Abstract
The recently discovered kagome superconductors AV_{3}Sb_{5} (A = K, Rb, Cs) exhibit unusual chargedensitywave (CDW) orders with timereversal and rotational symmetry breaking. One of the most crucial unresolved issues is identifying the symmetry of the superconductivity that develops inside the CDW phase. Theory predicts a variety of unconventional superconducting symmetries with signchanging and chiral order parameters. Experimentally, however, superconducting phase information in AV_{3}Sb_{5} is still lacking. Here we report the impurity effects in CsV_{3}Sb_{5} using electron irradiation as a phasesensitive probe of superconductivity. Our magnetic penetration depth measurements reveal that with increasing impurities, an anisotropic fullygapped state changes to an isotropic fullgap state without passing through a nodal state. Furthermore, transport measurements under pressure show that the double superconducting dome in the pressuretemperature phase diagram survives against sufficient impurities. These results support that CsV_{3}Sb_{5} is a nonchiral, anisotropic swave superconductor with no sign change both at ambient and under pressure.
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Introduction
The kagome lattice, a motif consisting of cornersharing triangles and hexagonal holes, provides a platform for a rich variety of novel quantum phases of matter. Due to its strong geometrical frustration, it has long been studied in quantum spin systems as a playground for realizing quantum spin liquids^{1}. Recently, however, significant efforts have been devoted to exploring topological metals and semimetals in kagomelattice systems, in which unique band structures such as flat bands, Dirac cones, and van Hove singularities (vHSs) can lead to Dirac/Weyl fermions^{2,3}, spin/charge ordering, and unconventional superconductivity^{4,5,6}. Although various topological kagome materials have been reported so far^{2,3}, superconductors with kagome lattices are rarely found ^{7}.
The recently discovered AV_{3}Sb_{5} (A = K, Rb, Cs) is a new family of kagome superconductors with the superconducting transition temperature T_{c} of 0.9–2.5 K^{8,9,10}. The alkali A atoms are intercalated between sheets consisting of the twodimensional (2D) kagome networks of V atoms and triangular and hexagonal networks of Sb atoms (Fig. 1a, b). The electronic band dispersions near the Fermi energy E_{F} share characteristic features predicted for an ideal kagomelattice system such as a vHS at the M point and a Dirac point at the K point^{8,10,11,12,13}. In these materials, E_{F} is located near the vHS point, and multiple Fermi surfaces are formed by the V dorbitals and Sb porbitals (Fig. 1c). Such unique band structures in AV_{3}Sb_{5} give rise to unconventional chargedensitywave (CDW) orders with the transition temperature T ^{*} ~ 78–103 K^{8,9,10} driven by electron correlation^{5,14,15,16,17}. The CDW transition is accompanied by a 2a_{0} × 2a_{0} × 2c_{0} or 2a_{0} × 2a_{0} × 4c_{0} superlattice composed of modulated starofDavid and inverse starofDavid patterns (where a_{0} and c_{0} indicate the lattice constants above T ^{*}), which breaks translational symmetry^{13,18,19,20}. More intriguingly, it has been reported that additional symmetries, such as timereversal symmetry (TRS) and rotational symmetry (RS), can be broken below T^{*}^{18,21,22,23,24,25}. Since the superconducting transition takes place inside the unusual CDW phase, a fundamental question arises as to whether the superconducting pairing mechanism in AV_{3}Sb_{5} is conventional or not^{26}.
Theories on the kagome lattice near van Hove filling have proposed that unconventional superconductivity beyond the electron–phonon mechanism can be realized by electron correlation effects^{4,5,6,15,27}. Spin and charge fluctuations can lead to spintriplet p and fwave^{6,15,27} and chiral dwave superconductivity^{15}, whereas bondorder fluctuations can promote anisotropic swave^{27} and chiral dwave superconductivity^{5}. In support of the above, firstprinciple calculations have pointed out that the BardeenCooperSchrieffer (BCS) theory (electron–phonon mechanism) cannot explain the experimental T_{c} values, suggesting an unconventional pairing mechanism in AV_{3}Sb_{5}^{28}.
Experimentally, however, the superconducting gap symmetry of AV_{3}Sb_{5} is highly controversial, and whether TRS is broken or not is still elusive. Thermal conductivity measurements in CsV_{3}Sb_{5}^{29} and μSR measurements in Rb/KV_{3}Sb_{5}^{30} have suggested a nodal gap structure. In contrast, magnetic penetration depth^{31} and scanning tunneling spectroscopy (STS)^{32} studies in CsV_{3}Sb_{5} have suggested a nodeless gap structure. Nuclear magnetic/quadrupole resonance (NMR/NQR) measurements in CsV_{3}Sb_{5}^{33} have shown a finite HebelSlichter coherence peak in 1/T_{1}T and a decrease in Knight shift below T_{c}, which exclude spintriplet superconductivity. Regarding the TRS in the superconducting state, Josephson STS measurements in CsV_{3}Sb_{5}^{34} have suggested a possible roton pairdensity wave, corresponding to an unconventional superconducting state with TRS breaking. Contrastingly, μSR studies in CsV_{3}Sb_{5}^{30} have reported that TRS is not broken in the superconductivity state. In addition to the above results at ambient pressure, highpressure studies^{35,36} have revealed that the CDW phase is suppressed by the application of pressure, accompanied by the emergence of a superconducting dome, indicating the close relationship between the CDW and superconductivity. Moreover, recent μSR experiments under pressure^{37,38} have suggested that TRS is broken in the superconducting state when the CDW phase is suppressed by applying pressure. Therefore, to clarify the pairing mechanism of the kagome superconductors, it is crucial to pin down the superconducting gap symmetry of AV_{3}Sb_{5} both at ambient and high pressure, including whether TRS is broken or not.
In general, the conventional phononmediated pairing mechanism leads to a superconducting gap opening all over the Fermi surface, while unconventional pairing mechanisms, such as spin fluctuations, can lead to an anisotropic gap with nodes where the superconducting gap becomes zero. Thus, experimental observations of the lowenergy quasiparticle excitations determine whether the gap structure has nodes or not. In addition to clarifying the presence or absence of nodes in the gap, determining the sign of the gap function also provides a strong constraint on the superconducting pairing symmetry. Especially in kagome superconductors, theories have predicted that two degenerated superconducting order parameters, \({d}_{{x}^{2}{y}^{2}}\) and d_{xy}, give rise to chiral \({d}_{{x}^{2}{y}^{2}}\pm i{d}_{xy}\)wave symmetry, where a finite gap opens all over the Fermi surface, but the phase of the order parameter changes by 4π in momentum k space^{39}. Therefore, phasesensitive probes are highly required to determine the pairing symmetry of AV_{3}Sb_{5}.
There are several experimental probes that are sensitive to the sign of gap functions, such as Josephson junction^{40}, quasiparticle interference^{41}, and neutron scattering techniques^{42}. In general, however, the analysis of such interference effects is complicated in multiband systems due to the complexity of the scattering processes. In addition, most of them require good surface/interface conditions. In contrast, the nonmagnetic impurity effect, on which we focus here, is one of the phasesensitive probes that are applicable to multiband systems and reflect the bulk superconducting properties^{43,44}. In swave superconductors with no signchanging order parameter, the Cooper pairs are not destroyed by nonmagnetic impurity scattering, and both T_{c} and quasiparticle density of states (DOS) are little affected by disorder (the socalled Anderson’s theorem^{45}) (Fig. 2e). In contrast, in the case of nodeless superconductors with signchanging order parameters, such as chiral dwave and s_{±}wave superconductors (note that considering the electronic structure of the present kagome system, one cannot expect sufficient interband interactions to induce the s_{±}wave superconductivity), the Cooper pairs are destroyed by impurity scattering, which suppresses T_{c} rapidly and induces impurity states associated with the Andreev bound state (Fig. 2f). In this case, additional lowenergy quasiparticle excitations appear near the zero energy, e.g., leading to a change in the temperature dependence of the magnetic penetration depth λ from exponential to T^{2} ^{43,46}.
Here, we show that superconductivity in CsV_{3}Sb_{5} is robust against impurities both at ambient and under high pressure. Our magnetic penetration depth measurements reveal that with increasing impurities, a highly anisotropic fully gapped superconducting state changes gradually to an isotropic fullgap state without showing impurityinduced Andreev bound states, which excludes any of signchanging symmetries. Moreover, transport measurements under high pressure show that the superconducting dome in the pressuretemperature (PT) phase diagram survives against sufficient impurities. These results suggest that the superconducting gap function in CsV_{3}Sb_{5} is nonchiral and nonsignchanging swave.
Results
Electron irradiation effects on the CDW and superconducting transition temperatures
In this study, we used electron irradiation to systematically introduce nonmagnetic impurities into CsV_{3}Sb_{5} single crystals (see Methods and Supplementary Information Sec. I). In this method, highenergy electron beam irradiation creates vacancies in the crystal^{43}, acting as point defects without changing the electronic structure and lattice constants (see Supplementary Information Sec. II). Figure 1d, e shows the temperature dependence of resistivity ρ(T) at ambient pressure in samples with irradiated doses of 0 (pristine), 1.3, 3.3, and 8.6 C/cm^{2}. The residual resistivity ratio (RRR) of the pristine sample is ~84, indicating the high quality of our crystals. As the dose increases, the residual resistivity ρ_{0} increases (also see Fig. 1h), and the RRR value decreases. The change in ρ(T) with impurities is successive for the irradiation dose in the whole temperature range. Furthermore, our Xray structural analysis and Hall coefficient measurements show the absence of any change in the lattice parameters and carrier density induced by electron irradiation. These results indicate that the nonparallel shift with impurities in ρ(T) is most likely due to the multiband nature of the present kagome system (for more details, see Supplementary Information Sec. III). Along with this, both the CDW and superconducting transition temperatures T^{*} and T_{c} shift to a lower temperature (Fig. 1g). In general, nonmagnetic impurity scattering can suppress longrange orders because the introduced defects shorten the coherence length. Indeed, the suppression of CDW order by impurities has been theoretically studied^{47}. The suppression of T_{c} has also been confirmed by the Meissner effect measured by the normalized frequency shift of a tunnel diode oscillator (TDO) (Fig. 1f). Note that the superconducting transition becomes sharper with increasing dose, which may be related to the suppression of superconducting phase fluctuations^{7,48} or the change in skin depth due to impurity scattering. The sharp superconducting transition width in the 8.6 C/cm^{2} irradiated sample with sufficient disorder indicates that the defects are introduced quite uniformly inside the crystals.
Nonmagnetic impurity effects on lowenergy quasiparticle excitations in the superconducting state
Next, we turn to the impurity effect on lowenergy quasiparticle excitations in the superconducting state. Magnetic penetration depth λ is one of the most fundamental properties of superconductors sensitive to lowenergy quasiparticle excitations^{7,31,43}. In this study, we measured the magnetic penetration depth of the pristine and irradiated CsV_{3}Sb_{5} single crystals down to 50 mK by using the TDO in a dilution refrigerator (see Methods). Figure 2a–d shows the change in the magnetic penetration depth δλ(T) ≡ λ(T) − λ(0) (where λ(0) is the absolute value of the penetration depth at 0 K) at low temperatures for the pristine and 1.3, 3.3, and 8.6 C/cm^{2} irradiated samples. In the pristine sample, δλ(T) shows a flat temperature dependence at low temperatures below 0.1T_{c} (Fig. 2a), indicating a fully gapped superconducting state in CsV_{3}Sb_{5}. To examine the lowenergy quasiparticle excitations in the pristine sample, we applied a powerlaw fit δλ(T) ∝ T^{n} to the experimental data. In general, in the case of nodal superconductors with line and point nodes, the exponent value n gives 1 and 2 in the clean limit, respectively. We obtained n ~ 2.8 from the fitting (Fig. 2a), indicating the absence of nodes in the gap (or conversely, the presence of a finite gap). Then, to quantitatively evaluate the gap value, we tried to fit the data with the fully gapped swave model \(\delta \lambda (T)\propto {T}^{1/2}\exp ({\Delta }_{0}/{k}_{{{{{{{{\rm{B}}}}}}}}}T)\), where k_{B} is the Boltzmann constant and Δ_{0} is the superconducting gap. We obtained an extremely small gap value Δ_{0} = 0.47k_{B}T_{c} (which is consistent with the previous study^{31}), suggesting the existence of gap minima \({\Delta }_{\min }\) coming from the anisotropic gap nature of CsV_{3}Sb_{5}, as discussed later. One of our key findings is that the fully gapped behavior in δλ(T) is robust against disorder (Fig. 2b–d). The flat temperature region at low temperatures expands to a higher temperature region with increasing dose. In the case of fully gapped superconductors with signchanging order parameters, δλ(T) is expected to change from an exponential to a T^{2} dependence with increasing impurities because of the impurityinduced DOS (Fig. 2f)^{43}. In sharp contrast, our experimental observations show that Δ_{0} and n obtained from the fitting rather increase with increasing dose (Fig. 2g, h), indicating no impurityinduced DOS in the superconducting gap. These results provide strong bulk evidence that the superconducting gap structure of CsV_{3}Sb_{5} is nodeless without a signchanging gap.
For a more detailed analysis of the superconducting gap structure, we derived the normalized superfluid density ρ_{s}(T) ≡ λ^{2}(0)/λ^{2}(T). We used λ(0) = 387 nm for the pristine sample estimated in the previous study^{31} and calculated λ(0) for the irradiated samples by using the relation λ(0) = λ_{L}(0)(1+ξ/l)^{1/2} (Fig. 1h), where the London penetration depth λ_{L}(0) is assumed to be equal to λ(0) = 387 nm for the pristine sample, and l and ξ are the mean free path and coherence length, respectively (for more details, see Supplementary Information Sec. IV). Figure 3a shows the obtained ρ_{s}(T) curve as a function of T/T_{c} for each sample. In all the samples, ρ_{s} shows a flat temperature dependence at low temperatures, which extends to a higher temperature region with increasing dose. This is again inconsistent with a nodal gap structure.
Here, we consider a multigap model to analyze the overall temperature dependence of ρ_{s}. In CsV_{3}Sb_{5}, the Fermi surfaces are formed by two different orbitals: one is derived from the dorbitals of V, forming a hexagonal Fermi surface around the Γ point and two triangular Fermi surfaces around the K point, while the other is from the porbitals of Sb, forming a circular Fermi surface around the Γ point (Fig. 1c)^{13}. The Fermi surfaces derived from the V dorbitals determine the physical properties of this material, and three equivalent q vectors^{18,21,23,24} are considered to give rise to anisotropic pairing interactions^{5,27}. Indeed, recent STM measurements^{32} have reported the emergence of an anisotropic superconducting gap as well as an isotropic gap below T_{c}. We therefore consider a multigap model with an anisotropic but nodeless superconducting gap with sixfold symmetry (\({\Delta }_{1}\propto 1+\alpha \,\cos (6\phi )\)) and an isotropic superconducting gap (\({\Delta }_{2}={{{{{{{\rm{const.}}}}}}}}\)) on two cylindrical Fermi surfaces (Fig. 3b). Note that an isotropic twogap model cannot produce reasonable results for irradiated samples (for more details, see Supplementary Information Sec. V). We fitted the experimental data with the anisotropic multigap model (Fig. 3a) and obtained the gap values Δ_{1} and Δ_{2} as a function of dose (Fig. 3c). As the dose increases, the difference between the maximum and minimum values of Δ_{1} decreases, and eventually, all the gaps become almost identical. This is due to the averaging effect between the two gaps introduced by impurityinduced intra/interband scattering, and a very similar behavior has been observed in the prototypical multigap superconductor MgB_{2}^{49}. This evidences nodeless multigap superconductivity with a signpreserving order parameter in CsV_{3}Sb_{5}, which excludes the possibility of spintriplet p and fwave and chiral dwave superconductivity.
Pairbreaking effect
To discuss the impurity effect on T_{c} more quantitatively, we next introduce a pairbreaking parameter g = ℏ/(τ_{imp}k_{B}T_{c0}), where \({\tau }_{{{{{{{{\rm{imp}}}}}}}}}={\mu }_{0}{\lambda }_{{{{{{{{\rm{L}}}}}}}}}^{2}(0)/{\rho }_{0}\) is the impurity scattering time and T_{c0} is the superconducting transition temperature of the pristine sample^{44,50}. The suppression of T_{c} is plotted as a function of g and compared to other superconductors with and without signchanging order parameters (Fig. 3d). T_{c} of CsV_{3}Sb_{5} is rapidly suppressed at a low irradiation dose but starts to saturate at moderate irradiation doses. The initial rapid suppression of T_{c} is considered to be related to the reduction of the anisotropy of Δ_{1} (Fig. 3c), as discussed later. The T_{c} suppression above 1.3 C/cm^{2} irradiation dose is much slower than those in superconductors with signchanging order parameters such as dwave, rather similar to those in swave superconductors without signchanging gaps. These results also support that multigap swave superconductivity with no sign change is realized in CsV_{3}Sb_{5} at ambient pressure.
Impurity effects on the highpressure superconducting phase
To further investigate the nonmagnetic impurity effect on the superconducting phase of CsV_{3}Sb_{5} under pressure, we constructed the PT phase diagram in the pristine sample and 4.8 and 8.6 C/cm^{2} irradiated samples. Figure 4a shows the ρ(T) curve of the pristine sample at several pressures. The CDW transition temperature T^{*}, which is determined from a jump or dip in dρ(T)/dT (Fig. 4b), decreases monotonically with increasing pressure (Fig. 4d). In contrast, the superconducting transition temperature T_{c} shows a nonmonotonic pressure dependence (Fig. 4c), and a double superconducting dome is observed, as reported in previous highpressure studies^{35,36} (Fig. 4d). The first peak of the superconducting double dome locates at P_{1} ~ 0.7 GPa inside the CDW phase, while the second peak locates at P_{2} ~ 2 GPa near the CDW endpoint. We conducted the same experiments for the 4.8 and 8.6 C/cm^{2} irradiated samples (Fig. 4e–l) and obtained the PT phase diagrams as shown in Fig. 4h, l. T^{*} is suppressed in the 4.8 and 8.6 C/cm^{2} irradiated samples, and the CDW endpoint shifts to lower pressure with increasing irradiation dose. T_{c} is also suppressed by disorder, but the superconducting dome survives even after 8.6 C/cm^{2} irradiation.
As already discussed in Fig. 3d, the irradiation dose of 8.6 C/cm^{2} introduces enough defects to suppress superconductivity with a signchanging order parameter. To investigate the impurity effect on the highpressure superconducting phase, we evaluated the pairbreaking parameter g at P_{2} for each dose (Fig. S5). Our results display that the suppression of T_{c} at the second dome is slower than that of the dwave case with a signchanging order parameter; it rather traces the trend in MgB_{2}. Therefore, these results suggest that the superconducting gap symmetry of CsV_{3}Sb_{5} at high pressure is also nonsignchanging swave.
Discussion
Recent μSR measurements under pressure^{37} have reported that the superconducting pairing symmetry near P_{2} has a finite superconducting gap across the Fermi surface and breaks TRS. As a possible symmetry, chiral \({d}_{{x}^{2}{y}^{2}}\pm i{d}_{xy}\) or p_{x} ± ip_{y}wave superconductivity has been discussed^{37,38}. However, such unconventional superconductivity is expected to be sensitive to disorder, because the chiral states have signchanging order parameters which would produce Andreev bound states by impurities. Our present results show that the superconducting state under high pressure is robust against disorder. These findings seem to be inconsistent with the chiral \({d}_{{x}^{2}{y}^{2}}\pm i{d}_{xy}\) and p_{x} ± ip_{y}wave states. To fully understand the relationship between our impurity effects and the μSR results, further theoretical and experimental studies on the highpressure phase in the kagome systems are highly desired. We note that the TRS breaking has been observed in the CDW phase, and thus the possible fluctuations of chiral CDW order in the highpressure phase on the time scale of μSR measurements may be an important issue.
Another important aspect of our findings is that the CDW endpoint shifts to lower pressure with irradiation, followed by the second peak of the superconducting double dome (Fig. 4d, h, l), suggesting that the CDW is closely related to the superconductivity in the present system. Recent theoretical calculations in AV_{3}Sb_{5} ^{27} have proposed that bondorder fluctuations originating from the tripleq vectors corresponding to the (inverse) star of David pattern induce anisotropic pairing interactions, leading to anisotropic swave superconductivity. This theory can explain the relatively high T_{c} in AV_{3}Sb_{5} that cannot be reproduced by the BCS theory ^{28} and the anisotropic superconducting gap structure in CsV_{3}Sb_{5} obtained in the present study. Moreover, in such anisotropic swave superconductivity, the introduction of impurity scattering averages out the anisotropic gap, changing to the isotropic gap. In this case, T_{c} drops rapidly at an initial introduction of impurities, but as the gap becomes isotropic, the reduction of T_{c} saturates and becomes much slower than that expected in the AbrikosovGor’kov (AG) theory. These expectations are in good agreement with our observations of the T_{c} suppression in CsV_{3}Sb_{5}. We note that a possible transition from a pwave to an swave state caused by impurities^{27} may explain the initial rapid suppression of T_{c}. However, our present results exclude the possibility of a nodal superconducting state in the pristine sample, which is at odds with the pwave state. The gradual change in the superconducting gap inferred from the temperature dependence of the superfluid density suggests that an impurityinduced transition from a fullgap chiral state to a nonchiral swave state is also unlikely. This is reinforced by the μSR measurements at ambient pressure^{30}, which report that chiral superconductivity in the pristine sample at ambient pressure can be ruled out. Thus, our present results support a new type of unconventional superconductivity due to bondorder fluctuations on the kagome lattice in CsV_{3}Sb_{5}, where the gap function is nonsignchanging swave. In the present kagome superconductors, the possibility of a loopcurrent phase with broken TRS and a nematic phase with broken RS has been pointed out above the superconducting phase^{18,21,22,23,24,25}. Therefore, elucidating the intertwining of these unusual normal and superconducting phases, which is commonly seen in highT_{c} cuprates and ironbased superconductors, will pave the way to understanding novel quantum phases of matter in condensed matter physics.
Methods
Single crystal growth
Highquality single crystals of CsV_{3}Sb_{5} were synthesized using the selfflux method. All sample preparations are performed in an argon glovebox with oxygen and moisture < 0.5ppm. The flux precursor was formed through mechanochemical methods by mixing Cs metal (Alfa 99.98%), V powder (Sigma 99.9%), and Sb beads (Alfa 99.999%) to form a mixture which is ~50 at.% Cs_{0.4}Sb_{0.6} (near eutectic composition) and 50 at.% VSb_{2}. Note that prior to mixing, asreceived vanadium powders were purified inhouse to remove residual oxides. After milling for 60 min a preseasoned tungsten carbide vial, flux precursors are extracted and sealed into 10 mL alumina crucibles. The crucibles are nested within stainless steel jackets and sealed under argon. Samples are heated to 1000 °C at 250 °C/h and soaked for 24 h before dropping to 900 °C at 100 °C/h. Crystals are formed during the final slow cool to 500 °C at 1 °C/h before terminating the growth. Once cooled, the crystals are recovered mechanically. Samples are hexagonal flakes with a brilliant metallic luster. The elemental composition of crystals was assessed using energydispersive Xray spectroscopy (EDS) using an APREOC scanning electron microscope.
Electron irradiation
Electron irradiation with the incident electron energy of 2.5 MeV was performed on SIRIUS Pelletron accelerator operated by the Laboratoire des Solides irradiés (LSI) at École Polytechnique. To prevent defect migration and agglomeration, the sample temperature was kept at ~20 K during irradiation which produces stable vacancyinterstitial Frenkel pairs. Subsequent warming to room temperature causes annealing of interstitials, which have a lower migration energy, leaving a uniform population of vacancy type defects. The electron irradiation of 1.3, 3.3, and 8.6 C/cm^{2} was performed at the same beam time (run#1), while the irradiation of 4.8 C/cm^{2} was conducted at another beam time (run#2).
Electrical resistivity measurements
The electrical resistivity was measured at ambient and high pressure by the 4terminal method using a Physical Property Measurement System (PPMS) from Quantum Design with the lowest temperature of about 1.8 K. The resistivity under pressure was measured using a piston cylinder cell to generate pressure up to ~2.5 GPa and daphne 7373 as a pressure medium in PPMS. The pressure value in the sample was determined from the superconducting transition temperature T_{c} of Pb under pressure, using the relation P = (7.20 − T_{c})/0.365. Note that when the 4.8 C/cm^{2} irradiated sample was set in the piston cell, even before pressure was applied, the resistivity value changed, probably due to cracks, so we have corrected it to the value before pressure was applied.
Magnetic penetration depth measurements
The temperature variation of the inplane magnetic penetration depth δλ(T) = λ(T) − λ(0) was measured by using a tunnel diode oscillator technique (TDO) with the resonant frequency of ~13.8 MHz in a dilution refrigerator down to ~50 mK. The sample was mounted on a sapphire rod with Apiezon N grease, then inserted into a copper coil in the LC circuit. The frequency shift δf in the TDO is related to the change of magnetic susceptibility δχ by the following equation, δf = − (f_{0}V_{s}/(2V_{c}(1 − N)))δχ, where f_{0} is the resonant frequency without the sample, V_{s} and V_{c} are the sample and coil volume, respectively, and N is the demagnetization factor. δχ is related to δλ by the following equation, δχ = δλ/R, where R is a constant determined by the geometry of the sample from the calculation. Thus, δf is related to δλ by the following equation, δf = − (f_{0}V_{s}/(2RV_{c}(1 − N)))δλ.
Data availability
All data supporting the findings of this study are available within the paper and its Supplementary Information. Source data are provided with this paper.
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Acknowledgements
We thank R. Tazai, Y. Yamakawa, S. Onari, and H. Kontani for fruitful discussions. This work was supported by GrantsinAid for Scientific Research (KAKENHI) (Nos. JP22K20349, JP22K18681, JP22K18683, JP22J21896, JP22H00105, JP21H01793, JP21KK0242, JP20H02600, JP19H00649, JP19H00648, JP18H05227, JP18KK0375), GrantinAid for Scientific Research on innovative areas “Quantum Liquid Crystals” (No. JP19H05824) and GrantinAid for Scientific Research for Transformative Research Areas (A) “Condensed Conjugation” (No. JP20H05869) from Japan Society for the Promotion of Science (JSPS), and CREST (No. JPMJCR19T5) from Japan Science and Technology (JST). S.D.W. and B.R.O. acknowledge support via the UC Santa Barbara NSF Quantum Foundry funded via the QAMASEi program under award DMR1906325. Electron irradiation was conducted at the SIRIUS accelerator facility at École Polytechnique (Palaiseau, France) and was supported by EMIR&A French network (FR CNRS 3618) (proposal No. 228950).
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K.H. and T.S. conceived the project. M.R., K.I., K. Ogawa, K. Okada, Y.M., K.H., and T.S. performed magnetic penetration depth measurements and analyzed the data. Y.T., K. Okada, K.M., and Y.U. carried out highpressure measurements. M.R. and S.L. performed electrical transport and Xray diffraction measurements. R.G. and M.K. conducted electron irradiation experiments. B.R.O. and S.D.W. carried out sample preparation. M.R., K.H., and T.S. prepared the manuscript with inputs from R.G., M.K., B.R.O., and S.D.W. All authors discussed the experimental results.
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Roppongi, M., Ishihara, K., Tanaka, Y. et al. Bulk evidence of anisotropic swave pairing with no sign change in the kagome superconductor CsV_{3}Sb_{5}. Nat Commun 14, 667 (2023). https://doi.org/10.1038/s4146702336273x
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DOI: https://doi.org/10.1038/s4146702336273x
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