Efficient hysteresis loop simulations of nanoparticle assemblies beyond the uniaxial anisotropy

We propose a modified Stoner-Wohlfarth model combined with the geometrical approach of the coherent rotation of magnetization for simulating the hysteresis loops of an assembly of magnetic nanoparticles. The temperature and the size distribution are taken into account. This combined model enables the computation of hysteresis loops at low temperatures for assemblies of particles having an arbitrary type of anisotropy. The applicability of this model for fitting experimental data is discussed and results are compared to the zero-field-cooled and field-cooled fits. As an application, the hysteresis loops measured on Co clusters embedded in carbon and germanium matrices are fitted revealing unambiguously the presence of a biaxial anisotropy.

Figure 1

Usual axis system for fine particles. The easy axis of the magnetization is along $z$.

Figure 2

Theoretical hysteresis loops at 2, 4, 6, 8, 10, and 12 K without size distribution (a) and with a log-normal size distribution (b).

Figure 3

Variation of $H_c$ with $T$ for a sample with no size distribution (solid circles), with log-normal size distribution (circles), and deduced from Eq. (7) (solid line).

Figure 4

Theoretical hysteresis loops at 2, 4, 6, 8, 10, and 12 K for a biaxial anisotropy $|K_2/K_1|=0.5$ without size distribution (a) and with a log-normal size distribution (b).

Figure 5

Theoretical hysteresis loops at 0 K for an uniaxial ($K_2=0$) and a biaxial anisotropy ($|K_2/K_1|=0.5$). The corresponding astroids are in inset.

Figure 6

Zero-temperature switching field distributions for a uniaxial and a biaxial anisotropy.

Figure 7

Magnetization trajectory of a nanoparticle with a biaxial anisotropy ($|K_2/K_1|=0.5$) during half a hysteresis loop (left) and switching field distribution (right). The magnetic field is applied from A to B (black arrow) and crosses the critical surface at C. Projection in the plane $xy$ (b) and in the plane $yz$ (c).

Figure 8

The switching fields of a 3 nm Co nanoparticle, measured in the $\mu$-SQUID plane, can be fitted well with either hexagonal (a) or cubic anisotropies (b). Corresponding 3D astroids for the hexagonal (c) and cubic (d) anisotropy. The plane crossing the 3D surface corresponds to the $\mu$-SQUID plane.

Figure 9

ZFC/FC magnetization curves and superparamagnetic $m\left(H\right)$ at 300 K (in inset) of the Co : C (a) and the Co : Ge (b) sample. The solid lines correspond to the fits.

Figure 10

Hysteresis loops at 2 K for Co nanoparticles embedded in C (a) and in Ge (b) matrices. The solids line correspond to the fits. The astroids associated with the biaxial fits are shown in inset together with an enlargement of the merging region (a).