Combined micromagnetic simulation and machine learning approach to analysis of polycrystalline bilayer system with exchange bias

Abstract

The exchange interaction between two magnetic materials may lead to a symmetry breaking manifesting itself as a unidirectional anisotropy induced in one layer (usually ferromagnetic material F) by another one less susceptible to an external magnetic field (usually antiferromagnetic material AF). The unidirectional anisotropy in the F layer leads to a hysteresis loop shifted along the magnetic field axis and the phenomenon is called exchange bias (EB). Despite its seeming simplicity, the EB has an intricate microscopic mechanism, which has been approached by many models of different complexity dealing with perfect or imperfect interface, polycrystalline structure of F and AF, and bulk spin configuration in the AF layer [1]. For real-life applications e.g. sensors, memory cells or spintronic devices, the number of significant parameters affecting coercivity and EB field of the F layer could become too large to be accounted by simplified models, so computer simulation becomes a method of choice. The increasing complexity leads to more realistic and universal models but makes it difficult to evaluate their performance and limitations, study interconnections between model’s parameters, and make a fast prediction on how the system with a given set of parameters would behave. In this study we approach this problem by applying methods of machine learning to a dataset obtained by a micromagnetic model representing a polycrystalline bilayer film consisting of F and AF layers. As AF state cannot be readily set in the continuous media approach, we adopted the model described in [2] by substituting AF with F material, for which contributions to Zeeman and magnetostatic energies were disabled. The example of the simulated hysteresis loop along with the geometry is shown in Fig. 1.Micromagnetic simulation was performed using MuMax3 software with GPU acceleration [3]. As an income several material and structural parameters have been chosen including saturation magnetization (same for F and AF), uniaxial magnetic anisotropy constants of F and AF, exchange constants in F and AF, attenuation coefficient of exchange constant between F and AF, thickness of the F layer, and average crystallites size (8 variables in total). As a result of approximately 1800 hysteresis loops simulations, for each configuration (each parameter has been varied randomly within a realistic range) two outcome parameters have been extracted: coercivity and EB field. The model has been validated with a smaller set of simulations where single parameter has been varied while fixing others at realistic values typical for a popular FeNi/FeMn system.Principal component analysis (PCA) allowed identifying readily input parameters having the strongest positive and negative correlations with the output parameters. Correlation coefficients for saturation magnetization, the thickness of the F layer, and the attenuation coefficient were significant for both coercivity and EB field, whereas the coefficient for AF anisotropy constant was among the largest only for EB field. Both PCA and T-SNE (t-distributed stochastic neighbor embedding) data visualizations suggested the possibility of building a data approximator. As EB field and coercivity are known to have complex nonlinear dependencies on some parameters, we chose the gradient boosting, which is a suitable algorithm for the current task. In this work Catboost open source library based on decision trees was used. The dataset was split into two parts for training the model and testing it using the root mean square error (RMSE) metric. The trained model could predict EB field for a test set of 500 examples rather well with RMSE = 0.04, whereas higher RMSE = 0.13 was achieved for coercivity, which could also be considered a good result. To demonstrate the applicability of the model for fast search of optimal coercivity and EB field, we employed Broyden–Fletcher–Goldfarb–Shanno and differential evolution algorithms realized in SciPy library to find sets of input parameters giving the largest values of output parameters. This approach allowed us to identify several areas of the highest coercivity and EB field values exceeding those in the training set, which were confirmed by additional micromagnetic simulations. Although the RMSE was large, the model can be further improved by including more examples of computer simulation to resolve the areas of interest.In conclusion, machine learning algorithms were successfully used for in-depth analysis of a bilayer micromagnetic model with exchange bias. The model capable of predicting values of coercivity and exchange bias using material and structural parameters as an input was successfully created and trained. The proposed approach was demonstrated to be suitable for building optimization algorithms allowing to find areas of interest in multidimensional space of income parameters giving the desired values coercivity and EB field. The demonstrated approach can be useful for the analysis of computer models as well as real systems with exchange bias.This work was supported by the Russian Science Foundation (project No 19-72-00141). **