Question

Explain why an increase in temperature increases the conductivity of semiconductors while it decreases the conductivity in metals.

Answer 1

For semiconductors, conductivity is limited by the number of electrons excited into the conduction band, well above the Fermi level, and that number increases with temperature, according to the basic principle of Boltzmann statistics (probability of being in state j proportional to exp(-E_j/kT)).

On the other hand, for metals there is no problem with finding charge carriers, since there is no band gap, meaning there are conduction MOs so barely above the Fermi energy even the tiniest bit of energy is enough to promote electrons into them. For metals, the limitation on conductivity tends to be electron-phonon scattering, which you can think of as a lattice vibration momentarily pushing an atomic core out of place, and "breaking" the perfect symmetry that allows the MO in which the electron is traveling to extend over a very large distance. The result is that the electron can't travel as far, before requiring some movement between MOs. It can get stuck on some atom for a little time.

Since lattice vibration increases wtih temperature, electron-phonon scattering increases with temperatuer, and conductivity decreases.