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

Department of Mechanical Engineering, City University of Hong Kong, Hong Kong SAR, China Institute of Theoretical Physics and IQST, Albert-Einstein-Allee 11, Ulm University, 89069 Ulm, Germany Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest, Hungary Eötvös Science University, Pázmány Péter Sétány 1/A, H-1117 Budapest, Hungary Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary


Introduction
Quantum emissions from two-dimensional (2D) materials has recently received considerable and rapidly rising interest of researchers in both condensed matter and quantum optics 1-3 as these systems provide a potential basis for emerging technologies such as quantum nanophotonics 4-6 , quantum sensing [7][8][9] , and quantum information processing 10,11 . 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 materials 12 . 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 substrates 13 . The experiments for exploring the nature of the emission of these color centers started with their first observations 10, 14-25 and recent theoretical works have provided evidence that the SPEs are indeed color centers, i.e. local point defects [26][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 them 31,32 motivates the investigation on the effect of strain on these emitters. Furthermore, the quantum emitters have shown magnetic properties in some experiments [33][34][35][36] , while in other experiments non-magnetic behaviour was found 21 .
The most commonly observed quantum emitters exhibit emission in the visible, where competing theoretical models 27,37 exists in the literature. It has been suggested recently, based on the similarities of the optical lifetimes of the observed quantum emitters 38 , 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 region 39 . Yet another experiment also found a relatively small shift upon applied stress on a given emitter 40 .
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 understand-ing 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 coupling 11,41,42 . 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 the optical emission and strain. So far the electron-strain coupling in h-BN emitters has been studied with limited accuracy 40 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, V N N B , which was first suggested as a candidate of the observed single-photon emissions 10 and then thoroughly studied as one of the feasible quantum emitter in the visible with S = 1/2 spin state 27 . The optical properties of this defect have been examined from various theoretical methods and point of view 27,[43][44][45] . The defect has a C 2v symmetry before relaxation of the atoms in monolayer h-BN, and introduces three levels in the energy band gap that are labeled as a 1 , b 2 and b ′ 2 owing to their irreducible symmetry representation. These energy levels are occupied by three electrons resulting a 2 B 2 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 b 2 may be promoted to b ′ 2 or in the spin minority channel the electron from a 1 may be promoted to b 2 . It is found that the a 1 ↔ b 2 has lower energy than b 2 ↔ b ′ 2 27, 43, 44 . However, for the defect with the C 2v symmetry the a 1 ↔ b 2 optical transition has a very small optical transition in-plane dipole moment 44 . Indeed, we also find this behavior in our own DFT calculation (see Supplementary Figure S1). We indeed notice that b 2 and b ′ 2 states have wavefunctions that extend out-of-plane [see Fig. 1(b)], and therefore, can couple to phonon modes that drive the atoms outof-plane, the membrane modes. There is an unpaired electron in both B 2 ground state and B ′ 2 excited state placed on the b 2 and b ′ 2 orbital, respectively, that induce an out-of-plane geometry distortion. These phonon modes strongly couple the 2 B 2 ground state and the 2 A 1 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 C s [see Fig. 1(a)]. This geometry is about 100 meV lower in energy than the C 2v configuration which reveals the strong coupling of the defect electrons to the membrane mode phonons. This result basically agrees with previous DFT calculations 46 . In C s symmetry, all the one-electron defect levels belong to a ′ irreducible representation. Despite the vibronic mixing, the correspondence to the high symmetry orbitals can be observed in Fig. 1(b) and (c). When the hole is left at a 1 orbital in the A 1 excited state, then the coupling to the membrane phonons is negligible and the defects C 2v symmetry is retrieved. For the sake of simplicity, here we use the C 2v symmetry labels for both configurations. The transformation of the point symmetry of the defect is discussed in Supplementary Note 1.
Owing to the rearrangement of the ions, the B 2 (C s ) ↔ A 1 (C 2v ) optical transition assumes a dipole moment symmetry similar to that of  components out-of-plane, it can be seen that in C s symmetry the a 1 state also gains an out-of-plane component.
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 V N N B quantum emitter.
The analysis is performed for the C 2v symmetry as the excited state transforms within C 2v symmetry whereas the C s ground state follows the same analysis taking into account the fact that C s is a subgroup of the C 2v symmetry. Within C 2v symmetry for the axial strain, the ZPL shift upon strain is given by whereε A 1 is the strain tensor applying the strain parallel to the C 2 symmetry axis (axial strain) and ∆ A 1 is associated with the energy shifts for the corresponding electronic states (see Methods). The 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. 2(c)-(f). Upon applying tensile axial strain, the a 1 level in the ground state shifts down whereas the occupied b 2 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 a 1 level shifts up steeply whereas the occupied b 2 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 exhibit a linear dependence on the applied strain. whereas the excited state remains stable at C 2v configuration. This suggest a strong pseudo Jahn-Teller (PJT) 47 system which is illustrated in Fig. 3(a). We depict the adiabatic potential energy surface (APES) of the 2 B 2 ground state of V N N B in Fig. 3

(b) as obtained by HSE DFT calculations.
We note that DFT calculation with a less accurate semilocal functional than HSE (see Method) obtained a similar APES in a previous study 46 . The Jahn-Teller energy is 95 meV. The solution of this strongly coupled electron-phonon system 47 is where Q is the normal coordinate of the effective phonon mode ω with the corresponding mass M, ∆ is the energy gap between the ground state and excited state at the high symmetry point (C 2v configurations) and F is the strength of electron-phonon coupling. In the dimensionless generalized coordinate system we obtain F =178 meV and ω=23 meV. This electron-phonon coupling parameter is about 2.5× larger than that of NV center in diamond 48   and ω=23 meV. The standard deviation is less than 2%.
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 2 B 2 is a static PJT system, and the ground state indeed exhibits low C s 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 center 49 . 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 single photon emitters. 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 N e valance electrons that are mostly isolated from the rest of lattice and gather around N n 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 states 27 . The attractive Coulomb energy imposed from nuclei on the electrons is then given by where R jk = |x k − x j | with x k position of the nuclei, while x j denotes location of the jth electron.
Here, Z ′ 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 where δx is the infinitesimal displacement of the nuclei imposed by the local strain. The value of displacement is obtained by δx = X ·ε with the strain tensorε. The electron-strain interaction Hamiltonian is then sum over all such first order variational terms where we have introduced∆ j = x j Ξ j , a dyadic whose components have different group sym- ∂R jk ] 0 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 force [51][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 configuration label symmetry predict that the only non-zero irreducible representations of the∆ when sandwiched between two single-electron orbitals are: 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 δ 0 , δ 1 , and δ 2 , 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 In the main text we have adopted the approximation thatε A ′ ·|∆ A ′ |· ≈ε A 1 ·|∆ A 1 |· which is reasonable owing to the fact that C s is a subgroup of C 2v and that the molecular orbitals retain their form.

Details on DFT calculations
The calculations are performed based on the density functional theory (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 plane-wave basis set was set to 450 eV which is enough to provide 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 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 C 2 axis, a 9 × 5 √ 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 less than 0.01 eV/Å. The excited state was calculated within ∆SCF method 57 that we previously applied to point defects in h-BN too 27 . 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 3.37Å with DFT-D3 method of Grimme 58 .
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.
Acknowledgements Author contribution SL carried out the DFT calculations under the supervision of JPC, AH, and AG.
MA developed the group theory analysis with MBP. PU and GT developed and applied the electron-phonon coupling theory on the defect under the supervision of AG. All authors contributed to the discussion and writing the manuscript. AG conceived and led the entire scientific project.
Competing Interests The authors declare that there are no competing interests..