## Mathematical analysis of graphene grid structures with defects

2022
##### 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.
##### 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
##### Keywords
Mathematical analysis, Graphene grid structures