Roughening of the Pb (110) surface: combined Monte Carlo and molecular-dynamics study

ELSEVIER

Roughening of

COMPUTATIONAL MATERIALS SCIENCE

Computational Materials Science 7 (1997) 257-270

the Pb ( 110) surface: combined Monte Carlo and molecular-dynamics study @I

A. Landa * , P. Wynblatt Department of Materials Science und Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Received 7 September 1996; accepted 3 October 1996

Abstract

The roughening transition on the Pb (110) surface has been studied using a combination of lattice-gas Monte Carlo and molecular-dynamics methods in conjunction with a many-body glue potential. Lattice-gas Monte Carlo simulations yield a roughening transition temperature of approximately TkGMC = 1100 K. Molecular-dynamics simulations, which account for surface relaxation and lattice vibrations, detect the roughening transition at TED = 545 K, above the high-resolution low-energy diffraction measurements of Tzxp = 415 K. The anisotropic body-centered solid-on-solid model is used in the

interpretation of these results. The time scale of local roughening at 545 K is about 0.6 ns. The time evolution of the Pb (110) surface from an initially smooth to a rough configuration is illustrated by the use of modem visualization techniques. Means for improving the theoretical results are discussed.

1. Introduction

Surface roughening and surface melting (or premelting) have been the subject of a number of theoretical and experimental investigations over the past several years. Despite the apparent similarity of these disordering phenomena, their atomic mecha- nisms are different. On the rough surface, atoms continue to occupy sites of the crystal lattice even though the interfacial width diverges on an atomic scale. In contrast, surface premelting may be distin- guished from roughening by a lack of registry of the

* Corresponding author.

101 A video accompanies this article, see ordering infor-

mation at the end of this issue.

atoms in the premelted surface region with respect to the underlying crystal, which results in liquid-like properties of the surface region.

Among low-Miller-index metal surfaces, FCC (110) surfaces show both roughening and premelting. In the temperature range from T = 0 K up to the bulk melting point (TM), these (110) surfaces may exhibit several structures. In the case of the heavier noble metals (e.g. Au, Pt, Ir), the (110) surface undergoes the so called ‘missing row’ 2 X 1 reconstruction at low temperatures by formation of (111) microfacets. Upon heating, this reconstructed surface undergoes an equilibrium order-disorder ‘deconstruction’ phase transition which belongs to the 2D-Ising universality class [l-5]. On the other hand, most of the other FCC (110) metal surfaces (e.g. Ag, Cu, Ni, Pb) do not reconstruct at low temperatures. Upon heating, the nonreconstructed surfaces undergo a so called

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258 A. Lada, P. Wynblatt/Computationcrl Materials Science 7 (1997) 257-270

‘preroughening’ transformation [6-81 from an or- ments [lo]. The Pb (110) surface is the only one dered flat (OF) phase to a thermally disordered flat where the preroughening transition mentioned above (DOF) phase characterized by long-range ‘up- has been experimentally observed (also by means of down-up-down’ order of steps. The critical expo- HRLEED measurements) at T,, = 380 K [19]. nents of this transformation differ significantly from Around this temperature, the Pb (110) OF phase, 2D-Ising values [9]. Generally speaking, both recon- which contains single-atomic height steps localized structed and nonreconstructed surfaces disorder via within the top surface layer, transforms into the DOF an intermediate DOF phase and both of these trans- phase, which contains double-atomic-height steps lo- formations take place somewhat below the roughen- calized within the top two surface layers. Upon ing transition temperature: Pb (110) (T, = 415 K, heating (as the temperature approaches T, = 415 K),

[lo]), Au (110) (T, = 690 K, [ll]), Ag (110) (T, = the Pb (110) surface thermally transforms into the 900 K, [12]), Cu (110) (Ta = 1070 K, [13]), and Ni rough surface, in which several layers of atoms are

(110) (ra ‘- 1300 K, [ 141). involved and steps have single-atomic-height (a so Both deconstruction and preroughening transfor- called ‘hill-valley-like landscape’).

mations involve a change of the in-plane sublattice order parameter (on the sublattices defined in [9] for the (110) surface). The sublattice order parameter changes from a value of 1 at T = 0 K to a value of 0 at T= T,, the preroughening (deconstruction) temperature. At the roughening transition temperature, the step formation free energy becomes equal to zero. This leads to the disappearance of the nucle- ation barrier for crystal growth, the appearance of strong fluctuations (i.e., out-of-plane registry) in the height of the surface, and the formation of large adatom clusters on the surface. Macroscopically, this transformation causes the disappearance of a facet of a given orientation from the equilibrium crystal

shape.

The appearance of the DOF phase can be post- poned by sputtering the Pb (110) surface to introduce a small amount of impurity (oxygen) [20]. The authors of this paper suggest that the presence of oxygen impurities on the Pb (110) surface increases the repulsion between neighboring steps making the free energy of the two-level regular terrace (ID) structure lower than that of the DOF phase. The competition between these two phases ‘shifts’ the Pb (110) roughening transition temperature from 415 K to 480 K.

For the Pb (110) surface, both roughening and premelting take place (see [ 15,161 for details). Ever since the first surface melting studies [ 171, which used medium energy ion scattering (MEIS), this surface has been the most widely investigated experimentally for surface melting and roughening phenomena. Heyraud and Metois found that for lead, only (111) and (100) facets can be observed at a temperature above 390 K during growth [18l. The { 110) facet also appears when crystal growth takes place at temperatures below 390 K. This temperature was therefore identified by these authors as the Pb (110) roughening transition temperature. The (110) facet is difficult to observe on the equilibrium crystal form because of the slowing down of the equilibration processes at low temperatures. Later, Yang, Lu and Wang detected the Pb (110) surface roughening transition at 415 K by means of high-resolution low-energy electron diffraction (HRLEED) measure-

As mentioned above, it is difficult to observe the Pb (110) facets on the equilibrium crystal form. Recent experiments, performed by Cheng and Wyn- blatt [21], showed that cosegregation of Bi and Ni to the surfaces of small crystals of a dilute Pb-Ni-Bi alloy changes the crystalline equilibrium form in relation to that of pure Pb and leads to the reversible formation of { 112) and { 110) facets at roughening transition temperatures of 515 K and 510 K, respectively. At these temperatures the kinetics of equilibration are fast enough for the observation of the reversible appearance and disappearance of { 110) and 1112) facets. These authors have also reported that the above mentioned phenomenon is coupled with a compositional surface phase transition (an abrupt change in the surface concentration of Bi and Ni solute).

Despite the large amount of experimental infor- mation about premelting and roughening of the Pb (110) surface, for a long time no efforts were devoted to the computer simulation of these phenomena on that surface. Recently, simulations using the so called ‘glue’ potential [15,16], have yielded re-

A. Lundu, P. Wynblatr/ Computational Morerids Science 7 (1997) 257-270 259

sults on disordering and premelting of that surface, in good agreement with MElS and HRLEED measurements. The success of those studies has encour- aged us to perform more detailed computer simulations of the roughening transition on the Pb (110) surface employing the same many-body ‘glue’ interatomic potential.

This paper is organized as follows. Pertinent details of the glue model and the simulation are described in Section 2. Estimates of the roughening transition temperature by means of lattice-gas Monte Carlo (LGMC) simulations and the body-centered

solid-on-solid model (BCSOS) are described in Sec- tion 3. Results of large-scale molecular-dynamics (MD) simulations of the roughening transition on the relaxed Pb (110) surface, and its subsequent melting, are given in Section 4. Section 5 is devoted to a study of the time scale of Pb (110) surface local roughening. This section is accompanied by a video tape animation of the results, entitled: “Roughening of the Pb (110) Surface: Large-Scale Molecular Dy- namics Simulation.” (performed using the Pittsburgh 3-Dimensional [P3D] graphics package developed at the Pittsburgh Supercomputing Center [PSCI by Welling). The tape presents a visualization of the evolution of the Pb (110) surface from the initial smooth unrelaxed configuration to a locally rough one. Our conclusions are given in Section 6.

2. Modeling approach

To describe the interatomic interactions in pure Pb we have used the so-called glue model developed by Ercolessi [22]. This is a semi-empirical model in which the total potential energy of N atoms in an arbitrary arrangement, (izj], i = 1,. . . , N, is written in the spirit of other embedding schemes, e.g. the embedded atom method (EAM) and the effective medium theory (EMT), as

E=; ,t Wjj) + 5 U(n,), (1) L.,’ I i= 1

where rij = ]yi - i-,1, the primed summation denotes omission of terms with i = j, @(t-j is a short-ranged pair-potential, and U(n;) is a many-body term de- pending on the local ‘generalized’ coordination of

atom i (including the ‘gluing’ effect of the conduc-

tion electrons), defined as

N ’

n,= c &-ii), (2) j=i _

where p(r) is a short-ranged monotonically decreas- ing function of distance. All three functions @, U and p are undetermined initially and are obtained

empirically. The scheme has been optimized for Pb using experimental data on cohesive, elastic, vibra-

tional, thermal expansion, and surface properties, and has been used rather successfully in studies of lead clusters, the Pb (111) surface where the non-

melting of that surface was demonstrated, and the Pb (110) surface, where surface melting takes place [l&16,23,24].

The structure of the Pb (110) surface has been studied here by using a combination of LGMC and MD methods initially suggested by HZkkinen and Merikoski [25] in simulations of the roughening transition on the Cu (110) surface. This novel approach, incorporating the EMT, allowed the determination of a roughening transition around Tp = 1000 K, in good agreement with experimental results [13].

In our LGMC simulations we have used a computational system and methodology similar to that developed by Daw and Foiles to study the structure and order-disorder transformations at Au (110) and Pt (110) surfaces [26]. We have constructed a slab consisting of 12 X L X L sites, where L is the number of translations in both of the in-plane directions based on the ‘glue’ model equilibrium bulk lattice constant at zero temperature (a. = 4.9095 A). Ini- tially, the slab consists of four layers of vacant sites, and four layers of ‘dynamic’ Pb atoms on top of four layers of ‘static’ Pb atoms. Atoms in the dynamic layers are allowed to exchange with vacant sites in the top part of the slab. This LGMC scheme allows for the exchange of atoms with vacancies, both sitting on ideal lattice sites (the so called ‘restricted’ solid-on-solid model [27]). For each isothermal run, the number of MC steps varied from 20,000 to 200,000 MC steps per lattice site (12 5 f, I 30). These calculations have been performed on the CRAY C90 and J90 plafforms at the PSC, using the MC code MONTE developed by Foiles and Daw of Sandia National Labs (Livermore).

260 A. Lada, P. Wynblatt/Computationl Materials Science 7 (1997) 257-270

In order to account for the effects of lattice vibrations and surface relaxation, we have performed large scale MD simulations in conjunction with the same glue potential. These calculations have been performed either on the CRAY C90 and J90 plat- forms using the MD code DYNAMO, also developed by Foiles and Daw, or on the Cray T3D

platform at the PSC using the code ParaDyn (a parallelized version of DYNAMO) developed by

Plimpton of Sandia National Labs (Albuquerque). In our MD simulations we use a system and

methodology similar to that of our previous study of premelting of the Pb (110) surface [16]. However, the ‘small’ slab which was successfully used in that paper (consisting of 26 dynamic layers on top of 4 static layers, with 88 atoms per layer) was far too small for reliable modeling of the roughening phenomenon. For this purpose, we have constructed a ‘large’ slab consisting of 8 dynamic (110) layers with 2500 atoms per layer (20,000 dynamics atoms) on top of 4 static substrate layers (resulting in 30,000 atoms in total, including the static substrate). Peri- odic boundary conditions were imposed only in the directions parallel to the surface. The lattice constant of the static substrate at each temperature was ad- justed to the value obtained from independent zero- pressure simulations of bulk Pb. At each temperature, equilibration was performed by a sequence of temperature control and constant energy runs (see [16] for details).

3. MC results

Roughening is an infinite order Kosterlitz-Thou- less-Berezinsky (K-T-B) phase transition [28,29] characterized by a logarithmic divergence of the mean-square height difference (height-height correlation function) [30,31]:

(3) where R,, = Izi - Rjl --) 00. The (K-T-B) universality class of the roughening transition ‘prescribes’ at the roughening temperature an essential singularity in the free energy and a divergent correlation length [ with the following asymptotic behavior [5,9]

G(R) - 2A(T)ln( 6). T< TR,

2A(T)ln( R), T2 T,, (4)

for R -+ cc). The coefficient A(T) behaves in a rather characteristic way for a K-T-B transition,

T- TR,

A(T) =

+u2(T-TR) + . ..). T+T,+,

(5)

where the value ACT,) and the square-root behavior are universal features. The subtle nature of the divergence requires large simulation cells. As was mentioned in [32], this divergence is so weak, that the height fluctuation amounts to about one lattice spacing for a distance R of 140 lattice spacing.

For a finite size system, it is more convenient to study the finite size behavior of the interfacial width 6h2 [5,9,33]:

(~h*)=;g([h(‘i) -7q2L I

where h = (l/N)Cih(ri) is the nominal surface height for a given configuration, the sum is carried out over the lattice sites, and angular brackets denote an ensemble average. The height fluctuations behave as

( 6h2) = A(T)ln( t), T< T,,

A(T)ln(L), T2 TR, (7)

thus saturating to a constant value for temperatures below TR (smooth surface), while diverging logarith-

L ” ‘, I,, I,

0.136 - 4

1100 1150 1200 1250 1300

‘UK)

Fig. 1. Coeffkient A as a function of temperature obtained from

the interfacial width of the LGMC model.

A. Landa, P. Wynblatt/ Computational Materials Science 7 (1997) 257-270 261

N

0.2 “.‘.,‘I”‘I”“.,,‘.” 2.4 2.6 2.8 3 3.2 3.4 3.6

wu

Fig. 2. Size dependence of the interfacial width in the LGMC

model at different temperatures.

mically with the system linear size L for T 2 TR

[5,9,331. Fig. 1 shows the coefficient A as a function of

temperature for L = 12, 18, 24,, and 30. Square root fits of the numerical results in the temperature range 1100 K to 1250 K is shown by the solid line. A = l/rr ’ at the LGMC roughening transition temperature of TkGMC = 1100 K.

Fig. 2 shows the size dependence of the squared interfacial width for different temperatures. The ac- tual data points are for .I, = 12, 18, 24, and 30. Above the roughening transition temperature, the lines are almost straight.

Fig. 3 shows snapshots of the surface configura- tionofPb(110)at T= llOOK, 1200K,and 1300K (L = 30). The fractional occupancy of the adlayers reaches 0.18, 0.20, and 0.24 respectively. The pic- ture shows the anisotropy of the adatom clusters directed along the [liO] direction and reflects an absence of repulsion between monoatomic adatom steps (see discussion below).

As was first mentioned in [23](a), the surface energies for fully relaxed Pb surfaces are 37.5, 38.0, and 41.8 meV/i* for the (11 l), (lOO), and (110) orientations respectively. It was also shown [34], that a geometrical criterion for the ratio of surface ener-

gies: u ’ ’ ‘/CT ‘lo < 6 = 0.82, must be fulfilled in

order for reconstruction of the (110) surface to take place. For the glue potential model, (T ” ‘/(T ‘lo = 0.9 and the Pb (110) surface does not reconstruct (i.e.,

the surface energy anisotropy is not high enough to cause the reconstruction). Under normal conditions no reconstruction occurs on the Pb (110) surface and the glue potential reflects this. (Observation of a c(2 X 4) reconstruction of the Pb (110) surface at T < 425 K [35] was apparently caused by the strains induced between the sample and holder or by plastic deformation as discussed in [19]>.

We have mapped the solid-gas interface to an anisotropic antiferroelectric version of the exactly solved six-vertex solid-on-solid (or BCSOS) model [36-381. There are two important characteristics for this model 1391:

a2 + h2 - c* A=-- - -,

2ab (8) which is a measure of ferroelectric (A = 1) or antiferroelectric (A = - 1) long range order sets, and

v=;,

which measures the preference for (1, 2) vertices over (3, 4) vertices and thus reflects the intrinsic anisotropy. Here, using Lieb’s notation [36], a = exp(-Be’), b=exp(-fie,), and c=exp(-pe,) are the weights of vertices (1, 21, (3, 41, and (5, 6), respectively, with vertex energies e, = e2, es = e4, e5 = e6 [4O]. The six allowed vertices, which obey the ‘ice’ rule (i.e., the local electrical neutrality condition), are shown in Fig. 4.

The low temperature phase of the (K-T-B) roughening transition corresponds to the antiferroelectric ground state (vertices 5 and 6 are favorable):

A(Ti’) = - 1, (10) where %iv is the roughening transition temperature within the six-vertex model.

In our calculation we have used the method suggested in [38,41] to determine the vertex energies. Vertices 5 and 6 correspond to the ground state, i.e. to the flat (110) surface. Their energies are:

es = e6 = (+ “‘A, (11)

where A is the area per vertex ( A = $a X (a/ fi) = a2/(2&)), and a is a lattice constant.

There are two pairs of ‘excited’ vertices in the system (1,2) and (3,4). As was shown in [38], vertices 1 and 2 correspond to tilting the surface by an

angle 8, = cos-’ - = 35” in the [OOll direction,

262 A. Landa, P. Wynblatt/ Computational Materials Science 7 (1997) 257-270

Fig. 3. Snapshot of the surface configuration of Pb (110) from the LGMC simulations with L = 30 at (a) T = 1100 K, (b) T = 1200 K, and

(c) T = 1300 K. Adatoms arc colored red.


(010) (110) (110)

Fig. 4. Tbe allowed six vertices for tbe FCC (110) surface viewed

along a (110) direction. The crossed circles indicate atoms in tbe

top plane (‘black’ atoms). The open circles indicate atoms in tbe

next plane down (‘white’ atoms). A minus sign indicates a

vacancy at tbe corresponding atomic site.

which is equivalent to forming a (111) facet. In this

case, the vertex energies are

u “‘A e,=e,=-=

cos 8, (12)

Vertices 3 and 4 correspond to tilting the surface by an angle e3 = 45” in the [liO] direction, which is equivalent to forming (100) facets. In this case, the vertex energies are

e3 = e4 = s = fi,‘OOA_ (13) 3

For the lattice constant and surface energies in the ground state (T = 0 K), the vertex energies are e, = eZ = 0.3918 eV, e3 = e4 = 0.4578 eV, and e5 = e6 = 0.3563 eV. Using these vertex parameters, we have calculated the roughening transition temperature in the BCSOS model, in the thermodynamic limit, (L + 03) to be: 7’lv( L + m) = 1050 K. The difference between TkGMC = 1100 K and T,.f’(L + w) = 1050 K is mainly caused by the finite size effect in LGMC calculations (L ~2 30).

It is not surprising that both these calculated values of the roughening transition temperature on the Pb (110) surface greatly exceed the experimental value of 415 K. LGMC simulations cannot account for lattice vibrations and surface relaxation. The outermost layer of the Pb (110) surface displays a large inward relaxation (see [15,16] for details). These simulations also do not account for the relaxation effects near defects (adatoms and vacancies). All of

these factors could significantly decrease the adatom-vacancy formation energy and thereby re- duce the roughening transition temperature. In the BCSOS calculations we have accounted for relaxation of the outermost surface layer (as the surface energies were calculated for relaxed surfaces). How-

ever, we have used the values of surface energies calculated for zero temperature. At high temperatures, these values are different, as are the anisotropy ratios. To account for these effects, we have performed large-scale MD simulations.

4. MD results

Equilibration runs were started from the ideal smooth surface at 460 K (see Fig. 5a). This temperature was identified in our previous calculations [42] (with a ‘small’ slab) as the lowest temperature of adatom-cluster formation, which is an important feature of the roughening transition. In Fig. 5, the time c indicated on the individual subfigures denotes the total time from the beginning of the equilibration runs at 460 K. Fig. 5b shows a snapshot of the equilibrated slab at 464 K (total time t = 300 ps). From this plot we can see the formation of 2 to 5 adatom clusters which are distributed randomly. The average equilibrium occupation of the adatom layers (i.e., the fraction of adlayer sites occupied by atoms) reaches - 10.5% at this temperature, indicating that the temperature is still far from the roughening transition point (which is characterized by an adatom occupancy of about 20% [43]). As the temperature increases, the average size of adatom clusters also increases (see Fig. 5c, T = 501 K and total time t = 740 ps) and the equilibrium average occupation of the adatom layer reaches - 13.2%. The roughening transition temperature (TFD = 545 K) was defined by studying the behavior of the interface width (more precisely, by the behavior of the pre-logarithmic coefficient A in Eq. (7) for L = 50), see Fig. 6). A snapshot of the slab at T = 545 K and total time t = 1.5 ns is shown in Fig. 5d. At the roughening transition temperature, the equilibrium average occupation of adatom layers reaches - 19% which is a reasonable value (see discussion in [43]). As can be seen from this figure, the tendency of adatoms to create large clusters (15-20 atoms) is clearly dis- played.

264 A. Landa, P. Wynblatt/ComputationaI Materials Science 7 (1997) 257-270

Fig. 7 shows the average equilibrium occupation of the adatom layers plotted versus temperature. The adatom concentration reaches its maximum value of 20% at T = 570 K, total time t = 2.7 ns. Fig. 5e is a snapshot of the slab at this temperature, which shows

the maximum observed occupation of the adatom layers (- 24%). The configuration of the monoatomic adatom steps reflects a lack of repulsion between them (see Section 3). With further temperature increase, the concentration of adatoms begins to

Fig. 5. Snapshots of (a) the Pb (110) initially smooth slab at the staring configuration; the equilibrated Pb (110) slab at (b) T = 464 K

(t = 300 ps); (c) T = 501 K (t = 740 ps); (d) T = 545 K (t = 1457 ps); (e) T = 570 K (t = 2693 ps) (notations as in Fig. 3); (f) T = 600 K (t = 5 ns). Melted layer is colored yellow.


q q

q q

0.ooo-I 1 ’ 1 1 I=. ’ ’ 460 480 500 520 540 =, 560

T (K)

Fig. 6. Coefficient A as a function of temperature obtained from

the interfacial width of the MD model (L = 50).

decrease and adatoms disappear in the temperature

range between 590 K and 600 K. In Fig. 8 we plot the logarithm of the average

equilibrium occupation of the adatom layer versus (k,T)-’ in the temperature interval 460 K < T < 540 K. From the slope of this curve, we estimate a vacancy-adatom formation energy of EAv - 0.147 eV. This value is higher than that ( EAv - 0.116 eV) estimated in [44] from the same kind of plot for the ‘small’ slab in the temperature interval 360 K < T < 400 K (see also Fig. 2 of [42] for details).

Fig. 9 shows the square of the layer structure factor plotted versus layer number at the roughening transition temperature. For layer #8 (the surface layer), order is completely lost in the [OOl] direction which is perpendicular to the close packed rows.

0.14 - 6 q

0.12 - q

0.10-m. ’ . ’ ’ n . ’ ’ . ’ 0 . 460 480 500 520 540 560 580 600

VW

Fig. 7. Average equilibrium occupation number of the

layer in the MD simulations plotted versus temperature.

21 22 23

Fig. 8. Natural logarithm of the average equilibrium occupation

number of the adatom layer plotted versus (k, T)- ’ .

Some residual order still remains along the closed packed [ liO] direction, and perpendicular to the (110) surface. The same kind of behavior is observed for the adatom layer (layer #9).

Fig. 10 shows the square of the layer structure factors at the temperatures indicated. Some level of in-plane order is still maintained for layers #8 and #9 at 560 K and 580 K. The outermost layer of the slab (# 8) is completely melted at T - 600 K indicating disappearance of its in-plane order. The phenomenon of surface melting, which was successfully described in [16], is suppressed in this case by the influence of the static substrate layers which are

0.4 - I .

m .

0.2 - a l

. .

q l

o.o+.n’n’,‘,‘,.,.:. 1 2 3 4 5 6 7 8 9

LAYER

Fig. 9. Squared layer structure factor 1 S,( g, )I * (g, = (2rr / aX2,

5, 0). g2 = (27r/ aX0, 0, 2) and 1s = (2rr/ aX2, 2, O)), plotted versus layer number at the MD roughening transition temperatum

(TrD = 545 K).

266 A. Lada, P. Wynblrrt/ Compuratiod Materials Science 7 (1997) 257-270

. :

8 l

: I

m CI

: . 0.0 -,.“,.,‘,‘z.:, 0.0’

! j 1 * 1 r 5 1 0.0 8 I I I I ? :

1 2 3 4 5 6 I 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8

LAYER LAYER LAYE

Fig. 10. Squared layer structure factor IS,( g, )I* plotted versus layer number at temperatures indicated. Notation as in Fig. 9.

9

R

located very close (only 4 layer spacings) to the outermost layer. A snapshot of the Pb (110) surface at T = 600 K is shown in Fig. 5f (the total time of all runs performed in the temperature interval 460 K to 600 K was I = 5 ns).

We have also used a modification of the BCSOS model [25] with energy parameters extracted from the surface excitation energies of the glue model to estimate the roughening transition temperature. In this case, e, = e2 = J, and es = e4 = J, are the nearest and the next nearest neighbor coupling constants, respectively (i.e., the energy cost of broken bonds in the [liO] and [OOl] directions) while the energy of a flat (110) surface is zero, i.e. e5 = e6 = 0 [401. For this assumption, Eqs. (8) and (lo), at T = TR, become

u2 + h2 - 1

2ab = -1, (14)

where a = exp( - /3 Jy ) and b = exp( - p J,). Fig. 11 shows the energy of an atom in the glue

model lattice gas simulations, E(C,, C,), where Cj. is the number of the j-th neighbors of an atom in the lattice. The energy costs of broken bonds (J, and J,) are found from the energy function E = E(C,, C,) for a coordination corresponding to the outermost atom layer for which C, = 7 and C, = 4. As can be seen from Fig. 11:

J, = E(7,4) - E(6,4) (15)

is the derivative of E(C,, C,) with respect to C,, and

J, = E(7,4) - E(7,3) (16)

is the distance between the E(C,, C,) curves. The calculated anisotropy ratio, Jy: J, = 3.23, is close to the value of 3.46, which was extracted from HRLEED experiments [lo]. The calculated vertex energies were then scaled so as to reproduce the correct energy for an adatom-vacancy pair (EAv = 2( J, + J,) = 0.147 eV). This leads to: Jy = 0.0561 eVand J,= 0.0174 eV. Using these energy parameters, the roughening temperature given by the glue potential six-vertex model is Ti& = 552 K in fair agreement with TFD = 545 K, and exceeds both the experimental result Tfxp = 415 K [lo] and our previous estimation of the roughening transition temperature (436 K, [44]). As was mentioned above, the calculated value of ‘the adatom-vacancy formation

energy, EAv = 0.147 eV, for the ‘large’ slab (8 dynamic layers with 2500 atoms per layer) is higher that the value of EAv = 0.116 eV estimated by means of a significantly ‘smaller’ slab (26 dynamic layers with 88 atoms per layer) [44]. We believe that due to

-1.05 - C2=6

g - c2=5

-1.25 - CL?=4

w - a=3

-1.45 - a=2

- CT=1 -1.65

-2.05 6 8

Cl lo 12

Fig. Il. Energy of an atom in the glue model lattice gas simula-

tions versus the number of j-th neighbors.


the size of the computational slab, the adatom- vacancy formation energy (0.147 eV) calculated here is a more accurate value.

We now address the discrepancy between the calculated value, Tillue = 552 K, of the glue potential six-vertex model roughening transition temperature on the Pb (110) surface with experimental results. The glue potential used here has a cutoff radius between the 2nd and 3rd neighbors [23Xa). Using SOS Hamiltonian notations, this case corresponds to so called ‘K,’ model [9] where the Hamiltonian becomes the same as that appropriate for the anisotropic version of the BCSOS model. If the interaction beyond 3rd neighbors is absent, the sublattice order parameter vanishes continuously at Tpr = TR, so that sublattice disordering and roughening do coincide in the BCSOS (K,) model [45]. This appears to have occurred in the present calculations.

The DOF (long range antiferromagnetic up- down-up-down) structure is stabilized by the presence of further-next-neighbor coupling (NNN interactions) in addition to interactions of the BCSOS model. This interaction acts as an effective hard core repulsion between two up-up and down-down monoatomic steps preventing bonding of steps of the same sign in pairs (clusters) and promoting the gen- eration of the DOF phase [9,46-481 (our simulations do not show this fine feature). Successful description of the preroughening and roughening transitions on the Ag (110) surface (using the so-called ‘K,’ model [9,47] in conjunction with an embedded atom potential having a cutoff radius between 3rd and 4th neighbors [49]) confirms this statement. Recently, the effect of extending interactions beyond 4th neighbors has been studied [48]. Also, extension of interactions in an EMT potential beyond 3rd neighbors [25] led to a sufficient increase in the anisotropy of the model to lower the roughening transition temperature in large-scale MD simulations.

5. Time scale of roughening

The MD simulations allowed not only considera- tion of the effects of surface relaxation and lattice vibration, but also determination of the time scale of local roughening. Such studies were first performed for the Cu (110) surface [25]. In those studies, the

4

ot.‘.““‘.‘~““““‘.“““‘~~ 0 100 200 300 400 500 600

t(Ps)

Fig. 12. Occupation number of the adatom layer as a function of

the simulation time in the MD run at TED = 545 K.

time scale of local roughening (the time required for the occupation number of adlayers to saturate to their ‘rough’ value of m 20%) was found to be 100 ps and 300 ps at 1150 K and 1100 K, respectively. Both of these temperatures were well above the calculated roughening transition temperature of the Cu (110)

surface (TR = 1000 K). Our simulations of the same phenomenon have been performed at the calculated roughening transition temperature of the Pb (110) surface (Tp = 545 K).

The present simulations used a smooth unrelaxed initial configuration and followed the change in the occupation number of adatom layers as a function of simulation time (see Fig. 12). The damped oscilla- tion of this value during the first 10 ps corresponds to the vibration of the outermost surface plane around its equilibrium (relaxed) position. This effect is clearly shown in the video tape of our results. After approximately 50 ps, adatoms form a small (3-5 atoms) cluster. As in the case of Cu (110) [25], we observed irregular fluctuations of a few tens of pi- coseconds superimposed on normal statistical fluctuations. These long-wavelength fluctuations are con- nected with the diffusion dynamics of adatom and vacancy clusters.

Fig. 13 shows the coefficient A (the pre-logarithmic coefficient in Eq. (7) for L = 50, calculated using interfacial width) as a function of simulation time. The rapid increase in interfacial width (and occupation of the adatom layers) begins after 400 ps of the simulation run. The animation of the results illustrate the creation of large adatom clusters via the coalescence of small ones. It takes about 560 ps for the surface to become rough locally via its intrinsic

268 A. Lunda, P. Wynblatt/ Compututional Materials Science 7 (1997) 257-270

9 T=.545K Y

q

0.008 q

Q

0.002 1

0.000 200 300 400 500

t R@O

tw Fig. 13. Coefficient A as a function of simulation time obtained

from the interfacial width of the MD run (L = 50) at TgD = 545

K.

dynamics which involve the creation of adatoms and their redistribution. This time is significantly higher then the 100 to 300 ps reported in [25]. However, in contrast to [25], the present calculations have been performed exactly at the roughening transition temperature. Increasing simulation time has been inter- preted in [25] as a ‘critical slowing down’ effect, which makes the observation times for the cluster dynamics in MD simulation extremely long as the transition temperature is approached.

6. Conclusions

The roughening of the Pb (110) surface has been studied via atomic-scale simulations using both the Monte Carlo and molecular dynamics methods in conjunction with a many-body (glue) potential description of interatomic interactions. In lattice-gas Monte Carlo simulations, the rough phase has been identified from the logarithmic behavior of the interface width The roughening transition temperature (~;oMc = 1100 K) was obtained from the universal square-root temperature-dependent coefficient of the logarithmic term. The solid-gas interface was also mapped to the body-centered solid-on-solid model with vertex energies being obtained from the surface energies of the (1111, (1001, and (110) facets. This approach yields a roughening transition temperature of about 1050 K in good agreement with lattice-gas

Monte Carlo results. In order to account for lattice vibration and surface relaxation effects, we have performed large-scale molecular-dynamics simulations in the temperature range from 460 K to 600 K (the duration of the total run is approximately 5 ns). The roughening transition temperature (TFD = 545 K) has been defined by the temperature behavior of the interface width. At the transition point, the average equilibrium occupancy of adatom layers reaches - 19%. The solid-gas interface has again been mapped to the anisotropic six vertex model with vertex parameters extracted from the surface excitation energies of the glue model and scaled to repro-

duce the calculated adatom-vacancy formation energy. This approach yields a roughening transition temperature of 553 K, in good agreement with the molecular-dynamics result. The time scale of the local roughening from an initially smooth surface at the roughening transition temperature (545 K) is about 0.6 ns. We have presented graphic results from our molecular dynamics simulations of the evolution of the Pb (110) surface from initially smooth to locally rough (in the form of an animated video tape). The outermost layer of the ‘large’ slab is completely melted at - 600 K. In this case, the premelting of the Pb (110) surface, which has been detected both experimentally 1171 and theoretically [ 15,161 at lower temperatures, has been suppressed in these calculations by the influence of the substrate layers which are located very close to the outermost layer.

Finally, we have concluded that the glue potential for Pb, which is cut off between the 2nd and 3rd neighbors, does not provide any repulsion between monoatomic adatom steps on the Pb (110) surface, and hence leads to a coincidence of the second order preroughening transition (formation of the disordered flat phase) with the infinite order roughening transition. This cut off also leads to an insufficient value of surface anisotropy on Pb (110) and an overesti- mated value of the roughening transition temperature.

Acknowledgements

We wish to thank Profs. U. Landman, R. Swend- sen, A. Patashinski, V. Vitek, and M. McHenry, Drs.

A. Lands, P. Wynhlutt/ Computatiwd Moterial.~ Science 7 (1997) 257-270 269

S. Foiles, S. Plimpton, R.N. Bamett, H. HZkkinen, J. Merikoski, A. Sutton, F. Ercolessi, and F. Di Tolla for helpful discussions and/or for providing us with computational codes. We acknowledge with thanks

support of this work by the National Science Foun- dation under grant DMR9530469, by the PSC under grant DMR930031P, and by NCSA under grant DMR950017N. Last but not least, we wish to thank Drs. J. Welling and A. Kar (PSC) for help in the

creation of the graphic images and their animation.

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Roughening of the Pb (110) surface: combined Monte Carlo and molecular-dynamics study

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