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The valence and conduction band of a semi-conductor are often drawn as here click.

This plot has essentially two features and I would like to understand them.

The peak and the valley of the two bands coincide, in what sense is this a characteristic feature for a semiconductor. I mean, I could have that they don't agree at all, would this necessarily mean that my material is not a semiconductor. I don't think so, as this should mostly depend on the energy gap. So why is this important in these drawings?

Both bands are drawn as parabolas ( with opposite curvatures), where does this come from?

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If the valence band maximum and the conduction band minimum are on the same position in k-space, this means that you have a direct gap semiconductor (GaAs, InAs, ...). If the CB minimum is at a finite k-value, it would be an indirect gap semiconductor (like Si, Ge, AlAs, ...)

The bands appear as parabolas due to the dispersion of a quasi-free electron/hole $$ E(k) = \frac{\hbar^2 k^2}{2 m^{\ast}} $$

Real bandstructures are not perfectly parabolic, but it serves as a good approximation around k = 0.

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