Asymmetric magnetic proximity effect in a Pd/Co/Pd trilayer system

In spintronic devices consisting of ferromagnetic/nonmagnetic systems, the ferromagnet-induced magnetic moment in the adjacent nonmagnetic material significantly influences the spin transport properties. In this study, such magnetic proximity effect in a Pd/Co/Pd trilayer system is investigated by x-ray magnetic circular dichroism and x-ray resonant magnetic reflectivity, which enables magnetic characterizations with element and depth resolution. We observe that the total Pd magnetic moments induced at the top Co/Pd interface are significantly larger than the Pd moments at the bottom Pd/Co interface, whereas transmission electron microscopy and reflectivity analysis indicate the two interfaces are nearly identical structurally. Such asymmetry in magnetic proximity effects could be important for understanding spin transport characteristics in ferromagnetic/nonmagnetic systems and its potential application to spin devices.

[ ] where λ is x-ray wavelength, r e is the classical electron radius, the summation is over constituent atoms, and ρ atom is the atom number per unit volume (atomic density). f'+ i f" is the complex atomic scattering factor, whose imaginary part is related to x-ray absorption, and the real part is related to the imaginary part by the Kramers-Kronig dispersion relation. The x-ray reflectivity intensities measured at grazing incidence angles can be calculated explicitly using the Parratt's recursive formalism 1 , which is an extension of optics theory to x-rays, with additional modification assuming a Gaussian distributed interfacial roughness. A Debye-Waller factor like form derived in the distorted wave Born approximation (DWBA) has been widely used 2 . Figure S1 shows the atomic scattering factors of the Pd atoms in Pd/Co/Pd near the Pd L 3 edge. The imaginary scattering factor ݂ ᇱᇱ was obtained from the measured XAS data ( Fig.   4(a)), which was then transformed into the real part ݂ ᇱ using the Kramers-Kronig relation 3 .
Both procedures use tabulated factors from away from the absorption edge for scaling (normalization) 4 . Since other constituent elements' scattering factors are almost constant, any change in reflectivity intensities at and away from the Pd L 3 edge results from the contrast in refractive indices at the Pd-Co and Pd-Ta interfaces. The vertical bars (blue) in Fig. S1 indicates that the contrast between the atomic scattering factors of Pd and Co atoms is remarkably enhanced at the Pd L 3 edge (E = 3.174 keV) when compared with those away from the edge (E = 3.160 keV). This energy dependent contrast results in a clear difference between the x-ray reflectivity curves measured at the two energies near the Pd L 3 edge, as shown in Fig. 3(a). Figure S1. Atomic factors of the Pd atoms in Pd/Co/Pd near the Pd L 3 edge.

X-ray reflectivity fitting
For simultaneous fitting of the two x-ray reflectivity curves in Fig. 3(a), an identical set of structural parameters are used. This is due to the fact that only a noticeable difference in the physical parameters describing the two reflectivity curves at the different x-ray energies is the Pd atomic scattering factors. Considering the Gaussian distributed interface profiles assumed in the DWBA, the total refractive index profile can be calculated using an error function, the derivative of which is a Gaussian function. Figure 3(b) shows the total refractive index profiles, explicitlyߩ ௧ ሺ‫ݖ‬ሻ · ݂' ሺE, zሻ, for the two different energies determined from the best fit structure. Since the difference between two profiles (dash-dotted blue lines) only results from different atomic factors of Pd layers, it determines an element-specific depth profile of the Pd layers, which corresponds to Similar to conventional x-reflectivity, x-ray resonant magnetic reflectivity is sensitive to the gradient of x-ray refractive index originating from the magnetic response. Typically the magnetic response in the x-ray refractive index is too small to be measurable, but it can be enhanced and observable when the incident x-rays are circularly polarized and the x-ray energy is tuned to the absorption edge of a specific magnetic element. Therefore, XRMR can determine laterally averaged depth profile of element-specific magnetic moments. Following Ref. 5, the magnetically sensitive part component is added to the complex x-ray atomic scattering factors for the magnetic element atoms. Then the atomic scattering factor in eq. (1) is given as where ݁̂ ሺሻ and ݉ ෝ are unit vectors of the incident(outgoing) x-ray polarization and magnetization of the magnetic layers, respectively. The magnetic atomic factors ݂Ԣ and factors ݂" are only detectable near the absorption edges of magnetic elements. Figure S2 shows the Pd magnetic atomic factor ݂ ᇱᇱ ሺ‫ܧ‬ሻ obtained from the measured XMCD in Fig. 4(a) with the same scaling factor used for the charge atomic factor ݂"ሺ‫ܧ‬ሻ. The real part ݂ ᇱ was obtained using the Kramers-Kronig relation.
Since the magnetic atomic factors are typically obtained from XMCD results, a homogeneous magnetization is assumed inside the magnetic layer, since XMCD is a bulk sensitive measurement. However, since Pd atoms are only magnetized near the Pd-Co interfaces, we added a factor r(z) with a relative spatial distribution to the magnetic atomic factors as Since only the saturation magnetization is of interest, it is assumed that ݉ ෝሺ‫ݖ‬ሻ ൌ ‫ݔ‬ ො, which is along the x-ray beam direction in the sample surface plane.
To consider explicitly the polarization dependence of magnetic atomic factors in eq. (2), a full dynamical formalism or optical formula extended for polarized x-rays and resonant scattering amplitudes should be used 6 . However, a more realistic model with both structurally and magnetically rough interfaces can be taken into account in the DWBA formula. The magnetic DWBA formula can be simplified for circularly polarized x-rays in an approximation of grazing incidence angles, relatively weak resonant scattering amplitudes, and dominant magnetization along the beam direction in the sample surface plane 7 . Since our experimental conditions meet this approximation, we have used the simplified DWBA formula to explain quantitatively the magnetic depth profile of induced Pd magnetic moments.
As in RXR, we assume a Gaussian distributed interfacial roughness in the DWBA formula used for the XRMR analysis to obtain the relative magnetic depth profile shown as the shaded area in Fig. 5(c).
. Figure S2. Magnetic atomic factors of the Pd atoms in Pd/Co/Pd near the Pd L 3 edge.

X-ray resonant magnetic reflectivity fitting with various models
To verify the validity of the asymmetric magnetic roughnesses determined from the best fit, we plot the calculated asymmetry ratios (green dotted lines in Fig. 5(a)) with identical magnetic roughness (1.5Å) at the top and bottom nonmagnetic Pd-magnetic Pd interface and other parameters unchanged. The green dotted line shows a clear deviation from the experimental data at high q z 's. Additionally, the calculated asymmetry ratio for the reverse case in which the parameters of the magnetic layers used for the best fit are switched for the top and bottom Pd layers (݀ ௧ ൏ ݀ ௧ ), is shown as the dashed blue lines in Fig. 5(b). The deviation from the measured data is obvious, in particular the reversed sign of the asymmetry ratio near q z ~ 0.23 Å -1 . Finally, the calculated asymmetry ratio based on the best fit using model B (symmetric magnetic thicknesses but asymmetric magnetic amplitude) are shown as the red dash-dotted lines in Fig. 5(a). The deviation from the experimental data is clear at high q z 's. It should be noted that this deviation is not improved by varying the fitting parameters. Since the magnetic moments of Pd induced by the Co atoms should result from the Pd 4d-Co 3d hybridization, the nearly identical chemical (electronic) structures at the Pd/Co and Co/Pd interfaces from the RXR analysis (Fig. 3)   the Pd layers were assumed to be fully magnetized 9 . It should also be noted that the calculated induced Pd magnetic moment in Fe/Pd(001) was found to be 0.36~0.38µ B /atom within two layers from the Fe-Pd interface 10