to narrow walls. this interaction
and the krypton-graphite
strong forces due to the direct krypton
and a shallow substrate potential
A relatively
is directly determined
give rise to broad walls.
interaction,
The temperature
in-
on the other hand,
dependent w e a k e n i n g
of
will therefore cause an increasing wall width for
higher temperatures. If the variation
of s(£)
is slow over the distance between neighbour-
ing rows of krypton atoms, static Sine-Gordon
Eq.(II.72)
may be approximated
by the usual
equation ~2s ~Z2 + B s i n ( g s ) = 0
(II.73)
73 It is well-known,
that this e q u a t i o n is an example for those n o n l i n e a r
d i f f e r e n t i a l equations, which allow soliton solutions
[110]
. Eg. (II.73)
is solved by s(£) ~ a r c t a n e 2£/d
(II.74)
This solution describes the i n c o m m e n s u r a t e domain wall. The t e m p e r a t u r e d e p e n d e n t w i d t h of these walls is d e n o t e d by d. It is p l o t t e d in Fig.41 in c o m p a r i s o n to the results of the m o l e c u l a r dynamics simulations.
The overall a g r e e m e n t between s i m u l a t i o n and theory is quite satisfactory. The slightly higher value of the t h e o r e t i c a l result is readily u n d e r s t o o d to be due to the lack of a c o m p l e t e l y s e l f - c o n s i s t e n t evaluation of the interatomic force constants
[94]
III. Nonequilibrium Phase Transitions The developments in the field of synergetics
[11,111
] provide signi-
ficant stimulation for investigation of systems far away from thermal equilibrium.
Some global features of such
'open systems'
and basic con-
cepts of nonequilibrium phase transitions have already been discussed in the introductory chapter I.I of this book. More details of these ideas will be elucidated in the following section dealing with dynamical aspects of nonequilibrium phase transitions in physical systems. Interestingly enough, one sometimes finds striking similarities to equilibrium systems. Especially the first discussed example,
the elec-
tron-hole droplet formation in laser excited semiconductors,
may be
treated as a generalization of classical nucleation theory. Merely the nonequilibrium effects due to energy dissipation in the open system have to be considered additionally.
The observed phenomena therefore
show some similarities to the equilibrium situation,
but one also finds
pronounced nonequilibrium modifications. The details of electron-hole droplet nucleation will be presented in the next chapter
(III.1). This is followed by a discussion of non-
equilibrium phase transitions in optical systems
(chapter III.2)
.
III.1 The Plasma Phase Transition in Highly Excited Semiconductors A highly laser excited semiconductor crystal constitutes an almost ideal example of an open system. A variety of nonequilibrium phase transitions may be observed,
several examples of which will be dis-
cussed in the following chapters.
Before that, a short introduction is
given into the relevant physical properties of this system. In the ground state of an intrinsic semiconductor,
all energy bands
are filled with electrons. At sufficiently low temperatures,
the con-
duction band is empty and the material acts as an insulator.
Laser
irradiation of the crystal generates a variety of electronic excitations.
If the frequency of the incident light is sufficiently
high, electrons from the valence band are lifted into the
75 c o n d u c t i o n band. The m i s s i n g valence band e l e c t r o n is c a l l e d a hole. In the framework of q u a n t u m field theory,
the c o n d u c t i o n band e l e c t r o n s
and valence band holes may be t r e a t e d as q u a s i - p a r t i c l e s .
T h e i r effec-
tive m a s s e s are d e t e r m i n e d by the curvature of the r e s p e c t i v e bands [112]
The system of q u a s i - p a r t i c l e s c o n s t i t u t e s a c o m p l i c a t e d q u a n t u m mechanical m a n y - b o d y problem. As e l e c t r i c a l l y charged units, (e) and holes
(h) interact via the Coulomb potential.
the e l e c t r o n s
The e-e and h-h
interaction is repulsive, whereas the e-h i n t e r a c t i o n is a t t r a c t i v e because of the opposite charge. At s u f f i c i e n t l y low densities,
this
gives rise to the formation of b o u n d e-h pair states, the e x c i t o n s If the so-called W a n n i e r a p p r o x i m a t i o n
is valid,
(x).
i.e.,if the spatial
e x t e n s i o n of a b o u n d e-h pair is large, the excitons e x h i b i t features, w h i c h are quite similar to that of a h y d r o g e n atom le, the e x c i t a t i o n s p e c t r u m of W a n n i e r e x c i t o n s
[112]
. For examp-
shows a h y d r o g e n - l i k e
series. Only the effective mass and the B o h r radius are d i f f e r e n t from the r e s p e c t i v e h y d r o g e n values.
The system of q u a s i - p a r t i c l e s
in h i g h l y laser e x c i t e d s e m i c o n d u c t o r s
is of great interest to theorists and e x p e r i m e n t a l i s t s . feature is, that changes of the e x c i t a t i o n
One very nice
intensity may cause changes
in the density of the e l e c t r o n i c excitations, w h i c h e x c e e d m a n y orders of magnitude.
At r e l a t i v e l y w e a k incident
light
intensity,
the quasi-
p a r t i c l e system has a low density and exists in a state, w h i c h may be called
'exciton gas phase'.
With i n c r e a s i n g e x c i t a t i o n intensity, m o r e
and more excitons are created.
In some materials,
the e x c i t o n d e n s i t y
may become s u f f i c i e n t l y high to cause a t r a n s i t i o n from the e x c i t o n gas to a plasma liquid.
Even though the classical e x p r e s s i o n s are used in this context,
'gas phase'
and
'liquid phase'
the system of e l e c t r o n i c e x c i t a t i o n s
is never-
theless a q u a n t u m system. One m a y h o w e v e r exploit its a n a l o g y to the classical g a s - l i q u i d system even further by s u b d i v i d i n g the t h e o r e t i c a l t r e a t m e n t into two m a j o r steps. Firstly,
one uses q u a n t u m m e c h a n i c a l
m a n y - b o d y theory to calculate the t h e r m o d y n a m i c quantities. e.g.,
These are,
the e q u i l i b r i u m value of the p l a s m a liquid d e n s i t y or an approxi-
m a t i o n to the equation of state.
In the second step, one may d e v e l o p
a classical theory for the d y n a m i c s of the n o n e q u i l i b r i u m phase transition.
78
The concept of c o m b i n i n g m i c r o s c o p i c c a l c u l a t i o n s of the system properties with a m a c r o s c o p i c d e s c r i p t i o n of the dynamic features is applied t h r o u g h o u t the f o l l o w i n g discussion.
It is also a m e a n i n g f u l
a p p r o x i m a t i o n for optical n o n e q u i l i b r i u m systems, w h i c h are investigated in chapter III.2 of this book. An i m p r o v e m e n t of this approach w o u l d be a fully q u a n t u m m e c h a n i c a l
theory of the dynamic effects. This may
be p o s s i b l e in the framework of n o n e q u i l i b r i u m Green's functions Such c a l c u l a t i o n s will be the subject of future investigations.
[113]. They
have not yet been a c c o m p l i s h e d for the effects d i s c u s s e d in this book.
Coming back to the topic of the p l a s m a phase t r a n s i t i o n in h i g h l y e x c i t e d semiconductors,
it is w o r t h w i l e to mention,
that the plasma
liquid does not consist of bound e-h pairs. At densities as high as the liquid density,
the attractive Coulomb i n t e r a c t i o n is screened consider-
ably causing i o n i z a t i o n of the excitons.
In some materials,
one ob-
serves at i n t e r m e d i a t e densities the c o e x i s t e n c e of excitons and droplets of e-h liquid. These e-h d r o p l e t s are the n o n e q u i l i b r i u m analogue £o liquid droplets in a classical s u p e r s a t u r a t e d vapour.
The dynamics
of e-h droplet n u c l e a t i o n m a y be treated as g e n e r a l i z a t i o n of classical n u c l e a t i o n theory.
Besides the q u a n t u m effects m e n t i o n e d earlier,
there is , however,
an-
other important d i f f e r e n c e b e t w e e n atoms or m o l e c u l e s in an o r d i n a r y v a p o u r - l i q u i d system and the q u a s i - p a r t i c l e s in h i g h l y excited semiconductors. This d i f f e r e n c e is due to the fact, that the e l e c t r o n i c e x c i t a t i o n s have a finite lifetime. E l e c t r o n s and holes may recombine,
thereby t r a n s f e r r i n g their e n e r g y to other e l e m e n t a r y excitations.
The f o l l o w i n g i n v e s t i g a t i o n s will show, that m a n y details of the p l a s m a phase t r a n s i t i o n depend on the actual value of the e-h lifetime.
There-
fore, one has a quite d i f f e r e n t s i t u a t i o n in s e m i c o n d u c t o r s with an indirect gap in c o m p a r i s o n to d i r e c t - g a p materials.
D i r e c t b a n d - g a p m e a n s that the v a l e n c e - b a n d m a x i m u m and the conductionband m i n i m u m are at the same k - v a l u e in the B r i l l o u i n - z o n e . b i n a t i o n by e m i s s i o n of a photon possible,
(radiative recombination)
E-h recomis thus
s i m u l t a n e o u s l y c o n s e r v i n g energy and total momentum.
This is
the reason for the r e l a t i v e l y short e-h pair lifetimes on the order of nanoseconds.
In i n d i r e c t - g a p semiconductors,
has the band extrema at d i f f e r e n t k-values.
on the other hand, one R a d i a t i v e e-h r e c o m b i n a t i o n
can occur only under p a r t i c i p a t i o n of another q u a s i - p a r t i c l e , phonon, which takes over the momentum.
e.g.
a
The e x c i t o n decay p r o b a b i l i t y is
77
therefore reduced causing r e l a t i v e l y long lifetimes on the order of microseconds.
III.la E l e c t r o n - H o l e D r o p l e t N u c l e a t i o n in I n d i r e c t - G a p S e m i c o n d u c t o r s In the past ten years,
the p r o p e r t i e s of e l e c t r o n - h o l e droplets
d i r e c t - g a p s e m i c o n d u c t o r s have been studied in great detail. be found,
e.g.,
pure germanium.
in Refs.
in in-
Reviews can
114-117. Most e x p e r i m e n t s have been done for
Here, one may m e a s u r e the size of the e-h d r o p l e t s by
means of the R a y l e i g h s c a t t e r i n g of i n f r a r e d light. T h r o u g h application of n o n e - u n i f o r m stress, one can g e n e r a t e energy m i n i m a via defor~tation potentials.
Large droplets with radii up to several h u n d r e d um
may be formed in these minima.
Photographic
images of the large e-h
drops gave a s p e c t a c u l a r d e m o n s t r a t i o n of the n o n e q u i l i b r i u m phase transition
[117]
. The dynamics of the droplet n u c l e a t i o n have been
m e a s u r e d rather d i r e c t l y by taking time r e s o l v e d l u m i n e s c e n c e spectra [118,119]
The t h e o r e t i c a l d e s c r i p t i o n of e-h droplet n u c l e a t i o n in i n d i r e c t - g a p s e m i c o n d u c t o r s may be a c c o m p l i s h e d by g e n e r a l i z i n g the c l a s s i c a l nuc l e a t i o n theory
[118-122]
cepts of surface tension, in mind,
. Even though, one uses the c l a s s i c a l cone v a p o r a t i o n current etc., one has to keep
that these q u a n t i t i e s have to be o b t a i n e d from q u a n t u m m e c h a n -
ical calculations.
This concept will be o u t l i n e d in the f o l l o w i n g para-
graphs. As starting point for the calculations, P l a n c k approximation,
Eq.
one may choose the Fokker-
(II.5), of the m a s t e r e q u a t i o n
c l a s s i c a l n u c l e a t i o n theory.
(II.1) of
In the e x p r e s s i o n s for the gain rate, Eq.
(II.3), one simply has to replace the d e n s i t y of m o n o m e r s n I by the time dependent exciton d e n s i t y nx(t).
In the loss rate, Eq.
(II.4),
one has to account not only for the e v a p o r a t i o n losses through the surface of the droplet but also for the losses due to e-h recombination. This leads to an a d d i t i o n a l term on the r i g h t - h a n d side of Eq. ~(n)=a(T)e°(T)/nl/3n2/3+n/Tn
where
,
(III.1)
Tn is the e-h pair lifetime in a cluster w i t h n pairs.
cludes radiative and n o n - r a d i a t i v e recombination. ditions,
(II.4)
It in-
For s t a t i o n a r y con-
78 n x (t )=nx=COnst.
9fstat ~jstat ~-~ =0= ~-H
These equations describe a one-dimensional problem with natural boundary conditions f (n=~) =0=J (n=~) Therefore, the stationary probality current ally. Evaluation of
J(n)
vanishes identic-
Eq. (II.6) for these conditions yields -¢(n,nx)/kBT fstat (n)=N e
(III.2)
Here, the generalized Ginzburg-Landau potential is given by
n
¢=2kBT ~1 dn'
{ [ £(n')-g(n') + .~1 an---~ ~ (9..(n,)+g(n,)l] x Ig(n')+~(n') ] -I}
Fig.42 shows
~ as function of the droplet radius R
4 ~___n ] R= ( ~ po
P0
(III.2a)
1/3
is the equilibrium density of the e-h liquid. The Ginzburg-Landau
potential is plotted for two values of the exciton density
n x , one
of it below and one of it above the critical density for two-phase coexistence,
respectively. Fig.42 demonstrates, that e-h droplet
nucleation in indirect-gap semiconductors is clearly a first-order nonequilibrium phase transition. For small cluster sizes the GinzburgLandau potential is qualitatively similar to the free energy in classical droplet nucleation
(see Fig.8). One may define a critical
79
radius
R c of the e-h
evaporate
and
larger
I
droplets,
such
ones
to grow.
I
l
I
tend
I J 111
that
I
I
smaller
E
I
l , ,ll
T =2.1
10 7 -
clusters
tend
1 1 , / 7 1
i
to re-
i
i
~ "
-
/ / 7x I0~ o
I03
.- __101 0
f ns
-103 0.01
I
I
I
I
I IIII
I
I
I
I
I IIII
0.1
,
I
,
,
,
,
,
1.0
10.0
Radius R [)urn]
Fig.42:
The
Generalized Ginzburg-Landau potential ~ versus radius of the e l e c t r o n - h o l e d r o p l e t s for two e x c i t o n d e n s i t i e s (n ~1.5 x I O l l c m 3 ) (From Ref. 123.) sg
cluster
growth
is in c o n t r a s t
in the
to the
R ~t 8
(see c h a p t e r
radius
of the
Rs
and thus
equating
the
e-h also gain
I.la).
droplets n s =R s3 rate
semiconductor
classical
case,
Under
stationary
approaches are
and the
system where
laser
rate
to a g o o d due
a growth
excitation,
a stationary
determined loss
is r e s t r i c t e d .
one has
to e-h
value
Rs>
R n
. x
This law the Rc .
approximation recombination
by
80
g(ns)~ns/T For still larger droplets,
the r e c o m b i n a t i o n losses
gain due to c o n d e n s a t i o n of excitons
(~ n) exceed the
(~ n2/3). The existence of the
s t a t i o n a r y radius R s is thus solely a c o n s e q u e n c e of the finite e-h lifetime.
It is unknown for classical systems. The actual value of R
is d e t e r m i n e d by the exciton density,
s i.e. by the e x c i t a t i o n intensity,
and by the p a r a m e t e r values of the respective semiconductors.
N e g l e c t i n g e x c i t o n i c effects,
the density profile of the stationary
droplets may be c a l c u l a t e d w i t h i n a h y d r o d y n a m i c theory
[124-127]
One solves the c o n t i n u i t y e q u a t i o n for the local e-h pair density p (.f,t)
~-~ p ( r , t ) = - V ( p v ) Here,
G(t)
(III.3)
T +G(t)+F2(t)
is the e-h pair g e n e r a t i o n rate due to the external laser
e x c i t a t i o n and v(r,t)
is the e-h v e l o c i t y field. F2 describes the shot
noise =0
= I p / Y + G ( t ) 1 6 (t-t')
The v e l o c i t y field obeys the g e n e r a l i z e d E u l e r e q u a t i o n d ~ ~ ÷ ÷ -at -- - m $~ (~) -Sv+F(t) In this equation,
d/dt stands for the substantial derivative, m is the
e f f e c t i v e e-h mass, pairs by phonons,
(III.4)
-~v describes the friction due to scattering of e-h
lattice i m p e r f e c t i o n s etc.,
and
~
is the p e r t i n e n t
noise ter~. The d i s s i p a t i v e terms in the viscous s t r e s s ~ t e n s o r of the u s u a l N a v i e r - S t o k e s e q u a t i o n may be n e g l e c t e d for the e-h system. They are c o n s i d e r a b l y smaller than - ~
[J26]
.
The e q u a t i o n of state for the e-h system has to be d e t e r m i n e d from q u a n t u m m e c h a n i c a l calculations.
A s s u m i n g local q u a s i - e q u i l i b r i u m ,
one obtains isotherms for i n d i r e c t - g a p semiconductors, w h i c h resemble the features of a classical Van der Waals system results,
one m a y c o n s t r u c t a m o d e l
[124,125]
the q u a l i t a t i v e features of the e-h droplets.
[128]
. Using these
, which allows to study
81
A numerical
integration
(III.3)
(III.4) yields the density and velocity profiles
and
of the stationary b a t h - a v e r a g e d
stable droplet . An example
for the results
15
/',
\
?19
///~
of a
is shown in Fig.43.
"r,To:o.9
I ,, e/%--o.s
V
I
equations
15
IVl
~\\\ \\\\\
/I11/
i 1
/
=
I
I
I
I
5
Fig.43:
Spherical density profile p(r) and velocity field v(r) for a stationary e-h droplet. Pc and T c are density and temperature at the critical point, respectively. (From Ref. 27.)
A depletion
zone is developed directly outside the droplet.
gion is rapidly traversed by the condensing phase.
In the stationary
state,
due to e-h recombination Close to the critical weakly of first order, [125]
droplet,
e-h pairs out of the gas
they directly
compensate
the losses
inside the droplet.
region,
where the plasma phase transition
one may proceed with an analytical
. As a first step, one performs
calculates
This re-
the eigenmodes
a linear stability
X to Eqs. (III.3)
and
they are found to be proportional
is only
treatment analysis
and
(III.4). For a spherical
to spherical Bessel
func-
tions j0(kr) Xik =J0(kr)
To treat the full problem one expands the nonlinear in terms of Xik and applies the so-called theory
[11]
solutions
generalized
All n o n - r e l e v a n t modes are e l i m i n a t e d
with an effective
equation
Ginzburg-Landau to end up
for the amplitude 7k of the unstable mode
BV ~t Yk = - ~Yk + Feff
82
Here, Fef f
is the effective noise c o n t r i b u t i o n and V is the potential
for the amplitude Yk " A value of Yk=0 indicates s t a b i l i t y of the homogeneous phase, whereas Yk~0 c h a r a c t e r i z e s phase separation. for V are plotted in Fig.44
for d i f f e r e n t mean e-h pair densities
~P
~
Illlq
\
T : .98 6, = .872 • .880
%, ,o4 = 1.08
~.11z
n;,,.oo
"~-
It1:1.1,6 ,
~06
-.04
-.02
0
.02
.04
~06
Amphtude i,kc
Fig.44:
i.e.
in the crystal.
\ '=
The results
for d i f f e r e n t e x c i t a t i o n intensities,
~l~
,
~ ~0Z
Amplitude
i
0
.02
.04
itKc
Potential for the amplit2de Yk of the u n d a m p e d mode (see text). The d i f f e r e n t values of n i indicate the respective e-h pair d e n s i t y P/Pc • The t e m p e r a t u r e is T=O.98 T c . Here, Pc and T c are the density and t e m p e r a t u r e at the critical point, respectively. (From Ref. 127.)
B e l o w the phase transition,
the h o m o g e n e o u s gas phase is stable. There-
fore, V has only one m i n i m u m at ¥k=0. With i n c r e a s i n g e x c i t a t i o n intensity,
a second m i n i m u m at Yk>0
in form
of
tion
c h a r a c t e r i z e s two-phase coexistence
e-h clusters w i t h i n the gas phase. The h o m o g e n e o u s solu-
(Yk=0) becomes unstable for still higher e-h pair densities.
a d d i t i o n a l m i n i m u m at Yk<0 develops
(curve 3 in Fig.44).
An
The two m i n i m a
at a p o s i t i v e and a n e g a t i v e value of Yk indicate coexistence of liquid droplets and gas bubbles.
For the p a r a m e t e r s of curve 7 in Fig.44,
has only bubbles in the e-h liquid. At the highest e-h density, homogeneous
liquid state is reached.
one
the
Thus, the p r e s e n t e d t r e a t m e n t
allows a m e a n i n g f u l q u a l i t a t i v e d e s c r i p t i o n of the whole phase transition in the v i c i n i t y of the critical point. In order to get results
for the d y n a m i c s of droplet n u c l e a t i o n outside
the critical region, one may apply a
suitable a p p r o x i m a t i o n scheme
83 for the distribution
function
f(n,t),
describing
the p r o b a b i l i t y
to
find a cluster with n e-h pairs at time t. The relevant equation Fokker-Planck additional
equation
equation
(II.5). However,
gouverning
e-h pairs in the system.
is the
it has to be supplement by an
conservation
N may be e x p r e s s e d
of the total number N of as moment of the distribu-
tion function N(t)= The conservation
equation
f an n f(n,t) I
is
-~ f an n f(n,t)= -f an ~t ~ I Equations
(II.5)
cription,
by integrating
ume
[]271
and
(III.5)
valent to the F o k k e r - P l a n c k equation
a complicated nonlinear the nucleation
dynamics
dinary differential different moments
from the hydrodynamic
equation
is s t o c h a s t i c a l l y
for the p h y s i c a l l y
equation
one subdivides
the generalized (see Fig.42).
n and for x [118,121] .
function
If two-phase
Ginzburg-Landau
potential
~(n c) is the potential fluctuations.
~ has a
barrier which of
larger than n c as droplets, size. On the
smaller than n c may be counted as part of the
Thus, one devides the d i s t r i b u t i o n
function
f(n,t)
an exciton part fx and a droplet part fd
f (n,t) =fx (nnc't) The density of all e-h pairs condensed
nd= f dn n fd(n,t) nc This
ex-
coex-
The existence
since they have the tendency to grow to the stationary the clusters
of
the system of e l e c t r o n i c
this barrier allows to consider the clusters
exciton systems.
How-
for the exciton concentration
has to be surmounted by the critical
other hand,
form
relevant properties
into an exciton part and a droplet part.
for n=nc(t)
(III.5)
equations.
from a much simpler set of or-
of the droplet d i s t r i b u t i o n
istence is occuring,
equi-
(II.5).
and the conservation
equations
des-
equation over the droplet vol-
may be obtained
To derive these equations,
maximum
equation
(III.5)
system of i n t e g r o - d i f f e r e n t i a l
ever, a good approximation
citations
can be derived
the continuity
. The resulting Langevin
The F o k k e r - P l a n c k
_~n f(n,t)+S(t) ~n
in droplets
is given by
.
is the first moment of the droplet
distribution
function
in
84 nd=Xl The exciton
density
is
approximated
n
by
c
nx= f
dn n fx(n,t) I
One may therefore
split the conservation ~
the density
dependent
x T
~n=
n
Td
n>n c
into
(III.5a)
+G(t) T
x
e-h lifetime
TX
(III.5)
nd
n
~-~ nx +Tt n d Here,
equation
d
has been approximated
by
N
In Ge, one has approximately of fd(n,t)
~d=4Tx
[118,119]
. The moments
x~ (t)
are given by x
(t)= f n
The zeroth moment x 0 measures x 2 one obtains
the relative
an n~f d
(III.6)
c
the density
mean square
of droplets,
x1=nd,
(&n) 2 of the droplet
and from distribu-
tion
(An)Z=[(x2/x0)-(Xl/X0)2]/(xl/x0) Differentiating Planck
equation
EqIIII.6) with respect (II.5),and
to time,
integrating
2
inserting
by p a r t s
yields
the Fokker-
the
equation
for
X
~ ~X0 ~--~ x =n c - ~
L-~--.~d (a (T) -nxb (T)) ~ FX
-a
- I/3]
+
(III.7) Here,
the surface
poration droplets.
tension
contribution
rate has been neglected. The droplet
density
This
in the expression
for the eva-
is well
for large
is determined
~n -~t x ° = J ( n c ' t ~ - ~ Equations
(III.5a),
9=2,3,...
a closed
(III.7),
and
(III.8)
set of equations.
justified
by
f(nc't)
(III.8)
form for all ~=9/3 with
For the evaluation
of the current
85 J(n c) over the p o t e n t i a l b a r r i e r ~(n c) one needs an e x p r e s s i o n for f(nc). The d i s t r i b u t i o n function may be approximated, c h a r a c t e r i s t i c times for e q u i l i b r a t i o n
a s s u m i n g that the
in the r e s p e c t i v e subsystems of
excitons and droplets are much faster than the r e l e v a n t time scale for v a r i a t i o n of x . If the p o t e n t i a l b a r r i e r is s u r m o u n t e d from the exciton side
(droplet formation and growth),
In the case of the droplet decay,
f(nc)
one replaces f(n c) by fx(nc).
is a p p r o x i m a t e d by fd(nc).
These assumptions are justified in situations, changes of the laser g e n e r a t i o n rate G(t) of the exciton lifetime.
in w h i c h the t e m p o r a l
are slow on the t i m e - s c a l e
For i n d i r e c t - g a p semiconductors,
tions may be r e a l i z e d e x p e r i m e n t a l l y
these condi-
[118,119]
For the n u m e r i c a l solution of the d i f f e r e n t i a l e q u a t i o n s a laser generation rate of the form -t/tr) G(t)=Gl(1-e has been assumed
[121]
,
(III.9)
. This simulates a s w i t c h - o n of the laser ex-
citation w i t h the risetime t r. For the m a t e r i a l p a r a m e t e r s of Ge, one o b t a i n ~ the results p l o t t e d in Fig.45. I
'
'
t''''i
'
G, :3.1Om~tcm"3 / |
T
'°'r
:2.1K
'
'
I''''I
'
'
'
.............
~
I''''I
'
~ X o - l O 4 c n ~
/
'
/
I''
-3
x 2 ..lOZ°cni 3
IY '/K
lo' - /
'
3
x
~
'
G "10t6{ICm3
IF/-
,u cm
,,, ..12
. . . . . .
-3
(~--~)z. 10"1
10°
102
103
104
105
106
time [ps ] Fig.45:
Time d e p e n d e n c e of the laser g e n e r a t i o n rate G,_the e x c i t o n density n x and the moments x0 , xl , and x2 . n = Xl/X0 ~ the average number of e l e c t r o n - h o l e pairs per droplet, is the v a r i a n c e of the d r o p l e t d i s t r i b u t i o n function . (From Ref. 121.)
88
This figure shows, that the exciton density has to e x c e e d a critical value, before e-h droplet formation
takes place. At early times, n
rises p r o p o r t i o n a l l y to G(t) until n u c l e a t i o n tion of many excitons
x sets in causing condensa-
into droplets. At this time,
the moments x 0 , Xl,
and x 2 steeply increase and the exciton d e n s i t y decreases.
Then,
a
stationary s i t u a t i o n is a p p r o a c h e d slowly. The f l u c t u a t i o n s become very large only during the n u c l e a t i o n process.
This temporal v a r i a t i o n has been tested by time r e s o l v e d luminescence measurements
[138]
. The theory is in q u a l i t a t i v e a g r e e m e n t w i t h the
e x p e r i m e n t a l findings of the i n t e g r a t e d exciton and e-h droplet luminescence,
respectively.
The s i t u a t i o n of droplet decay has also been
i n v e s t i g a t e d in this framework
[123] . The theory is found to reproduce
the results of the e x p e r i m e n t s d e s c r i b e d in Ref.129. At this stage, one may thus conclude,
that the dynamics of e-h droplet n u c l e a t i o n in in-
d i r e c t - g a p s e m i c o n d u c t o r s is well understood.
T h e o r y and e x p e r i m e n t
are in good q u a l i t a t i v e agreement.
III.Ib The P l a s m a Phase T r a n s i t i o n in D i r e c t - G a P Semiconductor_ss The c o m p o u n d s e m i c o n d u c t o r s CdS and GaAs are typical examples for the direct-gap materials
i n v e s t i g a t e d in this chapter. Here,
of the e-h pairs is only of the order of nanoseconds.
the lifetime
The recombina-
tion losses are therefore very high, causing s i g n i f i c a n t m o d i f i c a t i o n s of the plasma phase t r a n s i t i o n in c o m p a r i s o n to the situation in ind i r e c t - g a p materials.
Nevertheless,
in CdS and GaAs the e x i s t e n c e of
an exciton gas phase at low incident light intensities is e x p e r i m e n t a l ly assured, tion
as well as the o c c u r e n c e of an e-h plasma for high
[130]
. However,
excita-
no e x p e r i m e n t a l e v i d e n c e has been o b t a i n e d for
the d e v e l o p m e n t of two-phase c o e x i s t e n c e
in form of well defined e-h
droplets.
The t h e r m o d y n a m i c p r o p e r t i e s of the e-h plasma in d i r e c t - g a p semiconuctors have been c a l c u l a t e d q u a n t u m m e c h a n i c a l l y
[131-133]
. These
calculations assume that a q u a s i - e q u i l i b r i u m is r e a l i z e d in the system of e l e c t r o n i c excitations.
For these c o n d i t i o n s one obtains a phase
d i a g r a m for the e-h plasma w h i c h resembles that of a classical (see Fig.46).
liquid
87
50 40 I--
liquid
20
coexistence
01010 1012 , , i i I 101/" 1016 10 le 1020 (cm-3) F i g . 4 6 : T h e o r e t i c a l prediction for the phase d i a g r a m of the electronhole system in CdS. (From Ref. 132.) The phase d i a g r a m exhibits Consequently, However,
a well developed phase coexistence
one should observe e-h droplets
region.
in this parameter
region.
it is not clear to what extend these results are e x p e r i m e n t a l l y
relevant.
The calculations
electronic An estimate
did not consider the short lifetimes
of the
excitations. for the influence
of the e-h pair lifetime on the details
of the plasma phase transition may be obtained by evaluating ralized G i n z b u r g - L a n d a u
potential,
Eq.(III.2a),
meters of GaAs and CdS. The results
the gene-
for the m a t e r i a l
para-
are shown in Fig.47.
number of e-h pmrs
.
number of e-h pe,rs n
.~[CdS , .¢
6o As TISK . . . . . . n. [tO~c~ s] (D 867 ® 1,33 ~3.3 057 ®5?0
[ ©IZl, .,,0"*~"
Z
~ J
f
i0z ®12 I,
-10 -to
} 1 e'h d~p rodlus R (pm]
Fig.47:
e-h drop rod,us I Ipml
Generalized Ginzburg-Landau potential for various exciton densities. The results for GaAs at T=6 K are shown in the left figure for the exciton densities nx(1Ol4cm-3): (I) 0.67, (2) 1.33, (3) 3.3, (4) 6.7, (5) 67.0. The results for CdS at T=20 K are plotted in the right figure, n x (I016cm-3): (1) (I) 0.124 ( 2 ) 0 . 2 4 9 , ( 3 ) 0 . 6 2 3 , (4) 1.24, (5) 12.45. (From Ref. 134.)
88 In the d i r e c t - g a p s e m i c o n d u c t o r s GaAs and CdS, the r e c o m b i n a t i o n losses dominate over the e v a p o r a t i o n losses at p r a c t i c a l l y all e x c i t a t i o n intensities.
The size of the critical droplets is t h e r e f o r e e x t r e m e l y
small and the p o t e n t i a l b a r r i e r to n u c l e a t i o n is almost absent. Evaluation of the s t a t i o n a r y p r o b a b i l i t y distribution,
Eq. ~II.2),yields
the results shown in Fig.48.
.
.
.
.
f
.
.
.
.
l
.
.
.
.
i
.
.
.
.
r
.
.
.
.
M
,_~Cl5
/
~
/
G~Z.7.16'ec~/~
E I0
ZO
30
~0
50
number of e-h poirs n
Fig.48:
N o r m a l i z e d d i s t r i b u t i o n function for the p r o b a b i l i t y to find clusters w i t h n e-h pairs. The results are c a l c u l a t e d for CdS at T=15 K for d i f f e r e n t e-h pair g e n e r a t i o n rates G . (From Ref. 135.)
A n o t h e r c o n s e q u e n c e of the short e-h pair lifetime is the small size n s of the stable droplets. n s ~<30
(see Fig.48).
For e x p e r i m e n t a l l y relevant g e n e r a t i o n rates
This has to be c o m p a r e d
The e-h clusters in d i r e c t - g a p s e m i c o n d u c t o r s
to
ns ~IO6 in germanium.
are thus h a r d l y detect-
able e x p e r i m e n t a l l y and in fact, up to now no d i r e c t e x p e r i m e n t a l evidence has been obtained.
In the t h e o r e t i c a l t r e a t m e n t of formation and decay of these tiny e-h droplets one d i r e c t l y has to solve the d i s c r e t e m a s t e r equation of n u c l e a t i o n theory, Eq. (II.1). A c o n t i n u u m approximation, a F o k k e r - P l a n c k t r e a t m e n t , i s not well
i.e.
justified in this case. A numer-
ical solution of the m a s t e r e q u a t i o n and the e q u a t i o n for the exciton d e n s i t y n x has been r e p o r t e d in Ref. is
136. The d y n a m i c e q u a t i o n for n x
89
_ ~ n x = G ( t ) - n x / T x - 2 g ( n = 1 ) n x + 2 £ ( n = 2 ) f ( n = 2 )- [ Ig(n-1)f(n-1)-e(n)f(n)) ~t n=3
,
(III.10) where e(n)=£(n)- n T
The first two terms on the r i g h t - h a n d side of Eq. (III.IO)
are the rates
of exciton g e n e r a t i o n and decay. The third and fourth term d e s c r i b e fusion of two excitons into a m o l e c u l e and the reverse process of m o l e c u l e decay,
respectively.
The sum in the last term finally is the
net loss rate of excitons in the g e n e r a t i o n and e v a p o r a t i o n p r o c e s s e s of all cluster larger than a molecule. M o t i v a t e d by the s t a t i o n a r y results one may assume that during the n u c l e a t i o n process no large clusters are formed. authors solve the set of e q u a t i o n s t r u n c a t i n g fn(t)
at n=300
, i.e.
(II.1)
fn=0
and
In Ref.
136 the
(III.10) n u m e r i c a l l y
for n>300
. This a s s u m p t i o n is
j u s t i f i e d s e l f - c o n s i s t e n t l y w i t h i n the calculations. An example
for the
n u m e r i c a l results is r e p r o d u c e d in Fig.49.
No large e-h clusters are found under the i n v e s t i g a t e d conditions. droplet size
n
exceeds
n=50
even for very high excitations.
The
only in a short t r a n s i e n t t i m e - r e g i m e Under the assumed conditions,
these ex-
c i t a t i o n intensities are the h i g h e s t e x p e r i m e n t a l l y r e a l i z a b l e values before damage of the crystal sets in.
C a l c u l a t i n g the density of the e l e c t r o n i c e x c i t a t i o n s in the s e m i c o n ductor shows that the plasma liquid d e n s i t y has already been reached. Thus,
the state of a plasma liquid in d i r e c t - g a p m a t e r i a l s is not
a p p r o a c h e d by the growth of large droplets, p a c k i n g of tiny clusters.
but by i n c r e a s i n g l y dense
Due to the p r e d o m i n a n t n o n e q u i l i b r i u m effects,
the two- phase coexistence region is
h a r d l y developed.
This shows that
m o s t results of the q u a s i - e q u i l i b r i u m c a l c u l a t i o n s n e g l e c t i n g finite lifetime effects are e x p e r i m e n t a l l y not really relevant.
The absence of a well defined t w o - p h a s e region suggests the a p p l i c a t i o n of a h y d r o d y n a m i c d e s c r i p t i o n for the p l a s m a phase transition.
In in-
d i r e c t - g a p m a t e r i a l s this a p p r o a c h was found to be v a l i d only near the critical point. One may expect,
that the h y d r o d y n a m i c m o d e l has
a m u c h b r o a d e r range of a p p l i c a b i l i t y in d i r e c t - g a p s e m i c o n d u c t o r s
90 because here only tiny clusters are formed. These may be r e g a r d e d as local fluctuations of the e-h p a i r density.
fn(t)~
c¢s%-•
1o,: 40
2o
t Ins]
Q) n
0
f.(t) ], [cr6'
CdS Tp" 8K
1017I 10ls 2O
b)
~/~~
,o
I[ns]
1
f.Ct) CdS "rp- 8K
lO~S 10~ 2O ns'l
c)
6o\ n
~
1o
0
Fig.49: N u m e r i c a l results for the cluster d i s t r i b u t i o n function f(n,t) versus time and n u m b e r of e l e c t r o n - h o l e pairs per cluster. The p l a s m a t e m p e r a t u r e is TD=8 K . A 7 ns pulse has been assumed and a peak g e n e r a t i o n rate Gl(1027cm-3s-l): a) G/=0.5 , b) GI=I.0 , c) G/=2.O . (From Ref. 137.)
91
Eqs.(III.3) and theory shows,
(III.4)
[138,139]
are the starting point for the h y d r o d y n a m i c
. However,
a stability
that including the velocity
analysis
of these equations
field as a dynamic variable yields
nothing but a damped mode which always remains stable [27] . For simplification, one may thus eliminate the velocity adiabatically. density
This leads to an effective
equation
field
for the local e-h
0(r,t)
Here, M = I / S m and F2=VF/8 structure
. Mathematically,
as Eq. (II.23) of the classical
composition
in binary systems.
Eq. (III.11)
field theory
For example,
and Miller
[41]
droplet
the quite successful
has been g e n e r a l i z e d
formation
in direct-gap
theory by replacing the homogeneous Within the approximations
for a treatment
of
theory of Langer,
Bar-on,
to deal with the p r o b l e m of e-h
semiconductors
local chemical potential may be obtained
~(r,t).
de-
This analogy suggests the application
of m e t h o d s which are similar to those developed Eq. (II.23).
has the same for spinodal
[138,139]
. Here,
the
from quasi - e q u i l i b r i u m
e-h density by the local density
of Ref.
41 one constructs
a system
of equations which is solved numerically.
This allows the d e t e r m i n a t i o n
of the temporal
distribution
evolution of the density
gether with that of the moments of the p r o b a b i l i t y details see, e.g., Examples
the discussion
of the results
function
fl to-
distribution.
For
in chapter lI.Ib of this book.
are reproduced
in Figs.5Oa-d.
The calculations
have been done for CdS under pulse excitation with a pulse width of 1.2 ns
. Fig.5Oa
tribution system.
shows a typical example of the resulting
fl together with the chemical p o t e n t i a l
The temporal
density dis-
~h of the homogeneous
evolution of fl is p l o t t e d in Fig.5Ob.
It turns
out that the m a x i m u m of fl is always at the value ~ which is the mean e-h pair density determined by the external
laser g e n e r a t i o n
rate
G(t) ~d ~- G_( t ) With increasing
(111.12)
~?
times the distribution
due to prominent density fluctuations. double peaked structure
are found.
function broadens However,
no indications
This indicates
gative value of ~ = p - ~ .
for a
that no true separa-
tion into a liquid phase and a gas phase is taking place. be signified by two peaks of fl(u),
significantly
This would
one at a positive and one at a ne-
92
I -120I :ds f t,..w
~,,/ 1 /
_
-~
[ ~0zzx0"cm'
10"
Fig.5Oa:
..... z
t0's densityR [cm*J
I0'+
.5
0
Density d i s t r i b u t i o n fl for the g e n e r a t i o n rate G = 1 . 2 x 1 0 2 7 c m - 3 s -I and the q u a s i - e q u i l i b r i u m chemical p o t e n t i a l Ph as functions of the e l e c t r o n - h o l e density p . (From Ref. 137.)
-0.8
Fig.5Ob:
/,
0
1
2
u(1078c63)
3
T e m p o r a l e v o l u t i o n of the d e n s i t y d i s t r i b u t i o n fl(u,t), u=p-~. The times are given in ns after the onset of the e x c i t a t i o n pulse. (From Ref. 139.)
Cd 5
t2 If,.60K
Z,
/"\(d,
// ~\\
% , I0"¢m'
1,0
B
/
.5
f
\~
i
x.
• i
,
,
L
,
I
t Ins] Fig.5Oc:
~
W+,, ,
, ,
,
I
1.6
,
?
O~
.?
Time d e v e l o p m e n t of the average e l e c t r o n - h o l e density and of the second and third m o m e n t of the p r o b a b i l i t y distribution, and , respectively. (From Ref.137.)
g3
~
cd s
i,z HI B*l
S
tO
tZ
1,1~
16
IS
k [ Z,I" IOScm'~l
Fig.5Od:
Structure factor S(k,t) for various times after the onset of the excitation pulse (see text). (From Ref. 137.)
The temporal evolution of the mean density ~ is shown in Fig.5Oc together with that of the second and third moment of the distribution function. After 0.4 ns, ~(t) reaches the value corresponding to the instability point of the chemical potential.
For higher densities,
phase separation would occur in classical systems. This is however prohibited by the short e-h lifetimes in direct-gap semiconductors.
Only
a broadening of the distribution function takes place. This is also indicated by the second moment . Its time-development that
of
follows
~. The third moment measures the asyrmaetry of the
distribution function. Negative values indicate that most density fluctuations are in direction to lower densities.
An example for such a
situation is shown in Fig.5Oa. Finally,
in Fig.5Od the time dependent structure factor S(k,t)
is plot-
ted for several times after the onset of the exciation pulse. The wavelength at which S(k,t) has its maximum is the dominant wavelength of the expected density pattern
(see the discussion in chapters II.Ib
and II.lc). For CdS the wavelength is in the region IO-5cm>l>lO-Ucm. These values are compatible with the cluster model of e-h droplet nucleation, which predicts for temperatures well below T c the formation of e-h clusters with an average radius on the order of IO-6cm. The time development of S(k,t) synchrotron radiation
could be measured experimentally with pulsed [140]
. Up to now, however, no such measurements
have been reported. To summarize,
both the nucleation theory and the hydrodynamic model
predict significant nonequilibrium effects for e-h droplet nucleation
94
in d i r e c t - g a p semiconductors.
These effects lead to the c o n c l u s i o n
that the plasma phase t r a n s i t i o n in these m a t e r i a l s is only weakly of first order. Due to the smallness of the e-h density fluctuations,
a
direct experimental m e a s u r e m e n t is complicated. A c o m p a r i s o n between theorz and e x p e r i m e n t is t h e r e f o r e only i n d i r e c t l y possible. centrates on the theoretical result,
It con-
that a two phase c o e x i s t e n c e
region is only i n c o m p l e t e l y d e v e l o p e d for d i r e c t - g a p semiconductors. A lineshape analysis of the optical spectra of gain and a b s o r p t i o n in h i g h l y excited CdS and GaAs shows that the e-h system does not reach the plasma liquid d e n s i t y
[130,137]
. Moreover,
in a g r e e m e n t w i t h
t h e o r e t i c a l p r e d i c t i o n s the o b t a i n e d e-h d e n s i t y is found to be prop o r t i o n a l to the e x c i t a t i o n intensity.
This is in c o n t r a s t to the
s i t u a t i o n of a system in thermal equilibrium.
Here, the r e a l i z e d densi-
ty w o u l d e x c l u s i v e l y be d e t e r m i n e d by the t h e r m o d y n a m i c parameters.
III.2 Optical N o n e q u i l i b r i u m Systems M a n y optical systems w i t h suitable n o n l i n e a r i t i e s
exhibit nonequi-
librium phase transitions. A very important example is the laser w h i c h even shows a full h i e r a r c h y of those transitions.
Firstly,
at the laser
t h r e s h o l d coherent r a d i a t i o n d e v e l o p s out of the thermal light. For increasing pump power, up into short pulses.
e v e n t u a l l y the c o n t i n u o u s
laser r a d i a t i o n breaks
Still higher pumping leads to strongly fluc-
tuating m a n y - m o d e emission. A t h e o r e t i c a l analysis indicates that most of the n o n e q u i l i b r i u m transitions in lasers occur c o n t i n u o u s l y topic of this book.
[11 - 13]
In some special cases,
hibit first-order transitions.
. Thus they are not the
however,
a laser may also ex-
This happens for example if one inserts
an additional s a t u r a b l e absorber into the cavity. Hereby,
the total
losses of the light in the r e s o n a t o r become i n t e n s i t y dependent.
This
m a y lead to a d i s c o n t i n u o u s onset of laser action at the threshold. Details of this system will be d i s c u s s e d in the next chapter
(III.2a).
The very topical effect of optical b i s t a b i l i t y will be i n v e s t i g a t e d in the subsequent chapters.
Particularly,
array of two-level atoms in a cavity. optical bistability.
chapter III.2b deals w i t h an This is the classical m o d e l for
The relevant equations are the so-called optical
M a x w e l l - B l o c h equations, w h i c h will also be applied to the laser theory in chapter III.2a.
Some of the recent results for optical bistabi-
lity will be d i s c u s s e d in detail.
In this context,
it has been
85 shown, that a class of solutions of the Maxwell-Bloch equations exhibit limit cycle behaviour and even deterministic chaos. These features will be investigated in chapter III.2c.
Finally,
the last two chapters,
III.2d and III.2e, deal with optical bistability in semiconducting materials.
Such systems are especially interesting for possible device
applications,
e.g., as logical elements.
III.2a First-Order Nonequilibrium Phase Transitions
in Lasers
The optical effects of polarizable matter are often described by the so-called Maxwell-Bloch equations.
In the semi-classical approximation,
these equations are usually formulated for the electromagnetic (r,t), for the atomic polarization P(~,t)
field
and for the atomic inversion
o (r,t). To keep the analysis as transparent as possible,
in the follow-
ing the often complicated energy spectrum of the atomic system will be approximated by a simple two-level model. Only those two energy levels will be considered explicitly, cal transition.
which directly participate in the opti-
The influence of all other states is summarized in
form of dissipative contributions and noise terms in the dynamic equations of the relevant variables. The deterministic part of the optical Maxwell-Bloch equations is [141]
I ~2E
~2~
c ~ - ~ -AE= - c---/ 4~ ~-~/
,
(III.13)
82p +2Yi 3--~ 8P +92p~ =-2g2E~
,
(III.14)
and (III.15)
In these equations,
c is the velocity of light in vacuum,yiand y11are
the transversal and longitudinal damping constants, frequency of the atomic polarization,
g is the optical matrix element
and o 0 is the inversion due to external pumping, tem of partial differential equations fied considerably within the so-called imation'.
~ is the eigen-
(III.13)
respectively.
- (III.15)
The sys-
can be simpli-
'slowly varying enveloppe approx-
Here, one splits off the rapidly varying parts of E and P
98
÷ ÷ ei(Kr-~t%,,+c.c. (r,t)=E(r,t)
(r,t) =P(r,t)
+c.c °
The a m p l i t u d e s E and P are assumed to vary slowly in space and time. Inserting this ansatz into Eqs.
(III.13) -
the second d e r i v a t i v e s of the amplitudes
DE
(III.15)
and n e g l e c t i n g
leads to
~t -
'~iA~+KJE+~E0-igP+FE
'
(III.16)
~P 8t -
(iA~+yj)P+ig~E+Fp
,
(III.17)
2o ~t
where <
_
-YII (c-~0)-2ig(P~E-E~P)+Fc
is the loss rate and A~ , ~
,
(III.18)
are the d e t u n i n g s of the electric
field and the atomic p o l a r i z a t i o n w i t h respect to the frequency
~0 of
the incident field E 0 . In d e r i v i n g these equations a m e d i u m in an optical r e s o n a t o r has been assumed,
in w h i c h the s u p e r p o s i t i o n of forward
and b a c k w a r d t r a v e l l i n g waves has created a s p a t i a l l y homogeneous ation.
Details of this so-called
cussed,e.g., in Ref.
'mean-field'
situ-
a p p r o x i m a t i o n are dis-
141.
On the r i g h t - h a n d sides of Eqs.
(III.16)
-
(III.18)
random fluctuations
Fi, i=E, P, o, have been added to the d e t e r m i n i s t i c contributions. They d e s c r i b e the noise of the incident light and the thermal noise of the medium.
G a u s s - M a r k o v p r o p e r t i e s are assumed,
i.e.,
=0 and =2Qi~ij ~ (t-t') w h e r e i, j stand for E, P, and
The system of equations
c •
(III.16) - (III.18)
has been treated in a
97
v a r i e t y of approximations. in
[141 - 144]
Reviews of the results can be found,e.g.,
. In this book only two special cases w i l l be d i s c u s s e d
in some detail.
This is the s i t u a t i o n E0=O, d m = Av
b e t w e e n e l e c t r o m a g n e t i c and p o l a r i z a t i o n field)
(exact r e s o n a n c e
and
~11' ~ l >><
(iii.19)
The second limiting case w i l l be s p e c i f i e d in chapter III.2b d e a l i n g w i t h optical bistability.
Physically,
the inequality
(III.19)
implies slow r e l a x a t i o n of the
e l e c t r o m a g n e t i c field on the t i m e - s c a l e of the atomic v a r i a b l e s P and o
. The atomic system r a p i d l y follows the changing field.
rative sense,
it is
'slaved' by the field
role of an order parameter.
[11]
In a figu-
, w h i c h assumes the
The e f f e c t i v e dynamic e q u a t i o n for the or-
der p a r a m e t e r is o b t a i n e d by a d i a b a t i c a l l y e l i m i n a t i n g the atomic degrees of freedom. DE ~t Here, and
-
For the s i n g l e - m o d e laser this leads to
(K-~)E-yIEI2E+F
[11] (III.20)
~ is p r o p o r t i o n a l to ~0, F is the e f f e c t i v e noise c o n t r i b u t i o n
y is a c o m b i n a t i o n of the p a r a m e t e r s in
The e x p l i c i t expressions are g i v e n in
Eqs.
(III.17) .and (III.18).
[11] . The e q u a t i o n
(III.2o) m a y
be w r i t t e n as DE ~t
-
~ F ~-~+
(III.2oa)
'
w h e r e the g e n e r a l i z e d G i n z b u r g - L a n d a u p o t e n t i a l is ~=(<~)I+yI 2
,
(III.21)
and
i=l~l 2 If the d a m p i n g c o n s t a n t
~ is i n d e p e n d e n t of the field,
(I) r e s e m b l e s that of F(q) the role of the temperature.
p l o t t e d in Fig.
the shape of
3. The p a r a m e t e r
The p o t e n t i a l shown in Fig.
y assumes
3 is t y p i c a l
for a continuous phase transition.
A far m o r e s o p h i s t i c a t e d analysis of the laser e q u a t i o n s has b e e n presented,
e.g., in Ref.
141. Instead of the s i n g l e - m o d e laser, one m a y
also i n v e s t i g a t e the m u l t i - m o d e s i t u a t i o n or even a laser w i t h a c o n t i n u u m of modes.
These s y s t e m s e x h i b i t a d d i t i o n a l
instabilities
like the t r a n s i t i o n to pulsed r a d i a t i o n and to chaotic emission.
The
onset of u l t r a - s h o r t pulses may be a continuous or a d i s c o n t i n u o u s
98 nonequilibrium resonator
phase transition depending
[145,146
]. Other examples
lasers are systems with a saturable and a dye-laser with a special dye
on the length of the laser
for discontinuous absorber
dependence
light intensity.
loss constant
Here, ~0 describes
(III.22)
the resonator
< =<0 has been used in Eqs. (III.22)
losses without additional
(III.2o)
and
(III.21).
the saturable absorber. into
(III.2o)
I s is the saturation
intensity.
yields the g e n e r a l i z e d
Ginzburg-
~SA=(<0-a)I+<sIs£n(1+I/Is )+YI2. the logarithm
(III.23)
leads to
#SA=(<0-e+<s)I+(Y-<s/2Is)I2+(~s/3I~)I3 ~ If <s/2 Is> Y
absorber.
The second term in
,
Inserting Eq. (III.22) Landau potential
Expanding
T h e effect
< of the form
<=K°+<s/(1+I/Is)
characterizes
148]
is a
in a laser cavity can be modelled by the intensity
of an effective
< s / ( l + I / I s)
in
[147,
[149]. A saturable absorber
m a t e r i a l which is bleached by the increasing of such an absorber
transitions
inside the cavity
, the v a r i a t i o n of #SA(I)
curves shown in Fig.
....
(III.24)
for different ~ resembles
4. These are typical for a first-order
the
phase tran-
sition. Quite similar calculations may be done for the other d i s c o n t i n u o u s equilibrium phase transitions cial models
in laser systems.
are by far less important
This d i s c o n t i n u o u s III.2b Optical
non-
all these spe-
than optically bistable
effect will be discussed
Bistability
However,
systems.
in the following chapters.
of Two-Level Atoms
in a Resonator
Optical b i s t a b i l i t y
is an effect which can be observed
in light trans-
mission
using a suitably nonlinear medium.
Under certain
experiments
conditions
one finds two possible values of the transmitted
I t for one and the same incident sketched
in Fig.
51
intensity
intensity
I o. Such a situation
is
99
Fig.
51: S c h e m a t i c a l e x p l a n a t i o n of an e x p e r i m e n t showing o p t i c a l bistability. The intensity I of the light t r a n s m i t t e d t through the n o n l i n e a r m e d i u m is p l o t t e d as f u n c t i o n of the incident intensity I 0 .(From Ref. 150.)
Continuously
increasing the incident i n t e n s i t y I0 causes i n c r e a s i n g
t r a n s m i s s i o n until at I 0 = B ble b r a n c h is reached. discontinuous
(see Fig.
51) the end of the lower bista-
The t r a n s m i t t e d i n t e n s i t y I t exhibits an almost
jump to a high v a l u e on the upper b r a n c h of the h y s t e r e -
sis line. The upper branch is f o l l o w e d for d e c r e a s i n g I0
= C a discontinuous
I0
, u n t i l at
jump occurs b a c k onto the lower b i s t a b l e branch.
Optical b i s t a b i l i t y may be c l a s s i f i e d as d i s c o n t i n u o u s phase t r a n s i t i o n b e t w e e n o n - s t a t e and o f f - s t a t e , i . e . , transmission,
respectively.
b e t w e e n the state of high and low
A c o n s t a n t flow of energy t h r o u g h the
system is n e c e s s a r y for m a i n t a i n i n g the b i s t a b l e states. Thus all criteria are f u l f i l l e d for a f i r s t - o r d e r n o n e q u i l i b r i u m t r a n s i t i o n in a p h y s i c a l system. The effect of optical b i s t a b i l i t y has b e e n p r e d i c t e d in the year by Sz~ke et.al.
[151]
1969
. First n u m e r i c a l studies are due to M c C a l l
[152].
G i b b s / M c C a l l / V e n k a t e s a n s u c c e e d e d first in r e a l i z i n g optical b i s t a b l e behaviour experimentally
[153]. Since then,
optical b i s t a b i l i t y has
been i n v e s t i g a t e d for m a n y d i f f e r e n t c o n d i t i o n s materials.
Up to now,
to this effect.
in a large v a r i e t y of
several h u n d r e d p u b l i c a t i o n s have b e e n d e v o t e d
Surveys of the e x t e n s i v e l i t e r a t u r e may be found in the
c o n f e r e n c e p r o c e e d i n g s [142-144] and in [15o, 154,
155]. In the f o l l o w i n g
p a r a g r a p h s some topics w i l l be d i s c u s s e d w h i c h are of r e l e v a n c e to optical b i s t a b i l i t y as an example for a f i r s t - o r d e r n o n e q u i l i b r i u m phase transition.
100
N o n l i n e a r m a t e r i a l p r o p e r t i e s are the p h y s i c a l p r e r e q u i r e m e n t s of o p t i c a l l y b i s t a b l e behaviour.
M a t h e m a t i c a l l y they are c o n v e n i e n t l y
d e s c r i b e d by an intensity d e p e n d e n t d i e l e c t r i c function, e(m)=e' (~) +ie" (~) c h a r a c t e r i z i n g the medium.
In m o s t experiments,
this m e d i u m is placed
inside the cavity of an optical r e s o n a t o r b e c a u s e this causes an a d d i t i o n a l feedback for the light. The following d i s c u s s i o n will show, how an intensity d e p e n d e n t d i e l e c t r i c f u n c t i o n leads to m u l t i v a l u e d t r a n s m i s s i o n of such a nonlinear resonator.
An optical r e s o n a t o r
(Fabry-Perot resonator)
is a device consisting
of a cavity w i t h m i r r o r s at the end faces. Each m i r r o r has the reflectivity r and the t r a n s m i s s i v i t y t for a light w a v e of the a m p l i t u d e E. Light coming into this r e s o n a t o r is thus partly t r a n s m i t t e d and p a r t l y reflected.
The total t r a n s m i s s i o n c h a r a c t e r i s t i c s may be c a l c u l a t e d as
s u p e r p o s i t i o n of the d i r e c t l y t r a n s m i t t e d beam and the p a r t i a l l y reflected beams [156 ]. The a m p l i t u d e E t of the t r a n s m i t t e d light is Et=E0t2ei~ ~
(rei~) 2n
(III.25)
n=o
The first term in the sum
(n=o)
internal r e f l e c t i o n s etc.
. The q u a n t i t y 6 in Eq.
is the direct beam,
n=1 d e s c r i b e s two
(III.25)
is the com-
plex phase shift of the light passing through the m e d i u m inside the cavity ~=~'+i~"=D/~c ~/c
(III.26)
D is the total length of the resonator. t2=1-R
For m i r r o r s w i t h o u t
losses
,
where r2=R . Summing
up
the g e o m e t r i c a l series in Eq.
(III.25)
and c a l c u l a t i n g the
a b s o l u t e square yields the e q u a t i o n for the t r a n s m i t t e d i n t e n s i t y It=I0(1-R2)/[(e
It(~)
-Re
)2+4Rsin2S']
(III.27)
shows p r o n o u n c e d m a x i m a at certain values of ~
e i g e n f r e q u e n c i e s of the resonator. s e p a r a t e these maxima.
The r e l a t i o n
w h i c h are the
Broad regions of low t r a n s m i s s i o n (III.27)
establishes a nonlinear
101
d e p e n d e n c e of I t on I0 if the d i e l e c t r i c function, shift
and thus the phase
~ are functions of I0 • Under certain c o n d i t i o n s one finds bi-
stable t r a n s m i s s i o n c h a r a c t e r i s t i c s
similar to that shown in Fig.
51.
K n o w l e d g e of the d i e l e c t r i c f u n c t i o n ¢ yields the a b s o r p t i o n coefficient
~
of the m e d i u m e (~)=~¢"/n(~)c
(III. 28a)
as well as the index of r e f r a c t i o n n (~) ={~[ a '+ [( ¢')2+ (E") 21 1/2] } ~2 Here,
c is the speed of light in vacuum.
(III.28b) If
E is a f u n c t i o n of I 0 ,
one has also intensity d e p e n d e n t a b s o r p t i o n and refraction. 28a)
and
(III.28b)
show, that both effects are coupled.
some f r e q u e n c y range it may happen, dence of the optical p r o p e r t i e s tion.
Eqs.
However,
(III. in
that the d o m i n a n t intensity depen-
is due to either r e f r a c t i o n or absorp-
If optical b i s t a h i l i t y occurs in such a medium,
it is u s u a l l y
r e f e r r e d to as d i s p e r s i v e b i s t a h i l i t y or a b s o r p t i v e bistability,
res-
pectively,
Typically~
a b s o r p t i v e b i s t a b i l i t y m a y be caused by the s a t u r a t i o n
of an a b s o r p t i o n resonance. ches from high absorption,
Under light e x c i t a t i o n the m e d i u m switi.e.
low transmission,
i.e. high transmission.
Quite recently,
has also b e e n observed.
Here,
to low absorption,
exactly the o p p o s i t e effect
one is dealing w i t h m a t e r i a l s
in w h i c h
a b s o r p t i o n is induced through e x c i t a t i o n w i t h light. B i s t a b i l i t y can occur if the t r a n s m i s s i o n switches from low a b s o r p t i o n for low incident intensities to high a b s o r p t i o n for high intensities. been o b s e r v e d
in
the s e m i c o n d u c t o r m a t e r i a l CdS.
an external r e s o n a t o r feedback.
This effect has
It does not require
Some aspects of o p t i c a l b i s t a b i l i t y due
to induced a b s o r p t i o n w i l l be d i s c u s s e d in chapter III.2e.
In the case of d i s p e r s i v e bistability,
the a d d i t i o n a l r e s o n a t o r feed-
back is essential. At low incident intensities I 0 , the light is originally detuned w i t h r e s p e c t to an e i g e n m o d e of the r e s o n a t o r causing low transmission.
W i t h increasing i n t e n s i t y the r e f r a c t i v e index of
the m e d i u m changes so that the d e t u n i n g
is reduced and the internal
intensity increases. At a certain v a l u e of I 0
the state of low trans-
m i s s i o n becomes u n s t a b l e and the r e s o n a t o r switches to high transmission.
102
After
these
be d e v o t e d
introductory to optical
resonator
[157 - 162]
ted in chapters level m o d e l Eqs.
. More
III.2d
the r e m a i n d e r
-
complicated
and III.2e.
(III.18).
the d i e l e c t r i c
of this chapter will
of a system of t w o - l e v e l materials
The r e l e v a n t
are the optical M a x w e l l - B l o c h
(III.16)
to y i e l d
remarks,
bistability
Eqs.
Eq.
(III.16)
(III.29)
sult
-
(see Fig.
~sand
(III.29) 6s
(III.18).
(III.28)
they can be shown
function
are c o m b i n a t i o n s
The e x p l i c i t
establishes
into Eq.
for the two-
in the form of
(III.19),
i+A e=1+~ s 1+AZ+6si Here, A = A~/y~ and
in a
will be investiga-
equations
equations
For the c o n d i t i o n
atoms
the d e p e n d e n c e
yields
of the p a r a m e t e r s
expressions
of
bistability
~
are g i v e n
in
on I. Inserting
for some p a r a m e t e r
of [16o ] .
this revalues
52).
I
I
I
l
--
i
/~
/// /
/
//
_
,,'/
/J
///
5
10
15
20
Io Fig.
52: T r a n s m i t t e d intensity I t versus input i n t e n s i t y I 0 for a t w o - l e v e l system in a resonator, a) A = o = ¢ , b) A = I, ~ = -I , c) A = 3, # = 4 , w h e r e ¢ =A~ R/(I-R) (From Ref. 16o.)
In Ref.
159 the authors
DE _ ~t
(iA~+<)E+KE0_i~s
by a d i a b a t i c a l l y teresting
a maximum
(
that
formula
for this
system
the L a n g e v i n
i+A 1+Ak+6siEi 2 ) E+Fef f
eliminating
to note,
to the r e s o n a t o r
derive
the a t o m i c
degrees
the b a t h - a v e r a g e d (III.28)
of the transmission.
after
Eq.
equation
(III.3o) of freedom. (III.3o)
It is in-
is e q u i v a l e n t
it has been e x p a n d e d
around
103
A s s u m i n g effective fluctuations Fef f c h a r a c t e r i z e d by = 0
r
=Q~ (t-t')
,
one may w r i t e d o w n the F o k k e r - P l a n c k e q u a t i o n b e l o n g i n g to the Langevin e q u a t i o n
(III.3o)
~f = a (<x-Am ~--t ~-x
[159 ]
y-KE0+ ~ s
x-A ~ . )f + 1+A2+Bs(X2+y2)
92 + ~
Here,
92
s 1+A2+Ss (x2+y2)
the a m p l i t u d e E of the e l e c t r o m a g n e t i c field has b e e n split into
real part x and imaginary part y, E = x + i y. f(x,y,t)
is the joint
p r o b a b i l i t y to find a certain v a l u e of x and y at time t.
For the purely a b s o r p t i v e case, A=0=A~ the s t a t i o n a r y e q u a t i o n
(III.31) m a y be solved a n a l y t i c a l l y
[161]
The s o l u t i o n is
fSt=N(l+8 s(x2+y2)) -~s/Q exp{ K2I(x-E0)2+y2)/Q} , (III.32) w h e r e N is a n o r m a l i z a t i o n constant,
fst is p l o t t e d in Fig.
p a r a m e t e r s w i t h i n the b i s t a b l e regime.
53 for
The figure c l e a r l y shows a
d o u b l e - p e a k e d s t r u c t u r e of the p r o b a b i l i t y distribution.
The w e i g h t
of the m a x i m a m e a s u r e s the r e l a t i v e p r o b a b i l i t y for the r e a l i z a t i o n of the r e s p e c t i v e b i s t a b l e states. In the general case w i t h finite
A and Am , it has not yet been p o s s i b l e
to solve the s t a t i o n a r y e q u a t i o n
(III.31)
the w e a k noise limit, d e v e l o p e d in
analytically.
[_159 I. The authors m a k e the ansatz
fst (x,y) =N e- ¢
However,
in
a very e f f e c t i v e a p p r o x i m a t i o n scheme has b e e n
104
with (III.33)
~=~IQ
f st
i
Fig.
53: S t a t i o n a r y p r o b a b i l i t y d i s t r i b u t i o n fst of the a m p l i t u d e of the f l u c t u a t i n g light field. The s o l u t i o n has been o b t a i n e d for p u r e l y a b s o r p t i v e b i s t a b i l i t y of a system of two-level atoms in a resonator. Re E = x and Im E = y , see text. (From Ref. 161.)
I n s e r t i n g Eq.
(III.33)
into the stationary F o k k e r - P l a n c k equation
(III.31) one finds in the limit of small fluctuations
I (8~% 2
I ~
)2_
•~-~, + ~ ( ~
x-A x (Kx-<E0-Am y +a s
Y+~
~,
.,
- (Ky+Am X +a s I+A2+Ss(X2+y2)
For E0
) ~
1+A2+Ss(X2+y2)
)=-=o.
J ~
(III.34)
= o one knows an exact s o l u t i o n ~0 • 0 (x,y)=~(x,y)
E0=0
.
For r e l a t i v e l y small incident fields E 0
one can solve Eq.
(III.34)
w i t h the ansatz
• =~0+E0 Inserting
~I
(III.35)
(III.35) into
(III.34) yields a f i r s t - o r d e r partial diffe-
r e n t i a l e q u a t i o n for el (x,y). It can be solved easily w i t h the m e t h o d of characteristics.
Inserting the result into
(III.35) yields
[.159]
105
EO
)2 + (y+ ~Eo
2
1+ +2]
~ (x,y) = (x- - I+62
+ F 2 £ n ( l + x 2 + y 2)
-2E0 F 2 (6-A) Im (A(x,y) I Here,
~ =A m/~
tric function.
(III.36
and A is p r o p o r t i o n a l to
(x + i y) times a h y p e r g e o m e -
The d e t a i l e d e x p r e s s i o n is given in [159]
. For the
special case w i t h o u t detuning, 6 = o , the authors i n v e s t i g a t e the none q u i l i b r i u m a n a l o g u e to the M a x w e l l rule of t h e r m o d y n a m i c equilibrium. This can be done c a l c u l a t i n g the r e l a t i v e d e p t h of the two m i n i m a of (x,y). These m i n i m a d e s c r i b e the stable s t a t i o n a r y states of the system. Especially,
the a b s o l u t e m i n i m u m c h a r a c t e r i z e s the m o s t proba-
ble values of x and y. The second, stable state,
local m i n i m u m c o r r e s p o n d s to a meta-
the lifetime of w h i c h is m e a s u r e d by the height of the
p o t e n t i a l barrier to the a b s o l u t e minimum. depth,
If both m i n i m a have equal
the absolute s t a b i l i t y in the system is exchanged.
This criteri-
on may be v i s u a l i z e d as the n o n e q u i l i b r i u m a n a l o g u e to the M a x w e l l rule of e q u i l i b r i u m thermodynamics.
It is indicated as dashed
line in Fig.
54.
16
I
i
I
14 12 10 8 6 4 2
i '+ 10
i ~ 12
14
I0 Fig.
54: G e n e r a l i z e d M a x w e l l c o n s t r u c t i o n
(dashed lines) for the optical b i s t a b i l i t y of t w o - l e v e l atoms (see text). The full lines are the s t a t i o n a r y solutions of Eq. (III.31) for ~ = o = 6 They have b e e n o b t a i n e d as extrema of the G i n z b u r g - L a n d a u potential (III.36). The upper and lower b r a n c h e s of the full lines c o r r e s p o n d to the m i n i m a of the potential, w h e r e a s the i n t e r m e d i a t e line c o r r e s p o n d s to a r e l a t i v e maximum. It is t h e r e f o r e an u n s t a b l e steady s t a t e . ( F r o m Ref. 159.)
106
For ~ =
o the purely absorptive
ditions
of detailed
Maxwell
construction
finds deviations
Here,
[159 ] and the generalized
reduces to the equal area rule.
of optical bistability.
results have been obtained To calculate
an eigenvalue
For the eigenvalue
time dependent
analysis
result corresponds
formation on the dynamics
For ~ # o one
is contained
analysis
for the dynamics solutions,
of the F o k k e r - P l a n c k in the solutions
the F o k k e r - P l a n c k
equation
In @
therefore
is conveniently
(III.37) operator LFp is a d i f f e r e n t i a l
f. Usually
operator acting on
it is not a hermitian operator.
need not to be real and the eigenfunctions
orthogonal
set, neither.
mathematical
analysis.
thus the generalized
These features
However,
once the stationary
Ginzburg-Landau
The eigenvalues
need not form an
considerably
complicate solution
potential # are known,
Fokker-Planck
equation can be transformed
The resulting
hermitian F o k k e r - P l a n c k
into a hermitian
operator
the
fst and the
form
[ 16 ].
is
= e # / 2 L F p e -#/2 and Eq.
o .
equation
~f f ~-~=LFp The F o k k e r - P l a n c k
one
equation.
to the eigenvalue 10 = o and the in-
in the form of an operator
the function
the con-
from this rule.
usually performs
written
is reached.
balance are fulfilled
Only very few analytical
The stationary
situation
(III.38)
(III.37) may be w r i t t e n as in = i n f n
(III.39)
=e%/2f
(III.4o)
Here,
and the expansion f = ~ Cnf n e n have been used.
-I t n
(III.41)
The eigenvalues
The analogy of the transformed Schr6dinger
I are real and positive definite n Fokker-Planck
equation can be utilized
equation
problems of interest. In Ref. to calculate
to the
to apply the approved approxima-
tion schemes of single particle quantum m e c h a n i c s method
(III.39)
[16].
to the n o n e q u i l i b r i u m
162 the authors develop a variational
the first non-vanishing
eigenvalue
11 for the purely
107
absorptive limit of Eq. known
(III.32)
(III.31). Here, the stationary solution is
and L can be calculated analytically.
As mentioned above,
the bistable regime is characterized by two minima of the generalized Gi~zburg-Landau potential,
one of it corresponding to the absolutely
stable state and the other to the metastable this situation,
the first eigenvalue
state
(see Fig.
54). For
11 describes the rate at which
the potential barrier between the minima may be surmounted.
(XI) -I
measures the lifetime of the metastable state. An example for the results of Ref.
162 is shown in Fig. 55.
I
I
I
I
~%1 ~,1!: 0.5 [t~..~lt : I ~ ~,~t: 10 | %/ , I
0 Fig.
I
I
2
4
I
I
,
,.-.,
.,.:.~.~. .........~\ , I
I
I
G
I
,
I
B RelE
55: Approximate time dependent solution of the FokkerPlanck equation for purely absorptive bistability (.see text). The sum on the right-hand side of Eq. (III.41) has been restricted to n=o and n=l. The first non-vanishing eigenvalue 11 has been calculated var±ationally. (From Ref. 162.)
The probability distribution f(x,y,t)
is plotted versus x=Re(E)
for
various times after preparing the system in the metastable state at x=0.5
. The initial configuration decays and the system evolves to-
wards the equilibrium. tion is at
x=8
.
Here, the maximum of the probability distribu-
Thus the obtained approximate solution yields
qualitative results for the dynamic problem. However, restriction to a variational
because of the
calculation of the lowest non-vanishing
eigenvalue it does not allow for quantitative conclusions. In Refs.
163 and 164 the respective authors report numerical solu-
tions of the deterministic part of Eq. tive case. The results,
(III.3o)
for the purely absorp-
e.g., show the switching to the upper bistable
branch after the incident field has reached the instability point of the lower branch however,
(Io=B in Fig.
51). Negelcting the random fluctuations,
clearly over-estimates the stability of the metastable
states.
108
III.2c D e t e r m i n i s t i c Chaos in Bistabl e Systems
D e t e r m i n i s t i c chaos is the b e h a v i o u r of a p h y s i c a l system whose time d e p e n d e n c e a l t h o u g h being purely d e t e r m i n i s t i c n e v e r t h e l e s s m a y exhibit chaotic behaviour.
The study of these systems is a very topical subject
in the p h y s i c a l research.
I n t r o d u c t i o n s to the field and reviews of the
state of art are given, e.g.,
As a p r e r e q u i r e m e n t cal systems,
in [165-17o]
for the o c c u r e n c e of d e t e r m i n i s t i c
chaos in physi-
the m a t h e m a t i c a l d e s c r i p t i o n of the time e v o l u t i o n has to
be given by n o n l i n e a r equations of motion.
They need to be very sensi-
tive to small v a r i a t i o n s of the initial value conditions.
Then origin-
ally close phase space t r a j e c t o r i e s separate e x p o n e n t i a l l y fast.
Hsually,
chaotic solutions are not found for the c o m p l e t e region of
control parameters.
The system a p p r o a c h e s the chaotic regime in a very
c h a r a c t e r i s t i c manner.
One d i s t i n g u i s h e s b e t w e e n at least three so-
called routes to chaos. These are the F e i g e n b a u m or p e r i o d - d o u b l i n g route,
the i n t e r m i t t e n c y route,
and the R u e l l e - T a k e n s - N e w h o u s e route.
A d e t a i l e d d i s c u s s i o n of these m e c h a n i s m s w o u l d go far beyond the scope of this book. The interested reader is r e f e r r e d to the original literature,
e,g.
, [165-17o]
and the references given therein.
In the context of optical bistability, p r e d i c t e d first by Ikeda
d e t e r m i n i s t i c chaos has been
[173] for a ring r e s o n a t o r with a very long
r o u n d - t r i p time. The o b t a i n e d solutions have been i n v e s t i g a t e d in some detail,
e.g.,
in [172-174]
. They could be linked to a m u l t i - m o d e
instability of the M a x w e l l - B l o c h equations for this system. meantime,
In the
the Ikeda i n s t a b i l i t y has been d e m o n s t r a t e d e x p e r i m e n t a l l y
in several devices
[175,
176
]
For ring r e s o n a t o r s with a short r o u n d - t r i p time,
a single-mode
stability has been p r e d i c t e d to lead to chaotic solutions
in-
[177-182].
In these r e f e r e n c e s a system has been i n v e s t i g a t e d w h i c h consists of two-level atoms w i t h i n the ring resonator.
The model
is treated also
in the framework of the optical M a x w e l l - B l o c h equations. A s s u m i n g slowly v a r y i n g amplitudes
for the e l e c t r i c field and the p o l a r i z a t i o n
109
field, Eqs.
the spatially a v e r a g e d equations are f o r m a l l y identical to
(III.16)
-
(III.18) w h i c h have been f o r m u l a t e d for a F a b r y - P e r o t
etalon.
It has been found,
that these equations allow solutions d e s c r i b i n g the
c o n v e r s i o n of t e m p o r a l l y c o n s t a n t i n c i d e n t light into light w i t h a p e r i o d i c a l l y v a r y i n g amplitude.
For p a r a m e t e r s
corresponding
to the
upper b i s t a b l e b r a n c h of the steady state t r a n s m i s s i o n curve, a stability analysis of the d e t e r m i n i s t i c equations
indicates the e x i s t e n c e
of d i f f e r e n t p e r i o d i c solutions and of chaotic s o l u t i o n s
(see Fig.
56).
IEI !
9o ! 80 70 60 50 z,O
30
10 I
......r*"'7,
,
0o
Fig.
,
i
.
500
.
.
.
I
1000
,
.
,
,
I
~500
,
,
-
Eo
56: B i s t a b l e t r a n s m i s s i o n curve of the light field a m p l i t u d e IEI versus the incident field E . The s t a b i l i t y analysis of the d e t e r m i n i s t i c part of Eqs. ~ I I I . 1 6 ) - ( I I I . 1 8 ) yields the following solutions: stable solutions ( o ) , u n s t a b l e solutions ( D ) , solutions w i t h short time o s c i l l a t i o n s ( • ), p e r i o d i c solutions ( • ), and chaotic solutions ( x ). (From Ref. 182.)
A n u m e r i c a l i n t e g r a t i o n of the d e t e r m i n i s t i c parts of the full equations
(III.16)-(III.18)
tain these solutions,
yields the results shown in Fig.
57. To ob-
the initial values of E, P, and ~ have been
c h o o s e n as E (t=0) =Est+~E
,
(III.42)
P(t=0)=pSt+~p
,
(III.43)
110
and
Fig.
o(t=0)=ost+6~
57 a - g:
.
(III.44)
L e f t c o l u m n : Time d e p e n d e n t f i e l d i n t e n s i t y T(tlfor v a r i o u s i n c i d e n t f i e l d s E : a) ~ = 1 4 1 6 . 8 1 , b) P ---472.13, c) E = 3 7 9 . o 3 , d) E = 3 5 1 . 6 ~ , e) E ° = 3 4 6 . 6 7 , fl ~°=342.83, g) ~ = 3 4 2 . 6 1 . The ~hown s o l u t i o n ° have been o b t a i n e d by nume~ically integrating the deterministic part of the Maxwell-Bloch equations for parameters specified in [ 182], Right column: Phase space representation of the complex electric field for the situations shown in the respective left figures. The parameter of the curves is the time. (From Ref. 182.)
111
In Eqs.
(III.42)
-
(III.44),
on the upper b i s t a b l e 6P
E st, pSt,
branch
, 6a are small deviations.
rized
in Fig.
x stands
the solutions
input field E °
For the n u m e r i c a l
for E, P,
and
summa-
such that
,
a , respectively.
are i n d e p e n d e n t
values 6 E,
integrations
57 the q u a n t i t i e s 6 x have b e e n c h o o s e n x/xSt=1o-3
where
and a st are the s t a t i o n a r y
for the s p e c i f i e d
of 6x as long
It has been
found,
successive
period
that
as
l~xl<
of results
ling for d e c r e a s i n g
shown in Fig.
incident
57 e x h i b i t s
field E o.
lates w i t h a single basic p e r i o d w h i c h phase
space trajectory,
of a simple pectively. regime.
loop. Fig.
I(t)
e xhibit s the p h a s e
Figs.
Re(E)
versus
57a the i n t e n s i t y
is l a b e l l e d
Im(E),
has no p e r i o d i c shape.
to a v a l u e
structure
With
as period
is a limit
57 b - e show p e r i o d s
57f c o r r e s p o n d s
no closed
In Fig.
I. The in form
2, 4, 8, and 16, res-
of E ° w i t h i n
and the p h a s e
increasing
cycle
time
the c h a o t i c
space t r a j e c t o r y
it s u c c e s s i v e l y
space.
T 800 16
400
4
40
~ 3O0
Fig.
4O0
........ ___Z_..............._.1_.1 500
douboscil-
E"o
58: P e r i o d length T versus i n c i d e n t field E o. The numbers label the r e s p e c t i v e periods. (From Ref. 182.)
fills
112
The
regions
period
of the d i s c r e t e
length
T is p l o t t e d
T = t(ImP+1x) where
t ( I mxai
labels
) is the
time
the r e s p e c t i v e
The period-doubling To p r o v e structs
are
the
summarized
incident
in Fig.
field.
- t(Ilax)
,
at w h i c h
the
58.
The
T is d e f i n e d
as
(TII.45)
i-th maximum
of I occurs,
p
period.
route
unambigously a so-called
periods versus
to chaos
that
the
return
map
is k n o w n
system
as F e i g e n b a u m
follows
this
[17o ] . Here,
szenarium.
route,
one plots
one
the
con-
intensity
maxima gime
I n+l v e r s u s the p r e c e e d i n g m a x i m a I n . In the p e r i o d i c remax max this c r e a t e s a m a p in w h i c h the nulnber of p o i n t s c o i n c i d e s w i t h
the r e s p e c t i v e
period.
For
continuous.
example
is s h o w n
An
ii
0
chaotic
solutions,
in Fig.
e,
the
curve
becomes
quasi-
59.
•
o
%
@
0
% % 0
%
% o o 0 o Q
0:e
0:s
1:0
1:1
1:2
iLs
n Imax
Fig.
For
59:
the
R e t u r n m a p for a v a l u e E = 3 2 4 . 6 6 . n+1x c h a o t z•c regzme. T h e m a x l.o m a I__ v e r s u s the r e s p e c t l v e p r e c e e ~ i n g ( F r o m Ref. 182.) investigated
to e x h i b i t
always
system,
single
the
maxima
curves which
I. ,,E/-,
a
w i t h i n the re p l o t t e d maxzma I n max
I mn+1 ax
get
versus
flatter
I mn a x
are
found
for s u c c e s s i v e l y
113
decreasing
field E o. The existence of the single m a x i m u m
indicates
that the route to chaos is correctly described by the F e i g e n b a u m narium
[17o] . Consistently,
one should find windows
regime,
where again periodic
solutions
these oscillations
should be different
w i n d o w has been reported Fig.
57g. It has period
occur.
The basic period of
from 2. An example for such a
in [ 182 ] . The solution 12, belonging
is reproduced
(III.16)
- (III.18)
periodic and chaotic solutions has been investigated values of the noise strengths Qi corresponding situations,
most of the periodic
a broadening
for the higher periods.
to p h y s i c a l l y
realistic
and even the chaotic regime w h i c h becomes
lead to
successively
An example of the results
is
60.
_
Fig.
on the
in [ 182 ] . For
the random fluctuations
of the phase space trajectories
more prominent shown in Fig.
solutions
Basically,
in
to the basic period 3.
The influence of the noise terms in Eqs.
has been found to survive.
sze-
in the chaotic
I
60: The time dependent intensity (left column) is plotted without noise (full ~nes) and with noise (dashed lines) for the periods I, 2, and 4, respectively. The figures in the right column show the pertinent phase space trajectories. The values foe the noise strengths are QE=lO -6, Qp=5-Io-5 and Qo =5.1o -7 , respectively. (From Ref. 182.)
It is clearly visible that period period 4 is m o d i f i e d fluctuations,until
I is almost uninfluenced,
towards period
eventually
2. This trend continues
at very high noise
whereas for stronger
levels the solutions
114
on the upper b i s t a b l e b r a n c h are destabilized.
III.2d D i s p e r s i v e B i s t a b i l i k ~ in S e m i c o n d u c t o r Etalons
This chapter is d e v o t e d to the d i s c u s s i o n of optical b i s t a b i l i t y in s e m i c o n d u c t o r etalons. The d i s c u s s e d n o n l i n e a r i t i e s are m a i n l y due to d i s p e r s i v e effects even though in these real systems i n t e n s i t y dependent changes of the a b s o r p t i o n occur as well. An example for absorptive b i s t a b i l i t y
in s e m i c o n d u c t o r s w i l l be d i s c u s s e d in the next chap-
ter. The d i f f e r e n t n o n l i n e a r s e m i c o n d u c t o r m a t e r i a l s are of p o t e n t i a l technological
interest.
They may be f a b r i c a t e d as very thin slabs
having a size on the order of microns.
Thus they are well suited for
integrated optics.
A b i s t a b l e d e v i c e can simply be r e g a r d e d as an optical switch. The state of high transmission,
e.g.,
corresponds to the o n - s t a t e and
the state of low t r a n s m i s s i o n c o r r e s p o n d s to the o f f - s t a t e
.
S w i t c h i n g is p o s s i b l e for example by exciting the m a t e r i a l w i t h a temp o r a l l y c o n s t a n t input b e a m
(holding beam) w h i c h has an i n t e n s i t y I O
w i t h i n the b i s t a b l e region. A d d i t i o n or s u b t r a c t i o n of a short pulse w i t h s u f f i c i e n t l y large a m p l i t u d e leads to u p - s w i t c h i n g or downswitching,
respectively.
A s s o c i a t i n g the o n - s t a t e w i t h the numeral numeral O of the b i n a r y number system,
I and the o f f - s t a t e w i t h the
c o m b i n a t i o n s of b i s t a b l e ele-
m e n t s may be u t i l i z e d for algebraic operations.
Additionally,
it is
p o s s i b l e to c o n s t r u c t logical gates and even the a n a l o g u e to a transistor [183 ~. It has been s p e c u l a t e d that these d e v e l o p m e n t s m a y lead to the c o n s t r u c t i o n of a purely optical c o m p u t e r w i t h w a v e g u i d e s
in-
stead of wires and optical elements instead of e l e c t r o n i c a l elements. A tutorial r e v i e w of these c o n s i d e r a t i o n s is given in Ref.
Besides these a p p l i c a t i o n o r i e n t e d aspects,
184.
optical b i s t a b i l i t y in
s e m i c o n d u c t o r m a t e r i a l s is also of f u n d a m e n t a l p h y s i c a l interest. optical nonlinearities
The
in these m a t e r i a l s are due to m a n y - b o d y effects
among the d i f f e r e n t quasi-particles.
This leads to a d i e l e c t r i c func-
tion d e p e n d i n g on the density of electronic excitations. in turn is d e t e r m i n e d by the i n t e n s i t y of the exciting
Thus one has an intensity d e p e n d e n t d i e l e c t r i c function, sing optical b i s t a b i l i t y
This density
laser light. p o s s i b l y cau-
(see also the d i s c u s s i o n in chapter III.2b).
115
The d i e l e c t r i c f u n c t i o n has to be c a l c u l a t e d q u a n t u m m e c h a n i c a l l y .
The
results allow a c o m p a r i s o n w i t h e x p e r i m e n t a l m e a s u r e m e n t s of absorption, r e f l e c t i o n and t r a n s m i s s i o n spectra.
Moreover,
they may serve as basi-
cal i n f o r m a t i o n on the m a t e r i a l p r o p e r t i e s w i t h i n a t r e a t m e n t of optical b i s t a b i l i t y as a n o n e q u i l i b r i u m phase t r a n s i t i o n in an open system.
The binary compound d i r e c t - g a p s e m i c o n d u c t o r s GaAs, CuCl exhibit s u f f i c i e n t l y strong n o n l i n e a r i t i e s optical bistability.
Quite recently,
InSb, CdS,
and
to show the effect of
also the s o - c a l l e d m u l t i p l e
q u a n t u m well devices c o n s i s t i n g of a series of GaAs layers w i t h i n a GaAIAs substrate are being i n v e s t i g a t e d
in this respect.
Hi~torically,
the first e x p e r i m e n t a l r e a l i z a t i o n of b i s t a b i l i t y in a s e m i c o n d u c t o r e t a l o n is due to Gibbs et. al.
[185] . They used a r e s o n a t o r c o n s i s t i n g
of GaAs as active m a t e r i a l w i t h i n a GaAIAs s a n d w i c h w i t h e x t e r n a l ref l e o t i v i t y coatings. duced in Fig.
An example of the e x p e r i m e n t a l findings is repro-
61.
E~
0.5
0 0
Fig.
61:
I
i
I
0.5
1.0
1.5
E x p e r i m e n t a l result showing o p t i c a l b i s t a b i l i t y in GaAs at an e x c i t a t i o n f r e q u e n c y slightly below the e x c i t o n resonance. The t e m p e r a t u r e is T=15 K. (From Ref. 185.)
The d o m i n a n t optical n o n l i n e a r i t y in GaAs is due to the i o n i z a t i o n of the exciton. For increasing e x c i t a t i o n i n t e n s i t y an i n c r e a s i n g n u m b e r of e l e c t r o n - h o l e
(e-h) pairs is created causing screening of the attrac-
tive Coulomb i n t e r a c t i o n b e t w e e n e l e c t r o n s and holes.
This s c r e e n i n g
can b e c o m e strong enough to d e s t a b i l i z e the bound states.
The d i e l e c t r i c f u n c t i o n for the density d e p e n d e n t i o n i z a t i o n of the
116
exciton
has been
sorption
calculated
[186,
and of the r e f r a c t i v e
(III.28b)).
Examples
187]
. Thus,
index are k n o w n
for the t h e o r e t i c a l
. . . .
I
. . . .
I
also the spectra (see Eqs.
results
. . . .
r
of ab-
(III.28a)
are p l o t t e d
and
in Fig.
62.
. . . .
12 t0
BoAs l=lOK
~
08
I
-.-.,
5.,o"c,~'
t
c, Q2
~...:.~
,
. . . .
l
'
'
'
l
. . . .
I
. . . .
Go As /:tOK
38 -
......
.:0
----
n : 5,~0'Scm ~
/I
---
n : ],10~a~ a
/ !
3~
-
3~
i/
V
3(
1505
1500
Fig.
At
62: T h e o r e t i c a l spectra for for various (From Ref.
low d e n s i t i e s
1510 frequency [eV]
1515
1520
results for the a b s o r p t i o n and r e f r a c t i o n GaAs w i t h an e l e c t r o n i c t e m p e r a t u r e T=1o K e l e c t r o n - h o l e pair c o n c e n t r a t i o n s n. 187.)
one sees
the h y d r o g e n - l i k e
exciton
series
of absorp-
tion
lines together w i t h the i o n i z a t i o n continuum. The line for 15 -3 n=5.1o cm shows the a b s o r p t i o n s p e c t r u m at the c r i t i c a l d e n s i t y exciton called
ionization. excitonic
correlation state. tain
The
steep
front
enhancement.
This
of the e-h motion,
For high densities,
frequency
of the a b s o r p t i o n enhancement
even
though
the a b s o r p t i o n
Pe and
respectively. discussed
exhibit
form a b o u n d in a cer-
(III.46)
~h are the chemical
potentials indicates
absorption
is due to the i n v e r s i o n
in the c o n d u c t i o n
Fermi d i s t r i b u t i o n s
h" T h e s e
negative
,
Negative
system
The elctrons
they no longer
of the
range !P=Ue+P h
where
is a c o n s e q u e n c e
becomes
for
is due to so-
chemical
and holes,
gain w h i c h
of the free carrier
band and the holes
with
potentials
of the e l e c t r o n s optical
in the
population.
in the v a l e n c e
the q u a s i - c h e m i c a l
potentials
band
Pe and
are fixed by the number of e-h pairs
117
created by the laser excitation. m fulfilling c o n d i t i o n
(III.46)
An incident test b e a m w i t h a f r e q u e n c y is a m p l i f i e d due to s t i m u l a t e d emiss-
ion by the induced r a d i a t i v e r e c o m b i n a t i o n of e-h pairs.
The r e f r a c t i o n spectra
(lower part of Fig.
62) show p r o m i n e n t d e n s i t y
d e p e n d e n t changes in the region of w e a k a b s o r p t i o n b e l o w the e x c i t o n resonance.
It is b e l i e v e d that these d i s p e r s i v e changes are the under-
lying p h y s i c a l m e c h a n i s m for the e x p e r i m e n t a l l y o b s e r v e d b i s t a b i l i t y in GaAs
(see Fig.
61 ).
Changes of the e-h d e n s i t y also cause changes of the optical spectra due to b a n d - f i l l i n g and b a n d - g a p renormalization.
The b a n d - f i l l i n g
is
the d o m i n a n t reason for the optical b i s t a b i l i t y o b s e r v e d in InSb [155, 188]
. An example for the e x p e r i m e n t a l results is shown in Fig.
/
200 E
IO0
/ I
0
0
Fig.
63.
I
I
200 l.O0 Power In/roW
I
600
63: T r a n s m i t t e d power versus input power for InSb. The e x p e r i m e n t s have been done at 5 K for a f r e q u e n c y slightly b e l o w the u n r e n o r m a l i z e d band-gap. (From Ref. 188.)
Theoretically,
the m a n y - b o d y theory for InSb follows the same lines as
that for GaAs. The m a i n d i f f e r e n c e are the m a t e r i a l parameters. ally,
in InSb the b r o a d e n i n g
~Y
Especi-
of the e x c i t o n r e s o n a n c e is larger than
X
the exciton binding energy E b. E x p e r i m e n t a l l y the e x c i t o n line is thus x not resolved. (In GaAs E b > ~ T). The t h e o r e t i c a l results for InSb [ 187,
Both,
189 ]
show good a g r e e m e n t w i t h the e x p e r i m e n t a l findings.
for InSb and for GaAs, the i n t e n s i t y d e p e n d e n t d i e l e c t r i c f u n c t i o n
is only known n u m e r i c a l l y as the s o l u t i o n of an integral equation. analytic d e p e n d e n c e
e(1) has not b e e n determined.
The
Thus a L a n g e v i n or
F o k k e r - P l a n c k t r e a t m e n t of the n o n e q u i l i b r i u m effects c o n n e c t e d w i t h
118
optical bistability In wide-gap
semiconductors,
has been observed. i.e.
is hardly possible. such as CuCl,
It is due to the formation of excitonic molecules,
, of bound states of two e-h pairs.
cules can be generated dielectric
rather effectively
Experimentally, by two-photon
function E(~) for the process of m o l e c u l e
calculated, [ 19o,
another optical nonlinearity
and an approximate
expression
these moleabsorption.
The
creation can be
is available
analytically
191 ]
4n IM]2~..(~ -~)3-i e (=) =s=+ (E 0-~=) ~x2
Here, ~0 ~x and
and
~
(~2-~ 2) F ^ m x ~m (~m_2~_iYm )
(III.47)
are the low and high frequency dielectric
~m are the respective
molecules,
| j
frequencies
of the excitons
M is the matrix element for m o l e c u l e
formation,
constants, and of the Y m is
the width of the m o l e c u l e resonance, photons
in the medium,
i.e.
and n is the concentration of P , the polariton density. The real and
imaginary part of c(~) are plotted in Fig.
64 for the example of
CuCl.
10
I
I
CuCI
1./,
1.3
994
995 (,a/~T~
Fig.
996
.994
I .995
.995
r.a / r..a.r --.,,.
64: Calculated spectra of the real and imaginary part of the dielectric function ~ (~) for CuCI. The density of photons in the m e d i u m is n =4-1o 15 cm -3. The intensity dependent induced absorption occurs around ~m/2, where ~ m i S the frequency of the excitonic molecule. ~T is the frequency of the transverse excitons. The insert in the figure on the right-hand side shows the p o s i t i o n of ~m/2 relative to the exciton absorption frequency. (From ~ef. 19o.)
119 At s u f f i c i e n t l y high intensity,
an exciting
laser w i t h a f r e q u e n c y
around ~ m / 2 can create excitonic m o l e c u l e s by t w o - p h o t o n absorption. This leads to an a d d i t i o n a l a b s o r p t i o n r e s o n a n c e w h i c h is absent at low intensities.
This r e s o n a n c e a l s o i n t r o d u c e s a r e n o r m a l i z a t i o n of
the e i g e n m o d e s of the system. The s p e c t r u m genmodes,
i.e., of the polaritons,
~(k) of the t r a n s v e r s e ei-
is d e t e r m i n e d from Eq.
(III.47)
by means of the r e l a t i o n 2
~(k,~)=c2k 2
(III.48)
At high i n c i d e n t intensities this d i s p e r s i o n is c h a n g e d around because of the c r e a t i o n of e x c i t o n i c molecules.
m=~m/2
This r e n o r m a l i z e d po-
lariton s p e c t r u m can be m e a s u r e d by t w o - p h o t o n Raman s c a t t e r i n g experiments. The e x p e r i m e n t a l findings for CuCl
[192] and for CdS
[193]
can be fitted by the theoretical results
[19o,
The d e n s i t y of photons in the medium,
, may be e x p r e s s e d in terms
n
191 ] .
P of the internal
light intensity I. l n p = I i / ~ VE
'
(III.49)
w h e r e v E is the energy velocity.
The t r a n s m i t t e d intensity I t is re-
lated to the internal intensity I i It-~ (I-R) Ii/2
,
(III. 5o)
w h e r e R is the r e f l e c t i o n coefficient.
The t r a n s m i s s i o n c h a r a c t e r i s t i c s
of a F a b r y - P e r o t resonator w i t h CuCI as n o n l i n e a r m a t e r i a l is calculated by inserting the e x p r e s s i o n for the d i e l e c t r i c f u n c t i o n together with [ 194
(III.49)
and
(III.5o)
]. The result is shown in Fig.
3.185 3.1eseg'~
%
(III.47),
into the r e s o n a t o r formula
(III.27)
65.
i 7" \ ' ~
~.~ese~ \ " r " ~ ' ~ ' ~ ~I
/,o~. 31858~--
0.7
0 0.8
0.9
t.O
LI
10[10SWlcm2] Fig.
65: C a l c u l a t e d p h o t o n density n D ~ I for a CuCl r e s o n a t o r of I pm thickness as f u n c t i o n bf the i n c i d e n t i n t e n s i t y I o for v a r i o u s frequencies b e l o w ~m/2. (From Ref. 194.)
120
The calculations
yield dispersive
optical b i s t a b i l i t y
for frequencies
slightly below mm/2. Fig.
65 shows that the bistable
for
This is due to the increasing
m approaching
mm/2.
this spectral region.
In the m e a n t i m e
the b i s t a b i l i t y
[194 ] has been realized e x p e r i m e n t a l l y let resonators with an external In Ref.
and the off-state
stable systems, seconds.
The d o w n - s w i t c h i n g
nanoseconds
196]
that very fast switching between the in CuCI. Usually
occurs very fast,
in bi-
i.e. within pico-
times are on the order of the lifetimes semiconductors.
laser below ~m/2,
By choosing
excitonic molecules
their lifetime no longer determines 197 predict d o w n - s w i t c h i n g
the switch-off.
the frequency
cannot be genera-
in [ 198,
are directly transferable to for
199 ] . The reported function.
~ with a m a c r o s c o p i c
fluctuating
light field in the semiconductor
field.
(III.13)
a Langevin
field is decomposed
P (~) =[ (~£+~n£-I)/4~ ]E (~)
E£
respectively.
Inserting Eq.
resonator [ 198 ]. Here,
slowly varying amplitude
to (III.51)
part of the dielectric
(III.51)
into
(III.13)
leads to
to any other fields.
is contained
approximation
in
(see chapter
an effective equation for the spatially averaged
The in-
e . Employing III.2a),
r
r
- E
I- % ~ E + i I n
~+
~
~
1
the
one obtains
light field amplitude
E in the resonator
DE =< t~~-~
one
of the electromagnetic
.
an equation for E, which no longer couples formation on the nonlinear material
of the stochastic
according
and ~n£ are the linear and the nonlinear
function,
calculations
equation for the amplitude E of the
for the wave p r o p a g a t i o n
The p o l a r i z a t i o n
in
To combine the microsco-
treatment
one derives
Eq.
of
other nonlinear materials with analytically
properties, utilizes
The calculations
on the two-photon bistability
known intensity dependent dielectric pic calculations
that
times on the order of picoseconds.
The influence of random fluctuations CuCI has been investigated
of
This are several
ted. They may however be excited virtually which has the advantage Ref.
in
in
.
that give rise to the nonlinearity.
in most d i r e c t - g a p
of the exciting
absorption
predicted
laser excited CuCl plate-
should be possible
the up-switching
the quasi-particles
using
coating [ 195,
197 it has been estimated,
on-state
solutions vanish
E+F(t)
(III.52)
121
Here,
~ R is the eigenfrequency
to which the detuning
A mis being measured.
T r is the transmission perfectly
resonant
towards Eo/
/Tr
of the unexcited resonator with respect
coefficient.
field amplitude
< is the loss constant
Without nonlinear material,
inside the resonator would relax
' where E ° is the amplitude
The phenomenologically
and
the
of the incident
field.
added noise term F(t) has the properties
=0
,
(III.53)
=2Q 6(t-t')
The Fokker-Planck
equation
for the probability
distribution
f(E,E~,t)
of the complex field amplitude ~o E" ~f~t - ~E ~ E( <+ 2 -R~ n£]j - i E
+c.c.+2Q ~ is stochastically
(A ~+ mRen£ ~ ~
a2
allows to calculate
transmission
(~ 2 _
jf
(III.52)
of the resonator.
z.o
/4!%%
15
(III.52).
equation
! !
~, -
/
/ i
'
t
~,~
1
15
10
5
-----~F-/
167e910~
Fig.
(III.54)
to the Langevin equation
/
,-r
f
part of the stationary
the bistable
70
]
----~T r
f
equivalent
Solving the deterministic
<Eo
Z
...............................IO . . . . . . . . . .
3 t. ~67s9103
Z
3 4
66: Calculated mean intensity I and intensity correlation function g(2) together with the result of the deterministic part of the stationary equation (III.52), Imf. The unit of the intensity is 463.5 W/cm2. (From Ref. 198.)
Transmitted
intensity versus input intensity
I o is plotted
in Fig.
66.
122
Here,
the stationary
is indicated
as
solution of the d e t e r m i n i s t i c
'mean-field'
optical bistability. can d e t e r m i n e of noise.
It clearly
Solving the Fokker-
This has been done in Ref.
les I and ~ g i v e n
Transforming
part of Eq.
(III.52)
shows the effect of
Planck equation
the stability of the m e a n - f i e l d
tinued fractions.
yields
result Imf.
(III.54), one
result under the influence
198 with the help of matrix con-
the Fokker- Planck equation to the variab-
by E = / ~ e i~
(III.55)
3f 8--~=L f
(III.56)
for f(I, • , t)
with L=-2
<
E0/~(cos~ r
- fA ~
In Ref.
+
sin~ ~--] +2
~Ren£ (i) ~ ~ 1 32 2e~ ) ~-~ +2Q(~-f I -~ + ~ ~ )
198, the solution of Eq.
of g e n e r a l i z e d
f(x,~,t)=
Here,
-
Laguerre
(III.56)
polynomials
~
j
.
(III.56a)
has been expanded
Ln according m
~ (cn (t)e in~ e-XLn (x) +c. c. 1 m,n=o
x is the normalized
Fokker-Planck
equation,
recursion relations order differential
intensity.
applying
Inserting
equations
(III.57)
this expansion
the o r t h o g o n a l i t y
of the Laguerre polynomials
in terms
to
into the
relation and the
yields a system of first-
for the expansion coefficients
cmn
It has
the structure Cm=PmCm+1+QmCm+RnCm_ I
where c is the n-component ÷÷ m R m contain the coefficients explicit
expressions
An a p p r o x i m a t e
,
(III.58)
vector of c n and the matrices m of the expanded F o k k e r - P l a n c k
are given in Ref.
solution of Eq.
this relation at sufficiently
(III.58)
Pm' Qm' and equation.
The
198. can be obtained by truncating
high values of m and n, i.e. by putting
123
cn=o for m > M and n>N. This allows r e c u r s i v e c a l c u l a t i o n of all exm -p a n s i o n c o e f f i c i e n t s w i t h m < M and n < N. Usually the i t e r a t i v e evaluation is done n u m e r i c a l l y adjusting M and N to o b t a i n optimal accuracy. This s o l u t i o n p r o c e d u r e has b e e n d e v e l o p e d in Ref. cribed in detail in Risken's book
A p p l y i n g this method,
in Ref.
20o and it is des-
[16].
198 the s t a t i o n a r y F o k k e r - P l a n c k e q u a t i o n
(III.56) has been solved. Thus, one may calculate,
e.g.
, the expecta-
tion v a l u e of the i n t e n s i t y = fdI
I fst(I,~)
The result is p l o t t e d in Fig.
(III.59)
66. It r e p r e s e n t s the n o n e q u i l i b r i u m ana-
logue to the M a x w e l l c o n s t r u c t i o n of e q u i l i b r i u m t h e r m o d y n a m i c s the d i s c u s s i o n in chapter III.2b),
i.e.,
(see
it d i s c r i m i n a t e s b e t w e e n stable
and m e t a s t a b l e parts of the m e a n - f i e l d solution.
In Ref.
198, a relative-
ly high noise level Q has been assumed in order to have a smooth distrib u t i o n f u n c t i o n fst. The i n f l u e n c e of the noise is clearly seen in the o f f - b r a n c h of the b i s t a b l e t r a n s m i s s i o n curve in Fig.
66.
Knowing fst one can c a l c u l a t e all m o m e n t s = f d I
~fst(I,~)
(III.60)
I n t e r e s t i n g i n f o r m a t i o n on the p h o t o n statistics is c o n t a i n e d in the second moment,
n=2. It yields the v a r i a n c e of the d i s t r i b u t i o n function,
w h i c h c l a s s i c a l l y is given by g(2)=/2
(III.61)
A q u a s i - c l a s s i c a l a p p r o x i m a t i o n of the q u a n t u m m e c h a n i c a l e x p r e s s i o n is (2)-(-)/2 gqc~where < I > Fig.
(III.62)
(2) and gqcl _(2) are also plotted in = I . The results for gcl
66.
In Ref.
2ol the authors show that g(2) < 1 i n d i c a t e s a change of the
photon statistics, pally,
the so-called p h o t o n - a n t i c o r r e l a t i o n
this is a q u a n t u m m e c h a n i c a l
effect, but Fig.
may already be o b t a i n e d from the q u a s i - c l a s s i c a l
effect. Princi-
66 shows that it
expression
(III.62)
124 w h i c h simply takes into account the c o m m u t a t i o n r e l a t i o n s of the lightfield operators.
Details of the theory of a n t i c o r r e l a t e d photons are
summarized in a recent review article [ 202 ] .
l.s
10=100
=100
4~
10 =350
,.a
'
Io=350
o.i
-w-
~e E
~ 1°=550~ I
l
l
~
~ 10=550 ~
'
0~
,
]m[
I~[
Fig.
67: Left column: S t a t i o n a r y d i s t r i b u t i o n f u n c t i o n fst(E, E e ) for the i n d i c a t e d values of the incident i n t e n s i t y I A. Right column: Contour plots of the d i s t r i b u t i o n functions shown in the left column. The contours are equidistant. (From Ref. 198.)
Fig.
67 shows the c a l c u l a t e d s t a t i o n a r y d i s t r i b u t i o n functions fst for
three d i f f e r e n t values of the incident i n t e n s i t y I o. Io=IOo is still
125
b e l o w the b i s t a b l e regime only one maximum.
(see Fig.
66)
. Accordingly,
For I ° w i t h i n the b i s t a b l e regime,
fst e x h i b i t s
a double-peaked
s t r u c t u r e d e v e l o p s w i t h m a x i m a at the E - v a l u e s c o r r e s p o n d i n g to the upper and lower b i s t a b l e branches.
For small noise levels,
the s t a t i o n a r y F o k k e r - P l a n c k e q u a t i o n
has b e e n solved a n a l y t i c a l l y in Ref.
(III.56)
199 by applying the a p p r o x i m a t i o n
scheme d e s c r i b e d in chapter III.2b, Eqs.
(III.33) -
(III.36). The re-
sults have been found to be in good a g r e e m e n t w i t h a n u m e r i c a l l y cons t r u c t e d p r o b a b i l i t y distribution.
Time d e p e n d e n t solutions for the d i s c u s s e d b i s t a b i l i t y have b e e n obtained by a n u m e r i c a l solution of the L a n g e v i n e q u a t i o n
(III.52) [199 ] .
This e q u a t i o n has b e e n d i s c r e t i z e d according to
E ( t + A t ) = E ( t ) + A t G (E(t) ,Atl+ ~ G(E(t), Eq.
dF
(III.63)
At) is the d e t e r m i n i s t i c part of the r i g h t ~ h a n d side of
(III.52) e x p a n d e d in powers of At. The d i s c r e t e form of the
G a u s s i a n noise term dF
has been found to c o r r e c t l y r e p r o d u c e the d i s c r e t e form of the e q u a t i o n for the second m o m e n t c o n s t r u c t e d w i t h the help of the FokkerP l a n c k equation, dF c h a r a c t e r i z e s n o r m a l l y d i s t r i b u t e d random numbers w h i c h have b e e n o b t a i n e d n u m e r i c a l l y by a s t a n d a r d r a n d o m number generator. The n u m e r i c a l solution of Eq.
(III.63) w i t h one sequence of ran-
dom numbers yields one r e a l i z a t i o n of the stochastic process.
The ex-
p e c t a t i o n values of the i n t e r e s t i n g q u a n t i t i e s have b e e n o b t a i n e d by a v e r a g i n g over m a n y s t a t i s t i c a l l y i n d e p e n d e n t r e a l i z a t i o n s
[199]
.
T h e s e computer simulations yield i n t e r e s t i n g i n f o r m a t i o n s on the dynam i c s of the b i s t a b l e system. As an example, Fig.
68 shows the r e s p o n s e
of the i n t e n s i t y t r a n s m i t t e d t h r o u g h a CuCI r e s o n a t o r to an incident field I (x) w h i c h is i n c r e a s i n g l i n e a r l y from a value b e l o w o the b i s t a b l e region to a value above. For short times, IO(T) c o r r e s p o n d s to v a l u e s for w h i c h only the lower b i s t a b l e branch exists. branch
The t r a n s m i t t e d intensity rises a c c o r d i n g to the slope of that (see Fig.
mations.
66)
. Fig.
68 shows I t c a l c u l a t e d in d i f f e r e n t a p p r o x i -
126
O
~c %"
o.
I
o W
j ~
f
/
T
Fig.
68: Results of the computer simulations of the L a n g e v i n e q u a t i o n (III.63) for the s i t u a t i o n of a linearly increasing incident i n t e n s i t y I (~). As f u n c t i o n s of time are plotted: I (~)/Ioo ~. . . . ) , the q u a s i - s t a t i o n a r y i n s t a n t a n e o u s s ~ l u t i o n I( I (T) ) (-- --- ), the d e t e r m i n i s t i c s o l u t i o n of the full L ~ n g e v i n e q u a t i o n ( ), a typical realization of the stochastic e q u a t i o n (III.63) ( , thin line), and the a v e r a g e over I,ooo r e a l i z a t i o n s ( , thick line). The i n t e n s i t y is in units 463.5 W / c m 2 and the time T is measured in m u l t i p l e s of 6.3 X l o - 2 ps. (From Ref. 199.)
P l o t t e d are the i n s t a n t a n e o u s q u a s i - s t a t i o n a r y
s o l u t i o n I(Io(T) )
ob-
tained from the d e t e r m i n i s t i c part of the s t a t i o n a r y L a n g e v i n equation, the time d e p e n d e n t solution of the d e t e r m i n i s t i c part of the L a n g e v i n equation,
a typical r e a l i z a t i o n of the full L a n g e v i n equation,
and the
average over 1,ooo s t o c h a s t i c a l l y i n d e p e n d e n t realizations. All these results switch from low to high t r a n s m i s s i o n for i n c r e a s i n g Io( T ). The jump in the q u a s i - s t a t i o n a r y s o l u t i o n at • = 32.5 occurs exactly for IO(T)
r e a c h i n g the end of the lower branch.
The time d e p e n d e n t d e t e r m i -
nistic s o l u t i o n switches even later due to the inertia of the system. As already i n d i c a t e d by the g e n e r a l i z e d M a x w e l l construction,
I(Io)
in
Fig 66, the s t o c h a s t i c noise d e s t a b i l i z e s a p a r t of the lower branch. This can be seen from the s o l u t i o n
a v e r a g e d over 1,ooo reali-
zations of the full L a n g e v i n equation. A l r e a d y around T= 20, i.e.,
long
127
mI
~.~ Y
,~o ....:..: ,:0 oZ..,.. _,.. ~, "%X.~
~'", , Cl T,IO
Im[
Im[
I~o " 6;0 " s:o " |;o " 5:0 " o;o *-5',o"-s%o .Ir,o *~,o Im [
Fig.
69 a-e: Decay of a m e t a s t a b l e state. The numerically constructed time dependent p r o b a b i l i t y d i s t r i b u t i o n f(E,E~,~) is plotted for different times after preparing the system in a 8-function d i s t r i b u t i o n for Io=795 on the lower branch of the bistable transmission curve, a) T=o.2, b) T=o.5, c) T=I.o, d) ~=1.5, e) ~=2.5. All quantities are in dimensionless units given in Fig. 68. (From Ref. 199.)
128
b e f o r e Io(X) reaches the instability point of the lower branch,
shows s i g n i f i c a n t d e v i a t i o n s from the d e t e r m i n i s t i c result. A certain f r a c t i o n of the realizations has switched already to the state of high transmission.
A typical example is shown in Fig.
68. At about 3=3o,
reaches the v a l u e c o r r e s p o n d i n g to the upper b i s t a b l e branch. I n c r e a s i n g I (3) further, o
it then follows that branch.
The stability of the two b i s t a b l e states may be i n v e s t i g a t e d by the d i s c u s s e d s i m u l a t i o n p r o c e d u r e simply by p r e p a r i n g the system at the i n t e r e s t i n g stationary value It(Io) under the action of the noise. demonstrated,
and m o n i t o r i n g the t i m e - d e v e l o p m e n t
The decay of a m e t a s t a b l e state can be
e.g., by w a t c h i n g the time e v o l u t i o n of the p r o b a b i l i t y
d i s t r i b u t i o n function f(E,E~,~).
This d i s t r i b u t i o n has been o b t a i n e d in
Ref.
199 from 1o,ooo i n d e p e n d e n t r e a l i z a t i o n s of the stochastic equa-
tion
(III.63). The complex E-plane has been coarse grained and it has
b e e n counted at any instant of time, how many r e a l i z a t i o n s yield values of Re(E)
and Im(E)
shown in Fig.
Ones sees,
in a certain interval. An example for the results is
69
that already at 3=o.2 the p r o b a b i l i t y d i s t r i b u t i o n has re-
].axed from the initial
6-function.
Then a flow of p r o b a b i l i t y in the com-
plex E - p l a n e sets in. At i n t e r m e d i a t e times, of the p r o b a b i l i t y d i s t r i b u t i o n occurs.
a h o r s e s h o e - l i k e structure
Finally a new m a x i m u m d e v e l o p s
w h i c h is localized at E-values c o r r e s p o n d i n g to the upper b i s t a b l e branch.
A w h o l e series of those simulations has b e e n r e p o r t e d in Ref. results for the m e a n intensity
199. The
could be fitted by the e x p r e s s i o n
=I. e - l T + I f ( 1 - e - ~ ) 1
(III.64)
Here,
I. is the initial intensity at w h i c h the system has been p r e p a r e d l and If is the i n t e n s i t y of the final s t a t i o n a r y state. Several v a l u e s for I. have been choosen on both b i s t a b l e branches. The results are 1 s u m m a r i z e d in Fig. 70. The figure shows, that the upper b i s t a b l e b r a n c h is m o r e stable than the lower one. This is usually the case in o p t i c a l l y
b i s t a b l e systems because the r e l a t i v e s t r e n g t h of the fluctuations in the o n - s t a t e is m u c h w e a k e r than in the off-state.
The overall effect
of the r a n d o m fluctuations is to reduce the p r o n o u n c e d hysteresis phenomena.
However,
devices.
they still allow b i s t a b l e o p e r a t i o n of the i n v e s t i g a t e d
129
1.0,
-At
e +If(I-
I(t)
e-;~t)
,,,<:
c3
0.5
-I
tow~
(._)
upperbrench 0
200
400
6OO
BOO
10 Fig.
7o: Decay rate I for both bistable branches as function of I o, is in units ~-I. (From Ref, 299,)
III.2e Optical Bistability due to Induced Absorption In the following paragraphs a recently observed absorptive bistability will be discussed which is due to induced absorption in the spectral region below the exciton resonance in direct-gap semiconductors.
At
these frequencies the unexcited crystals are almost transparent.
Only
few e-h pairs are created by the weak incident light, The e-h generation takes place by two-photon absorption or by the small one-photon absorption in the exponential tail of the exciton resonance Nevertheless,
(Urbach tail).
if the exciting laser is sufficiently strong,
even these
relatively inefficient absorption channels lead to a density of electronic excitations high enough so that exchange and correlation effects among the electrons and holes can cause a reduction of the band-gap.
130
Strong one-photon absorption becomes possible if the band-edge shifts below the frequency of the exciting laser. The semiconductor switches from a state of weak absorption of strong absorption
at low incident intensities to a state
at high intensities.
Optical bistability due to
this effect has been observed in CdS crystals under pulsed laser exciation [203 ] . A theoretical explanation of the experiments has been given in Ref.
204 . The development of a spatially inhomogeneous e-h
density distribution has been predicted theoretically, cause substructures
in the bistable hysteresis
which should
loops[2o5].
A good approximation for the absorption coefficient ~ (~) in the spectral region below the exciton resonance is given by
(~)=
2e2Eg Y ~ nc~
I~ dx /x
[2o4]
(1-fe-fh) , (III.65) (~m-E'-x) 2+ (~Y) 2
0
where fi(eik)=
I (Eik_Pi)/kBT+1)_ I e
are the Fermi-distributions quasi-chemical potentials
~i
,
i=e,h
,
(III.66)
of the electrons and the holes with the and
ei--~2k2/2m i The broadening m y in Eq.
(III.65)
simulates the weak band-tail absorp-
tion. E' is the effective semiconductor band-gap which depends on the g e-h density N E' =E +&(N) g g
(III.67)
Here, E
is the band-gap of the unexcited crystal. The band-shift &(N) g takes into account the exchange and correlation energies of the elec-
trons and the holes. is plotted in Fig.
The resulting absorption coefficient
~ for CdS
71 as function of the e-h density N. The choosen
frequency is below the lowest exciton resonance.
One observes for low
e-h densities only the weak band-tail absorption. ~ (N) increases strongly at about N=1o 17 cm -3. Here, the band-edge shifts below the frequency of the exciting laser. This leads to strong one-photon absorption.
131
2.0
Cd S 1.5
T --30K ,.-.-,
11~ = 2.5~.8
' E 1.0 05
0
.
.
.
.
.
' ' ' '
10m
~
~
I IlllI
ld'
i
,
,
J
lo
,,
i0'9
N[cm "3 ] Fig.
71: Calculated absorption coefficient e (~,N) for a frequency below the exciton resonance as function of the electronhole density N. (From Ref. 204)
The described optical nonlinearity can be utilized to obtain absorptive optical bistability of the light transmitted through the semiconductor crystal. No external resonator feedback is required
[2o3]
. In CdS,
the nonlinearity is strong enough so that only a single pass of the light beam through the medium is necessary.
A theoretical description of
this effect assuming spatially homogeneous field distributions the crystal is therefore not appropriate. of spatial inhomogeneities
inside
One expects the development
in the direction of the light propagation.
Such effects have been investigated theoretically
in the framework of
transport equations for the e-h density distribution N(z,t) the distribution of the light intensity I(z,t)
and for
in the nonlinear medium.
These equations are [2o4, 205]
aa8 N + ~(~,N)I _--r N - ~ z D ~ z N = - -T
and
(III.68)
132 Here, light propagation in z - direction has been assumed. diffusion coefficient of the e-h pairs,
D is the
T is their lifetime,
and
n(~) is the refractive index. Eqs. (III.68) and (III.69) can be derived from the Maxwell-Bloch equations in the slowly varying amplitude approximation.
The polarization field has been eliminated adiabatically.
The diffusion dominated limit has been investigated in Ref.
2o4. If D
is very large, the density becomes spatially homogeneous N(z,t)=N(t)
,
(III.7o)
and the intensity decays exponentially into the crystal
i(z,t)=i 0(t)e-~ (~,N)Z
(III.71)
This situation is experimentally relevant in very thin crystal platelets with high e-h diffusion.
The effect of intrinsic bistability can
be described by the stationary solutions of the rate equations.
The
results for the diffusion dominated situation are reproduced in Fig. 72.
0.3 0.2 01
0
0.2
0.4
(26
08
1.0
lo [MW/cm2]
Fig.
72: Stationary results for the electron-hole density N (a) and the transmitted intensity I+ (b) versus incident intensity I for the bistability due ~o induced absorption in the fgequency range below the exciton resonance in CdS. (From Ref. 204.)
133
For a certain range of input intensities N and I t have been obtained. small incident fields.
renormalization
edge below the frequency of the exciting
effects
laser.
The highly absorbing
to dispersive clockwise
of the e-h density for decreasing
in a clockwise
The description full transport
reported in Ref.
inhomogeneities
(III.68)
and
requires
(III.69).
intensity
(several MW/cm2),
state excitation would damage the crystals. uses pulsed laser irradiation. is plotted
after travelling for comparison
Fig.
Therefore,
(D=o). This limiting
in the theoreti-
case has been i n v e s t i g a t e d
since it leads to the most p r o n o u n c e d
Results for D~o will be d i s c u s s e d
the excitation.
The local e-h density
immediately
in the short-time
loop
inhomoge-
increasing p r o p o r t i o n a l
regime after the onset of
(Fig.
73b)
increases
behind the front face of the crystal
which corresponds
stable hysteresis
spatial
later on.
light intensity
intensity
light intensity
with steady
one usually
through a CdS crystal of I ~m thickness without
73a shows the transmitted
reached,
high values of the re-
experiments
To mimic this situation
to the incident until
the solution of the
a typical pulse for I (t) has been assumed [205]. o in Fig. 73a. Also shown is the transmitted intensity
any carrier diffusion neities.
counter -
The results have been
205. Because of the relatively
cal calculations,
in contrast
61 and 63)
of spatial equations
quired excitation
Io(t~
I t as function
sense,
bistability where the loops are traversed
(see e.g. Figs.
is high
to a drop of the transmitted
state is m a i n t a i n e d
a loop which is passed
continuously
a value
is
to the end of the lower branch of the bi-
(see Fig.
72a). The still increasing
incident
causes the local e-h density to jump to the upper bi-
stable branch in the region behind the front face of the crystal. spatial d i s t r i b u t i o n N(z,t) in the insert to Fig. absorption
assumes the shape of a kink.
73b. Now,
The
This is shown
one has a high density and thus a high
in a part of the crystal.
in comparison
for
for
shift the band-
excitation until most of the e-h pairs have recombined. of I o exhibits
solutions
increases
Strong o n e - p h o t o n
sets in, leading to a sharp increase
and, because of the increased absorption, intensity.
intensity
Around Io=O.8 M W / c m 2 the e-h density
enough so that the many-body absorption
I° multi-valued
The transmitted
to a situation w i t h o u t
The total absorption a density kink.
has increased
This gives rise
to a drop of the transmission
causing the first peak in the temporal
dependence
light pulse
of the transmitted
( Fig.
73a).
134
o)
/\ /
\ \
'-incident pulse
~3
\
\
=E:
--2
/
/
/
\
",,
,
\
z :001
Id' i0m
\
,~ 10'7 1
h:,.~d
,ok-
Io" 04
zlluml
10
i0'~
10
20
30
40
50
t[ns]
Fig.
73: T h e o r e t i c a l results for a CdS crystal of I ~m thickness. As functions of time are p l o t t e d a) the i n c i d e n t pulse Io(t) and the pulse t r a n s m i t t e d through the crystal, b) the local electron-hole density i m m e d i a t e l y behind the front face of the crystal (z=o.ol ~m) and at z=l.o ~m. The insert shows the spatial d e n s i t y d i s t r i b u t i o n for different times. ( From Ref. 205.)
Since the incident i n t e n s i t y is still increasing, ses too. However,
the e-h density ri-
this occurs quite i n h o m o g e n e o u s l y t h r o u g h o u t the crys-
tal because the local values of N(z,t)
follow the two different b r a n c h e s
135
of the density hysteresis increases
according
loop.
Immediately behind the kink,
to the lower branch,
until
the density
it reaches the instabili-
ty point of that branch and it jumps to the upper branch.
Hereby another
fraction of the m e d i u m has switched to the state of high absorption. this m e c h a n i s m the density kink moves causing stepwise
stepwise deeper into the crystal
increasing total absorption,
successive
drops of the transmitted
structures
of It(t)
shown in Fig.
By
which in turn leads to
intensity
73a.
I t . This explains
It(t)
the kink has passed the crystal completely,
follows
again Io(t)
the after
which now is totally in
the state of high absorption. These substructures substructures constructed becoming
in the transmitted
in the dynamic bistable
by plotting
a parameter.
It(Io).
pulse m a n i f e s t hysteresis
Thereby,
themselves
loops.
in
Such a loop is
the time is eliminated, m e r e l y
The result for D=o is shown in Fig.
lib
L=2.7~Jm
116
a) O,O
b) O=0.1c~Is
c) O:lO c~/s
d) D=IO.Ocruz/5
74a.
0,4 02
~
Q8 D6 0.4 0.2 O0
i
J
1
?
3
I
4
0
I
Io[MW/cm2 ] Fig.
Z
3
4
5
74: Transmitted intensity I versus input intensity I for pulse excitation wi~h a pulse w i d t h of 5o ns ~nd a pulse height Imax=5 MW/cm 2 using different electronhole diffusion c~efficients D. A CdS crystal of 2.7 ~ m has been assumed. (From Ref. 205.)
The saw-tooth in It(t). tures.
structures
Electron-hole
The results
cally solving
in Fig. diffusion
shown in Fig.
74a are caused by the successive peaks leads to a reduction
of these struc-
74 b - d have been obtained by numeri-
the partial differential
for different values of the d i f f u s i o n
equations coefficient
(III.68) D.
and
(III.69)
186
I0m
I
f
I
I
I
i~Qx=5 MW
1018
f M =50 ns
t = 25 ns
~E 1017 z
10m
:°9
~,
D.~ cm'/s D:I0 cm21s:-
I 0~5 0
Fig.
I
i
1
2
i
3 z i/urn]
i
l
4
5
6
75: The density p r o f i l e s N(z,t) are p l o t t e d versus z for d i f f e r e n t e l e c t r o n - h o l e d i f f u s i o n c o e f f i c i e n t s D for a fixed time t=25 ns after the onset of the e x c i t a t i o n p u l s e I^(t) (see Fig. 73a). The pulse w i d t h is 5o ns and the pea~ intensity is 5 M W / c m 2. ( F r o m Ref. 205.)
W i t h increasing e-h d i f f u s i o n the d e n s i t y p r o f i l e becomes c o n s i d e r a b l y smoother.
The sharp kink for D = o
is altered into a continuous t r a n s i t i o n
b e t w e e n regions of high and low absorption. profiles
are shown in Fig.
practically
Examples
for the c a l c u l a t e d
75. D i f f u s i o n constants lager than D=1o cm2/s
lead to no a d d i t i o n a l changes of the dynamic h y s t e r e s i s
shown in Fig.
loop
74d. This shape is in q u a l i t a t i v e agreement with experi-
m e n t a l results for thin CdS p l a t e l e t s
[206] .
The c o e x i s t e n c e of regions of high and low e-h densities is an example of spatial t w o - p h a s e c o e x i s t e n c e in the f i r s t - o r d e r n o n e q u i l i b r i u m phase t r a n s i t i o n b e t w e e n the states of high and low transmission. spatial i n h o m o g e n e i t i e s are subpressed, forward and b a c k w a r d
e.g.
, by s u p e r p o s i t i o n of
t r a v e l l i n g light waves in a resonator,
r e a l i z e s t w o - p h a s e c o e x i s t e n c e by temporal fluctuations.
68 and 69).
the system
Examples
this s i t u a t i o n have been d i s c u s s e d in the p r e v i o u s chapter Figs.
If
for
(see, e.g.
,
137
Very recently,
the calculations
for absorptive optical bistability
have been extended considering the so-called multiple quantum-well structures
[207 ].
These devices are usually made of GaAs layers
in a Ga1_xAlxAS substrate.
The notation Ga1_xAlxAS
indicates that
x percent of Ga have been replaced by AI. Typically this percentage is choosen between 20 % and 3o % [208]
.
GaAs is the optically active material.
For the calculations in Ref.
2o7 a series of 3oo layers with a
thickness of 5o angstroems has been
assumed so that the total length of GaAs is LGaAs=O.5 ~m. Such multiple quantum-well devices
exhibit very large optical nonlinearities which
are still pronounced at room temperatures
[208]. The basic reason for
these effects is the almost two-dimensional
nature of the quasi-
particle system in the GaAs layers. Therefore,
the binding energy of
the excitons is increased in comparison to the three-dimensional system of bulk GaAs. Exciton resonances
in multiple quantum-well
tures are still present at room temperature. ties are observed if these materials ionize the excitons.
struc-
Large optical nonlineari-
are excited strongly enough to
Room temperature bistability due to this effect
has been observed experimentally
[2o9]
.
In Ref. 2o7 another room temperature bistability has been predicted theoretically.
!
The density dependent shift of the band-gap Eg, Eq.
(III.67), has been calculated and the transport equations
(III.68)
and
(III.69) have been solved for the conditions of the GaAs-GaAIAs devices. Pulsed laser excitation with a pulse width tm and a frequency below the exciton resonance has been assumed.
In Fig.
74 the calculated trans-
mitted intensity I t is plotted versus the input intensity I O. The time is the parameter of the different loops. One observes well-developed hysteresis material parameters
loops. Calculations with the
suitable for GaAs-GaAIAs multiple-quantum well
structures yield that the required intensities for room temperature optical bistability should be in the region of kW/cm 2 [207]. Fig.
76
shows, that in the case of short pulses Io(t) a superposition of dynamical effects onto the intrinsically bistable loops takes place. This enhances the hysteresis effects.
138
5a As - 5aAIAs -~\d) 2000 LGo
f 1000 ~a)
~
/
a)stationary
J
b) %:100"r
/./"
0
I
0
d) tm -- 5T I
~
I
I
2000
~
I
i
I
I
4000
I
i
i
i
0000
10--Fig.76:
Absorptive bistability in GaAs-GaAIAs multiple quantum-well structures. The calculations have been done for Ioo GaAs layers of total length o.5 pm. An exciting laser pulse I0(t) has been assumed with pulse widths t m indicated in the figure. T is the lifetime of the electron-hole pairs in the GaAs layers, m=1o-8s. (From Ref. 207.)
139
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Acknowledgements
The author has b e n e f i t e d g r e a t l y from m a n y d i s c u s s i o n s and active c o l l a b o r a t i o n w i t h Prof.
Dr. H. Haug, U n i v e r s i t y F r a n k f u r t
Germany).Furthermore,
he wants to thank Dr. F.F. Abraham,
San Jose
Dr. R.C. Desai, U n i v e r s i t y T o r o n t o
(USA), Prof.
(Fed. Rep. IBM Research,
(Canada),
and
his collegues at the I n s t i t u t e for T h e o r e t i c a l Solid State Physics, University Frankfurt
(Fed. Rep. Germany),
for stimulating collaboration.
Special thanks are due to Dr. H.H. Kranz for c a r e f u l l y reading and c r i t i c i z i n g the manuscript,
to Mrs. E. P f a n k u c h for expert typing and
to Mrs. E. B o f f o for the artwork.
Part of this w o r k has been supported f i n a n c i a l l y by the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t through the S o n d e r f o r s c h u n g s b e r e i c h furt-Darmstadt.
F i n a n c i a l support through IBM Germany and IBM San Jose
enabled the author to spent parts of the years R e s e a r c h Lab.
65 Frank-
, San Jose
1981 - 1983 at the IBM
(USA). During the last part of this work,
financial support through the F. T h y s s e n - S t i f t u n g , fully acknowledged.
K61n,
is great-
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