Abstract

The electronic and optical properties of 2D hexagonal boron nitride are studied using first principle calculations. GW and Bethe–Salpeter equation (BSE) methods are employed in order to predict with better accuracy the excited and excitonic properties of this material. We determine the values of the band gap (7.32 eV, indirect), optical gap (5.58 eV), and excitonic binding energies (2.19 eV) and analyze the excitonic wave functions. We also calculate the exciton energies following an equation of motion formalism and the Elliot formula and find good agreement with the GW+BSE method. The optical properties are studied for the TM and TE modes, showing that 2D hexagonal boron nitride (hBN) is a good candidate for polaritonics in the UV range. In particular, it is shown that a single layer of hBN can act as an almost perfect mirror for ultraviolet electromagnetic radiation.

© 2019 Optical Society of America

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 220–510Nm1 and Young’s modulus 1.0TPa) [1] and high thermal conductivity (varies between 1 and 2000Wm1K1) [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 π and π* 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 [1322]. 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 6.17eV (assuming an indirect gap), while in [26] an optical band gap >5.85eV 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.

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Table 1. Several Band Gaps Calculated in This Work and by Other Authors Using GW0, G0W0, and BSE, in eVa

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 [3032] and GW+BSE [2729] 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 Σ 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+BSE calculations, thus obtaining validation of the method. In Section 2, we describe the details of the G0W0 calculations and results. In Section 3, we show the results of the BSE calculations. Both G0W0 and BSE calculations were performed with the software package Berkeley GW [4042]. 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. G0W0 RESULTS

G0W0 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 G0W0 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 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 Σ=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].

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Table 2. Details of DFT Calculations

We found that, for DFT calculations, a grid of 6×6×1k-points is enough to reach convergence. For the GW calculations, a grid of 16×16×1 k-points and a cutoff energy of 22.6 Ry and 1100 bands were needed for the dielectric matrix calculations. For the Σ 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 Γ 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 G0W0 band structure, the lowest energy of the first conduction bands is at the Γ-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 G0W0 is not equal and a simple shift would not suffice.

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Table 3. G0W0 Gap Values for the Transitions KΓ, KK, and ΓΓ, and Optical Gap and Exciton Binding Energy (EBE) Obtained from BSE in This Worka

 figure: Fig. 1.

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 7eV. 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 π-band has its highest energy at the K-point, while the σ1 and σ2 have the highest energy at the Γ-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, G0W0 results differ from the experimental results by values equal or smaller than 0.1 eV.

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Table 4. Width of the Valence Bandsa

We also calculated the effective masses of the highest valence band and lowest conduction band using both G0W0 and DFT (Table 5). We found no differences between G0W0 and DFT, except for the effective mass at KΓ on the first conduction band (DFT value greater by 0.08me). Thus, we conclude that DFT calculations are reliable to obtain the values of the effective masses in this material.

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Table 5. Effective Masses [in Electron Mass (me) Units] for hBN Calculated Using G0W0a

Reference [32] also calculated the effective mass at the Γ-point for the conduction band and obtained a value of (0.95±0.05)me with only slight variations for different planar directions. In our work, we obtained differences of 0.3me between different directions in reciprocal space at the Γ-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 ϵ2(ω) is then [42]

ϵ2(ω)=8π2e2ω2S|e·0|v|S|2δ(ωΩS),
where ΩS is the energy for an excitonic state S, 0|v|S is the velocity matrix element, and e is the direction of the polarization of incident light with energy ω. e is the electron charge.

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

ϵ2(ω)=8π2e2ω2vck|e·vk|v|ck|2δ(ω(EckEvk)),
which is a random phase approximation (RPA). The labels v(c) denote valence (conduction) band states, and k denotes the single particle momentum (only vertical transitions are considered).

Figure 2 shows the imaginary part of the dielectric function calculated by BSE, done on top of a G0W0 calculation with a grid of 16×16×1 k-points. The convergence of the G0W0 band structure with a particular grid of k-points does not imply that BSE will be converged with the same grid. An interpolation with a fine grid of 120×120×1 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.

 figure: Fig. 2.

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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 figure: Fig. 3.

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 |ϕ(re,rh)|2obtained from the BSE for these eight excitonic states in Fig. 5. These plots show the probability to find an electron at position re if the hole is located at rh. We set the hole localized slightly above the nitrogen atom. The results were calculated using a coarse grid of 12×12×1 k-points and a BSE interpolation of 72×72×1 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.

 figure: Fig. 4.

Fig. 4. Excitonic energies for the lowest energy exciton states. The system has a C3v symmetry with three representations: A1, E, and A2. The states 1 to 4 have E symmetry and are valley degenerate; states 5 and 6 have A2 and A1 symmetries, respectively, and are nondegenerate (see [29]).

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 figure: Fig. 5.

Fig. 5. Probability density |ϕ(re,rh)|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 6.17eV and 5.85eV, 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 P^(t) after we introduce an external electric field of intensity E0 and frequency ω 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:

(ωω˜λk)pλ(k,ω)=(E0dλ(k)+Bkλ(ω))Δfk,
where λ=±, p±(k,ω) is the interband transition amplitude, ω˜λk is the transition energy renormalized by the exchange self-energy, and Bkλ(ω) is a term that renormalizes the Rabi-frequency, dλ(k) is the dipole matrix element, and Δfk is the occupation difference, given by the Fermi–Dirac distribution. See Appendix A for more details.

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:

σ(ω)σ0=4iωnpnωEn+iγ,
where n is the exciton state, γ is the exciton linewidth, En the exciton energy, pn the corresponding exciton weight, and σ0=e24. Figure 6 shows that the G0W0+BSE described in Section 3 fits well to the Elliot formula, with a good agreement in the real part and a small shift in the imaginary part. The energies and weights of the fit for the G0W0+BSEand the equation of motion method are compared in Table 6. We use the parameters from [49]: a0=2.51Å, t0=2.33eV vF=32t0a0, 2mvF2=3.92eV. The Keldysh potential parameter r0 was calculated in [29] to be r0=10Å. We can see excellent agreement between the exciton energies of both methods. The difference in the weights pn can be explained by the oversimplification of the Dirac Hamiltonian used for the Elliot formula and, consequently, the not-so-accurate dipole matrix elements that enter their calculation.

 figure: Fig. 6.

Fig. 6. Fit of the Elliot formula to the G0W0+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 γ=0.1eV. 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 G0W0+BSE Calculation and the Equation of Motion Approacha

Finally, we used the equation of motion to predict the behavior of the exciton energy and the KK 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 KK transition energy and an almost linear behavior, also decreasing, of the first exciton energy as the external dielectric constant increases.

 figure: Fig. 7.

Fig. 7. Exciton and KK 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 KK 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 ε1 and ε2, and that the hBN sheet is in the xy plane. Thus, the electromagnetic mode is evanescent in the z axis and proportional to eκiz(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],

ε1κ1+ε2κ2+iσ(ω)ε0ω=0,
and for the TE mode,
κ1+κ2iωμ0σ(ω)=0,
with σ(ω) the hBN optical conductivity, and
κi=q2εiω2c2,
where q is the exciton–polariton in-plane wave vector, and c is the velocity of light in vacuum. We shall consider the simplest case of ε1=ε2=1. A rule of thumb is that, when Iσ(ω)>0 (Iσ(ω)<0), TM (TE) modes are supported.

A. Complex q versus Complex ω Approaches to Polaritonics

First, we note that both Eqs. (5) and (6) are defined in the complex plane. Therefore, for a given q (ω) real, the solution will be a complex ω (q). Each of these approaches, complex q or complex ω, lead to different dispersion relations for the exciton–polaritons, as discussed elsewhere [5256]. Both complex q and complex ω approaches give the same results when an active media is used to balance the losses [56]. We note that the complex q or complex ω 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 ω 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 ω 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 G0W0+BSE calculation and a damping of γ=0.1eV. 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 κi 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.

 figure: Fig. 8.

Fig. 8. Exciton–polariton dispersion relation for complex frequency. The results are given as a function of the wavenumber ν˜=λq1. 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 ω approach, both excitons A and B support polaritons. This can be understood by examining Eq. (4). As ω approaches Eniγ, 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 En from the right or the left, thus supporting TM and TE modes, respectively. Figure 8 also shows that the electrostatic limit qω/c is approached near both exciton energies. In that limit, the lifetime τ of the TM exciton–polariton τ=1/Iω can be calculated from

τ1=γ+1pnI[bn]|(ε1+ε2)bn4παcq+1|2,
where α is the fine-structure constant and bn is the contribution that arises from the background conductivity provenient from interband transitions and other excitonic states. For a negligible background bn0, the exciton–polariton lifetime is proportional to the inverse of the exciton linewidth γ.

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

c2q2=ω2+c2κα2(ω),
with α=TM/TE and from Eqs. (5) and (6), we have
κTE(ω)=iε0ω2σ(ω),
κTM(ω)=iωμ0σ(ω)2.
The condition for the existence of polaritons is Rκα>0. These equations allowed us to calculate the dispersion relation shown in Fig. 9 for several values of the damping constant γ. The dependence of the γ parameter of excitons was studied for WS2 in [57] as a function of temperature, showing that the linewidth decreases as the temperature decreases. From Fig. 9 we can see that the TE mode is strongly supressed except when the damping has the very low value, but experimentally attainable, of 4 meV. The opposite happens for the TM mode, for which the dispersion relation is almost insensitive to the damping γ.

 figure: Fig. 9.

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 γ=4meV there is a different behavior.

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An important figure of merit is the ratio of the propagation length =Iq1 to the exciton wavelength λq=2π/Rq, as it indicates if a polariton can propagate before extinction, which is shown in Fig. 10 for several values of γ. The TM mode is highly supressed except for the low γ=4meV, while the TE mode has a higher propagation rate and two different qualitative behaviors. For larger γ, the propagation rate increases with the frequency while the opposite happens for γ=4meV. A better understanding of this behavior can be achieved if we consider the confinement ratio κα/λ0, with λ0 being the wavelength of the free-radiation (see Fig. 11). The confinement of the TM modes increases with increasing frequency and has a negligible γ dependence. On the other hand, the TE modes are poorly confined, with the confinement going to zero faster with increasing γ. 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.

 figure: Fig. 10.

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 γ=4meV. The peak at ω=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 γ=4meV, all other modes correspond to poorly confined modes (see Fig. 11 also). For γ=4meV and the TE mode, the propagation rate decreases with the increasing frequency.

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 figure: Fig. 11.

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 γ. The TE modes for the higher values of γ 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 ωapproach for both TM and TE modes. In the complex qapproach, the results show that the exciton–polariton can be observed only for γ=4meV.

B. UV Radiation Mirror

It was pointed out recently that excitons in MoSe2 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]

R=|παf(ω)2+παf(ω)|2,
where f(ω)=σ(ω)/σ0, α1371 is the fine structure constant, and σ0=e2/4. Figure 12 shows that the reflection can reach almost 100% for the value γ=4meV at the A exciton energy. This is a consequence of the high weights for hBN that appear in the Elliot formula (see Table 6). We emphasize that those results are for a freestanding hBN sheet. The γ value can be controlled by the temperature, as discussed in the previous sections. As shown in Fig. 7, the exciton energy and, therefore, the reflection peak can be controlled by varying the external dielectric constant.

 figure: Fig. 12.

Fig. 12. Reflection coefficient for monolayer hBN and different values of γ with the parameters from Table 6. Panels (a) and (c) show the A and B excitons, respectively, with the G0W0 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 G0W0 result.

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6. CONCLUSION

We calculated the band structure of 2D hexagonal boron nitride using DFT and the G0W0 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.

Appendix A: Formalism

The total Hamiltonian that we consider in the equation of motion approach is H=H0+H1+Hee, where we have the Dirac Hamiltonian,

H0(k)=vF(σ·k+σ3mvF2),
the dipole interaction Hamiltonian,
H^I(t)=eE(t)x^,
and the electron–electron interaction,
H^ee=e2dr1dr2ψ^(r1)ψ^(r2)V(r1r2)ψ^(r2)ψ^(r1),
where we used the field operator
ψ^(r,t)=1Lk,λϕλ(k)a^kλ(t)eik·r,
with the eigenvector of H0,
ϕλ(k)=Ek+λm2Ek(1kxikyλEk+m),
and eigenvalues
Ek=k2+m2,
where, for the matter of simplicity, we used units such vF==1.

We note that the electron–electron interaction for charges confined in a 2D material is given by the Keldysh potential [50,61]:

V(q)=e2ε01q(r0q+εm).
The expected value of the polarization operator for the 2D Dirac equation can be written as
P(ω)=igse2kλdλ(k)pλ(k,ω),
where gs=4 takes into account the spin and valley degeneracy, and λ=± labels the valence (−) or the conduction (+) band. The dipole matrix element dλ(k) is
dλ(k)=12Ek(sinθ+imEkcosθ).
The interband transition amplitude is defined as
pλ(k,ω)=dω2πeiωta^k,λ(t)a^k,λ(t),
where a^k,λ(t) (a^k,λ(t)) is the creation (annihilation) operator in band λ in the Heisenberg picture.

As explained in [48], from the equation of motion for the transition amplitude, we can derive the following Bethe–Salpeter equation:

(ωω˜λk)pλ(k,ω)=(E0dλ(k)+Bkλ(ω))Δfk,
where ω˜λk is the renormalized transition energy,
ω˜λk=2λEk+λΣk,λxc,
where the exchange self-energy is included as
Σk,λxc=dq(2π)2V(q)Δfkq[Fλλλλ(k,kq)Fλλλλ(k,kq)],
where Fλ1λ2λ3λ4 are defined in Eq. (A15). We define Δfλk=nF(λEk)nF(λEk), where nF is the Fermi–Dirac distribution and which gives us the difference in occupation between valence and conductance bands for a vertical transition. Finally, the integral term Bkλ(ω) is
Bkλ(ω)=dq(2π)2V(|kq|)[pλ(q,ω)Fλλλλ(k,q)+pλ(q,ω)Fλλλλ(k,q)].
The homogeneous part of Eq. (A11), obtained by setting E0=0, can be used to calculate the excitons wavefunctions and energies. From the inhomogeneous solution of Eq. (A11), pλ(k,ω) the macroscopic polarization P(ω) can be calculated using Eq. (A8) and from there it follows the optical conductivity, permittivity, and absorbance.

The overlap of four wavefunctions is given by the Fλ1,λ2,λ3,λ4(k1,k2) function:

Fλ1,λ2,λ3,λ4(k1,k2)=ϕλ1(k1)ϕλ2(k2)ϕλ3(k2)ϕλ4(k1).

Funding

European Commission (EC) (785219); Portugese Fundação para a Ciência e a Tecnologia (FCT)(UID/FIS/04650/2013; PTDC/FIS-NAN/3668/2013); European Regional Development Fund (ERDF) and the Portuguese Foundation for Science and Technology (FCT) (POCI-01-0145-FEDER-028114).

Acknowledgment

N.M.R.P. acknowledges support from the European Commission through the project “Graphene-Driven Revolutions in ICT and Beyond,” and the Portuguese Foundation for Science and Technology (FCT) in the framework of the strategic financing. Additionally, N.M.R.P. acknowledges COMPETE2020, PORTUGAL2020, FEDER, and the Portuguese Foundation for Science and Technology (FCT) through project PTDC/FIS-NAN/3668/2013 and FEDER and the Portuguese Foundation for Science and Technology (FCT) through project POCI-01-0145-FEDER-028114.

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References

  • View by:

  1. K. Zhang, Y. Feng, F. Wang, Z. Yang, and J. Wang, “Two dimensional hexagonal boron nitride (2d-hbn): synthesis, properties and applications,” J. Mater. Chem. C 5, 11992–12022 (2017).
    [Crossref]
  2. J. Bao, M. Edwards, S. Huang, Y. Zhang, Y. Fu, X. Lu, Z. Yuan, K. Jeppson, and J. Liu, “Two-dimensional hexagonal boron nitride as lateral heat spreader in electrically insulating packaging,” J. Phys. D 49, 265501 (2016).
    [Crossref]
  3. L. Banszerus, M. Schmitz, S. Engels, M. Goldsche, K. Watanabe, T. Taniguchi, B. Beschoten, and C. Stampfer, “Ballistic transport exceeding 28 in cvd grown graphene,” Nano Lett. 16, 1387–1391 (2016).
    [Crossref]
  4. J. Jung, A. M. DaSilva, A. H. MacDonald, and S. Adam, “Origin of band gaps in graphene on hexagonal boron nitride,” Nat. Commun. 6, 1 (2015).
    [Crossref]
  5. G. Shi, Y. Hanlumyuang, Z. Liu, Y. Gong, W. Gao, B. Li, J. Kono, J. Lou, R. Vajtai, P. Sharma, and P. M. Ajayan, “Boron nitride-graphene nanocapacitor and the origins of anomalous size-dependent increase of capacitance,” Nano Lett. 14, 1739–1744 (2014).
    [Crossref]
  6. K. Zimmermann, A. Jordan, F. Gay, K. Watanabe, T. Taniguchi, Z. Han, V. Bouchiat, H. Sellier, and B. Sacépé, “Tunable transmission of quantum Hall edge channels with full degeneracy lifting in split-gated graphene devices,” Nat. Commun. 8, 14983 (2017).
    [Crossref]
  7. J. Duan, X. Wang, X. Lai, G. Li, K. Watanabe, T. Taniguchi, M. Zebarjadi, and E. Y. Andrei, “High thermoelectric power factor in graphene/hbn devices,” Proc. Natl. Acad. Sci. 113, 14272–14276 (2016).
    [Crossref]
  8. A. Nagashima, N. Tejima, Y. Gamou, T. Kawai, and C. Oshima, “Electronic dispersion relations of monolayer hexagonal boron nitride formed on the Ni(111) surface,” Phys. Rev. B 51, 4606–4613 (1995).
    [Crossref]
  9. X. Li, S. Sundaram, Y. El Gmili, T. Ayari, R. Puybaret, G. Patriarche, P. L. Voss, J. P. Salvestrini, and A. Ougazzaden, “Large-area two-dimensional layered hexagonal boron nitride grown on sapphire by metalorganic vapor phase epitaxy,” Cryst. Growth Des. 16, 3409–3415 (2016).
    [Crossref]
  10. T. Q. P. Vuong, G. Nbois, P. Valvin, E. Rousseau, A. Summerfield, C. J. Mellor, Y. Cho, T. S. Cheng, J. D. Albar, L. Eaves, C. T. Foxon, P. H. Beton, S. V. Novikov, and B. Gil, “Deep ultraviolet emission in hexagonal boron nitride grown by high-temperature molecular beam epitaxy,” 2D Mater. 4, 021023 (2017).
    [Crossref]
  11. D. Basov, M. Fogler, and F. G. de Abajo, “Polaritons in van der Waals materials,” Science 354, aag1992 (2016).
    [Crossref]
  12. T. Low, A. Chaves, J. D. Caldwell, A. Kumar, N. X. Fang, P. Avouris, T. F. Heinz, F. Guinea, L. Martin-Moreno, and F. Koppens, “Polaritons in layered two-dimensional materials,” Nat. Mater. 16, 182–194 (2017).
    [Crossref]
  13. Y. Watanabe, W. Inami, and Y. Kawata, “Deep-ultraviolet light excites surface plasmon for the enhancement of photoelectron emission,” J. Appl. Phys. 109, 023112 (2011).
    [Crossref]
  14. N. Mattiucci, G. D’Aguanno, H. O. Everitt, J. V. Foreman, J. M. Callahan, M. C. Buncick, and M. J. Bloemer, “Ultraviolet surface-enhanced raman scattering at the plasmonic band edge of a metallic grating,” Opt. Express 20, 1868–1877 (2012).
    [Crossref]
  15. J. M. McMahon, G. C. Schatz, and S. K. Gray, “Plasmonics in the ultraviolet with the poor metals al, ga, in, sn, tl, pb, and bi,” Phys. Chem. Chem. Phys. 15, 5415–5423 (2013).
    [Crossref]
  16. Y. Yang, J. M. Callahan, T.-H. Kim, A. S. Brown, and H. O. Everitt, “Ultraviolet nanoplasmonics: a demonstration of surface-enhanced Raman spectroscopy, fluorescence, and photodegradation using gallium nanoparticles,” Nano Lett. 13, 2837–2841 (2013).
    [Crossref]
  17. G. Maidecchi, G. Gonella, R. Proietti Zaccaria, R. Moroni, L. Anghinolfi, A. Giglia, S. Nannarone, L. Mattera, H.-L. Dai, M. Canepa, and F. Bisio, “Deep ultraviolet plasmon resonance in aluminum nanoparticle arrays,” ACS Nano 7, 5834–5841 (2013).
    [Crossref]
  18. M. B. Ross and G. C. Schatz, “Aluminum and indium plasmonic nanoantennas in the ultraviolet,” J. Phys. Chem. C 118, 12506–12514 (2014).
    [Crossref]
  19. A. M. Watson, X. Zhang, R. Alcaraz de La Osa, J. M. Sanz, F. González, F. Moreno, G. Finkelstein, J. Liu, and H. O. Everitt, “Rhodium nanoparticles for ultraviolet plasmonics,” Nano Lett. 15, 1095–1100 (2015).
    [Crossref]
  20. R. Alcaraz de la Osa, J. Sanz, A. Barreda, J. Saiz, F. González, H. Everitt, and F. Moreno, “Rhodium tripod stars for UV plasmonics,” J. Phys. Chem. C 119, 12572–12580 (2015).
    [Crossref]
  21. Y. Gutierrez, D. Ortiz, J. M. Sanz, J. M. Saiz, F. Gonzalez, H. O. Everitt, and F. Moreno, “How an oxide shell affects the ultraviolet plasmonic behavior of ga, mg, and al nanostructures,” Opt. Express 24, 20621–20631 (2016).
    [Crossref]
  22. Y. Gutiérrez, R. Alcaraz de la Osa, D. Ortiz, J. M. Saiz, F. González, and F. Moreno, “Plasmonics in the ultraviolet with aluminum, gallium, magnesium and rhodium,” Appl. Sci. 8, 64 (2018).
    [Crossref]
  23. Y. Kumamoto, A. Taguchi, N. I. Smith, and S. Kawata, “Deep uv resonant Raman spectroscopy for photodamage characterization in cells,” Biomed. Opt. Express 2, 927–936 (2011).
    [Crossref]
  24. M. W. Knight, L. Liu, Y. Wang, L. Brown, S. Mukherjee, N. S. King, H. O. Everitt, P. Nordlander, and N. J. Halas, “Aluminum plasmonic nanoantennas,” Nano Lett. 12, 6000–6004 (2012).
    [Crossref]
  25. K. Ba, W. Jiang, J. Cheng, J. Bao, N. Xuan, Y. Sun, B. Liu, A. Xie, S. Wu, and Z. Sun, “Chemical and bandgap engineering in monolayer hexagonal boron nitride,” Sci. Rep. 7, 45584 (2017).
    [Crossref]
  26. Y. Stehle, H. M. Meyer, R. R. Unocic, M. Kidder, G. Polizos, P. G. Datskos, R. Jackson, S. N. Smirnov, and I. V. Vlassiouk, “Synthesis of hexagonal boron nitride monolayer: control of nucleation and crystal morphology,” Chem. Mater. 27, 8041–8047 (2015).
    [Crossref]
  27. L. Wirtz, A. Marini, and A. Rubio, “Excitons in boron nitride nanotubes: dimensionality effects,” Phys. Rev. Lett. 96, 126104 (2006).
    [Crossref]
  28. P. Cudazzo, L. Sponza, C. Giorgetti, L. Reining, F. Sottile, and M. Gatti, “Exciton band structure in two-dimensional materials,” Phys. Rev. Lett. 116, 066803 (2016).
    [Crossref]
  29. T. Galvani, F. Paleari, H. P. C. Miranda, A. Molina-Sánchez, L. Wirtz, S. Latil, H. Amara, and F. M. C. Ducastelle, “Excitons in boron nitride single layer,” Phys. Rev. B 94, 125303 (2016).
    [Crossref]
  30. N. Berseneva, A. Gulans, A. V. Krasheninnikov, and R. M. Nieminen, “Electronic structure of boron nitride sheets doped with carbon from first-principles calculations,” Phys. Rev. B 87, 035404 (2013).
    [Crossref]
  31. H. Şahin, S. Cahangirov, M. Topsakal, E. Bekaroglu, E. Akturk, R. T. Senger, and S. Ciraci, “Monolayer honeycomb structures of group-iv elements and iii-v binary compounds: first-principles calculations,” Phys. Rev. B 80, 155453 (2009).
    [Crossref]
  32. X. Blase, A. Rubio, S. G. Louie, and M. L. Cohen, “Quasiparticle band structure of bulk hexagonal boron nitride and related systems,” Phys. Rev. B 51, 6868–6875 (1995).
    [Crossref]
  33. L. Hedin, “New method for calculating the one-particle green’s function with application to the electron-gas problem,” Phys. Rev. 139, A796–A823 (1965).
    [Crossref]
  34. L. Hedin and S. Lundqvist, “Effects of electron-electron and electron-phonon interactions on the one-electron states of solids,” Solid State Phys. 23, 1–181 (1970).
    [Crossref]
  35. E. E. Salpeter and H. A. Bethe, “A relativistic equation for bound-state problems,” Phys. Rev. 84, 1232–1242 (1951).
    [Crossref]
  36. S. Albrecht, L. Reining, R. Del Sole, and G. Onida, “Ab Initio calculation of excitonic effects in the optical spectra of semiconductors,” Phys. Rev. Lett. 80, 4510–4513 (1998).
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  40. J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, “Berkeleygw: a massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures,” Comput. Phys. Commun. 183, 1269–1289 (2012).
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  48. A. Chaves, R. Ribeiro, T. Frederico, and N. Peres, “Excitonic effects in the optical properties of 2d materials: An equation of motion approach,” 2D Mater. 4, 025086 (2017).
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2018 (3)

Y. Gutiérrez, R. Alcaraz de la Osa, D. Ortiz, J. M. Saiz, F. González, and F. Moreno, “Plasmonics in the ultraviolet with aluminum, gallium, magnesium and rhodium,” Appl. Sci. 8, 64 (2018).
[Crossref]

P. Back, S. Zeytinoglu, A. Ijaz, M. Kroner, and A. Imamoğlu, “Realization of an electrically tunable narrow-bandwidth atomically thin mirror using monolayer MoSe2,” Phys. Rev. Lett. 120, 037401 (2018).
[Crossref]

G. Scuri, Y. Zhou, A. A. High, D. S. Wild, C. Shu, K. De Greve, L. A. Jauregui, T. Taniguchi, K. Watanabe, P. Kim, M. D. Lukin, and H. Park, “Large excitonic reflectivity of monolayer MoSe2 encapsulated in hexagonal boron nitride,” Phys. Rev. Lett. 120, 037402 (2018).
[Crossref]

2017 (8)

A. Chaves, R. Ribeiro, T. Frederico, and N. Peres, “Excitonic effects in the optical properties of 2d materials: An equation of motion approach,” 2D Mater. 4, 025086 (2017).
[Crossref]

F. Cadiz, E. Courtade, C. Robert, G. Wang, Y. Shen, H. Cai, T. Taniguchi, K. Watanabe, H. Carrere, D. Lagarde, M. Manca, T. Amand, P. Renucci, S. Tongay, X. Marie, and B. Urbaszek, “Excitonic linewidth approaching the homogeneous limit in MoSe2-based van der Waals heterostructures,” Phys. Rev. X 7, 021026 (2017).
[Crossref]

K. Ba, W. Jiang, J. Cheng, J. Bao, N. Xuan, Y. Sun, B. Liu, A. Xie, S. Wu, and Z. Sun, “Chemical and bandgap engineering in monolayer hexagonal boron nitride,” Sci. Rep. 7, 45584 (2017).
[Crossref]

F. Ferreira and R. M. Ribeiro, “Improvements in the gw and Bethe-Salpeter-equation calculations on phosphorene,” Phys. Rev. B 96, 115431 (2017).
[Crossref]

K. Zhang, Y. Feng, F. Wang, Z. Yang, and J. Wang, “Two dimensional hexagonal boron nitride (2d-hbn): synthesis, properties and applications,” J. Mater. Chem. C 5, 11992–12022 (2017).
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K. Zimmermann, A. Jordan, F. Gay, K. Watanabe, T. Taniguchi, Z. Han, V. Bouchiat, H. Sellier, and B. Sacépé, “Tunable transmission of quantum Hall edge channels with full degeneracy lifting in split-gated graphene devices,” Nat. Commun. 8, 14983 (2017).
[Crossref]

T. Q. P. Vuong, G. Nbois, P. Valvin, E. Rousseau, A. Summerfield, C. J. Mellor, Y. Cho, T. S. Cheng, J. D. Albar, L. Eaves, C. T. Foxon, P. H. Beton, S. V. Novikov, and B. Gil, “Deep ultraviolet emission in hexagonal boron nitride grown by high-temperature molecular beam epitaxy,” 2D Mater. 4, 021023 (2017).
[Crossref]

T. Low, A. Chaves, J. D. Caldwell, A. Kumar, N. X. Fang, P. Avouris, T. F. Heinz, F. Guinea, L. Martin-Moreno, and F. Koppens, “Polaritons in layered two-dimensional materials,” Nat. Mater. 16, 182–194 (2017).
[Crossref]

2016 (9)

X. Li, S. Sundaram, Y. El Gmili, T. Ayari, R. Puybaret, G. Patriarche, P. L. Voss, J. P. Salvestrini, and A. Ougazzaden, “Large-area two-dimensional layered hexagonal boron nitride grown on sapphire by metalorganic vapor phase epitaxy,” Cryst. Growth Des. 16, 3409–3415 (2016).
[Crossref]

D. Basov, M. Fogler, and F. G. de Abajo, “Polaritons in van der Waals materials,” Science 354, aag1992 (2016).
[Crossref]

J. Duan, X. Wang, X. Lai, G. Li, K. Watanabe, T. Taniguchi, M. Zebarjadi, and E. Y. Andrei, “High thermoelectric power factor in graphene/hbn devices,” Proc. Natl. Acad. Sci. 113, 14272–14276 (2016).
[Crossref]

J. Bao, M. Edwards, S. Huang, Y. Zhang, Y. Fu, X. Lu, Z. Yuan, K. Jeppson, and J. Liu, “Two-dimensional hexagonal boron nitride as lateral heat spreader in electrically insulating packaging,” J. Phys. D 49, 265501 (2016).
[Crossref]

L. Banszerus, M. Schmitz, S. Engels, M. Goldsche, K. Watanabe, T. Taniguchi, B. Beschoten, and C. Stampfer, “Ballistic transport exceeding 28 in cvd grown graphene,” Nano Lett. 16, 1387–1391 (2016).
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Y. Gutierrez, D. Ortiz, J. M. Sanz, J. M. Saiz, F. Gonzalez, H. O. Everitt, and F. Moreno, “How an oxide shell affects the ultraviolet plasmonic behavior of ga, mg, and al nanostructures,” Opt. Express 24, 20621–20631 (2016).
[Crossref]

P. Cudazzo, L. Sponza, C. Giorgetti, L. Reining, F. Sottile, and M. Gatti, “Exciton band structure in two-dimensional materials,” Phys. Rev. Lett. 116, 066803 (2016).
[Crossref]

T. Galvani, F. Paleari, H. P. C. Miranda, A. Molina-Sánchez, L. Wirtz, S. Latil, H. Amara, and F. M. C. Ducastelle, “Excitons in boron nitride single layer,” Phys. Rev. B 94, 125303 (2016).
[Crossref]

G. Cassabois, P. Valvin, and B. Gil, “Hexagonal boron nitride is an indirect bandgap semiconductor,” Nat. Photonics 10, 262–266 (2016).
[Crossref]

2015 (4)

Y. Stehle, H. M. Meyer, R. R. Unocic, M. Kidder, G. Polizos, P. G. Datskos, R. Jackson, S. N. Smirnov, and I. V. Vlassiouk, “Synthesis of hexagonal boron nitride monolayer: control of nucleation and crystal morphology,” Chem. Mater. 27, 8041–8047 (2015).
[Crossref]

J. Jung, A. M. DaSilva, A. H. MacDonald, and S. Adam, “Origin of band gaps in graphene on hexagonal boron nitride,” Nat. Commun. 6, 1 (2015).
[Crossref]

A. M. Watson, X. Zhang, R. Alcaraz de La Osa, J. M. Sanz, F. González, F. Moreno, G. Finkelstein, J. Liu, and H. O. Everitt, “Rhodium nanoparticles for ultraviolet plasmonics,” Nano Lett. 15, 1095–1100 (2015).
[Crossref]

R. Alcaraz de la Osa, J. Sanz, A. Barreda, J. Saiz, F. González, H. Everitt, and F. Moreno, “Rhodium tripod stars for UV plasmonics,” J. Phys. Chem. C 119, 12572–12580 (2015).
[Crossref]

2014 (3)

G. Shi, Y. Hanlumyuang, Z. Liu, Y. Gong, W. Gao, B. Li, J. Kono, J. Lou, R. Vajtai, P. Sharma, and P. M. Ajayan, “Boron nitride-graphene nanocapacitor and the origins of anomalous size-dependent increase of capacitance,” Nano Lett. 14, 1739–1744 (2014).
[Crossref]

M. B. Ross and G. C. Schatz, “Aluminum and indium plasmonic nanoantennas in the ultraviolet,” J. Phys. Chem. C 118, 12506–12514 (2014).
[Crossref]

A. Rodin, A. Carvalho, and A. C. Neto, “Excitons in anisotropic two-dimensional semiconducting crystals,” Phys. Rev. B 90, 075429 (2014).
[Crossref]

2013 (5)

Y. V. Bludov, A. Ferreira, N. Peres, and M. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27, 1341001 (2013).
[Crossref]

N. Berseneva, A. Gulans, A. V. Krasheninnikov, and R. M. Nieminen, “Electronic structure of boron nitride sheets doped with carbon from first-principles calculations,” Phys. Rev. B 87, 035404 (2013).
[Crossref]

J. M. McMahon, G. C. Schatz, and S. K. Gray, “Plasmonics in the ultraviolet with the poor metals al, ga, in, sn, tl, pb, and bi,” Phys. Chem. Chem. Phys. 15, 5415–5423 (2013).
[Crossref]

Y. Yang, J. M. Callahan, T.-H. Kim, A. S. Brown, and H. O. Everitt, “Ultraviolet nanoplasmonics: a demonstration of surface-enhanced Raman spectroscopy, fluorescence, and photodegradation using gallium nanoparticles,” Nano Lett. 13, 2837–2841 (2013).
[Crossref]

G. Maidecchi, G. Gonella, R. Proietti Zaccaria, R. Moroni, L. Anghinolfi, A. Giglia, S. Nannarone, L. Mattera, H.-L. Dai, M. Canepa, and F. Bisio, “Deep ultraviolet plasmon resonance in aluminum nanoparticle arrays,” ACS Nano 7, 5834–5841 (2013).
[Crossref]

2012 (3)

N. Mattiucci, G. D’Aguanno, H. O. Everitt, J. V. Foreman, J. M. Callahan, M. C. Buncick, and M. J. Bloemer, “Ultraviolet surface-enhanced raman scattering at the plasmonic band edge of a metallic grating,” Opt. Express 20, 1868–1877 (2012).
[Crossref]

M. W. Knight, L. Liu, Y. Wang, L. Brown, S. Mukherjee, N. S. King, H. O. Everitt, P. Nordlander, and N. J. Halas, “Aluminum plasmonic nanoantennas,” Nano Lett. 12, 6000–6004 (2012).
[Crossref]

J. Deslippe, G. Samsonidze, D. A. Strubbe, M. Jain, M. L. Cohen, and S. G. Louie, “Berkeleygw: a massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures,” Comput. Phys. Commun. 183, 1269–1289 (2012).
[Crossref]

2011 (5)

R. Ribeiro and N. Peres, “Stability of boron nitride bilayers: ground-state energies, interlayer distances, and tight-binding description,” Phys. Rev. B 83, 235312 (2011).
[Crossref]

P. Cudazzo, I. V. Tokatly, and A. Rubio, “Dielectric screening in two-dimensional insulators: Implications for excitonic and impurity states in graphene,” Phys. Rev. B 84, 085406 (2011).
[Crossref]

I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex-ω approach versus complex-k approach in description of gain-assisted surface plasmon-polariton propagation along linear chains of metallic nanospheres,” Phys. Rev. B 83, 115451 (2011).
[Crossref]

Y. Kumamoto, A. Taguchi, N. I. Smith, and S. Kawata, “Deep uv resonant Raman spectroscopy for photodamage characterization in cells,” Biomed. Opt. Express 2, 927–936 (2011).
[Crossref]

Y. Watanabe, W. Inami, and Y. Kawata, “Deep-ultraviolet light excites surface plasmon for the enhancement of photoelectron emission,” J. Appl. Phys. 109, 023112 (2011).
[Crossref]

2010 (3)

B.-C. Shih, Y. Xue, P. Zhang, M. L. Cohen, and S. G. Louie, “Quasiparticle band gap of zno: high accuracy from the conventional G0W0 approach,” Phys. Rev. Lett. 105, 146401 (2010).
[Crossref]

J. A. Berger, L. Reining, and F. Sottile, “Ab initio,” Phys. Rev. B 82, 041103 (2010).
[Crossref]

M. Conforti and M. Guasoni, “Dispersive properties of linear chains of lossy metal nanoparticles,” J. Opt. Soc. Am. B 27, 1576–1582 (2010).
[Crossref]

2009 (3)

A. Archambault, T. V. Teperik, F. Marquier, and J.-J. Greffet, “Surface plasmon fourier optics,” Phys. Rev. B 79, 195414 (2009).
[Crossref]

P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, “Quantum espresso: a modular and open-source software project for quantum simulations of materials,” J. Phys. 21, 395502 (2009).
[Crossref]

H. Şahin, S. Cahangirov, M. Topsakal, E. Bekaroglu, E. Akturk, R. T. Senger, and S. Ciraci, “Monolayer honeycomb structures of group-iv elements and iii-v binary compounds: first-principles calculations,” Phys. Rev. B 80, 155453 (2009).
[Crossref]

2006 (1)

L. Wirtz, A. Marini, and A. Rubio, “Excitons in boron nitride nanotubes: dimensionality effects,” Phys. Rev. Lett. 96, 126104 (2006).
[Crossref]

2004 (1)

K. Watanabe, T. Taniguchi, and H. Kanda, “Direct-bandgap properties and evidence for ultraviolet lasing of hexagonal boron nitride single crystal,” Nat. Mater. 3, 404–409 (2004).
[Crossref]

2000 (1)

M. Rohlfing and S. G. Louie, “Electron-hole excitations and optical spectra from first principles,” Phys. Rev. B 62, 4927–4944 (2000).
[Crossref]

1998 (1)

S. Albrecht, L. Reining, R. Del Sole, and G. Onida, “Ab Initio calculation of excitonic effects in the optical spectra of semiconductors,” Phys. Rev. Lett. 80, 4510–4513 (1998).
[Crossref]

1996 (1)

J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77, 3865–3868 (1996).
[Crossref]

1995 (2)

X. Blase, A. Rubio, S. G. Louie, and M. L. Cohen, “Quasiparticle band structure of bulk hexagonal boron nitride and related systems,” Phys. Rev. B 51, 6868–6875 (1995).
[Crossref]

A. Nagashima, N. Tejima, Y. Gamou, T. Kawai, and C. Oshima, “Electronic dispersion relations of monolayer hexagonal boron nitride formed on the Ni(111) surface,” Phys. Rev. B 51, 4606–4613 (1995).
[Crossref]

1986 (1)

M. S. Hybertsen and S. G. Louie, “Electron correlation in semiconductors and insulators: band gaps and quasiparticle energies,” Phys. Rev. B 34, 5390–5413 (1986).
[Crossref]

1984 (1)

P. Halevi and R. Fuchs, “Generalised additional boundary condition for non-local dielectrics. i. reflectivity,” J. Phys. C 17, 3869–3888 (1984).
[Crossref]

1976 (1)

H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integrations,” Phys. Rev. B 13, 5188–5192 (1976).
[Crossref]

1973 (1)

E. Arakawa, M. Williams, R. Hamm, and R. Ritchie, “Effect of damping on surface plasmon dispersion,” Phys. Rev. Lett. 31, 1127–1129 (1973).
[Crossref]

1970 (1)

L. Hedin and S. Lundqvist, “Effects of electron-electron and electron-phonon interactions on the one-electron states of solids,” Solid State Phys. 23, 1–181 (1970).
[Crossref]

1965 (1)

L. Hedin, “New method for calculating the one-particle green’s function with application to the electron-gas problem,” Phys. Rev. 139, A796–A823 (1965).
[Crossref]

1951 (1)

E. E. Salpeter and H. A. Bethe, “A relativistic equation for bound-state problems,” Phys. Rev. 84, 1232–1242 (1951).
[Crossref]

Adam, S.

J. Jung, A. M. DaSilva, A. H. MacDonald, and S. Adam, “Origin of band gaps in graphene on hexagonal boron nitride,” Nat. Commun. 6, 1 (2015).
[Crossref]

Ajayan, P. M.

G. Shi, Y. Hanlumyuang, Z. Liu, Y. Gong, W. Gao, B. Li, J. Kono, J. Lou, R. Vajtai, P. Sharma, and P. M. Ajayan, “Boron nitride-graphene nanocapacitor and the origins of anomalous size-dependent increase of capacitance,” Nano Lett. 14, 1739–1744 (2014).
[Crossref]

Akturk, E.

H. Şahin, S. Cahangirov, M. Topsakal, E. Bekaroglu, E. Akturk, R. T. Senger, and S. Ciraci, “Monolayer honeycomb structures of group-iv elements and iii-v binary compounds: first-principles calculations,” Phys. Rev. B 80, 155453 (2009).
[Crossref]

Albar, J. D.

T. Q. P. Vuong, G. Nbois, P. Valvin, E. Rousseau, A. Summerfield, C. J. Mellor, Y. Cho, T. S. Cheng, J. D. Albar, L. Eaves, C. T. Foxon, P. H. Beton, S. V. Novikov, and B. Gil, “Deep ultraviolet emission in hexagonal boron nitride grown by high-temperature molecular beam epitaxy,” 2D Mater. 4, 021023 (2017).
[Crossref]

Albrecht, S.

S. Albrecht, L. Reining, R. Del Sole, and G. Onida, “Ab Initio calculation of excitonic effects in the optical spectra of semiconductors,” Phys. Rev. Lett. 80, 4510–4513 (1998).
[Crossref]

Alcaraz de la Osa, R.

Y. Gutiérrez, R. Alcaraz de la Osa, D. Ortiz, J. M. Saiz, F. González, and F. Moreno, “Plasmonics in the ultraviolet with aluminum, gallium, magnesium and rhodium,” Appl. Sci. 8, 64 (2018).
[Crossref]

A. M. Watson, X. Zhang, R. Alcaraz de La Osa, J. M. Sanz, F. González, F. Moreno, G. Finkelstein, J. Liu, and H. O. Everitt, “Rhodium nanoparticles for ultraviolet plasmonics,” Nano Lett. 15, 1095–1100 (2015).
[Crossref]

R. Alcaraz de la Osa, J. Sanz, A. Barreda, J. Saiz, F. González, H. Everitt, and F. Moreno, “Rhodium tripod stars for UV plasmonics,” J. Phys. Chem. C 119, 12572–12580 (2015).
[Crossref]

Amand, T.

F. Cadiz, E. Courtade, C. Robert, G. Wang, Y. Shen, H. Cai, T. Taniguchi, K. Watanabe, H. Carrere, D. Lagarde, M. Manca, T. Amand, P. Renucci, S. Tongay, X. Marie, and B. Urbaszek, “Excitonic linewidth approaching the homogeneous limit in MoSe2-based van der Waals heterostructures,” Phys. Rev. X 7, 021026 (2017).
[Crossref]

Amara, H.

T. Galvani, F. Paleari, H. P. C. Miranda, A. Molina-Sánchez, L. Wirtz, S. Latil, H. Amara, and F. M. C. Ducastelle, “Excitons in boron nitride single layer,” Phys. Rev. B 94, 125303 (2016).
[Crossref]

Andrei, E. Y.

J. Duan, X. Wang, X. Lai, G. Li, K. Watanabe, T. Taniguchi, M. Zebarjadi, and E. Y. Andrei, “High thermoelectric power factor in graphene/hbn devices,” Proc. Natl. Acad. Sci. 113, 14272–14276 (2016).
[Crossref]

Anghinolfi, L.

G. Maidecchi, G. Gonella, R. Proietti Zaccaria, R. Moroni, L. Anghinolfi, A. Giglia, S. Nannarone, L. Mattera, H.-L. Dai, M. Canepa, and F. Bisio, “Deep ultraviolet plasmon resonance in aluminum nanoparticle arrays,” ACS Nano 7, 5834–5841 (2013).
[Crossref]

Arakawa, E.

E. Arakawa, M. Williams, R. Hamm, and R. Ritchie, “Effect of damping on surface plasmon dispersion,” Phys. Rev. Lett. 31, 1127–1129 (1973).
[Crossref]

Archambault, A.

A. Archambault, T. V. Teperik, F. Marquier, and J.-J. Greffet, “Surface plasmon fourier optics,” Phys. Rev. B 79, 195414 (2009).
[Crossref]

Avouris, P.

T. Low, A. Chaves, J. D. Caldwell, A. Kumar, N. X. Fang, P. Avouris, T. F. Heinz, F. Guinea, L. Martin-Moreno, and F. Koppens, “Polaritons in layered two-dimensional materials,” Nat. Mater. 16, 182–194 (2017).
[Crossref]

Ayari, T.

X. Li, S. Sundaram, Y. El Gmili, T. Ayari, R. Puybaret, G. Patriarche, P. L. Voss, J. P. Salvestrini, and A. Ougazzaden, “Large-area two-dimensional layered hexagonal boron nitride grown on sapphire by metalorganic vapor phase epitaxy,” Cryst. Growth Des. 16, 3409–3415 (2016).
[Crossref]

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Figures (12)

Fig. 1.
Fig. 1. Electronic band structure (left) and electronic density of states of hBN (right) for both DFT and G W calculations.
Fig. 2.
Fig. 2. Imaginary part of dielectric function of 2D hBN. The blue (red) lines represent the BSE (RPA) imaginary part of dielectric function.
Fig. 3.
Fig. 3. Real part of the dielectric function of 2D hBN. The blue (red) lines represent the BSE (RPA) real part of dielectric function.
Fig. 4.
Fig. 4. Excitonic energies for the lowest energy exciton states. The system has a C 3 v 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]).
Fig. 5.
Fig. 5. Probability density | ϕ ( r e , 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.
Fig. 6.
Fig. 6. Fit of the Elliot formula to the G 0 W 0 + 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 γ = 0.1 eV . The parameters of the fitting are shown in Table 6.
Fig. 7.
Fig. 7. Exciton and K 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 K K transition energy has a greater dependence on the dielectric constant.
Fig. 8.
Fig. 8. Exciton–polariton dispersion relation for complex frequency. The results are given as a function of the wavenumber ν ˜ = λ 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.
Fig. 9.
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 γ = 4 meV there is a different behavior.
Fig. 10.
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 γ = 4 meV . The peak at ω = 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 γ = 4 meV , all other modes correspond to poorly confined modes (see Fig. 11 also). For γ = 4 meV and the TE mode, the propagation rate decreases with the increasing frequency.
Fig. 11.
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 γ . The TE modes for the higher values of γ are poorly confined. For the value, we have a peak in the confinement below the exciton energy.
Fig. 12.
Fig. 12. Reflection coefficient for monolayer hBN and different values of γ 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.

Tables (6)

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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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Table 2. Details of DFT Calculations

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Table 3. G 0 W 0 Gap Values for the Transitions K Γ , K K , and Γ Γ , and Optical Gap and Exciton Binding Energy (EBE) Obtained from BSE in This Work a

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Table 4. Width of the Valence Bands a

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Table 5. Effective Masses [in Electron Mass ( m e ) Units] for hBN Calculated Using G 0 W 0 a

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Table 6. Comparison of the Elliot Formula Parameters Used in the G 0 W 0 + BSE Calculation and the Equation of Motion Approach a

Equations (27)

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ϵ 2 ( ω ) = 8 π 2 e 2 ω 2 S | e · 0 | v | S | 2 δ ( ω Ω S ) ,
ϵ 2 ( ω ) = 8 π 2 e 2 ω 2 v c k | e · v k | v | c k | 2 δ ( ω ( E c k E v k ) ) ,
( ω ω ˜ λ k ) p λ ( k , ω ) = ( E 0 d λ ( k ) + B k λ ( ω ) ) Δ f k ,
σ ( ω ) σ 0 = 4 i ω n p n ω E n + i γ ,
ε 1 κ 1 + ε 2 κ 2 + i σ ( ω ) ε 0 ω = 0 ,
κ 1 + κ 2 i ω μ 0 σ ( ω ) = 0 ,
κ i = q 2 ε i ω 2 c 2 ,
τ 1 = γ + 1 p n I [ b n ] | ( ε 1 + ε 2 ) b n 4 π α c q + 1 | 2 ,
c 2 q 2 = ω 2 + c 2 κ α 2 ( ω ) ,
κ TE ( ω ) = i ε 0 ω 2 σ ( ω ) ,
κ TM ( ω ) = i ω μ 0 σ ( ω ) 2 .
R = | π α f ( ω ) 2 + π α f ( ω ) | 2 ,
H 0 ( k ) = v F ( σ · k + σ 3 m v F 2 ) ,
H ^ I ( t ) = e E ( t ) x ^ ,
H ^ ee = e 2 d r 1 d r 2 ψ ^ ( r 1 ) ψ ^ ( r 2 ) V ( r 1 r 2 ) ψ ^ ( r 2 ) ψ ^ ( r 1 ) ,
ψ ^ ( r , t ) = 1 L k , λ ϕ λ ( k ) a ^ k λ ( t ) e i k · r ,
ϕ λ ( k ) = E k + λ m 2 E k ( 1 k x i k y λ E k + m ) ,
E k = k 2 + m 2 ,
V ( q ) = e 2 ε 0 1 q ( r 0 q + ε m ) .
P ( ω ) = i g s e 2 k λ d λ ( k ) p λ ( k , ω ) ,
d λ ( k ) = 1 2 E k ( sin θ + i m E k cos θ ) .
p λ ( k , ω ) = d ω 2 π e i ω t a ^ k , λ ( t ) a ^ k , λ ( t ) ,
( ω ω ˜ λ k ) p λ ( k , ω ) = ( E 0 d λ ( k ) + B k λ ( ω ) ) Δ f k ,
ω ˜ λ k = 2 λ E k + λ Σ k , λ xc ,
Σ k , λ xc = d q ( 2 π ) 2 V ( q ) Δ f k q [ F λ λ λ λ ( k , k q ) F λ λ λ λ ( k , k q ) ] ,
B k λ ( ω ) = d q ( 2 π ) 2 V ( | k q | ) [ p λ ( q , ω ) F λ λ λ λ ( k , q ) + p λ ( q , ω ) F λ λ λ λ ( k , q ) ] .
F λ 1 , λ 2 , λ 3 , λ 4 ( k 1 , k 2 ) = ϕ λ 1 ( k 1 ) ϕ λ 2 ( k 2 ) ϕ λ 3 ( k 2 ) ϕ λ 4 ( k 1 ) .

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