Abstract
The BCSBEC (BardeenCooperSchrieffer–BoseEinsteincondensate) crossover from strongly overlapping Cooper pairs to nonoverlapping composite bosons in the strong coupling limit has been a longstanding issue of interacting manybody fermion systems. Recently, FeSe semimetal with hole and electron bands emerged as a hightransitiontemperature (highT_{c}) superconductor located in the BCSBEC crossover regime, owing to its very small Fermi energies. In FeSe, however, an ordinary BCSlike heatcapacity jump is observed at T_{c}, posing a fundamental question on the characteristics of the BCSBEC crossover. Here we report on highresolution heat capacity, magnetic torque, and scanning tunneling spectroscopy measurements in FeSe_{1−x}S_{x}. Upon entering the tetragonal phase at x > 0.17, where nematic order is suppressed, T_{c} discontinuously decreases. In this phase, highly nonmeanfield behaviours consistent with BEClike pairing are found in the thermodynamic quantities with giant superconducting fluctuations extending far above T_{c}, implying the change of pairing nature. Moreover, the pseudogap formation, which is expected in BCSBEC crossover of singleband superconductors, is not observed in the tunneling spectra. These results illuminate highly unusual features of the superconducting states in the crossover regime with multiband electronic structure and competing electronic instabilities.
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Introduction
The crossover between the weakcoupling Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity and strongcoupling Bose–Einsteincondensate (BEC) describes the fundamentals of quantum bound states of particles, including information on the superconducting critical temperature (T_{c}) and the possible pseudogap formation^{1,2,3}. In the strongcoupling regime, the pairformation temperature T_{pair} and the actual superconducting transition temperature T_{c} are distinctly separated (Fig. 1a). Between T_{pair} and T_{c}, the socalled preformed Cooper pairs exist, which give rise to large superconducting fluctuations and possibly a depletion of the lowenergy density of states (DOS), namely the pseudogap. Almost all experimental studies on crossover physics have been performed in ultracold atomic systems for the past decades^{2} 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 highT_{c} cuprates, the pseudogap formation has been discussed in terms of the BCS–BEC crossover^{4}, 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 unresolved^{5}. On the other hand, very recent studies have successively pointed to the realization of BCS–BEC crossover in superconductivity in other strongly correlated compounds^{6,7,8,9}.
The superconducting semimetal FeSe^{10,11,12} with very small hole and electron Fermi surfaces are found to exhibit large ratios of the superconducting energy gap Δ and Fermi energy E_{F}, Δ/E_{F} ≈ 0.3–1.0^{7,13,14}, which appears to place FeSe in the crossover regime (Fig. 1a). Indeed, in highquality single crystals with long mean free paths of electrons^{7}, large superconducting fluctuations have been reported from the torque magnetometry, providing support for the presence of fluctuated Cooper pairs created above T_{c} in the crossover regime^{15}. These observations underline the importance of the BCSBEC 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 T_{c}^{16}.
These results imply that the superconductivity in FeSe is not simply described by a picture of the conventional BCSBEC 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/E_{F}. In fact, the Fermi surface of FeSe is composed of separated holes and electron pockets with different values of Δ/E_{F}^{7,13,17}. It has been pointed out that in multiband systems, the developments of superconducting fluctuations and pseudogap formation are sensitively affected by the interband coupling strength^{18,19,20}, indicating that the normal and superconducting properties associated with the crossover in multiband systems can be different from those in the single band systems^{21}. It is quite important, therefore, to clarify how the superconducting properties evolve when the multiband electronic structure is tuned.
Here, we focus on FeSe_{1−x}S_{x}, 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 x_{c} ≈ 0.17^{22,23}. The superconductivity persists in the tetragonal (nonnematic) phase above x_{c}. In the case of applying hydrostatic pressure, the suppression of nematic order is accompanied by the pressureinduced antiferromagnetic order^{24}. In contrast, S substitution suppresses the nematic order without inducing the antiferromagnetism that significantly changes the Fermi surface through band folding, even when crossing x_{c}. 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 ≈ 0.19^{25} demonstrate that the S ions do not act as strong scattering centers.
By using highresolution heat capacity, magnetic torque, and scanning tunneling spectroscopy measurements, we clarify the unusual evolution of the thermodynamic and spectroscopic properties in the FeSe_{1−x}S_{x} system, where the highly nonmeanfield 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 highresolution heat capacity, magnetic torque, and STS measurements on highquality single crystals, we investigate the evolution of superconducting and normal state properties with x in FeSe_{1−x}S_{x}. In Fig. 1b, we show the phase diagram obtained by the heat capacity measurements. The drastic change in T_{c} and superconducting fluctuations are found across x_{c}. 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 Tdependence of electronic heat capacity divided by T, C_{e}/T, for x = 0, 0.05, 0.10, and 0.13. Here we subtract the phonon contribution, which shows T^{3}dependence, as confirmed by the measurements in the normal state at high fields. In Fig. 2b, C_{e}/T(T) in FeSe (x = 0) exhibits an ordinary BCSlike jump at T_{c} = 9 K, consistent with the previous studies on larger samples^{26,27,28}. As the temperature is lowered below T_{c}, C_{e}/T decreases with decreasing slope. Below ~3 K, C_{e}/T shows Tlinear dependence, which is consistent with the strongly anisotropic superconducting gap^{7,28,29}. Similar C_{e}/T(T) is observed in orthorhombic phase x = 0.05, 0.10, and 0.13 with T_{c} of 9.7, 10, and 9.5 K, respectively (Fig. 2c–e).
Upon entering the tetragonal phase at x > x_{c}, C_{e}/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, C_{e}/T increases with the increasing slope in the normal state when approaching T_{c} from high temperatures. In addition, C_{e}/T exhibits a kink at T_{c} without showing a discontinuous jump. Such a continuous Tdependence at T_{c} is reminiscent of the BEC transition in the free Bose gas systems^{30}, although it is too simple to employ the free Bose gas model in our system. We note that a similar enhancement of C_{e}/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 regime^{31}. As the temperature is lowered below T_{c}, C_{e}/T(T) for x = 0.20 decreases with increasing slope, in stark contrast to C_{e}/T(T) in the orthorhombic crystals, as shown in Fig. 2b–e. Below T_{c}, C_{e}/T decreases sublinearly to T with large residual C_{e}/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 measurements^{32,33}. Similar C_{e}/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 T_{c} across x_{c}, 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 T_{c} has also been observed when nematicity is suppressed by pressure^{34}.
We stress that the observed peculiar C_{e}/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 FeSe_{1−x}S_{x} crystals reveals no evidence for the segregation of S atoms, indicating an excellent homogeneity^{33} (see also Fig. 3a, b). Second, the observation of the quantum oscillations of the crystals in the same batch^{25} 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 Xray 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 C_{e}/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 FeSe_{1−x}S_{x}. This is supported by the C_{e}/T(T) of the crystal at x ≈ 0.16, which is located in the very vicinity of x_{c}. As shown in Fig. 2f, C_{e}/T at x ≈ 0.16 exhibits two wellseparated peaks at T_{c1} = 4 K and T_{c2} = 7 K, indicating a phase separation (due to the small spatial variation of Δx ~ 0.01) at x_{c}. This Tdependence is well reproduced by the sum of C_{e}/T of x = 0.13 (T_{c} = 9.5 K) and 0.18 (T_{c} = 4.0 K), assuming that both contribute equally. Here, for the fitting, T_{c} values are shifted slightly. These results indicate a discontinuous change of T_{c} at x_{c}. Such a discontinuous change in the superconducting properties is consistent with the jump in Δ values at x_{c} recently reported by STS studies^{33}. The observed enhancement of C_{e}/T above T_{c} for the sample in the tetragonal phase, therefore, can be attributed to an intrinsic electronic property.
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 report^{33}. 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/T_{c} = 0.1). This large residual DOS is consistent with the large C_{e}/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 Tdependence of the gap depth plotted in Fig. 3d. The superconducting gap closes at ≈ 5 K, below which the Tdependence of magnetization shows a rapid decrease, as shown in Fig. 3e.
Nearly energyindependent DOS at E_{F} observed by STS above T_{c} indicates that the observed enhancement of C_{e}/T above T_{c} in the tetragonal phase is not caused by the DOS effect. Therefore, it is natural to consider that the enhancement of C_{e}/T stems from the superconducting fluctuations. Indeed, a similar enhancement of C_{e}/T toward T_{c} due to the strong fluctuations has been previously reported in other ironbased superconductors^{35,36,37}. In order to discuss the effect of the superconducting fluctuations in FeSe_{1−x}S_{x} quantitatively, we extract the superconducting contribution C_{sc}, which is obtained by subtracting the normal state electronic contribution γT from C_{e} for x = 0 (blue circles in Fig. 4a) and x = 0.20 (red circles in Fig. 4b). We compare C_{sc} with the conventional term of the meanfield Gaussian fluctuations^{38} (Supplementary Note 3). Obviously, the heat capacity contribution that significantly exceeds C_{Gauss} can be seen for x = 0.20. This extra heat capacity is observable up to t ~ 0.8. In Fig. 4c, we display the xdependence of ΔC/γT_{c}, where ΔC is the height of C_{sc} at T_{c}. For x < x_{c}, the magnitude of ΔC/γT_{c} is close to the BCS weak coupling value (1.43 for swave and 0.94 for dwave). On the other hand, ΔC/γT_{c} for x > x_{c} is reduced far below the meanfield BCS value. As a consequence of the unusual suppression of ΔC/γT_{c}, the lowT C_{e}/T is largely enhanced through the entropy balance. Figure 4d depicts the xdependence of the C_{e}/T at T/T_{c} = 0.1 taken from Fig. 2. Remarkable enhancement of C_{e}/T at x_{c} indicates the fundamental difference of the lowenergy excitations across x_{c}. We note that recent NMR measurements for x > x_{c} report no splitting of the spinecho signal in the crystal from the same batch^{39,40}, which excludes the microscopic phase separation of superconducting and nonsuperconducting regions.
Magnetic torque
The anomalous C_{e}/T in the tetragonal phase is reflected by the entropies in the normal and superconducting states, S_{n} and S_{s}, respectively, as shown in Fig. 4e, f. For x = 0, the Tdependence of S_{s} shows a kink at T_{c}, which is typical for the secondorder superconducting transition. For x = 0.20, on the other hand, owing to the absence of the jump in C_{e}/T at T_{c}, no kink anomaly is observed in the Tdependence of S_{s}. Moreover, the entropy is significantly suppressed from S_{n} even above T_{c}. In Fig. 4g, we show the x dependence of the lost entropy normalized by S_{n}, (S_{n} − S_{s})/S_{n}, just above T_{c}, T = 1.03T_{c}. Upon entering the tetragonal phase, an abrupt enhancement of (S_{n} − S_{s})/S_{n} 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 microcantilevers. The torque can sensitively detect the diamagnetic response through the anisotropy of magnetic susceptibility^{15}. By measuring the field angular variation of torque as a function of polar angle θ from the c axis, the anisotropy Δχ = χ_{c} − χ_{ab} is determined, where χ_{c} and χ_{ab} 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 Hdependence at low fields, as shown in Fig. 4h. It is well settled that the fluctuationinduced diamagnetic susceptibility of most superconductors, including multiband systems, can be well described by the standard Gaussiantype (Aslamasoz–Larkin, AL) fluctuation susceptibility χ_{AL} (Supplementary Note 3). Here we focus on the Hdependent diamagnetic susceptibility at low fields^{15}. Then, Δχ at low fields is much larger than the ∣χ_{AL}∣, which is added to high field 7 T data. This implies that the superconducting fluctuations in FeSe_{1−x}S_{x} are distinctly different from those in conventional superconductors, supporting the fluctuation effects observed in heat capacity measurements. We stress that although the multigap superconductivity may lead to a small value of ΔC/γT, it cannot give rise to the nonmeanfield behaviors observed in heat capacity and magnetic susceptibility above T_{c}.
Unusual crossover from BardeenCooperSchrieffer to BoseEinsteincondensate superconductivity
In FeSe, while large superconducting fluctuations that well exceed the Gaussian fluctuations are observed, the C_{e}/T around T_{c} does not exhibit a significant deviation from the meanfield behavior. This suggests that tetragonal FeSe_{1−x}S_{x} are closer to the BEC regime. However, with increasing S composition, the volume of the Fermi surface increases, as reported by quantum oscillation experiments^{25}, and Δ is reduced, according to STM measurements^{33}. Then Δ/E_{F} is expected to become smaller with increasing S concentration, which is opposite to the tendency of approaching the BEC regime in the conventional BCSBEC crossover. In fact, recent ARPES studies have reported BEClike behavior in the tetragonal FeSe_{1−x}S_{x}^{41}, which is in good agreement with our thermodynamic studies. These behaviors are in contrast to the BEClike dispersion observed for Fe_{1+y}Se_{x}Te_{1−x}^{14} with increasing Δ/E_{F}, as expected in the context of conventional BCS–BEC crossovers. On the other hand, in our case of FeSe_{1−x}S_{x}, 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 Fe_{1+y}Se_{x}Te_{1−x} system and our FeSe_{1−x}S_{x} system, leading to a change in the band structure and a significant change in pairing interactions. Another difference is that FeSe_{1−x}S_{x} holds the compensation condition, where both electron and hole Fermi energies are comparably small, whereas the Fe nonstoichiometry in Fe_{1+y}Se_{x}Te_{1−x} leads to the imbalance of Fermi energies of two carriers. This suggests that there is another important factor in the FeSe_{1−x}S_{x} system which has not been duly taken into consideration.
A key factor that controls the BCSBEC crossover in FeSe_{1−x}S_{x} is the multiband character. As shown previously^{16,21}, in FeSe, the strong interband interaction between electron and hole pockets prevents the splitting of the T_{pair} and the T_{c}, determined by the superfluid stiffness. We note that this strong interband interaction is also an important ingredient in the orbitalselective scenario^{29,42,43,44}, and therefore we expect that even taking orbital selectivity into account, the main result, i.e., meanfieldlike 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 x_{c} in FeSe_{1−x}S_{x}. Furthermore, recent experimental^{45} and theoretical^{46,47} works pointed out an interesting possibility of the substantial nonlocal nematic order on the d_{xy} orbital and/or interorbitalnematicity between d_{xy} and d_{yz} orbitals in FeSe. The most important consequence of this phenomenon would be an additional Lifshitz transition in FeSe_{1−x}S_{x} upon S substitution, which occurs close to x_{c}. This, in turn, implies the appearance of the incipient band near x_{c} (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 BCSBEC crossover. In addition, given a nearly equal mixture of the s and dwave symmetry components of the superconducting gap in FeSe^{48} to express such an anisotropic gap, it is natural to assume that they are competing in the tetragonal phase of FeSe_{1−x}S_{x}. In particular, the ratio of the pairing interactions in the dwave and swave channels \( {g}_{{{{{{{{\rm{ee}}}}}}}}}^{d} /{g}_{{{{{{{{\rm{eh}}}}}}}}}^{s}\) may, in addition to the incipient band, be another key parameter of the BCSBEC crossover, as follows from our model calculations (Supplementary Note 4). The pairformation temperature T_{pair} and condensation temperature T_{c} 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 E_{F}. This quantitatively explains the appearance of strong coupling superconductivity for x > x_{c} with no apparent enhancement of Δ/E_{F}. A remarkable and unexpected feature is the absence of pseudogap above T_{c} 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 ultracold atoms^{2}. In addition, the onset temperature of the pseudogap may differ from that of fluctuations^{31}. Moreover, it has been pointed out that the pseudogap phenomena in multiband systems are markedly altered from those in singleband systems. For instance, in multiband systems, the pseudogap formation is less prominent by the reduction of its onset temperature^{49}. In a recent study, the BEClike dispersion of the Bogoliubov quasiparticles has been reported in the tetragonal FeSe_{1−x}S_{x} by using ARPES measurements^{41}. 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 orbitaldependent.
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 stiffness^{50}, a Lifishitz transition near the orthorhombic to tetragonal transition^{46}, or nematic fluctuations. In FeSe_{1−x}S_{x}, the nematic fluctuations peaked at q_{nem} ≈ 0 because it is ordered state breaks the rotational symmetry while preserving translational symmetry. Enhanced nematic fluctuations^{22} play an important role in the normal state properties^{23,51}, particularly at x ~ x_{c}^{23}. It has also been pointed out theoretically that nematic fluctuations strongly influence the superconductivity^{52,53}. Therefore, it is tempting to consider that the nematic fluctuations and incipient bands in FeSe_{1−x}S_{x} 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 multiband nature affect the crossover physics.
Conclusion
Our thermodynamic studies have revealed a highly nonmeanfield behavior of superconducting transition in FeSe_{1−x}S_{x} when the nematic order of FeSe is completely suppressed by S substitution. This is consistent with the recent ARPES measurements, which report the BEClike dispersion of the Bogoliubov quasiparticles in the tetragonal regime. Our findings demonstrate that FeSe_{1−x}S_{x} 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 FeSe_{1−x}S_{x} 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 energydispersive Xray spectroscopy.
Heat capacity
The heat capacity of the crystals is measured using the long relaxation method^{54,55} 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 silvercoated 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 microcantilevers^{15}. 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 fieldangle dependence of torque is obtained by rotating the whole cryostat within the ac(bc) plane. Magnetic torque τ can be expressed as τ = μ_{0}VM × H, where μ_{0} 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 caxis. 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 = χ_{c} − χ_{ab} is anisotropy of magnetic susceptibility between the ac(ab) plane and the caxis.
Scanning tunneling microscopy/spectroscopy
The scanning tunneling microscopy/spectroscopy measurements have been performed with a commercial lowtemperature STM system. Fresh and atomically flat surfaces are obtained by in situ cleavage at liquid nitrogen temperature in an ultrahigh vacuum. All conductance spectra and topographic images are obtained by a PtIr tip. Conductance spectra are measured by a lockin 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 2ndorder polynomial fitting using outside superconducting gaps.
Theoretical calculation
The theoretical calculations were done within a simplified interacting twoband model in two dimensions, employed previously^{21} with hole and electron pockets with small Fermi energies. We assume superconductivity is due to repulsive interactions. While the dominant interband interaction favors Cooperpairing in the swave symmetry channel, the dwave projected intraband interaction yields attraction in the dwave channel. We obtained T_{pair} from the solution of the linearized meanfield gap equations, whereas T_{c} in each case is estimated from superfluid stiffness. Details of the calculations are given in the Supplementary Note 4.
Data availability
Most of the data in this study are available in the paper and supplementary information files. The other data, including raw data, can be provided upon request to the authors (Y.M.).
Code availability
The code of the analysis in this study is included in the supplementary information file.
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Acknowledgements
The authors thank A. Carrington, L. Malone, P. Walmsley, K. Ishida, K. Sugii, T. Taen, and T. Osada for experimental assistance. We also thank Y. Tada, H. Ikeda, K. Adachi, Y. Ohashi, and D. Inotani for their valuable discussion. This work has been supported by KAKENHI (Nos. JP23H01829, JP22H00105, JP22KK0036, JP21H04443, JP21KK0242, JP20H02600, JP20K21139, JP19H00649, JP18H05227, and JP18K13492), GrantinAid for Scientific Research on Innovative Areas “Quantum Liquid Crystals” (No. JP19H05824) from JSPS, and CREST (No. JPMJCR19T5) from Japan Science and Technology (JST). The work of J.B. and I.E. is supported by the German Research Foundation within the bilateral NSFCDFG Project ER 463/141. This work was partially carried out by the joint research in the Institute for Solid State Physics, the University of Tokyo.
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Y. Mizukami and T.S. conceived and supervised the project. Y. Mizukami and O.T. developed the setup of heat capacity measurement and performed heat capacity measurements. M.H. performed STM/STS measurements. Y. Mizukami, K.M., D.S., and S.K. synthesized highquality single crystals. J.B. and I.E. performed theoretical analysis. All authors discuss the results. Y. Mizukami prepared the paper with inputs from I.E., Y. Matsuda, and T.S.
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Mizukami, Y., Haze, M., Tanaka, O. et al. Unusual crossover from BardeenCooperSchrieffer to BoseEinsteincondensate superconductivity in iron chalcogenides. Commun Phys 6, 183 (2023). https://doi.org/10.1038/s42005023012898
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DOI: https://doi.org/10.1038/s42005023012898
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