Evolution of the Curie temperature for a substituted Cantor alloy

Knowledge of the magnetic ordering temperature is one of the fundamental physical properties characterizing material behavior. Magnetism is not only related to the magnitude of magnetic moments within specific magnetic ordering, but it also influences other physical properties. In the present work, we determine magnetic ordering temperatures for high-entropy alloys (HEAs), with examples of the Al- and Mo-based Cantor alloy derivatives. Nowadays, HEAs represent a promising class of materials in the sense of mechanical engineering. Regarding the paramagnetic state of the well-known and well-explored Cantor alloy, we deal with its Mo and Al derivatives, which offer higher possible ordering temperatures. We discuss the differences between $p$-type and $d$-type substitution as well as their influence on the magnetic behavior. We determine the ordering temperatures based on the ab initio calculated magnetic exchange interactions in terms of the mean-field approximation, random-phase approximation, and Monte Carlo simulations. Based on the thorough description of magnetic pair exchange interactions, a particular composition's influence on the achieved critical temperatures is described.

Figure 1

Comparison of DOSs related to Al and Mo substitutions in a nonmagnetic $X\mathrm{MnFeCoNi}$ alloy.

Figure 2

Distribution of the magnetic pair exchange interactions $J_{ij}^{\alpha \beta }$ as a function of the distance and the type of the element. Only exchange interactions within the sublattice of the same element are depicted. The distance is expressed in units of the nearest-neighbor distance $a_{\mathrm{NN}}$. The FM phase and bcc crystal structure are used. The insets depict exchange interactions' oscillations for larger distances.

Figure 3

Dependence of the MFA critical temperature $T_{\mathrm{C}}^{\mathrm{MFA}}$ on the number of included atomic shells $N_{\mathrm{sh}}$. The radius of the related sphere of the interacting neighbors in units of nearest-neighbor distance $a_{\mathrm{NN}}$ is depicted as well.

Figure 4

Evaluation of the Monte Carlo critical temperature $T_{\mathrm{C}}^{\mathrm{MC}}$ based on the Binder cumulant curves. The temperature dependence of the magnetic susceptibility $\chi$ is depicted for comparison.

Figure 5

Damping of long-range RKKY-like oscillations in the radial distribution of pair exchange interactions $J_{ij}^{\alpha \beta }$. The FM phase and bcc crystal structure are used. The distance is expressed in units of the nearest-neighbor distance $a_{\mathrm{NN}}$.

Figure 6

Comparison of evaluated critical temperatures $T_{\mathrm{C}}$ as a function of the composition and employed approach, where $M$ denotes type of the substitution and the superscript stands for the substituted element. Regarding the Mo-based alloy, $N_{\mathrm{sh}}=23$, and in the case of the Al-based alloys, $N_{\mathrm{sh}}=25$. Solid points depict MC simulations, whereas open ones show the MFA-ALM approach. The dashed curve serves only as an eye guide.

Figure 7

Mo-based derivatives. Lattice Fourier transformation of the pair exchange interaction along the path in the Brillouin zone. The most substantial interactions within the ferromagnetic phase are depicted.

Figure 8

Al-based derivatives. Lattice Fourier transformation of the pair exchange interaction along the path in the Brillouin zone. The most substantial interactions within the ferromagnetic phase are depicted.

Figure 9

DOS related to the nonmagnetic CrMnFeAlNi alloy.