Singlet levels of the NV$^{-}$ centre in diamond

The characteristic transition of the NV- centre at 637 nm is between ${}^3\mathrm{A}_2$ and ${}^3\mathrm{E}$ triplet states. There are also intermediate ${}^1\mathrm{A}_1$ and ${}^1\mathrm{E}$ singlet states, and the infrared transition at 1042 nm between these singlets is studied here using uniaxial stress. The stress shift and splitting parameters are determined, and the physical interaction giving rise to the parameters is considered within the accepted electronic model of the centre. It is established that this interaction for the infrared transition is due to a modification of electron-electron Coulomb repulsion interaction. This is in contrast to the visible 637 nm transition where shifts and splittings arise from modification to the one-electron Coulomb interaction. It is also established that a dynamic Jahn-Teller interaction is associated with the singlet ${}^1\mathrm{E}$ state, which gives rise to a vibronic level 115 $\mathrm{cm}^{-1}$ above the ${}^1\mathrm{E}$ electronic state. Arguments associated with this level are used to provide experimental confirmation that the ${}^1\mathrm{A}_1$ is the upper singlet level and ${}^1\mathrm{E}$ is the lower singlet level.


Introduction
The negatively charged nitrogen vacancy centre in diamond ( − NV ) [1] exhibits optically induced spin polarization. This property underpins many exciting applications of the − NV centre in fields such as magnetic sensing [2][3][4][5][6][7][8][9], biological imaging [10][11][12], and quantum information processing [13][14][15][16][17]. The principle zerophonon line (ZPL) associated with the centre is at 637 nm (1.945 eV, 15687 cm −1 ) and is found by uniaxial stress to involve a transition between a ground state of A symmetry and an excited state of E symmetry at a trigonal site [18]. Here we label this transition − NV vis since it is in the visible spectrum, and its fluorescence band is shown in figure 1. The ground and excited states are spin triplets [19][20][21][22][23] and optical excitation of this transition results in the spin being polarized into = m 0 s , although this does not arise from direct optical cycling as the optical transitions are spin-conserving [24]. When the triplet system is excited there is also relaxation via intermediate singlets and this decay causes the spin polarization. A weak emission band in the infrared (figure 1) with a ZPL at 1042 nm (1.19 eV, 9597 cm −1 ) is associated with decay between these two singlet levels [25].
A study of this emission (which we call − NV IR ) provides an opportunity to better understand the electronic levels in this important decay channel. Uniaxial stress is the experimental technique of choice [27][28][29]. In these experiments external stress close to the limits of fraction is applied to the diamond along well-defined crystal axes while the displacement and splitting of the ZPL are measured. The precise effects depend on the orientation of the impurity centre relative to the stress axis, and provide information about the orbital symmetry of the states involved with optical transitions. A previous uniaxial stress study has shown that the − NV IR transition is between levels of A and E symmetry [25], and this symmetry assignment is not in question. However, in that study the specific transitions were not correctly identified and this led to an inaccuracy of the stated stress parameters. Here the transitions are unambiguously identified and correct stress parameters are determined. In addition we experimentally resolve the long-standing contention regarding the order of the singlets [25,[30][31][32][33] and establish the E 1 to be the lower singlet. The magnitudes of the stress parameters are considered within the current electronic model of the centre. It is concluded that the interaction giving rise to the shift and splitting of the infrared ZPL is different from that giving rise to the shifts and splittings of the − NV vis ZPL and NV 0 ZPL.

Uniaxial stress theory
The theory for uniaxial stress applied to an ↔ A E transition at a site of trigonal symmetry in a cubic crystal has been given on several occasions [27][28][29]34] and has been developed by Davies and Hamer [18,35] for the case of the NV centre. In general, stress along the symmetry axis results in shifts of the energy levels while transverse stress lifts the degeneracy of the E state as illustrated in figure 2. The elements of the stress tensor s ij as applied to the cubic crystal can be expressed in terms of the irreducible representations appropriate for the trigonal site symmetry, and the stress perturbation at the NV site is given by  [18,29]. The stress s ij is given in terms of the lattice co-ordinates. The effects of this interaction on an ↔ A E transition have been described by Davies and Hamer [18] in terms of the following reduced matrix elements  NV IR infrared band lies on the tail of the visible emission and has an integrated area of 1 ± 0.2 × 10 −3 compared to that of the visible band. It is understood that the weakness of this fluorescence band is due to strongly competing non-radiative decay between the singlets illustrated by the wavy arrow in (a) [26].  For stress along 〈 〉 001 the splittings are the same for all orientations of the − NV centre. The ZPL is split into two components with one component σ polarized (electric field vector perpendicular to stress) and the other predominantly π polarized (parallel to stress). The splittings are determined by the value of the B parameter (see table 1) and the average shift is given by A1. It can be seen from comparing figures 3(a) and (b) that B IR is marginally larger than B vis whereas A1 IR is only about one third of A1 vis .
For 〈 〉 111 stress there are two subsets of centres (table 1). One subset contains the centres oriented along the stress direction, for which there is no change of symmetry. This means there is no splitting, but the transition is shifted by (table 1). NV − centres in this orientation are not excited when the electric field vector of the excitation is parallel to their axis, since the ↔ A E transitions do not involve a z dipole moment. Consequently this orientation does not contribute to the dashed traces of figures 3 (c) and (d) where this excitation polarization is adopted. The NV − centres in this orientation do give a line when transverse excitation is used. This 'extra' line is barely discernible in the case of the infrared spectrum as it overlaps the other features indicating a very small shift (A1 IR + A2 IR ). In contrast, there is a large shift of this line for the visible transition. Since A1 vis and A1 IR are known from the above 〈 〉 001 stress measurements, it can be readily deduced that A2 vis for the visible is large and negative whereas A2 IR for the infrared is small. This information is consistent with average shifts for the centres oriented at 70°to the 〈 〉 111 stress given by (table 2). The ZPL splitting for these centres depends on the C parameter, and it is apparent that C IR is about one third of C vis . 111 stress. Excitation was from 100 mW laser at 532 nm, and emission was observed at right angles and recorded separately in π (green) and σ (magenta) polarizations. Excitation polarization was perpendicular to the stress direction (σ ) with the exception of the dashed traces in (c) and (d) where the laser polarization was parallel to stress (π ) and so the axial centres were not excited. The sample temperature was ∼150 K. Variation of stress across the sample prevented the lines from being well resolved (and breakage prevented improvement of the data). However, identical stress settings allow the relative size of the shifts and splittings to be compared between − NV IR and − NV vis . The straight lines and annotations indicate the stress parameters calculated later (not direct fits). The − NV vis spectra are consistent with [18] and the stress parameters are those from [18].
The conclusion that A1 IR and C IR are a factor of three smaller than their − NV vis counterparts is consistent with the − NV IR strain parameters reported previously [25]. However, there is no consistency with the A2 IR and B IR parameters. Here we have established that A2 IR is an order of magnitude smaller than A2 vis (instead of the factor of 2.7 given previously), and that| | > | | B B IR vis (instead of the reverse). The previous values relied on the interpretation of spectra for stress along the 〈 〉 110 direction and, therefore, the spectra for this stress direction are re-investigated in the next section.

Uniaxial stress along 〈 〉
110 stress using selective excitation Stress along 〈 〉 110 causes the NV − centres to form two distinct sets of orientations, both of which have some component of transverse strain and therefore exhibit splitting (table 1). This produces a four-line structure in the spectrum, and the determination of strain parameters depends heavily on a correct assignment of each line to a transition in a given NV orientation. Here we use selective excitation techniques to provide reliable assignments.
A 200 mW tunable dye laser was swept through the − NV vis ZPL and the emission was detected in the vibronic sideband between 650 nm and 750 nm. Weak (1 mW) 532 nm laser light was applied simultaneously to inhibit loss of signal through hole burning. The polarized − NV vis excitation spectra for a 〈 〉 110 stress of1.4 GPa obtained in this way is shown in figure 4(a). This − NV vis excitation spectrum is consistent with the measurements of Davies and Hamer obtained in absorption [18]. The two higher energy lines in excitation (at 632.4 nm and 634.8 nm) are associated with centres at right angles to the stress (orientations ③ and ④, table 1), and the lower energy − NV vis lines are associated with orientations ① and ② which are at 36°to the stress [18].
The detection filter was changed to measure emission in the − NV IR band. Resonantly exciting the two higher energy − NV vis transitions gave the polarized − NV IR spectra shown in figure 4(b). Since these laser frequencies only excite the orientations ③ and ④ which are orthogonal to the stress, the − NV IR spectrum shows only two lines. These lines are clearly either predominantly π or σ polarized, enabling them to be assigned to the Γ 1 and Γ 2 components according to table 1. Tuning the laser to the lower energy − NV vis transitions caused only orientations ① and ② to be excited, producing the − NV IR spectra shown in figure 4(c). Again the lines are strongly polarized and readily assigned using table 1. There is always the equivalence between the visible and infrared spectra described in previous section but it is noted that the order of the π and σ lines for the 36°case are reversed between the visible and infrared spectra. This results from a reversal of the relative strengths of the B and C stress parameters between the visible and infrared cases with vis vis in the other. These selective excitation measurements provide the first unambiguous assignments of the infrared spectral features for 〈 〉 110 stress. It is now clear that the significantly different A2 IR and B IR spitting parameters given previously [25] resulted from an incorrect assignment of the − NV IR lines for 〈 〉 110 stress. In that work it was assumed that the four peaks were in the same order as for − NV , vis which does not turn out to be the case. Having established the identity of each line in the spectrum, more conventional photoluminescence measurements were made using the 532 nm non-resonant excitation. In this way the position of the four lines in the stress spectra were followed for stress values in the range 0-3 GPa, and the shifts and splittings are shown in figure 5. This figure also includes the results for stress along 〈 〉 001 and 〈 〉 111 , where there is less ambiguity in the assignments of the lines and therefore no advantage to adopting selective excitation techniques. It can be seen from the figures that the displacements with stress are not always linear and this requires consideration before the values of the stress parameters can be deduced.

4.3.
High stress and extra feature at 115 cm −1 At higher stress (>1 GPa) an extra feature was found to be induced 115 ± − 5 cm 1 to the low energy side of the ZPL. Its position is indicated in figures 5 and 6 shows spectra exhibiting this peak for 〈 〉 111 , 〈 〉 110 , and 〈 〉 001 transition. Uncertainty estimates for the ratios were calculated assuming a 7% uncertainty in the − NV vis parameters from [18]. The values for NV 0 are from [35] and N3 from [34].
1042.6 nm 637 nm ratio 575 nm 415 nm   stress. The feature appears with varying intensities but increases in strength with stress at the expense of a component of the ZPL with the same polarization. The measurements were made at higher resolution for the case of 〈 〉 110 uniaxial stress, and it is apparent that the extra feature gains at the expense of the line displaced nonlinearly. It also shifts slightly in the reverse direction, as shown in figure 5(c). This is typical for a situation where there are two interacting levels which have the same symmetry. From the analysis of the ZPL it has been established that the line shifting nonlinearly has Γ 1 in C s symmetry. The extra feature will, therefore, also have Γ 1 in C s and since it is not split it must have A 1 symmetry in C 3v .
This symmetry assignment is consistent with its occurrence for other stress directions. For 〈 〉 001 stress the line displaced to higher energy has π polarization and is assigned to a Γ 1 state, and this line mixes with the extra feature ( figure 6(a)). Even though the interacting ZPL component is shifting away from the 115 cm −1 feature, the displacement of this ZPL line becomes nonlinear as a result of the interaction, as shown in figure 5(a). Here it might be expected that the 115 cm −1 feature shifts in the reverse direction, but the effect is reduced owing to the proximity of the 320 cm −1 vibrational level. Indeed at the highest stress it is observed that there is a slight shift of the extra feature to shorter wavelength (higher energy) owing to the latter interaction. In the third case of 〈 〉 111 the effects are small but the feature again has the same polarization as that for the Γ 1 component of the split ZPL ( figure 6(b)).
Since the 115 cm −1 feature interacts with one component of a line that splits with stress, it must be associated with the E 1 electronic state. It occurs on the low energy side of the ZPL in the emission spectrum. Should the E 1 be the upper singlet level there will be relaxation to this level 115 cm −1 below the E 1 state and at cryogenic temperatures (<30 K) all the emission would be from this level. This is not the case and it is concluded that the E 1 is not the upper singlet level. The alternative is that the E 1 is the lower singlet level and the extra level lies In these cases the sample temperature was 150 K. (c) For 〈 〉 110 stress the sample temperature was 10 K and the higher resolution was obtained by detecting emission from a small volume using masking. In all cases the feature at 115 cm −1 is induced by the stress. It has the same polarization as the Γ 1 component of the ZPL (π in the upper two traces and σ in lowest trace-see table 1). The first vibrational sideband at 320 cm −1 can be seen to have the same polarization as the ZPL indicating the vibration has A 1 symmetry. 115 cm −1 above it. This confirms our previous report [36] and is consistent with the now generally accepted theoretical model [37,38].
The occurrence of low-energy vibronic levels in diamond is a fairly common observation and has been observed in previous uniaxial stress studies of diamond [35]. They are associated with a dynamic Jahn-Teller effect associated with an E state. Davies [35] has established five other cases of trigonal centres in diamond exhibiting this effect. The first vibrational state associated with a degenerate E vibration will involve the electronic and the vibration states, resulting in four vibronic states with symmetries × = + + E E A A E 1 2 . The E vibronic level is displaced up in energy and the + A A 1 2 down. Quadratic electron-vibration interaction will lift the A 1 and A 2 degeneracy and result in the low lying A 1 state at 115 cm −1 as observed here. A similar situation arises in the case of the E 2 ground state of NV 0 [35]. In this case the level occurs at 110 cm −1 and from the similarity in the situation it can be expected that the strength of the Jahn-Teller interaction is similar: This vibronic level has significant implications. It has been used above to establish the order of the singlet levels, resolving long-standing contention about this detail of the NV electronic structure [25,[30][31][32][33]. It should also be noted that one consequence of the dynamic Jahn-Teller interaction is a reduction of the effect of perturbations [39]. Thus the experimental measurements of the stress splittings will be slightly smaller than obtained from calculation unless such effects are included.

Stress parameters
The nonlinear shift of some spectral features with stress is, therefore, due to interaction with vibronic levels. Modelling these interactions is not straightforward as they can involve a distribution of vibrations and the distribution need not be simple. Consequently we have determined the stress parameters using shifts and splitting at levels of stress where the strength of this latter interaction is negligible, essentially using the asymptotic slopes at zero stress. The value of the parameters are given in table 2.

Discussion of the molecular model
The electronic model of the − NV centre has its foundations in the defect-molecule approach of Coulson and Kearsley [40] and has been given in detail by many authors [1,32,37,38,41]. The centreʼs electronic states are written in terms of symmetry-adapted molecular orbitals. There are four unbound sp 3 atomic orbitals adjacent to the vacancy and in C 3v symmetry these can be linearly combined to give two degenerate orbitals that transform as the E irreducible representation (denoted as e-orbitals) and two separate orbitals of A 1 symmetry (denoted as a 1 -orbitals). These are occupied by six electrons: one from each of the adjacent carbon atoms, two from the nitrogen, and one acquired from the lattice. The lower a 1 orbital is always occupied and need not be included in a description of the states. The occupancy of the other four electrons describe the multi-electron states.
The non-relativistic electronic Hamiltonian of the − NV centre may be defined as [37] where T e is the electronic kinetic energy, V Ne is the one-electron Coulomb interaction between the NV electrons and the lattice nuclei and electrons, V ee is the electron-electron Coulomb repulsion interaction of the NV electrons, ⃗ r are the collective coordinates of the NV electrons and ⃗ R are the collective coordinates of the lattice. Both T e and V Ne can be written as sums of one-electron operators, whereas V ee can be written as a sum of twoelectron operators. The molecular orbitals are defined as solutions of the one-electron terms + T V e N e . The a 1 and e molecular orbitals have energies that lie within the diamond band gap and are separated by ∼ h 2 eV ( figure 7(a)). The energies of the a 1 and e molecular orbitals define the energies of the electronic configurations. The four electrons occupying these molecular orbitals lead to three configurations a e 1 2 2 , a e 1 3 and e 4 , which are each separated by h ( figure 7(a)). The introduction of the electron-electron Coulomb interaction V ee separates the multi-electron states within a configuration into triplet and singlet levels. The separation can be of the order of eV and, hence, comparable in magnitude to that of the one-electron terms. For example, the lowest energy configuration a e 1 2 2 is split into equally separated states A ,  7(a)).The electron-electron Coulomb repulsion interaction can also give interaction between configurations and mix the singlet levels of the same orbital symmetry, thereby modifying the simple expressions for ϵ and ϵ′.
The above one-and two-electron Coulomb interactions give the dominant terms in determining the effects of stress, which are observed to be several meV. Other electronic interactions such a spin-orbit and spin-spin are less than meV and their effects are negligible compared to stress. Whilst electron-vibration interaction can be of the order of meV, it can not give stress splitting by itself. Although, as mentioned earlier, it can modify the magnitude of stress splittings in the case of dynamic Jahn-Teller interaction [39]. Hence, the analysis of the stress parameters can be largely restricted to consideration of the Coulomb interactions.
When stress is applied, the lattice coordinates ⃗ R change, which results in a change δV s [ ] Ne of the one-electron Coulomb interaction that in turn modifies the molecular orbitals and their energies. Furthermore, the modification of the molecular orbitals occupied by the electrons leads to a change in the electron-electron Coulomb repulsion interaction, which can be represented by the effective operator δV can be expanded in symmetry adapted form with terms that are in one to one correlation with those in equation (1). If the symmetry is not changed by the applied stress, δV s [ ] Ne will only alter the a 1 -e energy separation and this is denoted by Δh ( figure 7(b)). This will result in a change of the energy separation between configurations but cause no change within each configuration ( figure 7(b)). Where the applied stress lowers the symmetry of the centre, δV s [ ] Ne will result in a splitting of the e molecular orbitals by δh. The consequence is that the multi-electron E states with an odd number of electrons occupying the e molecular orbitals will be split by δh ( figure 7(c)). Significantly, no splitting occurs when an even number of electrons occupy the e molecular orbitals. For each pair of electrons, one of the e electrons is moved up in energy and the other down, such that there is no overall splitting.
It   . The first order changes are taken to be Δϵ and δϵ for axial and transverse stress, respectively (figures 7(b) and (c)). Such perturbations can account for the change of the A (a e ) 1 1 1 2 2 to E (a e ) 1  interaction. The splitting is that of a single e-electron and is expected to be of similar magnitude (but opposite sign) to that of the single e-hole in the case of E (a e ) 3  is minor. The − NV centre is the first colour centre in diamond where the stress parameters are known for two separate transitions, and this provides an ideal situation for testing theoretical calculations. For example it may help determine whether the separate contributions from one-electron Coulomb interaction and electron-electron Coulomb repulsion interaction, as outlined above, can be justified. Having similar information for a transition in the closely related neutral charge state NV 0 [35] is also valuable. The N3 centre is another nitrogen-related colour centre that has been studied by uniaxial stress [34]. It involves three nitrogen atoms and one carbon adjacent to a vacancy, rather than the three carbon and one nitrogen, and a similar molecular model is adopted for describing its electronic states. Despite the stress parameters being similar to those of − NV , vis in this case the molecular model has not successfully predicted all of the excited states [43]. Having the stress parameters for four related transitions as given in table 2 provides valuable information for ab initio calculations to test our understanding of the electronic model of nitrogen-related colour centres in diamond.

Summary and conclusions
The aim of the work was to use uniaxial stress techniques to better understand the singlet levels of the nitrogenvacancy centre in diamond. The 1042 nm ZPL is understood to be associated with the singlet to singlet transition between levels in the same configuration. The ZPL is spectrally narrow, the sideband is weak, and the symmetry maintaining stress shift parameters A1 and A2 are also relatively small and these are all characteristics of a transition between levels in the same electronic configuration. The ↔ A (a e ) E(a e ) 1 1 1 2 2 1 1 2 2 singlet-singlet transition is the only transition within the electronic model that satisfies this condition and these aspects all give confidence that the transition is correctly identified. However, the stress splitting parameters are large and comparable with those for the ↔ A E triplet and doublet transitions of − NV vis and NV 0 , respectively. These latter transitions involve a change of configuration and an E state with an odd number of e electrons. Consequently, one-electron Coulomb interaction can account for such effects. The singlet-singlet transition is different since the one-electron Coulomb interaction can not (in first order) split or shift the ZPL, and so it was anticipated the responses would be smaller. The strain parameters for − NV IR must arise from an alternative interaction and in this work it has been shown that they can be attributed to the two-electron Coulomb repulsion term. It is recognized within the Coulson and Kearsley [40] model that Coulomb repulsion always plays a significant role and in the case of the NV − centre this interaction is of comparable magnitude to the one-electron Coulomb term. It is, therefore, realistic that the one-electron and two-electron Coulomb interactions can result in similar energy changes in response to a distortion of the lattice. The conclusion is that there is overall consistency with the current electronic model of the NV − and it follows that there is an adequate understanding of the singlet states.
The present uniaxial stress studies have also established that there is a dynamic Jahn-Teller effect associated with the E 1 level. Combining this observation with previous reports of dynamic Jahn-Teller effect in the excited E 3 state, it is clear that electron-vibration interaction is significant within the NV − system. The presence of electron-vibration interaction has been determined from observations within the E 1 and E 3 degenerate electronic states independently, but the interaction can have more significant consequences between states. In particular, it can play a role in inter-system crossing between E 3 and A 1 1 and between E 1 and A 3 2 triplet and play a very important role in giving rise to the important spin polarization property of NV − .