Analytical Calculation of Frequency Spectrum and Group Velocities of Acoustic Phonons in Quasi-two-dimensional Nanostructures

Low-dimensional crystal structures that are attractive materials for the fabrication of modern nanoelectronic devices due to their unique physical properties are considered. Type of nanostructure (quantum well, quantum wire or quantum dot) and its sizes with respect to quasiparticle confined movement are determining factors for the formation of its mechanical, kinetic, optical and electronic properties and, since, main physical parameters. The processes of interaction between quasiparticles (in particular, electron-phonon interaction) also play an important role. Renormalized energy spectra of electrons and phonons define the physical processes, which are produced by electron-phonon interaction in nanostructure. In order to understand these peculiarities, it is necessary to study how the energy spectra of both these systems of quasiparticles are transformed in nanostructures of different types and sizes, as well as mechanisms of their interaction. Therefore, one should know the analytical form of dispersion relation for the electron energy and frequencies of all types of phonons in a particular nanostructure. In this paper, we propose the method how to obtain the explicit dispersion relations for all modes of confined acoustic phonons in plane quasi-two-dimensional nanostructures with hexagonal crystal lattice. Using the dielectric continuum model, the formulas describing frequency as a function of the phonon wave vector are derived and phonon group velocity, in its turn. Computer simulations are performed for GaN nanofilm with wurtzite structure.