 Solution

At low carrier concentration, $|S|$ can be large but electrical conductivity is too small. Increasing the carrier concentration usually increases $\sigma =ne\mu$, while $|S|$ decreases. Electronic thermal conductivity rises approximately as ${\kappa }_e=L\sigma T$ by the Wiedemann-Franz law; lattice conduction supplies an additional background. Thus neither an insulator nor an ordinary metal usually maximizes $Z$.

Heavily doped semiconductors and semimetals often give the useful compromise, near a maximum of $S^2\sigma /\kappa$ as doping is varied. The optimum is material- and temperature-dependent; there is no universal optimal concentration.