Theory of superconductors in coherent longitudinal phonon field

Abstract

The effect of injection of coherent longitudinal phonons into superconductors is investigated on the basis of the theory of nonequilibrium superconductors developed by Eliashberg. The two basic experiments about this effect, which were explained from different points of view, are explained for a unified point of view. They are (i) the change in the $I\ensuremath{-}V$ characteristics of $S\ensuremath{-}I\ensuremath{-}S$ tunnel junctions due to coherent phonon injection, i.e., phonon-induced tunneling, which was observed by Lax and Vernon, and Abeles and Goldstein, and (ii) an increase in the critical current ${I}_{c}$ in bridge and point-contact junctions, which was observed by Tredwell and Jacobsen. The phonon-induced tunneling was explained in terms of the theory by Tien and Gordon for the photon-induced tunneling. The increase in ${I}_{c}$ was attributed to an enhanced superconductivity, a theory which was first discussed by Eliashberg for superconductors in an electromagnetic field. We found that the Eliashberg effect always exists, but the phonon-induced tunneling is observed only when the space average of the phonon field, i.e., the deformation potential $\ensuremath{\phi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, t)$ is nonzero in the superconductor. The tunneling current is given by the superposition of both the effects. We have computed the excess tunneling current due to the Eliashberg effect. If we inject coherent longitudinal phonons parallel to the tunnel junction, they induce a space- and time-varying scalar (deformation) potential propagating with the phonons. We investigated whether we can extend the Josephson equation for the phase difference $\ensuremath{\varphi}$, i.e., $\ensuremath{\varphi}=\frac{2eV}{\ensuremath{\hbar}}$, to the space- and time-varying potential difference $V$ due to the phonons. We found that this type of extension is not justified, contrary to the photon-field case.