Gate-induced Dirac cones in multilayer graphenes

We study the electronic structures of $ABA$ (Bernal) -stacked multilayer graphenes in a uniform perpendicular electric field, and we show that the interplay of the trigonal warping and the potential asymmetry gives rise to a number of emergent Dirac cones nearly touching at zero energy. The band velocity and the energy region (typically a few tens of meV) of these gate-induced Dirac cones are tunable with the external electric field. In $ABA$-trilayer graphene, in particular, applying an electric field induces a nontrivial valley Hall state, where the energy gap at the Dirac point is filled by chiral edge modes which propagate in opposite directions between two valleys. In four-layer graphene, in contrast, the valley Hall conductivity is zero and there are no edge modes filling in the gap. A nontrivial valley Hall state generally occurs in asymmetric odd-layer graphenes, and this is closely related to a hidden chiral symmetry which exists only in odd-layer graphenes.