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

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.

💡 Research Summary

In this work the authors perform a systematic micromagnetic investigation of single‑domain magnetite (Fe₃O₄) nanoparticles, focusing on how particle shape influences the size at which the magnetic configuration changes from a quasi‑uniform (single‑domain) state to a vortex‑like state. To isolate shape effects, they employ the superball geometry defined by ((|x|/a)^{2p}+(|y|/b)^{2p}+(|z|/c)^{2p}\le1). By varying the exponent (p) from 1 (perfect sphere) to very large values (approaching a cube) they generate a continuous family of shapes, selecting four representative cases: (p=1) (sphere), (p=2) and (p=3) (intermediate rounded cubes), and (p=100) (cube). All particles are constructed to have the same magnetic volume, which allows a direct comparison of shape‑only effects.

The micromagnetic simulations are carried out with OOMMF using a 1 nm cubic mesh, well below the exchange length of magnetite ((L_{ex}\approx5.4) nm). Material parameters correspond to bulk magnetite: exchange stiffness (A=1.1\times10^{-11}) J m⁻¹, saturation magnetization (M_s=4.8\times10^{5}) A m⁻¹, and cubic anisotropy constant (K_c=-1.1\times10^{4}) J m⁻³. Two sets of calculations are performed: (i) with only exchange and magnetostatic energies (no crystalline anisotropy) and (ii) including the cubic magnetocrystalline anisotropy.

Shape‑only case (no anisotropy). For the smallest particles (side length (L\lesssim 50) nm) the equilibrium configuration is a quasi‑uniform ferromagnetic state with magnetization aligned along one of the Cartesian axes. As the size grows, the demagnetizing field induces a “flower” state where surface spins gradually tilt away from the axis. Beyond a critical size (L_c) the system adopts a vortex configuration: a core with magnetization parallel to an axis surrounded by a curling shell that reduces magnetostatic energy at the expense of exchange energy. By fitting linear trends in the magnetization versus size curves for the uniform and vortex regimes, the authors extract (L_c) for each shape. They find (L_c\approx 51) nm for spheres ((p=1)) and (L_c\approx 56) nm for cubes ((p=100)), with intermediate values for (p=2) and (p=3). This monotonic increase of (L_c) with (p) reflects the fact that faceted geometries lower the demagnetizing energy of the vortex state, delaying its onset. Energy analysis shows that at the transition the exchange and demagnetizing contributions become comparable, and that the vortex core always aligns with a Cartesian axis.

Effect of cubic magnetocrystalline anisotropy. Introducing the realistic cubic anisotropy of magnetite modifies the balance of energies. Because (K_c<0), the easy axes are the four equivalent (