6.2.1 The Effects of Boundary Scattering

Figure 6.3: 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. The dotted lines in (a) show for reference $ZT$ when only electron-phonon and phonon-phonon scattering is considered as in Fig. 6.2-a.

A profound increase in $ZT$ can be achieved when we consider the more realist case, in which we allow for electrons and phonons to undergo boundary scattering in addition to phonon scattering. The $ZT$ figure of merit in this case is shown in Fig. 6.3, (Fig.6.3-a for $n$ -type nanowires and Fig. 6.3-b for $p$ -type nanowires). The anisotropy, and the main features observed with diameter scaling are not altered significantly compared to Fig. 6.2. $ZT$ , however, increases by almost a factor of $4\mathrm{X}$ with the introduction of boundary scattering, both for $n$ -type and $p$ -type nanowires (the phonon-scattering only results of Fig. 6.2-a are shown in Fig. 6.3-a for reference). $ZT$ varies from $\sim 0.25$ to $\sim 0.75$ for $n$ -type nanowires ( $ZT \sim 0.75$ at larger diameters), also in agreement with other theoretical studies [145]. For $p$ -type nanowires it reaches a maximum of $ZT\sim 0.5$ for the $\textless 111\textgreater$ nanowires of smaller diameters.

This increase in $ZT$ indicates that the boundary scattering reduces the thermal conductivity by a factor of $\sim 4\mathrm{X}$ more than it reduces the power factor. As explained in Ref. [143], the introduction of surface roughness (SR) scattering, degrades the electrical conductivity significantly, but improves the Seebeck coefficient slightly. The degradation in the power factor, therefore, is dominated by the decrease in the electrical conductivity. Fig. 6.3 shows, however, that even in geometries with ultra-small feature sizes, in which the electrical conductivity is strongly degraded due to enhanced electron-phonon and electron-SR scattering, the thermal conductivity even stronger degrads. It suggests that phonon engineering techniques that further increase phonon scattering can help in the improvement of $ZT$ . This was considered to be the case for nanowires of larger diameters [20], but it appears to be the case also for nanowire diameters as thin as a few nanometers.