Aspect-ratio-dependent driving force for nonuniform alloying in Stranski-Krastanow islands

A semianalytical iterative method for determining the three-dimensional concentration profile minimizing the elastic energy in alloyed heteroepitaxial islands is developed. The method is shown to converge after only a few iterations (solutions of the elastic problem). Exploiting the reduced computational effort, aspect-ratio-dependent and average-concentration-dependent Si/Ge intermixing in alloyed SiGe/Si(001) islands is systematically investigated, considering realistic shapes. Islands providing the better elastic relaxation are also the ones able to lower the elastic load through nonuniform alloying more significantly. Universal behavior in terms of relaxation vs average concentration is demonstrated.

Figure 1

Typical concentration mesh, with focus on the external (a) and internal (b) nodes. When solving the elastic problem, a much finer mesh (c, d) is used. $\alpha$ and $\beta$ in panel b represent a possible pair of nodes randomly chosen during the MC-FEM procedure.

Figure 2

Scheme representing the improved MC-FEM method. An initial Ge distribution $c_0$ is chosen, and the elastic problem is solved yielding the elastic energy $E_0$ and the strain field. At this point, MC concentration changes are performed, by evaluating energy differences based solely on Eq. (7). In the figure, this corresponds to moving along the parabola passing through the point $\left(c_0,E_0\right)$. After $M$ accepted moves lowering the energy ($M=5$ in the figure where each filled circle represent an accepted move), a new concentration distribution $c_1$ is found. At this stage, the FEM solver is called again, leading to a proper estimate $E_1$ of the elastic energy (lower than the one found along the parabola). The procedure is iterated until convergence is reached, leading to the Ge distribution $c_{\text{min}}\left(x,y,z\right)$ minimizing the elastic energy $\left(E_{\text{min}}\right)$.

Figure 3

(Color online) Relative relaxation with respect to the uniform-concentration case as a function of the FEM calls during the iterative procedure. Results refer to a dome-shaped island with an average 50% Ge content. The final concentration profile minimizing the elastic energy (inset, where a top and a perspective view are shown) is found after $\sim 7000$ FEM calls using standard MC-FEM. Exploiting Eq. (5) and calling the FEM solver only every $M=1$ $\left(M=10\right)$ accepted moves, $\sim 2000$ $\left(\sim 200\right)$ FEM calls are sufficient to reach convergence.

Figure 4

Relative variation in the elastic energy with respect to the uniform case, as obtained using the semianalytical iterative procedure based on Eq. (10) for the very same system analyzed in Fig. 3. The dashed horizontal line corresponds to the convergent value obtained using standard MC-FEM.

Figure 5

Relative relaxation with respect to the uniform case produced by nonuniform intermixing minimizing the elastic energy for {105} pyramids (triangles), domes (full boxes), and barn-shaped islands (full circles). In panel a, results were obtained by considering $\overline{c}$-dependent elastic constants, while in panel b, pure-Ge ones were used.

Figure 6

(Color online) Distribution of Ge minimizing the elastic energy within a {105} pyramid with average Ge content $\overline{c}=0.3$ (left and right panel at the top), $\overline{c}=0.6$ (central), and $\overline{c}=0.9$ (bottom). Here and in the following three figures, a few concentration values are explicitly reported to make the plot clearer if printed in black and white.

Figure 7

(Color online) Distribution of Ge minimizing the elastic energy within a dome-shaped island with average Ge content $\overline{c}=0.3$ (left and right panel at the top), $\overline{c}=0.6$ (central), and $\overline{c}=0.9$ (bottom).

Figure 8

(Color online) Distribution of Ge minimizing the elastic energy within a barn-shaped island with average Ge content $\overline{c}=0.3$ (left and right panel at the top), $\overline{c}=0.6$ (central), and $\overline{c}=0.9$ (bottom).

Figure 9

(Color online) Distribution of Ge minimizing the elastic energy within a dome-shaped island, divided by the average concentration $\overline{c}$. Top left: $\overline{c}=0.1$. Top right: $\overline{c}=0.3$. Bottom left: $\overline{c}=0.6$. Bottom right: $\overline{c}=0.9$. Besides using a color scale, a few numerical values are reported.

Figure 10

(Color online) Left: elastic energy density $W$ in the full island $+$ Si-substrate system for a dome island with uniform Ge distribution $c\left(x,y,z\right)=0.3$. Right: $W$ for the same island, but with the Ge concentration minimizing the total elastic energy. In both cases, the island is artificially shown as slightly lifted from the substrate. We used this graphical artifact to make more evident that two different color scales (between 0 and $0.32\text{eV}/{\text{nm}}^3$ for the island, and between 0 and $0.1\text{eV}/{\text{nm}}^3$ for the substrate) are used to yield a clearer picture. The same scale, instead, is used to treat the uniform and the $c_{\text{min}}\left(x,y,z\right)$ case.

Figure 11

(Color online) Hydrostatic stress ${\sigma }_{xyz}$ for a dome-shaped island, after energy minimization. Four different average concentrations $\overline{c}$ are considered.