Understanding Freeze-Out in Semiconductors at Low Temperatures

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At very low temperatures, freeze-out in semiconductors leads to a breakdown of the Boltzmann approximation due to the emergence of quantum effects. The Boltzmann distribution, rooted in classical physics, fails to accurately describe particle behavior as thermal energy decreases. Instead, the Fermi-Dirac distribution is used for fermions, while the Bose-Einstein distribution applies to bosons. This shift reflects the necessity of quantum statistical mechanics in accurately modeling semiconductor behavior at low temperatures. Understanding these principles is crucial for studying semiconductor properties in low-temperature environments.
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Studying the semiconductor in equilibrium, i found a sentence which i don't understand.

"At the very low temperature, freeze-out occurs; the Boltzmann approximation is no longer valid."

I know that freeze-out occurs at the very low temperature, but why is it that the Boltzmann approximation is no longer valid?
 
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Because the Boltzmann distribution comes from a classical theory. At low temperatures "quantum effects" start to become noticeble and you have to use the Fermi-Dirac distribution (for Fermions) or the Bose-Einstein distribution (for bosons).
 

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