I have two questions, in fact, both involving 2D graphene: (1) How may I determine the number of nearest neighbours? (2) Given that graphene has linear energy dispersion near the fermi level and the dispersion is given by E=$\hbar$v$_F$|$\vec{K}$|, I would like to determine the density of states. I think it is equal to g(E)=E/2$\pi$$\hbar^2$v$_F^2$, but how may I show that?

I'd appreciate your help.

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  • $\begingroup$ Well you need to draw the crystal structure of graphene, then you can find the unit cell, primitive cell, all the info about Bravais lattice. Then you can see the number of nearest neighbors. For DoS you can search the google density of states for graphene, and 4th pdf has about this. Also look your literature: Kittle, Ashcroft Mermin... graphene is really popular and there is a lot of info on it. $\endgroup$ – dingo_d Jan 6 '14 at 19:34
  • $\begingroup$ I used the honey-comb structure to determine the number of nearest neighbours and came up with 3. Would you kindly confirm? $\endgroup$ – peripatein Jan 6 '14 at 19:37
  • $\begingroup$ And I have just read through some of the sources online and was still unable to determine how to derive the required expression for DoS. I could use some pointers. $\endgroup$ – peripatein Jan 6 '14 at 19:45
  • $\begingroup$ If I'm not mistaken, there are two carbon atoms per primitive cell, and each has 3 nearest neighbors, so that should be correct. For DoS you need the expression: $g_{2d}(E)=\frac{1}{A}\sum_k\delta(E-E(k))$ if I recall correctly. Then you assume that you have a large volume and go to continuous limit, from sum to integral. Then it's just solving the integral (transform the integral from $k$ to $E$ to integrate and that should be it) $\endgroup$ – dingo_d Jan 6 '14 at 22:55

So this is the hint:

Density of states in 2D:

$$g(E)=\frac{1}{A}\sum_k\delta(E-E(k))=\frac{1}{A}\frac{A}{(2\pi)^2}2\int_0^\infty\delta(E-E(k))2\pi k dk$$

Now use the dispersion relation and evaluate the integral.

You need to understand why the integral looks like that and what the variables mean. Since in three dimensions things change.

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  • $\begingroup$ I just recently started studying this so frankly I am not quite sure why the integral looks that way, nor how to interpret the delta function. Is that the dirac delta? $\endgroup$ – peripatein Jan 6 '14 at 23:14
  • $\begingroup$ In fact, all I am asking is an explanation why the following relation holds: N=total number of states=(A/2$\pi$$)*\int_0^{k(E)}$dkk in the vicinity of the dirac points. As other than that I have managed to prove the desired relation. Also, what are the units of g(E)? $\endgroup$ – peripatein Jan 7 '14 at 8:26
  • $\begingroup$ See this: en.wikipedia.org/wiki/Density_of_states It has basically all you need ;) $\endgroup$ – dingo_d Jan 7 '14 at 8:49
  • $\begingroup$ I did follow that link, however I wasn't able to understand how the relation for the total number of states (see my previous post) was derived. Could you please explain just that? $\endgroup$ – peripatein Jan 7 '14 at 8:59

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