Generally, in modeling a semiconductor device, a contact doping is specified from which the contact Fermi level is calculated. If the contact Fermi level is not calculated using the same band-structure model as that used for the transport calculations, the quantum-charge calculation in the device will be incorrect. We describe our method for calculating the semiclassical Fermi-level, charge, and Jacobian using the bandstructure generated from the Hamiltonian. Then we describe our quantum Jacobian for calculating the quantum charge.
We begin with the general expression for the electron density since our contact Hamiltonian contains a small imaginary potential which creates small band-tails, alters the density of states, and thus slightly alters the Fermi-level [3].
where the spectral function is given by (assuming spin degeneracy)
is the Fermi factor, 
is the dispersion relation
generated from the Hamiltonian, 
, and 
is the
energy-dependent broadening factor from the imaginary potential
[3].
For a spherically symmetric dispersion centered at the
valley, Eqs. (2) and (3)
become
The order of integration is chosen for numerical efficiency.
varies rapidly only around 
and the
spectral function is peaked at 
. Relatively
few k points can be used and the energy points are chosen
to resolve the regions around and 
.
The Newton-Raphson scheme for calculating the self-consistent
electrostatic potential requires an expression for
where 
is the electrostatic
potential. We use the approximation
which is exact in the absence of broadening.
In the absence of broadening, the spectral function is a delta function and Eq. (4) becomes
Integrating by parts and substituting
variables , Eq. (6) becomes
In Eq. (7), 
is the inverse of the
dispersion relation raised to the third power
valid if the dispersion is single-valued within
the domain of integration.
We cast Eq. (6) in the form of
Eq. (7) for numerical efficiency. The integrand
is only rapidly varying around , so that it is straightforward to
integrate for any temperature. For the Jacobian, we take
of Eq. (7).
So far, we have discussed the semi-classical
calculation of the Fermi-level,
charge, and Jacobian using
realistic bandstructure. We must also calculate the quantum charge
and a corresponding Jacobian. We have been using the the
semiclassical Jacobian for the quantum calculation [3],
but it is more efficient and even easier to compute an approximation for the
quantum Jacobian. In the equilibrium region of the leads, using
approximation (5), is
given by
where 
is the spectral function at layer L,
is the transverse wavevector, a is the layer thickness,
and the trace is over the cation and anion orbitals.
In the non-equilibrium region, in the absence of incoherent scattering, the expression for the electron density has two components resulting from injection from the left and right contacts [3]. Using approximation (5) on each component, we obtain
where 
and 
is the anti-Hermitian component of the
left boundary self-energy [3].
Eqs. (8) and (9) are
calculated in the same loop as the
quantum charge making the calculation very efficient.
Even in the presence of incoherent scattering,
the expression for 
can be broken
up into components contributed from the left and right
contacts;
however, seperating the components increases the computational
burden of the scattering calculation by a factor of two.
