Optical-phonon modes in a double heterostructure of polar crystals

The equation of motion for the polarization vector for a double heterostructure of polar crystals is solved exactly within the framework of the continuum model. There exist only two types of phonon modes, the interface modes and the confined bulk modes, whose eigenvectors are obtained explicitly. Dispersion relations are derived analytically for the interface modes, while the confined bulk modes are dispersionless, a fact consistent with the model. It is also found that in Raman scattering experiments the symmetric interface modes are predominantly longitudinal optical (LO) and the antisymmetric interface modes transverse optical (TO). In the central region of the Brillouin zone, however, they both split into two branches oscillating at LO and TO frequencies, respectively. Possible reinterpretation of various experiments is briefly discussed.