Transient regimes and crossover for epitaxial surfaces

We apply a formalism for deriving stochastic continuum equations associated with lattice models to obtain equations governing the transient regimes of epitaxial growth for various experimental scenarios and growth conditions. The first step of our methodology is the systematic transformation of the lattice model into a regularized stochastic equation of motion that provides initial conditions for differential renormalization-group (RG) equations for the coefficients in the regularized equation. The solutions of the RG equations then yield trajectories that describe the original model from the transient regimes, which are of primary experimental interest, to the eventual crossover to the asymptotically stable fixed point. We first consider regimes defined by the relative magnitude of deposition noise and diffusion noise. If the diffusion noise dominates, then the early stages of growth are described by the Mullins-Herring (MH) equation with conservative noise. This is the classic regime of molecular-beam epitaxy. If the diffusion and deposition noise are of comparable magnitude, the transient equation is the MH equation with nonconservative noise. This behavior has been observed in a recent report on the growth of aluminum on silicone oil surfaces [Z.-N. Fang et al., Thin Solid Films 517, 3408 (2009)]. Finally, the regime where deposition noise dominates over diffusion noise has been observed in computer simulations, but does not appear to have any direct experimental relevance. For initial conditions that consist of a flat surface, the Villain-Lai-Das Sarma (VLDS) equation with nonconservative noise is not appropriate for any transient regime. If, however, the initial surface is corrugated, the relative magnitudes of terms can be altered to the point where the VLDS equation with conservative noise does indeed describe transient growth. This is consistent with the experimental analysis of growth on patterned surfaces [H.-C. Kan et al., Phys. Rev. Lett. 92, 146101 (2004); T. Tadayyon-Eslami et al., Phys. Rev. Lett. 97, 126101 (2006)].

Figure 1

(Color online) RG trajectories of the random deposition-surface diffusion model obtained from Eqs. (21, 22, 23) with the initial conditions in Eqs. (8, 9, 10, 11, 12) for $d=2$ using the parameter values $E_S=0.9\text{eV}$, ${\nu }_0=5\times {10}^{12}{\text{s}}^{-1}$, and ${\tau }_0=13.5\text{s}$ with (a) $E_N=0.1\text{eV}$ and the substrate temperature $T=300\text{K}$ used in the experiments in Refs. [27, 28], (b) $E_N=0.1\text{eV}$ and $T=500\text{K}$, (c) $E_N=0.01\text{eV}$ and $T=300\text{K}$, and (d) $E_N=1\text{eV}$ and $T=300\text{K}$. In all plots, the RG flow is directed toward the ${\text{VLDS}}^{+}$ fixed point, and superimposed on the RG trajectories are points separated by a logarithmic scale of $\Delta \ell =1/3$.

Figure 2

Schematic representation of the RG flow of the random deposition-surface diffusion model.

Figure 3

(Color online) RG trajectories of the random deposition-surface diffusion model obtained from Eqs. (21, 22, 23) with the initial conditions in Eqs. (8, 9, 10, 11, 12) for $d=2$ using the parameter values [30] ${\tau }_0=2\text{s}$, ${\nu }_0=5\times {10}^{12}{\text{s}}^{-1}$, $E_S=1.58\text{eV}$, and $E_N=0.24\text{eV}$ for the indicated temperatures with (a) $\delta =10$ and (b) $\delta =0.01$. Superimposed on the RG trajectories are points separated by a logarithmic scale of $\Delta \ell =1/3$. In the upper panel of (a) and the upper and lower panels of (b), the RG flow is directed toward the ${\text{VLDS}}^{+}$ fixed point, whereas in the lower panel of (a), the RG flow takes the system away from the ${\text{VLDS}}^{+}$ fixed point.