= 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 $Z$.
\b[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.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2020/paper-1-thermoelectric.png]
{title=Qualitative competition between thermopower, conduction and heat leakage as carrier density rises}
{source=paper-1-thermoelectric.py}
<Complex thermoelectric materials by G. Jeffrey Snyder and Eric S. Toberer>[Carrier-concentration trade-off and thermoelectric optimization]{ref}
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