Envelope-function method for the conduction band in graded heterostructures

Abstract

The Kohn-Luttinger representation and canonical transformation of the all-band k p Hamiltonian have been used to derive an envelope-function equation with position-dependent effective mass for the conduction band of a structure with a graded heterojunction between related lattice-matched zinc-blende symmetry semiconductors. This equation should also include non-parabolicity correction terms and an interface spin-split-off term. Several forms of the effective-mass equation have been obtained that are at first sight different, but equivalent to one another. The boundary conditions for an envelope function at the heterointerface (in the case when the heterojunction might be treated as an abrupt one) are shown to depend on how the envelope functions are defined. Attention has also been paid to the problem of the influence of the position-dependent effective mass on the coefficient of optical absorption for transitions between conduction subbands in a quantum well. The absorption is shown to be negligibly small for the case of normal incidence of light on the quantum well.

Generalization of the effective mass method for semiconductor structures with atomically sharp heterojunctions

“…[28]. Formally, the resulting equation for a sharp heterojunction differs from that for a smooth heterojunction only by renormalization of the parameters entering into it.…”

“…(5) and as a result obtain the wellknown set of equations for the envelope functions [3,28]. In the case of an atomically sharp heterojunction, on the other hand, it is possible to proceed in the spirit of the method used in Ref.…”

“…It is possible to proceed otherwise. Reducing the fourth-order differential equation to a physically equivalent second-order equation [28], we obtain a discontinuous envelope function with characteristic relative discontinuity of order (k z λ)…”

“…For smooth heterojunctions the one-band equation for the envelope functions for the c band (conduction band) was derived in Ref. [28]. In the r representation it has the form:…”

“…Such indeterminate forms do not arise if we use a different, equivalent form of this operator where we separate out a term analogous to the relativistic Darwin term (see Ref. [28]):…”