Competing pairing interactions responsible for the large upper critical field in a stoichiometric iron-based superconductor, CaKFe$_4$As$_4$

The upper critical field of multiband superconductors is an important quantity that can reveal the details about the nature of the superconducting pairing. Here we experimentally map out the complete upper critical field phase diagram of a stoichiometric superconductor, CaKFe$_4$As$_4$, up to 90T for different orientations of the magnetic field and at temperatures down to 4.2K. The upper critical fields are extremely large, reaching values close to ~3$T_c$ at the lowest temperature, and the anisotropy decreases dramatically with temperature leading to essentially isotropic superconductivity at 4.2K. We find that the temperature dependence of the upper critical field can be well described by a two-band model in the clean limit with band coupling parameters favouring intraband over interband interactions. The large Pauli paramagnetic effects together with the presence of the shallow bands is consistent with the stabilization of an FFLO state at low temperatures in this clean superconductor.