4. Concluding Remarks
Simulations of magnetization processes can be very useful tools for designing new magnetic materials. It is of great importance to include atomic properties in considerations for studying magnetization processes of dimensionally real systems, i.e., mesoscopic in size. The presented MC method, with a disorder-based cluster approach, is suitable for multimagnetic systems, but the system size (counted in nodes) is restricted to 105–106 nodes, which means that only nano-objects can be analyzed. In order to enlarge this size, we propose re-scaling of the system using a concentration of the volume nr × nr × nr (n is the scaling factor, r is inter-node distance) into one node. The new system contains the same number of nodes but all its linear dimensions are enlarged by the n factor. Simultaneously, it is necessary to recalculate the system parameters occurring in a Hamiltonian, which is used for energy change calculation. On the whole, we propose scaling rules, which save energetic equivalence between rotation of the re-scaled magnetic moment and a group of spins in the nr × nr × nr volume. Additionally, we have found that the MC iterations can be carried out under the thermodynamic balance determined for the “reduced” system, similar to the initial one but with an exchange integral parameter equal to Jij/n.
The presented magnetic states for various objects with different values of the n parameter confirm the correctness of the proposed approach. The magnetic moment configurations revealed the transition from pure ferromagnetic to vortex or fingerprint structures, when the system size had been enlarged. The method was also shown to be useful for simulation magnetization processes of hard magnetic materials. In this case, the full hysteresis loop as well as magnetic moment configuration were determined, revealing some characteristic behaviors of the magnetic moments that are responsible for the shape of the reverse magnetization curve.
In conclusion, the proposed scaling rules facilitate simulation of large-scale systems based on the parameters determined for atomic level interactions. Our approach can be considered one way of “connecting” low-level or first principle calculations to finite element MC magnetic simulations.