Evidence for non-unitary triplet-pairing superconductivity in noncentrosymmetric TaRuSi and comparison with isostructural TaReSi

The gist: The study investigates the superconducting properties of noncentrosymmetric ternary superconductors TaXSi (X = Re, Ru) using muon spin rotation/relaxation and density functional theory calculations to determine if they exhibit conventional or unconventional pairing symmetries.

Superconducting States in TaReSi and TaRuSi

The transverse-field µSR measurements indicate isotropic s-wave superconductivity in TaReSi and multi-gap superconductivity in TaRuSi Zero-field µSR measurements find no evidence for spontaneous fields in the superconducting state of TaReSi, whereas small spontaneous fields that onset with superconductivity indicate broken time-reversal symmetry (TRS) superconductivity in TaRuSi Density functional theory calculations attribute the broken time-reversal symmetry (TRS) in TaRuSi to a non-unitary triplet pairing state This finding is consistent with the expectation that such a state is not allowed in the presence of strong spin-orbit coupling, as spin-orbit coupling is relatively weak in TaRuSi and strong in TaReSi.

Role of Crystal Structure and Spin-Orbit Coupling

The role of the underlying crystal structure has been recognized for its role in determining the allowed symmetries of the superconducting state In noncentrosymmetric superconductors (NCS), parity is not a good quantum number, and electronic antisymmetric spin-orbit coupling (ASOC) is allowed by symmetry. This relaxed symmetry in NCS systems, in the presence of strong ASOC, can give rise to the formation of a mixture of spin-singlet and spintriplet Cooper pairs Such a superconducting state can go beyond conventional BCS-like superconductivity with various exotic features like time-reversal symmetry (TRS) breaking, zeros or multiple gaps in the energy spectrum, and topologically protected nontrivial surface states. TaXSi (X = Ru/Re) is an ideal candidate system for such a study due to the presence of heavier Re atoms and lighter Ru atoms which leads to varying levels of spin-orbit coupling.

Muon Spin Rotation/Relaxation Analysis

Muon spectroscopy measurements were performed at M15 and M20 beamlines at TRIUMF’s Centre for Molecular and Material Science at Vancouver, which are equipped with a dilution refrigerator and He4 cryostat Transverse field µSR can be used to examine a superconductor in its vortex state to determine the temperature dependence of the magnetic penetration depth and, from that, the energy gap The asymmetry spectra were fitted using a twoterm sinusoidal decaying function. The depolarization rates for the sample and background signals are denoted as σ and ψ. For TaRuSi, the temperature dependence of relaxation rate (c) Λ and (f) σ measured on M15 and M20 spectrometers show an increment below Tc. This increase in relaxation is consistent with an onset temperature of the superconducting Tc, however, we cannot make a definitive statement on the precise onset temperature due to the relatively large error bars in the relaxation rate.

Superconducting Gap Symmetry and Multi-gap Nature

The fitting of data using s + s, s + p, and s + d wave models was performed to check for a possible multi-gap nature as inferred from susceptibility measurements The results of the fit are presented in Table I with the plots shown in Fig. 1(b) and (d) for TaReSi and TaRuSi, respectively. The measured superfluid density for TaRuSi produces the best goodness of fit for the double gap models with nodes, i.e., s + d or s + p. The presence of multiple gaps and the possibility of gap anisotropy makes it difficult to distinguish whether the gapless state is p or d-wave.

Comparison with Other Superconductors

The Uemura classification shows that TaRuSi and TaReSi lie close to the band of unconventional superconductors, including cuprates, iron-based superconductors, heavy fermion superconductors, etc., as shown in Fig. 3. The ratio Tc/TF is in the range 0.01 ≤ Tc/TF ≤ 0.1 for unconventional superconductors and Tc/TF ≤ 0.001 for conventional superconductors. The calculated superfluid densities are an order of magnitude smaller than the normal state carrier concentrations, which is the case with many unconventional superconductors.

Symmetry Analysis Conclusion

The results are in alignment with the symmetry analysis of Ref. [22] for superconductors with the C2v point-group symmetry. That is, we have only observed TRS breaking in the case of weak SOC, leading us to conclude that the superconducting state for TaRuSi is a non-unitary triplet-pairing state. TaRuSi would therefore be the first system other than LaNiC2 with the point group C2v to host such a nonunitary triplet-pairing state. The fits of the transverse field µSR data indicate fully-gapped superconductivity in TaReSi while we find superconductivity with gap nodes in TaRuSi, which is expected from symmetry analysis [22] for a broken time reversal symmetry state.

DFT Calculations and Electronic Structure

Fully-relativistic electronic structure calculations using the plane-wave pseudopotential formalism of density functional theory (DFT) were performed to characterize TaXSi For both compounds, we have found multiple Fermi surface sheets, allowing for the possibility of multi-gapped superconductivity. The effect of SOC on splitting the Fermi surfaces can be seen by comparing the top row to the bottom two rows in the case of (a) TaReSi and (b) TaRuSi. We have found that EASOC = 41 meV for TaRuSi and EASOC = 81 meV for TaReSi. Our results show that the effects of SOC on the band structure for TaRuSi (TaReSi) are relatively weak (strong) when compared to other NSC superconductors. The electron-phonon coupling strength is estimated through the renormalization factor, 1 + λe−ph [61] required to match the computed density of states at the Fermi energy, D(EF), to the values of 2.28 and 3.34 states eV f.u. obtained through measurements of the Sommerfeld constant for TaReSi and TaRuSi respectively. Both values are close to the values obtained previously from the McMillan formula [41] and indicate moderate electron-phonon coupling in both materials.

Final Conclusion on Pairing Mechanism

It is suspected that Hund’s coupling between electrons on Ni orbitals could provide the pairing mechanism for non-unitary triplet pairing in LaNiC2 Similar arguments could be made for describing the possible triplet-pairing mechanism in TaRuSi given Hund’s coupling is also significant in many Ru-based materials. Strong ASOC can drive superconducting systems to unconventional or conventional superconductivity for some underlying crystal structure symmetries. The results we have obtained for TaRuSi can be compared with what has been observed for LaNiC2. Both systems break TRS and share the same point group, C2v. Therefore, the symmetry analysis of Ref. [22] is applicable to TaRuSi as well. The study concludes that the superconducting state for TaRuSi is a non-unitary triplet-pairing state.

References

[1] M. Sigrist and K. Ueda, Phenomenological theory of unconventional superconductivity, Rev. Mod. Phys. 63, 239 (1991) <ref:2209.

1. Enhanced material property prediction for noncentrosymmetric superconductors like TaXSi by incorporating Density Functional Theory calculations to determine electronic states and spin-orbit coupling effects. This allows AI systems to predict whether a given crystal structure will host a non-unitary triplet pairing state based on calculated Asymmetry Energy of Spin-Orbit Coupling (EASOC).

2. Improved identification of superconducting pairing symmetries by training models on muon spin rotation relaxation (muSR) data, specifically distinguishing between isotropic s-wave superconductivity and multi-gap superconductivity in TaReSi versus TaRuSi. The AI system can now classify the observed magnetic field signatures as evidence for broken time-reversal symmetry (TRS) superconductivity based on the onset of small spontaneous fields.

3. Development of a predictive framework for superconducting gap structure by utilizing temperature dependence of muon depolarization rates, specifically fitting data to s+s, s+p, and s+d wave models. This allows the AI system to infer whether a superconducting state exhibits gap nodes, which is expected in TaRuSi but not in TaReSi based on the fitting results.

4. Refined assessment of spin-orbit coupling strength by analyzing DFT band structure calculations, specifically comparing band splitting for TaReSi (over 100 meV) versus TaRuSi (about 40 meV). This enables the AI to quantitatively state that the inclusion of ASOC in our DFT calculations significantly modifies the electronic structure of TaReSi when compared with DFT calculations neglecting ASOC.

5. Classification of unconventional superconductors using a Uemura plot based on the ratio Tc/TF derived from relaxation rates. The improved system can categorize materials like TaRuSi and TaReSi as lying close to the band of unconventional superconductors, suggesting they might exhibit behavior characteristic of cuprates or iron-based superconductors.