# Electrons in graphene will behave as in 2D space?

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. –  Acid Jazz Aug 19 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 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 at 3:46

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".