The nature of domain walls in ultrathin ferromagnets revealed by scanning nanomagnetometry
J.-P. Tetienne, T. Hingant, L. J. Martinez, S. Rohart, A. Thiaville, L. Herrera Diez, K. Garcia, J.-P. Adam, J.-V. Kim, J.-F. Roch, I. M. Miron, G. Gaudin, L. Vila, B. Ocker, D. Ravelosona, V. Jacques
TThe nature of domain walls in ultrathin ferromagnetsrevealed by scanning nanomagnetometry
J.-P. Tetienne , † , T. Hingant , † , L. J. Martinez , S. Rohart , A. Thiaville ,L. Herrera Diez , K. Garcia , J.-P. Adam , J.-V. Kim , J.-F. Roch , I. M.Miron , G. Gaudin , L. Vila , B. Ocker , D. Ravelosona and V. Jacques ∗ Laboratoire Aimé Cotton, CNRS, UniversitéParis-Sud and ENS Cachan, 91405 Orsay, France Laboratoire de Physique des Solides,Université Paris-Sud and CNRS UMR 8502, 91405 Orsay, France Institut d’Electronique Fondamentale,Université Paris-Sud and CNRS UMR 8622, 91405 Orsay, France INAC-SPINTEC, Université Grenoble Alpes,CNRS and CEA, 38000 Grenoble, France INAC, CEA and Université Grenoble Alpes, 38054 Grenoble, France Singulus Technology AG, Hanauer Landstrasse 103, 63796 Kahl am Main, Germany † ∗ [email protected] † These authors contributed equally to this work. a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t he recent observation of current-induced domain wall (DW) motion withlarge velocity in ultrathin magnetic wires has opened new opportunities for spin-tronic devices [1]. However, there is still no consensus on the underlying mech-anisms of DW motion [1–6]. Key to this debate is the DW structure, which canbe of Bloch or Néel type, and dramatically affects the efficiency of the differentproposed mechanisms [7–9]. To date, most experiments aiming to address thisquestion have relied on deducing the DW structure and chirality from its motionunder additional in-plane applied fields, which is indirect and involves strongassumptions on its dynamics [2–4, 10]. Here we introduce a general methodenabling direct, in situ , determination of the DW structure in ultrathin ferro-magnets. It relies on local measurements of the stray field distribution above theDW using a scanning nanomagnetometer based on the Nitrogen-Vacancy defectin diamond [11–13]. We first apply the method to a Ta/Co Fe B (1 nm)/MgOmagnetic wire and find clear signature of pure Bloch DWs. In contrast, we ob-serve left-handed Néel DWs in a Pt/Co(0.6 nm)/AlO x wire, providing directevidence for the presence of a sizable Dzyaloshinskii-Moriya interaction (DMI)at the Pt/Co interface. This method offers a new path for exploring interfacialDMI in ultrathin ferromagnets and elucidating the physics of DW motion undercurrent. In wide ultrathin wires with perpendicular magnetic anisotropy (PMA), magnetostaticspredicts that the Bloch DW, a helical rotation of the magnetization, is the most stable DWconfiguration because it minimizes volume magnetic charges [14]. However, the unexpectedlylarge velocities of current-driven DW motion recently observed in ultrathin ferromagnets [1],added to the fact that the motion can be found against the electron flow [2, 3], has castdoubt on this hypothesis and triggered a major academic debate regarding the underlyingmechanism of DW motion [4–9]. Notably, it was recently proposed that Néel DWs withfixed chirality could be stabilized by the Dzyaloshinskii-Moriya interaction (DMI) [7], anindirect exchange possibly occurring at the interface between a magnetic layer and a heavymetal substrate with large spin-orbit coupling [15]. For such chiral DWs, hereafter termedDzyaloshinskii DWs, a damping-like torque due to spin-orbit terms (spin-Hall effect andRashba interaction) would lead to efficient current-induced DW motion along a directionfixed by the chirality [7]. In order to validate unambiguously these theoretical predictions,2 direct, in situ , determination of the DW structure in ultrathin ferromagnets is required.However, the relatively small number of spins at the wall center makes direct imaging ofits inner structure a very challenging task. So far, only spin-polarized scanning tunnellingmicroscopy [16] and spin-polarized low energy electron microscopy [17] have allowed a directdetermination of the DW structure, demonstrating homochiral Néel DWs in Fe double layeron W(110) and in (Co/Ni) n multilayers on Pt or Ir, respectively. However, these techniquesare intrinsically limited to model samples due to high experimental constraints and thedebate remains open for widely used trilayer systems with PMA such as Pt/Co/AlO x [1],Pt/Co/Pt [4] or Ta/CoFeB/MgO [18].Here we introduce a general method which enables determining the nature of a DW invirtually any ultrathin ferromagnet. It relies on local measurements of the stray magneticfield produced above the DW using a scanning nanomagnetometer. To convey the basicidea behind our method, we start by deriving analytical formulas of the magnetic fielddistribution at a distance d above a DW placed at x = 0 in a perpendicularly magnetizedfilm [Fig. 1a]. The main contribution to the stray field, denoted B ⊥ , is provided by theabrupt variation of the out-of-plane magnetization M z ( x ) = − M s tanh( x/ ∆ DW ) [14], where M s is the saturation magnetization and ∆ DW is the DW width parameter. The resultingstray field components can be expressed as B ⊥ x ( x ) ≈ µ M s tπ dx + d B ⊥ z ( x ) ≈ − µ M s tπ xx + d , (1)where t is the film thickness. These approximate formulas are valid in the limit of (i) t (cid:28) d ,(ii) ∆ DW (cid:28) d and (iii) for an infinitely long DW along the y axis. On the other hand, thein-plane magnetization, with amplitude M (cid:107) ( x ) = M s / cosh( x/ ∆ DW ) , can be oriented withan angle ψ with respect to the x axis [Fig. 1b]. This angle is linked to the nature of theDW: ψ = ± π/ for a Bloch DW, whereas ψ = 0 or π for a Néel DW. The two possiblevalues define the chirality (right or left) of the DW. The spatial variation of the in-planemagnetization adds a contribution B (cid:107) cos ψ to the stray field, whose components are givenby B (cid:107) x ( x ) ≈ µ M s t ∆ DW x − d ( x + d ) B (cid:107) z ( x ) ≈ µ M s t ∆ DW xd ( x + d ) . (2)3he net stray field above the DW is finally expressed as B ψ ( x ) = B ⊥ ( x ) + B (cid:107) ( x ) cos ψ , (3)which indicates that a Néel DW ( cos ψ = ± ) produces an additional stray field owing toextra magnetic charges on each side of the wall. Using Eqs. (1) and (2), we find a maximumrelative difference in stray field between Bloch and Néel DWs scaling as ≈ π ∆ DW / d . Localmeasurements of the stray field above a DW can therefore reveal its inner structure, char-acterized by the angle ψ . This is further illustrated in Figs. 1(c,d), which show the strayfield components B ψx ( x ) and B ψz ( x ) for various DW configurations while using d = 120 nmand ∆ DW = 20 nm, which are typical parameters of the experiments considered below on aTa/CoFeB(1nm)/MgO trilayer system.We now demonstrate the effectiveness of the method by employing a single Nitrogen-Vacancy (NV) defect hosted in a diamond nanocrystal as a nanomagnetometer operatingunder ambient conditions [11–13]. Here, the local magnetic field is evaluated within anatomic-size detection volume by monitoring the Zeeman shift of the NV defect electronspin sublevels through optical detection of the magnetic resonance. After grafting thediamond nanocrystal onto the tip of an atomic force microscope (AFM), we obtain ascanning nanomagnetometer which provides quantitative maps of the stray field emanat-ing from nanostructured samples [19–21] with a magnetic field sensitivity in the range of µ T.Hz − / [22]. In this study, the Zeeman frequency shift ∆ f NV of the NV spin ismeasured while scanning the AFM tip in tapping mode, so that the mean distance be-tween the NV spin and the sample surface is kept constant with a typical tip oscillationamplitude of a few nanometers [20]. Each recorded value of ∆ f NV is a function of B NV , (cid:107) and B NV , ⊥ , which are the parallel and perpendicular components, respectively, of the localmagnetic field with respect to the NV spin’s quantization axis (Supplementary Section I).Note that a frequently found approximation is ∆ f NV ≈ gµ B B NV , (cid:107) /h , where gµ B /h ≈ GHz/T. This indicates that scanning-NV magnetometry essentially measures the projection B NV , (cid:107) of the magnetic field along the NV center’s axis. The latter is characterized by spheri-cal angles ( θ , φ ), measured independently in the ( xyz ) reference frame of the sample [Fig. 2a].We first applied our method to determine the structure of DWs in a 1.5- µ m-wide magnetic4ire made of a Ta(5 nm)/Co Fe B (1 nm)/MgO(2 nm) trilayer stack (SupplementarySection II). This system has been intensively studied in the last years owing to low dampingparameter and depinning field [23]. Before imaging a DW, it is first necessary to determineprecisely (i) the distance d between the NV probe and the magnetic layer and (ii) the prod-uct I s = M s t , which are both directly involved in Eq. (3). These parameters are obtainedby performing a calibration measurement above the edges of an uniformly magnetized wire,as shown in Fig. 2a. Here we use the fact that the stray field profile B edge ( x ) above an edgeplaced at x = 0 can be easily expressed analytically in a form similar to Eq. (10), which onlydepends on d and I s . An example of a measurement obtained by scanning the magnetometeracross a Ta/CoFeB/MgO stripe is shown in Fig. 2b. The data are fitted with a functioncorresponding to the Zeeman shift induced by the stray field B edge ( x ) − B edge ( x + w c ) , where w c is the width of the stripe (Supplementary Section III-A). Repeating this procedure for aset of independent calibration linecuts, we obtain d = 123 ± nm and I s = 926 ± µ A, ingood agreement with the value measured by other methods [24].Having determined all needed parameters, it is now possible to measure the stray fieldabove a DW [Fig. 2c] and compare it to the theoretical prediction, which only depends onthe angle ψ that characterizes the DW structure. To this end, an isolated DW was nucleatedin a wire of the same Ta/CoFeB/MgO film and imaged with the scanning-NV magnetometerunder the same conditions as for the calibration measurements. The resulting distributionof the Zeeman shift ∆ f NV is shown in Fig. 2d together with the AFM image of the magneticwire. Within the resolving power of our instrument, limited by the probe-to-sample distance d ∼ nm [20], the DW appears to be straight with a small tilt angle with respect to thewire long axis, determined to be ± ◦ (Supplementary Section III-B). Taking into accountthis DW spatial profile, the stray field above the DW was computed for (i) ψ = 0 (right-handed Néel DW), (ii) ψ = π (left-handed Néel DW) and (iii) ψ ± π/ (Bloch DW). Herewe used the micromagnetic OOMMF software [25, 26] rather than the analytical formuladescribed above in order to avoid any approximation in the calculation. The computedmagnetic field distributions were finally converted into Zeeman shift distribution taking intoaccount the NV spin’s quantization axis. A linecut of the experimental data across the DWis shown in Fig. 2e, together with the predicted curves in the three above-mentioned cases.Excellent agreement is found if one assumes that the DW is purely of Bloch type. The sameconclusion can be drawn by directly comparing the full two-dimensional theoretical maps5o the data [Fig. 2d and f]. As described in detail in the Supplementary Section III-C, allsources of uncertainty in the theoretical predictions were carefully analysed, yielding the 1standard error (s.e.) intervals shown as shaded areas in Fig. 2e. Based on this analysis,we find a 1 s.e. upper limit | cos ψ | < . . This corresponds to an upper limit for theDMI parameter D DMI , as defined in Ref. [7], of | D DMI | < . mJ/m (SupplementarySection III-C). This result was confirmed on a second DW in the same wire. In addition,the measurements were reproduced for different projection axes of the NV probe. Theresults are shown in Fig. 3 for four NV defects with different quantization axes, showingexcellent agreement between experiment and theory if one assumes a Bloch-type DW. Theseexperiments provide an unambiguous confirmation of the Bloch nature of the DWs in oursample, but are also a striking illustration of the vector mapping capability offered by NVmicroscopy, allowing for robust tests of theoretical predictions.We conclude that there is no evidence for the presence of a sizable interfacial DMI in aTa(5nm)/Co Fe B (1nm)/MgO trilayer stack. This is in contrast with recent experimentsreported on similar samples with different compositions, such as Ta(5nm)/Co Fe (0.6nm)/MgO [3,27] and Ta(0.5 nm)/Co Fe B (1nm)/MgO [18], where indirect evidence for Néel DWs wasfound through current-induced DW motion experiments. We note that contrary to thesestudies, our method indicates the nature of the DW at rest, in a direct manner, withoutany assumption on the DW dynamics. Our results therefore motivate a systematic study ofthe DW structure upon modifications of the composition of the trilayer stack.In a second step, we explored another type of sample, namely a Pt(3nm)/Co(0.6nm)/AlO x (2nm) trilayer grown by sputtering on a thermally oxidized silicon wafer (Sup-plementary Section II). The observation of current-induced DW motion with unexpectedlylarge velocities in this asymmetric stack has attracted considerable interest in the recentyears [1]. Here, the DW width is ∆ DW ≈ nm, leading to a relative field difference betweenBloch and Néel cases of ≈ at a distance d ≈ nm. We followed a procedure similarto that described above (Supplementary Section III). After a preliminary calibration of theexperiment, a DW in a 500-nm-wide magnetic wire was imaged [Fig. 4a,b] and linecutsacross the DW were compared to theoretical predictions [Fig. 4c]. Here the experimentalresults clearly indicate a Néel-type DW structure with left-handed chirality. The sameresult was found for two other DWs. This provides direct evidence of a strong DMI at the6t/Co interface, with a lower bound | D DMI | > . mJ/m . This result is consistent withthe conclusions of recent field-dependent DW nucleation experiments performed in similarfilms [28]. In addition, we note that the observed left-handed chirality, once combined witha damping-like torque induced by the spin-orbit terms, could explain the characteristics ofDW motion under current in this sample [8].In conclusion, we have shown how scanning-NV magnetometry enables direct discrim-ination between competing DW configurations in ultrathin ferromagnets. This method,which is not sensitive to possible artifacts linked to the DW dynamics, will help clarifyingthe physics of DW motion under current, a necessary step towards the development ofDW-based spintronic devices. In addition, this work opens a new avenue for studying themechanisms at the origin of interfacial DMI in ultrathin ferromagnets, by measuring the DWstructure while tuning the properties of the magnetic material [18, 29]. This is a key mile-stone in the search for systems with large DMI that could sustain magnetic skyrmions [30]. Aknowledgements . This research has been partially funded by the European Commu-nity’s Seventh Framework Programme (FP7/2007-2013) under Grant Agreement n ◦ iadems ) and n ◦ agwire ), the French Agence Nationale de la Recherchethrough the projects D iamag and E sperado , and by C’Nano Ile-de-France (N anomag ). Author contributions . S.R. and A.T. conceived the idea of the study. J.P.T., T.H.,L.J.M. and V.J. performed the experiments, analysed the data and wrote the manuscript.L.H.D, K.G., J.P.A., G.G., L.V., and B.O. prepared the samples. All authors discussed thedata and commented on the manuscript. 7 loch yz DW z cdba y x t Néel (right handed) x Intermediate ψ BlochNéel RightNéel Left B x [ m T ] -300 0 300x [nm] x (nm) B x ( m T ) -3-2-10123 B z [ m T ] -300 0 300x [nm] B z ( m T ) x (nm) BlochNéel RightNéel Left
Figure 1.
Determining the nature of a DW by scanning nanomagnetometry. a.
Schematicside view of a DW in a perpendicularly magnetized film. The black arrows indicate the internalmagnetization while the grey arrows represent the magnetic field lines generated above the film. b. Top view of the DW structure in a left-handed Bloch (top panel) or right-handed Néel (bottompanel) configuration, or an intermediate case characterized by the angle ψ . c,d. Calculated strayfield components B ψx ( x ) ( c ) and B ψz ( x ) ( d ) at a distance d = 120 nm above the magnetic layer,with a DW centered at x = 0 . Here we use M s = 10 A/m and ∆ DW = 20 nm. b z xz ! = 62° " = - 25° N V s p i n d x Data Left Néel Bloch Right Néel
ESR freq. (MHz)
Position x ( µ m) H e i gh t ( n m ) THEORY R-Néel d AFM f Z ee m an s h i ft ( M H z ) xyz H e i gh t ( n m ) c e x ESR freq. (MHz)
EXPERIMENT NV-image D W C a li b r a t i on D W m ea s u r e m en t BlochL-Néel - Height (nm) ! !
500 nm Z ee m an s h i ft ( M H z ) -1 0 1 Position x ( ! m) Position x ( µ m) Data Fit θ = 62 ◦ φ = − ◦ x z Z ee m an s h i ft ( M H z ) Figure 2.
Observation of Bloch DWs in a Ta/CoFeB/MgO wire. a.
The unknown param-eters (distance d and product I s = M s t ) are first calibrated by recording the stray field above auniformly magnetized stripe. The inset defines the spherical angles ( θ , φ ) characterizing the NVspin’s quantization axis, measured independently. b. Zeeman shift of the NV spin measured as afunction of x , across a 1.5- µ m-wide stripe of Ta/CoFeB(1 nm)/MgO. The data (markers) are fittedto the theory (solid line), yielding d = 123 ± nm and I s = 926 ± µ A. c. The stray field above aDW is then measured under the same conditions (same distance d , same NV spin). d. AFM image(left panel) and corresponding Zeeman shift map (right panel) recorded on a 1.5- µ m-wide stripecomprising a single DW. e. Linecut across the DW (see dashed line in the inset). The markersare the experimental data, while the solid lines are the theoretical predictions for a Bloch (red), aleft-handed Néel (blue) and a right-handed Néel DW (green). The shaded areas show 1 standarderror in the simulations due to uncertainties in the parameters (Supplementary Section III-C). f. Theoretical two-dimensional Zeeman shift maps for the same three DW configurations. In both e and f , the Bloch hypothesis is the one that best reproduces the data. z ! = 99° " = -65° xz ! = 102° " = 27° xz ! = 113° " = -81° xz ! = 42° " = -7° abcd . . . ESR freq. (GHz) Z ee m an s h i ft ( M H z ) Z ee m an s h i ft ( M H z ) Z ee m an s h i ft ( M H z ) Z ee m an s h i ft ( M H z ) Z ee m an s h i ft ( M H z ) φ = − ◦ φ = 27 ◦ φ = − ◦ φ = − ◦ . . . . ESR freq. (GHz) . . . ESR freq. (GHz) . . . ESR freq. (GHz) Z ee m an s h i ft ( M H z ) Z ee m an s h i ft ( M H z ) Z ee m an s h i ft ( M H z ) xyzxz xz xz xz Experiment Theory (Bloch)
500 nm
Figure 3.
Vector mapping of the DW stray field.
Zeeman shift maps above a DW in aTa/CoFeB/MgO wire recorded with different magnetometer’s projection axes (middle panels). Theright panels shows the corresponding calculated maps assuming a Bloch-type DW. The distance d is 116 nm ( a ), 122 nm ( b ), 195 nm ( c ), and 178 nm ( d ), respectively. The spherical angles ( θ , φ )characterizing the projection axis are indicated on the left panels. Z ee m an s h i ft ( M H z ) ! !
200 nm
AFM H e i gh t ( n m ) a b Z ee m an s h i ft ( M H z ) c Data Left Néel Bloch Right Néel
Position x ( µ m) NV-image
200 nm Pt / Co(0 . / AlO x Figure 4.
Observation of left-handed Néel DWs in a Pt/Co/AlO x wire. (a) AFM imageand (b) corresponding Zeeman shift map recorded by scanning the NV magnetometer above a DWin a 500-nm-wide magnetic wire of Pt/Co(0.6 nm)/AlO x . The spherical angles characterizing theNV defect quantization axis are ( θ = 87 ◦ , φ = 23 ◦ ) . From calibration measurements above edges ofthe sample, we inferred d = 119 . ± . nm and I s = 671 ± µ A (Supplementary Section III-A). c. Linecut extracted from b (markers), together with the theoretical prediction (solid lines) fora Bloch (red), a left-handed Néel (blue) and a right-handed Néel DW (green). The shaded areasindicates 1 s.e. uncertainty in the simulations (Supplementary Section III-C). upplementary Informations I. SCANNING-NV MAGNETOMETRY
The experimental setup combines a tuning-fork-based atomic force microscope (AFM)and a confocal optical microscope (attoAFM/CFM, Attocube Systems), all operating underambient conditions. A detailed description of the setup as well as the method to graft adiamond nanocrystal onto the apex of the AFM tip can be found in Ref. [22].
A. Characterization of the magnetic field sensor
The data reported in this work were obtained with NV center magnetometers hosted inthree different nanodiamonds, labeled ND74 (data of Figure 3 of the main paper), ND75(Figure 2) and ND79 (Figure 4). All nanodiamonds were ≈ nm in size, as measured byAFM before grafting the nanodiamond onto the AFM tip. The magnetic field was inferredby measuring the Zeeman shift of the electron spin resonance (ESR) of the NV center’sground state [13]. This is achieved by monitoring the spin-dependent photoluminescence(PL) intensity of the NV defect while sweeping the frequency of a CW radiofrequency (RF)field generated by an antenna fabricated directly on the sample.The Hamiltonian used to describe the magnetic-field dependence of the two ESR transi-tions of this S = 1 spin system is given by H = hDS Z + hE ( S X − S Y ) + gµ B B · S , (4)where D and E are the zero-field splitting parameters that characterize a given NV center, h is the Planck constant, gµ B /h = 28 . GHz/T [31], B is the local magnetic field and S is the dimensionless S = 1 spin operator. Here, the ( XY Z ) reference frame is defined bythe diamond crystal orientation, with Z being parallel to the NV center’s symmetry axis u NV , as shown in Figure 5(a).The two ESR frequencies are denoted f + and f − and the Zeeman shifts are definedby ∆ f ± = f ± − D [Fig. 5(b)]. In general, ∆ f ± are functions of B NV , (cid:107) = | B · u NV | and B NV , ⊥ = (cid:107) B × u NV (cid:107) . However, in the limit of small transverse fields ( gµ B B NV , ⊥ (cid:28) hD )[32], they depend only on the magnetic field projection along the NV axis B NV , (cid:107) , following12 z ! = 42° " = -7° z φ = − ◦ u NV a θ b m s − D f − f + − f − f + ∆ f NV E B = 0 B ! = 0 f − f + c ND75 B = 0 P L ( a . u . ) RF frequency (GHz)
Figure 5. (a) The quantization axis u NV of the NV center’s electron spin is characterized byspherical angles ( θ , φ ) in the ( xyz ) sample reference frame. (b) Structure of the spin sublevels inthe NV defect’s ground state. The ESR frequencies corresponding to the electron spin transitions m s = 0 → − and m s = 0 → +1 are denoted by f − and f + , respectively. (c) ESR spectrum ofND75 in zero external magnetic field. A fit to a sum of two Gaussian functions allows determiningthe parameters D and E that characterize the NV center. The values are given in Table I. the relation ∆ f ± ( B NV , (cid:107) ) ≈ ± (cid:113) ( gµ B B NV , (cid:107) /h ) + E . (5)The parameters D and E were extracted from ESR spectra recorded at zero magnetic fieldusing the fact that f ± ( B = ) = D ± E [see Fig. 5(b,c)]. In all the data shown in this work,only the upper frequency f + was measured. Thereafter, we will note the correspondingZeeman shift ∆ f NV = f + − D , the subscript ‘NV’ reminding that it depends on the direc-tion u NV [Fig. 5(b)]. The experimental measurements of ∆ f NV were compared to theoryby calculating the expected Zeeman shift through full diagonalization of the Hamiltoniandefined by Eq. (4), given the theoretical B map. However, we note that since the condition gµ B B NV , ⊥ (cid:28) hD is usually fulfilled in our measurements, the formula (5) is approximatelyvalid, so that in principle one could retrieve directly the value of B NV , (cid:107) with good accuracy( < . mT). 13he nanodiamonds were recycled several times to be used with different orientations u NV with respect to the ( xyz ) reference frame of the sample. The various orientations are labeledwith small letters: ND74a, ND74d, ND74e, ND74g, ND75c, ND79c. The spherical angles( θ , φ ) that characterize the direction u NV were obtained by applying an external magneticfield of known direction and amplitude with a three-axis coil system, following the proceduredescribed in Ref. [19]. The measurement uncertainty of ◦ (standard error) is related to theprecision of the calibration of the coils and their alignment with respect to the ( xyz ) referenceframe.Table I indicates the parameters D , E , θ and φ measured for each NV magnetometerused in this work, with the associated standard errors. ND74a ND74d ND74e ND74g ND75c ND79cFigure 3(a) 3(b) 3(c) 3(d) 2 4 D ( ± . MHz) 2867.1 2869.5 2866.6 E ( ± . MHz) 3.1 3.3 4.3 θ ( ± ◦ ) ◦ ◦ ◦ ◦ ◦ ◦ φ ( ± ◦ ) − ◦ ◦ − ◦ − ◦ − ◦ ◦ Table I. Summary of the parameters (
D, E, θ, φ ) measured for the different NV center magne-tometers used in this work. The second row mentions the figures of the main paper where themagnetometer is used.
B. Quantitative stray field mapping
The experimental Zeeman shift maps were obtained by recording ESR spectra whilescanning the magnetometer with the AFM operated in tapping mode. Each spectrum iscomposed of 11 bins with a bin size of 2 MHz, leading to a full range of 20 MHz. Theintegration time per bin is 40 ms, hence 440 ms per spectrum, that is, 440 ms per pixelof the image. As illustrated in Figure 6, only the upper frequency f + is probed, and themeasurement window is shifted from pixel to pixel in order to track the resonance. Eachspectrum is then fitted with a Gaussian lineshape to obtain f + and thus ∆ f NV . The full14 L ( a . u . ) . . . . ESR freq. (GHz)
500 nm
20 40 60 80Zeeman shift (MHz)
Zeeman shift (MHz)
Figure 6. Map of the Zeeman shift ∆ f NV obtained with ND74d above the domain wall (reproducedfrom Fig. 3b of the main paper), along with the raw ESR spectra corresponding to 3 differentselected pixels. Solid lines are Gaussian fits. width at half maximum (FWHM) is typically 5-10 MHz, and the standard error on f + is ≈ . MHz with the above-mentioned acquisition parameters.
II. SAMPLES
Two samples, Ta/CoFeB/MgO and Pt/Co/AlO x , were investigated in this work. TheTa/CoFeB/MgO trilayer was deposited on a Si/SiO (100 nm) wafer using a PVD Timaris de-position tool by Singulus Tech. The film stack composition is Ta(5)/CoFeB(1)/MgO(2)/Ta(5),starting from the SiO layer (units in nanometer). The stoichiometric composition ofthe as-deposited magnetic layer is Co Fe B . The second sample was fabricated fromPt(3)/Co(0.6)/Al(1.6) layers deposited on a thermally oxidized silicon wafer by d.c. mag-netron sputtering. After deposition, the aliminium layer was oxidized by exposure to anoxygen plasma. Both samples were patterned using e-beam lithography and ion milling. Asecond step of e-beam lithography was finally performed in order to define a gold striplinefor RF excitation, which is used to record the Zeeman shift of the NV defect magnetometer[cf. section I A].Figure 7 shows the general schematic of the samples, highlighting the regions used forcalibration linecuts (stripe of width w c ) and DW measurements (stripe of width w ) [cf.section III A]. The use of two perpendicular wires ensures that the DW is approximately15 ! calibration DW measurement ! ! DW measurement calibration ! ! RF excitation DWMOKE a Ta/CoFeB/MgOPt/Co/Pt b µ m10 µ m xyz MOKE µ m1.5 μ m DW w w c w w c MOKE
Figure 7. The samples were patterned into two perpendicular wires, one of width w c used for thecalibration, the other of width w for the DW study. Left panel: Schematic of the sample. Middlepanel: Scanning electron micrograph of the Ta/CoFeB/MgO sample, showing in color the magneticdomains (up in blue, down in red) and the RF antenna (yellow). Right panel: Magneto-opticalKerr microscopy image of the Ta/CoFeB/MgO sample after nucleation. parallel to the edges used for calibration. The final dimensions (height δd m , widths w c and w )of the patterned structures were measured with a calibrated AFM. For the Ta/CoFeB/MgOsample, δd m = 17 ± nm and w c = w = 1500 ± nm, whereas for the Pt/Co/AlO x sample δd m = 25 ± nm, w c = 980 ± nm and w = 470 ± nm. The nucleation was achieved byfeeding a current pulse through the gold stripline for the Ta/CoFeB/MgO sample, and byapplying pulses of out-of-plane magnetic field for the Pt/Co/AlO x sample. III. DATA ANALYSISA. Calibration linecuts
1. Fit procedure
As discussed in the main text, a preliminary calibration of the experiment is required inorder to infer the probe-sample distance d and the saturation magnetization of the sample M s . This calibration is performed by measuring the Zeeman shift ∆ f NV of the NV magne-tometer while scanning it across a stripe of the ferromagnetic layer in the x direction, as16epicted in Figure 8(a). Since d is of the order of 100 nm in our experiments, one has d (cid:29) t where t is the thickness of the magnetic layer, so that the edges can be considered to beabrupt, i.e. M z ( − w c < x <
0) = M s and M z = 0 otherwise, with w c the stripe width. Infact, due to the topography of the sample, the effective distance between the NV spin andthe magnetic layer varies during the scan [see Fig. 8(a)]. This position-dependent distancecan be written as d eff ( x ) = d + δd ( x ) , where δd ( x ) = 0 on average when the tip is above thestripe, and δd ( x ) = − δd m on average when the tip is above the bare substrate. Here δd m isthe total height of the patterned structures [cf. Section II]. Experimentally, one has accessto the relative variations of d eff ( x ) thanks to the simultaneously recorded AFM topographyinformation, hence one can infer the function δd ( x ) . Therefore, only the absolute distance,characterized by d , is unknown.The stray field components above a single abrupt edge parallel to the y direction, posi-tioned at x = 0 (magnetization pointing upward for x < ), are given by B edge x ( x ) = µ M s t π d eff ( x ) x + d ( x ) B edge y ( x ) = 0 B edge z ( x ) = − µ M s t π xx + d ( x ) . (6)These formulas correspond to the thin-film limit ( d (cid:29) t ) of exact formulas, but the relativeerror introduced by the approximation is < − in our case ( d/t ∼ ), which is negligiblecompared with other sources of error (see below). The field above a stripe is then obtainedby simply adding the contribution of the two edges, namely B stripe ( x ) = B edge ( x ) − B edge ( x + w c ) . (7)Using Eqs. (6) and (7), we obtain an analytical formula for the stray field above the stripe.A fit function ∆ f stripeNV ( x ) is then obtained by converting the field distribution into Zeemanshift of the NV defect after diagonalization of the Hamiltonian defined by Eq. (4), withthe characteristic parameters ( θ, φ, E, D ) of the NV magnetometer. The fit parameters arethe maximum distance d and the product I s = M s t . The geometric parameters of thestripe (width w c and height δd m ), measured independently, serve as references to rescalethe length scales x and z in the linecut data before fitting. Note that in assuming anuniformely magnetized stripe, we neglect the rotation of the magnetization near the edges17 Zee m a n s h i f t ( M H z ) Ta/CoFeB/MgO - ND75c
Zee m a n s h i f t ( M H z )
210 Position x (µm) -200
Pt/Co/AlO x - ND79c b c Z ee m an s h i ft ( M H z ) Z ee m an s h i ft ( M H z ) δ d ( n m ) Position x (µm) Position x (µm) ¤ ϕ z θ d δ d a δ d ( n m ) xz xy w w c w w c Figure 8. (a) Principle of the calibration experiment. The Zeeman shift ∆ f stripeNV ( x ) is measuredwhile scanning the NV magnetometer across a stripe of the ferromagnetic layer in the x direction.(b,c) Zeeman shift linecuts measured with ND75c across a stripe of Ta/CoFeB/MgO (b) and acrossa stripe of Pt/Co/AlO x with ND79c (c). The red solid line is the fit, as explained in the text. Theblue curve is the topography of the sample recorded simultaneously by the AFM and used to definethe distance change δd ( x ) in the fit function. induced by the Dzyaloshinskii-Moriya interaction (DMI) [26]. The effect of this rotation willbe discussed in Section III D.In the following, we focus on the experiments performed (i) with ND75c on the Ta/CoFeB/MgOsample and (ii) with ND79c on the Pt/Co/AlO x sample, corresponding to the experimentalresults reported in Figures 2 and 4 of the main paper, respectively. Typical calibrationlinecuts are shown in Figures 8(b,c) together with the topography of the sample. The redsolid line is the result of the fit, showing a very good agreement with the experimental data.18 . Uncertainties Uncertainties on the fit parameters X = { I s , d } come from those on (i) the NV center’sparameters ( θ, φ, E, D ) , (ii) the geometric parameters of the stripe ( w c , δd m ) and (iii) thefit procedure. There are therefore six independent parameters { p i } = { θ, φ, E, D, w c , δd m } which introduce uncertainties on the outcome of the fit. In the following, these parametersare denoted as p i = ¯ p i ± σ p i where ¯ p i is the nominal value of parameter p i and σ p i its standarderror. The uncertainties on θ , φ , E and D (resp. on w c and δd m ) are discussed in Section I A(resp. in Section II). The nominal values and the standard errors on each parameter p i aresummarized in Table II.The uncertainty and reproducibility of the fit procedure were first analyzed by fittingindependent calibration linecuts while fixing the parameters p i to their nominal values ¯ p i . Asan example, the histograms of the fit outcomes for X = { I s , d } are shown in Figure 9(a,b) fora set of 13 calibration linecuts recorded on the Ta/CoFeB/MgO sample with ND75c. Fromthis statistic, we obtain I s, ¯ p i = 926 . ± . µ A and d ¯ p i = 122 . ± . nm. Here the errorbar is given by the standard deviation of the statistic. The relative uncertainty of the fitprocedure is therefore given by (cid:15) d/ fit = 0 . for the probe-sample distance and (cid:15) I s / fit = 0 . for the product I s = M s t .We now estimate the relative uncertainty on the fit outcomes ( (cid:15) d/p i , (cid:15) I s /p i ) linked to eachindependent parameter p i . For that purpose, the set of calibration linecuts was fitted withone parameter p i fixed at p i = ¯ p i ± σ p i , all the other five parameters remaining fixed at theirnominal values. The resulting mean values of the fit parameters X = { d, I s } are denoted X ¯ p i + σ pi and X ¯ p i − σ pi and the relative uncertainty introduced by the errors on parameter p i is finally defined as (cid:15) X/p i = X ¯ p i + σ pi − X ¯ p i − σ pi X ¯ p i = ∆ X,p i X ¯ p i . (8)To illustrate the method, we plot in Figure 9(c,d) the histograms of the fit outcomes whilechanging the zero-field splitting parameter D from ¯ D − σ D to ¯ D + σ D . For this parameter,the relative uncertainties on d and I s are (cid:15) d/D = 1 . and (cid:15) I s /D = 1 . . The same analysiswas performed for all parameters p i and the corresponding uncertainties are summarized inTable II. The cumulative uncertainty is finally given by (cid:15) X = (cid:115) (cid:15) X/ fit + (cid:88) i (cid:15) X/p i , (9)19 O c u rr e n c e -9 c O c u rr e n c e -9 d (nm) O c u rr e n c e O c u rr e n c e ¯ D + σ D ¯ D − σ D ¯ D + σ D ¯ D − σ D O cc u rr en c e O cc u rr en c e d (nm) O cc u rr en c e O cc u rr en c e bd ∆ d/D I s = M s t ( µ A) I s = M s t ( µ A) ∆ I s /D Figure 9. (a,b) Histograms of the fit outcomes for the probe-sample distance d (a) and I s = M s t (b) obtained for a set of 13 calibration linecuts on the Ta/CoFeB/MgO sample with ND75c whilefixing the parameters p i to their nominal values ¯ p i . (c,d) Histograms of the fit outcomes with thezero-field splitting D fixed at D = ¯ D ± σ D and the other five parameters p i fixed at their nominalvalues. Notations are defined in the text. where all errors are assumed to be independent.Following this procedure, we finally obtain d = 122 . ± . nm and M s t = 926 ± µ A(or M s ≈ . MA/m) for the Ta/CoFeB/MgO sample, and d = 119 . ± . nm and M s t = 671 ± µ A (or M s ≈ . MA/m) for the Pt/Co/AlO x sample, in good agreementwith the values reported elsewhere for similar samples [34 and 35]. B. Micromagnetic calculations
While the calibration linecuts were fitted with analytic formulas, the predictions of thestray field above the DWs were obtained using micromagnetic calculations in order to accu-rately describe the DW fine structure. We first used the micromagnetic OOMMF software [2520 a) Ta/CoFeB/MgO with ND75c parameter p i nominal value ¯ p i uncertainty σ p i (cid:15) d/p i (%) (cid:15) I s /p i (%) w c δd m nm nm 1.0 0.2 θ ◦ ◦ φ − ◦ ◦ D . MHz . MHz 1.0 1.6 E . MHz . MHz 0.5 0.5 (cid:15) X = (cid:113) (cid:15) X/ fit + (cid:80) i (cid:15) X/p i (b) Pt/Co/AlO x with ND79c parameter p i nominal value ¯ p i uncertainty σ p i (cid:15) d/p i (%) (cid:15) I s /p i (%) w c
980 nm 20 nm 1.8 2.0 δd m nm nm 1.6 0.4 θ ◦ ◦ φ ◦ ◦ < . D . MHz . MHz 0.8 0.8 E . MHz . MHz < . < . (cid:15) X = (cid:113) (cid:15) X/ fit + (cid:80) i (cid:15) X/p i (cid:15) X/p i on the value of the fit parameter X ( X = d and X = I s )related to parameter p i for the experiments on Ta/CoFeB/MgO with ND75c (a) and on Pt/Co/AlO x with ND79c (b). The overall uncertainty (cid:15) X is estimated with Eq. (9), assuming that all errors areindependent. The standard deviations obtained from a series of 13 linecuts on Ta/CoFeB/MgO(resp. 9 linecuts on Pt/Co/AlO x ) are (cid:15) d/ fit = 0 . and (cid:15) I s / fit = 0 . (resp. (cid:15) d/ fit = 1 . and (cid:15) I s / fit = 0 . ). K = 5 . · J/m (obtained from the measured effective anisotropy field of 107 mT [33]), exchange constant A = 20 pJ/m, film thickness t = 1 nm, stripe width w = 1500 nm, cell size . × . × nm .For the Pt/Co/AlO x sample, we used: K = 1 . · J/m (measured effective anisotropy fieldof 920 mT), A = 18 pJ/m, t = 0 . nm, w = 470 nm, cell size . × . × . nm . The sat-uration magnetization M s was obtained from the product M s t determined from calibrationlinecuts [cf. Section III A].We considered a straight DW with a tilt angle φ DW with respect to the y axis [Fig. 10(a)].As illustrated in Figs. 10(b) and 10(c), this angle was directly inferred from the Zeemanshift images, leading to φ DW ≈ ± ◦ for the DW studied in Fig. 2 of the main paper, and φ DW ≈ ± ◦ for the DW studied in Fig. 4 of the main paper. The uncertainty on φ DW enables us to account for the fact that the DW is not necessarily rigorously straight. Thispoint will be discussed in Section III C.The calculation of the stray field was then performed with four different initializationsof the DW magnetization: (i) right-handed Bloch, (ii) left-handed Bloch, (iii) right-handedNéel and (iv) left-handed Néel. To stabilize the Néel configuration, the DMI at one of theinterfaces of the ferromagnet was added, as described in Ref. [26]. The value of the DMIparameter was set to | D DMI | = 0 . mJ/m , which is large enough to fully stabilize a NéelDW. The additional consequences of a stronger DMI will be discussed in Section III D.Once the equilibrium magnetization was obtained, the stray field distribution B ( x, y ) was calculated at the distance d by summing the contribution of all cells. Knowing theprojection axis ( θ , φ ), we finally calculate the Zeeman shift map ∆ f NV ( x, y ) by diagonalizingthe NV center’s Hamiltonian [cf. Section I A]. Under the conditions of Figs. 2 and 4 ofthe main paper, the difference of stray field near the maximum between left-handed andright-handed Bloch DWs is predicted to be < . [Fig. 10(d)]. Since this is much smallerthan the standard error [cf. Section III C], we plotted the mean of these two cases, whichis simply referred to as a Bloch DW, and added the deviation induced by the two possiblechiralities to the displayed standard error. 22 H e i gh t ( n m ) ¤
200 nm Z ee m an s h i ft ( M H z ) - Height (nm)
ESR freq. (MHz) a
500 nm ¤ Ta/CoFeB/MgO Pt/Co/AlO x ϕ DW xy xy Model xy y ¤ b c H e i gh t ( n m ) Z ee m an s h i ft ( M H z ) H e i gh t ( n m ) Z ee m an s h i ft ( M H z ) E x pe r i m en t S i m u l a t i on Bloch Left Néel Z ee m an s h i ft ( M H z ) Left-handed Néel Right-handed Néel Left-handed Bloch Right-handed Bloch Bloch average d Z ee m an s h i ft ( M H z ) Position x (µm) Figure 10. (a) The DW is assumed to be straight with a tilt angle φ DW with respect to the y axis,perpendicular to the wire’s long axis. (b,c) AFM image (top panel), Zeeman shift image (middlepanel) and associated simulation (bottom panel) corresponding to the DW studied in (b) Fig. 2 and(c) Fig. 4 of the main paper. The simulation assumes a straight DW with φ DW = 2 ◦ and ψ = π/ in (b), and φ DW = 6 ◦ and ψ = π in (c). (d) Linecuts taken from the simulation of (c), illustratingthe small effect of the chirality of the Bloch DW. Near the maximum, the field is changed by ± . with respect to the mean value. In the case of (b), the change is even smaller ( ± . ). C. Uncertainties on the DW stray field predictions
In this Section, we analyze how the uncertainties on the preliminary measurements affectthe final predictions of the Zeeman shift above the DW. To keep the analysis simple andinsightful, we use the approximate analytic expressions of the stray field of an infinitely longDW [Eqs. (1), (2) and (3) of the main paper]. Furthermore, we focus our attention on thepositions where the DW stray field is maximum, since this is what provides informationabout the DW nature [see Figs. 1(c) and 1(d) of the main paper]. Finally, we use the23pproximation ∆ f NV ≈ gµ B B NV , (cid:107) /h [cf. Section I A], which is quite accurate near the strayfield maximum and allows us to consider the magnetic field B NV , (cid:107) rather than the Zeemanshift ∆ f NV . For clarity the subscript (cid:107) will be dropped and the projected field will be simplydenoted B NV . ¤ ¤ Calibration DW measurement ϕ DW z θ ϕ d a b xz xy xz xy Figure 11. To estimate the uncertainty in the DW stray field prediction, we analyze how the erroron a calibration measurement above an edge (a) and on other parameters translates into an erroron the DW field (b). The calibration edge defines the ( xyz ) axis system. The DW is assumed tobe infinitely long, with its plane tilted by an angle φ DW with respect to the ( yz ) plane. The angle ψ defines the rotation of the in-plane magnetization of the DW with respect to the DW normal.Top panels: side view; Bottom panels: top view. . Out-of-plane contribution B ⊥ Let us first consider the out-of-plane contribution to the DW stray field, B ⊥ ( x ) . Thestray field components above the DW can be written, in the ( xyz ) axis system (Fig. 11), as B ⊥ x ( x ) = µ M s tπ d cos φ DW [( x − x DW ) cos φ DW ] + d B ⊥ y ( x ) = µ M s tπ d sin φ DW [( x − x DW ) cos φ DW ] + d B ⊥ z ( x ) = − µ M s tπ ( x − x DW ) cos φ DW [( x − x DW ) cos φ DW ] + d , (10)where x DW is the position of the DW (for a given y ). This is simply twice the stray fieldabove an edge [see Eq. (6)] expressed in a rotated coordinate system. The projection alongthe NV center’s axis is B ⊥ NV ( x ) = (cid:12)(cid:12) sin θ cos φB ⊥ x ( x ) + sin θ sin φB ⊥ y ( x ) + cos θB ⊥ z ( x ) (cid:12)(cid:12) (11) = µ M s tπ x − x DW ) cos φ DW ] + d | d sin θ cos( φ − φ DW ) − ( x − x DW ) cos φ DW cos θ | . (12)We now link B ⊥ NV ( x ) to the calibration measurement. For simplicity, we consider only oneof the two edges of the calibration stripe, e.g. the edge at x = 0 . We can thus write theprojected field above the edge, at a distance d , as B edgeNV ( x ) = (cid:12)(cid:12) sin θ cos φB edge x ( x ) + sin θ sin φB edge y ( x ) + cos θB edge z ( x ) (cid:12)(cid:12) (13) = µ M s t π x + d | d sin θ cos φ − x cos θ | . (14)Comparing Eqs. (12) and (14), one finds the relation B ⊥ NV (cid:18) x cos φ DW + x DW (cid:19) = 2 B edgeNV ( x )Θ d,θ,φ,φ DW ( x ) , (15)where we define Θ d,θ,φ,φ DW ( x ) = (cid:12)(cid:12)(cid:12)(cid:12) d sin θ cos( φ − φ DW ) − x cos θd sin θ cos φ − x cos θ (cid:12)(cid:12)(cid:12)(cid:12) . (16)Since B edgeNV ( x ) is experimentally measured, in principle one can use Eq. (15) to predict B ⊥ NV ( x ) by simply evaluating the function Θ d,θ,φ,φ DW ( x ) as defined by Eq. (16). As φ DW ∼ implies Θ d,θ,φ,φ DW ( x ) ∼ , it comes that, in a first approximation, B ⊥ NV ( x ) can be obtainedwithout the need for precise knowledge of any parameter. In other words, the calibrationmeasurement, performed under the same conditions as for the DW measurement, allows us25o accurately predict the DW field even though those conditions are not precisely known.This is the key point of our analysis.Strictly speaking, Θ d,θ,φ,φ DW ( x ) , hence B ⊥ NV ( x ) , does depend on some parameters as soonas φ DW (cid:54) = 0 , namely on { q i } = { d, θ, φ, φ DW } . To get an insight into how important theknowledge of { q i } is, we need to examine how sensitive Θ d,θ,φ,φ DW ( x ) is with respect toerrors on { q i } . Owing to the sine and cosine functions in Eq. (16), the smallest sensitivityto parameter variations (vanishing partial derivatives) is achieved when either (i) θ ∼ (projection axis perpendicular to the sample plane) or (ii) θ ∼ π/ (projection axis parallelto the sample plane) combined with φ ∼ and φ − φ DW ∼ . However, case (i) cannotbe achieved in our experiment, because the out-of-plane RF field cannot efficiently driveESR of a spin pointing out-of-plane. We therefore target case (ii), that is, θ ∼ π/ and φ − φ DW ∼ . For that purpose, we use a calibration edge that is as parallel to the DW aspossible ( φ DW → ) and we seek to have a projection axis that is as perpendicular to theDW plane as possible ( θ → π/ and φ → ). This is why we employ two perpendicular wiresfor the calibration and the DW measurements, respectively [cf. Section II]. Conversely, inthe worst case of φ DW ∼ π/ (calibration edge perpendicular to the DW) with θ ∼ π/ , onewould have Θ d,θ,φ,φ DW ( x ) ∼ φ − φ DW , directly proportional to the errors on φ and φ DW .To be more quantitative, we use Eq. (15) to express the uncertainty on the prediction B ⊥ NV ( x ) as a function of the uncertainties on the various quantities, which gives (cid:15) B ⊥ = (cid:115) (cid:15) B edge + (cid:88) i (cid:15) /q i . (17)Here, (cid:15) B edge is given by the measurement error of B edgeNV ( x ) , whereas (cid:15) Θ /q i is the uncertaintyon Θ { q i } introduced by the error on the parameter q i ∈ { d, θ, φ, φ DW } , the other parametersbeing fixed at their nominal values, as defined by (cid:15) Θ /q i = Θ ¯ q i + σ qi − Θ ¯ q i − σ qi ¯ q i . (18)The results are summarized in Table III for the cases considered in Figs. 2 (Ta/CoFeB/MgOsample) and 4 (Pt/Co/AlO x sample) of the main paper. (cid:15) Θ /q i is evaluated for x = x max ,which is the position where the field B ⊥ NV ( x ) is maximum. It can be seen that the dominatingsource of uncertainty, though small ( ≈ ), is the error on φ DW , while the errors on d , θ and φ have a negligible impact. 26n practice, to obtain the theoretical predictions shown in the main paper and in Fig.10, we do not use explicitly Eq. (15), but rather use the set of parameters { I s , d, θ, φ } determined following the calibration step, and put it into the stray field computation [cf.Section III B]. This allows us to simulate more complex structures than the idealized infinitelylong DW considered above [Fig. 11(b)], in particular the finite-width wires studied in thiswork. However, we stress that, as far as the uncertainties are concerned, this is completelyequivalent to using Eq. (15), since B edgeNV ( x ) is fully characterized by the set { I s , d, θ, φ } [cf.Section III A]. The main difference comes from the influence of the edges of the wire, ofwidth w , on the DW stray field. The standard error σ w then translates into a relative error (cid:15) B ⊥ /w on the DW field B ⊥ NV . For the Ta/CoFeB/MgO sample, w = 1500 ± nm, whichgives a negligible error (cid:15) B ⊥ /w < . for the field calculated at the center of the stripe. Forthe Pt/Co/AlO x sample, the stripe is narrower, w = 470 ± nm, leading to (cid:15) B ⊥ /w = 0 . .The overall uncertainty on the prediction B ⊥ NV , for a DW confined in a wire, then becomes (cid:15) B ⊥ = (cid:115) (cid:15) B ⊥ /w + (cid:15) B edge + (cid:88) i (cid:15) /q i . (19)The overall errors are indicated in Table III. For Ta/CoFeB/MgO (Fig. 2 of the main paper), the overall standard error is found to be ≈ . , whereas for Pt/Co/AlO x (Fig. 4) itis ≈ . , in both cases much smaller than the difference between Bloch and Néel DWconfigurations.
2. In-plane contribution B (cid:107) According to Eq. (2) of the main paper, the in-plane contribution to the DW strayfield, B (cid:107) ( x ) , is proportional to I s and to the DW width ∆ DW = (cid:112) A/K eff , where A is theexchange constant and K eff the effective anisotropy constant. The values of A reported inthe literature for Co and CoFeB thin films range from 10 to 30 pJ/m (see e.g. Refs. [36and 37]). Based on this range, we deduced a range for ∆ DW , namely 15-25 nm for theTa/CoFeB/MgO sample and 4.4-7.6 nm for the Pt/Co/AlO x sample. This amounts to arelative variation σ ∆DW ∆ DW ≈ around the mid-value of ∆ DW . Thus, (cid:15) B (cid:107) is dominated by theuncertainty on the DW width, that is, (cid:15) B (cid:107) ≈ σ ∆DW ∆ DW ≈ . All other errors can be neglectedin comparison. In the simulations [cf. Section III B], we used the value of A that gives themid-value of ∆ DW , that is A = 20 pJ/m for Ta/CoFeB/MgO ( ∆ DW = 20 nm) and A = 18 a) Ta/CoFeB/MgO with ND75c parameter q i nominal value ¯ q i uncertainty σ q i (cid:15) Θ /q i (%) d nm nm < . θ ◦ ◦ < . φ − ◦ ◦ φ DW ◦ ◦ (cid:15) B ⊥ = (cid:113) (cid:15) B ⊥ /w + (cid:15) B edge + (cid:80) i (cid:15) /q i (b) Pt/Co/AlO x with ND79c parameter q i nominal value ¯ q i uncertainty σ q i (cid:15) Θ /q i (%) d nm nm < . θ ◦ ◦ < . φ ◦ ◦ φ DW ◦ ◦ (cid:15) B ⊥ = (cid:113) (cid:15) B ⊥ /w + (cid:15) B edge + (cid:80) i (cid:15) /q i (cid:15) Θ /q i on the value of Θ related to parameter q i for theexperiments on Ta/CoFeB/MgO with ND75c (a) and on Pt/Co/AlO x with ND79c (b). The overalluncertainty (cid:15) B ⊥ is estimated with Eq. (19), assuming that all errors are independent. The relativeerror on the calibration field B edgeNV ( x ) is estimated to be (cid:15) B edge ≈ . in (a) and (cid:15) B edge ≈ . in (b). The effect of the stripe width uncertainty leads to an additional error (cid:15) B ⊥ /w < . in (a)and (cid:15) B ⊥ /w = 0 . in (b). pJ/m for Pt/Co/AlO x ( ∆ DW = 6 . nm).For an arbitrary angle ψ of the in-plane magnetization of the DW, the projected stray28eld writes B ψ NV ( x ) = B ⊥ NV ( x ) + cos ψB (cid:107) NV ( x ) , (20)where it is assumed that | B (cid:107) NV | < | B ⊥ NV | . We deduce the expression of the absolute uncer-tainty for B ψ NV σ B ψ = (cid:113) σ B ⊥ + cos ψσ B (cid:107) , (21)where σ B ⊥ = (cid:15) B ⊥ B ⊥ NV and σ B (cid:107) = (cid:15) B (cid:107) B (cid:107) NV . This is how the confidence intervals shownin Figs. 2 and 4 of the main paper were obtained. Finally, the confidence intervals for cos ψ were defined as the values of cos ψ such that the data points remain in the interval [ B ψ NV − σ B ψ ; B ψ NV + σ B ψ ] . The interval for the DMI parameter was deduced using therelation [7] D DMI = 2 µ M s t ln 2 π cos ψ , (22)which holds for an up-down DW provided that | cos ψ | < . D. Effects of a large DMI constant
So far, we have only considered, for simplicity and to avoid introducing additional pa-rameters, the effect of DMI on the angle ψ of the in-plane DW magnetization. In doingso, two other effects of DMI have been neglected: (i) the DMI induces a rotation of themagnetization near the edges of the ferromagnetic structure [26] and (ii) the DW profile inthe presence of DMI slightly deviates from the profile M z ( x ) = − M s tanh( x/ ∆ DW ) [7]. Thefirst (second) effect modifies the stray field above the calibration stripe (above the DW).Here we quantify these effects for the case of Pt/Co/AlO x , for which the DMI is expectedto be strong.Recently, Martinez et al. have estimated that a value D DMI = − . mJ/m associatedwith the spin Hall effect would quantitatively reproduce current-induced DW velocity mea-surements in Pt/Co/AlO x [8]. On the other hand, Pizzini et al. have inferred a similarvalue of D DMI = − . mJ/m from field-dependent DW nucleation experiments [28]. This is ≈ of the threshold value D c above which the DW energy becomes negative and a spinspiral develops. Taking D DMI = − . mJ/m , we predict that the magnetization rotation atthe edges reaches ≈ ◦ [26]. As a result, the field maximum above the edge is increased by29 Zee m a n s h i f t ( M H z ) Zee m a n s h i f t ( M H z ) D = 0 D = 2.5 mJ/m a Z ee m an s h i ft ( M H z ) Position x (µm) Zee m a n s h i f t ( M H z ) Z ee m an s h i ft ( M H z ) Position x (µm) D DMI = 2.5 mJ/m D DMI = 0.5 mJ/m D DMI = 2.5 mJ/m D DMI = 0 mJ/m b DMI-induced deviation of the DW profile DMI-induced magnetization rotation near the edges xz Figure 12. (a) In the presence of a strong DMI, the magnetization deviates from the out-of-planedirection near the edges of the stripe. The plot shows the Zeeman shift calculated under similarconditions as in Fig. 8(b), for two different values of D DMI . (b) The DMI also makes the DW profiledeviate from the profile M z ( x ) = − M s tanh( x/ ∆ DW ) . The plot shows the Zeeman shift calculatedunder similar conditions as in Fig. 4 of the main paper, for two different values of D DMI . ≈ . , under the conditions of Fig. 8(b) [Fig. 12(a)]. This is of the order of our measure-ment error, so that this DMI-induced magnetization rotation cannot be directly detected inour experiment. In fitting the data of Fig. 8(b), the outcome for I s and d is changed bya similar amount: we found d = 119 . ± . nm and I s = 671 ± µ A without DMI, ascompared with d = 121 . ± . nm and I s = 670 ± µ A if D DMI = − . mJ/m is included.The difference is below the uncertainty, therefore it does not affect the interpretation of thedata measured above the DW.To quantify the second effect, we performed the OOMMF calculation with two differentvalues of D DMI that stabilize a left-handed Néel DW: D DMI = − . mJ/m , as used for thesimulations shown in the main paper, and D DMI = − . mJ/m . The stray field calculations,under the same conditions as in Fig. 4 of the main paper, show an increase of the fieldmaximum by ≈ . for the stronger DMI [Fig. 12(b)]. Again, this is well below theuncertainty [cf. Section III C].Besides, it is worth pointing out that these two effects tend to compensate each other,30ince the first one tends to increase the estimated distance d , thereby decreasing the predictedDW field, while the second one tends instead to increase the predicted DW field. Overall,we conclude that neglecting the additional effects of DMI provides predictions for the NéelDW stray field that are correct within the uncertainty, even with a DMI constant as largeas of D c . We note finally that the predictions for the Bloch case, as plotted in Fig.2 and 4 of the main paper, are anyway not affected by the above considerations, since theBloch case implies no DMI at all. [1] I. M. Miron et al. , Fast current-induced domain-wall motion controlled by the Rashba effect, Nat. Mater. , 419 (2011).[2] K.-S. Ryu, L. Thomas, S.-H. Yang, and S. Parkin, Chiral spin torque at magnetic domainwalls, Nat. Nano. , 527 (2013).[3] S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D. Beach, Current-driven dynamics ofchiral ferromagnetic domain walls, Nat. Mater. , 611 (2013).[4] P. P. J. Haazen, E. Murè, J. H. Franken, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans,Domain wall depinning governed by the spin Hall effect, Nat. Mater. , 299 (2013).[5] J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, and H. Ohno,Layer thickness dependence of the current-induced effective field vector in Ta/CoFeB/MgO, Nat. Mater. , 240 (2013).[6] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Bl ¨u gel, S. Auffret, O.Boulle, G. Gaudin, and P. Gambardella, Symmetry and magnitude of spin-orbit torques inferromagnetic heterostructures, Nat. Nanotech. , 587 (2013).[7] A. Thiaville, S. Rohart, E. Jué, V. Cros, and A. Fert, Dynamics of Dzyaloshinskii domain wallsin ultrathin magnetic films, Europhys. Lett. , 57002 (2012).[8] E. Martinez, S. Emori, and G. S. D. Beach, Current-driven domain wall motion along highperpendicular anisotropy multilayers: The role of the Rashba field, the spin Hall effect, andthe Dzyaloshinskii-Moriya interaction,
Appl. Phys. Lett. , 072406 (2013).[9] A. Brataas, and K. M. D. Hals, Spin-orbit torques in action,
Nat. Nano. , 86 (2014).[10] S.-G. Je, D.-H. Kim, S.-C. Yoo, B.-C. Min, K.-J. Lee, and S.-B. Choe, Asymmetric magneticdomain-wall motion by the Dzyaloshinskii-Moriya interaction, Phys. Rev. B , 214401 (2013).
11] J. M. Taylor et al. , High-sensitivity diamond magnetometer with nanoscale resolution.
Nat.Phys. , 810 (2008).[12] G. Balasubramanian et al. , Nanoscale imaging magnetometry with diamond spins under am-bient conditions. Nature , 648 (2008).[13] L. Rondin, J.-P. Tetienne, T. Hingant, J.-F. Roch, P. Maletinsky, and V. Jacques, Magnetom-etry with nitrogen-vacancy defects in diamond.
Rep. Prog. Phys. , 056503 (2014).[14] A. Hubert, and R. Schäfer, Magnetic domains (Springer Verlag, Berlin) 1998.[15] A. Crépieux, and C. Lacroix, Dzyaloshinsky-Moriya interactions induced by symmetry break-ing at a surface,
J. Magn. Magn. Mater. , 341 (1998).[16] S. Meckler et al. , Real-space observation of a right-rotating inhomogeneous cycloidal spin spiralby spin-polarized scanning tunneling microscopy in a triple axes vector magnet,
Phys. Rev.Lett. , 157201 (2009).[17] G. Chen, T. Ma, A. T. N’Diaye, H. Kwon, C. Won, Y. Wu, and A. K. Schmid, Tailoring thechirality of magnetic domain walls by interface engineering,
Nat. Commun. , 2671 (2013).[18] J. Torrejon, J. Kim, J. Sinha, S. Mitani, M. Hayashi, M. Yamanouchi, and H. Ohno, In-terface control of the magnetic chirality in CoFeB/MgO heterostructures with heavy metalunderlayers, preprint arXiv:1401.3568 (2013).[19] L. Rondin et al. , Stray-field imaging of magnetic vortices with a single diamond spin, Nat.Commun. , 2279 (2013).[20] J.-P. Tetienne et al. , Quantitative stray field imaging of a magnetic vortex core, Phys. Rev. B
88, 214408 (2013).[21] J.-P. Tetienne, T. Hingant, J.-V. Kim, L. Herrera Diez, J.-P. Adam, K. Garcia, J.-F. Roch, S.Rohart, A. Thiaville, D. Ravelosona, V. Jacques, Nanoscale imaging and control of domain-wallhopping with a nitrogen-vacancy center microscope,
Science et al. , Nanoscale magnetic field mapping with a single spin scanning probe magne-tometer,
Appl. Phys. Lett. , 153118 (2012).[23] C. Burrowes et al. , Low depinning fields in Ta-CoFeB-MgO ultrathin films with perpendicularmagnetic anisotropy,
Appl. Phys. Lett. , 182401 (2013).[24] N. Vernier, J.P. Adam, S.Eimer, G. Agnus, T. Devolder, T. Hauet, B. Ockert, and D. Rav-elosona, Measurement of magnetization using domain compressibility in CoFeB films withperpendicular anisotropy,
Appl. Phys. Lett. , 122404 (2014).
25] M. J. Donahue, and D. G. Porter, OOMMF User’s Guide, Version 1.0 Interagency Re-port NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD.http://math.nist.gov/oommf (1999).[26] S. Rohart and A. Thiaville, Skyrmion confinement in ultrathin film nanostructures in thepresence of Dzyaloshinskii-Moriya interaction,
Phys. Rev. B , 184422 (2013).[27] S. Emori, E. Martinez, U. Bauer, S.-M. Ahn, P. Agrawal, D. C. Bono, G. S. D. Beach, SpinHall torque magnetometry of Dzyaloshinskii domain walls, preprint arXiv:1308.1432 (2013).[28] S. Pizzini, J. Vogel, S. Rohart, L. D. Buda-Prejbeanu, E. Jué, O. Boulle, I. M. Miron, C. K.Safeer, S. Auret, G. Gaudin and A. Thiaville, Chirality-induced asymmetric magnetic nucle-ation in Pt/Co/AlO x ultrathin microstructures, preprint arXiv:1403.4694 (2014).[29] K.-S. Ryu, S.-H. Yang, L. Thomas, and S. S. P. Parkin, Chiral spin torque arising fromproximity-induced magnetization, Nat. Commun. , 3910 (2014).[30] A. Fert, V. Cros, J. Sampaio, Skyrmions on the track, Nat. Nano. , 152 (2013).[31] M. W. Doherty, F. Dolde, H. Fedder, F. Jelezko, J. Wrachtrup, N. B. Manson, L. C. L.Hollenberg, Theory of the ground-state spin of the NV − center in diamond, Phys. Rev. B ,205203 (2012).[32] J.-P. Tetienne et al. , Magnetic-field-dependent photodynamics of single NV defects in diamond:an application to qualitative all-optical magnetic imaging. New J. Phys. , 103033 (2012).[33] T. Devolder et al. , Damping of Co x Fe − x B ultrathin films with perpendicular magneticanisotropy. Appl. Phys. Lett. , 022407 (2013).[34] N. Vernier et al. , Measurement of magnetization using domain compressibility in CoFeB filmswith perpendicular anisotropy.
Appl. Phys. Lett. , 122404 (2014).[35] I. M. Miron et al. , Current-driven spin torque induced by the Rashba effect in a ferromagneticmetal layer.
Nat Mater. , 230 (2010).[36] M. Yamanouchi et al. , Domain structure in CoFeB thin films with perpendicular magneticanisotropy. IEEE Magn. Lett. , 3000304 (2011).[37] C. Eyrich , Exchange Stiffness in Thin-Film Cobalt Alloys. MSc Thesis (2012, Simon FraserUniversity, Canada).(2012, Simon FraserUniversity, Canada).