Charge Hall effect driven by spin-dependent chemical potential gradients   and Onsager relations in mesoscopic systems
  We study theoretically the spin-Hall effect as well as its reciprocal phenomenon (a transverse charge current driven by a spin-dependent chemical potential gradient) in electron and hole finite size mesoscopic systems. The Landauer-Buttiker-Keldysh formalism is used to model samples with mobilities and Rashba coupling strengths which are experimentally accessible and to demonstrate the appearance of measurable charge currents induced by the spin-dependent chemical potential gradient in the reciprocal spin-Hall effect. We also demonstrate that within the mesoscopic coherent transport regime the Onsager relations are fulfilled for the disorder averaged conductances for electron and hole mesoscopic systems. 
E. M. Hankiewicz, Jian Li, Tomas Jungwirth, Qian Niu, Shun-Qing Shen,   and Jairo Sinova