Introduction

The Dzyaloshinskii–Moriya interaction (DMI) is an antisymmetric exchange interaction that occurs at interfaces between ferromagnetic and heavy metal layers with large spin–orbit coupling1,2,3. In magnetic systems, DMI generates chiral spin textures such as Néel domain walls (DWs)4,5,6,7 and magnetic skyrmions8,9,10. Because these chiral spin textures promise several potential applications4,7,8, it is crucial to quantify the strength of the DMI both to better understand its physical origin and for the technical optimization of ferromagnetic materials.

Several experimental schemes have been proposed to measure the DMI strength5,6,11,12,13,14,15. Using an optical microscope, it has been developed a method to estimate it based on the field-driven asymmetric DW speed with respect to an in-plane magnetic field11,16,17 and the current-driven asymmetric DW speed5,6. Moon et al.12 suggested another approach based on the frequency nonreciprocity, which provides a way to measure the DMI constant. Other measurement schemes relying on the nonreciprocal propagation of spin waves were demonstrated using Brillouin light scattering (BLS)18,19,20 and inductive ferromagnetic resonance (FMR)21. All these techniques are applicable to different ranges of DMI strength and have different sample requirements.

The optical microscopy technique11,16,17 based on asymmetric DW speed provides an easy and direct way to measure DMI-induced effective magnetic fields. However, its measurement range is limited by the maximum strength of the external in-plane magnetic field, which in turn is fundamentally constrained by the narrow space available inside the optical setup. Moreover, application of large external magnetic fields to the optical setup requires sophisticated care to prevent artifacts caused by stray fields from electromagnets as well as mechanical, optical, and thermal artifacts from such strong magnetic fields. In this study, we propose a way to overcome this field strength limit by utilizing DWs, inclined at an angle with respect to the direction of the in-plane magnetic field.

Results

DW energy model for a DW at an angle θ

The inset in Fig. 1 shows the case where a DW is placed at an angle θ with respect to an in-plane magnetic field, Hx. The DW energy density σDW can be expressed as a function of Hx and the angle ψ between the magnetization direction and the normal to the DW3,11:

Figure 1: Plot of H0 as a function of θ.
figure 1

The Red solid line is calculated by using Eq. (4). The inset is an schematic illustration of the measurement geometry.

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where σ0 is the Bloch-type DW energy density, λ is the DW width, KD is the DW-anisotropy energy density, MS is the saturation magnetization, and HDMI 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 third term to the Zeeman energy, including the DMI as an effective magnetic field. Note that Eq. (1) is identical to the Stoner–Wohlfarth equation22 for torque magnetometry with an additional unidirectional bias from HDMI.

For a given Hx, the equilibrium angle ψeq can be obtained by the minimization condition . Moreover, a numerical analysis of Eq. (1) shows that σDW has a maximum at Hx = H0, where H0 can be obtained from the maximization condition . By solving these minimization and maximization conditions simultaneously, one can readily obtain two coupled equations:

Equation (3) is identical to the relation , which implies that the DW magnetization stays perpendicular to the direction of H0. Inserting this value of ψeq into Eq. (2), it can be rewritten as

where HK (≡4KD/πMS) 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 HDMI, i.e., a plus sign for a positive HDMI and a minus sign for a negative HDMI. Note that the well-known relation H0 = −HDMI11,16,17 can be restored in the limit θ → 0. Figure 1 plots the value of H0(θ) obtained from Eq. (4). This plot shows that H0 has the clear angle dependence.

Equation (4) contains the key idea of this study: one can significantly reduce the value of H0 by increasing θ. With this scheme, the magnitude of H0 can be adjusted down to a small experimental range Hrange of the external magnetic field, which allows one to measure a large HDMI without upgrading the electromagnet. For example, by tilting the DWs up to about 80°, one can measure HDMI up to 1 T using an electromagnet with Hrange ~ 200 mT, which is easily achieved in conventional optical setups23. 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/AlOx films

To verify the feasibility of the present scheme, it was applied to ferromagnetic Pt/Co/AlOx films, for which HDMI is slightly smaller than Hrange. The procedure to measure H0 closely follows ref. 11, except for the initially tilted DWs. The tilted DWs were generated using a thermomagnetic writing technique7,24,25 (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 Hx (=120 mT) under the application of a fixed out-of-plane magnetic field Hz (=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 show the normalized DW speed v/vmin in the direction normal to the DW as a function of Hx (in creep regime), where vmin is the apparent minimum of v. It can be seen that v(Hx) is symmetric under inversion with respect to as shown in each plot. Here, indicates the inversion symmetry axis where v has a minimum. According to ref. 11, v follows the creep relation , where v0 is the characteristic speed. In the case of clear inversion symmetry with a constant v0, the experimental exactly matches H0, and thus, we will denote by H0 hereafter.

Figure 2: Plot of v/vmin as a function of Hx in a Pt/Co/AlOx film, for fixed Hz = 5.5 mT and under various values of θ.
figure 2

(a) 0° (b) 30° (c) 60°, and (d) 90°. The displacement of the DWs at each angle is driven by the external magnetic fields.

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Figure 3a plots the measured H0 as a function of θ. The red solid line shows the best fit to Eq. (4). The good agreement between the data and the fitting curve supports again the validity of the equation. The best-fit HDMI (=−132 ± 3 mT) matches well the experimental value (=−134 ± 6 mT) measured at θ = 0. Moreover, the best-fit value of HK (=−18 ± 5 mT) falls within the range of previous experimental reports11,26,27. The value of HK can be alternatively measured through independent measurements11,26,27 or estimated using the relation 28,29, where tf is the thickness of the magnetic layer.

Figure 3: Plot of the measured H0 as a function of θ in a Pt/Co/AlOx film.
figure 3

(a) Data collected for large θ range (from 0° to 90°). The red solid line shows the best fit to Eq. (4). (b) Data collected within a small θ range (from 70° to 90°). The blue solid line represents the best fit to Eq. (4) with the value of HK fixed. The blue dotted lines in (b) are the best fits for the cases with δHK =±10 mT.

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Application of present scheme to Pt/Co/AlOx and Pt/Co/MgO films

To reproduce a situation in which Hrange 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 HK obtained from Fig. 3a. This approach gives the best-fit value HDMI (=−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 HDMI in an experiment with limited Hrange. Note that the determined HDMI is more than twice larger than Hrange.

Because the fit in Fig. 3b was performed with a fixed HK, now we examine the effect of the inaccuracy δHK on HK. The blue dotted lines in Fig. 3b are the best fits when δHK =±10 mT. The error δHDMI is found to be slightly smaller than δHK, as expected from the relation δHDMI = δHKsinθ deduced from Eq. (4). Because HK is commonly within the range of a few tens of mT11,26,27,28,30, δHK typically will not exceed about ±10 mT, and thus one can confirm that the error induced by δHK 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 HDMI (≫δHK), significantly beyond the experimental Hrange.

Finally, the present scheme was applied to Pt/Co/MgO films, which exhibit DMI larger than Hrange. Figure 4a shows v as a function of Hx for θ = 0. From this plot, it is apparent that the inversion symmetry axis H0 lies far beyond the experimental Hrange (i.e., H0 ≫ 200 mT), and thus conventional optical schemes cannot be used to quantify HDMI. However, by applying the present method, Fig. 4b shows the measured H0 with respect to θ for large θ (≥80°). The black box in the figure indicates the measurable window for Hrange in the present setup. The best fit (blue solid line) of HK (=−30 ± 5 mT) indicates that HDMI = −483 ± 10 mT, which is more than twice larger than Hrange. The sign and magnitude are in good agreement with previously reported results21. The blue dotted lines in Fig. 4b are the best fits for the cases with δHK =±10 mT, and thus it is clearly demonstrated that the error becomes negligible in this case.

Figure 4: DMI determination in Pt/Co/MgO film.
figure 4

(a) Plot of the measured v as a function of the in-plane field Hx, for fixed Hz = 20 mT, at θ = 0. (b) Plot of the measured H0 as a function of θ, for values of θ between 80° and 90°. The blue solid line shows the best fit to Eq. (4) and the blue dotted lines in (b) are the best fits for the cases with δHK =±10 mT.

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Discussion

Additional asymmetry from chiral damping31 or asymmetric DW width variation28 may cause a shift δH0 in H0. However, because the asymmetric slope in v caused by these phenomena appears only during chirality variation occurring within the range of ±HK, |δH0| is essentially smaller than |HK|. Therefore, δH0-induced errors are negligible again in practice for large HDMI determination.

In conclusion, we proposed a scheme to measure HDMI over a wide range of values, overcoming the limitations caused by the small strength (Hrange) of the external magnetic fields typically used in experiments. By measuring the angular dependence of asymmetric DW motion, we found that H0 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/AlOx and Pt/Co/MgO films. The errors caused by additional asymmetry and inaccuracy of HK were found to be negligible in practice for large HDMI 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 AlOx 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 SiO2 layer by dc magnetron sputtering23. 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 technique7,24,25. 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 μm 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 Hx up to 200 mT, whereas the other created an out-of-plane magnetic field Hz 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 Hz calibration, where this ambient magnetic field can be estimated by measuring the DW speed along right direction when +Hx is applied and the DW speed along left direction when −Hx 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 Hz and/or Hx. The dependence of v on Hx exhibits asymmetries attributed to the variation of the DW energy density with Hx, and the DMI-induced effective field can be directly quantified at a local minimum11.

Additional Information

How to cite this article: Kim, D.-H. et al. Wide-Range Probing of Dzyaloshinskii–Moriya Interaction. Sci. Rep. 7, 45498; doi: 10.1038/srep45498 (2017).

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