  In this theoretical study, we examine the influence of dimensionality, size
reduction, and heat-transport direction on the phonon-drag contribution to the
Seebeck coefficient of silicon nanostructures. Phonon-drag contribution, which
arises from the momentum transfer between out-of-equilibrium phonon populations
and charge carriers, significantly enhances the thermoelectric coefficient. Our
implementation of the phonon drag term accounts for the anisotropy of
nanostructures, such as thin films and nanowires, through the boundary- and
momentum-resolved phonon lifetime. Our approach also takes into account the
spin-orbit coupling which turns out to be crucial for hole transport. We
reliably quantify the phonon drag contribution at various doping levels,
temperatures, and nanostructure geometries for both electrons and holes in
silicon nanostructures. Our results support the recent experimental findings,
showing that a part of phonon drag contribution survives in 100 nm silicon
nanostructures.
