# Conductivity of a crystalline solid

In a crystalline solid each atomic level 'splits' into n levels (n = number of atoms in the system). When the number of atoms is large each level becomes replaced by a band of closely spaced levels.

In a semi-conductor we have an empty "conduction band" and a fully occupied "valance band". Conductivity arises because electrons get excited to the conduction band.

Question: Why can't electron in the valence band freely move around and therefore conduct electricity? My question also applies to metals where the conduction band is already half-filled. What's special about this conduction band that allows electron to move around freely?

Every energy level in a band has an associated momentum, and the total momentum of all the levels in a band is zero. Because the total momentum is zero there can be no net movement of electrons and hence no current. In effect, for every electron with momentum $p$ there is another electron with momentum $-p$ and they cancel each other out.

You can't change the momentum of any of the electrons in a filled band by applying an external field because all the energy levels are full. There are no empty levels for you to move your electron into. That means an external field cannot cause a net movement of electrons.

When you excite an electron into the conduction band it will go into a low momentum state, but there are available states above it with higher momenta. Apply an external field and you will move the electron up into a state with higher momentum that is lined up with the field. In this state the electron has a velocity aligned with the field so there is a net movement of electrons and therefore a current.

The same effect causes a current in metals. There are empty states for the electrons in the band to move into.

• Could it be that you actually mean the conduction band at "When you excite an electron into the valence band"? But, maybe, I got this wrong. Mar 18 '14 at 17:00
• Brilliant, that makes things much clearer. Many thanks! Mar 18 '14 at 18:48
• @JohnRennie one more question: does it mean that the difference in energy between the levels within a band is due only to a difference in momentum (not in potential energy)? I just want a qualitative idea without stepping into the quantum mechanics math Mar 18 '14 at 18:51
• @JohnRennie ??? Mar 19 '14 at 20:18
• @Aegis: opps, soory, I'd missed this. Have a browse through this article on the nearly free electron model. The momentum is the momentum associated with the waves representing the electrons. For valance bands it's probably more appropriate to use the crystal momentum model instead. Mar 19 '14 at 20:23

Because when a band is full, as in undoped semiconductors and insulators, there is no state available into which an electron can move (in $k$ space) to produce a conduction of electricity. This is due to the Pauli Principle applied to the electronic states of a crystalline solid.

If you where to supply enough energy to one electron in the valence band to overcome the energy gap separating it from the conduction band, then it can access available states and then promote electrical conduction. Thus, what is special about this latter band is that has available (empty) states where the electrons can go to.

• Thanks for you answer! My confusion arises from the fact that both in the valence band and in the conduction band each electron has his own quantum-state (due to the Pauli Exclusion Principle). When the electron 'moves across' a crystalline solid to conduct electricity, is it moving from state to state? or is it staying in the same state? Mar 18 '14 at 16:35
• Don't confuse what's happening in the reciprocal space where this description "lives" and what's happening in real space where electrons "move". Remember that a localized state in k-space means a delocalized state in real space. An imbalance on k-space of electrons produces a net transport effect of charge on real space and thus what we observe as current. Mar 18 '14 at 16:41
• I'm sorry but could you explain this with a little less jargon? I am a freshman and to understand your answer I would need to ask at least 10 more questions :p Mar 18 '14 at 18:46