Analytical Maximum Energy Product (BH)max Model for Rare-Earth-Free Magnets: Core–Shell Nanostructure

This article presents an analytical model for the maximum energy product [ $(BH)_{\max }$ ] in core–shell structured magnetic exchange-coupled nanomagnets. The model was validated by comparing its results to the $(BH)_{\max }$ from core–shell magnets reported in the literature. This approach can serve as a universal model for designing core–shell magnets that achieve the desired $(BH)_{\max }$ . The $(BH)_{\max }$ was determined under two distinct nucleation field ( $H_{\mathrm {N}}$ ) conditions: $H_{\mathrm {N}} \le M_{\mathrm {r}}$ /2 and $H_{\mathrm {N}} \ge M_{\mathrm {r}}$ /2, where $M_{\mathrm {r}}$ is the remanent magnetization. In addition, two different values of magnetic hysteresis loop squareness (SQ $= M_{\mathrm {r}}$ / $M_{\mathrm {S}}$ ) were used: 1.0 and 0.7. The soft magnetic shell’s remanent magnetic flux density ( $B_{\mathrm {r}}$ ) ranged from 0.7 to 2.2 T, while the core diameter ( $D_{\mathrm {h}}$ ) varied between 50 and 250 nm in this $(BH)_{\max }$ model. The low-temperature phase (LTP) MnBi-core/soft-shell nanomagnet can achieve a $(BH)_{\max }$ of 40 MGOe at a $B_{\mathrm {r}}$ of 1.6 T, with a shell thickness ( $\delta _{\mathrm {S}}$ ) of 40 nm, a $D_{\mathrm {h}}$ of 250 nm, and a volume fraction of the hard-core ( $f_{\mathrm {h}}$ ) of 0.43. The $(BH)_{\max }$ of the hexaferrite (SrFe12O19)/soft-shell (1.9 T) nanomagnet can be improved from 5.8 (single hexaferrite phase) to 20 MGOe. This approach achieves the desired $(BH)_{\max }$ of the rare-earth(RE)-free permanent magnet, thereby tackling issues related to RE mineral security and unstable supply chains.