Absence of the ordinary and extraordinary Hall effects scaling in granular ferromagnets at metal-insulator transition

Universality of the extraordinary Hall effect scaling was tested in granular three-dimensional ${\mathrm{Ni}\text{−}\mathrm{SiO}}_2$ films across the metal-insulator transition. Three types of magnetotransport behavior have been identified: metallic, weakly insulating, and strongly insulating. Scaling between both the ordinary and the extraordinary Hall effects and material's resistivity is absent in the weakly insulating range characterized by logarithmic temperature dependence of conductivity. The results provide compelling experimental confirmation for recent models of granular metals predicting transition from logarithmic to exponential conductivity temperature dependence when intergranular conductance drops below the quantum conductance value and loss of Hall effect scaling when intergranular conductance is higher than the quantum one. The effect was found at high temperatures and reflects the granular structure of the material rather than low-temperature quantum corrections.

Figure 1

(a) The saturated extraordinary Hall effect resistivity ${\rho }_{\mathrm{EHE}}$ measured in a series of ${\mathrm{Ni}}_x\text{−}{\left({\mathrm{SiO}}_2\right)}_{1-x}$ samples as a function of resistivity at room temperature (solid circles) and at 77 K (open circles). Ni volume concentration $x$ ranges from 0.72 down to 0.2. The arrow indicates the resistivity of the material for which the mean intergranular (intercluster) junction resistance equals quantum resistance $R_Q$ (see the text). (b) The same data in the $\sigma$ presentation in which ${\sigma }_{\mathrm{EHE}}$ is shown as a function of the material's conductivity.

Figure 2

(a) Resistivity of a series of ${\mathrm{Ni}}_x\text{−}{\left({\mathrm{SiO}}_2\right)}_{1-x}$ samples as a function of temperature. The data are normalized at 77 K. Indices indicate Ni concentration. (b) Normalized conductivity of two plateau range samples as a function of logarithm of temperature. (c) Logarithm of resistivity of “beyond the plateau” samples as a function of $T^{-1/4}$.

Figure 3

Absolute values of the room-temperature ordinary Hall effect coefficient $R_{\mathrm{OHE}}$ and the EHE resistivity ${\rho }_{\mathrm{EHE}}$ as a function of resistivity. $R_{\mathrm{OHE}}$ of the highest resistivity sample beyond the plateau is multiplied by a factor of −0.5. Inset: $R_{\mathrm{OHE}}$ as a function of ${\rho }_{\mathrm{EHE}}$ for samples with resistivity below 1 Ω cm. The straight line is a guide to the eye.