Nonuniform Currents and Spins of Relativistic Electron Vortices in a Magnetic Field

We present a relativistic description of electron vortex beams in a homogeneous magnetic field. Including spin from the beginning reveals that spin-polarized electron vortex beams have a complicated azimuthal current structure, containing small rings of counterrotating current between rings of stronger corotating current. Contrary to many other problems in relativistic quantum mechanics, there exists a set of vortex beams with exactly zero spin-orbit mixing in the highly relativistic and nonparaxial regime. The well-defined phase structure of these beams is analogous to simpler scalar vortex beams, owing to the protection by the Zeeman effect. For states that do show spin-orbit mixing, the spin polarization across the beam is nonuniform rendering the spin and orbital degrees of freedom inherently inseparable.

Figure 1

The energy levels for a fixed value of $k$ sorted by their total angular momentum. The states with positive spin (red) have one quantum of squared energy more than the states with negative spin (blue). Thus the ground states are not degenerate with any opposite spin states and cannot spin-orbit mix as indicated by the arrows.

Figure 2

$j_{\varphi }$ for positive spin (red), negative spin (blue), and a spin 0 beam for comparison (dashed line) for $l=2$, $p=3$ (a), $l=-2$, $p=3$ (b), and $l=-2$, $p=0$ (c). The spin part of the current gives rise to a series of dips where the current flows in the opposite direction, which are absent when spin is neglected. For negative $l$, the azimuthal current is negative near the center but positive on the outside due to the interaction with the magnetic field. The most striking difference from a spin 0 vortex beam occurs for negative $l$ and $p=0$ where the negative spin is a Landau-Zeeman ground state and the azimuthal current is exactly zero everywhere.

Figure 3

The regions of negative ($=\text{clockwise}$) azimuthal electron current marked for the $p=3$, $l=2$ state (a) and $p=3$, $l=-2$ (b) for negative (left side, in blue) and positive spin (right side, in red) superposed on the beam profiles for a magnetic field of 1 T. The negative currents occur on the inner side of the dark fringes for positive spin and on the outer side for negative spin. Visible rearrangement of the electron density due to spin-orbit mixing only appears around 1 GT.