Chemical Physics Letters 378 (2003) 161–166

www.elsevier.com/locate/cplett

Capillary waves and thermodynamics of multisteps on Pt(1 1 1)

C.P. Flynn, M. Ondrejcek *, W. Swiech

Materials Research Laboratory, Physics Department, University of Illinois at Urbana-Champaign,

104 S. Goodwin, 1110 W. Green Street, Urbana, IL 61801-3080, USA

Received 8 May 2003; in final form 17 July 2003

Published online: 13 August 2003

Abstract

We describe observations of multisteps, up to n ¼ 5 steps high, that form on Pt(1 1 1) at temperatures above 1400 K.

Using step fluctuation spectroscopy, we determine the multistep stiffnesses, ~bbn, and so estimate their line free energies,

assumed isotropic. Measurements of mode relaxation times snq, for modes of wavevector q, reveal that ~bbnsnq=n2 depends

only weakly on step height, n, in agreement with capillary theory. From the measured energies, we infer that multisteps

have large free energies of internal motion, such that the net thermal free energy, including capillary modes, undergoes

only small changes when steps merge.

� 2003 Elsevier B.V. All rights reserved.

1. Introduction

In this Letter, we describe equilibrium fluctua-tions of multisteps on the Pt(1 1 1) surface. Little is

currently known about these structures, and their

behavior remain poorly explored. Multisteps form

when single steps bind together to make a step

complex several atomic planes high. The simplest

crystal surfaces consist of terraces broken by sur-

face steps where terraces terminate. Two steps may

then interact through their electronic, vibrationaland elastic energy [1,2], and these modify observed

step–step spacings as surfaces evolve towards

equilibrium. Two well-recognized complexes [3–5]

are facets, and bunches, which, respectively, com-

* Corresponding author. Fax: +1-217-244-2278.

E-mail address: ondrejce@uiuc.edu (M. Ondrejcek).

0009-2614/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/S0009-2614(03)01243-0

prise steps bound to form a crystallographic plane

and loose, non-crystallographic assemblies. Here,

we define a third category named multisteps.Multisteps are bound complexes that exhibit cap-

illary behavior as a unit. In this Letter, we explore

properties of multisteps on Pt(1 1 1) and use step

fluctuation spectroscopy to establish capillary re-

lationships among multisteps of heights 16 n6 5,

for relaxation modes with wavelengths >100 nm.

The measured step stiffnesses yield thermodynamic

information. We determine, in addition to line freeenergies, that the thermal free energies from in-

ternal and capillary degrees of freedom are large,

and are both of importance in step–step reactions.

Steps merging on clean vicinal surfaces have

been reported [6–8], and complexes induced by

adsorbates [9,10] or growth instabilities [11] are

known. Structures of multiple height usually occur

at low temperature, T , and break up at high T . The

ed.

162 C.P. Flynn et al. / Chemical Physics Letters 378 (2003) 161–166

opposite has been observed for W(4 3 0) where

single steps double reversibly at high T [6,7], and

for W(2 1 0) where, however, the doubling is not

reversible when T is lowered [12]. The stiffnesses of

complexes 1–4 steps high, as quenched from the

test temperature by rapid cooling, have been re-ported for Si(1 1 3) [13]. Studies of kinetics of

multisteps in thermal equilibrium are currently

lacking.

2. Experimental

In our research, low energy electron micros-copy (LEEM) is employed to record structure

and kinetics at 30 frames per second (fps) and

7 nm resolution, in situ, and at temperatures up

to 1500 K. The spatial and temporal behavior of

relaxation modes are thus directly accessible for

multisteps, and these offer valuable insight into

their evolution, kinetics and energetics. The

LEEM employed here was built by Tromp andReuter [14] and subsequently modified to include

other growth and analysis capabilities. A Pt

crystal was employed with a polished (1 1 1)

surface. After repeated sputtering at 300 and 900

K, annealing in oxygen at 900 K, and then an-

nealing at 1400 K, LEEM revealed a surface

unreconstructed and with no surface impurity

detectable at 300 and 1500 K by Auger analysis(<1%).

To explore capillary modes, we made fluctua-

tion measurements on single and multiple steps at

various accessible temperatures. Fourier modes

were determined from video frames, and their

amplitudes and relaxation times determined, as

detailed in earlier studies of Si(0 0 1) [15] and

elsewhere [16,17]. In all, 66 runs on single stepsand 25 runs on double steps were analyzed, often

several at a given temperature. At high T , the

study was limited to 1520 K by the power input to

the LEEM sample stage, and at low T the de-

creased fluctuation amplitudes and slower kinetics

became the limiting factors. For n > 2 results are

available only near the maximum accessible tem-

perature, owing to the small fluctuation ampli-tudes associated with ~bbn large. Elsewhere, we give

details of the case in which fluctuations of single

steps determine a quantitative step stiffness ~bb for

this crystal [17].

3. Results and discussion

Single steps on Pt(1 1 1) were observed to merge

into a single structure (multistep) of doubled

height at temperatures above 1400 K. The reverse

sequences in which steps peel or zip apart are ob-

served below 1400 K. The consequences of these

processes are illustrated in Fig. 1, as separate steps

imaged at 1400 K separate to form multisteps,

shown in (b) at 1505 K, and subsequently, at1295 K, dissociate into single steps with altered

spacings in (c). The steps join smoothly and os-

culate at contact, as expected for a contact at-

traction competing with component steps pinned

apart at a remote point. This type of step behavior

from contact interaction was anticipated earlier

[18]. Steps of greater height also form above

1400 K, and zip apart when the temperature islowered. The equilibrium distribution has not been

determined. Fig. 1b shows an example in which

multisteps of several heights are present. The

height, n, can be identified without ambiguity from

a video record of the multistep evolution. The re-

sults give strong evidence for a contact interaction

between steps and multisteps. Earlier research has

explored a long period reconstruction of Pt(1 1 1)(not visible at the present resolution) and has at-

tributed a tendency for step merging to this re-

construction [19,20]. Note that the apparent

multistep width, in images such as Fig. 1b, in-

creases with n, but is determined largely by inter-

ference between beams reflected from adjacent

terraces [21]. The observed widths have little con-

nection with internal structure.LEEM images make clear that the multisteps

exhibit fluctuations much like those of single steps.

Evidently, all component steps access defects dif-

fusing from the terraces. We report the observed

equilibrium amplitudes measured here for multi-

steps fluctuations. Fig. 2 presents, for T ¼ 1500 K,

the fluctuation amplitudes as functions of wave-

vector q for multisteps of heights 26 n6 5, andcompares them with results for single steps on the

same crystal, reported elsewhere [17]. Evidently,

Fig. 1. Views of same steps at different temperatures T . The dissociated steps at 1400 K form various multisteps at 1505 K. When

lowered to 1295 K the multisteps dissociate into parallel non-touching arrays with different grouping (the brackets enclose the same

steps; the magnification is lower for the third image; LEEM impact energy 18 eV).

C.P. Flynn et al. / Chemical Physics Letters 378 (2003) 161–166 163

the spatial and temporal bandpasses of the LEEM

can meet the demands of these experiments. The

observed amplitudes ynq decrease with n, owing to

increasing multistep stiffness (for clarity, the data

sets are drawn with offsets). In Fig. 2, raw data are

shown as open points, and the final result (solid

points) are corrected for space and time resolution[16]. The mode amplitudes for capillary objects

including single and multisteps of length L are

predicted [3,4] to obey hjynqj2i ¼ kBT =L~bbnq2, with L

the step length (here 2.2 lm) and with q ¼ 2pq=L,and q an integer. The q�2 prediction fits the data

quite well (solid lines).

Fig. 3 shows values of ~bbn determined for mul-

tisteps at various temperatures from the measuredamplitudes, using the same equation. Only at 1500

K are ~bbn obtained for all n ¼ 1, 2, 3, 4, 5. Shown

inset in Fig. 3 are values of ~bbn at 1500 K. The

stiffness increases with step height, as reported for

quenched samples [13]. Here, however, ~bbn < n~bb1,

which signals a tendency to step binding. Corre-

lation times snq also were obtained from the time

decay of the ynq using Fnqðt0 � tÞ ¼ hynqðtÞy�nqðt0Þ þcci=2hjynqj2i ¼ exp�ðt0 � tÞ=snq. Typical cases are

shown inset in Fig. 2 for n ¼ 4; data for n ¼ 1 are

given in [17]. The snq contain all available kinetic

information.

An important insight into multistep kinetics

now follows from the values of ~bbn and sq. Else-where [17] we show that capillary modes obey

s�1nq ¼ pDA2

nq2 ~bbn=XkBT ; ð1Þ

in which An is the area added per attached atom,

X is the atomic volume, and D is the effective

diffusion coefficient for the operative mechanisms

(see also [15]). This follows using the Nernst

Einstein equation to obtain the defect flux to the

steps from the terraces caused by the step chem-

ical potential. The same form applies with anappropriate effective D regardless of detailed

mechanism [17]. Now for multisteps, An ¼ A1=n,since n additions are required to move the mul-

tistep by A1. It follows for capillary modes in the

hydrodynamic limit, given identical diffusion

mechanisms, that the quantity

fnðqÞ � ~bbnsnq=n2; ð2Þ

must be independent of n. Our measurement per-mit the first precise assessment for multisteps of

this new prediction from capillary theory.

Fig. 4 shows fn as a function of q for 16 n6 5

for Pt(1 1 1) at 1500 K, using values of snq and ~bbn

determined as described above. The comparison

with prediction is quite satisfactory. The fn are

Fig. 2. Mean squared amplitudes of multistep modes at 1500 K,

shown as functions of q. For clarity the results for n < 5 are

displaced upwards (multiplied) byffiffiffiffiffi10

pbetween values of

multistep height n. One pixel equals 5.5 nm. Inset are examples

of time correlation data showing how the fluctuations relax

with time for n ¼ 4, for q ¼ 2, 4, 8, . . .; wavevector q ¼ 2pq=Lin nm; 30 f units (frames)¼ 1 s. When F ðtÞ 6¼ 0 at large t, theaverage step shape is not exactly straight.

Fig. 3. Measured values of ~bbn, for multisteps on Pt(1 1 1) with

16 n6 5. For n > 2 results are available only at the highest T .The half-tone point at 1400 K for n ¼ 2 corresponds to the

region of unstable steps. Inset shows the resulting dependence

of ~bbn on n at 1500 K (with 3-fold terms, the mean free energies

bn0 ¼ ~bbn0 may be �20% larger than the ~bbn).

Fig. 4. Variation of fnðqÞ with q and n, determined from

present measurements of ~bbn and snq. The line shows q2:5. Forsmall q the values are fairly independent of n, as predicted from

capillary theory. Inset is a sketch showing a multistep as a net

with sparse contacts.

164 C.P. Flynn et al. / Chemical Physics Letters 378 (2003) 161–166

independent of n to within a factor of 2, and vary

strongly with q as expected. This ability to predict

kinetics is a considerable success for the modeling

of multistep behavior. Specifically, multisteps are

seen to possess capillary behavior, regardless ofinternal motions and structures. One can infer

from the overlap of results for different n in Fig. 4

that internal structure causes little or no compli-

cation in the absorption and emission of the dif-

fusing defects that create changes of step shape.

Even for single steps the kinetics on Pt(1 1 1) are

not simple, because diffusion near 1500 K proceeds

by comparable bulk and surface contributions[17].

A final matter of some importance concerns the

energetics of multisteps and of step–step reactions.

C.P. Flynn et al. / Chemical Physics Letters 378 (2003) 161–166 165

The connection between step stiffness ~bbnðhÞ and

the step free energy bnðhÞ per unit length, is:~bbnðhÞ ¼ bnðhÞ þ d2bn=dh

2 þ � � �. With 3 mm sym-

metry, the Pt(1 1 1) surface must have stiffnesses

that follow ~bbnðhÞ ¼ ~bbn0 þ B cos 3hþ � � �, with Bconstant, whence from a Fourier representation ofeach side

bnðhÞ ¼ ~bbn0 � ðB=8Þ cos 3hþ � � � : ð3ÞA fit to our data, with steps in the range

4�6 h6 14�, yields b1 ¼ 200þ ð5� 2Þ cos 3h meV/

nm, with h measured from high energy close

packed direction. The average free energy and

stiffness for n ¼ 1 are thus �20% larger than thestiffness in the direction measured. A and B steps

(h ¼ 0 and p=3) differ in free energy [5] by �13% at

910 K, and here by 5% at 1400 K. From the

structure inferred below it seems likely that the

multistep free energies are still more isotropic, and

differ from the ~bbn by less than the 20% found here

for n ¼ 1. The ~bbn, shown inset in Fig. 3, may then

be used approximately as free energies to discussthe spontaneous formation of multisteps from

single steps at T > 1400 K.

To treat step reactions we must include the

thermal free energy of multistep motions both in-

ternal and of the line center. For example, the free

energy b1 of the straight step is augmented by the

free energy of capillary modes. Being quadratic in

its normal mode coordinate, each relaxation modehas a mean thermal energy of kBT=2. This is ex-

cited above a kink energy, e, where the thermal

free energy must resemble

f1 ¼1

2NkBT lnðe1=kBT Þ; ð4Þ

with N the number of modes per unit length of

step. This is analogous to the result f ¼ kBT� ln hm=kBT for vibrations of frequency m, at

T > hm=kB. Thus the specific heat c ¼ �To2f1=oT 2 ¼ NkB=2, as required. There are two points.

First, at 1500 K, and with one mode per atomlength, and given that lnðe1=kBT Þ � �1, we find

that f1 � �250 meV/nm, as compared to b1 � 200

meV/nm. We therefore deduce that the thermal

free energy must play an important role in step

reactions. The second point is that double steps

have the same free energy of capillary modes as

single steps, apart from a small term arising from

e1 6¼ e2. The result is that the free energy b2 þ f2 �80 meV/nm of a double step exceeds that, 2ðb1 þf1Þ � �100 meV of two single steps. This deficit is

much larger than uncertainties in the inferredvalues of the bn. Clearly, the net free energies are

not able to explain the observation that two single

steps react to form 2-steps above 1400 K, since the

free energy of 2-steps must then be less than the

free energy of two single steps. It would be rea-

sonable if e2 > e1, and this further increases the

disparity in free energy between a double step and

two single steps. Our data reveal unambiguouslythat the differences become still more difficult for

higher multisteps, such as four single steps forming

a 4-step. Use of higher harmonics in h in Eq. (3)

does not change the need for internal free energy.

It seems quite clear that the resolution of this

problem must lie in internal motion of the

multisteps. While no structure is seen at the

resolution of LEEM, detailed internal structureis visible in STM images of quenched samples

[13]. A simple but persuasive argument has the

component steps in contact only over limited

lengths, and otherwise spread apart to form a

network, as sketched inset in Fig. 4. This re-

sembles the zip structure suggested by Khare

et al. [18]. Note now that the thermal free energy

per unit length of free step is known [3,4] to beindependent of the free length L. When the

pinned points are relatively sparse, the thermal

free energy of a network of steps then remains

close to that of the n independent component

steps. If this were exactly true, and pinning

points were negligibly dilute, the thermal energy

would not enter into step reactions, which would

be determined entirely by the bn. It is thereforeof interest to note in the measured ~bbn (Fig. 3,

inset) that, for all n, the combination of ~bbn with~bb1 clearly exceeds ~bbnþ1, while

~bbn�m þ ~bbm otherwise

depends very little on m in the available results.

Taken alone, these results for the stiffness-related

free energy clearly are consistent with the ob-

served formation at 1500 K of multisteps for all

n. The various behaviors of different clean ma-terials [6–12] presumably result from differences

of the several contributing components of mul-

tistep free energies, which cannot be predicted at

166 C.P. Flynn et al. / Chemical Physics Letters 378 (2003) 161–166

this time. A final point is that the line center

defined as yn ¼P

i yi=n has a mean square am-

plitude hjynj2i ¼ hjynj2i=n when the components

fluctuate independently, so the effective stiffness

scales as multistep height, n. More realistically,

when cross links cause positive correlations be-

tween components, hjynj2i is increased and the

stiffnesses decrease below proportionality with n,perhaps much as in the data inset in Fig. 3.

Thus both the stiffnesses and the free energies of

multisteps seem qualitatively consistent with a

network model.

4. Summary

We have observed that steps on Pt(1 1 1) merge

above 1400 K to form multisteps with heights

16 n6 5, reversibly under cycling, with some

thermal hysteresis. The observed reaction mecha-

nisms are of interest in connection with the

strength of apparent step–step interactions at dif-

ferent distances and temperatures. From fluctua-

tion measurements, we have determined both thestiffnesses and the relaxation times for capillary

modes of multisteps with 16 n6 5. The experi-

mental results are interrelated by a diffusive mod-

eling of the step motions that successfully describes

the fluctuations of multisteps by an elaboration of

earlier treatments of single steps. With the as-

sumption of reasonable isotropy at high tempera-

tures, the measured step stiffnesses also allow thefree energies per unit length of the multisteps to be

estimated. Taken together with the capillary ther-

mal free energies determined here, these energies

fail badly to explain the observed coalescence of

single steps into multisteps above about 1400 K.

We deduce that the internal motions of the mul-

tistep structures contribute additional free energy

comparable to that of the dissociated steps, andsimilar to those of a net with sparse contact points.

Acknowledgements

This research was supported by the Department

of Energy through Grant DEFG02-91ER45439,

which also supports the Center for Microanalysisof Materials, in which the LEEM is maintained.

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