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μSR investigation of the Fe-doped Ca$_{3}$Ru$_{2}$O$_{7}$ polar metal

Ca$_{3}$Ru$_{2}$O$_{7}$ is a polar metal that belongs to the class of multiferroic magnetic materials. Here, tiny amounts of Fe doping in the Ru sites bring about dramatic changes in the electronic and magnetic properties and generate a complex H-T phase diagram. To date, not much is known about the ground state of such a system in the absence of magnetic field. By performing muon-spin spectroscopy ($μ$SR) measurements in 5% Fe-doped Ca$_{3}$Ru$_{2}$O$_{7}$ single crystals, we investigate its electronic properties at a local level. Transverse-field $μ$SR results indicate a very sharp normal-to-antiferromagnetic transition at T$_{N}$ = 79.7(1) K, with a width of only 1 K. Zero-field $μ$SR measurements in the magnetically ordered state allow us to determine the local fields B$_{i}$ at the muon implantation sites. By symmetry, muons stopping close to the RuO$_{2}$ planes detect only the weak nuclear dipolar fields, while those stopping next to apical oxygens sense magnetic fields as high as 150 mT. In remarkable agreement with the nominal Fe-doping, a $\sim$ 6% minority of the these muons feel slightly lower fields, reflecting a local magnetic frustration induced by iron ions. Finally, B$_{i}$ shows no significant changes across the metal-to-insulator transition, close to 40 K. We ascribe this surprising lack of sensitivity to the presence of crystal twinning.