Unstable epitaxy on vicinal surfaces

surface science ELSEVIER

Surface Science 369 (1996) 393-402

Unstable epitaxy on vicinal surfaces Martin Rost a,c,,, Pavel Smilauer b,1, Joachim Krug a,c a Institutfar FestkOrperforschung, Forschungszentrum Jiilich, D-52425, Jalich, Germany b HOchstleistungsrechenzentrum, Forschungszentrum Yalich, D-52425, Jfilich, Germany ¢ Faehbereich Physik, Universitat Gesamthochschule Essen, D-45117, Essen, Germany

Received 27 April 1996;accepted for publication 24 June 1996

Abstract

Epitaxial growth on a vicinal surface in the step flow regime, where the diffusion length exceeds the step spacing, is studied by simulation of a continuum equation and a solid-on-solid model. Such a surface is known to undergo a meandering instability if step edge barriers suppress downward interlayer transport. We show that the resulting ripple pattern is itself unstable, and evolves at long times into an essentially isotropic mound morphology which is qualitatively and quantitatively indistinguishable from that obtained on singular surfaces. Keywords: Computer simulations; Growth; Models of surface kinetics; Molecular beam epitaxy; Surface diffusion; Surface structure,

morphology, roughness, and topography; Vicinal single crystal surfaces

1. Introduction

Using molecular b e a m epitaxy ( M B E ) it is possible to g r o w structures with very high precision in the vertical direction, such as m o n o l a y e r - t h i n interfaces or atomically flat surfaces. The m a j o r challenge still remains of growing surfaces which are structured laterally. A promising w a y is to m a k e use of patterns evolving out of inherent instabilities in g r o w t h processes. A theorist's contrib u t i o n to controlled unstable g r o w t h can then lie in understanding its basic mechanisms. Here we focus on a regime which is governed by a nonequilibrium current of diffusing a d a t o m s on the growing surface. The particular interest * Corresponding author. Fax: +49 201 1832120; e-mail: [email protected] 1 On leave from Institute of Physics, CukrovamickA 10, 16200 Praha 6, Czech Republic.

focuses on the large-scale features at a late stage of growth. In m a n y experiments, e.g., for h o m o e p i taxy of G a A s [1,2]~ C u [ 3 ] , G e [-4] and Fe [ 5 , 6 ] , all g r o w n on singular (001) substrates, as well as for h o m o e p i t a x y on the R h ( l l l ) surface [ 7 ] , one observes the evolution of large, p y r a m i d or m o u n d like features. Their lateral size ~ is f o u n d to increase according to a power law in time, ¢ ~ t 1/z with z ~ 2 . 5 - 6 depending on the material and, possibly, deposition conditions used. A second characteristic is the slope of the m o u n d s ' hillsides s, which varying for different materials - is observed to remain constant (referred to as a "magic slope") or to increase with time as s g t ~ [ 8 ] . B o t h scenarios can be observed in c o m p u t e r simulations [ 9 , 1 0 ] . The origin of these instabilities is a growthinduced surface current caused by step edge barriers [ 11,12]. Since diffusing a d a t o m s preferably attach to steps f r o m the terrace below rather than from

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M. Rost et al./Surface Science 369 (1996) 393-402

Fig. 1. Three-dimensional representation of a configuration of the SOS model (parameter set I) after deposition of 80 monolayers, showing a partial view of 100 x 100 lattice sites. A larger part of the same configuration is shown in greyscales in Fig. 2.

continuum growth equation and a discrete solidon-solid model, that the meandering instability provides a pathway for the global destabilization of a vicinal surface, which eventually leads to the same type of mound morphology as has been found on singular surfaces. The destabilization proceeds through a secondary instability of the ripple pattern created by the Bales-Zangwill mechanism. Fig. 1 represents a configuration with ripples in the discrete model. Its further evolution is illustrated in Fig. 2 both for the continuum and the lattice model. In the late time regime we observe power-law coarsening of the mounds, with scaling exponents that appear to take the same values as on singular surfaces.

2. Continuum equation for MBE above 1-13,14], the current is uphill and destabilizing. The concentration of diffusing a d a t o m s is maintained by the incoming particle flux; thus, the surface current is a nonequilibrium effect. The instability of a singular surface due to step edge barriers can also be u n d e r s t o o d on a m o r e basic level: a d a t o m s form islands on the initial substrate and m a t t e r deposited on top of them is "caught" there by the step edge barrier. Thus a "wedding cake" structure of islands on top of islands develops. O n a vicinal surface growing in the step flow mode, such that the nominal step spacing lo is less than the typical spacing lo of islands formed by diffusion [ 1 5 ] , the step edge barriers have two opposing effects. A terrace wider than its upper neighbor receives m o r e mass out of the beam, which diffuses to the u p p e r step. The step then advances faster and the average terrace size is restored [ 14]. This a r g u m e n t has led to the conclusion that step flow g r o w t h should be generally stable [2,11,12]. However, as was first pointed out by Bales and Zangwill [ 1 6 ] , the same mechanism makes steps unstable towards transverse meandering, since matter is m o r e likely to attach to already a d v a n c e d parts in the step. Thus a vicinal surface is stable (unstable) with respect to fluctuations along (perpendicular to) the direction of the slope [ 1 7 ] ; completely stable epitaxial g r o w t h is n o t possible in the presence of step edge barriers. In this paper we show, t h r o u g h simulations of a

We base our discussion on a continuum equation for unstable growth of the type introduced by

t-lO00

80 ML

t~5000

400

Fig. 2. Configurations obtained by integrating Eq. (4) (left, grey level plots with the average tilt subtracted, size 100 x 100) and by simulating the SOS model with parameter set I (right, grey level plots with the initial staircase subtracted, size 150 x 150 lattice sites), at different times or number of deposited monolayers, respectively. Large mounds emerge and the system loses its anisotropy. In both models mounds are formed preferably at "dislocations" in the ripple pattern.

M. Rost et al./Surface Science 369 (1996) 393-402

395

several groups [2,6,12,18-20], which quantifies the intuitive picture presented above. Let H(x,t) denote the height of the surface above the substrate plane, measured in units of the monolayer thickness, As is appropriate for the experimental situation described above, desorption from the surface is neglected. The surface then evolves according to an equation of conservation type

by Johnson et al. [2,20], f(s)=(1 "Jvs2)-1, We do not consider sign changes in f, which would induce stable selected facets ("magic slopes") [ 10,12,18,19]. In dimensionless form the equation of interest then r e a d s

O,H = - ~c(VZ)ZH--V- J~,~ + F,

We first recall the stability properties of flat, vicinal or singular, surfaces under Eq. (4) [19,22]. With h(x,t)=m.x+e(x,t) the linearization of Eq. (4) yields

( 1)

where the first term on the right-hand side describes smoothening due to surface diffusion, the coefficient x being the product of the surface stiffness and the adatom mobility [21], F denotes the deposition rate, and JN~ is the uphill nonequilibrium mass current induced by the step edge barriers [12]. Under the assumption of in-plane isotropy the current is directed along the local tilt VH, and one can write

I ~ = ~ ( I V HI)V H.

(2)

A calculation [8,22] in the framework of Burton-Cabrera-Frank (BCF)theory [23] shows that

~(u)= F'{2f(uID),

(3)

where lD is the diffusion length or terrace size on the singular surface [15], andf(s) is a dimensionless shape function of order unity, with f(s~O),~ 1 (see below). The "effective" diffusion length [D is a combination of 1D and a capture length l_ ~ a exp (AE/kBT), which exceeds the lattice constant a by an activation factor reflecting the magnitude AE of the excess step edge barrier [8]; for l_ >>/v ("strong barriers") one finds IDa/V, while for weak barriers 1D~ ~ <
~?th= -

072)2h-

V •

Vh 1 + ([7 h) 2"

~t~= (VII~I-[-V±~2 __(V2)2)E,

(4)

(5)

where @ 0± denote derivatives parallel and perpendicular to the tilt vector m, and the coefficients are vll=(mZ-1)/(m2+l)2, v ± = - l / ( l + m 2) with m = [m[. For singular surfaces vii = v± < 0; the coefficient vii changes sign at m = 1, reflecting the transition to step flow growth. Since our unit of slope is a/ID, m = 1 implies that the average step spacing l0 = 1D. However, because v. < 0 for all m, the step flow regime is unstable against transverse fluctuations [16,17,19,22]. The pattern emerging from the linear instability are ridges or ripples running downhill (analogous to those obtained on a discrete lattice model represented in Fig. 1), with a lateral size given by the most unstable wavelength 2 = 2rcV2/I~/~] = 2n~V~S/1 +

m 2.

(6)

3. Numerical integration To follow the instability into the nonlinear regime, the continuum equation (Eq.(4)) was integrated numerically. The height function h(x,t) was discretized on a 2nx 2n grid (typically 512x512) with "helical" boundary conditions with a mean tilt along the x-direction, i.e. h(x + Ie~,t) = h(x,t) + mL. For the results presented in this work we set m = V ~ , so that vlt >0. We used a pseudospectral code, treating separately the linear and nonlinear terms of Eq. (4). Writing the equation as 0th = ~ h + JV'(h) we chose the iteration ht+~= (1 - (z/2)~q~)- 1( 1 + (z/2)Se)ht + zJg'(h0. One recognizes a semi-implicit scheme for the linear term, which was effectuated in momentum space

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where the operator ~qo is diagonal. The nonlinear term was added explicitly but its magnitude was controlled by varying the time step -c, such that at any grid point z~/'(h)<0.1. The derivatives entering Jg" were calculated by finite differences using the most simple expressions in order to keep the program fast. Eq. (4) is deterministic, and in the results presented here we did not add any temporal noise. As initial conditions we used fiat vicinal or singular surfaces with a small random variation e, h(x,O)= m . x + e ( x ) , independently chosen at each grid point x (typically le(x)l < 1/2000). For m = 0 the most unstable wavevectors are degenerate, and in an early stage we obtain a random pattern of buckles of nearly equal size. This corresponds to the development of the "wedding cake structure" described above. Here we focus on the behavior in the step flow regime Iml > 1, where the surface instability creates equidistant ripples parallel to m, as can be seen in Fig. 2. The central result of this work is the secondary instability of the ripples, which leads at long times to the formation of an essentially isotropic morphology of mounds. As shown in Fig. 2, the first mounds nucleate at the defects in the previously developed ripple structure. Such defects are generically present, because the ripple patterns emerging locally in different areas of the surface cannot be expected to have a globally coherent phase. Mounds appearing later arrange around these first nuclei and a mound pattern spreads across the surface, breaking up the ripples. An intermediate state is shown in the lower image of Fig. 3. However, even a perfect, defect-free ripple pattern is unstable towards mounding. This can be understood by considering profiles of the form h(x,t) = m . x + h.(x±,t), which vary only perpendicular to the tilt direction (apart from the overall vicinality). Inserting into Eq. (4) one finds that h . ( x . , t ) evolves, up to a constant factor, according to the one-dimensional version of Eq. (4). Since the one-dimensional equation is inherently unstable, the transverse profiles h± at different longitudinal positions x, tend to evolve apart, so that defects in the ripple pattern will be created, and finally mounds form. We checked this numerically by integrating Eq. (4) with the following initial condi-

Fig. 3. Representation of large systems, SOS model (parameter set II) of size 300 x 300 after deposition of 40 monolayers (top) and configuration of Eq. (4) at t = 5000 with size 500 x 500. Note the m o u n d s situated at defects in the ripple pattern, and coexistence of ripples a n d m o u n d s in different spatial areas.

tions: ripples of wavelength 2 as given in Eq. (6) with amplitude 1 and random fluctuations of amplitude 10 -1 , 10 - 2 , 10 - 3 . In all cases the ripples wiggled, broke up and mounds were created.

M. Rost et al.:/Surfaee Science 369 (1996) 393-402

On a more quantitative level the transition from ripples to mounds is reflected in the anisotropy of the height-height correlation function

397

100

G(x - y,t) = ( (h(x,t) - m "x)(h(v,t) - m "y)). We considered G in directions parallel and perpendicular to rh (the unit vector in the direction of the initial tilt), in the sequel denoted by G,(x,t)=-G(x~,t) and G±(x,t) for its counterpart. Data were averaged over two runs of a system of size 500 x 500. The typical lateral and perpendicular length scales ~1[(~±) correspond to the width of GII(Gj_), e.g. we chose lit such that G,(~ti/2,t)=G(O,t)/2 and the analogue for ~±. At late times the lengths ~, and ~± (see Fig. 4) approach each other and their logarithmic derivative d log ~/dlogt approaches 1/4. For earlier times ~l] takes large values but then decreases quite rapidly when the ripples break up. At the same time ~± increases more rapidly than at later times, since the appearing mounds are broader than the previously existing ripples, For late times the surface width w ( t ) = ( h ( x , t ) 2 ) ! / 2 = ~ approaches a power law w ~ t I/2 (Fig. 5). Again here the fastest increase occurs in the early stage of mound formation, a typical feature at the onset of unstable growth. Third we examined the typical 10o

lO

o

0 •

100

I000

10000

Time Fig. 5. Mean width w - = ( h 2 ) 1/2 of the growing surface. The inset shows d log w/d log t, approaching the power 1/2, as predicted for singular surfaces [24].

slope of the mounds s±(t)=w(t)/¢ll(t ) and sll, respectively (see inset of Fig. 4). For late time, both quantities approach each other, following a common power law s±=s, ~ t ~ with ~ 1/4. The same late time scaling laws 2;.,~'~stE~s±~ t 1/4, w ~ t 1/2 were found by numerically integrating Eq. (4) for singular surfaces. Analytic arguments indicating that these scaling laws are in fact exact will be presented elsewhere [24].

4. Solid-on-solid model

o.1

• '

° "

l 1000

10O00

Time Fig. 4. Different length scales of the surface features in direction along (~li) and across (4±) the tilt, obtained from simulations of the continuum equation. Long ripples formed at an early stage break up, and both lengths approach each other, as do the typical slopes (see inset). The lines show a power law t (1/4),which governs Eq. (4) on singular surfaces at late times [24].

To investigate the growth instability in more detail, we used a solid-on-solid model of epitaxial growth in which the crystal is assumed to have a simple cubic structure with neither bulk vacancies nor overhangs allowed. The basic processes included in our Monte Carlo simulation model are the deposition of atoms onto the surface at a rate F and the diffusion of surface adatoms modeled as a nearest-neighbor hopping process at the rate k(E,T)=ko exp (--E/kBT), where E is the hopping barrier, T is the substrate temperature, and kB is Boltzmann's constant. The prefactor ko is the attempt frequency of a surface adatom and is assigned the value ko = 2k~T/h where h is Planck's constant. The barrier to hopping is given by E-- Es + nEN + (ni-- nf)®(ni-- nf)EB, where Es, EN, and EL are model parameters, n is the number of

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M. Rost et al./Surface Science 369 (1996) 393-402

in-plane nearest neighbors before the hop, ni and nf are the number of next-nearest neighbors in the planes beneath and above the hopping atom before (hi) and after (nf) a hop, and ®(x) = 1 if x > 0, and 0 otherwise (see Ref. [-9] for a more detailed description of the model; note, however, that in all simulations reported here a random deposition was used without any search for a high-coordinated site). The simulations were carried out on square 150x 150 and 3 0 0 x 3 0 0 lattices with "helical" boundary conditions (cf. above) and 25 or 50 steps, respectively; thus the nominal step spacing is /o=6. Two basic sets of model parameters and growth conditions were used (following the study of unstable growth on a singular surface [9]): E s = 1.54 eV, EN=0.23 eV, EB =0.175 eV, T = 828 K, and F = 1/6 monolayer (ML) s -1 (set I), and Es=0.75 eV, EN=0.18 eV, EB=0.15 eV, T = 4 0 0 K and F = 1/40 M L s-1 (set II). Examples of the surface morphology evolution during growth on a vicinal surface are shown in Figs. 1 and 2. In this case, the parameter set I with a relatively weaker step-edge barrier was used. Initially, steps meander and soon organize themselve into "fingers". These fingers (ripples) undergo severe shape changes and eventually break up (first at defects such as ripple branches, see upper part of Fig. 3) creating mounds that subsequently coarsen similarly to those observed on a singular surface [9]. It is obvious that the surface morphology evolution closely resembles that obtained from integration of Eq. (4). We have carefully checked the robustness of this scenario with respect to the details of the simulation model. In particular, we have found that the strength of the step-edge barrier strongly affects the characteristic time to breakup of the ripples into mounds. Whereas for simulations with EB =0.175 e ¥ in Fig. 6 the surface ripples break up after ~ 300 M L are deposited, it takes only ~ 4 0 M L for EB =0.25 eV, while the ripples do not break even after ,-~3000 M L are deposited for EB = 0.05 eV. These results suggest that phenomena described in this paper could be observed for materials with strong step-edge barriers (e.g. (111) faces of some fcc metals). We also note that the basic scenario of the surface morphology evolution

I00

• i.o " i ii..'...... ....., . ..12.1. . . . . . . . . . . r

o•

0.1 t 5e AAA~ 1

10 Time

rl l 10

100 (ML)

•

100o . ,,~JOb

• OO~O oo O

o~x

oo

10

100

1000

T i m e (ML) Fig. 6. Same quantities as in Fig. 4 for the simulated SOS model. Power-law fits for late times are indicated.

for various modifications of the simulation model does not change [29]. For a quantitative study of the process of coarsening, we have,used parameter set II. In Fig. 6, ~ll and 4± are plotted as a function of the number of monolayers deposited. Similarly to the results of the numerical integration of Eq. (4), these characteristic lengths become equal at late times and their logarithmic derivative is ~0.21, close to the value of ~ 0.22 obtained in simulations on a singular surface [9]. The values of the typical slopes sll and sx increase at late times as t ~ with e~0.14. Consequently, w(t) grows ~ t ~ with fl~0.35 after a faster increase at short times. The value of fl is rather close to fl ~ 0.33 obtained in simulations on a singular surface in Ref. [-9] (cf. Fig. 7).

5. C o m p a r i s o n o f c o n t i n u u m and discrete m o d e l i n g

The morphologies shown in Figs. 2 and 3, as well as the shapes of the curves in Figs. 4-7, clearly demonstrate that both the formation of the ripple pattern and its subsequent destabilization proceed in an analogous manner in the continuum equation and the SOS model. While a quantitative comparison is not straightforward for the reasons described in the Appendix, a good match of time and length scales can be obtained by dividing the time variable of the continuum equation by a factor of 120 -

M. Rost et al./Surface Science 369 (1996) 393"402

~ ~

0 . 8

0 . 4

..,

. . . . . . . .

,

. . . . . . . .

,

. . . . . . . .

described by a continuum theory with a different shape function f in Eq.(3). For example, if f(s),.~s -4, for large s one obtains [24] scaling exponents e = 1/8 and /~= 3/8, quite close to the SOS estimates. However, we do not have any physical reason for assuming such a function.

,

oo

•

~ 0.2

'~o.o 00

399

lo lOO i ~ 0 Time (ML) • O'

6. Conclusions

r.¢l 1

•

10

I00

i000

Time (ML) Fig. 7. Width of S O S model as a function of time. Logarithmic derivative in the inset shows power law behavior w ~ tT M for late times.

that is, the growth of one monolayer (for set II of the SOS model) corresponds to the dimensionless time interval of 120. This rescaling aligns the times at which ~, attains its minimum in Figs. 4 and 6, and it gives rise to very similar morphologies. However some characteristic qualitative differences should also be noted. First, the length of ripples ~11is seen to be much larger initially in the continuum equation (Fig. 4), and it decreases by a much larger factor when the ripples break up. This is easily explained by the transverse fluctuations of the ripples in the SOS model (Fig. 2), which arise from the noisy nature of the deposition and diffusion processes. Second, the rise in 4± by about a factor of 2 at the onset of mounding in Fig. 4 is absent in the SOS model (Fig. 6). This reflects a "bulging out" of the ripples which occurs in the continuum equation, and which leads to mounds which are roughly twice as wide as the ripples prior to mounding. Possibly such a process is suppressed in the SOS model due to the anisotropy of the surface stiffness (see the Appendix). Quantitatively, the scaling exponent describing the asymptotic increase of lateral length scales is quite similar in the two models, 1/4 versus 0.21, while the exponents describing the steepening and the increase of the surface width are considerably smaller in the SOS model. This might indicate that the SOS model would be more appropriately

Homoepitaxy has been examined by two methods in this work: (i) by means of a continuum equation, and (ii) by a lattice (SOS) model. They are suitable for two opposing limits of the problem. The SOS model is easy to handle and yields good results in the limit of strong step-edge barriers where the time scales and lengths for unstable growth phenomena are quite small. In the other limit the capillarity length is large and a continuum description is appropriate. For both models the typical form of the surface (the correlation function) is similar, and the transition from ripples to mounds occurs in the same manner. Since the two models refer to rather different parameter regimes, this gives us confidence that the observed phenomenology is robust and relevant to real surfaces. It is encouraging that recent experimental results for metal homoepitaxy [30] seem to correspond to the scenario discussed in the present paper. After depositing 5 ML of P t / P t ( l l l ) at T=400 K and F = 0.005 ML s-1, a locally vicinal surface (a step bunch) breaks up into fingers and small mounds begin to form at some places (see Fig. 8). The pattern closely resembles those shown in Fig. 1, Figs. 2 and 3. Clearly, systematic experimental investigations of instabilities in step flow growth would be most welcome. Two final remarks are in order. First, we have seen that the (defective) transverse ripple morphology plays a central role in nucleating the global instability of a vicinal surface in two dimensions; this suggests that the noise-induced destabilization mechanism found for one-dimensional surfaces [22] is of minor importance here. Second, it should be emphasized that our results rely strongly on the absence of evaporation. Desorption introduces non-conservative nonlinearities into the continuum equations (Eq. (1) and

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M. Rost et al./Surface Science369 (1996) 393-402

Eq. (4)) [11], leading to a spatio-temporally chaotic behavior of the Kuramoto-Sivashinsky (KS) type [25,26] rather than to coarsening mound morphologies. More precisely, for vicinal surfaces one obtains an anisotropic KS equation which exhibits both chaotic and coarsening regimes [27]. Saito and Uwaha [28] have carried out growth simulations in the presence of desorption, which produce irregular step patterns consistent with a chaotic dynamics.

Acknowledgements M.R. benefited from discussions with M. Schimschak, G. Schuckert and W. Wintz on partial differential equations. P.S. gratefully acknowledges financial support from the Alexander yon Humboldt foundation as well as stimulating discussions with D.E. Wolf. We thank T. Michely for informing us about the unpublished results of Ref. [30], as well as for the kind permission to reproduce the scanning tunneling micrograph of a P t ( l l l ) surface (Fig. 8). M.R. and J.K. acknowledge partial support from D F G within SFB 237 Unordnung und grosse Fluktuationen.

Appendix: Continuum parameters for the SOS model In Section 5 we have discussed the qualitative similarities and differences between the phenomenologies displayed by the continuum equation and the SOS model. It is, however, important to realize that a rigorous, quantitative correspondence between the two levels of description can be established, at least in principle, provided the dimensionful parameters in Eq. (1) which correspond to the SOS model are correctly identified. This will be sketched below; see Refs. [31,32] for a detailed discussion of a simple case. The length scales associated with the surface current (Eq. (2)) are the diffusion length ID and the capture length I_. The former can be estimated from the island density in the submonolayer regime, which gives /Dgl2.1 for parameter set I and /Dg14.3 for set II. Comparing this to

Fig. 8. Homoepitaxy of Pt(lll) after deposition of five monolayers at T=400 K and deposition rate F=0.005 ML s-I (see text). Compare also to Fig. 1, Fig. 2 and Fig. 3. Courtesy of T. Michely [30]. the capture length l_=exp(EB/kBT), we find l_/lD~0.96 for set I and l_/ID~5.4 for set II. Thus both cases correspond to (moderately) strong barriers, stronger for set II, and we can assume 1D----1D. The dimensionless slope is m = 1D/lo----2.02 and 2.38, respectively. The capillarity parameter t¢ and the associated length scale l~=(~c/F) 1/4 are defined in terms of equilibrium properties of the SOS surface. Specifically, t~= o~7,where o- is the collective adatom mobility and f is the surface stiffness [21]. For simplicity we will ignore the effect of the step edge barrier on the mobility, and use the expression o-= (ko/4) e x p [ - (Es + 2EN)/kBT],

(7)

which is exact in the absence of barriers [31]. The stiffness of a vicinal surface below the roughening transition is highly anisotropic, taking different values '711for deformations along the direction of vicinality, and ~± for transverse modulations. Both can be expressed in terms of the step free energy, as [34] (TckBr)2

(8)

where /~ and /~ are the step free energy per unit length and the step stiffness, respectively. At low temperatures the steps can be approximately described by the 1 + 1 dimensional SOS model, which yields the following explicit expressions for

M. Rost et al./Surface Science 369 (1996) 393-402

steps parallel to the (01) direction [31,35]

fl/kBT= K + ln(tanhK/2), fl/kBT= c o s h K - 1,

401

corrected for. A more detailed continuum theory which takes these effects into account is needed. (9)

with K = EN/2kBT The stiffness anisotropy is not taken into account in the continuum description (Eq. (1)); however, it can easily be included at least on the level of linear stability analysis, One finds that the initial ripple spacing is still given by Eq. (6), multiplied by the scale factor 1= ?riD with the "transverse" capillarity length l~ = (a~.I./F) 1/4. The expressions given above yield l~ ~ 3.98 for set I and l~ ~ 2.25 for set II, hence the initial ripple spacings are predicted to be 2/~26.2 and 8.1, respectively. Comparing this to our numerical estimates of 22 and 12.5, the agreement is seen to be quite good for set I and reasonable for set II; as has been noted before [8,9,22], the continuum description is more accurate the larger the capillarity length and/or the weaker the barrier. With reference to the latter point, it is interesting to note that the large difference between the initial ripple spacing in the two cases can be explained by the continuum theory without taking into account the different barrier strengths, which do not enter into the expressions used above [33]. The straightforward extrapolation of these considerations into the nonlinear regime does not lead to satisfactory results. For example, using the previous estimate for l~ to determine the rescaling factors required to pass from Eqs. (1) to (4) we find, for parameter set II, A=0.0247, /=0.35 and Fz=6.1 × 10 -4. The last value implies that the growth of a single monolayer would correspond to a time interval of about 1600 in dimensionless units. This factor is more than one order of magnitude larger than the factor 120 determined in Section 5 by comparing typical morphologies. The failure is perhaps not too surprising, for two reasons. First, the temporal rescaling factor F'c=(l~/Io) 4 is very sensitive to uncertainties in our (crude) estimates of l~ and lo. Second, in the nonlinear regime the neglect of the stiffness anisotropy, in particular the divergence of ff at the singular orientation [34] in the continuum equation has severe consequences which are not easily

References [1] G.W. Smith, A.J. Pidduck, C.R. Whitehouse, J.L. Glasper and J. Spowart, J. Cryst. Growth 127 (1993) 996. C. Orme, M.D. Johnson, K.-T. Leung, B.G. On', P. Smilaner and D. Vvedensky, J. Cryst. Growth 150 (1995) 128. [2] M.D. Johnson, C. Orme, A.W. Hunt, D. Graft, J. Sudijouo, L.M. Sander and B.G. Orr, Phys. Rev. Lett. 72 (1994) 116. [3] H.-J. Ernst, F. Fabre, R. Folkerts and J. Lapujoulade, Phys. Rev. Lett. 72 (1994) 112. [4] J.E. Van Nostrand, S.J. Chey, M.-A. Hasan, D.G. Cahill and J.E. Greene, Phys. Rev. Lett. 74 (1995) 1127. [5] K. Tht~rmer, R. Koch, M. Weber and K.H. Rieder, Phys. Rev. Lett. 75 (1995) 1767. [6] J.A. Stroscio, D.T. Pierce, M. Stiles, A. Zangwill and L.M. Sander, Phys. Rev. Lett. 75 (1995) 4246. [7] F. Tsui, J. Wellman, C. Uher and R. Clarke, Phys. Rev. Lett. 76 (1996) 3164. [8] A review is given in J. Krug, Adv. Phys., to be published. [9] P. Smilauer and D.D. Vvedensky, Phys. Rev. B 52 (1995) 14263. [10] M. Siegert and M. Plischke, Phys. Rev. E 53 (1996) 307. [11] J. Villain. J. Phys. I 1 (1991) 19. [12] J. K_rug, M. Plischke and M. Siegert, Phys. Rev. Lett. 70 (1993) 3271. [13] G. Ehrlich and F.G. Hudda, J. Chem. Phys. 44 (1966) 1039. [14] R.L. Schwoebel and E.J. Shipsey, J. Appl. Phys. 37 (1966) 3682. R.L. Schwoebel, J. Appl. Phys. 40 (1969) 614. [15] J. Villain, A. Pimpinelli, L. Tang and D.E. Wolf, J. Phys. I France 2 (1992) 2107. [16] G.S. Bales and A. Zangwill, Phys. Rev. B 41 (1990) 5500. [17] A. Pimpinelli, I. Elkinani, A. Karma, C. Misbah and J. Villain, J. Phys.: Condens. Matter 6 (1994) 2661. [18] M. Siegert and M. Plischke, Phys. Rev. Lett. 73 (1994) 1517. [19] M. Siegert, in: Scale Invariance, Interfaces and NonEquilibrium Dynamics, Eds. A.J. McKane, M. Droz, J. Vannimenus and D.E. Wolf (Plenum, New York, 1995) p. 165. [20] A.W. Hunt, C. Orme, D.R.M. Williams, B.G. Orr and L.M. Sander, Europhys. Lett. 27 (1994) 611. [21] W.W. Mullins, J. Appl. Phys. 30 (1959) 77. [22] J. Krug and M. Schimschak, J. Phys. I France 5 (1995) 1065. [23] W.K. Burton, N. Cabrera and F.C. Frank, Philos. Trans. R. Soc. London A 243 (1951) 299. [24] M. Rost and J. Krug, to be published. [251 I. Bena, C. Misbah and A. Valance, Phys. Rev. B 47 (1993) 7408.

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M. Rostet al./Surface Science 369 (1996) 393-402

1,26] M. Sato and M. Uwaha, Europhys. Lett. 32 (1995) 639. 127] M. Rost and J. Krug, Phys. Rev. Lett. 75 (1995) 3894. 1,28] Y. Saito and M. Uwaha, Phys. Rev. B 49 (1994) 10677. 129] A different definition of the step-edge barrier such that instead of counting the next-nearest neighbors there is an additional barrier EB for all inter-layer hops, leads to a suppression of shape fluctuations of ripples and a postponement of the onset of mounding. Enhanced diffusion along step edges (P. Smilauer, M.R. Wilby, and D.D. Vvedensky, Suf. Sci. 291 (1993) L733) causes no observable changes. [30] S. Esch, T. Michely and G. Comsa, unpublished results. 131] J. Krug, H.T. Dobbs and S. Majaniemi, Z. Phys, B 97 (1995) 281. 1,32] S. Majaniemi, T. Ala-Nissila and J. Krug, Phys. Rev. B 53 (1996) 8071. 1'33] Of course, this is only true ff lD~ I'D.As shown in Section 2,

for weak enough barriers, [D decreases with the capture length l_ (and therefore with the barrier strength). Since we assume the capillarity length to be independent of the barrier, the lateral length scale l = ~/TDand the initial ripple spacing should increase with decreasing barrier strength. We verified this behavior in the SOS model for both parameter sets I and II, but with different (much smaller) values for EB. On the other hand, varying large values of EB changes the timescale z, whereas I remains practically unchanged. This cannot be explained simply by rescaling, we assume also the shape function f(s) for the nonequilibrium current (see Eq. (3)) to be influenced. 1,34] P. Nozi~res, in: Solids far from Equilibrium, Ed. C. Godrrche (Cambridge University Press, Cambridge, 1991). 1,,35] H.J. Leamy, G.H. Gilmer and K.A. Jackson, in: Surface Physics of Materials, Ed. J.M. Blakely (Academic Press, New York, 1975).