I have a rather naive question regarding DOS and dispersion.
We showed the existence of a band gap in class for a small, periodic perturbation in class last week. When drawing this, the professor took the parabolic dispersion relation and sort of "cut it" near the zone boundary to make the band gaps.
I am wondering in what sense we can look at the dispersion relation as the density of states, but with the axes flipped (or, alternatively, we can imagine rotating the curve 90 degrees)? It seems like the gaps in the DOS correspond to the same places as the gaps in the dispersion, but do the similarities end there?
My math in not strong enough to analyze this problem further on my own.
Thanks!
Best Answer
The relationship between the density of states (DOS) and the dispersion can be more easily visualized if you imagine a discrete dispersion rather than a continuous one (which is exaggerated here for visualization). The image on the left shows the dispersion $E(k)$ for a 1D parabolic band.
If we wanted to visualize the density of states, what you would do is sum the number of points in the left image for each energy slice. You can see this is effectively what is reproduced on the right. When the band is flat (i.e. at the bottom of the dispersion curve), there a lot of dots, which means there are a lot of states. This means that there is a high density of states for this energy. However, closer of the top of the curve, the points are sparse and the number of states per unit energy (or the density of states) is small.
Just as an aside: Where the density of states diverges, as at the bottom of the parabolic band below, it is called a van Hove singularity.