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My book states,

" At absolute zero temperature, the electrons are tightly bound by the atoms. At this temperature, the covalent bonds between silicon atoms remain very strong and all the valence electrons are engaged in forming covalent bonds. So, no free electrons are available. The valence band of the semiconductor crystal is completely filled, and the energy gap between the valence and conduction band becomes large.

Now, how does the energy gap increase? My intuition is that as the energy bands (energy ranges) become narrower due to the temperature decreasing, the book states the gap between the bands also increases. Am I correct?

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The band gap is the interval of energy where no one-electron state is available. If there is no change of volume and as long as the crystalline structure is not affected by temperature variations (for example by the creation of defects) the band structure remains the same and also the band gaps. What varies with the temperature is the occupancy of the bands, but that is a different thing that has nothing to do with the band gap.

Notice that the thermal expansion at constant pressure, if present, would go in the opposite direction (of increasing the band gap with when temperature increases).

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  • $\begingroup$ Of course for just about all materials as you cool them the interatomic spacing decreases and you do see the bandgap increase. Its not a huge change, but it matters for some applications. $\endgroup$
    – Matt
    Jun 1, 2021 at 22:16

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