Theory of charge-to-spin conversion under quantum confinement

TL;DR

Problem addressed: understanding spin-charge interconversion in confined, spin-orbit coupled multiterminal devices. Approach: develops a spin-dependent scattering-matrix framework that generalizes Büttiker formalism to compute charge currents, spin currents, and bias-induced spin densities, with explicit expressions such as $\langle J^j_c \rangle$, $\langle \vec{J}_s^j \rangle$, and $\langle \delta \vec{s}^j \rangle$. The method is applied to a Rashba Hall bar, implemented with Kwant, including orbital magnetic-field effects and disorder averaging to capture quasi-ballistic transport. Key results: nonlocal voltage signals $|V_{23}|$ show a bell-shaped dependence on the Zeeman energy and a decay with lead spacing, while spin currents exhibit Edelstein and SHE contributions, and an inverted Hanle signal arises from orbital effects. Significance: provides a microscopic, real-space framework bridging ballistic and diffusive spintronics and enabling orbitronics in SOC-confined devices, with relevance to oxide interfaces like LAO/STO.

Abstract

The interplay between spin and charge degrees of freedom in low-dimensional systems is a cornerstone of modern spintronics, where achieving all-electrical control of spin currents is a major goal. Spin-orbit interactions provide a promising mechanism for such control, yet understanding how spin and charge transport emerge from microscopic principles remains a fundamental challenge. Here we develop a spin-dependent scattering matrix approach to describe spin and charge transport in a multiterminal system in the presence of Rashba spin-orbit interaction. Our framework generalizes the Büttiker formalism by offering explicit real-space expressions for spin and charge current densities, along with the corresponding linear response function. It simultaneously captures the effects of quantum confinement, the orbital response to external magnetic fields, and the intrinsic (geometric) properties of the electronic bands, offering a comprehensive description of the spin-charge interconversion mechanisms at play in a Hall bar, in agreement with experiments.

Figures (10)

• Figure 1: (a) Schematic representation of a multiterminal scattering region through which spin-sensitive transport is measured. The illustration depicts a single scattering event originating from lead $i$, contributing to the charge current $\braket{J_c^i}$, the spin current $\braket{\vec{J}_s^j}$, and the bias-induced spin density $\braket{\vec{\delta s}^{k}}$, in leads $i$, $j$, and $k$, respectively. Spin-resolved incoming and outgoing modes are explicitly indicated by red arrows. (b) Hall bar with Rashba spin–orbit coupling in a current-biased configuration. A net current flows from lead 1 to lead 0. Non-local voltage signal is recorded using leads $2$ and $3$.
• Figure 2: (a) Nonlocal voltage, $|V_{23}|$, as a function of the external magnetic field applied along $y$, $M_y$. The blue gradient corresponds to increasing lead spacings $L$. (b) Vertical slices from panel (a) illustrating the exponential decay of $|V_{23}|$ when the lead spacing is increased. (c) $x$-polarized spin current, computed using Eqs. \ref{['Equation1']}, \ref{['Equation2']} for $\vec{M}=0$. The inset illustrates the generation of the $x$-polarized spin current via the Edelstein effect. (d) Computed $z$-polarized spin current, also obtained from Eqs. \ref{['Equation1']}, \ref{['Equation2']} for $\vec{M}=0$. The inset illustrates the generation of the $z$-polarized spin current via the spin-Hall effect. The source and drain leads are positioned at the center of the system, while the measuring probes are placed either to the left or right. The voltage and spin current curves are expressed in units of $V_0= 2\pi \hbar I / e^2$ and $J_{s}^0=e V_0/4\pi$, respectively, where $I$ is the applied current bias. For $\alpha = 0$, both $J_s^x$ and $J_s^z$ vanish identically.
• Figure 3: (a), (b) Charge current density in the scattering region for two configurations: (a) $\alpha = 10$, $M_z = 0$; (b) $\alpha = 0$, $M_z = -0.5$. The current is injected from lead 1 and collected at lead 0, with no net current through leads 2 and 3. We set $M_x = M_y = 0$. (c) $|V_{23}|$, expressed in units of $V_0= 2\pi \hbar I / e^2$, as a function of the external magnetic field $M_z$. The plot compares results with (green curve) and without (orange curve) the orbital Peierls contribution. The lead distance is fixed to $L = 72$.
• Figure 4: Schematic representation of the oxide Hall bar: (a) net current flux entering from lead $1$ and exiting through lead $0$ (current bias), and (b) fixed bias potentials applied to the four leads (voltage bias). (c) Sketch of the band structure of the LAO/STO $(001)$ interface in one dimension. The six $t_{2g}$$d$-orbitals, i.e., $d_{xy}$-like (blue and orange), $d_{yz}$-like (green and red), and $d_{xz}$-like (purple and brown), are shown. The yellow shaded area highlights the $d_{xy}$ band, matching the Rashba model investigated in the main text when $W_S=1$, as depicted in panel (d).
• Figure 5: (a) Impurity distribution within a single realization, featuring random potentials distributed with a $2 \%$ density across the strip geometry. (b) The lowest six energy bands of the Rashba SOC model, with the scattering energy $\mu$ explicitly highlighted.