We consider the anomalous, spin, valley, and valley spin Hall effects in a pristine ex-so-tic graphene-based van-der-Waals (vdW) heterostructure consisting of a bilayer graphene (BLG) between semiconducting van-der-Waals material with strong SOC (e.g., WS₂) and ferromagnetic and insulating vdW material (e.g. Cr₂Ge₂Te₆). Reducing the effective Hamiltonian derived by Zollner et al [Phys. Rev. Lett. 125(19), 196402 (2020)] to low-energy states, and using the Green function formalism, we derived analytical results for the Hall conductivities as a function of the Fermi level and gate voltage. Depending on these parameters, we found quantized valley conductivity.
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Wojciechowska et al. (2024) studied this question.
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