6.2 Thermoelectric Figure of Merit

Figure 6.2: 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 and the thermal conductivity, only electron-phonon scattering and phonon-phonon scattering are respectively considered.

To illustrate how the different scattering mechanisms affect $ZT$ , in Fig. 6.2 we show $ZT$ when only electron-phonon scattering is considered for the electronic system and only phonon-phonon scattering is considered for the phononic system. Results for $n$ -type nanowires are shown in Fig. 6.2-a, and results for $p$ -type nanowires in Fig.6.2-b. Further below, in Fig.6.3, we include the effect of boundary scattering for both systems. We consider nanowires of diameters from $D=3~\mathrm{nm}$ to $12~\mathrm{nm}$ in three different orientations, $\textless 100\textgreater$ (blue lines), $\textless 110\textgreater$ (red lines), and $\textless 111\textgreater$ (green lines). We assume a carrier concentration of $10^{19} \mathrm{cm}^{-3}$ , which is close to the concentration where the maximum of the power factor is observed. $ZT$ in Fig. 6.2 shows a considerable orientation and diameter dependence. For $n$ -type nanowires, the $ZT$ reaches values of $\sim 0.15$ in the best case (for the $\textless 111\textgreater$ nanowires and diameters above $6~\mathrm{nm}$ ). In the worst case, $ZT$ is significantly lower at only $\sim 0.05$ , which is achieved for the lowest diameters, as well as for the $\textless 110\textgreater$ nanowire in the entire diameter range. $ZT$ for these particular nanowires suffers from their large thermal conductivities as shown in Fig. 6.1 ($\sim 50\%$ higher compared to the other orientations). $P$ -type nanowires have lower $ZT$ values, below $0.05$ for most cases. A significant increase is observed in the cases of the $\textless 111\textgreater$ and $\textless 110\textgreater$ nanowires with decreasing diameter, which can be attributed to the improvement of their electrical conductivity [143] since the bandstructures of these nanowires undergo significant modifications with confinements and their hole effective mass is significantly reduced [144]. For both $p$ -type, and $n$ -type nanowires, the $\textless 111\textgreater$ orientation provides the best performance. The reduction in the thermal conductivity due to phonon confinement, therefore, increases $ZT$ by an order of magnitude compared to the bulk value ( $ZT_{\mathrm{bulk}}\approx 0.01$ ).