Single spin magnetometry and relaxometry applied to antiferromagnetic materials Open Access

Imaging the antiferromagnetic order with nanoscale spatial resolution is a notoriously difficult task,¹ but it is also a general requirement to understand the physics of antiferromagnetic materials and to use them for spintronics.^2,3 Two approaches can be followed for magnetic imaging: directly probing the magnetization direction or probing the magnetic stray field generated by the material. Since this stray field is very weak in antiferromagnets, most of the available imaging techniques measure the magnetization direction. Optical techniques rely on birefringence,⁴ nonreciprocal reflection,⁵ nonlinear processes, such as second harmonic generation (SHG),⁶ or nonreciprocal directional dichroism.⁷ The spatial resolution of these optical methods is however limited by diffraction at the micrometer scale. Synchrotron x-ray-based techniques, such as photoelectron emission microscopy combined with x-ray magnetic linear dichroism (XMLD-PEEM), allow reaching nanoscale resolution,⁸ but remain scarcely employed owing to the required large scale experimental facilities. Scanning probe microscopies, such as spin-polarized scanning tunneling microscopy⁹ or magnetic exchange force microscopy,¹⁰ offer an ultimate atomic resolution for conducting and insulating materials, respectively. Nevertheless, these techniques are only suited for the study of nearly perfect surfaces in ultrahigh vacuum, thus precluding the study of technologically relevant antiferromagnetic materials.

In this review, we will present another technique that uses the second path: probing the weak stray field coming from antiferromagnetic textures with the help of a quantum sensor, the nitrogen-vacancy (NV) center in diamond. Single NV centers can be used to detect static magnetic fields with a sensitivity in the μT range at a distance of a few tens of nanometers from the sample when integrated in a scanning probe microscope. In the last few years, this technique was employed to study many phenomena in condensed matter physics,^11–13 including non-collinear magnetic states in ferromagnets,^14–18 superconducting vortices,^19,20 or spin waves.^21–23 In this review, we will focus only on the applications of sensing with single NV centers to antiferromagnetic materials, from the imaging of complex magnetic textures^24,25 to the investigation of intrinsic spin transport phenomena through magnetic fluctuations.^26,27 We will start with an explanation of the principle of magnetic field measurements with NV centers and some technical details about diamond probes in Sec. II, before reviewing applications of scanning NV magnetometry to the quantitative imaging of antiferromagnetic textures in Sec. III. We will then focus on relaxometry experiments, which rely on the detection of magnetic fluctuations (see Sec. IV). To conclude, we will highlight recent developments that provide significant improvement of the magnetic sensitivity through the conversion of field gradients into oscillating signals in Sec. V.

NV centers are point defects in the carbon lattice of diamond made of a substitutional nitrogen atom next to a vacancy [Fig. 1(a)]. In their negatively charged state NV⁻ (which will be the only charge state considered here), they exhibit a spin triplet ground state, which can be initialized by optical pumping, manipulated with long coherence times, and read out optically even at room temperature. These properties make them powerful quantum sensors.²⁸ The direction defined by the positions of the nitrogen atom and the vacancy defines a quantification axis for the electronic spin, which is designated as the NV axis [see Fig. 1(a)]. This axis can be oriented along four different directions, corresponding to the [111] crystallographic direction in the diamond lattice. NV-based sensing can be performed using ensembles of NV centers,²⁹ which are thus randomly oriented along the 4 equivalent directions, or single NV defects. In this review, we will focus only on the use of single NV spins for sensing the properties of antiferromagnets, and we will write as m_s the projection of the spin along the NV axis.

Under optical illumination, NV centers produce a stable spin-dependent photoluminescence, which is usually detected using a confocal microscope.³⁰ A stronger photoluminescence is coming from the system in the state m_s = 0 (bright state) than from the states m_s = ±1 (dark states)¹¹ [see Fig. 1(b)]. In addition, under continuous green laser illumination, optical pumping polarizes NV centers in the bright state m_s = 0. From these two properties, it is possible to optically measure the position of the electron spin resonance (ESR) frequencies of NV centers: continuous optical excitation polarizes the spin in the bright m_s = 0 state, and the application of a resonant microwave field drives the transition to the darker m_s = ±1 states. As a result, one gets the Optically Detected Magnetic Resonance (ODMR) spectrum presented in Fig. 1(c). In the absence of external magnetic field, the states m_s = 1 and m_s = −1 are degenerate, and a single dip is found in the spectrum. However, the internal electric field in the diamond usually introduces a small splitting,³¹ as visible in the red curve in Fig. 1(c).

When a magnetic field is applied to the NV center, the Zeeman effect splits the states m_s = −1 and m_s = +1, and two dips at f₋ and f₊ are present in the spectrum. In the low field regime (B < 5 mT), which is going to be the relevant one to study antiferromagnets, this splitting depends linearly on the B_NV component of the field along the NV axis.¹¹ 

In order to combine this high magnetic field sensitivity with nanoscale spatial resolution, a single NV center can be integrated at the apex of the tip of a scanning probe microscope.^33,34 An ODMR spectrum is then recorded at each pixel of the scan in order to extract the value of the local magnetic field. In the linear low field regime, measuring the Zeeman shift of a single magnetic resonance is sufficient. Historically, scanning NV microscopy was achieved by attaching a nanodiamond hosting a single NV center on a standard atomic force microscopy (AFM) tip.^34,35 However, this method is rather difficult to implement and only allows a low photoluminescence signal from the NV center to be collected. To circumvent these drawbacks, all-diamond probes have been developed,^36–38 providing reliable and now commercially available probes for scanning NV center magnetometry.

All-diamond probes for scanning NV center microscopy have been introduced about ten years ago³⁶ to overcome the limitations of the use of nanodiamonds attached to AFM tips. In particular, NV centers in nanodiamonds have short spin coherence time because of the low quality of the diamond matrix and the vicinity to the diamond surface. In addition, the collection of the photoluminescence emitted by NV centers in nanodiamonds is inefficient, yielding a small signal and thus reducing the magnetic field sensitivity [see Eq. (1)]. Finally, nanodiamonds can easily detach from the AFM tips, making the probes mechanically fragile. The solution, which was proposed, is to attach a diamond cantilever to a quartz capillary, which is itself glued to a tuning fork used as the AFM probe. The diamond cantilever possesses a small pillar, with a diameter on the range of 100 nm at the end of which a single NV center is implanted [see Fig. 2(a)]. The bottom of the diamond pillar is then placed very close to the sample to perform an AFM scan. These probes are fabricated from high purity diamond in order to ensure good spin coherence properties, and the pillar guides the light emitted by the NV center in order to improve the collection efficiency.³⁷ Recently, further improvement have been made with the design of parabolic pillars to further increase the light collection efficiency.³⁸ 

As mentioned in the previous section, the orientation of the NV center is a very important parameter to determine before doing any measurement, as only the component of the magnetic field along the NV axis is measured. The direction of the NV axis can be described with its polar axis θ and its azimuthal angle ϕ, as defined in Fig. 2(b). These angles can be measured independently by applying a calibrated magnetic field of fixed intensity but variable orientation to the NV center and monitoring the ODMR spectrum.¹¹ 

Diamond probes are usually made from (001) diamond, and as a result, NV centers point in a tilted direction, with θ ∼ 54° from the sample normal. A difficulty arises from the fact that the bias field applied during the measurement should be aligned as much as possible with the NV center axis to avoid electron spin mixing. Such a mixing of the spin states would indeed reduce both the photoluminescence signal and the ESR contrast.³⁹ Aligning the field with this tilted direction can be experimentally challenging and even lead to unexpected effects, as such a tilted magnetic field can perturb the sample behavior.⁴⁰ To overcome this issue, recent developments have been made to fabricate diamond probes from (111)⁴¹ and (110)⁴² diamond, providing NV centers oriented, respectively, along the normal of the sample surface or in-plane.

Finally, as we will show in detail in Sec. III, knowing the distance d_NV between the NV sensor and the sample surface is crucial to quantitatively analyze experimental data. In addition, in a scanning NV microscopy experiment, d_NV is the parameter that sets the spatial resolution. When studying antiferromagnetic textures, it is desirable to use diamond tips with as short as possible NV-to-sample distances, to be able to distinguish nanoscale features and to probe larger fields, as the stray field amplitude decreases fast with the distance to the sensor. Using commercial all-diamond probes, d_NV values of 50 nm can routinely be achieved. This value of 50 nm thus corresponds to the standard spatial resolution of the technique, for any measurement mode. It is limited by the implantation of the NV center itself, which cannot be located directly at the surface of the diamond without losing its coherence properties,⁴³ and by the potential roughness of diamond at the apex of the pillar.

This calibration method usually yields a typical uncertainty of about 10% on the parameter d_NV. This uncertainty on d_NV is the major source of error during the quantitative analysis of images, like those described in Sec. III.

Scanning NV microscopy offers the possibility to quantitatively measure magnetic fields with a nanoscale spatial resolution and to thus obtain maps of the stray field produced by various types of magnetic textures. Although a uniform and perfectly compensated antiferromagnet—akin to a completely uniform ferromagnet—does not produce magnetic stray field, the presence of non-collinear textures generates a weak stray field. In addition, it is not rare that antiferromagnetic samples exhibit an uncompensated magnetic moment, also leading to a stray field at edges or above non-collinear magnetic objects. Such small fields can be probed using standard NV magnetometry procedures as presented in Sec. II A. The uncompensated moment can result either from a A-type antiferromagnetic state, like, for example, in Cr₂O₃,⁴⁵ or from a canting of the magnetic moments induced by the presence of a Dzyaloshinskii–Moriya interaction (DMI) like in α-Fe₂O₃,⁴⁶ BiFeO₃,⁴⁷ or Mn₃Sn.⁴⁸ 

In A-type antiferromagnets, when the magnetic moments are arranged ferromagnetically in each plane parallel to the surface, the last layer of magnetic moments can be uncompensated and therefore result in the presence of a detectable stray field a few tens of nanometers away from the surface. So far, Cr₂O₃ has been the most studied material of this type with NV center magnetometry, but similar effects are found in other compounds, and in particular bidimensional van der Waals magnets like CrI₃.⁴⁹ 

Cr₂O₃ is a magnetoelectric antiferromagnet, therefore, attracting attention for potential applications like the design of electrically controlled memory elements.²⁵ In (0001) oriented Cr₂O₃, below the ordering temperature $TNéel=308K$⁠, the symmetry breaking at the surface leads to a surface magnetic moment with a direction linked to the Néel vector below, enabling the NV imaging of the antiferromagnetic domains through the resulting stray field.^50–54 This uncompensated surface magnetic moment is normal to the surface plane, similarly to the Néel vector.

The morphology of the antiferromagnetic domains in this material strongly depends on the nature of the sample: granular thin films exhibit irregular domains with a size of a few hundreds of nanometers whereas single crystals host large micrometer size domains. The first magnetic images of Cr₂O₃ were obtained on the latter in 1995 by second harmonic generation (SHG).⁶ Besides scanning NV magnetometry, scanning probe techniques, such as Magnetic Force Microscopy (MFM)⁵⁵ and Magnetoelectric Force Microscopy (MeFM),⁵⁶ have been used to image Cr₂O₃ thin films and single crystals, respectively, as well as SHG,⁶ other optical techniques⁵⁷ and photoelectron emission microscopy combined with X-ray Magnetic Circular Dichroism (XMCD-PEEM).⁵⁵ However, none of these techniques can provide both the nanoscale spatial resolution and high sensitivity offered by scanning NV center magnetometry, together with the ability to perform quantitative measurements.

A significant amount of quantitative information can be extracted from scanning NV center microscopy experiments on Cr₂O₃. In thin films, the stray field maps allow the determination of the size of the antiferromagnetic domains, demonstrating the presence of intergranular coupling as the domains always comprise several structural grains.⁵⁰ In addition, looking at strained films revealed an unconventional flexomagnetic effect, as the strain gradient induces a modification of the Néel temperature.⁵⁴ This has been measured through the evolution of the stray field above the films at various temperatures across the transition. The surface magnetic moment density pattern can also be retrieved from stray field maps using Fourier reverse propagation methods.⁵⁸ This analysis technique is explained in Sec. III B 2.

In the case of single crystals, the images are very different as only isolated domain walls separating micrometer size domains are present [see Fig. 3(a)]. This situation offers the opportunity to study the domain walls themselves, as it can be done in ferromagnets,^15,16 and specifically the path they follow,⁵¹ and it gives hints about their morphology.^51,52 Establishing whether the domain walls are of Bloch or Néel type and determining their width is relevant for fundamental reasons, as it gives insight into the balance between the different magnetic energy contributions in the material, but also in the view of applications, since Bloch and Néel walls do not exhibit the same response to current-induced spin torques. This information could be extracted from stray field profiles measured across the domain walls, by fitting the experimental NV magnetometry data to analytical expressions of the stray field, as shown in Fig. 3(b). In Cr₂O₃, however, it requires a very good accuracy of the measurement, as Bloch and Néel profiles are hardly distinguishable.^51,52 A detailed discussion about the quantitative analysis of stray field maps from analytical models can be found in Sec. III B 1.

Synthetic antiferromagnets (SAF),⁵⁹ even if they possess a fully compensated magnetic moment, can also be probed similarly, as some magnetic stray field is always present above the sample surface. This happens because the distance between each ferromagnetic layer and the NV sensor is different. As a result, even if the total magnetic moment in the sample cancels, the stray fields from the different layers at the sensor position do not, enabling scanning NV microscopy imaging⁶⁰ while MFM experiments can be challenging for a two-layers SAF.⁶¹ 

Another mechanism leading to the presence of uncompensated magnetic moments in antiferromagnets is a DMI-induced canting, which is the origin of the weak ferromagnetism observed in many antiferromagnetic oxides, such as α-Fe₂O₃⁴⁶ or BiFeO₃.⁴⁷ We will focus here on BiFeO₃, which is the first canted antiferromagnet that has been studied with scanning NV center magnetometry,²⁴ but other materials can be imaged using NV centers (α-Fe₂O₃,^62–64 Mn₃Sn,^65,66 etc.). In particular, very recent studies demonstrated direct observation of electrically driven domain switching in α-Fe₂O₃⁶² and Mn₃Sn⁶⁶ using quantitative stray field maps measured with scanning NV magnetometry.

BiFeO₃ is multiferroic at room temperature and exhibits a G-type antiferromagnetic order with a cycloidal incommensurate modulation, which is coupled to the ferroelectric polarization $P⃗$⁠.⁶⁷ In bulk BiFeO₃, the period of the cycloid is about 64 nm and its propagation vector lies in the plane perpendicular to $P⃗$⁠. However, this cycloidal modulation alone does not result in the creation of an uncompensated magnetic moment. The weak ferromagnetism in BiFeO₃ comes from an additional DMI-induced effect that tilts the magnetic moments out of the cycloid plane.⁴⁷ This canting can be described as a spin density wave, which is locked onto the cycloid and therefore provides a way to image it owing to the stray field that it generates.⁶⁸ Scanning NV center magnetometry is so far the only technique capable of imaging the cycloid in BiFeO₃ in real space, which is shown in Fig. 3(c).

BiFeO₃ thin films are under intense study, as multiferroic materials at room temperature are especially rare. Imaging the cycloid directly with NV magnetometry leads to a clear demonstration that its propagation direction is coupled to $P⃗$ and can therefore be manipulated electrically.²⁴ Further investigations about the effect of epitaxial strain on BiFeO₃ thin films helped to refine the phase diagram of BiFeO₃ under strain and confirmed that an exotic cycloid type is stabilized when the strain strengthens.^69,70 Increasing the strain further destroys the cycloid and results in antiferromagnetic domains seen in SHG,⁷¹ which also possess a small DMI-induced uncompensated moment and could thus be observed as well with scanning NV center magnetometry.⁶⁹ 

In addition, the ability to image directly the non-collinear magnetic texture of BiFeO₃ at the nanoscale allowed the discovery of exotic antiferromagnetic objects. At chiral ferroelectric domain walls between regions where the cycloid propagates in different directions, some magnetic hotspots appear in the stitching of the cycloidal pattern.⁷² These can be seen as the embryos of topological multi-q states. In bulk BiFeO₃ single crystals, it is also possible to observe the antiferromagnetic cycloid and scanning NV center magnetometry revealed that, at the crystal surface, the direction of its wavevector rotates almost freely in the plane perpendicular to $P⃗$⁠. As a result, some defects form in the cycloidal structure, which can be described as topological defects typically found in lamellar ordered media.⁶⁸ 

Finally, a quantitative analysis of the stray field maps can also be performed when studying the antiferromagnetic cycloid in BiFeO₃ since the field originating from the cycloid and the spin density wave can be analytically computed. Owing to this type of analysis, one can, in particular, quantitatively extract the amplitude of the spin density wave. This topic is further discussed in the next section.

Scanning NV center magnetometry provides a quantitative map of the magnetic field component along the NV axis (see Fig. 1) at a fixed distance d_NV from the sample surface. In order to extract relevant information, such as the magnetization intensity, from these data, a careful calibration of the sensor is required. First, the orientation (θ, ϕ) of the NV axis should be determined, ideally using a vector magnet to sweep the direction of an external magnetic field of fixed intensity while monitoring the ODMR spectrum. Then, the distance between the NV center and the surface d_NV has to be measured, usually by recording a stray field profile over a well-known sample like a ferromagnetic stripe.⁴⁴ However, such a calibration is not sufficient to unambiguously derive the magnetization distribution at the source of the measured stray field since there is no bijective correspondence between them. Interpreting the stray field map thus requires to make some assumptions about the magnetization configuration. Two approaches can be followed to analyze the data: comparing them to a model or reconstructing the magnetization pattern using reverse propagation methods.⁵⁸ 

When comparing stray field data with a model, there are two options. First, if a specific magnetic configuration is expected or if the magnetic system is known well enough to perform micromagnetic or atomistic simulations, a magnetic state can be predicted. From this prediction, it is then possible to compute the corresponding field map at a distance d_NV from the surface and project it along the direction of the NV axis. The resulting image can be directly compared with the experimental data, as it was done to identify the chiral magnetic objects forming in the cycloid stitching at ferroelectric domain walls in BiFeO₃.⁷² This is the most generic approach, but the agreement between the model and the data will often be rather qualitative.

The other option is to use an analytical model of the stray field produced by the magnetic object of interest and fit it to the experiment in order to extract relevant parameters, such as the magnetization amplitude or the size of the object. This is the most accurate method to retrieve quantitative information from scanning NV center magnetometry measurements, but it is limited to specific “simple” cases, such as step edges, domain walls between out-of-plane oriented domains or homogeneous spirals, or spin density waves, which are described with a sine function. We will detail two examples below: a domain wall in a Cr₂O₃ single crystal and the cycloid in a low strained BiFeO₃ thin film (see Fig. 3).

The error on the fitted parameters can be quite large. This is related to the imperfect knowledge of the distance d_NV between the NV center and the surface.

Alternatively, based on assumptions on the magnetization distribution, it is possible to reverse propagate the measured magnetic field map to obtain a map of the magnetization density.⁵⁸ This method is well-suited for ferromagnets but can also be applied to antiferromagnets with uncompensated surface moments, such as Cr₂O₃⁵⁰ or the van der Waals magnet CrI₃.⁴⁹ The principle of this method is to invert the procedure used in the previous section to compute the field from the magnetization distribution, also by going to Fourier space. However, this procedure cannot be properly inverted, and therefore the reverse propagation approach requires some strong assumptions on the magnetic state of the sample: the magnetization $m⃗$ has to be confined to a plane, so it should depend only on x and y and its direction has to be assumed constant in the whole sample. Non-collinear magnetic textures cannot be retrieved from this method.

The best performances are obtained for out-of-plane magnetized samples, like on the illustration in Fig. 4(a). With in-plane magnetized samples, specific attention has to be given to noise in the experimental data, as it is systematically amplified during the calculation.⁵⁸ The reverse propagation method was successfully applied to compute the surface magnetization of Cr₂O₃, as well as to get the domain pattern in granular thin films.⁵⁰ It has also been widely used in the study of van der Waals magnets.^49,75,76

To avoid numerical artifacts that arise from the direct inversion of the problem and get better results with in-plane magnetized samples, it has been proposed to use a neural network to compute the magnetization map.⁷⁷ A diagram representing the procedure is shown in Fig. 4(b). Here, the network is not trained on a dataset, it rather minimizes an error function. This error function is obtained by computing the stray field expected from the output magnetization map and comparing it to the experimental data. This method has been successfully applied to in-plane magnetized samples but is still limited to samples with a fixed magnetization direction so far.

This property indicates that it is possible to localize sources of magnetic noise simply by monitoring the photoluminescence emitted by the NV center during a scan. An illustration of this is shown in Fig. 5(a). It shows a photoluminescence map recorded above a synthetic antiferromagnet, a metallic multilayer stack including two ferromagnetic Co layers that are antiferromagnetically coupled through a Ru/Pt spacer.⁶¹ Dark lines are visible on the image at the position of the domain walls in the magnetic layer, indicating an acceleration of the NV center relaxation. This is further confirmed by direct measurements of T₁ above the domains and above the domain walls, showing a significant drop, from about 120 to about 20 µs between both positions.⁶⁰ In addition, the observed photoluminescence contrast depends on the optical excitation power, which strengthens the NV center polarization in the bright state. The imaging contrast can thus be optimized by tuning the optical excitation power used.

The origin of the measured magnetic noise is the presence of thermally activated spin waves in the synthetic antiferromagnet, with frequencies in the vicinity of f_±. Calculations of the dispersion of spin waves in this sample are shown in Figs. 5(b) and 5(c). Panel (b) presents the spin wave dispersion in the domains, exhibiting a gap slightly larger than f_±. In contrast, the dispersion of spin waves channeled inside the domain walls is gapless, as displayed in panel (c). There are thus thermally activated modes at f_± inside the domain walls that create a magnetic noise probed by the NV center. This noise accelerates the NV center relaxation, resulting in the observed drop of photoluminescence.

The closing of the gap in the spin wave dispersion at a domain wall is a general phenomenon resulting from the Goldstone theorem,⁸⁰ which was predicted in ferromagnets⁸¹ and antiferromagnets⁸² and also measured using scanning transmission x-ray microscopy in synthetic antiferromagnets.⁸³ As a result, even though spin waves have usually much higher frequencies in antiferromagnets, rather in the THz range, which is not accessible directly with NV centers, noise from spin waves confined inside domain walls should be detectable in any antiferromagnetic material.

Besides magnetic imaging, NV-center-based relaxometry can also be used to probe other properties of antiferromagnets, and in particular spin transport, based on the fluctuation–dissipation theorem.²⁶ NV centers can probe non-perturbatively the fluctuations of the spin density along the Néel order parameter and thus allow the extraction of the intrinsic spin diffusion constant. An experimental demonstration of this method has been reported in α-Fe₂O₃²⁷ (see Fig. 6). While the imaging technique presented in Sec. IV A relies on fluctuations of the transverse spin density s_⊥ [see Fig. 6(a)], which are related to one-magnon scattering processes, here only the longitudinal fluctuations s_‖ are probed. Note that in this case we refer to the directions indicated in Fig. 6(a), and the reference direction is the one of the Néel vector, not the NV center axis. The longitudinal fluctuations, and the resulting magnetic noise, correspond to two-magnon scattering processes, where the frequency difference f_2m between the two magnons matches one of the magnetic resonance frequencies f_± of the NV center [see Fig. 6(b)]. As a result, the frequency of the longitudinal spin noise can be arbitrarily small and does not depend on the magnon bandgap. It can therefore be probed in the low GHz range using NV centers. Here, single NV centers inside a diamond nanobeam placed on the surface of the α-Fe₂O₃ sample are used, and the sensor is thus stationary and probes a specific location on the sample.

By measuring the evolution of the relaxation rate Γ of the NV center with the ESR frequency, i.e., under application of an external magnetic field, it is therefore possible to extract the value of D [see Figs. 6(c) and 6(d)], (6.6 ± 0.4) × 10⁶ m²/s at 300 K. This analysis was repeated at various temperatures, crossing at T_M the Morin transition between an out-of-plane antiferromagnetic order below T_M and an in-plane canted antiferromagnetic order above. The magnon gap is well above the NV detection frequencies on both sides of the transition, meaning that the analysis remains valid when changing the temperature. The data, presented in Fig. 6(d), show a smooth decrease in D with temperature, without any significant feature at T_M. This decrease in D is related to the increase in the inelastic scattering rate with temperature. The spin transport in equilibrium is mainly driven by thermal magnons governed by the exchange interaction. The changes in anisotropy and DMI occurring at the Morin transition correspond to small corrections and are therefore not significantly affecting D across the transition.²⁷ 

This type of experiments is especially interesting because they do not involve any currents or thermal gradients that can perturb the sample and be modified by interface effects. Probing spin fluctuations with NV centers gives thus access to fully intrinsic properties that can hardly be reached otherwise. Similar relaxometry experiments can also be achieved in a scanning probe microscope, offering the additional possibility to tune the distance d_NV between the NV center and the surface of the magnetic sample. Measuring the NV center relaxation rate as a function of d_NV should also provide insight into the spin diffusion length, without the need to apply a varying magnetic field during the experiment.²⁶ 

In order to increase the sensitivity of the quantitative measurement presented in Sec. III, one can use sequences of laser and microwave pulses to avoid power broadening of the ODMR spectra.⁸⁴ This measurement protocol called pulsed ODMR provides an improvement up to one order of magnitude in magnetic field sensitivity compared to continuous ODMR and can easily be implemented in a scanning NV microscope.¹¹ It is also very helpful when working at cryogenic temperatures since it reduces the heating induced by the laser and microwave powers. In this context, it has recently successfully be applied to the imaging of ferromagnetic domains in the van der Waals magnet CrBr₃.⁷⁵ 

Another more efficient solution to increase the sensitivity of the measurement with the goal of imaging fully compensated antiferromagnets is to use AC sensing.^85,86 When probing oscillating magnetic fields, dynamical decoupling protocols offer the possibility to go beyond the sensitivity offered by pulsed ODMR,³² down to $50nT/Hz$⁠. However, very few systems produce a field that can be periodically modulated. A method to circumvent this issue was proposed^87,88 and recently demonstrated on antiferromagnets by Huxter et al.:⁸⁹ convert spatial gradients of the magnetic field into an oscillating signal. This conversion is performed using the oscillating movement of the tuning fork which provides feedback for the AFM. The resonance frequency f_TF of these tuning forks lies around 32 kHz. The NV center is moving in a plane parallel to the sample surface, and the movement amplitude x_osc is fixed and can be controlled by adjusting the input excitation signal sent to the tuning fork, in a range between 10 and 70 nm. In the presence of a spatial gradient of magnetic field, the NV center experiences a periodically oscillating magnetic signal, which amplitude is the magnetic field gradient itself. The scan height d_NV is thus fixed, meaning that this oscillation does not affects the spatial resolution of the measurement.

First proof-of-principle experiments have been performed on the (0001) surface of a Cr₂O₃ single crystal,⁸⁹ and the obtained gradient image is shown in Fig. 7(b). This image reveals a regular pattern of stripes that correspond to the stray field gradient generated by atomic step edges. Correlations of gradient and topography measurements even allow the identification of single and double atomic steps at the surface of the crystal. Such a measurement is reminiscent of images of the A-type antiferromagnetic state of a Cr(0001) single crystal imaged using spin-polarized scanning tunneling microscopy.⁹⁰ Scanning NV gradiometry can thus provide access to atomic scale antiferromagnetism, which was so far restricted to spin-polarized scanning tunneling microscopy⁹¹ or magnetic exchange force microscopy,¹⁰ techniques requiring ultra-high vacuum and mainly cryogenic conditions.

Finally, the gradiometry technique is not limited to the measurement of magnetic fields. NV centers are also sensitive to electric field via the Stark effect,⁹² but their electric field sensitivity is much smaller than magnetic sensitivity. The sensitivity increase provided by the gradiometry approach, as well as its insensitivity to static parasitic fields, is an efficient technical solution. First measurements on patterned micrometer-sized electrodes⁹³ and more recently on ferroelectric domains⁹⁴ have shown that scanning NV electrometry can be achieved by gradiometry and open the perspective of combined non-perturbative nanoscale imaging of the antiferromagnetic and the ferroelectric order in multiferroics.

Within less than a decade, single NV centers have become an essential tool for probing antiferromagnetism at the nanoscale. Owing to its high magnetic sensitivity and its operation under ambient conditions, scanning NV center magnetometry is now routinely being used to image the magnetic state of many antiferromagnetic materials, from oxides, such as Cr₂O₃,^25,50–54 α-Fe₂O₃,^62–64 BiFeO₃^{24,68–70,72} to synthetic,⁶⁰ non-collinear^65,66 or van der Waals⁴⁹ antiferromagnets. We have reviewed here the main requirements to perform these measurements and analyze the obtained data, highlighting the need to carefully calibrate the NV sensor in order to extract quantitative information about the magnetization configuration. We also provide here a table gathering the three main measuring mode that we presented with their specifications (see Table I).

Recent developments relying on the detection of spin fluctuations open further perspectives to measure non-invasively spin waves⁶⁰ and spin transport properties,²⁷ suggesting that NV centers will also be of great help in the future to address antiferromagnetic magnonics. Finally, integrating AC sensing inside scanning NV center microscopy⁸⁹ leads to a significant improvement of the magnetic sensitivity, which will enable the imaging of perfectly compensated antiferromagnetic textures, so far inaccessible. This approach also opens the way to scanning NV center electrometry⁹⁴ for the imaging of ferroelectric states and further beyond to combined measurements of antiferromagnetic and ferroelectric textures at the nanoscale in multiferroics.

We are thankful for support from the French Agence Nationale de la Recherche (ANR) through the project TATOO (Grant No. ANR-21-CE09-0033-01) and the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement Nos. 964931 (TSAR) and 866267 (EXAFONIS).

The authors have no conflicts to disclose.

Aurore Finco: Conceptualization (equal); Writing – original draft (lead). Vincent Jacques: Conceptualization (equal); Writing – review & editing (lead).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.