For the purpose of discretization
one can expand the HAMILTONian, self-energies, and GREEN's functions
in terms of some basis functions to obtain the corresponding matrices.
In the tight-binding method one can take the basis functions to be any
set of localized functions, such as atomic s- and p-orbitals [116,122],
WANNIER functions [242], and so forth.
A common approximation used to describe the HAMILTONian of layered structures
consists of non-vanishing interactions only between nearest neighbor layers.
That is, each layer interacts only with itself and its
nearest neighbor layers and . Then the single particle
HAMILTONian of the layered structure is a block tri-diagonal matrix,
where diagonal blocks
represent the Hamiltonian of layer and
off-diagonal blocks
represent interaction between layers
and

(4.10)

where
. The matrix representation
of the kinetic equations (3.71) and (3.72) are

(4.11)

(4.12)

where
is the self-energy due to scattering
processes and
[60].

Figure 4.5:
Partitioning of the simulation domain into device
region and left and right contacts. Each point corresponds to a layer.

One can partition the layered structure into left contact with index ,
device region with index , and right contact with index (Fig. 4.5). The
device corresponds to the region where one solves the transport equations and
the contacts are the highly conducting regions connected to the device. While
the device region consists of only layers, the matrix equation
corresponding to (4.11) is infinitely dimensional due to the
semi-infinite contacts. As shown in Appendix G the influence
of the semi-infinite contacts can be folded into the device region by adding
a self-energy to the device region. This can be viewed as an additional
self-energy due to the transitions between the device and the contacts.
In the next sections the matrix representation of the HAMILTONian
and self-energies are discussed in detail.