= Solution
The material figure of merit is
$$
\boxed{ZT=\frac{S^2\sigma T}{\kappa_e+\kappa_l}}.
$$
Large <thermoelectric power factor> supplies electrical output, while low thermal conductivity limits heat leaking without useful conversion. For a constant-property ideal generator, a common approximation is
$$
\eta_{\max}=\frac{T_h-T_c}{T_h}
\frac{\sqrt{1+\overline{ZT}}-1}{\sqrt{1+\overline{ZT}}+T_c/T_h}.
$$
Increasing the appropriately averaged figure of merit raises efficiency toward, but not beyond, the Carnot limit. Actual devices also incur contact, geometric and heat-transfer losses, and a single peak material ZT does not specify their efficiency.
The supplied trends show the trade-off with carrier density: conductivity rises, the magnitude of the <Seebeck coefficient> tends to fall, and electronic thermal conductivity rises approximately with conductivity through the <Wiedemann-Franz law>. Thus neither arbitrarily high doping nor maximization of conductivity alone gives the best ZT; the power factor and thermal terms produce an intermediate optimum.
Two complementary strategies are:
* \b[Reduce lattice thermal conductivity while preserving useful charge transport.] Alloy disorder, grain boundaries and suitably designed nanoscale interfaces scatter phonons. Nanostructured bismuth–antimony telluride provides an experimental example in which microstructure and transport measurements supported reduced lattice heat conduction.<High-Thermoelectric Performance of Nanostructured Bismuth Antimony Telluride Bulk Alloys by Bed Poudel and others>[Science 320 (2008), pp. 634–638; grain-boundary/defect phonon scattering and enhanced ZT]{ref}
* \b[Improve the power factor through band engineering and carrier-density optimization.] <Band convergence> increases useful valley degeneracy and can support a larger Seebeck coefficient without sacrificing conductivity as severely as a simple low-density strategy. Composition and doping tuning in lead telluride–selenide alloys demonstrated this route.<Convergence of electronic bands for high performance bulk thermoelectrics by Yanzhong Pei and others>[Nature 473 (2011), pp. 66–69; experimentally engineered valley convergence]{ref}
Both strategies must be checked for mobility penalties and electronic heat transport. Simply suppressing every kind of transport would also suppress electrical output.
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