Polar optical vibrational modes in quantum dots

A macroscopic continuum model coupling the mechanical vibrational amplitude and electrostatic potential is applied to obtain the optical vibrational modes in quantum-dot structures. A unified method of solution (valid for any type of nanostructure) that reduces the four coupled second-order differential equations to Helmholtz’s and Laplace’s equations is described. Analytical solutions for the vibrational amplitude and the Fröhlich-type electron-phonon interaction are given for quantum dots with spherical geometry. The existence of surface modes and their relation to the matching boundary conditions are studied. A qualitative discussion of the ir and Raman activity of the calculated modes is given together with a comparison with the few existing experimental data.