7.3.3 Pinning on the Cell Structure

Then the influence of the cell boundary phase perpendicular to the domain wall has been studied. The geometry is shown in Fig. 7.8. It resembles the situation of a domain wall, which moves from the right to the left and gets trapped in the (softer) intercellular phase (``I''), where it is repelled by the cells (``II''). The interesting question is, if the pinning field is changed by the cellular structure as compared to the perfect planar interfaces discussed above. The results are given in Fig. 7.9, where the pinning field (normalized to the pinning field for $t=0$, which corresponds to the planar interface again) is given as a function of the relative thickness $t/T$ of the intercellular phase. $T$ is the sum of the edge length of a cell and the thickness of the intercellular phase $t$. In Fig. 7.8 the ``hard surface area'' of the cells is indicated by the shaded faces. Obviously, the pinning field depends linearly on the relative thickness of the intercellular phase. Even if the whole model is scaled to twice or three times its size (i.e. the cell size is increased by a factor of two or three) we find the same behavior, because the relative thickness remains constant (cf. data marked ``area x2'' and ``area x3'' in Fig. 7.9).

If we switch to the case of repulsive pinning again, we can assume that now the cells (``II'') are softer than the intercellular phase (``I'') in Fig. 7.8. In this case the domain wall moves from left to right and gets pinned in front of the intercellular phase. The pinning fields are also given in Fig. 7.9. They show the same linear behavior as in the case of attractive pinning. However, in a fully developed cell structure the possible range of values for the thickness $t$ are restricted by a minimum (cf. Fig. 7.7) below which the domain wall does not fit in and a maximum (cf. Fig. 7.21) above which the whole intercellular phase is reversed.

For Sm(Co,Fe,Cu,Zr)$_z$ precipitation hardened magnets, this means, that the pinning field increases with the cell size. However, it decreases linearly with the thickness of the intercellular phase, if it is larger than the domain wall width. Below this limit the pinning field is strongly reduced. These results are independent, whether attractive or repulsive pinning is dominating. Thus, the best magnetic properties should be found in magnets with large cells, thin (but still sufficiently thick) intercellular phases, and large differences in the domain wall energy (ideally a large difference in the exchange constants).

Figure 7.8: Model geometry for domain wall pinning on a coherent precipitation structure (with parallel anisotropy axes). The shaded areas indicate the faces of the cells, where the magnetic domain wall gets pinned.

Figure 7.9: Pinning field for attractive and repulsive pinning of a magnetic domain wall on the cell structure as a function of the relative thickness $t/T$. The data marked ``area x2'' and ``area x3'' have been obtained with a model scaled to twice and three times the initial size. The dashed line is just a guide to the eye.