Limits of thermoelectric performance with a bounded transport distribution

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