  The addition of porosity to thermoelectric materials can significantly
increase the figure of merit, ZT, by reducing the thermal conductivity.
Unfortunately, porosity is also detrimental to the thermoelectric power factor
in the numerator of the figure of merit ZT. In this manuscript we derive
strategies to recoup electrical performance in nanoporous Si by fine tuning the
carrier concentration and through judicious design of the pore size and shape
so as to provide energy selective electron filtering. In this study, we
considered phosphorus doped silicon containing discrete pores that are either
spheres, cylinders, cubes, or triangular prisms. The effects from these pores
are compared with those from extended pores with circular, square and
triangular cross sectional shape, and infinite length perpendicular to the
electrical current. A semiclassical Boltzmann transport equation is used to
model Si thermoelectric power factor. This model reveals three key results: The
largest enhancement in Seebeck coefficient occurs with cubic pores. The
fractional improvement is about 15% at low carrier concentration ($< 10^{20}\
\mathrm{1/cm^3}$) up to 60% at high carrier population with characteristic
length around $\sim 1\ \mathrm{nm}$. To obtain the best energy filtering effect
at room temperature, nanoporous Si needs to be doped to higher carrier
concentration than is optimal for bulk Si. Finally, in $n$-type Si
thermoelectrics the electron filtering effect that can be generated with
nanoscale porosity is significantly lower than the ideal filtering effect;
nevertheless, the enhancement in the Seebeck coefficient that can be obtained
is large enough to offset the reduction in electrical conductivity caused by
porosity.
