Unconventional superconductivity in Y5Rh6Sn18 probed by muon spin relaxation

Conventional superconductors are robust diamagnets that expel magnetic fields through the Meissner effect. It would therefore be unexpected if a superconducting ground state would support spontaneous magnetics fields. Such broken time-reversal symmetry states have been suggested for the high—temperature superconductors, but their identification remains experimentally controversial. We present magnetization, heat capacity, zero field and transverse field muon spin relaxation experiments on the recently discovered caged type superconductor Y5Rh6Sn18 ( TC= 3.0 K). The electronic heat capacity of Y5Rh6Sn18 shows a T3 dependence below Tc indicating an anisotropic superconducting gap with a point node. This result is in sharp contrast to that observed in the isostructural Lu5Rh6Sn18 which is a strong coupling s—wave superconductor. The temperature dependence of the deduced superfluid in density Y5Rh6Sn18 is consistent with a BCS s—wave gap function, while the zero-field muon spin relaxation measurements strongly evidences unconventional superconductivity through a spontaneous appearance of an internal magnetic field below the superconducting transition temperature, signifying that the superconducting state is categorized by the broken time-reversal symmetry.

Scientific RepoRts | 5:12926 | DOi: 10.1038/srep12926 (μSR) measurements in PrOs 4 Sb 12 (first unconventional superconductor amongst Pr-based metallic compounds) have revealed a significant upturn in the internal magnetic fields below the onset of superconductivity (T c = 1.82 K) 16 . The low-lying crystal field excitations of Pr ions play a vital role in the superconductivity 16 . On the other hand, PrV 2 Al 20 exhibits superconductivity below 50 mK in the antiferro-quadrupole ordered state, is an unusual specimen of a heavy-fermion superconductor based on strong hybridization among conduction electrons and nonmagnetic quadrupolar moments of the cubic Γ 3 ground doublet 17 . For PrV 2 Al 20 , the gapless mode associated with the quadrupole order is reflected in a cubic temperature dependence of electronic heat capacity. Non Fermi liquid behaviors are observed in PrIr 2 Zn 20 and PrRh 2 Zn 20 in the resistivity, specific heat and quadrupole ordering 18 . R 5 Rh 6 Sn 18 (R = Sc, Y, Lu) compounds also having a cage-like structure, crystallize in the tetragonal structure (see Fig. 1) with the space group I4 1 /acd and Z = 8, where R occupies two sites of different symmetry 19 , exhibit superconductivity 20 below 5 K (Sc), 3 K (Y), and 4 K (Lu). The crystal structure is analogous to the skutterudite structure with a Pr-based heavy fermion superconductor 21 . Lu 5 Rh 6 Sn 18 is a conventional BCS type superconductor, the gap structure of Y 5 Rh 6 Sn 18 is found to be strongly anisotropic as exposed from the specific heat (C P ) measurements; the heat capacity coefficient (γ) of Y 5 Rh 6 Sn 18 in the mixed state is found to follow a square root field dependence. This is a sign of anisotropic superconducting gapping. The superconducting properties of Y 5 Rh 6 Sn 18 thus have a correspondence with those of the anisotropic s-wave superconductor YNi 2 B 2 C except for the difference in superconducting transition temperature 22 . On the other hand, the γ of Lu 5 Rh 6 Sn 18 shows linear field dependence, which points to an isotropic superconducting gap. The unconventional superconductivity of nonmagnetic YNi 2 B 2 C draws considerable attention from several standpoints, such as high-T c superconductivity among intermetallics [23][24][25] and anisotropic superconducting gap 26,27 . Numerous measurements indicate that YNi 2 B 2 C has an anisotropic superconducting gap (point-node type). K. Izawa et al. 28 suggest that the superconducting gap structure of YNi 2 B 2 C has point nodes located along the a and b axes by thermal conductivity measurements. For Y 5 Rh 6 Sn 18 the field-angle-resolved specific heat C P (φ) measurements in a rotating magnetic field H exposes a clear fourfold angular oscillation of C P (φ), signifying that the field-induced quasiparticle density of state is strongly anisotropic 29 . Additionally no considerable angular oscillation was observed in C P (φ) of Lu 5 Rh 6 Sn 18 , a reference compound of an isotropic superconducting gap.
We have recently reported superconducting properties of the caged type compound Lu 5 Rh 6 Sn 18 using magnetization, heat capacity, and muon spin relaxation measurements 30 . ZF-μSR measurements reveal the spontaneous appearance of an internal magnetic field below the superconducting transition temperature, which indicates that the superconducting state in this material is characterized by the broken time-reversal symmetry 30 . From a series of experiments 22 on R 5 Rh 6 Sn 18 (R = Lu, Sc, Y and Tm), it was concluded that the superconducting gap structure is strongly dependent on the rare earth atom, but whose origin remains undetermined 29 . In this work, we address these matters by ZF-μSR measurements for the Y 5 Rh 6 Sn 18 system. The results unambiguously reveal the spontaneous appearance of an internal magnetic field in the SC state, providing clear evidence for broken time reversal symmetry.

Results and Discussion
The bulk nature of superconductivity in Y 5 Rh 6 Sn 18 was established by the magnetic susceptibility χ(T), as shown in Fig. 2(a). The low-field susceptibility displays a robust diamagnetic signal due to a superconducting transition at T c = 3.0 K. Figure 2(b) shows the magnetization M(H) curve at 2 K, which is typical for type-II superconductivity. Remarkably, the electrical resistivity ρ(T) of Y 5 Rh 6 Sn 18 shows uncommon (not shown here) non-metallic behavior 29 on cooling down to just above T c with a high residual resistivity ρ 0 of 350 μΩ cm. A moderately rare state in which the anisotropic superconductivity persists in an incoherent metallic state is suggested to occur in Y 5 Rh 6 Sn 18 . Figure 2(c) displays C P (T) at H = 0 and 6.0 T. Below 3.0 K in zero field a sharp anomaly is detected demonstrating the superconducting transition which matches well with χ(T) data. Subsequently the normal-state specific heat was found to be invariable under external magnetic fields. The normal-state electronic specific heat coefficient γ and the lattice specific heat coefficient β were deduced from the data in a field of 6.0 T by a least-square fitting of the C P /T data to C P /T = γ + βT 2 + δT 4 . This analysis provides a Sommerfeld coefficient γ = 38.13(3) mJ/(mol-K 2 ) and the Debye temperature Θ D = 183(2) K. We found the specific heat jump Δ C P (T c ) = 223 mJ/(mol K) and T c = 3.0 K, which yields Δ C/γT c = 1.95. Figure 2(d) shows the electronic specific heat C e which was obtained after subtraction of the phonon part, to illuminate the superconducting gap symmetry. In case of line-nodes in the superconducting gap structure, T 2 dependence of the C e is anticipated below T c . We find a power law C e = cT α with an exponent close to 3 (α = 2.93) over a comprehensive temperature region. This cubic temperature dependence suggest that Y 5 Rh 6 Sn 18 has an anisotropic superconducting gap with a point node, such as has been found for the borocarbides YNi 2 B 2 C and LuNi 2 B 2 C. Analogous behavior is also detected in the heavy fermion superconductor UBe 13 , which reveals a T 2.9 dependence of the specific heat below T c together with a higher fraction 31 of the ratio Δ C/γT c = 2.5. On the other hand, C e of Lu 5 Rh 6 Sn 18 behaves as an exponential dependence. We obtained the specific heat jump Δ C P (T c ) = 397 ± (3) mJ/(mol K) and T c = 4.0 ± (0.2) K, which yields Δ C/γT c = 2.06 ± (0.03) for Lu 5 Rh 6 Sn 18 . From the exponential dependence of C e , we obtained 2Δ (0)/k B T c to be 4.26 ± (0.04) for Lu 5 Rh 6 Sn 18 . Because this value is relatively larger than that of the theoretical BCS limit of weak-coupling superconductor (3.53), the Lu 5 Rh 6 Sn 18 compound is characterized as a strong-coupling superconductor 29,30 . In order to determine the superfluid density or superconducting gap structure of Y 5 Rh 6 Sn 18 we have carried out TF-μSR measurements at 400 G (well above lower critical field) applied magnetic field at 0.1 and 4.0 K as shown in Fig. 3(a,b). A clear difference is evident in the relaxation rate below and above T c . Below T c , the TF-μSR precession signal decays with time due to inhomogeneous field distribution of the flux-line lattice. The analysis of our TF-μSR asymmetry spectra was carried out in the time domain using the following functional form, where v s and v bg are the frequencies of the muon precession signal and background signal, respectively with phase angle φ i (i = 1, 2). The first term gives the overall sample relaxation rate σ; there are contributions from both the vortex lattice (σ sc ) and nuclear dipole moments (σ nm , which is expected to be constant over the whole temperature region The contribution from the vortex lattice was determined by quadratically subtracting the background nuclear dipolar relaxation rate obtained from spectra measured above T c . The relaxation rate from the vortex lattice is directly associated to the magnetic penetration depth, the superconducting gap function can be demonstrated by, where f e xp E k T Figure 3(c) shows the T dependence of the term σ sc which can be directly connected to the superfluid density. From this, the nature of the superconducting gap can be determined. The data can be well modeled using a single isotropic s-wave gap of 0.5 ± 0.05 meV. This gives a gap of 2Δ /k B T c = 3.91 ± 0.03. For Lu 5 Rh 6 Sn 18 , the analysis of temperature dependence of the magnetic penetration depth suggest an isotropic s-wave character for the superconducting gap with a gap value 2Δ /k B T c = 4.4 ± 0.02, which is significantly higher than the 3.53 expected for BCS superconductors. This is an indication of the strong electron-phonon coupling in the superconducting state. Y 5 Rh 6 Sn 18 is a type II superconductor, supposing that approximately all the normal state carriers (n e ) contribute to the superconductivity (i.e. n s ~ n e ), we have estimated the values of effective mass of the quasiparticles m * ≈ 1.21 m e and superconducting electron density ≈ 2.3 × 10 28 m −3 respectively. Additional details on these calculations can be found in Ref. [34][35][36]. Figure 4(a) shows the time evolution of the zero field muon spin relaxation asymmetry spectra in Y 5 Rh 6 Sn 18 at temperatures above and below T c . A constant background signal arising from muons stopping in the silver sample holder has been deducted from the data. Below T c , we observed that muon spin relaxation became faster with decreasing temperature down to lowest temperature, which is indicating the appearance of the spontaneous local field in the superconducting phase. No signature of muon spin precession is visible, which excludes internal fields produced by magnetic ordering as the origin of the spontaneous field. The only possibility is that the muon-spin relaxation is due to static, randomly oriented local fields associated with the nuclear moments at the muon site. The ZF-μSR spectra for Y 5 Rh 6 Sn 18 can be well described by the damped Gaussian Kubo-Toyabe (K-T) function, is the K-T functional form expected from an isotropic Gaussian distribution of randomly oriented static (or quasistatic) local fields at muon sites. λ is the electronic relaxation rate, A 1 is the initial asymmetry, A bg is the background. The parameters σ KT , A 1 , and A bg [not shown here] are found to be temperature independent all across the phase transition. Figure 4(b) shows the temperature dependence of λ. In the normal state above T c , λ is due to dipolar fields from neighboring nuclear magnetic moments. It is extraordinary that λ shows a substantial rise below T c , signifying the presence of a spontaneous internal field associated with the superconductivity. Further the temperature dependent λ exhibits nearly linear increase with decreasing temperature below T c . This observation provides unequivocal sign that TRS is broken in the SC state, below T c that may suggest a possibility of an unusual superconducting mechanism below 2 K, of Y 5 Rh 6 Sn 18 . Such a change in λ has only been detected in superconducting Sr 2 RuO 4 , LaNiC 2 , and Lu 5 Rh 6 Sn 18 4,11,30 . This rise in λ can be described in terms of a considerable internal field with a very small frequency as conferred by Luke et. al. 4 for Sr 2 RuO 4 . This proposes that the field distribution is Lorentzian in nature analogous to Sr 2 RuO 4 . The temperature dependence of λ compares quantitatively with that in in Sr 2 RuO 4 , Lu 5 Rh 6 Sn 18 , LaNiC 2 and Y 5 Rh 6 Sn 18 and thus we attribute this behavior of λ to the TRS breaking below T c in Y 5 Rh 6 Sn 18 . A longitudinal magnetic field of just 25 G [Fig. 4(c)] removes any relaxation due to the spontaneous fields and is sufficient to fully decouple the muons from this relaxation channel. This in turn shows that the associated magnetic fields are in fact static or quasistatic on the time scale of the muon precession. These observations further support the broken TRS in the superconducting state of Y 5 Rh 6 Sn 18 . The increase in the exponential relaxation for Lu 5 Rh 6 Sn 18 below T c is, 0.045 μs −1 , which corresponds to a characteristic field strength λ/γ μ = 0.5 G. For Y 5 Rh 6 Sn 18 , the rise in the exponential relaxation below T c is, 0.0184 μs −1 , which resembles a characteristic field strength λ/γμ = 0.21 G, where γ μ is the muon gyromagnetic ratio = 13.55 MHz/T. This is about half the value as observed in Lu 5 Rh 6 Sn 18 , the B phase of UPt 3 and Sr 2 RuO 4 7 . No theoretical estimates of the characteristic fields strength in Y 5 Rh 6 Sn 18 are yet presented; however, we presume them to be analogous to those in Sr 2 RuO 4 and UPt 3 as the fields should arise from an analogous mechanism.
Time reversal symmetry breaking in the superconducting state has significant consequences for the symmetry of pairing and for the quasi-particle spectrum. A typical symmetry analysis 37 carried out under the theory of strong spin orbit coupling, yields two possibilities, one with singlet pairing (d + id character) and additional one triplet pairing (non-unitary). The singlet pairing state has a line node and two point nodes, and the non-unitary triplet state has two point nodes. Below the superconducting transition temperature the thermodynamics of the singlet state would be dominated by the line node, yielding quadratic temperature dependence of the electronic specific heat. Likewise, the non-unitary triplet pairing state would be dominated by the point nodes, which happen to be shallow (a result protected by symmetry) and consequently also lead to C e ~ T 2 38 . Though, because of the location of the nodes in the triplet case, fully-gapped behavior may be recovered depending on the topology of the Fermi surface. Furthermore certain limiting cases of the triplet state correspond to regular, i.e. linear point nodes (cubic temperature dependence of the electronic heat capacity) as well as to a more unusual state with a nodal surface (gapless superconductivity, C e ~ T). The allowed pairing states and their quasiparticle spectra are discussed in detail in the Supplementary Online Material 39 .

Conclusions
In summary, we have investigated the vortex state in Y 5 Rh 6 Sn 18 by using zero field and transverse field muon spin relaxation measurements. Below the superconducting transition temperature, the ZF-μSR measurements revealed an appreciable increase in the internal magnetic fields which does not coincide with the superconducting phase transition at T c = 3.0 K, and which may indicate different nature of SC below and above 2 K. The appearance of spontaneous magnetic fields in our ZF-μSR measurements, deliver undoubted evidence for TRS breaking in this material and an unconventional pairing mechanism. TF-μSR measurements yield a magnetic penetration depth that is exponentially flat at low temperatures, and so our data can be fit to a single-gap BCS model. Symmetry analysis suggests either a singlet d + id state with a line node or, alternatively, nonunitary triplet pairing with point nodes, which may be linear or shallow and can become fully gapped depending on the Fermi surface topology 39 . The heat capacity below T c exhibits ~T 3 supporting point nodes scenario.

Methods
An important step to study the intrinsic, particularly anisotropic properties is to grow sizable single crystals. The single crystals of Y 5 Rh 6 Sn 18 were grown by liquefying the constituent elements in an excess of Sn-flux in the proportion of Y:Rh:Sn = 1:2:20. The quartz tube was heated up to 1323 K, kept at this temperature for about 3 h, and cooled down to 473 K at a rate of 5 °C/h, taking 168 hr in total. The excess flux was detached from the crystals by spinning the ampoule in a centrifuge 20 . Several single crystals of Y 5 Rh 6 Sn 18 were obtained having typical dimensions of 3 × 3 × 2 mm 3 . Laue patterns were recorded with a Huber Laue diffractometer. Well defined Laue diffraction spots indicates the high quality of the single crystals. The phase purity was inferred from the powder x-ray patterns, which were indexed as the Y 5 Rh 6 Sn 18 phase with the space group 20 I4 1 /acd [lattice parameters: a = 1.375(3) nm, c = 2.745(1)]. The magnetization data were measured using a Quantum Design Superconducting Quantum Interference Device. The heat capacity were measured down to 500 mK using Quantum Design Physical Properties Measurement System equipped with a 3 He refrigerator.
Muon spin relaxation is a dynamic method to resolve the type of the pairing symmetry in superconductors 40 . The mixed or vortex state in case of type-II superconductors gives rise to a spatial distribution of local magnetic fields; which demonstrate itself in the μSR signal through a relaxation of the muon polarization. We further employed the muon spin relaxation technique to examine the superconducting ground state. The zero field, transverse field (TF) and longitudinal field (LF) measurements were performed at the MUSR spectrometer at the ISIS Pulsed Neutron and Muon Facility located at the Rutherford Appleton Laboratory, United Kingdom. The ZF-μSR experiments were conducted in the longitudinal geometry. The unaligned single crystals were mounted on a high purity silver plate (99.995%) using diluted GE Varnish for cryogenic heatsinking, which was placed in a dilution refrigerator with a temperature range of 100 mK to 4.4 K. TF-μSR experiments were performed in the superconducting mixed state in applied field of 400 G, well above the lower critical field of 18 G of Y 5 Rh 6 Sn 18 . Data were collected in the field-cooled mode where the magnetic field was applied above the superconducting transition and the sample was then cooled down to base temperature. Using an active compensation system the stray magnetic fields at the sample position were canceled to a level of 0.01 G. Spin-polarized muon pulses were implanted into the sample and the positrons from the subsequent decay were collected in positions either forward or backwards of the initial muon spin direction. The asymmetry of the muon decay is