Josephson current in an irradiated Weyl semimetal junction*

Han Wang(王含) and Rui Shen(沈瑞)

1National Laboratory of Solid State Microstructures and School of Physics,Nanjing University,Nanjing 210093,China

2Collaborative Innovation Center of Advanced Microstructures,Nanjing University,Nanjing 210093,China

Keywords: Weyl semimetal,Josephson junction,FFLO pairing

1. Introduction

The Weyl semimetal is a three-dimensional topological material, where the conduction band and the valance band touch each other at two or more Weyl nodes and the paired Weyl nodes are connected by the unusual Fermi arcs.[1-7]The low-energy physics of the electrons around the Weyl node can be effectively described by the Weyl equation. Due to the nogo theorem,[8]the Weyl nodes always come in pairs with opposite chiralities.In order to realize the Weyl semimetal phase,the time reversal symmetry or the inversion symmetry are required to be broken.[9-11]

There are many interesting properties of the Weyl semimetals, such as the negative magnetoressitance,[12]the chiral anomaly,[13]and the anomalous Hall effect.[14,15]The superconducting pairing mechanism in the Weyl semimetals has also attracted much attention.[16-20]Two competing pairing states have been proposed,i.e.,the internode BCS pairing and the intranode Fulde-Ferrell-Larkin-Ovchinnikov(FFLO)pairing. The internode pairing combines two electrons from two Weyl cones of opposite chiralities and leads to the Cooper pair with zero-momentum. The intanode pairing combines two electrons from the same Weyl cone and leads to the Cooper pair with a finite momentum. It still remains a theoretical controversy that which pairing mechanism is favored in energy.[17,20]

Josephson current can be an effective method to probe the superconducting pairing mechanism.[21-31]Recently,there are several works focus on the Josephson effect and superconducting pairing mechanisms based on Weyl semimetal superconductor heterostructures. The influence of the chiral anomaly induced chiral charge imbalance on the Josephson current in a Weyl superconductor-Weyl semimetal-Weyl superconductor junction has been theoretically studied.[31]For BCS-like pairing, it has been predicted that the Josephson current exhibits 0-πtransitions and oscillates as a function of chirality imbalance. It has also been predicted[32]that inversion symmetric tilt will induce 0-πtransition and Josephsonφ0junction in BCS-like pairing.

In this paper, we investigate the dc Josephson current in the time-reversal broken superconducting Weyl semimetal junctions. A circularly polarized light is applied in the normal region of the junction and leads to an additional momentum shift to the electrons at the Fermi level. The momentum shifts are opposite for the electrons of the opposite chiralities. As a result, it is shown that the current phase relation remains unchanged for the BCS pairing and exhibits the 0-πtransition for the FFLO pairing, respectively. The different signals of the Josephson currents in the optical field may obtain a way to distinguish between the internode and intranode pairing cases.

The rest of this paper is organized as follows: our model is described in Section 2. The Andreev bound states and the Josephson current are obtained analytically in Section 3. The numerical results of the Josephson current are discussed in Setion 4. Finally,our results are briefly concluded in Section 5.

2. Model

The time-reversal broken Weyl semimetal junction under consideration is a normal Weyl semimetal sandwiched between two superconducing Weyl semimetal leads,as shown in Fig.1(a). The junction is along thezaxis and the width of the normal segment isd.There are at least one pair of Weyl nodes,namelyP±=±P0,located in the momentum space. The line connectingP±(axise3)is assumed in thex-zplane but has an angleβwith thezaxis,as shown in Fig.1(b).

Fig.1. (a)Schematic diagram of the Josephson junction. An irradiated Weyl semimetal with the incident light denoted by red arrows is sandwiched between two superconducting Weyl semimetals. The interfaces are at z=0 and z=d,respectively. (b)The illustration of the x-z coordinates and the e1-e3 coordinates.The line connecting two Weyl nodes(the e3 axis)is rotated from the z axis by an angle β. The direction of the optical field is measured from the z axis by an azimuth angle χ.

The minimal model of the Weyl semimetal without optical field is described by an effective two-band Hamiltonian,which is written in thee1-e3coordinates as[33]

withAx=E0cosχ,Ay=E0,Az=-E0sinχwhereωandE0are the frequency and the amplitude of the optical field,respectively,χis the the azimuth angle between the incident light and thezaxis,δrepresents light polarization andδ=0 (π)represents the right (left)-hand circularly polarized light. In the presence of optical light, the time-dependent Hamiltonian can be described by Peierls substitutionH(q,t)=H(q+eA(t)/ˉh). Here we focus on the high driving frequencyωgreater than the static energy bandwidth, so that an effective Hamiltonian reads[34-38]

In the superconducting region, both the internode BCS pairing term and the intranode FFLO pairing term are considered,which can be expressed as

whereT= iσ2Kis the time-reversal operator withKbeing the operator of complex conjugation.

For the convenience of the calculation,one can employ a unitary transformation[39]on Hamiltonian(8)and get

We note that the chemical potential and the pair potential are different in the different segments of the junction. The superconducting leads are heavily doped so that the Weyl nodes are well below the Fermi level. Therefore,the density of states at the Fermi level in the superconductor is large enough to form the condensation. While in the normal segment,the chemical potential remains finite and, in the rest of this paper, we useμNto denote the Fermi energy measured from the Weyl nodes in the normal region. We adopt the step approximation for the pair potential,which is assumed zero in the normal region and finite in the superconductors. A macroscopic phase of±φ/2 is assigned to the pair potentials in the right and left superconductors,respectively.

3. Andreev bound state

Based on the transfer matrix formalism, the Andreev bound states and the Josephson current can be obtained.[40]Taking the BCS pairing case as an example, the eigenwavefunction of the BdG Hamiltonian(9)for quasiparticles of energyεin the left superconductor reads

andqFis the Fermi wave vector in the superconductor. The wave vectors are properly chosen so that the wave functions decays exponentially in the infinity. Each four-component wave function can be divided into two parts,the first two components calledΨSeand the last two components calledΨSh. At the boundary,ΨSeandΨShare related by the transfer matrix given by As the case in the superconducting leads,the four-component wave functions in the normal region can also be divided into two-component wave functionsΨeandΨh, which are connected at the boundaries as

The continuity of the wave functions at the boundaries results in

which gives the Andreev bound energies.[40]For the short junction ofd ?ξwith the superconducting coherent lengthξ～ˉhvF/Δ0,one usually adopts the approximationΔ0?μN.Thus the energy dependence in anglesαe,handθe,hcan be neglected[40]and the Andreev bound energy is obtained analytically as

The Andreev bound energies in the FFLO pairing case can be obtained similarly. In contrast to the internode BCS pairing, the condensation in the FFLO case occurs in each Weyl cone independently. Therefore, the Andreev bound energies for the two Weyl nodes are different,which are written as

We note that the expression for theq‖,e(h)in the FFLO case is different from that in the BCS case(Eq.(19)). One obtains

for the calculation ofεF-,respectively.

With the help of the Andreev bound energies,the free energy of the junction is given by[41-43]

whereεbis the Andreev bound energy defined in Eqs.(25)and(27),the summation overbis restricted to all positive Andreev bound energies,andF0is a term independent of the phase differenceφ. Under the short junction approximation, only the contribution from the Andreev bound states is taken into account and that from the continuous spectrum is neglected in Eq.(31).[41-43]Finally,the Josephson current can be obtained as[41]

whereI+andI- are the current contributions from the two Weyl cones of opposite chiralities.

4. Results

First,we briefly discuss the basic concepts and the qualitative results on the Andreev bound states under the Bohr-Sommerfield quantization condition.[44,45]This will be very helpful in understanding the numerical results provided later.According to the Bohr-Sommerfield quantization condition,the Andreev bound state is determined bypdz+φA=2nπ,whereφAis the total phase shift obtained by the Andreev reflections andpdzis the phase shift accumulated by the quasiparticle moving in a loop in the normal region between two interfaces atz=0 andz=d,respectively. Under the circularly polarized light,the quasiparticle at the Fermi level acquires an additional momentum shift which results in an addition phase inpdzand the phase dependence of the Andreev bound energy.

In the BCS pairing case, the Cooper pair is composed by two electrons from the Weyl cones of opposite chiralities.From Eq.(6)one finds that the momentum shifts for the electrons in the two Weyl cones are opposite,namely,±A0cos(χ),respectively. The loop phase contains the contributions from a rightgoing electron and a leftgoing hole,which can be treated as the phase shift from two rightgoing electrons. Since the two electrons are from different Weyl cones and acquire the opposite phase shifts,the loop phase shift remains unchanged.Therefore, the phase dependence of the Andreev bound energy and the current phase relation of the Josephson junction remain unchanged.

In the FFLO pairing case, the Cooper pair is composed by two electrons of the same chirality. The accumulated phase shift of the rightgoing electron and that of the leftgoing hole in a loop are always the same. However, the total phase shift in a loop in theP+cone and that in theP- cone may be different. The phase shifts in a loop in theP+cone is 2φF+=2A0dcosχ,while that in theP- cone is 2φF-=-2A0dcosχ.The Josephson current includes the contributions from two Weyl cones, which take the form sin(φ-2A0dcosχ) and sin(φ+2A0dcosχ),respectively. Thus,the total current takes the form cos(2A0dcosχ)sinφ. As cos(2A0dcosχ) varying from positive to negative, the 0-πtransition occurs. As a result, the 0-πtransition can be expected in the FFLO pairing case withχ/=π/2.

Next,we give the numerical results of the Josephson current. The current phase relation in the BCS pairing Josephson junction is plotted in Fig. 2. The current is calculated at the temperatureT= 0.1TcwithTcthe superconducting transition temperature and at the width of the normal regiond/ξ= 0.025πwithξthe superconducting coherent length.The current is normalized byJ0=(eΔ0/ˉh)·(μ2NS),whereSis the section area andμ2Nrepresents the density of states at the Fermi level.Whenχ=0,it is shown in Fig.2 that the junction is always a 0 junction,as analyzed by the Bohr-Sommerfield quantization condition. Moreover, one finds that the Josephson currents with different optical field strengths coincide exactly.This is due to the widely adopted approximation that the energy dependences in anglesαe,handθe,hcan be neglected in short junctions.[40,46-49]Under this approximation, from Eq.(19), one finds that the wave vectorsq‖,e=q‖,hand consequently the angles for electrons (αeandθe) are exactly the same as those for holes(αhandθh)whenχ=0. The summation over the wave vectors in Eq.(32)is transformed into the integration over the anglesαeandθeand theA0-dependence in the current vanishes.

Fig. 2. Current phase relation in the BCS pairing Josephson junction. The parameters are assumed as μN =200Δ0 and β =π/4. The azimuth of the optical field is χ =0.

The current phase relation in the FFLO pairing Josephson junction is plotted in Fig 3. Whenχ=0, it is shown in Fig. 3 that the junction undergoes a 0-πtransition with increasing the optical field. The Josephson current presents a standard sinusoidal curve whenA0=nπ/(dcosχ) as shown by the black line in Fig. 3 which is a 0-junction. WhenA0= (2n+1)π/(2dcosχ), the current-phase relation is a negative sinusoidal curve as shown by the red line which is aπ-junction. IncreasingA0fromnπ/(dcosχ) to (2n+1)π/(2dcosχ), the current phase relationship deviates from the standard sinusoidal curve with an extra zero point other than 0 orπ, exhibiting a combined effect of two opposite Josephsonφ0junction contributing from two Weyl cones as described by the Andreev bound states in Eq.(27). Especially whenA0=(n+1/2)π/(2dcosχ),the current-phase relationship changes from sinφto sin2φ.

Fig.3. Current phase relation in the FFLO pairing Josephson junction. The parameters are assumed as μN =200Δ0 and β =π/4. The azimuth of the optical field is χ =0 for(a),χ =π/6 for(b),respectively.

In order to demonstrate the 0-πtransition more clearly,we also plot the free energy as a function of the optical field in Fig.4.One can easily find that,the minimum values of free energy changes with increasing optical field strength. IncreasingA0～(n+1/2)π/(2dcosχ),there are another dip/peak of free energy corresponding to extra zero points of the current-phase relationship. The relationship between the critical current andA0is plotted in Fig.5. Whenχ=0,there is a dip or some dips in theA0-dependence of the critical current as the signal of the 0-πtransition. Whenχ/=0,a similar behavior can be found and the critical current also decreases monotonically with the optical field.

Although our calculations are carried out with largeμN,the main results are still valid in the regime of smallμN. In the latter case the transport in the normal region is dominated by the evanescent modes. As a result,bothJandJ0decrease but the current phase relation remains qualitatively unchanged.[30]

Fig. 4. The free energy is plotted as a function of phase in the FFLO pairing Josephson junction. The parameters are assumed as μN =200Δ0 and β =π/4. The azimuth of the optical field is χ =0 for(a), χ =π/6 for(b),respectively.

Fig. 5. The critical current is plotted as a function of A0 in the FFLO pairing Josephson junction. The parameters are assumed as μN =200Δ0 and β =π/4.

5. Conclusion

In this paper,we have investigated the Josephson current in an irradiated Weyl semimetal junction. Both the internode BCS pairing and the intranode FFLO pairing are considered.The circularly polarized light-induced additional momentum shift modifies addition phase of the Andreev bound energy and the current-phase relation. It is shown that the circularly polarized light field will not alter the current phase relation for the BCS junction but can lead to a 0-πtransition for the FFLO junction. The 0-πtransition can only be observed for the FFLO junction and can be treated as an efficient signal for detecting the FFLO pairing in Weyl semimetals.