Experimental realization of a π/2 vortex mode converter for electrons using a spherical aberration corrector

In light optics, beams with orbital angular momentum (OAM) can be produced by employing a properly-tuned two-cylinder-lens arrangement, also called π/2 mode converter. It is not possible to convey this concept directly to the beam in an electron microscope due to the non-existence of cylinder lenses in commercial transmission electron microscopes (TEMs). A viable work-around are readily-available electron optical elements in the form of quadrupole lenses. In a proof-of-principle experiment in 2012, it has been shown that a single quadrupole in combination with a Hilbert phase-plate produces a spatially-confined, transient vortex mode. Here, an analogue to an optical π/2 mode converter is realized by repurposing a CEOS DCOR probe corrector in an aberration corrected TEM in a way that it resembles a dual cylinder lens using two quadrupoles. In order to verify the presence of OAM in the output beam, a fork dislocation grating is used as an OAM analyser. The possibility to use magnetic quadrupole fields instead of, e.g., prefabricated fork dislocation gratings to produce electron beams carrying OAM enhances the beam brightness by almost an order of magnitude and delivers switchable high-mode purity vortex beams without unwanted side-bands.

π/2 mode converters and vortex generators for electrons

In optics, mode conversion is an elegant way to switch between Hermite Gaussian and Laguerre Gaussian beam profiles and thereby impart orbital angular momentum onto the beam and to create vortices. In optics such vortex beams can be produced in a setup consisting of two identical cylinder lenses. In electron optics, quadrupole lenses can be used for the same purpose. Here we investigate generalized asymmetric designs of a quadrupole mode converter that may be realized within the constraints of existing electron microscopes and can steer the development of dedicated vortex generators for high brilliance electron vortex probes of atomic scale.

Entanglement and decoherence in electron microscopy

Interaction of the probe with the specimen in an electron microscope inevitably leads to entanglement between the probe and the scatterer. In spite of the importance of entanglement in many areas of modern physics, this subject has not been touched in the literature. Here, we develop some ideas about entanglement in electron microscopy for a number of scattering mechanisms. The relationship between entropy, density matrices, and coherence is discussed. In addition, we explore the questions "Why is Bragg scattering coherent and energy loss incoherent?" and "When does decoherence play a role?" It seems to be possible to measure decoherence on extremely short timescales of ∼10^-8s. This is especially important in view of recent developments in ultrafast electron microscopy.

EMCD with an electron vortex filter: Limitations and possibilities

We discuss the feasibility of detecting spin polarized electronic transitions with a vortex filter. This approach does not rely on the principal condition of the standard electron energy-loss magnetic chiral dichroism (EMCD) technique, the precise alignment of the crystal in order to use it as a beam splitter, and thus would pave the way for the application of EMCD to new classes of materials and problems, like amorphous magnetic alloys and interface magnetism. The dichroic signal strength at the L_{2, 3}-edge of ferromagnetic Cobalt (Co) is estimated on theoretical grounds using a single atom scattering approach. To justify this approach, multi-slice simulations were carried out in order to confirm that orbital angular momentum (OAM) is conserved in amorphous materials over an extended range of sample thickness and also in very thin crystalline specimen, which is necessary for the detection of EMCD. Also artefact sources like spot size, mask tilt and astigmatism are discussed. In addition, the achievable SNR under typical experimental conditions is assessed.

Spin polarisation with electron Bessel beams

The theoretical possibility to use an electron microscope as a spin polarizer is studied. It turns out that a Bessel beam passing a standard magnetic objective lens is intrinsically spin polarized when post-selected on-axis. In the limit of infinitely small detectors, the spin polarisation tends to 100%. Increasing the detector size, the polarisation decreases rapidly, dropping below 10^-4 for standard settings of medium voltage microscopes. For extremely low voltages, the Figure of Merit increases by two orders of magnitude, approaching that of existing Mott detectors. Our findings may lead to new desings of spin filters, an attractive option in view of its inherent combination with the electron microscope, especially at low voltage.

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.

Is magnetic chiral dichroism feasible with electron vortices?

We discuss the feasibility of detecting magnetic transitions with focused electron vortex probes, suggested by selection rules for the magnetic quantum number. We theoretically estimate the dichroic signal strength in the L₂,₃ edge of ferromagnetic d metals. It is shown that under realistic conditions, the dichroic signal is undetectable for nanoparticles larger than ∼1 nm. This is confirmed by a key experiment with nanometer-sized vortices.

Novel vortex generator and mode converter for electron beams

A mode converter for electron vortex beams is described. Numerical simulations, confirmed by experiment, show that the converter transforms a vortex beam with a topological charge m=±1 into beams closely resembling Hermite-Gaussian HG(10) and HG(01) modes. The converter can be used as a mode discriminator or filter for electron vortex beams. Combining the converter with a phase plate turns a plane wave into modes with topological charge m=±1. This combination serves as a generator of electron vortex beams of high brilliance.

Theory of free electron vortices

The recent creation of electron vortex beams and their first practical application motivates a better understanding of their properties. Here, we develop the theory of free electron vortices with quantized angular momentum, based on solutions of the Schrödinger equation for cylindrical boundary conditions. The principle of transformation of a plane wave into vortices with quantized angular momentum, their paraxial propagation through round magnetic lenses, and the effect of partial coherence are discussed.

Breakdown of the dipole approximation in core losses

The validity of the dipole approximations commonly used in the inelastic scattering theory for transmission electron microscopy is reviewed. Both experimental and numerical arguments are presented, emphasizing that the dipole approximations cause significant errors of the order of up to 25% even at small momentum transfer. This behavior is attributed mainly to non-linear contributions to the dynamic form factor due to the overlap of wave functions.

A software package for the simulation of energy-loss magnetic chiral dichroism

In this work, the theoretical framework for energy-loss magnetic chiral dichroism (EMCD) is reviewed and a new, fast, and easy-to-use software package is presented. This software is used to simulate the dependence of the EMCD signal on various parameters such as sample thickness and sample tilt. The results are found to be in excellent agreement with other theoretical predictions and experimental data. Furthermore, it is shown that the simulations are fast enough for the planning of experiments "on-the-fly" and for live didactic demonstrations.

Treating retardation effects in valence EELS spectra for Kramers–Kronig analysis

Retardation effects such as Cerenkov losses and waveguide modes alter the valence electron energy-loss spectrum of semiconductors and insulators as soon as the speed of the probing electron exceeds the speed of light inside the probed medium. This leads to the dilemma, that optical properties from these media cannot be determined correctly using electron energy-loss spectrometry (EELS) if no corrections are applied. In this work we present two ways out of this dilemma: a reduction of the beam energy and the application of an off-line correction. We demonstrate the accuracy of these two methods by using two similar layers of Si(x):H having slightly different refractive indices and discuss the impact of the normalization parameter during Kramers-Kronig analysis (KKA) on the obtained dielectric properties. We further demonstrate that KKA can be applied without the use of standard specimens, if thickness determination using transmission electron microscopy and EELS is accurate enough.

Optimal aperture sizes and positions for EMCD experiments

The signal-to-noise ratio (SNR) in energy-loss magnetic chiral dichroism (EMCD)--the equivalent of X-ray magnetic circular dichroism (XMCD) in the electron microscope--is optimized with respect to the detector shape, size and position. We show that an important increase in SNR over previous experiments can be obtained when taking much larger detector sizes. We determine the ideal shape of the detector but also show that round apertures are a good compromise if placed in their optimal position. We develop the theory for a simple analytical description of the EMCD experiment and then apply it to dynamical multibeam Bloch wave calculations and to an experimental data set. In all cases it is shown that a significant and welcome improvement of the SNR is possible.

Magnetic circular dichroism in electron energy loss spectrometry

The measurement of circular dichroism in the electron microscope is a new, emerging method and, as such, it is subject to constant refinement and improvement. Different ways can be envisaged to record the signal. We present an overview of the key steps in the energy-loss magnetic chiral dichroism (EMCD) experiment as well as a detailed review of the methods used in the intrinsic way where the specimen is used as a beam splitter. Lateral resolution up to 20-30 nm can be achieved, and the use of convergent beam techniques leads to an improved S/N ratio. Dichroic effects are shown for Ni and Co single crystal; as a counterexample, measurements were carried also for a non-magnetic (Ti) sample, where no dichroic effect was found.

The influence of relativistic energy losses on bandgap determination using valence EELS

Since monochromated transmission electron microscopes have become available, the determination of bandgaps and optical properties using electron energy loss spectrometry (EELS) has again attracted interest. The underlying idea is very simple: below the bandgap energy no transitions can contribute to the valence EELS signal. However, the bandgap cannot be directly read out from the recorded data. Therefore the optical properties cannot be determined correctly from the low loss using the Kramers-Kronig relations. We will discuss under which conditions relativistic effects may be suppressed. It is demonstrated that scanning TEM (STEM) geometry is not applicable for most bandgap measurements.

Inelastic electron holography as a variant of the Feynman thought experiment

Using a combination of electron holography and energy filtering, interference fringes produced after inelastic interaction of electrons with hydrogen molecules are examined. Surprisingly, the coherence of inelastic scattering increases when moving from the surface of a hydrogen-containing bubble to the vacuum. This phenomenon can be understood in terms of the Feynman two-slit thought experiment with a variable ambiguity of the which-way registration.

Imaging using inelastically scattered electrons in CTEM and STEM geometry

It is shown that energy filtered transmission electron microscopy images are closely related to energy spectroscopic scanning transmission electron microscopy images. For the case of a single atom, we explore this similarity using both the coupled channels and density matrix approaches. We extend the result to the crystal case and find that the similarity persists, the limiting effects due to energy differences in the scattered electrons being small for typical specimen thicknesses in high-resolution transmission electron microscopy.

ELNES at magic angle conditions

If one needs to cancel the effects of the anisotropy of the sample in a EELS experiment in the TEM, a particular couple of values for the collection and convergence angle must be used, called magic angle conditions (MAC). Recent developments in the theory have shown that a full relativistic treatment is mandatory to correctly describe this effect and that the MAC are strongly dependent on the acceleration voltage. We show how the analytical formula can be derived and give the exact analytical solution for the MAC which can then be easily applied to every practical case. We show the consequences of the energy dependence of the magic angle and that the parallelity of the beam will be the limiting factor for high acceleration voltages while for low acceleration voltages the contribution coming from Bragg spots may make it impossible to reach MAC.

Short note on parallel illumination in the TEM

Parallel illumination conditions are required for several experiments in the transmission electron microscope (TEM). The image rotation induced by the helical trajectory of electrons passing through the magnetic field of the TEM lenses inevitably induces an inclination of the beam relative to the optical axis in the object plane--even for an electron which travels parallel to the optical axis in the far field. This angle (shear angle) is vectorially added to the convergence angle; it depends both on the distance to the optical axis and the magnetic field. By using a beam tilt compensation method, the minimum shear angle is found to be of the order of 1 mrad for a field of view of 2 microm in a 200 kV TEM. In practice, "parallel illumination" can only be obtained for fields of view 1 microm.

Čerenkov losses: A limit for bandgap determination and Kramers–Kronig analysis

Measuring low energy losses in semiconductors and insulators with high spatial resolution becomes attractive with the increasing availability of modern transmission electron microscopes (TEMs) equipped with monochromators, C(s) correctors and energy filters. In this paper, we demonstrate that Cerenkov losses pose a limit for the interpretation of low energy loss spectra (EELS) in terms of interband transistions and bandgap determination for many materials. If the velocity of a charged particle in a medium exceeds the velocity of light, photons are emitted leading to a corresponding energy loss of a few electronvolt. Since these losses are strong for energies below the onset of interband transitions, they change the apparent loss function of semiconductors and insulators, with the risk of erroneous interpretation of spectra. We measured low energy losses of Si and GaAs with a monochromated TEM demonstrating the effect of sample thickness on Cerenkov losses. Angle resolved EELS and energy filtered diffraction patterns (taken without a monochromator) show the extremely narrow angular distribution of Cerenkov losses. The latter experiment provides a method that allows to decide whether Cerenkov radiation masks the very low loss signal in EELS.

Detection of magnetic circular dichroism using a transmission electron microscope

A material is said to exhibit dichroism if its photon absorption spectrum depends on the polarization of the incident radiation. In the case of X-ray magnetic circular dichroism (XMCD), the absorption cross-section of a ferromagnet or a paramagnet in a magnetic field changes when the helicity of a circularly polarized photon is reversed relative to the magnetization direction. Although similarities between X-ray absorption and electron energy-loss spectroscopy in a transmission electron microscope (TEM) have long been recognized, it has been assumed that extending such equivalence to circular dichroism would require the electron beam in the TEM to be spin-polarized. Recently, it was argued on theoretical grounds that this assumption is probably wrong. Here we report the direct experimental detection of magnetic circular dichroism in a TEM. We compare our measurements of electron energy-loss magnetic chiral dichroism (EMCD) with XMCD spectra obtained from the same specimen that, together with theoretical calculations, show that chiral atomic transitions in a specimen are accessible with inelastic electron scattering under particular scattering conditions. This finding could have important consequences for the study of magnetism on the nanometre and subnanometre scales, as EMCD offers the potential for such spatial resolution down to the nanometre scale while providing depth information--in contrast to X-ray methods, which are mainly surface-sensitive.

Plasmon holographic experiments: theoretical framework

A theoretical framework is described to understand the results of plasmon holography experiments leading to insight in the meaning of the experimental results and pointing out directions for future experiments. The framework is based on the formalism of mutual intensity to describe how coherence is transferred through an optical system. For the inelastic interaction with the object, an expression for the volume plasmon excitations in a free electron gas is used as a model for the behaviour of aluminium. The formalism leads to a clear graphical intuitive tool for understanding the experiments. It becomes evident that the measured coherence is solely related to the angular distribution of the plasmon scattering in the case of bulk plasmons. After describing the framework, the special case of coherence outside a spherical particle is treated and the seemingly controversial idea of a plasmon with a limited coherence length obtained from experiments is clarified.

The magic angle: a solved mystery

We resolve the long-standing mysterious discrepancy between the experimental magic angle in EELS--approximately 2theta(E)--and the quantum mechanical prediction of approximately 4theta(E). A relativistic approach surpassing the usually applied kinematic correction yields a magic angle close to the experimental value. The reason is that the relativistic correction of the inelastic scattering cross section in anisotropic systems is significantly higher than in isotropic ones.

Channeling, localization and the density matrix in inelastic electron scattering

Building on the relationship between the mixed dynamic form factor (MDFF) for inelastic electron scattering and the one-particle density matrices of the initial and final states of the scatterer it is shown that the MDFF contains information both on the spatial density and the spatial coherence of excitations. We discuss how the MDFF can-at least partly-be measured in scattering geometries invoking channeling conditions. Therefrom, the localization of inelastic events can be determined.

A proposal for dichroic experiments in the electron microscope

Building upon the similarities between inelastic electron scattering and X-ray absorption we show that dichroism can be observed in electron energy loss spectrometry (EELS) in the transmission electron microscope (TEM). Natural or magnetic linear dichroism can be studied in electron scattering experiment with definite wave vector transfer in the interaction.The detection of circular dichroism in the TEM relies on interferometric EELS in a particular scattering geometry that allows extraction of the mixed dynamic form factor from energy loss spectra. Similarities between dichroic signals in energy loss near edge structures and X-ray absorption near edge structures are discussed, and a new experimental setup for dichroic measurements in the TEM is proposed.

Improvement of energy loss near edge structure calculation using Wien2k

The density functional theory (DFT) is a recognised method for the calculation of electronic properties of materials. As such it can also be used for the calculation of energy loss near edge structures. Some care has to be taken since the DFT is intended for ground state calculation. The effect of the core hole left by the excited electron is different in an insulator and in a metal and can be observed in both cases. For an insulator (MgO, Si), a supercell calculation is needed while in the case of copper, extremely good agreement with experiment can be obtained with a partial core hole calculation. In the particular case of the WIEN code (APW method) we show that calculation of low lying edges (Si L at 99eV) where the initial state is not strongly localised can only be done within the dipole approximation and with some care. Random alloys (CuNi) have been calculated previously using a supercell; we show that a particular version of the virtual crystal approximation gives promising results.

The separation of surface and bulk contributions in ELNES spectra

We present a method to separate surface from volume contributions in the fine structure of ionization edges in electron energy loss spectrometry (ELNES). It is based on spectra taken at two positions with different surface-to-volume ratio. Contrary to the similar spatial difference method it uses well defined scaling factors, allowing an estimate of the errors propagated into the result.

Separation of pure elemental and oxygen influenced signal in ELNES

The energy loss near edge structure (ELNES) of many elements is strongly influenced by the presence of oxygen or other elements at surfaces, grain boundaries, or in the bulk material. The presented investigation deals mainly with the influence of oxygen at the surface. A method for the separation of both, the pure bulk signal and the oxidized surface signal, was evaluated and tested on Al, Cu, Mg, and Si. A comparison of experimental data with ab initio bandstructure calculations and other proofs of the accuracy of ELNES separation are presented. Influences of error propagations were tested and are exemplarily given for Al and Si.

Orientation dependence of ionization edges in EELS

Anisotropy in the density of unoccupied states can be detected in the fine structure of ionization edges in angle-resolved EELS. It is shown that in a crystal an interference term occurs in the inelastic signal, and how it relates to electron channeling and site selection. The combination of orientation and site selection induces subtle variations in the ELNES. It is shown how this technique can be used to analyze local anisotropy related to the point group of the target atom. A second example shows how to extract non-dipole transitions at small scattering angles.

Observation of ionization in a crystal interferometer

We present a new interferometric setup where interference of a fast probe electron affects the ionization cross section of an atom. Interference is detected in the intensity of the inelastically scattered electrons at the Bragg scattering angle in transmission. The crystal serves both as a target for core ionization and as a beam-splitting and phase-shifting device.