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

[Stefan Jeglinski <jeglin@4pi.com>]

> 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 argue that the split is already there to begin with. s- and 
p-states mix to form "hybrid" orbitals. When you mix the s and p, you 
are creating new hybrid wavefunctions (ultimately as a result of a 
new Hamiltonian that has been introduced as a result of an 
interaction), and the energy expectation values using those new 
wavefunctions are different. Every separated atom has the same energy 
levels. As they come together and the wave functions mix, new (mixed) 
states are created, and some energy levels are higher, and some are 
lower. In a ground state configuration, the electrons will tend to 
occupy the lower ones, and the total energy drops (cf "binding 
energy"). Here you have your "proto" bands already. As you create 
molecules, you mix the wavefunctions into "molecular orbitals," and 
refer to the LUMO (lowest unoccupied molecular orbital) and HOMO 
(highest occupied molecular orbital), analogous to the conduction and 
valence bands, respectively.

As you bring many many atoms together to form a bulk crystal, you can 
think of this happening on a finer and finer level. The energy levels 
in a band are still distinct, but very finely separated. In solid 
state physics, the assertion of periodicity in the Hamiltonian, 
V(x+a) = V(x), results in a new quantum number, usually denoted k, 
the wavevector. The band diagrams you see are energy plots as a 
function of k, E(k).


> 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?

They in fact do. And the band *is* huge (but be careful was how you 
envision "huge" - you are generally working in k-space, not 
x/y/z-space). There is overlap, but no 2 electrons can have the same 
quantum numbers. Not a problem, as you have a new quantum number 
(wavevector) for every 2 (spin up/down) electrons. The valence band 
is typically the set of highest energies occupied, "derived" from the 
HOMO, "derived" from the outer occupied shell of the atom in 
question. The conduction band is the set of lowest unoccupied 
energies, "derived" from the LUMO, "derived" from the outer 
unoccupied shell of the atom in question.

>  What is the mechanism for the later split.

Interaction. But the "splitting" occurs as soon as the interaction 
does. The split is not later. There is always an energy "gap" between 
any energy levels, by definition. The ultimate presence of "bands" is 
a convenient way to picture things when a large number of atoms have 
been brought together in periodic fashion.

My explanation is somewhat sophomoric as well, and there will be 
plenty of quibbles, but the essence is there I think. Interactions 
mix wavefunctions, and when they are mixed, they "move" to different 
energy levels (different soln to Schrodingers eqn). As you bring 
atoms together, the interactions increase, further changing the 
energy levels until they can be visualized as bands. The notion of 
periodicity (see Bloch's theorem) led to a brilliant way (k-space) to 
view bulk solids in the form of crystals. Amorphous solids 
(non-crystalline) makes for messier analysis. On the level you are 
trying to teach it, stick with crystals. But you will have to invoke 
quantum mechanics, I think.

As mentioned by JD, Ashcroft and Mermin is the standard text.


Stefan Jeglinski