# Density of states for graphene

I have seen a lot of plots for the density of states for graphene:

but have been unable to find the calculation explicetely. I know the dispersion relation for graphene is

$E_{\pm} (\textbf{k}) =\pm t \sqrt{1+4 \cos^2 (k_y a/2)+4\cos (k_y a/2) \cos (\sqrt{3} k_x a/2)}$

and I would like to calculate the DoS myself, but how do I begin?

• perhaps you might want to check The definition of Density of States Feb 8, 2016 at 18:13
• Are you asking for the equations that get you from the continuous dispersion relation to the DOS, or are you asking for methods to calculate the DOS numerically? Feb 8, 2016 at 21:00
• Hey, thanks. The first. I would like to recreate the plot I linked based on the dispersion relation.
– user103984
Feb 8, 2016 at 21:10
• @Henrymerrild : if You are interested in the full derivation, I may add it, since I haven't found it anywhere. Mar 26, 2016 at 20:12

The density of states is defined as $D(E)=\int_{1st BZ}\delta(E-E(\mathbf{k}) d\mathbf{k}$.
For numerical results just apply the definition of $D(E)$. Explicitly you should create two arrays containing values of $k_x$ and $k_y$ (say 100 for each going from their minimum values to their maximum values). Calculate its associated energies. Cut the 1st BZ into same-sized $\mathbf{k}$ regions. Finally calculate the number of states in each area (sum 1 for every $E(\mathbf{k})$ lying inside a given area bounded by the values of $E(\mathbf{k})$ evaluated at the boundaries of each of the $\mathbf{k}$ regions in the 1st BZ). Obviously, the dimensions of your $\mathbf{k}$ area should be greater than the difference between consecutive $k_x$ and $k_y$. Finally normalize your result.