Continuum treatment of phonon polaritons in semiconductor heterogeneous structures

Abstract

A phenomenological approach is applied to the theory of phonon polaritons in semiconductor heterogeneous structures, with special emphasis on semiconductor nanostructures. Applying the macroscopic approach to continuous media, seven coupled partial differential equations are derived for the fundamental quantities involved: the three components of the displacement field u, those of the magnetic potential A, and the electric potential \ensuremath{\varphi} in the Lorentz gauge. Our treatment is rather general in its conception: no assumptions on the system geometry and composition are made. We develop a general method allowing us to obtain the exact analytical solutions of the equations when the constituent materials can be assumed to be isotropic. The matching boundary conditions at the structure interfaces are derived from the differential equations and interpreted in physical terms. This theory leads to a phenomenological description of phonon polaritons valid in the long-wavelength limit. We apply it to the case of the double heterostructure, and calculate both the mechanical displacements u and the potentials A, \ensuremath{\varphi} of normal modes in the GaAs/AlAs prototype system. We also discuss the dispersion relations for these modes which are of transverse-electric and transverse-magnetic character. A comparison is made with some limiting cases: the unretarded case $(c\ensuremath{\rightarrow}\ensuremath{\infty})$ reproducing our previous results for polar-optical phonons, and the nondispersive case $({\ensuremath{\beta}}_{T}\ensuremath{\rightarrow}0),$ which leads to the Fuchs-Kliewer slab modes.