1. Introduction
Simulations of magnetization processes have significant meaning in regards to both scientific and practical points of view [1,2,3,4,5,6]. The observed progress in technologies utilizing magnetic materials requires new magnets with unique properties optimized for different applications. What is more, a permanent demand for soft and hard magnets with ultimate characteristics can be observed [7,8,9,10,11]. Designing such systems should include modeling of the magnetization processes using computer simulations, which enables the searching for and testing of their properties in a pre-lab state. In this field, two main directions can be indicated, i.e., simulations based on the Landau equation [12,13,14] or Monte Carlo (MC) rules [15,16,17,18,19]. Both methods have advantages and disadvantages depending on the intended use of the considered magnetic system. The first approach is rather dedicated to continuous large-scale systems in which phenomena occurring at the atomic level are represented by some average volumetric parameters. On the other hand, the MC simulation methods are based on inter-atomic properties, and therefore, modeling of mesoscopic systems results in significant consumption of computational resources. This problem can be solved using parallelization of the MC algorithm [20,21]. Unfortunately, in order to study magnetization processes of ferromagnetic systems, it is necessary to utilize one of the cluster MC methods, and therefore, the possible parallelization is rather problematic. One can imagine a situation in which a node in a system represents a “super-cell” instead of single spin or magnetic moment. In this way, the analyzed system can be enlarged by means of the super-cell size. The question is how to ensure thermodynamic as well as energetic equivalence between the initial (with spins) and the enlarged (with super-cells) systems, transferring properties on the atomic level to the approach based on the finite element method.
In this work, we present the scaling rules for MC methods and their application. In particular, we propose that the rules be applied to the disorder-based cluster MC algorithm [22], formulated and developed by our team, which is a promising tool for studying and designing new magnetic materials.