Surface Contribution to the Anisotropy of Magnetic Nanoparticles

We calculate the contribution of the Néel surface anisotropy to the effective anisotropy of magnetic nanoparticles of spherical shape cut out of a simple cubic lattice. The effective anisotropy arises because deviations of atomic magnetizations from collinearity and thus the energy depends on the orientation of the global magnetization. The result is second order in the Néel surface anisotropy, scales with the particle’s volume, and has cubic symmetry with preferred directions $[\pm 1,\pm 1,\pm 1]$.

Figure 1

Magnetic structure of a spherical nanoparticle of linear size $N=15$ with $L/J=2$ for the global magnetization directed along $[-1,1,0]$, showing atoms in the plane $z=0$.

Figure 2

Reduced shifted energy of the particle for different orientations of its global magnetization, obtained from Eq. (5). These curves manifest the cubic symmetry of the effective anisotropy; see Eq. (19).

Figure 3

Differences of the particle energies between main orientations of the global magnetization vs the particle size in the scaled form for $L/J=0.1$ and 0.01. The scaling is valid for $N\lesssim J/L$, and its violation for $L/J=0.1$ is seen in the right part of the figure.