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Surface Science 601 (2007) 13–23

The return of the kink

Gertjan S. Verhoeven 1, Joost W.M. Frenken *

Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands

Received 11 August 2006; accepted for publication 1 September 2006Available online 20 September 2006

Abstract

We present a simple theory for the statistics of subsequent passages of kinks at a particular position along a fluctuating step on acrystal surface. Two situations are treated, namely one where the waiting time until the next passage is measured starting from the pre-vious passage of a kink at the observation point, and one where the waiting time is measured starting from a random instant in time. Inboth cases the waiting time distribution is shown not to obey simple Poisson statistics. Monte Carlo simulations of kink motion showthat Poisson statistics emerges for longer waiting times only when the creation and annihilation of kink–antikink pairs are explicitlyincluded. Together, the theory and simulations provide an alternative interpretation of experimental results obtained by Giesen et al.with Scanning Tunneling Microscopy [M. Giesen-Seibert, H. Ibach, Surf. Sci. 316 (1994) 205].� 2006 Elsevier B.V. All rights reserved.

Keywords: Models of surface kinetics; Atomistic dynamics; Monte Carlo simulations; Surface diffusion; Scanning tunneling microscopy; Copper

1. Introduction

The fluctuations in time of the shape of atomic steps oncrystal surfaces contain detailed information on the micro-scopic mechanisms underlying surface mass transport. Forexample the precise scaling of the mean-square step dis-placement with elapsed time allows one to decide whetheror not the atoms involved in the fluctuations are exchangedwith the surrounding terraces [2–7]. From the temperaturedependence one can readily derive the activation energy in-volved in the process. Scanning tunneling microscopy isone of the most suitable techniques to measure these fluc-tuations with atomic-scale precision. However, one of thedrawbacks of this application of the STM is the limitationin time resolution inherent to this instrument. One partialremedy for this is to reduce the STM image to one shortsingle line, which is directed perpendicular to the step un-der investigation, and which is continually measured with

0039-6028/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.susc.2006.09.001

* Corresponding author. Tel.: +31 71 5275603; fax: +31 71 5275404.E-mail address: frenken@physics.leidenuniv.nl (J.W.M. Frenken).

1 Present address: FOM Institute for Atomic and Molecular Physics,P.O. Box 41883, 1009 DB Amsterdam, The Netherlands.

a high repetition frequency. In this way, the fluctuationsof a single position along the step can be followed with atime resolution well below 1 s. Every time that a kinkpasses the observation line, the measured step positionabruptly jumps backward or forward over one lattice spac-ing, as is illustrated in Fig. 1. One of the statistical func-tions that are very sensitive to the diffusion ‘activity’along the step is the time dependence of the mean-squarestep displacement [2–7]. Another function of interest isthe persistence probability distribution [8,9]. In its mostgeneral form, persistence is defined as the probability p(t)that a random variable (such as the position of a fluctuat-ing step) never crosses a chosen reference level (e.g., theposition of the step at t = 0) within the time interval t.For fluctuations of a step over a single lattice period, per-sistence is fully determined by the distribution P(t) of wait-ing times t between subsequent passages of kinks. For aCu(1119) surface, P(t) has been measured at temperaturesbetween 270 and 370 K in Ref. [1] by scanning the STMperpendicular to steps. This distribution forms the subjectof the present paper.

The naıve expectation for the waiting time distributionis that it would decay exponentially, i.e. P(t) = fexp(�ft),

Fig. 1. Panels A, B and C show schematics of three subsequent step configurations with an atom detaching from the left kink (A), traveling along the step(B) and ending up at the right kink (C). The location of the STM scan line is indicated by a dotted line perpendicular to the step. The resulting scan line inthe STM ‘‘time image’’ is shown next to each configuration. As is argued in Section 2, the duration of the adatom’s random walk along the step (B) is soshort that the STM records only the initial stage (A) and the final stage (C). Only after the adatom reattaches to a neighboring kink (going from A to C viaB) a change of step position is observed in the STM scan line. When the adatom reattaches to the kink from which it originated (from A via B to A again),the STM scan line shows no changes and no signature of a ‘failed’ attempt to move two kinks.

14 G.S. Verhoeven, J.W.M. Frenken / Surface Science 601 (2007) 13–23

where f is the frequency with which kinks pass the observa-tion line [1]. This expectation is based on the idea that thepassages of kinks through the observation line would beindependent, uncorrelated events. At short times, the mea-surements in Ref. [1] do not follow this exponential law.Initially, the probability density P(t) decays much morerapidly than the exponential behavior observed at longertimes. The explanation given in Ref. [1] is that most eventsin which a kink recedes over one lattice spacing are quicklyfollowed by the reverse event, in which the emitted atomreturns to the same kink. These ‘feints’ would lead to thelarge contribution of relatively short waiting times. The‘true’ motion of the kinks would result from events inwhich a second atom is emitted from the kink before thefirst one has returned, which was argued to result in a lossof correlation in the kink motion that would be reflected inthe exponential long-time part of the waiting timedistribution.

In the present paper we argue that the time scales onwhich adatoms diffuse and kinks jump are separated byat least two orders of magnitude. A theory, based on kinkmotion only, is developed that offers a more natural expla-nation for the observed waiting time distribution. We showthat one should take into account explicitly that (i) on ashort time scale the kink passages are by no means inde-pendent, uncorrelated events, while (ii) on a long time scalethe dynamics will be influenced by creation and annihila-tion of kink–antikink pairs. Various MC simulations areused to test the theory and some of its assumptions, andto incorporate the effect of the creation and annihilationof kinks. Also, the MC simulations allow us to simulatethe poor time resolution of the STM, because of which a

large portion of the kink jumps go completely unnoticed.Finally, by simulation of the temperature dependence ofthe waiting time distribution we show that it is not possibleto relate the short and long time scales of the distributionto separate microscopic processes, as is done in Ref. [1].Instead, each regime is determined by a non-trivial combi-nation of the activation energy for the emission of an ada-tom from a kink and the formation energy of a kink.

2. Definitions and assumptions

Our starting point will be to treat each kink in the stepas an independent random walker. By either ejecting anatom or receiving one, a kink moves backward or forwardover one lattice spacing.

The first simplification will be that we assume that thefrequency with which atoms manage to liberate themselvesfrom kinks is very much lower than that with which eachliberated atom subsequently performs its random walk.This random walk has a high probability to bring the atomback to the kink where it originated, and a small probabil-ity to make it join another kink, either in the same step orin a neighboring step. We expect that the ratio between theemission frequency of atoms from kinks and the hoppingfrequency of the emitted (ad)atoms is so low, that the ran-dom walk of a newly formed adatom is finished well beforethe next atom is emitted from the same kink. This has theconsequence that we need only consider those ‘events’ inwhich the atom does not return to the kink where it origi-nated. Only those non-return events really make the kinkmove. In practice, the time scales of the emission and thehopping are almost always sufficiently different, that the

G.S. Verhoeven, J.W.M. Frenken / Surface Science 601 (2007) 13–23 15

STM is ‘blind’ to the details of the faster process and it isable to ‘see’ only the outcome of each atom’s random walk:

(a) If the random walk brings the atom back to the kinkwhere it started, the STM sees no kink motion at all,since the duration of the random walk is too short.

(b) If the random walk brings the atom to a differentkink, the STM sees both the emitting and the receiv-ing kink move over a single lattice spacing.

The difference in activation energy between emission ofatoms from kinks and the subsequent hopping, required toreach the separation in time scales described above, can beestimated as follows. Contrary to naıve expectation, theaverage number of hops in a complete one-dimensionalrandom walk of an atom between two neighboring kinksis equal to L � 1, where L is the number of lattice spacingsbetween the kinks [10]. In the cases that we are consideringhere, the typical distances between kinks are a few tens ofatomic spacings, so that the average number of hops inthe random walk of a single atom will be below 100. Forthe full separation of time scales we thus have to requirea frequency ratio of at least 100 between the fast (hopping)and the slow (emission) processes. If both processes can bedescribed as simple activated processes with comparablepre-exponential factors, the required ratio corresponds toa minimum difference in activation energies of kBT ln(1 00),which amounts to 0.12 eV at room temperature. Based onthe information we have from theory (e.g. [11]) as well asexperiment (e.g. [12]), we expect that most steps easilysatisfy this criterion.

Before describing the calculation of the kink return sta-tistics, we address two further details of the elementarykink motion. A kink recedes by one atomic spacing whenit emits one atom. If the ejected atoms are forced to remainbound to the step, the probability for them not to return tothe kink from which they originated, and thus make thekink ‘really’ move, depends strongly on the distance tothe next kink.2 Thus, the effective jump frequency of a kinkis not constant, but it reflects the distances to the two near-est neighbors. Here, we average over all possible kink con-figurations and can treat the kink hopping frequency as asimple constant, as will be justified in Section 4.

Secondly, we start by also neglecting events in whichnew kinks are formed or existing ones are annihilated.The frequency of these events is observed to be only a verysmall fraction of the hopping frequency of kinks [12,13],which serves as additional evidence that most ejected kinkatoms find their new destination well before the next atomis emitted from the kink site (see discussion above). In Sec-tion 5 Monte Carlo simulation results will be presentedthat illustrate how such creation and annihilation eventsaffect the statistics at longer waiting times.

2 Even if the atoms are emitted onto the terrace, the degree ofcorrelation in the motion of neighboring kinks is substantial, unless thereis a significant attachment barrier for adatoms to kinks [12].

3. Calculation

For convenience, we simplify the motion of the kinks ina step to one in which they all make synchronized jumps (inrandom directions) at regular time intervals s = 1/C, whereC corresponds to the average jump rate of a kink.Although this greatly simplifies the mathematical treatmentof the kink motion, it does not significantly influence theresults at times larger than a few s, as will be shown inSection 4.

The STM measurements treated here record a singlescan line across a step with a high repetition frequency.Whenever a kink moves through this line, the STM detectsa sudden change in the step position. The quantity of inter-est is the time until the next event of a kink passing the line.Two different cases can be distinguished here, and we willtreat them both.

3.1. The clock starts when a kink crosses the STM line

This is the case considered in Ref. [1]. We will refer tothe kink that sets off the clock as the ‘special’ kink. Wechoose our space and time coordinates such that at t = 0,the special kink jumps from position 0 to position 1 alongthe step. The STM line is located somewhere between posi-tions 0 and 1, say at position 1/2. There are two possibili-ties for the next kink that crosses the line; it can be eitherthe special kink itself or a neighboring kink. We first treatthe limit of infinitely low kink density, for which we needonly consider the return of the special kink to the origin.In that case, the probability Pn � P(ns) for the next eventto take place at the nth time step, at time ns, is simply

P n ¼ P sn ¼ u1;n; ð1Þ

where P sn is the probability for the special kink to reach the

origin for the first time at the nth tick of the clock. Thefirst-passage probability uz,n denotes the probability for arandom walk, starting at position z, to reach the origin,i.e. cross the location of the STM line, for the first timeat the nth time step. This probability has been derivedbefore, and it can be cast in the following form [10]:

uz;n ¼zn

n12ðnþ zÞ

� �1

2n for even nþ z;

uz;n ¼ 0 for odd nþ z:

ð2Þ

This probability is shown on a semi-logarithmic scale inFig. 2 (curve for zero kink density). In order to removethe variation between non-zero and zero values for evenand odd n + z, which are an artifact due to our choice ofa regular time interval between successive kink jumps, alldistributions in Fig. 2 have been convoluted with a simplethree-point smooth function with weights (0.25,0.50, 0.25).Two time regimes can be distinguished in Fig. 2. At shorttimes, the probability shows a steep decrease with time.At longer times, Pn decreases much more slowly. Theshort-time behavior reflects the fact that directly after

Fig. 2. Probability distributions Pn for waiting times n between subsequent passages of a kink. The time is measured in units of the average time betweensubsequent kink jumps. The curves show the results obtained with Eq. (9) for kink densities ranging from m = 0 to m = 0.10. The result for m = 0 is identicalto that from Eq. (2). As explained in the text, the results have been smoothed to remove the artificial variation between odd and even n-values. The circlesare the result of the Monte Carlo simulation introduced in Section 4, for m = 0.02. The simulation included the effect of correlations in the kink motion dueto the exchange of atoms between kinks. The simulation did not have the artifacts of regular time intervals and synchronized jumps that were present in thetheory leading to Eqs. (2) and (9).

16 G.S. Verhoeven, J.W.M. Frenken / Surface Science 601 (2007) 13–23

starting the clock, the special kink is still close to the originand has a high, but quickly reducing, probability for jump-ing back to the origin. Although the memory of the startingconfiguration is lost increasingly for longer times, at notime the result converges to a simple exponential form. Infact, the distribution u1,n is known to converge quickly tou1,n / n�3/2.

If the kink density q is finite, we have to take intoaccount the possibility for another kink than the specialone to cross the STM line at time n. In that case we find

P n ¼ P snQd

n�1 þ QsnP d

n ¼ P snQd

n�1 þ QsnðQd

n�1 � QdnÞ: ð3Þ

The first term in Eq. (3) corresponds to the contribution ofthe special kink returning to the origin, possibly accompa-nied by other kinks. As before, P s

n is the probability for thespecial kink to reach the origin for the first time at the timen. Qd

n�1 is the probability that none of the other kinksreaches the origin within the first n � 1 time steps. The sec-ond term is the contribution from the other kinks reachingthe origin before the special kink. P d

n is the probability thatthe first ‘non-special’ kink reaches the origin at time n, pos-sibly accompanied by other non-special kinks. Qs

n is theprobability that the special kink does not reach the originwithin the first n time steps. We express Qs

n in terms ofthe first-passage probabilities introduced in Eq. (2):

Qsn ¼ 1�

Xn

m¼1

u1;m: ð4Þ

Qdn is the product of non-passage probabilities for all other

kinks

Qdn ¼

Yn

k¼1

X1i¼0

qk;iðQk;nÞi" #( )2

: ð5Þ

The product runs over the first n positions along the step,i.e. those positions from which a kink can reach the originwithin the first n time steps. The summation runs over allpossible numbers of kinks i at position k along the step.The probability for finding i kinks at position k at t = 0is denoted by qk,i. The quantity Qk;n in Eq. (5) is the prob-ability that a random walk starting at position k does notreach the origin within the first n time steps. This probabil-ity can be written as

Qk;n ¼ 1�Xn

l¼1

uk;l: ð6Þ

The square in Eq. (5) reflects the fact that the non-specialkink can cross the STM scan line from two opposite sides.

We assume the kinks to be distributed randomly alongthe step, with an average number density (average numberof kinks per lattice site along the step) m. The probabilityfor finding i non-special kinks at the same position k att = 0 is then given by the Poisson distribution

qk;i ¼mi

k

i!expð�mkÞ: ð7Þ

G.S. Verhoeven, J.W.M. Frenken / Surface Science 601 (2007) 13–23 17

In Eq. (7) we have retained the label k, indicating the posi-tion along the step. This is necessary, since the mere selec-tion of the starting event, the special kink jumping from 0to 1, has a direct influence on the probabilities of findingother kinks at positions 0 and 1 at t = 0. These probabili-ties deviate slightly from a Poisson distribution, but themain effect is that, in the limit of low average kink densitiesm along the step, the kink densities m0 = m1 at positions 0and 1 at t = 0 are lowered to approximately 0.75m.3 Theprobabilities for other positions along the step are notaffected.

With Eq. (5) we have tacitly assumed that there are nointeractions between the kinks, even when they occupythe same position along the step. Each time interval, everykink independently jumps in a direction chosen randomly.In the absence of interactions, we can treat ‘collisions’ be-tween kinks as if the kinks simply run through each other(by permutation of their labels), rather than reflect fromeach other. Since we do not discriminate which kink ‘stopsthe clock’, it is not necessary to re-label the kinks in such acrossing event. In reality, there are of course only two kinksthat can stop the clock: the special kink and one of its twoneighbors.

Combining Eqs. (5)–(7) we obtain the following expres-sion for Qd

n :

Qdn ¼

Yn

k¼1

exp 2mk Qk;n � 1� �� �

¼Yn

k¼1

exp �2mk

Xn

l¼1

uk;l

!

¼ exp �2Xn

k¼1

Xn

l¼1

mkuk;l

!: ð8Þ

We substitute Eqs. (1), (4) and (8) in Eq. (3), to obtain thecombined probability for a waiting time of n time steps.

P n ¼ u1;n exp �2Xn�1

k¼1

Xn�1

l¼1

mkuk;l

!þ 1�

Xn

m¼1

u1;m

!

� exp �2Xn�1

k¼1

Xn�1

l¼1

mkuk;l

!� exp �2

Xn

k¼1

Xn

l¼1

mkuk;l

!" #:

ð9Þ

3 The effective lowering of the densities of non-special kinks at positions0 and 1 at t = 0 can understood as follows: when the special kink jumps,by definition from 0 to 1, a non-special kink can jump to either position 0or 1 (at low kink densities, we can neglect configurations where three ormore kinks jump to positions 0 and 1). In half of these events the non-special kink does not cross the STM line at position 1/2. The other halfhowever, consists of events where both kinks cross the STM line atposition 1/2 and set off the clock. Since either kink can be labeled ‘special’,the ‘other’ kink contributes only half to the kink density at positions 0and 1. This leads to an effective kink density of 0.75m.

We have not obtained an analytical solution of the sum-mations in Eq. (9), but we have directly evaluated theexpression for Pn numerically. The result is shown inFig. 2 for kink densities m ranging from 0.02 to 0.10.

Comparing the calculations in Fig. 2 for non-zero kinkdensities with those for m = 0, we see that the effect of otherkinks only becomes noticeable for relatively high kink den-sities. The estimated kink density in steps on Cu(001)ranges from 0.011 to 0.036 for temperatures between290 K and 370 K [1]. This suggests that on Cu(001) both

the short and the long-time part of the waiting time distri-bution would be dominated almost completely by a single

kink, namely the special kink. Fig. 2 might give the impres-sion that for higher kink densities the integral under thecurves would fall below the integral for lower kink densi-ties. Actually, the integrals are all unity; at short times,the curves for high kink density run above those for lowkink density, which is difficult to see on the logarithmicscale of Fig. 2.

In order to assess whether the non-special kinks alonewould result in an exponential waiting time distribution,we next evaluate Pn in the absence of a special kink.

3.2. The clock starts at a random point in time

If we start the clock at a random instant in time, theconfiguration of starting positions of kinks with respectto the origin is completely random. In that case, there isno ‘special’ kink. The distribution of waiting times untilthe first passage of a kink through the origin is now givenby

P n ¼ P dn ¼ Qd

n�1 � Qdn

¼ exp �2mXn�1

k¼1

Xn�1

l¼1

uk;l

!� exp �2m

Xn

k¼1

Xn

l¼1

uk;l

!:

ð10Þ

Note that in this case mk � m for all k. Again, we have notfurther simplified the expression for Pn. However, it is clearthat even for long times, Pn will still not converge to a sim-ple exponential law. This is confirmed by Fig. 3, whichshows the numerical evaluation of Eq. (10). At no pointin time does the probability behave exponentially. At firstsight this may seem unexpected in view of the completelyrandom starting configuration and the complete absenceof correlation between the motion of the kinks. However,an exponential distribution should be expected only for atrue Poisson process, in which each event is completelyuncorrelated with previous or later events. Here, this con-dition is not fulfilled, since the probability for a kink toreach the origin for the first time at t = n depends on thefull history of that kink in time steps 1 to n � 1.

What is responsible for the continuous curving of ln(Pn)versus time is the following selection phenomenon. Config-urations that start with kinks close to the origin on averagelead to shorter waiting times than those starting off with

Fig. 3. Probability distributions for waiting times until the first passage of a kink, starting from a random point in time. The curves show the resultsobtained with Eq. (10) for kink densities ranging from m = 0.02 to m = 0.10.

18 G.S. Verhoeven, J.W.M. Frenken / Surface Science 601 (2007) 13–23

kinks far away from the origin. In other words, the distri-bution of configurations contributing to Pn changes as afunction of time, those with kinks originally close to theorigin ‘disappearing’ before the others.

4. Simplifications that have no noticeable effect:

synchronization, regular time intervals, and correlationsin the kink motion

In order to investigate whether the simplifying assump-tions underlying the analytical theory noticeably affect thetheoretical waiting time distributions, we have performedfour types of Monte Carlo (MC) simulations. Table 1 indi-cates the main differences between the simulations. AllMC simulations presented here have been performed ona 1-dimensional discrete lattice with periodic boundaryconditions. All results presented here are not significantly

Table 1Characteristics of the four simulations discussed in this paper

Simulationtype

Introducedin Section

All kinks jumpsimultaneously

Kinks juin pairs

1 Section 4 Yes No2 Section 4 No Yes3 Section 5 No Yes4 Section 6 No Yes

The simulations can be distinguished on four main points: (a) whether or not akinks are correlated because they result from the exchange of atoms; (c) whetheof the limited time resolution of the STM is taken into account.

influenced by choosing a different starting configuration,increasing the number of MC steps or increasing the latticesize.

In the first version of the simulation the kinks performeduncorrelated, synchronized jumps at regular time intervals,precisely as in the analytical theory of the previous section.This simulation, performed for kink densities ranging fromm = 0.01 to m = 0.10 on a lattice of 500 sites for 107 jumps,confirmed Eqs. (9) and (10). In the second MC simulationthe kink jumps were not synchronized and not regular intime. In addition, the second simulation included the corre-lation that results when kinks move by exchanging atomswith each other (see also Section 2). Each exchange takesplace via a one-dimensional random walk of an emittedatom along an (empty) step. After the emission, the atomstarts its random walk one position next to the kink i, fromwhich it originated. For an unbiased random walk, the

mp Kinks can be createdor annihilated

Limited timeresolution of STM

No NoNo NoYes NoYes Yes

ll kinks jump simultaneously; (b) whether or not the jumps of neighboringr or not kink pairs are created and annihilated; (d) whether or not the effect

Fig. 4. Creation or annihilation of a kink–antikink pair consisting of twoadatoms. Going from A to B, the kink–antikink pair is annihilated in thefollowing way. One of the two adatoms is emitted and attaches to theneighboring kink. This leaves the other adatom free to diffuse to eitherneighbor, at which point the total number of kinks has decreased by two.The reverse process (creation of a new kink–antikink pair, going from B toA) is also implemented in our simulations (see text). Note that in thisexample, the adatoms end up at (or originate from) different neighboringkinks. This is just one of the possibilities; both adatoms could also end upat (or originate from) the same neighboring kink. Creation and annihi-lation of structures with step vacancies (‘antikink–kink’ pairs) rather thanstep adatoms are implemented in our simulation completely equivalently.

G.S. Verhoeven, J.W.M. Frenken / Surface Science 601 (2007) 13–23 19

probability for this atom to reach the next kink j beforerevisiting its parent kink i is equal to 1/Lij, where Lij isthe distance (dimensionless: number of lattice sites) be-tween the two kinks [10]. This was accomplished in our sec-ond MC simulation by moving the kinks in pairs (i, j), withan atom exchange rate between the kinks varying propor-tional to the inverse distance between the kinks, 1/Lij(t).The direction in which each kink moved was determinedaccording to the kink orientation, inward or outward,which was chosen randomly for each kink. The exchangerate was chosen to be independent of the combination ofthe orientations of the two kinks involved in the exchangeprocess; i.e. we did not consider the possibility of an addi-tional (‘Ehrlich–Schwoebel’-type [14,15]) energy barrier forthe diffusing atoms at inwardly oriented kinks. The timescale of the MC simulation was linked to that of the theoryof Section 3 by equating the time step in the theory to theaverage time per kink jump in the simulation.

The circles in Fig. 2 are the results of the second MCsimulation, i.e. with asynchronous, non-regular kink jumps,and including the correlations described above, for a kinkdensity of m = 0.02 (109 simulated emission events). Overthe calculated number of time steps (see Fig. 2) the resultsof the MC simulation are identical to the curve calculatedwith Eq. (9) for this kink density, to within the statistical er-ror. This means that the correlation in the kink motion hasa negligible effect on the waiting time distribution. It furtherimplies that the effects of the synchronization of the kinkjumps and the regular time intervals in the derivation ofEq. (9) are also negligible.

5. Effect of spontaneous creation and annihilation of kinks

A third MC simulation was used to investigate the effectof the spontaneous creation and annihilation of kinks. Inthis simulation we kept track of the orientation of eachkink. As in Section 4, a MC step consists of an atom emis-sion event in which the atom either returns to the kink fromwhich it originated, leaving the step unchanged, or reachesa neighboring kink where it attaches, resulting in correlatedkink motion. In this simulation we assumed that a pair ofneighboring adatoms would form the shortest stable kink–antikink configuration along a step. When reduced inlength to a single adatom, such a configuration was madeto disappear via diffusion of that last adatom to one ofthe neighboring kinks, thus reducing the total number ofkinks by 2 (depicted in Fig. 4). To which neighboring kinkthe adatom would attach was determined by a weightedprobability based on the inverse distances between the ada-tom and the two nearest kinks (see previous section). Thesecond minimum kink–antikink configuration existed of apair of neighboring vacancies (missing atoms), which wasassumed to disappear in the same way, when one of thetwo vacancies was removed.

In the simulation, new kink pairs (adatom or vacancypairs) were created at a fixed rate per lattice spacing at ran-dom locations along the step. For each annihilation path-

way, the corresponding reverse process was included inthe simulation. For example, a new pair of adatoms canbe created at a randomly chosen location along the stepwhile two atoms are removed from one or two neighboringkinks. The probabilities involved for at which kink theatoms originate again depend on the inverse distances tothe neighboring kinks. Since our procedure makes theannihilation an automatic consequence of random kinkmotion, the ratio between the kink pair creation rate andthe kink hopping rate determined the average kink density;in other words, the desired kink density dictated the choiceof the kink pair creation rate used in the simulation. Thedependence of the average kink density on the creation rateis shown as an inset in Fig. 5. In all simulations presentedbelow the kink configuration has been equilibrated bystarting with an initially straight step with one kink–anti-kink pair on a lattice of 2000 sites and letting the stepevolve to the average kink density dictated by the creationrate (�108 MC steps).

During the simulation, the ratio between the total num-ber of kinks and the quantity 1

2h1=Li (each successful atom

exchange makes two kinks move) is calculated at every MCstep. The average of this quantity equals the number ofMC steps over which on average all kinks have jumpedonce. This allows us to again fix the time axis so that thesimulated waiting time distributions can be compareddirectly with the analytical curves presented in Fig. 2.

Note that this MC approach differs from an approach inwhich kink motion emerges through actual simulation ofemission and diffusion of adatoms. As discussed in Section2, the time scale of both the theory and the simulations pre-sented here is one where only the result of adatom diffusionis considered. Thus, the kink jump frequency, that depends

Fig. 5. Probability distributions Pn for waiting times n between subsequent passages of a kink. The circles are the result of the Monte Carlo simulationintroduced in Section 5, for an average kink density of m = 0.02. This simulation not only included the effect of correlations in the kink motion due to theexchange of atoms between kinks, but also the creation and annihilation of kink pairs. The theoretical curve for m = 0.02 is reproduced from Fig. 1. Thedashed, straight line shows that for long times, the distribution approaches an exponential form. Inset: Relation between the kink pair creation rate andthe average kink density (divide the creation rate by the lattice size of 2000 to obtain the creation rate per MC step per lattice site).

20 G.S. Verhoeven, J.W.M. Frenken / Surface Science 601 (2007) 13–23

on the kink density, implies an emission frequency of ada-toms from a kink,4 and the adatom jump frequency isabsent as parameter.

In Fig. 5 the modest effect of the creation and annihila-tion of kink–antikink pairs on the waiting time distributionis shown for a simulation with an average kink densityof m = 0.02 (2 · 108 MC steps). For short waiting timesthe simulation closely follows the theoretical curve form = 0.02, which was shown to be completely determinedby the special kink that started the clock. For longer wait-ing times the simulation is slightly above the theoreticalcurve and it approaches an exponential form (straight linein Fig. 5). This is because the addition and removal ofkinks along the step slowly erases the memory of the kinkconfiguration present when the clock was started.

In addition to the waiting time distribution shown inFig. 5, simulations have been performed for different aver-age kink densities. As expected, for short waiting times thesimulated distributions are nearly identical to the theoreti-cal curves (not shown). The exponential decay rate for eachdistribution was determined by fitting the long-waiting-time part (waiting times between 1000 and 2000 time steps)to a single exponential. The decay rates obtained are shownin Fig. 6. The dashed line is a linear fit to the obtained

4 The emission frequency of atoms from kinks is equal to Ce ¼12 Cj=h1=Li, where Cj is the jump frequency of the kinks, h1/Li is theaverage inverse distance between kinks, and the factor 1/2 appears becauseeach successfully exchanged atom makes two kinks move.

decay rates. The exponent (slope) of the long-time behavioris directly correlated with the average kink density.

6. Comparison with experiment

Finally, in Fig. 7, we reproduce the data for steps onCu(001) at 340 K from Ref. [1], in order to compare themwith our theory. The step position had been measured withregular time intervals of 90 ms. The probability that anindividual step position measurement is the end point (orstarting point) of a waiting period of n time intervals isplotted along the vertical axis of Fig. 7. Note that this def-inition of the vertical axis differs from that in Figs. 2, 3 and5. The area under each of the curves in Figs. 2, 3 and 5 isidentical to 1, whereas that in Fig. 7 is equal to the proba-bility that the measured step position is different from thatfound in the preceding measurement. Although Eq. (9) can,after proper scaling of the horizontal and vertical axes,mimic the main characteristics of the data in Fig. 7, sucha direct comparison between the theory and the data isnot appropriate.

The frequency of 11 Hz of the measurements of the stepposition is much lower than the expected kink jump rate[12]. The effect of this is more severe than that of a mereconvolution with an ‘instrument function’, accounting forthe limited time resolution, since a large fraction of the‘events’ of kinks crossing the observation line actually goescompletely unnoticed in the experiment!

We extended our MC simulation of Section 5 to takethis effect into account properly. This simulation goes

Fig. 7. Comparison between the experimental waiting time distribution for Cu(001) at 340 K from Ref. [1] and the best-fit Monte Carlo simulation resultfor a kink density of mavg = 0.025. The probability that an individual step position measurement is the end point (or starting point) of a waiting period of n

time intervals is plotted along the vertical axis. Note that this probability is differently normalized than the probabilities in the previous plots. The time perSTM scan line was 90 ms. The only free fitting parameter has been the ratio j between the kink hopping rate and the line rate of the STM. The best-fitvalue for this ratio is j = 150, which shows that a large fraction of kink motion events must have gone completely unnoticed in the experiment. Theory andexperiment are compared on an absolute scale, without any free scaling of either the probability or the time axis. Inset: Comparison between the best-fitMonte Carlo simulation (j = 150) with a simulation where the measurement frequency equals the average kink jump frequency (j = 1), i.e. where the timeresolution of the STM is sufficient to follow all kink jumps. The time axis of the experimental fit (j = 150) has been scaled accordingly.

Fig. 6. Exponential decay rates determined from the long-waiting-time part (between 1000 and 2000 time steps) of calculated waiting time distributions fordifferent average kink densities. The dashed line is a linear fit.

G.S. Verhoeven, J.W.M. Frenken / Surface Science 601 (2007) 13–23 21

beyond measuring the time of subsequent kink passages. Inaddition it records changes in step position, fully takinginto account the orientations of the passing kinks and thedirections in which they pass. If the combination of kinkpassages, over the time between subsequent observations,

adds up to zero step displacement, this kink activity goescompletely unnoticed. The two unknowns in this simula-tion, which serve as the fitting parameters, are the averagekink density m (determined by the creation rate of newkink–antikink pairs) and the ratio j between the (average)

22 G.S. Verhoeven, J.W.M. Frenken / Surface Science 601 (2007) 13–23

kink jump rate and the measurement frequency. Together,these two parameters completely fix the vertical scale andthe time scale, so that the comparison can be made on anabsolute basis. The effect of increasing m is very similar tothat of increasing j As a result, we obtain reasonableagreement between the theory and the experiment for a sur-prisingly wide range of (j,m) combinations (for example:m = 0.01–0.04, j = 50–300). This means that the waitingtime distribution of Ref. [1] at 340 K does not allow usto draw independent conclusions about the density of thekinks and their mobility.

The fit shown in Fig. 7 was obtained from a simulationfor a kink density of m = 0.025. This is the kink density at340 K, calculated from the kink formation energy ofEk = 0.128 eV, which was derived independently in Ref.[1] from the spatial meandering of steps on Cu(001) at290 K. For this choice of kink density the best-fit valueof the frequency ratio is j = 150 ± 20. This correspondsto an average jump frequency of the kinks of 1650 ±200 Hz, which in turn corresponds, at the given kink den-sity, to an emission frequency of atoms from kinks of9 ± 1 kHz.4 This allows us to estimate the emission activa-tion energy at 0.54 eV (6309 K) (assuming a pre-exponen-tial factor of 1012 Hz). These numbers compare favorablywith those obtained in a direct measurement of kink mobil-ity on a vicinal Ag(0 01) surface at room temperature [12].In that measurement an atom emission frequency wasfound of 700 ± 100 Hz, and the emission activation energywas estimated to be 0.6 ± 0.1 eV (6962 K). Thus, the emis-sion frequency expected for atoms from kinks on Ag(00 1)

Fig. 8. Temperature dependence of the two integrated probabilities Plong andparts of plots of the type shown in Fig. 4, calculated for a range of temperaturesPshort and Plong, in each case ignoring the highest-temperature data point (see

at 340 K is 19 ± 10 kHz, which is only a factor 2 differentfrom the emission frequency obtained here for Cu(001)at that temperature. The similarity between these two sur-faces was to be expected, since it was shown in Ref. [11]that the energetics of surface defects on Ag(001) andCu(001) are very similar. The MC simulation providesan excellent fit to the experimental waiting time distribu-tion in Fig. 7 over the entire time range, describing not onlythe steep initial decrease, but also the slower decrease atlonger times. MC simulations that do not include the crea-tion and annihilation of kinks provide a somewhat lessgood match with the experimental data of Ref. [1].

The inset in Fig. 7 illustrates the effect of the measure-ment frequency on the waiting time distribution. The sim-ulation used for fitting the experimental data (j = 150) iscompared to a simulation where the measurement fre-quency equals the average kink jump frequency (j = 1).The time axis of the experimental fit (j = 150) has beenscaled accordingly.

We complete our investigation with the temperaturedependence of the process. Combined with the 0.128 eV(1490 K) kink formation energy, the emission activationenergy estimated above at 0.54 eV (6309 K) allows us topredict the values of j and m for each temperature and sim-ulate, using the MC simulation of Section 5, the resultingtemperature dependence of the distribution in Fig. 7. Wesubject these distributions to the same analysis as was ap-plied in Ref. [1] to the experimental distributions. At eachtemperature, the integrated probability Ptot is divided intwo parts. One part, Plong, corresponds to the integrated

Pshort = Ptot � Plong, with Plong obtained from linear fits to the long-time. The two straight lines are Arrhenius fits to the temperature dependence oftext).

G.S. Verhoeven, J.W.M. Frenken / Surface Science 601 (2007) 13–23 23

probability below an exponential fit (linear fit in the semi-logarithmic plot of Fig. 7) to the tail of the distributionfor long-time intervals up to 50 scan lines (c.f. Fig. 4 inRef. [1]). The other part corresponds to the differencePshort = Ptot � Plong. The two integrated probabilitiesPshort and Plong are shown in Fig. 8 as a function of inversetemperature. At the highest temperature T = 370 K, onlythe first few data points differ significantly from the linearfit to the long-time behavior, making it difficult to deter-mine Pshort accurately. For all other, i.e. lower tempera-tures Pshort is higher than Plong, demonstrating that thespecial kink is responsible for the larger part of the totalprobability Ptot at the temperatures considered here.

From the slopes of the two straight-line fits in the Arrhe-nius plot the apparent activation energies for the short andlong-time regimes of the distribution are calculated to beEshort = 0.412 eV (4779 K) and Elong = 0.484 eV (5618 K)respectively. These numbers are close to the experimentalactivation energies derived in Ref. [1] of Eexp

short ¼ 0:43�0:04 eV (5010 ± 482 K) and Eexp

long ¼ 0:41� 0:04 eV (4780 ±490 K). We have verified that the outcome of this proce-dure is very insensitive to the precise choice of assumedpre-exponential factor. However, the value of Elong doesdepend critically on the range of times (no. of scan lines)to which the procedure is applied. In particular, on the timescale of the experiment of Ref. [1], the distributions do notconverge sufficiently to pure exponential behavior toobtain a proper value for Elong. When given more time,Elong increases from 0.484 to 0.697 eV (8089 K), whileEshort remains almost the same, 0.394 eV (4567 K).

From the numbers extracted from our Arrhenius analy-sis, we immediately see that neither of the two apparentactivation energies can be associated directly with one ofthe two microscopic properties that fully determine thekink dynamics in our calculations, namely the frequencyof adatom emission from kink sites (activation energy0.54 eV) and the density of kinks (formation energy of0.128 eV).

7. Summary

In summary, we have shown that the distribution ofwaiting times between subsequent passages of kinks atone position along an atomic step, as measured in Ref.[1], emerges as a combination of the kink dynamics, includ-ing creation and annihilation of kinks, and the limited timeresolution of the STM. The majority of time intervals arestarted and stopped by passages of the same kink. At lowkink density the resulting waiting time distribution is de-scribed well by P(t) / t�3/2. The presence of other kinks

makes the distribution slightly steeper. Finally, the effectof the creation and annihilation of kinks is to slowly erasethe memory of the starting configuration and make the dis-tribution approach an exponential form. However, thiseffect becomes noticeable only at long times and high kinkdensities.

The poor time resolution of the STM makes most of thekink passages go completely unnoticed, which has a signif-icant influence on the experimental waiting time distribu-tion. Rather than to depend on two activated processes,atom emission from kinks and atom diffusion along steps,the distribution only depends on the atom emission pro-cess, and its temperature dependence reflects a non-trivialcombination of the corresponding activation barrier andthe formation energy for kinks.

Acknowledgments

The authors are grateful for helpful commentsby S. Yu Krylov. This work was financially supportedby the ‘Nederlandse Organisatie voor WetenschappelijkOnderzoek’ (NWO).

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