Introduction

Much interest has been attracted to recent theoretical predictions that the superconducting pairing driven by strong electron correlation in materials with three- and sixfold rotational lattice symmetries favors topological chiral d + id symmetry. This symmetry is close to the d-wave superconducting pair in high temperature superconductors, but time-reversal broken. The materials which have been proposed to be chiral d-wave superconductors include graphene and silicene1,2,3, MoS24,5,6, In3Cu2VO97, SrPtAs8,9, and bilayer SrIrO310,11. But the experimental identification is still lacking. One of the experimental method to verify the chiral d-wave superconductivity is detecting the edge supercurrent induced by the topologically nontrivial superconducting order. But so far, no definite evidence of edge supercurrent is observed. Also, even theoretically, whether chiral d-wave supports an edge supercurrent still remains controversial12. Therefore, how to identify this novel superconducting order in experiments is of significant importance to further develop the microscopic theory of high temperature superconductors. The Josephson effect is another powerful method to identify the superconducting pair, should be promising to definitely identify the chiral d-wave superconductivity.

The efforts to detect the chiral d-wave order by Josephson effect are currently focusing on the signal of the critical current of the Josephson junction which varies with the superconducting order or other junction parameters. But the critical current, the amplitude of supercurrent, includes only indirect information on the pair symmetry, and can not definitely identify the chiral d-wave order. By contrast, the ground-state phase difference of the Josephson current, namely, the so-called Josephson φ0-states contains more ample and direct information on the pair potential, thus should be more promising to definitely identify the chiral d-wave pairing. Therefore, the theoretical and experimental investigation in the relation between the ground-state phase difference and the chiral d-wave pairing become important. To our best knowledge, the effort in this way is still blank Fig. 1.

Figure 1
figure 1

Schematic diagrams of Josephson junctions linked by a normal metal between a d + id superconductor and (a) another d + id superconductor, (b) a d-wave superconductor, and (c) a s-wave superconductor. The junctions lie in the x-y plane and the transport is along the x-direction.

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In the Josephson φ0-state, or the anomalous Josephson effect, the current-phase relation (CPR) has a phase shift φ0 compared with the conventional CPR, namely, I(φ) = Icsin(φ − φ0)13,14,15,16,17,18,19,20. The ground-state phase difference φ0 is neither 0 nor π in general, and tunable by the junction parameters. Such a φ0-state has been predicted in Josephson junctions with coexisting exchange field and spin-orbit coupling14,15,16,17,19, multilayer ferromagnets18,20, noncentrosymmetric superconductors21,22,23, and topological edge or interface states24,25,26. Very recently, the first experimental demonstration of a φ0-junction has been reported in a quantum dot junction by use of a quantum interferometer device27. These φ0-junctions with tunable ground-state phase difference may have applications in superconducting computer memory components28, superconducting phase batteries and rectifiers29, as well as flux- or phase-based quantum bits30. Among various ways to achieve a φ0-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 φ0 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 φ0-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

where is the kinetic energy measured from the chemical potential μ. The d + id pair potential is

with

Here θ is the injection angle satisfying tanθ = ky/kx, γ is is the angle between the x-direction and the α axis of the superconductor. Δ1 and Δ2 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) = [ΔLΘ(−x)eiφ/2 + ΔRΘ(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 ΔL (ΔR) reads 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 , and , with . Here is the Fermi wave vector, λ = 1 (−1) for the left (right) superconductor, τ = 1 (−1) for electron-like (hole-like) quasiparticles. Δλ± = Δλ(±θ) with Δλ+ (Δλ−) being the pair potential for right-going electron-like and left-going hole-like (left-going electron-like and right-going hole-like) quasiparticles. δλ± = δ(±θ, γλ) 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 R1 (R2) 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 R1 and R231. Then the Josephson current in the normal layer can be evaluated by

where is the Matsubara-Green’s function with the Matsubara frequencies ωn = πkBT(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 ΔL1 = ΔL2 = ΔR1 = ΔR2, the additional phase in the pairing potential for the left and right superconductor δλ(θ) = 2(θ − γλ) according to Eq. (3). Then the additional phase difference is 2(γR − γL), which means the ground-state phase difference is 2γL when γR = 0. It is exactly the case shown in Fig. 2(a) where the nearly sinusoidal CPR just moves to the right with increasing γL. The ground-state phase difference increases smoothly, while the shape and amplitude of the CPR keep unchanged. In the other limit ΔL2 = 0, ΔL(θ) = ΔL1cos[2(θ − γL)]. The left superconductor becomes a d-wave superconductor and the additional phase is 0. However, |ΔL(θ)| reaches its maximum at θ = γL, , or γL − 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 2γL because the additional phase for the right superconductor is 2θ with γR = 0. As shown in Fig. 2(b), the ground-state phase difference φ0 nearly keeps unchanged with varied ratio ΔL2/ΔL1 and just equals 2γL with increasing γL. The critical current Jc is shown in Fig. 2(c). With decreasing ratio ΔL2/ΔL1, the critical current decreases because |ΔL(θ)| decreases for all the incident angles θ except for the angles at which |ΔL(θ)| takes the maximum. Note that with a fixed ΔL2/ΔL1 smaller than 1, the critical current oscillates with a period of with increasing γL. This oscillation is due to the factor cosθ in Eq. (6) and the period is attributed to the four-fold rotational symmetry of d + id pairing. With decreasing ratio ΔL2/ΔL1 from 1 to 0, the amplitude of the oscillation increases from 0.

Figure 2: Josephson current for the Junction between two d + id superconductors with different directions of the α-axis.
figure 2

(a) CPR for various γL with fixed ΔL2/ΔL1 = 1. (b) The ground-state phase difference and (c) the critical current as functions of ΔL2/ΔL1 and γL are shown in the contour plots. The temperature T = 0.5Tc with Tc the critical temperature. ΔL1 = ΔR1 = ΔR2 = 10−3 μ, γR = 0. |ΔL|max = max(ΔL1, ΔL2).

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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 dxy-wave pairing by setting ΔR2 = 0 and . According to the above discussion, the right dxy-wave pairing decides that the dominant components in the Josephson current come from the incident angles . When ΔL2/ΔL1 = 1, the additional phases in ΔL(θ) for the two components are , which leads to the ground-state phase difference by taking into account of the sign change in dxy-wave pairing. The numerical results on φ0 shown in Fig. 3(a,b) verify this discussion. Because |ΔL(θ)| is independent of θ and γL for ΔL2/ΔL1 = 1, the critical current is also independent of γL as shown in Fig. 3(a,c). In the other limit ΔL2 = 0, the left superconductor reduces to a d-wave superconductor and its additional phase becomes 0. Then the anomalous Josephson effect disappears and φ0 must be either 0 or π. With increasing γL, 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 and turns into a π-junction for , accompanied with minima in Jc at the transition points γL = 0, , π. It is interesting to note the smooth transition between these two limiting cases ΔL2/ΔL1 = 0 and 1. With the changing of ΔL2/ΔL1 from 1 to 0, φ0 increases more and more nonuniformly with increasing γL, at the same time accompanied with more and more heavy oscillations in Jc, and finally evolves into a conventional 0-π transition.

Figure 3: Josephson current for the Junction between a d + id superconductors and a d-wave superconductor.
figure 3

(a) CPR for various γL with fixed ΔL2/ΔL1 = 1. (b) The ground-state phase difference and (c) the critical current as functions of ΔL2/ΔL1 and γL are shown in the contour plots. T = 0.5Tc, ΔL1 = ΔR1 = 10−3 μ, ΔR2 = 0, γR = π/4.

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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 ΔR(θ) = ΔR where the additional phase δR(θ) = 0. When ΔL2/ΔL1 = 1, the additional phase in ΔL(θ) is δL(θ) = 2(θ − γL). The angle-resolved Josephson current in the first harmonic approximation is I(θ) ∝ |ΔL|ΔRsin(φ + 2θ − 2γL). The total Josephson current is

This conclusion of φ0 = 2γL is consistent with the numerical result shown in Fig. 4(b). When ΔL2/ΔL1 = 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 γL. The evolution from the tunable φ0-state for ΔL2/ΔL1 = 1 to conventional 0-π transition for ΔL2/ΔL1 = 0 is similar to that discussed in Fig. 3. For the general case between ΔL2/ΔL1 = 1 and 0, the change of φ0 with increasing γL is simultaneously accompanied with the oscillation in Jc. The oscillation in Jc is achieved by second and higher harmonic terms as shown in Fig. 4(a).

Figure 4: Josephson current for the Junction between a d + id superconductors and a s-wave superconductor.
figure 4

(a) CPR for various γL with fixed ΔL2/ΔL1 = 0.3. (b) The ground-state phase difference and (c) the critical current as functions of ΔL2/ΔL1 and γL are shown in the contour plots. T = 0.5Tc, ΔL1 = ΔR = 10−3 μ.

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Finally, we comment on the experimental feasibility of observing the d + id superconductivity. In view of the fact that there are various possible paring mechanisms32,33 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 device27. 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 φ0 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 φ0-states advances the understanding of Josephson effect. And the simple φ0-junction consisting of the d + id superconductor is important to applications in superconducting electronics and superconducting quantum computation.

Additional Information

How to cite this article: Liu, J.-F. et al. Identifying the chiral d-wave superconductivity by Josephson φ0-states. Sci. Rep. 7, 43899; doi: 10.1038/srep43899 (2017).

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