Evidence for the coexistence of time-reversal symmetry breaking and Bardeen-Cooper-Schrieffer-like superconductivity in La 7 Pd 3
aa r X i v : . [ c ond - m a t . s up r- c on ] D ec Evidence for the coexistence of time-reversal symmetry breaking andBardeen-Cooper-Schrieffer-like superconductivity in La Pd D. A. Mayoh, ∗ A. D. Hillier, G. Balakrishnan, and M. R. Lees † Physics Department, University of Warwick, Coventry, CV4 7AL, United Kingdom ISIS Facility, STFC Rutherford Appleton Laboratory,Harwell Science and Innovation Campus, Oxfordshire OX11 0QX, United Kingdom
Time-reversal symmetry breaking (TRSB) with a Bardeen-Cooper-Schrieffer (BCS) -like super-conductivity occurs in a small, but growing number of noncentrosymmetric (NCS) materials, al-though the mechanism is poorly understood. We present heat capacity, magnetization, resistiv-ity, and muon spin resonance/relaxation ( µ SR) measurements on polycrystalline samples of NCSLa Pd . Transverse-field µ SR and heat capacity data show La Pd is a type-II superconductor witha BCS-like gap structure, while zero-field µ SR results provide evidence of TRSB. We discuss the im-plications of these results for both the La X (where X = Ni, Pd, Rh, Ir) group of superconductorsand other CS and NCS superconductors for which TRSB has been observed. I. INTRODUCTION
The discovery of time-reversal symmetry breaking(TRSB) in polycrystalline La Ir has provided a newgroup of superconductors in which to investigate uncon-ventional superconducting behavior [1]. The report ofan isotropic s -wave gap symmetry (nodeless) supercon-ductivity in La Ir , along with density functional theorycalculations suggesting that the enhanced specific heat inthis compound is due to electron-phonon coupling, haveraised questions about the origin of the TRSB in this ma-terial [1, 2]. Time-reversal symmetry breaking has alsobeen observed in La Rh [3] suggesting this is a com-mon feature of the La X (where X = Ni, Pd, Rh, Ir)series. Here we present evidence of time-reversal symme-try breaking in La Pd which further cements this claim.Noncentrosymmetric (NCS) superconductors have gar-nered considerable interest in recent years. Their lack ofa center of inversion means that parity is no longer a goodquantum number [4]. A Rashba-type anti-symmetricspin-orbit coupling is allowed, lifting the degeneracy ofthe Fermi surface. The superconducting Cooper pairsmay then form with a mixture of spin-singlet or spin-triplet components [5, 6]. The superconducting gap func-tion ∆ can be described by ∆ = i ˜ σ y [ ψ ( k ) + d ( k ) · ˜ σ ],where for a pure singlet d ( k ) = 0 and the triplet casecorresponds to ψ ( k ) = 0. It is important to note that ifthe triplet term is small, there will be a nodeless, nearisotropic gap, making it experimentally difficult to dis-tinguish from a Bardeen-Cooper-Schrieffer (BCS) -like s -wave superconductor.A noncentrosymmetric superconductor with a mix-ture of triplet and singlet states may be expected todisplay unconventional superconducting properties. Alarge number of NCS superconductors have now beenstudied and many have indeed been shown to exhibitexotic superconducting behavior [4, 7]. For example, ∗ [email protected] † [email protected] CePt Si, is an antiferromagnetic heavy fermion super-conductor [8], BiPd, has an unconventional order param-eter [9], Li Pt B, has a line-nodal gap structure [10–13],and (Ta/Nb)Rh B , are multigap superconductors withupper critical fields that violate the Pauli limit [14–16].The possibility of spin-triplet superconductivity orsinglet-triplet mixing make NCS superconductors primecandidates to exhibit time-reversal symmetry breaking.Muon spectroscopy studies of several NCS supercon-ductors, including LaNiC [17] and several members ofthe Re T ( T = Zr, Hf, Ti) α -Mn group of materi-als [18–22] have confirmed the presence of spontaneousmagnetic moments that arise below the superconductingtransition temperature, T c , when time-reversal symme-try is broken [23]. However, TRSB has also been ob-served in a number of centrosymmetric superconductorssuch as Sr RuO [24, 25], PrPt Ge [26], (Pr,La)(Os,Ru) Sb [27, 28], LaNiGa [29], UPt [30–32] and(U,Th)Be [33], and Lu Rh Sn [34] and even elemen-tal rhenium [20], while for a large number of NCS super-conductors, time-reversal symmetry is preserved [7]. Thisleaves open the important question of how the occurrenceof time-reversal symmetry breaking, the crystallographicstructure (NCS or CS), and the nature of the supercon-ducting pairing mechanism are related to one another.In this paper we discuss the properties of La Pd whichis one of a group of hexagonal NCS superconductors witha Th Fe -type structure (space group P mc ). By in-vestigating the superconducting ground state of La Pd using muon spectroscopy further insight can be gainedinto the unusual superconducting behavior of this groupof intermetallic compounds. Pedrazzini et al . previouslyreported some of the superconducting and normal-stateproperties of La Pd as well as other members of thisgroup of superconductors, however, the fact that thisfamily of materials has a noncentrosymmetric structurewas not emphasized [35]. Here, the superconducting andnormal-state properties of La Pd are investigated us-ing magnetization, heat capacity, and resistivity mea-surements. Transverse-field muon spin rotation data arepresented that indicate La Pd has a conventional s -wavesuperconducting gap symmetry. Zero-field muon spin re-laxation curves provide evidence of time-reversal symme-try breaking in La Pd . Finally, we compare our resultson La Pd with those obtained for other La X super-conductors and the Re-based α -Mn superconductors. II. EXPERIMENTAL DETAILS
Polycrystalline samples of La Pd were prepared fromstoichiometric quantities of La (3N) and Pd (3N) in anarc furnace under an argon atmosphere on a water-cooledcopper hearth. The sample buttons were melted andflipped several times to ensure phase homogeneity. Theobserved weight loss during the melting was negligible.The sample buttons were then sealed in an evacuatedquartz tube, and annealed for 5 days at 500 ◦ C. Thematerial is air sensitive and was observed to rapidly de-velop an orange surface discoloration if exposed to air.The samples were stored in a glove-box under an argonatmosphere. A Quantum Design Physical Property Mea-surement System was used to measure both the heat ca-pacity and electrical resistivity between 1.8 and 300 Kin applied fields up to 9 T. A Quantum Design He-3 in-sert was used to access temperatures down to 0.5 K. AQuantum Design Magnetic Property Measurement Sys-tem with iQuantum He-3 insert was used to measure themagnetization between 0.5 and 300 K in applied fields upto 7 T. Muon spin relaxation/rotation ( µ SR) measure-ments where performed using the MuSR spectrometer atthe ISIS Neutron and Muon Source at the RutherfordAppleton Laboratory, UK µ SR measurements were per-formed in both transverse-field (TF) and zero-field (ZF)modes. A full description of the detector geometries canbe found in Ref. 36. A crushed sample of La Pd wasmounted on a 99.995% silver plate and inserted into adilution refrigerator to measure at temperatures from50 mK to 4 K. III. CHARACTERIZATIONA. Magnetization and electrical resistivitymeasurements
To confirm bulk superconductivity in the La Pd sam-ples the dc volume magnetic susceptibility as a functionof temperature, χ dc ( T ), was measured in an applied fieldof 1.2 mT between 0.5 and 1.7 K as shown in Fig. 1(a).A rectangular sample of La Pd was cut from the samplebutton to give a well-defined shape with a demagnetiza-tion factor N = 0 .
13 [37]. A sharp change in the suscep-tibility marks the onset of superconductivity in La Pd at T onsetc = 1 . χ dc = −
1) is clearly visible indicating bulk su-perconductivity in La Pd . Several magnetization versusfield loops were collected at different temperatures from 0.5 to 1.5 K in fields up to 10 mT. The lower critical fieldis estimated by measuring the field at which the flux firstenters the sample, the first deviation from linearity inthe magnetization versus applied field [38]. Figure. 1(b)shows the lower critical field values extracted from thesemagnetization versus field loops. The temperature de-pendence of H c1 ( T ) can be described by the Ginzburg-Landau (GL) formula H c1 ( T ) = H c1 (0) (cid:2) − t (cid:3) , where t r = T /T c , giving µ H c1 (0) = 6 . ∼
50 K. An upturn in χ ( T ) below5 K is consistent with the presence of a small quantity ofparamagnetic impurities (other rare-earths with sizablelocalized magnetic moments) present in the La used toprepare the samples. The normal state behavior of χ ( T )is qualitatively similar to the only previous report [35].The temperature dependence of the electronic resistiv-ity in La Pd is shown in Fig. 2. The shape of ρ ( T )is characteristic of other NCS superconductors in theTh X series [35] as well as many intermetallic mate-rials [39, 40]. The resistivity at 300 K is 201.1(5) µ Ω cmand the residual resistivity ρ at 2 K, just above T c , is48 . µ Ω giving a residual resistivity ratio of ∼ . ℓ =12 . B. Specific heat measurements
The temperature dependence of the heat capacity inLa Pd between 0.45 and 2.75 K is shown in Fig. 3(a). Asharp jump in the heat capacity, ∆ C , is seen at 1 . Pd .The superconducting transition in La Pd is typical ofthat seen in a type-II superconductor. The normal-stateheat capacity at T > .
45 K can be modeled using C ( T ) /T = γ n + β T + β T , (1)to give the Sommerfeld coefficient γ n = 50 . / mol K and β = 4 . / mol K .Writing θ D = (cid:16) π N k B /β (cid:17) gives the Debye temper-ature Θ D = 159(2) K. The large γ n is quite unusual fornoncentrosymmetric superconductors, with the value forLa Pd surpassing some heavy-fermion superconductorssuch as CeCoGe ( γ n = 32 mJ/mol K ) [42]. Thissuggests that there is an enhanced density of states atthe Fermi level for La Pd . This value of γ n is consistentwith that seen in other La X [2, 3, 35, 43] compoundssuggesting that this enhancement is a common featureof this group of superconductors.The electronic heat capacity in zero applied field in thesuperconducting state can be used to look for evidencefor an unusual superconducting order parameter. Thenormalized entropy S el /γ n T c is written as S el ( T ) γ n T c = − π ∆ (0) k B T c Z ∞ [ f ln( f ) + (1 − f ) ln (1 − f )] dǫ, (2)where f ( E ) = [1 + exp ( E/k B T )] − is the Fermi-Diracdistribution function with energy E = q ǫ + [∆ ( T )] . ǫ is the energy of the normal state electrons and∆ ( T ) = ∆ (0) δ ( T ) where ∆ (0) is the magnitude ofthe superconducting gap at zero kelvin and the tem-perature dependence of the energy gap is approxi-mated using a single-gap BCS expression [44], δ ( T ) =tanh { .
82 [1 .
018 ( T c /T − . } . The specific heat inthe superconducting state is then calculated from [45] C el ( T ) γ n T c = T d ( S el /γ n T c ) dT . (3)Figure 3(b) shows that the normalized electronic heatcapacity C el /γ n T as function of the reduced tempera-ture T /T c . The data and the fit are in good agreementindicating that the material has an isotropic, largely s -wave superconducting gap. ∆ C/γ n T c = 1 .
27 (1) and∆ (0) /k B T c = 1 . Pd , however, see Section IV. C. Upper critical field calculations
Fig. 4(a) shows measurements of the resistivity as afunction of temperature, ρ ( T ), for La Pd in variousmagnetic fields. There is a sharp superconducting tran-sition at T c = 1 . T = 0 .
05 K inzero-applied field that is suppressed with increasing mag-netic field. The transition also broadens, ∆ T = 0 . T c values determined from these data are includedin Fig. 4(c). The temperature dependence of the heat ca-pacity around the superconducting transition in differentapplied magnet fields is shown in Fig. 4(b). Again, thetransition is suppressed and broadened with increasingmagnetic field.Values for the upper critical field, H c2 , of La Pd were estimated from the midpoint of the superconduct-ing transition in the resistivity data [see Fig. 4(a)] andthe heat capacity data [see Fig. 4(b)]. Figure 4(c) showsthe upper critical field values as a function of tempera-ture. The H c2 values exhibit a positive curvature closeto T c . This behavior can be captured by the Ginzburg-Landau phenomenological model for upper critical fields, H c2 ( T ) = H c2 (0) (cid:2)(cid:0) − t (cid:1) / (cid:0) t (cid:1)(cid:3) , giving a goodfit to the data and an estimated upper critical field µ H c2 (0) = 652(5) mT, almost 2.5 times the previouslyreported value [35], but still well below the Pauli limit which is similar to other superconductors in this series.Using H c2 (0) = Φ2 πξ gives ξ = 22 . Pd contains lanthanum and palladium and so itis anticipated that there may be a significant spin-orbitcoupling contribution to the physics of the superconduct-ing state of La Pd . The Werthamer-Helfand-Hohenberg(WHH) model allows for the inclusion of spin-orbit cou-pling in the upper critical field calculations [39, 46]. A fitto the H c2 ( T ) data was attempted using the WHH modelas shown by the orange dash-dotted line in Fig. 4(c) giv-ing a slightly smaller µ H c2 (0) = 620(3) mT. However,this model is unable to capture the curvature of the up-per critical field values, cf. La Ir where a WHH modelprovided a reasonable fit to the H c2 ( T ) data [2]. IV. µ SR MEASUREMENTS, THESUPERCONDUCTING ORDER-PARAMETER,AND TIME-REVERSAL SYMMETRYBREAKING
The macroscopic superconducting state of La Pd wasprobed using magnetization, resistivity, and heat capac-ity, however, in superconductors the microscopic mag-netic environment formed by the vortex lattice can pro-vide an essential insight into the superconducting state.Positive muons are an excellent probe of the local mag-netic environment when implanted into a superconduc-tor. The superconducting state of La Pd has been in-vestigated using transverse-field, longitudinal-field, andzero-field µ SR.Transverse-field spectra were collected at temperaturesbetween 0.1 to 2.75 K in applied fields ranging from 10to 50 mT, in the mixed state [ µ H c1 (0) = 6 . G TF ( t ) = A exp (cid:18) − σ t (cid:19) cos ( γ µ B t + φ )+ A cos ( γ µ B t + φ ) . (4) A and A are the sample and silver sample holder asym-metries, B and B are the average fields in the super-conductor and silver plate, φ is the shared phase offset, γ µ / π = 133 . − is the muon gyromagnetic ra-tio and σ is the total depolarization rate. By fitting thespectra collected at different temperatures and fields us-ing Eq. (4), the temperature dependence of σ can be de-termined as shown in Fig. 6(a). The total depolarizationrate, σ , is related to the depolarization due to the fluxline lattice, σ FLL , and the nuclear moments, σ N , by σ = σ + σ . (5)The nuclear depolarization rate is found to remain con-stant over all temperatures at σ N = 0 . µ s − . Sincethe upper critical field is comparable with the appliedfields used in these measurements σ FLL has a consider-able field dependence. This is due to significant shrinkingof the inter vortex distances within the flux line latticeas the applied field is increased. The effect of the vor-tex cores and the expected field dependence of the sec-ond moment of the field distribution have been calculatedusing different models. From calculations based on theGinzburg-Landau model the field dependence of σ FLL canbe described using σ FLL (cid:2) µ s − (cid:3) = 4 . × (1 − h r ) × { . (cid:16) − p h r (cid:17) } λ − (cid:2) nm (cid:3) (6)where h r = H/H c2 is the reduced field and λ − is theinverse square of the penetration depth [47]. By takingisothermal cuts of the data shown in Fig. 6(a) as denotedby the dashed line, Eq. (6) can be used to fit to the data,as shown in Fig. 6(b), and the penetration depth can beextracted. The resulting temperature dependence of λ − ,which reflects the variation in the superfluid density, isshown in Fig. 6(b) and this can be used to investigatethe nature of the superconducting gap in La Pd . Inthe clean limit, the magnetic penetration depth can bemodeled using λ − ( T ) λ − (0) = 1 + 2 Z ∞ ∆( T ) (cid:18) ∂f∂E (cid:19) EdE p E − ∆ ( T ) , (7)where f is the Fermi-Dirac distribution function and thetemperature dependence of the gap for an isotropic s -wave model is ∆ ( T ) = ∆ (0) δ ( T ) as in Section III B.The fit produced by this model is shown by the dashedline in Fig 6(c). The penetration depth at zero kelvin wascalculated to be λ (0) = 495(4) nm. The value of ∆ (0) =0 . /k B T c = 2 . /k B T c determined from heat capacity and µ SR data have been observed in other superconductors,e.g. [48]. Possible reasons for the difference include multi-gap superconductivity, a gap anisotropy, or a nodal gapdue to a small triplet component leading to a reducedanomaly in heat capacity at T c [49–51].Zero-field measurements were performed on La Pd tolook for evidence of time-reversal symmetry breaking inthe superconducting state. Examples of the asymme-try spectra collected at temperatures above (2.75 K) andbelow (0.1 K) the superconducting transition are shown in Fig. 7(a). These spectra exhibit considerable relax-ation. The absence of any oscillatory component in thesignals rules out the possibility of there being magneticordering in the sample. This observation is supportedby measurements of the temperature dependence of themagnetic susceptibility at magnetic fields above H c2 (0)(see inset in Fig. 2). It can be assumed that the majorityof the relaxation arises from the presence of static, ran-domly orientated nuclear moments, while the increasedrelaxation rate below T c indicates the presence of addi-tional small internal magnetic fields in the superconduct-ing state. These small magnetic fields are associated withthe onset of time-reversal symmetry breaking. To elim-inate any possibility of the relaxation coming from spinfluctuations a small longitudinal field (LF) of 5 mT wasapplied, as shown in Fig. 7(a). The complete decouplingof the muons from the proposed relaxation channel inthis small LF indicates that the spontaneous magneticfields are static or at least quasi-static over the lifetimeof the muon.The response of the muons to the nuclear moments canbe captured using the Kubo-Toyabe expression G KT ( t ) = 13 + 23 (cid:0) − σ t (cid:1) exp (cid:18) − σ t (cid:19) , (8)where σ ZF measures the width of the nuclear dipolar fieldexperienced by the muons. The asymmetry can then bemodeled by G ( t ) = A G KT ( t ) exp ( − Λ t ) + A bg , (9)where A and A bg are the sample and background asym-metries, respectively, and Λ measures the electronic re-laxation rate. The sample and background asymmetrieswere found to be constant at all temperatures. σ ZF wasfound to decrease linearly with increasing temperaturefrom 0.1 to 2.75 K across T c [see Fig. 7(c)] while Λ wasfound to be temperature independent above the super-conducting transition and to increase immediately below T c at ∼ . µ SR measurements. This temperature differencebetween the signal marking the onset of TRS breakingand T c is also seen in both La Ir [1] and La Rh [3],two other members of this group of materials, as well asother superconductors such as PrPt Ge [26]. V. COMPARISONS WITH OTHERSUPERCONDUCTORS EXHIBITING TRSB
Magnetization, heat capacity, resistivity, and µ SRmeasurements reveal that La Pd is type-II supercon-ductor with a T c = 1 . µ SR measurements indicate that the su-perconducting order parameter in La Pd is dominatedby a BCS-like s -wave component. The temperature de-pendence of the upper critical field is well fitted by aGinzburg-Landau model which provides further evidenceof conventional superconducting behavior.On the other hand, zero-field µ SR measurements revealan increase in Λ ( T ) at low temperature of 0 .
005 (1) µ s − .This is taken as evidence for the onset of time-reversalsymmetry breaking although this change in Λ ( T ) is onlyvisible below ∼ . Ir and La Rh (see Table II). It is possible that forall three materials TRS is broken at a second transitionjust below T c . Such a scenario has been suggested forLaNiC and LaNiGa [50]. However, to date, there areno indications of any additional transition or evidence fortwo gap superconductivity in the La X series of super-conductors.An increase in Λ ( T ) at T c in Sr RuO [24],LaNiC [17], and SrPtAs [52] is also attributed to time-reversal symmetry breaking. The ∆Λ and behaviorof σ ZF ( T ) are given in Table II for comparison. InSr RuO , TRSB is thought to arise due to a degeneracyin the superconducting phase brought about by non-zerospin and orbital moments. This in turn allows for the cre-ation of spontaneous moments near to grain boundariesand impurities due to variations in the superconductingorder parameter [24, 27, 53]. In LaNiC , the signatureof TRSB results from hyperfine fields made by nonuni-tary spin triplet pairs [17]. LaNiC and Sr RuO haveunconventional gap structures, while La Pd , La Rh ,and La Ir all appear to have BCS-like s -wave gaps. Aspin-split Fermi surface can look conventional if the mag-nitude of the two superconducting order parameters aresimilar, and singlet-triplet pairing would be difficult todifferentiate from conventional s -wave pairing if d ( k ) issmall. In all the noncentrosymmetric compounds stud-ied to date, the signature of TRSB is relatively weakwhen compared to the change in Λ ( T ) seen in the p -wave superconductor Sr RuO . Measurements on singlecrystals of La X , which should more readily reveal anyanisotropy or nodes in the gap, are essential to clarify theTRSB mechanism. The superconducting and normal-state properties of single crystal La Ni point to it beinga conventional superconductor [43], although no µ SR re-sults have yet been published to confirm whether time-reversal symmetry breaking is present in La Ni . Thisis important as the effects of spin-orbit coupling shouldbe more prominent in La X materials containing theheavier elements Ir, Pd, and Rh. Similar challenges are faced by those investigating theproperties of the Re-based α -Mn NCS superconductors.TRSB is reported for several compounds in this series,again with s -wave BCS-like gap structures [18–22, 39, 54–59] and rather small changes in σ ZF ( T ) (see Table II).The observation of time-reversal symmetry breaking inthe centrosymmetric rhenium exhibiting type-II super-conductivity is particularly interesting [22] and may in-dicate that TRSB in these Re-based compounds is notrelated to the noncentrosymmetric structure, cf. LaNiC and LaNiGa [17, 29]. Pristine rhenium is a type-I super-conductor and is driven type-II by shear strain. It wouldbe interesting to investigate whether the increase in de-fects that accompanies this strain plays any role in theTRSB, and whether other elemental superconductors, in-cluding in this context La, show any evidence for similareffects. Further studies of centrosymmetric La X ma-terials will also provide vital information on the role thecrystallographic structure plays in time-reversal symme-try breaking. In particular, studies of the centrosymmet-ric La Ru should indicate whether a lack of a center ofinversion is necessary for time-reversal symmetry break-ing in the La X compounds. VI. CONCLUDING REMARKS
Now that time-reversal symmetry breaking has beenobserved in La Pd it joins the small number of NCSsuperconductors that also show this phenomena. Singlecrystals of these superconductors are required in orderto distinguish parity mixing effects in the materials thatappear to have predominantly s -wave BCS-like supercon-ducting gap structures. Studies of elemental supercon-ductors and centrosymmetric superconducting analoguesare also needed to clarify how important properties suchas the lack of an inversion center and inhomogeneitiesare in driving time-reversal symmetry breaking in NCSsuperconductors. ACKNOWLEDGMENTS
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FIG. 1. (a) Temperature dependence of the dc volumemagnetic susceptibility χ dc for La Pd , in zero-field-cooledwarming (ZFCW) mode, measured in an applied magneticfield 1.2 mT showing a superconducting onset temperature T onsetc = 1 . H c1 versus tem-perature for La Pd . The dashed line shows a fit to thedata using H c1 ( T ) = H c1 (0) (cid:2) − t (cid:3) which gives µ H c1 (0) =6 . Pd .Parameter Value Unit µ H c1 (0) 6.9(2) mT µ H c2 (0) 652(5) mT λ (0) 495(4) nm ξ (0) 22.5(1) nm κ GL r ( m W c m ) Temperature (K) c ( · - e m u / m o l ) Temperature (K) m H = 0.5 T
FIG. 2. Normal state properties of La Pd : Temperaturedependence of electronic resistivity from 1 to 300 K with zeroapplied magnetic field. The residual resistivity ratio (RRR)for La Pd is approximately 4.2 and the residual resistivityjust above the transition, ρ (2 K) = 48 . µ Ω cm. Inset:Temperature dependence of the magnetic susceptibility in anapplied field of 0.5 T (in the normal state). An upturn at lowtemperature is consistent with the small quantity of paramag-netic impurities present in the La used to prepare the sample.Compound T c (K) ∆Λ ( µ s − ) ∆ σ ( µ s − ) Channel ofTRSB Behavior ofsecondarychannel Instrument Gapstructure CS or NCS Ref.La Pd s -wave NCS This work.La Ir s -wave NCS 1 and 2La Rh s -wave NCS 3Re Zr 6.75 - 0.008 σ Constant MuSR s -wave NCS 18, 39, 54–56Re Hf 5.98 - 0.005 σ Linear inc. MuSR s -wave NCS 19, 57, and 58Re Ti 6 - 0.009 σ Linear dec. MuSR s -wave NCS 21Re Ti s -wave NCS 22 and 59Re . Nb . σ - MuSR, GPS, LTF s -wave NCS 20Re 2.7 - 0.01 σ - GPS, LTF s -wave CS 20LaNiC RuO p -wave CS 24TABLE II. Selected properties of some noncentrosymmetric and centrosymmetric superconductors in which time-reversal sym-metry breaking has been observed. ∆Λ and ∆ σ are taken from the data presented in the listed references along with thechannel in which the signal indicating TRS breaking is observed and the muon spectrometer used. These instruments includeMuSR at ISIS, UK, the General Purpose Surface-Muon (GPS) and the Low Temperature Facility (LTF) instruments based atPSI, Switzerland, and the M15 beam line at TRIUMF, Canada. C / T ( m J / m o l K ) T (K )DC/g n T = 1.27 (1) C e l / g n T T/T c (a)(b) FIG. 3. (a) Temperature dependence of the zero-field heatcapacity for La Pd between 0.45 and 2.75 K showing a su-perconducting transition at T c = 1 . C versus T is indicative of a typical type-II superconductor.Fitting the data above T c in the normal-state using Eq. (1)gives γ n = 50 . . (b) Normalized electronicheat capacity C el /γ n T versus the reduced temperature T /T c in zero applied field. The dotted line shows a fit to the datafor an isotropic s -wave gap made using Eqs. 2 and 3. r vs T r vs H C vs T WHH GL m H c ( T ) Temperature (K) r ( m W c m ) Temperature (K)
450 mT 0 mT C / T ( m J / m o l K ) T (K ) (a) (b) (c) FIG. 4. (a) Temperature dependence of the resistivity of La Pd showing the suppression and broadening of the resistivesuperconducting transition in applied fields from 0 to 450 mT. (b) C/T versus T for La Pd showing the suppression andbroadening of the superconducting transition as the applied field is increased from 0 to 500 mT. (c) Temperature dependenceof the upper critical field for La Pd . The H c2 ( T ) values were extracted from the midpoints of the anomalies in C ( T ) /T andthe midpoints of the resistive transitions. The dotted and dash-dotted lines show fits to the µ H c2 ( T ) data using the GL andWHH models [39, 46], respectively. A y s mm e t r y Time (ms)
FIG. 5. Transverse-field µ SR spectra for La Pd collectedat 100 mK (top) and 2.25 K (bottom) in an applied mag-netic field of 20 mT. The solid lines are fits to data usingEq. (4). Below the superconducting transition temperaturethe field distribution of the FLL causes the spectra to be sig-nificantly depolarized. Above the superconducting transitiontemperature the randomly oriented array of nuclear magneticmoments continue to depolarize the muons but at a reducedrate. s-wave l - ( m m - ) Temperature (K) s F LL ( m s - ) Field (mT) (c)(b) s ( m s - ) Temperature (K)
10 mT 20 mT 30 mT 40 mT (a)
FIG. 6. (a) Temperature dependence of the total spin depolarization, σ , for La Pd collected in fields between 10 and 50 mT.Isothermal cuts (the dashed line shows the cut made at 0.8 K) were used to calculate the field dependence of σ FLL in La Pd .(b) Field dependence of the muon spin relaxation due to the flux line lattice, σ FLL , at different temperatures. The solid linesare fits to the data using Eq. (6). (c) Temperature dependence of the inverse square of the penetration depth, λ − . The dashedline is a fit to the data using Eq. (7). T c s ( m s - ) Temperature (K) L ( m s - ) Temperature (K) T c A s y mm e t r y Time (ms) (a) (b) (c)
FIG. 7. (a) ZF and LF- µ SR spectra collected at 0.1 (green) and 2.75 K (red), the data is fit using the Gaussian Kubo-Toyabemodel (dashed lines). (b) Temperature dependence of the electronic relaxation rate Λ can be seen to increase below 1.2 K justbelow T c . (c) Temperature dependence of the nuclear relaxation rate σ shows no change at T cc