Kittel figure 9.12 (I attached it)

[ehrenfest]

Homework Statement

This figure is driving me insane. How can the energy gradient point inwards! For the hole orbit. The shading represents filled orbits (I am pretty sure), so how can you possibly have orbits filled everywhere except the center of the First Brillioun zone. That is outrageous! What does a hole orbit physically mean??

Homework Equations

The Attempt at a Solution

 

[genneth]

Consider the 1D case. Your dispersion relations are in bands --- so you can get negative gradients even for +ve k. As the Fermi level goes up, you can start filling a band from high k's going towards k=0. It's hard to describe without drawing...

 

[ehrenfest]

Consider the 1D case. Your dispersion relations are in bands --- so you can get negative gradients even for +ve k.

What is "+ve k"?

As the Fermi level goes up, you can start filling a band from high k's going towards k=0.

How?

 

[genneth]

See http://venables.asu.edu/qmms/band.gif as an example (it was the first reasonable one I could find with google images)

Now imagine filling up from the lowest energies. As the Fermi level crosses the first band, you start filling from low k to high k. But when it gets to the 2nd band it starts from high k --- at the zone boundary.

 

[ehrenfest]

Just to be sure, you are referring to the energy versus k diagram on the right and using the fact that the first band is concave up and the second band is concave down? Although, they did not show it, I assume k=0 is below the minimum of the first band? Also, is k on the x-axis equal to the magnitude of the vector k or one of its components?

 

[genneth]

Yes to all, and k is on the x-axis since we are considering a 1D situation. For higher dimensions, the dispersion relation is a surface through some k-E space, but the general idea holds --- within a band, the edge of the BZ can be lower in energy than the middle.

 

[ehrenfest]

OK. Referring to the attachment, can you explain why the sense of the orbit is opposite in a) and b)? It is an electron that is doing the orbit in both cases, right?