Pisarenko's Formula for the Thermopower

The thermopower $\alpha$ (also known as the Seebeck coefficient) is one of the most fundamental material characteristics for understanding charge carrier transport in thermoelectric materials. Here, we revisit the Pisarenko formula for the thermopower, which was traditionally considered valid only for non-degenerate semiconductors. We demonstrate that regardless of the dominating scattering mechanism, the Pisarenko formula describes accurately enough the relationship between thermopower $\alpha$ and charge carrier concentration $n$ beyond the non-degenerate limit. Moreover, the Pisarenko formula provides a simple thermopower-conductivity relation, $\alpha = \pm \frac{k_{\mathrm{B}}}{e} (b - \ln \sigma)$, valid for materials with $\alpha > 90$ $\mu$V K$^{-1}$ when acoustic phonon scattering is predominant. This offers an alternative way to analyze electron transport when Hall measurements are difficult or inaccessible. Additionally, we show how the Pisarenko formula can be used to estimate the maximum power factor of a thermoelectric material from the weighted mobility of a single, not necessarily optimized, sample at any given temperature.