Nonreciprocal spin waves of helical magnetization states in CoFeB/NiFe bilayers

Abstract

We investigated the nonreciprocal spin-wave properties, including the frequency shift, of a helical equilibrium state in a versatile CoFeB/NiFe bilayer. Through an extension of the dynamic matrix formalism (developed in this work) to an arbitrary non-collinear configuration along a heterostructured multilayered system thickness, we explained the frequency shift via differences in the dynamic dipolar and interlayer exchange interactions arising from the distinct spin-wave mode profiles across the bilayer thickness for counterpropagating modes at the same wave vector. In contrast to recent literature wherein the frequency shift is attributed solely to the dipolar interaction, our results and explanations hereby presented involve a starring role of the interlayer exchange interaction not accounted in current literature. Furthermore, we also found a combination of large frequency shift values and sub-100 nm spin wave wavelengths that can be tuned or even enhanced with the twisting degree of the helical magnetization state by the application of the external field, and with the thickness of the NiFe sublayer, which might be highly relevant for magnonic applications. We validated our model and the physical mechanism that explains the frequency shift using recent simulations and experimental results.

Figures (5)

• Figure 1: Illustration of CoFeB/NiFe bilayer in a helical magnetization state with a total thickness $d_t$, NiFe thickness $d_{\text{Py}}$, and CoFeB thickness $d_{\text{CoFeB}}$. (a) Illustration of the system subdivided into N sublayers, where the n'th layer is positioned at the $y_n$, with thickness $d_n$, saturation magnetization $M_s^n$, and equilibrium magnetization $\boldsymbol{M}_n^0=M_{s_n}\hat{Z}_n$, where $(X_n,Y_n,Z_n)$ are local coordinates. (b) Illustration of the n'th layer with its equilibrium angle $\phi_n$ regarding the global coordinate $z$. Global coordinates are denoted by $(x,y,z)$ and the spin-wave wave vector is $\boldsymbol{k}_x=k_x\hat{x}$.
• Figure 2: (a) Coercive field as function of Permalloy thickness $d_{\text{NiFe}}$. Helical equilibrium states of the bilayer CoFeB/NiFe, with CoFeB (NiFe) thickness $d_{\text{CoFeB}} = 25 \, \text{nm}$ ($d_{\text{NiFe}} = 25 \, \text{nm}$), and $K_{\text{CoFeB}} = 500 \, \text{kJ/m}^3$. The equilibrium helical states are shown in (b) for $d_{\text{NiFe}} = 25 \, \text{nm}$ and (c) for $d_{\text{NiFe}} = 47 \, \text{nm}$.
• Figure 3: Dispersion relation of an insulated and homogeneously saturated NiFe layer at different propagation directions $\phi_r$ and zero applied magnetic field $\mu_0 H=0$ mT of thickness (a) $d_{\text{NiFe}}=25$ nm and (b) $d_{\text{NiFe}}=47$ nm, respectively. (c) Damon-Eschbach dispersion relation of an insulated and homogeneously saturated CoFeB layer of thickness $d_{\text{CoFeB}}=25$ nm and in-plane uniaxial anisotropy $K_{\text{CoFeB}}$ = 500 $\text{kJ/m}^3$, at different magnetic fields applied in $-\hat{z}$ direction. (d) Dispersion relation and (e) frequency shift $\Delta f=(f[k_x]-f[-k_x])$ of CoFeB/NiFe system with $d_{\text{NiFe}} = 25 \, \text{nm}$. (f) Dispersion relation and (g) frequency shift $\Delta f=(f[k_x]-f[-k_x])$ of CoFeB/NiFe system with $d_{\text{NiFe}} = 47 \, \text{nm}$. (h-i) and (j-k) Mode profiles at $|k_x| =40$ rad$/\mu$m of and a set of applied field $\mu_0 H\in\{0, 90, 111, 150\}$ mT for $d_{\text{NiFe}} = 25$nm and $d_{\text{NiFe}} = 47$ nm, respectively, where top (bottom) line of modes are for $k_x=40$ rads$/\mu$m ($k_x=-40$ rads$/\mu$m).
• Figure 4: (a,c) Surface charge ratio of the CoFeB/NiFe bilayer bottom surface and (b,d) its frequency shift as a function of the applied magnetic field. (a) and (b) corresponds to a NiFe sublayer thickness $d_{\text{NiFe}}=25$ nm, and (c) and (d) for $d_{\text{NiFe}}=25$ nm. A wave vector magnitude $|k_x|=40$ rads$/\mu$m was taken for calculations.
• Figure 5: (a-c) Frequency shift extreme values $\Delta f^{\text{ext}}$ at (b-d) the corresponding wave vector $k_x^{\text{ext}}$ of a CoFeB/NiFe bilayer, as a function of the applied magnetic field. The CoFeB sublayer thickness is set to $d_{\text{CoFeB}}=25$ nm, whereas three different thicknesses of the NiFe sublayer were considered.