Abstract
A theory for long-wavelength optical phonons in quantum wires and quantum dots is developed under very general conditions. Confined and interface modes are obtained as solutions of the full system of four coupled second-order differential equations for the vibrational amplitudes and the electrostatic potential. These equations are solved without incompatibility between the mechanical and electrostatic matching boundary conditions. The eigensolutions and eigenmodes for cylindrical quantum wires and spherical quantum dots are found in terms of the longitudinal and transverse parts of the mechanical vibrational field and the electrostatic field, all appropriately coupled. The pure surface phonons (Fröhlich phonons) are obtained as a particular case. The orthogonality and the general conditions under which the set of eigensolutions form a complete basis are established. The electron-phonon interaction Hamiltonians are given in explicit form.
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