Abstract: Electron microscopy is an excellent tool which provides resolution down to the atomic scale with up to pm precision in locating atoms. The characterization of materials in these length scales is of utmost importance to answer questions in biology, chemistry and material science. The successful implementation of aberration-corrected microscopes made atomic resolution imaging relatively easy, this could give the impression that the development of novel electron microscopy techniques would stagnate and only the application of these instruments as giant magnifying tools would continue. This is of course not true and a multitude of problems still exist in electron microscopy. Two of such issues are discussed below. One of the biggest problems in electron microscopy is the presence of beam damage which occurs due the fact that the highly energetic incoming electrons have sufficient kinetic energy to change the structure of the material. The amount of damage induced depends on the dose, hence minimizing this dose during an experiment is beneficial. This minimizing of the total dose comes at the expense of more noise due to the counting nature of the electrons. For this reason, the implementation of four dimensional scanning transmission electron microscopy (4D STEM) experiments has reduced the total dose needed per acquisition. However, the current cameras used to measure the diffraction patterns are still two orders of magnitude slower than to the conventional STEM methods. Improving the acquisition speed would make the 4D STEM technique more feasible and is of utmost importance for the beam sensitive materials since less dose is used during the acquisition. In TEM there is not only the possibility to perform imaging experiments but also spectroscopic measurements. There are two frequently used methods: electron energy-loss spectroscopy (EELS) and energy dispersive x-ray spectroscopy (EDX). EELS measures the energy-loss spectrum of the incoming electron which gives information on the available excitations in the material providing elemental sensitivity. In EDX, the characteristic x-rays, arising from the decay of an atom which is initially excited due to the incoming electrons, are detected providing similar elemental analysis. Both methods are able to provide comparable elemental information where in certain circumstances one outperforms the other. However, both methods have a detection limit of approximately 100-1000 ppm which is not sufficient for some materials. In this thesis, two novel techniques which can make significant progress for the two problems discussed above.

Abstract: We present a Ginzburg-Landau theory for a two-band superconductor with emphasis on MgB2. We propose experiments which lead to identification of the possible scenarios: whether both sigma- and pi-bands superconduct or sigma-alone. According to the second scenario a microscopic theory of superconducting MgB2 is proposed based on the strongly interacting or-electrons and non-correlated pi-electrons of boron ions. The kinematic and Coulomb interactions of sigma-electrons provide the superconducting state with an anisotropic gap of s(*)-wave symmetry. The critical temperature T-c has a non-monotonic dependence on the distance r between the centers of gravity of sigma- and pi-bands. The position of MgB2 on a bell-shaped curve T-c (r) is identified in the overdoped region. The derived superconducting density of electronic states is in agreement with available experimental and theoretical data. It is argued that the effects of pressure are crucial to identify the microscopic origin of superconductivity in MgB2. Possibilities for increase of T, are discussed.