7.4.2 Influence of the Thickness of the Intercellular Phase

However, not only the material composition and material parameters, but also the geometry of the cellular structure has an important influence on the magnetic properties. The influence of the cell size has been studied previously [96] and it showed a strong increase with increasing size of the cells (when the composition of the magnet was kept constant by increasing the cell boundary, too). We have investigated solely the influence of the thickness of the cell boundary phase on the domain wall pinning and the pinning field, but kept the cell size constant. This time we have used a larger model of $3 \times 3 \times 3$ cells and larger cells with $e=100 \mathrm{nm}$ and $\beta=60^\circ$, which gives $D \approx 250 \mathrm{nm}$ (cf. Fig. 7.17). The thickness has been varied from $t=2.5 \mathrm{nm}$ to $t=40 \mathrm{nm}$ and the material parameters for the ``2:17'' cells and the ``1:5'' intercellular phase at 300 K given above have been used.

Figure 7.17: Finite element model with $3 \times 3 \times 3$ cells.

As we are varying the thickness, the ratio of the volume of the cells $V_{ 2:17}$ to the volume of the cell boundary phase $V_{ 1: 5}$ changes. Thus, the composition and the $z$-value changes. These data are summarized in Tab. 7.2, where a volume of $V_\mathrm{e}^{2:17}=0.24853 \mathrm{nm}^3$ and $V_\mathrm{e}^{1:5}=0.0859 \mathrm{nm}^3$ for the elementary cells of the ``2:17-type'' cells and the ``1:5-type'' cell boundary phase, respectively, have been assumed [97]. The $z$-value is determined by

$$z=\frac{17 \cdot V_{ 2:17}/V_\mathrm{e}^{ 2:17} + 5 \cdot ... ...{e}^{ 2:17} + 1 \cdot V_{ 1: 5}/V_\mathrm{e}^{ 1: 5} }\quad,$$

where all additives have been neglected.

Table 7.2: $z$-values for different thickness $t$ of the cell boundary phase around cells with $D=250 \mathrm{nm}$.

$t (\mathrm{nm})$	$V_\mathrm{2:17} (\mathrm{nm}^3)$	$V_\mathrm{1: 5} (\mathrm{nm}^3)$	ratio	$z$
2.5	$28 738$	$2 358$	12.187	8.13
5	$28 738$	$4 843$	5.934	7.81
10	$28 738$	$10 202$	2.817	7.31
20	$28 738$	$22 570$	1.273	6.64
40	$28 738$	$54 643$	0.526	5.93

The demagnetization curves given in Fig. 7.19 show, that for a very thin cell boundary phase ( $2.5 \mathrm{nm}$, $5 \mathrm{nm}$) the effect of domain wall pinning vanishes, because the domain wall width is larger than the thickness of the cell boundary phase. For a thickness of 10 and $20 \mathrm{nm}$ we find strong domain wall pinning. When an external field of about 2500 kA/m is applied, the domain wall can overcome the energy barrier and cross the cell boundary phase (Fig. 7.20). For a cell boundary phase with a thickness of more than 4 times the domain wall width, the analysis of the magnetization distribution (Fig. 7.21) reveals a new behavior: The whole cell boundary phase reverses starting from the original position of the domain wall, because the curvature of the domain wall [98] allows it to propagate through the whole cell boundary phase (Fig. 7.18). This effect is a result of the competition between Zeeman energy and domain wall energy. The system can lower its Zeeman energy by domain wall bending, because the domain with its magnetization parallel to the external field increases its volume. However, this happens at the expense of domain wall energy, because the bending increases the area of the domain wall. For very thin intercellular phases the domain wall is very flat because the reduction in Zeeman energy would be very low. For thick intercellular phases the domain wall bending gets more and more pronounced until it can reverse the whole intercellular phase. This leads to the second plateau in the demagnetization curve for $t=40 \mathrm{nm}$ in Fig. 7.19. Only at higher fields the magnetization reversal of the cells starts with the nucleation of a reversed domain in a corner of the rhomboidal cells.

Figure 7.18: Domain wall bending of a magnetic domain wall in the (softer) intercellular phase (attractive pinning).

[$t=20$ nm]

[$t=30$ nm]

[$t=40$ nm]

Figure 7.19: Demagnetization curves for varying thickness $t$ (values in the legend in nm) of the intercellular phase around large cells with $D=250 \mathrm{nm}$.

Due to pinning on the computational grid an external field of 1500 kA/m is required to move the domain wall from its initial position. This effect has to be attributed to the size of the large model with $3 \times 3 \times 3$ cells, the larger size of the cells and the lower resolution of the finite element mesh. However, it has been verified that similar pinning fields are obtained for the smaller model with a proper high resolution mesh.

Figure: Magnetization distribution for $D=250 \mathrm{nm}$ and $t=10 \mathrm{nm}$. The green surface indicates the domain wall, which separates the two domains (red and blue areas) with antiparallel magnetization.

[Demagnetization curve]

[ $H_\mathrm{ext}\hspace*{1.5pt}=-1360 \mathrm{kA/m} \newline \hspace*{4 ex} J/J_\mathrm{s}=-0.03$]

[ $H_\mathrm{ext}\hspace*{2.5pt}=-2040 \mathrm{kA/m} \newline \hspace*{4 ex} J/J_\mathrm{s}=-0.35$]

[ $H_\mathrm{ext}\hspace*{1.15pt}=-2500 \mathrm{kA/m} \newline \hspace*{4 ex} J/J_\mathrm{s}=-0.53$]

Figure: Magnetization distribution for $D=250 \mathrm{nm}$ and $t=20 \mathrm{nm}$. The green surface indicates the domain wall, which separates the two domains (red and blue areas) with antiparallel magnetization.

[ $H_\mathrm{ext}\hspace*{2pt}=-1360 \mathrm{kA/m} \newline \hspace*{4 ex} J/J_\mathrm{s}=-0.01$]

[ $H_\mathrm{ext}\hspace*{1.5pt}=-2120 \mathrm{kA/m} \newline \hspace*{4 ex} J/J_\mathrm{s}=-0.41$]

[ $H_\mathrm{ext}\hspace*{2.5pt}=-2420 \mathrm{kA/m} \newline \hspace*{4 ex} J/J_\mathrm{s}=-0.56$]

[ $H_\mathrm{ext}\hspace*{1.5pt}=-2720 \mathrm{kA/m} \newline \hspace*{4 ex} J/J_\mathrm{s}=-0.86$]