Finite Size Effects of Thermal Conductivity for One-Dimensional Mesoscopic Systems

The finite size effects of the thermal conductivity $\kappa$ have been studied in the phonon space. It is found that only a few phonon modes are selected to take part in the thermal transport when the size $L$ of the system is decreased. The amount of the selected phonon modes is proportional to the $L$. In this way, $\kappa$ decreases with the decreasing of $L$. Such mechanism for the size effect of $\kappa$ found in this work is beyond the Phonon-Boundary scattering. The exponent $\alpha$ of the power law $\kappa \sim L^{\alpha}$ has been fitted, showing that the exponent is not universal.