Why does the density of states of a semiconductor show no asymptotic behaviour compared to other bands?

Physics Q&A
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When trying to construct the density of states for a semiconductor, I used the approach from the many atoms per unit cell lecture. For me this looks very similar to a semiconductor with two bands, but the semiconductors have no asymptotic behaviour. What causes this difference?

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The top density of states is for a 3D system, the bottom one is for a 1D system. It makes a big difference, check out the lecture notes here: Sommerfeld model - Open Solid State Notes.

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Thank you! So the trick with drawing the density of states from the dispersion only works in 1D then I imagine?

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Indeed. The same conceptual idea would apply in higher dimensions (sampling every point in the higher dimensional k-space and making a histogram), but sketching using a plot of a 1D dispersion doesn’t.