Langevin equations for fluctuating surfaces

Exact Langevin equations are derived for the height fluctuations of surfaces driven by the deposition of material from a molecular beam. We consider two types of model: deposition models, where growth proceeds by the deposition and instantaneous local relaxation of particles, with no subsequent movement, and models with concurrent random deposition and surface diffusion. Starting from a Chapman-Kolmogorov equation the deposition, relaxation, and hopping rules of these models are first expressed as transition rates within a master equation for the joint height probability density function. The Kramers–Moyal–van Kampen expansion of the master equation in terms of an appropriate “largeness” parameter yields, according to a limit theorem due to Kurtz [Stoch. Proc. Appl. 6, 223 (1978)], a Fokker-Planck equation that embodies the statistical properties of the original lattice model. The statistical equivalence of this Fokker-Planck equation, solved in terms of the associated Langevin equation, and solutions of the Chapman-Kolmogorov equation, as determined by kinetic Monte Carlo (KMC) simulations of the lattice transition rules, is demonstrated by comparing the surface roughness and the lateral height correlations obtained from the two formulations for the Edwards-Wilkinson [Proc. R. Soc. London Ser. A 381, 17 (1982)] and Wolf-Villain [Europhys. Lett. 13, 389 (1990)] deposition models, and for a model with random deposition and surface diffusion. In each case, as the largeness parameter is increased, the Langevin equation converges to the surface roughness and lateral height correlations produced by KMC simulations for all times, including the crossover between different scaling regimes. We conclude by examining some of the wider implications of these results, including applications to heteroepitaxial systems and the passage to the continuum limit.

Figure 1

Relaxation rules of the Edwards-Wilkinson model, with contributions to (a) $w_i^{\left(1\right)}$, (b) $w_i^{\left(2\right)}$, (c) $w_i^{\left(3\right)}$, and (d) to $w_i^{\left(2\right)}$ and $w_i^{\left(3\right)}$. The corresponding expressions are given in Eqs. (52, 53, 54). The arrows indicate the incident and deposition sites. In (d), both of the deposition sites are equally likely. The broken lines show where greater heights do not affect the deposition site.

Figure 2

Surface roughness obtained from the Langevin equations (23, 25) with Eqs. (52, 53, 54) and from KMC simulations for a system of length $L=1000$ for $\Omega =1$, 2, 20. Time is measured in units of monolayers (ML) deposited. The data were averaged over 500 independent realizations. The slopes of the straight lines correspond to the growth exponent of random deposition $(\beta =1∕2)$ and that of the Edwards-Wilkinson model $(\beta =1∕4)$.

Figure 3

Lateral height correlation function obtained from the Langevin equations (23, 25) with Eqs. (52, 53, 54) and KMC simulations up to $r=500$ for a system of size $L=1000$ at (a) $t=100\mathrm{ML}$ and (b) $t=5000\mathrm{ML}$ for $\Omega =1$, 2, 20. Data were obtained by averaging over 500 independent realizations. The slope of the straight line in (b) has the Edwards-Wilkinson value of the roughness exponent $(\alpha =1∕2)$.

Figure 4

Local height configurations that contribute to $w_i^{\left(1\right)}$ for the Wolf-Villain model. Arrows indicate the incident and deposition sites. Column heights strictly greater than and strictly equal to $h_i$ are as indicated; those less than or equal to $h_i$ are shown with broken lines.

Figure 5

Local height configurations that contribute to $w_i^{\left(2\right)}$ for the Wolf-Villain model. Arrows indicate the incident and deposition sites. Where more than one deposition site is obtained, both are equally likely. Column heights strictly greater than and strictly equal to $h_i$ are as indicated: those less than or equal to $h_i$ are shown with broken lines.

Figure 6

Comparison of surface roughness obtained from the lattice Langevin equation with $\Omega =1$ and KMC simulations for the Wolf-Villain model for a system of length $L=40000$. Scaling regimes [24, 62] are shown by straight lines whose slopes have the indicated values of the growth exponent $\beta$.

Figure 7

The correlation function $C\left(r\right)$ defined in Eq. (47) calculated from KMC simulations and the Langevin equation with $\Omega =1$ for the Wolf-Villain model for a system size of $L=40000$ at times (a) ${10}^4\mathrm{ML}$, (b) ${10}^6\mathrm{ML}$, and (c) ${10}^8\mathrm{ML}$.

Figure 8

Comparison between surface roughness obtained from the lattice Langevin equation with the indicated values of $\Omega$ and KMC simulations for a model with random deposition and surface diffusion for a system of length $L=100$. The results have been averaged over 500 independent realizations.

Figure 9

Relaxation of a sinusoidal profile in the absence of deposition at 600 K modeled by KMC simulations (solid lines) and the Langevin equation with $\Omega =20$ (broken lines) for a system with $L=40$ on which periodic boundary conditions are imposed. The initial profile is indicated by dots in (a). Profiles are shown at times (a) 3000, (b) 6000, (c) 9000, (d) 12000, (e) 15000, and (f) 18000. In each panel, the abscissa is the spatial position $(1⩽i⩽40)$ and the ordinate is the height $h_i$. The energy parameters are the same as those used in Fig. 8 and each result has been averaged over 50 independent realizations.

Figure 10

(a) The step function in Eq. (13), and (b) the regularization in Eq. (A1). The function in (b) has the same values for integer arguments, which are indicated by dots, as the function in (a) for $0<a⩽1$. The shaded regions show how the abrupt threshold behavior in (a) is smoothed by the regularization in (b).

Figure 11

Local deposition probabilities for the Edwards-Wilkinson model using (a) $\theta (x;1)$ and (b) $\theta (x;0.01)$, where $\theta (x;a)$ is defined in Eq. (A1). The randomly chosen site is denoted by $i$ and the deposition probabilities to each site are written at the top of the corresponding column. Height differences between the central and nearest-neighbor heights are shown at the sides of each configuration.

Figure 12

The roughness in Eq. (44) for the Edwards-Wilkinson model obtained by solving the Langevin equation with $\Omega =50$ for a system of size $L=20$ using the regularization $\theta (x;1)$ defined in Eq. (A1) and the original threshold function $\theta \left(x\right)$ in Eq. (13). Each data set was obtained from an average of 1500 realizations.

Figure 13

Local deposition probabilities for the Wolf-Villain model using (a) $\theta (x;1)$ and (b) $\theta (x;0.01)$. The randomly chosen site is denoted by $i$ and the transition probabilities to each site are written at the top of the corresponding column. Height differences between the central and nearest-neighbor heights are indicated at the sides of each configuration.

Figure 14

The roughness in Eq. (44) for the Wolf-Villain model obtained by solving the Langevin equation with $\Omega =50$ for a system of size $L=20$ using $\theta (x;1)$ and $\theta \left(x\right)$. Each set of data was obtained from an average of 1500 realizations.