Minimal model for vortex nucleation and reversal in spherical magnetic nanoparticles

TL;DR

The paper tackles the lack of a transparent analytical framework for vortex-mediated reversal in spherical magnetic nanoparticles by introducing a semi-analytical reduced model based on a hyperbolic vortex Ansatz. A single width parameter $\nu$ and a global rotation angle $\tau$ yield a two-parameter description of vortex states, leading to a reduced Hamiltonian $\mathcal{H}'$, and, to capture hysteresis, a minimal Hamiltonian $\mathcal{H}''$ that omits a specific anisotropy term. The authors derive analytic expressions for the vortex-nucleation radius $R_{\mathrm{nuc}}$ and field $B_{\mathrm{nuc}}$, extending Brown’s classic results within a variational framework, and demonstrate that $\mathcal{H}''$ reproduces the hysteretic behavior observed in micromagnetic simulations for small fields. This work provides a bridge between analytical theory and numerical micromagnetics, offering fast, physically transparent insights into vortex nucleation and reversal, with potential extensions to other particle shapes and anisotropy orientations; data and code are publicly available.

Abstract

Magnetic nanoparticles beyond the single-domain limit often develop vortex-like magnetization textures arising from the competition between exchange and magnetostatic energies. While such states are routinely studied using micromagnetic simulations, transparent analytical descriptions of vortex-mediated hysteresis and nucleation remain scarce. Here, we develop a semi-analytical minimal framework for vortex states in spherical magnetic nanoparticles. Guided by micromagnetic simulations, we introduce a parametrized vortex magnetization Ansatz based on hyperbolic functions that continuously interpolates between uniform and vortex states. In this way, we achieve a complexity reduction leading to a minimal Hamiltonian, which enables the efficient computation of magnetization curves and provides insight into vortex-mediated magnetization reversal. As an application, we derive analytical estimates for the critical vortex nucleation radius and field, recovering the functional form of Brown's classic result and extending it within a variational framework.

Figures (4)

• Figure 1: Micromagnetic simulation results (using MuMax3) for the remanent state of a spherical iron nanoparticle with a diameter of $D = 2R = 40 \, \mathrm{nm}$. The particle volume is discretized into cubical cells with a side length of $2 \, \mathrm{nm}$. (a) Visualization of a stable vortex configuration; color encodes the $x$ component of the magnetization. (b) Radial magnetization profiles obtained from several particles, shown in cylindrical coordinates $\{m_{\rho}, m_{\phi}, m_z\}$. The in-plane azimuthal component $m_{\phi}$ (solid line) is well described by a $\operatorname{tanh}$ profile, whereas the out-of-plane component $m_z$ (dashed line) follows a hyperbolic $\operatorname{sech}$ profile. Here, the vortex-profile parameter is $\nu = 4.443$.
• Figure 2: Comparison between the semi-analytical modeling approach (solid blue curves) and the micromagnetic simulation results obtained with MuMax3 (solid orange curves) for a spherical magnetic nanoparticle in the vortex state. The magnetization response is computed from the effective Hamiltonian $\mathcal{H}'$ [Eq. \ref{['eq:ReducedHamiltonian1']}] by minimizing the total energy with respect to the vortex profile parameter $\nu$ and the inclination angle $\tau$ at each applied field value $B_0$. The right-hand panels show the corresponding field-dependent evolution of $\tau$ (b) and $\nu$ (c). The MuMax3 simulation was carried out for a spherical particle of diameter $D = 40 \, \mathrm{nm}$, using a cubic discretization with a cell side length of $2 \, \mathrm{nm}$.
• Figure 3: Same as Fig. \ref{['fig2']}, but with the minimal-model Hamiltonian $\mathcal{H}"$ defined by Eq. \ref{['eq:VortexReducedSimplifiedHamiltonian']}.
• Figure 4: Magnetization reversal and vortex nucleation obtained from the minimal vortex Hamiltonian $\mathcal{H}"$ [Eq. \ref{['eq:VortexReducedSimplifiedHamiltonian']}]. (a) Field-dependent average magnetization $\langle m_z \rangle$ for different particle radii $R$ (see inset). For $R < R_{\mathrm{nuc}} \cong 9.2 \,\mathrm{nm}$ [Eq. \ref{['eq:VortexNucleationRadius']}], the system exhibits single-domain Stoner-Wohlfarth hysteresis (rectangular loop for $R = 8 \,\mathrm{nm}$) with abrupt switching at $B_{\mathrm{SW}} = 2 K_{\mathrm{u}}/M_{\mathrm{s}} \cong 0.056 \, \mathrm{T}$. For $R > R_{\mathrm{nuc}}$, vortex formation enables a smooth, nonuniform magnetization reversal. (c) Vortex-nucleation field $B_{\mathrm{nuc}}$ versus particle radius $R$. The analytical expression \ref{['eq:VortexNucleationField']} (dashed line) corresponds to the linear stability (spinodal) limit, while the hysteresis data reflect the loss of metastability along the reversal path.