Changing growth conditions during surface growth.

Abstract

Motivated by a series of experiments that revealed a temperature dependence of the dynamic scaling regime of growing surfaces, we investigate theoretically how a nonequilibrium growth process reacts to a sudden change of system parameters. We discuss quenches between correlated regimes through exact expressions derived from the stochastic Edwards-Wilkinson equation with a variable diffusion constant. Our study reveals that a sudden change of the diffusion constant leads to remarkable changes in the surface roughness. Different dynamic regimes, characterized by a power-law or by an exponential relaxation, are identified, and a dynamic phase diagram is constructed. We conclude that growth processes provide one of the rare instances where quenches between correlated regimes yield a power-law relaxation.

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arXiv:0912.0062v1 [cond-mat.stat-mech] 1 Dec 2009
Changing growth conditions during surface growth
Yen-Liang Chou,1Michel Pleimling,1and R. K. P. Zia1
1Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435, USA
(Dated: December 2, 2009)
Motivated by a series of experiments that revealed a temperature dependence of the dynamic
scaling regime of growing surfaces, we investigate theoretically how a nonequilibrium growth process
reacts to a sudden change of system parameters. We discuss quenches between correlated regimes
through exact expressions derived from the stochastic Edwards-Wilkinson equation with a variable
diffusion constant. Our study reveals that a sudden change of the diffusion constant leads to
remarkable changes in the surface roughness. Different dynamic regimes, characterized by a power-
law or by an exponential relaxation, are identified, and a dynamic phase diagram is constructed. We
conclude that growth processes provide one of the rare instances where quenches between correlated
regimes yield a power-law relaxation.
PACS numbers: 81.15.Aa,68.35.Md,64.60.Ht,05.70.Np
I.INTRODUCTION
Because of their ubiquity in nature, nonequilibrium
growth processes have been the subject of numerous stud-
ies during the last two decades [1, 2, 3]. Remarkably,
these studies revealed very general properties of grow-
ing interfaces that are encountered in a large variety of
growth processes, ranging from crystal growth to tumor
growth. In this context, due to its obvious technological
relevance, thin film growth has been one of the major
research thrusts.
From the theoretical point of view, many important
insights into the universal aspects of nonequilibrium
growth processes have been obtained through the study
of stochastic differential equations and of simple model
systems [4, 5]. One of the simplest approach is due to
Edwards and Wilkinson [6] who described the surface
growth due to particle sedimentation by ∂h(x,t)/∂t =
ν∇2h(x,t)+u, where h(x,t) is the surface height at time
t at a site x of a d-dimensional substrate (of area A) and
u represents a constant flux of deposition. The physi-
cal origin of the “diffusion constant” ν can be traced to
the surface tension as well as T, the temperature of the
substrate. When noise is added to this and the process
is viewed from a co-moving frame of the steady state
(i.e., h(x,t) − ut → h(x,t)), we arrive at the stochastic
Edwards-Wilkinson (EW) equation
∂h(x,t)
∂t
= ν∇2h(x,t) + η(x,t) (1)
where η(x,t) is a Gaussian white noise with zero mean
and covariance ?η(x,t)η(y,s)? = Dδd(x − y)δ(t − s). A
microscopic realization of the Edwards-Wilkinson equa-
tion was soon proposed by Family [3, 7] (see also Ref.
[8] for a recent more in-depth comparison of that model
with the EW equation). In the random deposition with
surface relaxation (RDSR) process particles drop from
randomly chosen sites over the surface. Instead of stick-
ing to the surface at the point of impact, the particles are
allowed to explore the nearest vicinity of that point and
are finally incorporated into the surface at the neighbor-
ing site with lowest height. This diffusion step smoothes
the surface and yields correlated growth.
The roughness of a growing surface is characterized by
the time dependent mean interface width
W(t) =
?
??h − h?2? (2)
where h(t) ≡?h(x,t)ddx/A is the mean surface height
(after an initial regime of random surface growth when
starting from a flat initial condition) displays a power-
law dependence on time, W(t) ∼ tβ, before saturating
at a value W ∼ Lαwhere L is the size of the system.
The growth exponent β and the roughness exponent α
are universal quantities that characterize large classes of
growth systems belonging to the same growth universal-
ity class. Thus for the Edwards-Wilkinson universality
class one finds for a one-dimensional substrate β = 1/4
and α = 1/2. Other universality classes have also been
found [4, 5], some of which, as for example the Kardar-
Parisi-Zhang (KPZ) [9] and the conserved KPZ univer-
sality classes [10, 11], are of direct relevance for thin film
growth.
The morphology of growing structures can depend cru-
cially on experimental parameters as for example the
flux of deposited particles or the substrate tempera-
ture [12]. Different experimental groups have reported
a temperature dependence of the roughness of growing
or sputtered surfaces, and temperature dependent values
of the growth exponents have been found in some sys-
tems [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Systems
for which this has been observed include homoepitax-
ial growth on Cu(001) [13, 21], Ag(100) [16, 19], and
Ag(111) [16], amorphous thin-film growth [22], growth of
CdTe polycrystalline films [23], molecular-beam epitaxy
growth of Si/Si(111) [15], as well as ion-sputtered Si(111)
[14], Ge(001) [17, 18] or Pt(111) [20] surfaces. The de-
pendence of the roughness on temperature can be rather
involved, yielding different types of behavior for differ-
ent systems, depending on how diffusion takes place and
at time t. In many instances the mean interface width

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whether additional smoothening mechanisms are present.
In many instances one observes that growth becomes in-
creasingly rougher with decreasing temperature, yielding
an increase of the value of β. On the other hand, some
experiments [22, 23] revealed an increase of the global
roughness with temperature.
The observation of a transition between different dy-
namic scaling regimes due to a change of experimental
conditions is very intriguing and raises the question how
a growing surface evolves from one regime to another af-
ter a sudden change of, for example, the substrate tem-
perature. We are not aware of any experimental study
of growth processes where this kind of protocol has been
implemented. However, we expect that the intriguing
results reported here will motivate future experimental
studies along similar lines. In this paper we discuss exact
results derived from the EW equation with a variable dif-
fusion constant, which allows us to investigate systemati-
cally the changes in the surface roughness in case growth
conditions are changed during the growth process. In-
terestingly, different dynamic regimes are encountered,
some of which are characterized by a power-law relax-
ation. Here, we do not have a specific experimental sys-
tem in mind. Instead, our interest is broader, namely, to
understand universal responses of a growing surface to
sudden changes of experimental parameters through the
study of simple models.
It is to be noted that our study is complementary to an
earlier investigation due to Majaniemi, Ala-Nissila, and
Krug [24]. Similarly to our work, these authors studied
the impact of a change of growth conditions on processes
described by linear growth equations. Whereas we dis-
cuss in the following the effects of a variable diffusion
constant, Majaniemi et al. analyzed how the growing
surface reacts to a change of the noise in the system.
There is also a second, theoretical motivation for our
study. Sudden changes of external conditions, as for ex-
ample due to a temperature quench, have been investi-
gated in recent years in a large variety of systems, rang-
ing from magnetic systems to glasses [25, 26, 27]. In the
most common setting, a system initially prepared in some
equilibrium state is suddenly brought out of equilibrium
through a temperature quench. If the system is charac-
terized by slow, non-exponential relaxation (as it is the
case for a ferromagnet quenched to or below the critical
point), interesting nonequilibrium processes and aging
phenomena are observed [27]. Some studies also focused
on slow relaxation encountered in up-quenches in which a
magnetic system initially in the ground state is quenched
to the critical point [28, 29]. Interestingly, however, slow
relaxation is usually not observed when both the initial
and final temperatures are below the critical point. In-
deed, for models with a discrete global symmetry such as
Ising or Potts models a competing ordered state can not
be reached if the starting point is close to one of the min-
ima of the equilibrium free energy. Consequently, non-
exponential relaxation is only encountered in this type
of quench if the system has a continuous symmetry, as
it is for example the case of the XY model [30, 31]. As
we will show in this paper, surface growth processes con-
stitute an interesting class of systems where a change of
parameters yields a transition from one correlated state
to another characterized by a power-law relaxation.
The paper is organized in the following way. In Sec-
tion II we discuss the dependence of the surface width,
derived from the EW equation, on the value of the diffu-
sion constant. Section III is devoted to the study of the
time evolution of the surface roughness following a sud-
den change of the diffusion constant. We thereby identify
different dynamic regimes and present a dynamic phase
diagram that summarizes the possible responses of the
growing interface. Finally, we end with a summary and
outlook in Section IV.
II.INTERFACE WIDTH
In order to study the effect of a change of external
conditions on simple growth processes we consider the
Edwards-Wilkinson equation (1) with a variable diffu-
sion constant ν (a microscopic realization of this process
has recently been discussed in [32]). When describing a
deposition or growth process by this equation, one implic-
itly assumes that ν and the noise amplitude D depend
on the experimental parameters, as for example the tem-
perature T. In the following we will not need to know
the explicit dependence of ν and D on these parameters
(which would be system dependent), and study how the
interface width changes when changing the value of ν.
The reaction of the growing surface to a change of the
noise has been studied in [24].
The stochastic EW equation (1), can be solved exactly
to give us for a fixed value of ν the width (squared)
W2(t) =
D
2νL
?
n
1 − e−2νtk2
k2
n
n
(3)
where kn= 2πn/L and the sum is over [−L/2,L/2] but
excluding the zero mode: n = 0. In Figure 1a,c we show
the time dependence of the surface width for, respec-
tively, the case with fixed L = 1000 at different ν’s and
the case with a fixed ν = 0.1768 and various L’s. As for
the RDSR process one distinguishes three regimes sepa-
rated by two crossover points: a random deposition (RD)
regime, followed by a EW regime, with a final crossover
to the saturation regime. In contrast to Family’s origi-
nal model, the initial RD process is not confined to very
early times t ≤ 1 but might extend to larger times. In
fact, the crossover time t1between the RD and the EW
regimes is shifted to higher values for decreasing diffu-
sion constants and diverges in the limit of vanishing ν.
As the crossover is smeared out, we identify the crossover
point with the intersection point of the straight lines fit-
ted to the two linear regimes in the log-log plots. We
have the identity W2= t in the RD regime, yielding
the width W1=√t1at the crossover point. In the EW

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regime the relation between width and deposition time
changes to W ∝ t1/4. This regime extends up to a second
crossover point (t2,W2), whose precise location depends
on the values of ν and D and beyond which the final
saturation regime prevails. The crossover between the
different regimes is further illustrated in Fig. 1b,d where
we show the time evolution of the effective exponent
βeff=dlogW
dlogt
=
νt?
?1 − e−2νtk2
n
e−2νtk2
n
?
n
n?/k2
n
. (4)
for the two cases.
FIG. 1: (a) Log-log plot of the surface width vs time for a sys-
tem of size L = 1000 and different diffusion constants. The
dotted lines have the slopes 1/2 and 1/4 expected in the ran-
dom deposition and EW regimes, respectively. The locations
of both crossover points depend on the diffusion constant.
The data are obtained from the exact solution of the EW
stochastic equation. (b) Time evolution of the effective ex-
ponent (4) for the data shown in (a). (c) Log-log plot of the
surface width vs time for systems of different sizes evolving at
the same diffusion constant ν = 0.1768. (d) Time evolution
of the effective exponent (4) for the data shown in (c). The
data in this and the following figures have been obtained for
D = 1.
We end this Section with a few remarks:
• The (“first”) crossover from the RD to the EW
regime, denoted by (t1,W1), depends only on ν.
For the range of L’s we explored, we find W2
t1∼= τ/ν, with a constant τ∼= 0.148. In the L → ∞
limit, τ → 1/2π (as shown in the Appendix, see also
[33]).
1=
• The (“second”) crossover from the EW to the sat-
uration regimes, denoted by (t2,W2), depends on
both ν and L. W2may be identified with the sat-
uration width, i.e.,?DL/8π2ν??
EW regime, we arrive at t2= (L/24τ)2t1(see Ap-
pendix).
nn−2[33]. Com-
bining this result with the line drawn through the
III. SUDDEN CHANGE OF GROWTH
CONDITIONS
With a clear picture of the properties of a surface grow-
ing under a constant diffusion constant, we proceed to
study the time evolution of the roughness when ν is sud-
denly changed. To investigate the response of the growth
process to this change, we use the following protocol: We
start at t = 0 with a flat surface and let the surface grow
at νi until time t = s, at which point we change the
diffusion constant to the final value νf. The change of
roughness is then monitored through the time evolution
of W2.
As our system displays three different roughness
regimes (RD, EW, and saturation), there can be in prin-
ciple nine scenarios for the change to be arranged. They
can be distinguished conveniently by (1) νi, the diffusion
constant of the initial growth, (2) νf, the final value of
the diffusion constant, and (3) s, the time at which ν
is suddenly changed. By choosing these three controls
judiciously, we can access all the scenarios. However,
covering all cases in detail is not the aim of this paper.
Instead, we are interested in the new phenomena asso-
ciated with the t-dependence of the width, W2(t,s), af-
ter (up- or down-) quenches. Obviously, for t ≫ s, the
surface roughness will settle into the value in an “unper-
turbed” system, grown at νf from the start. Denoted by
W2
u(t), it is just expression (1) with ν = νf. We also
refer to this as the “reference system.” To highlight the
changes, we also study the difference (with t > s)
∆W2(t,s) = |W2(t,s) − W2
u(t)| (5)
between quenched system and the reference. Clearly, this
quantity reveals how the roughness of the growing surface
adapts itself to the new “experimental” condition, and
behaves very differently for the various cases. Our goal is
to map out the regions in the νi-νf-s space corresponding
to the novel behavior following a quench.
To be precise, we will evolve the height h(x,t) starting
from h(x,0) = 0 with νi to time s and then continue
with νf until time t. At that point, we compute the
width squared and denote it by W2(t,νf;s,νi).
Our starting point is the exact solution of (1)
˜h(kn,t) = e−νfk2
n(t−s)
s
?
0
dt′e−νik2
n(s−t′)˜ η(kn,t′)
+
t ?
s
dt′′e−νfk2
n(t−t′′)˜ η(kn,t′′) (6)
written here in terms of the Fourier amplitudes for h(x,t)
and η(x,t):
˜h(kn,t) =
?
dxeiknxh(x,t), etc.Since
the noise is delta-correlated, ?|˜h|2? simplifies so that the

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width square is
W2(t,νf;s,νi) =
D
L
?
t ?
s
n

e−2νfk2
dt′e−2νfk2
n(t−s)
s
?
0

dt′e−2νik2
n(s−t′)
+
n(t−t′)
. (7)
Since the width square W2
tem is given by
u(t,νf) of the unperturbed sys-
W2(t,νf;s,νi) =D
L
?
n
t
?
0
dt′e−2νfk2
n(t−t′), (8)
we arrive at the exact result for the difference ∆W2=
??W2(t,νf;s,νi) − W2
∆W2=
L
n
s
?
0
?????
2νik2
n
u(t,νf)??:
e−2νfk2
?????
D
?
n(t−s)
dt′?
e−2νik2
n(s−t′)− e−2νfk2
n(s−t′)?
??????
=
D
L
?
?
n
e−2νfk2
n(t−s)
1 − e−2νik2
ns
−1 − e−2νfk2
2νfk2
ns
n
??????. (9)
For later convenience, we define Ω ≡ νf∆W2(t,νf;s,νi)
and note that it depends only on three scaling variables:
µ ≡ νi/νf,σ ≡ νfs,ρ ≡ t/s .
Explicitly, we have
Ω(µ,σ,ρ) ≡ νf∆W2
(10)
=
?????
D
L
?
?
n
e−2k2
nσ(ρ−1)
1 − e−2k2
2k2
nµσ
nµ
−1 − e−2k2
nσ
2k2
n
??????. (11)
To set the stage for discussions, we begin with the data
for some typical cases, all with s = 105, shown in Fig.
2. In order to be able to discuss the different cases for
a fixed s, we must work with a relatively small system:
L = 400. The dashed lines in (a-c) represent W2
an unperturbed system. With νf= 4.7 · 10−10, 0.00047,
and 0.18, the surface is, at the time of the quench, in the
(a) RD, (b) EW, and (c) saturation regimes, respectively.
The corresponding differences, ∆W2(t,s), are shown in
Fig. 2d-f.
The effects of two up-quenches into the RD regime,
from the EW (νi= 0.0046) and the saturation (νi= 0.23)
u(t) in
regimes, are displayed in Fig. 2a. As Fig. 2d shows, for
quenches into RD, the width W2(t,s) cannot reach that
of the reference system, W2
can achieve is a constant. The physical origin of this be-
havior lies in the linear growth of W2in the RD regime.
Thus, for two unperturbed systems started at different
times (say, t = t0and t1), the difference W2
a constant: t1−t0. In our case, the correlated growth up
to time s endowed our surface with a smaller W2(s) than
the reference W2
u(s). Immediately after the quench, cor-
related growth is simply replaced by independent growth
of different columns and the width W2(s) is “frozen” in
as a kind of “initial condition” (at t = s). As a result,
the difference ∆W2= W2
value W2
systems further in time, ∆W2will eventually vanish.
Turning next to quenches to the EW regime, we found
the most interesting behavior (Fig. 2b,e). The difference
∆W2initially decreases rapidly, before crossing over to
a slower, power-law decay at larger times (t ≫ s):
∆W2∼ t−γ
For example, for the cases shown in Fig. 2e, we mea-
sure close to the end of the time interval the exponents
γ = 0.72 for νi = 0.23, γ = 1.24 for νi = 0.0046, and
γ = 1.65 for νi= 4.7 · 10−10. As we argue below, these
are effective values, the asymptotic values of γ being 1/2
or 3/2. Below we will also discuss in more detail the
conditions under which these values can be expected. Fi-
nally, for a quench to the saturation regime, ∆W2decays
exponentially:
u(t). The “best” ∆W2(t,s)
0−W2
1is just
u(t) − W2(t,s) remains at the
u(s) − W2(s). Of course, if we follow these two
(12)
∆W2∼ exp(−κt) (13)
with a decay constant κ that depends both on the value
of the final diffusion constante νf and on the system size
L.
Up to now, we have shown only the simplest situation
where the system is well within a given initial regime at
the moment of the quench and, in addition, that it has
time to relax into a well defined regime of the reference
system. Clearly, as we let the final system evolve further,
it may crossover to a different regime (e.g., in case of Fig.
2a,d a crossing over to the EW and saturation regimes
will take place for larger t). Therefore, we should expect
the general relaxation process to be quite complex.
The closed forms (9,11) are not particularly transpar-
ent, as they involve all possible crossover behaviors. To
shed some light on the various scenarios, we consider
some limiting cases where simple properties (exponen-
tials and powers) can be extracted. A straightforward
case is νf(t − s) ≫ L2, so that νfk2
large. Then, the leading decay is exponential, namely
e−8π2νf(t−s)/L2, since the other terms in the sum will be
much smaller:
n(t − s) is always
?
e−8π2νf(t−s)/L2?4
≪ e−8π2νf(t−s)/L2.
,
?
e−8π2νf(t−s)/L2?9
,···
(14)

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FIG. 2: (Color online) (a-c) Time evolution of the width
square in case the diffusion constant is changed after 105time
steps. The dashed lines show W2
face growing at constant νf. In (a) the quench is to the RD
regime, whereas in (b) and (c) the quenches are to the EW
and saturation regimes, respectively. (d-f) The same cases as
shown in (a-c), but now the difference ∆W2, see Eq. (5), is
plotted. Qualitative different behavior is observed, depending
on the regime that the unperturbed system has at the quench
time.
u(t) for an unperturbed sur-
In the opposite limit, where νfs,νis ≪ 1, we have
?
2νik2
n
2νfk2
1 − e−2νik2
ns
−1 − e−2νfk2
ns
n
?
∼= (νf− νi)s2k2
n
(15)
to leading order. The summation over n then yields, for
t − s ≫ 1, a power-law decay: t−3/2, i.e., γ = 3/2.
In order to explore the three-dimensional parameter
space (µ,σ,ρ) in a comprehensive way, we evaluate nu-
merically the closed form (11). The exponent γ can be
defined effectively as −dlog(Ω)/dlog(t). In Fig. 5 we
show the contour plot of γ as a function of µ and σ for
ρ = 64. This plot reveals four different regimes: the
regime where Ω or ∆W2is constant (labeled by γ = 0),
two power-law regimes with values γ = 1/2 and γ = 3/2,
and finally a regime of exponential decay for large σ.
The different regimes are separated by crossover regions
where the effective exponent does not lock-in into one of
FIG. 3: Contour plot of γ as a function of νfs and νi/νf for
t/s = ρ = 64. Four different regimes, separated by crossover
regions, are identified.The two dashed lines separate the
three qualitatively different types of behavior encountered
when plotting the effective exponent as a function of t, see
Fig. 5.
the values 0, 1/2, or 3/2.
Fig. 4a shows how the extensions of the four regimes
depend on the value of ρ in cases where ρ ≫ 1. In-
terestingly, an increase of the value of ρ mainly shifts
the contours in the log(σ) vs.
(−1,1) direction. This is shown in Fig. 4b where we plot
log(νft) = log(σρ) vs. log(νis/νft) = log(µ/ρ). This
way of plotting indeed leads to an approximate data col-
lapse, which gets better for larger values of ρ. That this
data collapse is only approximate also follows from in-
spection of the exact solution (11). Still, Fig. 4b nicely
allows us to visualize the extent of the different dynamic
regimes for large ratios ρ.
Finally, in Fig. 5, we discuss the change of the effective
exponent γ as a function of t for various values of µ (and
s = 105). Note that an increasing time t corresponds in
Fig. 3 approximately to a cut along the (−1,1) direction,
so that we can distinguish three typical scenarios, sepa-
rated by the dashed lines there. Along the upper dashed
line, Fig. 5a shows the effective exponent rising to the
γ = 1/2 plateau where it remains for a long time before
crossing over to the regime where the difference ∆W2
vanishes exponentially fast (“γ = ∞”). For the region
above this line in Fig. 3, we can expect similar results.
Along the lower dashed line, the same behavior is seen,
except that the plateau value is now γ = 3/2 (Fig. 5c).
This can also be expected for the region below this lower
dashed line in Fig. 3. Between these two protocols, a
log(µ) plot along the