Temperature-dependent thermal conductivity in silicon nanostructured materials studied by the Boltzmann transport equation

Nanostructured materials exhibit low thermal conductivity because of the additional scattering due to phonon-boundary interactions. As these interactions are highly sensitive to the mean free path (MFP) of phonons, MFP distributions in nanostructures can be dramatically distorted relative to bulk. Here we calculate the MFP distribution in periodic nanoporous Si for different temperatures, using the recently developed MFP-dependent Boltzmann transport equation. After analyzing the relative contribution of each phonon branch to thermal transport in nanoporous Si, we find that at room temperature optical phonons contribute $17\%$ to heat transport, compared to $5\%$ in bulk Si. Interestingly, we observe a constant thermal conductivity over the range $200\mathrm{K}<T<300\mathrm{K}$. We attribute this behavior to the ballistic transport of acoustic phonons with long intrinsic MFP and the temperature dependence of the heat capacity. Our findings, which are in qualitative agreement with the temperature trend of thermal conductivities measured in nanoporous Si-based systems, shed light on the origin of the reduction of thermal conductivity in nanostructured materials and demonstrate the necessity of multiscale heat transport engineering, in which the bulk material and geometry are optimized concurrently.

Figure 1

(a) Periodically aligned square pores (with porosity $\varphi =0.25$ and $L_c=10\mathrm{nm})$ subjected to a difference of temperature. The magnitude of thermal flux, which is normalized to its maximum value, shows that phonons travel mostly in areas between pores along the direction of the temperature gradient. (b) Thermal conductivity versus temperature for bulk Si and np-Si with porosities $\varphi =0.25$ and $\varphi =0.05$.

Figure 2

Cumulative thermal conductivity at $T=300\mathrm{K}$ for (a) bulk Si, (b) np-Si with porosity $\varphi =0.05$, and (c) np-Si with porosity $\varphi =0.25$. All the values are normalized to the total thermal conductivity. The largest MFP contributing to heat transport for a given branch can been seen from the point where the relative cumulative thermal conductivity becomes flat.

Figure 3

MFP distributions of bulk Si at (a) $T=300\mathrm{K}$ and (b) $T=200\mathrm{K}$. In both panels, the shaded area shows the region of MFPs up to ten times the characteristic size of the nanostructure in the case of $\varphi =0.25$. At low temperatures, the contribution to the thermal conductivity from long-MFP acoustic phonons rises. However, the ballistic regime constrains these MFPs to be equal to $L_c$.

Figure 4

(a) Specific heat capacity for different phonon branches and temperatures. Contribution of each phonon branch to the total thermal conductivity for (b) bulk Si and (c) np-Si for $\varphi =0.25$. In the range $200\mathrm{K}<T<300\mathrm{K}$, the contributions to the thermal conductivity from each phonon branch in np-Si do not change significantly with temperature. Within the ballistic regime, the temperature dependence of thermal conductivity is mainly dictated by the heat capacity.