Control of the noncollinear interlayer exchange coupling

Metallic spacer layers enable precise control of the angle between the magnetic moments of two ferromagnetic layers.

the spacer layer thickness, as shown in Fig. S1A. This means that, in these structures, the ratio J 2 /|J 1 | is also constant for each Fe concentration x and is equal to This linear dependence is evident from Fig. S1C, where II. SUPPLEMENTAL DATA 2 Figure S2 shows the saturation magnetization of (2)  film roughness (4), loose spins (25), and spatial fluctuations (26). It is shown in the manuscript that spatial fluctuations are the source of the large J 2 in our multilayers. In this section and in Supplemental Data 4, we will discuss the first three mechanisms in detail.
Pin-holes: Fe is soluble over a large composition range in Ru; thus, the existence of pin-holes in our films is not expected. Microstructures of Co|Ru multilayers have been extensively studied in both academia (30) and industry (7), and the presence of pin-holes has not been reported.
Uncorrelated film roughness: When the film roughness is uncorrelated, the orange-peel coupling is zero, and the biquadratic coupling contribution can be estimated from a simple model that assumes one smooth ferromagnetic/spacer layer interface and the other interface with roughness described by the sinusoidally varying function δ · cos(2πx/L). J 2 can then be calculated as (4) The period and the root mean square roughness of our films are L ∼ 30 nm and δ = 0.12 nm, respectively. The thickness of our spacer layers is roughly D = 0.7 nm, with the exchange stiffness of 2 nm Co layers being A ex (Co) = 13 pJ/m. The roughness-induced biquadratic coupling is then J 2 = 0.1 mJ/m 2 . This is forty times smaller than the largest measured J 2 in our films.  The M s of a single Co(4) layer increases from 1314 to 1338 kA/m as the temperature is reduced from 298 to 10 K (Fig. 4). Assuming that the M s of Co in Loose spins: In this model, the exchange coupling constants are calculated as where ρ = c/( √ 3/2 a 2 ) is the areal density of the loose spins, c is the fractional concentration of loose spins, a = 2.51 × 10 −10 m, and f (T, θ) is the free energy per loose spin given by In (4), U (θ) = [U 2 1 + U 2 2 + 2U 1 U 2 cos(θ)] 0.5 , U 1 and U 2 are exchange coupling fields on a loose spin from the surrounding ferromagnetic layers, θ is the angle between magnetic moments of surrounding ferromagnetic layers, k B is the Boltzmann constant, and S is the spin quantum number, which we assume to be 1 in our calculations (25).   order of magnitude larger than the experimentally obtained J 1 (T ) (Fig. S5A).
Additionally, the experimental data for J 1 (T ) and The size of spatial fluctuations. The spatial fluctuation mechanism (26) is based on a magnitude change of the bilinear coupling term, J 1 , across the film's plane.
These fluctuations are usually due to a spatial variation of the spacer layer thickness (33,34). The competition between the coupling and stiffness energies leads to an orthogonal (90 • ) alignment between the magnetic moments. Following the theory proposed by Slonczewski (26), the strength of the biquadratic coupling between two identical magnetic layers of thickness t across a spacer layer can be determined from the following relation:  (5), if L is less than the magnetic layer thickness t, coth(π t L ) = 1 and, thus, J 2 is independent of t. Fig. S6 shows the measured and fitted M (H) curves of Co(t)|Ru 27 Fe 73 (0.7)|Co(t) (t = 1.5, 2, 3.8, 6 nm). It is found that the structures with t = 2, 3.8, and 6 nm have similar J 1 and J 2 values, and that J 1 and J 2 decrease in the structure with t = 1.5 nm. Due to J 2 being independent of t, for t > 2, the size of the spatial fluctuations, L, in (5) should be about 2 nm. This is in agreement with the micromagnetic modelling results in Fig. 5A and B, which also predict that the size of spatial fluctuation is below 2 nm. This is promising for nanometer sized devices, as 2 nm is well below the current lithography process node size used in the fabrication of spintronic devices. x VI. SUPPLEMENTAL DATA 6 The interlayer exchange coupling between two ferromagnetic layers across the majority of 3d, 4d, and 5d nonmagnetic metallic spacer layers oscillates between antiferromagnetic and ferromagnetic as a function of the spacer layer thickness (2). The period of oscillation and strength of the exchange coupling depends on the crystal growth orientation of the ferromagnetic|spacer layer|ferromagnetic film structures. However, for films grown along the same crystal orientation, the bilinear coupling constant J 1 is isotropic with respect to the direction of magnetization in the ferromagnetic layers. For example, the dependence of J 1 on the spacer layer thickness in Co|Ru|Co, grown along the 0001 crystal orientations, is the same for magnetizations in Co layers parallel (Fig. 3A) and perpendicular to the film surface (35). The origin of the large J 2 in our structures is the spatial fluctuation of J 1 . J 1 is isotropic and, thus, J 2 is also expected to be isotropic.
In our manuscript, we have studied Co|RuFe|Co structures with Co layers having in-plane magnetization.
To show that the coupling across RuFe is indepen-