Posted by Steve Simon on April 9, 2014, 8:52 pm, in reply to "density of states"
Hi n, Yes,this problem is hard. Generally we are concerned here with dispersions like E(kx,ky) = cos(kx)cos(ky) (and I'm dropping factors of a and prefactors etc). Near the bottom of the band (here kx=ky=0) the dispersion is a quadratic parabolic bowl E = (kx^2 + ky^2) + Constant just like a free electron. However as you go further from the bottom of a band it is no longer a parabola. As we get near the top of the band (at kx=ky=pi) it is actually an inverted parabolic bowl. E = (kx^2  ky^2) + Constant. But again as we go further from this point, it loses the perfect parabola shape (and indeed, it has to connect up with the other parabolic bowl at kx=ky=0). The first thing we might do is to imagine the case of the free electron (which is correct near the bottom of the band  and also at the top of the band). As we derived in some other exercises, in 2D, the density of states for free electrons is a CONSTANT independent of energy. So the density of states below the bottom of the band is zero, then it jumps up to a constant above the bottom of the band. Near the top of the band similarly it is a constant (then drops to zero above the top of the band). Now this result, that the density of states is constant is strictly correct as long as we have a parabolic dispersion (correct near the top or bottom of the band). But in the middle of the band it will fail. We would like to find out if there is a peak in the density of states in the middle of the band or a minimum. Consider the following. If you compare the actual cos+cos density of states to the parabolic approximation, it is clear that as you go away from the bottom of the band, the cos+cos increases LESS fast than a pure parabola (you can see this in several ways, one way is to expand the cosine to higher order and see what the fourth order term does). As a result there are MORE eigenstates per unit energy than you would have for the pure parabola (you can think of starting with the parabola and having to push energies down and together in order to match the cos+cos spectrum  and the further you go away from the minimum the more you have to push the energies around). Thus, compared to the free electron parabola, where the density of states is constant the cos+cos has a higher density of states as you move away from the minimum of the band. Thus there must be a peak in the density of states. Does this make sense? 
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