Below are videos on:
1) Energy bands in crystals
2) Effective mass
Let's discuss these here and, then also have a live discussion on Friday at 5:30 PM.
This is a video on bands and bandwidth. An isolated atom has bound states. When an infinite number of atoms are arranged to form a spatially periodic crystal, we find that there are an infinite number of electron (energy) eigenstates, for the crystal, coming from each single atom eigenstate. Please post any questions, thoughts comments, associations, etc. here.
Here is a video on effective mass:
On the energy band discussion: Since cos(ka) is symmetric and gives the same value for any any k and -k , doesn't that mean we're considering the energy states twice over? Is that where spin comes in?
ReplyDeleteAaron: very pertinent questions.
DeleteThose k and -k states are actually two different states with the same energy. (degenerate states) One represents a left moving wave with (crystal) momentum -\(\hbar k\), the other a right moving wave with momentum \(\hbar k\).
Spin, once we include it, doubles the number of states. With spin 1/2 included there are two identical overlapping bands. One for spin up, one for spin down.
Here is an aside that your comment reminds me of: in a superconductor, an electron in a state -k pairs with an electron in the opposite state, +k, and also, the two paired electrons generally have opposite spin, one electron spin up and the other electron spin down. wacky...
One more thing about spin:
DeleteSpin, once we include it, doubles the number of states. With spin 1/2 there are two identical overlapping bands. One for spin up, one for spin down.
Ferromagnetism occurs when something causes those overlapping spin bands to split apart and then only one is filled and the other is empty. (That "something" is the coulomb interaction (repulsion) between electrons, by the way.)
Ah, that makes sense. And on the effective mass discussion: I was under the impression that effective mass could also take on negative values? I suppose I'll ask that during the discussion.
DeleteI thought this video was fun and little bit helpful.
ReplyDeletehttp://upload.wikimedia.org/wikipedia/commons/transcoded/8/81/Metals_and_insulators%2C_quantum_difference_from_band_structure.ogv/Metals_and_insulators%2C_quantum_difference_from_band_structure.ogv.480p.webm
I'm confused? Do you mean the linear term? I am not sure what the 2t term is. Perhaps I made a mistake?, but I am just not sure what you mean.
ReplyDeleteI see. We are doing two different problems. I am only doing an expansion around k=0. You are working on generalizing that. What you are doing is more difficult. My definition of m* is based only on the curvature at the bottom of the band.
ReplyDeleteMaybe one could. You can try.
ReplyDeleteMy definition of m* is based only on the curvature at the bottom of the band.
"But if we were to pick a single point to describe the whole band structure, I suppose it would be best to pick the bottom point."
ReplyDeleteThe thing is, we are not really trying to describe the whole band structure. Effective mass is just an aside for us, a cool thing to notice that happens to be relevant for semiconductors cause often their carriers (electrons) are all pretty much in states near k=0 (the bottom of the band).
I don't know if what I am thinking is the same thing, but I think you mean around 8 minutes of the 2nd video when the Professor Taylor expands E_3. I think the reason the first derivative term is dropped is because sign at k=0 is zero, so we ignore the term.
ReplyDeletePerhaps I missed it, but in the first video when we are looking for t, what happened to the integral?
ReplyDeletePlease make up a name to post under.
DeleteAlso, I think your question is about how to integrate with delta functions. What you describe is basically what happens. The integral just sort of disappears. (That is because the delta function is zero except at one value of x, in this case x=a.
DeleteI see now. You are correct! it is just as you said. I was confused. I thought you were keeping a linear term. my apologies. Yes, at k=0 it starts at E1-2t, as you said.
ReplyDeleteThis comment has been removed by the author.
ReplyDeleteWhen you plotted psi(x) and psi(x-a) they overlapped rather than combining. This kind of overlapping function is new to me. Why does it overlap, how is it important?
ReplyDelete