Muon Spin Spectroscopy Study of the Noncentrosymmetric

Superconducting and normal state properties of noncentrosymmetric superconductor NbOs2 investigated by muon spin relaxation and rotation

D. Singh,1 Sajilesh K.P,1 Sourav Marik,1 A. D. Hillier,2 and R. P. Singh1, ∗ 1Indian Institute of Science Education and Research Bhopal, Bhopal, 462066, India 2ISIS facility, STFC Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Oxfordshire, OX11 0QX, UK (Dated: January 18, 2019)

Noncentrosymmetric superconductors with α-manganese structure has attracted much attention recently, after the discovery of time-reversal symmetry breaking in all the members of Re6X (X = Ti, Hf, Zr) family. Similar to Re6X, NbOs2 also adopts α-Mn structure and found to be superconducting with critical temperature Tc ' 2.7 K. The results of the resistivity, magnetization, specific heat and muon-spin relaxation/rotation measurements show that NbOs2 is a weakly coupled type- II superconductor. Interestingly, the zero-field muon experiments indicate that the time-reversal symmetry is preserved in the superconducting state. The low-temperature transverse-field muon measurements and the specific heat data evidence an conventional isotropic fully gapped superconductivity. However, the calculated electronic properties in this material show that the NbOs2 is positioned close to the band of unconventionality of the Uemura plot, indicating that NbOs2 potentially borders an unconventional superconducting ground state.

I. INTRODUCTION In these superconductors, additional complications come due to the presence of strong electronic correlation effects The discovery of exotic superconducting properties in and quantum criticality, which often hinders the research the heavy fermion noncentrosymmetric superconductor aimed to understand the interplay between crystal sym- (NCS) CePt3Si [1], sparked renewed research interest metry and superconductivity. As a result, many new both from both experimental and theoretical perspec- noncentrosymmetric superconductors with weak correla- tives, to understand the role of structural asymmetry tion have been targeted to look for singlet-triplet mixing. in superconductivity. Theoretically, in the Bardeen- Some of these compounds are LaNiC2 [16, 17], Li2Pd3B Cooper-Schrieffer (BCS) superconductors, the pairing [18–23], Li2Pt3B[20–25], Ru7B3 [26], Nb0.18Re0.82 [27], state is protected by the crystal inversion symmetry. Re3W[28], Y2C3 [29], Mo3Al2C[30, 31]. A particularly When a superconductor has an inversion center, the interesting example is Li2M3B (M = Pd, Pt) [22–25, 32– pairing states can be unambiguously classified into de- 39]. Li2Pd3B exhibits a isotropic fully gapped supercon- generated states of spin-singlet (even parity) and spin- ductivity [21–23], whereas the gap of Li2Pt3B has line triplet (odd parity). However, this scenario changes in nodes [23–25]. Here the substitution of Pd with Pt el- noncentrosymmetric superconductors, where the lack of ement strengthens the ASOC and enhances the relative inversion symmetry induces odd parity antisymmetric pairing mixing ratio with a dominant spin-triplet compo- spin-orbital coupling (ASOC). A strong ASOC results in nent [23]. Hence, the change from fully gapped to nodal mixed parity states and develops complicated spin struc- superconductivity is directly dependent on the magni- tures [2–4]. Such an admixture of spin states reflects in tude of ASOC. Indeed, 11B NMR experiments also con- a variety of unique properties, such as: anomalous upper clude that the Cooper pair included about 60% of the critical field values [5–7], line nodes [8,9], and magne- spin-triplet pairing in Li2Pt3B. Accordingly, many non- toelectric effects [10, 11] etc. Besides, recently it was centrosymmetric superconductors with wide variety of proposed that some NCS can also be a potential candi- SOC have been studied, yet most of them such as LaMSi3 date for topological superconductivity, of which known (M = Pd, Pt)[40], LaMSi3 (M = Rh, Ir)[41, 42], BiPd examples are PbTaSe2 [12, 13] and BiPd [14, 15], which [43, 44], Nb0.18Re0.82 [27], Re3W[28] found to be com- patible with fully gapped superconductivity. Thus, the arXiv:1901.05630v1 [cond-mat.supr-con] 17 Jan 2019 shows topological surface states and potentially lead to signatures of Majorana fermions. importance of ASOC strength in the degree of mixing ra- Since CePt3Si, several more heavy fermion superconduc- tio is not fully understood, and therefore requires to find tors have been found to exhibit a wide variety of un- more examples where the effects of modified spin-orbit usual superconducting properties [2]. But it was soon coupling on the singlet-triplet mixing can be studied di- realized that heavy fermion noncentrosymmetric super- rectly. conductors are not the simplest of systems to study. Another noteworthy feature of unconventional superconductivity is time-reversal symmetry (TRS) breaking, which is an extremely rare phenomenon has been found only in few noncentrosymmetric superconductors, such ∗ [email protected] 2 as: LaNiC2 [16], La7Ir3 [45], SrPtAs [46] and Re6X (X = Ti,Hf,Zr) [47–49]. It is the latter family of compounds that are particularly interesting owing to the frequent occurrence of TRS breaking among all the members. In Nb Re X, the TRS breaking effects seem very similar irre- 6 Os spective of the substitution at X-sites, suggesting a negligible effect on the ASOC and hints towards its common c b origin. Interestingly, the band-structure calculations of a the Re-based binary alloys [50, 51], indicate that the density of states (DOS) at the Fermi level to be dominated by the d-bands of Re and could be the critical factor leading to TRS breaking. Indeed, recent observation

of TRS breaking in pure Re metal (centrosymmetric), Intensity (arb. units) strongly suggests that the sizable Re spin-orbit coupling to be the origin of TRS breaking in the Re6X family [52]. However, to strengthen the above conclusion one need to study other Re-free materials, whilst still having the 30 60 90 same α-Mn type structure. In this context, we report 2θ(°) a detailed investigation of the superconductivity of NCS NbOs2, isostructural to the Re6X family. NbOs2 exhibits bulk superconductivity around 2.7 K [53, 54], character- FIG. 1. The powder x-ray diffraction pattern of NbOs2 at ized via electrical resistivity, specific heat, magnetic sus- room temperature. The line is a Rietveld refinement to the ceptibility, and muon spectroscopy. The principal goal of data. The inset shows the crystal structure of NbOs2. the present work is to study the superconducting properties of NbOs2 and search for TRS breaking in a Re-free material with α-Mn structure. tation (TF-µSR) measurements were performed on the MuSR beam line at the ISIS pulsed muon source. A full description of the µSR technique may be found in Ref. II. EXPERIMENTAL DETAILS [55]. In ZF-µSR, the contribution from the stray fields at the sample position due to neighbouring instruments and the Earth’s magnetic field is cancelled to within ∼ 1.0 µT The polycrystalline samples of NbOs were prepared 2 using three sets of orthogonal coils. TF-µSR measure- by arc melting a stoichiometric mixture of Nb (99.95 %) ments are performed to measure the temperature depen- and Os (99.95 %) under high purity argon gas atmosphere dence of the magnetic-penetration depth λ(T). λ−2(T) is on a water cooled copper hearth. The as-cast ingot was proportional to the superfluid density, and can provide flipped and remelted several times to ensure the phase ho- information on the symmetry of the gap structure. mogeneity. The observed mass loss was negligible. The sample was then sealed inside an evacuated quartz tube and annealed at 800 ◦C for one week. To verify the phase purity we performed room temperature (RT) powder x- III. RESULTS AND DISCUSSION ray diffraction (XRD) using a X’pert PANalytical diffrac- a. Normal and superconducting state properties tometer (Cu-Kα1 radiation, λ = 1.540598 Å). The superconducting properties of NbOs2 were measured using magnetization M, ac susceptibility χac, electrical resis- Figure1 shows the room temperature powder X-ray tivity ρ, specific heat C, and muon relaxation/rotation diffraction pattern. Rietveld refinement of the RT XRD (µSR) measurements. The dc magnetization and ac sus- pattern shows that our sample is in the single phase, ceptibility data were collected using a Quantum Design crystallizing with a cubic α-Mn structure (space group superconducting quantum interference device (SQUID). I43¯ m) with lattice cell parameter a = 9.654(3) Å. A The electrical resistivity and specific heat measurements schematic view of the crystal structure of NbOs2 is shown were performed on a Quantum Design physical property in the inset of Fig.1. Refined lattice parameters are measurement system (PPMS). The µSR measurements shown in TableI and are in good agreement with the were conducted at the ISIS Neutron and Muon facil- published literature [53]. In the crystal structure except ity, in STFC Rutherford Appleton Laboratory, United for the Nb(1) site (2a), no other sites have an inversion Kingdom using the MUSR spectrometer. The powdered center. The sample used here is found to have a refined NbOs2 sample was mounted on a silver holder and placed stoichiometry of NbOs1.98(3), close to the target compo- in a sorption cryostat, which we operated in the temper- sition NbOs2, with the possibility of some site mixing ature range 0.3 K - 3.0 K. Zero-field muon spin relax- between Nb and Os sites [56]. ation (ZF-µSR) and the transverse-field muon spin ro- Temperature dependence of the electrical resistivity 3

TABLE I. Crystal structure parameters obtained from the 132 Rietveld reﬁnement of the room temperature powder x-ray NbOs2 (a) diﬀraction of NbOs2 130 Structure Cubic 130 Space group I43¯ m cm )

Lattice parameters cm ) 128 onset a(Å) 9.654(3) 65 Tc = 2.7 K ρ (µΩ ρ (µΩ

Atomic Coordinates 126 0 1.5 2.0 2.5 3.0 3.5 Temperature (K)

Atom Wyckoﬀ position x y z 124 Nb1 2a 0 0 0 0 50 100 150 200 250 300 Temperature (K) Os1 8c 0.322(1) 0.322(1) 0.322(1) 0.0 Nb2 8c 0.322(1) 0.322(1) 0.322(1) (b) Os2 24g 0.3536(6) 0.3536(6) 0.0371(8) onset Tc = 2.7 K Nb3 24g 0.3536(6) 0.3536(6) 0.0371(8) -0.4 Os3 24g 0.0912(3) 0.0912(3) 0.2828(4)

πχ H = 1 mT 4

-0.8

ZFCW FCC ρ(T ) in the temperature range 1.8 K to 300 K in zero applied magnetic field is shown in Fig.2 (a). The resid- -1.2 1 2 3 4 5 ual resistivity ratio (RRR) is 1.02(1). This low value for Temperature (K) the RRR is comparable to other α-Mn structure compounds such as Nb0.18Re0.82 ('1.3)[27], Re3W('1.15) [28], Re6Hf ('1.08) [57], Re6Zr ('1.09) [58], and Re3Ta FIG. 2. (a) Temperature dependence of resistivity over the ('1.04) [59]. These compounds are widely known to have range 1.8 K ≤ T ≤ 300 K, with the inset showing a drop onset strong electronic scattering, with a large residual resis- in resistivity at the superconducting transition, Tc = 2.7 tivity due to occupational site disorder. This means that ± 0.2 K. The normal state resistivity fitted with the Bloch- the resistivity of the materials with α-Mn structure is Gru¨neisen model shown by solid red line.(b) Temperature dependence of the χ(T) is shown over the range of 1.8 K ≤ T sensitive to disorder which is responsible for the poor onset ≤ 5 K, displaying strong diamagnetic signal around Tc = conductivity. A superconducting transition can be seen 2.7 ± 0.1 K in H = 1 mT. clearly in the inset of Fig.2(a) with a onset temperature onset Tc = 2.7 ± 0.2 K. The normal-state resistivity for a non-magnetic metallic (6.2 ± 3.2) µΩcm and residual resistivity ρ0 = (130 ± 2) crystalline solid is analyzed using the Bloch-Grüneisen µΩcm. The value of the Debye temperature ΘR is close (BG) model, which describes the resistivity arising due to that extracted from the specific heat data. to electrons scattering from acoustic phonons. The tem- Figure2(b) displays the temperature dependence of mag- perature dependence of the resistivity, ρ(T), is modeled netic susceptibility χ(T ) measured in an applied field of as 1 mT. Both zero-field cooled warming (ZFCW) and field- cooled cooling (FCC) regimes show a sharp diamagnetic ρ(T ) = ρ0 + ρBG(T ) (1) onset transition around Tc = 2.7 ± 0.1 K, indicating the onset of bulk superconductivity. In the FCC data, the where ρ0 is the residual resistivity due to the defect scattering and is essentially temperature independent superconductor does not return to a full Meissner state, whereas ρ is the BG resistivity given by [60] indicating strong flux pinning. The calculated Meissner BG volume fraction (4πχ) is close to 100 %, suggesting com- n T Z ΘR/T xn plete superconductivity within the sample. ρBG(T ) = C x −x dx. (2) To estimate the lower critical field Hc1, low-field mag- ΘR (e − 1)(1 − e ) 0 netization data M(H) curves were measured at differ- Here ΘR is the Debye temperature obtained from resis- ent fixed temperatures from 1.8 K to 2.5 K as shown in tivity measurements, while C is a material dependent Fig.3(a). The M(H) data follows a linear relation with pre-factor and n takes values between 2-5 depending on the applied magnetic field below Hc1, whereas, above this the nature of the electron scattering [61]. A fit to the data point, magnetization starts to deviate from the straight employing this model is shown in Fig.2(a) and gave n = line behavior due to flux penetration. The point of devia- (4.2 ± 0.2), Debye temperature ΘR = (376 ± 2) K, C = tion was computed for different isotherm curves to obtain 4

0.00 3 0.04 2.5 K 1.8 K 4 K H 0.02 c1

(mT) 2

-0.08 0.00 emu/g) (emu/g) M ( M 1 -0.02 M = -H HIrr(1.8 K) = 0.38 T

(a) 1.8 K Hc1(0) = 2.98 ± 0.11 mT (b) -0.16 0 -0.04 0 1 2 3 0.5 1.0 -1.8 -1.2 -0.6 0.0 0.6 1.2 1.8 H (mT) T/Tc H (T) 0.014 4 (c) (d) 0.0 α M = 0 ' -0.4 0.007

3 4πχ -0.8 0.05 T to 1.5 T

0.000 -1.2 0.02 1.5 2.0 2.5 3.0 3.5 T(K) Hc2 (T) 2

emu/g) α 0.01 c2 M = 0.95 -0.007 H M(

1.95 emu/g)

( 0.00 1.97 M 1 -0.014 2.0 -0.01 H(AC) 2.1 H(M(H)) 2.3 0 1 2 2.5 H(T) -0.021 0 0 1 2 0 1 2 3 H(T) Temperature (K)

FIG. 3. (a) The M(H) curves at different fixed temperatures as a function of applied magnetic fields. The lower critical field Hc1 is determined by the GL formula is 2.98 ± 0.11 mT. (c) Zero-field cooled magnetic isotherms at T = 1.8 K and T = 4.0 K. (c) Magnetization versus magnetic field is shown at several fixed temperatures. The inset shows the detailed view of M(H) curve, where Hc2 is determined from the discontinuity in the gradient. (d) Upper critical field versus temperature of NbOs2 determined from the high-field M(H) curves and ac susceptibility data. The black and blue dotted line shows the prediction of Hc2(0) using Eq. (4). The inset shows the ac susceptibility data measured at different applied magnetic field.

the temperature dependence of Hc1(T), as presented in behavior clearly corresponds to a type-II superconductor. the Fig.3(a). The resulting graph is modeled using the The irreversibility field, HIrr, is defined as the field where Ginzburg-Landau relation the magnetic hysteresis collapses. At a given temperature for field H < HIrr, the magnetization is irreversible ow- " 2# T ing to pinned vortices, whereas for H > HIrr, it becomes Hc1(T ) = Hc1(0) 1 − . (3) Tc reversible as the applied field is strong enough to depin the vortices. As shown in Fig.3(b) at T = 1.8 K, HIrr = 0.38 ± 0.02 T. The magnetization data were also col- When fitted to the experimental data, it yields Hc1(0) = 2.98 ± 0.11 mT. lected in the normal state at T = 4 K and we conclude Figure3(b) presents the high-field magnetization hystere- that there is no evidence for a magnetic impurity phase sis loop collected in the superconducting state at T = 1.8 in our sample. K, in the magnetic field range below ± 2 T. The magnetic The magnetic hysteresis loop at different temperatures is 5 displayed in Fig.3(c). The hysteresis in magnetization The value for Hc2 can therefore be calculated using the ∆M decreases with increasing temperature and magnetic relation [64] field, characteristic of a conventional type-II supercon- Horbital ductor. The gradient has a clear discontinuity at a field H = c2 , (7) c2 p 2 that is identified as the upper critical field Hc2 [see inset 1 + αM Fig.3(c)]. The resulting values of H are determined c2 where it is considered λ = 0. For α = 0, H = using this discontinuity at different magnetic isotherms M c2 Horbital(0) = (3.36 ± 0.07) T. plotted in Fig.3(d) (solid triangle markers). The tem- c2 The value for H used to determine the Ginzburg Lan- perature dependence of H (T) is also determined from c2 c2 dau coherence length ξ using the relation [65] the ac susceptibility measurements (solid square mark- GL ers Fig.3(d)), where the shift in Tc is evaluated under Φ0 different applied magnetic fields up to 1.5 T [see inset of Hc2(0) = 2 , (8) 2πξGL Fig.3(d)]. The Hc2 vs. T curve can be described by the −15 2 Werthamer-Helfand-Hohenberg (WHH) model [62, 63] by where Φ0 (= 2.07 × 10 T m ) is the magnetic flux including the orbital breaking, the effect of Pauli spin quantum. For Hc2(0) = (3.36 ± 0.07) T, we find ξGL(0) paramagnetism (α) and spin-orbit scattering parameter = (99 ± 1) Å. (λSO). The value of α measures the relative strengths Consequently, the values of Hc1 and ξGL can be em- of the orbital and Pauli-limiting field, while λSO is dom- ployed to calculate the Ginzburg-Landau penetration inated by the atomic numbers of the elements. In the depth λGL(0) from the expression [65] WHH model Φ0 λGL(0) Hc1(0) = ln . (9) λ ! 2 1 1 iλ 1 h¯ + SO + iγ 4πλGL(0) ξGL(0) ln = + SO ψ + 2 t 2 4γ 2 2t Integrating the values of Hc1(0) = (2.98 ± 0.11) mT and ! ξ (0) = (99 ± 1) Å in Eq. (9), yields λ (0) = (4608 ± 1 iλ 1 h¯ + λSO + iγ 1 GL GL + − SO ψ + 2 − ψ , (4) 88) Å. The Ginzburg-Landau parameter κ = λ /ξ = 2 4γ 2 2t 2 √ GL GL 46.5 ± 0.4 > 1/ 2, therefore, NbOs2 classified as strong type-II superconductor. where t is the reduced temperature T/Tc, ψ is the di- The thermodynamic critical field Hc(0) is given by 2 ¯ 2 λSO 2 ¯ agamma function, γ = (αh) − ( 2 ) , and h is the dimensionless form of the upper critical field given by h¯ Φ H (0) = √ 0 , (10) = (4π2)(H |dH (T )/dT | ). c c2 c2 T =Tc 2 2πλGLξGL(0) The Maki parameter which measures the relative strengths of the orbital and Pauli-limiting field is cal- and found to be Hc(0) = (51 ± 1) mT. culated using the relation Recently unconventional vortex dynamics has been observed in some noncentrosymmetric superconductors √ orb HC2 (0) where unusual vortex pinning mechanism and flux creep αM = 2 p . (5) HC2(0) rates were proposed [66, 67]. Therefore, it is necessary to measure the stability of vortex system against the various For a type-II superconductor in the dirty limit, the or- factors unsettling the vortex equilibrium. The measure orbital bital limit of the upper critical field Hc2 (0) is given of stability against the thermal fluctuations can be given by the WHH expression by by Ginzburg number Gi. This is basically the ratio of thermal energy kBTc to the condensation energy associ- orbital −dHc2(T ) H (0) = −0.693Tc (6) ated with the coherence volume. Ginzburg number, Gi, c2 dT T =Tc is given by [68]

−1 For initial slope of −1.80±0.04 TK near Tc calculated 1 k µ τT 2 orbital G = B 0 c , (11) from Hc2-T plot, it gives Hc2 (0) = (3.36 ± 0.07) T. i 3 2 p 2 4πξ (0)Hc (0) The Pauli limiting field is given by Hc2(0) = 1.86Tc = (5.02 ± 0.18) T. The Maki parameter is then calculated where τ is anisotropy parameter which is 1 for the cubic to be αM = 0.95. NbOs2. For ξ(0) = (99 ± 1) Å, Hc(0) = (51 ± 1) mT Figure3(d) show the temperature dependence of Hc2 for and Tc = (2.7 ± 0.1) K, we obtained Gi = (1.1 ± 0.1) −6 two cases: αM = 0.95, λ = 0 and αM = 0, λ = 0, dis- × 10 . In conventional low Tc superconductors, pin- played by dotted black and blue lines. It is clear from ning is strong, whereas thermal fluctuations are weak, −8 the graph that the measured data is best described with with Gi ∼ 10 . In high temperature superconductors, αM = 0, whereas the calculation with αM = 0.95 fails to Tc is high and hence the coherence volume is small, mak- −2 account for the experimental data. This certainly implies ing it sensitive to thermal fluctuations, Gi ∼ 10 . In that Hc2 is limited by orbital critical field and the Pauli our compound Gi value is more towards the low Tc su- limiting appears to have a limited effect if any at all. perconductors, suggesting that thermal fluctuations may 6

3 0.3 25 s -wave fit ) 2 20 ∆(0) /kBTc = 1.77 3.0 K 15 0.5 K

2 10 2 5 /T(mJ/molK γ +β Τ +β Τ T C/T = n 3 5 C n 5 γ

/ 0 25 50 75 100 0.2 2 2 el T (K ) C

1 Asymmetry

Tc = 2.7 K

0 0.1 0 1 2 3 4 0 5 10 15 Temperature (K) Time (µs)

FIG. 4. The specific heat data in superconducting regime FIG. 5. Zero-field µSR spectra collected below (0.5 K) and fitted for single-gap s-wave model [Eq.(15)] for a fitting pa- above (3 K) the superconducting transition temperature. The rameter α = ∆(0)/kB Tc = 1.77 ± 0.01. Inset: C/T vs solid lines are the fits to Guassian Kubo-Toyabe (KT) func- T 2 data was fitted between 3 K ≤ T 2 ≤ 100 K by Eq. tion given in Eq.(16). 2 4 C/T = γn + β3T + β5T to measure electronic and phononic contribution to specific heat. (Cph) from total specific heat data.

3 5 not be playing any important role in vortex unpinning in Cel = C − Cph = C − (β3T + β5T ). (13) this material. Figure4 shows the low temperature specific heat data Once the phononic contribution is subtracted, an equal measured in zero applied field. The plot C/T vs T 2 is entropy conservation line is drawn to estimate the nor- shown in the inset of Fig.4, in the temperature range malized specific heat jump. The value for the specific 2 heat jump ∆C ' 1.45, which is remarkebly similar to 3 K ≤ T ≤ 100 K. A jump at around Tc = 2.7 K, γnTc apparently confirms bulk superconductivity in NbOs2. the value for a BCS isotropic gap superconductor (= The normal state specific heat data above Tc is con- 1.43) in the weak coupling limit, indicating that NbOs2 tributed by both electronic and phononic parts given by: is a weakly coupled superconductor, consistent with the 2 4 C/T = γn + β3T + β5T , where γn is the Sommerfeld λe−ph value obtained above. coefficient and β3, β5 are phononic contributions. The The temperature dependence of the specific heat data in solid red line represents the best fit to the data with γn the superconducting state can best be described by the −1 −2 = 8.58 ± 0.01 mJ mol K , β3 = 0.123 ± 0.002 mJ BCS expression for the normalized entropy S written as mol−1 K−4, and β = 0.43 ± 0.04 µJ mol−1 K−6 [see 5 S 6 ∆(0) Z ∞ inset Fig.4]. The value for γ was used to determine n = − 2 [f ln(f) + (1 − f) ln(1 − f)]dy, γnTc π kBTc the density of states at the Fermi level Dc(EF) using the 0 2 2 (14) relation γn = (π kBDc(EF))/3, where EF is the Fermi −1 energy. For γ = 8.58 ± 0.01 mJ mol−1 K−2, it yields where f(ξ) = [exp(E(ξ)/kBT )+1] is the Fermi function, n p 2 2 D (E ) = 3.64 ± 0.02 states eV−1 f.u.−1. The Debye E(ξ) = ξ + ∆ (t), where ξ is the energy of normal elec- c F trons measured relative to the Fermi energy, y = ξ/∆(0), 4 1/3 temperature θD = 12π RN/5β3 , where using R = t = T /T , and ∆(t) = tanh[1.82(1.018((1 /t)-1))0.51] is −1 −1 c 8.314 J mol K and N = 3, yields θD = 362 ± 1 the BCS approximation for the temperature dependence K. The value of θD = 362 K can be used to calculate of the energy gap. The normalized electronic specific heat the electron-phonon coupling constant λe−ph using the is calculated by McMillan formula [69], Cel d(S/γnTc) ∗ = t . (15) 1.04 + µ ln(θD/1.45Tc) γnTc dt λe−ph = ∗ , (12) (1 − 0.62µ )ln(θD/1.45Tc) − 1.04 Fitting the low temperature specific heat data using this where µ∗ represents the repulsive screened Coulomb po- model as shown by the solid red line in Fig.4, yields ∗ tential, usually given by µ = 0.13. With Tc = 2.7 K α = ∆(0)/kBTc = 1.77 ± 0.01. This is consistent with and θD = 361.67 K, we obtained λe−ph ' 0.52, which the value for a BCS superconductor αBCS = 1.764 in is similar to other weakly coupled NCS superconductors the weak coupling limit. Therefore, good agreement be- [27, 70]. tween the measured data (black symbols) and the BCS fit The electronic contribution (Cel) to the specific heat can (solid red line), confirms an isotropic fully gapped BCS be calculated by subtracting the phononic contribution superconductivity in NbOs2. It should be noted that in 7 order to extract the true nature of the superconducting 0.25 gap, the specific heat data up to the low temperature 3.0 K 0.20 0.4 K region need to be analyzed. It can give a complete un- 0.15 derstanding whether the superconducting gap is isotropic 0.10 (exponential) or have nodes (power law). A summary of 0.05 all the experimentally measured and estimated parame- 0.00 ters is given in TableII. -0.05 To further inspect the superconducting ground state -0.10 of NbOs2, we performed the muon spin relaxation and Asymmetry rotation measurements. Firstly, ZF-µSR measurements -0.15 were carried out to investigate the occurrence of TRS -0.20 (a) -0.25 breaking in NbOs2. ZF-µSR being extremely sensitive to 0 1 2 3 4 5 6 -1 tiny changes in the internal magnetic fields, can unam- Time (µs ) biguously detect the presence of a TRS breaking signal. 0.27 This technique was proved to be very crucial in establish- H = 20 mT ing the TRS breaking in several superconductors such as Tc La7Ir3 [45], Re6(Ti,Hf,Zr) [47–49], Sr2RuO4 [71, 72] etc. Figure5 shows the muon spin relaxation time spectra, 0.18 ) 20.1 collected below (T = 0.5 K < Tc) and above (T = 3.0 -1 s 20.0 Bbg µ

K > Tc) the superconducting transition temperature Tc ( = 2.7 K. The absence of precessional signals suggests sc 19.9 σ 0.09 the absence of coherent internal ﬁelds which is generally 19.8 (mT) associated with long range magnetic ordering. In the 19.7 0 1 2 3 (b) absence of atomic moments muon-spin relaxation in zero T (K) 0.00 ﬁeld can best be described by the Gaussian Kubo-Toyabe 0 1 2 3 (KT) function [73] together with a non-decaying constant Temperature (K) background, ABG, 2.7 1 2 −σ2 t2 G (t) = A + (1 − σ2 t2)exp ZF exp(−Λt) KT 1 3 3 ZF 2 Tc + ABG, (16) 1.8 ) -2

where A1 is the initial asymmetry and σZF accounts for m µ the relaxation due to static, randomly oriented local fields ( -2 0.9 associated with the nuclear moments at the muon site λ and Λ is the electronic relaxation rate. Near identical µSR data Dirty s-wave model relaxation signals, as seen in Figure5, on the either side (c) of the superconducting transition suggest the absence of 0.0 any additional magnetic moments in the superconduct- 0 1 2 3 ing ground state, usually associated with exotic phenom- Temperature (K) ena such as TRS breaking. This clearly suggests that the time-reversal symmetry is preserved in NbOs2 within the detection limit of µSR. FIG. 6. (a) Representative TF-µSR signals collected at 3.0 K Figure6(a) shows the TF- µSR precessional signals for and 0.4 K in an applied magnetic field of 20 mT.(b) Tempera- ture dependence of the superconducting contribution to depo- NbOs2 collected above (3.5 K) and below (0.4 K) Tc in larization σsc in an applied magnetic field of 20 mT. The inset an applied magnetic field of 20 mT. The measurements shows the internal field variation with temperature where a were done in the field-cooled mode where the field 20 clear diamagnetic signal appears around Tc. (c) Temperature mT was applied above the transition temperature. The dependence of λ−2 follow a single gap s-wave model in dirty sample was then subsequently cooled to the base tem- limit for ∆(0) = 0.43 ± 0.02. perature of 0.4 K. In the normal state there is an almost homogeneous field distribution throughout the sample, The TF-µSR time spectra, for all temperatures, is best where the weak depolarization is due to nuclear dipo- described by the sinusoidal oscillatory function damped lar field. The depolarization rate in the superconducting with a Gaussian relaxation and an oscillatory background state becomes more prominent, due to the formation of term arising from the muons implanted directly into the an inhomogeneous field distribution in the flux line lat- silver sample holder that do not depolarize, tice (FLL) state. The TF-µSR spectra in6(a) show a very small difference between above and below T , indi- −σ2t2 c G (t) = A exp cos(γ B t + φ) cating large magnetic penetration depth. TF 0 2 µ int 8

+ A cos(γ B t + φ). (17) 1 µ bg TABLE II. Experimentally measured and estimated superconducting and normal-state properties for the noncentrosymmetric superconductor NbOs2 Here A0 and A1 are the initial asymmetries of the sample and background signals, B and B are the in- int bg Properties unit value ternal and background magnetic fields, φ is the initial phase offset, γµ/2π = 135.53 MHz/T is the muon gy- romagnetic ratio and σ is the depolarization rate. The Tc K 2.7 ± 0.2 depolarization rate σ comprised of the following terms: Hc1(0) mT 2.98 ± 0.11 2 2 2 σ = σsc + σN, where σsc is the depolarization arising Hc2(0) T 3.36 ± 0.07 due to the field variation across the flux line lattice and Hc(0) mT 51 ± 1 P σN is the contribution due to nuclear dipolar moments. Hc2(0) T 5.02 ± 0.18

The superconducting contribution to the depolarization ξGL Å 99 ± 1 rate σ is calculated by using the above relation. Figure sc λGL Å 4608 ± 88 6(b) shows the temperature dependence of σsc where be- κ 46.5 ± 0.4 low Tc, σsc increases systematically as the temperature γ mJmol−1K−2 8.58 ± 0.01 is lowered. The inset of Fig.6(b) shows the temperature β mJmol−1K−4 0.123 ± 0.002 dependence of internal magnetic field, where the flux ex- θD K 362 ± 1 pulsion at Tc is evident from the reduction of average feld < B > inside the superconductor, and the correspond- λe−ph 0.52 ± 0.02 DC (Ef ) states/ev f.u 3.64 ± 0.02 ing background field Bbg is approximately constant over the temperature range. In an isotropic type-II super- ∆Cel/γnTc 1.45 ± 0.03 conductor with a hexagonal Abrikosov vortex lattice the ∆(0)/kB Tc 1.77 ± 0.01 magnetic penetration depth λ is related to σsc by the equation [74]:

√ −1 4 3 −2 IV. ELECTRONIC PROPERTIES σsc(µs ) = 4.854 × 10 (1 − h)[1 + 1.21(1 − h) ]λ , (18) where h = H/Hc2(T ) is the reduced field. The above The Sommerfeld coefficient is related to the quasipar- equation is valid for systems κ > 5. For NbOs2, κ = (46.5 ticle number density (n) per unit volume given by the ± 0.4) and the temperature dependence of Hc2(T ) is relation [75] shown in Fig.3(d). Using the data of Hc2(T ) for NbOs2, −2 2/3 2 ∗ 1/3 the temperature dependence of λ was extracted from π kBm Vf.u.n γn = 2 (20) Eq. (18), as displayed in Fig.6(c). The temperature de- 3 ~ NA −2 pendence of λ nearly constant below Tc/3 ' 0.9 K. This possibly suggests the absence of nodes in the super- where kB is the Boltzmann constant, NA is the Avogadro ∗ conducting energy gap at the Fermi surface. The solid number, Vf.u. is the volume of a formula unit and m is line in Fig.6(c) represent the temperature dependence of the effective mass of quasiparticles. The residual resis- the London magnetic penetration depth λ(T ) within the tivity, ρ0, can be calculated, using the equation local London approximation for a s-wave BCS superconductor in the dirty limit: 3π2~3 l = 2 ∗2 2 (21) e ρ0m vF λ−2(T ) ∆(T ) ∆(T ) while the Fermi velocity vF can be written in terms of −2 = tanh , (19) λ (0) ∆(0) 2kBT effective mass and the carrier density by

∗ 3 0.51 1 m vF where ∆(T )/∆(0) = tanh{1.82(1.018(Tc/T − 1)) } is n = . (22) the BCS approximation for the temperature dependence 3π2 ~ of the energy gap, where ∆(0) is the gap magnitude at zero temperature. Dirty-limit expression was used in ac- For superconductors in the dirty-limit, where ξ0/l >> cordance with the calculation done in the section below 1, the properties are aﬀected due to the scattering of electrons. The dirty limit expression for the penetration where we found ξ0 > l. The ﬁt yields energy gap ∆(0) = (0.43 ± 0.02) meV and λ (0) = (6190 ± 45) Å. The depth λ(0) is then given by [65] gap to Tc ratio ∆(0)/kBTc = 1.82 is close to the value 1/2 of 1.764 expected from the BCS theory, suggesting that ξ0 λ(0) = λL 1 + (23) NbOs2 is a weakly-coupled superconductor. l 9

100 8 TABLE III. Electronic properties calculated of NbOs2 for 6 High Tc Hc1 µ 4 λ ' 4608 Å and λ ' 6190 Å T F = T c Chevrel 2 T B = c T Mo3Al2C Properties unit Hc1 µSR 10 Re Hf Re6Zr 8 6 Re W Li2Pd3B 6 3 Re24Ti5 4 Re Ta Organics 3 Mg10Ir19B16 λ Å 4608 6190 (K) LaNiC2 GL c 2 ∗ T NbOs 2 Li2Pt3B m /me 11.6 ± 0.3 13.4± 0.3 La7Ir3 ∗ 1 Heavy m /me 7.63 ± 0.13 8.8 ± 0.2 8 Fermion band 6 n 1027m−3 4.16 ± 0.23 2.74 ± 0.17 4 Conventional Superconductors ξ0 Å 64 ± 5 90 ± 6 2 l Å 37± 4 50 ± 3 0.1 1 2 3 4 5 λ Å 2809 ± 130 3720 ± 164 10 10 10 10 10 L v 104m s−1 4.95 ± 0.23 3.75 ± 0.16 TF (K) F TF K 1060 ± 30 620 ± 20

Tc/TF 0.0025 ± 0.0001 0.0044 ± 0.0001 FIG. 7. The Uemura plot showing the superconducting transition temperature Tc vs the Fermi temperature TF , where NbOs2 is shown as a solid red circle just outside the range of band of unconventional superconductors. TB is the Bose- Einstein condensation temperature. The orange region represents the band of unconventionality, other points with broken V. UEMURA PLOT TRS such as LaNiC2, La7Ir3 etc. and Re-based α-Mn structure compounds were obtained from Ref. [59, 76–78]. For a 3D system the Fermi temperature TF is given by the relation where ξ0 is the BCS coherence length. The London pen- 2 n2/3 k T = ~ (3π2)2/3 , (26) etration depth, λL, is given by B F 2 m∗ where n is the quasiparticle number density per unit vol- m∗ 1/2 λ = (24) ume. According to Uemura et al. [76–78] superconduc- L 2 T µ0ne tors can be conveniently classiﬁed according to their c TF ratio. It was shown that for the unconventional super-

The BCS coherence length ξ0 and the Ginzburg-Landau conductors such as heavy-fermion, high- Tc, organic su- coherence, ξGL(0), at T = 0 K in the dirty limit is related perconductors, and iron-based superconductors this ratio falls in the range 0.01 ≤ Tc ≤ 0.1. In Fig.7, the by the expression TF orange region represents the band of unconventional superconductors. Uemura plot is presented in accordance ξ (0) π ξ −1/2 GL = √ 1 + 0 (25) with Ref. [59, 76–78], where it shows the superconduc- ξ l 0 2 3 tors with unconventional properties such as Mo3Al2C, Mg10Ir19B16 Li2Pt3B, LaNiC2, La7Ir3 and the family of Using Eq.(20)-(25) form a system of equations which can Re-based α-Mn structure superconductors with broken ∗ ∗ be used to estimate the parameters m , n, l, and ξ0 as TRS. Using the estimated value of n and m for NbOs2 Tc done in Ref.[58, 59]. The system of equations was solved in Eq. 26, it yields TF = 620 K, giving = 0.004. TF simultaneously using the values γn = (8.58 ± 0.01) mJ NbOs2 is located just outside the range of unconven- −1 −2 mol K , ξ(0) = (99 ± 1) Å and ρ0 = (130 ± 5) µΩcm. tional superconductors as shown by a solid red marker in The values were estimated for λHc1 ' 4608 Å and λµ ' Fig.7, potentially borders an unconvnetional supercon- 6190 Å, tabulated in TableIII. It is clear that ξ0 > l, ducting ground state. Interestingly, all the Re6X[47–49] indicating that NbOs2 is in the dirty limit as previously noncentrosymmetric superconductors with TRS breaking asserted. This also accounts for the low RRR and resid- are located in the same phase space in the Uemura plot. ual resistivity that have been measured. The bare-band The close proximity of the entire family might also be ∗ ∗ effective mass mband can be related to m , which contains pointing towards the common origin of TRS breaking, enhancements from the many-body electron phonon in- which seems to be Re SOC. However, other Re-based ∗ ∗ teractions [60] m = mband(1+λe−ph), where λe−ph is compounds for example Re3X (X =W, Ta)[28, 59] show the electron-phonon coupling constant. Using this value preserved TRS. This may be due to the reduced Re com- ∗ of λe−ph = 0.52, a value for mband can be found, given in position in these compounds which may have a major TableIII. effect on the underlying electronic characteristics of the spin-triplet channel. In addition, our ZF-µSR measure- 10 ments on the Re-free NbOs2 also advocate the above conclusion, since TRS is found to be preserved in the superconducting state of this material. Therefore, this makes [1] E. Bauer, G. Hilscher, H. Michor, Ch. Paul, E. W. the role of Re element even more interesting. However, Scheidt, A. Gribanov, Yu. Seropegin, H. NoÂšel, M. to make a generic comment about the contribution of Re Sigrist, and P. Rogl, Phys. Rev. Lett. 92, 027003 (2004). in the observed TRS breaking in this family, it is impor- [2] E. Bauer and M. Sigrist, Non-centrosymmetric supercon- tant to study more Re-rich α-Mn materials and more ductors:Introduction and Overview, Vol. 847 (Springer importantly Re itself. Science & Business Media, 2012). [3] L. P. Gor’kov and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001). [4] P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, VI. SUMMARY AND CONCLUSIONS Phys. Rev. Lett. 92, 097001 (2004). [5] E. Bauer, G. Hilscher, H. Michor, C. Paul, E. W. Scheidt, A. Gribanov, Y. Seropegin, H. NoÃ«l, M. Sigrist, and P. Detailed investigation of the normal and superconduct- Rogl, Phys. Rev. Lett. 92, 027003 (2004). ing phase properties of NbOs2 was carried out using [6] N. Kimura, K. Ito, H. Aoki, S. Uji, and T. Terashima, XRD, magnetization, resistivity, specific heat and µSR Phys. Rev. Lett. 98, 197001 (2007). measurements. Sample was prepared by standard arc- [7] I. Sugitani, Y. Okuda, H. Shishido, T. Yamada, A. melting technique, where the phase purity and noncen- Thamizhavel, E. 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