Spin–orbit coupling effect on energy level splitting and band structure inversion in CsPbBr3

The band structures and density of states (DOS) of all the three structural configurations of CsPbBr3 without spin–orbit coupling (SOC = 0) and with the addition of spin–orbit coupling (SOC ≠ 0) effects were calculated, using density functional theory. Upon the inclusion of the spin–orbit coupling, the bandgaps exhibit reductions of 1.27 eV, 1.16 eV and 1.08 eV for the cubic, tetragonal and orthorhombic phases, respectively. These calculations provide a positive split-off energy value of Δso = 1.69 eV for the simple cubic phase. For the lower symmetry phases, the p-like fourfold degenerate $$\varGamma_8v⁽⁴⁾$$ band has been observed to split to form two bands, in addition to the $$\varGamma_6v⁽²⁾$$ split-off band. The calculated splitting energies between these bands are found to be in close agreement with previous experimentally measured values. The calculated electronic band structures show that CsPbBr3 has a negative ‘inversion energy’ (Δi < 0). The magnitude of the inversion energy for the cubic phase is 2.36 eV for SOC = 0, which increased by 0.4–2.76 eV with the addition of the spin–orbit coupling. The arrangement of Bloch levels in the band structure of CsPbBr3 has been found to resemble that of a typical topological semimetal, but with a nonzero bandgap opening, due to the presence of the inversion asymmetry within its molecular structure.