Renormalization of stochastic lattice models: Epitaxial surfaces

We present the application of a method [C. A. Haselwandter and D. D. Vvedensky, Phys. Rev. E 76, 041115 (2007)] for deriving stochastic partial differential equations from atomistic processes to the morphological evolution of epitaxial surfaces driven by the deposition of new material. Although formally identical to the one-dimensional (1D) systems considered previously, our methodology presents substantial additional technical issues when applied to two-dimensional (2D) surfaces. Once these are addressed, subsequent coarse-graining is accomplished as before by calculating renormalization-group (RG) trajectories from initial conditions determined by the regularized atomistic models. Our applications are to the Edwards-Wilkinson (EW) model [S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London, Ser. A 381, 17 (1982)], the Wolf-Villain (WV) model [D. E. Wolf and J. Villain, Europhys. Lett. 13, 389 (1990)], and a model with concurrent random deposition and surface diffusion. With our rules for the EW model no appreciable crossover is obtained for either 1D or 2D substrates. For the 1D WV model, discussed previously, our analysis reproduces the crossover sequence known from kinetic Monte Carlo (KMC) simulations, but for the 2D WV model, we find a transition from smooth to unstable growth under repeated coarse-graining. Concurrent surface diffusion does not change this behavior, but can lead to extended transient regimes with kinetic roughening. This provides an explanation of recent experiments on Ge(001) with the intriguing conclusion that the same relaxation mechanism responsible for ordered structures during the early stages of growth also produces an instability at longer times that leads to epitaxial breakdown. The RG trajectories calculated for concurrent random deposition and surface diffusion reproduce the crossover sequences observed with KMC simulations for all values of the model parameters, and asymptotically always approach the fixed point corresponding to the equation proposed by Villain [J. Phys. I 1, 19 (1991)] and by Lai and Das Sarma [Phys. Rev. Lett. 66, 2899 (1991)]. We conclude with a discussion of the application of our methodology to other growth settings.

Figure 1

(Color online) RG flow trajectory of the 1D EW model obtained from Eqs. (38, 39, 40, 41, 42) with the initial conditions in Table with ${\nu }_4={\nu }_6=0$: (a) RG flow of the deterministic part of Eq. (20) and (b) RG flow of the amplitude of fluctuations with ${\Gamma }_2=0$ at every point along the RG trajectory. After a transient regime the EW fixed point is approached asymptotically. The curves in panels (a) and (b) are obtained with $0\le \ell \le 4$. Superimposed on the RG trajectories are points separated by a logarithmic scale of $\Delta \ell =1/5$.

Figure 2

(Color online) RG flow trajectory of the 2D EW model obtained from Eqs. (38, 39, 40, 41, 42) with the initial conditions in Table with ${\nu }_4={\nu }_6=0$: (a) RG flow of the deterministic part of Eq. (20) and (b) RG flow of the amplitude of fluctuations with ${\Gamma }_2=0$ at every point along the RG trajectory. After a transient regime the EW fixed point is approached asymptotically. The curves in panels (a) and (b) are obtained with $0\le \ell \le 4$. Superimposed on the RG trajectories are points separated by a logarithmic scale of $\Delta \ell =1/5$.

Figure 3

Plan view of the local environment relevant for the 2D WV model. The initial deposition site is $\left(i,j\right)$, and any of the sites marked with a circle may be selected as the final deposition site.

Figure 4

(Color online) RG flow trajectory of the 2D WV model obtained from Eqs. (38, 39, 40, 41, 42) with the initial conditions in Table : (a) RG flow of the deterministic part of Eq. (20) and (b) RG flow of the amplitude of fluctuations with ${\Gamma }_2=0$ at every point along the RG trajectory. The trajectory starts off not too far from the MH fixed point, but eventually crosses over to a regime for which $r>1$ and $u_2>0$, and becomes unstable. The range of the flow parameter $\ell$ is the same in panels (a) and (b). Superimposed on the trajectory are points separated by a logarithmic scale of $\Delta \ell =1/6$.

Figure 5

Atomic force micrographs of the surface of Ge(001) during low-temperature homoepitaxial growth at $T=155°\text{C}$ and a growth rate of $0.5\text{Å}{\text{s}}^{-1}$. Film thicknesses $\mathit{h}$ and black-to-white grayscales $\Delta z$ are (a) $h=70\text{Å}$, $\Delta z=40\text{Å}$, (b) $h=500\text{Å}$, $\Delta z=70\text{Å}$, (c) $h=3500\text{Å}$, $\Delta z=150\text{Å}$, (d) $h=7500\text{Å}$, $\Delta z=250\text{Å}$, (e) $h=8100\text{Å}$, $\Delta z=400\text{Å}$, and (f) $h=1.1\mu \text{m}$, $\Delta z=1000\text{Å}$. The insets show 2D slope histograms, ranging over $\pm 25°$ in the $\mathit{x}$ and $\mathit{y}$ directions, indicating the directions of surface normal vectors. After Ref. [30].

Figure 6

Plan view of a local height configuration producing an upward jump in the 2D WV model. The initial deposition site is the central site indicated by the dashed circle, the final deposition site is indicated by the solid circle, and the numbers in each square denote the surface height at this lattice site disregarding the newly deposited particle.

Figure 7

(Color online) RG trajectories for random deposition with surface diffusion obtained from Eqs. (38, 39, 40, 41, 42) with the initial conditions in Eqs. (67, 68, 69, 70, 71) for $d=2$ with [63] ${\nu }_0=5\times {10}^{12}{\text{s}}^{-1}$, $E_S=1.3\text{eV}$, $E_N=0.24\text{eV}$, ${\tau }_0=1\text{s}$, and (a) $T=500\text{K}$ and (b) $T=750\text{K}$. We have $r=1$ and $u_1=0$ everywhere along the RG trajectories. The RG flow is always directed toward the VLDS fixed point and the points superimposed on the trajectories are separated by a logarithmic scale of $\Delta \ell =1/4$.

Figure 8

(Color online) RG flow trajectories of the 1D WV model with surface diffusion for (a) $T=300\text{K}$ and (b) $T=500\text{K}$ obtained from Eqs. (38, 39, 40, 41, 42) with the initial conditions in Table and Eqs. (67, 68, 69, 70, 71) with ${\tau }_0\to \infty$ and $\delta ={10}^{-4}$. For thermally activated diffusion we use ${\nu }_0=5\times {10}^{12}{\text{s}}^{-1}$, $E_S=0.9\text{eV}$, and $E_N=0.2\text{eV}$. In both (a) and (b) the RG flow is directed away from the MH or cMH fixed point, and superimposed on the RG trajectories are points separated by $\Delta \ell =1/3$. The range of the flow variable $\ell$ is identical in the left and right panels, with the exception of the plots of ${\Gamma }_2$ vs ${\mathit{u}}_2$, in which the RG trajectories are only plotted up to the turning point away from the VLDS fixed point.

Figure 9

(Color online) RG flow trajectories of the 2D WV model with surface diffusion for (a) $T=400\text{K}$ and (b) $T=500\text{K}$ obtained from Eqs. (38, 39, 40, 41, 42) with the initial conditions in Table and Eqs. (67, 68, 69, 70, 71) with ${\tau }_0\to \infty$ and $\delta =1$. For thermally activated diffusion we use ${\nu }_0=5\times {10}^{12}{\text{s}}^{-1}$, $E_S=0.9\text{eV}$, and $E_N=0.2\text{eV}$. In both (a) and (b) the RG flow is directed toward increasing values of $\mathit{r}$, and superimposed on the RG trajectories are points separated by $\Delta \ell =1/6$. The range of the flow variable $\ell$ is identical in the left and right panels. The shaded region corresponds to $r>1$ and $u_2>0$, for which Eqs. (33, 35) imply ${\nu }_2<0$ and ${\nu }_4>0$.

Figure 10

Plan view of the relevant height variables for the 2D WV model. The initial deposition site has height $\left(H_{i,j},D\right)$, and any of the sites marked with a circle may be selected as the final deposition site.