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Fluctuations of steps on crystal surfaces

a r X i v :c o n d -m a t /0108391v 1 [c o n d -m a t .s t a t -m e c h ] 24 A u g 2001

Fluctuations of steps on crystal surfaces

W.Selke a ,?,F.Szalma a ,b ,and J.S.Hager c

a Institut

f¨u r Theoretische Physik,Technische Hochschule,D–52056Aachen,Germany

b Institute for Theoretical Physics,Szeged University,H–6720Szeged,Hungary

c Institute of Physical Science an

d Technology,University of Maryland,Colleg

e Park,MD,20742USA

1.Introduction

In recent years,the dynamics of steps of monoatomic height on crystal surfaces has at-tracted much interest,both experimentally and theoretically.Experimentally,?uctuations of iso-lated steps as well as trains of steps on vicinal surfaces have been studied thoroughly [1].Theoretically,three distinct mechanisms driv-ing the step ?uctuations have been identi?ed,step di?usion,evaporation–condensation and terrace di?usion.Predictions of Langevin descriptions [2–5]have been checked,con?rmed,and extended in Monte Carlo simulations on discrete SOS models [2,6,7].Recently,extensive simulations on step di?usion and evaporation–condensation have been

fusion,and,in the following,only the idealized kind of kinetics will be called terrace di?usion. Obviously,step positions are uniquely de?ned for terrace,but not for surface di?usion.One of the aims of the present Monte Carlo study is to com-pare simulational data,in the framework of SOS models,for terrace di?usion and surface di?usion.

Above the roughening transition,steps are def-initely no longer microscopically well de?ned;so the analysis will be restricted to surface di?u-sion.One may study the evolution of the pro?le of initially straight steps.Indeed,the correspond-ing equilibration problem,at T>T R,has been described by Mullins many years ago[8],and we shall compare our Monte Carlo?ndings to that classical theory.Likewise,some of our?ndings at T

2.Below roughening

We simulate square surfaces with isolated and pairs of steps of monoatomic height.Initially,at time t=0,the steps are perfectly straight,and the bordering terraces are perfectly?at;pairs of steps are usually separated by at most one lattice spacing.Step?uctuations result from terrace or surface di?usion.

In case of terrace di?usion,the acceptance rates of detaching and attaching atoms at steps,with a random walk in between,is assumed to be given by the Boltzmann factor of the change in the kink energies as described by the one–dimensional SOS model.There the kink energy is proportional to the number of missing bonds to the neighbouring step sites,i.e.?|u s(l)?u s(l±1)|,u s(l)being the position of step s,(s=1,2),at site l.The time unit,one Monte Carlo step(MCS),is assigned to L(or2L)attempted elementary moves for isolated (pairs of)steps of length L.In case of pairs of steps no crossing of steps is allowed.–To speed up

simulations,one may replace the actual random walk by a probability distribution[3,6].

In case of surface di?usion,the acceptance rates for jumps of surface atoms to neighbouring sites

will be given by the Boltzmann factor of the cor-responding energy change of the two–dimensional SOS model,where the local energy is given by

?|h(i,j)?h(i′,j′)|,with(i,j)and(i′,j′)being neighbouring surface sites.The roughening tran-sition is known to occur at k B T R/?≈1.25.The

time unit,one MCS,is assigned to LM attempted jumps,where L is the step length,say,j=1,...,L,

and M refers to the other direction of the surface. To monitor the step?uctuations,we recorded the step pro?le,z(i,t)= h(i,j) /L,summing over j and averaging over N Monte Carlo realizations with di?erent random numbers.To stabilize the steps,the heights at the boundary lines parallel to

the initial straight steps are kept constant during the simulation,e.g.for pairs of steps at h=0and h=2.

The step dynamics may be described,both for surface and terrace di?usion,by the time evolution of the step pro?le z(i,t),and,for terrace di?usion, by the average step positions,u s0(t),and the?uc-tuation function w s(t)=

-24

-18

-12

-6

06

12

18

24

i

-0.2

0.00.20.40.60.81.01.21.41.61.82.02.2z (i ,t )

Fig.1.Simulated step pro?les z (i,t )of the two-dimensional SOS model at k B T/?=1.0with 44×1000sites at t =0(circle),6000(square),30000(diamond),and 120000(triangle)MCS,using surface di?usion.Averages have been taken over 70realizations.

fusion and at 1.0for terrace di?usion.

For pairs of steps,in case of terrace and surface di?usion,the e?ective exponent b f is observed to approach 1/5at large times of up to several 105MCS (well before the step ?uctuations saturate due to the ?nite step length),when z f approaches 0or 2.For reasons of symmetry,one may restrict the discussion to 0

d (t )=u 10(t )?u 20(t ),and th

e ?uctuations w

1,2

(t )follow closely the power–law w,d ∝t 1/5

.How-ever,in that time regime,for terrace and surface di?usion,the e?ective exponent b f changes signif-icantly with z f ,increasing with z f from about 1/5at small heights rather slowly up to about 0.23at z f ≈0.8,and then more rapidly to roughly 1/3as z f approaches 1,depending only weakly on time.It remains to be seen whether full scaling of the step pro?le,with,possibly,b =1/5,holds in the limit L,t ?→∞.Note that the continuum theory of Spohn on the equilbration of steps due to sur-face di?usion [10]does not provide an easy answer

to this question.There the oscillatory character of the pro?le is emphasized,which may a?ect the scaling behaviour.In fact,oscillations show up,but with very small amplitudes;see Fig.1and next section.–The value 1/5for the exponent de-scribing the separation of the two steps,d (t ),has been argued before to follow from the continuum theory of Rettori and Villain [2,9].

Terrace di?usion for pairs of short steps may be described,in the limit L ?→1,by two points on a line emitting particles which execute one–dimensional random walks.When the emitted particle returns to the emitter,that point will stay at its original position,while the other point will move by one when the particle hits that point.The description may also be applicable to the motion of a pair of kinks along a smooth step in the case of step–edge di?usion when there are only those two kinks.From our simulations of such random walks,we infer that the distance d between the two steps of length L =1(or the two kinks)in-creases as d ∝t 1/3at large times.

For isolated long steps,the step pro?les tend to scale,at su?ciently large times,with b ≈1/6,for surface di?usion as well as for terrace di?usion.In-deed,the critical exponent of the power–law de-scribing the step ?uctuations w (t )at those times is about 1/6as well,in accordance with previous sim-ulations for terrace di?usion [6].That result con-?rms the validity of Langevin descriptions for step dynamics at late stages [3,4].

3.Above roughening

We studied isolated and pairs of ascending steps in the framework of two–dimensional SOS mod-els at T >T R ,or one–dimensional SOS models,which are rough at all temperatures T >0,com-puting the step pro?les z (i,t ),applying surface di?usion.All cases lead to similar results,because above roughening individual steps are smeared out completely.

In particular,the step pro?les scale,already

3

-30

-25-20

-15-10-50

i

0.50

0.60

0.70

0.80

0.901.00

1.10

z (i ,t )

Fig. 2.Part of simulated step pro?le for isolated steps of monoatomic height above roughening.Data for the one–dimensional SOS model at k B T/?=1.0with 152sites have been taken at 1(circle),1000(square)and 5000(di-amond)MCS,averaging over 106realizations.

at moderate times of typically a few 104MCS,with the critical exponent b ≈1/4,the e?ective exponent b f depending only very weakly on z f .The pro?les show oscillations,with the amplitude decreasing rapidly with distance from the center of the surface,see Fig.2.The onset of the oscil-lations may be readily understood by calculating the energetics of the ?rst few excitations,starting from ?at terraces and straight steps.Already the ?rst move leads to an overshooting of the pro?le at the next–nearest distance from the center,i =0,which again triggers an undershooting at further distance,and so on.This e?ect has been described before by Mullins in a continuum theory of surface equilibration above roughening [8].Note that the oscillations persist to temperatures below rough-ening,as discussed above;however,the amplitudes become much smaller,at least at the times used in our Monte Carlo study,see also Fig.1.In Mullins’theory,the basic equation reads dz/dt =?A (d 4z/dx 4),where A is a temperature dependent coe?cient;the continuum variable x corresponds to i in the discrete description.The equation may easily be solved by Fourier analysis [8].The resulting step pro?les resemble closely those found in the simulations;di?erences are

expected to show up only at early stages of equi-libration,as observed before for other surface defects such as periodic grooves [13].From the basic equation,it follows that the amplitudes of the oscillations settle at ?xed values,independent of temperature.We con?rmed these features by simulating steps at various temperatures.Actu-ally,the continuum description of Mullins leads to a perfect scaling of the step pro?les with b =1/4.F.Sz.thanks the Hungarian National Research Fund,under grant number OTKA D32835,and the Deutsche Forschungsgemeinschaft for ?nancial support.References

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