Negative Electronic Compressibility and Tuneable Spin Splitting in WSe$_{2}$

Recently, semiconducting transition metal dichalcogenides have gained attention for their extraordinarily large exciton-binding energies [1,2] and locking of the spin with valley and layer pseudospins [3,4]. Through sub-monolayer deposition of alkali metals onto the surface of WSe$_{2}$, analogous to the gating in a field-effect transistor, we create a 2DEG at the sample surface with tuneable carrier concentration [5]. Counter-intuitively, we find that the addition of carriers induces a reduction of the chemical potential in the near-surface. We attribute this to negative electronic compressibility [6] where strong Coulomb effects lead to the lowering of the chemical potential with band filling, which we find persists to remarkably high electron densities. Simultaneously, we show this is accompanied by a giant tuneable spin-splitting of the valence band states and a reduction of the quasiparticle band gap. [1] Ugeda, \textit{et al., Nature Mat.} \textbf{13} (2014) 1091 [2] Ye \textit{et al}., \textit{Nature} \textbf{513} (2014) 214 [3] Xu \textit{et al., Nature Phys. }\textbf{10} (2014) 343 [4] Riley \textit{et al., Nature Phys.} \textbf{10} (2014) 835 [5] Riley \textit{et al.} doi:10.1038/nnano.2015.217 [6] Eisenstein \textit{et al. Phys. Rev. Lett. }\textbf{68} (1992) 674