  We investigated the thermal conductivity of surface plasmon polaritons (SPPs)
propagating along thin Au and Ag films on a SiO$_2$ substrate with a Ti
adhesive layer. To determine the propagation length and skin depth of SPPs
along Au and Ag thin films, we numerically solved the dispersion relation while
considering the size effect of the permittivity of metal. Additionally, we
derived the spatial distribution of SPPs along the film thickness to analyze
the effect of the Ti adhesive layer on the plasmon thermal conductivity of Au
and Ag thin films. Our theoretical predictions revealed a decrease of
approximately 30\% in plasmon thermal conductivity when considering the size
effect of the permittivity of thin metal films. Furthermore, this causes the
film thickness at which maximum thermal conductivity occurs to increase by
about 30\%. Taking these factors into account, we calculated the optimal
thickness of Au and Ag films, along with Ti adhesive layers, on SiO$_2$
substrates to be approximately 20 nm. By fabricating a sample with the optimal
thickness of Au and Ag films, we experimentally demonstrated that the plasmon
thermal conductivity of Au and Ag films can be as high as about 20\% of their
electron contribution. This research will broaden the thermal design
applications of ballistic thermal transport by SPPs propagating along thin
metal coatings in microelectronics.
