Anomalous Spin-Resolved Point-Contact Transmission of Holes due to Cubic Rashba Spin-Orbit Coupling

Evidence is presented for the finite wave vector crossing of the two lowest one-dimensional spin-split subbands in quantum point contacts fabricated from two-dimensional hole gases with strong spin-orbit interaction. This phenomenon offers an elegant explanation for the anomalous sign of the spin polarization filtered by a point contact, as observed in magnetic focusing experiments. Anticrossing is introduced by a magnetic field parallel to the channel or an asymmetric potential transverse to it. Controlling the magnitude of the spin splitting affords a novel mechanism for inverting the sign of the spin polarization.

Figure 1

(a) Voltage across the detector QPC as a function of magnetic field for zero tilt angle. Insets: AFM micrograph of a sample, where arrows schematically show the cyclotron motion for the two spin orientations; the bars are $0.5\mu \mathrm{m}$ scales. (b) Signal for the first focusing peak in a tilted magnetic field. Curves are offset for clarity. The values of $G_{\mathrm{inj}}$ for dashed blue (solid red) curves are within the smaller (larger) rectangles in the injector QPC characteristic in (c). (d) Relative population of the spin subbands, estimated for $G_{\mathrm{inj}}=2e^2/h$ (blue squares) and $G_{\mathrm{inj}}\sim 0.3\times 2e^2/h$ (red dots).

Figure 2

Energy subbands of 1D channels obtained from a 15 nm quantum well grown in the [113] direction. An electric field ${\mathcal{E}}_z=1\mathrm{V}/\mu \mathrm{m}$ along [113] is present. The lateral confinement has width $W=40\mathrm{nm}$. Upper panel: Wire along $[33\overline{2}]$. The inset shows the energy splitting of the two lowest subbands at several values of $B_{\parallel }$. The solid curve is for $B_{\parallel }=0$ and the dashed curves for $B_{\parallel }=0.5,1,1.5,2\mathrm{T}$. Lower panel: Wire along $[1\overline{1}0]$. The inset shows the energy splitting with a lateral electric field. The solid curve is for ${\mathcal{E}}_y=0$ and the dashed curves for ${\mathcal{E}}_y=\pm 12.5,\pm 37.5\mathrm{V}/\mathrm{mm}$ (the splitting is reduced for positive values of ${\mathcal{E}}_y$).

Figure 3

Total conductance $G$ (black solid curves) and spin-resolved conductances $G_{+}$ (blue, long-dashed curves) and $G_{-}$ (red, short-dashed curves), plotted in units of $2e^2/h$ as functions of the minimum width $W_0$ of the QPC [see Eq. (4)]. We used parameters appropriate for the experimental setup: $m=0.14m_0$ [20], where $m_0$ is the bare electron mass, $g^{*}=0.8$ [5], $\gamma {\hslash }^3=0.45\mathrm{eV}{\mathrm{nm}}^3$, $\Delta x=0.3\mu \mathrm{m}$, and ${ϵ}_F=2.3\mathrm{meV}$.