Low-Field Phase Diagram of the Spin Hall Effect in the Mesoscopic Regime

When a mesoscopic two dimensional four-terminal Hall cross bar with 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 $G_{\mathrm{sH}}$ and its fluctuations $\mathrm{rms}(G_{\mathrm{sH}})$ vanish identically on even numbered IQHE plateaus, they have finite values on those odd numbered plateaus induced by SOI, and they have values $G_{\mathrm{sH}}=1/2$ and $\mathrm{rms}(G_{\mathrm{sH}})=0$ on those odd numbered plateaus induced by Zeeman energy. At larger disorders, the system crosses over into a regime where both $G_{\mathrm{sH}}$ and $\mathrm{rms}(G_{\mathrm{sH}})$ are finite, a chaotic regime, and finally a localized regime.

Figure 1

Transmission coefficient $T_{21}$ vs $E(t)$ or $B(T)$, where $t$ is the hopping energy. (a),(b) Setup I. (c),(d) Setup II. Inset of (a) schematic plot of setup I; inset of (b) the corresponding flow of edge states. Inset of (c) schematic plot of setup II; inset of (d) the corresponding flow of edge states.

Figure 2

(a) The edge-state plateaus in ($B,W$) plane. (b). The transmission coefficient $T_{21}$ for setup I and setup II, as well as direct transmission coefficient $T_{31}$ as a function of $B$ in the absence of disorder. (c). The spin Hall conductance in ($B,W$) plane. (d). The spin Hall conductance fluctuation in ($B,W$) plane.

Figure 3

The low-field “phase diagram” of mesoscopic spin Hall effect in ($B,W$) plane. Note that for disorder strength between $W=2$ to $W=20$ the system is the spin Hall liquid.