Spin Hall and Edelstein effects in a ballistic quantum dot with Rashba spin-orbit coupling

TL;DR

This work analyzes spin-resolved transport in a ballistic quantum dot with tunable Rashba SOC, linking quantum interference-driven localization phenomena to charge-to-spin conversion mechanisms. Using a tight-binding Hamiltonian and a multi-terminal scattering-matrix approach, the authors demonstrate a WL to WAL crossover at $\alpha_c \approx 19$ meV when the dot size is comparable to the Fermi wavelength, accompanied by gate-tunable Edelstein and spin Hall currents. A key finding is the inflection in the Edelstein current at $\alpha_c$, tying spin-charge conversion directly to the quantum interference regime, and a transition in magnetoresistance angular periodicity from $\pi$ to $2\pi$ under in-plane fields, reflecting the evolving spin textures. The results illuminate how quantum coherence, SOC, and confinement govern spin-dependent transport in mesoscopic devices, with implications for oxide-based quantum dot implementations.

Abstract

We study spin-resolved transport in a ballistic quantum dot with Rashba spin-orbit coupling, focusing on charge-to-spin conversion and spin Hall effect. In the regime where the dot size is comparable to the Fermi wavelength, we identify a clear crossover from weak localization to weak antilocalization as the Rashba coupling increases. This transition is accompanied by gate-tunable spin currents of Edelstein and spin Hall type, whose behavior reflects the underlying electron wavefunction interference. Notably, the Edelstein current shows an inflection point at the critical Rashba strength, signaling the crossover from weak localization to weak antilocalization. In the presence of an in-plane magnetic field we also report a transition in angular periodicity of the magnetoresistance -- from $π$ to $2π$ -- arising from the interplay between spin-orbit interaction and Zeeman coupling. These results establish a direct link between quantum coherence, charge-to-spin conversion, and geometric confinement in mesoscopic systems.

Figures (7)

• Figure 1: (a) Tight-binding lattice used in our model, implemented via Kwant Groth_2014. The central blue region defines the quantum dot (scattering region), composed of a circular core of radius $r$ connected to two symmetric compound arms, each consisting of two adjacent rectangles. The four red regions indicate the semi-infinite normal-metal leads, labeled from 0 to 3. Geometric parameters $L_1$, $L_2$, $W_1$, and $W_2$ define the arm sizes, as shown. (b) Density of states (DOS) of the dot. The vertical dashed red line marks the chemical potential considered here.
• Figure 2: (a) Magnetoresistance MR between leads 2 and 0 as a function of Zeeman energy $M_z$, for $\alpha$ ranging from 10 to 25 meV. The curvature reversal around $M_z = 0$ highlights a crossover from WL to WAL as $\alpha$ increases. This transition is also reflected in the color shift from warm tones (WL) to cool tones (WAL). (b)–(c) Resistance $R_{20}$ vs $M_z$ for selected values of $\alpha$ from panel (a), illustrating the curvature evolution that characterizes the WL-to-WAL transition. (d) Resistance $R_{20}$ as a function of $\alpha$ for $\vec{M} = 0$, showing a monotonic decrease due to the enhanced suppression of backscattering by spin-orbit interaction. The inset illustrates the four-terminal setup and current injection scheme. The resistance curves are expressed in unit of $R_0=h/e^2$.
• Figure 3: (a) $x$-, $y$-, and $z$-polarized components of the spin current in lead 2 as a function of $\alpha$ for $\vec{M} = 0$. (b)–(d) Comparison of the spin currents in leads 2 (solid lines) and 3 (dashed lines) as a function of $\alpha$, showing how spin-polarized transport evolves between the two leads. (e) Voltage differences between the current-injecting lead (lead 1) and all other leads, illustrating how the internal potential landscape evolves with $\alpha$ and reflects the redistribution of current paths within the scattering region. (f) Comparison between the derivative of the magnetoresistance, $\partial$MR$/\partial M_z$, and the second derivative of the Edelstein spin current, $\partial^2 J_{s_x}/\partial \alpha^2$, both evaluated at $M_z = 0$ and plotted as a function of $\alpha$. The sign change in both quantities identifies the crossover from WL to WAL. In the plot, $\partial^2 J_{s_x}/\partial \alpha^2$ is shown scaled by 100 for readability. The voltage and spin current curves are expressed in units of $V_0=h I/e^2$ and $J^0_s=e V_0/4 \pi$, respectively, where $I$ is the applied current bias.
• Figure 4: (a) Magnetoresistance MR as a function of the direction $\theta$ of an in-plane Zeeman field $\vec{M} = (M \cos \theta, M \sin \theta, 0)$, for $\alpha$ ranging from 16 to 22 meV. The field amplitude is kept constant at $M = 0.05\ \mathrm{meV}$. (b)–(d) MR for three selected values of $\alpha$, showing a transition in angular periodicity from $\pi$ to $2\pi$ as $\alpha$ increases. This behavior reflects the growing dominance of Rashba spin-orbit coupling over the external Zeeman field. The numerical data are well described by a fitting function including $\cos(\theta)$ and $\cos(2\theta)$ harmonics. (e)–(g) $x$-, $y$-, and $z$-polarized components of the spin current in lead 2, calculated for the same parameter sets as in panels (b)–(d). At low $\alpha$, the spin currents exhibit harmonic oscillations induced by the in-plane field, particularly in $J_{s_x}$ and $J_{s_z}$, while $J_{s_y}$ remains negligible. At larger $\alpha$, the spin currents become nearly constant with $\theta$, indicating a regime dominated by the spin-orbit interaction. The spin current curves are expressed in units of $J^0_s=e V_0/4 \pi$, with $V_0=h I/e^2$ and $I$ is the applied current bias. For readability, all y-axis values are scaled by $10^2$.
• Figure 5: (a)–(c) Magnetoresistance MR as a function of $M_z$, for increasing values of the dot radius $r = 10$, 20, and 50, and several Rashba coupling strengths $\alpha$. For small $r$, MR shows a clear crossover from WL to WAL as $\alpha$ increases. As $r$ grows, this transition is suppressed, and the MR response becomes more complex. (d)–(f) Probability densities $|\psi|^2$ of the scattering wavefunction for $\alpha = 0$ and $\vec{M} = 0$, with an electron injected from lead 1. These panels represent single scattering processes and illustrate how the spatial distribution of the wavefunction changes with $r$. For $r = 10$, backscattering dominates, consistent with the WL regime. As $r$ increases, transmission becomes more prominent.