Modeling the effect of Schottky contacts requires a model for the metal that is compatible with the localized orbital bandstructure model for the semiconductor. The microscopic physics of the semiconductor - metal interface is complex. The electrical characteristics of the standard model [4] depend on the presence of an interfacial layer of atomic dimensions. Rather than attempt a microscopic model of the interface, we present a model which reproduces the essential macroscopic electric properties of the interface. The metal layer enters Poisson's equation and Schrödinger's equation as a boundary condition on the semiconductor region.
For Poisson's equation, the metal is an equipotential region with a Fermi-level fixed by the applied potential. The Fermi-level of the metal is assumed to be pinned relative to the valence band of the semiconductor. The metal fixes the electrostatic potential of the adjacent semiconductor atomic layer resulting in a Dirichlet boundary condition on Poisson's equation. A metal also gives rise to an image potential which is added to the electrostatic potential of the semiconductor:
where 
is the position
of the left/right metal - semiconductor interface
defined as a/2
to the left of the first semiconductor layer and
a/2 to the right of the last semiconductor layer,
respectively.
For Schrödinger's equation, the metal region acts as
a source and sink of electrons for the semiconductor.
Since we do not know a priori into which semiconductor bands
and from which semiconductor bands electrons will be sourced
or sunk, we create a boundary self-energy which allows sourcing
and sinking of electrons from all semiconductor bands.
A general form of the boundary self energy is given by
where 
is the block
matrix coupling the device to the left lead, 
is the
matrix of Bloch states, and Z is the diagonal matrix of propagation factors
[3].
For a given energy and momentum, most of the bands will be
evanescent resulting in propagation
factors 
for which 
is
complex with a large imaginary component.
To allow electrons to be absorbed by the metal contacts
from any semiconductor band, we set the elements of
the matrix of propagation factors to be
independent of energy and momentum.
Since Z is now a constant times the identity matrix,
the boundary self energy becomes
To obtain an estimate for the magnitude of 
,
we consider a metal Fermi level around mid-band
with 
resulting in a propagation
factor 
which gives a value for
of 1 and a boundary self energy of
.
In summary, we have presented several improvements and enhancements of the charge models, bandstructure models, and contact models described in [3] which increase the comprehensiveness and numerical efficiency of our approach.
