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.
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