Anti-localization of graphene under the substrate electric field

A simple criterion is provided on how the (anti-)localization properties of graphene are determined in the presence of inter-valley scattering, Kane-Mele topological mass term, and Rashba spin-orbit interaction (SOI). A set of (pseudo) time-reversal operations show that the number of effective internal degrees of freedom, such as spin and pseudo-spins distinguishing the sublattice and the valley, is the crucial parameter for localization. It is predicted that the perpendicular electric field due to the gate voltage of the substrate drives the system to anti-localization by enhancing the Rashba SOI.