Dodecagonal bilayer graphene quasicrystal and its approximants

Abstract

Dodecagonal bilayer graphene quasicrystal has 12-fold rotational order but lacks translational symmetry which prevents the application of band theory. In this paper, we study the electronic and optical properties of graphene quasicrystal with large-scale tight-binding calculations involving more than ten million atoms. We propose a series of periodic approximants which reproduce accurately the properties of quasicrystal within a finite unit cell. By utilizing the band-unfolding method on the smallest approximant with only 2702 atoms, the effective band structure of graphene quasicrystal is derived. Novel features, such as the emergence of new Dirac points (especially the mirrored ones), the band gap at M point and the Fermi velocity are all in agreement with recent experiments. The properties of quasicrystal states are identified in the Landau level spectrum and optical excitations. Importantly, our results show that the lattice mismatch is the dominant factor determining the accuracy of layered approximants. The proposed approximants can be used directly for other layered materials in honeycomb lattice, and the design principles can be applied for any quasi-periodic incommensurate structures.

Electronic structure of 30∘ twisted double bilayer graphene

“…In our calculations, all the hoppings with the neighbor distance less than 5 Å are under consideration. For 30 • TBG, the Fermi velocity and effective band structure calculated using this tight-binding model fit well the experimental results [28]. This tight-binding model has also been justified by comparing results with several experiments [44][45][46].…”

“…In this paper, the 30 • TDBG is approximated by a moiré pattern obtained by introducing slight stress to the top bilayer, which changes the lattice constant of the top bilayer from a = 2.456 toã = 2.454 Å. The moiré pattern is named a 15/26 approximant [28] due to the commensurate period 15 × √ 3a = 26 ×ã along the x axis, where √ 3a andã are the basic periods of the bottom and top bilayers, respectively. Accordingly, the elementary unit cell of the 15/26 approximant contains 5404 sites, including 1350 × 2 sites from the bottom bilayer and 1352 × 2 sites from the top bilayer.…”

“…The 30 • TBG, an incommensurate bilayer configuration, has been grown successfully on H-SiC(0001) [17], Pt(111) [18], Cu-Ni(111) [19], and Cu [20,21] surfaces. As the first two-dimensional quasicrystal based on graphene, 30 • TBG has received increasing attention both experimentally and theoretically [22][23][24][25][26][27][28]. A method to grow high-quality 30 • TBG epitaxially on SiC using borazine as a surfactant has also been proposed [22].…”

Electrical band flattening, valley flux, and superconductivity in twisted trilayer graphene

“…In particular, the different quantum states in twisted graphene systems stem from the possibility of controlling the ratio between kinetic and interaction terms. In twisted graphene bilayers, such tunability has allowed the realization of superconductivity [1,2,4], correlated insulators, topological networks [5,6], Chern insulators [11], and quasicrystals [12][13][14][15]. As a result, current experimental efforts are focusing on exploring new twisted van der Waals materials, with the aim of finding platforms that allow for an even higher degree of control [16,17].…”

Layer-dependent topological phase in a two-dimensional quasicrystal and approximant

“…4 A and B), the local bonding is consistent, and, finally, its unit cell is large enough to capture the relevant physics of the quasicrystalline phase, and the superlarge approximant Ta 181 Te 112 , yet small enough to be feasible for full ab initio treatment. We note that the use of approximants for the theoretical investigation of quasicrystals has been explored, for example, in twisted graphene bilayers, and their applicability has been proven by direct comparison with full quasicrystal lattices (34,35). To deduce the layer dependence of the material's electronic structure, we first classified the topological properties of three distinct structures comprising Ta 21 Te 13 units: a bulk structure, a monolayer with a 10-Å vacuum, and a bilayer with a 10-Å vacuum.…”