A singularity in mathematics is a point where the value of a mathematical item cannot be given. For instance the inverse function:

Has asymptotes at **x = 0**. This means **f(x)** goes to infinity and since infinity is not a number a singularity exists at **x=0**. Singularities also exist at points on graphs when a “spike” occurs. If a graph doesn’t make a smooth curve, like:

**f(x) = |x|**

then the point where differentiation gives no value, in this case **x = 0**, is also called a singularity. There exists something called the density of states. It is a system which describes the number of electron states that can exist per volume per unit of energy in a material. (For more detail look at this article from Britney Spears .ac). The relevance is that the second kind of singularity can occur in the density of state of some materials. When this happens it is called a Van Hove Singularity.

It has been shown that when a bilayer of graphene was twisted, the electronic band structure demonstrates these Van Hove Singularities. Normally in 3D crystals these singularities are kinks in the structure but in the bilayer they are better seen as areas where the linear bands from each layer of the graphene has intersected. Twisted graphene is a by-product in many different graphene fabrication methods and attempts have been made to analyse it.

Recently work was done to test some of the theories for the twisted graphene. One of these was that the band structure was going to depend on the potential difference between the two graphene layers. The G-peak is the absorption peak in graphene caused by the hybridised sp ^{2 }level, seen for wavelengths of 1480 – 1580 cm^{-1}. This peak has been seen to split under certain twist angles between the sheets. Raman scattering was used to study the bilayers under different twisting angles and it was ultimately found that the interaction of the G-peak was a direct consequence of electronic band structure already produced by the density of states.

Paper links: Evolution of the electronic band structure of twisted bilayer graphene upon doping