Energy-based multichannel formulation of tunneling times and Hartman effect in coupled vibrational systems

We investigate vibrational energy transport, tunneling times, and Hartman effect in semiconductor thin films supporting coupled longitudinal and transverse modes. When mode conversion occurs at interfaces, the scattering process becomes intrinsically multichannel, requiring a consistent treatment of the associated energy flow and stored energy. Within the elastic continuum framework, we derive analytical expressions for the vibrational energy density and energy flux, explicitly identifying the interference contributions arising from longitudinal–transverse coupling. Based on these energy considerations, we formulate dwell and interference times that account for both channel-resolved and cross-channel effects. Using a stationary-phase analysis of quasi-monochromatic wave packets, we evaluate group delays at oblique incidence and analyze their dependence on frequency, angle, and layer thickness. The influence of the Goos-Hänchen shift on tunneling times has also been demonstrated. Numerical results for isotopically engineered germanium heterostructures illustrate how mode conversion influences interference times and leads to saturation of the tunneling time in the evanescent regime. The results provide a physically transparent energy-based description of tunneling dynamics in layered vibrational systems.