Surface roughness and thermal conductivity of semiconductor nanowires: Going below the Casimir limit

By explicitly considering surface roughness at the atomic level, we quantitatively show that the thermal conductivity of Si nanowires can be lower than Casimir's classical limit. However, this violation only occurs for deep surface degradation. For shallow surface roughness, the Casimir formula is shown to yield a good approximation to the phonon mean free paths and conductivity, even for nanowire diameters as thin as $2.22\mathrm{nm}$. Our exact treatment of roughness scattering is in stark contrast with a previously proposed perturbative approach, which is found to overpredict scattering rates by an order of magnitude. The obtained results suggest that a complete theoretical understanding of some previously published experimental results is still lacking.

Figure 1

(Color online) $D=2.22\mathrm{nm}$ Si nanowire: perfect (left), with shallow surface roughness (middle), and with deeper surface cavities (right)

Figure 2

(Color online) Relaxation times, averaged over all the available modes for each frequency, in a $D=2.22\mathrm{nm}$ wire with half of its outermost atoms removed, calculated using the exact ${\mathbf{t}}^{+}$ matrix and the Born approximation. The Casimir limit is also plotted for comparison. The shadowed area corresponds to the part of the spectrum for which the resistance of the contacts would be the dominating factor, as explained in the text.

Figure 3

(Color online) Thermal conductivity of a $D=2.22\mathrm{nm}$ nanowire with half of its outermost atoms removed, as a function of temperature, calculated using Eq. (1) with the exact $t$ matrix, the Born approximation, or the Casimir MFP.

Figure 4

(Color online) (Bottom) Thermal conductivity at $T=1000\mathrm{K}$, relative to the Casimir value, of a $D=2.22\mathrm{nm}$ nanowire with half of its outermost atoms and an additional cluster of atoms removed, as a function of temperature, calculated using Eq. (1) with the exact $t$ matrix (solid line) or the Born approximation (dashed line). (Top) Minimum cross-sectional area of each nanowire, relative to the perfect system.