Abstract
Absorption and emission of pristinelike semiconducting monolayers of BN, AlN, GaN, and InN are systematically studied by abinitio methods. We calculate the absorption spectra for inplane and outofplane light polarization including quasiparticle and excitonic effects. Chemical trends with the cation of the absorption edge and the exciton binding are discussed in terms of the band structures. Exciton binding energies and localization radii are explained within the RytovaKeldysh model for excitons in two dimensions. The strong excitonic effects are due to the interplay of low dimensionality, confinement effects, and reduced screening. We find exciton radiative lifetimes ranging from tenths of picoseconds (BN) to tenths of nanoseconds (InN) at room temperature, thus making 2D nitrides, especially InN, promising materials for lightemitting diodes and highperformance solar cells.
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Introduction
Since the discovery of graphene, the research on alternative twodimensional (2D) materials has gained enormous interest. Silicene, germanene, phosphorene, and transition metal dichalcogenides are just a few examples of classes of novel lowdimensional systems which may serve as building blocks for efficient nanoelectronic and nanooptical devices^{1,2,3,4}. On the other hand, bulk groupIII nitrides such as GaN, AlN and InN are important materials for solid state lighting, as witnessed by the Nobel prize awarded in 2014 to Akasaki, Amano and Nakamura^{5}. The possibility to play with dimensionality to enhance the already exceptional properties of bulk nitrides has caused many attempts to grow 2D nitrides, although the strong tendency for \(sp^3\) bonding makes their preparation extremely difficult. Only monolayer BN can, in principle, be easily prepared by exfoliation from hexagonal bulk BN^{6}. Promising experimental attempts to obtain 2D AlN and GaN recently appeared^{7,8,9}. Measurements of GaN sheets encapsulated in graphene seem to suggest a fundamental optical gap of about 5 eV^{10}. Gaps even larger than in BN have been found for lowdimensional AlN structures^{9}. These successful experiments pave the way to new photovoltaic and optical nanodevices based on 2D nitrides^{11}.
Experimental activities to synthetize monolayer and fewlayer groupIII nitride sheets are accompanied by theoretical studies. For graphenelike, hexagonal 2D AlN and GaN, their dynamical stability has been explored and verified by phonon or even moleculardynamics calculations^{12,13,14,15,16}. The progress in theoretical growth predictions has been recently summarized in a review paper^{11}. Although the crystal structure of freestanding AlN and GaN is under debate, most studies favor a planar, honeycomb structure^{11,12,17,18}.
The recent progress in achieving growth of ultrathin 2D AlN and GaN layers on substrates, as well as the exfoliation of BN monolayers, suggest the possibility to prepare also 2D InN. Therefore, a theoretical understanding of the monolayer properties of BN, AlN, GaN, and InN, in particular of their electronic structure as well as optical absorption and emission properties, may push forward the experimental activities on 2D nitrides and their possible optoelectronic applications. Several studies have been focused on the tunability of the electronic gap^{13,14,15,16,19,20}. Excitonic properties have been recently investigated for GaN^{15,16,16,21} and InN^{22,23}, in addition to what was already found on the wellknown BN layer (^{18,24} and references therein).
In this letter we study the excitons of the 2D groupIII nitrides BN, AlN, GaN and InN relying on data from ab initio calculations of the electronic structure and optical properties. We focus on the onset of optical absorption and emission spectra for both in and outofplane light polarization. We study the main bound excitons with regards in particular to the influence of spatial confinement effects and reduced screening of the electronhole interaction. For the lowest exciton we calculate the binding energy and the excitonic radius and compare the results with those of model calculations. Finally we determine the excitonic radiative lifetime to characterize the optical emission.
Methods
Structural and electronic properties are calculated from ab initio with a three steps procedure: first, the equilibrium geometry of 2D honeycomb sheets , e.g. the lattice constants and the atomic coordinates, is determined by minimizing the total energy within the densityfunctional theory (DFT)^{25} in the localdensity approximation (LDA) using the Quantum Espresso code^{26,27}. The isolated 2D crystals are modeled by a supercell approach with a vacuum layer of thickness L of about 15 Å between the periodic images. Second, starting from the DFT Kohn–Sham electronic structure we calculate the quasiparticle energies \(\varepsilon ^{G_0W_o}_{n\mathbf{k}}\) in the \(G_0W_0\) approximation for the selfenergy by the chisig code^{28}. For the exchange part of the selfenergy, we use a \(102\times 102\times 1\) \(\mathbf {k}\)point mesh centered on \(\Gamma\) in the Brillouin Zone (BZ). For the correlation part of the selfenergy and for the screened Coulomb interaction W, we use a \(51\times 51\times 1\) \(\mathbf {k}\)point mesh and 300 bands. To avoid interaction between spurious replica of the monolayers, we apply to the Coulomb potential a cutoff at the vacuum layer boundary.
Electronhole pairs
In the third step, we solve the homogeneous Bethe–Salpeter equation (BSE)^{29} in the Tamm–Dancoff approximation by the dp4exc code^{30}. To account for the electronhole attraction, the BSE kernel is approximated by an electronhole static screened interaction term W beyond the electronhole exchange term v. Electronhole pair (exciton) states \(S\mathbf{Q}\rangle\) with energies \(E_S(\mathbf{Q})\), labeled by the quantum number S and the translation momentum \(\hslash \mathbf{Q}\), are the eigenstates of the excitonic Hamiltonian \(H_{exc}\) and can be obtained by diagonalizing it. The optical spectra are calculated using a mesh of 51 \(\times\) 51 \(\times\) 1 kpoints. The number of occupied (empty) bands are 3 (3 for BN, 4 for AlN, 5 for GaN and InN). A cut in the Coulomb potential was applied.
For the lowestenergy, at the onset of the absorption spectrum, groundstate bound excitons \(S=0\), \(\mathbf{Q}=0\) binding energy \(E_b\) and radius \(r_\text{exc}\), obtained within the accurate, but computationally heavy BSE calculations^{31}, are compared with predictions of an analytical model of excitons in two dimensions^{32,33}. The model Schrödinger equation in (SM1) relies on the effective mass approximation (EMA), with the reduced exciton mass \(\mu =m_em_h/(m_e+m_h)\) in terms of the electron (hole) mass \(m_e\) (\(m_h\)) extracted from the dispersion of the lowest conduction (highest valence) band. The screened Coulomb interaction between the electron and the hole is described by a RytovaKeldysh potential^{33,34,35} whose distance dependence is ruled by the 2D static electronic polarizability of the sheet, \(\alpha _{2D} = L \cdot ({\text{ R }e} \epsilon _{\parallel }(\omega = 0)  1) / 4 \pi\), which is computed from the inplane dielectric function \(\epsilon _{\parallel }\) in the limit of vanishing wave vector and frequency. The lowest exciton state is determined by a variational approach (see Supplementary Material for more details).
Optical properties
With the eigenstates of the BSE excitonic Hamiltonian \(E_S(\mathbf{Q}=0)\), \(S,\mathbf{Q}=0\rangle = \sum _{vc\mathbf {k}} A_S^{vc\mathbf {k}} v\mathbf {k}\rangle c\mathbf {k}\rangle\) at zeromomentum transfer \(\mathbf {Q}=0\) (the photon wave vector is negligible with respect to crystal momenta), we can calculate the optical conductivity \(\sigma _{\Vert /\bot }(\omega )\)^{36} or equivalently the frequencydependent dielectric tensor \(\epsilon _{\Vert /\bot }(\omega )\) for light polarization parallel/perpendicular to the crystal sheets,
with \(\langle c\mathbf{k}p_{\Vert /\bot }v\mathbf{k}\rangle\) as optical transition matrix elements between KS valence \((v\mathbf{k}\rangle )\) and conduction \((c\mathbf{k}\rangle )\) states and the unit area A. The imaginary part \(\text{Im} \epsilon _{\Vert /\bot }(\omega )\) is related to the optical absorption for in and outofplane light polarization.
Finally, following Ref.^{37}, we calculate the radiative decay rate \(\gamma _S(0)\) and lifetime \(\tau _S(0)\) of an exciton in state S with vanishing wavevector \(\mathbf{Q}=0\) as^{38}
where \(\mu _S^2\) is the square modulus of the exciton dipole transition matrix element. \(A_{uc}\) is the area of the unit cell and \(E_S(0)\) is the exciton excitation energy. To compute the thermally averaged radiative lifetimes \(\langle \tau _S\rangle\) of excitons in excitonic bands \(E_S\)(Q) at temperature T we apply the formula obtained in the classical hightemperature limit^{38}
with \(M=m_e+m_h\) being the translational (total) mass of the exciton.
Results
Geometry and electronic properties
2D sheets of groupIII nitrides have as equilibrium geometry a flat honeycomb structure similar to graphene^{39} but with reduced D\(_{3h}\) symmetry. As in graphene and BN, also in AlN, GaN and InN the firstrow element N tends toward a \(sp^2\) hybridization and inplane IIIN bonds. Therefore, the geometries are only characterized by the lattice constant a. The lattice parameters, listed in Table 1, follow a clear chemical trend with increasing atomic number of the cation. The lattice constants are smaller than those for the 3D bulk counterparts, consistently with the \(sp^2\) bonding being stronger than the \(sp^3\) one.
The electronic properties of the 2D IIIN monolayers, conversely to graphene, do not show any Dirac cone in the electronic band structure^{16,19,20,39}. The band structures, see Supplemental Material (SM) Fig. SM1, are characterized by indirect gaps with the valence band maximum (VBM) at K and the conduction band minimum (CBM) at \(\Gamma\). Apart from BN with a pronounced lowest direct band gap at K, the other nitrides exhibit a direct gap at \(\Gamma\). While BN and AlN tend to be indirect semiconductors, GaN and InN are direct 2D crystals^{19}. Table 1 indicates that the valence band maxima at \(\Gamma\) and K are not far away from each other. Consequently, for GaN, the direct and indirect gaps of 4.53 and 4.57 eV are close to each other. Indirect gaps of 4.55 eV^{14}, 4.38^{15} and 4.44 eV^{16} have also been computed. Fully selfconsistent QP calculations^{15} seem to strengthen the tendency for a direct gap at \(\Gamma\) but with a larger value of 5.39 eV. Experimental values of GaN layers embedded in AlN cover a wide range of 4.76–5.44 eV^{40} not too far from all the QP gaps. Moreover, the QP gap values in Table 1 are close to those found in the literature for BN^{18}, AlN^{19}, GaN^{15,16,19,21} and InN^{19,20,22}.
The gap values \(E_g\), listed in Table 1 are due to VBM composed primarily of N2\(p_z\) orbitals, while the CBM is a hybridization of groupIII s and some N2s states. The gaps arise from the presence of two different kinds of atoms, bearing different electronegativity, in the two hexagonal sublattices. A charge transfer occurs from the cation (B, Al, Ga, In) to the anion (N) with a subsequent opening of the electronic QP gaps, which span from about 7 eV for 2D BN to 1.7 eV for 2D InN, respecting the chemical trend. The electronic gaps shown by 2D IIIN monolayers are much larger than those of their 3D bulk counterparts (see collections in^{13,24,41}). Quantum confinement of the electrons is mainly responsible for this behavior. In addition, the reduced 2D screening also modifies the QP corrections.
Excitonic effects in optical spectra
The absorption spectra including the QP and excitonic effects are displayed in Fig. 1. Indeed the real part of the optical conductivity represents the absorbance for vanishing reflectance. The inclusion of electronhole interaction through BSE leads to a redshift of the absorption spectra and seems to compensate the quasiparticle effects, restoring a qualitative agreement with the position of DFT spectra but not their lineshape (see Figs. SM2 to SM5). For inplane light polarization strong peaks appear due to bound excitons, well below the fundamental direct gaps. For GaN and InN, and especially InN, such bound exciton peaks also occur above the fundamental QP gap belonging to highenergy interband transitions and, hence, being resonant with the continuum of scattering states of higher valenceconduction band pairs. The striking difference between the two light polarizations is the significant blueshift of the absorption spectra \(\text{ Re } \sigma _\bot (\omega )\) mainly due to localfield or depolarization effects described by the electronhole exchange interaction in the twoparticle Hamiltonian \(H_\text{exc}\)^{18,36,37,42}. Also their intensities are reduced, in particular for BN.
In Fig. 1 we focus on the onset of the absorption spectra for inplane light polarization. For BN the onset is characterized by three lowenergy peaks of decreasing intensity. For the first exciton at 5.2 eV, usually identified as optical gap \(E^\text{opt}_g\) reduced by the binding energy \(E_b\) with respect to the QP gap \(E_g\), we find a large exciton binding energy of 2.0 eV. A second and a weak third peak appear at 6.1 eV and 7.2 eV (see Fig. SM2), respectively. The first structure originates from transitions at the fundamental gap at K between the highest valence band and the lowest conduction band. The second exciton peak has a similar origin (see Fig. SM1). The first two bandedge excitons are related to \(\pi \rightarrow \pi ^*\) transitions. The third peak belongs mainly to transitions near \(\Gamma\) with some hybridization with \(\sigma\) and \(\sigma ^*\) states. Although the position of the peaks is slightly different, our value of the binding energy is in agreement with other predictions^{18,24,43}. The AlN inplane conductivity exhibits two prominent structures of increasing intensity, located at 4.6 eV and 5.9 eV. The binding energy of the first exciton is 1.9 eV. In a previous theoretical work^{44} three excitons peaks at 4.12 eV, 4.97 eV and 5.4 eV were found with a similar binding energy of 1.88 eV. Also for GaN we register the presence of two main peaks for the inplane component of the BSE sheet polarizability near the absorption edge. The first peak appears at 3.3 eV while the second is at 4.7 eV (see also SM), similarly to other calculations^{16}. The first exciton presents a large binding energy of 1.2 eV in agreement with Refs.^{15,16,21}. According to Fig. SM1(c), the first exciton is built by allowed, mainly \(\sigma \rightarrow \sigma ^\star\), optical transitions between the highest two valence bands and the lowest conduction band near \(\Gamma\). The second broader excitonic feature in Fig. 1 and Fig. SM4(a) with partial resonant character is mainly related to the lowest energy \(\pi \rightarrow \pi ^ \star\) transitions near K, combined with transitions along the \(K \rightarrow M\) line in the BZ, because of the flatness of the lowest conduction and highest valence bands. However, also contributions between the highest valence bands and the second lowest conduction band near \(\Gamma\), with \(\sigma \rightarrow \pi ^\star\) character, do contribute. Two bound excitons appear for InN in the BSE optical conductivity for inplane light polarization below photon energies of 6 eV. The first exciton, at 1.2 eV, has a binding energy of 0.6 eV and originates VBMCBM transitions near \(\Gamma\). A second bound but resonant exciton peak, visible at 3.9 eV, arises from valenceconduction band transitions near M and K at the BZ boundary.
Exciton binding and localization
In all 2D nitrides the first strong excitonic peak originates from transitions at the fundamental direct gap between the highest valence band and the lowest conduction band near \(\Gamma\) (K in the BN case). Their binding energies \(E_b\) are strongly related to the exciton radii \(r_{exc}\) as expressed by the chemical trends in Table 1 and Fig. 3. Thereby, the exciton radius has been derived abinitio from the expectation value with the computed exciton wavefunctions. The corresponding exciton wave functions are plotted in Fig. 2. They represent the probability to find the electron, having fixed the position of the hole near one nitrogen atom. Their localization around the chosen hole position clearly indicates the 2D character of the excitons, which extend above and below the basal plane just for a length of the order of the lattice parameter. The lateral electronhole distance covers instead a wider range in Fig. 2. By defining the excitonic radius \(r_{exc}\) as the first moment
where \(\psi (\mathbf{r}_h,\mathbf{r}_e)\) denotes the exciton eigenstate \(S\mathbf{Q}\rangle\) for the state \(S=0\) and \(\mathbf{Q}=0\) in realspace representation, we find an excitonic radius as small as 3.8 Å for 2D BN, and as large as 15.5 Å for 2D InN. Comparing the values \(r_{exc}\) with the lattice constants a in Table 1, a clear tendency for the formation of 2D excitons in the ground state from BN to InN becomes visible. The averaged electronhole distances \(r_\text{exc}\) in BN and AlN are not significantly larger than the lattice parameters. Their lowest energy excitons may be therefore interpreted to possess a Frenkellike character. Electron and hole in the excitons are found close to nearestneighbor atoms in the plane with the highest probability. In the case of GaN and InN, however, the larger electronhole distances \(r_\text{exc}\) suggest a character, which tends to be closer to that of 2D hydrogenlike Wannier–Mott excitons. In any case, the excitonic features in the 2D nitrides are much stronger compared to their 3D counterparts^{41}, because of two main reasons: the confinement of the motion of electrons and holes in the 2D honeycomb structure and the weak 2D screening of the electronhole attraction.
The chemical trends of the exciton parameters \(E_b\) and \(r_\text{exc}\) are qualitatively in line with the fundamental QP gaps \(E_g\) and the lattice constant a (see Table 1). As summarized in Fig. 3, with increasing cation atomic number we find an increasing lattice constant, accompanied by a decrease of the QP gaps. The exciton binding energy \(E_b\) qualitatively follows the gap energy, which characterizes the intrinsic screening in the sheet as visible from the polarizability \(\alpha _{2D}\).
Model 2D excitons
Using the polarizability \(\alpha _{2D}\) and mass parameters \(m_e\) and \(m_h\) from Table 1, the variational approach to 2D excitons^{33,34} yields model values for \(E_b\) and \(r_{exc}\) which are also listed in the same table. They are in good agreement with the BSE results. Moreover, we list in Table SM1 the results obtained by solving the model Hamiltonian in the two opposite (Wannier–Mott and logarithmic^{34}) approximations. The results clearly demonstrate that neither the logarithmic limit nor, in particular, the unscreened 2D Coulomb potential for the electronhole attraction^{34} are fit to describe the lowest bound for the 2D nitrides. A quantitative relationship between the two exciton parameters can be approximately given by \(E_b\sim \hslash ^2/(2\mu r^2_\text{exc})\) with \(\mu\) as the reduced exciton mass. Using in the above relation our BSE results give a prefactor of about 0.27, thereby confirming that the true abinitio computed excitons are far from a 2D hydrogenlike model, which would suggest a prefactor 1. The variational solution of the 2D model Hamiltonian, instead, gives a prefactor 0.3, in good agreement with the BSE results. Finally, a linear interpolation using the values obtained within the logarithmic limit gives a prefactor of about 0.5. In other words, the intermediate range is realized, i.e., neither the Wannier–Mott nor the logarithmic behavior gives a reasonable description of the screened potential, although in the logarithmic limit the error is smaller. The reliability of the variational solution of the 2D model Hamiltonian, previously tested on graphane, silicane, and germanane in^{33}, is demonstrated by comparing the abinitio and 2D model results in Table 1. For both exciton binding energy and exciton radius the chemical trends and the absolute values are in astonishing agreement. The energy deviations are only of the order of 0.1–0.2 eV and the radii differences are of the order or less than 20%. For the strongest bound exciton in BN practically the same values are obtained within abinitio and model descriptions. The rough description of the lowest exciton as a 2D 1s variational state seems hence to cover the correct physics.
Analyzing the variational results in Table 1 we can affirm that the model solutions are in good agreement with the results of full abinitio but cumbersome BSE calculations. The general validity of this conclusion is underlined in Fig. 4 including the results for the hydrogenated groupIV materials, graphane, silicongraphane, silicane, and germanane^{33,45,46}, which are in line with the findings for nitride monolayers.
Exciton radiative lifetimes
Important material characteristics for lightemitting diodes and laser devices are the rates of different electronhole recombination processes. Not much is known at the moment about these processes in 2D nitrides. Here, we focus on the exciton radiative lifetimes and compare the results with the corresponding lifetimes of 3D nitrides. Again, we consider the lowestenergy excitons \(S=0\) and \(\mathbf{Q}=0\) to find the radiative decay time (1) or its thermal average (2). The lifetimes \(\tau _S(0)\) of BN, AlN, and GaN are very similar to each other, while for 2D InN it is significantly larger (see Table 1). The lifetimes and the exciton binding energies in Table 1 show opposite trends, in accordance with the Heisenberg uncertainty principle.
The recombination times of the lowest energy excitons, are by more than one order of magnitude smaller than the corresponding lifetimes of excitons of transition metal dichalcogenides at the corner points K of the hexagonal BZ^{38}. The main reason for this difference is the stronger transition matrix elements in the case of the nitrides. The radiative lifetimes of the lowestenergy excitons clearly indicate the excellent emission properties of 2D nitrides. They seem to be more appropriate for LED and laser applications than the transition metal dichalcogenides.
The averaged lifetimes for excitons in the lowestenergy exciton band \(\langle \tau _S\rangle\) at room temperature still give rise to shorter values than the transition metal dichalcogenides^{38}. In general, they possess an averaged radiative lifetime \(\langle \tau _S\rangle\), which is three orders of magnitude larger than \(\tau _S(0)\). This is a consequence of the much faster recombination of the excitons with vanishing translational energy. Comparing the averaged lifetimes \(\langle \tau _S\rangle\) at room temperature in Table 1 with the values for 3D nitrides^{47,48}, differences of one to two orders of magnitude are observed. This fact, again, underlines the outstanding emission properties of 2D nitrides. Due to the quantum confinement, the reduced screening and, hence, the huge exciton binding energies, the sheet crystals appear as promising materials for active optoelectronic applications.
Finally, for a better understanding of the exciton radiative decay rates, we also apply the variational solutions for the lowest 1s excitons with a pure 2D screening derived in Ref.^{33}. With the parameters summarized in Table 1 we find the values given in parenthesis in columns 8 and 9. They are close to those computed abinitio. These findings indicate that the material dependence, the chemical trend, can be described by \(\tau _S(0) \sim E_g^{opt}r^2_{exc}/\langle v \mathbf{k} p_{\parallel } c \mathbf{k} \rangle ^2\). For the thermally averaged quantities it holds \(\langle \tau _S\rangle \sim \tau _S(0)M/(E_g^{opt})^2\). These relations are in qualitative agreement with the variation of the abinitio exciton lifetimes with the 2D crystal discussed above. However, Fig. 5 also clearly indicates that the material dependence on the optical matrix element square \(\langle v\mathbf{k} p_\parallel  c\mathbf{k} \rangle ^2\) is not negligible, at least going from GaN to InN. Along the row BN, AlN and GaN, this dependence is weak. The exciton lifetime \(\langle \tau _S\rangle\) at 300 K computed to about 73 (36) ps within the abinitio (model) scheme for the first exciton in GaN (see also Table 1) has to be related to similarly estimated values of 20 ps^{15} or 600 ps^{21}. The weak difference in the first case^{15} is astonishing because of the used different approaches. The stronger deviation in the second case^{21} is probably due to the applied larger exciton mass M estimated by bulk particle values and the different optical transition matrix elements taken into account.
Summary and conclusions
In conclusion, the optoelectronic properties of groupIII nitride monolayers have been here investigated abinitio. The quasiparticle electronic structures are computed within the \(G_0 W_0\) approximation. The twoparticle excitations and the accompanying optical properties are predicted by solving the Bethe–Salpeter equation with screened electronhole attraction and unscreened electronhole exchange. In addition to the absorption of the lowest bound electronhole pairs, their emission properties are also studied by computing the exciton radiative lifetimes. The characteristic parameters of the bound excitons dominating both absorption and emission have been also investigated in a variational framework by applying static sheet polarizabilities to describe the screening and the effective mass approximation. Abinitio and model excitonic parameters are compared. We have demonstrated a significant influence of the electronhole interactions on the optical spectra of the 2D nitrides, in particular on the absorption spectra. Strong bound exciton peaks appear below the direct quasiparticle gap. The lowest energy excitons exhibit huge exciton binding energies of 0.6–2.0 eV going from InN to BN. Thereby, the average electronhole distances vary between 3.8 and 15.5 Å from BN to InN, indicating a transition from a Frenkellike to a Wannier–Mottlike behavior along the row BN, AlN, GaN, and InN.
The emission properties of the 2D nitrides are illustrated by the radiative lifetimes of the lowestenergy excitons. We observed lifetimes of the order of 30–60 fs, much faster than for other 2D semiconductors. In the case of the thermal average of the excitons with finite translation vector, the radiative lifetimes increase substantially. Their low values indicate that groupIII nitride sheets appear to be promising materials for active optoelectronic applications in the visible up to the vacuum ultraviolet region.
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Acknowledgements
O.P. and I.K. acknowledge financial support from the EU MSCA HORIZON2020 projects ‘CoExAN’ (GA644076) and DiSeTCom (GA 823728). F.B. acknowledges travel support by INFN Tor Vergata. We thank Prof. Maurizia Palummo for useful discussions.
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M.S.P., D.G., O.P. and I.K. performed the calculations. F.B., V.O. and O.P. conceived the project. All authors wrote the paper. All authors analyzed the results. All authors reviewed the manuscript.
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Prete, M.S., Grassano, D., Pulci, O. et al. Giant excitonic absorption and emission in twodimensional groupIII nitrides. Sci Rep 10, 10719 (2020). https://doi.org/10.1038/s41598020676672
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DOI: https://doi.org/10.1038/s41598020676672
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