Universal Limits of Thermopower and Figure of Merit from Transport Energy Statistics

TL;DR

This paper derives universal upper limits for thermopower and figure of merit in thermoelectric materials based on transport energy statistics, independent of microscopic details, and confirms these limits with various system models.

Contribution

It introduces a general theoretical framework to estimate universal upper bounds for thermopower and ZT, independent of microscopic transport mechanisms.

Findings

Universal thermopower limit of about 160 μV/K.

Universal ZT limit of approximately 1.11.

Limits agree with experimental thermoelectric systems.

Abstract

The search for new thermoelectric materials aims at improving their power and efficiency, as expressed by thermopower $S$ and figure of merit $ZT$. By considering a very general transport spectral function $W(E)$, expressions for $S$ and $ZT$ can be derived, which contain the statistical weights of an effective distribution function only, see Refs. \cite{eltschka16,mahan96,matveev00}. We assumption of a Lorentzian shape with width $k_BT$ resulting from the electron-phonon coupling allows to estimate an upper limit of $S$ and $ZT$ independent on the microscopic mechanisms of the transport process. A simple estimate for an upper limit of the thermopwer $S$ is derived from {} formula. It is given by 3 times the unit of the thermopower $k_b/e$ which is about 250~$\mu V/K$. We consider different systems which represent the general features of the electronic structure of thermoelectric relevant materials very well. The transport integrals were evaluated varying the band gap size and the chemical potential position. For all cases upper limits for both, the thermopower and the figure of merit, are obtained. The universal limit of $|S|$ is given by 1.88 in units of $k_B/e$, which is about 160~$\mu V/K$. The universal limit for $ZT$ is obtained by about 1.11, which is in good agreement with available thermoelectric systems and devices.