Calculation of surface phonons and resonances: the matching procedure revisited: I

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Abstract
A detailed account is given of the matching procedure devised to calculate surface phonon and resonance dispersion curves. Each kind of phonon symmetry is studied separately. A secular equation is first derived which is shown to contain in particular all the information on the dispersion curves and density of states of the bulk phonons along the reciprocal space direction not contained in the two-dimensional subspace associated with the surface. The dispersion curves of the surface phonons and resonances are obtained within a single framework by matching the evanescent and travelling solutions, respectively, of the secular equation, satisfying the boundary conditions brought about by the surface. A surface phonon is found to be the sum of evanescent waves. Bulk phonons of the same frequency travelling to and away from the surface are related to one another by reflection coefficients, for which sum rules are derived. An initial condition describing the scattering of an incoming bulk wave by the surface is introduced. The resonance frequency and the behaviour of the surface displacement spectral density in the vicinity of a bulk Van Hove singularity are shown to depend upon this initial condition.