Electronic structure of turbostratic graphene

Abstract

We explore the rotational degree of freedom between graphene layers via the simple prototype of the graphene twist bilayer, i.e., two layers rotated by some angle $\theta$. It is shown that, due to the weak interaction between graphene layers, many features of this system can be understood by interference conditions between the quantum states of the two layers, mathematically expressed as Diophantine problems. Based on this general analysis we demonstrate that while the Dirac cones from each layer are always effectively degenerate, the Fermi velocity $v_F$ of the Dirac cones decreases as $\theta\to 0^\circ$; the form we derive for $v_F(\theta)$ agrees with that found via a continuum approximation in Phys. Rev. Lett., 99:256802, 2007. From tight binding calculations for structures with $1.47^\circ \le \theta < 30^\circ$ we find agreement with this formula for $\theta \gtrsim 5^\circ$. In contrast, for $\theta \lesssim 5^\circ$ this formula breaks down and the Dirac bands become strongly warped as the limit $\theta \to 0$ is approached. For an ideal system of twisted layers the limit as $\theta\to0^\circ$ is singular as for $\theta > 0$ the Dirac point is fourfold degenerate, while at $\theta=0$ one has the twofold degeneracy of the $AB$ stacked bilayer. Interestingly, in this limit the electronic properties are in an essential way determined \emph{globally}, in contrast to the 'nearsightedness' [W. Kohn. Phys. Rev. Lett., 76:3168, 1996.] of electronic structure generally found in condensed matter.Comment: Article as to be published in Phys. Rev B. Main changes: K-point mapping tables fixed, several changes to presentation

Many‐body effects in optical response of graphene‐based structures

“…[77] As the twist angle decreases, the size of the unit cell enlarges [78] and the number of atoms comprising it may reach a few thousands. [79] The calculations of such configurations require invoking of the TB approaches, whereas band electronic structures of bilayers with reasonably small cells were obtained within DFT. [76,80], [81] Both types of computational approaches gave qualitatively similar results indicating that for the twist angles above 108 up to the symmetric point of 308, the dispersion in a twisted bilayer is linear as in single-layer graphene at the K point.…”

“…[80] At an angle of less than 58, the p bands are flattened that is strictly different from their behavior in the monolayer and AB stacked bilayer. [79,82] However, after the value of $1.58, the bands acquire a parabolic dispersion. [82] The critical twist angle of $108 causes the formation of the AA and AB stacking regions in a bilayer and electron wave senses this local inhomogeneity.…”

Long-range atomic ordering and variable interlayer interactions in two overlapping graphene lattices with stacking misorientations

“…From a theoretical perspective, twisted bilayer graphene (TBG) has been examined extensively using both continuum [6][7][8][9] and ab-initio [10][11][12][13][14] approaches. These studies have shown that the individual monolayers become significantly decoupled by the rotational faults.…”

Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene

“…The electronic properties of moiré crystals depend sensitively on the ratio of the interlayer hybridization strength, which is independent of twist angle, to the band energy shifts produced by momentum space rotation (5)(6)(7)(8)(9)(10)(11)(12). In bilayer graphene, this ratio is small when twist angles exceed about 2° (10,13), allowing moiré crystal electronic structure to be easily understood using perturbation theory (5).…”