  Due to their extremely long propagation lengths compared to the wavelengths,
surface plasmon polaritons (SPPs) have been considered as a key in enhancing
thermal conductivity in thin metal films. This study explores the conditions at
which the plasmon thermal conductivity is maximized, considering the
thickness-dependent metal permittivity. We derived the analytical solutions for
the plasmon thermal conductivity in both the thin-film and thick-film limits to
analyze the effect of the permittivities of metals and substrates. From the
analytical solutions of plasmon thermal conductivity, we deduced that the
plasmon thermal conductivity is proportional to the electron thermal
conductivity based on the Wiedemann-Franz law. Additionally, we analyzed the
conditions where the enhancement ratio of the thermal conductivity via SPPs is
maximized. Metals with high plasma frequency and low damping coefficient are
desirable for achieving the maximum plasmon thermal conductivity as well as the
maximum enhancement ratio of thermal conductivity among metals. Significantly,
10-cm-long and 14-nm-thick Al film demonstrates most superior in-plane heat
transfer via SPPs, showing a 53.5\% enhancement in thermal conductivity
compared to its electron thermal counterpart on a lossless glass substrate.
