Effect of spin disorder on the specific loss power of a nanomagnet

Abstract

Spin non-collinearities in magnetic nanostructures arise from a variety of sources, including structural defects, finite-size effects, boundary or surface effects, Dzyaloshinskii-Moriya exchange coupling, and magnetic vortex formation. While strong forms of spin disorder generally require a numerical treatment, relatively weak non-collinearities induced by surface anisotropy are amenable to the analytical framework of the effective one-spin problem (EOSP). In this work, we exploit this framework to present a qualitative, semi-analytical study of the effect of spin disorder on the specific loss power (SLP) of a single nanomagnet within linear-response theory. Surface-induced spin misalignment mainly manifests as an additional quartic (cubic-symmetry) contribution to the anisotropy energy, parametrized by the ratio $ζ\equiv K_4/K_2$. We derive a semi-analytical expression for the SLP as a function of $ζ$ by combining the $ζ$-dependent equilibrium susceptibility and the relaxation rate obtained within Langer's approach. Our results show that, for systems in the slow-relaxation regime, the SLP is enhanced by spin misalignment, predominantly through the increase of the relaxation rate caused by the lowering of the effective energy barrier. Retaining the full Debye factor reveals that for moderate reduced barriers $σ$, where the system is close to the superparamagnetic regime, the SLP can actually \emph{decrease} with increasing spin disorder. The enhancement is asymmetric with respect to the sign of $ζ$ and depends on the nanomagnet shape (sphere versus cube) through the geometric prefactors in the EOSP mapping.

Figures (3)

• Figure 1: (a) SLP normalized to its $\zeta=0$ value as a function of the surface-anisotropy parameter $\zeta$ for three values of the reduced barrier $\sigma$. The inset magnifies the $\sigma = 7$ curve, which decreases monotonically with $|\zeta|$ (fast-relaxation regime, $\eta_0 < 1$). (b) Equilibrium susceptibility $\chi_{\text{eq}}$, normalized to its $\zeta=0$ value, for $\sigma = 10$. (c) Relaxation rate $\Gamma_0$, normalized to its $\zeta=0$ value, for $\sigma = 10$. The shaded band near $\zeta=0$ marks the crossover region where Langer's approach is not applicable and the Néel-Brown rate is used, which is independent of $\zeta$. Parameters: $\lambda = 1$, $\tau_0 = 10^{-9}$ s, $f = 100$ kHz, $\mu_0 H_0 = 10$ mT.
• Figure 2: Normalized SLP as a function of the dimensionless surface-anisotropy constant $K_{\mathrm{s}}$ for spherical (solid) and cubic (dashed) nanomagnets on an fcc lattice ($\zeta > 0$), for three values of $\sigma$. The inset magnifies the $\sigma = 7$ curves, showing the monotonic SLP decrease characteristic of the fast-relaxation regime. The effective cubic coefficient $k_4(K_{\mathrm{s}})$ is computed from Eq. (\ref{['eq:k2k4']}): $k_4 = \kappa\,K_{\mathrm{s}}^2/z$ for the sphere (with $\kappa = 0.535$) and $k_4 = (1-0.7/\mathcal{N}^{1/3})^4\,K_{\mathrm{s}}^2/z$ for the cube. Parameters: $\mathcal{N} = 1500$, $z = 12$ (fcc), $k_\mathrm{c} = 0.01$, $N_\mathrm{c}/\mathcal{N} \approx 0.47$. Curves are clipped at $\zeta = 0.35$ (EOSP validity limit).
• Figure 3: SLP of a maghemite nanoparticle ($D = 12$ nm) as a function of temperature for several values of $\zeta$ (see inset). The top axis shows the corresponding reduced barrier $\sigma = K_2 V/k_\mathrm{B} T$. Parameters: $\lambda = 1$, $\tau_0 = 10^{-9}$ s, $f = 100$ kHz, $\mu_0 H_0 = 10$ mT.