Calculation of phonon dispersion in superlattices using the matching procedure

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Abstract
A method for calculating phonon dispersion curves of three-dimensional superlattices is presented. The eigenvalues of the dynamical matrix are obtained by matching the eigenvectors of two adjacent media along their common interface. The dispersion curves are obtained by solving a polynomial equation, the degree of which does not depend on the thickness of each medium, but is determined only by the range of interlayer coupling. This makes this method very competitive for large-period superlattices. The formalism is applied to the linear superlattice with two kinds of atoms coupled by nearest-neighbor interactions, for which the dispersion equation can be solved analytically. A formula for sound velocity is also included.