Magnetism and its coexistence with superconductivity in CaK(Fe$_{0.949}$Ni$_{0.051}$)$_4$As$_4$: muon spin rotation/relaxation studies

The magnetic response of CaK(Fe$_{0.949}$Ni$_{0.051}$)$_4$As$_4$ was investigated by means of the muon-spin rotation/relaxation. The long-range commensurate magnetic order sets in below the N{e}el temperature $T_{rm N}= 50.0(5)$~K. The density-functional theory calculations have identified three possible muon stopping sites. The experimental data were found to be consistent with only one type of magnetic structure, namely, the long-range magnetic spin-vortex-crystal order with the hedgehog motif within the $ab-$plane and the antiferromagnetic stacking along the $c-$direction. The value of the ordered magnetic moment at $Tapprox3$ K was estimated to be $m_{rm Fe}=0.38(11)$ $mu_{rm B}$ ($mu_{rm B}$ is the Bohr magneton). A microscopic coexistence of magnetic and superconducting phases accompanied by a reduction of the magnetic order parameter below the superconducting transition temperature $T_{rm c}simeq 9$ K is observed. Comparison with 11, 122, and 1144 families of Fe-based pnictides points to existence of correlation between the reduction of the magnetic order parameter at $Trightarrow 0$ and the ratio of the transition temperatures $T_{rm c}/T_{rm N}$. Such correlations were found to be described by Machidas model for coexistence of itinerant spin-density wave magnetism and superconductivity [Machida, J. Phys. Soc. Jpn. 50, 2195 (1981) and Budko et al., Phys. Rev. B 98, 144520 (2018)].