Landau-Level Degeneracy and Quantum Hall Effect in a Graphite Bilayer

We derive an effective two-dimensional Hamiltonian to describe the low-energy electronic excitations of a graphite bilayer, which correspond to chiral quasiparticles with a parabolic dispersion exhibiting Berry phase $2\pi$. Its high-magnetic-field Landau-level spectrum consists of almost equidistant groups of fourfold degenerate states at finite energy and eight zero-energy states. This can be translated into the Hall conductivity dependence on carrier density, ${\sigma }_{xy}(N)$, which exhibits plateaus at integer values of $4e^2/h$ and has a double $8e^2/h$ step between the hole and electron gases across zero density, in contrast to $(4n+2)e^2/h$ sequencing in a monolayer.

Figure 1

Left: schematic of the bilayer lattice (bonds in the bottom layer $A,B$ are indicated by solid lines and in the top layer $\tilde{A},\tilde{B}$ by dashed lines) containing four sites in the unit cell: $A$ (white circles), $\tilde{B}$ (hashed), $\tilde{A}B$ dimer (solid). Right: the lattice of a monolayer.

Figure 2

Schematic of the Fermi line in the valley $K$, $\xi =1$, for high (light shading) and low density (dark shading). Note that the asymmetry of the Fermi line at valley $\tilde{K}$, $\xi =-1$, is inverted.

Figure 3

Numerically calculated Landau levels for various values of $v_3$ for valley $K$, $\xi =1$. (a) for $B=0.1T$ ($\hslash {\omega }_c=0.216\mathrm{meV}$ assuming ${\gamma }_1=0.39\mathrm{eV}$, $v=8.0\times {10}^5\mathrm{m}/\mathrm{s}$), (b) for $B=1T$ ($\hslash {\omega }_c=2.16\mathrm{meV}$). Broken lines show groups of four consecutive levels that become degenerate at large $v_3$, with each group separated from the next by two solid lines representing levels that are not degenerate.

Figure 4

Landau levels for a bilayer (left) and monolayer (right). Brackets ($n,\xi$) indicate LL number $n$ and valley index $\xi =\pm 1$. In the center the predicted Hall conductivity ${\sigma }_{xy}$ (center) as a function of carrier density for bilayer (solid line) is compared to that of a monolayer (dashed line).