Scaling of the Hall effects beyond the quantum resistance threshold in oxidized CoFeB

The ordinary and the extraordinary Hall effects were studied in gradually oxidized amorphous CoFeB ferromagnets over six orders of resistivity from the metallic to the strongly insulating regime. Polarity of the extraordinary Hall effect reverses, and the amplitude of both the ordinary and the extraordinary Hall effects increases quadratically with resistivity when resistance exceeds the quantum resistance threshold $h/2e^2.$ The absolute value of the extraordinary Hall effect scales linearly with the ordinary one in the entire range over eight orders of magnitude between the metallic and the insulating states. The behavior differs qualitatively and quantitatively from the theoretically predicted and experimentally known behavior in other materials.

Figure 1

Hall resistance $R_{xy}$ of two 10-nm-thick samples with lower (a) and higher (b) degrees of oxidation as a function of applied magnetic fields. Resistivity is $11.5\times {10}^{-3}\mathrm{\Omega }\mathrm{cm}$ (a) and $1.4\mathrm{\Omega }\mathrm{cm}$ (b).

Figure 2

The ordinary Hall effect coefficient as a function of longitudinal resistivity $\rho$. M-I indicates the metal-insulator transition. Inset: resistivity of two typical samples as a function of temperature, taken from Ref. [12]. Straight lines are a guide to the eye.

Figure 3

The effective mobility of a series of 50-nm-thick samples as a function of sheet resistance. Straight lines are a guide to the eye.

Figure 4

The absolute value of the EHE conductivity as a function of longitudinal conductivity. Straight lines are a guide to the eye. Inset: magnetization of 50-nm-thick samples as a function of resistivity.

Figure 5

The absolute value of the EHE resistivity as a function of the ordinary Hall coefficient for multiple series of different thickness. Open symbols correspond to the annealed polycrystalline samples. The slope of the straight line is 1. $R_Q$ marks the crossover threshold.