Coercivity limits in nanoscale ferromagnets

It has been a puzzle for a century about how ``hard'' (coercive) a ferromagnet can be. Seven decades ago, W. Brown gave his famous theorem to correlate coercivity of a ferromagnet to its magnetocrystalline anisotropy field. However, the experimental coercivity values are far below the calculated level given by the theorem, which is called Brown's Coercivity Paradox. The paradox has been considered to be related to the complex microstructures of the magnets in experiments because coercivity is an extrinsic property that is sensitive to any imperfections in the specimens. To date, coercivity cannot be predicted and calculated by quantitative modeling. In this investigation, we carried out a case study on the high magnetic coercivity of Co nanowires exceeding the magnetocrystalline anisotropy field as predicted by Brown's theorem. It is found that the aspect ratio and diameter of the nanocrystals have a strong effect on the coercivity. When the nanocrystals have an increased aspect ratio, the coercivity is significantly higher than the magnetocrystalline anisotropy field of a hcp Co crystal. Micromagnetic simulations give a coercivity aspect-ratio dependence that is well consistent with the experimental results. It is also revealed that a coercivity limit exists based on the geometrical structures of the nanocrystals that govern the demagnetizing process. The quantitative correlation obtained between the structure and coercivity enables material design of advanced permanent magnets in the future.