Exchange Energy of the Ferromagnetic Electronic Ground State in a Monolayer Semiconductor

Mobile electrons in the semiconductor monolayer MoS_{2} form a ferromagnetic state at low temperature. The Fermi sea consists of two circles: one at the K point, the other at the K[over ˜] point, both with the same spin. Here, we present an optical experiment on gated MoS_{2} at low electron density in which excitons are injected with known spin and valley quantum numbers. The resulting trions are identified using a model which accounts for the injection process, the formation of antisymmetrized trion states, electron-hole scattering from one valley to the other, and recombination. The results are consistent with a complete spin polarization. From the splittings between different trion states, we measure the exchange energy Σ, the energy required to flip a single spin within the ferromagnetic state, as well as the intervalley Coulomb exchange energy J. We determine Σ=11.2  meV and J=5  meV at n=1.5×10^{12}  cm^{-2} and find that J depends strongly on the electron density n.