Disorder, itinerant ferromagnetism and the anomalous Hall effect in two dimensions

This talk will describe research motivated by the lack of consensus on what happens in a band ferromagnet such as iron when the itinerancy of the electrons, which carry spin information, is compromised by disorder. We address this challenging problem by performing \textit{in situ} studies of magnetotransport in a series of films having sheet resistances varying from 50 to 1,000,000 Ohms. In the weakly disordered regime of this two-dimensional system, where the quantum corrections to the conductivity have logarithmic temperature dependence, we find a surprising scaling of the longitudinal and anomalous Hall (transverse) resistances. For higher disorder the scaling breaks down and the anomalous Hall resistance $R_{xy}$ saturates at a constant value near 100 Ohms. These results imply the presence of an \textit{anomalous Hall insulating} state where the longitudinal $L_{xx}$ and transverse $L_{xy}$ conductivities approach zero with a ratio $R_{xy}~=~L_{xy}/L_{xx}^{2}$ that remains constant.