Optical-phonon modes of circular quantum wires

The optical-phonon modes of a circular wire are studied by means of the generalized Born-Huang equation. Its eigensolutions are found to be confined bulk and interface modes for almost all wave vectors parallel to the wire axis. At particular wave vectors hybridization of the two mode branches occurs. The eigenfrequencies and displacement eigenfields are derived for confined and interface modes. Three branches of confined modes exist, two TO and one LO. For modes confined to the wire the radial wave numbers are quantized, and determined by the zeros of Bessel functions of the first kind and of their derivatives. The twofold TO degeneracy is removed from reasons of symmetry. The displacement fields may be understood in terms of superpositions of linearly polarized plane waves with wave vectors forming a cone around the wire axis. For modes confined to the surroundings of the wire radial wave vectors remain continuous. Two types of interface modes are found. The dispersion of their frequencies with wave-vector components parallel to the wire is given in terms of modified Bessel functions. The same dispersion relations are obtained from electrostatic matching conditions. The Fröhlich interaction is studied for confined LO and interface modes. The corresponding interaction Hamiltonians are given in explicitly quantized form.