Shape and Composition Map of a Prepyramid Quantum Dot

We present a theory for the shape, size, and nonuniform composition profile of a small prepyramid island in an alloy epitaxial film when surface diffusion is much faster than deposition and bulk diffusion. The predicted composition profile has segregation of the larger misfit component to the island peak, with segregation enhanced by misfit strain and solute strain but retarded by alloy solution thermodynamics. Vertical composition gradients through the center of the island due to this mechanism are on the order of $2\%/\mathrm{nm}$ for ${\mathrm{Ge}}_X{\mathrm{Si}}_{1-X}/\mathrm{Si}$ and $10–15\%/\mathrm{nm}$ for ${\mathrm{In}}_X{\mathrm{Ga}}_{1-X}\mathrm{As}/\mathrm{GaAs}$.

Figure 1

Island shape and composition map. The scaled island shape is $H_0(r)/L$. The composition map is a contour plot of $C_B^1/\mathcal{C}$, the scaled deviation from the reference composition. The step between contours is 0.25, vertical lines are the zero contours, and dark corresponds to negative values. Results are for $\nu =0.276$.

Figure 2

Island width for ${\mathrm{Ge}}_X{\mathrm{Si}}_{1-X}$ on Si. The misfit is $ϵ=-\eta X$ and parameters used for GeSi are $\eta =0.042$, $\nu =0.276$, $E=11.6\times {10}^{11}\mathrm{erg}/{\mathrm{cm}}^3$, $\gamma =2220\mathrm{erg}/{\mathrm{cm}}^2$. Data from Refs. 9, 13, 17, 26, 27, 28, 29, 30, 31.

Figure 3

Vertical composition gradient $\mathcal{G}$ along the island center line for $\mathrm{GeSi}/\mathrm{Si}$ and $\mathrm{InGaAs}/\mathrm{GaAs}$. For $\mathrm{GeSi}/\mathrm{Si}$, $\Omega =3.47\times {10}^9\mathrm{erg}/{\mathrm{cm}}^3$; for $\mathrm{InGaAs}/\mathrm{GaAs}$, $\eta =0.069$, $\nu =0.332$, $E=6.82\times {10}^{11}\mathrm{erg}/{\mathrm{cm}}^3$, $\gamma =1650\mathrm{erg}/{\mathrm{cm}}^2$, and $\Omega =4.12\times {10}^9\mathrm{erg}/{\mathrm{cm}}^3$.