Wide-Range Probing of Dzyaloshinskii–Moriya Interaction

The Dzyaloshinskii–Moriya interaction (DMI) in magnetic objects is of enormous interest, because it generates built-in chirality of magnetic domain walls (DWs) and topologically protected skyrmions, leading to efficient motion driven by spin–orbit torques. Because of its importance for both potential applications and fundamental research, many experimental efforts have been devoted to DMI investigation. However, current experimental probing techniques cover only limited ranges of the DMI strength and have specific sample requirements. Thus, there are no versatile methods to quantify DMI over a wide range of values. Here, we present such an experimental scheme, which is based on the angular dependence of asymmetric DW motion. This method can be used to determine values of DMI much larger than the maximum strength of the external magnetic field strength, which demonstrates that various DMI strengths can be quantified with a single measurement setup. This scheme may thus prove essential to DMI-related emerging fields in nanotechnology.


DW energy model for a DW at an angle θ. The inset in
shows the case where a DW is placed at an angle θ with respect to an in-plane magnetic field, H x . The DW energy density σ DW can be expressed as a function of H x and the angle ψ between the magnetization direction and the normal to the DW 3 where σ 0 is the Bloch-type DW energy density, λ is the DW width, K D is the DW-anisotropy energy density, M S is the saturation magnetization, and H DMI is the DMI-induced effective magnetic field in the direction normal to the DW. The second term in the right-hand side of the equation corresponds to the DW-anisotropy energy and the Equation (3) is identical to the relation ψ θ π =  /2 eq , which implies that the DW magnetization stays perpendicular to the direction of H 0 . Inserting this value of ψ eq into Eq. (2), it can be rewritten as where H K (≡ 4K D /πM S ) is the DW-anisotropy field, which usually is small. Hence, in practice, the sign of the first term on the right-hand side of Eq. (4) coincides with that of H DMI , i.e., a plus sign for a positive H DMI and a minus sign for a negative H DMI . Note that the well-known relation H 0 = − H DMI 11,16,17 can be restored in the limit θ → 0. Figure 1 plots the value of H 0 (θ) obtained from Eq. (4). This plot shows that H 0 has the clear angle dependence. Equation (4) contains the key idea of this study: one can significantly reduce the value of H 0 by increasing θ. With this scheme, the magnitude of H 0 can be adjusted down to a small experimental range H range of the external magnetic field, which allows one to measure a large H DMI without upgrading the electromagnet. For example, by tilting the DWs up to about 80°, one can measure H DMI up to 1 T using an electromagnet with H range ~ 200 mT, which is easily achieved in conventional optical setups 23 . It is also worth noting that by using this approach we can prevent a number of artifacts caused by large magnetic fields, such as mechanical instability produced by the induced magnetic moment in the optical setup, magneto-optical effects in the objective lens, and large Joule heating caused by the huge currents passing through the electromagnet.

Verification of θ-dependence in Pt/Co/AlO x films.
To verify the feasibility of the present scheme, it was applied to ferromagnetic Pt/Co/AlO x films, for which H DMI is slightly smaller than H range . The procedure to measure H 0 closely follows ref. 11, except for the initially tilted DWs. The tilted DWs were generated using a thermomagnetic writing technique 7,24,25 (see Methods). The images in the right panel of Fig. 2 show the displacements of the DWs for various values of θ with respect to the direction of in-plane magnetic field H x (= 120 mT) under the application of a fixed out-of-plane magnetic field H z (= 5.5 mT) bias. Each image was obtained by adding several images, sequentially acquired during the DW displacement with a constant time step (= 500 ms). Thus, each image simultaneously shows several DWs moving from brighter to darker interfaces as time goes by. One can then measure the DW speed v for each image. The plots in the left panel of Fig. 2    alternatively measured through independent measurements 11,26,27 or estimated using the relation π λ ≅ H M t (4 ln 2/ ) / K 2 S f 28,29 , where t f is the thickness of the magnetic layer.

Application of present scheme to Pt/Co/AlO x and Pt/Co/MgO films. To reproduce a situation in
which H range is limited (< 50 mT), the fit was also performed only for the data (box in the plot) with large θ (≥ 70°) as shown in Fig. 3b. The blue solid line indicates the best fit to Eq. (4), using the fixed value of H K obtained from Fig. 3a. This approach gives the best-fit value H DMI (= − 138 ± 12 mT), which again matches the previous values within the experimental accuracy. It is therefore demonstrated that the present approach enables one to measure large H DMI in an experiment with limited H range . Note that the determined H DMI is more than twice larger than H range . Because the fit in Fig. 3b was performed with a fixed H K , now we examine the effect of the inaccuracy δH K on H K . The blue dotted lines in Fig. 3b are the best fits when δH K = ± 10 mT. The error δH DMI is found to be slightly smaller than δH K , as expected from the relation δH DMI = δH K sinθ deduced from Eq. (4). Because H K is commonly within the range of a few tens of mT 11,[26][27][28]30 , δH K typically will not exceed about ± 10 mT, and thus one can confirm that the error induced by δH K error is not significantly large as compared to other experimental errors. Moreover, this error becomes negligible in practical cases because the present approach is designed for the determination of large H DMI (≫ δH K ), significantly beyond the experimental H range .
Finally, the present scheme was applied to Pt/Co/MgO films, which exhibit DMI larger than H range . Figure 4a shows v as a function of H x for θ = 0. From this plot, it is apparent that the inversion symmetry axis H 0 lies far beyond the experimental H range (i.e., H 0 ≫ 200 mT), and thus conventional optical schemes cannot be used to quantify H DMI . However, by applying the present method, Fig. 4b shows the measured H 0 with respect to θ for large θ (≥ 80°). The black box in the figure indicates the measurable window for H range in the present setup. The best fit (blue solid line) of H K (= − 30 ± 5 mT) indicates that H DMI = −483 ± 10 mT, which is more than twice larger than H range . The sign and magnitude are in good agreement with previously reported results 21 . The blue dotted lines in Fig. 4b are the best fits for the cases with δH K = ± 10 mT, and thus it is clearly demonstrated that the error becomes negligible in this case.

Discussion
Additional asymmetry from chiral damping 31 or asymmetric DW width variation 28 may cause a shift δH 0 in H 0 . However, because the asymmetric slope in v caused by these phenomena appears only during chirality variation occurring within the range of ± H K , |δH 0 | is essentially smaller than |H K |. Therefore, δH 0 -induced errors are negligible again in practice for large H DMI determination.
In conclusion, we proposed a scheme to measure H DMI over a wide range of values, overcoming the limitations caused by the small strength (H range ) of the external magnetic fields typically used in experiments. By measuring the angular dependence of asymmetric DW motion, we found that H 0 is strongly correlated with θ, which means that large DMI can be quantified in a robust manner by setting large values of θ. The feasibility of the present approach is experimentally demonstrated for various DMI strengths using ferromagnetic Pt/Co/AlO x and Pt/Co/MgO films. The errors caused by additional asymmetry and inaccuracy of H K were found to be negligible in practice for large H DMI determination. The present scheme enhances the experimental range of optical measurement techniques without the need upgrade electromagnets. Our findings represent a novel and straightforward way to explore materials and systems with large DMI, and thus surmounts the key obstacle to design new devices in which the DMI is tailored to achieve for topological stability and efficient manipulation, as required for next-generation nanotechnology.

Methods
Sample preparation. For this study, we prepared 5.0-nm Ta/3.0-nm Pt/0.6-nm Co/1.6-nm AlO x and 5.0-nm Ta/3.0-nm Pt/0.6-nm Co/2.0-nm MgO films, which were deposited on a Si wafer with a 100 nm SiO 2 layer by dc magnetron sputtering 23 . To enhance the sharpness of the layer interfaces, we set a small deposition rate (= 0.25 Å/s) through adjustment of the Ar sputtering pressure (~2 mTorr) and power (~10 W). All the films exhibited strong perpendicular magnetic anisotropy and circular domain expansion with weak pinning strength.
Thermomagnetic writing of tilted domain walls. To create tilted DWs, we adopted a thermomagnetic writing technique 7,24,25 . The film was first saturated by a magnetic field (= − 10 mT) and then, a laser beam (= 75 mW) was focused onto a small spot (5 μ m in diameter) of the film under a reversed magnetic field (= 3.3 mT) smaller than the coercive field (= 8 mT). At this instant, the sample stage was moved along a desired direction, resulting in formation of a tilted straight DW. Alternatively, tilted DWs can be obtained from small arcs of larger circular domains.
Experimental setup and measurement. The magnetic domain images were acquired using a magneto-optical Kerr effect (MOKE) microscope equipped with a charge-coupled device (CCD) camera. To apply the magnetic field onto the films, two electromagnets were attached to the sample stage. One of them was used to produce an in-plane magnetic field bias H x up to 200 mT, whereas the other created an out-of-plane magnetic field H z up to 35 mT. The possible effect from the small misalignment of the in-plane magnet as well as the ambient magnetic field is included in the H z calibration, where this ambient magnetic field can be estimated by measuring the DW speed along right direction when + H x is applied and the DW speed along left direction when − H x is applied. This field difference between them is come from the small misalignment of the in-plane magnet. Using this system, the field-driven DW speed v was measured in the creep regime. To do so, a linear DW was initially placed at a tilted angle θ, as shown in the inset of Fig. 1, and then the DW displacement in the normal direction of the DW was monitored by the MOKE microscope under application of constant H z and/or H x . The dependence of v on H x exhibits asymmetries attributed to the variation of the DW energy density with H x , and the DMI-induced effective field can be directly quantified at a local minimum 11 .