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

TL;DR

This paper develops a self-consistent theory to analyze the nonlinear response of $d$-wave superconductors with a subdominant $s$-wave component under voltage bias, revealing how non-equilibrium effects influence conductance features.

Contribution

It introduces a current-conserving, self-consistent framework for studying non-equilibrium superconducting states with mixed $d$ and $s$ wave order parameters.

Findings

Zero-bias conductance peak due to interface Andreev bound states.

Appearance of $s$-wave component near interfaces in $d+is$ state.

Voltage-induced suppression of $s$-wave component and peak splitting.

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 $eV\rightarrow\Delta_\mathrm{s}$. 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-B\"uttiker scattering approach.