Rashba coupling and Lifshitz transition in monolayer graphene

We take a wide-angle view of the problem of monolayer graphene (MLG) where spin-degeneracy lifting is assumed to be possible by wedging in the tunable Rashba spin-orbit coupling(RSOC) and the sub-lattice staggered potential. We next consider the AB-stacked bi-layer graphene (BLG) system (the A-carbon of the upper sheet lying on top of the B-carbon of the lower one) and assume that a perpendicular electric field is created by the external gates deposited on the BLG surface. This system exhibits the occurrence of trigonal warping due to a (skew) interlayer hopping leading to the well-known Lifshitz transition(LT)[Y. Lemonik, I.L. Aleiner, C. Toke, and V.I. Fal'ko; arXiv:1006.1399]. We do not observe the replication of the features associated with BLG-LT in MLG in the presence of RSOC.