# Fully-gapped superconductivity in single crystals of noncentrosymmetric ReZr with broken time-reversal symmetry

###### Abstract

The noncentrosymmetric superconductor ReZr has attracted much interest due to the observation of broken time-reversal symmetry in the superconducting state. Here we report an investigation of the superconducting gap structure of ReZr single crystals by measuring the magnetic penetration depth shift and electronic specific heat . exhibits an exponential temperature dependence behavior for , which indicates a fully-open superconducting gap. Our analysis shows that a single gap -wave model is sufficient to describe both the superfluid density and results, with a fitted gap magnitude larger than the weak coupling BCS value, providing evidence for fully-gapped superconductivity in ReZr with moderate coupling.

## I Introduction

Noncentrosymmetric superconductors (NCS) have attracted a great deal of attention due to the influence of antisymmetric spin-orbit coupling (ASOC) on the superconducting properties, which is induced by the antisymmetric potential gradient arising due to broken inversion symmetry smidman2017superconductivity (); BauerNCS (). Strong ASOC may lift the spin degeneracy of the conduction bands, allowing for superconducting states which are an admixture of spin-singlet and spin-triplet pairing, giving rise to a variety of unique properties smidman2017superconductivity (); BauerNCS (); gorkov2001superconducting (); frigeri2004superconductivity (). Unconventional superconductivity was evidenced in the first heavy fermion NCS CePtSi, where a linear temperature dependence of the magnetic penetration depth at low temperatures and a constant Knight shift across were observedbonalde2005evidence (); yogi2006evidence (). While studies of heavy fermion NCS revealed a range of unusual findings, disentangling the role played by broken inversion symmetry from the effects of strong electronic correlations and magnetism is highly challenging, which spurred an interest in looking for weakly correlated NCS with singlet-triplet mixing. Evidence for such an admixture was demonstrated in Li(Pd,Pt)B, where LiPdB has a fully open gap Takeya2005 (); Nishiyama2005 (); yuan2006s (), but the gap of LiPtB exhibits line nodes yuan2006s (); Nishiyama2007 (); Takeya2007 (). This change from nodeless to nodal superconductivity upon switching Pd for Pt was explained using a model with a mixture of singlet and triplet states, where there is a relative increase in the size of the spin-triplet component of the order parameter, as the ASOC is increased yuan2006s (). The Knight shift also shows a marked difference between the two compounds, corresponding to a decrease of the spin susceptibility below for LiPdB, while for LiPtB this remains constant, indicating greater influence of the ASOC on the superconductivity Nishiyama2005 (); Nishiyama2007 ().

Subsequently, the order parameters of a wider range of NCS have been studied, where evidence for nodal superconductivity was also found in quasi-one-dimensional KCrAspang2015evidence () and the low carrier system YPtBi YPtBinode (). Meanwhile, although studies of YC at higher temperatures were accounted for by fully gapped superconductivity Y2C3NMR (); Y2C3HC (); Y2C3MuSR (), very low temperature penetration depth measurements also indicate the presence of a nodal gap structure chen2011evidence (). However, most NCS have been found to be fully-gapped superconductors, such as BaPtSibauer2009a (), BiPdSun2015 (); Matano2013 (); jiao2014anisotropic (); yan2016nodeless (), PbTaSePbTaSe2SpecH (); pang2016nodeless (); wang2016nodeless (), LaIr barker2015unconventional () and ReZrsingh2014detection (); matano2016full (); khan2016complex (); Mayoh2017 (). In particular, many NCS have been found compatible with single-gap -wave superconductivity, indicating that the size of any triplet component is very small, and the relationship between the ASOC strength and the degree of singlet-triplet pairing is not entirely understood. Moreover, it has been proposed that some NCS can exhibit topological superconductivity. Both theoretical calculations and experimental studies reveal topological surface states in the NCS PbTaSe and BiPd Bian2016topological (); Guan2016 (); Sun2015 (); Neupane2016observation (), which may lead to possible Majorana fermions in the vortex cores when these states are sufficiently close to the Fermi level.

Another notable feature of several NCS is the breaking of time-reversal symmetry (TRS). While broken TRS was previously discovered in the unconventional triplet superconductor SrRuO luke1998time (); xia2006high (), evidence for this phenomenon has also been found in a number of NCS with fully gapped superconducting states, such as LaNiChillier2009evidence (), LaIrbarker2015unconventional (), and ReZrsingh2014detection (). In the case of LaNiC, the symmetry analysis shows that for TRS to be broken at , the effect of the ASOC on the superconductivity should be weak, indicating a lack of significant mixed parity pairing quintainilla2010relativistic (). On the other hand, ReZr crystallizes in the cubic -Mn structure, where the presence of three dimensional irreducible representations of the point group potentially allows for TRS breaking with singlet-triplet mixing singh2014detection (). It is therefore of particular importance to characterize the superconducting order parameter of ReZr. Previous measurements of the gap structure of polycrystalline samples are accounted for by nodeless single gap -wave superconductivitysingh2014detection (); matano2016full (); khan2016complex (); Mayoh2017 (), but different conclusions are drawn from recent point contact spectroscopy results from single crystal measurements, which give evidence for multiple gaps pradnya2017multiband (). As such, it is important to perform further measurements of ReZr single crystals sensitive to low energy excitations to clarify this issue. In this paper, we study the superconducting order parameter of single crystalline ReZr by measuring the magnetic penetration depth and specific heat of single crystals, which are both consistent with a single nodeless isotropic gap with a moderate coupling strength.

## Ii Experimental details

Single crystals of ReZr were synthesized using the Czochralski method, as described in Ref. pradnya2017multiband, . Magnetization measurements were performed using a superconducting quantum interference device (SQUID) magnetometer (MPMS) with both field-cooling (FC) and zero-field-cooling (ZFC), with a small applied magnetic field of 10 Oe. The electrical resistivity was measured by using a standard four-probe method from room temperature down to 0.3 K in a He refrigerator. The specific heat was measured using a Physical Property Measurement System (PPMS) with a He insert down to around 0.6 K. The temperature dependence of the magnetic penetration depth shift was measured using a tunnel-diode-oscillator (TDO) technique in a He cryostat down to 0.35 K. The operating frequency of the TDO setup is about 7 MHz, with a noise level as low as 0.1 Hz. For the TDO measurements, the sample was cut into a regular shape with dimensions of 700500200 m and mounted onto a sapphire rod so that it may be placed into the coil without making contact. The sample experiences a very small ac field of about 20 mOe along the [100] direction induced by the coil, which is much smaller than the lower critical field , ensuring that the sample remains in the Meissner state, so that the change of the magnetic penetration depth is proportional to the frequency change . Here , where the calibration constant is determined by the geometry of the sample and coil prozorov2000meissner ().

## Iii Results and discussion

Single crystals of ReZr were characterized by the measurements of the electrical resistivity and magnetic susceptibility , which are displayed in Fig. 1. exhibits metallic behavior in the normal state with a residual resistivity of = 0.3 m cm, just above the superconducting transition. This yields a mean free path of 2.1 nmorlando1979critical (), with a coherence length of = 5.5 nm deduced from the upper critical field, and a Sommerfeld coefficient of = 27.4 mJ mol K from our specific heat measurements described below. The calculated mean free path is less than half the coherence length, indicating that the sample is closer to the dirty limit. The inset of Fig. 1 (a) displays an enlargement of at low temperatures, where a sharp superconducting transition occurs at = 6.8 K, with a transition width less than 0.2 K. In addition, the temperature dependence of the magnetic susceptibility in Fig. 1(b) shows that the zero-field-cooled (ZFC) curve exhibits full diamagnetism, providing evidence for bulk superconductivity in the ReZr single crystals.

Figure 2 displays the temperature dependence of the penetration depth shift at low temperatures from TDO-based measurements. The frequency change is plotted in the inset from about 7.5 K down to the base temperature of 0.35 K, which has been normalized by the normal state value. Here a sharp superconducting transition occurs at around 6.8 K, in line with the resistivity, and magnetic susceptibility measurements. decreases rapidly upon reducing the temperature, before becoming flat below around 1.5 K, indicating a fully open superconducting gap in ReZr. For an -wave superconductor, the change of magnetic penetration depth shows exponentially activated behavior at , as

(1) |

where and are the penetration depth and superconducting gap magnitude at zero temperature, respectively. As shown by the solid red line, our experimental data can be well described by a single -wave gap with = 200 nm and = 1.96 . The value of and the temperature dependence of are comparable with those in Ref. khan2016complex, , suggesting that both single crystal and polycrystalline samples are consistent with a similar nodeless gap structure. We note that for a superconductor with line nodes in the dirty limit, a quadratic dependence would be expected at low temperatures instead of a linear Goldenfeld1993 (). However, upon analyzing with a power law dependence , values of of 4.4 and 6.1 are obtained fitting from the base temperature up to 3 K and 2.2 K, respectively. Since these exponents are significantly larger than two, such a dirty nodal scenario can be ruled out.

Moreover the magnetic penetration depth was also converted into the normalized superfluid density via , which is displayed as a function of the reduced temperature in Fig. 3. For a clean superconductor, can be calculated using

(2) |

where is the Fermi-Dirac distribution function and the superconducting gap function is defined as g. This contains an angular dependent part g and a temperature dependent component , which can be approximated as = Carrington2003 (). Here the zero temperature gap magnitude is the only fitted parameter. As shown by the green dashed-dotted line in Fig. 3, a single gap -wave model with g=1 can well reproduce the experimental data with = 2.23 . This gap magnitude is larger than the value of 1.76 for weakly-coupled BCS superconductors, suggesting moderately strong coupling in ReZr. On the other hand, since as discussed above the samples are near to the dirty limit, a dirty -wave model was also applied, where Tinkham1996introduction (). The results are shown by the solid red line, which also well accounts for the experimental data with = 2.1 . In order to compare with nodal superconducting scenarios, both a -wave model with g=sin and a -wave model g=cos2 are also displayed, where is the polar angle and is the azimuthal angle. It is obvious that the superfluid density of and wave superconductors change significantly with temperature, even at very low temperatures, which is in contrast with the data showing near temperature independence below around 0.25 . Furthermore, we also compare our results with measurements of polycrystalline samples using transverse-field muon-spin rotation (SR) measurements from Ref. singh2014detection, . These are displayed in Fig. 3 and show that comparable superconducting gap structures are inferred from measurements of single crystal and polycrystalline samples.

The specific heat measurements of ReZr single crystals were also analyzed, to further characterize the superconducting order parameter. In the inset of Fig. 4, the total specific heat as is displayed, which contains both electronic and phonon contributions. In the normal state, the data were fitted using , where and represent the electron and phonon contributions respectively, and is the Sommerfeld coefficient. The fitting results are shown by the dashed line with fitted parameters of = 0.39 mJ mol K, = 1 J mol K, and = 27.4 mJ mol K, which are consistent with measurements of polycrystalline samplessingh2014detection ().

The main panel of Fig. 4 displays the temperature dependence of the electronic specific heat as in the superconducting state after subtracting the phonon contribution. A sizeable jump is observed around of =1.76, which is larger than the value of 1.43 for a weakly-coupled BCS superconductor, again indicating an enhancement of the coupling strength in ReZr. In the superconducting state, the entropy can be expressed asbouquet2001phenmonoligical ():

(3) |

and therefore the electronic specific heat in the superconducting state can be calculated as . Due to becoming flat with decreasing temperature for , the data were fitted using an isotropic -wave model, and the results are shown in Fig. 4 by the solid red line. It can be clearly seen that this model well describes the experimental data, with a fitted parameter of = 1.95 . This is consistent with the penetration depth and superfluid density results and also indicates a moderately enhanced superconducting gap magnitude.

Since ReZr is a NCS, the ASOC will lift the spin degeneracy of the electron bands, which can potentially give rise to a mixture of spin singlet and spin triplet pairing states, leading to a two-gap structure = , where and represent the singlet and triplet components smidman2017superconductivity (). Therefore the experimental results being consistent with a single nodeless gap implies that the spin triplet component is very small relative to the spin singlet. The origin of TRS breaking in ReZr still remains difficult to account for. In LaNiC, because of the low symmetry of the orthorhombic crystal structure, the breaking of TRS at implies that singlet-triplet mixing is weak, and moreover that the pairing corresponds to a non-unitary triplet state hillier2009evidence (); quintainilla2010relativistic (). Here the lack of a role played by ASOC was corroborated by the breaking of TRS in the similar but centrosymmetric LaNiGa hillier2012nonunitary (), and the observation of two-gap superconductivity in both LaNiC and LaNiGa allowed for the proposal of a nodeless even-parity, non-unitary triplet state, where the overall pairing wave function is antisymmetric upon particle exchange due to an antisymmetric orbital index weng2016two (). Such a scenario is not readily applied to ReZr, since the higher symmetry of the cubic -Mn crystal structure lifts the constraint that TRS breaking at must correspond to non-unitary pairing, and moreover most studies only show evidence for single-gap behavior. As the symmetry analysis of ReZr also allows for mixed singlet-triplet pairing with broken TRS singh2014detection (), further studies are highly desirable to understand the origin of TRS breaking and the role of the ASOC in the superconducting state of ReZr.

## Iv Conclusions

To summarize, we have measured the magnetic penetration depth and electronic specific heat of single crystals of the noncentrosymmetric superconductor ReZr. Exponential behavior of was observed at , which indicates the absence of low energy excitations, and a fully gapped superconducting state. Both the superfluid density and specific heat can be well accounted for by a single gap -wave model across the whole temperature range with a gap magnitude larger than the weak coupling BCS value, providing strong evidence for fully-gapped superconductivity with moderately strong coupling in ReZr.

## V Acknowledgments

We thank X. Lu for helpful discussions and suggestions.This work was supported by the National Natural Science Foundation of China (No. 11474251, No. U1632275), the National Key R&D Program of China (No. 2017YFA0303100, No. 2016YFA0300202), and the Science Challenge Project of China (No. TZ2016004). R.P.S. acknowledges the Science and Engineering Research Board, Government of India for the Young Scientist Grant No. YSS/2015/001799.

## References

- (1) M. Smidman, M. B. Salamon, H. Q. Yuan, and D. F. Agterberg, Superconductivity and spin-orbit coupling in non-centrosymmetric materials: a review. Rep. Prog. Phys. 80, 036501 (2017).
- (2) E. Bauer and M. Sigrist, Non-Centrosymmetric Superconductors: Introduction and Overview, Lecture Notes in Physics (Springer-Verlag, Berlin, 2012).
- (3) L. P. Gorkov and E. I. Rashba, Superconducting 2D System with Lifted Spin Degeneracy: Mixed Singlet-Triplet State. Phys. Rev. Lett. 87, 037004(2001).
- (4) P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Superconductivity without Inversion Symmetry: MnSi versus CePtSi. Phys. Rev. Lett. 92, 097001 (2004).
- (5) I. Bonalde, W. Bramer-Escamilla, and E. Bauer, Evidence for Line Nodes in the Superconducting Energy Gap of Noncentrosymmetric CePtSi from Magnetic Penetration Depth Measurements. Phys. Rev. Lett. 94, 207002 (2005).
- (6) M. Yogi, H. Mukuda, Y. Kitaoka, S. Hashimoto, T. Yasuda, R. Settai, T. D. Matsuda, Y. Haga, Y. Onuki, P. Rogl, and E. Bauer, Evidence for Novel Pairing State in Noncentrosymmetric Superconductor CePtSi: Si-NMR Knight Shift Study. J. Phys. Soc. Jpn. 75, 013709 (2006).
- (7) H. Takeya, K. Hirata, K. Yamaura, K. Togano, M. El Massalami, R. Rapp, F. A. Chaves, and B. Ouladdiaf, Low-temperature specific-heat and neutron-diffraction studies on LiPdB and LiPtB superconductors. Phys. Rev. B 72, 104506 (2005).
- (8) M. Nishiyama, Y. Inada, and G. -Q. Zheng, Superconductivity of the ternary boride LiPdB probed by B NMR. Phys. Rev. B 71, 220505(R) (2005)
- (9) H. Q. Yuan, D. F. Agterberg, N. Hayashi, P. Badica, D. Vandervelde, K. Togano, M. Sigrist and M. B. Salamon, -Wave spin-triplet order in superconductors without inversion symmetry: LiPdB and LiPtB. Phys. Rev. Lett. 97, 017006 (2006).
- (10) M. Nishiyama, Y. Inada, and G. -Q. Zheng, Spin Triplet Superconducting State due to Broken Inversion Symmetry in LiPtB. Phys. Rev. Lett. 98, 047002 (2007).
- (11) H. Takeya, M. ElMassalami, S. Kasahara, and K. Hirata, Specific-heat studies of the spin-orbit interaction in noncentrosymmetric Li(PdPt)B () superconductors. Phys. Rev. B 76, 104506 (2007).
- (12) G. M. Pang, M. Smidman, W. B. Jiang, J. K. Bao, Z. F. Weng, Y. F. Wang, L. Jiao, J. L. Zhang, G. H. Cao, and H. Q. Yuan, Evidence for nodal superconductivity in quasi-one-dimensional KCrAs. Phys. Rev. B 91, 220502(R) (2015).
- (13) H. Kim, K. Wang, Y. Nakajima, R. Hu, S. Ziemak, P. Syers, L. Wang, H. Hodovanets, J. D. Denlinger, P. M. R. Brydon, D. F. Agterberg, M. A. Tanatar, R. Prozorov. and J. Paglione, Beyond triplet: Unconventional superconductivity in a spin-3/2 topological semimetal. Sci. Adv. 4, eaao4513 (2018).
- (14) A. Harada, S. Akutagawa, Y. Miyamichi, H. Mukuda, Y. Kitaoka, J. Akimitsu, Multigap Superconductivity in YC: A 13C-NMR Study, J. Phys. Soc. Jpn. 76, 023704 (2007).
- (15) S. Akutagawa, and J. Akimitsu, Superconductivity of YC Investigated by Specific Heat Measurement, J. Phys. Soc. Jpn. 76, 024713 (2007).
- (16) S. Kuroiwa, Y. Saura, J. Akimitsu, M. Hiraishi, M. Miyazaki, K. H. Satoh, S. Takeshita, and R. Kadono, Multigap Superconductivity in Sesquicarbides LaC and YC, Phys. Rev. Lett. 100, 097002 (2008).
- (17) J. Chen, M. B. Salamon, S. Akutagawa, J. Akimitsu, J. Singleton, J. L. Zhang, L. Jiao, and H. Q. Yuan, Evidence of nodal gap structure in the noncentrosymmetric superconductor. YC, Phys. Rev. B 83, 144529 (2011).
- (18) E. Bauer, R. T. Khan, H. Michor, E. Royanian, A. Grytsiv, N. Melnychenko-Koblyuk, P. Rogl, D. Reith, R. Podloucky, E. W. Scheidt, W. Wolf, and M. Marsman, BaPtSi: A noncentrosymmetric BCS-like superconductor, Phys. Rev. B 80, 064504 (2009).
- (19) K. Matano, S. Maeda, H. Sawaoka, Y. Muro, T. Takabatake, B. Joshi, S. Ramakrishnan, K. Kawashima, J. Akimitsu, and G. -Q. Zheng, NMR and NQR Studies on Non-centrosymmetric Superconductors ReB, LaBiPt, and BiPd, J. Phys. Soc. Jpn. 82, 084711 (2013).
- (20) Z. Sun, M. Enayat, A. Maldonado, C. Lithgow, E. Yelland, D. C. Peets, A. Yaresko, A. P. Schnyder, and P. Wahl, Dirac surface states and nature of superconductivity in Noncentrosymmetric BiPd, Nat. Commun. 6, 6633 (2015).
- (21) L. Jiao, J. L. Zhang, Y. Chen, Z. F.Weng, Y. M. Shao, J. Y. Feng, X. Lu, B. Joshi, A. Thamizhavel, S. Ramakrishnan, and H. Q. Yuan, Anisotropic superconductivity in noncentrosymmetric BiPd. Phys. Rev. B 89, 060507(R) (2014).
- (22) X. B. Yan, Y. Xu, L. P. He, J. K. Dong, H. Cho, D. C. Peets, Je-Geun Park and S Y Li, Nodeless superconductivity in the noncentrosymmetric superconductor BiPd. Supercond. Sci. Technol. 29, 065001 (2016).
- (23) C. -L. Zhang, Z. Yuan, G. Bian, S. -Y. Xu, X. Zhang, M. Z. Hasan, and S. Jia, Superconducting properties in single crystals of the topological nodal semimetal PbTaSe.Phys. Rev. B 93, 054520 (2016).
- (24) G. M. Pang, M. Smidman, L. X. Zhao, Y. F. Wang, Z. F. Weng, L. Q. Che, Y. Chen, X. Lu, G. F. Chen, and H. Q. Yuan, Nodeless superconductivity in noncentrosymmetric PbTaSe single crystals. Phys. Rev. B 93, 060506 (2016).
- (25) M. X. Wang, Y. Xu, L. P. He, J. Zhang, X. C. Hong, P. L. Cai, Z. B. Wang, J. K. Dong, and S. Y. Li, Nodeless superconducting gaps in noncentrosymmetric superconductor PbTaSe with topological bulk nodal lines, Phys. Rev. B 93, 020503(R) (2016).
- (26) J. A. T. Barker, D. Singh, A. Thamizhavel, A. D. Hillier, M. R. Lees, G. Balakrishnan, D McK. Paul, and R. P. Singh, Unconventional Superconductivity in LaIr Revealed by Muon Spin Relaxation: Introducing a New Family of Noncentrosymmetric Superconductor That Breaks Time-Reversal Symmetry. Phys. Rev. Lett. 115, 267001 (2015).
- (27) R. P. Singh, A. D. Hillier, B. Mazidian, J. Quintanilla, J. F. Annett, D. M. Paul, G. Balakrishnan, and M. R. Lees, Detection of Time-Reversal Symmetry Breaking in the Noncentrosymmetric Superconductor ReZr Using Muon-Spin Spectroscopy. Phys. Rev. Lett. 112, 107002 (2014).
- (28) K. Matano, R. Yatagai, S. Maeda, and Guo-qing Zheng, Full-gap superconductivity in noncentrosymmetric ReZr, ReZr, and ReZr. Phys. Rev. B 94, 214513 (2016).
- (29) M. A. Khan, A. B. Karki, T. Samanta, D. Browne, S. Stadler, I. Vekhter, A. Pandey, P. W. Adams, D. P. Young, S. Teknowijoyo, K. Cho, R. Prozorov, and D. E. Graf, Complex superconductivity in the noncentrosymmetric compound ReZr. Phys. Rev. B 94, 144515 (2016).
- (30) D. A. Mayoh, J. A. T. Barker, R. P. Singh, G. Balakrishnan, D. McK. Paul, and M. R. Lees, Superconducting and normal-state properties of the noncentrosymmetric superconductor ReZr. Phys. Rev. B 96, 064521 (2017).
- (31) G. Bian, T. -R. Chang, R. Sankar, S. -Y. Xu, H. Zheng, T. Neupert, C. -K. Chiu, S. -M. Huang, G. Q. Chang, I. Belopolski, D. S. Sanchez, M. Neupane, N. Alidoust, C. Liu, B. K. Wang, C. -C. Lee, H. -T. Jeng, C. L. Zhang, Z. J. Yuan, S. Jia, A. Bansil, F. C. Chou, H. Lin, and M. Z. Hasan, Topological nodal-line fermions in spin-orbit metal PbTaSe. Nat. Commun. 7, 10556 (2016).
- (32) S. Y. Guan, P. J. Chen, M. W. Chu, R. Sankar, F. Chou, H. T. Jeng, C. S. Chang, and T. M. Chuang, Superconducting topological surface states in the noncentrosymmetric bulk superconductor PbTaSe. Sci. Adv. 2, e1600894 (2016).
- (33) M. Neupane, N. Alidoust, M. M. Hosen, J. -X. Zhu, K. Dimitri, S. -Y. Xu, N. Dhakal, R. Sankar, I. Belopolski, D. S. Sanchez, T. -R. Chang, H. -T. Jeng, K. Miyamoto, T. Okuda, H. Lin, A. Bansil, D. Kaczorowski, F. C. Chou, M. Z. Hasan, and T. Durakiewicz, Observation of the spin-polarized surface state in a noncentrosymmetric superconductor BiPd. Nat. Commun. 7, 13315 (2016).
- (34) G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merrin, B. Nachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori et al., Time-reversal symmetry-breaking superconductivity in SrRuO. Nature (London) 394, 558 (1998).
- (35) J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, High resolution polar Kerr effect measurements of SrRuO: evidence for broken time-reversal symmetry in the superconducting state. Phys. Rev. Lett. 97, 167002 (2006).
- (36) A. D. Hillier, J. Quintanilla, and R. Cywinski, Evidence for Time-Reversal Symmetry Breaking in the Noncentrosymmetric Superconductor LaNiC. Phys. Rev. Lett. 102, 117007 (2009).
- (37) J. Quintanilla, A. D. Hillier, J. F. Annett, and R. Cywinski, Relativistic analysis of the pairing symmetry of the noncentrosymmetric superconductor LaNiC. Phys. Rev. B 82, 174511 (2010).
- (38) Pradnya Parab, Deepak Singh, Harries Muthurajan, R. P. Singh, Pratap Raychaudhuri, Sangita Bose, Multiband Superconductivity in the time reversal symmetry broken superconductor ReZr. arXiv:1704.06166.
- (39) R. Prozorov, R. W. Giannetta, A. Carrington, and F. M. Araujo-Moreira, Meissner-London state in superconductors of rectangular cross section in a perpendicular magnetic field. Phys. Rev. B 62, 115 (2000).
- (40) T. P. Orlando, E. J. McNiff, S. Foner, and M. R. Beasley, Critical fields, Pauli paramagnetic limiting, and material parameters of NbSn and VSi. Phys. Rev. B 19, 4545(1979).
- (41) P. J. Hirschfeld and N. Goldenfeld, Effect of strong scattering on the low-temperature penetration depth of a -wave superconductor. Phys. Rev. B 48, 4219 (1993).
- (42) A. Carrington and F. Manzano, Magnetic penetration depth of MgB, Physica C 385, 205 (2003).
- (43) M. Tinkham, Introduction to Superconductivity, 2nd (McGraw Hill, New York, 1996).
- (44) F. Bouquet, Y. Wang, R. A. Fisher, D. G. Hinks, J. D. Jorgensen, A. Junod and N. E. Phillips, Phenomenological two-gap model for the specific heat of MgB. Europhys. Lett. 56, 856 (2001).
- (45) A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, and R. Cywinski, Nonunitary Triplet Pairing in the Centrosymmetric Superconductor LaNiGa. Phys. Rev. Lett. 109, 097001 (2012).
- (46) Z. F. Weng, J. L. Zhang, M. Smidman, T. Shang, J. Quintanilla, J. F. Annett, M. Nicklas, G. M. Pang, L. Jiao, W. B. Jiang, Y. Chen, F. Steglich, and H. Q. Yuan, Two-Gap Superconductivity in LaNiGa2 with Nonunitary Triplet Pairing and Even Parity Gap Symmetry. Phys. Rev. Lett. 117, 027001 (2016).