Intrinsic anomalous, spin and valley Hall effects in ’ex-so-tic’ van-der-Waals structures

Abstract We consider the anomalous, spin, valley, and valley spin Hall effects in a pristine graphene-based van-der-Waals (vdW) heterostructure consisting of a bilayer graphene (BLG) sandwiched between a semiconducting van-der-Waals material with strong spin-orbit coupling (e.g., $$\hbox {WS}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>WS</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> ) and a ferromagnetic insulating vdW material (e.g. $$\hbox {Cr}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>Cr</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> $$\hbox {Ge}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>Ge</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> $$\hbox {Te}_6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>Te</mml:mtext> <mml:mn>6</mml:mn> </mml:msub> </mml:math> ). Due to the exchange proximity effect from one side and spin-orbit proximity effect from the other side of graphene, such a structure is referred to as graphene based ’ex-so-tic’ structure. First, we derive an effective Hamiltonian describing the low-energy states of the structure. Then, using the Green’s function formalism, we obtain analytical results for the Hall conductivities as a function of the Fermi energy and gate voltage. For specific values of these parameters, we find a quantized valley Hall conductivity.