Elastic propagation of fast electron vortices through crystals

Abstract

The propagation of electron beams carrying angular momentum in crystals is studied using a multislice approach for the model system Fe. It is found that the vortex beam is distorted strongly due to elastic scattering. Consequently, the expectation value of the angular momentum as well as the local vortex components change with the initial position of the vortex and the propagation depth, making numerical simulations indispensable when analyzing experiments.

On the electron vortex beam wavefunction within a crystal

“…Indeed Schattschneider et al [4] have shown that in order to detect a strong dichroic signal the vortex probe should be localised at the excited atom. A further complication is that even for a coordinate system centred about the beam propagation direction, the winding number is a constant of motion only when the electrostatic potential is either zero (free space propagation) or has cylindrical symmetry [5]. This condition is not satisfied for a zone-axis crystal, so that <L z > shows an oscillatory pendellösung-type behaviour [5].…”

“…A further complication is that even for a coordinate system centred about the beam propagation direction, the winding number is a constant of motion only when the electrostatic potential is either zero (free space propagation) or has cylindrical symmetry [5]. This condition is not satisfied for a zone-axis crystal, so that <L z > shows an oscillatory pendellösung-type behaviour [5]. At sufficiently large depths therefore the vortex probe can be in a mixed state, even for zero impact parameter.…”

Magnetic effects in the paraxial regime of elastic electron scattering

“…However, the spin of the electron is fixed while orbital angular momentum of an electron vortex beam can be deliberately increased by beam shaping techniques. and, furthermore, varies significantly as the beam scatters through a crystal 9,26 . In a magnetic material this term will be significant if the beam carries large angular momentum, which is possible with electron vortex beams 8,14,27 .…”

“…Via a Gordon decomposition 21,28 it is possible to calculate the spin current density j S (r) = ∇ × m(r) (26) and in the further considerations the orbital current density is neglected as we focus on ferromagnetic transition metals with magnetism dominated by the spin, so the index S is dropped in the notation of current density. From the current density, the periodic part of the magnetic vector potential A p is given by the Poisson equation, since Maxwell's equations tell us that…”

EMCD with an electron vortex filter: Limitations and possibilities

“…In spite of much progress in the production and application of vortex beams [6,[10][11][12][13][14][15], it soon became clear that atom-sized vortices incident on the specimen are needed for EMCD experiments [16][17][18]. Attempts to produce such beams and to use them for EMCD measurements did not show an effect so far [19].…”