Green’s function-based multiscale modeling of defects in a semi-infinite silicon substrate

We have developed a Green’s function (GF) based multiscale modeling of defects in a semi-infinite silicon substrate. The problem—including lattice defects and substrate surface, i.e., an extended defect, at different length scales—is first formulated within the theory of lattice statics. It is then reduced and solved by using a scale-bridging technique based on the Dyson’s equation that relates a defect GF to a reference GF and on the asymptotic relationship of the reference lattice-statics GF (LSGF) to the continuum GF (CGF) of the semi-infinite substrate. The reference LSGF is obtained approximately by solving the boundary-value problem of a super-cell of lattice subject to a unit point force and under a boundary condition given by the reference CGF. The Tersoff potential of silicon, germanium and their compounds is used to derive the lattice-level force system and force constants and further to derive the continuum-level elastic constants (of the bulk silicon, needed in the reference CGF). We have applied the method to solve for the lattice distortion of a single vacancy and a single germanium substitution. We have further calculated the relaxation energy in these cases and used it to examine the interaction of the point defects with the (traction-free) substrate surface and the interaction of a single vacancy with a relatively large germanium cluster in the presence of the substrate surface. In the first case, the point defects are found to be attracted to the substrate surface. In the second case, the single vacancy is attracted to the germanium cluster as well as to the substrate surface.