Exact effective-mass theory for heterostructures

Abstract

An exact effective-mass differential equation is derived for electrons in heterostructures. This equation is exactly equivalent to the Schr\"odinger equation, and is obtained by applying a k-space transformation of variables to the Burt envelope-function theory in which the Brillouin zone is mapped onto the infinite real axis. The mapping eliminates all nonlocal effects and long-range Gibbs oscillations in the Burt theory, producing an infinite-order differential equation in which interface effects are strongly localized to the immediate vicinity of the interface. A general procedure is given for obtaining finite-order boundary conditions from the infinite-order equation; the second-order theory reduces to the BenDaniel-Duke model with a \ensuremath{\delta}-function potential at the interface. The derivation is presented for a simple one-dimensional crystal but can easily be generalized for more complex situations.