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Modified Transport-of-Intensity Approach for Mapping In-situ Magnetic Induction

Lorentz tranmission electron microscopy (LTEM) is ideally suited for quantitative analysis of magnetic domains at the nanometer length scale. The ability to study both the microstructure and magnetic domain structure simultaneously in functional materials allows for direct understanding of the fundamental role of inhomogeneities in microstructure as well as the effect of shape and size of nanostructures on the magnetic domain behavior. Quantitative information about the magnetization is contained in the phase shift of electrons as given by the Aharonov-Bohm phase shift.[1] There are several ways to recover this phase shift such as using the transport-of-intensity equation (TIE) formalism or off-axis electron holography. The phase shift consists of electrostatic and magnetic components, which are related to the projection of the electrostatic potential and the magnetic vector potential respectively. To study the magnetization state of an object, it is necessary to separate these phase shifts, which is achieved using time reversal symmetry i.e., retrieveing the phase shift with the sample in upright orientation and 180◦ flipped over orientation and then substracting one from the other. This method is not suitable for in-situ experiments, where the sample flip at every stage of the in-situ experiment is not practical. In this work, we present a modified version of the TIE formalism to retrieve the magnetic phase shift during an in-situ experiment.