Polar optical modes and electron-phonon interaction in semiconductor nanostructures

The basic analytical properties of the linear differential operator describing the long-wave phenomenological model of polar optical modes in semiconductor nanostructures are studied under very general conditions. While full account is taken of the coupling between longitudinal and transverse parts of the mechanical vibration amplitude u and of u with the electrostatic potential field {φ}, the analysis holds for interfaces of arbitrary geometry and for any type of nanostructure. The Hermiticity of the linear differential operator, the orthogonality of the eigenvectors u, and the consequent completeness relation are established (i) without need to resort to the explicit form of the eigenvectors (ii) in a way which bears out the role of the matching rules at the interfaces. From this the amplitude of {φ} and the consequent electron-phonon interaction Hamiltonian are obtained for arbitrary geometry and structure under general conditions.