The influence of phonon softening on the superconducting critical temperature of Sn nanostructures

The increase in superconducting transition temperature (T_C) of Sn nanostructures in comparison to bulk, was studied. Changes in the phonon density of states (PDOS) of the weakly coupled superconductor Sn were analyzed and correlated with the increase in T_C measured by magnetometry. The PDOS of all nanostructured samples shows a slightly increased number of low-energy phonon modes and a strong decrease in the number of high-energy phonon modes in comparison to the bulk Sn PDOS. The phonon densities of states, which were determined previously using nuclear resonant inelastic X-ray scattering, were used to calculate the superconducting transition temperature using the Allen-Dynes-McMillan (ADMM) formalism. Both the calculated as well as the experimentally determined values of T_C show an increase compared to the bulk superconducting transition temperature. The good agreement between these values indicates that phonon softening has a major influence on the superconducting transition temperature of Sn nanostructures. The influence of electron confinement effects appears to be minor in these systems.