Chiral decomposition in the electronic structure of graphene multilayers

We show that the low-energy electronic structure of arbitrarily stacked graphene multilayers with nearest-neighbor interlayer tunneling consists of chiral pseudospin doublets. Although the number of doublets in an $N$-layer system depends on the stacking sequence, the pseudospin chirality sum is always $N$. $N$-layer stacks have $N$ distinct Landau levels at $E=0$ for each spin and valley, and quantized Hall conductivity ${\sigma }_{xy}=\pm (4e^2∕h)(N∕2+n)$, where $n$ is a non-negative integer.

Figure 1

(Color online) (a) Energetically favored stacking arrangements for graphene sheets. The honeycomb lattice of a single sheet has two triangular sublattices, labeled by $\alpha$ and $\beta$. Given a starting graphene sheet, the honeycomb lattice for the next layer is positioned by displacing either $\alpha$ or $\beta$ sublattice carbon atoms along a honeycomb edge. This results in three distinct two-dimensional (2D) sheets, labeled by A, B, and C. Representative $\alpha$ and $\beta$ sublattice positions in A, B, and C layers are identified in this illustration. It is also possible to transform between layer types by rotating by 60° about one of the carbon atoms. (b) Each added layer cycles around this stacking triangle in either the right-handed or the left-handed sense. Reversals of the sense of this rotation tend to increase the number of low-energy pseudospin doublets $N_D$. In graphite, the Bernal $\left(\mathrm{A}\mathrm{B}\right)$ stacking corresponds to a reversal at every step and orthorhombic $\left(\mathrm{A}\mathrm{B}\mathrm{C}\right)$ stacking corresponds to no reversals.

Figure 2

(Color online) Stacking sequences and linkage diagrams for $N=3,4,5$ layer stacks. The low-energy band and the Landau level structure of a graphene stack are readily read off these diagrams as explained in the text. Shaded ovals link $\alpha$ and $\beta$ nearest interlayer neighbors.

Figure 3

(a) Band structure and (b) low-energy Landau levels of tetralayer graphene with $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{B}$ stacking. These bands were evaluated for nearest intralayer neighbor hopping $t=3\mathrm{eV}$ and nearest interlayer neighbor hopping $t_{\perp }=0.1t$.

Figure 4

Landau level spectrum at the $K$ valley as a function of ${\gamma }_3$ for an $\mathrm{A}\mathrm{B}$ stacked bilayer for (a) $B=0.1\mathrm{T}$ and (b) $B=1\mathrm{T}$, and as a function of ${\gamma }_2$ for an $\mathrm{A}\mathrm{B}\mathrm{A}$ stacked trilayer for (c) $B=1\mathrm{T}$ and (d) $B=10\mathrm{T}$. Here $t=3\mathrm{eV}$, $t_{\perp }=0.1t$, and ${\omega }_c=eB∕mc$, with $m=t_{\perp }∕2v^2$, were used.

Figure 5

(Color online) Noninteracting Hall conductivity as a function of the Fermi energy for all inequivalent four-layer graphene stacks when $B=10\mathrm{T}$, $t=3\mathrm{eV}$, and $t_{\perp }=0.1t$.