Mound formation and coarsening from a nonlinear instability in surface growth

22 Pages·0.44 MB·English

by  Buddhapriya Chakrabarti

Mound formation and coarsening from a nonlinear instability in surface growth
Buddhapriya Chakrabarti and Chandan Dasgupta∗
Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, INDIA
and
3 Condensed Matter Theory Unit, JNCASR, Bangalore 560064, INDIA
0 (February 1, 2008)
0
Westudy a class of one-dimensional, nonequilibrium, conserved growth equations for both non-
2
conserved and conserved noise statistics using numerical integration. An atomistic version of these
n
growth equations is also studied using stochastic simulation. The models with nonconserved noise
a
statistics arefoundtoexhibitmoundformation andpower-lawcoarseningwithslopeselection fora
J
range of values of the model parameters. Unlike previously proposed models of mound formation,
5
theEhrlich-Schwoebelstep-edgebarrier,usuallymodeledasalinearinstabilityingrowthequations,
1
is absent in our models. Mound formation in our models occurs due to a nonlinear instability in

whichtheheight(depth)ofspontaneouslygeneratedpillars (grooves) increasesrapidlyiftheinitial
]
h height (depth) is sufficiently large. When this instability is controlled by the introduction of an
c infinite number of higher-order gradient nonlinearities, the system exhibits a first-order dynamical
e
phasetransitionfromaroughself-affinephasetoamoundedoneasthevalueoftheparameterthat
m
measures theeffectivenessof control is decreased. Wedefine a new“order parameter” that may be
- usedtodistinguishbetweenthesetwophases. Inthemoundedphase,thesystemexhibitspower-law
t
a coarseningofthemoundsinwhichaselectedslopeisretainedatalltimes. Thecoarseningexponents
t forthecontinuumequationandthediscretemodelarefoundtobedifferent. Anexplanationofthis
s
. difference is proposed and verified by simulations. In the growth equation with conserved noise,
t
a we findthecurious result that thekinetically rough and moundedphases areboth locally stable in
m a region of parameter space. In this region, the initial configuration of the system determines its
- steady-state behavior.
d
n PACS numbers: 05.70.Ln, 64.60.Ht, 81.10.Aj, 81.15.Hi
o
c
[ I. INTRODUCTION tially increases with time and eventually saturates at a

2 value comparableto the sample size. In the second case,
v the characteristic length is the typical mound size R(t)
Theprocessofgrowingfilmsbythedepositionofatoms
6 whose time-dependence is qualitatively similar to that
4 on a substrate is of considerable experimental and theo- of ξ(t). However, the interface in this case looks well-
6 reticalinterest[1]. Whiletherehasbeenalotofresearch
orderedat lengthscales shorterthan R(t). Nevertheless,
2 on the process of kinetic roughening [1–3] leading to a
there are certain similarities between the gross features
1 self-affine interface profile, there has been much recent
of these two kinds of surface growth. First consider the
2
experimental[4–12]andtheoretical[4,6,13–25]interestin
0 simpler situation in which the slope of the sides of the
a different mode of surface growth involving the forma-
/ mounds remains constantin time. Simple geometrytells
t tion of “mounds” which are pyramid-like or “wedding-
a usthatifthe systemevolvesto asingle-moundstructure
m cake-like” structures. The precise experimental condi- at long times, then the “roughness exponent” α must
tions that determine whether the growth morphology
- be equal to unity. Also, the height-difference correla-
d would be kinetically rough or dominated by mounds are tion function g(r) is expected to be proportionalto r for
n presently unclear. However,many experiments show the
r R(t). This is consistent with α = 1. If the mound
o formationofmoundsthatcoarsen(thetypicallateralsize siz≪e R(t) increases with time as a power law, R(t) tn,
c ofthe moundsincreases)withtime. During this process, ∼
: during coarsening, then the interface width W, which
v the typicalslope ofthe sides of the pyramid-likemounds
is essentially the height of a typical mound, should also
i may or may not remain constant. If the slope remains
X increase with time as a power law with the same expo-
constantintime,thesystemissaidtoexhibitslopeselec-
r nent n. Thus, dynamic scaling with “growth exponent”
a tion. Asthemoundscoarsen,thesurfaceroughnesschar- β equal to n, and “dynamical exponent” z equal 1/n is
acterizedby the root-mean-squarewidth of the interface
recovered. Ifthe moundslopes(t)increaseswithtime as
increases. Eventually, at very long times, the system is a power law, s(t) tθ (this is known in the literature as
expected to evolve to a single-mound structure in which ∼
steepening),thenoneobtainsbehaviorsimilarto anoma-
the mound size is equal to the system size.
lous dynamical scaling [26] with β =n+θ, z =1/n.
There are obvious differences between the structures
of kinetically rough and mounded interfaces. In the first These similarities between the grossscaling properties
case, the interface is rough in a self-affine way at length of kinetic roughening with a large value of α and mound
scales shorter than a characteristic length ξ(t) that ini- formation with power-law coarsening make it difficult to
1
experimentally distinguish between these two modes of in a 1d model with both discrete and continuum fea-
surface growth. This poses a problem in the interpreta- tures. We also note that a new mechanism for mound-
tionofexperimentalresults[11,12]. Existingexperiments ing instability has been discoveredrecently [17,18]. This
on mound formation show a wide variety of behavior. instability, generated by fast diffusion along the edges
Withoutgoingintothedetailsofindividualexperiments, of monatomic steps, leads to the formation of quasi-
we note that some experiments show mound coarsening regularshapedmoundsintwoorhigherdimensions. The
with a time-independent “magic” slope, whereas other effects of this instability have been studied in simula-
experiments do not show any slope selection. The de- tions[17,18,20,24,25]. Thewidevarietyofresults[17–25]
tailed morphology of the mounds varies substantially obtained from simulations of different models, combined
from one experiment to another. The reported values with similar variations in the experimental results, have
of the coarsening exponent n show a large variation in made it very difficult to identify the microscopic mecha-
the range 0.15-0.4. nism of mound formation in surface growth.
Traditionally, the formation of mounds has been In this paper, we show that mound formation, slope
attributed to the presence of the so-called Ehrlich- selection and power-law coarsening in a class of one-
Schwoebel(ES)step-edgebarrier[27,28]thathindersthe dimensional (1d) continuum growth equations and dis-
downwardmotionofatomsacrosstheedgeofastep. This creteatomistic models canoccurfromamechanismthat
step-edge diffusion bias makes it more likely for an atom is radically different from the ones mentioned above.
diffusingonaterracetoattachtoanascendingstepthan Our study is based on the conserved nonlinear growth
to a descending one. This leads to an effective “uphill” equation proposed by Villain [29] and by Lai and Das
surface current [29] that has a destabilizing effect, lead- Sarma [30], and an atomistic version [31] of this equa-
ingto the formationofmoundedstructuresasthe atoms tion. We have studied the behavior of the continuum
on upper terraces are prevented by the ES barrier from equation by numerical integration, and the behavior of
coming down. the atomistic model by stochastic simulation. Previous
This destabilizing effect is usually represented in con- work [32–34] on these systems showed that they exhibit
tinuum growth equations by a linear instability. Such a nonlinear instability, in which pillars (grooves) with
growth equations usually have a “conserved” form in height (depth) greater than a critical value continue to
which the time-derivative of the height is assumed to be grow rapidly. This instability can be controlled [32–34]
equal to the negative of the divergence of a nonequilib- bythe introductionofaninfinite number ofhigher-order
rium surface current j. The effects of an ES barrier are gradient nonlinearities. When the parameter that de-
modeled in these equations by a term in j that is pro- scribes the effectiveness of control is sufficiently large,
portional to the gradient of the height (for small values the controlled models exhibit [32–34] kinetic roughen-
of the gradient)with a positive proportionalityconstant. ing, characterized by usual dynamical scaling with ex-
Such a term is manifestly unstable, leading to unlimited ponent values close to those expected from dynamical
exponential growth of the k = 0 Fourier components of renormalization group calculations [30]. As the value of
6
the height. This instability has to be controlledby other the control parameter is decreased, these models exhibit
nonlineartermsinthegrowthequationinordertoobtain transient multiscaling [32–34] of height fluctuations. For
aproperdescriptionofthelong-timebehavior. Anumber yet smaller values of the control parameter, the rapid
of different choices for the nonlinear terms have been re- growth of pillars or grooves causes a breakdown of dy-
portedintheliterature[4,6,13,14,22,23]. Ifthe“ESpart” namicalscaling,withthe widthversustimeplotshowing
ofjhasoneormorestablezerosasafunctionoftheslope asharpupwarddeviation[33]fromthepower-lawbehav-
s, then the slope of the mounds that form as a result of iorfoundatshorttimes(beforetheonsetofthenonlinear
the ES instability is expected to stabilize at the corre- instability).
sponding value(s) of s at long times. The system would Wereportheretheresultsofadetailedstudyofthebe-
then exhibit slope selection. If, on the other hand, this haviorofthesemodelsintheregimeofsmallvaluesofthe
partofjdoesnothaveastablezero,thenthemoundsare controlparameterwhereconventionalkineticroughening
expected to continueto steepen with time. Analytic and is not observed. We find that in this regime, the inter-
numerical studies of such continuum growth equations face self-organizes into a sawtooth-like structure with a
have produced a wide variety of results, such as power- seriesoftriangular,pyramid-likemounds. Thesemounds
law coarsening and slope selection with n = 1/4 [13] or coarsen in time, with larger mounds growing at the ex-
n 0.17[6]in two dimensions,power-lawcoarseningac- penseofsmallerones. Inthiscoarseningregime,apower-
≃
companied by a steepening of the mounds [4,14,16], and law dependence of the interface width on time is recov-
acomplexcoarseningprocess[22,23]inwhichthegrowth ered. The slope of the mounds remains constant during
of the mound size becomes extremely slow after a char- the coarsening process. In section II, the growth equa-
acteristic size is reached. tion and the atomistic model studied in this work are
There are several atomistic, cellular-automaton-type definedandthe numericalmethods wehaveusedtoana-
models[19–21]thatincorporatetheeffectsofanESdiffu- lyzetheirbehavioraredescribed. Thebasicphenomenol-
sionbarrier. Formationandcoarseningofmounds inthe ogyofmound formationandslope selectionin these sys-
presence of an ES barrier have also been studied [22,23] tems is described in detail in section III. Specifically, we
2
show that the nonlinear mechanism of mound formation obtained for the dynamics of mounds in the atomistic
in these systems is “generic” in the sense that the qual- model are consistent with this explanation.
itative behavior does not depend on the specific form of In section VI, we consider the behavior of the con-
the function used for controlling the instability. In par- tinuum growth equation for “conserved”noise statistics.
ticular, we find very similar behavior for two different The nonlinearinstabilityfound inthe nonconservedcase
forms of the control function: one used in earlier stud- is expected to be present in the conserved case also.
ies [32–34] of these systems, and the other one proposed However, there is an important difference between the
by Politi and Villain [23] from physical considerations. two cases. The nonconserved model exhibits anomalous
Since the linear instability used conventionally to model dynamical scaling, so that the typical nearest-neighbor
the ESmechanismisexplicitly absentinourmodels,our height difference continues to increase with time, and
workshowsthat the presenceofstep-edge barriersisnot the instability is always reached [33] at sufficiently long
essentialformoundformation. Theslopeselectionfound times, even if the starting configuration is perfectly flat.
in our models is a true example of nonlinear patternfor- Since the continuum model with conserved noise statis-
mation: since the nonequilibrium surface current in our tics exhibits [35] usual dynamic scaling with α < 1, the
models vanishes for all values of constant slope, the se- nearest-neighborheightdifferenceisexpectedtosaturate
lectedvalueoftheslopecannotbepredictedinanysim- atlongtimes iftheinitialstateofthe systemisflat. Un-
ple way. This is in contrast to the behavior of ES-type der these circumstances, the occurrence of the nonlinear
models where slope selection occurs only if the surface instability in runs started from flat states would depend
current vanishes at a specific value of the slope. on the values of the parameters. Specifically, the insta-
Next, in section IV, we show that the change in the bility may not occur at all if the value of the nonlinear
dynamical behavior of the system (from kinetic rough- coefficientinthegrowthequationissufficientlysmall. At
ening to mound formation and coarsening) may be de- the same time, the instability can be initiated by choos-
scribed as a first-order nonequilibrium phase transition. ing an initial state with sufficiently high (deep) pillars
Since the instability in our models is a nonlinear one, (grooves). Sincemoundformationinthesemodelsiscru-
the flat interface is locally stable in the absence of noise cially dependent on the occurrence of the instability, the
for all values of the model parameters (the strength of arguments above suggest that the nature of the long-
the nonlinearityandthe valueofthe controlparameter). time steady state reached in the conserved model may
The mounded phase corresponds to a different station- depend on the choiceof the initial state. Indeed, we find
arysolutionofthedynamicalequationsintheabsenceof fromsimulationsthatinaregionofparameterspace,the
noise. Weusealinearstabilityanalysistofindthe“spin- mounded and kinetically rough phases are both locally
odal boundary” in the two-dimensional parameter space stable and the steady state configuration is determined
across which the mounded stationary solution becomes by the choice of the initial configurationof the interface.
locallyunstable. Weshowthattheresultsofthisnumer- These results imply the surprising conclusion that the
ical stability analysis can also be obtained from simple long-time, steady-state morphology of a growing inter-
analyticarguments. Toobtainthephaseboundaryinthe face,as wellasthe dynamicsofthe processby whichthe
presenceofnoise,wefirstdefine anorder parameterthat steady state is reached may be “history dependent” in
is zero in the kinetically roughphase and nonzero in the the sense that the behavior would depend crucially on
mounded phase. We combine the numerically obtained the choice of the initial state. A summary of our find-
results forthis orderparameterfor differentsample sizes ings and a discussion of the implications of our results
with finite-size scaling to confirm that this order param- are provided in Sec.VII. A summary of the basic results
eter exhibits the expected behavior in the two phases. of our study was reported in a recent Letter [36].
The phase boundary that separates the mounded phase
from the kinetically rough one is obtained numerically.
Thephaseboundariesforthecontinuummodelwithtwo
II. MODELS AND METHODS
different forms of the control function and the atomistic
model are found to be qualitatively similar.
The results of a detailed study of the process ofcoars- Conservedgrowthequations(deterministic partofthe
ening of the mounds are reported in section V. Surpris- dynamics having zero time derivative for the k = 0
ingly, we find that the coarsening exponents of the con- Fourier mode of the height variable) with nonconserved
tinuum equation and its atomistic version are different. noisearegenerallyused[2]to modelnonequilibriumsur-
We propose a possible explanation of this result on the facegrowthinmolecularbeamepitaxy(MBE).Thecon-
basisofananalysisofthecoarseningprocessinwhichthe servation is a consequence of absence of bulk vacancies,
problemismappedtothatofofaBrownianwalkerinan overhangs and desorption (evaporation of atoms from
attractiveforcefield. Inthismapping,theBrownianwalk the substrate) under optimum MBE growth conditions.
issupposedtodescribethenoise-inducedrandommotion Thus, integrating over the whole sample area gives the
of the peak of a mound, and the attractive “force” rep- number of particles deposited. This conservation is not
resentstheinteractionbetweenneighboringmoundsthat strictly valid because of “shot noise” fluctuations in the
leads to coarsening. We show that the numerical results beam. The shot noise is modeled by an additive noise
3
term η(r,t) in the equation of motion of the interface. results that are very similar to those obtained from cal-
The noise η is generally assumed to be delta-correlated culations in which these simple definitions are used. We
in both space and time: have also checked that the results obtained in the deter-
ministic limit (η = 0) by using a more sophisticated in-
η(r,t)η(r′,t′) =2Dδd(r r′)δ(t t′), (1) tegration routine [37] closely match those obtained from
h i − −
the Euler method with sufficiently small values of the
where r is a point on a d-dimensionalsubstrate. Thus, a integration time step.
conserved growth equation may be written in a form Wehavealsostudiedanatomisticversion[31]ofEq.(3)
in which the height variables h are integers. This
i
∂h { }
= j+η, (2) model is defined by the following deposition rule. First,
∂t −∇· a site (say i) is chosen at random. Then the quantity
where h(r,t) is the height at point r at time t, and j is
K ( h )= ˜2h +λ ˜h 2 (6)
the surface current density. The surface current mod- i { j} −∇ i |∇ i|
els the deterministic dynamics at the growth front. As
is calculated for the site i and all its nearest neighbors.
mentioned in section I, the presence of an ES step-edge
Then,aparticleisaddedtothesitethathasthesmallest
barrier is modeled in continuum equations of the form
valueofK amongthe siteiandits nearestneighbors. In
of Eq.(2) by a term in j that is proportionalto the slope
the case of a tie for the smallest value, the site i is cho-
s= h,withapositiveconstantofproportionality. This
∇ sen if it is involved in the tie; otherwise, one of the sites
makestheflatsurface(h(r)constantforallr)linearlyun-
involved in the tie is chosen randomly. The number of
stable. This instability is controlled by the introduction
depositedlayersprovidesameasureoftimeinthismodel.
of terms involving higher powers of the local slope s and
It was found in earlier studies [32–34] that both these
higher-order spatial derivatives of h.
models exhibit a nonlinear instability in which isolated
We consider the conserved growth equation proposed
structures (pillars for λ > 0, grooves for λ < 0) grow
byVillain[29]andLaiandDasSarma[30]fordescribing
rapidly if their height (depth) exceeds a critical value.
MBE-typesurface growthin the absence of ES step-edge
This instability can be controlled [32–34] by replacing
barriers. This equation is of the form
˜h 2 in Eqns.(4) and (6) by f( ˜h 2) where the non-
i i
|∇ | |∇ |
∂h′(r,t′)/∂t′ = ν 4h′+λ′ 2 h′ 2+η′(r,t′), (3) linear function f(x) is defined as
− ∇ ∇ |∇ |
1 e−cx
whereh′(r,t′) representsthe heightvariableatthe point f(x)= − , (7)
r at time t′. This equation is believed [2] to provide c
a correct description of the kinetic roughening behavior
c > 0 being a control parameter. We call the result-
observed in MBE-type experiments [12].
ing models “model I” and “model II”, respectively. This
In our study, we numerically integrate the 1d version
replacement, amounts to the introduction of an infinite
of Eq.(3) using a simple Euler scheme [33]. Upon choos-
series of higher-order nonlinear terms. The time evolu-
ingappropriateunits oflengthandtime anddiscretizing
tion of the height variables in model I is, thus, given by
in space and time, Eq.(3) is written as [33]
h (t+∆t) h (t)=∆t˜2[ ˜2h (t)
h (t+∆t) h (t)=∆t˜2[ ˜2h (t)+λ ˜h (t)2] i − i ∇ −∇ i
i − i ∇ −∇ i |∇ i | +λ(1 e−c|∇˜hi(t)|2)/c]+√∆tη (t). (8)
+√∆tη (t), (4) − i
i
In model II, the quantity K is defined as
i
where h (t) represents the dimensionless height variable
i
at the lattice point i at dimensionless time t, ˜ and ˜2 K ( h )= ˜2h +λ(1 e−c|∇˜hi|2)/c. (9)
are lattice versionsof the derivative and Lapla∇cianop∇er- i { j} −∇ i −
ators, and ηi(t) is a random variable with zero average While the function f(x) was introduced in the earlier
and variance equal to unity. These equations, with an work purely for the purpose of controlling the nonlin-
appropriate choice of ∆t, are used to numerically follow ear instability, it turns out that the introduction of this
the time evolution of the interface. In most of our stud- functioninthegrowthequationisphysicallymeaningful.
ies, we have used the following definitions for the lattice Politi and Villain [23] have shown that the nonequilib-
derivatives: rium surface current that leads to the 2 h′ 2 term in
Eq.(3) should be proportional to h∇′ 2|∇whe|n h′ is
˜hi =(hi+1 hi−1)/2, small, and should go to zero whe∇n|∇ h|′ is larg|e∇. T|he
∇ − |∇ |
∇˜2hi =hi+1+hi−1−2hi. (5) introduction of the “controlfunction” f(|∇˜hi|2) satisfies
thisphysicalrequirement. Wehavealsocarriedoutstud-
Wehavecheckedthattheuseofmoreaccurate,left-right ies of a slightly different model (which we call “model
symmetricdefinitionsofthelattice derivatives,involving IA”) in which the function f(x) is assumed to have a
more neighbors to the left and to the right [33], leads to form suggested by Politi and Villain:
4
x
f(x)= , (10) time t. The time at which this departure occurs varies
1+cx
from run to run. This behavior for model I with λ=4.0
and c=0.02 is shown by the dash-dotted line in Fig.1.
where c is, as before, a positive control parameter. This
This instability leads to the formation of a large num-
function has the same asymptotic behavior as that of
ber of randomly distributed pillars of height close to
the function defined in Eq.(7). As we shall show later,
h . As the system evolves in time, the interface self-
theresultsobtainedfromcalculationsinwhichthesetwo max
organizestoformtriangularmoundsofafixedslopenear
different forms of f(x) are used are qualitatively very
these pillars. These mounds then coarsen in time, with
similar. In fact, we expect that the qualitative behavior
largemoundsgrowinglargerattheexpenseofsmallones.
of these models would be the same for any monotonic
In this coarsening regime, a power-law growth of W in
function f(x) that satisfies the following requirements:
time is recovered. The slope of the sides of the triangu-
(i) f(x) must be proportional to x in the small-x limit;
larmoundsremainsconstantduringthisprocess. Finally,
and (ii) it must saturate to a constant value as x .
→∞ the systemreachesa steadystate with one peak andone
We have carried out extensive simulations of both
trough (if periodic boundary conditions are used) and
these models for different system sizes. The results
remains in this state for longer times. The interface pro-
reported here have been obtained for systems of sizes
filesinthekineticallyroughphase(obtainedforrelatively
40 L 1000. There is no significant dependence of
≤ ≤ large values of c) and the mounded phase (obtained for
theresultsonL. Thetimestepusedinmostofourstud-
small c) are qualitatively different. This difference is il-
ies of models I and IA is ∆t = 0.01. We have checked
lustrated in Fig.2 that shows typical interface profiles in
thatverysimilarresultsareobtainedforsmallervaluesof
the two differentphases. This figure alsoshowsa typical
∆t. We used both unbounded (Gaussian) and bounded
interface profile for model IA in the mounded regime, il-
distributions for the random variables η in our simula-
i lustrating the fact that the precise choice of the control
tionsofmodelsIandIA,withnosignificantdifferencein
function f(x) is notcrucialfor the formationofmounds.
the results. For computational convenience, a bounded
The evolution of the interface structure in the mounded
distribution (uniform between +√3 and √3) was used
− regime of model I is illustrated in Fig.3 which shows
in most of our calculations. Unless otherwise stated, the
the interface profiles obtained in a typical L = 200 run
results described in the following sections were obtained
starting from a flat initial state at three different times:
using periodic boundary conditions. The effects of using
t=200(beforetheonsetoftheinstability),t=4000(af-
other boundary conditions will be discussed in the next
tertheonsetoftheinstability,inthecoarseningregime),
section.
and t = 128000 (in the final steady state). This figure
alsoshows the steady-stateinterface profile ofa L=500
sample with the same parameters, to illustrate that the
III. MOUND FORMATION AND SLOPE
results do not depend on the sample size.
SELECTION
Very similar behavior is found for model II. Since the
heights in this atomistic model can increase only by dis-
It has been demonstrated earlier [32,33] that if the crete amounts in each unit of discrete time, the increase
controlparameter c is sufficiently large, then the nonlin- ofW attheonsetoftheinstabilityislessrapidherethan
ear instability is completely suppressed and the models in the continuum models I and IA. Nevertheless, the oc-
exhibit the usual dynamical scaling behavior with the currence of the instability for small values of c shows
expected [30] exponent values, β 1/3, z 3, and up as a sharp upward deviation of the W versus t plot
≃ ≃
α = βz 1. This behavior for model I is illustrated from the initial power-law behavior with β 1/3. This
≃ ≃
by the solid line in Fig.1, which shows a plot of the is illustrated by the dash-dotted line in Fig.4, obtained
width W as a function of time t for parameter values from simulations of model II with λ = 2.0, c = 0.005.
λ=4.0 and c = 0.06. As the value of c is decreased with This behavior is to be contrasted with that for λ = 2.0,
λ held constant, the instability makes its appearance: c = 0.015, shown by the full line in Fig.4, where the
the height h of an isolated pillar (for λ > 0) increases nonlinear instability is absent. The difference between
0
in time if h (λ,c) < h < h (λ,c). The value of the surface morphologies in the two regimes of model II
min 0 max
h is nearly independent of c, while h increases is illustrated in Fig.5. The kinetically rough, self-affine
min max
as c is decreased [33]. If c is sufficiently large, h is morphologyobtainedforc=0.02isclearlydifferentfrom
max
small and the instability does not affect the scaling be- themoundedprofilefoundforc=0.005. Thetimeevolu-
haviorofglobalquantitiessuchasW,althoughtransient tionoftheinterfaceinthemoundedregimeofthismodel
multiscaling atlength scalesshorterthan the correlation is illustrated in Fig.6. The general behavior is clearly
length ξ t1/z may be found [32,33] if c is not very similar to that found for models I and IA. This figure
∼
large. As c is decreased further, h becomes large, also shows a properly scaled plot of the interface profile
max
and when isolated pillars with h > h are created at ofa L=500sample with the same parametersat a time
0 min
an initially flat interface through fluctuations, the rapid in the coarsening regime. It is clear from this plot that
growth of such pillars to height h leads to a sharp the nature of the interface and the value of the selected
max
upwarddeparture from the power-lawscaling of W with slope do not depend on the sample size.
5
The occurrence of a peak and a symmetrically placed model I in the mounded and kinetically rough phases.
troughin the steady-state profiles shownin Figs 3 and 6 A bimodal distribution is seen for the mounded phase,
is a consequence of using periodic boundary conditions. thetwopeakscorrespondingtothe valuesoftheselected
The deterministic part of the growth equation of Eq.(8) slopeandthe heightofthe pillarsatthe topandbottom
strictlyconservestheaverageheightifperiodicboundary of the pyramids. The kinetically rough phase, on the
conditions are used. So, the averageheight remains very other hand, exhibits a distribution peaked atzero. Fig.9
close to zero if the initial state is flat, as in most of our shows the values of the selected slope at different times
simulations. The steady-state profile must have at least inthecoarseningregimeofmodelI.Theconstancyofthe
one peak and one trough in order to satisfy this require- slopeisclearlyseeninthisplot. Allthesefeaturesremain
ment. Also, it is easy to show that if the slopes of the trueforthediscretemodel. PlotsofthedistributionP(s)
”uphill” and ”downhill” parts of the steady-state profile at two different times in the coarsening regime of model
are the same in magnitude (this is true for our mod- II are shown in Fig.10. The peak position shows a small
els), then the two extrema must be separated by L/2. shift in the positive direction as t is increased, but this
≃
Therefore, it is clear that the steady state obtained in shift is smallcomparedto the width ofthe distributions,
simulationswithperiodicboundaryconditionsmusthave indicatingnearconstancyoftheselectedslope. Thevalue
a peak and a symmetrically placed trough separated by ofthe selected slope depends onthe parametersλ andc.
distance L/2. This is discussed in the next section.
≃
To check whether the basic phenomenology described
above depends on the choice of the boundary condition,
we have carried out test simulations using two other IV. DYNAMICAL PHASE TRANSITION
boundary conditions: “fixed” boundary condition, in
which the height variable to the left of i=1, and to the The instability that leads to mound formation in our
right of i=L are pinned to zero at all times; and “ zero models is a nonlinear one,so that the perfectly flatstate
flux” boundary condition with vanishing first and third oftheinterfaceisalocallystablesteady-statesolutionof
derivatives of the height at the two ends of the sample. the zero-noise growth equation for all parameter values.
For these boundary conditions, the deterministic part of When the instability is absent (e.g. for large values of
the growth equation does not strictly conserve the av- the control parameter c), this “fixed-point” solution of
erage height. As a result, the symmetry between the the noise-free equation is transformed to the kinetically
mound and the trough, found in the long-time steady roughsteadystateinthepresenceofnoise. Themounded
state obtained for periodic boundary condition, is not steady state obtained for small values of c must corre-
present if one of the other boundary conditions is used. spond to a different fixed point of the zero-noise growth
Inparticular,it is possible to stabilize a single moundor equation. Such nontrivial fixed-point solutions may be
asingletroughinthesteadystatefortheotherboundary obtained from the following simple calculation.
conditions. Sincetheheightsatthetwoendsmustbethe The profile near the top (i = i ) of a triangular
0
same for fixed boundary condition, the two extrema in a mound may be approximated as h = x +x , h =
i0 0 1 i0+j
configuration with one mound and one trough must be x (j 1)x , where x is the height of the pillar at
0 2 1
separatedby L/2,asshowninFig.7. Thetwoextrema the−top| |o−f the mound and x is the selected slope. This
≃ 2
wouldnotbeseparatedby L/2forthezero-fluxbound- profile would not change under the dynamics of Eq.(8)
≃
ary condition. These effects of boundary conditions are with no noise if the following conditions are satisfied:
illustrated in Fig.7 which shows profiles in the mounded
regime obtained for the three different boundary condi- ∇˜2hi0 −λ(1−e−c|∇˜hi0|2)/c
tinionthsismfiengutiroentehdaatbtohvee.baItsiicspclheeanrofmroemnotlhoegyr,esi.uel.tsthsehofworn- =∇˜2hi0±1−λ(1−e−c|∇˜hi0±1|2)/c
mation and coarsening of mounds and slope selection, is =∇˜2hi0±2−λ(1−e−c|∇˜hi0±2|2)/c. (11)
notaffectedbythechoiceofboundaryconditions. Inpar-
ticular, the values of the selected slope and the heights These conditions lead to the following pair of non-linear
of the pillars at the top of a mound and the bottom of equationsforthevariablesx1 andx2 usedtoparametrize
a trough remain unchanged when boundary conditions the profile near the top of a mound:
other than periodic are used.
The selection of a “magic slope” during the coars- 2x1 λ[1 e−cx22]/c=0,
− −
ening process is clearly seen in the plots of Fig.3 and 3x x λ[1 e−c(x1+x2)2/4]/c=0. (12)
1 2
Fig.6. More quantitatively, the probability distribution − − −
of the magnitude of the nearest-neighbor height differ- These equations admit a non-trivial solution for suffi-
ences s h h is found to exhibit a pronounced ciently small c, and the resulting values of x and x are
i i+1 i 1 2
≡ | − |
peak at the selected value of the slope, and the position found to be quite close to the results obtained from nu-
of this peak does not change during the coarsening pro- mericalintegration. Asimilaranalysisfortheprofilenear
cess. Fig.8 shows a comparisonof the distribution of the the bottom of a trough (this amounts to replacing x by
2
magnitude of the nearest-neighbor height difference for x inEq.(12))yieldsslightlydifferentvaluesforx and
2 1
−
6
x . The full stable profile (a fixed point of the dynamics The local stability of the mounded fixed point may be
2
withoutnoise)withonepeakandonetroughmaybeob- determined from a calculation of the eigenvalues of the
tained numerically by calculating the values of h for stability matrix, M = ∂g /∂h , evaluated at the fixed
i ij i j
{ }
which g , the term multiplying ∆t in the right-hand side point. We find that the largesteigenvalue of this matrix
i
of Eq.(8), is zero for all i. The fixed-point values of h (disregarding the trivial zero eigenvalue associated with
i
{ }
satisfy the following equations: anuniformdisplacementofthe interface,h h +δ for
i i
→
all i) crosses zero at c = c (λ) (see Fig.(12)), signaling
1
g = ˜2[ ˜2h +λ(1 e−c|∇˜hi|2)/c]=0 for all i. (13) an instability of the mounded profile. The structure of
i i
∇ −∇ − Eq.(8) implies that c (λ) λ2. Thus, for 0<c<c (λ),
1 1
∝
Anumericalsolutionofthesecouplednonlinearequations the dynamics of Eq.(8) without noise admits two locally
shows that the small mismatch between the values of x2 stable invariantprofiles: a trivial, flat profile with hi the
near the top and the bottom is accommodated by creat- sameforalli,anda non-trivialone withonemoundand
ing afew ripples nearthe top. The numericallyobtained one trough. Depending on the initial state, the noise-
fixed-point profile for a L = 500 system with λ = 4.0, less dynamics takes the system to one of these two fixed
c=0.02 is shown in Fig.11, along with a typical steady- points. For example, an initial state with one pillar on
state profile for the same system. The two profiles are a flat background is driven by the noiseless dynamics to
foundtobenearlyidentical,indicatingthatthemounded the flat fixed point if the height of the pillar is smaller
steady state in the presence of noise corresponds to this thana criticalvalue,and to the mounded one otherwise.
fixed-point solution of the noiseless discretized growth The “relevant” perturbation that makes the mounded
equation. fixed point unstable at c = c1 is a uniform vertical rela-
Fixed-point solutions of the continuum equation, tivedisplacementofthesegmentoftheinterfacebetween
Eq.(3),withν =1and h2 replacedbyf( h2)where the peak and the trough of the fixed-point profile. This
f(x)hastheformshown|∇in|Eq.(10)mayalso|∇be|obtained can be seen by numerically evaluating the right eigen-
by a semi-analytical approach following Racz et al. [38]. vector corresponding to the eigenvalue of the stability
We consider stationarysolutions of the continuum equa- matrix that crosses zero at c=c1. This is demonstrated
tionthatsatisfythefollowingfirst-orderdifferentialequa- intheinsetofFig.12. Also,examinationofthetime evo-
tion with appropriate boundary conditions: lution of the mounded structure for values of c slightly
higher than c shows that the instability of the struc-
1
ds s2 ture first appears at the bottom of the trough. Taking
+λ =A, (14)
−dx 1+cs2 cuefromtheseobservations,thevaluec1 canbeobtained
from a simple calculation. We consider the profile near
where s(x)=dh(x)/dx is the local slope of the interface the bottom of a trough at i = i . As discussed above,
0
and A is a constant that must be positive in order to theprofileneari maybeparametrizedash =x +x ,
0 i0 0 1
obtain a solution that resembles a triangular mound. At h = x + (j 1)x , and the values of x and x
i0+j 0 | |− 2 1 2
large distances from the peak of the mound, the slope may be obtained by solving a pair of nonlinear equa-
s would be constant, so that ds(x)/dx would vanish, tions, Eq.(12) with x replaced by x . We now con-
2 2
−
whereas the second term would give a positive contri- sider a perturbation of this profile, in which the heights
butionifλispositive. Atthepeakoftheprofile,thesec- on one side of i are all increased by a small amount δ
0
ond term would be zero because s is zero, but ds(x)/dx (i.e. h = x +(j 1)x +δ, h = x +(j 1)x
i0+j 0 − 2 i0−j 0 − 2
would be negative, making the left-hand side of Eq.(14) with j > 0), and use Eq.(8) to calculate how δ changes
positive. While a closed-form solution of this differen- with time, assuming its value to be small. The require-
tial equation cannot be obtained, the value of s(x) at ment that δ must decrease with time for the fixed-point
anypointxmaybecalculatedwithanydesireddegreeof structureto belocallystableleadstothe followingequa-
accuracyby numericallysolvingasimple algebraicequa- tion for the value of c at which the structure becomes
tion. The height profile is then obtained by integrating unstable:
s(x)withappropriateboundaryconditions. Inourcalcu-
lation, we used the procedure of Racz et. al. [38] to take λ(x x )e−c(x1−x2)2/4 =1, (15)
1 2
2 −
into account periodic boundary conditions. In Fig. 11,
wehaveshownatypicalsteady-stateprofileofaL=200 Bysubstitutingthenumericallyobtainedvaluesofx and
1
sample of model IA with λ = 4.0 and c = 0.01, and x inthisequation,thecriticalvalue,c (λ),oftheparam-
2 1
a fixed-point solution of the corresponding continuum etercisobtainedasafunctionofλ. Thevaluesobtained
equation. The value of the constant A in Eq.(14) was thiswayareingoodagreementwiththoseobtainedfrom
chosentoyieldthesameslopeasthatofthesteady-state our full numerical calculation of the eigenvalues of the
profileofthediscretemodel. Theseresultsshowthatthe stability matrix. The “spinodal” lines (i.e. the lines in
steady-statepropertiesforthetwoformsoff(x)arevery the c λ plane beyond which the mounded fixed point
−
similar, and the continuum equation admits stationary is unstable) for models I and IA are shown in Fig.13.
solutions that arevery similar to those of the discretized Both these lines have the expected form, c (λ) λ2. It
1
∝
models. wouldbe interestingtocarryoutasimilarstabilityanal-
7
1 1 1
ysisforthemoundedstationaryprofile(seeFig.11)ofthe f(m)= am2 bm3+ um4. (17)
continuum equation corresponding to model IA. Such a 2 − 3 4
calculationwouldhavetobeperformedwithout discretiz-
ing space if we want to address the question of whether ItiseasytoshowthatforT <T <T =T +b2/(4a u),
0 s 0 0
the behavior of the truly continuum equation is similar the function f(m) has two local minima, one at m = 0,
to that of the discretized versions considered here. We andtheotheratapositivevalueofm. Thesetwominima
have not succeeded in carrying out such a calculation: represent the two phases of the system. This system ex-
since the mounded stationary profiles for the continuum hibits a first order equilibrium phase transition from the
equation are obtained from a numerical calculation, it disordered phase (m=0) to an ordered phase with pos-
would be extremely difficult, if not impossible, to carry itive m as the temperature is decreased. The transition
outalinearstabilityanalysisforsuchstationarysolutions temperatureT liesbetweenT andT . Thetemperature
c 0 s
without discretizing space. T at which the minimum corresponding to the ordered
s
In the presence of the noise, the perfectly flat fixed phasedisappearsiscalledthe“spinodal”temperaturefor
point transforms to the kinetically rough steady state, theorderedphase. Thespinodaltemperatureforthedis-
and the non-trivial fixed point evolves to the mounded ordered phase is T .
0
steady state shown in Fig.11. A dynamical phase tran- Now consider the dynamics of this system according
sition at c = c2(λ) < c1(λ) separates these two kinds to the following time-dependent Ginzburg-Landauequa-
of steady states. To calculate c2(λ), we start a system tion [39]:
at the mounded fixed point and follow its evolution ac-
cording to Eq.(8) for a long time (typically t = 104) to
∂ψ(r,t) δF[ψ]
checkwhetheritreachesakineticallyroughsteadystate. = Γ +η(r,t), (18)
∂t − δψ(r,t)
Byrepeatingthisproceduremanytimes,theprobability,
P (λ,c),ofatransitiontoakineticallyroughstateisob-
1
tained. For fixed λ, P increases rapidly from 0 to 1 as where Γ is a kinetic coefficient and η represents Gaus-
1
c is increased above a critical value. Typical results for sian delta-correlated noise whose variance is related to
P as a function of c for model I with λ=4.0 are shown Γ and the temperature T via the fluctuation-dissipation
1
in the inset of Fig.13. The value of c at which P = 0.5 theorem [39]. In the absence of noise, this equation con-
1
providesanestimate of c . Another estimate is obtained verges to local minima of the functional F. So, the
2
fromasimilarcalculationofP (λ,c),theprobabilitythat noiselessdynamicsexhibitstwolocallystablefixedpoints
2
a flatinitial state evolvesto a mounded steadystate. As for T0 < T < Ts, corresponding to the two minima of
expected, P increases sharply from 0 to 1 as c is de- f(m) that represent the disordered and uniformly or-
2
creased (see inset of Fig.13), and the value of c at which dered states. This is analogous to the two locally stable
this probability is 0.5 is slightly lower than the value at fixedpointsofournonequilibriumdynamicalsystemsfor
which P1 = 0.5. This difference reflects finite-time hys- c < c1(λ). If we identify the flat and mounded fixed
teresis effects. The value of c is taken to be the average points as the “disordered” and “ordered” states, respec-
2
of these two estimates, and the difference between the tively,andthecontrolparameterctoplaytheroleofthe
two estimates provides a measure of the uncertainty in temperature T, then the noiseless dynamics of our mod-
the determination of c . The phase boundary obtained els would look similar to that of Eq.(18) for η = 0, with
2
this way is shown in Fig.13, along with the results for c1 playing the role of the spinodaltemperature Ts of the
c (λ) obtained for the discrete model II from a similar equilibrium problem.
2
analysis. In the presence of noise, the system described by
The general behavior found for all the models as the Eq.(18) exhibits a first-order phase transition at T that
c
parameters λ and c are varied is qualitatively very simi- lies between T and T : the system selects one of the
s 0
lar to that in equilibrium first order phase transitions of phasescorrespondingtothetwofixedpointsofthenoise-
two- and three-dimensional systems as the temperature less dynamics, except at T where both phase coexist.
c
and other parameters, such as the magnetic field in spin The local stability of the mean-field ordered and disor-
systems, are varied. To take a standard example of an dered states in a small temperature-interval around T
c
equilibrium first order transition, we consider a system is manifested in the dynamics as finite-time hysteresis
with a scalar order-parameter field ψ(r), described by a effects. The behavior we find for out nonequilibrium
Ginzburg-Landau free energy functional [39] that has a dynamical models is qualitatively similar: the system
cubic term: selects the steady state corresponding to the mounded
(“ordered”) fixed point of the noiseless dynamics as the
1 1 1
F[ψ]= dr aψ2(r) bψ3(r)+ uψ4(r) , (16) control parameter c (analogous to the temperature T of
Z (cid:20)2 − 3 4 (cid:21)
the equilibrium system) is decreased below c which is
2
wherebanduarepositiveconstants,a=a (T T )with smaller than the “spinodal” value c . The growth mod-
0 0 1
−
a > 0, and T is the temperature. Considering uniform els do not exhibit a “spinodal” point for the kinetically
0
states, ψ(r) = m, the free energy per unit volume may rough (“disordered”) phase: the flat fixed point of the
be written as noiselessdynamics is locally stable for allpositive values
8
of the control parameter c. If this analogy with equi- γ 0.2. Thefinite-sizescalingdatafortheorderparam-
≃
librium first order transition is correct, then our mod- eter m for models I and II for both faceted and kineti-
els should show hysteresis and coexistence of kinetically callyroughphasesisshowninFig.15. ItisseenthatmL
rough and mounded morphologies for values of c near varies linearly with the system size L in the mounded
c (λ). As mentioned above, we do find hysteresis (see phase,whereasmL L1−γ withγ 0.2formodelIand
2
∼ ≃
insetofFig.13)infinite-timesimulationswithvaluesofc γ 0.15 for model II in the the kinetically rough phase.
≃
near c . Evidence for two-phase coexistence is presented So,intheL limit,theorderparameterwouldjump
2
→∞
in Fig.14, where a snapshot of the interface profile for a from zero to a value close to unity as c is decreased be-
L = 500 sample of model I with λ = 4.0, c = 0.42 is low c (λ). This is exactly the behavior expected at a
2
shown. This value of c is very close to the critical value first-order phase transition.
c2 forλ=4.0(seeinsetofFig.13). Thisplotclearlyillus- The occurrence of a first-order phase transition in our
trates the simultaneous presence of mounded and rough 1dmodelswithshort-rangeinteractionsmayappearsur-
morphologies in the interface profile. prising–itiswell-known[39]that1dsystemswithshort-
The results described above suggest that our growth rangeinteractionsdonotexhibitanyequilibriumthermo-
models exhibit a first-order dynamical phase transition dynamic transitionatanon-zerotemperature. The situ-
at c = c (λ). To make this conclusion more concrete, ationis,however,differentfornonequilibriumphasetran-
2
we need to define an order parameter, analogous to the sitions: In contrast to equilibrium systems, a first-order
quantity m in the equilibrium problem discussed above, phase transition may occur in one-dimensional nonequi-
that is zero in the kinetically rough phase, and jumps librium systems with short-range interactions. Several
to a non-zero value as the system undergoes a transition such transitions have been well documented in the lit-
to the mounded phase at c = c . The identification of erature [41]. So, there is no reason to a priori rule out
2
such an order parameter would also be useful for dis- the occurrence of a true first-order transition in our 1d
tinguishing between these two different kinds of growth nonequilibriumsystems. Asdiscussedabove,ournumer-
in experiments – as mentioned in the Introduction, it is icalresultsstronglysuggestthe existenceofatrue phase
difficult to experimentally differentiate between kinetic transition. However, since all our results are based on
roughening and mound formation with coarsening from finite-time simulations of finite-size systems, we can not
measurements of the usual bulk properties of the inter- claim to have established rigorously the occurrence of a
face. A clear distinction between the two morphologies truephasetransitioninourmodels. Thecrucialquestion
maybeobtainedfrommeasurementsoftheaveragenum- in this context is whether the order parameter m would
berofextremaoftheheightprofile[40]. Thesteady-state be nonzero in the mounded phase in the L limit if
→∞
profile in the mound-formation regime exhibits two ex- the time-averagein Eq.(19)is performedoverarbitrarily
trema for all values of the system size L. In contrast, long times. Since the steady-state profile in this phase
the number of extrema in the steady state in the kinetic hasasinglemoundandasingletrough(thisisclearfrom
rougheningregimeincreaseswithLasapowerlaw[40]– oursimulations),the onlywayinwhichmcangotozero
wefind thatforvaluesofcfor whichthe systemiskinet- is through strong “phase fluctuations” corresponding to
ically rough, e.g. for λ = 4.0, c = 0.05 for model I, the lateralshifts ofthe positions ofthe peak andthe trough.
averagenumberofextremainthesteadystateispropor- We do not find any evidence for such strong phase fluc-
tional to Lδ with δ 0.83. This observation allows us tuations. We have calculated the time autocorrelation
≃
to define an “order parameter” that is zero in the large- function of the phase of the order parameter for small
c, kinetic roughening regime and finite in the small-c, samples over times of the order of 107 and found that it
mound-formationregime. Letσ beanIsing-likevariable, remainsnearlyconstantatavalueclosetounityoverthe
i
equal to the sign of the slope of the interface at site i. entire range of time. So, if such phase fluctuation even-
An extremumin the heightprofile then correspondsto a tually make the order parameter zero for all values of c,
“domainwall” in the configurationof the σ variables. then this must happen over astronomically long times.
i
{ }
Since there are two domain walls separated by L/2 Our finite-time simulations can not, of course, rule out
∼
in the steady state in the mound-formation regime, the this possibility.
quantity
1 L V. COARSENING OF MOUNDS
m= σ e2πij/L , (19)
j
L|h i|
Xj=1
During the late-stage evolution of the interface, the
where ... representsatime-averageinthesteadystate, mounds coarsen with time, increasing the typical size
h i
would be finite in the L limit. On the other hand, of the triangular pyramidal structures. The process of
→∞
m would go to zero for large L in the kinetically rough coarseningoccursbylargermoundsgrowinglargeratthe
regime because the number of domains in the steady- expense of the smaller ones while always retaining their
state profile would increase with L. We find numeri- “magic” slope. Snapshots of the system in the coarsen-
cally that in the kinetically rough phase, m L−γ with ing regime are shown in Figs 16 and 17 for model I and
∼
9
modelII,respectively. Theconstancyoftheslopeduring ble for c < c (λ). Since structures with several mounds
1
the coarseningprocessisclearlyseeninthesefigures. As andtroughsapproachthissteady-statestructurethrough
discussed in the Introduction, the constancy of the slope the coarsening process, it is reasonable to expect that
impliesthatifthetypicallateralsizeofamoundincreases fixedpointsofthenoiselessequationwithmorethanone
intimeasapowerlawwithexponentn(R(t) tn),then moundsandtroughswouldbeunstable. Wehavenumer-
∝
the width of the interface would also increase in time as ically obtained fixed points of Eq.(8) with two mounds
a power law with the same exponent (W(t) tβ with andtroughsfordifferentvaluesofthesamplesizeandthe
∝
β = n). Therefore, the value of the coarsening expo- separation between the peaks of the two mounds. The
nent n may be obtained by measuring the width W as a slope of the mounds at these fixed points is found to be
function of time in the coarsening regime. In Fig.18, we the same as that in the fixed point with one mound and
show a plot of the width as a function of time for model one trough. We find that the stability matrix for such
I with λ = 4.0, c = 0.02. It is clear from the plot that fixed points always has a real, positive eigenvalue, indi-
the time-dependence of the width is well-described by a catingthatthe structureis unstableandwouldevolveto
power law with β = n = 0.34 0.01. A similar plot for thestableconfigurationwithonemoundandonetrough.
±
the discrete model II with λ = 2.0, c = 0.005, shown The magnitude ofthe positive eigenvalue of the stability
in Fig.19, also shows a power-law growth of the width matrixfortwo-moundedfixedpointsdependsonthesam-
in the long-time regime, but the value of the coarsen- plesize,theseparationbetweenthepeaksofthemounds
ing exponent obtained from a power-law fit to the data and the relative heights of the mounds in a complicated
is β = n = 0.50 0.01, which is clearly different for way. We have not been able to extract any systematic
±
the value obtained for model I. This is a surprising re- quantitative information from these dependences. We
sult: model II was originally defined [31] with the spe- find a qualitative trend indicating that the magnitude
cific purpose of obtaining an atomistic realization of the ofthe positive eigenvaluedecreasesasthe separationbe-
continuum growth equation of Eq.(3), and earlier stud- tweenthepeaksofthetwomoundsisincreased. Sincethe
ies [31–33] have shown that the dynamical scaling be- time scale of the development of the instability of two-
havior of this model in the kinetic roughening regime is moundedstructuresisgivenbytheinverseofthepositive
the same as that of model I. Also, we have found in the eigenvalue, this result is consistent with the expectation
presentstudythatthedynamicalphasetransitioninthis thatthetimerequiredfortwomoundstocoalesceshould
modelhas the samecharacteras thatin modelI. So,the increase with the separation between the mounds.
difference in the values of the coarsening exponents for These results suggest that the coarsening of the
these two models is unexpected. As noted earlier, there mounds in model I reflects the instability of structures
is some evidence suggesting that the typical slope of the with multiple mounds and troughs. If this is true, then
mounds in model II increases very slowly with time (see coarsening of mounds should be observed in this model
Fig.10). However,this“steepening”,ifitactuallyoccurs, even when the noise term in Eq.(8) is absent. To check
istooslowtoaccountforthelargedifferencebetweenthe this, we have carriedout numericalstudies of coarsening
values of the coarsening exponents for models I and II. in the noiseless version of Eq.(8). In these studies, the
Inordertounderstandthesenumericalresults,wefirst time evolution of an initial configurationwith a pillar of
address the question of why the mounds coarsen with heighth >h (λ,c) atthe centralsite ofanotherwise
0 min
time. This problem has certain similarities with domain flat interface is followed numerically in the presence of
growth in spin systems [42]. Using the Ising variables noise until the instability caused by the presence of the
σ defined in the preceding section, each height profile pillar is well developed. The profiles obtained for differ-
i
{ }
can be mapped to a configuration of Ising spins. The entrealizationsofthenoiseusedintheinitialtimeevolu-
coarsening of mounds then corresponds to a growth of tionarethenusedasinitialconfigurationsforcoarsening
the typical size of domains of these Ising spins. There runs without noise. The dotted line in Fig.18 shows the
is, however, an important difference between the coars- width versus time data obtained from this calculation.
ening of mounds in our models and the usual domain Thecoarseningexponentinthe absenceofnoiseisfound
growth problem [42] for Ising spins. Domain growth in to be the same (n 1/3) as that of the noisy system,
≃
spinsystemsistheprocessthroughwhichthesystemap- indicating that the coarsening in this model is driven by
proaches equilibrium from an out-of-equilibrium initial processes associated with the deterministic part of the
state. The dynamics of this process may be understood growth equation.
intermsofargumentsbasedonconsiderationsofthefree We have examined the details of the process by which
energy (at finite temperatures) or energy (at zero tem- two mounds coalesce to form a single one. The different
perature). Such argumentsdo not apply to our nonequi- stepsinthisprocessareillustratedinthesnapshotsofin-
librium growthmodels. The reasonfor the coarseningof terfaceprofilesshowninFig.17whereonecanseehowthe
moundsinourmodelsmustbesoughtintherelativesta- twomoundsnearthe centercometogethertoformasin-
bilityofdifferentstructuresundertheassumeddynamics gleone astime progresses. First, the separationbetween
and the effects of noise. the peaksofthe moundsdecreaseswithtime. Whenthis
As discussed in the preceding section, the fixed point distance becomes sufficiently small, the “V”-shaped seg-
of Eq.(8) with one mound and one trough is locally sta- ment that separates the peaks of the mounds “melts” to
10