The study of electronic structure topology of crystalline materials has been one of the major fields in condensed matter physics during the last decade. The main motivation for this research came from the discovery of topological insulators, although the focus has recently been shifted towards topological semimetals and even metals. Topological materials are a new class of quantum matters whose special electronic structure in the momentum space leads to the formation of a topological invariance. Theoretical predictions, particularly first-principles electronic structure calculations based on the density functional theory (DFT), have played an important role in the exploration of almost all topological electronic materials. Up to now, different topological phases have been investigated and new materials have been discovered in the field. In a recent study using experimental techniques, Aluminum Diboride (AlB 2 ) has been introduced as a topological semimetal. In the current work, electronic and topological features of AlB 2 semimetal were investigated in both equilibrium structure and also under high pressure. Some other compounds with similar structures were studied in this thesis as well. For this purpose, the first step was to calculate the electronic characteristics of AlB 2 using DFT calculations and then extracting its Wannier functions, in order to construct a tight-binding hamiltonian. This hamiltonian was then used to compute the topological invariant and surface states, which are the manifestations of non-trivial topology in bulk materials. To study the effects of pressure on topological properties of AlB 2 , first, using the evolutionary algorithm, the stable structure of this material was searched in different pressures and then these new structures were investigated more thoroughly by calculating their electronic features. Afterward, Wannier functions were calculated for each structure and as a result, the calculation of the Berry phase for all the new structures became possible. In the end, using the calculated electronic structures and Berry phases, different topological stable phases were analyzed. In addition, the results on the calculation of electronic features and topological invariants of some similar materials were also offered.
