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Conventional superconductivity follows Bardeen-Cooper-Schrieffer(BCS) theoryof electrons-pairing in momentum-space, while superfluidity is theBose-Einstein condensation(BEC) of atoms paired in real-space. These propertiesof solid metals and ultra-cold gases, respectively, are connected by theBCS-BEC crossover. Here we investigate the band dispersions inFeTe$_{0.6}$Se$_{0.4}$($T_c$ = 14.5 K $\sim$ 1.2 meV) in an accessible rangebelow and above the Fermi level($E_F$) using ultra-high resolution laserangle-resolved photoemission spectroscopy. We uncover an electron band lyingjust 0.7 meV ($\sim$ 8 K) above $E_F$ at the $\Gamma$-point, which shows asharp superconducting coherence peak with gap formation below $T_c$. Theestimated superconducting gap $\Delta$ and Fermi energy $\epsilon_F$ indicatecomposite superconductivity in an iron-based superconductor, consisting ofstrong-coupling BEC in the electron band and weak-coupling BCS-likesuperconductivity in the hole band. The study identifies the possible route toBCS-BEC superconductivity.

Superconductivity in an electron band just above the Fermi level: possible route to BCS-BEC superconductivity