  BaSnO$_3$ (BSO) is a promising transparent conducting oxide (TCO) with
reported room-temperature (RT) Hall mobility exceeding 320
cm$^{2}$V$^{-1}$s$^{-1}$. Among perovskite oxides, it has the highest RT
mobility, about 30 times higher than that of the prototypical SrTiO$_3$. Using
first-principles calculations based on hybrid density functional theory, we
elucidate the physical mechanisms that govern the mobility by studying the
details of LO-phonon and ionized impurity scattering. A careful numerical
analysis to obtain converged results within the relaxation-time approximation
of Boltzmann transport theory is presented. The ${\bf k}$ dependence of the
relaxation time is fully taken into account. We find that the high RT mobility
in BSO originates not only from a small effective mass, but also from a
significant reduction in the phonon scattering rate compared to other
perovskite oxides; the origins of this reduction are identified. Ionized
impurity scattering influences the total mobility even at RT for dopant
densities larger than $5\times10^{18}$ cm$^{-3}$, and becomes comparable to
LO-phonon scattering for $1\times10^{20}$ cm$^{-3}$ doping, reducing the drift
mobility from its intrinsic LO-phonon-limited value of $\sim$594
cm$^{2}$V$^{-1}$s$^{-1}$ to less than 310 cm$^{2}$V$^{-1}$s$^{-1}$. We suggest
pathways to avoid impurity scattering via modulation doping or polar
discontinuity doping. We also explicitly calculate the Hall factor and Hall
mobility, allowing a direct comparison to experimental reports for bulk and
thin films and providing insights into the nature of the dominant mechanisms
that limit mobility in state-of-the art samples.
