Fast computation of magnetostatic fields by nonuniform fast Fourier transforms

Spin electronics devices are based on the control of magnetization in nanometer-sized elements, so that the knowledge of the spatial distribution of magnetization is of prime importance. Owing to the continuous decrease of the size of devices while the power of computers increases, most systems pertaining to nanomagnetism and spin electronics are now at reach from micromagnetic numerical simulation. Thus these now play a leading role in understanding experiments and even predicting their properties. In practice the speed of the simulations is mostly limited by the evaluation of the dipolar energy, which by nature is long-ranged and thus involves the mutual interactions of all spins of a system.

In most cases micromagnetism is based on the assumption that the spatial variation of magnetization is slow at the scale of atoms, in which case a continuous medium description can be used. However to be handled numerically magnetic systems have to be discretized in cells of finite size, typically a few nanometers. There are essentially two approaches concerning the meshing, both yielding advantages and drawbacks :

  • Finite Differences Methods (FDM). The sample is meshed using a periodic prismatic grid. The translational invariance allows one to make use of Fast Fourier Transforms for the evaluation of the dipolar energy, with a NLog N efficiency. One drawback of FDM is that curved boundaries, often found in devices by design of because of the limited resolution of lithography, are not faithfully described.
  • Finite Elements Methods (FD). The sample is meshed non-periodically using tetrahedrons. The boundaries are then much better described, however FFTs can no longer be applied. Dipolar fields require a time N4/3 in the best case
A typical mesh in a two-dimensional code of Finite Elements
FIG 1 : A typical mesh in a two-dimensional code of Finite Elements

We developed a micromagnetic code, FEM-NFFT, combining the advantages of FDM and FEM, i.e. a FEM mesh faithfully describing curved boundaries, with a processing efficiency scaling as NLog N like for FDM. This is made possible by the implementation of Fast Fourier Transforms on an irregular lattice, a theory recently developed and assessed in Mathematics. We gave a proof of principle on a two-dimensional code. NFFT become more efficient while suffering from no loss of precision starting from a few thousands of cells. We are now developing the three-dimensional version for tackling real magnetic nanodevices, where the advantage of FEM-NFFT is expected to become decisive against conventional FEM techniques (real devices must often be meshed using tens of thousands up to millions of cells).

Contributors

  • MNM : J. C. Toussaint, E. Kritsikis, O. Fruchart, H. Szambolics.
  • Spintec (Grenoble) : L. Buda-Prejbeanu

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Corresponding author : J. C. Toussaint

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