The ratio of charge carrier densities in a pure silicon crystal at $27^\circ \text{C}$ and $57^\circ \text{C}$ is 1.25. The energy gap of the semiconductor is:

Question

The energy of a photon of sodium light ($\lambda = 589 \, \text{nm}$) is equal to the band gap of a semiconducting material. The minimum energy required to create a hole-electron pair is:

Answer 1

The energy gap is linked to the charge carrier density ratio via:
\[\frac{n_2}{n_1} = \exp\left(-\frac{E_g}{2k}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)\right)\]
Using the given values: $\frac{n_2}{n_1} = 1.25$, $T_1 = 27 + 273 = 300 \text{ K}$, $T_2 = 57 + 273 = 330 \text{ K}$, and $k = 8.617 \times 10^{-5} \text{ eV/K}$:
\[1.25 = \exp\left(-\frac{E_g}{2 \times 8.617 \times 10^{-5}}\left(\frac{1}{330} - \frac{1}{300}\right)\right)\]
Applying the natural logarithm and solving for $E_g$:
\[E_g \approx 1.11 \text{ eV}\]

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The ratio of charge carrier densities in a pure silicon crystal at \(27^\circ \text{C}\) and \(57^\circ \text{C}\) is 1.25. The energy gap of the semiconductor is:

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The Correct Option is A

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