  Monolayer transition metal dichalcogenide (TMDC) semiconductors exhibit
strong excitonic optical resonances which serve as a microscopic, non-invasive
probe into their fundamental properties. Like the hydrogen atom, such excitons
can exhibit an entire Rydberg series of resonances. Excitons have been
extensively studied in most TMDCs (MoS$_2$, MoSe$_2$, WS$_2$ and WSe$_2$), but
detailed exploration of excitonic phenomena has been lacking in the important
TMDC material molybdenum ditelluride (MoTe$_2$). Here, we report an
experimental investigation of excitonic luminescence properties of monolayer
MoTe$_2$ to understand the excitonic Rydberg series, up to 3s. We report
significant modification of emission energies with temperature (4K to 300K),
quantifying the exciton-phonon coupling. Furthermore, we observe a strongly
gate-tunable exciton-trion interplay for all the Rydberg states governed mainly
by free-carrier screening, Pauli blocking, and band-gap renormalization in
agreement with the results of first-principles GW plus Bethe-Salpeter equation
approach calculations. Our results help bring monolayer MoTe$_2$ closer to its
potential applications in near-infrared optoelectronics and photonic devices.
