Magnetostatic field computation in thin films based on k-space fast convolution with truncated Green's function

Abstract

The numerical studies of magnetization dynamics in micromagnetic systems with spatially distributed magnetization crucially rely on the computation of magnetostatic fields generated by the ferromagnet. The solution of the magnetostatic problem can be written as convolution of an appropriate Green’s function with the magnetization vector field. Since the Green’s function is long-range, after discretization of the problem in N cells, the direct numerical computation of the convolution integral has a computational cost scaling as N2.In this work, we use, as it is often done in micromagnetics, the Fast Fourier Transform (FFT) method to reduce the computational cost scaling to N logN. We consider micromagnetic systems with thin-film geometry where the magnetization can be considered constant across the thickness of the magnet. In these situations, only the field averaged across the thickness is relevant to the magnetostatic interactions and both the magnetization and the averaged field inside the magnet depend only on two spatial variables. The usual approach to deal with this type of problem is to use FFT to compute the discretized version of the convolution integral relating the field and the magnetization [1]. The dual approach to the usual one, is based on the discretization of the integral relation between the field (averaged across the thickness) and the magnetization in the reciprocal space (k-space). The connection in the k-space is given by the Fourier transform of Green function of the magnetostatic problem which, in case of infinite thin-films, can be written in a special form taking into account the thin-film geometry [2]. The analysis in the k-space has been extensively used in connection with linear spin wave dynamics in thin films working as magnonic waveguide, where all the relation of practical use (e.g dispersion relations), are easily obtained by algebraic manipulations. When the dimensions of the thin film are finite, the reconstruction of the magnetostatic field with the infinite film Green's function, is altered by the presence of replicas of the thin film in the ordinary space due to the sampling operation in the k-space. This effect is usually smoothed by using zero-padding, which increases the distance between two interacting replicas weakening their coupling.In this work, a fast convolution algorithm in the k-space able to remove the influence of replicas in the reconstruction of the magnetostatic field in thin films of finite size, is proposed and applied to study linear spin wave dynamics. It is based on the use of the Fourier transform of truncated Green function, which has been recently used to compute volume potentials defined on free-space in a finite (truncated) domain [3]. Such method, guarantees a regularization of the Green function by cutting off the interaction in physical space beyond the domain of interest and this in turn produces spectral accuracy [5] for the Fourier transform evaluation by FFT algorithm. The accuracy of the proposed method is studied by comparing the results with analytical calculations and with those obtained by fast convolution with the infinite thin film Green's function as a function of mesh size. For example, in the figure, the computation of the demagnetizing factor along the x direction for a thin film of dimensions Lx = 300nm, Ly = 100nm and Lz = 3nm as a function of the mesh dimension (kxmax = kymax = kmax) and discretization step-size (Δkx = Δky = Δk) in the k-space is considered. With solid lines, the curves in the infinite film approximation (infin.) are indicated, while curves with symbols results from the proposed approach (trunc.). The horizontal dotted line, represents the analytical value of the demagnetizing factor evaluated with Newell’s formula [6]. Computation with Δk< π/300 nm-1, for the duality property of the Fourier transform, are equivalent to use zero padding in the ordinary space. This produces a substantial improving of estimation accuracy with respect to the analytical value, in case of infinite film approximation. As a benchmark of the proposed approach, it results that the curves computed with the truncated Green function, show a weak dependence from Δk for Δk>π/R and collapse into the same one for Δk<π/R, where R is the radius of a disk region which defines the compact support of the truncated Green function. This, as a confirmation of the fact, that when the region that exceeds the support of the truncated Green's function is zero padded, there is no effect on the computation of the magnetostatic field. Moreover, for sufficient value of kmax the asymptotic value tends towards analytical one, since when Δk<π/R the influence of replicas on the magnetostatic field computation is removed. **