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Will electrons in graphene behave as in pure 2D space, that is they interact with eachother by a Coulomb potential ~ $\ln r$ instead of $1/r$? I think many force lines will "leak" out of graphene membrane $M(x,y)$ so 2D Poisson's equation for charge distribution in graphene $\Delta V(x,y) = - \varepsilon _0^{ - 1}\rho (x,y)$ never hold. It should be $\Delta V(x,y,z) = - \varepsilon _0^{ - 1}\rho (x,y)\delta (z)$. Is there a 'modified 2D Poisson's equation' for graphene?

Sorry for my English!

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No offence intended, but that's a big claim, imo. Have you a published source for a 2D field, or am I possibly misunderstanding your question. – user81619 Aug 19 '15 at 3:09
I suspect the answer might depend on (or at least require knowledge of) boundary conditions, though I could be wrong. – Kyle Kanos Aug 19 '15 at 3:22
I have to solve Poisson's equation for graphene. But if i work with 3D version, i meet a very small 3rd dimension so i ask for a 2D version of Poisson's eq for graphene. I dont think it's just simple $\Delta V(x,y) = - \varepsilon _0^{ - 1}\rho (x,y)$. – Fandroid Aug 19 '15 at 3:46

Well, I'm no expert on this, but a quick literature search suggests that you are sort of correct. Here's a paper on arXiv which is about this. You shouldn't trust everything you read on arXiv because it is not peer reviewed but this paper has subsequently been published in Phys. Rev. B as

1/N expansion in correlated graphene

By: Kotov, Valeri N.; Uchoa, Bruno; Castro Neto, A. H.

PHYSICAL REVIEW B Volume: 80 Issue: 16 Article Number: 165424

So it is probably good. The readers digest summary is that at small distances the interaction potential is the standard Coulomb $1/r$ but for larger distances it is $\sim -\ln{(r)}$ because of "weak confinement of the electric field in the graphene plane".

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