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How is graphene a 2D substance? It has length, width and some thickness to it, else it would be invisible. Why is it considered a 2D substance?

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I bet someone could write a really good and comprehensive answer to this explaining wave propagation in atomic layers. Unfortunately I don't have time right now. –  David Z Jul 12 '13 at 23:10

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Because it's structure displays translational symmetry in 2D. Atoms themeselves are 3D as in other materials, but their are placed on a 2D flat plane. Compare to 1D fullerenes.

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The dimensionality of a system in practice means the number of dimensions in which objects confined to that system are free to move. For graphene we are generally talking about the motion of electrons within it (though I guess we could be talking about phonons).

Anyhow, the thickness of the sheet is around one atom, which means that in the direction normal to the sheet the electrons are confined within a potential well of around 1 atom in width. Consequently the energy spacings for excitations normal to the layer are going to be around atomic energy levels, which are typically a few eV. At room temperature electron energies are going to be of order $kT$, or about 0.025eV, so no excitations in the direction normal to the sheet are possible. That's why the electrons can be treated as moving in a two dimensional system.

If you increase the temperature then $kT$ increases as well, and at some point the material will stop behaving as a purely 2D material. However I'd guess you'd need several thousand degrees before this became significant.

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