High field magneto-spectroscopy of excitons in monolayer WSe$_{\mathrm{2}}$

We have performed circularly polarized photoluminescence (PL) experiments on monolayer WSe$_{\mathrm{2}}$ in magnetic fields up to \textpm 31T and at temperatures between 2K and 45K, focusing on the emission from the neutral (X$^{\mathrm{0}})$ and negatively charged (X$^{\mathrm{-}})$ excitons. A parallel magnetic field does not affect the exciton energy. At 45K, a perpendicular magnetic field (Faraday geometry) induces linear shift of about 0.12 meV/T$\approx $2$\mu $B for both X$^{\mathrm{0}}$ and X$^{\mathrm{-}}$ peaks indicating lifting of the valley degeneracy. The magnitude of this valley Zeeman shift agrees with the valence band edge lifting due to atomic orbital contribution. The change of the X$^{\mathrm{-\thinspace }}$PL intensity with the magnetic field suggests that the intravalley configuration is the lower energy state of the trion in WSe$_{\mathrm{2}}$. At lower temperatures, the X$^{\mathrm{0}}$ exhibits the same shift with the magnetic field as at 45K, while the X$^{\mathrm{-}}$ shows a more pronounced and non-linear shift with respect to magnetic field.