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Fully-gapped superconductivity and topological aspects of
the noncentrosymmetric TaReSi superconductor

T. Shang [email protected] Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China Chongqing Key Laboratory of Precision Optics, Chongqing Institute of East China Normal University, Chongqing 401120, China    J. Z. Zhao [email protected] Co-Innovation Center for New Energetic Materials, Southwest University of Science and Technology, Mianyang 621010, China    Lun-Hui Hu Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA    D. J. Gawryluk Laboratory for Multiscale Materials Experiments, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland    X. Y. Zhu Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    H. Zhang Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    J. Meng Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Z. X. Zhen Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    B. C. Yu Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Z. Zhou Key Laboratory of Nanophotonic Materials and Devices & Key Laboratory of Nanodevices and Applications, Suzhou Institute of Nano-Tech and Nano-Bionics (SINANO), CAS, Suzhou 215123, China    Y. Xu Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Q. F. Zhan Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    E. Pomjakushina Laboratory for Multiscale Materials Experiments, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland    T. Shiroka Laboratory for Muon-Spin Spectroscopy, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland Laboratorium für Festkörperphysik, ETH Zürich, CH-8093 Zürich, Switzerland
Abstract

We report a study of the noncentrosymmetric TaReSi superconductor by means of muon-spin rotation and relaxation (µSR) technique, complemented by electronic band-structure calculations. Its superconductivity, with TcT_{c} = 5.5 K and upper critical field μ0​Hc2​(0)\mu_{0}H_{\mathrm{c2}}(0) ∼\sim 3.4 T, was characterized via electrical-resistivity- and magnetic-susceptibility measurements. The temperature-dependent superfluid density, obtained from transverse-field µSR, suggests a fully-gapped superconducting state in TaReSi, with an energy gap Δ0\Delta_{0} = 0.79 meV and a magnetic penetration depth λ0\lambda_{0} = 562 nm. The absence of a spontaneous magnetization below TcT_{c}, as confirmed by zero-field µSR, indicates a preserved time-reversal symmetry in the superconducting state. The density of states near the Fermi level is dominated by the Ta- and Re-5dd orbitals, which account for the relatively large band splitting due to the antisymmetric spin-orbit coupling. In its normal state, TaReSi behaves as a three-dimensional Kramers nodal-line semimetal, characterized by an hourglass-shaped dispersion protected by glide reflection. By combining nontrivial electronic bands with intrinsic superconductivity, TaReSi is a promising material for investigating the topological aspects of noncentrosymmetric superconductors.

††preprint: Preprint: , 10:00

I Introduction

In crystalline solids, a suitable combination of space-, time-reversal-, and parity symmetries often gives rise to exotic quasiparticles, analogous to the particles predicted in high-energy physics, such as Dirac-, Weyl-, or Majorana fermions [1, 2, 3, 4, 5, 6, 6, 7, 8]. In particular, materials which lack an inversion center are among the best candidates for studying topological phenomena, since most of them also exhibit nonsymmorphic symmetry that can often generate unusual types of fermionic excitations. For instance, Weyl fermions were experimentally discovered as quasiparticles in noncentrosymmetric tantalum- and niobium pnictides [9, 10, 11, 12]. Noncentrosymmetric materials can also host exotic fermions with an hourglass-shaped dispersion protected by glide reflection [13, 14, 15], which are known to exhibit interesting topological properties. In addition, Kramers nodal-line (KNL) fermions were recently forecasted to occur in noncentrosymmetric metals with a sizable spin-orbit coupling (SOC) [16, 17]. To date, research on topological materials has been primarily focused on the case of non-interacting electronic bands. On the contrary, the interplay between topology and correlated electronic states, such as superconductivity or magnetism, remains largely unexplored.

Many noncentrosymmetric topological materials also exhibit superconductivity (SC) and, in view of their structure, are known as noncentrosymmetric superconductors (NCSCs). In NCSCs, the antisymmetric spin-orbit coupling (ASOC) allows, in principle, the occurrence of admixtures of spin-singlet and spin-triplet superconducting pairing, whose degree of mixing is generally believed to be determined by the strength of ASOC [18, 19, 20]. This sets the scene for a variety of exotic superconducting properties, e.g., nodes in the energy gap [21, 22, 23, 24], multigap SC [25], upper critical fields beyond the Pauli limit [26, 27, 28], and breaking of time-reversal symmetry (TRS) in the superconducting state [24, 29, 30, 31, 32, 33, 34, 35].

Noncentrosymmetric superconductors also provide a fertile ground in the search for topological SC and Majorana zero modes, with potential applications to quantum computation [36, 37, 38, 39, 40, 41, 42, 43]. Among the many routes attempted to realize it, one approach consists in combining a conventional ss-wave superconductor with a topological insulator to form a heterostructure. The proximity effect between the resulting surface states can lead to an effective two-dimensional SC with pp+i​pip pairing, known to support Majorana bound states at the vortices [44, 45, 46]. One can also consider introducing extra carriers (e.g., via chemical doping) into a topological insulator to achieve topological superconductivity [47, 48]. A more elegant and clean route to attain topological SC is that of combining a nontrivial electronic band with intrinsic superconductivity in the same compound [49]. Some of the materials with nontrivial electronic band structures display topological surface states with spin-polarized textures [1, 2, 3, 4, 5, 6, 6, 7, 8]. When the bulk of the material transitions into the superconducting state, the proximity effect can give rise to topological superconducting surface states. Such protected surface states have been proposed, for instance, in noncentrosymmetric β\beta-Bi2Pd and PbTaSe2 superconductors, both considered as suitable platforms for investigating topological SC [50, 51]. Clearly, to pursue the “intrinsic” route, it is of fundamental interest to identify new types of superconductors with a nontrivial band topology.

Recently, NSCSs have become one of the most investigated superconducting classes due to their unconventional- and topological nature. To this superconducting family belong also the TiFeSi-type materials, such as TTRuSi and TTReSi (with TT a transition metal). The normal states of TaRuSi and NbRuSi are three-dimensional KNL semimetals, characterized by large ASOCs and by hourglass-like dispersions [52]. Both compounds spontaneously break the TRS in the superconducting state and adopt a unitary (s+i​ps+ip) pairing, reflecting a mixture of spin singlets and spin triplets.

Refer to caption
Figure 1. : The temperature-dependent volume magnetic susceptibility χV​(T)\chi_{\mathrm{V}}(T) (a) and electrical resistivity ρ​(T)\rho(T) (b) for TaReSi. The results of an as-cast sample (S1) and of samples annealed at 900∘C (S2) and 1100∘C (S3) are shown. While ρ​(T)\rho(T) was measured in a zero-field condition, χV​(T)\chi_{\mathrm{V}}(T) data were collected in a magnetic field of 1 mT. The susceptibility data were corrected to account for the demagnetization factor. The inset in (a) shows the crystal structure of TaReSi viewed along the cc-axis and its mirror plane, while the black lines mark the unit cell. Green, blue, and red spheres are Ta, Re, and Si atoms, respectively. The inset in (b) shows the ρ​(T)\rho(T) of S2 up to 300 K.

TaReSi also belongs to the TiFeSi family, and becomes a superconductor below 5.5 K [53]. Although certain properties of TaReSi have been previously investigated [54], its superconducting properties, in particular, the superconducting order parameter, have not been explored at a microscopic level. In this paper, by combining muon-spin relaxation and rotation (µSR) measurements with electronic band-structure calculations, we show that TaReSi exhibits a fully-gapped superconducting state with a preserved TRS. It shares similar band topology with TaRuSi and NbRuSi, whose Kramers- and hourglass fermions can be easily tuned towards the Fermi level by chemical substitutions. TaReSi serves as another candidate material for investigating the interplay between topological states and superconductivity.

II Experimental details

Polycrystalline TaReSi samples were prepared by arc melting stoichiometric Ta slugs (Alfa Aesar, 99.98%), Re powders (ChemPUR, 99.99%), and Si chunks (Alfa Aesar, 99.9999%) in a high-purity argon atmosphere. To improve sample homogeneity, the ingots were flipped and re-melted more than six times. The resulting samples were then separated and annealed at 900∘C and 1100∘C for two weeks, respectively. An as-cast sample (denoted as S1) and samples annealed at 900∘C (S2) and 1100∘C (S3) were studied. As shown in the inset of Fig. I, TaReSi crystallizes in an orthorhombic structure with a space group of I​m​a​2Ima2 (No. 46) [54]. All samples were characterized by electrical-resistivity- and magnetization measurements, performed on a Quantum Design physical property measurement

Refer to caption
Figure 2. : (a) Field-dependent magnetization curves collected at various temperatures after cooling the S2 sample in zero field (the other samples behave similarly). (b) Lower critical fields Hc1H_{\mathrm{c1}} vs. temperature. Solid lines are fits to μ0​Hc1​(T)=μ0​Hc1​(0)​[1−(T/Tc)2]\mu_{0}H_{\mathrm{c1}}(T)=\mu_{0}H_{\mathrm{c1}}(0)[1-(T/T_{c})^{2}]. For each temperature, Hc1H_{\mathrm{c1}} was determined as the value where M​(H)M(H) starts deviating from linearity (see dashed line).

system (PPMS) and a magnetic property measurement system (MPMS), respectively. The bulk µSR measurements were carried out at the multipurpose surface-muon spectrometer (Dolly) on the π\piE1 beamline of the Swiss muon source at Paul Scherrer Institut, Villigen, Switzerland. The samples were mounted on a 25-µm thick copper foil which ensures thermalization at low temperatures. The time-differential µSR data were collected upon heating and then analyzed by means of the musrfit software package [55].

First-principles calculations were performed based on the density functional theory (DFT), as implemented in the Quantum ESPRESSO package [56, 57]. The exchange-correlation function was treated with the generalized gradient approximation using the Perdew-Burke-Ernzerhof (PBE) realization [58]. The projector augmented wave pseudopotentials were adopted [59]. We considered 13 electrons for Ta (5s2s^{2}6s2s^{2}5p6p^{6}5d3d^{3}), 15 electrons for Re (5s2s^{2}6s2s^{2}5p6p^{6}5d5d^{5}), and 4 electrons for Si (3s2s^{2}3p2p^{2}) as valence electrons. The calculations use the measured lattice parameters a=7.002a=7.002 Å, b=11.614b=11.614 Å, and c=6.605c=6.605 Å, and coordinates Ta1 (0.2500, 0.2004, 0.2964), Ta2 (0.2500, 0.7793, 0.2707), Ta3 (0.2500, 0.9979, 0.9178), Re1 (0, 0, 0.25), Re2 (0.0295, 0.3764, 0.1200), and Si1 (0.25, 0.9747, 0.5055), Si2 (0.0060, 0.1675, 0.9953) for TaReSi and include also the spin-orbit coupling effects [60]. The kinetic energy cutoff for the wavefunctions was set to 60 Ry, while for the charge density it was fixed to 600 Ry. For the self-consistent calculations, the Brillouin zone integration was performed on a Monkhorst-Pack grid mesh of 10×10×1010\times 10\times 10 kk-points, which ensures their unbiased sampling. The convergence criterion was set to 10−710^{-7} Ry. The Hf- and W doping effects were simulated by the virtual crystal approximation (VCA) [61] implemented in the Vienna Ab initio Simulation Package (VASP) [62, 63].

III Results and discussion

The bulk superconductivity of the TaReSi samples was first characterized by magnetic-susceptibility measurements, using both field-cooling (FC) and zero-field-cooling (ZFC) protocols in an applied magnetic field of 1 mT. As shown in Fig. I(a),

Refer to caption
Figure 3. : (a) Temperature-dependent electrical resistivity for various applied magnetic fields. TcT_{c} was determined as the onset temperature where the resistivity drops to zero. (b) Field-dependent magnetization (up to 4 T) collected at various temperatures below TcT_{c}. The inset shows the high-field range of the M​(H)M(H) curve at 2 K. Hc2H_{\mathrm{c2}} was chosen as the field where the diamagnetic response vanishes (indicated by an arrow). The reported data refer to the sample S2 — samples S1 and S3 show similar features.

a clear diamagnetic response appears below the superconducting transition at TcT_{c} = 5.5 K for S2. The samples S1 and S3 show a slightly lower transition temperature, i.e., TcT_{c} ∼\sim 5.0 K. After accounting for the demagnetization factor, the superconducting shielding fraction of TaReSi samples is close to 100%, indicative of bulk SC.

The temperature-dependent electrical resistivity ρ​(T)\rho(T) of TaReSi samples was measured from 2 K up to room temperature. It reveals a metallic behavior, without any anomalies associated with structural, magnetic, or charge-density-wave transitions at temperatures above TcT_{c} [see lower inset in Fig. I(b)]. The electrical resistivity in the low-TT region is plotted in Fig. I(b), clearly showing the superconducting transition of all the samples. A Tconset=6.1T^{\mathrm{onset}}_{c}=6.1, 7.0, and 7.5 K, and Tczero=4.8T^{\mathrm{zero}}_{c}=4.8, 5.6, and 4.6 K were identified for the S1, S2, and S3 samples, respectively. The TczeroT^{\mathrm{zero}}_{c} values are consistent with the transition temperatures determined from the magnetic susceptibility [see Fig. I(a)]. In view of its higher TcT_{c} and narrower Δ​Tc\Delta T_{c} transition, most of the µSR measurements were performed on the TaReSi sample S2.

To determine the lower critical field Hc1H_{\mathrm{c1}}, to be exceeded (at least twice) when performing µSR measurements on type-II superconductors, the field-dependent magnetization M​(H)M(H) of TaReSi was measured at various temperatures. Here, the M​(H)M(H) data of the S2 sample are shown in Fig. II(a), with the other samples showing a similar behavior. The estimated Hc1H_{\mathrm{c1}} values at different temperatures (accounting for a demagnetization factor), determined from the deviations of M​(H)M(H) from linearity, are summarized in Fig. II(b). The solid lines are fits to μ0​Hc1​(T)=μ0​Hc1​(0)​[1−(T/Tc)2]\mu_{0}H_{\mathrm{c1}}(T)=\mu_{0}H_{\mathrm{c1}}(0)[1-(T/T_{c})^{2}] and yield the lower critical fields μ0​Hc1​(0)\mu_{0}H_{\mathrm{c1}}(0) = 6.6(1), 6.4(1), and 5.5(1) mT for S1, S2, and S3 samples, respectively.

Refer to caption
Figure 4. : Upper critical field Hc2H_{\mathrm{c2}} vs. the reduced temperature T/Tc​(0)T/T_{c}(0) for the different TaReSi samples. The TcT_{c} and Hc2H_{\mathrm{c2}} values were determined from the measurements shown in Fig. III. Full symbols refer to magnetization, while empty symbols to resistivity measurements. Note the systematically higher values in the latter case. Solid lines represent fits to the GL model.

To investigate the upper critical field Hc2H_{\mathrm{c2}} of TaReSi, we measured the temperature-dependent electrical resistivity ρ​(T,H)\rho(T,H) at various applied magnetic fields, as well as the field-dependent magnetization M​(H,T)M(H,T) at various temperatures. As shown in Fig. III(a), upon increasing the magnetic field, the superconducting transition in ρ​(T)\rho(T) shifts to lower temperatures. Similarly, in the M​(H)M(H) data, the diamagnetic signal vanishes once the applied magnetic field exceeds the upper critical field Hc2H_{\mathrm{c2}} [see inset in Fig. III(b)]. Figure III summarizes the upper critical fields Hc2H_{\mathrm{c2}} vs. the reduced superconducting transition temperatures Tc/Tc​(0)T_{c}/T_{c}(0) for all the TaReSi samples, as identified from the ρ​(T,H)\rho(T,H) and M​(H,T)M(H,T) data. To determine the upper critical field at 0 K, the Hc2​(T)H_{\mathrm{c2}}(T) data were analyzed by means of a semiempirical Ginzburg-Landau (GL) model, Hc​2=Hc​2​(0)​(1−t2)/(1+t2)H_{c2}=H_{c2}(0)(1-t^{2})/(1+t^{2}), where t=T/Tc​(0)t=T/T_{c}(0). As shown by the solid lines, the GL model gives μ0\mu_{0}Hc2​(0)=3.7​(1)H_{\mathrm{c2}}(0)=3.7(1), 3.4(1), and 3.3(1) T for the TaReSi samples S1, S2, and S3, respectively. As for the electrical-resistivity data, the derived Hc2​(0)H_{\mathrm{c2}}(0) values are much larger than the bulk values determined from magnetization data. The different TcT_{c} or Hc2H_{\mathrm{c2}} values might be related to a strongly anisotropic upper critical field, or to the appearance of surface/filamentary superconductivity above bulk TcT_{c}. Moreover, although the magnetization- and electrical-resistivity measurements reveal different sample qualities, the superconducting properties of TaReSi seem to be robust. Such an insensitivity of SC to nonmagnetic impurities or disorder implies an ss-wave pairing in TaReSi, as further evidenced by the µSR measurements (see below).

In the GL theory of superconductivity, the coherence length ξ\xi can be calculated from ξ\xi = Φ0/2​π​Hc​2\sqrt{\Phi_{0}/2\pi\,H_{c2}}, where Φ0=2.07×103\Phi_{0}=2.07\times 10^{3} T nm2 is the quantum of magnetic flux. With a bulk μ0​Hc​2​(0)=3.4​(1)\mu_{0}H_{c2}(0)=3.4(1) T (for S2 sample), the calculated ξ​(0)\xi(0) is 9.8(1) nm. The magnetic penetration depth λ\lambda is related to the coherence length ξ\xi and the lower critical field μ0​Hc​1\mu_{0}H_{c1} via μ0Hc​1=(Φ0/4πλ2)[\mu_{0}H_{c1}=(\Phi_{0}/4\pi\lambda^{2})[ln(κ)+0.5](\kappa)+0.5], where κ\kappa = λ\lambda/ξ\xi is the GL parameter [64]. By using λ0\lambda_{0} = 562(3) nm, we find μ0​Hc​1=2.4​(1)\mu_{0}H_{c1}=2.4(1) mT, which is smaller than the value determined from the magnetization data (see Fig. II). Such difference in Hc1H_{\mathrm{c1}}, as well as the unusual behavior in Hc2H_{\mathrm{c2}} might be attributed to the anisotropic TaReSi superconductivity. To clarify this, studies on the single crystals are required.

Since certain rhenium-based superconductors are known to break time-reversal symmetry in their superconducting state [32, 33, 34, 35, 65], to verify the possible breaking of TRS in TaReSi, we performed zero-field (ZF-) µSR measurements in its normal- and superconducting states. This technique is very sensitive to the weak spontaneous fields expected to arise in these cases [65]. As shown in Fig. III, the ZF-µSR spectra of TaReSi lack any of the features associated with magnetic order or magnetic fluctuations. Indeed, in the datasets collected above- (8 K) and below TcT_{c} (0.3 K), neither coherent oscillations nor fast decays could be identified. In the absence of an external magnetic field, the muon-spin relaxation is mainly determined by the randomly oriented nuclear moments. As a consequence, the ZF-µSR spectra can be modeled by means of a phenomenological relaxation function, consisting of a combination of a Gaussian- and a Lorentzian Kubo-Toyabe relaxation [66, 67], i.e., AZF=As​[13+23​(1−σZF2​t2−ΛZF​t)​e(−σZF2​t22−ΛZF​t)]+AbgA_{\mathrm{ZF}}=A_{\mathrm{s}}[\frac{1}{3}+\frac{2}{3}(1-\sigma_{\mathrm{ZF}}^{2}t^{2}-\Lambda_{\mathrm{ZF}}t)\,\mathrm{e}^{(-\frac{\sigma_{\mathrm{ZF}}^{2}t^{2}}{2}-\Lambda_{\mathrm{ZF}}t)}]+A_{\mathrm{bg}}. Here, AsA_{\mathrm{s}} and AbgA_{\mathrm{bg}} are the initial asymmetries for the sample and sample holder, and σZF\sigma_{\mathrm{ZF}} and ΛZF\Lambda_{\mathrm{ZF}} represent the zero-field Gaussian and Lorentzian relaxation rates, respectively. The solid lines in Fig. III are fits to the above equation, yielding σZF=0.231​(1)\sigma_{\mathrm{ZF}}=0.231(1) µs-1 and ΛZF=0.005​(2)\Lambda_{\mathrm{ZF}}=0.005(2) µs-1 at 8 K and σZF=0.234​(1)\sigma_{\mathrm{ZF}}=0.234(1) µs-1 and ΛZF=0.003​(2)\Lambda_{\mathrm{ZF}}=0.003(2) µs-1 at 0.3 K, respectively. The relaxation rates in the normal- and the superconducting states of TaReSi are almost identical, visually confirmed by the overlapping ZF-µSR spectra in Fig. III. The absence of an additional µSR relaxation below TcT_{c} excludes the breaking of TRS in the superconducting state of TaReSi. On the contrary, the enhanced σZF\sigma_{\mathrm{ZF}} below TcT_{c} in TaRuSi and NbRuSi provides clear evidence of the occurrence of spontaneous magnetic fields, which break the TRS at the superconducting transition [52]. Such a selective occurrence of TRS breaking, observed also in other superconducting families [65], independent of ASOC, is puzzling and not yet fully understood, clearly demanding further investigations. Future ZF-µSR measurements on the TaRu1-xRexSi series could potentially clarify this issue.

Refer to caption
Figure 5. : ZF-µSR spectra collected in the superconducting- (0.3 K) and in the normal state (8 K) of TaReSi. The practically overlapping datasets indicate the absence of TRS breaking, whose occurrence would have resulted in a stronger decay in the 0.3-K case.
Refer to caption
Figure 6. : (a) TF-µSR spectra of TaReSi collected in the superconducting (0.3 K)- and normal (7 K) states in an applied magnetic field of 40 mT. Dashed- and solid-lines are fits to Eq. (1) using one- and two oscillations. In the latter case, each contribution is shown separately as dash-dotted lines, together with a background contribution. Fits with two oscillations show a goodness-of-fit value χr2∼1.0\chi_{\mathrm{r}}^{2}\sim 1.0, smaller than the one-oscillation fits (χr2\chi_{\mathrm{r}}^{2} ∼\sim 1.6).

To investigate the superconducting pairing in TaReSi, we carried out systematic temperature-dependent transverse-field (TF-) µSR measurements in an applied field of 40 mT. Representative TF-µSR spectra collected in the superconducting- and normal states of TaReSi are shown in Fig. III(a). In the superconducting state (e.g., at 0.3 K), the development of a flux-line lattice (FLL) causes an inhomogeneous field distribution and, thus, it gives rise to an additional damping in the TF-µSR spectra [67]. In such case, the TF-µSR spectra are generally modeled using [68]:

ATF​(t)=∑i=1nAi​cos⁡(γμ​Bi​t+ϕ)​e−σi2​t2/2+Abg​cos⁡(γμ​Bbg​t+ϕ).A_{\mathrm{TF}}(t)=\sum\limits_{i=1}^{n}A_{i}\cos(\gamma_{\mu}B_{i}t+\phi)e^{-\sigma_{i}^{2}t^{2}/2}+A_{\mathrm{bg}}\cos(\gamma_{\mu}B_{\mathrm{bg}}t+\phi). (1)
Refer to caption
Figure 7. : Temperature dependence of the superfluid density of TaReSi. The inset shows the muon-spin relaxation rates σi​(T)\sigma_{i}(T) vs. temperature. The solid, dashed, and dash-dotted lines represent fits to the ss-, pp-, and dd-wave model, with χr2∼1.1\chi_{\mathrm{r}}^{2}\sim 1.1, 1.8, and 5.2, respectively.

Here AiA_{i}, AbgA_{\mathrm{bg}} and BiB_{i}, BbgB_{\mathrm{bg}} are the initial asymmetries and local fields sensed by implanted muons in the sample and sample holder, γμ\gamma_{\mu}/2π\pi = 135.53 MHz/T is the muon gyromagnetic ratio, ϕ\phi is a shared initial phase, and σi\sigma_{i} is the Gaussian relaxation rate of the iith component. In general, the field distribution p​(B)p(B) in the superconducting state is material dependent. In case of a symmetric p​(B)p(B), one oscillation (i.e., n=1n=1) is sufficient to describe the TF-µSR spectra, while for an asymmetric p​(B)p(B), two or more oscillations (i.e., n≥2n\geq 2) are required. Here, we find that Eq. (1) with n=2n=2 can describe the experimental data quite well [see solid lines in Fig. III(a)]. The derived muon-spin relaxation rates σi\sigma_{i} are small and temperature-independent in the normal state, but below TcT_{c} they start to increase due to the onset of FLL and the increased superfluid density [see inset in Fig. III(b)]. Then, the effective Gaussian relaxation rate σeff\sigma_{\mathrm{eff}} can be calculated from σeff2/γμ2=∑i=12Ai​[σi2/γμ2−(Bi−⟨B⟩)2]/Atot\sigma_{\mathrm{eff}}^{2}/\gamma_{\mu}^{2}=\sum_{i=1}^{2}A_{i}[\sigma_{i}^{2}/\gamma_{\mu}^{2}-\left(B_{i}-\langle B\rangle\right)^{2}]/A_{\mathrm{tot}} [68], where ⟨B⟩=(A1​B1+A2​B2)/Atot\langle B\rangle=(A_{1}\,B_{1}+A_{2}\,B_{2})/A_{\mathrm{tot}} and Atot=A1+A2A_{\mathrm{tot}}=A_{1}+A_{2}. Considering the constant nuclear relaxation rate σn\sigma_{\mathrm{n}} in the narrow temperature range investigated here, confirmed also by ZF-µSR measurements (see Fig. III), the superconducting Gaussian relaxation rate can be extracted using σsc=σeff2−σn2\sigma_{\mathrm{sc}}=\sqrt{\sigma_{\mathrm{eff}}^{2}-\sigma^{2}_{\mathrm{n}}}.

In TaReSi, the upper critical field μ0​Hc2​(0)\mu_{0}H_{\mathrm{c2}}(0) ∼\sim 3.4 T is significantly larger than the applied TF field (40 mT). Hence, we can ignore the effects of the overlapping vortex cores when extracting the magnetic penetration depth from the measured σsc\sigma_{\mathrm{sc}}. The effective magnetic penetration depth λeff\lambda_{\mathrm{eff}} can then be calculated by using σsc2​(T)/γμ2=0.00371​Φ02/λeff4​(T)\sigma_{\mathrm{sc}}^{2}(T)/\gamma^{2}_{\mu}=0.00371\Phi_{0}^{2}/\lambda_{\mathrm{eff}}^{4}(T) [69, 64]. Figure III(b) summarizes the temperature-dependent inverse square of magnetic penetration depth, which is proportional to the superfluid density, i.e., λeff−2​(T)∝ρsc​(T)\lambda_{\mathrm{eff}}^{-2}(T)\propto\rho_{\mathrm{sc}}(T). The ρsc​(T)\rho_{\mathrm{sc}}(T) was analyzed by applying different models, generally described by:

ρsc​(T)=1+2​⟨∫Δk∞EE2−Δk2​∂f∂E​dE⟩FS.\rho_{\mathrm{sc}}(T)=1+2\,\Bigg{\langle}\int^{\infty}_{\Delta_{\mathrm{k}}}\frac{E}{\sqrt{E^{2}-\Delta_{\mathrm{k}}^{2}}}\frac{\partial f}{\partial E}\mathrm{d}E\Bigg{\rangle}_{\mathrm{FS}}. (2)

Here, f=(1+eE/kB​T)−1f=(1+e^{E/k_{\mathrm{B}}T})^{-1} is the Fermi function and ⟨⟩FS\langle\rangle_{\mathrm{FS}} represents an average over the Fermi surface (assumed to be an isotropic sphere, for ss-wave superconductors) [70]; Δk​(T)=Δ​(T)​δk\Delta_{\mathrm{k}}(T)=\Delta(T)\delta_{\mathrm{k}} is an angle-dependent gap function, where Δ\Delta is the maximum gap value and δk\delta_{\mathrm{k}} is the angular dependence of the gap, equal to 1, cos⁡2​ϕ\cos 2\phi, and sin⁡θ\sin\theta for an ss-, dd-, and pp-wave model, respectively, with ϕ\phi and θ\theta being the azimuthal angles. The temperature dependence of the gap is assumed to follow Δ​(T)=Δ0​tanh​{1.82​[1.018​(Tc/T−1)]0.51}\Delta(T)=\Delta_{0}\mathrm{tanh}\{1.82[1.018(T_{\mathrm{c}}/T-1)]^{0.51}\} [70, 71], where Δ0\Delta_{0} is the gap value at 0 K. Three different models, including ss-, pp-, and dd-wave, were used to describe the λeff−2\lambda_{\mathrm{eff}}^{-2}(T)(T) data. For an ss- or pp-wave model [see solid and dashed lines in Fig. III(b)], the best fits yield the same zero-temperature magnetic penetration depth λ0=562​(3)\lambda_{\mathrm{0}}=562(3) nm, but different superconducting gaps Δ0\Delta_{0} = 0.79(2) and 1.05(2) meV, respectively. While for the dd-wave model, the gap size is the same as pp-wave model, but the λ0\lambda_{\mathrm{0}} = 510(3) nm is much shorter. As can be seen in Fig. III(b), the temperature-independent λeff−2​(T)\lambda_{\mathrm{eff}}^{-2}(T) for T<2T<2 K strongly suggests a fully-gapped superconducting state in TaReSi. As a consequence, λeff−2​(T)\lambda_{\mathrm{eff}}^{-2}(T) is more consistent with the ss-wave model, here reflected in the smallest χr2\chi_{\mathrm{r}}^{2}. In the case of a pp- or dd-wave model, a less-good agreement with the measured λeff−2​(T)\lambda_{\mathrm{eff}}^{-2}(T) is found, especially at low temperatures. Although the unitary (s+i​ps+ip) pairing [52] can also describe the nodeless SC in TaReSi, its preserved TRS excludes such a possibility. In summary, TF-µSR combined with ZF-µSR data, indicate that TaReSi behaves as a conventional fully-gapped superconductor with preserved TRS.

Refer to caption
Figure 8. : (a) Calculated total- and partial (Ta-5dd, Re-5dd, and Si-3pp orbitals) density of states for TaReSi. (b) Electronic band structure of TaReSi, calculated by ignoring (red) and by considering (blue) the spin-orbit coupling. Several bands cross the Fermi level.

We also note that due to the lack of inversion symmetry in TaReSi, a mixing of spin-singlet and spin-triplet pairing is allowed. Such mixing not only can be consistent with a fully-gapped superconducting state but, more importanly, it can lead to unconventional or even topological SC. Indeed, our TF-µSR results clearly suggests a fully-gapped superconducting state, here fitted by using an ss-wave model (see Fig. III). However, this does not imply that ss-wave pairing is the only possibility. A mixed singlet-triplet pairings also allow a fully-gapped superconducting state, which in principle is allowed by the presence of ASOC [72]. Furthermore, topological SC can occur when the pairing gap changes sign on different Fermi surfaces according to the topological criterion [73]. For a minimal single-band model, there are two spin-split Fermi surfaces, whose gaps are given by Δs\Delta_{\mathrm{s}} ±\pm Δt\Delta_{\mathrm{t}}(kFk_{\mathrm{F}}), which implies that a sign change occurs when Δs\Delta_{\mathrm{s}} ¡ Δt\Delta_{\mathrm{t}}(kFk_{\mathrm{F}}).

Refer to caption
Figure 9. : (a) Illustration of Kramers Weyl points and Kramers nodal lines in TaReSi. The KWP are marked by red circles, while the KNL are depicted by green- (along Γ\Gamma–ZZ) or orange lines (along RR–WW), respectively. (b) Illustration of hourglass-shaped dispersion (purple lines) for TaReSi along the Γ\Gamma–RR–ZZ direction. The analogous results for TaReSi doped with 5% Hf and 50% W are shown in panels (c)-(d) and (e)-(f), respectively.

To gain further insight into the electronic properties of the TaReSi superconductor, we also performed band-structure calculations using the density-functional theory. The electronic band structure of TaReSi, as well as its density of states (DOS) are summarized in Fig. III. Close to the Fermi level EFE_{\mathrm{F}}, the DOS is dominated by the Ta- and Re-5dd orbitals, while the contribution from Si-3pp orbitals is negligible. The dominance of high-ZZ orbitals might lead to a relatively large band splitting. In TaReSi, the estimated DOS at EFE_{\mathrm{F}} is about 1.1 states/(eV f.u.) [= 6.5 states/(eV cell)/ZZ, with Z=6Z=6 the number of atoms per primitive cell]. This is comparable to the experimental value of 2.3 states/(eV f.u.), determined from the electronic specific-heat coefficient [54]. The electronic band structure of TaReSi, calculated by ignoring and by considering the spin-orbit coupling, is shown in Fig. III(b). When taking SOC into account, the electronic bands split due to the lifting of degeneracy, with one of them ending up closer to the Fermi level. The band splitting EASOCE_{\mathrm{ASOC}} caused by the antisymmetric spin-orbit coupling is clearly visible in TaReSi, e.g., near the XX (X1X_{1}), YY, and WW points. The estimated band splitting in TaReSi is EASOC∼300E_{\mathrm{ASOC}}\sim 300 meV, which is much larger than that of NbRuSi (∼\sim 100 meV), but comparable to TaRuSi (∼\sim300 meV) [52]. Though smaller than the band splitting in CePt3Si [74], it is comparable to that of CaPtAs and Li2Pt3B [24, 75], and much larger than that of most other weakly-correlated NCSCs [19]. The EASOCE_{\mathrm{ASOC}} of TaReSi is almost twice larger than that of the analog NbReSi compound (EASOCE_{\mathrm{ASOC}} ∼\sim 150 meV) [28, 76]. The latter crystallizes in a ZrNiAl-type noncentrosymmetric structure (P​6¯​2​mP\overline{6}2m, No. 189) and exhibits features of unconventional superconductivity, e.g., its Hc2H_{\mathrm{c2}} exceeds the Pauli limit. However, the Hc2H_{\mathrm{c2}} of TaReSi is much smaller than that of NbReSi, the former being mostly determined by the orbital limit. Since Ta has a much larger atomic number than Nb (and, hence, a larger SOC), it is not surprising that TaReSi exhibits a larger EASOCE_{\mathrm{ASOC}}, in particular, considering that its Ta-5​d5d (instead of Nb-4​d4d) orbitals contribute as much as Re-5​d5d orbitals to the DOS at the Fermi energy [see Fig. III(a)].

According to the topological-materials database [5, 6, 77, 7, 78, 79] and from our own band-structure calculations, TaReSi can be classified as a symmetry-enforced semimetal, which shares a similar band topology with NbRuSi and TaRuSi [52]. In the presence of spin-orbit coupling, owing to its nonsymmorphic space group (I​m​a​2Ima2, No. 46), TaReSi hosts Kramers Weyl points (KWP) at the high-symmetry points and Kramers nodal lines along the high-symmetry lines of its Brillouin zone. These features are marked by red circles (KWP) and green/orange lines (KNL) in Fig. III(a). The high-symmetry points at SS and TT are time-reversal symmetry invariant. As a consequence, the respective energies exhibit a twofold Kramers degeneracy protected by TRS. At the same time, due to the lack of inversion symmetry in TaReSi, these points cannot achieve the fourfold degeneracy of Dirac points and, hence, they are Weyl points. As for the high-symmetry lines along the Γ\Gamma-ZZ and RR-WW directions, the bands form a two-dimensional representation, i.e., twofold degenerate, indicating the occurrence of KNL in TaReSi. Since most of KNLs occur near the EFE_{\mathrm{F}}, with a few of them even crossing it, similarly to NbRuSi and TaRuSi [52], TaReSi can be classified as a Kramers nodal-line semimetal (KNLS). At the high-symmetry SS and TT points, the KWP in TaReSi are closer to EFE_{\mathrm{F}} than in NbRuSi and TaRuSi [52]. Since the Ru atoms have one more electron than Re, the KWP in NbRuSi and TaRuSi is shifted further below EFE_{\mathrm{F}}.

More interestingly, as shown by purple lines in Fig. III(b), due to its nonsymmorphic space-group symmetry, TaReSi also exhibits 3D bulk hourglass-type fermions, characterized by an hourglass cone with five doubly degenerate points [13, 15]. The I​m​a​2Ima2 nonsymmorphic space group contains the generator of a glide mirror reflection My={m010|1/2,0,0}M_{y}=\{m_{010}|1/2,0,0\}: (x,y,z)(x,y,z)→\to(x+1/2,−y,z)(x+1/2,-y,z) [see inset in Fig. I(a)]. Here, the ky=0k_{y}=0- and π\pi-planes are MyM_{y}-invariant planes, where all the states along the Γ\Gamma–RR–ZZ line carry the MyM_{y}-index ±i​e−i​kx/2\pm ie^{-ik_{x}/2} and give rise to the 3D bulk hourglass fermions, protected by the MyM_{y} operator [80, 81]. In this case, at a high-symmetry point, the k-vectors are 𝐤Γ=(0,0,0){\bf k}_{\Gamma}=(0,0,0), 𝐤R=(0,1/2,0){\bf k}_{R}=(0,1/2,0) and 𝐤Z=(1/2,1/2,−1/2){\bf k}_{Z}=(1/2,1/2,-1/2). Therefore, the MyM_{y}-index is ±1\pm 1 at the RR point, and ±i\pm i at Γ\Gamma and ZZ points. In agreement with the Kramers theorem, each state is twofold degenerate, i.e., pairs of doubly degenerate states exhibit identical energies, but carry opposite MyM_{y}-indexes. In the presence of a strong SOC, these doubly degenerate states split along the R→ΓR\to\Gamma or R→ZR\to Z directions. Despite this SOC-induced splitting of bands with different MyM_{y}-indexes, a residual degeneracy remains, which could give rise to the non-interacting hourglass fermions in TaReSi. To date, hourglass fermions were experimentally observed only in very few materials, as e.g., the KHgSb and Nb3(Si,Ge)Te6 topological insulators [82, 83]. Here, we establish that similar to the NbRuSi and TaRuSi compounds [52], also TaReSi belongs to this restricted class of materials, where Kramers Weyl points and hourglass fermions exist and can be tuned toward EFE_{\mathrm{F}} by Hf- or W- chemical substitutions on the Ta site [see Fig. III(c)-(d) for 5%-Hf substitution and Fig. III(e)-(f) for 50%-W substitution]. At the same time, we could show that neither chemical substitution on the Si site (here introduced via Si-to-Ge substitution), nor physical pressure have appreciable effects on the band structure of TaReSi. Besides exhibiting nontrivial electronic bands, TaReSi shows also intrinsic SC at low temperatures. This remarkable combination makes it a promising candidate material for investigating topological properties.

IV Conclusion

To summarize, we studied the noncentrosymmetric TaReSi superconductor by means of µSR measurements and band-structure calculations. The superconducting state of TaReSi is characterized by a TcT_{c} of ∼\sim5.5 K and an upper critical field μ0​Hc​2​(0)\mu_{0}H_{c2}(0) of ∼\sim3.4 T. The temperature-dependent superfluid density reveals a fully-gapped superconducting state in TaReSi. The lack of spontaneous magnetic fields below TcT_{c} indicates a preserved time-reversal symmetry in the superconducting state of TaReSi. Electronic band-structure calculations reveal that TaReSi shares a similar band topology to NbRuSi and TaRuSi, which also belong to the three-dimensional Kramers nodal-line semimetals. It, too, features hourglass fermions, protected by the nonsymmorphic space-group symmetry. Our results demonstrate that TaReSi represents a potentially interesting system for investigating the rich interplay between the exotic electronic states of Kramers nodal-line fermions, hourglass fermions, and superconductivity. It will be also interesting to explore the Zeeman-field-induced Weyl superconductor in this material. Considering the nontrivial band structure near the Fermi level and its intrinsic superconductivity, TaReSi represents one of the promising platforms for investigating the topological aspects of noncentrosymmetric superconductors.

Acknowledgements.
This work was supported by the Natural Science Foundation of Shanghai (Grants No. 21ZR1420500 and 21JC1402300), Natural Science Foundation of Chongqing (Grant No. CSTB2022NSCQ-MSX1678), and the Schweizerische Nationalfonds zur Förderung der Wissenschaftlichen Forschung (SNF) (Grants No. 200021_188706 and 206021_139082). Y.X. acknowledges support from the Shanghai Pujiang Program (Grant No. 21PJ1403100). We acknowledge the allocation of beam time at the Swiss muon source (Dolly µSR spectrometer).

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