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

enter image description here

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?

  • $\begingroup$ perhaps you might want to check The definition of Density of States $\endgroup$ Feb 8, 2016 at 18:13
  • $\begingroup$ 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? $\endgroup$
    – lnmaurer
    Feb 8, 2016 at 21:00
  • $\begingroup$ Hey, thanks. The first. I would like to recreate the plot I linked based on the dispersion relation. $\endgroup$
    – user103984
    Feb 8, 2016 at 21:10
  • $\begingroup$ @Henrymerrild : if You are interested in the full derivation, I may add it, since I haven't found it anywhere. $\endgroup$
    – Name YYY
    Mar 26, 2016 at 20:12

1 Answer 1


Not sure if its too late for this answer, but just in case.

The density of states is defined as $D(E)=\int_{1st BZ}\delta(E-E(\mathbf{k}) d\mathbf{k}$.

For analytic results the following paper was the first one to calculate the DOS in a honeycomb structure with nearest-neighbor hopping (whose dispersion relation is the one that you mention): http://journals.aps.org/pr/abstract/10.1103/PhysRev.89.662 .

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.


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