Thickness-dependent structural, vibrational, and superconducting properties of Bi ₂ Sr ₂ CaCu ₂ O <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow/> <mml:mrow> <mml:mn>8</mml:m

We report a systematic study of Bi₂Sr₂CaCu₂O8+x(Bi-2212) thin films grown on SrTiO₃substrates by pulsed laser deposition, with varying thicknesses to induce different levels of residual strain. The structural, morphological, vibrational, and superconducting properties of the films were thoroughly characterized by x-ray diffraction, AFM, Raman spectroscopy, and electrical transport. We observe that increasing<i>c</i>-axis strain correlates with a softening of selected phonon modes and a reduction in the superconducting critical temperatureTc. Although most vibrational shifts are consistent with<i>ab initio</i>predictions for this compound, the in-planeB1goxygen mode shows an anomalous response under strain, suggesting a strong electron-phonon coupling. A linear correlation betweenTcand the frequency of this mode further supports its relevance to the superconducting mechanism. A theoretical model based on quantum phase fluctuations and charge screening is applied to account for the observed suppression ofTc. However, the experimentally measured slope of theTcversus<i>c</i>-axis strain far exceeds the model prediction. This discrepancy is attributed to non-uniform strain and possible strain-induced changes in the dielectric environment. To address this, we propose an extension of the model considering spatial variations in the dielectric constant. Our findings provide new insights into the interplay between strain, phonon dynamics, and superconductivity in cuprates.