In the semiconductor structures which originally motivated this work the charge carriers whose motion we seek to describe are really quasiparticles whose properties are determined by the energy band structure (or energy-momentum dispersion relation) of the semiconductor material. These carriers usually occupy states near an extremum of a band, and thus for the simpler cases of interest the band structure can be approximated as
where 
is the energy at the edge of the band and is just the
heterostructure potential used in Appendix 9, 
is the
wavevector at which this extremum occurs, and 
is the ``effective
mass'' which characterizes the curvature of the dispersion relation.
This dispersion relation may be modeled by the effective mass
Schrödinger equation
where 
is the Hartree potential, which is assumed to be slowly
varying. The
wavefunction 
in (13.153) is strictly an envelope function
for the true wavefunction. In the Wannier-Slater approach to
effective-mass theory (Slater, 1949) 
is a discrete function
(defined on the lattice points) giving the amplitude of the Wannier
function at each point [though 
is approximated by a continuous
function to derive the differential equation (13.153)]. In the
approach of Luttinger and Kohn (1955) 
is a continuum but
band-limited function which is multiplied by a perfectly periodic Bloch
function to obtain the complete wavefunction.
A semiconductor heterostructure is a single crystal which includes
(deliberately introduced) local changes in the chemical composition.
These introduce changes in the ``local band structure'' which must be
incorporated into the effective mass equation (13.153) to obtain
an accurate model of the quasiparticle dynamics in a heterostructure.
For the sake of concreteness let us consider an abrupt heterojunction.
The local band-edge energy 
will be shifted across the
heterojunction, and this effect is easily incorporated into
(13.153) by making 
a function of position. In general, the
value of the effective mass will also change across a heterojunction, and
this requires a more careful treatment of the kinetic energy term.
(Another way to view this problem is to state the conditions for
matching 
across an interface with discontinuous 
. Because
the matching condition follows uniquely from the form of the
Hamiltonian, we will focus upon the latter.) The
problem is that many of the expressions one might write down [such as
that which appears in (13.153)] become non-Hermitian when 
is taken to be a function of position. The simplest manifestly
Hermitian form is:
although other, more complicated expressions have been suggested (see
Morrow and Brownstein, 1984). In general, it appears that
(13.154), which might be termed the ``minimal Hermitian form,'' is
an adequate approximation when the magnitude of the change in 
is small, as is typically true of equivalent energy bands in closely
related materials. When the discontinuity is of a larger magnitude, as
when inequivalent bands are involved, one probably needs to explicitly
solve the multiband problem and infer the form of the effective-mass
equation from the results (see, for example, Grinberg and Luryi, 1989).
We can obtain different discrete approximations to (13.154) depending upon where we assume the heterojunction is actually located with respect to the mesh points. The most consistent scheme is to assume that the junction is located midway between two adjacent meshpoints. The discrete Hamiltonian (3.24) then becomes (Mains, Mehdi, and Haddad, 1989)
which was used in all of the tunneling calculations presented here.
If we use (13.154) to construct the kinetic-energy
superoperator 
, how is the form of this superoperator (in
the Wigner-Weyl representation) affected? We might hope that a simple
expression would result, such as
(This is the expression which was actually used in the calculations
presented here.)
Unfortunately, (13.156) holds only if 
, which holds only if the band structure
varies slowly as a function of position. In general, a
position-dependent effective mass will produce a nonlocal form for the
kinetic-energy superoperator in the Wigner-Weyl representation
(Barker, Lowe, and Murray, 1984). A more complete treatment, expressing
the Wigner-Weyl transformation in terms of the Wannier and Bloch
representations (rather than the position and momentum representations)
has been developed by Miller and Neikirk (1990). This analysis also
demonstrates a nonlocal kinetic-energy term.
