Engineering skyrmions in transition-metal multilayers for spintronics

Magnetic skyrmions are localized, topologically protected spin structures that have been proposed for storing or processing information due to their intriguing dynamical and transport properties. Important in terms of applications is the recent discovery of interface stabilized skyrmions as evidenced in ultra-thin transition-metal films. However, so far only skyrmions at interfaces with a single atomic layer of a magnetic material were reported, which greatly limits their potential for application in devices. Here we predict the emergence of skyrmions in [4d/Fe2/5d]n multilayers, that is, structures composed of Fe biatomic layers sandwiched between 4d and 5d transition-metal layers. In these composite structures, the exchange and the Dzyaloshinskii–Moriya interactions that control skyrmion formation can be tuned separately by the two interfaces. This allows engineering skyrmions as shown based on density functional theory and spin dynamics simulations.

Interlayer distances obtained after structural relaxation of the Rh/Pd/2Fe/2Ir multilayer system and the ultra-thin film 2Pd/Fe/Ir(111) from Ref. [4]. All distances are given in Å.

Supplementary Note 2 | Energy dispersion of spin spirals
We consider flat spin spirals in which the magnetic moments are confined in a plane with a constant angle between moments at adjacent lattice sites propagating along high symmetry directions of the surface 6 . Such a spin spiral can be characterized by a wave vector from the two-dimensional We have fitted the exchange dispersion curve to the spin Hamiltonian where the first sum runs over sites within each Fe layer and ‖ are the intralayer exchange coupling constants and the second sum runs over sites in different layers with ⊥ parameterizing the exchange between the two Fe layers. The intralayer contributions, ‖ , are obtained as in the monolayer case 7 .
Due to the loss of inversion symmetry which results from the presence of the second iron layer, the derivation for the interlayer part, ⊥ , differs slightly due to a phase shift to the atoms in the neighboring plane as given below. We can write the expression of the interlayer exchange energy contribution by inserting the magnetization for a flat spin spiral where is the position of the atom in the th nearest neighbor shell and is the propagation We can continue this procedure in order to calculate the first four ⊥ . This yields the expressions: The best fit of the energy dispersion obtained with four ‖ and four ⊥ is given in Supplementary

Supplementary Note 3 | Exchange and DMI in Rh/Pd/2Fe/2Ir
We have calculated the energy dispersion ( ) of spin spirals for Rh/Pd/2Fe/2Ir along the high-  Fig. 3 of the main paper, we find a skyrmion which is embedded in the two Fe layers with a profile as in Fig. 3. The only difference is a slight distortion from the circular shape. Therefore, our approximation of treating DMI at only one interface is well fulfilled for our systems.
Note, that the four-spin interaction which couples spin spirals into the nanoskyrmion lattice of Fe/Ir(111) 9 is short ranged. Therefore, it is most effective if the energy minimum of spin spirals occurs for short periods, e.g. = 1 nm in the case of Fe/Ir(111). For the multilayer systems discussed in this paper the period or in an external field the skyrmion diameter is controlled by the 4d/Fe interface, e.g.
we obtain = 2.25 nm for Rh/Pd/2Fe/2Ir. Thereby, we go into a regime in which the four-spin interaction, which acts on the atomic scale and becomes strong for fast rotating spirals, plays a minor role and cannot enforce a zero-field atomic-scale skyrmion lattice.

Supplementary Note 4 | Variation of intralayer exchange couplings in the bilayer
In  We have performed Monte-Carlo simulations using the intra-and interlayer exchange constants from Supplementary Table 3

Supplementary Note 5 | Stability with respect to variations of Jeff and K
Intermixing at the 4d/Fe interface can affect the exchange interactions in the Fe layers. We have performed DFT calculations for the Rh/Pd0.66Fe0.33/Pd0.33Fe0.66/Fe/2Ir multilayer system as an ordered alloy in a √3 × √3 unit cell. By performing spin spiral calculations we have obtained the effective exchange constant which is eff ≈ 9 meV. Compared to the multilayer Rh/Pd/2Fe/2Ir with a perfect interface with eff = 1 meV the system apparently becomes much more ferromagnetic. In the spirit of i.e. = 0, and the energy minimum disappears. Thus the ferromagnetic state becomes the ground state. Therefore, this value of eff represents the limiting case in which a skyrmion lattice does not appear in the phase diagram as shown below. We conclude that for all systems in Fig. 1  In the other scenario, we kept = 0.6 meV but we increased 1 ‖,Fe@Pd to 10 meV which corresponds to an effective exchange interaction of eff = 4 meV (triangles in Supplementary Figure 7). As a consequence, the period of the spin spiral increases dramatically as the exchange energy increases and compensates the DMI. Then there is an evolution from the blue curve towards the black curve with triangles. As compared with the case of strong anisotropy, we do not go through two degenerate minima when the exchange is increased.
Analysis of the energy contributions. Even in the limiting cases discussed above in which there is no stable skyrmion lattice in the phase diagram, metastable skyrmions can form. We have analyzed the energetical stability of such metastable isolated skyrmions in both cases, i.e. of increased exchange stiffness and of increased magnetic anisotropy. We focus on the situation of vanishing external magnetic field where the energy difference with respect to the ferromagnetic state is smallest.
Supplementary Figure 8 shows the site-resolved total energy of the isolated skyrmion per Fe atom in the case of 1 ‖,Fe@Pd = 10 meV (left column) and = 2.15 meV (right column). In both cases the site-resolved energy has a similar behavior with a minimum at the center of the skyrmion, then a maximum due to exchange, and finally a small negative contribution due to the DMI. Although the stabilizing forces have the same behavior as in the case shown in Fig. 4 of the main text, the total energy averaged over the th neighbor shell, shown the middle row of panels of Supplementary Figure   8, shows a strongly frustrated behavior in the case of strong anisotropy: While the anisotropy strongly disfavors the magnetic structure near the center, due to the DMI some energy is gained at the edge of the skyrmion.  Figure   8(e,f)), the energy per spin is 14 eV and 11 eV, respectively. While the total energy of the skyrmion at = 0 T is lower in the case of high anisotropy, the frustration in the spin lattice is higher.

Supplementary Note 6 | Inter-bilayer exchange coupling and transition temperature
Using DFT and Monte-Carlo (MC) simulations we have also studied the effect of the exchange coupling between adjacent Fe bilayers in our multilayers, i.e. the inter-bilayer exchange coupling. In order to take the exchange coupling between adjacent Fe bilayers into account we have extended the The c extracted from the peak of the specific heat or the total energy inflection point is in relatively good agreement (±10 K) with the method described above.
Supplementary Figure 9 shows the susceptibility as a function of temperature for various values of the inter-bilayer exchange coupling . Clearly the peak of shifts to higher temperatures with increasing