Posted by Steve Simon on May 30, 2013, 10:30 am, in reply to "Density of States in 2D tight binding"
Good question.
You are right, there is no closed form analytic expression. However, there are a few points that you should be able to get:
(1) For *free* electrons in 2D the DOS is indep of energy. So near the bottom of the band and top of the band, where the dispersion is basically parabolic, the DOS should be flat.
(2) As you go deeper into the band (away from the band edge) the dispersion increases LESS fast than the pure parabola. This means the DOS must deviate upwards from the flat answer.
(3) In the middle of the band, there are saddle points in the dispersion, this means there is a peak in the DOS. In fact, it is a log divergence -- and this requires a short calculation to see (I would not expect many students to have figured out the detailed form) -- but if you are interested, you can show it as follows
First, focus in on one of the saddles. Let qx = (kx - pi/a) and qy = ky then expanding around the saddle
E = c (qy^2 - qx^2)
with c some constant you can calculate. Now write the DOS as
DOS(epsilon) = integral dkx dky delta(E(kx,ky) - epsilon)
ie, you integrate over the BZ and use a delta function to count all of the k points that have energy epsilon.
Again focusing on one saddle this is
DOS(epsilon)
~ integral dqx dqy delta( c ( qy^2 - qx^2) -epsilon)
Do the qy integral first to get
~ integral dqx 1/sqrt{c qx^2 + epsilon}
Which diverges as a log from the contribution near qx=0.
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