Trigonal warping and Berry’s phase $N\pi$ in ABC-stacked multilayer graphene

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 $N\pi$ 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 $N\pi$. 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.

Figure 1

(a) Schematic of the ABC-stacked trilayer lattice containing six sites in the unit cell, $A$ (white circles) and $B$ (black circles) on each layer, showing the Slonczewski-Weiss-McClure parameterization (Ref. 39) of relevant couplings ${\gamma }_0$ to ${\gamma }_4$. (b) Schematic of the hexagonal Brillouin zone with two inequivalent valleys $K_{\pm }$ showing the momentum $\mathbf{p}$ measured from the center of valley $K_{+}$. (c) Schematic of the unit cell of ABC-stacked trilayer graphene, (d) ABA-stacked trilayer graphene, and (e) bilayer graphene. In (c), ${\gamma }_2$ describes a vertical coupling between sites $B3$ and $A1$ in different unit cells.

Figure 2

(Color online) Band dispersion of ABC-stacked trilayer graphene in the vicinity of $K_{+}$ along $p_x$ axis. Parameter values are ${\gamma }_0=3.16\text{eV}$, ${\gamma }_1=0.39\text{eV}$, ${\gamma }_2=-0.020\text{eV}$, ${\gamma }_3=0.315\text{eV}$, and ${\gamma }_4=0.044\text{eV}$ (Ref. 39).

Figure 3

(Color) (a) Equienergy contour plots of the lowest electron band of ABC trilayer graphene at (a) ${\Delta }_1=0$ and (b) $0.4{\gamma }_1$. Numbers on the contours indicate energy in units of ${\gamma }_1$. Filled and empty triangles represent local minima and maxima, respectively, of the energy band.

Figure 4

(Color online) Landau levels of ABC trilayer graphene, plotted against (a) $B^{1/2}$ at fixed ${\Delta }_1=0$, and (b) ${\Delta }_1$ at fixed magnetic field ${\left(2\hslash v^{2}eB\right)}^{1/2}=0.1{\gamma }_1$ $\left(B\sim 1\text{T}\right)$. The region in which Landau levels are triply degenerate is highlighted by shading.

Figure 5

Low-energy band structure of ABC-stacked multilayer graphene for several different layer numbers $N$, at ${\Delta }_1={\Delta }_2=0$. Solid and dashed curves are calculated using Eq. (13) and its approximation Eq. (14), respectively. Insets show the equienergy lines at $ϵ=0.04{\gamma }_1$. The black and white arrows (circles in insets) represent Dirac points having Berry’s phases $\xi \pi$ and $-\xi \pi$, respectively.