Field-dependent anisotropic magnetoresistance and planar Hall effect in epitaxial magnetite thin films

A systematic study of the temperature and magnetic field dependence of the longitudinal and transverse resistivities of epitaxial thin films of magnetite (Fe${}_3$O${}_4$) is reported. The anisotropic magnetoresistance (AMR) and the planar Hall effect are sensitive to the in-plane orientation of current and magnetization with respect to crystal axes in a way consistent with the cubic symmetry of the system. We also show that the AMR exhibits a sign reversal as a function of temperature, and that it shows significant field dependence without saturation up to 9 T. Our results provide a unified description of the anisotropic magnetoresistance effects in epitaxial magnetite films and illustrate the need for a full determination of the resistivity tensor in crystalline systems.

Figure 1

Resistivity ($\rho$), in log scale, as a function of temperature. Inset: Sketch of the relative orientations of the current density J, magnetization M, and the crystallographic axes. ${\rho }_{\mathrm{long}}$ is measured between A and B, ${\rho }_{\mathrm{trans}}$ is measured between B and C.

Figure 2

${\rho }_{\mathrm{long}}$ and ${\rho }_{\mathrm{trans}}$ vs $\alpha$, the angle between the magnetization M and [100], for different angles $\theta$ (the angle between the current direction J and [100]) at $T=150$ K with an applied magnetic field of 9 T for two different samples. The solid lines are fits to Eqs. (3) and (4).

Figure 3

The parameters in Eqs. (3) and (4) normalized by the parameter $D$ as a function of temperature with an external field of 4 T. The lines are guides for the eye.

Figure 4

(a) The parameter $D$ (right axis) and the other parameters (left axis) from Eqs. (3) and (4) normalized by $D$ as a function of the external field at $T=150$ K. The lines are guides to the eye. (b) AMR normalized to its maximum value as a function of $\alpha$ for two different external magnetic fields at $T=150$ K. The solid lines are fits to Eq. (3).

Figure 5

(a) The field dependence of $a_2$ (the numerical average of $\cos 2\beta$) and $a_4$ (the numerical average of $\cos 4\beta$) for the whole domain and (inset) for the vicinity of the boundary alone (see text). (b) An illustration of spin orientation of a ferromagnetic chain with antiferromagnetic coupling at the origin, taken from Ref. 18.