Trigonal warping and Berry’s phaseNπin ABC-stacked multilayer graphene

Abstract

The electronic band structure of ABC-stacked multilayer graphene is studied within an effective mass approximation. The electron and hole bands touching at zero energy support chiral quasiparticles characterized by Berry's phase $N\ensuremath{\pi}$ for $N$ layers, generalizing the low-energy band structure of monolayer and bilayer graphene. We investigate the trigonal-warping deformation of the energy bands and show that the Lifshitz transition, in which the Fermi circle breaks up into separate parts at low energy, reflects Berry's phase $N\ensuremath{\pi}$. It is particularly prominent in trilayers, $N=3$, with the Fermi circle breaking into three parts at a relatively large energy that is related to next-nearest-layer coupling. For $N=3$, we study the effects of electrostatic potentials which vary in the stacking direction, and find that a perpendicular electric field, as well as opening an energy gap, strongly enhances the trigonal-warping effect. In magnetic fields, the $N=3$ Lifshitz transition is manifested as a coalescence of Landau levels into triply degenerate levels.