2.1

Electronic band structure of graphene in the tight binding approximation

We begin with a brief review of the electronic structure of graphene to establish a common notation for discussing the interlayer electronic structure of twisted bilayer graphene (tBLG) in the later sections. The atomic structure of single layer graphene in Figure 1a shows distinct carbon atoms, A and B, in the unit cell. The lattice vectors are: where a₀ is the nearest neighbor distance, a₀ = 1.42 Ź⁵. Out of the four valence electrons, three are used for sp², bonds in carbon. The fourth electron is in a p_zorbital, and there are two electrons per unit cell in a p_zorbital. Therefore, there are two orbitals, one for atom A, and one for atom B for a unit cell (there are two π- bands, one for each electron). The tight-binding model Hamiltonian: Hermitian Conjugate (HC)¹⁶ can be expanded over the graphene unit cell site basis to obtain,

here t represents the transfer integral corresponding to electronic hopping between atoms and the Fourier-transformed creation and annihilation operators are defined as,

Alternatively, this can be expressed in a matrix form, with an oveall phase factor of e^{–ik′·(r–r′)} as

The diagonal term H_AA corresponds to the orbital interactions between two A atoms, H_BB corresponds to the orbital interactions between two B atoms. H_AB, and H_BA represent hopping between the two sites A, and B. To find the dispersion relation, setting onsite energies, ε_A = ε_B = 0, and diagonalizing the Hamiltonian;

, expressing in (x, y) components of k, , where with lattice constant .

Figure 1.

(a) Lattice structure of single layer graphene; Two carbon atoms per unit cell are labeled as 1, and 2 (grey, and orange). Primitive vectors are , and , (b) Lattice structure of bilayer graphene. Sub lattices for the lower layer are A1, and B1, and those for the top layer are A2, and B2, with relative positions of atoms projected onto the x-y plane (c) Lattice structure of twisted bilayer graphene for rotation θ, (d) tBLG θ=21.8°, with [(m,n)=(1,2)], highlighted unit cell for (e) θ=9.4°, (f) θ=3.9°

This well-known dispersion relation is illustrated graphically in Figure 2a. At Brillouin zone boundaries, the function f(k) goes to zero.¹⁵ At K points, the solutions of the dispersion relation are degenerate giving zero bandgap between the valence band, and conduction band. Near K points, the transfer Hamiltonian is like the Dirac Hamiltonian, and the dispersion relation is approximately linear, which describes massless quasiparticles.²

Figure 2.

(a) Representation of the band structure for single layer graphene. (b) tBLG free electron band-structure, (c) the band dispersion near the K-point in bilayer graphene, (d) cross-sectional view of tBLG shows vHS transitions X₁₃, X₂₄ between avoided crossing regions.

2.2

Electronic band structure of bilayer graphene in the tight binding approximation

For AB stacked or Bernal bilayer graphene, there are interlayer 2p_z orbital interactions between the atoms of the unit cell of the upper layer (1) and lower layer (2). In Figure 1b we illustrate the lattice structure of bilayer graphene along with the relative positions of atoms projected onto the x-y plane. In this bilayer structure, there are now clearly four distinct atoms (A1, B1, A2, and B2). Their electronic coupling can be approximated by considering the intralayer contributions (A1/B1, B1/B2), and the new nearest-neighbor interlayer contributions corresponding to A1/A2 and B1/B2 shown in Figure 1b.¹⁶ Taking four distinct atoms into account (A1, B1, A2, B2) in the tight binding model, Castro et al. and Neto et al. show that the Hamiltonian can be re-expressed as:

Here, ε represents the onsite energies, is the intralayer transfer integral for electronic hopping between A1 to B1, and A2 to B2 and -γ represents the hopping between A1 to A2 , and B1 to B2 .

Using the Fourier transformed operators; , and defining vectors it has been shown the approximate Hamiltonian for bilayer graphene is, ; which can alternately be expressed in matrix form as,

Where, , and ε_A1, ε_B1, ε_A2, and ε_B2 are onsite energies on atomic sites A1, B1, A2, and B2 respectively. The nearest neighbor vectors are: , r₄ = b₀(0,0,1); giving

The 2 × 2 blocks in the upper-left side and the lower-right side of the Hamiltonian in equation (2) represents the intralayer coupling, which looks similar to the monolayer Hamiltonian terms. The 2 × 2 blocks in the upper-right side, and the lower-left side of the Hamiltonian represents interlayer coupling, where the parameter γ describes the coupling between orbitals A1, A2, and/or B1, B2¹⁸. The parameters γ = 0.14 eV,t = 3.16 eV,and ε_A1,ε_B1,ε_A2,ε_B2 = 0 in equation (2) have been obtained using infrared spectroscopy²⁰. The well-known free electron band structure of AB stacked bilayer graphene is roughly illustrated in the Figure 2b.

Without bias V, 2p_z orbital interactions between four atomic sites are responsible for four bands in the dispersion relation as shown in Figure 2c, two for the conduction band, and two for the valence band. The valence band pairs and conduction band pairs are split by the amount of about 0.4 eV, which is an interlayer coupling energy. The splitting occurs due to the strong bonding, and anti-bonding between atomic orbitals A1, A2, and/or B1, B2. The hopping between A1, A2, and/or B1, B2 is responsible for low energy bands. The tight binding model is in good agreement with DFT calculations in the low energy limit^21,22,23. Wang et al. showed that with a potential bias V between two layers of graphene, it is further possible to see the shift in electrochemical potential between the layers³².

2.3

Effective free electron model: Electronic band structure of tBLG

Lastly, we complete our review of the free electron graphene models by considering of the structure of twisted bilayer graphene defined by two layers of AA stacked bilayer graphene rotated around a common site by an arbitrary angle θ. The two lattice vectors for AA stacked bilayer graphene with θ = 0° are just a₁ = a₀ (1,0) and . For non-zero angles the lattice vectors are more complicated because they must span the new superlattice, and are given stacking angle indexed vectors; ,and .²⁵ Rotation angle θ = 0° produces AA stacked bilayer graphene and θ = 60° corresponds to AB bilayer graphene stacking.

Rotation angle θ and – θ produce a similar band structure. tBLG does not have a periodic structure in general because of mismatch between the uniformity of layers with respect to each other. The structure is periodic only for some special angles with a well-defined unit cell. A periodic lattice structure of tBLG for rotation angles θ = 21.8°, and θ = 9.4°, θ =3.9° is shown in Figure 1d, e and f. When the rotation angle is small, the interference between two lattice vectors in two layers gives rise to the pattern called the ‘Moiré pattern’^25,26. The unit cell is relatively big for small rotation angles as seen in Figures 1e and f. The lattice vectors for a given angle are: ; (for layer 1),; (for layer 2) and the lattice constant is ; where m, n are integers.²⁴

For electronic states near the Fermi level, only p_z orbitals needs to be taken into account in the tight binding model. For twisted bilayer graphene, neighbors are rotated with respect to each other (unlike Bernal AB stacking). Therefore, we now have to consider both the ppπ and the ppσ terms to account for the new interlayer interactions between atomic orbitals. As shown by Shallcross et al.^24,25, the Hamiltonian can be written as,

where, t(R_i–R_j)is the transfer integral. The transfer integral matrix equations for each type of interlayer orbital interactions are given by: , and , with the ratio Where, a₀ is the nearest neighbor distance, a₁is the interlayer distance, d is the distance between two orbitals, γ₀is the transfer integral between nearest neighbors, and γ₁ is the transfer integral between nearest neighbor atoms that are located vertically. Combining these terms, the combined transfer integral is with, a₀ ≈ 0.14 nm, a₁ ≈ 3.35 Å, d ≈ 0.335 nm, γ₀ ≈ 2.7 eV, γ₁ ≈ 0.48 eV, as obtained from literature²⁶. Using the above transfer integral, the dispersion relation can be obtained from the determinant of the Hamiltonian of equation (3).¹³ A rough illustration of the resulting free electron interlayer band structure of tfiLG is shown in Figure 2b. There is an anti-crossing where bands of two layers meet. This anti-crossing is a result of interlayer interactions between the layers.

Figure 2d sketches the cross-section view of the band structure, showing the van Hove Singularity (vHS) transitions X₁₃ and X₂₄.The bands near K, and K′ come from single layer graphene (layer 1, and layer 2), and they are folded into the same Brillouin zone. These bands don’t mix with each other giving rise to the linear dispersion relation near K, and K′. Near the M point, there is a splitting in energy due to anti-crossing of the bands. The allowed optical transitions are also shown. The band structure of tBLG for different rotation angles look similar, however, the overall energy decreases by minimizing the rotation angle. The splitting energy near the M-point is of the order of interlayer coupling between the two layers, which is about 0.2 eV for all possible rotation angles.¹⁵ For rotation angles < 1°, parabolic dispersion is obtained. For rotation angles around 1°, there bands are flat near the Fermi level. There is a transition from parabolic to linear dispersion for rotation angles 1° ≤ θ ≤ 2°. For angles above 10° and up to the symmetric point of 30°, the dispersion is linear as in single layer graphene at the K point²⁷.

2.4

Review of bound exciton model for tBLG

Upon photoexcitation of tBLG, exciton effects may manifest themselves owing to Coulombic interactions of e-h pairs. To theoretically model the many-body excitonic effects, it is necessary to solve the BSE as shown by Rohlfing et al.²⁸: is the exciton wave function in k-space, Ω^s is the exciton eigenenergy, K^eh is the e-h interaction kernel, and |vk⟩, |ck⟩ are the hole and electron states, respectively. In order to predict possible bound exciton states, it is important to know the e-h attractive energy , which roughly corresponds to the binding energy, and recently the first-principle BSE simulations in Liang et al.¹⁴ reported that for 21.8° tBLG.

Specifically, Liang et al. solved for , which is the net exciton kinetic energy plus the weighted single-particle band energy difference between electrons and holes¹⁴. To simplify the direct analysis of the first principle BSE simulation, low-energy effective model is often further assumed, giving a simpler perturbed Hamiltonian that is analogous to Equation (1), specfically²⁹;

The average interlayer interaction between AB and BA stacking order is described by matrix T with interlayer coupling strength Δ. The low-energy effective model is best for small-twisting angles because the linear Dirac-Fermion dispersion is preserved. Using the BSE and the low effective model, Liang et al.¹⁴ found that upon resonant excitation of tBLG, you can access degenerate Fano resonant transitions X₁₃, X₂₄ shown in Figure 2d which then effectively rehybridize as depicted in Figure 3. The symmetric superposition of the two degenerate transitions give rise to a higher energy bright state and anti-symmetric superposition give rise to the lower energy dark state (‘ghost’ state, see figure 3). The constructive interference couples with the lower lying continuum which broadens the resulting resonant bright excitonic state. The antisymmetric superposition cancels the coupling between the lower lying continuum making it a localized bound excitonic state. The so-called ghost Fano resonance effect was found in quantum dot molecules.³⁰ This model is appropriate for small twist angle, however, the double resonance of transitions and the destructive interference plays crucial role in existence of strongly bound excitons.¹⁴ The formation mechanism of this dark state X_A is very general, and may hold exciting promise for other twisted stacked 2D vdW materials.

Figure 3.

Upon resonant optical excitation, degenerate Fano resonance states X₁₃ and X₂₄ rehybridize producing a higher energy resonant exciton state X_s and lower energy bound exciton state X_A.