Magnetoelastic Interactions Reduce Hysteresis in Soft Magnets

The width of the magnetic hysteresis loop is often correlated with the material's magnetocrystalline anisotropy constant $\kappa_1$. Traditionally, a common approach to reduce the hysteresis width has been to develop alloys with $\kappa_1$ as close to zero as possible. However, contrary to this widely accepted view, we present evidence that magnetoelastic interactions governed by magnetostriction constants, elastic stiffness, and applied stresses play an important role in reducing magnetic hysteresis width, despite large $\kappa_1$ values. We use a nonlinear micromagnetics framework to systematically investigate the interplay between material constants $\lambda_{100}$, $c_{11}$, $c_{12}$, $\kappa_1$, applied or residual stresses $\sigma_{\mathrm{R}}$, and needle domains to collectively lower the energy barrier for magnetization reversal. A distinguishing feature of our work is that we correlate the energy barrier governing the growth of needle domains with the width of the hysteresis loop. This energy barrier approach enables us to capture the nuanced interplay between anisotropy constant, magnetostriction, and applied stresses, and their combined influence on magnetic hysteresis. We propose a mathematical relationship on the coercivity map: $\kappa_1 = \alpha(c_{11}-c_{12})(\lambda_{100}+\beta\sigma_{11})^2$ for which magnetic hysteresis can be minimized for a uniaxial residual stress $\sigma_\mathrm{R} = \sigma_{11}\hat{\mathbf{e}}_1\otimes\hat{\mathbf{e}}_1$ (and for some constants $\alpha$, $\beta$). These results serve as quantitative guidelines to design magnetic alloys with small hysteresis, and potentially guide the discovery of a new generation of soft magnets located beyond the $\kappa_1 \to 0$ region.