9.3.3 Hysteresis

Figure: Hysteresis curve of a nanodot with $L/R=20 \mathrm{nm}/100 \mathrm{nm}=0.2$ for an in-plane external field. The circles mark the position on the hysteresis curve at which the snapshots in Fig. 9.10 have been taken.

Fig. 9.9 shows the hysteresis curve for a circular nanomagnet with in-plane external field. For very high external fields (applied in the plane of the nanodot), the magnetization is almost uniform and parallel to the external field (Fig. 9.10(a)). As the field decreases (solid line in Fig. 9.9) the magnetization distribution becomes more and more non uniform, which is caused by the magnetostatic stray field. Upon further decrease of the external field, the symmetry of the magnetization distribution breaks and a ``C'' state (Fig. 9.10(b)) develops. At the nucleation field (about 5 kA/m for our example) a vortex nucleates on the circumference and quickly moves towards its equilibrium position (close to the center of the nanodot). As a result we find a sudden drop in the average magnetization. When the external field is reduced to zero the vortex moves into the center of the nanodot (Fig. 9.10(c)). If the external field is increased in the opposite direction, the vortex is forced out of the center of the dot. For about $-70$ kA/m the vortex is pushed out of the nanodot (annihilation: Fig. 9.10(d)) and we find the second jump in the hysteresis curve to (almost) saturation.

Figure 9.10: Typical magnetization distributions along the hysteresis loop. The snapshots have been taken at the corresponding position on the hysteresis curve indicated in Fig. 9.9.

[(Almost) homogeneous magnetization.]

[``C'' state before the vortex nucleates.]

[Centered vortex in zero field.]

[Magnetization distribution before annihilation of the vortex.]

This characteristic behavior has also been found experimentally using Hall-micromagnetometry by Hengstmann et al. [115], who measured the stray field of individual permalloy disks using a sub-$\mu$m Hall magnetometer. The hysteresis loops of arrays of Supermalloy nanomagnets have been measured by Cowburn et al. [110] using the Kerr effect. Their characteristic loop shape has then been used to identify the single-domain in-plane and the vortex phase.

The rigid vortex model can describe very well the susceptibility, magnetization distribution, and vortex annihilation field for low fields as well as the vortex nucleation field for a wide range of dot sizes [116,117,118,119]. The experimentally observed nucleation fields appear to be bigger than those predicted by the rigid vortex model [120]. This is probably due to the fact, that the simplest ``C-shape'' nucleation is not always an appropriate approach to describe the magnetization reversal in circular dots.

In Fig. 9.11 the total energy is plotted as a function of the external field for the branch of decreasing field of the hysteresis curve. The solid line indicates the hysteresis branch for decreasing external field and the dashed line that for increasing field. For very high fields we have an almost uniformly magnetized nanodot. For decreasing field the total energy increases (almost) linearly. The dashed line for positive field values indicates the total energy for the vortex state. At the intersection of the solid and the dashed line (at a value of about 35 kA/m for the external field) the vortex state and the uniform magnetization have equal energy. However, they are separated by an energy barrier, which arises from the magnetostatic energy, which in turn is caused by the stray field on the circumference of the nanodot as the vortex is pushed out of the center. Thus, the vortex state is a metastable state for external fields higher than 35 kA/m and the uniform state is metastable for external fields below 35 kA/m.

Figure 9.11: Total energy as a function of the external field for both branches (solid line for decreasing field - dashed line for increasing field) of the hysteresis loop.

The field dependence of exchange and magnetostatic energy are given in Fig. 9.12. The exchange energy remains approximately constant for negative external fields until the annihilation field is reached. Since all exchange energy is stored in the vortex core, this indicates that the vortex core remains undisturbed for even very large vortex shifts.

Figure 9.12: Exchange and magnetostatic energy and their sum as a function of the external field (for decreasing external field).

For a twice as large nanodot with $R=200 \mathrm{nm}$ and $L=40 \mathrm{nm}$ we find a nucleation field of 28 kA/m and an annihilation field of 84 kA/m. The corresponding hysteresis loop is given in Fig. 9.13.

Figure 9.13: Hysteresis curve of a nanodot with a radius of 200 nm and a thickness of 40 nm.

In general, the initial susceptibility, the vortex nucleation, and the annihilation fields depend on the dot's saturation magnetization $M_\mathrm{s}$ and should scale universally as a function of the dimensionless dot-aspect ratio $L/R$ [116,117].