Single-particle electron-tunneling investigation of the phonon spectra and superconductivity of indium-tin alloys

Single-particle electron tunneling through an oxide barrier, between two metal films, Al—oxide—${\mathrm{In}}_x{\mathrm{Sn}}_{1-x}$, has been investigated in a study of the lattice and superconducting properties, in particular the phonon density of states, of the four distinct single-phase alloys of the In-Sn system. Measurements of the dynamic resistance of the junction, and second-harmonic response to first-harmonic stimulation across the junction, with the metal films in their superconducting state, have been analyzed using the Eliashberg gap equations and the McMillan computer program. The product function ${\alpha }^2\left({\omega }_q\right)F\left({\omega }_q\right)$, which exhibits the structure of the density of phonon states, $F\left({\omega }_q\right)$, has been determined for nine intermediate In-Sn compositions. Here $F\left({\omega }_q\right)$ is the phonon density of states, and ${\alpha }^2\left({\omega }_q\right)$ is an electron-phonon coupling parameter which varies slowly with ${\omega }_q$. In the determination of the phonon spectra, the following normal-state properties of In and Sn, and their alloys have been computed: the Coulomb pseudopotential ${\mu }^{*}$; the electron-phonon interaction strength $\lambda$; and an average phonon energy $〈{\omega }_q〉$. These three normal-state parameters were used to calculate the superconducting transition temperatures using the McMillan $T_c$ equation. These calculated transition temperatures are compared with our directly measured $T_c$'s and with previously measured values.