This study explores the link between voltage-controlled architectural engineering and the resulting lattice dynamics in Cu–Fe functionalized TiO \(_2\) nanotubes. By systematically tuning the anodization potential (30-40 V) and time (40-60 min), we successfully tailored the nanotubular morphology and interfacial structure, as verified by XRD, Raman spectroscopy, and SEM analyses. This work is to elucidate how these structural modifications influence govern the fundamental thermodynamic behavior of the system. Through temperature-dependent heat capacity ( \(C_p\) ) measurements in the 2-300 K range, we reveal a complex vibrational beyond conventional. In the low-temperature regime, deviations from ideal Debye behavior by a Schottky-type contribution indicate a defect rich interfacial environment, where oxygen vacancies and structural disorder introduce discrete energy-level splittings. As temperature increases (50-210 K), a hybrid Debye-Einstein model suggests structural stiffening in the functionalized nanotubes. This effect is reflected in the upward shift of the characteristic phonon temperatures ( \(\theta _D\) and \(\theta _E\) ), indicating increased lattice rigidity and modified sound velocity within the nanotubular framework. Thermodynamic integration up to 298.15 K further demonstrates that the system’s free-energy balance is predominantly governed by vibrational entropy rather than internal energy accumulation. These results suggest that the performance of functionalized semiconductors arises not solely from composition, but from nanoscale architectural features that influence phonon-related behavior.
