# How is graphene a 2D substance?

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?

• 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

Because its structure displays translational symmetry in 2D. Atoms themselves are 3D as in other materials, but they are placed on a 2D flat plane. Compare to 1D Fullerenes.

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.

It is 2D from the standpoint that it's a great marketing scheme to call it 2D. Call something the most advanced substance on earth, and people will go nuts wanting to buy anything made out of it.

From an actual physics standpoint, it's not 2D...and thankfully so. The smallest-known distance is a Planck length. In order to have a distance smaller than a Planck length, it requires incredible amounts of gravitational force...the kind you find in a black hole. It's generally accepted that at the center of a black hole is a singularity, a 1D object. So theoretically there may be truly 2D objects within a black hole, maybe somewhere beyond the event horizon. In fact, some have speculated that the entire black hole may be a 2D object. So you can see the problem: if graphene was truly a 2D object...well, we wouldn't be here right now.

• Seems a bit harsh to call it a marketing scheme. Can't get much flatter than that with atoms. The term "2D" is certainly used outside of black holes in all kinds of fields. Here it does make sense to point out its 2D-ness since graphene has some unique properties because of that. – user1583209 Jan 5 '17 at 0:42