Spin-wave study of magnetic perpendicular surface anisotropy in single crystalline MgO/Fe/MgO films

Broadband ferromagnetic resonance is measured in single crystalline Fe films of varying thickness sandwiched between MgO layers. An exhaustive magnetic characterization of the films (exchange constant, cubic, uniaxial and surface anisotropies) is enabled by the study of the uniform and the first perpendicular standing spin wave modes as a function of applied magnetic field and film thickness. Additional measurements of nonreciprocal spin-wave propagation allow us to separate each of the two interface contributions to the total surface anisotropy. The results are consistent with the model of a quasi-bulk film interior and two magnetically different top and bottom interfaces, a difference ascribed to different oxidation states.

Figure 1

Ferromagnetic resonance spectrum measured at 40 GHz for the t = 20 nm film. The main panel shows the real and imaginary components of the change of effective permeability as function of the in-plane applied magnetic field (along an the [100] direction of Fe). One distinguishes both a main peak (uniform resonance, $n$ = 0) and a satellite one (perpendicular standing spin wave, $n$ = 1). The inset contains a photograph of the CPW loaded with a sample at its center (top) and a cross-section sketch showing the tapered CPW, the magnetic film, and the thickness profiles of the dynamic magnetization for the two resonance modes (bottom, not to scale).

Figure 2

Measured frequency vs resonance magnetic field of both the uniform mode ($n$ = 0) and first perpendicular standing spin wave mode ($n$ = 1) for the entire thickness series. In the case of in-plane applied field, squares (triangles) correspond to mode $n$ = 0 ($n$ = 1). For the case of out-of-plane applied field, circles (stars) correspond to mode $n$ = 0 ($n$ = 1). The lines are the corresponding Kittel fits [see Eqs. (5)].

Figure 3

(a) Stiffness fields as a function of Fe thickness obtained from Kittel-like fits of the experimental resonance frequencies (open circles) compared with the analytical model of Table 1 (solid lines) and the values obtained from Kittel-like fits of SWIIM simulations (dotted lines). (b) Stiffness field $H_{X1}$ as a function of $1/t^2$. (c) Stiffness fields $H_{Y0}$ and $H_{Z0}$ as a function of $1/t$.

Figure 4

Measured imaginary component of the mutual inductance between antennas due to counterpropagating spin waves (with wave numbers: $k_1=3.9\text{rad}/\text{µm}$ and $k_2=1.57\text{rad}/\text{µm}$) in a strip of $t=20$ nm Fe film under a 120-mT magnetic field (Damon-Eshbach geometry). The inset shows an electronic microscope image of the experimental device (top) and a sketch of the asymmetric modal profiles of counterpropagating waves across the thickness of the film and its interplay with asymmetric magnetic surface anisotropies (bottom).

Figure 5

Frequency nonreciprocity measurements as a function of applied magnetic field for devices $t=10$ nm, 20 nm, and 20 nm* and each wave numbers $k_1=3.9\text{rad}/\text{µm}, k_2=1.57\text{rad}/\text{µm}$. The lines correspond to the average value of $f_{\text{NR}}$ over the experimental field range.

Figure 6

Frequency nonreciprocity as a function of wave number for the three films under study. The points represent the average experimental values in the applied field range 30–200 mT (see Fig. 5). The solid lines are the corresponding linear fits to Eq. (8). The dashed lines show the mean value as calculated with SWIIM while using values for $\mathrm{\Delta }{K}_S$ that best adjust the experimental points (see text for details).

Figure 7

Second-order correction functions for the stiffness fields [Eq. (B1)] as calculated with $\mathrm{\Delta }{K}_{\text{S}}=1.65$ mJ/${\mathrm{m}}^2$ to obtain a sizable effect. Functions $g_{Yn}$ and $g_{\text{Z}}$ are the stiffness field corrections for in-plane applied field, while $g_{\text{Z}}$ applies to the out-of-plane case. The shaded zones around the lines account for the variations of $g_{\alpha n}$ with applied magnetic field in the range 0–1.3 T.

Figure 8

Stiffness fields $H_{\text{Y}0}$ (red) and $H_{\text{Z}0}$ (blue) as functions of the Fe film thickness $t$. The open circles are the experimentally determined values and the lines are predictions of our analytical approach. Continuous, dashed, and dotted lines correspond, respectively, to first-order approximation, second-order approximation with $\mathrm{\Delta }{K}_{\text{S}}=0.8$ mJ/${\mathrm{m}}^2$, and second-order approximation with $\mathrm{\Delta }{K}_{\text{S}}=2.3$ mJ/${\mathrm{m}}^2$.