The pyramid-to-dome transition in Ge$_{x}$Si$_{1-x}$ on Si(100) initiated by step bunching on pyramidal quantum dots is atomistically simulated using a novel multi-state lattice model incorporating effective surface reconstructions. Results are explained by a simple theory based on a shallow island approximation. Under given deposition conditions in $d$ dimensions, the shape transition is shown to occur at island size $n_c$ following $n_c^{1/d} propto x^{-zeta}$ independent of temperature and deposition rate, where $zetaalt 2$ and $x$ is the actual Ge concentration in the island. The transition has an energy barrier dominated by the facet interface energy. Fast deposition however can out-run and delay the transition to larger island sizes.
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