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dissertation entitled

Atomic Diffusion and Step Dynamics
On Vicinal Surfaces

presented by

Branislav D. Blagoj evic

has been accepted towards fulﬁllment
of the requirements for

 

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1m chlRCIDutoOupﬁS—p.“

ATOMIC DIFFUSION AND STEP DYNAMICS ON VICINAL
SURFACES

By

Branislav D. Blagojevic

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

1999

ABSTRACT
ATOMIC DIFFUSION AND STEP DYNAMICS ON VICINAL SURFACES
By

Branislav D. Blagojevic

Steps are the most fundamental extended defects on crystal surfaces. Their dynamics
mediate the annealing of crystal surfaces, the growth of single crystals and many other
surface processes. Step dynamics are in turn mediated by atom motion, usually either
attachment and detachment from a vapor, diffusion across terraces or facets, or by
diffusion along step edges. The high resolution provided by. Scanning Tunneling
Microscopy (STM) and Reﬂection Electron Microscopy (REM) is providing atomic scale
images of stochastic step ﬂuctuations. Understanding of this data requires a quantitative
analysis of step ﬂuctuations in terms of the atomic processes producing them. Such
theory, in combination with high quality step ﬂuctuation data, provides a unique method
for determining the dominant modes of atomic mass transport across surfaces, and for
estimating the energy baITiers which impede the various types of atomic processes. A
complementary probe of atomic diffusion processes is the study of the decay of specially
prepared surface gratings, which in the small SlOpe limit are described by a closely
related theory. Here, a comprehensive formalism, based on atomic diffusion, for treating

step ﬂuctuations and the decay of surface gratings is presented.

ACKNOWLEDGMENTS

I would like to thank my advisor, Prof. Phil Duxbury, for his knowledgeable, relaxed and

professional guidance.

I would also like to thank professors Wolfgang Bauer, Norman Birge, Mark Dykman, and

Daniel Stump for their time and effort while serving in my Ph.D. committee.

I thank Prof. Julius Kovacs, Janet King, Stephanie Holland, and Debbie Simmons for

their assistance.

TABLE OF CONTENTS

LIST OF TABLES ................ ~ ............................................................................................. vi
LIST OF FIGURES .......................................................................................................... vii
CHAPTER 1

INTRODUCTION ................................... 1
1.1 Research in Surface Physics .......................................................................................... 1
1.2 Basic Concepts in Surface Physics ................................................................................ 2
1.3 Vicinal Surfaces ............................................................................................................. 9
1.4 Overview of Chapters .................................................................................................. 10
CHAPTER 2

AN OVERVIEW ............................................................................................................... 13
2.1 Average Height and Fluctuation Width ............................................... - ........................ 1 3
2.2 Scaling Concepts .......................................................................................................... 15
2.3 Correlation Length ....................................................................................................... 22
2.4 Analysis of Simulation and Experimental Data ........................................................... 23
2.5 Fluctuation Equation and Symmetry Principles .......................................................... 27
2.6 Step-Edge Diffusion ..................................................................................................... 30
2.7 Attachment-Detachment .............................................................................................. 33
CHAPTER 3

AN ISOLATED STEP ....................................................................................................... 39
3.1 Description of the Model ............................................................................................. 39
3.2 Step-Edge Diffusion ..................................................................................................... 45
3.3 Attachment-Detachment Model ................................................................................... 46
3.4 Terrace Diffusion ......................................................................................................... 47
3.5 Anomalous Diffusion ................................................................................................... 55
CHAPTER 4

ARRAY OF STEPS ........................................................................................................... 58
4.1 Introduction .................................................................................................................. 58
4.2. The Model ................................................................................................................... 59
4.3 Analysis of Various Time Regimes ............................................................................. 65
4.4. Terrace Diffusion ........................................................................................................ 73
CHAPTER 5

SURFACE EVOLUTION DUE TO STEP INTERACTIONS ......................................... 88
5.1 Introduction .................................................................................................................. 88
5.2 Chemical Potential of Interacting Steps ....................................................................... 88
5.3 Continuum Limit, The Entropic Repulsion ................................................................. 92

iv

CHAPTER 6

MONTE-CARLO SIMULATIONS AND NUMERICAL ANALYSIS ......................... 100
6.1 Introduction ................................................................................................................ 100
6.2 Monte-Carlo Simulations ........................................................................................... 100
6.3 Integration of Stochastic Equations ........................................................................... 106
6.4 Method for Checking Probabilities ............................................................................ 109
CHAPTER 7

VLSI DESIGN ................................................................................................................. 113
7.1 Introduction ................................................................................................................ 113
7.2 MOS Transistors ........................................................................................................ 113
7.3 Design Steps ............................................................................................................... 117
7.4 Simulation .................................................................................................................. 120
CHAPTER 8

CONCLUSION ................................................................................................................ 122
APPENDICES ................................................................................................................. 124
Appendix A: Calculation of Equilibrium ........................................................................ 125
Appendix B: Solution of Langcvin Equation ................................................................... 128
Appendix C: Random Walk Probabilities ........................................................................ 131
Appendix D: Overview of Other Models ......................................................................... 136
BIBLIOGRAPHY ............................................................................................................ 141

LIST OF TABLES

Table 4.1 Limiting behaviors for G(O,t) ........................................................................... 85
Table 4.2 Limiting behaviors for C , a,{,(O,t) ....................................................................... 86
Table 4.3 Equations for limiting timescales ...................................................................... 87

vi

LIST OF FIGURES

Figure 1.1 STM image showing the steps on a surface near Si(001) .................................. 3'
Figure 1.2 A scanning tunneling microscope image of the phase separation of a Vicinal Si

surface ................................................................................................................ 4
Figure 1.3 Two step trains and an isolated step as viewed by REM ................................... 5
Figure 1.4 A STM derivative-mode image of a Si(OOl) Vicinal surface miscut O.3°

towards (100). .................................................................................................... 6
Figure 1.5 Sample conﬁgurations of the surfaces in Monte-Carlo

simulation ........................................................................................................... 8
Figure 2.1 Isolated step edge ............................................................................................. 14
Figure 2.2 Fluctuation width vs. time ................................................................................ 16
Figure 2.3 A typical trajectory of a random walker ........................................................... 17
Figure 2.4 Saturation width for different system sizes ...................................................... 18
Figure 2.5 Illustration of steps involved in rescaling the time dependent ﬂuctuation

width ................................................................................................................ 21
Figure 2.6 Illustration of three possible local geometries .................................................. 31
Figure 2.7 Monte Carlo simulation of an isolated step evolving through step-edge

diffusion ........................................................................................................... 34
Figure 2.8 Correlation function averaged over space and time ......................................... 35
Figure 2.9 Monte Carlo Simulation of an isolated step evolving through attachment

detachment ....................................................................................................... 37
Figure 2.10 Squared ﬂuctuation width for Au(l 10) at 556K ............................................. 38
Figure 3.1 Terrace diffusion with diffusion kernel ........................................................... 42
Figure 3.2 Key energy barriers at a step edge .................................................................... 49
Figure 3.3 Diffusion kernel vs. I ..................................................................................... 51
Figure 3.4 The correlation function in terrace diffusion case ............................................ 52

. Figure 3.5 The ﬂuctuation exponent ,6 is plotted vs. a ................................................... 57

Figure 4.1 Diffusion kernels on the Vicinal surface ........................................................... 60
Figure 4.2 Four spectrums of inverse relaxation times in the terrace diffusion case ......... 75
Figure 4.3 The autocorrelation function in the terrace diffusion limit .............................. 76
Figure 4.4 The step-step correlation function in the terrace diffusion T3 limit ................. 80
Figure 5.1 Geometry of the step train ................................................................................ 89
Figure 5.2 Slope evolution ................................................................................................. 95
Figure 5.3 The SIOpe in the center of the train vs. time ..................................................... 97
Figure 5.4 Comparison of steady shapes ........................................................................... 98
Figure 5.5 Evolution of the proﬁle .................................................................................... 99
Figure 6.1 A snap-Shot of the step shape in the equilibrium ........................................... 102
Figure 6.2 Isolated step Monte-Carlo Simulation for the terrace diffusion with perfect

sticking ........................................................................................................... 103

vii

Figure 6.3 Isolated step Monte-Carlo simulation for weak sticking terrace diffusion ....104

Figure 6.4 The Simulation of terrace diffusion with perfect sticking .............................. 105
Figure 6.5 The ﬂuctuation width of an isolated step obtained doing numerical

integration ...................................................................................................... 107
Figure 6.6 The distance between steps in 2-step train obtained doing numerical

integration ...................................................................................................... 108
Figure 6.7 The probability that an atom is at the site (i, j) ............................................. 111
Figure 6.8 The matrix method ......................................................................................... 112
Figure 7.1 A 3-dimensional view of the (a) PMOS; and (b) NMOS ............................... 114
Figure 7.2 The schematic symbols for the (a) NMOS; and (b) PMOS ............................ 115
Figure 7.3 An inverter ...................................................................................................... 116

viii

Chapter 1

Introduction

1.1 Research in Surface Physics

There has been a large amount of work on the equilibrium properties and the dynamics of
crystal surfaces and interfaces [22,28,37,44]. To gain comprehension of surface
phenomena, it is critical to understand the complexity arising from the termination of a

crystal by a surface.

The manifestations of this complexity can be studied from two different aspects. Firstly,
the atomic arrangement in, the surface region. Two particularly interesting effects may
take place due to the presence of a surface: (1) the distance between the surface plane and
those parallel to the surface deviate from their bulk values; (2) some lateral structural
change such as formation of steps, islands, and ripples on the surface may also occur.
Secondly, dynamic properties, for instance vibrations and diffusions, which are of
considerable importance to the elucidation of surface phenomena and understanding of

experimental results.

1.2 Basic Concepts in Surface Physics

To understand complex surface problems, one must address several fundamental
problems: atomic and electronic structure, diffusion along surfaces or across interfaces,
the energy and chemistry of surface regions and the response to external forces. Existing
techniques for investigating surfaces have reached maturity and are increasingly being
applied to systems of practical relevance. Here, four important ideas are brieﬂy

discussed.

Scaling concepts: One of the modern concepts used to study various toughening
processes on surfaces is scaling [37]. It has a surprising power of prediCtion and simple
manipulations that enable connection of apparently independent quantities and exponents.
As will be shown later, many measurable quantities obey simple scaling relations. For

example, for a large number of systems it is found that the interface width, w, increases

as a power of time, w(t) ~ t” . The width eventually saturates at a value that increases as a

power law of the system size, w(L) ~ L" .

Experiments: In many branches of physics, experiments are the driving force that leads to
new problems for theoretical investigation. Sometimes, the reverse occurs. For example,
the roughening process was an unwanted experimental artifact that one attempted to
minimize. It was only recently that a number of experimentalists became interested in it,
and now they are taking the lead showing that there are more mechanisms leading to

toughening than theoretically anticipated [3535]. Thanks to the innovation and widening

 

Figure 1.1 STM image showing the steps on a surface near Si(001) [35].

 

Figure 1.2 A scanning tunneling microscope image of the phase separation of a Vicinal Si
surface [6].

 

Figure 1.3 Two step trains and an isolated step as viewed by REM [5].

 

Figure 1.4 A STM derivative-mode image of a Si(OOl) Vicinal surface miscut 0.3°
towards (100). The surface steps down from leﬁ to right [3].

applications of reﬂection electron microscopy (REM) and scanning tunneling microscopy
(STM), surface research has clearly shifted its interest from the macroscopic to the

nanometer scale (see Figures 1.1-1.4).

Discrete models: Theoretical discrete modeling provided a substantial fraction of the
driving force behind early investigations of interface and surface morphology. Besides,
many phenomena can be modeled successfully using elementary computer algorithms
[8,16,23,31]. Computer simulation is adding a new dimension to scientiﬁc investigation,
establishing a role of equal importance with the traditional approaches of theory and
experiment. Moreover, Simulation studies represent an essential link between theory and
experiments, and can allow us to separate the essential ingredients determining the
morphology from the unnecessary details. As the speed and accuracy of arithmetic
computers improve tremendously, it is feasible and advantageous to can-y out quantitative

investigation through detailed computer simulation (see Figure 1.5).

Continuum equations: A very successful tool for understanding the behavior of the
various surface relaxation processes is stochastic differential equations [37]. Such
equations typically describe the interface at large length scales. which means that short
length scale “details” are neglected and focus is only on the asymptotic coarse-grained
properties. If there already is a discrete model for the system, then it should be possible to
derive the corresponding continuum equation directly from this model. There is an
alternative method for deriving such equations: exploiting symmetry principles. Every

interface ﬂuctuation process obeys some simple symmetries. For example, the laws of

 

Figure 1.5 Sample conﬁgurations of the surfaces used to study step motion. (a) The
initial conﬁguration of 5 steps separated by 6 lattice constants. (b) The surface
after 106 Monte Carlo steps per site. (0) The ﬁnal conﬁguration after 3 x 106
MCS/Site [16].

physics are independent of where one deﬁnes the zero height of the interface, so the
continuum equation must have this same invariance. In most cases the scaling properties
of the system and the continuum equations are unambiguously determined by the
symmetries of the system. Once an equation is obtained one must ﬁnd its predictions,
determining, e.g., the scaling exponents and scaling functions. If the relaxation process is
described by a sufﬁciently Simple equation, it can be solved exactly. However, for many
equations no exact solutions exist, so one must apply various approximations to uncover

the scaling behavior.

1.3 Vicinal Surfaces

The work presented here is focused on special surfaces, called Vicinal surfaces. Vicinal
surfaces have pronounced steps and terraces between them (see Figures 1.1-1.5). Steps
are the most fundamental extended defects on crystal surfaces. Their dynamics mediate
the annealing of crystal surfaces, the growth of single crystals and many other surface
processes. Step dynamics are in turn mediated by atom motion, usually either attachment
and detachment from a vapor, diffusion across terraces or facets, or by diffusion along
Step edges. The high resolution provided by Scanning Tunneling Microscopy (STM) and
Reﬂection Electron Microscopy (REM) is providing atomic scale images of Stochastic
step ﬂuctuations. Understanding of this data requires a quantitative analysis of step
ﬂuctuations in terms of the atomic processes producing them. Such theory, in
combination with high quality step ﬂuctuation data. provides a unique method for

detemiining the dominant modes of atomic mass transport across surfaces, and for

estimating the energy barriers which impede the various types of atomic processes. A
complementary probe of atomic diffusion processes is the study of the decay of Specially
prepared surface gratings, which in the small Slope limit are described by a closely

related theory.

The equilibrium structure [1-6] and dynamics [7- 20] of steps on Vicinal surfaces are now
routinely probed and modeled with atomic precision. Traditionally, the relaxation of
these structures is treated using linear relaxation theory [21-23]. However as realized
even in those early seminal papers, the linear theory usually needs Signiﬁcant
modiﬁcation below roughening [24-33] where steps are well deﬁned. Actually, when the
steps are well deﬁned, their (capillary) ﬂuctuations are usually treated linearly, and the
non-linear evolution is though the center-of—mass motion of the steps. In Chapters 3-5, a
complimentary formalism based on atomic diffusion is presented, the atomic diffusion
processes which lead to the various limiting behaviors treated by previous workers [8-17]
are explicitly Shown. It also predicts when “non-universal” behavior can occur, and it

provides a formalism which describes the crossover between the various limits.

1.4 Overview of Chapters

This dissertation is arranged as follows. In Chapter 2, an overview of the general
approach to treating step-edge ﬂuctuations is given. It starts by deﬁning the average step
height and ﬂuctuation width. Then, scaling concepts are introduced. It is shown that the

ﬂuctuation width of an initially ﬂat step increases with time as a power law until it

reaches a saturation width. The concept of correlation and correlation length is also
introduced. Correlation between the parts of the ﬂuctuating step plays an important role
in describing the mechanisms that lead to saturation. Also, ﬁve methods of extracting the
scaling parameters from computer simulation, in order to compare them with
experimental and analytical results, are described. A general continuum stochastic
equation that describes step ﬂuctuations and which satisﬁes symmetry principles is
derived. Finally, two previously introduced models, step-edge diffusion and attachment-
detachment, are analyzed. Analytical, simulation and experimental results for those two

models are given.

In Chapter 3, a complimentary fomialism to treat the isolated step ﬂuctuations is
introduced. It is based on atomic diffusion through a Langevin (rate) theory. The different
atomic mechanisms are mathematically described by different diffusion kernels. This
allows one to smoothly interpolate between the step-edge diffusion limit and the

attachment-detachment limit.

In Chapter 4, the same method is expanded to an array of steps. Step-step interaction
terms in the step free energy are ignored. There are diffusion kernels between steps as
well as a diffusion kernel to the same step. This enables one to analyze the correlation
between the ﬂuctuation along the same step as well as mutual correlation between any

two steps on the surface.

In Chapter 5, the formalism is extended to include step-step interaction terms. That leads

to the spreading of a step array (train) and is related to the relaxation of the whole

surface.

In Chapter 6, the methods for Monte-Carlo Simulation of atomic diffusion and numerical
integration of stochastic differential equations are summarized. The results obtained by

those two methods are then compared to analytical results.

In Chapter 7, an overview of VLSI electrical design is given. It describes the
computational modeling experience gained while doing practical training in industry (a

ﬁnal part of Ph.D. program).

Chapter 8 contains briefconclusion.

There are four appendices. Appendix A derives saturation width for two cases of
boundary conditions at the ends of the step: pinned and periodic. Appendix B summarizes
the calculation of correlation functions from the linear Langevin equation. Appendix C
derives diffusion kernels used in the model. Appendix D summarizes recent analytical

work of two other research groups and compares the approach presented here with theirs.

Chapter 2

An Overview

2.1 Average Height and Fluctuation Width

Consider a Single isolated step at the surface (see Figure 2.1 (a) or (b)). The length of the
step is L. The surface is deﬁned with the axis x (along step edge) and height h(x,t)
describing the position of the step edge at time t. x has discrete values I to L. The step
edge ﬂuctuates due to the atoms detaching from the step edge and attaching to the step
edge on both sides. In order to describe step ﬂuctuations quantitatively, two functions are

introduced:

(i) The average height of the step, (12(0) , deﬁned by

<h(t)> a %Zh(x,t) (2.1)

If the ﬂux of atoms detaching equals the ﬂux that is reattaching, then (h(t)) is constant in

time.

(ii) The ﬂuctuation width, w(t), which characterizes the roughness of the step edge, is

deﬁned by the rms ﬂuctuation in the height,

w: (z) a %Z[h(x,t) — {12(0)}2 = (123(1)) — (12(1))2 (2.2)

(a)

 

(b)

 

Figure 2.1 Isolated step edge. Two mechanisms for atomic processes which mediate step
ﬂuctuations. a) Step-edge diffusion; b) Attachment-detachment.

l4

 

To monitor the roughening process quantitatively, one measures the ﬂuctuation width as
a function of time. By deﬁnition, the step edge at time zero is simply a straight line, with

zero width. As detachment and attachment occur, the step edge gradually roughens.

2.2 Scaling Concepts

A typical plot of the time evolution of the width has two regions separated by a

“crossover” time ts," (see Figure 2.2).

(i) Initially, the width increases as a power of time,

w(t) ~ z” (2.3)
The exponent ,8, called the ﬂuctuation exponent, characterizes the time-dependent
dynamics of the toughening process. The driving force for the increase of the step edge
ﬂuctuation is its tendency to increase entropy - the larger the ﬂuctuation, the larger the
entropy of the system. If one imagines each step edge as a one dimensional random
walker along the h-axis, it would give the ﬂuctuating line with exponent ﬂ =1/ 2 (see
Figure 2.3). But, sites of the step edge are not completely free to move perpendicularly to
x because the motion of each site is constrained by attachment to its neighboring sites.

Thus, the exponent ,8 is always less that 7.. Its value depends on the speciﬁc processes

that drives the step-edge ﬂuctuations and that will be discussed later in this chapter.

(ii) The power-law increase in width does not continue indeﬁnitely, but is followed by a

saturation regime (the horizontal region of Figure 2.4) during which the width reaches a

 

 

 

 

 

 

10 a - w - . - .
. o
/O-.OOeGOOr-x
U V
O
D O
O
O
O
B
it"
1 ”L2 A ‘ A 13 l A 4
10 10 10 10
t

Figure 2.2 Fluctuation width vs. time. There are two characteristic regimes.

l6

 

 

 

 

 

1000 A 2000 I 3000

Figure 2.3 A typical trajectory of a random walker. The dashed lines are proportional to
1/2
t [37]. .

l7

 

 

 

 

l T ' f' ' FF“ '*
AAA
AA
AA
A
A O
A . O

10* . A 000000 ‘

r // Aoo ‘

/ A0 [3
S // A0
D
// AODDDDDUDD C]
0/ 30 D
. ,/ (ﬂaw
’ 0433 000000 00000
// 033000 i
/ C
,, e
W .
10 100 1000 10000

Figure 2.4 Saturation width for different system sizes.

18

saturation value, WW. The step ﬂuctuation width saturates when the tendency of the
system to keep its free energy smaller (by suppressing the increase of ﬂuctuations)
equilibrates with the tendency of the system to increase its entropy. As L increases, the

saturation width, wsa, increases as well, and the dependence also follows a power law,
w..,(L> ~ L“ (2.4)
The exponent or, called the roughness exponent, is a second critical exponent that

characterizes the roughness of the saturated step edge ﬂuctuation. As shown in Appendix

A, for the free energy expression used in models described in Chapters 3 and 4, a = 1/ 2 .

(iii) The crossover time t,,,, (sometimes called saturation time) at which the step edge
crosses over from the behavior of (2.3) to that of (2.4) depends on the system size

tm, ~ L‘ (2.5)
where z is called the dynamic exponent. The construction illustrated in Figure 2.2 is a

simple way to estimate t,,,,.

Although the scaling exponents are interesting, perhaps the main reason for studying
them is to extract the correct prefactors from the expressions for w(t). These prefactors
are related to step-edge kinetic barriers, and the study of w(t) provides an important

technique for estimating these critical parameters.

The scaling exponents a , ,0 , and z are not independent, and there exists a simple way

to “collapse" the data of Figure 2.4 onto a single curve.

(A) Plotting w(t)/wm,(L) as a function of time will result in curves that saturate at the
same value, independent of the system size L (see Figure 2.5).
(B) Plotting the width as a function of t/tm, will cause the curves to saturate at the

same characteristic time (see Figure 2.5).

These two observations suggest that w(t)/wm/L) is a function of t/tsa, only, i.e

w(t) ~ _t_
w...(L) fit ]’ (2'6)

30!

 

where ﬂu) is called a scaling function. Replacing WW and t“, in (2.6) with their scaling

fomis (2.4) and (2.5) one obtains the Family-Vicsek scaling relation [36]

t

w(L,t) ~ L“f{?). (2.7)

The general form of the scaling function f(u) can be read off from Figure 2.5. There are

two different scaling regimes depending on its argument u s t/tm, .

(a) For small u, the scaling function increases as a power law. Since during the
rescaling shown in Figure 2.5 one did not rotate the curves, but only shifted them,
it can be seen that in this regime one has

f(u) ~ u”, u << 1 (2.8)

(b) As t —-> oo , the width saturates. Saturation is reached for t> >t,,,,, i.e. the argument

ofthe scaling function it >> 1. In this limit we have
f(u) = const (2.9)
The validity of the scaling relation (2.7) can be tested numerically by replotting the data

of Figure 2.4 in a fashion suggested by (2.8) and Figure 2.5.

20

 

 

 

 

 

 

A A p ,_ A
log... loam/L") log(w/L°)
ﬂ ......... ' ﬂ
9 9 >
logt logt 1080/“)
V
(3)

Figure 2.5 Illustration ofsteps involved in rescaling the time dependent ﬂuctuation width
[37].

 

It can be next Shown that the exponents a , ,6 , and z are not independent. In Figure 2.2
if one approaches the crossover point (t,a,,w(t,m)) from the left one ﬁnds, according to

(2.3), that w(tm) ~ I” However, approaching the same point from the right, one has,

sat '

from (2.4), w(t )~ L“. From these two relations follows is“, ~ L“ which, according to

sat

(2.5), implies that

2:3. (2.10)

13
Equation (2.10), a scaling law linking the three exponents, is valid for any grth process
that obeys the scaling relation (2.7). It will be seen in the following chapter that this

relation is also satisﬁed in the presented formalism.

2.3 Correlation Length

The fact that the saturation time, tsat, and the saturation width, wsat increase with the
system size suggests that saturation phenomena constitute a ﬁnite Size effect [37]. One
property of the ﬂuctuation process is that correlations develop along the step edge, which
implies that the ﬂuctuations of the different parts of the step are not completely
independent. Although the process of attaching and detaching on the step edge is local,
the “information” about the height of each of the Sites spreads globally. The typical
distance over which the heights “know about” each other is the characteristic distance

over which they are correlated and is called the correlation length, and is denoted by 5“.

22

At the beginning of the ﬂuctuation, the Sites are uncorrelated. Then 4‘“ grows with time.
For a ﬁnite system 4‘“ cannot grow indeﬁnitely, because it is limited by the size of the

system, L. When 511 reaches the size of the system, the entire step edge becomes
correlated, resulting in the saturation of the ﬂuctuation width. Thus at saturation
6.1% . t>>tsm (2.11)

According to (2.7), saturation occurs at a time rm, given by (2.5). Replacing L with 4‘“,

It:
so! ’

one obtains 4‘. ~t which in fact holds fort < tm, as well,

5112'”, ’<< Isa: (2.12)

2.4 Analysis of Simulation and Experimental Data

Two main methods to check scaling laws are to analyze the data from computer
simulation and experimental measurement. Computer simulation is usually done using
the Monte-Carlo method on discrete models. Experimental data are obtained through
STM and REM images of the surface. The analysis of data requires special care due to
ﬁnite size effects. and slow crossover behavior in computer simulations, and unstable
condition, and inadequate resolution in experimental measurements. In this section an
overview of ﬁve methods that are useful in obtaining a thorough analysis of the scaling

properties is given [37]. In the ideal case, a and ﬂdo not depend on the method one

uses. But in practice, a sometimes varies from method to method, and it is useful to use
as many methods as possible in order to get a feeling for the range of values of a that are

consistent with the data.

23

Method 1. Assume that the saturation width w,a,(L) has been calculated for different
system Sizes L. Then the roughness exponent a can be estimated using (2.4) by plotting
the data on a log-log plot, and measuring the slope of the straight line. To use this
method, one must be able to obtain w,a,(L) for various system sizes L. In most
experimental situations one does not have results for different systems Sizes, or in
addition there may be strong boundary effects inﬂuencing wm,(L). Thus Method 1 is more
useful for computer simulations, where one can simulate systems of arbitrary sizes, and
can require periodic boundary conditions to reduce the boundary effects. Method 1
cannot always be used, since for sufﬁciently large systems it is not possible to saturate
the step edge ﬂuctuation. According to (2.5), the time necessary for saturation increases
with the system size. If the dynamic exponent z is large, this increase can be quite

dramatic requiring a prohibitive computation time.

Method 2. In many situations, there is no information on the dynamics of the growth, nor
is there the possibility of producing interfaces with different system Sizes. Suppose the
only data one has are collected at the ﬁnal stage of an experiment consisting of the value
of the height at different points. In this situation, one can study the sealing of the local

width w,(t) deﬁned by
w; (z) :- <[h,(x,t) - (h,(t))]2>. (2.13)

The subscript 1 means that one selects a portion (“window”) of length l on the interface

and measures the width and average height (h,(t)) in this window. The brackets (...)

24

denote spatial (over x) and ensemble averages - one chooses many different windows

along the surface and averages over the obtained results. For small I
Wl(t)~la, l<<€n : (2.14)
where a is the same roughness exponent deﬁned in (2.4). One can verify that (2.14)

reduces to (2.4) in the long time limit, by replacing l with the system size L. Method 2 is

also useful for the determination of a in numerical Simulations, and is complementary to

Method 1.

Method 3. If, in addition to Wsat one has the full time evolution of the width, as shown in
Figure 2.4, one can use (2.7) to determine both or and z. This can be done by attempting
to rescale the data as it is illustrated in Figure 2.5. If the correct exponents for the
rescaling is used, it should give good data collapse. Any signiﬁcant deviation would lead

to the breakdown of data collapse.

Method 4. Another quantity that scales in the same way as the step ﬂuctuation width is

the correlation function,
G(x,z)s<(h(xl +x,t, +t)—h(x,,t,))2> (2.15)

The correlation function G(x,t) deﬁned in (2.15) is sensitive to bias in the data, i.e., if the
step edge has an overall slope, the scaling is affected. If the step edge is obtained from
numerical simulations with periodic boundary conditions, there should be no bias in the
data so (2.15) gives the correct scaling. However, step edges obtained from experimental
data may have an overall Slope. To overcome the effect of the bias, one can use a slightly

different height variable from which the overall average slope has been subtracted.

25

If one is interested only in a, one takes t = 0 (Le, there is no time delay), and

G(x,0) ~ x2“ , x << 5”. (2.16)
If one wishes to determine ,6, one can measure the correlation with a time delay.
Provided one studies times shorter than the crossover time t3“, where the width saturates,

one ﬁnds that G(0,t) has the same scaling as w2 (t) (see Appendix B)

G(0,t) ~t2” . t<<t. er...- (2.17)

Method 5. Some experiments measure the power Spectrum of the step edge, and not the
height h(x, t). Thus one can examine how the dynamic scaling relation (2.7) is modiﬁed

in Fourier space. Consider the structure factor
Cqu) s (Izq(:)h_,(z)) , (2.18)
where

hq(t) '—:— Z[h(x,t)—<h(t)>]e"" (2.19)

The dynamic scaling hypothesis (2.7) can be translated to the structure factor, with result

(1,0) =q“’2“g(z/q'=). (2.20)

(Zavdl :

Here g(u) ~ It for u << I and g(u)=const for u >> 1.

Note that w2(t) = (l/L)Zq Cq(t) gives a simple relation between the structure factor and

the width ofthe step edge ﬂuctuation. The calculation ofthe structure factor provides an

alternative way to estimate the scaling exponents. It has the advantage over the real space

26

methods that only the long wavelength modes contribute to its scaling. The scaling of the
real space form (2.7) is determined by all modes, including the short wavelength modes.

Thus (2.7) is expected to have stronger ﬁnite size effects than (2.20) [37].

2.5 Fluctuation Equation and Symmetry Principles

Symmetry principles can be used to derive the equation that describes the equilibrium
step edge [37]. By equilibrium one means that the ﬂuctuation of the step edge is not
driven by an external ﬁeld. Thus an equilibrium step separates two terraces of the surface
that are in “equilibrium” in the sense that one terrace is not growing at the expense of the

other.

As assumed at the beginning of this chapter, the position of the step edge is characterized
by its height h(x,t). Beside that, it should be assumed that h(x,t) is single valued - i.e.,
there are no “overhangs”. The goal is to derive a general ﬂuctuation equation. It is

expected that equation has the form

ah t
———g:’ )= F(h,x,t) + n(x,t). (221)
Here F (h,x,t) is the deterministic term, while 77(x,t) is the noise term. It reﬂects the

random ﬂuctuations in the detachment and attachment process and is an uncorrelated

random number that has zero conﬁgurational average
(r7(x.t)) = 0. (2.22)

The second moment of the noise is given by

27

(77(x.t)n(x’,t'))°c5(x-x')5(t—t') (2.23)

for nonconservative and
<77(x, t)r7(x' , t' )) oc -V25(x — x' )5(t — t') (2.24)
for conservative noise. Relations (2.25) and (2.24) imply that noise has no correlations in
Space and time, Since averaging over the product (77(x,t)n(x',t')) produces zero, except
for the Special case in which t=t’ and x=x’. Conditions (2.22) and (2.23) are
automatically satisﬁed if the noise variable 77 is chosen from a Gaussian distribution.

Another type of noise that also satisﬁes both (2.22) and (2.23) and is often used in

numerical simulations is “bounded noise”, in which 77 =+I and r] =-1 are chosen with

equal probability.

As a ﬁrst step in obtaining the ﬂuctuation equation, one should consider the basic

symmetries of the problem :

(i) Invariance under translation in time. The equation should not depend on where one
deﬁnes the origin of time, so the equation must be invariant under the transformation

t —> t + 6 t . This symmetry rules out an explicit time dependence of F.

(ii) Translation invariance along the x direction. The equation should not depend on the
actual value of x, having the symmetry x —> x + 6 x . This excludes explicit x dependence

ofF.

28

(iii) Translation invariance along the h direction. The ﬂuctuation rule Should be
independent of where one deﬁnes h=0, so the ﬂuctuation equation should be invariant

under the translation h —) h +6 h . This symmetry rules out the explicit h dependence of

F. So that the equation must be constructed from combinations of Vh, Vzh,..., V"h .

(iv) Inversion symmetry in x direction. The equation must be invariant under

transformation x—>—x. This rules out odd order derivatives in the coordinates,

excluding vectors such as Vh, V(V2h) , etc.

(v) Inversion symmetry in h direction. The equation must be invariant under
transformation It —> —h. This rules out even powers of VI: , terms such as (Vh)2,(Vh)“,

etc. This symmetry is intimately connected to the equilibrium nature of the step edge, and

for nonequilibrium problems this symmetry may be broken.

To ﬁnd the ﬁnal form of the ﬂuctuation equation. one considers all terms that can be

fonned from combinations of powers of WI: . One by one, one can eliminate all those
that violate at least one of the symmetries listed above. Thus,

611(x,t)

a = V312 + V‘h + + V3"h + (Vzh)(Vh)3 + + (VI‘hXVhf’ + r](x,t) , (2.25)
t

 

where n, k, j can take any positive integer value. For simplicity of notation, the

coefﬁcients in front ofthe terms are not explicitly indicated.

29

In the rest of the chapter, the continuum equations for two existing models, step edge
difﬁrsion and attachment-detachment, will be derived, analyzed and compared with

simulation and experimental results

2.6 Step-Edge Diffusion

In the step edge diffusion model, atoms are only allowed to move along a step edge by
reattaching to one of the two closest neighboring sites (see Figure 2.1(a)). That diffusion
generates a macroscopic current j(x,t), parallel to the average step edge direction. The
local changes in the step height are the result of the nonzero current along the edge. Since
the total number of particles remains unchanged during the diffusion process, the current
must obey the continuity equation

Lg”) = mat). (2.26)

On the other hand, the step edge current is driven by the differences in the local chemical

potential ,u(x,t)
j(x.t) cc —V#(x.t). (2.27)
To determine the local chemical potential, one may use either (a) physical reasoning or

(b) symmetry arguments.

(a) Step edge diffusion is an activated process. The motion of an atom does not depend
on the local height of the step, but only on the number of bonds that must be broken
for diffusion to take place. The number of bonds a particle may form increases with

the local curvature of the step edge at that point (see Figure 2.6). If the local radius of

30

 

 

 

a
( ) (b)‘ (C)
I, I, \
.' H / ‘~
I ’ ‘
I I ‘
'. " ‘
‘ I
\ l I'
\\ I l
- I
, .r"""1~.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘-u-§

Figure 2.6 Illustration of three possible local geometries [3 7].

31

curvature R is positive, the atom has a large number of neighbors, and moving away from
the Site will be difﬁcult. In contrast, if R is negative, the atom has a few neighbors, and is
able to diffuse easily. The Simplest assumption is that the chemical potential controlling

the diffusion probability is proportional to —1/R, which in turn is proportional to

V2h(x,t) . Hence

,u(x,t) oc —V2h(x,t). (2.28)

(b) Symmetry arguments can lead to the same result (2.28) [37]. The diffusion
probability of an atom should not depend explicitly on the height h(x,t) of the step
edge, since it does not depend on where one deﬁnes the origin. Similarly, if the
chemical potential were to depend on a power of (Vh), then the sticking probability
would be different if the system of coordinates were tilted. Thus one is left with

(2.28) as the lowest order form for the chemical potential.

Combining (2.26)-(2.28), one obtains the equation describing relaxation by step edge
diffusion [38]

ah
6(— = -KV4/z + 77(x,t) (2.29)

Noise 77(x,t) must be added, otherwise equation (2.29) would be deterministic. During

the detachment process there is an inherent randomness in the system. Noise is

uncorrelated, but conserved (2.24).

32

 

By applying a Fourier transform to (2.29) one gets a Langevin equation in the form (B.1)
where 1“," at q“. Thus, using solution (8.12) derived in Appendix B, it gives the

ﬂuctuation width (2.2) as

w2(t) or t"4 (2.30)
That result has been conﬁrmed with Monte Carlo simulation [8] (see Figure 2.7) and was

experimentally observed [1 1] (see Figure 2.8).

2.7 Attachment-Detachment

In this model, atoms are randomly detaching to the terrace and reattaching from the
terrace (see Figure 2.1(b)). Thus, the terrace is a particle reservoir that randomly
exchanges atoms with the step edge. The process is governed by the difference between

the average chemical potential on the terrace ,um and the local chemical potential on the

surface ,u(x, t)

012(6):» : 4,010,, I) _ “m )+ W, ,) (2.31)

 

where randomness of the process is represented by nonconservative noise (2.23). Setting

the average chemical potential on the terrace to be the. = O and using Equation (2.28),

one gets

(311(x, t)

a = vv3h(x,i) + r](x,t). (2.32)
t

 

33

 

I09(w(t))

 

 

 

 

Figure 2.7 Monte Carlo simulation of an isolated step evolving through step-edge
diffusion. The dashed line shows the t”8 [8].

34

 

5 i I v I V I r I

Cu(1 1 19)
4_ T=,362K ,

V

    
   

all('alnt((lltlla
. ‘1' -|.'..""|"l|

 
 

4

< (Axm- Axtu)2>

:1 conventional images
0 time images

‘

 

 

l

6
t[sec]

_

 

C

NP-
A
WP
op

Figure 2.8 Correlation function G(0,t) at: w2 (t) (see Equation 2.17) averaged over space
and time denoted by squares and circles, respectively. The full line represents
a ﬁt to at”4 law [11].

35

By applying a Fourier transform to (2.32) one also gets a Langevin equation in the form
(B.1) where 2': oc qz. Thus, using again solution (B12) derived in Appendix B, it gives
the ﬂuctuation width (2.2) as

w2(t) act“2 (2.33)
That result has been conﬁrmed with Monte Carlo Simulation [8] (see Figure 2.9) and was

also experimentally observed [10] (see Figure 2.10).

36

 

1 5 rrrrrrrrrrrtrrrrvIrrtrrrrrrrrrrrrirrrirlttrrrrrr
1 a)
F
C
y

  
    

 

AL 444

'09(W(t))
0
01

'0.5 ILLIIIIIIIiIIIIIIILIIIIIIIIIIIIIIrIIIIIIILILIIIIi
1O 7T1ITYTWTTTTTTIITTI'IIYTTTTITIITiTTITTYTTTYTITYTTT

Eb)

b-

 

 

 

  

.4
' l
. LILJILIIIIILIIILIIJIIIIILMLLLUIILIIIIILILLLL I

0 1 2 3 4 5
mm

 

 

 

Figure 2.9 Monte Carlo simulation of an isolated step evolving through attachment-
detachment . The dashed line shows the t"4 [8].

37

 

 

 

 

 

 

0.0 L 1 A
0.00 0.10

10-2 I LJ ILILJ‘ l l r ILLIL

 

 

 

10" 10"2 10"

t(s)

Figure 2.10 Squaired ﬂuctuation width for Au(l 10) at 556K. The full line represents a ﬁt
tot [10].

38

Chapter 3
An Isolated Step
3.1 Description of the Model

Consider a single isolated step at the surface. Its energy is often modeled by,
L
E=JZ(h.- —h.-..)" (3.1)
i=1

where h,- are in discrete units of oi, the lattice constant perpendicular to the step edge.
p =1 is the solid-on-solid (SOS) model, p = 2 is the discrete Gaussian model, while

p = 00 is the restricted solid-on-solid model (RSOS).

In analytic work it is more convenient to use a model which takes on continuum values of
the variable [1,. Moreover, it can be shown that for an isolated step, the free energy

derived from the energy (3.1) is often well approximated (at small slopes) by,

E L/z 2
F~§J (Vh) dx (3.2)

-L/2
where the “stifﬁiess” E = 2(0) + 2"(0) depends on the value of p in (3.1). The “small-
slope approximation” corresponds to w/L << 1. It is assumed that this holds. 2(6) is the

angle-dependent interface free energy derived from (3.1). h(x) is a continuous height

variable. Three cases in which the stiffness can be calculated exactly [39-41] are,

if = ZkBTsinhZU/ZkBT) (3.3)

39

for the SOS model,

1+ 2e"”‘”T
for the RSOS model
2 = k,Tsinh(2J/k,r) (3.5)

for the Ising model. J is the kink energy, and in the low temperature and large L limit,

Equations (3.3)-(3.5) all reduce to

kT

E = ”Temir for kBT << J (3.6)

From the Hamiltonian (3.2), it is straightforward to ﬁnd the saturation width,

2 _

wm < 113(t —> 3c) > — <h(t —> so) >2 (one takes h(x,0)=0 henceforth), of the step edge

(see Appendix A),

kBTL . . LO , ,

_—~ " z: — ‘- 3.7

122 ) (123') ( )
where E = EQ/kBT with Q a a (11 as the area ofthe surface element (a and al are the

lattice spacings perpendicular and parallel to the step edge respectively). As it is seen

from expression (3.7), periodic boundaries at the end of the step are assumed.

The chemical potential, y(x,t) can be found from Equation (3.2) in the following way.

Using the formula for the variation of an integral, (3.2) gives
~ +1. 2 ,
arm) = ~22]! ,v-h 5h dx.

On the other hand, variation of the energy can be expressed through the local chemical

potential [22] as

40

6Fh —1 “’2 6hd
()-5J_m# x.

Finally, equating those equations gives the expression for the chemical potential as
,u(x,t) = ~QEV2h(x,t). (3.8)
Note that this agrees with assumptions (2.28). Within non-equilibrium thermodynamics,
the driving force for relaxation is provided by deviations in the local chemical potential
from its equilibrium value. The rate at which such deviations relax is determined by the
dominant kinetics in the physical system of interest. In addition, the thermal noise in the

system randomly generates ﬂuctuations. One thus describes the dynamics of a step edge

 

by the equation,
Elr(.r,t) I",
= J x,t + x,t . 3.9
51 k8?" ( ) nt .) ( )
where:
L 2
J(x.t)= I“ P,(l){,u(x+l,t)-2,u(x,t)+,u(x—l,t)}dl (3.10)

L is the length of the step along the x-direction and 1“,, is the rate
1". =a./r.. (3.11)
where r,I is the time between detachment events from a site on the step edge. r](x,t) is

the noise term, which must reproduce the equilibrium ﬂuctuations of the step at long
times (see Appendix B). Equation (3.9) states that the relaxation rate of a chemical
potential difference between two step positions (separated by distance I) is a product of
the chemical potential difference between those Sites and the ﬂux of atoms which is

exchanged between the two positions. Thus, P,(I) is the probability that an atom will be

4]

 

Figure 3.1 Terrace diffusion with diffusion kemel P,(l).

42

exchanged between sites separated by distance I along the isolated step edge (see Figure

3.1). It is normalized

L12
2 [0 13(1) d1 =1

A Fourier transform of (3.9) and (3.10) using normalization yields Langevin equation,

612,10) _ hq(t)
T _ _

 

+nq(t) (3.11)

q

where

~ 2 ,.
r" =5—‘1—"—i{1-2]0“12(1)cosq1d1} (3.12)
T

‘7
h
and (periodic boundaries),
h(x,t) = th(t)e‘r , with q = :27: n /L , n =1,...,L/a“.

‘1

Note that if the noise is removed from the right hand Side of Equation (3.11), one has
hq (t) = hq(0) ed,” with no noise (3.13)

A general expression for I: was found by Bales and Zangwill [14] (corrected by

Pimpinelli et al. [13]). Their expressions will be derived from a different physical

perspective later in this chapter.

In the presence of Gaussian white noise

(n, (t. m; (t. 1) = ftq. .q. >6.,,,,6(t. — t2).

the solution to the linear Langevin Equation (3.11) is (see Appendix B)

43

 

3 Q - m
<lhq<t>l >= 23.12 “-82 “>- (3.15)

An inverse transform of lhq(t)l2 then gives,

Q

3 2

 

1.220) =< h2(x,t) > -(h(x,t>>2 =< “0"”) >= 2

q

(1 — e""'°) (3.16)

since (h(x,t)) = 0 (center of the mass of the step does not move from its initial position

at x = 0) Besides, w2(t) = G(2t) , where (see Appendix B)

G(t) = %<(h(x,t, + z) — h(x,t, ))2), t << t, z :50, (3,17)

and t5“, is the saturation time. w(t) is a fluctuation width, while G(t) is a time dependent
equilibrium correlation function. Within the linear Langevin Equation (3.11) they are

essentially equivalent. At long times, the exponential part in Equation (3.16) is dominated

by the n = 1 term in the sum. One thus deﬁnes the saturation time tm, , by the time it takes
for the (n = 1) exponential term to decay to e‘1 so that,
tsat : z'q=22r/L ' (3'18)

This gives a Simple and accurate estimate of the saturation time. Thus the “natural”

scaled variables are G(t)/ wig, and t/tm.

In several cases, one ﬁnds that
-1

in which case, the continuum limit of Equation (3.16) yields (see (B.12))

1

Q r(1— l)(2.«1t)7, (3.20)
7

7:?

w2 (t) = G(2t) ~

44

where F is the Gamma integral as deﬁned for example in [34]. Thus, scaling assumed in

(2.3) holds if(3.19) holds.
3.2 Step-Edge Diffusion

The diffusion is restricted to nearest neighbors along the step edge (see Section 2.6 and

Figure 2.1(a)) by deﬁning,
P.(0=%{6(I—a.,)+6 (I+a.)} (3.21)

The characteristic time rh is now the time between atomic hopping events between

nearest neighbors. Let us assume simple activated behavior

I; = vexp(—ESE /kBT){exp(—E, /kBT)+exp(—E, /kBT)} , where v is a characteristic
phonon frequency, E SE is a step-edge diffusion activation energy, while E, and E, are
the energy barriers that an atom, once activated, has to overcome to move to the left or to
the right site respectively. In the simplest simetric case E, = E, = 0, thus r: = 22;;
where r; = vexp(—ESE /k,,T). Using Equation (3.12) then yields

.._23al
q _

 

T q2 {1 - 0086101} - (3.22)

TSE
An expansion of rJ' at small q yields the behavior (3.19), with A = Ea“2 ”55 , 7 = 4 for
[glam]. The region |q|>l/all is very small and has negligible inﬂuence to the

correlation function. Thus, in the intemiediate time regime one ﬁnds (using Equation

(320)),

45

(25 J. 1
Wm) = G(zt) =;:-r(§-)(§,—f'“r(f—’r. (3.23)

This agrees with Equation (2.30) and case F of [13]. It is the essentially the same as the

result previously obtained in [8].

A behavior similar to Equation (3.23) occurs for any short range diffusion kernel. For

 

example, taking
1 -L
13(1) = —2—e :1 , with aH << 5” << L (3.24)
It
yields
gzt 1/4
w2(t) = G(2t) = const(§sr ] . (3.25)
SE

A little later in this chapter, it will be shown that if P,(l) = 8/1“ , with a 23, then a

similar behavior pertains.

3.3 Attachment-Detachment Model

The attachment-detachment (see Section 2.7) assumes that a step is randomly exchanging
atoms with the terrace (see Figure 2.1(b)). It can be described by using the probability

function Equation (3.24) but now assuming that the correlation length, 15H>>L.

Physically this means that inﬁnite range conserved-order-parameter dynamics is the same
as non-conserved order parameter dynamics. To simplify the calculation, one can use the

normalized uniform probability P.3(l) as,

46

12(1) = const =% (3.26)

The characteristic time I, is the time between atomic detachment events from a step

edge onto the terrace. In the simplest case one would expect activated behavior
I: = 1;], = vexp(—EAD / kBT) , where v is a characteristic phonon frequency and E AD is

an attachment-detachment activation energy. However, this simpliﬁed theory remains
valid even if many different processes (and barriers) play a role, provided they can be

treated as independent. Equation (3.12) then yields,

r; = ”i qz, (3.29)
TAD.

 

for all values of q. Thus, the q dependence (3.19) holds with A = 3ai ”5c and y = 2.

Using (3.20), one has

W20) = G(2t) = Miami)“ . (3.28)
”S

TAD
This agrees with Equation (2.33) and case A of [13]. It is essentially the same result as

the one obtained by prior workers in [8] and [42].

3.4 Terrace Diffusion

Over broad temperature regimes, terrace diffusion is the dominant kinetic mechanism. In

this model, an atom detaches onto the terrace below the step edge with probability p L ,
and onto the terrace above the step edge with probability pU. It then diffuses on the

terrace until it reattaches to the same or one of the adjacent step edges. Let us deﬁne aL

to be the sticking coefﬁcient of an atom which approaches a step edge from the lower

47

terrace and on, to be the sticking coefﬁcient on approach to the step edge from the upper
terrace. One assumes Simple behavior where Em is the terrace diffusion activation
energy, and E L and EL, are the energy barriers that an atom, once activated, has to
overcome to move to the lower or to the upper terrace respectively (see Figure 3.2). E L
and E U are also energy barriers that an atom has to overcome to stick to the Step edge on
its approach from the lower and upper terrace respectively. EU is known as the
“Schwoebel barrier” [45] and one expects in general EU >> E L. This is understood by

noting that the atom that sticks to a step on its approach from the upper terrace has to go
through the position where it has fewer neighbors. This is unfavorable energetically. One

then ﬁnds that probabilities of detaching

 

 

 

 

e'EL/kar e-EU/kar
pL = e—ELrisr +e-£,,,./k,,r ’ pU = e-EL/kHT +e—ELJkBT (3'29)
and sticking coefﬁcients (see Appendix C)
-E /k T -E.-/It T
e L 8 e L 8
= = . (3.30)

a . , a l _ l
L (1_ e-EL/kﬂr)ai L (1_ 8 EL' kBT )ai
are related to the step edge barriers. Sticking coefﬁcients can have any value in the range

[0.00) where 0 corresponds to no sticking and co to perfect sticking.

In terms of the energies deﬁned in Figure 3.2, the characteristic time r,l is,
1;! = 2?; {e-EL/ksl' +e-EC/k3T}, (3.31)
where r;,', = vexp(-—Em / kBT) and v is a characteristic phonon frequency. Therefore, for

the terrace diffusion case, the fundamental activation energies are E TD, EL and E U, as

48

 

 

Figure 3.2 Key energy barriers at a step edge.

49

 

deﬁned in Figure 3.2, while the “natural” parameters appearing in the dynamics are rm ,

61,, and aL.

P,-(I) for terrace diffusion is
13(1) = pL,P'(aU,aL,I) +pLP'(aL,aU,l) (3.32)
where P'= P with d —> 00 given by Equation (C.14) derived in Appendix C (see also

Equation (C.17)), from which one ﬁnds,

 

a cos kl e"""*dk 0: cos kl e"""*dk
P. l = i -—————+ 4 3.33
1() pU 7r 0 k'l'aU pL 7r k+aL ( )
-ka‘.
On Figure 3.3, P' = 3 row vs. 1 is plotted for four different values of the

it 0 k + a
sticking coefﬁcient a . Note that for the large values of a (perfect sticking) P’ at 1/12

for all I , while for the small values of a (weak sticking) P' at —-ln(l) for small I and

P'ocl/l2 for large 1.

Using expression (3.33) in (3.12) leads to,

~ 2
-1 Sq aJ.
rq =—

 

 

e'iqi'ai ”4'10;
l-aUpU -aLpL (3.34)
av + 9' at. + ‘1

Th
Using this inverse time and doing numerical summations over q, G(t), deﬁned by
Equation (8.10), is plotted on Figure 3.4 for a, =aU =10‘2/ a 1. (weak sticking),

a, =103/ai, (1,, =10"2 /ai (Schwoebel barrier), and a, = aU =103/al (perfect

sticking). As will be shown below, G(t) oc t"3 for perfect sticking, Schwoebel ban'ier and

50

Single Step : probability

for different values of sticking coefﬁcient alpha

0.0 1 . . ~. 4 -
\

 

   

 

 

 

 

 

 

 

-10.0 ~
In P'
-~-— ~2510pe \\\
'15'0 _ *~-—-'- alpha = 0.0] ‘
--—-— alpha = 0.1
- ---—~ alpha = l
-—- alpha = 10
_200 1 . 1 1 L 1 1 1
0.0 2.0 4.0 6.0 8.0 10.0
In I

Figure 3.3 Diffusion kernel (C.17) is plotted vs. I for four different values of sticking
coefﬁcient (from the top): aL =aU =0.0l/al, aL =aU =0.1/a,_, aL =aU =1/al, and
(IL = (1,, = 10/ a i . Note that for large values of a (perfect sticking) P' 0: 1/ l2 for all I ,
while for small values of a (weak sticking) P’ 0: -—ln(l) for small I and P' at 1/ l2 for
large I .

51

 

 

 

 

10° —

10‘1 - _
”TS-10’2 — ~
3.

2
E5103 — ~
10“ — _
10-5 0 l2 l4 16 18 I10 112 I14
10 10 10 10 10 10 10 10
t/t

Figure 3.4 The correlation function G(t) for (from the top): at = a” =103 la];
a, =103/a], a” =10'2/al; a, =aU =10‘2/ai. In all cases L=10’a”, kBT/E=l. It
is plotted using inverse times (3.34) and doing numerical summations over q , deﬁned by
Equation (B.10), As shown in Equations (3.39)-(3.41), G(t) at t”3 for perfect sticking,

Schwoebel barrier, and weak sticking case at large times, while G(t) 0: t”2 for weak
sticking case at small times.

52

weak sticking case (the last one at large times only), while G(t) oc t"2 for weak sticking

case at small times.

For |q| < l/ai , (3.34) reduces to,

~ 2

r = — q +
TTo “U + M at + M

 

q

). (3.35)

From Equation (3.35) it is seen that there are three possible regimes:
(i) when a“ << |q| then

Ta" = him... + 61,.qu /Tm (3.36) r
(q2 dependence for small sticking coefﬁcients has been also suggested in [16]);

(ii) when aU << |q| << aL (assuming a(, << aL) then

_ ~ 1 3
rq' =sa;|q| Hm; (3.37)

(iii) and when a“ >> |q| then

r" = 2§ailql3 / rm (3.38)

(I

(last expression has been derived using different formalism in [16]).

Thus, extreme values of sticking coefﬁcients leads to three different time dependences of
the width and correlation function. In the case of small sticking coefﬁcients, an atom has
to bounce off the step edge many times before it sticks. Thus, terrace diffusion with weak
sticking has the same effect as attachment-detachment process (see Equation 3.28). After

substituting (3.36) to (3.20) one has

53

 

1/2 1’2
W2 (t) = G(2t) = Qal(aU;aL) [A] , for (1“ << 1 . (3.39)

[TD
In the large Schwoebel barrier case, one of the terraces (the upper one for the case

all << aL) is not contributing Since approach to the step edge from that terrace has such a

small sticking coefﬁcient. After substituting (3.37) to (3.20) one has,

w2(t) = G(2t) =M["i
72'

1!} 2t “3
:7 — , for aU <<l<<aL (3.40)

However, for very small values of q (3.36) and (3.37) do not hold. Instead, (3.38) then

holds for all values of sticking coefﬁcients. Thus at large enough times, weak sticking
and Schwoebel barrier case have the same correlation function as perfect sticking ease for

all times (after substituting (3.38) to (3.20)),

2 1/3 1/3
,,.2(,)=G(2,)=“Ff/3)[Zf;] (-24) . for au>>1 (3.41)

S TTD

 

That is visible on Figure 3.4, where correlation function of weak sticking (bottom line)
and Schwoebel barrier (middle line) at certain time cross to (3.41) from (3.39) and (3.38)

respectively.

In experiment, the step-edge, terrace diffusion and attachment-detachment mechanisms
all occur, albeit with different activation energies. It is straightforward to combine the

diffusion kernels (3.21 ), (3.26) and (3.33), with arbitrary weights and hence ﬁnd,

, a2 1 a, a
If =§alq‘ a—”—-q2 +b—+c 0* q[ L‘ + L j (342)
SE 7.11) Tri) all + l‘ll a]. +l‘ll

 

 

54

with a+b+c=l to preserve the normalization of the kernel P,-(I). The weights a,b,c are
expected to have an activated form, dependent on the various step-edge and kink
energies, so the crossover between the various mechanisms as a function of temperature
is quite Sharp. At present atomistic simulations are not sufﬁciently precise to predict in

detail the relative importance of the various terms in Equation (3.42).

3.5 Anomalous Diffusion

It is interesting to note that there are cases in which the step dynamics is non-universal.

One can illustrate this point through use of the probability distribution
B >
P,(l) = 70— for l_ a“ (3.43)

where a =0 recovers the attachment-detachment limit, a = 2 corresponds to terrace

diffusion with perfect sticking, while a > 3 (see below) corresponds to “short-range”

hopping and recovers the step—edge diffusion behavior. B”1 = 2((L/ 2)” —a|'|"’)/(l —a) '
normalizes the probability on the interval (”w”)- By evaluating the integral Equation

(3.12), one ﬁnds that:

2

Ma 2-.
”) ‘0) for OSa<l.0; (3,44)

-1 Sai 2
r z———— 1+0
., Th q( (( L

 

 

 

_,z§ai 2 1——':—-C£I"l—cr cos(—1——a—)£ a “"+ l—a a 2 forl<a<3 3.45
Tq Th q [ 2 ( ) 2 (q u) 4(3-0’) (q ll) ( )

(Note that for integer a , rq" is ﬁnite despite the divergence in F(l - a) );

2 _ a ‘1 4
(an) —_48(a _ 5) (an) J for a > 3 (3.46)

 

 

_, Eai 3 62-1
rq 2: q
4(a—l)

55

Using Equation (3.20), one then ﬁnds

w2(t) = G(2t) ~ t”2 for O < a <1, (3.47)
7 J— ’
w'(t) = G(2t) ~ tCHI for IS a S 3, (3.48)
and
w2(t) = G(2t) ~ t'” for a 2 3. (3.49)

Equations (3.47-3.49) Show a continuous variation in the dynamical exponent for
1 S a S 3 , while the attachment-detachment universality class holds for 0 < a <1 and the
step-edge universality class holds for a 2 3. Figure 3.5 shows ﬂuctuation exponent ,8
(see (2.3)) vs. a . The points on the graph are obtained using diffusion kernel (3.43) in
(3.12) to numerically ﬁnd q dependence and than using (3.20) to get exponent ﬂ. As it
can be seen, there is a continuous variation for 0 S a S 3. Thus, (347-349) holds as an

approximate dependence around a = l . This is due to complex forms of (3.44-3.46).

56

 

 

 

9.0 [ f r r . . 1

8.0 r / +

7.0 -

 

6.0 e . s

5.0 L / _

 

 

 

 

 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 A 3.5 A 4.0 A 4.5 A 5.0

Figure 3.5 The ﬂuctuation exponent ,B (see (2.3)) is plotted vs. a from (3.43). The
points on the graph are obtained using diffusion kernel (3.43) in (3.12) to numerically
ﬁnd q dependence and than using (3.20) to get exponent ,6.

57

Chapter 4

Array of Steps

4.1 Introduction

In this chapter, the step train (array) is treated (see Figure 4.1). The model introduced in

Chapter 3 is extended to include the exchange of atoms between neighboring steps. The

pair correlation function between two steps C k(x,t) is introduced [48]. It is an averaged

product of two step positions separated by distance x parallel to the step edge and located

k steps away. The relaxation time qu for steps which are modulated along the steps (with

wave number q), and perpendicular to the steps (with wave number p) is calculated. By

summing over all modes, one derives C ,(x,t). It is shown that the pair correlation
function between two different steps, C,,O(x,t) , can often differentiate between terrace
diffusion and attachment-detachment or step-edge diffusion. C ,,0(x,t) is identically zero
for the cases of attachment-detachment and step—edge diffusion. In contrast, C M(x,t) is

ﬁnite and quite large in the case of terrace diffusion. This means that measurement of a

ﬁnite value for C ,,0( x,t) implies that terrace diffusion is important. Since G(t) quantiﬁes

the cumulative effect of all mass transport mechanisms, by combining the results of G(t)

and C M(x,t) it is then possible to determine the relative importance of terrace diffusion

to the other types of mass transport in surface dynamics. In addition, G(t) used in Chapter

3 is generalized by introducing G(x, t) as the averaged square ﬂuctuation of the difference

58

between two step positions separated by distance x parallel to the step edge and located at
the same step. It enables one to derive initial time-independent ﬂuctuation for nonzero x.

The currently used correlation function is included as G(t) E G(O, t).

The next section contains the real-space formulation of step dynamics, extended from the

one in Chapter 3. The key equations are Equation (4.17) for the relaxation time qui and
Equation (4.24) for the correlation function Ck(x,t). Section 4.3 contains an analysis of

Ck(x,t) and G(x,t) in a variety of time regimes. This section can be skipped by those

uninterested in the detailed analysis. Section 4.4 treats the terrace diffusion case. Since
there is no correlated mass exchange between steps in attachment-detachment and step-
edge diffusion case, in these cases the step train model essentially gives the same results
as for an isolated step model. For that reason, these two models are not treated in Section

4.4. Analytic results for G(x,t) and C,(0,t)for all three solvable limiting cases are

summarized in Tables 4.1 and 4.2 respectively. Examples illustrating some of the more

interesting crossovers between these limiting cases are illustrated in Figures 4.3 and 4.4.
4.2. The Model

Consider a train of N steps, all with length L (see Figure 4.1). Steps are numbered with
k=1,2,...,N and periodic boundary conditions both along the step train, and along each
step are assumed. (this simpliﬁes the analysis and is typical of Steps away from the edges

of a ﬁnite step train). Let hk(x,t)describe the random motion of the k"' step in the train

about its center of mass, which is assumed to be ﬁxed - there are no “direct" interaction

terms to produce center of mass dynamics. The average distance between centers of mass

59

 

 

Figure 4.1 Diffusion kernels on the Vicinal surface.
60

of adjacent steps is the same everywhere on the surface and equals d. The existence of
direct interaction would cause the centers of mass of the steps to move. That would lead

to the change in surface morphology. That is analyzed in Chapter 5.

Let us deﬁne the equilibrium (t << t1 —3 oo) correlation functions
G(x,t) = g < [72," (x1 + x,tl + t) — hi. (x,,tl )]2 > (4.1)
and
C, (x,t) =< hkﬁ,‘ (x1 + x,tl + t) hi. (x,,t,)>. (4.2)
While G(x,t) measures the ﬂuctuation width, the pair correlation function C , (x,t)

quantiﬁes the correlation between the ﬂuctuation of two points. Because of the periodic

boundaries, G(x,t) and Ck(x,t) do not depend on x1 and k1. By squaring the bracket in

Equation (4.1) and taking an average of each term individually, one ﬁnds that these two

functions are related as

G(x,t) = C,(0,0) - c, (x,t) (4.3)

where Co(0,0) is actually the squared equilibrium width of the step ﬂuctuation.

The local chemical potential, ,uk(x,t) , one associates with k'II step is,
,uk(x,t) = -Oiv2h, (x,t). (4.4)
It is derived using (see derivation of Equation (3.8))

i NZ 2
F, A .2. [w(vm) (a (4.5)

as the energy cost of Gaussian ﬂuctuations of the step edge about its mean position. As

introduced before, Q = a la” is the area of a surface element, with a 1 and all being the

61

lattice spacings perpendicular and parallel to the step edge respectively. Time dependent

ﬂuctuations of the steps is described using a coupled set of Langevin equations,

 

ah (x,t) I“ l 1
tat = thT {SJ/Hi (X,t)+Jk.,, (x,t)+§J,‘+M (x,t)} +771! (x,t). (4.6)

J u describes the healing of the step ﬂuctuation due to mass transport along a step, while
J 14.1 and J (+1.1. describe the healing of such ﬂuctuations due to mass transport between

step k and steps k — 1 and k +1 respectively. The healing of ﬂuctuations is driven by
differences in the step chemical potential, and the rate of that healing is controlled by the
rate at which mass may be transferred in order to heal unfavorable chemical potential

differences. The rate
rh = _i (4.7)

where r, is the time between detachment events, depends on the energy ban'iers which

exist at the step edges. This will be discussed further in the context of speciﬁc types of

mass transport.

Considering first healing due to mass transport along a step edge, the integral J u: is
given by

J,,(x,z) = LLAAR,(I){,uk(x+l,t)—2y,(x,t)+,uk(x—l,t)}dl (4.8)
where the chemical potential difference between sites separated by distance I is
contained in the curly brackets, and the “mobility” is described by Po(l). This mobility is

the probability that an atom is exchanged between two sites separated by distance I

along x -axis and located on the same step edge. The form of Po(l) depends on the mass

transport mechanism as it is elucidate later in this section. As in Chapter 3, one can do the
62

analysis to a large degree without knowing the explicit form for this quantity. The
integral describing the way in which ﬂuctuations heal due to mass transport between
steps is very Similar to Equation (4.8)
L/2
JktlJt (x,t) = I0 [31(1) {#k:1(x + [,0 - Zﬂk (x,t) +.uk11(x " l,t)}dl (49)
P, (l) is the probability that an atom is exchanged between two sites separated by distance

I along x -axis but located on the adjacent step edges. The probability functions are even

and normalized,
L/2
2 i. (401+ 8(1)) =1 (4.10)
Step perturbations or ﬂuctuations are randomly generated by the noise term qk(x,t) and

then healed by the processes described in the integrals J u and J rm . The noise must be

constructed so that the time independent equilibrium properties of the steps are
reproduced. Since it is assumed that there are no interactions between steps, the

equilibrium correlations of the step train are simply those of non-interacting steps.

A Fourier transform

h, (x,t) = Emmet" (4.11)

and similarly for 77,,(x,t) with q = 27m/L, n = il,i2,... iL/a“ in Equation (4.6) yields

the N -dimensional set of linear ﬁrst-order differential equations,

 

aiiqu) ~ — -
a: =-—M(q)h.,(t)+rr.,(t) (4.12)

with M given by,

63

f 1 1 \

 

 

80(4) “581(4) 10 —§g.(q)
--2-gi(q) 80(9) —§g.(q) 0
M(q)= 5 1A.' 2 (4.13)

1

0 __ __
1 2 81(9) qotq) 281(6))
(381“!) 0 -§g.(q) 80(9) )

where
80(61): 5% {1 — 2 JUL/AR,(I)cosqldl}

h

. (4.14)

2:; 2a U2
g.(q)=——q i] Pl(l)cosqldl
r, 0

with the reduced stiffness E 2 E0 / kBT having units of length.

In order to decouple the system of equations (4.12), one needs to diagonalize the matrix

M . Since this matrix is circulant, it is also diagonalized by a Fourier transform [46]

{34(2) =0‘ii'q(i) with U,” = e”*""’ (4.15)

at:

where p = Zmn/N, m = 0.1.2.....N —1. When this is completed one arrives at a set of

decoupled linear Langevin equations,

6vp,(t) _ v.0)
at —

 

+r7pq(t) (4.16)

m

where each mode (p,q) has a relaxation time rm given by,
r; = g,(q) — g,(q)cos,2 (4.17)
and is associated with the noise 77mm. The statistics of the noise is unaltered and

therefore one can keep the same symbol I}. In the absence of noise, equation (4.16)

64

describes the relaxation of a periodic mode (e.g. sinusoidal with wavenumber q along
the steps and p perpendicular to the steps) with relaxation time qu' This mode relaxes
33,

qu(t) = vm(0)e"”~, (4.18)

where qu(0) is the initial (small) amplitude of the mode.

In the presence of Gaussian white noise

< 17..., (I, ML”, 0.) >= f (p. ,1). .q. .612 )5“, 5,,,,,5 (t. — t.) (4.19)

the correlation function becomes (see Equation (8.18))

Q ‘. £14”
G(x,t) = mzméf——Zcos pk emmw (4.20)
q

p

where p = an/N , m = 0,1,2,...,N -l , and q = 2m1/L , n = i1,i2,...,iL/a”.

The next Section 4.3 presents an analysis of Equation (4.20) in a variety of solvable
limits. This section is used in Section 4.4 which discusses physical cases, but may be

skipped by those uninterested in the detailed analysis.
4.3 Analysis of Various Time Regimes

Taking the continuum limit ofthe p sum gives (p376-9.6.l9 of[34]),

20 cosrxe’gw”I
C.<.u>=-: —’—.———I.(g.(q)z), (4.21)

q>0 (I
where l,(:) is the modified Bessel function of order k and argument 2. To make the

analysis in this section simpler, summation over positive q only will be considered.
65

At t = 0 , Equation (4.21) reduces (because [w(O) = Oand 10(0) = 1, p375-9.6.7 of [34])
to (p39-l 443(3) of [47])

C.x( thg l—i+("r) (4.22)

which is the equilibrium correlation function along a step, while the equilibrium

correlations between steps are zero (C , w(x,0) = 0 ). At x = 0 , Equation (4.22) reduces to

the squared equilibrium width 11% of the step edge. The equilibrium correlations
s

between steps Ck,0(x,0) are zero due to the fact that there are no direct energy terms
between steps in this model. Nevertheless, as demonstrated below, there are strong time
dependent correlations C h0(x,t > 0) between steps when there is correlated mass
transport across the terraces (e.g. terrace diffusion). These correlations are absent in the
case of attachment-detachment and step-edge diffusion. The analysis of the correlations
between different steps thus provides a method to distinguish between attachment-

detachment and terrace diffusion, which have very similar trends in the correlation

function on the same step.

Due to the difference between the equilibrium behavior of the correlation functions
between the same and different steps, these two cases differently will be treated

differently. For correlations between different steps C “G(x,t) is used. It increases from
zero at short time. For the step self-correlation, G(x,t) is used. It also increases from

zero at the Short time, rather then C0(x,t) that starts at its equilibrium value. This way is

66

more convenient for analyzing log-log plots to test the power law behaviors so typical of

step dynamics.

To proceed to an analytic analysis of the time dependence of C “G(x,t) and G(x,t) , one

needs more detailed expressions for go(q) and gl (q). It will shown later in this section

that the important asymptotic behaviors of these functions can be expressed in relatively
simple forms. There are two cases, one for terrace diffusion (case A) and the other
applies to attachment-detachment, step edge diffusion or terrace diffusion with an inﬁnite

Schwoebel barrier (case B).

CASE A (terrace diffusion)
80(9) z Athl" + qulp, 81(4) 9 A.|q|" for Iql < qo
80(4) z Azlqlhv g.(q) = 0 for lql > 90. (4.23)
For [(1] < qo, BM” << A,|q|" is always satisﬁed and temi 8|q|” will be neglected in the

following calculation except in expression go(q) — gI (q).

CASE B (attachment-detachment, step-edge diffusion, terrace diffusion with an inﬁnite

Schwoebel barrier)

800]) z Allqlr' , g,(q) = 0 for lg] < q0

 

g0(q)z A2 (1172, g,(q)=0 for |q|>q0- (4-24)

The details of the system of interest (terrace width, energies, diffusion constants, etc.) are

primarily contained in the prefactors A1,: , the crossover wavenumber q0 and lower limit

67

of q = 227/ L. The exponents 71.2 take on quite universal values depending of the mass

transport mechanism. Nevertheless, a complete asymptotic analysis will now be done
without knowing these details, and then those results will be used in the next section. In
the next section these asymptotic results will also be compared with direct numerical

evaluations of Equation (4.20) for physically interesting cases.

There are three important asymptotic time regimes in the step dynamics:
- (1) Early times t -—> 0
- (ii) Intermediate times 0 << t << tm,
- (iii) Very long times t>> tm,
Here, the saturation time,
rm, = (L / 2n)" /Al (4.25)
is the decay time of slowest mode in the system. The asymptotic fomrs for C k w(x,t)and
G(x,t) in these three regimes will be presented, though frequently (ii) is the most

experimentally accessible one.

Correlations between different steps - C , w(x, t) (only case A)

Starting from zero at t=0. the correlation between steps C “G(x,t) grows due to

correlated mass transport between steps. Neither step-edge diffusion, attachment-
detachment nor terrace diffusion with an inﬁnite Schwoebel barrier (case B) produce this

correlation, SO C,,0(x,t) = Oin these cases. Considering case A (4.23), the sum (4.21)

68

ceases to contribute for q > qo , due to the fact that the Bessel function 1“,,(0) = 0. One

then has (combining (4.21) and (4.23))

20 cos x e“""’"'
C,,,(x,t) = —~— Z—q2——1,(A,q"t). (4.26)
LS q<4o q

(i) Short times t —) 0

At short times, one expands Bessel function for small argument (p375-9.6.7 of [34]),
1()—(5)*l —>0 427
" Z — 2 k1’ Z ( ‘ )
and sets the exponential to one, which yields,

C x,t=—
"‘°( ) LEM q2 2*k1

 

20 A " "
2“)qu “1 t) . (4.28)

This is valid provided t<<1/(A,qg'). Turning the sum into an integral and evaluating

shows that at small x, one has,

kn-l
CM“) z ~2___qo___(ﬂ
szr k!(kyl —l) 2

)" , qox —+ O. (4.29)

(ii) Intermediate times 0 << t << tm,

In this regime, the sum (4.21) is dominated by small q , so one can take the upper limit to
00. However all values of q contribute, so one can take the continuum limit, and

rewriting in terms of more convenient variables one then ﬁnds,

'1' _y_l"71
$40., [”6 9080414") 11.61

~ 1 1”
IIS 7. ° )’+ r'

Ckx0('\A’I) :

 

 

dy. (4.30)

69

For small x, the cosine tends to unity, and the integral may be written in terms of

Gamma functions (p486-l 1.4.13 of[34]),

2'21 mt —i)r(l+i)
c,,,(x,t)= ~ 3,, 7' 1 7‘ (21,1)“4, x—>0. (4.31)
3” y,r(k+—+1)

 

(iii) Long time limit t >> tm
One may then use the large argument expansion for the Bessel function (p377-9.7.l of

[34])

2

e
Ik(Z)~(—2'ﬂz—)lj—2-.

(4.32)

Now, small correction PM” is the only term that survives in (4.21). The ﬁrst term in the

sum is dominant, thus for small x

2.?
-B-_IB!
(L)

29 L 2171 ,‘2 e x

C ,1e—— —, —
‘0‘) L§(27t) (27rA,t)" L

—+ 0 , (4.33)

A surprising feature of this expression is that it is independent of k indicating that
asymptotically all of the steps are correlated in the same way. Besides, it is also valid for

k = O , i.e. both correlations on the same and different steps behave in the same manner.

Correlations on the same step - G(x,t) (Both cases A and B)

One considers (combining (4.3) and (4.21)),

_ ”80(9)!
G(x,t)zzfig' 1 cosqxe qz 10(gl(q)t).
q

(4.34)

 

70

First, this equation for small x (when cosine tends to unity) is analyzed for three time

regimes.

(i) Short times t —> 0
At short times, using [0(0) = l, g0(q) = Alq“ for q < qo, go(q) = Azq’2 for q > q0 , and
expanding the exponential in (4.34) to the second order, one ﬁnds

r.-I 2 / rz-i_ rz-l
G(x—>O,t):_g_3—(ﬁi_q_0_+A ( It a“) qo
7Z5

 

t 4.35
7H 2 724 1 ( 1

for both the case A and the case B.

(ii) Intermediate times 0 << t << tm,

This regime actually breaks up into two regimes. The ﬁrst 0 << t << t, is dominated by
the modes in the regime q >> qo, while the second t, << t << I, is dominated by modes
in the regime q << q“. These regimes are quite distinct due to the fact that In only

appears in the exponential. It is then quite a good approximation to treat the ﬁrst early
time regime using Azq” and the later time regime using A,q"'. One can do an analysis

Similar to that leading to Equation (4.30) to ﬁnd,

 

G(x —> 0,t) = 753—111 — l )(Azt)Z , t << t, (4.36)
sn .
for both cases A and B, and
O 1 ;'-
G(x —> 0,t) = C~—-F(l — —)(A,t) ', t >> tx, (4.37)
sn 7,

where C = (2' ’"7IF(1 /2 + 1/7,))/(\/;F(l /yl )) (integration by parts using p486-l 1.4.13

of [34]) for the case A, and C=l (p333-3.434(l) of [47]) for the case B. The crossover

71

time t, is found by equating (4.36) to (4.37), and solving for the time. In the physical

cases, there may be several different crossovers in this intermediate time regime, but the

same general principles can be applied. Naturally this complexity makes experimental

data difﬁcult to analyze in general.

(iii) Long times t > tm,
Co(x —> 0,t) for long times is ﬁrst found. In case A, it is given by Equation (4.33). In

case B, the argument of the Bessel function in (4.21) is zero. One then has [0(0) = l and

since ﬁrst term in the sum is dominant,
20 L -,1,
C0(x —> 0,t) z —:(—)2e ' “" , (4.38)
Ls 27:

whose time scale is set by the slowest mode in the system, t3“. Then, using Equation

(4.3),

G(x —> 0,!) e {‘22 - C0(x —> 0,t). (4.39)
S

For larger x, one ﬁnds combining (4.3) and (4.22)
G(x $0.1) =gf-(1—i) (4.40)
25 L

for short times. On the other hand, one ﬁnds G(x ¢O,t) = G(0,t) for longer times.
Therefore, G(x :6 0,t) is initially constant in time until the correlation length along the
step increases approximately to value x. If Aq’ is the dominant regime, one ﬁnds that

Equation (4.40) is valid for t<< x" / A. This time depends on the kind of dynamics

through 7 and A and deﬁnes the propagation speed of the nonlinear ﬂuctuation. As

72

expected, initial ﬂuctuation (4.40) depends on x and S , but not on the kind of dynamics

i.e. 7 and A.

4.4. Terrace Diffusion

Assumptions of the terrace diffusion model are explained in Section 3.4. Since atoms

diffuse on the terrace, P0(l) and P,(l) are the probabilities for random walk on the

surface between two ﬂuctuating semi-sticking walls. The calculation of these

probabilities for straight step edges is done in Appendix C,
Po(l) = pUP(aU,aL,d,d —ai,l) + pLP(aL,aU,d,d —al,l) (4.41)
and
P,(I) = pUP(aL,aU,d,ai,l) + pLP(aU,aL,d,ai,l) (4.42)

where

 

P(a..a2 ,d,b.l) = -2—1- [mail a' (1‘ cos“ kb + “2 5‘“ kb) coskl. (4.43)
71'

r- (k2 +aUaL)sinhkd +k(aU +aL)coshkd
These results for two straight edges can be used as the diffusion kernels mediating the

dynamics of the ﬂuctuations provided the ﬂuctuations are relatively weak (the small

slope limit). The integrals (4.14) may be evaluated to ﬁnd

~

 

 

SCI ‘1
go(q)= A q- {l—pi’g(auvaud2d_a12q)—pLg(aLaaU2d2d_ai9q)} (444)
I:

So
81(9) = T * qztpigmmaudﬂhq) + pug(ai,at,.d,a..q)} (4.45)

h

where

g(a,,a,,d,b,q) : a,(qcoshqb+a2 smh qb) (4.46)

 

(q2 +a,,aL)sinh qd +q(a,, +aL)cosh qd'

73

Here, the characteristic time is related to step edge energies by

- _ —1 -E/kT —E,/I(T
Th ‘Tmie L a +3 ‘ A9}

3 (4.47)
where 1;}, = vexp(—Em /kBT) and v is a characteristic phonon frequency. On Figure
4.2, the spectrum of inverse relaxation times is plotted using Equations (4.44) and (4.45)

in (4.17) for four different cases: aL = 61,, =103 / a 1 (perfect sticking),
aL zaU =10‘2 /ai (weak sticking), otL = 103 /al , aU =10'3/al (ﬁnite Schwoebel
barrier E L << EU ), and (IL = 103 / a l, a), = 0 (inﬁnite Schwoebel barrier EU —> 00).

Note on the Figure 4.2 that, except in the inﬁnite Schwoebel barrier case when there in no
atom exchange between the steps, the spectrum of relaxation times is spreaded for small

values of q. This means that there is a mutual step-step correlation of the large wave
length ﬂuctuations. On the other hand, the spectrum shrinks to one value for larger values

of q, since there is much less mutual step-step correlation of small wave length

ﬂuctuations. As it will be derived later, for large q , r; o: q3 for perfect sticking, ﬁnite

and inﬁnite Schwoebel barrier case , while rgq or q2 for weak sticking case. For small q ,

spectrum 1;; spreads between q2 and q4 for ﬁnite Schwoebel barrier, perfect and weak
sticking case, while rm; at q4 for inﬁnite Schoebel banier case. Using this spectrum from

Figure 4.2 and doing numerical summations over q and p , G(x,t) and C ,(0,t) , deﬁned

by Equations (4.3) and (4.20), are plotted on Figures 4.3 and 4.4. G(x,t) are plotted for

all four different cases of sticking coefﬁcient, deﬁned above for Figure 4.2. Each case has
different scaling and is shown for a couple of values of x (for .r it 0 time independent

regime (4.40) holds). C ,(O,t) is plotted just for perfect sticking case for a couple of

nonzero values of k. The other cases of non-zero sticking coefficients are not plotted

74

10

 

 

 

 

10 I' IUTIYVII I YITIIIII I TYIIYYII U VIIITIII
5
10 — —
0
10 .. —
E
P
\Q
in —5
v 10 — ~
-'10
10 — —
'15 . 1 1 ...rii U... 1 a 1 ....I ..1
1O ‘—-2 -1 l0 1 2 3
10 10 10. 10 10 10

Figure 4.2 Four spectrums of inverse relaxation times in the terrace diffusion case for the
surface with N = 10 steps (there are only up to 6 visible lines in each spectrum since four

pairs overlap); d = 3001 , L = lO‘a". Different ratios 3" a l / a; for each spectrum are used
to avoid overlapping. They are plotted for the following sticking coefﬁcients (from the
top): aL =aU =10'2/al, Sal/a: =109, a, =aU =103/al, Sal/a"2 =10‘,

a: =103/a,, a; =10‘3/a,, raga: =10, a: =10’/a,. a: =0, Sal/a“: =1.

 

 

 

 

 

 

 

10"
10‘5
10.6 0 12 14 A6 18 I10 12
10 10 10 10 10 10 10
t/tm

Figure 4.3(a) The autocorrelation function in the terrace diffusion limit. N =10,
L =10’a", d =100al and kaT IE=1. Solid lines are Equation (4.3) plotted doing
numerical Stunmation (4.20) using (4.44)-(4.47), dashed lines are equations listed in
Table 4.1 and horizontal dashed-dotted line is d2 , all divided by Co(0,0) , for the sticking

coefﬁcients aL = av =10”2 /al (case T3), 1: = O , x = 5, x = 500 from the bottom.

76

 

 

 

 

 

 

 

12

10 10 1010 10

t/r

10

TD

Figure 4.3(b) The autocorrelation function in the terrace diffusion limit. N =10,
L=105a“, d =100a l and kBT /E=1. Solid lines are Equation (4.3) plotted doing
numerical summation (4.20) using (4.44)-(4.47), dashed lines are equations listed in
Table 4.1 and horizontal dashed-dotted line is d 2, all divided by C0(O,0) , for the sticking

coefﬁcients a, = a). =103 “1.1 (case T3), x = O, x =10, x = 1000 from the bottom.

77

 

 

 

 

 

 

 

12

10

Figure 4.3(c) The autocorrelation function in the terrace diffusion limit. N =10,
L=105a“, d =100al and kBT/ i=1. Solid lines are Equation (4.3) plotted doing

numerical summation (4.20) using (4.44)-(4.47), dashed lines are equations listed in
Table 4.1 and horizontal dashed-dotted line is d2 , all divided by Co(0,0) , for the sticking

coefﬁcients a, =103/ai, at“ =10'3/al (case T3), x =0, x = 10, x =1000 from the
bottom.

78

 

 

 

 

 

 

 

10 l I I l l l l
10‘1
8.10“2
9.
DO
\
L310
10"
10-5 “AoAAA 12 A4 6 I8 110 12... A14;
10 10 10 10 10 10 1O 10
t/tn,

Figure 4.3(d) The autocorrelation function in the terrace diffusion limit. A" = l0.

L=105a”, d =100‘11 and kﬂl'. i=1. Solid lines are Equation (4.3) plotted doing

numerical summation (4.20) using (4.44)-(4.47), dashed lines are equations listed in

Table 4.1 and horizontal dashed-dotted line is d 2 , all divided by Co(0,0) , for the sticking

coefﬁcients a, = 103 /al , a), = 0 (case T2), x = 0, x = 10, x = 1000 from the bottom.
79

 

 

 

 

 

 

 

 

 

 

 

 

10 l l I I j I I I
-2 _ ————————————— - '
10 - '1
-4 / ///
a 10 — 100 -
Q.
Q .
Us 106 l 8 10-.
\ P 9: 'l
7: / 9°
6 E
\732 ‘ I E1040
D 10'8 — U
10'15 ”#1...
40 0 5e+16 1e+17
10 ”‘ t/‘Cm
/
10-12 0 112 14 A6 18 110 I12 114 116
10 10 10 10 10 10 10 10 10
{/1310

Figure 4.4 The step-step correlation function in the terrace diffusion T3 limit. N = 10.
L =105a,, d = 30a], kBT/E =1 and a, =aU =103/al. Solid lines are plotted doing
numerical summation (4.20) using (4.44)-(4.47), dashed lines are equations listed in
Table 4.2 and horizontal dashed-dotted line is d2, all divided by C0(0,0) , for k = 1,2,3
(from the top). Inner graph is linear-log plot that shows exponential time dependence at
large times.

80

since they Show the same scaling. These numerical results are then compared to
approximate analytical forms plotted as dashed lines on Figures 4.3 and 4.4. Those

analytical forms, listed in Tables 4.1 and 4.2, are obtained substituting expressions for

relaxation times derived below, to Equations (4.36), (4.37), (4.29) and (4.31). d 2 as an
horizontal dashed-dotted line is also plotted to show when step ﬂuctuation width exceeds

the distance between the steps. Since this model does not include the direct step-step

interaction, it fails above the d 2 line.

In the limit q <<1/a.I expressions (4.44)-(4.46) reduce to,

 

 

 

 

Eai 3 2a,a cosh d+ a,+a: sinh d
g0(q)= l l 2 L L q ( L L)q q (448)
rm (q + aUaL)smh qd + q(aU +aL)cosh qd
and
( )_ Fail '3 ZauaL (4 49)
g, q rm q (q2 +aUaL)sinhqd+q(a'U +aL)coshqd ° A

These expressions then lead to the relaxation time of a mode of (p, q) according to
1;; = g0(q) — g,(q)cos p. Approximate expressions for this relaxation time have been
found in [13] in a variety of limiting cases, though they implicitly assume “in phase
motion". Khare and Einstein [17] consider only in phase and out of phase motion
(A, i 28q = g0(q) i gl (q) ). while Ihle et a1. [43] consider the full spectrum (all values of
p). In Appendix D, an brief overview of those two models [17,43] is given. The

following is the discussion of limiting cases in which the time dependent correlations

may be explicitly evaluated.

Tl - Terrace diﬂ'usion 1 (d -> oo)

81

The large d limit eliminates the ﬂux from neighboring steps, so the steps act
independently. This is essentially the isolated step case discussed in Chapter 3. In this
limit, Equations (4.48) and (4.49) reduce to,

S'lqlBai( +

rm aU+|q| 02+

“U a].

 

go(q) = lql)’ 8. (q) = 0 (4.50)

Since g,(q) = 0, 1;; = g0 (q) and Cm(x,t) = 0 (case B). From Equation (4.50) it is seen
that even isolated steps have some of three possible regimes: when |q| >> aw then
go(q) = Soﬁa), + crL)q2 Mm; when or), << |q| << a, (assuming aU << aL) then

go(q)=§ai|q|3/rm; and when |q|<< a“ then g0(q) =2§aiqu3/rm. This leads to
three different time regimes in the correlation function G(0,t) as summarized in Table
4.1. After the non-interacting diffuser regime t —> 0, there is an early time 0 << t << tlr'
(large q), in which short range processes must overcome the sticking coefﬁcients to

proceed. For this reason aU + 61,, appears as a prefactor. This regime is not well deﬁned
if a“ —) 00. In the regime trl <<t << t5l , one of the terraces (the upper one for the case

(1,, << a L ) is not contributing since approach to the step edge from that terrace has such a

small sticking coefﬁcient. This regime is only well deﬁned if the two sticking coefﬁcients
are very different (e.g. the large Schwoebel barrier case, when 01,, << a, ). In the regime

T1

sat ’

tf' <<t <<t both of the terraces are contributing equally and small sticking
coefﬁcients do not play a role. It is rather interesting that in this limit only rm appears in
the time dependent correlation function. This regime is not well deﬁned if or), or a, z 0.
Finally at long enough time the relaxation is controlled by the slowest mode in the

system.

82

 

Note that these forms apply, at sufﬁciently short times, even for quite closely spaced

steps in a step train, as discussed below.

T2 - Terrace diffusion 2 (d ﬁnite, aU = 0 or (1U = 0 ).

Setting aU = O or aU = 0 implies that no mass is transported from one step to either of
its neighbors (“large barrier limit”, e.g. the large Schwoebel barrier corresponds to
ca, =0). At short time (large q dominant), the isolated terrace behavior occurs,

especially for well separated steps. For small q , Equations (4.48) and (4.49) reduce to,

2

sdai

 

86(51): 9‘, 81(61): 0 (4.51)

TTD
thus, Ck,O (x,t) = 0 (case B). The q4 dependence leads to the long times behavior like

that of step—edge diffusion, as seen by comparing the correlation functions for this case
with that of step edge diffusion (see Table 4.1). The results in Table 4.1 are found using

Equations (4.37), (4.39) and (4.51).

T3 - Terrace diffusion 3 (d ﬁnite, or“ at 0 ).
In this case the T1 (isolated step) processes occur at short times (large q dominant). For
analysis of the long times behavior (small q dominant), it is useful to deﬁne the length

: at; + a1.

d, (4.52)

aUaL
If (10 is ﬁnite (d ——> 00 leads to T2 limit), mass transport between steps is dominant. The

approximate forms of Equations (4.48) and (4.49) for this case are,

83

~ 2 ~ 2

sai 2 2 4 2sa 2
+d , =____1__ .

110(d+doq q) g.(q) 7T0(d+do)q

 

 

go(q) = (4.53)

Due to the mass transport occurring between steps g1 (q) at 0, and this is the
mathematical origin of a ﬁnite value of C 1 ”(x,t). Results for this limit are summarized
in Tables 4.1 and 4.2, where q0 = 2/d is used. (see Figure 4.2). This case is probably

very important in many experiments and leads to a correlation between step ﬂuctuations
on neighboring steps which becomes the “signature” of the T3 process. The auto-

correlation function G(0,t) (Table 4.1) can have a wide variety of different behaviors.

But the pair correlation function only measures the T3 process, as it depends on having a

ﬁnite g,(q) corresponding to correlated mass transfer between steps. This makes the pair

correlation function a useful diagnostic of the dominant surface transport modes.

84

Table 4.1 — Limiting behaviors for G(0,t)

Mass transport mechanism

A ttachment-detachment (AD)

Step-edge diffusion (SE)

Terrace diffusion 1 (T1)

(1 —) 00
(isolated step)

Terrace diffru‘ion 2 (T2)

(1 ﬁnite. or). =0 or a, = 0

(e.g. Schwoebel barrier = 00)

Terrace diffusion 3 (T3)

(1 ﬁnite, (1,1 at 0

Time regime
t—>0
0<<t<<gm

AD
sat

t>>t
t—+0

O<<t<<gm

t>>gj

t—>O

0<<t<<tlrl

tf'<<t<<t?

T1 TI
5 <<t<<gm
TI
t>>gm
t<<t,’2

T2

tl <<t<<tT2

sat
T2
t>>gm

t<<tIT3

T3
sat

tf3<<t<<t

t>>gm

85

G(0,t)
~t

Q(_a_:_)ll2(_t_)ll2

TAD
——:O—-6e
”2

dug?
123 )

~t

523,1»...(11 ,.,.

TSE

6 an“
“)

——-1-——e
123 ( n2
~t

a), +a, ___,-). ,

W1( %( )m

QF(— )
(.92— 21% )1/3
Qr(— 22) t
—( 2~2—_A2L-) [3(—)“3

TTD

L 6 r1
——-l———eW“'
12:( n2 )

as for isolated step (Tl)

Or(— —) d
a / /
4(__1;t__14( )14
S3 TTD
LE. -1i W41
12? n2

as for isolated step (Tl)

 

i . t .
401.2 1/2
“A n3§(d+d )) (rm)
3L d+d 2 - ,
——%1---( 0' (— L)'W
2H3 no?” {TD
—(—7)‘Ea’d—
8 fm)

Table 4.2 — Limiting behaviors for C w(O, t)

 

 

 

Mass transport mechanism Time regime no (t)

Terrace dzﬂusion 3 (T3) Qd 430:

d ““6, “w i 0 t ” 0 ka!(2k— 1) ((d +d 0)d2)rm (_)

49 m m
0<<t<<tfm 4k2—1((d+d: )7f3~) ( 1:0)
2:: 4~ 2 r
t>> If; 014232“! +d° I/2(— )-'/Ze-(—L-) “id;
85 Mt mis rm

86

Table 4.3 — Equations for limiting timescales

Characteristic time

 

eESE/kBT
Tse :
V
SE _ Ts]; ( L )4
sat _ ~ 2
sala 2a
1
eEm/kiiT
Tm =
V

Tl_ Tn) L 3

’“ 23al 2n

n_ Tm L 4

3“ dE'ai 2;.

[T3 : Tm(d+d0)(i)2
’” 23ai 2a
t'Tl,t2TI

I‘T2

[In

Physical origin

A ttachmen t-detachment (AD)
Time related to the binding energy (E A D) at the step edge

Saturation time

Step-edge diffusion (SE)
Time related to the binding energy (ESE) at the step edge

Saturation time

Terrace diffusion

Time related to the binding energy (Em) at the step edge

Saturation time: terrace diffusion Tl (isolated step limit)
Saturation time: terrace diffusion T2 (large barrier limit)
Saturation time: terrace diffusion T3 (step to step limit)

Crossover times for isolated step limit (Tl)
Crossover time for large barrier limit (T2)

Crossover time for step to step limit (T3)

87

Chapter 5

Surface Evolution Due to Step Interactions

5.1 Introduction

The model that describes the dynamics of a step train with direct step-step interaction is
introduced in this chapter [49]. The steps exchange atoms through terrace diffusion. The
direct interaction term is included in the expression for the free energy (Equation 4.5).
There are many possible origins of step-step interactions: the fact that steps cannot cross
(delta function repulsion, which leads to the so called “entropic” repulsion), strain terms
(either due to dipole terms or applied stress terms) and electronic structure effects (which
sometime are oscillatory). Due to direct interactions, the centers of mass of steps are not
ﬁxed. The evolution of large scale surface morphology is mediated by this step motion.
Here, repulsive step-step interaction is assumed. At long times the morphology of the
surface approaches a steady state proﬁle shape, which spreads with a characteristic

exponent
5.2 Chemical Potential of Interacting Steps

Let us assume that the free energy F of N steps (see Figure 5.1) is given by,

N L’Z "'
szz I{ii—(VH‘UJDZ+g(HM(x,t)-Hk(x,t))}d.r (5.1)

88

where the “stiffness” E = 2(0) + 2"(0) depends on 2(6), the angle-dependent interface
free energy (see Equations (3.l)-(3.6)). H k (x,t) is the position of site at x on the kth step
at the time t. It is the sum of its center of mass motion T11 (t) and its ﬂuctuation h,‘(x,t) ,

Hk(x,t) =T1-k(t)+hk(x,t). (5.2)
the term g( y) describes the interaction between steps. In this section, analysis is done
without specifying form of g( y). Using the similar manipulation as for Equation (4.5),
the chemical potential of the k'” step yk(x,t)is found to be,

[1, (x,t) = —Q§V2Hk(x,t) + Qg’(Hk(x,t) — Hk_,(x,t)) — Qg'(HM(x,t) — Hk(x,t)) (5.3)
where primes denote derivatives with respect to argument. Expanding (5.3) to ﬁrst order
in the ﬂuctuations 6d, and using Equation (5.2) (with V7710) = 0) yields,

)1, (x,t) z —Q§V2hk (x,t) + Qg'(dH(t)) + 06d“ (x,t)g"(d,‘_l(t)) -—
- Qg'(di (0) - Q561,.(x,t)g"(di(t)) (5.4)
where 6d, (x,t) = h,HI (x,t) — hk(x,t) describes ﬂuctuations of the terrace spacing.
d,(t) =i’7,(:)—I7,_,(z) (5.5)
is the average distance between two adjacent steps (i.e. the average terrace width).

Comparing (5.4) with Equation.(4.5) for non-interacting steps, one sees that, in the ﬁrst

order approximation, the direct interaction g adds four additional terms. While the third

and ftﬁh term in (5.4) are corrections to the chemical potential of step ﬂuctuations due to

interaction g , the second and fourth terms lead to center-of—mass motion of the steps.

Equation (5.4) can be rewritten in the following simpler form,

mm ..., #306,!) + 5#f(x,t) wftr) (5.6)

90

where substitutions

yf(x,t) E —Q§V3hk (x,t)
Mo,» 2 05d.-. (x,t>g"<d.-, (t>> - 06d. (x,t)g"(d. (t>> (5.7)
#f (t) E Qg'(dr-.(t)) - Qg'(dk(t))

are introduced.

As explained in Chapter 4, one can describe step dynamics with a coupled set of
Langevin equations,

6H,(x,t) _ 1“,,
at kBT

 

{éJk_Lk(x,t) + J”(x,t) + éJk+,'k(x,t)} + 77, (x,t) (5.8)

where

J...<x,t) = [0"2a(l,d<t»{#.(x +1.0 — 2#.(x,t) + y.<x — Midi

JktlJr (x,t) = IOL12R(l,d(t)){)u/:il(x + lat) — 2yk(x,t) + ,ukﬂ(x - l,t)}dl ' (59)

Aﬁer substituting (5.2) and (5.6) in (5.8) and averaging it, one obtains an equation that
describes the center—of-mass motion as

dill—,0) __ _ 1“,!)
dt kBT

 

{m.(t)(g'(d...(t))-2g'(d.(t»+g'<d.-.(t»)—

— m.-.(t>(g'(d.(t» — 2g'(d.-,(z» + g'(d.-2(z»)} (5.10)
where (see Equation (4.40) for P,)

“[210 (1 d( ))dl 2ai 1
m (t) = , t =
k 0 l TTDFII dk(t)+d0

 

(5.11)

and ato =(aU +aL)/aUaL. Subtracting equation (5.10) from (5.8) gives the coupled

system of equations for step ﬂuctuations. Similarly to Equation (4.13) in Chapter 4, the

91

Fourier transform can be applied on that system. Then, it is possible to numerically

decouple the system and ﬁnd the correlation function.

In the following section, short range entropic (contact) repulsion for g is assumed. In
that case, 5y: << ,uf and results for step ﬂuctuation derived in Chapter 4 hold as a good

approximation. For that reason, the rest of the chapter will deal only with the analysis of

the center-of mass equation (5.10).
5.3 Continuum Limit, The Entropic Repulsion

In the continuum k limit, Equation (5.10) becomes

aﬁk(t)=_ ZaiQ _6_ 1 52g'(dk(t)) . (5.12)
at kBTrm ak d,‘(t)+do ak2

By writing the similar equation for 171+: (t) and subtracting (5.12) from it, one has

ad,(t)=_ Zaiﬂ a2 1 a’g'(d,(z)) (5.13)
a: kBTrm 5k2 d,‘(t)+do ak2

which is the general equation for the proﬁle evolution.

The entropic repulsion between adjacent steps is given by,

e-y/Zwm C
gER(y)=C 2 “‘3‘ (5.14)
y y

 

The exponential ensures that the repulsion “cuts-off” when two steps are separated by

more than twice their saturation width Wm. In the following it is assumed that w”, —> oo ,

so the last expression in Equation (5.14) will be used. By substituting (5.14) to (5.13),

92

and introducing the slope uk(t)Ea/dk(t), where a is the height of the step edge

(difference in height between two adjacent terraces), one gets

MPJEW—uzmﬁi “1“) 52‘4“). (5.15)
at a‘k,rt,,,d, " ak2 u,(z)+a/d, ak2

Equation (5.15) was obtained by Ozdemir and Zangwill [29] using a different approach.
It is easy to show by separation of variables that Equation (5 .15) has shape preserving
solution in time in two limiting cases:

(1') perfect sticking (alo —> O)

 

 

 

2 2 2 3
M = __‘_:§_‘1Luf(,)a_z “ma my) (5.16)
a: 0 km”, 6k 6k
with solution
u,(t) = N’s/4,, (5.17)
where Ak satisﬁes
2 2 2 3
lzifcﬁ‘gAkL’T Ak" “ft . (5.18)
a k,rr,,, dk dk‘
(ii) weak sticking ((10 —> oo)
2 4 3
au,(z):_ 44cm, “ma tuft) (5.19)
at a kBTrmdo 6k
with solution
u,(t) = 1"”8, (5.20)

where Bk satisﬁes

_ 16CQaf 8 (1‘3:

_ 1 (5.21)
a‘kBTrmdo dk‘

 

93

On Figure 5.2, slope evolution obtained doing numerical integration of Equation (5.15)
for (i) aL = aU = 103 / a i and (ii) aL = aU = 10‘3 / a 1 for uniform initial proﬁle is plotted.
It shows interesting oscillatory shape in the transition period until it assumes the steady
proﬁle. It is interesting to note that the slope in the center of the train actually increases
for small times. This means that steps in the center are bunching at small times despite
the repulsive potential between them. The slope in the center of the train vs. time is
plotted on Figure 5.3. It shows that the slope in the center in initially unchanged, then
rises and ﬁnally decreases with the power law. The Figure 5.4 compares the steady form

of 14, for large time for weak and prefect sticking. On Figure 5.5, the evolution of the

proﬁle for perfect sticking is shown.

94

 

 

 

 

 

 

 

 

M W

1 _ —

0.8 — _

i

ii 0.6 ~ —
3
2
:1

0.4 — -

0.2 ~ _

O 1 l I 1
0 10 20 30 40 50
k

Figure 5.2(a) Slope evolution at times t/ rm = 0,102,103,10“,105,10°,107obtained doing
numerical integration of Equation (5.15) for at =aU =103/ai and 4CQai/a5kBT =1
for uniform initial proﬁle 19(0) = const . Number of steps in the train is N = 50.

95

 

 

 

 

 

 

 

 

 

1 _ m _
0.8 F —
i 0.6 — -
3
\
~ 0.4 — -
0.2 7 ~
1
2
O ‘ 1 ' 1
0 10 20 30 40 50
k

Figure 5.2(b) Slope evolution at times t/ rm = 0.10:,103,104,105,10",107obtained doing
numerical integration of Equation (5.15) for (1L = a), =10“3 / a 1 and

4(‘Qai /a‘k,,Tal0 =1 for uniform initial proﬁle uk(0) = const. Number of steps in the
train is N = 50.

96

 

 

uN/2(t)/uN/2(0)

 

 

 

 

Figure 5.3 The slope in the center of the train vs. time for ca = av = 103 / a 1’
4CQai/askBT =1 (dashed line), and aL = aU =10'3/ai, 4CQai /a“k1,Tdo = 0.1 (solid
line) for uniform initial proﬁle a, (0) = const . Number of steps in the train is N = 50. For

large times u,‘ approaches ~ {”5 for perfect sticking and ~ t"” for weak sticking.

97

 

w(t)/Lima)

 

 

 

30 40 50

Figure 5.4 Comparison of steady shapes of uk for a,_ = aU = 103 /al , 4CQai /a5kBT :1

(dashed line), and aL = aU =10""/ai , 4CQai /a‘k,3Td0 = 0.1 (solid line) at time
t/rm = 107 . Number of steps in the train is N = 50.

98

step-train

profile, number of steps = 10

r—Fﬂu- “v Y’ﬁ‘?’ r T v r r v *— ﬁ—‘Y' *—-w—=r- r r r r I v w v T I v v v v ~‘r

 

 

9.0

7.0 ~

 

3.0 -

1.0 L

 

 

 

I J. l A I I

40.0

 

- 0 A 4 14 J J A¥L AL A A L L 1

-30.0 -20.0 -10.0 0.0 10.0 L200 30.0

Figure 5.5 Evolution of the proﬁle for or) = aU = 103 / a 1 , 4CQai laskBT =1 of N =10
steps in the train.

99

Chapter 6

Monte-Carlo Simulations and Numerical Analysis

6.1 Introduction

In this chapter, some results of the Monte-Carlo simulation and numerical analysis are
summarized. The purpose of performing the simulation is to check whether the
continuum equation is properly formed based on initial assumptions. On the other hand,
the numerical analysis is the nice way to check whether the analytical analysis of the

continuum equation is correctly done.

6.2 Monte-Carlo Simulations

The most common type of Monte-Carlo simulation of surface evolution is the SOS
model. As stated at the beginning of Chapter 2, it assumes the step energy in the form
(3.1) with p =1 . The way the simulation is performed is to ﬁrst randomly choose the
possible change of the step shape according to model (as it is discussed below). Then the
random number p 6 [0,1] with uniform distribution is generated. If the following relation

between Bolzmann factor and p is satisﬁed,

e‘(E"-E')/kBT

2p, (6.1)

100

the change is done (where E, and E f are the initial and ﬁnal step energy, according to

(3.1), before and after the change respectively). Otherwise, the step shape stays
unchanged. Then the other possible change of the step shape is chosen and so on. Due to
the stochastic ﬂuctuation, the results have to be averaged over many possible

conﬁgurations to smooth the obtained curves.

In attachment-detachment case, one randomly (with uniform distribution) picks one site
on one of the steps at the surface. Then with uniform probability it is randomly chosen to
either detach or attach an atom to that particular step site. Finally, after generating
random number p , the change is done if (6.1) is satisﬁed. In the step-edge diffusion
model, after randomly picking the site on one of the steps, one chooses with uniform
probability to either move the atom from that site to the left or to the right along the step-

edge. Again, whether or not to do that particular change is determined according to (6.1).

In terrace diffusion case, after randomly picking the site on the step edge, one chooses

upper or lower terrace according to probabilities pU and pL (sec (3.29)). Then the

second site on the same step or on the step across the chosen terrace is picked according

to diffusion kernels p1 and p2 (see (C.14)). It is uniformly chosen in which direction to

move the atom from one site to the other. Then according to (6.1) the change is made or

not.

On the Figure 6.1, a snap-shot of the step shape in the equilibrium is shown. It is obtained

by doing SOS Monte-Carlo of terrace diffusion with perfect sticking. Figure 6.2 is the

101

 

10 . . . ,

 

0 50 100

Figure 6.1 A snap-shot of the step shape in the equilibrium.
Monte-Carlo of terrace diffusion with perfect sticking.

102

 

 

 

[t is obtained by doing SOS

 

0.8 T ' I I Y I f T If I

log (width) [lattice constant]

 

 

 

 

 

 

 

 

-0_1 ' —-—- Monte - Carlo H
. — - - 1/6 slope .
-O.2 - - - — log (equilibrium width) _
_03 I 1 1 1 1 1 1 1 1 1 1 g .
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

log (time) [jumps/site]

Figure 6.2 Isolated step Monte-Carlo simulation for the terrace diffusion with perfect
sticking. It agrees with theoretical prediction (3.40) that w(t) at t'“5 (compare also with

Figure 3.4).

103

log (width) [lattice constant]

Sticking coefﬁcient alpha = 0.1

 

Monte - Carlo simulation. step length = 200. kT = J

1.0 . r e . T . . , - , -
0.8~
0.6-
0.4~
0.2—

0.0 -

 

 

——-— Monte - Carlo
- - — l/6 slope
I/4 slope

 

 

 

 

 

J

 

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
log (time) [iumps/site]

Figure 6.3 Isolated step Monte-Carlo simulation for weak sticking terrace difﬁision
aL =aU =0.1/ a 1- As predicted by (3.39), there is the tendency toward w(t) act'“ for

small times, while w(t) cc t'“5 for large times (compare also with Figure 3.4).

104

Two Steps - Monte Carlo Simulation
step length = 200, [J = J. nr=I60

 

 

 

 

 

 

 

 

 

3.0 ' I V F V I r 1' v 1 v T v I , I Y r v I
2.5 ~ _
2;: / ,.
o /
.2 ///
g 1 5 " , l/mﬂ . -
’6 '//’::.-
0g [xi/7"
.9 /.-'
3 1.0 - , //' -
U) ...-2”
2 .g/
‘ -—-—- Monte- Carlo
05 P l/4slope ‘
............ 1/5 slope 4
00 . L I 1 1 1 1 1 I 1 . 1 . 1 . 1 1 1 1 1 .
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0

log (time) [jumps/site]

l/S

Figure 6.4 The simulation of terrace diffusion with perfect sticking. It gives d (t) oct
where d (t) is the distance between two steps. It agrees with theoretical prediction (5.17).

105

isolated step Monte-Carlo result for the terrace diffusion with perfect sticking. It agrees
with theoretical prediction (3.40) that w(t) at t'”5 (compare also with Figure 3.4). Figure
6.3 shows same isolated step Monte-Carlo simulation but for weak sticking terrace

diffusion aL =aU =0.1/a 1- As predicted by (3.39), there is the tendency toward

w(t) oct“4 for small times, while w(t) act"6 holds for large times (compare also with

Figure 3.4). The step-train results from Chapter 5 are checked with a 2-step train with
contact repulsion. The best way to do it would be to take at least 10 steps in order to
avoid boundary effects. But that would even more increase already tremendously huge
simulation time due to the large step size. Despite the approximation of taking only 2

steps, Figure 6.4 shows that the simulation ofterrace diffusion with perfect sticking gives
d (t) at t'ls where d (t) is the distance between two steps. That agrees with the theoretical

prediction (5.17) (change in slope is inversely related to the change in distance between

the steps).

6.3 Integration of Stochastic Equations

The solution of Langevin equations (3.9) and (5.8) are checked by doing the discrete
numerical integration of those equations. The noise is randomly generated according to
correlation equations (8.15). On Figure 6.5, the ﬂuctuation width of an isolated step is
obtained doing numerical integration of (3.9) for terrace diffusion case with perfect
sticking. It agrees with w(t) at t' 6 theoretical prediction (3.40) (compare also with Figure
3.4 and Figure 6.2). Also, on Figure 6.6, the distance between steps in 2-step train is

obtained doing numerical integration of(5.8) for terrace diffusion case with perfect

106

 

 

log (width) [lattice constant]

 

, ---- numerical integration
-004 ~ , // - - - 1/6 slope ~
" log (equilibrium width)

 

 

 

 

 

 

_06 1 1 1 1 1 1 . 1 I 1 . 1 1 1 . 1 1
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

log (time) [jumps/site]

Figure 6.5 The ﬂuctuation width of an isolated step obtained doing numerical integration
of (3.9) for terrace diffusion case with perfect sticking. It agrees w1th

w(t) cc t'“5 theoretical prediction (3.40) (compare also with Figure 3.4 and Figure 6.2).

107

Two Steps - Numerical Integration

step length = 200. kT = .I
30 ‘ i ' r r r T i v 1 v r 1 I f T v I , I ,

 

N
o
1

..s
O
1

log (distance) [lattice unit]
I;

 

 

 

 

 

 

 

 

 

_ " / numerical integration
0'5 ’ . 1/4 slope ‘
............ 1/5 slope
0.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 L 1 , 1 ,
0.0 1.0 2.0 3.0 4.0 . 5.0 6.0 7.0 8.0 9.0 10.0 11.0
log (time) [jumps/site]

Figure 6.6 The distance between steps in 2-step train obtained doing numerical
integration of (5.8) for terrace diffusion case with perfect sticking. It agrees with

d (t) at t"5 theoretical prediction (5.17) (compare also with Figure 6.4).

108

l/S

sticking. It agrees with d (t) oct theoretical prediction (5.17) (compare also with Figure

6.4).
6.4 Method for Checking Probabilities

This is an short overview of two methods used to numerically check analytically obtained
probabilities p1 and p2 given in (CM). The simplest method is to do a simulation of

random walk on the discrete lattice. But, that approach takes a lot of simulation time

(especially for large system sizes).

The other alternative method is to use (see Figure 6.7)

1 1 1 1
3,,(1411): 212-..,(1) + Z P...) (t) + Z P1,,-. (t) + 20.1.10) (63-)

where PM“) is the probability that an atom is on the discrete lattice site (i, j). Or, one

can write
P(1+1)= MP(t) (6.3)
where the matrix M is formed in such a way to satisfy (6.2) and all assumed boundary

conditions. If P(0) is the initial distribution of the probabilities (say

Pic-[0(0) =1, Ram/“0(0) = 0 ifan atom starts from site (i0,j0) ), then

P(t) = M'P(0) (6.4)
Thus, in order to ﬁnd static probabilities used here, one has to run (6.3) to the times large

enough for P(t) to become saturated.

109

On Figure 6.8, this method is performed on the terrace with the large distance between

the steps (isolated step case) for perfect sticking in order to conﬁrm 1‘2 dependence (see

the comment after (C. 17)).

110

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6.7 The probability that an atom is at the site (1', j) at time t +1 is the sum of the

probabilities that an atom is at one of the neighbor site at time t multiplied by l/ 4 (there
are four directions for atom to move).

1]]

 

0.0 ‘17,\ I I ' I ' r
\

 

 

 

-10.0 J L 1 1 . 1 .
0.0 1.0 2.0 3.0 4.0

 

Figure 6.8 The matrix method (6.3) (bottom line) to ﬁnd the diffusion kernel for isolated

step in the perfect sticking case. It shows 1‘2 dependence (top line) and agrees with the
analytical result (see the comment after (C. 1 7)).

112

Chapter 7

VLSI Design

7.1 Introduction

VLSI stands for Very Large Scale Integration andrefers to those integrated circuits that
contain more than 105 transistors (in current-day technologies, circuits of 108 transistors
are already produced). Designing such a circuit is a difﬁcult task and requires use of
computational modeling. This chapter gives an overview of steps involved in VLSI
design [51]. It starts with the description of MOS technology transistors. Then, it goes to

chip design methods. Finally, the description of simulation methods is given.

7.2 MOS transistors

Design starts from transistor level cells, logic gates. Logic gates are usually composed of
2 to 6 transistors. They are almost always made using MOS (Metal-Oxide-
Semiconductor) technology. Figure 7.1 shows 3-dimensional physical view of two
existing types: n-channel (NMOS) and p-cliannel (PMOS). Figure 7.2 shows schematic
symbols for these two types of transistors. The MOS transistor can be seen as a voltage-
controlled switch. It is a device with three terminals. Value of voltage signal at the gate

determines whether there is a conducting connection between the source and the drain.

113

source gate drain

—polysilicon
silicon oxide
diffusion

 

 

 

 

n-type substrate
(a)

source gate drain

11‘1“

 

p-type substrate H
(b)

Figure 7.1 A 3-dimensional view of the (a) PMOS; and (b) NMOS.

114

drain

gate | E:
source

(a)

drain
gate l C
source

(b)

Figure 7.2 The schematic symbols for the (a) NMOS; and (b) PMOS.

115

 

.. a
—1

Figure 7.3 An inverter.

116

NMOS behaves as a closed connection when its gate signal is ‘1’ (positive voltage), and
as an open connection when its gate signal is ‘0’ (zero voltage). PMOS has the reverse

behavior. Figure 7.3 shows the simplest logic gate: inverter. Vdd is the positive supply
voltage while V“ is a ground. A signal ‘1’ applied to the input In will result in a
connection of Out to V” (signal ‘0’). Similarly, signal ‘0’ at the input will lead to the

output signal ‘1’ as a result of the connection between the voltage supply Vdd and Out.

7.3 Design Steps

Suppose that one has the layout of the cells available, together with the list of
interconnections to be made. From an abstract point of view, these cells are rectangles (or
polygons) with terminals at their periphery to which wires can be connected. The
problem is to compose the layout of the entire integrated circuit. It is often solved in two
stages. First, a position in the plane is assigned to each cell, trying to minimize the area to
be occupied by interconnections between these cells. This is called the placement
problem. The next step is to generate the wiring patterns that realize the correct
interconnections between these cells. This is called the routing problem. The goal of
placement and routing is to generate the minimal chip area, while possibly satisfying
some constraints. Constraints may e.g. be derived from timing requirements. As the
length of a wire affects the propagation time of a signal along the wire, it may be
important to keep speciﬁc wires short in order to guarantee an overall execution speed of

the circuit. Layout design with timing constraints is called timing-driven layout.

117

The partitioning problem concerns the grouping of the cells in a structural description
such that those cells that are tightly connected are put in the same group while the
number of connections from one group to the other is kept low. This problem is not
strictly a layout problem. It is encountered when some design is too large to ﬁt in a single
integrated circuit and has to be distributed among several chips. Partitioning can also help

to solve the placement.

The simultaneous development of structure and layout is called ﬂoorplanning. In a top-
down design methodology, when making a transition of a behavioral description to a
structure, one also ﬁxes the relative positions of the cells. Through ﬂoorplanning, layout
information becomes available at early stages of the design. It gives early feedback on
e.g. long wires in the layout and may lead to a reconsideration of the decisions on
structural decomposition. The ﬂoorplanning problem is closely related to the placement
problem with the difference that detailed layout information is available in placement

whereas ﬂoorplanning has mainly to deal with estimations.

A cell compiler generates the layout of cells (consisting of, say, at most 100 transistors).
One could follow a placement and routing approach in this case. However, such a two-
stage method does not work very well here. As a transistor occupies the same area as a
short wire segment, it is very difﬁcult to estimate the area necessary for wiring at the
placement stage. Most cell compilers therefore do not exploit the full freedom of being

able to place and interconnect transistors arbitran'ly. They target regular arrangements of

HS

transistors, such as linear or matrix orderings, and use speciﬁc optimization techniques

suitable for the particular regular arrangement.

A problem somewhat related to cell compilation is module generation. A module is
normally understood to be a hardware block, the layout of which can be composed by an
arrangement of cells from a small subset. These elementary cells are sometimes called
microcells. They have a complexity of around 10 transistors. Given some parameters, the
module generator puts the right cells at the right place and composes a module in this

way.

When designing chips, the designer should have the possibility to modify the layout at
the level of mask patterns. The computer tool that supports this action is called a layout
editor. This tool is essential and any set of tools meant for full-custom design includes it.
Its basic function is to allow the insertion, deletion and modiﬁcation of patterns in
speciﬁc layers. Most layout editors have additional functions, like the possibility of
designing hierarchically and duplicating subcells, which speed up the tedious work of

operating at the mask level.

Working at the mask level gives the freedom of manipulating the layout at the lowest
level, but the increased freedom is as well a source of errors. In a correct design, the mask
patterns should obey some rules, e.g. on minimal distances and the minimal widths,
called design rules. Tools that analyze a layout to detect violations of these rules are

called design-rule checkers. A somewhat related tool that also takes the mask patterns as

ll9

its inputs is the circuit extractor (see also Section 7.4). It constructs a circuit of
transistors. Both design-rule checking and circuit extraction lean on knowledge from the

ﬁeld called “computational geometry”.

7.4 Simulation

Depending on the accuracy required, transistors can be simulated at different levels. At
the switch level, transistors are modeled as ideal bidirectional switches (as explained in
Section 7.2) and the signals are essentially digital, although the model is often augmented
to deal with different signal strengths, capacitances of nodes, etc. At the timing level,
analog signals are considered, but the transistors have simple models (e.g. piecewise
linear ﬁmctions). At the circuit level, more accurate models of the transistors are used
which often involve nonlinear differential equations for the currents and voltages. The
equations are then solved by numerical integration. The more accurate the model, the

lower the maximum size of the circuit that can be simulated in reasonable time.

The fact that an integrated circuit will be realized in a mainly two-dimensional physical
medium has implications for design decisions at many levels. This is certainly true for the
transistor level. The exact values of (parasitic) capacitances and resistors depend on the
shapes of the patterns in the different layers of material. Therefore, it is the custom to
extract the circuit from the layout data, i.e. to construct the network of transistors,

resistors and capacitances taking the mask patterns as inputs. The extracted circuit can

120

then be simulated at the circuit or switch level especially to ﬁnd out how parasitic

capacitances and resistors affect the circuit behavior.

lZl

Chapter 8

Conclusion

The key physics of the model presented here is contained in the “non-local” diffusion
kernels (see Equations (3.10), (4.8), and (4.9)) which occur after integrating over the
atomic processes which produce step ﬂuctuations. These kernels have been calculated for
a variety of physically interesting cases and have related the parameters in those kernels
to atomic energy barriers. The model used here is close in spirit to the work of Pimpinelli
et al. [13], who developed a scaling analysis based on diffusion ideas. The theory of
Einstein and co-workers [1,2,6,8,12,l6,17,42] and Bales and Zangwill [14] is based on an
“equilibrated gas” of atoms on each terrace. The concentration of this gas of atoms obeys
Laplace’s equation just as probability 13 does here (see (C .2)). To make complete contact
between the two methods however, one needs to treat the effect of a gas of atoms on the
diffusion probabilities that have been studied here. Actually there are two effects that
could be included: (1) the effect of step roughness on F - it is checked numerically and
found to be quite small, and (2) the effect of atom interactions on the terrace - this leads
to the “tracer" diffusion problem. It is known that in the presence of interactions,
Laplace's equation still holds for the calculation of l3 , but there is a concentration
dependence in the prefactors. 1n the small “gas” concentration limit, this analysis and that

of Einstein et al. are equivalent (see Appendix D).

122

In addition, the link between atomic diffusion and step ﬂuctuations on Vicinal surfaces
has been elucidated. It was shown how atomic diffusion leads to the limiting results
(step-edge and attachment-detachment) found by the previous workers (see Sections 3.2
and 3.3). It was also shown that in terrace diffusion cases the ﬂuctuation exponent may
be non-universal (Table 4.1). It is relatively straightforward to include different types of
terrace diffusion, for example as occurs due to parallel and perpendicular dimerization on

Si(001), by using different diffusion kernels P0 and PI on alternating terraces. A more

sophisticated, but straightforward, generalization is necessary to include the possibility of

transmission across a step edge, a processes which may be important at many step edges

[50].

The pair correlation function between two steps C,,o(x,t) is introduced (see (4.2)). It is

identically zero for the cases of attachment-detachment and step-edge diffusion. In
contrast, it is ﬁnite and quite large in the case of terrace diffusion. This means that

measurement of a ﬁnite value for C ”G(x,t) implies that terrace diffusion is important.
Since G(x,t) (see (4.1)) quantiﬁes the cumulative effect of all mass transport
mechanisms, by combining the results of G(x,t) and Ck*o(x,t) it is then possible to

determine the relative importance of terrace diffusion to the other types of mass transport

in surface dynamics.

123

APPENDICES

124

Appendix A: Calculation of Equilibrium Width

The saturation width wm,(t) of step ﬂuctuation depends on the form of the step free

energy and boundary conditions. However, it does not depend on atomic mechanisms.
The convenient way to derive the saturation width is by using the Fourier transform of
h(x) in the expression for the free energy. The form of the Fourier transform of
hdepends on the boundary conditions. Here, two cases will be analyzed: periodic
boundaries (the case used in model presented in previous chapters) and pinned
boundaries (the case in many experimental measurements when the measured part of the

step is effectively pinned by the rest of the step). As shown below, the saturation width of

the periodic step is by a factor J2 smaller than pinned one. Beside periodic and pinned,
there can be a step with free boundaries with even greater saturation width than pinned
step.

(i) periodic boundaries:

In this case, h(x) = 11(x + nL) has to be satisﬁed. Thus, the following Fourier transform
can be used

h(x) = the‘q‘ with qL = 12m, 11 = 1,2,... (A.1)
q

where

l1_q = h; (A.2)

has to be satisﬁed in order to make h(x) real. Then, the free energy (4.5) becomes

"' [.12 ~
‘7

-Lt2 q q'

125

where

L/2

1 (X I
-L-je"e“’dx= —6_ (A.4)

—L/2

and (A3) are used. The partition function for the system is

:0 EL '2' 2
___q ,, 'l
__ I I ZkBT ”
q' 0

Then,

 

<|h|2=::> 1"” . = I‘BT . (A.5)

 

 

.n
't
a N
l
A
IV
to
V
|
A
:r
v
N
H
A
zr
N
V
II
14—.
E
A
:r
:zr
v
m5
NR
N5
H
1;
>1
9
2r
Q
"'15
V
ll
3?
h
in

(A.6)

where 211/112 = n2 /6 , (A.l), and (A.4) are used.

(ii) pinned boundaries
In this case, 11(0) 2 h(L) = 0 has to be satisﬁed. Thus, the following Fourier transform

can be used

h(x) = thsinqx with qL = WI, 11 =1,2,... (A.7)
q

where liq is real. Then, free energy (3.2) becomes

NHIMI

L
:lzthhq qq 'cosqx cosq'x dx— — ithjqz (A.8)
0 q

126

where

L l.
1 . . 1
— cos xcos ' dx=— s1n xsm 'xdx=—6 .
L! q q L0 q q 2 4,1,

is used. The partition function for the system is

2

2:1] Idhqe 1 .

 

 

 

4' ~00
Then,
+oo :14 2 2 +00 EL .2 2
- 4 h - q 1
2 41 r ” 41: r "
Idh h e 9 .e 5
q q 9 2k T
(h2> : -Q ~ ql$q —Q ~ = B
4 ”0 "4:114“: ”0 4k T114113. 2:qu .
jdh e ” .e "
q 11
—oo 0"“? -oo

Finally, by doing the both spatial and thermal averaging

wfa, = (112) —(h)’ = (112) = %]’ZZ(hqhq.)sinqx sinq'x dx = 1201;) = £83

0441' 24 62

where Z] /n2 = n2 /6 , (A7), and (A9) are used.

127

(A9)

(A.lO)

(A.ll)

Appendix B: Solution of Langevin Equation

The linear Langevin equation (3.11) in q-space has the form

6114(2)

61 =—r;‘hq(t)+nq(t), (B.l)
Note that by symmetry this sets <hq(t)> = <h(x,t)> = 0 for L —-) 00. In order to reproduce

the thermal ﬂuctuations, the noise is assumed to satisfy the relation,
(12,0072;(t1>)=f(q.,q1>6, ,60— > (13.2)
The general solution of(B.l) is,
129(1) = a“ [e""vnq(1')dt' (3.3)

The height-height correlation function in Fourier space is then,

'1 ’2

<th (4)11” (12)) = e'“ " [ [e’W (17,10; )0,“ (1', ))dz', di', . (13.4)

00

Using (B.2), one ﬁnds (assuming t2 2 t,)

 

(h,,(z1)h,,(r1>)= f“1:15(e“”""”‘" —e""'”°”’”') (13.5)
The Kronecker delta function imposes q 1 = - q 2 = q, and since h_q = h q , one has
(h (t )h; (t, ))= “(I)" ———"(e (W) — e“’2*"”’1) (B.6)
When t,=t2=t, one gets
<lhq (0'2 >= f(Q)Tq ___:11(_ 6-2111,) (B7)

128

Since Equation (8.7) must saturate (when t -—> co) to the value of the saturation width

(A.5)

 

<Ih<w>l>=§31=“:” a...)

From Equation (8.8), then one must impose the condition f (q)=(2kBT )/(§quz'q).

Fluctuation width is then (similarly to derivation of (A6»

w2(t) =<h2(x,t) >= k8 T2 cam" ‘)

 

       

(3.9)

4
On the other hand, by squaring and taking the average of each term individually in (3.17),

using (8.6), and keeping in mind that tl -> oo , one gets

 

k3 T2 —urq
G(t): Zi'quTZ— ZE—Zz—e (13.10)

Thus, comparing (13.9) to (B. 10), yields w2(t) = G(2t) .

In all ofthe cases analyzed in Chapter 3
2'" z Aq’ , (3.11)

in which case. the continuum limit of Equations (8.9) and (8.10) yields (p333-3.434(1)

of[47])

w2(t) = G(2t

               

(3.12)

where F is the Gamma integral as deﬁned for example in [34].

Analysis of linear Langevin equation (4.16) in Chapter 4 can be done similarly as (8.3),

129

qu(t) = e-l/tm £61'/fmnpq(tl )dtl . (3.13)
Then,
< vPIqI (t! )vl’zqz (t2) >=

~l'/t -! It [I [I t' /t +t' /r
p q 2 I 2
e l I me: 8 pm me:

< I)”. (t'l )nm2 (t'2 ) > dt', dt'z. (8.14)
77 is Gaussian white noise, in which case

< 77p“,l (’1 ML”, (’2) >= f(P1’pz’qt’q2)5p,,p,§q,,q,6(ti —t2) ' (3.15)

Evaluating (B. 14) using this form of the noise leads to,

‘ f(p’q)T (pr; r -(+( It
<qu(tl)qu(tz)>:_2__ﬂ(e( )’m_e (I 2) m). (B.16)
Taking the limit t -—> co and setting t2 — tl =t one gets,
‘ 9 T -l f
<qu(tl)qu(tl +t) >=£-(—E-éq—)—‘1'Le I”. (8.17)

One can reconstruct the correlation function Ck(x,t) (4.2) by inverting the Fourier
transforms (4.11) and (4.15) as

k T 10(4)!
G(x,t):ﬁL—E cosqxe ECOSp/(eg‘wmosp (8.18)

 

2

q q P

where p = Zion/N, m = O,l,2...N — l , and q = 27m/L , n = i1,i2,...i L/a, ln ﬁnding

Equation (B.18), one must set f ( p,q) = ZkBT/(iqurpq). As shown Appendix A, this

choice is necessary so that the formalism reproduces the equilibrium behavior (A5).

130

Appendix C: Random Walk Probabilities

Consider the case of two straight parallel steps 1 and 2 at the distance d from each other.
Terrace between the steps is the discrete square lattice deﬁned by x and y axis that are
parallel and perpendicular to the steps. Origin (x, y)=(0,0) is located at the distance a to

the right from the step I and at the distance d-a to the left from the step 2. Sticking
coefﬁcients for an atom on its approach from the terrace to steps I and 2 are a: and a; ,

respectively. An atom starts the random walk from the origin. Let us call p((l) and p;([)
the probabilities that this atom will be absorbed at x=l by the step I and step 2,

respectively.

In order to ﬁnd p1(1) and p2(l), one can assume that there is a source of atoms at the
origin. After emerging from the source with magnitude F, atoms do random walk on the
terrace until they sink on the step edges. In a static case, it is possible to deﬁne a

probability density function for the distribution of the walking atoms at the terrace as

P(x,y) . By knowing 15(x,y) , p1(l) and p2(l) can be found as
p,(1)= alalll~J(I,—a +01), p,(l) = ala;ﬁ(l,d —a —al). (C.l)
P(x, y) must satisfy Poisson equation
ana,V2ﬁ(.t-,y) = -—Fc5(.t)c5(y). ((12)

Beside that, it has to satisfy certain boundary conditions at the step edges. These

conditions can be derived with the following simple argument. Probability that an atom,

sitting on the site (x,-a+ai), will stick to site (.r.-a) on the step edge 1 is a,’ . The

131

probability that it will not stick and bounce back is l-al'. Thus, effectively
P‘(x,-a)2(1—a{)'ﬁ(x,—a+a,). Using that and the fact that
P(x,—a+al)—P(x,—a)—>al6}‘5(x,y=—a)/ay,one can obtain

aﬁtw = —a)

0y = aII~’(x,y = —a) (C3)

 

where notation a, s a,’ /(l — a,’ )al . Similarly, on the other step edge,

6ﬁ(x,y =d —a)
3y

 

= azﬁ(x,y = d — a) (C4)

where a, E a; /(1—a§)ai.

F(x, y) can be found by solving Equation (C2) with conditions (C3) and (C4) in the

following way. Using the Fourier transform,

 

~ _ 1 mm ,k,

P(x, y) _ 3; L Pk (x)e dk (C5)
and,

_ 1 a: ”0'
6(y)—2—”—J:oe dk, (C.6)
one ﬁnds that,
dzﬁ x 7 ~
dig ) — k‘P,‘(x) = —F5(x). (C.7)

One can take the solutions to this equation to be,

13;: (x) = ake"u + dke'“ for d — a 2 x > 0 (CS)

if(x) = Ake'“ + Dke‘“ for - a s x < 0 (C9)

132

One must ensure that the beside (C3) and (C4), the following boundary conditions are

also satisﬁed,

may) =E‘(O.y) (010)

and

E—ij‘dx P2(x) _F__§'_15_*_ . ap-
8x

               

(CH)

The ﬁrst of these equations ensures continuity, while the fourth equation ensures ﬂux
conservation near the source. From the boundary conditions, one gets equations for the

coefﬁcients Ar, Dk, a), and (1,. as
A, +Dk =0,, +d,‘ ,
A,(k — aL)e""“ = D,‘(k + aL)e"“,
a, (k — aU )e‘*“"“’ = d, (k + aU )eW‘“ (C.12)

and

k(Ak —Dk +ak —dk)=F.

Solving these equations yields Em that through Equations (CS) gives 13(x, y) .

After calculating P(x, y) , by using Equation (Cl) and normalization

L12
2 I, (p.(1)+p.(1»d1=1. (C.13)
that gives F = af , p1(l) and p2(l) can be found as
pl“) = P(a,,a,,d,d—a,l) a p2(l)=P(al’a2’daa,1) (C.14)

where

133

(k cosh kb+ a2 sinhkb)
(k2 +aUaL)sinhkd +k(aU +aL)coshkd

 

P(a,,oz2,d,b,l) = 3'— [dk coskl (C.15)
27: _Q

with k = i 2 mn/L where m = 1,...,L/a.

It is interesting to note that after manipulation with above equations, it can be obtained
that

2 aﬁtm = -—a)

p‘(l)=ai 6y a P2(l)=-ai aP(x,y:d_a)

5y

 

(C.16)

 

as another interesting relationship between EU, y) , and p1(l) and p2(l), besides (C.l).

Finally, one can obtain results in some limiting cases which are relevant to the discussion

in the text.

(i) An isolated step with arbitrary sticking coefﬁcient

Take (1 —> oo , then p2 ——) O, and (from Equation (C.14)),

e""* cos kl dk

a].
pl(l)-—ﬂ-f kmL (C.17)

 

There are two limiting behaviors. If I >>1/a,_ , then 131(1) ~ 1/12, while if I <<1/aL , then

Plﬂ) ~ constant - ln(l).

(ii) Both steps perfectly absorbing aL, aU —> 00
Then the sum (C.15) can be evaluated (for a i —> O) and gives

1
d2

 

pt (1) ~ csch2 (ﬁ) (C.18)

134

and
1 7d
1 ~—sech2 — .
P2() d2 (2d)

For 1 small compared to the step separation, one has,

2

1 7r
NIP—21—2 and EUPE for1<<d.

while when l is large compared to the step separation, one has,

It!

Pt(l)zP2(l)~ e7 for I>>d

 

2d2
(iii) Step 1 perfectly absorbing aL —> oo , step 2 repelling aU = 0

Then

1 7r! 7d
1 ~—csch -—— coth —

135

(019)

(C .20)

(C21)

(C22)

Appendix D: Overview of Other Models

In order to compare the results with other groups, an overview of two models (Khare and
Einstein [17], and T.Ihle et al.[43]) for the dynamics of step ﬂuctuation is given in this
appendix. Both models have similar approach that is different from the one presented
here in previous chapters. However, they obtain approximately same results as the one
found for the terrace diffusion case in Chapter 4. Different models have different notation
for the same physical quantities. To make the comparison simpler, notation used here will
be assigned to all common quantities in those two models. Of course, the notation of the

quantities that are unique for the model will be kept in its original form.

Khare and Einstein assume the same free energy form (4.5) for non-interacting steps as

here. The surface is deﬁned with axis x parallel to the steps and axis y perpendicular to
the steps. Instead of sticking coefﬁcients, they introduce kinetic coefﬁcients k_ (k, ) for

the mass exchange with the upper (lower) terrace. Instead of diffusion kernels, they

deﬁne the chemical potential ,uhm(x, y) of the terrace between steps k and k +1 . The

step edge moves as

612k(x,t)_ k+ — L _
at -kBT(#t.t..(x,kd.) #i(x,t))+kBT(,uM_,(x,kd_) p,(x,t)) (D.l)

 

where pk‘,,,(x,kd,) ( y,‘_k_,(x,kd-)) is the terrace chemical potential just right (left) of

step edge k . p, (x,t) is the step chemical potential as deﬁned in (4.4). On the other hand,

mass conservation at the step edge leads to the equation

136

 

 

5ht(x,t) _ 1 a D azluk,k+1(x’kd+) azﬂk.k-l(x’kd-)
— L 3! 2 + 2 —
5t kBT 0x 6x

a J“! t k—I , -
—D3u[ luk,k+l(x +) __ 5,11 , (x kd )J} (D2)

6y 6x
where D3, is the diffusion constant of atoms moving only along the step edge and D“ is

the diffusion constant of atoms on the terrace. Equating (D.l) to (D2) gives the following

boundary condition for ,uw,+l (x, y)

k1(#k,k1|(x’kd1)_ I'll: (x,t)) = 0105:

 

 

52/411195,de 'T D” alukjtl(x9kdt) . (13.3)

ﬁx by

Note that for D3, =0 (no step edge diffusion coupled with terrace diffusion) this

resembles Equations (OB) and (C4). Beside boundary condition (D.3), chemical

potential has to satisfy
Vzpk‘mtny) =0. (D.4)
After ﬁnding chemical potential luk'M(x,y) by solving (D.4) with boundary condition

(D.3), substituting it in (DI), and taking Fourier transform, they also obtain equations

(4.48) and (4.49) with diagonal elements
gem) = §q2(k.1,' +k-I; ) (0.5)
and off-diagonal elements
gl (q) = §q2(k,a; + k_a; )/(2Eq cosh(| q | (1)) (D6)
where

E E[a;(1+ bq‘) +a;(1 + b;)]+ [a;a; + (1 + 1);)(1 + b; )Jtanhq q Id), (13.7)

137

1; s—El—{a:(l+2b:)+[a;a; +bj(1+b;)]tanh(|q1d)}, (D.8)

and
a: a D... 1q 1 /k. . b: a D..a.q2 /k.. (0.9)
In order to compare this with (4.48) and (4.49), one has to set D“ = O, k_ /D,u —) aU ,

k+ / D,“ —) ozL and oi / rm —> D,“ . Khare and Einstein did not go further then to decouple

the system and ﬁnd correlation functions by summing over all modes.

T. Ihle et al. applied the similar principle, but went further to decouple the system.
Similarly to terrace chemical potential (see Equation (D4)) in the Khare and Einstein
model and diffusion kernel (see Equation (C.2)) used here in Chapters 3 and 4, they

introduce atom concentration c(x, y,t) on the terrace that satisﬁes

£36£=Dv2c_£+p+f—vq. (0.10)
I 1'

Here, steps interact and thus atom concentration on the terrace depends on time. D is the
diffusion constant. Deposition of atoms on the surface is assumed and F is the rate of
that deposition. Atoms can also evaporate from the surface after an average time 1.

Noise f describes the fluctuation of the deposition rate while conserved noise 'q' is

related to ﬂuctuations of the diffusion current.

The interaction between steps is assumed to be entropic, thus step chemical potential
,u,(x,t) is given by (5.3) with (5.14). Similarly to (D.l) in Khare and Einstein’s model,

step edge motion is described with

138

6h, 6 6,11
—=Q — — _ _ ~
at (NC. C..+n.)+V-(c- c.,+n-))+as( 63+0q] (D.11)

where v+ (v_) are the attachment coefﬁcients for an atom on its approach from lower
(upper) terrace, 77, (IL) are the random attachment-detachment forces, and c+ (c_) are
the atom concentrations on a lower (upper) terrace just next to the step edge. on is the
equilibrium concentration at a step. The last term in (D.11) corresponds to ﬂuctuation of

the mass ﬂux along the step edge. B is the transport coefﬁcient, s is the arc length along

the step and 5 is the noise related to that fluctuation. As in Khare and Einstein’s model

(D.3), concentration has to satisfy boundary conditions
Tﬁ-(—DVc+2j)t=vt(c—ceq+ni). (D.12)

By comparing to (D.3), one sees that T.Ihle et a1. do not assume coupling between terrace

and step-edge diffusion.

After linearization and diagonalization by Fourier transform, system (D.10)—(D.12) gives
hwpq : Wwpqlwpq (D°13)
where the Fourier transform of h,(x,t) (like any other quantity) is deﬁned as

=+oo+°°

hm = *2 I dedt h,(x,t)e"‘“’”"”*""’. (D.14)

(HM,
1,1/ is a noise resulting from the random forces introduced above, and x is, so called,
linear susceptibility of the step. The inverse of the susceptibility is
1;}; = iw+ 1»qu (D.15)

with

139

_ ~ 3 C
EWLq = [Xq‘ +12?(1—c03p))x

Dc° (22A (1116)
x [BQqZ + __w {2(cosh(Awd) - cosp) + Awq(d+ + d_)sinh(Awqd)}]
B

where c; is the equilibrium concentration in front of an isolated straight step, C is the

constant from interaction term (5.14),

 

Aw =,/tw/D+q2+1/xf, (13.17)
R = (d, + d_ )Aw cosh(Awqd) + (d+d_l\2wq +1)sinh(Awqd), (D.18)
d, = D/vt (D19)
and diffusion length
x, = JE . (p.20)

Susceptibility here corresponds to relaxation time (4.17) with (4.48) and (4.49). In order
to compare results, one must set C = 0 (step interactions are not assumed in Chapter 4),

B = 0 (there is no step-edge diffusion), w = 0 (static case) and x, -—) 00 (since I —> 00,

no evaporation) thus Awq =| q I, D —) oi / rm, and d, = chQ/aq .

140

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