  Through the Non-Equilibrium Green's Function (NEGF) formalism, quantum-scale
device simulation can be performed with the inclusion of electron-phonon
scattering. However, the simulation of realistically sized devices under the
NEGF formalism typically requires prohibitive amounts of memory and computation
time. Two of the most demanding computational problems for NEGF simulation
involve mathematical operations with structured matrices called semiseparable
matrices. In this work, we present parallel approaches for these computational
problems which allow for efficient distribution of both memory and computation
based upon the underlying device structure. This is critical when simulating
realistically sized devices due to the aforementioned computational burdens.
First, we consider determining a distributed compact representation for the
retarded Green's function matrix $G^{R}$. This compact representation is exact
and allows for any entry in the matrix to be generated through the inherent
semiseparable structure. The second parallel operation allows for the
computation of electron density and current characteristics for the device.
Specifically, matrix products between the distributed representation for the
semiseparable matrix $G^{R}$ and the self-energy scattering terms in
$\Sigma^{<}$ produce the less-than Green's function $G^{<}$. As an illustration
of the computational efficiency of our approach, we stably generate the
mobility for nanowires with cross-sectional sizes of up to 4.5nm, assuming an
atomistic model with scattering.
