CONTINUUM TREATMENT OF PHONON POLARITONS IN SEMICONDUCTOR HETEROGENEOUS STRUCTURES

A phenomenological approach is applied to the theory of phonon polarit ons in semiconductor heterogeneous structures, with special emphasis o n semiconductor nanostructures. Applying the macroscopic approach to c ontinuous media, seven coupled partial differential equations are deri ved for the fundamental quantities involved: the three components of t he displacement field u, those of the magnetic potential A, and the el ectric potential phi in the Lorentz gauge. Our treatment is rather gen eral in its conception: no assumptions on the system geometry and comp osition are made. We develop a general method allowing us to obtain th e exact analytical solutions of the equations when the constituent mat erials can be assumed to be isotropic. The matching boundary condition s at the structure interfaces are derived from the differential equati ons and interpreted in physical terms. This theory leads to a phenomen ological description of phonon polaritons valid in the long-wavelength limit. We apply it to the case of the double heterostructure, and cal culate both the mechanical displacements u and the potentials A, phi o f 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 limi ting cases: the unretarded case (c-->infinity) reproducing our previou s results for polar-optical phonons, and the nondispersive case (beta( T)-->0), which leads to the Fuchs-Kliewer slab modes.