Nitrogen related paramagnetic defects: Decoherence source of ensemble of NV⁻ center

Negatively charged nitrogen vacancy center (NV⁻) ensemble with S = 1 is a promising color center for highly sensitive magnetometers.¹ In an earlier study, weak magnetic signal detection such as detection of single-neuron action potential,² malarial hemozoin,³ biological tissue,⁴ and paleomagnetic in rocks⁵ were demonstrated using the spin ensemble at room temperature. For weak signal detection, diamond with [N_s⁰], which is a concentration of neutral substitutional nitrogen atoms and which works as an electron source to form an NV⁻ center, was 1–30 ppm; also, [NV⁻] was found empirically as 0.1–4 ppm. A world-record highest DC magnetic-field sensitivity of 15 pT Hz^−1/2 has been achieved using diamond containing [N_s⁰] of 27 ppm and [NV⁻] of 1.7 ppm.² We also succeeded in magnetocardiographic measurements of a live rat using single-crystalline diamond with [N_s⁰] of 15 ppm and [NV⁻] of 1.8 ppm.⁶ At that time, DC magnetic-field sensitivity of 140 pT Hz^−1/2 was achieved.

The magnetic sensitivity of NV⁻ center-based sensors is inversely proportional to both the square root of [NV⁻] and NV⁻ center spin coherence time T₂.¹ The T₂ is assumed as the NV⁻ center spin manipulation time in a sensing protocol. When effects of static magnetic noise such as strain gradients, electric field noise, magnetic-field gradients, and temperature variation are negligibly small as a source of decoherence mechanism, the dipole–dipole interaction (DDI) between the NV⁻ center spin and paramagnetic defects such as impurity and point defects becomes the main source of decoherence for T₂. In diamond samples, the major paramagnetic defect is N_s⁰.¹ Because increasing [NV⁻] is essential for high magnetic sensitivity, decoherence contribution from NV⁻ center and NV⁰ center, which is the neutral charge state of NV center and usually coexist with NV⁻ center,⁷ is expected to be non-negligible. We need to study the DDI of NV⁻ center and NV⁰ center in diamond with [NV_T] (=[NV⁻] + [NV⁰]), which is comparable to [N_s⁰]. Furthermore, the types of paramagnetic defects formed during diamond growth are different in two representative diamond synthesized methods, high-pressure and high-temperature (HPHT) method and chemical vapor deposition (CVD) method. Specifically, the incorporation of hydrogen atoms into diamond frequently occurs only in CVD growth. Hydrogen-related negatively charged NV (NVH⁻) center is formed in CVD-grown diamond and is not negligible to [N_s⁰] in the CVD diamond.⁸ Therefore, we need to understand the DDI of various paramagnetic defects including NVH⁻ center.

Although an increase of both [NV⁻] and [N_s⁰] improves the magnetic sensitivity and stabilizes NV centers in the negative charge state, the paramagnetic defect being a decoherence source,¹ such as N₂⁺, N₂V⁻, N₃V⁰, or NVH⁻ centers and named as “other defects” in this paper, is thought to increase at the same time during diamond growth, and spin coherence time T₂ of the NV⁻ center will be shortened. Elucidating the effects of paramagnetic defects in diamond on spin decoherence of the NV⁻ center is crucially important for quantum sensing applications.

Theoretical⁹ and experimental reports^1,10–14 indicate that the decoherence rate is attributed to paramagnetic defects. If the paramagnetic defect is X, the decoherence rate is written as 1/T₂{X}. The decoherence rate is, therefore, proportional to the product of the DDI coefficient of X and [X]. When T₂ for the spin of interest involves a decoherence contribution from paramagnetic defects of various types and when the noise spectrum for each paramagnetic defect has a shape of a Lorentzian function, then T₂ can be expressed as a linear summation of the decoherence rate for each defect.¹ The relation between T₂ for the spin of interest and the concentration of paramagnetic defects is formulated as¹ 

where [X_N] stands for the concentration of Nth paramagnetic defect X_N and $DXN$ denotes the DDI coefficient between the spin of interest and paramagnetic defect X_N. Here, we assume that the components of Eq. (1) are [N_s⁰], [NV⁻], [NV⁰], and [other defects]. Using [N_s⁰], [NV⁻], [NV⁰] and [other defects], Eq. (1) can be transformed as

where C represents the coherence time of the single NV⁻ center, with [N_s⁰], [NV⁻], [NV⁰], and [other defects] of less than approximately 0.01 ppb.^1,11,14 When the paramagnetic defect is the main decoherence source for the sensor spins, shot noise δB is expressed as¹ 

where V denotes the sensor volume. Herein, we consider an ideal situation for quantum sensing under the assumption that [other defects] is negligibly smaller than the total nitrogen concentration [N_T] through the improvement of crystal growth technology.^15,16 This assumption is reasonable for [N_T] > 1 ppm because T₂ is reportedly limited by [N_T].^1,11,13,14 When the ideal situation can be realized, Eqs. (1′) and (2) are transformed as

where V is assumed as a constant. It is apparent that δB is determined by the concentrations of [N_s⁰], [NV⁻], and [NV⁰]. The DDI coefficients of N_s⁰$DNs0$⁠, and NV⁻ center $DNV−$ and NV⁰ center $DNV0$ also determine δB. Actually, N_S⁰ with spin S = 1/2 is known as a decoherence source for NV⁻ center.^1,11–14 The value of $DNs0$ is reportedly 160 μs ppm.^1,11 The DDI coefficients of other nitrogen-related paramagnetic centers including NV⁻ center and NV⁰ center have not been reported.

This study is aimed at investigating the DDI coefficient of the NV⁻ center. The DDI coefficient of the NV⁻ center was evaluated using diamond single crystals with substantial amounts of NV⁻ center being comparable to that of N_S⁰. We also investigated the DDI coefficient of the NVH⁻ center by comparing CVD diamond with HPHT diamond.

Diamond crystals grown using HPHT method and CVD method were used for this study. The HPHT and CVD crystal sizes were, respectively, typically 1.5 × 1.5 × 0.4 mm³ and 3 × 3 × 0.5 mm³. The [N_s⁰] of diamond crystals was in the range of 1–50 ppm. Details of the diamond sample preparation method are described below. The [N_s⁰] of HPHT crystals was controlled by tuning the amount of titanium in the metal solvent.¹⁶ The ¹⁴N isotope is dominant in examined HPHT diamond samples because nitrogen in the air is a nitrogen source for HPHT synthesis and nitrogen in its natural abundance consists primarily of ¹⁴N (99.6%). After these HPHT crystals were cut parallel to the {111} crystal plane, both the top and bottom surfaces were mirror polished. The [N_s⁰] of CVD-grown crystals was controlled by changing the flow rate ratio of nitrogen gas to the methane gas. The nitrogen gas used for growing the examined CVD diamond samples was isotopically enriched ¹⁵N₂ (>98%). After CVD growth, laser cutting was used to remove the (001) substrates from the CVD-grown layers. Both the top and bottom surfaces of the CVD-grown free-standing layers were mirror polished.

The NV centers were formed using the following procedure. The grown HPHT and CVD diamonds were irradiated with a 2.0 MeV electron beam with a total fluence range of 1 × 10¹⁷–2 × 10¹⁸ e/cm², with subsequent annealing at 1000 °C for 2h in vacuum. Hereinafter, we designate the post process for NV center formation as “the e-beam/anneal process.” The [NV_T] (=[NV⁻] + [NV⁰]) examined for this study was 0.1–4 ppm. The value was controlled by the nitrogen concentration and the total fluence of an electron beam.

[N_s⁰] and [NV⁻] before and after the e-beam/anneal process were estimated using electron paramagnetic resonance (EPR) measurements. The EPR spectrum was measured at about 300 K using an X-band EPR spectrometer (FA-100; JEOL). The microwave excitation frequency, microwave power, magnetic-field modulation, and the modulation amplitude were, respectively, 9.428 GHz, 0.02 μW, 100 kHz, and 0.02 mT. Because field modulation and lock-in detection were used, the EPR spectra were obtained in terms of derivative absorption. The applied magnetic field was set as parallel to the [111] or [001] direction of single-crystalline diamond. The sweep range of the magnetic field was 2 mT. Figure 1(a) shows the EPR spectra for the CVD sample used for this study. The EPR spectrum in the left side panel shows the fine-structure splitting peak for one symmetry-related site of the ¹⁵NV⁻ center with the NV axis, which is parallel to the [111] direction of the CVD sample. The double peaks represent the hyperfine interaction of ¹⁵N (I = 1/2) of the ¹⁵NV⁻ center. The EPR spectrum in the central panel shows half of EPR spectra, corresponding to an overlap of the four hyperfine peaks that originate from the four symmetry-related sites of ¹⁵N_s⁰. The EPR spectrum in the right panel shows the ¹⁵NVH⁻ center. Here, we can observe a clear EPR spectrum of NVH⁻ center using ¹⁵N enriched CVD diamond samples due to avoid overlap between EPR spectrum of ¹⁴NVH⁻ center and ¹⁴N_s⁰. Figure 1(b) shows the EPR spectra for the HPHT sample used for this study. The EPR spectrum of the left side panel shows the fine-structure splitting peak for one symmetry-related site of the ¹⁴NV⁻ center with the NV axis, which is parallel to the [111] direction of the HPHT sample. The triple peaks represent the hyperfine interaction of ¹⁴N (I = 1) of the ¹⁴NV⁻ center. The EPR spectrum in the right panel shows 1/3 of EPR spectra, corresponding to an overlap of the four hyperfine peaks that originate from the four symmetry-related sites of ¹⁴N_s⁰. We evaluated [N_s⁰], [NV⁻], and [NVH⁻] by calculating the ratio of double integration of the showed EPR spectra of N_s⁰, NV⁻, or the NVH⁻ center to that of a reference sample of CuSO₄ • 5H₂O. For consideration of the number of hyperfine splitting peaks of N_s⁰, we multiplied the calculated [N_s⁰] by a factor of 2(CVD) or 3(HPHT). The factor of 2 and 3 is the number of hyperfine splitting for ¹⁵N_s⁰ and ¹⁴N_s⁰, respectively. Note that the number of hyperfine splitting peaks of one nitrogen atom for each symmetry-related site of ¹⁵N_s⁰ (or ¹⁴N_s⁰) is calculated from (2 × I + 1). To consider the number of fine-structure splitting peaks of the NV⁻ center, we multiplied [NV⁻] by a factor of 8. The factor of 8 corresponds to the multiplied number of symmetry-related sites of NV⁻ center (4) and the number of fine-structure splitting peaks for one symmetry-related site of NV⁻ center (2).

To evaluate [NV⁰], photoluminescence (PL) measurements were conducted. The confocal micro-PL mapping system (Nanofinder FLEX; Tokyo Instruments, Inc.) was used for collecting PL from NV⁰ centers and NV⁻ centers. Our earlier work presented a detailed evaluation scheme.⁷ 

To evaluate the T₂ values limited by DDI decoherence, the decoherence induced by static magnetic noise components was eliminated using the Hahn echo microwave protocol.¹⁷ Spin-echo measurements were conducted using a home-built PL system with a pulsed microwave module (microwave source—SynthNV PRO; Windfreak Technologies, LLC/arbitrary waveform generator—M3202A; Keysight Inc.). A 532 nm laser (gem 532; Laser Quantum Inc.) was pulsed using an acoustic–optic modulator (No. 35 250-0.2-0.53-XQ; Gooch & Housego). Red fluorescence emitted from the NV⁻ center by the pulsed green laser excitation was collected through an achromatic lens (AC254-030-AB-ML; Thorlabs, Inc.) and was detected using an avalanche photodiode (APD410A/M; Thorlabs, Inc.). A DC magnetic field of approximately 2.5 mT at the measurement position was applied parallel to the [111] crystallographic axis of diamond. Figure 2(a) shows a typical optically detected magnetic resonance (ODMR) spectrum of ensemble NV⁻ center in an examined diamond sample. Two fine-structure splitting cases are presented. The outer fine-structure splitting corresponds to one symmetry-related site of the NV⁻ centers, of which the NV axis is parallel to the [111] diamond direction. The inner fine-structure splitting corresponds to an overlap of three symmetry-related sites of NV⁻ centers for which the NV axis is not parallel to the [111] direction. Operation of the spin was conducted using one site of the NV⁻ center with the outer fine-structure splitting. We defined this site of the NV⁻ center as “on resonance NV⁻ center” and defined the other three sites of NV⁻ center as “off resonance NV⁻ center.” For this study, spin-echo measurements were conducted by following the pulse sequence presented in Fig. 2(b). First, to initialize the spin states of NV⁻ centers, a pulsed green laser with a duration time of 180 μs was used. The green laser power density was 15 mW/μm². Second, two sets of π/2 and one set of π pulsed microwave, as determined by the period of Rabi oscillation for ensemble of NV⁻ center spins, were irradiated during the free precision time of τ/2. Note that “common mode noise” from fluctuation of the microwave pulse should be subtracted using 3π/2 pulse instead of π/2 pulse in the second half of pulse sequence of Fig. 2(b). Actually, the difference between T₂ measured using π/2 pulse and T₂ measured using 3π/2 pulse is within T₂ fitting error. In this study, we applied π/2–π–π/2 sequence instead of π/2–π–3π/2 because the noise is negligibly small. Diamond samples examined for this study were mounted at the center of a home-built microwave resonator circuit.¹⁸ At that point, the pulsed microwave power was 47 dBm. Third, for collecting PL fluorescence (650–850 nm) with information on the NV⁻ center coherence state phase, the diamond was irradiated with a pulsed green laser with 5 μs duration time. Figure 2(c) shows the typical PL fluorescence intensity against the free precision time τ during the readout process. This fluorescence decay is designated as the spin-echo intensity. The spin-echo intensity $S(τ)$ of NV⁻ center found from this measurement was fitted with the function¹⁷ below as

where p denotes the stretched exponential factor. It is known that p parameterizes the NV⁻ spin resonance line shape.¹ For example, a Gaussian line shape was p = 3; a Lorentzian line shape was p = 1. The solid line in Fig. 2(c) shows the curve of the equation with T₂= 2.4 μs and p = 1.

First, we discuss the effects of nitrogen-related paramagnetic defects on decoherence of the ensemble of the NV⁻ center. Figure 3 presents T₂ of the ensemble of the NV⁻ center as a function of concentration of the nitrogen-related paramagnetic defect N_s⁰ and the NV center. In Fig. 3, circles and squares, respectively, correspond to HPHT diamond samples and CVD diamond samples. The color scales in Figs. 3(a), 3(b), 3(d), and 3(f) show ratios of [NV_T] to [N_s⁰]. Color scales in Figs. 3(c) and 3(e) show [N_s⁰]. Figure 3(a) portrays the relation between T₂ and [N_s⁰]. These data are distributed along a line, indicating a good inverse proportionality between T₂ and [N_s⁰]. The HPHT samples and CVD samples exhibit almost identical trends of the relation between T₂ and [N_s⁰]. The orange curve in Fig. 3(a) represents the relation between T₂ and [N] measured by SIMS, as reported by Bauch et al.¹¹ Because [NV⁻] and [NV⁰] of the diamond samples examined in that study are much smaller than [N_s⁰], that is because [N] is nearly equal to [N_s⁰], the orange line was drawn with an assumption of [N] = [N_s⁰]. As this figure shows, most of the data are on the reported line, with some exceptional data shown with some deviation from the line. A larger ratio of [NV_T] to [N_s⁰] would be associated with a tendency to greater deviation. This feature can be interpreted as follows. When [NV_T] is nearly equal to or larger than [N_s⁰], the decoherence contribution of the spin bath of NV⁻ and NV⁰ centers to the NV⁻ center spin of interest cannot be ignored. For observing the decoherence contribution of spin bath of the NV center, we study the relation between T₂ and the concentration of the NV center. Figure 3(b) depicts the relation between T₂ and [NV⁻]. Correlation between T₂ and [NV⁻] was small when using all of the data. Figure 3(c) portrays a similar plot, but while using data with a ratio of [NV_T] to [N_s⁰] of more than about 0.3. We can observe a tendency of inverse proportionality in the relation between T₂ and [NV⁻], as depicted in Fig. 3(c). The decoherence contribution of NV⁻ center becomes greater with an increase in the ratio of [NV_T] to [N_s⁰]. Similar to Fig. 3(b), less inversely proportional correlation was visible between T₂ and [NV⁰], as depicted in Fig. 3(d). When we specified data with a ratio of [NV_T] to [N_s⁰] greater than about 0.3, the relation between T₂ and [NV⁰] tends to have inverse proportionality, as depicted in Fig. 3(e). The decoherence contribution of NV⁰ center is larger when the ratio of [NV_T] and [N_s⁰] is large. The T₂ value depicted in Figs. 3(c) and 3(e) involves a decoherence contribution of N_s⁰. The T₂ value tends to decrease concomitantly with increasing [N_s⁰], similar to that shown in Fig. 3(a). The tendency of inverse proportionality, as depicted in Figs. 3(c) and 3(e), results not only from the NV center but also from N_s⁰. To evaluate decoherence effects of both N_s⁰ and the NV center on the NV⁻ center spin of interest, the correlation diagram presented in Fig. 3(a) was redrawn as depicted in Fig. 3(f) using [N_PM] as the x axis. Here, [N_PM] is defined as a summation of [N_s⁰], [NV⁻], and [NV⁰]. The NVH⁻ center is excluded from N_PM. The relation between T₂ and [N_PM] shows good inverse proportionality in [N_PM] of 1–50 ppm. Actually, T₂ can be expressed by following Eq. (5), inferred from the relation between T₂ and the concentration of nitrogen-related paramagnetic defects as

The slope of line $DNPM$ shows the “average” DDI coefficient for nitrogen-related paramagnetic defects: N_s⁰, the NV⁻center, and the NV⁰ center. The black curve presented in Fig. 2(f) shows the line of Eq. (5) with $DNPM$ of about 138 μs ppm. The slope is slightly less than the orange line slope of 160 μs ppm.^1,11,13,14 The higher spin multiplicity of NV⁻ center spin S = 1 than that of N_s⁰ spin S = 1/2 might explain the differences in slopes between those of the present work and those of earlier reports.^1,11,13,14

This subsection presents the discussion of the effects of the e-beam/anneal process on the paramagnetic center concentration [N_PM] and T₂ of ensemble of the NV⁻ center. To elucidate the effects of the e-beam/anneal process, after we prepared three nitrogen-doped {001} CVD diamond crystals, we labeled three CVD diamonds as CVD-I, CVD-II, and CVD-III. Subsequent EPR and PL measurements revealed that these crystals contain point defects in the form of N_s⁰, the NV⁻ center, the NV⁰ center, and the NVH⁻ center (NV⁻ center with a hydrogen atom). Here, an NVH⁻ center is typically observed in as-grown CVD diamond crystals.^8,19,20 The NVH⁻ center, with a spin of 1/2, is a candidate as a decoherence source of the NV⁻ center. Table I shows summarization of [N_s⁰] _initial (concentration of N_s⁰ before the e-beam/anneal process), [NV⁻], [NV⁰], and [NVH⁻] for each prepared sample. Figure 4(a) shows changes of [NV⁻], [NV⁰], [NVH⁻], and [N_s⁰] of a CVD diamond sample CVD-I through the e-beam/anneal process with a total electron beam fluence of 3.3 × 10¹⁷ e/cm². A small change of [N_PM] through the e-beam/anneal process comes from the EPR measurement error. Through the e-beam/anneal process, [Ns⁰] decreased, but [NV_T] (=[NV⁻] + [NV⁰]) increased. The change of the concentration ratio of [NV_T] with respect to [N_s⁰] is attributable to the formation of NV centers from N_s⁰ through diffusion of monovacancies created by the electron beam irradiation. In contrast to the changing of [NV_T] and [N_s⁰] through the e-beam/anneal process, [NVH⁻] is constant. The robustness of the NVH⁻ center during 1000 °C annealing is consistent with that described in an earlier report.²⁰ Figure 4(a) also shows T₂ values of ensemble of NV⁻ center when un-irradiated and with a dose of 3.3 × 10¹⁷ e/cm². Results show that T₂ is constant irrespective of the abundance of three associated nitrogen-related paramagnetic defects when [N_PM] is constant.

Next, an effect of additional irradiation of an electron beam on T₂ of the ensemble of NV⁻ center was examined. Figures 4(b) and 4(c) show changes of [NV⁻], [NV⁰], and [N_s⁰] of the other two CVD diamond samples CVD-II and CVD-III through the e-beam/anneal process with total fluences of 1.6 × 10¹⁷ and 4.9 × 10¹⁷ e/cm². The small change of [N_PM] which occurred through the e-beam/anneal process derives from the EPR measurement error. The creation yields of [NV⁻] and [NV⁰] through additional electron beam irradiation are comparable to the case of CVD-I discussed in Fig. 4(a). Again, T₂ was constant through electron beam irradiation. This result demonstrates that T₂ of the NV⁻ center of interest has no dependence on structural changes among three associated paramagnetic defects when [N_PM] is constant. Given electron irradiation fluence of up to 4.9 × 10¹⁷ e/cm², the e-beam/anneal process does not strongly affect T₂ of ensemble of NV⁻ center. Bogdanov et al.²¹ reported that T₂ of NV⁻ center is independent of the total fluence of electron beam irradiation.

This section presents the discussion of effects of NVH⁻ center paramagnetic defects on decoherence of ensemble of NV⁻ center. For estimation of the DDI of the NVH⁻ center, [NVH⁻] formed in diamond must be comparable to [NV_T] and [N_s⁰]. We prepared two nitrogen-doped {001} CVD samples and labeled them as CVD-II and CVD-III. We also prepared an e-beam/anneal processed {111} HPHT sample, labeled as HPHT-I, for comparison to the CVD samples. The e-beam/anneal process was applied for these samples for creating ensemble of the NV⁻ center. Figure 5 shows the defect concentrations of N_s⁰, NV⁻, NV⁰, and the NVH⁻ center for these three samples. Each paramagnetic defect concentration, except for [NVH⁻], is nearly the same among three samples, as shown in Table I. We can expect that the decoherence contribution of each defect, except for the NVH⁻ center, is nearly the same among three samples: CVD-II, CVD-III, and HPHT-I. This situation is suitable for evaluating the decoherence contribution of the NVH⁻ center. Typically, no residual hydrogen atoms exist in HPHT synthetic diamond. In fact, no EPR signal attributed to the NVH⁻ center was detected from HPHT-I. For CVD-II and CVD-III, nearly one order of difference was observed in [NVH⁻], as presented in Fig. 5. For the present study, the ratio of [NVH⁻] to [N_s⁰]_initial is 4%–27%, as shown in Table I. This value is not very different from the reported value of 10%.⁸ At this moment, we have no technical means of controlling [NVH⁻] by CVD growth parameters. Additional study is necessary to control [NVH⁻]. Although the differences of [N_PM] among the three samples were within 10% of the EPR measurement error, T₂ changes only slightly despite the large differences in [NVH⁻]. No difference exists in T₂ among the three samples CVD-II, CVD-III, and HPHT-I, even though [NVH⁻] of sample CVD-II is comparable to [NV⁻] and [NV⁰]. This fact indicates that the DDI coefficient of the NVH⁻ center is significantly weaker than that of N_s⁰, the NV⁰ center, or the NV⁻ center. We also studied the spin relaxation time for nitrogen-related paramagnetic center with rapid passage (RP) EPR mode.²² Using this mode, we can selectively detect paramagnetic centers with a long spin relaxation time. Figure 6 shows EPR spectra measured using slow passage (SP) EPR mode (top panel) and RP EPR mode (bottom panel) for the NV⁻ center, N_s⁰, and the NVH⁻ center in CVD-II. Although the EPR signal being attributed to the NV⁻ center and N_s⁰ was detected in both SP and RP EPR mode spectra, the EPR signal for NVH⁻ center was observed only in the SP EPR mode spectrum. The result indicates that the spin relaxation time for the NVH⁻ center is shorter than that for the NV⁻ center and N_s⁰. Here, strength of DDI between NV⁻ center and a spin bath can be described as a product of concentration of spin bath and correlation function.^11,12 The correlation function is characterized by correlation time $(τc)$ for each spin bath. The $τc$ is a timescale of decaying of coherence spin precession in a spin bath. Spin relaxation time correlates to the $τc$ because $1/τc$ indicates the spin flip-flop rate between spin pairs in spin bath.^11,12 Short $τc$ is leading to the small strength of the DDI because correlation function form is $e−|t|τc$⁠.^11,12 The $τc$ of spin bath of NVH⁻ center is expected to be shorter than that of spin bath of NV⁻ center and N_s⁰ due to shorter spin relaxation time for NVH⁻ center than that for the NV⁻ center and N_s⁰. In CVD diamonds with [NVH⁻] ranging from 0.2 to 1.2 ppm, examined in this study, the short spin relaxation time of NVH⁻ centers was not sufficiently long to disturb the coherence of NV⁻ centers. But, there is a possibility of increasing decoherence contribution from NVH⁻ center by increasing [NVH⁻]. The RP EPR mode spectrum (lower right panel) is not assigned to the NVH⁻ center because this spectrum has no multiple peaks related with the NVH⁻ center.¹⁹ The SP EPR mode spectrum for the unknown center (UC) was hidden in the SP EPR mode spectrum for the NVH⁻ center: it was not resolved. Observation of the SP EPR mode spectrum related with the NVH⁻ center revealed that the UC concentration is lower than that of the NV⁻ center, N_s⁰, or NVH⁻ center. This study did not clarify the physics origin of the great difference in DDI coefficient between the NVH⁻ center and other nitrogen-related paramagnetic defects (N_s⁰, NV⁰ center, and NV⁻ center). Direct investigation of the spin character such as the Rabi frequency, T₁, and T₂ for each nitrogen-related paramagnetic defect paves the way for better characterization of decoherence.

Here, we estimate the DDI coefficient of the NV⁻ center from Eq. (1″) using the experimentally obtained T₂ value of the ensemble NV⁻ center. The DDI coefficient of NV center $DNV−$comprises components of two types: one is $(DNV−)on−reso$ attributed to the “on resonance NV⁻ center” and the other is $(DNV−)off−reso$ from the “off resonance NV⁻ center.” The words “on resonance” and “off resonance NV⁻ center” are defined in Sec. II and Fig. 2(a). The first component $(DNV−)on−reso$ is DDI between two “on resonance NV centers.” The second component is DDI between an “on resonance NV⁻ center” and “off resonance NV⁻ center.” Numbers of “on resonance” and “off resonance” NV⁻ center are, respectively, 1/4 and 3/4 of the total number of NV⁻ center. Equation (1″) is transformed to the formulation below using individual constituent concentrations of the nitrogen-related paramagnetic defects as

where $(DNV−)on−reso$ can be evaluated using instantaneous diffusion (ID) method.^12,23 For this study, instead of performing ID method, we used the averaged DDI coefficient of NV⁻ center $DNV−$ according to the following equation:

Consequently, using Eq. (8), Eq. (7) is transformed to the equation below as

The DDI coefficient is determined by the spin number (S) and g factor of a paramagnetic center.²⁴ We assume $DNV0$ as the value of $DNs0$ because the spin number and g factor of the NV⁰ center^1,25 are nearly the same as those of N_s⁰.^1,26 Using $DNs0$⁠, Eq. (9) is transformed into the following equation:

Actually, T₂ in Fig. 3(c) is inversely proportional to [NV⁻] when choosing data with [NV_T] /[N_s⁰] of over about 0.3. We selected these experimentally obtained data for estimating the value of $DNV−$⁠. In Eq. (10), we applied values of 160 μs ppm^1,11,13,14 and 694 μs,^1,11,13,14 respectively, to $DNs0$ and C. Using the values of coefficients and the experimentally obtained values of T₂, [N_s⁰], [NV⁻], and [NV⁰], determination of the value of $DNV−$ was conducted. Figure 7 shows the relation between $T2{NV−}$ and [NV⁻]. The solid line represents the fitting function curve with respect to plotting data. The fitting function is defined as $1T2{NV−}=1DNV−×[NV−]$⁠. We obtained $DNV−$ of 89 μs ppm through data fitting. This value is nearly equal to that of $DNV−$ of 98 μs ppm estimated using the relation between DDI coefficient and spin multiplicity²⁴ below:

This good agreement between theoretical estimation and experimentally obtained data indicates the validity of Eq. (1″), assuming $DNV0$ as the reported value of $DNs0$⁠.

From the discussion presented herein, we have the following three experimentally obtained results:

1. T₂ is constant irrespective of the relative abundance of three nitrogen-related paramagnetic defects that are N_s⁰, NV⁻, and NV⁰ when [N_PM] is constant.

2. The DDI coefficient of the NVH⁻ center is negligibly small compared to those of three nitrogen-related paramagnetic defects.

3. The DDI coefficient of the NV⁻ center, $DNV−$ of 89 μs ppm, as experimentally obtained, is comparable to the value obtained when estimated using Eq. (11).

Results (1) and (2) show that the decoherence rate is dominated by that of N_s⁰, the NV⁻ center, and the NV⁰ center. Actually, Eq. (1″), which describes the relation between T₂ of ensemble of the NV⁻ center and the concentration of each nitrogen-related paramagnetic defect, has been reported.^1,11,27 Good agreement of $DNV−$ between the theoretical and experimental obtained values suggests that Eq. (1″) used for estimation of $DNV−$ is reasonable in a real system.

We studied spin-echo coherence time T₂ of an ensemble of the NV⁻ center in single-crystalline diamond. Specifically, we investigated T₂ change with respect to nitrogen-related paramagnetic defects when [NV⁻] is comparable to [N_s⁰]. Results indicated that the T₂ of the ensemble of the NV⁻ center was inversely proportional to [N_PM] = [N_s⁰] + [NV⁻] + [NV⁰]. From the inverse proportionality between T₂ and [N_PM], we elucidated that nitrogen-related paramagnetic defects of three kinds are a major decoherence source of electron spins of an ensemble of the NV⁻ centers in single-crystalline diamond. We also found that T₂ was unchanged even though each fraction of these nitrogen-related paramagnetic defect concentration changes if [N_PM] is constant. This result indicates that T₂ of the ensemble of the NV⁻ center has no dependence on structural change among three associated paramagnetic defects. By comparing a CVD diamond containing NVH⁻ centers and HPHT diamonds where NVH⁻ centers are not observed, we also elucidated that the DDI coefficient of NVH⁻ center is significantly smaller than that of N_s⁰ and NV center. Finally, we estimated the average DDI coefficient of the NV⁻ center $DNV−$ as 89 μs ppm from experimentation using the equation, indicating that the average decoherence rate is determined by the linear sum of the decoherence rates of each nitrogen-related paramagnetic defect. The values of $DNV−$ obtained from this study are useful for specifying the appropriate concentrations of NV⁻ center and N_s⁰ for enhancing the magnetic sensitivity of diamonds.

This work was supported by MEXT Q-LEAP (Nos. JPMXS0118068379 and JPMXS0118067395). T. Teraji acknowledges the support of JST CREST (No. JPMJCR1773), MIC R&D for construction of a global quantum cryptography network (No. JPMI00316), and JSPS KAKENHI (Nos. 20H02187, 20H05661, and 19H02617). T. Teraji and S. Onoda acknowledge the support of JST Moonshot R&D (No. JPMJMS2062).

The authors have no conflicts to disclose.

Chikara Shinei: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Writing – original draft (lead). Yuta Masuyama: Methodology (equal); Writing – review & editing (supporting). Masashi Miyakawa: Resources (lead); Writing – review & editing (supporting). Hiroshi Abe: Methodology (lead); Writing – review & editing (supporting). Shuya Ishii: Methodology (lead); Writing – review & editing (supporting). Seiichi Saiki: Writing – review & editing (supporting). Shinobu Onoda: Funding acquisition (lead); Writing – review & editing (supporting). Takeshi Ohshima: Funding acquisition (lead); Writing – review & editing (supporting). Takashi Taniguhi: Resources (lead); Writing – review & editing (lead). Tokuyuki Teraji: Funding acquisition (lead); Resources (lead); Supervision (lead); Writing – review & editing (lead).

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