Features of the scaling of the anomalous Hall effect in (CoFeB)$_x$(LiNbO$_3)_{100-x}$ nanocomposite films below the percolation threshold: manifestation of the cotunneling Hall conductance?

A scaling behavior of the anomalous Hall effect resistivity $\rho_{\mathrm{AHE}}$ versus the longitudinal resistivity $\rho$ in (CoFeB)$_x$(LiNbO$_3)_{100-x}$ nanocomposites with a low content of dispersed Co and Fe atoms ($N_d\sim4\times 10^{20}$cm$^{-3}$) in an amorphous LiNbO$_3$ matrix is studied in the range $x= 40$–$48$ at $\%$ below the percolation threshold $x_p\approx 49$ at $\%$. A logarithmic temperature dependence of the conductivity $\sigma\sim \ln T$ has been observed in the range $x\approx 44{-}48$ at $\%$, which is transformed in the range $x\approx 40{-}42$ at $\%$ to a square root law $\ln\sigma\propto- (T_0/T)^{1/2}$, which is characteristic of cotunneling transport processes in nanocomposites. It has been found that the exponent $n\approx 0.24$ in the scaling law $\rho_{\mathrm{AHE}}/x\propto [\rho(x)]^n$ coincides within an accuracy of $5\%$ with the exponent n in a similar dependence for nanocomposites based on the (CoFeB)$_x$(LiNbO$_3)_{100-x}$ matrix with a high content $N_d\sim10^{21}{-}10^{22}$cm$^{-3}$ and with the exponent $n$ in a parametric dependence $\rho_{\mathrm{AHE}}\propto [\rho(T)]^n$ for the samples with the minimum content $x\approx40$ at $\%$. The found features are attributed to the correlated change in the probability of cotunneling transitions in a set of more than three centers under the effect of spin–orbit coupling. The manifestation of the barrier tunneling anomalous Hall effect at granule interfaces is also possible.