Magnetostatic dipolar energy of large periodic Ni fcc nanowires, slabs and spheres

Abstract

The computational effort to calculate the magnetostatic dipolar energy, MDE, of a periodic cell of N magnetic moments is an O(N 2) task. Compared with the calculation of the Exchange and Zeeman energy terms, this is the most computationally expensive part of the atomistic simulations of the magnetic properties of large periodic magnetic systems. Two strategies to reduce the computational effort have been studied: An analysis of the traditional Ewald method to calculate the MDE of periodic systems and parallel calculations. The detailed analysis reveals that, for certain types of periodic systems, there are many matrix elements of the Ewald method identical to another elements, due to some symmetry properties of the periodic systems. Computation timing experiments of the MDE of large periodic Ni fcc nanowires, slabs and spheres, up to 32000 magnetic moments in the periodic cell, have been carried out and they show that the number of matrix elements that should be calculated is approximately equal to N, instead of N 2 /2, if these symmetries are used, and that the computation time decreases in an important amount. The time complexity of the analysis of the symmetries is O(N 3), increasing the time complexity of the traditional Ewald method. MDE is a very small energy and therefore, the usual required precision of the calculation of the MDE is so high, about 10 −6 eV/cell, that the calculations of large periodic magnetic systems are very expensive and the use of the symmetries reduces, in practical terms, the computation time of the MDE in a significant amount, in spite of the increase of the time complexity. The second strategy consists on parallel calculations of the MDE without using the symmetries of the periodic systems. The parallel calculations have been compared with serial calculations that use the symmetries.

Magnetostatic dipolar anisotropy energy and anisotropy constants in arrays of ferromagnetic nanowires as a function of their radius and interwall distance

“…An implementation of the traditional Ewald method that uses the symmetries of the periodic magnetic system to reduce substantially the computation time have been used in the present research. Details of this implementation and its application to any type of lattice, and especially to nanowire arrays, can be found elsewhere [48].…”

“…The convergence of the MDAEs and MDEs as a function of the real and reciprocal space cutoffs, r c and g c , respectively, was studied on a previous publication. It was found that the MDAE converges faster than the MDEs and that to obtain the mentioned precision of 10 −6 eV/cell for nanowire arrays, r c should be 38a and g c should be at least 9/a radian/Å, where a is the lattice parameter of bulk Ni or Co [48].…”