Abstract

Exchange energy is especially sensitive to the numerical representation selected. We compare three discretized exchange energy formulations for 3D numerical micromagnetics on rectangular grids. Explicit formulae are provided for both Neumann and Dirichlet boundary conditions. Results illustrate the convergence order of these methods as a function of discretization cell size and the effect of cell size on vortex pinning.

Author Keywords: Micromagnetics; Exchange; Convergence

PACS classification codes: 75.40.Mg; 75.30.Et; 75.75.ta

Article Outline

1. Introduction

2. Theory

2.1. Integration techniques

2.2. Six- and 12-neighbor methods

2.3. Twenty-six-neighbor exchange energy

3. Simulation results

4. Conclusions

References

(6K)
Fig. 1. Exchange energy error for a two-period uniform magnetization spiral. The fitted lines are, top to bottom, 10⁻³γ, 3×10⁻⁵γ², and 10⁻⁹γ⁴.

(5K)
Fig. 2. Total energy error for μMAG standard problem No. 3. The fitted lines are, top to bottom, 0.02Δ², 0.01Δ², and 5.5×10⁻³Δ², where Δ=h/l_ex. The reference energy is obtained by extrapolating to h=0.

(5K)
Fig. 3. Applied field required to unpin a vortex as a function of discretization cell size.

Table 1. Corner coefficients for O(h⁴) interaction matrices, for ∂m/∂Not-found=0 Neumann and Dirichlet boundary conditions with constant exchange coefficient A

Table 2. Coefficients for the 26-neighbor exchange energy formulation, Eq. (20)
Here Not-found=i±1, Not-found=j±1, Not-found=i±1, and c_{ijki′j′k′}=0 otherwise.

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