Signatures of spin-orbit coupling in scanning gate conductance images of electron flow from quantum point contacts

Electron flow through a quantum point contact in the presence of spin-orbit coupling is investigated theoretically in the context of the scanning gate microscopy (SGM) conductance mapping. Although in the absence of the floating gate the spin-orbit coupling does not significantly alter the conductance, we find that the angular dependence of the SGM images of the electron flow at the conductance plateaus is substantially altered as the spin-orbit interaction mixes the orbital modes that enter the quantum point contact. The radial interference fringes that are obtained in the SGM maps at conductance steps are essentially preserved by the spin-orbit interaction as backscattering by the tip preserves the electron spin although the effects of the mode mixing are visible.

Figure 1

Sketch of the system under consideration. The gray area depicts the confinement potential of the input channel. The contours show the potential of a QPC for $\hslash \omega =3$ meV for which we obtain the first plateau of conductance $G=2e^2/h$. The real part of the spin-up component of the scattering wave function for the electron incident from the channel is presented with the blue-red color map. The tip is located at $x=100$ nm and $y=2300$ nm. The crossed parts at the input and output channels denote the regions used for resolution of the transmitted and reflected plane waves. The red lines at the edges depict the regions where transparent boundary conditions are introduced.

Figure 2

(a) Dispersion relation in the input channel without (black curves) and with (green curves) SO interaction included. (b) Dispersion relation in the output channel in the absence of SO coupling. (c) Conductance calculated in the absence (black curve) or in the presence (green curve) of SO coupling. The symbols mark the points in the conductance for which the SGM maps are calculated.

Figure 3

Maps of the conductance changes $\Delta G$ as a function of the SGM tip position for the QPC tuned to three plateaus; the symbols correspond to the ones in Fig. 2. The upper (bottom) row corresponds to the results obtained without (with) SO interaction.

Figure 4

(a), (b) Color maps show the probability density for the transmitted electron. The arrows present the probability current and the contours present amplitude of the probability current. Results obtained for the plateau $G=4e^2/h$ for (a) $\alpha =0$ and for (b) $\alpha =11.44$ meV nm. (c) Probability densities obtained for the transport from the first five modes of the input channel without SO coupling. (d) Same as (c) but for SO interaction included.

Figure 5

(a), (b) Cross sections of probability densities above QPC for $y=1896$ nm. (c), (d) Charge density corres-ponding to the eigenmodes of the input channel. (e), (f) Densities from (c), (d) multiplied by the corresponding transfer probability. (g), (h) Sum of the densities from (e), (f) correspondingly.

Figure 6

Kinetic energy of the modes of channel with transverse potential corresponding to the distance $D$ from the center of the QPC for $E_f=3$ meV. (a) Case without SO interaction. (b) SO interaction included. Black dots correspond to the case with full SO Hamiltonian included and blue dots corresponds to the case of neglected last term of Eq. (1).

Figure 7

Maps of conductance changes obtained with the scanning gate tip potential for the QPC tuned to three conductance steps; the symbols correspond to the ones in Fig. 2. The upper (bottom) row corresponds to the results obtained without (with) SO interaction.

Figure 8

(a) Conductance as a function of scanning gate position along $x=0$ axis obtained for the QPC tuned to the last conductance step for $\hslash \omega =6.1$ meV without (black curve) and with SO interaction (red curve). The vertical solid and dashed lines depict $l$ and $l_{\mathrm{SO}}$, respectively (see text). (b) Dispersion relation of the 2DEG in the presence of SO interaction. Colors of the curves represent spin polarization antiparallel (blue) and parallel (red) to the $x$ direction. The curved arrows presents the transitions between the modes that occur during the backscattering from the tip.

Figure 9

(a) Conductance as a function of scanning gate position along $x=0$ axis obtained for the QPC tuned to the last conductance step for $\hslash \omega =6.1$ meV in the presence of Zeeman term for $B=0.3$ T. Black (red) curve corresponds to the case $\alpha =0$ ($\alpha =11.44$ meV nm). (b) Dispersion relation for Zeeman effect present with $\alpha =0$. The colors of the curves depict positive and negative spin polarization in the $z$ direction. (c) Dispersion relation for Zeeman effect present with $\alpha =11.44$ meV nm. The colors of the curves depict the sign of the $\langle s_x\rangle$.

Figure 10

Maps of the conductance changes $\Delta G$ as a function of the SGM tip position for the QPC tuned to $G=4e^2/h$ obtained for QPC potential of different length. The upper (bottom) row corresponds to the results obtained without (with) SO interaction.

Figure 11

Maps of the conductance changes $\Delta G$ for varied width $W$ of the input channel for the QPC tuned to $G=4e^2/h$.

Figure 12

Maps of the conductance changes $\Delta G$ for (a) $\alpha =5.72$ meV nm and (b) $\alpha =17.16$ meV nm for the QPC tuned to $G=4e^2/h$.