Why do extrinsic semiconductors behave intrinsically at high temperatures?

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Is it because at high temperatures quasi-all electrons due to the doping are in the conduction band such that only the intrinsic behaviour is left?

Or is it something else?
 
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At high temperature the number of electrons in the conduction band (and holes in the valence band) due to thermal excitations (Fermi-Dirac statistics) is much larger than the number due to doping. You can then neglect the extra doped electrons/holes which takes you back to the intrinsic behavior.

The criteria for this should be kT >> | E_F,intrinsic - E_F,doped |
 

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