Surface Science Letters

Effect of kink-rounding barriers on step edge fluctuations

Jouni Kallunki, Joachim Krug *

Fachbereich Physik, Universit€aat Essen, 45117 Essen, Germany

Received 26 August 2002; accepted for publication 9 October 2002

Abstract

The effect that an additional energy barrier Ekr for step adatoms moving around kinks has on equilibrium step edge

fluctuations is explored using scaling arguments and kinetic Monte Carlo simulations. When mass transport is through

step edge diffusion, the time correlation function of the step fluctuations behaves as CðtÞ ¼ AðT Þt1=4. At low temper-

atures the prefactor AðT Þ shows Arrhenius behavior with an activation energy ðEdet þ 3�Þ=4 if Ekr < � and ðEdet þ Ekr þ2�Þ=4 if Ekr > �, where � is the kink energy and Edet is the barrier for detachment of a step adatom from a kink. We point

out that the assumption of an Einstein relation for step edge diffusion has lead to an incorrect interpretation of step

fluctuation experiments, and explain why such a relation does not hold. The theory is applied to experimental results on

Pt(1 1 1) and Cu(1 0 0).

� 2002 Elsevier Science B.V. All rights reserved.

Keywords: Models of surface kinetics; Atomistic dynamics; Surface diffusion; Stepped single crystal surfaces; Monte Carlo simulations

1. Introduction

In thin film growth the detailed knowledge of

the microscopic elementary processes is essentialsince the large scale morphology is determined by

the competition between the nonequilibrium de-

position flux and the different relaxation mecha-

nisms [1]. Clear demonstrations of this are for

example growth instabilities on high symmetry or

vicinal crystal surfaces, which are known to pro-

duce self-organized nanoscale patterns [2,3]. In

both cases, mound formation on singular [4] andstep meandering on vicinal surfaces [5], the size of

the structures is set by the relation between the

time scales of deposition and relaxation kinetics

[6]. Thus the knowledge of the relaxation kinetics

opens a possibility to control the size of the struc-tures by controlling the external parameters such

as deposition rate and temperature.

The most important elementary process on

surfaces is the hopping of individual atoms. The

motion of an adatom from one lattice site to an-

other is a thermally activated process and takes

place at rate

Ci ¼ Ci;0 expð�bEiÞ; ð1Þwhere b ¼ 1=kBT is the inverse temperature, Ci;0 is

an attempt frequency, and Ei is the activation en-

ergy of the process i, which depends on the envi-

ronment of the hopping atom, e.g. at a step. Thus

knowledge of the activation energies gives access

*Corresponding author. Tel.: +492011833443; fax:

+492011834578.

E-mail address: jkrug@theo-phys.uni.essen.de (J. Krug).

0039-6028/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0039-6028 (02 )02435-4

Surface Science 523 (2003) L53–L58

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to the elementary rates. Unfortunately the activa-

tion energies are very rarely accessible by direct

measurement; apart from the case of a single ad-

atom moving on a flat surface [7], it is extremely

difficult to follow the evolution of individualatomic configurations in real time. Thus one has to

rely on measurements of mesoscopic quantities

and compare these with theoretical predictions in

order to extract the microscopic parameters. Ex-

amples of such an approach are the determination

of the adatom diffusion barrier from the island

density [8], the estimate of interlayer diffusion

barriers from second layer nucleation experimentsand mound shapes [9,10], and the extraction of

activation energies for processes at step edges from

the characteristic length scales of growth instabil-

ities on vicinal surfaces [11–13].

An elegant method for measuring energetic and

kinetic parameters of atomistic processes at steps

exploits the time correlation function of equilib-

rium step fluctuations, see [14,15] for recent re-views. In this paper we revisit the theoretical basis

of these experiments for the case of mass transport

dominated by step edge diffusion, and take into

account the possibility of a significant kink round-

ing barrier, which prevents adatoms migrating

along a step from hopping around a kink. The

kink-rounding barrier is the one-dimensional an-

alog of the well-known Ehrlich–Schwoebel (ES)barrier [16] suppressing the inter-layer mass

transport on crystal surfaces. It is worth noting

that also a three-dimensional analog of the ES

barrier, inhibiting atoms from going around facet

edges, has been observed [17]. The existence of the

kink rounding barrier is still under debate; nu-

merical calculations support its existence [18,19],

but until now experimental observations are few[21,20]. If present, the kink-rounding barriers have

great impact on the patterns formed under un-

stable epitaxial growth [12,13,22–27] as well as

on the shape relaxation of islands and other

nanostructures [28,29]. In this paper we show how

kink-rounding barriers affect equilibrium step

fluctuations, thus providing an alternative way of

determining the barriers experimentally. In addi-tion, in Section 4 we clarify a misconception in the

theory of step fluctuations which has lead to an

incorrect data analysis in some cases.

2. Step fluctuations and the adatom mobility

On a vicinal surface the monoatomic steps,

separating high symmetry terraces, are not per-

fectly straight but wander due to thermal fluctua-tions. Here we will consider a situation where

adatoms cannot detach from the steps, thus the

only possible mass transfer process is migration of

adatoms along the steps. The Langevin theory of

step fluctuations then yields the expression [14,15]

CðtÞ � h fðx; tÞ½ � fðx; 0Þ2i ¼ a2?Cð3=4Þ~ccpbX1=2

ð2r~cctÞ1=4

ð2Þfor the time correlation function of the step edge

position fðx; tÞ with the initial condition of a flat

step, fðx; 0Þ � 0. Here ~cc is the step stiffness, a? the

lattice spacing perpendicular to the step, X is the

atomic area, Cð3=4Þ � 1:2254 . . ., and r denotes

the adatom mobility along the step edge. It is de-

fined through the relation

j ¼ �roxl ¼ roxX~ccoxxf ð3Þbetween the mass current along the step and the

chemical potential gradient driving it. In the last

equality the Gibbs–Thomson relation has been

used.

To make use of the expression (2) for theanalysis of experimental data, the parameters ~ccand r of the continuum description must be ex-

pressed in terms of the rates of the elementary

processes. This can be done exactly if the step is

modeled as a one-dimensional solid-on-solid (SOS)

surface with energy barriers proportional to the

number of lateral bonds in the initial state

(Arrhenius kinetics); see Section 3 for a precisedefinition. In this case one obtains [30]

r ¼ aC0b2

expð�bEdetÞ ð4Þ

and

b~cc ¼ a�1ðcoshðb�Þ � 1Þ; ð5Þ

where Edet denotes the energy barrier for detach-ment of a step atom from a kink site, the kink

energy � is the energy cost of creating a kink, a is

the lattice constant parallel to the step, and C0 is

the attempt frequency, which is assumed to be the

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same for all processes. Since the detachment of akink atom creates two kinks, within the Arrhenius

model Edet ¼ Est þ 2�, where Est is the energy bar-

rier for diffusion along the straight (unkinked)

step. Fig. 1 shows a cartoon of a step displaying

the relevant processes. The relation r e�bEdet has

also been derived within a Kubo formalism [31,32].

Putting everything together yields

CðtÞ=a2? ¼ gtsst

� �1=4

; ð6Þ

where g is a numerical constant of order unity, and

sst ~cc3=r is the characteristic time for the step to

fluctuate one lattice constant. In the low temper-

ature limit b� � 1 one finds

sst exp½bðEdet þ 3�Þ: ð7Þ

Eqs. (6) and (7) form the basis of the experimental

determination of kinetic barriers from step edge

fluctuations. Measuring the coefficient of the t1=4-behavior of the correlation function (6), the acti-

vation energy of the characteristic time (7) can be

obtained. Provided the kink energy � is knownfrom other sources (e.g., from the analysis of static

step fluctuations [14,15]), this yields an estimate of

the detachment barrier Edet.

To see how (7) has to be modified in the pres-

ence of kink rounding barriers, we first rederive sstfrom a scaling argument. The elementary process

driving the step fluctuations is the transport of an

atom from one kink to another, which allows thekinks to diffuse along the step. In order to move

the step by a distance a?, a kink must diffuse over

a distance of the order of the mean kink spacing

‘k ¼ ð1=2Þaeb�. The detachment rate of atoms from

a kink site is Cdet ¼ C0 expð�bEdetÞ. The proba-

bility Patt for an emitted adatom to reach another

kink at distance ‘k before it reattaches to the

original kink can be calculated from a ran-

dom walk theory [33], yielding Patt � a=‘k. Thusthe diffusion rate of a kink is CdetPatt and the

characteristic time for a step to fluctuate over a

single lattice constant reads sst ‘2k=ðCdetPattÞ exp½bð3�þ EdetÞ, in agreement with (7).

When atoms diffusing along the step experience

an extra barrier Ekr for going around a kink site, as

drawn in Fig. 1, the probability Patt of the adatomto attach to a kink at distance ‘k is altered. After

reaching the distant kink the step adatom still has

to overcome the kink rounding barrier in order to

attach to it. This yields [33]

Patt � ð‘k=aþ 1=pkrÞ�1; ð8Þ

where pkr � expð�bEkrÞ is the probability for go-

ing around a kink. Comparing ‘k=a with p�1kr it is

obvious that the kink rounding barriers are rele-

vant if Ekr > �. In this case (7) has to be replaced

by

sst exp½bð2�þ Edet þ EkrÞ: ð9ÞAs shown before the characteristic time is gener-

ally a combination of the step adatom mobilityand the step stiffness, sst ~cc3=r. Since the step

stiffness does not depend on the dynamics of the

adatoms along the step, but only on the energetics,

we may conclude that the adatom mobility is re-

duced to

r expð�bðEdet þ Ekr � �ÞÞ ð10Þ

by the kink-rounding barrier, when Ekr > �. Theexpression (8) suggests the interpolation formula

r ¼ 1

2aC0

e�bEdet

1þ ebðEkr��Þ ð11Þ

for the mobility, which recovers (4) for Ekr � �.

3. Monte Carlo simulations

In order to confirm the validity of the argu-

ments of the previous section we have conducted

Monte Carlo simulations of a simple one-dimen-

sional SOS model. The position of the step at site iis hi and the atoms may hop along the step to

neighboring sites (i ! i� 1) with rate

Fig. 1. A schematic picture of a monoatomic step with the

activation energies of the elementary processes.

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Ci;i�1 ¼ C0 expð�bEi;i�1Þ; ð12Þwhere the activation energy depends on the local

configuration as

Ei;i�1 ¼ Est þ 2�ni þ ½1� dðhi � hi�1 � 1ÞEkr:

ð13ÞHere ni ¼ 0; 1; 2 is the number of lateral nearestneighbors of the atom at initial site i and Ekr is an

extra barrier suppressing kink rounding; whenever

the hop from i ! i� 1 is not along flat step i.e.

hi � hi�1 6¼ 1, the extra barrier Ekr is added. In the

simulations we used a lattice of size L ¼ 131072,

starting with a straight step hið0Þ � 0. The hopping

rate C0 exp½�bEst of a free step edge atom on an

unkinked step segment determines the time scale ofthe model, and can be set to unity in the simula-

tion. For the kink energy we used the value � ¼ 0:1eV. The kink-rounding barrier Ekr was varied be-

tween 0 and 0.24 eV, and the inverse temperature

in the interval b� ¼ 1:25–3.5, corresponding to

T ¼ 331–928 K.

The time-correlation function

CðtÞ � L�1XL

i¼1

hiðtÞ2 ð14Þ

measured from the simulations is shown in Fig. 2.

It has a clear t1=4 time dependence and the pre-

factor in Eq. (2) was determined by fitting the data

in the long time limit with CðtÞ ¼ At1=4 þ B. Using

Eqs. (2) and (5), the mobility r can be extracted

from the prefactor. The mobility obtained from

the simulation results is shown in Fig. 3 and the

activation energy for the mobility Er, determined

from a fit to the Arrhenius plot, is shown in Fig. 4.

There is a cross-over in the behavior of the acti-vation energy Er as the kink-rounding barrier

0 1 2 3t / 10

7 t0

0

1

2

C(t)

Fig. 2. The time correlation function (Eq. (14)) for �b ¼ 2:5

and Ekr=� ¼ 0:0=0:4=1:2=1:6=2:0 (from top to bottom). The

dashed line is the best fit At1=4 þ B. Time is measured in units of

the inverse diffusion rate along flat a step, t0 � 1=ðC0 exp

½�bEstÞ.

1.0 2.0 3.0 4.0βε

10-4

10-3

10-2

σ/β

Fig. 3. The adatom mobility along the step edge obtained from

the prefactor of the correlation function in Fig. 2, with � ¼ 0:1

eV and kink rounding barrier Ekr=� ¼ 0:0 ð�Þ; 1:2 ð}Þ; 1:6ð4Þ; 2:4 ð�Þ. The full lines are best fits to an Arrhenius form.

0 0.1 0.2 0.3Ekr

0.1

0.2

0.3

0.4

Eσ

Fig. 4. The activation energy for the adatom mobility obtained

from the fits in Fig. 3. A clear cross-over in the behavior is seen

at Ekr � � ¼ 0:1. The full lines are the theoretical predictions

Eqs. (4) and (10).

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roughly equals the kink energy Ekr � �. Thus the

simulation results are in good agreement with the

analytical results (Eqs. (4) and (10)) of the previ-

ous section.

4. No Einstein relation for step edge diffusion

In the literature [15,34,35] the interpretation of

experimental step fluctuation data is often based

on an Einstein relation [36–38]

r ¼ nstDst

kBTð15Þ

for the mobility, where nst is the (one-dimensional)

concentration of step adatoms and Dst denotes the

tracer diffusion coefficient for an adatom migrat-

ing along a kinked step. The latter can be esti-

mated by considering the motion of an adatom in

a model potential where kink sites are represented

as traps of depth Edet spaced at the mean kinkdistance [34,35]. The resulting activation energy

for Dst is Edet � �. Since a step adatom can be

viewed as a double kink, the concentration of step

adatoms in equilibrium is n0 e�2�=kBT , and hence

the activation energy of r is predicted by (15) to be

Edet þ �, in disagreement with the exact result (4).

The problem with (15) was already noted in

[32]. Since step adatoms are continually absorbedand emitted at kinks, they do not constitute a

conserved species, and it is difficult to consistently

define Dst and nst. In fact, strictly speaking Dst � 0:

It can be shown [39,40] that the mean square

displacement of a marked step adatom moving

along a kinked step grows sublinearly, as hðxðtÞ�xð0ÞÞ2i t7=8. This reflects the fact that a trapped

adatom runs a considerable risk of being over-grown by a large step fluctuation, and hence the

probability distribution of trapping times has a

very broad tail. Representing the migration of an

adatom along a kinked step by the diffusion of a

particle in an external potential neglects both the

possibility of long term trapping due to step fluc-

tuations, and the fact that a kink site is not actu-

ally ‘‘filled’’ when an adatom attaches to it––it ismerely shifted. Nevertheless a relation of the form

(4) does hold, if Dst is replaced by the diffusion

coefficient ð1=2Þa2C0 exp½�bEst for the migration

along a straight close packed step, and the bond

counting relation Edet ¼ Est þ 2� is assumed. Then

(15) simply expresses the balance between the de-

tachment and attachment of step adatoms at the

kinks.

5. Conclusions

We have studied the time fluctuations of a

monoatomic step when the mass transport is re-

stricted to migration along the step. The scaling

arguments presented in this paper (Eqs. (9) and(10)) show how the adatom mobility along the step

is reduced if the kink-rounding hops are sup-

pressed with an extra barrier. The results of our

Monte Carlo simulations confirm the validity of

the scaling arguments. The results presented here

show that kink-rounding barriers have to be taken

into account in the analysis of time-dependent step

fluctuation experiments [14,15]. Provided the kinkenergy � is known, the temperature dependence of

the prefactor of the time correlation function CðtÞgives access to the activation energy Er of the

adatom mobility. In general, Eqs. (4) and (10)

show that Er is an upper bound on the detachment

barrier Edet.

For example, the interpretation of the results

presented in [35] in the light of our work showsthat Edet 6 1:50� 0:16 eV for close-packed steps

on Pt(1 1 1). When additional information on the

energy barriers is available, the analysis of time

fluctuations may be used to determine the strength

of the kink-rounding barrier itself. For the close-

packed [1 1 0]-step on Cu(1 0 0), the re-interpreta-

tion of the experimental results of [34] for the time

fluctuations, supplemented with a result of [11] forthe diffusion barrier Est at a straight step, yields the

estimate Ekr � 0:41 eV [13].

Acknowledgements

Useful discussions with H.J. Ernst, M. Giesen,

M. Rusanen, S.V. Khare and T. Michely aregratefully acknowledged. This work was sup-

ported by DFG within SFB 237.

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