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



Stefan & John,

Thanks for your herculean efforts to bring me up to speed on this!

On Tue, Mar 2, 2010 at 11:17 AM, Stefan Jeglinski <jeglin@4pi.com> wrote:

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.


Thank you for this! You somehow intuited where I was coming from and stepped
back to pick me up from there. The idea that I could get lead to be a
semiconductor if only I could get the atoms closer together is *exactly* what
I had previously thought. The fact that this isn't the case certainly helps
me understand the later problem of the "appearance" of a band gap for
silicon.


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.


Hugely helpful from where I'm standing. Viewing this in terms of 3D k-space
was something I knew we did, but somehow I failed to make the connection
that you lose something when you try to view it in 1D. Excellent insight
here.

I also really liked this next bit and your description of each material that
follows.

Thanks for all your help. I will try to dig at the QM of this a bit more,
trying to see (as best I can) how the gap between 3s and 3p is eliminated
and then how that "superband" is later split into two equal parts for
silicon. That last step is still the one for which I am seeking a
conceptual/qualitative description of. It seems likely that you have given
me that answer in these discussions, but I haven't yet processed them well
enough to see it.

Cheers,

Jeff

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

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