7.3.1 Pinning on a Planar Interface

If a perfect interface with a discontinuity in the material parameters is assumed, a one-dimensional micromagnetic model can be calculated analytically. Schrefl [90] has calculated the nucleation field for two neighboring grains with 90$^\circ$ misoriented easy axes. Kronmüller and Goll [91] have presented a model for grains with parallel easy axes but a step-like change in the material parameters at the interface. Finally, Della Torre et al. [92] have assumed a variation of the exchange energy as a function of position and they have obtained results for a square well and Gaussian decrease.

Our simplest geometrical model of the pinning process includes two different materials (different saturation polarization $J_\mathrm{s}$, uniaxial magnetocrystalline anisotropy $K_1$ and exchange constant $A$) with a perfectly planar interface assuming a step like change [91] of the material parameters (cf. Fig. 7.4).

Figure 7.4: Model geometry for domain wall pinning on a perfectly planar interface with parallel anisotropy axes. The chain of arrows indicates the magnetization distribution of a pinned domain wall.

Typical values for Sm(Co,Fe,Cu,Zr)$_z$ magnets can be found in Tab. 7.1. However, the simulations are independent of the choice of $A$ and $K_1$, only the exchange length $l_\mathrm{ex}=\sqrt{A/K_1}$ is relevant. If an ideal Bloch wall is assumed, the pinning field, which is required to force the domain wall into the hard material (with higher domain wall energy), can be calculated with the 1D analytical model of Kronmüller and Goll [91]. The micromagnetic simulation is initialized with a Bloch wall in the softer material (I) and the external field pushes it towards the interface into the harder material (II) (cf. Fig. 7.4).

The comparison with the micromagnetic simulations shows, that the edge length of the finite elements has to be smaller than the exchange length (e.g. 1.8 nm in Sm$_2$Co$_{17}$ [91]) of the harder material in order to avoid ``artificial pinning'' on the finite element mesh (cf. Sec. 7.3.4). Fig. 7.5 shows the dependence of the pinning field $H_\mathrm{pin}$ (in units of the anisotropy field $H^\mathrm{II}_\mathrm{ani}=2 K^\mathrm{II}_1/J_\mathrm{s}^\mathrm{II}$) on the ratio of exchange ( $\varepsilon_A=A^\mathrm{I}/A^\mathrm{II}$) and anisotropy constants ( $\varepsilon_K=K^\mathrm{I}_1/K^\mathrm{II}_1$). The thick curve represents $\varepsilon _A \cdot \varepsilon _K = 1$, where the two materials have equal domain wall energy and therefore exhibit no domain wall pinning ( $H_\mathrm{pin}=0$). Only that part with $H_\mathrm{pin}>0$ is physically relevant. The pinning field is always smaller than the anisotropy field. For given $\varepsilon _A$ the pinning field is proportional to $\varepsilon _K$. However, if $\varepsilon _A$ is reduced (which decouples the two materials), the coercive field shows a steep increase towards the anisotropy field. Thus, in order to reach high pinning fields, a low $\varepsilon _A$ ratio has to be achieved.

Figure: Dependence of the pinning field $H_\mathrm{pin}$ (in units of the anisotropy field of material II) on the ratio of exchange ($\varepsilon _A$) and anisotropy constants ($\varepsilon _K$) for given $A^\mathrm{II}$ and $K^\mathrm{II}_1$ according to a 1D analytical model [91]. The thick curve represents $\varepsilon _A \cdot \varepsilon _K = 1$, where the two materials have equal domain wall energy and $H_\mathrm{pin}=0$.

When the external field is switched on, the domain wall moves towards the interface and gets pinned. As the external field increases the Bloch wall is more and more forced into the ``harder material'' until it depins and propagates further through the ``harder material''. The analytical result has been calculated with the one dimensional model of Kronmüller and Goll [91], which gives the pinning field as

$$H_\mathrm{pin}= \frac{2 K^{\mathrm{II}}_1}{J_\mathrm{s}^\m... ...psilon_K} {(1+\sqrt{\varepsilon_A \varepsilon_J})^2} \quad ,$$ (7.1)

where

$$\varepsilon_J = \frac{J_\mathrm{s}^\mathrm{I}}{J_\mathrm{s}^... ... , \quad \varepsilon_K = \frac{K^I_1}{K^{\mathrm{II}}_1} \quad$$ (7.2)

and ($I$) denotes the material parameters of the softer material and ($\mathrm{II}$) those of the harder material.

Table 7.1: Material parameters of typical Sm(Co,Fe,Cu,Zr)$_z$ permanent magnets [80,93,94,91].

	``2:17'' type	``1:5'' type
	cells	boundary phase
$J_\mathrm{s}$ (T)	1.3	0.8
$A$ (pJ/m)	14.0	14.0
$K_1$ (MJ/m$^3$)	5.0	9.0
$l_\mathrm{ex}=\sqrt{A/K_1} \mathrm{(nm)}$	1.7	1.3
$\delta=l_\mathrm{ex}\cdot\pi \mathrm{(nm)}$	5.3	3.9
$H_\mathrm{ani}= 2 K_1/J_\mathrm{s} \mathrm{(kA/m)}$	7692	22500