Donuts and Spin Vortices at the Fermi Surfaces of Hybrid Lead-Iodide CH₃NH₃PbI₃ Perovskites

We show that spin–orbit coupling in methylammonium lead-iodide perovskite generates Fermi surfaces of peculiar topology: for the low charge-injection regime of interest for photovoltaic applications, the Fermi surfaces are donuts (ring tori), which evolve into apples (spindle tori) as the band population is raised, with a vortex spin texture that indicates the dominance of the Rashba effect. This material is a significant example of a bulk system, where despite the lack of any macroscopic field, a strong Rashba effect shows up, originated by a local dipole field on Pb and I orbitals, with the field direction locked to the vortex point loci in k-space. Remarkably, the same Fermi surface topology and spin helicity characterize both electrons and holes: this makes the presence of Rashba compatible with the direct band gap behavior described by photoluminescence experiments.