Stray-field imaging of magnetic vortices with a single diamond spin

L. Rondin1,†, J.-P. Tetienne1,2,†, S. Rohart3, A. Thiaville3, T. Hingant2, P. Spinicelli1, J.-F. Roch2, and V. Jacques1,2∗1Laboratoire de Photonique Quantique et Moléculaire,

Ecole Normale Supérieure de Cachan and CNRS UMR 8537, 94235 Cachan Cedex, France2Laboratoire Aimé Cotton, CNRS, Université Paris-Sud and ENS Cachan, 91405 Orsay, France. and

3Laboratoire de Physique des Solides, Université Paris-Sud and CNRS UMR 8502, 91405 Orsay, France†

Despite decades of advances in magnetic imaging, obtaining direct, quantitative information withnanometer scale spatial resolution remains an outstanding challenge. Recently, a new techniquehas emerged that employs a single nitrogen-vacancy (NV) defect in diamond as an atomic-sizemagnetometer. Although NV magnetometry promises significant advances in magnetic imaging,the effectiveness of the technique, when applied to realistic magnetic nanostructures, remains to bedemonstrated. Here we use a scanning NV magnetometer to image a magnetic vortex, which is oneof the most iconic object of nanomagnetism, owing to the small size (∼ 10 nm) of the vortex core.We report three-dimensional, vectorial, and quantitative measurements of the stray magnetic fieldemitted by a vortex in a ferromagnetic square dot, including the detection of the vortex core. Wefind excellent agreement with micromagnetic simulations, both for regular vortex structures and forhigher-order magnetization states. These experiments establish scanning NV magnetometry as apractical and unique tool for fundamental studies in nanomagnetism.

Introduction. Although a remarkable number of mag-netic microscopy techniques have been developed overthe last decades, imaging magnetism at the nanoscaleremains a challenging task since it requires a combi-nation of high spatial resolution and sensitivity1. Afirst approach consists in directly mapping the sam-ple magnetization, which implies sending and collect-ing back test particles whose interaction with matterhas a magnetization-dependent term. This approachprovides the highest spatial resolution to date, downto the atomic-scale for spin-polarized scanning tunnel-ing microscopy2 and about 10 nm in transmission X-raymicroscopy3. However, these techniques require highlycomplex experimental apparatus and a dedicated samplepreparation so that the particles can reach and escapefrom the tested region without perturbations. To ob-serve magnetic samples in their real, unprepared state,a more suited approach consists in mapping the mag-netic stray field generated outside the sample, even ifthis method cannot uniquely determine the actual sam-ple magnetization4. Furthermore, the spatial resolutionis then limited both by the probe size and its distanceto the sample. Among many stray field microscopy tech-niques5, magnetic force microscopy (MFM) has becomeubiquitous, as it provides a spatial resolution below 50nm6 and operates under ambient conditions without anyspecific sample preparation. It was for instance the firstmethod that allowed the observation of the core of a mag-netic vortex in a thin ferromagnetic disk7. However, dueto the intrinsic magnetic nature of the probe, MFM isknown to be perturbative and not easily quantitative8,therefore limiting its field of applications.

Recently, a magnetometer based on the magnetic re-sponse of a single nitrogen-vacancy (NV) defect in dia-mond has been proposed9–12, which promises significantadvances in magnetic imaging. Indeed, it provides non-perturbing and quantitative measurements of the straymagnetic field, with an unprecedented access to low fields

combined with an atomic-sized detection volume13,14.In this letter, we use NV-based magnetometry to mea-sure the stray field emanating from magnetic vorticesin nanostructured ferromagnetic thin films. Such struc-tures, which are of great interest both for fundamentalstudies in nanomagnetism15 and for applications such asnon-volatile magnetic storage16,17 and microwave gener-ation18, have been investigated using a wide range of mi-croscopy techniques2,7,19–22. However, obtaining quanti-tative information that is directly comparable to theoryremains a challenging task. Further, imaging the vortexcore, which can be as small as 10 nm and plays a crucialrole in the vortex dynamics, is a long-standing goal thathas been reached by very few methods only2,3,7. Here weshow that scanning NV magnetometry enables to quan-titatively map the stray field above a thin ferromagneticsquare in a vortex state, revealing in 3D the full structureof the magnetic field distribution, including the detectionof the vortex core. Furthermore, we demonstrate thatthe vectorial and quantitative nature of the measurementprovide direct comparisons with micromagnetic simula-tions. This work thus opens new avenues for fundamentalstudies in nanomagnetism and spintronics.

Principle of scanning NV magnetometry. Thescanning NV magnetometer combines an optical confocalmicroscope and an atomic force microscope (AFM), alloperating under ambient conditions (see Methods). Assketched in Figure 1(a), a diamond nanocrystal hostinga single NV defect is grafted at the apex of the AFMtip and used as an atomic-sized magnetic sensor13.The NV defect ground state is a spin triplet with azero-field splitting D = 2.87 GHz between a singlet statems = 0 and a doublet ms = ±1, where ms denotesthe spin projection along the intrinsic quantizationaxis of the NV defect uNV (Fig. 1(a)). Under opticalillumination, the NV defect is efficiently polarized intothe ms = 0 spin sublevel and exhibits a spin-dependentphotoluminescence (PL)23. These combined properties

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Figure 1. Principle of the experiment. (a)-The apex of anAFM tip is functionalized with a 20-nm diamond nanocrystalhosting a single NV defect. A microscope objective placed ontop of the AFM tip is used both for exciting and collectingthe NV defect spin-dependent PL. A microwave (MW) fieldis generated by an antenna approached in the vicinity of theNV defect. The NV defect quantization axis uNV, which givesthe measured magnetic field component, is described by theangles θ and ϕ in a spherical coordinate system. (b)-ESRspectrum of the magnetic probe recorded by monitoring theNV defect PL intensity while sweeping the frequency of theMW field. The ESR splitting is proportional to the projectionof the magnetic field along the NV axis, here BNV = 0.9 mT.(c)-Typical AFM image of a thin square of Fe20Ni80. Thewhite arrows indicate the curling magnetization of the vortexstate. Scale bar, 1 µm.

enable the detection of electron spin resonance (ESR)by optical means24. A typical ESR spectrum of the NVsensor placed in a static magnetic field is shown in Fig-ure 1(b). In the limit of weak magnetic fields (< 3 mT),the spin quantization axis remains fixed by the NVdefect axis, and the two ESR frequencies are given byνR = D ± γeBNV/2π, where BNV is the magnetic fieldprojection along the NV axis and γe is the electrongyromagnetic ratio (Fig. 1(b)). Note that strain-inducedsplitting of the ms = ±1 spin sublevels has been omittedfor clarity purpose (see Supplementary Information).Measurement of the magnetic field through Zeemanshifts of the ESR frequency is therefore quantitative andnon perturbing, since the dipolar field from a single NVdefect is as low as 1 µT at 10 nm distance9.

Stray field imaging of magnetic vortices inferromagnetic films. In the following, we use thescanning NV magnetometer to map the magnetic strayfield generated by magnetic vortices in ferromagneticthin films. More precisely, we investigate squares ofFe20Ni80 with a thickness of 50 nm and a 5 µm sidelength. This magnetic structure is characterized by acurling in-plane magnetization with a vortex core in thecenter where the magnetization points out of the plane

(Fig. 1(c)). Such a magnetic vortex is an ideal object forevaluating the sensitivity and spatial resolution of anyadvanced magnetic microscopy technique since the sizeof the vortex core can reach 10 nm [3]. Furthermore thismagnetic structure is topologically stable and thereforeappears as an interesting candidate for memory cells innon-volatile data-storage devices16,17.

The NV probe is first approached at a distanced ≈ 300 nm above the magnetic vortex structure.Magnetic field mapping is performed by scanning thesample while measuring the difference of NV defectPL intensity for two fixed microwave (MW) frequen-cies ν1 and ν2, applied consecutively at each pointof the scan13. This signal is positive if νR = ν1, i.e.when the local field experienced by the NV defect isBNV,1 = ±2π(ν1 − D)/γe, and negative if νR = ν2, i.e.for BNV,2 = ±2π(ν2 −D)/γe (see Supplementary Infor-mation). The resulting image thus exhibits positive andnegative signal regions corresponding to iso-magneticfield contours (BNV,1,BNV,2), as well as zero-signalregions for any other field projections. An example ofsuch a dual-iso-B image recorded above a ferromag-netic square is presented in Figure 2(a), revealing aflower-shaped magnetic field distribution. Independentmeasurement of the NV defect orientation with acalibrated magnetic field indicates that the NV defect isoriented mainly along the y axis in this experiment (seeSupplementary Information). Hence the magnetic mapshown in Figure 2(a) corresponds to the field component

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Figure 2. Stray field imaging of a square of Fe20Ni80.(a),(b)-Dual-iso-B images recorded with a probe-to-sampledistance d = 400 nm in (a) and d = 300 nm in (b). The redsquares depict the 5 µm magnetic structure. In (a), the NVis oriented mainly along the y-axis (θ = 85◦, ϕ = 85◦) and in(b) the NV defect has a strong component along the z-axis(θ = 37◦, ϕ = 135◦). Images correspond to 200× 200 pixels,with a 30-nm pixel size and a 60 ms acquisition time perpixel. (c),(d)-Simulated magnetic field maps correspondingto (a) and (b), respectively.

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that is parallel to the sample surface and along a sideof the ferromagnetic square. By properly selecting theorientation of the single NV defect grafted at the apexof the AFM tip, any field component can be similarlymeasured. For instance, Figure 2(b) is essentially a mapof the z-component, i.e. the out-of-plane magnetic field.These experiments establish NV-based magnetometry asa unique instrument providing quantitative and vectorialmagnetic field images at the nanoscale.

The measured stray field arises from Néel domainwalls at the square diagonals which induce volumecharges with opposite sign on each side of the wall. Thegeneral lobe structure of the magnetic distribution cantherefore be qualitatively understood by consideringthe magnetic field created by an assembly of magneticdipoles placed along the diagonals. To get more precisepredictions, the sample magnetization distribution wasfirst calculated through micromagnetic simulations usingOOMMF software25, with a cell size of 5 × 5 × 5 nm3.Once the equilibrium magnetization state found, themagnetic field generated by the structure is computedby summing the contribution of all magnetizationcells. This magnetic field is then projected along theexperimentally measured NV axis, and the NV defectESR response is finally applied to get the simulateddual-iso-B image (see Supplementary Information). Asshown in Figure 2(c)&(d), the simulations reproducewell the experimental data, including apparent asym-metries which stem from the imperfect alignment ofthe corresponding NV defects with respect to the yand z axis. By varying the probe-to-sample distanced, it is even possible, as shown in the SupplementaryInformation, to obtain an overview of the stray field inthe half-space above the sample, thus providing a 3Dmapping of the magnetic field distribution.

Imaging the vortex core. The magnetic imagesshown in Fig. 2(a)&(b) are recorded with a probe-to-sample distance of several hundreds of nanometers. Atsuch distances, the stray field linked to the out-of-planevortex core magnetization cannot be detected. Thevortex center therefore appears dark (zero field) bothin the images and in the simulations. With the aim ofobserving the vortex core, the NV probe is brought asclose as possible to the sample with the AFM operatingin tapping mode. Figure 3(a) shows the dual-iso-B imageof the whole square obtained with the NV defect probeused in Fig. 2(b), i.e. mainly oriented along the z axis.By zooming in the center of the structure, the vortexcore is revealed with an apparent size Lcore ≈ 100 nm(Fig. 3(b)). Although the spatial resolution of scanningNV magnetometry is ultimately given by the atomic-sized detection volume, the effective resolving poweris rather limited by the probe-to-sample distance, acommon feature of any stray field microscopy technique.Indeed, numerical calculations indicate that the size ofthe out-of-plane vortex core magnetization is around 20nm at the sample surface. The probe-to-sample distance

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Figure 3. Observation of the vortex core. (a)-Dual-iso-Bimage of the whole square obtained with the AFM operatingin tapping mode. The same NV probe as in Fig. 2(b) is used(θ = 37◦, ϕ = 135◦). Scale bar, 1 µm. (b)-Zoom into thecenter of the structure enabling the detection of the stray fieldemanating from the vortex core magnetization. The imagecorresponds to 110× 110 pixels, with a 4.7 nm pixel size anda 60 ms acquisition time per pixel. Scale bar, 100 nm. (c),(d)-Simulated magnetic field maps corresponding to (a) and (b),respectively, with a probe-to-sample distance d = 100 nm.(e)-Line-cut taken along the red dashed line in (b) (markers)and (d) (solid line). Scale bar, 50 nm. (f),(g)-Dual-iso-Bimage of the square center while applying a bias field witha projection along the NV axis Bext = −0.8 mT in (f) andBext = +0.8 mT in (g). Scale bar, 500 nm.

can be estimated by comparing the experimental resultsto magnetic images simulated at different distances fromthe sample surface. As shown in Figures 3(c),(d)&(e), agood agreement is obtained for a distance d ≈ 100 nm.We attribute this relatively large value to an imperfectpositioning of the diamond nanocrystal at the apex

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of the AFM tip. A more precise control of the NVmagnetic sensor position could be achieved by usingdiamond nanopillar-probes14. Although the simulationfairly agrees with experimental data, we note thatthe sharp structures lying along the square diagonals(Fig. 3(c)) are not observed in the experimental images(Fig. 3(a)). In those regions, the stray field generated byNéel domain walls is strong enough to induce a mixing ofthe NV defect electron spin sublevels which results in anoverall reduction of ESR contrast26 (see SupplementaryInformation).

Chirality and polarity of the vortex state. Thevortex structure is commonly characterized by two inde-pendent binary properties. The chirality c determineswhether the in-plane magnetization is curling clockwise(c = +1) or counterclockwise (c = −1), while thepolarity p of the vortex indicates the upward (p = +1)or downward (p = −1) magnetization of the core. Theapparent pairing of the zero-field contours around thevortex core observed in Figure 3(b) directly enables us todetermine that c× p = +1 (see Supplementary Informa-tion). We note that only an absolute value of the field ismeasured, preventing a direct measurement of the corepolarity without additional measurements. This limi-tation could be overcome by addressing selectively oneof the two ESR transitions through circularly polarizedMW excitation27. Here, we gain further insights into thevortex structure by applying an external bias magneticfield along the z-axis with a projection Bext along theNV axis. The amplitude of this external magnetic fieldis chosen weak enough to avoid modifying the samplemagnetization, so that Bext just adds up to the regularstray field of the vortex structure. Figures 3(f)&(g)show magnetic images obtained with Bext = −0.8 mTand Bext = +0.8 mT, respectively. The stray fieldof the vortex core turns dark – the net magnetic fieldvanishes – with the negative bias field, which reveals thatthe vortex core magnetization points upward p = +1,and indicates a stray field emanating from the coreBcore ≈ +0.8 mT. Conversely, the core field is increasedwith a positive bias field since it adds up to Bcore. Fromthis polarity measurement, the chirality c = +1 canfinally be deduced leading to a full characterization ofthe vortex structure.

Imaging higher-order magnetization distribution.As a final experiment, we study a higher-order mag-netization distribution on a Fe20Ni80 square dot. Asshown in Figure 4(a), highly non trivial magnetic fielddistributions are observed above some square ferromag-netic structures of the sample. Obviously these partic-ular squares are not in the regular vortex distributionstudied previously. One way to determine the magne-tization distribution is to make an assumption aboutit, compute the expected dual-iso-B-image, and com-pare to experimental data. Here the four near-symmetrypoints distributed around the square center suggest a

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Figure 4. Higher-order magnetization distribution.(a)-Magnetic field distribution recorded above a square ofFe20Ni80 for a probe-to-sample distance d = 300 nm. The NVdefect orientation is given by θ = 65◦ and ϕ = 30◦. (b)-AFMimage of the square of Fe20Ni80. The white arrows depictthe higher-energy magnetization structure, with four mag-netic vortices (red dots) distributed around one anti-vortexat the center (blue dot). (c)-Simulated image considering amagnetization following the structure shown in (b). Scale bar,1 µm.

magnetization state comprising four vortices distributedaround one anti-vortex at the center (Fig. 4(b)), whichis a known equilibrium state in such ferromagnetic struc-tures28. Comparison between the simulated (Fig. 4(c))and measured magnetic images unambiguously validatesour hypothesis that this particular square exhibits a four-vortex-one-antivortex magnetization state29.

Discussion. These experiments illustrate that scanningNV magnetometry enables robust comparisons withmicromagnetic simulations and therefore appears asa powerful tool for fundamental studies in nanomag-netism and, more generally, nanoscience. Indeed, thedirect measurement of weak magnetic fields, withoutperturbation and with a nanoscale resolution, mayallow clearing some important issues, such as thenature of magnetic domain walls in ultrathin filmswith perpendicular anisotropy, which controls theircurrent-induced motion30. Bloch- and Néel-type domainwalls exhibit distinct stray field distributions whichcould be discriminated by scanning NV magnetometry.As an example, micromagnetic simulations indicatethat the typical stray field 50 nm above a Bloch wallin a 0.6 nm Co film with perpendicular anisotropy is≈ 5 mT, that changes by ±0.5 mT for a Néel wall (see

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Supplementary Information). Such a difference is largeenough to allow NV magnetometry to experimentallydetermine the nature of the domain wall and analysethe relevance of Dzyaloshinskii-Moriya interactions inultrathin magnetic films in contact with non magneticlayers6, which is expected to stabilize domain wall intothe Néel configuration32.

The detection of exotic magnetic structures known asskyrmion lattices33 is another example of a challengethat may be met by scanning NV magnetometry. On abroader perspective, the observation of the stray fieldscreated by persistent currents in micro- and nanos-tructures where the phase coherence of the carriers ispreserved, a quantum realization of the naive Amperiancurrents proposed two centuries ago to explain themagnetism of matter, appears also within reach ofscanning NV magnetometry.

The authors acknowledge C. Dal Salvio and K. Karraifor fruitful discussions. This work was supported by theAgence Nationale de la Recherche (ANR) through theproject Diamag, and by C’Nano Île-de-France (contractsMagda and Nanomag)

METHODS

Experimental setup and sample fabrication. The ex-perimental setup combines a tuning-fork-based atomic forcemicroscope (AFM) and a confocal optical microscope (at-

toAFM/CFM, Attocube Systems), all operating under am-bient conditions. A detailed description of the setup as wellas the method to graft a diamond nanocrystal at the apex ofthe AFM tip can be found in Ref.13. We use commerciallyavailable diamond nanocrystals (SYP 0.05, Van Moppes SA,Geneva), in which single NV defects were created throughhigh energy electron irradiation (13.6 MeV) followed by ther-mal annealing at 800◦C. The irradiated nanocrystals were fi-nally oxidized in air at 550◦C during two hours. This pro-cedure enables to reduce the size of the nanodiamonds andleads to an efficient charge state conversion of the created NVdefects into the negatively-charged state34. For the experi-ments reported in the main article, the size of the nanodia-monds were in the ∼ 20 nm range, as verified using AFM im-ages before grafting the nanodiamond at the apex of the tip.The unicity of the NV defect was checked through measure-ments of antibunching in the second order correlation func-tion g(2)(τ) of the NV defect photoluminescence (PL), usinga standard Hanbury Brown and Twiss interferometer.

For electron spin resonance (ESR) spectroscopy, a mi-crowave (MW) excitation is applied through a 20 µm cop-per wire directly spanned on the magnetic sample. The5 × 5 µm2 square dots were patterned on a silicon substrateusing electron-beam lithography followed by evaporation of50 nm of Fe20Ni80 and lift-off. All experiments were per-formed at zero external magnetic field except the measure-ment of the vortex core polarity which required to apply abias field. The acquisition time of all magnetic images wasset to 60 ms per pixel corresponding to a total acquisitiontime of 10 mn for a 100× 100 pixels image.

∗ vjacques@ens-cachan.fr† These authors contributed equally to this work.1 Freeman, M. R. & Choi, B. C. Advances in magnetic mi-croscopy. Science 294, 1484-1488 (2001).

2 Wachowiak, A, Wiebe, J., Bode, M., Pietzsch, O., Morgen-stern, M. & Wiesendanger, R. Direct observation of internalspin structure of magnetic vortex cores. Science 298, 577-580 (2002).

3 Fischer, P., Im, M.-Y., Kasai, S., Yamada, K., Ono, T.& Thiaville, A. X-ray imaging of vortex cores in confinedmagnetic structures. Phys. Rev. B 83, 212402 (2011).

4 Beardsley, I. A. Reconstruction of the magnetization in athin film by a combination of Lorentz microscopy and ex-ternal field measurements. IEEE Trans. Magn. 25, 671-677(1989).

5 Kirtley, J. R. Fundamental studies of superconductors usingscanning magnetic imaging. Rep. Prog. Phys. 73, 126501(2010).

6 Martin, Y. & Wickramasinghe, H. K. Magnetic imaging byforce microscopy with 1000 Å resolution. Appl. Phys. Lett.50, 1455-1457 (1987).

7 Shinjo, T., Okuno, T., Hassdorf, R., Shigeto, K. & Ono,T. Magnetic Vortex Core Observation in Circular Dots ofPermalloy. Science 289, 930-932 (2000).

8 Garcia, J. M., Thiaville, A., Miltat, J., Kirk, K. J., Chap-man, J. N. & Alouges, F. Quantitative interpretation ofmagnetic force microscopy images from soft patterned ele-

ments. Appl. Phys. Lett. 79, 656 (2001).9 Taylor, J. M. et al. High-sensitivity diamond magnetometerwith nanoscale resolution. Nature Phys. 4, 810-816 (2008).

10 Degen, C. L. Scanning magnetic field microscope with adiamond single-spin sensor. Appl. Phys. Lett. 92, 243111(2008).

11 Balasubramanian, G. et al. Nanoscale imaging magnetom-etry with diamond spins under ambient conditions. Nature455, 648-651 (2008).

12 Maze, J. R. et al. Nanoscale magnetic sensing with an in-dividual electronic spin in diamond. Nature 455, 644-647(2008).

13 Rondin, L., Tetienne, J.-P., Spinicelli, P., Dal Savio, C.,Karrai, K., Dantelle, G., Thiaville, A., Rohart, S., Roch,J.-F. & Jacques, V. Nanoscale magnetic field mapping witha single spin scanning probe magnetometer. Appl. Phys.Lett. 100, 153118 (2012).

14 Maletinsky, P., Hong, S., Grinolds, M. S. Hausmann, B.,Lukin, M. D., Walsworth, R. L., Loncar, M. & Yacoby, A.A robust scanning diamond sensor for nanoscale imagingwith single nitrogen-vacancy centres. Nat. Nano. 7, 320-324(2012).

15 Antos, R., Otani, Y. & Shibata, J. Magnetic vortex dy-namics. J. Phys. Soc. Jap. 77, 031004 (2008).

16 Drews, A., Krüger, B., Meier, G., Bohlens, S., Bocklage,L., Matsuyama, T. & Bolte, M. Current- and field-drivenmagnetic antivortices for nonvolatile data storage. App.

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Phys. Lett. 94, 062504 (2009).17 Pigeau, B., de Loubens, G., Klein, O., Riegler, A.,Lochner, F., Schmidt, G., Molenkamp, L. W., Tiberkevich,V. S. & Slavin, A. N. A frequency-controlled magnetic vor-tex memory. Appl. Phys. Lett. 96, 132506 (2010).

18 Pribiag, V. S., Krivorotov, I. N., Fuchs, G. D., Bra-ganca, P. M., Ozatay, O., Sankey, J. C., Ralph, D. C. &Buhrman, R. A. Magnetic vortex oscillator driven by d.c.spin-polarized current. Nat. Phys. 3, 498-503 (2007).

19 Raabe, J., Pulwey, R., Sattler, R., Schweinböck, T., Zweck,J. & Weiss D. Magnetization pattern of ferromagnetic nan-odisks. J. Appl. Phys. 88, 4437 (2000).

20 Chung, S.-H., Pierce, D.T. & Unguris, J. Simultaneousmeasurement of magnetic vortex polarity and chirality us-ing scanning electron microscopy with polarization analysis(SEMPA). Ultramicroscopy 110, 177-181 (2010).

21 Choe, S. B., Acremann, Y., Scholl, A., Bauer, A., Doran,A., Stohr, J. & Padmore, H. A. Vortex core-driven magne-tization dynamics. Science 304, 420-422 (2004).

22 Vansteenkiste, A., Chou, K. W., Weigand, M., Curcic, M.,Sackmann, V., Stoll, H., Tyliszczak, T., Woltersdorf, G.,Back, C. H., Schutz, G. & VanWaeyenberge, B. X-ray imag-ing of the dynamic magnetic vortex core deformation. Nat.Phys. 5, 332-334 (2009).

23 Manson, N. B., Harrison, J. P. & Sellars, M. J. Nitrogen-vacancy center in diamond: Model of the electronic struc-ture and associated dynamics. Phys. Rev. B 74, 104303(2006).

24 Gruber, A., Drabenstedt, A., Tietz, C., Fleury, L. &Wrachtrup, J. & von Borczyskowski C. Scanning confocaloptical microscopy and magnetic resonance on single defectcenters. Science 276, 2012-2014 (1997).

25 Donahue, M. J. & Porter, D. G. OOMMF User’s Guide,Version 1.0 Interagency Report NISTIR 6376, National In-stitute of Standards and Technology, Gaithersburg, MDhttp://math.nist.gov/oommf (1999).

26 Tetienne, J.-P., Rondin, L., Spinicelli, P., Chipaux, M.,Debuisschert, T., Roch, J.-F.& Jacques, V. Magnetic-field-dependent photodynamics of single NV defects in diamond:an application to qualitative all-optical magnetic imaging.New J. Phys. 14, 103033 (2012).

27 Alegre, T. P. M., Santori, C., Medeiros-Ribeiro, G. &Beausoleil, R. G. Polarization-selective excitation of nitro-gen vacancy centers in diamond. Phys. Rev. B 76, 65205(2007).

28 Shigeto, K., Okuno, T., Mibu, K., Shinjo, T. & Ono, T.Magnetic force microscopy observation of antivortex corewith perpendicular magnetization in patterned thin film ofpermalloy. Appl. Phys. Lett. 80, 4190 (2002).

29 Higher-energy magnetization states might have appearedduring the sample growth. We verified that applying astrong rotating magnetic field restored all ferromagneticsquares in their minimum-energy single-vortex state.

30 Khvalkovskiy, A. V., Cros, V., Apalkov, D., Nikitin, V.,Krounbi, M., Zvezdin, K. A., Anane, A., Grollier, J. &Fert, A. Matching domain-wall configuration and spin-orbittorques for efficient domain-wall motion. Phys. Rev. B 87,020402(R) (2013).

31 Bode, M. et al., Chiral magnetic order at surfaces drivenby inversion asymmetry, Nature 447, 190-193 (2007).

32 Thiaville, A., Rohart, S., Jué, E., Cros, V., & Fert, A.,Dynamics of Dzyaloshinskii domain walls in ultrathin mag-netic films, Europhysics Lett. 100, 57002 (2012).

33 Heinze, S., von Bergmann, K., Menzel, M., Brede, J., Ku-

betzka, A., Wiesendanger, R., Bihlmayer, G. & Blugel, S.Spontaneous atomic-scale magnetic skyrmion lattice in twodimensions. Nat. Phys. 7, 713-718 (2011).

34 Rondin L., et al. Surface-induced charge state conversionof nitrogen-vacancy defects in nanodiamonds. Phys. Rev. B82, 115449 (2010).

SUPPLEMENTARY INFORMATION

A. ESR frequencies vs. magnetic field

In this section, we give the analytical expression of theNV defect ESR frequencies as a function of the appliedmagnetic field, while taking into account strain effects ofthe diamond lattice.

The ground-state spin Hamiltonian of the NV defectelectron spin (S = 1) reads

H = hDS2Z + hE(S2

X − S2Y ) + hγeB · S (1)

where h is the Planck constant, D is the zero-field split-ting, E the strain splitting, Z the NV defect axis, γethe electron gyromagnetic ratio, and B the local mag-netic field [Fig. 5(a)]. By numerically computing theeigenenergies of H, the two ESR frequencies νR,± canbe calculated for any magnetic field B. For instance,the ESR frequencies νR,± are plotted in Figure 5(b) asa function of the magnetic field amplitude B = ‖B‖ forvarious angles α between B and the NV defect axis.

With the aim of obtaining a simple analytical expres-sion for νR,±, the Hamiltonian H is written

H = H‖ +H⊥ +Hstrain (2)

with H‖ = hDS2Z + hγeBNVSZ

H⊥ = hγe(BXSX +BY SY )Hstrain = hE(S2

X − S2Y ) ,

(3)

where we introduced the notation BNV = BZ as usedin the main article. Typical values for the zero-field pa-rameters in diamond nanocrystals are D ≈ 2.87 GHzand E ≈ 5 MHz. One can therefore distinguish differ-ent regimes depending on the value of B‖ = |BNV| andB⊥ =

√B2

X +B2Y , which are the parallel and transverse

components of the magnetic field with respect to the NVaxis, respectively.

Focusing on weak magnetic field amplitudes, we con-sider that H⊥ � H‖. The ESR frequencies are thensimply given by

νR,±(BNV) = D ±√

(γeBNV/2π)2 + E2 . (4)

This formula can be used whenever the transverse com-ponent of the magnetic field B⊥ is much smaller than2πD/γe ≈ 100 mT. This is illustrated in Figure 5(b),where the prediction of Equation (4) is plotted in dot-ted lines together with the full calculation (solid lines).

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a b

C

a b

c d

Figure 5. (a)-Schematic of the NV defect. A magnetic field Bis applied with an angle α with respect to the NV defect axis.(b)-ESR frequencies νR,± as a function of the magnetic fieldamplitude B = ‖B‖ for different angles α. The solid lines areobtained by diagonalizing the full Hamiltonian described byEquation (1), while using D = 2.87 GHz and E = 5 MHz.The dashed lines are obtained by using Equation (4). (c)-Optically-detected ESR spectra of a single NV defect placedat the apex of the AFM tip for BNV = 0 mT and BNV = 1.2mT (down-shifted for clarity). The solid lines are data fit-ting with Equation (6) which allows us to determine the ESRlinewidth ∆ν = 13 MHz and the ESR contrast C = 0.11, forthis particular NV defect. (d)-ESR frequencies as a functionof BNV, with α = 44◦. The solid lines are the predictions ofEquation (4) with the zero-field parameters D = 2869.5 MHzand E = 10.5 MHz.

Obviously the approximate formula is relevant for fieldamplitudes in the range 0-2 mT, which is the range weaddress in this work. In addition, if γeB‖/2π � E, thenHstrain can be neglected and the ESR frequencies become

νR,±(BNV) = D ± γeBNV/2π , (5)

as indicated in the main paper.Experimentally, the ESR frequencies are inferred from

the optically-detected ESR spectrum, i.e. the NV defectPL rate R(ν,BNV) as a function of the MW frequency ν.Once normalized to unity, it can be expressed as

R(ν,BNV) = 1− C ×[G(ν − νR,+(BNV)

∆ν/2√

ln 2

)+ (6)

G(ν − νR,−(BNV)

∆ν/2√

ln 2

)](7)

where G(x) = e−x2

denotes a Gaussian function, ∆ν isthe ESR linewidth (FWHM) and C its contrast. Twomeasured ESR spectra are shown in Figure 5(c) for asingle NV defect attached to the AFM tip, together with

data fitting using Equation (6). Varying BNV enables usto verify Equation (4) experimentally, as illustrated inFigure 5(d). Note that the vanishing slope at very lowmagnetic fields – i.e. when γeB/2π � E – is responsiblefor the fact that in our dual-iso-B images, the iso-B linescorresponding to zero field are always wider that thosecorresponding to a non-zero field.

B. Determining the orientation of the NV defect

The orientation uNV of the NV defect gives theprojection axis of the measured magnetic field in allmagnetometry experiments, i.e. BNV = B · uNV. Todetermine the spherical angles θ and ϕ that characterizeuNV [Fig. 6(a)], we apply a calibrated magnetic fieldBcoil = Bcoilucoil using a three-axis coil system [Fig. 6(b)]while monitoring one of the two ESR frequencies of theNV defect (e.g. νR,+). The procedure we employed isas follows. We first set Bcoil = 2 mT and θcoil = 90◦,and sweep the angle ϕcoil from 0 to 90◦. This yields theangle ϕ by fitting Equation (4) to the data, using thefact that BNV ∝ cos(ϕcoil − ϕ) [Fig. 6(c)]. We next setϕcoil = ϕ and sweep θcoil from 0 to 90◦. The angle θ isdeduced by fitting Equation (4) to the data, given thatBNV ∝ cos(θcoil − θ) [Fig. 6(d)]. This procedure enablesus to determine the angles θ and ϕ in the range [0, π],which completely defines the direction of uNV.

a b

c d

Figure 6. (a)-The axis of the NV defect uNV is described bythe angles θ and ϕ in the spherical coordinate system of thelaboratory reference frame. (b)-An external magnetic field isapplied using a three-axis coil system. Its direction, of unitvector ucoil, is described by the angles θcoil and ϕcoil in thesame coordinate system. (c)-Measured ESR frequency νR,+

as a function of the angle ϕcoil of the applied magnetic field,while fixing θcoil = 90◦. The solid line is a fit to the data usingEquation (4), which yields the angle ϕ. (d)-Measured ESRfrequency νR,+ as a function of the angle θcoil while keepingϕcoil = ϕ. The solid line is a fit to the data using Equation (4),which yields the angle θ.

8

C. Magnetization and stray field simulations

The magnetization distribution inside the square filmof Fe20Ni80 was computed using OOMMF software1. Theparameters used for Fe20Ni80 were: spontaneous mag-netization Ms = 8 105A/m and an exchange constantA = 10−11 J/m. A magnetization cell size 5× 5× 5 nm3

was used, which is below the micromagnetic exchangelength in order to describe the vortex core structure pre-cisely. From the calculated equilibrium magnetization[Fig. 7(a)], the stray magnetic field is computed by sum-ming the contribution of all magnetization cells, and thenprojected along the NV defect axis in order to reach amap of BNV. As an example, Figure 7(b) shows the cal-culated z-axis component of the stray field for a probe-to-sample distance d = 200 nm.

a b

c d e

x

yz B

z(m

T)

Figure 7. (a)-Map of the in-plane magnetization of a 5 µmFe20Ni80 square dot, as computed by OOMMF software. (b)-Calculated distribution of the Bz component of the stray mag-netic field at a distance d = 200 nm above the sample surface.(c)-ESR spectrum measured at BNV = 0. The microwave fre-quencies applied for the dual-iso-B images are chosen to beν1 = 2905 MHz and ν2 = 2880 MHz, as indicated with bluearrows. (d)-PL difference signal S(ν1, ν2, BNV) as a func-tion of BNV, calculated from the ESR spectrum in (c) andEquation (8). (e)-Dual-iso-B image calculated by applyingS(ν1, ν2, BNV) to the magnetic field map shown in (b). Herewe assumed the NV axis to be parallel to the z axis.

In the following, we take into account the strain-induced splitting, which was omitted in the main textfor clarity purpose. We therefore use Equation (4) forthe simulation of the magnetic field maps recorded withthe scanning NV magnetometer. The dual-iso-B imagesshown in the main article are obtained by measuring thedifference of NV defect PL intensity for two fixed MWfrequencies ν1 and ν2. The resulting signal reads

S(ν1, ν2, BNV) = R(ν2, BNV)−R(ν1, BNV) , (8)

where R(ν,BNV) is given by Equation (6). The sig-nal S is positive (resp. negative) when ν1 (resp. ν2)matches one of the two ESR frequencies νR,± – i.e.when BNV = BNV,1 = ±2π

√(ν1 −D)2 − E2/γe (resp.

BNV,2 = ±2π√

(ν2 −D)2 − E2/γe) – and is null other-wise.

As an illustration, we consider the same NV defectas in Figures 5(c)&(d), with an ESR spectrum at zeromagnetic field reproduced in Figure 7(c). We chooseν1 = 2905 MHz and ν2 = 2880 MHz, which correspondsto BNV,1 = 1.2 mT and BNV,2 = 0. Having fully charac-terized the ESR response, i.e. its linewidth and contrast,the resulting dual-iso-B signal S(ν1, ν2, BNV) can be com-puted as a function of BNV, as shown in Figure 7(d). Thesimulated magnetic field maps are then obtained by ap-plying the S(ν1, ν2, BNV) response to the calculated strayfield once projected along the NV defect axis. An exam-ple of a calculated dual-iso-B image is shown in Fig. 7(e),where we assumed d = 200 nm and the NV axis to bealong the z axis, i.e. Z = z.

D. Distance dependence

Magnetic field images recorded above a 5 µm Fe20Ni80square dot at several probe-to-sample distances, varyingbetween d = 100 and d = 600 nm, are shown in Figure 8.For these experiments, the probe-to-sample distance iscontrolled by using the AFM in a dual-pass mode witha constant lift height during the second pass. The sameNV defect probe as in Figure 3 of the main article is used,i.e. mainly oriented along the z-axis. A good agreementwith the simulations is found, which illustrates the three-dimensional stray field mapping capabilities of scanningNV magnetometry. The slight asymmetries observed inboth the experimental and simulated images result fromthe imperfect alignment of the NV defect axis with re-spect to the z-axis.

±0.8 mT

0 mT

5 µm

d

! = 135!! = 37!

d 0 +1-1

Experim

ent

Simulation

Figure 8. Experimental dual-iso-B images (top panels) ob-tained for a NV defect oriented mainly along the z axis(θ = 37◦, ϕ = 135◦) at different probe-sample distances d, to-gether with the simulations (bottom panels). The red squaredepicts the position of the 5 µm square dot.

9

E. Magnetic-field-induced fluorescence quenching

Although the simulations fairly agree with exper-imental data, the sharp structures lying along thesquare diagonals are not observed in the experimentalimage recorded at the smallest probe-to-sample distanced = 100 nm [Fig. 8]. As explained in the main text, astrong stray field is generated by Néel domain walls inthese regions. Indeed, the calculated magnetic field mapsat d = 100 nm indicate that the stray field componentsalong the y- and z-axis reach nearly ∼ 10 mT at thesquare diagonals, as illustrated in Figures 9(a)&(b).In this condition, the approximation H⊥ � H‖ isnot fulfilled [see Eq. (2)] and the quantization axis isdetermined by the local magnetic field rather than bythe NV defect axis. As a result, the spin projectionalong the NV axis ms is no longer a good quantumnumber and the eigenstates of the spin Hamiltonian aregiven by superpositions of the ms = 0 and ms = ±1spin sublevels2. In this regime, the contrast of opticallydetected ESR vanishes3. Magnetic field imaging throughoptically-detected ESR is therefore inefficient in theregime of ‘high’ off-axis magnetic field. This explainswhy the zero-field lines are not visible at the square’sdiagonals for d = 100 nm, where a strong off-axismagnetic field is experienced by the NV defect.

It was also shown that the decreased ESR contrastis accompanied by a reduction of the NV defect PLintensity when the magnitude of B⊥ increases2,4. Sucha magnetic-field-dependent PL can be exploited as analternative method to map high off-axis magnetic fieldregions3,5. This is illustrated in Figure 9(c) where darklines are observed along the square’s diagonals in the PLimage for a probe-to-sample distance d = 100 nm. Inthis experiment, we record the NV defect PL intensitywithout applying any MW field. The dark lines indicatea PL quenching induced by the off-axis component ofthe magnetic field generated by Néel domain walls. Nearthe center of the magnetic structure, the magnetic fieldamplitude decreases [see Fig. 9(c)], which enabled us toimage the vortex core using the ESR response of the NVdefect.

We note that the resulting PL quenching imageexhibits obvious similarities with the one recorded witha magnetic force microscope (MFM), which is sensitiveto the field gradient [Fig. 9(d)].

F. Vortex polarity and chirality

The product of the vortex chirality c and the vortexcore polarity p can be determined by simply examiningthe zero-field contours around the vortex core in theexperimental dual-iso-B images. Indeed, zero-field linesseparate the regions of positive and negative magnetic

By (mT) Bz (mT)

a 50-5 50-5b

PL (a.u.)

6 8 10 12

1 µm1 µm

c dMFM image

Figure 9. (a),(b)-Calculated By and Bz components of thestray field distribution at a distance d = 100 nm. (c)-PLquenching image obtained without applying any MW field andwith the AFM operating in tapping mode. Zooming into thecenter shows that the PL quenching vanishes in the vicinity ofthe center of the structure, where the magnetic field amplitudeis expected to decrease. (d)-MFM image of a similar squareof Fe20Ni80.

field projection. It is then possible to determine which“lobes” have the same sign as the vortex core field.Figure 10(a) shows the magnetic field map calculatedat d = 100 nm from the sample surface and projectedalong the z-axis, for all possible combinations of c andp. As expected, the pairing of the zero-field contoursaround the vortex core (black dotted lines) are fixedby the product c × p. Comparing the experimentalresults to simulations directly enables to determine thatc × p = +1 for the vortex state studied in Figure 3 ofthe main text [reproduced in Fig. 10(b)].

By applying an external magnetic field Bext that can-cels out the net field above the vortex core, the polarityof the vortex core can be determined as shown in themain paper. In Figures 10(c), we rather show dual-iso-Bimages obtained for a bias field Bext antiparallel to the

10

c = +1 , p = +1

c = !1 , p = !1

c = !1 , p = +1

c = +1 , p = !1

±0.8 mT

0 mT

0

-2

2

Bz (mT)

a bc ! p = +1

Bext

c

±1.4 mT0 mT

Bext = !1.3 mTBext = !0.6 mT

100 nm100 nm

100 nm

Figure 10. (a)-Calculated Bz component of the stray field produced by the center of the magnetic structure at d = 100 nm,for the four possible combinations of chirality c and polarity p of the vortex state. The black dotted lines indicate the zero-field contours, separating positive (upward) and negative (downward) field regions. (b)-Magnetic field map of the stray fieldemanating from the vortex core magnetization, as shown in Figure 3 of the main text. (c)-Dual-iso-B image of the squarecenter while applying a bias field with a projection along the NV axis Bext = −0.6 mT and Bext = −1.3 mT.

vortex core’s field Bcore such that |Bcore| − |Bext| isslightly positive or slightly negative. In the latter case,the net field above the vortex core is now of oppositesign compared to the zero-field case, which reverses thepairing of the zero-field lines in the image.

I. MICROMAGNETIC SIMULATIONS FORDOMAIN WALL IMAGING

As indicated in the conclusion of the main manuscript,the direct quantitative measurement of weak magneticfields, without perturbation and with a nanoscale res-olution, may allow clearing some important issues innanomagnetism. As an example, we show in this fi-nal section how scanning-NV magnetometry might allowto clearly identify the nature of magnetic domain walls(DW) in ultrathin ferromagnetic films with perpendicu-lar anisotropy.

The magnetism of ultrathin films has attracted a re-newed interest since the experimental discovery of therelevance of Dzyaloshinskii-Moriya interactions (DMI) inultrathin magnetic films in contact with non magneticlayers6. In particular, it has been shown that the DMIaffects the nature of the lowest-energy magnetic domainwall structures in such films7, which in turn controls theircurrent-induced motion8. In the following, we presentmicromagnetic simulations of domain walls (DWs) in athin film with perpendicular anisotropy, the typical sam-ple being Pt/Co(0.6nm)/AlOx

9.As shown in Figure 11(a), we consider a film paral-

lel to the x-y plane, with a DW centered at x = 0 thatextends to infinity along the y direction. The magneti-zation distribution in the case of a Bloch- and Néel-type

DW was obtained using OOMMF software, from whichthe magnetic stray field was computed [Figure 11(b)&(c)]with typical probe-to-sample distances d = 100 nm and

-2

-1

0

1

B [m

T]

-1.0 -0.5 0.0 0.5 1.0x [!m]

Bx,Bloch

Bz,Bloch

Bx,Neel I

Bz,Neel I Bx,Neel II Bz,Neel II -6

-4

-2

0

2

B [m

T]

-1.0 -0.5 0.0 0.5 1.0x [!m]

Bx,Bloch

Bz,Bloch

Bx,Neel I

Bz,Neel I Bx,Neel II Bz,Neel II

0.20

0.15

0.10

0.05

0.00

|!B

| [m

T]

-1.0 -0.5 0.0 0.5 1.0x [!m]

|Bx,Bloch - Bx,Neel| |Bz,Bloch - Bz,Neel|

0.8

0.6

0.4

0.2

0.0

| !B

| [m

T]

-1.0 -0.5 0.0 0.5 1.0x [!m]

|Bx,Bloch - Bx,Neel| |Bz,Bloch - Bz,Neel|

x

y

z

d = 100 nm d = 50 nm

a b

c d

a

b c

d e

Figure 11. Simulated stray field of a domain wall(DW). (a)-Scheme of a domain wall in an ultrathin ferro-magnetic film with perpendicular anisotropy in contact withnon magnetic layers. (b),(c)-Stray field generated by a DWin a Bloch configuration or in a Néel configuration at (b)d = 100 nm and (c) d = 50 nm from the sample. The nota-tions Néel I and Néel II indicate the different Néel wall chi-rality. (d),(e)-Absolute difference of the stray field betweena Bloch and a Néel domain wall, at (d) d = 100 nm and (e)d = 50 nm from the sample. The sign of the difference isgiven by the Néel wall chirality.

11

d = 50 nm. Figure 11(d)&(e) shows the absolute dif-ference in the magnetic stray field produced by the twotypes of DWs at the same probe-to-sample distances. Ob-viously the calculated stray fields, reaching a few millites-las right above the DW, are in the operation range of the

NV magnetometer, and the ±10% difference between thetwo types of DW is large enough to allow the NV mag-netometer to experimentally determine the nature of theDW.

∗ vjacques@ens-cachan.fr† These authors contributed equally to this work.1 Donahue M. J. and Porter D. G. OOMMF User’s Guide,Version 1.0 Interagency Report NISTIR 6376, National In-stitute of Standards and Technology, Gaithersburg, MDhttp://math.nist.gov/oommf (1999).

2 Epstein, R. J., Mendoza, F. M., Kato, Y. K. & Awschalom,D. D. Anisotropic interactions of a single spin and dark-spinspectroscopy in diamond. Nature Phys. 1, 94-98 (2005).

3 Tetienne J.-P., Rondin L., Spinicelli P., Chipaux M., De-buisschert T., Roch J.-F. and Jacques V.,Magnetic-field-dependent Photodynamics of Single NV Defects in Dia-mond: An Application to Qualitative All-optical MagneticImaging. New J. Phys. 14 103033 (2012).

4 Lai, N. D., Zheng, D., Jelezko, F., Treussart F. & Roch,J.-F. Influence of a static magnetic field on the photolumi-nescence of an ensemble of nitrogen-vacancy color centers

in a diamond single-crystal. Appl. Phys. Lett. 95, 133101(2009).

5 Rondin L., Tetienne J.-P., Spinicelli P., Dal Savio C., KarraiK., Dantelle G., Thiaville A., Rohart S., Roch J.-F. andJacques V., Nanoscale magnetic field mapping with a singlespin scanning probe magnetometer, Appl. Phys. Lett. 100,153118 (2012).

6 Bode, M. et al., Chiral magnetic order at surfaces driven byinversion asymmetry, Nature 447, 190 (2007).

7 Thiaville, A., Rohart, S., Jué, E., Cros, V., & Fert, A., Dy-namics of Dzyaloshinskii domain walls in ultrathin magneticfilms, Europhysics Lett. 100, 57002 (2012).

8 Khvalkovskiy, et al., Matching domain-wall configurationand spin-orbit torques for efficient domain-wall motion,Phys. Rev. B 87, 020402(R) (2013).

9 Miron, I. M. et al., Fast current-induced domain-wall mo-tion controlled by the Rashba effect, Nat. Mater. 10, 419(2011).