Exchange-dominated origin of spin-wave nonreciprocity in planar magnetic multilayers

TL;DR

This work addresses the origin of spin-wave nonreciprocity in planar magnetic multilayers without Dzyaloshinskii–Moschik interaction (DMI). By introducing a frequency-shift dynamic matrix (FSDM) and an interaction-resolved dynamic energy-density formalism, the authors decompose the nonreciprocal frequency shift $\Delta\omega$ into contributions from dipolar, intralayer exchange, and, critically, interlayer exchange. Across realistic parameter ranges and two representative systems (the magnonic diode MD and the graded-magnetization layer GML), interlayer exchange dominates the shift by up to 2–3 orders of magnitude, even when dipolar interactions are substantial. The work also links the symmetry properties of spin-wave orbits, via eccentricity $\epsilon_n$, tilting $\varphi_n^0$, phase $\tau_n$, and rotational matrices $\mathbb{R}_{11}$, $\mathbb{R}_{12}$, to the emergence of nonreciprocity, providing a unified, exchange-centered mechanism for designing large nonreciprocal effects in multilayer magnonic devices.

Abstract

Spin-wave nonreciprocity, manifested as a frequency difference between counterpropagating modes, underpins many proposed magnonic devices. While this effect is commonly attributed to dipolar interactions or interfacial chirality, the microscopic origin of nonreciprocal dispersion in magnetic multilayers remains under debate. Here, we analyze nonreciprocal spin-wave dispersion in planar multilayer heterostructures without Dzyaloshinskii-Moriya interaction. Using a frequency-shift dynamic matrix and an interaction-resolved dynamic energy-density formalism, we show that the frequency asymmetry cannot generally be ascribed to dipolar effects alone. Instead, once counterpropagating modes differ in their geometric structure along the thickness, interlayer exchange dominates the frequency shift. Applied to representative multilayer systems, we find that the interlayer exchange contribution exceeds dipolar and intralayer exchange effects by up to two to three orders of magnitude over a broad wave-vector range. Our results establish interlayer exchange as the primary mechanism controlling nonreciprocal dispersion in multilayer magnonic systems and provide a quantitative framework for engineering large frequency shifts in nonreciprocal magnonic devices.

Figures (7)

• Figure 1: (a) Illustration of multilayer in a homogeneous magnetization state with a total thickness $d_{t}$. (b) Spin-wave orbits of the third excitation mode. (c) The time trajectory of the magnetization, which describes elliptical orbits due to precession, is illustrated.
• Figure 2: Dipolar strength interaction ($f_{\text{dip}}^{(1)}$ and $f_{\text{dip}}^{(2)}$), and maximum and minimum strengths of the interlayer exchange interaction ($f_{\text{int}_{\text{min}}},f_{\text{int}_{\text{max}}}$) and intralayer exchange interaction ($f_{\text{ex}_{\text{min}}},f_{\text{ex}_{\text{max}}}$), as a function of the wave vector $k_{x}$. (a) Geneneral multilayered film, (b) graded magnetization NiFe layer, and (c) Magnonic diode CoFeB/NiFe bilayer.
• Figure 3: (a), (b) and (c) correspond to the dispersion relation $f$ v.s $k$, the frequency shift $\Delta f\equiv f[k]-f[-k]$ and total energy density shift $\Delta\varepsilon\equiv \varepsilon[k]-\varepsilon[-k]$ of a graded magnetization NiFe multilayer, respectively. (d), (e) and (f) present the same quantities of a CoFeB/NiFe bilayer. (g), (h) and (i) show the energy frequency shift contribution from dipolar interaction, intra-layer interaction, inter-layer interaction of the graded magentization NiFe multilayer, respectively, while (j), (k) and (l) show the same quantities of the CoFeB/NiFe bilayer. The blue-circle and green open circle points in (b) and (d) correspond to the frequency shift calculated with the Frequency Shift Dynamic Matrix, whereas the full blue line and dot-dashed green line correspond to the frequency shift calculated with the Dynamic Matrix. The blue and dot-dashed green curves shows the frequency, frequency shift and energy shift of the fundamental and first order spin wave modes. The planar NiFe layer has a graded magnetization saturation along the thickness whose profile changes from $M_s$ = 800 $\text{kA/m}$ to $M_s$ = 1600 $\text{kA/m}$, a thickness $d_{t,\text{NiFe}}=60$ nm and a stiffness constant $A_{\text{NiFe}}$ = 11 $\text{pJ/m}$. For this system, the applied field is $\mu_0 H=1.5$ mT along the saturation magnetization direction. The CoFeB/NiFe bilayer has a saturation magnetization, stiffness constant, thickness and uniaxial anisotropy constant as $M_s^{\text{CoFeB}}$ = 1270 $\text{kA/m}$ ($M_s^{\text{NiFe}}$ = 845 $\text{kA/m}$), $A_{\text{CoFeB}}$ = 17 $\text{pJ/m}$ ($A_{\text{NiFe}}$ = 12.8 $\text{pJ/m}$), $d_{t,\text{CoFeB}}=25$ nm ($d_{t,\text{NiFe}}=25$ nm) and $K_{\text{CoFeB}}=0$ ($K_{\text{NiFe}}=0$), respectively, with an applied magnetic field $\mu_0H=30$ mT The parameter $\beta$ is defined as $\mu_{0}\pi /2$.
• Figure 4: Dispersion relation and spin wave modes distribution at the same wavevector $k_x=20~\text{rads}/\mu\text{m}$ of a homogeneous NiFe layer with in-plane homogeneous magnetization aligned along $\hat{z}$ axis, and saturation magnetization, stiffness constant, thickness and uniaxial anisotropy constant given as $M_s^{\text{NiFe}}$ = 845 $\text{kA/m}$, $A_{\text{NiFe}}$ = 12.8 $\text{pJ/m}$, $t_{\text{NiFe}}=50$ nm, and $K_{\text{NiFe}}=0$, respectively. The first, second and third columns of plots correspond to our results at $100\%$, $10\%$ and $0\%$ of the full interlayer exchange interaction, respectively. (a)-(c) correspond to the dispersion relation, (d)-(f) and (g)-(i) correspond to the dynamic magnetization distribution along the thickness $\mathbf{m}_n[t=0]=\text{Re}[\tilde{\mathbf{m}}_{x_n}]$ and $\mathbf{m}_n[t=\pi/2]=-\text{Im}[\tilde{\mathbf{m}}_{y_n}]$, respectively. Blue-circle, green-square and red-triangle dots correspond to the fundamental, first and second order spin wave modes.
• Figure 5: A single layer of NiFe with in-plane homogeneous magnetization, and saturation magnetization, stiffness constant, thickness and uniaxial anisotropy constant given as $M_s^{\text{NiFe}}$ = 845 $\text{kA/m}$, $A_{\text{NiFe}}$ = 12.8 $\text{pJ/m}$, $t_{\text{NiFe}}=50$ nm, and $K_{\text{NiFe}}=0$, respectively.