Theory of strain relaxation in heteroepitaxial systems

We introduce a general approach to calculating the morphological consequences of coherent strain relaxation in heteroepitaxial thin films based on lattice statics using linear elasticity. The substrate and film are described by a simple cubic lattice of atoms with localized interactions. The boundary conditions at concave and convex corners that appear as a result of this construction, those along straight interfacial segments, and the governing equations are obtained from a variational calculation applied to a discretized form of the total elastic energy. The continuum limit of the equations and the boundary conditions along straight boundaries reproduces standard results of elasticity theory, but the boundary conditions at corners have no such analog. Our method enables us to calculate quantities such as the local strain energy density for any surface morphology once the lattice misfit and the elastic constants of the constituent materials are specified. The methodology is illustrated by examining the strain, displacement, and energies of one-dimensional strained vicinal surfaces. We discuss the effects of epilayer thickness on the energy of various step configurations and suggest that coupling between surface and substrate steps should affect the equilibration of the surface toward the bunched state.