Role of nanoparticle shape on the critical size for quasi-uniform ordering: from spheres to cubes through superballs

TL;DR

This work uses micromagnetic simulations to quantify how nanoparticle shape, encoded by the superball exponent $p$, controls the transition from single-domain to vortex states in magnetite nanoparticles, while keeping magnetic volume fixed across shapes. Incorporating cubic magnetocrystalline anisotropy $K_c$ shifts the critical size to smaller volumes and biases magnetization toward $[111]$ easy axes, with the transition also modulated by aspect ratio. The results show that shape anisotropy from geometry and intrinsic crystal anisotropy compete to determine ground-state configurations and vortex-core morphology, highlighting the need to account for realistic shape distributions when modeling nanoparticle ensembles. Overall, the study provides quantitative mappings of critical sizes as a function of shape and elongation, and characterizes how vortex cores adapt to faceted geometries, informing applications in magnetic data storage, hyperthermia, and nanomagnetism.

Abstract

The equilibrium states of single-domain magnetite nanoparticles (NPs) result from a subtle interplay between size, geometry, and magnetocrystalline anisotropy. In this work, we present a micromagnetic study of shape-controlled magnetite NPs using the superball geometry, which provides a continuous interpolation between spheres and cubes. By isolating the influence of shape, we analyze the transition from quasi-uniform (single-domain) to vortex-like states as particle size increases, revealing critical sizes that depend on the superball exponent p. Our simulations show that faceted geometries promote the stabilization of vortex states at larger sizes, with marked distortions in the vortex core structure. The inclusion of cubic magnetocrystalline anisotropy, representative of magnetite, further lowers the critical size and introduces preferential alignment along the [111] easy axes. For isotropic shapes, the critical size for this transition increases with p, ranging from ~49 nm for spheres to ~56 nm for cubes, in agreement with experimental trends. In contrast, the presence of slight particle elongation increases the critical size and induces another preferential alignment direction. These results demonstrate that even small deviations from sphericity or aspect ratio significantly alter the magnetic ordering and stability of equilibrium magnetic states.

Figures (12)

• Figure 1: Schematic representation of superellipsoidal shapes corresponding to different exponents $p$ and axial ratios $r = a/c$. It can be seen how small deviations from spherical or cubic symmetry give rise to more flattened or elongated shapes, similar to those observed experimentally. TEM images modified from salazar2008cubic with the permission of American Chemical Society.
• Figure 2: Schemes showing the different geometries studied in this work: $p= 1$ (sphere), $p= 2$, $p= 3$ and $p= 100$ (cube).
• Figure 3: Size dependence of the normalized equilibrium magnetization modulus of NPs with no anisotropy and different shape exponents $p$ corresponding to spherical ($p= 1$, circles), superballs ($p= 2, 3$, rhombus and triangles) and cubic ($p= 100$, squares) shapes. The insets show schematic snapshots of central planes perpendicular to the $x$ and $y$ axes (upper and lower subpanels, respectively) of typical magnetic configurations of cubic NPs: (a) flower state for $L= 48$ nm; vortex state for (b) $L= 58$ nm and (c) $L= 68$ nm.
• Figure 4: Energy contributions of the equilibrium configurations of NPs without (dashed lines) and with cubic magnetocrystalline anisotropy (symbols) as a function of the equivalent NP size $V^{1/3}$. $E_{\text{d}em}$ (circles), $E_{\text{e}xc}$ (triangles), and $E_{\text{a}ni}$ (squares) stand for the demagnetizing, exchange and anisotropy energies, respectively. $E_{\text{a}ni}$ has been rescaled by a factor $-5$ for better visibility. Results are shown for $4$ different shapes corresponding to $p= 1, 2, 3, 100$ as indicated in each subpanel.
• Figure 5: Size dependence of the normalized equilibrium magnetization of magnetite NPs with cubic anisotropy and different shape exponents ($p= 1, 2, 3, 100$) corresponding to shapes from spherical ($p= 1$), superballs ($p= 2, 3$) to cubic ($p= 100$). The inset shows an expanded region of the main panel around the critical size. The subpanels in the left display schematic snapshots of central planes perpendicular to the $x$ and $y$ axes (upper and lower subpanels, respectively) of the magnetic configuration of a cubic NP with the critical size, marked with an orange circle.