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



On 03/01/2010 01:14 PM, Jeff Loats wrote:

I am teaching a modern physics course and we are discussing the way in which
very similar valence electron configuration (2 electrons in a p-shell) leads
to very different conduction behaviors for carbon, silicon, germanium, etc.
My solid state is pretty rusty so I am seeking some help.

I don't understand this question about "similar" and
"different". Different relative to what? Relative
to metals? To a sophomore-level approximation, and
even better than that, band-structure-wise, diamond
is a carbon copy of silicon.

But let's move on from that issue; the following
questions don't seem to depend on it.

As we bring many C atoms together, the 2s and 2p bands mix together.
However, if you bring them even closer together a split occurs, creating the
valence and conduction bands for C, SI, etc.

I am having trouble finding a good sophomore-level description of what
causes the splitting to occur. That is, why don't the 2s and 2p bands just
continue to overlap, forming one huge band? What is the mechanism for the
later split.

That seems kinda backwards. In the hypothetical lattice
where the atoms are far apart, i.e. zero overlap, the
bands are split. They start out split. This is just the
atomic s-p splitting.

So the question of "continuing to overlap" doesn't arise.

The width of each band is proportional to the overlap,
i.e. to the overlap integral between the wavefunctions
of the neighboring atoms. So to explain a band gap in
the tight-binding model (which is I assume is far as
the sophomores are going to go), all you need to do is
not bring the atoms too close.

There's a lot more that could be said, but I'll stop
here.

A fine graduate-level text covering all such things is
Ashcroft & Mermin.

There is also a nice discussion in Feynman volume III.
Review chapter 13 ("propagation in a crystal lattice")
before diving into chapter 14 ("semiconductors").

Note that Feynman does all this *before* mentioning
the Schrödinger equation. Indeed he uses the results
of chapter 13 as the basis for deriving the Schrödinger
equation. It's a thing of beauty.