Conductivity vs temperature

Conductivity σ = n e μ, where n is the density of free charge carriers, e the electron charge and μ their mobility.

Metals

In a metal the number of free electrons is fixed by the atoms and does not change with temperature. When the temperature rises, the lattice atoms vibrate more strongly, so electrons collide more often and their mobility μ falls. Hence conductivity decreases and resistivity increases with temperature. Over normal ranges ρ = ρ₀[1 + α(T − T₀)] with a positive temperature coefficient α. Copper and silver are common examples.

Semiconductors

In a semiconductor, few carriers exist at low temperature. Heat breaks covalent bonds and generates more electron-hole pairs, so n rises very rapidly (exponentially). The fall of mobility due to lattice vibration is much smaller than this rise. Hence conductivity increases with temperature, and resistivity decreases (negative temperature coefficient). Silicon and germanium are examples.

At very low temperatures, σ of a pure semiconductor approaches zero, while that of a metal rises to a high value limited by impurities and defects (a superconductor's resistance drops to zero below its critical temperature).