Thermal conductivity in metallic nanostructures at high temperature: Electrons, phonons, and the Wiedemann-Franz law

The Boltzmann transport equation is used to calculate thermal and electrical conductivity of metal nanostructures with characteristic dimensions in the 25–500 nm range, near to and above the Debye temperature. Thermal conductivity contributions from phonons and electrons are considered. The intrinsic effects of electron-phonon, phonon-phonon, and phonon-electron scattering, and grain boundary and surface interactions are addressed. Excellent agreement is found between model results and available data reporting direct measurements of thermal conductivity of nanowires, ribbons, and thin films in Al, Pt, and Cu, respectively. The Wiedemann-Franz (W-F) law and Lorenz factor are examined with decreasing size; their applicability is found to degrade in nanowires due mainly to increased relative phonon contribution. The effect of differences in the electron mean-free path for thermal gradient versus electrical field is also examined. A modified version of W-F is presented, corrected for these two factors and valid from macroscale to nanoscale provided characteristic sizes exceed the phonon mean-free path.

Figure 1

(Color online) Thermal conductivity of Al nanowires (Ref. 16) with 100 nm thickness and varied width. Dashed curve is for electrons only, solid curve is combined electrons and phonons. Also shown as horizontal lines are measured values of $\kappa$ for bulk (Ref. 26) and 100-nm-thick film of Al (Ref. 25).

Figure 2

(Color online) Thermal conductivity of Pt nanoribbons (Ref. 6). Dashed curve is calculated thermal conductivity due to electrons, solid curve is for electrons and phonons. The dotted curve is calculated $\kappa$ using reported Pt grain sizes. Also shown is the measured value of $\kappa$ for bulk Pt (Ref. 26).

Figure 3

(Color online) Measured thermal conductivity of Copper thin films (Ref. 7), Dashed curve is for electrons only, solid curve is for electrons and phonons combined. The measured bulk value of $\kappa$ is also shown for Cu (Ref. 26).

Figure 4

(Color online) Thermal conductivity of selected metallic nanowires with square cross section at room temperature. (a) and (b) ${\kappa }_e$, (c) and (d) ${\kappa }_{ph}$.

Figure 5

(Color online) Ratio of phonon to total thermal conductivity for metallic nanowires with square cross section at room temperature.

Figure 6

(Color online) Size dependence of Lorenz factor due to ${\lambda }_e^{\nabla T}\ne {\lambda }_e^E$ and ${\kappa }_{ph}$.