We propose that the carbon dimer defect $ C B C N$ in hexagonal boron nitride gives rise to the ubiquitous narrow luminescence band with a zero-phonon line of 4.08 eV (usually labeled the 4.1 eV band). Our first-principles calculations are based on hybrid density functionals that provide a reliable description of wide bandgap materials. The calculated zero-phonon line energy of 4.3 eV is close to the experimental value, and the deduced Huang-Rhys factor of $ S \u2248 2.0$, indicating modest electron-phonon coupling, falls within the experimental range. The optical transition occurs between two localized *π*-type defects states, with a very short radiative lifetime of 1.2 ns, in very good accord with experiments.

Layered materials with interlayer van der Waals (vdW) bonding have recently attracted a lot of interest due to their distinct chemical and physical properties.^{1} Among this class of systems, hexagonal boron nitride (hBN) stands out because of its large bandgap of 6.08 eV.^{2} Advances in growth techniques have improved the material quality^{3} and enabled the growth of single-layer hBN, opening up applications in electronic and optoelectronic devices. Currently, hBN is mainly used in a passive role, for example, as a substrate and/or insulating layer in electronic devices made of graphene and other vdW materials^{1} or as a dielectric for photonic crystal cavities.^{4} However, hBN can also be used as an *active* optoelectronic material, e.g., as an electron-pumped ultraviolet (UV) laser.^{3} The recent discovery^{5} that hBN can host bright and stable single-photon emitters in the visible spectral range has sparked huge interest in the application of hBN as a light source in quantum optics applications.

Photoluminescence (PL), cathodoluminescence (CL), and electroluminescence experiments dating back to the 1950s^{6} already revealed a strong emission band between 3.3 and 4.1 eV in bulk hBN. This near-UV emission was so prevalent in some early samples that the bandgap of hBN was sometimes erroneously assumed to be just above 4 eV.^{7} However, more careful spectroscopic experiments^{8–10} on better-quality material revealed that the 4 eV luminescence is defect-related and is composed of at least two bands with very distinct properties. One is a broad featureless band centered around 3.9 eV.^{9} The other is a much narrower band with a clearly distinguishable zero-phonon line (ZPL) at 4.08 eV (typically called the 4.1 eV band in the literature), which is accompanied by a few phonon replicas.^{8–10} The dimensionless Huang-Rhys parameter, which quantifies electron-phonon coupling during optical transitions,^{11} was estimated to fall in the range of $ S = 1 \u2013 2$ for this band.^{12} The PL of this structured band appears when excitation energies exceed the ZPL of 4.1 eV. At variance, the broad 3.9 eV band appears only at excitation energies larger than 5.0 eV.^{9} Furthermore, time-dependent luminescence associated with these bands possesses very distinct characteristics. The structured narrow band shows very fast single-exponential decay with a lifetime of $ \tau = 1.1 \u2013 1.2$ ns,^{8,9} while the wide band exhibits multiexponential dynamics with the slowest components having decay times of a few 100 ns.^{9} All these results indicate a very distinct origin of the two bands, and from now on we will only discuss the structured 4.1 eV band.

Recently, single-photon emission associated with the 4.1 eV band has been reported.^{13} Fast electrons in a transmission electron microscope^{14} were used to excite luminescence at *T *=* *150 K. The measurement of the second-order correlation function confirmed that photons originate at a single emitter. The lineshape and the lifetime^{14} of the CL band were identical to those in ensemble measurements, confirming that in both experiments luminescence was caused by the same defect. These experiments have renewed the interest in the 4.1 eV band due to its potential use in quantum optics.

Despite the ubiquity of the 4.1 eV line, the microscopic nature of the defect that causes the luminescence is still not known. The intensity of the band increases drastically in both bulk crystals^{8} and epitaxial layers^{15} when carbon is purposely introduced during growth. Therefore, the involvement of carbon has been naturally assumed.^{8,10} It has been suggested^{10,15,16} that the 4.1 eV emission is caused by a transition from either a shallow donor (the so-called donor-acceptor-pair or DAP transition) or the conduction band (free-to-bound transition) to the neutral carbon acceptor on the nitrogen site, $ C N$. There are strong arguments against these scenarios. First, the time dynamics of DAP and free-to-bound transitions are inconsistent with the measured lifetime of $ \tau = 1.1 \u2013 1.2$ ns. For DAP transitions, the variation in donor-acceptor pair distances usually leads to marked nonexponential decay dynamics with very long tails,^{9} at odds with the single-exponential decay of the 4.1 eV line.^{8,9} Regarding radiative free-to-bound transitions, these occur on a millisecond time scale at typical excitation conditions (carrier densities $\u223c$ $ 10 17$ cm^{−3}),^{11} significantly slower than the dynamics of the 4.1 eV line. An additional argument comes from our recent first-principles study,^{17} where the acceptor level of $ C N$ was found to be at 2.9 eV above the valence band maximum (VBM). Since the bandgap of hBN is ∼6.1 eV, DAP and free-to-bound transitions should therefore have energies smaller than 3.2 eV, i.e., they should not appear in the UV region at all. The fast nanosecond radiative decay dynamics of the 4.1 eV line^{8,9,14} indicates that this is a transition where the ground state and excited state are localized in close proximity, likely on the same defect. Recently, Korona and Chojecki^{18} used quantum chemistry calculations to suggest that carbon clusters made from two to four atoms give luminescence in the range of 3.9 to 4.8 eV in monolayer hBN. However, different UV lines were not discriminated in that study, and neither the stability of clusters nor parameters (lifetime and electron-phonon coupling) of optical transitions were investigated.

In this Letter, we use first-principles density functional theory to show that the $ C B C N$ complex, in which two carbon atoms substitute on nearest-neighbor sites in the hBN lattice, accounts for all known experimental facts about the 4.1 eV luminescence: the involvement of carbon, the energy of the transition, the very short radiative lifetime, and moderate electron-phonon coupling.

Our calculations are based on the hybrid density functional of Heyd *et al.*^{19} In this approach, a fraction *α* of screened Fock exchange is admixed to the short-range exchange potential described by the generalized gradient approximation of Perdew *et al.*^{20} We use $ \alpha = 0.40$, for which calculations yield a bandgap of 6.42 eV, consistent with the experimental gap^{2} when zero-point renormalization due to electron-phonon interactions^{21} is accounted for. We used the projector-augmented wave approach^{22} with a plane wave energy cutoff of 500 eV, and van der Waals interactions were included via the Grimme D3 empirical correction scheme.^{23} With these settings, the calculated lattice parameters of hBN (*a *=* *2.49 Å and *c *=* *6.51 Å) and the enthalpy of formation for hBN (2.96 eV per formula unit) are in very good agreement with experimental values.^{24} Defect calculations have been performed in orthorhombic supercells containing 240 atoms^{17} and with lattice vectors $ 5 ( a + b ) , \u2009 3 ( a \u2212 b ) , \u2009 2 c$, where $ a , \u2009 b , \u2009 c$ are vectors of the primitive hBN lattice. The Brillouin zone was sampled at the Γ point. Ionic relaxation was carried out until Hellman-Feynman forces were less than 0.005 eV/Å. Calculations have been performed using the Vienna Ab-initio Simulation package (vasp).^{25}

We start by calculating the formation energy^{26} of the carbon dimer $ E f ( C B C N )$, which is given by

where $ E tot ( C B C N )$ is the total energy of the supercell containing one dimer and $ E tot ( BN )$ is the total energy of a pristine supercell. $ \mu N$ and $ \mu B$ are chemical potentials of nitrogen and boron; $ \mu N + \mu B = \mu BN = E BN$, where $ E BN$ is the total energy of bulk BN per formula unit. $ \mu C$ is the chemical potential of carbon, set to the per-atom energy of a diamond crystal. In Eq. (1), *q* is the charge of the defect and *E _{F}* is the Fermi level, referenced to the VBM

*E*. Δ

_{V}_{q}is a finite-size electrostatic correction term.

^{27}We note that the formation energy of the dimer does not depend on individual chemical potentials $ \mu N$ and $ \mu B$, as $ \mu N + \mu B = \mu BN$. The calculated formation energy is shown in Fig. 1, together with formation energies of $ C B$ and $ C N$ defects.

^{17}For these two latter defects, formation energies do depend on the chemical potentials of boron and nitrogen; only two limit cases are shown in Fig. 1. For N-rich conditions, $ \mu N rich = 1 / 2 E tot ( N 2 )$, half the energy of the N

_{2}molecule; for B-rich (N-poor) conditions, $ \mu B rich = E tot ( B )$, the energy of the B atom in elemental boron.

We find that $ C B C N$ has three possible charge states, *q* = −1, *q *=* *0, and $ q = + 1$ (Fig. 1). The neutral charge state is the most stable one throughout most of the bandgap, with a formation energy of 2.2 eV. As can be seen in Fig. 1, the formation energy of the dimer is not lower than those of simple substitutional defects for the two limiting cases of atomic chemical potentials. However, there is a wide range of chemical potentials ( $ \mu N rich \u2212 2.5 \u2009 eV < \mu N < \mu N rich \u2212 0.7 \u2009 eV$) and Fermi levels for which $ C B C N$ is more stable than either $ C B$ or $ C N$. In addition, if both $ C B$ and $ C N$ are present in the material (e.g., as a result of nonequilibrium growth), the formation of $ C B C N$ is expected. For example, when $ C B +$ binds to $ C N \u2212$ to form ( $ C B C N$)^{0}, an energy of 3.1 eV is released, indicating an exothermic reaction. We conclude that whenever carbon is present during the growth of hBN, $ C B C N$ should be a common defect.

We now turn to electronic properties of the dimer. In the neutral state, which is the one we will consider here, the dimer is nonmagnetic (*S *=* *0). The Kohn-Sham electronic state diagram [Fig. 2(a)] shows that there are two defect states in the bandgap. The lower-lying state is a *p _{z}* orbital localized on the “acceptor” site $ C N$, while the higher-lying state is a

*p*orbital on the “donor” site $ C B$ [Fig. 2(b)]. The defect geometry belongs to the $ C 2 v$ point group, and both states can be labeled according to the irreducible representation

_{z}*b*

_{2}. To distinguish the two states, we label the upper one $ b 2 *$.

In the ground state of the neutral dimer, the *b*_{2} state is doubly occupied, while the $ b 2 *$ state is empty, resulting in electronic configuration $ | b 2 b \xaf 2 \u27e9$ (symbols without a bar are for spin-up electrons, and symbols with a bar are for spin-down). This is a singlet state $ A 1 1$. In the ground state, the lengths of the C–C, C–N, and C–B bonds are 1.361, 1.391, and 1.497 Å, respectively (cf. the nearest-neighbor distance of 1.435 Å in bulk hBN). The excited state is obtained when one *b*_{2} electron is promoted to the $ b 2 *$ state, yielding the wave function $ 1 2( | b 2 b \xaf 2 * \u27e9 \u2212 | b \xaf 2 b 2 * \u27e9)$, also a $ A 1 1$ state. We calculate the energy and the resulting geometry of the defect in the excited state using the so-called delta self-consistent field approach (ΔSCF)^{28} with constrained orbital occupations, as explained in the supplementary material. In the excited state, there is a slight geometry rearrangement: the C–C bond elongates by 7% to 1.456 Å, while C–N and C–B bond lengths change by less than 1.5% (to 1.372 and 1.499 Å, respectively).

The calculated one-dimensional configuration coordinate diagram^{11} is shown in Fig. 3(a). We obtain a ZPL energy of $ E ZPL = 4.31$ eV. The Franck-Condon shifts are 0.22 eV in the excited state and 0.24 eV in the ground state. To quantify electron-phonon coupling, we calculate^{29} the Huang-Rhys factor *S*, which is a measure of the average number of phonons emitted during the optical transition.^{11} We find an effective phonon frequency of $ \u210f \Omega = 120$ meV, yielding the Huang-Rhys factor $ S = 0.24 / 0.12 = 2.0$, which is consistent with the experimental estimate $ S = 1 \u2212 2$ reported in Ref. 12. The Huang-Rhys factor is related to the Debye-Waller factor $ w ZPL$ (the fraction of light emitted into the ZPL) via $ w ZPL \u2248 e \u2212 S$. Our calculated value of $ w ZPL \u2248 0.14$ is smaller than the experimental value of 0.26 reported in Ref. 30. However, note that the exponential dependence of $ w ZPL$ on *S* small errors in the latter can lead to large errors in the former.

Apart from the excited-state singlet $ A 1 1$, there is also a triplet state $ A 3 1$ with configuration $ | b 2 b 2 * \u27e9$. $ A 3 1$ is 3.22 eV above the ground state, i.e., 1.09 eV lower than the excited-state singlet due to the exchange interaction between the two electrons. The energy-level diagram of the neutral $ C B C N$ dimer is shown in Fig. 3(b).

The rate of the radiative transition between the two singlet states is given by (in SI units)^{11}

Here, $ \epsilon 0$ is the vacuum permittivity, *n _{D}* is the refractive index of the host ( $ n D \u2248 2.6$ for an energy of $ E \u2248 4$ eV

^{31}), and $ \mu = 1.06$ $ e \xc5$ is the computed transition dipole moment for the transition $ b 2 \u2192 b 2 *$. Using the calculated value of $ E ZPL = 4.31$ eV, we obtain the rate of $ \Gamma rad = 8.6 \xd7 10 8$ s

^{−1}, corresponding to $ \tau rad = 1.2$ ns.

The $ b 2 \u2192 b 2 *$ transition is a strong dipole transition (so-called $ \pi \u2192 \pi *$ transition) with the polarization along the C–C bond. Experimental measurements of polarization would be really valuable but have not yet been performed so far. The calculated $ \tau rad = 1.2$ ns is in good agreement with the experimental value of $ 1.1 \u2212 1.2$ ns,^{8,9} but one should exercise caution comparing the two. The lifetime of the excited state $ A 1 1$ is governed by two decay mechanisms [Fig. 3(b)]: the radiative transition to the ground state $ \Gamma rad$ and the intersystem crossing (ISC) to the triplet state $ \Gamma ISC$: $ \tau = 1 / ( \Gamma rad + \Gamma ISC )$. However, in the supplementary material, we show that $ \Gamma ISC \u226a \Gamma rad$, and this justifies the comparison of the calculated rate with the measured one. Since $ \Gamma ISC \u226a \Gamma rad$, we also conclude that the quantum efficiency of the radiative transition is close to unity.

Our results show that the calculated optical properties of the $ C B C N$ defect are in very good agreement with the known properties of the 4.1 eV line. In fact, carbon dimers *have* been observed by annular dark field (ADF) electron microscopy in boron nitride monolayers^{32} exfoliated from *bulk* hBN. Carbon atoms have distinct intensity in ADF images, and this allowed a direct identification of C–C pairs.^{32} This experimental proof of the existence of $ C B C N$ defects in bulk hBN ions is in excellent agreement with our conclusions regarding the stability of carbon dimers.

In summary, we have reported the results of hybrid functional calculations for the carbon dimer in hexagonal boron nitride. These calculations allow us to conclude that $ C B C N$ is the defect, which is responsible for the 4.1 eV emission in hBN. The carbon dimer is expected to form whenever carbon is present during growth, explaining the observed correlation between the presence of carbon and the 4.1 eV line. The calculated zero-phonon line of the intradefect optical transition of 4.31 eV is close to the experimental value. Moreover, the theoretical Huang-Rhys factor of *S *=* *2.0 is consistent with the experimental estimate $ S = 1 \u2212 2$, and the radiative lifetime of $ \tau rad = 1.2$ ns is close to the experimental value of $ \tau rad = 1.1 \u2013 1.2$ ns. Identification of the chemical nature of the defect will enable more controlled experiments involving the 4.1 eV line, in particular using the $ C B C N$ defect as a single photon emitter.^{13} Our analysis shows that the quantum efficiency of this emitter should be close to unity. Combined with a very short radiative lifetime, this results in a very high photon yield. Together with a modest value of the Huang-Rhys factor (large weight of the ZPL) and a well-defined polarization axis, this makes the carbon dimer a very interesting quantum emitter in the near UV.

See the supplementary material for the calculation of the excited-state singlet and the intersystem crossing rate from the excited-state singlet $ A 1 1$ to the $ A 3 1$ state.

We acknowledge useful discussions with Marcus W. Doherty, Lukas Razinkovas, and Mark Turiansky. A.A. was funded by Grant No. 9.3.3.-LMT-K-712-14-0085 from the Research Council of Lithuania. M.M.S. acknowledges funding from the European Union's Horizon 2020 Research and Innovation Programme under Grant Agreement No. 820394 (project Asteriqs). The research reported here was also partially supported by the National Science Foundation (NSF) through the Materials Research Science and Engineering Center at UC Santa Barbara, No. DMR-1720256 (Seed). Calculations were performed at the High Performance Computing Center “HPC Saulėtekis” in the Faculty of Physics, Vilnius University. Computational resources were also provided by the Extreme Science and Engineering Discovery Environment (XSEDE), supported by the NSF (No. ACI-1548562).