Something we mentioned in tutorial.
In the Drude and Sommerfeld our electrons were non-interacting, so we just had our sea of electrons and our Fermi surfaces were just circles (or spheres). We only had one circle as well as we did not have a condition of periodicity.
In the Bloch model, there is periodicity, so there is a Fermi surface in each unit cell (since each unit cell must be identical).
The Bloch model also introduces interaction between electrons, so our Fermi surfaces are no longer spheres, but can be oddly shaped things (as we've seen throughout the later part of the course).
Consider a closed Fermi surface (in 2D) that extends past the boundaries of a unit cell, arbitrary, take the First Brillouin Zone.
Now this Fermi surface also has to be repeated in the neighbouring cell.
Suppose we go from the repeated diagram (left) to the reduced diagram (right).
How are they equivalent?
What happens at the overlap?
There still has to be periodicity in each unit cell as we are in the Bloch model.
Ross suggested that the top would come down and take a bite out of the bottom.
Dave pointed out that the top position is equivalent to the bottom position (due to periodicity).
Dave pointed out that the top position is equivalent to the bottom position (due to periodicity).
Somewhere in there, there was a mention of how they could be considered as from electrons or holes...
I'm probably just mixing bands and Fermi surfaces.
(Thanks Joseph and David).
I guess this just goes to show how it is important to note whether one is working in the repeated scheme or the reduced scheme. Because whilst initially when dealing with the Bloch Model the Fermi levels appeared relatively identical between the two schemes, now we can develop two quite different looking objects that describe the same energy level.
ReplyDeleteJust a quick, cautionary note: the Bloch model doesn't explicitly introduce electron interactions! These get added on later. Of course, periodicity doesn't really make sense for the Drude model :), and not really for the Sommerfeld model either. (:
ReplyDeleteThe effective electron-hole distinction is simply due to which direction the electron is travelling (which is due to whether it is on the bottom of a 'happy' parabola bit of a band—electron—or the top of a 'sad' parabola bit of a band—hole). Also, remember that the bands can be drawn in a repeated way too!
So, if you're using a reduced picture, you just have to remember that the reduced bit is actually reversed and on the bottom of the next band, so that if you are drawing the 'hole' regions of the band in the reduced picture, you're actually talking about the region of the next cell that impinges on the current cell.
The reduced scheme is a short-hand way of not needing a periodic picture, whereas the repeated scheme is an actual drawing of the periodic k-space band structure. So, the reduced picture is NOT actually periodic, it just represents a periodic idea! Hence, if you wanted to get a real picture of the band structure, you should draw a picture of one cell surrounded by a few buddy cells :).
Did my rambling help Ann? I think Dale is right: as long as you are aware of which scheme you're looking at, you should be okay. Just remember that the repeated scheme is 'real' and the reduced scheme is not. :)
Thanks Dale and Josh
ReplyDeleteYeah, I think I was just getting band structures and Fermi surfaces mixed up.
Periodicity really only makes sense in the Bloch model when we require the symmetry.
We could think of something in the neighbouring cell as (equivalently) being in our original cell -- they have equivalent wavevector.
Sidenote: this periodic stuff - remind anyone of modular arithmetic?