The nature of domain walls in ultrathin ferromagnets revealed by scanning nanomagnetometry

The recent observation of current-induced domain wall (DW) motion with large velocity in ultrathin magnetic wires has opened new opportunities for spintronic devices. However, there is still no consensus on the underlying mechanisms of DW motion. Key to this debate is the DW structure, which can be of Bloch or N\'eel type, and dramatically affects the efficiency of the different proposed mechanisms. To date, most experiments aiming to address this question have relied on deducing the DW structure and chirality from its motion under additional in-plane applied fields, which is indirect and involves strong assumptions on its dynamics. Here we introduce a general method enabling direct, in situ, determination of the DW structure in ultrathin ferromagnets. It relies on local measurements of the stray field distribution above the DW using a scanning nanomagnetometer based on the Nitrogen-Vacancy defect in diamond. We first apply the method to a Ta/Co40Fe40B20(1 nm)/MgO magnetic wire and find clear signature of pure Bloch DWs. In contrast, we observe left-handed N\'eel DWs in a Pt/Co(0.6 nm)/AlOx wire, providing direct evidence for the presence of a sizable Dzyaloshinskii-Moriya interaction (DMI) at the Pt/Co interface. This method offers a new path for exploring interfacial DMI in ultrathin ferromagnets and elucidating the physics of DW motion under current.

The recent observation of current-induced domain wall (DW) motion with large velocity in ultrathin magnetic wires has opened new opportunities for spintronic devices [1]. However, there is still no consensus on the underlying mechanisms of DW motion [1][2][3][4][5][6]. Key to this debate is the DW structure, which can be of Bloch or Néel type, and dramatically affects the efficiency of the different proposed mechanisms [7][8][9]. To date, most experiments aiming to address this question have relied on deducing the DW structure and chirality from its motion under additional in-plane applied fields, which is indirect and involves strong assumptions on its dynamics [2][3][4]10]. Here we introduce a general method enabling direct, in situ, determination of the DW structure in ultrathin ferromagnets. It relies on local measurements of the stray field distribution above the DW using a scanning nanomagnetometer based on the Nitrogen-Vacancy defect in diamond [11][12][13]. We first apply the method to a Ta In wide ultrathin wires with perpendicular magnetic anisotropy (PMA), magnetostatics predicts that the Bloch DW, a helical rotation of the magnetization, is the most stable DW configuration because it minimizes volume magnetic charges [14]. However, the unexpectedly large 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 cast doubt on this hypothesis and triggered a major academic debate regarding the underlying mechanism of DW motion [4][5][6][7][8][9]. Notably, it was recently proposed that Néel DWs with fixed chirality could be stabilized by the Dzyaloshinskii-Moriya interaction (DMI) [7], an indirect exchange possibly occurring at the interface between a magnetic layer and a heavy metal substrate with large spin-orbit coupling [15]. For such chiral DWs, hereafter termed Dzyaloshinskii DWs, a damping-like torque due to spin-orbit terms (spin-Hall effect and Rashba interaction) would lead to efficient current-induced DW motion along a direction fixed by the chirality [7]. In order to validate unambiguously these theoretical predictions, a 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 of its inner structure a very challenging task. So far, only spin-polarized scanning tunnelling microscopy [16] and spin-polarized low energy electron microscopy [17] have allowed a direct determination of the DW structure, demonstrating homochiral Néel DWs in Fe double layer on W(110) and in (Co/Ni) n multilayers on Pt or Ir, respectively. However, these techniques are intrinsically limited to model samples due to high experimental constraints and the debate 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 in virtually any ultrathin ferromagnet. It relies on local measurements of the stray magnetic field produced above the DW using a scanning nanomagnetometer. To convey the basic idea behind our method, we start by deriving analytical formulas of the magnetic field distribution at a distance d above a DW placed at x = 0 in a perpendicularly magnetized film [ Fig. 1a]. The main contribution to the stray field, denoted B ⊥ , is provided by the abrupt 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 resulting stray field components can be expressed as where t is the film thickness. These approximate formulas are valid in the limit of (i) t d, (ii) ∆ DW d and (iii) for an infinitely long DW along the y axis. On the other hand, the in-plane magnetization, with amplitude M (x) = M s / cosh(x/∆ DW ), can be oriented with an angle ψ with respect to the x axis [ Fig. 1b]. This angle is linked to the nature of the DW: ψ = ±π/2 for a Bloch DW, whereas ψ = 0 or π for a Néel DW. The two possible values define the chirality (right or left) of the DW. The spatial variation of the in-plane magnetization adds a contribution B cos ψ to the stray field, whose components are given The net stray field above the DW is finally expressed as which indicates that a Néel DW (cos ψ = ±1) produces an additional stray field owing to extra magnetic charges on each side of the wall. Using Eqs. (1) and (2), we find a maximum relative difference in stray field between Bloch and Néel DWs scaling as ≈ π∆ DW /2d. Local measurements of the stray field above a DW can therefore reveal its inner structure, characterized by the angle ψ. This is further illustrated in Figs. 1(c,d), which show the stray field components B ψ x (x) and B ψ z (x) for various DW configurations while using d = 120 nm and ∆ DW = 20 nm, which are typical parameters of the experiments considered below on a Ta/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 operating under ambient conditions [11][12][13]. Here, the local magnetic field is evaluated within an atomic-size detection volume by monitoring the Zeeman shift of the NV defect electron spin sublevels through optical detection of the magnetic resonance. After grafting the diamond nanocrystal onto the tip of an atomic force microscope (AFM), we obtain a scanning nanomagnetometer which provides quantitative maps of the stray field emanating from nanostructured samples [19][20][21] with a magnetic field sensitivity in the range of 10 µT.Hz −1/2 [22]. In this study, the Zeeman frequency shift ∆f NV of the NV spin is measured while scanning the AFM tip in tapping mode, so that the mean distance between the NV spin and the sample surface is kept constant with a typical tip oscillation amplitude of a few nanometers [20]. Each recorded value of ∆f NV is a function of B NV, and B NV,⊥ , which are the parallel and perpendicular components, respectively, of the local magnetic 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, /h, where gµ B /h ≈ 28 GHz/T. This indicates that scanning-NV magnetometry essentially measures the projection B NV, of the magnetic field along the NV center's axis. The latter is characterized by spherical 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-µm-wide magnetic wire made of a Ta(5 nm)/Co 40 Fe 40 B 20 (1 nm)/MgO(2 nm) trilayer stack (Supplementary Section II). This system has been intensively studied in the last years owing to low damping parameter and depinning field [23]. Before imaging a DW, it is first necessary to determine precisely (i) the distance d between the NV probe and the magnetic layer and (ii) the product I s = M s t, which are both directly involved in Eq. (3). These parameters are obtained by 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 edge placed at x = 0 can be easily expressed analytically in a form similar to Eq. (10), which only depends on d and I s . An example of a measurement obtained by scanning the magnetometer across a Ta/CoFeB/MgO stripe is shown in Fig. 2b. The data are fitted with a function corresponding 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 a set of independent calibration linecuts, we obtain d = 123 ± 3 nm and I s = 926 ± 26 µA, in good agreement with the value measured by other methods [24].
Having determined all needed parameters, it is now possible to measure the stray field above a DW [ Fig. 2c] and compare it to the theoretical prediction, which only depends on the angle ψ that characterizes the DW structure. To this end, an isolated DW was nucleated in a wire of the same Ta/CoFeB/MgO film and imaged with the scanning-NV magnetometer under the same conditions as for the calibration measurements. The resulting distribution of the Zeeman shift ∆f NV is shown in Fig. 2d together with the AFM image of the magnetic wire. Within the resolving power of our instrument, limited by the probe-to-sample distance d ∼ 120 nm [20], the DW appears to be straight with a small tilt angle with respect to the wire long axis, determined to be 2 ± 1 • (Supplementary Section III-B). Taking into account this DW spatial profile, the stray field above the DW was computed for (i) ψ = 0 (righthanded Néel DW), (ii) ψ = π (left-handed Néel DW) and (iii) ψ ± π/2 (Bloch DW). Here we used the micromagnetic OOMMF software [25,26] rather than the analytical formula described above in order to avoid any approximation in the calculation. The computed magnetic field distributions were finally converted into Zeeman shift distribution taking into account the NV spin's quantization axis. A linecut of the experimental data across the DW is 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 same conclusion can be drawn by directly comparing the full two-dimensional theoretical maps to the data [ Fig. 2d and f]. As described in detail in the Supplementary Section III-C, all sources of uncertainty in the theoretical predictions were carefully analysed, yielding the 1 standard 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| < 0.07. This corresponds to an upper limit for the DMI parameter D DMI , as defined in Ref. [7], of |D DMI | < 0.01 mJ/m 2 (Supplementary Section 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. The results are shown in Fig. 3 for four NV defects with different quantization axes, showing excellent agreement between experiment and theory if one assumes a Bloch-type DW. These experiments provide an unambiguous confirmation of the Bloch nature of the DWs in our sample, but are also a striking illustration of the vector mapping capability offered by NV microscopy, allowing for robust tests of theoretical predictions.
We conclude that there is no evidence for the presence of a sizable interfacial DMI in a Ta(5nm)/Co 40 Fe 40 B 20 (1nm)/MgO trilayer stack. This is in contrast with recent experiments reported on similar samples with different compositions, such as Ta(5nm)/Co 80 Fe 20 (0.6nm)/MgO [3,27] and Ta(0.5 nm)/Co 20 Fe 60 B 20 (1nm)/MgO [18], where indirect evidence for Néel DWs was found through current-induced DW motion experiments. We note that contrary to these studies, our method indicates the nature of the DW at rest, in a direct manner, without any assumption on the DW dynamics. Our results therefore motivate a systematic study of the 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(0.6 nm)/AlO x (2nm) trilayer grown by sputtering on a thermally oxidized silicon wafer (Supplementary Section II). The observation of current-induced DW motion with unexpectedly large velocities in this asymmetric stack has attracted considerable interest in the recent years [1]. Here, the DW width is ∆ DW ≈ 6 nm, leading to a relative field difference between Bloch and Néel cases of ≈ 8% at a distance d ≈ 120 nm. We followed a procedure similar to that described above (Supplementary Section III). After a preliminary calibration of the experiment, a DW in a 500-nm-wide magnetic wire was imaged [ Fig. 4a,b] and linecuts across the DW were compared to theoretical predictions [ Fig. 4c]. Here the experimental results clearly indicate a Néel-type DW structure with left-handed chirality. The same result was found for two other DWs. This provides direct evidence of a strong DMI at the Pt/Co interface, with a lower bound |D DMI | > 0.1 mJ/m 2 . This result is consistent with the conclusions of recent field-dependent DW nucleation experiments performed in similar films [28]. In addition, we note that the observed left-handed chirality, once combined with a damping-like torque induced by the spin-orbit terms, could explain the characteristics of DW motion under current in this sample [8].
In conclusion, we have shown how scanning-NV magnetometry enables direct discrimination between competing DW configurations in ultrathin ferromagnets. This method, which is not sensitive to possible artifacts linked to the DW dynamics, will help clarifying the physics of DW motion under current, a necessary step towards the development of DW-based spintronic devices. In addition, this work opens a new avenue for studying the mechanisms at the origin of interfacial DMI in ultrathin ferromagnets, by measuring the DW structure while tuning the properties of the magnetic material [18,29]. This is a key milestone 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- -300 0 300  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.

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 under ambient conditions. A detailed description of the setup as well as the method to graft a diamond 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 in three different nanodiamonds, labeled ND74 (data of Figure 3 of the main paper), ND75 ( Figure 2) and ND79 ( Figure 4). All nanodiamonds were ≈ 50 nm in size, as measured by AFM before grafting the nanodiamond onto the AFM tip. The magnetic field was inferred by measuring the Zeeman shift of the electron spin resonance (ESR) of the NV center's ground 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 transitions of this S = 1 spin system is given by where D and E are the zero-field splitting parameters that characterize a given NV center, h is the Planck constant, gµ B /h = 28.03(1) 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 by the diamond crystal orientation, with Z being parallel to the NV center's symmetry axis u NV , as shown in Figure 5 [32], they depend only on the magnetic field projection along the NV axis B NV, , following   Table I. the relation The parameters D and E were extracted from ESR spectra recorded at zero magnetic field In all the data shown in this work, only the upper frequency f + was measured. Thereafter, we will note the corresponding The 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 labeled with 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 magnetic field of known direction and amplitude with a three-axis coil system, following the procedure described in Ref. [19]. The measurement uncertainty of 2 • (standard error) is related to the precision of the calibration of the coils and their alignment with respect to the (xyz) reference frame. Table I indicates the parameters D, E, θ and φ measured for each NV magnetometer used in this work, with the associated standard errors.   The stray field components above a single abrupt edge parallel to the y direction, positioned at x = 0 (magnetization pointing upward for x < 0), are given by These formulas correspond to the thin-film limit (d t) of exact formulas, but the relative error introduced by the approximation is < 10 −5 in our case (d/t ∼ 100), which is negligible compared with other sources of error (see below). The field above a stripe is then obtained by simply adding the contribution of the two edges, namely Using Eqs. (6) and (7), we obtain an analytical formula for the stray field above the stripe.
A fit function ∆f stripe NV (x) is then obtained by converting the field distribution into Zeeman shift of the NV defect after diagonalization of the Hamiltonian defined by Eq. induced by the Dzyaloshinskii-Moriya interaction (DMI) [26]. The effect of this rotation will be discussed in Section III D.   Table II.
The uncertainty and reproducibility of the fit procedure were first analyzed by fitting independent calibration linecuts while fixing the parameters p i to their nominal valuesp i . As an example, the histograms of the fit outcomes for X = {I s , d} are shown in Figure 9(a,b) for a set of 13 calibration linecuts recorded on the Ta/CoFeB/MgO sample with ND75c. From this statistic, we obtain I s,p i = 926.3 ± 2.8 µA and dp i = 122.9 ± 0.7 nm. Here the error bar is given by the standard deviation of the statistic. The relative uncertainty of the fit procedure is therefore given by d/fit = 0.6% for the probe-sample distance and Is/fit = 0.3% for the product I s = M s t.
We now estimate the relative uncertainty on the fit outcomes ( d/p i , Is/p i ) linked to each independent parameter p i . For that purpose, the set of calibration linecuts was fitted with one parameter p i fixed at p i =p i ± σ p i , all the other five parameters remaining fixed at their nominal values. The resulting mean values of the fit parameters X = {d, I s } are denoted Xp i +σp i and Xp i −σp i and the relative uncertainty introduced by the errors on parameter p i is finally defined as To illustrate the method, we plot in Figure 9(c,d) the histograms of the fit outcomes while changing the zero-field splitting parameter D fromD − σ D toD + σ D . For this parameter, the relative uncertainties on d and I s are d/D = 1.0% and Is/D = 1.6%. The same analysis was performed for all parameters p i and the corresponding uncertainties are summarized in Table II. The cumulative uncertainty is finally given by where all errors are assumed to be independent.
Following this procedure, we finally obtain d = 122.9 ± 3.1 nm and M s t = 926 ± 26 µA

B. Micromagnetic calculations
While the calibration linecuts were fitted with analytic formulas, the predictions of the stray field above the DWs were obtained using micromagnetic calculations in order to accu-  Table II. Summary of the uncertainty 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 X is estimated with Eq. (9), assuming that all errors are independent. The standard deviations obtained from a series of 13 linecuts on Ta/CoFeB/MgO (resp. 9 linecuts on Pt/Co/AlO x ) are d/fit = 0.6% and Is/fit = 0.3% (resp. d/fit = 1.4% and Is/fit = 0.5%).
and 26] to obtain the equilibrium magnetization of the structure. For the Ta/CoFeB/MgO sample, the nominal values used in OOMMF are: anisotropy constant K = 5.9 · 10 5 J/m 3 (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 2.5 × 2.5 × 1 nm 3 .
For the Pt/Co/AlO x sample, we used: K = 1. 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 Zeeman shift images, leading to φ DW ≈ 2 ± 1 • for the DW studied in Fig. 2 of the main paper, and φ DW ≈ 6 ± 2 • 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. This point will be discussed in Section III C.
The calculation of the stray field was then performed with four different initializations of the DW magnetization: (i) right-handed Bloch, (ii) left-handed Bloch, (iii) right-handed Néel and (iv) left-handed Néel. To stabilize the Néel configuration, the DMI at one of the interfaces of the ferromagnet was added, as described in Ref. [26]. The value of the DMI parameter was set to |D DMI | = 0.5 mJ/m 2 , which is large enough to fully stabilize a Néel DW. 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) right-handed Bloch DWs is predicted to be < 0.5% [ Fig. 10(d)]. Since this is much smaller than the standard error [cf. Section III C], we plotted the mean of these two cases, which is simply referred to as a Bloch DW, and added the deviation induced by the two possible chiralities to the displayed standard error.  Fig. 4 of the main paper. The simulation assumes a straight DW with φ DW = 2 • and ψ = π/2 in (b), and φ DW = 6 • and ψ = π in (c). (d) Linecuts taken from the simulation of (c), illustrating the small effect of the chirality of the Bloch DW. Near the maximum, the field is changed by ±0.5% with respect to the mean value. In the case of (b), the change is even smaller (±0.3%).

C. Uncertainties on the DW stray field predictions
In this Section, we analyze how the uncertainties on the preliminary measurements affect , which is quite accurate near the stray field maximum and allows us to consider the magnetic field B NV, rather than the Zeeman shift ∆f NV . For clarity the subscript will be dropped and the projected field will be simply denoted B NV .

¤ ¤
x z x y x z x y Figure 11. To estimate the uncertainty in the DW stray field prediction, we analyze how the error on a calibration measurement above an edge (a) and on other parameters translates into an error on the DW field (b). The calibration edge defines the (xyz) axis system. The DW is assumed to be 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). The stray field components above the DW can be written, in the (xyz) axis system (Fig. 11), as where x DW is the position of the DW (for a given y). This is simply twice the stray field above an edge [see Eq. (6)] expressed in a rotated coordinate system. The projection along the NV center's axis is We now link B ⊥ NV (x) to the calibration measurement. For simplicity, we consider only one of the two edges of the calibration stripe, e.g. the edge at x = 0. We can thus write the projected field above the edge, at a distance d, as Comparing Eqs. (12) and (14), one finds the relation where we define Since B edge NV (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 ∼ 0 implies Θ d,θ,φ,φ DW (x) ∼ 1, it comes that, in a first approximation, B ⊥ NV (x) can be obtained without the need for precise knowledge of any parameter. In other words, the calibration measurement, performed under the same conditions as for the DW measurement, allows us to 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 soon as φ DW = 0, namely on {q i } = {d, θ, φ, φ DW }. To get an insight into how important the knowledge of {q i } is, we need to examine how sensitive Θ d,θ,φ,φ DW (x) is with respect to errors on {q i }. Owing to the sine and cosine functions in Eq. (16), the smallest sensitivity to parameter variations (vanishing partial derivatives) is achieved when either (i) θ ∼ 0 (projection axis perpendicular to the sample plane) or (ii) θ ∼ π/2 (projection axis parallel to the sample plane) combined with φ ∼ 0 and φ − φ DW ∼ 0. However, case (i) cannot be achieved in our experiment, because the out-of-plane RF field cannot efficiently drive ESR of a spin pointing out-of-plane. We therefore target case (ii), that is, θ ∼ π/2 and φ − φ DW ∼ 0. For that purpose, we use a calibration edge that is as parallel to the DW as possible (φ DW → 0) and we seek to have a projection axis that is as perpendicular to the DW plane as possible (θ → π/2 and φ → 0). This is why we employ two perpendicular wires for the calibration and the DW measurements, respectively [cf. Section II]. Conversely, in the worst case of φ DW ∼ π/2 (calibration edge perpendicular to the DW) with θ ∼ π/2, one would 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 Here, B edge is given by the measurement error of B edge NV (x), whereas Θ/q i is the uncertainty on Θ {q i } introduced by the error on the parameter q i ∈ {d, θ, φ, φ DW }, the other parameters being fixed at their nominal values, as defined by The results are summarized in Table III for the cases considered in Figs. 2 (Ta/CoFeB/MgO sample) and 4 (Pt/Co/AlO x sample) of the main paper. 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 dominating source of uncertainty, though small (≈ 1%), is the error on φ DW , while the errors on d, θ and φ have a negligible impact.
In 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 infinitely long DW considered above [ Fig. 11(b)], in particular the finite-width wires studied in this work. However, we stress that, as far as the uncertainties are concerned, this is completely equivalent to using Eq. (15), since B edge NV (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, of width w, on the DW stray field. The standard error σ w then translates into a relative error B ⊥ /w on the DW field B ⊥ NV . For the Ta/CoFeB/MgO sample, w = 1500 ± 30 nm, which gives a negligible error B ⊥ /w < 0.1% for the field calculated at the center of the stripe. For the Pt/Co/AlO x sample, the stripe is narrower, w = 470 ± 20 nm, leading to B ⊥ /w = 0.9%.
The overall uncertainty on the prediction B ⊥ NV , for a DW confined in a wire, then becomes 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 ≈ 1.5%, whereas for Pt/Co/AlO x (Fig. 4) 1   Table III. Summary of the uncertainty Θ/q i on the value of Θ related to parameter q i for the experiments on Ta/CoFeB/MgO with ND75c (a) and on Pt/Co/AlO x with ND79c (b). The overall uncertainty B ⊥ is estimated with Eq. (19), assuming that all errors are independent. The relative error on the calibration field B edge NV (x) is estimated to be B edge ≈ 1.0% in (a) and B edge ≈ 1.5% in (b). The effect of the stripe width uncertainty leads to an additional error B ⊥ /w < 0.1% in (a) and B ⊥ /w = 0.9% in (b).
For an arbitrary angle ψ of the in-plane magnetization of the DW, the projected stray field writes where it is assumed that |B NV | < |B ⊥ NV |. We deduce the expression of the absolute uncertainty for B ψ where σ B ⊥ = B ⊥ B ⊥ NV and σ B = B B NV . This is how the confidence intervals shown in 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 The interval for the DMI parameter was deduced using the relation [7] D DMI = 2µ 0 M 2 s t ln 2 π 2 cos ψ , which holds for an up-down DW provided that | cos ψ| < 1.

D. Effects of a large DMI constant
So far, we have only considered, for simplicity and to avoid introducing additional parameters, the effect of DMI on the angle ψ of the in-plane DW magnetization. In doing so, two other effects of DMI have been neglected: (i) the DMI induces a rotation of the magnetization near the edges of the ferromagnetic structure [26] and (ii) the DW profile in the presence of DMI slightly deviates from the profile M z (x) = −M s tanh(x/∆ DW ) [7]. The first (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 expected to be strong.  [28]. This is ≈ 70% of the threshold value D c above which the DW energy becomes negative and a spin spiral develops. Taking D DMI = −2.5 mJ/m 2 , we predict that the magnetization rotation at the edges reaches ≈ 20 • [26]. As a result, the field maximum above the edge is increased by  Fig. 8(b) [Fig. 12(a)]. This is of the order of our measurement error, so that this DMI-induced magnetization rotation cannot be directly detected in our experiment. In fitting the data of Fig. 8(b), the outcome for I s and d is changed by a similar amount: we found d = 119.0 ± 3.4 nm and I s = 671 ± 18 µA without DMI, as compared with d = 121.0 ± 3.4 nm and I s = 670 ± 17 µA if D DMI = −2.5 mJ/m 2 is included.
The difference is below the uncertainty, therefore it does not affect the interpretation of the data measured above the DW.
To quantify the second effect, we performed the OOMMF calculation with two different values of D DMI that stabilize a left-handed Néel DW: D DMI = −0.5 mJ/m 2 , as used for the simulations shown in the main paper, and D DMI = −2.5 mJ/m 2 . The stray field calculations, under the same conditions as in Fig. 4 of the main paper, show an increase of the field maximum by ≈ 0.5% for the stronger DMI [ Fig. 12(b)]. Again, this is well below the uncertainty [cf. Section III C].
Besides, it is worth pointing out that these two effects tend to compensate each other, since the first one tends to increase the estimated distance d, thereby decreasing the predicted DW 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 Néel DW stray field that are correct within the uncertainty, even with a DMI constant as large as 70% 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 the Bloch case implies no DMI at all.