Right, it's friday night, time to do some blogging!
So now that we've got our Bloch structure for metals, we're starting to study the dynamics of particles using this theory. But I just wanted to post some important requirements and restrictions of the theory here (most of what I'm saying comes from chapter 12 of Ashcroft and Mermin).
So by applying a semiclassical approach, we neglect various quantum features (such as position-momentum uncertainty principles) whilst maintaining the quantum structure of the Bloch model. Furthermore, three rules of motion are enforced in this description:
1. There are no interband transitions (n stays fixed)
2. The position and momentum of a particle are known and related via
dr/dt = (1/hbar)(de/dk)
(hbar)k = -e[E + (1/c)v x H]
(I apologise about the terrible notation on here! They're equations 12.6a and 12.6b respectively)
3. Two electrons described by the labels n, r, k and n, r, k + K where K is the reciprocal lattice vector are in fact not distinct electrons. This is of course a consequence of the lattice theory used to derive the Bloch model.
The consequences of the semiclassical theory are the many-carrier theory (due to rule 1), crystal momentum is not momentum (due to rule 2), and that filled bands are inert (due to all three rules).
Subscribe to:
Post Comments (Atom)
Crystal momentum is interesting. This is phonons right? So if an electron absorbed a photon in a material the would gain crystal momentum... But then what of conservation of momentum?
ReplyDeleteDoes the entire object shift?
Dale, I'm not sure what you mean by neglect quantum feautres - we still have position-momentum, don't we?
ReplyDeleteIs it that n is quantum number, r position, k momentum completely specify our electron => we know "exactly" position, momentum and band?
Andy, what is a phonon?
I've never really understood this.
Ann, essentially that's the case. In our equations for the dynamics of the particles (holes or electrons), we use both its position and momentum. Obviously knowing both exactly contradicts the uncertainty principle, hence the description isn't completely quantum. Ashcroft and Mermin say we just use the position and momentum of the wavepacket, what values exactly they are it doesn't specify, but I would just imagine the most probabilistic.
ReplyDeleteFrom what I can understand about crystal momentum is that it arises due to our definition of the wave vector. For a simple particle in a box described by a wavevector k, it just happened that when you multiplied by hbar the units coincided with momentum, so it was a very natural description of it. For the Bloch model, k is no longer proportional to momentum, but the quantity hbar*k is still quite useful and is now referred to as the crystal momentum. It first comes up on page 139.
As for its physical significance, wiki says that it plays a part in the total kinetic energy of the particle http://en.wikipedia.org/wiki/Crystal_momentum. It comes up a lot in the index of the book too, so I'm sure things might become a tad clearer about its proper uses, although I don't think it is related to phonons.... could very well be wrong though!!!
Ann, a phonon is pretty much a quasi-particle of sound. Think of it like waves moving through a medium. In the same way as a photon can be thought of as an excitation in the EM field, a phonon can be thought of as an excitation in the "matter field" of the crystal or what have you.
ReplyDeleteThere are some pretty awesome animations on the wiki page: http://en.wikipedia.org/wiki/Phonon
Yeah, quanta of vibration!
DeleteThat's pretty much how I'm thinking about it at the moment (like our working definition of Fourier tramsform of being space -> momentum...)
From wikipedia, it seems like sound wave in material is the historic interpretation!