The Kelvin Formula for Thermopower

TL;DR

This paper introduces the Kelvin formula for thermopower, relating the Seebeck coefficient to entropy and particle number derivatives, showing its effectiveness especially in strongly correlated systems and exactness in certain quantum states.

Contribution

The paper reexamines and derives a concise Kelvin formula for thermopower, linking entropy derivatives to the Seebeck coefficient, with applications to strongly correlated materials and quantum Hall states.

Findings

Kelvin formula provides a competitive approximation for thermopower.

The Kelvin formula is exact for non-Abelian fractional quantum Hall states.

The approach simplifies understanding thermopower in complex systems.

Abstract

Thermoelectrics are important in physics, engineering, and material science due to their useful applications and inherent theoretical difficulty, especially in strongly correlated materials. 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 the Seebeck coefficient. According to this formula, the Seebeck coefficient is given as the particle number $N$ derivative of the entropy $\Sigma$, at constant volume $V$ and temperature $T$, $S_{\text{Kelvin}}=\frac{1}{q_e}\{\frac{\partial {\Sigma}}{\partial N} \}_{V,T}$. 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.