Question

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

${\mathbf{n}}_{\mathbf{e}}\mathbf{=}\left(\mathbf{4}\mathbf{.}\mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{15}} {\mathbf{cm}}^{\mathbf{-}\mathbf{3}} {\mathbf{K}}^{\mathbf{-}\mathbf{3}\mathbf{/}\mathbf{2}}\right){\mathbf{T}}^{\mathbf{3}\mathbf{/}\mathbf{2}}{\mathbf{e}}^{\mathbf{-}{\mathbf{E}}_{\mathbf{g}}\mathbf{/}\left(\mathbf{2}\mathbf{RT}\right)}$

Where, T is the temperature in kelvins and Eg the band gap in joules per mole. The band gap of diamond at 300 K is $\mathbf{8}\mathbf{.}\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{19}}\mathbf{J}$. How many electrons are thermally excited to the conduction band at this temperature in a$\mathbf{1}.\mathbf{00}-{\mathbf{cm}}^{\mathbf{3}}$ diamond crystal?

Short answer

The number of electrons excited to conduction band in diamond at 300 K is $n_e=2.49\times {10}^{19}mol/c{m}^3$.

Step-by-step solution

The given data

The given temperature is 300 K.

The bandgap energy $E_g=8.7\times {10}^{-19} J$

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)

$$\begin{gathered} n_e=\left(4.8\times {10}^{15}c{m}^{-3}{K}^{-3/2}\right)T^{3/2}{e}^{-Ea/2RT} \\ =\left(4.8\times {10}^{15}c{m}^{-3}{K}^{-3/2}\right)\times {\left(300K\right)}^{3/2}{e}^{-8.7\times {10}^{-19}J/\left(2\times 8.314 J/mol.K\times 300K\right)} \\ =2.49\times {10}^{19}mol/c{m}^3 \end{gathered}$$

The number of electrons excited to conduction band in diamond at 300 K is $n_e=2.49\times {10}^{19}mol/c{m}^3$.