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dissertation entitled

" The Island—To—Percolat ion Transit ion

During Growth of Metal Films"

presented by

George William Jeffers, Jr.

has been accepted towards fulﬁllment
of the requirements for

Ph D degree in Phy§ics

 

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Date 10/24/94

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THE ISLAND — To - PERCOLATION TRANSITION
DURING GROWTH OF METAL FILMS

By

George William Jeﬁ‘ers, Jr.

A DISSERTATION

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

DOCTOR OF PH]LOSOPHY

Department of Physics and Astronomy

1994

 

 

 

ABSTRACT

THE ISLAND - TO - PERCOLATION TRANSITION
DURING GROWTH OF METAL FILMS

By

George William J effers, Jr.

The formation of percolating Pb ﬁlms thermally evaporated onto Si02 substrates
has been experimentally studied using electron microscopy. Pb does not wet Si02 so that
compact, three-dimensional islands are formed during the initial stages of ﬁlm growth. As
these islands grow in size they eventually touch their neighbors and coalesce due to
surface diﬂilSiOll. Islands smaller than a critical radius, RC, fully coalesce and form a single
island with a. compact shape. However, islands larger than this critical radius undergo a
partial coalescence to form an elongated island. This crossover to partial coalescence
produces a network of connected metal structures which increase in size until the
percolation threshold is crossed. The main topics studied in this work are: 1) the ﬁlm's
area coverage at the percolation threshold, pc, is often quite high (pC z 80%) and 2) the
factors controlling the size at which the crossover from ﬁill to partial coalescence occurs.
The high values of pC result from the combination of wiping during coalescence and the
coalescence crossover which cause the larger droplets to effectively repel one another.
The coalescence crossover is described by a Kinetic Freezing Model in which the
crossover is due to a competition between the time for two touching islands to coalesce
and the time for one of the islands to grow in size, due to continuing deposition, and make

contact with a neighboring island. The predictions of this model are tested and some of

the absent features are discussed.

 

 

 

 

Computer simulations of the proﬁle decay of a corrugated crystalline surface due
to surface diffusion are also investigated. A fast Monte Carlo algorithm is used to solve a

(1+ 1)D system at low temperature. The results are signiﬁcantly diﬂerent than those

predicted by the high-temp erature continuum model.

 

 

 

 

 

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ACKN OWLEDGlVIEN TS

I would like to thank my advisor, Michael Dubson, for his advice and guidance
during the course of this research. His knowledge and interest in physics and science in
general have helped me to learn much over the last number of years.

Professor Phillip M. Duxbury and his graduate student, Xinhua Yu, have provided
much in the way of theoretical justiﬁcation dealing with this research.

I am thankﬁrl to my fellow labmates for much help in both research and non-
research related matters as well as making the lab a ﬁiendly environment. These people
are: Jeeseong Hwang, Evstatin Krastev, Lowell McCann, Richard Smalley, David
Wollman, and Qiﬁi Zhu.

Many of my fondest memories of graduate school at MSU involve the network of
students and staff who helped one another maintain our respective levels of sanity while
learning some physics at the same time. Most notable of this group are: Glenn Alers, Erik
Hendrickson, Lilian Hoines, Mike J aeger, Shang-Fan Lee, Ninamarie Levinsky, Catherine
Mader, Paul "Rooster" McConville, Carl "Spike" Nelson, Raman Pfaﬁ‘, and Mike Wilson.

I also wish to thank a small group of friends who were among the most important
in my adjustment to MSU. Many enjoyable times were had with Crystal Cheatham, Kathy
McCloud, and Geri Palka. I owe special thanks to Kathy McCloud for many reasons. Her
love and support both during and after her time here at MSU have had a major impact on
my life. Her presence in my life has profoundly aﬂ‘ected my perception and understanding
of life and people. Kathy will always occupy a special place in my heart and to her I

extend my thanks, appreciation, and love.

 

 

 

 

 

Erik Hendrickson and Rooster McConville deserve extra acknowledgment for their
support as friends, roommates, and colleagues. Their tolerance of my sometimes eccentric
behavior has been greatly appreciated. Together we had a unique balance of work and
play. For constant calming of my nerves I am indebted to Dirk, Lerxt, and Pratt.

Thanks are due Professors Norman 0. Birge, Eugene Capriotti, Phillip M.
Duxbury, and Michael R Thoennessen for serving on my Guidance Committee.

For their skill, advice, and humor I am thankful to the staff of the PA Machine
Shop staff: Tom Palazzolo, Tom Hudson, and Jim Muns. I am thankful to Stephanie
Holland for her willingness to assist us sometimes helpless graduate students and for
making our lives in graduate school less worrisome.

My deepest acknowledgments are extended to my parents, brother, and sister. It is
largely due to their support throughout my life that I have completed this goal.

Finally, this work was supported by The MSU Physics and Astronomy
Department, The MSU Center for Fundamental Materials Research, The National Science
Foundation under Grant No. DMR—8857364, and by the Donors of The Petroleum
Research Fund, administered by the American Chemical Society.

 

 

 

 

rat .4 i;—

 

 

 

 

 

TABLE OF CONTENTS
LIST OF FIGURES ................................................................................................... x
1. INTRODUCTION
Background and outline of thesis ............................................................................ l

 

2. EXPERIMENTAL EQUIPMENT AND RIETHODS

 

 

 

 

Sample preparation, observation, and analysis ........................................................ 5
2.1 Introduction .............................................................................................. 5
2.2 Thermal Evaporation System ..................................................................... 5
2.3 Substrate Preparation ................................................................................ 15 ,
2.4 Film Deposition ......................................................................................... l6
2.5 Sample Observation ................................................................................... 18
2.6 Digitized SEM Image Acquisition .............................................................. 19
2.7 Image Analysis Programs .......................................................................... 23
2.7.1 Measurement of pc ...................................................................... 23
2.7.2 Determination of Island Width Rc ............................................... 26
2.7.3 Compression Programs ............................................................... 31
2.8 Summary ................................................................................................... 31
Chapter 2 References ....................................................................................... 34 '.
3. THE ISLAND - TO - PERCOLATION TRANSITION f
DURING GROWTH OF METAL FILMS i
Experimental observations of Pb on Si02 ﬁlm growth ............................................ 36 l
3.1 Introduction .............................................................................................. 36

vii

 

3.2 Film Growth Sequence and Features .......................................................... 37
3.3 Interrupted Coalescence Model ................................................................. 42
3.4 Film Morphology and Percolation .............................................................. 47
3.5 Kinetic Freezing Model ............................................................................. 61
3.6 Predictions of the Kinetic Freezing Model ................................................. 65
3.6.1 Rc Dependence on Deposition Time ............................................ 65

3.6.2 Inferred Surface Diﬁusion Constants DS ...................................... 67

3.7 Missing Ingredients of the KFM ................................................................ 69

3.8 Summary ................................................................................................... 72

Chapter 3 References ....................................................................................... 7 5

4. POSSIBLE FUTURE DIRECTIONS

Further experiments related to thin ﬁlm growth and structure ................................. 77
4.1 Introduction .............................................................................................. 77
4.2 Substrate Inhomogeneity ........................................................................... 77
4.3 Grain Boundary and Coalescence Studies .................................................. 80
4.4 Summary ................................................................................................... 83
Chapter 4 References ....................................................................................... 84

5. PROFILE DECAY DUE TO SURFACE
DIFFUSION AT LOW TEMPERATURE

Computer simulations of mass transport on a stepped surface ................................. 85
5.1 Introduction .............................................................................................. 85
5.2 Basis of the Simulation Algorithm ............................................................. 86
5.3 Movement Trajectories and Probabilities ................................................... 90
5.4 Simulation Algorithm ................................................................................ 99
5. 5 Continuum Model and Previous Results .................................................... 100
5.6 Simulation Results ..................................................................................... 105

 

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5.7 Parallel Resistor Model for P1, P2, Pr, and Pt ............................................. 124
5.8 Summary ................................................................................................... 126
Chapter 5 References ....................................................................................... 128

 

 

 

 

LIST OF FIGURES

Figure 2.1 Diagram of thermal evaporation system, described in the text,
used in sample preparation. Diagram is not drawn to scale. ............................ 7

Figure 2.2 Diagram of heatable Cu substrate stage. 11

Figure 2.3 Substrate temperature rise due to exposure to a heated, empty
evaporation boat. The diﬁ‘erent traces correspond to different
substrate temperatures at time zero as shown in the legend. ............................. 13

Figure 2.4 Schematic diagram of digitized image acquisition circuit. .......................... 21

Figure 2.5 Sample digitized image of a 400 nm Pb ﬁlm on a 423 K substrate
using the digitization technique outlined in Section 2.6. ................................... 24

Figure 2.6 Intensity histogram for the Pb sample shown in Figure 2.5. The arrow
determines the threshold for creating a binary image. ....................................... 27

Figure 2.7 Binary image of the top 384 lines of Figure 2.5 using the intensity
histogram shown in Figure 2.6. 28

Figure 2.8 Percent of sites burned during the forest ﬁre algorithm vs. cycle
number for the binary image shown in Figure 2. 7. 30

Figure 3.1 Schematic illustration of three-dimensional Volmer-Weber
island growth. 37

Figure 3.2 Coalescence and wiping process. 39

Figure 3.3 Growth evolution of Pb 011 insulating Si02 substrates.
(a) Compact islands to elongated structures (from thin (right) to
thicker (left) deposits). (b) Percolation threshold which occurs
at a higher ﬁlm thickness than in (a), with pC w 63%. ...................................... 40

Figure 3.4 "Breath Figure" stage which occurs in the early stages of metal
ﬁlm growth. This image is of a Pb ﬁlm on a 483 K Si02 substrate. ................. 44

Figure 3.5 Percolation pattern resulting for 3D droplets on a 2D substrate
using the ICM with L/RO = 200 and Mo = 8. The coverage is 82%. .......... 45

 

 

Figure 3.6 Percolation pattern of a 500 nm thick Pb ﬁlm deposited at 6 nm/s
onto a 453 K Si02 substrate. This ﬁlm was quenched immediately
following deposition. The coverage is m84% and the average metal
island width is l um

 

Figure 3.7 Area coverage at percolation, pc, as a ﬁmction of substrate

 

temperature for Pb ﬁhns deposited at 6 nm/s.

The structure is discussed in the text.

 

Figure 3.9 A ﬁlm prepared with T = 498 K which is well above To. The

Figure 3.10 Sequence of images illustrating the effects of annealing on the

(a) 0 and (b) 1.2 minutes.

 

 

prepared on 423 K substrates.

Figure 3.12 Crossover radius Rc vs. substrate temperature for Pb samples

.48
49
Figure 3.8 Film morphology ofa 1600 nm Pb ﬁlmprepared at T = 463 K > To.
........ 5 1
ﬁlm is 3470 nm thick with a deposition rate of 6 nm/s. ..................................... 53
ﬁhn's ﬁnal morphology and percolation coverage. Each ﬁhn is 400 nm
of Pb deposited at 6 nm/s onto a 423 K substrate. Annealing times are
55
Figure 3.10 (cont) Images with annealing times of (c) 2.3 and (d) 4.5 minutes.
The deposition time for each sample was 1.1 minutes. ..................................... 56
Figure 3.11 Percolation coverage vs. ratio of anneal time to deposition time
for Pb samples with different deposition rates. All samples were
58
6O

 

evaporated at 6 nm/s onto Si02 substrates.

Figure 3.13 Log-log plot of the radius RC vs. deposition time tc needed to
reach percolation for Pb ﬁlms. The lines are power law ﬁts to Rc ~ (tcﬂ/l3

66

 

with B e 7.4 (408 K) and 13 e 7.6 (423 K).

Figure 3.14 Plot of inferred DS values as a function of Tmemng/Tsubstmte.
The solid lines show the range of DS values measured on fcc metal
surfaces [BONZS3].

68

 

Figure 3.15 Surface energy of two coalescing droplets with an interior grain
boundary having a dihedral angle, ‘1’, of 78 degrees. The coalescence

stage is shown in the accompanying schematic diagrams. ................................. 71

Figure 3.16 Plot of substrate temperature vs. RC showing the different

coalescence regimes. These Pb samples were deposited at 6 nm/s. .................. 74

Figure 5.1 Binding energy diagrams near a monatomic step for the triangular
and square lattices. A is the Schwoebel barrier, Ek and Ft are
described in the text. 87

 

 

 

Figure 5.2 Typical environment of an activated kink site atom (shown as
the solid square). The four possible movement trajectories and the

 

terrace widths, as discussed in the text, are also shown. ................................... 91
Figure 5.3 Coordinate system for an adatom on a terrace of width L. ......................... 93
Figure 5.4 Environments of kink site atoms in the topmost (left) and bottom

(right) terraces. 95
Figure 5.5 Creation and disappearance of an abandoned adatom on a top terrace. ...... 96

Figure 5.6 Equivalency of triangular and square lattices along with the
respective adatom binding energy curves for each. 98

Figure 5.7 Amplitude decay, A(t)/Amax, vs. Time/A35 for step edge reﬂection
coefﬁcient k = 0.9 and wavelengths A. = 64, 128, and 256. .............................. 107

Figure 5.8 Amplitude decay, A(t)/Amax, vs. Time/M-0 for step edge reﬂection
coefﬁcient k = 0.9 and wavelengths A = 64, 128, and 256. .............................. 108

Figure 5.9 Time evolution of a system with A = 256, A0 = 40.7, and k = 0.9
shown at various stages of evolution. Each proﬁle is the average of a
system containing 32 periods. Beginning with curve one the simulation
times of the different average proﬁles are 0, 1.35x105, 3.82x105, 9.80x105,
and 1.45x106. 110

Figure 5.10 Curves two through ﬁve of Figure 5.9 normalized to have the same
area as curve two in Figure 5.9. The self-similarity of the curves indicates
the shape preservation quality of the proﬁles. 111

Figure 5.11 Amplitude decay, A(t)/Amax, vs. simulation time for a system with
A = 256, A0 = 10.2, and k = 0.9 showing a decay which behaves nearly
linearly with time. The solid line has a slope of —3.85x10‘7. .......................... 116

Figure 5.12 Amplitude fraction A/AO vs. Time/7&275 for A0 = 5.1 and 10.2 at
various wav ‘ g‘hs 7t. 118

Figure 5.13 Amplitude fraction A/Ao vs. scaled time at ﬁxed AO/Amax = 1/2
for k E {0, 0.70, 0.90, 0.997}. The original time values were multiplied
by 26.6, 30.0, 18.5, and 1.0 for the low to high values of k, respectively. ......... 119

Figure 5.14 Movement probability Pr vs. terrace width L for various values of k. ....... 121

Figure 5.15 Decay time plotted vs. 1 — k for AO/Amax = 1/2 and A = 128.
The decay time is deﬁned as the time at which A/Ao = 1/4. ............................. 123

 

 

 

 

CHAPTER 1

INTRODUCTION
Background and outline of thesis

Surface diffusion plays the determining role in many areas of materials physics.
The process of surface diﬁ‘usion refers to the random migration of single atoms or clusters
of atoms amongst the binding Sites on a surface. This occurs during both the growth and
decay of metallic structures and is the dominant mode of mass transfer for temperatures
signiﬁcantly below the melting temperature. Currently, a complete understanding of the
surface diffusion process is lacking.

There is presently much activity, both experimentally and theoretically, directed at
elucidating the role, and possible control, of surface diﬁ'usion. Much of this effort is
centered on the growth of epitaxial ﬁlms via Molecular Beam Epitaxy. These ultrathin
ﬁhns may be exploited to form electronic devices as well as formation of well-deﬁned
multilayered samples. The quality and roughness of these layers and interfaces is often
determined by surface diﬂ’usion. However, surface diﬂirsion plays a dominant role
throughout the growth of metallic ﬁlms. During growth of a metal ﬁlm, surface diifusion
eﬁects the interaction of the incident vapor atoms with both the bare substrate and already
existing deposit. This is particularly true when, during the course of growth, small islands
of either metal or liquid deposit come into contact.

However, epitaxial ﬁlm growth is the exception rather than the rule. Quite often,
the deposit and substrate form a non-wetting system. That is, the deposit forms three-
dimensional structures on the substrate rather than growing as successive monolayers. In
such a case, the ﬁlm proceeds through the following growth stages. The smaller, three-
dimensional islands grow in size until contact is made with similar islands nearby. At this

point the touching droplets either fully coalesce and maintain their compact nature, or they

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partially coalesce and result in an elongated metal structure. Which coalescence behavior
results depends on the size of the touching islands. In the partial coalescence case, a
network of islands results which has a high degree of lateral connectivity. Electrical
conduction across the sample becomes possible at the percolation threshold. With
continuing deposition, the sample ﬁlls in any remaining empty channels and quickly
completes formation of a continuous, hole-free ﬁlm The sequence just described
generally occurs for metals deposited onto insulating substrates.

The stage of ﬁlm growth near the point where a percolating structure occurs, is
discussed in the bulk of this thesis. While much work has previously been performed by
many groups regarding the initial nucleation and early stages of ﬁlm growth, less work has
concerned the intermediate to late stages of growth. The research described herein deals
with the coalescence and percolation regimes just prior to the formation of a continuous
ﬁlm. The experimental techniques used are discussed in Chapter 2. Nucleation is not
dealt with for two reasons. First, the features to be discussed occur well alter initial island
nucleation. Secondly, the experimental equipment used oﬁ‘ers insufﬁcient resolution to
accurately study island sizes and densities near the onset of ﬁlm growth.

There are two main features which will be discussed in relation to ﬁlm growth.
The ﬁrst is the fact that when metal ﬁlms deposited on insulating substrates form an
electrically conducting structure there is an anomalously high metal coverage on the
substrate. This area coverage, pc, is the fraction of substrate covered by deposited metal.
The coverages observed are substantially higher than the values predicted by a wide
variety of numerical simulations of ﬁlm growth. The second feature is the existence of a
crossover in the structure of the ﬁlm Prior to this crossover the ﬁlm contains many
isolated, compact crystallites of deposit material. After the crossover the ﬁlm contains
many elongated, intermingling metal structures which form a connected, percolating
structure. This crossover is signaled by a change in the coalescence behavior of two

touching droplets. Smaller islands fully coalesce and maintain their compact appearance.

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However, the larger islands only partially coalesce, thus forming structures which have an
oblong shape. Many of these randomly oriented structures result in the percolating
structure. In Chapter 3, I discuss these observations and simple models, the Interrupted
Coalescence Model (ICM) and Kinetic Freezing Model (KFM), are presented which
qualitatively explain these coupled features.

Our models explaining the large area coverage at percolation and the coalescence
crossover are incomplete. Factors are absent which are believed to be very important to a
thorough understanding of the growth process. Chapter 4 contains some thoughts on
possible experiments which could be performed to quantify certain of these growth
inﬂuences. One main effect on the ﬁlm growth is the inﬂuence of any topological or
chemical inhomogeneity of the substrate. This may result from either physical damage
(such as scratching or ion bombardment of the substrate) or contamination of the substrate
surface. The second important ingredient missing from the Kinetic Freezing Model
discussed in Chapter 3 is the eﬁ'ect of grain boundaries which are known to be trapped
within the interior of two coalescing islands. These grain boundaries are observed in all
ﬁlms which are past the crossover from full to partial coalescence.

Finally, extensive computer simulations, detailed in Chapter 5, were performed to
study the elfects of surface diffusion on the decay of a corrugated surface structure. For a
given proﬁle shape, there is only one adjustable parameter in these simulations, the step
edge reﬂection coeﬂicient, k. This temperature dependent parameter accounts for the
asymmetry in the potential energy on a crystal surface near a monatomic step. This
parameter alone determines the decay dynamics of the (1+1)D system studied. These
simulations possess two unique features over most similar studies. First, these simulations
were performed below the toughening temperature. Previously, theoretical and
experimental efforts concentrated on the T > Troughening regime. Our results are
markedly diﬁ‘erent than in this higher temperature region. Second, our simulations are

novel in the sense that they bypass the normal Monte Carlo algorithm Movement of

 

 

 

 

4

atoms is still random, however, the movement is "instantaneous." Once an atom is
selected and given the opportunity to move, its actual movement is based on a speciﬁed
prob ability distribution depending up on the atom's environment. The movement of any
atom need not be continually followed throughout its entire random walk in moving from
the initial to ﬁnal positions. While many of these simulations required a large amount of
computing time, the avoidance of tracking the entire movement yields a large reduction in
computational time. The method was applied to a one- dimensional system with height
z = z(x, t) with t representing the simulation time. However, extension of this algorithm
to three dimensions where z = z(x, y, t) is not straightforward since each activated atom
has a complicated terrain in which to move. Calculation of the relative probabilities in
such a case is very complicated in analogy to the diﬂiculty in calculating the resistance

between any two points on a planar random resistor network.

 

 

 

 

CHAPTER 2

EXPERIMENTAL EQUIPMENT AND METHODS
Sample preparation, observation, and analysis

2.1 INTRODUCTION

In this chapter I will discuss the experimental details used in sample preparation,
sample observation, image acquisition, and image analysis. A slightly modiﬁed
commercial thermal evaporation system, described in Section 2.2, was used to deposit thin
ﬁlms of metal as outlined in Sections 2.3 and 2.4. Post-deposition annealing and cooling
were performed prior to removal from the vacuum chamber. The area coverage and
average size of the connected metal islands at percolation, the main quantities of interest,
were obtained from Scanning Electron Microscope (SEM) images using the image
digitization technique described in Section 2.6. The digitized images were analyzed, using
programs written in the QuickBASIC programming language. The analysis procedure will

be discussed in Section 2.7.

2.2 THERMAL EVAPORATION SYSTEM

Many different techniques exist for producing thin ﬁlms of metallic, insulating,
magnetic, and superconducting materials. The simplest and perhaps oldest of these
methods is thermal evaporation. The basic idea of this method is that the material to be
deposited is heated to a temperature to cause signiﬁcant evaporation. When performed in
a chamber where the pressure is low enough that the mean free path of the evaporated
material is comparable to, or larger than, the source to sample distance, the evaporated

material travels ballistically and will condense upon the substrate (the material which the

5

 

 

 

 

sample is grown on). This condensation process clearly depends on such parameters as:
chamber pressure, supersaturation of the vapor above the substrate (related to the
deposition rate), substrate material, and substrate temperature. In subsequent sections of
this chapter, details of the experimental control of these parameters will be described.

All samples to be discussed in this thesis were prepared via thermal evaporation in
our deposition system [EDWA] which is shown in Figure 2.1. As there are many elements
to this system, the following paragraphs will describe these individually.

The source of metal to be deposited (in wire or pellet form) rests in the dimple of a
high melting point metal (generally W or Mo) boat [MATH]. Passage of a large current,
generally in the range 50 to 100 amps through the boat results in Joule heating sufﬁcient
to evaporate the metal in the boat, but not the boat itself This current and, hence, the
source temperature and deposition rate, are manually controlled using a variable
transformer connected to the metal boat. Care must be taken so that the boat and metal
do not form an alloy (such as in the case of Al and W) which would result in a ﬁlm of
unknown stoichiometry. As the ﬁlms to be discussed were Pb, a W boat was used for
which alloying was not relevant [HAN SS 8]. Other sources of heat to evaporate the
deposit, such as a crucible (boron nitride or quartz), which sits in and is heated by a wire
basket, may also be used. This technique is possible with Pb due to the low melting point
of Pb. Higher melting point materials are more difﬁcult to deposit using a crucible setup
due to a poor thermal link between the wire basket and the crucible. Our source holds
enough Pb so that up to ﬁve samples, of varying thicknesses, could be prepared in one
pumpdown cycle without breaking vacuum. One problem encountered with our sources
was stability of the deposition rate. The deposition rate was generally constant to within
roughly 5%. These ﬂuctuations could often be lowered by means of a longer pre-melt of
the deposit during steadying of the deposition rate. An enclosed, isothermal effusion
crucible cell source (Knudsen cell) offering extreme deposition rate control could also

have been used. Efforts to construct such a cell will be mentioned in Chapter 4.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7
Rotaryz Electrical Removable lid
feedthrough feedthrough\v
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S 1 cold ﬁnger
amp e stage,
heater, and ,/ /
thermometer \\ ,9, N... Film thickness
. M; .. monitor (F TM)
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Figure 2.1 Diagram of thermal evaporation system, described in the text, used in sample
preparation. Diagram is not drawn to scale.

 

 

 

 

8

One difﬁculty associated with thermal sources is that the high temperatures
required to evaporate the deposit metal also heat regions of the evaporator near the
source. The outgassing of these heated parts causes a pressure rise in the chamber.
Depending upon the composition and cleanliness of these nearby surfaces, foreign species
may be desorbed and integrated into the sample. This might happen if, for example,
following the deposition of a low melting point material, a high melting point material was
evaporated from the same position. To minimize contamination of the vacuum chamber
and samples, Pb ﬁlms were prepared with the source surrounded by an A1 sheet metal
chimney extending to just below the sample. This chimney restricts the evaporated
deposit to only strike the sample and ﬁlm thickness monitor. Additionally, the area
directly below the source was covered with Al foil. These simple measures served the
dual purpose of shielding the sample from stray material as well as covering the nearby
surfaces from being coated with Pb, which has a fairly low melting temperature of
TM = 601 K= 328 °C.

A ﬁlm thickness monitor (F TM) is used to measure the deposition rate and total
ﬁlm thickness during deposition. A FTM consists of a vibrating piece of quartz which has
its resonant frequency changed due to adsorption of material upon its surface, similar to a
mass on a spring. This measured ﬁequency shift is combined with parameters (density and
acoustic impedance) of the deposit material and a geometry factor (to account for the
position of the FTM relative to the sample) to yield the deposition rate and total thickness.
Most thicknesses quoted in this thesis are nominal thicknesses. That is, they are generally
correct relative to one another. The absolute thickness, however, is only known to an
accuracy of roughly ﬁve to ten percent. The Physics and Astronomy Department has a
surface proﬁlometer [SLOA] which was used to measure some ﬁlm thicknesses and

calibrate the FTM. This apparatus Operates by moving a sharp stylus of radius ~5 um
over the surface to be measured and recording the vertical deﬂection. However, because

Pb is a very soft material the stylus tends to "plow" into the Pb, resulting in meaningless

 

 

 

 

 

 

thickness measurements. The FTM was, however, calibrated by depositing harder
materials such as Ag in the same geometry and measuring their thicknesses with the
proﬁlometer. However, as our evaporation system was often used by people who
required different sample geometries from my own, the FTM was repeatedly moved.
While care was taken during setup to return the FTM to the same position, thickness
variations undoubtedly existed from run to run. As a check on the calibration of the FTM,
several ﬁlms were viewed "edge on" in the Scanning Electron Microscope (SEM) with a
resolution of 10 to 20 nm This method, where the sample is viewed with the incident
electron beam parallel to the substrate plane, allows determination of the average ﬁlm
thickness. To do this, samples were either tilted or the substrate is cleaved through the
sample spot and mounted vertically inside the SEM. Thicknesses measured in this fashion
were found to be within roughly 20% of the indicated thickness using the FTM. This 20%
discrepancy is partly due to the uncertainty in determining the average thickness of a
discontinuous ﬁlm when viewed "edge-on. "

The pumping system on our thermal evaporator consists of a rotary vane
mechanical pump and a diﬁusion pump. The mechanical pump serves two purposes: initial
pumping of the chamber (from atmosphere down to ~5x10—3 mbar) and backing of the
diﬂirsion pump. The diffusion pump is equipped with a "cold trap" which could be ﬁlled
with liquid nitrogen (LNZ) to prevent backstreaming of pump oil into the chamber. Typical
base pressures prior to deposition were in the low to mid 10‘7 mbar range. Often, the
pressure was less than the 10—7 mbar lower limit of the Penning pressure gauge.

Prior to sample deposition, the source was outgassed during the process of setting
the deposition rate. This process of steadying the deposition rate generally took about
ﬁve minutes. After degassing the source, the deposition took place in a vacuum of
5x10—7 to 5x10"6 mbar. The "cold trap" was always ﬁlled prior to, and during, sample
deposition in order to reduce sample contamination from hydrocarbons present in the

pump oil and other compounds in the chamber.

 

 

 

 

 

10

Nearly all of the samples were prepared while mounted on a substrate
holder/heater stage suspended from a rotary feedthrough in the removable lid of the
evaporation system The heatable sample holder will be described below. The rotary
feedthrough enables preparation of multiple samples during one pump down and deposition
cycle. Samples consist of small (N3 mm diameter) spots on substrates to be detailed
further in Section 2.3. Roughly 10 cm directly above the source is located an Al plate
with a small aperture deﬁning the sample pattern. The substrate holder is located about
1 cm above this aperture. Due to the separation of the aperture and the substrate, the
samples have a thickness gradient. The thickness proﬁle is such that the average quoted
thickness occurs at the center of the sample and decreased monotonically to zero along a
radius of the sample spot. This "shadow effect" occurs because different positions on the
substrate are exposed to varying amounts of the source. The advantage of this scheme is
that for most samples all stages of growth were present on the same sample. That is,
different portions of the sample range from zero thickness up to a percolating region near
the center of the spot. In Section 2.5 this useful feature will be exploited.

A diagram of the heatable substrate holder is shown in Figure 2.2. Substrates are
clamped to the smoothed surface of a Cu block which has a resistive heater inside and a Pt
resistance thermometer on the surface. The heater is a 35 W resistive cartridge heater
which is more than adequate to reach the desired sample temperatures. The Cu block,
with a mass of 50 g, may be heated to roughly 675 K with 7 W of power. The block was
designed such that the heater-to-block contact could be tightened prior to every sample
preparation to ensure a good thermal link between the heater and sample stage. The
thermometer, a Pt Thin Film Detector (TFD) [OMEG], was also routinely tightened to the
Cu block's surface by pressure from a stiff stainless steel clip. The sample temperature
was monitored and controlled by a standard temperature controller [LAKE] which allows
temperature control to much better than 1 K Two terminal measurements of the

thermometer's resistance are adequate as the lead resistance corresponds to less than 1 K

 

 

 

 

11

TOP VIEW

  
 
      

2-56 x 1/2"
'ainless steel screws
3 SW resistive
cartridge heater
with leads

§WWV Stainless steel clip

 

Pt thin ﬁlm resistance thermometer and leads

4(-__-

2H

Thermometer
1/2"

Substrate
Old thermometer hole
Stainless steel strips

Figure 2.2 Diagram of heatable Cu substrate stage.

 

 

 

 

12

The sample is mounted to the Cu block by two stiff stainless steel bars which are tightened
against the sample to push it into contact with the heated block. These practices of
ensuring good heater, thermometer, and sample contact to the sample stage were a result
of early problems encountered with the stability and reproducibility of the sample
temperature during and after deposition. The above design was found to yield
reproducible and easily controlled sample temperatures.

The temperature of real concern here is the temperature of the substrate surface
because it governs the interaction of the incident vapor and the substrate. Simple heat
ﬂow calculations indicate that the temperature of the surface of the substrate and the Cu
block should diﬁ‘er by much less than 1 K Experimentally this temperature difference was
found to be a few degrees. This difference was measured by placing two TFD's on the
sample stage. One TFD was placed on the surface of the Cu block while the second TFD
was located atop the substrate, and their resistance's were measured simultaneously during
heating and cooling.

Since the substrate temperature is so important, it is necessary to make sure that
the temperature is stable during sample deposition. A plot of substrate surface
temperature increase due to radiative heating caused by exposure to a heated, empty boat
is shown in Figure 2.3. It can be seen from Figure 2.3 that the incident radiation from the
empty boat has a small effect on the sample temperature. A similar test with metal deposit
incident upon the substrate was also performed. The temperature rise in this case was
roughly the same as for the case of an empty boat in agreement with calculations of the
heating due to the incident metal vapor. Thus, the sample temperature during deposition

was constant to within #5 K

 

 

 

 

 

 

 

 

 

 

 

 

 

l3
1.0 . 1 . I r I F I F 1
A P 0 1
é a. 121908 . ..
0
g - 270 C . .4
D: 6‘- -i
9 I o l
:3 I I A
C3 4" I . ‘ "
cu __ ‘ .
CE). A
<12 2’ ‘
l" - _
0 r J L L r L L I r I

0 300 600 900 1200 1500
Exposure Time (see)

Figure 2.3 Substrate temperature rise due to exposure to a heated, empty evaporation
boat. The diﬁ‘erent traces correspond to different substrate temperatures at time zero as
shown in the legend.

 

 

 

 

 

 

l4

Mounted inside the chamber, near the sample, is a "cold ﬁnger" consisting of a Cu
block connected to a LNZ reservoir. This "cold ﬁnger" is used for cooling samples
following deposition and as a small cryopump near the sample. Because the sample
morphology is very sensitive to the rate of sample cooling following deposition, the "cold
ﬁnger" is used primarily for the former purpose. In early experiments, samples were
allowed to 0001 via radiation, in vacuum, by turning off the heater's power supply. This
resulted in a cooling rate of about 1 K/minute. This slow cooling rate allowed the samples
to anneal as will be discussed further in Chapter 3. To achieve a faster cooling rate, a new
method of cooling was implemented. Immediately after deposition, the hot sample block
was rotated over the LN2 cooled "cold ﬁnger. " Within a few seconds of this, the source
power was terminated, the diﬁusion pump isolated from the chamber, and the chamber
backﬁlled with He2 gas to a pressure of roughly 10“3 mbar. The presence of the Hez gas,
which has a very high thermal conductivity, provides a good thermal link between the hot
Cu block and the "cold ﬁnger." The separation of the sample block and the cooled surface
was roughly 1 cm, which is approximately the mean free path at 10”3 mbar. This simple
scheme increased the cooling rate to roughly 20 K/minute. This increased cooling rate
produced a dramatic change in the sample morphology as will be discussed in Section 3.3.
Once the samples were cooled to near room temperature (about 10 to 20 minutes after the
"quench" began), the chamber was vented with N2 gas and the samples removed. Since
Pb oxidizes rapidly, exposure of the samples to air resulted in the rapid growth of a native
oxide which "freezes in" the morphology and prevents any further evolution of the ﬁlm
morphology. During deposition, the "cold ﬁnger" was cooled with LN2 and located near
the sample to act as a cryopump. Because of this, the chamber pressure fell. However,
the pressure (and hence the contamination rate) near the sample were decreased by an

unknown amount.

 

 

 

 

 

 

15

2.3 SUBSTRATE PREPARATION

The substrates used were single crystal Si wafers with a thin native oxide of
amorphous SiOZ. Substrates were cleaved from p-type Si (100) wafers [ZITI] with the
following characteristics: test grade, resistivity of 14 — 22 Q - cm, and a thickness of
19 — 21 mils. Optical ellipsometry [OHRI92] was performed, in the Michigan State
University Engineering Department, on a bare substrate to determine the amorphous Si02
layer thickness. This thickness was found to be roughly 3.5 —— 4.0 nm Another test piece
was imaged using an Atomic Force Microscope [AFM] in the Center For Electron Optics
at Michigan State University. The rms roughness, according to the AFM image, was
0.1 nm Since we were interested in metal on insulator growth no attempt was made to
remove the amorphous oxide layer. Two identical samples were grown on Si (100) and
Si (1 l 1) with the results (both qualitatively and quantitatively) being indistinguishable.
Thus, it is only the insulating layer which controls the morphology and not the structure of
the underlying semiconducting substrate. For instance, any effects such as strain at the Si
surface are invisible to the deposit due to the presence of the oxide layer. Glass slides
cannot be used as substrates since an electrically conducting substrate is needed to allow
sample viewing with an SEM. An insulating all- glass substrate would become charged
under the electron beam However, the insulating oxide layer on our Si wafers was thin
enough to be transparent to the high energy incident electron beam

The substrates were careﬁrlly cleaned, after cleavage, without removing the oxide,
according to the following procedure:

6 Rinse in acetone

0 Ultrasonic cleaning in acetone for roughly 2 minutes
9 Rinse in methyl alcohol

6 Rinse in de ionized water

0 Soak in hot MICRO [MICRO] for 10 minutes at above 80 °C

 

 

 

 

———

16

0 Rinse repeatedly with de ionized water to remove the MICRO cleaner
9 Dry with N2 gas and optically inspect for signs of debris
6 One of two ﬁnal steps before insertion in the vacuum chamber

a.) Pass through the ﬂame of a Bunsen burner to remove
residual organic contaminants

b) Place in a UV/ ozone cleaner for 10 to 15 minutes
After cleaning, the samples were mounted on the sample stage and placed in the

vacuum chamber as soon as possible. This was generally ﬁve minutes or so after the ﬂame

 

or UV/ ozone cleaning step was completed. The ﬂame cleaning and the UV/ozone
cleaning methods produced ﬁlms which were nearly identical in appearance when viewed
with both the naked eye and the SEM. As it was faster and easier, the ﬂaming method
was usually employed. Once the substrates were in vacuum and the chamber pressure
below about 10‘5 mbar, the heater was turned on and the substrate was heated to near
67 5 K for anywhere from 2 to 12 hours. The Si02 layer is stable to far higher
temperatures. The duration of this bake cycle had no effect upon the resultant ﬁlms.
Thus, it was reduced to roughly 2 hours in the interests of saving time. This was the ﬁnal
substrate cleaning step prior to ﬁlm deposition.

Surface contamination is a very important issue in surface controlled processes.

 

Unfortunately, we have no in situ surface analytical equipment, such as Auger Electron
Spectroscopy or a residual gas analyzer with which to characterize the surface
contamination present on the samples. Our only estimate of contamination is based on the

pressure during deposition and sample appearance from visual and SEM inspection.

2.4 FILM DEPOSITION

Following the substrate preparation procedure outlined in Section 2.3 the sample
temperature is lowered to the desired deposition temperature a couple of hours prior to

ﬁlm deposition. These few hours are needed to allow the heater stage to cool to the

 

—71

l7

deposition temperature. This deposition temperature range was 300 K (room
temperature) to 500 K This highest temperature is 83% of the bulk melting temperature,
601 K, of Pb. After allowing the sample to achieve the desired temperature, setting the
FTM parameters, and degassing of the source (Pb, In, or Sn usually), the sample was
ready to be prepared. By moving a shutter the substrate is exposed to the molten metal
source. The substrate is exposed to the deposit until the desired sample thickness is

obtained. Alter deposition, or alter a post-deposition annealing, the samples were quench

 

cooled as described in Section 2.2. For most samples, the deposition rate was a constant
value between 0.2 nm/s and 20 nm/s and the total thickness was generally 100 to 4000 nm
Thus, the total deposition time was usually between ten seconds and ten minutes for each
sample. During all sample preparations the deposition rate was held nearly constant.
Variations often occurred, but the average deposition rate was usually within about ﬁve
percent of the desired value. A complete sample preparation cycle typically required
between Six and sixteen hours depending up on the time necessary to achieve a satisfactory
vacuum in the evaporation chamber.

Depending upon the thicknesses, up to ﬁve samples could be prepared in one
pump down cycle with control over the temperature and deposition rate of the individual
sample Spots. The size of the substrate and the need to have all the samples on a single

substrate, yet separated from one another, limited the number of samples per run. Before

 

the importance of the sample cooling rate was realized, several samples were prepared on
one substrate at diﬁ‘ering temperatures. This was accomplished by depositing the highest
temperature sample, cooling to the next highest temperature, depositing sample number
two, and so on. While feasible, this method proved unsatisfactory because of the effective
annealing due to the Slow cooling already mentioned. Thus, few of these results will be
mentioned. Once samples began being prepared using the Hez gas "quenching" method,
described in Section 2.2, all the samples prepared in a single pumpdown were prepared at

the same temperature. By holding the deposition temperature constant, the parameter

 

   

 

 

—i—

18

varied in such sample sequences is the annealing time following deposition. The results
obtained from samples prepared in this manner will be discussed in Chapter 3.

Sample thicknesses were chosen so that the center of the ﬁlm was slightly past
percolation and the percolation threshold occurred just off-center of the sample. These
thicknesses, generally found by trial and error, were nominally 100 nm to 150 nm for Pb
on a 300 K substrate and roughly 4000 nm or more for substrates above 475 K Typical
deposition rates for most samples were 5 to 6 nm/s. To achieve these deposition rates,

with the W boats we use, a power of roughly 250 to 500 W of Joule heating in the boat is

 

required.

After cooling the samples to near room temperature, they were removed from the
vacuum chamber. If necessary the substrate could be cleaved to separate the samples
ﬁom one another. In any case, the samples were placed in a desiccator until inserted into
the SEM for observation. Samples were also refrigerated after removal from the
evaporator and] or after viewing with the SEM to lower any time induced morphology
change. With these methods, the appearance of the samples did not change over a period

of months.

2.5 SAlVIPLE OBSERVATION

 

In this section I will give a brief description of the Scanning Electron Microscope

(SEM) used in all sample observations to be discussed. Many factors make the SEM a

 

logical instrument to use in this type of work. Most importantly an SEM allows a wide
range of sizes to be investigated with a single instrument. Our instrument, a Model SX-
40A manufactured by International Scientiﬁc Instruments [ISI], has a resolution of
roughly 6 nm, at best, and the maximum ﬁeld of view is a few millimeters. Equipped with
a motorized stage for sample movement, this SEM can effectively map an area of several

square centimeters. This feature was especially beneﬁcial when viewing multiple samples

 

 

:—

 

 

 

 

19

covering the ﬁrll extent of a single substrate. All such samples could be imaged in one
pumpdown cycle. Typical operating parameters were a beam energy of between 15 and
30 keV, a spatial resolution of roughly 10 nm, specimen current ~10 pA, and a working
distance of 8 to 10 mm. The SEM is pumped by a mechanical pump and a diﬁ‘usion pump.
The diffusion pump was equipped with an LN2 cold trap. The base pressure of the
instrument was in the high 10“6 mbar range.

Usually, observations were made with the plane of the samples perpendicular to
the electron beam This allowed views of the sample along the same direction as the
incident vapor ﬂux, which were necessary to determine the percolation and island size
values desired. Typically, four images were taken at or near the percolation threshold, of
different positions on each sample to obtain "average" information. The location of these
pictures was determined visually on the SEM monitor screen. Samples could also be
rotated about both the polar and azimuthal axes. The polar rotation ability was especially
useful for orienting the substrate plane nearly parallel to the incident electron beam
(perpendicular to the incident vapor ﬂux during deposition). Normally these observations
were done on samples which had been cleaved after deposition (though sometimes before)
to obtain cross-sectional views of the morphology, ﬁlm thicknesses, and
crystallite/ substrate contact angles.

This instrument was adequate for routine sample viewing. However, it severely
restricted quantitative analysis of the samples. As described in the following sections, this

difﬁculty was overcome via computerized image acquisition and analysis.

2.6 DIGITIZED SEM IlVIAGE ACQUISITION

The biggest drawback of older, analog SEM's is that they produce analog images,
with hard copies in the form of black and white Polaroid photographs. While such

pictures are easy to create and store, many disadvantages exist. These photographs limit

 

 

 

———

2O

analysis to being performed by hand. Another option for image analysis, which we
considered until the following method was implemented, was to digitize the Polaroid
photographs with a CCD camera and use commercially available analysis packages. These
Polaroid pictures are also non- erasable and relatively expensive. Once the picture is
exposed, it can‘t be changed. Thus, any errors in contrast or brightness setting, focus, or
astigmatism may result in a useless picture. Our instrument uses Polaroid Type 52 ﬁlm
which costs roughly $1.50 per picture. For these reasons, especially the difﬁculty of
obtaining quantitative information, we implemented a technique to obtain digitized images.

Scanning electron microscopes create images using secondary electrons created in
the interaction of the incident electron beam and the sample. After an electron from the
incident beam strikes the sample, secondary electrons, with energies less than 50 eV, are
emitted ﬁom a small region near the surface of the sample. Inside the SEM chamber is a
secondary electron detector (SED). This detector consists of a ﬂuorescent material
covered by a thin Al ﬁlm held at a positive 10 kV and a photomultiplier tube. Secondary
electrons are attracted to the Al ﬁlm and upon striking the ﬂuorescent material create one
or more photons which are directed down a light pipe onto the photomultiplier tube.
Because the number of emitted secondary electrons is related to the tilt of the sample
surface relative to the incident beam and not the sample height, SEM micrographs

generally lack three dimensional information.

 

The signal from the SED is used to create identical images on two separate
monitor screens. One of these screens is viewed directly by the SEM operator. The
second screen is a short persistence monitor inside the SEM console. When a Polaroid
photograph is taken, it is of this second screen.

To obtain digitized images we intercept the SED voltage signal between the
photomultiplier tube and the photograph screen. This signal is digitized using a 12-bit
A/D board [DATA] in a personal computer and stored on diskette [DUB S92]. Care must

be taken to sample the signal at the correct ﬁequency so that each image line is

 

 

 

 

 

21

synchronized with its neighboring scan lines. Thus, proper record must be kept for the

beginning and end of the image, as well as the start of a new line. A schematic diagram is

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

shown in Figure 2.4.
x-tri er
.gg > Delay > Am
SEM y-trigger \ Board
f
image signal Low-pass
‘1’ V + Filter
Photograph Viewing Computer
Screen Screen

 

 

 

 

 

 

 

 

 

Figure 2.4 Schematic diagram of digitized image acquisition circuit.

When in "photo mode" the SEM produces a 2048 line scan in roughly 70 seconds.
This corresponds to a rate of 33 ms/line. The 12-bitA/D board used can digitize points at
a rate of up to 20 kHz which corresponds to 512 points in 25.6 ms. This is very
convenient given the 33 ms/line scan rate. The A/D board was installed in a 286 class PC,
running DOS with 640 kB of RAM. After loading DOS and the QuickBASIC (the
programming language used) program in RAM, only enough RAM remained to hold
about a 512 (x) by 480 (y) image (at 2 bytes/pixel). This corresponds to an image size of
480 kB. The originally 2048 line scan is cut to 480 lines by ignoring the ﬁrst 128 lines of
the scan and saving the rest of the image in four line bunches. That is, each line in the
recorded image is, in fact, the sum of four (of 1920) scan lines. In the horizontal
direction, a short (~3 ms) delay is placed at the start of every "new line" trigger. This acts
to postpone the recording at the start of each new line. The resulting image is actually a

nearly square portion of the originally rectangular image.

 

 

 

 

 

 

 

22

As illustrated in Figure 2.4, there are two trigger signals which control when the
A/D board is digitizing the SED image signal, which is between —5 V and +5 V. Prior to
being digitized by the A/D board, this signal is ﬁltered by a low pass RC ﬁlter with a
corner frequency of 10 kHz. Because we are sampling at 20 kHz, the sampling theorem
[PRES86] allows a maximum signal frequency of 10 kHz without loss of information.
The y-trigger ﬁrst signals the start of a new frame for recording. It is this signal which
begins the A/D board's digitization after a built in software delay to center the digitized
image relative to that displayed on the viewing monitor. The start of a new scan line is
signaled by the x-trigger. After the y—delay, the software begins summing four line
bundles together before sending this data to the computer's RAM. The y-trigger signal
also terminates the digitization process by providing an end of frame signal. The delay
box in the schematic acts similarly to the software delay mentioned above. To obtain a
digitized image of the center portion of the SEM image a certain amount must be ignored
before recording data for each line. This delay box has a maximum delay of roughly 6 ms.
A 3 ms delay serves to capture the center image portion. Once the end of frame trigger is
received the program is terminated and SEM operation is returned to normal.

Some important computing problems were overcome in the course of developing
this digitization technique. The QuickBASIC programming language running on an
8 MHz 286 class PC is too slow to process data produced by the A/D board at 20 kHz.
Therefore, the code to record the data to RAM was written in Assembly language. Were
the program run on a faster 386 PC or a 486 PC, this might not have been necessary as
these machines (16 MHz to 66 MHz) may be quick enough to write the data directly to
RAM. Another problem with using QuickBASIC is its diﬂiculty manipulating arrays
larger than 64 kB. As the images are 480 kB, this is clearly a problem To overcome this
limitation the image is split into eight segments. The ﬁrst seven of these segments are
64 kB each with the eighth segment being 32 kB. The beneﬁts of this segmenting are

twofold. First, the time required to save the segmented image from RAM to ﬂoppy disk

 

 

 

ll

 

 

 

 

23

was roughly 15 seconds, compared to the nearly three minutes required when the image
was saved as a sequential binary ﬁle. Secondly, the bottom portion of the image could
easily be ignored when performing the image analysis. The image produced by the SEM
contains a region at the bottom which contains image information such as beam
accelerating voltage, image magniﬁcation, scale bar, and photograph number. This
bottom portion of the image must be dropped for analysis purposes.

With images successfully digitized from the SEM and saved on disk, computerized
analysis could be performed. An example of such a digitized image of one of my samples
is shown in Figure 2.5. Figure 2.5 shows a 400 nm Pb ﬁlm evaporated on a 423 K
substrate at 6 nm/s. The analysis programs used to determine various image properties

will be described in the next section.

2.7 IlVIAGE ANALYSIS PROGRAMS

Simple computer programs written in the QuickBASIC programming language
were utilized to extract quantitative information ﬁom the digitized SEM images. The
original 512 (x) by 480 (y) image was reduced to a size of 510 by 382 for analysis
purposes by removing the status bar and a small portion of the image area. The desired
information ﬁom most images were values for the area coverage of metal on the substrate
at percolation and the average width of the connected metal islands. This section will

describe these programs.

2.7.1 MEASUREMENT OF pc

Percolation is the point at which a connected path spans the extent of a sample.
This connected path would allow electrical conduction in the case of a thin metal ﬁhn

The methodology used to obtain images of our samples near the percolation threshold was

 

 

 

 

 

24

 

Figure 2.5 Sample digitized image of a 400 nm Pb ﬁlm on a 423 K substrate using the
digitization technique outlined in Section 2.6.

 

 

  

 

25

described in Sections 2.5 and 2.6. In order to determine if a connected path exists a.
criterion is needed to determine if a section belongs to this connected path.

Images were taken near regions of the sample where the percolation threshold had
just been reached. Since the samples have a thickness gradient, it is possible to ﬁnd
locations on the sample where the scan area changes from not having a connected path of
metal islands to a stage where this path is present. Due to array size limitations, the
analysis programs are restricted on how complicated the image can be, in terms of number
of occupied pixels and perimeter length of the islands. Thus, after some experience I
determined an Optimum magniﬁcation to use on the SEM depending on the size of the
structures being observed Once this magniﬁcation was determined, ﬁnding image
locations was a matter of looking around the sample until a percolating region was
located. The actual determination of whether or not a spot was percolating was
performed visually. To obtain average results, typically images were taken of four
different percolating regions on each sanque.

During the digitization process, pixel intensity values from 0 to 4095 are produced.

A distinction must be made as to which pixels are part of the metal structure. An intensity
histogram of the image enables this distinction to be made. Such a histogram is shown in
Figure 2.6. The original image from which this histogram is produced is shown in Figure
2.5. The horizontal axis is pixel intensity grouped into 512 equal bins while the vertical
axis is the number of pixels falling within the corresponding intensity bin. The striking
feature of this plot is the double peaked structure. This structure is easily understandable
in terms of the substrate (low intensity) and the metal deposit (high intensity). Consider a
checkerboard where all squares are of two different colors. The intensity histogram of
such a structure consists of two peaks, one at the intensity of each color. In the real

images, however, these peaks are spread out due to the range of intensities near each

peak. In addition to black (the substrate) there are light grays, dark grays, etc.... The

horizontal scale is set with intensity bins 0 and 511 determined by the minimum and

 

 

 

 

 

 

 

26

maximum pixel intensities in the image. Using this histogram, it is easy to set a criterion
for separating substrate ﬁom deposit. This is conveniently done by setting a threshold at
the minimum intensity bin, as indicated by the arrow in Figure 2.6, lying between the two
peaks. This threshold is used to convert the image into a binary image. Such a binary
image is shown in Figure 2.7 which is created from the original SEM image in Figure 2.5.
Pixels with intensity larger than the threshold setting are automatically set to a value of
one (deposit) while those pixels with intensities below the threshold setting are set to zero
(substrate). In Figure 2.7 black sites represent the deposit. The resulting two color image
is used for subsequent analysis.

Before ﬁrrther analysis, single pixel noise in the images is eliminated. If a pixel is
assigned value zero (unoccupied) in the binary image but has four nearest neighbors which
are all value one (occupied), this lone pixel is changed to be occupied. The opposite of
this is handled as well.

A simple counting of occupied pixels is then used to determine the area coverage
pc. The coverage at percolation, pc, is deﬁned as the number of occupied pixels divided
by the total number of pixels in the image. A quoted value of pC for a given sample is the
average coverage determined from two to four different images of the same sample, taken

at various positions on the sample.

2.7.2 DETERNIINATION OF ISLAND WIDTH Rc

We deﬁne R,: as the average half width of the metal islands near the percolation
threshold. This dimension is close to the radius of compact islands at the full to partial
coalescence crossover. More importantly, this is a consistent, reproducible criterion for
RC. With the binary image resulting ﬁom the determination of pc this quantity is easy to
measure. The determination of Rc relies on the use of the well-known "forest ﬁre" or

"tree burning" algorithm

 

 

 

 

 

 

27

 

 

 

 

 

1 400 _r u 1 ' r u 1 T r 1‘
1200 — ,v --
P '. . L 'l
1000 - 5" ‘1, a
n: ’ Binary 5’ '1. J
NJ 800 *- Threshold .1
[I] L .° ‘, 4
E 600 _ .. ‘1. a
a L :2... I... 1. 'l
400 r- . 1.5 f: .4
L. : this. v ‘f‘: 1
2 O 0 — : ° «sz’V-r"? '2 "
0 L3; m I L l n 1 L L “KN-T.—
O 1 00 200 300 400 500
INTENSITY
Figure 2.6 Intensity histogram for the Pb sample shown in Figure 2.5. The arrow

determines the threshold for creating a binary image.

 

 

 

 

28

 

Figure 2.7 Binary image of the top 384 lines of Figure 2.5 using the intensity histogram
shown in Figure 2.6.

 

 

 

 

29

Beginning with the binary image the perimeter of the connected metal structures
must be found. This is done by checking every occupied pixel in the image for the
following condition: are there one, two, or three occupied nearest neighbors? If this
condition is true, the selected site is a perimeter site and its pixel value changed to three.
Having one occupied nearest neighbor assures the site belongs to a cluster, while it must
have less than four occupied neighbors to ensure the site is located on an island perimeter.
This step results in selection of all the sites lying along the cluster perimeters.

The R,2 determination is based upon counting the number of cycles needed for the
perimeters to "burn" the metal islands. After setting the perimeter on ﬁre, as described in
the previous paragraph, a cycle consists of the following sequence. Pixels are labeled as
unoccupied (pixel value 0), occupied and "unbumt" (value 1), "burning" (value 3) or
"burnt" (value 0). In each cycle of the ﬁre, pixels which are burning become burnt, and
unbumt pixels become burning. This cycle continues until all unbumt sites become burnt.
Since, ﬁom our determination of pc, we know the total number of occupied sites to begin
with, we merely need monitor the number of these sites which are burnt during each
sub sequent cycle. From this information we may form a consistent deﬁnition of RC. A
plot of the percentage of burned sites versus cycle number is shown in Figure 2.8. The
conversion from cycle number to physical distance in the real ﬁlm is accomplished by
knowing the magniﬁcation of the original image. This information is speciﬁed in the status
bar area which is created with every SEM image. This relation between pixel distance and
physical distance is: 1 pixel = [170.8/(image magniﬁcation)] um The condition used to
deﬁne Rc is when 95% of the occupied sites have been burned by the forest ﬁre as
indicated by the intersection of the two solid lines shown in Figure 2.8. The interpolated

cycle number at this point is converted to a physical distance and set equal to RC.

 

 

 

 

 

 

 

 

30

1.0 .3 ” ' T‘ r‘ 'J‘lllllﬁliiltii‘
a ’ " “
;I_-_' 0.8 " . -

O
E 06 r . Island Wldth _
3 ' .
m r q
q_ o
O 0.4 r— . I
C
.9 ’ - ‘
IE; 0.2 " o _.
E
r. d

U.

0.0h-JLLLlrmherr LLLIIIIIIIIIIILL_

 

 

 

 

0 5 10 15 20 25 30
Burn Cycle Number

 

Figure 2.8 Percent of sites burned during the forest ﬁre algorithm vs. cycle number for the
binary image shown in Figure 2.7.

 

 

 

 

 

31

2.7 .3 COD/IPRESSION PROGRAMS

Although the raw binary image ﬁles are 480 kB, this size may be reduced by a
factor of roughly six, by saving the image in a different ﬁle format, for archival purposes.
Once the image is saved to ﬂoppy disk it may be easily read into various QuickBASIC
programs. Once the image is displayed on a VGA color monitor, commercial programs
called GRAB SCRN and RAW2GIF [GRAB] are used to convert the screen image into a
GIF ﬁle. GIF refers to a standard image ﬁle format. The GRABSCRN program
effectively takes a raw "snap shot" of the computer monitor (actually the video memory).
This raw monitor image information is converted to the GIF format using the RAW2GIF
program Once in this standard format, the image can be read into a desktop publishing
program (we use Publisher's Paintbrush [PUBP]). Such a program has image editing
capabilities as well as the ability to print high quality images on a compatible printer. In
our case, the printer used was a Hewlett-Packard [HEWL] HP4 laser printer. This
software/printer combination produces 600 dpi publication quality images.

Using these methods the original SEM images are obtained in hard copy form and
as a roughly 80 kB ﬁle. For archival purposes and later analysis, the original image ﬁles
are also saved to ﬂ0ppy disk. This is because, without knowing the compression format
used by GIF ﬁles, the original images cannot be fully reconstructed from the compressed
ﬁles. Even if the GIF format were known, some image information might be lost during

image reconstruction.

2.8 SUNﬂVIARY

In this chapter I have discussed all aspects of sample preparation relevant to the
results to be presented and discussed in Chapter 3 of this thesis. Although the

experimental techniques used are not overly complicated or difﬁcult, signiﬁcant problems

 

 

 

 

32

were encountered. The most important of these problems were: rapid sample cooling,
obtaining quantitative information from the samples, and production of high quality image
output for both storage and hard copy production. These obstacles required a non—trivial
amount of time and effort to overcome.

The importance of sample annealing due to slow sample cooling is one which
should have been appreciated earlier in the course of this research. However, once this
effect was realized, a very simple solution was instituted. The sample was "quench
cooled" at a selected time after deposition was completed. The simple method used
involved heater power termination, vacuum chamber isolation, and the high thermal
conductivity of He2 gas at a pressure of roughly 10"3 mbar. This technique is described
in Section 2.2 and its results will be discussed in Chapter 3. This improvement in sample
preparation produced reliable and reproducible results.

Overcoming the difﬁculties inherent in the Polaroid photographs produced by our
SEM was a major advancement for the purposes of this research. As described in Section
2.6 the technique employed was direct digitization of the secondary electron detector
signal and recording of this output to computer. Not only did this method allow
subsequent computerized analysis, as outlined in Section 2.7, but the throughput of image
production increased. Images were created, quantitatively analyzed, and printed in a
matter of minutes. The beauty of this arrangement is that all of the components used were
either already present or produced here in our laboratory. Qiﬁr Zhu handled perhaps the
most difﬁcult portion of implementing this technique -- coding of the Assembly language
program needed to record data quickly enough.

Some of the experimental diﬂiculties present in this research will be further
discussed in Chapter 4. Chapter 4 will deal with some of our planned remedies to various
problems encountered as well as future directions of this research which could be pursued.

Since the stage of ﬁlm growth studied in this research has received little attention

previously, the goal was to obtain a qualitative understanding of the physics controlling

 

 

 

 

33

the coalescence and percolation of ﬁlm growth. As the results of Chapter 3 will show, this

goal was achieved.

 

 

 

 

 

——f

 

34

CHAPTER 2 REFERENCES

AFM The AFM used was a Nanoscope III Scanning Probe Microscope by
Digital Instruments, 520 Montecito St., Santa Barbara, CA 93 103.

DATA Data Translation A/D-D/A board, model DT-2811-PGH.

DUBSQZ M. A. Dubson and Q. Zhu, Rev. Sci. Instrum 63, 4461 (1992).

EDWA Edwards High Vacuum, Manor Royal, Crawley,
West Sussex RH10 2LW, England.

GRAB GRAB SCRN and RAW2GIF are shareware programs.

HANSS8 M. Hansen and K Anderko, Constitution of Binary Alloys,
(McGraw-Hill, New York, 1958), p. 1117.

HEWL Hewlett-Packard Company, 11311 Chinden Boulevard, Boise, ID, 83714,
(208) 323-6000.

181 International Scientiﬁc Instruments, Inc., now Topcon Technologies,
6940 Koll Center Parkway, Pleasanton, CA 94566, (510) 462-2212.

LAKE Temp erature controller Model DRC 91C from Lake Shore Cryotronics, Inc.,
4 E. Walnut Street, Westerville, OH 43081, (614) 891-2243.

MATH Evaporation boats obtained ﬁom R D. Mathis, 2840 Gundry Ave,
Long Beach, CA 90806, (310) 426-7049.

MICRO MICRO is a cleaning solution available from International Products
Corporation, P. O. Box 70, Burlington, NJ 08016, (609) 386-8770.
A typical mixture is 5% MICRO in deionized water, heated to about 90 °C.

OHRI92 M. Ohring, The Materials Science of Thin Films,
(Academic Press, San Diego, 1992).

OMEG Pt thin ﬁlm resistance thermometer ﬁom OMEGA Engineering, Inc.,
P. 0. Box 4047, Stamford, CT 06907-0047, (800) 826-6342.

PRES86 W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,
Numerical Recipes: The Art of Scientiﬁc Computing, (Cambridge University
Press, Cambridge, 1986).

PUBP Publisher's Paintbrush by ZSoft Corporation, 450 Franklin Road, Suite 100,
Marietta, GA 30067 (404) 428-0008.

SLOA Model DEKTAK 11A from Sloan Technology Corporation, P. O. Box 4608,

602 East Montecito St., Santa Barbara, CA 93103, (805) 963-4431.

 

 

 

 

 

‘l

 

 

————

35

ZITI Si wafers obtained from Ziti Incorporated, 97 0-A University Ave., Los
Gatos, CA 95030, (408) 395-2115). This company is now known as
Silicon Quest International, 2904 Scott Boulevard, Santa Clara, CA 95054,

(408) 496-1000.

 

 

 

 

 

‘1

 

 

 

 

CHAPTER 3

THE ISLAND - TO - PERCOLATION TRANSITION
DURING GROWTH OF METAL FILMS
Experimental observations of Pb on Si02 film growth

3.1 INTRODUCTION

Thin ﬁlm growth is, in most cases, a poorly understood process. One such
important case is that of metal ﬁlms deposited by physical vapor deposition on insulating
substrates. It is this situation which forms the majority of this chapter's contents. In this
chapter I will summarize my experiments concerning the growth of metal ﬁlms on
insulating substrates as well as some of the many computer simulations which motivated
this research.

The results presented here are for Pb ﬁlms thermally evaporated onto clean SiOz
substrates at different substrate temperatures and deposition rates. The two major effects
studied are: l) the very high area coverage which these metal ﬁlms exhibit at the
percolation threshold, and 2) the crossover from full coalescence of smaller crystalline
islands to partial coalescence of larger islands. These two features and the general
sequence of events during ﬁlm growth are discussed in Section 3.2.

Qualitative descriptions of both these features are presented and compared to
experimental results. These two effects are not independent, however. It is shown, for
instance, that the large area coverage is a direct result of the existence of the crossover in
coalescence behavior. This is supported by computer simulations of an Interrupted
Coalescence Model (ICM) described in Section 3.3. Section 3.4 contains information
regarding the ﬁlm morphologies, percolation coverages, and ﬁrll to partial coalescence

crossover sizes. A Kinetic Freezing Model (KFM) is presented in Section 3.5 and its

36

 

 

 

 

 

 

 

 

37

predictions are compared in Section 3.6 to my experimental results. The KFM views the
crossover as being due to a competition between droplet growth (due to adsorption ﬁom
the vapor) and the coalescence of two touching droplets. The differences between the
KFM‘s predictions and the experimental results, as well as some weaknesses of the KFM,
are discussed in Section 3.7 in the context of what features are missing from the KFM.

Section 3.8 is a brief summary of our ﬁndings.

3.2 FILM GROWTH SEQUENCE AND FEATURES

Metals deposited on insulating substrates do not immediately form a continuous
monolayer ﬁlm Instead, the ﬁlm often grows in what is known as the Volmer-Weber
growth mode [LEWI78, PASH64, 65]. This growth mode is characterized by the
formation of isolated three- dimensional structures (here termed "islands") which

eventually touch as illustrated in Figure 3.1.

 

// // //// /////.3

—-
///

//,

Deposit
\\

— 1

Figure 3.1 Schematic illustration of three- dimensional Volmer-Weber island growth.

This growth process, as all others, proceeds ﬁom an early stage of nucleation.
Here, small, stable clusters of material are formed which act as preferential growth sites on

the substrate. As the average ﬁlm thickness increases, these structures accumulate atoms

 

 

 

 

 

—f—

38

and increase in height and width. The island size and density increase until they begin
touching nearby islands. When these islands come into contact they undergo either full or
partial coalescence, resulting in an increase in the average island size and a decrease in the
total number of islands.

Full coalescence is the process whereby two compact islands touch and grow
together to form a larger, but still compact, island. An example of this process is the
touching of two water droplets. While the material volume is conserved, the surface area
changes in order to minimize the total surface energy. For the case of liquid droplets,
where the surface energy is isotropic, on a substrate, the shape which minimizes the
surface energy is the cap of a sphere with the angle of contact between the droplet and the

substrate given by Young's equation [OI-IRI92]. Non-liquid islands behave similarly

 

except that in a crystal system the surface energy is anisotr0pic and the equilibrium shaped
islands contain facets.

Under the experimental conditions studied here, coalescence of solid metal islands
is predominantly due to surface diffusion which is driven by gradients in the surface
curvature (or, equivalently the chemical potential) [PASH66] of the material. Other
methods of mass ﬂow exist such as volume diffusion, evap oration-condensation, and
viscous ﬂow. Volume difﬁrsion and evaporation-condensation are negligible until close to

the melting point. Calculations [HERRS 1] indicate, that for particle sizes larger than

 

roughly 50 nm, the surface stress is much smaller than the yield stress needed to cause
viscous ﬂow of dislocations. For the case of two touching islands, the curvature gradient
is strongest at the connection point of the two islands so that material is drawn into this
"neck region." Intuitively, this can be understood in terms of where a wandering atom
would feel the most binding. This location is at the bottom of a groove.

One important feature of the full coalescence process is wiping. Wiping refers to
the fact that a portion of the substrate is wiped clean during the coalescence event.

Regions of the substrate which were originally covered by one of the two dr0plets are left

 

 

 

 

———

39

bare following coalescence. If deposition is taking place, then this clean substrate now
provides a location in which new islands may nucleate. Thus, this cycle of growth and
coalescence is a recurring event. Wiping always creates empty regions with a size
comparable to the radius of the initial islands involved in the coalescence event.

The coalescence and wiping process is shown in Figure 3.2. Two droplets which
touch quickly grow together, by formation of a neck, to form a cigar shaped island. This
partial coalescence stage then slowly proceeds on to full coalescence as shown at the right
in Figure 3.2. The shaded regions in the full coalescence portion of Figure 3.2 indicate the

regions of the substrate which were initially covered by the two touching droplets.

”—e FAST C D—‘a‘ SLOW /

Touching droplets Partial coalescence Full coalescence

 

 

 

Figure 3.2 Coalescence and wiping process.

This full coalescence process does not continue until the ﬁlm is hole-free. Instead,

at some point during grth the islands fail to fully coalesce once they are touching. In

 

this case, typically only the partial coalescence stage shown in Figure 3.2 is completed.

An effect of this crossover from full to partial coalescence is illustrated by the two
images in Figure 3.3. These two images are of the same sample which is roughly 50 nm of
Pb on oxidized Si (100) at room temperature. However, as mentioned in Chapter 2 all of
our samples have a thickness gradient so that the right side of Figure 3.3 (a) is thinner than
the left side. By how much the average thickness differs in these two regions is unknown.
The point of this image is to illustrate, qualitatively, the stages of ﬁlm growth ﬁom

compact islands to a percolating structure. The right side of Figure 3.3 (a) shows a large

 

 

 

 

a?

rake

 

 

Figure 3.3 Growth evolution of Pb on insulating Si02 substrates. (a) Compact islands to
elongated structures (from thin (right) to thicker (left) deposits). (b) Percolation threshold
which occurs at a higher ﬁlm thickness than in (a), with pc m 63%.

 

 

 

 

 

41

number of small islands with nearly uniform sizes and shapes. Actually, there exist many
smaller islands between these which are unresolved in the micrograph. This stage of
growth is known as the "Breath Figure" state in analogy to the patterns resulting from the
condensation of warm water vapor on a cold surface [BEYS86, 91]. This structure is
produced by many generations of complete coalescence and wiping. As the thickness
increases (to the left in Figure 3.3 (a)) full coalescence of islands ceases. This results in
the elongated "wormy" structure becoming apparent in the left side of the image. With
their larger size, these islands quickly join together with their neighbors and cause the ﬁlm
to percolate as shown in Figure 3.3 (b) which was taken further to the left of the region in
Figure 3.3 (a).

Two important features are present in Figure 3.3. The ﬁrst is the area coverage of
pC z 63%, at the percolation threshold for the image in Figure 3.3 (b). This value of pC is
in the range of that predicted by various random percolation models and, as such, is not
particularly high. Samples will be discussed which have a much larger value of pc as they
were prepared under different conditions. As this is what we ﬁrst noticed and interested
us in ﬁlm growth it will be discussed ﬁrst in this chapter. The second feature is the
crossover from full to partial coalescence of the islands. This transition is interesting since
it determines the characteristic length scale of the percolating ﬁlm Knowledge of the
factors controlling this crossover would be useﬁil in advancing our understanding of ﬁlm
growth. The crossover will be discussed in the latter portions of this chapter. Neither of
these issues in ﬁlm growth has been adequately addressed.

Many authors have studied the geometrical and transport exponents of real metal
ﬁlms and compared them to the predictions of percolation models [STAU85]. These
exponents include those governing the correlation length, the ﬁactal dimension, and the
electrical conductivity. Measurement of these exponents by ﬁlm morphology analysis
yields values which are in agreement with those found in two- dimensional pure percolation

models [KAP182, LAIB82, VOSS82]. While these experimental results show agreement

 

 

 

—’—

42

with computer simulations, in terms of quantitative exponents, they fail, on the other hand,
to exhibit certain characteristics of real ﬁlms. In particular, they do not exhibit important
features such as wiping, ﬁrll coalescence, and a high value of pc.

Extensive computer simulations have been performed in our laboratory over the
last number of years in efforts to develop simulations which accurately mimic the growth
of real metal ﬁlms. . The majority of this work was aimed at determining the scaling
exponents of various models of growth [WOLL93]. Simulations incorporating attractive
nearest-neighbor interactions with different deposition and post-deposition annealing

scenarios were studied. One of the main results of these studies is that the percolation

 

coverage never exceeded roughly 60% for a wide class of simulations. This is surprising
since pure percolation models also have pc in the 60 to 70% range. As stated earlier, the
geometric and transport exponents agreed very well with simple two-dimensional pure
percolation models with no interactions.

A simple model was developed which shows that the high values of pC are a direct
result of the crossover from full to partial coalescence [YU91]. This model is described in

the following section.

3.3 INTERRUPTED COALESCEN CE MODEL

 

In this section we argue that the high value of pc observed in real ﬁlms (higher than
that predicted by random percolation models) arises naturally from the existence of the ﬁll

to partial coalescence crossover. ‘ A general model of ﬁlm growth involving only

 

immediate, full coalescence was developed by Family and Meakin [FAB/1188a, 88b, 89]. I
will refer to this as the Family and Meakin Model (FMM). Hemispherical droplets of
radius R0 are deposited at random on a two—dimensional substrate. If two droplets, of
radii R1 and R2 touch they instantaneously coalesce with conservation of volume and

center of mass according to the rule,

 

 

 

 

43

(3.1) R3 =Rf+Rg
which results in a single droplet of radius K This model predicts a bimodal droplet size
distribution which agrees with that observed by Carr et al [CARR] for the size distribution
of Sn droplets thermally deposited onto a heated sapphire substrate. This model is
applicable to the early stages of ﬁlm growth when the "Breath Figure" structure is still
present as shown in Figure 3.4. However, there is a serious feature absent from this
model. The PM has no percolation transition. Since coalescence is always present,
percolation will only occur when one large droplet spans the entire sample surface.

It is for this reason that Duxbury and Yu developed the Interrupted Coalescence
Model (ICM) [YU91] as an extension of the FMM. As the name implies there is an
interruption in the coalescence behavior of two droplets. In this model individual droplets
of radius R0 are still deposited at random on a substrate. Imagine two touching droplets
of radii R1 and R2. If R1 and/or R2 < R,2 (Rc is a parameter to be described shortly) then
coalescence takes place according to the rule expressed in equation (3.1). However,
should R1 and R2 be larger than Rc then there is no coalescence. Thus, RC corresponds to
the critical droplet radius for the transition ﬁom full to partial coalescence. The pure
percolation case corresponds to Rc/RO < 1. Indeed, for Mo < 1 the ICM predicts that
pC = 0.676 1- 0.002 which is in agreement with the predicted result [GAWL81]. The radii
R1, R2, R2, and the system size L may be expressed in terms of the unit length R0.

A sample morphology produced by the ICM for the case RC/RO = 8 is shown in
Figure 3.5. The major difference between Figure 3.5 and the image in Figure 3.3 (a)
occurs in the neck region where two droplets touch. In real ﬁlms these indentations ﬁll in
very quickly as they are regions of high binding energy and curvature gradient. Also, real
ﬁlms and the ICM exhibit wiped regions near the perimeters of the larger droplets. The
simulation in Figure 3.5 is at the percolation threshold and has a p0 of roughly 82%. Only
droplets of size R > RC lie along the percolation path (the inﬁnite cluster).

 

 

 

44

 

Figure 3.4 "Breath Figure" stage which occurs in the early stages of metal ﬁlm growth.
This image is of a Pb ﬁlm on a 483 K Si02 substrate.

 

 

 

 

Figure 3.5 Percolation pattern resulting for 3D droplets on a 2D substrate using the ICM
with L/RO = 200 and Rc/RO = 8. The coverage is 82%.

 

 

 

 

 

 

 

46

Since only the large droplets lie along the percolation path we can understand the
relatively high value of pc obtained using this model. In the "Breath Figure" morphology,
the large droplets appear to act as repelling islands -- they tend to be as far apart as
possible. This apparent repulsion is not due to any direct energetic interactions. It is
caused by many generations of coalescence and wiping which leave gaps, between
droplets, which are comp arable in size to the droplet radius. As an example of the high
values of pc, consider a square lattice with identical disks at each vertex. If we now
simultaneously increase the radius of all these disks by the same amount, eventually each
disk will come into contact with its neighbors. At this point the coverage will be
pC = 75/4 2: 79%. If we use a triangular lattice, the most densely packed structure,
instead, the disks will touch when pc N 91%. Since these simulations are performed on a
continuum substrate and have droplets of varying sizes, a quantitative prediction of pC is
not possible. However, the large droplets still behave as a system of repelling objects
causing pC to increase.

For the case of three-dimensional droplets on a two-dimensional substrate the ICM
predicts that pc saturates at roughly 80% for L/RO = 66 and Rc/RO 2 3 when averaged
over 100 trials. This length scale is much different than that of real ﬁlms. Real ﬁlms
exhibit the crossover from full to partial coalescence at sizes ~ 300R0 with R0 an atomic
radius. Were Rc equal to roughly 300R0 the simulations would require an intolerable
amount of time to complete. The pattern shown in Figure 3.5 required a few hours of
CPU time on a Convex mainframe computer. Thus, although quantitatively inaccurate,
the ICM provides a simple intuitive explanation of the experimentally observed high pc

based on the observed existence of a crossover from full to partial coalescence in actual

ﬁlms.

 

 

 

 

 

 

47

3.4 F EM MORPHOLOGY AND PERCOLATION

As mentioned in Section 3.2 real ﬁlms exhibit larger values of pC than predicted by
various computer simulations. An example of this is shown in Figure 3.6 which shows a
500 nm thick Pb ﬁlm, deposited at a rate of 6 nm/s, onto a 453 K amorphous Si02
substrate. This sample was "quench cooled" within a few seconds after completion of the
deposition. This particular sample has p0 = 83.9 i 0.3%. This ﬁlm consists of metal
"wormy" structures which are of nearly uniform height separated by long, narrow,
intertwining channels. These channels are populated by many small islands which are not
resolved in this micrograph. Additionally, the islands app ear to repelling their neighbors
on all sides. It is the long channels and the apparent "repulsion" which result in the values
of pc being so high. Similar growth behavior and ﬁlm structure has been observed for Ag
ﬁlms on Mica [BASK94] where networked island morphologies with coverages of roughly
66% to 89% were found.

The same general appearance shown in Figure 3.6 occurs near percolation for
substrate temperatures from room temperature up to about To == 460 K = 0-77Tmelting-
The major difference between samples prepared in this temperature range is the length
scale. Lower temperature samples require a smaller thickness to reach percolation and the
width of the metal islands at the percolation point is also less than for higher substrate
temperatures. Figure 3.7 shows a plot of pc versus substrate temperature for samples
deposited at 6 -l_: 0.3 mn/s. It is clear from this ﬁgure that there is no overall trend to pC
with temperature. Individual points have uncertainties which are roughly the same size as
the symbols and were averaged over two to ﬁve images taken of each sample near the
percolation threshold. The anomalously large values of pC observed for samples prepared

with Tsubstrate Z 460 K is, as we shall see, a result of a signiﬁcant difference in the sample

morphology.

 

 

 

 

 

48

 

Figure 3.6 Percolation pattern of a 500 nm thick Pb ﬁlm deposited at 6 nm/s onto a 453 K
Si02 substrate. This ﬁlm was quenched immediately following deposition. The coverage
is m84% and the average metal island width is 1 um

 

 

 

 

 

 

 

49

1.00 1 r I r I r 1 r I l I 1 I T r l 1
0.96 *- . ..
0.92 F- "‘

0
Q l- at
0.88 - ‘ 4
O C
l- . . .4
. Q

0.84 _ 0 . _
- '1

080 L 1 I L L L L L I r J I J 4 I 4 I

 

300 325 350 375 400 425 450 475 500
Substrate Temperature (K)

 

Figure 3.7 Area coverage at percolation, pc, as a function of substrate temperature for Pb
ﬁlms deposited at 6 nm/s.

 

 

 

 

 

 

50

An example of the difference in morphology above and below To is seen by
comparing Figure 3.6 to Figure 3.8. Figure 3.8 is a 1600 nm thick Pb ﬁlm evaporated at a
rate of 6 nm/s onto a substrate held at 463 K This image contains regions which appear
similar to both the early "Breath Figure" stage as well as other portions which more
resemble the percolation structure shown in Figure 3.6. It is important to notice the scale
difference between Figures 3.6 and 3.8. This complicated morphology complicates
determination of the percolation point. For this reason, values of pC and Rc for samples
with T > T0 are very uncertain.

There have been numerous studies on the equilibrium crystal shape (EC S) of Pb
crystallites [HEYR83, 87, 89, METO82a, 82b, 89a, 89b] as well as of Au and In crystals.
There is a difference in the resultant crystal shape depending on the starting shape of the
crystallites. Pb condensed on a room temperature substrate and then annealed results in
ECS‘s which are cubo-octahedra with a contact angle between the substrate and the
crystallite of roughly 110°. On the other hand, Pb condensed onto graphite substrates at
473 K form ﬂat. triangularly shaped, tabular crystallites which require hundreds of hours
of annealing before showing substantial morphological change. This symmetric shape
results ﬁom the Pb crystallites being epitaxially oriented with their {111} faces parallel to
the {001} planes of the substrate.

For substrate temperatures below To my ﬁhns show a ﬂat, tabular island shape
with an average height-to-radius ratio near 1.0 and a contact angle 0C w 135° to be
present during the early stages of ﬁlm grth and through the percolation crossover.
However, for T > To droplets form spherical caps with a height-to-radius average of 1.7
and 0c m 145° which are present until near the percolation threshold. These structures
resemble the "Breath Figure" shown in Figure 3.4. Near percolation, the ﬁhn structure
contains regions of "near-equilibrium shaped" islands as well as areas of connected metal

islands.

 

 

 

 

ll

 

 

 

51

 

Figure 3.8 Film morphology of a 1600 nm Pb ﬁlm prepared at T = 463 K > To. The
structure is discussed in the text.

 

 

 

52

Examples of this dual morphology are shown in Figures 3.8 and 3.9. Figure 3.9
shows a. 3470 nm Pb ﬁhn deposited at 6 nm/s on a 498 K Si02 substrate. The large
faceted droplets appear quite similar to the cubo-octahedral ECS of Pb. These large
crystals are not all of uniform orientation since the substrate is amorphous. However,
x-ray scattering measurements of a few of my samples indicates that the (111) face is
generally parallel to the substrate. This is expected with the growth of face centered cubic
materials, such as Pb, since the (111) face has the lowest surface energy.

An interesting aspect of Figure 3.9 is that most of the larger crystals are extremely
faceted. Although some of these crystallites resemble the ECS it is not possible, without
more orientation information, to claim whether or not these are ECS crystals. We don't
believe these are ECS crystals. The cause of the faceting is becauSe these crystals have
not coalesced recently. Thus, they have had time to begin approaching the ECS but not
enough time to reach the ECS. Other dr0p1ets of similar size appear to be spherical as
they have recently coalesced with a. nearby droplet, as evidenced by the denuded zone near
their perimeters. This combined with continuing deposition results in the very faceted
crystals shown in Figure 3.9. It has been shown that crystals approach their ECS more
quickly if incident vapor is still present due to the constant ﬂux of material. This trend has
been observed in my samples since the tops (exposed to the incident vapor) of crystallites
appear more faceted than do the lower portions (which are effectively shielded from the
vapor). Metois and Heyraud [METO82a] found that on the order of four hours is needed
to anneal roughly 5 um Pb crystallites to their cubo-octahedral ECS. The deposition time
for the sample in Figure 3.9 was 9.7 minutes. Thus, we should not, perhaps, even expect
the crystallites to resemble the ECS. Also, if these crystals are approaching the ECS,
smaller crystallites should more closely resemble the ECS which they do not.
Additionally, contamination is known to strongly inﬂuence surface diffusion [MET082b].
The studies of Heyraud and Metois were performed under UHV conditions. In particular,

the presence of 02 was found to signiﬁcantly increase the surface diffusion of Pb. On the

 

 

 

 

53

38kt)

 

Figure 3.9 A ﬁlm prepared with T = 498 K which is well above To. The ﬁlm is 3470 nm
thick with a deposition rate of 6 nm/s.

 

 

 

54

other hand, my samples are prepared in a vacuum of roughly 10“7 to 5x10‘6 mbar. The
samples of the above authors were also prepared in a diﬁerent method: deposition at room
temperature followed by an annealing process (instead of observing as-grown ﬁlms) onto a
very different substrate. Thus, the faceted crystallites in Figure 3.9 are not ECS crystals,
but are a result of the growth conditions.

As mentioned in Chapter 2 the sample cooling rate following deposition was found
to strongly inﬂuence the resultant ﬁlm morphology. This effect is illustrated by the four
images shown in Figure 3.10. This sequence of images shows samples which are identical
in thickness, deposition rate, and substrate temperature, but with various annealing times,
tmeal. Each sample is 400 nm of Pb evaporated onto a 423 K substrate. Immediately
following deposition, but prior to quench cooling at roughly 20 K/minute, each sample
was allowed to sit for a) 0, b) 1.2, c) 2.3, and d) 4.5 minutes at the deposition
temperature. The deposition time, tc, of each of these samples was roughly 1.1 minutes
so that these annealing times correspond to 0, l, 2, and 4 times the deposition time. The
four images shown are of four different samples prepared successively in one pumpdown
cycle. Each of these images was taken at the center of the sample spot. Similar behavior
is seen at the percolation thresholds of each sample. The height of the islands in the later
images are probably thicker than those in the earlier images if we imagine each image
evolving from the structure of the previous sample. This is because to decrease their
lateral extent, as is evident in Figure 3.10, the islands must grow in the vertical direction.
Were this annealing allowed to continue the islands would eventually break up into
isolated droplets which would then approach the ﬁnal ECS.

From Figure 3.10 it is clear that pc is strongly effected by annealing. The image in
Figure 3.10 (d) is reminiscent of the morphology observed in early samples which were
cooled slowly. During annealing, surface diffusion and coalescence events continue to

occur. Thus, slow cooling and annealing produce similar results.

 

 

 

 

 

 

 

Figure 3.10 Sequence of images illustrating the effects of annealing on the ﬁhn's ﬁnal
morphology and percolation coverage. Each ﬁlm is 400 nm of Pb deposited at 6 nm/s
onto a 423 K substrate. Annealing times are (a) 0 and (b) 1.2 minutes.

 

 

 

 

Figure 3.10 (cont) Images with annealing times of (c) 2.3 and (d) 4.5 minutes. The
deposition time for each sample was 1.1 minutes.

 

 

 

57

The percolation coverages of the four samples in Figure 3.10 (but taken at the
percolation threshold), and others, at a substrate temperature of 423 K, are shown in
Figure 3.11 as a plot of pC vs. the ratio tanneal/tc where tanned is the annealing time
following deposition and tc is the deposition time to reach the percolation threshold. The
three different symbols in Figure 3.11 correspond to samples prepared at different
deposition rates. In addition, each of these different deposition rates requires a different
thickness to reach percolation. This will be discussed further in a later section of this
chapter. Higher deposition rate samples require a lower thickness to reach percolation.
Thus, although the incident vapor ﬂux is larger, the total amount of deposit is lower.
However, pc turns out to be fairly independent of deposition rate. Since samples receive
no incident vapor ﬂux during annealing it is not surprising that samples with different
deposition rates show a common behavior in how pc drops as tanneal/tc increases.

The percolation threshold is a result of the existence of the crossover from full to
partial coalescence of touching islands. The important length scale at which this crossover
occurs is denoted by RC. Islands having a radius smaller than RC will undergo full
coalescence whereas islands larger than Rc only complete the partial coalescence stage.
Thus, this size should be given strictly by the radius of the largest droplets in "Breath
Figure" images such as that shown in Figure 3.4. We decided, however, to determine Rc
using the ﬁlm morphology at the percolation threshold since using "Breath Figure" images
could be ambiguous. The reasons for this decision were many. First, the half width of the
connected structures of deposit material at percolation is not signiﬁcantly different from
the radius of the largest droplets in the "Breath Figure" stage. Secondly, since we were
already interested in the percolation threshold, the analysis program used to ﬁnd pc was
easily modiﬁed to include determination of R,: as discussed in Section 2.7. Finally, the

method used is simple and yields consistent results.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

58
0.90 r I l | j r l 1 I
o_35 — Deposition Rate 1
e o _ 18 nm/s
5 A 6 nm/s
0.80 L- n 2 nm/s 4
0
Q i
0.75 +- E i _
0.70 _ T = 423 K I A _
Cl
0.65 I Lg I I | I L # J r
0 1 2 3 4 5

tanneal/to

Figure 3.11 Percolation coverage vs. ratio of anneal time to deposition time for Pb
samples with different deposition rates. All samples were prepared on 423 K substrates.

 

 

 

 

 

 

59

From the same images used to produce the pc vs. substrate temperature data,
shown in Figure 3.7, the resultant metal structure ("worms") half widths vs. substrate
temperature data is displayed in Figure 3.12. The uncertainties in RC, determined ﬁom
multiple images of each sample, for T < 460 K are much smaller than the data symbols. In
fact, the uncertainties are generally ~ 0.01 um However, the uncertainties in Rc for the
higher temperature samples is a result of the already mentioned morphology change near
460 K As illustrated in Figure 3.9, the simultaneous presence of percolating regions and
"Breath Figure" areas obstructs the percolation point and causes our algorithm to be
inappropriate in such situations.

At a constant deposition rate the crossover radius Rc increases with rising
substrate temperature. This behavior is in agreement with the Kinetic Freezing Model
(KFM) to be presented in the following section. In the KFM, the ﬁrll to partial
coalescence crossover is a result of the competition between ﬁlm growth and droplet
coalescence. Surface diffusion is an activated process which increases at higher
temperatures. A constant deposition rate holds the droplet growth rate steady, allowing
surface diffusion to become more inﬂuential in coalescence. That is, increased surface
diﬂirsion allows droplet coalescence to occur with less opposition from vapor deposition.
As will be discussed in Sections 3.6 and 3.7, the KFM is incomplete. Other inﬂuences are
present during ﬁlm growth which effect the coalescence process and crossover. Aside
from the competition between coalescence and deposition these other effects can act to
pin, or inhibit, the coalescence process.

The dramatic change in the measured values of Rc above To is deﬁnitely a result of
the morphology transformation at this temperature. However, we do not fully understand
the cause of this morphology change. Were this change due to an increased solubility of
Pb in the Si02 then the Pb would tend to spread out more on the substrate rather than
"balling up" as is observed. Simple calculations indicate that Pb is not easily soluble inside

the amorphous Si02 network. Si02 is also extremely stable (Tmelting Z 1300 K) over the

 

 

 

 

60

 

2.2 T T I— I I j H T W T r I M

2.0
1 .8
1 .6
1 .4
1 .2
1 .0
0.8
0.6
0.4
0.2 O

00 rLgliJiliIiig

300 330 360 390 420 450 480 510
Substrate Temperature (K)

lGl

RC (um)

00
o 000

iJmeJmJn In LLLII_1_I_Ll 1

I—I I I I I I I I—I I I I I I I’I I I_I I
O

 

 

 

Figure 3.12 Crossover radius RC vs. substrate temperature for Pb samples evaporated at
6 nm/s onto Si02 substrates.

 

 

 

 

61

range of temperatures studied. Another possible cause of this morphology change is due
to substrate contamination. Prior to sample deposition, the substrate is heated to 675 K
However, the deposition temperatures are lower than this so that residual contamination,
such as pump oils, may be readsorbed on the substrate. The temperature To may
correspond to a temperature where some of this contamination is desorbed from the
substrate. This would leave the substrate surface cleaner, thus reducing pinning of the
coalescence process. Unfortrmately, with no quantitative contamination information
available on our vacuum system, this is only speculation. This was one reason for

pursuing the UHV deposition system which will be discussed in Chapter 4.

3.5 KINETIC FREEZING MODEL

I will now discuss our model [JEFF94] to explain the ﬁrll to partial coalescence
crossover. This model is based on the fact that during ﬁlm growth there are two main
competing processes occurring. These two processes are coalescence and deposition.
Coalescence tends to decrease the system connectivity (recall the argument about the large
droplets acting as though they repel each other). On the other hand, deposition of further
material acts to increase the lateral extent (termed "spreading") of the dIOplets already
present on the substrate. Both of these actions have a characteristic time scale which
depends on the droplet size. The process having the shorter completion time will be
dominant. The crossover from ﬁrll to partial coalescence occurs when the coalescence
time changes from being smaller than the deposition time scale to being larger. At this
transition, when two islands are touching they will grow to touch neighboring islands
before they have fully coalesced. This results in a connected, percolating system The
deposition of material is effectively freezing the coalescence kinetics, thus creating a non-

equilibrium morphology.

 

 

62

For two touching droplets to coalesce via surface diffusion requires a time Tcoal- A
complete analytical theory covering the entire coalescence process is lacking at present,
but enough is known for our purposes. The dynamics of the early "neck-ﬁlling" stage
have been studied [KING55] as well as the regime near the ﬁnal stages of coalescence.
This work begins with some of the ideas to be presented in Chapter 5, most importantly
that a surface curvature gradient is the driving force for surface diﬂusion. The result most
germane to my purposes here is that due to Nichols and Mullins [NICH65, 66]. These
authors numerically solved the relevant partial differential equations, as ﬁrst derived by
Mullins [MULL57, 5 9], governing the entire coalescence process. Their results during the
initial "neck—ﬁlling" stage are that the time required to reach the intermediate "neck-ﬁlling"
stage x/RO = 0.3 is very rapid, ~10”5(R0)4/B where B = DsyoNOQZ/kBT, x is the radius
of the metal bridge between the two droplets and R0 is the initial droplet radius. In
performing these studies the surface diffusion constant Ds and the surface energy yo were
both assumed to be isotropic. The number of surface sites per unit area No is related to
the atomic volume (2 through N0 = 9—2/ 3. To reach the stage labeled "partial
coalescence" in Figure 3.2 requires a time of 0. 166(RO)4/B while a time of 0.890(R0)4/B
is needed to complete ﬁrll coalescence.

We can estimate order of magnitude values for these times by taking Au as an
example. For Au, yo = 1450 erg/cmz, Q m 10‘23 cm3, and DS ~ 10‘7 cmZ/s when
T m 675 K For An droplets of radius 100 nm, neck elimination and complete
coalescence each require much less than one second to complete. Smaller metal islands
will have correspondingly shorter coalescence times. This has been veriﬁed
experimentally. Full coalescence of droplets has been observed using in situ electron

microscopy during ﬁlm growth. Coalescence events do, indeed, occur on the scale of

ﬁactions of a second for the smaller islands.

From the times given above for the various stages of coalescence it is obvious that

the ﬁnal approach to equilibrium during the late stages of coalescence is very slow. The

 

 

 

 

63

majority of the material movement occurs until shortly after the neck is eliminated (the
partial coalescence stage in Figure 3.2). Thus, we take a reasonable value of 0.4 as the
prefactor. That is, we deﬁne the effective coalescence time to be

kBT

4
D 7094/3 R '
s

(3.2) Tcoal : 0.4

We now consider the second major inﬂuence on the droplets in the ﬁlm, namely
growth due to adsorption from the vapor. We View this from the "Breath Figure" stage
shown in Figure 3.4. That is, since it is the large droplets which dominate the percolating
inﬁnite cluster we consider their growth. In particular, the growth time of interest is that
required for these large droplets to increase in size enough to touch neighboring droplets
of similar size. Since these larger droplets generally exhibit a denuded zone near their
perimeters due to the wiping which occurred during their last coalescence event, accretion
of smaller islands at their perimeters is small. In addition, to take this effect properly into
account would require one to make ﬁirther assumptions about the nucleation rate and
island size distribution in these wiped regions.

Up to the percolation threshold we have observed the ﬁlms prepared below To to
maintain a constant height-to-radius ratio 0t e: 1. After many generations of coalescence,
a "Breath Figure" stage is reached where the large droplets have a coverage p m 0.57
[DUXB]. The average thickness of deposited material after a time t at deposition ﬂux J
(#lcmz-sec) of atoms having atomic volume 0 is JQt = havg- However, since the ﬁlm
forms three dimensional islands the average thickness is less than the maximum droplet
height. A ﬁlm, consisting of islands of radius R, with coverage p has an average height of
roughly pork

Since it is the percolation threshold which attracts our interest we concentrate on
the case when R = RC. Further, we envision the situation where two droplets of radius Rc
have just come into contact. The time we seek is that needed for one of these two

droplets to increase its radius to the point where it contacts more neighbors. When this

 

 

 

 

64

happens a connected network of metal "worms" will form. The time to reach the stage

where the islands have an average radius R is given by

0‘P
3.3 t = ——R
( ) J!)

where p is the substrate coverage fraction and takes into account the fact that the ﬁlm is
not of constant height. In the "Breath Figure" stage p m 0.57. Now, in order for the large
islands to touch their neighbors, their projected area on the substrate must increase by

roughly 1/p which corresponds to a radius increase to R' = Rp‘l/Z. Thus,

I._ apc |_ apc
3.4 t —-——R————R
( ) J0 JQ,/p

is the time needed to grow to a size R'. The spreading time Tspr is the difference between

t' and t, under the assumption of constant deposition rate, so that

Of
3.5 t z 0.5——R.
( ) 58' LE)

At the crossover between full and partial coalescence the premise of the KFM is
that tspr = Tcoat Equating the expressions in equations (3.2) and (3.5) with R = RC

yields.

1/3
(3.6) R2 2,. M.
kBTJ

This equation may be rearranged and expressed in a manner more easily compared to the
experimental measurements. During sample deposition the time required to deposit each
ﬁlm, to, is recorded and the ﬁlms are prepared to slightly past the percolation threshold.
All positions on each sample have the same deposition time, but, due to the thickness
gradient mentioned in Chapter 2, different locations experience different deposition ﬂuxes

J. Since, from equation (3.3), Jth = orpcR,2 equation (3.6) may be expressed as

(3.7)

m to.

 

 

 

 

65

Given the approximations involved in arriving at this expression it is estimated to be
correct to within roughly a factor of two. This relatively large uncertainty has little impact

on the comparisons to be drawn with the experimental results.

3.6 PREDICTION S OF THE KINETIC FREEZING MODEL

Equation (3.7) carries two predictions which may be tested experimentally. These
predictions are: (1) the deposition time dependence of Rc and (2) that average surface
diffusion constants DS may be inferred. These two predictions will be discussed separately

in the following two sub sections.

3.6.1 Rc DEPENDENCE ON DEPOSITION TIlVIE

The most evident feature of equation (3.7) is the prediction that RC ~ (tc)l/ 4 at
constant temperature. To test this prediction, Rc was measured for samples prepared with
various deposition times to. Films prepared with higher deposition rates percolate at lower
thicknesses than do ﬁlms deposited at lower rates. Some of the samples required on the
order of ten seconds to deposit. This short deposition time combined with the importance
of careﬁrl quenching of the sample would have made preparation of these samples much
more diﬂicult if the He "quench" method had not been used. The deposition time
dependence of R.: was measured carefully at two temperatures. Both temperatures were
below To and yielded similar results. Figure 3.13 shows a log-log plot of our Rc vs. tC
data at T = 408 K and 423 K for different deposition rates which vary over one decade
ﬁom about 2 to 20 nm/s. With Rc ~ (tc)1/l3, the KFM predicts B = 4. The data of Figure
3.13 displays a weaker dependence with [3an z 7.5. We attribute this discrepancy to two

Mportant factors missing from the KFM, namely grain boundaries and substrate

roughness, as will be discussed in Section 3.7.

 

 

 

 

 

66

 

 

 

 

 

 

 

'50 j I I I ' 1 rljl'IT'rI'l‘I fr! I j I ' I 'j1 I 'TrI'T'I'I'I I I
i- "l
45 L- .1
L 'l
E .40 r- _
3-
o P J
C:
+ 408K
-35 5 43— 423K 4
L 7
.30 n l I 14 u‘mllillﬂd J_I LI 4 LI Li LLlLliILLLIIliI IL

 

101 2 3 455 102 2 3 456 103

tc (sec)

Figure 3.13 Log-log plot of the radius Rc vs. deposition time tC needed to reach
percolation for Pb ﬁlms. The lines are power law ﬁts to R‘: ~ (tc)1/l3 with B m 7.4
(408 K) and B m 7.6 (423 K).

 

 

 

 

 

67

3.6.2 INFERRED SURFACE DIFFUSION CON STAN TS Ds

Using equation (3.7) and the experimentally measured values of RC, tc, and T we
may solve for the surface diffusion constant Ds. These values of DS are averaged over all
crystal faces present on the crystallites during growth. Different crystal surfaces possess
different values of DS due to the varying surface structures. For instance some faces are
isotropic in two dimensions while other faces consist of "channeled" structures so that DS
on these surfaces may be anisotropic as well. If the KFM is qualitatively reasonable then
our estimated values of DS should roughly agree with other experimental measurements on
speciﬁc crystal faces.

In order to calculate DS we need values of Q and yo in equation (3.7). The atomic
volume of Pb is 3.0x10—23 cm3. While the surface energy of a crystal is, in general,
anisotropic, in the KFM the surface energy is assumed to be isotropic with a value yo. We
have chosen to use the surface energy yo = 480 erg/cm2 [MELF56, ZANG88] which is
the surface energy of liquid Pb in equilibrium with its vapor.

Equation (3.7), the above values for Q and yo, along with my experimentally
measured values of RC and tc yield the estimated values of Ds shown in Figure 3.14.
These data are obtained from the same samples studied to make Figure 3.12. DS is plotted
vs. Tmelting/T since surface diffusion is an activated process which is expected to follow
an Arrhenius type behavior of the form Ds ~ DoeXP(“aTmelting/T) with aTmeltmg
proportional to the surface diffusion energy barrier. The two solid lines in Figure 3.14
indicate the range of previously measured values of DS on fcc metals [BONZ83].
Considering the estimations involved in the KFM there is surprisingly good agreement of
DS with other studies. This is especially true since some of the previous studies used UHV
environments with well cleaned and characterized samples. The upturn in my DS values

for T 2. 460 K is a result of the inapplicability of the KFM in this temperature range.

 

 

 

 

68

 

 

T(K)
601 550 500 450 400 350 300
10'3_1 r T r r I
o
10-4 0°
07: 10-5 .-
O
3 06 ‘ . o
a) 1'
D o
10'7 c
10-3 4 1 1 1 1 1 1 L L L
1.01.1 1.21.31.41.51.61.71.8 1.9 2.0
Tmelting/T

Figure 3.14 Plot of inferred DS values as a ﬁmction of Tmelting/Tsubstrate The solid lines
show the range of DS values measured on fcc metal surfaces [BONZ83].

 

 

 

 

69

3.7 MISSING INGREDIENTS OF THE KFM

The Kinetic Freezing Model is merely a "zeroeth order" approach. There are many
important factors which have been neglected. As already mentioned, both the wrface free
energy yo and the surface diffusion constant DS are, in general, actually anisotr0pic. These
were taken to be isotropic in the KFM. Two important elements missing ﬁom the KFM
are the presence of substrate inhomogeneity and the trapping of grain boundaries within
the interior of two coalescing droplets.

Roughness of the substrate can act to pin the droplet perimeters, thus impeding the
coalescence. This is because during coalescence the droplet perimeter is effectively
"dragged" across the substrate surface. Ifthe surface is rough then more energy will be
required to move the perimeter. Thus, unless the system has a high enough temperature to
overcome this pinning action, coalescence will cease at a stage earlier than predicted by
the KFM. This pinning may be caused by either topological or chemical inhomogeneity on
the substrate surface. Since the amorphous substrates we used were determined, via
AFM, to have a mean roughness of roughly 0.1 nm, chemical contamination is most likely
the more important factor in these samples.

Some experimental observations indicate chemical contamination of the substrates
to be present. Occasionally, samples which were prepared otherwise identically (cleaning,
deposition rate, pressure, and thickness) exhibited different morphologies when viewed
either optically or with the SEM. Also, some samples appeared to be strongly
inhomogeneous. For example, regions of some ﬁlms appeared to be percolating (or at
least very well connected) while other regions of the same ﬁlm were still in the compact
island stage. This was not due to positions of differing thickness resulting from the
thickness gradient on the samples. This effect is, we believe, due to nonuniform cleaning
of the substrate. Prior to insertion of the cleaned substrates in the vacuum chamber,

regions could be occasionally noticed which were obviously contaminated. This might

 

 

 

 

70

have been caused by incomplete dissolution of one of the cleaning agents used in cleaning
the substrates, such as acetone or the MICRO cleaner.

Substrate inhomogeneity is only partly responsible for the formation of the
metastable elongated metal structures. In forming a comp act island from an initially
elongated structure the surface energy gain (assuming isotropic yo) scales as R2 with R
being the island radius whereas the pinning energy is proportional to R, the perimeter
length. Thus, it is doubtful that substrate pinning alone prevents full coalescence.

However, we know that grain boundaries are present during coalescence. The
droplets form predominantly with the (111) face parallel to the substrate. But, the
azimuthal orientation (about an axis normal to the substrate) is random due to the
amorphous substrate. Thus, when two crystalline metal islands come into contact and
begin to coalesce, a grain boundary forms in the interior of the composite island which
extends through the bulk and down to the substrate. This causes the formation of a grain
boundary groove to form at the surface of the metal islands. These grain boundaries are
visible in images such as those in Figure 3.10. The boundaries are evidenced by regions of
the elongated metal islands which narrow slightly near the grain boundary groove location.
Removal of such a grain boundary from the intersection of two droplets requires a certain
amount of energy. Often, the system temperature and the gain in surface energy if
coalescence were completed is insuﬂicient. In such a case, the grain boundary halts the
coalescence process.

A large number of the grain boundaries created between two coalescing islands are
high angle boundaries. Such high angle grain boundaries have a surface energy, yb, of
roughly one-third the surface energy, 7, of the droplet material. It is possible to calculate
the total energy of two compound islands with such an interior high angle grain boundary
[DUXB94]. Such a plot is shown in Figure 3.15 which shows the total energy of two
coalescing droplets, initially of the same radius but subject to a constant volume

constraint, with a grain boundary trapped in the middle. In this plot, the dihedral angle, LP,

 

 

 

 

 

7l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3: 1
w ..
E; 1
(I) ..
Li

a, 1
o _
5‘1 1
5 1
'5

3 J
F d

1 .00 1.02 1.04 1.06 1.08 1.1 0

Radius of Larger Island F12

Figure 3.15 Surface energy of two coalescing droplets with an interior grain boundary
having a dihedral angle, ‘1’, of 78 degrees. The coalescence stage is shown in the
accompanying schematic diagrams.

 

I1

 

 

 

72

of the grain boundary is 78 degrees. The x—axis, labeled R2, is the radius of the larger
droplet in terms of the initial radius of each droplet. The equilibrium condition
Yb = 2ycos‘l’ causes the formation of the grain boundary groove and results in yb m y/3
for ‘1’ m , 80°. The important feature of this plot is the very slight gradient near R2 m 1.
This produces a small driving force for coalescence which can, thus, be pinned by weak
substrate pinning along the island perimeter. A similar plot of surface energy vs. droplet
size for the case of no grain boundary has a much larger driving force for coalescence
since there are no impeding factors. Thus, substrate pinning and grain boundary pinning

are not totally separate and a complete theory incorporating both effects is still lacking.

3.8 SUNINIARY

In this chapter I have attempted to describe the growth and structure of thin metal
ﬁlms evaporated onto insulating substrates. A complete understanding of ﬁlm growth in
these systems requires explanation of many factors. These include the anomalously high
area coverage at the percolation threshold, the island size for crossover from full to partial
coalescence, sample annealing effects, as well as the effects of substrate inhomogeneity
and grain boundaries.

Through a variety of computer simulations it was found that the keys to achieving
a large value of pc (as is observed in actual ﬁlms) are coalescence, wiping, and the
existence of the transition from full to partial coalescence. This ingredient yields values of
pc larger than % 0.6 which could not be produced with normal lattice simulations.

Once the cause of high pC was qualitatively understood, the size at which full
coalescence ended was studied. For T S 0-75Tmelting this size, RC, was found to have a
weak temperature dependence. A simple Kinetic Freezing Model was presented which
offers an intuitive understanding of this ﬁrll to partial coalescence crossover. This

vieWpoint, that the coalescence time and the droplet growth time change dominance, is

 

 

 

 

 

73

not complete in that substrate and grain boundary effects are neglected. For temperatures
above To z 460 K the ﬁlm morphology undergoes a change we do not fully understand
The signiﬁcant change ﬁom a fairly uniform appearance to ﬁhns with a dual structure may
be due to an alteration in the substrate behavior (e. g. contamination or surface energy).

It was also found that the percolation coverage as well as the ﬁhn appearance is
dependent on the sample cooling rate. While even short annealing treatments have a
signiﬁcant inﬂuence on pc, surprisingly little effect on R: is observed. While pC decreases
during annealing the perimeter of the connected metal islands becomes much smoother as
coalescence events continue after the end of deposition if the sample is still hot.

Finally, some ideas were presented on how the missing ingredients of the KFM
might effect the results. These absent features are the presence of substrate roughness or
inhomogeneity and grain boundaries which are trapped in the interior between two
coalescing droplets. Neither of these issues is easily taken into account. However, we
have made some initial progress in this direction.

A simple plot may help to sum up much of what has been discussed in this chapter.
This plot, shown in Figure 3.16, shows the same data presented in Figure 3.12 in a
different form Here, R2 is taken as the independent variable and the corresponding
substrate temperature TC is that temperature at which the sintering stage (neck-ﬁlling)
gives way to full coalescence. At a given Rc a certain substrate temperature is needed to
overcome the effects of deposition and allow completion of coalescence. While
coalescence of two solid droplets has been termed "liquid-like" since it is so rapid, Figure
3.16 shows that bulk melting of the droplets is negligible. Only dr0p1ets of size S 10 nm

have their melting points signiﬁcantly reduced below the bulk melting temperature of

601K

 

 

 

 

 

 

74

 

 

 

 

 

 

650 l I l r W T I
BULK LIQUID BULK MELTING: 601K
:8 T
9’. 550 — BULK SOLID —
3 FULL COALESCENCE
C6
5 _
‘5‘ L
a, 450
I—-
52 - SINTERING -
‘3 . (NECK FILLING)
175 350 P _
‘3
co - l
250 1 I 1 I 1 I 1
0.0 0.5 1.0 1.5 2.0

Critical Radius RC (um)

Figure 3.16 Plot of substrate temperature vs. R,2 showing the different coalescence
regimes. These Pb samples were deposited at 6 nm/s.

 

 

 

 

75

 

CHAPTER 3 REFERENCES

BASK94 A. A. Baski and H. Fuchs, Surface Science 313, 275 (1994).

BEYS86 D. Beysens and C. M. Knobler, Phys. Rev. Lett. 57, 1433 (1990).

BEYSQl D. Beysens, A. Steyer, P. Guenon, D. Fritter, and C. M. Knobler,
Phase Transitions 31, 219 (1991).

BONZ83 H. P. Bonze], in Surface Mobilities on Solid Materials: Fundamental
Concepts and Applications, edited by V. T. Binh, NATO ASI Series,
Series B, Physics, Vol. 86, (Plenum Press, New York, 1983), p. 195.

CARR G. L. Carr, B. Caldwell, F. Family, and P. Meakin, (unpublished).

DUXB P, M. Duxbury and X. Yu, (unpublished).

DUXB94 P. M. Duxbury, M. Dubson, X. Yu, and G. Jeﬁers,
Europhys. Lett. 26, 601 (1994).

FAMIS8a F. Family, in Random Fluctuations and Pattern Growth: Experiments and
Models, edited by H E. Stanley and N. Ostrowsky, NATO ASI Series,
Series B, Applied Sciences, no. 157, (Kluwer Academic Publishers,
Netherlands, 1988), p. 345.

FAMI88b F. Family and P. Meakin, Phys. Rev. Lett. 61, 428 (1988).

FAB/[[89 F. Family and P. Meakin, Phys. Rev. A 40, 3836 (1989).

GAWL81 E. Gawlinski and H. E. Stanley, J. Phys. A: Math. Gen., 14, L291 (1981).

HERRSI C. Herring, in The Physics of Powder Metallurgy, edited by W. E. Kingston,
(McGraw-Hill, New York, 1951), Chapter 8.

HEYR83 J. C. Heyraud and J. J. Metois, Surf Sci. 128, 334 (1983).

HEYR87 J. C. Heyraud and J. J. Metois, J. Crystal Growth 82, 269 (1987).

HEYR89 J. C. Heyraud, J. J. Metois and J. M. Bermond,
J. Crystal Growth 98, 355 (1989).

JEFF 94 G. Jeffers, M. A. Dubson, and P. M. Duxbury,
J. Appl. Phys. 75, 5016 (1994).

KAPI82 A. Kapitulnik and G. Deutscher, Phys. Rev. Lett. 49, 1444 (1982).

KING55 W. D. Kingery and M. Berg, J. Appl. Phys. 26, 1205 (1955).

LAIB82 R B. Laibovvitz, E. I. Alessandrini, and G. Deutscher,

Phys. Rev. B 25, 2965 (1982).

 

 

 

 

 

LEWI78

MELF 56
METO82a
lVIETO82b

MET089a
MET089b
MULL57
MULL59
NTCH65
NTCH66
OHRI92

PASH64

PASH65
PASH66
STAU85

VOSS82

WOLL93
YU9 l

ZANG88

76

B. J. Lewis and J. C. Anderson, Nucleation and Growth of Thin Films,
(Academic Press, New York, 197 8).

D. A. Melford and T. P. Hoar, J. Inst. Metals 85, 197 (1956).
J. J. Metois and J. C. Heyraud, J. Crystal Growth 57, 487 (1982).

J. J. Metois, G. D. T. Spiller, and J. A. Venables,
Phil. Mag. A 46, 1015 (1982).

J. J. Metois, S. Nitsche, and J. C. Heyraud, Ultramicros00py 27, 349 (1989).
J. J. Metois and J. C. Heyraud, Ultramicroscopy 31, 73 (1989).

W. W. Mullins, J. Appl Phys. 28, 333 (1957).

W. W. Mullins, J. Appl. Phys. 30, 77 (1959).

F. A Nichols and W. W. Mullins, J. Appl. Phys. 36, 1826 (1965).

F. A Nichols, J. Appl. Phys. 37, 2805 (1966).

M. Ohring, The Materials Science of Thin Films,
(Academic Press, San Diego, 1992).

D. W. Pashley, M. J. Stowell, M. H. Jacobs, and T. J. Law,
Phil. Mag. 10, 127 (1964).

D. W. Pashley, Adv. Phys. 14, 569 (1965).
D. W. Pashley and M. J. Stowell, J. Vac. Sci T echnol. 3, 156 (1966).

D. Stauﬁ‘er, Introduction to Percolation Theory,
(Taylor and Francis, London, 1985).

R F. Voss, R B. Laibowitz, and E. I. Alessandrini,
Phys. Rev. Lett. 49, 1441 (1982).

D. A. Wollman, M. A. Dubson, and Q. Zhu, Phys. Rev. B 48, 3713 (1993).

X. Yu, P. M. Duxbury, G. Jeffers, and M. A. Dubson,
Phys. Rev. B 44, 13163 (1991).

A. Zangwill, Physics At Surfaces,
(Cambridge University Press, Cambridge, 1988).

 

 

 

 

 

CHAPTER 4

POSSIBLE FUTURE DIRECTIONS
Further experiments related to thin film growth and structure

4.1 INTRODUCTION

Clearly, the ﬁlm growth work presented in this thesis is incomplete. A small
portion of a very large parameter space has been studied with much remaining. The two
most obvious directions which could be pursued as extensions of this work concern the
factors absent in the Kinetic Freezing Model. These are substrate inhomogeneity and the
inﬂuence of grain boundaries upon the sample growth and morphology. Other issues such

as deposit material and deposition rate are more speciﬁc and, thus, of lower priority.

4.2 SUBSTRATE INHOMOGENEITY

One very imp ortant factor in this research has been the sample preparation
conditions. The background pressure during sample deposition was in the range 10‘6 to
10“7 mbar. A pressure of 10‘6 mbar corresponds to roughly one monolayer of
contamination coating the sample surface every second [OHAN89]. This issue is mainly
relevant at the deposit-substrate interface. The deposition rates of generally 2 2 nm/s
correspond to at least six monolayers of deposit material being added to the sample
surface every second which is larger than the contamination rate. Thus, it is possible that
a signiﬁcant number of impurities are trapped within the ﬁlm during growth. In addition,
although the deposit materials are usually of 99.999% purity, this small impurity
concentration may be signiﬁcant if it is able to migrate to the surface of the growing ﬁlm

Contamination from regions of the vacuum chamber which are heated by the deposition

77

 

 

 

 

 

78

source, as discussed in Section 2.2, probably inﬂuence the ﬁlm's composition. However,
metals of higher purity are more diﬂicult and much more costly to obtain. In order to
signiﬁcantly reduce the background contamination present in the vacuum chamber, use of
an improved vacuum system is necessary.

Modern vacuum technology enables the background pressure to be signiﬁcantly
diminished with the proper equipment. Design and construction of such a system was
pursued during this research and will be described brieﬂy below.

Ultrahigh vacuum can be fairly easily achieved these days. Diffusion pumps such
as that on our evaporation system have a lower ultimate pressure typically ~ 10“8 mbar.
Completely oil-free environments require the use of either ion pumps or turbomolecular
pumps. I designed a UHV sample deposition system centered around the ideas of using a
stainless steel chamber and an ion pump to achieve pressures below 10‘9 mbar, thus
reducing the present contamination rate by roughly a factor of 1000. The stainless steel
chamber is necessary since baking of the system (to outgas the Chamber's interior) is
required to reach such low pressures. In addition to the pumping elements, this system
was to contain similar features as the present evaporator. These were to include two
deposition sources, a ﬁlm thickness monitor, and sample stages. Very precise control of
the deposition rate can be obtained through use of a Knudsen eﬂ’usion cell such as those
used in Molecular Beam Epitaxy (MBE) systems. However, this extreme controllability is
quite expensive, on the order of $10,000. Since this was outside our available funds, we
were going to attempt building a small enclosed source in hopes of mimicking the
isothermal environment of the effusion cell. Since ﬂux control was not a major problem in
the existing setup, this approach was deemed adequate. Due to the long times required
for pumpdown to the ultimate pressure before deposition (roughing, baking, pumping) it
was desired to have multiple sample stages located in the chamber. Additionally, these
stages would have a small thermal mass and very rapid heating and cooling rates

compared to the present heater stage. This feature could also prove useful for ﬁrture

 

 

 

 

79

experiments requiring ﬁne temperature control. A technique [SCHW 93] whereby two
thin ﬁlm thermometers (TFD's) are used on each stage was tested. In such an
arrangement, one TFD is used as the sample heater while the other is used as the
thermometer. Because of the small size of these thermometers (10.2 x 3.3 x 0.5 m3),
use of two of these enabled a very small, light sample stage to be fabricated. Rather than a
"big Cu block" aﬂ that is needed is a small support structure, the two TFD's, and the
substrate. A prototype stage was assembled with a mass of only 5 grams. The thermal
response of this structure was much better than the present stage. Starting at room
temperature this heater could be stabilized at a temperature of 100 °C in roughly half a
minute with an input power of roughly 0.1 W. The cooling rate was about 30 K/minute
without use of a "cold ﬁnger" as in the present quenching method. With an improved
design these numbers could undoubtedly be improved. The plan was to suspend at least
three of these sample stages in the chamber on a rotary feedthrough to enable each sample
to be prepared independently of the others. After deposition the samples would be
analyzed in the same manner presently used.

Since substrate inhomogeneity can be topological as well as chemical, an in situ
device for sample roughening and cleaning was also considered. Using an ion mill gun,
the sample could be bombarded by ion doses of various ionic species, energy, and duration
times. Again, this equipment was quite expensive. Fortunately, an ion milling system
already exists within the department which we planned to use, instead, for substrate
surface roughening prior to insertion of the samples into the evaporation chamber. Tests
were performed with various ion treatments using this system Surface topology
deformations ~ 0.1 11m were easily formed on Si02 substrates as observed with an SEM.
Much smaller dimensions are undoubtedly achievable with more detailed tests. Typical

dimensions in the percolating ﬁlms are presently ~ 0.1 to 1 11m so that smaller dimensions

would be needed.

 

 

 

 

80

These above ideas could also be applied to the case of an insulating, crystalline
substrate such as quartz (SiOZ) or sapphire (A1203). Generally, metals do not wet
insulating surfaces. However, if the lattice parameters are close enough, more nearly

layer-by-layer growth than that observed on amorphous substrates might be achieved.

4.3 GRAIN BOUNDARY AND COALESCEN CE STUDIES

The second major issue of concern is that of grain boundaries and the effect of
their presence on the coalescence process. A few ideas as to how to study these. issues are
presented below.

Since the substrates used are amorphous, crystalline islands grow with random
orientations. Variations exist in the crystallite‘s orientation both in terms of what face is
parallel to the substrate and the azimuthal rotation. Both of these factors determine the
orientation of the grain boundary which is formed when two crystallites touch and begin to
coalesce. If pinning by such a grain boundary is important, quantitative information on the
boundary’ s mobility would be useful in learning to control ﬁlm growth.

Two single crystal regions with an interior grain boundary have an electron
diﬂiaction pattern characteristic of the two orientations present. These two crystal
orientations may be observed using a Transmission Electron Microscope (TEM). Since
different composite islands will contain different grain boundaries and orientations, data on
many different islands would be required to obtain information regarding the average
orientation. This might be achieved, inside a TEM, through the use of selected area
diﬂiaction where the diffraction information from a small area of the sample is
investigated. There are two problems with this approach. First, the random orientation
will complicate determination of one or both crystal alignments. Secondly, use of a
(TEM) would limit the material thicknesses which could be studied. Sample thicknesses
of roughly 10 nm or greater are not compatible with the TEM's such as that in the Physics

 

 

 

 

 

81

Astronomy Department since the maximum beam energy of roughly 100 keV is
insufﬁcient to penetrate such thicknesses, particularly in a dense material such as Pb. A
similar measurement could be made using electron channeling. However, channeling, also
requires further initial orientation information. We also have no access to a microscope
with channeling cap abilities.

Another method, involving the use of x-ray scattering, might be used to obtain
information regarding the crystallite orientation distribution, but not that of the grain
boundaries. The x-ray system in the Michigan State University Chemistry Department has
rotation capability about one axis which allows normal 0 - 20 scans. Thus, two scans
might be needed with the sample rotated 90° about the substrate normal during the
successive scans. Through measurement of the relative intensities of different Bragg
peaks, quantitative information could be obtained regarding the orientation distribution
which could be correlated with the ﬁlm morphology. A drawback of this technique is that
long collection times might be necessary. Because the ﬁlm is not continuous and contains
many crystals (of very small size relative to the x—ray beam diameter) of different
orientations, the signal to noise ratio would be small. Access to an x-ray goniometer with
rotation capability about the polar and azimuthal axes would simplify this approach
considerably.

A method also exists whereby information about the grain boundary pinning
strength and diffusion may be obtained. The idea is to observe the coalescence process
inside an electron microscope. Lead is a good candidate for this type of experiment due to
its low vapor pressure; little material would contaminate the SEM chamber, especially the
electron beam column. Using photolithography, a sample could be prepared which
consists of two metal islands which just touch. This sample would be mounted on a
heatable stage inside an SEM. By adjusting the temperature, the process of coalescence
could be controlled while imaging the process. The grain boundary formed between these

two crystals can be easily detected in the SEM image. Once formed, observations could

 

 

 

 

 

 

82

be made of the boundary‘ s movement, particularly the time and temperature dependencies
for movement out of the compound island. This could provide information about the
grain boundary's pinning and migration energies. As the process of coalescence is
generally quite fast, a few precautions must be taken. The droplet size could be made
large enough so that the coalescence occurs on a time scale compatible with observation.
This would also be a continuous process so that a more efﬁcient imaging scheme would be
needed than the present snapshots which are taken. This could be done by splitting the
electron detector signal to a video system (or another monitor) which could be videotaped
at speeds of roughly 1/30 second. There are also potential problems with this method.
The sample would have to be cleaned once inside the SEM chamber to eliminate
contamination produced after removal from the evaporation chamber. This could be done
using either in situ UV/ozone cleaning or gas jet cleaning. Gas jet cleaning relies on the
interaction of a gas jet, which is directed onto the sample, with the incident electron beam
to clean the sample surface. Because the SEM reaches an ultimate pressure comparable to
that inside the evaporation chamber, the coalescence dynamics should be similar if the
initial contamination can be eliminated.

An alternative approach for studying some aspects of grain boundary dynamics is
to use an STM and compatible materials. This could be done for Au or Ag on a graphite
substrate. For a given substrate temperature during deposition these metals will have a
smaller grain size than Pb which would enable inspection of many more crystallites using
the STM. Also, graphite is not an insulating, amorphous substrate. The ﬁlm growth
would be different since ordered island structures might be produced as have been
observed for Pb on graphite [METO82]. The STM we have in our laboratory would limit

real time observations of the coalescence process because the image acquisition time is at

least 15 seconds or more.

 

 

 

 

 

83

4.4 SUNIMARY

The possible experiments presented in this chapter are insuﬂicient to ﬁrlly
understand the growth of metal ﬁlms on insulating substrates. However, they are
experiments which we have considered performing as we feel the results could prove
beneﬁcial in light of some of the results observed during the course of this research.
Additionally, we are unaware of experiments of this nature having been previously
performed.

Numerous experiments were performed in the 1960's dealing with ﬁlm nucleation
and structure [PASH66, INUZ69]. These were mainly in situ experiments where the
sample was observed during deposition inside the chamber of an SEM or TEM. The
presence of the electron beam was found to increase the island nucleation rate most likely
due to the increased contamination produced by the interaction of the electron beam with
hydrocarbons in the chamber [CHRI60]. Other studies have dealt with the crystallite
alignment's relative to the substrate and its behavior during coalescence. The resulting
structure of ﬁlms prepared in situ or observed after removal ﬁom a thermal evaporation
system were found to be similar [PASH64]. These types of experiments are more
complicated than observing ex situ to the deposition chamber and yield similar results.

Were the experiments discussed in this chapter carried out they may help to

quantify various parameters controlling continuous ﬁlm formation. Controlled

experimentation may lead to the production of high quality, perhaps technologically
useful, ﬁlms.

 

 

 

 

84

CHAPTER 4 REFERENCES
CHRI60 R W. Christy, J. Appl. Phys. 31, 680 (1960).
METO82 J. J. Metois and J. C. Heyraud, J. Crystal Growth 57, 487 (1982).
lNUZ69 T. Inuzuka and K Ueda, J. Vac. Sci. Technol. 6, 379 (1969).
OHAN 89 J. F. O'Hanlon, A User’s Guide To Vacuum Technology, Second edition
(John Wiley & Sons, New York, 1989).
PASH64 D. W. Pashley, M. J. Stowell, M. H. Jacobs, and T. J. Law,
Phil Mag. 10, 127 (1964).
PASH66 D. W. Pashley and M. J. Stowell, J. Vac. Sci. Technol 3, 156 (1966).
SCHW93 R B. Schwarz and J. B. Rubin, Rev. Sci. Instrum 64, 1292 (1993).

 

 

 

CHAPTER 5

PROFILE DECAY DUE TO SURFACE DIFFUSION AT
LOW TEMPERATURE
Computer simulations of mass transport on a stepped surface

5.1 INTRODUCTION

We have performed computer simulations of the decay of a sinusoidal proﬁle in
(1+1)D in an attempt to obtain a better understanding of the dynamics controlling surface
diffusion on a corrugated surface below the toughening temperature. While Monte Carlo
simulations of proﬁle relaxation have been previously performed [JIAN89, SELK93] in
this temperature regime, neither conclusive evidence of faceting nor the decay time scaling
relation has been found. Monte Carlo simulations which rely up on analytical results for a
one-dimcnsional random walk are discussed. These analytical results, derived via rate
equations in Section 5.3, or using an electric circuit analogy in Section 5.7, determine the
various movement probabilities for individual atoms depending up on their local
environment. Thus, atom motion is "instantaneous," eliminating the need to track an
atom's entire random walk. The simulations are a novel attempt to study surface diffusion
dynamics below the roughening temperature. Long-ranged step-step interactions are
absent in our model. However, a contact interaction is present because steps of like sign
cannot cross in this solid-on- solid model. This interaction leads to an entropic repulsion
between steps. The results differ from those for the high temperature (above the

toughening temperature) regime. Similarities between this model and other studies are

discussed.

 

 

 

 

 

86

5.2 BASIS OF THE SIlVIULATION ALGORITHTVI

Most simulations of ﬁlm growth and/ or surface diﬂilsion are based upon the
Monte Carlo algorithm The method is called Monte Carlo as it is based upon randomly
chosen events. As an illustration of this method consider the motion of a single atom,
initially at the origin, random walking among the sites of a square lattice, which are
allowed to have different binding energies. Furthermore, restrict the atom's motion to its
nearest neighbor sites which determine the local energy landscape. One possible question
of interest is how long this single adatom will take to move a certain distance ﬁom the
origin or to a given position. At each movement opportunity four possible jump directions
exist. The direction in which to attempt a jump is chosen randomly. The actual decision
to jump is based upon the Boltzmann factor, e‘PAE, where B = (kBT)“1 and AB is the
binding energy difference between the ﬁnal and initial sites, AB = Eﬁnal — Einitial' Should
this energy difference be less than or equal to zero the walker is allowed to move to the
ﬁnal position. That is, movement to a position of equal or higher binding energy occurs
with probability one if that direction is chosen. However, some sites have lower binding
energies so that these sites are energetically costly to occupy. This is where the
Boltzmann factor becomes necessary. If AB 2 0 the walker moves to the ﬁnal position
with a probability of e—BAE. Comparison of a random number, which is uniformly
distributed in the range 0 < r < l, to this Boltzmann factor determines whether the
movement occurs. If the random number is less than 6‘ I3AE the move is allowed. This
cycle of allowing the walker to move is continued until the adatom reaches the desired
location. This algorithm is easily extended to the case with more than one adatom present.
Then, individual walkers are interrogated in random order after the completion of the
previous walker's journey or after a single movement attempt by the previous adatom

Monte Carlo methods may be very ineﬂicient in cases where the only points of

interest are the starting and ﬁnishing positions. The reason for this ineﬂiciency arises ﬁom

 

 

 

 

 

87

the need to track the entire random walk of the diﬂirsing atom Our simulation algorithm
avoids the need to follow each random walker through its entire journey. Instead, the
relevant probabilities (to be discussed in Section 5.3) for diﬁerent types of movement can
be calculated for every walker based on its local environment. This feature results in a
large decrease in the necessary computational time.

The basis of our model [DUBS94a] is that a wandering adatom on a crystal
surface experiences a different local environment near a monatomic step than when
located on a ﬂat terrace. This may be understood simply by the difference in numbers of
nearest neighbors (NN) and next nearest neighbors (NNN) between these respective
positions. Figure 5.1 demonstrates this for both a triangular and a square lattice. Also

shown are the position dependent binding energies for adatoms on both surfaces.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.1 Binding energy diagrams near a monatomic step for the triangular and square
lattices. A is the Schwoebel barrier. Ek and E are described in the text.

A barrier, denoted by A, exists between the last binding position on the top side of
the step and one position to its right. This so-called Schwoebel barrier [SCHW66, 69]

makes it more difﬁcult for an atom to move down to the next terrace. Embedded atom

 

 

 

 

 

 

88

calculations [LIU93] conﬁrm the qualitative form of this adatom potential. Also shown in
Figure 5.1 are the terrace corrugation energy barrier, E, and the binding energy of a kink
site, Ek. The Schwoebel barrier arises ﬁom a missing N to the last binding site on the
upper terrace relative to a site one position to its left.

Some controversy exists over the nature and size of the Schwoebel barrier. Field '
Ion Microscopy (FIM) has been the primary experimental tool for observing the inﬂuence
of this barrier upon surface diffusion of single adatoms. In 1966 Ehrlich and Hudda
[EHRL66a] were the ﬁrst to observe an adatom reﬂecting from the top side of a step
edge. This occurred for W adatoms on different crystal faces of a W surface. The amount
of reﬂection depended upon the step direction and it was concluded that an energy barrier
of 430.06 eV more than the energy barrier for open-terrace diffusion could cause the
observed behavior. Tsong measured [TSON72] A and E for Re diffusion on W(123) at
364 K and found A m 0.05 eV and E m 0.88 eV. Thus, in this case A is small compared
to the diﬂirsion barrier Et which is usually 20.5 eV. Fink and Ehrlich [FINK84]
discovered that for W and Re adatoms on W(211) there exists a preferential binding of the
adatom at the ﬁrst or second top side binding position from the step. However, they did
not look at adatoms descending down a step so that no information regarding the
Schwoebel barrier was determined. In a series of papers [WANG91, 93a, 93b] Wang and
Ehrlich studied the behavior of It and W adatoms near the boundaries of Ir clusters of
differing sizes using FIM. Their results show that a step acts as a funnel for migrating
adatoms. That is, adatoms approaching an ascending step experience a gradually
decreasing energy barrier as they approach the step. This causes an increased ﬂux of
atoms to the step edge On the other hand, an atom approaching a descending step also
feels a decreasing energy barrier close to the step edge, but not at the edge. At the step
edge the barrier to move down the step is larger. The authors above also observed that
the size of the terrace is important. Small terraces show complicated behavior near their

centers due to possible long range effects of the step edges. Kunkel et al [KUNK90] and

 

 

 

 

 

89

Bott et al [BOTT92] postulated Schwoebel barrier effects to explain their observations of
reentrant layer-by-layer growth of Pt on Pt( 1 1 l) homoepitaxial ﬁlms. While being far
from completely understood, the Schwoebel barrier has been experimentally shown to
inﬂuence adatom behavior near step edges.

The system used for the simulations to be discussed here is a (1+1)D solid-on-solid
(SOS) square lattice at low temperature. The (1+1)D means the height 2 of the system is
characterized by z(x, t) where x is the horizontal direction in Figure 5.1 and t is time.
Such a system is physically unattainable as it would require a stack of matter one atom
wide, but is meant to mimic the behavior of a full three-dimensional system having
straight, parallel grooves in the y-direction. Our algorithm is most applicable to a (1+1)D
system Here, low temperature means: 1) terrace atoms (those with two in-plane NN) are
not allowed to activate and become walkers on the surface and 2) there is only one walker
present on the surface of the sample at any time. These two restrictions prohibit
roughening of the surface and island nucleation.

Kink site atoms are the only atoms in these simulations which are allowed to
activate and become random walkers on the surface. Kink site atoms are those located at
the end of a terrace and have one less in-plane NN than other surface sites. Neglecting the
activation of atoms embedded in terraces (with two in-plane NN's) keeps this (1+1)D
system from roughening which would otherwise occur at non-zero temperatures for a one-
dimensional system The toughening temperature TR is that temperature at which a given
crystal face can no longer maintain a faceted surface. At and above TR terrace atoms
activate without effecting the total system free energy so that the facet which was present
below TR becomes lost. The roughening temperature concept will be discussed in more
detail later. Since the model is at low temperature and we are neglecting the effects of
evaporation-condensation, the total number of atoms in the system is conserved.
Evap oration-condensation and volume diffusion are also negligible [PASH6 6] for

temperatures well below the melting temperature as is the case here.

 

 

 

 

 

 

90

The energies controlling the temporal dynamics are the terrace diffusion energy,
E, and kink site activation energy, Ek, as shown in Figure 5.1. Since Ek >> Et, kink site
activation times are much longer than the diﬁusion times of atoms on open terraces. In
order for the random walk time of a single adatom on a ﬂat, Open terrace, deﬂ‘, to be very
fast compared to the kink site activation time the condition which must be satisﬁed is
Lze'3Et << eBEk where L is the terrace width in units of the lattice constant. This
condition leads to B Z ln(L2)/Ek since Et is small compared to Ek. With L m 10 — 100
this yields kBT < Ek/ 10. Recent calculations [LIU92] estimate, for low-index Ni surfaces,
Ek z 0.7 — 0.8 eV so that T S Tmelting/Z corresponds to a single walker on a terrace at
any given time. Experimentally, detachment times are found [BART93, POEN92] to be
much longer (roughly 100 to 1000 times) than diffusion times. Thus, an atom random
walks over a terrace in a time much faster than the time between subsequent kink
detachment events, Tdet- In addition, these two processes occur simultaneously so that the
time unit may be taken as Tdet- If there are N active sites (kink sites which may move) the
average time between kink site activation events is 'c = Ida/N = MN. Thus, after an atom

is moved the simulation time is incremented by an amount I = l/N.

5.3 MOVElVIENT TRAJECTORIES AND PROBABILITIES

Figure 5.2 shows a typical portion of a stepped surface encountered during the
decay process along with the relevant distances and possible movement trajectories of the
solid kink site atom shown. The position one lattice spacing to the right of the kink site is
a well site as it is a position of higher binding energy than anywhere else on a terrace.
Well sites are empty surface sites with two NN and two NNN's. L, and LI denote the
terrace widths to the left and right, respectively, of the activated atom The trajectories
labeled 1, 2, r, and 5 indicate the four diﬂ‘erent possible paths which might be taken by the

activated atom

 

 

 

91

 

,__. f 1 (r) = right

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L (1)1de ll
I Ll l IJJ
Lg 9% LI’

 

 

 

 

 

 

 

 

 

 

Figure 5.2 Typical environment of an activated kink site atom (shown as the solid
square). The four possible movement trajectories and the terrace widths, as discussed in
the text, are also shown.

An activated atom such as that shown in Figure 5.2 will follow one of four
trajectories. Path number 1 involves activation of the atom out onto the terrace, a
subsequent random walk on this terrace, including possible reﬂections due to the
Schwoebel barrier at +Lr, and ﬁnally a return to the original site. Path 2 is similar to path
1 except that here the kink site activates up onto the top terrace. After a meandering
random walk on this upper terrace the atom ﬁnally overcomes the barrier near where it
started and returns to its original position. Paths 1 and 2 result in no contribution to the
net mass ﬂow of the system as structurally the system is unchanged after either of these
two movements. Path r results in the activated atom passing over the barrier at +Lr and
incorporation into the step edge at +Lr. Finally, path 1 entails movement onto the upper
terrace and inclusion at the well site located at —L¢. In all that follows the four paths 1, 2,
r, and twill be designated by the subscripts 1, 2, r, and a, respectively.

Due to the presence of the Schwoebel barrier near a kink site, activation out onto
the lower terrace (to the right in Figure 5.2) is more likely than activation up onto the
upper terrace (to the left in Figure 5.2). As can be seen in Figure 5.1, activation to the

upper terrace requires the atom to overcome a barrier which is larger, by an amount A,

 

 

 

92

than that to activate onto the lower terrace. Activation out onto a terrace occurs with
probability 0 while activation onto the upper terrace has a probability U. Clearly, U and
0 must satisfy the condition that U + O = l. The barrier presence causes 0 2 U with the
ratio of the two probabilities given by the Boltzmann factor, U/O = e_l3A. This barrier
also acts to reﬂect walkers incident from the upper terrace side since at the step edge the
walker sees a larger barrier, again by an amount A, to move down the step than to reﬂect.
Thus, we deﬁne the reﬂection coeﬂicient, k, via U/O = e‘BA = 1 — k. Since U + O = 1,
O = l/(2 — k) andU = (1 — k)/(2 — k). If, for example, A = 0 thenU = O =1/2. For
the measured value [TSON72] ofA = 0.052 eV for Re on W(123) at 364 K, k w 0.8 is
found. A

The behavior of a one-dimensional random walker in the presence of reﬂecting
and! or absorbing barriers has been solved exactly by Chandrasekhar [CHAN43,
EHRL66b]. Our method is much simpler than Chandrasekhar's in that straightforward
rate equations are used to determine the effect of these barriers on a random walker as
opposed to a detailed statistical analysis. Prior to development of the rate equation
method described below, Professor Hetherington [HETH93] formulated an equivalent,
albeit more mathematical, scheme of determining the various movement probabilities.
This formulation treats each walker as a Markov process and used the transfer matrix
method [DOYL84] to ﬁnd the relative movement probabilities. This Markov chain
method was much more complicated than the rate equation approach and, thus, dropped
in favor of a simpler, but equivalent, approach. As fate would have it, during the writing
of this thesis yet another, even simpler, method was discovered. This scheme will be
discussed brieﬂy in Section 5.7.

Consider paths 1 and r as shown in Figure 5.2. This situation is shown along with
a coordinate system in Figure 5.3. Path 1 involves movement of the black activated atom
from x = 0 out onto the terrace and a subsequent return to x = 0. Path r, however, occurs

if the atom makes it over the barrier at the step edge between x = L — 1 and x = L.

 

 

 

 

93

 

 

Figure 5.3 Coordinate system for an adatom on a terrace of width L.

I will now set up and solve the rate equations needed to ﬁnd the movement
probabilities for paths 1 and r. These probabilities are denoted by P1 and Pr, respectively.
Following activation of a kink site at x = 0 we have the case depicted in Figure 5.3. In
determining the movement probabilities it is useful to envision a continuous stream of non-
interacting atoms emanating from a source at x = 1 and walking amongst scattering
centers at x = 1, 2,..., L —- 1 prior to capture at x = 0 or x = L. The probabilities P1 and Pr
will be given by the fractions of atoms absorbed at the sinks, x = 0 and x = L, respectively.
Let c denote the rate of walker production at x = 1. By Ni we represent the number of
atoms arriving per unit time at x = i. The absorption rates at x = 0 and x = L are a0 and
aL, respectively. Conservation of matter requires that c = a0 + aL. The probabilities are Pr
= 0(aL/c) and P1 = 0(aO/c) with 0 being the probability of activation onto the lower
terrace as described earlier in this section. The rate equation for x = 0 is

C N
( ) 0 2 2

This is because half of all the atoms at site x = 1 will move to x = 0. For x = 1 only atoms

scattered from x = 2 are incident so that
(5.2) N1 = —-—.

For x = 2 we have

(5.3) N2 =

 

 

 

 

94

and forx= 3,

£2,311,

5.4 N --=
( ) 3 2 2
This pattern may be continued for subsequently higher values of x. The absorption rate at

x = L is given by
N
(5.5) aL =—sz—1(l—k)

since only a fraction (1 — k) of atoms at site x = L — 1 make it to x = L. Equations (5.1)
to (5.4) may be combined to yield the following recursion relation for Ni,

(5.6) Ni =2iaO—2(i—l)c, i=2, 3, 4, ...,L— 1.
Insertion of (5.6) into (5.5) yields

(5.7) aL =(1—k)[(L—l)a0—(L—2)c].

Finally, since c = a0 + aL, equation (5.7) may be solved for aL/c,

(5.8) 11:: l’k .
c 1+(l—k)(L—1)

 

This was for a general terrace of width L whereas in the simulations this width is denoted
by Lr. Ifwe now deﬁne nr = 1 — ( UL), and multiply equation (5.8) by 0, then Pr may be
expressed. as

__0_ <1—k>

(5'9) " ’ Lr (l—nrk)‘

Now, since PIf + P1 = 0, equation (5.9) may be used to ﬁnd

(510) P =0m(1-k)+k(1-nr).
1 l—nrk

Thus, we have the two probabilities involving kink site movement out onto the

lower terrace. Using the procedure outlined above the probabilities for trajectories 2 and 8

in Figure 5.2 may also be calculated. These probabilities turn out to be

THU-k)
(1-mk)

(5.11) P2 =U

 

 

 

95

and

-2 1
Lia—116k).

(5.12) P)
Equations (5.9) to (5.12) obey P1 + P2 + Pr + Pa = 1 as they must.

Other environments occur in which an activated kink site atom does not have an
environment such as that shown in Figure 5.2. For example, at the extrema of the
corrugated surface atoms may, instead, be caught between two reﬂecting or two absorbing
barriers as illustrated in Figure 5.4. These environments are simply taken care of in the
rate equation approach by effectively changing the boundary conditions (similar to a0 and
aL above).

Consider the two situations shown in Figure 5.4. The case on the left corresponds
to activation of a kink site onto the topmost terrace, which subsequently moves down the

step at the other end with a probability of

P _ U
t_1+k+(1—k)L,'

 

(5.13)

The case shown on the right of Figure 5.4 corresponds to a walker trapped between two

absorbing barriers. In this case the movement probability is,

 

(5.14) 1)r =

 

 

 

 

J i ——_l-I:———>f
" IF ral

 

 

Figure 5.4 Environments of kink site atoms in the topmost (left) and bottom (right)
terraces.

 

96

The ﬁnal environment that must be considered is that of an isolated walker atop a
terrace. This occurs as the ﬁnal step in the disappearance of a terrace. When a dimer
rests upon a terrace, eventually one of the two will be activated and may move away from
its partner. When this happens, the single remaining atom on the terrace will immediately
diffuse off the terrace as shown in Figure 5.5.

Here, the single walker is trapped between two reﬂecting barriers. Using the

appropriate rate equations this single atom has a probability

_ l+LA1—k)
r"2+(L,+L,)(1—k)

 

(5.15)

of being incorporated at the step to its right. The walker will move to the left side step

with probability Pr = 1 — Pr,

 

l—l

 

 

L

 

 

I? lit

K _
I -
L, L
a

 

 

 

 

 

V
LLeelle

Figure 5.5 Creation and disappearance of an abandoned adatom on a top terrace.

 

 

 

Thus all activated atom environments have been considered and the appropriate

probabilities found. These probabilities, equations (5.9) to (5.15), govern the evolution of
the entire system For a given proﬁle the only user-speciﬁed parameter in these

ent k. The parameters 0 and U are only
k)/(2 — k).

probabilities is the step edge reﬂection coeﬁici

dependent upon k and are given by O = 1/(2 — k) and U = (1 —

 

97

The proﬁle does not always consist of widely spaced steps. Situations may arise,
usually with the beginning proﬁle, where narrow terraces exist. For example, the (11) face
is a stepped surface with terraces of width one and steps of height one. These situations
must be handled careﬁilly. In these simulations the (10) face is stable since no embedded
terrace atoms (those having two in-plane NN sites) are allowed to activate. The (l 1) face
is taken to be stable as well in order to imitate the behavior of a triangular lattice. This
also excludes formation of steps of height two which are energetically costly. Atoms
diffuse over the (11) face just as they do over the (10) face. Thus, atoms lying entirely
within the (11) face are not kink sites. These conditions result in this square lattice
mimicking a triangular lattice. A schematic of the two equivalent lattices is shown in
Figure 5.6. The two well sites have the same binding energy so that on average they will
have equal occupation probabilities. Thus, with the (11) face stable, the probabilities of
movement are not based on the distances to the nearest steps, but, rather to the nearest

well site locations. That is, Lr and Lt are actually the distances from the activated site to

the nearest well site.

 

 

98

 

W
00000000

000000

000000 X
0.000.000...
OOOOOOOO

 

= kink atom

x = well site

 
  
  

 

 

 

 

 

 

 

 

 

 

 

>X

Figure 5.6 Equivalency of triangular and square lattices along with the respective adatom
binding energy curves for each.

 

 

 

 

 

99

5.4 SIlVIULATION ALGORITHM

With the dynamical rules outlined above I now discuss the Operational structure of
the simulations. Brieﬂy, an active kink site is chosen at random and given the opportunity
to move. If the movement occurs the program up dates the proﬁle information and system
time before another site is chosen. This continues until the proﬁle's evolution is complete.

The program consists of two arrays containing all of the required information. The
program begins by ﬁlling a proﬁle array with the initial height vs. position information,
z(x, t = 0). In the environment array, all kink site and nearby well site positions are stored.
This array has the following information for every kink site: the closest well site positions
on each side of the kink atom and whether these well sites are up steps (absorbing barrier)
or down steps (absorbing barrier alter a step down). This is all the information needed to
determine an atom's movement if chosen.

Once the initial information is determined, the system begins the evolution process.
Using a very reliable random number generator [CHAW94], a kink site atom is chosen at
random The local environment of this site is found in the environment array and the
probabilities for its available types of motion are calculated. A second random number, r,
0 < r < I, is generated to determine which trajectory the chosen atom follows. Ifr < Pr,
path I is ch0sen, whereas if r > 1 — Pt path 5 is taken. Otherwise the atom remains
stationary (since paths 1 and 2 result in no movement). The chosen atom is then moved to
its ﬁnal position at +Lr or —L3. Two things happen prior to movement of the next atom
First, the position of the original kink position is deleted as a kink is no longer there and
the ﬁnal position is also removed as a well site since it is now occupied. Second, the
environment array must be corrected to reﬂect the changes caused by the last movement.
This includes updates for possible new kink and/or well sites near the initial and ﬁnal
positions. Regardless of movement, the simulation time is increased by T = (number of

kinks)_1, which is the mean time between detachment events in the entire system in units

 

 

 

 

 

100

of the kink detachment time. After every movement a check occurs to see if the previous
movement resulted in an abandoned atom being left on a terrace. Ifthis is true, equation
(5.15) is used to move this isolated atom and the arrays are updated as necessary. After
these updates, and checking for abandoned adatoms, the system is ready to move another
kink site. This cycle is repeated until the simulation is a) manually ended or b) the proﬁle
reaches a preset amplitude and stops automatically.

Periodically throughout the course of a run the proﬁle is analyzed. The recorded
information includes the average peak-to-peak amplitude and the Fourier transform
coefﬁcients of the average proﬁle (both averaged over the m periods in each system),
which were recorded at preset times along with the simulation time. The area difference
between the proﬁle and a ﬂat surface as well as the system roughness are also recorded.
In addition, the average proﬁle is recorded at ﬁve diﬂerent stages of the decay. These

data provide the results presented in Section 5.6.

5.5 CON T INUUM MODEL AND PREVIOUS RESULTS

For systems above their roughening temperature a continuum approach may be
used to solve the time dependence of the proﬁle evolution. To promote matter tranSport a
driving force must be present. For example, the driving force for heat conduction is a
thermal gradient. In the case of surface diffusion the driving force is a gradient in the
chemical potential or, equivalently, the surface curvature "seen" by a surface atom By J

we denote the surface current (number/cm2 - s) of diffusing atoms. For an areal density

(number/cmz) n and an average drift velocity v we have the following relationship,
(5.16) J = nv.

Now, the drift velocity v can be related to the driving force F through

(5.17) v = umF

 

 

 

 

 

101

where pm is the particle mobility. The Nemst-Einstein relationship [FEYN63] relating the

mobility to the diffusion constant and temperature is,

(5.18) D = ukaT.
Thus,
(5.19) v = (D / kBT)F = (D / kBT)Vu

where u is the position dependent chemical potential. As v refers to the average velocity
of atoms over the surface, Vii = du/ds with 3 representing the coordinate along the
surface. Since we are dealing with a closed system we must conserve matter. To do this,
we begin with the continuity equation,

(5.20) VJ+E= 0.
dt

Applying equation (5.20) to equations (5.16) and (5.19) leads to

2
(5.21) D” d“ ——= —9.
kBT (152 dt

It is possible to relate the chemical potential u(x) to the surface curvature K(x).
The calculation [MULL63] uses a Lagrangian formalism to minimize the ﬁee energy for a
given proﬁle shape. Consider a two-dimensional curve, 2 = z(x), its associated curvature
K00, and surface energy 7(0), where tanﬂ = dszx. The quantity minimized is fy(s) - ds
where s is the coordinate along the surface. This energy is also expressed in terms of the
integrated chemical potential per atom along the surface and the chemical potential

it = u(x) which minimizes this energy is calculated. The result is,

62 0
(5.22) MK): QK(X){7(9)+ 5,43% ) .
where Q is the atomic volume. This result is just a special case of the result for the
chemical potential of an ideal gas. For an ideal gas, the chemical potential is expressed
using the Gibbs-Thomson result,

(5.23) u(x) = NO + QK(X)Y(9),

 

 

 

 

 

102

where #0 is the constant chemical potential of a vapor above a ﬂat (K(x) = 0) surface.
Thus, for materials with isotropic 7(0), such as liquids, the two expressions agree.

Since the proﬁle height is the experimentally measured quantity we need to express
the governing equations in terms of z(x) rather than u(x). The curvature K(x) is, in terms

of the function z(x),

—Z

. K = .
(5 24) (X) (1+(z.)2)3/2

 

where 2' denotes the derivative of z with respect to x. Since the gradients of u are with
respect to the surface coordinate s we change variables from this coordinate to x by using
the projection of a surface element onto the x— axis,

(5.25) ds= dxj/1+(z')2

and the chain rule

d dx (1
5.26 ——= -— —.
( ) ds ((1 dx

Multiplying equation (5.21) by Q and applying equations (5.24) to (5.26) we obtain the
following differential equation describing the proﬁle's evolution [BONZ84a]

25g}; 62:1- at
at kBT ’1+(Z')2 6X2 (1+(zr)2)3/26x a

with u(x) given by equation (5.22). This equation is, understandably, unsolvable in closed

(5.27)

 

 

form for a reasonable choice of anisotropic 7(0) let alone the simpler case of isotropic
7(0). However, in the small slope approximation, 2' << 1, and for the case of isotr0pic
surface energy, 7(0) = 7, equation (5.27 ) simpliﬁes to the following form,

22. _._ 1113027

(5.28) at k T
B

 

(_Z1HI) = _lelll.

The solution to this separable differential equation is a shape preserving exponential decay.
For an initially sinusoidal proﬁle, z(x, t = O) = Aosin(kx), 20% t) is given by
(5.29) z(x,t) = e_Bk4tAOsin(kx) = z(x, 0)e—Bk4t

 

 

 

 

103

where k = 21:0» is the wavenumber and not the reﬂection coefficient. Thus, a plot of
1n(A(t)/AO) versus time t should result in a straight line of slope —Bk4 so that the "decay
time" scales as 1' ~ 2.4. Equation (5.28) can also be applied to other geometries by
changing the boundary conditions. For example, this same exact method is used to derive
the time dependent depth and width of a grain boundary groove under the action of
surface diﬁusion and agrees with experimental observations.

Proﬁle decay has received much more theoretical attention than experimental. The
main reason for this discrepancy is that reliable surface diffusion measurements are very
difﬁcult to perform Surface contamination and impurities are known to signiﬁcantly
effect surface diffusion rates and surface structure. Thus, stringent experimental
conditions such as UHV and extensive sample cleaning are required. Experimental
techniques such as in situ optical interference micros00py and laser light diffraction, as
well as ex situ electron microscopy have been employed to observe proﬁle relaxation.
However, most of these experiments have been aimed at obtaining information on the
static proﬁle shape instead of quantitative information regarding the temporal dynamics.
One instance where the decay time was observed to follow a 2.4 dependence was the laser
light scattering experiment of Keefe et al [KEEF93]. Many of the theoretical treatments,
but few experimental studies, have dealt with the case below TR. As will be discussed
below, the difﬁculties in properly treating the situation on a lattice are formidable and
many problems remain.

Another important prediction of the continuum model is that the proﬁles are shape
preserving as indicated by equation (5.29). That is, the initial proﬁle shape should be
completely preserved throughout the proﬁle's evolution if only one wavelength is present.
Proﬁles created on macroscopic surfaces both above and below their roughening
temperatures have, indeed, exhibited shape preservation throughout the decay process.
For example, Yamashita et a] [Y AMA81] studied gratings produced on the (110), ( 100),
and (111) faces of a Ni crystal. The low density Ni(110) face has the largest surface

 

 

 

 

 

104

energy of these three and, thus, the lowest roughening temperature. Proﬁles prepared on
the (l 10) face maintained their initially sinusoidal shape. In contrast, gratings prepared on
the other two faces resulted in trapezoidal proﬁles with ﬂattened tops and bottoms. The
(100) and (111) macroscopic samples were studied below their respective roughening
temperatures. The ﬂattened tops and bottoms correspond to the macroscopic orientations
which are also directions of lower surface energy. These same shapes were observed
throughout the relaxation process.

The reason for the appearance of these ﬂat portions is that different crystal faces
possess different surface free energies. The low energy faces are represented by cusps in
the Wulff plot. The Wulﬂ‘ plot [WULF01, HERR51, MULL63] is a polar plot of the
surface free energy as a function of surface orientation. Certain orientations correspond to
cusps in 7 where the derivative 87(0)/80 becomes discontinuous. Since these orientations
have a smaller contribution to the overall free energy of the sample, a crystal will
preferentially form areas of these orientations which are as large as possible. These cusps,
and hence, the facets, persist for temperatures up to the roughening temperature TR of the
face under consideration at which point facets of this orientation disappear. As T
approaches TR from below the facet size shrinks according to a Kosterlitz-Thouless type
behavior [VANB87]. Above TR the cusps in the surface free energy disappear in favor of
blunted minima.

The presence of the cusps in the Wulif plot can be simply understood. Consider an
atomically ﬂat crystal surface (a facet) below the roughening temperature. This facet has a
surface energy 70. In order to change the overall macr0500pic orientation of this surface,
a periodic spacing of steps must be created. The formation of such a vicinal surface
requires an amount of energy proportional to the density and length of the steps. This
energy is expressed in terms of an energy per unit length of the step edge. To produce a
small orientation change (equivalent to a large step spacing) a non-zero amount of energy

is required. This is the origin of the sharp cusp in the surface energy. The reason for the

 

 

 

 

 

105

disappearance of facets at TR is due to the disappearance of the step free energy per unit
length at TR [WEEK80]. This means that steps of all lengths can be thermally excited
with no affect on the system's surface free energy.

Below the roughening temperature the details of proﬁle decay are not well
understood. The discontinuity in 07(0)/80 leads to a singularity in 627(0)/602. This
singularity makes equation (5.27 ) unsolvable. Bonzel et al shed much light on the proﬁle
shape below the roughening temperature in a series of papers [BONZ84a, 84b, PREU86].
Their approach was to handle the divergent 627(0)/6‘02 singularity through a mathematical
approximation. While they were able to explain the experimentally observed trapezoidal
shapes they were unable to explain the ob served relaxation dependence on wavelength.
However, the observed relaxation of a faceted, trapezoidal, proﬁle was found to be non-
exponential and shape preserving.

This decay time dependence on k4 can be simply understood by recalling some of
the results presented in Section 3.5 regarding the coalescence time for two spheres of
radius R The coalescence time Tcoal was proportional to R4 so it is not surprising that
here the time scale behaves as the fourth power of the typical length scale 2» of the
problem The argument outlined above showed that theoretically the amplitude decays

exponentially with a characteristic time I ~ 7&4 in the high temperature regime.

5.6 SINIULATION RESULTS

The time equal zero proﬁle is a sinusoid: z(x, t = 0) = CINT[A0sin(21tx/?t)] where
A0 is the initial amplitude and 7» is the wavelength. The total system size is T and
0 s x s T— l. The CINT(y) function takes the value y and rounds it off to the nearest
integer value. For example, ClNT(—2.4) = -—2. Thus, the system consists of m = T/l
periods. The initial amplitude A0 is chosen so that no face steeper than the (11) face is

present. This corresponds to a maximum slope of one. For a given wavelength 2 the

 

 

106

maximum allowable amplitude consistent with this condition is 2nAmax/9t = 1 so that

Amax = M21r. The only other parameter speciﬁed is the step edge reﬂection coefﬁcient k.

This coefﬁcient k is related to the temperature by A/(kBT) = -—ln(l — k), which comes
from the relation 1 — k = e‘pA. Usually, the total system size T was 8192. The ratio of
AO/Amax studied was typically l/4 to l for the wavelength range 32 s 7» s 256. This
range of AO/Amax was chosen to avoid initial proﬁles with amplitudes of one or two. For
it = 256 this range included Ao/Amax 2 1/16. Wavelengths larger than 256 were not
used due to the intolerably long computation times required. With such a large system
size our data is averaged over many periods. The reﬂection coefﬁcient k used was
generally 0.9 which corresponds to kBT = 0.434A. In the remainder of this section I will
discuss the inﬂuences of A0, 7», and k upon the system decay. I will also attempt to
discuss some of these results and how they relate to other studies by various authors.

The most useﬁrl of the data is the peak-to-peak amplitude versus time. This
amplitude will be denoted by 2A = 2A(t). The parameters A0, 7», and k are all
independent. This section will cover much of the available parameter space of these three
variables. However, the main results presented cover the amplitude evolution as a
function of time while varying 9» and k.

One striking result of these simulations is shown by the data in Figure 5.7. This
ﬁgure is a plot of the peak-to-peak amplitude vs. time/k3-5 for three different wavelengths
it and three different starting values of Ao/Amax = 27tA0/9t for k = 0.9. It is apparent from
this ﬁgure that, at ﬁxed ratio of initial amplitude to wavelength, all the curves collapse
onto one "universal" curve as time/(k355iO-1). Figure 5.8 is the same data plotted vs.

time/7L4 which is the scaling prediction [MULL59] for the T > Troughem-ng regime

described in Section 5.5.

 

 

 

 

107

 

 

 

 

 

 

 

 

 

 

>< _
cu
E _.
<1:
2 _
\(I —r
0.000 0.004 0.008

TIME/k3-5

Figure 5.7 Amplitude decay, A(t)/Amax, vs. Time/2.3-5 for step edge reﬂection coefﬁcient
k = 0.9 and wavelengths A = 64, 128, and 256.

 

 

 

 

 

 

108

 

 

 

 

 

 

 

 

 

 

 

0.8 :33 k = 0'9 -*
0 A i
E 0.6 l‘ l300 A A 64 "
< D000 A 0 1 28
’3 P no 00 AAA '3 256 -
IE 0.4 43939 on 0o A ‘-
D A I: 0o A
" DO A an o A ‘J
u o o A
P D o A no 00 AA
0.2 FEB DU OOAADD%AOO A A
o
k 8830000 0000 A5209 A AA 1
0.0 L 008° 0003330 AA AA;
0.0000 0.0004 0.0008

TIME/7.4

Figure 5.8 Amplitude decay, A(t)/Amax, vs. Time/76‘-O for step edge reﬂection coefﬁcient
k = 0.9 and wavelengths 7» = 64, 128, and 256.

 

 

 

 

109

Figures 5. 7, 5.9, and 5.10 show that our simulations exhibit a non-exp onential
decay of a trapezoidal shape which shows shape preservation as well. Figure 5.9 shows
the time evolution of a sample proﬁle. The shape preservation is exhibited in Figure 5.10.
This plot shows curves 2 through 5 of Figure 5.9 scaled to have the same area. Since all
ﬁve traces lie nearly atop one another, the trapezoidal shape is preserved throughout the
decay process. The tops and bottoms of the proﬁles are ﬂat to about one lattice constant
over a width of roughly 20 lattice constants. As can be seen in Figure 5.9, the ﬂat regions
are smoothly joined to the sloped hillsides by rounded regions. The shapes near the edge
of a ﬂattened extrema follow a power law z(5x) ~ (8x)°‘ with on w 1.6 over a distance of
roughly 10 to 15 lattice constants. Experimental observations [I-IEYR87, MET089]
indicate that faceted regions on equilibrium shaped metallic crystals meet at sharp corners
only for temperatures near the roughening temperature of the facet. Below TR the proﬁle
shape near the facet edge is found experimentally to behave as z(ﬁx) ~ (8x)°‘ with or m 1.6
[ROTT84, MET087] in agreement with our observed morphology. This prediction of
or m 1.6 may be derived from a surface energy function containing step-step interactions
(the G3 term mentioned below). Our simulations lack long-ranged interactions between
steps and it is, thus, probably coincidental that near the edge of the ﬂattened tops of our
trapezoidal proﬁles we observe rounded regions which follow the or m 1.6 prediction.

An important difference between our simulations and others [RETT88, OZDE90]
is how step-step interactions are treated. Prior efforts include such interactions directly in
the chemical potential and surface free energy expressions. These repulsive interactions
provide a continuous driving force for proﬁle decay. This can be seen by looking at the
expression for the free energy per unit area projected on the x-y plane for the case of a
(2+1)D system where z = z(x, y). Let 7(z') denote the free energy per unit area normal to
the proﬁle and G(z‘) denote the projection of 7(z’) onto the x-y plane. Thus, 7(z') is,

G( 2')

(5-30) 7(2')=

 

 

 

 

 

110

 

J

L

]

 

 

LIJLJLlJJ

l

 

 

 

-50 4 l r l r l m I
0 64 128 192 256

X

 

Figure 5.9 Time evolution of a system with 2» = 256, A0 = 40.7, and k = 0.9 shown at
various stages of evolution. Each proﬁle is the average of a system containing 32 periods.
Beginning with curve one the simulation times of the different average proﬁles are 0,
1.35x105, 3.82x105, 9.80x105, and 1.45x106.

 

 

 

 

 

lll

 

 

O

 

 

 

-50 I r l r l r l r I
0 64 128 192 256

X

Figure 5.10 Curves two through ﬁve of Figure 5.9 normalized to have the same area as
curve two in Figure 5.9. The self-similarity of the curves indicates the shape preservation
quality of the proﬁles.

 

 

 

 

 

 

112
Below TR, G(z') takes the Gruber-Mullins-Pokrovski-Talapov form [GRUB67, RETT88],
(5.31) G(z’)=G0 +g|z‘l+%G3lz’|3+....

Here, Go is the free energy of the macroscopic (ﬂat) surface. In the second term, g is the
step energy per unit length and lz‘l is the step density on the surface. The |z'|3 term is the
step-step interaction term This term may be due to direct energetic interactions such as
strain or dipole-dipole type interactions or in a (2+1)D system it also arises from entropic
repulsion between steps. Consider the case of three steps of the same sign which are free
to move about, but unable to cross, and free of direct energetic interactions. The center
step has a higher entropy when the outer steps are far apart. This arises from the ability of
the center step to wander, thus having a larger conﬁgurational entropy. With the y-
direction into the page in Figure 5.2 this simply means that the step position x may vary
with position y along the step edge. When the two outer steps are close, this available
wandering of the center step is decreased. Since the free energy has the entropy in the
form —TS, a larger entropy is energetically favorable. Thus, the outer two steps are
effectively repelled. All previous simulations of proﬁle decay have included this repulsive
term which serves to drive the system to widely spaced steps, i.e. a ﬂat surface.
Experimental studies of terrace width distributions yield results which are consistent with
the existence of repulsive interactions amongst steps with a lz'l3 dependence [WILL91].

In the present simulations long-ranged interactions between steps are absent.
However, there is a contact interaction between adjacent steps (for step separations of one
lattice constant) since double height steps and overhangs are not allowed. Thus, two steps
are unable to cross.

Consider two completely independent random walkers on a line which may pass
through one another. The absolute value of their average separation (averaged over all
possible paths) will increase as (time)1/2. Now, if these two walkers are restricted so that

they cannot pass through one another (they simply bounce off one another like billiard

 

 

 

 

 

 

113

balls) then the average separation still increases as (time)1/2. This behavior is the same
since the process of passing through one another and reﬂecting off each other are identical
except for an exchange of identity labels.

Two adjacent steps which cannot pass through one another behave in a manner
similar, but not identical, to random walkers. Simulations of the separation of two
isolated steps (with k = 0) using the present algorithm yield an average terrace width
between the two steps which grows as (time)“ 3. This weaker time dependence results
from the fact that often, when a kink site activates it returns to its original site after a
random walk, and the step does not move. Only when the activated walker makes it to an
adjacent step without returning home does the step move. This step movement becomes
less frequent as the steps separate. The cause of the separation increase is that the
separation is prohibited from being less than one lattice constant. This skews the
likelihood of the steps to have a larger average separation. Occasionally, two steps collide
and have no option but to increase their separation from one to two lattice constants. This
effect is, however, small. So, while there is an entropic repulsion present in our
simulations, it is a weak driving force for the proﬁle decay. A long time is needed for
adjacent steps to explore phase space and discover the effects of reﬂecting away from
neighboring steps. Additionally, this step-step repulsion is only present on the sloping
hillsides of the corrugated proﬁle. Steps of unlike sign on the top terraces and bottom
valleys do not have any repulsion since when two steps attempt to cross in these
situations, they simply annihilate. This annihilation is irreversible. Once a terrace or valley
disappears, it cannot re-form Thus, although this small repulsive effect pushes the top
terraces and bottom valleys to be eliminated, it is the lack of island nucleation which
makes it permanent. We have been unable to relate the (time)1/ 3 spreading to the I ~ A35
decay time scaling.

As the exchange of atoms between any two terraces is random, we should expect

the top terrace to decay in a random fashion. For a top terrace of width W, ~W2

 

 

 

 

ll4

exchanges should be needed before the top terrace disappears. This same behavior is true
for the bottom terrace (which is ﬁlling in rather than decaying). This type of behavior is
observed in our simulations. Since the simulations take place at low temperature island
nucleation is forbidden. Once a terrace disappears, or ﬁlls in for the case of a layer at the
bottom of the proﬁle, it cannot reappear. With no energetic interactions, the system
energy is determined entirely by the number of steps. The disappearance of a terrace
reduces the number of steps by two. Thus, the energy decreases in a discontinuous
manner until the zero energy, ﬂat, state is reached.

As already noted a true continuum approach cannot be used below the toughening
temperature due to the singularities in 7(0). Instead, the problem must be treated on a
lattice. This causes great difﬁculty in applying ideas such as derivatives and curvature. It
is for this reason we were unable to formulate an analytical treatment describing the
dynamics of our model. Such a treatment could help immensely in understanding the
temporal behavior and the proﬁle shape dependencies upon various parameters. Since no
"energy based" concepts were used in these simulations this may be impossible. Thus, the
rest of this section will discuss the remaining main results of these simulations and how
they relate to some of the other analytical and numerical results of other investigators. No
attempt at a ﬁrll mathematical discussion such as those of [RETT88, OZDE90] will be
made.

First I will discuss the scaling of the decay time with wavelength. Figure 5.7
shows that the amplitude decay scales as I ~ 23-5 in a non-exponential way. As noted
earlier, Bonzel et al [BONZS4a, 84b, PREU86] attempted to solve the proﬁle decay
problem below the roughening temperature using a continuum approach. Their results

consisted of a numerical solution of the following differential equation,

(5.32)

a_z_ _(2___n§22DG3)62___(ax2 |z|z")
kBT

 

 

 

 

 

l 15
Though not explicitly stated in the paper, the data in Figure 7 of [PREU86] yield a decay
time which behaves as roughly 23-7. Additionally, their numerical solutions exhibited
trapezoidal shapes similar to those shown in Figure 5.9.
Ozdemir and Zangwill [OZDE90] attempted to solve equation (5.32) numerically
for the case of unidirectional grooves, z(x, y, t = 0) = AOsin(27tx/?t). Using separation of
variables, z(x, y, t) = u(x)w(t), the temp oral and spatial variations of z(x, t) are solutions

of the following equations

 

 

(5.33) i—Yhawz,

and

5.34 21102ch d2<lu(x)2|u"(x))

( > ( kBT ) =au<x).

The separation constant e, parametrizes a family of solutions. Integration of the ﬁrst of

these equations yields,

_U__(X)
(Hit)

Thus, the proﬁle should decay with time as t‘1 for ﬁt >> 1. Our simulations instead

(5.35) 2(x,t)=

exhibit the temporal behavior shown in Figure 5.11 which is a plot of A(t)/Amax vs. time
for the case of AO/Amax = 0.25, k = 0.9, and 7» = 256. For the majority of the decay the
amplitude behaves nearly as (—time) as shown by the straight line which has been offset
from the data, although there is a slight curvature. The deviations of the data and a
straight line are generally $0.01. This behavior is also seen for other values of AO/Amax

and 7t. Careful inspection of Figure 5.7 shows that this behavior occurs over the majority

of the decay cycle for nearly all chosen values of 7» and AO/Amax, especially for smaller

A00» values.

 

 

 

 

 

 

116

 

A/Amax

  

 

l

 

 

 

0'05: (-3.85x10'7)TIME D :
0.00 ' 4 ‘I
0.0 3.0 6.0

TIME (105)

Figure 5.11 Amplitude decay, A(t)/Amax, vs. simulation time for a system with 7» = 256,
A0 = 10.2, and k = 0.9 showing a decay which behaves nearly linearly with time. The
solid line has a slope of—3.85x10—7.

 

 

 

117

In addition to holding the ratio of initial amplitude to wavelength (AG/7t) constant
we have studied the behavior at constant initial amplitude A0 and constant reﬂection
coeﬂicient value while varying the wavelength. The temp oral evolution of this sequence
should be different for two reasons. By holding the amplitude constant and varying A. the
system is forced to change 1) the average terrace width and 2) the initial number of kink
sites. At constant A00» the average terrace width and number of initial kink sites is fairly
constant for different K‘s since the ratio AO/lt determines the slope of the system The
scaling behavior of the amplitude decay at constant A0 turns out to be signiﬁcantly
different than at constant A00». As shown in Figure 5.12 the data at different 7» collapse
to a single curve when scaled as time/99754501. The two curves shown correspond to
two different values of A0. Each wavelength corresponds to averaging over a different
number of periods since the total system size was always held constant at 8192. However,
the standard deviations of A/AO were generally $0.01 which is smaller than the symbol
sizes used in Figure 5.12.

We can also look at the proﬁle decay at constant AO/Amax and 7» while varying k.
The major difference which results from varying k is the time taken to reach a given stage
of evolution. It turns out that simply multiplying the time values by differing amounts
leads to the A/AO curves falling very nearly atop one another when plotted versus time.
This behavior is shown in Figure 5.13 which is a graph of A/AO vs. time with 7» = 128, a
constant A0 = 10.2, and k E {0, 0.7, 0.9, 0.997}. In making this graph, the time values
of the lower k value plots have been scaled so the curves all nearly coincide with the
k = 0.997 curve. The multiplication factors used were; 26.6 for k = 0, 30.0 for k = 0.70,
18.5 for k = 0.90, and 1.0 for k = 0.997. Thus, the runs using larger values of k required

a much longer simulation time to complete. These runs also took signiﬁcantly longer in

total CPU time to complete as well.

 

 

 

 

118

 

 

 

 

 

 

 

1.0 —g
A
- ‘5 -
_ 6‘ o 7). = 64
0.8 8g“ A 1 28 A
‘ 98‘: n 256 ‘
O 0.6 - rzrEB ’6 0 1 28 -
S _ ~ A 256 _
< 8 ‘A.
0.4 — 0° 5“ -
- 423a,, ‘,<— A0 =10.2 -
0.2 — / ‘3 4
- A0 = 5.1 may ”he“. -
M3 “‘bbmo
0'0 I . I . l . L . L-

 

0.00 0.05 0.1 0 0.1 5 0.20
TIME/r275

Figure 5.12 Amplitude fraction A/AO vs. Time/7&2-75 for A0 = 5.1 and 10.2 at various
wavelengths it.

 

 

 

 

 

 

 

 

 

 

 

 

 

119
1.0—é TS ' ' . ' IT IT IV
r O -
0.8 F gm ‘
_ “In 0 k =O%.(9)97 -
V .
(C) 0.6 - %% A 0.70 -
2 r 5 ° ° ‘
0.4 r °e -1
, e... _
$651
0.2 L 8R? ‘
L 03v An
0 Vega :1 ‘
00L qDO fvgNAADDCID
l r r r I r r L l r r r I
0.0 1 0 2.0 3.0

SCALED TIME (1 06)

Figure 5.13 Amplitude fraction A/AO vs. scaled time at ﬁxed AO/Amax = 1/2 for
k E {0, 0.70, 0.90, 0.997}. The original time values were multiplied by 26.6, 30.0, 18.5,
and 1.0 for the low to high values of k, respectively.

 

 

 

 

 

120

This behavior as well as other properties to be discussed can be qualitatively
understood through an analysis of equations (5.9) through (5.12) which express the
movement probabilities P1, P2, Pr, and P). In doing this I will assume the terrace widths Lr
and L7 to be equal. While this is deﬁnitely not true during the decay it will allow us to gain
some insight into the decay behavior. For this case Lr = L) and Pr = P), for all values of k.
Since these are the only two movements which advance the system's decay, it is suﬁicient
to investigate their behavior and not be concerned with P1 and P2 for the time being.

Figure 5.14 is a plot of Pr vs. terrace width L for different values of the reﬂection
coefﬁcient. One major feature of this plot is the k-dependence of Pr. For example, the
k = 0.99 curve is very nearly independent of the terrace width L. This case is what I will
call reﬂection coefﬁcient limited. The L independence means that even for small terraces
an atom is unlikely to strike the Schwoebel barrier ~ 100 times (since 1 — k ~ 1/ 100)
without returning to its origin. The very small value of Pr (31%) means that many more
attempts are required for an atom to move away from its original position. This increase
in number of attempts required explains why the k = 0.997 curve takes nearly 30 times
longer to decay than the lower k runs in Figure 5.13. More generally we can understand
the need for excessive times needed to complete these simulations. Many of the runs
required anywhere from an hour up to 22 days of CPU time on a 486 class personal
computer. This is because the total probability to move (path 1' or l) is, from Figure 5.14,
in the range of 2 to 30%. Thus, the total probability to not move is generally k70%. The
majority of the time in the simulations is taken in giving various activated sites the
opportunity to move without actually resulting in a movement. This is why sometimes 106
or more inquiries are needed to ultimately move ~104 atoms. An improvement in this
respect could be provided by using a more efﬁcient scheme for moving atoms [CHAS91].
In this algorithm, every movement is one which results in a movement away from the
original site. However, the time increment is not just l/N, but, rather, a properly weighted

time. This weighted time accounts for the average time necessary for various movements

 

 

 

 

 

OJC)

(105

(100

H1

 

 

 

 

 

 

 

WululullllrlTlTrﬁrlTlrll
F 5 ‘
o 35 0 k=0
- , A a .10 1
_ , g: . .30
, no v .50 1
. “9 0 .70 -
F 0.8A90 0 .90
P 083:90 I .8; _
L 8‘290 ‘ ' j
QB‘Z$°0
- 86552929
IIIIIIII Baaaﬁgiggsssl
Ill-Ill... B
T ""lEeegg
r-AAAAAAAAAAAAAAAAAAAAAAAAAAAA
rrrrlrrrrIrrrrlmrrrlrrrrlrrrr

 

0 5 10

15 20 25 3O

TERRACE WIDTH L

Figure 5.14 Movement probability Pr vs. terrace width L for various values of k.

 

 

 

 

 

122

to be successﬁrl. This scheme has been employed in the (2+ 1)D extension of this work
[DUB S94b]. These three-dimensional simulations are still forced to track the entire
random walk of the activated atom, however. Ifthis system is run on a sinusoidal proﬁle
of width one (making it identical to our (1+1)D system), the amplitude decay is found to
decay as time/76‘-5 in agreement with Figure 5.7. While this three-dimensional system is
also somewhat idealized, the agreement between the two methods is reassuring
considering the signiﬁcantly different algorithms used.

Another feature of Figure 5.14 results in the data shown in Figure 5.15. This latter
ﬁgure shows the decay time as a ﬁmction of l — k at constant 7» = 128 and A0 = 10.2.
The decay time is deﬁned as the simulation time when A/AO has fallen to one quarter of its
initial value. For a system of wavelength 7» and amplitude A we can deﬁne an average
terrace width Lavg = lt/4A since there are 4A terraces in a length it. At A/AO = 1/4, this
yields Lavg m 13. In Figure 5.14, with L = 13, Pr has a maximum for the case k = 0.7.
Figure 5.15 indicates that the minimum decay time occurs for k z 0.7. These two results
are in agreement. The higher value of P1. means a slightly higher probability for an atom to
move than for k = 0, for example. Moving down the values of Pt(L = 13) in Figure 5.14
we see that the highest Pr (k = 0.7) corresponds to the lowest decay time in Figure 5.15.
On the other hand, higher k values lead to smaller Pr and longer decay times. A smaller
value of Pr leads to more atoms being interrogated (i.e. given a chance to move) before a
ﬂuctuation occurs which is large enough for a terrace to decay or ﬁll in, thus moving the
decay process further along. Not too surprisingly, the ratio between Pr at two different k
values from Figure 5.15 is, to within about 10%, the same as the inverse ratio of the two
decay times from Figure 5.14 for the same two k values. As mentioned earlier, the higher
values of k correspond to the case of reﬂection coefﬁcient limited motion. The three
curves in Figure 5.14 for k 2 0.9 are nearly independent of L, especially above L = 10.
Thus, we might expect the decay time to depend up on the transmission coefﬁcient 1 — k

for these values of k. The three high k points in Figure 5.15 follow a Tdecay ~ (1 — k)—a

 

 

 

 

 

123

 

. .nurr [Tgml . urrmrrrrﬁlﬁ Iﬁ'rrl TITIrrI .
A

106

 

le/Amax =1/2l

 

N Cal-Fm

U1
' V'Tj'qj'u lﬁlrr ﬁ'lﬁ'l'l'l ITIITF

10

 

DECAY TIME

|r=128J A H‘

 

M 03-501

I LrIrILLrJrI IIJJII J_l rlrlrlrlrl I IIIII

 

 

104 ...r...r 111ml ...mn rurrrl ...mu 111ml

10‘3 2 3 10‘2 2 3 10'1 2 3 10°
1-k

 

Figure 5.15 Decay time plotted vs. 1 — k for Ac/Amax = 1/2 and A = 128. The decay
time is deﬁned as the time at which A/AO = 1/4.

 

 

124

behavior with a m 0.9. This non-linear behavior differs from the linear behavior, a = 1,
one might expect if the decay were not random in nature. The slight upturn of the decay
time when k approaches one is also explained by Figure 5.14. Comparing the curves for
k = 0 and k = 0.1 we see that PI (and thus the total probability to move away from the
original site) increases as k becomes non-zero. This increase in movement likelihood
results in a decrease in the decay time. As is evident from Figure 5 . 15 this effect is small,

but noticeable.

5.7 PARALLEL RESISTOR MODEL FOR P1, P2, Pr, AND Pg

While writing this thesis a method even simpler in nature than the rate equation
approach was discovered. It can be shown [DOYL84] that a random walker trapped
between two absorbing barriers, possibly with repulsive barriers in between, is identical to
an electrical circuit problem consisting Of a battery and a set Of appropriately chosen
resistors. The principal idea of this method is to place a resistor (with a resistance which
is dependent upon the site's neighbors) between adjacent surface sites and determine the
amount Of current ﬂowing in the appropriate directions. This current is generated by a
battery connected to the site Of the activated kink atom The nearest well sites on either
side of the activated kink atom are connected to ground. The amount of current ﬂowing
in a particular direction is directly related to the probability Of a random walker to move in
that direction. Terrace sites with two in-plane NN atoms (such as x = 2 and x = 3 in
Figure 5.3) are connected by a unit resistance. The connection between the top side of a
step and the well site on the terrace below (x = L — 1 and x = L in Figure 5.3) is given a
resistance of (1 —- k)-1, where k is still the step edge reﬂection coeﬂicient. The reason
for this choice of resistance values is related to the movement probability between adjacent
sites. On an Open terrace with two equivalent neighboring sites a walker is equally likely

to move right or move left. Movement down a step occurs with a transmission probability

 

 

 

 

 

 

125

of 1 — k. As will be explained below, the resistance is reciprocal in nature to the
movement probability, thus the (l - k)“1 value Of a step edge resistor.

When the kink site atom in Figure 5.1 activates it ﬁrst moves to a position one
lattice constant away from its starting site. These positions are shown in Figure 5.3 as
x = l for activation out onto the lower terrace or what would be labeled as position
x = —1 if the atom activates onto the upper terrace. With the site-to-site resistance
values designated as above, we imagine the following scenario. At position x = 1 or
x = — 1 (depending upon which direction was chosen for activation) we connect a battery
at some voltage V0. The adjacent well sites are connected to ground. The movement
probabilities we seek are precisely determined by the ratio of the currents which ﬂow
through either path to ground. However, this method can't take care of the activation
probabilities O and U. Given that the atom activates, for example, onto the lower terrace
this method determines the probabilities Of return to x = 0 or to move to the adjacent
terrace.

As a simple illustration of this method consider the situation of paths 1 and r
involving activation onto the lower terrace. The adjacent site pairs (0,1), (1,2), (2,3) ...,

(L — 2, L — 1) are all connected by unit resistors. The (L — l, L) pair has a resistance Of

(1 — k)“1. The resistance between x. = l and x = 0 is, thus, R_ = l and that between
x = 1 andx = LisR+ = (L — 2) + (1 — k)_1. DeﬁningPl = O(p1) andPr = O(pr)the
ratio pl/pr is given by R+/R_. This is because the probabilities p1 and pr are just the
currents 11 and 1,, respectively, which will ﬂow in the plus or minus directions. Since the
voltage drop between x = 1 and x = 0 is the same as that between x = 1 and x = L, the
reciprocal relation exists. Ohm's law leads to IlR_ = IrR+ which is precisely the same

relationship. Thus, the probabilities p 1 and pr are related by the following equation

(5.36) B"—=L—2+—1—-=L—1+—i-.
p1 l—k l—k

 

 

 

126

Combining this result with p1 + pr = 1 and multiplying by the activation factor 0 we ﬁnd
the same results for pr and p1 as those in equations (5.9) and (5.10). This method may
also be used for all other environments discussed in Section 5.3. This method, however, is
much simpler and more intuitive than the rate equation approach.

Computationally this scheme would not offer any increase in eﬂiciency since the
same equations are being calculated and compared to random numbers. However, it is a
third method which yields the same results for the movement probabilities. This method is
not easily extended to the full three- dimensional case mentioned earlier. The mathematical
problem of determining the resistance between an interior point and any perimeter point
on a complicated surface is presently unsolved. Even in cases of simple geometry and/or

symmetry this is generally true.

5.8 SUlVIlVIARY

This chapter has dealt with extensive simulation results Of the surface diffusion
controlled decay of a corrugated surface at low temperature. The results show qualitative
agreement with previous theoretical and experimental studies. The idealized system used
could be extended to the high temperature region by including two effects: island
nucleation and embedded terrace atom activation. The difﬁculties involved in extending
the algorithm to three dimensions have been discussed and require some ingenuity to
overcome.

An analytical treatment Of the decay process was avoided for various reasons.
Appropriate differential equations are difﬁcult to formulate on a lattice. Even if this could
be done, the random nature of the decay would be troublesome to take into account.
Finally, no direct energy concepts such as surface energy or chemical potentials were used
in the course of this work. Without these concepts, formulation of a driving force which

would be necessary for a proper analytical treatment, might be impossible to achieve.

 

 

 

 

 

 

 

 

 

127

One area of possible future exploration using this algorithm might be to study ﬁlm
grth by applying the rules to deposited atoms. Quantities such as island size and
density could be compared to other theoretical and experimental results of epitaxial
growth. To obtain non layer-by-layer grth would require inclusion Of island nucleation
atop an existing island or the substrate. This would also necessitate extension to three
dimensions. While not at all easy, this could prove useﬁrl, especially in light Of the

tremendous amount Of work presently dealing with surface diﬂusion and various metal

ﬁlm growth modes.

 

 

128

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OZDE90
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