Hanle effect in vertical Ohmic spin valves

Hanle effect measurements in lateral spin valves provide valuable information on spin lifetimes and diffusion lengths in the transport channel. However, their interpretation is made difficult by the fact that current distribution near the injector substantially deviates from the assumptions of a simple 1D model. These difficulties can be partially alleviated by using the "three terminal" measurement method. A logical continuation of this line of thought is the usage of a “current perpendicular to plane” (vertical) F/N/F spin valve, where electric current distribution is strictly uniform. Additional advantage of the vertical valve is a better experimental control of the normal spacer thickness. The complication is the presence of two F/N interfaces, both contributing to the total voltage drop. Here we derive theoretical expressions for the valve magnetoresistance due to Hanle effect experienced by spins in the normal spacer. We assume diffusive regime, transparent boundaries with Ohmic conductivity, and allow for the back-action of spin accumulation in N on the injection process. Magnetoresistance is calculated for arbitrary angle between the magnetizations of the F-layers and any thickness of the N-spacer.