Question

A semiconductor having electron and hole mobilities ${\mu }_n$ and ${\mu }_p$ respectively.
if its intrinsic carrier density is $n_i$, then what will be the value of hole concentration $P$ for which the conductivity will be minimum at a given temperature?

• A. $n_i\sqrt{\frac{{\mu }_n}{{\mu }_p}}$
• B. $n_h\sqrt{\frac{{\mu }_n}{{\mu }_p}}$
• C. $n_i\sqrt{\frac{{\mu }_p}{{\mu }_n}}$
• D. $n_h\sqrt{\frac{{\mu }_p}{{\mu }_n}}$

Answer 1

Answer: (A)

$n_i\sqrt{\frac{{\mu }_n}{{\mu }_p}}$

Let the hole concentration be $n_h$.

Then the electron concentration $=n_e=\frac{n_i^2}{n_h}$

The total conductivity of the semiconductor material $=\sigma ={\sigma }_e+{\sigma }_h$

$=e{n}_e{\mu }_n+e{n}_h{\mu }_p$

$=e\frac{n_i^2}{n_h}{\mu }_n+e{n}_h{\mu }_p$

Hence, the condition for maximum conductivity is found by $\frac{\sigma }{n_h}=0$

$⟹e{n}_i^2{\mu }_n(-\frac{1}{n_h^2})+e{\mu }_p=0$

$⟹n_h=n_i\sqrt{\frac{{\mu }_n}{{\mu }_p}}$