Nodeless superconductivity in the noncentrosymmetric Mo 3 Rh 2 N superconductor: a μ SR study
T. Shang, Wesen Wei, C. Baines, J. L. Zhang, H. F. Du, M. Medarde, M. Shi, J. Mesot, T. Shiroka
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Preprint: November 6, 2018, 2:12.
Nodeless superconductivity in the noncentrosymmetric Mo Rh N superconductor: a µ SR study
T. Shang,
1, 2, 3, ∗ Wensen Wei, C. Baines, J. L. Zhang, H. F. Du, M. Medarde, M. Shi, J. Mesot,
6, 3, 7 and T. Shiroka
7, 6 Laboratory for Multiscale Materials Experiments, Paul Scherrer Institut, Villigen CH-5232, Switzerland Swiss Light Source, Paul Scherrer Institut, Villigen CH-5232, Switzerland Institute of Condensed Matter Physics, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne CH-1015, Switzerland. Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions,High Magnetic Field Laboratory of the Chinese Academy of Sciences, Hefei 230026, People’s Republic of China Laboratory for Muon-Spin Spectroscopy, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland Laboratorium für Festkörperphysik, ETH Zürich, CH-8093 Zurich, Switzerland
The noncentrosymmetric superconductor Mo Rh N, with T c = β -Mn-type structure (spacegroup P Al C. Its bulk superconductivity was characterized by magnetizationand heat-capacity measurements, while its microscopic electronic properties were investigated by means ofmuon-spin rotation and relaxation ( µ SR). The low-temperature superfluid density, measured via transverse-field (TF)- µ SR, evidences a fully-gapped superconducting state with ∆ = k B T c , very close to 1.76 k B T c – the BCS gap value for the weak coupling case, and a magnetic penetration depth λ =
586 nm. Theabsence of spontaneous magnetic fields below the onset of superconductivity, as determined by zero-field(ZF)- µ SR measurements, hints at a preserved time-reversal symmetry in the superconducting state. BothTF-and ZF- µ SR results evidence a spin-singlet pairing in Mo Rh N. Introduction.
The current research interest in super-conductivity (SC) involves either studies of high tempera-ture superconductors (such as cuprates or iron pnictides),or investigations of unconventional superconducting states.Superconductors with centrosymmetric crystal structuresare bound to have either pure spin-singlet or spin-tripletpairings. On the other hand, due to the relaxed space-symmetry requirement, noncentrosymmetric superconduc-tors (NCSCs) may exhibit unconventional pairing.
A lackof inversion symmetry leads to internal electric-field gra-dients and, hence, to antisymmetric spin-orbit coupling(ASOC), which lifts the spin degeneracy of the conduction-band electrons. As a consequence, the superconducting or-der can exhibit a mixture of spin-singlet and spin-tripletpairing.
Of the many NCSCs known to date, however, only a fewexhibit a mixed singlet-triplet pairing. Li Pt B and Li Pd Bare two notable examples, where the mixture of singlet andtriplet states can be tuned by modifying the ASOC through aPd-for-Pt substitution. Li Pd B behaves as a fully gapped s -wave superconductor, whereas the enhanced ASOC turnsLi Pt B into a nodal superconductor, with typical featuresof spin-triplet pairing. Other NCSCs may exhibit uncon-ventional properties besides mixed pairing. For instance,CePt Si, CeIrSi , and K Cr As , exhibit line nodes inthe gap, while others such as LaNiC and (La,Y) C , show multiple nodeless superconducting gaps. In addi-tion, due to the strong influence of ASOC, their upper crit-ical fields can exceed the Pauli limit, as has been found inCePt Si and (Ta,Nb)Rh B . Mo Al C forms a β -Mn-type crystal structure with spacegroup P
32. Muon-spin rotation / relaxation ( µ SR), nu-clear magnetic resonance (NMR), and specific heat stud-ies have revealed that Mo Al C is a fully-gapped, strongly-coupled superconductor, which preserves time-reversalsymmetry (TRS) in its superconducting state.
The re-cently synthesized Mo Rh N NCSC, a sister compound toMo Al C, has been studied via transport and specific-heatmeasurements. Yet, to date the microscopic nature of itsSC state remains largely unexplored. DFT calculations sug-gest a strong hybridization between the Mo and Rh 4 d -orbitals, reflecting the extended nature of the latter. Thedensity of states (DOS) at the Fermi level E F , arising from the Rh and Mo 4 d -orbitals, are comparable. This is in strongcontrast with the Mo Al C case, where the DOS at E F ismostly dominated by Mo 4 d -orbitals. In the Mo Rh Ncase, the SOC is significantly enhanced by the replacementof a light element, such as Al, with one with a strong SOC,such as Rh. Considering that already Mo Al C exhibits un-usual properties, we expect the enhanced SOC to af-fect the superconducting properties of Mo Rh N, too. InRe T ( T = transition metal) alloys, whose DOS is dom-inated by the Re 5 d -orbitals (with negligible contributionsfrom the T metal orbitals), even a robust increase in SOC— from 3 d Ti to 5 d Ta — is shown to not significantly affectthe superconducting properties. Conversely, similarly to theLi (Pd,Pt) B case, SOC effects are expected to be more im-portant in Mo Rh N. Therefore, a comparative microscopicstudy of Mo Rh N vs. Mo Al C is very instructive for under-standing the (A)SOC effects on the superconducting prop-erties of NCSCs. Another goal of this study was the searchfor a possible TRS breaking in the superconducting state ofMo Rh N.In this paper, we report on the systematic magnetiza-tion, thermodynamic, and µ SR investigation of the re-cently discovered Mo Rh N NCSC. In particular, zero- (ZF)and transverse-field (TF) µ SR measurements allowed us tostudy the microscopic superconducting properties and tosearch for a possible TRS breaking below T c in Mo Rh N. Experimental details.
Polycrystalline Mo Rh N sampleswere synthesized by solid-state reaction and reductive ni-tridation methods, whose details are reported elsewhere. The room-temperature x-ray powder diffraction confirmedthe β -Mn-type crystal structure, with no detectable extraphases. The magnetization and heat capacity measure-ments were performed on a 7-T Quantum Design MagneticProperty Measurement System (MPMS) and a 9-T Physi-cal Property Measurement System (PPMS). The bulk µ SRmeasurements were carried out using the general-purposesurface-muon (GPS) and the low-temperature facility (LTF)instruments of the π M3 beamline at the Swiss muon sourceof Paul Scherrer Institut, Villigen, Switzerland. For mea-surements on LTF, the samples were mounted on a silverplate using diluted GE varnish. The µ SR data were ana-lyzed by means of the musrfit software package. Characterizing bulk superconductivity . The magnetic sus- m H = 1 mT FC c v ZFC (a) m H c ( m T ) T (K)
0T 1 2 3 4 (b) C / T ( m J / m o l - K ) T (K) m H c ( T ) T (K)
FIG. 1. (a) Temperature dependence of magnetic susceptibility χ ( T ) and (b) of specific heat C ( T ) / T for Mo Rh N. The inset in(a) shows the estimated µ H c vs. temperature up to T c , the solid-line being a fit to µ H c ( T ) = µ H c ( )[ − ( T / T c ) ] . For eachtemperature, µ H c was determined from the value where M ( H ) deviates from linearity. The inset in (b) shows µ H c ( T ) , as deter-mined from heat-capacity measurements in various applied fields,with the solid-line being a fit to the WHH model without spin-orbitscattering. ceptibility of Mo Rh N was measured using both field-cooled (FC) and zero-field-cooled (ZFC) protocols in anapplied field of 1 mT. As shown in Fig. 1(a), the ZFC-susceptibility indicates bulk superconductivity below T c = Rh N, consistent with the previously reportedvalue. The lower critical field µ H c was determined fromthe field-dependent magnetization M ( H ) , measured at vari-ous temperatures below T c . The estimated µ H c ( T ) valuesare shown in the inset of Fig. 1(a). The solid-line repre-sents a fit to µ H c ( T ) = µ H c ( )[ − ( T / T c ) ] and yieldsa lower critical field µ H c ( ) = ( ) mT. The bulk super-conductivity of Mo Rh N was further confirmed by heatcapacity measurements [ see Fig. 1(b) ] . The specific heat,too, exhibits a sharp transition at T c , which shifts towardslower temperature upon increasing the magnetic field. Thesharp transitions ( ∆ T ∼ T c values vs. the applied field are summa-rized in the inset of Fig. 1(b), from which the upper crit-ical field µ H c was determined following the Werthamer-Helfand-Hohenberg (WHH) model. The solid-line in theinset of Fig. 1(b) represents a fit to the WHH model, withoutconsidering spin-orbital scattering, and gives µ H c ( ) = Transverse-field µ SR.
To explore the microscopic super-conducting properties of Mo Rh N, TF- µ SR measurementswere performed down to 0.02 K. In order to track the ad-ditional field-distribution broadening due to the flux-line-lattice (FLL) in the mixed superconducting state, a mag-netic field of 30 mT [ i.e., larger than the lower critical field µ H c (0) ] was applied at temperatures above T c . The TF- µ SR time spectra were collected at various temperatures up (b) A m p li t ude FFT
Field (mT) A sy mm e t r y Time (ms) (a)
TF-30mT
FIG. 2. (a) The Mo Rh N TF- µ SR time spectra, collected at 0.02 Kand 6.4 K in an applied field of 30 mT, show very different relax-ation rates. Fourier transforms of the above time spectra at 6.4 K(b) and 0.02 K (c). The solid lines are fits to Eq. (1) using a singleGaussian relaxation; the dashed lines indicate the applied mag-netic field. Note the clear diamagnetic shift below T c in (c). to T c , following a field-cooling protocol. Figure 2(a) showstwo representative TF- µ SR spectra collected above (6.4 K)and below T c (0.02 K) on GPS and LTF, respectively. Theobserved phase shift between the two datasets is due to in-strumental effects. The faster, FLL-induced decay in the su-perconducting state is clearly seen in the second case. Thetime evolution of the µ SR-asymmetry is modeled by: A TF = A s cos ( γ µ B s t + φ ) e − σ t / + A bg cos ( γ µ B bg t + φ ) . (1)Here A s and A bg represent the initial muon-spin asymme-tries for muons implanted in the sample and sample holder,respectively, with the latter not undergoing any depolariza-tion. The A s / A TF ratios were determined from the long-timetail of TF- µ SR spectra at base temperature [ see Fig. 2(a) ] , and fixed to 0.88 (GPS) and 0.90 (LTF) for all the tem-peratures. B s and B bg are the local fields sensed by im-planted muons in the sample and sample holder, γ µ = π × / T is the muon gyromagnetic ratio, φ isthe shared initial phase, and σ is a Gaussian relaxationrate. The Gaussian nature of relaxation is clearly evincedfrom the fast-Fourier-transform (FFT) spectra shown inFig. 2(b) and (c). In the mixed superconducting state, thefaster decay of muon-spin polarization reflects the inho-mogeneous field distribution due to the FLL, which causes – 2 – .00.10.20.30.4 0 1 2 3 4 5 6 7 8 9-0.20-0.15-0.10-0.050.00 GPS LTF s ( m s - ) TF-30mT (a) T c = 4.6 K(b) D B ( m T ) T (K)
FIG. 3. Temperature dependence of (a) the muon-spin relaxationrate σ ( T ) and (b) diamagnetic field shift ∆ B ( T ) for Mo Rh Nmeasured in an applied field of 30 mT. Here ∆ B = B s − B bg , where B bg is the same as the applied magnetic field. the additional distribution broadening in the mixed state [ see Fig. 2(c) ] . In the superconducting state, the measuredGaussian relaxation rate includes contributions from botha temperature-independent relaxation due to nuclear mo-ments ( σ n ) and the FLL ( σ sc ). The FLL-related relaxationcan be extracted by subtracting the nuclear contribution ac-cording to σ sc = Æ σ − σ . The derived Gaussian relax-ation rate and the diamagnetic field shift as a function oftemperature are summarized in Fig. 3. The relaxation rate,shown in Fig. 3(a), is small and independent of tempera-ture for T > T c , but it starts to increase below T c , indicatingthe onset of FLL and an increase in superfluid density. Con-comitantly, a diamagnetic field shift appears below T c [ seeFig. 3(b) ] .Since σ sc is directly related to the magnetic penetrationdepth and the superfluid density ( σ sc ∝ /λ ), the super-conducting gap value and its symmetry can be determinedfrom the measured σ sc ( T ) . For small applied magneticfields [ H appl / H c ∼ ≪ ] , the magnetic penetrationdepth λ can be calculated from: σ ( T ) γ µ = Φ λ ( T ) . (2)Figure 4 shows the inverse square of the magnetic pene-tration depth (proportional to the superfluid density) asa function of temperature for Mo Rh N. To gain insightinto the SC pairing symmetry in Mo Rh N, its temperature-dependent superfluid density ρ sc ( T ) was further analyzedby using different models, generally described by: ρ sc ( T ) = + (cid:28) Z ∞ ∆ k E q E − ∆ ∂ f ∂ E d E (cid:29) FS , (3)where ∆ k is an angle-dependent gap function, f = ( + e E / k B T ) − is the Fermi function, and 〈〉 FS represents an av-erage over the Fermi surface. The gap function can be written as ∆ k ( T ) = ∆ ( T ) g k , where ∆ is the maximum gapvalue and g k is the angular dependence of the gap, equalto 1, cos 2 ψ , and sin θ for an s -, d -, and p -wave model,respectively. Here ψ and θ are azimuthal angles. The d-wave p-wave s-wave-clean s-wave-dirty GPS LTF l - ( m m - ) T (K)
FIG. 4. Superfluid density vs. temperature, as determined from TF- µ SR measurements. The different lines represent fits to variousmodels, including s -, d -, and p -wave pairing (see text for details). temperature dependence of the gap is assumed to follow ∆ ( T ) = ∆ tanh { [ ( T c / T − )] } , where ∆ ,the gap value at zero temperature, is the only adjustableparameter. Note that the function ∆ ( T ) is practically inde-pendent of the different models.Three different models, including s -, d -, and p waves,were used to describe the temperature-dependent super-fluid density λ − ( T ) . By fixing the zero-temperature mag-netic penetration depth λ = ( ) nm, the estimatedgap values for the s - and p -wave model are 0.76(1) and1.07(1) meV, respectively; while for the d -wave model, theestimated λ and gap value are 536(3) nm and 1.11(1) meV.As can be seen in Fig. 4, the temperature dependence of thesuperfluid density is clearly consistent with a single fully-gapped s -wave model. In case of d - or p -wave models, apoor agreement with the measured λ − values is found, es-pecially at low temperature. The s -wave nature of SC isfurther confirmed by the temperature-independent behav-ior of λ − ( T ) for T < / T c , which strongly suggests anodeless superconductivity in Mo Rh N. Such conclusionis supported also by low- T specific-heat data. Unlike the clean-limit case [ see Eq. (3) ] , in the dirty limitthe coherence length ξ is much larger than the electronicmean-free path l e . In this case, in the BCS approximation,the temperature dependence of the superfluid density isgiven by: ρ sc ( T ) = ∆ ( T ) ∆ tanh • ∆ ( T ) k B T ˜ . (4)Following the above equation, the estimated gap valueis 0.68(1) meV, slightly smaller than the clean-limit value,yet still in excellent agreement with the gap values ex-tracted from low- T specific-heat (0.67 meV) and Andreev-reflection spectroscopy data (0.59 meV). Such ‘dirty’-nature of SC might reflect the large electrical resistivity( ρ = Ω cm) and the small residual resistivity ratio(RRR ∼
1) of Mo Rh N. The 2 ∆/ k B T c ratios of about 3.46(dirty limit) and 3.84 (clean limit) are both comparable to3.53, the ideal value expected for a weakly-coupled BCSsuperconductor. – 3 – A sy mm e t r y Time (ms)
FIG. 5. Coinciding ZF- µ SR spectra in the superconducting (1.5 K)and the normal state (8 K) show that in Mo Rh N the TRS is pre-served. Both spectra show only a weak muon-spin depolarization,but no visible differences. The solid line is a fit to the 1.5-K spectraby means of Eq. (5), as described in the text.
Zero-field µ SR.
We performed also ZF- µ SR measurements,in order to search for a possible TRS breaking in the su-perconducting state of Mo Rh N. The large muon gyro-magnetic ratio, combined with the availability of 100%spin-polarized muon beams, make ZF- µ SR a very sensi-tive probe for detecting small spontaneous magnetic fields.This technique has been successfully used to detect theTRS breaking in the superconducting states of differenttypes of materials.
Normally, in the absence of ex-ternal fields, the onset of SC does not imply changes inthe ZF muon-spin relaxation rate. However, if the TRSis broken, the onset of tiny spontaneous currents givesrise to associated (weak) magnetic fields, readily detectedby ZF- µ SR as an increase in muon-spin relaxation rate.Given the tiny size of such effects, we measured the ZF- µ SR with high statistics in both the normal and the su-perconducting phases. Representative ZF- µ SR spectra col-lected above (8 K) and below (1.5 K) T c for Mo Rh N areshown in Fig. 5. For non-magnetic materials, in the ab-sence of applied fields, the relaxation is mainly determinedby the randomly oriented nuclear moments, which can bedescribed by a Gaussian Kubo-Toyabe relaxation function G KT = h + ( − σ t ) e ( − σ t ) i . The ZF- µ SR spectraof Mo Rh N can be modeled by adding a Lorentzian relax-ation Λ to the Kubo-Toyabe function: A ZF = A s G KT e − Λ t + A bg . (5)Here A s and A bg are the same as in the TF- µ SR case [ seeEq. (1) ] . The resulting fit parameters are summarized inTable I. The weak Gaussian and Lorentzian relaxation ratesreflect the small value of Mo Rh N nuclear moments. Therelaxations show very similar values in both the normal andthe superconducting phase, as demonstrated by a lack of vis- ible differences in the ZF- µ SR spectra above and below T c .This lack of evidence for an additional µ SR relaxation be-low T c , implies that TRS is preserved in the superconduct-ing state of Mo Rh N. Since TRS is preserved also in theMo Al C sister compound, this explains the many commonfeatures shared by these two β -Mn-type NCSCs. Discussion.
Since the admixture of spin-singlet and spin-triplet pairing depends on the strength of ASOC, the latterplays an important role in determining the superconduct- Table I. Fit parameters extracted from ZF- µ SR data for Mo Rh N(collected above and below T c ) by using the Eq. (5) model.Temperature 1.5 K 8 K A s σ ( µ s − ) 0.0366(69) 0.0379(58) Λ ( µ s − ) 0.0069(32) 0.0047(28) A bg ing properties of NCSCs. An enhanced ASOC can turn afully gapped s -wave superconductor into a nodal supercon-ductor, with typical features of spin-triplet pairing, as exem-plified by the Li (Pd,Pt) B case. However, a larger SOC isnot necessarily the only requirement for a larger ASOC andan enhanced band splitting E ASOC , since the latter two de-pend also on the specific crystal- and electronic structures.All 4 d -Rh, -Ru and 5 d -Ir are heavy SOC metals, but theirASOC-related band splittings E ASOC are relatively small insome materials. For example, the expected E ASOC valuesfor Ce(Rh,Ir)Si , LaRhSi , Rh Ga , and Ru B are less than20 meV (i.e., ten times smaller than in CePt Si or Li Pt B). Therefore, their pairing states remain in the spin-singletchannel and all of them behave as fully-gapped supercon-ductors. In β -Mn-type materials, like Mo Rh N, the replace-ment of a light metal such as Al by the heavy Rh does indeedincrease the SOC, yet the E ASOC still remains weak. Hence,the superconducting pairing is of spin-singlet type, in goodagreement with both TF- and ZF- µ SR results. Further bandstructure calculations, which explicitly take into account theSOC effects, are needed to clarify this behavior.
Summary.
We perfomed comparative µ SR experimentsto study the superconducting properties of NCSC Mo Rh N.Bulk superconductivity with T c = Rh N, which is well de-scribed by an isotropic s -wave model and is consistent witha spin-singlet pairing. The lack of spontaneous magneticfields below T c indicates that time-reversal symmetry is pre-served in the superconducting state of Mo Rh N.This work was supported by the Schweizerische Natio-nalfonds zur Förderung der Wissenschaftlichen Forschung,SNF (Grants 200021-169455 and 206021-139082) and theNational Natural Science Foundation of China (Grant No.11504378). ∗ Corresponding authors:[email protected] P. W. Anderson, “Structure of “triplet” superconducting energygaps,” Phys. Rev. B , 4000–4002 (1984). E. Bauer and M. Sigrist, eds.,
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