Temporal Evolution Of Step-Edge Fluctuations
Under Electromigration Conditions
P.J. Rous∗ and T.W. Bole
Department of Physics,
University of Maryland Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
(Dated: October 31, 2018)
The temporal evolution of step-edge fluctuations under electromigration conditions is analysed
using a continuum Langevin model. If the electromigration driving force acts in the step up/down
direction, and step-edge diffusion is the dominant mass-transport mechanism, we find that significant
deviations from the usual t1/4 scaling of the terrace-width correlation function occurs for a critical
time τ which is dependent upon the three energy scales in the problem: kBT , the step stiffness,
γ, and the bias associated with adatom hopping under the influence of an electromigration force,
±∆U . For (t < τ ), the correlation function evolves as a superposition of t1/4 and t3/4 power laws.
For t ≥ τ a closed form expression can be derived. This behavior is confirmed by a Monte-Carlo
simulation using a discrete model of the step dynamics. It is proposed that the magnitude of the
electromigration force acting upon an atom at a step-edge can by estimated by a careful analysis of
the statistical properties of step-edge fluctuations on the appropriate time-scale.
PACS numbers: 05.40.-a, 66.30.Qa, 68.35.Ja
I. INTRODUCTION
During the past decade, continuum models and dis-
crete lattice simulations have been applied to understand
direct imaging observations of the thermal fluctuations of
step edges in which the step position is monitored as a
function of time [1, 2]. Of particular interest has been
the study of the dynamics of the equilibration of terrace
width distributions where the temporal evolution of step
edge fluctuations are driven by the exchange of atoms
between the step and the adjacent terrace and/or by mo-
tion of adatoms along the step edge itself [3, 4, 5, 6, 7, 8].
It is well known that the position of the step edge, as de-
scribed by its temporal correlation function, has a time
dependence that scales as a power law with an expo-
nent characteristic of specific atomic processes driving
the step fluctuations; tβ . In cases where mass transport
at the step is dominated by diffusion of atoms along the
step edge; β = 1/4. When mass transport proceeds via
exchange of atoms between the step edge and the terrace
β = 1/2. Careful experiments allow a crossover from t1/4
to t1/2 scaling to be observed as a function of the sam-
ple temperature [9]. Further, experimental measurement
of the correlation functions have been used to determine
thermodynamic properties of the steps, such as the step
stiffness [1, 2].
In this paper we investigate how the scaling of the step
edge fluctuations is changed by the presence of an electro-
migration force [10] acting upon atoms diffusing along the
step edges. The primary motivation for this study is the
possibility of using measurements of these changes to ob-
∗Electronic address: rous@umbc.edu
tain information about the electromigration force itself.
In conducting materials, a surface electromigration force
can be generated by passing an electrical current through
the sample [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].
In terms of a simple discrete model, the presence of the
electromigration force introduces a small bias in the dif-
fusion of atoms at the step edge, parallel to the current
(and field). By convention, this bias can be expressed as
an energy difference between atoms diffusing parallel or
anti-parallel to the current ∆U = Z∗eEa⊥ where E is
the electric field applied to the sample, Z∗ is the effec-
tive valence of adatoms and a⊥ is the lattice spacing per-
pendicular to the step edge. A characteristic property of
surface electromigration is that the electromigration bias,
∆U , is several orders of magnitude smaller than the other
energy scales in the problem: γa⊥, where γ is the step
stiffness and the energy associated with thermal fluctua-
tions; kBT . This suggests that thermal fluctuations will
completely dominate the short-time behavior of the step
fluctuations with the effect of the electromigration bias
emerging only on much longer time scales. Nevertheless,
such time scales (of the order of seconds) are accessible to
experiment offering the possibility that the observation of
step fluctuations under electromigration conditions may
allow us to obtain quantitative information about the
magnitude of the force itself; a quantity that, to date,
has been hidden from experimental study.
This paper is organized as following. In section 2 we
present a continuum theory of step edge fluctuations un-
der electromigration conditions. In section 3 we test the
fidelity of the theory by showing the results of a Monte-
Carlo simulation of the temporal evolution step correla-
tion function. Concluding remarks are contained in sec-
tion 4.
http://arxiv.org/abs/0704.0624v1
mailto:rous@umbc.edu
II. THEORY
In order to describe the dynamics of a step-edge evolv-
ing under electromigration conditions we employ the
usual Langevin formalism where each degree of freedom
diffuses towards lower energy with a velocity that is pro-
portional to the energy gradient, subject to random ther-
mal fluctuations. The position of step edge is described
by it’s edge profile y(x, t) where the x-axis is oriented
parallel to the step edge. y(x, t) is the position of the
step edge at x and at time t.
In this paper we consider the limiting case where the
step motion occurs most easily by adatom diffusion along
the step edge itself. Adatom exchange between the step-
edge and the adjacent terraces, via attachment and de-
tachment, is neglected. It is well known that atomic diffu-
sion along a step edge is driven by the step curvature [23],
which generates a flux
Dsδsγ
∇sκ (1)
where Jc is the curvature-driven flux, Ds = a
||/τh is
the surface diffusion constant, a|| is the atomic diameter,
δs = a⊥ the width of an atom perpendicular to the step,
τh is the average time between hopping events and γ is
the step stiffness. ∇sκ is the gradient of the curvature
along the step edge. Mass conservation determines the
normal velocity of the step edge,
n̂ = −Ω∇s · Jc (2)
where Ω is the area of a single atom and n̂ is normal to
the step edge.
The inclusion of a thermal noise term, η and lineariza-
tion of the above equation leads to the well-known equa-
tion of motion for a step edge [3]
− Γhγ
y(x, t) = η(x, t). (3)
where we have defined
a3||a
. (4)
We model the effects of electromigration by adding to
this equation of motion a term generated by a constant
force, F = Z∗eE, felt by atoms at the step edge, which
arises from the application of a electric field, E, to the
material oriented parallel to the y-axis. Z∗ is the effective
valence of an atom at the step edge. The electromigration
force generates an additional flux,
JEM =
DsδsZ
∗eE||
DsδsZ
1 + y2x
where E|| is component of the electric field along the step
edge. yx indicates an x derivative.
The energy of any step configuration, relative to the
energy of the straight step, is determined by the step stiff-
ness γ and the electromigration force, F , felt by atoms
at the step edge. If the force acts perpendicular to the
step edge (i.e. when F > 0, the force acts in the +y,
or step down, direction) then, for small fluctuations, one
can linearize the stochastic equation of motion for the
step edge to obtain the following Langevin equation for
the step dynamics,
− Γhγ
− ΓhF
kTa||a⊥
y(x, t) = η(x, t) (6)
where a|| and a⊥ are the lattice parameter parallel and
perpendicular to the step edge. The noise term, η, de-
scribes the thermal fluctuations and is correlated be-
cause, in our model, of the random hopping of atoms
occurs only between between adjacent step edge sites.
〈η(x, t)η(x′, t′)〉 = 2Γhδ(t− t′)
δ(x − x′) (7)
To first order, the electromigration force does not al-
ter the correlation properties of the noise so that in the
single-jump regime equation 4 remians valid.
Before proceeding, for notational simplicity we rewrite
eqn. 6 as
− α ∂
∓ αq2c
y(x, t) = η(x, t) (8)
where
α ≡ Γhγ
, qc ≡
a||a⊥
The parameter qc depends only on the magnitude of the
electromigration force and in eqn. 6 the ∓ denotes the
force acting in the step down(-) or step up (+) direction.
In the case where there is no random thermal noise
(i.e. T → 0), eqn. 6 predicts that the amplitude of a
small sinusoidal fluctuation of the step-edge profile, with
wavevector q, evolves according to the following disper-
sion relation,
iω = αq2
q2 ± q2c
This is the same dispersion relation as the one that de-
scribes step flow under growth conditions [24, 25, 26]. If
the electromigration force acts in the step up (+) direc-
tion the step fluctuations are linearly stable. For a force
acting in the step-down direction there exists a linear in-
stability which initiates the well-known phenomenon of
electromigration-induced step-wandering for fluctuations
with wavevector smaller than qc [27, 28, 29]. The critical
wavevector is an important parameter in our Langevin
model that, from eqn. 9, is determined by the ratio of
the electromigration force and the step stiffness.
In order to determine the statistical properties of the
solutions of eqn. 6 we take the usual approach and first
derive the Green’s function for the problem using stan-
dard Fourier transform methods,
G(x, t|x′, t′) = i
e−α(k
k2)(t−t′)eik(x−x
In terms of the Green’s function, the displacement of the
step at time t is:
y(x, t′′) = y(x, t′) +
∫ t′′
G(x, t′′|x′, t)η(x′, t′)dx′dt
We can now compute the time correlation function of the
step edge, G(t), after time t = t′′ − t′ has elapsed,
g(t) ≡< (y(x, t′′)− y(x, t′))2 > (13)
Substituting eqn. 11 into 12 and using the correlation
properties of the noise (eqn. 7) we obtain we find, after
some calculation,
g±(t) =
21/4πα
(k2 ± q2c )
1− e−2α(k
or, using substitution,
g±(t) = t
(kT )
× (15)
(u2 ±
αtq4c )
1− e−2u
When there is no electromigration force (i.e. |F | =
0, qc = 0), the integral in eqn. 16 is clearly time-
independent and we recover the result derived by Bartelt
et al. [3] where the g(t) the step edge fluctuations evolves
with the well-known t1/4 scaling law characteristic of
step-edge limited diffusion.
g±(t) ≡ g0(t) = t1/4
(kT )
It is helpful to re-express eqn. 16 in terms of the average
time, τ0, that it takes for the mean-squared width of an
initially straight step to reach a value equal to g20(τ0) =
a2⊥ (i.e. one square lattice spacing):
g0(t) = a
where
τ0 ≡ τh
2kTa||
4Γ (3/4)
When the electromigration force is finite ( i.e. |F | > 0),
it is apparent from eqn. 16 that this scaling behavior is
modified by the appearance of an explicit time depen-
dence in the integral. This can be seen more clearly by
defining a critical time, τ , and a rescaled time, ζ = t/τ ,
where,
= 2τh
2kTa||
, (19)
Then, eqn. 16 can be rewritten as
g±(t) = a
where
I± (ζ) ≡
21/4Γ(3/4)
× (21)
(u2 ±
1− e−2u
I± is a universal function of the rescaled time, ζ and
is normalized such that I±(t → 0) = 1. The integral
appearing in eqn. 22 is easily evaluated numerically and
is shown in fig. 1 in which we display I± plotted as a
function of the rescaled time ζ (i.e. time is expressed in
units of τ). The solid curves show I± obtained for F > 0
(step up) , F < 0 (step down) electromigration forces
(For F = 0, I±(ζ) = 1).
From fig. 1. it is apparent that g±(t) (eqn. 20) deviates
very significantly from the t1/4 scaling behavior of g0(t)
(eqn. 17) as t approaches τ (ζ → 1). These deviations be-
gin to appear at earlier times, t ∼ 0.1τ , when the effect of
the force on the evolution of the step fluctuations begins
to be felt. This can be seen more clearly in fig. 2 which
displays the correlation function of a step plotted as a
function of time for τ0 = 5s and τ = 10
+4s. The values
for these parameters were chosen so that τ/τ0 ∼ 104, a
ratio typical of accelerated electromigration experiments.
As noted above, deviations from t1/4 scaling start to ap-
pear when t ∼ 0.1τ = 100s. The results shown in figs. 1
and 2 have a simple qualitative interpretation; the short
time behavior of the step fluctuations (t ≪ τ) is com-
pletely dominated by the thermal fluctuations and the
effect of the electromigration bias emerges only at later
FIG. 1: The integral function I(ζ) (eqn. 22) plotted as a
function of the rescaled time ζ = t/τ for no electromigration
force (F = 0), the electromigration force in the step down
direction (F > 0) and in the step-up direction (F < 0).
FIG. 2: The time corrleation function, g(t), of the step-
edge distribution predicted by the continuum Langevin model
(eqns. 20 and 22), plotted as a function of time for τ0 = 5s
and τ = 104s.
times. Such behavior is typical of the dynamics of diffu-
sion driven by weak external forces.
It is instructive to perform a power series expansion of
the integral about ζ = 0 such that eqn. 20 becomes,
g±(t) = a
1∓ a 1
. . .
The expansion coefficents can be obtained analytically,
= 0.3487, a1 =
= 0.1500,
Shown as the dashed lines in fig. 1 are the results of this
series approximation for I±(ζ) (eqn. 22), evaluated up
to, and including the terms linear in time. Clearly, this
truncated expression is a reasonable approximation for
t ≤ 0.4τ .
III. SIMULATION
In order to test the predictions of the continuum
Langevin model described above, we developed a Monte
Carlo simulation of step edge fluctuations in the pres-
ence of an external force. In this model, atomic diffu-
sion was restricted to the step edges with atoms jump-
ing between adjacent step sites only. Only nearest-
neighbor interactions on a square lattice were permit-
ted (a⊥ = a|| = a = 1) and were modelled by a single
bond energy ǫ. In order to describe the electromigration
force, the atomic jump probabilities for motion parallel
and anti-parallel to the force were biased by a small en-
ergy differential ∆U . In terms of the electromigration
force, F , and the lattice spacing perpendicular to the
step edge, a; ∆U = ±Fa.
Simulations were performed for steps of length ℓ =
10000a|| fluctuation on a square lattice. Periodic bound-
ary conditions were employed. The bond energy was set
to ǫ = 2.0kBT and the magnitude of the electromigra-
tion bias was |∆U | = 0.01kBT , a factor of 103 smaller
than the typical binding energy of an atom to the step
edge. This value was chosen to generate statistically sig-
nificant deviations from the (no-force) t1/4 scaling within
reasonable simulation times. If ǫ = 0.1eV and it is as-
sumed that a ∼ 1.5Å (typical of metals) then this bias
corresponds to an electric field with a magnitude of or-
der 1000V cm−1 acting on an atom with effective valence
of Z∗ ∼ ±10e. In actual accelerated electromigration
experiments, a field of 0.1− 1V cm−1 is typical.
Figure 3 shows the results of the simulation where the
correlation function of an isolated step is plotted as a
function of the time measured in Monte-Carlo steps per
step-edge site (MCS). Shown is g(t) obtained when the
electromigration force acts in the step-up and step-down
directions and when ∆U = 0 (i.e. no electromigration
force is acting at the step edge).We define one Monte
FIG. 3: Results of the a kinetic Monte Carlo simulation where
the correlation function of an isolated step is plotted as a
function of the time (measured in Monte-Carlo steps per step-
edge site, the lattice spacing is a = 1). Shown is g(t) obtained
when the electromigration force acts in the step-up and step-
down directions and when ∆U = 0 (i.e. no electromigration
force is acting at the step edge). The curves shown were
obtained by averaging the data from 200 randomly generated
replicas after each was equilibrated for 105 Monte Carlo steps.
Carlo step to be equal to the average time needed for ev-
ery atom at the step edge to attempt a hop. The results
shown in fig. 3 were obtained by averaging the data from
100 randomly generated replicas after each was equili-
brated for 105 Monte Carlo iterations per site. Compar-
ing the simulation results (fig. 3.) to the predictions of
the Langevin theory (fig 4.) it is apparent that the qual-
itative features of the continuum theory are reproduced
by the Monte Carlo simulation. These same data are pre-
sented in the form of a log-log plot in fig. 4. In the ab-
sence of the force (F = 0 , ∆U = 0), a least-squares fit of
the no-force simulation data shows that g0(t) is very well
fit by a tβ power-law where β = 0.25± 0.01. Therefore,
when there is no electromigration force present in the
simulation, the correlation function of the step evolves
according well-known t1/4 scaling, as predicted by the
Langevin analysis (eqn. 17). By least squares fitting the
simulation results to eqn. 17 we obtain a value of τ0 = 5
We now consider the results of the simulation obtained
for a finite electromigration force, also shown in figs. 3
and 4. Equation 22 suggests that the value of τ can be
FIG. 4: A log-log plot of the simulation data shown in fig. 3.
The dashed line shows the best fit of a power law, (t/τ0)
β, to
the no-force (F = 0) data; β = 0.25, τ0 = 5 MCS.
extracted from the simulation results by considering the
scaling of the difference between the correlation functions
for the force in the up and down step directions:
∆(t) =
g+ − g−
= 2a 1
. . . (24)
For the simulation results, this normalized difference is
plotted in fig. 5. The behavior of this quantity is well
fit by the leading order term in eqn. 24 from which we
obtain a value of τ = 62000± 10000 MCS.
For comparison with the fits to the continuum
Langevin model , we can estimate τ from the microscopic
parameters employed in the discrete Monte Carlo model
used to generate the simulation data. Combining eqn. 19
with the step stiffness appropriate for our model
2kTa||
2kBTa||
we obtain an estimate of τ in units of MCS:
τ = 2
2kBTa||
The ratio of the hopping time in the Langevin model ,
τh to the time between hopping attempts in the Monte
Carlo simulation τa can be obtained by monitoring the
FIG. 5: For the simulation results shown in figs. 3 and 4; a
log-log plot of the normalized difference (eqn. 24) plotted as a
function of time (MCS). The behavior of this quantity is well
fit by the leading order term in eqn. 24 (t1/2, dashed curve)
from which we obtain a value of τ = 62000 ± 10000 MCS.
success rate of hops between adjacent lattice sites in the
simulation. We find that τa/τh = 3.6 ± 0.1. Thus our
estimate for the value of τ in the Monte Carlo simu-
lations, used to generate the data shown in figs 3-5, is
τ = 62000±10000 MCS. Clearly, the agreement between
the continuum Langevin theory (τ = 71000 MCS ) and
the microscopic model (τ = 62000± 10000 MCS) is rea-
sonable. Finally we note that in the high temperature
limit (kBT ≫ ǫ) the ratio of the electromigration bias to
the binding energy at a step edge, ǫ is related directly to
τ that would be obtained from experiment:
where τh can be determined from eqn. 18, if the step
stiffness is known.
IV. CONCLUSIONS
The temporal evolution of step-edge fluctuations un-
der electromigration conditions has been analyzed using
a continuum Langevin model for the case where diffu-
sion is limited by mass transport along the step edge.
We find that the presence of the electromigration force,
felt by atoms at the step edge, causes deviations from
the usual t1/4 scaling of the terrace-width distribution
driven by thermal fluctuations alone. We have identified
a critical time τ that is a function of the three energy
scales in the problem: kBT , the step stiffness, γ, and the
bias associated with adatom hopping under the influence
of an electromigration force, ±∆U . For (t < τ), the cor-
relation function evolves, to a good approximation, as a
superposition of t1/4 and t3/4 power laws. For all τ a
closed form expression was derived. This behavior was
confirmed by a Monte-Carlo simulation using a discrete
model of the step dynamics. Finally we propose that the
magnitude of the electromigration force acting upon an
atom at a step-edge could be determined directly by care-
ful measurement and analysis of the statistical properties
of step-edge fluctuations on the appropriate time-scale.
Acknowledgments
We acknowledge helpful discussions with E.D.
Williams.
This work has been supported by the US Department
of Energy grant DE-FG02-01ER45939 and by the NSF-
Materials Research Science and Engineering Center un-
der grant DMR-00-80008.
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  The temporal evolution of step-edge fluctuations under electromigration
conditions is analysed using a continuum Langevin model. If the
electromigration driving force acts in the step up/down direction, and
step-edge diffusion is the dominant mass-transport mechanism, we find that
significant deviations from the usual $t^{1/4}$ scaling of the terrace-width
correlation function occurs for a critical time $\tau$ which is dependent upon
the three energy scales in the problem: $k_{B}T$, the step stiffness, $\gamma$,
and the bias associated with adatom hopping under the influence of an
electromigration force, $\pm \Delta U$. For ($t < \tau$), the correlation
function evolves as a superposition of $t^{1/4}$ and $t^{3/4}$ power laws. For
$t \ge \tau$ a closed form expression can be derived. This behavior is
confirmed by a Monte-Carlo simulation using a discrete model of the step
dynamics. It is proposed that the magnitude of the electromigration force
acting upon an atom at a step-edge can by estimated by a careful analysis of
the statistical properties of step-edge fluctuations on the appropriate
time-scale.

Temporal Evolution Of Step-Edge Fluctuations
Under Electromigration Conditions
P.J. Rous∗ and T.W. Bole
Department of Physics,
University of Maryland Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
(Dated: October 31, 2018)
The temporal evolution of step-edge fluctuations under electromigration conditions is analysed
using a continuum Langevin model. If the electromigration driving force acts in the step up/down
direction, and step-edge diffusion is the dominant mass-transport mechanism, we find that significant
deviations from the usual t1/4 scaling of the terrace-width correlation function occurs for a critical
time τ which is dependent upon the three energy scales in the problem: kBT , the step stiffness,
γ, and the bias associated with adatom hopping under the influence of an electromigration force,
±∆U . For (t < τ ), the correlation function evolves as a superposition of t1/4 and t3/4 power laws.
For t ≥ τ a closed form expression can be derived. This behavior is confirmed by a Monte-Carlo
simulation using a discrete model of the step dynamics. It is proposed that the magnitude of the
electromigration force acting upon an atom at a step-edge can by estimated by a careful analysis of
the statistical properties of step-edge fluctuations on the appropriate time-scale.
PACS numbers: 05.40.-a, 66.30.Qa, 68.35.Ja
I. INTRODUCTION
During the past decade, continuum models and dis-
crete lattice simulations have been applied to understand
direct imaging observations of the thermal fluctuations of
step edges in which the step position is monitored as a
function of time [1, 2]. Of particular interest has been
the study of the dynamics of the equilibration of terrace
width distributions where the temporal evolution of step
edge fluctuations are driven by the exchange of atoms
between the step and the adjacent terrace and/or by mo-
tion of adatoms along the step edge itself [3, 4, 5, 6, 7, 8].
It is well known that the position of the step edge, as de-
scribed by its temporal correlation function, has a time
dependence that scales as a power law with an expo-
nent characteristic of specific atomic processes driving
the step fluctuations; tβ . In cases where mass transport
at the step is dominated by diffusion of atoms along the
step edge; β = 1/4. When mass transport proceeds via
exchange of atoms between the step edge and the terrace
β = 1/2. Careful experiments allow a crossover from t1/4
to t1/2 scaling to be observed as a function of the sam-
ple temperature [9]. Further, experimental measurement
of the correlation functions have been used to determine
thermodynamic properties of the steps, such as the step
stiffness [1, 2].
In this paper we investigate how the scaling of the step
edge fluctuations is changed by the presence of an electro-
migration force [10] acting upon atoms diffusing along the
step edges. The primary motivation for this study is the
possibility of using measurements of these changes to ob-
∗Electronic address: rous@umbc.edu
tain information about the electromigration force itself.
In conducting materials, a surface electromigration force
can be generated by passing an electrical current through
the sample [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].
In terms of a simple discrete model, the presence of the
electromigration force introduces a small bias in the dif-
fusion of atoms at the step edge, parallel to the current
(and field). By convention, this bias can be expressed as
an energy difference between atoms diffusing parallel or
anti-parallel to the current ∆U = Z∗eEa⊥ where E is
the electric field applied to the sample, Z∗ is the effec-
tive valence of adatoms and a⊥ is the lattice spacing per-
pendicular to the step edge. A characteristic property of
surface electromigration is that the electromigration bias,
∆U , is several orders of magnitude smaller than the other
energy scales in the problem: γa⊥, where γ is the step
stiffness and the energy associated with thermal fluctua-
tions; kBT . This suggests that thermal fluctuations will
completely dominate the short-time behavior of the step
fluctuations with the effect of the electromigration bias
emerging only on much longer time scales. Nevertheless,
such time scales (of the order of seconds) are accessible to
experiment offering the possibility that the observation of
step fluctuations under electromigration conditions may
allow us to obtain quantitative information about the
magnitude of the force itself; a quantity that, to date,
has been hidden from experimental study.
This paper is organized as following. In section 2 we
present a continuum theory of step edge fluctuations un-
der electromigration conditions. In section 3 we test the
fidelity of the theory by showing the results of a Monte-
Carlo simulation of the temporal evolution step correla-
tion function. Concluding remarks are contained in sec-
tion 4.
http://arxiv.org/abs/0704.0624v1
mailto:rous@umbc.edu
II. THEORY
In order to describe the dynamics of a step-edge evolv-
ing under electromigration conditions we employ the
usual Langevin formalism where each degree of freedom
diffuses towards lower energy with a velocity that is pro-
portional to the energy gradient, subject to random ther-
mal fluctuations. The position of step edge is described
by it’s edge profile y(x, t) where the x-axis is oriented
parallel to the step edge. y(x, t) is the position of the
step edge at x and at time t.
In this paper we consider the limiting case where the
step motion occurs most easily by adatom diffusion along
the step edge itself. Adatom exchange between the step-
edge and the adjacent terraces, via attachment and de-
tachment, is neglected. It is well known that atomic diffu-
sion along a step edge is driven by the step curvature [23],
which generates a flux
Dsδsγ
∇sκ (1)
where Jc is the curvature-driven flux, Ds = a
||/τh is
the surface diffusion constant, a|| is the atomic diameter,
δs = a⊥ the width of an atom perpendicular to the step,
τh is the average time between hopping events and γ is
the step stiffness. ∇sκ is the gradient of the curvature
along the step edge. Mass conservation determines the
normal velocity of the step edge,
n̂ = −Ω∇s · Jc (2)
where Ω is the area of a single atom and n̂ is normal to
the step edge.
The inclusion of a thermal noise term, η and lineariza-
tion of the above equation leads to the well-known equa-
tion of motion for a step edge [3]
− Γhγ
y(x, t) = η(x, t). (3)
where we have defined
a3||a
. (4)
We model the effects of electromigration by adding to
this equation of motion a term generated by a constant
force, F = Z∗eE, felt by atoms at the step edge, which
arises from the application of a electric field, E, to the
material oriented parallel to the y-axis. Z∗ is the effective
valence of an atom at the step edge. The electromigration
force generates an additional flux,
JEM =
DsδsZ
∗eE||
DsδsZ
1 + y2x
where E|| is component of the electric field along the step
edge. yx indicates an x derivative.
The energy of any step configuration, relative to the
energy of the straight step, is determined by the step stiff-
ness γ and the electromigration force, F , felt by atoms
at the step edge. If the force acts perpendicular to the
step edge (i.e. when F > 0, the force acts in the +y,
or step down, direction) then, for small fluctuations, one
can linearize the stochastic equation of motion for the
step edge to obtain the following Langevin equation for
the step dynamics,
− Γhγ
− ΓhF
kTa||a⊥
y(x, t) = η(x, t) (6)
where a|| and a⊥ are the lattice parameter parallel and
perpendicular to the step edge. The noise term, η, de-
scribes the thermal fluctuations and is correlated be-
cause, in our model, of the random hopping of atoms
occurs only between between adjacent step edge sites.
〈η(x, t)η(x′, t′)〉 = 2Γhδ(t− t′)
δ(x − x′) (7)
To first order, the electromigration force does not al-
ter the correlation properties of the noise so that in the
single-jump regime equation 4 remians valid.
Before proceeding, for notational simplicity we rewrite
eqn. 6 as
− α ∂
∓ αq2c
y(x, t) = η(x, t) (8)
where
α ≡ Γhγ
, qc ≡
a||a⊥
The parameter qc depends only on the magnitude of the
electromigration force and in eqn. 6 the ∓ denotes the
force acting in the step down(-) or step up (+) direction.
In the case where there is no random thermal noise
(i.e. T → 0), eqn. 6 predicts that the amplitude of a
small sinusoidal fluctuation of the step-edge profile, with
wavevector q, evolves according to the following disper-
sion relation,
iω = αq2
q2 ± q2c
This is the same dispersion relation as the one that de-
scribes step flow under growth conditions [24, 25, 26]. If
the electromigration force acts in the step up (+) direc-
tion the step fluctuations are linearly stable. For a force
acting in the step-down direction there exists a linear in-
stability which initiates the well-known phenomenon of
electromigration-induced step-wandering for fluctuations
with wavevector smaller than qc [27, 28, 29]. The critical
wavevector is an important parameter in our Langevin
model that, from eqn. 9, is determined by the ratio of
the electromigration force and the step stiffness.
In order to determine the statistical properties of the
solutions of eqn. 6 we take the usual approach and first
derive the Green’s function for the problem using stan-
dard Fourier transform methods,
G(x, t|x′, t′) = i
e−α(k
k2)(t−t′)eik(x−x
In terms of the Green’s function, the displacement of the
step at time t is:
y(x, t′′) = y(x, t′) +
∫ t′′
G(x, t′′|x′, t)η(x′, t′)dx′dt
We can now compute the time correlation function of the
step edge, G(t), after time t = t′′ − t′ has elapsed,
g(t) ≡< (y(x, t′′)− y(x, t′))2 > (13)
Substituting eqn. 11 into 12 and using the correlation
properties of the noise (eqn. 7) we obtain we find, after
some calculation,
g±(t) =
21/4πα
(k2 ± q2c )
1− e−2α(k
or, using substitution,
g±(t) = t
(kT )
× (15)
(u2 ±
αtq4c )
1− e−2u
When there is no electromigration force (i.e. |F | =
0, qc = 0), the integral in eqn. 16 is clearly time-
independent and we recover the result derived by Bartelt
et al. [3] where the g(t) the step edge fluctuations evolves
with the well-known t1/4 scaling law characteristic of
step-edge limited diffusion.
g±(t) ≡ g0(t) = t1/4
(kT )
It is helpful to re-express eqn. 16 in terms of the average
time, τ0, that it takes for the mean-squared width of an
initially straight step to reach a value equal to g20(τ0) =
a2⊥ (i.e. one square lattice spacing):
g0(t) = a
where
τ0 ≡ τh
2kTa||
4Γ (3/4)
When the electromigration force is finite ( i.e. |F | > 0),
it is apparent from eqn. 16 that this scaling behavior is
modified by the appearance of an explicit time depen-
dence in the integral. This can be seen more clearly by
defining a critical time, τ , and a rescaled time, ζ = t/τ ,
where,
= 2τh
2kTa||
, (19)
Then, eqn. 16 can be rewritten as
g±(t) = a
where
I± (ζ) ≡
21/4Γ(3/4)
× (21)
(u2 ±
1− e−2u
I± is a universal function of the rescaled time, ζ and
is normalized such that I±(t → 0) = 1. The integral
appearing in eqn. 22 is easily evaluated numerically and
is shown in fig. 1 in which we display I± plotted as a
function of the rescaled time ζ (i.e. time is expressed in
units of τ). The solid curves show I± obtained for F > 0
(step up) , F < 0 (step down) electromigration forces
(For F = 0, I±(ζ) = 1).
From fig. 1. it is apparent that g±(t) (eqn. 20) deviates
very significantly from the t1/4 scaling behavior of g0(t)
(eqn. 17) as t approaches τ (ζ → 1). These deviations be-
gin to appear at earlier times, t ∼ 0.1τ , when the effect of
the force on the evolution of the step fluctuations begins
to be felt. This can be seen more clearly in fig. 2 which
displays the correlation function of a step plotted as a
function of time for τ0 = 5s and τ = 10
+4s. The values
for these parameters were chosen so that τ/τ0 ∼ 104, a
ratio typical of accelerated electromigration experiments.
As noted above, deviations from t1/4 scaling start to ap-
pear when t ∼ 0.1τ = 100s. The results shown in figs. 1
and 2 have a simple qualitative interpretation; the short
time behavior of the step fluctuations (t ≪ τ) is com-
pletely dominated by the thermal fluctuations and the
effect of the electromigration bias emerges only at later
FIG. 1: The integral function I(ζ) (eqn. 22) plotted as a
function of the rescaled time ζ = t/τ for no electromigration
force (F = 0), the electromigration force in the step down
direction (F > 0) and in the step-up direction (F < 0).
FIG. 2: The time corrleation function, g(t), of the step-
edge distribution predicted by the continuum Langevin model
(eqns. 20 and 22), plotted as a function of time for τ0 = 5s
and τ = 104s.
times. Such behavior is typical of the dynamics of diffu-
sion driven by weak external forces.
It is instructive to perform a power series expansion of
the integral about ζ = 0 such that eqn. 20 becomes,
g±(t) = a
1∓ a 1
. . .
The expansion coefficents can be obtained analytically,
= 0.3487, a1 =
= 0.1500,
Shown as the dashed lines in fig. 1 are the results of this
series approximation for I±(ζ) (eqn. 22), evaluated up
to, and including the terms linear in time. Clearly, this
truncated expression is a reasonable approximation for
t ≤ 0.4τ .
III. SIMULATION
In order to test the predictions of the continuum
Langevin model described above, we developed a Monte
Carlo simulation of step edge fluctuations in the pres-
ence of an external force. In this model, atomic diffu-
sion was restricted to the step edges with atoms jump-
ing between adjacent step sites only. Only nearest-
neighbor interactions on a square lattice were permit-
ted (a⊥ = a|| = a = 1) and were modelled by a single
bond energy ǫ. In order to describe the electromigration
force, the atomic jump probabilities for motion parallel
and anti-parallel to the force were biased by a small en-
ergy differential ∆U . In terms of the electromigration
force, F , and the lattice spacing perpendicular to the
step edge, a; ∆U = ±Fa.
Simulations were performed for steps of length ℓ =
10000a|| fluctuation on a square lattice. Periodic bound-
ary conditions were employed. The bond energy was set
to ǫ = 2.0kBT and the magnitude of the electromigra-
tion bias was |∆U | = 0.01kBT , a factor of 103 smaller
than the typical binding energy of an atom to the step
edge. This value was chosen to generate statistically sig-
nificant deviations from the (no-force) t1/4 scaling within
reasonable simulation times. If ǫ = 0.1eV and it is as-
sumed that a ∼ 1.5Å (typical of metals) then this bias
corresponds to an electric field with a magnitude of or-
der 1000V cm−1 acting on an atom with effective valence
of Z∗ ∼ ±10e. In actual accelerated electromigration
experiments, a field of 0.1− 1V cm−1 is typical.
Figure 3 shows the results of the simulation where the
correlation function of an isolated step is plotted as a
function of the time measured in Monte-Carlo steps per
step-edge site (MCS). Shown is g(t) obtained when the
electromigration force acts in the step-up and step-down
directions and when ∆U = 0 (i.e. no electromigration
force is acting at the step edge).We define one Monte
FIG. 3: Results of the a kinetic Monte Carlo simulation where
the correlation function of an isolated step is plotted as a
function of the time (measured in Monte-Carlo steps per step-
edge site, the lattice spacing is a = 1). Shown is g(t) obtained
when the electromigration force acts in the step-up and step-
down directions and when ∆U = 0 (i.e. no electromigration
force is acting at the step edge). The curves shown were
obtained by averaging the data from 200 randomly generated
replicas after each was equilibrated for 105 Monte Carlo steps.
Carlo step to be equal to the average time needed for ev-
ery atom at the step edge to attempt a hop. The results
shown in fig. 3 were obtained by averaging the data from
100 randomly generated replicas after each was equili-
brated for 105 Monte Carlo iterations per site. Compar-
ing the simulation results (fig. 3.) to the predictions of
the Langevin theory (fig 4.) it is apparent that the qual-
itative features of the continuum theory are reproduced
by the Monte Carlo simulation. These same data are pre-
sented in the form of a log-log plot in fig. 4. In the ab-
sence of the force (F = 0 , ∆U = 0), a least-squares fit of
the no-force simulation data shows that g0(t) is very well
fit by a tβ power-law where β = 0.25± 0.01. Therefore,
when there is no electromigration force present in the
simulation, the correlation function of the step evolves
according well-known t1/4 scaling, as predicted by the
Langevin analysis (eqn. 17). By least squares fitting the
simulation results to eqn. 17 we obtain a value of τ0 = 5
We now consider the results of the simulation obtained
for a finite electromigration force, also shown in figs. 3
and 4. Equation 22 suggests that the value of τ can be
FIG. 4: A log-log plot of the simulation data shown in fig. 3.
The dashed line shows the best fit of a power law, (t/τ0)
β, to
the no-force (F = 0) data; β = 0.25, τ0 = 5 MCS.
extracted from the simulation results by considering the
scaling of the difference between the correlation functions
for the force in the up and down step directions:
∆(t) =
g+ − g−
= 2a 1
. . . (24)
For the simulation results, this normalized difference is
plotted in fig. 5. The behavior of this quantity is well
fit by the leading order term in eqn. 24 from which we
obtain a value of τ = 62000± 10000 MCS.
For comparison with the fits to the continuum
Langevin model , we can estimate τ from the microscopic
parameters employed in the discrete Monte Carlo model
used to generate the simulation data. Combining eqn. 19
with the step stiffness appropriate for our model
2kTa||
2kBTa||
we obtain an estimate of τ in units of MCS:
τ = 2
2kBTa||
The ratio of the hopping time in the Langevin model ,
τh to the time between hopping attempts in the Monte
Carlo simulation τa can be obtained by monitoring the
FIG. 5: For the simulation results shown in figs. 3 and 4; a
log-log plot of the normalized difference (eqn. 24) plotted as a
function of time (MCS). The behavior of this quantity is well
fit by the leading order term in eqn. 24 (t1/2, dashed curve)
from which we obtain a value of τ = 62000 ± 10000 MCS.
success rate of hops between adjacent lattice sites in the
simulation. We find that τa/τh = 3.6 ± 0.1. Thus our
estimate for the value of τ in the Monte Carlo simu-
lations, used to generate the data shown in figs 3-5, is
τ = 62000±10000 MCS. Clearly, the agreement between
the continuum Langevin theory (τ = 71000 MCS ) and
the microscopic model (τ = 62000± 10000 MCS) is rea-
sonable. Finally we note that in the high temperature
limit (kBT ≫ ǫ) the ratio of the electromigration bias to
the binding energy at a step edge, ǫ is related directly to
τ that would be obtained from experiment:
where τh can be determined from eqn. 18, if the step
stiffness is known.
IV. CONCLUSIONS
The temporal evolution of step-edge fluctuations un-
der electromigration conditions has been analyzed using
a continuum Langevin model for the case where diffu-
sion is limited by mass transport along the step edge.
We find that the presence of the electromigration force,
felt by atoms at the step edge, causes deviations from
the usual t1/4 scaling of the terrace-width distribution
driven by thermal fluctuations alone. We have identified
a critical time τ that is a function of the three energy
scales in the problem: kBT , the step stiffness, γ, and the
bias associated with adatom hopping under the influence
of an electromigration force, ±∆U . For (t < τ), the cor-
relation function evolves, to a good approximation, as a
superposition of t1/4 and t3/4 power laws. For all τ a
closed form expression was derived. This behavior was
confirmed by a Monte-Carlo simulation using a discrete
model of the step dynamics. Finally we propose that the
magnitude of the electromigration force acting upon an
atom at a step-edge could be determined directly by care-
ful measurement and analysis of the statistical properties
of step-edge fluctuations on the appropriate time-scale.
Acknowledgments
We acknowledge helpful discussions with E.D.
Williams.
This work has been supported by the US Department
of Energy grant DE-FG02-01ER45939 and by the NSF-
Materials Research Science and Engineering Center un-
der grant DMR-00-80008.
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