Long-range electrostatic contribution to electron-phonon couplings and mobilities of two-dimensional and bulk materials

Charge transport plays a crucial role in manifold potential applications of two-dimensional materials, including field-effect transistors, solar cells, and transparent conductors. At most operating temperatures, charge transport is hindered by scattering of carriers by lattice vibrations. Assessing the intrinsic phonon-limited carrier mobility is thus of paramount importance to identify promising candidates for next-generation devices. Here we provide a framework to efficiently compute the drift and Hall carrier mobility of two-dimensional materials through the Boltzmann transport equation by relying on a Fourier-Wannier interpolation. Building on a recent formulation of long-range contributions to dynamical matrices and phonon dispersions [Phys. Rev. X 11, 041027 (2021)], we extend the approach to electron-phonon coupling including the effect of dynamical dipoles and quadrupoles. We identify an unprecedented contribution associated with the Berry connection that is crucial to preserve the Wannier-gauge covariance of the theory. This contribution is not specific to two-dimensional crystals, but also concerns the three-dimensional case, as we demonstrate via an application to bulk SrO. We showcase our method on a wide selection of relevant monolayers ranging from ${\mathrm{SnS}}_{2}$ to ${\mathrm{MoS}}_{2}$, graphene, BN, InSe, and phosphorene. We also discover a nontrivial temperature evolution of the Hall hole mobility in InSe whereby the mobility increases with temperature above 150 K due to the Mexican-hat electronic structure of the InSe valence bands. Overall, we find that dynamical quadrupoles are essential and can impact the carrier mobility in excess of 75%.