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Re: [Phys-l] Band splitting in Carbon (diamond)...



As you bring the atoms closer and closer (starting from a large distance)
you first have no overlap between 3s and 3p, then the bonding and
antibonding states from the huge number of atoms create bands which spread
out and overlap, making a huge (as in many states as well as wide energy
range) band that makes lead a conductor, for example. As you continue to
reduce the separation this "huge" band splits again into two equal parts,
which gives us band gaps for Ge, Si and C. It is that final split that I am
fuzzy on.

Huge (statewise/energywise) bands don't make lead a conductor per se. Also, with a nod to JD's response, large energy gaps don't make a material an insulator per se.

You started above with lead -> conductor and then jumped to "reduction in separation" causing a "split into band gaps," as if you would see a similar effect in lead if only you could get the atoms closer together. I recommend stepping back a bit.

In the spirit of our discussion, bringing atoms together mixes states, shifts energy levels. Bring enough atoms together as bulk crystalline solids, and you get band structures. Unequivocally true for both lead and silicon, and elsewhere in the periodic table.

However, as you do the quantum mechanical calculations for lead as opposed to silicon, you see they are different, based on the "boundary conditions" (pardon the loose terminology) of their electronic structures. As you refine the calculations, you see that both materials have intersecting bands in k-space, and also non-intersecting "gaps" (I don't know about Pb specifically, but Al for example has non-intersecting gaps in certain symmetry directions).

What you find specifically though, about a crystalline collection of Pb atoms, as opposed to that of Si, is that there is always *some* direction (actually, most) in k-space where you can find a band intersection, whereas in Si, there is a region in the E(k) family of curves where there are no intersections at all in any direction. Aka THE band gap.

If you are not skilled at computerized QM calculations, you are reduced to making approximations about the Hamiltonian (or dimensionality) so as to create mathematical plausibility arguments. The free electron approximation, for example, essentially predicts intersecting bands in all k-space directions, a self-fulfilling prophecy if there ever was one. The tight-binding approximation, on the other side, leads to no intersecting bands in all k-space directions, in certain regions on the energy axis.

A "full" QM calculation for Pb leads (not only to another pun but also) to a band structure that looks qualitatively like the free electron approximation, whereas that for Si leads to a band structure qualitatively like the tight-binding approximation. Eureka. It's not that you continue to bring the atoms closer and closer together and suddenly you have bands and then band gaps. The gaps were there all along - with the greater complexity of the calculation (ie interactions) and the introduction of bands, some gaps remained small, others didn't. Sooner or later you have to account for the fact that Pb != Si.

Another comment, also a nod to JD's answer: band structure says nothing per se about conductivity. You have your band structure. You then pour in all the electrons to fill the bands as they can. Conduction, semiconduction, or insulation, then is a function of electron population and mobility in those bands. In the lowest approximation, valence bands exhibit limited mobility, conduction bands do not. However, mobility alone doesn't do much good as a concept without considering population.

*Metals are relatively boring at this level - pour the electrons in and given the free-electron-like intersecting bands, you're pretty well good to go. High mobility, high population.

*Semiconductors are the canonical example of interest. Pour the electrons in, then encounter a rich world, frameworked by band structures, wherein population mechanisms (eg absorption of a radiation quantum, temperature, donors via dopants) and mobility lead to a plethora of interesting effects.

*Depopulate and/or demobilize, and you have an insulator. As was mentioned, there's more to this than band structure.

Ashcroft and Mermin has been highlighted as the standard text. To me it's a lot like Jackson - all the basic answers are there if only you can see them. Allow me to suggest another book (though not undergrad level): Wave Mechanics of Electrons in Metals, by Stanley Raimes. Out of print, and although obviously concerned with metals, a really clear exposition of many solid state concepts - one of the better pedagogical physics texts I've read.


Stefan Jeglinski