Explain the recombination coefficient of an intrinsic semiconductor and derive the relation $n_i^2 = n_e n_h$.

(A) In a semiconductor,the generation of electron-hole pairs occurs when electrons gain enough energy to jump from the valence band to the conduction band.
These electron-hole pairs are not permanently stable. Due to thermal motion,electrons and holes collide. When an electron encounters a hole,it falls back into the hole,a process known as recombination.
In a semiconductor at thermal equilibrium,the rate of generation of electron-hole pairs is exactly equal to the rate of their recombination.
The rate of recombination is proportional to the product of the electron density $(n_e)$ and the hole density $(n_h)$:
$\text{Recombination rate} \propto n_e n_h$
$\text{Recombination rate} = R n_e n_h$
where $R$ is the recombination coefficient.
For an intrinsic semiconductor,the number density of electrons equals the number density of holes,denoted as $n_i$ $(n_e = n_h = n_i)$.
Thus,the recombination rate in an intrinsic semiconductor is $R n_i^2$.
At thermal equilibrium,the rate of generation $(G)$ must equal the rate of recombination:
$G = R n_e n_h$
For an intrinsic semiconductor,$G = R n_i^2$.
Equating the rates for the intrinsic case:
$R n_i^2 = R n_e n_h$
Therefore,$n_i^2 = n_e n_h$.