Necessary Conditions on Electric Potential for the Formation of Spectral Touching Points

Martin, Reese
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In recent years special attention has been given to graphene as building material and as an energy storage medium. The main property that make graphene uniquely suited to both of these concerns is that it possesses mass-less fermions, which enable loss-less electron transfer across a graphene sheet. Dirac conical points, a specialized kind of spectral touching point, are believed to be responsible for the mass-less fermions found in graphene, meaning Dirac points could be responsible for graphenes most highly sought after properties. In this work I was able to show that touching points form in graphene for all values of electrical potential, as long as certain symmetry conditions were maintained on the graph. I was then able to show how touching point formation changes as symmetry conditions were removed.

Mathematics, Spectral Graph Theory