More on Arrhenius plots

Axel Lorke’s Quick Study (Physics Today, May 2021, page 66 ) discusses underlying connections between seemingly unrelated phenomena. His brief reminder of Svante Arrhenius’s major contributions to science—and in particular, his now-famous empirical relation describing the rate of thermally activated processes—is thought-provoking.

There is, however, a minor point appearing in the caption of figure 2, showing electron density versus inverse temperature, that needs clarification. If we concentrate on the high-temperature (left) portion of the graph, the “reaction” involved is the generation of an electron–hole pair (analogous to electron–positron pair production in particle physics), for which the Gibbs free-energy change is what is known as the energy gap $E_{\mathrm{g}}$. For that reaction, the product of the electron density $n$ and hole density $p$ is given by $np\propto \exp (-E_{\mathrm{g}}/k_{\mathrm{B}}T)$, where $k_{\mathrm{B}}$ is the Boltzmann constant and $T$ is absolute temperature. ¹ At the high temperatures on the left side of the graph, the semiconductor is nearly intrinsic, so $n\approx p\propto \exp (-E_{\mathrm{g}}/2k_{\mathrm{B}}T)$, which is where the factor of two in the denominator of the slope originates.

The caption states that the “factor of two in the denominator arises because electrons obey Fermi–Dirac statistics rather than classical Boltzmann distribution.” But at such high temperatures, Fermi–Dirac statistics approximates the Boltzmann distribution, so it cannot be the reason for the factor of two as the caption states. A similar explanation applies to the ionization of neutral donors. ²