Abstract
We present a rigorous analysis of the thermal conductivity of bulk silicon (Si) and Si nanowires (Si NWs) which takes into account the exact physical nature of the various acoustic and optical phonon mechanisms. Following the Callaway solution for the Boltzmann equation, where resistive and nonresistive phonon mechanisms are discriminated, we derived formalism for the lattice thermal conductivity that takes into account the phonon incidence angles. The phonon scattering processes are represented by frequency-dependent relaxation time. In addition to the commonly considered acoustic three-phonon processes, a detailed analysis of the role of the optical phonon decay into acoustic phonons is performed. This optical phonon decay mechanism is considered to act as acoustic phonon generation rate partially counteracting the acoustic phonon scattering rates. We have derived the analytical expression describing this physical mechanism which should be included in the general formalism as a correction to the resistive phonon-point-defects and phonon-boundary scattering expressions. The phonon-boundary scattering mechanism is taken as a function of the phonon frequency, incidence angles, and surface roughness. The importance of all the mechanisms we have involved in the model is demonstrated clearly with reference to reported data regarding the isotopic composition effect in bulk Si and Si NW samples. Namely, our model accounts for previously unexplained experimental results regarding (i) the isotope composition effect on the thermal conductivity of bulk silicon reported by Ruf et al. [Solid State Commun. 115, 243 (2000)], (ii) the size effect on κ(T) of individual Si NWs reported by Li et al. [Appl. Phys. Lett. 83, 2934 (2003)], and (iii) the dramatic decrease in the thermal conductivity for rough Si NWs reported by Hochbaum et al. [Nature (London) 451, 163 (2008)]