# Anomalous Josephson Effect in magnetic Josephson junctions with noncentrosymmetric superconductors

## Abstract

We show that the two-band nature of noncentrosymmetric superconductors leads naturally to an anomalous Josephson current appearing at zero phase difference in a clean noncentrosymmetric superconductor/ferromagnet/noncentrosymmetric superconductor junction. The two-band nature provides two sets of Andreev bound states which carry two supercurrents with different amplitudes. When the magnetization direction of the ferromagnet is suitably chosen, two supercurrents experience opposite phase shifts from the conventional sinusoidal current-phase relation. Then the total Josephson current results in a continuously tunable ground-state phase difference by adjusting the ferromagnet parameters and the triplet-singlet ratio of noncentrosymmetric superconductors. The physics picture and analytical results are given on the basis of the + wave, while the numerical results are reported on both + and + waves. For the + wave, we find novel states in which the supercurrents are totally carried by continuous propagating states instead of discrete Andreev bound states. Instead of carrying supercurrent, the Andreev bound states which here only appear above the Fermi energy block the supercurrent flowing along the opposite direction. These novel states advance the understaning of the relation between Andreev bound states and the Josephson current. And the ground-state phase difference serves as a tool to determine the triplet-singlet ratio of noncentrosymmetric superconductors.

###### pacs:

74.50.+r, 74.70.Tx, 74.20.Rp## I Introduction

The noncentrosymmetric superconductor (NCS) has attracted much attention for the coexistence of spin-singlet and spin-triplet superconductivity and the possibility of topologically nontrivial surface states(1); (2); (3); (4); (5); (6); (7). The unusual properties of NCS originate from the absence of inversion symmetry from the crystal structure, which permits an antisymmetric spin-orbit coupling (SOC) odd in electron momentum, and leads to a chiral ground state. The SOC mixes the spin-singlet (even-parity) component and the spin-triplet (odd-parity) component in the superconducting pairing potential (8); (9); (10). The list of NCSs has grown to include dozens of materials such as LiPdPtB (11); (12), YC (13), and the heavy-fermion compounds CePtSi (14), CeRhSi (9), and CeIrSi (10). These compounds have various crystal structures and hence various forms of SOC and mixed triplet-singlet pair symmetry. The actual superconducting pairing mechanics and pairing state symmetry realized in these NCSs are still unclear.

In CePtSi and several other Ce-based NCSs, the spin-singlet and the spin-triplet components are expected to appear in comparable magnitudes (14), which may lead to more exotic effects. As a result, the NCS has two effective superconducting gaps that are relevant to two sets of spin-split Andreev bound states. The relative magnitude of the two parity components determines the relative size of the two gaps and the main property of a NCS. The question of how to determine the triplet-singlet ratio has attracted many theoretical and experimental efforts (15); (16); (17); (18); (19); (20); (21); (22). As one of the main experimental methods in the field of superconductivity, the Josephson effect has been widely studied in junctions based on NCSs to investigate the characteristic triplet-singlet ratios. There were efforts focusing on the investigations of the steps in the current-voltage characteristics (15), the low-temperature anomaly in the critical current (16), and the transition from a junction to a junction (17) in Josephson junctions with NCSs. The magnetic Josephson junction with NCSs has also been studied (18) and the authors focused on the high-order harmonics in the charge and spin current-phase relations and the possibility of - transitions.

On the other hand, much attention has also been paid to the anomalous Josephson effect which means a -junction with arbitrary ground-state phase difference other than or . Usually, the supercurrent in a Josephson junction vanishes when the phase difference between the two superconductors becomes zero and the current-phase relation (CPR) is sinusoidal in the tunnelling limit (23). Whereas an anomalous Josephson current flowing even at zero phase difference () has recently been predicted in various types of Josephson junctions (24); (25); (26); (27); (28); (29); (30); (31); (32). The anomalous supercurrent is equal to a phase shift in the conventional CPR, i.e., . In general, there are two prerequisites for the emergence of a -junction: (i) two sets of spin-split Andreev bound states (ABS) with opposite phase shifts compared with the conventional CPR, and (ii) different amplitudes of the supercurrents carried by the two sets of ABS. Especially, the Josephson junction formed on the surface of topological insulators by the proximity effect can be considered as the limiting case where only one set of ABS remain (33).

In Josephson junctions based on NCSs, the second prerequisite is naturally met due to the two-band nature of superconductivity in NCSs. The ABS are naturally spin-split into two sets and the two gaps with different sizes ensure that the supercurrents carried by two sets of ABS have different amplitudes. To meet the first prerequisite, we introduce a ferromagnet into the Josephson junction which brings phase shifts to ABS. In this study, we investigate the anomalous Josephson effect in a noncentrosymmetric superconductor/ferromagnet/noncentrosymmetric (NCS/F/NCS) junction. It is shown that the ground-state phase difference is sensitive to the triplet-singlet ratio of NCSs. Therefore, the anomalous Josephson effect serves as a mechanism to determine the unknown triplet-singlet ratio of a NCS. The physics picture and analytical results are given on the basis of the + wave, while the numerical results are reported on both + and + waves. For + wave, we find novel states in which the supercurrents are totally carried by continuous propagating states instead of discrete Andreev bound states.

The paper is organized as follows. In Sec. II we present the model Hamiltonian and introduce the numerical method based on the lattice Green’s function to solve the CPR. In Sec. III we present the analytical results of the normal incidence component in the case of s+p wave. The numerical results and relevant discussion on three types of pair potentials s+p, d+p, d+f will be given in Sec. IV. Finally, the conclusion will be given in Sec. IV.

## Ii Model and Numerical Methods

We consider a two-dimensional NCS/F/NCS junction in the clean limit. A schematic diagram of the junction under study is shown in Fig. 1. The ferromagnetic layer F has a finite width , and an exchange field whose direction is in the - plane and makes an angle with the -axis.

The numerical method used to evaluate the supercurrent is the lattice Green’s function technique. We consider the junction in a square lattice with the lattice constant . The lattice lies in the - plane. The ferromagnetic layer F sandwiched by the two NCS electrodes locates in the region with is the number of columns. During the tunnelling processes in the -direction, the transverse momentum is assumed to be conserved. In this context, the Hamiltonian of the hybrid junction reads

(1) |

where () is the creation (annihilation) operator of an electron in column with spin (= or ) and transverse momentum . The on-site energy has the form with . The Fermi energy and the nearest-neighbor hopping integral are assumed to be the same in the whole junction. The hopping coefficients of the exchange interaction and the Rashba SOC are given by the matrices in the spin space

(2) |

where and denote the exchange field and the Rashba strength respectively. The pair potential coefficient reads(42); (43)

(3) |

where is the BCS gap function which takes at zero temperature and vanishes at critical temperature . For simplicity, the pair potentials in the two NCS leads are set to be equal in amplitude. with , , are the Pauli matrices. The parameter () characterizes the percentage of spin-singlet(triplet) component which reads and in the left and right NCS respectively. In Eq. , we also omit the phase difference between the two NCSs for simplicity. The phase difference is set to be . Note that the exchange field exists only in the F layer while the Rashba SOC and the pair potential only exist in the two NCSs.

In the F region (, the charge operator in column with momentum is defined as

(4) |

where , is the unite matrix, and is the time. By using the Heisenberger equation , the operator of supercurrent is found. Then we can construct the Green’s function to calculate the supercurrent through column as follows (44)

(5) |

where and denote the hopping matrix and the charge matrix respectively. is the Pauli matrix in Nambu space and is the unit charge. In equilibrium, the lesser-than Green’s function equals

(6) |

where is the Fermi-Dirac distribution function. The retarded (advanced) Green’s function can be numerically calculated by the recursive method.

Besides the supercurrent, the ABS spectra can be also calculated numerically. It is known that the ABS results in the peaks of particle density within the superconducting gap. Therefore, by searching the peaks of particle density in column

(7) |

at a given phase difference , the energies of ABS can be located. Then we scan and obtain the ABS spectrum which is useful for understanding the behavior of supercurrent.

## Iii Analytical Results

Before we discuss our numerical results, we present the analytical results of the normal incidence component in the + wave case, which is very helpful to understand how the junction under consideration becomes a -junction. We start with the Bogoliubov-de Gennes (BdG) Hamiltonian for a NCS in the momentum space

(8) |

Here, is the spin-independent part of band dispersion with the chemical potential, and is the vector of Pauli matrices. is the antisymmetric SOC with and the SOC strength. The superconducting gap function is , where and are spin-singlet and spin-triplet superconducting gaps respectively with and turns between purely spin-triplet () and purely spin-singlet () pairings. We assume , , and are positive constants, and the orbital-angular-momentum pairing state is described by the structure factor . In the case of s+p wave, . The spin-triplet pairing vector is aligned with the polarization vector of the SOC where is the Fermi wave vector and taken as the unit of the wave vector. When the SOC splitting is much less than the chemical potential, we can use the Andreev approximation with the spin-split Fermi wave vectors.

To diagonalize the kinetic term, it is convenient to express the Hamiltonian in the so-called helicity basis. We introduce the following spin rotation transformation

(9) |

where is the incident angle of quasiparticles. Under the rotation , the normal part of the Hamiltonian is diagonalized and the superconducting gap function is transformed to

(10) |

with . The factor is the same for the left and right NCSs and thus has no net effect on the Josephson effect in the first harmonic approximation where the normal reflection is absent. The Hamiltonian of a NCS in the hilicity basis reads

(11) |

where . It is clear that the Hamiltonian shows a two-band nature that there are two bands with different superconducting gaps are uncoupled in the helicity basis. One band is for the Cooper pair made of spin-up electron and spin-down hole with gap , the other band is for the pair of spin-down electron and spin-up hole with gap with respect to the helicity basis. Since the critical Josephson current is linear in the gap, the two bands provide two supercurrents with different amplitudes. Thus the second prerequisite for the emergence of a -junction is naturally reached.

The first prerequisite for a -junction is easy to meet by utilizing a ferromagnetic interlayer. The middle F layer can bring opposite phase shifts to the two supercurrents provided by the two bands because the spin-triplet component of the gap function is also spin-opposite pairing in the helicity basis. For simplicity, we consider only the normal incidence component with which is dominant in the Josephson current for the case of orbital wave. We consider a F interlayer with the exchange filed strength and the width . The magnetization direction is chosen to be aligned with the polarization direction of the SOC, i.e., the -direction. In this situation, the two helical bands keep uncoupled from each other and the Hamiltonian of the whole NCS/F/NCS junction for the two bands is respectively

(12) |

where the helicity index is for the pair of spin-up electron and spin-down hole while is for the pair of spin-down electron and spin-up hole with respect to the helicity basis, with , when and with the macroscopic phase difference of the two NCSs. The F layer is expected to bring phase shifts to the set of ABS with index and the corresponding supercurrent. In the dimensionless units, the phase shift is approximately (45) with the wave vectors for spin-up or spin-down . Note that the right-going and left-going Cooper pairs with the same helicity index actually have opposite real spin with respect to the -direction. It is interesting that these two pairs have the same phase shift induced by the F layer (45), and correspond to the same gap at the same time.

After the two prerequisites are reached, we come to discuss the supercurrent carried by the two bands in the NCS/F/NCS junction by assuming that the ratio of the supercurrent to the gap is a constant for the two bands. In the first harmonic approximation, we have two supercurrents

(13) |

The total Josephson current is the sum . It is noticeable that the Josephson current depends on only and it does not matter whether or in the first harmonic approximation. We refer to the bigger (smaller) one of and as (). Then the total Josephson current can be written as

(14) | |||||

with

(15) |

It is seen that the anomalous ground-state phase difference depends not only on the F layer induced phase shift but also the triplet-singlet ratio of the two NCSs. Thus the arbitrary -junction can be obtained by tuning the ferromagnet parameters and the triplet-singlet ratio. We can also determine the triplet-singlet ratio of NCSs by detecting the ground-state phase difference of such a NCS/F/NCS junction.

In the above discussion, we assume that the left and right NCSs have the same singlet percentage . If the two NCSs have opposite parameters, i.e., with the parameter of left (right) NCS, the situation is a little different. For the band, the exchange of and changes the sign of the gap function as shown in Eq. (11). Therefore, there will be equivalently an additional phase difference between the left and right NCSs. For the band, the situation keeps unchanged. Then the total Josephson current is

(16) | |||||

with

(17) |

For the special case of and , the result of a junction for the triplet-ferromagnet-singlet Josephson junction is recovered as the same as in Ref. (46).

## Iv Numerical Results and Discussion

### iv.1 s+p wave

For the + wave case, the momentum dependence of the gap function . The magnetization direction of the F layer is chosen to be aligned with the -direction because the contribution from normal incidence is dominant. In Fig. 2, we show the numerically solved ABS at normal incidence in the NCS/F/NCS junction. With increasing ferromagnet induced phase shift (from left to right in Fig. 2), two sets of ABS with different helicity index depart from each other more heavily. The set of ABS with index experience a phase shift from the original degenerate ABS. With increasing singlet percentage , the energy span of the set of ABS with helicity index shrinks due to the decreasing minor gap . That means the corresponding supercurrent with has a smaller amplitude. The special case of is noticeable because the minor gap closes completely. Only the ABS with index is the remaining supercurrent-carried ABS. Then a -junction is easily realized by taking , which agrees well with Eq. (15). The evolution of ABS with varying and shown in Fig. 2 is fully consistent with the above analytical results.

It is also noteworthy that the ABS is almost the same for and as discussed in the analytical results. Because of the presence of weak normal reflections at NCS/F interfaces in the numerical results, there opens a small gap at some ABS crossing points due to the coupling of two ABS bands with the same spin and the opposite travelling direction (thus with the opposite helicity index ). What is interesting is that the anti-crossing effect for the case of is much weaker than that for . We can understand this effect easily in two limit cases and . For the coupled two bands with the opposite and the opposite , Eq. (12) shows that changes sign while does not. That is the essential difference between the singlet and triplet pairing. Thus, the normal reflection induced coupling results in an anti-crossing gap opening for the triplet-dominant case but no obvious effect for the singlet-dominant case at ABS crossing points. The anti-crossing effect gives rise to high-order harmonics of the CPR, which is sensitive to the triplet-singlet ratio as reported in Ref. (18). In this paper, we focus on the first harmonic of the CPR.

The CPR of the total Josephson current is shown in Fig. 3. The temperature is taken as so that only the first harmonic is remaining. When the F layer is absent, the CPR shows a normal -junction for all the values of as shown in Fig. 3 (a). While the critical current decreases with increasing firstly till and then increases when increases from to . The change of the critical current coincides well with the change of the minor gap in Fig. 2 because the contribution from normal incidence is dominant. Although the ABS are almost the same for and in the case of normal incidence, the total Josephson current is not exactly the same for and because of the contribution from inclined incidence. For oblique incidences, the spin-splitting of the Fermi surface is enhanced and will modify the magnitude of triplet pairing.

When the phase shift , the CPR is shown in Fig. 3 (b) for various . For from to , the ground-state phase difference reduces firstly to the minimum at and then goes back up to . The critical current also reaches its minimum at . All these features are qualitatively consistent with the ABS in Fig. 2 and Eq. (15). Unfortunately, we cannot distinguish the triplet-dominant pairing from the singlet-dominant pairing just by the ground-state phase difference. However, such is not the case when the phase shift . It is shown in Fig. 3 (c) that the ground-state phase difference of the CPR decreases monotonically from to when goes up from to . It is important that we can determine the triplet-singlet ratio just by the ground-state phase difference of the CPR. This special feature of can be understood by considering three limit cases . The case of is simple. The gap with closes and the remaining set of ABS with experience a phase shift . Obviously we obtain a junction. For the cases of pure triplet or pure singlet pairing (, or ), the supercurrents carried by two sets of ABS have the opposite phase shift and the same amplitude. Then the two supercurrents cannel each other out. Thus the contribution to supercurrent from large incidence angles is dominant instead of from normal incidence. For large incidence angle, the magnetization direction (-direction) is not aligned with the spin direction of helicity basis any more. The singlet pairing is always opposite-spin pairing (independent of the direction of the spin quantization axis) while the triplet pairing is equal-spin pairing in the spin quantization axes perpendicular to the helicity basis. It is well-known that the F layer cannot bring a phase shift for the equal-spin pairing. And when , the wave vector difference for . So the larger the incident angle becomes, the less (more) the absolute value of the phase shift is than for (). For pure pairings (, or ), the CPR should be either a -junction or a -junction because of the presence of two supercurrents with the opposite phase shift and the same amplitude. Therefore the CPR tends to become a -junction for while a -junction for . When goes up from to , the CPR naturally exhibits a - transition with smoothly changed ground-state phase difference. When , the ground-state phase difference varies between and , which is similar to the case of and also consistent with the ABS in Fig. 2 and Eq. (15). The difference is that the critical current for is now larger than that for because of the contribution from inclined incidence.

### iv.2 + wave

For the + wave NCS, the momentum dependence of the gap function . The dominant contributions come from the two components with the incident angle . We cannot choose a single magnetization direction which is aligned with both spin polarization directions of these two components. That is to say, the coupling of two helical bands is unavoidable for at least one of the two components. For simplicity, we began with the special case where only one gap survives and the other is closed. The numerical results show that this can occur at when . The singlet percentage deviates from due to the lattice model and the spin-splitting of Fermi surface. In Fig. 4, the particle density in the F layer at incidence angle shows the evolution of ABS with varied strength and direction of the exchange field. Because the minor gap is closing, there are only two ABS remaining. The right-going (left-going) ABS consists of a right-going (left-going) electron and a left-going (right-going) hole. It is noticeable that two ABS have different eigen spin direction because is fixed and finite.

When , the magnetization direction makes an angle of with both spin-down directions ( with respect to the helical basis) of two ABS. On the one hand, the F layer precesses the spin of electron and hole by an angle . Some spin-down particles are flipped to spin-up and enter into the NCSs without Andreev reflection because the spin-up gap is closed. The remaining spin-down particles proceed to finish the cycle of two Andreev reflections to form a ABS. So the particle density of two ABS shrinks with increasing . On the other hand, the F layer also brings a phase shift to the two ABS. The phase shift is opposite for right-going and left-going ABS. This is opposite to the previous case of the + wave where the phase shift is the same for right-going and left-going ABS. In that case, the magnetization direction along the -axis is parallel to the spin-down direction of left-going ABS but antiparallel to the spin-down direction of right-going ABS. When , the situation is similar. But the phase shift is now the same for two ABS because the -axis makes an angle of with the spin-down direction of right-going ABS while with that of left-going ABS.

When and , the situation is particularly interesting. Now the magnetization direction is parallel (or antiparallel) to the spin-down direction of one ABS but perpendicular to that of the other ABS. So one ABS experiences only a phase shift while the other not only experiences a phase shift but also shrinks a little. It is interesting that the shrinking ABS disappears totally at , which means that the spin-down particles are flipped totally to spin-up and propagate into NCSs as a continuous propagating state. For example, when and , the supercurrent is totally carried by continuous propagating states instead of discrete ABS for the phase difference range . It is shown that the ABS of and that of are symmetric to each other with respect to the axis for the reason of symmetry. Similarly, the ABS for is symmetric to that for with respect to the axis.

The corresponding total Josephson currents are shown in Fig. 5. When , the CPR exhibits a - transition with increasing . The anomalous Josephson effect does not occur because the two ABS have opposite phase shifts as discussed above. When , , and , the anomalous Josephson current appears. The CPR is the same for and as the symmetry between their ABS shows. And the CPR for is only a bit different from that for . As a whole, the ground-state phase difference is continuously tunable by adjusting . When and , the contribution from to the supercurrent (not shown here) show that the propagating state carried supercurrent (, see Fig. 4) equals to that carried by the ABS ().

Now we discuss the general case of arbitrary where the two gaps are both open. We choose to let the exchange field align with the spin polarization axis of left-going (right-going) ABS of () component. The CPRs for various and are shown in Fig. 6. The case of is similar to Fig. 3 (a) and thus not shown here. For and , the situation is similar to that in the case of s+p wave. Especially, the case of is still important to determine the triplet-singlet ratio of NCS because the ground-state phase difference varies monotonically with increasing while the critical current changes little. What is different from the case of s+p wave is that is not zero even at or . That is due to the spin-splitting of the momentum factor of gap function . For and , the evolution of critical current is similar to that in the + wave case. But the evolution of is very different from that for + wave. Here changes monotonically with increasing . As discussed previously, that is because the exchange field is perpendicular to the spin direction of right-going (left-going) ABS of () component. These ABS with perpendicular spin direction makes difference between singlet and triplet pairing. Thus as well as the critical current are different for singlet-dominant and triplet-dominant pairing.

### iv.3 + wave

For the + wave case, the momentum dependence of the gap function . The gap is maximum at or . However, the contributions from to the supercurrent are small in comparison with that from . The contribution from normal incidence is still dominant as in the case of s+p wave. Therefore the situation for d+f wave is similar to that for s+p wave and then the numerical results for d+f wave are not presented here.

## V Conclusion

In summary, we predicted the appearance of anomalous Josephson effect with nonzero ground-state phase difference in a NCS/F/NCS junction. The ground-state phase difference is proposed to serve as a tool to determine the triplet-singlet ratio of NCS. The physics picture and analytical results are given on the basis of + wave, while the numerical results and discussion are given on both + and + waves. For + wave, reaches the extremum when the singlet and triplet components have equal magnitude and there is no difference between singlet-dominant case and triplet-dominant case generally. But in the special case of , changes monotonically with increasing singlet percentage . For d+p wave, the monotonic change of with increasing is much more general if only is not too small. Interestingly, in the case of d+p wave, we also find novel states in which the supercurrents are totally carried by continuous propagating states instead of discrete ABS. Instead of carrying supercurrent, the ABS which here only appear above the Fermi energy block the supercurrent flowing along the opposite direction. These novel states advance the understanding of the relation between ABS and the Josephson current.

###### Acknowledgements.

The work described in this paper is supported by the National Natural Science Foundation of China (NSFC, Grant Nos. 11204187, 11204185, and 11274059).### References

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