For state-of-the-art In-based resonant tunneling diodes (RTDs)
[1], transport takes place solely within the 
valley of the conduction band. However, the non-parabolicity is
so large that we are forced to use a 10-band sp3s* model
[2].
Modeling transport in a single valley of a single band with
a full 10-band model is inefficient.
Therefore, we describe a parameterized 1-band tight-binding model
that mimics the the full-band -valley non-parabolicity.
To make quantitative
comparisons with experimental results, a self-consistent
quantum charge calculation is also generally required.
As part of that calculation, we describe our improved method
for calculating the
semi-classical and quantum electron charge, Fermi-level, and
Jacobian that is
consistent with the full bandstructure model.
Finally, we describe our Schottky contact model that is
compatible with any localized orbital bandstructure model.
To mimic the -valley non-parabolicity with a single-band
model, we parameterize as functions of kinetic energy and transverse momentum
the hopping elements and site energies of the 1-band, tight-binding
Hamiltonian according to
where 
is the kinetic energy, 
is the transverse (real) wavevector,
is the longitudinal (in general complex) wavevector,
t is the hopping element,
and D is the correction to the site energy.
The arrays 
and 
are created from the
bulk 
dispersion relation
and then used for interpolation in the transport
calculation.
The arrays are created from the following algorithm.
For each , the kinetic energy, , is swept from a minimum
to maximum value. At each value, is found from the full-band
dispersion relation, .
Near the conduction band edge, is found from
where the derivative is obtained
from . At higher energies,
the dispersion becomes fairly linear and we find
from the first derivative: 
.
Thus, we choose t to give the correct effective mass or group
velocity. With t set, we then choose D to give the correct
energy, i.e. 
.
At each point of the actual dispersion we are choosing a tight-binding
dispersion with the correct slope or curvature. We then add a
site potential to move the tight-binding dispersion up or down to get
the right energy.
For the transport calculations, we work with total energy, E, and transverse
momentum, [3]. Foreach E, and site i,
we calculate
the kinetic energy, 
, and then interpolate from
and to obtain the Hamiltonian elements,
and 
.
The averaged value of t is used between
sites, and the site energy at site i is given by
where 
.
This reproduces the standard tight-binding Hamiltonian for a cosine
dispersion.
