Boron dangling bonds in a monolayer of hexagonal boron nitride

Point defects in semiconductor crystals are being investigated as a potential platform to realize quantum information applications, such as quantum computing, quantum communication, and metrology.¹ The nitrogen-vacancy center in diamond is a testament to the power of these “quantum defects,” with applications including room-temperature qubit operation,² long-distance quantum communication,³ and nanoscale sensing.⁴ Despite its success, the performance of the nitrogen-vacancy center is hampered by decoherence from defects on the diamond surface.⁵ Quantum defects in two-dimensional (2D) materials may avoid these problems and enable extreme sensing, where the defect is brought atomically close to the sample.

Hexagonal boron nitride (h-BN) has emerged as an excellent candidate material; it is a layered, van-der-Waals-bonded material with an ultra-wide bandgap⁶ and excellent stability.⁷ Several single-photon emitters have been observed in h-BN; carbon dimers are the suspected origin of the emission near 4 eV.⁸ In this work, we will focus on the single-photon emitters in the visible spectrum whose quantum nature was first observed in 2016.⁹ Since then, characterization efforts have uncovered a trove of interesting properties. The so-called 2-eV emitters have a main optical transition that is notoriously heterogeneous, ranging in energy from 1.6 eV to 2.2 eV.¹⁰ Coupling to phonons is modest, characterized by a Huang–Rhys factor of $∼1–2$⁠.¹¹ The transition is linearly polarized; however, the alignment of the absorptive and emissive dipoles depends on the excitation energy.¹² For resonant excitation, the absorptive and emissive dipole are expected to be aligned, but a large misalignment may occur due to the presence of a third electronic state at higher energies. Magnetic field dependence has been observed, and a singlet ground state has been suggested.¹³ 

The microscopic origin of the emitters has been heavily debated. A negatively charged boron vacancy (⁠$VB$⁠) has been suggested to explain the emission.¹⁴ However, this defect has recently been shown to have very broad luminescence with peak emission at 1.46 eV;¹⁵ it is more consistent with the spin qubits observed in Ref. 16 rather than with the 2-eV emitters. Another proposal was for a complex between a nitrogen vacancy and a nitrogen antisite (⁠$VN$-$NB$⁠);^9,14 recent calculations demonstrate a large strain dependence that was suggested to explain the range of emission energies observed experimentally.¹⁷ However, this center suffers from a lack of stability: for Fermi-level positions where the neutral $VN$-$NB$ is stable, the $VB$ defect is lower in energy.¹⁴ The $VN$-$NB$ defect is, therefore, unlikely to be seen in the experiment. A complex with carbon impurities, $VN$-$CB$⁠, has also been proposed to explain the 2-eV emission.¹⁸ Similar to other vacancy complexes, this defect has a high formation energy, and the neutral charge state (corresponding to the $S=0$ spin state) is stable over only a narrow range of Fermi-level energies,^19,20 making it unlikely to be seen in experiment.

We recently proposed a very different type of center to explain the 2-eV emission, namely, boron dangling bonds (DBs), and demonstrated that their properties are consistent with the experimental observations.²¹ In our model calculations, doubly occupied (negatively charged) boron DBs exhibit an optical transition from the ground state to a localized $pz$ state at an energy of 2.06 eV, with a Huang–Rhys factor of 2.3; the transition energy is sensitive to the local geometry, explaining the heterogeneity in emission wavelengths. The transition is linearly polarized, and the proximity of the excited state to the conduction band explains the misalignment of the absorptive and emissive dipoles. A metastable triplet state exists, which can explain the magnetic-field dependence of the emission.

Our study in Ref. 21 addressed the DB in a bulk crystal of h-BN, consistent with the thick flakes that are typically used in the experiment. For applications, monolayers are being envisioned, and it is therefore important to examine the optical properties of the boron DB in a single monolayer. Here, we report first-principles calculations based on density functional theory (DFT) with a hybrid functional for boron DBs in a monolayer of h-BN. We find that the optical transition has the same character as in bulk and occurs at roughly the same energy (2.02 eV). The coupling to phonons is characterized by a Huang–Rhys factor of 2.4, again close to the value for the bulk defect. We also find that a metastable triplet state and thus an intersystem crossing (ISC) exists. While the properties of the B DB in a monolayer are quite similar to those for the DB in bulk h-BN, we provide details about both the singlet and triplet states that can affect the ISC rates.

Our work is based on hybrid DFT as implemented in the VASP code.^22,23 Projector augmented wave potentials²⁴ and a plane-wave basis set with an energy cutoff of 520 eV are used. We utilize the hybrid functional of Heyd et al.^25,26 to ensure accurate energetics, electronic structure, and charge localization. For consistency with our previous study,²¹ the fraction of non-local Hartree–Fock exchange $α$ is set to 0.40. The value of $α$ is related to the dielectric constant of the system,²⁷ and the reduced dielectric environment of a monolayer might require a higher value of $α$⁠. We have checked that our results are not sensitive to the value of $α$ and all our qualitative conclusions are robust. $α=0.40$ results in an indirect (⁠$K→Γ$⁠) fundamental bandgap of 6.31 eV. The direct optical gap is at K with a value of 7.06 eV.

We employ the point-defect formalism based on a supercell with periodic boundary conditions.²⁸ The B DB is investigated using a 60-atom supercell, constructed by building an orthorhombic cell (⁠$a×a3$⁠) and scaling by $5×3$⁠. A vacuum region of 20 Å is used to separate periodic images along the $z^$ direction. The Brillouin zone is sampled using a single special k point at (0.25, 0.25, 0.25). Lattice vectors are held fixed, and atomic coordinates are relaxed until forces are below 0.01 eV/Å. Spin polarization is explicitly taken into account. The B DB is modeled using a small void,²⁹ as described in detail in Ref. 21.

Our boron DB is negatively charged; the treatment of charged defects in reduced dimensions requires special care in the handling of the compensating background charge. In plane-wave DFT, the divergent $G=0$ Fourier component of the electrostatic potential is set to zero. For a charged system, this corresponds to adding a homogeneous background charge, sometimes referred to as the jellium background. This procedure works very well in three-dimensional solids, as long as appropriate correction terms are included to compensate for the interaction of the defect with the background in a finite cell.³⁰ Problems arise, however, in simulations of 2D materials that require a large vacuum region: the jellium background is present in the vacuum, which is unphysical. We contrast this with the actual physical situation, in which a charged defect in a monolayer would be compensated by oppositely charged dopants or defects associated with the same monolayer, rendering the system net neutral.

The unphysical nature of the standard treatment, in which a jellium background is present in the vacuum, is illustrated in Fig. 1. The conduction-band minimum of monolayer h-BN is a nearly-free electron (NFE) state at $Γ$⁠. The NFE state has a large spatial extent above and below the h-BN plane. When a negatively charged defect is introduced into the system, the NFE state is attracted to the positively charged jellium background through the Coulomb interaction. The NFE state spreads out over the entire vacuum region and lowers in energy; as a result, a spurious reduction of the bandgap by 1.0 eV is observed.

One method to correct for this effect would be to only apply a homogeneous background in the plane of the material, which requires modifying the underlying DFT implementation.³¹ Here, we use a different approach, in the spirit of the virtual-crystal approximation.³² The compensating background is applied by modifying the valence charge $Z$ of the B and N atoms in the cell by an amount $ΔZ=−q/Nat$⁠, where $q$ is the charge of the defect and $Nat$ is the number of atoms that the compensating charge is spread over. The result of this procedure is shown in Fig. 1, where the charge distribution of the NFE state more closely resembles that of the pristine supercell, with only minor perturbations due to the presence of the defect. Furthermore, the bandgap in the defect supercell now agrees with the value obtained from a primitive-cell calculation.

In this paper, we will focus on internal transitions of the dangling bond, which are transitions between states that are localized on the defect (as opposed to involving the delocalized bulk bands). To examine internal transitions we utilize the $Δ$SCF approach,³³ in which excitation energies are determined from total-energy differences between calculations with constrained occupations. Full atomic relaxations for the excited state are taken into account. Configuration coordinate diagrams are used to examine the coupling to phonon modes during an electronic transition¹ and to quantify the electron–phonon coupling by determining the Huang–Rhys factor in a one-dimensional approximation.³⁴ 

The relaxed structure of the B DB is shown in Figs. 2(a) and 2(b). For our model, the B DB assumes $C2v$ symmetry in the monolayer; we emphasize that based on experimental conditions, it is likely that the B DB will attain a lower symmetry, like $C1h$⁠, due to out-of-plane distortions. This argument is motivated by the fact that the emitters tend to be found near flake edges or extended defects.^35,36 Indeed, the sensitivity of the DB to its local environment explains why a wide range of emission energies have been observed in experiments.¹⁰ 

We assume a coordinate system where the $x$ axis is along the high-symmetry axis of the defect and the $z$ axis is orthogonal to the h-BN plane. The coordinate axes are shown in Fig. 2. With these definitions, the irreducible representation $A1$ of $C2v$ transforms like the vector $x$⁠, $B1$ like $z$⁠, and $B2$ like $y$⁠.

We focus on the negative charge state of the defect, as this was found to host an internal transition corresponding to the 2-eV emission in bulk h-BN.²¹ A spin-degenerate Kohn–Sham state associated with the negatively charged B DB is located in the gap, as shown in Fig. 2(c). The charge-density isosurface for this state is shown in Fig. 3(a). This state transforms like the $A1$ irreducible representation and is labeled as $a1$⁠.

We now explore, using the constrained DFT approach, whether an internal transition can take place in the negatively charged B DB. We find that, upon excitation, an electron becomes localized in a $pz$ orbital, as shown in Fig. 3(b). This state transforms like the $B1$ irreducible representation and is labeled as $b1$⁠.

In the ground state, two electrons occupy the B DB in a singlet state; this many-body state is labeled as $1A1$ where the superscript is the spin multiplicity $2S+1$ for total spin $S$⁠. The excited state has one electron in the B DB and one in the B $pz$ state, giving a many-body state of $1B1$⁠. From our calculations, we find that the transition $1A1→1B1$ has an energy of 2.02 eV, which corresponds to the zero-phonon line. The relevant energetics after considering coupling to phonons are displayed as a configuration coordinate diagram in Fig. 3(c). We extract a Huang–Rhys parameter of 2.4.

These values for the transition energy and for the Huang–Rhys factor differ only slightly from the case of a B DB in bulk h-BN (a zero-phonon line of 2.06 eV and a Huang–Rhys factor of 2.3) and are actually within the computational error bar. The value of the Huang–Rhys parameter is related to the fraction of light going into the zero-phonon line compared to the phonon sideband. Our extracted Huang–Rhys parameter compares favorably with experiment, albeit slightly larger than the reported values of 1–2;¹¹ however, it has been suggested that experiments may underestimate the value due to the difficulty in distinguishing between photons from the zero-phonon line and those coupled to low-energy acoustic phonons.³⁷ Experimentally, it has been observed that the linewidth of the zero-phonon line broadens significantly in a monolayer compared to bulk flakes,⁹ but we do not attempt to capture this effect in our calculations.

The transition examined in Fig. 3 is spin-conserving. We can also examine the high-spin configuration of the excited state, where the electrons in the DB and in the $pz$ state have parallel spins, i.e., $S=1$⁠. We label this state $3B1$ and show its energetic position with respect to $1A1$ and $1B1$ in Fig. 4. The state $3B1$ is a metastable triplet state that is accessible through an ISC from the singlet states. The energies are again close to those calculated for the defect in bulk h-BN, but the Huang–Rhys factors are slightly different: for the monolayer, we have 0.2 and 1.5 for the upper and lower ISC, while for the bulk the values are 0.3 and 1.1.²¹ The difference arises from small differences in the geometry of the triplet state: the triplet state in the monolayer is closer in geometry to the excited singlet state, compared to the bulk case. Further investigations of how this impacts the ISC rates could be fruitful. The presence of the metastable triplet state would manifest experimentally through a magnetic-field dependence in the photoluminescence spectrum. When one considers a lower symmetry, such as $C1h$ (also shown in Fig. 4), the level structure is consistent with the experimental observations of magnetic-field dependence in bulk samples.¹³ Such symmetry lowering is quite plausible in realistic samples, in which the emitters tend to be found near flake edges or near extended defects.^35,36 Based on our calculations, we expect a similar dependence will be observed for monolayer samples.

In summary, we described comprehensive hybrid-functional calculations for the boron DB in monolayer h-BN. The handling of the compensating background charge was of key importance for correctly assessing the behavior of defects in monolayer h-BN. An internal transition where an electron is excited from the doubly occupied DB into a localized $pz$ state gives rise to optical emission at 2.02 eV. The coupling to phonon modes is characterized by a Huang–Rhys factor of 2.4. A metastable triplet state exists that enables an ISC and may give rise to a magnetic-field-dependent photoluminescence spectrum. While, overall, the properties of the B DB in the monolayer are similar to those in bulk h-BN, distinct differences exist that will impact the ISC rates. Our results provide essential information toward the utilization of the B DB in monolayer h-BN for extreme sensing and quantum information applications.

We gratefully acknowledge A. Alkauskas and L. C. Bassett. This work was supported by the National Science Foundation (NSF) through the Materials Research Science and Engineering Centers (MRSEC) Program under Grant No. DMR-1720256 (Seed Program) and through Enabling Quantum Leap: Convergent Accelerated Discovery Foundries for Quantum Materials Science, Engineering and Information (Q-AMASE-i) under Award No. DMR-1906325. Computational resources were provided by the Extreme Science and Engineering Discovery Environment (XSEDE), supported by the NSF (No. ACI-1548562).

The data that support the findings of this study are available from the corresponding author upon reasonable request.