Fig. 3: Band spin splitting enabled by a defect gradient.

a The band diagram of the optical selection rules for the symmetry-broken PtTe₂. The dashed line denotes the Fermi level (E_F). The blue and red arrows denote the electronic states with the opposite spins. b, c Band structures for the bulk samples without defects and with a defect gradient, respectively. The high-symmetry points Γ, M, and K in the (k_x, k_y) plane of the reciprocal space are shown in the inset of b. d Schematic diagram of an 8-layer-thick PtTe₂ film with a defect gradient. The V_Te gradient is indicated by the color scheme on the right. e, f Band structures for a concrete 8-layer-thick PtTe₂ sample without defects and with a defect gradient, respectively. Notably, spin degeneracy is preserved in both defect-free bulk in b and defect-free 8-layer-thick PtTe₂ sample in e. However, a band spin splitting occurs in c, f where the number of subbands increases. The colors denote the proportion ρ contributed by the upper and lower surface layers. The white double arrows schematically show the electron transition range under 800-nm excitation (~1.55 eV) near the Dirac point. The black and red crossing curves in e, f denote the Dirac point and spin-splitting bands, respectively.