# Formation of Bands in Semi-Conductors

At school, we are learning about semi-conductors and their applications in modern electronics. One of the features of semi-conductors is that there is a small energy gap between the valence and conduction band. My teacher explains that he bands occur because in the compound, since there are millions of atoms close together, due to the Pauli Exclusion Principle the energy levels of the atoms slightly change in value, forming band. When electrons are excite form the valence to conduction band, the substance is able to conduct an electric current.

However I fairly certain this is a superficial explanation as it doesn't really explain what the conduction and valence band is and doesn't explain why electrons in the conduction band are able to conduct electricity while electrons in the valence band can't.

After some researching on the web this is my rough understanding of semi-conductors now:

The s and p orbitals of the metals combine together to form MOs. As the AOs of millions of atoms combine together, these MOs become a continuum, forming two bands. One band is the valence band which consists of 2s sigma bonding and anti-bonding orbitals while the other band is the conduction band which consists of 2p pi bonding and anti-bonding orbitals. This explains why electrons in the conduction band are able to conduct an electric current since they are conjugated (not sure if this is the correct term) across the entire substance and hence explains why electrons in metals are described as a 'sea of de-localised electrons'.

This is just my understanding which isn't that solid. Could someone please offer a more in-depth explanation into how these bands form and correct any wrong things that I said. Also I still have some questions.

One website stated that 'it is only when these bands become filled with 2p electrons that the elements lose their metallic character'. Which I interpret as that when the conduction band is completely filled with electrons, the substance isn't able to conduct electricity any more. However why is this so? Aren't there still electrons in the pi MOs which are 'de-localised' along the substance?

Also the above explanation using MOs seems to explain why these bands form. Does this mean that the explanation that my teacher gave me about the Pauli Exclusion Principle being responsible for the formation of bands is wrong and not needed?

doesn't explain why electrons in the conduction band are able to conduct electricity while electrons in the valence band can't

Electrons in the valence band can conduct electricity, as long as the valence band is not completely filled. This occurs in semiconductors, when you thermally promote electrons from the valence to the conduction band, you are leaving behind some holes in the valence band. The electrons in the valence band can then conduct electricity as well by "jumping" through the holes. The conductivity of semiconductors arises therefore not only from electrons in the conduction band, but also holes in the valence band.

The origin of conductivity lies in a partially-filled band, and it doesn't matter which band it is. For a more rigorous treatment google for "free electron model". The idea is that each crystal orbital (i.e. MO of the crystal) has an associated value of electron momentum. For each orbital with a positive electron momentum, there is an orbital with an equal but negative momentum. When the band is fully filled, all of the crystal orbitals are filled, and there is no net electron momentum because the positives and negatives cancel out exactly. When the band is incompletely filled, the application of a potential difference in a certain direction across the metal causes the electrons to preferentially populate orbitals with positive momentum in that direction. This leads to a net positive electron momentum in that direction, meaning that electrons "flow".

The s and p orbitals of the metals combine together to form MOs. [...]

Yes. Bands are simply a collection of crystal orbitals that have such small energy gaps between them such that they are almost continuous. I don't see how the Pauli exclusion principle comes into play except for when you are discussing the population of the orbitals (obviously any one orbital can have a maximum of two electrons). The two ways that I know of to obtain the bands (nearly free electron model, and the LCAO/tight-binding approach) both do not invoke the PEP. However I will add a disclaimer that physicists approach solid state chemistry from a very different direction and there could very well be some explanation that I do not know of.

However, yes, the electrons populate orbitals that are "delocalised" across the entire solid and this is the origin of the "sea of delocalised electrons" description of metals.

'it is only when these bands become filled with 2p electrons that the elements lose their metallic character'

I honestly don't see how that is relevant or even correct at all. Let's use the example of sodium versus chlorine. The band structure of Na metal is vastly different from that of chlorine since the molecular structures are different to begin with. Sodium is a metal and chlorine is a molecular gas, you wouldn't use a band structure to describe chlorine. You cannot just take a diagram for $\ce{Na}$ and say that it applies to $\ce{Cl2}$ and try to make statements such as "filling of the 3p bands leads to decrease in metallic character" because those bands don't even exist in chlorine. Band structures are just large MO diagrams, it is almost like trying to compare the physical properties of sodium and chlorine using MO diagrams for $\ce{Na2}$ and $\ce{Cl2}$, which is plain absurd.

However, if chlorine were to adopt a structure that has close-packed atoms like in sodium, then you would be right that it would still be metallic due to the partially-filled 3p bands.

If you want to read further I would recommend finding a solid state chemistry textbook, all of them would cover basic band theory. In particular I find this book by Moore and Smart clear and accessible.

• It is perhaps also worth noting that the atomic orbitals may not simply merge in to the solid state bands - odd things can happen in the process. It has been some time since I looked at a system in detail, but more than one had fairly deep AOs end up above the Fermi level while a nominally unoccupied AO dropped into the valence band. – Jon Custer May 28 '16 at 14:52
• Thanks for the great answer! This might be completely irrelevant but is the combination of AOs to form bands of MOs also responsible for high pressure/density gases having a continuous spectrum? I asked this question on Physics Stack Exchange and they said the energy levels become bands due to the PEP, allowing for different electron transitions (my school teacher said the same thing as well). – Nanoputian May 29 '16 at 1:15
• @Nano Sorry but you'll have to ask someone else, I don't know about that. Seems my hunch was right, it is a physicist's explanation.. – orthocresol May 29 '16 at 2:08