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Fully-gapped superconductivity with preserved time-reversal symmetry in
NiBi3 single crystals

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. Meng Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    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    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. X. Zhen Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Y. H. Wang Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, 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    D. J. Gawryluk 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 NiBi3 single crystals by means of electrical-resistivity-, magnetization-, and muon-spin rotation and relaxation (μ\muSR) measurements. As a single crystal, NiBi3 adopts a needle-like shape and exhibits bulk superconductivity with Tc≈4.1T_{c}\approx 4.1 K. By applying magnetic fields parallel and perpendicular to the bb-axis of NiBi3, we establish that its lower- and upper critical fields, as well as the magnetic penetration depths show slightly different values, suggesting a weakly anisotropic superconductivity. In both cases, the zero-temperature upper critical fields are much smaller than the Pauli-limit value, indicating that the superconducting state is constrained by the orbital pair breaking. The temperature evolution of the superfluid density, obtained from transverse-field μ\muSR, reveals a fully-gapped superconductivity in NiBi3, with a shared superconducting gap Δ0\Delta_{0} = 2.1 kBk_{\mathrm{B}}TcT_{c} and magnetic penetration depths λ0\lambda_{0} = 223 and 210 nm for H∥bH\parallel b- and H⟂bH\perp b, respectively. The lack of spontaneous fields below TcT_{c} indicates that time-reversal symmetry is preserved in NiBi3. The absence of a fast muon-spin relaxation and/or precession in the zero-field μ\muSR spectra definitely rules out any type of magnetic ordering in NiBi3 single crystals. Overall, our investigation suggests that NiBi3 behaves as a conventional ss-type superconductor.

††preprint: Preprint: , 9:15

I Introduction

Topological superconductors offer many attractive properties, which range from the possibility to host Majorana quasiparticles to enabling topological quantum computing [1, 2, 3]. This has spurred researchers to explore different routes and/or materials to realize them. For instance, superconductors with p+i​pp+ip pairing have been predicted to support Majorana bound states at their vortices [4]. Besides relying on bulk superconductors with nontrivial electronic bands, another approach towards topological superconductivity (SC) consists in combining conventional ss-wave superconductors with topological insulators or ferromagnets, so as to form heterostructures. In both cases, proximity effects at the interface may lead to a two-dimensional superconducting state with an unconventional pairing [5, 6, 7].

Among the different types of heterostructures, epitaxial Bi/Ni bilayers have been proposed as a promising candidate for hosting chiral pp-wave SC [8]. In fact, tunneling experiments in such bilayers have shown the coexistence of ferromagnetism (FM) and SC below the superconducting transition, at Tc≈4T_{c}\approx 4 K [9]. More recently, surface magneto-optic Kerr-effect measurements revealed spontaneous magnetic fields occurring below the onset of SC, which suggest the breaking of time-reversal symmetry (TRS) in the superconducting state of Bi/Ni bilayers [10]. Finally, time-domain terahertz spectroscopy experiments found a nodeless SC state in Bi/Ni bilayers [11], more consistent with a chiral pp-wave SC [12].

At the same time, there is mounting evidence that the occurrence of SC in Bi/Ni bilayers could be related to the formation of a superconducting NiBi3 phase (with Tc≈T_{c}\approx 4 K [13, 14, 15]) during the thin-film growth [16, 17, 18, 19, 19, 20, 21, 22, 23]. This because NiBi3-free Bi/Ni bilayers show no signs of superconductivity [20, 21]. These observations have triggered the researchers’ interests to study the superconducting properties of pure NiBi3. Similar to the Bi/Ni bilayers [9], also bulk NiBi3 shows the coexistence of SC and FM [24, 25, 26, 27]. Thus, magnetization data in its superconducting state, show a clear loop of ferromagnetic hysteresis [25, 26, 15]. Generally, superconductivity and ferromagnetism are antagonistic ground states. However, akin to certain U-based materials [28, 29, 30], NiBi3, too, belongs to the rare class of ferromagnetic superconductors, potentially able to host spin-triplet pairing.

To date, the superconducting properties of NiBi3 have been mostly investigated via magnetic- and transport measurements [13, 14, 15, 24, 25, 26, 27]. Yet, the microscopic nature of its SC, in particular, the superconducting order parameter, has not been explored and awaits further investigation. In addition, it is not yet clear if the observed breaking of TRS and the unconventional SC of epitaxial Bi/Ni bilayers can be attributed to the formation of the NiBi3 phase or not [10].

To clarify the above issues, we synthesized NiBi3 single crystals using the flux method, and studied their superconducting properties by means of electrical-resistivity-, magnetization-, and muon-spin rotation and relaxation (μ\muSR) measurements. We found that NiBi3 exhibits a fully-gapped superconducting state with a preserved TRS. The absence of any magnetic order, on the surface or in the bulk of NiBi3 single crystals, was also confirmed. Our results suggest that NiBi3 behaves as a conventional ss-type superconductor and, thus, the unconventional superconducting properties observed in Bi/Ni bilayers cannot be attributed to the NiBi3 phase.

Refer to caption
Figure 1. : (a) Room-temperature x-ray powder diffraction pattern and Rietveld refinement for NiBi3. The red crosses and the solid black line represent the experimental pattern and the Rietveld refinement profile, respectively. The blue line at the bottom shows the residuals, i.e., the difference between the calculated and experimental data. The vertical bars mark the calculated Bragg-peak positions for NiBi3 (green) and Bi (orange). The refinement RR-factors are RpR_{\mathrm{p}} ∼\sim 3.71 %, RwpR_{\mathrm{wp}} ∼\sim 5.82 %, and RexpR_{\mathrm{exp}} ∼\sim 1.37 %. Note that, the minority Bi phase comes from the remaining flux on the surface of crystals. (b) XRD pattern of a NiBi3 single crystal. The unit cell is shown in the inset, with the blue and orange spheres depicting the Bi and Ni atoms.

II Experimental details

The NiBi3 single crystals were grown from molten bismuth flux. High-purity Bi pieces (Alfa Aesar, 99.99%) and Ni powders (Alfa Aesar, 99.9%) with a 10:1 ratio were loaded in an Al2O3 crucible, which was sealed in a quartz ampoule under high vacuum. The quartz ampoule was heated up to 1050 ∘C and then kept at this temperature for over 24 hours. Finally, it was slowly cooled down to 360 ∘C at a rate of 2 ∘C/h, with the remaining Bi flux being removed via centrifugation. The typical dimensions of the obtained needle-like single crystals were about ∼\sim3 mm ×\times 0.3 mm ×\times 0.3 mm (see below). The crystals were checked by powder x-ray diffraction (XRD), recorded using a Bruker D8 diffractometer. Further characterization involved electrical-resistivity- and magnetization measurements, performed on a Quantum Design physical property measurement system (PPMS) and a magnetic property measurement system (MPMS), respectively. For the electrical-resistivity measurements, the dc current was applied along the bb-axis.

The bulk μ\muSR 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 crystals were aligned and mounted on a 25-μ\mum thick copper foil, which ensured thermalization at low temperatures. The magnetic fields were applied both parallel and perpendicular to the bb-axis of the NiBi3 single crystal. The time-differential μ\muSR data were collected upon heating and then analyzed by means of the musrfit software package [31].

Refer to caption
Figure 2. : Temperature-dependent magnetic susceptibility χV​(T)\chi_{\mathrm{V}}(T) (a) and electrical resistivity ρ​(T)\rho(T) (b) for a NiBi3 single crystal. The dashed lines mark the TcT_{c} value. While ρ​(T)\rho(T) was measured in zero-field conditions, χV​(T)\chi_{\mathrm{V}}(T) data were collected in a magnetic field of 1 mT, applied perpendicular to the bb-axis. The susceptibility data were corrected to account for the demagnetization factor. The upper inset depicts a needle-like NiBi3 single crystal, with the arrow indicating the bb-axis, while the lower inset shows ρ​(T)\rho(T) up to 300 K. In the latter case, the solid black line is a fit to the equation discussed in the text.

III Results and discussion

III.1 Crystal structure

The crystal structure and phase purity of NiBi3 single crystals were checked via XRD at room temperature. As shown in Fig. I, we performed Rietveld refinements of the XRD pattern by using the Fullprof suite [32]. Consistent with previous results, we confirm that NiBi3 crystallizes in a centrosymmetric orthorhombic structure with a space group P​n​m​aPnma (No. 62) [13, 14, 15, 24, 25, 26, 27]. In this structure, bismuth atoms form an octahedral array, while nickel atoms form part of linear chains [25]. The crystal structure of a unit cell is shown in the inset of Fig. I(b). The refined lattice parameters, aa = 8.8799(1) Å, bb = 4.09831(6) Å, and cc= 11.4853(1) Å, are in good agreement with the results reported in the literature [13, 14, 15, 24, 25, 26, 27]. According to the refinements in Fig. I, a tiny amount of elemental bismuth (∼\sim1%) was also identified, here attributed to the remanent Bi-flux on the surface of single crystals. Considering its small amount and the fact that Bi is not superconducting in the studied temperature range [33], the bismuth presence has negligible effects on the superconducting properties of NiBi3. We also performed XRD measurement on a NiBi3 single crystal. As shown in Fig. I(b), only the (004) reflection [the strongest among the (00ll)-reflections] was detected, confirming the single-crystal nature of the NiBi3 sample.

III.2 Bulk superconductivity

Refer to caption
Figure 3. : Field-dependent magnetization curves collected at various temperatures after cooling the NiBi3 single crystal in zero field for (a) H∥bH\parallel b and (b) H⟂bH\perp b. (c) The respective 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. The lower critical fields were corrected by a demagnetization factor NN, which is 0.5 for H⟂bH\perp b. Since the NiBi3 crystal is needle-like, no demagnetization factor is required for H∥bH\parallel b (i.e., NN = 0).

The bulk superconductivity of NiBi3 single crystals was first characterized by magnetic-susceptibility measurements, using both field-cooling (FC) and zero-field-cooling (ZFC) protocols in an applied field of 1 mT. As shown in Fig. II(a), a clear diamagnetic response appears below the superconducting transition at TcT_{c} = 4.05 K. Similar results were obtained when applying the magnetic field parallel to the bb-axis. After accounting for the demagnetization factor NN, which is estimated to be 0.5 for H⟂bH\perp b and 0 for H∥bH\parallel b [34], the superconducting shielding fraction of NiBi3 was found to be close to 100%, indicative of bulk SC.

The temperature-dependent electrical resistivity ρ​(T)\rho(T) of NiBi3 single crystals was measured from 2 K up to room temperature. It reveals a metallic behavior, without any anomalies associated with phase transitions in the normal state [see inset in Fig. II(b)]. The electrical resistivity in the low-temperature region is plotted in the main panel of Fig. II(b), where the superconducting transition (with Tconset=4.16T_{c}^{\mathrm{onset}}=4.16 K, Tcmid=4.08T_{c}^{\mathrm{mid}}=4.08 K and Tczero=4.0T_{c}^{\mathrm{zero}}=4.0 K) is clearly seen. As shown by the dashed-line in Fig. II, the TcmidT^{\mathrm{mid}}_{c} value is consistent with the onset of the superconducting transition in the magnetic susceptibility. Therefore, the TcmidT^{\mathrm{mid}}_{c} values were used to determine the upper critical field of NiBi3. The ρ​(T)\rho(T) curve can be described by the Bloch-Grüneisen-Mott (BGM) formula ρ​(T)=ρ0+4​A​(TΘDR)5​∫0ΘDRTz2​d​z(ez−1)​(1−e−z)−α​T3\rho(T)=\rho_{0}+4A\left(\frac{T}{\Theta_{\mathrm{D}}^{\mathrm{R}}}\right)^{5}\int_{0}^{\frac{\Theta_{\mathrm{D}}^{\mathrm{R}}}{T}}\!\!\frac{z^{2}\mathrm{d}z}{(e^{z}-1)(1-e^{-z})}-\alpha T^{3} [35, 36]. Here, the first term ρ0\rho_{0} is the residual resistivity due to the scattering of conduction electrons on the static defects of the crystal lattice, while the second term describes the electron-phonon scattering, with ΘDR\Theta_{\mathrm{D}}^{\mathrm{R}} being the characteristic (Debye) temperature and AA a coupling constant. The third term represents a contribution due to ss-dd interband scattering, α\alpha being the Mott coefficient [37, 38]. The fit in Fig. II (solid black line) results in ρ0=4.5​(2)\rho_{0}=4.5(2) µΩ\Omegacm, A=32​(1)A=32(1) µΩ\Omegacm, ΘDR=100​(3)\Theta_{\mathrm{D}}^{\mathrm{R}}=100(3) K, and α=8.0​(2)\alpha=8.0(2) ×\times10-7 µΩ\OmegacmK-3. The derived ΘDR\Theta_{\mathrm{D}}^{\mathrm{R}} is comparable with the value determined from low-TT specific heat [13]. The fairly large residual resistivity ratio (RRR), i.e., ρ\rho(300 K)/ρ0\rho_{0} ∼\sim 18, and the sharp superconducting transition (Δ\DeltaTcT_{c} = 0.15 K) both indicate a good sample quality.

III.3 Lower and upper critical fields

To determine the lower critical field Hc1H_{\mathrm{c1}}, the field-dependent magnetization M​(H)M(H) of a NiBi3 single crystal was measured at various temperatures up to 3.75 K. Figure III.2(a)-(b) plots the M​(H)M(H) for H∥bH\parallel b and H⟂bH\perp b, respectively. The estimated Hc1H_{\mathrm{c1}} values at different temperatures (accounting for a demagnetization factor) are summarized in Fig. III.2(c). 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) = 7.4(1) and 6.2(1) mT for H∥bH\parallel b and H⟂bH\perp b, respectively. The zero-temperature Hc1H∥bH_{\mathrm{c1}}^{H\parallel b}/Hc1H⟂bH_{\mathrm{c1}}^{H\perp b} ratio (∼\sim1.2) suggests a weakly anisotropic superconductivity in the NiBi3 single crystal.

We also performed temperature-dependent electrical-resistivity measurements ρ​(T,H)\rho(T,H) at various applied magnetic fields, as well as studied the field-dependent magnetization M​(H,T)M(H,T) at various temperatures by applying the magnetic field either parallel or perpendicular to the bb-axis of the NiBi3 single crystal. As shown in Fig. III.3(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}}. For H∥bH\parallel b, no zero resistivity is observed down to ∼2\sim 2 K for μ0​H>\mu_{0}H> 0.35 T. For H⟂bH\perp b, the zero resistivity already vanishes for μ0​H>\mu_{0}H> 0.25 T, implying that, in NiBi3, the Hc2H_{\mathrm{c2}} value along the bb-axis is larger than that perpendicular to it.

Figure III.3(e) and (f) summarizes the upper critical fields Hc2H_{\mathrm{c2}} vs. the reduced superconducting transition temperature Tc/Tc​(0)T_{c}/T_{c}(0) for both H∥bH\parallel b and H⟂bH\perp b, as identified from the ρ​(T,H)\rho(T,H) and M​(H,T)M(H,T) data. The temperature evolution of the upper critical field Hc2​(T)H_{\mathrm{c2}}(T) is well described by the Werthamer-Helfand-Hohenberg (WHH) model [39]. As shown by the dash-dotted lines in Fig. III.3(e)-(f), the WHH model agrees remarkably well with the experimental data and provides μ0​Hc2H∥b​(0)\mu_{0}H_{\mathrm{c2}}^{H\parallel b}(0) = 0.70(2) T and μ0​Hc2H⟂b​(0)\mu_{0}H_{\mathrm{c2}}^{H\perp b}(0) = 0.41(1), respectively. These values are comparable to previous results [13, 15, 27] and both are much smaller than the Pauli-limiting field (i.e., 1.86TcT_{c} ∼\sim 7.4 T). Consequently, the orbital pair-breaking mechanism seems to be dominant in NiBi3. Similarly to the lower critical fields, also the Hc2H∥bH_{\mathrm{c2}}^{H\parallel b}/Hc2H⟂bH_{\mathrm{c2}}^{H\perp b} ratio (∼\sim 1.7) confirms the weakly anisotropic SC in NiBi3.

In the Ginzburg-Landau (GL) theory of superconductivity, the magnetic penetration depth λ\lambda is related to the coherence length ξ\xi, and the lower critical field via μ0​Hc1=(Φ0/4​π​λ2)\mu_{0}H_{\mathrm{c1}}=(\Phi_{0}/4\pi\lambda^{2})ln(κ)(\kappa), where Φ0=2.07×103\Phi_{0}=2.07\times 10^{{\color[rgb]{0,0,1}3}} T nm2 is the quantum of magnetic flux, κ\kappa = λ\lambda/ξ\xi is the GL parameter [40]. By using μ0​Hc1\mu_{0}H_{\mathrm{c1}} and ξ\xi values [calculated from μ0​Hc2​(0)\mu_{0}H_{\mathrm{c2}}(0) = Φ0\Phi_{0}/2π​ξ​(0)2\pi\xi(0)^{2}], the resulting λGL​(0)=229​(2)\lambda_{\mathrm{GL}}(0)=229(2) and 238(2) nm for H∥bH\parallel b and H⟂bH\perp b are both compatible with the experimental value determined from μ\muSR data. All the superconducting parameters are summarized in Table IV.

Refer to caption
Figure 4. : Temperature-dependent electrical resistivity ρ​(T,H)\rho(T,H) for various applied magnetic fields (a) and field-dependent magnetization curves M​(H,T)M(H,T), collected at various temperatures below TcT_{c} (b), for the H∥bH\parallel b case. The analogue results for H⟂bH\perp b are shown in panels (c) and (d). In ρ​(T,H)\rho(T,H), TcT_{c} was chosen as the mid-point of the superconducting transition (see details in Fig. II). While in M​(H,T)M(H,T), Hc2H_{\mathrm{c2}} was determined as the field where the diamagnetic response vanishes (here indicated by an arrow). Upper critical field Hc2H_{\mathrm{c2}} vs. the reduced transition temperature Tc/Tc​(0)T_{c}/T_{c}(0) for H∥bH\parallel b (e) and H⟂bH\perp b (f), respectively. The TcT_{c} and Hc2H_{\mathrm{c2}} values were determined from the measurements shown in panels (a)-(d). Circles refer to magnetization, while squares refer to resistivity measurements. The dash-dotted lines represent fits to the WHH model.

III.4 TF-μ\muSR and fully-gapped superconductivity

Refer to caption
Figure 5. : TF-μ\muSR spectra in the normal- and superconducting states of NiBi3 single crystals in an applied magnetic field of 14.5 mT for H∥bH\parallel b (a) and H⟂bH\perp b (b), respectively. The fast Fourier transforms of the TF-μ\muSR spectra in panels (a) and (b) are shown in panels (d) and (e), respectively. For H⟂bH\perp b-axis, the TF-μ\muSR and FFT spectra for an applied magnetic field of 30 mT are shown in panels (c) and (f), respectively. Solid lines are fits to Eq. (1) using two oscillations, here also shown separately as dash-dotted lines in (c)-(f), together with a background contribution. Note the clear field-distribution broadening due to the onset of FLL below TcT_{c}.

To investigate the single-crystal anisotropy of superconducting pairing in NiBi3, we carried out systematic transverse-field (TF-) μ\muSR measurements, where we applied the field along different orientations. Representative TF-μ\muSR spectra collected in a field of 14.5 mT in the superconducting- and normal states of NiBi3 are shown in Fig. III.4(a) for H∥bH\parallel b-axis and in Fig. III.4(b) for H⟂bH\perp b-axis. In the latter case, the TF-μ\muSR spectra were also collected in a field of 30 mT [see Fig. III.4(c)]. In case of a type-II superconductor (see, e.g., the 0.3-K data in Fig. III.4), 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-μ\muSR spectra [41]. These are generally modeled using [42]:

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)

Here AiA_{i} (∼\sim85%), AbgA_{\mathrm{bg}}(∼\sim15%) and BiB_{i}, BbgB_{\mathrm{bg}} are the initial asymmetries and local fields sensed by the implanted muons in the sample and sample holder (i.e., Cu) or in the residual bismuth, 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. Note that, in the studied temperature range, the effects of the residual bismuth are negligible due to its non-magnetic and non-superconducting nature. We find that two oscillations (i.e., n=2n=2) are required to describe the TF-μ\muSR spectra for both H∥bH\parallel b- and H⟂bH\perp b-axis (see solid lines in Fig. III.4). Indeed, as shown in Fig. III.4(a)-(c), oscillations with two distinct frequencies can be clearly identified and, generally, the model with two oscillations provides a better fit than that with a single oscillation. For example, in Fig. III.4(c), the two-oscillation model yields a goodness-of-fit parameter χr2\chi_{\mathrm{r}}^{2} ∼\sim 1.4, twice smaller than the single-oscillation fit (χr2\chi_{\mathrm{r}}^{2} ∼\sim 3.2). The fast Fourier transforms (FFT) of the TF-μ\muSR spectra at 0.3 K are shown by dash-dotted lines in Fig. III.4(d)-(f), which illustrate the two components (A1A_{1} and A2A_{2}) and the background signal (AbgA_{\mathrm{bg}}). The two-peak FFT in NiBi3 might be related to two different muon-stopping sites, a feature to be confirmed by future DFT calculations.

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. At the same time, a diamagnetic field shift, Δ​B​(T)=⟨B⟩−Bappl\Delta B(T)=\langle B\rangle-B_{\mathrm{appl}}, appears below TcT_{c}, with ⟨B⟩=(A1​B1+A2​B2)/Atot\langle B\rangle=(A_{1}\,B_{1}+A_{2}\,B_{2})/A_{\mathrm{tot}}, where Atot=A1+A2A_{\mathrm{tot}}=A_{1}+A_{2} and BapplB_{\mathrm{appl}} is the applied external field (see, e.g., the inset of Fig. III.4). The effective Gaussian relaxation rate 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}} [42]. Then, the superconducting Gaussian relaxation rate σsc\sigma_{\mathrm{sc}}, can be extracted by subtracting the nuclear contribution according to σsc=σeff2−σn2\sigma_{\mathrm{sc}}=\sqrt{\sigma_{\mathrm{eff}}^{2}-\sigma^{2}_{\mathrm{n}}}. Here, σn\sigma_{\mathrm{n}} is the nuclear relaxation rate, almost constant in the covered temperature range and relatively small in NiBi3, as confirmed also by the ZF-μ\muSR data (see Fig. III.5). Since the applied TF fields (14.5 and 30 mT) are not really small compared to the modest upper critical fields of NiBi3 [see Fig. III.3(e) and (f)], to calculate the magnetic penetration depth λ\lambda from σsc\sigma_{\mathrm{sc}} we had to consider the overlap of the vortex cores. Consequently, in our case, λ\lambda was calculated by means of σsc=0.172​γμ​Φ02​π​(1−h)​[1+1.21​(1−h)3]​λ−2\sigma_{\mathrm{sc}}=0.172\frac{\gamma_{\mu}\Phi_{0}}{2\pi}(1-h)[1+1.21(1-\sqrt{h})^{3}]\lambda^{-2}, where h=Bappl/Bc2h=B_{\mathrm{appl}}/B_{\mathrm{c2}} [43, 44].

Refer to caption
Figure 6. : Superfluid density vs. reduced temperature T/TcT/T_{c} for NiBi3 single crystals. Inset shows the temperature-dependent muon-spin relaxation rate σi​(T)\sigma_{i}(T) (left axis) and the diamagnetic shift Δ\DeltaB​(T)B(T) (right-axis) for the TF-30 mT-μ\muSR spectra. The solid line in the main panel represents a fit to the fully gapped ss-wave model with a single energy gap, while the dashed line is a fit using the pp-wave model to the H∥bH\parallel b-data.

Figure III.4 summarizes the temperature-dependent inverse square of the magnetic penetration depth [proportional to the superfluid density, i.e., λ−2​(T)∝ρsc​(T)\lambda^{-2}(T)\propto\rho_{\mathrm{sc}}(T)] for both H∥bH\parallel b and H⟂bH\perp b. For H⟂bH\perp b, the data collected in a field of 30 mT are also presented. 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 [40]; 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 and sin⁡θ\sin\theta for an ss- and pp-wave model, respectively, with 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}\} [40, 45], where Δ0\Delta_{0} is the gap value at 0 K. As can be clearly seen in Fig. III.4 (see dashed line), the pp-wave model exhibits a significant deviation from the experimental data. While the temperature-invariant superfluid density below TcT_{c}/3 strongly suggests the absence of low-energy excitations and, hence, a fully-gapped superconducting state in NiBi3. Consequently, the ρsc​(T)\rho_{\mathrm{sc}}(T) is more consistent with the ss-wave model. As shown by the solid line in Fig. III.4, in both cases (i.e., H∥bH\parallel b- and H⟂bH\perp b-axis), the ss-wave model describes ρsc​(T)\rho_{\mathrm{sc}}(T) very well across the entire temperature range and yields a zero-temperature gap Δ0\Delta_{0} = 2.10(5) kBk_{\mathrm{B}}TcT_{c}. Such isotropic gap value along different crystal directions is consistent with previous Andreev-reflection spectroscopy results [46]. For H∥bH\parallel b, the estimated λ0ac\lambda_{0}^{\mathrm{ac}} is 223(2) nm, while for H⟂bH\perp b, we find λ0b\lambda_{0}^{\mathrm{b}} = 216(2) and 210(2) nm for TF-14.5 mT and TF-30 mT, respectively. The small λ0ac\lambda_{0}^{\mathrm{ac}}/λ0b\lambda_{0}^{\mathrm{b}} ratio (here, ∼\sim 1.03) and the similar gap sizes along different crystal directions confirm once more the weakly anisotropic SC of NiBi3 single crystals. Note that, although the single-oscillation model also gives a temperature-independent superfluid density below 1/3TcT_{c} as the two-oscillation model, the estimated λ0\lambda_{0} values by using the former model are significantly different from the values estimated from the magnetization data.

III.5 ZF-μ\muSR and preserved time-reversal symmetry

Refer to caption
Figure 7. : Representative ZF-μ\muSR spectra of a NiBi3 single crystal in the superconducting- (0.3 K) and in the normal states (4.5 K) (a). The inset shows an enlarged plot of 0.3 K-ZF-μ\muSR on a short time scale. Solid lines are fits to the equation described in the text, with the derived temperature-dependent Lorentzian- ΛZF​(T)\Lambda_{\mathrm{ZF}}(T) and Gaussian σZF​(T)\sigma_{\mathrm{ZF}}(T) relaxation rates being shown in panels (b) and (c), respectively. The dashed line at 4 K marks the bulk TcT_{c} and serves as a guide to the eyes. None of the reported fit parameters shows a distinct enhancement below TcT_{c}, thus suggesting a preserved TRS.

To search for possible ferromagnetism and breaking of the time-reversal symmetry in a NiBi3 single crystal, we performed zero-field (ZF-) μ\muSR measurements covering both the normal- and superconducting states. As shown in Fig. III.5, neither coherent oscillations nor fast decays could be identified in the spectra collected above- (4.5 K) and below TcT_{c} (0.3 K), hence implying the lack of any magnetic order or fluctuations. As clearly demonstrated in the inset of Fig. III.5(a), the ZF-μ\muSR spectra are almost flat on a short time scale (i.e., tt ¡ 0.5 μ\mus), confirming again the absence of fast oscillations which would be caused by a possible ferromagnetic ordering, as instead reported in previous work [24, 25]. Thus, in the absence of external fields, the muon-spin relaxation in NiBi3 is mainly due to the randomly oriented nuclear moments, which can be modeled by a Gaussian Kubo-Toyabe relaxation function GKT=[13+23​(1−σZF2​t2)​e−σZF2​t2/2]G_{\mathrm{KT}}=[\frac{1}{3}+\frac{2}{3}(1-\sigma_{\mathrm{ZF}}^{2}t^{2})\,\mathrm{e}^{-\sigma_{\mathrm{ZF}}^{2}t^{2}/2}]  [47, 41]. Here, σZF\sigma_{\mathrm{ZF}} is the zero-field Gaussian relaxation rate. The ZF-μ\muSR spectra were fitted by considering also an additional electronic contribution. The solid lines in Fig. III.5(a) represent fits to AZF​(t)=As​GKT​e−ΛZF​t+AbgA_{\mathrm{ZF}}(t)=A_{\mathrm{s}}G_{\mathrm{KT}}\mathrm{e}^{-\Lambda_{\mathrm{ZF}}t}+A_{\mathrm{bg}}, where ΛZF\Lambda_{\mathrm{ZF}} is the zero-field exponential muon-spin relaxation rate, AsA_{\mathrm{s}} and AbgA_{\mathrm{bg}} are the same as in TF-μ\muSR [see Eq. (1)]. The derived ΛZF\Lambda_{\mathrm{ZF}} and σZF\sigma_{\mathrm{ZF}} values as a function of temperature are shown in Fig. III.5(b) and (c), respectively. Neither ΛZF​(T)\Lambda_{\mathrm{ZF}}(T) nor σZF​(T)\sigma_{\mathrm{ZF}}(T) show a systematic enhancement below TcT_{c}. Here, the jump of σZF\sigma_{\mathrm{ZF}} at 3.5 K [Fig. III.5(c)] is not related to a TRS-breaking effect, but to the correlated decrease of ΛZF\Lambda_{\mathrm{ZF}} [Fig. III.5(b)]. The lack of an additional relaxation below TcT_{c} excludes a possible TRS-breaking effect in the superconducting state of the NiBi3 single crystal, here also reflected in the practically overlapping datasets shown in Fig. III.5(a).

IV Discussion

Refer to caption
Figure 8. : Uemura plot showing the superconducting transition TcT_{c} against the effective Fermi temperature TFT_{\mathrm{F}} for different kinds of superconductors. The grey region 1/100<Tc/TF<1/101/100<T_{c}/T_{\mathrm{F}}<1/10 delimits the band of unconventional superconductors, including heavy fermions, organics, fullerenes, pnictides, and high-TcT_{c} cuprates. A few selected superconductors (see Refs. 48, 49, 50, 51, 52) are shown here with different symbols. The dotted line corresponds to Tc=TFT_{c}=T_{\mathrm{F}}, where TFT_{\mathrm{F}} is the temperature associated with the Fermi energy, EF=kB​TFE_{\mathrm{F}}=k_{\mathrm{B}}T_{\mathrm{F}}.

First, we discuss the absence of ferromagnetism in NiBi3. Previous works found evidence for either extrinsic or intrinsic ferromagnetism in NiBi3 single crystals. In the extrinsic case, the amorphous Ni impurities in NiBi3 lead to a clear drop in magnetization at temperatures close to their Curie temperature [15]. In NiBi3, such FM can even coexist with SC, as demonstrated by the ferromagnetic hysteresis loops observed in the magnetization data [24, 25, 26]. Besides bulk crystals with Ni impurities, also submicrometer-sized NiBi3 particles, or quasi-one-dimensional nanoscale strips of NiBi3 undergo a ferromagnetic order [26]. In addition, below 150 K, electron-spin-resonance data reveal ferromagnetic-like fluctuations on the surface of NiBi3 crystals [27]. The above evidence suggests that NiBi3 is a possible ferromagnetic superconductor, which might enable spin-triplet SC pairing, analogous to that found in other ferromagnetic superconductors [28, 29, 30]. Here, we applied the ZF-μ\muSR technique to detect a possible ferromagnetic order in NiBi3 single crystals. Due to their large magnetic moment, muons represent one of the most sensitive magnetic probes, able to sense very low internal fields (∼\sim10−210^{-2} mT), and thus, to detect local magnetic fields of either nuclear or electronic origin [41, 53]. In contrast to previous studies, our ZF-μ\muSR results do not evidence any ferromagnetic order or fluctuations in NiBi3, thus implying that the previously reported ferromagnetism is most likely of extrinsic nature. In addition, our ZF-μ\muSR results indicate that time-reversal symetry is preserved in the superconducting state of NiBi3. This conclusion is further supported by recent high-resolution surface magneto-optic Kerr effect measurements in NiBi3 single crystals, which also do not find any traces of a spontaneous Kerr signal in the superconducting state [54].

Second, we discuss the weakly anisotropic superconductivity in NiBi3. Although NiBi3 single crystals exhibit a quasi-one dimensional needle-like shape [see inset in Fig. II(a)], their crystal structure is definitely three dimensional, with aa =8.8799 Å, bb = 4.09831 Å, and cc= 11.4853 Å (see Fig. I). Such an intrinsically three-dimensional structure could naturally explain the almost isotropic- (or weakly anisotropic) electronic properties of NiBi3. As confirmed by our study, the lower critical field Hc1H_{\mathrm{c1}} (Fig. III.2), the upper critical field Hc2H_{\mathrm{c2}} (Fig. III.3), the magnetic penetration depth λ\lambda, and the superconducting gap Δ0\Delta_{0} (Fig. III.4), all show comparable values for H∥bH\parallel b- and H⟂bH\perp b-axis. These results clearly indicate a weakly anisotropic SC in NiBi3 single crystals, as also observed in NiBi3 thin films [18]. Moreover, electrical-resistivity measurements reveal almost identical upper critical fields when the magnetic field is rotated in the a​cac-plane of a NiBi3 single crystal [13]. Future electronic band-structure calculations could provide further hints about the observed weak anisotropy.

Finally, let us compare the superconducting parameters of NiBi3 with those of other superconductors. By using the SC parameters obtained from the measurements presented here, we calculate an effective Fermi temperature TFT_{\mathrm{F}} ∼\sim 1.5 ×\times103 K for NiBi3 (here, we consider the H∥bH\parallel b case). We recall that TFT_{\mathrm{F}} is proportional to ns2/3n_{\mathrm{s}}^{2/3}/m⋆m^{\star}, where nsn_{\mathrm{s}} and m⋆m^{\star} are the carrier density and the effective mass. The estimated m⋆m^{\star} ∼\sim 8 mem_{e}, indicates a moderate degree of electronic correlations in NiBi3. Different families of superconductors can be classified according to their TcT_{c}/TFT_{\mathrm{F}} ratios into a so-called Uemura plot [49]. In such plot, many conventional superconductors, as e.g., Al, Sn, and Zn, are located at Tc/TF<∼T_{c}/T_{\mathrm{F}}<\sim10-4. Conversely, several types of unconventional superconductors, including heavy fermions, organics, high-TcT_{c} iron pnictides, and cuprates, all lie in the 10-2 <Tc/TF<<T_{c}/T_{\mathrm{F}}< 10-1 band. Between these two categories lie a few different types of superconductors, including multigap- and noncentrosymmetric superconductors, etc. [48, 49, 50, 51, 52]. Although located clearly far off the conventional band, our TF- and ZF-μ\muSR results suggest that the NiBi3 superconductivity is more consistent with a conventional fully-gapped SC with preserved time-reversal symmetry. Such conclusion is also supported by the linear suppression of TcT_{c} under applied pressure. For instance, the TcT_{c} of NiBi3 decreases from 4 K to 3 K when the pressure increases from 0 to 2.2 GPa [55].

Table 1.: Superconducting parameters of NiBi3 single crystal, as determined from electrical-resistivity, magnetization, and TF-14.5 mT-μ\muSR measurements.
Property Unit H∥bH\parallel b H⟂bH\perp b
TcχT_{c}^{\chi} K 4.05(5) 4.05(5)
Tcμ​SRT_{c}^{\mu\mathrm{SR}} K 3.9(1) 3.9(1)
μ0​Hc​1\mu_{0}H_{c1} mT 7.4(1) 6.2(1)
μ0​Hc​2\mu_{0}H_{c2} T 0.70(1) 0.41(1)
Δ0\Delta_{0}(ss-wave) kB​Tck_{\mathrm{B}}T_{c} 2.10(5) 2.10(5)
Δ0\Delta_{0}(pp-wave) kB​Tck_{\mathrm{B}}T_{c} 3.00(5) 3.00(5)
λ0\lambda_{0} nm 223(2) 216(2)
ξ​(0)\xi(0) nm 21.7(2) 28.3(3)
λGL​(0)\lambda_{\mathrm{GL}}(0) nm 229(2) 238(2)
κ\kappa – 10.5(2) 8.4(2)

V Conclusion

To summarize, we investigated the superconducting properties of flux-grown NiBi3 single crystals by means of electrical resistivity-, magnetization-, and μ\muSR measurements. NiBi3 was shown to exhibit bulk SC with TcT_{c} ∼\sim 4.1 K. By applying magnetic fields along different crystal directions, we obtained the lower critical field Hc1H_{\mathrm{c1}}, the upper critical field Hc2H_{\mathrm{c2}}, and the magnetic penetration depth λ\lambda for both the H∥bH\parallel b- and H⟂bH\perp b cases. Both Hc2H∥b​(0)H_{\mathrm{c2}}^{H\parallel b}(0) and Hc2H⟂b​(0)H_{\mathrm{c2}}^{H\perp b}(0) values are much smaller than the Pauli-limit field, implying that the orbital pair-breaking mechanism is dominant in NiBi3. Although the NiBi3 single crystals show needle-like shapes, their superconducting properties are only weakly anisotropic, as reflected in the small ratios of Hc1H∥bH_{\mathrm{c1}}^{H\parallel b}/Hc1H⟂bH_{\mathrm{c1}}^{H\perp b}, Hc2H∥bH_{\mathrm{c2}}^{H\parallel b}/Hc2H⟂bH_{\mathrm{c2}}^{H\perp b}, and λ0ac\lambda_{0}^{\mathrm{ac}}/λ0b\lambda_{0}^{\mathrm{b}}. The temperature dependence of the NiBi3 superfluid density reveals a nodeless SC, well described by an isotropic ss-wave model. The relatively large gap value, here, Δ0\Delta_{0} = 2.1 kB​Tck_{\mathrm{B}}T_{c}, suggests strong electron-phonon interactions in NiBi3. Further, the lack of spontaneous magnetic fields below TcT_{c} indicates that the time-reversal symmetry is preserved in the NiBi3 superconductor. The absence of a fast relaxation and/or oscillations in the ZF-μ\muSR spectra excludes a magnetic order on the surface or in the bulk of NiBi3 single crystals, thus implying that the previously observed ferromagnetism is most likely of extrinsic origin. Overall, our systematic results suggest that, in contrast to the unconventional superconductivity observed in Bi/Ni bilayers, NiBi3 behaves as a conventional ss-type superconductor.

Acknowledgements.
This work was supported by the Natural Science Foundation of Shanghai (Grants No. 21ZR1420500 and 21JC1402300), Natural Science Foundation of Chongqing (Grant No. 2022NSCQ-MSX1468), 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 μ\muSR spectrometer).

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