Limits of thermoelectric performance with a bounded transport distribution

With the goal of maximizing the thermoelectric (TE) figure of merit $ZT$, Mahan and Sofo [Proc. Natl. Acad. Sci. USA 93, 7436 (1996)] found that the optimal transport distribution (TD) is a $\ensuremath{\delta}$ function. Materials, however, have TDs that appear to always be finite and nondiverging. Motivated by this observation, this study focuses on deriving what is the optimal bounded TD, which is determined to be a boxcar function for $ZT$ and a Heaviside function for power factor. From these optimal TDs, upper limits on $ZT$ and power factor are obtained; the maximum $ZT$ scales with ${\mathrm{\ensuremath{\Sigma}}}_{\mathrm{max}}T/{\ensuremath{\kappa}}_{l}$, where ${\mathrm{\ensuremath{\Sigma}}}_{\mathrm{max}}$ is the TD magnitude and ${\ensuremath{\kappa}}_{l}$ is the lattice thermal conductivity. These results help establish practical upper limits on the performance of TE materials and provide target TDs to guide band and/or scattering engineering strategies.