Schottky anomaly and short-range antiferromagnetic correlations in filled skutterudites Pr$_{1-x}$Eu$_x$Pt$_4$Ge$_{12}$

TL;DR

This study investigates how Eu substitution affects magnetic correlations and Schottky anomalies in filled skutterudites, revealing the interplay between external magnetic fields, antiferromagnetic ordering, and short-range magnetic correlations.

Contribution

It provides new insights into the magnetic behavior and Schottky anomalies in Eu-doped PrPt4Ge12, highlighting the effects of external magnetic fields on magnetic ordering and correlations.

Findings

Schottky anomalies persist across all doping levels.

External magnetic fields shift Schottky peaks differently depending on Eu content.

Short-range AFM correlations are induced by Eu magnetic moments.

Abstract

By performing a series of thermodynamic measurements in an applied magnetic field $H_{\textrm{ext}}$, we investigated the effects of Eu substitution on the Pr sites in filled skutterudite compound Pr$_{1-x}$Eu$_x$Pt$_4$Ge$_{12}$ ($ 0 \leq x \leq 1$). A heat capacity Schottky anomaly is present over the whole doping range. For the samples with $x > 0.5$, these Schottky anomaly peaks shift to lower temperature with increasing $H_{\textrm{ext}}$. We argue that this behavior reflects the antiferromagnetic (AFM) ordering of the Eu moments, as the AFM transition is suppressed by $H_{\textrm{ext}}$. The Schottky peaks in the samples with $x \leq 0.5$ shift to higher temperatures with increasing magnetic field, signaling the presence of an internal magnetic field due to short-range AFM correlations induced by magnetic moments of neighboring Eu sites. In low $H_{\textrm{ext}}$, the Schottky gaps show a non-linear relationship with $H_{\textrm{ext}}$ as the magnetic moments become weakly magnetized. In high $H_{\textrm{ext}}$, the magnetic moments of Eu sites become completely aligned with $H_{\textrm{ext}}$. Thus, increasing $H_{\textrm{ext}}$ does not further increase the magnetization, hence the Schottky gaps increase linearly with $H_{\textrm{ext}}$

Equations

$$\begin{array}{rl}{C}_{\text{Sch}}& =r(x)\frac{R}{T^2}\left[\frac{f_2(T)}{f_0(T)}-\frac{f_1^2(T)}{f_0^2(T)}\right],\\ f_m(T)& =\sum _{j=0}^7{\Delta }_j^m\exp \left(-\frac{{\Delta }_j}{k_{B}T}\right),m=0,1,2\end{array}$$

$$Z={\int }_0^{2\pi }d\varphi {\int }_0^{\pi }d\theta \sin \theta e^{\beta m{H}_{\text{ext}}\cos \theta }=4\pi \frac{\sinh \beta m{H}_{\text{ext}}}{\beta m{H}_{\text{ext}}}.$$

$$⟨{\text{H}}_{\text{eff}}^2⟩=$$

$$[(H_{\text{ext}}-H_{\text{int}}(x)\cos \theta {)}^2+(H_{\text{int}}(x)\sin \theta {)}^2]$$

$$=$$

$$\Delta =(g{\mu }_B/k_B)\sqrt{A(x)H_{\text{ext}}^2+H_{\text{int}}^2(x)},$$

$$\Delta =(g{\mu }_B/k_B)\left(H_{\text{ext}}-H_{\text{int}}(x)\right).$$

Full text

I Introduction

Filled skutterudite compounds with the chemical formula $M$Pt4Ge12 (where $M$ denotes alkaline earth, lanthanide, or actinide) belong to the family of heavy-fermion superconductors. These materials have attained a renewed experimental and theoretical interest recently due to their potential to host unconventional electronic many-body ground states, such as various multipolar orders which are driven by magnetic degrees of freedom at low temperatures. A competition and/or possible coexistence between unconventional superconductivity and various - possibly nontrivial - magnetic ground states provides a major motivation for further exploring the physics of these compounds.

PrPt4Ge12 is a heavy-fermion superconductor which has a moderately high - relative to other heavy-fermion superconductors - critical temperature of $T_{c}\simeq{7.9}$ K and smaller $\gamma\sim 60$ mJ/(mol$\cdot$K2), corresponding to a medium enhancement of the conduction electron effective mass Maisuradze et al. (2009). It is important to keep in mind that PrPt4Ge12 is also multiband superconductor with two Fermi surfaces having one nodal and one nodeless gap indicating the unconventional nature of superconductivity (SC). Specifically, its multiband nature may be the main reason why superconductivity remains fairly robust with respect to introducing disorder by chemical substitutions. Indeed, the substitution of Ce to the filler cites of Pr leads to the overall suppression of $T_{c}$ with the nodal gap being gradually suppressed as well Singh et al. (2016). Generally, the unconventional SC in Pr-based scutterudites is likely driven by system’s proximity to magnetic ground states. In fact, several Eu-based compounds have been found to exhibit antiferromagnetism (AFM) Ren et al. (2008); Midya et al. (2016).

Compounds containing Eu2+ (or Gd3+) ions have large total angular momentum $J=S=7/2$ per Eu (Gd) and hence are expected to host large magnetic moments. Consequently, these compounds typically exhibit magnetic ordering. Some of the filled skutterudite systems with Eu2+ electronic configuration such as EuFe4Sb12 and EuFe4As12, indeed, show magnetic ordering at temperatures $T_{C}\sim$ 88 K and $\sim$ 152 K, respectively, where the higher $T_{C}$ has been attributed to the existence of an additional, albeit small, magnetic moment ($\sim$ 0.21$\mu_{B}$ for the Fe-Sb cage) on the Fe ion Bauer et al. (2004); Sekine et al. (2009). Furthermore, EuPt4Ge12 has AFM with the Néel critical temperature $T_{N}\sim$ 1.78 K, an effective magnetic moment of $\mu_{\textrm{eff}}=7.4\mu_{B}$, and a Curie-Weiss temperature $\Theta_{CW}\sim-11$ K Nicklas et al. (2011); Grytsiv et al. (2008). The lower magnetic ordering temperature for EuPt4Ge12 may be due to strong fluctuations and the absence of a magnetic moment on Pt in the Pt-Ge cage, causing a decrease in the exchange coupling determined by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the Eu2+ localized magnetic moments and the conduction-electron spins Grytsiv et al. (2008); Krishnamurthy et al. (2007).

Heat capacity measurements of Eu- or Gd-based samples have been found to exhibit upturn in $C_{e}/T$ upon lowering the temperature. This upturn happens to be due to Schottky anomaly resulting from the splitting of the ground state octet of Eu/Gd by the internal molecular and externally applied magnetic field Naugle et al. (2006); Leigh and Hampshire (2003); Köhler et al. (2006). Further studies have shown upturns in $C_{e}/T$ vs $T$ data containing Pr and have been attributed to crystalline electric field (CEF) splitting of the ground state of Pr ions Frederick et al. (2005); Takeda and Ishikawa (2000). Thus, at least in principle, it should be possible to probe the signatures of magnetic correlations by analyzing the Schottky contribution to the heat capacity, since the CEF structure of the Pr ion is well known.

We performed low-temperature specific heat measurements on samples of Pr1-xEuxPt4Ge12 in magnetic field. Our detailed and systematic analysis indicates that the upturns in the heat capacity are caused by the splitting of the octet degenerate states of Eu2+ due to the internal magnetic field that co-exists with superconductivity Adhikari (2018). We have also systematically analyzed the effect of magnetic field on the temperature where the Schottky peaks appear. Our analysis shows that the temperature where the Schottky peaks appears increases linearly with increasing magnetic field in the high Eu content and high magnetic field region. For the low Eu-substituted samples and in the low magnetic field region, the temperature of the Schottky peak show a non-linear magnetic field dependence since more and more Eu magnetic moments become weakly magnetized with increasing applied field.

II Experimental Details

Pr1-xEuxPt4Ge12 samples were synthesized by arc-melting and annealing from high purity Pr ignots, Eu ignots, Pt sponge, and Ge pieces according to the procedure described in detail in Ref. Jeon et al. (2016). The crystal structure was determined through X-ray powder diffraction using a Bruker D8 Discover X-ray diffractometer with Cu-Kα radiation, and the XRD patterns were analysed through Rietveld refinement Toby (2001). A detailed sample characterization of the series of these polycrystalline samples through X-ray diffraction, electrical resistivity, and magnetic susceptibility, as described elsewhere Jeon et al. (2017), shows the purity of the samples used in this study.

The two surfaces of each sample were polished with sand paper to improve the contact between the sample and the specific heat platform. We performed a series of specific heat measurements on the polycrystalline samples of Pr1-xEuxPt4Ge12 sith $x=$ 0, 0.05, 0.10, 0.15, 0.20, 0.30, 0.38, 0.50, 0.70, 0.80, 0.90, and 1.00 in applied magnetic field $H_{\textrm{ext}}$ up to 14 T over the temperature $T$ range $0.50$ K $\leq T\leq 10$ K. The specific heat measurements were performed via a standard thermal relaxation technique using the He-3 option of a Quantum Design Physical Property Measurement System (PPMS).

III Results and Discussion

The measured specific heat in the normal state $C(T)$ = $\gamma_{n}T$+$BT^{3}$ is the sum of electronic $C_{\rm e}\equiv\gamma_{n}T$ ($\gamma_{n}$ is the normal-state Sommerfeld coefficient) and phonon $C_{\textrm{ph}}\equiv BT^{3}$ contributions; hence, we did a least-square fit of $C/T$ vs $T^{2}$ data in the normal state ($T_{c}<T\leq 15$ K) for different Eu concentrations, as described and shown in Fig. 1 of ref. Adhikari (2018), in order to determine $\gamma_{n}$ and $B$. We then subtracted the phonon contribution to the specific heat for all the measured samples of Pr1-xEuxPt4Ge12. We subtracted the same phonon contribution from the specific heat data measured in an applied magnetic $H_{\textrm{ext}}$, thus assuming that the phonon contribution is field independent.

Figures 1(a), 1(b), and 1(c) display the $C-C_{ph}$ vs $T$ data for the $x=0.05$, 0.38, and 0.80, respectively, samples measured in different applied magnetic fields $H_{\textrm{ext}}$. These figures reveal upturns present in the specific heat data below 2 K. For the samples with $x\leq 0.15$, the specific heat shows an upturn without reaching a maximum [Fig. 1(a)], whereas the samples with larger $x$ values show a clear peak present in the low-temperature region [Figs. 1(b) and 1(c)].

With increasing applied magnetic field $H_{\textrm{ext}}$, the samples with $0.05\leq x\leq 0.15$ begin to show a distinct peak [Fig. 1(a)], while the samples with $x\leq 0.5$ show that the peak becomes broader, shifts to higher temperatures, and decreases in amplitude [Fig. 1(b)]. The Schottky peaks for the $0.7\leq x\leq 1$ samples reveal a long-range AFM transition at $T_{N}$, see inset Fig. 1(c), that, as expected, shift to lower temperatures with increasing $H_{\textrm{ext}}$. Nevertheless, once the AFM transition is suppressed below the lowest measured temperature of 0.5 K, the Schottky peak shifts to higher temperatures with further increase $H$ [Fig. 1(c)], as also observed in the $x<0.5$ samples [Figs. 1(a) and 1(b)].

We note that a superconducting transition at a temperature $T_{c}$ is clearly seen in the heat capacity of the samples with $0\leq x\leq 0.3$ Adhikari (2018). The superconducting jump becomes broader and $T_{c}$ shifts to lower temperatures, as expected, with increasing $H_{\textrm{ext}}$ and increasing Eu concentration Adhikari (2018). The samples with $0.3<x\leq 0.5$ show a superconducting transition only in resistivity measurements Jeon et al. (2017), while the samples with $x>0.5$ do not display a superconducting transition for temperatures down to $T=0.5$ K.

We have attributed the Schottky anomaly present in the heat capacity data of the $x\leq 0.5$ samples in $H_{\textrm{ext}}$ to the splitting of the degenerate ground state of Eu2+ into eight equally-spaced energy levels, as shown schematically in Fig. 2, by the internal field $H_{\textrm{int}}$ due to the net magnetic moment m present as a result of short-range antiferromagnetic correlations between the nearest-neighbor Eu ions Adhikari (2018). These short-range antiferromagnetic correlations co-exist with superconductivity in these lower Eu substituted samples (see Fig. 7 of Ref. Adhikari (2018)). We note that one may expect short-range antiferromagnetic correlations between the Eu ions to be present in the alloys Pr1-xEuxPt4Ge12 since EuPt4Ge12 orders antiferromagnetically.

The Schottky heat capacity anomaly for an eight-energy-levels system with the degeneracy fully lifted by magnetic field is given by Gopal (1966)

[TABLE]

where $r(x)$ is one of the fitting parameters and it turns out that it represents the concentration of Eu ions Adhikari (2018), $R=8.31$ J/mol-K is the universal gas constant, $\Delta_{j}=j\cdot\Delta$ is the energy gap between the lowest energy level ($j=0$) and the $j^{th}$ energy level, $\Delta\equiv g\mu_{B}H_{\textrm{eff}}/k_{B}$, $g=2$ ($L=0$), $\mu_{B}$ is the Bohr magnetron, and $H_{\textrm{eff}}$ is the effective magnetic field. Hence, the value of the Schottky gap $\Delta$ fully depends on $H_{\textrm{eff}}$. In the presence of an applied magnetic field $H_{\textrm{ext}}$, both $H_{\textrm{int}}$ and $H_{\textrm{ext}}$ split the degenerate ${}^{8}S_{7/2}$ ground state of Eu into 8 equally-spaced energy levels. Hence, the effective field has two contributions: $\textbf{H}_{\text{eff}}\equiv\textbf{H}_{\textrm{int}}+\textbf{H}_{\textrm{ext}}$.

As just mentioned, the lower Eu-doped samples ($0.05\leq x\leq 0.5$) have two contributions to the heat capacity in the low field and low temperature ($0.5\leq T<T_{c}$) region: Schottky and superconducting contributions. Specifically, for the samples with Eu concentrations in the range $0.05\leq x\leq 0.15$, the superconducting contribution is best described by $(C-C_{ph})\propto T^{2}$, i.e., line nodes in the superconducting gap, while for $0.2\leq x\leq 0.5$ by $(C-C_{ph})\propto\exp^{-\delta/T}$, i.e., by an isotropic gap; this is perfectly expected since the nodal gap is quickly suppressed by scattering on lattice imperfections. In the high field region where $T_{c}<0.5$ K or for the $x\geq 0.7$ samples, the total specific heat is the sum of Schottky and normal-state electronic ($\gamma_{n}T$) contributions. Least-square fits of the data are shown by the solid curves in Figs. 1(a) through 1(c). The fitted curves are in excellent agreement with the measured specific heat data.

The values of the Schottky gaps obtained from these fits are plotted as a function of applied magnetic field $H_{\textrm{ext}}$ for various Eu substitutions in Figs. 3 and  4. We note that the Schottky gap $\Delta$ for a system with eight energy levels and the degeneracy fully lifted is also given by the temperature corresponding to the Schottky peak, i.e., $\Delta=T_{\textrm{peak}}$. The values of $\Delta$ obtained from fits (as discussed above) and from $T_{\textrm{peak}}$ for the samples for which the Schottky peak is clearly observed are in excellent agreement. Hence, some $\Delta$ values shown in these figures for some $x$ and/or $H_{\textrm{ext}}$ values were also extracted from $T_{\textrm{peak}}$. Notice that the Schottky gaps are non-linear in $H_{\textrm{ext}}$ for small values of $H_{\textrm{ext}}$ and increase linearly with $H_{\textrm{ext}}$ with a doping-independent slope for large values of $H_{\textrm{ext}}$.

We gained an understanding of the physics that governs the behavior of these materials in the presence of an applied magnetic field as follows. Taking $\textbf{H}_{\textrm{ext}}=H_{\textrm{ext}}\hat{z}$, and witting $\textbf{H}_{\textrm{int}}=H_{x}\hat{x}+H_{y}\hat{x}+H_{z}\hat{z}$, gives $\langle\textbf{H}^{2}_{\text{eff}}\rangle=\langle H^{2}_{x}+H_{y}^{2}+(H_{\textrm{ext}}+H_{z})^{2}\rangle$. To obtain a function that describes the effective field at intermediate field values, we treat the magnetic degrees of freedom as classical, freely-rotating magnetic moments m. The energy of the magnetic moment is $E=-mH_{\textrm{ext}}\cos\theta$, where $\theta$ is the angle between $\textbf{H}_{\textrm{ext}}$ and m with $\textbf{H}_{int}=-\textbf{m}$. The partition function is then:

[TABLE]

The expectation value of $\textbf{H}^{2}_{\text{eff}}$ is

[TABLE]

where the Langevin function is $L(\beta mH_{\textrm{ext}})=\coth(\beta mH_{\textrm{ext}})-1/(\beta mH_{\textrm{ext}})$ and $H_{\textrm{int}}(x)$ is proportional to the concentration of Eu atoms. In the limit $H_{\textrm{ext}}\to 0$, we find $\langle H_{\text{eff}}^{2}\rangle\approx H_{\textrm{int}}^{2}(x)$, so for $\beta mH_{\textrm{ext}}\ll 1$ it follows that $L(\beta mH_{\textrm{ext}})\approx(\beta mH_{\textrm{ext}})/3$ and $\langle H_{\text{eff}}^{2}\rangle=H_{\textrm{ext}}^{2}-2/3H^{2}_{\textrm{ext}}H_{\textrm{int}}(x)\beta m+H_{\textrm{int}}^{2}(x)$. $\Delta\equiv g\mu_{B}H_{\textrm{eff}}/k_{B}$ becomes

[TABLE]

where $A(x)=1-2\beta mH_{\textrm{int}}(x)/3$. In the limit $\beta mH_{\textrm{ext}}\gg 1$, $L(\beta mH_{\textrm{ext}})=1$ and $\langle H_{\text{eff}}^{2}\rangle=\left(H_{\textrm{ext}}-H_{\textrm{int}}(x)\right)^{2}$. Hence,

[TABLE]

We fitted the $\Delta(x)$ vs $H_{\textrm{ext}}$ data of Figs.  3 and  4 with Eqs. 4 and 5, respectively, with $A(x)$ and $H_{\textrm{int}}(x)$ as fitting parameters. Notice the excellent fits obtained especially for the non-linear $\Delta(x)$ vs $H_{\textrm{ext}}$ data of Fig.  3. When the externally applied magnetic field is large enough, it aligns the net magnetic moment in its direction, hence, $\Delta$ becomes proportional to the resultant magnetic field. In this high field region where the superconducting contribution of the $0.05\leq x\leq 0.5$ samples or the antiferromagnetic contribution of the $0.7\leq x\leq 0.9$ samples is suppressed below 0.5 K, $\Delta$ increases linearly with $H_{\textrm{ext}}$ with a doping-independent slope [see Fig. 4]. Least-square linear fits of $\Delta$ vs $H_{\textrm{ext}}$ in this higher field region give $g\mu_{B}/k_{B}\approx 1.4$, which corresponds to $g=2$ as expected for the ground state ${}^{8}S_{7/2}$ of Eu.

IV Conclusions

We analyzed the low-temperature specific heat data in order to investigate the interaction between magnetism and superconductivity and reveal the effect of magnetic field on the nature of the superconducting and antiferromagnetic orders in the Pr1-xEuxPt4Ge12 filled skutterudite system. Our data show the presence of short range AFM correlations between Eu ions under the superconducting dome for $x\leq 0.50$. These short range AFM correlations produce a local internal magnetic field, which lifts the eight-fold degeneracy of the Eu ground state and gives rise to a Schottky peak in heat capacity that shifts to higher temperature with increasing $H_{\textrm{ext}}$. The low temperature heat capacity data can be fitted with the sum of a superconducting/normal state term and a Schottky term. In low values of $H_{\textrm{ext}}$, the internal and external magnetic fields are comparable, hence the Schottky gap shows a non-linear dependence on $H_{\textrm{ext}}$. In high magnetic field, the applied magnetic field aligns the internal moment in its direction, hence the Schottky anomaly increases linearly with $H_{\textrm{ext}}$ with a doping-independent slope, as normally expected.

V Acknowledgments

This work was supported by the National Science Foundation grants DMR-1505826 and DMR-1506547 at KSU and by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Grant No. DE-FG02-04ER46105 at UCSD. The work of M. D. and P. S. was financially supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences under Award No. DE-SC0016481.