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Dzyaloshinskii–Moriya interaction in noncentrosymmetric superlattices

Dzyaloshinskii–Moriya interaction in noncentrosymmetric superlattices www.nature.com/npjcompumats ARTICLE OPEN Dzyaloshinskii–Moriya interaction in noncentrosymmetric superlattices 1 2,3✉ 4 5 5 2 1 Woo Seung Ham , Abdul-Muizz Pradipto , Kay Yakushiji , Kwangsu Kim , Sonny H. Rhim , Kohji Nakamura , Yoichi Shiota , 5 1 ✉ ✉ Sanghoon Kim and Teruo Ono Dzyaloshinskii–Moriya interaction (DMI) is considered as one of the most important energies for specific chiral textures such as magnetic skyrmions. The keys of generating DMI are the absence of structural inversion symmetry and exchange energy with spin–orbit coupling. Therefore, a vast majority of research activities about DMI are mainly limited to heavy metal/ferromagnet bilayer systems, only focusing on their interfaces. Here, we report an asymmetric band formation in a superlattices (SL) which arises from inversion symmetry breaking in stacking order of atomic layers, implying the role of bulk-like contribution. Such bulk DMI is more than 300% larger than simple sum of interfacial contribution. Moreover, the asymmetric band is largely affected by strong spin–orbit coupling, showing crucial role of a heavy metal even in the non-interfacial origin of DMI. Our work provides more degrees of freedom to design chiral magnets for spintronics applications. npj Computational Materials (2021) 7:129 ; https://doi.org/10.1038/s41524-021-00592-8 INTRODUCTION space, and it has been vastly utilized to interpret a number of 11,18–20 magnetic phenomena , in particular those well understood The lack of inversion symmetry at the interface between a heavy to originate from the ISB, such as the DMI responsible for exotic metal (HM) and a ferromagnet (FM) induces the antisymmetric magnetic textures such as skyrmions and chiral domain walls, and exchange interaction so-called Dzyaloshinskii–Moriya interaction 1–4 spin–orbit torque. It should be noted that amorphous ferrimagnet (DMI) . Recently, DMI has been intensively studied in the GdFeCo exhibits bulk DMI feature, which is independent of material combinations possessing perpendicular magnetic aniso- interface but due to inhomogeneous distribution of elemental tropy (PMA) due to their necessities in creating magnetic chiral content . In this context, further study to distinguish DMI textures, such as magnetic skyrmions for the new type of racetrack 5–8 interface and bulk origin would be interesting topic. memory device . Generally, in order to stabilize skyrmions at In this study, we investigate DMI of the [Co/Pd/Pt]-SL arising room temperature, multilayer structures with repetitive stacking of from bulk spin-momentum locking. First-principles calculations FM/HM bilayer are utilized because multi-stacking of the bilayer reproduce such DMI enhancement in the SL, showing that the unit easily provide the PMA and the sizable DMI at the same time, asymmetry of bands around Fermi energy level induced by ISB. both of them arising from the same physical origin, i.e., interfacial 9–13 The observed behavior of DMI upon increasing the repetitions of SOC . In this respect, Co/Pd and Co/Pt interfaces are one of the the ABC-layer unit in the SL suggests that while the interfacial and well-known material combinations providing both the PMA and bulk DMI co-exist with small N, the enhancement of DMI with the DMI originating from interfaces, resulting in stable magnetic 14,15 larger N can be attributed to the bulk-type asymmetric band skyrmions . With the same manner of such an AB-type multi- formation around the Fermi level. stacking structure composed of several nanometer-thick layers, a layer structure with ABC-type repetitive stacking of a few atomic monolayers is interesting system as illustrated in Fig. 1a. An RESULTS AND DISCUSSION epitome is the [Co/Pd/Pt] superlattice (SL) possessing PMA DMI of the noncentrosymmetric SLs generated by the bulk-type spin momentum locking due to absence of inversion symmetry in stacking order . Note that not Estimation of the magnetocrystalline energy (MCA) and DMI is 22–24 interfaces but asymmetry of bulk-type band formation in the [Co/ done by following the steps outlined previously . Here, we Pd/Pt]-SL as illustrated in Fig. 1b is essential to give rise to such a consider 1–4 [Co/Pt/Pd] units, and anticlockwise rotation of the chiral phenomenon, resulting in strong PMA. spin-spiral structures as shown in Fig. 2a–c. The detailed process is Such inversion symmetry breaking (ISB) in the SL with ABC-type explained in the “Method” section. The calculation results are stacking order would traditionally be accounted for by involving summarized in Fig. 2d and e. The odd terms of the MCA energy odd the Rashba model Hamiltonian, H ¼ αðÞ k ´ ^ z  σ, which was (E ), which quantifies ISB, are summarized in Fig. 2d. This quantity R R MCA initially proposed for a surface, where z is the direction of is related to the ISB-induced shift of the band structure along the k inversion-symmetry-breaking-induced potential gradient . The direction due to the magnetization along x direction. We note that odd oddness of the SOC in the k space due to the ISB is shown by the E increases with the repetition number N of [Co/Pt/Pd] unit MCA odd dependence of the Hamiltonian on the linear terms in k, although layers. The total MCA and E in the [Pt/Co/Pd] increases shown MCA higher odd-order may in principle also appear. Rashba effect in Fig. 2d, and the total MCA value can be expected to reach −2 manifests most immediately into a spin-splitting within the k 0.043 meV Å obtained for the bulk [Co/Pt/Pd]-SL, i.e. the infinitely 1 2 3 Institute for Chemical Research, Kyoto University, Uji, Kyoto, Japan. Department of Physics Engineering, Mie University, Tsu, Mie, Japan. Faculty of Mathematics and Natural 4 5 Sciences, Institut Teknologi Bandung, Bandung, Indonesia. National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, Japan. Department of Physics, University of Ulsan, Ulsan, South Korea. email: a.m.t.pradipto@gmail.com; sanghoon.kim@ulsan.ac.kr; ono@scl.kyoto-u.ac.jp Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences 1234567890():,; W.S. Ham et al. Fig. 1 SL structure with symmetry breaking in stacking order. a SLs with AB-type and ABC-type stacking-order, which is composed of nm-thick and sub-nm-thick layers, respectively. b Schematic image of an asymmetric band structure in terms of direction of magnetization in the SL with ABC-stacking order. asym periodic system along c direction [Pt/Co/Pt] . When the DMI E , the contribution of a particular layer L to the asymmetric q¼ ±0:25 constants are further extracted by utilizing the polynomial energy can be given as expression of the frozen magnon energy (Eq. 2 in “Method” asym asym asym E ðÞ L ¼ E ðÞ tot  E ðL Þ (1) off q¼ ±0:25 q¼±0:25 q¼ ±0:25 section), we obtain a DMI energy density (D) value of the asym asymmetric [Pt/Co/Pd] structure which increases with N [see where E ðtotÞ is the asymmetric energy which includes the q¼±0:25 −2 asym Fig. 2e]. Here, we use the unit J m for DMI values of the structures contribution of all layers, and E ðL Þ is the asymmetric off q¼ ±0:25 in order to compare with experimental results, shown later in this energy when the SOC contribution of layer L is switched off. The study. It is important to note that the D value increases with N and results are summarized in Fig. 4, clearly showing that, first of all, converges to the bulk value. Such linear dependence of DMI cannot despite being the carrier of the magnetic moments, Co gives very be simply explained by the interfacial origin. If the increase in DMI small contribution to the DMI. In fact, the largest contribution to with N is fully originated from the interfacial spin–orbit coupling, the asymmetric energy, hence to the DMI, is coming from the DMI should be independent of N because the gains in the DMI neighboring Pt layers, which can be understood to dominate the energy by increasing the number of interfaces are compensated by band around Fermi level, as shown in Fig. 3d–f. It should be noted the volume of Co layers. The detailed mechanism of this result that for all N, Pd gives smaller and opposite contribution to the asym is explained in the following paragraph with anatomy of the DMI, in comparison to Pt. Additionally, the evaluated total E q ±0:25 k-dependent band structures. for all N, as plotted in the inset of Fig. 4, shows considerable The two-dimensional MCA contour map of [Pt/Co/Pd] is shown 1 deviations for all contributions, then, converges to that of the bulk in Fig. 3a. This map shows the typical characteristic of the MCA in with infinite periodicity. This implies the existence of a non- the systems with ISB. Since the in-plane magnetization is oriented interfacial origin, but bulk spin-momentum locking for the DMI along the +x direction, i.e., m ¼ m x, the MCA is asymmetric in the considered [Pt/Co/Pd]-SL systems. þx along the k axis, which is along the ẑ × m direction, and shows mirror symmetry along k direction. Similar behavior is obtained Experimental observation of DMI of the noncentrosymmetric for other N. The Fermi surface of [Pt/Co/Pd] is visibly shifted SLs towards negative k as shown by using the red color in Fig. 3b for To experimentally confirm the bulk spin-momentum locking for m magnetization, and is shifted symmetrically towards negative +x the DMI, the [Co 0.4/Pd 0.4/Pt 0.4 (nm)] SLs, hereafter [Co/Pd/Pt]- k when the magnetization is along the m x direction, as plotted y x SL, was prepared as the noncentrosymmetric SL varying N as by using the blue color in Fig. 3b. In order to check the band explained in the “Method” section [see also Fig. 5a]. The X-ray contribution to the Rashba splitting, we plot Fermi surface by reflection pattern of the film with N = 5 clearly shows the 1st- switching off the spin–orbit coupling in each atomic layer. Bragg-like peak, implying that the [Co/Pd/Pt] unit structure is well Comparing the results shown in Fig. 3d–f, it is visible that preserved without severe atomic intermixing in spite of such ultra- 26,27 switching off SOC in Pt layer alters the shape of the Fermi contour thin thickness range . All [Co/Pd/Pt]-SLs are confirmed to have most considerably compared to that in Co and Pd. As the change PMA as shown in Fig. 5c and d. In Particular, all magnetic in the Fermi contour indicates the modification of the band hysteresis curves measured with the easy-axis field sweep show structure around the Fermi level, the result implies the crucial role squareness of unity and sharp switching properties of the SLs. In of Pt bands to the band splitting, which should be attributed to case of the hard-axis loops taken with in-plane field sweep, any the strong SOC constant of the Pt atom. signal jump in the loops was not observed, showing saturation at The anatomy of the DMI can be decomposed in a similar higher than 1 T. When N > 2, the saturation field becomes about fashion. For this purpose, one can extract the DMI using the 2 T. Thus, the films have strong PMA property with good polynomial expression (Eq. 1 the in “Method” section) from the coherency even though they have complexed layer structures. spiral structures built for several q vectors by switching on/off In order to estimate the DMI energy of each SL, the the SOC of a particular layer. In this work, however, we choose a measurement based on the extended droplet model was 28,29 simpler approach, i.e., by calculating the asymmetry between the conducted . Figure 6a shows the schematic illustration of the energies of spiral structures with q = ±0.25, of which the droplet measurement. Since the nucleated magnetic droplet degeneracy is lifted due to the ISB. When the asymmetric energy possesses domain wall magnetizations with two opposite radial in a total structure, which is correlated with DMI and defined here direction. As illustrated in Fig. 6a, we can consider two domain as the energy difference between q = ±0.25 states, are denoted as wall magnetizations (M and M ) with respect to H .If we DW1 DW2 x npj Computational Materials (2021) 129 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences 1234567890():,; W.S. Ham et al. Fig. 2 ISB-related MCA energy and DMI of the [Co/Pt/Pd]-SL. a The side view and b the top view of the model crystal structure of Co/Pt/Pd- based multilayer system. The M direction for the propagation of the spin-spiral wave is indicated by the red arrow in (b). c Anticlockwise odd rotation of the spin-spiral structures. The calculated N dependence of d the odd term of MCA energies E and e the DMI for the [Pt/Co/Pd] MCA model systems, while those obtained for the bulk system are shown using straight lines. The N dependence of the total MCA energies E of MCA the [Pt/Co/Pd] model systems and the corresponding bulk value are shown in (d) using dashed lines. approximate that the two domain wall magnetizations dominate of complete switching via domain wall propagation initiated from the total domain wall energy under H , the total domain wall the nucleation, we can consider the relation of μ H = x 0 n,z energy of nucleated magnetic droplet can be obtained from sum μ H cosθ. The angular dependence of the H was obtained 0 SW SW of the domain wall energies: σ = σ (+H ) + σ (−H ), total DW1 x DW2 x from the anomalous Hall effect measurement in terms of θ [see where σ (+H ) and σ (−H ) are the domain wall energies DW1 x DW1 x Fig. 6b]. Every SLs shows sharp switching behavior as shown in with respect to H . The nucleation field through the relation is as Fig. 6b. With this approach, we could obtain the quantitative pffiffiffiffiffiffiffiffi follows: μ H = πσ t =2μ M pk T, where t is the ferro- FM S B 0 n 0 FM DW;total information of DMI by plotting the H dependence of H . x n;z magnet thickness, μ M is the saturation magnetization, 0 S H values for all N values with each SL are clearly DMI pffiffiffiffiffiffiffiffi p represents the thermal stability factor, k is the Boltzmann determined with the H vs: H plotsasshown in Fig. 6cwith x n;z constant, and T denotes temperature . Therefore, square root of arrows. The measured data show that H values of [Co/Pd/Pt]- DMI nucleation fields in terms of H should follow the blue curve in pffiffiffiffiffiffiffiffi SLs are N-dependent. Here, D should be discussed for precise Fig. 6a. Note that σ is proportional to H when H > H , total n;z x DMI discussion with excluding effects of K and M , based on the eff S otherwise σ becomes constant. The crossing point of the two total relation D = μ M ΔH ,where Δ denotes the domain wall width 0 s DMI linear curves can be defined as DMI-induced effective field (H ). DMI which is determined by the exchange stiffness constant (A)and pffiffiffiffiffiffiffiffiffiffiffiffi Based on this model, the angular dependence of coercivity was pffiffiffiffiffiffiffiffi −1 K by Δ = A=K . In all samples, A = 10 pJ m is assumed eff eff measured to obtain H of the SLs as illustrated in inset of Fig. 6 n;z since A in ultra-thin Co layers are usually an order of a few (b). In this measurement concept, applied magnetic field can be −131,32 (typically 1–15) pJ m . decomposed into H and H . The detailed measurement scheme x n,z Note that difference of DMI between Pt/Co and Co/Pd interfaces is as follows; first of all, the sample is saturated to the +z direction. in the [Co/Pd/Pt]-SL can also enhance total DMI energy and the Then, the magnetic field is swept from the positive to negative value should be larger than other symmetric structures. In other field in terms of various polar angles (θ) in order to obtain the θ dependence of switching field (H ). Considering the measure- words, the interfacial origin should be also considered to explain SW ment time scale (ramping rate ~1 T/min.) is much slower than that our observation. Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2021) 129 W.S. Ham et al. Fig. 3 k-dependent MCA energy and Fermi surface in the [Pt/Co/Pd] unit structure. a Two-dimensional (k , k ) contour map of k-dependent x y MCA energy and b Fermi surface for different in-plane magnetization directions along +x and –x axes, plotted using respectively red and blue lines, of the [Pt/Co/Pd] model system depicted in (c). d–f Fermi surface of the [Pt/Co/Pd] model system in the absence of SOC on the Pt1, 1 1 Co1, and Pd1 layer, respectively. In order to confirm the characteristics of DMI originating from Our experimental and theoretical studies demonstrate that the the interface, the dependence of repetition number (N) was bulk spin-momentum locking in a SL can be made with studied in all series of the SLs. Figure 6d shows that the [Co/Pd/ asymmetric atomic stacking and structural coherency. Especially, Pt]-SL shows N-dependence of DMI. Such increase of DMI in the the N-dependence of DMI of the [Co/Pd/Pt]-SL is an important phenomenon arising from the band asymmetry. So far, material [Pt/Co/Pd] is fully consistent with our theoretical works. selection has been limited to several cases for the development of Especially, DMI of the [Co/Pd/Pt]-SL linearly increases by more skyrmion-based devices. On the other hand, bulk DMI can be than 300% when N increases from 2 to 10. Such behavior cannot made with such ABC-type material combination within atomic be simply explained by the interfacial origin. We should note here scale, which is larger than interfacial contribution. Hence, our that in order to reproduce the calculation results in our experimental and theoretical findings can provide more options to experiment, we used the smallest possible thickness of each layer design chiral magnet for spintronic devices. As a final remark, in which is about thickness of two atomic monolayers. Our XRR study two-dimensional system with C symmetry, only non-Lifshitz in Fig. 4 proves well-defined layer structures. One atomic 3v invariant DMI can exist . Our observation is related to this non- monolayer as shown in our DFT calculation is difficult to have Lifshitz invariant but Lifshitz invariant terms dominate as the SL is continuous film in reality. Therefore, there should be both the three-dimensional object with strong two-dimensional feature conventional interface and Rashba-type bulk contributions inherent from interface. The noncentrosymmetric SL in this sense simultaneously for our experimental observation, while the latter would invite further study on Lifshitz invariant contribution in DMI. is dominant for the calculation especially with larger repetition numbers. Although quantitative comparison between calculation and experimental results is limited in this study, both results METHODS manifest that DMI in such an ABC-type structure cannot be Experiment explained with conventional approach with interfacial origin, The [Co/Pd/Pt]-SL were deposited by the UHV magnetron sputtering indicating formation of asymmetric band structure. We also note system. The thickness of magnetic layer (Co) is varied in each SL. The static that D has the linear relation with uniaxial magnetic anisotropy magnetic properties such as M and magnetic anisotropy energy are (K ) estimated from the magnetic hysteresis loops shown in Fig. 5c investigated by the vibrating sample magnetometer (VSM). In order to 24–26 and d as shown in Fig. 6e. Our results suggest that the quantify DMI energy, we used the extended droplet method . All the enhancements of both D and PMA have the same origin in the films were patterned into a microstrip with a Hall bar structure by E-beam Rashba SLs, namely the bulk spin-momentum locking . lithography to prevent the nucleation of domain at the rough microstrip npj Computational Materials (2021) 129 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences W.S. Ham et al. Fig. 4 Origin of the N-dependence of DMI in the [Pt/Co/Pd]-SL. The contribution of different atomic layer L to the asymmetric energy asym between the spiral structures with wave vectors q = ±0.25, E ðÞ L . The corresponding atomic layer contributions for bulk system, i.e. with q¼ ±0:25 asym N = ∞, are shown with black lines. The sequence of the layers is illustrated. The total asymmetric energy E ðÞ tot is also shown. q¼ ±0:25 Fig. 5 Magnetic properties of the [Co/Pd/Pt] SLs. a A layer structure of the Co/Pd/Pt SLs. b XRR measurement result with the Co/Pd/Pt SL with N = 5. Magnetic hysteresis curves with c out-of-plane and d in-plane magnetic fields. edge. Ti (5 nm)/Au (100 nm) electrodes are defined to make electrical Augmented Plane Wave (FLAPW) code . The MCA energy E has been MCA contacts with μm-scale Hall bars by photolithography and lift-off process. defined in our calculation as the energy difference between the in- and out-of-plane magnetization direction, i.e., E = E − E , where E and MCA ip op ip E refer respectively to the total energy of the in- and out-of-plane op The first principles calculations for MCA and DMI of SLs magnetization. The inclusion of SOC is done via the second variational method . The in-plane magnetization direction has been chosen to be Our first-principles DFT calculations are based on the Generalized Gradient along the x direction. We found that our calculated MCA energy has Approximation as implemented in the Full-Potential Linearized Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2021) 129 W.S. Ham et al. Fig. 6 N-dependent DMI of the [Co/Pd/Pt]-SL. a Illustration of droplet nucleation field profile in terms of in-plane field. b H of the [Co/Pd/ SW Pt]-SL with N = 6 measured using anomalous Hall effect in terms of θ. Inset shows the configuration of a device with external field (H ), ex pffiffiffiffiffiffiffiffi components of the H (H and H ), and M . Solid lines are the fitting results before and after the threshold field. c Normalized H vs. H n;z ex x z s x plots in terms of N. The solid lines indicate the fitted lines when H < H and H > H . H values can be obtained from crossing points x DMI X DMI DMI indicated by the arrows. dD values of the SLs in terms of N. e K vs. D plots. The red solid line is the linearly fitted line. The error bars represent standard deviations from the best fit. converged at a relatively large two-dimensional k-point mesh of 100 × 100. the two-dimensional Brillouin zone. As in the case of MCA energy, we have In addition to the total MCA energy, we also virtually decomposed also computed the DMI for the bulk CoPtPd system with the Néel xz out-of- þk k ± k y y y plane rotation type magnetic structures along the M direction, and a three- the k-dependent MCA energy into E and E in which E ¼ MCA MCA MCA dimensional 40 × 40 × 20 k-point mesh has been used. Em ; ± k Em ; ± k and calculated the odd term of the MCA x y z y odd energy E by following the recipe in our previous work ,as MCA þky ky odd E ¼ E  E . This quantity, despite the fact that it contains no MCA MCA MCA DATA AVAILABILITY physical meaning, illustrates the physics of Rashba spin–orbit coupling and Data that support the plots within this paper and other findings of this study are provides an estimation of the degree of ISB by the shift of the Fermi available from the corresponding authors upon reasonable request. surface. Comparison with bulk values has been done with a three- dimensional k-point mesh of 75 × 75 × 45 for the bulk [Co/Pt/Pd]-SL . The estimation of D values was done by following the steps outlined 23,24 CODE AVAILABILITY previously We start from the ferromagnetic configuration which is found to be the ground state of all model systems at the scalar relativistic Code that supports the findings of this study are available from the corresponding approximation, i.e., without the SOC. Next a set of spiral spin structures authors on reasonable request. with wave vectors q = a/λ along the M = (1, 1, 0) direction (see Fig. 3b), where λ is the wavelengths of the spin-spiral structures, are generated by Received: 19 January 2021; Accepted: 6 July 2021; 35,36 utilizing the generalized Bloch theorem . When the SOC is then included, the spin-spiral structures have been assumed to be the Néel xz out-of-plane rotation type. The frozen magnon energy, E(q), for q = 0, ±0.1, ±0.2, ±0.25 has been fitted with a polynomial expansion; 2 3 4 EqðÞ¼ C þ C q þ C q þ C q þ C q (2) 0 1 2 3 4 REFERENCES where the odd terms C and C occur due to the presence of ISB. The DMI 1. Dzyaloshinskii, I. E. Thermodynamic theory of “Weak” ferromagnetism in anti- 1 3 discussed in this work is extracted as the antisymmetric exchange ferromagnetic substances. Sov. Phys. JETP. 5, 1259 (1957). stiffness constant C , i.e., first order in q,or D ≡ C . Convergence of the 2. Moriya, T. 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B. 95, 220402 (2017). article’s Creative Commons license and your intended use is not permitted by statutory 29. Kim, S. et al. Correlation of the Dzyaloshinskii-Moriya interaction with Heisenberg regulation or exceeds the permitted use, you will need to obtain permission directly exchange and orbital asphericity. Nat. Commun. 9, 1648 (2018). from the copyright holder. To view a copy of this license, visit http://creativecommons. 30. Pizzini, S. et al. Chirality-Induced asymmetric magnetic nucleation in Pt/Co/AlOx org/licenses/by/4.0/. ultrathin microstructures. Phys. Rev. Lett. 113, 047203 (2014). 31. Eyrich, C. et al. Effects of substritution on the exchange stiffness and magneti- zation of Co films. Phys. Rev. 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Dzyaloshinskii–Moriya interaction in noncentrosymmetric superlattices

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www.nature.com/npjcompumats ARTICLE OPEN Dzyaloshinskii–Moriya interaction in noncentrosymmetric superlattices 1 2,3✉ 4 5 5 2 1 Woo Seung Ham , Abdul-Muizz Pradipto , Kay Yakushiji , Kwangsu Kim , Sonny H. Rhim , Kohji Nakamura , Yoichi Shiota , 5 1 ✉ ✉ Sanghoon Kim and Teruo Ono Dzyaloshinskii–Moriya interaction (DMI) is considered as one of the most important energies for specific chiral textures such as magnetic skyrmions. The keys of generating DMI are the absence of structural inversion symmetry and exchange energy with spin–orbit coupling. Therefore, a vast majority of research activities about DMI are mainly limited to heavy metal/ferromagnet bilayer systems, only focusing on their interfaces. Here, we report an asymmetric band formation in a superlattices (SL) which arises from inversion symmetry breaking in stacking order of atomic layers, implying the role of bulk-like contribution. Such bulk DMI is more than 300% larger than simple sum of interfacial contribution. Moreover, the asymmetric band is largely affected by strong spin–orbit coupling, showing crucial role of a heavy metal even in the non-interfacial origin of DMI. Our work provides more degrees of freedom to design chiral magnets for spintronics applications. npj Computational Materials (2021) 7:129 ; https://doi.org/10.1038/s41524-021-00592-8 INTRODUCTION space, and it has been vastly utilized to interpret a number of 11,18–20 magnetic phenomena , in particular those well understood The lack of inversion symmetry at the interface between a heavy to originate from the ISB, such as the DMI responsible for exotic metal (HM) and a ferromagnet (FM) induces the antisymmetric magnetic textures such as skyrmions and chiral domain walls, and exchange interaction so-called Dzyaloshinskii–Moriya interaction 1–4 spin–orbit torque. It should be noted that amorphous ferrimagnet (DMI) . Recently, DMI has been intensively studied in the GdFeCo exhibits bulk DMI feature, which is independent of material combinations possessing perpendicular magnetic aniso- interface but due to inhomogeneous distribution of elemental tropy (PMA) due to their necessities in creating magnetic chiral content . In this context, further study to distinguish DMI textures, such as magnetic skyrmions for the new type of racetrack 5–8 interface and bulk origin would be interesting topic. memory device . Generally, in order to stabilize skyrmions at In this study, we investigate DMI of the [Co/Pd/Pt]-SL arising room temperature, multilayer structures with repetitive stacking of from bulk spin-momentum locking. First-principles calculations FM/HM bilayer are utilized because multi-stacking of the bilayer reproduce such DMI enhancement in the SL, showing that the unit easily provide the PMA and the sizable DMI at the same time, asymmetry of bands around Fermi energy level induced by ISB. both of them arising from the same physical origin, i.e., interfacial 9–13 The observed behavior of DMI upon increasing the repetitions of SOC . In this respect, Co/Pd and Co/Pt interfaces are one of the the ABC-layer unit in the SL suggests that while the interfacial and well-known material combinations providing both the PMA and bulk DMI co-exist with small N, the enhancement of DMI with the DMI originating from interfaces, resulting in stable magnetic 14,15 larger N can be attributed to the bulk-type asymmetric band skyrmions . With the same manner of such an AB-type multi- formation around the Fermi level. stacking structure composed of several nanometer-thick layers, a layer structure with ABC-type repetitive stacking of a few atomic monolayers is interesting system as illustrated in Fig. 1a. An RESULTS AND DISCUSSION epitome is the [Co/Pd/Pt] superlattice (SL) possessing PMA DMI of the noncentrosymmetric SLs generated by the bulk-type spin momentum locking due to absence of inversion symmetry in stacking order . Note that not Estimation of the magnetocrystalline energy (MCA) and DMI is 22–24 interfaces but asymmetry of bulk-type band formation in the [Co/ done by following the steps outlined previously . Here, we Pd/Pt]-SL as illustrated in Fig. 1b is essential to give rise to such a consider 1–4 [Co/Pt/Pd] units, and anticlockwise rotation of the chiral phenomenon, resulting in strong PMA. spin-spiral structures as shown in Fig. 2a–c. The detailed process is Such inversion symmetry breaking (ISB) in the SL with ABC-type explained in the “Method” section. The calculation results are stacking order would traditionally be accounted for by involving summarized in Fig. 2d and e. The odd terms of the MCA energy odd the Rashba model Hamiltonian, H ¼ αðÞ k ´ ^ z  σ, which was (E ), which quantifies ISB, are summarized in Fig. 2d. This quantity R R MCA initially proposed for a surface, where z is the direction of is related to the ISB-induced shift of the band structure along the k inversion-symmetry-breaking-induced potential gradient . The direction due to the magnetization along x direction. We note that odd oddness of the SOC in the k space due to the ISB is shown by the E increases with the repetition number N of [Co/Pt/Pd] unit MCA odd dependence of the Hamiltonian on the linear terms in k, although layers. The total MCA and E in the [Pt/Co/Pd] increases shown MCA higher odd-order may in principle also appear. Rashba effect in Fig. 2d, and the total MCA value can be expected to reach −2 manifests most immediately into a spin-splitting within the k 0.043 meV Å obtained for the bulk [Co/Pt/Pd]-SL, i.e. the infinitely 1 2 3 Institute for Chemical Research, Kyoto University, Uji, Kyoto, Japan. Department of Physics Engineering, Mie University, Tsu, Mie, Japan. Faculty of Mathematics and Natural 4 5 Sciences, Institut Teknologi Bandung, Bandung, Indonesia. National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, Japan. Department of Physics, University of Ulsan, Ulsan, South Korea. email: a.m.t.pradipto@gmail.com; sanghoon.kim@ulsan.ac.kr; ono@scl.kyoto-u.ac.jp Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences 1234567890():,; W.S. Ham et al. Fig. 1 SL structure with symmetry breaking in stacking order. a SLs with AB-type and ABC-type stacking-order, which is composed of nm-thick and sub-nm-thick layers, respectively. b Schematic image of an asymmetric band structure in terms of direction of magnetization in the SL with ABC-stacking order. asym periodic system along c direction [Pt/Co/Pt] . When the DMI E , the contribution of a particular layer L to the asymmetric q¼ ±0:25 constants are further extracted by utilizing the polynomial energy can be given as expression of the frozen magnon energy (Eq. 2 in “Method” asym asym asym E ðÞ L ¼ E ðÞ tot  E ðL Þ (1) off q¼ ±0:25 q¼±0:25 q¼ ±0:25 section), we obtain a DMI energy density (D) value of the asym asymmetric [Pt/Co/Pd] structure which increases with N [see where E ðtotÞ is the asymmetric energy which includes the q¼±0:25 −2 asym Fig. 2e]. Here, we use the unit J m for DMI values of the structures contribution of all layers, and E ðL Þ is the asymmetric off q¼ ±0:25 in order to compare with experimental results, shown later in this energy when the SOC contribution of layer L is switched off. The study. It is important to note that the D value increases with N and results are summarized in Fig. 4, clearly showing that, first of all, converges to the bulk value. Such linear dependence of DMI cannot despite being the carrier of the magnetic moments, Co gives very be simply explained by the interfacial origin. If the increase in DMI small contribution to the DMI. In fact, the largest contribution to with N is fully originated from the interfacial spin–orbit coupling, the asymmetric energy, hence to the DMI, is coming from the DMI should be independent of N because the gains in the DMI neighboring Pt layers, which can be understood to dominate the energy by increasing the number of interfaces are compensated by band around Fermi level, as shown in Fig. 3d–f. It should be noted the volume of Co layers. The detailed mechanism of this result that for all N, Pd gives smaller and opposite contribution to the asym is explained in the following paragraph with anatomy of the DMI, in comparison to Pt. Additionally, the evaluated total E q ±0:25 k-dependent band structures. for all N, as plotted in the inset of Fig. 4, shows considerable The two-dimensional MCA contour map of [Pt/Co/Pd] is shown 1 deviations for all contributions, then, converges to that of the bulk in Fig. 3a. This map shows the typical characteristic of the MCA in with infinite periodicity. This implies the existence of a non- the systems with ISB. Since the in-plane magnetization is oriented interfacial origin, but bulk spin-momentum locking for the DMI along the +x direction, i.e., m ¼ m x, the MCA is asymmetric in the considered [Pt/Co/Pd]-SL systems. þx along the k axis, which is along the ẑ × m direction, and shows mirror symmetry along k direction. Similar behavior is obtained Experimental observation of DMI of the noncentrosymmetric for other N. The Fermi surface of [Pt/Co/Pd] is visibly shifted SLs towards negative k as shown by using the red color in Fig. 3b for To experimentally confirm the bulk spin-momentum locking for m magnetization, and is shifted symmetrically towards negative +x the DMI, the [Co 0.4/Pd 0.4/Pt 0.4 (nm)] SLs, hereafter [Co/Pd/Pt]- k when the magnetization is along the m x direction, as plotted y x SL, was prepared as the noncentrosymmetric SL varying N as by using the blue color in Fig. 3b. In order to check the band explained in the “Method” section [see also Fig. 5a]. The X-ray contribution to the Rashba splitting, we plot Fermi surface by reflection pattern of the film with N = 5 clearly shows the 1st- switching off the spin–orbit coupling in each atomic layer. Bragg-like peak, implying that the [Co/Pd/Pt] unit structure is well Comparing the results shown in Fig. 3d–f, it is visible that preserved without severe atomic intermixing in spite of such ultra- 26,27 switching off SOC in Pt layer alters the shape of the Fermi contour thin thickness range . All [Co/Pd/Pt]-SLs are confirmed to have most considerably compared to that in Co and Pd. As the change PMA as shown in Fig. 5c and d. In Particular, all magnetic in the Fermi contour indicates the modification of the band hysteresis curves measured with the easy-axis field sweep show structure around the Fermi level, the result implies the crucial role squareness of unity and sharp switching properties of the SLs. In of Pt bands to the band splitting, which should be attributed to case of the hard-axis loops taken with in-plane field sweep, any the strong SOC constant of the Pt atom. signal jump in the loops was not observed, showing saturation at The anatomy of the DMI can be decomposed in a similar higher than 1 T. When N > 2, the saturation field becomes about fashion. For this purpose, one can extract the DMI using the 2 T. Thus, the films have strong PMA property with good polynomial expression (Eq. 1 the in “Method” section) from the coherency even though they have complexed layer structures. spiral structures built for several q vectors by switching on/off In order to estimate the DMI energy of each SL, the the SOC of a particular layer. In this work, however, we choose a measurement based on the extended droplet model was 28,29 simpler approach, i.e., by calculating the asymmetry between the conducted . Figure 6a shows the schematic illustration of the energies of spiral structures with q = ±0.25, of which the droplet measurement. Since the nucleated magnetic droplet degeneracy is lifted due to the ISB. When the asymmetric energy possesses domain wall magnetizations with two opposite radial in a total structure, which is correlated with DMI and defined here direction. As illustrated in Fig. 6a, we can consider two domain as the energy difference between q = ±0.25 states, are denoted as wall magnetizations (M and M ) with respect to H .If we DW1 DW2 x npj Computational Materials (2021) 129 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences 1234567890():,; W.S. Ham et al. Fig. 2 ISB-related MCA energy and DMI of the [Co/Pt/Pd]-SL. a The side view and b the top view of the model crystal structure of Co/Pt/Pd- based multilayer system. The M direction for the propagation of the spin-spiral wave is indicated by the red arrow in (b). c Anticlockwise odd rotation of the spin-spiral structures. The calculated N dependence of d the odd term of MCA energies E and e the DMI for the [Pt/Co/Pd] MCA model systems, while those obtained for the bulk system are shown using straight lines. The N dependence of the total MCA energies E of MCA the [Pt/Co/Pd] model systems and the corresponding bulk value are shown in (d) using dashed lines. approximate that the two domain wall magnetizations dominate of complete switching via domain wall propagation initiated from the total domain wall energy under H , the total domain wall the nucleation, we can consider the relation of μ H = x 0 n,z energy of nucleated magnetic droplet can be obtained from sum μ H cosθ. The angular dependence of the H was obtained 0 SW SW of the domain wall energies: σ = σ (+H ) + σ (−H ), total DW1 x DW2 x from the anomalous Hall effect measurement in terms of θ [see where σ (+H ) and σ (−H ) are the domain wall energies DW1 x DW1 x Fig. 6b]. Every SLs shows sharp switching behavior as shown in with respect to H . The nucleation field through the relation is as Fig. 6b. With this approach, we could obtain the quantitative pffiffiffiffiffiffiffiffi follows: μ H = πσ t =2μ M pk T, where t is the ferro- FM S B 0 n 0 FM DW;total information of DMI by plotting the H dependence of H . x n;z magnet thickness, μ M is the saturation magnetization, 0 S H values for all N values with each SL are clearly DMI pffiffiffiffiffiffiffiffi p represents the thermal stability factor, k is the Boltzmann determined with the H vs: H plotsasshown in Fig. 6cwith x n;z constant, and T denotes temperature . Therefore, square root of arrows. The measured data show that H values of [Co/Pd/Pt]- DMI nucleation fields in terms of H should follow the blue curve in pffiffiffiffiffiffiffiffi SLs are N-dependent. Here, D should be discussed for precise Fig. 6a. Note that σ is proportional to H when H > H , total n;z x DMI discussion with excluding effects of K and M , based on the eff S otherwise σ becomes constant. The crossing point of the two total relation D = μ M ΔH ,where Δ denotes the domain wall width 0 s DMI linear curves can be defined as DMI-induced effective field (H ). DMI which is determined by the exchange stiffness constant (A)and pffiffiffiffiffiffiffiffiffiffiffiffi Based on this model, the angular dependence of coercivity was pffiffiffiffiffiffiffiffi −1 K by Δ = A=K . In all samples, A = 10 pJ m is assumed eff eff measured to obtain H of the SLs as illustrated in inset of Fig. 6 n;z since A in ultra-thin Co layers are usually an order of a few (b). In this measurement concept, applied magnetic field can be −131,32 (typically 1–15) pJ m . decomposed into H and H . The detailed measurement scheme x n,z Note that difference of DMI between Pt/Co and Co/Pd interfaces is as follows; first of all, the sample is saturated to the +z direction. in the [Co/Pd/Pt]-SL can also enhance total DMI energy and the Then, the magnetic field is swept from the positive to negative value should be larger than other symmetric structures. In other field in terms of various polar angles (θ) in order to obtain the θ dependence of switching field (H ). Considering the measure- words, the interfacial origin should be also considered to explain SW ment time scale (ramping rate ~1 T/min.) is much slower than that our observation. Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2021) 129 W.S. Ham et al. Fig. 3 k-dependent MCA energy and Fermi surface in the [Pt/Co/Pd] unit structure. a Two-dimensional (k , k ) contour map of k-dependent x y MCA energy and b Fermi surface for different in-plane magnetization directions along +x and –x axes, plotted using respectively red and blue lines, of the [Pt/Co/Pd] model system depicted in (c). d–f Fermi surface of the [Pt/Co/Pd] model system in the absence of SOC on the Pt1, 1 1 Co1, and Pd1 layer, respectively. In order to confirm the characteristics of DMI originating from Our experimental and theoretical studies demonstrate that the the interface, the dependence of repetition number (N) was bulk spin-momentum locking in a SL can be made with studied in all series of the SLs. Figure 6d shows that the [Co/Pd/ asymmetric atomic stacking and structural coherency. Especially, Pt]-SL shows N-dependence of DMI. Such increase of DMI in the the N-dependence of DMI of the [Co/Pd/Pt]-SL is an important phenomenon arising from the band asymmetry. So far, material [Pt/Co/Pd] is fully consistent with our theoretical works. selection has been limited to several cases for the development of Especially, DMI of the [Co/Pd/Pt]-SL linearly increases by more skyrmion-based devices. On the other hand, bulk DMI can be than 300% when N increases from 2 to 10. Such behavior cannot made with such ABC-type material combination within atomic be simply explained by the interfacial origin. We should note here scale, which is larger than interfacial contribution. Hence, our that in order to reproduce the calculation results in our experimental and theoretical findings can provide more options to experiment, we used the smallest possible thickness of each layer design chiral magnet for spintronic devices. As a final remark, in which is about thickness of two atomic monolayers. Our XRR study two-dimensional system with C symmetry, only non-Lifshitz in Fig. 4 proves well-defined layer structures. One atomic 3v invariant DMI can exist . Our observation is related to this non- monolayer as shown in our DFT calculation is difficult to have Lifshitz invariant but Lifshitz invariant terms dominate as the SL is continuous film in reality. Therefore, there should be both the three-dimensional object with strong two-dimensional feature conventional interface and Rashba-type bulk contributions inherent from interface. The noncentrosymmetric SL in this sense simultaneously for our experimental observation, while the latter would invite further study on Lifshitz invariant contribution in DMI. is dominant for the calculation especially with larger repetition numbers. Although quantitative comparison between calculation and experimental results is limited in this study, both results METHODS manifest that DMI in such an ABC-type structure cannot be Experiment explained with conventional approach with interfacial origin, The [Co/Pd/Pt]-SL were deposited by the UHV magnetron sputtering indicating formation of asymmetric band structure. We also note system. The thickness of magnetic layer (Co) is varied in each SL. The static that D has the linear relation with uniaxial magnetic anisotropy magnetic properties such as M and magnetic anisotropy energy are (K ) estimated from the magnetic hysteresis loops shown in Fig. 5c investigated by the vibrating sample magnetometer (VSM). In order to 24–26 and d as shown in Fig. 6e. Our results suggest that the quantify DMI energy, we used the extended droplet method . All the enhancements of both D and PMA have the same origin in the films were patterned into a microstrip with a Hall bar structure by E-beam Rashba SLs, namely the bulk spin-momentum locking . lithography to prevent the nucleation of domain at the rough microstrip npj Computational Materials (2021) 129 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences W.S. Ham et al. Fig. 4 Origin of the N-dependence of DMI in the [Pt/Co/Pd]-SL. The contribution of different atomic layer L to the asymmetric energy asym between the spiral structures with wave vectors q = ±0.25, E ðÞ L . The corresponding atomic layer contributions for bulk system, i.e. with q¼ ±0:25 asym N = ∞, are shown with black lines. The sequence of the layers is illustrated. The total asymmetric energy E ðÞ tot is also shown. q¼ ±0:25 Fig. 5 Magnetic properties of the [Co/Pd/Pt] SLs. a A layer structure of the Co/Pd/Pt SLs. b XRR measurement result with the Co/Pd/Pt SL with N = 5. Magnetic hysteresis curves with c out-of-plane and d in-plane magnetic fields. edge. Ti (5 nm)/Au (100 nm) electrodes are defined to make electrical Augmented Plane Wave (FLAPW) code . The MCA energy E has been MCA contacts with μm-scale Hall bars by photolithography and lift-off process. defined in our calculation as the energy difference between the in- and out-of-plane magnetization direction, i.e., E = E − E , where E and MCA ip op ip E refer respectively to the total energy of the in- and out-of-plane op The first principles calculations for MCA and DMI of SLs magnetization. The inclusion of SOC is done via the second variational method . The in-plane magnetization direction has been chosen to be Our first-principles DFT calculations are based on the Generalized Gradient along the x direction. We found that our calculated MCA energy has Approximation as implemented in the Full-Potential Linearized Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2021) 129 W.S. Ham et al. Fig. 6 N-dependent DMI of the [Co/Pd/Pt]-SL. a Illustration of droplet nucleation field profile in terms of in-plane field. b H of the [Co/Pd/ SW Pt]-SL with N = 6 measured using anomalous Hall effect in terms of θ. Inset shows the configuration of a device with external field (H ), ex pffiffiffiffiffiffiffiffi components of the H (H and H ), and M . Solid lines are the fitting results before and after the threshold field. c Normalized H vs. H n;z ex x z s x plots in terms of N. The solid lines indicate the fitted lines when H < H and H > H . H values can be obtained from crossing points x DMI X DMI DMI indicated by the arrows. dD values of the SLs in terms of N. e K vs. D plots. The red solid line is the linearly fitted line. The error bars represent standard deviations from the best fit. converged at a relatively large two-dimensional k-point mesh of 100 × 100. the two-dimensional Brillouin zone. As in the case of MCA energy, we have In addition to the total MCA energy, we also virtually decomposed also computed the DMI for the bulk CoPtPd system with the Néel xz out-of- þk k ± k y y y plane rotation type magnetic structures along the M direction, and a three- the k-dependent MCA energy into E and E in which E ¼ MCA MCA MCA dimensional 40 × 40 × 20 k-point mesh has been used. Em ; ± k Em ; ± k and calculated the odd term of the MCA x y z y odd energy E by following the recipe in our previous work ,as MCA þky ky odd E ¼ E  E . This quantity, despite the fact that it contains no MCA MCA MCA DATA AVAILABILITY physical meaning, illustrates the physics of Rashba spin–orbit coupling and Data that support the plots within this paper and other findings of this study are provides an estimation of the degree of ISB by the shift of the Fermi available from the corresponding authors upon reasonable request. surface. Comparison with bulk values has been done with a three- dimensional k-point mesh of 75 × 75 × 45 for the bulk [Co/Pt/Pd]-SL . The estimation of D values was done by following the steps outlined 23,24 CODE AVAILABILITY previously We start from the ferromagnetic configuration which is found to be the ground state of all model systems at the scalar relativistic Code that supports the findings of this study are available from the corresponding approximation, i.e., without the SOC. Next a set of spiral spin structures authors on reasonable request. with wave vectors q = a/λ along the M = (1, 1, 0) direction (see Fig. 3b), where λ is the wavelengths of the spin-spiral structures, are generated by Received: 19 January 2021; Accepted: 6 July 2021; 35,36 utilizing the generalized Bloch theorem . When the SOC is then included, the spin-spiral structures have been assumed to be the Néel xz out-of-plane rotation type. The frozen magnon energy, E(q), for q = 0, ±0.1, ±0.2, ±0.25 has been fitted with a polynomial expansion; 2 3 4 EqðÞ¼ C þ C q þ C q þ C q þ C q (2) 0 1 2 3 4 REFERENCES where the odd terms C and C occur due to the presence of ISB. The DMI 1. Dzyaloshinskii, I. E. Thermodynamic theory of “Weak” ferromagnetism in anti- 1 3 discussed in this work is extracted as the antisymmetric exchange ferromagnetic substances. Sov. Phys. JETP. 5, 1259 (1957). stiffness constant C , i.e., first order in q,or D ≡ C . Convergence of the 2. Moriya, T. 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Financial current-driven dynamics in ultrathin metallic ferromagnets. Nat. Mater. 15, support from the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. 501–506 (2016). 15H05702 is gratefully acknowledged. T.O. was supported by the Collaborative Research 10. Jiang, W. et al. Skyrmions in magnetic multilayers. Phys. Rep. 704,1–49 (2017). Program of the Institute for Chemical Research, Kyoto University. S.K. was supported by 11. Yang, H. et al. Significant Dzyaloshinskii–Moriya interaction at graphene–ferromagnet the Basic Research Laboratory Program through the National Research Foundation of interfaces due to the Rashba effect. Nat. Mater. 17, 605 (2018). Korea (NRF) funded by the MSIT(NRF-2018R1A4A1020696, and NRF-2019R1C1C1010345). 12. Kundu, A. & Zhang, S. Dzyaloshinskii-Moriya interaction mediated by spin- A.-M.P. is also supported by Institut Teknologi Bandung through P3MI-ITB 2020 research polarized band with Rashba spin-orbit coupling. Phys. Rev. B 92, 094434 (2015). grant. Computations were performed at Research Institute for Information Technology, 13. Di, K. et al. Direct observation of the Dzyaloshinskii-Moriya interaction in a Pt/Co/ Kyushu University. Ni film. Phys. Rev. Lett. 114, 047201 (2015). 14. Pollard, S. D. et al. Observation of stable Neel skyrmions in cobalt/palladium multilayers with Lorentz transmission electron microscopy. Nat. Commun. 8, AUTHOR CONTRIBUTIONS 14761 (2017). S.K., A.M.P., and T.O. designed the research, K.Y. fabricated the SLs, W.S.H. carried out 15. Gilbert, D. A. et al. Realization of ground-state artificial skyrmion lattices at room the device fabrication, XRR and electrical measurements, W.S.H, K.K., S.K., and T.O. temperature. Nat. Commun. 6, 8462 (2015). analyzed experimental data. A.M.P., S.R., and K.N. carried out and analyzed the DFT 16. Pradipto, A.-M. et al. Enhanced perpendicular magnetocrystalline anisotropy calculations. W.S.H., A.M.P., S.K., and T.O. wrote the manuscript with input from all the energy in an artificial magnetic material with bulk spin-momentum coupling. authors. Phys. Rev. B 99, 180410 (2019). 17. Bychkov, Y. A. & Rashba, E. I. Properties of a 2D electron gas with lifted spectral degeneracy. JETP Lett. 39, 78 (1984). COMPETING INTERESTS 18. Kim, K.-W. et al. Chirality from interfacial spin-orbit coupling effects in magnetic bilayers. Phys. Rev. Lett. 111, 216601 (2013). The authors declare no competing interests. 19. Barnes, S., Ieda, J. & Maekawa, S. Rashba spin-orbit anisotropy and the electric field control of magnetism. Sci. Rep. 4, 4105 (2015). 20. Yang, H. et al. Anatomy and giant enhancement of the perpendicular magnetic ADDITIONAL INFORMATION anisotropy of cobalt–graphene heterostructures. Nano Lett. 16, 145 (2016). Correspondence and requests for materials should be addressed to A.-M.P., S.K. or T.O. 21. Kim, D. H. et al. Bulk Dzyaloshinskii–Moriya interaction in amorphous ferrimag- netic alloys. Nat. Mater. 18, 685–690 (2019). Reprints and permission information is available at http://www.nature.com/ 22. Nakamura, K. et al. Enhancement of magnetocrystalline anisotropy in ferro- reprints magnetic Fe films by intra-atomic noncollinear magnetism. Phys. Rev. B 67, 014420 (2003). Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims 23. Yamamoto, K. et al. Interfacial Dzyaloshinskii-Moriya interaction and orbital in published maps and institutional affiliations. magnetic moments of metallic multilayer films. AIP Adv. 7, 056302 (2017). 24. Nakamura, K. et al. Symmetric and asymmetric exchange stiffnesses of transition-metal thin film interfaces in external electric field. J. Magn. Magn. Mater. 457, 97 (2018). 25. Bahramy, M. S. & Ogawa, N. Bulk Rashba semiconductors and related quantum Open Access This article is licensed under a Creative Commons phenomena. Adv. Mater. 29, 1605911 (2017). Attribution 4.0 International License, which permits use, sharing, 26. Yao, B. & Coiffey, K. R. The influence of periodicity on the structures and prop- adaptation, distribution and reproduction in any medium or format, as long as you give erties of annealed [Fe/Pt] multilayer films. J. Mag. Mag. Mater. 320, 559 (2008). appropriate credit to the original author(s) and the source, provide a link to the Creative 27. Yu, Y. S. et al. Structure and magnetic properties of magnetron-sputtered FePt/Au Commons license, and indicate if changes were made. The images or other third party superlattice films. J. Phys. D.; Appl. Phys. 41, 245003 (2008). material in this article are included in the article’s Creative Commons license, unless 28. Kim, S. et al. Magnetic droplet nucleation with a homochiral Neel domain wall. indicated otherwise in a credit line to the material. If material is not included in the Phys. Rev. B. 95, 220402 (2017). article’s Creative Commons license and your intended use is not permitted by statutory 29. Kim, S. et al. Correlation of the Dzyaloshinskii-Moriya interaction with Heisenberg regulation or exceeds the permitted use, you will need to obtain permission directly exchange and orbital asphericity. Nat. Commun. 9, 1648 (2018). from the copyright holder. To view a copy of this license, visit http://creativecommons. 30. Pizzini, S. et al. Chirality-Induced asymmetric magnetic nucleation in Pt/Co/AlOx org/licenses/by/4.0/. ultrathin microstructures. Phys. Rev. Lett. 113, 047203 (2014). 31. Eyrich, C. et al. Effects of substritution on the exchange stiffness and magneti- zation of Co films. Phys. Rev. B 90, 235408 (2014). © The Author(s) 2021 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2021) 129

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