Introduction

The crossover between the weak-coupling Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity and strong-coupling Bose–Einstein-condensate (BEC) describes the fundamentals of quantum bound states of particles, including information on the superconducting critical temperature (Tc) and the possible pseudogap formation1,2,3. In the strong-coupling regime, the pair-formation temperature Tpair and the actual superconducting transition temperature Tc are distinctly separated (Fig. 1a). Between Tpair and Tc, the so-called preformed Cooper pairs exist, which give rise to large superconducting fluctuations and possibly a depletion of the low-energy density of states (DOS), namely the pseudogap. Almost all experimental studies on crossover physics have been performed in ultracold atomic systems for the past decades2 because of the difficulty in tuning the attractive interaction between electrons in solids. Therefore, it is extremely challenging to realize such strongly coupled pairs in electron systems. In underdoped high-Tc cuprates, the pseudogap formation has been discussed in terms of the BCS–BEC crossover4, but recent experiments support a phase transition of different kinds at the onset of the pseudogap, and the relevance of BCS–BEC crossover in cuprates remains unresolved5. On the other hand, very recent studies have successively pointed to the realization of BCS–BEC crossover in superconductivity in other strongly correlated compounds6,7,8,9.

Fig. 1: Phase diagrams of BCS–BEC (Bardeen–Cooper–Schrieffer–Bose–Einstein-condensate) crossover in single-band systems and multiband FeSe1−xSx superconductors.
figure 1

a Theoretical phase diagram of BCS-BEC crossover generally obtained for single-band systems1,2. The temperature T is normalized by the Fermi temperature TF, and the strength of attraction is given by dimensionless coupling constant 1/(kFas). Here, kF and as are Fermi wavenumber and s-wave-scattering length, respectively. Below the pairing temperature Tpair, bosonic pairs form, while the superconducting coherence is acquired when the condensation occurs at superconducting transition temperature Tc. In the weak-coupling BCS limit, the ratio of the superconducting gap and Fermi energy Δ/EF is much smaller than unity, and the coherence length ξ is much longer than the interparticle distance (~1/kF), and Tpair is very close to the actual Tc. In the BCS–BEC crossover regime Δ/EF ~1, Tc/TF reaches a maximum, and preformed Cooper pairs are expected to exist in an extended temperature region between Tpair and Tc. b Experimentally determined T–x phase diagram of FeSe1−xSx. The superconducting transition temperature Tc (red squares) is determined by the present heat capacity measurements using small crystals. The nematic transition temperature (blue diamonds) determined by transport measurements22 is also plotted. The abrupt change in Tc indicates a significantly different superconducting ground state between SC (superconducting state)1 and SC2 divided by the nematic end point at xc ≈ 0.17. The error bars are determined by the distribution of the chemical composition in the samples.

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The superconducting semimetal FeSe10,11,12 with very small hole and electron Fermi surfaces are found to exhibit large ratios of the superconducting energy gap Δ and Fermi energy EF, Δ/EF ≈ 0.3–1.07,13,14, which appears to place FeSe in the crossover regime (Fig. 1a). Indeed, in high-quality single crystals with long mean free paths of electrons7, large superconducting fluctuations have been reported from the torque magnetometry, providing support for the presence of fluctuated Cooper pairs created above Tc in the crossover regime15. These observations underline the importance of the BCS-BEC crossover physics both in the superconducting and normal states of FeSe. On the other hand, scanning tunneling spectroscopy (STS) measurements do not observe the pseudogap formation above Tc16.

These results imply that the superconductivity in FeSe is not simply described by a picture of the conventional BCS-BEC crossover. An important aspect that has not been taken into account in the previous studies is the effects of multiband electronic structure, which may have other tuning factors of the crossover in addition to Δ/EF. In fact, the Fermi surface of FeSe is composed of separated holes and electron pockets with different values of Δ/EF7,13,17. It has been pointed out that in multiband systems, the developments of superconducting fluctuations and pseudogap formation are sensitively affected by the interband coupling strength18,19,20, indicating that the normal and superconducting properties associated with the crossover in multiband systems can be different from those in the single band systems21. It is quite important, therefore, to clarify how the superconducting properties evolve when the multiband electronic structure is tuned.

Here, we focus on FeSe1−xSx, where the orthorhombic (nematic) phase transition at 90 K in FeSe can be tuned by isovalent sulfur (S) substitution. The nematic transition that distorts the Fermi surface is completely suppressed at a critical concentration xc ≈ 0.1722,23. The superconductivity persists in the tetragonal (non-nematic) phase above xc. In the case of applying hydrostatic pressure, the suppression of nematic order is accompanied by the pressure-induced antiferromagnetic order24. In contrast, S substitution suppresses the nematic order without inducing the antiferromagnetism that significantly changes the Fermi surface through band folding, even when crossing xc. This enables us to investigate the evolution of the superconducting state in the BCS–BEC crossover with little influence of antiferromagnetic fluctuations. Moreover, the observations of quantum oscillations up to x ≈ 0.1925 demonstrate that the S ions do not act as strong scattering centers.

By using high-resolution heat capacity, magnetic torque, and scanning tunneling spectroscopy measurements, we clarify the unusual evolution of the thermodynamic and spectroscopic properties in the FeSe1−xSx system, where the highly non-mean-field behaviors are found in tetragonal phase x > 0.17 with no signature of pseudogap formation. These results illuminate the unusual feature of BCS–BEC crossover with the multiband electronic structure.

Results and Discussion

Heat capacity

Through the high-resolution heat capacity, magnetic torque, and STS measurements on high-quality single crystals, we investigate the evolution of superconducting and normal state properties with x in FeSe1−xSx. In Fig. 1b, we show the phase diagram obtained by the heat capacity measurements. The drastic change in Tc and superconducting fluctuations are found across xc. To ensure sample homogeneity, we used very small crystals (10–190 μg) for the thermodynamic measurements and employed the long relaxation method with a homemade cell (Fig. 2a) designed for small single crystals. Figure 2b–e depicts the T-dependence of electronic heat capacity divided by T, Ce/T, for x = 0, 0.05, 0.10, and 0.13. Here we subtract the phonon contribution, which shows T3-dependence, as confirmed by the measurements in the normal state at high fields. In Fig. 2b, Ce/T(T) in FeSe (x = 0) exhibits an ordinary BCS-like jump at Tc = 9 K, consistent with the previous studies on larger samples26,27,28. As the temperature is lowered below Tc, Ce/T decreases with decreasing slope. Below ~3 K, Ce/T shows T-linear dependence, which is consistent with the strongly anisotropic superconducting gap7,28,29. Similar Ce/T(T) is observed in orthorhombic phase x = 0.05, 0.10, and 0.13 with Tc of 9.7, 10, and 9.5 K, respectively (Fig. 2c–e).

Fig. 2: Heat capacity in FeSe1−xSx superconductors.
figure 2

a The schematic of the experimental setup. A Cernox small thermometer chip is used as a heater, and large current pulses are applied. b–i T-dependence of electronic heat capacity divided by T, Ce/T in FeSe1−xSx single crystals at low temperatures for x = 0 (b), x = 0.05 (c), x = 0.10 (d), x = 0.13 (e) in the orthorhombic phase and x = 0.18 (g), x = 0.20 (h), x = 0.21 (i) in the tetragonal phase. f T-dependence of Ce/T for x = 0.16 near the nematic end point. The data (orange dots) are fitted with a weighted sum (black line) of the x = 0.13 orthorhombic data (dashed line) and x = 0.18 tetragonal data (dashed-dotted line) with adjusted Tc values.

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Upon entering the tetragonal phase at x > xc, Ce/T exhibits a dramatic change not only in the superconducting state but also in the normal state. As represented by x = 0.20 in Fig. 2h, Ce/T increases with the increasing slope in the normal state when approaching Tc from high temperatures. In addition, Ce/T exhibits a kink at Tc without showing a discontinuous jump. Such a continuous T-dependence at Tc is reminiscent of the BEC transition in the free Bose gas systems30, although it is too simple to employ the free Bose gas model in our system. We note that a similar enhancement of Ce/T in the normal state is obtained in the calculations for the strongly interacting Fermi systems near the unitary limit in the BCS–BEC crossover regime31. As the temperature is lowered below Tc, Ce/T(T) for x = 0.20 decreases with increasing slope, in stark contrast to Ce/T(T) in the orthorhombic crystals, as shown in Fig. 2b–e. Below Tc, Ce/T decreases sublinearly to T with large residual Ce/T at the lowest temperatures, which is indicative of a large DOS at zero energy. The large residual DOS is consistent with the recent thermal conductivity and STS measurements32,33. Similar Ce/T(T) is also observed for x = 0.18 and 0.21 in the tetragonal phase, as shown in Fig. 2g, i. These results, together with the sharp change in Tc across xc, demonstrate that the superconducting properties in the tetragonal phase are drastically different from those in the orthorhombic phase. We note that a similar broadening of C/T at Tc has also been observed when nematicity is suppressed by pressure34.

We stress that the observed peculiar Ce/T(T) in the tetragonal phase does not stem from chemical inhomogeneity within the small sample for the following reasons. First, the topographic image in STM of FeSe1−xSx crystals reveals no evidence for the segregation of S atoms, indicating an excellent homogeneity33 (see also Fig. 3a, b). Second, the observation of the quantum oscillations of the crystals in the same batch25 demonstrates a long mean free path of the carriers, ensuring the high quality of our crystals. Third, according to the elemental mapping of the energy dispersive X-ray spectroscopy (EDX) at different scales of distance (Supplementary Note 1), the typical spatial variation of the composition Δx is ~ 0.01, and there are no discernible segregations or large inhomogeneity of the chemical compositions which can explain our observation in Ce/T(T). Fourth, we measured the heat capacity on two different crystals for x = 0.20 (Supplementary Note 2) and two other concentrations, x = 0.18 and 0.21 (Fig. 2g, i), and confirmed the reproducibility of the behavior. We also note that our results are consistent with the previous data of the crystal with a much larger volume within the error bar (Supplementary Note 2). Fifth, most importantly, there is no concentration of the sample, which shows the superconducting transition between ≈ 4 K and ≈ 7 K in the phase diagram of FeSe1−xSx. This is supported by the Ce/T(T) of the crystal at x ≈ 0.16, which is located in the very vicinity of xc. As shown in Fig. 2f, Ce/T at x ≈ 0.16 exhibits two well-separated peaks at Tc1 = 4 K and Tc2 = 7 K, indicating a phase separation (due to the small spatial variation of Δx ~ 0.01) at xc. This T-dependence is well reproduced by the sum of Ce/T of x = 0.13 (Tc = 9.5 K) and 0.18 (Tc = 4.0 K), assuming that both contribute equally. Here, for the fitting, Tc values are shifted slightly. These results indicate a discontinuous change of Tc at xc. Such a discontinuous change in the superconducting properties is consistent with the jump in Δ values at xc recently reported by STS studies33. The observed enhancement of Ce/T above Tc for the sample in the tetragonal phase, therefore, can be attributed to an intrinsic electronic property.

Fig. 3: Scanning tunneling spectroscopy in the tetragonal phase.
figure 3

a Topographic STM (scanning tunneling microscopy) image (10 mV, 100 pA) of FeSe0.75S0.25 surface over a 10 nm by 10 nm area. b A line profile of the normalized conductance along the AB is shown in (a). The data are taken at 2 K. c Normalized tunneling conductance spectra are taken at several temperatures. The spectra are vertically shifted for clarity. The tip is stabilized at bias voltage V = 6 mV and current I = 1 nA. d The T-dependence of gap depth, which is defined by the difference between normalized conductance values at V = 0 and 6 mV. The red dashed line is a guide for the eye. e The T-dependence of magnetization measured at a zero field cool condition.

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Scanning tunneling microscopy (STM)/spectroscopy (STS)

To elucidate whether the depletion of DOS associated with the pseudogap formation occurs, the STS measurements on the crystal of x = 0.25 are performed. By counting the number of S atoms in the STM topographic image, we determined the x value accurately (Fig. 3a). Figure 3b shows the line profile of the normalized conductance along line AB depicted in Fig. 3a. Spectra show little variation within the 10 nm scale, demonstrating the uniform spatial distribution of the superconducting gap. We also note that the gap value is consistent with the previous report33. Figure 3c depicts the STS spectra normalized by the conductance above the superconducting gap in a wide temperature range. Large residual DOS, which is more than half of the normal state value outside the superconducting gap (Δ ≈ 1 meV), is observed even at T = 0.4 K (T/Tc = 0.1). This large residual DOS is consistent with the large Ce/T at low temperatures. Here, the STS spectrum becomes nearly energy independent at high temperatures above ~5 K, showing no signs of gap formation. This is also seen by the T-dependence of the gap depth plotted in Fig. 3d. The superconducting gap closes at ≈ 5 K, below which the T-dependence of magnetization shows a rapid decrease, as shown in Fig. 3e.

Nearly energy-independent DOS at EF observed by STS above Tc indicates that the observed enhancement of Ce/T above Tc in the tetragonal phase is not caused by the DOS effect. Therefore, it is natural to consider that the enhancement of Ce/T stems from the superconducting fluctuations. Indeed, a similar enhancement of Ce/T toward Tc due to the strong fluctuations has been previously reported in other iron-based superconductors35,36,37. In order to discuss the effect of the superconducting fluctuations in FeSe1−xSx quantitatively, we extract the superconducting contribution Csc, which is obtained by subtracting the normal state electronic contribution γT from Ce for x = 0 (blue circles in Fig. 4a) and x = 0.20 (red circles in Fig. 4b). We compare Csc with the conventional term of the mean-field Gaussian fluctuations38 (Supplementary Note 3). Obviously, the heat capacity contribution that significantly exceeds CGauss can be seen for x = 0.20. This extra heat capacity is observable up to t ~ 0.8. In Fig. 4c, we display the x-dependence of ΔC/γTc, where ΔC is the height of Csc at Tc. For x < xc, the magnitude of ΔC/γTc is close to the BCS weak coupling value (1.43 for s-wave and 0.94 for d-wave). On the other hand, ΔC/γTc for x > xc is reduced far below the mean-field BCS value. As a consequence of the unusual suppression of ΔC/γTc, the low-T Ce/T is largely enhanced through the entropy balance. Figure 4d depicts the x-dependence of the Ce/T at T/Tc = 0.1 taken from Fig. 2. Remarkable enhancement of Ce/T at xc indicates the fundamental difference of the low-energy excitations across xc. We note that recent NMR measurements for x > xc report no splitting of the spin-echo signal in the crystal from the same batch39,40, which excludes the microscopic phase separation of superconducting and non-superconducting regions.

Fig. 4: Comparisons of superconducting transition properties between the orthorhombic and tetragonal phases.
figure 4

a, b The superconducting contribution to the electronic heat capacity Csc = Ce − γT as a function of reduced temperature \(t\equiv \frac{T-{T}_{{{{{{{{\rm{c}}}}}}}}}}{{T}_{{{{{{{{\rm{c}}}}}}}}}}\) for x = 0 (a) and x = 0.20 (b). The conventional Gaussian fluctuation contribution CGauss is estimated (solid lines). c, d x dependence of ΔC/γTc (c), and Ce/T at T/Tc = 0.1 (d). e, f Entropy calculated from Ce/T(T) in the normal state (Sn) and superconducting state (Ss) as a function of t for x = 0 (e) and x = 0.20 (f). Sn is estimated from the high-field data where superconductivity is suppressed. The entropy for x = 0.20 is linearly extrapolated below 1.0 K. g x dependence of the relative entropy difference between zero and high fields (Sn − Ss)/Sn at T = 1.03Tc. h The T-dependence of the anisotropy of the magnetic susceptibility Δχ = χc − χab for several fields in x = 0.20. Each data is obtained by fitting the field-angle dependence of the torque τ(θ). The inset shows the τ(θ) at 1.0 T and 41 K as the typical signal of torque, where the red markers and the solid black line show the raw data and fitting curve, respectively. The error bars are determined by the errors in estimating the volume of the sample and the error in the fitting procedure.

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Magnetic torque

The anomalous Ce/T in the tetragonal phase is reflected by the entropies in the normal and superconducting states, Sn and Ss, respectively, as shown in Fig. 4e, f. For x = 0, the T-dependence of Ss shows a kink at Tc, which is typical for the second-order superconducting transition. For x = 0.20, on the other hand, owing to the absence of the jump in Ce/T at Tc, no kink anomaly is observed in the T-dependence of Ss. Moreover, the entropy is significantly suppressed from Sn even above Tc. In Fig. 4g, we show the x dependence of the lost entropy normalized by Sn, (Sn − Ss)/Sn, just above Tc, T = 1.03Tc. Upon entering the tetragonal phase, an abrupt enhancement of (Sn − Ss)/Sn can be seen, indicating a drastic change of the superconducting fluctuations associated with the preformed Cooper pairs.

To obtain further insight into the superconducting fluctuations in the tetragonal phase, the magnetic torque was measured by using micro-cantilevers. The torque can sensitively detect the diamagnetic response through the anisotropy of magnetic susceptibility15. By measuring the field angular variation of torque as a function of polar angle θ from the c axis, the anisotropy Δχ = χc − χab is determined, where χc and χab are the magnetic susceptibilities along the c axis and in the ab plane, respectively. As shown in the inset of Fig. 4h, τ(θ) shows an almost perfect sinusoidal curve, \(\tau (\theta )\propto \Delta \chi \sin \theta\). Although Δχ(T) at high fields is almost independent of T and H, it exhibits strong T and H-dependence at low fields, as shown in Fig. 4h. It is well settled that the fluctuation-induced diamagnetic susceptibility of most superconductors, including multiband systems, can be well described by the standard Gaussian-type (Aslamasoz–Larkin, AL) fluctuation susceptibility χAL (Supplementary Note 3). Here we focus on the H-dependent diamagnetic susceptibility at low fields15. Then, Δχ at low fields is much larger than the ∣χAL∣, which is added to high field 7 T data. This implies that the superconducting fluctuations in FeSe1−xSx are distinctly different from those in conventional superconductors, supporting the fluctuation effects observed in heat capacity measurements. We stress that although the multi-gap superconductivity may lead to a small value of ΔC/γT, it cannot give rise to the non-mean-field behaviors observed in heat capacity and magnetic susceptibility above Tc.

Unusual crossover from Bardeen-Cooper-Schrieffer to Bose-Einstein-condensate superconductivity

In FeSe, while large superconducting fluctuations that well exceed the Gaussian fluctuations are observed, the Ce/T around Tc does not exhibit a significant deviation from the mean-field behavior. This suggests that tetragonal FeSe1−xSx are closer to the BEC regime. However, with increasing S composition, the volume of the Fermi surface increases, as reported by quantum oscillation experiments25, and Δ is reduced, according to STM measurements33. Then Δ/EF is expected to become smaller with increasing S concentration, which is opposite to the tendency of approaching the BEC regime in the conventional BCS-BEC crossover. In fact, recent ARPES studies have reported BEC-like behavior in the tetragonal FeSe1−xSx41, which is in good agreement with our thermodynamic studies. These behaviors are in contrast to the BEC-like dispersion observed for Fe1+ySexTe1−x14 with increasing Δ/EF, as expected in the context of conventional BCS–BEC crossovers. On the other hand, in our case of FeSe1−xSx, a structural phase transition from orthorhombic to tetragonal occurs as x increases. This leads to modification in the band structure and a significant change in the pairing interaction. This can be regarded as the main difference between the Fe1+ySexTe1−x system and our FeSe1−xSx system, leading to a change in the band structure and a significant change in pairing interactions. Another difference is that FeSe1−xSx holds the compensation condition, where both electron and hole Fermi energies are comparably small, whereas the Fe non-stoichiometry in Fe1+ySexTe1−x leads to the imbalance of Fermi energies of two carriers. This suggests that there is another important factor in the FeSe1−xSx system which has not been duly taken into consideration.

A key factor that controls the BCS-BEC crossover in FeSe1−xSx is the multiband character. As shown previously16,21, in FeSe, the strong interband interaction between electron and hole pockets prevents the splitting of the Tpair and the Tc, determined by the superfluid stiffness. We note that this strong interband interaction is also an important ingredient in the orbital-selective scenario29,42,43,44, and therefore we expect that even taking orbital selectivity into account, the main result, i.e., mean-field-like jump of the specific heat, will remain the same. At the same time, much less is known about the evolution of the ratio of the interband versus intraband interactions upon approaching xc in FeSe1−xSx. Furthermore, recent experimental45 and theoretical46,47 works pointed out an interesting possibility of the substantial non-local nematic order on the dxy orbital and/or inter-orbital-nematicity between dxy and dyz orbitals in FeSe. The most important consequence of this phenomenon would be an additional Lifshitz transition in FeSe1−xSx upon S substitution, which occurs close to xc. This, in turn, implies the appearance of the incipient band near xc (see, for example, Fig. 6 in Rhodes et al. 46). On general grounds, the presence of the incipient band would enhance the tendency towards the BCS-BEC crossover. In addition, given a nearly equal mixture of the s- and d-wave symmetry components of the superconducting gap in FeSe48 to express such an anisotropic gap, it is natural to assume that they are competing in the tetragonal phase of FeSe1−xSx. In particular, the ratio of the pairing interactions in the d-wave and s-wave channels \(| {g}_{{{{{{{{\rm{ee}}}}}}}}}^{d}| /{g}_{{{{{{{{\rm{eh}}}}}}}}}^{s}\) may, in addition to the incipient band, be another key parameter of the BCS-BEC crossover, as follows from our model calculations (Supplementary Note 4). The pair-formation temperature Tpair and condensation temperature Tc can be split more by increasing the ratio of intraband and interband interactions \(| {g}_{{{{{{{{\rm{ee}}}}}}}}}^{d}| /{g}_{{{{{{{{\rm{eh}}}}}}}}}^{s}\) (Supplementary Note 4) without changing EF. This quantitatively explains the appearance of strong coupling superconductivity for x > xc with no apparent enhancement of Δ/EF. A remarkable and unexpected feature is the absence of pseudogap above Tc in the STS spectra, despite the presence of giant superconducting fluctuations in thermodynamic quantities. It should be noted that the relationship between the preformed pairs and the pseudogap formation is still elusive even in the ultra-cold atoms2. In addition, the onset temperature of the pseudogap may differ from that of fluctuations31. Moreover, it has been pointed out that the pseudogap phenomena in multiband systems are markedly altered from those in single-band systems. For instance, in multiband systems, the pseudogap formation is less prominent by the reduction of its onset temperature49. In a recent study, the BEC-like dispersion of the Bogoliubov quasiparticles has been reported in the tetragonal FeSe1−xSx by using ARPES measurements41. Although the direct comparison of the ARPES data with the present thermodynamic results is not straightforward because ARPES measures only the portion of the hole Fermi pocket, it provides spectroscopic evidence of the BCS–BEC crossover. The ARPES measurements also report the presence of a distinct pseudogap in the hole pocket. However, we point out that this is not an apparent contradiction to the present STM results. It has been suggested that the DOS of electron pockets is much larger than that of hole pockets. As STM measures the DOS integrated over all bands, STM is insensitive to the pseudogap formation in the hole pocket. Therefore, the STM spectra, combined with the ARPES results, suggest that the pseudogap formation is also orbital-dependent.

It still, however, cannot fully explain the reduction of the gap magnitudes found experimentally, and further mechanisms may be needed, such as the influence of magnetic fluctuations, which would further reduce the superfluid stiffness50, a Lifishitz transition near the orthorhombic to tetragonal transition46, or nematic fluctuations. In FeSe1−xSx, the nematic fluctuations peaked at qnem ≈ 0 because it is ordered state breaks the rotational symmetry while preserving translational symmetry. Enhanced nematic fluctuations22 play an important role in the normal state properties23,51, particularly at x ~ xc23. It has also been pointed out theoretically that nematic fluctuations strongly influence the superconductivity52,53. Therefore, it is tempting to consider that the nematic fluctuations and incipient bands in FeSe1−xSx upon approaching a tetragonal phase can enhance the pairing interaction, leading the system to approach the BEC regime. Further theoretical and experimental studies are required to uncover whether the nematic fluctuations and multi-band nature affect the crossover physics.

Conclusion

Our thermodynamic studies have revealed a highly non-mean-field behavior of superconducting transition in FeSe1−xSx when the nematic order of FeSe is completely suppressed by S substitution. This is consistent with the recent ARPES measurements, which report the BEC-like dispersion of the Bogoliubov quasiparticles in the tetragonal regime. Our findings demonstrate that FeSe1−xSx offers a unique playground to study the superconducting properties in the BCS–BEC crossover regime of multiband systems.

Methods

Crystal growth and characterization

The single crystals of FeSe1−xSx were grown by the chemical vapor transport technique as described in Böhmer et al. and Hosoi et al. 11,22. The composition x is determined by energy-dispersive X-ray spectroscopy.

Heat capacity

The heat capacity of the crystals is measured using the long relaxation method54,55 whose experimental setup is illustrated in the inset of Fig. 1a. A single bare chip of Cernox resistor is used as the thermometer, heater, and sample stage, which is suspended from the cold stage by silver-coated glass fibers in order that the bare chip has a weak thermal link to the cold stage and electrical connection for the sensor reading. The mass of the samples measured in this study is 81, 41, 171, 137, 179, 50, 107, 192, and 10 μg for x = 0, 0.05, 0.1, 0.13, 0.16, 0.18, 0.20#1, 0.20#2, and 0.21, respectively. The samples are mounted on the bare chip using Apiezon N grease. The heat capacity of the crystals is typically obtained by subtracting the heat capacity of the bare chip and grease from the raw data.

Magnetic torque

The torque measurements are performed using piezoresistive micro-cantilevers15. The small single crystals with a typical size of 150 μm × 100 μm × 10 μm are mounted on the tip of microcantilevers with a small amount of Apiezon N grease. The magnetic field is applied inside the ac(bc) plane by vector magnet, and the field-angle dependence of torque is obtained by rotating the whole cryostat within the ac(bc) plane. Magnetic torque τ can be expressed as τ = μ0VM × H, where μ0 is vacuum permeability, V is sample volume, M is magnetization, and H is an external magnetic field. Here the H is rotated within the ac (bc) plane, and θ is the polar angle from the c-axis. In this configuration, the τ is expressed as \({\tau }_{2\theta }(T,H,\theta )=\frac{1}{2}{\mu }_{0}{H}^{2}V\Delta \chi \sin (\theta )\). Here Δχ = χc − χab is anisotropy of magnetic susceptibility between the ac(ab) plane and the c-axis.

Scanning tunneling microscopy/spectroscopy

The scanning tunneling microscopy/spectroscopy measurements have been performed with a commercial low-temperature STM system. Fresh and atomically flat surfaces are obtained by in situ cleavage at liquid nitrogen temperature in an ultra-high vacuum. All conductance spectra and topographic images are obtained by a PtIr tip. Conductance spectra are measured by a lock-in technique with a modulation frequency of 997 Hz and a modulation voltage of 200 μV. The backgrounds of raw conductance data are normalized by curves which are derived by 2nd-order polynomial fitting using outside superconducting gaps.

Theoretical calculation

The theoretical calculations were done within a simplified interacting two-band model in two dimensions, employed previously21 with hole and electron pockets with small Fermi energies. We assume superconductivity is due to repulsive interactions. While the dominant interband interaction favors Cooper-pairing in the s-wave symmetry channel, the d-wave projected intraband interaction yields attraction in the d-wave channel. We obtained Tpair from the solution of the linearized mean-field gap equations, whereas Tc in each case is estimated from superfluid stiffness. Details of the calculations are given in the Supplementary Note 4.