Giant shift upon strain on the fluorescence spectrum of VNNB color centers in h-BN

Abstract

We study the effect of strain on the physical properties of the nitrogen antisite-vacancy pair in hexagonal boron nitride (h-BN), a color center that may be employed as a quantum bit in a two-dimensional material. With group theory and ab initio analysis we show that strong electron–phonon coupling plays a key role in the optical activation of this color center. We find a giant shift on the zero-phonon-line (ZPL) emission of the nitrogen antisite-vacancy pair defect upon applying strain that is typical of h-BN samples. Our results provide a plausible explanation for the experimental observation of quantum emitters with similar optical properties but widely scattered ZPL wavelengths and the experimentally observed dependence of the ZPL on the strain.

Introduction

Quantum emissions from two-dimensional (2D) materials have recently received considerable and rapidly rising interest of researchers in both condensed matter and quantum optics1,2,3 as these systems provide a potential basis for emerging technologies such as quantum nanophotonics4,5,6, quantum sensing7,8,9, and quantum information processing10,11. The observation of single-photon emitters (SPEs) in hexagonal boron nitrite (h-BN) has added a fascinating new facet to the research field of layered materials12. The wide band gap of h-BN makes it an insulator that can host high-quality emitters and allows for combination with other materials as substrates13. The experiments for exploring the nature of the emission of these color centers started with their first observations10,14,15,16,17,18,19,20,21,22,23,24,25 and recent theoretical works have provided evidence that the SPEs are indeed color centers, i.e., local point defects26,27,28,29,30.

Despite the considerable efforts that have been directed at the experimental exploration of these SPEs so far, a thorough theoretical understanding of the properties of the emitters that have been experimentally observed remains to be developed. Especially, the fact that many emitters appear at the edges of h-BN flakes and wrinkles on them31,32 motivates the investigation on the effect of strain on these emitters. Furthermore, the quantum emitters have shown magnetic properties in some experiments33,34,35,36, while in other experiments non-magnetic behavior was found21.

The most commonly observed quantum emitters exhibit emission in the visible, where competing for theoretical models27,37 exists in the literature. It has been suggested recently, based on the similarities of the optical lifetimes of the observed quantum emitters38, that two types of quantum emitters in the visible region may occur in h-BN, and the widely scattered zero-phonon-line (ZPL) energies might be attributed to external perturbations. One of the key candidates for this external perturbation is the local strain that may vary significantly in the h-BN samples, in particular, in polycrystalline h-BN samples. In another recent work, on the other hand, four different types of emitters have been found by means of combined chatodoluminescence and photoluminescence study where the applied stress did not change the brightness of emitters and the shift in ZPL was in the 10 meV region39. Yet another experiment also found a relatively small shift upon applied stress on a given emitter40.

These seemingly contradictory observations and the lack of a conclusive theoretical prediction motivates the study of the effect of strain on point defects in h-BN based on the assumption that point defects are the origin of the observed quantum emitters. Furthermore, a thorough understanding of the electron–strain coupling properties also forms a rigorous theoretical basis for proposed spin- and electro-mechanical systems for control and manipulation of a mechanical resonator by means of spin-motion coupling11,41,42. Group-theory analysis in combination with ab initio Kohn–Sham density functional theory (DFT) simulations can be a very powerful tools for understanding the coupling between optical emission and strain. So far the electron-strain coupling in h-BN emitters has been studied with limited accuracy40 by monitoring only the change in Kohn–Sham levels and the states that do not directly provide the ZPL energy in the PL spectrum monitored in the experiments.

Here we study the effect of strain on the ZPL emission of a key color center in h-BN, the nitrogen antisite-vacancy pair defect, by means of group theory and advanced DFT calculations, which can act as a quantum emitter with exhibiting a ZPL emission at around 1.9 eV (645 nm)27. We report a giant, 12 eV/strain ZPL-strain coupling parameter for this quantum emitter which results in about 100 nm scattering of the ZPL emission with ±1% strain in h-BN sample. The physical origin of this giant effect is the strong electron–phonon coupling in h-BN. This result implies that local perturbations for vacancy type defects can seriously affect their optical spectrum and provide an explanation for the zoo of reported quantum emitters in the visible region.

Results and discussion

Group theory and DFT calculation analysis

The sensitivity of the optical ZPL emission to the applied local strain greatly depends on the microscopic configuration of the point defect in h-BN. Here, we put our focus on the neutral nitrogen antisite-vacancy pair defect, VNNB, which was first suggested as a candidate of the observed single-photon emissions10 and then thoroughly studied as one of the feasible quantum emitter in the visible with S = 1/2 spin state27. The optical properties of this defect have been examined from various theoretical methods and point of view27,43,44,45. The defect has a C2v symmetry before the relaxation of the atoms in monolayer h-BN, and introduces three levels in the energy band gap that are labeled as a1, b2 and \({b}_{2}^{\prime}\) owing to their irreducible symmetry representation. These energy levels are occupied by three electrons resulting a 2B2 many-electron ground state (see Figs. 1 and 2). In this paper, we use majuscule and minuscule to labels the symmetry of the many-elctron states and one-electron levels, respectively. The two lowest-energy spin-preserving optical transitions are the following: in the spin majority channel the electron from b2 may be promoted to \({b}_{2}^{\prime}\) or in the spin minority channel the electron from a1 may be promoted to b2. It is found that the a1b2 has lower energy than \({b}_{2}\leftrightarrow {b}_{2}^{\prime}\)27,43,44. However, for the defect with the C2v symmetry, the a1b2 optical transition has a very small optical transition in-plane dipole moment44. Indeed, we also find this behavior in our own DFT calculation (see Supplementary Fig. S1). We indeed notice that b2 and \({b}_{2}^{\prime}\) states have wavefunctions that extend out-of-plane (see Fig. 1b), and therefore, can couple to phonon modes that drive the atoms out-of-plane, the membrane modes. There is an unpaired electron in both B2 ground state and \({B}_{2}^{\prime}\) excited state placed on the b2 and \({b}_{2}^{\prime}\) orbital, respectively, that induce an out-of-plane geometry distortion. These phonon modes strongly couple the 2B2 ground state and the 2A1 excited state leading to the vibronic instablility of the ground state. Therefore, the defect does not preserve the planar structure and the nitrogen antisite moves out from the plane in the ground state reducing the symmetry of the defect to Cs (see Fig. 1a). This geometry is about 100 meV lower in energy than the C2v configuration which reveals the strong coupling of the defect electrons to the membrane mode phonons. This result basically agrees with previous DFT calculations46. In Cs symmetry, all the one-electron defect levels belong to \({a}^{\prime}\) irreducible representation. Despite the vibronic mixing, the correspondence to the high symmetry orbitals can be observed in Fig. 1b and c. When the hole is left at a1 orbital in the A1 excited state, then the coupling to the membrane phonons is negligible and the defects C2v symmetry is retrieved. For the sake of simplicity, here we use the C2v symmetry labels for both configurations. The transformation of the point symmetry of the defect is discussed in Supplementary Note 1.

Fig. 1: Defect orbitals of VNNB defect.
figure1

a The geometry of VNNB defect in the ground state. Top view (left) and side view (right). The dashed circle denotes the impurity nitrogen atom. The defect is out-of-plane and exhibits Cs symmetry. The red arrows indicate the direction of the phonon vibration. b The a1, b2 and \({b}_{2}^{\prime}\) defect states in C2v geometry. c The defect states in the Cs symmetry with the same energy order as those in (b). While in C2v symmetry only b2 state has components out-of-plane, it can be seen that in Cs symmetry the a1 state also gains an out-of-plane component.

Owing to the rearrangement of the ions, the B2(Cs) ↔ A1(C2v) optical transition assumes a dipole moment symmetry similar to that of \({B}_{2}({{\rm{C}}}_{{\rm{s}}})\leftrightarrow {B}_{2}^{\prime}({{\rm{C}}}_{{\rm{s}}})\) transition (see Supplementary Note 2). This becomes clear from Fig. 1c where the wavefunctions b2 and a1 are shown for the Cs configuration. In fact, both of them transform as \({A}^{\prime}\) in the Cs configuration, and therefore, are coupled via the in-plane (the stronger component) polarization. As a consequence, the lowest-energy fluorescence is expected to occur as a radiative decay from the A1(C2v) excited state to the B2(Cs) ground state because A1 excited state has lower energy than that of \({B}_{2}^{\prime}\) excited state27. This result constitutes the optical activation of a color center in h-BN by means of a strong electron–phonon coupling. Consequently, it becomes important to study the strain dependence of the ZPL emission for the B2(Cs) ↔ A1(C2v) optical transition. Details on the strain dependence for the higher energy transition is shown in the Supplementary Note 3.

It is intriguing to carry out group theory analysis before starting the numerical ab initio calculations. The detailed general description about the multi-electron configurations and their interaction with strain can be found in the “Methods”, that we apply to the VNNB quantum emitter. The analysis is performed for the C2v symmetry as the excited state transforms within C2v symmetry whereas the Cs ground state follows the same analysis taking into account the fact that Cs is a subgroup of the C2v symmetry. Within C2v symmetry for the axial strain, the ZPL shift upon strain is given by

$$\delta ={\hat{\varepsilon }}^{{A}_{1}}\left[\langle {b}_{2}| {\hat{\Delta }}^{{A}_{1}}| {b}_{2}\rangle -\langle {a}_{1}| {\hat{\Delta }}^{{A}_{1}}| {a}_{1}\rangle \right],$$
(1)

where \({\hat{\varepsilon }}^{{A}_{1}}\) is the strain tensor applying the strain parallel to the C2 symmetry axis (axial strain) and \({\hat{\Delta }}^{{A}_{1}}\) is associated with the energy shifts for the corresponding electronic states (see “Methods”). The Cs configuration can then be described as an out-of-plane distortion due to a built-in strain that acts perpendicular to the basal plane. This mixes a1 and b2 orbitals through B2 component of the strain as explained in “Methods”. Since the energy spacing between the a1 and b2 levels is much larger than the typical deformation values this mixture should not significantly alter the energy shift of δ upon applying basal uniaxial strain. Hence, the energy shift in Eq. (1) is a good approximation. This ZPL energy shift is expected to depend linearly on strain for the B2(Cs) ↔ A1(C2v) optical transition. The magnitude of the strain-ZPL coupling as a function of the orientation of the applied uniaxial strain cannot be determined by means of group theory. We, therefore, apply DFT simulations to quantify the strength of the strain-ZPL coupling for VNNB emission.

We calculate the ZPL energy as the total energy difference between the excited state and the ground state in the global energy minimum of the corresponding electronic configuration (see “Methods”). The strain is modeled by changing the lattice constant of the employed supercell (see Fig. 2a, where the parallel (red) and perpendicular (black) components of the strain are depicted). The corresponding curves for the ZPL shift are plotted in Fig. 2b as a function of the applied strain. The calculated ZPL of VNNB without strain is 1.90 eV for the B2(Cs) ↔ A1(C2v) optical transition, which is very close to the observed photoemission at 1.95 eV of certain SPEs14,17. As expected, the curves are quasi-linear within (−1; +2)% strain region, where we use − and + for compressive and tensile strain, respectively. For both strain directions, we obtain a giant, 12 eV/strain shift in the ZPL energy which results in a huge variation in the emission wavelength as a function of strain. For this quantum emitter, the tensile axial (perpendicular) uniaxial strain causes blueshift (redshift) in the ZPL emission. The variation of the ZPL energy with strain is about 100 nm and we believe our proposed mechanism yields the most plausible explanation for the observed variation of ZPL energies for certain SPEs.

Fig. 2: Zero-phonon-line energies upon strain for VNNB defect.
figure2

a The simplified cartoon of VNNB defect in h-BN. The defect is not planar and exhibits Cs symmetry in the ground state. The black and red arrow denote the directions of uniaxial strain perpendicular () and parallel () to the C2 main axis, respectively. b The ZPL evolution as a function of external strain for a1 to b2 transition. The black and red lines denote the directions of strain perpendicular () and parallel () directions. The energy level at ground state with Cs symmetry evolution as a function of external strain for perpendicular (c) and parallel (d) directions, respectively. The energy level at excited state (a1 to b2) with C2v symmetry evolution as a function of external strain for perpendicular (e) and parallel (f) directions, respectively. The left and right panels show, respectively, the results for the spin-up channel and spin-down channels.

In order to further study these results, we plot the shift of the defect levels upon the applied strain in the corresponding ground and excited states in Fig. 2c–f. Upon applying tensile axial strain, the a1 level in the ground state shifts down whereas the occupied b2 level shifts up in the ground and excited electronic configurations, respectively. This will result in a blueshift in the ZPL. Upon applying tensile perpendicular strain, the a1 level shifts up steeply whereas the occupied b2 level moderately shifts up in the ground and excited electronic configurations, respectively. As a consequence, the two levels approach each other upon applying this strain which results in a redshift in the ZPL energy. This is in agreement with the observations reported in ref. 40 where the ZPL energy shift of the studied emitters exhibits a linear dependence on the applied strain. Furthermore, the three different possible orientations of the defect axis and our results on the blueshift (redshift) of the emission line for armchair (zigzag) strain explains the experimentally observed behavior40.

Role of membrane phonons

We find that the phonon modes with atoms moving out-of-plane, i.e., membrane modes, play a crucial role in the optical activation of the VNNB defect in h-BN (see Supplementary Note 4). The modes are mainly contributed from the substitutional nitrogen atom. These membrane B2 phonons couple the 2A1 excited state and 2B2 ground state where the ground state will be unstable at the C2v symmetry configuration and is distorted to Cs configuration, whereas the excited state remains stable at C2v configuration. This suggests a strong pseudo Jahn–Teller (PJT)47 system which is illustrated in Fig. 3a.

Fig. 3: Electron–phonon coupling of VNNB defect.
figure3

a Pseudo Jahn–Teller (PJT) effect between the excited state (red) and ground state (blue). Dashed line: before electron–phonon coupling; straight line: after electron–phonon coupling. ϵ is the total energy of the system whereas Q is the selected configuration coordinate. b The calculated adiabatic potential energy surface of the 2B2 ground state. The line is a fit of PJT model (see text), where the resultant values are F = 178 meV and \(\hbar\)ω = 23 meV. The standard deviation is <2%.

We depict the adiabatic potential energy surface (APES) of the 2B2 ground state of VNNB in Fig. 3b as obtained by HSE DFT calculations. We note that DFT calculation with a less accurate semilocal functional than HSE (see “Methods”) obtained a similar APES in a previous study46. The Jahn–Teller energy is 95 meV. The solution of this strongly coupled electron–phonon system47 is

$${\epsilon }_{\pm }(Q)=\frac{1}{2}M{\omega }^{2}{Q}^{2}\pm {\left({\Delta }^{2}+{F}^{2}{Q}^{2}\right)}^{\frac{1}{2}},$$
(2)

where Q is the normal coordinate of the effective phonon mode ω with the corresponding mass M, Δ is the energy gap between the ground state and excited state at the high symmetry point (C2v configurations) and F is the strength of electron–phonon coupling. In the dimensionless generalized coordinate system, we obtain F = 178 meV and \(\hbar\)ω = 23 meV. This electron-phonon coupling parameter is about 2.5× larger than that of NV center in diamond48. This indicates a giant electron-phonon interaction for the vacancy type defects in h-BN. We solved the electron-phonon PJT system quantum mechanically and found that the jumping rate between the two minima is 8.4 kHz. This is a relatively slow rate where the optical Rabi-oscillation between the ground and excited states should be more than two orders of magnitude faster (see Supplementary Note 5). This means that the ground state of 2B2 is a static PJT system, and the ground state indeed exhibits low Cs symmetry.

We note here that strong electron-phonon interaction with membrane phonons play an important role in the activation of intersystem crossing process in boron-vacancy optically detected magnetic resonance center49. This type of phonon modes can be found only in 2D solid-state systems. These findings demonstrate that the membrane phonon modes are major actors in the magneto-optical properties of solid-state defect quantum bits and SPEs.

Comparison to known SPEs

Many SPEs were reported in multilayer h-BN structures. The physics of the membrane phonons and their effects on the optical properties for VNNB defect are mainly discussed here based on the results achieved in monolayer h-BN, which directly models the single sheet h-BN flakes and can provide a tentative insight to the top layer of multilayered h-BN structures. We extend our study to the bulk h-BN model (see “Methods”) which corresponds to such VNNB defects that are buried deep in the multilayer h-BN structures. We find that the physics of the membrane phonons is the same: PJT occurs in the presence of van der Waals interaction but it suppresses the Jahn–Teller energy to 50.5 meV. Nevertheless, the resulting electron-phonon coupling remains strong with F = 193 meV (see Supplementary Note 6). The weak interlayer interaction has little effect on the quantum emission of VNNB as the ZPL energies and the ZPL energy shifts upon strain change <0.01 eV compared to the results obtained in the monolayer h-BN.

Most of the SPEs in h-BN were first observed at room temperature10,14 which emit in the visible region with various wavelength. Based on the ZPL emission region and the contribution of the phonon sideband (PSB) to the total emission, the visible SPEs were categorized into two groups according to early measurements14: Group-1 with ZPL energies at 1.8–2.2 eV and with significant PSB contribution; Group-2 with ZPL energies at 1.4–1.8 eV with small PSB contribution. Group-1 emitters often showed an asymmetry in the ZPL lineshape at room temperature that was attributed to electron-phonon effects10. Recently, low-temperature observation challenged this idea where they could decompose the PL spectra into two emitter components that could naturally explain the asymmetry of the spectrum at elevated temperatures where the spectra of the two emitters cannot be resolved50. They found that the two emitters have ZPL energies with about 15 nm apart but very similar PSBs and also with a large variation of the ZPL wavelength of these pairs between 600 nm (2.06 eV) and 720 nm (1.72 eV) where the variation was tentatively attributed to strain50. Based on the calculated ZPL energy and strong electron-phonon coupling of the VNNB defect, the VNNB definitely belongs to the Group-1 emitters and the calculated strain dependence of the calculated ZPL energies can cover the range of the observed ZPL energies by assuming about ±2% strain in the h-BN lattice. Parallel to our study, nanobeam electron diffraction has been applied to correlate the emitter optical emission with the emitter’s local in-plane strain and found that about ±2% local strain can appear in h-BN flakes39. They have found SPEs at 630 nm and 705 nm ZPL energies that are most likely connected by strain39. The VNNB defect can produce this giant shift of ZPL wavelength upon about 1.5% strain which is not far from the experimental observations claiming about 1%39. We note that unambiguous identification of the SPEs require further work from experiments and theory. Nevertheless, our study shows by means of accurate DFT calculations that the optical response of VNNB defect to strain is indeed very sensitive. Further theoretical studies might reveal other defects in h-BN with similar properties.

Implications towards quantum information processing applications

The strong electron–phonon interaction implies that the optical properties of vacancy type quantum emitters in h-BN can vary significantly with strain. Indeed, our DFT simulations show a giant shift in the ZPL emission at strain values that can appear in h-BN. Compared to the SPEs in three-dimensional (3D) materials such as diamond and SiC, the SPEs in 2D h-BN are not buried by high refractive index medium which makes collection efficiency of the emitted light much higher than that for 3D materials. Integration of h-BN based SPEs with nanophotonic devices offers a promising path to engineer quantum gates and circuitry which are key building blocks of quantum information processing and our work provides crucial implications in this field. First, we report the activation of the forbidden optical transition due to the strong electron-phonon coupling. The electron-phonon coupling reduces the symmetry of the ground state of the defect and it is a static PJT system. Second, we find the giant shift of the ZPL spectrum with external strain and propose a particular explanation for the phenomena. Until now, the spectral shift on SPEs in h-BN observed in our work is the largest known so far. Third, our result provides an analysis method for similar quantum emitters with varying ZPL energies and emission intensities in h-BN. Strain might be a reason for the spectral broadening and can influence the optical contrast and quantum efficiency. This might be harnessed to use this quantum emitter for realizing stress detector at the nanoscale as well as nanomechanical devices for quantum technologies.

In this paper, we performed a thorough group theory analysis and DFT calculations on the effect of strain on nitrogen antisite-vacancy color center in h-BN. We find a very strong electron-phonon interaction that can activate photoluminescence, and is responsible for the giant ZPL shift upon the applied strain. The behavior of the strain-induced energy shift is correlated with the experimental observations further revealing their microscopic nature.

Methods

Group theory analysis on strain

In this section, we provide a formulation that describes the effect of a local strain of the point defects. The derived Hamiltonian is general and is then applied to the VNNB quantum emitter in h-BN.

When considering point defects as candidates of the color centers in h-BN, the local strain manifests itself in modifying the atomic distances, which in turn, leads to the modification of the molecular orbitals (MOs) around the defect. Any change in the properties of MOs result-in a redistribution of the energy states in the band structure of the host solid. Therefore, the strain directly couples to the electronic degrees-of-freedom of the color center. The coupling strength is called deformation potential.

We derive the Hamiltonian of the defect under local strain in the following way: We start by assuming that the color center is composed of Ne valance electrons that are mostly isolated from the rest of lattice and gather around Nn nuclei forming up the defect. Note that this is a fairly good assumption by looking at the MOs drawn by DFT belonging to the defect states27. The attractive Coulomb energy imposed from nuclei on the electrons is then given by

$${V}_{{N}_{{\rm{e}}}}=\mathop{\sum }\limits_{j = 1}^{{N}_{{\rm{e}}}}\mathop{\sum }\limits_{k = 1}^{{N}_{{\rm{n}}}}{V}_{jk}({{\bf{x}}}_{k},{{\bf{x}}}_{j})\approx -\mathop {\sum}\limits_{j}\mathop {\sum}\limits_{k}\frac{{Z}_{k}^{\prime}{{\rm{e}}}^{2}}{{R}_{jk}},$$
(3)

where Rjk = xkxj with xk position of the nuclei, while xj denotes the location of the jth electron. Here, \(Z^{\prime}_k\) is the effective atomic number (screened nuclear charge) of the ion. The local strain displaces ions involved in the point defect x → x + δx and thus their Coulomb interaction. In the first order of accuracy we get \(V(| {\bf{x}}-{\bf{x}}| )\,\longrightarrow \,V(| {\bf{x}}+\delta {\bf{x}}-{\bf{x}}| )\approx V(R)+{\left[{\boldsymbol{\nabla }}V(R)\right]}_{0}\cdot \delta {\bf{x}}\), where δx is the infinitesimal displacement of the nuclei imposed by the local strain. The value of displacement is obtained by \(\delta {\bf{x}}={\bf{X}}\cdot \hat{\varepsilon }\) with the strain tensor \(\hat{\varepsilon }\). The electron–strain interaction Hamiltonian is then summed over all such first-order variational terms

$${H}_{{\rm{str}}}=\mathop {\sum}\limits_{j,k}{\left[{\boldsymbol{\nabla }}V({R}_{jk})\right]}_{0}\cdot \hat{\varepsilon }\cdot {{\bf{X}}}_{k}\equiv -\mathop {\sum}\limits_{j}\mathop {\sum}\limits_{\alpha }{\hat{\Delta }}_{j}^{\alpha }{\hat{\varepsilon }}^{\alpha },$$
(4)

where we have introduced \({\hat{\Delta }}_{j}={{\bf{x}}}_{j}{{\mathbf{\Xi }}}_{j}\), a dyadic whose components have different group symmetries denoted by α, while \({{\mathbf{\Xi }}}_{j}={\sum }_{k}{\left[\frac{1}{{R}_{jk}}\frac{\partial {V}_{jk}}{\partial {R}_{jk}}\right]}_{0}{{\bf{x}}}_{k}\) is the deformation potential. Here, we have assumed that the radial component of the gradient is the dominant one and neglected an irrelevant constant term. The former is a valid assumption as the total Coulomb attraction of the ions is more or less a central force51,52,53.

The electronic configuration of this defect are given in Table 1. Given the orbital symmetries of these states and the following table of symmetry for strain components the group theory can predict that the only non-zero irreducible representations of the \(\hat{\Delta }\) when sandwiched between two single-electron orbitals are:

Table 1 The configuration, symmetry, and spin multiplicity of the ground and two lowest excited states of neutral VNNB.

The effect of strain on multi-electron states is a non-equal shift in their energy levels imposed by the axial components of the strain as well as inducing an interaction between the states via the axial and non-axial strain components. The amount of shift only depends on the electronic states and its relations for the ground and excited states are δ0, δ1, and δ2, respectively. The strain prompted inter-state interactions are much smaller than the energy difference between the levels, hence one neglects them in an adiabatic manner. The explicit form of the energy shifts are

$$\begin{array}{rcl}{\delta }_{0}&=&{\hat{\varepsilon }}^{A^{\prime} }\left[2\langle {a}_{1}| {\hat{\Delta }}^{A^{\prime} }| {a}_{1}\rangle +\langle {b}_{2}| {\hat{\Delta }}^{A^{\prime} }| {b}_{2}\rangle \right],\\ {\delta }_{1}&=&{\hat{\varepsilon }}^{{A}_{1}}\left[2\langle {b}_{2}| {\hat{\Delta }}^{{A}_{1}}| {b}_{2}\rangle +\langle {a}_{1}| {\hat{\Delta }}^{{A}_{1}}| {a}_{1}\rangle \right],\\ {\delta }_{2}&=&{\hat{\varepsilon }}^{A^{\prime} }\left[2\langle {a}_{1}| {\hat{\Delta }}^{A^{\prime} }| {a}_{1}\rangle +\langle b^{\prime} | {\hat{\Delta }}^{A^{\prime} }| b^{\prime} \rangle \right].\end{array}$$

In the main text we have adopted the approximation that \({\hat{\varepsilon }}^{A^{\prime} }\langle \cdot | {\hat{\Delta }}^{A^{\prime} }| \cdot \rangle \approx {\hat{\varepsilon }}^{{A}_{1}}\langle \cdot | {\hat{\Delta }}^{{A}_{1}}| \cdot \rangle\) which is reasonable owing to the fact that Cs is a subgroup of C2v and that the molecular orbitals retain their form.

Details on DFT calculations

The calculations are performed based on the DFT implemented in Vienna ab initio simulation package (VASP)54,55. Projector augmented wave (PAW) is used to separate the valence electrons from the core part. The energy cutoff for the expansion of the plane-wave basis set was set to 450 eV which is enough to provide an accurate result. The screened hybrid density functional of Heyd, Scuseria, and Ernzerhof (HSE)56 is used to calculate to band gap and defect levels. Within this approach, the short-range exchange potential is described by mixing with part of nonlocal Hartree–Fock exchange and this also provides reasonable geometry optimization of the dynamic Jahn–Teller system. The HSE hybrid functional with mixing parameter of 0.32 closely reproduces the experimental band gap at 5.9 eV. To apply the strain along the parallel and perpendicular directions to the C2 axis, a \(9\times 5\sqrt{3}\) supercell is constructed through changing the basis to the orthorhombic structure. The perfect supercell contains 160 atoms which is sufficient to avoid the defect-defect interaction, and the single Γ-point scheme is converged for the k-point sampling for the Brillouin zone. The coordinates of atoms are allowed to relax until the force is <0.01 eV/Å. The excited state was calculated within ΔSCF method57 that we previously applied to point defects in h-BN too27. For the bulk simulation, a periodic model containing two layers are used, where one perfect layer is placed above the defective layer. The optimized interlayer distance is 3.37 Å with DFT-D3 method of Grimme58.

Data availability

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

Code availability

The codes that were used in this study are available upon request to the corresponding author.

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Acknowledgements

A.G. acknowledges the Hungarian NKFIH grant No. KKP129866 of the National Excellence Program of Quantum-coherent materials project, the National Quantum Technology Program (Grant No. 2017-1.2.1-NKP-2017-00001), and the EU H2020 Quantum Technology Flagship project ASTERIQS (Grant No. 820394). M.B.P. acknowledges support from the ERC Synergy grant BioQ (Grant No. 319130), the EU H2020 Quantum Technology Flagship project ASTERIQS (Grant No. 820394), the EU H2020 Project Hyperdiamond (Grant No. 667192), a DFG Reinhart Koselleck project and the BMBF via NanoSpin and DiaPol. M.A. acknowledges support by INSF (Grant No. 98005028).

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S.L. carried out the DFT calculations under the supervision of J.P.C., A.H., and A.G. M.A. developed the group theory analysis with M.B.P. P.U. and G.T. developed and applied the electron–phonon coupling theory on the defect under the supervision of A.G. All authors contributed to the discussion and writing the manuscript. A.G. conceived and led the entire scientific project.

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Correspondence to Adam Gali.

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Li, S., Chou, JP., Hu, A. et al. Giant shift upon strain on the fluorescence spectrum of VNNB color centers in h-BN. npj Quantum Inf 6, 85 (2020). https://doi.org/10.1038/s41534-020-00312-y

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