Mesoscopic spin Hall effect along a potential step in graphene

We consider a straight one-dimensional potential step created across a graphene flake. Charge and spin transport through such a potential step are studied in the presence of both intrinsic and extrinsic (Rashba) spin-orbit coupling (SOC). At normal incidence electrons are completely reflected when the Rashba interaction (with strength $\lambda_R$) is dominant whereas they are perfectly transmitted if the two types of SOC are exactly balanced. At normal incidence, the transmission probability of the step is thus controlled continuously from 0 to 1 by tuning the ratio of the two types of SOC. Besides the transport of charge in the direction normal to the barrier, we show the existence of a spin transport along the barrier. The magnitude of the spin Hall current is determined by a subtle interplay between the height of the potential step and the position of Fermi energy. It is demonstrated that contributions from inter-band matrix elements and evanescent modes are dominant in spin transport. Moreover, in the case of vanishing extrinsic SOC ($\lambda_R =0$), each channel carries a conserved spin current, in contrast to the general case of a finite $\lambda_R$, in which only integrated spin current is a conserved quantity. Finally, we provide a quasi-classical picture of the charge and spin transport by imaging flow lines over the entire sample and Veselago lensing (negative refraction) in the case of a $p-n$ junction.