The conductivity of a semiconductor increases with increase in temperature because

For a semiconductor,

$\mathrm{\sigma }={\mathrm{n}}_{\mathrm{e}}{\mathrm{e\mu }}_{\mathrm{e}}+{\mathrm{n}}_{\mathrm{p}}{\mathrm{e\mu }}_{\mathrm{h}}$

$\mathrm{\sigma }=\frac{\mathrm{e}\left({\mathrm{n}}_{\mathrm{e}}{\mathrm{v}}_{\mathrm{e}}+{\mathrm{n}}_{\mathrm{p}}{\mathrm{v}}_{\mathrm{h}}\right)}{\mathrm{E}}$

With increase in temperature, motion as well as number density of the current carrier increases. Increase in number density increases the conductivity while increase in motion decreases the relaxation time and increases the collisions which decreases the conductivity. But, increase in conductivity due to increase in number density of charge carrier is dominated over decreases in conductivity due to decrease in relaxation time.