  The intrinsic weak and highly non-local dielectric screening of
two-dimensional materials is well known to lead to high sensitivity of their
optoelectronic properties to environment. Less studied theoretically is the
role of free carriers on those properties. Here, we use ab initio GW and
Bethe-Salpeter equation calculations, with a rigorous treatment of dynamical
screening and local-field effects, to study the doping-dependence of the
quasiparticle and optical properties of a monolayer transition metal
dichalcogenide, 2H MoTe$_2$. We predict a quasiparticle band gap
renormalization of several hundreds meV for experimentally-achievable carrier
densities, and a similarly sizable decrease in the exciton binding energy. This
results in an almost constant excitation energy for the lowest-energy exciton
resonance with increasing doping density. Using a newly-developed and
generally-applicable quasi-2D plasmon-pole model and a self-consistent solution
of the Bethe-Salpeter equation, we reveal the importance of accurately
capturing both dynamical and local-field effects to understand detailed
photoluminescence measurements.
