Modelling of magnetic structure using interior point optimization given exchange parameters

Abstract

In this report, a model is presented to alleviate some of the computational work that goes into the effort of finding the magnetic properties of magnetocaloric materials. The model utilizes an interior point optimization routine to solve for the minimal exchange energy configuration of a system, given the exchange interactions of the material. The model is tested against four materials (Ni, MnO, Fe\textsubscript{2}P and Mn\textsubscript{2}Sb). For Ni and MnO, the exchange interactions are also computed. Three iterations of the model are compared. The base model, which only considers exchange interactions inside a chosen supercell, the base model with the inclusion of boundary conditions, and the base model with boundary conditions and the addition of an algorithm to find optimal solutions. \\ The algorithm analyzes the found results by the optimization routine, and if the result is considered not properly symmetric, runs the optimization routine another time, from a symmetrical starting point obtained from the outcome of the previous run. \\ In all versions of the model, effectiveness (percent of runs that resulted in the optimal configuration) and average run times were recorded. Three initialization methods for the model were used, and also tested for their effectiveness. For the algorithm, a parameter $\gamma$ is introduced that changes the size of some of the moments for the new starting points. Six different values for $\gamma$ were tested for their effectiveness against a test set of suboptimal solutions.
The model with the addition of boundary conditions and the algorithm performed the best out of the three iterations of the model, with an effectiveness of 99.895\%, and an average run time ranging from 0.62 s for 2$\times$2$\times$2 Ni, to 94.64 s for 3$\times$3$\times$3 Fe\textsubscript{2}P, in the case of $\gamma = 0.3$. To conclude, the model with the inclusion of the boundary conditions and the algorithm proves to be a robust method to evaluate the magnetic configuration of a material, especially for smaller systems.