Spontaneous Ordering of Arrays of Coherent Strained Islands

The energetics of an array of three-dimensional coherent strained islands on a lattice-mismatched substrate is studied. The contribution of the edges of islands to the elastic relaxation energy always has a minimum as a function of the size of an island $L$, and the total energy $E(L)$ may have a minimum at an optimum size ${L}_{\mathrm{opt}}$. Among different arrays of islands on the (001) surface of a cubic crystal, the total energy is minimum for the 2D periodic square lattice with primitive lattice vectors along ``soft'' directions [100] and [010]. This is a stable array of islands which do not undergo ripening.