# What is it about the “conduction band” of a material that is distinct from the valence band?

I'm taking a course in nanotech and we're discussing nanoelectronics. This has led to a discussion of conductors, semiconductors, and insulators. I have a number of lovely diagrams explaining the fact that there is a band gap of energy states that fall in-between quantum energy levels that are "forbidden" for electrons to occupy. I understand the concept of quanta and why an electron cannot take on an energy state in this band gap.

What I don't understand is simply this: Why is it that when an electron is in the valence band it doesn't conduct but when it is in the conduction band it does?

In other words, what, physically speaking, is special about the conduction band that is not characteristic of the valence band or any lower filled bands?

In other words, I have read and heard that the electrons in the conduction band are "free to roam in a sea of electrons." But from what are they free? The pull of an atomic nucleus? Some other kind of intermolecular or interatomic force(s)? Is there something physically unique about conduction band electrons that is distinct from electrons not in the conduction band?

I guess in the final analysis I am trying to get a grasp for a quintessential "conduction-ness" that will help me understand the fundamental principle of conductivity. I will be taking a course in Electricity and Magnetism in 2 semesters and will no doubt learn a great deal more about this, but for now, any light you can shed would be helpful not only in this course but in understanding superconductance as well because that is all about getting electrons into the conduction band and then keeping them there by dropping the resistance to 0.

• I suppose 'free' is a bad word to use here. The entire calculation of bands was already based on a free fermion system. What is meant is about how easy the transitions between states in those bands are. These are caused by perturbations like interactions. – AHusain Jul 4 '16 at 9:46
• What would conduction in the valence band actually look like? Keeping in mind that the valence shells are fully occupied in all of the atoms, where would the electron actually hop to? – lemon Jul 4 '16 at 9:47

An example of a doped semiconductor might give an intuitive picture of some aspects of this topic:

• Consider a material like Germanium. Atoms are structured in a lattice. All valance electrons are "used" in the crystal structure to form bonds to neighbours; none are more "free" than others.
• Now dope it with another atom of one higher electron number, in this case for example Arsenic. In its effort to take a lattice position in the Ge-structure and behave like Ge, the one extra electron it has is "pushed away" to a higher energy-level.
• This electron is now less strongly bound to a location and so is not in the valence band but also still not all the way up in the conduction band:
• A tiny bit of energy will do the last push and rip the almost-free electron all the way free, bringing it to an energy state within the conduction band.

• Here in the conduction band, any infinitely small amount of energy bias will let the electron move around from energy state to energy state within the conduction band. Think of a band as a collection of extremely many closely packed possible energy states.

Now, Pauli's exclusion principle forbids two electrons to occupy the exact same energy state. If one electron takes one energy state, then other electrons in the same band now have less unoccupied states to move to.

So the more electrons that are brought to the conduction band, the less freedom each of them has to move around within this band. If it becomes totally full, they are all again "fixed" at their current energy state, and must again be added a large portion of energy to be ripped (to make the large jump) to an even higher band to reach freedom again.

At this point the now full conduction band has become "the new" valence band. And "the new" conduction band is the nearest band higher up (energy-wise). There is thus no difference between valence and conduction bands other than how occupied they are.

• The diagram really helps me understand this better. Thank you for that. So it seems that what is "special" about the conduction band (and you'll forgive my somewhat anthropomorphic term I hope) is simply that it is filled by electrons that are no longer bound to an atom and can therefore move freely. In other words, what makes an electron a 'free electron' is that the atomic forces holding it in an orbital have been overcome and the electron has broken free from the atom? So is electrocution just a form of electron-mitigated ionization akin to a linear accelerator electron beam? – 1John5vs7 Jul 6 '16 at 15:51
• @1John5vs7 Don't say that a conduction band is "filled by electrons". It wouldn't be a conduction band if it was full. – Steeven Jul 7 '16 at 7:45
• @1John5vs7 The forces holding it in an orbital are overcome, yes, and the more energized electron is moved to another orbital. This all simply means that it is more energized (at a higher level energy state) and therefor less energy is required to give it the remaining "strength" to rip itself free from the original atom and move it om to the next. – Steeven Jul 7 '16 at 7:48
• @1John5vs7 Electrocution is another question to ask. I cannot answer that. – Steeven Jul 7 '16 at 7:49

A band is essentially a (near) continuous collection of momentum eigenstates. Within the band the electrons can be treated as free to a reasonable approximation, so their eigenstates are just plane waves. The symmetry means that for every eigenstate there is another with an equal and opposite momentum. So if we populate every momentum eigenstate the net momentum is zero.

And that's why the electrons in the valence band won't conduct electricity. It isn't that they can't move, it's because for every electron moving one way there is another moving the opposite way so there can be no net motion.

By contrast the conduction band contains unpopulated eigenstates so when an electric field is applied the electrons can redistribute themselves among the available momentum states to produce a non-zero net velocity.

• This is true for completely filled valence band. But, as in the conductivity band there always exist electrons (at $T>0$), valence band always has holes. These holes allow for some redistribution of electrons, which results in hole current on application of electric field. – Ruslan Jul 5 '16 at 7:43
• Ah, so what is important is not so much that electrons are moving but that there is NET movement in some direction of the electrons? – 1John5vs7 Jul 6 '16 at 15:55
• @1John5vs7: yes exactly. The net motion is the drift velocity needed to sustain a current. – John Rennie Jul 6 '16 at 16:01

No band is special. A partially full valence band does conduct, just like a partially full conduction band.

On the other hand, a perfectly full band conducts just as well as a perfectly empty band: No conduction at all.

Now, nobody is surprised when you say an empty band can't conduct, but at first it seems surprising that a full band is the same way. After all, we are given analogies of water pipes and such, and there's nothing wrong in a pipe full of water, right? Well, all analogies have their limits. As John Rennie points out in his answer, the problem is that a full band contains equal populations of left-moving electrons and right-moving electrons, and so no net motion occurs. It's not that space is filled with electrons getting in each others' way, but rather, the single-particle phase space is fully packed and has no remaining degrees of freedom. The idea that phase space can be packed is something very peculiar to fermions, with no classical analogy.

What we can do is to add an electron to an otherwise-empty band, or to remove an electron from an otherwise-full band, and then conduction will be possible.

(By the way, you will probably not find this answered in your Electricity and Magnetism course. Nanoelectronics is the closest you'll get to answering this question about what conduction really means, microscopically.)

• Hmmm. Well then I will have to talk to my nanotech professor about this to get more insight. Thanks for sharing this, it's interesting, though I don't fully understand it yet. This is a lot of stuff to wrap one's head around, and it is humbling to know there is so much to know and our own abilities are so limited. – 1John5vs7 Jul 6 '16 at 15:49