  Low dimensional structures have demonstrated improved thermoelectric (TE)
performance because of a drastic reduction in their thermal conductivity,
{\kappa}l. This has been observed for a variety of materials, even for
traditionally poor thermoelectrics such as silicon. Other than the reduction in
{\kappa}l, further improvements in the TE figure of merit ZT could potentially
originate from the thermoelectric power factor. In this work, we couple the
ballistic (Landauer) and diffusive linearized Boltzmann electron transport
theory to the atomistic sp3d5s*-spin-orbit-coupled tight-binding (TB)
electronic structure model. We calculate the room temperature electrical
conductivity, Seebeck coefficient, and power factor of narrow 1D Si nanowires
(NWs). We describe the numerical formulation of coupling TB to those transport
formalisms, the approximations involved, and explain the differences in the
conclusions obtained from each model. We investigate the effects of cross
section size, transport orientation and confinement orientation, and the
influence of the different scattering mechanisms. We show that such methodology
can provide robust results for structures including thousands of atoms in the
simulation domain and extending to length scales beyond 10nm, and point towards
insightful design directions using the length scale and geometry as a design
degree of freedom. We find that the effect of low dimensionality on the
thermoelectric power factor of Si NWs can be observed at diameters below ~7nm,
and that quantum confinement and different transport orientations offer the
possibility for power factor optimization.
