Analytical Calculations of Electron's Phase Shift Due to Interactions with Magnetized Materials

Differential phase contrast, through the use of transmission electron microscopes, allows accessing information on magnetic configurations or textures with resolutions smaller than nanometers and even close to angstroms depending on the characteristics of the microscope. This phenomenon is due to the Aharonov-Bohm effect, that predicts a shift in the relative phase (of the electron wave) between two beams traveling through different regions of the material, with electric or magnetic potentials. This shift can be used to make insights into the local magnetic configuration- contained in the region bounded by the beam paths. Here, we show how Toroidal Harmonics functions can be used to compute the phase shift generated by the magnetic vector potential for different geometries and magnetization textures with cylindrical symmetric geometries.