Polar Optical Phonon Modes and Fröhlich Electron-Phonon Interaction Hamiltonians in Coaxial Cylindrical Quantum Cables

Authors: LI ZHANG, HONG-JING XIE

Abstract: By using the dielectric continuum approximation, the polar optical phonon modes of coaxial cylindrical quantum cables with arbitrary layer-number were studied. In order to describe the vibrating of the longitudinal-optical (LO) phonons, a set of legitimate eigenfunctions for LO phonon modes are constructed and adopted. In order to deal with the interface optical (IO) phonon modes, the transfer matrix method is employed. The quantized LO and IO phonons fields, as well as their corresponding Fröhlich electron-phonon interaction Hamiltonians, are also derived. Numerical calculations on a four-layer GaAs/Al_xGa_{1-x}As QC have been performed. Results reveal that there are six branches of IO phonon modes. When the wave vector k_z in the z direction and the azimuthal quantum number m are small, the dispersion frequencies of IO modes sensitively depend on k_z and m, and the frequency forbidden behaviors of IO phonon modes were observed and the reason was analyzed. When k_z and m are relatively large, with increasing k_z and m, the frequency for each mode converges to the limit frequency value of IO mode in a single heterostructure, and the electrostatic potential distribution of each branch of IO mode tends to be more and more localized at some interface; meanwhile, the coupling between the electron-IO phonon becomes weaker and weaker. The calculation also shows that the phonon modes with higher frequencies have more significant contribution to the electron-phonon interaction. At last, it is found that k_z and m have analogous influences on the frequencies and the electrostatic potentials of the IO phonons.

Keywords: Optical Phonon Modes; Coaxial Cylindrical Quantum Cables.

Full Text: PDF