Suppose that an intrinsic semiconductor has a band gap energy of $0.5 \, \text{eV}$ and that the Fermi level (the hypothetical "natural" energy level for an electron in this material) lies at the top of the conduction band. What is the probability that the lowest-energy state of the conduction band is occupied at room temperature ($300 \, \text{K}$)?

Use the fact that that electrons are distributed according to the Fermi-Dirac distribution

$f(E) = \frac{1}{e^{(E-E_F)/(kT)} + 1},$

where $E_F$ is the energy of the Fermi level, $T$ is the temperature, and $k$ is the Boltzmann constant.

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