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Because atoms in a material are close together, electron wave function from different atoms interact, and they couple. But as Pauli exclusion says, two electrons cannot have the same quantum state, and electrons do obtain a slightly difference in energy. If we consider the 10$^{23}$ atoms a material will have, then these energy levels close together will generate a continuum, creating a band. The furthest band from the atomic nucleii occupied is the valence band, which can be totally or partially filled. The next empty band is the conduction band. An electron can jump from the valence to the conduction if has enough energy and then it will be "free to move", conducting electricity.

Now, this is what I understood, and perharps this is another person asking the same question, but I couldn't answer my doubts with similar questions. The problem I have is that, to me, even if you are in the conduction band you are still attached to the nucleii potentials, so you are "not free". Furthermore, electrons in the valence band are at undetermined places. Precisely, because electron waves interacted, as in a single atom you cannot say at which position and time the electron is, now you cannot say even if it is around atom 1 or atom 300. Thus, to me you are also "free" in this sea of electrons for the valence band. Then, how can you differentiate from a "free electron in the conduction band" from one in the valence band?

Another way to understanding this to me is that if the valence band is full, all possible electron states are present in all the atoms, so at the end is like not having any fluxes, so you do not have current. But in the conduction band, this does not apply, so you will have current. But then, what happens if you have a valence band not totally filled? Wouldn't that be the same as the conduction band? And even in the case of a valence band fully populated, if you promote one electron to the conduction, wouldn't you also have a net flux in the valence band?

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  • $\begingroup$ Bands are Bloch functions across the whole crystals, not atomic energy levels. Solid state physics not atomic physics. $\endgroup$
    – Jon Custer
    Feb 24, 2021 at 0:33

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Congratulations! You have discovered the concept of "holes". An empty state in a otherwise full valence band can act like a positive charge moving in the lattice.

A quibble on vocabulary: an electron in a lattice potential isn't a "free particle" in the sense that I am used to seeing the term in classical and quantum mechanics. (In electromagnetism there is the distinction between "free" and "bound" charge, but I don't think that is what you mean.)

I think that you are asking about non-interacting electrons in a lattice potential. In this case you can differentiate between an electron in a conduction band state versus one in a valence band state because their wave functions are different. (For example, they may have different crystal momenta). It's not bound to any single atom, it's in an extended state across the entire lattice.

However, this is also a bit of a shorthand we use to describe the system, because electrons are indistinguishable. The true many-body wave function is an anti-symmetric combination of all of those single electron states. When they are non-interacting (i.e. turning off their Coulomb interaction) you can show that you can often just treat the problem by thinking of it as putting "an" electron into each state.

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But then, what happens if you have a valence band not totally filled? Wouldn't that be the same as the conduction band?

Yes, this is correct. When constructing a solid, the valence electrons are of main interest. If there are enough valence electrons to fill an entire band, the material is an insulator and we call this last band filled the valence band. If there are enough valence electrons to only fill part of a band, then the material is a conductor and we call this last band the conduction band. [In a semiconductor, the distance from the filled valence band to the empty band above it is small enough so that electrons can be thermally excited into the empty band, and we call that a conduction band as well.]

In either case, the electron wavefunctions of the valence or conduction bands extend throughout the crystal. To first order the main difference between the wavefuction of an electron in the conduction band and one in the valence band, is that in the former the wavefunction looks very much like a plane wave that has only been slightly distorted by the atomic potentials of the ion cores, while in the latter, the wavefunctions are heavily distorted plane waves that are strongly localized around the ion cores.

Thus electrons in the conduction band can indeed move very freely throughout the entire crystal, while the electrons in the valence band tend to hop from one ion to the next.

Both of the wavefunctions described above would be termed "Bloch Functions" because they can both be written as a plane wave multiplied by a function that is periodic with the lattice, the difference lying entirely in what the periodic function looks like.

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