Question

The band gap of pure crystalline germanium is $\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{19}}\mathbf{J}$ at 300 K. How many electrons are excited from the valence band to the conduction band in a$\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{-}{\mathbf{cm}}^{\mathbf{3}}$ crystal of germanium at 300 K? Use the equation given in the preceding problem

Short answer

The number of electrons excited to conduction band in Germanium at 300 K is $n_e=2.49\times {10}^{15}mol/c{m}^3$.

Step-by-step solution

The given data

The given temperature is 300 K.

The bandgap energy $E_g=1.1\times {10}^{-19} J$

The value of universal gas constant (R) is $R=8.314 J/mol.K$

Explanation

The number of electrons excited to the conduction band per cubic centimeter in a semiconductor can be estimated from the following equation:

$n_e=\left(4.8\times {10}^{15}c{m}^{-3}{K}^{-3/2}\right)T^{3/2}{e}^{-Ea/2RT}$ -----(1)

Calculation

Substitute the given values in formula (1)

$$\begin{gathered} n_e=\left(4.8\times {10}^{15}c{m}^{-3}{K}^{-3/2}\right)T^{3/2}{e}^{-Ea/2RT} \\ =\left(4.8\times {10}^{15}c{m}^{-3}{K}^{-3/2}\right)\times {\left(300K\right)}^{3/2}{e}^{-1.1\times {10}^{-19}J/\left(2\times 8.314 J/mol.K\times 300K\right)} \\ =2.49\times {10}^{15}mol/c{m}^3 \end{gathered}$$

The number of electrons excited to conduction band in diamond at 300 K is $n_e=2.49\times {10}^{15}mol/c{m}^3$.