ABSTRACT
Title of dissertation: A MACROSCALE PERSPECTIVE
OF NEAR-EQUILIBRIUM RELAXATION
OF STEPPED CRYSTAL SURFACES
John Quah, Ph.D. 2009
Dissertation directed by: Professor Dionisios Margetis
Department of Mathematics
Institute for Physical Science and Technology
Crystal surfaces serve a crucial function as building blocks in small electronic
devices, especially for mobile communications technology and photovoltaics. In the
history of computing, for example, a crucial innovation that hastened the demise
of vacuum tube computers was the etching of patterns on surfaces of semiconduc-
tor materials, which led to the integrated circuit. These early procedures typically
worked with the materials at very high temperatures, where the thermally rough
surface could be modeled from the perspective of continuum thermodynamics. More
recently, with the drive towards smaller devices and the accompanying reduction of
the lifetime of surface features, manufacturing conditions for the shaping of crys-
tal surfaces have shifted to lower temperatures. At these lower temperatures the
surface is no longer rough. In order to describe an evolving surface under typical
experimental conditions today, we need to consider the processes that take place at
the nanoscale.
Nanoscale descriptions of surface evolution start with the motion of adsorbed
atoms (adatoms). Because of their large numbers, the concentration of adatoms is
a meaningful object to study. Restricted to certain bounded regions of the surface,
the adatom concentration satisfies a diffusion equation. At the boundaries between
these regions, the hopping of adatoms is governed by kinetic laws. Real-time obser-
vation of these nanoscale processes is difficult to achieve, and experimentalists have
had to devise creative methods for inferring the relevant energy barriers and kinetic
rates. In contrast, the real-time observation of macroscale surface evolution can be
achieved with simpler imaging techniques. Motivated by the possibility of experi-
mental validation, we derive an equation for the macroscale surface height, which
is consistent with the motion of adatoms. We hope to inspire future comparison
with experiments by reporting the novel results of simulating the evolution of the
macroscale surface height.
Many competing models have been proposed for the diffusion and kinetics
of adatoms. Due to the difficulty of observing adatom motion at the nanoscale,
few of the competing models can be dismissed outright for failure to capture the
observed behavior. This dissertation takes a few of the nanoscale models and sys-
tematically derives the corresponding macroscopic evolution laws, of which some
are implemented numerically to provide data sets for connection with experiments.
For the modeling component of this thesis, I study the effect of anisotropic adatom
diffusion at the nanoscale, the inclusion of an applied electric field, the desorption
of adatoms, and the extension of linear kinetics in the presence of step permeability.
Analytical conjectures based on the macroscale evolution equation are presented.
For the numerical component of this thesis, I select a few representative simulations
using the finite element method to illustrate the most salient features of the surface
evolution.
A MACROSCALE PERSPECTIVE
OF NEAR-EQUILIBRIUM RELAXATION
OF STEPPED CRYSTAL SURFACES
by
John Quah
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2009
Advisory Committee:
Professor Dionisios Margetis, Chair/Advisor
Professor Ricardo Nochetto
Professor Manoussos Grillakis
Professor Theodore Einstein
Professor Janice Reutt-Robey
Dedication
To M. Z., who taught me to leave bridges intact and encouraged me to repair
the ones I burned in moments of impulsiveness.
ii
Acknowledgments
I owe my gratitude to all the people who have made this thesis possible and
without whom my perseverence in the graduate program might not have endured
six turbulent years.
First and foremost I?d like to thank my advisor, Professor Dionisios Margetis
(Dio), who presented me with challenging research problems and equipped me with
the specialized techniques needed to solve them. If I had not been daring enough
to enroll in his course on advanced analytical methods the first time he taught it
at Maryland (Spring 2007), this collaboration and the resulting thesis might never
have taken off. Apart from being an invaluable source of mathematical knowledge,
Dio also made a special effort to involve me in the conferences and workshops he
organized. These meetings brought me in contact with a larger network of research
scientists and mathematicians, whose examples inspired me to pursue my own the-
sis problem with more alacrity. The professional training I received under Dio?s
supervision greatly exceeded any expectations I had upon entering the graduate
program.
Other faculty at the University of Maryland deserve credit for their role in the
development of this thesis. Thanks to Professor Georg Dolzmann, whose course in
numerical analysis imparted an appreciation for the beauty of the field. Thanks to
Professors Eugenia Kalnay, Da-Lin Zhang, and Jim Carton for making accessible to
applied mathematics students the world of atmospheric and oceanic science, along
with the quantitative tools needed to understand the interactions among processes
iii
at different time and length scales. Professors Stuart Antman, Manoussos Grillakis,
and Matei Machedon conveyed their knowledge of partial differential equations and
nonlinear analysis with exceptional clarity.
I am particularly indebted to several faculty in physics, chemistry, and ma-
terials science. Professor Ted Einstein of the Physics Department co-organized a
research interaction team (RIT) with Dio and Professor Tzavaras in the spring of
2008. This series featured many informative lectures and also gave me the opportu-
nity to practice my own blackboard presentation skills. Professor Einstein contin-
ued to provide guidance and physical intuition as my thesis research progressed into
new areas. Similarly, conversations with Professor John Weeks of the Chemistry
Department and IPST1, and with Professor Ray Phaneuf of the Materials Science
and Engineering Department, helped point out some of the problems I would face in
connecting my theoretical investigations with real materials. Professor Janice Reutt-
Robey of the Chemistry Department has been gracious with her time by offering to
serve as the Dean?s Representative on my thesis defense committee.
I also appreciate the financial support of the Monroe Martin Graduate Fel-
lowship in summer 2008, which would not have come to my attention were it not
for Professor Jim Yorke of the Mathematics Department and IPST. Professors Ein-
stein and Margetis wrote the letters to nominate me for this stipend. During that
summer, collaboration with Dio and the REU student Jerrod Young led to my first
publication in Physical Review E.
My own undergraduate research experience, working out a problem in the
1Institute of Physical Science and Technology
iv
mathematicsofcompositematerialsundertheguidanceofProfessorYuryGrabovsky
of Temple University, was decisive in encouraging me to pursue a graduate degree
in applied mathematics. Little did I know at the time that a colleague of one
of Professor Grabovsky?s collaborators would eventually supervise my thesis, after
first introducing me to interesting mathematical problems motivated by a different
branch of materials science. Other faculty who deserve mention for their helpful dis-
cussions are Bob Kohn of New York University, Joachim Krug of the Institute for
Theoretical Physics (Universit?at K?oln), John Venables of Arizona State University,
Jonah Erlebacher of Johns Hopkins University, Jim Evans of Iowa State University,
and Richard James of the University of Minnesota.
I am grateful to Dr. Alex Prasertchoung of MRSEC2 for pushing me to be
a mentor for the AIP3 Student Science Conference (a truly rewarding experience,
not least because I got to see how much talent and fluid intelligence today?s middle
school students possess), and for pushing me around on the tennis court. I also had
the good fortune of receiving financial support from MRSEC through grant NSF-
MRSEC DMR0520471 during the spring of 2009, for which I thank Professor Ellen
Williams, director of the UMD MRSEC. By sheer coincidence, I happened to be
finalizing my thesis precisely when the NSF4 was scheduled to conduct a site visit
at the UMD MRSEC. Professor Williams invited me to present a poster at the site
visit, which gave me the rare opportunity to have my work scrutinized in person by
scientists from outside Maryland.
2Materials Research Science and Engineering Center
3American Institute of Physics
4National Science Foundation
v
Among all the faculty behind this thesis, Professor Ricardo Nochetto deserves
recognition for sharing his extensive knowledge of the finite element method, both
in private discussions and in a formal classroom context. For his patience with me
in the latter setting, in a semester which challenged my management of time among
such disparate tasks as thesis writing, teaching and grading, work at the food co-op,
organizing numerical simulations, and finishing up a paper for publication, Professor
Nochetto can hardly be thanked enough. Collaboration with him and his former
postdoc, Andrea Bonito, always went smoothly, despite the differing perspectives
we brought as mathematical modelers and numerical analysts.
The staff of the math department also offered just the right degree of support,
without drowning me in paperwork or other bureaucratic hurdles. Alverda McCoy of
the AMSC5 program has been ever helpful with registration issues and completion of
forms along the road to graduation. Mark Tilmes graciously applied his computer
expertise to maintain all the software I needed for the numerical portion of this
thesis. Bill Schildknecht accommodated several last-minute requests to change my
teaching assignment, no small feat considering how many courses and other graduate
teaching assistants were potentially affected.
For their support in my personal life, the following people deserve a word of
thanks. David Shoup, Carter Price, J. C. Flake, Charles Glover, Chris Manon, Jeff
Heath, Avanti Athreya, Ryan Janicki, Toni Watson, I-Kun Chen, and other occu-
pants of ?the Zoo? in 2003 helped with my acclimation to graduate life. Chris Zorn,
David Bourne, Seba Pauletti, J. T. Halbert, and other more advanced graduate stu-
5Applied Mathematics, Statistics, and Scientific Computation
vi
dents showed by example how to balance work and play while still making adequate
progress toward a finished thesis. Dr. Jonathan Kandell of the Counseling Cen-
ter offered some much-needed relief when relations among peers became strained.
Under the care of Dr. Stephen Feinberg, my astigmatic eyes received the proper
corrective lenses and vision therapy to enable two years of intense near-focus work.
Dr. Robert Delong and the neurosurgeons at Duke University Medical Center were
instrumental in curing me of epilepsy. Dr. James Wainer of Raleigh, NC, and Dr.
Eric Griffey of Lewiston, ME, kept me on psychotropic drugs long enough to delay
until graduate school the broadening of social and intellectual horizons that I should
have experienced in my undergraduate years.
I also thank the badminton club and its dedicated faculty sponsor, Professor
Peter Teuben, for providing access to a fun-filled team sport and a supportive en-
vironment on the many occassions when I needed to release frustrations. On my
very first visit to the badminton club, I made contact with one person in particular,
whose friendship led to many learning experiences in disciplines ranging from the
humanities to the fine arts. This friendship finally culminated in the healing of a
rift between me and the Maryland Food Co-op, where I have worked or volunteered
since summer 2007. The community of workers and volunteers at the food co-op has
kept me anchored and well-nourished throughout the hectic two years of working on
my thesis.
Finally, I?d like to thank my family for providing a loving and nurturing envi-
ronment throughout my formative years. Their continued support during graduate
school has not gone unnoticed, although I have not always made the time to recip-
vii
rocate and keep in touch with them. When I emerge from the tunnel vision that
will allow me to see the thesis through to the end, my parents deserve to be the first
to hear that I?m still alive.
viii
Table of Contents
List of Abbreviations xiii
1 Introduction 1
1.1 Chapter overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Physical description of surfaces at various length scales . . . . . . . . 1
1.3 Mathematical models of various length scales . . . . . . . . . . . . . . 2
1.4 Connecting step models to experiments . . . . . . . . . . . . . . . . . 10
1.5 Macroscale limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Macroscopic equations and numerical solutions 16
2.1 Macroscopic theories of crystal surfaces . . . . . . . . . . . . . . . . . 16
2.1.1 Effect of different mass transport mechanisms . . . . . . . . . 17
2.1.2 Macroscopic theory above the roughening temperature . . . . 18
2.1.3 Macroscopic theory below the roughening temperature . . . . 21
2.2 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 One-dimensional nanoscale models of interacting steps 27
3.1 Geometry of straight steps . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 BCF model for interacting straight steps . . . . . . . . . . . . . . . . 28
3.3 1D interaction energy and the step chemical potential . . . . . . . . . 30
3.4 BCF model for interacting concentric circular steps . . . . . . . . . . 37
3.5 Interaction energy and step chemical potential for circular steps . . . 40
4 Elastic-dipole interactions between steps in 2 dimensions 43
4.1 From electrostatics to linear elasticity . . . . . . . . . . . . . . . . . . 43
4.1.1 Field and potential of an electric dipole . . . . . . . . . . . . . 44
4.1.2 Energy of a charge distribution in an electric field . . . . . . . 45
4.1.3 Interaction energy of two force dipoles . . . . . . . . . . . . . 47
4.2 Integral for the energy of interacting steps . . . . . . . . . . . . . . . 49
4.2.1 Mellin transform of the axisymmetric integral . . . . . . . . . 52
4.2.2 Asymptotic expansion using the inverse Mellin transform . . . 54
4.2.3 Corrections to the leading-order term under axisymmetry . . . 55
4.3 Elastic interaction energy between steps without axisymmetry: A
brief discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 Review of equations for interacting steps in 2+1 dimensions, and macroscale
limit 65
5.1 Geometry of a step train in 2+1 dimensions . . . . . . . . . . . . . . 66
5.2 BCF model with step interactions in 2+1 dimensions . . . . . . . . . 68
5.3 Approximations for slowly varying step train . . . . . . . . . . . . . . 73
5.4 Macroscale limit with isotropic diffusion in 2+1 dimensions . . . . . . 75
5.4.1 Adatom flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
ix
5.4.2 Macroscale step chemical potential . . . . . . . . . . . . . . . 78
5.4.3 Mass conservation for adatoms . . . . . . . . . . . . . . . . . . 81
5.4.4 Evolution equation for surface height . . . . . . . . . . . . . . 82
5.5 Scaling ansatz for the height profile . . . . . . . . . . . . . . . . . . . 83
5.6 Presence of facets in the variational formulation . . . . . . . . . . . . 84
6 Macroscale equation with terrace anisotropy and step edge diffusion 86
6.1 Approximations for fast and slow step variables . . . . . . . . . . . . 87
6.2 Macroscale adatom flux . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3 Alternative approach to macroscale limit: Taylor expansions . . . . . 92
6.4 Mass conservation law and total surface flux . . . . . . . . . . . . . . 93
6.5 Evolution equation for the surface height . . . . . . . . . . . . . . . . 95
6.6 New scaling laws with terrace anisotropy and edge diffusion . . . . . 96
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7 Step permeability and macroscale limit 103
7.1 Step permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.2 Macroscale limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.2.1 Straight steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2.2 Circular steps with axisymmetry . . . . . . . . . . . . . . . . 110
7.2.3 Steps in 2+1 dimensions . . . . . . . . . . . . . . . . . . . . . 114
7.3 Discussion and interpretation . . . . . . . . . . . . . . . . . . . . . . 118
8 Electromigration and desorption 123
8.1 Macroscale limit and assumptions . . . . . . . . . . . . . . . . . . . . 125
8.2 One-dimensional step models . . . . . . . . . . . . . . . . . . . . . . 127
8.2.1 Straight steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.2.1.1 Electric field with no desorption . . . . . . . . . . . . 129
8.2.1.2 Desorption with no electric field . . . . . . . . . . . . 130
8.2.2 Concentric circular steps . . . . . . . . . . . . . . . . . . . . . 130
8.2.2.1 Electric field with no desorption . . . . . . . . . . . . 132
8.2.2.2 Desorption with no electric field . . . . . . . . . . . . 134
8.3 2-dimensional step geometry . . . . . . . . . . . . . . . . . . . . . . . 135
8.3.1 Step geometry and formulation . . . . . . . . . . . . . . . . . 136
8.3.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . 136
8.3.3 Approximate solution of diffusion equation . . . . . . . . . . . 137
8.4 Evolution equations at the macroscale . . . . . . . . . . . . . . . . . 142
8.4.1 Macroscopic Fick?s law with drift . . . . . . . . . . . . . . . . 143
8.4.2 Evolution equation in Cartesian system . . . . . . . . . . . . . 146
8.4.3 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . 147
8.5 Extensions of macroscopic limit . . . . . . . . . . . . . . . . . . . . . 149
8.5.1 Exponential law for step chemical potential . . . . . . . . . . 150
8.5.2 Adatom desorption . . . . . . . . . . . . . . . . . . . . . . . . 151
8.5.2.1 Solution for microscale diffusion . . . . . . . . . . . . 152
8.5.2.2 Limit a? 0 . . . . . . . . . . . . . . . . . . . . . . . 154
x
8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
9 Facets and boundary layers 158
9.1 Faceting on crystal surfaces . . . . . . . . . . . . . . . . . . . . . . . 159
9.2 Formulation of faceted relaxation as a free-boundary problem . . . . . 159
9.3 Natural boundary conditions of the variational problem for periodic
profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.4 Scaling of the boundary layer width . . . . . . . . . . . . . . . . . . . 169
9.5 Effect of an electric field on the slope profile near a facet . . . . . . . 174
9.5.1 Straight-step morphology . . . . . . . . . . . . . . . . . . . . . 174
9.5.2 Axisymmetric structure . . . . . . . . . . . . . . . . . . . . . 176
9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10 Numerical results 179
10.1 Review of the finite element method . . . . . . . . . . . . . . . . . . . 182
10.2 Software and computers . . . . . . . . . . . . . . . . . . . . . . . . . 184
10.3 Zero line tension (?ideal? case) . . . . . . . . . . . . . . . . . . . . . 185
10.3.1 Simulations with zero E-field . . . . . . . . . . . . . . . . . . . 187
10.3.2 Simulations with nonzero E-field . . . . . . . . . . . . . . . . . 203
10.4 Continuous dependence on line tension of the 2D to 1D transition . . 207
10.5 Conclusions and open issues . . . . . . . . . . . . . . . . . . . . . . . 210
11 Epilogue: conclusions and open questions 213
11.1 Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . 215
11.2 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
A Brief review of the Mellin transform 220
B Nondimensionalization of the evolution equations 224
Bibliography 227
xi
List of Abbreviations
symbol meaning dimensions
? aspect ratio of a biperiodic height profile
?x, ?y characteristic lengths at the macroscale length
a atomic height length
h macroscale surface height length
r position vector in the (x,y) plane length
? transverse step coordinate
?i transverse coordinate of the ith step
? longitudinal step coordinate
?? metric coefficient along ? length
?? metric coefficient along ? length
mi discrete step density
E surface free energy energy
? free energy density energy/length2
? step stiffness energy/length
g1 line tension coefficient energy/length2
g3 step interaction coefficient energy/length2
Ci adatom density on the ith terrace 1/length2
C macroscale adatom density 1/length2
Ceqi equilibrium adatom density at the ith
step
1/length2
Cs straight step equilibrium adatom den-
sity
1/length2
D, Dad terrace adatom diffusivity (bold-
face=tensor)
length2/time
Ded edge atom diffusivity length2/time
Jad terrace adatom flux 1/(length?time)
Jed edge atom flux 1/(length?time)
J macroscale adatom flux 1/(length?time)
k, ku, kd kinetic rates of attachment/detachment length/time
M macroscale mobility (boldface=tensor) length2/(energy?time)
? dimensionless mobility tensor
?i chemical potential of the ith step energy
? macroscale chemical potential energy
q material parameter comparing diffusion
to attachment/detachment
T temperature temperature
v adatom drift velocity length/time
u nondimensionalized drift velocity
E electric field energy/(charge?length)
Z?e effective charge of adatoms charge
? atomic volume length3
xii
Chapter 1
Introduction
1.1 Chapter overview
In this chapter, we will introduce the basic principles of the near-equilibrium
morphological evolution of crystal surfaces. Depending on the length scale under
consideration, these surfaces present different physical pictures to the viewer. To
capture the relevant physics of these experimental pictures, mathematical mod-
els make certain simplifying assumptions about the operative mechanisms at each
length scale. The different mathematical models are typically analyzed indepen-
dently of one another. In this chapter, we emphasize the connections between mod-
els at different length scales. These connections constitute the primary focus of the
next few chapters. Finally, we outline the main results of this thesis, indicating how
this work contributes to the ongoing effort to understand and exploit the physical
mechanisms that govern the evolution of crystal surfaces.
1.2 Physical description of surfaces at various length scales
There are three distinct scales for crystal surfaces: atomistic scale, nanoscale,
and macroscale. A typical picture on the atomistic scale has individually distin-
guishable atoms arranged periodically in a crystal lattice. The atomistic picture
1
of a surface has atoms with missing bonds at the solid-vapor interface. The more
missing bonds these surface atoms have, the rougher the resulting height profile
looks.
At the nanoscale, the viewer can identify not the individual atoms, but only the
extended surface features they form in aggregate. Just as a viewer from an airplane
can see the Great Wall of China but not its individual stones, the nanoscale viewer
of crystal surfaces can see the meandering lines formed by a large number of surface
atoms at one height, adjacent to a large number of surface atoms at a different height.
These line defects are called steps, since they appear at the boundary between two
regions whose heights differ by a lattice constant. From the perspective of the
nanoscale viewer, the steps resemble smooth curves. For this viewer, the fact that
steps can be counted individually is the only clue that the microscope is showing an
essentially discrete object.
At the macroscale, all clues of discreteness are taken away, and the viewer is
left with what looks like a smooth topography. Until the mid-20th century, this
resolution was the only one accessible to experimentalists. However, the mental
picture of a discrete lattice structure was always available, and it helped establish
a framework for thinking about crystal surfaces.
1.3 Mathematical models of various length scales
One possible mathematical approach to the atomistic scale is to seek approx-
imate solutions of the many-body Schroedinger equation for electrons and nuclei
2
[121]. These approximate solutions are valid over prohibitively short time scales,
which impedes most attempts to reconcile the motion of individual atoms with
coarser descriptions of the surface. By allowing for atomic motions governed by
classical mechanics, rather than quantum mechanics, Vvedensky and Haselwandter
[121] sought a connection between atomistic models and the nanoscale. This con-
nection lies outside our present scope.
In this thesis, we study analytical and numerical aspects of the connection
between nanoscale and macroscale. The nanoscale model assumes that steps move
in response to (i) adatom attachment/detachment at step edges, (ii) diffusion of
adatoms on terraces, and (iii) energetic interactions between steps. This description
of the microscopic physics, introduced in a simplified form by Burton, Cabrera,
and Frank (BCF) [10], serves as the basis for our present discussion of surface
morphological evolution. The BCF model and its generalizations support a rich
variety of step dynamics. The patterns that emerge during surface evolution have
motivated analytical and numerical attempts to understand what experimentalists
observe based on the behavior of steps.
The temperature below which the BCF model applies is called the rough-
ening temperature, TR. Thermal wandering of steps is allowed below TR, but the
steps themselves are stable objects, not momentary deviations from surface flatness.
Above TR, creation of a step has zero free energy cost, and steps are created and
destroyed spontaneously. Not only steps, but also point defects such as vacancies
can be present on the surface. This roughening indicates that a description of the
surface in terms of steps is not appropriate. A rough surface is more easily visualized
3
Figure 1.1: The stepped surface of Si(001) as seen through an STM, taken from [99].
at the atomistic scale, where bonds between atoms at the surface are broken and
formed at random.
The macroscopic theory of Herring and Mullins [34, 75], based on continuum
thermodynamics and mass conservation, accurately predicts the observed behavior
of surfaces above the roughening temperature. However, the fabrication of crys-
tal surface morphologies is no longer restricted to temperatures above TR, which
challenges the validity of the continuum theory of Herring and Mullins.
Following the trend toward miniaturization of semiconductor devices, the
length scales on which crystal surfaces are patterned have grown smaller, neces-
sitating lower temperatures so that the desired pattern is more stable. These lower
temperatures require a macroscale theory consistent with the flow of thermally sta-
ble steps.
Figure 1.1 shows the surface of Si(001) as seen through a Scanning Tunneling
Microscope (STM). We can identify terraces where the surface is locally flat, except
4
Figure 1.2: Schematic of curved crystal steps on a vicinal surface. The arrows
represent the different physical processes through which adatoms affect the flow of
steps.
for a few isolated bumps and vacancies of size about 3 ?A(one lattice constant). The
line defects bounding the terraces are steps about one lattice constant high. The
widths of terraces range from one to a thousand lattice constants.
The presence of kinks and edge atoms, visible in Figure 1.3, reminds us that
continuum steps are onlyan approximation of a fundamentallydiscrete phenomenon.
To use an image with the same resolution as Figure 1.3 to generate initial conditions
for a simulation of continuum steps, one would have to smooth out the kinks and
edge atoms. Jeong and Williams [43] suggest the use of an integral mollifier for this
purpose. The result is a representation of the surface by smooth steps as in Figure
1.2.
In Figure 1.2, the relevant near-equilibrium physical processes are (i) adatom
diffusion on terraces, (ii) diffusion of edge adatoms along steps, and (iii) attach-
ment/detachment of adatoms at step edges. Diffusion processes such as (i) and (ii)
arise when each terrace or edge adatom is assumed to perform a random walk; the
5
Figure 1.3: Image of a silicon (100) surface, taken from [99], showing kinks and edge
atoms over many steps. Solid curves along the diagonal have been superimposed
to illustrate the approximation of smooth steps. Note two kinds of steps on this
surface.
6
resulting step-continuum equations represent the homogenized (averaged) effect of a
large number of these adatoms moving stochastically throughout their domains. The
overall evolution of nanoscale averages, such as the adatom density, is predictable
according to the law of large numbers. The kinetic process (iii) is a manifestation
of a near-equilibrium law, whose forcing term is similar to those that appear in dif-
ferential equations for chemical reaction rates. In the case of surface steps, the rate
at which adatoms attach or detach at step edges is proportional to a difference of
concentrations. A step advances when adatoms attach to the step edge; it retreats
when adatoms detach from the step edge.
We now discuss a few of the atomic processes that change the shape of a step. A
curved step, such as those depicted by the solid diagonal paths in Figure 1.3, actually
features kinks that ?zig-zag? along the step edge. The expansion of kinks can result
from nucleation of terrace adatoms, if the difference of concentrations is favorable to
adatom attachment at the ends of a kink. The opposite effect, when concentration
differences favor adatom detachment, consists of a step emitting adatoms. This
change in the step shape reduces the free energy associated with unbonded edge
atoms. Such energy-minimizing mass transport can be interpreted as an effect of
step line tension. In general, a step can change its shape by adsorbing or emitting
adatoms. It remains to explain how the equilibrium shape of a step is communicated
to the adatoms.
Absorption and emission of adatoms depend on the differences between con-
centrations. One of these, the terrace adatom concentration, is in principle directly
observable. The other concentration is an equilibrium quantity that depends on the
7
surface geometry via the step curvature and energetics of interacting steps. Intu-
itively, steps with high curvature and small separations from their nearest neighbors
have higher energy than straight steps widely separated from their nearest neigh-
bors. We expect a highly curved step to require more energy because a greater
number of broken bonds are associated with regions of high curvature, as one can
see in Figures 1.1 and 1.3. The effect of step separation is more subtle and involves
two different kinds of interaction, as we see shortly.
The energy of a step train is conveniently analyzed using the concept of step
chemical potential. The step chemical potential, reflecting the contribution of step
line tension and step-step interactions, measures the propensity of a step to incor-
porate adatoms. This physical interpretation follows from the monotone relation
between adatom equilibrium density and step chemical potential; see the discussion
around (3.10).
Interactions between steps of the same sign (both either step-up or step-down)
are of two types, entropic and elastic-dipole. Entropic repulsions enforce the non-
crossing of steps, by restricting how far each step at nonzero temperature can deviate
from its average position. This thermal wandering is an unavoidable consequence
of nonzero temperature. However, the presence of other steps imposes a penalty
for any wandering that brings a step into collision with one of its neighbors. This
penalty takes the form of a loss in entropy, making the step?s free energy greater
than it would be in the absence of neighboring steps. Elastic-dipole interactions
are mediated by the strain fields that steps produce in the crystal [63, 76]. Any
defect on the surface, including a step, induces a force distribution that in principle
8
has a multipole expansion of all orders. Comparison between predictions of this
multipole expansion and data from classical atomistic simulations led Najafabadi
and Srolovitz [76] to conclude that the dipole-dipole interaction is dominant over
higher-order contributions for the typical range of terrace widths. The combination
of elastic-dipole and entropic interactions yields the step chemical potential as the
superposition of two terms, each term reflecting an inverse-square dependence on
the distance between the step and one of its neighbors. This superposition is not
linear, appearing in the equation for interaction energy as a renormalized parameter
for the strength of repulsions [43].
The approximation of continuum steps, as noted above, deliberately glosses
over the details of point defects on the steps themselves. Kinks and edge atoms,
always present to some extent on steps at nonzero temperature, are not directly
incorporated into the formula for step chemical potential. Some variations of the
BCF model account for kinks and edge atoms by introducing auxiliary fields, such
as kink density. Equations for these auxiliary fields are coupled with the other
equations for step motion in a way that respects mass conservation [3, 11, 68]. We
do not treat coupled field equations in this thesis, instead opting for the simplicity
of a semi-empirical formula for the step chemical potential. By ?semi-empirical? we
mean that the scaling of the step chemical potential with nearest-neighbor distance
can be derived analytically, but in general the prefactor can only be approximated
by comparisons with data, numerical or experimental.
9
1.4 Connecting step models to experiments
The feedback loop between step models and experiments began with an obser-
vation that puzzled the mid-20th century crystal growth community. According to
the accepted models of crystal nucleation and growth in solution, the concentration
of crystal atoms had to be sufficiently large in order for crystals to precipitate on
the growing surface. When the threshold concentration was observed to be much
lower than predicted, Burton, Cabrera, and Frank [10] hypothesized a new mech-
anism for crystal growth in solution to explain the discrepancy. This explanation
proposed the existence of steps, which serve as convenient nucleation sites for the
atoms adsorbed from solution. In the BCF model, step motion is governed by the
diffusion of adatoms across terraces and the attachment/detachment of adatoms at
step edges. Their formulation of equilibrium density differs from the one we adopt
following [37, 64, 113], in that the BCF model omits step interactions. The simple
picture inspired by the BCF model enabled many successful predictions of surface
phenomena, such as Spohn?s prediction of step density in the neighborhood of a
facet [109], and the inverse linear decay of peak-to-valley distances found by Rettori
and Villain [98] for certain biperiodic profiles.
More recently, the BCF model has been expanded to account for step-step in-
teractions and anisotropies in the material parameters. These extensions of the BCF
model are often introduced to provide better interpretations of the observations of
crystal surfaces. For example, the experimental determination of certain coefficients
in the formula for surface free energy leads to interesting analogies between the ter-
10
race width distribution and other probabilistic phenomena. The successful interplay
between step models and experiments confirms that step-based models still hold a
valuable place among the various approaches to surface science.
1.5 Macroscale limit
Advances in imaging techniques since the early investigations of epitaxial phe-
nomena have expanded the range of time and length scales over which we can study
crystal surfaces. Today with Atomic-Force Microscopy (AFM) and other imaging
devices, it is common to observe samples with characteristic lengths of the order
of 100 ?m or larger for durations of minutes, hours or more. Samples of this size
comprise hundreds or thousands of steps, and the AFM probe will not resolve all the
details fast enough to follow the individual steps as the surface relaxes. A meaning-
ful connection between observations at this scale and the predictions of a step-based
theory requires a systematic way to bridge the gap between the limited capabilities
of an AFM probe and the difficulties of understanding large-scale surface morpho-
logical evolution via the flow of steps. In applications it is also often desired to
control the evolution of crystal surfaces, and this control is more readily achieved at
larger scales. Because of the below-roughening temperatures under which modern
small devices are made, the presence of steps at the nanoscale must be taken into
account when attempting to control the surface evolution at the macroscale.
It turns out that we can bypass a complete simulation of step motion in 2+1
dimensions by using a formal procedure known as coarse-graining. This method
11
regards the equations of step flow as a discrete scheme for some evolution law gov-
erning macroscale quantities, such as the surface height. In the same way that
decompilation of computer code reconstructs a plausible version of the higher-level
source code, coarse-graining takes the low-level ?instructions? given by a nanoscale
model and determines a consistent macroscale ?program? for which the nanoscale
equations are one implementation. This analogy fails in several ways, most impor-
tantly by suggesting that the output of a coarse-graining procedure is not unique (up
to isomorphism), since different computer code decompilers can yield structurally
different source code programs. In fact, coarse-graining is a well-defined operation,
producing a unique partial differential equation (PDE) given a deterministic system
for the finer-scale variables.
The PDE generated by a coarse-graining operation describes the relation be-
tween space and time derivatives of the surface height. This PDE is also known
as the full continuum limit (or macroscale limit) of the nanoscale equations. This
terminology reflects the facts that (i) a limit is taken when going to the coarser
scale, and (ii) the resulting PDE is a fully continuum equation, making no explicit
reference to the nanoscale variables from which it was derived. After verifying the
hypotheses that justify using the macroscale limit, one might make direct compar-
isons between the predictions of this PDE and the experimental results.
Inthecaseofsurfacerelaxation, themacroscalelimitisafourth-orderparabolic
nonlinear PDE for the height decay in the region of space where ?hnegationslash= 0 [37, 64, 96].
If the complementary region {x : ?h(x) = 0}?i.e., the set of macroscopically flat
facets?has positive measure, then we need to determine the appropriate bound-
12
ary conditions at facet edges to obtain a well-posed problem from the macroscale
limit [31, 109]. The choice of boundary conditions must weigh the sometimes con-
flicting demands of easy numerical implementation on the one hand, and accurate
microscale physics on the other hand.
1.6 Overview of this thesis
The main results of this thesis are fully continuum equations for the surface
height based on step models, and investigation of these equations from numerical
and analytical perspectives. In Chapter 2, we introduce the basic ideas common
to all macroscopic theories, and we review the numerical method we apply later to
approximate the solutions of macroscale equations. In Chapter 3, we introduce the
assumptions of near-equilibrium step flow in the context of one-dimensional step
trains, to lay the groundwork for subsequent generalizations in 2+1 dimensions. In
Chapter 4, we take a brief detour to derive the formula for elastic interaction energy
between steps. The generalization of this formula is used in Chapter 5, where we
review the (2+1)-dimensional macroscopic limit of [66].
The modeling component of this thesis extends the (2+1)-dimensional setting
of [66] in several ways. First, the diffusion of adatoms on terraces is allowed to
be anisotropic, which allows for a more realistic representation of surfaces where
adatoms hop more readily perpendicular to steps than parallel to them, or vice
versa. Second, we include the effect of step edge diffusion, where mass transport
along the step edge is modeled as the result of longitudinal variations in the step
13
chemical potential. The macroscale limit with anisotropic terrace diffusion and step
edge diffusion is presented in Chapter 6. Different scalings of approximate, separable
solutions of the PDE are made possible by the additional parameters corresponding
to step edge diffusion and anisotropic terrace diffusion.
In Chapter 7, we generalize the boundary conditions at step edges to address
the possibility of permeable steps. Adatoms approaching a permeable step are free to
hop directly to the next terrace. In principle, step permeability can lead to nonlocal
effects, as adatoms diffuse over many terraces before attaching to a step edge. The
macroscale limit, however, reflects step permeability only through a renormalized
kinetic rate. Our derivation in 2+1 dimensions generalizes the work of Pierre-Louis
[88] for straight steps.
Another extension of (2+1)-dimensional step models introduces a drift velocity
in the terrace diffusion equation, which corresponds physically to an applied elec-
tric field. The study of crystal surfaces evolving under the influence of an electric
field began with an experiment by Latyshev et al. [59], who reported the onset of
step bunching. This surface instability is characterized by wide terraces between
clusters of closely-spaced steps. Electromigration-induced step bunching has been
used as a practical method for fabricating quantum wires. More recently, the appli-
cation of an electric field parallel to step edges has been shown to produce a step
meandering instability. This phenomenon features the growth of harmonic pertur-
bations of straight steps, as predicted by Dufay et al. [15] and observed by Degawa
et al. [14]. Motivated by these possible applications, we introduce in Chapter 8 the
physics of electromigration at the nanoscale. Then we use the BCF-type model with
14
electromigration to derive the corresponding macroscale equation.
In Chapter 9, we discuss the inclusion of facets in the macroscopic theory. A re-
laxing surface with a facet can be modeled as a free-boundary problem [64, 109]. The
influence of step-step interactions on the slope profile depends on the distance from
the facet edge. We apply boundary-layer ideas, similar to those employed in [64],
to determine how the slope profile behaves near a facet edge in 2+1 dimensions.
The boundary-layer solution is qualitatively different if we introduce an appreciably
large electric field. We study this effect analytically in the case of straight steps,
axisymmetry, and also more general step geometries with slowly varying curvature.
In Chapter 10, we use the finite element method to simulate the macroscopic
equations. Numerical simulations of the PDEs derived in Chapters 5?8 illustrate the
rich variety of relaxation phenomena predicted by the model. The first noteworthy
result, evident even in the case of isotropic terrace diffusion, is the morphological
transition exhibited by an initially biperiodic profile, which relaxes to become almost
one-dimensional [7]. We discuss the plausible mechanisms behind this transition.
To isolate the effect of longitudinal fluxes, we simulate the macroscopic equation
with electromigration for an initially biperiodic profile. In this case, the acceler-
ated transition to an almost one-dimensional profile offers insight into the driving
mechanism behind the observed morphological changes.
15
Chapter 2
Macroscopic equations and numerical solutions
In this chapter, we provide the necessary background for the subsequent in-
vestigation of macroscale crystal surface evolution. Our research relies on PDEs for
the macroscale surface variables, as well as numerical simulations of these PDEs.
These components of our research program build on previous work in modeling and
numerics. In the following review of the relevant background material, we make
special note of the limitations of previous macroscopic theories and simulations, in
order to motivate the new topics of the following chapters.
2.1 Macroscopic theories of crystal surfaces
In a fully continuum picture of crystal surfaces, we smooth out the kinks and
approximate the discrete height by a continuous graph. The immediate advantage
of this approach is the reduced number of dependent variables. The height profile
in the macroscopic model is a single function of time and the two spatial variables.
The PDE satisfied by the macroscale height function lends itself more readily to
numerical simulations on modest hardware, whereas atomistic models require much
more memory to store the positions of all the atoms in a sample. Fully continuum
theories also have the advantage of enabling global predictions, such as scaling laws
and similarity solutions, more easily than discrete models. These global predictions
16
are better suited for comparisons with experimental results, because the time scale
of experimental observation is realized by macroscopic simulations but is typically
unattainable by atomistic methods.
Macroscopic theories are not without their limitations, however. One should be
cautious in applying a fully continuum theory when detailed resolution of atomistic
or nanoscale processes is desired. The configuration of kinks and edge atoms, for
example, cannot be recovered from a macroscopic simulation. Moreover, kinetic
rates manifest themselves in macroscopic theories only through combinations of
material parameters. To determine any particular kinetic rate is outside the scope
of our fully continuum models.
2.1.1 Effect of different mass transport mechanisms
In total, four different physical mechanisms have been identified for mass trans-
port in a crystal material. In [33], Herring considers separately the processes of
(i) viscous flow, (ii) evaporation and condensation, (iii) volume diffusion, and (iv)
surface diffusion. Each of these processes leads to a different scaling between the
lifetime and size of a surface feature. For example, two geometrically similar sur-
face features with characteristic length scales L1 and L2 will change their shape by
evaporation/condensation with lifetimes ?1 and ?2, which satisfy ?2/?1 = (L2/L1)2.
The exponent 2 appears because evaporation/condensation works at the solid-vapor
interface, where total surface area (proportional to the square of the characteristic
length) is relevant. In general, a surface feature with characteristic length ?L1 de-
17
caying via one of the mechanisms (i)?(iv) has a lifetime ? given by ? = ?1?p, where
p ? {1,2,3,4} is associated with the operative mechanism of mass transport. The
largest possible power, p = 4, is associated with surface diffusion.
At a given temperature, any of the four possible mass transport mechanisms
may be present in a crystal material. The lifetime of a structure varies most rapidly
with characteristic size when surface diffusion is present. The trend toward fabricat-
ing ever smaller surface features, as required for mobile electronic devices, would im-
pose severe time constraints at the temperatures commonly used in previous decades
for crystal surface patterning. For example, a miniaturized surface feature, half as
large as one produced currently, might decay 16 times faster at the same tem-
perature. To mitigate the difficulties of shorter lifetimes in structures subject to
surface diffusion, experimentalists and manufacturers have brought their equipment
to lower operating temperatures. At these lower temperatures, the continuum ther-
modynamic description of crystal surface evolution is challenged. The appropriate
theory, based on the motion of steps below roughening, is thus enjoying a resurgence
in the research community.
2.1.2 Macroscopic theory above the roughening temperature
Before describing the extension of macroscale theories to temperatures below
roughening, we review the ingredients of surface evolution models that first appeared
indescriptions above therougheningtemperature. Macroscale descriptionsofcrystal
surfaces above roughening apply the ideas of continuum thermodynamics and mass
18
conservation, obtaining PDEs or variational principles for the surface height. A
classic example is the fourth-order PDE,
?th? div?(?h), (2.1)
derived by Mullins and Herring. This result holds if
1. the chemical potential ? is proportional to the mean curvature, ???1 +?2,
2. the surface flux is proportional to the gradient of chemical potential, J ???,
and
3. mass conservation is enforced via ?th??divJ.
In this derivation, Mullins introduces only one chemical potential,?, which is associ-
ated with the surface geometry through the mean curvature. A second chemical po-
tential for the vapor phase would be needed in the case of evaporation/condensation,
but Mullins assumes only surface diffusion in his analysis. As noted in Chapter 1,
it is reasonable to expect the mean curvature to determine the likelihood of adatom
attachment/detachment on the surface. In the regime of small slope (|?h| lessmuch 1),
the mean curvature definition of chemical potential yields
?? ?h, (2.2)
where ? is the 2D Laplacian. An alternative formulation of the chemical potential
is in terms of the surface energy E, via the variational formula
? = ?E?h, (2.3)
19
where
E =
integraldisplay
R2
?(|?h|)dx, ? = g0 + 12g2|?h|2 +o(|?h|2). (2.4)
We remark that the above-roughening free energy density? is smooth for any surface
slope |?h|.
The relation between surface flux and the gradient of chemical potential is the
macroscale analog of Fick?s law1. At the level of steps and terraces we have a flux
Jad defined by Jad = ?Dad ??C, where C is the adatom concentration and Dad is
the adatom diffusivity. At the macroscale we have the similar relation
J ??M???, (2.5)
where M is the mobility. Both Dad and M are in principle tensor-valued functions
of the slope, restricted on physical grounds to be positive definite. The case of scalar
Dad and M is perhaps easiest to understand intuitively, but we shall soon encounter
regimes where this intuition is challenged.
Finally, the law of mass conservation is written so that a net flux of adatoms
across a level set for h is exactly balanced by the local time derivative of the height.
In the absence of deposition, the material derivative (or total derivative) of the level
set h = const. yields the mass conservation law,
?th??divJ. (2.6)
When deposition is present, a forcing term must be added to the right hand side of
(2.6). This case is not studied in this thesis.
1See the discussion around (5.3).
20
2.1.3 Macroscopic theory below the roughening temperature
The three ingredients of macroscale models of surface morphological evolu-
tion, first introduced in the study of materials above roughening, also carry over to
temperatures below TR. Care must be taken in writing the analogs of (2.2)?(2.6)
below roughening, in order to respect the presence of steps. For example, according
to the BCF model, there are N different step chemical potentials for an array of
N steps. It is not immediately obvious how one would obtain a fully continuum
chemical potential ? in this setting, much less a relation analogous to (2.2). Sim-
ilarly, the dependence of terrace flux Jad on adatom concentration is not readily
connected with the macroscale relation (2.5). In Chapter 5 we review the details
of this macroscale limit procedure for the simple case of isotropic terrace diffusion,
where Dad is a scalar constant.
The idea of a macroscale limit has a long history in the literature, producing
PDEs based on BCF models alone [2, 51, 86, 94, 119], BCF models coupled with
adatom density [16, 62, 104], or kinetic Monte Carlo models [120]. We follow the
first approach and consider BCF models alone, allowing for some interesting gen-
eralizations to account for different possible physics at the nanoscale. In Chapter
6 we include the effects of step edge diffusion and anisotropic terrace diffusion. In
Chapter 7 we derive the macroscale limit for permeable steps, which allow adatoms
to jump from one terrace to its neighbor without first attaching to the step between
them. In Chapter 8 we include the effects of a drift velocity caused by an applied
electric field, and desorption of adatoms from a terrace into the vapor phase. These
21
effects are easily accommodated in the below-roughening analogs of (2.2)?(2.6) with-
out disrupting the structure of the macroscopic equations. In every case we study
here, the macroscale limit can be written in the form
?th = div(M???), (2.7)
? = ?E?h (2.8)
for some appropriate mobility M and surface energy E.
Below roughening, the singular surface energy E can be expanded as a series
in |?h|:
E = g0 +g1|?h|+ g33 |?h|3. (2.9)
By interpreting |?h| as the macroscale analog of the discrete step density, the co-
efficients of the expansion for E reflect the relative contribution of energies (i) pro-
portional to the number of steps (g1 term), (ii) independent of the number of steps
(g0 term), or (iii) stemming from interactions that decay as the inverse square of
step separation (g3 term).
Because ? is not directly observable, but only serves as an auxiliary variable
in the fully continuum PDE, it will sometimes be convenient to change variables so
that the variational structure (2.7),(2.8) remains intact.
2.2 Numerical simulations
In this section, we discuss the numerical treatment of evolution equations for
the macroscale surface height. Our approach takes advantage of the variational
structure (2.7),(2.8) to implement a weak form of the macroscale PDE using finite
22
elements. If the mobility M and surface energy E are chosen appropriately, we
expect the solution of the macroscale PDE to approximate the near-equilibrium
motion of steps. However, a full comparison between the predictions of nanoscale
and macroscale models is outside the scope of this research. Lacking a numerical
implementation of step flow equations in 2+1 dimensions, the only data available
for comparison come from simulations of the coupled differential equations for the
positions of straight or concentric circular steps [25, 45].
We now turn to the implementation of fully continuum models for the surface
height. To simulate these macroscale PDEs, we must decide on a discretization,
which introduces computable quantities to approximate the fully continuum data.
The vector of step positions is, of course, one such discretization. However, the
equations of step flow are in general too unwieldy to simulate directly. We take
advantage of the flexibility afforded by the fact that any given differential equation
has more than one possible discretization.
Amongthepossiblediscretizationchoicesavailabletous, twoinparticularhave
proven useful for the simulation of crystal surface relaxation. The first (Galerkin)
method approximates the surface height by an expansion in trigonometric basis
functions [108], which yields coupled differential equations for the coefficients after
substitution in (2.7),(2.8). This system of ordinary differential equations is then
integrated numerically. Due to its easy formulation and capacity to accommodate
higher-order discretizations in time, the Galerkin method is a natural choice for
simulating the evolution equation.
We adopt the finite element method, which partitions the domain into non-
23
overlapping elements (e.g., triangles) and approximates at time n the surface height
hn and the macroscale chemical potential ?n as a sum of basis functions ?i (some-
times called hat functions), each of which is supported on a finite, contiguous cluster
of elements, viz.,
hn =
summationdisplay
i
Hni ?i, (2.10)
?n =
summationdisplay
i
mni?i. (2.11)
The approximations (2.10),(2.11) can only resolve the spatial dependence ofhand?
over distances not less than the mesh size. By choosing the mesh size appropriately,
we can resolve all the physically meaningful spatial variations in h, without intro-
ducing unmanageably many degrees of freedom in the triangulation T . We denote
by VT the finite element space spanned by the hat functions ?i for the triangula-
tion T . The weak formulation of the evolution law, expressed in the space VT , will
finally allow us to find approximate solutions of (2.7),(2.8). In order to write this
weak formulation, we must first choose a discretization for the time derivative.
The finite element method works by solving a linear system for the coefficients
Hni and mni in (2.10),(2.11). The exact definition of ? in terms of h is, in general,
nonlinear, so the first task of a time discretization is to find an approximate, linear
version of the relation (2.8). We choose a semi-implicit Euler time step [32, 77],
replacing (2.7) by
hn ?hn?1
?t = divM(h
n?1)???n, (2.12)
?n = ?E?h
vextendsinglevextendsingle
vextendsinglevextendsingle
hn?1
. (2.13)
24
This discretization allows us to find the unknown coefficients at any time by
solving a matrix equation. To determine which linear system the coefficientsHni and
?ni satisfy, we take the inner product of (2.12) with an arbitrary test functiong ?VT ,
and also the inner product of (2.13) with an arbitrary test function? ?VT . Because
VT is spanned by the basis functions ?j, it suffices to consider inner products with
each ?j individually. The linear system that emerges is
summationdisplay
i
(Hni ?Hn?1i )
integraldisplay
B
?i?j = ?t
summationdisplay
i
mni
integraldisplay
B
M(hn?1)??i ???j, (2.14)
summationdisplay
i
mni
integraldisplay
B
?i?j =
integraldisplay
B
?E
?h
vextendsinglevextendsingle
vextendsinglevextendsingle
hn
?j, ??j, (2.15)
where B is the domain over which the PDE (2.7),(2.8) is assumed to hold.
The integral on the right-hand side of (2.15) is more commonly rewritten using
integration by parts, as we shall see in Chapter 5 when considering the formula for
macroscale chemical potential in more detail. Integration by parts has already been
performed on the right-hand side of (2.14), with vanishing boundary terms2.
We alert the reader that the surface energyE has a singularity at|?h| = 0. For
the integration in (2.15) to make sense, we add a regularization parameter ? lessmuch 1
to the denominator |?h| that appears in the g1 term of ?E/?h. The qualitative
behavior of the approximate solution is not significantly affected by ? [7].
Finally, the tensor mobility in the fixed coordinate system might also have a
singularity at |?h| = 0. Different regularization schemes are possible to remove
this singularity. For mathematical consistency we might wish that the regularized
2Namely, ?th and the flux J are both zero on the boundary of B; or, we apply periodic boundary
conditions (assuming steps lie on a torus in R2).
25
mobility approaches a scalar as ?h ? 0. However, we find that the approximate
solution is rather robust with respect to the mobility regularization, as discussed in
Chapter 10.
A number of linear solvers are available to compute the coefficients Hni , ?ni
using (2.14),(2.15). Once a solution is obtained, we can quantify its usefulness by
estimating the size of the truncation errors and interpolation errors (due to dis-
cretization in time, the choice of mesh, and the choice of polynomial space for
the basis functions). In the present work, we do not focus on a posteriori analy-
sis in this vein. Our emphasis is on the global behavior of solutions to the PDE
(2.7),(2.8), which we expect to be well captured by any reasonable numerical solution
of (2.14),(2.15).
The preceding illustration of the finite element method gives only a flavor of
the mathematics behind this powerful tool. A much richer description appears in
[9]. Evans [24] serves as a good reference for the necessary background in Sobolev
spaces.
26
Chapter 3
One-dimensional nanoscale models of interacting steps
In this chapter we review an extended version of the BCF step model. This
classical picture of step motion, updated to include entropic and elastic-dipole step
interactions, describes the relevant nanoscale physics of stepped surfaces. Here
nanoscale reminds us that the motion of steps, which we use to derive a macroscale
limit, is itself a coarse approximation of atomistic processes.
3.1 Geometry of straight steps
We consider a descending train of steps with atomic heighta, separated by flat
regions called terraces; see Figure 3.1. The step positions xi are treated as moving
boundaries for the adatom diffusion of each terrace. To account for the fact that
no overhangs or step crossings are allowed, we require that at each time t, the step
position xi(t) increase monotonically with the step number i, xi+1(t) > xi(t). If
this condition is satisfied for the initial step positions xi(0), then the subsequent
evolution is expected to respect the monotonicity, due to the repulsive step-step
interactions introduced in the equations below. We regard the monotonicity of
step positions for t > 0 as a basic assumption, although an elementary proof is
27
Figure 3.1: Geometry of straight steps.
possible for certain mass transport mechanisms.1 For each time t, the ith terrace
{x : xi < x < xi+1} is a well-defined domain on which we seek solutions of the
equation of motion for adatom diffusion. For the present case of one independent
spatial variable, the elliptic problem corresponding to adatom diffusion reduces to
an ordinary differential equation that admits an explicit solution.
3.2 BCF model for interacting straight steps
A quantitative discussion of the BCF theory begins with the adatom density,
Ci, on the ith terrace, xi < x < xi+1. This Ci satisfies the diffusion (or heat)
equation [10],
?tCi = ?x(Dad ??xCi) , (3.1)
where Dad is the diffusivity for adsorbed atoms on the terrace. Note that we have
omitted from (3.1) terms that describe atom desorption, electromigration and ma-
terial deposition from above, considering only the effect of surface relaxation. Equa-
tion (3.1) can be simplified significantly by elimination of the time derivative, for the
following reason. The time scale for step motion is much larger than the time scale
1For instance, by considering mass transport dominated by evaporation-condensation, we obtain
a second-order PDE for the slope, to which the maximum principle applies.
28
for terrace diffusion. Thus, we simplify (3.1) via the ?quasi-steady approximation?,
?tCi ? 0; the time dependence in Ci enters through the boundary conditions at step
edges. Solving the resulting differential equation for Ci by successive integrations
yields
Ci = Ai(x?xi) +Bi, xi <x<xi+1. (3.2)
The approximation ?tCi ? 0 might break down in the case of fast-moving steps,
for example, in a surface grown by molecular beam epitaxy using a large deposition
flux [29]. Since we discuss only surface relaxation, material deposition from above
is not relevant, and the quasi-steady approximation can safely be invoked.
The adatom flux on the ith terrace is defined by
Jadi = ?Dad?xCi, (3.3)
which is Fick?s law of diffusion.
Mixed (or Robin) boundary conditions at the step edges x = xi and x = xi+1
complement (3.1) to yield a unique solution for Ci. These conditions emerge from
linear kinetics [10, 37, 43]:
?Jadi (xi,t) = ku[Ci(xi,t)?Ceqi (t)] , (3.4)
Jadi (xi+1,t) = kd[Ci(xi+1,t)?Ceqi+1(t)] , (3.5)
whereku,kd are kinetic rates that account for the Ehrlich-Schwoebel barrier [17, 105],
and Ceqi (t) is the equilibrium density at the ith step edge. Substituting for Ci and
Ji using (3.2) and (3.3), we find a linear system of equations for the integration
29
constants. ?
??
?
Dad ?ku
?Dad ?kd?xi ?kd
?
??
??
?
??
?
Ai
Bi
?
??
?= ?
?
??
?
kuCeqi
kdCeqi+1
?
??
?. (3.6)
This matrix equation is solved for Ai and Bi to obtain
?
??
?
Ai
Bi
?
??
?=
1
Dad(ku +kd) +kukd?xi
?
??
?
kukd(Ceqi+1 ?Ceqi )
Dad(kuCeqi +kdCeqi+1) +kukd?xiCeqi
?
??
? (3.7)
Knowing the explicit values of Ai and Bi, we write the formula for the mass
flux Ji on the ith terrace:
Jadi = ?DadAi
= ?Dad kukd(C
eq
i+1 ?C
eq
i )
Dad(kd +ku) +kukd?xi, (3.8)
where ?xi := xi+1 ?xi is the terrace width.
We then compute the velocity of the ith step according to the law of mass
conservation:
?xi = dxidt = ?a[Jadi?1 ?Jadi ]. (3.9)
Together with the initial step positions and a formula for determining Ceqi
based on the step geometry, the system (3.8),(3.9) governs the subsequent motion
of steps.
3.3 1D interaction energy and the step chemical potential
In order to render (3.8)?(3.9) a closed system of equations for the step po-
sitions, we need a formula for the equilibrium densities in terms of the xi. Our
30
starting point is the near-equilibrium thermodynamics law [37, 43]
Ceqi = Cs exp ?ik
BT
?Cs
bracketleftbigg
1 + ?ik
BT
bracketrightbigg
, (3.10)
where ?i is the chemical potential of the ith step. We consider this ?i to depend on
the energy of interactions with other steps [37, 43, 66], in contrast to the original
BCF model [10], which omitted step interactions. The linearization in (3.10) is
permissible under typical experimental conditions, where |?i|lessmuchkBT [114].
The usual interpretation of chemical potential refers to the change in a system?s
energy upon addition or removal of a single atom. In the case of a straight step,
we can only add or remove atoms one row at a time. The step chemical potential
accounts for this geometric restriction by appropriate scaling of the relevant energy
changes.
We denote by Vi,j the repulsive interaction energy between steps i and j. This
interaction is repulsive if the steps have the same sign, e.g., both are step-down. The
mechanisms underlying this repulsion between steps of the same sign deserve some
attention. We build towards an understanding of elastic interactions between one-
dimensional line defects (steps) by starting with the strain field of zero-dimensional
point defects (e.g., vacancies or impurities). Then we address entropic interactions
using ideas from statistical physics.
From the theory of crystal lattice mechanics [50], we know that a point defect
in the bulk (e.g., a vacancy or an impurity) creates an elastic strain field that acts
over distances far from the point defect. This elastic strain field can be modeled by
31
a point distribution of forces
Fi = Aik ??x
k
?(r), i,k = 1,2,3; (3.11)
where r is a three-dimensional radius vector and the point defect is located at r = 0;
Aik is some symmetric tensor; the overall effect of such a distribution is zero total
force and zero moment.
The analogous point distribution formula in the case of a point defect located
on the surface is
f? = A?? ??x
?
?(?), ?,? = 1,2, (3.12)
where the indices ?,? correspond to a coordinate system in the tangential plane; ?
is a two-dimensional radius vector in the plane of the surface (the defect is located
at ? = 0), and A?? is some symmetric tensor. The calculation of elastic interaction
energy between two such point defects [58] uses the field of elastic strains resulting
from the force distribution (3.12). In an isotropic solid those defects with isotropic
force distributions (A?? = A???) exert on each other a mutual repulsive force F
given by [63]
F(?) = 1??
2
piY
A2
?3 , (3.13)
whereY is the Young modulus and? is the Poisson ratio. The??3 decay is predicted
even in the case of an arbitrary symmetric tensor A??.
An isolated step differs from a point defect in that its associated force dis-
tribution has nonzero total moment. To analyze the step-step interactions using
continuum elasticity, we model the displacement field associated with an isolated
step by an array of force dipoles oriented normal to the step in the plane of the
32
terrace [63]:
ux(x,z = 0) = fx ??x?(x), (3.14)
where fx is the strength of the dipole force and the step is identified with the y-
axis. This formula greatly oversimplifies the true force distribution, which in fact has
finite width, has components normal the surface plane, and has in-plane components
which are not pure dipoles [76]. Rather than include all these effects systematically,
Najafabadi and Srolovitz [76] take an empirical approach and invoke a multipole
expansion for the true force distribution due to an isolated step. The coefficients
of this multipole expansion are fit to data from simulations of interacting steps.
To compare the data with the theory, we need to know what repulsive forces are
predicted by the multipole expansions.
The interaction between two steps is calculated in a manner similar to the
calculation of repulsive forces between point defects. Using a multipole expansion
for the force distribution due to each step, Najafabadi and Srolovitz derived an
expansion for the total interaction energy in a one-dimensional step train. The
starting equation is a volume integral of the elastic interactions between the strain
fields epsilon1 due to individual steps.
Eint = 12
integraldisplay
C
bracketleftBiggsummationdisplay
i
epsilon1i(r)
bracketrightBiggbracketleftBiggsummationdisplay
j
epsilon1j(r)
bracketrightBigg
dV (3.15)
By distributing the product over the sums inside the integral, Najafabadi and
Srolovitz expand (3.15) to separate the contributions from self-energy and inter-
action energy:
33
Eint = N[?self(w) +?int], (3.16)
where the interaction energy is found by summing over all steps,
?int(w) =
?summationdisplay
j=1
?s-s(jw) (3.17)
and the contribution from steps of separationjw is given by the multipole expansion
?s-s(w) =
?summationdisplay
k=1
?k
parenleftBiga
w
parenrightBigk
. (3.18)
The substitution of (3.18) into (3.17) yields
?int(w) =
?summationdisplay
j=1
bracketleftBigg ?summationdisplay
k=1
?k
parenleftbigg a
jw
parenrightbiggkbracketrightBigg
=
?summationdisplay
k=1
?k?k
parenleftBiga
w
parenrightBigk
, (3.19)
where?k =summationtext?j=1j?k are material independent and easily calculated. In this formula
for total interaction energy, series truncation and curve fitting suffice to determine
the coefficients in the multipole expansion of a typical step?s force distribution.
By fitting the coefficients of this expansion to the step simulation data, Na-
jafabadi and Srolovitz conclude that for step spacingswlarger than 3a, the dominant
term is the dipole-dipole interaction energy. Truncating the series for interaction
energy at a higher power ofa/w does not significantly improve the fit between (3.19)
and the simulation data [76]. Therefore, the repulsive interaction between steps is,
for all practical purposes, the same as if each step were viewed as an array of force
dipoles. Considering, for example, the pair of neighboring steps indexed by i and
i+ 1, we find an elastic interaction energy per unit length Vi,i+1 given by
Vi,i+1 = g3
parenleftbigg a
xi ?xi+1
parenrightbigg2
, (3.20)
34
where g> 0 measures the strength of repulsive elastic interactions between steps.
Another reason for steps to repel each other is the fact that steps are thermally
rough. Just as steps are line defects on the surface, steps themselves can have point
defects (edge atoms) or kinks, which slightly perturb their straight shape. The
formation of an edge atom requires only the energyepsilon1? associated with broken bonds,
while the creation of an entire step requires energy in proportion to its length. At
any nonzero temperature T, it is more energetically favorable for a step to wander
(due to formation of kinks and edge atoms) than to advance or retreat as a whole.
This thermal wandering does not proceed unimpeded. Invoking the non-crossing
condition for steps, Gruber and Mullins [30] observed that collisions with neighbors
would restrict the thermal wandering of a given step in an array of parallel steps.
This restriction leads to a smaller entropy for the step in question, which then has
larger free energy. The overall effect is that of entropic repulsion between steps.
To quantify the entropic interaction between steps, it is helpful to consider the
in-plane correlation function
?|x(y)?x(yprime)|2? = kBT?? |y?yprime|, (3.21)
where the angle brackets denote a thermal average with respect to the equilibrium
distribution, the steps are aligned with the y-axis, and ?? is the step stiffness derived
from the Ising lattice gas model [73],
?? = kBT
a e
epsilon1?/kBT. (3.22)
Neighboring steps are assumed to have an average separation w (measured in
the x-direction), and the average distance between collisions is Lc (measured in the
35
y-direction). Setting the left hand side of (3.21) equal to w2 and substituting Lc for
|y?yprime|, we find Lc ?w2??/(kBT). If each collision decreases the entropy by a fixed
amount kBC, then the step?s free energy per unit length is increased by the amount
?? ? CkBTL
c
= C(kBT)
2
w2?? . (3.23)
To determine the coefficient C, it is useful to think of the steps as world lines of
non-interacting fermions, with the y-axis playing the role of time [42, 44]. This
representation yields C = pi2/6.
The decay of entropic repulsion strength asw?2 is the same scaling we found for
elastic-dipole interactions. The combination of both forces is easily accommodated
in (3.20) by renormalizing the coefficient g.
Here we consider ?i stemming from nearest-neighbor step interactions. Thus,
?i depends only on the distances |xi ?xi?1|. (In the macroscopic limit, adding up
all the pairwise interactions has the same effect as renormalizing the parameter of
interaction strength when considering only nearest-neighbor forces [66].) To deter-
mine the chemical potential of the ith step, it suffices to compute the elastic-dipole
forces exerted by neighboring steps only, not taking into account any interactions
beyond nearest-neighbor.
Suppose the ith step advances by an amount ?xi. This step advance requires
the attachment of an additional ?N = a?xi/? atoms per unit length, where ? is
the atomic volume. By analogy with the usual definition of chemical potential, we
define the step chemical potential as the change in interaction energy upon addition
36
or removal of ?N atoms per unit length. Formally letting ?xi ? 0, we write
?i := ?a ddx
i
[Vi,i+1 +Vi,i?1]. (3.24)
The formulas (3.20) and (3.24), together with (3.8) and (3.9), constitute a
system of ordinary differential equations governing the flow of straight steps.
A typical question of interest for this system is whether an equilibrium solution
(i.e., that for equally-spaced steps or constant surface slope) is stable. If not, we
try to characterize the types of instabilities that emerge as the surface relaxes.
Stability analysis of the step flow equations in this geometry employs techniques
from perturbation theory, such as the phase equation method [49], which studies the
evolution equation satisfied by the phase ? of a given initial pattern. Research in
this vein has led to general conclusions about the properties of steady state solutions
for a large class of nonlinear PDEs [95]. The focus of our work is on step systems
in 2+1 dimensions, where such a stability analysis is not practical. The reader is
directed to [15, 95] for more details about the dynamics of (1+1)-dimensional step
trains.
3.4 BCF model for interacting concentric circular steps
The restriction that step positions be expressible in terms of only one spa-
tial variable allows yet another geometry. Whereas the geometry of straight steps
emerges from translational invariance, we obtain the geometry of concentric circu-
lar steps (Figure 3.2) by requiring rotational invariance. Polar coordinates (r,?)
are appropriate in this setting, where the dependence on the angular variable ? is
37
Figure 3.2: Geometry of concentric circular steps.
eliminated due to the assumed axisymmetry.
The point r = 0 is singled out as the center of the mound or valley. To
establish some parallelism with the case of straight steps, we restrict our attention
to axisymmetric structures where the height is a decreasing function of r (mounds).
At any position inside the innermost step, in particular at r = 0, the height is a
constant; h(r,t) = hf(t) for 0 <r<r0, where r0 is the radius of the innermost step.
The subscript f recalls the picture of a facet, a macroscopically flat region about
r = 0 characterized by a length scale significantly larger than the typical terrace
width.
The analog of (3.1) for axisymmetric surface profiles reads
d2Ci
dr2 +
1
r
dCi
dr = 0, (3.25)
38
where again we have invoked the quasi-steady approximation, ?tCi ? 0.
We solve (3.25) by successive integrations to find
Ci(r) = Bi +Ai ln(r/ri), ri <r<ri+1. (3.26)
The terrace adatom flux, directed radially outward to satisfy the symmetry
constraints, is calculated using Jadi = ?Dad?rCi. In terms of the integration con-
stants, the adatom flux is
Jadi = ?D
adAi
r , ri <r<ri+1. (3.27)
Boundary conditions analogous to (3.4) and (3.5) complement (3.25) to yield
the adatom density field on the ith terrace. Again assuming linear kinetics at step
edges, we have
?Jadi (ri,t) = ku[Ci(ri,t)?Ceqi (t)], (3.28)
Jadi (ri+1,t) = kd[Ci(ri+1,t)?Ceqi+1(t)], (3.29)
where Ceqi is the equilibrium adatom density at the ith step edge.
Substituting (3.26),(3.27) for Ci and Ji, we find the integration constants by
solving the matrix equation
?
??
?
?Dadkuri 1
Dad
kdri+1 + ln
ri+1
ri 1
?
??
?
?
??
?
Ai
Bi
?
??
?=
?
??
?
Ceqi
Ceqi+1
?
??
?. (3.30)
Solving for Ai and Bi allows us to write the explicit solution of the requisite
39
boundary value problem on each terrace.
Ai = C
eq
i+1
Dad
parenleftBig
1
kuri +
1
kdri+1
parenrightBig
+ ln ri+1ri
(3.31)
Bi =
Ceqi+1 +Ceqi
parenleftBig
1 + kurik
dri+1
+ kuriDad ln ri+1ri
parenrightBig
1 + kurik
dri+1
+ kuriDad ln ri+1ri . (3.32)
In these coefficients for the solution of the diffusion equation, we identify
certain fractions that indicate the rate-limiting process. The kinetics are called
attachment-detachment limited (ADL) if Dad(k?1u +k?1d )/a greatermuch 1, i.e., adatoms dif-
fuse much faster than they attach and detach at step edges. The kinetics are called
diffusion limited (DL) if Dad(k?1u +k?1d )/a lessmuch 1, i.e., adatoms diffuse much slower
than they attach and detach at step edges. ?Mixed kinetics? is the intermedi-
ate regime where Dad(k?1u +k?1d )/a = O(1). Simplifications of (3.31),(3.32) in the
regimes of DL or ADL kinetics will be useful when discussing scaling laws in Chapter
5.
The difference of fluxes from neighboring terraces at the ith step edge yields
the step velocity ?ri = dridt according to the law of mass conservation:
?ri = ?a[Jadi?1(ri)?Jadi (ri)]. (3.33)
3.5 Interaction energy and step chemical potential for circular steps
It remains to derive the dependence of Ceqi on step positions. This calculation
reveals one key difference between the axisymmetric geometry and the straight-
step geometry. In the axisymmetric setting, the equilibrium density Ceqi contains
information not only about the energetic interactions analogous to those of straight
40
steps, but also the energetic cost of step curvature. For the circular step with radius
ri, the equilibrium density is given by (3.10) and the inclusion of curvature in the
formula for step chemical potential:
?i = ?a
bracketleftbigg?
ri +
1
ri?ri(riU
int
i )
bracketrightbigg
, (3.34)
Uinti = [V(ri,ri+1) +V(ri,ri?1)], (3.35)
V(r,rprime) = g3 2r
prime
r+rprime
parenleftbigg a
r?rprime
parenrightbigg2
, (3.36)
where ? is the step line tension (energy/length) and Uinti is the interaction energy
per unit length of the ith step.
A geometric argument for the appearance of step curvature and line tension
(the term ?/ri in (3.34)) is as follows. Again we measure the change in the system?s
energy ?Ui and count the number ?N of additional atoms that must attach to the
step at r = ri to accomplish this energy change. For circular steps, the length of
the ith step is 2piri, which imposes an energetic cost of 2pi?ri due to line tension.
To advance the ith step by a distance ?ri will increase the line tension energy
by an amount ?Ui = 2pi??ri. The number ?N of additional atoms required to
advance this step is proportional to the area of an annulus, approximately 2piri?ri
in the limit ?ri ? 0. To be precise, ?N = 2piri?ri/(?/a), where ?/a is the cross-
sectional area of an atom. Taking the ratio ?Ui/?N, we find that the change in
line tension energy for each added atom is (?/a)(?/ri).
We remark that formula (3.36) for the interaction energy has the same de-
pendence on the local step density as in the case of straight steps, but with a
shape-dependent prefactor 2rprimer+rprime. This shape factor appears when we calculate the
41
interaction energy between concentric arrays of dipoles by integrating along each
step. To our knowledge, (3.36) first appeared in [113], with little explanation of
its origin. An independent derivation of this formula, emphasizing the geometric
assumptions under which it holds, is given in Chapter 4. The presence of a shape
factor arising from integration of the dipole energy formula is characteristic of 2-
dimensional steps, as we see in section 4.3 when considering arbitrary step geometry.
Equations (3.10),(3.36),(3.31), and (3.33) suffice to determine the subsequent
evolution of the step positions in the axisymmetric setting.
Due to the inclusion of step curvature in the formula for the step chemical
potential, an axisymmetric surface profile can exhibit step bunching instabilities [26].
This phenomenon results from the competition between nearest-neighbor repulsions
and the tendency of an isolated circular step to minimize its line tension energy
by expanding. Step bunching has been proposed as a possible means of growing
nanowires, a setting in which the axisymmetric model applies.
42
Chapter 4
Elastic-dipole interactions between steps in 2 dimensions
In this chapter, we pause to examine the origin of formula (3.36) for the elastic-
dipole interaction energy between two steps. As noted above, equation (3.36) first
appeared in [113], accompanied only by a brief plausibility argument. Subsequent
uses of the formula in [25, 36, 37, 64, 118] offered neither a derivation nor a clear set
of assumptions under which (3.36) is expected to hold. The purpose of this chapter
is to derive equation (3.36) starting from an interaction energy formula based on
linear elasticity. In the course of this derivation it will become apparent to what
extent (3.36) is an approximation, and under which conditions we can expect it to
provide correct values for the interaction energy. Finally, this calculation is extended
to more general step configurations by appropriate local definitions of the relevant
geometric parameters. We use the result of this derivation when describing the fully
(2+1)-dimensional relaxation problem in the next chapter.
4.1 From electrostatics to linear elasticity
The presence of a line defect (step) in a crystal surface introduces an elastic
strain field, as noted in Chapter 3. This field can be understood as stemming from
a line of force dipoles normal to the step. In this section we appeal to knowledge
of electrostatics to motivate the expression for the field of an elastic force dipole.
43
We caution the reader not to expect an exact analogy between the equations of
electrostatics and those of linear elasticity. In contrast to electric fields, an elas-
tic displacement field cannot exist without a medium. This medium introduces
material-dependent quantities, such as the Young modulus and the Poisson ratio,
which might lead to correction terms in the formula for dipole interaction energy.
It turns out that we find the same leading-order term as if the steps consisted of
electric dipoles, since the characteristic scaling of the field strength as the inverse
cube of separation is common to both types of dipoles. We begin by reviewing the
electrostatic equations required to describe interacting dipoles.
4.1.1 Field and potential of an electric dipole
A common calculation in electrostatics is the long-range behavior of the electric
potential due to a collection of point charges q?. The expression for this potential is
a series in negative powers of the distancerfrom the center of the distribution, which
is known as a multipole expansion [40]. This series is only expected to yield correct
values at distancesrmuch greater than the finite extentRof the charge distribution.
Often only the first two terms in the series make a significant contribution to the
electric potential. The monopole term decays as r?1 and is proportional to the total
charge. The dipole term decays as r?2 and depends on the magnitude and direction
of the dipole moment p. This dipole moment is calculated by [40]
p =
summationdisplay
?
q?R?, (4.1)
where R? is the position of charge q? in some appropriate coordinate system.
44
At positions r far from the center of the dipole, the potential ? can be written
as
? = r?pr3 . (4.2)
Note that we choose CGS units, where the usual SI prefactor (4piepsilon10)?1 is set to 1,
so that the eventual transition to elastic interactions is not hindered by lingering
dimensional constants.
To calculate the electric field E at position r, we apply the gradient operator
to (4.2).
E = ??? = 1r3[3?r(p??r)?p]. (4.3)
The analog of the electric field in the context of continuum mechanics is the
strain field, which exerts a force on point and line defects in the surface. We deter-
mine the energy associated with a given configuration of surface defects by comput-
ing a volume integral over a region where the potential ? admits an expansion in
Taylor series.
4.1.2 Energy of a charge distribution in an electric field
Assembling a given collection of charges in an ambient electric field requires
an amount of work given by
W =
integraldisplay
?
??d3x, (4.4)
where ? is the charge density defined at each point in the domain ?, ? is the electric
potential, and d3x is the volume element.
We assume that the spatial variation of ? within ? is accurately captured by
45
a Taylor series expansion about x0 ? ?:
?(x) = ?(x0) + (x?x0)???|x0 +O(|x?x0|2), x ? ?. (4.5)
Because the energy of a charge configuration does not depend on the origin of
the coordinate system, we choose x0 = 0 for convenience. Substituting (4.5) into
(4.4), we find
W =
integraldisplay
?
d3x??(0) +?x???(0) +O(|x|2). (4.6)
The evaluation of the electric potential and its derivatives at 0 can now be
taken outside the integral.
W = q?(0) +p???(0) +..., (4.7)
where q =integraltext??d3x is the total charge, p =integraltext??xd3x is the dipole moment about the
origin of the given charge distribution, and the omitted terms account for energy
contributions from higher-order poles.
In combining formula (4.3) for the field of an electric dipole and (4.7) for the
work done to assemble a collection of charges, we need to distinguish the two dipole
moments, even though they play symmetric roles in the final formula for interaction
energy. We use pA for the dipole that creates the ambient electric field, and pB for
another dipole that we bring to a desired location within this field. Using (4.7), the
term accounting for dipole-dipole interactions is
Wdip = 1r3
AB
[pA ?pB ?3(pA ??rAB)(?rAB ?pB)], (4.8)
where rAB is the separation vector between dipoles A and B.
46
4.1.3 Interaction energy of two force dipoles
To check whether a direct adaptation of (4.8) would yield plausible results, we
consider first the special one-dimensional case of electric dipoles A and B aligned
with the separation vector rAB. If pA and pB are parallel and point in opposite
directions, then the force between them is repulsive, and we should expect that a
positive amount of work is needed to assemble this charge configuration. Indeed, the
overall sign of (4.11) is positive in this case. However, if pA and pB are parallel and
point in the same direction, then they experience an attractive force, so it should
take a negative amount of work to assemble this charge configuration. The sign
predicted by (4.8) is again consistent with this intuition.
The nature of elastic step-step interactions implies that two steps of the same
sign exert a repulsive force on each other. To illustrate this effect, it helps to think
of a stepped surface as a continuous elastic medium. In the absence of any other
steps, a single step on such a medium would decay to a smoother, flat profile,
just as an initial injection of dye into a clear solution would eventually diffuse to
uniform concentration. The relaxation of a single step is frustrated by the presence
of neighboring steps. Effectively, each pair of same-sign steps experiences a mutually
repulsive force. This repulsive interaction between steps of the same sign, in marked
contrast to the attractive interaction between parallel electric dipoles, indicates that
(4.8) needs to be replaced by its counterpart from linear elasticity before proceeding.
The reader who is already familiar with the theory of defect interactions in
linear elasticity should consult [21, 56, 93] to find a more thorough discussion of the
47
correct formulas describing interacting line defects on a crystal surface. We appeal
to the calculation given by Pimpinelli and Villain [93] for the interaction energy
of two force dipoles. By analogy with (4.1), they define a force dipole moment m
resulting from a set of forces FR acting at points R:
m?? =
summationdisplay
R
R?F?R (?,? = x,y,z). (4.9)
Two such force dipoles, mandmprime, are calculated to have an interaction energy
Wint = 1??
2
piYr3
bracketleftBigg
mmprime ? ?1??(mmprimezz +mzzmprime) +
parenleftbigg ?
1??
parenrightbigg2
mzzmprimezz
bracketrightBigg
, (4.10)
where Y is the Young modulus, ? is the Poisson coefficient, and r is the separation
between the two dipoles. The dominant term in (4.10) is the product mmprime, since
the Poisson coefficient is always smaller than 1/2 [93]. This inequality suggests that
(4.10) can be viewed as a series expansion in powers of the small material parameter
?/(1??). For the purposes of calculating the total interaction energy between two
steps, we restrict our attention to the truncated version of (4.10), which reads
Wint = 1??
2
piYr3 mm
prime +O[?/(1??)]. (4.11)
In order to apply (4.11), the dipole moments m and mprime are measured with
respect to a fixed coordinate system. In particular, the interaction energy between
non-parallel step segments involves the projection (or inner product) of one dipole
moment onto the other.
48
4.2 Integral for the energy of interacting steps
In light of the rotational symmetry (see Figure 3.2), it suffices to fix our
attention on the interaction between a line element on the inner step and the net
strain field induced by the outer step. By superposition, we can calculate the energy
per unit length of the inner step by applying (4.11) for each line element on the outer
step. We recall that the dipole moment is proportional to the length of the line
element, and its contribution to elastic interactions arises through the step-normal
component (radially outward). Using Cartesian coordinates, the interacting dipoles
are located at (r1,0) and (r2 cos?2,r2 sin?2). Their respective xx dipole moments
are
m = Pr1d?1
mprime = P cos?2r2d?2, (4.12)
whereP is the dipole moment per unit length associated with an atomic-height step.
The squared distance between the two dipoles is
r2 = r21 +r22 ?2r1r2 cos?2. (4.13)
We calculate the product required by (4.11) for step-step interactions:
mmprime = P2 cos?2r1d?1r2d?2. (4.14)
The interaction energy per unit length is given by the integral
dW
r1d?1 = P
2
integraldisplay pi
?pi
cos?2 ?r2d?2
(r21 +r22 ?2r1r2 cos?2)3/2. (4.15)
49
Motivated by the physically reasonable assumption that the step separation is
much less than the two individual step radii, we seek a simplified integral in which
the dependence on step separation becomes more obvious. We setr2 = r1+?, where
the inner radius r1 = epsilon1?1 defines a ?large? length scale, and the step separation ? is
considered only O(1). Our task now is to compute
I = 2
integraldisplay pi
0
cos?2
(epsilon1?2 + (epsilon1?1 +?)2 ?2epsilon1?1(epsilon1?1 +?)cos?2)3/2r2d?2. (4.16)
In the following calculation we omit the subscript of ?2 for brevity, keeping in
mind that the only varying angle refers to the position on the step of radius r2. We
expand the denominator of (4.16), arriving at the simplified formula
I =
integraldisplay pi
0
cos?
[?2 + 2(epsilon1?2 +epsilon1?1?)(1?cos?)]3/2r2d?. (4.17)
Note that this integral can be computed in terms of complete elliptic integrals, but
the resulting formulas are not instructive for our purposes.
Now we use the substitution s := sin(?/2), so that
cos? = 1?2s2, (4.18)
d? = 2ds?1?s2. (4.19)
After this change of variables, the interaction energy is given by
I = 2r2
integraldisplay 1
0
ds?
1?s2
1?2s2
[4(epsilon1?2 +epsilon1?1?)s2 +?2]3/2. (4.20)
We define the macroscopic length parameter ? by
?2 := 4(epsilon1?2 +epsilon1?1?). (4.21)
50
Notice that in the limit of small step separation (epsilon1? lessmuch 1), applying the binomial
expansion to the square root of (4.21) yields
? = 2epsilon1?1(1 +epsilon1?)1/2
? 2epsilon1?1(1 +epsilon1?/2)
= 2epsilon1?1 +? = r1 +r2. (4.22)
Already we see a possible origin of the denominator r1 +r2 in the formula (3.36),
and the geometric condition under which the replacement for ? is valid. We proceed
with the calculation by substituting ? into (4.20).
I = 2r2
integraldisplay 1
0
ds?
1?s2
1?2s2
[?2s2 +?2]3/2 (4.23)
= 2r2
integraldisplay 1
0
ds?
1?s2
1?2s2
?3
bracketleftBig
s2 +parenleftbig??parenrightbig2
bracketrightBig3/2. (4.24)
Now we introduce the small parameter ? := ?/?. The integral becomes
I = 2r2?3
integraldisplay 1
0
ds?
1?s2
1?2s2
[s2 +?2]3/2
= 2r2?3
integraldisplay 1
0
ds?
1?s2
?2(s2 +?2) + 2?2 + 1
[s2 +?2]3/2
= 2r2?3
integraldisplay 1
0
ds?
1?s2
bracketleftbigg 2?2 + 1
(s2 +?2)3/2 ?
2
(s2 +?2)1/2
bracketrightbigg
. (4.25)
Notice that by regarding ? and s as independent, the partial fraction decom-
position (4.25) reduces our problem to that of calculating a single integral, namely,
Ic(?2) =
integraldisplay 1
0
ds?
1?s2
1?
s2 +?2. (4.26)
The subscript c recalls the underlying circular geometry. The desired interaction
energy per unit length is then given by the identity
I = 2r2?3
bracketleftbigg
?2(2?2 + 1) dd(?2) ?2
bracketrightbigg
Ic(?2). (4.27)
51
Our motivation for writing I in this way stems from the method of asymptotic
expansion we want to use in the limit ? ? 0. The integral for elastic interaction
between circular steps can actually be given in terms of complete elliptic integrals,
which have well-studied asymptotic expansions for small values of the parameter
? [19]. Keeping in mind the eventual application of our method to settings without
axisymmetry, we opt to follow a more general procedure, using the Mellin transform
with respect to?2. When we apply this technique, we must evaluate double integrals
with respect to s and ?2. The calculation of these double integrals using the Fubini
theorem (coming next) is greatly simplified if we omit at the very beginning any
trivial ?-dependence, such as multiplicative factors.
4.2.1 Mellin transform of the axisymmetric integral
To approximate the behavior of Ic(?2) for ? lessmuch 1, we resort to the Mellin
transform in ?2. This variant of the Laplace transform has proved useful in many
fields of applied mathematics since the 1980s, especially in asymptotic evaluation
of integrals; see [100] for a review. The Mellin transform ?f(?) of a function f(x) is
given by
?f(?) =
integraldisplay ?
0
x??f(x)dx, a< Re? <b, (4.28)
where the domain of ?f is restricted to ensure convergence of the integral; see Ap-
pendix A for details.
The inversion formula is
f(x) = 12pi?
integraldisplay ?+??
????
x??1 ?f(?)d?, (4.29)
52
where a<?<b so that ?f is defined for ? along the integration path.
We apply these formulas to the function Ic as follows.
?Ic(?) =
integraldisplay ?
0
d?2Ic(?2)(?2)?? (4.30)
=
integraldisplay ?
0
d?2(?2)??
integraldisplay 1
0
ds?
1?s2
1?
s2 +?2 (4.31)
=
integraldisplay 1
0
ds?
1?s2
parenleftbiggintegraldisplay ?
0
d?2 (?
2)??
?s2 +?2
parenrightbigg
. (4.32)
For the inner integral, we pull out a factor of s from the denominator and let
? := ?2/s2, treating s2 as a fixed nonzero parameter. This change of variables leads
to
?Ic(?) =
integraldisplay 1
0
ds?
1?s2(s
2)1/2??
integraldisplay ?
0
d? ?
??
?1 +?, (4.33)
and the rightmost integral can be evaluated in terms of Gamma functions [1]. We
have the formula
integraldisplay ?
0
d? ?
??
?1 +? = B(1??,??1/2), (4.34)
where the Beta function B is more conveniently written as
B(z,w) = ?(z)?(w)?(z+w), (4.35)
and ? denotes the Gamma function.
Applying these formulas to the Mellin transform ?Ic yields
?Ic(?) = ?(1??)?(??1/2)
?(1/2)
integraldisplay 1
0
ds(s
2)1/2??
?1?s2 (4.36)
= ?(1??)?(??1/2)?(1/2)
integraldisplay 1
0
d(s2)
2s
(s2)1/2???
1?s2 (4.37)
= ?(1??)?(??1/2)?(1/2)
integraldisplay 1
0
d(s2) (s
2)??
?1?s2, (4.38)
53
and again the rightmost integral admits a representation in terms of the Beta func-
tion. From p. 258 of [1],
integraldisplay 1
0
dt t
??
?1?t = B(1??,1/2) = ?(1??)?(1/2)?(3/2??) . (4.39)
So the Mellin transform of Ic(?2) is concisely expressed as
?Ic(?) = 1
2
?(1??)2?(??1/2)
?(3/2??) , (4.40)
a meromorphic function of ?. The fundamental strip where the integral (4.32)
converges is found to be 12 < Re? < 1.
4.2.2 Asymptotic expansion using the inverse Mellin transform
Recall that the only singularities of ?(z) are poles at the nonpositive integers.
Thus, the line integral prescribed by the inverse Mellin transform can be found by
applying the residue theorem. The sum of these residues is a power series in ?2,
with the possible presence of logarithmic terms.
The inverse Mellin transform applied to ?Ic reads
Ic(?2) = 12pi?
integraldisplay ?+??
????
d?(?2)??1?Ic(?), 12 <?< 1. (4.41)
The leading-order term comes from the residue at the closest pole to the right
of the fundamental strip, or ? = 1 in the present case. To approximate the Gamma
functions near ? = 1, we make the substitution epsilon1 = 1?? ?? = 1?epsilon1. Then
?Ic(?) = ?(epsilon1)2?(12 ?epsilon1)
?(12 +epsilon1) . (4.42)
54
As epsilon1? 0, we have
(?2)?epsilon1?Ic(?) ? (1?epsilon1ln?2)
parenleftbigg1
epsilon1 ??
parenrightbigg2 ?(1
2)[1?epsilon1?(
1
2)]
?(12)[1 +epsilon1?(12)] (4.43)
? (1?epsilon1ln?2)
parenleftbigg1
epsilon12 ?
2?
epsilon1
parenrightbigg
[1?2epsilon1?(12)] (4.44)
? (1?epsilon1ln?2)
bracketleftbigg1
epsilon12 ?2
?(12) +?
epsilon1
bracketrightbigg
, (4.45)
where ? is Euler?s constant and ? denotes the logarithmic derivative of ?, i.e.,
?(z) = d/dz(ln?(z)) [1].
Reading off the coefficient of epsilon1?1 in the Laurent series given by (4.45), we find
the residue contribution from ? = 1 to be ?2?(12) ? 2? ? ln?2 = 4ln2 ? ln?2 =
ln(16/?2). To leading order, the original integral Ic behaves like ln(16/?2) for small
values of ?.
Substituting ln(16/?2) for Ic in (4.27), we find
I(?2) ??4r2?3
bracketleftbigg
ln 16?2 ? 1?2 ?2
bracketrightbigg
? 4r2?
3?2
. (4.46)
In terms of the step radii, (4.46) reads
dW
r1d?1 ?
4P2r2
(r2 +r1)(r2 ?r1)2, (4.47)
in agreement with the inverse-square dependence on r2 ?r1 and the shape factor
r2/(r2 +r1) given in [113].
4.2.3 Corrections to the leading-order term under axisymmetry
It is of interest to note that corrections to the leading-order term (4.47) in
principle arise from several different places. First, the inverse Mellin transform
55
could be better approximated by including the residues from poles beyond ? = 1.
This direction is summarized below. Secondly, by retaining higher-order terms in
(4.10) or (4.46), we obtain other corrections, whose magnitude in relation to the
first correction is not known a priori. Finally, the superposition of strain fields from
different line elements on the two steps is justified only in the context of linear
elasticity. We have not addressed the validity of this linear regime in real crystal
surfaces.
The contribution from the pole ? = 2 is found by writing expansions in epsilon1 :=
2??, which parallel the derivation above. We find
?(1??)2?(?? 12)
?(32 ??) ?
bracketleftbigg1
epsilon12 +
2(1??)
epsilon1
bracketrightbigg ?(3
2)[1?epsilon1?(
3
2)]
?(?12)[1 +epsilon1?(?12)]
??14
bracketleftbigg1
epsilon12 +
2(1??)
epsilon1
bracketrightbigg
(1?epsilon1[?(3/2) + ?(?1/2)])
?? 14epsilon12 ? 2ln2?12epsilon1 . (4.48)
Multiplying (4.48) by (?2)??1 = (?2)1?epsilon1 ??2?(1?epsilon1ln?2) furnishes the residue
contribution from ? = 2. This residue contribution,
1
4?
2
parenleftbigg
ln ?
2
16 + 2
parenrightbigg
, (4.49)
is added to the previous expansion for Ic, which upon substitution in (4.27) yields
I ? 4r2?3?2 + 5r2?3 ? 4r2?
2
?3 ?
3r2?2
?3 ln
?2
16 +
3r2
?3 ln
?2
16. (4.50)
We see that in addition to the inverse-square dependence on ?, algebraic and
logarithmic correction terms are needed to characterize the asymptotic behavior
of the integral for interaction energy between concentric circular steps. We alert
56
the reader that pursuing further corrections may be inappropriate: higher-order
corrections may be comparable or smaller in magnitude than the terms omitted
from our starting point, (4.10). Thus we choose not to seek more corrections of the
integral I at this point.
4.3 Elastic interaction energy between steps without axisymmetry:
A brief discussion
Our Mellin transform manipulations of the axisymmetric integral (4.26) could
have been bypassed in favor of elliptic integrals and their asymptotic expansions, if
our only goal was to compare the result with the interaction energy formula given
in [113]. The worth of the Mellin transform is better demonstrated by indicating how
this transform can be applied to the integral for elastic dipole interactions between
steps of reasonably general shape, i.e., without axisymmetry. We still regard the
steps as small perturbations of a circle, which allows them to be described as polar
graphsr = r(sin(?/2)),?: polar angle. The substitutions := sin(?/2) is used exactly
as above to simplify the integrand. In Figure 4.1 we illustrate two neighboring steps
restricted to a circular sector around ? = 0. The inner step admits the polar
representation r = r1(s), while the outer step is given by r = r2(s).
In order to quantify the deviation of these steps from circles, we let
r2(s) = r1(s) +?r(s), (4.51)
r1(s) = epsilon1?1?(s), (4.52)
where ? = O(1) > 0, r(s) = O(1), ?(s) = O(1), and epsilon1 lessmuch 1. Here epsilon1?1 denotes
57
Figure 4.1: Geometry of neighboring steps without axisymmetry, parametrized by
r1(s) and r2(s). We define s := sin(?/2) to convert trigonometric functions of ? into
algebraic functions of s.
58
the characteristic length scale, roughly analogous to the radius of circular steps. In
addition, we assume ?(s),r(s) ?C1(?1,1), so that the tangent vector at each point
on the two steps is well defined.
A crucial quantity to calculate is the distance R between points on the two
steps, which enters as R3 in the denominator of the elastic interaction energy inte-
grand. We have
R2 = r2(s)2 +r1(0)2 ?2r1(0)r2(s)cos? (4.53)
= epsilon1?2{[?(s)??(0)]2 + 4s2?(0)?(s)}+ 2epsilon1?1?r(s)[?(s)??(0) + 2?(0)s2] +?2r(s)2.
(4.54)
Because ?,r are assumed to be C1, using Taylor series about s = 0 we can
define the C0 functions ?,r by
?(s)??(0) =: s?(s), (4.55)
r(s)?r(0) =: sr(s). (4.56)
After adding and subtracting ?(0),r(0) as needed, we find
R2 = epsilon1?24?(0)2s2 + 4epsilon1?1?r(0)?(0)s2 +?2r(0)2
+epsilon1?2[?(s)2 + 4s?(s)?(0)]s2 + 2epsilon1?1?s{r(s)[s?(s) + 2?(0)s2] +r(0)?(s)}
+?2s[2r(0)r(s) +sr(s)2]. (4.57)
In order to factor out the coefficient of s2, we define
A(s) := {4[epsilon1?2?(0) +epsilon1?1?r(0)]?(0)}?1?
parenleftbigepsilon1?2[?(s)2 + 4s?(s)?(0)]s+ 2epsilon1?1?{r(s)[s?(s) + 2?(0)s2] +r(0)?(s)}
+?2[2r(0) +sr(s)]r(s)parenrightbig. (4.58)
59
Then
R2 = 4[epsilon1?2?(0)+epsilon1?1?r(0)]?(0)
braceleftbigg
s2 +sA(s) + ?
2r(0)2
4?(0)[epsilon1?2?(0) +epsilon1?1?r(0)]
bracerightbigg
, (4.59)
where A(s) ?O(1).
To follow as closely as possible the notation of the axisymmetric case, we define
the parameters
?2 := ?
2r(0)2
4?(0)[epsilon1?2?(0) +epsilon1?1?r(0)], (4.60)
?2 := 4[epsilon1?2?(0) +epsilon1?1?r(0)]?(0), (4.61)
which feature in the asymptotic expansion of the integral for elastic interaction en-
ergy. In terms of?and?, the squared distance between points with (s,r) coordinates
(0,r1(0)), (s,r2(s)) is
R2 = ?2{s2 +?2 +sA(s)}. (4.62)
We note that the breaking of axisymmetry enters here through the termsA(s), which
accounts for the curvature variation when steps deviate from circles. The geometry
of Figure 4.1 is deceptive in its simplicity. We have chosen a coordinate system such
that the vector normal to step 1 ats = 0 is aligned with thex-axis. By following the
procedure of the axisymmetric case, we still obtain an elastic interaction energy per
unit length, but now we cannot immediately generalize the resulting formula to other
points along step 1. In principle, a different calculation with different definitions
of A(s),?, and ? is required for each point on step 1. Despite this shortcoming,
we expect the qualitative features of our interaction energy formula, in particular
the inverse-square dependence on local step separation, to persist no matter which
60
point on the inner step is used to orient the Cartesian coordinate system.
Below we outline the calculation for reasonably general 2D steps. A complete
exposition of all the details would distract from the main point of this thesis (and
chapter) and is hence omitted for the sake of clarity.
For simplicity, we write the integral over step 2 (which in principle requires s
to range over [?1,1]) as two separate integrals,
dWint
r1d?1 =
1??2
piY P
2
parenleftbiggintegraldisplay pi
0
+
integraldisplay 0
?pi
parenrightbigg sin?dr
2/d?+r2 cos?
(r21 +r22 ?2r1r2 cos?)3/2d?, (4.63)
where P is the dipole moment per unit length of an atomic-height step. The first
integral considers the energetic contributions given by (4.11) for ? ranging from 0 to
pi. The evaluation of the second integral is essentially not different and is therefore
omitted. We now face the task of calculating
I2D =
integraldisplay pi
0
sin?dr2/d?+r2 cos?
(r21 +r22 ?2r1r2 cos?)3/2d?, (4.64)
The numerator in (4.64) stems from the inner product of the two vectors normal to
step 1 at s = 0 and normal to step 2 at s = sin(?/2).
We expand the numerator in (4.64), substituting the definitions of ?2,?2, and
A(s) wherever possible. After grouping terms according to the highest power of s
they contain as a factor, we reduce our problem to calculating integrals of the form
Ik(?2) := 2
integraldisplay 1
0
skBk(s)
[A(s)s2 +?2r(s)2]3/2
ds?
1?s2, (4.65)
where Bk(s? 0) negationslash= 0 and A(s? 0) negationslash= 0 and r(s? 0) negationslash= 0.
We alert the reader that (4.65), unlike (4.26), in general cannot be evaluated in
terms of elliptic integrals, especially since we don?t know much about the behavior
61
of A(s), r(s), or Bk(s). However, the Mellin transform with respect to ?2 does not
need to know about these functions, since the main dependence on ?2 is already
accounted for in (4.65). We apply (4.28) to (4.65), which yields
?Ik(?) = 2
integraldisplay ?
0
d(?2)(?2)??
integraldisplay 1
0
skBk(s)
[A(s)s2 +?2r(s)2]3/2
ds?
1?s2. (4.66)
We apply Fubini?s theorem to (4.66) in order to switch the order of integration.
The integral with respect to?2 can be evaluated in terms of Gamma functions, which
entails
?Ik(?) = 2?(1??)??(32 +?)
?(52)
integraldisplay 1
0
ds?
1?s2s
k?2??3Bk(s)A(s)
???3/2
[r(s)2]??+1 . (4.67)
This integral converges for
{? ?C : Re(?k+ 2? + 3) < 1} = {? ?C : Re? <?1 +k/2}, (4.68)
assuming that A(s),r(s) negationslash= 0 as s ? 1 and similarly for Bk, A, r as s ? 0. For each
integral Ik, the leading-order term comes from expanding the Mellin transform near
? = ?k = k?22 . In the case where ?Ik has a simple pole at ?k, we have to compute the
residue at the next pole on the right. This residue requires us to find an analytical
continuation of ?Ik outside the fundamental strip of convergence, which is achieved
via integration by parts.
After the required residues have been computed, we apply the inverse Mellin
transform to determine the expansion of Ik as an extended power series in ?. We
find again that to leading order dWint/r1d?1 depends on ??2, the inverse square of
nearest-neighbor separation between steps. The correction terms involve logarithms
and higher powers of ?. We reiterate that the magnitude of these corrections in
62
relation to the omitted terms in (4.10) involving the Poisson ratio is not known a
priori. Adding the corrections from the residues at more distant poles of ?Ik might not
be enough to ensure agreement of our formula with the elastic energy of a stepped
surface. The cross-terms of (4.10) might prove more important in the correction of
our leading-order result.
For the purposes of this thesis, it is sufficient to note that the elastic inter-
action energy between two neighboring steps decays as the inverse square of their
separation along the transverse direction. The prefactor of this leading-order term
depends only on the values of the local coordinates and their derivatives, e.g., the
factor B0(0)A(0)?1/2/r(0)4 which appears in the expansion of I0(?) [71]. This re-
sult is similar to the step interaction formula used by Margetis and Kohn [66], who
postulated a geometrical factor ?(?,?) to account for the possible variation of step
curvature. Here we do not attempt to make an explicit connection between the ?
of [66] and the corresponding coefficient of our extended power series in ?. This
coefficient in our extended power series depends in principle on which point of step
1 we use to orient our Cartesian coordinates. In contrast, the ? of [66] depends only
on the transverse coordinates ?i,?i?1. Compatibility conditions between these two
formulas are not pursued further, since the macroscopic limit eventually averages
out any microscopic discrepancies in the interaction energy.
In this section we have summarized a rather substantial calculation, which
derives the leading-order term of the elastic interaction energy between reasonably
arbitrary 2D steps. The macroscale limits we derive in the next three chapters are
not sensitive to the exact coefficient of this leading-order term. It suffices to note
63
that the calculations above offer a convincing demonstration of the step separation
dependence typically assumed for elastic interaction energy. This inverse-square
dependence is used in Chapter 5 to derive a formula for the macroscale chemical
potential, which remains invariant under changes in the kinetic processes operating
at the nanoscale.
64
Chapter 5
Review of equations for interacting steps in 2+1 dimensions, and
macroscale limit1
In this chapter, we describe a model of interacting steps in 2+1 dimensions
when the adatom diffusion is isotropic on each terrace. Then, we review the deriva-
tion of a macroscale evolution equation for the relaxation of a stepped crystal sur-
face. A remarkable feature of the macroscopic description is that the effective sur-
face mobility (or diffusivity) is strictly a tensor. The step-normal and step-parallel
components of the macroscale surface flux scale differently as functions of the corre-
sponding components of the chemical potential gradient. This prediction of a tensor
mobility appears only as a result of coarse-graining the equations for step flow. Pro-
ceeding by analogy with the macroscale equation of 1-dimensional stepped surfaces
would suggest a scalar mobility instead [64, 108].
We note that the tensor character of the mobility only manifests itself in
the regimes of attachment-detachment limited (ADL) or mixed kinetics. Our pre-
dictions for surface relaxation subject to diffusion-limited (DL) kinetics would be
indistinguishable from the predictions of a theory with scalar mobility.
1Material in this chapter appeared previously in Quah and Margetis, J. Phys. A: Math. Theor.
41, 235004/1?18.
65
Figure 5.1: Orthogonal projection of step edges.
5.1 Geometry of a step train in 2+1 dimensions
In the spirit of the BCF theory [10], the edges of steps are projected to closed,
noncrossing, and non-self-intersecting curves in a reference (basal) plane, where
positions are indicated using fixed coordinates (x,y); see Fig. 5.1. These curves
must possess x- and y-derivatives up to second order, so that the curvature, tangent
vector, and normal vector are well-defined at each point on a curve. We treat the
curves corresponding to step edges as moving boundaries for the adatom diffusion
of each terrace.
Numbering the stepsi = 0,1,2,..., starting from the topmost step, we identify
66
the ith step with the boundary of the compact set {(x,y) : h(x,y,t) ?hf(t)?ia}.
(Recall that hf(t) denotes the maximum height at time t, whether achieved at an
isolated point or on a facet.) The ith step is in general a one-dimensional level
set, and defining it in terms of fixed coordinates x and y is not always the most
convenient option. We introduce local coordinates (?,?) associated with the steps
to describe the surface profile. The local coordinate ? is roughly analogous to the
radius in polar coordinates and increases monotonically as we descend the step train.
In these new coordinates, the ith step corresponds to the set {? = ?i} in the basal
plane. The variable ? serves as a continuous analog of the discrete step index i.
Positions between step i and step i + 1 can be assigned ?-coordinates from the
set ?i < ? < ?i+1, which is a local-coordinate description of the ith terrace. The
local coordinate ?, analogous to the polar angle ?, measures the distance along a
step edge; to be definite, we choose ? to increase counterclockwise (top view). By
construction, the level sets of ? and ? intersect at right angles; the associated unit
vectors e? and e? are orthogonal, e? ?e? = 0. The reader is referred to [66] for more
details.
For a fixed surface height profile, positions in the basal plane can be specified
by their ? and ? coordinates. If the height profile evolves in time, the mapping
from local coordinates to fixed coordinates via r : (?,?) mapsto? (x,y) ? R2 can be
viewed as a function of the time t. From this coordinate transformation we obtain
two scale factors (metric coefficients) associated with the step coordinates. These
metric coefficients, which appear when we compute spatial derivatives in (?,?), are
67
[5]
?? := |??r|, ?? := |??r|. (5.1)
A typical geometry of interest occurs when a mound or valley described by a given
collection of steps projects onto a bounded rectangular region B in the basal plane.
Periodically extending the height profile from B to all of R2 creates a mathemat-
ical representation of a regular surface corrugation. In this periodic setting, the
monotonicity of the step train is violated in certain regions. We expect on physical
grounds that solutions of the macroscale evolution law, known to be valid where
the step train is monotonic, can be connected continuously across regions where
monotonicity is violated. We consider periodic surface profiles in our subsequent
numerical investigations in Chapter 10. The step geometry outlined here remains
unaltered when we consider terrace anisotropy in Chapter 6.
5.2 BCF model with step interactions in 2+1 dimensions
Proceeding in the same vein as in the case of everywhere parallel steps, we
denote by Ci the adatom density on the ith terrace, ?i <?<?i+1. This Ci satisfies
a diffusion equation, which under the quasi-steady approximation reads
0 = div(Dad ??Ci), (5.2)
where Dad is in principle a tensor (2 ? 2 matrix) diffusivity and ? = e???1? ?? +
e???1? ??) is the gradient on the basal plane. We define the adatom flux as
Jadi = ?Dad ??Ci, (5.3)
68
which is Fick?s law of diffusion in the absence of drift.
Note that (5.2) amounts to the most elementary description of surface relax-
ation by adatom diffusion, leaving out the effects of atom desorption, electromigra-
tion, and material deposition from above. These effects can be modeled by additive
terms in (5.2), as we discuss in Chapter 8 (for desorption and electromigration). For
the purposes of this chapter, (5.2) along with requisite boundary conditions at step
edges are enough to illustrate the contrast between (1+1)- and (2+1)-dimensional
surface morphological evolution.
Robin boundary conditions at the boundaries (step edges) of the ith terrace
complement (5.2) to yield a unique solution for Ci. By analogy with the boundary
conditions for everywhere parallel steps, we expect that the flux normal to a step
edge is proportional to the difference between the actual adatom density and the
equilibrium adatom density. So, we consider only the component Jadi,? := e? ? Jadi
when writing the linear kinetic boundary conditions:
?Jadi,?(?i,?,t) = ku[Ci(?i,?,t)?Ceqi (?,t)], (5.4)
Jadi,?(?i+1,?prime,t) = kd[Ci(?i+1,?prime,t)?Ceqi+1(?prime,t)], (5.5)
whereku,kd are kinetic rates that account for the Ehrlich-Schwoebel barrier [17, 105]
and Ceqi (?,t) is the equilibrium density at the ith step edge.
The relation betweenCeqi and the chemical potential of theith step is precisely
the same as in 1+1 dimensions, but now the chemical potential ?i depends on the
coordinate ? as well:
Ceqi (?) = Cs exp?i(?)k
BT
?Cs
bracketleftbigg
1 + ?i(?)k
BT
bracketrightbigg
. (5.6)
69
The step chemical potential formula for arbitrary surface morphology, derived in
[66] by a differential geometry calculation, expresses in 2+1 dimensions the combi-
nation of interaction energy and step line tension, which we first saw together in the
axisymmetric setting. The result reads [66]
?i = ?a
parenleftbigg1
????iUi +?iUi
parenrightbigg
, (5.7)
where ? is the atomic volume, Ui is the total energy per length of the ith step edge
and ?i is the step edge curvature. We identify two contributions to Ui [66]:
Ui = ? +Uinti , (5.8)
where ? is the step line tension, assumed here to be constant, and Uinti is the
interaction term which depends on ? and the step positions {?j}. For a surface with
sufficiently small slope and nearest-neighbor interactions that decay as the inverse
square of the terrace width, Uinti is [66]
Uinti = Vi,i+1 +Vi,i?1, (5.9)
Vi,i+1 = g3m2i?(?i,?i+1), ?i :=
integraldisplay ?i
?0
??d?, mi := a?
i+1 ??i
, (5.10)
where g > 0 is a constant quantifying the strength of step interactions, ?i corre-
sponds to the radial distance in polar coordinates, mi is the discrete step density,
and ? is a shape factor; note that ?(?i,?i) is a constant, independent of the distance
?i [66]. The important feature of (5.10) is the inverse-square dependence on step
separation, which follows from the elastic interaction formulas of Chapter 4. As
we saw in Chapter 4, the calculation of a shape factor even in the special case of
axisymmetry requires a substantial amount of analysis. We alert the reader that
70
(5.10) is an approximation, omitting the ? dependence of the prefactor to avoid
unwarranted complications in the subsequent coarse-graining.
An alternative definition of the step chemical potential invokes the total energy,
Esteps, of a step train in 2+1 dimensions. Each step contributes a line tension energy
in proportion to its total arclength, because it takes energy to create the line defect
of a step on a crystal surface. Moreover, the repulsive interaction between a step and
its same-sign neighbors also adds to the total energy Esteps. Recall from Chapter 3
that the existence of a step introduces a mechanical strain field in the crystal. This
strain field can be modeled as the result of a force dipole located at each point along
the step. Each pair of neighboring step segments contributes an elastic interaction
energy given by the energy of two interacting force dipoles, for example the dipoles
dsie?|i, dsi+1e?|i+1. In addition, neighboring steps exert an entropic repulsion on
each other. We consolidate these contributions into a single integral over the ith
step, and then sum over the step train to obtain
Esteps =
summationdisplay
i
integraldisplay
Li
(? +Vi,i+1)ds. (5.11)
To determine the step chemical potential ?i, we need to know the number
of additional atoms required at the ith step to effect a given change in step train
energy. The normal displacement vi(?)?t at each position ? along the ith step will
require the addition of enough atoms to cover an area ofintegraltextLi vi(?)?tds. The typical
cross-sectional area of an atom is ?/a. According to the definition of step chemical
potential, these additional atoms at the ith step will change the total energy by an
amount 1?/aintegraltextLi vi(?)?i(?)ds?t. Summing up the contributions from every step to
71
determine the change in total energy, ?E = ?Esteps?t, yields a weak formulation of
the step chemical potential, ?i. We equate the sum over all steps with the direct
calculation of ?Esteps using (5.11). The result is
?Esteps = 1
?/a
summationdisplay
i
integraldisplay
Li
vi?i(?)ds. (5.12)
We will use this weak formulation of the step chemical potential when deriving the
macroscale limit in Section 5.4.2.
Before giving the final equations connecting the chemical potential to the step
velocities, we take a moment to identify which of the preceding relations will carry
through unchanged when we consider extensions of the terrace adatom kinetics. The
definitions of local coordinates and metric coefficients, as noted above, will persist in
their present form. In addition, the definitions of Ceqi and ?i are independent of the
adatom kinetics on terraces, so their formulation here will carry through unaltered to
the study of anisotropic terrace diffusion (Chapter 6) and electromigration (Chapter
8).
Lastly, we present the step velocity law in 2+1 dimensions. By including
diffusion of atoms along the step edge with constant edge diffusivityDed, the normal
velocity of the ith step edge is [82]
vi = e? ? dridt = ?a(Jadi?1,? ?Jadi,?) +a?s
parenleftbigg
Ded?s ?ik
BT
parenrightbigg
, (5.13)
where ?s is the space derivative along a step edge; ?s = ??1? ??. The first term
in (5.13) is the contribution of terrace adatom fluxes. The second term is due to
step edge diffusion and stems from the variation of the step chemical potential, ?i.
We reason that the same ?i has to be used both in edge diffusion and in Ceqi by
72
virtue of the fact that ?i controls the equilibrium shape of a step. This equilibrium
shape should be the same regardless of which kinetic pathway (edge diffusion or
attachment-detachment) is operative in the deformation of the step shape. So, if
mass exchange with the terrace is turned off and relaxation occurs via edge diffusion,
the step attains the same shape as in the case where edge diffusion is turned off and
relaxation is allowed only by attachment-detachment kinetics. This property implies
that the thermodynamic driving force has to be the same chemical potential, ?i, in
both cases [54].
Equations (5.2)?(5.13) in principle constitute a system of coupled differential
equations for the step positions. As a discrete scheme of step flow, this system
has been implemented numerically only in the case of straight or axisymmetric step
trains. In this chapter we focus on (2+1)-dimensional settings where
Ded = 0 (5.14)
and Dad is a scalar constant.
5.3 Approximations for slowly varying step train
We see from (5.13) that the adatom flux Jadi plays a pivotal role in connecting
the step normal velocity vi to the step chemical potential. By solving approxi-
mately the diffusion equation (5.2) following [66], we write an explicit formula for
the adatom flux.
We review the assumptions underlying the approximate solution of (5.2) sub-
ject to (5.4),(5.5). We assume that the boundary data Ceqi varies more rapidly with
73
the step index i than the longitudinal coordinate ?. The slow variation in ? should
also be expected of the solution for adatom density. We treat the derivative ?? as
O(epsilon1) in comparison to the derivative ??, which is treated as O(1); epsilon1lessmuch 1. A possible
geometric interpretation of epsilon1 is the ratio a/R where R = O(?) is a typical radius of
curvature for steps and ? is a suitable macroscopic length [66]. Once the macroscale
surface flux is derived, the assumptions for the ? and ? derivatives are relaxed: both
derivatives are allowed to be O(1).
By neglecting the? derivatives in (5.2), for constant Dad the diffusion equation
for Ci reduces to
??
parenleftbigg?
?
????Ci
parenrightbigg
? 0 , (5.15)
which has the explicit solution
Ci ?Ai(?,t)
integraldisplay ?
?i
??
?? d?
prime +Bi(?,t) ?i <?<?i+1 , (5.16)
where Ai and Bi are integration constants to be determined via the boundary con-
ditions (5.4), (5.5). A more systematic description of this procedure is offered in
Chapter 8.
For isotropic adatom diffusion [66] with (scalar) diffusivity Dad the vector-
valued adatom flux is computed by
Jadi = ?Dad?Ci . (5.17)
74
By use of (5.4) and (5.5), the flux components evaluated at ? = ?i are
Jadi,? = ?D
adCs
kBT
1
??|i
?i+1 ??i
Dad
parenleftBig
1
ku??|i +
1
kd??|i+1
parenrightBig
+integraltext?i+1?i ????d?
, (5.18)
Jadi,bardbl = ?D
ad
??|i??
?
?
?
Dad
parenleftBig Ceq
i+1
ku??|i +
Ceqi
kd??|i+1
parenrightBig
+Ceqi integraltext?i+1?i ????d?
Dad
parenleftBig
1
kd??|i+1 +
1
ku??|i
parenrightBig
+integraltext?i+1?i ????d?
?
?
?, (5.19)
where Jadi,bardbl := e? ?Jadi .
Analternativeyetequivalentderivationoftheapproximatesolutionto(5.2),(5.4),(5.5),
based on Taylor expansions at adjacent step edges [67] is reviewed in Chapter 6.
5.4 Macroscale limit with isotropic diffusion in 2+1 dimensions
In this section we present the macroscale limit of the discrete model (5.2)?
(5.13) when the physics of each terrace is isotropic (Dad = Dad: scalar) and there
is no step edge diffusion (Ded = 0). This macroscale limit is a PDE for the surface
height, one of whose possible discretizations coincides with the equations of step
flow, (5.2)?(5.13). The surface height will be shown to obey a nonlinear fourth-
order PDE.
First, we summarize the main assumptions underlying the derivation in [66].
The macroscale limit corresponds formally to taking a/? ? 0 where ? is a macro-
scopic length. We note that it is a dimensionless ratio whose magnitude tends to
zero; the macroscale limit makes no claim about physical lengths in an absolute
sense, only in relation to other lengths relevant to the crystal surface. The metric
coefficients ?? and ?? are O(?), while the terrace width ??i is O(a). Therefore, we
have ??i = ?i+1 ??i ? ??i??1? = O(a/?) ? 0. In this limit, we must keep as fixed,
75
O(1) quantities the step density mi = a/??i and the kinetic parameters Dad/(kla)
where l = u or d.
To obtain the macroscale limit, we must identify any discrete variable Qi
defined at a step edge (? = ?i) with the evaluation of a continuous, sufficiently
differentiable function tildewideQ(?) at ? = ?i. Thus, Qi+1 ?Qi ? (??i)??tildewideQ|i where A|i is
used in place of A(?i) throughout. The following geometric limits made in [66] carry
through for the macroscale limit of Chapters 6, 7, and 8 as well.
(i) The step density approaches the surface slope,
mi ?m = |?h||i = O(1).
(ii) The unit vector normal to the ith step edge becomes
e?|i ? e? = ? ?h|?h|.
(iii) The step curvature, ?i = ??e?|i, approaches
?i ?? = ???
parenleftbigg ?h
|?h|
parenrightbigg
.
(iv) The step normal velocity, vi = e? ?dri/dt, becomes
vi ?v(r,t) = ?th|?h|,
the velocity of the level set with height h.
Justasinthecaseofrelaxationabovetherougheningtemperature, themacroscale
evolution equation consistent with the behavior of steps has three ingredients. We
consider separately (i) the macroscopic analog of Fick?s law, (ii) the formula for
76
macroscale step chemical potential, and (iii) the macroscopic law of mass conser-
vation. Combining these three ingredients yields the desired PDE for the surface
height h.
5.4.1 Adatom flux
We derive the macroscale limit of the flux components (5.18) and (5.19). The
terms on the right-hand sides of these equations are replaced by series expansions
as ??i ? 0. In particular, we use the expansions
integraldisplay ?i+1
?i
??
??d? =
??|i
??|i??i +O[(??i)
2], (5.20)
?i+1(?)??i(?) = ??|i??i???|i +O(??2i ), (5.21)
1
ku
1
??|i +
1
kd
1
??|i+1 =
parenleftbigg 1
ku +
1
kd
parenrightbigg 1
??|i[1 +O(??i)], ??i = ?i+1 ??i ? 0. (5.22)
The resulting macroscale limit has the form of a matrix equation involving the
adatom mobility Mad,
Jadi |i ? Jad(r,t) =
?
??
?
Jad?
Jadbardbl
?
??
?= ?CsMad ?
?
??
?
???
?bardbl?
?
??
? , (5.23)
where
Mad = D
ad
kBT
?
??
?
1
1 +q|?h| 0
0 1
?
??
? , (5.24)
?? = ??1? ??, ?bardbl = ??1? ?? and the kinetic parameter q is defined by
q := 2D
ad
ka , k
?1 := (k?1
u +k
?1
d )/2. (5.25)
Evidently, normal and longitudinal variations in the chemical potential contribute
with unequal weight to the respective components of the surface flux. The tensor
77
mobility M reflects the fact that adatom flux normal to steps is inhibited by the
attachment-detachment process, whereas longitudinal fluxes face no such barrier.
The mobility M in (5.24) has dimensions ((length)2(time)?1(energy)?1), and the
macroscale step chemical potential ? has dimensions (energy/length).
In the basal plane?s Cartesian coordinate system, the matrix elements Mij are
found by applying the change-of-basis matrix [66, 67]
S = (e?e?) = 1|?h|
?
??
?
??xh ?yh
??yh ??xh
?
??
?, (5.26)
which yields
Mxx = D
ad
kBT
(?xh)2
|?h|2
bracketleftbigg 1
1 +q|?h| +?
2
bracketrightbigg
, (5.27)
Mxy = Myx = ?D
ad
kBT
q|?h|
1 +q|?h|
(?xh)2
|?h|2 ?, (5.28)
Myy = D
ad
kBT
(?xh)2
|?h|2
bracketleftbigg ?2
1 +q|?h| + 1
bracketrightbigg
, (5.29)
where ? := ?yh?xh.
Equation (5.23) is complemented by a mass conservation statement for the
height profile h and a formula for the macroscale step chemical potential ?.
5.4.2 Macroscale step chemical potential
Next, we invoke (5.11)?(5.12) for the step chemical potential ?i. First we
multiply both sides of (5.11) by the step height a:
aEsteps =
summationdisplay
i
a
integraldisplay
Li
ds
bracketleftBig
? + g3m2i?(?i,?i+1)
bracketrightBig
. (5.30)
78
The sum over all steps in (5.30) can be interpreted as a quadrature scheme for the
following integral with respect to the surface height h:
lima?0aEsteps =
integraldisplay
dh
integraldisplay
h=const.
ds
bracketleftBig
? + g3?0|?h|2
bracketrightBig
. (5.31)
Notice that the integrand in (5.31) is a function of the step indexiand position
? along the step. Because h depends monotonically on the step index, we could
consider the integrand in (5.31) as a function of h and ?. The sum over all steps
in (5.30) formally approaches the Riemann integral with respect to h as a? 0. In
addition, the integrand of (5.11) is recast into its macroscale height counterpart,
with |?h| in place of mi and ?0 in place of ?(?i,?i+1).
The integral in (5.31) can be evaluated more easily by a change of variables.
We apply the ?coarea formula? [24], which allows us to integrate with respect to the
area element dA in the basal plane.
For any integrable function ?(r), we have
integraldisplay
dh
integraldisplay
h=const.
ds?(r) =
integraldisplay integraldisplay
dA|?h|?(r). (5.32)
Using (5.32) and (5.31), we find
lima?0aEsteps =
integraldisplay integraldisplay
dA[?|?h|+ g3?0|?h|3]. (5.33)
Similarly, the sum over all steps in (5.12) can be interpreted as a quadrature
scheme for the following integral with respect to surface height:
79
?Esteps = 1
?
summationdisplay
i
a
integraldisplay
Li
dsvi?i(?) (5.34)
lima?0 ?Esteps = 1?
integraldisplay
h=const.
dh
integraldisplay
Li
ds ?th|?h|? (5.35)
= 1?
integraldisplay integraldisplay
dA|?h| ?th|?h|?, (5.36)
where the last equality follows from the coarea formula.
An alternative formula for ?Esteps is obtained by differentiating (5.33) with
respect to time.
lima?0 ?Esteps =
integraldisplay integraldisplay
dA??t[?|?h|+ g3?0|?h|3]
= 1a
integraldisplay integraldisplay
dA ??m[?m+ g3?0m3]?m?t , m := |?h|
= 1a
integraldisplay integraldisplay
dA ??m[?m+ g3?0m3] ?h|?h| ???th. (5.37)
We integrate (5.37) by parts to separate space and time derivatives of h. The
integration region extends to the far-field where ?th = 0; hence, boundary terms
vanish and (5.37) reduces to
lima?0a ?Esteps = ?
integraldisplay integraldisplay
dAdiv
braceleftbigg
?m[?m+ g3?0m3] ?h|?h|
bracerightbigg
?th. (5.38)
Now we compare (5.36) and (5.38), which must agree for any ?th. We can
view ?th as an arbitrary test function in L2(B) that vanishes on the boundary ?B.
In none of these calculations did we assume that h solved a particular PDE. As far
as the energetic formulas are concerned, ?th can vary independently of ?h. In order
80
for the two formulas to agree, the macroscale chemical potential must be given by
? = ??a div
braceleftbigg
? ?h|?h| +g?0|?h|?h
bracerightbigg
. (5.39)
For ease of notation we define g1 := ?/a and g3 := g?0/a. An alternative derivation
of (5.39) using differential geometry on the basis of (5.7) appears in [66].
5.4.3 Mass conservation for adatoms
ForDed = 0 the step velocity law (5.13) reduces to the usual mass conservation
statement foradatoms[66]. Asnotedabove, thestep velocityvi approaches?th/|?h|
in the macroscale limit. To derive the mass conservation law in the limit a/?? 0,
we first define a functional I[?] which integrates the test function ? against the
normal velocity vi and sums over all steps:
I[?] :=
summationdisplay
i
integraldisplay
Li
dsvi?|i. (5.40)
Viewing the sum (5.40) as a quadrature scheme for an integral with respect
to h, we apply the coarea formula (5.32) to obtain
I[?] ? 1a
integraldisplay integraldisplay
dA|?h|v? = 1a
integraldisplay integraldisplay
dA?th?, a/?? 0. (5.41)
On the other hand, if in (5.40) we replace vi by a difference of normal fluxes
according to (5.13), the result is
I[?] = ??a
summationdisplay
i
integraldisplay
Li
ds[J?e?]|i?|i, (5.42)
where [J?e?]|i denotes the jump (Ji(?i,?)?Ji?1(?i,?))?e?|i,?. Recalling that the
adatom surface current J(r,t) on the basal plane interpolates the nodal values Jadi |i,
and divJadi = 0 on each terrace, we have
81
I[?] = ??a
integraldisplay integraldisplay
dAdivJ?. (5.43)
Equating (5.41) and (5.43) for arbitrary test functions ?, we find the weak
formulation of adatom mass conservation:
?th = ??divJ. (5.44)
Analternativederivationof(5.44)proceedsbysuccessiveintegrationof divJadi?1 =
0 over a path in the (i?1)th terrace from (?i?1,?+??) to (?i,?), in order to sub-
stitute for Ji?1,?|i,? in terms of Ji?1,?|i?1,?+??. The resulting formula for the jump
in the normal flux is easily recognized as a discrete scheme for calculating the di-
vergence of Jad in local coordinates, and the limit (5.44) follows in the strong sense
[66].
5.4.4 Evolution equation for surface height
By combining the equations (5.23), (5.39), and (5.44), we find a PDE for the
surface height h(r,t) [66, 67]:
?th = ?Bdiv
braceleftbigg
?ad ??
bracketleftbigg
div
parenleftbigg ?h
|?h| +
g3
g1|?h|?h
parenrightbiggbracketrightbiggbracerightbigg
, (5.45)
where
?ad := kBTDad Mad, g1 := ?/a, g3 := 3g?0/a, B := D
adCsg1?2
kBT . (5.46)
TheconsolidationofconstantsischosensothatBhasdimensions(length)4/time
and ?ad is dimensionless.
82
5.5 Scaling ansatz for the height profile
One of the motivations for deriving the macroscale limit (5.45) is its suitabil-
ity for answering questions about the global surface morphological evolution under
certain initial data for the surface height profile. In particular, we seek to classify
the possible scaling laws permitted by the PDE for h. Here, the term ?scaling law?
describes the time-dependent part A(t) of a separable solution,
h(r,t) ?H(r)A(t). (5.47)
This variable separation, called a ?scaling ansatz?, is consistent with step simula-
tions in 1D and kinetic Monte Carlo simulations in 2D , both for initial sinusoidal
profiles. However, the scaling ansatz satisfies the PDE only approximately; we find
a consistent equation by regarding as negligible some of the terms in ? and Mad.
It is not obvious that a single power of A will remain dominant in the equation for
long times. Indeed, a given initial-boundary value problem with (5.45) need not
have a separable solution. This property relies crucially on the initial data.
In order to extract a dominant power of A from the mobility matrix, we
must compare the kinetic term (1 + q?)?1 with the square of the aspect ratio ?;
here ? = ?|?h|? is a typical slope. Depending on the relative magnitudes of these
kinetic-geometric parameters, the elements of the mobility matrix might scale as A0
or A?1.
The macroscale step chemical potential ? scales with amplitude A according
to which contribution to the energy dominates. If line tension (g1 term) is dominant,
then ? scales as A0. If step interactions dominate, then ? scales as A2. Combin-
83
Table 5.1: Decay laws for the amplitude A(t) in ADL kinetics, taken from [67]. Top
row: kinetic-geometric conditions on mobility. Leftmost column: dominant effects
in ?. The constants C, c, and t?(t? >t) depend on A(0) = A0 and H. The constant
B3 is defined by B3 := DadCsg3?2k
BT
.
?2 lessmuch (1 +q?)?1 lessmuch 1 (1 +q?)?1 lessmuch?2 < 1
Step interaction A0e?CB3t A0(1 +cB3A0t)?1
Line tension A0radicalbig1?t/t? A0(1?t/t?)
ing these two possibilities with the kinetic-geometric conditions on the mobility,
Margetis [67] obtained the possible scaling laws listed in Table 5.1.
We do not address analytically the spatial part of the approximate separable
solution, H(r), which solves a nonlinear PDE. Numerical solutions of the PDE,
such as those given in Chapter 10, remain our most viable option for visualizing the
spatial dependence of a separable solution.
5.6 Presence of facets in the variational formulation
The surface evolution law derived here conforms more closely to the behav-
ior of steps than the PDEs obtained phenomenologically or by analogy with 1D
macroscale models. However, the behavior of steps near a facet edge, where the sur-
face energy has a singularity, is not respected by the PDE derived in this chapter.
The variational formulation imposes its own natural boundary conditions at the facet
edge, which in general are not identical to the boundary conditions stemming from
84
a microscale drop in height when the top step collapses [38]. We outline the natural
boundary conditions in Chapter 9, in order to interpret the results of simulating the
variational formulation in terms of strong PDE solutions informed by conditions at
the free boundary. Only in the limiting case g1 ? 0 can we expect the variational
formulation to respect the flow of steps near |?h| = 0. A hybrid approach, which
informs the variational equations of the discrete height drop when the topmost step
collapses, is beyond our scope in 2+1 dimensions, where an implementation of the
step flow model is lacking. Comparison of the variational and hybrid approaches
in the axisymmetric case [65] suggests that the discrepancy in boundary conditions
is not likely to affect the observed scaling laws. To be cautious, however, we focus
mainly on the case g1 = 0 in our subsequent numerical simulations.
85
Chapter 6
Macroscale equation with terrace anisotropy and step edge diffusion1
In this chapter we extend the theory of chapter 5 to cases with a tensor-
valued terrace diffusivity Dad and a nonzero edge diffusivity Ded. These extensions
offer a more realistic description of diffusion processes on terraces and steps, while
remaining firmly based in the framework of continuum steps. Again we seek a PDE
for the surface height. The main contribution of this chapter is the surface mobility
M, which extends (5.24) by allowing for off-diagonal elements even in the local
coordinate representation.
For convenience we rewrite the quasi-steady equation (5.2) describing adatom
diffusion on the ith terrace:
div(Dad ??Ci) = 0 . (6.1)
Equation (6.1) is complemented by the mixed boundary conditions dictated
by linear kinetics:
?Jadi (?i,t) = ku[Ci(?i,t)?Ceqi (t)] , (6.2)
Jadi (?i+1,t) = kd[Ci(?i+1,t)?Ceqi+1(t)] . (6.3)
In this chapter the terrace diffusivity Dad is assumed to have the tensor form
Dad = D11e?e? +D12e?e? +D21e?e? +D22e?e?. For the sake of some generality,
1Material in this chapter appeared previously in Quah and Margetis, J. Phys. A: Math. Theor.
41, 235004/1?18.
86
we do not impose the symmetry condition D12 = D21, although in most physical
settings this equality holds. The surface flux Jadi depends on the adatom density Ci
through the linear relation
?
??
?
Jadi,?
Jadi,bardbl
?
??
?= ?
?
??
?
D11 D12
D21 D22
?
??
??
?
??
?
??1? ??Ci
??1? ??Ci
?
??
? ?i <?<?i+1, (6.4)
which is Fick?s law of diffusion in two spatial dimensions. This definition of surface
flux is chosen so that the continuity relation
contintegraldisplay
L
Jadi ??ds = ddt
integraldisplay
intL
CidA (6.5)
holds for every closed loop L lying in the ith terrace.
6.1 Approximations for fast and slow step variables
In this subsection we provide relations for the adatom flux components at step
edges for slowly varying step trains. The starting point is the boundary value prob-
lem for terrace adatom concentration Ci, which satisfies (6.1) subject to boundary
conditions (6.2),(6.3). We note that the boundary conditions involve only the equi-
librium concentrations at neighboring step edges, allowing us to solve the diffusion
equation on each terrace independently. In local coordinates, the diffusion equation
(6.1) becomes
?
??
parenleftbigg?
?D11
??
?Ci
??
parenrightbigg
+ ???
parenleftbigg
D12?Ci??
parenrightbigg
+ ???
parenleftbigg
D21?Ci??
parenrightbigg
+ ???
parenleftbigg?
?D22
??
?Ci
??
parenrightbigg
= 0 , (6.6)
where ?i <?<?i+1.
87
For a slowly varying step train we invoke the separation of the variables (?,?)
into fast and slow as outlined in Chapter 5. A more rigorous justification of this
procedure is postponed until Chapter 8, after we have developed some intuition
about the solutions we obtain. Dropping ? derivatives in accordance with slow/fast
variable separation, (6.6) reduces to
??
parenleftbigg?
?
????Ci
parenrightbigg
? 0 , (6.7)
which is solved by
Ci ?Ai(?,t)
integraldisplay ?
?i
??
?? d?
prime +Bi(?,t) ?i <?<?i+1 , (6.8)
for some ?-independent functions Ai,Bi. In order to determine the integration con-
stants Ai and Bi, we evaluate the flux components at the step edges and apply the
boundary conditions (6.2),(6.3). The integration constants have a natural interpre-
tation in terms of the adatom concentration and its gradient at the step edge. We
eliminate Ai and Bi in favor of the more physical quantities Ci|i, J|i by applying
Fick?s law, (6.4).
Using (6.4), the corresponding flux components are
Jadi,? ? ?D11?
?
Ai(?,t)? D12?
?
??
bracketleftbigg
Bi(?,t) +Ai(?,t)
integraldisplay ?
?i
??
??d?
prime
bracketrightbigg
, (6.9)
Jadi,bardbl ? ?D21?
?
Ai(?,t)? D22?
?
??
bracketleftbigg
Bi(?,t) +Ai(?,t)
integraldisplay ?
?i
??
??d?
prime
bracketrightbigg
. (6.10)
Equations (6.9) and (6.10) are simplified when we evaluate Jadi at ? = ?i. The
resulting matrix equation is
???|i
?
??
?
Jadi,?|i
Jadi,bardbl|i
?
??
?=
?
??
?
D11 D12
D21 D22
?
??
?
?
??
?
Ai
??Bi
?
??
? . (6.11)
88
By inspection of (6.11), the term ??Bi must be treated on equal footing with
Ai,since both terms make comparable contributions to the surface flux. We proceed
to invert the matrix equation (6.11), viewing Ai and ??Bi as integration constants
that we have to eliminate from the boundary conditions (6.2) and (6.3). Thus, we
obtain the formula
?
??
?
Ai
??Bi
?
??
?= ?
??|i
|Dad|
?
??
?
D22 ?D12
?D21 D11
?
??
?
?
??
?
Jadi,?|i
Jadi,bardbl|i
?
??
? , (6.12)
where |Dad| := D11D22 ?D12D21 denotes the determinant of Dad.
Next, we apply the boundary conditions (6.2) and (6.3) for atom attachment-
detachment at step edges. By substituting the solution for the adatom density Ci
into these conditions, we find the relations
?Jadi,?(?i,?,t) = ku[Bi(?,t)?Ceqi (?,t)] (6.13)
Jadi,bardbl(?i+1,?prime,t) = kd
bracketleftbigg
Bi(?prime,t) +Ai(?prime,t)
integraldisplay ?i+1
?i
??
?? d??C
eq
i+1(?
prime,t)
bracketrightbigg
. (6.14)
We eliminate Bi by setting ?prime = ? in equation (6.14), multiplying (6.13) by
kd/ku and subtracting the resulting equation from (6.14). Substituting for Ai from
(6.12), we arrive at the first desired relation between the surface flux components:
parenleftbigg 1
ku +
??|iD22
|Dad|
integraldisplay ?i+1
?i
??
??d?
parenrightbigg
Jadi,?|i + 1k
d
Jadi,?|i+1
? ??|iD12|Dad|
parenleftbiggintegraldisplay ?i+1
?i
??
??d?
parenrightbigg
Jadi,bardbl|i = Ceqi ?Ceqi+1 . (6.15)
We obtain a second relation by exploiting variations in?, which can be taken to
be arbitrarily small; in contrast, changes in? are restricted byaand the requirement
of finite slope. Therefore, we differentiate (6.13) with respect to? and substitute for
89
??Bi from (6.12). Subsequently, we neglect ??Jadi,?, consistent with the hypothesis
of slowly varying step edge curvature. Thus, the second desired relation of the flux
components reads
??|i
|Dad|(D21J
ad
i,?|i ?D11J
ad
i,bardbl|i)???C
eq
i = 0 ,
which in turn becomes
D21Jadi,?|i ?D11Jadi,bardbl|i = Cs|D
ad|
??|i
???i
kBT =
Cs|Dad|
kBT ?bardbl?i . (6.16)
Equations (6.15) and (6.16) suffice for the purpose of taking the macroscale limit.
6.2 Macroscale adatom flux
In this section we derive the analogue of (5.23) and (5.24), the relation between
macroscale adatom flux and step chemical potential. The resulting mobility tensor,
Mad, accounts for the macroscale flux arising from adatom diffusion on terraces. An
additional term, derived in Section 6.4, provides the mobility element corresponding
to step edge diffusion.
First, we simplify relations (6.15) and (6.16) forJadi . Considering??i = ?i+1??i
as small, we make the approximations
1
kuJ
ad
i,?|i +
1
kdJ
ad
i,?|i+1 =
parenleftbigg 1
ku +
1
kd
parenrightbigg
Jadi,?|ibracketleftbig1 +O(??i)bracketrightbig, (6.17)
integraldisplay ?i+1
?i
??
??d? =
??|i
??|i??i
bracketleftbig1 +O(??
i)
bracketrightbig. (6.18)
We consolidate the kinetic rates ku, kd into the parameter k = 2/(k?1u + k?1d ) of
(5.25). Thus, (6.15) reduces to
bracketleftbiggparenleftbigg2
k +
??|iD22
|Dad| ??i
parenrightbigg
Jadi,?|i ? ??|iD12|Dad| ??iJadi,bardbl|i
bracketrightbiggbracketleftbig
1 +O(??i)bracketrightbig= Ceqi ?Ceqi+1 . (6.19)
90
We multiply (6.19) by |Dad|/(??|i??i) and thereby obtain
parenleftbigg
D22 + 2|D
ad|
k??|i??i
parenrightbigg
Jadi,?|i ?D12Jadi,bardbl|i = |Dad|C
eq
i ?C
eq
i+1
??|i??i . (6.20)
As ??i ? 0, the right-hand side of (6.20) approaches Cs|Dad|???/kBT. On the
other hand, the ratio of parameters in the prefactor of Jadi,?|i has the limiting value
2|Dad|
k??|i??i ?
2|Dad|
ka |?h| = D
ad|?h| , Dad := 2|D
ad|
ka , (6.21)
where Dad has dimensions of diffusivity [(length)2/time].
A matrix equation for the macroscale surface flux Jad = (Jad? ,Jadbardbl )T in terms
of the step chemical potential ? comes from combining (6.16), (6.20) and (6.21):
?
??
?
D22 +Dad|?h| ?D12
?D21 D11
?
??
?
?
??
?
Jad?
Jadbardbl
?
??
?= ?
Cs|Dad|
kBT
?
??
?
???
?bardbl?
?
??
? . (6.22)
By solving (6.22) for Jad we obtain
Jadi |i ? Jad(r,t) =
?
??
?
Jad?
Jadbardbl
?
??
?= ?CsMad ?
?
??
?
???
?bardbl?
?
??
? , (6.23)
where the macroscale adatom mobility is
Mad = 1k
BT (1 +q|?h|)
?
??
?
D11 D12
D21 D22 +Dad|?h|
?
??
? , q :=
2D11
ka . (6.24)
This formula reduces to the equation with diagonal Mad found in [66] when
D11 = D22 = Dad and D12 = D21 = 0; cf (5.24). In contrast to the case with scalar
diffusivity, all matrix elements of the mobility in (6.24) depend on the slope. This
dependence is quite pronounced in the kinetic regime of attachment-detachment
limited (ADL) kinetics, which we discuss in Section 6.6.
91
6.3 Alternative approach to macroscale limit: Taylor expansions
For the sake of completeness, we re-derive (6.23) and (6.24) via an alternative
yet equivalent route. This is based on expansions of the boundary conditions (6.2)
and (6.3) for atom attachment-detachment in appropriate Taylor series when ??i =
?i+1 ??i ? 0 and ?? = ?prime ?? ? 0. Instead of finding the general solution of (6.7)
and imposing boundary conditions, we proceed directly to the physical variables
Ci|i and Jad|i. By assuming sufficient differentiability, the values of Ci and Jad at
?i+1,?prime can be approximated using Taylor series about ?i,?, as outlined in [67].
We first expand Ci|i+1 and Jadi,?|i+1 in (3.5) to first order in ?? and ??i:
kuparenleftbigJadi,?|i +??Jadi,?|i??i +??Jadi,?|i??parenrightbig = kukdbracketleftbigCi|i +??Ci|i??i
+??Ci|i???Ceqi (?+??,t)bracketrightbig. (6.25)
Second, we multiply (6.2) by kd and subtract the resulting equation from (6.25), so
as to eliminate Ci. By neglecting the ?- and ?-derivatives of Jadi,?, we find
(ku +kd)Jadi,?|i = kukdbraceleftbig??Ci|i??i +??Ci|i??
? Csk
BT
[?(?i+1,?+??)??(?i,?)]bracerightbig. (6.26)
Next, we solve for ??Ci and ??Ci by applying the matrix equation (6.4). The
substitution of ??Ci and ??Ci into (6.26) and subsequent expansion of the difference
?(?i+1,?+??)??(?i,?) about (?i,?) yields a relation between Jadi and the gradient
of the macroscale step chemical potential ?(r,t) :
parenleftbigg 1
ku +
1
kd +
D22????i
|Dad|
parenrightbigg
Jadi,?|i ? ??D12??i|Dad| Jadi,bardbl|i + Csk
BT
???|i??i
=
bracketleftbigg ?
?
|Dad|
parenleftbigD
12Jadi,? ?D11Jadi,bardbl
parenrightbig|
i ?
Cs
kBT???|i
bracketrightbigg
?? . (6.27)
92
Setting ?? = 0 in (6.27) and taking the macroscale limit provides our first
equation for the components of the surface flux in terms of ? :
parenleftbigg
1 + 2|D
ad|
kaD22|?h|
parenrightbigg
Jad? ? D12D
22
Jadbardbl = ?Cs|D
ad|
kBTD22??? . (6.28)
The macroscale limit of (6.27) still applies when ?? negationslash= 0. By (6.28), we know
that the left-hand side of (6.27) tends to zero in that limit. Therefore, the term
proportional to ?? must also vanish as ??i ? 0. Thus, we have
D21Jad? ?D11Jadbardbl = Cs|D
ad|
kBT ?bardbl? . (6.29)
Bysolvingsimultaneously(6.28)and(6.29)forthecomponentsofthemacroscale
surface flux, we find
?
??
?
Jad?
Jadbardbl
?
??
?=
?Cs
kBT (1 +q|?h|)
?
??
?
D11 D12
D21 D22 +Dad|?h|
?
??
??
?
??
?
???
?bardbl?
?
??
? , (6.30)
which is directly identified with the combination of (6.23) and (6.24).
6.4 Mass conservation law and total surface flux
In this subsection we define the total surface flux J so that the mass conser-
vation law for atoms is satisfied in the presence of step edge diffusion. The surface
mobility is defined accordingly through the relation of J and ?.
At a given location ? on the ith step edge, the step normal velocity vi must
respect conservation of mass, taking into account all possible sources and sinks of
atoms; see (5.13). By Section 5.4.3, in the macroscale limit (5.13) reduces to
?th = ????Jad + a|?h|?
?
??
braceleftbiggDed
?? ??
parenleftbigg ?
kBT
parenrightbiggbracerightbigg
, (6.31)
93
where the adatom flux Jad is described by (6.23) and (6.24).
Since the terrace is a level set for the height, we have h = H(?,t); in other
words, h does not vary in the step-longitudinal (?-) direction. Thus, |?h| =
??1? |??H| and the factor |??H| can be passed through the ? derivative in (6.31).
It follows that
?th = ????Jad + 1?
???
??
braceleftbigg
aDed|?h|???
?
??
parenleftbigg ?
kBT
parenrightbiggbracerightbigg
. (6.32)
We recognize the second term on the right-hand side of (6.32) as the divergence of
aDed|?h|?bardbl(?/kBT)e?. (Recall ?bardbl = ??1? ?? as in Chapter 5.) Hence, we refer to the
term ?aDed? |?h|?bardbl(?/kBT)e? as the edge atom flux, denoted by Jed. Combining the
two divergence terms into one term, we obtain the mass conservation law
?th = ????(Jad +Jed) = ????J , (6.33)
where
J = Jad +Jed , Jed := ?aD
ed
? |?h|?bardbl
parenleftbigg ?
kBT
parenrightbigg
e? . (6.34)
Recall that ? here is the same macroscale chemical potential derived in Chap-
ter 5, a function of the local slope according to the formula
? = ??div
braceleftbigg
g1 ?h|?h| +g3|?h|?h
bracerightbigg
, (6.35)
which remains unchanged in the presence of terrace anisotropy and step edge diffu-
sion.
The matrix equation (6.30) involving the mobility tensor can be updated to
94
reflect the additional source term in the overall surface flux:
J(r,t) =
?
??
?
J?
Jbardbl
?
??
?= ?Cs
?
??
?
M?? M??
M?? M??
?
??
??
?
??
?
???
?bardbl?
?
??
?= ?CsM???, (6.36)
where
M =
?
??
?
M?? M??
M?? M??
?
??
? , (6.37)
M?? = D11/(kBT)
1 + 2D11ka |?h|
, M?? = D12/(kBT)
1 + 2D11ka |?h|
,
M?? = D21/(kBT)
1 + 2D11ka |?h|
, M?? = 1k
BT
?
??
?
D22 + 2|D
ad|
ka |?h|
1 + 2D11ka |?h|
+ aD
ed
?Cs |?h|
?
??
? .(6.38)
Thistotalmobilitytensorexpressesthegeometricdependenceofthemacroscale
surface flux on the chemical potential via (6.36), the continuum analog of Fick?s law
when step edge diffusion is present. In the above formulation, we assume that
D11
ka = O(1),
|Dad|
D22ka = O(1),
a
?Cs
Ded
D22 = O(1). (6.39)
6.5 Evolution equation for the surface height
We now combine the mass conservation law (6.33) with the effective surface
flux (6.36) and the formula for the macroscale step chemical potential (5.39) in order
to derive a PDE analogous to (5.45) for the surface height profile, h(r,t). With the
substitutions for ? and J by (5.39) and (6.36), the mass conservation law (6.33)
becomes
?th = ??
2Cs
a div
braceleftbigg
M??
parenleftbigg
div
bracketleftbigg
(? + 3g?0|?h|2) ?h|?h|
bracketrightbiggparenrightbiggbracerightbigg
. (6.40)
95
To consolidate the physical parameters, we again use the definitions g1 = ?/a,
g3 = 3g?0/a, and B = ?2Csg1; see (5.46). Accordingly, we obtain (5.45) with Mad
replaced by the effective total mobility M.
6.6 New scaling laws with terrace anisotropy and edge diffusion
In this section we derive approximate, separable solutions of PDE (6.40). Our
goal is to find plausible connections between macroscale solutions and decay laws
observed in biperiodic profiles, e.g. observations reported in [4, 20, 48, 83]. Our
discussion is heuristic; the relation of PDE solutions to experiments is not well un-
derstood at the moment. Questions to consider when comparing PDE solutions to
experimental observations include (i) whether the experimental conditions satisfy
the assumptions under which the PDE was derived, (ii) what values of the material
parameters are appropriate for a given material and temperature, and (iii) what
time scale is common to the PDE solution and the decay of the observed material.
When seeking separable PDE solutions, the issue of material parameters is partic-
ularly relevant, as we see in the following argument when deciding which terms are
negligible in the free energy formula and the mobility matrix elements.
We start with the scaling ansatz h(r,t) ? A(t)H(r). This separation of vari-
ables is consistent with previously reported step flow simulations in 1D [38] and
kinetic Monte Carlo simulations in 2D [108], both with initial sinusoidal profiles.
The amplitude A(t) can be obtained formally from an ordinary differential equation
(ODE) by direct substitution in (6.40). We alert the reader that conditions on the
96
initial data and material parameters for having separable solutions and recovering
an ODE for A(t) are currently elusive, requiring detailed numerical studies. Such
studies are not addressed in this thesis.
Additive terms in the driving force ?? and in the total mobility M scale
differently with A. We need to retain in the right-hand side of the PDE terms
proportional to the same power of A and thus resort to approximations. It should
be borne in mind that the nonlinearities in M and ? lead to spatial-frequency
coupling for biperiodic height profiles; accordingly, evolution is in principle more
complicated than the one implied here by our simple scaling scenario.
Depending on the powers of A that possibly prevail in the evolution equation,
we find several plausible behaviors of h with time, including the exponential decay
and inverse linear decay reported in related experiments [4, 20, 48, 83]. By (6.35)
the driving force ?? scales as A0 if the dominant term is step line tension. If
step interactions are dominant, then ?? scales as A2. To determine the scaling of
the mobility tensor, it is convenient to introduce the ?aspect ratio? ? := ?yh/?xh;
it is plausible yet not compelling to estimate ? by ?x/?y where ?x and ?y are
wavelengths in the x and y directions. We also define the slope-dependent quantity
b := (1+ 2D11ka |?h|)?1. Note that ? scales as A0. When step edge diffusion is absent
(Ded = 0), the possible scalings found for A with nonzero D12 and D21 are not
different from those for isotropic adatom diffusion (where D12 = D21 = 0) [67].
With these definitions, the elements Mij = (kBT)?1|?h|?2btildewiderMij (i, j = x, y)
97
from the Cartesian representation (5.27)?(5.29) of M read
Mxx = b(?xh)
2
kBT|?h|2
bracketleftbigD
11 ??(D12 +D21) +?2D22
+2|D
ad|
ka ?
2|?h|+ aD
ed?2|?h|
b?Cs
bracketrightbigg
,
Mxy = b(?xh)
2
kBT|?h|2
bracketleftbigD
12 +?(D11 ?D22)??2D21
?2|D
ad|
ka ?|?h|?
aDed?|?h|
b?Cs
bracketrightbigg
,
Myx = b(?xh)
2
kBT|?h|2
bracketleftbigD
21 +?(D11 ?D22)??2D12
?2|D
ad|
ka ?|?h|?
aDed?|?h|
b?Cs
bracketrightbigg
,
Myy = b(?xh)
2
kBT|?h|2
bracketleftbigD
22 +?(D12 +D21) +?2D11
+2|D
ad|
ka |?h|+
aDed|?h|
b?Cs
bracketrightbigg
. (6.41)
We restrict attention to ADL kinetics which closely correspond to relevant
experimental situations [4, 20, 48, 83]. It follows that blessmuch 1 where b scales as A?1;
by the scaling ansatz for h, the prefactor b(?xh)2k
BT|?h|2
also scales as A?1. For the sake
of simplicity we consider weak anisotropy, |Dad| ? D11D22 (i.e., if the off-diagonal
diffusivity elements D12,D21 are small in comparison to the diagonal elements) and
|Dad|/(ka) greatermuchaDed/(b?Cs). The dominant terms in M scale as:
(i) A0 if blessmuch min{(D22/D11)?2,(D22/D11)??2,D22/D11}; and
(ii) A?1 if bgreatermuch max{(D22/D11)?2,(D22/D11)??2,D22/D11}.
In the presence of step edge diffusion with|Dad|/(ka) lessmuchaDed/(b?Cs),the dominant
terms in the mobility tensor scale as A1. Note that in all these cases the matrix M
tends to become singular since the lowest eigenvalue acquires a small value. Hence,
correction terms in M, which strictly spoil the scalings reported here, are physically
98
important; solutions of the form A(t)H(r) should be thought of as leading-order
terms of appropriate asymptotic expansions for h.
Next, we combine the three possible scalings of M with the two possible scal-
ingsof??.Eachcombinationyields anODEof theform ?A??Ap forsome exponent
p; the minus sign here is assumed for achieving profile decay. In the case of ADL
kinetics, outlined above, we have p ? {?1,0,1}?{1,2,3}, where the first set cor-
responds to dominant step line tension and the second set corresponds to dominant
step interactions in ??. Since p = 1 is common to both sets, the associated scaling
law A = A0 exp(?t/?) could perhaps be observed in a wide range of experimental
situations. On the other hand, the scaling law A = A0/radicalbig1 +t/? associated with
p = 3 and dominance of step edge diffusion may not be physical; to our knowledge,
this last decay law has not been observed.
We illustrate the procedure of finding A for weak anisotropy under condition
(ii) above and dominant step interactions; thus, p = 1. The PDE has the form
?A(t)H(r) ??A(t) div
braceleftBigg
(?xH)2
|?H|3
?
??
?
mxx mxy
myx myy
?
??
???
bracketleftbigdivparenleftbig|?H|?Hparenrightbigbracketrightbig
bracerightBigg
, (6.42)
where the prefactor is positive and the elements {mij}yi,j=x are constants that stem
from M(x,y) after factoring outA(but notH); the precise definition ofmij is omitted
here.
To satisfy (6.42) for all t and r, we require that the time-dependent part A(t)
solve ?A(t) = ?CA for some positive constant C (C > 0). The height profile H(r)
99
Table 6.1: Decay laws for height amplitude A(t) in ADL kinetics. Leftmost column
indicates plausible conditions. Next two columns list decay laws for line tension and
step interaction dominated ??. The time constant ? depends on A(0) and H.
Line tension Step interaction
|Dad|?D11D22
bgreatermuch max{(D22/D11)?2,D22/D11,(D22/D11)??2} A0radicalbig1?t/? A0 exp(?t/?)
blessmuch min{D22/D11)?2,D22/D11,(D22/D11)??2} A0(1?t/?) A0/(1 +t/?)
|Dad|/(ka) lessmuchaDed/(b?Cs) A0 exp(?t/?) A0/radicalbig1 +t/?
solves a nonlinear PDE of the form
CH ? div
braceleftBigg
(?xH)2
|?H|3
?
??
?
mxx mxy
myx myy
?
??
???
bracketleftbigdivparenleftbig|?H|?Hparenrightbigbracketrightbig
bracerightBigg
. (6.43)
The solution for A(t) is given in terms of the separation constant C and the initial
amplitude A0: A(t) = A0e?Ct. Using a similar procedure, we derive other possible
scaling laws for ADL kinetics under different restrictions. Our results are summa-
rized in Table 6.1.
We do not address the issue of solving (6.43) in this analysis. Particularly in-
teresting is the case with facets. The macroscale limit breaks down near facet edges,
and associated boundary conditions for H must take into account the discrete step
flow equations [65]. A numerical scheme to implement these boundary conditions
within a macroscopic simulation is a worthy subject of future research.
A similar analysis can be carried out if terrace diffusion is the slowest process,
i.e., q|?h| = |?h|D11/(ka) lessmuch 1. Then, b is approximately a constant, b ? 1. The
100
dominant terms in the mobility tensor scale as A0 or A1. Thus, we obtain ?A??Ap
for p ? {0,1,2,3}, which yields four of the five decay laws already found for ADL
kinetics.
6.7 Conclusion
By interpreting a (2+1)-dimensional step flow model for a relaxing surface as
a discretization of a macroscale evolution equation, we derived the relevant PDE for
the surface height profile. The starting point is a step velocity law that accounts
for anisotropic adatom diffusion on terraces, diffusion of atoms along step edges
and atom attachment-detachment at steps. In the macroscale limit we obtained
a relation between the surface flux and the step chemical potential. This relation
involves a tensor surface mobility with nonzero off-diagonal elements even in the
local coordinate representation.
Combining the step velocity law with the constitutive relation between the sur-
face flux and the step chemical potential resulted in a fourth-order nonlinear PDE
for the surface height. Transforming the mobility tensor from local step coordinates
to fixed coordinates introduced a dependence on the height partial derivatives. This
dependence offers a plausible scenario of how an epitaxial surface can exhibit dif-
ferent decay laws. We found separable solutions for the height that approximately
satisfy the evolution equation under certain conditions. These separable solutions
exhibit different decay and may be used as a guide in interpreting experimental
observations from a macroscale perspective.
101
Our PDE only accounts for a part of the possible microscopic physics. In
Chapters 7?8 we discuss the macroscale consequences of additional effects, such as
step permeability, adatom drift under the influence of an electric field, and desorp-
tion of adatoms. The calculations of this chapter serve as a prototypical example
for later macroscale limits.
102
Chapter 7
Step permeability and macroscale limit 1
In this chapter, we extend the model of continuum steps in 2+1 dimensions
to include the possibility of permeable steps. Also known as step transparency,
this phenomenological mechanism allows adatoms to jump from one terrace to a
neighboring terrace without first attaching and detaching at the intervening step.
In principle, step permeability can lead to long-range interactions, since an adatom
might cross many terraces before attaching to a step and influencing the step motion.
If such an effect is realized in the equations of step flow, we might naturally wonder
whether the macroscopic limit also shows evidence of long-distance interactions. The
main achievement of this chapter is the derivation of a macroscale limit consistent
with permeable steps in 2+1 dimensions.
The macroscale PDE we derive here leaves unchanged the local character of
step interactions, finding only a renormalized kinetic parameter as the result of
including step permeability. This renormalized kinetic parameter generalizes to
(2+1)-dimensions a formula in [88], which can be interpreted via an electric-circuit
analog of the paths by which adatoms can jump between terraces.
In this chapter we continue to assume that mass transport on terraces is due
to diffusion only. We omit from the diffusion equation terms corresponding to de-
1Material in this chapter appeared previously in Quah, Young and Margetis, Phys. Rev. E 78,
042602/1?4.
103
position flux, electromigration current, and desorption of adatoms. The latter two
effects will be addressed in Chapter 8. Here we discuss only the changed boundary
conditions in the case of permeable steps.
7.1 Step permeability
According to the original BCF framework for studying step motion, mass
transport of adatoms between adjacent terraces requires the intermediate processes
of attaching and detaching at the common step edge. By decomposing the mass
transport mechanism in this way and allowing for an Ehrlich-Schwoebel barrier,
we have the freedom to choose two kinetic rates ku and kd, which in principle can
explain such phenomena as the appearance of different island sizes during multilayer
growth [23]. Motivated more by intellectual curiosity than by discrepancies between
theory and experiment, Ozdemir and Zangwill imagined the possibility of direct
exchange of adatoms and vacancies from one terrace to another with kinetic rate
p [81]. This exchange mechanism is called step permeability or step transparency.
Pierre-Louis used this theoretical construct in his attempt to resolve the question of
temperature-dependent conditions for electromigration-induced step bunching [88].
The effect of step permeability is to change the linear kinetic boundary con-
ditions (3.4) and (3.5) to
?Ji,? = ku(Ci ?Ceqi ) +p(Ci ?Ci?1), ? = ?i, (7.1)
Ji,? = kd(Ci ?Ceqi+1) +p(Ci ?Ci+1), ? = ?i+1, (7.2)
where p is the permeability rate and Ji,? is the adatom flux component in the step-
104
normal direction; see Chapter 5 to review the geometry and notation. The same p
appears in both boundary conditions by Onsager reciprocity relations [89].
7.2 Macroscale limit
In order to derive a macroscale limit from (7.1),(7.2), and the terrace diffusion
equation, we impose the same geometric conditions as in chapter 6, along with the
conditions on the kinetic rates given in the preceding sections. The terrace widths
are assumed to be much smaller than [66]: (i) the length ? over which the step
density varies; and (ii) the step radius of curvature, if applicable. The discrete
step density a/??i with ??i = ??|?i??i is assumed to remain fixed while we formally
let ??i ? 0. Recall that ? is a dimensionless coordinate with discrete isolines
corresponding to steps.
The restriction on the kinetic rates is that D/(?a) = O(1) for ? = ku,kd,p.
We used a similar assumption in Chapter 5 to derive our first macroscale limit.
Here the condition D/(?a) = O(1) plays the same role, allowing us to keep fixed
the dimensionless parameters D/(?a) as various limits are taken.
If we wanted to proceed in the same vein as our first macroscale limit in Chap-
ter 5, we would begin by solving the diffusion equation ?2Ci = 0 on the ith terrace,
subject to the boundary conditions (7.1),(7.2). Immediately we notice that step per-
meability explicitly couples the resulting adatom concentrationCi with that of other
terraces. For a step train with a finite number of terraces, the system of boundary
conditions can be solved simultaneously by inverting a circulant matrix [43]. This
105
coupling at the step level suggests that the macroscopic limit of Ci must also be
incorporated into the macroscale equation. A similar approach was taken in [16],
although the authors provide only a brief discussion of the connection to the per-
meability rate p.
A direct solution of the terrace diffusion equation, in any case, would yield two
integration constants that have to be found by applying the boundary conditions.
We can bypass some tedious algebra by letting the values of adatom concentration
and its normal derivative play the role of integration constants, for which the bound-
ary conditions will serve as a linear system of equations. Accordingly, we assume
the existence of the interpolating functions C and J for adatom concentration and
transverse flux, respectively, which agree with Ci and Ji,? when evaluated at the
ith step edge. We take C and J as our primary variables. Whether we evaluate the
terrace concentration and its derivative at the upper step edge (?i) or the lower step
edge (?i+1) is irrelevant in the macroscale limit, since ?i and ?i+1 approach the same
value, corresponding to the transverse coordinate of a single level set. Our final goal
is a system of equations for C and J?, which emerge as the macroscale limits of C
and J, respectively.
106
7.2.1 Straight steps
In the straight step setting with no Ehrlich-Schwoebel barrier, the boundary
conditions (7.1),(7.2) reduce to
?Ji = k(Ci ?Ceqi ) +p(Ci ?Ci?1), x = xi (7.3)
Ji = k(Ci ?Ceqi+1) +p(Ci ?Ci+1), x = xi+1. (7.4)
We now identify Ci,Ci+1,Ci?1 with values of a continuous adatom concentra-
tion function at the left step edge of the corresponding terrace. In particular, we
define the interpolant C by Ci|xi =: C(i). As usual, the subscript xi to the right
of the vertical bar denotes the point of evaluation. Similarly, the flux Ji|xi at the
left step edge is identified with the value J(i) of an interpolating flux function. If
a value of the terrace adatom concentration Ci at a point other than the left step
edge is desired, then an approximation in terms of the concentration C(i) and flux
J(i) can be found by applying a Taylor expansion along with Fick?s law:
Ci|x = C(i)?D?1J(i)(x?xi) +O[(x?xi)2]. (7.5)
For the purposes of the macroscale limit, only the first-order term in the
expansion (7.5) needs to be retained. Using (7.5) atx = xi+1 and a similar expansion
for Ci?1|xi, the boundary conditions now read
?J(i) = k[C(i)?Ceqi ] +p[C(i)?C(i?1) +D?1(xi ?xi?1)J(i?1)] (7.6)
J(i) = k[C(i)?D?1(xi+1 ?xi)J(i)?Ceqi+1]
+p[C(i)?D?1(xi+1 ?xi)J(i)?C(i+ 1)]. (7.7)
107
In the absence of an Ehrlich-Schwoebel barrier, direct subtraction of (7.6) and
(7.7) leads immediately to
?2J(i) = k[Ceqi+1 ?Ceqi +D?1(xi+1 ?xi)J(i)]
+p[C(i+ 1)?C(i?1) +D?1(xi+1 ?xi)J(i) +D?1(xi ?xi?1)J(i?1)]. (7.8)
Bringing all the flux terms to the left side, we find
J(i)
bracketleftbigg
2 + kD(xi+1 ?xi) + pD(xi+1 ?xi?1)
bracketrightbigg
+ pD(xi ?xi?1)J(i?1) = ?k[Ceqi+1 ?Ceqi ]?p[C(i+ 1)?C(i?1)]. (7.9)
Assuming thatC and the macroscale equilibrium concentrationCeq both admit
Taylor series expansions about xi, we can replace the differences Ceqi+1 ?Ceqi and
C(i+ 1)?C(i?1) according to
Ceqi+1 ?Ceqi = (xi+1 ?xi) ?C
eq
?x
vextendsinglevextendsingle
vextendsinglevextendsingle
xi
+o(xi+1 ?xi), (7.10)
C(i+ 1)?C(i?1) = (xi+1 ?xi?1) ?C?x
vextendsinglevextendsingle
vextendsinglevextendsingle
xi
+o(xi+1 ?xi?1). (7.11)
Since the macroscale height profile is required to be differentiable, the mean-
value theorem allows us to substitute 2a/|?xh|forxi+1?xi?1 anda/|?xh|forxi+1?xi,
where the evaluation point of the x-derivative is left unspecified for now. The
constitutive relation (7.9) then becomes
J(i)
bracketleftbigg
2 + kaD|?
xh|
+ 2paD|?
xh|
bracketrightbigg
+ paD|?
xh|
J(i?1) = ? ka|?
xh|
?xCeq? 2pa|?
xh|
?xC. (7.12)
In the limit as a/?? 0, the positions of steps i and i?1 are assumed to tend
to a common point x, so the equation for the macroscale flux J is
J(x)
bracketleftbigg
2 + (k+ 2p)aD|?
xh|
bracketrightbigg
= ka|?
xh|
?xCeq + 2pa|?
xh|
?xC. (7.13)
108
To eliminate the explicit dependence on C, we take the macroscale limit of the
boundary condition (7.6) directly and then substitute for J(x) using (7.13). The
macroscale limit of (7.6) reads
?J(x) = k(C?Ceq) +p
bracketleftbigg
?xC a|?
xh|
+ J(x)aD|?
xh|
bracketrightbigg
. (7.14)
Combined with the macroscale flux from (7.13), we find an equation for C ?
Ceq.
k(C?Ceq) + pa|?
xh|
?xC =
ka
|?xh|?xC
eq + 2pa
|?xh|?xC
2 + (k+2p)aD|?xh|
bracketleftbigg
1 + paD|?
xh|
bracketrightbigg
parenleftbigg
1 + (k+ 2p)a2D|?
xh|
parenrightbigg
(C?Ceq) = a2|?
xh|
parenleftbigg
1 + paD|?
xh|
parenrightbigg
?xCeq + p
2a2
kD|?xh|2?xC.
(7.15)
In principle, we regard (7.15) as a series in powers of a, which holds identically
asa/?? 0. By equating the coefficients of like powers ofa, we obtain conditions on
the coefficients. The dominant balance of terms in (7.15) indicates that we can take
C = Ceq in the macroscopic limit. The factor multiplying C?Ceq on the left-hand
side of (7.15) remains O(1) as a/? ? 0, while the right-hand side vanishes since
p/k = O(1) does not scale with a. Replacing C by Ceq in (7.13), we find
J(x) = ? D?xC1 + 2D
(k+2p)a|?xh|
= ?DCsk
BT
?x?
1 +qeff|?xh|, (7.16)
whereqeff := 2D/(keffa)andkeff := k+2p. Thus, the effective attachment-detachment
rate keff appears as an additive renormalization of k to k+ 2p.
We pause to mention a few previous derivations of this result in the literature.
The unifying theme of these derivations is a replacement of the step-terrace system
by a simpler mass transport mechanism with renormalized parameters. In this vein,
109
Figure 7.1: Electric circuit analogous to a terrace and a permeable step, adapted
from [88].
Nozi`eres proceeds by analogy with an electric circuit [78], where the attachment
and detachment processes are regarded as a series of resistors, and the effect of
step transparency is to provide a parallel resistor through which an adatom current
can flow across the step. The simplest illustration of this analogy appears in the
subsequent paper by O. Pierre-Louis [88], adapted here as Figure 7.1. A more
complicated electric circuit analogy, incorporating several steps and terraces, is given
by Jeong and Williams [43].
7.2.2 Circular steps with axisymmetry
We use the familiar setting of axisymmetry to introduce the additional algebra
needed to deal with an Ehrlich-Schwoebel barrier. This bias of adatom flux toward
a preferred direction appears in the boundary conditions as an inequality in the
kinetic rates: kd > ku if downhill drift is preferred, or ku > kd if uphill drift is
110
preferred. The boundary conditions (7.1),(7.2) read
?Ji = ku(Ci ?Ceqi ) +p(Ci ?Ci?1), r = ri (7.17)
Ji = kd(Ci ?Ceqi+1) +p(Ci ?Ci+1), r = ri+1. (7.18)
Again we identify Ci,Ci+1,Ci?1 with values of a continuous interpolating
function C at the inner step edge of the corresponding terrace. This convention
says C(i) := Ci|ri. Similarly, we define an interpolating flux function J(i) by
J(i) := Ji|ri. An approximation of the adatom concentration at the outer step
edge can be found in terms of the concentration C(i) and flux J(i) by applying
Fick?s law:
Ci|ri+1 = C(i)? J(i)D (ri+1 ?ri) +o(ri+1 ?ri). (7.19)
Retaining only the terms of the expansion (7.19) that are linear in ri+1 ?ri,
the boundary conditions (7.17),(7.18) become
?J(i) = ku[C(i)?Ceqi ] (7.20)
+p[C(i)?C(i?1) + J(i?1)D (ri ?ri?1)], (7.21)
J(i) +?rJi|ri(ri+1 ?ri) = kd[C(i)? J(i)D (ri+1 ?ri)?Ceqi+1]
+p
bracketleftbigg
C(i)? J(i)D (ri+1 ?ri)?C(i+ 1)
bracketrightbigg
. (7.22)
Eventually we plan to take a macroscale limit from these equations, which
allows us to consider a/ri lessmuch 1. The nondimensional parameter kda/D, however,
remains O(1) in the macroscale limit. Hence we have the inequality
a
ri lessmuch
kda
D . (7.23)
111
Multiplying (7.23) by |J(i)/a| and identifying the left-hand side as
vextendsinglevextendsingle
vextendsingle?rJi|ri
vextendsinglevextendsingle
vextendsingle by use
of the diffusion equation, we find
vextendsinglevextendsingle
vextendsingle?rJi|ri
vextendsinglevextendsingle
vextendsinglelessmuch
vextendsinglevextendsingle
vextendsinglevextendsinglekdJ(i)
D
vextendsinglevextendsingle
vextendsinglevextendsingle. (7.24)
Therefore it is justified to neglect ?rJi|ri from (7.22), since our final goal is a
macroscale limit relating the variables J and C. These macroscale variables sat-
isfy
C(i)?C(i?1) = (ri ?ri?1)?rC|ri +o(ri ?ri?1), (7.25)
C(i+ 1)?C(i) = (ri+1 ?ri)?rC|ri +o(ri+1 ?ri), (7.26)
J(i)?J(i?1) = o(1). (7.27)
The discrete step density a/(ri+1 ?ri) is assumed to approach the macroscale
surface slope |?h|. In fact, replacement ofri+1?ri (orri?ri?1) bya/|?rh|evaluated
somewhere on [ri,ri+1] (or [ri?1,ri]) is justified by the mean value theorem and the
assumed smoothness of the macroscale surface height h. The desired system of
equations for the macroscale variables is
?J
parenleftbigg
1 + pD a|?
rh|
parenrightbigg
= ku(C?Ceq) + pa|?
rh|
?rC, (7.28)
J
parenleftbigg
1 + p+kdD a|?
rh|
parenrightbigg
= kd(C?Ceq)? kda|?
rh|
?rCeq ? pa|?
rh|
?rC, (7.29)
where Ceq is the macroscale limit of Ceqi . To solve for the radial flux J, we divide
(7.28) by ku and (7.29) by kd; the difference of the two resulting equations retains
only the concentration gradients, not their pointwise values. We obtain the formula
J = ?D(2p/k)?rC +?rC
eq
1 + 2p/k+q|?rh| , (7.30)
112
where k := 2(k?1u +k?1d )?1 and q := 2D/(ka); see section 5.4.1.
Unlike [16], we do not treat the macroscopic limit of adatom concentration
as a variable in its own right. Indeed, we have seen above in the case of straight
steps that the macroscale limit renders such a treatment unnecessary. A similar
argument in the axisymmetric setting allows us to replace C by Ceq in (7.30). With
this substitution, (7.30) becomes the desired relation between flux and chemical
potential:
J = ?D ?rC
eq
1 +qeff|?rh| = ?
DCs
kBT
?r?
1 +qeff|?rh|, (7.31)
where qeff = D/(keffa) and keff = k + 2p. Evidently, the effect of step permeability
in the axisymmetric setting is precisely the same renormalization of the kinetic pa-
rameter that we saw in the straight step case. The additional mass transport mech-
anism of adatoms hopping directly between terraces is reflected in the macroscale
equations by a modified attachment-detachment coefficient. Under the conventional
interpretation of step permeability, where p > 0, the new attachment-detachment
coefficient is greater than k. However, no physical principle is violated if p < 0.
This generalization has been suggested as a means of introducing different kinetic
regimes as the temperature or the supersaturation is varied [90]. We discuss below
the implications of negative p in relation to a two-region model of step dynamics by
Weeks et al. [125].
113
7.2.3 Steps in 2+1 dimensions
The geometry considered here is identical to that of Chapter 5, where? is used
as the transverse coordinate and ? is used as the longitudinal coordinate. Starting
with the boundary conditions (7.1),(7.2), we replace the adatom concentrations
and the normal flux components with either the macroscale interpolants, e.g., C(i),
J?(i), or series expansions about the upper step edge of the appropriate terrace. For
example, the domain of the adatom concentration Ci?1 is ?i?1 <?<?i, and in (7.1)
we want to express this concentration at ?i in terms of the macroscale interpolant
C(i?1). We use the expansion
Ci?1|?i = C(i?1) +??i?1??Ci?1|?i?1 +o(??i?1), (7.32)
which now forces us to find an expression for ??Ci?1|?i?1 in terms of the primary
variables. Fortunately, Fick?s law allows us to write this normal derivative as a
multiple of the corresponding flux component. The resulting microscale expansion
is
Ci?1|?i = C(i?1)? ??DJ?(i?1)??i +o(??i?1). (7.33)
In the case of tensor diffusivity, we could still replace the normal derivative of
Ci?1 by a linear combination of flux components evaluated at the upper step edge.
The flexibility of this series method offers a streamlined procedure to derive the
macroscale limit even when an explicit solution of the terrace diffusion problem is
inaccessible.
Similar series expansions apply to the other quantities that do not correspond
114
immediately to the macroscale interpolants. We have
Ci|?i+1 = C(i) +??i??Ci|?i +o(??i), (7.34)
Ji,?|?i+1 = J?(i) +??i??Ji,?|?i +o(??i). (7.35)
Again, Fick?s law allows us to replace the normal derivative in (7.34) by a multiple
of the normal flux component. For the normal derivative in (7.35), appeal to the
diffusion equation itself, along with the separation of variables into slow and fast; see
Chapter 5. Keeping in mind these considerations and the eventual macroscale limit,
we may safely neglect ??Ji,? from (7.35). After substituting the primary variables,
the boundary conditions now read:
?J?(i) = ku[C(i)?Ceqi ] +p[C(i)?C(i?1) + ????i?1D J?(i?1)], (7.36)
J?(i) = kd[C(i)?Ceqi+1 ? ????iD J?(i)] +p[C(i)?C(i+ 1)? ????iD J?(i)]. (7.37)
Next, we introduce the (macroscale) variables C and J? via the expansions
C(i)?C(i?1) = ??i?1??C|?i +o(??i?1), (7.38)
C(i+ 1)?C(i) = ??i??C|?i +o(??i), (7.39)
J?(i)?J?(i?1) = o(1). (7.40)
The discrete step density a/(????) is assumed to approach the macroscale surface
slope |?h|, independent of the evaluation point (?i or ?i?1). We therefore replace
????i and ????i?1 by a/|?h|. The desired system of equations for the macroscale
115
variables is
?J?
parenleftbigg
1 + pD a|?h|
parenrightbigg
= ku(C?Ceq) + pa|?h|??C, (7.41)
J?
parenleftbigg
1 + p+kdD a|?h|
parenrightbigg
= kd(C?Ceq)? kda|?h|??Ceq ? pa|?h|??C, (7.42)
where ?? = ??1? ?? and Ceq(r) is the macroscale limit of Ceqi . We solve this system
of equations for J? to obtain the formula
J? = ?D(2p/k)??C +??C
eq
1 + 2p/k+q|?h| , (7.43)
where k := 2(k?1u +k?1d )?1 and q := 2D/(ka) as before.
We now need a relation between C and Ceq. First, the sum of the boundary
conditions (7.41),(7.42) yields
kda
D|?h|J? = (ku +kd)(C?C
eq)? ka
|?h|??C
eq. (7.44)
By substituting for J? according to (7.43), we find
0 =
parenleftbigg
1 + 2pk + 2Dka|?h|
parenrightbigg
(C?Ceq) + 2pak|?h| kdk
u +kd
??C. (7.45)
In order to isolate the dominant terms, it helps to multiply by the O(1) quantity
ka/(2D|?h|). The resulting equation,
0 =
parenleftbigg
1 + (k+ 2p)a2D|?h|
parenrightbigg
(C?Ceq) + a|?h|2 paD kdk
u +kd
??C, (7.46)
can be viewed as an expansion in powers of a. For this equality to hold in the
macroscale limit a/? ? 0, the zeroth-order term must vanish, which implies C =
Ceq. Returning to (7.43), we have
J? = ? D??C
eq
1 +qeff|?h| = ?
DCs
kBT
???
1 +qeff|?h|, (7.47)
116
whereqeff := 2D/(keffa) andkeff = k+2p. Again, as far as the macroscale limit of the
normal flux is concerned, the kinetic rate undergoes only an additive renormalization
in the presence of permeability.
The parallel flux Jbardbl is not expected to require modification in the case of
permeable steps, which reduces the barrier only for the motion of adatoms across
terrace boundaries. Motion of adatoms parallel to a step edge is an independent
process, subject to the same fully continuum treatment as in Chapter 5. Accordingly,
we differentiate the boundary condition (7.36) with respect to ?:
a?bardblJ?(i) = kuaD [Jbardbl(i)+D?bardblCeqi ]+paD[Jbardbl(i)?Jbardbl(i?1)??bardbl(????i?1J?(i?1))], (7.48)
where ?bardbl := ??1? ??. Here we have used Fick?s law to write ?bardblC(i) in terms of the
parallel flux component. From here we let a/?? 0 to obtain
Jbardbl = ?D?bardblCeq = ?DCsk
BT
?bardbl?, (7.49)
which is identical to the macroscale longitudinal flux derived for p = 0 in Chapter
5. Again we emphasize the following
Result 7.2.1. Step permeability affects only the transverse flux component, via the
boundary conditions (7.1),(7.2), while the longitudinal flux remains unchanged for
nonzero p.
The relations (7.47), (7.49) can be combined into the matrix equation J =
?CsM ? ??, where M = M(?h) is the same surface mobility tensor derived in
Chapter 5 (diagonal when expressed in local coordinates). Writing M = m??e?e? +
m??e?e?, we observe that only the matrix element m?? is affected by the perme-
ability rate p, which enters through the nondimensional parameter qeff. We obtain
117
a PDE for h by combining (7.47), (7.49) with the mass conservation law (5.44) and
formula (5.39) for the macroscale step chemical potential. The resulting evolution
law is essentially (5.45), with M modified to account for step permeability through
the renormalized kinetic parameter k.
7.3 Discussion and interpretation
In this chapter, we found the macroscale limit of the step flow equations in the
case of transparent steps with permeability coefficientp. For each of the step geome-
tries studied, only the transverse flux experiences the effect of step transparency,
through a renormalized kinetic parameter in the macroscale analog of Fick?s law.
Unchanged in this chapter are the relation between chemical potential and
local slope, Eq. (5.39), and the mass conservation law, Eq. (9.11). These two in-
gredients, along with an appropriate macroscale analog of Fick?s law, constitute a
closed set of equations for the macroscale surface relaxation. Here we observe that
only the macroscale analog of Fick?s law needs to be modified, just as in Chapter
6 with the introduction of anisotropic diffusivity on terraces. This modification of
Fick?s law in the macroscale limit is required whenever the adatom concentration
satisfies a different boundary value problem. In Chapter 6 we assumed a tensor diffu-
sivity and retained the usual linear kinetic boundary conditions; here we considered
a scalar diffusivity and extended the boundary conditions to include a dependence
on neighboring terrace adatom concentrations. In general we can modify both the
terrace diffusion equation and the boundary conditions at step edges, which leads
118
to an extension of the mobility M in the resulting macroscale analog of Fick?s law.
The simple result of this chapter has possible limitations. Negative values of
p have been suggested to explain electromigration-induced step pairing on Si(111).
The case p < ?k/2 challenges our macroscale limit by allowing qeff to take on
negative values, leading to a singularity in the equations at slopes |?h| = ?1/qeff.
Since the slope smoothly approaches zero near facet edges or local height extrema,
the singularity where |?h| = ?1/qeff can only be avoided by imposing unwarranted
restrictions on the magnitude of the slope. However, if the permeability rate p is
large and positive, we expect the adatoms to jump without difficulty from one terrace
to the next, being limited in their motion only by the finite terrace diffusivity. The
macroscale limit derived here ought to hold in this case of diffusion-limited kinetics.
We might also interpret the renormalized kinetic rate in the macroscale equa-
tions as aresultofhomogenization. Insteadof seeingacrystalsurfaceasterracessep-
arated by sharp steps, the two-region model treats the surface as a layered medium,
with two types of regions distinguished by their diffusivities and their widths [125].
Since surface reconstruction might take place more readily near a line defect, it is
natural to postulate the existence of a tubular region around each step, in which
the adatom diffusivity differs from that of terraces. The effective rate of adatom
transport over a mesoscale consisting of many terraces is now seen to depend on the
different diffusivity Da due to reconstruction in the tubular region corresponding to
a step, and the width la of this region relative to the typical terrace width l; see
Figure 7.2. We appeal to the calculation of [70] to find the averaged (or effective)
diffusivity Dav for a train of N terraces spanning a total transverse distance L. This
119
Figure 7.2: Illustration of the two-region model in contrast to the sharp step picture
it replaces (adapted from [125]).
120
Dav is given by
Dav = 1
1
L
parenleftBig
la
Da +
l
D
parenrightBig = D
1 +
parenleftBig
D
Da ?1
parenrightBig
la
am0
, (7.50)
where m0 is the local slope in the mesoscale region. In hindsight, the same result
can be derived by mapping the two-region model to an electric circuit consisting of
resistors in series. The effective conductance ?eff = 1/Reff of a two-resistor unit is
given by the harmonic average of the two conductances, i.e.,
1
?eff =
1
? +
1
?a. (7.51)
We identify the conductance?with the terrace diffusivity per unit length,D/l, while
?a corresponds to the ?step region? diffusivity per unit length, Da/la. Equation
(7.50) ensues.
Zhao, Weeks and Kandel [125] obtained (7.50) without recourse to homoge-
nization theory or electric circuit analogies. Their interpretation allows for different
sign of the coefficient (D/Da?1)la/a depending on which inequality holds between
the diffusivities in the two regions. Faster diffusion in the step region (Da > D)
leads to a negative coefficient of m0. Similarly, a permeability coefficient p sat-
isfying p < ?k/2 leads to a negative q in the macroscale limit of this chapter.
Evidently, the two-region model and the picture of permeable sharp steps both have
enough flexibility in the choice of parameters to match a range of coefficients in
the slope-dependent mobility. The agreement between these two nanoscale models
at the level of macroscale equations is noteworthy, in view of the strikingly differ-
ent pictures they present at the nanoscale. The persistence of a slope-dependent
mobility with (possibly negative) material parameter q indicates just how robust
121
the macroscale limit is, emerging in the same form out of two conceptually distinct
nanoscale models.
Our interpretation of permeability, as an additive renormalization of the ki-
netic rate in the macroscale equations, is simple enough to allow meaningful compar-
isons between predictions of the macroscale PDE (chapter 10) and Monte-Carlo sim-
ulations of stepped surfaces. An alternative interpretation by O. Pierre-Louis [89],
in which the equilibrium concentration of the step flow equations is renormalized
through a suitable combination of the boundary conditions, also admits direct com-
parison with atomistic simulations. The question of whether step transparency itself
is exhibited by realistic surfaces cannot be addressed without a further understand-
ing of the physical mechanisms which combine with permeability to produce different
morphological changes. Our discussion of electromigration in Chapter 8 helps com-
plete the picture of possible nanoscale physics, in order to categorize the kinetic
regimes under which a stepped surface can evolve. By studying those regimes where
permeability would couple with electromigration to yield a qualitatively different
evolution, we might in principle be able to detect the presence and strength of step
transparency for realistic materials.
122
Chapter 8
Electromigration and desorption1
In this chapter, we consider another set of extensions to the BCF picture
of terrace adatom diffusion. First we introduce an electric field, which biases the
motion of adatoms in a prescribed direction. Next we allow desorption of adatoms
into the vapor phase. Combined with our results for anisotropic terrace diffusion
and step edge diffusion, the inclusion of electromigration and desorption provides a
comprehensive model of the mass transport processes over terraces and steps.
Our analysis aims to complement previous works in surface electromigration;
see, e.g., [12, 15, 27, 28, 35, 52, 53, 55, 59, 60, 61, 72, 74, 88, 90, 92, 101, 102,
103, 111, 112, 124, 125]. These focus on the development of instabilities (e.g. step
meandering and bunching) across scales, often from a dynamical system viewpoint.
We, on the other hand, place electromigration in the context of connecting discrete
schemes for interacting steps to global PDE laws and variational principles in 2+1
dimensions [16, 64, 66, 78, 80, 96, 98, 106, 109, 123]. The PDE we derive can be
studied using local analysis, to see the possible control that an electric field might
have on the spatial dependence of the slope profile near a facet. Investigating the
possible global control of the surface evolution by an electric field is within the scope
of our numerical method, as we discuss in Chapter 10.
1Parts of this chapter appeared previously in Quah and Margetis, Macroscopic Electromigration
on Stepped Surfaces, submitted to SIAM Multiscale Modeling and Simulation.
123
The study of electromigration began with the observation by Latyshev et
al. [59] that direct current heating of stepped Si(111) resulted in a step bunching
instability. The appearance of step bunches depends both on the direction of the
applied current and on the surface temperature. An early model by Stoyanov [111]
offered a possible explanation for the link between electromigration current and step
bunching. Stoyanov?s model works only up to 850?C, above which temperature the
direction of electric current required for step bunching undergoes two distinct rever-
sals. In particular, direct current in the step-down direction produces step bunching
at temperatures around 935?C and 1275?C, but at 1190?C step bunching requires
current in the step-up direction.
More recent proposals to address the relation between electromigration and
step bunching (although not directly relevant to our focus here) include but are not
limited to the work by Pierre-Louis and M?etois [88, 90], and by Zhao, Weeks, and
Kandel [124, 125]. Experimental studies found the bias in adatom motion?the drift
velocity?to be always parallel to the imposed electric field E. This drift velocity v
is related to E by [15, 28]:
v = D(Z
?e)E
T , (8.1)
where D is the adatom diffusivity, Z?e is the effective adatom charge,2 and T is the
Boltzmann energy3. At the level of steps, the electric field amounts to the addition
of a convective (linear in the density gradient) term in the diffusion equation for the
2Here, we follow the notation of [15]. The symbol Z? (with an asterisk) should not be confused
with the usual symbol for the conjugate of a complex number.
3In this chapter only we use units where kB = 1, thus replacing by T the kBT of other chapters.
124
adatom density; see (8.35). Regarding (8.1), |Z?| is larger than unity for metals but
can be much smaller than unity for semiconductors [15]. We note in passing that E
influences the temperature in an experimental setup, since it directly controls the
electric current that flows through and eventually heats up the sample.
In addition to electromigration, we will allow adatoms to desorb into the vapor
with characteristic time ?. So, the term C/? is added in the adatom diffusion equa-
tion where C is the adatom density. We study if and how this addition at the BCF
level influences the macroscopic limit. Note that a phenomenological approach to
desorption in crystal morphological evolution is offered by Villain [119]. We derive
?-dependent corrections of the large-scale flux, and describe how large ? needs to
be so that desorption can be neglected in the macroscopic laws.
8.1 Macroscale limit and assumptions
The macroscopic limit aims to describe a continuous surface at sufficiently
large scales. Previous studies of electromigration and desorption apparently have not
focused on global evolution laws for the surface height, and hence carry a perspective
different from ours. These works (i) focus mainly on stability issues, (ii) make use
primarily of nanoscale models where steps are everywhere parallel, and (iii) allow
steps to interact mainly through terrace diffusion (with less emphasis on step-step
entropic or elastic dipole interactions).
Our derivation uses the multiscale expansion reviewed in Chapter 5, which
handles a fully (2+1)-dimensional geometry just as easily as a straight-step geom-
125
etry. The expansion parameter is the step height, a. This formulation relies on
the separation of local variables for interacting steps into fast and slow, and can
encompass additional physical effects such as desorption and electromigration. As a
result, we derive a fully continuum model in 2+1 dimensions that extends the model
of Chapter 5 to a greater variety of experimentally relevant situations. The addi-
tional effect of anisotropic terrace diffusion can also be incorporated subsequently,
following the calculation of Chapter 6.
We recall the geometric assumptions underlying the macroscopic limit. The
(microscale) terrace width is assumed to be of the order of the step height, a, and
small compared to: (i) the step radius of curvature; (ii) the length over which the
step curvature varies; and (iii) the macroscopic length over which the step density
varies. A step train satisfying these conditions is referred to as ?slowly varying?, cf.
Sections 5.3 and 6.1. The macroscopic limit is reached formally as a/? ? 0 where
? is a macroscopic length. The step density is fixed and approaches the surface
slope. We assume that the initial (at t = 0) ordering of steps is preserved by the
flow (for t> 0),4 and that step trains are monotonic (say, with descending steps in
the direction of increasing transverse coordinate).
Furthermore, we require that certain kinetic parameter groups involving the
step height, a, remain O(1) (as a/? ? 0). In particular, we take D/(ka) = O(1)
where k is any relevant kinetic rate for attachment-detachment of adatoms at a step
edge.
4This property should be derivable from microscale physics, such as the entropic repulsions that
enforce the non-crossing condition. This aspect is not studied here.
126
Figure 8.1: Schematic of 1D steps (cross section) with an applied electric field, E.
Steps are descending with increasing x; E is shown to be in the step-up direction
(E < 0); and ku (kd) is the kinetic rate for atom attachment-detachment from a
terrace to an up- (down-) step edge.
8.2 One-dimensional step models
In this section, we review briefly ingredients of step motion, including repul-
sive entropic and elastic dipole step interactions, for 1D geometries: straight and
concentric circular steps. Related versions of the adatom diffusion equation with
either an electric field or desorption are solved explicitly.
The step configuration is shown in Figure 8.1. Both the diffusion of adatoms
(density gradient) and electric field (drift velocity) contribute to the flux in each
terrace. Each step advances or retreats in response to the net (normal) flux from
the neighboring terraces. The surface evolves as the steps move, lowering its free
energy.
127
8.2.1 Straight steps
The position of the ith step is denoted by xi(t) (t is time), where xi+1 > xi
for all t, and Ci(x,t) is the adatom density on the ith terrace, i.e., the region
xi < x < xi+1. A constant electric field is applied externally in the x direction
(perpendicular to steps), introducing a drift velocity v according to (8.1). The
diffusion equation satisfied by Ci(x,t) is
D?2xCi ?v?xCi ???1Ci = ?tCi ? 0 xi <x<xi+1 , (8.2)
where ? is the characteristic desorption time. We use the quasi-steady approxima-
tion, ?tCi ? 0, by assuming that step motion is much slower than diffusion.
The adatom flux on the ith terrace, Ji(x,t), is defined by Fick?s law with drift:
Ji(x,t) = ?D?xCi +vCi . (8.3)
Linear kinetics prescribes the familiar boundary conditions at the step edges:
?Ji = ku(Ci ?Ceqi ) x = xi , (8.4)
Ji = kd(Ci ?Ceqi+1) x = xi+1 ; (8.5)
ku (kd) is the kinetic rate for an up- (down-) step edge, and Ceqi is the equilibrium
adatom density at the ith step.
This Ceqi expresses step interactions via the relation
Ceqi = Cs exp
parenleftBig?i
T
parenrightBig
?Cs
parenleftBig
1 + ?iT
parenrightBig
, |?i|lessmuchT , (8.6)
where ?i is the ith-step chemical potential and Cs is the equilibrium adatom density
at an isolated step. In Section 8.5.1 we consider the consequences of not linearizing
128
the exponential dependence on ?i.5
To illustrate some simple computations, we next distinguish the limits (i)
? ? ? (no desorption) and constant nonzero v, and (ii) v = 0 with finite ?.
The combination of finite ? and nonzero v can be described explicitly, but to keep
the presentation uncluttered we postpone this calculation until Section 8.5.2 on 2D
geometry.
8.2.1.1 Electric field with no desorption
Solving (8.2) for ? ?? yields
Ci(x) = Aievx/D +Bi xi <x<xi+1 . (8.7)
The adatom flux on the ith terrace, Ji(x), is computed by (8.3):
Ji(x) = vBi . (8.8)
Substituting for Ci and Ji in (8.5) via (8.7) and (8.8) leads to the system
?
??
?
evxi/D 1 +v/ku
evxi+1/D 1?v/kd
?
??
?
?
??
?
Ai
Bi
?
??
?=
?
??
?
Ceqi
Ceqi+1
?
??
?. (8.9)
This matrix equation is solved explicitly to yield
Ji(x) = Ji(xi) = ?v C
eq
i+1 ?e
v(?xi)/DCeq
i
ev(?xi)/D(1 +v/ku)?(1?v/kd) , (8.10)
where ?xi := xi+1 ?xi = O(a) is the terrace width.
5This extension also applies to the macroscale limits of previous chapters, but the formal ma-
nipulations used here are better motivated by first calculating the macroscale limit with electro-
migration.
129
8.2.1.2 Desorption with no electric field
Now consider diffusion equation (8.2) with v = 0 and finite ?. The solution
reads
Ci(x) = Aiebx +Bie?bx , b := (D?)?1/2 . (8.11)
The mass flux isJi(x) = ?D?xCi = ?Db(Aiebx?Bie?bx). Boundary conditions (8.5)
yield the matrix equation?
??
?
ebxi(1??u) e?bxi(1 +?u)
ebxi+1(1 +?d) ?e?bxi+1(1??d)
?
??
?
?
??
?
Ai
Bi
?
??
?=
?
??
?
Ceqi
Ceqi+1
?
??
? , (8.12)
where ?l := Db/kl (l = u, d). The mass flux on the ith terrace at x = xi is
Ji(xi) = ?Db C
eq
i+1 ?[cosh(b?xi) +?d sinh(b?xi)]C
eq
i
(1 +?u?d)sinh(b?xi) + (?u +?d)cosh(b?xi) . (8.13)
The flux at x = xi+1 can be obtained by the interchanges u?d and Ceqi ??Ceqi+1.
Remark 8.2.1. The fluxes in (8.10) and (8.13) are not simply proportional to
Ceqi+1 ?Ceqi (which in the macroscopic limit, ?xi = xi+1 ?xi = O(a) ? 0, becomes
proportional to the gradient of the macroscale chemical potential, ?). In section 8.4
we argue more generally that (8.10) and (8.13) reduce to different macroscopic laws
for the flux: The electric-field drift has a distinct macroscopic signature in the result-
ing Fick?s law; in contrast, the desorption effect is of higher order (in the expansion
parameter a), and is deemed negligible for a broad class of materials and conditions.
8.2.2 Concentric circular steps
In this section, we apply the BCF formulation to an axisymmetric setting.
Our goal is to study the interplay of drift and step curvature. So, consider steps
130
that are concentric circles of radii ri(t) with ri+1 >ri. The external electric field E
is taken to be radial, viz., E = erE (er: radial unit vector) where E may vary with
the polar coordinate, r. To simplify the exposition, we set E = E(r)/r.6
The electric field produces the radial drift velocity v = erv = er V/r according
to (8.1), where V = DZ?eE/T. The radial adatom flux, Ji(r) = er ?Ji(r), is
Ji(r) = ?D?rCi +vCi ri <r<ri+1 . (8.14)
The terrace diffusion equation under the quasi-steady approximation thus reads
0 ??divJi???1Ci = D(?2rCi+r?1?rCi)?r?1V?rCi?r?1(?rV)Ci???1Ci (8.15)
[where r?1(?rV)Ci = (divv)Ci]. In the following, we neglect the term r?1(?rV)Ci
assuming that, for v negationslash= 0, |?rV|Ci lessmuch |V?rCi|; this holds, for example, if V(r) varies
sufficiently slowly.7 Hence, (8.15) simplifies to
0 ?D(?2rCi +r?1?rCi)?r?1V?rCi ???1Ci . (8.16)
In Section 8.5.2.2 we discuss the effect of retaining (divv)Ci in the diffusion equation.
Boundary conditions analogous to (8.5) that complement (8.16) follow from
linear kinetics at the step edges:
?Ji(ri) = ku[Ci(ri)?Ceqi ] , Ji(ri+1) = kd[Ci(ri+1)?Ceqi+1] . (8.17)
6In experimental setups, E must of course be generated and sustained by feasible charge dis-
tributions. The issue of sources for E is not addressed here. These practical concerns suggest
the view that this radial case serves as a toy model for studying the interplay of drift and step
curvature.
7This condition is also expected to hold for V of the form V = cr? with sufficiently large r.
131
The difference of fluxes from neighboring terraces at the ith step edge yields
the step velocity ?ri = dridt according to mass conservation,
?ri = ?a[Ji?1(ri)?Ji(ri)] . (8.18)
The coupled step flow equations with drift and desorption form an extension of the
axisymmetric model by Israeli and Kandel [37] and Fok [25].
By analogy with section 8.2.1, we distinguish the cases with (i) ? ? ?, and
(ii) v = 0, where explicit solutions are relatively simple.
8.2.2.1 Electric field with no desorption
First, we solve (8.16) for ? ? ?. By successive integrations we obtain the
formula
Ci(r) = Bi +Ai
integraldisplay r
ri
1
z exp
bracketleftbigg
D?1
integraldisplay z
ri
v(rprime)drprime
bracketrightbigg
dz ri <r<ri+1 . (8.19)
By (8.14), we compute the corresponding (radial) flux:
Ji(r) = Ai
braceleftbigg
?Dr exp
bracketleftbigg
D?1
integraldisplay r
ri
v(rprime)drprime
bracketrightbigg
+v(r)
integraldisplay r
ri
1
z exp
bracketleftbigg
D?1
integraldisplay z
ri
v(rprime)drprime
bracketrightbigg
dz
bracerightbigg
+v(r)Bi . (8.20)
Substituting (8.19) and (8.20) for Ci and Ji in conditions (8.17), we find
?
??
??
? Dk
uri
1 + vik
uparenleftbigg
1? vi+1k
d
parenrightbiggintegraldisplay ri+1
ri
?i(z)dz+ Dk
d
?i(ri+1) 1? vi+1k
d
?
??
??
?
??
?
Ai
Bi
?
??
?=
?
??
?
Ceqi
Ceqi+1
?
??
? ,
(8.21)
132
where vj := v(rj)8 and
?i(r) := 1r exp
bracketleftbigg
D?1
integraldisplay r
ri
v(rprime)drprime
bracketrightbigg
. (8.22)
Solving for Ai and Bi we have
Ai = DADv , Bi = DBDv , (8.23)
Dv := Dk
uri
parenleftbigg
1? vi+1k
d
parenrightbigg
+
bracketleftbiggD
kd?i(ri+1) +
parenleftbigg
1? vi+1k
d
parenrightbiggintegraldisplay ri+1
ri
?i(z)dz
bracketrightbigg
?
parenleftbigg
1 + vik
u
parenrightbigg
, (8.24)
DA :=
parenleftbigg
1 + vik
u
parenrightbigg
Ceqi+1 ?
parenleftbigg
1? vi+1k
d
parenrightbigg
Ceqi , (8.25)
DB := Dk
uri
Ceqi+1 +
bracketleftbiggD
kd?i(ri+1) +
parenleftbigg
1? vi+1k
d
parenrightbiggintegraldisplay ri+1
ri
?i(z)dz
bracketrightbigg
Ceqi . (8.26)
The above formulas are simplified considerably if rv is treated as a constant.
For r = ri the radial flux reads
Ji(ri) = ?DAir
i
+viBi . (8.27)
The valueJi?1(ri) can be obtained similarly. The step velocity ?ri is written explicitly
by substituting the values for the integration constants into (8.27) and evaluating
the net flux at ri; cf. (3.33)?(3.36).
We conclude this subsection by noting in passing that the term (divv)Ci,
which was neglected above, can be included in the diffusion equation without diffi-
culty (although the algebra is lengthier and less transparent). An idea is to treat the
term in question as a forcing, apply Duhamel?s principle and thus derive an integral
8This distinction is not necessary if v(r) is macroscopic, i.e., it varies slowly.
133
equation for Ci, which can be solved via a Born-Neumann series by iteration. This
approach is undertaken in Section 8.5.2 for 2D. Alternatively, Ci can be expressed
in terms of confluent hypergeometric functions [18].
8.2.2.2 Desorption with no electric field
Next, we turn our attention to (8.16) with v = 0 and finite ?. The solution
reads
Ci(r) = AiI0(br) +BiK0(br) , b = (D?)?1/2 , ri <r<ri+1 , (8.28)
where I0 and K0 are modified Bessel functions [19]. The flux at the ith terrace is
Ji(r) = ?D?rCi(r) = ?Db[AiIprime0(br) +BiKprime0(br)] ri <r<ri+1 . (8.29)
By applying boundary conditions (8.17) we find the matrix equation
?
??
?
I0(bri)? Dbk
u
Iprime0(bri) K0(bri)? Dbk
u
Kprime0(bri)
I0(bri+1) + Dbk
d
Iprime0(bri+1) K0(bri+1) + Dbk
d
Kprime0(bri+1)
?
??
?
?
??
?
Ai
Bi
?
??
?=
?
??
?
Ceqi
Ceqi+1
?
??
?.
(8.30)
The microscale flux on the ith terrace at r = ri is
Ji(ri) = DbMiC
eq
i ?(bri)
?1Ceq
i+1
Pi , (8.31)
where
Mi =Iprime0(bri)K0(bri+1)?Kprime0(bri)I0(bri+1)
+ Dbk
d
[Iprime0(bri)Kprime0(bri+1)?Kprime0(bri)Iprime0(bri+1)] , (8.32)
134
Pi =I0(bri+1)K0(bri)?I0(bri)K0(bri+1)
+ Dbk
u
[Iprime0(bri)K0(bri+1)?I0(bri+1)Kprime0(bri)]
+ Dbk
d
[Iprime0(bri+1)K0(bri)?I0(bri)Kprime0(bri+1)]
? (Db)
2
kukd [I
prime
0(bri+1)K
prime
0(bri)?I
prime
0(bri)K
prime
0(bri+1)] . (8.33)
In the above relations, the prime denotes differentiation with respect to argument.
The step velocity is expressed in terms of (ri?2,ri?1,ri,ri+1,ri+2) via (3.33)?(3.36).
8.3 2-dimensional step geometry
In this section, we consider steps of reasonably arbitrary shape and formulate
their equations of motion under a macroscopic electric field E without desorption
(? = ?, i.e., b = (D?)?1/2 = 0). Of particular interest are unidirectional electric
fields with magnitudes that vary over a length scale large compared to the scale over
which the adatom density varies. So, we impose
|(divv)Ci|lessmuch|v??Ci| . (8.34)
Condition (8.34) is satisfied trivially (by divv ? 0) if E is constant, which corre-
sponds to many experimentally relevant situations.9 With regard to the actual steps,
we assume that their edges are represented by sufficiently smooth, closed curves on
the reference plane (x,y), which do not cross or self-intersect, as ensured physically
by the step-step repulsive (entropic and elastic dipole) interactions.
9If the three-dimensional E depends only on (x,y), (8.34) is trivially satisfied (by divv ? 0) in
the absence of charge distribution.
135
8.3.1 Step geometry and formulation
We remind the reader of our notation for the geometry of 2D steps given in
Chapter 5. The step train is monotonic, with steps descending outward from a
top terrace (surface peak). Their edges are numbered 1,2,...,N, starting from the
topmost step. The local coordinates (?,?) measure the distances transverse and
parallel to steps, respectively, with ? increasing in the step-down direction and ?
increasing counterclockwise (top view). The unit vectors e?,e? are orthogonal, in
the direction of increasing ?,?, respectively. The relevant metric coefficients (r:
position vector) are defined by
?? := |??r| , ?? := |??r| .
In the following, ?? and ?? are treated as O(1) and slowly varying with ?.
8.3.2 Equations of motion
In view of (8.34) and the quasi-steady approximation, the adatom density field
Ci in the ith-terrace region (?i <?<?i+1) solves
0 ??tCi = div(D??Ci)?v??Ci , ? = (?x,?y) = (??1? ??,??1? ??) , (8.35)
where D is in principle a tensor function of position r. Here, we consider a scalar
constant, D = D; anisotropic terrace diffusion is a straightforward application of
the methods outlined in Chapter 6.10 The (vector-valued) adatom flux Ji is
Ji = ?D?Ci +vCi . (8.36)
10The details of this extension are given in [97].
136
Robin-type boundary conditions for (8.35) inform Ci of: (i) attachment and
detachment of atoms at steps; and (ii) step energetics, especially step-step interac-
tions, via the equilibrium concentration, Ceqi . These boundary conditions read
?Ji,?(?i,?) = ku[Ci(?i,?)?Ceqi (?)] , (8.37)
Ji,?(?i+1,?prime) = kd[Ci(?i+1,?prime)?Ceqi+1(?prime)] ; Ji,? := e? ?Ji , ku/kd = O(1) .
(8.38)
8.3.3 Approximate solution of diffusion equation
By use of the local coordinates (?,?), diffusion equation (8.35) for Ci(?,?)
reads
0 = D?
???
bracketleftbigg
??
parenleftbigg?
?
????Ci
parenrightbigg
+??
parenleftbigg?
?
????Ci
parenrightbiggbracketrightbigg
?v???1? ??Ci ?vbardbl??1? ??Ci , (8.39)
where ?i <?<?i+1; v? := e? ?v and vbardbl := e? ?v.
By analogy with the case of zero electric field [66], we solve (8.39) with recourse
to the following proposition (which puts the derivations of Chapters 5,6, and 7 on
a more rigorous footing).
Proposition 8.3.1. Consider the boundary value problem consisting of PDE (8.39)
on the terrace Ui = {(?,?)|?i < ? < ?i+1}, and conditions (8.37), (8.38) on the
boundary, ?Ui, of Ui. Suppose that: (i) the boundary data exhibits a scale separation
in the sense that, for some geometric parameter epsilon1lessmuch 1,Ceqi is a fixed, O(1) function
of the slow variable ? := epsilon1?, i.e., Ceqi = Ceqi (?), while the rates ku, kd are epsilon1-
independent; (ii) the metric coefficients ??, ?? depend on (?,?) and are O(1) as
137
epsilon1 ? 0; and (iii) curlv = 0 (i.e., ?xvy = ?yvx)11 and min{ku,kd} > (1/2)|v|. Let
Ci be a C2 (twice continuously differentiable) function on Ui and C1 (continuously
differentiable) function on Ui, the closure of Ui. Then, for ? = O(1), Ci(?,?) =
C(0)i (?,?) +o(1) where
C(0)i (?,?) = Bi(?) +Ai(?)
integraldisplay ?
?i
dz ??|z?
?|z
exp
bracketleftbiggintegraldisplay z
?i
(v???)|?prime
D d?
prime
bracketrightbigg
; (8.40)
o(1) ? 0 as epsilon1? 0. The integration constants Ai,Bi are given by
Ai(?) = (1 +v?|?i/ku)C
eq
i+1 ?(1?v?|?i+1/kd)C
eq
i
D
ku??|?i
parenleftbigg
1? v?|?i+1k
d
parenrightbigg
+
parenleftbigg
1 + v?|?ik
u
parenrightbiggbracketleftbigg D
kd????fi|?i+1 +
parenleftbigg
1? v?k
d
parenrightbigg
fi|?i+1
bracketrightbigg ,
(8.41)
Bi(?) =
parenleftbigg
1 + v?|?ik
u
parenrightbigg?1bracketleftbigg
Ceqi (?) + Dk
u??|?i
Ai(?)
bracketrightbigg
, (8.42)
where fi(?,?) is defined by
fi(?,?) :=
integraldisplay ?
?i
??|z
??|z exp
bracketleftbiggintegraldisplay z
?i
(v???)|?prime
D d?
prime
bracketrightbigg
dz (?,?) ?Ui . (8.43)
In the above, Q|z denotes the value Q(? = z). The role of parameter epsilon1 is
to express small variations of the step edge curvature, which in turn cause slow
variations of Ci = Cepsilon1i(?,?) with respect to ?, permitting a perturbative treatment.
In the limit epsilon1 ? 0, the step edges become 1D, approaching concentric circles or
straight lines (depending on limits of ??, ??). We also assume that ku and kd are
positive (as is typical in crystalline materials). For comments on the condition
|v|/kl < 2 where l = u, d, see Remark 8.3.2.
Proof. For convenience and notational economy, in this proof we set D = 1 (or,
define the inverse length hatwidev := v/D and drop the hat) and suppress the terrace
11Alternatively, it can be assumed that v(r) varies slowly, e.g., v = v(epsilon11?,epsilon11?), epsilon11 lessmuch epsilon1.
138
index i unless noted otherwise. Assuming that a solution C(r) = Cepsilon1(r) exists and
is unique, as implied below, we partly separate scales in PDE (8.39) for ? = O(1).
So, we expand formally Cepsilon1(r) in an epsilon1-power series; each coefficient, C(j), is allowed
to be a function of (?,?,?), which are treated as independent variables:12
Cepsilon1(?,?) = C(0)(?,?,?) +
summationdisplay
j?1
epsilon1j C(j)(?,?,?) = C0(?,?,?) +o(1) , (8.44)
where the remainder approaches 0 as epsilon1? 0, assuming continuity of the solution with
epsilon1.13 In addition, the operator ?? is replaced by the linear combination
?? ??? +epsilon1?? .
By dominant balance of O(epsilon10) terms, (8.39) entails the (zeroth-order) PDE
braceleftbigg 1
????
bracketleftbigg
??
parenleftbigg?
?
????
parenrightbigg
+??
parenleftbigg?
?
????
parenrightbiggbracketrightbigg
?v? ???
?
?vbardbl ???
?
bracerightbigg
C(0) = 0 (r ?U) , (8.45)
which must be solved under conditions (8.37), (8.38) for C(0) (for epsilon1-independent ku,
kd).
Next, we show that, for given continuous Ceq(?), the above boundary value
problem forC(0) has at most one solution. We apply a standard energy method [24];
the same argument carries through for proving uniqueness of Cepsilon1. First, by the
transformation C(0) = ?Ce(1/2)
R r
ri v?dr
prime, PDE (8.45) is converted to the Helmholtz
equation ?r ?C ? 14(|v|2 ?2divv) ?C = 0 (r ? U: terrace), where ?C obeys the Robin
12The use of the extra slow variable ? = epsilon1?, although formally justifiable, is not deemed necessary.
13This physical property is invoked in conjunction with the assumption that eliminating the
curvature variation (as epsilon1 ? 0) results in a well-defined adatom density, as suggested by Section 8.2.
139
boundary condition ?????C =: ??? ?C = K(r) ?C?k(r)Ceqe?(1/2)
R r
ri v?dr
prime (r ??U);
? is the unit outward normal vector, K(r) := k(r)?(1/2)? ?v, and k(r) = ku for
an up-step edge (? = ?i where ? = ?e?) while k(r) = kd for a down-step edge
(? = ?i+1 where ? = e?). Consider sufficiently small |v| so that K(r) > 0, yet
|v|2 ? 2divv, consistent with the neglect of the term (divv)Cepsilon1 in the diffusion
equation. Now suppose there exist two solutions, say ?C1 and ?C2, of the boundary
problem for C(0), with ? := ?C1 ? ?C2; this ? satisfies the given Helmholtz equation
with boundary condition ???? = K?. Second, define the non-negative energy
E[?] = 12 integraltextU |??|2 +pi12?2 where pi12 := (1/4)(|v|2 ? 2divv). By Green?s identity,
E[?] =integraltext?U(????)? = ?integraltext?U K?2 ? 0; thus, ?? 0 which entails C1 ?C2.
Based on this uniqueness assertion, we construct solution (8.40)?(8.42). The
?-dependence of Ceq in the boundary data suggests that we look for a C(0)(r) that
depends on ? and ? (but not ?). Such a C(0)(r), if it can be constructed plausibly,
is interpreted as the leading-order, unique solution of the boundary value problem.
Hence, we solve (8.45) by dropping the ?-derivatives. Two successive integra-
tions with respect to the variable ? immediately yield
C(0)(?,?) = B(?)+ ?A(?)
integraldisplay ?
?i
exp
braceleftbigg
?
integraldisplay z
?i
bracketleftbigg?
?prime
?? ??prime
parenleftbigg?
?
??prime
parenrightbigg
?(v?|?prime)??primeD
bracketrightbigg
d?prime
bracerightbigg
dz, (8.46)
where ?A and B are integration constants. Equation (8.46) readily reduces to (8.40)
by direct integration (in ?prime) of the first term in the exponent and the subsequent
substitution A(?) := (??/??)|?i ?A(?).
The coefficients A(?) and B(?) are determined through boundary conditions
140
(8.37), (8.38) with ? = ?prime. Accordingly, we obtain the system?
??
??
? Dk
u??|?i
1 + v?|?ik
u
D
kd????f|?i+1 +
parenleftbig1?v
?/kd
parenrightbigf|
?i+1 1?
v?|?i+1
kd
?
??
??
?
??
?
A
B
?
??
?=
?
??
?
Ceqi
Ceqi+1
?
??
? ,
where f(?,?) is defined by (8.43). Solving this system leads to (8.41) and (8.42).
Henceforth, we drop the dependence of v? = v?e? (and vbardbl = e??v) on i, since
v is considered macroscopic and e?, e? vary slowly with ?. It can be verified that
(8.40) reduces to the solutions for 1D settings of sections 8.2.1.1 and 8.2.2.1.
Remark 8.3.2. Thus far, we invoked the condition |v|/klscript < 2 (lscript = u, d). By
definition (8.1) for v, this restriction amounts to imposing
|v|
2klscript =
D
2klscripta
|Z?e||E|a
T < 1 , lscript = u, d .
In typical experimental situations, D/(2klscripta) is of the order of 102 or smaller [43],
|eE|a/T is of the order of 10?5, and for semiconductors |Z?| ranges from 10?4 (or
even smaller values) to about 10 [15, 28]. Hence, |v|/klscript would not exceed values of
the order of 10?2.
From the perspective of perturbation theory adopted here, D/(klscripta) is treated as
an O(1) quantity whereas |v|/klscript will be considered as o(1) (see Proposition 8.4.1)
when a ? 0. This view furnishes the distinct physical contribution of drift in the
large-scale surface flux. Practically, setting |v|lessmuchklscript seems to more closely describe
semiconductor surfaces, where |Z?| can be much smaller than unity [28]. Mathemat-
ically, this view is a choice for obtaining the distinguished-limit contribution of v as
a? 0.
141
In Section 8.4.1 we impose |v|/kl lessmuch 1; as a result, Fick?s law for the large-scale
flux does not distinguish between up- and down-step edges, being symmetric in ku
and kd.
8.4 Evolution equations at the macroscale
In this section we find the evolution equations for the macroscale height in the
presence of an electric field. The modification of Fick?s law (8.36) at the nanoscale
leads to a convective term in the constitutive relation between the macroscale flux
and chemical potential, as we show in Proposition 8.4.1. Unchanged from Chapters
5,6, and 7 are the mass conservation law and the formula for the macroscale chemical
potential.
The law of mass conservation, in the absence of edge atom diffusion and ma-
terial deposition from above, reads
?th+ ?divJ = 0 ; (8.47)
see (5.44).
The macroscale chemical potential ? is regarded as a smooth interpolation,
through a suitable Taylor expansion, of the microscale step chemical potential ?i.
The presence of an electric field is not expected to change the surface energy due to
steps, which experience the same line tensions and repulsive interactions as in the
case of zero electric field. We adopt the formula from Chapter 5, which reads
? = ??div
braceleftbigg
?m[m(g1 +g3m2)] ?h|?h|
bracerightbigg
, m := |?h| . (8.48)
142
8.4.1 Macroscopic Fick?s law with drift
In this section, we derive the constitutive relation between macroscale flux
and chemical potential in full 2D. The restrictions to 1D geometries then follow as
special cases. We choose units where the macroscopic length ? is unity (? = 1) for
convenience.
Proposition 8.4.1. Suppose that in the macroscopic limit, a ? 0, the step density
mi = a/(????i) and the kinetic parameters D/(kla) are O(1) while v/kl = o(1),
where l = u, d and ??i := ?i+1 ??i. Then, the solution to diffusion equation (8.39),
under boundary conditions (8.37), (8.38), gives rise to the macroscale constitutive
relation
J =
?
??
?
J?
Jbardbl
?
??
?= ?CsM?
bracketleftbigg
??? TD v
parenleftBig
1 + ?T
parenrightBigbracketrightbigg parenleftbigg
v = DZ
?eE
T
parenrightbigg
, (8.49)
where the (E-independent) mobility M (with units of length 2/energy/time) is a
second-rank tensor. In the coordinate system (?,?), this M has the familiar repre-
sentation of Chapter 5, namely,
M = DT 11 +q|?h|
?
??
?
1 0
0 1 +q|?h|
?
??
? , q :=
2D
ka , k :=
2
k?1u +k?1d . (8.50)
Proof. The starting point is the solutionC(0)i derived in Proposition 8.3.1; see (8.40).
We drop the superscript (denoting perturbation order) in C(0)i for ease of notation.
The microscale adatom flux components are obtained by the formulas
Ji,bardbl = ?D??1? ??Ci +vbardblCi ,
Ji,? = ?D??1? ??Ci +v?Ci .
143
The plan is to consider the restrictions of these components to ? = ?i, and view
these restrictions as interpolations of (continuous) smooth functions, J?(r,t) and
Jbardbl(r,t).
First, we compute the requisite derivatives of Ci by (8.40):
??Ci ???Bi +??Ai
integraldisplay ?
?i
??|z
??|z exp
bracketleftbiggintegraldisplay z
?i
??prime v?|?prime
D d?
prime
bracketrightbigg
dz ,
??Ci = Ai ???
?
exp
bracketleftbiggintegraldisplay ?
?i
??prime(v?|?prime)
D d?
prime
bracketrightbigg
.
It follows that
Ji,bardbl = ?D??1? ??Bi +vbardblBi , (8.51)
Ji,? = ?D??1? Ai +v?Bi ? = ?i . (8.52)
Consider (8.41), (8.42) (in Proposition 8.3.1) for Ai, Bi. In the limit a? 0, or
??i ? 0 with????i = O(a), these formulas simplify via the expansionintegraltext?i+1?i F(?)d? =
F(?i)??i +o(??i), where F(?) is any continuous function. After some algebra and
neglect of o(??i) terms, (8.51) and (8.52) become
Ji,bardbl|?i ?
vbardbl
bracketleftbigg D
????i
parenleftbiggCeq
i
kd +
Ceqi+1
ku
parenrightbigg
+Ceqi
bracketrightbigg
?D
bracketleftbigg
?bardblCeqi + D?
???i
parenleftbigg?
bardblC
eq
i
kd +
?bardblCeqi+1
ku
parenrightbiggbracketrightbigg
D
????i
parenleftbigg 1
ku +
1
kd
parenrightbigg
+
parenleftbigg
1 + v?k
u
parenrightbigg ,
(8.53)
Ji,?|?i ?
D
????i(C
eq
i ?C
eq
i+1) +v?C
eq
i
D
????i
parenleftbigg 1
ku +
1
kd
parenrightbigg
+
parenleftbigg
1 + v?k
u
parenrightbigg as ??i ? 0 ; ?bardbl := ??1? ?? . (8.54)
In the above, all variables are evaluated at (the same) ? along the ith step edge.
We seek further simplification of (8.53) and (8.54). In the macroscopic limit,
we invoke the (assumed as well defined) C1 function Ceq(r) where Ceq(r)|?i ?Ceqi ,
144
Ceqi+1 ? Ceq(r)|?i + (??Ceq)|?i??i +o(??i) and ?bardblCeq|?i ? ?bardblCeqi = ?bardblCeqi+1 +O(??i).
We keep only those combinations of microscopic parameters that remain O(1). For
example, the step density mi = a/(????i) approaches the positive surface slope, i.e.,
lima?0mi = |?h|. On the other hand, the ratio v?/ku, involved in the denominator
of Ji,bardbl and Ji,?, is treated as negligibly small (compared to unity) by virtue of our
hypothesis; see also Remark 8.3.2.
Without further ado, we make the substitutions Ceqi+1 ?Ceqi = (??Ceq)????i +
o(??i) and Ceq ? Cs(1 +?/T) (|?| lessmuch T), where ?? := ??1? ??. The resulting limits
for the flux components read
lim
a?0
Ji,bardbl|?i =: Jbardbl(r)|?i = ?CsDT ?bardbl?(r) +Csvbardbl
bracketleftbigg
1 + ?(r)T
bracketrightbigg
, (8.55)
lim
a?0
Ji,?|?i =: J?(r)|?i = ?CsDT
???? TDv?
parenleftBig
1 + ?T
parenrightBig
1 +q|?h| ? = ?i . (8.56)
These relations are identified with (8.49) under definition (8.50) for the mobility M.
It can be verified directly that the same limits emerge if the evaluation point
of fluxes is at ? = ?i+1 [66]. Two remarks on the results of Proposition 8.4.1 are in
order.
Remark 8.4.2. In the absence of electromigration (v = 0), we recover the relation
J = ?CsM ? ?? of Chapter 5. For nonzero drift (v negationslash= 0), the additional term
derived above is affine with ?. This new relation is recast to the zero-drift form via
an exponential transformation of ?; see Section 8.4.3.
Remark 8.4.3. An alternate proof of Proposition 8.4.1 makes direct use of the
145
adatom concentration, Ci, and the normal flux component, Ji,?, avoiding entirely
the use of integration constants Ai and Bi. This approach treats boundary conditions
(8.37), (8.38) for ?prime = ? as a system of equations for Ci(?i) and Ji,?(?i). This argu-
ment is also applicable to situations where the boundary conditions couple densities
of different terraces, e.g., in the case with step permeability (see Chapter 7).
By Proposition 8.4.1, we state the following corollary for cases of symmetry.
Corollary 8.4.4. Consider 1D settings with v negationslash= 0, where translational or rotational
symmetry causes all dependent variables to have zero?-derivatives. The macroscopic
surface flux corresponding to (8.49) is
J = J(?)e? , J(?) = ?CsDT ???1 +q|?
?h|
+ Csv1 +q|?
?h|
parenleftBig
1 + ?T
parenrightBig
, (8.57)
where ? = x for straight steps and ? = r for concentric circular steps; q = 2D/(ka).
Recall that, in the microscale model underlying the limit of this section, the
contribution (divv)Ci is left out from the terrace diffusion equation (see Section
8.2.2). For a discussion on a correction to the macroscopic limit, see Remark 8.5.4.
8.4.2 Evolution equation in Cartesian system
In this section, we describe the PDE for the surface height by combining
ingredients (8.47) and (8.48)?(8.50) and making use of Cartesian coordinates, (x,y).
First, we recall the non-singular orthogonal matrix S, Eq. (5.26).
S(?xh,?yh) = (e? e?) = 1|?h|
?
??
?
??xh ?yh
??yh ??xh
?
??
? (?hnegationslash= 0) . (8.58)
146
The mobility tensor M, which is defined by (8.49) in the (?,?) coordinate system,
is now expressed in the Cartesian coordinate system (x,y) by
M(x,y) = SM(?,?)ST ; (8.59)
as usual, ST denotes the transpose of S (ST = S?1). The Cartesian components of
the surface flux (8.49) read (elscript: orthonormal vectors, lscript = x, y):
Jx =? Cs1 +q|?h|
braceleftbiggD
T
bracketleftbiggparenleftbigg
1 +q (?yh)
2
|?h|
parenrightbigg
?x??q (?xh)(?yh)|?h| ?y?
bracketrightbigg
?
parenleftbigg
1 + ?T
parenrightbiggbracketleftbiggparenleftbigg
1 +q (?yh)
2
|?h|
parenrightbigg
vx ?q (?xh)(?yh)|?h| vy
bracketrightbiggbracerightbigg
, (8.60)
Jy =? Cs1 +q|?h|
braceleftbiggD
T
bracketleftbiggparenleftbigg
1 +q(?xh)
2
|?h|
parenrightbigg
?y??q(?xh)(?yh)|?h| ?x?
bracketrightbigg
?
parenleftbigg
1 + ?T
parenrightbiggbracketleftbiggparenleftbigg
1 +q (?xh)
2
|?h|
parenrightbigg
vy ?q (?yh)(?xh)|?h| vx
bracketrightbiggbracerightbigg
; vlscript = elscript ?v (lscript = x, y) .
(8.61)
The PDE for the surface height follows from the mass conservation statement.
Using the Cartesian representation of (8.47), we have
?th = ??(?xJx +?yJy) . (8.62)
This relation leads to a nonlinear, fourth-order parabolic PDE for h after substitu-
tion for Jx,Jy, and ? from (8.60), (8.61), and (8.48).
8.4.3 Change of variables
We show that law (8.49) is recast to the form
J = ?cM??? (c = const.) , (8.63)
147
leading to the PDE
??t?h = div?[M????] , (8.64)
with suitable choices of the tensor M and variable ?[??], the transformed chemical
potential; the dimensionless variables ?t, ?h, ?? and operators div?, ?? are defined in
Appendix B.
Constitutive relation (8.49) for the surface flux reads
J = ?CsD?
x
tildewiderM?[?????u(1 + ??)] , (8.65)
where the dimensionless mobilitytildewiderM ? in the (x,y) representation ? and drift velocity
u are defined by
tildewiderM(x,y) := tildewideS?
?
??
??
1
1 + ?q|???h| 0
0 1
?
??
???tildewideST , u := ?xD v ; ?q := qh0?
x
. (8.66)
The corresponding change-of-basis matrix, tildewideS, is tildewideS ? S(??x?h,???y?h); cf. (8.58).
Consequently, the PDE for the nondimensional height, ?h, reads
??t?h = div?{tildewiderM?[?????u(1 + ??)]} . (8.67)
Next, we show (8.63) and (8.64). By inspection, we start with the transfor-
mation
? = (1 + ??)f? , (8.68)
where the (nonzero) f? = f?(?x,?y) is to be determined. By virtue of M???? =
(Mf?)?[????+ (??f?/f?)(1 + ??)] and (8.65), we have the consistency relations
Mf? = tildewiderM , (??f?)/f? = ?u ,
148
the second one of which yields
f?(?x,?y) = Ae?u?(?x,?y/?) ? M = A?1tildewiderM(???h)eu?(?x,?y/?) , A = const.negationslash= 0 . (8.69)
Relations (8.63), (8.64) ensue. For definiteness, take A = 1.
The transformed chemical potential ? and mobility M are
?(?x,?y) =
braceleftbigg
1? ??0 div?
bracketleftbigg
?g ??
?h
|???h| +|??
?h|???h
bracketrightbiggbracerightbigg
e?(ux?x+uy?y/?) , (8.70)
M(x,y) = eux?x+uy?y/?tildewideS?tildewiderM?tildewideST = eux?x+uy?y/?tildewideS?
?
??
??
1
1 + ?q|???h| 0
0 1
?
??
???tildewideST . (8.71)
The finite element scheme introduced in Chapter 2 can easily accommodate the
transformed chemical potential and mobility. The change-of-variables is not deemed
likely to introduce numerical instability if the grid Peclet number Pe = |v|?x/D is
much less than 1 [77].14 For easier readability of the code, however, we opted to
include the convective term directly when simulating the macroscopic effect of an
electric field. A thorough comparison of the two methods is still lacking.
8.5 Extensions of macroscopic limit
In an attempt to formulate a reasonably general theory of macroscopic surface
relaxation, we enrich the BCF model with additional microscale effects. These
are: (i) the exponential law Ceqi = Cs exp(?i/T) in the place of its linearization,
and (ii) atom desorption. Our broader goal with these extensions is to reconcile
the macroscopic theory with realistic situations where an electric field is present.
14Pe compares the relative importance of convection to diffusion; ?x : mesh size.
149
We show that (i) can modify significantly the constitutive relation between surface
flux and chemical potential. In contrast, effect (ii) arguably has vanishingly small
influence on the macroscopic evolution law.
8.5.1 Exponential law for step chemical potential
To derive the constitutive relation between large-scale flux and chemical po-
tential, we approximated the difference of equilibrium concentrations Ceqi by use of
the linearized form (8.6). The quantitative justification for this approximation is not
clear in the literature, particularly since the chemical potential ?i is not measured
directly. However, the PDE is easily modified to accommodate the complete, expo-
nential law. For a similar modification in 1+1 dimensions withv = 0 and long-range
step interactions, see [123].
By skipping details irrelevant to the modification at hand, we start from (8.54).
By setting ?i = ?(r)|?i we expand the difference Ceqi+1 ?Ceqi as follows:
Ceqi+1 ?Ceqi = Cs
bracketleftbigg
exp
parenleftbigg?
i+1
T
parenrightbigg
?exp
parenleftbigg?
i
T
parenrightbiggbracketrightbigg
= aCs|?h| exp
parenleftbigg?(??)
T
parenrightbigg?
??(??)
T , ?? ? [?i,?i+1] .
Hence, the normal flux component Ji,? at (?i,?) becomes
Ji,?|?i =
? D?
???i
aCs
|?h| exp
parenleftbigg?(??)
T
parenrightbigg?
??(??)
T +v?Cs exp
parenleftbigg?(?
i)
T
parenrightbigg
D
????i
parenleftbigg 1
ku +
1
kd
parenrightbigg
+
parenleftbigg
1 + v?k
u
parenrightbigg . (8.72)
Similarly, the ?-derivative of Ceqi appearing in (8.53) for the longitudinal flux com-
150
ponent, Ji,bardbl, now acquires exponential factors when expressed in terms of ?i :
Ji,bardbl|i =
bracketleftbigg
? D?
???i
parenleftbigg 1
ku +
1
kd
parenrightbigg
?1
bracketrightbigg?1braceleftbigg
?CsD?
???i
vbardbl
bracketleftbigg 1
kd exp
parenleftBig?i
T
parenrightBig
+ 1k
u
exp
parenleftBig?i+1
T
parenrightBigbracketrightbigg
?vbardblCs exp
parenleftBig?i
T
parenrightBig
+CsDexp
parenleftBig?i
T
parenrightBig?bardbl?i
T
+ D
2Cs
kd????i exp
parenleftBig?i
T
parenrightBig?bardbl?i
T +
D2Cs
ku????i exp
parenleftBig?i+1
T
parenrightBig?bardbl?i+1
T
bracerightbigg
. (8.73)
The coarse-graining procedure carries through as in Proposition 8.4.1. The
components of the flux J(r) are found to satisfy the equations
J?(r)
parenleftbigg2
k +
a
D|?h|
parenrightbigg
= exp
bracketleftbigg?(r)
T
bracketrightbiggbraceleftbigg
?aCs|?h|???(r)T +v? aCsD|?h|
bracerightbigg
, (8.74)
Cs exp
bracketleftbigg?(r)
T
bracketrightbigg?
bardbl?(r)
T = ?
1
D
braceleftbigg
Jbardbl(r)?vbardblCs exp
bracketleftbigg?(r)
T
bracketrightbiggbracerightbigg
. (8.75)
The last two equations result in the effective constitutive relation
J = ?Cse?/TM?
parenleftbigg
??? TD v
parenrightbigg
. (8.76)
By inspection of (8.76), we have the following remark:
Remark 8.5.1. The modified constitutive relation for the macroscopic surface flux
results from the invariant under the law Ceq = Ceq[?] form (8.64) and the definition
?(r) := e?(r)/T?v?r/D , (8.77)
which is a direct extension of the linearized version, (8.68) or (8.70).
8.5.2 Adatom desorption
The effect of desorption is expected to affect only the relation between the
large-scale surface flux and chemical potential. In this section, we show that, under
151
certain conditions, desorption does not appear in the macroscopic laws to leading
order in a.
8.5.2.1 Solution for microscale diffusion
Following the rationale of slow and fast step variables of section 8.3.3, we write
the (quasi-steady) diffusion equation on the ith terrace with desorption time ? and
a drift velocity v (v? = e? ?v) as
??
parenleftbigg?
?
????Ci
parenrightbigg
? v???D ??Ci ? ?????DCi (?i <?<?i+1) . (8.78)
A first observation is that, for v? negationslash= 0 and ? negationslash= ?, this equation does not admit a
relatively simple solution, i.e., in terms of elementary functions. (The radial case of
Section 8.2.2 certainly alludes to the same conclusion.)
It is of interest to note that there are at least two ways of solving (8.78). First,
it can be converted to a canonical ordinary differential equation (ODE) solvable by
confluent hypergeometric functions [18]. This route is not particularly informative.
Alternatively, (8.78) can be recast to a Volterra integral equation, which can be
solved by iterations through a (convergent) Born-Neumann series [116].
We now focus on the (simpler) integral-equation formulation for (8.78).15
Proposition 8.5.2. Suppose Ci is a solution of the PDE (8.78). Then, Ci satisfies
the following Volterra equation on ?i <?<?i+1 :
Ci(?,?) = [Bi(?) +Ai(?)fi(?,?)] + [ui,1(?,?) +ui,2(?,?)fi(?,?)] , (8.79)
15For notational simplicity, we use ? in the place of the slow variable ?.
152
where
ui,1(?,?) = ui,1[Ci] = ?(?D)?1
integraldisplay ?
?i
fi(?prime,?)?2?primeCi(?prime,?)
W(?prime) d?
prime , (8.80)
ui,2(?,?) = ui,2[Ci] = (?D)?1
integraldisplay ?
?i
?2?prime Ci(?prime,?)
W(?prime) d?
prime , (8.81)
W(?) = ??fi = ???
?
exp
bracketleftbigg
D?1
integraldisplay ?
?i
??prime (v?|?prime)d?prime
bracketrightbigg
, (8.82)
fi(?,?) is defined by (8.43), and the coefficients Ai(?) and Bi(?) are determined
through boundary conditions (8.37), (8.38) for atom attachment-detachment at ? =
?i, ?i+1. Note that W(?) is the Wronskian of the two homogeneous (for ? = ?)
solutions of ODE (8.78), namely, the functions 1 and fi(?,?).
Proof. We provide a sketch of the proof since this relies on standard techniques for
linear ODEs and PDEs. Consider the (?-dependent) term in the right hand side
of (8.78) as a forcing. Apply Duhamel?s principle (or the method of ?variation of
parameters?) to construct a particular solution of the ODE. This approach yields
Ci as a sum of: (i) a linear combination of the two homogeneous solutions, 1 and
fi; and (ii) a ??1-scaled particular solution, which involves two distinct integrals
of Ci/? (one for each homogeneous solution). The coefficients Ai and Bi in the
aforementioned linear combination are determined via enforcement of the boundary
conditions.
The integral equation (8.79) can be solved by the conventional iteration (Born-
Neumann) scheme [116]: First, set Ci = 0 under the integral (in ui,1 and ui,2) and
obtain Ci ? Bi + Aifi; second, replace Ci by Bi + Aifi under the integral; next,
repeat successively with the updated Ci. Because the kernel is L2 and smooth, the
153
series generated by iterations converges to the solution Ci for all ? > 0, and yields a
sufficiently smooth Ci. Since the kernel is proportional to ??1, the procedure yields
a power series in 1/?, whose convergence rapidity is thus enhanced by increasing ?.
The value of the Born-Neumann scheme becomes evident for ? ? ?i lessmuch 1.
Since integraltext??i F(?prime)d?prime = F(?i)(???i)+o(???i) for any continuous F, the associated
iterated integrals contribute respective ascending powers of ? ??i. So, evidently,
in the limit maxi??i = maxi{?i+1 ??i} ? 0, constructing a solution to (8.79) via a
Born-Neumann series corresponds to producing a Taylor series in ???i for Ci(?,?).
This result is directly applicable to the macroscopic limit of the step system.
8.5.2.2 Limit a? 0
Next, we derive a macroscopic Fick?s law with desorption and external electric
field. We follow the main procedure of Proposition 8.4.1 by use of formula (8.79)
(in Proposition 8.5.2) for Ci. Suppressing the ? dependence, we define
Cvi (?) := Bi +Aifi(?) , (8.83)
which is the solution form without desorption.16 Thus, Ci satisfies
Ci(?)?Cvi (?) = (?D)?1
integraldisplay ?
?i
?2?prime fi(?)?fi(?
prime)
??primefi Ci(?
prime) d?prime . (8.84)
By differentiation of (8.84) with respect to ? we obtain the normal flux com-
ponent, Ji,? = e? ?(?D?Ci +vCi):
Ji,??Jvi,? = ???fi?
integraldisplay ?
?i
?2?prime Ci(?prime)
??primefi d?
prime + v?
?D
integraldisplay ?
?i
?2?prime fi(?)?fi(?
prime)
??primefi Ci(?
prime) d?prime , (8.85)
16Ai and Bi entering Cvi in principle depend on ? via boundary conditions at step edges.
154
where Jvi,? := ?D??Cvi +v?Cvi ; recall that ?? = ??1? ??.
Now consider the limit ?? := ???i ? 0. By Taylor expanding the right hand
sides of (8.84) and (8.85), we readily obtain
Ci(?)?Cvi (?) = (2D?)?1Cvi (?i)?2?i ??2 +O(??3) , (8.86)
Ji,?(?)?Jvi,?(?) = ???1Cvi (?i)??i ??+O(??2) as ?? ? 0 . (8.87)
Next, we apply conditions (8.37) and (8.38) for?prime = ?. With recourse toCvi (?i+1) =
Cvi (?i) + (??Cvi )|?i??i +o(??i), and likewise for Ji,?(?i+1), we find
?k?1u Jvi,? = Cvi ?Ceqi , (8.88)
k?1d
bracketleftbiggparenleftbigg
1 + kdD?wi
parenrightbigg
Jvi,? + (??Jvi,?)?wi
bracketrightbigg
?
bracketleftbigg
1 +
parenleftbiggv
?
D +
1
kd? +
?wi
2D?
parenrightbigg
?wi
bracketrightbigg
Cvi ?Ceqi+1 , (8.89)
where ?wi := ??i ??i, and Cvi , Jvi,? and ??Jvi,? are evaluated at ?i. Note that (8.88)
does not involve ? explicitly. In contrast, (8.89) manifests desorption terms in the
right hand side. If these, ??1-scaled, terms can be neglected appropriately, the
resulting system becomes identical to that without desorption; the corresponding
terms in Ci and Ji,? can then be dropped.
Remark 8.5.3. By inspection of the (attachment-detachment) boundary conditions
(8.88) and (8.89) for the adatom flux and density, sufficient conditions for neglecting
the desorption effect in the macroscopic limit [assuming k/klscript = O(1), lscript = u, d] are
|v|
D greatermuch
1
k? and
a
k? lessmuch 1 . (8.90)
155
Then, the large-scale adatom flux is not affected by ? uniformly in space coordinates.
Both of these conditions are expected to be met in a wide range of physical situations.
In particular, the first condition amounts to having |v|? greatermucha when D/(ka) = O(1).
From the viewpoint of coarse graining and perturbation theory, formulas (8.90) sim-
ply state that the effect of desorption is of higher order (in a).
We conclude this section with the following observation.
Remark 8.5.4. The (thus far neglected) term (divv)Ci in the terrace diffusion
equation for the adatom density Ci can be treated on the same footing as the des-
orption term Ci/?. Hence, in assessing the validity of dropping the former, it is
reasonable to repeat the above procedure with ??1 being replaced by divv, in an
appropriate sense.
8.6 Conclusions
Starting with the BCF model of step flow, we derived macroscopic evolu-
tion laws for the surface height in settings with an electric field. We considered
microscopic processes of isotropic adatom diffusion on terraces, attachment and de-
tachment of atoms at step edges with an Ehrlich-Schwoebel barrier, and desorption
of atoms to the surrounding vapor. Energetic effects such as entropic and elastic
dipole step interactions were included.
Our central contribution is Fick?s law (8.49) relating surface flux J, chemical
potential ?, and drift velocity v. The linear combination of ? and ?? of this law
is consolidated into a single variable, ?, by an exponential transformation. Accord-
156
ingly, the PDE for the macroscale height is recast to form (8.64), which has the
same structure as the evolution law ?th = ?Cs div{M???} derived in the absence
of an electric field. We showed that desorption is a higher-order effect in a sense
dictated by multiscale expansions in the step height, a.
Many effects are absent from our analysis. For instance, time-dependent,
coupled electromagnetic fields are not accounted for. In the same vein, the control
of evolving surface morphologies by electric fields was barely touched upon. Our
assumption of a slowly-varying step train restricts the validity of the PDE to regions
where the steps do not experience drastic instabilities. We also leave out long-range
step interactions, e.g., interactions mediated by bulk stress. We postpone until
Chapter 9 a more thorough discussion of the appropriate boundary conditions for
the derived PDE, and the effect of an electric field on the height profile near this
boundary.
157
Chapter 9
Facets and boundary layers
In this chapter, we discuss analytical aspects of faceted relaxation of crystal
surfaces. As we see in the next chapter, the results of applying the finite element
method to the previously derived PDEs have been validated only in the limiting case
g1/g3 lessmuch 1. For larger values of g1/g3, our numerical scheme is challenged by the
appearance of facets, which expand during the surface relaxation. The variational
approach on which our numerics are based still offers a possible connection with
the physically realistic case of nonzero line tension. To see this connection, we turn
to analytical methods, focusing on the local behavior of the height profile near a
facet edge. We study how the slope far from the facet approaches continuously its
boundary value (|?h| = 0 at the facet edge), for different values of the material
parameters and the electric field. This local analysis of this chapter complements
the global analysis we used to derive decay rates for approximately separable PDE
solutions, which are assumed to exist based on experimental observations and 1D
step simulations. Our analysis is based on a conjecture (or ansatz) about the sep-
arability of space and time dependence of the slope profile, which entails a locally
factorizable solution of the PDE.
158
9.1 Faceting on crystal surfaces
The presence of facets indicates clearly that a given surface orientation is below
roughening, which is the context we assume when deriving a PDE for h from the
motion of steps. However, strictly speaking the domain of this PDE does not include
the facet itself, where the surface energy is singular due to the vanishing slope |?h|.1
We then face the problem of determining the appropriate boundary conditions at
the facet edge. Analysis of the PDE and the boundary conditions near the facet
edge provides insight into the local behavior of the slope profile. This analytically
predicted behavior could be experimentally testable [115].
9.2 Formulation of faceted relaxation as a free-boundary problem
Spohn [109] first studied facet evolution as a free-boundary problem for nonlin-
ear PDEs. Here, we summarize the main idea and extensions. The moving boundary
is the facet edge, a curve we denote by rf(t) ? L0(t). More precisely, for a train of
descending steps in the vicinity of a facet with height hf (Fig. 9.1), we identify the
facet with the set {r = (x,y) : h(x,y,t) = hf(t)}, and rf(t) ?L0(t) is the boundary
of this set. (Recall that Li was used in Chapter 5 to denote the ith step edge. Al-
though, strictly speaking, the facet edge does not correspond to the topmost step,
we use L0 for lack of a more suggestive notation.)
1The reason for this pathology is that step positions near the facet edge do not satisfy the usual
assumptions of the macroscopic limit. Mathematically, the PDE can be extended on the facet by
the ?subgradient (subdifferential) formulation? [24], as outlined below.
159
Figure 9.1: Orthogonal projection of a 2D step train in the vicinity of a facet.
160
The motion of the facet edge is found by imposing the requisite boundary
conditions on the PDE solution. The slope profile at perpendicular distanced? from
the facet edge has a (characteristic) d1/2? behavior related to a Pokrovsky-Talopov
phase transition, as predicted by Jayaprakash et al. [41, 42] for equilibrium crystal
shapes. Margetis, Aziz and Stone [64] applied the free-boundary approach in an
axisymmetric setting (circular steps) to determine how the width of a boundary layer
near the facet edge scales with the ratiog3/g1 of step interaction strength to the step
line tension, when g3/g1 lessmuch 1. In the case of straight steps, Odisharia used the free-
boundary approach to establish the time decay of self-similar profiles [79]. Shenoy
et al. applied nonlinear Galerkin schemes to incorporate facets into a variational
formulation of surface relaxation [108].
9.3 Natural boundary conditions of the variational problem for pe-
riodic profiles
To investigate the effect of the free energy coefficients g1 and g3 on the ob-
served slope profile near a facet, one must in principle write down the boundary
conditions at the facet edge. A complete set of boundary conditions emerges ?nat-
urally? by considering the variational approach in more detail. We alert the reader
that not all of the natural boundary conditions play a role in our subsequent local
analysis. For the sake of completeness, we present the full set of natural bound-
ary conditions, so that the numerical results of Chapter 10 with g1 negationslash= 0 can be
interpreted from the perspective of strong PDE solutions informed by conditions at
161
the free boundary. Not all of these boundary conditions are physically meaningful,
but they possess the advantage of being enforced automatically by any numerical
scheme based on the variational formulation. An ?unphysical? condition is that the
chemical potential extends continuously across the facet edge. In fact, this latter
condition can be replaced by a boundary condition respecting the discrete sequence
of collapse times for the top step [37]. In this case, the resulting set of boundary
conditions, while consistent with the microstructure of the crystal surface, unfor-
tunately requires feedback from discrete step simulations to implement a numerical
solution of the free-boundary problem.
We now develop a brief argument for the natural boundary conditions arising
from the variational formulation for evolution of a spatially periodic profile h. This
approach follows closely the presentation of [79] in 1+1 dimensions. We restrict
our attention to diffusion-limited kinetics, where the mobility M = CsD/(kBT) is
a scalar constant. To address ADL kinetics we would need to invoke the semi-
implicit time stepping of Chapter 2, which yields a weighted H?1 inner product
in the extended gradient (subgradient) formulation. By using the height profile
at a previous time step in the evaluation of the tensor mobility M, we determine
explicitly the inner product through which the surface evolution is interpreted as
a subgradient flow. This issue has been addressed in the doctoral dissertation by
Odisharia [79].
The macroscale equations have the familiar structure of three key relations:
(i) mass conservation, (ii) Fick?s law for surface diffusion, and (iii) the equation for
chemical potential in terms of slope. We restate these three ingredients under the
162
assumption of diffusion-limited kinetics.
?h
?t = ??divJ,
J = ?CsDk
BT
??,
? = ?g3?E?h = ?g3?
x
div?N,
where div? is the divergence operator with respect to nondimensional spatial vari-
ables2, ?x is a characteristic length in the basal plane, and E = integraltext ?(?h)dx with
?(?h) = g1g3|?h|+13|?h|3. To emphasize the importance of the ratio g1/g3, we choose
to treatE andNas nondimensional; the chemical potential acquires the correct units
through the prefactor ?g3/?x. The auxiliary vector N is introduced to write the
variation of E more explicitly: outside the facet the formula N = g1g3 ?h|?h| +|?h|?h
holds. We take the height h to be a periodic function of the nondimensional spatial
variables x,y, which entails
[0,1]?[0,1]?[0,?) owner (x,y,t) mapsto?h(x,y,t) ? (?h0 ??,h0 +?);
see Appendix B.3
Because the surface energy density ?(?h) has a corner singularity at ?h = 0,
we need to make use of a generalized notion of differentiation when combining the
three ingredients of the macroscopic theory to incorporate facets. This generaliza-
tion of a gradient flow begins with the definition of subgradient for a real-valued
2Following the notation of Appendix B; in the calculations below we drop the subscript ? for
the sake of readability.
3The range for h is chosen to accommodate the transient behavior in which the height slightly
exceeds its initial amplitude maxh(?,?,0) = h0 at the beginning of a relaxation experiment.
163
function on R2. Here we follow the presentation of Odisharia [79] to introduce the
subgradient in the context of surface relaxation. We use??,??to denote the Euclidean
inner product in the following
Definition 9.3.1. The subgradient of ? : R2 ?R, denoted ??, is the set
?? = {a?R2 : ?a,????(?h+?)??(?h) ?? ?R2}.
To illustrate this definition, we consider first the singular case ?h = 0. Then
?? = {a?R2 : ?a,??? g1g
3
|?|+|?|3/3},
which is the ball of radius g1/g3 around 0.
In the smooth case (?hnegationslash= 0), we have
?? = {a?R2 : ?a,??? g1g
3
|?h+?|? g1g
3
|?h|+ |?h+?|
3
3 ?
|?h|3
3 },
which reduces to the single element
a = g1g
3
?h
|?h| +|?h|?h,
after expanding the norms in Taylor series about ? = 0.
Evidently, the subgradient agrees with the usual notion of gradient when the
free energy density is smooth enough. We note that the calculation of a subgradient
depends on which inner product ??,?? is used in the definition. For the free energy
density?, the natural choice for??,??is the Euclidean inner product. For a functional
such as E[?h], we must use the inner product of the H?1 Sobolev space [24]. The
essential property of the H?1 inner product is an identity that follows (formally)
164
from integration by parts:
?u,v?H?1 = ????1u,v?L2, (9.1)
where??,??L2 denotestheusualinnerproductofsquare-integrablefunctions: ?f,g?L2 =
integraltext
[0,1]2 fgdA. The idea of the H
?1 inner product becomes more transparent by using
the Fourier transform ?u of periodic u with zero mean.
The variation of E must be expressed using the language of subgradients so
that the chemical potential ? can be defined in the case of facets. In this vein, we
compute the subgradient of the functional E[h] = integraltext ?(?h)dx with respect to the
H?1 inner product. By definition,
u??H?1E ?E[h+?]?E[h] ??u,??H?1 ??. (9.2)
Having at our disposal the calculation of ??(?h) in both the smooth and the non-
smooth cases, we let N(x,y) ???(?h) and consider u = ?divN.
For such N we have
?(?h+??)??(?h) ??N(x,y),??? (9.3)
pointwise, which can be integrated to obtain
E[h+?]?E[h] ?
integraldisplay
?N(x,y),???dx = ?
integraldisplay
?divN,??dx (9.4)
= ??divN,??L2 (9.5)
= ??divN,??H?1. (9.6)
Therefore u = ?divN ? ?H?1E. In fact, the subgradient of the functional E
consists precisely of these elements [47]; i.e.,
?H?1E = {?divN : N(x,y) ???(?h) for each (x,y)}.
165
Returning to the problem of surface evolution, with DL kinetics we have
?th = ?CsD?
2g3
kBT ?divN globally, (9.7)
where N is the canonical restriction of the subgradient ?H?1E, i.e., the element
?nearest? to the origin with respect to the H?1 norm. As we observed above, for
?h negationslash= 0 the subgradient consists of a single element, and the canonical restriction
does not come into play. Only in the singular case ?h = 0 do we need to invoke the
canonical restriction to determine ht uniquely. The auxiliary function N ? ?H?1E
can be found by solving the minimization problem
min
N???
bardbl?divNbardbl2H?1 ? min
N???
bardbl?divNbardbl2L2. (9.8)
The canonical restriction N which solves the minimization problem (9.8) must
have a square integrable derivative ?divN. This condition implies that N and
? = ?g3 divN are continuous on [0,1]2. We are now in a position to state the
natural boundary conditions arising from the variational formulation.
The first boundary condition states that the slope |?h| approaches zero at the
facet edge L0(t). Since N is restricted to the ball of radius g1/g3 on the facet, while
|N| = g1g3(1 + |?h|2) outside the facet, continuity of N dictates that |?h| ? 0 as
r ? rf ?L0, i.e.,
?h = 0, (x,y) ?L0. (9.9)
The second boundary condition in DL kinetics follows from enforcing continu-
ity of surface flux (?divN in the discussion above) at the facet edge. Outside the
166
facet, we have a surface flux given by
J = ?CsDk
BT
?? = CsD?g3k
BT
?div
parenleftbiggg
1
g3
?h
|?h| +|?h|?h
parenrightbigg
. (9.10)
The flux on the facet follows from mass conservation, ?hf +?divJf = 0, where
hf(t) is the facet height. Solving for the divergence of Jf, we write
divJf = ?
?hf
? on the facet. (9.11)
Fix t, and integrate (9.11) over the interior of L0(t). We apply the divergence
theorem to conclude
contintegraldisplay
L0(t)
Jf ??ds = ???1?hfAf(t), (9.12)
where Af(t) is the area of the facet at time t.
Now we invoke continuity of J at L0(t) to replace Jf in (9.12) with its coun-
terpart (9.10) restricted at L0(t). The result is
?hfAf(t) = ?
contintegraldisplay
L0(t)
CsD?2g3
kBT ?div
parenleftbiggg
1
g3
?h
|?h| +|?h|?h
parenrightbigg
??ds. (9.13)
The next boundary condition follows from continuity of the surface height
h = h(x,y,t) at the facet edge. The height continuity dictates
hf(t) = h(x,y,t)|r?L0(t). (9.14)
We now differentiate with respect to t to obtain
?hf(t) = d
dth(r,t) =
?h
?t
vextendsinglevextendsingle
vextendsinglevextendsingle
r?L0(t)+
??h|r ? ?r?t. (9.15)
The value of ?th as (x,y) ? L0(t) from outside the facet is governed by the
relaxation PDE for DL kinetics.
?h
?t = ?
CsD?2g3
kBT ?div
parenleftbiggg
1
g3
?h
|?h| +|?h|?h
parenrightbigg
. (9.16)
167
Using (9.16) together with?h|L0(t) = 0,we have the third boundary condition,
?hf = ?CsD?2g3
kBT ?div
parenleftbiggg
1
g3
?h
|?h| +|?h|?h
parenrightbiggvextendsinglevextendsingle
vextendsinglevextendsingle
L0(t)
. (9.17)
We now elaborate on the boundary conditions stemming from continuity of
the chemical potential ? and the vector field N. These quantities have physical
meaning outside the facet, where ?h negationslash= 0 and the step positions satisfy the usual
assumptions conducive to a macroscale limit. Following Spohn [109], to determine
how N and ? should be defined on the facet, we reverse the calculations that lead
to surface flux outside the facet. In this vein, we start with the mass conservation
law, ?hf = ??divJf, assuming that ?hf is a known function of t. Extending the
constitutive relation (9.10) to the facet, we want to solve
Jf = ?CsDk
BT
??f
for ?f. Due to the scalar proportionality constant in DL kinetics, the solution for
?f can be given as a line integral of Jf over a path contained in the facet:
?f(x,y,t) = ?0 ? kBTC
sD
integraldisplay (x,y)
(x0,y0)
Jf ?dx. (9.18)
Then, continuity of ? implies ?f(rf(t),t) = ?(rf(t),t), or
?0 ? kBTC
sD
integraldisplay (x,y)
(x0,y0)
Jf ?dx = ??g3 div
bracketleftbiggg
1
g3
?h
|?h| +|?h|?h
bracketrightbigg
, (x,y) ?L0(t). (9.19)
Outside the facet we have ? = ?g3 divN, so the natural extension of N to the
facet is a vector field Nf satisfying
CsD?2g3
kBT ?divNf =
?hf. (9.20)
168
Continuity of N at the facet edge gives us Dirichlet boundary conditions to
complement the third-order PDE (9.20):
Nf = ?
bracketleftbiggg
1
g3
?h
|?h| +|?h|?h
bracketrightbigg
, (x,y) ?L0(t). (9.21)
Conditions (9.9), (9.13), (9.14), (9.19), (9.20), and (9.21) for J, ?, N, and h
follow naturally from the treatment of macroscopic height evolution in terms of the
variational framework. With this perspective, the smoothing of the height profile is
described by a path of steepest descent for the energy with respect to the H?1 inner
product. At the risk of redundancy, we emphasize that conditions (9.19), (9.20), and
(9.21), for continuity of ? and N at the facet edge, rely on a nonphysical chemical
potential defined on the facet.4 Implementation of the weak formulation for g1 negationslash= 0
must therefore be interpreted cautiously, since the boundary conditions it enforces
are not entirely consistent with the discrete character of processes occurring on a
facet [37].
9.4 Scaling of the boundary layer width
While studying numerical simulations of relaxing axisymmetric steps, Mar-
getis, Aziz and Stone [64] noticed a rapid variation of the slope in the vicinity of
the top facet (in contrast to the slowly varying slope farther away, in the bulk of
the step train). This observation prompted them to investigate the possibility of a
boundary layer near the facet edge. Intuitively, one might argue for the existence of
a boundary layer by noting that a slowly varying slope profile outside the facet must
4This inconsistency has been reported previously by Israeli and Kandel [37].
169
still connect continuously with the slope |?h| = 0 at the facet edge, for any solution
that respects the boundary conditions given above. In the boundary layer, the step
interaction (g3) term plays the role of a perturbation, influencing the slope profile
through the balance among derivatives of different orders. This local analysis uses
only the PDE for h and continuity of slope, not the other boundary conditions at the
facet edge. As a result, the scaling we obtain is not altered if we replace ?-continuity
by a more physical, microscale condition. To quantify the perturbation effect of the
g3 term, Margetis et al. [64] introduced a small parameter
epsilon1 = g3g
1
lessmuch 1. (9.22)
The slope profile F := ?h? e? that solves the boundary value problem is then a
function of space, time and epsilon1. The PDE for F is found by computing the directional
derivative of (9.16) along the local normal:
?F
?t =
CsD?2g1
kBT ?
parenleftbig?div?bracketleftbige
? +epsilon1F2e?
bracketrightbigparenrightbig?e
?. (9.23)
Assuming a long-time similarity solution, which depends separately on t and
the rescaled local coordinate, Margetis et al. found that the boundary layer width
? scales as epsilon11/3, independently of the axisymmetric initial conditions.
In this section we expand the details of this calculation to accommodate ge-
ometries in 2+1 dimensions. Allowing the facet to evolve in time, we assume that
a boundary layer of (possibly time-dependent) width ? moves with the facet edge.
Within this boundary layer, the slope profile is assumed to have an approximate
factorization, which then leads to an ODE in the ?fast? spatial variable. The re-
170
sults we obtain for the boundary layer width take the form of inspired conjectures
or speculations, awaiting verification by experiments and rigorous analysis.
The fully 2-dimensional geometry is handled using local analysis, which sepa-
rates the spatial coordinates into slow (?) and fast (?). The terminology reflects the
separation of scales (short distances along the step normal direction, larger distances
along the step tangential direction) over which comparable variations in the slope
profile are observed. Suppose that the rapid variation of the slope F, from 0 at the
facet edge ? = w(t) to its value where step interactions are less important, takes
place within a boundary layer of width ?(t). We then define a rescaled transverse
coordinate ? according to
? = ??w(t)?(t) , (9.24)
so that?remainsO(1) inside the boundary layer. We express the spatial dependence
of the slope profile in terms of this rescaled coordinate:
F(?,?,t;epsilon1) = F(?,?,t;epsilon1) = F
parenleftbigg??w(t)
?(t) ,?,t;epsilon1
parenrightbigg
. (9.25)
By substituting F for F in the PDE, we explicitly account for the rapid variation
of the slope within the boundary layer. The time derivative ?tF is replaced by
?tF = ?
parenleftBigg
???
? +
?w
?
parenrightBigg
??F +?tF, (9.26)
while the normal derivative is replaced by
??F = ??F ? ???? = 1???F. (9.27)
Motivated by analogy with observations of the step density from axisymmetric
step simulations, we make the conjecture of a long-time solution that obeys the
171
following local ansatz to leading-order in epsilon1: 5
F(?,?,t;epsilon1) ?a0(?,t)?f0(?;epsilon1), |??w|<O(?). (9.28)
Outside the boundary layer (? greatermuch 1), we expect that the slope becomes independent
of the distance from the facet, i.e., f0(?) ? 1 and df0/d? ? 0 as ? ??.
We compute the required derivatives of F and then substitute into the PDE
(9.23). Thus, we have
?tF ?
parenleftBigg
???
? +
?w
?
parenrightBigg
??F = ?B?parenleftbig?divbracketleftbige? +epsilon1F2e?bracketrightbigparenrightbig?e?, (9.29)
where B = CsD?2g1/(kBT).
If we neglect ? derivatives in accordance with slow-fast variable separation,
then the PDE becomes
?tF ?
parenleftBigg
???
? +
?w
?
parenrightBigg
??F = ? Bepsilon1?
??4
?? 1?
???
??
parenleftbigg?2
?
????
1
??????[(epsilon1
?1 +F2)??]
parenrightbigg
. (9.30)
There are two small parameters in (9.30): the ratio epsilon1 of step interactions to
step line tension, and the boundary layer width ?, which is assumed to scale as a
power of epsilon1. To decide systematically which terms dominate the PDE for small epsilon1,
we multiply (9.30) by ?4/(Bepsilon1) and expand the nested derivatives to retain only the
highest derivative of F2. This calculation yields
?3 ?w
Bepsilon1??F ?
1
?4??
4
?F
? = O
parenleftBigg
?4Ft
Bepsilon1 ,
?3 ??
Bepsilon1
parenrightBigg
, (9.31)
where the terms on the right-hand side are shown to be negligible [64].
5By ? we mean asymptotic equivalence as epsilon1 ? 0, i.e., F ? g ? limepsilon1?0 F?g
g = 0.
172
Now the separation ansatz (9.28) indicates that f0 satisfies an ordinary differ-
ential equation, whose coefficients are time-independent. We substitute (9.28) into
(9.31) to obtain
?w?3
Bepsilon1a0??f0 ?
1
?4??
4
?(f
2
0) = O
parenleftBigg
?4Ft
Bepsilon1 ,
?3 ??
Bepsilon1
parenrightBigg
(9.32)
Considering the left-hand side of 9.32, the requirement of time-independent coeffi-
cients for the ODE satisfied by f0 implies
? = O(epsilon11/3). (9.33)
The same local analysis can be applied in the case of a fixed drift velocity
u, the macroscale effect of an electric field (see Chapter 8). Again, the dominant
balance of terms as epsilon1 ? 0 would involve only the highest derivatives of f0 with
respect to ?. In particular, the drift term could be made negligible in comparison
to the other terms by taking ? ?w small enough. This asymptotic treatment is
complementary to the analysis of the next section, in which we discuss the local
behavior of the slope itself when a fixed drift velocity is present.
We emphasize that the power-law scaling of the boundary layer width?(t) with
epsilon1 arises only from the form of the PDE and the condition of a long-time similarity
solution satisfying (9.28). The existence of a solution with this particular form is
not easily established for an arbitrary facet shape.
Further studies of step-flow models in 2+1 dimensions should be pursued to
support the scaling ansatz assumed above. We would then have graphical confirma-
tion that even a weak step-step interaction strength can lead to a rapid variation of
step density in the neighborhood of a facet edge. Far away from the facet, line ten-
173
sion is expected to dominate so that the step density varies much more slowly. These
remarks generalize the observations found for axisymmetric step flow simulations,
as reported by [64].
9.5 Effect of an electric field on the slope profile near a facet
Another question that we might treat using local analysis concerns the appli-
cation of a direct current through a faceted crystal shape. The destabilizing effect
of an electric field on a uniform step train, producing either step bunching or step
meandering, is well-known [15, 28, 125]. For a faceted material, even in the sim-
ple geometry of straight steps, the effect of an electric field has received far less
attention. We conclude this chapter with a brief discussion of this effect, bringing
together the local analysis of the previous section and the macroscale PDE with
a drift term from Chapter 8. Our derivation assumes diffusion-limited kinetics in
order to facilitate an approximate solution of the macroscale PDE, i.e., we take
q|?h|lessmuch 1 and replace 1 +q|?h| by 1 wherever possible.
9.5.1 Straight-step morphology
We consider the continuous graph y = h(x) with the facet {(x,y) ? R2|h =
const.} in x ? 0, while ?xh < 0 in x > 0; so the facet edge is at x = 0 as
shown in Figure 9.2. We seek to characterize stationary solutions of the macroscale
evolution equation. Setting ?th? 0 in the conservation law ?th+ ??xJ = 0, where
174
Figure 9.2: Cross-section of a facet and a descending train of straight steps.
J = ?[?x??D?1vkBT(1 +?/(kBT))] entails the ODE
?x??l?1v ? = l?1v kBT?J0 ? ?(x) = (?0+kBT?lvJ0)ex/lv ?(kBT?lvJ0) x> 0 ;
(9.34)
wherelv := D/v,J0 := J(0)and?0 := ?(0). Bysubstituting? = ??g3?x(|?xh|?xh),
g3 > 0, we obtain the relation
?g3(?xh)2 = (?kBT +lvJ0)x+lv(?0 +kBT ?lvJ0)parenleftbigex/lv ?1parenrightbig x> 0 ; ?g3 = ?g3 .
(9.35)
The above formula is simplified for 0 <xlessmuch|lv| (weak drift) and forxgreatermuch|lv| (strong
drift). Accordingly, we obtain the approximation
|?xh|? ?g?1/23
?
???
???
?
???
???
?
??
0x , xlessmuch|lv| ,
radicalbig
lv(?0 +kBT ?lvJ0)ex/(2lv) , xgreatermuchlv > 0 ,
radicalbig(?|l
v|J0 ?kBT)x , xgreatermuch?lv > 0 .
(9.36)
175
For xgreatermuch?lv > 0, a compatibility condition is
?DJ0 >kBT|v| . (9.37)
By (9.36), changes of the electric field magnitude and direction can cause a drastic
qualitative change in the slope behavior. In particular, a strong electric force Z?eE
in the step-down direction causes an increase of the slope, as steps tend to bunch.
(Ultimately, of course, m approaches O(?x) as x? 0.) This behavior is reversed by
a strong electric force in the step-up direction, which restores the familiar O(?x)
behavior.
9.5.2 Axisymmetric structure
The radial case provides a model for the interplay of step edge curvature
and electric field. Suppose the surface is axisymmetric, h = h(r), with the facet
{(r,h) ? R+ ?R|h = const.} in 0 ? r ? rf while ?rh < 0 for r > rf; so, the
facet edge is at r = rf. Consider a constant drift velocity v. The ODE for ?(r) is
?r??l?1v ? = l?1v kBT ?rfJ0/r, with solution
? = (?0 +kBT)e(r?rf)/lv ?kBT ?rfJ0
integraldisplay r
rf
e(r?rprime)/lv
rprime dr
prime r>rf , (9.38)
where?0 := ?(rf),J0 := J(rf). Takingintoaccountthat??1? = g1/r+g3r?1?r[r(?rh)2],
by direct integration we have
?g3(?rh)2 ? [lvrfJ0 (1?lv/rf)? ?g1 ?rfkBT](r?rf) + (kBT +?0)lvbracketleftbig(r?lv)e(r?rf)/lv
?rf +lvbracketrightbig?lv(rfJ0)(r?lv)
integraldisplay r
rf
e(r?rprime)/lv
rprime dr
prime , 0 < r?rf
rf lessmuch 1 ;
(9.39)
176
?gl := ?gl (l = 1, 3). Here, we consider distances r?rf from the facet boundary
that are small compared to the facet radius of curvature. By analogy with Section
9.5.1, we simplify the formula for ?rh by imposing weak or strong drift:
|?rh|? ?g?1/23
?
???
???
?
???
???
?
radicalBig
(?0 ? ?g1/rf)(r?rf) , 0 <r?rf lessmuch|lv| ,
[lv(?0 +kBT ?lvJ0)]1/2e(r?rf)/(2lv) , r?rf greatermuchlv > 0 ,
radicalbig(|l
v||J0|?kBT ? ?g1/rf)(r?rf) , r?rf greatermuch?lv > 0 ,
(9.40)
The approximation for r?rf greatermuch?lv > 0 is compatible with the condition
?DJ0 > (kBT + ?g1/rf)|v| . (9.41)
Approximations analogous to the radial case can be worked out in 2D by
invoking the separation of local variables into fast (?) and slow (?). Variations
with respect to ? are dominant in the vicinity of the facet, which allows us to find
stationary solutions by solving an ODE in ?. This approach provides formulas that
directly generalize the radial case and is not further discussed here.
9.6 Conclusions
In this chapter, we considered the inclusion of facets in a weak solution of
the macroscale evolution equation. A weak solution with one or more facets can be
interpreted as a strong solution of the PDE informed by natural boundary conditions
at the moving facet edge. These natural boundary conditions include continuity of
slope, continuity of flux, and continuity of chemical potential across the facet edge.
To satisfy the boundary condition that the positive surface slope |?h| ap-
proaches zero at the facet edge, while varying more slowly away from the facet, we
177
hypothesized the existence of a boundary layer near the edge of the facet. Under
the assumption that the slope profile in this boundary layer satisfies a separability
ansatz, we derived a scaling of the boundary layer width in terms of the ratio g3/g1
of step interactions to step line tension.
Finally, we discussed the effect of an electric field on stationary solutions of
the evolution equation in 1D step geometries. This approach yields qualitatively
different slope profiles near the facet edge, depending on the strength and direction
of the external electric field.
178
Chapter 10
Numerical results1
In this chapter, we illustrate numerically the rich variety of relaxation behav-
iors corresponding to solutions of the macroscale PDEs derived above. Of particular
interest are the simulations that demonstrate a morphological transition, from a 2-
dimensional profile at the start to an essentially 1-dimensional profile at finite times.
Factors affecting the onset time of this transition include (i) the material parameter
q = D/ka, (ii) the aspect ratio ? = ?x/?y of the initial sinusoidal profile, where
?x and ?y are the wavelengths in x and y, respectively, and (iii) the presence of an
external electric field. By varying these three factors independently, we tentatively
conclude that transition phenomena can be explained as the result of longitudinal
fluxes comparable in size to the transverse fluxes. We alert the reader, however, that
we still lack a complete understanding of the reason for this behavior. Also, some
of the results we obtain might not apply to more general periodic profiles. Initial
data with additional harmonics superimposed on the fundamental sinusoidal profile
might exhibit nonlinear mode coupling effects, which could destroy the separable
behavior reported below.
1Material in this chapter appeared previously in Bonito, Nochetto, Quah and Margetis 2009,
Phys. Rev. E 79, 050601 (R), and relies heavily on the finite element algorithms and codes gra-
ciously contributed by Andrea Bonito and Ricardo Nochetto.
179
The height evolution is governed by the PDE
?h
?t = ?Cs div
braceleftbigg
M?
bracketleftbigg
??? kBTD v
parenleftbigg
1 + ?k
BT
parenrightbiggbracketrightbiggbracerightbigg
,
? = ??div
braceleftbigg
g1 ?h|?h| +g3|?h|?h
bracerightbigg
,
which we nondimensionalize prior to numerical implementation. This nondimension-
alization (carried out in Appendix B) serves two main purposes: (i) to focus on the
structure of the equations without the distraction of dimensional coefficients, and
(ii) to isolate the relevant combinations of material parameters, in particular those
that have significant temperature dependence. The second purpose furnishes five
combinations of parameters to describe the experimental setting. These parameter
combinations are:
? q = Dka, which compares diffusion and attachment/detachment rates;
? g1/g3, which compares the step line tension to step-step interactions;
? ? = ?x/?y, the aspect ratio of the initial data;
? h0/?x, the amplitude of the initial data;
? u = v?x/D, the nondimensional drift velocity (in the presence of an electric
field).
The last three parameters can in principle be adjusted independently for any given
material and temperature. On physical grounds we expect some restrictions on the
physically attainable profile dimensions and drift velocity magnitude. For example,
a corrugated surface with aspect ratio ?x/?y = 1/10000 might be indistinguishable
180
from a one-dimensional profile, if imperfections in the patterning process ended up
obscuring the long-wavelength period. Also, a sufficiently strong electric field will
heat the sample, thereby changing the temperature unless the material is kept in
contact with a suitably cold reservoir. The latter practical consideration is not
accounted for in our simulations.
The first two parameters depend strongly on the temperature and the choice of
material. The parameterq is the ratio of two thermally-activated quantities with dif-
ferent energetic barriers. If the energy barrier for adatom diffusion is larger (smaller)
than that of attachment/detachment, then q is expected to increase (decrease) as
temperature increases. The inequality between these energetic barriers need not
be consistent from one material to another. In principle we can have one material
where q increases with temperature, and another material where q decreases with
temperature. Even two different orientations of the same material might exhibit
contrasting temperature dependence of q. A good starting point for enumerating
these possibilities is the table of energy barriers in [43] and the references cited
therein.
The ratio g1/g3 of line tension to step interactions is also sensitive to the
choice of material and temperature. Roughly speaking, the line tension decreases as
temperature increases, reaching g1 = 0 precisely at the roughening transition tem-
perature. A more precise description of the temperature dependence of g1 is given
by invoking the Ising model [73]. The strength of repulsive step interactions, quan-
tified by g3, has two different contributions. Entropic repulsions become stronger
as temperature increases, because steps have a greater tendency to wander and
181
collision distance is shorter. The strength of elastic dipole interactions, which are
mediated through the crystal, varies with temperature mainly through the temper-
ature dependence of the Young modulus for the material. Entropic repulsions tend
to dominate at higher temperatures. As T approaches TR from below, we therefore
expect g3 to increase. A formula for the step-step interaction coefficient g3 is given
in [43].
We note that a meaningful connection with experiments requires physically
attainable values for the material parameters. In particular, crystal surfaces well
below the roughening transition temperature typically fail to satisfy g1/g3 lessmuch 1. To
obtain a numerically reliable result, and to circumvent the difficulty of enforcing
boundary conditions at the moving facet edge, we were forced to consider only
g1/g3 = 0 or g1/g3 = O(10?8) in our simulations. Note that small (O(10?8)) values
of g1/g3 were used in the numerical simulations in order to check the continuity of
the observed phenomena withg1, as discussed below. Our preliminary investigations
with g1/g3 lessmuch 1 merit careful scrutiny, in order to avoid erroneous conclusions about
the relaxation of real materials.
10.1 Review of the finite element method
With these limitations in mind, we now describe our numerical method, start-
ing from the introductory material of Chapter 2. After nondimensionalization (see
182
Appendix B), the PDE we want to solve is
?th = divM???, (10.1)
? = ?E?h (10.2)
for some appropriate mobility M and surface energy E. The spatial domain of this
PDE is taken to be a rectangular region B = [0,1]?[0,1], and h,? at each time t
are required to be spatially periodic. Given the initial data h(?,t) at t = 0, we want
to determine the height profile at t> 0.
We exploit the variational structure of (10.1),(10.2) to simulate surface relax-
ation by using the FEM in space [9] and finite differences in time. The fourth-order
PDE for h is conveniently written as two second-order equations: one for the height
h, using mass conservation and the constitutive law for J, and another for the chem-
ical potential ?, which is essentially the variation with respect to h of the energy
functional E[h]. We apply a semi-implicit Euler scheme [77] to express (10.1),(10.2)
as a system in the updated variables (hn+1,?n+1) where hn ?h(?,n?n) and ?n is the
(adaptive) time step. The mobility M and the energy E[h] are evaluated by use of
(hn,?n) to ensure linearity of the finite difference equations with (hn+1,?n+1).
The equations for hn+1 and ?n+1 are recast to their ?weak form? via multi-
plication by a periodic test function ? and integration by parts over B. Restricting
hn+1,?n+1, and ? to the finite-dimensional space VT of continuous piecewise linear
functions associated with the triangulation T , we obtain the FEM equations for
(hn+1,?n+1), which are required to hold for all ??VT :
183
integraldisplay
B
parenleftbiggg
1
g3
?hn+1
|?hn| +|?h
n|?hn+1
parenrightbigg
??? =
integraldisplay
B
?n+1?, (10.3)
integraldisplay
B
bracketleftbighn+1?+?nM?(?hn)??n+1 ???bracketrightbig=integraldisplay
B
hn?. (10.4)
One major concern is the singularity of the g1 term and the similarity trans-
formation Mx,y = SM?,?S?1, Eqs. (5.24)?(5.26), when |?h| = 0. Computing the
fraction ?hn+1/|?hn| which multiplies g1/g3 is numerically unstable when |?hn| is
close to zero. The elements of the change-of-basis matrix S given by (5.26) have
|?h| in the denominator; consequently, the computation of S near ?h = 0 is also
numerically unstable. We eliminate the singularity in these terms by adding a small
regularization parameter so that the problematic denominators remain nonzero as
?h? 0; see Table 10.1 for possible regularization schemes. Mathematically it can
be argued that the mobility becomes a scalar (identity tensor) wherever ?h = 0.
Ideally, this limit should be respected by the regularization we choose, in order
to avoid any spurious results arising from a mathematically inconsistent mobility.
However, we do note that in a particular instance where one of our regularization
schemes did not respect this limit, the numerical results were practically the same,
as we discuss below.
10.2 Software and computers
Our implementation of (10.3),(10.4) uses the FreeFem++ program,
http://www.freefem.org/ff++/, developed by Olivier Pironneau, Fr?ed?eric Hecht, and
Jacques Morice. The code specific to our PDE (10.3),(10.4) was written by Andrea
184
Bonito following the algorithm suggested by Ricardo Nochetto. Simulations were
performed on a Dell desktop PC with 2.0GHz Intel Core 2 processor, 1GB of RAM,
running Fedora Core 7 and Linux kernel 2.6.21.
The following auxiliary libraries and programs were also used:
? The UMFPACK library, http://www.cise.ufl.edu/research/sparse/umfpack, was
usedfordirectsolvers. UMFPACKimplementstheUnsymmetricMultiFrontal
Method to solve unsymmetric sparse linear systems.
? The utility Gnuplot http://www.gnuplot.info/ was used to plot graphs. Gnu-
plot offers a command-line interface for visualizing data and functions in 2D
and 3D plots.
? The MATLAB software suite, http://www.mathworks.com/, was used for post-
processing the FreeFem++ output files, when Gnuplot did not offer the needed
capabilities for certain plots.
10.3 Zero line tension (?ideal? case)
The macroscale equation we simulate for the height evolution has been shown
in Chapter 5 to stem from the motion of steps, which exist as stable objects only
below the surface roughening temperature. Temperatures below roughening dictate
g1 > 0; otherwise there would be no free energy cost to create surface defects, and
these would appear and disappear spontaneously. However, once we have found a
PDE consistent with step flow, we are free to ask how its solutions behave in certain
limiting cases. As a test case, we take g1 = 0 to circumvent the difficulty of treating
185
height evolution as a free boundary problem; see Chapter 9 for a discussion of the
open problems in this direction. We also alert the reader that the case g1 negationslash= 0 with
sufficiently high g1/g3 leads to an ill-conditioned linear system for our numerical
method [6], which makes the results of a simulation unreliable. Thus far, we have
not been able to overcome the numerical hurdles of larger values for g1/g3.
The limit g1 ? 0 is a starting point, with the same capacity to illustrate
the contributions of our model as the plausibly more physical choice with g1/g3 =
O(1). In this limiting case we can see the effect of a tensor mobility and an electric
field. One major concern addressed within this perspective is whether observables
of the macroscopic theory (e.g., height, surface energy) are continuous solutions as
functions of g1. In particular, we want to check whether the numerically observed
transition for 2D to almost 1D profiles persists for nonzero (even small) g1/g3. In
other words, we want to make sure that no ?spurious? singular behavior is developed
by the macroscale theory as g1 ? 0. We begin this section by contrasting our tensor
mobility predictions with the scalar mobility simulations of [108], both with g1 = 0.
By analogy with 1D macroscale models, these simulations set
M = Dk
BT
1
1 +q|?h|, (10.5)
i.e., a scalar mobility which does not distinguish between step-normal and step-
parallel fluxes. Our purpose in conducting these tests is to see whether our finite
element code generates height profiles consistent with the results of previous varia-
tional methods, namely the Fourier series expansions within a Galerkin scheme used
by Shenoy et al [107, 108].
186
10.3.1 Simulations with zero E-field
Our first test of the finite element implementation of the relaxation PDE
used a scalar mobility, in order to validate our code against previously published
results [107, 108]. In [107], Shenoy and co-authors outline their method of Galerkin
expansions of the surface height profile. They start from a variational formulation of
the height evolution equation and derive a system of ODEs for the coefficients in the
Galerkin expansion of h. They simulate the relaxation with different values for the
material parameters q and g1/g3, finding approximately separable solutions which
decay either exponentially or algebraically in time when the initial height profile is
a perfect sinusoid [107, 108].
We simulated a decaying bidirectional modulation, with different wavelengths
in x and y. The height profiles from this simulation are plotted in Figure 10.1, with
time increasing to the right.
187
Figure 10.1: Height profiles computed using scalar mobility M = Dk
BT
1
1+q|?h|, for
initial data h = h0 cos(k1x)cos(k2y), h0/?x = 0.03, k2/k1 = 11/24, and material
parameters q = 104, g1/g3 = 0.
The biperiodicity of the initial sinusoidal profile evidently persists throughout
the relaxation, suggesting that the spatial dependence can be factored from the
height in the approximate sense of (5.47). Assuming this factorization holds, a plot
of the height peak versus time would reveal which decay law is exhibited by the
solution. We plot in Figure 10.2 the evolution of energy and height peak using data
from the same numerical experiment as Figure 10.1.
With logarithmic axes for height and energy, the two decay curves in Fig-
ure 10.2 are well-approximated by straight lines for most of the relaxation. To the
eye, this trend is enough to classify the decay law as exponential. This finding is
qualitatively consistent with [107, 108]. We are inclined to interpret their agreement
with our plots as good evidence that Bonito?s code [7] is reliable for the parameters
under consideration. However, we have not carried out comparisons of our finite
element-based numerical results with Shenoy?s Fourier series-based numerics. Thus,
our discussion here is restricted to qualitative (rather than precise quantitative)
features of simulations.
188
Figure 10.2: Decay of height peak and energy computed using scalar mobility, for
initial data h = h0 cos(k1x)cos(k2y), h0/?x = 0.03, k2/k1 = 11/24, and material
parameters q = 104, g1/g3 = 0. t? and E? are measured in units where kBT?5xCsDg3?2h0
and g3h30?y?2
x
are taken to be 1.
189
Table 10.1: Different regularization schemes for the tensor mobility. The first
(?na??ve?) scheme was originally coded by Bonito. To respect the limit of scalar
mobility as |?h|? 0, we discussed and implemented scheme 2. Nochetto suggested
scheme 3, which takes a convex combination of the identity with the unregularized
M.
1. M? = 1|?h|2 +?2
?
??
?
h2x
1+q|?h| +h
2
y ?
q|?h|hxhy
1+q|?h|
?q|?h|hxhy1+q|?h| h2y1+q|?h| +h2x
?
??
?
2. M? =
?
??
?
(hx+?)2+(1+q|?h|)(hy+?)2
(1+q|?h|)(|?h|+?2?)2
?q|?h|(hx+?)(hy+?)
(1+q|?h|)(|?h|+?2?)2
?q|?h|(hx+?)(hy+?)
(1+q|?h|)(|?h|+?2?)2
(hy+?)2+(1+q|?h|)(hx+?)2
(1+q|?h|)(|?h|+?2?)2
?
??
?
3. M? = (1??2)I +?2M, ? = min(?,|?h|)/?.
We now present the results of implementing a tensor mobility in the finite
element code. Because the tensor mobility distinguishes between step-normal and
step-parallel fluxes, its representation in a fixed coordinate system relies on the
change-of-basis matrix S given by (5.26), which is singular at ?h = 0 as discussed
above. We ran the simulations with different choices of regularization for the tensor
mobility (see Table 10.1) , finding good agreement even between regularizations that
gave different limits for M as ?h ? 0 [8]. This robustness with respect to regu-
larization suggests that the novel phenomena reported below are insensitive to the
exact behavior of fluxes around peaks and valleys. We expand on this interpretation
after showing a few more plots.
190
Figure 10.3: Height profiles computed using tensor mobility, starting from the initial
data h = h0 cos(k1x)cos(k2y), h0/?x = 0.03, k2/k1 = 11/24, and material parame-
ters q = 104, g1/g3 = 0.
Figure 10.3 shows the computed height profiles at three selected times during
a relaxation with tensor mobility. The initial data is identical to that of Figure 10.1,
biperiodic with different wavelengths in x and y. At later times (middle and right
surface plots), the x-dependence becomes less pronounced. Even before the height
peak decays to 1% of its initial amplitude, the computed height profile looks essen-
tially like a 1-dimensional modulation.
Because the spatial dependence of the height profile does not remain the
same throughout the simulation, we suspect that a different separation ansatz
h?H(x,y)A(t) is obeyed for different time intervals, e.g.,
h?H1(x,y)A(t), 0 <t<t1
h?H2(x,y)A(t), t1 <t<t2
...
We find it instructive to show the decay of height peak and energy. These curves,
we hope, would give us a clue about what time scales are relevant for observing the
transition.
191
In Figure 10.4 we compare the predictions for decay of peak height using scalar
and tensor mobilities. The simulation with tensor mobility appears to exhibit an
exponential decay of the height peak until we reach t? ? 0.1, after which the height
peak drops sharply into another decay regime. This sharp drop on the graph of
height peak vs. time coincides with the onset of a transition, as confirmed visually
by plotting the height profiles for t? before and after t? = 0.1.
To give the reader a feel for what physical time might correspond to t? = 0.1,
we compute the magnitude of the time unit kBT?5xCsDg3?2h0 for the hypothetical material
that has an extremely small value of g1/g3, corresponding to the simulation above.
To this end we seth0/?x = 0.03 as dictated by the simulation, and chooseD/(ka) =
q = 104 and g1/g3 = 0. The other parameters we take from Si(001) at 700 K, which
Erlebacher et al. used in the first demonstration of nonclassical smoothing [20]. We
originally considered Si(001) because of its largeq value, but the non-negligible ratio
g1/g3 makes it unsuitable for a simulation using finite elements. For now its main
purpose is to furnish plausible values for the undetermined parameters in our time
unit.
The possibility of constructing the ?metamaterial? which agrees with all of
the parameters of our simulation is not addressed here. One promising direction is
to look for materials that exhibit a preroughening transition, such as GaAs, in which
g1/g3 ? 0 before the BCF theory becomes inapplicable [85]. Based on the meager
literature, it appears that kinetic parameters such as ku,kd, and D are difficult to
determine for GaAs, due to the high deposition flux needed to keep a sample under
observation [85]. Accordingly, we fall back on the more fully-documented Si material
192
Figure 10.4: Decay of height peak, as computed using scalar and tensor mobilities,
starting from initial data h = h0 cos(k1x)cos(k2y), h0/?x = 0.03, k2/k1 = 11/24,
with material parameters q = 104 and g1/g3 = 0. t? is measured in units where
kBT?5x
CsDg3?2h0 is taken to be 1.
193
in order to calculate a possibly reasonable time unit.
After substituting the values from [20, 39], we determine our time unit to be
kBT?5x
CsDg3?2h0 = 3?10
?8(?x/?A)4 seconds. The wavelength ?x of the surface feature is
still free to vary, which allows an appreciably large range of time scales that this one
simulation can address. In particular, if we could pattern a bidirectional corrugation
in our hypothetical material with 100 nm wavelength along the shorter direction,
then relaxation at 700 K would need to proceed for at least 45 minutes before
our predicted transition could occur. However, a wavelength of 100 nm might not
comprise enough steps for the macroscale limit to be justified. Biperiodic profiles
with larger period (e.g., 363 nm [20]) would need to be observed much longer: at
least 12 hours to see the transition computed by our simulation above.
If the height profile did indeed decay in a separable manner until about
t? = 0.1, then the surface energy E = integraltextBg1|?h| + g33 |?h|3 should also show an
exponential dependence on time. Since g1/g3 was taken to be zero for this simula-
tion, a globally approximately factorizable solution h(x,y,t) = H(x,y)A(t) would
entail E = A(t)3integraltextB g33 |?H|3, whose time dependence is exponential precisely when
A(t) is exponential. We plot in Figure 10.5 the decay of surface energy as predicted
by simulations with scalar and tensor mobilities. Note that this factorizability of E
is strictly speaking spoiled if both g1 and g3 are nonzero.
194
With E? plotted on a logarithmic axis in Figure 10.5, the curve corresponding
to either mobility appears as a straight line at least until t? = 0.1. The slope of
this line is approximately three times as large as the slope in Figure 10.4, which is
what we would expect from an exactly separable solution. However, after t? = 0.1,
the curve corresponding to tensor mobility deviates from a straight line, dropping
sharply into another decay law. The simulation described here does not continue
long enough to say anything definitive about this second decay law, owing to the
stopping criterion that terminated the simulation after an essentially 1D profile
had been reached [7]. Earlier simulations, without such a stopping criterion in
place, indicate that the 1D profile after the transition follows another separable,
exponential decay.
195
Figure 10.5: Decay of surface energy, as computed using scalar and tensor mobilities,
starting from initial data h = h0 cos(k1x)cos(k2y), h0/?x = 0.03, k2/k1 = 11/24,
with material parameters q = 104 and g1/g3 = 0. t? and E? are measured in units
where kBT?5xCsDg3?2h0 and g3h30?y?2
x
are taken to be 1.
196
We now show contour plots of the height profile at various stages of the relax-
ation, in order to elucidate a possible mechanism for the appearance of a transition.
Figure 10.6 shows level sets of height (top row) and streamlines for the vector adatom
flux (bottom row) at four different times during the transition. The streamlines are
defined by the property of being tangent to the vector adatom flux at each point
in B. Our use of streamlines is adopted from fluid mechanics, where they serve as
a visual aid in describing the velocity field of a fluid. To plot the streamlines at a
given time t, we first compute the height profile and macroscale chemical potential,
which yields by (8.49) the vector-valued flux J. We then solve Poisson?s equation
for?, i.e., ?? = ?xJy??yJx, so that J = (Jx,Jy) is orthogonal to ?? = (?x?,?y?).
Level sets of ? then correspond to streamlines.
In contrast to the case of scalar mobility, the adatom flux is not orthogonal
to the level sets, but instead has an appreciable component in the longitudinal
direction. This longitudinal component can be comparable in size to the transverse
component of flux, despite the slower variation of the chemical potential along the
former direction than the latter. Again, this effect can be traced back to the tensor
mobility, which for q greatermuch 1 inhibits the macroscale flux in the transverse direction
but not in the longitudinal direction. To illustrate the effect of smaller q (and
hence a lower energy barrier for uphill and downhill mass transport), we show in
Figure 10.7 the height level sets and the streamlines corresponding to q = 5000.
The other parameters from the previous simulation are kept unchanged.
197
Figure 10.6: Snapshots during the transition from the initial data h =
h0 cos(k1x)cos(k2y), h0/?x = 0.03, k2/k1 = 11/24 to an eventually 1D profile, taken
from [7]. The simulation uses tensor mobility and material parameters q = 104,
g1/g3 = 0. Top row: level sets of height. Bottom row: streamlines, which are locally
tangent to the vector adatom flux. Plot window is [0,?x/2]?[0,?y/2].
198
As q is increased, the transverse flux is diminished, and downhill mass trans-
port proceeds more slowly. The decay of height peak would then be expected to
take longer, as confirmed in Figure 10.8. Meanwhile, the longitudinal flux now has
a greater relative contribution, since adatoms flowing along level sets of height en-
counter the same energetic barrier as for smaller q. This flow of adatoms along the
step-parallel direction happens fast enough that steps with large perimeter have a
chance to straighten, eventually aligning themselves with the x-axis as more mass
flows downhill (uphill) from the peaks (valleys). We now have a possible explana-
tion for the robustness of a transition with respect to regularization of the mobility.
If the transition results from sizable contributions of the longitudinal adatom flux
on terraces of large perimeter, then the fluxes on smaller-perimeter terraces near
|?h| = 0 are of secondary importance, serving only to transport mass downhill (or
uphill) from peaks (or valleys). As long as the tensor mobility yields an adatom
flux at peaks of the surface profile that points approximately downhill (and flux at
valleys that points uphill), then the accurate calculation of flux at other points is
sufficient to induce a transition.
The error in the computed flux arising from the regularized mobility can be
quantified by comparing the regularization parameter ? with the computed denom-
inator |?h| to which it is added. If the mesh size is 10?2, then the height difference
between neighboring gridpoints, say |h(xi,yj)?h(xi+1,yj)|, must be much greater
than 10?2? in order to avoid an inaccurate flux calculation. In practice, this in-
equality is only violated very close to the peaks and valleys of a profile, or for very
flat profiles. We find that an ?outward-pointing? flux is respected by all of the
199
Figure 10.7: Snapshots during the transition from the initial data h =
h0 cos(k1x)cos(k2y), h0/?x = 0.03, k2/k1 = 11/24 to an eventually 1D profile. The
simulation uses tensor mobility and material parameters q = 5000, g1/g3 = 0. Top
row: level sets of height. Bottom row: streamlines, which are locally tangent to the
vector adatom flux. Plot window is [0,?x/2]?[0,?y/2].
200
Figure 10.8: Decay of the height peak happens more slowly as q is increased, due
to the increased attachment-detachment barrier that must be surmounted for mass
to be transported uphill or downhill.
201
regularization schemes we tried, and they all predicted a transition event. We infer
that the tensor mobility manifests a transition more through the fluxes on level sets
close to the mean height, than through the fluxes on terraces of shorter perimeters.
202
10.3.2 Simulations with nonzero E-field
We were fortunate to have finished calculating the macroscale limit of step flow
in the presence of an electric field, just when the numerical results of implementing
the tensor mobility were available. The presence of an electric field at the BCF
level, as we saw in Chapter 8, leads to a convective term in the macroscale adatom
flux, Eq. (8.49). The evolution equation is given by modifying (10.3),(10.4) with the
addition of a convective term in (10.3). We solve numerically the following system
within the space of piecewise linear functions on VT (cf. Section 2.2):
integraldisplay
B
parenleftbiggg
1
g3
?hn+1
|?hn| +|?h
n|?hn+1
parenrightbigg
??? =
integraldisplay
B
?n+1?, (10.6)
integraldisplay
B
bracketleftbighn+1?+?nM?(?hn)parenleftbig??n+1 + (1 +?n+1)uparenrightbig???bracketrightbig=integraldisplay
B
hn?; (10.7)
see Appendix B for a nondimensional interpretation of these equations. Again,
the relevant geometric ratios and parameter combinations are ?x/?y, h0/?x, u =
?xD?1v, q, and g1/g3.
By tuning the electric field (and hence the drift velocity u), we hoped to gain
some control over the relative contributions of transverse and longitudinal fluxes,
which were hypothesized as the driving mechanisms behind a transition. This con-
trol would allow us to inhibit or accelerate a transition, by reinforcing the fluxes
along a specified direction. We find that by choosing a drift velocity in the direction
of the longer wavelength, the time to observe a transition is drastically reduced.
In Figure 10.9 we plot the decay of the height peak and surface energy, com-
puted with Bonito?s finite element code using tensor mobility, with and without
an electric field. The exponential transformation of Chapter 8 is not implemented
203
Figure 10.9: Decay of the height peak and surface energy using tensor mobility,
with and without an electric field. The electric field is aligned with the y-axis and
appears to accelerate a transition.
204
in this code, due to unresolved issues with the validation. We see that the height
peak and surface energy follow an exponential decay in both cases, until the sudden
drop corresponding to a transition. The transition happens earlier in the presence
of an electric field, suggesting that the reinforced adatom fluxes are a plausible ex-
planation for this phenomenon. The nondimensional time of the transition is now
t? ? 0.05 in the presence of an electric field, corresponding to about half of the
previously indicated time for the hypothetical material above.
For completeness we also show in Figure 10.10 the contour plots and stream-
lines computed with the electric field. Again we see that fluxes are not normal to
level sets of height, so that large-perimeter steps have a chance to elongate before
downhill or uphill mass transport causes them to collapse. In addition, the pres-
ence of an electric field seems to direct the fluxes into a pattern more conducive to
straightening the large-perimeter steps than the u = 0 simulation presented above.
The reinforcement of adatom fluxes in the y-direction eventually assists the uphill
and downhill mass transport as the profile tends to become 1-dimensional.
205
Figure 10.10: Snapshots during the transition from the initial data h =
h0 cos(k1x)cos(k2y), h0/?x = 0.03, k2/k1 = 11/24 to an eventually 1D profile,
calculated using tensor mobility, drift velocity (u1,u2) = (0,10?3), and material
parameters q = 104, g1/g3 = 0. Top row: level sets of height. Bottom row:
streamlines, which are locally tangent to the vector adatom flux. Plot window
is [0,?x/2]?[0,?y/2].
206
10.4 Continuous dependence on line tension of the 2D to 1D transi-
tion
Now that we have a basic idea of the macroscale surface evolution in the
limiting case g1 = 0, we address the question of whether our observations carry over
to the case of nonzero g1. Another possibility is that some of our observations arise
only as a singular limit for g1 ? 0, while others do persist and vary continuously
with g1.
The disappearance of facets at g1 = 0 has been interpreted as a singular
limit, prompting some materials scientists to view g1 as an O(1) parameter that
determines whether the Mullins above-roughening theory or the BCF step-based
framework should be adopted [107]. Our perspective is more inclusive, treating the
PDE derived from step flow as a tool with predictive value over a large range of g1.
The reliability of these predictions, however, depends in large part on the accuracy
of our numerics.
In our approach, the g1 = 0 case serves as a testbed from which to draw
tentative conclusions about g1 > 0, where simulations are currently incomplete. In
addition, the PDE simulated here is the limit of step motion. Thus, the limitg1 ? 0
of the macroscopic theory is essentially the limg1?0 lima?0 of the step flow equations.
In principle, this limit is expected to be different from lima?0 limg1?0, in which the
BCF theory is questionable or not valid.
While the results forg1/g3 = 0 were still being collected and discussed, we also
began preliminary trials with g1/g3 nonzero. Returning to the idea of partitioning
207
the parameter space into regions based on their eventual evolution, we wondered
about the shape of the separatrix between transition-supporting and transition-
suppressing regions. One cross-section that we considered trying to plot was the
restriction of this separatrix by keeping fixed the aspect ratio ?x/?y, the initial am-
plitude h0/?x, and the drift velocity u. The resulting cross-section would be a curve
in the q?g1/g3 plane, which we hypothesized to look something like Figure 10.11.
We conjectured a threshold q that increases with g1/g3, because the facet
size grows more quickly for larger g1/g3, which would increase the contribution
of transverse fluxes in the steeply-sloped regions between facets. The longitudinal
fluxes can still be given time to effect a transition, if a larger value ofq is used to slow
down the uphill and downhill mass transport. The threshold q has been determined
only for the smallest values of g1/g3 in Figure 10.11. The trend of threshold q for
largerg1/g3 is merely speculated. As it turned out, the calculation of these threshold
q values for larger g1/g3 was never completed, due to the enormous computational
time our algorithm seemed to require. After further investigation, we attributed
the uncharacteristically slow convergence of the method to an ill-conditioned linear
system [6]. Work to address this ill-conditioning and to extend confidently our finite
element implementation in the case of large g1/g3 is still an open problem [6, 77].
The more modest goal of considering only a single nonzero value of g1/g3,
small enough for the numerical method to yield reliable results, led us to take
(g1/g3)(?x/h0)2 = 10?4. We simulated the relaxation using both (i) scalar and
tensor mobilities, (ii) zero and nonzero drift velocities. Again the use of a tensor
mobility produced a transition, which is marked by the abrupt change from pure
208
Figure 10.11: Speculation for the separatrix between transition-supporting and
transition-suppressing regions, with fixed aspect ratio ? = 2/3, initial amplitude
h0/?x = 0.03, and drift velocity u = 0. The upper (lower) curve would connect
the computed (g1/g3,q) pairs for which a transition does (does not) occur. The
uncertainty (vertical distance between the two curves) of the separatrix ordinates
can in principle be decreased using bisection. In practice, a good preconditioner is
needed to stabilize the numerics when g1/g3 is large [6].
209
exponential behavior in the height peak and energy curves; see Figure 10.12. Note
also that the presence of an electric field accelerates the transition. Evidently, the
two major observations we made for zero g1 are also true for this particular choice
of nonzero g1. It is tempting to conjecture that the transition and its acceleration
by an electric field are not singular limits as g1/g3 ? 0.
10.5 Conclusions and open issues
The two major contributions of our modeling in this thesis are (i) a tensor
mobility, which distinguishes between step-parallel and step-normal components of
the macroscale flux J, and (ii) an electric field, which leads to a convective term
in the constitutive relation for J. These contributions inspired and led directly
to the (hopefully) novel phenomena predicted here by finite element simulations
of the macroscale evolution equation. A tensor mobility, which for ADL kinetics
makes longitudinal fluxes predominant, leads to a transition from initially biperiodic
data to an eventually one-dimensional profile. The inclusion of an electric field
reinforces adatom fluxes along a specified direction, thereby causing the transition
to be observed earlier. These two effects appear to persist continuously as g1/g3 is
allowed to take nonzero values.
The connection between our simulations and real materials is still elusive. The
first reason is computational: at larger values of g1/g3 corresponding to real ma-
terials, our finite element implementation tries to solve an ill-conditioned matrix
problem. The second reason is physical: the boundary conditions enforced by ap-
210
Figure 10.12: Log plots for maximum height h?m = maxh/h0 (left axis) and sur-
face energy E? (right axis), taken from [7]. These simulations used tensor mo-
bility, material parameters (g1/g3)(?x/h0)2 = 10?4, q = 104, and initial data
h = h0 cos(k1x)cos(k2y), k2/k1 = 11/24, h0/?x = 0.03. A transition still occurs
in the presence of a facet.
211
plying the finite element method over the entire domain B are incompatible with the
boundary conditions stemming from the flow of steps [65]. Even if we had trustwor-
thy numerical results for larger values of g1/g3, the weak formulation itself would
be questionable.
We leave it for future numerical work to characterize how the predicted phe-
nomena vary with increasing g1/g3. Pending issues include (i) finding a good pre-
conditioner, so that the matrix problem given by our finite element equations is
well-posed; (ii) enforcing more realistic boundary conditions, by restricting the do-
main of the PDE and informing the solution of microscale effects at the facet edge;
and (iii) developing a numerical scheme for step flow in 2+1 dimensions, in order to
determine the discrete sequence of collapse times for the top step [37].
The most promising route for connecting our macroscale theory to experiments
is analytical, at least until the ill-conditioning problem is resolved and a hybrid
scheme is adopted to incorporate step collapse times into the weak formulation.
We also hope that ongoing research in metamaterials might lead to an engineered
material with the precise set of parameters that numerical constraints forced us
to use in our simulations. Lacking a suitable experimental setting that can be
simulated reliably by our numerics, we leave it for future research to make a more
specific connection with real materials. For now we can only report the noteworthy
numerical results for a hypothetical material, which appear precisely because of the
novel contributions of our macroscale model; namely, a tensor mobility and the
macroscale drift due to electromigration.
212
Chapter 11
Epilogue: conclusions and open questions
In this thesis we have studied the macroscale consequences of different physical
processes at the nanoscale, deriving PDEs to describe macroscopic surface relaxation
in the absence of material deposition. Our main result is a macroscale analog of
Fick?s law relating surface flux to the chemical potential. We revisit these results
by discussing correspondences between the terms of this relation and the underlying
nanoscale processes. For convenient reference, we give once again the macroscale
analog of Fick?s law in the most general form found so far.
J = ?CsM?
bracketleftbigg
???kBTD?1v
parenleftbigg
1 + ?k
BT
parenrightbiggbracketrightbigg
(11.1)
Energetic considerations yield the formula for the macroscale step chemical
potential, which remains invariant under changes of the nanoscale kinetic processes
considered in this thesis:
? = ??div
braceleftbigg
g1 ?h|?h| +g3|?h|?h
bracerightbigg
. (11.2)
The inclusion or suppression of the different possible nanoscale processes is
reflected in the macroscale equation (11.1) by a linear superposition of effects, or
a renormalization of parameters. As an example, consider the effect of step edge
diffusion. By allowing for adatoms to diffuse along the step edge, we saw in chapter
6 that the?? mobility element acquired an additional term proportional to the edge
213
Table 11.1: Macroscale consequences of several physical processes at the nanoscale.
An asterisk next to ?none? alerts the reader that the macroscale effects are deemed
negligible, being of higher order in the expansion parameter a.
nanoscale process effect on M effect on v
isotropic diffusion
tensor character dis-
tinguishes between
transverse and longi-
tudinal fluxes, with
diagonal form in
the local coordinate
representation
none
anisotropic diffusion
tensor form even in
the local coordinate
representation
none
step edge diffusion additional term inthe m
?? element
none
step transparency
kinetic parame-
ter q = 2D/(ka)
renormalized via
k mapsto?k+ 2p
none
electric field E none proportional to E
adatom desorption none? none?
diffusivity. Later, when studying the results of electromigration current, we saw that
the macroscale equation acquired a convective term featuring the drift velocity v.
These correspondences are summarized in Table 11.1.
Although the description of step flow below roughening can vary appreciably
depending on which nanoscale processes we include in the model, the macroscale
limit is much more robust, taking the form of a standard conservation equation
together with (11.1) and (11.2). Finding the same macroscale limit via homogeniza-
tion theory [70] serves as further confirmation of the coarse-graining results shown
here.
214
11.1 Summary of contributions
One macroscale PDE, with appropriate modification of the parameters, en-
compasses a wide range of step flow models within the BCF framework. We have
studied representative versions of the macroscale evolution equation both numeri-
cally and analytically. These complementary approaches address different regimes in
the parameter space, and different questions about the evolving surface morphology.
Numerically, we have simulated the macroscale PDE using a semi-implicit time
scheme and the finite element method in space. We observe a 2D?1D transition
under ADL kinetics, indicating that the tensor mobility distinguishes between step-
normal and step-parallel fluxes. This transition appears to be accelerated by the
application of an electric field in the direction of the longer initial wavelength. Our
understanding of the transition phenomena is incomplete, but we hypothesize that a
key contribution is the initially enhanced magnitude of longitudinal fluxes relative to
transverse fluxes under ADL kinetics. The relevance of our numerical results to real
materials is not immediate, owing to the restricted subset of the parameter space
(g1/g3 lessmuch 1) that the finite element method could handle reliably. Preliminary
computations with nonzero but small g1/g3 suggest that the 2D?1D transition
persists continuously as g1/g3 is allowed to take on small positive values. This
continuous dependence on g1/g3 offers the promise of an eventual connection with
real materials, or even engineered metamaterials.
Analytically, we have looked for approximately separable solutions of the
macroscale PDE for surface height, predicting novel decay laws for some regions
215
of the parameter space. We have studied the possibility of a boundary layer near
a facet edge, finding a boundary layer width that scales as a power of g1/g3 in the
regime where g1/g3 is large. We have also characterized the effect of an electric field
on the slope profile near a facet. These latter two predictions offer the most promis-
ing connection with real materials, whose parameter values challenge the reliability
of our numerical method.
11.2 Open questions
An ambitious research program inevitably raises more questions than it an-
swers. We list below some of the pending issues that this thesis could not address.
? Stabilizing the numerics for larger values of g1/g3. As indicated in
Chapter 10, we lack a reliable simulation of the macroscale PDE for values of
the parameter g1/g3 of the order of unity. The usual way to address this nu-
merical instability is by means of a preconditioning matrix. An open problem
is to develop an accurate and efficient preconditioner for the matrix equation
given by the finite element method.
? Higher-order time stepping. Much of the discrepancy between different
simulations could be attributed to the choice of adaptive time step [6]. In
contrast, the calculated decay of height peak and energy were rather robust
with respect to mesh size [6]. Perhaps a fruitful direction to pursue is the
modification of our algorithm to reduce the time discretization errors.
? Comparison of macroscale simulations with step flow in 2D. Valida-
216
tion of the macroscale theory is commonly done by comparing its predictions
with those of an atomistic simulation, as in the work of Shenoy et al. [107, 108].
Since our derivation of the macroscale PDE begins with the generalized BCF
model rather than atomistic physics, it might be instructive to compare a step
flow simulation with the results of this thesis. The step-flow model of Weeks
et al. [122], which has been implemented numerically by Kan et al. [45], offers
a promising means of comparison. A crucial question is whether the material
parameters used in simulations of Weeks? model are within the scope of our
finite element method. (Note that extensive comparisons of step flow with
PDEs have been carried out for one-dimensional geometries [25, 79].)
? Boundary conditions at facet edges. As discussed in Chapter 9, the vari-
ational formulation of our evolution equation does not respect the boundary
conditions arising from the microstructure at a facet edge. An open problem
is the development of a hybrid approach, which would couple the variational
implementation of a fully continuum PDE with collapse time data from the
flow of steps near facets.
? Further analysis of the macroscale PDE. All of our numerics assumed
a spatially periodic solution whose period remained unchanged throughout
the simulation. The analytical justification of this assumption is still lacking.
Other qualitative features of the computed surface morphology, such as the
non-monotonic decay of height peak, could likewise be subject to analysis.
The question of backwards uniqueness in time is also of physical significance,
217
as indicated in the following.
? Inverse problem. To make our modeling more attractive to the materials
engineering community, we might want to analyze whether a desired surface
pattern can emerge from self-organization of an initial profile, which relaxes
near equilibrium according to the flow of steps below roughening. This ques-
tion is an obvious analog for our macroscale PDE of the backwards uniqueness
property for the heat equation. We essentially want to characterize which ?fi-
nal data? can successfully be evolved backward in time by our fourth-order
nonlinear PDE. This characterization might involve the Fourier components of
the final data (as in the case of the heat equation), or perhaps a more restric-
tive set of conditions. If we manage by analytical means to address backwards
uniqueness satisfactorily, we might then try to implement a numerical algo-
rithm to approximate the solution of the inverse problem.
? Stochastic effects. In this thesis we performed coarse-graining for a com-
pletely deterministic BCF-type model. Stochastic effects (especially the effect
of noise at the nanoscale) were not studied. A more realistic picture of the
nanoscale physics would allow for the fluctuation of step edges by adding noise
terms to the step flow equations. An interesting problem is then to derive the
macroscale limit from these stochastic differential equations.
? Far-from-equilibrium conditions. Throughout this thesis we assumed the
surface to evolve near equilibrium. In this vein we have focused entirely on
relaxation phenomena, omitting the effect of material deposition. The under-
218
standing we gain from studying relaxation is expected to carry through to
far-from-equilibrium conditions, because relaxation by surface diffusion is al-
ways present in evolving surfaces. However, the far-from-equilibrium setting is
found to yield qualitatively different macroscale equations [11, 68, 69], which
are of interest in their own right and for the connection they promise with
modern experiments in epitaxial growth.
219
Appendix A
Brief review of the Mellin transform
In this Appendix, we introduce the Mellin transform starting from the more
familiar Laplace integral transform. Motivated by the eventual use of the Mellin
transform in asymptotic expansion of integrals, we include only those technical
details needed to facilitate this application. The extension to iterated integrals and
inversion formulas in higher dimensions are not needed for our purposes; the reader
is directed to [100] for a more thorough discussion of these issues.
Many problems in engineering and applied mathematics are cumbersome to
solve when expressed in the relevant physical variables. Integral transforms were
developed to address this difficulty. The role of an integral transform is to map
an equation from the original domain (e.g., physical space or time) into another
domain (e.g., wavenumber or frequency space) where the computations are simpler.
The solution in the transformed domain is then pulled back to the physical domain
by the inverse transform.
The Laplace transform is well-known for its use in ordinary differential equa-
tions (ODEs): by taking the Laplace transform of a linear ODE, we obtain an
algebraic equation for the transform of the ODE solution. After solving the result-
ing algebraic equation, we can apply the inverse Laplace transform to express the
solution in the original variable. This idea is made rigorous by the following
Definition A.0.1. For a function f : R ? R, the two-sided Laplace transform
?f(s) is given by:
?f(s) =
integraldisplay ?
??
f(t)e?stdt. (A.1)
220
The original function f(t) is recovered from ?f(s) by the inverse Laplace transform:
f(t) = 12pi?
integraldisplay ?+??
????
?f(s)estds, (A.2)
where ? is a real number so that the integration path lies in the region of convergence
of ?f(s).
For asymptotic expansion of integrals, it turns out to be more convenient to use
an integral transform whose kernel is a power function rather than an exponential.
In this vein, we let x = et, dx = xdt in (A.1), which yields
?f(s) =
integraldisplay ?
0
f(lnx)x?sdxx . (A.3)
We introduce F(x) to denote the composite function f(lnx), and ?? = ?s?1 to
denote the power of x. The resulting integral, viewed as a function of ?, is called
the Mellin transform and written ?F(?), i.e.,
?F(?) =
integraldisplay ?
0
F(x)x??dx. (A.4)
We break up the integral (A.4) into two parts when studying its convergence
properties:
?F(?) =
parenleftbiggintegraldisplay 1
0
+
integraldisplay ?
1
parenrightbigg
F(x)x??dx. (A.5)
The restrictions on ? for convergence of (A.5) emerge from enforcing boundedness
of the two integrals separately. We have a bound for the first integral,
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay 1
0
F(x)x??dx
vextendsinglevextendsingle
vextendsinglevextendsingle?
integraldisplay 1
0
|F(x)|x?Re?dx, (A.6)
where Re? denotes the real part of ?. Suppose that F(x) = O(xp) as x ? 0. This
first integral converges if p?Re? >?1, or Re? < 1 +p.
221
On the other hand, we have for the second integral
vextendsinglevextendsingle
vextendsinglevextendsingle
integraldisplay ?
1
F(x)x??dx
vextendsinglevextendsingle
vextendsinglevextendsingle?
integraldisplay ?
1
|F(x)|x?Re?dx. (A.7)
If F(x) = O(xq) as x ? ?, then the convergence of the second integral requires
q?Re? < ?1, or 1 +q < Re?. Together with the first condition on ?, we find the
fundamental strip 1+q< Re? < 1+p, which indicates the region in the complex ?
plane where the integral defining the Mellin transform converges.
The inverse Mellin transform is obtained from ?F(?) by the same contour
integral as (A.2):
F(x) = f(lnx) = 12pi?
integraldisplay ?+??
????
?F(?)x??1d?. (A.8)
For the asymptotic evaluation of integrals, whereF(x) is defined by an integral
with a ?large? or ?small? variable x, this formulation is useful when the Mellin
transform ?F(?) is meromorphic. Then the only singularities of ?F are poles in the
complex ? plane, and an asymptotic expansion for F can be calculated by applying
the Cauchy residue theorem. Care must be taken to ensure that the integration
path is deformed to account for the sign of lnx and the strip of convergence.
The Mellin transform is useful when F(x) is defined by a definite integral with
?large? or ?small? parameterx, and the main contribution to this integral cannot be
attributed to isolated values of the integration variable. Indeed, classical asymptotic
approaches, such as the steepest descent or the stationary phase methods, are usually
sufficient in such cases. When the contribution to F(x) comes from an extended
interval in the domain of integration, the Mellin transform helps by mapping this
contribution to an isolated singularity in the transformed ? space. The required
222
calculation is often easier in the transformed domain, as we see in Chapter 4 when
studying the integral for elastic interaction energy.
223
Appendix B
Nondimensionalization of the evolution equations
In this chapter we rewrite the evolution equations for the macroscale sur-
face height in nondimensional form. These equivalent representations allow for a
more systematic numerical treatment of relaxation experiments. We find that for
relaxation of initially sinusoidal profiles, any collection of material and geometric
parameters can be thought of as a point p in R6, with coordinates ?x/?y, h0/?x,
?xD?1v, q, g1/g3. If a certain crystal surface is specified, then q and g1/g3 are not
independent; both quantities are related through the temperature T. If the crystal
surface is not specified, then varying q and g1/g3 independently has the same effect
as considering many different possible materials over a wide range of temperatures.
The subset of R6 corresponding to physically realizable parameters can be fur-
ther partitioned according to the distinct morphological changes that the macroscale
PDE allows. One such partition, distinguishing between initial data that supports a
2D ? 1D transition and initial data that prohibits such a transition, has been illus-
trated in [7] as cross-sections in the q,? plane, where ? = ?x/?y is the aspect ratio.
To extract from these plots the initial conditions for an experimental comparison
with real materials, we need to undo the nondimensionalization of the simulated
equations. Undoing this nondimensionalization will determine all the terms of the
224
original macroscale PDE, rewritten here in the universal form
?h
?t = ?CsDdiv
braceleftbigg
??
bracketleftbigg??
kBT ?D
?1v
parenleftbigg
1 + ?k
BT
parenrightbiggbracketrightbiggbracerightbigg
, (B.1)
? = ??g3 div
braceleftbiggg
1
g3
?h
|?h| +|?h|?h
bracerightbigg
, (B.2)
where ? is the dimensionless mobility tensor.
The initial data has characteristic lengths ?x, ?y in the basal plane, and h0 in
the vertical direction. Accordingly, we define nondimensional spatial variables by
?x = x/?x, (B.3)
?y = y/?y, (B.4)
?h = h/h0. (B.5)
To compute spatial derivatives, we define the operators ?? and div? by
?? = (??x,???y), (B.6)
div? = ??x +???y, (B.7)
which are related to the usual operators ? and div in an obvious way:
?? = ?x? and div? = ?x div. (B.8)
The nondimensional chemical potential ?? and drift velocity u are defined as
?? = ?k
BT
, (B.9)
u = ?xD?1v. (B.10)
Applying (B.3)?(B.8) to (B.2), we find
? = ??g3h
2
0
?3x div?
parenleftBigg
g1?2x
g3h20
???h
|???h| +|??
?h|???h
parenrightBigg
. (B.11)
225
The prefactor in (B.11) requires us to find ?? by solving
?? = ? ?g3h
2
0
kBT?3x div?
parenleftBigg
g1?2x
g3h20
???h
|???h| +|??
?h|???h
parenrightBigg
. (B.12)
Substituting (B.3)?(B.10) into the PDE (B.1), we find
??h
?t =
?CsD
?2xh0 div?{??[?????u(1 + ??)]}. (B.13)
The prefactor in (B.13) suggests that we choose a time unit t0 given by
t0 = h0?
2
x
?CsD, (B.14)
so that ?t = t/t0 is nondimensional.
In light of (B.12), (B.13), the numerical implementation of the macroscale
height evolution with drift u negationslash= 0 requires that we rescale three coefficients in the
code: the line tension coefficientg1/g3, the material parameterq appearing in ?, and
the prefactor in (B.12). The nondimensional time can absorb only the prefactor in
(B.13). In contrast, for u = 0 the nondimensional time can also absorb the prefactor
of (B.12), thereby reducing the number of rescaled coefficients to two: g1/g3 and
q. This procedure is adopted in Chapter 10, where our time unit is chosen as
t0 = kBT?5x?2Csg3Dh0.
226
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