## Mathematical analysis of graphene grid structures with defects

 dc.contributor.author Hlawe, Phinda K. dc.date.accessioned 2023-11-13T07:17:18Z dc.date.available 2023-11-13T07:17:18Z dc.date.issued 2022 dc.description A dissertation submitted in fulfilment of the requirements for the degree of Master of Science to the Faculty of Science, University of the Witwatersrand, Johannesburg, 2022 dc.description.abstract In this dissertation, we explore the work by M. Archibald, S. Currie and M. Nowaczyk in their paper “Finding the hole in a wall.” In this paper, the authors solve the inverse problem of locating the position of a single vacancy break using lengths of closed paths on an infinite hexagonal grid structure. In order to do this they transform the infinite hexagonal grid structure that models graphene into a brick wall structure. When a single vacancy break occurs, polygons of odd length are introduced into the grid structure. First, we explore lemmas that state which polygon the closed paths of shortest odd length circumnavigate. We then use these to provide a rigorous proof of the two main theorems in “Finding the hole in a wall”. These depend on the region that the path originates from in the brick walls and tell us what the path is congruent to modulo 4. Finally, we study the algorithm for determining the exact position of the defect, and sometimes, provide alternative formulae for locating the defect. When this is the case, we show the formulas are equivalent to those in their work. Also provided are potential future studies in this area. dc.description.librarian PC(2023) dc.faculty Faculty of Science dc.identifier.uri https://hdl.handle.net/10539/36958 dc.language.iso en dc.school Mathematics dc.subject Mathematical analysis dc.subject Graphene grid structures dc.title Mathematical analysis of graphene grid structures with defects dc.type Dissertation
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