Induced spin-orbit coupling in twisted graphene–transition metal dichalcogenide heterobilayers : Twistronics meets spintronics

We propose an interband tunneling picture to explain and predict the interlayer twist-angle dependence of the induced spin-orbit coupling in heterostructures of graphene and monolayer transition metal dichalcogenides (TMDCs). We obtain a compact analytic formula for the induced valley Zeeman and Rashba spin-orbit coupling in terms of the TMDC band structure parameters and interlayer tunneling matrix elements. We parametrize the tunneling matrix elements with few parameters, which in our formalism are independent of the twist angle between the layers. We estimate the value of the tunneling parameters from existing density functional theory calculations at zero twist angle and we use them to predict the induced spin-orbit coupling at nonzero angles. Provided that the energy of the Dirac point of graphene is close to the TMDC conduction band, we expect a sharp increase of the induced spin-orbit coupling around a twist angle of ${18}^{\ensuremath{\circ}}$.