Identifying the chiral d-wave superconductivity by Josephson φ(0)-states

We propose the Josephson junctions linked by a normal metal between a d + id superconductor and another d + id superconductor, a d-wave superconductor, or a s-wave superconductor for identifying the chiral d + id superconductivity. The time-reversal breaking in the chiral d-wave superconducting state is shown to result in a Josephson φ(0)-junction state where the current-phase relation is shifted by a phase φ(0) from the sinusoidal relation, other than 0 and π. The ground-state phase difference φ(0) and the critical current can be used to definitely confirm and read the information about the d + id superconductivity. A smooth evolution from conventional 0-π transitions to tunable φ(0)-states can be observed by changing the relative magnitude of two types of d-wave components in the d + id pairing. On the other hand, the Josephson junction involving the d + id superconductor is also the simplest model to realize a φ(0)- junction, which is useful in superconducting electronics and superconducting quantum computation.