Dynamics of First-Order Phase Transitions in Equilibrium and Nonequilibrium Systems

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 TT 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 VE 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]

~

r2

(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 m0 ,) [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 Ti

+(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

Fnc'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=

nn 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 ~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