Single-spin wavy brim bucket hat in the band edge of monolayer GdIH

Exotic electronic bands, such as flat, linear crossing, and spontaneous valley- or spin-polarized bands, in two-dimensional materials have been a hot topic in condensed-matter physics. In this paper, we first proposed a general dispersion model for possible hatlike electronic bands. Then, we identified a single-spin wavy brim bucket hat in the valence-band edge of a stable ferromagnetic semiconducting electrene, the Janus GdIH monolayer, which can be well described by a simplified two-band Hamiltonian model. Specifically, the hat band exhibited a wavy brim with six valleys along the boundary of the first Brillouin zone and held a flat top close to the Fermi level. This resulted in the emergence of single-spin van Hove singularity divergence and Lifshitz transitions. Due to the breaking of both time-reversal and space-inversion symmetries, a sizable spontaneous valley polarization was observed between adjacent brim valleys. This provided the opportunity to realize the high-temperature anomalous valley Hall effect. Particularly, via modest strains and carrier doping, various conductive bipolar states (spin-up versus spin-down, K valley versus $\ensuremath{-}\text{K}$ valley, and ultra-low-speed versus ultra-high-speed) can be modulated from the distorted wavy brim bucket hat of GdIH monolayer.