4.1.1 Anisotropy in Ultra-Narrow Silicon Nanowires

The phononic dispersions of silicon nanowires of $2~\mathrm{nm} \times2~\mathrm{nm}$ cross section for different transport orientations are shown in Fig. 4.1. There are differences in the dispersions, especially in the low energy, low momentum region, which indicate that the thermal properties could be orientation-dependent as well. Figure 4.2 shows the ballistic transmission function of nanowires with side sizes $W=H=6~\mathrm{nm}$ for three orientations. The transmission function of the $\textless 110\textgreater$ nanowires is the highest in most part of the energy spectrum, whereas the transmission of the $\textless 111\textgreater$ nanowires is the lowest in almost the entire energy spectrum. As a result, the ballistic lattice thermal conductance of the nanowires (see Fig. 4.3-a) shows that the $\textless 110\textgreater$ nanowires has the highest thermal conductance compared to the $\textless 100\textgreater$ and $\textless 111\textgreater$ nanowires.

Figure 4.1: Phononic dispersions of silicon nanowires of square cross sections with $W=H=2~\mathrm{nm}$ for a) $\textless 100\textgreater$ , b) $\textless 110\textgreater$ , and c) $\textless 111\textgreater$ transport orientations.

The thermal conductance is larger for the $\textless 110\textgreater$ nanowires of all the side sizes we have considered, up to $10~\mathrm{nm}$ as shown in Fig. 4.3-a. The difference between the thermal conductances of the $\textless 110\textgreater$ nanowire, which has the highest, and the $\textless 111\textgreater$ nanowire which has the lowest, is a factor of $\sim 2$ . Another observation is that the thermal conductance increases as the cross section of the nanowire increases. This is expected since the larger nanowires contain more transport modes. The increase is close to linear. Once the conductances are normalized by the cross section area of the nanowires, however, the resultant normalized conductances remain almost constant. This is indicated in the Fig. 4.3-b. In this case, again, the $\textless 111\textgreater$ nanowire has clearly the lowest conductivity, almost 2 times lower than the $\textless 110\textgreater$ nanowire for all cross section sizes. In the next section, the reasons behind the anisotropy of the thermal conductance and conductivity are elucidated in terms of the nanowires’ phonon bandstructure and its related quantities.

Figure 4.2: Transmission function versus energy for nanowires in different transport orientations. $\textless 110\textgreater$ nanowires have the highest and $\textless 111\textgreater$ nanowires the lowest transmission.

Figure 4.3: (a) Ballistic lattice thermal conductance versus nanowires cross section side size for nanowires in different orientations. (b) The thermal conductance normalized by the cross section area.