Relationship between magnetic ordering and gigantic magnetocaloric effect in ${\mathrm{HoB}}_2$ studied by neutron diffraction experiment

Recently, ${\mathrm{HoB}}_2$ was discovered to undergo a gigantic magnetic entropy change; the largest $|\mathrm{\Delta }{S}_M|=40.1\mathrm{J}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ ($0.35\mathrm{J}{\mathrm{cm}}^{-3}{\mathrm{K}}^{-1}$) in magnetocaloric materials studied for hydrogen liquefaction [P. Baptista de Castro et al., NPG Asia Mater. 12, 35 (2020)]. In this study, in order to investigate the origin of the gigantic $|\mathrm{\Delta }{S}_M|$ in ${\mathrm{HoB}}_2$, we studied the magnetic orderings by neutron diffraction with an enriched powder sample of ${\mathrm{Ho}}^{11}{\mathrm{B}}_2$. Our experimental results showed the onset of ferromagnetic long-range order below $T_C=15\mathrm{K}$. For the second phase transition at $T_2=11\mathrm{K}$, the spin reorientation occurs, and the temperature dependence of the total magnetic moment of ${\mathrm{Ho}}^{3+}$ shows a kink anomaly, which is typically seen in compounds with quadrupolar ordering. In addition, the temperature dependence of the ferromagnetic short-range order above $T_C$ is strongly related to that of the magnetic entropy change in ${\mathrm{HoB}}_2$. These results infer that the gigantic $|\mathrm{\Delta }{S}_M|$ in ${\mathrm{HoB}}_2$ is caused by (i) the spin reorientation transition, which is likely associated with quadrupolar ordering, giving the additional contribution to $|\mathrm{\Delta }{S}_M|$ at the lower phase transition temperature, $T_2=11\mathrm{K}$, and (ii) the large FM fluctuation above $T_C$ with origins in the strong suppression of the FM long-range order.

Figure 1

Illustration of crystal structure of ${\mathrm{HoB}}_2$. Ho is placed at Wyckoff position 1a (0,0,0), and B is at 2d $(1/3,2/3,1/2)$ in space group $P6/mmm$. The lattice constants are $a=b=3.2849\left(8\right)\AA$, $c=3.8183\left(16\right)\AA$ at room temperature.

Figure 2

Temperature dependence of the powder neutron diffraction pattern in ${\mathrm{HoB}}_2$.

Figure 3

Temperature dependence of the (a) integrated intensities of the magnetic 001 and 100 reflections and (b) tilting angle of the refined spin direction from the $a$ axis toward the $c$ axis, (c) total magnetic moment, and (d) specific heat in ${\mathrm{HoB}}_2$.

Figure 4

Results of Rietveld refinements of the data measured at (a) $T=12\mathrm{K}$ and (b) $T=2.0\mathrm{K}$ for ${\mathrm{HoB}}_2$. The insets in (a) and (b) show the refined parameters and the reliable factor for the magnetic Bragg reflections. The upper and lower vertical bars denote the positions of nuclear and magnetic reflections, respectively.

Figure 5

Possible magnetic structure represented by the combination of the two magnetic irreducible representations, ${\mathrm{\Gamma }}_2^{+}\oplus {\mathrm{\Gamma }}_6^{+}$. The spin components are allowed by magnetic moments along the hexagonal [101] in (a), $\left[1\overline{1}1\right]$ in (b), and general direction in (c). For the first two cases in (a) and (b), the resultant magnetic space group is $C{2}^{\prime }/m^{\prime }$. For the case in (c), it becomes $P\overline{1}$. The brown plane denotes a plane on which spins are lying.

Figure 6

(a) Temperature dependence of diffuse scattering intensity integrated for the scattering angle between ${5.3}^{\circ }$ and ${11}^{\circ }$ in the neutron diffraction pattern of ${\mathrm{HoB}}_2$. The inset shows the neutron diffraction patterns at typical temperatures, 2.0, 12, 20, and 150 K. (b) Temperature dependence of the magnetic entropy changes between zero field and at 5 T in ${\mathrm{HoB}}_2$. The entropy was estimated from the magnetization curves plotted using measurements recorded in the present work. Dashed lines show $T_C$ and $T_r$ at which the maximum $|\mathrm{\Delta }{S}_M|$ becomes the half value above $T_C$.

Figure 7

Comparison between observed and calculated neutron intensities for the data at (a) 150 K and (b) 20 K. The data at $T=2\mathrm{K}$ is subtracted from those at $T=20\mathrm{K}$, apart from the Bragg reflection positions masked by shading. The calculated curves represent the paramagnetic (dashed line) and short-range order (solid line), which are represented by the first and second terms in Eq. (1), respectively. For $T=150\mathrm{K}$, the refined magnetic moment is ${\mu }_{\mathrm{eff}}=7.3\left(1\right){\mu }_{\mathrm{B}}$. For $T=20\text{K}$, $J_1=-0.22\left(1\right)k_{\mathrm{B}}$, $J_2=-0.16\left(4\right)k_{\mathrm{B}}$, $J_3=-0.28\left(1\right)k_{\mathrm{B}}$, and ${\mu }_{\mathrm{eff}}=6.75{\mu }_{\mathrm{B}}$.

Figure 8

Ionic radius (trivalent) dependence of the Curie-Weiss temperature ${\mathrm{\Theta }}_{CW}$, Curie temperature $T_C$, and de Gennes factor ${(g-1)}^{2}J(J+1)$ ($g$ is the Landé $g$ factor, and $J$ is the total angular momentum quantum number). The de Gennes factor was normalized to the value of ${\mathrm{TbB}}_2$.

Figure 9

Temperature dependence of specific heat at typical magnetic fields of 0, 2, and 5 T. The data were taken from Ref. [5].