Interlayer hybridization in graphene quasicrystal and other bilayer graphene systems

Abstract

The incommensurate 30 • twisted bilayer graphene possesses both relativistic Dirac fermions and quasiperiodicity with 12-fold rotational symmetry arising from the interlayer interaction [Ahn et al., Science 361, 782 (2018) and Yao et al., Proc. Natl. Acad. Sci. 115, 6928 (2018)]. Understanding how the interlayer states interact with each other is of vital importance for identifying and subsequently engineering the quasicrystalline order for the applications in future electronics and optoelectronics. Herein, via symmetry and group representation theory we unravel an interlayer hybridization selection rule for D 6d bilayer consisting of two C6v monolayers no matter the system size, i.e., only the states from two C6v subsystems with the same irreducible representations are allowed to be hybridized with each other. The hybridization shows two categories including the equivalent and non-equivalent hybridizations with corresponding 12-fold symmetrical and 6-fold symmetrical antibonding (bonding) states, which are respectively generated from A1 + A1, A2 + A2, E1 + E1, and E2 + E2 four paring states and B1 + B1 and B2 + B2 two paring states. With the help of C6 and σx symmetry operators, calculations on the hybridization matrix elements verify the characteristic of the zero non-diagonal and nonzero diagonal patterns required by the hybridization selection rule. In reciprocal space, a vertical electric field breaks the 12-fold symmetry of originally resonant quasicrystalline states and acts as a polarizer allowing the hybridizations from two E1, E2 and B2 paring states but blocking others. Our theoretical framework also paves a way for revealing the interlayer hybridization for bilayer system coupled by the van der Waals interaction.

Polarization-dependent selection rules and optical spectrum atlas of twisted bilayer graphene quantum dots

“…(B2) and Eq. (B3) has been widely used to well describe the electronic structures in twisted bilayer graphene systems [36,40,45,[58][59][60][61]. The low-energy physics from the p z -orbital based tight-binding model is also consistent with that from Wannier-function based tight-binding method [62] and density functional theory [61].…”

“…(B3) has been widely used to well describe the electronic structures in twisted bilayer graphene systems [36,40,45,[58][59][60][61]. The low-energy physics from the p z -orbital based tight-binding model is also consistent with that from Wannier-function based tight-binding method [62] and density functional theory [61]. For the correlated sp 2 carbon systems with a moderate local U , the mean-field approximation can successfully capture the low-energy physics occurring in the systems [53][54][55]63].…”