Weak-Localization Correction to the Anomalous Hall Effect in Polycrystalline Fe Films

In situ transport measurements have been made on ultrathin ($<100\AA$ thick) polycrystalline Fe films as a function of temperature and magnetic field for a wide range of disorder strengths. For sheet resistances $R_{xx}$ less than $\sim 3k\Omega$, we find a logarithmic temperature dependence of the anomalous Hall conductivity ${\sigma }_{xy}$, which is shown for the first time to be due to a universal scale dependent weak-localization correction within the skew-scattering model. For higher sheet resistance, granularity becomes important and the break down of universal behavior becomes manifest as the prefactors of the $\ln T$ correction term to ${\sigma }_{xx}$ and ${\sigma }_{xy}$ decrease at different rates with increasing disorder.

Figure 1

$\ln (T)$ dependence of $R_{xx}$ and $R_{xy}$ for $T<20\mathrm{K}$ for a type $A$ sample with $R_0=2733\Omega$. Inset: $R_0=49\mathrm{k}\Omega$.

Figure 2

Coefficients $A_R$ and $A_{\mathrm{AH}}$ as defined in Eq. (1) for type $A$ (triangles) and type $B$ (circles) samples. Solid triangles represent type $A$ samples that have undergone surface modification and conductance changes associated with postdeposition ion milling as described in text.

Figure 3

Weak-localization diagrams. Solid lines are impurity averaged Green’s functions, broken lines are impurity scattering amplitudes [5, 14]), shaded cross is the Cooperon $C(Q,i{\omega }_l)=(2\pi N_0{\tau }^2{)}^{-1}[|{\omega }_l|+D^p{Q}^2+{\tau }_{\phi }^{-1}{]}^{-1}$, where $D^p$ is the diffusion constant in the particle-particle channel [21], and the triangles are current vertex corrections. There are two diagrams of type (b) and four diagrams of type (c).

Figure 4

$\ln (T)$ dependence of relative changes in the AH resistance for type $A$ films with three different $R_0$.