Hysteresis and energy barriers in soft magnets

Past research has often correlated a magnet's hysteresis width with its magnetocrystalline anisotropy constant ${\ensuremath{\kappa}}₁$. Based on this relationship, a common approach to reducing magnetic hysteresis has been to develop alloys with ${\ensuremath{\kappa}}₁$ as close to zero as possible. Contrary to this view, we show that magnets with large ${\ensuremath{\kappa}}₁$ can have small hysteresis for specific combinations of magnetoelastic constants and applied stresses. Using our recently developed and validated micromagnetics framework, we systematically investigate the interplay of material constants (${\ensuremath{\lambda}}₁₀₀,{c}₁₁,{c}₁₂$, ${\ensuremath{\kappa}}₁$), applied stresses (${\ensuremath{\sigma}}_{\mathrm{R}}$), and needle domains to collectively lower the energy barrier for magnetization reversal. This energy barrier approach enables us to quantify the effects of magnetoelastic interactions, in addition to magnetocrystalline anisotropy, on magnetic hysteresis. Our findings show a mathematical relationship: ${\ensuremath{\kappa}}₁=\ensuremath{\alpha}({c}₁₁\ensuremath{-}{c}₁₂){({\ensuremath{\lambda}}₁₀₀+\ensuremath{\beta}{\ensuremath{\sigma}}₁₁)}²$ for which magnetic hysteresis can be minimized for a uniaxial residual stress ${\ensuremath{\sigma}}_{\mathrm{R}}={\ensuremath{\sigma}}₁₁{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{e}}}₁\ensuremath{\bigotimes}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{e}}}₁$ (and for some constants $\ensuremath{\alpha},\ensuremath{\beta}$). These results serve as quantitative guidelines to design magnetic alloys with small hysteresis, and potentially guide the discovery of soft magnets outside the traditional ${\ensuremath{\kappa}}₁\ensuremath{\rightarrow}0$ region.