Generation of Current Vortex by Spin Current in Rashba Systems

Employing unbiased large-scale time-dependent density-matrix renormalization-group simulations, we demonstrate the generation of a charge-current vortex via spin injection in the Rashba system. The spin current is polarized perpendicular to the system plane and injected from an attached antiferromagnetic spin chain. We discuss the conversion between spin and orbital angular momentum in the current vortex that occurs because of the conservation of the total angular momentum and the spin-orbit interaction. This is in contrast to the spin Hall effect, in which the angular-momentum conservation is violated. Finally, we predict the electromagnetic field that accompanies the vortex with regard to possible future experiments.

Figure 1

Sketch of the setup described by Eqs. (1)–(4). A spin current (purple arrow) polarized perpendicular to the Rashba plane is induced in the spin chain by switching on a spin voltage in the lead. This spin current is injected into the Rashba system, where it causes the formation of a charge-current vortex (red and blue arrows). The orange segments denote the coupling between the spin chain and the lead and Rashba systems. In an experiment, the magnetic field induced by the charge current may be detected using scanning probe microscopy.

Figure 2

Tensor-network-state ansatz for the numerical simulations. The vertical lines denote the physical indices, i.e., the basis states of the local Hilbert spaces. Here, they correspond to the occupation numbers $n_{j,\sigma }$ of the fermions in the Lanczos basis or, in the spin chain, the $z$ components $S_j^z$ of the spins. The remaining lines indicate the bond indices of the tensor network. On the left side, the one-dimensional lead is similarly split into two branches (not shown).

Figure 3

Snapshot of the charge-current densities ${\mathit{j}}_{\mathit{r}}^c$ at different times $t$. For easier visualization, each arrow corresponds to the average value of the currents in a square of $3\times 3$ sites. The length and color of the arrows indicate the magnitude and direction of the current, respectively. Black arrows show that the current points in the radial direction, while blue (red) arrows denote current in the clockwise (counterclockwise) azimuthal direction.

Figure 4

Radial dependence of the azimuthal component of the charge current at time $t=45$. The solid lines are according to Eqs. (9) and (10).

Figure 5

Time evolution of the $z$ component of the spin and orbital angular momentum in the Rashba system. The total spins in the lead and the spin chain are denoted by $S_L^z$ and $S_S^z$, respectively. Therefore, the black line indicates the injected spin angular momentum. It would match the blue line if the total angular momentum was conserved.