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Question
The band gap for silicon is 1.1 eV. (a) Find the ratio of the band gap to kT for silicon at room temperature 300 K. (b) At what temperature does this ratio become one tents of the value at 300 K? (Silicon will not retain its structure at these high temperatures.)
(Use Planck constant h = 4.14 × 10-15 eV-s, Boltzmann constant k = 8·62 × 10-5 eV/K.)
Solution
Given:-
Band gap of silicon, E = 1.1 eV
Temperature, T = 300 K
Boltzmann's constant, k = \[8 . 62 \times {10}^{- 5} e\]V/K
(a) We need to find out the ratio of the band gap to kT.
Ratio \[= \frac{1 . 1}{kT}\]
\[= \frac{1 . 1}{8 . 62 \times {10}^{- 5} \times 3 \times {10}^2}\]
\[ = 42 . 53 = 43\]
(b) The new ratio is `1/10` th of the earlier ratio.
i.e. New ratio = 4.253
We know,
Ratio = (Band gap)/(kT)
\[\Rightarrow 4 . 253 = \frac{1 . 1}{8 . 62 \times {10}^{- 5} \times T}\]
\[ \Rightarrow T = \frac{1 . 1}{8 . 62 \times {10}^{- 5} \times 4 . 253}\]
\[ \Rightarrow T = 3000 . 4 K \approx 3000 K\]
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