Nunn and co-workes studied in [1] the interlayer exchange coupling of transition metal multilayers. They applied the Eyrich’s model and developed a modified version to successfully estimate the interlayer exchange coupling constant of Cobalt-Ruthenium-Cobalt multi-layers. They measured initial magnetization curves and compared them with their established approximation models. The Cobalt layer is polycrystalline. One major assumption of Eyrich’s model is that the magneto-crystalline anisotropy is averaged throughout the layer and therefore neglected. This work investigates this assumption with micromagnetic simulations [2] by introducing a deviation in the magneto-crystalline easy-axis distribution applied to the setup shown in Figure 1. Figure 1a shows a equilibrated C-state magnetization configuration with strongly coupled Co layers whereas in Figure 1b a complex multi-domain state appears with lesser coupling between the films.

We investigate two different easy-axis distributions with micromagnetic simulations of the Co/Ru/Co stack. Cobalt has a hexagonal close-packed (hcp) crystal structure and an out-of-plane uni-axial magneto-crystalline anisotropy constant of 0.21 MJ/m³. We use a random distribution within a specific cone angle which describes the deviation from the perfect out-of-plane direction. A cone angle of 10° and 60° is investigated. Additionally to the variation of the easy-axis distribution we varied the interlayer exchange constant between the two Cobalt layers. These have been estimated with ab-initio calculations. Depending on the number of Ruthenium monolayers three different exchange constants are depicted in Figure 1. The larger the cone angle of the easy-axis distribution the more external field is required to overcome the higher exchange and anisotropy energy to saturate the multi-layer system.

The two Co films have the dimensions of 100 nm x 100 nm x 2 nm and are separated by one, two or three monolayer of Ruthenium, with  coupling constants between the to Co layers of -0.109 J/m², -0.056 J/m², and 0.023 J/m², respectively, which were obtained from spin-polarized density-functional theory calculation. An average grain size of 5 nm is assumed for the Co layer. For simplicity it is assumed that the Ruthenium layer does not affect the microstructure so the granular structure of the first and the second Co layer is equivalent and tessellated with the polycrystal generation software Neper [3]. The Co films saturation polarization and exchange stiffness constant are assumed to be 1.51 T and 15 pJ/m respectively, following [1]. The grains are initialized with randomly chosen up or down pointing magnetization. We minimize the system's energy to compute first the relaxed magnetization configurations in zero-field and increase the field strength up to 12 T in in-plane direction.

Work supported by the Austrian Science Fund ( I5712).

[1] Nunn et al., Sci. Adv. 6 (2020)

[2] Exl et al. Computer Physics Communications 235 (2019) 179-186

[3] Quey et al., Comput. Methods Appl. Mech. Eng. 330 (2018) 308-333