Simulation of momentum resolved Electron Energy Loss Spectroscopy in the low loss region using model band structures

One of the key limiting factors to progress within nano science is the ability to measure properties on the relevant length scale. The probe size provided by optical measurements is often larger than the individual nanoscale structures, and the resulting measurement is an average over some large volume, thus other methods must be applied. Electron Energy Loss Spectroscopy (EELS) in a Transmission Electron Microscope (TEM) provides a probe size suitable for measuring on nanoscopic structures, but the physics of the probe change when using electrons instead of photons. The fast electrons passing through the sample carry a significant momentum in addition to energy, and both can be transferred to an electron in the sample. The possible transfer of momentum in addition to energy increases the number of possible excitations immensely thus making the spectra of EELS more complex than its optical equivalent. The EELS-spectra also provide useful information about properties earlier methods could not measure such as excitations resolved by momentum, and a straight measure of transitions across indirect band gaps. However, simulations are key in interpretation of EELS where transitions with momentum transfer contribute. Most simulation software for EELS focus on the optical limit and a production ready software for momentum resolved simulations is so far missing. In the present project a simulation software for EELS is developed for a momentum and energy resolved spectrum. Based on existing theory, a full framework for EELS simulation is developed in the dielectric formulation, strongly depending on the dielectric permittivity. The framework has been implemented with focus on a interactive visualization and interpretation of the result which should be easy to handle. Some limitations have been encountered when it comes to computational cost when mapping both momentum and energy. To limit the computational cost, the permittivity was heavily simplified by treating only its longitudinal component. When applying the software on parabolic bands it was found that the calculated joint density of states reproduced analytically derived results. In calculations of joint density of states of parabolic bands with indirect band gap it was found that the intensity onset had different shape when probing a range of momentum transfers opposed to single momentum transfers. When applied to electronic structure models from tight binding calculations, it was found that the longitudinal permittivity was not sufficient to describe the full response of the systems. The longitudinal permittivity is found insufficient in the presence of transverse electric fields and in non-isotropic systems, thus a correction to the permittivity has been presented, this has not been implemented. To conclude, the developed software indicates that momentum resolved calculations can provide useful information in its simplest manner, and be comparable to experiment with further development.