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Which one of the following figures corresponds to the density of states g(E) of a typical intrinsic semiconductor? (E represents the energy level of a charge carrier)

  • A.E against g(E): one continuous curve bulging to the right with a single dashed level at its widest point
  • B.E against g(E): two curves leaving the energy axis at the edges of a gap with two dashed levels between them
  • C.E against g(E): two separate lobes, each rising from zero at its band edge to a maximum and back, with a gap between
  • D.E against g(E): two curves that run toward each other and overlap between two dashed levels

Answer: C

The correct figure is (C).

An intrinsic semiconductor has a filled valence band and an empty conduction band separated by a gap with no allowed states, so the plot must show two bands with a gap. Near each band edge the density of states is proportional to \(\sqrt{E - E_{edge}}\), so \(E - E_{edge} \propto g^{2}\): plotted with E up and g(E) across, each curve leaves the edge with a horizontal tangent and then bends towards the vertical. Each band is finite, so g(E) rises to a maximum and returns to zero at the other edge, which gives the two separate lobes of (C).

(A) has one continuous band and no gap (a metal-like picture). (D) shows the two bands overlapping, which has no gap. (B) does have a gap, but its curves leave the axis steeply (the wrong curvature for a square-root edge) and have no upper limit to the band, so it is not the usual picture.

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