6.2.3 Diffusive Thermoelectric Figure of Merit

Figure 6.5: The ZT figure of merit for (a) $n$ -type and (b) $p$ -type silicon nanowires versus diameter at $T=300~\mathrm{K}$ . Results for nanowires in the $\textless 100\textgreater$ (blue-circle), $\textless 110\textgreater$ (red-square), and $\textless 111\textgreater$ (green-triangle) orientations are shown. In the calculation of the power factor electron-phonon plus electron-boundary scattering is considered. In the calculation of the thermal conductivity phonon-phonon and phonon-boundary scattering are considered, but in this case the boundary is assumed to be fully diffusive. The dotted lines in (a) show for reference the $ZT$ when only electron-phonon and phonon-phonon scattering is considered as in Fig. 6.2-a.

This improvement in the power factor further proves the point that boundary scattering is more effective in reducing the thermal conductivity than reducing the power factor. As a result, phonon engineering techniques that cause additional reductions in the thermal conductivity could provide improvements of $ZT$ , despite the consequent reduction in the electrical conductivity. For example, in the case of a fully diffusive boundary for phonons, either by special engineering of the roughness [148,149,21], by decorating the surfaces with various species [150,59,151], or by modulating the nanowire's diameter [124], the $ZT$ performance could be increased. This is illustrated in Fig. 6.5, showing $ZT$ for the same nanowires as before in Fig. 6.3, but now we assume a fully diffusive boundary for phonons, e.g. the specularity parameter is set to $p=0$ for all wavevectors. In this case, $ZT$ is increased to values close to $ZT\sim 1.3$ for both $n$ -type and $p$ -type nanowires (in the best cases). This is almost a factor of $2\mathrm{X}$ improvement compared to the case we present in Fig.6.3 where we employ the $q$ -dependent specularity parameter $p(q)$ , rather than fully diffusive boundaries for all phonons.