Calculate the intrinsic carrier concentration in silicon at T=250 \mathrm{~K} and at T=400 \mathrm{~K}.
The values of N_{c} and N_{v} for silicon at T=300 \mathrm{~K} are 2.8 \times 10^{19} \mathrm{~cm}^{-3} and 1.04 \times 10^{19} \mathrm{cm}^{-3}, respectively. Both N_{c} and N_{v} vary as T^{3 / 2}. Assume the bandgap energy of silicon is 1.12 \mathrm{eV} and does not vary over this temperature range.
Using Equation (4.23), we find, at T=250 \mathrm{~K}
\begin{array}{c} n_{i}^{2}=N_{c} N_{v} \exp \left[\frac{-\left(E_{c}-E_{v}\right)}{k T}\right]=N_{c} N_{v} \exp \left[\frac{-E_{g}}{k T}\right] \\ \end{array} (4.23)
\begin{array}{c}n_{i}^{2} =\left(2.8 \times 10^{19}\right)\left(1.04 \times 10^{19}\right)\left(\frac{250}{300}\right)^{3} \exp \left[\frac{-1.12}{(0.0259)(250 / 300)}\right] \\ =4.90 \times 10^{15} \end{array}
or
n_{i}=7.0 \times 10^{7} \mathrm{~cm}^{-3}
At T=400 \mathrm{~K}, we find
\begin{aligned} n_{i}^{2} &=\left(2.8 \times 10^{19}\right)\left(1.04 \times 10^{19}\right)\left(\frac{400}{300}\right)^{3} \exp \left[\frac{-1.12}{(0.0259)(400 / 300)}\right] \\ &=5.67 \times 10^{24} \end{aligned}
or
n_{i}=2.38 \times 10^{12} \mathrm{~cm}^{-3}
Comment
We may note from this example that the intrinsic carrier concentration increased by over 4 orders of magnitude as the temperature increased by 150^{\circ} \mathrm{C}.