Ultrafast Orbital Hall Effect in Metallic Nanoribbons

TL;DR

This work addresses how the orbital Hall effect (OHE) manifests on ultrafast timescales in a confined metallic system. Using a tight-binding Hamiltonian with spin–orbit coupling and laser coupling, the authors solve the density-matrix dynamics to resolve spatio-temporal evolution of orbital angular momentum and its currents in a Cu(001) monolayer nanoribbon. They find that the ultrafast OHE retains the qualitative OHE picture but exhibits distinct phase relations and differences between OAM- and charge-carrying quantities, including edge-localized OAM accumulation and persistent OAM currents driven by the laser. The results lay a foundation for exploring ultrafast Hall phenomena in nanoscale metals and motivate further studies across materials and experimental probes such as THz emission.

Abstract

The orbital Hall effect can generate currents of angular momentum more efficiently than the spin Hall effect in most metals. However, so far, it has only been understood as a steady state phenomenon. In this theoretical study, the orbital Hall effect is extended into the time domain. We investigate the orbital angular momenta and their currents induced by a femtosecond laser pulse in a Cu nanoribbon. Our numerical simulations provide detailed insights into the laser-driven electron dynamics on ultrashort timescales with atomic resolution. The ultrafast orbital Hall effect described in this work is consistent with the familiar pictorial representation of the static orbital Hall effect, but we also find pronounced differences between physical quantities that carry orbital angular momentum and those that carry charge. For example, there are deviations in the time series of the respective currents. This study lays the foundations for investigating ultrafast Hall effects in confined metallic systems.

Figures (6)

• Figure 1: Snapshots of the ultrafast orbital Hall effect in a two-dimensional sample (gray rectangular solid). (a) A linearly polarized femtosecond laser pulse impinges perpendicular to the surface (along the $z$-axis) onto the sample. The laser's electric field $E$ (orange), oscillating along the nanoribbon ($\pm x$-direction), causes an oscillating longitudinal charge current $j$, which -- at the moment depicted here -- is oriented in $+x$-direction (black arrow) and is deflected toward the ribbon's edges; confer the three representative pairs of current filaments (bent blue and red arrows). Hence, orbital angular momentum (OAM) $L^{z}$ is transported across the ribbon, giving rise to a transverse OAM current $j_{L}$ (yellow arrow along $+y$-direction). As a result, $L^{z}$ is accumulated with opposite orientation at the edges (upward red and downward blue arrows). (b) Half a laser's period later, the reversal of $E$ reverses the orientation of $j$, $j_{L}$, and $L^{z}$. The periodic field switching creates an ultrafast (on the femtosecond scale) orbital Hall effect.
• Figure 2: Longitudinal current. (a) Amplitude of the laser pulse with a width of $\unit[10]{fs}$ and centered at $t = \unit[0]{fs}$. (b) Mean longitudinal current $\langle \langle j \rangle \rangle_{x}(t)$; see text. Vertical lines mark maxima of the laser amplitude. (c) Profile of currents of $x$-oriented links (along the ribbon), depicted as color scale. The $y$-average of these currents gives the data shown in panel b.
• Figure 3: Transversal currents. (a) Profile of transversal link currents depicted as color scale. (b) As panel a but for the $L^{z}$-polarized OAM currents. (c) and (d) display the data of (a) and (b), respectively, but averaged over the two $y$-regions indicated in panel a. In panel d, data for the lower and the upper $y$-region are identical. Vertical lines indicate maxima of the laser's amplitude.
• Figure 4: Accumulation of occupation and orbital angular momentum. (a) Occupation profile $\langle \Delta p_{k} \rangle$ across the ribbon (i.e., in $y$-direction) at $t = \unit[0]{fs}$ relative to the equilibrium profile (black; left abscissa). The respective OAM profile $\langle L_{k}^{z} \rangle$ is shown in red (right abscissa). (b) As panel a, but $\langle \Delta p_{k} \rangle$ versus time $t$ for two edge sites (blue: lower edge; green: upper edge) and a central site (orange). Both edges have identical occupation (cf. panel a). (c) As panel b but for $\langle L_{k}^{z} \rangle$. Opposite edges exhibit OAM with opposite signs. Vertical lines mark the maxima of the laser's amplitude.
• Figure 5: Snapshots of the orbital angular momentum (OAM) dynamics for a full period of the laser pulse. Small magenta spheres represent the Cu sites on a square lattice [fcc(001) monolayer]. Horizontally oriented small arrows along either $+x$- or $+y$-direction (connecting neighboring Cu atoms) represent the link direction and, thus, the direction of the OAM currents; their color encodes the magnitude and orientation of the $L^{z}$-polarized currents (orientation: red positive, blue negative, gray zero; magnitude: color saturation). Panels a--e show situations at selected times $t$ (as indicated). Vertically oriented arrows in panels a, c and e display the induced OAM (red positive $L^{z}$, blue negative $L^{z}$.)