  With the goal of maximizing the thermoelectric (TE) figure of merit $ZT$,
Mahan and Sofo [Proc. Natl. Acad. Sci. U.S.A. 93, 7436 (1996)] found that the
optimal transport distribution (TD) is a delta function. Materials, however,
have TDs that appear to always be finite and non-diverging. 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 $\Sigma_{\rm max} T /\kappa_l$,
where $\Sigma_{\rm max}$ is the TD magnitude and $\kappa_l$ 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/scattering
engineering strategies.
