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.