Dipole-exchange spin waves and mode hybridization in magnetic nanoparticles

Abstract

We investigate spin-wave modes in confined ferromagnetic resonators with spherical and cylindrical geometries across the exchange-dominated, dipole-exchange, and dipolar interaction regimes. Starting from the linearized Landau-Lifshitz-Gilbert equation, we show that the projection of the total angular momentum and mirror parity are conserved quantities in the problem of axially symmetric resonators. These symmetries provide a natural classification of spin-wave modes and explain the degeneracy of exchange modes, as well as its lifting by dipolar interactions. Numerical analysis shows that the nonlocal dipolar interaction removes the exchange degeneracy and hybridizes modes, leading to avoided crossings between modes that belong to the same symmetry sector. To describe this behavior, we develop a coupled-mode theory formulated directly in terms of dynamical magnetization, which reduces the dipole-exchange problem to a finite system of interacting modes. The resulting framework provides a unified description of spin-wave spectra in confined magnetic particles from the exchange limit to the dipolar regime.

Figures (5)

• Figure 1: Schematic evolution of spin-wave modes in a ferromagnetic resonator as the system size increases from the exchange- to dipole-dominated regimes. Dipolar interactions lift the exchange degeneracy of modes labeled by $l$ (total orbital momentum) and induce hybridization, producing avoided crossings, while the uniform $l=0$ Kittel mode remains unchanged in spherical particle.
• Figure 2: Spin-wave spectrum of a uniformly magnetized sphere as a function of volume $V$ normalized by the exchange volume $l_{\mathrm{exch}}^3$, shown in three interaction regimes: (a) exchange, (b) dipole-exchange, and (c) dipole. (a) Exchange-dominated regime consists of degenerate multiplets characterized by $(l,l_z,p)$ denoting the orbital angular momentum, its projection, and the radial number. The Kittel mode $(0,0,0)$ is highlighted in green. (b) Dipole--exchange regime, the nonlocal dipolar interaction lifts the exchange degeneracy and induces hybridization between modes (gray rectangle; see FIG.\ref{['fig:CMT']}). Only a subset of modes is shown to improve readability. The Kittel mode does not hybridize with other modes, as discussed in Sec.\ref{['subsec:cmt:c']}. (c) Dipole dominated regime shows the continuation of the dipole–exchange modes for large volume, where the spectrum evolves into the well‑known Walker‑mode structure characterized by the indices $[n,m,r]$walker1958fletcher1959. (d) Representative mode profiles. Spatial distributions of $|\mathbf m(\mathbf r)|^2$ and dynamical magnetization flow (red arrows) are shown for lowest-energy exchange modes from panel a). In all panels the external field is $\mathbf H^{\mathrm{ext}} = 13.3 \cdot 10^4 \ A\cdot m$ and is applied along $\mathbf{z}$ axis.
• Figure 3: Spin-wave spectrum of ferrimagnetic cylinder as a function of volume $V$, normalized by the exchange volume $l_{\mathrm{exch}}^3$, shown in three interaction regimes: (a) exchange, (b) dipole-exchange, and (c) dipole. The top axis indicates the corresponding cylinder radius $r_0$ with fixed aspect ratio $2r_0/h_0=1$. (a) Exchange regime consists of degenerate multiplets labeled by $(l_z,n,p)$, denoting the orbital‑momentum projection, the number of $\mathbf{z}$-axis nodes, and the radial index. Only the homogeneous demagnetizing field $\overline{\mathbf{H}}_0^{\mathrm{dm}}$(black arrows) is included. (b) Dipole--exchange regime, the spatially varying demagnetizing field $\delta \mathbf{H_{\mathnormal{0}}^{\mathrm{dm}}}(\mathbf{r})$ (blue arrows) and the resulting nonlocal dipolar interaction $\mathcal{H}_{\rm{dip}}$ lift the exchange degeneracy and induce hybridization between modes (gray rectangle; see Fig.\ref{['fig:CMT']}). Only a subset of modes is shown to improve readability. (c) Dipole regime represents the continuation of the dipole–exchange modes at large sample volumes. Among the modes undergoing multiple anti‑crossings, we show only the Kittel mode, indicated by the upper green dash‑dotted line. (d) Representative mode profiles. Spatial distributions of $|\mathbf m(\mathbf r)|^2$ together with the dynamical magnetization flow (red arrows) are presented for the lowest-energy exchange modes shown in panel a). In all panels, the external field is $\mathbf H^{\mathrm{ext}} = 5.4\cdot 10^4 \ A\cdot m$ and is oriented along the $\mathbf{z}$ axis.
• Figure 4: Plots of the spin‑wave spectrum (logarithmic axis) calculated using magnonic coupled‑mode theory (CMT), compared with the numerical solution of the dipole–exchange problem (filled circles). Crossing (dashed black) lines show the CMT solution with self‑energy terms (non‑interacting limit), while anti-crossing (solid black) lines include both self‑energy and interaction terms. Panel (a) shows dipolar‑induced coupling between the spin‑wave modes originating from the exchange modes $(1,0,0)$ and $(3,0,1)$ in sphere. Panel (b) shows coupling between the $(0,0,0)$ and $(0,2,0)$ modes in cylinder; anti-crossing (dashed gray) lines indicate CMT results including only the non‑uniform static‑field operator $\delta \mathcal{H_{\mathrm{exch}}}$. Insets show the evolution of the mode profiles before, near, and after the interaction at the volume marked by vertical gray lines.
• Figure 5: In zero external field, magnetic state energy $E$ is shown as a function of the sample radius. The uniformly magnetized state is indicated as the purple line with energy $E_{\text{uni.}}$. (a) For a sphere, the vortex state (blue line) forms at radii above the critical value $r_{sph} = 90~nm$ and has lower energy than the uniform state (purple line). (b) For a cylinder, the unstable uniform state (purple line) transforms into a more energetically favorable quasi-uniform state (green line). The vortex state (blue line), with its core oriented along the cylinder axis, attains the lowest energy for cylinder radii above $r_{cyl} = 100~nm$ .