Numerical solution of the interrelated differential equation of motion in phonon engineering

Abstract

In this work, we study numeric calculations of phonon modes in nanostructures. The motion equation of atoms in a crystal with some simplification, results in a second order ordinary differential equation and two interrelated second order differential equations for 3 polarizations according to 3 dimensions. Although first equation can easily be solved, the next two interrelated equations cannot be solved by usual numerical methods. Based on discretization, a new technique is proposed for studying the motion equations. The results are presented by dispersion curves for shear, dilatational, and flexural modes of phonons.