INTERFACE OPTICAL-PHONON MODES IN A 4-LAYER HETEROSTRUCTURE OF POLAR CRYSTALS

The equations of motion for the p-polarization field in a four-layer h eterostructure (FLHS) of polar crystals are solved exactly for the int erface optical-phonon modes. The eigenvectors and the interface charge densities are obtained explicitly. The dispersion relations and their plots for a FLHS and its special cases, an asymmetric trilayer hetero structure (asymmetric single quantum well) and a step quantum well, ar e given and discussed. We find that there are six (not eight) frequenc y solutions for the interface optical-phonon modes in a FLHS and that, in the long-wavelength limit, the longitudinal and transverse modes i n the two side materials 1 and 4 with frequencies (omega(L1), omega(L4 ), omega(T1), and omega(T4)) are forbidden (the four-layer structure c omprises layers 1-4); two frequency solutions are obtained in their st ead. These results are due to the asymmetry of the structure. Moreover , we also find that the situation in an asymmetric trilayer heterostru cture is similar to that of a FLHS. This work can be regarded as a gen eralization of the formalism of Chen, Lin, and George [Phys. Rev. B 41 , 1435 (1990)].