7.3.4 Artificial Pinning

The influence of the space discretization using the finite element mesh has been investigated in more detail, because the magnetic domain walls in permanent magnets like SmCo are very thin. Thus, a high resolution mesh with very small elements is required where domain walls occur.

A very homogeneous finite element mesh with 59491 elements and 11085 vertices has been used. The minimum edge length of any edge connecting two vertices in the finite element mesh is 0.29, the maximum edge length is 1.02 and the average is 0.56. The simulations are initialized with a magnetic domain wall in the center of the softer material (cf. Fig. 7.4). Then the magnetization is relaxed in zero field, so it can obtain its true minimum energy state. The domain wall energy as a function of the average edge length in units of the exchange length is shown in Fig. 7.10. The result for the domain wall energy remains almost constant. However, as the average edge length increases (the mesh is scaled up) the exchange energy increases while the magnetocrystalline anisotropy energy decreases and cancels the error in the former.

Figure 7.10: Domain wall energy as a function of the average edge length of the finite element model. The average edge length is given in units of the exchange length. The solid line indicates the analytical Bloch wall energy $4\sqrt {A K_1}$.

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 pushed against the interface until it depins and propagates further through the harder material. The simulation results for different scaling are given in Fig. 7.11. The analytical result has been calculated with the one dimensional model of Kronmüller and Goll [91] (cf. Eq. (7.1)).

Figure 7.11: Demagnetization curves for different average edge length of the finite element model. The analytical pinning field has been calculated with the model presented in Sec. 7.3.1.

Fig. 7.12 summarizes the pinning fields, which have been obtained from the demagnetization curves in Fig. 7.11. The error bars indicate the step size by which the external field has been increased, when the magnetization distribution reached equilibrium. Thus, the finite element simulation gives the correct pinning fields within its numerical limits.

Figure 7.12: Pinning fields for different average edge length of the finite element model. As the mesh size increases the domain wall gets artificially pinned on the finite element mesh.

In conclusion, the (average) edge length of a very homogeneous finite element mesh has to smaller or at most equal to the exchange length of the material. For inhomogeneous finite element meshes, the maximum edge length has to be considered because domain walls can get stuck in very coarse parts of the mesh.