Self-consistent perturbation theory for the vibrational spectra of certain disordered systems

The effect of coupling parameter fluctuations (quantitative disorder) on the vibrational spectrum of a solid is investigated. Since small deviations from crystalline order already turn out to lead to considerable broadening of the crystalline spectrum, a self-consistent perturbation scheme is developed similar to the coherent potential approximation for electronic systems. An approximate equation is obtained in which disorder is described in terms of a single dimensionless parameter which is a measure of the mean square fractional deviations of the coupling parameters from their average values. The equation is solved self-consistently by iteration. The final result is a convolution expression for the amorphous spectrum in which all van Hove singularities are smeared out except the ones at zero frequency and at the upper cutoff of the spectrum. Numerical results are obtained for silicon using a Keating crystalline density of states and assuming that the average values of the coupling parameters are identical to the crystalline ones. The maxima of the crystalline spectrum are broadened but qualitatively retained to a larger extent than in a simple convolution of the spectrum with a Gaussian.