Theory of the Three-Dimensional Quantum Hall Effect in Graphite

We predict the existence of a three-dimensional quantum Hall effect plateau in a graphite crystal subject to a magnetic field. The plateau has a Hall conductivity quantized at $\frac{4e^2}{\hslash }\frac{1}{c_0}$ with $c_0$ the $c$-axis lattice constant. We analyze the three-dimensional Hofstadter problem of a realistic tight-binding Hamiltonian for graphite, find the gaps in the spectrum, and estimate the critical value of the magnetic field above which the Hall plateau appears. When the Fermi level is in the bulk Landau gap, Hall transport occurs through the appearance of chiral surface states. We estimate the magnetic field necessary for the appearance of the effect to be 15.4 T for electron carriers and 7.0 T for holes.

Figure 1

(a) Graphite in Bernal stacking. (b) Under strong magnetic field, graphite is gapped in the bulk and exhibits chiral surface sheet states. (c) Idealized Brillouin zone for graphite. (d) Predicted 3D Hall conductivity, quantized in units of $1/c_0$. Only one plateau is observable in graphite.

Figure 2

Zero mode spectrum and Landau levels of our toy model in $B=10\mathrm{T}$ interpolated between graphene ($x=0$), with no $k_z$ dispersion, and graphite ($x=1$). The zero mode (blue line) is doubly degenerate, giving a 3D Hall conductance of $\frac{e^2}{h}\frac{1}{c_0}$ per independent $K$ point per spin.

Figure 3

Hopping matrix elements and flux assignment. Sites $A$ and $B$ belong to different graphene layers than $C$ and $D$. Bernal stacking corresponds to $A$ and $C$ differing by a $c$-axis translation. Not shown are further-neighbor in-plane hoppings $t_{AB}^{\parallel ,2}$ and $t_{AB}^{\parallel ,3}$. $\varphi =2\pi p/q$ is the magnetic flux per hexagon in units of $\hslash c/e$, and $n$ runs from 1 to $q$ the size of the magnetic unit cell.

Figure 4

Clockwise from upper left: (a) $B=5\mathrm{T}$, no gap present in the full spectrum. (b) $B=12\mathrm{T}$, clear gap in the hole LL spectrum. (c) $B=20\mathrm{T}$ clear gaps for both hole and electron LL. Spin-up (blue line) and spin-down (red line) bands are shown. (d) Principal energy gaps surrounding $n=0$ LLs, including effects due to Zeeman splitting. The particle gap collapses at $B_c^e=15.4\mathrm{T}$ and the hole gap at $B_c^h=7.0\mathrm{T}$. All energies have been shifted upward by 100 meV.

Figure 5

Zero mode, first electron Landau level, and surface state spectrum over the first Brillouin zone for one spin species at $B=40\mathrm{T}$. (a) Surface state spectrum, (b) surface state and bulk $n=0,1$ Landau levels where the $n=1(n=0)$ levels are shifted by $+0.1\mathrm{eV}$ $(-0.1\mathrm{eV})$ for clarity. (Inset) $n=0,1$ Landau levels unshifted without surface states.