Axel Rother

The statistics of the thermal motion of the atoms during imaging process in transmission electron microscopy and related techniques.

The concern of this work is the influence of the thermal motion of the atoms on electron scattering simulations, used for quantitative interpretation of results in high-resolution electron microscopy. We distinguish between the influence of inelastic phonon excitation and the effect of a moving lattice on images generated by elastically scattered electrons. It is shown that, analog to aberrations, the impact of a moving lattice differs substantially with respect to different imaging conditions and cannot be described by the Debye-Waller damping applicable in XRD. We derive a new formalism, based on the frozen lattice and multislice approach, to incorporate the statistics of the thermal motion into elastic TEM imaging simulations, taking into account different imaging conditions. The averaging over different atom positions is generally performed within a density matrix framework, which can be linearized in the special case of off-axis electron holography. All findings are supported by explicit numerical simulations: molecular dynamics simulations are performed to get a realistic thermal motion and the electron scattering simulations are performed within the new multislice algorithm.

Relativistic effects in elastic scattering of electrons in TEM.

Transmission electron microscopy typically works with highly accelerated thus relativistic electrons. Consequently the scattering process is described within a relativistic formalism. In the following, we will examine three different relativistic formalisms for elastic electron scattering: Dirac, Klein-Gordon and approximated Klein-Gordon, the standard approach. This corresponds to a different consideration of spin effects and a different coupling to electromagnetic potentials. A detailed comparison is conducted by means of explicit numerical calculations. For this purpose two different formalisms have been applied to the approaches above: a numerical integration with predefined boundary conditions and the multislice algorithm, a standard procedure for such simulations. The results show a negligibly small difference between the different relativistic equations in the vicinity of electromagnetic potentials, prevailing in the electron microscope. The differences between the two numeric approaches are found to be small for small-angle scattering but eventually grow large for large-angle scattering, recorded for instance in high-angle annular dark field.

High-Resolution Electron Holography on Ferroelectrics

Within the continuing process of miniaturization in information technology analytical tools are required to characterize materials in terms of electric and magnetic fields on the nanometer scale. In this discipline, off-axis electron holography has proven to be a measuring talent. The holographic reconstruction offers an access to the complete complex object wave, i.e. amplitude and phase of the object-modulated electron wave with all details from largest area information up to the resolution limit of the microscope [1].

Electron Microscope Object Reconstruction: Confidence Criteria of Inverse Solutions

The retrieval of local object information can be performed directly from the electron microscope exit wave function without using trial-and-error iterative matching [1,2]. The present algorithm allows the direct analysis of variations of the object thickness and the beam orientation, local bending, and changes of the scattering potential within the lateral object extension. In principle, extensions are possible also to include local structural variations and special lattice defects into the reconstruction algorithm. However, the object retrieval requires the solution of the inverse scattering problem, which can be gained by linearizing the solution of the dynamical theory and constructing regularized and generalized inverse matrices. Such an inverse scattering problem is in mathematical sense ill-posed and needs special techniques to get well-posed solutions. The retrieval procedure may be summarized as follows. Starting e.g. from an electron hologram, where all reflections g are separately reconstructed, the moduli and phases for each g of the experimental exit plane wave Φ exp are determined. Moduli and phases up to the maximum resolution are necessary to get sufficient a priori data. Theoretical waves Φ th are then calculated using the dynamical scattering matrix M for an a priori model characterized by the number of beams and the scattering potential represented by the potential coefficients V o g. With a suitable experimentally predetermined trial average beam orientation Koxy and a sample thickness to as a free parameter, a perturbation approximation yields both Φ th and M as linear functions of parameters to be retrieved.

Electron Holography on the charge modulated structure In2O3(ZnO)m in comparison with DFT-calculations

The structure - properties relation of modern materials requires fundamental understanding of the interaction between charge structure and relaxation processes e.g. induced by doping metal oxides. In this context, In2O3(ZnO)m provides a well defined arrangement of ZnO -like domains influenced by intersecting layers with fully occupied In3+-ions and unoccupied metal sites. These layers are ordered periodically in a strictly alternating manner, with spacing controllable by the quantity m. An inversion of structural polarity of the domains [1] is forced by the octahedral coordination of In3+-ions in so-called inversion domain boundaries (IDB) (Figure 1).

(Multi-)ferroic domain walls- a combined ab-initio and microscopical investigation

(Multi-)ferroic materials attracted growing interest during the last decade due to their interesting (multiple)-ordering phenomena and the resulting applications (i.e. nonvolatile memories). Physical properties of boundaries are of particular importance as electronic device dimensions shrink and multiferroic bulk materials have not revealed a sufficient magneto-electric coupling so far. We will combine Density Functional calculations and microscopic techniques to examine basic properties of model boundaries, i.e. BiFeO3 71°/109°/180° domain walls (Fig. 1) and BaTiO3 90°/180° domain walls. The BaTiO3 180° domain wall is considered in the lower energetic parallel and in the higher energetic head-to-head configuration. The DFT calculations are performed within LDA+U on a plane wave basis set. PAW pseudopotentials have been incorporated to represent core states. Both, unit cell dimensions and ion positions have been relaxed to yield minimal energy structures. Transmission Electron Microscopy and in particular Electron Holography are applied to probe electric potential distributions and structure properties at the domain boundaries. Structural changes at the boundary occur due to lattice misfits and reconstruction of electronic orbitals. The thereby produced polarization change at the boundary leads to depolarization fields (Fig. 2). We particularly calculate and investigate such fields and discuss the prospects and problems of depolarization field measurements with TEM techniques. Moreover, the reconfiguration of the band structure at the boundary can lead to completely new physical properties like reduction of the band gap, magnetization change, etc [1, 2]. We calculate boundary band structures predicting ferromagnetic BiFeO3 71°/180° domain walls, band gap reduction, etc.

A Relativistic Formalism and the Small Angle Approximation of Elastic Electron Scattering in TEM

A main part of the electrons scattered at the specimen in Transmission Electron Microscopy (TEM) are elastically scattered. The scattering is commonly described by a so-called relativistically corrected Schrodinger equation [1]. The original derivations of this equation from the Dirac equation are rather involved and lengthy. We will present a short derivation, which furthermore will shed some additional light on the approximations necessary. In the quadratic form of the Dirac equation [ ] [ ]Ψ ⋅ + + = Ψ ⋅ − + − σ α r r h r r r h B c e c m p c E c ie V EV E 2 4 2 0 2 2 2 2 2 . (1)