Silicon has a bandgap of ~1.1 eV whereas the room temperature thermal energy is ~0.04 eV. But we still find electrons in the conduction band for a pure silicon wafer, away from any radiation. I understand that Fermi-Dirac statistics predicts the existence of these conduction electrons in high energy states, but I'm not sure as to how do these electrons physically obtain this energy to excite, since the thermal energy is just too small too small.

I think the conceptual difficulty you're having is this: ~0.04 eV is only the average energy per degree of freedom. This energy, however, is randomly distributed. So at any given instant, some particles will have more than their fair share of energy while others will have less. It follows that some fraction will have enough energy to reach the conduction band.

  • And that fraction is very very small, the intrinsic carrier concentration in Si is about $10^{-12}$ per atom at room temperature. – Pieter Aug 10 at 8:31

Well the answer why is in these equation:

$ n_i \approx \exp{\frac{E_g}{2kT}}$ Even in GaN there exist the electrons in conduction band in intrinsic semiconductor but a much less. The reason is in the statictical behavior of electrons. The probability of left the valence level by electrons are not zero in temperature above 0K. Only in OK the semiconductor acts like insulator, no free electrons in conduction band, no free states to occupy in conduction band. Im not sure but in 0K exist only the hopping transport in valence band (like in polymers and insulator).

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