## Introduction

The kagome lattice, a motif consisting of corner-sharing 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 liquids1. Recently, however, significant efforts have been devoted to exploring topological metals and semimetals in kagome-lattice systems, in which unique band structures such as flat bands, Dirac cones, and van Hove singularities (vHSs) can lead to Dirac/Weyl fermions2,3, spin/charge ordering, and unconventional superconductivity4,5,6. Although various topological kagome materials have been reported so far2,3, superconductors with kagome lattices are rarely found 7.

The recently discovered AV3Sb5 (A = K, Rb, Cs) is a new family of kagome superconductors with the superconducting transition temperature Tc of 0.9–2.5 K8,9,10. The alkali A atoms are intercalated between sheets consisting of the two-dimensional (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 EF share characteristic features predicted for an ideal kagome-lattice system such as a vHS at the M point and a Dirac point at the K point8,10,11,12,13. In these materials, EF is located near the vHS point, and multiple Fermi surfaces are formed by the V d-orbitals and Sb p-orbitals (Fig. 1c). Such unique band structures in AV3Sb5 give rise to unconventional charge-density-wave (CDW) orders with the transition temperature T* ~ 78–103 K8,9,10 driven by electron correlation5,14,15,16,17. The CDW transition is accompanied by a 2a0 × 2a0 × 2c0 or 2a0 × 2a0 × 4c0 superlattice composed of modulated star-of-David and inverse star-of-David patterns (where a0 and c0 indicate the lattice constants above T*), which breaks translational symmetry13,18,19,20. More intriguingly, it has been reported that additional symmetries, such as time-reversal 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 AV3Sb5 is conventional or not26.

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 effects4,5,6,15,27. Spin and charge fluctuations can lead to spin-triplet p- and f-wave6,15,27 and chiral d-wave superconductivity15, whereas bond-order fluctuations can promote anisotropic s-wave27 and chiral d-wave superconductivity5. In support of the above, first-principle calculations have pointed out that the Bardeen-Cooper-Schrieffer (BCS) theory (electron–phonon mechanism) cannot explain the experimental Tc values, suggesting an unconventional pairing mechanism in AV3Sb528.

Experimentally, however, the superconducting gap symmetry of AV3Sb5 is highly controversial, and whether TRS is broken or not is still elusive. Thermal conductivity measurements in CsV3Sb529 and μSR measurements in Rb/KV3Sb530 have suggested a nodal gap structure. In contrast, magnetic penetration depth31 and scanning tunneling spectroscopy (STS)32 studies in CsV3Sb5 have suggested a nodeless gap structure. Nuclear magnetic/quadrupole resonance (NMR/NQR) measurements in CsV3Sb533 have shown a finite Hebel-Slichter coherence peak in 1/T1T and a decrease in Knight shift below Tc, which exclude spin-triplet superconductivity. Regarding the TRS in the superconducting state, Josephson STS measurements in CsV3Sb534 have suggested a possible roton pair-density wave, corresponding to an unconventional superconducting state with TRS breaking. Contrastingly, μSR studies in CsV3Sb530 have reported that TRS is not broken in the superconductivity state. In addition to the above results at ambient pressure, high-pressure studies35,36 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 pressure37,38 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 AV3Sb5 both at ambient and high pressure, including whether TRS is broken or not.

In general, the conventional phonon-mediated 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 low-energy 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 dxy, 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 space39. Therefore, phase-sensitive probes are highly required to determine the pairing symmetry of AV3Sb5.

There are several experimental probes that are sensitive to the sign of gap functions, such as Josephson junction40, quasiparticle interference41, and neutron scattering techniques42. 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 non-magnetic impurity effect, on which we focus here, is one of the phase-sensitive probes that are applicable to multiband systems and reflect the bulk superconducting properties43,44. In s-wave superconductors with no sign-changing order parameter, the Cooper pairs are not destroyed by non-magnetic impurity scattering, and both Tc and quasiparticle density of states (DOS) are little affected by disorder (the so-called Anderson’s theorem45) (Fig. 2e). In contrast, in the case of nodeless superconductors with sign-changing order parameters, such as chiral d-wave 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 Tc rapidly and induces impurity states associated with the Andreev bound state (Fig. 2f). In this case, additional low-energy 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 T2 43,46.

Here, we show that superconductivity in CsV3Sb5 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 full-gap state without showing impurity-induced Andreev bound states, which excludes any of sign-changing symmetries. Moreover, transport measurements under high pressure show that the superconducting dome in the pressure-temperature (P-T) phase diagram survives against sufficient impurities. These results suggest that the superconducting gap function in CsV3Sb5 is non-chiral and non-sign-changing s-wave.

## Results

### Electron irradiation effects on the CDW and superconducting transition temperatures

In this study, we used electron irradiation to systematically introduce non-magnetic impurities into CsV3Sb5 single crystals (see Methods and Supplementary Information Sec. I). In this method, high-energy electron beam irradiation creates vacancies in the crystal43, 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/cm2. 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 ρ0 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 X-ray 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 non-parallel 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 Tc shift to a lower temperature (Fig. 1g). In general, non-magnetic impurity scattering can suppress long-range orders because the introduced defects shorten the coherence length. Indeed, the suppression of CDW order by impurities has been theoretically studied47. The suppression of Tc 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 fluctuations7,48 or the change in skin depth due to impurity scattering. The sharp superconducting transition width in the 8.6 C/cm2 irradiated sample with sufficient disorder indicates that the defects are introduced quite uniformly inside the crystals.

### Non-magnetic impurity effects on low-energy quasiparticle excitations in the superconducting state

Next, we turn to the impurity effect on low-energy quasiparticle excitations in the superconducting state. Magnetic penetration depth λ is one of the most fundamental properties of superconductors sensitive to low-energy quasiparticle excitations7,31,43. In this study, we measured the magnetic penetration depth of the pristine and irradiated CsV3Sb5 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/cm2 irradiated samples. In the pristine sample, δλ(T) shows a flat temperature dependence at low temperatures below 0.1Tc (Fig. 2a), indicating a fully gapped superconducting state in CsV3Sb5. To examine the low-energy quasiparticle excitations in the pristine sample, we applied a power-law fit δλ(T) Tn 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 s-wave model $$\delta \lambda (T)\propto {T}^{-1/2}\exp (-{\Delta }_{0}/{k}_{{{{{{{{\rm{B}}}}}}}}}T)$$, where kB is the Boltzmann constant and Δ0 is the superconducting gap. We obtained an extremely small gap value Δ0 = 0.47kBTc (which is consistent with the previous study31), suggesting the existence of gap minima $${\Delta }_{\min }$$ coming from the anisotropic gap nature of CsV3Sb5, 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 sign-changing order parameters, δλ(T) is expected to change from an exponential to a T2 dependence with increasing impurities because of the impurity-induced DOS (Fig. 2f)43. In sharp contrast, our experimental observations show that Δ0 and n obtained from the fitting rather increase with increasing dose (Fig. 2g, h), indicating no impurity-induced DOS in the superconducting gap. These results provide strong bulk evidence that the superconducting gap structure of CsV3Sb5 is nodeless without a sign-changing gap.

For a more detailed analysis of the superconducting gap structure, we derived the normalized superfluid density ρs(T) ≡ λ2(0)/λ2(T). We used λ(0) = 387 nm for the pristine sample estimated in the previous study31 and calculated λ(0) for the irradiated samples by using the relation λ(0) = λL(0)(1+ξ/l)1/2 (Fig. 1h), where the London penetration depth λL(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 ρs(T) curve as a function of T/Tc for each sample. In all the samples, ρs 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 ρs. In CsV3Sb5, the Fermi surfaces are formed by two different orbitals: one is derived from the d-orbitals 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 p-orbitals of Sb, forming a circular Fermi surface around the Γ point (Fig. 1c)13. The Fermi surfaces derived from the V d-orbitals determine the physical properties of this material, and three equivalent q vectors18,21,23,24 are considered to give rise to anisotropic pairing interactions5,27. Indeed, recent STM measurements32 have reported the emergence of an anisotropic superconducting gap as well as an isotropic gap below Tc. We therefore consider a multigap model with an anisotropic but nodeless superconducting gap with six-fold 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 two-gap 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 Δ1 and Δ2 as a function of dose (Fig. 3c). As the dose increases, the difference between the maximum and minimum values of Δ1 decreases, and eventually, all the gaps become almost identical. This is due to the averaging effect between the two gaps introduced by impurity-induced intra/interband scattering, and a very similar behavior has been observed in the prototypical multigap superconductor MgB249. This evidences nodeless multigap superconductivity with a sign-preserving order parameter in CsV3Sb5, which excludes the possibility of spin-triplet p- and f-wave and chiral d-wave superconductivity.

### Pair-breaking effect

To discuss the impurity effect on Tc more quantitatively, we next introduce a pair-breaking parameter g = /(τimpkBTc0), where $${\tau }_{{{{{{{{\rm{imp}}}}}}}}}={\mu }_{0}{\lambda }_{{{{{{{{\rm{L}}}}}}}}}^{2}(0)/{\rho }_{0}$$ is the impurity scattering time and Tc0 is the superconducting transition temperature of the pristine sample44,50. The suppression of Tc is plotted as a function of g and compared to other superconductors with and without sign-changing order parameters (Fig. 3d). Tc of CsV3Sb5 is rapidly suppressed at a low irradiation dose but starts to saturate at moderate irradiation doses. The initial rapid suppression of Tc is considered to be related to the reduction of the anisotropy of Δ1 (Fig. 3c), as discussed later. The Tc suppression above 1.3 C/cm2 irradiation dose is much slower than those in superconductors with sign-changing order parameters such as d-wave, rather similar to those in s-wave superconductors without sign-changing gaps. These results also support that multigap s-wave superconductivity with no sign change is realized in CsV3Sb5 at ambient pressure.

### Impurity effects on the high-pressure superconducting phase

To further investigate the non-magnetic impurity effect on the superconducting phase of CsV3Sb5 under pressure, we constructed the P-T phase diagram in the pristine sample and 4.8 and 8.6 C/cm2 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 Tc shows a non-monotonic pressure dependence (Fig. 4c), and a double superconducting dome is observed, as reported in previous high-pressure studies35,36 (Fig. 4d). The first peak of the superconducting double dome locates at P1 ~ 0.7 GPa inside the CDW phase, while the second peak locates at P2 ~ 2 GPa near the CDW endpoint. We conducted the same experiments for the 4.8 and 8.6 C/cm2 irradiated samples (Fig. 4e–l) and obtained the P-T phase diagrams as shown in Fig. 4h, l. T* is suppressed in the 4.8 and 8.6 C/cm2 irradiated samples, and the CDW endpoint shifts to lower pressure with increasing irradiation dose. Tc is also suppressed by disorder, but the superconducting dome survives even after 8.6 C/cm2 irradiation.

As already discussed in Fig. 3d, the irradiation dose of 8.6 C/cm2 introduces enough defects to suppress superconductivity with a sign-changing order parameter. To investigate the impurity effect on the high-pressure superconducting phase, we evaluated the pair-breaking parameter g at P2 for each dose (Fig. S5). Our results display that the suppression of Tc at the second dome is slower than that of the d-wave case with a sign-changing order parameter; it rather traces the trend in MgB2. Therefore, these results suggest that the superconducting gap symmetry of CsV3Sb5 at high pressure is also non-sign-changing s-wave.

## Discussion

Recent μSR measurements under pressure37 have reported that the superconducting pairing symmetry near P2 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 px ± ipy-wave superconductivity has been discussed37,38. However, such unconventional superconductivity is expected to be sensitive to disorder, because the chiral states have sign-changing 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 px ± ipy-wave states. To fully understand the relationship between our impurity effects and the μSR results, further theoretical and experimental studies on the high-pressure 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 high-pressure 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 AV3Sb5 27 have proposed that bond-order fluctuations originating from the triple-q vectors corresponding to the (inverse) star of David pattern induce anisotropic pairing interactions, leading to anisotropic s-wave superconductivity. This theory can explain the relatively high Tc in AV3Sb5 that cannot be reproduced by the BCS theory 28 and the anisotropic superconducting gap structure in CsV3Sb5 obtained in the present study. Moreover, in such anisotropic s-wave superconductivity, the introduction of impurity scattering averages out the anisotropic gap, changing to the isotropic gap. In this case, Tc drops rapidly at an initial introduction of impurities, but as the gap becomes isotropic, the reduction of Tc saturates and becomes much slower than that expected in the Abrikosov-Gor’kov (AG) theory. These expectations are in good agreement with our observations of the Tc suppression in CsV3Sb5. We note that a possible transition from a p-wave to an s-wave state caused by impurities27 may explain the initial rapid suppression of Tc. However, our present results exclude the possibility of a nodal superconducting state in the pristine sample, which is at odds with the p-wave state. The gradual change in the superconducting gap inferred from the temperature dependence of the superfluid density suggests that an impurity-induced transition from a full-gap chiral state to a non-chiral s-wave state is also unlikely. This is reinforced by the μSR measurements at ambient pressure30, 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 bond-order fluctuations on the kagome lattice in CsV3Sb5, where the gap function is non-sign-changing s-wave. In the present kagome superconductors, the possibility of a loop-current phase with broken TRS and a nematic phase with broken RS has been pointed out above the superconducting phase18,21,22,23,24,25. Therefore, elucidating the intertwining of these unusual normal and superconducting phases, which is commonly seen in high-Tc cuprates and iron-based superconductors, will pave the way to understanding novel quantum phases of matter in condensed matter physics.

## Methods

### Single crystal growth

High-quality single crystals of CsV3Sb5 were synthesized using the self-flux 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.% Cs0.4Sb0.6 (near eutectic composition) and 50 at.% VSb2. Note that prior to mixing, as-received vanadium powders were purified in-house to remove residual oxides. After milling for 60 min a pre-seasoned 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 energy-dispersive X-ray spectroscopy (EDS) using an APREO-C scanning electron microscope.