Universal scaling and Implications of the apparent Anomalous Hall voltage of magnetic layer with the adjacent conducting layer

The anomalous Hall voltage of magnetic heterostructures plays a key role via the interplay with a variety of spintronic effects. Under a given electric field (E), the anomalous Hall voltage of a magnetic layer should remain unaffected by adjacent layers unless the latter could modify or contribute to the anomalous Hall conductivity. In this letter, we report the observation, universality, mechanism, and impacts of the dramatic scaling of the apparent anomalous Hall voltage per electric field (VAH/E) of magnetic heterostructures with the thickness and resistivity of the nonmagnetic conducting layers by combining transport experiments, analytical derivation, and finite-element analyses. We identify that the scaling arises simply from the misreading of the invariant internal anomalous Hall voltage due to the serial resistor effect of the magnetic and non-magnetic layers within the magnetic heterostructures. The scaling remains strong regardless of the strengths of the spin-orbit torques and spin and orbital Hall conductivities of the normal metals, suggesting that interfacial angular momentum transport plays a minimal role in the anomalous Hall effect. We also show that the harmonic Hall voltage analyses of magnetic heterostructures remain robust against the considerable under-measuring of the anomalous Hall voltage and other transverse voltages due to the serial-resistor effect. These findings provide crucial information for understanding a variety of spintronic experiments involving the anomalous Hall and other transverse voltages.