Identifying the chiral d-wave superconductivity by Josephson φ0-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 φ0-junction state where the current-phase relation is shifted by a phase φ0 from the sinusoidal relation, other than 0 and π. The ground-state phase difference φ0 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 φ0-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 φ0- junction, which is useful in superconducting electronics and superconducting quantum computation.

as flux-or phase-based quantum bits 30 . Among various ways to achieve a ϕ 0 -junction, the junctions involving chiral d-wave superconductors should be the simplest one because the time-reversal symmetry is already broken in chiral d-wave pairing.
In this study, we investigate 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 (see Fig. 1). Anomalous Josephson effect appears as a result of the broken time-reversal symmetry in the d + id superconductor. The ground-state phase difference ϕ 0 other than 0 and π should be the definite evidence of the d + id pairing. Furthermore, the ground-state phase difference and the critical current can be used to determine the ratio between two types of d-wave components in the d + id superconductor. The demonstration of a smooth evolution from conventional 0-π transitions to tunable ϕ 0 -states is also interesting.
The paper is organized as follows. In Sec. II we present the model Hamiltonian and introduce the method to solve the CPR. The numerical results and relevant discussion of three types of junctions will be given in Sec. III. Finally, the conclusion will be given in Sec. IV.

Model and Methods
We begin with the BdG Hamiltonian of a two-dimensional d + id superconductor with parabolic spectrum in the normal state. The particular form of the spectrum may not take much effect in the ground-state phase difference of the Josephson junction, but the phase of the pair potential does. The spin degree of freedom is also not important and ignored here. Then the simplest BdG Hamiltonian of a d + id superconductor can be written as is the kinetic energy measured from the chemical potential μ. The d + id pair potential is Here θ is the injection angle satisfying tanθ = k y /k x , γ is is the angle between the x-direction and the α axis of the superconductor. Δ 1 and Δ 2 are two positive real numbers, denoting the amplitudes of two kinds of d-waves. It is clearly shown that an additional phase δ(θ) emerges in the pair potential and depends on the injection anlge. For a Josephson junction between two d + id superconductors, the pair potential can be approximately described by two step functions Δ (x) = [Δ L Θ (− x)e iϕ/2 + Δ R Θ (x − L)e −iϕ/2 ] where L is the length of the normal layer and ϕ is the macroscopic phase difference between two superconductors. Similar to Eq. (delta), the left (right) pair potential ( ) with λ = L or R denoting the left or right superconductor respectively. For simplicity, we assume that the momentum component in the y-direction is conserved. Then the eigen wavefunctions in two superconductors can be written as for electron-like quasiparticles and for hole-like quasiparticles where being the pair potential for right-going electron-like and left-going hole-like (left-going electron-like and right-going hole-like) quasiparticles. δ λ± = δ(± θ, γ λ ) according to Eq. (3).
The eigen wavefunctions can be easily solved for the normal layer. Then the scattering problem can be solved by considering the boundary conditions at two interfaces. Each interface gives a scattering matrix, from which the reflection matrix of the right-going (left-going) incident particles R 1 (R 2 ) can be abstracted in the normal layer.
To calculate the Josephson current, we can work out the Green's function G(z, z' , E) in the normal layer which is made of the reflection matrices R 1 and R 2 31 . Then the Josephson current in the normal layer can be evaluated by n is the Matsubara-Green's function with the Matsubara frequencies ω n = πk B T(2n + 1), n = 0, ± 1, ± 2, . By integrating over the injection angle γ, the total Josephson current in two dimension is where W is the transversal width of the junction.

Results and Discussion
Next, we present the numerical results and relevant discussion of three types of junctions involving chiral d-wave superconductors. We focus on the ground-state phase difference and critical current of the CPR. The interface transparency is found to neither change the ground-state phase difference, nor change the relative magnitude of critical current, but the absolute value of critical current. Therefore, the results about the effect of interface barriers is not presented here.

Junction between two d + id superconductors. For the Junction between two d + id superconductors
with different directions of the α-axis, the Josephson current is shown in Fig. 2. In the simple case Δ L1 = Δ L2 = Δ R1 = Δ R2 , the additional phase in the pairing potential for the left and right superconductor δ λ (θ) = 2(θ − γ λ ) according to Eq. (3). Then the additional phase difference is 2(γ R − γ L ), which means the ground-state phase difference is 2γ L when γ R = 0. It is exactly the case shown in Fig. 2(a) where the nearly sinusoidal CPR just moves to the right with increasing γ L . The ground-state phase difference increases smoothly, while the shape and amplitude of the CPR keep unchanged. In the other limit The left superconductor becomes a d-wave superconductor and the additional phase is 0. However, |Δ L (θ)| reaches its maximum at θ = γ L , γ ± π L 2 , or γ L − π. It implies that the Josephson current is dominated by the components related to these two incident angles. For these two components, the pair potential changes sign for both left and right superconductors. Then the ground-state phase difference is nearly still 2γ L because the additional phase for the right superconductor is 2θ with γ R = 0. As shown in Fig. 2(b), the ground-state phase difference ϕ 0 nearly keeps unchanged with varied ratio Δ L2 /Δ L1 and just equals 2γ L with increasing γ L . The critical current J c is shown in Fig. 2(c). With decreasing ratio Δ L2 /Δ L1 , the critical current decreases because |Δ L (θ)| decreases for all the incident angles θ except for the angles at which |Δ L (θ)| takes the maximum. Note that with a fixed Δ L2 /Δ L1 smaller than 1, the critical current oscillates with a period of π 2 with increasing γ L . This oscillation is due to the factor cosθ in Eq. (6) and the period π 2 is attributed to the four-fold rotational symmetry of d + id pairing. With decreasing ratio Δ L2 /Δ L1 from 1 to 0, the amplitude of the oscillation increases from 0. Junction between a d + id superconductor and a d-wave superconductor. Figure 3 shows the Josephson current through the junction between a d + id superconductor and a d-wave superconductor. The d-wave pairing in the right superconductor is chosen to be a d xy -wave pairing by setting Δ R2 = 0 and γ = π R 4 . According to the above discussion, the right d xy -wave pairing decides that the dominant components in the Josephson current come from the incident angles θ = ±  Fig. 3(a,b) verify this discussion. Because |Δ L (θ)| is independent of θ and γ L for Δ L2 /Δ L1 = 1, the critical current is also independent of γ L as shown in Fig. 3(a,c). In the other limit Δ L2 = 0, the left superconductor reduces to a d-wave superconductor and its additional phase becomes 0. Then the anomalous Josephson effect disappears and ϕ 0 must be either 0 or π. With increasing γ L , the junction experiences conventional 0-π transitions which is accompanied with heavy oscillations in the critical current. The oscillations in the critical current come from the varying angles between the α-axes of two d-wave superconductors. Specifically, the junction is a 0-junction for γ ∈       π 0, L 2 and turns into a π-junction for γ π ∈       π , L 2 , accompanied with minima in J c at the transition points γ L = 0, π 2 , π. It is interesting to note the smooth transition between these two limiting cases Δ L2 /Δ L1 = 0 and 1. With the changing of Δ L2 /Δ L1 from 1 to 0, ϕ 0 increases more and more nonuniformly with increasing γ L , at the same time accompanied with more and more heavy oscillations in J c , and finally evolves into a conventional 0-π transition. Junction between a d + id superconductor and a s-wave superconductor. The situation is similar for the junction between a d + id superconductor and a s-wave superconductor. The s-wave pairing in the right superconductor is described by setting Δ R (θ) = Δ R where the additional phase δ R (θ) = 0. When Δ L2 /Δ L1 = 1, the additional phase in Δ L (θ) is δ L (θ) = 2(θ − γ L ). The angle-resolved Josephson current in the first harmonic approx- This conclusion of ϕ 0 = 2γ L is consistent with the numerical result shown in Fig. 4(b). When Δ L2 /Δ L1 = 0, the junction reduces to a d-wave/normal metal/s-wave junction. Only the conventional 0-π transition is possible due to the sign change in the d-wave pairing with increasing γ L . The evolution from the tunable ϕ 0 -state for Δ L2 /Δ L1 = 1 to conventional 0-π transition for Δ L2 /Δ L1 = 0 is similar to that discussed in Fig. 3. For the general case between Δ L2 /Δ L1 = 1 and 0, the change of ϕ 0 with increasing γ L is simultaneously accompanied with the oscillation in J c . The oscillation in J c is achieved by second and higher harmonic terms as shown in Fig. 4(a).
Finally, we comment on the experimental feasibility of observing the d + id superconductivity. In view of the fact that there are various possible paring mechanisms 32,33 in graphene and graphene-like materials, we suggest that SrPtAs should be the most promising candidate to observe the d + id pairing. The experimental challenge lies in the observing the ground-state phase difference, and can be met by employing the superconducting quantum interference device 27 . We argue that the slight nonmagnetic impurities do not affect the ground-state phase difference qualitatively if only the d + id pairing is not destroyed. But the interplay between the magnetic impurities and the superconductivity is complicated and beyond the topic of this paper, and may be the next aim in our further study. Otherwise, we employ a quadratic dispersion relation to describe the quasiparticles in the normal state for the d + id paired superconducting leads. This quadratic dispersion is obviously invalid for graphene and graphene-like materials. But the result for the ground-state phase difference will not be changed because it is only affected by the phase of the pair potential. The critical current will be qualitatively unchanged but quantitatively modified if the linear dispersion is adopted.

Conclusion
In conclusion, we study the anomalous Josephson effect in junctions with chiral d + id superconductor induced by the time-reversal breaking of superconducting order. The ground-state phase difference ϕ 0 other than 0 and π is predicted and should be the definite evidence of the d + id pairing. The ground-state phase difference and the critical current are shown to depend on the ratio between two types of d-wave components and the direction of the α-axis of the d + id superconductor. The demonstration of a smooth evolution from conventional 0-π transitions to tunable ϕ 0 -states advances the understanding of Josephson effect. And the simple ϕ 0 -junction consisting of the d + id superconductor is important to applications in superconducting electronics and superconducting quantum computation.