Minimal model for vortex nucleation and reversal in spherical magnetic nanoparticles

This repository contains supplementary data and code to the article Minimal model for vortex nucleation and reversal in spherical magnetic nanoparticles. This version of the repository carries an optimized implementation for the energy profiles of the micromagnetic model. The Hamiltonian of the problem was defined as follows: $$\mathcal{H}(\mathbf{m})=A\int_V (\nabla \mathbf m)^2\,\mathrm dV-K_u\int_V (\mathbf m\cdot \mathbf e_u)^2\,\mathrm dV-M_{\mathrm{s}}\int_V \mathbf B_{\mathrm{ext}}\cdot \mathbf m\,\mathrm dV-\frac{1}{2}\int_V \mathbf M\cdot\mathbf B_{\mathrm d}\,\mathrm dV$$ The local hyperbolic-vortex Ansatz was used: $$\mathbf{M}'(\mathbf{r}') = M_{\mathrm{s}} [\tanh(\nu \rho' /R) \mathbf{e}_{\phi}' + \operatorname{sech}(\nu\rho'/R)\mathbf{e}_z']$$ Where the global hyperbolic-vortex Ansatz is: $$\mathbf{M}(\mathbf{r}) = \mathbf{R}(\tau, \omega)\mathbf{M}'(\mathbf{R}^T(\tau, \omega)\mathbf{r})$$, where $\tau$ is the polar rotation angle and $\omega$ the azimuthal rotation angle, and the rotation matrix is defined as $\mathbf{R}(\tau, \omega) = \mathbf{R}_z(\omega) \mathbf{R}_y(\tau)$. Under this assumption the complexity reduced Hamiltonian is given by: $$\mathcal{H}(\nu,\tau, B, \beta)=4\pi A R\, g_{\mathrm{ex}}(\nu)-K_u V_s\left[g_u^x(\nu)\sin^2(\tau-\beta)+g_u^z(\nu)\cos^2(\tau-\beta)\right]-M_s B V_s\, g_z^z(\nu)\cos\tau-\mu_0 M_s^2 V_s\, g_{\mathrm{dem}}(\nu) $$ The dimensionless energy profiles depending on the hyperbolic-vortex parameter $\nu$ are then in their reduced form defined as: $$g_u^z(\nu)=3\int_0^1x\sqrt{1-x^2}\,\operatorname{sech}^2(\nu x)\,\mathrm{d}x$$ $$g_z^z(\nu)=3\int_0^1x\sqrt{1-x^2}\,\operatorname{sech}(\nu x)\,\mathrm{d}x$$ $$g_u^x(\nu)=4\int_0^1x\sqrt{1-x^2}\,\tanh^2(\nu x)\,\mathrm{d}x$$ $$g_{\mathrm{ex}}(\nu)=3\int_0^1\sqrt{1-x^2}\bigg[\nu^2 x\,\operatorname{sech}^2(\nu x)+\frac{\tanh^2(\nu x)}{x}\bigg]\mathrm{d}x$$ $$g_{\mathrm{dem}}(\nu)=\frac{1}{2}-\frac{3}{2}\sum_{\substack{\ell=1\\ \ell\ \mathrm{odd}}}^{\infty}\left[\int_0^1x\,\operatorname{sech}(\nu x)\,P_\ell\!\left(\sqrt{1-x^2}\right)\,\mathrm{d}x\right]^2, \; \; g_{\mathrm{dem}}(0) = \frac{1}{3} $$ The reduced Hamiltonian in the article is defined as: $$\mathcal{H}^{\prime}(\nu,\tau,B, \beta)=\mathcal{H}^{\prime\prime}(\nu,\tau,B, \beta)-\frac{K_{\mathrm u}R^2}{A}g_u^x(\nu)\sin^2(\tau-\beta),$$ and the minimal Hamiltonian: $$\mathcal{H}^{\prime\prime}(\nu,\tau,B, \beta)=g_{\mathrm{ex}}(\nu)-\frac{K_{\mathrm u}R^2}{A}g_u^z(\nu)\cos^2(\tau-\beta)-\frac{M_{\mathrm s}R^2B}{A}g_z^z(\nu)\cos\tau-\frac{\mu_0M_{\mathrm s}^2R^2}{A}g_{\mathrm{dem}}(\nu),$$ where the second variation for the vortex stability is given by: $$\left.\frac{\partial^2\mathcal{H}^{\prime\prime}}{\partial\nu^2}\right|_{\nu=0,\tau=0,\beta=0}=4+\frac{4}{5}\frac{K_{\mathrm u}R^2}{A}+\frac{2}{5}\frac{M_{\mathrm s}BR^2}{A}-\frac{2}{15}\frac{\mu_0M_{\mathrm s}^2R^2}{A}.$$ With the optimized profile computation presented here, the exact vortex-nucleation field becomes: $$B_{\mathrm{nuc}} = \frac{1}{3} \mu_0 M_{\mathrm{s}} - \frac{2 K_{\mathrm{u}}}{M_{\mathrm{s}}} - \frac{10A}{M_{\mathrm{s}} R^2}.$$ The prefactor of the demagnetization term proportional to $M_{\mathrm{s}}$ is only approximate in the published article to the article Minimal model for vortex nucleation and reversal in spherical magnetic nanoparticles, where the factor $1/3$ is the exact prefactor. Of further interest may be the configuration for $B_{\mathrm{nuc}}=0$, which is given for $$K_{\mathrm{u}} = \frac{1}{6}\mu_0 M_{\mathrm{s}}^2 \left(1 - 15 \frac{l_{\mathrm{ex}}^2}{R^2}\right), \quad l_{\mathrm{ex}} = \sqrt{\frac{2A}{\mu_0 M_{\mathrm{s}}^2}}.$$ The formula for the critical vortex nucleation radius with updated coefficients is given by: $$R_{\mathrm{nuc}}=\sqrt{15}\,\ell_{\mathrm{ex}}\left[1-6\frac{K_{\mathrm u}}{\mu_0 M_{\mathrm s}^2}\right]^{-1/2}.$$ For the evaluation of the hysteresis loop the average magnetization projected along the field axis is defined as: $$\langle m_z \rangle = g_z^z(\nu)\cos\tau$$ The classical Stoner-Wohlfarth limit coercive field / switching field $B_{\mathrm{sw}}$ can be reproduced by $$\left.\frac{\partial^2\mathcal{H}^{\prime\prime}}{\partial\tau^2}\right|_{\nu=0,\tau=\{0,\pi\},\beta=0} = 0 \quad \rightarrow \quad B_{\mathrm{sw}} = \pm\frac{2K_{\mathrm{u}}}{M_{\mathrm{s}}}$$ For the hysteresis calculations, the field-dependent metastable states were obtained by a field-following procedure. At each field step and for each tabulated value of the vortex parameter \(\nu\), the stationary angles \(\tau\) were determined analytically from the angular stationarity condition of the reduced Hamiltonian. This condition can be written as a polynomial equation on the unit circle and is solved using a companion-matrix formulation. Stable angular branches are selected from the sign of the second angular variation. The remaining minimization over \(\nu\) is performed by following the local minimum connected to the previous field step, rather than by selecting the global minimum independently at each field. This preserves the metastable branches required for hysteresis. $$\eta =\begin{cases}0, & \mathcal{H}^{\prime\prime},\\1, & \mathcal{H}^{\prime}.\end{cases}$$ $$\frac{\partial \mathcal{H}}{\partial \tau}=\frac{K_uR^2}{A}\left[g_u^z(\nu)-\eta g_u^x(\nu)\right]\sin\left[2(\tau-\beta)\right]+\frac{M_sBR^2}{A}g_z^z(\nu)\sin\tau=0.$$ $$\frac{\partial^2 \mathcal{H}}{\partial \tau^2}=2\frac{K_uR^2}{A}\left[g_u^z(\nu)-\eta g_u^x(\nu)\right]\cos\left[2(\tau-\beta)\right]+\frac{M_sBR^2}{A}g_z^z(\nu)\cos\tau>0.$$ $$b(\nu,B)=\frac{M_sB}{2K_u}\frac{g_z^z(\nu)}{g_u^z(\nu)-\eta g_u^x(\nu)}.$$ For \(\beta=0\), the angular stationarity condition reduces to \(\sin\tau[\cos\tau+b(\nu,B)]=0\). For arbitrary \(\beta\), the equivalent unit-circle polynomial is solved via the companion matrix. The stability criterion above is evaluated with the full prefactor \(g_u^z(\nu)-\eta g_u^x(\nu)\), so that sign changes of this factor are treated correctly.