Muon spin rotation and relaxation study on Nb$_{1-y}$Fe$_{2+y}$

We present a detailed study of the magnetic properties of weakly ferromagnetic/quantum critical Nb$_{1-y}$Fe$_{2+y}$ using muon spin rotation and relaxation ($\mu$SR). By means of an angular dependent study of the muon spin rotation signal in applied magnetic fields on a single crystal in the paramagnetic state we establish the muon stopping site in the crystallographic lattice of NbFe$_2$. With this knowledge we develop models to describe the muon spin rotation and relaxation signals in the weakly ferromagnetic, spin density wave and quantum critical phases of Nb$_{1-y}$Fe$_{2+y}$ and fit the corresponding experimental data. In particular, we quantify the $\mu$SR response for quantum critical behavior in Nb$_{1.0117}$Fe$_{1.9883}$ and extract the influence of residual weak structural disorder. From our analysis, Nb$_{1-y}$Fe$_{2+y}$ emerges to be uniquely suited to study quantum criticality close to weak itinerant ferromagnetic order.