The graphene twist bilayer represents the prototypical system for investigating the stacking degree of freedom in few-layer graphenes. The electronic structure of this system changes qualitatively as a function of angle, from a large-angle limit in which the two layers are essentially decoupled—with the exception of a 28-atom commensuration unit cell for which the layers are coupled on an energy scale of ≈8meV—to a small-angle strong-coupling limit. Despite sustained investigation, a fully satisfactory theory of the twist bilayer remains elusive. The outstanding problems are (i) to find a theoretically unified description of the large- and small-angle limits, and (ii) to demonstrate agreement between the low-energy effective Hamiltonian and, for instance, ab initio or tight-binding calculations. In this article, we develop a low-energy theory that in the large-angle limit reproduces the symmetry-derived Hamiltonians of Mele [Phys. Rev. B 81, 161405 (2010)], and in the small-angle limit shows almost perfect agreement with tight-binding calculations. The small-angle effective Hamiltonian is that of Bistritzer and MacDonald [Proc. Natl. Acad. Sci. (U.S.A.) 108, 12233 (2011)], but with the momentum scale ΔK, the difference of the momenta of the unrotated and rotated special points, replaced by a coupling momentum scale g(c)=8π3asinθ2. Using this small-angle Hamiltonian, we are able to determine the complete behavior as a function of angle, finding a complex small-angle clustering of van Hove singularities in the density of states (DOS) that after a “zero-mode” peak regime between 0.90∘<θ<0.15∘ limits θ<0.05∘ to a DOS that is essentially that of a superposition DOS of all bilayer stacking possibilities. In this regime, the Dirac spectrum is entirely destroyed by hybridization for −0.25<E<0.25eV with an average band velocity ≈0.3vF(SLG) (where SLG denotes single-layer graphene). We study the fermiology of the twist bilayer in this limit, finding remarkably structured constant energy surfaces with multiple Lifshitz transitions between K– and Γ-centered Fermi sheets and a rich pseudospin texture.
- Received 8 September 2015
DOI:https://doi.org/10.1103/PhysRevB.93.035452
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