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# Nanophotonics

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# The fine structure of the neutral nitrogen-vacancy center in diamond

Michael S.J. Barson
/ Elmars Krausz
/ Neil B. Manson
/ Marcus W. Doherty
Published Online: 2019-07-20 | DOI: https://doi.org/10.1515/nanoph-2019-0142

## Abstract

The nitrogen-vacancy (NV) center in diamond is a widely utilized system due to its useful quantum properties. Almost all research focuses on the negative charge state (NV) and comparatively little is understood about the neutral charge state (NV0). This is surprising as the charge state often fluctuates between NV0 and NV during measurements. There are potentially under-utilized technical applications that could take advantage of NV0, either by improving the performance of NV0 or utilizing NV directly. However, the fine structure of NV0 has not been observed. Here, we rectify this lack of knowledge by performing magnetic circular dichroism measurements that quantitatively determine the fine structure of NV0. The observed behavior is accurately described by spin-Hamiltonians in the ground and excited states with the ground state yielding a spin-orbit coupling of λ = 2.24 ± 0.05 GHz and a orbital g-factor of 0.0186 ± 0.0005. The reasons why this fine structure has not been previously measured are discussed and strain-broadening is concluded to be the likely reason.

This article offers supplementary material which is provided at the end of the article.

## 1 Introduction

The nitrogen-vacancy (NV) center is a promising color center in diamond for quantum technology. Applications include nanoscale quantum sensing and quantum information processing. This remarkable utility is due to the useful spin-dependent photodynamics of the center’s negative charge state (NV). Despite the intense interest in NV, surprisingly little is known about the other common charge state of the NV center, the neutrally charged NV0. It is important to understand NV0 for a number of reasons. In particular, there is a long standing puzzle as to why the ground state electron paramagnetic resonance (EPR) signal [1] has not been seen in either NV0 or similar 2E ground state centers, such as the negatively charged silicon-vacancy [2], [3], [4] (SiV) and the negatively charged germanium-vacancy [3] (GeV) centers.

There are similarities [5], [6] between the Jahn-Teller (JT) induced vibronic structure of the NV0 ground state and of the poorly understood singlet states of NV. The properties of these singlet states and their associated inter-system crossings underpin all of the desirable spin-dependent photodynamics of NV, such as spin-polarization and readout. Greater understanding of the NV singlet states will lead to the discovery of ways to enhance these properties of NV or engineer other defects with improved properties. Due to their similarities, the clues to understanding the behavior of the NV singlet levels may lead from a better understanding of the NV0 ground state.

There are also technical applications using NV0 in conjuncture with NV. One application is controlling the NV charge state to limit the dephasing effect of the electron spin on nearby nuclear spins [7]. A better understanding of the ground state electron spin of NV0 could also lead to improved control and longer nuclear spin coherence. Another application is the ability to detect the electron ejected to the conduction band during ionization from NV to NV0 for spin-to-charge readout of the NV spin [8]. This is used as an alternative spin readout process for quantum applications using NV. A further application is using the long-lived photoionization of NV to NV0 for classical data storage [9].

The only previously detected EPR [1] of NV0 was for the metastable quartet state 4A2. As no ground or excited fine structure is determined for NV0 either by photo-luminescence excitation (PLE) or by EPR, an alternative means of measurement is required. The optical resonance of NV0 is well known [5]. This provides a starting point and by probing the magneto-optical properties of this resonance, new information regarding the electronic structure of both ground and excited states can be obtained. One such magneto-optical measurement is magnetic circular dichroism (MCD) spectroscopy. MCD is a differential optical spectroscopic technique that determines the difference of left and right circularly polarized light in the presence of an axial magnetic field [10], [11]. This provides quantitative measurements of the magnetic behavior of optical transitions. In particular, the fine structure of the ground and excited states can be extracted from the MCD measurements.

In this paper, we determine the fine structure of NV0 and develop a simple electronic model that fully describes the observed behavior. The MCD experiment is introduced and it is shown how the electronic model relates the resulting spectra. The results and model allow for the fine structure parameters of the system to be determined. We then discuss why such fine structure behavior has not been observed in previous measurements.

## 2 Theory

The NV center consists of a substitutional nitrogen impurity and an adjacent vacancy. The defect in its neutral charge state has five electrons, three from the dangling carbon bonds and two from the dangling nitrogen bond. The negatively charged state gains an additional electron from nearby charge donors (usually nitrogen) [12], [13]. The NV axis (line between nitrogen and vacancy sites) is aligned along a 〈111〉 direction and the defect has three-fold rotational symmetry (C3v), as shown in Figure 1A.

Figure 1:

The atomic and electronic description of the NV0 center.

(A) Unit cell of diamond containing a nitrogen vacancy (NV) center with crystallographic coordinates (X, Y, Z)=([100], [010], [100]) and NV coordinates chosen as (x, y, z)=([112̅], [1̅10], [111]). The carbon atoms are depicted as gray, the nitrogen atom as green, and the vacancy site as pink. (B) Energy levels of the NV0 ground 2E (a1e2) and excited 2A2 $\left({a}_{1}^{2}e\right)$ states, where the brackets signify the molecular orbitals [14]. The orbital states are denoted by +,−and spin states by ↑, ↓. Left and right circularly polarized optical transitions are denoted by the red and green arrows. The gradients of the ground state resonances with respect to magnetic field are unequal due to the orbital Zeeman competing with spin Zeeman effects.

The electronic structure of NV0 consists of a 2E ground state, 2A2 optical excited level, and an intermediate meta-stable 4A2 level. The zero phonon line (ZPL) of the 2E2A2 occurs at 2.16 eV (575 nm). The 2A2 has no orbital degeneracy and exhibits no zero field fine structure. The ground state 2E has orbital degeneracy that gives rise to spin-orbit fine structure as shown in Figure 1B. The 2E fine structure can be described by the Hamiltonian

$H=gμBS→⋅B→+lμBLzBz+2λLzSz,$(1)

where μB is the Bohr magneton, g~2 is the spin g-factor and l is the orbital g-factor, λ is the spin-orbit interaction parameter, the orbital Li=σi and spin ${S}_{i}=\frac{1}{2}{\sigma }_{i}$ operators are defined respectively in terms of the Pauli matrices σi and z is along the NV axis (see definition of coordinate system in Figure 1A). Note that the effect of strain is not included in the Hamiltonian as measurements using MCD are insensitive to strain, see the Supplementary information for a proof of this. In the presence of a large magnetic field, it is convenient to transform the spin coordinate system so that Sz is parallel to the applied field (i.e. SzSz cos θ+Sx sin θ), whilst the orbital operators remain unchanged because the orbitals remain defined by the crystal axes. The transformed Hamiltonian is then

$H=μBgSzB+μBlLzB cos θ +2λLz(Sz cos θ+Sx sin θ),$(2)

For our measurements, the diamond sample had a <100> face that was aligned parallel to the magnetic field, and thus $\mathrm{cos}\text{\hspace{0.17em}}\theta =\stackrel{^}{z}\cdot \stackrel{^}{Z}=1/\sqrt{3}.$ For this field orientation, all possible NV orientations have an equivalent magnetic field projection. The eigenenergies of the transformed Hamiltonian are thus

$E±↓=±μBlB cos θ−124λ2+(gμBB)2±4λgμBB cos θE±↑=±μBlB cos θ+124λ2+(gμBB)2±4λgμBB cos θ,$(3)

where the ↑, ↓ signifies spin-up-down states and the ± signifies orbital states with well-defined positive and negative units of orbital angular momentum about the NV axis. The optical selection rules are such that the ± orbital states participate in R/L circularly polarized transitions, respectively. Care was taken to transform the circularly polarized light from the lab coordinate system into the NV coordinate system, see Supplementary information.

We use MCD to determine the magnetic splitting of the 2E state. This allows for the spin-orbit parameter λ and orbital g-factor l to be determined. As these measurements are performed at low temperature and high magnetic fields (kBT~BB), the thermal populations of the 2E ground state sub-levels must also be included into the model. The optical transition being measured is spin allowed and the energy levels of the excited 2A2 must also be considered. We have assumed a simple spin-1/2 system with a g-factor of 2. Thus the excited state eigenenergies are simply EA↑↓=EZPL±BB/2, where EZPL is the optical ZPL energy.

## 3 Materials and methods

The experimental apparatus is described in Ref. [15]. White light is passed through a double monochromator so as to provide monochromatic light that can be swept in frequency. This light is then passed through a chopper wheel to modulate the intensity (488 Hz modulation frequency). Next the light is filtered to remove second-order diffracted light and is then linearly polarized and passed through a photoelastic modulator which is driven at 50 kHz to create alternatively left and right circularly polarized light. The combination of the chopper wheel and periodic circularly polarized light allows for a heterodyne-based detection process so as to measure small differences in left and right circularly polarized absorption. In this work, the differential absorption is about 102 times weaker than the total absorption. The sample is placed in a temperature controlled (1.46–300 K) liquid helium immersion cryostat with a 0–6 T superconducting magnet.

The sample was a CVD grown (Element six) crystal with ~ppm concentration of nitrogen that was irradiated with electrons at a fluence of 1017/cm2 and energy of ~2 MeV and then annealed at 750°C for 2 h to create NV centers. The NV0 ZPL is on top of the NV absorption phonon sideband, however, only the difference in absorption ZPL relative to the background is considered. As such, the results are not affected by the presence of the NV ZPL. The absorption of NV0 is weak with about a 7% reduction in transmittance at the peak of the ZPL.

## 4 Results

The total absorption (ZL+R) and the differential absorption (ZLR), as shown in Figure 2 are simultaneously acquired. The differential signal ZLR=ZLZR, is the difference of two nondegenerate absorptive lineshapes and the total absorption ZL+R=(ZL+ZR)/2 is the arithmetic mean of the two absorptive lineshapes. By comparing the two spectra using either parametric curve fitting or moment analysis [11], [16] the separation and magnitude of the two circularly polarized absorptions can be determined. Moment or curve fitting analyses each have separate benefits and drawbacks and both methods of analysis were pursued in parallel. For curve-fitting, Gaussian lineshapes were used with the free parameters A1, A2, σ, μ and d. A1 and A2 are the amplitudes of the lines, 2d is the separation of the two lines and μ and σ are the shared central position and width of the two lines.

Figure 2:

Examples of MCD spectra.

The total (ZL+R) and differential (ZLR) absorption spectra at B=5 T and T=1.55 K (A) and at B=5 T and T=10.2 K (B). Notice the change in sign of the ZLR signal for the higher temperature data. Plots are in units of absorbance $A={\mathrm{log}}_{10}\frac{{\Phi }_{i}}{{\Phi }_{t}},$ where Φi,t are the incident and transmitted radiant fluxes. The fits are of the same form as shown in (4).

$ZL±R(E)=A12πσe(E−μ−d)22σ2±A22πσe(E−μ+d)22σ2$(4)

The moment analysis is performed as described in Ref. [10].

The first MCD quantity we consider describes the probability of absorbing a left or right circularly polarized photon. This can be expressed by the ratio of the zeroth ZL+R and ZLR spectral moments or the amplitudes (A1 and A2) obtained from the parametric curve fitting

$〈ZL−R〉0〈ZL+R〉0=2(P−↓+P−↑)−(P+↓+P+↑)P−↓+P−↑+P+↓+P+↑ =A1−A2A1+A2.$(5)

In the above, the angle brackets 〈ZL+Rn represent the nth spectral moment. The second MCD quantity is probability weighted transition energies of absorbing a left or right circularly polarized photon. As such, it also provides information regarding the energy separation of the left and right circularly polarized transitions. This quantity can be determined from the ratio of the first and zeroth moments of the ZLR and ZL+R spectra respectively. This quantity can also be determined by curve fitting using the energy separation (2d) and amplitudes (A1 and A2) of two lineshapes.

$〈ZL−R〉1〈ZL+R〉0=[(P−↓(E−↓−EA↓)+P−↑(E−↑−EA↑)) −(P+↓(E+↓−EA↓)+P+↑(E+↑−EA↑))] ×2P−↓+P−↑+P+↓+P+↑ =4dA1A2(A1+A2)2.$(6)

In the above, the probabilities Pi are determined from the thermal occupations ${P}_{i}={e}^{-{E}_{i}/{k}_{B}T}/{\sum }_{j}\text{\hspace{0.17em}}{e}^{-{E}_{j}/{k}_{B}T}$ of the fine structure states of the 2E level.

The results for a variety of temperatures and magnetic fields are shown in Figure 3. The change in sign of the zeroth and first-order moments for increasing temperature is due to the thermal populations of the ground state sub-levels changing, flipping the likelihood of absorbing a left or right circularly polarized photon.

Figure 3:

Summary of the MCD results.

The MCD values from (6) for varying magnetic field (A) and temperature (B). MCD values from (5) for varying magnetic field (C) and temperature (D). The points represent the values from curve fitting, the moment analysis points are removed for clarity. The dashed curve is the Hamiltonian model using the average parameter values, λ=2.24±0.05 GHz and l=0.0186±0.0005, the shaded regions represent the 1σ uncertainty in the average parameter values.

## 5 Analysis

The data are fitted against the Hamiltonian (2) [via (5) and (6)] using only two free parameters, the spin-orbit coupling λ and the orbital g-factor l, the other parameters g=2 and $\theta ={\mathrm{cos}}^{-1}\left(1/\sqrt{3}\right)$ were kept fixed. This fit obtains the values λ=2.24±0.05 GHz and l=0.0186±0.0005. Due to the difference in methods, the error analysis for both the curve fitting and the moment analysis was performed via a Monte-Carlo method. The raw spectra was modulated with random normally distributed noise with an amplitude equal to the standard deviation of the signal to the sides of the main feature. This was chosen as it represents the noise in the photocurrent signal for a particular spectra. The entire fitting procedure was re-run multiple times to obtain a distribution of the extracted parameters with an associated standard error and mean.

These results show that the orbital g-factor l is reduced by an order of magnitude from the NV− 3E level, of which there are three published values (0.10(1) [17], 0.22 [18], and 0.164(6) [19]). There is only one published value of the orbital g-factor for the NV0 2E state of 0.017(2) [19]. This is in reasonable agreement with our value of l=0.0186(5). The spin-orbit parameter λ is also reduced from its corresponding value in the NV− 3E level (λ=5.3 GHz [20], [21]). However, the reduction is not as large as for the orbital g-factor (a factor of ~1/2 rather than ~1/10).

## 6 Discussion

There are a number of possible explanations for the differing reductions of the orbital g-factor and spin-orbit parameter from NV to NV0. Both parameters are affected by the same Ham reduction factor [22], [23] arising from the JT interactions of the NV0 2E or NV− 3E levels. If the JT interaction of NV0 is larger than of NV (as indicated by features observed in piezospectroscopy of NV0 [5]), then this would explain the reduced size of these parameters in NV0. However, this explanation would imply that both parameters should be reduced by the same factor.

An alternate or complimentary explanation can be found in how the parameters depend on the defect’s molecular orbital structure and local electrostatic potentials. The orbital g-factor is proportional to the reduced matrix element $〈e\left(\stackrel{\to }{r};\stackrel{\to }{R}\right)||{l}_{z}||e\left(\stackrel{\to }{r};\stackrel{\to }{R}\right)〉,$ whereas the spin-orbit parameter is $\propto 〈e\left(\stackrel{\to }{r};\stackrel{\to }{R}\right)||\left[\stackrel{\to }{\nabla }{V}_{Ne}\left({\stackrel{\to }{r}}_{i},\stackrel{\to }{R}\right)×\stackrel{\to }{p}{\right]}_{z}||e\left(\stackrel{\to }{r};\stackrel{\to }{R}\right)〉,$ where lz is the orbital angular momentum operator along the axis of the NV center, ${\left[\stackrel{\to }{\nabla }{V}_{Ne}\left({\stackrel{\to }{r}}_{i},\stackrel{\to }{R}\right)×\stackrel{\to }{p}\right]}_{z}$ is similarly the axial component of the orbital operator of the spin-orbit interaction, VNe is the electrostatic interaction between the defect electrons and the nuclei, $\stackrel{\to }{r}$ are the electron coordinates, $\stackrel{\to }{R}$ are the nuclear coordinates, and $e\left(\stackrel{\to }{r};\stackrel{\to }{R}\right)$ are e molecular orbitals in the Born-Oppenheimer approximation [14].

Owing to the different charge states of NV0 and NV, their nuclear coordinates and molecular orbitals differ [24], [25]. Thus, the reduced matrix elements demonstrate that both the orbital g-factor and spin-orbit parameter will differ between the two charge states due to the molecular orbital differences, and the spin-orbit parameter will additionally differ due to differences in the electrostatic potential VNe. This additional dependence of the spin-orbit parameter over the orbital g-factor is the likely reason why the parameter reduction differ when comparing NV0 and NV.

Our observations do not immediately explain why fine structure has not been observed in EPR or PLE measurements of the NV0 ground-state. Based on the spin-orbit parameter determined here and assuming the use of a 9.6 GHz X-band EPR spectrometer, the NV0 2E EPR features are expected to be at B=±(±fλ)/μBg=±0.16T, ±0.50T which are within the available field range of a typical X-band EPR spectrometer. Additionally, it has been shown that strong illumination with a green laser can photo-convert NV to NV0. Thus, samples could be conveniently pumped to contain more NV0 centers [12], [13]. Additionally, a tunable high-resolution laser should also be able to see these features in PLE.

We believe that there are three major reasons why the EPR signals of the 2E were previously observed: (1) A reduction in angular momentum could also be influenced by fast averaging over the orbital states by a weak JT coupling to a bath of acoustic E modes. This is seen in the NV− 3E state at room temperatures [26], [27] as a removal of spin-orbit splittings. The remaining EPR signal would then be obscured by other spin-1/2 paramagnetic spins in the diamond sample, primarily the substitutional nitrogen or P1 center. (2) Another reason would be if there is a large-strain broadening of the resonances. For the limit of a strain distribution width (Γ) which is larger than spin-orbit splitting (Γ≫λ), only a central spin-1/2 resonance remains. This will be obscured for the same reason mentioned above. (3) The final reason is that measurements have simply not been performed in the correct spectral range or the signal has been overlooked.

We have modeled the electron-phonon interactions and have found them to produce a negligible effect on the NV0 spin resonances (see Supplementary information), ruling out the first reason. For explanation 3, it seems unlikely that the signal was overlooked, given targeted efforts to look for it [1], leaving explanation 2. We have modeled the second explanation (see Supplementary information) and we find that a wide strain distribution significantly reduces the contribution of the strained centers to the total EPR spectrum. We model this by introducing a strain shift to the spin resonances , resulting in the spin-resonances (neglecting l=0.0186) at $ℬ$ and $ℬ±\Delta$ where $ℬ=g{\mu }_{B}{B}_{z}$ and $\Delta =\sqrt{{\lambda }^{2}+{ℰ}^{2}}.$ These spin resonances have the associated transition amplitudes cos2φϕ and sin2φ, where φ=tan (/λ). By constraining the oscillator strength to be conserved over the integrated spectral band, the intensity of the above resonances are ${I}_{ℬ}=\frac{\Gamma }{\Gamma +\lambda }$ and ${I}_{ℰ}=\frac{\lambda }{\Gamma +\lambda },$ where Γ is the width of the strain distribution, here assumed to be Lorentzian. In the limit that Γ≫λ, then I→0 and ${I}_{ℬ}\to 1,$ resulting in spectra with a single resonance only at the free spin−1/2 resonance frequency, this behavior is demonstrated in Figure 4. The limit of Γ≫λ is reasonable, as the stress susceptibility and resulting width of the NV0 orbital electronic states are both of order 100 s of GHz [5]. Note that the ensemble ZPL width in absorption of this particular sample (~50 GHz) is about 20 times narrower than the ensemble emission ZPL width presented by Davies [5] and Manson et al. [12].

Figure 4:

Behavior of 2E spin-levels for increasing strain () in a nonzero magnetic field.

In the high strain limit (λ) the system is better represented by the {|X〉, |Y〉} strain basis than the {|+〉, |−〉} orbital basis. The colored arrows represent the observable electron paramagnetic resonance (EPR). For example, EPR spectra for increasing strain are shown at the bottom of the figure, with increasing strain from left to right. The spectra changes from exhibiting spin-orbit, orbital Zeeman, and spin Zeeman behavior to simple spin Zeeman behavior.

The above arguments for EPR also apply to PLE measurements on a single NV0. PLE should still observe some structure even in the presence of large strain. However, similar to EPR, in the large strain limit the PLE spectra will be modified to just show two transitions separated by a splitting ℰ, as such the fine-structure parameters cannot be extracted. This problem is general to measuring the PLE of any centre that has a Kramers doublet ground-state. For a Kramers doublet the degeneracy of the spin-levels will not be lifted by the strain and the eigenenergies will only add in quadrature with the applied strain splitting. This is not the same for NV− 3E as it is a not a Kramers doublet and PLE is routinely observed [20] (see figure 4 in the Supplementary information). Our fine structure parameters will put a lower bound on the expected PLE splitting, allowing for low-strain NV0 single sites to be found via PLE.

## 7 Conclusion

This work has determined the fine structure of the NV0 electronic states using low temperature and high magnetic field MCD measurements and found agreement by using simple spin-Hamiltonians to describe the ground and excited states. The spin-orbit interaction in the NV0 ground state λ=2.24±0.05 GHz is found to be different to that of the NV excited state. This difference is attributed to the change in electrostatic potential associated to variations in the nuclear coordinates between the two charge states. There is significant quenching of the angular momentum with an orbital g-factor of l=0.0186±0.0005. We have discussed why measurement of NV0 ground-state fine structure has been elusive in previous EPR and PLE measurements and conclude that significant strain broadening of the 2E resonances obscure the spin resonances.

## 8 Supplementary material

The Supplementary information can be found with the online version of the article.

## Acknowledgments

We would like to acknowledge various funding sources for support during this work. In particular, NM would like to acknowledge the Australian Research Council through grants DP170103098. MD would like to acknowledge the Australian Research Council through grants DP170103098 and DE170100169. EK would like to acknowledge the Australian Research Council through grants DP110104565 and DP150103137.

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## Supplementary Material

Revised: 2019-06-20

Accepted: 2019-06-23

Published Online: 2019-07-20

Citation Information: Nanophotonics, Volume 8, Issue 11, Pages 1985–1991, ISSN (Online) 2192-8614,

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