Posted by Steve Simon on April 17, 2015, 1:01 pm, in reply to "Question 4.4 (c): Density of States in 2D Band theory"
Yeah, good question. It requires a bit of detailed thinking. You should know what E(kx,ky) is (just tight binding in 2d). You want to know, given some particular E0, how many different (kx,ky) have E(kx,ky) = E0. This is the density of states. So how does one go about this. Well, first, you might look at energies near the top or bottom of a band. Here, the dispersion E(kx,ky) is more or less a pure parabola in 2d, so you can use the result that you derive for density of states in 2d for free electrons (with only m modified to m*). I think we did this in the first homework assignment. The more difficult thing to think about is what happens in the middle of the band  where it is neither a simple parabola near the top nor a simple parabola near the bottom. One possible way to start is to compare the band dispersion to that of the free electrons with mass m* (which is exactly what you have near top and bottom of band) and see if, as you move towards the middle of the band, can you figure out if the density of states in the band should be more or less than what you would get if you assumed the band were free electrons with mass m*. It takes a bit of thought, but you should get this far, and this should get you a pretty good cartoon of what the DOS looks like. If you want to get things more precise still, you can write down an explicit expression for the DOS DOS(E0) = (L^2/ (2pi)^2) integral dkx dky delta(E0  E(kx,ky)) (DOS per unit area would not have the L^2 out front). In other words, you just sum over all states in the band, and the delta function counts all states where E0=E(kx,ky). The problem with this approach is that the integral is pretty nasty. Nonetheless, it gives you an expression to work with. This representation is particularly useful when you are very near the center of the band where it is hard to guess exactly what is going on without more detailed calcualtion. 
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