Finding nearest neighbours in graphene and deriving density of states

[peripatein]

Hi,

Homework Statement

I have two questions, in fact, both involving 2D graphene:
(1) How may I determine the number of nearest neighbours in a primitive cell of graphene?
(2) Given that graphene has linear energy dispersion near the fermi level and the dispersion is given by E=(hbar)v_F|k|, I would like to determine the density of states. I think it is equal to g(E)=E/2π$(hbar)₂v_F², but how may I show that?<br /> <br /> I'd appreciate your help.<br /> <br /> <br /> <h2>Homework Equations</h2><br /> <br /> <br /> <br /> <h2>The Attempt at a Solution</h2>$

 

[sandy.bridge]

I suppose I will aid in getting the ball rolling with this. There are a number of ways that you can approach this problem, however, it is rather difficult to help you without a gauge on your level of quantum mechanics.

 

[peripatein]

Hi,
I believe I have managed through most of this. I have found the number of near neighbours to be 3. I even managed to nearly prove the required relation. However, what I am lacking is an explanation, or an algebraic proof, why N=total number of states=(A/2π)∗∫[between 0 and k(E)] dkk in the vicinity of the dirac points?
Also, how may I show that the units of g(E) in the above expression (my first post) are number of states per area per energy?