  Structural phase transitions and soft phonon modes pose a longstanding
challenge to computing electron-phonon (e-ph) interactions in strongly
anharmonic crystals. Here we develop a first-principles approach to compute
e-ph scattering and charge transport in materials with anharmonic lattice
dynamics. Our approach employs renormalized phonons to compute the
temperature-dependent e-ph coupling for all phonon modes, including the soft
modes associated with ferroelectricity and phase transitions. We show that the
electron mobility in cubic SrTiO$_{3}$ is controlled by scattering with
longitudinal optical phonons at room temperature and with ferroelectric soft
phonons below 200~K. Our calculations can accurately predict the temperature
dependence of the electron mobility in SrTiO$_{3}$ between 150$-$300~K, and
reveal the microscopic origin of its roughly $T^{-3}$ trend. Our approach
enables first-principles calculations of e-ph interactions and charge transport
in broad classes of crystals with phase transitions and strongly anharmonic
phonons.
