Electrical measurement of spin Hall current in mesoscopic ring: inhomogeneous Rashba effect

We investigate theoretically the spin Hall current in an inhomogeneous Rashba mesoscopic ring attached to four terminals. It is shown that a voltage drop can be tuned by adjusting the gate voltage due to the inhomogeneous Rashba effect, and provides us a tool to measure spin Hall current electrically. The spin Hall current and the ratio of the probe voltages can survive and keep their obvious relationship even in the presence of disorder. The regular relationship between the spin Hall conductance and the ratio of the probe voltages will be destroyed by the interference between different channels in multi-channel ring.