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Magnetization dynamics: Fast coarse graining approach agrees with all-atom simulations
Cooperative Research Team on
Magnetic Materials Bridging Basic and Applied Science
Abstract: Large-scale processes in real magnets necessitate the use of a collective variables description to circumvent complex and computationally intensive large-scale all-atom calculations. Here we show that a micromagnetics approach with variable cell size using the Landau-Lifshitz-Gilbert equation does not work, but a new coarse graining scheme reproduces the results from simulations taking into account the dynamics of all the atoms.
Figure 1. Magnetization reversal in a 1-D chain: dependence of the total magnetization of the chain on time. The micromagnetics-based approach (small black circles) is compared with the coarse graining simulations (crosses); the results of exact atomistic simulations (large white circles) are shown for reference. Since the coarse-graining results almost coincide with the atomistic ones, the crosses appear mostly in the centers of the white circles.
The magnetization dynamics in real magnets is often governed by defects, which serve as nucleation centers for magnetization or as pinning centers for domain walls. The torques acting on individual atomic spins in the vicinity of the defect are strongly non-uniform at the atomic-scale and can result in large-scale inhomogeneities in the static and dynamical magnetization distribution affecting properties such as the coercive field and magnetization reversal. First principles or atomic level modeling can be used for regions near the defects, but for large-scale processes occurring far from the defects a description in terms of collective variables (e.g. micromagnetics) must be used. There is no need and no possibility of following the vast majority of atomic spins in bulk systems. The simplest solution would be to use the standard micromagnetics (MM) approach with gradually increasing resolution near the defect, until every micromagnetic cell contains a single atomic spin, and the micromagnetic Landau-Lifshitz-Gilbert (LLG) equation coincides with the equation of precession of an individual atomic spin. By comparing this approach on simple models where exact solutions can be obtained with all-atom simulations, we have shown that this approach does not work, particularly for dynamical problems. The basic issue is the standard MM approach does not properly account for the small-scale fluctuations and results in fictitious torques and magnon scattering. As part of the CMSN magnetism project we have developed a theoretical framework and computational scheme which overcomes the difficulties, and we have tested it with excellent results against all-atom simulations. [Dobrovitski 03]
The approach is based on statistical coarse graining extended to the consideration of nonlinear systems (this is an important difference between modeling magnetic systems and lattice dynamical systems). One defines collective variables for large unit cells (in which the atomic spins have been suitably averaged). The main assumption is that the atomic degrees of freedom are in local equilibrium, and the equilibrium conditions are determined by the gross variables. With some care in the treatment of the zero frequency (Goldstone) mode, LLG type of equations of motion can be obtained for the collective or gross variables, with renormalized "exchange" interactions. The formalism and several test examples can be found in cond-mat/0111324 v1, which has been accepted for publication in Physical Review Letters. Two examples are given below.
We can use a simple 1-dimensional model to illustrate the performance
of the coarse graining method (see Figure 1). Consider
a chain of N=465 spins with ferromagnetic nearest-neighbor coupling J=25, and antiferromagnetic next-nearest-neighbor coupling J' = -γ J = -2.5. The spins possess single-ion anisotropy of
easy-axis
type K = 0.01. For simplicity, we assume zero temperature, no
dissipation, and neglect the dipole-dipole interactions. At the ends of
the chain, atomic-scale defects are located; this can represent, e.g., a
nanowire with the growth defects at the ends. Six spins at the ends of the
chain have the parameters J, J' and K different from their
bulk values. Initially, the chain is magnetized to saturation along the
easy axis (y-axis). At t = 0, the external field H = 0.02 is
applied at the angle φ = -0.4π to the x-axis. We have modeled the
system's dynamics by (1) atomic simulations which give the exact solution,
(2) by a micromagnetic-based multiscale approach, and (3) by our coarse
graining method, with the same grid as used in the micromagnetic
simulations. The computational time step, along with all other relevant
parameters, has been kept the same for all schemes. The temporal
dependence of the chain's total magnetization 
is shown in Figure 1. Since an energy-conserving
case is studied, in the absence of the defects at the ends of the chain,
all spins would rotate in unison. However, in the presence of the
perturbation caused by the defects, different spins rotate with slightly
different rates, and the system's motion becomes stochastic. Therefore,
after some time, the Zeeman energy is transferred to the energy of
short-wavelength magnons, leading to a gradual decrease of the system's
total magnetization. Also, one can see that the micromagnetics-based
approach does not always describe the behavior of the system correctly.
The coarse graining method performs much better, giving results very close
to the exact atomistic solution.
Figure 2. Magnetization distribution in a strip: (left) - with defects; (right) - without defects. Blue (red) color represents negative (positive) values of the magnetization components. Note the difference in the lower part of the strip between the "clean" system and the system with defects.
The CG method can easily be extended to two- and three-dimensional systems. One particularly interesting problem which can be studied using a simplified 2-D model is the problem of the interface between the ferromagnetic film and the substrate. We have applied the CG method to modeling a domain wall (DW) in a thin ferromagnetic strip, focusing on the influence of the interface on the DW dynamics. In real materials, the interface contains a large number of atomic-scale defects appearing due to various stacking faults (in the substrate and on the interface itself). As a result, in a thin ferromagnetic film, several atomic layers near the interface are different from the bulk (the exchange parameters, anisotropy constants, etc.). These surface defects, even reasonably small, can lead to a nucleation of the Bloch lines in the moving DW [A.P. Malozemoff, and J. C. Slonczewski, Magnetic domain walls in bubble materials (Academic Press, New York, 1979)], thus drastically changing the effective mass of the domain wall. Modern experimental techniques allow the study of the depinning of a DW from a single pinning site [K. S. Novoselov et al., IEEE Trans. Magn. 38, 2583 (2002)] produced by interface defects. In order to consider basic processes, we have considered a 2-D cross-section of a ferromagnetic strip, which initially contains a domain wall. The parameters (exchange, anisotropy, etc.) of the four lowest (closest to the interface) atomic layers include random variations caused by the interface defects. After an external field is applied, the DW starts moving. As our simulations show, the interface defects induce flexural deformations of the domain wall, thus leading to irregular oscillations of the DW profile in time. This is in agreement with the qualitative picture suggested in [A.P. Malozemoff, see the ref. above]. For comparison, the simulations performed on a "clean" system (with no interface defects) show that the magnetization distribution in a "clean" system is very smooth, without any small-scale features.
Contact: Bruce Harmon, Ames Laboratory
