Accurate finite-difference micromagnetics of magnets including RKKY interaction: Analytical solution and comparison to standard micromagnetic codes

Within this paper we show the importance of accurate implementations of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions for antiferromagnetically coupled ferromagnetic layers with thicknesses exceeding the exchange length. In order to evaluate the performance of different implementations of RKKY interaction, we develop a benchmark problem by deriving the analytical formula for the saturation field of two infinitely thick magnetic layers that are antiparallelly coupled. This benchmark problem shows that state of the art implementations in commonly used finite-difference codes lead to errors of the saturation field that amount to more than 20% for mesh sizes of 2 nm which is well below the exchange length of the material. In order to improve the accuracy, we develop higher order cell-based and nodal-based finite-difference codes that significantly reduce the error compared to state of the art implementations. For the second order cell-based approach and first order nodal-based approach the error of the saturation field is reduced by about a factor of 10 (2% error) for the same mesh size of 2 nm.

Automated summary

Within this paper, we demonstrate the significance of accurate implementations of RKKY interactions for antiferromagnetically coupled ferromagnetic layers with thicknesses larger than the exchange length. We develop a benchmark problem to evaluate different implementations of RKKY interaction by deriving an analytical formula for the saturation field of two infinitely thick antiparallelly coupled magnetic layers. Our benchmark problem reveals that current implementations in commonly used finite-difference codes lead to errors in the saturation field, exceeding 20% for mesh sizes of 2 nm, which is below the material's exchange length. To enhance accuracy, we introduce higher order cell-based and nodal-based finite-difference codes that significantly reduce errors compared to existing implementations. With a mesh size of 2 nm, the second-order cell-based approach and the first-order nodal-based approach reduce the error in the saturation field by a factor of 10 (2% error).