Pseudogap and strong pairing induced by incipient and shallow bands in quasi-two-dimensional KCa2Fe4As4F2

The optical properties of $\mathrm{K}{\mathrm{Ca}}_{2}{\mathrm{Fe}}_{4}{\mathrm{As}}_{4}{\mathrm{F}}_{2}$ (K12442, ${T}_{c}=33.5$ K) and $\mathrm{K}{\mathrm{Ca}}_{2}{({\mathrm{Fe}}_{0.95}{\mathrm{Ni}}_{0.05})}_{4}{\mathrm{As}}_{4}{\mathrm{F}}_{2}$ (Ni-K12442, ${T}_{c}=29$ K) have been examined at a large number of temperatures. For both samples, a nodeless superconducting gap is clearly observed in the optical conductivity at 5 K. The superconducting gap $\mathrm{\ensuremath{\Delta}}\ensuremath{\simeq}8.7$ meV ($2\mathrm{\ensuremath{\Delta}}/{k}_{\text{B}}{T}_{c}\ensuremath{\simeq}6.03$) in K12442, pointing towards strong-coupling Cooper pairs, but in sharp contrast, $\mathrm{\ensuremath{\Delta}}\ensuremath{\simeq}4.6$ meV ($2\mathrm{\ensuremath{\Delta}}/{k}_{\text{B}}{T}_{c}\ensuremath{\simeq}3.68$) in Ni-K12442, which agrees with the BCS weak-coupling pairing state. More intriguingly, below ${T}^{*}\ensuremath{\simeq}75$ K, the optical conductivity of K12442 reveals a pseudogap that smoothly evolves into the superconducting gap below ${T}_{c}$, while no such behavior is detected in the electron-doped Ni-K12442. The comparison between the two samples suggests that the pseudogap and strong-coupling Cooper pairs in K12442 may be intimately related to the shallow and incipient bands. We provide arguments supporting a preformed pairing mechanism of the pseudogap, but at the moment a magnetic scenario cannot yet be excluded.