Twisted Bilayer Graphene: Low‐Energy Physics, Electronic and Optical Properties

Abstract

Van der Waals (vdW) heterostructures -formed by stacking or growing two-dimensional (2D) crystals on top of each other-have emerged as a new promising route to tailor and engineer the properties of 2D materials. Twisted bilayer graphene (tBLG), a simple vdW structure where the interference between two misaligned graphene lattices leads to the formation of a moiré pattern, is a test bed to study the effects of the interaction and misalignment between layers, key players for determining the electronic properties of these stackings. In this chapter, we present in a pedagogical way the general theory used to describe lattice mismatched and misaligned vdW structures. We apply it to the study of tBLG in the limit of small rotations and see how the coupling between the two layers leads both to an angle dependent renormalization of graphene's Fermi velocity and appearance of low-energy van Hove singularities. The optical response of this system is then addressed by computing the optical conductivity and the dispersion relation of tBLG surface plasmon-polaritons.

Electronic Observables for Relaxed Bilayer 2D Heterostructures in Momentum Space

“…We now describe practical details on the implementation of our momentum basis model for relaxed TBG [8], and present results on band structure and convergence. At energies near the Fermi energy, monolayer graphene's band structure near the Brillouin zone corners K and K can be described by the Dirac equation…”

“…with q = (q x , q y ) and σ = (σ 1 , σ 2 ), the first two Pauli matrices [7,8]. Importantly, the only free parameter is the Fermi velocity v F , which sets the dispersion (slope) of the bands, and it is linearly dependent on the nearest-neighbor hopping parameter in the graphene tightbinding model.…”

“…In Fig. 6, we show results of our model for a single valley of TBG [8] for three angles, both relaxed and unrelaxed. We see that at large angles (θ = 3.0 • ), the Dirac cones of graphene are still clearly visible.…”

“…These twisted bilayer graphene systems form a large scale moiré pattern [12,13,23] that is incommensurate [11,14,18,21], or aperiodic, which has motivated the development of methods to overcome the theoretical and computational challenge posed by the lack of periodicity. Most current physics investigations overcome the lack of periodicity by utilizing the lowenergy approximation of Bistritzer and MacDonald [1,9] that restricts interlayer scattering to nearest neighbor reciprocal superlattice vectors. Recent work has developed theory and efficient computational methods for studying the electronic structure of incommensurate 2D heterostructures without such approximations via configuration space representations [2,4,15,20,21].…”

Efficient computation of Kubo conductivity for incommensurate 2D heterostructures

“…Real space electronic approaches have recently been developed that directly compute electronic observables such as the density of states or conductivity [2,10,13]. There is also extensive literature on momentum space or k · p approaches, which use the monolayers' Bloch bases [1,7].…”