OMPUTER IMULATION TUDIES FOR OLLECTIVE ADATOM DIFFUSION STEPPED

COMPUTER SIMULATION STUDIES FOR

COLLECTIVE ADATOM DIFFUSION

STEPPED SURFACES

M. Masın1,2, I. Vattulainen2,3, T. Ala-Nissila1,4 and Z. Chvoj21Laboratory of Physics, Helsinki University of Technology,

P.O. Box 1100, FIN–02015 HUT, Espoo, Finland2Institute of Physics, Academy of Sciences, Czech

Republic, Na Slovance 2, 182 21, Praha 8, Czech Republic3Helsinki Institute of Physics, Helsinki University of

Technology, P.O. Box 1100, FIN–02015 HUT, Espoo, Finland4Department of Physics, Brown University, Providence,

R.I. 02912–1843, U.S.A.

AbstractWe present a comprehensive summary of our recent work on computer simulation

of diffusion on stepped surfaces. We focus on the mobility and collective diffusionof mutually interacting adparticles on regulary stepped surfaces through Monte Carlosimulations. To this end, we consider a model system characterized by binding at thestep edge, complemented by either nearest–neighbor repulsive or attractive interac-tions between two adparticles. This approach allows us to study the interplay of stepbinding and adparticle interaction effects in a systematicmanner, thus providing in-sight into the influence of steps on surface diffusion phenomena on a rather genericlevel. We show that the coverage dependence of diffusion across the steps is sensitiveto the adparticle interaction energy. For strong step binding, it is known that the cor-responding collective diffusion coefficientDx,c(θ) has a maximum at a coverage ofθ = 1/L for a given step spacingL. We show that this maximum moves to highercoverages with increasing repulsion, strongly enhancing diffusion. In the case of at-tractive adparticle interactions, the local maximum is smoothed out with increasingattraction, strongly reducing diffusion. For weak step binding, these effects are evenmore pronounced. We also find that with a given adparticle interaction, increasingthe step binding energy slows down diffusion across the steps but does not change theoverall coverage dependence ofDx,c(θ). As to diffusion along the steps, the presenceof steps is found not to be as important. The behavior of the corresponding collec-tive diffusion coefficientDy,c(θ) remains qualitatively unchanged, being only ratherweakly affected by steps at intermediate coverages. When onefurther includes an ad-ditional activation barrier for diffusion along the lower step edge, thus either enhancingor slowing down diffusion along the step edge, one findsDy,c(θ) to be strongly en-hanced or reduced respectively at low coverages, while at high coverages the behaviorof Dy,c(θ) remains unchanged. Implications of these findings are briefly discussed.

320 M. Masın, I. Vattulainen, T. Ala-Nissila et al.

1 Introduction

In surface systems, diffusion is one of the most interesting processes that arise from the dy-namics of adparticles. Many present-day technological applications suchas surface growth,formation of nano-scale dots and clusters, and spreading as well as lubrication phenomenaare essentially dynamic processes that involve the transport of adatoms and molecules onsolid surfaces. Diffusive transport on surfaces furthermore playsa vital role in chemicalreactions where diffusion may be the rate-limiting factor, thus its role in technologicallyand environmentally important processes such as catalysis should be appreciated.

Due to the above reasons, surface diffusion has attracted a great deal of attention duringthe last decade [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. This highly positive effort hasallowed one to gain insight into a wide variety of issues where surface diffusion plays a role,the most thoroughly studied subject being related to single-particle and collective diffusionon flat surfaces under equilibrium conditions. While this part of the subject is currentlywell understood, as we feel, there are also a number of challenging issues that are muchless understood, if at all.

As a matter of fact, despite major efforts during the last decade, it is rather surprisinghow little is known about a number of different issues related to conditions used in practicalapplications. For example, there is a great need for an understanding ofdiffusive processesunder non-equilibrium conditions, which is the case in surface growth conditions and phaseordering phenomena, among other examples. Recent studies have been able to clarify thisissue in part (see, e.g., Refs. [15, 16, 17, 18, 19, 20]). Also, in contrast to flat, ideal-likesurfaces, the situation is obviously far more complicated in the case of spatiallyinhomoge-neous surfaces which exist in many experimental systems. Some insight hasbeen gainedrecently through studies of systems where the influence of impurities and surfactants on sur-face diffusion has been investigated (see Ref. [21] and referencestherein). Another exampleof very intriguing recent work concerns the diffusion of vacancies and vacancy-mediateddiffusion of atoms in the surface layer, which has shown highly unusual, concerted type ofmotion of surface atoms [22].

The above examples highlight the complex nature of surface diffusion in strongly in-teracting adparticle systems. Additionally, they illustrate the fact that the understanding ofdynamic phenomena associated with realistic surface systems is on a very limited ground,thus further attention is needed to address questions related to this exciting field.

In the present work, our aim is to shed some light on an issue that is an inherent partof essentially all realistic surface systems. Namely, most surfaces used in experiments aswell as practical applications are not flat. Rather, they are characterized by steps. Vicinalsurfaces [9, 23, 24, 25] with equally spaced steps, in particular, constitute a highly relevantspecial case of stepped surfaces and play an important role in e.g. surface growth andcatalysis. Despite this, the understanding of diffusion on such surfacesis still very limited.It has been shown through many experiments that the diffusion of adatoms on steppedsurfaces may be distinctly different from diffusion on flat terraces, thediffusion along andclose to step edges being recent examples of this issue [26, 27, 28]. Therole of steps

Computer Simulation Studies for Collective Adatom Diffusion Stepped Surfaces 321

on chemical reactions and catalysis has also been recognized [23, 29].Further, it has beenfound through a number of experiments that the role of impurities and surfactants in surfacegrowth is commonly related to steps (see Refs. [9, 21] and references therein). Thus, as itseems clear that the influence of steps on surface diffusion is prominent and of generalnature, there is an obvious need for studies where the issue is considered from a theoreticalpoint of view [20, 30, 31, 32, 33, 34].

In the present brief review, our objective is to illustrate some of the exciting theoreticalaspects of diffusion on stepped surfaces. We focus on recent theoretical progress in thefield, the emphasis being on a stepped model system with migrating adatoms interactingthrough a nearest–neighbor repulsive or attractive interaction in equilibrium conditions. Wehave chosen this model to allow a comparison to previous analytical studies for diffusionin the absence of direct adatom–adatom interactions [30]. Further comparison to numericalstudies in the case of an ideal, flat surface [5, 35] is also possible. We focus on the inter-play between step effects and repulsive or attractive adparticle interactions, and considerdifferent scenarios where either the step–binding or the adparticle interaction dominatesdiffusion. We show that with an appropriate combination of the step and adparticle interac-tion parameters, it is possible to systematically influence the rate of adparticle mobilityandcollective diffusion on stepped surfaces. Our results for adparticle mobility can also yieldinformation of tracer diffusion in the limit of a small adparticle concentration. About onehalf of the present discussion is a summary of our previous published works [20, 32, 33],and the remaining parts are new, unpublished results.

Overall, the present results show that in stepped systems there is a strong interplaybetween step binding and adparticle interactions. This is exemplified by the case of strongstep binding for collective diffusion across the steps. Then, we find thatsteps lead to apronounced maximum in the collective diffusion coefficient at an intermediatecoveragewhich depends on the relative strength of step binding and adparticle interactions. In thecase of weak step binding, on the other hand, the effects due to steps arenot pronounced, asexpected. The nature of adparticle interactions also plays a substantial role, since our resultssuggest that attractive and repulsive adparticle interactions may lead to distinctly differentcoverage dependence in stepped surface systems. The present results comprise the firstsystematic study of surface diffusion on stepped surfaces and provideinsight into furtherwork in this context, including possible applications related to e.g. diffusion–controlledcatalysis or nanostructure growth on vicinal surfaces.

2 Model

We consider the nearest–neighbor (NN) repulsive and attractive lattice–gas model on a 2Dlattice. The Hamiltonian for this model without steps is given by

H/kBT = K∑

〈ij〉

ninj , (1)


E > 02

E < 02

ES

EB

E0

x1

1y

1V(y )

V(x )1

b)

E0

a)

Figure 1: Geometry and activation barriers for the model of the stepped surface studied inthis work. (a) Activation energies for jumps in the direction across the steps. E0 denotesthe activation energy on a terrace,ES is the Ehrlich–Schwoebel barrier, andEB is theadditional binding energy at the lower step edge positions. (b) Activation energies forjumps along the steps at the lower step edge.E2 is the additional activation energy forjumps along the lower step edge which is set to zero here.

whereK = ±J/kBT is the NN interaction strength, and the sum over the lattice occupa-tion numbersni = 0, 1 goes over the NN sites of a square lattice. Within the lattice–gasapproximation this model corresponds to the NN Ising model with a coupling constant±J .The phase diagram of this model is well–known, with a critical temperature (coupling) ofKc = 1.76, and its diffusion properties have been previously studied e.g. in Refs. [5, 35] inthe case of an ideal, flat surface.

Here we apply this model to a system with regularly spaced steps (see Fig. 1). The stepparameters are the same as in Refs. [30, 31], and also in our previous works [20, 32]. Theterraces are separated by parallel steps with regular spacingL, whereL is typically 5 or 10lattice sites.1 The diffusion rates of adparticles are determined by three relevant activationbarriers: the activation energy for jumps on the terraceE0, the activation energy for jumpsalong a lower step edgeE0 + E2, and the activation energy for jumps away from the stepedgeE1. We note that here the Ehrlich–Schwoebel barrierES is taken to be zero becauseit does not bring any new physical properties into the present study [30, 31]. We considerthe system in the disordered phase, i.e. for|K| < |Kc|. Due to the spatial anisotropy of

1For larger values ofL the behavior of diffusion in the present system is in most cases very similar to thecase of a flat surface.


the steps, diffusion along and across steps need to be studied separately. In this work, westudy both cases of diffusion, i.e. across the steps since there the step and interaction effectsare most pronounced and along the steps. To model this scenario, an additional bindingenergyEB is added to the step edge sites. We have chosenEB ≥ 0 to describe commonlyobserved enhanced binding [23, 9, 24] at the step edge. Thus,EB describes the increase inthe saddle point for jumps away from step edges which typically act as sinksfor adatoms,and thusE1 = E0 + EB here. Finally, an additional activation energy for jumps along thelower step edge,E2, is used to describe enhanced (E2 < 0) or reduced (E2 > 0) diffusionrates along the step edges with respect to diffusion on the terrace.

We perform standard Monte Carlo simulations in equilibrium using the TransitionDy-namics Algorithm of Ref. [37]. The saddle–point energyES/kBT = 3|K| has been chosensuch that it is always larger than or equal to the initial and final state energiesEi andEf ,respectively. Thus the single-particle transition probability is always proportional to thefactor exp[−(ES − Ei)/kBT ], from which the effect of the (arbitrary) energyES can bescaled out in computingDc. Using this algorithm allows us to directly compare the smoothsurface results with a finite NN interaction parameter to those in Refs. [5, 35].

To evaluate the collective diffusion coefficientDc(θ) versus surface coverageθ we de-compose it in the standard way [11, 38, 39, 40] into the center–of–mass diffusion coefficient(mobility) Dcm(θ) and the thermodynamic factorξT (θ) as

Dc(θ) = ξT (θ)Dcm(θ), (2)

whereξT (θ) = 1/S0(θ). The static structure factorS0 is calculated as the zero–wavevectorlimit of the full dynamic structure factorS(~q, t) at t = 0 asS0 = lim~q→0 S(~q, t = 0).For calculation ofS0 we have used a single large system of500 × 500 lattice sites, andS0

has been calculated for subsystems of sizes 50, 100, 150, 200, 220 and 240 lattice sites,from which the final value ofS0 has been extracted. The mobility is calculated from thecenter-of-mass displacement

Dcm = limt→∞

1

2Ntd〈~R2(t)〉, (3)

where~R(t) =∑N

i [~ri(t) − ~ri(0)], ~ri(t) is the position of the particlei at timet, andd = 2is the spatial dimension. As usual, we computedDcm(θ) by using the memory expansionmethod [11, 41].

While the thermodynamic factor is a scalar quantity, the mobility and thusDc dependon the spatial direction. In our geometry (see Fig. 1), the steps run in they direction and thusdiffusion across the steps is in thex direction. All the diffusion coefficients and mobilitieshere have been normalised by the isotropic single–particle diffusion coefficient on an idealsurfaceD0 ≡ Dc(θ → 0). We conclude that, despite its simplicity, the present modelallows us to study generic effects in systems governed by step binding and adatom-adatominteractions.

In the case of a stepped surface where there is binding at the step edge, the total coverageθ is spatially non–uniform. One can define the partial coveragece andct on the step edge


and the terrace, such thatθ = ct + (ce − ct) /L (4)

whereL is terrace spacing. For the Langmuir gas, an exact result can be derived from thedetailed balance condition that

ce (1 − ct)

ct (1 − ce)= eEB/kBT . (5)

We will compare this result to the cases where there are finite adparticle interactions.

3 Results

We consider here systems with repulsive and attractive interactions. In both cases we focusto three main typical combinations of the coupling strengthK and binding at lower stepedgeEB/kBT : (i) case where interaction dominates, (ii) case where step binding dominatesand (iii) case whereK andEB/kBT are balanced. We present results of diffusion in bothprincipial directions, i.e. across and along the steps.

3.1 Diffusion Across the Steps – Repulsive Interactions

0 0.2 0.4 0.6 0.8 1θ

0

5

10

15

20

25

Dx,

c(θ

)/D

0

EB/k

BT = 0

EB/k

BT = 1, L = 10

EB/k

BT = 1, L = 5

(a)

0 0.2 0.4 0.6 0.8 1θ

0

0.5

1

1.5

2

2.5

3

3.5

4

Dx,

cm(θ

)/D

0

EB/k

BT = 0

EB/k

BT = 1, L = 10

EB/k

BT = 1, L = 5

(b)

Figure 2: Comparison of (a)Dx,c(θ)/D0 and (b)Dx,cm(θ)/D0 for diffusion on flat surface(whereEB = 0) and for diffusion across the steps (whereEB/kBT = 1, L = 5 andL = 10). In all casesK = 1.

We first consider the case where the step binding energy and the adparticle interactionsare of equal strength, i.e.K = 1 andEB/kBT = 1 [32]. In Fig. 2 we showDx,c(θ)/D0

andDx,cm(θ)/D0 for this case and for the case of no steps (solid lines). Our results forthe flat surface withK = 1 are in good agreement with diffusion coefficients published inRefs. [5, 35]. As can be seen from the figure, both the mobility and the diffusion coefficientare reduced in value in the presence of steps, but they display the same type of dependenceon θ as in the flat surface case. The mobility has a rather broad maximum aroundθ ≈


0.7, but the strong increase of the thermodynamic factor at higher coverages leads to amaximum ofDc(θ) aroundθ ≈ 0.8. As can be seen from Fig. 3,ξT increases somewhatwith increasingK.

0 0.2 0.4 0.6 0.8 1θ

0

2

4

6

8

10

12

ξ T

K = 1.0, EB/k

BT = 0

K = 1.0, EB/k

BT = 1

K = 0.5, EB/k

BT = 1

K = 0.2, EB/k

BT = 1

K = 0.0, EB/k

BT = 1

(a)

0 0.2 0.4 0.6 0.8 1θ

0

2

4

6

8

10

12

ξ T

K = 1.0, EB/k

BT = 4

K = 0.9, EB/k

BT = 4

K = 0.5, EB/k

BT = 4

K = 0.2, EB/k

BT = 4

K = 0.0, EB/k

BT = 4

(b)

Figure 3: The thermodynamic factor for cases with (a)EB/kBT = 1 and (b)EB/kBT = 4.The step spacingL = 5. Solid lines represent the analytic solution forK = 0 fromRef. [30].

0 0.2 0.4 0.6 0.8θ

0

0.5

1

1.5

2

2.5

Dx,

c(θ

)/D

0

K = 1.0K = 0.9K = 0.5K = 0.2K = 0.0

(a)

0 0.2 0.4 0.6 0.8 1θ

0

0.1

0.2

0.3

0.4

0.5

Dx,

cm(θ

)/D

0

K = 1.0K = 0.9K = 0.5K = 0.2K = 0.0

(b)

Figure 4: (a)Dx,c(θ)/D0 and (b)Dx,cm(θ)/D0 for the case of strong step binding, i.e.EB/kBT = 4. The analytic solution forK = 0 is taken from Ref. [30]. HereL = 5 in allcases.

Next, we consider the case where the step binding energy dominates over the adpar-ticle interactions. In Fig. 4 we showDx,c(θ)/D0 andDx,cm(θ)/D0 for the case whereEB/kBT = 4 while K varies between 0.2 and 1.0. The analytic solution for the caseK = 0 is shown with a solid line [30]. We see that increasing the NN interaction stronglyenhances collective diffusion (see Fig. 4(a)) through both the mobility (Fig. 4(b)) and thethermodynamic factor (Fig. 3). This is in analogy with many other systems whererepulsiveinteractions dominate in diffusion [11]. However, what is highly intriguing is thesystematicshift of the maximum inDc(θ) as a function ofK. ForK = 0, it has been shown [30] that


this maximum occurs at a coverage ofθ = 1/L corresponding to the ideally full partial oc-cupation of the step binding sites. For increasingK, this maximum moves towards highercoverages. ForK = 1 the maximum occurs atθ = 0.6, while for the mobility it is at aslightly lower value ofθ. The shift in the maximum is a direct consequence of the com-petition between step edge filling and the repulsive NN interactions. We have analysed thefractional concentration of adparticles at the step edgesce and find that forEB/kBT = 4,the maximum inDx,cm(θ) coincides with the valuece = 0.8 for anyK.

0 0.2 0.4 0.6 0.8 1θ

0

1

2

3

4

Dx,

c(θ

)/D

0

K = 0.5K = 0.2K = 0.0

(a)

0 0.2 0.4 0.6 0.8 1θ

0

0.2

0.4

0.6

0.8

1

Dx,

cm(θ

)/D

0

K = 0.5K = 0.2K = 0.0

(b)

Figure 5: Dx,c(θ)/D0 andDx,cm(θ)/D0 in the intermediate case whereEB/kBT = 1,L = 5, andK = 0.2 and 0.5. The solid lines represent an analytic solution forK = 0 [30].

Finally, we consider the so–called intermediate case where the step binding energy ismoderate (EB/kBT = 1) and the NN interaction parameter is somewhat smaller (K = 0.2and 0.5 here). The data forDx,c(θ)/D0 andDx,cm(θ)/D0 are presented in Fig. 5. In thiscase the step binding effect is very weak, andDx,c(θ) is almost coverage independent forK = 0 as in the limit ofEB → 0 whereDc(θ) = D0 for a Langmuir gas [2]. However,even a relatively small NN repulsion dramatically changes diffusion.Dx,c(θ) develops amaximum which shifts to higher coverages with increasingθ (see also Fig. 2), and themobility develops a strong peak, too. From Fig. 3 it can also be seen thatξT (θ) is rathersensitive to the NN repulsion and further contributes to the increase ofDx,c(θ) at interme-diate coverages.

3.2 Diffusion Across the Steps – Attractive Interactions

Monotonic decrease ofDx,c(θ) andDx,cm(θ) is characteristic for systems with attractiveinteractions. The case of weak binding,EB = 1 here, does not change this behaviorqualitatively but only slows down diffusion, see Fig. 6. Decrease inDx,c(θ) comes onlyfrom decrease inDx,cm(θ), ξT (θ) remains unchanged due to the weak binding. Thereforeonly values forEB/kBT = 1 are plotted in the Fig. 7(a). Our results for the flat surfacewith K = −1 are in good agreement with diffusion coefficients published in Refs. [5, 36].

Qualitatively different behavior occurs in the case where the step bindingenergy dom-inates over the adparticle interactions, see Fig. 8(a).Dx,c(θ) for the system of non–


0 0.2 0.4 0.6 0.8 1θ

0

0.1

0.2

0.3

0.4

Dx,

c(θ

)/D

0

EB/k

BT = 0

EB/k

BT = 1

(a)

0 0.2 0.4 0.6 0.8 1θ

0

0.1

0.2

0.3

0.4

0.5

Dx,

cm(θ

)/D

0

EB/k

BT = 0

EB/k

BT = 1

(b)

Figure 6: Comparison of (a)Dx,c(θ)/D0 and (b)Dx,cm(θ)/D0 for diffusion on flat surface(whereEB = 0) and for diffusion across the steps (whereEB/kBT = 1, L = 5 andL = 10). In all casesK = −1 (attractive).

0 0.2 0.4 0.6 0.8 1θ

0

2

4

6

8

10

12

ξ T

K = 0.0, EB/k

BT = 1

K = -0.2, EB/k

BT = 1

K = -0.5, EB/k

BT = 1

K = -1.0, EB/k

BT = 1

(a)

0 0.2 0.4 0.6 0.8 1θ

0

2

4

6

8

10

12

ξ T

K = 0.0, EB/k

BT = 4

K = -0.2, EB/k

BT = 4

K = -0.5, EB/k

BT = 4

K = -1.0, EB/k

BT = 4

(b)

Figure 7: The thermodynamic factor for cases with (a)EB/kBT = 1 and (b)EB/kBT = 4.The spacingL = 5. Solid lines represent the analytic solution forK = 0 from Ref. [30].

interacting particles has a local maximum at coverage1/L, i.e. θ = 0.2 in our particularcaseL = 5. Diffusion is slowed down by the attraction as in the previous case but thelocal maximum is smoothed out. Local minimum atθ = 0.1 appears forK ≈ 0.1 andbecames more apparent for large attraction, see Figs. 7(b) and 8(b). The local maximum atθ = 0.2 in Dx,cm(θ) disappears for relatively small interactions already – forK = −0.2the mobility is a monotonically decreasing function of coverage. We find that themobil-ity for attractive case is more sensible for interaction effects than in the repulsive case atθ = 1/L. However, this is compensated for in the thermodynamic factor, where for cov-erages around1/L changes are very small andξT (θ) increases slightly whereas for othercoverages it decreases as a result of attractions. This behavior comesfrom changes in thespatial concentration profiles of adatoms. Atθ = 0.2 the step edge atoms become almostimmobile for strong step binding. ThusDx,cm(θ) aroundθ = 1/2 is determined mainlyby the extra adatoms on the terrace. For attractive interactions their mobility is strongly


0 0.2 0.4 0.6 0.8 1θ

0

0.05

0.1

0.15

0.2

0.25

0.3

Dx,

c(θ

)/D

0

K = 0.0K = -0.2K = -0.5K = -1.0

(a)

0 0.2 0.4 0.6 0.8 1θ

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Dx,

cm(θ

)/D

0

K = 0.0K = -0.2K = -0.5K = -1.0

(b)

Figure 8: (a)Dx,c(θ)/D0 and (b)Dx,cm(θ)/D0 for the case of strong step binding, i.e.EB/kBT = 4. The analytic solution forK = 0 is taken from Ref. [30]. HereL = 5 in allcases.

reduced, leading to the vanishing of the peak inDx,cm(θ).

0 0.2 0.4 0.6 0.8 1θ

0

0.2

0.4

0.6

0.8

Dx,

c(θ

)/D

0

K = 0.0K = -0.2K = -0.5

(a)

0 0.2 0.4 0.6 0.8 1θ

0

0.2

0.4

0.6

0.8

Dx,

cm(θ

)/D

0

K = 0.0K = -0.2K = -0.5

(b)

Figure 9: Dx,c(θ)/D0 andDx,cm(θ)/D0 in the intermediate case whereEB/kBT = 1,L = 5, andK = −0.2 and−0.5. The solid lines represent an analytic solution forK = 0[30].

Finally, we consider the so–called intermediate case where the step binding energy ismoderate (EB/kBT = 1) and the NN interaction parameter is somewhat smaller (K =−0.2 and−0.5 here). The data forDx,c(θ)/D0 andDx,cm(θ)/D0 are presented in Fig. 9.In this case the step binding effect is very weak, andDx,c(θ) is almost coverage independentfor K = 0 as in the limit ofEB → 0 whereDc(θ) = D0 for a Langmuir gas [2]. However,even a relatively small NN repulsion dramatically changes diffusion. The local maximumin Dx,c(θ) moves to lower coverages and disappears. Decrease inDx,c(θ) is a result ofdecrease in bothDx,cm(θ) andξT (θ).


0 0.2 0.4 0.6 0.8 1θ

0

2

4

6

8

10

12

14

16

18

20

22

24

Dy,

c(θ

)/D

0

K = 1.0, EB/k

BT = 0.0

K = 1.0, EB/k

BT = 4.0

K = 0.9, EB/k

BT = 4.0

K = 0.5, EB/k

BT = 4.0

K = 0.2, EB/k

BT = 4.0

K = 0.0, EB/k

BT = 4.0

(a)

0 0.2 0.4 0.6 0.8 1θ

0

0.5

1

1.5

2

2.5

3

Dy,

cm(θ

)/D

0

K = 1.0, EB/k

BT = 0.0

K = 1.0, EB/k

BT = 4.0

K = 0.9, EB/k

BT = 4.0

K = 0.5, EB/k

BT = 4.0

K = 0.2, EB/k

BT = 4.0

K = 0.0, EB/k

BT = 4.0

(b)

Figure 10: (a)Dy,c(θ)/D0 and (b)Dy,cm(θ)/D0 for the case of strong step binding,i.e.EB/kBT = 4. The analytic solution forK = 0 is taken from Ref. [30], and the flat surfacecase withK = 1 andEB = 0 is also given for the purpose of comparison. HereL = 5 andE2 = 0 in all cases.

3.3 Diffusion Along the Steps – Repulsive Interaction

We first consider the case where the step binding energy dominates over the adparticle inter-actions. In Fig. 10 we showDy,c(θ)/D0 andDy,cm(θ)/D0 for the case whereEB/kBT =4 andK varies between 0 and 1. WhenK = 0, we note thatDy,c(θ)/D0 is coverage inde-pendent as on a flat surface. When direct adparticle interactions are present (K > 0), therepulsion gives rise to a local maximum inDy,c(θ)/D0 aroundθ = 0.8. These features arequalitatively similar to the flat surface case, as the comparison in Fig. 4 reveals for the caseK = 1. Nevertheless, quantitative differences between the stepped and flat systems arerather substantial, and are due enhancement in to both the mobility and the thermodynamicfactorξT .

Next, we briefly discuss the case where the step binding energy and the adparticle inter-actions are of equal strength,i.e. K = 1 andEB/kBT = 1. We find that in this case (datanot shown) bothDy,c(θ)/D0 andDy,cm(θ)/D0 are almost identical with the correspond-ing results on a flat surface presented in Ref. [32]. This simply implies that for weak stepbinding, diffusion and mobility along the steps are only weakly influenced by steps.

The influence of the additional activation energy for jumps along the lower step edge,E2, is demonstrated in Fig. 11. First, we find that diffusion is slowed down forE2 > 0,since then diffusion along step edges is slower as compared to diffusion onterraces. ForE2 < 0, the situation is the opposite, and diffusion is enhanced. Second, the effect of E2 isminor at large coverages. This is essentially due to the fact that the fractional concentrationof adparticles at the step edgesce is almost one forθ > 0.5. Thus, the most prominentinfluence ofE2 on Dy,c(θ)/D0 andDy,cm(θ)/D0 is observed at small coverages, whereE2 can essentially change the nature of the collective diffusion coefficient as its qualitativebehavior is completely different for positive and negativeE2. The results in Fig. 11 haveimportant consequences fore.g. in systems where impurities and surfactants tend to migrate


0 0.2 0.4 0.6 0.8 1θ

0

1

2

3

4

5

6

7

8

9

10

11

Dy,

c(θ

)/D

0

K = 0.5, E2/k

BT = -2

K = 0.5, E2/k

BT = 0

K = 0.5, E2/k

BT = 2

K = 0.0, E2/k

BT = -2

K = 0.0, E2/k

BT = 0

K = 0.0, E2/k

BT = 2

(a)

0 0.2 0.4 0.6 0.8 1θ

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Dy,

cm(θ

)/D

0

K = 0.5, E2/k

BT = -2

K = 0.5, E2/k

BT = 0

K = 0.5, E2/k

BT = 2

K = 0.0, E2/k

BT = -2

K = 0.0, E2/k

BT = 0

K = 0.0, E2/k

BT = 2

(b)

Figure 11: (a) The influence ofE2 on Dy,c(θ)/D0 and (b)Dy,cm(θ)/D0 for the case ofstrong step binding, i.e.EB/kBT = 4. We consider three cases, whereE2 = 0, or ±2.The analytic solution forK = 0 is taken from Ref. [30]. HereL = 5 in all cases.

into the lower step edge position and influence adatom motion.

3.4 Diffusion Along the Steps – Attractive Interaction

0 0.2 0.4 0.6 0.8 1θ

0

0.2

0.4

0.6

0.8

1

Dy,

c(θ

)/D

0

K = 0.0, EB/k

BT = 4.0

K = 0.2, EB/k

BT = 4.0

K = 0.5, EB/k

BT = 4.0

K = 1.0, EB/k

BT = 4.0

K = 1.0, EB/k

BT = 0.0

(a)

0 0.2 0.4 0.6 0.8 1θ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dy,

cm(θ

)/D

0

K = 0.0, EB/k

BT = 4.0

K = -0.2, EB/k

BT = 4.0

K = -0.5, EB/k

BT = 4.0

K = -1.0, EB/k

BT = 4.0

K = -1.0, EB/k

BT = 0

(b)

Figure 12: (a)Dy,c(θ)/D0 and (b)Dy,cm(θ)/D0 for the case of strong step binding,i.e.EB/kBT = 4. The analytic solution forK = 0 is taken from Ref. [30]. HereL = 5 andE2 = 0 in all cases.

We first consider the case where the step binding energy dominates over the adparti-cle interactions. In Fig. 12 we showDy,c(θ)/D0 andDy,cm(θ)/D0 for the case whereEB/kBT = 4 and K varies between 0 and−1. Results for diffusion on flat surface,EB/kBT = 0, andK = −1 are included for comparison. WhenK = 0, we note thatDy,c(θ)/D0 is coverage independent as on a flat surface. ForK < 0 diffusion is slower, es-pecially for high coverages. These features are qualitatively similar to the flat surface case,as the comparison in Fig. 12 reveals for the caseK = 1 except coverages aroundθ = 0.2where diffusion is enhanced in a way similar to diffusion across the steps. Nevertheless,


quantitative differences between the stepped and flat systems are rathersubstantial, and aredue to enhancement in both the mobility and the thermodynamic factorξT .

0 0.2 0.4 0.6 0.8 1θ

0

0.5

1

1.5

2

2.5

3

Dy,

c(θ

)/D

0

K = 0.0, E2/k

BT = -2

K = 0.0, E2/k

BT = 0

K = 0.0, E2/k

BT = 2

K = -0.5, E2/k

BT = -2

K = -0.5, E2/k

BT = 0

K = -0.5, E2/k

BT = 2

(a)

0 0.2 0.4 0.6 0.8 1θ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Dy,

cm(θ

)/D

0

K = 0.0, E2/k

BT = -2

K = 0.0, E2/k

BT = 0

K = 0.0, E2/k

BT = 2

K = -0.5, E2/k

BT = -2

K = -0.5, E2/k

BT = 0

K = -0.5, E2/k

BT = 2

(b)

Figure 13: (a) The influence ofE2 on Dy,c(θ)/D0 and (b)Dy,cm(θ)/D0 for the case ofstrong step binding, i.e.EB/kBT = 4. We consider three cases, whereE2 = 0, or ±2.The analytic solution forK = 0 is taken from Ref. [30]. HereL = 5 in all cases.

The influence of the additional activation energy for jumps along the lower step edge,E2, demonstrated in Fig. 13, is qualitatively similar to repulsive case. Diffusion issloweddown forE2 > 0 and enhanced forE2 < 0. The effect disappears at lower coverages thanin the case of repulsion, as can be seen by comparing Figs. 13 and 11.

3.5 Analysis of Partial Coverages

0 0.2 0.4 0.6 0.8 1θ

0

0.2

0.4

0.6

0.8

1

c e, ct

EB/k

BT = 4, K = 0

EB/k

BT = 4, K = 0.2

EB/k

BT = 4, K = 1

EB/k

BT = 1, K = 0.5

ce

ct

(a)

0 0.2 0.4 0.6 0.8 1θ

0

0.2

0.4

0.6

0.8

1

c e, ct

EB/k

BT = 4, K = 0

EB/k

BT = 4, K = -0.2

EB/k

B T= 4, K = -1

EB/k

BT = 1, K = -0.5

ce

ct

(b)

Figure 14: Coverage dependence of partial concentration at step edge ce and on terracect

for several combinations of step bindingEB/kBT and coupling strengthK.

As discussed in the introduction, the distribution of adatoms on the stepped surface isinhomogeneous. In the case of strong step binding, particles prefer to occupy the step edgesites up toθ = 1/2. In the case ofK = 0, this lead to a small local maximum inξT andθ = 1/2. As can be seen in Fig. 14, this behavior is suppressed by repulsive interactions,


and enhanced slightly by attractive interactions, because of changes in the partial coveragesct andce.

On the other hand, the mobilityDx,cm(θ) is strongly influenced by interactions. ForK = 0, there is a small maximum in the mobility atθ = 1/2 (see Fig. 4(b)). For increasingrepulsion, this peak moves to higher coverages due to the reduction ince apparent in Fig.14(a). This reductions allows more particles to cross the steps, leading to higher mobility.In the attractive case, however, the opposite situation occurs. Enhancement of ce leads tomore efficient blocking of particles crossin the steps, andDx,cm(θ) is significantly reduced,in particular close toθ = 1/2 (see Fig. 8(a)).

4 6 8 10 12 14x

0

0.2

0.4

0.6

0.8

1

θ

0.90.70.50.30.1

(a)

4 6 8 10 12 14x

0

0.2

0.4

0.6

0.8

1

θ

0.90.70.50.30.1

(b)

4 6 8 10 12 14x

0

0.2

0.4

0.6

0.8

1

θ

0.90.70.50.30.1

(c)

Figure 15: Spatial coverages profilesθ(x) in the direction across the steps for the casewith L = 5 and repulsion: (a)EB/kBT = 4, K = 0.2, (b) EB/kBT = 4, K = 1, (c)EB/kBT = 1, K = 0.5.

In the Figs. 15 and 16 we plot spatial coverages profilesθ(x) in the direction acrossthe steps. As expected, the distributions are spatially inhomogeneous on the terraces due toadparticle interaction. Repulsive interaction reduce the occupation of positions which arenearest neighbour to the step edge due to the preferential filling of the stepedge. In thecase of attraction, this effect is reversed which in part explain the strongreduction in themobility Dx,cm(θ).

In Figs. 17 and 18 we present some examples of configurations of the systems withstrong bindingEB/kBT = 4 and coupling strengthK = 0.2, andK = −0.2. We observedecoration of the steps at coverageθ = 0.2 with only a few atoms on the terraces. At


4 6 8 10 12 14x

0

0.2

0.4

0.6

0.8

1

θ

0.90.70.50.30.1

(a)

4 6 8 10 12 14x

0

0.2

0.4

0.6

0.8

1

θ

0.90.70.50.30.1

(b)

4 6 8 10 12 14x

0

0.2

0.4

0.6

0.8

1

θ

0.90.70.50.30.1

(c)

Figure 16: Spatial coverages profilesθ(x) in the direction across the steps for the case withL = 5 and attraction: (a)EB/kBT = 4, K = −0.2, (b) EB/kBT = 4, K = −1, (c)EB/kBT = 1, K = −0.5.

coverageθ = 0.5 the step edges are almost occupied and diffusion proceeds mainly alongterraces.

4 Concluding Remarks

Surfaces of solid materials are seldom flat. Rather, they are typically characterized byvicinal steps, steps due to nanoclusters and vacancies migrating on surfaces, various kindsof impurities, and surfactants. In the midst of all these “imperfections”, there may be ideal-like terraces, too.

Nevertheless, the diffusion and dynamics of adatoms is best known in the case of ideal,flat surfaces. This is of course evident since this is the obvious starting point when oneaims to understand how the dynamics takes place on surfaces in the first place. However,due to major progress in the field during the last few decades, the secrets of ideal-likesurface systems are at present reasonably well uraveled, and one isnow confronted by anew challenge to take a major leap forward and to clarify the physical mechanisms andlaws associated with surface diffusion phenomena under realistic conditions characterizedby non-ideal features such as steps. Obviously, this challenge is relatedto a number oftechnologically important processes such as thin film growth and the formationof nanosized


0 10 20 30 40 50X

0

10

20

30

40

50

Y

0 10 20 30 40 50X

0

10

20

30

40

50

Y

Figure 17: Examples of adparticle onfigurations of the system studied for the case withstrong step bindingEB/kBT = 4 and weak repulsive coupling strengthK = 0.2, forcoverages (a)θ = 0.2, (b) θ = 0.5.

clusters and islands.

In the present work, we have summarized and extended our recent theoretical work ofstepped model systems and discussed the behavior of surface diffusionunder the influenceof a wide range of different interactions. The model studied here is based on a surface witha regular array of parallel steps, including the effect of step binding atthe step edge, andalso a varying concentration of adparticles interacting through either repulsive or attractivenearest–neighbor interactions. Despite its simple and idealised nature, the present modelincludes the key features of stepped surface systems and is able to demonstrate the roleof steps on surface diffusion phenomena. In particular, the results discussed in this articleillustrate the strong interplay between the adparticle interactions and binding atthe stepedges. Roughly speaking, one can conclude that step binding effects are most prominent atsmall adparticle concentrations, leading to strong ordering behavior closeto the step edges.The role of adparticle interactions in turn becomes more prominent at larger adparticleconcentrations. In any case, this simple picture should currently be taken as a starting pointsince the overall behavior of surface diffusion does indeed depend on the subtle interplayof step binding and adparticle interaction effects. For example, the role of ordering effectsamong adparticles at low temperatures might lead to features not found so far. Also, arelated and long-standing issue is the influence of impurities attached to step edges onsurface diffusion. Thus, as these ideas are just a few of the many topical questions thatshould be addressed by both theoretical and experimental studies, we are confident thatone will not run out of challenging and technologically important problems. At the end,one would like to understand how surface dynamics takes place under realistic conditions,and then, at the same time, to see if this opens up new means to control different kinds ofprocesses on stepped surfaces.


0 10 20 30 40 50X

0

10

20

30

40

50

Y

0 10 20 30 40 50X

0

10

20

30

40

50

Y

Figure 18: Examples of adparticle onfigurations of the system studied for the case withstrong step bindingEB/kBT = 4 and weak attractive coupling strengthK = 0.2, forcoverages (a)θ = 0.2, (b) θ = 0.5.

AcknowledgementsThis work has been supported in part by the Academy of Finland through itsCenter of

Excellence program, by the Academy of Finland grant No. 80246 (I.V.), and by the GrantAgency of Academy of Sciences of the Czech Republic, No. IAA1010207.

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Documents

OMPUTER IMULATION TUDIES FOR OLLECTIVE ADATOM DIFFUSION STEPPED