Critical role of orbital hybridization in the Dzyaloshinskii-Moriya interaction of magnetic interfaces

11:31 19 junio in Artículos por Website

Sample details

For this study, magnetic bilayers of Au1−xPtx 4 nm/Co 2–7 nm (x = 0, 0.25. 0.5, 0.65, 0.75, 0.85, and 1) were sputter-deposited onto oxidized Si substrates and capped with MgO 2 nm/Ta 1.5 nm protective layers (see more details in the “Method” Section). The layer thicknesses are estimated using the calibrated deposition rates and the deposition time and then verified by scanning transmission electron microscopy (STEM) measurements. The Au1−xPtx/Co samples have (111)-orientated face-centered-cubic (fcc) polycrystalline texture, and strong, tunable SOTs20. Cross-sectional dark-field scanning transmission electron microscopy (STEM) imaging in Fig. 1a and Supplementary Fig. S1a (higher magnification view of the Au1−xPtx/Co interface) reveal the reasonably sharp Au1−xPtx/Co interfaces (minimal intermixing as we discuss further below), the coherent growth of Co and Au1−xPtx, and the nearest-neighbor atomic distance of 0.212 nm in the (111) plane. The bilayers were patterned into 10 × 20 μm2 microstrips for measuring the interfacial perpendicular magnetic anisotropy energy density (\({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\)) via spin-torque ferromagnetic resonance (ST-FMR)21.

Fig. 1: Interfacial spin-orbit coupling.
figure 1

a Cross-sectional dark-field scanning transmission electron microscopy imaging of a typical Pt/Co/MgO sample (the Co layer is 2.5 nm, the heavy Pt atoms appear to be brighter than the light Co atoms), indicating reasonably sharp Pt/Co and Co/MgO interfaces and negligible interfacial alloying within the measurement resolution. b The inverse thickness (tCo−1) dependence of the effective demagnetization field 4πMeff (x = 1, 0.75, and 0.65) of the Au1−xPtx/Co bilayers. The error bars determined from the standard deviation of the fit parameters are smaller than the points. The solid lines present the best linear fits. c Interfacial magnetic anisotropy energy density for the Au1−xPtx/Co interface (\({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\)) and for the two interfaces of Au1−xPtx/Co/MgO (Ks) plotted as a function of x. The arrow indicates a factor of 1.8 variation of \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\). The error bars represent the standard deviation of the fit parameters. The dashed lines in (c) are to guide the eyes. d The perpendicular orbital momentum for the single ferromagnetic interface layer (\({m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }\)) as determined from x-ray magnetic circular dichroism experiments22,23,28 and e The strength of interfacial spin-orbit coupling (ξ) for the Au1−xPtx/Co interface as determined using the \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) in (c) following Eq. (2). In (d) and (e), the solid points show the experimental data and the dashed lines represent the simple case that \({m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }\) of the Au1−xPtx /Co interfaces varies linearly with the composition of the Au1−xPtx.

Tuning interfacial spin-orbit coupling by composition

It has been well established that \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) of HM/FM interfaces originates from SOC-enhanced perpendicular orbital magnetic moments (\({m}_{{{{\rm{o}}}}}^{\perp }\)) localized at the first FM atomic layer adjacent to the interface22,23. For a single HM/FM interface with ξ the bandwidth of the FM, Bruno’s model23,24,25 predicts

$${K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}=\frac{1}{4{\mu }_{{{{\rm{B}}}}}}\frac{G}{H}\xi ({m}_{{{{\rm{o}}}}}^{\perp }-{m}_{{{{\rm{o}}}}}^{\parallel })t_{{{\rm{FM}}}}$$


where the G/H ratio is a band structure parameter dependent on the type of the FM and is estimated to be ≈0.2 for Co by theory and experiments23,26,27, tFM and \({m}_{{{{\rm{o}}}}}^{\parallel }\) are the layer thickness and the in-plane orbital magnetic moment of the FM layer, respectively. X-ray magnetic circular dichroism (XMCD) experiments22,23,28 have consistently reported an interface-insensitive \({m}_{{{{\rm{o}}}}}^{\parallel }\) of 0.12 ± 0.02 µB/Co for Co layers with different neighboring metals (Au23, Pt22,28,29, Pd29, Cu30, Ni29), which reasonably approximates the bulk orbital magnetic moment value for Co (0.12 µB/Co as determined by experiments22,23,29 and first-principle calculations31) particularly when the Co layers are thicker than 2 nm as is in the case of our samples. Using the relation22,23\({m}_{{{{\rm{o}}}}}^{\perp }\) = \({m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }\)/tFM + \({m}_{{{{\rm{o}}}}}^{\parallel }\), where \({m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }\) is the \({m}_{{{{\rm{o}}}}}^{\perp }\) value for the single FM interface layer, we obtain

$${K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}=\frac{1}{4{\mu }_{{{{\rm{B}}}}}}\frac{G}{H}{\xi m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }$$


Note that this relation assumes a sharp HM/FM interface and may lose accuracy for a strongly inter-mixed interface. Since Bruno’s model is suggested to apply only to interfaces with a relatively weak ISOC (e.g., ξ ≈ 46 meV) and/or a relatively weak dd orbital hybridization25, Eqs. (1) and (2) may also become less accurate in the limit that ISOC and orbital hybridization are very strong at the same time.

To quantify \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) for the Au1−xPtx/Co interface, we first determined the total interfacial perpendicular magnetic anisotropy energy density (Ks) of the two Co interfaces of the Au1−xPtx/Co/MgO samples from the fits of the effective demagnetization field (4πMeff) vs tCo−1 (see Fig. 1b and Supplementary Fig. S2) to the relation 4πMeff ≈ 4πMs + 2Ks/MstCo. We obtain the saturation magnetization Ms ≈ 1200–1300 emu cm−3 and Ks ≈ 1.3–2.1 erg/cm2 (Fig. 1c) for Au1−xPtx/Co/MgO samples with different x. From a control sample of MgO/Co/MgO (Supplementary Fig. S3), we determine \({K}_{{{{\rm{s}}}}}^{{{{\rm{Co}}}}/{{{\rm{MgO}}}}}\)≈ 0.32 erg cm−2, which is consistent with ≈ 0.34 erg/cm2 in Ta/Co/MgO sample with negligible two-magnon scattering and thus magnetic roughness32 and ≈0.4 erg cm−2 in [MgO/Co]n multilayers with strong superlattice peaks in the low-angle x-ray diffraction spectra33. Using \({K}_{{{{\rm{s}}}}}^{{{{\rm{Co}}}}/{{{\rm{MgO}}}}}\) = 0.32 erg cm−2, we estimate \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) for the Au1−xPtx/Co interfaces in Fig. 1c. \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) increases from 1.02 erg cm−2 for x = 0 to 1.69 erg cm−2 for x = 0.25–0.75, and then gradually decreases to 0.95 erg cm−2 for x = 1. Any small variation in \({K}_{{{{\rm{s}}}}}^{{{{\rm{Co}}}}/{{{\rm{MgO}}}}}\) can only add a minor uncertainty to \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) and will not influence our conclusions below because the dominant contributions to the magnitudes of Ks come from the Au1−xPtx/Co interfaces rather than the Co/MgO interface (\({K}_{{{{\rm{s}}}}}^{{{{\rm{Co}}}}/{{{\rm{MgO}}}}}\ll \) Ks,\({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\)).

Considering that previous XMCD experiments have determined that \({m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }\) for the Au/Co interface (≈0.36 µB/Co)23 is twofold of that for the Pt/Co interface (≈0.18 µB/Co)22,28, the similar values of \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) for x = 0 and 1 would indicate ξPt/Co/ξAu/Co ≈ 2. Provided that \({m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }\) for the Au1−xPtx/Co interfaces for x = 0.25–0.75 is not significantly smaller than that of the Pt/Co interface, the 1.8 times variation of \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) with x would indicate a strong, but less than 3.6 times, tuning of ξ. In a simple case that \({m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }\) for the Au1−xPtx/Co interfaces varies approximately linearly with the composition x (the dashed line in Fig. 1d), the values of \({m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }\) and ξ (= 20µB\({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\)/\({m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }\)) can be estimated for all the Au1−xPtx/Co interfaces. As shown in Fig. 1e, in this case, the strength of ISOC is varied only by a factor of 2.5, from 14.2 meV to 36 meV. Thus, ISOC of all the samples in this study is sufficiently weak to allow application of Bruno’s model25.

It is interesting to note that, despite the wide tuning of ISOC, the bulk SOC strength for the Au1−xPtx layer, which should vary in between that of pure Au and pure Pt34, is expected to be approximately invariant with x because Au and Pt have almost the same SOC strength (410 meV)35. Note that the bulk SOC of the Au1−xPtx is also approximately 11–29 times stronger than ISOC of the Au1−xPtx/Co interface (Fig. 1e). We have also previously observed a threefold enhancement in ISOC of Pt/Co heterostructures by thermal engineering of the spin-orbit proximity, without varying the composition and thus the bulk SOC of the HM36. These observations consistently demonstrate the distinct difference between the interfacial and the bulk SOCs, with the former being very sensitive to the local details of the interfaces.

Tuning interfacial DMI by composition

We determined the interfacial DMI of the Au1−xPtx 4 nm/Co 3.6 nm bilayers by measuring the DMI-induced frequency difference (ΔfDMI) between counter-propagating Damon–Eshbach spin waves using Brillouin light scattering (BLS)11,12,13,14,15,16,17,18,19,37. Fig. 2a shows the geometry of the BLS measurements. The laser wavelength (λ) is 532 nm. The light incident angle (θ) with respect to the film normal was varied from 0° to 32° to tune the magnon wave-vector (k = 4πsinθ/λ). A magnetic field (H) of ±1700 Oe was applied along the x direction to align the magnetization of the Co layer. The anti-Stokes (Stokes) peaks in BLS spectra (Fig. 2b) correspond to the annihilation (creation) of magnons with k (−k), while the total in-plane momentum is conserved during the BLS process. In Fig. 2c, we plot ΔfDMI as a function of |k | for Au1−xPtx/Co interface with different x. Here ΔfDMI is the frequency difference of the ±k peaks and averaged for H = ± 1700 Oe (see ref. 18 for more details). The linear relation between ΔfDMI and k for each x agrees with the expected relation11,38 ΔfDMI ≈ (2γ/πμ0Ms)Dk, where γ ≈ 176 GHz T−1 is the gyromagnetic ratio as determined from ST-FMR measurements and D is the volumetric DMI constant.

Fig. 2: Interfacial Dzyaloshinskii-Moriya interaction (DMI).
figure 2

a Brillouin light scattering measurement geometry; b Brillouin light scattering spectrum (wave-vector k = 8.1 μm−1, magnetic field H = 1700 Oe, Pt concentration x = 0.85), c k dependence of ΔfDMI (x = 0, 0.85, and 1), d The volumetric DMI constant D, the total DMI constant Ds, and e D for Au1−xPtx 4 nm/Co 3.6 nm bilayers with different x plotted as a function of interfacial magnetic anisotropy energy density \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) (black) and the strength of interfacial spin-orbit coupling ξ (red). In (b) the blue solid curves represent fits to the Lorentzian function. In (c) the solid lines refer to the best linear fits. In (d) the blue arrow indicates a fivefold variation of the DMI strength and the dashed line is just to guide the eyes. In (e) the ξ values for the red dashed line are taken from Fig. 1e. The error bars represent the standard deviation of the fit parameters.

As summarized in Fig. 2d, with increasing x, D for the Au1−xPtx/Co interface varies by a factor of ~5, from −0.08 ± 0.02 for x = 0 to −0.45 ± 0.01 erg cm−2 for x = 0.85. Taking into account the inverse dependence of D on the FM thickness14,15,17 due to the volume averaging effect, the total DMI strength for HM/FM interface can be estimated as Ds = DtFM. For the Au1−xPtx/Co interfaces, Ds changes gradually from –30 ± 6 nerg cm−1 (1 nerg cm−1 = 10−9 erg cm−1) at x = 0 to –144 ± 8 nerg cm−1 at x = 0.85, and then drops slightly to –114 ± 6 nerg cm−1 at x = 1. We first note that the DMI for the Au1−xPtx/Co interfaces is very strong compared to those for other HM/FM systems. Even the relatively small Ds ≈ −30 nerg/cm for the Au/Co interface is already greater than all the reported BLS results (typically |Ds | ≤ 20 nerg cm−1)13,15,16,18,39 for (W, Ta, or Au)/(Co, Fe, or Fe–Co–B) interfaces (there was a report of Ds ≈ 40–44 nerg cm−1 for W/FeCoB from asymmetric domain wall motion and magnetic stripe annihilation, but not from BLS40). The maximum value of Ds ≈ −144 nerg cm−1 for Au0.85Pt0.15/Co is comparable to the highest reported values for Pt/FeCoB (from −80 to −150 nerg cm−1)18,41,42, Pt/Co (< −160 nerg cm−1)17,18, and Ir/Co (from 30 to 140 nerg cm−1)18,19. The large DMI amplitude and the fivefold tunability provided by the Au1−xPtx composition make Au1−xPtx/Co an especially intriguing system for chiral spintronics (e.g., magnetic domain-wall logic and skyrmion memories). Here, we note that the DMI contribution from the Co/MgO interface (\({K}_{{{{\rm{s}}}}}^{{{{\rm{Co}}}}/{{{\rm{MgO}}}}}\)≈ 0.32 erg cm−2) should be negligble because the DMI is found to be essentially zero at FeCoB/MgO interface where ISOC is much stronger (≈0.5 erg cm−2)41. This conclusion is in line with the wide consensus11,12,17,18,19,26 that the DMI in HM/FM/MgO heretostructures is dominantly from the HM/FM interface and is neglible at the FM/MgO interfaces. We note that some studies suggested a sizable DMI from delibrately oxidized magnetic surfaces (e.g., by chemiabsorption of oxygen43, electric field-driven oxygen migration44, or depostion in oxygen atomosphere45), which is, however, distinctly different from the case of our HM/FM/MgO samples in which the FM/MgO interfaces are sharp and the FM layers are unoxidized as indicated by the results of STEM46,47 (see Fig. 1a for Co/MgO interface), electron energy loss spectrum46, magnetic damping measurements32, and the thickness-dependent study of magnetic moment (see Supplementary Fig. S2 for an example of Co/MgO interfaces).

The fivefold variation of interfacial DMI with x for the Au1−xPtx/Co interfaces is clearly not in linear proportion to the 1.8 times (<3.6 times) variation of \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) (ξ). Indeed when we plot D as a function of \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) and ξ (Fig. 2e), it becomes quite apparent that there is no linearly proportional correlation between D and \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) or between D and ξ. This is a striking observation as it indicates that there must be another composition-sensitive effect that is, at least, as critical as ISOC for interfacial DMI.

Critical role of interfacial orbital hybridization

As we discuss below, the non-monotonic and exceptionally strong tunability of the DMI (stronger than that can be provided solely by ISOC) can be understood by taking into account the essential role of the varying orbital hybridization at the Au1−xPtx/Co interface. Theoretical calculations have shown that, besides ISOC, the 3d orbital occupations and their spin-flip scattering with the spin-orbit active 5d states collectively control the overall DMI48. The strength of the interfacial orbital hybridization is expected to be inversely proportional to the on-site energy difference of the 3d and 5d states49. This is because the FM 3d states and the HM 5d states are the main sources of the total density of states (DOS) of conduction bands around the Fermi level (EF). As we show in Fig. S4 of the Supplementary Materials, EF of bulk Au (5d106s1) is located ~2 eV above the top of 5d band so that there are no 5d orbitals at the Fermi surface50. In contrast, EF of bulk Pt (5d96s1) is located in the top region of its 5d band. In both bulk Au and Pt, the density of states (DOS) has a sharp peak at the top region of the 5d band51. First-principles calculations52 have indicated that, in magnetic multilayers consisting of repeats of HM (2 monolayers)/Co (monolayer) multilayers where the interfacial orbital hybridization becomes very important, both the HM 5d bands and the Co 3d bands are significantly broadened compared to their bulk properties. In Fig. 3a, we compare the calculated results51 for the local DOS of the Au/Co and Pt/Co systems with sharp interfaces, in consistence with our samples32. For Au/Co, the top of the Au 5d band is located ≈1.2 eV below EF, the minority spin band of Co 3d is centered at EF. For the Pt/Co system, the top of the Pt 5d band is 0.7 eV above EF, and the minority spin band of Co 3d is 0.63 eV above EF for the Pt/Co. In both cases, the majority spin band of Co 3d is located well below EF due to the exchange splitting.

Fig. 3: Critical role of interfacial orbital hybridization.
figure 3

a Calculated local density of states (DOS) for Au/Co (up) and Pt/Co (down) interfaces (blue for the DOS of Au and Pt, red for that of Co)51, highlighting a substantial shift of the 5d (3d) band from below (around) to beyond the Fermi level (EF) as a function of x. The two dashed lines refer to the top of the 5d bands of Au and Pt. Note that the Co 3d states and the Au1−xPtx 5d states are the main sources of the total DOS of conduction bands around EF. b Schematic depict of the evolution of interfacial orbital hybridization at the Au1−xPtx/Co interface with the composition. At x = 0 (i.e., Au/Co), the 5d band of Au1−xPtx (blue) has little orbital hybridization with the 3d minority band of the Co (red) at the EF such that the Dzyaloshinskii-Moriya interaction is the weakest. With increasing x, both bands are increasingly shifted upwards, with the 5d band is shifted faster than the 3d band. As a result, the 3d-5d orbital hybridization increases gradually and reaches the maximum at an intermediate composition of x = 0.85. As x further increases to 1 (Pt/Co), the peaks of both the 3d and 5d bands are shifted far above EF, resulting in a moderate orbital hybridization at EF and a reduced Dzyaloshinskii-Moriya interaction.

To more clearly demonstrate the role of the interfacial orbital hybridization, the evolution of the 3d–5d orbital hybridization with the composition is schematically depicted in Fig. 3b. At x = 0 (i.e., Au/Co), the 5d band of Au1−xPtx (blue) has little orbital hybridization with the 3d minority band of the Co (red) at the EF such that the DMI is the weakest. As x increases, the 5d band of the Au1−xPtx can be expected to be lifted continuously and pass through the Fermi level (i.e., from −1.2 eV for x = 0 to +0.7 eV for x = 1 with respect to EF). Meanwhile, the top of the minority spin band of Co 3d should also be shifted to be well above EF, with a lower shifting rate than the 5d band of the Au1−xPtx. As a consequence, with increasing x, the hybridization of the 5d orbitals of the Au1−xPtx with the 3d orbitals of Co would be first strengthened (mainly due to the shift of 5d band), then be maximized at an intermediate composition where the DOS peak of the 5d band is approximately at EF and is well aligned with the DOS peak of the 3d minority spin band of Co, and then finally decrease slightly as x approaches 1 (the DOS peak of the minority spin band Co 3d is moved away from EF). This is quite consistent with our experimental observation that, as x increases, the DMI is first increasingly enhanced by, we propose, 5d–3d hybridization, then is maximized at about x = 0.85, and finally decrease (Fig. 2d). The moderate reduction of ISOC for x > 0.85 (Fig. 1e) is also a likely contributor to the decrease of the DMI in this composition range.

In our previous study of Pd1−xPtx/Fe60Co20B20 interfaces41, we found that the DMI does vary proportionally with \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\), which suggests a negligible variation of the interfacial orbital hybridization in that particular material system (Eq. (2)). This seems reasonable because, in the bulk, the energy distribution of the 4d DOS of Pd (4d106s0) is rather analogous to that of Pt 5d, with the first DOS peak of the Pd 4d band lying at about EF (Supplementary Fig. S4). Since the orbital hybridization at the Pd1−xPtx/Fe60Co20B20 interfaces can further broaden the 5d and 3d bands, the DOS distribution and thus the strength of the interfacial 5d-3d orbital hybridization should be reasonably similar as a function of the Pd1−xPtx composition. This conclusion is very well supported by a first-principles calculation52. Pd also has the same electronegativity (2.2)53,54,55 and valence electron number (10)35 as Pt. Therefore, in the case of Pd1−xPtx/Fe60Co20B20, ISOC is left as the only important variable that determines the variation of the interfacial DMI with the Pd1−xPtx composition.

Finally, the lack of a linearly proportional correlation between interfacial DMI and ISOC in our samples cannot be attributed to any effects of electronegativity4, intermixing12, proximity magnetism12,56, or orbital anisotropy28. Since Au and Pt have quite similar electronegativities (2.2 for Pt and 2.4 for Au)53,54,55, a substantial composition-induced electronegativity variation in Au1−xPtx seems unlikely to occur at the Au1−xPtx/Co interface. As revealed by the STEM imaging (Fig. 1a and Supplementary Fig. S1) and the thickness-dependent magnetic moment measurements (Supplementary Fig. S2), our samples have reasonably sharp interfaces and no obvious intermixing or magnetic dead layer within the experimental resolution. Previous studies have suggested that interfacial alloying, if significant, may substantially degrade both the DMI12 and \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\)41,57. However, our Au1−xPtx/Co bilayers have very high Ds and \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\). In previous work, we have reported on the structural characterizations of HM/FM samples prepared with similar conditions by X-ray reflectivity, and secondary ion mass spectrometry36, all of which have indicated minimal interface alloying. The relatively low values of Ms (1200–1300 emu cm−3) also indicate a rather minimal proximity magnetism at these Au1−xPtx/Co interfaces58. The DMI in Pt/Co bilayers was also suggested to correlate linearly with orbital anisotropy, with the latter being quantified by the \({m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }\)/\({m}_{{{{\rm{o}}}}}^{\parallel }\) ratio28. By using Eq. (2) and the approximately invariant \({m}_{{{{\rm{o}}}}}^{\parallel }\) of Co22,23,28,29,30, we can expect \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\propto \xi {m}_{{{{\rm{o}}}},{{{\rm{i}}}}}^{\perp }/{m}_{{{{\rm{o}}}}}^{\parallel }\), due to which the combined effect of the orbital anisotropy and ISOC would be a linear dependence of D on \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\). Thus, from the lack of a linear correlation between D and \({K}_{{{{\rm{s}}}}}^{{{{\rm{ISOC}}}}}\) in Fig. 2e we conclude that the variation of the DMI at the Au1−xPtx/Co interfaces with x cannot be attributed to the change of orbital anisotropy.

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