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Why is the fermi level half-way between the conduction band and valence band?

user44840
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    Contemplate the inverse question: how could the Fermi level NOT be mid gap in an intrinsic semiconductor? – Jon Custer Jun 05 '15 at 17:14
  • Fermi energy is the highest occupied energy state at T=0. Why can't it be 1/3 or 1/4 the way? – user44840 Jun 05 '15 at 17:27
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    Two things: there are no states in the gap, so clearly the definition of Fermi energy is not that simple. Second, for an intrinsic semiconductor at absolute zero all electrons are in the valence band. Now, raise the temperature just enough so that one, and only one, electron moves from the valence band to the conduction band. Where is the Fermi level? Raise it a bit more so a second electron moves from the valence to the conduction band. How much did the Fermi energy change? – Jon Custer Jun 05 '15 at 17:37
  • At low, non-zero temperature With one electron in the conduction band, the fermi energy is in the conduction band. Exactly at 0 temperature, the fermi energy is at the valence band. How do I find the average energy? – user44840 Jun 05 '15 at 17:42
  • Both answers are incorrect, pointing back to a misunderstanding around point one above. – Jon Custer Jun 05 '15 at 18:16
  • Ok, now I'm really confused. I thought fermi energy is the highest occupied state at near T=0? – user44840 Jun 05 '15 at 18:21
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    When there is a state there to occupy. The Fermifunction is convoluted with the density of states. – Jon Custer Jun 05 '15 at 19:28
  • Ok..so I still don't get it why must it be half-way – user44840 Jun 05 '15 at 22:05
  • Because otherwise the number of electrons would not be conserved, when heating the system. (Or stated in another way, you get that relation from the conservation of the electron number). – Sebastian Riese Jun 05 '15 at 23:44
  • At non-zero temperature, the Fermi level will be where the Fermi function is 50%. Now from that reconsider point two above. – Jon Custer Jun 05 '15 at 23:55

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