Higher-order ferromagnetic resonances in out-of-plane saturated Co/Au magnetic multilayers

Artificial ferromagnetic (FM)/nonmagnetic multilayers, with large enough FM thickness to prevent the dominance of interface anisotropies, offer a straightforward insight into the understanding and control of perpendicular standing spin wave (PSSW) modes. Here we present a study of the static and dynamic magnetic properties of ${\left[\mathrm{Co}\left(3.0\mathrm{nm}\right)/\mathrm{Au}\left(0.6\mathrm{nm}\right)\right]}_{1\le N\le 30}$ multilayers. Magnetometry reveals that the samples exhibit magnetization reversal properties typical of an effective single layer with weak perpendicular anisotropy, with the distinctive thickness-dependent magnetization reorientation transition from uniform in-plane to out-of-plane stripe domains at remanence. However, when such multilayer systems are out-of-plane saturated, the dynamic response reveals the existence of several different ferromagnetic resonances in the form of PSSW modes that strongly depend on the material modulation characteristics along the total thickness. These modes are induced by the layer stacking itself as the effective single layer model fails to describe the observed complex dynamics. In contrast to most systems considered in the past, described by a dynamic model of a single effectively homogeneous thick layer, the specific structures investigated here provide a unique platform for a large degree of tunability of the mode frequencies and amplitude profiles. We argue that the combination of periodic magnetic properties with vertical deformation gradients, arising from heteroepitaxial strain relaxation, converts the Au interlayer regions into a vertical regular array of magnetic pinning planes for the PSSW modes, which promotes the complex dynamics observed in this system.

Figure 1

(a) XRR scans of all ${\left[\mathrm{Co}\left(3.0\mathrm{nm}\right)/\mathrm{Au}\left(0.6\mathrm{nm}\right)\right]}_N$ multilayer samples. The (red) solid lines represent the least-squares fits obtained with the genx software. The top-left inset shows a schematic drawing of the layer stack. The top right inset displays the average total thickness of the single Co/Au bilayer as a function of $N$ as obtained from XRR measurements. (b) Determined thicknesses from the XRR data fit for the Au, Co, and their sum as (black) squares, (red) circles, and (blue) triangles, respectively. (c) Determined thicknesses from the XRR data fit for the Au cap and Au buffer layers as (orange) squares and (green) circles, respectively.

Figure 2

(a) X-ray θ-2θ scans of all samples. Each scan has been normalized to the intensity of its Si (400) substrate peak. (b) shows the $N$ dependence of the Co/Au* ($n=0$) diffraction peak position (squares), while (c) displays the associated average out-of-plane interplanar distance $d_{\mathrm{Co}/\mathrm{Au}}$ of the Co/Au heterostructure vs $N$. [(d)−(f)] Cross-sectional TEM image of the ${\left[\mathrm{Co}\left(3.0\mathrm{nm}\right)/\mathrm{Au}\left(0.6\mathrm{nm}\right)\right]}_{30}$ sample using different magnifications. The (white) rectangle in (d) shows the magnified area that is displayed in (e), whereas the (white) rectangle in (e) indicates the magnified area shown in (f).

Figure 3

[(a)−(l)] VSM room-temperature (RT) magnetization reversal curves with the applied field along the IP (black dashed lines) and OOP (red full lines) directions for the entire set of ${\left[\mathrm{Co}\left(3.0\mathrm{nm}\right)/\mathrm{Au}\left(0.6\mathrm{nm}\right)\right]}_N$ samples. Data are normalized to its $M_S$ in each case. [(m)−(r)] Remanent MFM images recorded after IP [(m), (o), and (q)] and OOP [(n), (p), and (r)] demagnetization processes for the samples [(m) and (n)] $N=2$, [(o) and (p)] 14, and [(q) and (r)] 30. (s) $N$ dependence of the in-plane remanence ratio ${\mathrm{M}}_{\mathrm{r}}^{\mathrm{IP}}/M_S$ [obtained from the magnetometry data of (a)−(l)]. (t) RT saturation magnetization $M_S$ as a function of the Co/Au bilayer repetitions $N$. (u) RT in-plane saturation field ${\mu }_0{\mathrm{H}}_{\mathrm{s}}^{\mathrm{IP}}$ as a function of the inverse total multilayer thickness $t_{\mathrm{ML}}=N\times \mathrm{\Lambda }$, with $\mathrm{\Lambda }=3.6$ nm being the Co/Au bilayer period. The solid line represents the least-squares fits to Eq. (1).

Figure 4

[(a)−(l)] Frequency vs. magnetic field dependences of the ferromagnetic resonances for each sample. The external OOP magnetic field was always swept from −1.7 to −1 T, ensuring saturated OOP states through the whole measurement. The vertical dashed (gray) lines mark the value ${\mu }_{0}H=-1.2$ T corresponding to the vertical cut that is used to construct Fig. 5, whereas the horizontal dotted (gold) lines in (c) and (l) correspond to the explicit measurements shown in (m) and (n). (m) and (n) show the normalized microwave absorption lines (amplitude of the ${\mathrm{S}}_{12}$ parameter) as a function of the applied field for the samples $N=3$ and 30, respectively, which were measured with an excitation frequency of 25 GHz. (o) Contribution of the inhomogeneous broadening to the total field linewidth vs $N$.

Figure 5

[(a)−(c)] FMR measurements (dots) and theory (lines) evaluated at ${\mu }_{0}H=-1.2$ T. Cases (a)−(c) show different approximations of the theoretical model (see text for details) that are summarized by the layer sketches on the right.

Figure 6

Anisotropy fields as a function of $\nu$ for a sample with $N$ Co/Au bilayers (1 ≤ $\nu \le N$). The bulk ($H^b$, red squares) as well as the bottom-most and top-most sublayer anisotropy fields ($H_a^o$, black pentagon and $H_a^o\left(N\right)$, light blue rhomb) are assumed fixed, while the anisotropy field at the boundaries of the inner Co layers $[H_a\left(\nu \right)$, dark blue dots] increases as its location moves towards the top part of the multilayers stack. The overall average anisotropy fields $\overline{H_K}\left(\nu \right)$ are shown as green triangles. The inset associates each anisotropy field to the corresponding region within the sample.

Figure 7

In-plane dynamic magnetization orbits in $z$ direction across the sample thickness as a function of $N$. In cases (a)–(d) the first mode is illustrated for different $N$ values, while the first four excited modes are shown in (e)–(h) for the case $N=30$. Dynamic magnetization components $m_{x,y}$ are expressed in arbitrary units. The red dots mark the excitation amplitude and phase for all modes at the same time ($t=0$).