  The high room temperature mobility and the electron effective mass in
BaSnO$_3$ are investigated in depth by evaluation of the free carrier
absorption observed in infrared spectra for epitaxial films with free electron
concentrations from $8.3 \times 10^{18}$ to $7.3 \times 10^{20}$~cm$^{-3}$.
Both the optical band gap widening by conduction band filling and the carrier
scattering mechanisms in the low and high doping regimes are consistently
described employing parameters solely based on the intrinsic physical
properties of BaSnO$_3$. The results explain the current mobility limits in
epitaxial films and demonstrate the potential of BaSnO$_3$ to outperform
established wide band gap semiconductors also in the moderate doping regime.
