Why doesn’t a conductor have hole current?

Question

I understand how holes are treated as independent mobile charges by imagining each cation left behind when an electron jumps to an adjacent hole as “moving” positive charges (holes).

I also understand that conventional current refers to the flow of “extra” electrons not part of a covalent bond pair (like in an n-type semiconductor) flowing in the conduction band, and that the flow/jumping of electrons from one bond pair to a hole in an adjacent bond pair refers to the hole current in the valence band.

But I don’t understand why hole currents are neglected in conductors. In conductors, valence bands and conduction bands are said to overlap, but what exactly does this mean for the movement of electrons? In solid states, we say that metals consist of a sea of mobile electrons in a lattice of fixed positive ions. Is this the reason we say there’s no hole current? Because the leftover cations can’t really move (even virtually, since there’s no adjacent bond pair “jumps”)?

Am I confused about what hole current is or what conduction current is? Because the same question for conductors could be asked for the majority charge carriers in n-type semiconductors. How exactly does not being part of a covalent bond and lying in an independent orbital give these majority charge carriers more “freedom”? The cations that these extra electrons leave behind when they flow freely are similarly fixed in the lattice, but why can’t that one empty orbital from which the extra conduction electron emerged be treated as a hole too? Why reserve that term only for vacancies created in bond pairs? Is it because no electrons can actually jump into these lone vacancies, thereby ensuring that the vacancy stays with the same atom?

I feel like I’m very close to the answer but I can’t quite get it. Hopefully, my chain of reasoning is clear.

Answer 1

Shortest answer: in reality, conduction by holes is just movement by all the electrons in the band, causing the hole to move, and causing net / overall conduction in the material. In a conductor / conduction band, the electrons are already able to move almost-freely, so we don't need to think about the hole concept

Longer version: when you calculate / determine the band structure of a material, you determine both the possible energy levels of the electrons as a function of the wavevector, as well as the occupancy of those levels. For the three major classes of electrical materials there are three situations of how the electrons fill up the bands:

• insulator: electrons fill up a band completely. There is a large energy gap (> ~ 5 eV) between the top of the filled band and the next energy states
• semiconductor: electrons fill up a band completely, but the energy gap to the next energy states (from doping) or band is relatively small
• conductor: electrons partially fill up an existing band

A hole provides conductivity in an insulator or a semiconductor because it appears within a previously completely filled band, and allows conduction w/in that band. In a crude picture, imagine before the hole is present that electrons occupy every orbital within each atom in a solid. When a hole is created, one of these is now empty and electrons can fill in this empty position sequentially, causing a net effect of conduction.

However in the conduction band things are very different. Since it starts out only partially filled, there are plenty of empty states that electrons can occupy with only a very small gain in energy that allow them to move. There's no need to use the concept of the hole moving because it's easier to just think / calculate the effect of the electrons themselves moving.