Nonlinear effect of stress and wetting on surface evolution of epitaxial thin films

An epitaxial thin film can undergo surface instability and break up into discrete islands. The stress field and the interface interaction have profound effects on the dynamics of surface evolution. In this work, we develop a nonlinear evolution equation with a second-order approximation for the stress field and a nonlinear wetting potential for the interface. The equation is solved numerically in both two-dimensional (2D) and three-dimensional configurations using a spectral method. The effects of stress and wetting are examined. It is found that the nonlinear stress field alone induces “blow-up” instability, leading to cracklike grooving in 2D and circular pitlike morphology in 3D. For thin films, the blow-up is suppressed by the wetting effect, leading to a thin wetting layer and an array of discrete islands. The dynamics of island formation and coarsening over a large area is well captured by the interplay of the nonlinear stress field and the wetting effect.

Figure 1

(Color online) Schematic of surface evolution of an epitaxial film. (a) Reference state. (b) Evolving state.

Figure 2

Two-dimensional simulation of surface evolution based on the linear analysis.

Figure 3

Three-dimensional simulation of surface evolution based on the linear analysis. (a)–(e) are gray scale contour plots of the thickness profile $h(x_1,x_2)$, white for crests and dark for troughs. (a) Random perturbation at $t=0$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=5.77\times {10}^{-5}$; (b) $t=20$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=7.40\times {10}^{-5}$; (c) $t=50$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=1.64\times {10}^{-3}$; (d) $t=75$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=2.39\times {10}^{-2}$; (e) $t=100$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=3.60\times {10}^{-1}$; (f) the Fourier spectrum of the surface pattern, which remains the same for (b)–(e).

Figure 4

Two-dimensional simulation of surface evolution by the nonlinear analysis with no wetting effect.

Figure 5

(Color online) Three-dimensional simulation of surface evolution based on the nonlinear analysis with no wetting effect. (a)–(e) are gray scale contour plots of the thickness profile $h(x_1,x_2)$, white for crests and dark for troughs. (a) Random perturbation at $t=0$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=5.77\times {10}^{-5}$; (b) $t=20$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=7.40\times {10}^{-5}$; (c) $t=50$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=1.64\times {10}^{-3}$; (d) $t=75$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=2.44\times {10}^{-2}$; (e) $t=88$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=1.24\times {10}^{-1}$; (f) the local 3D view of a circular pit at $t=88$.

Figure 6

(Color online) Linear analysis of the wetting effect: the growth rate versus the wave number for different film thickness. The critical thickness $h_c$ is defined by Eq. (36).

Figure 7

Two-dimensional simulation of surface evolution based on the nonlinear analysis of both stress and wetting effect: (a) stable growth $(0<t<200)$; (b) coarsening $(t>200)$.

Figure 8

Three-dimensional simulation of surface evolution based on the nonlinear analysis of both stress and wetting effect. (a)–(f) are gray scale contour plots of the thickness profile $h(x_1,x_2)$, white for crests and dark for troughs. (a) Random perturbation at $t=0$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=5.77\times {10}^{-5}$; (b) $t=20$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=2.07\times {10}^{-5}$; (c) $t=50$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=6.27\times {10}^{-5}$; (d) $t=220$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=4.86\times {10}^{-2}$; (e) $t=500$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=9.11\times {10}^{-2}$; (f) $t=5000$, $\mathrm{r}.\mathrm{m}.\mathrm{s}.=1.18\times {10}^{-1}$.

Figure 9

Comparison of the evolution of surface roughness from three-dimensional simulations using the linear equation (I), the nonlinear equation with no wetting (II), and the nonlinear equation with wetting (III).