Green's-function calculation of the lattice response near the vacancy in silicon

The lattice response near a vacancy in covalent systems is determined using a Green's-function technique. The phonon Hamiltonian is taken from a valence-force-field description. The calculation is done by two different techniques, a fully numerical one and an analytic approximate one. This last method is shown to lead to a clear understanding as well as to a good precision. A first application is made to the vibration entropy of the vacancy in an undistorted environment, found to be of order $3k$. Then the effective force constants describing the response to forces applied on the neighbors of the vacancy are determined. They are found to lie midway between those of two previous cluster calculations which were differing by a factor of 2 to 3. Finally the influence of local changes in force constants is studied, and it is shown that they can lead to a dramatic increase in the lattice response and in the formation entropy.