Confined optical phonons in Ge/Si/Ge semiconductor quantum-wells: An improved continuum approach

Confined optical phonons are discussed for a semiconductor quantum-well based on Si/Ge by using a theory developed some years ago. In the present treatment this theory is adapted to a nonpolar material and generalized to the case when the phonon dispersion law involves both linear and quadratic terms in the wave vector. We apply a continuous medium model leading to a system of coupled differential equations describing oscillations of mixed nature. The solutions of the fundamental equations are found by taking advantage of analogous works on the subject. We are thus led to a description of long wavelength optical phonons allowing close comparisons with both experimental data and calculations along atomistic models. Dependence of the phonon frequency on the wave vector, the phonon dispersion curves, are discussed in detailed form considering two different matching approximations: the complete and the incomplete matching problems.