Optical properties of twisted bilayer graphene with magnetic defects

Even when fabricated under ideal conditions twisted bilayer graphene (TBG) inevitably contains various defects which may significantly affect its physical properties. Here we comprehensively analyze the impact of typical point defects, represented by adsorbed hydrogen atoms, on the electronic and optical properties of TBG. It is shown using self-consistent tight-binding Hamiltonians that such point defects make TBG ferromagnetic, and that its ground state comprises a pair of nearly dispersionless spin-polarized energy bands around the Fermi level. Transitions to and from these bands strongly modify the infrared absorption of TBG and result in a sharp low-energy peak in its spectrum. It is also revealed that the adsorption of hydrogen atoms suppresses the circular dichroism of TBG due to the weakening of the electronic coupling between the graphene layers. Our findings will guide future experimental studies on the optical properties of TBG in realistic, impurity-rich environments.


Introduction
Twisted bilayer graphene (TBG) is a highly promising material with unique electronic and optical properties, fabricated by mutually twisting two graphene layers stacked on top of each other [1][2][3]. Such twisting results in a two-dimensional moiré pattern that has a significant impact on the electronic properties of TBG, introducing Van-Hove singularities to the energy spectrum of the material [4,5]. At small twist angles ≈1 • , for example, TBG features almost flat energy bands around the Fermi level, which enables unconventional superconductivity and Mott insulating states [6][7][8].
Optical properties of TBG are sharply different from those of normal monolayer graphene, whose optical absorption is almost constant over the visible spectral range [9]. In contrast to that, interlayer coupling in TBG leads to the formation of new absorption peaks, whose positions depend on the twist angle [10][11][12][13][14]. The luminescence peaks of TBG, in particular, coincide with the Van-Hove singularities in its electronic spectrum [15]. Angle tunability can be used for selective enhancement of the photocurrent in TBG [16] or for engineering photonic crystals in the moiré pattern of the material [17]. The valley-symmetry breaking in TBG at the magic angle was shown to result in a strong nonlinear optical response and anomalous Hall conductivity [18]. On top of that, magic-angle TBG has ultra-sensitive calorimetric properties which enable fine monitoring of its electronic temperature under laser heating and can potentially be used in single-photon detectors of THz radiation [19,20]. Importantly, TBG is an intrinsically chiral material with strong circular dichroism (CD)-a differential absorption of left-and right-circularly polarized light [21]. This opens up prospects of using TBG for chiral sensing of organic molecules [22][23][24][25][26].
Even in the best experimental conditions, the crystal structure of TBG is inevitably characterized by various defects, which can significantly affect electronic properties of the material. Typical types of point defects in TBG include vacancies of carbon atoms [27,28], Stone-Wales defects [29], and adsorption of atoms such as carbon, fluorine, and hydrogen on the surface of TBG [30][31][32]. Hydrogenation of TBG, in particular, distorts chemical bonds at the absorption sites [32], modifies the bandgap of the material [33], and induces magnetism in TBG [34,35]. Hydrogen atoms on the surface of TBG can form during the fabrication process such as the bottom-up synthesis of graphene from small hydrocarbon molecules. They can also be introduced into TBG intentionally via gaseous or wet-chemical approaches [36,37]. To date, research has been primarily focused on electronic and magnetic properties of TBG with defects, while their impact on optical properties of TBG remains mostly unexplored.
In this paper, we model optical properties of TBG with point defects in its crystal structure, focusing on the example of single hydrogen atoms absorbed on the surface of TBG. We assume that adsorption of hydrogen removes electrons from the π-network of TBG and use the tight binding Hamiltonian to describe electronic, magnetic, and optical properties of TBG at different twist angles and defect concentrations. We reveal that the inclusion of point defects significantly modifies the absorption spectrum of TBG by introducing a sharp absorption peak at lower energies, which persists even at smaller concentrations of the defects due to magnetic properties of the ground state. In contrast to that, we find that the CD signal of TBG is merely suppressed due to the adsorption of hydrogen atoms, which is attributed to the partial decoupling of the two graphene layers around the defects. The results of our study will aid the design of new optoelectronic and chiral devices based on TBG.

Methods
Stacking two monolayer sheets of graphene with a twist generally yields a non-periodic two-layer structure-TBG. This structure can be viewed as a two-dimensional crystal for a specific set of commensurate twist angles [22,38] cos The number of atoms in the unit cell of a TBG crystal is given by N = 4(3i 2 + 3i + 1). Figure 1(a) shows TBG for i = 1 (θ 1 = 21.78 • ), highlighting the regions of high-symmetry stacking within the Moiré pattern of the structure. The stacking of the graphene sheets in these regions resembles AA, AB, and BA stacking in the bilayer graphene without twist. There are several types of point defects that can form in TBG. Here, we focus on the simple case of hydrogen adsorption onto the surface of TBG. Figure 1(b) shows the crystal structure of TBG with hydrogen atoms adsorbed on the top layer. We assume that the atomic orbitals of hydrogen simply bind to the p z -orbital of the carbon atom at the adsorption site, thus removing the corresponding p z -electron from the π-network of TBG [39,40]. Note that this assumption only holds for the adosprtion of single hydrogen atoms, whereas extensive hydrogenation of TBG is known to significantly distort the sp 2 -orbitals of carbon atoms near the defects, resulting in the formation of local islands of sp 3 -hybridization [32,34,35].
To analyze the electronic properties of TBG with and without defects, we write the tight-binding Hamiltonian as the sum of three contributionsĤ =Ĥ in +Ĥ out +Ĥ e−e . Here the first term describes the interaction of atoms of the same layer and has the form [41,42] where t 1 = −2.88 eV, t 2 = 0.20 eV and t 3 = −0.25 eV are the hopping integrals between the nearest, second-nearest, and third-nearest neighbors and σ =↑, ↓ denotes electron spin. The interaction between atoms in different layers is described by the Hamiltonian in the Slater-Koster approximation [43] where the hopping integrals are given by and where δ ϕ,90 • is the Kronecker delta, a = 1.42 Å is the length of the C-C bond, d = 3.35 Å is the interlayer distance, ϕ is the angle between r and the z axis, t ⊥ = 0.25 eV is the hopping energy between the atoms located on top of each other, and γ π = 3.14 and γ σ = 7.41 are the spatial decay rates of the hopping integrals.
Since the inclusion of point defects in graphene can give rise to a magnetic ground state [44], we also take into account electron-electron repulsion described by the Hubbard term in the mean-field approximation [45]: where U is the repulsive energy of two electrons residing on the same atom. We take U = 3 eV, which is in good agreement with the density functional theory (DFT) calculations and is within the applicability domain of the mean-field approximation used in this study [46]. The Hubbard Hamiltonian thus describes interaction of the spin-up electron density,n i↑ =ĉ † i↑ĉ i↑ , with the average density of electrons with spin down, ⟨n i↓ ⟩, and vice versa. Diagonalization of the Hubbard Hamiltonian is a self-consistent problem, since the density of electrons is used in calculating the matrix elements of the Hamiltonian. This problem is solved iteratively, starting from a randomized initial distribution of electrons.
Upon diagonalization of the Hamiltonian, we obtain energies of the electronic bands E αkσ , where α is the band number and k is the electron momentum, as well as eigenvectors a αkσi , with i labeling sublattices of the material. The average densities of electrons of different spins are calculated by the summation over all occupied states as where Magnetic properties of the system are characterized through the magnetization on each atom, defined as with the full spin of the unit cell being S = ∑ i M i . Optical absorption in atomically flat TBG is characterized using the optical conductivity calculated through the Kubo formula [24], where µ, ν = x, y label Cartesian coordinates, Ω is the area of the material, η is the spectral broadening, andĴ is the current density operator. In order to obtain CD of the system, we need to take into account the spatial dispersion of light between the two layers of TBG by writing the current operator in the form where q is the wave vector of radiation andĴ T(B) is a current operator in the top (bottom) layer, calculated as Since the material is relatively thin, qd ≪ 1, exponential factors in equation (11) can be decomposed into the Taylor series up to the first order in qd. Absorption of radiation by a two-dimensional material is determined by the diagonal components of the optical conductivity tensor, Re σ xx + Re σ yy . By symmetry, absorption does not depend on q and can be essentially calculated by setting q = 0, yielding On the other hand, CD is determined by the off-diagonal component of the conductivity tensor, Im σ xy , which is linear in q: Before we proceed to modelling electronic, magnetic, and optical properties of TBG with adsorbed hydrogen atoms, it is important to discuss the computational complexity of our calculations. The key parameters affecting the cost of the calculations are the number of k-points, N k , and the number of atoms per unit cell, N A , which is also equal to the number of energy bands of Given that for small twist angles N A ∝ θ −2 , the cost of calculating the optical spectra grows approximately as ∝ θ −8 . Such a steep raise of the computational cost prevents us from assessing the impact of the defects on the optical properties of TBG with small θ, including the magic-angle TBG with θ ≈ 1.05 • . The optical properties of TBG with smaller twist angles can potentially be obtained with continuum models which have much smaller computational costs [2].

Results and discussion
We begin by calculating electronic properties of pristine TBG, which have also been reported in numerous previous studies [48][49][50]. Figure 2(a) shows band structures and density of states (DoSs) for TBG with i = 1, 2, 3. In the absence of defects, energy bands of electrons with opposite spins coincide and the total spin of the unit cell is S = 0, indicating the absence of magnetism in TBG. Near the Fermi level, energy bands of TBG resemble Dirac cones found in monolayer graphene. Coupling of the two layer manifests itself away from the Fermi level, where the coupled energy bands produce distinct peaks in the DoS spectrum. Figure 2(b) shows band structures and DoS for TBG samples with one hydrogen atom per unit cell, adsorbed in the AA-stacking region of the moiré pattern. We have explicitly verified that varying the adsorption site does not lead to significant changes in the electronic and optical properties of TBG. One sees that the energy bands of TBG with adsorbed atoms cease to be degenerate with respect to spin and the system acquires a non-zero magnetic moment of S = 1/2 per defect. Most notably, adsorption of impurities leads to the formation of two relatively flat energy bands of opposite spins near the Fermi level of TBG. One of these bands lies entirely below the Fermi level and is filled with electrons, while the other band is above the Fermi level and completely empty. This imbalance of spin-up and spin-down electron densities is what leads to ferromagnetism of TBG with point defects. The band structures and magnetic moments predicted by our tight-binding approach are found to be in a good qualitative agreement with those obtained using the DFT calculations on TBG with adsorbed hydrogen atoms [34]. Upon decreasing the twist angle, these energy bands become flatter, while the corresponding peaks in the DoS spectrum decrease in intensity. The latter is because increase in the size of the TBG's unit cell reduces the concentration of defects. In addition to the two bands near the Fermi level, two pairs of bands above and below the Fermi level appear in the band structure of TBG with defects. For example, in the case of TBG with i = 1, one can notice the presence of two new conduction bands at 1.5 eV and two new valence bands at −1.2 eV. As the parameter i increases, these bands become closer to the Fermi level. The peculiarity of the new conduction bands is that they are particularly flat inside the Dirac cone, which leads to the formation of additional peaks of DoS at these energies. The rest of the band structure practically coincides with that of pristine TBG. Dirac cones maintain their structure but shift slightly towards negative energies. This means that even at zero temperatures, the conduction band contains a finite number of electrons, making TBG with adsorbed hydrogen atoms a conductor.
In order to study the real-space character of the new energy bands, we introduce local density of states (LDoSs), which is the projection of DoS on a particular atom of the material [51]. Figure 3 shows LDoS of TBG with i = 2 and one defect per unitcell, plotted for different valence and conduction bands. We also depict relative magnetization on each atom, defined as the spin-polarized contribution to LDoS normalized by its absolute value. One can see that far from the Fermi level, at E = −1.0 and 1.0 eV, LDoS is evenly distributed between the two layers of the material and there is no preferential direction of electron spin. At these energies, the band structure of TBG with defects is very similar to that of pristine TBG, hence the effect of defects is almost negligible.
At E = −0.5 and 0.5 eV, electrons are predominantly localized in the bottom layer of the material, also featuring no noticeable spin polarization. At these particular energies, the band structure of TBG is represented by Dirac cones, almost identical to those in TBG without defects. Finally, electrons in the two flat bands near the Fermi level, E = −0.1 and 0.1 eV, are almost exclusively localized in the top layer, with the largest LDoS on the three atoms closest to the adsorption site. It is peculiar that LDoS of these two bands does not occupy the entire layer of hexagonally packed C atoms, but forms a triangular pattern, in which only the sublattice containing no defects is occupied. Naturally, these energy bands are completely spin-polarized, with all electrons having spin either up or down depending the particular energy band. If the adsorption sites are moved away from the AA stacking region, the LDoS peaks of the two energy bands at the Fermi level naturally follow the impurities in space. However, the localization of states at other energies is found to be independent of the positions of the impurities.
We can therefore see that the presence of defects leads to the spatial separation of electron density in TBG. Electrons in the energy bands close to the Fermi level tend to strongly localize in the layer that contains defects. This happens because the Fermi level in pristine TBG coincides with the onsite energy ε pz of the p z -electrons in the sp 2 -hybridized carbon. As the adsorption of hydrogen leads to the removal of the p z -electrons from the adsorption sites, it is natural that the electrons of energies around ε pz are most strongly affected by the presence of impurities. On the other hand, electrons at energies slightly further away from the Fermi level (−0.5 and 0.5 eV in figure 3) predominantly occupy the defect-free layer of the material. This is in sharp contrast to pristine TBG, in which all electrons occupy both layers of the material. Adsorption of atoms on the surface of TBG thus reduces electronic coupling of its layers, which must significantly affect its optical and chiral properties. Figure 4 shows absorption and CD spectra for pristine TBG and TBG with defects for three different twist angles. Both absorption and CD are measured in conduction quanta σ 0 = e 2 /(4h). We first revisit the optical properties of pristine TBG without defects, whose spectra are shown by dashed gray curves in the figure. One can see that absorption of TBG at low energies is a constant plateau with a value of 2σ 0 , which is twice the value of absorption in one graphene layer. Starting from 2.7, 1.8, and 1.3 eV in the case of TBG with i = 1, 2 and 3, respectively, we observe a series of peaks associated with transitions between the bands with strong hybridization between graphene layers [10,24]. As shown in figure 2(a), the transitions contributing to the first absorption peak in the optical spectrum of TBG occur mostly at the M point of the Brillouin zone, where the energy bands become flat.
The CD spectra of TBG without defects are characterized by a series of alternating peaks starting at 2.2, 1.5 and 1.0 eV in the case of i = 1, 2 and 3, respectively. The practically constant absorption at low energies does not produce a CD signal. This is due to the fact that at low energies, absorption of light occurs almost independently in the two layers of the material, so that its chirality does not manifest itself in the optical response. On the contrary, absorption at high-energy peaks occurs between the energy bands that are strongly hybridized between the two layers, meaning that the twist of the material plays an important role. Upon decreasing the angle of rotation, the CD values gradually decrease, implying that the chirality of TBG is directly proportional to the twist angle.
The presence of point defects leads to significant changes in the absorption spectra of TBG (solid red curves in figure 4). In the case of TBG with i = 1, a pronounced peak at 1.6 eV appears in the visible region of the spectrum. The absorption intensity at lower energies is reduced to about σ 0 , meaning that absorption mainly occurs only in the graphene layer that does not contain defects. As the twist angle decreases, the new absorption peak shifts to the infrared range, 0.9 and 0.6 eV for i = 2 and i = 3, respectively, while the higher-energy peaks become smoother. As shown in figure 2(b), the new peaks in the absorption spectra of TBG with defects are mostly formed by the transitions of electrons of one spin. These transitions occur between the quasi-flat valence band close to the Fermi level and a relatively flat conduction band inside the Dirac cone.
The CD spectra of TBG with adsorbed atoms demonstrate significant signal suppression in the region of low-energy peaks. For example, in the case of TBG with i = 1, CD signal at 2.3 eV practically disappears. The  reason why the CD signal reduces in the presence of defects is due to the partial decoupling of the two graphene layers, revealed in figure 3. Since some electrons occupy either top or bottom layer of the material, the chirality of TBG does not manifest itself for these electrons. In particular, this explains why sharp low-energy peaks, appearing in the absorption spectrum of TBG with defects, do not lead to the formation of additional CD peaks. These sharp peaks come from the nearly flat energy bands close to the Fermi level, which are localized only in the top layer of the material.
In addition to the twist angle, optical spectra of TBG with defects are dependent on the surface concentration of adsorbed atoms. The previously obtained optical spectra correspond to the structures with one defect per unit cell. This relatively high concentration of impurities, which we denote by N 0 , leads to the observed large differences between the absorption and CD spectra of pristine TBG and TBG with defects. However, actual experimental samples are likely to have lower concentrations of impurities. In order to understand how the decrease in the defect concentration affects the optical spectra of TBG, we place one defect per supercell consisting of multiple unit cells. Figure 5 shows absorption and CD spectra of TBG with five different defect concentrations N: N 0 , N 0 /2, N 0 /3, N 0 /4, and N 0 /5. These concentrations are achieved by using supercells of size 1 × 1, 1 × 2, 1 × 3, 2 × 2, and 1 × 5, respectively.
One can notice straight away that reducing the concentration of adsorbed atoms naturally makes optical spectra of TBG with defects similar to those of pristine TBG. In the absorption spectra, shown in the top panels of figure 5, the defect concentration mainly affects the intensity and position of the low-energy absorption peak. At lower concentrations, this peak moves to lower energies and decreases in intensity. It is peculiar, however, that this peak remains pronounced even at the smallest defect concentration N = N 0 /5, while the rest of the absorption spectrum becomes almost identical to the spectrum of pristine TBG. Even for small densities of adsorbed atoms, the ground state of the material remains spin-polarized, featuring two flat bands around the Fermi level of TBG. The low-energy absorption peak originates from these strongly localized energy bands and thus persists even at smaller defect concentrations.
In the CD spectra of TBG with defects, shown in the bottom panels of figure 5, the effect of the concentration is much more straightforward. Reducing the concentration of the defects gradually restored the intensity of the CD signal and makes the spectrum more similar to the CD spectrum of pristine TBG. Unlike in the case of absorption, magnetic properties of TBG with the defects do not manifest in the CD spectra, leading to their trivial dependence on the defect concentration.
Overall, our findings suggest that the hydrogenation of TBG is a promising avenue for tailoring optical properties of this remarkable material. Even low concentrations of adsorbed hydrogen atoms were shown to induce profound changes in the optical spectra of TBG, amplifying absorption of radiation in the infrared and visible spectral regions and making TBG more transparent at low frequencies. On top of that, hydrogenation can be used to suppress optical activity of TBG for photonic applications where it is desirable to avoid the circular polarization of light. The opposite approach is also possible-by measuring the optical spectra of TBG and comparing them to the theoretical predictions, one can establish the presence of adsorbed atoms in the fabricated samples and find their concentration.
It is important to stress that adsorption of hydrogen atoms considered in this work is a relatively simple example of point defects in TBG. Other imperfections of the crystal structure, such as vacancies of carbon atoms or Stone-Wales defects, present more challenges for theoretical modelling. These defects break σ-bonds around them and lead to the formation of new energy bands near the Fermi level, consisting of σ-electrons [52,53]. It would be very difficult to incorporate these features within the framework of the tight-binding approximation employed in our work. In future studies, other types of defects and their effect on optical properties of TBG can be addressed using DFT calculations.

Conclusion
We have comprehensively analyzed electronic and optical properties of TBG with point defects, using the example of adsorbed hydrogen atoms. In the tight binding formalism employed in this study, such defects were described as the vacancies in the electronic π-network of TBG. It was found that the inclusion of point defects makes energy bands of TBG spin-polarized and induces a total spin of 1/2 per unit cell of the material. Band structures of TBG with defects revealed the presence of two relatively flat energy bands of opposite spin near the Fermi level. Electrons in these flat bands are strongly localized around the point defects in the top layer of the structure. In the absorption spectra, adsorption of hydrogen atoms leads to the formation of a sharp peak in the infrared spectral range, originating from electrons localized at the point defects. These peaks, however, do not produce a new CD signal, since the localized electrons are decoupled from the second layer of the material and thus are not affected by the twist. Instead, adsorption of hydrogen suppresses the overall CD signal of TBG. Decreasing the concentration of defects naturally makes optical spectra of the material more similar to those of pristine TBG. Our results will be instrumental in designing optical devices based on TBG in realistic experimental conditions, in which it is impossible to completely eliminate impurities.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).