1. INTRODUCTION

Two-dimensional hexagonal boron nitride (hBN), also called by some “white graphene,” is an electrical insulator in which the boron (B) and nitrogen (N) atoms are arranged in a honeycomb lattice and are bounded by strong covalent bonds. As with graphene, hBN has good mechanical properties (elastic constant of $220\mathrm{–}510\mathrm{N}{\mathrm{m}}^{-1}$ and Young’s modulus $\sim 1.0\mathrm{TPa}$) [1] and high thermal conductivity (varies between 1 and $2000{\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$) [2]. Especially interesting is the possibility of using hBN as a buffer layer in van der Waals heterostructures, namely, ones comprised by layers of hBN/graphene. A hexagonal boron nitride layer can serve as a dielectric or a substrate material for graphene in order to improve its mobility [3] and open a gap [4]. It was shown that graphene and hBN heterostructures have potential applications on nanocapacitors [5] and also quantum point contact devices [6]. It can also be used to improve the thermoelectric performance of graphene [7].

Yet, its electronic properties differ significantly from those of graphene. Graphene $\pi$ and ${\pi }^{*}$ electronic bands have a linear dispersion at the K-point, whereas in hBN there is a lift of the degeneracy at the same point, and a wide band gap [8] is formed. That would in principle make it ideal for optoelectronic devices in the deep ultraviolet region [9,10]. As we will see, however, excitonic effects play an important role in this material: excitonic peaks are created at the near UV, and this is a much more useful electromagnetic spectral range, when compared with deep UV.

The optical properties of monolayer hBN at the UV range are characterized by the exciton with a corresponding optical band gap calculated in the 5.30–6.30 eV range (see Section 2). The presence of the exciton in this range can be used to excite exciton–polaritons that share some properties with surface plasmon–polaritons [11,12]. Therefore, the UV optical properties of hBN can be used as an alternative to the emerging field of UV plasmonics [13–22]. The plasmonics in the UV range also attract interest in biological tissue [23] as a consequence of the resonances in nucleotide bases and aromatic amino acids. Plasmonics in this range rely on poor metals [15,17,18,24] and rhodium [19,20,22].

Because of the difficulty of its synthesis, few experimental works have been done on an hBN single layer. It is also necessary to work in the UV range to study and probe its electronic and optical properties. To our knowledge, only one experimental work [8] has been produced that studies the electronic properties of monolayer hBN. The authors observed the band structure of hBN monolayer on a Ni(111) surface by using an angle-resolved ultraviolet-photoelectron spectroscopy and angle-resolved secondary-electron-emission spectroscopy. Because the bond between the interface of hBN and Ni(111) is weak, the band structure of the monolayer hBN is not significantly altered by the Ni(111) substrate [8]. The band gap was determined to be; however, a comparison with experimental values for bulk hBN (5–6 eV) led the authors to conclude that they may have overestimated the binding energies of the valence bands, and the actual value should be within the range of 4.6–7.0 eV. We think the authors would not lower the range of possible values if they considered that, usually, bulk band gaps are smaller than their monolayer counterparts.

To our knowledge, there are two experimental works to measure the optical gap of an hBN monolayer [25,26]. Reference [25] determined an optical gap of $\sim 6.17\mathrm{eV}$ (assuming an indirect gap), while in [26] an optical band gap $>5.85\mathrm{eV}$ was estimated for the hBN monolayer produced by the authors.

From the theoretical point of view, there are several works that calculate the fundamental and the optical band gap (see Table 1). It is clear from Table 1 that there is a dispersion in results, even higher than the few experimental results mentioned earlier. Even the question of whether the band gap is direct or indirect is still not clear: half of the previous works claim a direct gap, while the other half claims an indirect gap. Our results gave an indirect gap.

Table 1. Several Band Gaps Calculated in This Work and by Other Authors Using $G{W}_0$, $G_0{W}_0$, and BSE, in eV^a

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One way of calculating the fundamental gap is to include many-body effects on top of a first principles calculations, e.g., density functional theory (DFT). DFT frequently uses an approximation to the exchange-correlation functional (e.g., LDA and GGA) that is simple to implement and does not require heavy calculations but underestimates the band gap of semiconductors and insulators. A typical way of adding the many-body effects and correct the value of the band gap is the $GW$ approximation [33,34]. The optical gap needs a further step to be calculated, especially in 2D materials where excitonic effects are important. To include excitonic properties, the Bethe–Salpeter equation (BSE) [35,36] is usually used. There are several works in the literature that have used the $GW$ approximation [30–32] and $GW+\mathrm{BSE}$ [27–29] on 2D hBN. The results from these works vary significantly, as can be seen in Table 1.

Convergence can be an issue in $GW$ and BSE calculations, as can be seen in [37] and [38]. It is likely that the works summarized in Table 1 use different criteria for convergence, and that may explain the differences. A small number of bands used in the calculation [31,32] or not using a truncation to avoid interaction with periodic images [30] may also explain some differences. Sometimes there is ambiguity between the value stated for the gap and the one that can be obtained from the absorption spectrum presented [27]. More difficult to explain are the values obtained in [29]. They differ significantly from our work and others’, although they seem to have converged the calculations carefully. One explanation may be that they fixed the lattice constant at the experimental value instead of relaxing the unit cell. The experimental lattice constant may not match the value that actually optimizes the system and can influence the values of the gaps in the electronic band structure. An effective energy technique [39] was adopted in [28]. That technique allowed the calculation of the screened Coulomb interaction $W$ to be converged with only 90 bands and 60 bands for the self-energy $\mathrm{\Sigma }$ calculation. The use of such technique certainly will produce differences in the final results.

In this work, we confirm that the gap in hBN is indirect and determine its value and the exciton energies. We also calculate the excitonic spectra using an equation of motion formalism and the Elliot formula, fitting it with the $GW+\mathrm{BSE}$ calculations, thus obtaining validation of the method. In Section 2, we describe the details of the $G_0{W}_0$ calculations and results. In Section 3, we show the results of the BSE calculations. Both $G_0{W}_0$ and BSE calculations were performed with the software package Berkeley $GW$ [40–42]. Section 4 presents the equation of motion formalism and the results for the excitonic properties of monolayer hBN. In Section 5, we study the properties of exciton–polaritons of hBN, and we show that a monolayer of hBN can be used as a UV mirror. We draw the conclusions in Section 6.

2. $G_0{W}_0$ RESULTS

$G_0{W}_0$ calculations were done on top of DFT calculations with a scalar-relativistic norm-conserving pseudopotential. The software package Quantum ESPRESSO [43] was used for the DFT calculations. Details of the DFT calculations are summarized in Table 2. For $G_0{W}_0$ calculations, a truncation technique is needed due to the nonlocal nature of this theory. Our criteria and convergence methods are the same as those used in [37]. A grid of $\mathbf{k}$-points has to be chosen for the $GW$ calculation. Then, a convergence study for the screened Coulomb interaction $W$ is done. For that, we need to analyze the dielectric matrix calculations, where the dielectric matrix cutoff and the summation of bands are the parameters to be converged. After that, we study the convergence of the self-energy $\mathrm{\Sigma }=GW$ calculation, using the same cutoff as for the dielectric matrix calculation because it is the largest value that we can set. The bands summation of this calculation has also to be converged. These paramaters are all defined and carefully explained in [37].

Table 2. Details of DFT Calculations

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We found that, for DFT calculations, a grid of $6\times 6\times 1$$\mathbf{k}$-points is enough to reach convergence. For the $GW$ calculations, a grid of $16\times 16\times 1$ $\mathbf{k}$-points and a cutoff energy of 22.6 Ry and 1100 bands were needed for the dielectric matrix calculations. For the $\mathrm{\Sigma }$ self-energy calculation, we used a cutoff energy of 22.6 Ry and 1000 bands. The results obtained for the electronic band gap are summarized in Table 3. They show that a monolayer of hBN is a wide band gap indirect-gap material. Figure 1 shows the electronic band structure and electronic density of states for DFT and $GW$ calculations, and they present differences. We conclude that a simple shift applied by using a scissors operator would therefore not provide reliable results. It is clear that the energy of the first conduction band at the K-point is very close to the energy at the $\mathrm{\Gamma }$ point on the DFT band structure. This explains why the question about the indirect or direct nature of the band gap has different answers in previous theoretical works. On the $G_0{W}_0$ band structure, the lowest energy of the first conduction bands is at the $\mathrm{\Gamma }$-point with a difference of approximately 0.3 eV when compared with the K-point. We can also see that the shape of the bands is different on both the valence and conduction zones. This is why the density of states of DFT and $G_0{W}_0$ is not equal and a simple shift would not suffice.

Table 3. $G_0{W}_0$ Gap Values for the Transitions $\mathrm{K}\to \mathrm{\Gamma }$, $\mathrm{K}\to \mathrm{K}$, and $\mathrm{\Gamma }\to \mathrm{\Gamma }$, and Optical Gap and Exciton Binding Energy (EBE) Obtained from BSE in This Work^a

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Fig. 1. Electronic band structure (left) and electronic density of states of hBN (right) for both DFT and $GW$ calculations.

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As mentioned in the introduction, the only experimental work on the electronic properties that we aware of is the one from [8], which in fact measures a fundamental band gap of $\sim 7\mathrm{eV}$. We can see that 7 eV is actually close to the ones obtained by works referred to in Table 1, including our own. Experimental studies on hBN crystal with multilayers show band gaps values close to 6.0 eV [46,47]. For a monolayer, the value is likely to be higher because the quantum confinement has the effect of increasing the band gap as well as the optical gap.

Reference [8] also calculated the width of the valence bands and found no good agreement with contemporary theoretical works. Table 4 shows the width of the valence bands as calculated with DFT, $GW$, and the experimental determination of [8]. The $\pi$-band has its highest energy at the K-point, while the ${\sigma }_1$ and ${\sigma }_2$ have the highest energy at the $\mathrm{\Gamma }$-point. Table 4 shows that DFT results differ from the experimental ones by values greater than 0.5 eV in all cases. On the other hand, $G_0{W}_0$ results differ from the experimental results by values equal or smaller than 0.1 eV.

Table 4. Width of the Valence Bands^a

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We also calculated the effective masses of the highest valence band and lowest conduction band using both $G_0{W}_0$ and DFT (Table 5). We found no differences between $G_0{W}_0$ and DFT, except for the effective mass at $\mathrm{K}\to \mathrm{\Gamma }$ on the first conduction band (DFT value greater by $0.08m_e$). Thus, we conclude that DFT calculations are reliable to obtain the values of the effective masses in this material.

Table 5. Effective Masses [in Electron Mass ($m_e$) Units] for hBN Calculated Using $G_0{W}_0$^a

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Reference [32] also calculated the effective mass at the $\mathrm{\Gamma }$-point for the conduction band and obtained a value of $(0.95\pm 0.05)m_e$ with only slight variations for different planar directions. In our work, we obtained differences of $0.3m_e$ between different directions in reciprocal space at the $\mathrm{\Gamma }$-point.

3. BSE RESULTS

After determining the conduction and valence band states, the electron-hole pair states are determined using the BSE. The imaginary part of the dielectric function ${ϵ}_2(\omega )$ is then [42]

If we do not consider excitonic effects, the expression becomes a transition between single particle states [42]

Figure 2 shows the imaginary part of the dielectric function calculated by BSE, done on top of a $G_0{W}_0$ calculation with a grid of $16\times 16\times 1$ $\mathbf{k}$-points. The convergence of the $G_0{W}_0$ band structure with a particular grid of $\mathbf{k}$-points does not imply that BSE will be converged with the same grid. An interpolation with a fine grid of $120\times 120\times 1$ $\mathbf{k}$-points was needed to achieve convergence. Figure 2 also shows the imaginary part of dielectric function without excitonic effects, which are given by the RPA. It is clear that the excitonic effetcs are important in this material. In the RPA spectrum, the dielectric function is broad, and it does not show any notable peak in the beginning of the absorption zone. Besides, the beginning of the absorption zone coincides with the fundamental band gap obtained, which was 7.77 eV. On the other hand, in the BSE calculation, two dominant peaks emerge below the conduction band. These peaks have the highest oscillator strength and correspond to the first and second excitonic states. The first peak is at an energy of 5.58 eV, and the second peak is at an energy of 6.48 eV. In Table 3, we summarize the gap values of the band structure, the optical gap, and the excitonic binding energy. Figure 3 shows the real part of the dielectric function calculated with and without excitonic effects.

 

Fig. 2. Imaginary part of dielectric function of 2D hBN. The blue (red) lines represent the BSE (RPA) imaginary part of dielectric function.

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Fig. 3. Real part of the dielectric function of 2D hBN. The blue (red) lines represent the BSE (RPA) real part of dielectric function.

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We also calculated the eigenvalues of the two particle states. Figure 4 shows the energies of the eight lowest-energy excitonic states. From now on, we label each state by the corresponding energy in an ascending order. The pairs of states (1,2), (3,4), and (7,8) are degenerate. States 1 and 2 are the degenerated ground state. We plot the probability density ${|\varphi ({\mathbf{r}}_e,{\mathbf{r}}_h)|}^2$obtained from the BSE for these eight excitonic states in Fig. 5. These plots show the probability to find an electron at position ${\mathbf{r}}_e$ if the hole is located at ${\mathbf{r}}_h$. We set the hole localized slightly above the nitrogen atom. The results were calculated using a coarse grid of $12\times 12\times 1$ $\mathbf{k}$-points and a BSE interpolation of $72\times 72\times 1$ $\mathbf{k}$-points. The complementarity of the degenerate states can be noted. For instance, if one adds the probability density of states 3 and 4, the symmetry of the lattice is recovered. The same can be seen for the other degenerate states.

 

Fig. 4. Excitonic energies for the lowest energy exciton states. The system has a $C_{3v}$ symmetry with three representations: $A_1$, $E$, and $A_2$. The states 1 to 4 have $E$ symmetry and are valley degenerate; states 5 and 6 have $A_2$ and $A_1$ symmetries, respectively, and are nondegenerate (see [29]).

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Fig. 5. Probability density ${|\varphi ({\mathbf{r}}_e,{\mathbf{r}}_h)|}^2$ for the exciton states 1 to 8. The hole is localized slightly above the nitrogen atom (light color) at the center of the lattice.

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The work of [29] has also studied the excitonic states, and their results are in good agreement with the ones obtained from this work.

On the other hand, the experimental works of [25] and [26], which obtained an optical gap of $\sim 6.17\mathrm{eV}$ and $\sim 5.85\mathrm{eV}$, respectively, differ but not much from our results and the other theoretical works. The experimental values were obtained with the hBN on a substrate (not suspended), and that should change the screening of the electric field and, consequently, the binding energy of the excitons, reducing it, and so giving an excitonic peak at higher energies. More experimental works directed to the determination of the optical properties of monolayer hBN are needed.

4. BSE IN THE EQUATION OF MOTION FORMALISM AND THE ELLIOT FORMULA

In this section, we will follow the approach of the equation of motion derived in [48] and detailed in the Appendix A. The formalism is grounded on the calculation of the expected value of the polarization operator $\widehat{P}(t)$ after we introduce an external electric field of intensity ${\mathcal{E}}_0$ and frequency $\omega$ that couples with the electron gas in the 2D material. The optical conductivity and other properties can be obtained from the macroscopic relations. The starting point of our model is an effective Dirac Hamiltonian [49], which can be obtained from a power series expansion of the tight-binding Hamiltonian. The electron–electron interaction for a 2D material is given by the Keldysh potential [50]. This effective model only considers the top valence band and the bottom conductance band.

From the equation of motion, we derive the following BSE:

From the homogeneous part of Eq. (3), we can obtain the exciton energies and the wave functions. Using the procedure explained in [48], we can obtain the corresponding Elliot formula for the optical conductivity:

 

Fig. 6. Fit of the Elliot formula to the $G_0{W}_0+\mathrm{BSE}$ result. There is a very good agreement for the real part and a small shift in the imaginary part. The exciton linewidth used was $\gamma =0.1\mathrm{eV}$. The parameters of the fitting are shown in Table 6.

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Table 6. Comparison of the Elliot Formula Parameters Used in the $G_0{W}_0\mathbf{+}\mathrm{BSE}$ Calculation and the Equation of Motion Approach^a

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Finally, we used the equation of motion to predict the behavior of the exciton energy and the $\mathrm{K}\to \mathrm{K}$ transition energy as a function of the environment dielectric constant. The result can be seen in Fig. 7. There is a strong decrease in the $\mathrm{K}\to \mathrm{K}$ transition energy and an almost linear behavior, also decreasing, of the first exciton energy as the external dielectric constant increases.

 

Fig. 7. Exciton and $\mathrm{K}\to \mathrm{K}$ transition energy as function of the environment dielectric constant. We can see that the dependence of the first exciton energy is almost linear, while the $\mathrm{K}\to \mathrm{K}$ transition energy has a greater dependence on the dielectric constant.

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5. EXCITON–POLARITONS

In this section, we discuss the exciton–polariton modes in 2D hBN. Those modes are electromagnetic evanescent waves along the direction perpendicular to the hBN sheet. We assume that the hBN monolayer is cladded between two uniform, isotropic media with dielectric constants ${\epsilon }_1$ and ${\epsilon }_2$, and that the hBN sheet is in the $xy$ plane. Thus, the electromagnetic mode is evanescent in the $z$ axis and proportional to ${\mathrm{e}}^{-{\kappa }_{i}z}(i=1,2)$. These modes can be classified as transverse magnetic or transverse electric (TM/TE).

The dispersion relation for the TM mode is given by the solution given in [51],

A. Complex $q$ versus Complex $\omega$ Approaches to Polaritonics

First, we note that both Eqs. (5) and (6) are defined in the complex plane. Therefore, for a given $q$ ($\omega$) real, the solution will be a complex $\omega$ ($q$). Each of these approaches, complex $q$ or complex $\omega$, lead to different dispersion relations for the exciton–polaritons, as discussed elsewhere [52–56]. Both complex $q$ and complex $\omega$ approaches give the same results when an active media is used to balance the losses [56]. We note that the complex $q$ or complex $\omega$ approaches to polaritonics describe different experimental conditions. The complex $q$ approach is suitable when the polariton is excited in a finite region of space with a monochromatic wave, while the complex $\omega$ approach is valid when the entire sample is excited by a pulsed light [54].

The dispersion relation for both the TE and TM modes in the complex $\omega$ approach was obtained by solving Eqs. (5) and (6) and using the Elliot formula in Eq. (4) with the parameters of Table 7 for the ${\mathrm{G}}_0{\mathrm{W}}_0+\mathrm{BSE}$ calculation and a damping of $\gamma =0.1\mathrm{eV}$. The result is shown in Fig. 8, where $A$ and $B$ denote the first two excitonic energies. Both TE and TM modes can have a large localization (high ${\kappa }_i$ or $q$) in this case. The TE mode has a flat dispersion relation that approaches the exciton energy as $q$ goes to infinity. We note that the strong localization of the TE polaritons contrasts with the same type of mode in graphene, with the TE mode showing a poor degree of localization in this material. As expected, the TM mode has a higher frequency than the exciton energy, while the TE mode has a lower frequency.

 

Fig. 8. Exciton–polariton dispersion relation for complex frequency. The results are given as a function of the wavenumber $\tilde{\nu }={\lambda }_q^{-1}$. The gray dashed-dot line represents the light cone in air. In this approach, the wavenumber can reach large values for both TE and TM modes for either $A$ or $B$ exciton energies. Detail around excitons $A$ and $B$ is shown in the right panels.

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In the complex $\omega$ approach, both excitons $A$ and $B$ support polaritons. This can be understood by examining Eq. (4). As $\hslash \omega$ approaches $E_n-\mathrm{i}\gamma$, the corresponding contribution to the optical conductivity diverges. This quantity can be infinitely negative or positive depending on the real part of the frequency approaching $E_n$ from the right or the left, thus supporting TM and TE modes, respectively. Figure 8 also shows that the electrostatic limit $q\gg \omega /c$ is approached near both exciton energies. In that limit, the lifetime $\tau$ of the TM exciton–polariton $\tau =-1/\mathfrak{I}\omega$ can be calculated from

Next, we shall consider the case of complex $q$. There will then be a simple relation to obtain $q$ for a given frequency (assuming ${\epsilon }_i=1$):

 

Fig. 9. Exciton–polariton dispersion relation in the complex wavenumber approach. Panel A (B) shows the TM (TE) mode. The TM mode has a dispersion almost insensitive to the relaxation rate while the TE mode changes significantly: the wavenumber is close to the free-light one and only for $\gamma =4\mathrm{meV}$ there is a different behavior.

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An important figure of merit is the ratio of the propagation length $\ell =\mathfrak{I}{q}^{-1}$ to the exciton wavelength ${\lambda }_q=2\pi /\mathfrak{R}q$, as it indicates if a polariton can propagate before extinction, which is shown in Fig. 10 for several values of $\gamma$. The TM mode is highly supressed except for the low $\gamma =4\mathrm{meV}$, while the TE mode has a higher propagation rate and two different qualitative behaviors. For larger $\gamma$, the propagation rate increases with the frequency while the opposite happens for $\gamma =4\mathrm{meV}$. A better understanding of this behavior can be achieved if we consider the confinement ratio ${\kappa }_{\alpha }/{\lambda }_0$, with ${\lambda }_0$ being the wavelength of the free-radiation (see Fig. 11). The confinement of the TM modes increases with increasing frequency and has a negligible $\gamma$ dependence. On the other hand, the TE modes are poorly confined, with the confinement going to zero faster with increasing $\gamma$. This explains the large propagation rate in this case: the poorly confined field is essentially attenuated free radiation, i.e., there are no more excitons being excited, but the radiation field is attenuated by the material free charges.

 

Fig. 10. Exciton–polariton propagation ratio. Panel A (B) shows the TM (TE) mode. The propagation rate of the TM mode is very low except for $\gamma =4\mathrm{meV}$. The peak at $\omega =5.48$ corresponds to the propagation of radiation. As can be seen in Fig. 9, the wavenumber tends to the free-light wavenumber. The same result appears in the propagation rate for the TE modes: except for $\gamma =4\mathrm{meV}$, all other modes correspond to poorly confined modes (see Fig. 11 also). For $\gamma =4\mathrm{meV}$ and the TE mode, the propagation rate decreases with the increasing frequency.

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Fig. 11. Exciton–polariton confinement ratio. Panel A (B) shows the TM (TE) mode. The confinement of the TM mode increases with the frequency and has a small dependence with the relaxation rate $\gamma$. The TE modes for the higher values of $\gamma$ are poorly confined. For the value, we have a peak in the confinement below the exciton energy.

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The overall conclusion is that 2D hBN is a good platform for exciton–polaritons when we consider the complex $\omega$approach for both TM and TE modes. In the complex $q$approach, the results show that the exciton–polariton can be observed only for $\gamma =4\mathrm{meV}$.

B. UV Radiation Mirror

It was pointed out recently that excitons in ${\mathrm{MoSe}}_2$ can lead to high reflection of electromagnetic radiation [58,59]. In this section, we show that the same occurs with hBN but in a different spectral range. We consider a freestanding hBN monolayer. In this case, the reflection is given by [60]

 

Fig. 12. Reflection coefficient for monolayer hBN and different values of $\hslash \gamma$ with the parameters from Table 6. Panels (a) and (c) show the A and B excitons, respectively, with the $G_0{W}_0$ parameters, while panels (b) and (d) show the result from the equation of motion formalism. As the equation of motion formalism predicts higher excitonic weights, in this case we have broader reflectance peaks around the excitons energies in comparison with the $G_0{W}_0$ result.

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6. CONCLUSION

We calculated the band structure of 2D hexagonal boron nitride using DFT and the $G_0{W}_0$ approximation. Then, the Bethe–Salpeter equation was used to determine the excitonic energies of hBN. We determined the values of the band gap, optical gap, and excitonic binding energies using a first principles approach. The results are in good agreement with the ones obtained using a different approach, namely, the equation of motion formalism and the Elliot formula, which are also presented in this paper. This latter formalism allowed us to study the optical properties for both the TM and TE modes. Our results show that 2D hBN is a good candidate for polaritonics in the UV range. We also show that a single-layer hBN can act as an almost perfect mirror for ultraviolet electromagnetic radiation.