Effective-modulus approximation for surface-vibration modes

Abstract

The effective-mass approximation enables one to obtain the energies and wave functions of impurity levels, in a slowly varying perturbing potential, associated with extrema in the electronic band structure of the unperturbed crystal. We have developed an analogous method for obtaining the dispersion relation and displacement field of acoustical or optical surface vibration modes associated with extremal points in the surfaces of constant frequency of the corresponding, infinitely extended crystal. By factoring out the rapidly varying part of the displacement field corresponding to each of the normal modes associated with such an extremal point, a set of partial difference equations is obtained for the slowly varying amplitudes. These equations are converted into a set of coupled partial differential equations which explicitly incorporate the stress-free boundary conditions at the crystal surface. The solution of these equations yields the surface-mode dispersion relation and displacement field. The method is illustrated by applying it to one- and three-dimensional semi-infinite models.