  Through the integration of the power spectral density, we obtain temperature
profiles of both multi-segment harmonic and anharmonic systems, showing the
presence of an anomalous negative temperature gradient inside the interfacial
segment. Via investigating patterns of the power spectral density, we found
that the counterintuitive phenomenon comes from the presence of interfacial
localized phonon modes. Two out-band localized modes of the harmonic model,
which make no contributions to local temperature due to the absence of phonon
interactions, result in the concave temperature profile and over-cooling
effect. For the anharmonic model, thanks to the phonon-phonon interactions, the
localized modes are excited and make considerable contributions to interfacial
temperature, which is clearly shown by examining the temperature accumulation
function. When anharmonicity is considerably large, the negative temperature
gradient is absent since the localized phonon modes are fully mixed. The
presence of localized modes are evidently demonstrated by the inverse
participation ratio and normal mode analysis for the isolated harmonic model.
