The quality of being self similar is when a pattern manages to contain a smaller, and possibly distorted, version of the entire pattern. Mathematicians can generate these shapes to their heart’s content but in physics these patterns only appear sparingly. Below is a picture of the Hofstadter-Butterfly, a geometric pattern representing the theorised energy levels for electrons in a magnetic field.
This shape was just a theoretical prediction until it was actually demonstrated to be accurate thanks to the old favourite, graphene. The electrons in graphene act like relativistic particles, an essential component of the self similar set production, as well as the fact that graphene lends itself to easy observations of charge carriers within it. The aim of this study is to investigate the existence of scaling rules, literally the ability for a quality to scale, for conductance within the graphene. This would make it much more likely, if results can be enlarged from each other, that self similar patterns would appear in the conductance characteristics. It turns out that one of the key features required for self similar patterns in the conductance is an approximately self similar structure in the graphene (although this is logically true it was not necessarily true). Nanostructuring of the graphene was employed, however this method is challenging to implement due to the specific requirements related to exact interacting strengths required and how these change over the graphene’s surface. It is suggested that other options may be available such as using external magnetic and electric fields to control the graphene properties or depositing the entire graphene layer onto a base substrate such as silicon carbide (SiC) in order to alter the band gaps in the compound material. The information provided in this paper will be very important until the mechanics of Dirac electrons are understood in more complex materials.