Graphene bilayer and trilayer moiré lattice with Rashba spin-orbit coupling

We study the behavior of equilibrium spin currents near the magic angles of twisted bilayer and trilayer graphene in the presence of Rashba spin-orbit coupling. There is a substantial difference in the properties of local observables in twisted graphene layers, as compared with those in single and/or untwisted graphene layers. Remarkably, when plotted as a function of the twist angle θ , the electronic charge density and the equilibrium spin currents are nonanalytic at angles that are close to within 1% of the magic angles. In addition to the occurrence of rich spin texture patterns, these findings enable the determination of magic angles within an accuracy of less than 0 .01∘ in terms of a scanning tunneling microscopy measurement of the local density and spin-resolved measuring devices for measuring equilibrium spin currents.