Nanoscale zero-field electron spin resonance spectroscopy

The basic essence of our method is shown in Fig. 1a, where the sensors are shallow NV centers embedded in a bulk diamond. The diamond is adhered on a coplanar waveguide, and observed by a home-built confocal microscope from the other side. As illustrated in Fig. 1b, the NV centers are surrounded by a spin mixture consisting of target spins and background spins. The target spins can be either defects inside the diamond or paramagnetic targets on the diamond surface. The background spins are dangling bonds on the diamond surface. The NV center is a kind of color defect in diamond consisting of a substitutional nitrogen atom and an adjacent vacancy (see Fig. 1c). The electrons around the defect form an effective electron spin with a spin triplet ground state ($S=1$) and a zero-field splitting of $D=2\pi\times 2.87$ GHz [21]. The NV electron spin can be polarized into the $|m_{\text{s}}=0\rangle$ state ($|0\rangle$, henceforth) by 532 nm laser excitation. The photoluminescence (PL) rate of NV center is state-dependent, i.e., $|0\rangle$ is ‘brighter’ than $|\pm 1\rangle$, then the populations of the states can be directly read out by the photoluminescence rate. Furthermore, the spin transition between $|0\rangle$ and $|\pm 1\rangle$ can be driven by the microwave. As shown in Fig. 1d, under resonant microwave ($B_{1}\cos{ft}$, $f=D$) radiation, the NV electron spin can be driven to the dressed states, which are

$$\begin{split}&|-1\rangle_{\text{d}}=\frac{1}{2}|1\rangle-\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{2}|-1\rangle,\\ &|0\rangle_{\text{d}}=-\frac{1}{\sqrt{2}}|1\rangle+\frac{1}{\sqrt{2}}|-1\rangle,\\ &|1\rangle_{\text{d}}=\frac{1}{2}|1\rangle+\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{2}|-1\rangle,\end{split}$$

(1)

in the rotating reference frame with eigenenergies $-\Omega/2$, 0 and $\Omega/2$, respectively. Here $\Omega$ is the driving power valued by the corresponding Rabi frequency with $\Omega=\gamma_{\text{e}}B_{1,\perp}$ proportional to the perpendicular amplitude of the driving microwave, where $\gamma_{\text{NV}}=-2\pi\times 2.803$ MHz/G is the gyromagnetic ratio of the NV electron spin. Similar to Hartmann-Hahn double resonance [22, 23], the dipole-dipole interaction between the NV center and the target spins will induce a flip-flop process at the resonance condition

$$\Omega=2\Delta\omega_{ij},$$

(2)

where $\Delta\omega_{ij}=\omega_{j}-\omega_{i}$ ($i<j$) is the energy level splitting of the target spins corresponding to an allowed magnetic dipole transition. If the NV center is initially polarized to one of the dressed states given by Eq. 1, the polarization will be transferred from the NV center to the target spins, then the populations of dressed states will change. Therefore, a resonance spectrum of the target spins can be measured by sweeping the driving power and measuring the populations of the dressed states.

In zero magnetic field, all the Zeeman terms are disappeared and the energy level structure of the target spins is determined solely by the intrinsic spin-spin interactions. To see how these interactions can be resolved by ZF-ESR, we take a free radical with hyperfine interaction between the electron spin $\mathbf{S}$ and the nuclear spin $\mathbf{I}$ as an example. For clarity, here we deal with $S=1/2$ and $I=1/2$ (for $I=1$ nuclei, see Supplementary Note 5). The Hamiltonian of the target spin can be written as [24]

$$H_{0}=\mathbf{S}\cdot\mathbb{A}\cdot\mathbf{I},$$

(3)

where $\mathbb{A}$ is the hyperfine tensor, which is diagonal

$$\mathbb{A}=\left(\begin{array}[]{ccc}A_{xx}&&\\ &A_{yy}&\\ &&A_{zz}\end{array}\right)$$

(4)

in the principal axis frame. The eigenstates of Eq. 3 are

$$\begin{split}&|\phi_{1}\rangle=\frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle),\\ &|\phi_{2}\rangle=\frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle),\\ &|\phi_{3}\rangle=\frac{1}{\sqrt{2}}(|\uparrow\uparrow\rangle-|\downarrow\downarrow\rangle),\\ &|\phi_{4}\rangle=\frac{1}{\sqrt{2}}(|\uparrow\uparrow\rangle+|\downarrow\downarrow\rangle),\\ \end{split}$$

(5)

where $|\uparrow\rangle$ and $|\downarrow\rangle$ denote spin-up and spin-down respectively with the electron spin on the left and the nuclear spin on the right. The corresponding eigenenergies are

$$\begin{split}&\omega_{1}=\frac{1}{4}(-A_{xx}-A_{yy}-A_{zz}),\\ &\omega_{2}=\frac{1}{4}(A_{xx}+A_{yy}-A_{zz}),\\ &\omega_{3}=\frac{1}{4}(-A_{xx}+A_{yy}+A_{zz}),\\ &\omega_{4}=\frac{1}{4}(A_{xx}-A_{yy}+A_{zz}).\end{split}$$

(6)

Six allowed magnetic dipole transitions between these states are expected with frequencies

$$\begin{split}\Delta\omega_{12}=\frac{1}{2}|A_{xx}+A_{yy}|,\Delta\omega_{34}=\frac{1}{2}|A_{xx}-A_{yy}|,\\ \Delta\omega_{13}=\frac{1}{2}|A_{yy}+A_{zz}|,\Delta\omega_{24}=\frac{1}{2}|A_{yy}-A_{zz}|,\\ \Delta\omega_{14}=\frac{1}{2}|A_{xx}+A_{zz}|,\Delta\omega_{23}=\frac{1}{2}|A_{xx}-A_{zz}|.\end{split}$$

(7)

Note that a rotational transformation does not change the eigenenergies, which means these transition frequencies are independent of the orientation of the radicals. While the transition speed determined by the dipole-dipole interaction is orientation-dependent. A detailed discussion can be found in the Discussion section below.

We perform the measurement on a similar electron-nuclear system named P1 center, which is another kind of defects in diamond consisting of only a substitutional nitrogen atom. For ¹⁵N P1 center, the principal values of the hyperfine tensor are $A_{xx}=A_{yy}=A_{\perp}=114$ MHz and $A_{zz}=159.9$ MHz [25]. Due to the Jahn-Teller effect, one of the nitrogen-carbon bonds is distorted, and the orientation of P1 center can change between the four kinds of N-C bonds [26]. The NV and P1 centers are generated by ion implantation. To make the P1 centers close enough to the NV centers, the diamond we used is implanted with high dose of nitrogen ions (see Methods). So the experiment is performed on an ensemble of NV centers with each NV center surrounded by dozens of P1 centers. As demonstrated by Hall et al. [18], the NV center is a highly local sensor. Although ensembles of NV and P1 centers are involved, the response of each NV center is dominated by only those P1 centers resided in its nanoscale detection area (a quantitative discussion is given below). Therefore, each NV center is an isolated sensor, and the detected ZF-ESR spectrum is the average spectrum of P1 centers in each nanoscale region. Since the resonance frequencies of difference P1 center are the same, using of ensembles of NV centers can speed up the measurement.

Different from the frequency-sweeping method [13, 14, 16, 17, 19], where the microwave frequency can be easily tuned, the microwave power is limited by the microwave amplifier and the transform efficiency of the waveguide. Here by using a proper designed coplanar waveguide, the Rabi frequency up to $\sim 400$ MHz was realized (see Fig. 2a,c), enough to capture all the interaction information of P1 centers. We note by further employing a high-power microwave amplifier and shaped pulses, the Rabi frequency up $\sim 1$ GHz is possible [27], which is theoretically limited by the zero-field splitting of NV centers. Then this method is applicable to more high-spin paramagnetic targets. On the other hand, the lower limit of Rabi frequency is bounded by the dephasing time of NV centers. The NV centers have four kinds of directions, so the $B_{1,\perp}$ on different NV sites may be different, which induces an extra sine modulation on the Rabi oscillation. Comparing with the Rabi oscillation, where the NV state is rotating between $|0\rangle$ and $|\pm 1\rangle$, the NV state is locked in $|-1\rangle_{\text{d}}$ by applying a $\pi/2$ pulse with 90^∘ phase difference before the continuous driving (see Methods and Fig. 2b). The experimental result in Fig. 2d shows that the NV state is successfully locked with a relaxation time of $T_{1\rho}=70\pm 2$ $\mu$s, which is much longer than the dephasing time of $T_{2}^{*}\sim 0.1$ $\mu$s.

The ZF-ESR spectrum is obtained by sweeping the driving power while fixing the driving length $\tau=10$ $\mu$s, as illustrated in Fig. 3a. Since the hyperfine tensor $\mathbb{A}$ of P1 centers is cylindrically symmetrical, the eigenstate $|\phi_{3}\rangle$ and $|\phi_{4}\rangle$ are degenerate, thus only three magnetic dipole transitions are observable (see Fig. 3b). Figure 3c gives the measured spectrum with three characteristic peaks at $44.0\pm 3.6$ MHz, $221.7\pm 3.4$ MHz and $270.8\pm 4.8$ MHz, where the error bars are given by half the linewidths of the Gaussian peaks. It seems like there are other peaks at $\sim 80$ MHz and $\sim 130$ MHz. The reason is unknown yet. A possible source of these extra signals is other kinds of defects, due to the numerous electron spin defects in diamond [28]. As the spin-locking relaxation time $T_{1\rho}$ depends on the driving power $\Omega$, the baseline of the original ZF-ESR spectrum is uneven (Supplementary Note 3), which has been calibrated to unity as shown in Fig. 3c. Combining with Eq. 2 and Eq. 7, the principal values of the hyperfine tensor can be directly obtained from any two peaks or a least-squares fitting of all the three peaks. The least-squares fitting gives $A_{\perp}^{\text{exp}}=110.7\pm 3.5$ MHz and $A_{zz}^{\text{exp}}=155.0\pm 7.1$ MHz, consistent with the hyperfine coupling of ¹⁵N P1 center. We have also measured the ZF-ESR spectrum of ¹⁴N P1 center which can be found in Supplementary Note 5.

The energy level shifts induced by the environmental fluctuations on either the NV center or the P1 center will destroy the energy exchange between them, which acts to broaden the resonance frequencies, as characterized by the transverse spin relaxation rate, $\Gamma_{\text{tot}}=\Gamma_{\text{NV}}+\Gamma_{\text{P1}}$. The transverse spin relaxation of the NV center can be significantly suppressed by continuously driving [29], so $\Gamma_{\text{tot}}\approx\Gamma_{\text{P1}}=1/T_{2,\text{P1}}^{*}$. The dephasing time of the P1 centers can be estimated by the dephasing time of the NV center, which is $\sim$ 0.1 $\mu$s, because of the similar environment of the NV and P1 centers. Thus the linewidth induced by the dephasing is $\Delta\Omega\sim 1/(\pi T_{2,\text{P1}}^{*})\sim 3.2$ MHz [15], not enough to explain the observed linewidth of $7\sim 10$ MHz. Besides, the fluctuations of the driving power will also induce the broadening of zero-field lines. It was recorded and calibrated during the measurement with fluctuation $<$ 0.5% (Supplementary Note 4). Moreover, the residual static magnetic field (mainly contributed by the geomagnetic field) will induce line splitting of $\sim$ 2.3 MHz (Supplementary Note 4), which will behave as line broadening if the line splitting is smaller than the linewidth. The residual field can be further avoided by placing the setup in a mu-metal box or actively compensating with an electromagnetic field.

To confirm that the detected spectrum is indeed dominated by the P1 centers in a nanoscale detection area, we have calculated the signal induced by all the P1 centers residing at a distance of $>15$ nm from the NV center, which is $\leq 0.29\%$ (see Methods). While the signal contrast in Fig. 3c is $\sim 3\%$, indicates more than $90\%$ of the signal is attributable to the P1 centers residing in the detection area with radius of 15 nm. The mean number of P1 centers in this detection area is $\sim 4$. Note it is a conservative estimate, as the peaks in Fig. 3c are broadened by some additional effects which will lower the signal contrast.

In some practical nanoscale applications, such as detection of nitroxide radicals dispersed on the diamond surface, the NV center-based ZF-ESR can outperform the NV center-based ESR (here we take DEER as an example) for several reasons, including: (1) The ZF-ESR method, which does not require an external magnet and any manipulations of the target spins (see Fig. 4a,b), will be easily used. (2) Different from the inner P1 centers which are restricted by the diamond lattice and have only four orientations, the external nitroxides are much more disordered and can be any direction-oriented [17, 19]. The simulated spectra with different orientations of nitroxide are given in Fig. 4c,d. Since the DEER spectrum is orientation-dependent, the peaks will be averaged if differently oriented nitroxides are included. For conventional induction-based ESR, the statistical average of the orientations over a large ensembles of spins induces an inhomogeneously broadened powder spectrum, where the hyperfine values can still be extracted even with limited accuracy. But for nanoscale ESR, the detected spins are much fewer ($\sim 10^{1}$), then the average spectrum (see Fig. 4e) still has a certain randomness. Without the prior knowledge of the orientations of each nitroxide, extraction of the hyperfine values will be nearly impossible. However, for the ZF-ESR spectrum, different orientations will lead to different peak intensities rather than peak positions, so the characteristic peaks remain clear in the zero-field ESR spectrum (see Fig. 4f) all the time. (3) There exists a paramagnetic background, which consists of dangling bonds on the diamond surface with $g$ factor similar to free radicals [30, 31, 32]. Due to the dense distribution and the short correlation time [30, 33, 34], the background spins always behaved as a broad central peak in the DEER spectrum (see Fig. 4e and Supplementary Note 2), which can be hardly distinguished from the nitroxides. While in zero field, the background peak will move to the zero frequency and only a characterless low-frequency signal can be observed (see Fig. 4f and Supplementary Note 2), thus the fingerprint-like ZF-ESR signal of target spins can be naturally distinguished.

On the other side, different from the conventional induction-based ESR, where the signal intensity is proportional to the polarization of the electron spins (i.e., the magnetic field), the signal of NV center-based ESR is induced by the statistically polarized electron spins. So the sensitivity of this nanoscale ESR does not depend on the magnetic field. Specifically, the nanoscale detection is based on dipole-dipole coupling between the probe and the target and limited by the coherence properties of the probe. For DEER measurement, only $zz$ coupling contributes to the signal, and the sensitivity is limited by the $T_{2}$ of the NV center [12]. For our ZF-ESR measurement, all the $zx$, $zy$ and $zz$ couplings contribute to the signal (detail in Supplementary Note 1), and the sensitivity is limited by the $T_{1\rho}$ of the NV center, which is usually much longer than $T_{2}$. Therefore, the sensitivity of our nanoscale ZF-ESR is comparable with and even potentially better than the nanoscale ESR.

The setup is based on a home-built confocal microscopy. A temperature-stabilized diode laser (CNI MGL-III-532) is used for the polarization and readout of the NV centers. The photoluminescence is detected by an Avalanche photodiode (Perkin Elmer SPCM-AQRH-14) through a 60X, 0.7 NA objective (Olympus LUCPLFLN). The driving microwave is generated by an arbitrary waveform generator (Agilent M8190a) and amplified by a microwave amplifier (Mini-circuits ZHL-16W-43+). The diamond is obtained commercially, 100-oriented, and electronic-grade with a thickness of 500 $\mu$m. The NV and P1 centers was created by implantation of 5 keV ¹⁵N⁺ ions with dose of $5.5\times 10^{11}$ $\text{cm}^{-2}$. The mean spacing of N atoms is 13.5 nm. The production efficiency of NV centers is $\sim 1\%$, and then the mean spacing of NV centers is $\sim 135$ nm. The mean depth of N atom is estimated to be $8\pm 3.2$ nm by SRIM, while the actual depth may be larger [35].

Since the NV and P1 centers are created by the implantation of N⁺ ions, we suppose both of them are located in a thin layer and only the transverse distance is considered. It is reasonable when the transverse distance is much larger than the longitudinal straggling. Here we consider the sum of the signal contributed by all the P1 centers out of the detection area, which can be given by (see Supplementary Note 1)

$$\begin{split}S^{\text{sum}}(r>r_{0})&=\sum_{r>r_{0}}\frac{C_{0}^{2}\langle\eta^{2}\rangle\tau}{8\Gamma_{\text{P1}}r^{6}}\\ &=\int_{r_{0}}^{\infty}\frac{C_{0}^{2}\langle\eta^{2}\rangle\tau}{8\Gamma_{\text{P1}}r^{6}}\cdot\sigma\cdot 2\pi rdr\\ &=\frac{\pi\sigma C_{0}^{2}\langle\eta^{2}\rangle\tau}{16\Gamma_{\text{P1}}r_{0}^{4}},\end{split}$$

(8)

where $\sigma=5.5\times 10^{11}$ $\text{cm}^{-2}$ is the area density of P1 centers, $C_{0}=2\pi\cdot 52$ MHz$\cdot$nm³ is the dipolar coupling constant, $\eta$ is the coefficient of dipole-dipole coupling depending on both the direction of the spatial separation $\mathbf{r}$ and the P1 center. After averaging over all the possible directions, $\langle\eta^{2}\rangle$ is calculated to be 5/4 or 3/4 corresponding to the left/right or middle peak in Fig. 3c. Other parameters in this experiment are $\Gamma_{\text{P1}}=10$ MHz and $t=10$ $\mu$s. Hence, the sum of signals contributed by those P1 centers outer the detection area decreases dramatically with the radius of the area. For example, if $r_{0}=15$ nm, then $S^{\text{sum}}$ can be calculated to be $0.29\%$, $0.17\%$ and $0.29\%$ for the left, middle and right peaks, respectively.

The Hamiltonian of NV center in zero magnetic field is

$$H_{0}=DS_{z}^{2},$$

(9)

where $\mathbf{S}(S=1)$ is the NV electron spin operators, $D=2\pi\times 2.87$ GHz is the zero-field splitting. The NV center can be driven by the microwave of the form

$$H_{1}=\Omega\cos(ft+\phi)S_{x},$$

(10)

where $\Omega$, $f$ and $\phi$ are the power, frequency and phase of the microwave respectively, $f=D$ in the resonance condition. In rotating reference frame, the Hamiltonian becomes

$$\begin{split}H^{\text{rot}}&=e^{iDS_{z}^{2}t}(H_{0}+H_{1})e^{-iDS_{z}^{2}t}-DS_{z}^{2}\\ &=\frac{\Omega}{2\sqrt{2}}\left(\begin{array}[]{ccc}0&e^{-i\phi}&0\\ e^{i\phi}&0&e^{i\phi}\\ 0&e^{-i\phi}&0\end{array}\right),\end{split}$$

(11)

where the high frequency items are ignored. By defining $\phi=0$, $\pi/2$ and $-\pi/2$ for $x$, $y$ and $-y$ phase, respectively, we can write

$$\begin{split}H_{x}^{\text{rot}}&=\frac{\Omega}{2}S_{x},\\ H_{y}^{\text{rot}}&=\frac{\Omega}{2}S_{yy},\\ H_{-y}^{\text{rot}}&=-\frac{\Omega}{2}S_{yy}.\end{split}$$

(12)

Note here

$$S_{yy}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-i&0\\ i&0&i\\ 0&-i&0\end{array}\right)$$

(13)

is slightly different from $S_{y}$. The eigenstates and eigenenergies of $H_{x}^{\text{rot}}$ can be directly calculated as given in the main text Eq. 1. The spin-locking sequence is performed as following:
(1) The initial state of the NV center is polarized to $|\psi_{0}\rangle=|0\rangle$ by a laser pulse. Then a $y$-phase $\pi/2$ pulse with power $\Omega_{0}$ and length $t_{\pi/2}=1/(4\Omega_{0})$ is applied and the NV state is prepared to

$$\begin{split}|\psi_{1}\rangle&=e^{-i2\pi H_{y}^{\text{rot}}t_{\pi/2}}|\psi_{0}\rangle\\ &=\frac{1}{2}|1\rangle-\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{2}|-1\rangle\equiv|-1\rangle_{\text{d}}.\end{split}$$

(14)

(2) A $x$-phase pulse with power $\Omega$ and length $\tau$ is applied. Since $|\psi_{1}\rangle$ is an eigenstate of $H_{x}^{\text{rot}}$, the state will be locked in $|\psi_{1}\rangle$. However, it can be transferred to other dressed states by $T_{1\rho}$ relaxation or coupling with nearby P1 centers. Supposing the transition probability is $P$, then the state becomes

$$\begin{split}\rho_{2}=&(1-P)|-1\rangle_{\text{d}}\langle-1|+\\ &P(p_{1}|0\rangle_{\text{d}}\langle 0|+p_{2}|1\rangle_{\text{d}}\langle 1|).\end{split}$$

(15)

where $p_{1}+p_{2}=1$.
(3) A $-y$-phase $\pi/2$ pulse with power $\Omega_{0}$ and length $t_{\pi/2}=1/(4\Omega_{0})$ is applied. The state $|-1\rangle_{\text{d}}$ will be rotated back to $|0\rangle$, while $|0\rangle_{\text{d}}$ remains unchanged and $|1\rangle_{\text{d}}$ will be rotated to $|0\rangle_{\text{d}}$. Therefore, the final state is

$$\begin{split}\rho_{3}=&(1-P)|0\rangle\langle 0|+\\ &P(|1\rangle\langle 1|-|1\rangle\langle-1|-|-1\rangle\langle 1|+|-1\rangle\langle-1|),\end{split}$$

(16)

with the normalized photoluminescence of $1-P$.

Data supporting the findings of this study are available within the article and its Supplementary Information file, and from the corresponding authors upon reasonable request.