Itinerant magnetism in the half-metallic Heusler compound ${\mathrm{Co}}_2\mathrm{HfSn}$: Evidence from critical behavior combined with first-principles calculations

${\mathrm{Co}}_2$-based Heusler alloys show diverse magnetic behavior, extending from the localized Heisenberg to the delocalized itinerant, and thus cannot be generalized with one specific model. Therefore, we bring together experiments and first-principles calculations to reveal the origin of long-range ferromagnetism in the ${\mathrm{Co}}_2$-based Heusler alloy ${\mathrm{Co}}_2\mathrm{HfSn}$. The precise value of the critical exponents $\beta =0.471\left(4\right), \gamma =1.02\left(4\right)$, and $\delta =3.273\left(5\right)$ with the Curie temperature ($T{}_c=430$ K) were determined experimentally by means of different analytical methods such as modified Arrott plot analysis, the Kouvel-Fisher method, and critical isotherm analysis. The deduced critical exponents belong to the theoretical prediction of the three-dimensional mean-field model. The magnetic exchange distance is found to decay as $J\left(r\right)\sim r^{-4.52}$, indicating the long-range magnetic interaction in ${\mathrm{Co}}_2\mathrm{HfSn}$. Moreover, the magnetic entropy change $-\mathrm{\Delta }{S}_M$ features a maximum at $T_c$, i.e., $-\mathrm{\Delta }{S}_M^{max}\sim$ 29.87(6) J ${\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ at 5 T, whereas power law fitting of $-\mathrm{\Delta }{S}_M^{max}$ with $H$ gives $n^{\prime }=0.663\left(5\right)$, also confirming the mean-field-type magnetic interaction in ${\mathrm{Co}}_2\mathrm{HfSn}$. Meanwhile, in order to explore the origin of long-range ferromagnetic ordering in ${\mathrm{Co}}_2\mathrm{HfSn}$, spin-polarized density functional theory calculations were employed in this study. No stable antiferromagnetic solution with stable local magnetic moments at the Co or Hf site was observed in our fixed spin moment simulation, revealing that ${\mathrm{Co}}_2\mathrm{HfSn}$ is a highly itinerant magnet with a long-range ferromagnetic solution together with robust half-metallic conduction stable against thermal excitation higher than room temperature. Furthermore, our theoretical analysis shows that ${\mathrm{Co}}_2\mathrm{HfSn}$ belongs to an unconventional family of itinerant magnetism. Both the 100% spin-polarized metallicity and ferromagnetism arise from the Co $d$ orbitals, causing it to be unique compared to the conventional itinerant family of Ni, Co, or Fe, where magnetism and metallicity originate from different orbitals characters.

Figure 1

(a) Refined powder XRD pattern of ${\mathrm{Co}}_2\mathrm{HfSn}$ at room temperature. The solid black spheres represent the experimental data, and the solid red line shows the calculated result. The vertical purple lines show the Bragg positions, and the solid green line at the bottom corresponds to the difference between the experimental and fitted intensities.

Figure 2

(a) Temperature dependence of magnetization $M\left(T\right)$ on the left and $M^{-1}\left(T\right)$ on the right. The inset shows the field dependence of magnetization measured at 5 K. (b) The derivative magnetization $dM/dT$ vs $T$.

Figure 3

(a) The initial isothermal magnetization curves around $T{}_c$. (b) Arrott plot of $H/M$ vs $M^2$ around $T{}_c$.

Figure 4

The modified Arrott plot around $T_c$ for the optimum fitting with $\beta =0.471$ and $\gamma =1.02$.

Figure 5

(a) The spontaneous magnetization $M_s$ (left) and inverse initial susceptibility ${\chi }_0^{-1}$ (right) vs $T$ with the fitting solid curves. (b) KF plots for $M{}_s\left(T\right)$ (left) and ${\chi }_0^{-1}\left(T\right)$ (right; solid lines are fitted). (c) Scaling plots around $T{}_c$ using $\beta$ and $\gamma$ determined by the KF method. (d) The renormalized magnetization and field potted as $m^2$ vs $h/m$.

Figure 6

Field dependence of the magnetization isotherm at $T{}_c=430$ K for ${\mathrm{Co}}_2\mathrm{HfSn}$. The inset shows the same plot in ${\log }_{10}-{\log }_{10}$ scale, where the solid line is the linear fit following Eq. (3) that gives the critical exponent $\delta$.

Figure 7

(a) The magnetic entropy change $-\mathrm{\Delta }{S}_M$ obtained from magnetization at various magnetic fields. (b) Normalized $-\mathrm{\Delta }{S}_M$ as a function of the rescaled temperature $\theta$. The inset shows the magnetic field dependence of the maximum magnetic entropy change $-\mathrm{\Delta }{S}_M^{max}$ with power law fitting given by the solid red line.

Figure 8

These plots demonstrate our theoretical insights: Atomic resolved spin-polarized band structure for the (a) majority- and (c) minority-spin components. Atomic resolved and total spin-polarized density of states (DOS) in the (b) majority and (d) minority components. (e) The calculated energy difference between the spin-polarized and non-spin-polarized solutions under the constraint of fixing the total spin magnetic moment ranging from $0{\mu }_B$ to $6{\mu }_B$. The non-spin-polarized solution with vanishing local magnetic moments of Co under the constraint $M=0{\mu }_B$ can be noted for ${\mathrm{Co}}_2\mathrm{HfSn}$. In comparison, the ${\mathrm{Co}}_2\mathrm{MnSn}$ system is shown, for which the spin polarization under the constraint of $M=0{\mu }_B$ exists. (f) The integrated DOS for ${\mathrm{Co}}_2\mathrm{HfSn}$ where the system's magnetic moment can be understood from the difference of up- and down-spin electrons. (g) The spin moments' variations under the constraint of total magnetization ranging from $0{\mu }_B$ to $6{\mu }_B$ are shown for two Co atoms (in ${\mathrm{Co}}_2\mathrm{HfSn}$ and ${\mathrm{Co}}_2\mathrm{MnSn}$) and Hf in ${\mathrm{Co}}_2\mathrm{HfSn}$. Distribution of spin-charge density for the ${\mathrm{Co}}_2\mathrm{HfSn}$ system under the fixed total magnetic moment (h) $M=2{\mu }_B$ and (i) $M=0{\mu }_B$. The vanishing of the local spin density can be noticed in the $M=0{\mu }_B$ solution. (j) For comparison, the spin-charge density under the constraint of zero total magnetic moments $M=0{\mu }_B$ is also shown for the ${\mathrm{Co}}_2\mathrm{MnSn}$ system.