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Neamen's book states that the Fermi-Dirac distribution can be interpreted as the ratio of filled to total quantum states at any energy. For intrinsic materials, the value of Fermi-Dirac distribution at the Fermi level, which is in the middle of the conduction band and valence band, is 0.5. Does that mean quantum states and electrons exist in the forbidden energy gap? This seems to contradict with the definition of the forbidden energy gap. Please help! Thanks.

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    $\begingroup$ No, there are no states in the gap. That first sentence is actually incorrect - the F-D distribution is an occupancy factor if there are available states. $\endgroup$
    – Jon Custer
    Aug 12, 2020 at 15:12
  • $\begingroup$ Thanks! That makes much more sense. $\endgroup$
    – user207787
    Aug 13, 2020 at 0:10

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You need two ingredients to have electrons in some state

  1. Non void density of states (DOS)
  2. Non zero probability

If any of those ingredients fail, you'll have no electrons there.

In the case you mention, Fermi-Dirac's probability indicates that there is some probability of having electrons in the gap. However, the density of state says that there are no states there. So no matter how high the probability is, if there are no allowed states there, you'll have no electrons.

In the same way, if the DOS says that you have a band there, but the probability is zero, then that band is empty. If the probability changes then it can be filled.

In sum, you need both things, and they are independent.

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