Dual quantum confinement and anisotropic spin splitting in the multi-valley semimetal PtSe$_2$

We investigate the electronic structure of a two-dimensional electron gas created at the surface of the multi-valley semimetal 1T-PtSe$_2$. Using angle-resolved photoemission and first-principles-based surface space charge calculations, we show how the induced quantum well subband states form multiple Fermi surfaces which exhibit highly anisotropic Rashba-like spin splittings. We further show how the presence of both electron- and hole-like bulk carriers causes the near-surface band bending potential to develop an unusual non-monotonic form, with spatially-segregated electron accumulation and hole accumulation regions, which in turn amplifies the induced spin splitting. Our results thus demonstrate the novel environment that semimetals provide for tailoring electrostatically-induced potential profiles and their corresponding quantum subband states