The refractive index in electron microscopy and the errors of its approximations.

In numerical calculations for electron diffraction often a simplified form of the electron-optical refractive index, linear in the electric potential, is used. In recent years improved calculation schemes have been proposed, aiming at higher accuracy by including higher-order terms of the electric potential. These schemes start from the relativistically corrected Schrödinger equation, and use a second simplified form, now for the refractive index squared, being linear in the electric potential. The second and higher-order corrections thus determined have, however, a large error, compared to those derived from the relativistically correct refractive index. The impact of the two simplifications on electron diffraction calculations is assessed through numerical comparison of the refractive index at high-angle Coulomb scattering and of cross-sections for a wide range of scattering angles, kinetic energies, and atomic numbers.

No surprise in the first Born approximation for electron scattering.

In a recent article it is argued that the far-field expansion of electron scattering, a pillar of electron diffraction theory, is wrong (Treacy and Van Dyck, 2012). It is further argued that in the first Born approximation of electron scattering the intensity of the electron wave is not conserved to first order in the scattering potential. Thus a "mystery of the missing phase" is investigated, and the supposed flaw in scattering theory is seeked to be resolved by postulating a standing spherical electron wave (Treacy and Van Dyck, 2012). In this work we show, however, that these theses are wrong. A review of the essential parts of scattering theory with careful checks of the underlying assumptions and limitations for high-energy electron scattering yields: (1) the traditional form of the far-field expansion, comprising a propagating spherical wave, is correct; (2) there is no room for a missing phase; (3) in the first Born approximation the intensity of the scattered wave is conserved to first order in the scattering potential. The various features of high-energy electron scattering are illustrated by wave-mechanical calculations for an explicit target model, a Gaussian phase object, and for a Si atom, considering the geometric conditions in high-resolution transmission electron microscopy.

Effective object planes for aberration-corrected transmission electron microscopy.

In aberration-corrected transmission electron microscopy, the image contrast depends sensitively on the focus value. With the point resolution extended to an information limit of below 0.1nm, even a focus change of as small as one nanometer could give a significant change in image contrast. Therefore, it is necessary to consider in detail the optimum focus condition in order to take full advantage of aberration-correction. In this study, the thickness dependence of the minimum contrast focus has been investigated by dynamical image simulations for amorphous model structures of carbon, germanium, and tungsten, which were constructed by molecular dynamics simulations. The calculation results show that the minimum contrast focus varies with the object thickness, supporting the use of an effective object plane close to the midplane instead of the exit plane of a sample, as suggested by Bonhomme and Beorchia [J. Phys. D: Appl. Phys. 16, 705 (1983)] and Lentzen [Microscopy and Microanalysis 12, 191 (2006)]. Thus supported particles and wedge-shaped crystals with symmetrical top and bottom surfaces could be imaged at a focus condition independent of the uneven bottom face. Image simulations of crystalline samples as a function of focus and thickness show: for an object thickness of less than 10nm, the optimum focus condition is matched better if the midplane of the object, instead of the exit plane, is chosen as reference plane.

The tuning of a Zernike phase plate with defocus and variable spherical aberration and its use in HRTEM imaging.

With the advent of the double-hexapole aberration corrector in transmission electron microscopy the spherical aberration of the imaging system has become a tunable imaging parameter like the objective lens defocus. Now Zernike phase plates, altering the phase of the diffracted electron wave, can be approximated more perfectly than with the lens defocus alone, and the amount of phase change can be adjusted within wide limits. The tuning of the phase change allows an optimum contrast transfer in high-resolution imaging even for thick crystalline objects, thus surpassing the limits of the well-known Scherzer lamda/4 phase plate to the imaging of thin objects. The optimum values for the spherical aberration and the lens defocus are derived, and the limits and imperfections of the approximation explored. A mathematical link to the channelling approximation of high-energy electron diffraction shows how the image contrast of atomic columns can be improved systematically within wide thickness limits. Depending on the specimen thickness different combinations of spherical aberration and defocus are favourable: positive spherical aberration with an underfocus, zero spherical aberration with zero defocus, as well as negative spherical aberration with an overfocus.

Atomic-resolution imaging of oxygen in perovskite ceramics.

Using an imaging mode based on the adjustment of a negative value of the spherical-aberration coefficient of the objective lens of a transmission electron microscope, we successfully imaged all types of atomic columns in the dielectric SrTiO3 and the superconductor YBa2Cu3O7. In particular, we were able to view the oxygen atoms which, due to their low scattering power, were not previously accessible, and this allowed us to detect local nonstoichiometries or the degree of oxygen-vacancy ordering. This technique offers interesting opportunities for research into oxides, minerals, and ceramics. In particular, this holds for the huge group of perovskite-derived electroceramic materials in which the local oxygen content sensitively controls the electronic properties.

High-resolution imaging with an aberration-corrected transmission electron microscope.

Recently an electromagnetic hexapole system for the correction of the spherical aberration of the objective lens of a 200 kV transmission electron microscope has been constructed by Haider and coworkers. By appropriately exciting the hexapole elements it is possible to adjust specific values of the spherical aberration coefficient ranging from the value of the original uncorrected instrument over zero even to negative values. In the first part of the paper the consequences of the tunable spherical aberration are investigated. New imaging modes are available: By adjustment of an optimum value for the spherical-aberration coefficient, the point resolution of phase-contrast imaging can be extended to the information limit. Phase-contrast imaging can be improved by a reduced level of contrast delocalisation. For zero aberration contrast delocalisation does not occur. In this case high-resolution investigations are carried out under amplitude-contrast conditions, where the local image intensity of crystalline objects is controlled by electron diffraction channelling. The defocus and spherical aberration values related to the new imaging modes are given. In the second part novel applications of the instrument to semiconductor heterostructures and ceramic grain boundaries are examined.

Reconstruction of the projected crystal potential in transmission electron microscopy by means of a maximum-likelihood refinement algorithm

The projected crystal potential is reconstructed from a nonperiodic high-resolution transmission electron microscopy exit wave function using a maximum-likelihood refinement algorithm. The convergence and the accuracy of the algorithm are investigated using simulated exit wave functions of SiGe, a Shockley partial dislocation in Ge and an area containing randomly distributed Ge columns at different specimen thicknesses. The performance of two different start models for the projected crystal potential is investigated: the weak-phase-object model and a model based on the electron-channelling approximation. The reconstruction is successful even under the strongly nonlinear dynamical diffraction conditions at larger specimen thicknesses, relevant for high-resolution work, and on specimen areas large enough to cover defects in crystalline materials.

Impact of column bending in high-resolution transmission electron microscopy on the strain evaluation of GaAs/InAs/GaAs heterostructures

The accuracy of strain profiles obtained by a quantitative analysis of lattice fringe spacings from high-resolution micrographs is discussed. Focusing on highly lattice mismatched GaAs/InAs/GaAs heterostructures the local strain distribution of the layers is calculated by finite element simulations to determine the atom positions in elastically relaxed transmission electron microscopy specimens. By analysing simulated images a significant decoupling between the layer structure and the contrast pattern motifs is found for relevant imaging conditions, which may result in an incorrect determination of strain profiles and layer compositions when examining experimental micrographs.

Reconstruction of the projected crystal potential from a periodic high-resolution electron microscopy exit plane wave function.

A method based on a simulated annealing algorithm is applied for the reconstruction of the projected crystal potential belonging to a periodic high-resolution electron microscopy exit plane wave function. Using simulated exit plane wave functions of GaAs at different specimen thicknesses, the convergence behaviour and the accuracy of the algorithm are investigated. It is demonstrated that the reconstruction is possible even under strongly non-linear scattering conditions at small specimen thicknesses. Further, the convergence of the algorithm to an ambiguous solution beyond a certain specimen thickness is discussed.