Interface optical-phonon modes in a four-layer heterostructure of polar crystals

The equations of motion for the p-polarization field in a four-layer heterostructure (FLHS) of polar crystals are solved exactly for the interface 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 heterostructure (asymmetric single quantum well) and a step quantum well, are given and discussed. We find that there are six (not eight) frequency solutions for the interface optical-phonon modes in a FLHS and that, in the long-wavelength limit, the longitudinal and transverse modes in the two side materials 1 and 4 with frequencies (${\mathrm{\omega }}_{\mathit{L}1}$,${\mathrm{\omega }}_{\mathit{L}4}$,${\mathrm{\omega }}_{\mathit{T}1}$, and ${\mathrm{\omega }}_{\mathit{T}4}$) are forbidden (the four-layer structure comprises layers 1–4); two frequency solutions are obtained in their stead. These results are due to the asymmetry of the structure. Moreover, we also find that the situation in an asymmetric trilayer heterostructure is similar to that of a FLHS. This work can be regarded as a generalization of the formalism of Chen, Lin, and George [Phys. Rev. B 41, 1435 (1990)].