Electron Vortex Research Articles

Creating electron vortex beams with light.

We propose an all-optical method of creating electron vortices utilizing the Kapitza-Dirac effect. This technique uses the transfer of orbital angular momentum from photons to free electrons creating electron vortex beams in the process. The laser intensities needed for this experiment can be obtained with available pulsed lasers and the resulting electron beams carrying orbital angular momentum will be particularly useful in the study of magnetic materials and chiral plasmonic structures in ultrafast electron microscopy.

Interactions of nonlinear electron-acoustic solitary waves with vortex electron distribution

In the present work, based on a one dimensional model, we consider the head-on-collision of nonlinear electron-acoustic waves in a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions. The analysis is based on the use of extended Poincare, Lighthill-Kuo method [C. H. Su and R. M. Mirie, J. Fluid Mech. 98, 509 (1980); R. M. Mirie and C. H. Su, J. Fluid Mech. 115, 475 (1982)]. It is shown that, for the first order approximation, the waves propagating in opposite directions are characterized by modified Korteweg-de Vries equations. In contrary to the results of previous investigations on this subject, we showed that the phase shifts are functions of both amplitudes of the colliding waves. The numerical results indicate that the waves with larger amplitude experience smaller phase shifts. Such a result seems to be plausible from physical considerations.

Electron vortex beams subject to static magnetic fields

The properties of electron vortex beams are examined when subject to static magnetic fields. The fields are assumed to be applied after the electron vortex beam carrying a well-defined orbital angular momentum has been created as a result of using a holographic mask. The shifts in the electron vortex beam energy momentum as well as its angular momentum due to the presence of an axial uniform magnetic field are evaluated. Order-of-magnitude estimates of the shifts are given with reference to typical electron vortex beams subject to moderate magnetic fields.

Electron vortex magnetic holes: A nonlinear coherent plasma structure

We report the properties of a novel type of sub-proton scale magnetic hole found in two dimensional particle-in-cell simulations of decaying turbulence with a guide field. The simulations were performed with a realistic value for ion to electron mass ratio. These structures, electron vortex magnetic holes (EVMHs), have circular cross-section. The magnetic field depression is associated with a diamagnetic azimuthal current provided by a population of trapped electrons in petal-like orbits. The trapped electron population provides a mean azimuthal velocity and since trapping preferentially selects high pitch angles, a perpendicular temperature anisotropy. The structures arise out of initial perturbations in the course of the turbulent evolution of the plasma, and are stable over at least 100 electron gyroperiods. We have verified the model for the EVMH by carrying out test particle and PIC simulations of isolated structures in a uniform plasma. It is found that (quasi-)stable structures can be formed provided that there is some initial perpendicular temperature anisotropy at the structure location. The properties of these structures (scale size, trapped population, etc.) are able to explain the observed properties of magnetic holes in the terrestrial plasma sheet. EVMHs may also contribute to turbulence properties, such as intermittency, at short scale lengths in other astrophysical plasmas.

Rotating structures in low temperature magnetized plasmas—insight from particle simulations

The EXB configuration of various low temperature plasma devices is often responsible for the formation of rotating structures and instabilities leading to anomalous electron transport across the magnetic field. In these devices, electrons are strongly magnetized while ions are weakly or not magnetized and this leads to specific physical phenomena that are not present in fusion plasmas where both electrons and ions are strongly magnetized. In this paper we describe basic phenomena involving rotating plasma structures in simple configurations of low temperature EXB plasma devices on the basis of PIC-MCC (Particle-In-Cell Monte Carlo Collisions) simulations. We focus on three examples: rotating electron vortices and rotating spokes in cylindrical magnetrons, and azimuthal electron-cyclotron drift instability in Hall thrusters. The simulations are not intended to give definite answers to the many physics issues related to low temperature EXB plasma devices but are used to illustrate and discuss some of the basic questions that need further studies.

Electron vortex beams prepared by a spiral aperture with the goal to measure EMCD on ferromagnetic films via STEM.

X-ray magnetic circular dichroism is a well established method to study element specific magnetic properties of a material, while electron magnetic circular dichroism (EMCD), which is the electron wave analogue to XMCD, is scarcely used today. Recently discovered electron vortex beams, that carry a discrete orbital angular momentum (OAM) L, are also predicted to reveal dichroic signals. Since electron beams can be easily focused down to sub-nanometer diameters, this novel technique promises the possibility to quantitatively determine local magnetic properties with unrivalled lateral resolution. As the spiralling wave front of the electron vortex beam has an azimutally growing phase shift of up to 2π and a phase singularity in its axial center, specially designed apertures are needed to generate such non-planar electron waves. We report on the preparation and successful implementation of spiral apertures into the condenser lens system of an aberration-corrected FEI Titan(3) 80-300 transmission electron microscope (TEM). This setup allows to perform scanning TEM (STEM) with vortex beams carrying user-selected OAM. First experiments on the interaction of the vortex beam with a poly-crystalline sample are presented. Within the achieved signal to noise ratio no EMCD signal has been detected. This finding is supported by simulations of inelastic scattering of a beam generated by spiral aperture.

Transition radiation of electrons with a nonzero orbital angular momentum

Several experimental groups have recently obtained the socalled vortex electrons (electrons with orbital angular momentum (OAM) of ≤ 100) with energies of ~ 300 keV. The angular momentum of such elec� trons becomes proportional to the OAM value, which leads to the corresponding increase in the electron magnetic moment. In this work, we investigate the transition radiation from the "charge + magnetic moment" system by means of the classical electrodynamics theory. The circular polarization of optical tran� sition radiation reaches its maximum level to ~ 70%, which allows using this effect for the independent mea� surement of the electron orbital momentum value.

Modeling of dislocations in a CDW junction: Interference of the CDW and normal carriers

We derive and study equations for dissipative transient processes in a constraint incommensurate charge density wave (CDW) with remnant pockets or a thermal population of normal carriers. The attention was paid to give the correct conservation of condensed and normal electrons, which was problematic at presence of moving dislocation cores if working within an intuitive Ginzburg–Landau like model. We performed a numeric modeling for stationary and transient states in a rectangular geometry when the voltage V or the normal current are applied across the conducting chains. We observe creation of an array of electronic vortices, the dislocations, at or close to the junction surface; their number increases stepwise with increasing V. The dislocation core strongly concentrates the normal carriers but the CDW phase distortions almost neutralize the total charge. At other regimes, the lines of the zero CDW amplitude flash across the sample working as phase slips. The studies were inspired by, and can be applied to experiments on mesa-junctions in NbSe3 and TaS3.

Dynamic scattering of electron vortex beams--a Bloch wave analysis.

Two important applications of electron vortex beams are in electron magnetic chiral dichroism (EMCD) measurements and nanoparticle manipulation. In both cases orbital angular momentum (〈Lz〉) transfer between the vortex beam and the specimen due to dynamic scattering is critical. In general the 〈Lz〉 pendellösung consists of short and long wavelength oscillations. The former is due to interference between the tightly bound 1s and more dispersive non-1s Bloch states, while the latter is due to interference between the non-1s states. For EMCD experiments with ±ħ angular momentum beams, momentum transfer can be minimised by selecting the appropriate aperture size, so that the probe wavefunction approximately matches that of the 2p-type Bloch states. For manipulating nanoparticles with large angular momentum beams small apertures are required to excite the 1s state and thereby enhance the short wavelength oscillations in 〈Lz〉. This enables efficient momentum transfer to the specimen, provided the nanoparticle dimension corresponds to a minimum in the 〈Lz〉 pendellösung.

Nonquasineutral electron vortices in nonuniform plasmas

Electron vortices are observed in the numerical simulation of current carrying plasmas on fast time scales where the ion motion can be ignored. In plasmas with nonuniform density n, vortices drift in the B × ∇n direction with a speed that is on the order of the Hall speed. This provides a mechanism for magnetic field penetration into a plasma. Here, we consider strong vortices with rotation speeds Vϕ close to the speed of light c where the vortex size δ is on the order of the magnetic Debye length λB=|B|/4πen and the vortex is thus nonquasineutral. Drifting vortices are typically studied using the electron magnetohydrodynamic model (EMHD), which ignores the displacement current and assumes quasineutrality. However, these assumptions are not strictly valid for drifting vortices when δ ≈ λB. In this paper, 2D electron vortices in nonuniform plasmas are studied for the first time using a fully electromagnetic, collisionless fluid code. Relatively large amplitude oscillations with periods that correspond to high frequency extraordinary modes are observed in the average drift speed. The drift speed W is calculated by averaging the electron velocity field over the vorticity. Interestingly, the time-averaged W from these simulations matches very well with W from the much simpler EMHD simulations even for strong vortices with order unity charge density separation.

Three-wave electron vortex lattices for measuring nanofields.

It is demonstrated how an electron-optical arrangement consisting of two electron biprisms can be used to generate three-wave vortex lattices with effective lattice spacings between 0.1 and 1 nm. The presence of vortices in these lattices was verified by using a third biprism to perform direct phase measurements via off-axis electron holography. The use of three-wave lattices for nanoscale electromagnetic field measurements via vortex interferometry is discussed, including the accuracy of vortex position measurements and the interpretation of three-wave vortex lattices in the presence of partial spatial coherence.

Efficient diffractive phase optics for electrons

Electron diffraction gratings can be used to imprint well-defined phase structure onto an electron beam. For example, diffraction gratings have been used to prepare electron beams with unique phase dislocations, such as electron vortex beams, which hold promise for the development of new imaging and spectroscopy techniques for the study of materials. However, beam intensity loss associated with absorption, scattering, and diffraction by a binary transmission grating drastically reduces the current in the beam, and thus the possible detected signal strength it may generate. Here we describe electron-transparent phase gratings that efficiently diffract transmitted electrons. These phase gratings produce electron beams with the high current necessary to generate detectable signal upon interaction with a material. The phase grating design detailed here allows for fabrication of much more complex grating structures with extremely fine features. The diffracted beams produced by these gratings are widely separated and carry the designed phase structure with high fidelity. In this work, we outline a fabrication method for high-efficiency electron diffraction gratings and present measurements of the performance of a set of simple prototypical gratings in a transmission electron microscope. We present a model for electron diffraction gratings that can be used to optimize the performance of diffractive electron optics. We also present several new holograms that utilize manipulation of phase to produce new types of highly efficient electron beams.

Imaging the dynamics of free-electron Landau states.

Landau levels and states of electrons in a magnetic field are fundamental quantum entities underlying the quantum Hall and related effects in condensed matter physics. However, the real-space properties and observation of Landau wave functions remain elusive. Here we report the real-space observation of Landau states and the internal rotational dynamics of free electrons. States with different quantum numbers are produced using nanometre-sized electron vortex beams, with a radius chosen to match the waist of the Landau states, in a quasi-uniform magnetic field. Scanning the beams along the propagation direction, we reconstruct the rotational dynamics of the Landau wave functions with angular frequency ~100 GHz. We observe that Landau modes with different azimuthal quantum numbers belong to three classes, which are characterized by rotations with zero, Larmor and cyclotron frequencies, respectively. This is in sharp contrast to the uniform cyclotron rotation of classical electrons, and in perfect agreement with recent theoretical predictions.

Dichroism in the interaction between vortex electron beams, plasmons, and molecules.

We study the transfer of orbital angular momentum between vortex electron beams and chiral samples, such as staircase plasmonic nanostructures and biomolecules. Inelastic electron scattering from these samples produces large dichroism in the momentum-resolved electron energy-loss spectra. We illustrate this phenomenon with calculations for chiral and nonchiral clusters of silver spheres using both focused and extended electron beams, which exhibit ∼10% difference between channels of opposite angular momentum. In addition to its fundamental interest, this remarkably high dichroism suggests a way of spatially resolving chiral optical excitations, including dark plasmons. We also predict a dichroic response when probing a chiral biomolecule, which suggests the use of these electron beams for resolving different enantiomers.

Superposition of Fraunhofer Diffractions from Fork-Shaped Gratings and their Openings with Electron Vortex Beam

Superposition of Fraunhofer Diffractions from Fork-Shaped Gratings and their Openings with Electron Vortex Beam

Quantitative measurement of orbital angular momentum in electron microscopy

Electron vortex beams have been predicted to enable atomic scale magnetic information measurement, via transfer of orbital angular momentum. Research so far has focussed on developing production techniques and applications of these beams. However, methods to measure the outgoing orbital angular momentum distribution are also a crucial requirement towards this goal. Here, we use a method to obtain the orbital angular momentum decomposition of an electron beam, using a multi-pinhole interferometer. We demonstrate both its ability to accurately measure orbital angular momentum distribution, and its experimental limitations when used in a transmission electron microscope.

Toward single mode, atomic size electron vortex beams.

We propose a practical method of producing a single mode electron vortex beam suitable for use in a scanning transmission electron microscope (STEM). The method involves using a holographic "fork" aperture to produce a row of beams of different orbital angular momenta, as is now well established, magnifying the row so that neighboring beams are separated by about 1 µm, selecting the desired beam with a narrow slit, and demagnifying the selected beam down to 1-2 Å in size. We show that the method can be implemented by adding two condenser lenses plus a selection slit to a straight-column cold-field emission STEM. It can also be carried out in an existing instrument, the monochromated Nion high-energy-resolution monochromated electron energy-loss spectroscopy-STEM, by using its monochromator in a novel way. We estimate that atom-sized vortex beams with ≥ 20 pA of current should be attainable at 100-200 keV in either instrument.

Scattering of electron vortex beams on a magnetic crystal: Towards atomic-resolution magnetic measurements

Use of electron vortex beams (EVB), that is convergent electron beams carrying an orbital angular momentum (OAM), is a novel development in the field of transmission electron microscopy. They should allow measurement of element-specific magnetic properties of thin crystals using electron magnetic circular dichroism (EMCD)---a phenomenon similar to the x-ray magnetic circular dichroism. Recently it has been shown computationally that EVBs can detect magnetic signal in a scanning mode only at atomic resolution. In this follow-up work we explore in detail the elastic and inelastic scattering properties of EVBs on crystals, as a function of beam diameter, initial OAM, acceleration voltage and beam displacement from an atomic column. We suggest that for a 10 nm layer of bcc iron oriented along (001) zone axis an optimal configuration for a detection of EMCD is an EVB with OAM of $1\hbar$ and a diameter of 1.6 \AA, acceleration voltage 200 keV and an annular detector with inner and outer diameters of $G$ and $5G$, respectively, where $\mathbf{G}=(100)$.

A perturbation theory study of electron vortices in electromagnetic fields: The case of infinitely long line charge and magnetic dipole

The novel discovery of electron vortices carrying quantized orbital angular momentum motivated intensive research of their basic properties as well as applications, e.g. structural characterization of magnetic materials. In this paper, the fundamental interactions of electron vortices within infinitely long atomic-column-like electromagnetic fields are studied based on the relativistically corrected Pauli–Schrödinger equation and the perturbation theory. The relative strengths of three fundamental interactions, i.e. the electron–electric potential interaction, the electron–magnetic potential/field interaction and the spin–orbit coupling are discussed. The results suggest that the perturbation energies of the last two interactions are in an order of 103–104 smaller than that of the first one for electron vortices. In addition, it is also found that the strengths of these interactions are strongly dependant on the spatial distributions of the electromagnetic field as well as the electron vortices.

The geometric phase and the geometrodynamics of relativistic electron vortex beams

We have studied here the geometrodynamics of relativistic electron vortex beams from the perspective of the geometric phase associated with the scalar electron encircling the vortex line. It is pointed out that the electron vortex beam carrying orbital angular momentum is a natural consequence of the skyrmion model of a fermion. This follows from the quantization procedure of a fermion in the framework of Nelson's stochastic mechanics when a direction vector (vortex line) is introduced to depict the spin degrees of freedom. In this formalism, a fermion is depicted as a scalar particle encircling a vortex line. It is here shown that when the Berry phase acquired by the scalar electron encircling the vortex line involves quantized Dirac monopole, we have paraxial (non-paraxial) beam when the vortex line is parallel (orthogonal) to the wavefront propagation direction. Non-paraxial beams incorporate spin–orbit interaction. When the vortex line is tilted with respect to the propagation direction, the Berry phase involves non-quantized monopole. The temporal variation of the direction of the tilted vortices is studied here taking into account the renormalization group flow of the monopole charge and it is predicted that this gives rise to the spin Hall effect.