  We report first-principles density-functional study of electron-phonon
interactions and thermoelectric transport properties of full-Heusler compounds
Sr$_{2}$BiAu and Sr$_{2}$SbAu. Our results show that ultrahigh intrinsic bulk
thermoelectric performance across a wide range of temperatures is physically
possible and point to the presence of multiply degenerate and highly dispersive
carrier pockets as the key factor for achieving it. Sr$_{2}$BiAu, which
features ten energy-aligned low effective mass pockets (six along $\Gamma-X$
and four at $L$), is predicted to deliver $n$-type $zT=0.4-4.9$ at
$T=100-700$~K. Comparison with the previously investigated Ba$_{2}$BiAu
compound shows that the additional $L$-pockets in Sr$_{2}$BiAu significantly
increase its low-temperature power factor to a maximum value of
$12$~mW~m$^{-1}$~K$^{-2}$ near $T=300$~K. However, at high temperatures the
power factor of Sr$_{2}$BiAu drops below that of Ba$_{2}$BiAu because the $L$
states are heavier and subject to strong scattering by phonon deformation as
opposed to the lighter $\Gamma-X$ states that are limited by polar-optical
scattering. Sr$_{2}$SbAu is predicted to deliver lower $n$-type of $zT=3.4$ at
$T=750$~K due to appreciable misalignment between the $L$ and $\Gamma-X$
carrier pockets, generally heavier scattering, and slightly higher lattice
thermal conductivity. Soft acoustic modes, responsible for low lattice thermal
conductivity, also increase vibrational entropies and high-temperature
stability of the Heusler compounds, suggesting that their experimental
synthesis may be feasible. The dominant intrinsic defects are found to be Au
vacancies, which drive the Fermi level towards the conduction band and work in
favor of $n$-doping.
