Giant superconducting fluctuations in the compensated semimetal FeSe at the BCS–BEC crossover

The physics of the crossover between weak-coupling Bardeen–Cooper–Schrieffer (BCS) and strong-coupling Bose–Einstein condensate (BEC) limits gives a unified framework of quantum-bound (superfluid) states of interacting fermions. This crossover has been studied in the ultracold atomic systems, but is extremely difficult to be realized for electrons in solids. Recently, the superconducting semimetal FeSe with a transition temperature Tc=8.5 K has been found to be deep inside the BCS–BEC crossover regime. Here we report experimental signatures of preformed Cooper pairing in FeSe, whose energy scale is comparable to the Fermi energies. In stark contrast to usual superconductors, large non-linear diamagnetism by far exceeding the standard Gaussian superconducting fluctuations is observed below T*∼20 K, providing thermodynamic evidence for prevailing phase fluctuations of superconductivity. Nuclear magnetic resonance and transport data give evidence of pseudogap formation at ∼T*. The multiband superconductivity along with electron–hole compensation in FeSe may highlight a novel aspect of the BCS–BEC crossover physics.

a, Temperature dependence of the Nernst coefficient ν in the zero-field limit (blue circles) compared with the expected superconducting fluctuation contribution calculated from the AL theory Eq. (S1).
b, ν/T as a function of temperature. c, Seebeck coefficient divided by temperature S/T as a function of temperature at zero field. Dashed lines are the linear fits at high temperatures. The arrows mark the psudogap temperature T * . in the main text. We note that the torque signal above T c is completely reversible without hysteresis, which excludes any ferromagnetic impurity as a source for the enhanced |∆χ| below T * .

SUPPLEMENTARY NOTE 2. DOMINANT QUASIPARTICLE CONTRIBUTION IN NERNST SIGNAL
We estimate the Gaussian-type superconducting fluctuation contribution to the Nernst signal by ν AL ≈ α AL xy ρ xx /(µ 0 H), where α AL xy is the Peltier coefficient [30]. In the Aslamasov-Lakin (AL) theory, this coefficient is given by whereξ i (T ) = ξ i (0)/ ln(T /T c ) (i = ab or c) are the fluctuation coherence lengths parallel to the ab plane and c axis, and ℓ H = /2eµ 0 H is the magnetic length. The red line in Supplementary Fig. 4a represents the AL calculation for the superconducting fluctuation contribution ν AL (T ) by using the resistivity data of the same sample and Eq. (S2).
Just above T c , the measured data are almost four orders of magnitude larger than the AL estimate. Although the diamagnetic signal is one order of magnitude larger than the AL estimate (Fig. 2f), both the four-orders-of-magnitude difference and the absence of a diverging behaviour near T c in ν(T ) demonstrate that in our samples of FeSe the normal quasiparticle contribution dominates over the superconducting fluctuation contribution in the normal state.
As shown in Supplementary Figs. 4b and 4c, ν/T and S/T deviate from the hightemperature linear extrapolation below T * ∼ 20 K. At lower temperatures, ν/T and |S|/T both decrease with decreasing T before superconductivity sets in, which is not expected in a Fermi-liquid metal where ν/T and S/T should become constant in the low-temperature limit.

ITY AND NERNST SIGNAL
In our clean crystals, a huge magnetoresistance is observed as shown in Fig. 1a of the main text. Due to the large value of ω c τ , where ω c is the cyclotron frequency and τ is the scattering time, ρ xx under strong magnetic fields exhibits a semiconducting behaviour at low temperatures. In strong fields, superconducting fluctuations, or excess conductivity, are observed more clearly as they compete with the increase of ρ xx . To see this, we plot the temperature derivative of the in-plane resistivity, dρ xx /dT , for several magnetic fields ( Supplementary Fig. 5a). For each field, dρ xx /dT exhibits a clear minimum at T * ∼ 20 K, indicating a slope change due to the emergence of superconducting fluctuations.
Supplementary Figure 5b shows the temperature dependence of the Nernst coefficient for different fields. Similar to dρ xx /dT , ν(T ) at each field exhibits a clear peak at T * ∼ 20 K, suggesting a change of the energy dependence of the scattering time at the Fermi level. We note that the anomalies in dρ xx /dT and ν(T ), as well M dia are less sensitive to magnetic field, suggesting that they represent the amplitude fluctuations, while the non-linear behaviour in magnetic torque is readily suppressed by a strong magnetic field. This suggests that the non-linear diamagnetic response in torque is directly related to the phase fluctuations arising from the mode coupling of superconducting fluctuations.