Pisarenko's Formula for the Thermopower

TL;DR

This paper revisits Pisarenko's formula for thermopower, demonstrating its broader applicability across different scattering mechanisms and providing a simple relation useful for analyzing thermoelectric materials.

Contribution

It extends the validity of Pisarenko's formula beyond non-degenerate semiconductors and introduces a practical thermopower-conductivity relation for thermoelectric analysis.

Findings

Pisarenko's formula accurately describes thermopower beyond the non-degenerate limit.

A simple thermopower-conductivity relation is valid for high thermopower materials with acoustic phonon scattering.

The formula can estimate maximum power factor from a single sample's mobility.

Abstract

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.