⇦Chinese Physics Letters, 2019, Vol. 36, No. 10, Article code 107402 Broken Time-Reversal Symmetry in Superconducting Partially Filled Skutterudite Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ * Jia-Wei Zang (臧佳伟)1, Jian Zhang (张建)1,2, Zi-Hao Zhu (朱子浩)1, Zhao-Feng Ding (丁兆峰)1, Kevin Huang1, Xiao-Ran Peng (彭小冉)1, Adrian D. Hillier3, Lei Shu (殳蕾)1,4** Affiliations 1State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433 2Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA 3ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Chilton, Didcot, Oxon OX11 0QX, United Kingdom 4Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093 Received 13 August 2019, online 21 September 2019 *Supported by the National Key Research and Development Program of China under Grant Nos 2017YFA0303104 and 2016YFA0300503, the National Natural Science Foundation of China under Grant No 11774061, and the Chinese Government Scholarship of China Scholarship Council.
**Corresponding author. Email: leishu@fudan.edu.cn Citation Text: Zang J W, Zhang J, Zhu Z H, Ding Z F and Huang K et al 2019 Chin. Phys. Lett. 36 107402    Abstract Time reversal symmetry (TRS) is a key symmetry for classification of unconventional superconductors, and the violation of TRS often results in a wealth of novel properties. Here we report the synthesis and superconducting properties of the partially filled skutterudite Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$. The results from x-ray diffraction and magnetization measurements show that the [Pt$_{4}$Ge$_{12}$] cage-forming structure survives and bulk superconductivity is preserved below the superconducting transition temperature $T_{\rm c}=7.80$ K. The temperature dependence of both the upper critical field and the electronic specific heat can be described in terms of a two-gap model, providing strong evidence of multi-band superconductivity. TRS breaking is observed using zero field muon-spin relaxation experiments, and the magnitude of the spontaneous field is nearly half of that in PrPt$_{4}$Ge$_{12}$. DOI:10.1088/0256-307X/36/10/107402 PACS:74.25.Bt, 76.75.+i, 81.30.Dz © 2019 Chinese Physics Society Article Text One of the key challenges in the study of unconventional superconductivity is to discover more materials with extra broken symmetries in addition to gauge symmetry breaking.[1] Broken time reversal symmetry (TRS) often leads to novel properties in superconductors and the possible existence of multiple superconducting phases.[2] For example, the existence of a second superconducting phase was found in the TRS breaking superconductors UPt$_{3}$ and U$_{1-x}$Th$_{x}$Be$_{13}$.[3–5] The filled skutterudite compounds $RT_{4}X_{12}$ ($R$=rare-earth or alkaline-earth metals, $T$=Fe, Ru, Os, and $X$=P, As, Sb),[6] including the first discovered Pr-based heavy fermion superconductor PrOs$_{4}$Sb$_{12}$ and the iso-structural PrPt$_{4}$Ge$_{12}$,[7,8] provide a remarkable playground for studying broken TRS superconductivity. Moreover, the 4$f$ electron-filled skutterudite family displays an astonishing diversity of physical properties, including unconventional superconductivity, quadrupolar ordering, heavy fermion, and non-Fermi liquid behaviors.[9–11] PrPt$_{4}$Ge$_{12}$ has a much higher superconducting transition temperature ($T_{\rm c}=7.9$ K) than PrOs$_{4}$Sb$_{12}$ ($T_{\rm c}=1.8$ K),[9] but shows no heavy-fermion behavior.[8] Specific-heat and transverse filled muon-spin relaxation ($\mu$SR) experiments suggested strongly coupled unconventional superconductivity with point-like nodes in the energy gap.[12] A spontaneous static local magnetic field $B_{\rm s}$ below $T_{\rm m} \simeq 6.8$ K ($T_{\rm m} < T_{\rm c}$) was observed by performing a zero-field $\mu$SR experiment, signaling a TRS breaking superconducting state.[8] Remarkably, due to the observation of the point-like nodes and broken TRS, PrOs$_{4}$Sb$_{12}$ and PrPt$_{4}$Ge$_{12}$ were proposed to be the candidates to host 3D gapless Majorana fermions.[13] However, a $^{73}$Ge-NQR study revealed that PrPt$_{4}$Ge$_{12}$ is a weakly coupled BCS superconductor,[14] while analysis on the critical fields[15] and photoemission spectroscopy[16] provided evidence of multi-band superconductivity. Thus the superconducting gap structure in PrPt$_{4}$Ge$_{12}$ remains controversial. The behavior that TRS breaks at temperature $T_{\rm m}$, which is below $T_{\rm c}$, is also puzzling. It is crucial to shed new light on the superconducting order parameter of PrPt$_{4}$Ge$_{12}$ and clarify the origin of the broken TRS state. It is confirmed that in ternary skutterudite compounds the polyanionic [$T_{4}X_{12}$] host structure remains stable despite marked deviations from the required 72 electrons count per formula unit, providing a great deal of freedom to explore skutterudite structures composed with partially filled host elements.[17,18] Following this idea, we report a study of an imperfectly filled skutterudite Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ to investigate (i) whether the cage-forming structure and superconductivity survive with partially filled $^{141}$Pr nuclei, (ii) the effect of the insufficiently filled $^{141}$Pr nuclei on TRS breaking in PrPt$_{4}$Ge$_{12}$, and (iii) the gap symmetry of Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ and implications of the superconducting order parameter of PrPt$_{4}$Ge$_{12}$. Polycrystalline samples of Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ were synthesized by mixing Pr ingots (Alfa Aesar 99.9%), Pt sponge (99.9999%), and Ge pieces (Alfa Aesar 99.9999%) with the ratio of $y$:4:12 ($y=0.5$, 0.6, 0.7, 0.8, and 1).[19] The samples were annealed at 800$^{\circ}\!$C for 14 days in a sealed quartz tube (containing 200 Torr Ar at room temperature). XRD measurements were performed by a Bruker D8 x-ray diffractometer using a Cu $K_\alpha$ source. Rietveld refinements were conducted on powder XRD patterns using GSAS and EXPGUI.[20,21] Magnetization measurements were performed by a Quantum Design superconducting quantum interference device. Heat capacity measurements were performed on a Quantum Design physical properties measurement system, which employs a standard thermal relaxation technique. ZF-$\mu$SR experiments were carried out at the ISIS Neutron and Muon Facility, Rutherford Appleton Laboratory, Chilton, UK.
cpl-36-10-107402-fig1.png
Fig. 1. A representative powder XRD pattern of the grown polycrystalline sample. Orange, green and red tick marks below the pattern indicate the positions of expected Bragg peaks for the refined ordered crystal structures of Ge, PtGe$_{2}$, and Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$, respectively. Inset: the lattice parameter $a$ of Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ derived from the fitting of the PrPt$_{4}$Ge$_{12}$ structure. The abscissa is the nominal Pr concentration.
XRD phase analysis for the obtained powder patterns reveals the presence of three phases in the samples: Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$, Ge and PtGe$_{2}$. A representative XRD pattern (for the nominal Pr concentration $y=0.5$ sample) is displayed in Fig. 1, with the weight ratios of Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$, Ge and PtGe$_{2}$ equalling to 37.0%, 24.3%, and 38.7%, respectively. The patterns and their refinement with the three phases agree well with each other for all the samples (fits not shown). The inset of Fig. 1 indicates that the lattice parameter $a$ derived from the fitting of the PrPt$_4$Ge$_{12}$ structure has a roughly linear relation with the nominal Pr concentration $y$, strongly suggesting that the $^{141}$Pr nuclei are partially filled in [Pt$_{4}$Ge$_{12}$] cage. Since Ge and PtGe$_{2}$ are nonmagnetic and nonsuperconducting,[22] they have a slight influence on the study of superconducting transition and the analysis of ZF-$\mu$SR experiments. The $y=0.5$ sample most likely has the lowest Pr filling fraction, so we choose this sample to study the effect of the insufficiently filled $^{141}$Pr nuclei on superconductivity and TRS breaking in the following experiments. The magnetization measurement indicates that the dc-susceptibility displays a strong diamagnetic signal due to the superconducting transition provided by Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ in the sample at $T_{\rm c}=7.80$ K, where zero-field-cooled (ZFC) and field-cooled (FC) data deviate from each other, as shown in Fig. 2(a). The value of $T_{\rm c}$=7.80 K derived from the midpoint of the transition at zero field, corresponding well to the result from the magnetization measurements, is nearly the same as that of PrPt$_{4}$Ge$_{12}$. The temperature dependence of the specific heat coefficients $C_{\rm p}/T$ of the sample measured at different magnetic fields are displayed in Fig. 2(b), showing the suppression of $T_{\rm c}$ by magnetic fields.
cpl-36-10-107402-fig2.png
Fig. 2. Magnetic susceptibility and specific heat data of the partially filled skutterudite Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$. The molar mass was calculated using the mass ratio of Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$, Ge and PtGe$_{2}$. (a) Magnetic susceptibility was measured in an applied field of $\mu_{0}H=1$ mT. FC and ZFC stand for the field-cooled and zero-field-cooled measurements, respectively. (b) Specific heat data of the sample displayed as $C_{\rm p}/T$ versus $T^2$ in different magnetic fields.
cpl-36-10-107402-fig3.png
Fig. 3. Temperature dependence of the electronic specific heat coefficient of Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ at zero field. The curves represent the fits using six different gap models listed in Table 1. Inset: the critical fields $H_{\rm c2}$ derived from the midpoints of the jump in $C_{\rm p}/T$. The green curve is the fit of Eq. (2) to the data.
The normal-state $C_{\rm p}$ of the sample is well described by the expression $C_{\rm p}=\gamma T+\beta T^3$, where the first and second terms correspond to the electronic and phononic contributions, respectively. The fit in the range of 2.5–10 K gives the value of coefficients: $\gamma=3.35$ mJ$\cdot$mol$^{-1}\cdot$K$^{-2}$ and $\beta$=0.28 mJ$\cdot$mol$^{-1}\cdot$K$^{-4}$.
Table 1. Fitting models.
Model $f$ ${\it \Delta}/k_{\rm B}T_{\rm c}$ Correlation coefficient ${\it \Delta}_{C_{\rm p}}/\gamma T_{\rm c}$
A BCS 1 1.57 0.974 1.11
B Line-node 1 0.913 1.07
C Point-node 1 0.912 1.73
D BCS+line-node 0.60 2.20 0.998 1.30
E BCS+point-node 0.26 1.12 0.997 1.37
F Line+point-node 0.51 0.993 1.44
The electronic contribution $C_{\rm e}$ is calculated by subtracting the phonon part from the total heat capacity, and is plotted as $C_{\rm e}/\gamma T$ versus $T^2$, as displayed in Fig. 3. Ge is a semiconductor and its electronic contribution can be ignored. The electronic $C_{\rm p}$ of PtGe$_{2}$ is extremely small compared with Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ in the low temperature, thus it can be assumed that the electronic $C_{\rm e}$ with $T < 7.8$ K is provided by Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$. To investigate the gap symmetry of Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$, the data of $C_{\rm e}(T)$ are fitted using the six models, as listed in Table 1. A single gap function cannot fit the data, thus we use the phenomenological two-gap $\alpha$ model with a weighing factor $f$,[23] $$\begin{align} C_{\rm e}=fC_{\rm e,1}+(1-f)C_{\rm e,2},~~ \tag {1} \end{align} $$ where $C_{{\rm e},x}$ ($x=1$, 2) is the electronic contribution of one of the gaps. This model is often used to describe multi-band superconductors like MgB$_2$.[24] Here $C_{{\rm e},x} \propto e^{-{\it \Delta}/k_BT}$, $\propto T^2$, and $\propto T^3$ refer to the weakly coupled BCS gap, the point-node gap, and the line-node gap, respectively. The data are well fitted to a combination of any two gaps (D, E, F), suggesting that Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ is a multi-band superconductor. On the other hand, since the jump ${\it \Delta}_{C_{\rm p}}/\gamma T_{\rm c}$ at $T_{\rm c}$ is small, the most likely situation is E, a weakly coupled BCS gap (${\it \Delta}/k_{\rm B}T_{\rm c}\sim1.12$) combined with a strongly coupled gap with point nodes. However, an exact expression of the multi-band function cannot be determined due to the simple form of the two-gap $\alpha$ model, the one-gap symmetry can be ruled out because of the poor fit to $C_{\rm p}$. It is noted that the multi-band feature is also observed in iso-structural Pr-based superconductors, including PrOs$_{4}$Sb$_{12}$, PrRu$_{4}$As$_{12}$, and PrRu$_{4}$Sb$_{12}$.[25,26] Considering the similarity in the structures of PrPt$_{4}$Ge$_{12}$ and Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$, PrPt$_{4}$Ge$_{12}$ is presumably also a multi-band superconductor. This may provide an explanation for the discrepancy of different experimental results in PrPt$_{4}$Ge$_{12}$: point-like nodes in the gap by $\mu$SR and specific heat experiments while weakly coupled BCS superconductivity by the $^{73}$Ge-NQR study.[12,14] It is possible that the smaller gap is rather subtle and can be easily destroyed by external effects, e.g., a magnetic field, and then the other gap is predominant.[22] The inset of Fig. 3 shows the temperature dependence of the upper critical field $\mu_0H_{\rm c2}$, which is determined from the midpoints of the jumps in $C_{\rm p}/T$ in different fields. Data are best fitted by the Ginzburg–Landau (GL) two-band equation[27] $$\begin{align} \mu_0H_{\rm c2}(T)=\mu_0H_{\rm c2}{(0)}\frac{1-t^2}{1+t^2},~~ \tag {2} \end{align} $$ where $t=T/T_{\rm c}$, giving $\mu_0H_{\rm c2}(0)=2.91$ T. The coherence length $\xi_0$ estimated from $H_{\rm c2}(0)$ is about 11 nm. The reported value of the electron mean free path $l$ estimated from different samples is controversial, i.e., 10–14 nm for polycrystalline PrPt$_{4}$Ge$_{12}$, and 103 nm for single crystalline PrPt$_{4}$Ge$_{12}$.[15,22] To give a heuristic estimation, we use the Wethamer–Helfand–Hohenberg (WHH) formula in the clean limit,[28] $H_{\rm c2}(0)=-0.73T_{\rm c}(dH_{\rm c2}/dT)_{T=T_{\rm c}}$, and derive $\mu_0H_{\rm c2}(0)=2.14$ T.
cpl-36-10-107402-fig4.png
Fig. 4. (a) Zero-field $\mu$SR time spectra at 12 K (red circles) and 0.6 K (blue squares) for Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$. A background signal has been subtracted from the data. The corresponding solid lines are the fits according to Eq. (3), where $\lambda$ is fixed at 0 µs$^{-1}$. (b) Temperature dependence of the muon spin relaxation rates $\sigma$ (red circles) and ${\it \Lambda}$ (blue triangles), where $\sigma$ is derived from the fitting of Eq. (3) with ${\it \Lambda}$ fixed at 0.08 µs$^{-1}$. The red curve is the fit of Eq. (5). The blue line denotes the average of ${\it \Lambda}$ data from 0 to 13 K.
Further investigation of the superconducting order parameter in Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ was performed by ZF-$\mu$SR experiments. Figure 4(a) shows the time evolution of the decay positron count asymmetry, which is proportional to the muon spin polarization, at temperatures above and below $T_{\rm c}$ in Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$. A constant background contribution, which originates from muons that stop in the silver sampler holder and nonmagnetic impurities, has been subtracted from the data. The data was initially fitted with the Voigtian function[29] $$\begin{align} P_\mu (t)=G_{\rm z}^{\rm KT}(\sigma,\lambda,t)\exp(-{\it \Lambda}t).~~ \tag {3} \end{align} $$ The term $$\begin{align} G_{\rm z}^{\rm KT}(\sigma, \lambda,t)=\,&\frac{1}{3}+\frac{2}{3}(1-\sigma^2t^2-\lambda t)\\ &\cdot\exp\Big(-\frac{1}{2}\sigma^2t^2-{\rm \lambda}t\Big)~~ \tag {4} \end{align} $$ describes a muon spin relaxation by randomly oriented static muon local fields, with Cartesian components that vary according to a convolution of Gaussian and Lorentzian distributions with distribution widths $\delta B_{\rm G}$=$\sigma/\gamma_\mu$ and $\delta B_{\rm L}$=$\lambda /\gamma_\mu$, where $\sigma$ and $\lambda $ are relaxation rates and $\gamma_\mu=2\pi \times135.53$ MHz/T is the muon gyromagnetic ratio.[8] The value of $\lambda$ is extremely small and could be fixed at zero without changing the results. The value of $\lambda$ is also found to be extremely small thus fixed at zero for Pr$_{1-x}$Ce$_{x}$Pt$_{4}$Ge$_{12}$, Pr(Os,Ru)$_{4}$Sb$_{12}$, and (Pr,La)Os$_{4}$Sb$_{12}$.[30,31] The term $\exp(-{\it \Lambda} t)$ indicates an additional fluctuating relaxation process, where ${\it \Lambda}$ is the damping rate. The value of ${\it \Lambda}$ has a very weak temperature dependence and thus is fixed at its temperature average value of 0.08 µs$^{-1}$, as shown in Fig. 4(b). The small dynamic contribution ${\it \Lambda}$ is presumably due to additional spin-lattice relaxation mechanisms, probably mediated by the Korringa relation of the local fluctuations from the hyperfine-enhanced $^{141}$Pr nuclear dipolar field or the conduction-electron hyperfine interactions.[30] The value of $\sigma(T)$ is derived after ${\it \Lambda}$ is fixed at 0.08 µs$^{-1}$. An obvious increase of $\sigma(T)$ is observed below $T_{\rm c}$, which signalizes the onset of a spontaneous internal field $B_{\rm s}$, indicating that TRS is broken in partially filled Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$. The contributions to $\sigma(T)$ from nuclear dipolar and electrons in the superconducting state are uncorrelated in most skutterudite alloys[7,30] and added in quadrature, $$ \sigma(T)=\left\{\begin{matrix} \sigma_n, & T > T_{\rm c}\\ {[\sigma_n^2+{\sigma_{\rm e}(T)}^2]}^{1/2}, & T < T_{\rm c} \\ \end{matrix},\right.~~ \tag {5} $$ where $\sigma_n/\gamma_\mu$ is the width of the nuclear dipolar field distribution, and ${\sigma_{\rm e}}(T)/\gamma_\mu$ is the width of the spontaneous field distribution from broken TRS. Provided that ${\sigma_{\rm e}}$ follows the temperature dependence of the BCS order parameter, the data are fitted with an empirical expression,[30] $$\begin{align} {\sigma_{\rm e}}(T)={\sigma_{\rm e}(0)\tanh}\Big(b\sqrt{\frac{T_{\rm m}}{T}-1}\Big),~~ \tag {6} \end{align} $$ where $b$ is a dimensionless coefficient, $b=1.74$ for BCS order parameter in the weak-coupling limit,[32] and $T_{\rm m}$ is the transition temperature determined by ZF-$\mu$SR experiments. The fitting results are listed in Table 2.
Table 2. Parameters from the fit of Eq. (6).
$\sigma_{\rm e}(0)$ (µs$^{-1}$) ${\sigma_{\rm e}(0)}/\gamma_\mu$ (mT) $\sigma_n$ (µs$^{-1}$) $b$ $T_{\rm m}$ (K)
0.065(5) 0.077(4) 0.171(0) 1.79(3) 6.9(2)
The increase of $\sigma(T)$ was not observed in the spin-singlet superconductor LaPt$_{4}$Ge$_{12}$,[8] suggesting that the dominant origin of broken TRS is the existence of $^{141}$Pr nuclei. A previous study on Pr$_{1-x}$Ce$_x$Pt$_{4}$Ge$_{12}$ reported that ${\sigma_{\rm e}(0)}/\gamma_\mu$ has a linear relation with $^{141}$Pr nuclei concentration, indicating that Pr–Pr intersite interactions are responsible for broken TRS.[30] The value of ${\sigma_{\rm e}(0)}/\gamma_\mu$ in Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ is 0.077(4) mT, nearly half of that (0.141 mT) in PrPt$_{4}$Ge$_{12}$.[30] Thus we speculate that the broken TRS in PrPt$_{4}$Ge$_{12}$ originates from either the $^{141}$Pr nuclei or the Pr–Pr intersite interactions. Intriguingly, similar to the case of the parent compound PrPt$_{4}$Ge$_{12}$ and Pr-related alloys including Pr$_{1-x}$Ce$_x$Pt$_{4}$Ge$_{12}$,[8,30] $T_{\rm m}=6.9$ K is smaller than $T_{\rm c}=7.8$ K in Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$, indicating the possibility of a second phase, which could be either a distinct magnetic phase coexisting with superconductivity, or a subdominant superconducting phase with a magnetic (time-reversal violating) ground state, such as the case of U$_{1-x}$Th$_x$Be$_{13}$.[5] However, for Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$, the possibility of a distinct magnetic phase is excluded from the magnetization and the specific heat experiments because there is no anomaly around $T_{\rm m}$. ZF-$\mu$SR experiments also found no long-range order expected from a magnetic phase. TRS breaking usually occurs in the triplet state with a degenerate representation,[33] including UPt$_3$ and Sr$_2$RuO$_4$.[34,35] A study on the superfluid density of PrPt$_4$Ge$_{12}$ also suggests a chiral p-wave gap function,[12] while this speculation lacks further evidence in other experiments. On the other hand, spin-singlet pairing is also plausible. In this case, broken TRS will require additional phase transitions admixing other pairing channels with the superconducting phase, e.g., the $d+is$ state.[36] Thus the difference between $T_{\rm m}$ and $T_{\rm c}$, as well as the multi-band features in $H_{\rm c2}$ and $C_{\rm p}(T)$, could be attributed to the appearance of a subdominant fully gapped component in the superconducting order parameter. It then leads to the occurrence of an intrinsic consecutive phase.[36,37] Such a proposal is not against the absence of multiple superconducting $C_{\rm p}$ jumps. A possible spin-singlet scenario has been proposed for PrOs$_4$Sb$_{12}$ considering broken TRS in the second phase and point nodes in the gap.[38] This scenario is compatible with our results on Pr$_{1-\delta}$Pt$_4$Ge$_{12}$. In such a state, the supercurrents are induced around nonmagnetic imperfections and produce the TRS breaking magnetic field.[36] In summary, partially filled skutterudite Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ samples have been successfully synthesized, and a study has been carried out by magnetization, specific heat, and ZF-$\mu$SR experiments. The [Pt$_{4}$Ge$_{12}$] cage-forming structure survives and superconductivity is observed below $T_{\rm c}=7.80$ K. The temperature dependence of $H_{\rm c2}$ and the electronic specific heat are well described by the two-band model. Intriguingly, the onset of broken TRS is observed at $T_{\rm m} < T_{\rm c}$, possibly due to the appearance of a second phase, while no obvious specific jump is observed around $T_{\rm m}$. The value of ${\sigma_{\rm e}(0)}/\gamma_\mu$ in Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ is half of that in PrPt$_{4}$Ge$_{12}$, indicating that the $^{141}$Pr nuclei or Pr–Pr intersite interactions are responsible for broken TRS. These results suggest that Pr$_{1-\delta}$Pt$_{4}$Ge$_{12}$ holds a complicated superconductivity order parameter and is a unique playground for the study of unconventional superconductivity. We are grateful to T. P. Ying for numerous discussions. We thank the support team at the ISIS facility, and the ISIS Cryogenics Group for invaluable help during the experiments. References The Challenge of Unconventional SuperconductivitySuperconductivity in non-centrosymmetric materialsSpecific heat of UPt 3 : Evidence for unconventional superconductivityObservation of broken time-reversal symmetry in the heavy-fermion superconductor UPt3New phase diagram for (U,Th) Be 13 : A muon-spin-resonance and H C 1 studyLaFe 4 P 12 with filled CoAs 3 -type structure and isotypic lanthanoid–transition metal polyphosphidesTime-Reversal Symmetry-Breaking Superconductivity in Heavy-Fermion P r O s 4 S b 12 Detected by Muon-Spin RelaxationEvidence for time-reversal symmetry breaking in superconducting PrPt 4 Ge 12 Superconductivity and heavy fermion behavior in PrOs 4 Sb 12 Novel strongly correlated electron states in filled skutterudite lanthanide osmium antimonidesPenetration depth, multiband superconductivity, and absence of muon-induced perturbation in superconducting PrOs 4 Sb 12 Superfluid Density and Energy Gap Function of Superconducting PrPt 4 Ge 12 Topology of chiral superfluid: Skyrmions, Weyl fermions, and chiral anomalySuperconducting state of filled-skutterudite RPt4Ge12 (R=La, Pr):73Ge-NQR studiesEvidence for two superconducting gaps in the unconventional superconductor PrPt 4 Ge 12Comparative photoemission studies on the superconducting gap of the filled skutterudite superconductors LaPt 4 Ge 12 and PrPt 4 Ge 12 Filled platinum germanium skutterudites MPt4Ge12 (M = Sr, Ba, La—Nd, Sm, Eu): crystal structure and chemical bondingSKUTTERUDITES: A Phonon-Glass-Electron Crystal Approach to Advanced Thermoelectric Energy Conversion ApplicationsSuperconductivity in the Platinum Germanides MPt 4 Ge 12 ( M = Rare − Earth or Alkaline-Earth Metal) with Filled Skutterudite StructureGSAS-II : the genesis of a modern open-source all purpose crystallography software packageEXPGUI , a graphical user interface for GSASMultiband superconductivity in PrPt 4 Ge 12 single crystalsPhenomenological two-gap model for the specific heat of MgB 2Specific heat of Mg11B2Multiband Order Parameters for the PrOs 4 Sb 12 and PrRu 4 Sb 12 Skutterudite Superconductors from Thermal Conductivity MeasurementsSuperconducting Properties of Pr-Based Filled Skutterudite PrRu 4 As 12The critical magnetic field of anisotropic two-band magnetic superconductorsSuperconductivity at 2.5 K in the new transition-metal chalcogenide Ta 2 PdSe 5A stochastic theory of spin relaxationBroken time-reversal symmetry in superconducting Pr 1 − x Ce x Pt 4 Ge 12 Muon spin relaxation and hyperfine-enhanced Pr 141 nuclear spin dynamics in Pr ( Os , Ru ) 4 Sb 12 and ( Pr , La ) Os 4 Sb 12 Anomalous temperature dependence of the magnetic field penetration depth in superconducting UBe13Phenomenological theory of unconventional superconductivityEvidence for a Two Component Magnetic Response in UPt 3 Time-reversal symmetry-breaking superconductivity in Sr2RuO4Broken time reversal symmetry in unconventional superconductorsSuperconductivity with broken time-reversal symmetrySuperconducting states in the tetrahedral compound PrOs 4 Sb 12
[1] Norman M R 2011 Science 332 196
[2] Sigrist M, Agterberg D, Frigeri P, Hayashi N, Kaur R, Koga A, Milat I, Wakabayashi K and Yanase Y 2007 J. Magn. Magn. Mater. 310 536
[3] Fisher R A, Kim S, Woodfield B F, Phillips N E, Taillefer L, Hasselbach K, Flouquet J, Giorgi A L and Smith J L 1989 Phys. Rev. Lett. 62 1411
[4] Schemm E R, Gannon W J, Wishne C M, Halperin W P and Kapitulnik A 2014 Science 345 190
[5] Heffner R H, Smith J, Willis J O, Birrer P, Baines C, Gygax F N, Hitti B, Lippelt E, Ott H, Schenck A, Knetsch E, Mydosh J and MacLaughlin D 1990 Phys. Rev. Lett. 65 2816
[6] Jeitschko W and Braun D 1977 Acta Crystallogr. Sect. B 33 3401
[7] Aoki Y, Tsuchiya A, Kanayama T, Saha S R, Sugawara H, Sato H, Higemoto W, Koda A, Ohishi K, Nishiyama K and Kadono R 2003 Phys. Rev. Lett. 91 067003
[8] Maisuradze A, Schnelle W, Khasanov R, Gumeniuk R, Nicklas M, Rosner H, Leithe-Jasper A, Grin Y, Amato A and Thalmeier P 2010 Phys. Rev. B 82 024524
[9] Bauer E, Frederick N, Ho P C, Zapf V and Maple M 2002 Phys. Rev. B 65 100506
[10] Maple M, Frederick N, Ho P C, Yuhasz W, Sayles T, Butch N, Jeffries J and Taylor B 2005 Physica B 359–361 830
[11] Shu L, MacLaughlin D, Beyermann W, Heffner R, Morris G, Bernal O, Callaghan F, Sonier J, Yuhasz W, Frederick N et al 2009 Phys. Rev. B 79 174511
[12] Maisuradze A, Nicklas M, Gumeniuk R, Baines C, Schnelle W, Rosner H, Leithe-Jasper A, Grin Y and Khasanov R 2009 Phys. Rev. Lett. 103 147002
[13] Kozii V, Venderbos J W F and Fu L 2016 Sci. Adv. 2 e1601835
[14] Kanetake F, Mukuda H, Kitaoka Y, Sugawara H, Magishi K, Itoh K M and Haller E 2010 Physica C 470 S703
[15] Chandra L S, Chattopadhyay M and Roy S 2012 Philos. Mag. 92 3866
[16] Nakamura Y, Okazaki H, Yoshida R, Wakita T, Takeya H, Hirata K, Hirai M, Muraoka Y and Yokoya T 2012 Phys. Rev. B 86 014521
[17] Gumeniuk R, Borrmann H, Ormeci A, Rosner H, Schnelle W, Nicklas M, Grin Y and Leithe-Jasper A 2010 Z. Kristallogr. 225 531
[18] Nolas G, Morelli D and Tritt T M 1999 Annu. Rev. Mater. Sci. 29 89
[19] Gumeniuk R, Schnelle W, Rosner H, Nicklas M, Leithe-Jasper A and Grin Y 2008 Phys. Rev. Lett. 100 017002
[20] Toby B H and Von D R B 2013 J. Appl. Crystallogr. 46 544
[21] Toby B H 2001 J. Appl. Crystallogr. 34 210
[22] Zhang J, Chen Y, Jiao L, Gumeniuk R, Nicklas M, Chen Y, Yang L, Fu B, Schnelle W, Rosner H et al 2013 Phys. Rev. B 87 064502
[23] Bouquet F, Wang Y, Fisher R, Hinks D, Jorgensen J, Junod A and Phillips N 2001 Europhys. Lett. 56 856
[24] Fisher R, Li G, Lashley J, Bouquet F, Phillips N, Hinks D, Jorgensen J and Crabtree G 2003 Physica C 385 180
[25] Hill R, Li S, Maple M and Taillefer L 2008 Phys. Rev. Lett. 101 237005
[26] Namiki T, Aoki Y, Sato H, Sekine C, Shirotani I, Matsuda T D, Haga Y and Yagi T 2007 J. Phys. Soc. Jpn. 76 093704
[27] Changjan A and Udomsamuthirun P 2011 Solid State Commun. 151 988
[28] Zhang J, Dong J, Xu Y, Pan J, He L, Zhang L and Li S 2015 Supercond. Sci. Technol. 28 115015
[29] Kubo R 1981 Hyperfine Interact. 8 731
[30] Zhang J, MacLaughlin D, Hillier A, Ding Z, Huang K, Maple M and Shu L 2015 Phys. Rev. B 91 104523
[31] Shu L, MacLaughlin D, Aoki Y, Tunashima Y, Yonezawa Y, Sanada S, Kikuchi D, Sato H, Heffner R, Higemoto W et al 2007 Phys. Rev. B 76 014527
[32] Gross F, Chandrasekhar B, Einzel D, Andres K, Hirschfeld P, Ott H, Beuers J, Fisk Z and Smith J 1986 Z. Phys. B 64 175
[33] Sigrist M and Ueda K 1991 Rev. Mod. Phys. 63 239
[34] Yaouanc A, P D De Reotier, Gygax F, Schenck A, Amato A, Baines C, Gubbens P, Kaiser C, A De Visser, Keizer R et al 2000 Phys. Rev. Lett. 84 2702
[35] Luke G M, Fudamoto Y, Kojima K, Larkin M, Merrin J, Nachumi B, Uemura Y, Maeno Y, Mao Z, Mori Y et al 1998 Nature 394 558
[36] Sigrist M 2000 Physica C 341 695
[37] Sigrist M 2000 Physica B 280 154
[38] Sergienko I and Curnoe S 2004 Phys. Rev. B 70 144522