Why does the “counting rule” of band theory fail to predict the conduction properties of some materials?

I'm a little confused by the description I commonly hear about the electron counting rule in band theory. The general statement I find is that a "solid with an odd number of electrons per unit cell is a metal, while an even number of electrons could be an insulator or a metal". However, in materials such as CuO and VO2, there are two or more (even number) of formula units per unit cell, so regardless of, e.g., VO2 having a 3d1 configuration, two formula units would mean two d1 electrons and therefore an even number of electrons per unit cell. For some reason, "counting arguments" still imply that these materials should be metals, but I'm not sure how this is the case if the number of electrons per unit cell is the determining factor (according to band theory). I think this rule must be misstated. Can someone clear this up for me?

Note: I am aware that these are NOT metals - I'm just trying to understand why the band theory "counting argument" would suggest that they are.

Band theory does not properly describe systems with strong electron-electron interactions. Specifically, the transition metal oxides you have mentioned are both examples of strongly correlated electron systems, which show all kinds of novel behavior which escapes description of band theory. In general strong interactions can induce phases such as Mott insulators, Kondo insulators, unconventional superconductors, and exotic magnetic states such as spin liquids.

Mott insulators are the direct answer to your general question about how can a system with an odd number of electrons per unit cell be an insulator. In models of these systems there is often one valence electron per atom. If the coulomb repulsion of two electrons on the same atom is much stronger than the kinetic energy of the electrons then it will be energetically unfavorable for an electron to "hop" from atom to atom, as it would only be able to move to sites with one electron already, and it wouldn't have the kinetic energy required to overcome the coulomb repulsion. It is believed that by this type of mechanisms, systems with an odd number of electrons per site can still be insulating.

Specifically, $\mathrm{VO}_2$ I know to have a metal-insulator transition that is still being widely studied. The cause is unknown, although there are arguments for both a Mott-like transition (which relies on electron-electron interactions) and a Peierls-like structural transition (which causes the unit cell to double and takes you to an even number per cell - so band theory works with the Peierls).

• In order to complete the answer, one can refer to the book by Ashcroft and Mermin, which contains a nice discussion about the difference between the band picture of solid and other concepts, like the tight binding method. It's only introductory materials, but it might be of interest for someone. Thanks for this answer. – FraSchelle Apr 2 '15 at 5:27

Are the bands filled for these materials?

Filled bands do not contribute to transport. If a Band is filled at t=0 it remains filled for all times (Consequence of Liouville Theorem). Of course no transport also means the material has to be an insulator. I hope this helps.

• That's part of the issue - I think they both have metal-insulator transitions (VO2 definitely does). Below 340 K, VO2 has no occupied states at the Fermi level and is insulating, but above 340 K it has partially filled bands and is metallic. The idea, from band theory, is that an odd number of electrons per unit cell indicates a partially filled band and therefore a metal. Reportedly, that's why it's unusual for both of these materials to be insulators, although as far as I can tell, they only have an odd number of electrons per formula unit, but not unit cell. – SMPAT Nov 13 '14 at 1:56