5. Anomalous Diameter Dependence of Thermal Conductivity in Ultra-Thin Silicon Nanowires

In this chapter, we present atomistic valence force field calculations of thermal conductivity in silicon nanowires of diameters from $12~\mathrm{nm}$ down to $1~\mathrm{nm}$ at room temperature. We show that as the diameter is reduced, the phonon density-of-states and transmission function acquire a finite value at low frequency, in contrast to approaching zero as in the bulk material. It turns out that this effect results in what Ziman described as the problem of long longitudinal waves [61], which states that the thermal conductivity of a material increases as its length is increased due to the vanishing scattering for long-wavelength phonons. We show that this effect also appears in nanowires as their diameter is decreased below $D=5~\mathrm{nm}$ . These features persist even in the presence of phonon-boundary scattering. Due to this increase in the contribution of long wavelength phonons, we show that the boundary becomes effectively more specular, and strikingly, the thermal conductivity increases with diameter decrease.