Abstract
Chiral superconductors have been proposed as one pathway to realize Majorana normal fluid at its boundary. However, the longsought 2D and 3D chiral superconductors with edge and surface Majorana normal fluid are yet to be conclusively found. Here, we report evidence for a chiral spintriplet pairing state of UTe_{2} with surface normal fluid response. The microwave surface impedance of the UTe_{2} crystal was measured and converted to complex conductivity, which is sensitive to both normal and superfluid responses. The anomalous residual normal fluid conductivity supports the presence of a significant normal fluid response. The superfluid conductivity follows the temperature behavior predicted for an axial spintriplet state, which is further narrowed down to a chiral spintriplet state with evidence of broken timereversal symmetry. Further analysis excludes trivial origins for the observed normal fluid response. Our findings suggest that UTe_{2} can be a new platform to study exotic topological excitations in higher dimension.
Introduction
Topological insulators, with nonzero topological invariants, possess metallic states at their boundary^{1}. Chiral superconductors, a type of topological superconductors with nonzero topological invariants, possess Majorana fermions at their boundary^{2,3,4,5}. Majorana fermions are an essential ingredient to establish topological quantum computation^{6}. Hence, great effort has been given to search for chiral superconducting systems. So far, evidence for the 1D example has been found from a semiconductor nanowire with endpoint Majorana states^{7}. However, 2D and 3D chiral superconductors with a surface Majorana normal fluid has not been unequivocally found^{3}. Recently, a newly discovered heavyfermion superconductor UTe_{2} (ref. ^{8}) is proposed to be a longsought 3D chiral superconductor with evidence of the chiral ingap state from a scanning tunneling microscopy (STM) study^{9}. This raises a great deal of interest in the physics community to independently establish the existence of the normal fluid response, determining whether or not the response is intrinsic, and identifying the nature of the pairing state of UTe_{2}.
To address these three questions, the microwave surface impedance of a UTe_{2} crystal was measured by the dielectric resonator technique (see Supplementary Notes 1–3). The obtained impedance was converted to the complex conductivity, where the real part is sensitive to the normal fluid response and the imaginary part is sensitive to the superfluid response. By examining these results, here we confirm the existence of the significant normal fluid response of UTe_{2}, verify that the response is intrinsic, and identify that the gap structure is consistent with the chiral spintriplet pairing state.
Results
Anomalous residual normal fluid response
Figure 1 shows the surface impedance Z_{s} = R_{s} + iX_{s} of the sample as a function of temperature, measured from the disk dielectric resonator setup (Supplementary Note 2 and ref. ^{10}). The surface resistance R_{s} decreases monotonically below ≈1.55 K and reaches a surprisingly high residual value R_{s}(0) ≈ 14 mΩ at 11.26 GHz. This value is larger by an order of magnitude than that of another heavyfermion superconductor CeCoIn_{5} (R_{s}(0) ≈ 0.9 mΩ at 12.28 GHz)^{11}. Subsequently its electrical resistance was determined by transport measurement and a midpoint T_{c} of 1.57 K was found.
With the surface impedance, the complex conductivity \(\tilde{\sigma }={\sigma }_{1}i{\sigma }_{2}\) of the sample can be calculated. In the local electrodynamics regime (Supplementary Note 4), one has \({Z}_{\mathrm{s}}=\sqrt{i{\mu }_{0}\omega /\tilde{\sigma }}\). Figure 2a shows σ_{1} and σ_{2} of the sample as a function of temperature. Here, an anomalous feature is the monotonic increase of σ_{1}(T) as T decreases below T_{c}. Note that σ_{1} is a property solely of the normal fluid. For superconductors with a topologically trivial pairing state, most of the normal fluid turns into superfluid and is depleted as T → 0. As a result, in the lowtemperature regime, σ_{1} shows a strong decrease as temperature decreases, and is expected to reach a theoretically predicted residual value σ_{1}(0)/σ_{1}(T_{c}) = 0 (for fully gapped swave^{12}), <0.1 (for the bulk state of a point nodal pwave^{13}), and 0.1–0.3 (for line nodal \({d}_{{x}^{2}{y}^{2}}\)wave^{14,15}). As shown in Fig. 2b, this behavior is observed for the case of Ti^{16} (swave), as well as CeCoIn_{5} (ref. ^{11}, \({d}_{{x}^{2}{y}^{2}}\)wave). In contrast, the UTe_{2} crystal shows a monotonic increase in σ_{1} as the temperature decreases and reaches a much larger σ_{1}(0)/σ_{1}(T_{c}) = 2.3, implying the normal fluid conduction channel is still active and provides a significant contribution even at the lowest temperature.
Axial triplet pairing state from superfluid response
Another property one can extract from the complex conductivity is the effective penetration depth. The imaginary part of the complex conductivity σ_{2}(T) determines the absolute value of the effective penetration depth at each temperature as \({\sigma }_{2}(T)=1/{\mu }_{0}\omega {\lambda }_{{\rm{eff}}}^{2}(T)\). Once the absolute value of the penetration depth is known, the normalized superfluid density can be calculated as \({\rho }_{\mathrm{s}}(T)={\lambda }_{{\rm{eff}}}^{2}(0)/{\lambda }_{{\rm{eff}}}^{2}(T)\) (see “Methods” for how λ_{eff}(0) is determined), and its lowtemperature behavior is determined by the lowenergy excitations of the superconductor, which is sensitive to the pairing state^{17}. The s and dwave pairing states, representative spinsinglet pairing states, are inconsistent with our penetration depth data (see Supplementary Note 5). More crucially, singlet states cannot explain either the reported upper critical field H_{c2}, which is larger than the paramagnetic limiting field^{8}, or the absence of a change in the Knight shift across the T_{c} (refs. ^{8,18}). Thus, only the spintriplet pairing states are discussed below.
For a spintriplet pairing state, ρ_{s}(T) follows different theoretical lowtemperature behaviors depending on two factors^{17,19}. One is whether the magnitude of the energy gap \( {{\Delta }}(\widehat{{\bf{k}}},T)\) follows that of an axial state \({{{\Delta }}}_{0}(T) \widehat{{\bf{k}}}\times \widehat{{\bf{I}}}\) (Fig. 3a) or a polar state \({{{\Delta }}}_{0}(T) \widehat{{\bf{k}}}\cdot \widehat{{\bf{I}}}\) (Fig. 3b), where Δ_{0}(T) is the gap maximum. The other is whether the vector potential direction \(\widehat{{\bf{A}}}\) is parallel or perpendicular to the symmetry axis of the gap \(\widehat{{\bf{I}}}\). Figure 3c shows fits of ρ_{s}(T) to the theoretical behavior (refs. ^{17,19} and Supplementary Note 8) of the various triplet pairing states. Apparently, the data follows the behavior of the axial pairing state with the direction of the current aligned to \(\widehat{{\bf{I}}}\). The axial state can be either chiral or helical depending on the presence or absence of timereversal symmetry breaking (TRSB) of the system^{3}. Recently, direct evidence for TRSB in this system was found by a finite polar Kerr rotation angle developing below T_{c} (ref. ^{20}). Also, a specific heat study, a sensitive probe to resolve multiple superconducting transitions, showed two jumps near 1.6 K (ref. ^{20}). This implies that two nearly degenerate order parameters coexist, allowing the chiral pairing state from the group theory perspective^{20}. Therefore, one can argue that UTe_{2} shows ρ_{s}(T) consistent with the chiral triplet pairing state. In addition, since the symmetry axis connects the two point nodes of the chiral pairing order parameter, and the measurement surveys the abplane electrodynamics, one can further conclude that the point nodes are located near the abplane. The lowtemperature asymptote of ρ_{s}(T) in this case is given as \({\rho }_{\mathrm{s}}(T)=1{\pi }^{2}{({k}_{\mathrm{B}}T/{{{\Delta }}}_{0}(0))}^{2}\). The fitting in Fig. 3c yields an estimate for the gap size Δ_{0}(0) = 1.923 ± 0.002k_{B}T_{c} ≈ 0.265 meV. Note that a recent STM study^{9} measures a similar gap size (0.25 meV).
Examination of extrinsic origins
Our study shows evidence for the chiral triplet pairing state and a substantial amount of normal fluid response in the ground state of UTe_{2}. Before attributing this residual normal fluid response to an intrinsic origin, one must first examine the possibilities of an extrinsic origin. One of the possible extrinsic origins would be a large bulk impurity scattering rate Γ_{imp}. However, if one fits the temperature dependence of the normalized superfluid density to ρ_{s}(T) = 1 − aT^{2} and compares the estimated coefficient a with the modified theoretical asymptote for the chiral triplet state, which includes Γ_{imp} (ref. ^{19}),
one obtains a quadratic equation for (Γ_{imp}, Δ_{0}(0)), and two solutions: Γ_{imp} ≤ 0.06Δ_{0}(0) and Γ_{imp} ≥ 4.36Δ_{0}(0) for the range of Δ_{0}(0) ≤ 0.280 meV. For a nodal superconductor, the large Γ_{imp} ≥ 4.36Δ_{0}(0) suppresses T_{c} more than 80% (refs. ^{21,22}). Such suppression was not observed for the samples grown by the chemical vapor transport method^{8,20}. They show a consistent T_{c} ~ 1.6 K, including the sample in this study. In contrast, the samples prepared by the fluxgrowth method^{23} show suppression in T_{c} all the way down to <0.1 K, as their normal state dc resistivity is much higher. This observation leaves the small Γ_{imp} < 0.06Δ_{0} solution as the only physically reasonable choice in our sample. Note that this small bulk Γ_{imp} is consistent with the absence of the residual linear term in the thermal conductivity^{24}, both implying the clean limit Γ_{imp} ≪ 2Δ_{0}(0). These results are inconsistent with the impurityinduced bulk normal fluid scenario. Note that the temperature independent Γ_{imp} here is much different from the scattering rate above T_{c} (see Supplementary Note 4), possibly suggesting the presence and dominance of highly temperaturedependent inelastic scattering due to spin fluctuations in the normal state.
Another possibility is a quasiparticle response excited by the microwave photons of the measurement signal. However, this scenario is also improbable because the maximum energy gap Δ_{0}(0) = 265 μeV is much larger than that of the microwave photon E_{ph} = 45 μeV used here. At this low E_{ph}/Δ_{0}(0) ratio, even when the upper limit of Γ_{imp} = 0.06Δ_{0}(0) ~ 0.1k_{B}T_{c} is assumped, a theoretical estimate^{13} predicts only a small residual normal response from the bulk states σ_{1}(0)/σ_{n} < 0.1, which cannot explain the measured value of σ_{1}(0)/σ_{n} = 2.3.
Possible intrinsic origin
With several candidates for extrinsic origin excluded, now one can consider the possibility of an intrinsic origin. One important constraint to consider is the recent bulk thermal conductivity measurement in UTe_{2} (ref. ^{24}). It revealed the absence of a residual linear term as a function of temperature in the thermal conductivity, implying the absence of residual normal carriers, at least in the bulk (see Supplementary Note 7). This result suggests the microwave conductivity data can be explained by a combination of a surface normal fluid and bulk superfluid. Topologically trivial origins for the surface normal fluid (e.g., pair breaking rough surface) are first examined, and shown to be inconsistent with the monotonic increase of σ_{1} as T → 0 in Fig. 2b (see Supplementary Note 11). Instead, considering the evidence of the chiral triplet pairing state from the superfluid density analysis and polar Kerr rotation measurement^{20}, a more pluasible source of the surface normal fluid is the gapless chiraldispersing surface states of a 3D chiral superconductor^{4}. Point nodes of a chiral superconductor can possess nonzero Chern number with opposite sign. These point nodes in the superconducting gap are analogous to the Weyl points in the bulk energy bands of a Weyl semimetal. The nodes are predicted to introduce gapless surface Majorana arc states that connect them^{5}. Evidence for these surface states in UTe_{2} is seen in a chiral ingap density of states from an STM study^{9}.
If this scenario is true, the anomalous monotonic increase in σ_{1} down to zero temperature from this surface impedance study (Fig. 2b) may be understood as the enhancement of the scattering lifetime of the surface normal fluid. With its chiral energy dispersion, direct backscattering can be suppressed. As the temperature decreases, the superconducting gap Δ_{0}(T) that provides this topological protection increases, while thermal fluctuations k_{B}T which poison the protection decrease. As a result, the suppression of the backscattering becomes stronger as T → 0, which can end up enhancing the surface scattering lifetime (and σ_{1}). Although this argument is speculative at the moment, we hope the anomalous behavior of the σ_{1}(T) reported in this work motivates quantitative theoretical investigation for the microwave response of the topological surface state of chiral superconductors.
In conclusion, our findings imply that UTe_{2} may be the first example of a 3D chiral spintriplet superconductor with a surface Majorana normal fluid. With topological excitations in a higher dimension, this material can be a new platform to pursue unconventional superconducting physics, and act as a setting for topological quantum computation.
Methods
Growth and preparation of UTe_{2} single crystals
The singlecrystal sample of UTe_{2} was grown by the chemical vapor transport method, using iodine as the transport agent^{8}. For the microwave surface impedance measurement, the top and bottom abplane facets were polished on a 0.5 μm alumina polishing paper inside a nitrogenfilled glove bag (O_{2} content < 0.04%). After polishing was done, the sample was encapsulated by Apeizon Ngrease (see Supplementary Note 10) before being taken out from the bag, and then mounted to the resonator so that the sample is protected from oxidization. Longterm storage of the sample is done in a glove box with O_{2} content < 0.5 p.p.m. Note that the electrical properties of oxidized uranium are summarized in Supplementary Note 9. The sample size after polishing is about ~1.5 × 0.7 × 0.3 mm^{3} with the shortest dimension being along the crystallographic caxis of the orthorhombic structure. The midpoint T_{c} of the sample from DC transport measurements was 1.57 K.
Microwave surface impedance measurement
Due to its large volume, the measurement setup, data processing procedure, and interpretation are decribed in detail in the Supplementary Notes 1 and 2.
Determining of the value of the zero temperature absolute penetration depth and comparison to other uraniumbased superconductors
The effective penetration depth at each temperature (T ≥ 50 mK) can be obtained from \({\sigma }_{2}(T)=1/{\mu }_{0}\omega {\lambda }_{{\rm{eff}}}^{2}(T)\), where σ_{2}(T) is obtained by the surface impedance Z_{s}(T) data. The effective penetration depth at zero temperature can be obtained by extrapolating the data with a power law fit λ_{eff}(T) − λ_{eff}(0) = aT^{c} over the lowtemperature regime T < 0.3T_{c}, resulting in λ_{eff}(0) = 791 nm and c = 2.11. This value is similar to those found in the uraniumbased ferromagnetic superconductor series, such as UCoGe (λ_{eff}(0) ~ 1200 nm)^{25} and URhGe (λ_{eff}(0) ~ 900 nm)^{26}, where UTe_{2} represents the paramagnetic end member of the series^{8}. This result is also consistent with recent muonspin rotation measurements on UTe_{2}, which concluded λ_{eff}(0) ~ 1000 nm (ref. ^{27}).
Error bar of the fitting parameters of the normalized superfluid density
In the fitting of the normalized superfluid density, the error bar of the fitting parameter (e.g., Δ_{0}) was determined by the deviation from the estimated value, which increases the rootmeansquare error of the fit by 1%.
Data availability
The datasets generated and analysed in this work are provided with the paper in the Source data tab. Source data are provided with this paper.
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Acknowledgements
This work is supported by NSF grant No. DMR1410712 (Support of S.B.), DMR2004386 (data analysis), DOE grants No. DESC 0017931 (support of S.A.), DESC 0018788 (support of S.B. for lowtemperature surface impedance measurements), DESC0019154 (lowtemperature transport measurements), Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF9071 (singlecrystal synthesis), NIST (crystal synthesis and sample characterization), and the Maryland Quantum Materials Center (facilities support). Identification of commercial materials does not imply endorsement by NIST.
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S.B., N.P.B., and S.M.A. conceived the project. S.B. polished the sample, conducted the microwave surface impedance measurements, and carried out analysis on the complex conductivity. H.K. and Y.S.E. performed the transport measurements. S.R., I.L., and W.T.F. grew the UTe_{2} single crystal used in this study. S.B., H.K., J.P., N.P.B., and S.M.A. interpreted the results. S.B. and S.M.A. wrote the manuscript with input from other authors.
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Bae, S., Kim, H., Eo, Y.S. et al. Anomalous normal fluid response in a chiral superconductor UTe_{2}. Nat Commun 12, 2644 (2021). https://doi.org/10.1038/s41467021229066
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DOI: https://doi.org/10.1038/s41467021229066
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