Spin-dependent edge-channel transport in a $\mathrm{Si}∕\mathrm{SiGe}$ quantum Hall system

We study the edge-channel transport of electrons in a high-mobility $\mathrm{Si}∕\mathrm{SiGe}$ two-dimensional electron system in the quantum Hall regime. By selectively populating the spin-resolved edge channels, we observe suppression of the scattering between two edge channels with spin-up and spin-down. In contrast, when the Zeeman splitting of the spin-resolved levels is enlarged with tilting magnetic field direction, the spin orientations of both the relevant edge channels are switched to spin-down, and the inter-edge-channel scattering is strongly promoted. The evident spin dependence of the adiabatic edge-channel transport is an individual feature in silicon-based two-dimensional electron systems, originating from a weak spin-orbit interaction.

Figure 1

(a) Energy diagrams of Landau levels between $N=0$ and $N=1$ for $\hslash {\omega }_{\mathrm{c}}>\mathrm{\Delta }{E}_{\mathrm{z}}$ (left) and $\hslash {\omega }_c<\mathrm{\Delta }{E}_z$ (right). The Fermi level $E_F$ is located at the filling factor of $\nu =4$. The valley splittings $\left({\Delta }_V\right)$ are also depicted. (b) Edge channel dispersions for $\hslash {\omega }_{\mathrm{c}}>\mathrm{\Delta }{E}_z$ (left) and $\hslash {\omega }_c<\mathrm{\Delta }{E}_z$ (right) for $\nu =4$ (a two-channel case). The spin orientation of the relevant edge channels switches from ($0\downarrow$, $0\uparrow$) to ($0\downarrow$, $1\downarrow$) through the coincidence angle.

Figure 2

(a) A schematic illustration of the Hall bar sample. (b) The enlarged figure of $L_{\mathrm{edge}}$ region for a two-channel case. The arrows indicate the direction of electron drift in the edge channels.

Figure 3

$R_{xy}$ as a function of $V_{\mathrm{G}1}$ at $({\nu }_{\mathrm{B}},{\nu }_{\mathrm{G}2})=(4,2)$ at $23\mathrm{mK}$. (b) Temperature dependence of $R_{xy}$ at $({\nu }_{\mathrm{B}},{\nu }_{\mathrm{G}1},{\nu }_{\mathrm{G}2})=(4,2,2)$.

Figure 4

(a) Plots of $R_{xy}$ vs $B_{\perp }$ for various $\theta$ at around ${\nu }_B=4$ at $28\mathrm{mK}$. $R_{xy}-B_{\perp }$ curves include an offset by 0.05 $h∕e^2$ for each curve. (b) and (c) $R_{xy}$ as a function of $B_{\Vert }$ at $28\mathrm{mK}$ for $({\nu }_{\mathrm{B}},{\nu }_{\mathrm{G}1},{\nu }_{\mathrm{G}2})=(4,2,2)$. The arrows illustrated represent the spin orientation of the relevant edge channels in each $B_{\Vert }$ range.

Figure 5

$R_{xy}^{\prime }$ vs $I_{\mathrm{dc}}$ for different $\theta$ at $28\mathrm{mK}$ for $({\nu }_{\mathrm{B}},{\nu }_{\mathrm{G}1},{\nu }_{\mathrm{G}2})=(4,2,2)$. The data traces of $R_{xy}^{\prime }$ include an offset by 0.1 $h∕e^2$ for each curve, and the major ticks are presented at every 0.1 $h∕e^2$.