Phonon Critical Points Reflected in Superconducting Tunneling Characteristics

Van Hove showed that there are mathematical singularities in the phonon distribution in frequency $g\left(\omega \right)$ associated with stationary points in $\omega \mathrm{}\left(q\right)\mathrm{}$ versus $q$; $\frac{\mathrm{dg}}{d\omega }$ has at least two infinite discontinuities, and approaches $-\infty$ at the upper end of the spectrum. Additional weaker singularities have been discussed by Phillips. The current voltage characteristic of superconducting tunnel diodes exhibits structure related to the phonon spectrum; we investigate theoretically the nature of the mathematical singularities in the tunnel characteristic associated with the Van Hove and Phillips singularities, using the Eliashberg gap equation. Discontinuities and logarithmic singularities are predicted for the metal superconductor junction at $T=0\mathrm{}$, and inverse square-root singularities as well in two-superconductor junctions. Analysis of Rowell's experimental curves for Pb and Sn in conjunction with neutron data on phonons in Pb provides definite evidence for the existence of such singularities and thus detailed confirmation of the theory.