Thermodynamic and transport studies of the ferromagnetic filled skutterudite compound ${\text{PrFe}}_4{\text{As}}_{12}$

A variety of thermodynamic and transport measurements were made on high-quality single crystals of the Pr-based filled skutterudite compound ${\text{PrFe}}_4{\text{As}}_{12}$. Abrupt features in magnetization, ac susceptibility, specific heat, resistivity, thermoelectric power, and ultrasonic velocity reveal the onset of long range ferromagnetic order below ${\Theta }_C=18\text{K}$. The low-temperature magnetic susceptibility is characterized by a Curie–Weiss law with an effective moment of $3.52{\mu }_B/\text{f}\text{.u}\text{.}$ and a saturation magnetization of $2.3{\mu }_B/\text{f}\text{.u}\text{.}$, which is consistent with a magnetic ${\Gamma }_5$ triplet ground state. A gaplike reduction of the large electronic specific heat coefficient of $340\text{mJ}/\text{mol}{\text{K}}^2$ and several other features point to a strongly correlated electron behavior that is likely coupled to a change in magnetic and/or structural order near $T^{\ast }\simeq 12\text{K}$. Furthermore, this complex magnetic state is found to be strongly field dependent, as evidenced by a change in the easy axis at low fields and an additional contribution to thermal conductivity appearing only at high fields.

Figure 1

(Color online) Inverse dc magnetic susceptibility ${\chi }_{\text{dc}}^{-1}$ vs temperature $T$ data for ${\text{PrFe}}_4{\text{As}}_{12}$ from 1.8 to 300 K in a 0.5 T magnetic field that is applied along the [100] crystallographic direction. Curie–Weiss fits from 70 to 300 K (solid line in the main panel) and from 20 to 40 K (dashed line in the inset) result in quantities as displayed.

Figure 2

Magnetization $M$ vs magnetic field $H$ for 2 K (squares) and 15 K (circles) isotherms in fields that are applied along the [111] (solid symbols) and [100] (open symbols) crystallographic directions. The 2 K isotherms show a change in the easy axis at $H=0.8\text{T}$ from the [100] to the [111] directions. This change is not observed in the 15 K data, which indicates that the easy axis does not change at higher temperatures.

Figure 3

(a) Arrott plots of $M^2$ vs $H/M$ isotherms for ${\text{PrFe}}_4{\text{As}}_{12}$ between 17 and 19 K. The solid lines are linear fits to the isotherms and are used to extrapolate ${\chi }_0^{-1}=H/M$ to $M=0$ and $M$ to $H=0$. (b) Temperature dependence of the $M^2$ intercepts (open circles) from the Arrott plots in (a) are shown with linear extrapolation (dashed-dotted line) of the data above 10 K that are used to estimate $M\left(0\right)=2.7{\mu }_B/\text{f}\text{.u}\text{.}$ Temperature dependence of the inverse susceptibility (solid circles) is also shown with a linear Curie–Weiss fit (dashed line) of the data from 18 to 25 K.

Figure 4

Temperature $T$ dependence of the real ${\chi }_{\text{ac}}^{\prime }$ and imaginary ${\chi }_{\text{ac}}^{″}$ parts of the ac magnetic susceptibility of ${\text{PrFe}}_4{\text{As}}_{12}$ for three driving frequencies of 10, 100, and 500 Hz with a driving field of 0.1 mT that is applied along the [100] direction.

Figure 5

(Color online) (a) Zero-field-cooled and field-cooled magnetization data for ${\text{PrFe}}_4{\text{As}}_{12}$ for constant magnetic fields between 5 and 5 T that are applied along the [100] crystallographic direction. (b) Temperature derivatives of the field-cooled magnetization. Local minima in $dM/dT$ are identified with characteristic temperatures $T^{\ast }$ and ${\Theta }_C$ in the low-field limit.

Figure 6

Temperature dependence of the total specific heat of ${\text{PrFe}}_4{\text{As}}_{12}$, with Debye function fit (solid line) of Eq. (2) plus a constant electronic term. Inset: electronic plus magnetic portions of the specific heat that are obtained from subtracting lattice and nuclear contributions [see text (circles)]. The solid line is the associated entropy $s_{e+m}$.

Figure 7

(Color online) Temperature dependence of electrical resistivity $\rho$ for ${\text{PrFe}}_4{\text{As}}_{12}$ for magnetic fields of up to 8 T that are applied along the [100] direction. Inset: (zero-field) temperature dependence of resistivity under applied pressures from 1 to 23.5 kbar. (Note that the values for 6.7 and 11.7 kbar overlap and are difficult to distinguish.)

Figure 8

(Color online) Thermal conductivity of ${\text{PrFe}}_4{\text{As}}_{12}$ in zero and applied fields with $H\parallel \left[100\right]$. The top-left panel presents the ratio of heat and charge conductivities, which is plotted as the Lorenz number $L=\frac{\kappa \rho }{T}$ normalized to the WF law expectation $L_0=2.44\times {10}^{-8}\Omega \text{W}/{\text{K}}^2$. The top-right panel plots the thermal conductivity remaining after subtraction of the electronic contribution estimated using the WF law (see text).

Figure 9

Thermoelectric power of ${\text{PrFe}}_4{\text{As}}_{12}$ with heat currents directed along the principal [100] axis. The inset shows the low-temperature behavior, which highlights the coincidence of the ferromagnetic transition ${\Theta }_C$ and a kink in $S\left(T\right)$. The dotted line is a linear extrapolation of data below 2 K.

Figure 10

(Color online) (a) Thermoelectric power of ${\text{PrFe}}_4{\text{As}}_{12}$ taken in various fields $\left(H\parallel \left[100\right]\right)$, with the arrows indicating the position of the kink in $S\left(T\right)$ associated with ${\Theta }_C$ (see text). (b) Dimensionless thermoelectric figure of merit $ZT$ that is obtained from the data in (a) (see text).

Figure 11

(Color online) Relative change in the elastic constants $\Delta C_{44}/C_{44}$ and $\Delta C_{11}/C_{11}$ as a function of temperature for ${\text{PrFe}}_4{\text{As}}_{12}$. The inset shows the detailed behavior below 30 K. The vertical dashed lines indicate the characteristic temperatures $T^{\prime }$, $T^{\ast }$, and ${\Theta }_C$ (see text).

Figure 12

(Color online) A comparison of low-temperature specific heat, electrical resistivity, magnetization, ac susceptibility, thermopower, and ultrasound data that are used to correlate features that are observed at the ferromagnetic ordering temperature ${\Theta }_C$ (dotted line) and the characteristic temperature $T^{\ast }$ (dashed line). (a) Temperature derivative of resistivity and electronic plus magnetic contributions to specific heat, which shows a striking resemblance of the temperature dependence of each throughout the entire range. (b) Low-field (5 mT) dc magnetization and the imaginary part of the ac susceptibility, highlighting the coincidence of both the irreversibility temperature in $M\left(T\right)$ and the peak in ${\chi }_{\text{ac}}^{″}\left(T\right)$ with $T^{\ast }$. (c) Zero-field thermoelectric power and transverse sound velocity data.