Persistent spin current in nano-devices and definition of the spin current

We investigate two closely related subjects: (i) existence of a pure persistent spin current in semiconducting mesoscopic device with a spin-orbit interaction's (SOI), and (ii) the definition of the spin current in the presence of SOI. Through physical argument from four physical pictures in different aspects, we provide strong evidences that the persistent spin current does exist in a device with SOI in the absence of any magnetic materials. This persistent spin current is an analog of the persistent charge current in a mesoscopic ring threaded by a magnetic flux, and it describes the real spin motion and can be measured experimentally. We then investigate the definition of the spin current. We point out that (i) the non-zero spin current in the equilibrium SOI device is the persistent spin current, (ii) the spin current is in general not conserved, and (iii) the Onsager relation is violated for the spin transport no matter what definition of the spin current is used. These issues, the non-zero spin current in the equilibrium case, the non-conserved spin current, and the violation of the Onsager relation, are intrinsic properties of spin transport. We note that the conventional definition of the spin current has very clear physical intuition and describes the spin motion very well. Therefore we feel that the conventional definition of the spin current makes physical sense and there is no need to modify it. In addition, the relationship between the persistent spin current and transport spin current, the persistent linear and angular spin currents in the SOI region of the hybrid ring, are discussed. Finally, we show that if the spin-spin interaction is included into the Hamiltonian, the persistent spin current is automatically conserved using the conventional definition.