Electronic spectrum of twisted bilayer graphene

Abstract

We study the electronic properties of twisted bilayers graphene in the tight-binding approximation. The interlayer hopping amplitude is modeled by a function, which depends not only on the distance between two carbon atoms, but also on the positions of neighboring atoms as well. Using the Lanczos algorithm for the numerical evaluation of eigenvalues of large sparse matrices, we calculate the bilayer single-electron spectrum for commensurate twist angles in the range 1• . We show that at certain angles θ greater than θc ≈ 1.89• the electronic spectrum acquires a finite gap, whose value could be as large as 80 meV. However, in an infinitely large and perfectly clean sample the gap as a function of θ behaves non-monotonously, demonstrating exponentially-large jumps for very small variations of θ. This sensitivity to the angle makes it impossible to predict the gap value for a given sample, since in experiment θ is always known with certain error. To establish the connection with experiments, we demonstrate that for a system of finite sizeL the gap becomes a smooth function of the twist angle. If the sample is infinite, but disorder is present, we expect that the electron mean-free path plays the same role asL. In the regime of small angles θ < θc, the system is a metal with a well-defined Fermi surface which is reduced to Fermi points for some values of θ. The density of states in the metallic phase varies smoothly with θ.

Model for the metal-insulator transition in graphene superlattices and beyond

“…This is, however, a nontrivial task that has not been accomplished so far. Thanks to extensive studies using various methods [6,[11][12][13][14][15][16][17][18][19][20][21][22][23], the band structure of TBG at small twist angle is known to be rather complex and depends sensitively on microscopic details such as lattice relaxation. Near the socalled magic twist angle, various methods find four nearly flat minibands at low energy, but differ significantly on important features such as their bandwidth and the gap to excited bands.…”

Spatially resolved electronic structure of twisted graphene

“…Both Vσ and Vπ decay rapidly when the distance between the two sites is larger than the lattice parameter a0 = 2.46 Å, and the contribution of is negligible in the interlayer hoppings in multilayer graphene [23,24]. Here we use 0.24 eV as the maximum value of (for two sites with A-A stacking, the same value as used in Ref.…”

“…Wong et al [16] found, besides the coexistence of moiré patterns and moiré super-superlattices, also a very rich an interesting electronic structure. Despite the fact that the electronic structure of twisted bilayer graphene has been extensively studied [5,[11][12][13][14][15][16][17][18][19][20][21][22][23][24], the spatial variation of electronic structure within the unit cell of the moiré pattern did not receive a lot of attention yet.…”

Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle

“…In real materials, the difference between an incommensurate and a commensurate twist-angle is less clear, as the presence of imperfections (strain, tears, ripples) may make even a commensurate system sample Ω continuously. The effect of disorder on twisted bilayer graphene's electronic properties has begun to be investigated theoretically 20,21 , but we do not study it here.…”