Strained islands formed in heteroepitaxy sometimes change shape during grow
th. Here we show that there is typically a first-order shape transition wit
h island size, with the discontinuous introduction of steeper facets at the
island edge. We present a phase diagram for island shape as a function of
volume and surface energy, showing how surface energy controls the sequence
of island shapes with increasing volume. The discontinuous chemical potent
ial at the shape transition drastically affects island coarsening and size
distributions. [S0031-9007(99)08789-X].