Possible anisotropic superconducting pairing in cubic ThPt{}_{4}Ge{}_{12}

Possible anisotropic superconducting pairing in cubic ThPtGe

V. H. Tran , A. D. Hillier , D. T. Adroja , D. Kaczorowski Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P. O. Box 1410, 50-950 Wrocław, Poland
ISIS Facility, Rutherford Appleton Laboratory, Chilton, Oxfordshire OX11 0QX, United Kingdom

Transverse-field muon-spin-rotation measurements have been carried out for polycrystalline ThPtGe. The magnetic penetration depth and the superconducting coherence length in the vortex state of this compound were found to be 112(5) nm and 32(2) nm, respectively. We have estimated the effective mass of the quasiparticles , the superfluid carrier density carriers/m, and the zero-temperature superconducting gap 0.67 meV, corresponding to the ratio: 2 = 3.76. We found markable difference between the temperature dependence of the vortex state muon-spin relaxation rate ZFC- and FC- below the irreversibility temperature 2.5 K. Linear field dependence of , small ratio , and power law behavior of the temperature dependencies of seem to be consistent with anisotropic superconducting pairing in the compound studied. The analysis of correlation between the superconducting transition temperature and the effective Fermi temperature within the Uemura classification scheme reveals that ThPtGe belongs to the same class of exotic superconductors.

74.70.Dd; 74.25.Op;74.20.Rp;76.75.+i

I Introduction

Recently, two research groups have independently reported discovery of a novel class of ternary compounds with the filled skutterudite structure MPtGe, where M=Sr, Ba, La, Ce, Pr, Nd, and Eu.Bauer07 ; Gumeniuk The compounds with Sr and Ba have been found to be superconducting below = 5.10 and 5.35 K,Bauer07 respectively, whereas those with La and Pr have been found with = 8.3 and 7.9 K,Gumeniuk respectively. We also reported on the formation and the superconducting properties of another Ge-based skutterudite ThPtGe with = 4.62 K, which is the actinoid bearing representative of the MPtGe series.KT08 The compound can be characterized by the Ginzburg-Landau coherence length of 35 nm, the penetration depth of 150 nm, electronic mean free path 435 nm, and the upper critical field = 0.29 T. A large ratio signifies ThPtGe to be in the clean limit. Subsequently, the superconductivity in this material has been confirmed by Bauer et al.Bauer08 Thus, like the compounds with M = Sr, Ba, La, and Pr, the ThPtGe compound exhibits type-II superconductivity at low temperatures. Owing to the fact that ThPtGe and other MPtGe exhibit similar values of the critical temperatures, which range from 5 to 8 K, one expects a minor role of the filler M element, but a dominating contribution to the superconducting properties from interactions within the Pt-Ge framework. In natural consequence, it is believed that superconductivity in these skutterudites is most likely mediated by phonons. However, some experimental data on ThPtGe,KT08 indicated possibility of an unconventional character of the electron pairing, as in the superconducting state both the temperature dependence of the electronic specific heat and the magnetic field variation of the Sommerfeld coefficient exhibit some behavior characteristic for the point-like node superconductivity. For ThPtGe, NMR experiments and electronic band structure calculations,Tran09b have suggested that the physical properties of this compound may be anisotropic. Furthermore, the Hall effect and thermoelectrical power data have revealed the existence of both electrons and holes,Tran09a manifesting different behavior from those of conventional BCS ordinary superconductors, which usually exhibit one-band conduction. The unclear mechanism of electron pairing in ThPtGe shows that further experimental and theoretical studies of this compound are needed.

In this paper, we report a study of the vortex state in ThPtGe by means of transverse-field muon-spin rotation (TF-SR) technique, which is very sensitive measurement of the microscopic field distribution inside type-II superconductors.Amato ; Sonier1 ; Kadono ; Sonier2 From the measurements, we have estimated the magnetic penetration depth and the coherence length , and then we have derived an effective mass of the quasiparticles and superconducting carrier density . Applying zero-field cooling and field-cooling sample modes we have investigated the flux pining phenomenon in ThPtGe, and have found the irreversibility temperature . Interestingly, the temperature dependencies of the magnetic penetration depth and field dependence of exhibit deviations from those expected for the isotropic s-wave superconductors. A small ratio and close relationship of ThPtGe to unconventional superconductors within the Uemura classification scheme,Uemura are further evidences of unusual behavior of the investigated superconductor. Very recently the transverse-field (TF)-muon spin rotation experiments for PrPtGe have established the presence of point-like nodes in the superconducting energy gap.Maisuradze

Ii Experimental details

Polycrystalline sample of about 5 g used in this study was prepared and characterized as reported previously.KT08 measurements were carried out using MuSR spectrometer installed at the ISIS Facility of the Rutherford Appleton Laboratory, Chilton, UK, where pulses of muons are produced every 20 ms with FWHMs of 70 ns. For the measurements, pulverized sample was mixed with GE varnish and glued onto a high purity silver (4N) holder of 30 mm diameter and 1 mm thick. We have carried out the measurements at temperatures ranging from 0.3 to 5 K in magnetic field up to 0.06 T, applying two different modes of cooling the sample: in zero magnetic field (ZFC) and applied magnetic field (FC).

Iii Results and discussion

Typical TF- time spectra for ThPtGe, taken above and below , are shown in Fig. 1. Comparison of the spectra indicates clear difference in the relaxation rate above and below . This difference is related to the change in magnetic field distribution inside the sample, which can be obtained using the algorithm of fast Fourier transformation, and it is displayed on the right- hand side of the spectra. In the normal state, the field distribution corresponds to the applied external magnetic field of = 40 mT and the Gaussian-like peak with some broadening is attributed to the nuclear magnetic moments. Below , due to the formation of the flux line lattice state associated with the superconductivity, the magnetic field distribution becomes inhomogeneuos and two Gaussian components are observed. The larger component at ascribes the field distribution of nonsuperconducting signal, whereas the smaller peak corresponds to the second moment of the field distribution , arising due to the formation of the flux line lattice.

Figure 1: (color online) Transverse-field SR time spectra and internal magnetic field distributions of ThPtGe taken above (upper panel) and below (lower panel) in applied magnetic of 0.04 T. The solid lines represent the best fit with eq. 1. The dashed lines are the fits to Gaussian functions. Open and close symbols in TF-SR time spectra denote real and and imaginary components, respectively.

To elucidate details of the vortex state we have analyzed the data using the expression:


where the first term is the contribution from the sample and the second term from the background. and , and , and , are the initial asymmetries, muon precession frequencies, and phase angles for the sample and background, respectively. and are the muon-spin relaxation rates in the superconducting and normal state, respectively. = 135.53 MHz/T is the gyromagnetic ratio of the muon.

Fits to the TF- time spectra, shown in Fig. 1 as the solid lines, have been obtained as follows. First we have fitted the normal state data, both the real and imaginary components, keeping = 0. In this way the values of , , , , , and have been determined. For the data below , we have kept , , as the same values as in the normal state, and from the fits we have determined the parameters , , , and . The obtained values of the parameters and as functions of the magnetic field strength and the temperature are shown in Figs. 2 - 4 and will be discussed below.

Fig. 2 a shows the field dependence of the muon-spin relaxation rate in ThPtGe obtained at 0.3 K. We see that depends strongly on applied fields. A similar behavior has been observed in PrOsSb,MacLaughlin KOsO,Koda and PrRuSb.Androja05 From the field dependence of the muon relaxation rate, we have evaluated the magnetic penetration depth and the coherence length using the modified London equation:Brandt


where is the lattice sum over the hexagonal flux line lattice. The best fit (see the solid line) gave = 112(5) nm and = 32(2) nm, which corroborate the values derived from the specific heat measurements with 150 nm and 35 nm, respectively.KT08

Generally, the magnetic penetration depth is related to the superconducting carrier density , effective mass , coherence length and the mean free path . In clean limit, , is given as:


where c and q have the usual meaning. The estimation of and is impossible using the SR data alone, and hence it is helpful assuming that all the normal state carriers contribute to the superconductivity, i.e., the coefficient of the electronic specific heat is related to and via the relation:


Taking = 112 nm and = 42 mJ mol K from Ref. KT08, , and combining Eqs. 3 and 4, one evaluates = 4.6 and = 1.0210 carriers/m. The obtained values substantiate that ThPtGe is a good metal with enhanced mass carriers.

Figure 2: (color online) a) Field dependence of the TF-muon spin-relaxation rate of ThPtGe at 0.3 K. The solid line is the fit of eq. 2 to the experimental data. b) The magnetic penetration depth vs. reduced field . The closed square is the fitted value with eq. 2. The dashed line indicates linear dependence of for 0.1 0.2 with the slope of 1.6.

It is known that the field dependence of the magnetic penetration depth can provide information on the degree of anisotropy of the superconducting order parameter.Kadono In fact, the -dependence is entirely different for conventional BCS-type superconductors and unconventional superconductors with anisotropic energy gaps. Therefore, it is tempting to consider the dependence for ThPtGe. It can be derived from the relation:Brandt2


Note that this equation holds for 5 and 0.25/. Because ThPtGe has = 3.5 then the description with Eq. 5 may have an error larger than 5 %. Taking temperature dependence of ,KT08 and using Eq. 5 we evaluated and plot the -dependence in Fig. 2 b. Apparently, in a similar manner as found for various unconventional superconducting materials, of ThPtGe is increased almost linearly with the field in a certain range of the strength of applied field. A linear fit of the data in the field interval 0.1 0.2 to the function yields = 1.6(1). Here, one should recall that unconventional superconductors takes values between 1 and 6, in contrast to the case of isotropic gap BCS-type superconductors for which it is equal to zero. The value determined for ThPtGe is close to that found for isotropic superconductors KOsO (n = 2.58),Koda and NbSe ( = 1.61),Sonier97 but smaller than that derived for the high- superconductor with d-wave pairing YBaCuO, ( 6).Sonier97b This comparison suggests possible anisotropic pairing in ThPtGe.

Figure 3: (color online) Temperature dependence of the normalized superfluid density of ThPtGe in an applied magnetic field of 0.04 T. The lines are the fits discussed in the text.

Fig. 3 shows the normalized superfluid density in the FC regime for ThPtGe in an applied magnetic field of 0.04 T versus temperature . The temperature dependence of the magnetic penetration depth has been inferred data using Eq. (5) and taking the ,KT08 and data. According to the Gorter-Casimir two-fluid model for s-wave isotropic gap superconductors, is proportional to . As shown in Fig. 3, the TF-data of ThPtGe exhibit significant deviation from this relation (dotted line). This observation indicates that ThPtGe might be not an isotropic BCS superconductor. It turns out that the experimental data can be fitted to the relation:


with = 130(5) nm, n = 2.7(1), = 4.1 K (note the solid line). We would mention that the observed n-value is close to n = 2, predicted for nodal superconductors.Annett Similar reducing in n-value has been observed in KOsO (n = 2.39),Koda and in PrRuSb (n =1.44).Androja05

Since ThPtGe is in the clean limit, the relation:


should be tested for the experimental data. In the above equation, is the Fermi function, is the zero-temperature value of the superconducting gap, and ),carrington represents the temperature dependence of the gap. Fitting the experimental data with = 130 nm, taken equal to the extrapolated value at T = 0 one obtains = 4.1 K and = 0.67 meV. These values lead to the ratio: 2 = 3.76, which is slightly larger than the weak-coupling BCS value of 3.52. Within the experimental error limit, the fit with eq. 7 is quite good for 2.5 K but becomes little ambiguous for the temperature range 0.8 - 2.2 K (see the dashed line).

To check the possibility of unconventional superconducting order parameters in ThPtGe, the magnetic penetration depth data are fitted to point-node model, because this model predicted the proportionality of the electronic specific heat ,Bash as that we have previously found for ThPtGe. We used the following equations in evaluating the data:Prorozov


where , and are the polar and the azimuthal coordinates in the k-space. In fitting we assumed for the point node and the theoretical values are the average of the partial superfluid densities. The subscripts aa and cc indicate principal axis. The fit of the data to this point-node model, shown as the dashed line in Fig. 3, is very good for 1.5 K but little poor for the temperature range 1.5 - 3.0 K. We would like to note that deduced from the SR measurements is fairly consistent with that extrapolated from -dependence for a field of 0.04 T ( 4.0(1)). KT08 Moreover, examining in Fig. 3, we see that the -curve has negative curvature around . This behavior may indicate that the electronic properties in the vortex state of ThPtGe differ from those of isotropic BCS superconductors. Such a curvature of was previously found for anisotropic superconductors YBaCuO,Mao and predicted theoretically for anisotropic p- or d-wave superconductors.PC

In Fig. 4 a we show the temperature dependencies of the TF-muon spin-relaxation rate taken in the ZFC and FC modes. These data point to the existence of distinct irreversibility with characteristic temperature 2.5 K and below which being larger than . The enhancement of at temperatures below implies that there is an additional inhomogeneity of local-field distribution, which is related to flux pinning in the superconducting state. In other words, when the flux is strongly pinned below , the flux vortex in the ZFC mode avoids the formation of the equillibrium flux vortex lattice. In contrast, at temperatures higher than , where the thermal fluctuation of the flux vortices overcomes the flux pinning to form the equillibrium flux lattice, the the field distribution in the ZFC regime is the same as that in the FC mode. This means that the temperature is a result of the competition between the thermal fluctuation and the pinning of vortices. From the experimental values of the second moment of the field distribution, we have calculated the muon precession frequencies (Fig. 4 b). In a similar manner as in the case of the relaxation rate, the irreversible effect appears in the -curves below 2.5 K, namely measured in the FC mode is larger than that obtained in the ZFC mode. We may recall that SR studies of flux pining phenomena for numbers of superconductors,Le ; Wu have shown that there is a relationship between the ratio and the anisotropy of superconducting characteristics, namely the more anisotropic superconductor has a lower value of this ratio. Since the observed ratio of 0.54 is rather small, the superconducting gap in ThPtGe is likely anisotropic.

Figure 4: (color online) Temperature dependence of the TF-muon a) spin-relaxation rate and b) muon precession frequency from the ZFC and FC measurements in a field of 0.04 T.
Figure 5: (color online) Uemura plot, vs. , for some selected superconductors including unconventional, isotropic s-wave,Uemura and studied here ThPtGe. The dotted and dashed lines 0.01 0.2 outline positions of unconventional superconductors from conventional BCS superconductors with solid line 0.001.

Let us now discuss data of ThPtGe within the frameworks of the Uemura classification scheme.Uemura According to the author, there exists a close relationship between the superconducting transition temperature and the effective Fermi temperature , which can be determined from the SR and heat capacity data using the formula , where in K, in , and in mJ Kcm. Interestingly, Uemura et al. was able to distinguish HTc, heavy-fermion and some exotic superconductors from the conventional BCS superconductors like Nb, Sn and Al. The first class of superconductors should fall in the range 0.01 0.1, while the latter class of superconductors have 0.001.Uemura We re-plot in Fig. 5 the correlation between the superconducting temperature and the Fermi temperature for some selected superconductors, including the data for PrPtGe (calculated from data of Ref. Maisuradze, ) and the data for ThPtGe. In Fig. 5 we have drawn two lines with the ratio of 0.2 and 0.01, intending to show the area in which unconventional superconductors are situated. For ThPtGe  the ratio = 0.053, much larger than 0.001, suggests that the pairing mechanism in this compound is different than that in isotropic s-wave superconductors but could be similar to that in exotic superconductors.

Iv Conclusions

We have investigated the vortex state in ThPtGe by performing the transverse-field muon-spin-rotation in magnetic fields up to 0.06 T upon cooling the sample in zero and finite field (the ZFC and FC modes). The difference between the ZFC and FC data has been found to occur below the irreversibility temperature 2.5 K. From the field dependence of the vortex state muon-spin relaxation rate , we have estimated the magnetic penetration depth 112(5) nm, the coherence length 32(2) nm. Combining the and specific heat data we have evaluated the effective mass of the quasiparticles and the superfluid carrier density carriers/m, confirming that ThPtGe is good metal with enhanced mass carriers. We have analyzed the temperature dependence of the superfluid density using various superconducting symmetry order parameters. Within the BCS clean limit approach, the fit of the experimental data revealed zero-temperature superconducting gap = 0.67 meV corresponding to the ratio: 2 = 3.76. The phenomenological two-fluid model with the exponent of 2.7 and as well as the point-node model describe reasonably the dependence. Furthermore, we have observed the linear field dependence of and the small ratio , which together with the behavior around might be taken as an evidence for an anisotropic order parameter. Comparison of the data with other superconductors in terms of the Uemura classification scheme suggests that ThPtGe and PrPtGe may share the pairing mechanism similar to that of exotic superconductors.

The authors are grateful to J. Sonier and A. Karen for an interesting discussion. The work was supported by the Polish Ministry of Science and Higher Education through the research grants N202 082/0449 and N202 116 32/3270. The measurement of SR at ISIS is made possible due to support from EU within NMI3 programme.


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