Graphene can be used to create circuits that are almost superconducting, potentially speeding electronic components by as much as 1000 times [1]. Such blazing speed might also help produce ever-tinier computing devices with more power than your clunky laptop [2]. Graphite is a polymorph of the element carbon [3]. Graphite is made up of tiny sheets of graphene. Graphene sheets stack to form graphite with an interplanar spacing of 0.335 nm, which means that a stack of 3 million sheets would be only one millimeter thick. [1] This nano scale 2 dimensional sheet is graphene. Novoselov and Geim's discovery is now the stuff of scientific legend, with the two men being awarded the Nobel Prize in 2010 [4]. In 2004, two Russian-born scientists at the University of Manchester stuck Scotch tape to a chunk of graphite, then repeatedly peeled it back until they had the tiniest layer possible [2]. Graphene has exploded on the scene over the past couple of years. "Six years ago, it didn't exist at all, and next year we know that Samsung is planning to release their first mobile-phone screens made of graphene." - Dr Kostya Novoselov [4]. It is a lattice of hexagons, each vertex tipped with a carbon atom. At the molecular level, it looks like chicken wire [4]. There are two common lattice formations of graphene, armchair and zigzag. The most studied edges, zigzag and armchair, have drastically different electronic properties. Zigzag edges can sustain edge surface states and resonances that are not present in the armchair case Rycerz et al., 2007 [5]. This research focused on the armchair graphene nanoribbon formation (acGNR). Graphene has several notable properties that make it worthy of research. The first of which is its remarkable strength. Graphene has a record breaking strength of 200 times greater than steel, with a tensile strength of 130GPa [1]. Graphene has a Young's modulus of 1000, compared to just that of 150 for silicon [1]. To put it into perspective, if you had a sheet of graphene as thick as a piece of cellophane, it would support the weight of a car. [2] If paper were as stiff as graphene, you could hold a 100-yard-long sheet of it at one end without its breaking or bending. [2] Another one of graphene's attractive properties is its electronic band gap, or rather, its lack thereof. Graphene is a Zero Gap Semiconductor. So it has high electron mobility at room temperature. It's a Superconductor. Electron transfer is 100 times faster than Silicon [1]. With zero a band gap, in the massless Dirac Fermion structure, the graphene ribbon is virtually lossless, making it a perfect semiconductor. Even in the massive Dirac Fermion structure, the band gap is 64meV [6]. This research began, as discussed in Chapter 2, with an armchair graphene nanoribbon unit cell of N=8. There were 16 electron approximation locations (ψ) provided per unit cell that spanned varying Fermi energy levels. Due to the atomic scales of the nanoribbon, the carbon atoms are separated by 1.42Å. The unit vector is given as, ~a = dbx, where d = 3αcc and αcc = 1.42°A is the carbon bond length [5]. Because of the close proximity of the carbon atoms, the 16 electron approximations could be combined or summed with their opposing lattice neighbors. Using single line approximation allowed us to reduce the 16 points down to 8. These approximations were then converted into charge densities (ρ). Poisson's equation, discussed in Chapter 3, was expanded into the 3 dimensional space, allowing us to convert ρ into voltage potentials (φ). Even though graphene is 2 dimensional; it can be used nicely in 3 dimensional computations without the presence of a substrate, due to the electric field lines and voltage potential characteristics produced being 3 dimensional. Subsequently it was found that small graphene sheets do not need to rest on substrates but can be freely suspended from a scaffolding; furthermore, bilayer and multilayer sheets can be prepared and characterized.