PHENOMENOLOGICAL TREATMENT OF LONG-WAVELENGTH POLAR OPTICAL OSCILLATIONS IN SEMICONDUCTOR NANOSTRUCTURES

In recent work a correct phenomenological approach to long-wavelength polar optical phonons in semiconductor nanostructures has been settled. In such treatment the vector displacement u and the electric potential phi are considered coupled quantities satisfying a coupled system of differential equations. In the present work we give a brief description of the theory. In a systematic way we discuss the differential equations of the theory, the derivation of the matching boundary conditions together with their physical interpretation, and the rigorous proof of the completeness and orthogonality of the oscillation modes. A general method for the solution of the coupled equations is presented and applied to concrete systems: quantum wells, quantum dots and quantum wires. The dispersion relation curves are obtained, as well as the displacement fields and potentials as a function of the coordinates. The Frohlich-like electron-phonon Hamiltonian is derived by the methods of quantum field theory and applied to the different cases.