Electron Vortex Research Articles

Elastic Scattering of Electron Vortex Beams in Magnetic Matter.

Elastic scattering of electron vortex beams on magnetic materials leads to a weak magnetic contrast due to Zeeman interaction of orbital angular momentum of the beam with magnetic fields in the sample. The magnetic signal manifests itself as a redistribution of intensity in diffraction patterns due to a change of sign of the orbital angular momentum of the electron vortex beam. While in the atomic resolution regime the magnetic signal is most likely under the detection limits of present transmission electron microscopes, for electron probes with high orbital angular momenta, and correspondingly larger spatial extent, its detection is predicted to be feasible.

The possibility of amplification and generation of higher harmonics in a microwave device with turbulent electron flow (Natural experiment and phenomenological model)

A generator of turbulent electron beams in an independent mode and in the mode of amplification of an external single-frequency signal is experimentally investigated. The formation of higher harmonic components in the spectrum of output radiation of the generator is shown experimentally. The phenomenological model of the turbulent electron flow is constructed on the basis of a chain of interacting “electron vortices” described by the modified nonlinear Van-der-Pol differential equations. The modes of independent dynamics and the mode of amplification of an external harmonic signal are considered. The results of numerical modeling and experimental investigations are compared. The qualitative correspondence of the behavior of the dependences obtained from the experimental investigation of the generator of turbulent electron beams and the results obtained from numerical modeling is shown.

Influence of nuclear quantum effects on frozen phonon simulations of electron vortex beam HAADF-STEM images

We have evaluated atomic resolution high-angle annular dark field images with ordinary beams and electron vortex beams for thin crystals of bcc iron, explicitly considering the atomic vibrations using molecular dynamics. The shape of the image representing an atomic column depends on the orbital angular momentum, sample thickness and temperature. For electron vortex beams we observe characteristic doughnut-shaped images of atomic columns. It is shown how the thermal diffuse scattering reduces the depth of their central minima, which get further smeared by finite source size effects. In addition, it is shown that in calculations of HAADF-STEM images at low temperatures one has to explicitly consider the nuclear quantum effects (zero point vibrations), otherwise the effect of atomic vibrations is strongly underestimated.

Relativistic Electron Vortex Beams in a Laser Field.

The orbital angular momentum Hall effect and the spin Hall effect of electron vortex beams (EVBs) have been studied for the EVBs interacting with a laser field. In the scenario of a paraxial beam, the cumulative effect of the orbit-orbit interaction of EVBs and laser fields drives the orbital Hall effect, which in turn produces a shift of the center of the beam from that of the field-free case towards the polarization axis of the photons. In addition, for nonparaxial beams one can also perceive a similar shift of the center of the beam owing to the spin Hall effect involving spin-orbit interaction. Our analysis suggests that the shift in the paraxial beams will always be larger than that in the nonparaxial beams.

Focused electron beam induced deposition as a tool to create electron vortices

Focused electron beam induced deposition (FEBID) is a microscopic technique that allows geometrically controlled material deposition with very high spatial resolution. This technique was used to create a spiral aperture capable of generating electron vortex beams in a transmission electron microscope (TEM). The vortex was then fully characterized using different TEM techniques, estimating the average orbital angular momentum to be ∼0.8ℏ per electron with almost 60% of the beam ending up in the ℓ=1 state.

Electron vortices in photoionization by circularly polarized attosecond pulses.

Single ionization of He by two oppositely circularly polarized, time-delayed attosecond pulses is shown to produce photoelectron momentum distributions in the polarization plane having helical vortex structures sensitive to the time delay between the pulses, their relative phase, and their handedness. Results are obtained by both abinitio numerical solution of the two-electron time-dependent Schrödinger equation and by a lowest-order perturbation theory analysis. The energy, bandwidth, and temporal duration of attosecond pulses are ideal for observing these vortex patterns.

Parallel axis theorem for free-space electron wavefunctions

We consider the orbital angular momentum of a free electron vortex moving in a uniform magnetic field. We identify three contributions to this angular momentum: the canonical orbital angular momentum associated with the vortex, the angular momentum of the cyclotron orbit of the wavefunction, and a diamagnetic angular momentum. The cyclotron and diamagnetic angular momenta are found to be separable according to the parallel axis theorem. This means that rotations can occur with respect to two or more axes simultaneously, which can be observed with superpositions of vortex states.

A note on the cylindrical solitary waves in an electron-acoustic plasma with vortex electron distribution

In the present work, we consider the propagation of nonlinear electron-acoustic non-planar waves in a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions. The basic nonlinear equations of the above described plasma are re-examined in the cylindrical coordinates through the use reductive perturbation method in the long-wave approximation. The modified cylindrical Korteweg–de Vries equation with fractional power nonlinearity is obtained as the evolution equation. Due to the nature of nonlinearity, which is fractional, this evolution equation cannot be reduced to the conventional Korteweg–de Vries equation. An analytical solution to the evolution equation, by use of the method developed by Demiray [Appl. Math. Comput. 132, 643 (2002); Comput. Math. Appl. 60, 1747 (2010)] and a numerical solution by employing a spectral scheme are presented and the results are depicted in a figure. The numerical results reveal that both solutions are in good agreement.

Disentangling multipole contributions to collective excitations in fullerenes

Angular resolved electron energy-loss spectroscopy (EELS) gives access to the momentum and the energy dispersion of electronic excitations and allows to explore the transition from individual to collective excitations. Dimensionality and geometry play thereby a key role. As a prototypical example we analyze theoretically the case of Buckminster fullerene C60 using ab initio calculations based on the time-dependent density-functional theory. Utilizing the non-negative matrix factorization method, multipole contributions to various collective modes are isolated, imaged in real space, and their energy and momentum dependencies are traced. A possible experiment is suggested to access the multipolar excitations selectively via EELS with electron vortex (twisted) beams. Furthermore, we construct an accurate analytical model for the response function. Both the model and the ab initio cross sections are in excellent agreement with recent experimental data.

Electron Singularities, Matter Wave Catastrophes, and Vortex Lattice Singularimetry

Singularities in wave fields, such as phase vortices, are of substantial interest for light optics, matter waves and x-ray optics. New theoretical insights [1] and cutting edge nanofabrication techniques have recently created a wide variety of electron vortex beams [2, 3] for the transmission electron microscope (TEM), which can provide spectroscopic information pertaining to the magnetic properties of nanoscale specimens [2]. The rich diffraction physics of singular optics is vast and many concepts can be directly translated to electron microscopy. Interesting physics has been discovered by highlighting differences between electron and optical vortex beams, such as a remarkable interplay between Zeeman coupling, Landau levels and Gouy phases in the TEM [4] or the Faraday effect for electron vortex beams propagating in a magnetic field, which interestingly occurs in vacuum [5].

The Surprising Dynamics of Electron Vortex Beams

The Surprising Dynamics of Electron Vortex Beams

At-Focus Observations of High Quality Electron Vortex Beams Created from Ferromagnetic Rods

Journal Article At-Focus Observations of High Quality Electron Vortex Beams Created from Ferromagnetic Rods Get access Arthur M Blackburn, Arthur M Blackburn Hitachi Cambridge Laboratory, Hitachi Europe Ltd., Cambridge, U. K Search for other works by this author on: Oxford Academic Google Scholar James C Loudon, James C Loudon Department of Materials Science and Metallurgy, University of Cambridge, Cambridge, U. K Search for other works by this author on: Oxford Academic Google Scholar Rodney Herring, Rodney Herring Department of Mechanical Engineering, University of Victoria, Victoria, BC, Canada Search for other works by this author on: Oxford Academic Google Scholar Ales Hrabec, Ales Hrabec School of Physics and Astronomy, University of Leeds, Leeds, U.K Search for other works by this author on: Oxford Academic Google Scholar David Hoyle David Hoyle Hitachi High-Technologies Canada, Toronto, Canada Search for other works by this author on: Oxford Academic Google Scholar Microscopy and Microanalysis, Volume 21, Issue S3, 1 August 2015, Pages 501–502, https://doi.org/10.1017/S143192761500330X Published: 23 September 2015

Peculiar rotation of electron vortex beams

Standard electron optics predicts Larmor image rotation in the magnetic lens field of a TEM. Introducing the possibility to produce electron vortex beams with quantized orbital angular momentum brought up the question of their rotational dynamics in the presence of a magnetic field. Recently, it has been shown that electron vortex beams can be prepared as free electron Landau states showing peculiar rotational dynamics, including no and cyclotron (double-Larmor) rotation. Additionally very fast Gouy rotation of electron vortex beams has been observed. In this work a model is developed which reveals that the rotational dynamics of electron vortices are a combination of slow Larmor and fast Gouy rotations and that the Landau states naturally occur in the transition region in between the two regimes. This more general picture is confirmed by experimental data showing an extended set of peculiar rotations, including no, cyclotron, Larmor and rapid Gouy rotations all present in one single convergent electron vortex beam.

On the electron vortex beam wavefunction within a crystal

Electron vortex beams are distorted by scattering within a crystal, so that the wavefunction can effectively be decomposed into many vortex components. Using a Bloch wave approach equations are derived for vortex beam decomposition at any given depth and with respect to any frame of reference. In the kinematic limit (small specimen thickness) scattering largely takes place at the neighbouring atom columns with a local phase change of π/2rad. When viewed along the beam propagation direction only one vortex component is present at the specimen entrance surface (i.e. the ‘free space’ vortex in vacuum), but at larger depths the probe is in a mixed state due to Bragg scattering. Simulations show that there is no direct correlation between vortex components and the 〈Lz〉 pendellösung, i.e. at a given depth probes with relatively constant 〈Lz〉 can be in a more mixed state compared to those with more rapidly varying 〈Lz〉. This suggests that minimising oscillations in the 〈Lz〉 pendellösung by probe channelling is not the only criterion for generating a strong electron energy loss magnetic circular dichroism (EMCD) signal.

Uncertainty relation between angle and orbital angular momentum: interference effect in electron vortex beams

The uncertainty relation between angle and orbital angular momentum had not been formulated in a similar form as the uncertainty relation between position and linear momentum because the angle variable is not represented by a quantum mechanical self-adjoint operator. Instead of the angle variable operator, we introduce the complex position operator $ \hat{Z} = \hat{x}+i \hat{y} $ and interpret the order parameter $ \mu = \langle \hat{Z} \rangle / \sqrt{ \langle \hat{Z}^\dagger \hat{Z} \rangle} $ as a measure of certainty of the angle distribution. We prove the relation between the uncertainty of angular momentum and the angle order parameter. We prove also its generalizations and discuss experimental methods for testing these relations.

Properties and origin of subproton‐scale magnetic holes in the terrestrial plasma sheet

AbstractElectron‐scale magnetic depressions in the terrestrial plasma sheet are studied using Cluster multispacecraft data. The structures, which have an observed duration of ~5–10 s, are approximately 200–300 km wide in the direction of propagation, and they show an average reduction in the background magnetic field of 10–20%. A majority of the events are also associated with an increase in the high‐energy high pitch angle electron flux, which indicates that the depressions are presumably generated by electrons with relatively high velocity perpendicular to the background magnetic field. Differences in the recorded electron spectra in the four spacecraft indicates a possible nongyrotropic structure. Multispacecraft measurements show that a subset of events are cylindrical, elongated along the magnetic field, and with a field‐parallel scale size of at a minimum 500 km. Other events seem to be better described as electron‐scale sheets, about 200–300 km thick. We find that no single formation mechanism can explain this variety of events observed. Instead, several processes may be operating in the plasma sheet, giving rise to similar magnetic field structures in the single‐spacecraft data, but with different 3‐D structuring. The cylindrical structures have several traits that are in agreement with the electron vortex magnetic holes observed in 2‐D particle‐in‐cell simulations of turbulent relaxation, whereas the sheets, which show nearly identical signatures in the multispacecraft data, are better explained by propagating electron solitary waves.

Probing Chirality with Electron Vortices

Probing Chirality with Electron Vortices

Using electron vortex beams to determine chirality of crystals in transmission electron microscopy

We investigate electron vortex beams elastically scattered on chiral crystals. After deriving a general expression for the scattering amplitude of a vortex electron, we study its diffraction on point scatterers arranged on a helix. We derive a relation between the handedness of the helix and the topological charge of the electron vortex on one hand, and the symmetry of the Higher Order Laue Zones in the diffraction pattern on the other for kinematically and dynamically scattered electrons. We then extend this to atoms arranged on a helix as found in crystals which belong to chiral space groups and propose a new method to determine the handedness of such crystals by looking at the symmetry of the diffraction pattern. Contrary to alternative methods, our technique does not require multiple scattering which makes it possible to also investigate extremely thin samples in which multiple scattering is suppressed. In order to verify the model, elastic scattering simulations are performed and an experimental demonstration on Mn$_2$Sb$_2$O$_7$ is given where we find the sample to belong to the right handed variant of its enantiomorphic pair. This demonstrates the usefulness of electron vortex beams to reveal the chirality of crystals in a transmission electron microscope and provides the required theoretical basis for further developments in this field.

Modulation of Electron-Acoustic Waves in a Plasma with Vortex Electron Distribution

Abstract In the present work, employing a one-dimensional model of a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution and stationary ions, we study the amplitude modulation of electron-acoustic waves by use of the conventional reductive perturbation method. Employing the field equations with fractional power type of nonlinearity, we obtained the nonlinear Schrödinger equation as the evolution equation of the same order of nonlinearity. Seeking a harmonic wave solution with progressive wave amplitude to the evolution equation it is found that the NLS equation with fractional power assumes envelope type of solitary waves.

Electron vortex beams in a magnetic field and spin filter

We investigate the propagation of electron vortex beams in a magnetic field. It is pointed out that when electron vortex beams carrying orbital angular momentum propagate in a magnetic field, the Berry curvature associated with the scalar electron moving in a cyclic path around the vortex line is modified from that in free space. This alters the spin-orbit interaction, which affects the propagation of nonparaxial beams. The electron vortex beams with tilted vortices lead to a spin Hall effect in free space. In the presence of a magnetic field in time-space we have spin filtering such that either positive or negative spin states emerge in spin Hall currents with clustering of spin $\frac{1}{2}$ states.