Foundations of the envelope-function theory for electrons and phonons in semiconductor heterostructures

Envelope-function models form a valuable tool for the analysis of electrons and phonons in semiconductor heterostructures because they provide a high degree of physical insight at a low computational cost. This dissertation investigates the fundamental physical principles underlying the envelope-function approach to heterostructures through the direct derivation of long-wavelength envelope-function models from basic microscopic theory. The derivation starts with the establishment of an exact envelope-function formalism for electrons and phonons, following the work of Burt and Kunin, and proceeds through a series of approximations to reduce the exact equations of motion to simple, second-order differential equations with spatially varying coefficients. This provides the first direct proof of the validity of second-order differential equations at an abrupt heterojunction and resolves a long-standing controversy in the literature.; The effective-mass theory for electrons derived here gives a straightforward analytical justification for the approximations used by Burt in his treatment of the problem (J. Phys. Condens. Matter, 6651 (1992)). Burt showed that by assuming the envelope functions are slowly varying, one can derive an effective-mass theory which reproduces very accurately the properties of the solutions to the exact theory; however, he offered no analytical justification for why such an approximation should be valid at an abrupt junction. The present work provides this justification, showing that the envelopes can be treated as slowly varying so long as any rapid interface fluctuations are confined to a region small on the macroscopic scale.; The long-wavelength theory of phonons derived here reproduces the boundary conditions of classical elasticity for acoustic vibrations, but it also shows that these boundary conditions are not adequate to describe optical modes. In general, the change in vibrational properties at the interface leads to the appearance of delta-function terms in the reduced mass and zone-center force constant, which tend to dominate the optical boundary conditions. Thus the phenomenological models currently used for optical phonons in GaAs/AlAs heterostructures must be modified if they are to describe the strong interface effects that exist in InAs/GaSb structures.