5.2 The Long-Wavelength Problem

In bulk semiconductors and insulators the thermal resistance arises from phonon- phonon scattering due of the anharmonicity of the inter-atomic potential. At room temperature, phonon-phonon scattering processes are strong and dominate the behavior of the thermal conductivity ( $\kappa_{\mathrm{ph}}$ ). A large part of the heat in semiconductors is carried by long wavelength longitudinal phonons, which have extremely long mean-free-paths (MFP) for scattering as well. In fact, it was pointed out that the MFP of the long wavelength longitudinal phonons diverges as their frequency tends to zero, resulting in the thermal conductivity to be a function of the size of the bulk solid, and diverging as the size of the solid increases [61,117]. The most commonly employed single-mode-relaxation-time (SMRT) approximation for the solution the Boltzmann transport equation (BTE) for phonons [60,62], uses an $\omega^2$ -dependent phonon-phonon (Umklapp) scattering rate [118,119]. This model, in combination with the 3D bulk density-of-states, which is proportional to $\omega^2$ at low frequencies, removes this ambiguity, and successfully explains the thermal conductivity of various semiconductors over a wide range of temperatures (also after appropriately including other common scattering mechanisms, such as defect scattering and grain-boundary scattering).

The thermal conductivity of 1D channels such as carbon nanotubes (CNTs), nanoribbons, and silicon nanowires has also been addressed in several recent studies [120,72,121,122,98,123,111,124], since such channels are attractive for heat management and thermoelectric applications [20,59]. The divergence of $\kappa_{\mathrm{ph}}$ with the size of the solid, or the problem of long longitudinal waves as referred to by Ziman [61], is stronger in this case, since the density-of-states in 1D structures is no longer $\omega^2$ -dependent, but has a finite value even at $\omega=0$  [125], which increases the importance of low wavevector phonons. Indeed, several theoretical and experimental works have pointed out that thermal conductivity in 1D systems deviates from Fourier's law, or even increases with increasing channel length, either linearly, logarithmically, or following some power law [120,126,72,121,127,128]. By including additional scattering mechanisms to the 3-phonon Umklapp mechanism usually employed, such as 3-phonon processes of second order [120], highly anharmonic potentials [129,130], employing the exact solution of the Boltzmann equation [126], or molecular dynamics (MD) [122], the divergence is reduced, but it is still persistent, especially at low temperatures.

Here, we revisit this problem for ultra-thin silicon nanowires of diameters below $12~\mathrm{nm}$ using the atomistic MVFF method for the calculation of the phonon modes and the BTE for phonon transport. We show that the issue of long-wavelength phonons turns out to be much more significant in 1D systems compared to the bulk material: This holds not only for low temperatures, but also at room temperature, and not only as the length of the channel is increased, but also as the diameter is reduced.