  In this work, we present an atomistic first-principles framework for modeling
the low-temperature electronic and transport properties of disordered
two-dimensional (2D) materials with randomly distributed point defects
(impurities). The method is based on the $T$-matrix formalism in combination
with realistic density-functional theory (DFT) descriptions of the defects and
their scattering matrix elements. From the $T$-matrix approximations to the
disorder-averaged Green's function (GF) and the collision integral in the
Boltzmann transport equation, the method allows calculations of, e.g., the
density of states (DOS) including contributions from bound defect states, the
quasiparticle spectrum and the spectral linewidth (scattering rate), and the
conductivity/mobility of disordered 2D materials. We demonstrate the method by
examining these quantities in monolayers of the archetypal 2D materials
graphene and transition metal dichalcogenides (TMDs) contaminated with vacancy
defects and substitutional impurity atoms. By comparing the Born and $T$-matrix
approximations, we also demonstrate a strong breakdown of the Born
approximation for defects in 2D materials manifested in a pronounced
renormalization of, e.g., the scattering rate by the higher-order $T$-matrix
method. As the $T$-matrix approximation is essentially exact for dilute
disorder, i.e., low defect concentrations ($c_\text{dis} \ll 1$) or density
($n_\text{dis}\ll A_\text{cell}^{-1}$ where $A_\text{cell}$ is the unit cell
area), our first-principles method provides an excellent framework for modeling
the properties of disordered 2D materials with defect concentrations relevant
for devices.
