Electron energy spectrum and the Berry phase in a graphite bilayer

We emphasize that there exist four Dirac-type points in the electron energy spectrum of a graphite bilayer near the point $K$ of its Brillouin zone. One of the Dirac points is at the point $K$, and three Dirac points lie nearby. Each of these three points generates the Berry phase $\pi$, while the Dirac point at $K$ gives the phase $-\pi$. It is these four points that determine the Berry phase in the bilayer. If an electron orbit surrounds all these points, the Berry phase is equal to $2\pi$.

Figure 1

Shown is the dispersion law $E(k_x,k_y)$ of the electrons $\left(e\right)$ and holes $\left(h\right)$ in the graphite bilayer near the point $K$. The spectrum contains four Dirac points. At the energy $E_c\sim 2\mathrm{meV}$ all the electron cones merge. Note that without neglecting the parameter ${\gamma }_4\approx 0.044\mathrm{eV}$, the spectrum $E(k_x,k_y)$ is slightly asymmetric relative to the plane $E=0$: The Dirac points in the central and side cones are at energies $E=0$ and $E=2{\gamma }_4{\gamma }_1{\gamma }_3^2∕{\gamma }_0^3\approx 0.15E_c$, respectively. Thus, in contrast with graphene, the graphite bilayer is a semimetal, and changing the Fermi level in it, one cannot obtain the concentration of the charge carriers less than $\sim {10}^8{\mathrm{cm}}^{-2}$. Scales of the plot are distorted for clarity.

Figure 2

The field $\mathbf{\Omega }(k_x,k_y)$ (arrows) for the electron band $e$ in the bilayer. The length of the arrows is proportional to $\mid \mathbf{\Omega }\mid$. Small circles mark the positions of the Dirac points in the $k_x\text{−}{k}_y$ plane, and the solid line shows an electron orbit with $E>E_c$. Note that near the central and the side Dirac points the vector $\mathbf{\Omega }$ circulates in opposite directions. The components $k_i$ are given in the dimensionless units ${\zeta }_i=(2\pi ∕\sqrt{3})(k_i∕\mid \Gamma K\mid )$ where $\mid \Gamma K\mid$ is the distance between the point $K$ and the center of the Brillouin zone, $\Gamma$.