# 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!

• 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. 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 $\Delta V(x,y) = - \varepsilon _0^{ - 1}\rho (x,y)$. Aug 19 '15 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".