Shot noise of charge current and spin current in the presence of Rashba interaction

We study fluctuations of spin current and charge current of two-terminal systems in the presence of spin-orbit interaction. Based on the scattering matrix theory, we obtain the general expressions for the fluctuations of various currents ( charge or spin current). It is well known that, for a two-terminal system, the auto-correlation function is enough to characterize the fluctuations for the charge current. Our result shows that, even for a conserved spin current, both auto-correlation and cross-correlation functions are needed to characterize the fluctuations of spin current when spin-orbit interaction are present. As a result, the cross-correlation of spin current may become positive, exhibiting bosonic behaviour. Indeed, our calculations on simple models show that the sign of the cross-correlation can be tuned by either the gate voltage or intra-dot coupling.