Let us consider a one-dimensional, single-band model. In a semiconductor heterostructure, the electron wavefunctions are described most simply by the effective-mass Schrödinger equation:
The form of the kinetic-energy operator in (1) is the simplest Hermitian form one can use if the materials parameters (such as the effective mass ) vary with position [7,8]. Now, let us assume that all variation of the potential and of the materials parameters are confined to an interval , so that outside of this interval the form of Schrödinger 's equation is translationally invariant. Thus, in the regions and (the ``asymptotic regions''), the solutions of Schrödinger 's equation are superpositions of plane waves, and the energies of these plane waves are described by a well-defined dispersion relation . We will refer to quantities in the asymptotic regions by the subscripts l and r for the left- and right-hand regions, respectively.
Now, for any energy and , there will be two independent solutions to Schrödinger 's equation representing electrons incident from the left and the right, respectively. In the asymptotic regions, these solutions will have the form:
In general, because . There exist several rigorous relationships between the transmission and reflection amplitudes and . Invoking Green's identity leads to the current-continuity equations
and an orthogonality condition
One may also invoke time-reversal symmetry to find the relationship between and . Noting that , say, is a solution of Schrödinger 's equation with energy E, it must be possible to write as a linear combination of and . With a bit of manipulation, one finds
In most textbooks, the relations (1--7) are presented with the wavenumbers in place of the velocities . Such expressions are derived within the assumptions that the dispersion relation (or band structure) is perfectly parabolic and does not depend upon position. Neither assumption is warranted in semiconductor heterostructures, as electrons in heterostructure devices frequently explore non-parabolic regions of the energy-band structure, and the band structure itself (particularly the effective mass) will vary with semiconductor composition and thus position. The expressions (1--7) are valid for nonparabolic and spatially varying dispersion relations and should thus always be used. The velocity is the electron group velocity, given by
using the dispersion relation appropriate to the given semiconductor material.
One conventionally defines the transmission probability T as the ratio of the transmitted to the incident flux, or
so that T is same for both directions of incidence. One can also show, using equations (1--7),
(Because the reflection probability is measured on the same side of the system as the incident flux, there is no velocity correction.) Also note that, from (1),
as one would expect.
Finally, we investigate the normalization and orthogonality properties of the scattering states. Because we are dealing with a continuum of states over which we must integrate to evaluate any physical observable, a ``delta-function'' normalization is appropriate. With such a normalization convention, any finite contribution to the inner product, such as the integral over , may be neglected in comparison to the integrals over and . Thus,
A relationship similar to (12) can be written for . The -functions can be rewritten in terms of the energy E using
and similarly for . We then obtain
from equations (1).