Low field phase diagram of spin-Hall effect in the mesoscopic regime

When a mesoscopic two dimensional four-terminal Hall cross-bar with Rashba and/or Dresselhaus spin-orbit interaction (SOI) is subjected to a perpendicular uniform magnetic field $B$, both integer quantum Hall effect (IQHE) and mesoscopic spin-Hall effect (MSHE) may exist when disorder strength $W$ in the sample is weak. We have calculated the low field `phase diagram' of MSHE in the ($B, W)$ plane for disordered samples in the IQHE regime. For weak disorder, MSHE conductance $G_{sH}$ and its fluctuations \textit{rmsG}$_{sH}$ vanish identically on even numbered IQHE plateaus, they have finite values on those odd numbered plateaus induced by SOI, and they have values $G_{sH}=1/2$ and \textit{rmsG}$_{sH}=0$ on those odd numbered plateaus induced by Zeeman energy. For moderate disorder, the system crosses over into a regime where both $G_{sH}$ and \textit{rmsG}$_{sH}$ are finite. A larger disorder drives the system into a chaotic regime where $G_{sH}=0$ while \textit{rmsG}$_{sH}=0$ is finite. Finally at large disorder both $G_{sH}$ and \textit{rmsG}$_{sH}$ vanish. We present the physics behind this `phase diagram'.