Extraordinary Hall Effect in (Ni80Fe20)x(SiO2)1-x Thin Films

The extraordinary Hall effect (EHE) in ferromagnetic samples is generally attributed to scatterings of iterant electrons in the presence of spin-orbit interactions. In this work, study on the thickness dependence of the EHE in the $(\mbox{Ni}_{\mbox{80}} \mbox{Fe}_{\mbox{20}} )_x (\mbox{SiO}_{\mbox{2}})_{1-x} $ system showed the spontaneous Hall resistivity, $\rho _{sy}^S$, to be a constant while the Hall coefficient, $R_S (\equiv \rho _{sy}^S /M_S$ where $M_S $ is the saturated magnetization) increased monotonically owing to a depression in $M_S $. We propose the constancy of $\rho _{sy}^S $ with reducing thickness to arise from the sample morphological structure becoming two-dimensional with decreasing film thickness, expected from classical percolation theory. We also find in this system with varying $x$ that $\rho _{sy}^S \propto \rho _{xx}^\gamma $, with $\gamma =0.53$ to 1 in disagreement with the value 2 frequently attributed to the side jump effect, but explainable in terms of the more general form $\rho _{sy}^S =\rho _{xx} \Delta y_e /\Lambda _{SO} $, where $\Delta y_e $ is the side jump displacement and $\Lambda _{SO} $ the spin-orbit mean-free-path.