Higher-order magnetic anisotropy in soft-hard magnetic materials

We have computationally studied the properties of higher-order magnetic anisotropy constants in an $L{1}_0/A1\text{−}\mathrm{FePt}$ core-shell system which is characterized by a strong second-order two-ion Fe-Pt anisotropy component. We show that the core-shell structure induces an unexpected fourth-order anisotropy constant $K_2$, the magnitude of which varies nonmonotonically with the core-size ratio $R$ reaching a peak at $R\approx 0.50$. Furthermore, we find that $K_2$ scales with the normalized magnetization by ${(M/M_s)}^{2.2}$ at temperatures below the Curie temperature, a remarkable deviation from the established Callen-Callen theory which instead predicts a scaling exponent of 10. We construct an analytic model which demonstrates $K_2$ arises from the canting of the core and shell magnetization, and successfully reproduces and justifies the scaling exponent obtained from numerical simulation.

Figure 1

The crystal structures of FePt: (a) the disordered $A1$-fcc bulk-alloy ${\mathrm{Fe}}_{0.5}{\mathrm{Pt}}_{0.5}$ and (b) ordered $L{1}_0$-fct. Dark and light spheres indicate Fe and Pt atoms, respectively. Cross-sectional views of the simulated $L{1}_0/A1$ core-shell grains with core-size ratios (c) $R=0.80$, (d) $R=0.60$, and (e) $R=0.40$ showing different volume fractions of the ordered and disordered phases. The core ordered region in each cross section is indicated by the shaded area.

Figure 2

(a) Fitting to the torque $\tau$ for a core size $R=0.50$ at 10 K displays a clear deviation from simulation data (solid symbols) if including only a second-order anisotropy term (dashed line), but matches better if adding a fourth-order anisotropy term (solid line). (b) The canting of the core and shell magnetization with analytic fits.

Figure 3

The dependence of the anisotropy magnitude ratio $K_2/K_1$ on the core size $R$. Symbols are the numerically determined values from atomistic simulations and the solid line is the predicted analytic value

Figure 4

Variation of the scaling exponent $\beta$ of the fourth-order anisotropy constant $K_2$ as a function of the core size $R$ with an inset showing an example of $K_2$ scaling to the normalized magnetization $M/M_s$ for $R=0.70$ with a scaling exponent $\beta =2.326\pm 0.013$.