Graphene bilayer and trilayer Moir\'e lattice with Rashba spin-orbit coupling

Abstract

We consider twisted bilayer and trilayer graphene in the presence of Rashba spin-orbit coupling and explore the physics of Moir\'e spintronics. The electronic charge density has a sharp step right at the magic angles $\theta_m$. As a result, local spin observables (polarization and equilibrium spin currents) have sharp peaks (of width about a small fraction of 1$^\circ$) as a function of the twist angle $\theta$, and abrupt sign reversals at $\theta_m$. Thereby, the magic angle can be determined in an unprecedented accuracy. In the first chiral limit, the spin currents vanish, but the peculiar pattern of the polarization at $\theta_m$ persists. Major differences result in spintronics of twisted bilayer graphene at magic angles as compared with the spintronics of single and/or {\it un-twisted} bilayer graphene. Thus, in addition to the numerous spectacular physical phenomena already reported in twisted bilayer graphene at magic angles, new phenomena also occur in twistronic spintronics.