Why does the resistivity of a semiconductor decrease with an increase in temperature?

(N/A) The resistivity of a semiconductor is given by the relation $\rho = \frac{m}{n e^2 \tau}$,where $m$ is the mass of the electron,$n$ is the number density of charge carriers,$e$ is the charge of the electron,and $\tau$ is the relaxation time.
In semiconductors,as the temperature increases,the thermal energy provided to the valence electrons increases.
This causes a large number of electrons to jump from the valence band to the conduction band,leading to a significant increase in the charge carrier density $(n)$.
Although the relaxation time $(\tau)$ decreases slightly due to increased collisions,the increase in the number density $(n)$ is exponential and dominates the effect.
Since $\rho \propto \frac{1}{n}$,the significant increase in $n$ results in a net decrease in the resistivity $(\rho)$ of the semiconductor.