In the family of the iron-based superconductors, the $RE$FeAsO-type compounds (with $RE$ being a rare-earth metal) exhibit the highest bulk superconducting transition temperatures ($T_{mathrm{c}}$) up to $55 textrm{K}$ and thus hold the key to the elusive pairing mechanism. Recently, it has been demonstrated that the intrinsic electronic structure of SmFe$_{0.92}$Co$_{0.08}$AsO ($T_{mathrm{c}}=18 textrm{K}$) is highly nontrivial and consists of multiple band-edge singularities in close proximity to the Fermi level. However, it remains unclear whether these singularities are generic to the $RE$FeAsO-type materials and if so, whether their exact topology is responsible for the aforementioned record $T_{mathrm{c}}$. In this work, we use angle-resolved photoemission spectroscopy (ARPES) to investigate the inherent electronic structure of the NdFeAsO$_{0.6}$F$_{0.4}$ compound with a twice higher $T_{mathrm{c}}=38 textrm{K}$. We find a similarly singular Fermi surface and further demonstrate that the dramatic enhancement of superconductivity in this compound correlates closely with the fine-tuning of one of the band-edge singularities to within a fraction of the superconducting energy gap $Delta$ below the Fermi level. Our results provide compelling evidence that the band-structure singularities near the Fermi level in the iron-based superconductors must be explicitly accounted for in any attempt to understand the mechanism of superconducting pairing in these materials.
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