Nanoscale magnetic field mapping with a single spin scanning probe magnetometer
L. Rondin, J.-P. Tetienne, P. Spinicelli, C. Dal Savio, K. Karrai, G. Dantelle, A. Thiaville, S. Rohart, J.-F. Roch, V. Jacques
NNanoscale magnetic field mapping with a single spin scanning probe magnetometer
L. Rondin , J.-P. Tetienne , P. Spinicelli , C. Dal Savio , K. Karrai ,G. Dantelle , A. Thiaville , S. Rohart , J.-F. Roch , and V. Jacques ∗ Laboratoire de Photonique Quantique et Mol´eculaire,Ecole Normale Sup´erieure de Cachan and CNRS UMR 8537, 94235 Cachan Cedex, France Attocube systems AG, Koeniginstrasse 11A RGB, Munich 80539, Germany Laboratoire de Physique de la Mati`ere Condens´ee,Ecole Polytechnique and CNRS UMR 7643, 91128 Palaiseau, France and Laboratoire de Physique des Solides, Universit´e Paris-Sud and CNRS UMR 8502, 91405 Orsay, France
We demonstrate quantitative magnetic field mapping with nanoscale resolution, by applying alock-in technique on the electron spin resonance frequency of a single nitrogen-vacancy defect placedat the apex of an atomic force microscope tip. In addition, we report an all-optical magnetic imagingtechnique which is sensitive to large off-axis magnetic fields, thus extending the operation range ofdiamond-based magnetometry. Both techniques are illustrated by using a magnetic hard disk as atest sample. Owing to the non-perturbing and quantitative nature of the magnetic probe, this workshould open up numerous perspectives in nanomagnetism and spintronics.
The ability to map magnetic field distributions withhigh sensitivity and nanoscale resolution is of crucial im-portance for fundamental studies ranging from materialscience to biology, as well as for the development of newapplications in spintronics and quantum technology [1–3]. In that context, an ideal scanning probe magnetome-ter should provide quantitative magnetic field mappingat the nanoscale under ambient conditions. In addition,the magnetic sensor should not introduce a significantmagnetic perturbation of the probed sample.Over the last decades, different roads have been takentowards ultra-sensitive detection of magnetic fields in-cluding superconducting quantum interference devices(SQUIDs) [1], semiconductor-based Hall probes [1] andoptical magnetometers [2]. Even though extremely highsensitivity has been achieved with these devices, theirspatial resolution remains limited at the micron-scale.Prominent approaches to reach nanoscale resolution arescanning-tunneling microscopy [4], mechanical detectionof magnetic resonance [5], nanoSQUIDs [6], X-ray mi-croscopy [7] and magnetic force microscopy (MFM) [8].Since the latter technique operates under ambient con-ditions without any specific sample preparation, it isnow routinely used for mapping magnetic field gradientsaround magnetic nanostructures. However, besides in-troducing an inevitable perturbation of the studied mag-netic sample owing to the intrinsic magnetic nature ofthe probe [7], MFM does not provide quantitative infor-mation about the magnetic field distribution.Here we follow a recently proposed approach to mag-netic sensing based on optically detected electron spinresonance (ESR) [10]. It was shown that this method ap-plied to a single nitrogen-vacancy (NV) defect in diamondcould provide an unprecedented combination of spatialresolution and magnetic sensitivity under ambient con-ditions [6, 11, 12, 14]. The principle of the measurement ∗ Electronic address: [email protected] is similar to the one used in optical magnetometers basedon the precession of spin-polarized atomic gases [2]. Theapplied magnetic field is evaluated by measuring the Zee-man shifts of the NV defect spin sublevels. In this articlewe demonstrate quantitative magnetic field mapping withnanoscale resolution, by applying a lock-in technique onthe ESR frequency of a single NV defect placed at theapex of an atomic force microscope (AFM) tip. In addi-tion, we report an all-optical magnetic imaging techniquewhich is sensitive to large off-axis magnetic fields, and re-lies on magnetic-field-dependent photoluminescence (PL)of the NV defect sensor, induced by spin level mixing.As depicted in Fig. 1(a), the scanning probe magne-tometer combines an optical confocal microscope placedon top of a customized tuning-fork-based AFM (At-tocube Systems), all operating under ambient conditions.The atomic-sized magnetic sensor was first engineered bygrafting a diamond nanocrystal hosting a single NV de-fect at the end of the AFM tip [3, 4, 15]. After func-tionalization, a PL raster scan of the AFM tip shows anisolated spot at its apex (Fig. 1(b)), which corresponds tothe collected light emitted by a single NV defect, as ver-ified using photon correlation measurements (Fig. 1(c)).The magnetic response of the probe was first charac-terized by sweeping the frequency of a driving microwave(MW) field through the spin resonance while recordingthe NV defect PL. Owing to spin dependent PL, ESRappears as a drop of the PL signal (Fig. 1(d)) [18]. Thespin Hamiltonian of the NV defect reads as [6] H = hDS z + hE ( S x − S y ) + (cid:126) γ e B · S , (1)where h is the Planck constant, D and E are the zero-field splitting parameters, z is the NV defect quantizationaxis, γ e the electron gyromagnetic ratio, and B the localmagnetic field applied to the NV defect electron spin S =1. For magnetic fields smaller than roughly 10 mT, i.e. such that (cid:126) γ e B (cid:28) hD , the quantization axis is fixed bythe NV defect axis itself and the ESR frequencies aregiven by ν R = D ± (cid:112) ( γ e B z / π ) + E , where B z is themagnetic field amplitude along the NV axis. The spin a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r (a) MW AFM tip Objective antennaNV defect Magnetic structure P L ( kc n t s / s ) g ( ) ( τ ) (c) μ m (d) B z = 0 . B z = 1 mT (b) N o r m a li z ed P L MW frequency (GHz)
Delay ! (ns) FIG. 1: (a)-Simplified scheme of the scanning probe magne-tometer. A 20 nm diamond nanocrystal hosting a single NVdefect is grafted at the end of an AFM tip. A confocal mi-croscope placed on top of the tip allows us both to excite andcollect the NV defect spin-dependent PL. A microwave field(MW) is generated by an antenna approached in the vicin-ity of the NV defect. (b)-PL raster scan of the AFM tipshowing a bright PL spot at its apex. White dashed linesindicate the shadow of the tip. (c)- Second-order autocorre-lation function g (2) ( τ ) recorded from the bright emission spotusing a Hanbury Brown and Twiss interferometer. A stronganticorrelation effect, g (2) (0) ≈ .
1, is observed at zero de-lay. More than 1 × photons per second can be detectedfrom the single NV defect when excited at saturation, with asignal-to-background ratio greater than 10. (d)-Optically de-tected ESR spectra recorded for different magnetic field mag-nitudes. For these experiments, the ESR contrast is C ≈
ESR ≈ R ≈ × cnt/s, with an optical pumpingpower P opt = 300 µ W. The zero-field splitting parameters are D = 2 .
87 GHz and E = 5 MHz. orientation of the magnetic sensor was determined byrecording ESR frequencies as a function of the orientationand the magnitude of a calibrated magnetic field [15].The shot-noise-limited sensitivity to d.c. magnetic field η B is linked to the minimum detectable magnetic fieldalong the NV axis B min z during an acquisition time ∆ t through the relation [19] η B (T / √ Hz) = B min z √ ∆ t ≈ γ e × π ∆ ESR
C√R , (2) where R is the rate of detected photons, C the ESR con-trast and ∆ ESR the associated linewidth (FWHM). Fromthe optically detected ESR spectra shown in Fig. 1(d),we infer η B ≈ µ T/ √ Hz. We note that this sensitiv-ity could be significantly enhanced by using pulsed-ESRtechniques [14, 19] or by engineering a single NV defectin an ultra-pure diamond nanostructure [20].The performances of the NV scanning probe magne-tometer were characterized by mapping the magneticfield distribution created by a commercial magnetic harddisk. The magnetic bit geometry of this simple test sam-ple is depicted in Fig. 2(a). As a first experiment, themagnetic sensor was approached at roughly 250 nm fromthe sample surface. At such a distance, the random orien-tation of the magnetic bits provides non-trivial magneticfield patterns with a typical magnetic field magnitudein the mT range [15]. The simplest way to map sucha magnetic field distribution is borrowed from magneticresonance imaging [6]. A MW field was applied with afixed frequency corresponding to the zero-field ESR fre-quency. Iso-magnetic field images were then recorded bymonitoring the NV defect PL while scanning the harddisk [6]. Dark areas corresponding to zero magnetic fieldlines can be observed (Fig. 2(b)). However, the image ex-hibits a varying contrast related to inhomogeneous back-ground luminescence from the sample. This limitationcan be overcome by measuring the difference of PL fortwo fixed MW frequencies applied consecutively at eachpoint of the scan (Fig. 2(c)). In this dual iso-magneticfield image, the contrast is significantly improved sincebackground luminescence from the sample is suppressed.Even with two iso-magnetic field lines recorded atonce, the information remains incomplete. To determinethe full magnetic field distribution, an ESR spectrumcould be measured for each pixel of the scan. However,since a few seconds are required to record such a spec-trum with a reasonable signal-to-noise ratio, this methodwould be extremely slow. To circumvent this limitation,a lock-in method was developed in order to track the shiftof the ESR frequency while scanning the magnetic sam-ple [21]. This technique allows us to measure the strengthof the magnetic field along the NV axis over the full scan-ning area with a typical sampling time of 110 ms [15]. Asshown in Fig. 2(d), the complete magnetic field map isin clear agreement with iso-magnetic field experiments.In addition, we note that the recorded pattern can bequalitatively reproduced by numerical simulations, bothin shape, size, and magnetic field amplitude [15]. Thisconfirms the reliability of NV-based scanning probe mag-netometry for imaging non trivial magnetic field distri-butions at the nanoscale within a reasonable data acqui-sition time.From the pixel-to-pixel noise in Fig. 2(d), an uncer-tainty δB ≈ µ T is deduced, which corresponds to asensitivity η exp B ≈ µ T/ √ Hz, given the 110 ms samplingtime [15]. This value is in good agreement with the shot-noise-limited value calculated using Eq. (2). Regardingspatial resolution, the magnetic field is experienced by
400 nm 400 nm 400 nm
15 25 35 -1 0 1 (b)
PL (kcounts/s) PL diff (a.u.) (c) (d) | B z | (mT) | B z | (mT) d NV n m n m Random distribution of bits (a) NV . . µ m FIG. 2: Measurements of the magnetic field distribution created by a commercial magnetic hard disk for a probe-to-sampledistance d = 250 nm. All images correspond to 150 ×
138 pixels, with a 13-nm pixel size. (a)-Geometry of the magneticbits. (b)-Single iso-magnetic field image. Dark areas correspond to a null | B z | component and the red arrow indicates the NVdefect axis, along which the magnetic field projection is measured. (c)-Dual iso-magnetic field image recorded by measuringthe PL difference for two fixed MW frequencies applied consecutively. Dark (resp. bright) areas correspond to | B z | = 0(resp. | B z | = 0 . the NV defect electron spin wavefunction, resulting in asubnanometric probe volume. Experimentally the spa-tial resolution is rather limited by the pixel size (13 nm)and the positioning accuracy. The B -map depicted inFig. 2(d) is thus assumed to closely follow the actualfield distribution – within the noise δB – with an overallspatial resolution of a few tens of nm. The line-cut inFig. 2(d) further illustrates the ability to follow abruptvariations, up to 0.4 mT ( (cid:29) δB ) between two adjacentpixels, corresponding to a field gradient of 3 × T/m.We emphasize the fact that no other instrument is ableto quantitatively map magnetic field distributions withsuch a resolution.Yet, many material science studies are interested inthe sample magnetization M rather than the stray field B . In this context, the spatial resolution is limited notonly by the probe volume but also by its distance tothe sample. Here for instance, simulations indicate thatthe magnetic bits could be resolved by using a probe-to-sample distance smaller than 40 nm (Fig. 3(a)) [15].However, the magnetic field amplitude then reaches sev-eral tens of mT, including a component B ⊥ orthogonalto the NV defect axis [15]. In such a situation, the quan-tization axis becomes determined by the local magneticfield rather than the NV defect axis. The eigenstatesof the spin Hamiltonian are therefore given by super-positions of the m s = 0 and m s = ± >
10 mT). How-ever, it was also shown that the NV defect PL drops byroughly 30% when the magnitude of B ⊥ increases from0 to approximately 30 mT [22, 23]. This property can be exploited to perform a direct, microwave-free, magneticfield imaging using the single NV defect sensor.Figure 3(b) shows a PL image recorded while scanningthe magnetic sample without applying any microwavefield. In this experiment, an AFM feedback loop is ap-
400 nm
40 nm P L ( kc n t s / s )
25 30 350 17 35
400 nm400 nm (b)
PL (kcounts/s) (a) B ⊥ (mT) (c)(d) - - - - - FIG. 3: (a)-Simulation of the off-axis magnetic field compo-nent B ⊥ for a random distribution of bit magnetizations and aprobe-to-sample distance d = 30 nm. (b)-PL image recordedwith the NV scanning probe magnetometer operating in tap-ping mode, without applying any microwave field. The imagecorresponds to 200 ×
200 pixels, with a 8-nm pixel size and a20-ms acquisition time per pixel. (c)-Line-cut correspondingto the white dashed line in (b). A possible configuration ofthe bit magnetization is schematically depicted in (d). plied to maintain the mean probe-to-sample distance to aconstant value. Comparison with simulations (Fig. 3(a))suggests that this distance is around 30 nm, limited bythe diamond size and its position on the tip as well asby the tip oscillation. The PL image exhibits dark ar-eas with a contrast greater than 20% which reveal thetracks and the bits of the magnetic hard disk. We notethat the contrast depends on the field strength, whichitself depends on the number of bits that are consecu-tively parallel. For instance, from the known bit-to-bitspacing of 55 nm, we deduce that the line-cut in Fig. 3(b)corresponds to a sequence 001100, where 0 and 1 denotethe two possible bit magnetizations (Fig. 3(b)-(c)). Eachbit inversion gives rise to a stray field that appears darkon the PL image. We thus demonstrate a mode of NV-based magnetic field sensing that does not require theuse of microwave and is sensitive to the off-axis magneticfield magnitude.Summarizing, quantitative nanoscale magnetic field mapping with a sensitivity below 10 µ T / √ Hz was demon-strated using NV-based scanning probe magnetometry.Moreover, we reported a microwave-free magnetic imag-ing technique that extends the range of application ofNV-based magnetometry to large off-axis magnetic fields.Improvement of the sensitivity using pulsed-ESR tech-niques [19], together with a better control of the NV-to-sample distance, should enable imaging of few-spins sys-tems, which would be of great interest for fundamentalstudies in nanomagnetism and spintronics.The authors acknowledge L. Mayer, O. Klein, F. Maza-leyrat, M. Lo Bue, P. Bertet, A. Slablab, T. Gacoin,F. Treussart, S. Huant, O. Mollet and O. Arcizet forfruitful discussions. This work was supported by theAgence Nationale de la Recherche (ANR) through theproject D iamag , by C’Nano ˆIle-de-France and by RTRA-Triangle de la Physique (contract 2008-057T). [1] J. R. Kirtley, Rep. Prog. Phys. , 126501 (2010).[2] D. Budker and M. Romalis, Nature Phys. , 227 (2007).[3] L. Bogani and W. Wernsdorfer, Nature Mater. , 179-186(2008).[4] S. Heinze, M. Bode, A. Kubetzka, O. Pietzsch, X. Nie, S.Blgel, R. Wiesendanger, Science , 1805 (2000).[5] D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui,Nature , 329-332 (2004).[6] A. Finkler, Y. Segev, Y. Myasoedov, M. L. Rappaport, L.Neeman, D. Vasyukov, E. Zeldov, M. E. Huber, J. Martinand A. Yacoby Nano Lett. , 1046-1049 (2010).[7] W. Chao, J. Kim, S. Rekawa, P. Fischer, and E. H. An-derson, Opt. Exp. , 17669 (2009).[8] Y. Martin and H. K. Wickramasinghe, Appl. Phys. Lett. , 1455 (1987).[9] J. M. Garcia, A. Thiaville, J. Miltat, K. J. Kirk, J. N.Chapman, and F. Alouges, Appl. Phys. Lett. , 656(2001).[10] B. M. Chernobrod and G. P. Berman, J. Appl. Phys. ,014903 (2005).[11] C. L. Degen, Appl. Phys. Lett. , 243111 (2008).[12] J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D.Budker, P. R. Hemmer, A. Yacoby, R. Walsworth and M.D. Lukin, Nature Phys. , 810-816 (2008).[13] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hem-mer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Brats-chitsch, F. Jelezko and J. Wrachtrup, Nature , 648-651(2008).[14] J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M.Taylor, P. Cappellaro, L. Jiang, M. V. Gurudev Dutt, E. Togan, A. S. Zibrov, A. Yacoby, R. L. Walsworth and M.D. Lukin, Nature , 644-647 (2008).[15] See supplementary material at [URL will be inserted byAIP] for experimental details and numerical simulationsof the magnetic field distribution above the magnetic harddisk.[16] A. Cuche, A. Drezet, Y. Sonnefraud, O. Faklaris, F.Treussart, J.-F. Roch, and S. Huant , Opt. Exp. , 19969(2009).[17] L. Rondin, G. Dantelle, A. Slablab, F. Grosshans, F.Treussart, P. Bergonzo, S. Perruchas, T. Gacoin, M.Chaigneau, H.-C. Chang, V. Jacques and J.-F. Roch ,Phys. Rev. B , 115449 (2010).[18] A. Gr¨uber, A. Drabenstedt, C. Tietz, L. Fleury, J.Wrachtrup, and C. von Borczyskowski, Science , 2012-2014 (1997).[19] A. Dr´eau, M. Lesik, L. Rondin, P. Spinicelli, O. Ar-cizet, J.-F. Roch and V. Jacques, Phys. Rev. B, Preprint arXiv:1108.4437 .[21] R. S. Schoenfeld and W. Harneit, Phys. Rev. Lett. ,030802 (2011).[22] R. J. Epstein, F. M. Mendoza, Y. K. Kato, and D. D.Awschalom, Nature Phys. , 94-98 (2005).[23] N. D. Lai, D. Zheng, F. Jelezko, F. Treussart, and J.-F.Roch, Appl. Phys. Lett. , 133101 (2009). I. SUPPLEMENTARY METHODSA. Experimental setup
The experimental setup combines an atomic force microscope (AFM) and a confocal optical microscope (AFM/CFM,Attocube System) as shown in Fig. 4). Two sets of piezoelectric actuators and scanners are used for positioning thesample and the AFM tip. The AFM cantilever is an Akiyama-Probe (Nanosensors) corresponding to a quartz tuningfork equipped with a micromachined silicon cantilever. The cantilever ends with a sharp tip whose nominal curvatureradius is ≈
15 nm. A nanodiamond hosting a single nitrogen-vacancy (NV) defect is placed at the end of the tip asexplained in the next section.A continuous laser source operating at 532 nm wavelength (Spectra-Physics Excelsior) is tightly focused at the endof the AFM tip through a ×
100 microscope objective with a 0 . µ m diameter pinhole and directed to silicon avalanche photodiodes (Perkin-Elmer, SPCM-AQR-14) operating in the single-photon counting regime. Independently of the AFM scanners, precise positioningof the NV defect at the focus of the confocal microscope is achieved by scanning the laser beam with a fast steeringmirror (Newport, FSM-300-02) combined with a pair of telecentric lenses. The focal length is adjusted by using thepiezoelectric positioner of the AFM tip.The unicity of the emitter was checked by using a Hanbury Brown and Twiss (HBT) interferometer consistingof two avalanche photodiodes placed on the output ports of a 50 /
50 beamsplitter. The HBT detection system isassociated with a fast multichannel analyser (Picoquant, TimeHarp 300) for recording the histogram of time delaysbetween two consecutive single-photon detections. After normalization to a Poissonnian statistics, the recordedhistogram is equivalent to a measurement of the second-order autocorrelation function g (2) ( τ ) [1]. The observationof an anticorrelation effect at zero time delay, g (2) (0) < .
5, is the signature that a single NV defect is addressed (seeFig. 1(c) of the main text).For the electron spin resonance (ESR) measurements, a microwave synthesizer (Rohde & Schwarz, SMR20) is
FIG. 4: Schematic of the experimental setup, showing both the confocal optical microscope (upper part) and the tuning-forkbased AFM microscope (lower part). L stands for lens, F for filter, λ/ (resp. F ) is a 10 nm (resp. 75nm) width interference filter centered at 532 nm (resp. 697 nm). followed by a power amplifier (Mini-Circuits, ZHL-42) and connected to a 20- µ m-diameter copper wire directlyspanned on the sample surface.Scanning probe magnetometry is performed by monitoring the spin-dependent NV defect PL while scanning themagnetic sample. The tip-to-sample distance is adjusted using the piezoelectric scanner of the sample. For tappingmode operation, usual tuning-fork based AFM operation is used with a feedback loop adjusting the tip-to-sampledistance. B. Grafting a single NV defect at the end of the AFM tip
We started from commercially available diamond nanocrystals (SYP 0.05, Van Moppes SA, Geneva). Such nanocrys-tals are produced by milling type-Ib high-pressure high-temperature diamond crystals with a high nitrogen content([ N ] ≈
200 ppm). The formation of NV defects was carried out using high energy (13 . ◦ C under vacuum during two hours. The electron irradiation creates vacancies in thediamond matrix while the annealing procedure activates the migration of vacancies to intrinsic nitrogen impurities,leading to NV defect bonding. After annealing, the irradiated nanocrystals were oxidized in air at 550 ◦ C during twohours. This procedure reduces the size of the nanodiamonds [2] and leads to an efficient charge state conversion of thecreated NV defects into the negatively-charged state [3]. After sonication and washing with distilled water, aqueouscolloidal solutions of dispersed nanodiamonds, with a mean size of 20 nm, were obtained and finally deposited byspin-coating onto a silica coverslip.A nanodiamond (ND) hosting a single NV defect was grafted at the apex of the AFM tip following the methodintroduced by Cuche et al. [4, 5]. The tip was first dipped in a solution of poly-L-lysine (EMS, molecular weight30000-70000 u) for a few minutes. The tip as well as the sample of dispersed nanodiamonds were then mounted inthe experimental set-up, and the NDs were optically characterized. Once a ND hosting a single negatively-chargedNV defect was found, the sample was scanned with the AFM tip in tapping mode in a small area around the prese-lected ND. Electrostatic interactions between the polymer and the ND allow reproducible and robust grafting of thenanodiamond at the apex of the AFM tip.
C. Real-time tracking of the ESR frequency
The lock-in method allowing to track the shift of the ESR frequency ν R while scanning the magnetic sample isdescribed in Fig. 5. Two slightly different microwave frequencies ν + and ν − , defined by ν ± = ¯ ν ± ∆ ν/
2, are appliedconsecutively and the PL difference D (¯ ν ) = S ( ν + ) − S ( ν − ) is computed (Fig. 5(a) and (b)). This signal can beused as an error signal for tracking the ESR frequency [21] . Indeed, if ¯ ν = ν R + δ with δ smaller than the ESRlinewidth (FWHM), the error signal is at first order proportional to the frequency detuning δ , i.e. to the magnetic fieldvariation (Fig. 5(b)). Monitoring D (¯ ν ) thus enables quantitative magnetic field mapping by continuously tracking theESR frequency and measuring the frequency detuning from point to point, with a typical sampling time of 100 ms.Reliability of this method is demonstrated by monitoring in real-time the magnetic field created by an AC currentpassing through a coil with a frequency up to 1 Hz, as shown in Fig. 5(c). D. Estimation of the magnetic field sensitivity
Since the magnetic stray field of the investigated hard disk is a priori not known, the measured field map (Fig. 2(c)in the main text) cannot be directly compared to theory. However, the field is assumed to vary slowly with respectto the spatial sampling, i.e. the pixel size, so that high-frequency components are attributed to measurement noise.Therefore, we introduce a pixel-to-pixel noise to estimate the experimental error δB . If { B i,j } is the matrix cor-responding to the magnetic field map, with 1 ≤ i ≤ p and 1 ≤ j ≤ q , then δB can be defined as the standarddeviation δB = (cid:80) p − i =1 (cid:80) qj =1 (cid:16) B i +1 ,j − B i,j (cid:17) + (cid:80) pi =1 (cid:80) q − j =1 (cid:16) B i,j +1 − B i,j (cid:17) ( p − × q + p × ( q −
1) (3)which takes into account the pixel-to-pixel noise along both axis. For Fig. 2(c) of the main text, for which p = 138and q = 150, we find δB = 27 µ T. Given the acquisition time per pixel ∆ t = 110 ms, an experimental field sensitivity η expB = δB √ ∆ t = 9 µ T / √ Hz is obtained.
Estimate (a)(b)(c) ν + ν − D (¯ ν ) ¯ ν = ν R sourceMW ν R trackingcycle B z B z ( m T ) ¯ ν P L ( a . u . ) S ( ν + ) S ( ν − ) S ( ν ) -10010 P L d i ff e r en c e MW frequency (GHz) δ D ( ¯ ν )( k c n t s / s ) Time (s) ES R f r equen cy ( M H z ) FIG. 5: Real-time tracking of the ESR frequency. (a)-Optically detected ESR spectrum S ( ν ) and schematic of the field-frequency lock-in. (b)-Error signal D (¯ ν ) used for the feedback loop. Provided that δ < ∆ ESR , D (¯ ν ) = A × δ at first order,where A is experimentally determined and corresponds to the first derivative of D (¯ ν ) evaluated at the resonance frequency ν R .For each cycle of the tracking algorithm, D (¯ ν ) around the former ESR frequency is measured and the local magnetic field B z is deduced. (c)-Real-time measurement of the magnetic field created by a 0 . ν = 5 MHz and A = 5 × cnts.s − .MHz − . E. Determination of the NV defect axis
The NV defect axis, characterized by the unit vector ˆ z , can be determined by measuring the ESR spectrum whileapplying a static magnetic field along several well-chosen directions. Denoting θ and φ the polar and azimuthal anglesof the applied magnetic field in the reference frame xyz of the NV defect, the diagonalization of the electron spinHamiltonian (equation (1) of the main text) yields to the relation [6]∆ = 7 D + 2( ν + ν ) (cid:0) (cid:0) ν + ν (cid:1) − ν ν − E (cid:1) − D (cid:0) ν + ν − ν ν + 9 E (cid:1) ν + ν − ν ν − D − E ) , (4)where ν and ν are the Zeeman-shifted ESR frequencies, ∆ = D cos 2 θ + 2 E cos 2 φ sin θ while D and E are thezero-field splitting parameters of the NV ground-state electron spin. Since E (cid:28) D , one can use the approximation∆ ≈ D cos 2 θ .For a magnetic field applied along a given direction ˆ K , Eq. (4) gives the angle | θ K | between ˆ z and ˆ K . Repeatingthe measurement for two other directions of the magnetic field enables to completely determine ˆ z . Since the magneticimages presented in the main manuscript are XY maps of the magnetic field component | B z | , it is important to knowˆ z with respect to the reference frame XY Z of the laboratory. The orientation of the NV defect used in Fig. 2 and 3 ofthe main text was determined following the latter method. Fig. 6(b) shows the evolution of the measured resonancefrequencies ν and ν as a function of a static magnetic field produced by two coils in an Helmholtz configuration.This field was first aligned along the X -axis (filled circles) and then along the Y -axis (open circles). From thesemeasurements and Eq. (4), the angles | θ X | = 37 ± ◦ and | θ Y | = 53 ± ◦ were obtained. As | θ X | + | θ Y | = 90 ◦ , onededuces that ˆ z lies in the XY plane. The actual orientation of the NV axis in the reference frame XY Z is illustratedin Fig. 6(a). The other orientation compatible with the set ( | θ X | , | θ Y | ) was ruled out by a measurement along a thirddirection. (cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3) (cid:5) (cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:5)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:6) (cid:1) (cid:6) (cid:1) (cid:5) (cid:14)(cid:15)(cid:10)(cid:11)(cid:12)(cid:13)(cid:1)(cid:16)(cid:17)(cid:3)(cid:18) (cid:19) (cid:20) (cid:21)(cid:22) (cid:21)(cid:23)(cid:22)(cid:24)(cid:23)(cid:22)(cid:24)(cid:20)(cid:25)(cid:24)(cid:18)(cid:25)(cid:24)(cid:22) (cid:26) (cid:27) (cid:28) (cid:13) (cid:29) (cid:30) (cid:31) (cid:32) (cid:33) (cid:31) (cid:34) (cid:15) (cid:11) (cid:31) (cid:35) (cid:13) (cid:1) (cid:36) (cid:37) (cid:38) (cid:3) (cid:1) (cid:5) (cid:39)(cid:40)(cid:25) (cid:43)(cid:21) (cid:21)(cid:41) (cid:1) (cid:6) (cid:39)(cid:25)(cid:42) (cid:43)(cid:21) (cid:21)(cid:41) (a) (b) ES R f r equen c i e s ( G H z ) FIG. 6: Measurement of the NV defect orientation using Helmholtz coils. (a)-Measuring the ESR spectrum while applying astatic magnetic field along X (resp. Y ) gives access to the angle | θ X | (resp. | θ Y | ) between the NV axis (red arrow) and the X -axis (resp. Y -axis). (b)-ESR frequencies ν and ν as a function of the magnetic field magnitude. The filled (resp. open)black circles are the experimental data for a static magnetic field applied along X (resp. Y ). The angles | θ X | and | θ Y | wereestimated using Eq. (4). The solid lines are the theoretical curves using the full diagonalization of the spin Hamiltonian andthe calculated values of | θ X | and | θ X | , considering a field oriented along X (green lines) and Y (blue lines), respectively. II. MODELLING OF THE MAGNETIC FIELD DISTRIBUTION ABOVE THE HARD DISK
The magnetic sample used for the demonstration of magnetic field mapping is a piece of a commercial magnetichard disk (Maxtor, 20 Gb). The geometry of the magnetic bits is depicted in Fig. 7. AFM topography of the sampleallows to infer the geometrical orientation of the bits since grooves imprinted by the read/write head indicate thedirection of the tracks (Fig. 7(a)). This information is used to determine the NV orientation with respect to the bitsorientation.The hard disk was modeled as an array of uniformly magnetized parallelepipeds (the ‘bits’). The dimensions ofeach ‘bit’ are 2 a = 40 nm, 2 b = 400 nm and 2 c = 10 nm, along X , Y and Z , respectively, and their magnetizationis M = ± M s ˆ X depending whether the bit is at ‘0’ or ‘1’ (Fig. 7(b)-(c)). M s is the average magnetization of theferromagnetic material. Typical value for the studied hard disk is µ M s = 0 . µ is the permeability of freespace. The pitch of the array is 55 nm along X and 450 nm along Y . (b) (c)(a) FIG. 7: Modelling the magnetic hard disk. (a)-AFM Topography of a 2 × µ m area of the hard disk. (b)-Model used for the20 Gb hard disk. Each bit is magnetized along ± ˆ X and has dimensions 400 x 40 x 10 nm. The bits are spaced by 450 nmalong Y (track width) and 55 nm along X . (c)- A parallelepiped uniformely magnetized along X is equivalent to a magnetic“capacitor” with two charged surfaces located at X = ± a . The magnetic field distribution produced by a single bit (Fig. 7(c)) can be calculated using magnetostatics theory.Outside the parallelepiped, the three components of the magnetic field read as B X = − µ M s π (cid:88) i = ± (cid:88) j = ± (cid:88) k = ± ijk tan − ( Y + jb )( Z + kc )( X + ia ) (cid:112) ( X + ia ) + ( Y + jb ) + ( Z + kc ) (5) B Y = − µ M s π ln (cid:89) i = ± (cid:89) j = ± (cid:89) k = ± (cid:32) ( Z + kc ) − (cid:112) ( X + ia ) + ( Y + jb ) + ( Z + kc ) ( Z + kc ) + (cid:112) ( X + ia ) + ( Y + jb ) + ( Z + kc ) (cid:33) ijk (6) B Z = − µ M s π ln (cid:89) i = ± (cid:89) j = ± (cid:89) k = ± (cid:32) ( Y + jb ) − (cid:112) ( X + ia ) + ( Y + jb ) + ( Z + kc ) ( Y + jb ) + (cid:112) ( X + ia ) + ( Y + jb ) + ( Z + kc ) (cid:33) ijk . (7)The magnetic fields produced by each bit of the array were then summed up in order to compute the whole fielddistribution above the hard disk. Since the exact distribution of magnetic bit orientations is unknown, we computethe magnetic field distributions along X , Y and Z directions for a random distribution of magnetic bits (Figure 8(a)).Figure 8(b) indicates the magnetic field distribution at 250 nm from the hard disk, once projected along the axis ofthe NV defect used in the scanning-probe magnetometry experiments. The magnetic field follows rounded non trivialpatterns with characteristics size and amplitude similar to the one observed experimentally (Fig. 2(c) of the maintext). We note that the magnetic field distribution at another distance or along the Y axis follows a clearly different (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:3)(cid:4)(cid:5)(cid:6)(cid:8)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:4)(cid:5)(cid:6) (cid:1) (cid:9)(cid:2)(cid:3)(cid:3)(cid:2)(cid:3)(cid:9)(cid:1)(cid:3)(cid:3)(cid:1)(cid:3)(cid:2)(cid:3)(cid:9)(cid:2)(cid:7)(cid:3)(cid:9)(cid:7) (cid:3)(cid:10)(cid:2)(cid:3)(cid:10)(cid:3)(cid:9)(cid:3)(cid:10)(cid:2)(cid:8)(cid:1)(cid:3)(cid:9)(cid:1)(cid:7)(cid:3)(cid:3)(cid:9)(cid:7)(cid:3)(cid:8)(cid:3)(cid:1)(cid:3)(cid:3)(cid:9)(cid:1)(cid:3) (cid:2)(cid:3)(cid:3)(cid:9)(cid:2)(cid:3)(cid:1)(cid:3)(cid:3)(cid:9)(cid:1)(cid:3)(cid:2)(cid:3)(cid:9)(cid:2)(cid:7)(cid:3)(cid:9)(cid:7) (cid:1)(cid:2)(cid:2)(cid:3)(cid:4)(cid:5) (cid:1) (cid:4)(cid:4)(cid:11)(cid:6)(cid:12)(cid:13) (cid:2) (cid:1) (cid:4)(cid:4)(cid:11)(cid:6)(cid:12)(cid:13) (cid:3) (cid:1) (cid:4)(cid:4)(cid:11)(cid:6)(cid:12)(cid:13) (cid:4) (a) (b) | B z | ( m T ) FIG. 8: (a)-Calculated magnetic field distributions above a randomly written area of the hard disk. The three components B X , B Y and B Z (columns 1, 2 and 3, respectively) are calculated at various distances d from the surface, from 10 nm to 250nm (from bottom to top). The plots are 1.8 x 1.8 µ m in size, but the array used for the calculation is 9 times larger (5.4 x 5.4 µ m) in order to avoid edge effects. (b)-Simulation of the magnetic field created by a random distribution of bit magnetizationsfor d = 250 nm, once projected along the NV defect axis used in the experiments described in the main text of the article. Themagnetic bits are oriented as shown in the AFM topography (Fig. 4(a)). [1] R. Brouri, A. Beveratos, J.-P. Poizat, and P. Grangier, Opt. Lett. , 1294 (2000).[2] T. Gaebel, C. Bradac, J. Chen, P. Hemmer, and J. R. Rabeau, preprint arXiv:1104.5075 (2011).[3] L. Rondin et al. , Phys. Rev. B , 115449 (2010).[4] A. Cuche et al. , Opt. Exp. , 19969 (2009).[5] A. Cuche, A. Drezet, J.-F. Roch, F. Treussart, and S. Huant, J. Nanophoton. , 043506 (2010).[6] G. Balasubramanian et al. , Nature , 648-651 (2008).[7] J. M. Garcia, A. Thiaville, J. Miltat, K. J. Kirk, J. N. Chapman, and F. Alouges, Appl. Phys. Lett.79