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(A) The probability $P(E)$ of finding an electron in an energy state $E$ is given by the Fermi-Dirac distribution function:$P(E) = \frac{1}{1 + e^{(E

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Decreases exponentially with increasing band gap

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(A) The probability $P(E)$ of finding an electron in an energy state $E$ is given by the Fermi-Dirac distribution function:$P(E) = \frac{1}{1 + e^{(E - E_F) / kT}}$where $E_F$ is the Fermi level,$k$ is the Boltzmann constant,and $T$ is the absolute temperature.For an intrinsic semiconductor,the conduction band starts at energy $E_c$. The probability of finding an electron in the conduction band is proportional to $e^{-(E_c - E_F) / kT}$.Since the band gap $E_g = E_c - E_v$ is related to the Fermi level (approximately $E_g \approx 2(E_c - E_F)$),the probability depends on the term $e^{-E_g / 2kT}$.As the band gap $E_g$ increases,the term $e^{-E_g / 2kT}$ decreases exponentially.Therefore,the probability of electrons being found in the conduction band decreases exponentially with an increasing band gap.

The probability of electrons to be found in the conduction band of an intrinsic semiconductor at a finite temperature:

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