Influence of electrons-per-atom ratio and phonon frequencies on the superconducting transition temperature of lead alloys

Abstract

The dependence of the superconducting transition temperature $T_c$ of binary and ternary alloys in the Tl-Pb-Bi family on the electron-per-atom ratio ($\mathfrak{z}$) and on the average phonon frequencies has been determined using electron tunneling. These alloys exhibit a variation from weak ($T_c=2.3\mathrm{}$ K) to strong ($T_c=8.95\mathrm{}$ K) coupling behavior as $\mathfrak{z}$ increases from 3 to 4.35. Over a substantial range of this variation the crystal structure remains fcc and it is found that $T_c$ is a monotonic function of $\mathfrak{z}$ in this phase. This is attributed to the increase in electron concentration, rather than to the decrease in phonon frequencies that was inferred from earlier neutron scattering measurements. In fact it is found that, although the electron-phonon coupling strength changes quite substantially throughout this series ($\lambda$ from 0.8 to 2.1), the phonon spectrum $F\left(\omega \right)$ does not vary appreciably and the average phonon energies remain relatively constant. We point out that the "softening" of the lattice observed by neutron scattering has only a small effect on the total phonon spectrum. Our data serve as a critical test for the various calculations of the electron-phonon interaction and superconducting parameters. Good agreement with the extensive free-electron calculations for $\lambda$ is obtained. The McMillan equation for $T_c$ works well throughout this series with the largest deviations occurring at highest $\lambda$, beyond the region considered by McMillan. It is found that the product $N\left(0\right)\mathrm{}〈{\mathcal{I}}^2〉$, which is approximately constant for the fcc transition elements, is not constant in this case and we question whether this assumption should be applied indiscriminately. For simple metals McMillan suggested that a better choice of constant would be $\frac{\lambda 〈{\omega }^2〉}{{\Omega }_p^2}$, where ${\Omega }_p$ is the ionic plasma frequency, and for this alloy series we note that this appears to be more closely obeyed. Hence we derive an alternative "maximum $T_c$ expression" by assuming that the phonon frequencies are fixed but that the electron density can be increased indefinitely. This expression agrees particularly well with experiment at smaller values of $\lambda$. Finally, we use the data to test the relationships between $\lambda$ and the phonon energy renormalization ${\Omega }^2-{\omega }^2$ and it is found that quite good agreement can be achieved in this case.