A unified macroscopic continuum theory for the treatment of optical-phonon modes in quantum-wire structures is established. The theory is based on a Lagrangian formalism from which the equations of motion are rigorously derived. They consist of four coupled second-order differential equations for the vibrational amplitude and electrostatic potential. The matching boundary conditions are obtained from the fundamental equations. It is shown that no incompatibility exists between mechanical and electrostatic matching boundary conditions when a correct mathematical treatment of the problem is given. The particular case of a GaAs quantum wire buried in AlAs, where the phonons can be considered completely confined, is analyzed and the vector displacement and electron-phonon interaction potential are illustrated for several modes.

• Received 11 September 1992

DOI:https://doi.org/10.1103/PhysRevB.47.7602