Transfer lengths and spin injection from a three-dimensional ferromagnet into graphene

In graphene spintronic devices, spin currents are injected and probed by three-dimensional ferromagnetic electrodes on a two-dimensional graphene layer. Here we develop a unified theory to understand the observed spin-injection signals and their dependence on a dc-bias current in these devices by different groups. The spin-dependent transfer length, which measures the length required for a current to change from lateral in graphene to vertical in the electrode, is found to play a central role in determining the spin-current distribution as well as the signal fidelity. The theory consistently explains the experiments and suggests that the observed dc-bias dependence is due not to change in the spin-injection efficiency but to redistribution of the spin current near the ferromagnetic electrode.