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.

中文翻译：

---

纳米铁磁体的矫顽力极限

一个世纪以来，关于铁磁体有多“硬”（强制）一直是个谜。七十年前，W. Brown 给出了他著名的定理，将铁磁体的矫顽力与其磁晶各向异性场相关联。然而，实验的矫顽力值远低于定理给出的计算水平，这被称为布朗矫顽力悖论。这个悖论被认为与实验中磁体的复杂微观结构有关，因为矫顽力是一种外在特性，对试样中的任何缺陷都很敏感。迄今为止，矫顽力无法通过定量建模来预测和计算。在这项研究中，我们对 Co 纳米线的高矫顽力进行了案例研究，该高矫顽力超过了布朗定理所预测的磁晶各向异性场。发现纳米晶体的纵横比和直径对矫顽力有很强的影响。当纳米晶体具有增加的纵横比时，矫顽力显着高于hcp Co晶体的磁晶各向异性场。微磁模拟给出了与实验结果非常一致的矫顽力纵横比依赖性。还揭示了基于控制退磁过程的纳米晶体的几何结构存在矫顽力极限。在结构和矫顽力之间获得的定量相关性使未来能够设计先进的永磁体。矫顽力显着高于hcp Co晶体的磁晶各向异性场。微磁模拟给出了与实验结果非常一致的矫顽力纵横比依赖性。还揭示了基于控制退磁过程的纳米晶体的几何结构存在矫顽力极限。在结构和矫顽力之间获得的定量相关性使未来能够设计先进的永磁体。矫顽力显着高于hcp Co晶体的磁晶各向异性场。微磁模拟给出了与实验结果非常一致的矫顽力纵横比依赖性。还揭示了基于控制退磁过程的纳米晶体的几何结构存在矫顽力极限。在结构和矫顽力之间获得的定量相关性使未来能够设计先进的永磁体。