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 δ 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 ΣₘₐₓT/κₗ, where Σₘₐₓ is the TD magnitude and κₗ 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.