Impact of surface anisotropy on the spin-wave dynamics in a thin ferromagnetic film

The spin-wave dynamics in the thin CoFeB film in Damon-Eshbach geometry are studied in three cases of boundary conditions—free boundary conditions, symmetrical surface anisotropy, and one-sided surface anisotropy. The analytical model created by Wolfram and De Wames was extended to include perpendicular surface anisotropy in boundary conditions. Its comparison with numerical simulations demonstrates perfect agreement between the approaches. The analysis of the dispersion relation indicates that the presence of surface anisotropy increases the avoided crossing size between the Damon-Eshbach mode and perpendicular standing modes. Additionally, asymmetrical one-sided surface anisotropy induces nonreciprocity in the dispersion relation. In-depth analysis of the avoided crossing size is conducted for systems with different boundary conditions, different thicknesses, surface anisotropy constant values, and external magnetic fields. It shows the significant role of the strength of surface localization of the Damon-Eshbach mode and the symmetry of perpendicular standing modes in the avoided crossing broadening. Interestingly, for a specific set of parameters, the interaction between the particular modes can be suppressed, resulting in a mode crossing. Such a crossing, which occurs only for one direction of the wave vector in a one-sided surface anisotropy system, can be utilized in nonreciprocal devices.

Figure 1

(a) A general schematic of the system and the coordinate system. (b)–(d) Schematics of the boundary conditions investigated in the manuscript: (b) free boundary conditions, (c) symmetrical surface anisotropy, and (d) one-sided surface anisotropy.

Figure 2

(a)–(d) Dispersion relations of six lowest modes of a 100-nm-thick CoFeB film with (a) FBC, (b) SSA with $K_{\mathrm{s}}^{\mathrm{t}}=K_{\mathrm{s}}^{\mathrm{b}}=-700\mu \mathrm{J}/{\mathrm{m}}^2$, and (c),(d) OSA with $K_{\mathrm{s}}^{\mathrm{t}}=0$ and $K_{\mathrm{s}}^{\mathrm{b}}=-1500\mu \mathrm{J}/{\mathrm{m}}^2$ for (c) negative and (d) positive wave vectors in the external magnetic field ${\mu }_0{H}_0=50\mathrm{m}\mathrm{T}$. The plots present the comparison between the analytical model (orange lines) and numerical simulations (blue lines). Avoided crossings (ACs) are marked with labels. (e) The frequency difference between neighboring modes in FBC system. In plots (a)–(e) wave vector $k$ on the $x$-axis is presented in the logarithmic scale. (f)–(i) Closeup on the ACs: (f) AC1, (g) AC2, (h) AC3, and (i) AC4. The plot axes are showing the wave vector and frequency values relative to the AC position calculated from Eqs. (20) and (24), respectively. Plots present numerical simulation results only, which are in agreement with analytical results.

Figure 3

Distribution across the film thickness of dynamic magnetization components $m_x$ (solid lines) and $m_z$ (dashed lines) for (a) a first mode for wave vector $k=0$, (b) a first mode for wave vector $k=-5\times {10}^8\mathrm{rad}/\mathrm{m}$, (c) a third mode for wave vector $k=-2.5\times {10}^6\mathrm{rad}/\mathrm{m}$, and (d) a third mode for wave vector $k=2.5\times {10}^6\mathrm{rad}/\mathrm{m}$. Mode profiles are presented for a system with FBC (blue lines), SSA (orange lines), and OSA (green lines). Plots present numerical simulation results only, which are in agreement with analytical results.

Figure 4

The AC size $\mathrm{\Delta }{f}_{\mathrm{AC}}$ as a function of film thickness $L$ for the system with (a) FBC, (b) SSA, and OSA for (c) negative and (d) positive wave vector $k$. Odd-numbered ACs are shown with solid lines, even-numbered ACs with dashed lines. The $y$-axis is in the logarithmic scale. Plots present results of numerical simulations.

Figure 5

(a) The AC1 size $\mathrm{\Delta }{f}_{\mathrm{AC}1}$ as a function of film thickness $L$ for the system with OSA for positive wave vector $k$—the close-up of Fig. 4 to the AC1 local minimum in a small-thickness range. Inset plots show the magnetization profiles of first (orange line) and second (green line) mode at $k_{\mathrm{AC}1}$ for the thickness of 39 nm (on the left) and 46 nm (on the right). (b) Dispersion relation of three lowest modes for the system with OSA for positive wave vector $k$ for a thickness of 42.4 nm. Inset in the bottom-right corner: the close-up to the AC1 with marking of three wave vectors: $k_{\mathrm{L}}=4.853\times {10}^6\mathrm{rad}/\mathrm{m}, k_{\mathrm{AC}}=5.033\times {10}^6\mathrm{rad}/\mathrm{m}$, and $k_{\mathrm{R}}=5.2\times {10}^6\mathrm{rad}/\mathrm{m}$. Inset at top: magnetization profiles of first (orange line) and second (green line) mode at $k_{\mathrm{L}}$ (left), $k_{\mathrm{AC}}$ (center), and $k_{\mathrm{R}}$ (right). Plots present results of numerical simulations.

Figure 6

(a) Dispersion relation of the lowest six modes of the system with SSA for $K_{\mathrm{s}}=2500\mu \mathrm{J}/{\mathrm{m}}^2$. (b) Frequency difference between the neighboring modes of the system with SSA presented in (a). The $x$-axis is in the logarithmic scale. (c)–(e) The AC size $\mathrm{\Delta }{f}_{\mathrm{AC}}$ as a function of the surface anisotropy $K_{\mathrm{s}}$ for the system with (c) SSA and OSA for (d) negative and (e) positive wave vector $k$. Odd-numbered ACs are shown with solid lines, even-numbered ACs with dashed lines, and second-order ACs with dotted lines. The $y$-axis is in the logarithmic scale. Plots present results of numerical simulations.

Figure 7

The AC size $\mathrm{\Delta }{f}_{\mathrm{AC}}$ as a function of the external magnetic field $B_0$ for the system with (a) FBC, (b) SSA, and OSA for (c) negative and (d) positive wave vector $k$. Odd-numbered ACs are shown with solid lines, even-numbered ACs with dashed lines. The $y$-axis is in the logarithmic scale. Plots present results of numerical simulations.

Figure 8

Wave vector ${\tilde{q}}_1=q_{1}L$ as a function of wave vector $k$ of the six lowest modes of a CoFeB film of thickness $L=100\mathrm{n}\mathrm{m}$ with (a) FBC, (b) SSA with $K_{\mathrm{s}}^{\mathrm{t}}=K_{\mathrm{s}}^{\mathrm{b}}=-700\mu \mathrm{J}/{\mathrm{m}}^2$, and (c),(d) OSA with $K_{\mathrm{s}}^{\mathrm{t}}=0$ and $K_{\mathrm{s}}^{\mathrm{b}}=-1500\mu \mathrm{J}/{\mathrm{m}}^2$ for (c) negative and (d) positive wave vector $k$ in the external magnetic field ${\mu }_0{H}_0=50\mathrm{m}\mathrm{T}$. Horizontal dashed black lines represent the values $q_1=n\pi /L$ for $n=1,2,3\cdots$ . To make a comparison with ${\tilde{q}}_1$, on the top axis we show the modified wave vector $\tilde{k}=kL$. Plots present analytical results.

Figure 9

The absolute value of overlapping integral $I$ as a function of film thickness $L$ for the system with (a) FBC, (b) SSA, and OSA for (c) negative and (d) positive wave vector $k$. Odd-numbered ACs are shown with solid lines, even-numbered ACs with dashed lines. The $y$-axis is in the logarithmic scale.