The wave mechanics of semiconductor heterostructures is complicated by the fact that the electron dispersion relation (or energy band structure) is generally non-parabolic at modest energies and necessarily varies with position. Under such circumstances, the form of the current density operator is no longer simply a symmetrized gradient.
The form of the particle current density operator J is clearly constrained by the group-velocity theorem [1], so that the expectation value of J on a state of definite wavevector k is
This equation, together with the band energies and eigenstates, in principle determines the form of J. It is, however, much more convenient to directly derive J from the Hamiltonian for a given problem. One does so by evaluating the time derivative of the probability density:
Green's identity
[
],
or a generalization thereof,
is then invoked to write the right-hand side of (2)
as the divergence of the current density.
Heterostructures are most often described at a ``mesoscopic'' level where the microscopic (smaller than the atomic diameter) behavior of the wavefunction can be factored out. In order to realistically describe the non-parabolic dispersion relation, the resulting effective Hamiltonian must be more elaborate than a simple Laplace operator, and Green's identity must be correspondingly be generalized to derive the form of J. The commonly-used mesoscopic models can be classified as effective-mass or tight-binding approaches.
