Crystal structure and thermodynamic properties of the non-centrosymmetric PrRu$_4$Sn$_6$ caged compound

TL;DR

This study investigates the crystal structure and thermodynamic properties of PrRu4Sn6, revealing its non-centrosymmetric tetragonal structure, magnetic behavior, and phonon interactions, with implications for understanding caged compounds.

Contribution

The paper provides the first detailed analysis of PrRu4Sn6's structure, magnetic properties, and phonon modes, highlighting its non-centrosymmetric nature and thermal behavior.

Findings

No long-range magnetic order down to 2 K

Effective magnetic moment of 3.34 μB/Pr

Presence of optical-phonon mode and glass-like thermal conductivity

Abstract

PrRu$_4$Sn$_6$ is a tetragonal, non-centrosymmetric structure compound. It is isostructural to the extensively studied Kondo insulator CeRu$_4$Sn$_6$ which crystallizes in the YRu$_4$Sn$_6$-type structure with space group $I$\={4}2$m$. In this structure, the Pr atom fills the void formed by the octahedral Ru$_4$Sn$_6$ units which results in a tetragonal body-centred arrangement. Here we present reports on the physical and magnetic properties of PrRu$_4$Sn$_6$. The temperature dependences of specific heat, $C_p(T)$, electrical resistivity, $\rho(T)$, and magnetic susceptibility, $\chi(T)$, reveal the absence of a long-range magnetic ordering down to 2 K. $\chi(T)$ follows a Curie-Weiss behaviour above 100 K with an effective magnetic moment, $\mu_\mathrm{eff}$ = 3.34 $\mu_B$/Pr and paramagnetic Weiss temperature, $\theta_p$ = $-$19.47 K indicating a dominant antiferromagnetic interaction. The magnetization at 2 K is quasi-linear in nature and attains a value of 0.86 $\mu_B$/Pr at 7 T which is well reduced compared to the calculated value of 3.32 $\mu_B$/Pr expected for a free Pr$^{3+}$ ion. This is attributed to possible magneto-crystalline anisotropy in the system. $C_p(T)$ indicates the presence of a optical-phonon mode which is supported by a glass-like thermal conductivity above $\sim$45 K. This observation is associated with caged structured compounds where the low-frequency optical-phonon mode of the guest atom interacts with the host lattice, resulting in the scattering of heat-carrying quasiparticles.

Tables

Table 1. Table 1: The atomic positions and lattice parameters of PrRu 4 Sn 6 obtained from a Rietveld refinement of the XRD pattern.

Site notation	Atom	Wyckoff site	Point symmetry	$x$	$y$	$z$
Sn(1)	Sn	8$i$	$m$	0.17635	0.17635	0.28771
Ru	Ru	8$i$	$m$	0.32788	0.32788	0.08126
Sn(2)	Sn	4$c$	222	0	1/2	0
Pr	Pr	2$a$	$-$42$m$	0	0	0
$a$ (Å)	$c$ (Å)	$V$ (Å3)	formula units ($Z$)	$R_{wp}$ (%)	$R_p$ (%)	${\chi }^2$
6.870(3)	9.761(2)	461.5(9)	2	8.588	7.295	5.210

Equations

$$C_p(T)=mD\left(\frac{{\theta }_D}{T}\right)+nE\left(\frac{{\theta }_E}{T}\right),$$

$$D\left(\frac{{\theta }_D}{T}\right)=9R{\left(\frac{T}{{\theta }_D}\right)}^3{\int }_0^{{\theta }_D/T}\frac{x^4\exp (x)}{(\exp (x)-1{)}^2}dx,$$

$$E\left(\frac{{\theta }_E}{T}\right)=3R{\left(\frac{{\theta }_E}{T}\right)}^2\cdot \frac{\exp \left({\theta }_E/T\right)}{(\exp \left({\theta }_E/T\right)-1{)}^2},$$

$$\rho (T)={\rho }_0+\frac{4K}{{\Theta }_R}{\left(\frac{T}{{\Theta }_R}\right)}^5{\int }_0^{{\Theta }_R/T}\frac{x^{5}dx}{(e^x-1)(1-e^{-x})},$$

Full text

1 Introduction

The $R$Ru4Sn6 ($R$ = Y, La-Nd, Sm, Gd-Ho) series are intermetallic compounds which crystallize in the tetragonal YRu4Sn6-type structure with a non-centrosymmetric space group $I$4̄2$m$ (No. 121) [1]. The structure was first reported by Venturini et al [2]. The crystal structure is made up of an octahedral Ru4Sn6 unit enclosing the guest $R$ atom. Crystal structures of this nature have generated much interest lately especially in the search for new superconductors [3, 4]. Also, the non-centrosymmetric nature of the space group is characteristic of certain superconductors where the mixing of the spin-singlet and spin-triplet Cooper pairing channels have been found to give rise to a two-component order parameter [5, 6, 7]. CeRu4Sn6 is a Kondo insulator, and it is the most extensively studied member of the series [8, 9, 10, 11]. Other studies by Koch and Strydom reveal a magnetic ordering for the isostructural compounds of $R$Ru4Sn6, with $R$ = Sm, Gd and Dy compounds at low temperatures while those of Nd, Tb and Ho compounds are paramagnetic down to 2 K [12].

As part of our search for Pr-based systems exhibiting novel ground states, we have synthesized a polycrystalline sample of PrRu4Sn6 and investigated its physical and magnetic properties. It is noted that the existence of PrRu4Sn6 was first reported by Zumdick and Pöttgen [1] but no physical or magnetic properties have been reported thereafter. The Pr atom in this structure has a tetragonal site symmetry of $D_{2d}$ similar to those of the Pr${}_{3}T_{4}X_{13}$ compounds, resulting in the crystal electric field splitting of the $J$ = 4 multiplet into seven levels consisting of five singlets and two non-Kramers doublets.

2 Experimental methods

A polycrystalline sample of PrRu4Sn6 was prepared by arc melting stoichiometric amounts of high-purity elements (wt.% $\geq$ 99.9) on a water-cooled Cu plate under a purified static argon atmosphere in an Edmund Buehler arc furnace. The weight loss after melting was $\sim$ 0.05%. The arc-melted pellet was wrapped in Ta foil, placed in an evacuated quartz tube and annealed at 900∘C for 21 days. A powder X-ray diffraction (XRD) pattern was recorded on a pulverized sample using a Rigaku diffractometer employing Cu-K$\alpha$ radiation. The obtained powder XRD pattern was refined using the Rietveld method [13] employing the FullProf suite of programs [14]. We found that the compound was phase-pure within the limits of the resolution of the instrument. In Table 1, the atomic positions and lattice parameters obtained from the refinement are presented and are comparable with a previous report [1]. The refined XRD pattern and the crystal structure are shown in Fig. 1.

Magnetic properties were measured using the Magnetic Property Measurement System (Quantum Design Inc., San Diego) between 2 K and 300 K with an external magnetic field up to 7 T. The four-probe DC electrical resistivity, specific heat and thermal transport measurements between 2 K and 300 K were measured using the Physical Property Measurement System also from Quantum Design.

3 Magnetic properties

The temperature dependence of magnetic susceptibility, $\chi(T)$, of PrRu4Sn6 in an external field of 0.1 T and in the temperature range of 2 K to 300 K is presented in Fig. 2. $\chi(T)$ shows a paramagnetic behaviour down to low temperatures with no indication of a long-range magnetic ordering observed. The white-solid line is a Curie-Weiss fit based on the expression: $\chi(T)=N_{A}\mu_{\mathrm{eff}}^{2}/(3k_{B}(T-\theta_{p}))$ for data above 100 K with values of effective magnetic moment, $\mu_{\mathrm{eff}}=3.34~{}\mu_{\mathrm{B}}$/Pr and Weiss temperature, $\theta_{p}=-19.47~{}$K. The observed $\mu_{\mathrm{eff}}$ is close to the calculated value of 3.58 $\mu_{\mathrm{B}}$/Pr expected for a free Pr3+ ion. At low temperatures, a Van-Vleck paramagnetic behaviour in $\chi(T)$ suggests a nonmagnetic ground state in PrRu4Sn6. The isothermal magnetization at 2 K is presented in the inset (b) of Fig. 2. The magnetization follows a quasi-linear behaviour up to 7 T and attains a value of 0.86 $\mu_{\mathrm{B}}$/Pr at 7 T which is well reduced compared to the saturation moment of 3.32 $\mu_{\mathrm{B}}$/Pr expected for a free Pr3+ ion implying a possible magneto-crystalline anisotropy in the compound.

4 Specific heat

The temperature dependence of specific heat, $C_{p}(T)$, of PrRu4Sn6 studied between 2 K and 300 K is presented in Fig. 3. Inset (a) of Fig. 3 shows a plot of $C_{p}/T^{3}$ against $T$. Such a plot is important in determining the possible presence of low-frequency Einstein modes in $C_{p}(T)$ through the occurrence of a local maximum in $C_{p}/T^{3}$. A local minimum is observed in the plot as indicated by the arrow at $T_{max}$ = 6 K which confirms the presence of low-frequency Einstein modes in PrRu4Sn6. $T_{\mathrm{max}}$ is the temperature below which the Einstein modes are frozen out. By using a model incorporating both the Debye and Einstein terms, the experimental specific heat is fitted as shown by the red line in Fig. 3. The Debye-Einstein model is given by:

[TABLE]

[TABLE]

[TABLE]

where $\theta_{D}$ and $\theta_{E}$ are the Debye and Einstein temperatures with values of 241.73(9) K and 32.431(3) K, respectively. It is observed that $T_{max}$ $\simeq$ 0.2$\theta_{E}$ which is in agreement with the observation in Ce3Rh4Sn13 [15]. In Inset (b), a plot of $C_{p}/T$ against $T^{2}$ is shown together with a least-square fit (red line) based on the expression: $C_{p}/T=\gamma+\beta T^{2}$ and $\beta=12\pi^{4}nR/(5\theta^{3}_{D}$), where $n$ and R are the number of atoms per formula unit and universal gas constant, respectively, $\gamma$ is the Sommerfeld coefficient and $\theta_{D}$ is the Debye temperature. Values obtained from the fit are: $\gamma=38.60~{}$mJ/(K2 mol) and $\theta_{D}$ = 154.50 K. The $\gamma$ observed for PrRu4Sn6 is about 10 times the values found in ordinary metals.

5 Transport properties

To further understand the physical properties of PrRu4Sn6, a thermal transport measurement was carried out between 2 K and 300 K. The temperature dependences of thermoelectric power, $S(T)$, and thermal conductivity, $\kappa(T)$, were measured simultaneously on a bar-shaped sample. As shown in Fig. 4 (a), $S(T)$ is positive throughout the temperature range investigated and attains a value of 18.81 $\mu$V/K at room temperature. The red and black-dashed lines suggest two areas of linear-in-$T$ behaviour on either side of $\sim$135 K. At 2 K, $S(T)$ has a value of $\sim$1 $\mu$V/K indicating a significant drop in the carrier concentration between room temperature and 2 K. The change in slope of $S(T)$ at about 145 K is consistent with the anomaly observed in $C_{p}(T)$ around the same temperature. The origin of such an observation is not immediately clear and further measurements are needed to resolve the physics at play. A plot of $S(T)/T$ is shown in the inset of Fig. 4 (a). For $T\leq 100$ K the slope of $S(T)/T$ is  $\sim$ 0.7 $\mu$ V/K2 which is slightly above those of ordinary metals. The general feature of $S(T)$ suggests a hole-type charge carriers near the Fermi level.

The total thermal conductivity, $\kappa_{T}(T)$, of PrRu4Sn6 is presented in Fig. 4 (b) on a $\log$-$\log$ axes. $\kappa_{T}(T)$ is nearly temperature independent from room temperature down to about 45 K (as shown by the black-dashed line) which is characteristic of a glassy behaviour in thermal conductivity. The observation of a glass-like thermal conductivity in a crystalline compound is often associated with caged systems. The low-frequency optical-phonon mode of the guest atom scatters heat-carrying quasiparticles thus leading to a reduction in lattice thermal conductivity. Using the Wiedemann-Franz relation [16] given as: $\kappa=L_{0}T/\rho(T)$, where $L_{0}$ is the Lorentz number given by: $L_{0}$ = $\pi^{2}k_{B}^{2}/3e^{2}$ = $2.45\times 10^{-8}~{}$W$\Omega$/K2, the electronic contribution to the thermal conductivity, $\kappa_{E}(T)$ is extracted and it is also presented in Fig. 4 (b). Also shown in the plot is $\kappa_{Ph}(T)$ obtained by subtracting $\kappa_{E}(T)$ from $\kappa_{T}(T)$. Below about 10 K, $\kappa_{T}(T)$ and $\kappa_{Ph}(T)$ show power-law behaviour of $T^{1.5}$ while $\kappa_{E}(T)$ is linear-in-$T$ as indicated by the green, brown and blue-dashed lines. This indicates a good metallic behaviour. $\kappa_{Ph}(T)$ $>$ $\kappa_{E}(T)$ in the whole temperature range studied revealed that the heat transport is not charge-carrier dominated.

The temperature dependence of electrical resistivity, $\rho(T)$, of PrRu4Sn6 is presented in the inset of Fig. 4 (b) between 2 K and 300 K. $\rho(T)$ follows a typical metallic behaviour down to low temperature with residual resistivity ratio $\approx$ 5 which indicates a good crystalline quality. No signature of long-range magnetic or any type of ordering is observed in the temperature range studied in support of the observations in $\chi(T)$ and $C_{p}(T)$. To further understand the electrical transport properties of PrRu4Sn6, the Bloch-Grüneisen (BG) expression [17] was fitted to the data in the whole temperature range (shown as a red line). The BG expression is given as:

[TABLE]

where $\rho_{0}$ is the residual resistivity due to defect scattering in the crystal lattice, $K$ is the electron-phonon coupling constant and $\Theta_{R}$ is the resistivity Debye temperature. Values of $\rho_{0}$ = 102.8(2) $\mu\Omega$ cm, $K$ = 90.19(1) $\mu\Omega$ cm K, and $\Theta_{R}=39.20(1)$ K are obtained from the least-square fit.This observation here further supports a metallic behaviour of PrRu4Sn6.

6 Conclusion

We have studied the physical and magnetic properties of the non-centrosymmetric PrRu4Sn6 compound. A paramagnetic ground state is inferred from the magnetic susceptibility results down to 2 K. The presence of low-frequency Einstein modes are observed in $C_{p}(T)$. This observation is further supported by the glass-like thermal conductivity for temperatures above 45 K. $S(T)$ undergoes a change in slope at $\sim$ 145 K, which is around the same temperature an anomaly in $C_{p}(T)$ is observed. Further measurements are expected to help clarify the origin of the observations in $C_{p}(T)$ and $S(T)$.

Acknowledgement

MOO acknowledges the UJ-URC bursary for doctoral studies in the Faculty of Science. AMS thanks the SA-NRF (93549) and UJ-URC for financial support.