Giant Dzyaloshinskii-Moriya interaction in Rashba superlattices

Dzyaloshinskii-Moriya interaction (DMI) is considered as one of the most important energy for specific chiral texture such as magnetic skyrmions. The key of generating DMI is absence of structural inversion symmetry and exchange energy with spin-orbit coupling. Therefore, a vast majority of researches about DMI is mainly limited to heavy metal/ferromagnet bilayer systems, only focusing on their interfaces. Here, we report that asymmetric band formation in an artificial superlattice arises from inversion symmetry breaking in stacking order of atomic layers, resulting in bulk DMI. 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. Such Rashba superlattices can be a new class of material design for spintronics applications.


Introduction
The lack of inversion symmetry at the interface between a heavy metal (HM) and a ferromagnet (FM) induces the antisymmetric exchange interaction so-called Dzyaloshinskii-Moriya interaction (DMI) [1][2][3][4]. Recently, DMI has been intensively studied in the material combinations possessing perpendicular magnetic anisotropy (PMA) due to their necessities in creating magnetic chiral textures, such as magnetic skyrmions for the new type of racetrack memory device [5][6][7][8]. Generally, in order to stabilize skyrmions at room temperature, multilayer structures with repetitive stacking of FM/HM bilayer are utilized because multistacking of the bilayer unit easily provide the PMA and the sizable DMI at the same time, both of them arising from the same physical origin, i.e., interfacial SOC [9,10]. In this respect, Co/Pd and Co/Pt interfaces are one of the well-known material combinations providing both the PMA and the DMI originating from interfaces, resulting in stable magnetic skyrmions in [Co/Pd] superlattices [11,12]. With the same manner of such a AB-type multi-stacking structure composed of several nanometer-thick layers as illustrated in Fig. 1(a, left), a superlattice with ABC-type repetitive stacking of a few atomic mono-layers [ Fig. 1

(b, right)] is interesting
system. An epitome is the [Co/Pd/Pt] superlattice (SL) possessing PMA generated by the bulktype spin momentum locking due to absence of inversion symmetry in stacking order [13].
Note that not interfaces but asymmetry of bulk-type band formation in the [Co/Pd/Pt]-SL as illustrated in Fig. 1(b) is essential to give rise to such a chiral phenomenon, resulting in strong PMA.
Such inversion symmetry breaking (ISB) in the SL with ABC-type stacking order would traditionally be accounted for by involving the Rashba model Hamiltonian, ℋ = ( ×) • , which was initially proposed for a surface, where ̂ is the direction of inversion-symmetrybreaking-induced potential gradient [14]. The oddness of the SOC in the space due to the ISB is shown by the dependence of the Hamiltonian on the linear terms in , although higher odd-order may in principle also appear. Rashba effect manifests most immediately into a spinsplitting within the k space, and it has been vastly utilized to interpret a number of magnetic phenomena [15][16][17][18], in particular those well understood to originate from the ISB, such as the Dzyaloshinskii-Moriya interaction responsible for exotic magnetic textures such as skyrmions and chiral domain walls, and spin-orbit torque.

DMI of superlattices
The magnetic superlattices of [Co/Pt], [Co/Pd], and [Co/Pd/Pt] were prepared varying the Co thickness as shown in Fig. 1 [19,20]. factor, is the Boltzmann constant and T denotes temperature [21]. Therefore, square root of nucleation fields in terms of x should follow the blue curve in Fig. 2 (a). Note that total is proportional to √ n, when x > DMI , otherwise total becomes constant. The crossing point of the two linear curves can be defined as DMI-induced effective field ( DMI ). Based on this model, the angular dependence of coercivity was measured to obtain √ n, of the superlattices as illustrated in inset of Fig. 2 (b). In this measurement concept, applied magnetic field can be decomposed into x and n, . The detailed measurement scheme is as follows; first of all, the sample is saturated to the +z direction. Then, the magnetic field is swept from the positive to negative field in terms of various polar angles (θ) in order to obtain the θ dependence of switching field ( 0 SW ). Considering the measurement time scale (ramping rate ~ 1 T/min.) is much slower than that of complete switching via domain wall propagation initiated from the nucleation, we can consider the relation of 0 n,z = 0 SW cos . With this approach, we could obtain the quantitative information of DMI by plotting the dependence of √ n, .
In order to study the relation between DMI and the structural coherency, DMI values of superlattices was obtained using above method in terms of Co . The x vs. √ n, plots in Fig.   2  direction. This will be discussed in detail at the end of this session.
Obtained values of all superlattices are plotted in Fig. 2  have. In other words, the interfacial origin should be also considered to explain our observation.
In order to confirm the characteristics of DMI originating from the interface, the dependence of repetition number (N) was studied in all series of the SLs. Figure 2 (

Theoretical Consideration about DMI of the Superlattices
Estimation of the magnetocrystalline energy (MCA) and DMI is done by following the steps outlined previously [13,26] Here, we consider 1~4 [Co/Pt/Pd] units, and anticlockwise rotation of the spin spiral structures as shown in Fig. 3

(a)-(c). Detail process is explained in the Method
Section. The calculation results are summarized in Fig. 3 (d) and (e). The odd terms of the MCA energy ( ), which quantifies ISB, are summarized in Fig. 3(d). This quantity is related to the ISB-induced shift of the band structure along the ky direction due to the magnetization along x direction. We note that increases with the repetition number N of [Co/Pt/Pd] unit layers.
The total MCA and in the [Pt/Co/Pd]n increases shown in Fig. 3(d), and the total MCA  for these systems is difficult due in parts to the idealized structures, i.e. without in-plane structural asymmetry, used in calculations. However, the increase of DMI in the [Pt/Co/Pd]N is fully consistent with our experimental observations. Atomic intermixing should be also considered in calculations as observed in our experiments. However, both studies manifest that DMI in such an ABC-type structure with coherent strain cannot be explained with conventional approach with interfacial origin.
The two-dimensional MCA contour map of [Pt/Co/Pd]1 is shown in Fig. 4(a). This map shows the typical characteristic of the MCA in the systems with ISB. Since the in-plane magnetization is oriented along the +x direction, i.e. = +̂ , the MCA is asymmetric along the ky axis, which is along the ̂× direction, and shows mirror symmetry along kx direction. Similar behavior is obtained for other N. The Fermi surface of [Pt/Co/Pd]1 is visibly shifted towards negative ky as shown by using the red color in Fig. 4(b) for m+x magnetization, and is shifted symmetrically towards negative ky when the magnetization is along the −̂ direction, as plotted by using the blue color in Fig. 4(b). In order to check the band contribution to the Rashba splitting, we plot Fermi surface by switching off the spin-orbit coupling in each atomic layer. Comparing the results shown in Figs. 4(d)-(f), it is visible that switching off SOC in Pt layer alters the shape of the Fermi contour most considerably compared to Co and Pd, implying crucial role of Pt bands to the band splitting, which should be attributed to the strong SOC constant of the Pt atom.
The anatomy of the DMI can be decomposed in a similar fashion. For this purpose, one can extract the DMI using the polynomial expression (Eq. 1 in Method Section) from the spiral structures built for several q vectors by switching on/off the SOC of a particular layer. In this work, however, we choose a simpler approach, i.e. by calculating the asymmetry between the energies of spiral structures with = ±0.25, of which the degeneracy is lifted due to the ISB. When the asymmetric energy in a total structure, which is correlated with DMI and defined here as the energy difference between = ±0.25 states, are denoted as =±0. 25 , th e contribution of a particular layer L to the asymmetric energy can be given as where =±0.25 (tot) is the asymmetric energy which includes the contribution of all layers, and =±0.25 ( off ) is the asymmetric energy when the SOC contribution of layer L is switched off. The results are summarized in Fig. 5, clearly showing that, first of all, despite being the carrier of the magnetic moments, Co gives very small contribution to the DMI. In fact, the largest contribution to the asymmetric energy, hence to the DMI, is coming from the neighboring Pt layers, This implies a non-interfacial origin, but bulk spin-momentum locking for the DMI in the considered [Pt/Co/Pd]-SL systems 30 .

Conclusion
Our experimental and theoretical studies demonstrate that the bulk spin-momentum locking in a superlattice can be made with asymmetric atomic stacking and structural coherency.
Especially, the N-dependence of DMI of the [Co/Pd/Pt]-SL is important phenomenon arising from the band asymmetry. Because interfacial DMI in a ferromagnet/heavy metal bilayer has been only considered so far, material selection has been limited to several cases for development of skyrmion-based devices. On the other hand, bulk DMI can be made with such ABC-type material combination within atomic scale, which is larger than interfacial contribution. Our experimental and theoretical findings can provide a new class of material design for spintronic devices with chiral magnets.

Experiment
The The thickness of magnetic layer (Co) is varied in each superlattice. The static magnetic properties such as saturation magnetization ( S ) and magnetic anisotropy energy are investigated by the vibrating sample magnetometer (VSM). In order to quantify DMI energy, we used the extended droplet method [19][20][21]. All the films were patterned into a microstrip with a Hall bar structure by E-beam lithography to prevent the nucleation of domain at the rough microstrip edge. Ti (5 nm)/Au (100 nm) electrodes are defined to make electrical contacts with μm-scale Hall bars by photolithography and lift-off process.

The 1 st principle calculation for MCA and DMI of SLs
The MCA energy EMCA has been defined in our calculation as the energy difference between the in-and out-of-plane magnetization direction, i.e. MCA = ip − op , where ip and op refer respectively to the total energy of the in-and out-of-plane magnetization. The inclusion of SOC is done via the second variational method [5]. The in-plane magnetization direction has been chosen to be along the x direction. We found that our calculated MCA energy has converged at a relatively large two-dimensional k−point mesh of 100 × 100. In addition to the total MCA energy, we also virtually decomposed the k-dependent MCA energy into MCA + and MCA − in which ± = ( x , ± y ) − ( z , ± y ) and calculated the odd term of the MCA energy MCA by following the recipe in our previous work [13], as odd = + − − . This quantity, despite the fact that it contains no physical meaning, illustrates the physics of Rashba spin-orbit coupling and provides an estimation of the degree of ISB by the shift of the Fermi 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 [13].
The estimation of D values was done by following the steps outlined previously [13,26]. We start from the ferromagnetic configuration which is found to be the ground state of all model systems at the scalar relativistic approximation, i.e. without the SOC. Next a set of spiral spin structures 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 utilizing the generalized Bloch theorem [28,29]. 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; where the odd terms C1 and C3 occur due to the presence of ISB. The DMI discussed in this work is extracted as the antisymmetric exchange stiffness constant C1, i.e. first order in q, or D ≡ C1.
Convergence of the calculated DMI has been obtained with a 40 × 40 k−point mesh within the two-dimensional Brillouin zone. As in the case of MCA energy, we have also computed the DMI for the bulk CoPtPd system with the Néel xz out-of-plane rotation type magnetic structures along the M direction, and a three-dimensional 40 × 40 × 20 k−point mesh has been used.