Multiband superconductivity and a deep gap minimum evidenced by specific heat in KCa2_2(Fe1−x_{1-x}Nix_x)4_4As4_4F2_2

Specific heat can explore low-energy quasiparticle excitations of superconductors, so it is a powerful tool for bulk measurement on the superconducting gap structure and pairing symmetry. Here, we report an in-depth investigation on the specific heat of the multiband superconductors KCa$_2$(Fe$_{1-x}$Ni$_x$)$_4$As$_4$F$_2$ ($x$ = 0, 0.05, 0.13) single crystals and the overdoped non-superconducting one with $x$ = 0.17. For the samples with $x$ = 0 and $x$ = 0.05, the magnetic field induced specific heat coefficient $\Delta\gamma(H)$ in the low temperature limit increases rapidly below 2 T, then it rises slowly above 2 T. Using the non-superconducting sample with $x$ = 0.17 as a reference, and applying a mixed model that combines Debye and Einstein modes, the specific heat of phonon background for various superconducting samples can be fitted and the detailed information of the electronic specific heat is obtained. Through comparative analyses, it is found that the energy gap structure including two $s$-wave gaps and an extended $s$-wave gap with large anisotropy can reasonably describe the electronic specific heat data. According to these results, we suggest that at least one anisotropic superconducting gap with a deep gap minimum should exist in this multiband system. With the doping of Ni, the $T_c$ of the sample decreases along with the decrease of the large $s$-wave gap, but the extended $s$-wave gap increases due to the enlarged electron pockets via adding more electrons. Despite these changes, the general properties of the gap structure remain unchanged versus doping Ni. In addition, the calculation of condensation energy of the parent and doped samples shows the rough consistency with the correlation of $U_0 \propto {T_c}^n$ with $n$ = 3-4, which is beyond the understanding of the BCS theory