Relaxation of compressed elastic islands on a viscous layer

A recent technique can fabricate SiGe thin film islands on a glass layer, which itself lies on a silicon wafer. The islands initially have an inplane compressive strain. Upon annealing, the glass flows, and the islands relax. The resulting strain-free islands are used as substrates to grow epitaxial optoelectronic devices. This paper models the annealing process. A small island relaxes by inplane expansion. The glass being viscous, the relaxation starts at the island edges, and propagates to the island center. A large island, however, wrinkles at the center before the inplane relaxation arrives. Further annealing gives rise to one of two outcomes. The wrinkles may disappear when the inplane relaxation arrives, leading to a flat, strain-free island. Alternatively, the wrinkles may cause significant tensile stress in the island, leading to fracture. We model the island by the von Karman plate theory, and the glass layer by the Reynolds lubrication theory. The solid and the fluid couple at the interface by the continuous traction and displacement. Numerical simulations evolve the inplane expansion and the wrinkles simultaneously. We determine the critical island size, below which inplane expansion prevails over wrinkling. This critical island size depends on several experimental variables, and is much larger than the Euler buckling wavelength.