Snakelike trajectories of electrons released from quantum dots driven by the spin Hall effect

TL;DR

The paper investigates how electrons released from a gate-defined quantum dot into a spin-orbit-coupled InSb channel follow snake-like trajectories due to the spin Hall effect. By combining time-dependent quantum simulations with semiclassical models, it shows that spin precession in the effective spin-orbit field drives spin-dependent lateral deflections that encode the initial quantum state. A T-junction readout leverages the state-dependent partitioning of the electron wave packet into two drains, achieving readout fidelities above ~82% under optimized channel widths and confinement. The work highlights the potential for all-electrical spin-state detection in spin-orbit qubit architectures and provides design guidelines for geometry and field strength to optimize readout.

Abstract

Using time dependent simulations, we analyze the trajectories of electrons released from a quantum dot in a waveguide made of a spin-orbit-coupled material (InSb). An electron released from the quantum dot, when driven by an electric field follows a trajectory that is deflected by spin-orbit interaction and undergoes spin precession that results in a spin-dependent, snake-like trajectory. The trajectory strongly depends on the initial state of the electron, enabling detection of the electron quantum state in the dot when connected to the T-junction. Notably, we show that the snake-like trajectory persists even under a small external magnetic field with low, incomplete initial electron spin polarization. Our findings are supported by semiclassical calculations of the electron trajectory, which show good agreement with full quantum mechanical simulations

Figures (17)

• Figure 1: Schematics of the QD embedded within the chanel of width $d$. The electrostatic confinement of the quantum dot is switched off at $t=0$ and replaced by a homogenous electric field oriented in the $y$ direction. The split in the channel is located at a distance $b$ from the QD. The red arrow schematically indicates the trajectory of the electron released from the QD.
• Figure 2: (a) The energy spectrum for the electron confined in a harmonic oscillator quantum dot potential with $\hbar\omega=0.15625$ meV, the width of the InSb channel $d=175$ nm and the magnetic field oriented parallel to the $y$ axis. The color of the lines indicates the spin $y$ component of the eigenstates. In (b) we show the average $y$ component of the spin for the ground-state (red line) and the first-excited state (black line). (c,d) same as (a,b) only for $d=100$ nm.
• Figure 3: (a) The probability density for ground state of the Kramers doublet, together with (b) the spin density $s_y$ - the spin density for the second state of the Kramers pair is opposite. (c) and (d) the spin density $s_y$ divided by the probability density $\rho$, determined for both states of the Kramers doublet. The results correspond to the QD confinement energy $\hbar\omega=0.15625$ meV and the channel width $d=175$ nm.
• Figure 4: Snaphots of the probability density (a,c) and spin-y component (b,d), calculated as a ratio of the $s_y$ component density to the probability density, for selected moments in time listed on top of panel (a). The initial condition set in one of the states of the Kramers doublet in the QD; (a,b) for the lower energy state and (c,d) for the higher energy state. The results for the InSb channel of width $d=175$ nm and the magnetic field $B_y=10~\mu$T. The color scale for (a,b) as well as (c,d) is the same.
• Figure 5: (a) Trajectory and (b–d) average spin components of the Kramers doublet states flowing throughout the channel. Black and red lines correspond to the first and second states of the Kramers doublet for $d=175$ nm. In (a) and (b) additionally blue line markes results for the lowest state and a thinner channel $d=125$ nm. Results for for $B_y=10$$\mu$T.