Equilibrium and dynamics of strained islands

We focus in this work on the effect of the surface energy anisotropy on an elastically strained semiconductor film and in particular on its role on the coarsening dynamics of elastically strained islands. To study the dynamics of a strained film, we establish a one-dimensional nonlinear and nonlocal partial differential equation which takes into account the elastic, capillary, wetting, and anisotropic effects. We first construct an approximate stationary solution of our model using a variational method and an appropriate ansatz. This stationary solution is used to compute the chemical potential dependence on the island height. In particular, we find that the surface energy anisotropy increases the convexity of the chemical potential and this is shown to have an effect on the driving force for the coarsening. Second, we study the coarsening dynamics of an islands pair by means of numerical simulations. We find that the presence of the surface energy anisotropy may increase or decrease the coarsening time of the system. We show that this phenomenon depends on the initial heights of island pairs. We thus highlight that the driving force for the coarsening is due to the variation of the chemical potential with respect to the islands height and that two different regimes are possible.

Figure 1

Sketch of the system: $h(x,t)$ is the height of the free surface of the film with respect to the substrate.

Figure 2

Plot of the surface energy anisotropy $\gamma (h,h_x)/{\gamma }_f$ given in Eqs. (5) and (6) as a function of the slope $h_x$. The wetting layer potential ${\gamma }_w\left(h\right)$ having no dependance on $h_x$ is not represented here. The anisotropy parameters are $\alpha =0.01$ and ${\theta }_e=\pi /9$. The minimum of the surface energy is chosen at a value of $h_x=\pm tan\left({\theta }_e\right)$. The anisotropy strength $\alpha$ represents the amplitude of the perturbation. The vertical axis is in unit of ${\gamma }_f$.

Figure 3

Surface stiffness $\tilde{\gamma }=[\gamma (h,h_x)+{\gamma }^{″}(h,h_x)]//{\gamma }_f$ obtained using Eqs. (5) and (6) as a function of $h_x$. The wetting layer potential ${\gamma }_w\left(h\right)$ having no dependance on $h_x$ is not represented here. For $h_x<0$ the surface stiffness is an even function of the slope $h_x$. The horizontal (blue color online) curve represents the surface stiffness for the isotropic case ($\alpha =0$). The (orange color online) curve represents the stiffness for the anisotropic case ($\alpha =0.01$ and ${\theta }_e=\pi /9$). The surface stiffness is always positive so that no missing orientations takes place. The vertical axis is in unit of ${\gamma }_f$.

Figure 4

Stationary solutions obtained by the numerical simulation of Eq. (15) (dots) compared to the ansatz proposed in Eq. (23) (continuous curve) for the isotropic case $\alpha =0$. The height of the island is represented by $h_0$, the height of the wetting layer is represented by $h_w$ and $x_1$ is the half-width of the island. The initial condition is given by a small random perturbation around a constant value of $h=0.1$. The value of the surface is $S_T=3.25$. We use as wetting parameters $c_w=0.05$ and $\delta =0.005$. The system size is $L=16$. The horizontal and vertical axes are in units of $l_0$.

Figure 5

Energy of the system given in Eq. (3) as a function of the island height $h_0$ and the island width $x_1$. The wetting parameters are $c_w=0.05$ and $\delta =0.005$, and the anisotropic parameters are $\alpha =0.01$ and ${\theta }_e=\pi /9$. The system size is $L=16$ and the total surface is $S_T=3$. The horizontal and vertical axes are in units of $l_0$. The energy is in unit of ${\gamma }_f^2/{\mathcal{E}}_0$.

Figure 6

Isotropic case: $\alpha =0$, $x_1$ is the half-width of the island is almost constant with respect to $h_0$. Blue-curve bottom ($S_T=1.5$), orange ($S_T=3.9$), color online. The islandlike solutions (dots) resulting from the numerical simulations of Eq. (15) are compared to the ansatz proposed in Eq. (23) represented by the continuous curve. The wetting parameters are $c_w=0.05$ and $\delta =0.005$, the system size is $L=16$. The horizontal and vertical axes are in units of $l_0$. Inset: Island half-width $x_1$ as a function of the island height $h_0$ for the isotropic case ($\alpha =0$). The blue dots represents the numerical results of Eq. (15) surface value between $S_T=1.45$ and $S_T=8.21$. The solid blue curve is the prediction obtained using Eqs. (28) and (29) for the island half-width $x_1$. The straight line $x_1=\frac{9\pi }{8}$ is the value obtained in Ref. [32] in which we have used a linear approximation of the film curvature. The horizontal and vertical axes are in units of $l_0$.

Figure 7

Anisotropic case: $x_1$ increases with respect to $h_0$. Islandlike solutions resulting from the time evolution of Eq. (15). Numerical simulations of Eq. (15) represented by dots, compared to the ansatz proposed in Eq. (23) represented by the continuous curve. We use as wetting parameters $c_w=0.05$ and $\delta =0.005$. The anisotropic strength is $\alpha =0.01$ and ${\theta }_e=\pi /9$. The system size is $L=16$. From bottom to top: blue-curve bottom ($S_T=1.5$), orange ($S_T=3.9$), color online. The horizontal and vertical axes are in units of $l_0$. Inset: Island half-width $x_1$ as a function of the island height $h_0$ for the anisotropic case ($\alpha =0.01$ and ${\theta }_e=\pi /9$). The results of the numerical results of Eq. (15) are represented by dots for different values of the surface. The surface varies from a value of $S_T=1.45$ to a value of $S_T=8.21$. The solid curve is the prediction obtained using Eqs. (28) and (29) for the island half-width $x_1$. The horizontal and vertical axes are in units of $l_0$.

Figure 8

Chemical potential $\mu$ as a function of the island height $h_0$ for the isotropic and anisotropic case. The black dots are obtained by the numerical simulations of Eq. (15) for $\alpha =0$ and the red squares are obtained by the numerical simulations for $\alpha =0.01$. The continuous black curve and the red dashed curve are obtained using the analytical predictions of Eqs. (28, 29, 30), respectively, for $\alpha =0$ and $\alpha =0.01$. As shown in this figure, the convexity of the chemical potential is larger for $\alpha =0.01$ (red curve) than for $\alpha =0$ (black curve). The value of the preferential slope is ${\theta }_e=\pi /9$. The horizontal axis is in units of $l_0$ and the vertical axis is in units of ${\gamma }_f$.

Figure 9

Coarsening driving force: absolute value of the derivative of the chemical potential $\mu$ with respect to the island height $h_0$. The black disks and the red squares are obtained respectively from the numerical simulations of Eq. (15) for $\alpha =0$ and $\alpha =0.01$. The continuous black curve ($\alpha =0$) and the red dashed curve ($\alpha =0.01$) are obtained using Eqs. (28, 29, 30). The critical height $h_c$ is defined as the point at which the black curve and the red dashed curve intersect. At this point $h_c$, the two tangents to the the curves displayed on Fig. 8, have the same slope. The value of the preferential slope is ${\theta }_e=\pi /9$. The horizontal axis is in units of $l_0$ and the vertical axis is in units of ${\gamma }_f$.

Figure 10

Island height $h_0$ as a function of the surface $S_T$. The black dots and the red squares represent, respectively, the results of the numerical simulations of Eq. (15) for $\alpha =0$ and $\alpha =0.01$. The black solid curve ($\alpha =0$) and the red dashed curve ($\alpha =0.01$) are obtained using the resolution of Eqs. (24), (28), and (29). The system size is $L=16$ for all the simulations. The horizontal axis is in units of $l_0^2$ and the vertical axis is in units of $l_0$.

Figure 11

Spatiotemporal evolution of two islands computed by the numerical simulation of Eq. (15). The initial condition are two islands of height $h_1=0.51$ and $h_2=0.49$ separated by a distance $d=L/2$, where $L=64$ represents the system size. The anisotropic parameters are $\alpha =0.01$ and $\theta =\pi /9$. After a time $t_c=69$, only one island remains in the system. The horizontal and vertical axes are in units of $l_0$.

Figure 12

Time evolution of the islands heights $h_1\left(t\right)$ and $h_2\left(t\right)$. We perform five numerical simulations of Eq. (15) for different values of the anisotropy strength $\alpha$ (blue dot $\alpha =0$, orange square $\alpha =0.0025$, green rhombus $\alpha =0.005$, red up-pointing triangle $\alpha =0.0075$, and violet down-pointing triangle $\alpha =0.01$). We observe that as the anisotropy increases, the coarsening time decreases. The initial system surface is $S_T=2.91$ for the five numerical simulations. The horizontal axis is in units of $t_0$ and the vertical axis is in units of $l_0$.

Figure 13

Time evolution of the island heights $h_1\left(t\right)$ and $h_2\left(t\right)$. We perform five numerical simulations of Eq. (15) for different values of the anisotropy strength $\alpha$ (blue dot $\alpha =0$, orange square $\alpha =0.0025$, green rhombus $\alpha =0.005$, red up-pointing triangle $\alpha =0.0075$, and violet down-pointing triangle $\alpha =0.01$). We observe that as the anisotropy increases, the coarsening time increases. The initial system surface is $S_T=4.15$ for the five numerical simulations. The horizontal axis is in units of $t_0$ and the vertical axis is in units of $l_0$.

Figure 14

Coarsening time $t_c$ as a function of the anisotropy strength $\alpha$ for two different values of $S_T$. The system under study consists of two islands similar to the one shown in Fig. 11. The decreasing curve (orange square) is obtained using the numerical simulations of Eq. (15) for $S_T=2.91$ (small islands). The blue curve (blue disk) is obtained using the numerical simulations of Eq. (15) for $S_T=4.15$ (large islands). The vertical axis is in units of $t_0$.