Unified ab initio description of Fröhlich electron-phonon interactions in two-dimensional and three-dimensional materials

\\textit{Ab initio} calculations of electron-phonon interactions including the\npolar Fr\\"ohlich coupling have advanced considerably in recent years. The\nFr\\"ohlich electron-phonon matrix element is by now well understood in the case\nof bulk three-dimensional (3D) materials. In the case of two-dimensional (2D)\nmaterials, the standard procedure to include Fr\\"ohlich coupling is to employ\nCoulomb truncation, so as to eliminate artificial interactions between periodic\nimages of the 2D layer. While these techniques are well established, the\ntransition of the Fr\\"ohlich coupling from three to two dimensions has not been\ninvestigated. Furthermore, it remains unclear what error one makes when\ndescribing 2D systems using the standard bulk formalism in a periodic supercell\ngeometry. Here, we generalize previous work on the \\textit{ab initio}\nFr\\"ohlich electron-phonon matrix element in bulk materials by investigating\nthe electrostatic potential of atomic dipoles in a periodic supercell\nconsisting of a 2D material and a continuum dielectric slab. We obtain a\nunified expression for the matrix element, which reduces to the existing\nformulas for 3D and 2D systems when the interlayer separation tends to zero or\ninfinity, respectively. This new expression enables an accurate description of\nthe Fr\\"ohlich matrix element in 2D systems without resorting to Coulomb\ntruncation. We validate our approach by direct \\textit{ab initio}\ndensity-functional perturbation theory calculations for monolayer BN and\nMoS$_2$, and we provide a simple expression for the 2D Fr\\"ohlich matrix\nelement that can be used in model Hamiltonian approaches. The formalism\noutlined in this work may find applications in calculations of polarons,\nquasiparticle renormalization, transport coefficients, and superconductivity,\nin 2D and quasi-2D materials.\n