I'm studying solid-state physics and it says that if the solid is thin enough we can view it as a 2D solid but it doesn't say how thin it should be that the electrons are restricted in one dimension. Does anyone know how thin a solid should be that it can be viewed as 2D?
Think about a particle in an infinite square well. As the width of the well is decreased, the separation between energy levels increases.
The same thing happens to an electron in a thin layer of material. As the thickness of the layer decreases the energy levels associated with the electron's motion perpendicular to plane become more widely spaced. If the layer is thin enough the separation between the ground state and first excited level for the perpendicular motion may become much larger than the other energies in the problem (in particular much larger than $k_BT$) and this degree of freedom freezes out, leaving a 2D electron gas.