  The large variety of complex electronic structure materials and their alloys,
offer highly promising directions for improvements in thermoelectric (TE) power
factors (PF). Their electronic structure contains rich features, referred to as
'surface complexity', one of them being the highly anisotropic warped energy
surface shapes with elongated features and threads in some cases. In this work
we use Boltzmann transport simulations to quantify the influence that the shape
of the electronic structure energy surfaces has on the PF. Using both
analytical ellipsoidal bands, as well as realistic bands from the group of
half-Heuslers, we show that band shape complexity alone can offer an advantage
to the PF of ~3x in realistic cases. The presence of anisotropic scattering
mechanisms such as ionized impurity or polar optical phonon scattering,
however, can reduce these improvements by up to ~50%. We show that expressions
based on the simple ratio of the density-of-states to the conductivity
effective masses, mDOS/mC, together with the number of valleys, can capture the
anisotropy shape with a moderate to high degree of correlation. For this, we
use a convenient way to extract these masses by mapping the complex
bandstructures of materials to parabolic electronic structures, without the
need for Boltzmann transport codes. Despite the fact that the PF depends on
many parameters, information about the benefits of the band shape alone, would
be very useful for identifying and understanding the performance of novel
thermoelectric materials.
