Non-conserved dynamics of steps on vicinal surfaces during electromigration-induced step bunching

0.94 MB·English

by  Marian Ivanov

EPJ manuscript No.
(will be inserted by the editor)
Non-conserved dynamics of steps on vicinal surfaces during
electromigration-induced step bunching
2
1
0 M. Ivanov1 and J. Krug1
2
1
n Institut fu¨r Theoretische Physik,Universit¨at zu K¨oln - 50937 K¨oln, Germany
a
J

6 Received: date/ Revised version: date

]
h
c Abstract. Wereportnewresultsonthenon-conserveddynamicsofparallelstepsonvicinalsurfacesinthe
e
m
caseofsublimationwithelectromigrationandstep-stepinteractions.Thederivedequationsarevalidinthe
-
at quasistatic approximation and in the limit f−1 ≫ lD ≫ l± ≫ li, where f is the inverse electromigration
t
s
. length, lD the diffusion length, l± the kinetic lengths and li the terrace widths. The coupling between
t
a
m crystalsublimation andstep-stepinteractionsinducesnon-linear,non-conservativetermsin theequations
-
d ofmotion.Dependingontheinitialconditions,thisleadstointerruptedcoarsening,anticoarseningofstep
n
o bunchesor periodic switching between step trains of different numbersof bunches.
c
[

2
PACS. XX.XX.XX No PACS code given
v
6
4
0 1 Introduction change of the crystal volume is a result of the movement
4
. of the steps. On this scale we can reduce a surface with
9
For the theoretical study of homoepitaxial growth and
0
1 straightstepstoaone-dimensionalsteptrain.Suchasur-
sublimation of a crystal in contact with the gas phase it
1
: face may undergo step bunching, an instability where the
v is important to have a model, which includes the kinetic
i
X steps move close to each other and form groups, called
processes and the different effects existing on the crystal
r
a
step bunches [2,3,4,5].
surface. The classical model for the evolution of vicinal
surfaces was introduced by Burton, Cabrera and Frank The theoretical description of step bunching instabili-
(BCF) [1]. It is based on the observation that the kink tieswithintheframeworkoftheBCF-modelanditsexten-
sites are those positions at the surface steps where the sionshasbeenthesubjectofmuchrecentinterest[6,7,8,9,
exchange between the adatom layer on the terraces and 10,11,12,13,14].Herewefocusspecificallyontheeffectof
the solid phase takes place. On the mesoscopic scale the non-conservativeprocessesonthenon-linearevolutionofa
2 M. Ivanovand J. Krug: Non-conserved dynamicsduringelectromigation-induced step bunching
steptrain.Aswereportedin[14]fortheproblemofsubli- h F −→ F
el 1
mationinthepresenceofEhrlich-Schwoebel(ES)barriers D τ
s
[15,16], non-conservative terms violating volume conser-
vation in the co-moving frame arise generically from the x x x
i i+1
interplay of sublimation and step-step interactions, and
cause the interruption of the coarsening of the growing Fig. 1. (Color online) Sketch of the elementary processes in
bunches or splitting of a large bunch into several smaller theBurton-Cabrera-Frank model.
bunches. In the present paper we expand this analysis to
threshold [5]. In this setting the non-conserved dynamics
include the experimentally relevant effect of surface elec-
isdescribedonlargescalesbytheBenneyequation,which
tromigration[2,5,17,18,19,20,21,22,23,24,25,26,27].In1989,
displays either spatio-temporal chaos or an ordered array
Latyshev and collaborators discovered that by changing
of bunches, but no coarsening [28,29]. This macroscopic
thedirectionofthedirectheatingcurrent,avicinalSi(111)
behavior is consistent with the complex mesoscopic step
surface switches between bunching and debunching [17].
dynamics revealed in the present work.
Additionally, they observed several distinct temperature
The paperis organizedas follows.First, we sketchthe
regimes. In the so called regimes I and III [5] the bunch-
derivation of the discrete step equations for the case of
ing instability occurs only if the heating current is ap-
attachment-detachment limited kinetics and present the
plied in the down-step direction. On the other hand, for
result of the linear stability analysis for very large wave
the same direction in regime II debunching occurs, and
lengths. Additionally, for comparison, we write down the
bunching requires an up-step current. Here, we consider
corresponding equations for the case of growth. We then
thefirsttemperatureregime,wherethetemperatureislow
discuss the dicrete equations and their continuum limits
enough in order to neglect step transparency (the motion
for two special cases, where the kinetic asymmetry be-
of adatoms across steps) [5,8,25].
tween ascending and descending steps is caused solely by
Interruptedcoarseningofelectromigration-inducedstep an ES-effect or by electromigration, respectively. Finally,
bunches in the presence of sublimation was previously we show the results of numerical simulations of the dis-
observed numerically by Sato and Uwaha [6], however crete step equations for the case with electromigration.
a detailed analysis of the phenomenon was not carried
out due to the complexity of their model. Other stud-
2 Model
ies haveapproachedthe problemwithin the frameworkof
weakly nonlinear amplitude equations, which can be sys- Weconsideranascendingone-dimensionalsteptrainwith
tematicallyderivedbyanexpansionaroundtheinstability step edges located at positions x (Fig.1). The starting
i
M. Ivanovand J. Krug: Non-conserved dynamicsduringelectromigation-induced step bunching 3
pointfor the derivationofthe equationsofmotionfor the musthold,whereΩisthecrosssectionofanatomicsiteat
steps is the balance equation the step. The labels +/ refer to quantities correspond-
−
ing to the lower/upper terrace of a step. The fluxes f±
∂n ∂2n ∂n n
i i i i !
=D f +F =0 (1)
∂t s ∂x2 − ∂x − τ depend on both the difference of the adatom concentra-
(cid:20) (cid:21)
tion n(x) compared to its equilibrium value n and on
for the concentration of adatoms n (x,t) on the i-th ter- eq
i
the attachment/detachment to the steps with kinetic co-
race of width l =x x . Here D is the surface diffu-
i i+1 i s
−
sion constant,and τ is the averagelife time of an adatom efficientsk±.Iftheconditionk+ >k− isfulfilledwespeak
about a standardES effect [15]. It induces an asymmetry
before desorption. Together these two quantities define
intheconcentrationprofilesn (x)quantifiedbytheasym-
the diffusion length l = √D τ, which sets the scale of i
D s
metry parameter
spatial variation for the adatom concentration. The ter-
race is losing adatoms due to desorption at rate 1/τ and k+ k− l− l+
b − = − , (3)
ES
≡ k++k− l−+l+
gaining adatoms because of deposition with constant flux
where l± =Ds/k± are called kinetic lengths.
F. The applied direct heating force F causes a drift of
el
Apart from the attachment kinetics, a second effect
adatoms for which we can use the Einstein relation and
incorporatedintothe boundaryconditions(2)isthe step-
define a further length scale, the so called electromigra-
step repulsion. The equilibrium concentration n is de-
tion length f−1 = k T/F [18,6]. As is common in the eq
B el
termined by the chemical potential ∆µ at the ith step
i
field, we assume that the adatom concentration adjusts
through the relation n n0 (1+ µ /k T), and µ
instantaneously to the slowly moving steps, an assump- eq ≈ eq △ i B △ i
depends on the widths of the two neighboring terraces l
i
tion that is know as the quasistatic approximation and
and li−1 according to [2,31]
amountstosetting∂ n (x,t)=0in(1).Anapproachthat
t i
µ l3 l3
goesbeyondthisapproximationwasrecentlypresentedby △ i = g = :gν . (4)
k T − l3 − l3 i
B (cid:18) i i−1(cid:19)
Ranguelov and Stoyanov [24,25].
where l is the meanterrace spacing andg is a dimension-
The general solution n (x) of the ordinary differential
i
less measure for the strength of repulsion between the
equation (1) can be specified using mass conservation at
steps [2,32].
the steps as boundary conditions. A terrace of width l is
bounded by two steps with positions x = l/2, at which
±
3 Step equations of motion
the flux continuity conditions
Using Eqs. (1,2,4) we find the concentrations n (x) for
f− ∂n l i
=Ds fn =+k−[n neq], x= ,
Ω (cid:20)∂x − (cid:21) − −2 all terraces. The velocity of the i th step is then given
−
f ∂n l
+
=D fn = k [n n ]x=+ , (2)
Ω s ∂x − − + − eq 2 by the superposition of the fluxes coming from the two
(cid:20) (cid:21)
4 M. Ivanovand J. Krug: Non-conserved dynamicsduringelectromigation-induced step bunching
neighboring terraces as dxi/dt=f−+f+. Since the non- (in the absence of non-linear, non-conservative terms, see
conservativetermsofprimaryinterestherearisefromsub- below) and U =(glD2)/(l−+l+).
limation, we discuss separately the limiting cases of pure
sublimation(F =0, 1 >0)andpuregrowth(F >0, 1 =
τ τ 4 Linear stability
0); of course, in a typical experimental setup both pro-
cesses may proceed simultaneously. For the case of pure Equations similar to (6) can be derived when the sur-
sublimation and in the limit f−1 l , we obtain the face is subject to a growth flux but sublimation is absent
D
≫
non-linear system (F > 0, τ1 = 0). In that case the factor γi in front of
R−1dxi = ll2D+ + f2 si+ l1Dci γi− l1De−f2liγi+1 the square bracket on the right hand side of (6), which
dt h(cid:16) f(l(cid:17)−−l+) +1 is + l−+l+c depends nonlinearly on the step coordinates, is replaced
2 i lD i
h i
by the constant
+ (ll2D− − f2)si−1+ l1Dci−1 γi− l1Defli2−1γi−1, (5) Fτ
h f(l−2−l+) +1 si−1i+ l−l+Dl+ci−1 γ =−n0eq. (8)
h i
whereR=n0eqΩDs,si =sinh(li/lD),ci =cosh(li/lD)and Analogous to the problem considered in [14], this implies
γi =1+gνi. The result (5) contains all four length scales qualitativelydifferentinstabilityconditionsforgrowthand
and illustrates the complicated functional dependence for sublimation.Performingastandardlinearstabilityanaly-
a simple one-dimensional step train. sis, in the limit of large wavelength perturbations we find
Tosimplifytheseexpressionsweusetheapproximation the instability conditions
of attachment-detachment limited kinetics, lD l± l
≫ ≫ bsub 2b +b >6g forsublimation (9)
el ES
≡
[32,33]. After some calculations along the lines of [14] we
Fτ
bgr 2b b >0 forgrowth. (10)
arrive at ≡ el− n0 ES
eq
dx (1 b ) (1+b ) In the case of growth step bunching merely requires the
Re−1 dti ≈ γi −2ES li+ 2ES li−1
(cid:20) (cid:21) compoundasymmetryparameterbgr tobepositive,whereas
+ U(2νi νi+1 νi−1)
− −
for sublimation the corresponding quantity bsub needs to
b
el
[(γi+γi+1)li (γi+γi−1)li−1]. (6)
− 2 − exceed a positive threshold value 6g. This is an impor-
Here a second asymmetry parameter tant consequence of the qualitatively different contribu-
tionstothebalanceeq.(1)thatarisefromdesorptionand
fl2
b D (7)
el
≡−l−+l+ deposition, respectively. Note that in a general situation
incorporating the strength of electromigration has been the instability conditions (9,10) canbe combined into the
introduced, R =(n0 ΩD )/l2 =n0 Ω/τ is the constant formb=(1 Fτ/n0 )b +2b >6g,whichwas already
e eq s D eq − eq ES el
ratewithwhichthesurfacechangesvolumeinaunittime obtained in [12].
M. Ivanovand J. Krug: Non-conserved dynamicsduringelectromigation-induced step bunching 5
5 Conservative and nonconservative dynamics undersummationoveri,theircontributionisconservative
andtheonlynon-conservativecontributionsineq.(11)are
Beyond the linear stability properties, a fundamental dif-
the first terms multiplied by γ .
i
ference between the scenarios of pure growth and subli-
For comparison, setting b = 0 eq.(6) reduces to the
el
mationisthatthesurfacedynamicsisconservative during
equations derived in [14],
growth but not during sublimation [8,14]. Here conserva-
tive dynamics implies that the rate of volume change of dx γ
Re−1 dti = 2i (li+li−1)+U(2νi−νi+1−νi−1)
thecrystal,obtainedbysummingtheequationsofmotion
b gb
ES ES
(li li−1)+ (νili−1 νili). (12)
− 2 − 2 −
over all steps x , is independent of the surface configura-
i
tion [34]. Indeed, replacing the γi in front of the square The difference between the two cases is that the terms
brackets on the right hand side of (6) by the constant (8) proportionalto gb on the RHS of eq. (12) do not cancel
el
and summing over i, one readily obtains ix˙i =−FΩL, under summation with respect to i, and thus are non-
where L is the total length of the crystal.P conservative. As will be shown in the next section, this
It is instructive to compare the structure of the non- givesrisetodistinctcontributionsinthecontinuumlimit.
conservativecontributions induced during sublimationby
the configuration-dependent factors γ in (6) for the two
i
6 Continuum equations
stepbunchinginstabilities drivenbyelectromigrationand
by an ES-effect, respectively. First we neglect the ES-
effect, setting b =0, which simplifies (6) into the form In previous work a systematic method for deriving con-
ES
tinuum equations of motion from the discrete step dy-
dx γ
Re−1 dti = 2i (li+li−1)+U(2νi−νi+1−νi−1)
namics was developed [9,10] which was applied to the
bel(li li−1) (11)
− − model (12) in [14]. Briefly, the method can be seen as
gb
el
+ 2 (νili−1−νili+νi−1li−1−νi+1li). akindofLagrangetransformation[35]whichreplacesthe
The second group of terms on the RHS of eq. (11) with ‘Lagrangian’ dynamics of particle-like steps by the ’Eu-
prefactor U arises from equilibrium step-step interactions lerian’ evolution of the step density m(x,t). The latter
andstabilizestheregularsteptrain,whilethethirdgroup in turn defines a continuous height profile h(x,t) through
of terms describes the effect of electromigration, which is m(x,t)= ∂h.
∂x
stabilising or destabilising depending on the sign of b , Here we wish to compare the two instability mecha-
el
i.e., the direction of F . The last group of terms arises nisms described by eqs.(11) and (12), respectively, on the
el
from the interplay of electromigration and step-step in- continuum level. Following the procedure outlined in [14]
teractions. Since the terms in this group cancel pairwise for both models, we find that the continuum evolution
6 M. Ivanovand J. Krug: Non-conserved dynamicsduringelectromigation-induced step bunching
equation takes the general form comeclearthatsmallnon-conservativetermsmayqualita-
∂h ∂ 3gm2 m′ 3U m2 ′′ tivelychangethenonlineardynamicsofsurfacesteps[14],
+ + J +1=
∂t ∂x"− 2 − 6m3 2(cid:0)m (cid:1) − b# and the fact that these terms are of different form for the
3g m2 ′ m′ ′
Φ , (13) two instability mechanisms implies that their equivalence
− 2 6m3 − b
(cid:0) (cid:1) (cid:20) (cid:21)
needs to be reexamined. In the following we therefore ex-
where primes denote spatial derivatives. Here time t is
plorethenonlinearbehavioroftheelectromigrationmodel
rescaled by R , length x by the average step distance l,
e
(11) using numerical simulations.
andheighth is measuredin units ofthe monoatomicstep
height. The terms inside the square brackets on the LHS
7 Nonlinear step dynamics
are conservative, and the non-conservative contributions
are collected on the RHS of eq.(13). The two models (11) Numerical simulations of eq. (11) were carried out using
and (12) differ in the form of the contribution J to the anodeint-typeprocedure[37]forsystemsofM stepswith
b
conserved surface flux, and of the non-conservative term periodic boundary conditions. We consider the following
Φ . Labeling the contributions due the ES-effect by ES ranges for the four independent parameters of the model:
b
and those due to electromigration by el, respectively, the bel [0,0.5], U [0,0.5], g [0,0.1] and M < 100.
∈ ∈ ∈
conservative terms are given by Another degree of freedom is provided by the choice of
the initial condition. In general, we start the simulations
b b
JES = ES, Jel = el +3gb m′, (14)
b 2m b m el
with two types of initial step train configurations: either
and the non-conserved contributions are a randomly disturbed equidistant step train, or an ini-
ΦES = 3gbES m2 ′ 1 ′, Φel 0. (15) tial shock of closely spaced steps and a single large ter-
b 2 2m b ≡
(cid:0) (cid:1) (cid:20) (cid:21) race. Step trajectores are shown in the co-moving coor-
As wasdiscussedabove,the termsineq.(11)proportional dinate system x˜ (t) = x (t) lt, and we normalize both
i i
−
to gbel give rise to a conservative contribution, whereas theheightofthe (monoatomic)steps andthe averageter-
the terms in (12) proportional to gbES contribute to the race width l to unity. The time t is rescaled by Re and
non-conservative part of the continuum equation. we measure the integration time in time units (t.u.). For
In earlier work based on the continuum approach [9, the description of the bunch geometry we use two mea-
10] the non-conservative contributions were generally ne- sures:themaximalslopem max m andthemin-
max i i
≡ { }
glectedbecauseofthesmallnessofg[14],anditwasthere- imal curvature κ min κ , where m = 1/l and
min i i i i
≡ { }
foreconcludedthatstepbunchingphenomenainducedby κ = 8(l l )/(l +l )3 respectively. A step is de-
i i+1 i i+1 i
− −
electromigration and by the ES-effect belong to the same fined to belong to a bunch, if its distance to the next
universalityclass [9,36]. However,it has subsequently be- closest step of the bunch is smaller than l =1.
M. Ivanovand J. Krug: Non-conserved dynamicsduringelectromigation-induced step bunching 7
50 -60
a) -135 b)
0 -145
-80
1 -155
˜x
-50 -165
-175 -100
-185
-100 11000 11500 12000
1
˜x time ˜x -120
-150
-140
-200
-250 -160
-300 -180
0 2000 4000 6000 8000 10000 12000 14000 16000 11280 11300 11320 11340 11360 11380 11400 11420 11440 11460
time time
160 0 3.4
min. curvature
3.2
150 c) d) 3
-0.2
2.8
140
max. slope 2.6
-0.4 2.4
130 min no. of bunches 2.2 max
h 120 κ -0.6 2 m
1.8
110 1.6
-0.8
1.4
100
1.2
6000 t.u. -1 1
90 11500 t.u.
13000 t.u. 0.8
15000 t.u.
80 -1.2 0.6
-150 -100 -50 0 50 6000 8000 10000 12000 14000 16000
x˜ time
Fig. 2. (Color online) An example for the splitting of a large bunch in a system of 80 steps with parameters bel = 0.4, g =
0.05, U = 0.05. a) Time evolution of one of the steps. The inset shows a blowup around the onset of splitting. The period of
oscillations prior to the breakup reflect the time required for the step to once traverse the (single) bunch. b) Plot of all step
trajectories between 11280 t.u. and 11470 t.u. c) Comparison of the profiles after 6000 t.u., 11500 t.u., 13000 t.u. and 15000
t.u. d) Time evolution of theglobally maximal slope, the globally minimal curvatureand the numberof bunches.
Animportantconsequenceofthenon-conservativechar- is reflected in a clear change in the velocity of the step;
acterofthedynamicsisthephenomenonofanti-coarsening, smallerbunches movefaster.Focusing onthe splitting re-
whereaninitiallargestepbunchsplitsintosmallerbunches giononefindsthattheinitialsmallbunchdisappearsagain
[14]. In fig. 2 we show an example of this behavior for after 4 t.u., an event which is repeated at 11258 t.u. with
parameter values b = 0.4, g = 0.05, U = 0.05, and a small bunch life time of 19 t.u. In fig. 2b) we show the
el
M = 80. Figure 2a) displays the movement of the first trajectories of all steps in the train in the time window
step in the train. After 11180 t.u. the large bunch splits 11280 11470 t.u. At 11333t.u. the small bunch appears
−
forthefirsttimeintoalargeandaverysmallbunch,which for the third time and at 11415 t.u. a third bunch arises
8 M. Ivanovand J. Krug: Non-conserved dynamicsduringelectromigation-induced step bunching
8
0.5 a) 7 b)
0.4 6
x
a 5
m
el 0.3 m
b 4
0.2 3
2
lin. stab.
0.1
1 bunch
1 or 2 bunches 1
2 bunches
0 0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 1000 2000 3000 4000 5000 6000 7000 8000
g time
Fig. 3. (Color online) System with U=0.2, M=40 and fluctuating initial condition. a) Stability/instability diagram, showing
thenumberof bunchesin the final state, for different combinations of bel and g - (cid:3): 1 bunch,•: 1 or 2 bunches,×: 2 bunches,
belowthelinebel =3g:stability.b)Timeevolutionofthemaximalslopemmax withbel=0.35and(fromtoptobottom)g=0.00,
0.01, 0.02, 0.05, and 0.09.
for the first time. The switching between different num- tions with one single bunch, with two bunches, and time-
bers of bunches continues until the step train relaxes into dependent solutions that switch periodically between one
four bunches, as seen in the height profiles in fig. 2c). and two bunches. In fig. 3b) we plot the time evolution
Figure 2d) shows the corresponding time evolution of the of the maximal slope m at five points along the line
max
minimal curvature κ and the maximal slope m , b =0.35. For g=0 we see the usual coarsening behavior,
min max el
along with the number of bunches. Both κ and m in which the number of bunches decreases in a step-wise
min max
showclearchangesinthe regionofthe splitting;however, fashion until a single bunch configuration is reached and
whereas the maximal slope remains essentially the same the system relaxes to a stationary periodic state with a
after the splitting event, there is a significant decrease in clearly bounded maximal slope. The remaining temporal
the minimal curvature. periodicityofm isduetothepermanentstepexchange
max
betweenthefrontandbackendofthebunch(seealsoinset
In fig. 3 we summarize results obtained from simula-
infig.2a)).Increasinggthemaximalslopedecreases(while
tionsstartingfromarandomlydisturbedequidistantstep
still maintaining the single bunch configuration), until at
array of 40 steps. Figure 3a) shows the phase diagram
g =0.05the regimeofperiodicswitchingisreached,lead-
of the system in the g-b plane at fixed U = 0.2. Be-
el
ing to a complex periodic pattern in m .
max
low the line b = 3g the system is linearly stable and
el
m = m(x) 1. Above this line we see three qualita- Finally, in fig. 4 we plot the behavior of the maximal
max
≡
tively different types of long-time behavior: steady solu- slope as a function of the number of steps for two differ-
M. Ivanovand J. Krug: Non-conserved dynamicsduringelectromigation-induced step bunching 9
8 Conclusion
2.2
2.1
In this work we have extended the non-conservative step
2
x bunching model presented in [14] to include the effect of
a 1.9
m
m
1.8 electromigration.The model applies to the firstofthe ex-
1.7
perimentallyobservedtemperatureregimesontheSi(111)
1.6
surface, where step transparency can be neglected. The
1.5
strong disturbance
small disturbance general step equations of motion incorporating sublima-
1.4
10 20 30 40 50 60 70 80
M
tion, the Ehrlich-Schwoebel effect, electromigration and
step-stepinteractionswerederivedfromtheclassicalBCF
Fig. 4. (Color online) Dependence of the globally maximal
slope mmax on the number of steps M for g = U = 0.04, model in the quasistatic approximation. For the case of
bel =0.2andforlargeandsmallnoiseamplitudeintherandom attachment-detachmentlimited kinetics we comparedthe
initial condition. equations for growth and sublimation. In previous pub-
lications[9,10,11,32] non-conservativecontributions were
neglected,becauseoftheexperimentallysmallprefactorg
[2,14].Thosetermswerenowtakenintoaccountandsome
importantconsequenceswereidentified.First,onthelevel
ent amplitudes of the initial disturbance, g=U=0.04,and oflinearstabilityanalysis,theyshifttheinstabilitycondi-
b =0.2.Herem istheglobalmaximalslope,measured tion on the dimensionless asymmetry parameter b by 6g,
el max
as its largest value for the last 500 t.u. of the simulation. aswasfirstpointedoutin[12].Thisshiftispresentinthe
Weseethatm generallyincreaseswithM,butthisbe- case of sublimation, but not in the case of growth [14].
max
havior is interrupted by downward jumps every time the Moreover, in the case of sublimation the structure of the
number of bunches that can fit into the system increases non-conservative terms differs depending on the underly-
by one. This shows that the existence of stationary solu- ing mechanism inducing the asymmetry between ascend-
tions with multiple bunches can be seen as a consequence ing and descending steps. This leads to different contin-
of the fact that, in the presence of non-conservative pro- uum equations for step bunching caused by an ES-effect
cesses,themaximalslopeisboundedfromabove[14].Near or by electromigration,respectively.
the transition between different numbers of bunches the Nevertheless,the numericalintegrationof the discrete
system behavior depends very sensitively on the ampli- step equations for the case with sublimation and electro-
tude of fluctuations in the initial configuration, an effect migrationreproducesqualitativelythe resultsof[14].The
that is particularly pronounced around M =70. non-linear, non-conservative terms supply a richness of
10 M. Ivanovand J. Krug: Non-conserved dynamicsduringelectromigation-induced step bunching
dynamical behaviors in this simple one-dimensional step 3. J. Krug, in Multiscale Modeling of Epitaxial Growth, Int.
model. There are steady solutions which contain more Ser.Num.Math.149,editedbyA.Voigt(Birkh¨auser,Basel
than one bunch, periodic switching between step trains 2005), p.69
of different numbers of bunches, and a sensitive depen- 4. O.Pierre-Louis, C.R. Physique6, 11 (2005)
5. C. Misbah, O.Pierre-Louis, Y. Saito, Rev.Mod. Phys. 82,
dence on the initial condition.This showsthat the notion
981 (2010)
of universality between different types of step bunching
6. M. Sato and M. Uwaha,Surf. Sci. 442, 318 (1999)
mechanisms,whichwasoriginallyformulatedonthe basis
7. M. Sato and M. Uwaha, Surf. Sci. 493, 494 (2001)
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