Role of Debye temperature in achieving large adiabatic temperature changes at cryogenic temperatures: A case study on ${\mathrm{Pr}}_2\mathrm{In}$

The excellent magnetic entropy change ($\mathrm{\Delta }{S}_T$) in the temperature range of 20 $\sim$ 77 K due to the first-order phase transition makes ${\mathrm{Pr}}_2\mathrm{In}$ an intriguing candidate for magnetocaloric hydrogen liquefaction. As an equally important magnetocaloric parameter, the adiabatic temperature change ($\mathrm{\Delta }{T}_{ad}$) of ${\mathrm{Pr}}_2\mathrm{In}$ associated with the first-order phase transition has not yet been reported. In this work, the $\mathrm{\Delta }{T}_{ad}$ of ${\mathrm{Pr}}_2\mathrm{In}$ is obtained from heat capacity measurements: 2 K in fields of 2 T and 4.3 K in fields of 5 T. While demonstrating a $\mathrm{\Delta }{T}_{ad}$ that is not as impressive as its remarkable $\mathrm{\Delta }{S}_T, {\mathrm{Pr}}_2\mathrm{In}$ exhibits a low Debye temperature ($T_D$) of around 110 K. Based on these two observations, an approach that combines the mean-field and Debye models is developed to study the correlation between $\mathrm{\Delta }{T}_{ad}$, one of the most important magnetocaloric parameters, and $T_D$, one important property of a material. The role of $T_D$ in achieving large $\mathrm{\Delta }{T}_{ad}$ is revealed: materials with higher $T_D$ tend to exhibit larger $\mathrm{\Delta }{T}_{ad}$, particularly in the cryogenic temperature range. This discovery explains the absence of an outstanding $\mathrm{\Delta }{T}_{ad}$ in ${\mathrm{Pr}}_2\mathrm{In}$ and can serve as a tool for designing or searching for materials with both a large $\mathrm{\Delta }{S}_T$ and a $\mathrm{\Delta }{T}_{ad}$.

Figure 1

(a) Magnetization of ${\mathrm{Pr}}_2\mathrm{In}$ as a function of temperature. (b) $\mathrm{\Delta }{S}_T$ of ${\mathrm{Pr}}_2\mathrm{In}$ from magnetization measurements. The inset shows the exponent $n$ ($|\mathrm{\Delta }{S}_T|\propto H^n$) vs $T$. (c) $\mathrm{\Delta }{S}_T$ of ${\mathrm{Pr}}_2\mathrm{In}$ from $MT$ measurements and heat capacity measurements. (d) $\mathrm{\Delta }{T}_{ad}$ from heat capacity measurements. (e), (f) $\mathrm{\Delta }{S}_T$ and $\mathrm{\Delta }{T}_{ad}$ for light and heavy rare-earth-based ${\mathrm{R}}_2\mathrm{In}$ [18, 19, 22, 28, 36, 37, 38], ${\mathrm{RAl}}_2$ (Pr, Nd, Gd, Tb, Dy, Ho, Er) [1, 2, 16] in magnetic fields of 5 T. The shadows mark the range of 77 $\sim$ 20 K.

Figure 2

(a) Heat capacity of ${\mathrm{Pr}}_2\mathrm{In}$ as a function of temperature in magnetic fields of 0, 1, 2, 5, and 10 T. (b) Volumetric lattice heat capacity from Debye model with $T_D$ varying from 110 to 410 K with a step of 50 K.

Figure 3

(a) $\mathrm{\Delta }{S}_T$ calculated from the mean-field approach. (b) $\mathrm{\Delta }{T}_{ad}$ calculated from the mean-field theory with $T_C$ and $T_D$ varying. The inset compares the $\mathrm{\Delta }{T}_{ad}$ for $T_D=110\mathrm{K}$ and 360 K with both $T_C=56.5\mathrm{K}$. (c) Lattice entropies for different $T_D$. (d) Illustration of how the slope of the entropy curve influences $\mathrm{\Delta }{T}_{ad}$.