Resonant spin Hall and Nernst effect in a nanoribbon of a spin-orbit coupled electronic system

We present a theoretical study of spin Hall phenomenon in a nanoribbon of a two-dimensional electronic system with Rashba and Dresselhaus spin-orbit coupling. We model the electronic system by a square lattice in real space. We show that such nanoribbon can give rise to a number of additional spin degeneracy points as well as anticrossing points, apart from the $\Gamma$ point, between two opposite spin subbands. We compute the SHC and demonstrate that it diverges and gives rise to a resonance when the chemical potential passes through those spin degenerate or anticrossing points. Contrary to the previous studies, here such resonance emerges even without any external perturbation like magnetic field or light. We also examine the spin Nernst effect and find that it shows clear peaks at the anticrossing and spin degeneracy points, consistent with the Mott relation at low temperature. Finally, we also investigate the signature of such additional spin degeneracy and anticrossing points in the longitudinal conductance by using the retarded Green function approach in lattice model. The finite width induced subbands are reflected in the longitudinal conductance, which takes quantized values of $2n e^{2}/{h}$ where $n$ denotes the number of bands occupied by the chemical potential with each band having spin split subbands. We also note that anticrossing that occurs at low energy between two opposite spin subbands could be also detected via longitudinal conductance.