Electronic Structure of Multilayer Graphene with Arbitrary Stackings

Abstract

Stacking geometry in multilayer graphene (MLG) provides an interesting degree of freedom to engineer its electronic structure near the Fermi level, wherein the linear bands in single layer graphene could retain or evolve into parabolic or flat bands. Using a tight-binding model, we carried out a detailed analytical analysis of the electronic band structures for arbitrarily stacked MLGs. We show that their low energy band dispersions near the Fermi level may be deduced from its substacks in isolation. The analytical solutions of the momenta with zero eigenvalue for an AA stacking allows us to generalize the results of the zero energy momenta for arbitrarily stacked MLGs. Moreover, we find that an interplay of parallel and rhombohedral stackings allows for flat band engineering and enhancement in arbitrarily stacked MLGs. The existence of flat bands in MLGs might offer another interesting platform for exploring the superconductivity in graphene systems beyond the twisted bilayer graphene.

Figures (7)

• Figure 1: (Color online)(a) Contour map of the positive branch of the band dispersion for monolayer graphene in the whole hexagonal Brillouin zone. The corners of the hexagon are the points of zero energy. Examples of the $K$ and $K'$ points have been shown. (b) Schematic of AAA, ABA, and ABC graphene stacking geometry in the Bloch basis. The $a$ and $b$ bases (gray), hopping integrals (cyan), and stacking letters (black) have been shown, which naturally constructs the Hamiltonian.
• Figure 2: Calculated electronic band dispersions for (a) tetralayer AA stacked graphene, (b) tetralayer AB stacked graphene, and (c) tetralayer ABC stacked graphene. The approximate low-energy band analytical solution (red) are also shown for comparison in (c). Note the accuracy at small momentum. The maximum flat band range, when $N$ is large, is bounded by the momentum values plotted via purple vertical lines.
• Figure 3: The low energy charge density for tetralayer ABC stacked graphene with sufficiently low momentum in the flat band. The stack is copied over horizontally to show the periodic nature of the crystal lattice. Each dot represents a Bloch site (labeled in a picture for a given stack), and the electronic $\pi$ orbitals are highlighted with a given intensity based on the eigenvectors to our Hamiltonian. Note the edge state localization at sites $a_1$ and $b_4$. These are the low energy basis states in our Schrieffer-Wolf transform.
• Figure 4: (a) Band dispersions for AAABCCC (blue). The dispersion for an ABC trilayer graphene is shown in red for comparison. Note how the flat band flattens the points of zero energy. (b) Band dispersions for AAAAAABCABCCCCCC (blue). The band dispersions for ABCABC are shown in red. (c) Band dispersions for ABABCBC (blue). The band dispersions for ABC trilayer graphene are shown in red. (d) Band dispersions for BABABABCABCBCBCB (blue) and for ABCABC (red).
• Figure 5: Sufficiently low energy flat band eigenstate for (a) ABABCBC and (b) BABABABCABCBCBCB. For clarity, we have highlighted the stacking sequence of the carbon atoms using green dots. Note the localization at the edges of the embedded ABC substack. We do not see this for an ABC substack embedded within a parallel stacking.