Kelvin formula for thermopower

Abstract

Thermoelectrics are important in physics, engineering, and material science due to their useful applications and inherent theoretical difficulty. Recent experimental interest has shifted to strongly correlated materials, where the calculations become particularly difficult. Here we reexamine the framework for calculating the thermopower, inspired by ideas of Lord Kelvin from 1854. We find an approximate but concise expression, which we term as the Kelvin formula for the Seebeck coefficient. According to this formula, the Seebeck coefficient is given as the particle number $N$ derivative of the entropy $\mathcal{S}$, at constant volume $V$, and temperature $T$, ${S}_{\text{Kelvin}}=\frac{1}{{q}_{e}}{{\frac{\ensuremath{\partial}\mathcal{S}}{\ensuremath{\partial}N}}}_{V,T}$. This formula is shown to be competitive compared to other approximations in various contexts including strongly correlated systems. We finally connect to a recent thermopower calculation for non-Abelian fractional quantum-Hall states, where we point out that the Kelvin formula is exact.