So Chapter 9 tells us that WPP works well for metals in group I,II,II and IV of the periodic table because the valence electrons are not tightly bonded to the ion core because of screening and pauli exclusion principle. And we were told in chapter 10 that tight binding works well for transition metals. Is there a more concrete reason why it works well for transition metals which have partially filled d orbitals? If WPP and TB are the two extremes, what about those that are not either extreme case?
An extension to the tight binding model would be to include electron-electron interaction, the simplest case being the Hubbard model where we only consider on-site electron interaction, i.e. if there is an electron pair on the same lattice site then the Hamiltonian would include an extra energy term due to the repulsion between the two electron with opposite spins.
So Chapter 9 goes through WPP.
ReplyDeleteChapter 10 goes through TB.
The only difference between them, really, is the potential. These tell us what the band structures will be.
From the band structures, we can find properties, such as those derived from considering location/filling of electrons and holes.
Chapter 11 goes through a few more methods to finding the band structure.
Quote from book (p192):
the purpose of this chapter is therefore to describe some of the more common methods actually used in the calculation of real band structures.
I like how they write actually.
It seems reasonable. That for real crystals, there isn't going to be a nice potential described by the extreme cases, as Joseph has pointed out.
They certainly look messier - the bands aren't really (at a more global level) the nice parabolas or sine curves from the WPP and TB.
Then again, we don;t have the nice potentials from before.
We often approximate things in physics by extremes. Whatever makes the maths easiest. For example when finding antenna fields we do near and far field approximations.
ReplyDeleteI think the reason d fields are appropriate for tight binding models is because of the shape. If you have a look at the general shape
http://en.wikipedia.org/wiki/D-orbital
you can see the wave function of the electron is has maximums away from the center.
I believe this is essential to the tight binding approximation where we assume the wave function is small when the potential is large and vice versa.
As for the Huckel model. It seems to make sense since electrons that are far away shouldn't effect each other much. Especially considering there are electrons doing the same thing from all directions.
I think at the end of the day, the Bloch model whilst being extremely successful, still isn't perfect. Chapter 21 in the book actually lists the various flaws of the Bloch model, and then goes on to describe other models. So I would imagine the only practical aspects of this model would be for these extreme cases. Any other intermediate cases must be described by the other features, which as you rightly said would include electron-electron interaction, as well as the introduction of lattice vibrations and phonons.
ReplyDeleteAlso, on the topic of electron-electron interactions, another important case to consider is Cooper pairs forming. I'd imagine this would come up more in the superconductivity chapter, but having a quick read through the wiki article http://en.wikipedia.org/wiki/Cooper_pair brings up an interesting point about this phenomenom. Essentially, whilst Cooper Pairs are a quantum feature, the explanation of the fermions pairing up is very classical. Just found that correleation interesting