Free surface relaxation of a hexagonal network of misfit dislocations

Thin epitaxial layers deposited on a single crystalline substrate can be associated with networks of misfit dislocations. Using two different isotropic elasticities for the epilayer and the semi-infinite substrate, and appropriate boundary conditions at the heterointerface, it is shown how to derive closed analytic expressions for the complete three-dimensional elastic field of a hexagonal network of misfit dislocations, possibly non-regular, versus the thickness h of the epilayer. The boundary conditions are expressed by a set of nine linear equations with nine unknowns. To obtain an idea of the effects of inhomogeneities in μ and v and thickness h on the elastic field of the bimaterial, some normal stresses and elastic dilations have been investigated numerically for three heterotwin systems based on (111)CoSi2/(111)Si. These systems only differ by h ranging from 1 nm to 120nm and by the shear modulus of the epilayer taken as μCoSi2, 2μSi or μSi/2. The full bimaterial solutions are compared with the results obtained when the eleastic constants are taken to be those of Si throughout. In addition, two-dimensional cross-sections through the surface profiles of these heterosystems have been calculated for h varying from 2 to 12nm. They show that, for small h the free surface has pronounced bumps directly above the dislocation lines and triple nodes.