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In this paper, we address a closed-form analytical solution of the Joule-heating equation for metallic single-walled carbon nanotubes (SWCNTs). Temperature-dependent thermal conductivity κ has been considered on the basis of second-order three-phonon Umklapp, mass difference, and boundary scattering phenomena. It is found that κ , in case of pure SWCNT, leads to a low rising in the temperature profile along the via length. However, in an impure SWCNT, κ reduces due to the presence of mass difference scattering, which significantly elevates the temperature. With an increase in impurity, there is a significant shift of the hot spot location toward the higher temperature end point contact. Our analytical model, as presented in this study, agrees well with the numerical solution and can be treated as a method for obtaining an accurate analysis of the temperature profile along the CNT-based interconnects.