Effects of band gap on the magic-angle of twisted bilayer graphene

Band flattening has been observed in various materials with twisted bilayer structures, such as graphene, MoS2, and hexagonal boron nitride (hBN). However, the unique phenomenon of magic-angle has only been reported in the twisted bilayer graphene (tBG) and not in the twisted bilayer semiconductors or insulators. We aim to investigate the impact of gap opening and interlayer coupling strength on the magic-angle in the tBG. Our results based on the continuum model Hamiltonian with mass term indicate that the presence of a band gap hinders the occurrence of the magic-angle, but strengthening the interlayer coupling tends to restore it. By introducing layer asymmetry, such as interlayer bias or mass difference between layers, the flat bands become more dispersive. Furthermore, we have explored the influence of the Moiré’s potential due to the hBN substrate by calculating the quasi-band-structure of the hetero-structure tBG/hBN. Our findings indicate that the conclusions drawn from using the mass term remain valid despite the presence of the Moiré’s potential due to the hBN substrate.

The presence of flat bands has also been observed in other materials with twisted bilayer structures, such as hexagonal boron nitride (hBN) [38], MoS 2 [39], and γ-graphyne [40].However, unlike graphene, these semiconductors and insulators do not exhibit magic angles.Instead, the bands become flat when the twist angle is sufficiently small, and the bandwidth does not undergo a further increase as the twist angle decreases.It seems that the magic angle only occurs in twisted bilayer semimetals where the band gap is zero.Our previous research [41] has confirmed that the hybridization between the valence bands of one layer and the conduction bands of the other layer (VB-CB hybridization) is crucial for the appearance of the magic angle in the tBG.However, the VB-CB hybridization in twisted bilayer insulators is weak and negligible due to the presence of the big band gap, which prevents the magic angle from occurring.However, it is conceivable that the VB-CB hybridization may have a non-negligible effect in twisted bilayer semiconductors with a relatively small band gap and strong interlayer coupling.This raises a natural question: does the magic angle occur in twisted bilayer semiconductors with a non-zero band gap, and how does the band gap and interlayer coupling strength influence the magic angle?
In this paper, we analyze the impact of band gap on the magic-angle phenomenon of the tBG.We begin by using the mass term [42] to induce a band gap and investigate how the magic-angle is influenced by the band gap and interlayer coupling strength.Our findings suggest that the band gap hinders the occurrence of the magic-angle, but increasing the interlayer coupling strength helps restore it.Furthermore, introducing the layer asymmetry can widen the bandwidth and weaken correlation effects.Additionally, we examine the impact of the Moiré's potential arising from the mismatch in lattice constant and orientation between graphene and the hBN substrate.Our results indicate that the presence of additional Moiré's potential does not alter our conclusions based solely on mass terms.

Structure and methods
The structure of the tBG is illustrated in figure 1(a), where the blue and red hexagonal networks represent the bottom and top layers of graphene, respectively.Starting from AA stacked bilayer graphene with lattice vectors 2 ) and reciprocal lattice vectors , θ-tBG forms if the bottom and top layers rotate −θ/2 and θ/2, respectively, where a = 2.46 Å is the lattice constant of graphene.The corresponding vectors for the two layers become ) with η = − and + for the bottom (l = b) and top layer (l = t).R θ is the rotation anticlockwise by θ degree.The positions of sublattice X (A or B) in layer l are R l X = ma l 1 + na l 2 + τ l X with m and n integers, where The reciprocal lattice vectors of the tBG are Valley index ξ is a good quantum number for the tBG.For small twist angle, its continuum model Hamiltonian is given by [1,[43][44][45][46][47]] where σ = (ξσ x , σ y ), v f = √ 3ta/2h = 10 6 m s −1 is the Fermi velocity, corresponding to the intralayer coupling strength t ≈ 3.1 eV, and K ξ l 's are the valley positions as shown in figure 1(b).The interlayer coupling is expressed as with If we introduce the mass terms (m 1 and m 2 for the bottom and top layers) and interlayer bias (U b ), the additional Hamiltonian that needs to be included is where σ 0 is 2 × 2 identity matrix, σ x , σ y , and σ z are Pauli matrices acting on the sublattice, and ω and ω ′ are the interlayer coupling strengths for the coupling between the same and different types of sublattices.Throughout the paper, we assume ω = ω ′ unless specified otherwise.The value of ω within the range of 0.11-0.6eV is physically attainable, as demonstrated in appendix A. The low-energy eigenstates of Hamiltonian in equation ( 1) can be solved in reciprocal space.Using the plane waves as the basis, the Hamiltonian can be written as where k is the wavevector of the tBG restricted to its Brillouin zone (BZ), G and G ′ are the reciprocal points of the tBG, and |k; l⟩ = (|k, A⟩ , |k, B⟩) T is the plane wave of layer l with wavevector k and the two components correspond to the sublattices.During the calculations, we consider G's that are within the circle |G − K ξ mid | ⩽ 6|g 1 | for the summation to obtain the converged band structure of the tBG, where

Effects of Fermi velocity, mass term and interlayer coupling strength on magic-angle
To begin with, we examine the impact of the Fermi velocity on the magic-angle.Figure 2(a) illustrates the relationship between the twist angle and the width of the VB 0 band (the closest valence band to the Fermi energy) for various Fermi velocities.Our findings demonstrate that the magic-angle phenomenon persists regardless of the Fermi velocity, while adjusting the Fermi velocity merely alters the specific value of the magic-angle.This result is consistent with previous analytical work [1], which indicated that the renormalized velocity v * due to interlayer coupling is related to the Fermi velocity of graphene monolayer, interlayer coupling strength, and twist angle by . When twist angle is at the magic-angle θ = θ M , v * = 0, leading to ω v f θM being a constant, and larger v f results in smaller θ M .
Next, we explore the relationship between the width of the VB 0 band and the twist angle for several mass terms (m 1 = m 2 = m) while keeping the interlayer coupling strength ω fixed at 0.135 eV. Figure 2(b) presents our results, indicating that the VB 0 band widens at the magic-angle as the mass term increases.A small mass term, such as m = 0.03 eV, does not significantly affect the magic-angle phenomenon.However, a sufficiently large mass term, such as 0.4 eV, can completely disrupt the magic-angle.Therefore, the opening of the band gap does tend to deteriorate the magic-angle, making it more challenging to observe in twisted bilayer semiconductors compared to semimetals.
Finally, let us discuss the impact of the interlayer coupling strength ω on the magic-angle.Figure 2(c) depicts the width of the VB 0 band as a function of the twist angle, with the mass term fixed at m 1 = m 2 = m = 0.3 eV.When ω is equal to 0.1 eV, the magic-angle phenomenon is disrupted.However, as ω increases to more than 0.2 eV, the width of the VB 0 band initially decreases and then increases again via a critical angle as the twist angle is reduced.This means that the magic-angle phenomenon is recovered, and we refer to this critical angle as the magic-angle of the tBG with non-zero mass.So increasing the interlayer coupling strength can restore the magic-angle phenomenon despite the presence of the non-zero mass term (band gap).However, compared with the case of m = 0 eV, the VB 0 band becomes wider when twist angle is at magic angle for m > 0 eV.Therefore, the strongly-correlated effect is less pronounced in the tBG with non-zero band gap.
The interlayer coupling can be divided into three components, as shown in figure 3: the hybridization between two layers' valence bands (VB-VB hybridization), the hybridization between two layers' conduction bands (CB-CB hybridization), and the VB-CB hybridization.We can imagine that the Hilbert space for the tBG can be categorized into the valence band space (including the VB-VB hybridization) and the conduction  band space (including the CB-CB hybridization), with the two spaces being interconnected through the VB-CB hybridization.The VB-CB hybridization is always weaker than the VB-VB and CB-CB hybridizations because the two component layers have identical energy spectra, but their valence and conduction bands are separated by a band gap.The interlayer coupling can be understood in two steps, as explained in appendix B. Firstly, in the valence (conduction) band space, the VB-VB (CB-CB) hybridizations cause the valence (conduction) bands of the two layers around the Fermi energy to shift upward (downward).Then, the VB-CB hybridization, primarily contributing near the Fermi energy, leads to the formation of the band structure of the tBG.
Based on our previous work [41], we can explain the emergence of the magic-angle in the tBG as follows.For large twist angles, such as θ = 30 • , the interlayer coupling is weak because there is minimal overlap between the interlayer p z orbitals.Consequently, the band structure of the tBG at large twist angles resembles that of graphene.As the twist angle decreases, the interlayer coupling gradually strengthens, reaching its maximum when the twist angle is reduced to θ = 0 • , where the interlayer p z orbitals are arranged vertically.Compared to large twist angles, the increased interlayer coupling at small twist angles, including the VB-VB, CB-CB, and VB-CB hybridizations, leads to a band inversion between the VB 0 and CB 0 (the conduction band closest to the Fermi energy) bands at the Γ point.When the twist angle reaches the critical value, known as the magic-angle, the VB 0 and CB 0 bands flatten.The opening of the band gap weakens the VB-CB hybridization and hampers the occurrence of the magic-angle, as supported by our results in figure 2(b).However, enhancing the interlayer coupling strength can boost the VB-CB hybridization, which favors the manifestation of the magic-angle, as shown in figure 2(c).

Layer-asymmetry
Our results and previous works indicate that adjusting the twist angle is a valid method to change the bandwidth of the flat bands of the magic-angle tBG.Additionally, we suggest that introducing layer asymmetry, specifically breaking the degeneracy of the two layers, is an alternative approach to tune the bandwidth.This can be achieved by introducing different masses to the two layers (m 1 ̸ = m 2 ) or applying an interlayer bias (U b ̸ = 0).Both of these methods can be implemented experimentally.For instance, placing the tBG on the hBN substrate can induce a non-zero mass term for the bottom layer while keeping the mass term of the top layer at zero.Applying a vertical electric field to the tBG can create different local potentials vertically and lead to an interlayer bias in the tBG.
Figure 4 shows the influence of the mass difference between layers and interlayer bias on the band structure.Here, ω is fixed to 0.135 eV, and magic angle is at θ M = 1.2 • when m 1 = m 2 = 0 eV, for which there are four flat bands around the Fermi energy as shown in figure 4(a).Figures 4(a In figure 4, we illustrate the band structures of the tBG represented by the solid black lines, along with the curves for only the VB-VB and CB-CB hybridizations shown as dashed blue and red lines respectively.They correspond to cases where the coupling between the valence and conduction band spaces is considered and ignored, respectively.We focus on three systems without layer-asymmetry, namely m 1 = m 2 = 0, 0.1 and 0.2 eV as shown in figures 4(a), (f), and (h).Without the coupling between the two spaces, namely without the VB-CB hybridization, the dashed blue and red bands around the Fermi energy exhibit dispersion, but they become flat when the VB-CB hybridization is taken into account.This suggests that unlike in twisted bilayer transition metal dichalcogenides and hBN, where the VB-CB hybridization can be neglected due to their large band gaps [41], the VB-CB hybridization in the tBG cannot be disregarded and remains crucial for the formation of flat bands around the Fermi energy, even with the introduction of the mass term of 0.1 or 0.2 eV.

Signature of the magic-angle
When m 1 = m 2 = U b = 0 eV, the Hamiltonian for each valley in equation ( 1) has the symmetry of magnetic space group P6 ′ 2 ′ 2 [48].The little group at k = Γ is D 3 point group.Its generators are C 3z and C 2x , and the axes of all two-fold rotations are shown in figure 1(c) (see appendix C for the symmetry operations).The magic-angle phenomenon is characterized by the inversion of the irreducible representation (irrep) of the states at Γ around the Fermi energy.Namely that the irreps change from A 1 and A 2 to A 2 and A 1 when the twist angle decreases from larger than to less than magic-angle [48].However, C 2x cannot keep the Hamiltonian invariant for m 1 , m 2 or U b ̸ = 0 eV.The little group at k = Γ reduces to C 3 point group, and its generator is only C 3z .The character tables of D 3 and C 3 point groups are given in table 1. Different from D 3 , C 3 point group has only one-dimensional irreps, so there are not degenerate energy states due to the symmetry.In figure 5(a), we present the evolution of the irreps with twist angle for the four bands at Γ around the Fermi energy.Our results indicate that the irrep inversion occurs at some twist angle, instead of the magic-angle.Therefore, the inversion of the irrep cannot be used to characterize the magic-angle.
Table 1.The character tables of the little group at Γ.For the case of zero mass term (D3 point group), E, C3, and C ′ 2 represent the conjugation classes generated from identity, C3z, and C2x, respectively; For the case of non-zero mass term (C3 point group), the little group is an Abelian group, and each group member forms an independent conjugation class.Fortunately, the band inversion remains a characteristic of the magic-angle phenomenon for a non-zero mass term, and it can be reflected by a k-dependent quantity β.Given an eigenstate of the tBG |φ(k)⟩, β is defined by [41] The Hamiltonian of the tBG consists of the Hamiltonians of two isolated two layers and their coupling, namely 6) is the ith eigenstate of H l at k with eigen energy ε l,i , namely The summations in the first (second) term of equation ( 6) contains only the conduction (valence) band states of the two isolated layers.Please refer to appendix B for the detailed description of the process to calculate β(k) when plane waves are used as the basis.Positive and negative values of β signify that the conduction bands of the two component layers contribute more strongly and weakly than the valence bands of the two layers, respectively.In figure 5 ).However, the contributions from the valence and conduction band states of the two layers to the states of the tBG reverse when the twist angle decreases to less than θ M .Therefore, our results confirm that band inversion, characterized by the difference in contributions between valence and conduction band states of two layers, is always a signature of the magic-angle phenomenon.

Effects of Moiré's perturbation potential due to substrate hBN
The main effect of the hBN substrate is to induce the mass term.Additionally, there is the lattice mismatch between graphene and hBN, namely a hBN = (1 + δ)a with δ ≈ 0.018, where a hBN is the lattice constant of hBN.In addition to the Moiré's potential resulting from the relative twist between two graphene layers, the lattice mismatch and orientation mismatch between graphene and hBN substrate lead to an additional Moiré's perturbation potential.In this section, we explore the impact of the interlayer bias and the mass difference between layers on the flat bands of the magic-angle tBG in the presence of the additional Moiré's perturbation potential.Here, we specifically focus on the hetero-structure tBG/hBN, which has two interfaces, namely G/G and G/hBN interfaces, with G denoting graphene.The twisted angles associated with the two interfaces are denoted as θ and θ hBN , measured with respect to the bottom graphene layer.The continuum Hamiltonian is [49][50][51][52] H = H tBG + ∆H + δV (r) , (7) where H tBG is the Hamiltonian of the tBG as given in equation ( 1), ∆H contains the mass term and interlayer bias as given in equation ( 4), and δV(r) is the periodic perturbation potential due to the G/hBN interface, which is and U P,j e iG j •r U P,j = u P 0 P j+ 1 2 + (−P) where σ ′ = (−ξ σ x , σ y ), e j = (cos jπ 3 , sin jπ 3 ), and ).In this paper, we adopt the parameters from [51]: .775, 3.609, −12.43, 0.017, −6.849) meV.Here, m 2 is set to 0 eV because there is not mass term induced to the top graphene layer for the tBG/hBN.
The G/hBN and G/G interfaces exhibit incommensurability for most twist angles.As a result, the periodicity disappears and momentum is no longer a good quantum number for the tBG/hBN, making it challenging to define the band structure.A previous study [51] addressed this issue by introducing a dual lattice in the reciprocal space of the tBG and defined the quasi-band-structure where wavevector k is restricted to the BZ of the tBG.In this paper, we adopt this method but use plane waves instead of the eigenstates of the tBG as the basis.The advantage of our choice using planes waves as the basis is that it requires only one step of calculation, whereas the original version using eigenstates of the tBG as the basis required at least two steps of calculations.Given wavevector k, the Hamiltonian has the tight-binding form in reciprocal space where G is the reciprocal point corresponding to the periodic potentials of the G/hBN interfaces.In order to obtain the converged low-energy quasi-band-structure, we consider all G's and G's within the circles , respectively, in the summation.Throughout our calculations of the quasi-band-structure, we use the parameters ω = 0.0797 eV and ω ′ = 0.0975 eV, which are commonly used in literature [46,47,51].Compared to the choice of ω = ω ′ = 0.11 eV, these new parameters with a difference between ω and ω ′ can capture the out-of-plane corrugation of the tBG and separate the four flat bands around the Fermi energy from other valence and conduction bands.When the tBG is placed on the hBN substrate, the four flat bands around the Fermi energy evolve into a broad structure.Using different ω and ω ′ , the broad structure around the Fermi energy is more distinguishable from others than using the same ω and ω ′ .In addition, we adopt v f = 7.98 × 10 5 m s −1 as [46,47], which corresponds to a magic-angle of around 1.05 • .
In figure 6, we present the quasi-band-structure of the tBG/hBN at θ = 1.05 • , as well as θ hBN = 0.0 • and θ hBN = 2.0 • .Consistent with previous research [51], the four flat bands around the Fermi energy (the dashed red lines) evolve into a broad structure consisting of a number of bands (the blue lines) after considering the additional Moiré's potential, which is separated from others and remain distinguishable.So the width in energy of the isolated broad structure around the Fermi energy is adopted to reflect the magic-angle phenomenon of the tBG/hBN [51].Our results indicate that its width is larger for θ hBN = 0 • compared to θ hBN = 2.0 • because the BZ size of the G/G interface at θ = 1.05 • is similar to that of the G/hBN interface at θ hBN = 0 • , while there is a significant difference in BZ size for θ hBN = 2.0 • and θ = 1.05 • [51].Importantly, our previous findings using only the mass term hold true.For instance, the interlayer bias U b = 0.1 eV or the mass difference between layers (m 1 , m 2 ) = (0.1, 0.0) eV increases the band width, no matter whether θ hBN = 0.0 • or 2.0 • .It is noteworthy that, in the presence of the additional Moiré's potential, the width of the broad structure around the Fermi energy for large θ hBN , such as 2.0 • , is almost equal to the width of the four flat bands without substrate.This highlights that the magic-angle phenomenon should persist regardless of the hBN substrate.

Conclusion
In this paper, we investigate the impact of gap opening on the magic-angle behavior of the tBG by means of the continuum model Hamiltonian with mass terms.Our findings suggest that increasing the band gap tends to disrupt the magic-angle phenomenon, but enhancing the interlayer coupling strength can restore it.Therefore, we expect the magic-angle phenomenon to occur in twisted bilayer systems where the monolayer has a small band gap.Moreover, introducing the layer asymmetry, such as the mass difference between layers or the interlayer bias, can cause the flat bands to become dispersive.We further study the effect of the mass difference between layers and interlayer bias on the electronic structure, taking into account the additional Moiré's potential originating from the hBN substrate.It turns out that the additional Moiré's potential does not change our conclusions drawn by using only the mass term model.
Here, H l+ and H l− are diagonal matrices with eigenvalues belonging to the valence and conduction bands of layer l, respectively.U b+t+ (U b−t− ) represents the couplings between the valence (conduction) bands of the two layers, referred to as the VB-VB (CB-CB) hybridization.U b+t− and U b−t+ represent the couplings between the valence band of one layer and the conduction band of the other layer, known as the VB-CB hybridization.If we group the blocks of H according to energy instead of layer, we have

Appendix C. Symmetry operations
Following [48], the symmetry of the Hamiltonian of the tBG is shown below.By defining Q l = K ξ l − G, the positions of all Q b 's and Q t 's for ξ = − are depicted in figure 1(c), and they interchange their positions for ξ = +.Consequently, the Hamiltonian in equation ( 5) can be expressed as The Hamiltonian matrix element is We use Q to denote any of the Q b 's and Q t 's, and the matrix element is

Figure 1 .
Figure 1.(a) The structure of the tBG with the bottom and top layers shown in blue and red hexagonal network.(b) The BZs of two component layers with a twist.The gray hexagonal network shows the BZ of the tBG with reciprocal lattice vectors of g1 and g2.The positions of Dirac points of the two layers are pointed out by K ξ l for ξ = ± valley and layer l.(c) Positions of Q b 's (blue dots) and Qt's (red dots) for ξ = −.Three BZ corners of the two layers are connected by vectors d i 's.Three dashed lines are the two-fold rotation axes.

Figure 2 .
Figure 2. The dependence of the width of VB0 band on (a) Fermi velocity v f with v f in units of v f0 = 10 6 m s −1 with m = 0 eV and ω = 0.135 eV, (b) mass term m with w = 0.135 eV, and (c) interlayer coupling strength ω with m = 0.3 eV.In (b) and (c), Fermi velocity is v f = v f0 .In all panels, two layers possess the same mass term (m1 = m2 = m).

Figure 3 .
Figure 3.A schematic representation of the band alignment between two layers as well as the VB-VB, CB-CB, and VB-CB hybridizations.
)-(d) are the band structures with the interlayer bias U b increasing from 0 to 0.3 eV, which indicate clearly that the interlayer bias can noticeably increase the bandwidth of the flat bands, and larger interlayer bias always leads to larger bandwidth.As depicted in figures 4(e)-(h), compared with the cases without mass difference between layers, such as m 1 = m 2 = 0 eV, m 1 = m 2 = 0.1 eV, and m 1 = m 2 = 0.2 eV, the mass difference between layers, (m 1 , m 2 ) = (0.0, 0.1) eV and (m 1 , m 2 ) = (0.0, 0.2) eV, yield wider bands.And larger mass difference results in wider bands.

3 e −i 2π 3 Figure 5 .
Figure 5. (a) The energy of the four states at Γ around the Fermi energy with twist angle changing in the range of 4 • ∼ 8 • .Different colors denote different irreps.Black and purple dashed vertical lines show the magic-angle (θM) and the twist angle where the irrep inversion occurs.(b) The β values for the band structure of the tBG with the twist angle within the range of 4.1 • -5.1 • .In (a) and (b), we only show the results for ξ = − and we choose the parameters m1 = m2 = 0.3 eV and ω = 0.53 eV, which gives the magic-angle ∼4.65 • .
(b), we show the β values for m 1 = m 2 = 0.3 eV, ω = 0.53 eV, and twist angles ranging from 4.1 • to 5.1 • .Our results clearly demonstrate the band inversion phenomenon, indicating that the valence (conduction) band state of the tBG closest to the Fermi energy at Γ originates more from the valence (conduction) band states of the two component layers when the twist angle is larger than the magic-angle (θ M = 4.65 •

Figure 6 .
Figure 6.Quasi-band-structures of the tBG/hBN (blue lines) at θ = 1.05 • as well as (a) θ hBN = 0 • and (b) θ hBN = 2 • for different mass terms m1 and interlayer bias U b .The dashed red lines are the band structures of the tBG without the Moiré's potential from the hBN substrate.m2 = 0.0 eV for all panels.

3 )
where Q's and Q ′ 's are confined within the circle around Γ. We specifically focus on the case of k = Γ for the symmetry operations.When m 1 = m 2 = U b = 0 eV, the Hamiltonian exhibits the symmetry of the point group ofD 3 = {E, C 3z , C −1 3z , C 2x , C 3z C 2x C −1 3z , C −1 3z C 2x C 3z }with generators C 3z and C 2x .The matrix elements of the symmetry operations C 3z and C 2x are given byD Q ′ ,Q (C 3z ) = e iξ 2π 3 σz δ Q ′ ,C3zQ and D Q ′ ,Q (C 2x ) = σ x δ Q ′ ,C2xQ.However, when either m 1 , m 2 , or U b is not 0 eV, C 2x is no longer the symmetry operation, and its symmetry reduces to the point group of C 3 = {E, C 3z , C −1 3z }.See table1for the character tables of point groups D 3 and C 3 .
.3) By now, we have the Hamiltonian matrix H of the tBG with the eigenstates of the two isolated layers as the basis, and the bases are arranged in the order: VB of bottom layer, VB of top layer, CB of bottom layer, CB of top layer.H + (H − ) is the Hamiltonian containing only the valence (conduction) band parts of the two layers and their coupling.(1) Diagonalizing H + and H − allows us to obtain the bands arising from the VB-VB and CB-CB hybridizations; (2) To calculate β(k), we first diagonalize H to obtain its eigen states.For any eigenstate vector ϕ, it has the form ϕ = (C 1 b+ , C 2 b+ , . .., C 1 t+ , C 2 t+ , . .., C 1 b− , C 2 b− , . .., C 1 t− , C 2 t− , . ..),where C i l+ (C i l− ) represents the component on the ith valence (conduction) band of layer l.So β(k) = il |C i l− | 2 − il |C i l+ | 2 .