Analysis of the London penetration depth in Ni-doped CaKFe4As4

We report combined experimental and theoretical analysis of superconductivity in CaK(Fe$_{1-x}$Ni$_x$)$_4$As$_4$ (CaK1144) for $x=$0, 0.017 and 0.034. To obtain the superfluid density, $\rho=\left(1+\Delta \lambda_L(T)/\lambda_L(0) \right)^{-2}$, the temperature dependence of the London penetration depth, $\Delta \lambda_L (T)$, was measured by using tunnel-diode resonator (TDR) and the results agreed with the microwave coplanar resonator (MWR) with the small differences accounted for by considering a three orders of magnitude higher frequency of MWR. The absolute value of $\lambda_L (T \ll T_c) \approx \lambda_L(0)$ was measured by using MWR, $\lambda_L (\mathrm{5~K}) \approx 170 \pm 20$ nm, which agreed well with the NV-centers in diamond optical magnetometry that gave $\lambda_L (\mathrm{5~K}) \approx 196 \pm 12$ nm. The experimental results are analyzed within the Eliashberg theory, showing that the superconductivity of CaK1144 is well described by the nodeless s$_{\pm}$ order parameter and that upon Ni doping the interband interaction increases.