Self-consistent theory of current injection into $d$ and $d+is$ superconductors

Abstract

We present results for the steady-state nonlinear response of a $d_{x^2-y^2}$ superconducting film connected to normal-metal reservoirs under voltage bias, allowing for a subdominant $s$-wave component appearing near the interfaces. Our investigation is based on a current-conserving theory that self-consistently includes the non-equilibrium distribution functions, charge imbalance, and the voltage-dependencies of order parameters and scalar impurity self-energies. For a pure $d$-wave superconductor with [110] orientation of the interfaces to the contacts, the conductance contains a zero-bias peak reflecting the large density of zero-energy interface Andreev bound states. Including a subdominant $s$-wave pairing channel, it is in equilibrium energetically favorable for an $s$-wave order parameter component ${\Delta }_{\mathrm{s}}$ to appear near the interfaces in the time-reversal symmetry breaking combination $d+is$. The Andreev states then shift to finite energies in the density of states. Under voltage bias, we find that the non-equilibrium distribution in the contact area causes a rapid suppression of the $s$-wave component to zero as the voltage $eV\to {\Delta }_{\mathrm{s}}$. The resulting spectral rearrangements and voltage-dependent scattering amplitudes lead to a pronounced non-thermally broadened split of the zero-bias conductance peak that is not seen in a non-selfconsistent Landauer-Büttiker scattering approach.

15 pages, 11 figures

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