  This paper presents an ab initio methodology to account for electron-phonon
interactions in 2D materials, focusing on transition metal dichalcogenides
(TMDCs). It combines density functional theory and maximally localized Wannier
functions to acquire material data and relies on the linearized Boltzmann
transport equation (LBTE) and the non-equilibrium Green's functions (NEGF)
method to determine the transport properties of materials and devices,
respectively. It is shown that for MoS$_2$, both LBTE and NEGF return very
close mobility values, without the need to adjust any parameter. The excellent
agreement between both approaches results from the inclusion of non-diagonal
entries in the electron-phonon scattering self-energies. The NEGF solver is
then used to shed light on the "current vs. voltage" characteristics of a
monolayer MoS$_2$ transistor, highlighting how the interactions with phonons
impact both the current magnitude and its distribution. The mobility of other
TMDCs is considered as well, demonstrating the capabilities of the proposed
technique to assess the potential of 2D channel materials in next-generation
logic applications.
