Introduction

In conventional Bardeen-Cooper-Schrieffer (BCS) superconductors, the electron-lattice coupling leads to the formation of coherent (2e) Cooper pairs and the opening of an isotropic s-wave gap at the Fermi level1. In comparison, a key signature of unconventional superconductivity in materials such as copper oxides and iron-pnictides is that the pairing may be mediated by spin fluctuations2,3 and associated with intertwined charge-spin-lattice degrees of freedom4. The discovery of superconductivity in the layered kagome AV3Sb5 (A = K, Rb, Cs) metals5,6,7 is interesting because the superconducting state develops in the presence of a charge density wave (CDW)8,9,10,11,12,13, and the competition between these two ordered states may give rise to unconventional superconductivity14,15,16,17,18. However, no spin fluctuations are reported, thus raising the question of whether these materials are conventional BCS or unconventional superconductors like cuprates and iron pnictides. From pressure and Ta-doping dependence of the Cs(V1−xTax)3Sb5 phase diagram, it is clear that CDW order competes with superconductivity, and optimal superconductivity of Tc = 5.3 K appears around Cs(V0.86Ta0.14)3Sb5 with vanishing CDW order19,20.

The majority of experimental evidence in the AV3Sb5 materials indicates singlet s-wave pairing with a nodeless but possibly anisotropic superconducting gap19,21,22,23. Furthermore, multiband superconductivity with a large difference in the gap size on different Fermi surface sheets has been reported12,15,22,23,24,25. An experimental determination of electron-lattice coupling strength supports conventional BCS superconductivity26, although there are questions about whether the relation between the gap and critical temperature is consistent with weak coupling14. There are several reports of broken time-reversal symmetry (BTRS) both in the normal state8,27,28,29 as well as the superconducting state30,31,32, although it is not observed consistently in all experiments23,33,34. In addition, possible Majorana bound states were observed in scanning tunneling spectroscopy studies9. Finally, a transition from conventional 2e-pairing towards (vestigial) 4e- and 6e-pairing35 has been reported upon heating towards the transition to the normal state36,37,38,39,40.

To determine the microscopic origin of superconductivity in AV3Sb5, it is critical to separate the effects of the CDW as this may be associated with an electronic nematic phase and affect the electron pairing41,42. Here, we report on small-angle neutron scattering (SANS) studies of the vortex lattice (VL) induced by an applied magnetic field in Cs(V0.86Ta0.14)3Sb5 where charge ordering is suppressed19,20. The vortices produce singularities in the order parameter and may be used as probes of the superconducting state in the host material43,44. Our results indicate a highly conventional bulk superconducting state, and thus suggest that the reported exotic behavior in other members of the AV3Sb5 family of superconductors are not relevant to the microscopic origin of superconductivity. Importantly, the SANS technique provides information about the bulk superconducting state, whereas more exotic phenomena such as BTRS may occur only at the sample surface.

Results

Vortex lattice imaging

The sample used for the SANS experiments, shown in Fig. 1a, consisted of a mosaic of co-aligned Cs(V0.86Ta0.14)3Sb5 single crystals. Two SANS experiments (#1/2) were performed, exploring different temperature and field ranges as indicated in the phase diagram in Fig. 1b. In all cases, the magnetic field was applied perpendicular to the six-fold symmetric (kagome) lattice planes. Figures 1c–e shows VL diffraction patterns for Cs(V0.86Ta0.14)3Sb5 at three different applied magnetic fields. The system was prepared by a field cooling to the measurement temperature from above Tc, followed by a damped field oscillation with an initial amplitude of 5% of the measurement field. As witnessed by the well-defined defined Bragg peaks, this produces an ordered VL, which is consistent with weak pinning45.

Fig. 1: Vortex lattice diffraction patterns.
Fig. 1: Vortex lattice diffraction patterns.The alternative text for this image may have been generated using AI.
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a Cs(V0.86Ta0.14)3Sb5 sample mosaic used for the SANS experiments, with the arrows indicating the in-plane crystalline directions. b Field-temperature phase diagram indicating where SANS were performed. c Cs(V0.86Ta0.14)3Sb5 VL diffraction pattern at 1.7 K and 0.1 T. The arrows indicate the crystalline orientation. d Same at 0.25 T and e 0.5 T. In panel (e) only Bragg peaks at the lower half of the detector were rocked through the Ewald sphere and the positions of symmetry equivalent peaks are indicated by circles. f CsV3Sb5 VL diffraction pattern at 35 mK and 50 mT. For all diffraction patterns, background measurements obtained at zero field are subtracted and the region near q = 0 is masked off. A separate color scale is used for each diffraction pattern.

A triangular VL is observed at all fields and temperatures as expected for a superconductor with a six-fold symmetric basal plane, and oriented with Bragg peaks along the \([\overline{1}\,2\,0]\) crystalline direction. The same symmetry and orientation is observed in a reference measurement on undoped CsV3Sb5 as shown in Fig. 1f, although VL imaging is only possible at a low field and temperature due to the much larger penetration depth (182.7 nm vs 106.8 nm) and much lower upper critical field (0.3 T vs 2.2 T)46. In contrast, vortex imaging by scanning tunneling spectroscopy (STS) found a VL that undergoes a 15° rotation as the applied field is increased from 75 mT to 200 mT, and where the high field orientation is orthogonal to the one reported here47. This highlights how different results, reflecting the bulk properties, may be obtained using SANS compared to surface probes with a limited field of view, such as STS.

The VL orientation is determined by anisotropies within the screening current plane, which may arise from the Fermi surface (FS)48 or the superconducting gap49,50,51. Considering reports of an isotropic gap on all three FS sheets in Cs(V0.86Ta0.14)3Sb519, the former scenario is the most likely. A direct correlation between the VL orientation and the band structure requires an evaluation of Fermi velocity averages as well as a directionally resolved Density of States at the Fermi level48,52, which has presently not been carried out for CsV3Sb5 or Cs(V0.86Ta0.14)3Sb5. Nevertheless, the morphology of almost perfectly nested hexagonal (β) and triangular (δ) FS sheets with flat sections perpendicular to the Γ-K direction13,19,53,54, is consistent with the observed VL orientation. This commonality, together with the isotropic (α) FS sheet, may also explain the absence of a field-driven VL rotation transition observed in other multiband superconductors such as MgB255 and UPt356,57.

Scattering vector magnitude

The scattering vector magnitude for a triangular VL is

$${q}_{ne}(B)=2\pi \sqrt{\frac{2B}{\sqrt{3}{\Phi }_{ne}}},$$
(1)

where B is the magnetic induction, Φne = h/ne is the flux quantum and n is an even integer. For regular 2e-pairing the flux quantum is given by Φ2e = Φ0 = 2068 Tnm2 58. Figure 2 shows the measured VL scattering vector (q) versus applied magnetic field (μ0H). This follows the behavior expected for 2e-pairing, assuming B = μ0H. Furthermore, q can be reliably determined at temperatures up to \(\sim \frac{2}{3}{T}_{c}\) at 0.1 T and shows no deviation from q2e. In contrast, q4e and q6e are not compatible with the data within the experimental error. While agreement could in principle be achieved for n ≠ 2, it requires \(B=\sqrt{n/2}\,{\mu }_{0}H\) which is inconsistent with magnetization measurements on Cs(V0.86Ta0.14)3Sb546. The SANS results do thus not provide evidence for 4e- and 6e-pairing at the bulk level that has been reported for CsV3Sb5 as one approaches Tc39. The inset to Fig. 2 shows the magnetic induction inferred from the measured q and using eqn. (1) with n = 2. A linear fit to the data yields a slope of 1.033 ± 0.015 and an ordinate intercept of B = 7.4 mT ±  4.2 mT. The latter provides an upper limit of a few milliTesla on any net spontaneous field.

Fig. 2: Field dependence of the vortex lattice scattering vector.
Fig. 2: Field dependence of the vortex lattice scattering vector.The alternative text for this image may have been generated using AI.
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Full, dashed and dot-dashed lines indicate q expected for 2e, 4e and 6e-pairing respectively. Inset shows the magnetic induction determined from eqn. (1) assuming 2e-pairing. The line is a linear fit. Data for x = 0.14 was measured at 1.7 K and for x = 0 at 35 mK. For both the main figure and the inset, error bars represent one standard deviation.

Form factor field dependence

The field dependence of the scattered intensity provides information about the superconducting penetration depth (λ) and coherence length (ξ). This requires a measurement of the integrated intensity, obtained by rotating the VL diffraction peak through the Bragg condition as shown in Fig. 3a. Normalizing by the incident neutron flux one obtains the VL reflectivity

$$R=\frac{2\pi {\gamma }_{n}^{2}\,{\lambda }_{n}^{2}\,{t}_{s}}{16{\Phi }_{ne}^{2}\,q}{\left\vert F(q)\right\vert }^{2},$$
(2)

where F(q) is the VL form factor, γn = 1.913 is the neutron magnetic moment in units of the nuclear magneton, and ts is the sample thickness59,60. Figure 3(b) shows the form factor obtained from the SANS measurement which is found to decrease exponentially with increasing field. This is consistent with the London model

$$F(q)=\frac{B}{1+{q}^{2}{\lambda }^{2}}\,{e}^{-{q}^{2}{\xi }^{2}/2},$$
(3)

with a Gaussian core cut-off to account for a finite coherence length61,62. The numerical factor of \(\frac{1}{2}\) in the exponent has been found to yield reasonable values for the coherence length in a range of superconductors, although other values have been used in the literature43. For qλ ≫ 1 the only field dependence is through q2 ∝ B in the exponent. A fit to the data yields λ = (113.7 ± 4.8) nm from the zero field intercept, and ξ = (14.5 ± 0.6) nm from the slope. This agrees well with values for λ = 106.8 nm and ξ = 12.2 nm inferred from measurements of the lower and upper critical fields46. Notably, the data in Fig. 3b shows no deviation from a purely exponential behavior which may arise from multiband superconductivity55 or Pauli paramagnetic effects effects63,64,65.

Fig. 3: Vortex lattice form factor.
Fig. 3: Vortex lattice form factor.The alternative text for this image may have been generated using AI.
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a Scattered intensity as the VL Bragg is rotated through the Ewald sphere. The curve is a Lorentzian fit and the full-width-half-maximum (w) is indicated by the arrow. b VL form factor versus applied magnetic field. The dashed line is a fit to eqn. (3), with the penetration depth and coherence length indicated in the plot. Inset shows the full-width-half-maximum rocking curve widths obtained from Lorentzian fits. The solid line is a guide to the eye and the dashed line shows the experimental resolution. For all panels, error bars indicate one standard deviation.

The rocking curve width (w) is inversely proportional to the longitudinal VL correlation length. As shown in the Fig. 3b inset, w decreases with increasing field, approaching the experimental resolution. This gradual ordering is commonly observed in superconductors with low pinning and attributed to an enhanced vortex-vortex interactions, and resulting increasing VL tilt modulus, as the density increases66. This provides additional support for weak vortex pinning previous reported for Cs(V0.86Ta0.14)3Sb546.

Temperature dependence of scattering intensity

The form factor is proportional to the superfluid density (ρs), which is dominated by the lowest gap values for a reduced temperature \(t=T/{T}_{c}\lesssim \frac{1}{3}\). Figure 4 shows the normalized superfluid density versus temperature for two different applied fields, which display a clear saturation as t → 0. In the simplest case of an s-wave superconductor with a uniform gap

$${\rho }_{s}(t)=1-\frac{1}{2t}\int_{\!\!\!\!0}^{\infty }{\cosh }^{-2}\left(\frac{\sqrt{{\varepsilon }^{2}+{\Delta }^{2}(t)}}{2t}\right)d\varepsilon,$$
(4)

where Δ(t) is the temperature-dependent superconducting gap in units of kBTc67. In the weak coupling limit

$$\Delta (t)={\Delta }_{0}\tanh \left(\frac{\pi }{{\Delta }_{0}}\sqrt{\frac{1}{t}-1}\right),$$
(5)

and Δ0 is the zero-temperature gap amplitude68. Curves in Fig. 4 show fits to the SANS data across the entire measured temperature range, using eqs. (4) and (5) and the measured values for Tc. This yields a superconducting gap of 2Δ0 = (1.11 ± 0.04)meV (0.1 T) and (1.08 ± 0.04)meV (0.2 T) or 2Δ0/kBTc = 2.74 ± 0.09 and 2.85 ± 0.11 respectively, somewhat lower than the BCS prediction of 3.53 and confirming weak coupling superconductivity. This is consistent with the uniform gap across all FS sheets obtained from ARPES19, although our values are roughly 25% smaller. Estimates of the gap obtained from ρs(t) determined by measurements of the lower critical field for \(T\ge \frac{1}{3}{T}_{c}\) yielded an even larger gap, 2Δ0 = (3.6 ± 0.8)meV46. We note that a uniform (nodeless) gap does not preclude e.g., s ± is pairing that would break time-reversal symmetry or multiband superconductivity with a s++ or s+− state69, but is inconsistent with d-wave or p-wave pairing.

Fig. 4: Temperature dependence of the superfluid density.
Fig. 4: Temperature dependence of the superfluid density.The alternative text for this image may have been generated using AI.
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a 0.1 T and b 0.2 T. Error bars indicate one standard deviation. The curves are fits to an s-wave model as described in the text, with gap values and critical temperatures indicated in the plots.

In summary, our SANS studies of the Cs(V0.86Ta0.14)3Sb5 VL indicates a wholly conventional superconducting state. Furthermore, the optimal superconductivity without CDW order is likely a conventional BCS superconductor, where electron pairing is induced by electron-lattice coupling. Our results on Cs(V0.86Ta0.14)3Sb5 thus suggest that the exotic properties reported for CDW related phenomena in other members of the AV3Sb5 superconductors are likely not related to the microscopic origin of superconductivity.

Methods

Single crystals of Cs(V1−xTax)3Sb5 were grown by by the self-flux method5,19,70,71. These materials form in a layered kagome structure with a P6/mmm space group5, and with facets that allow for an easy determination of the in-plane crystalline axes. For the SANS measurements a mosaic of co-aligned Cs(V0.86Ta0.14)3Sb5 single crystals with a critical temperature Tc = 5.3 K and a total mass of 200 mg was used, oriented with the [1 0 0]-axis horizontal and the \([\overline{1}\,2\,0]\)-axis vertical. The co-alignment of the individual crystals in the mosaics is confirmed by the six sharp VL Bragg peaks observed in Figs. 1c–f and the absence of significant scattered intensity in between those. This also excludes twinning within the single crystals.

The SANS measurements were carried out using the SANS-I instrument at the Swiss Spallation Neutron Source (SINQ) at the Paul Scherrer Institute. Two experiments were performed, using a pumped 4He cryomagnet for measurements down to 1.7 K (#1) or a dilution refrigerator (DR) for measurements between 67 mK and Tc (#2). A reference measurement was performed on a mosaic of undoped CsV3Sb5 sample (280 mg, Tc = 3.0 K) using the GP-SANS instrument at the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory (ORNL). These measurements were carried out at a temperature of 35 mK using a DR.

In all cases, a neutron wavelength λn = 1.4 nm and bandwidth of Δλn/λn = 10% was used, and the diffracted neutrons were detected using a position-sensitive detector placed at 11 m from the sample. The horizontal magnetic field was applied along the crystalline [0 0 1]-direction and near-parallel to the incident neutron beam. The sample and cryomagnet were rotated together about the horizontal axes perpendicular to the beam direction to satisfy the Bragg condition for the different VL peaks. Small-angle background measurements were collected in zero field at the base temperature for the respective experiment, and subtracted from the data.

All SANS data was analyzed using the GRASP graphical reduction and analysis software for small-angle neutron scattering available at https://www.ill.fr/grasp/72.