Which one of the following is the correct expression for the Fermi-Dirac distribution ?
\(f(\mathrm{E}) = \frac{1}{e^{(\mathrm{E}-\mathrm{E}_f)/\mathrm{KT}}+1}\)
The Fermi-Dirac function gives the probability that an allowed electron state at energy E is occupied at temperature T. It follows from the Pauli exclusion principle applied to indistinguishable fermions:
\(f(\mathrm{E}) = \frac{1}{1 + e^{(\mathrm{E}-\mathrm{E}_f)/\mathrm{KT}}}\)
Checks: at E = Ef, f = 1/2; for E much greater than Ef, f tends to 0; for E much less than Ef, f tends to 1 — all physically required.
Swapping the exponent to (Ef − E) gives the probability of a state being empty (the hole occupancy, 1 − f). Using −1 in the denominator gives the Bose-Einstein distribution, which applies to bosons, not electrons. The last form drops the +1 and misplaces KT, so it is not a valid probability function.
Hence, the Fermi-Dirac distribution is \(f(\mathrm{E}) = \frac{1}{e^{(\mathrm{E}-\mathrm{E}_f)/\mathrm{KT}}+1}\).
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