Anomalous Josephson effect between d-wave superconductors through a two-dimensional electron gas with both Rashba spin–orbit coupling and Zeeman splitting

Bin-Hao Du(杜彬豪), Mou Yang(楊謀), and Liang-Bin Hu(胡梁賓)

Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,School of Physics and Telecommunication Engineering,South China Normal University,Guangzhou 510631,China

Keywords: anomalous Josephson effect,d-wave pairing,Rashba spin–orbit coupling,Zeeman splitting

1.Introduction

The Josephson effect in hybrid mesoscopic systems involving conventional or unconventional superconductors have been extensively studied for decades.[1–5]Such hybrid mesoscopic systems provide ideal platforms for studying various aspects of the superconducting proximity effect.For example, in a conventional superconductor/topological insulator/conventional superconductor junction, the proximity effect between the superconducting paring and the helical spin–momentum locking of the topological insulator surface states may lead to the formation of Majorana zero modes[6]and oddfrequency pairing,[7]which have been viewed as rarely occurring in nature.The Josephson effect in various types of hybrid mesoscopic systems is not only of theoretical interest but also of practical importance,which are the key elements of superconducting electronics and also the bases of some conceptually new types of superconducting electronic components proposed in recent years,such as superconducting qubits,[8]cryogenic memories,[9]superconducting phase batteries,[10]phase qubits,[11]rectifiers,[12]etc.

The dissipationless Josephson currentIJflowing in a Josephson junction is related to the macroscopic phase differenceφof the superconducting order parameters between two superconductor leads.This relation is described by the current–phase relationship (CPR).The CPR is an important characteristic of a Josephson junction and its features link with the symmetries of the junction.For instance,in the presence of time-reversal symmetry or inversion symmetry,the constraintIJ(φ)=?IJ(?φ)is satisfied,which results in the vanishing of the Josephson current atφ=0.If both time-reversal symmetry and inversion symmetry are preserved,the CPR in its simplest form,readsIJ(φ)=ICsin(φ),withICthe critical current.Such a Josephson junction is called a 0-junction, the Josephson free energy of which is minimum atφ=0.However, if time-reversal symmetry or inversion symmetry is broken in a Josephson junction, the junction can turn into aπ-junction,the Josephson free energy of which is minimum atφ=πbut not atφ=0.In aπ-junction, the direction of the flow of the Josephson current at a phase biasφis reversed as compared with that in a 0-junction at the same phase bias.It is predicted thatπ-junctions and controllable 0–πtransitions may find some important applications in superconducting electronics, such as in the design of noise-immune superconducting qubits[8]and cryogenic memory.[9]The 0–πtransitions in conventional superconductor/ferromagnet/conventional superconductor junctions have been intensively studied in the past decades, which may result from Zeeman splitting by exchange interaction in the ferromagnetic layers sandwiched between two conventional s-wave superconductors.If a single ferromagnetic layer is sandwiched between two superconductor leads, the 0–πtransition can be induced by tuning the thickness of the ferromagnetic layer.[13]If multiple ferromagnetic layers are sandwiched between two superconductor leads, the 0–πtransition can also be induced by tuning the relative orientations of the magnetizations in different ferromagnetic layers.[14]In some recent studies, it was also found that when two conventional s-wave superconductors are linked by a bridge with strong intrinsic spin–orbit coupling(SOC),which may originate from either bulk inversion asymmetry (such as Dresselhaus SOC[15]) or structural inversion asymmetry(such as Rashba SOC[16]),the 0–πtransition can also be induced by tuning the strength of SOC in the bridge region.[2]Moreover,when strong SOC is in appropriate combination with Zeeman splitting in the bridge region,an anomalous phase shift may be induced in the Josephson current, which can take an intermediate valueφ0between 0 andπ.Such a Josephson junction is called aφ0-junction, the minimum of the Josephson free energy of which occurs at the phase biasφ=φ0but not atφ=0 or atφ=π.The presence of this anomalous phase shift implies that the Josephson current can flow between two superconductor leads even at zero phase bias.Such phenomenon is called the anomalous Josephson effect.Experimentally, the anomalous Josephson effect has recently been observed in Josephson junctions through a quantum dot[17]and Bi2Se3interlayer.[4]It is predicted that the anomalous Josephson effect may also find some important applications in superconducting electronics,such as superconducting phase batteries,[10]phase qubits,[11]and rectifiers,[12]etc.Due to these possibilities, various different ways have been proposed in recent years focusing on breaking both the time-reversal and the parity symmetries of a Josephson junction to create anomalous Josephson current at zero phase bias.

From the theoretical points of view, the anomalous Josephson effect is relate to the inverse Edelstein effect observed in some normal metals or semiconductors with strong SOC.While the Edelstein effect consists in the generation of an out-of-equilibrium spin polarization by a charge current,[18–21]the inverse Edelstein effect,also called the spingalvanic effect, consists in the generation of a charge current by an out-of-equilibrium spin polarization.[22–24]Both these two kinds of magnetoelectronic effects have been observed experimentally in some normal metals and semiconductors with strong SOC.[19–21,23,24]From the theoretical points of view,if in a Josephson junction two superconductor leads are linked by a bridge with strong SOC, similar magnetoelectronic effects may also occur in the junction,and in particular,if Zeeman splitting coexists with strong SOC in the bridge region,a finite Josephson current may also be generated in the junction even at zero phase bias due to the occurrence of out-ofequilibrium spin polarization in the bridge region, similar to the spin-galvanic effect found in some normal metals or semiconductors with strong SOC.Due to this similarity,the anomalous Josephson effect in such types of Josephson junctions may also be viewed as a kind of spin-galvanic effect.[1,25,26]

In this paper we study theoretically the Josephson effect in a ballistic Josephson junction in which two d-wave superconductor leads are linked by a bridge of two-dimensional electron gas(2DEG)with both Rashba SOC and Zeeman splitting.Up to now,while the features of the Josephson effect in junctions consisting of conventional superconductor leads and various types of bridges with strong SOC and Zeeman splitting have been intensively studied in recent years,[27–37]what peculiar features the Josephson effect may exhibit in junctions consisting of unconventional superconductor leads linked by bridges with both strong SOC and Zeeman splitting remains to be further investigated in some detail.From the theoretical points of view, since pairing potentials in unconventional superconductors are anisotropic and depend on the directions of quasi-particles’motions,the interplay of strong SOC and Zeeman splitting on the Josephson effect in junctions consisting of unconventional superconductor leads may be more subtle as compared with that in junctions consisting of conventional s-wave superconductor leads.For instance,in the d-wave case,the influence of SOC and Zeeman splitting on the propagations of quasi-particles may depend not only on the relative orientations and the relative strengths of the spin–orbit field and the Zeeman field in the bridge region but also depend on the relative orientations between theaaxes in two d-wave superconductor leads,[38]and due to such complicated dependences,it is possible that more rich features may be exhibited in the Josephson effect in such junctions.In this paper we take such complicated orientation dependences into account and present a systematical theoretical investigation on the features of the Josephson effect between two d-wave superconductor leads linked by a bridge of 2DEG with the simultaneous presence of both Rashba SOC and Zeeman splitting.Our studies are based on the McMillan’s Green’s function formalism,[39]which is a suitable formalism for studying quasi-particle transport in ballistic superconducting junctions.[39–41]We extend this formalism to include the interplay of Rashba SOC and Zeeman splitting and d-wave pairing,and based on the extended formalism,we calculate the Josephson current and the current–phase relation under various different situations.We show that due to the coexistence of multiple complicated factors, the interplay of Rashba SOC and Zeeman splitting on the Josephson effect in such heterostructures may exhibit a series of novel features,and in particular, we show that anomalous Josephson current may occur at zero phase bias under various different situations in such a junction, which depends sensitively not only on the strengths of Rashba SOC and Zeeman splitting but also on the relative orientations of the Zeeman field in the bridge region and the relative orientations of theaaxes in two d-wave superconductor leads.

The paper is organized as follows.In Section 2 we present the model Hamiltonian and construct the McMillan’s Green’s function formalism for calculating the Josephson current and the current-phase relation.The numerical results for the Josephson current and the current-induced relation are presented and discussed in Section 3.Finally,a summary is given in Section 4.

2.Model and formulation

We consider a ballistic Josephson junction in which two d-wave superconductor leads are linked by a bridge of twodimensional electron gas with both Rashba SOC and Zeeman splitting.The junction lies in they–zplane.The two interfaces between the bridge region and the two superconductor leads extend along theydirection and are located atz=0 andL,respectively,withLthe width of the bridge region.For simplicity,the junction interfaces are assumed to be perfectly flat and transparent.The direction of the superconducting phase bias is along thezdirection.To open a gap at the crossing point of two split bands by Rashba SOC, we assume that a Zeeman field?=(hx,hy,hz) is applied in the bridge region.The low-energy electron and hole excitations of the system can be described by the following Bogoliubov–de Gennes (BdG)Hamiltonian[42]

whereξk=ˉh2k2/2mis the kinetic energy of an electron with wave vectork= (0,ky,kz) and effective massm,μis the chemical potential,σ=(σx,σy,σz) represent the Pauli matrices acting on the real-spin degrees of freedom,σ0stands for the 2×2 identity matrix,Θ(z)is the Heaviside step function,andαRis strength of Rashba SOC in the bridge region.The gap matrix ??(k)= i?(θ)σyis a 2×2 matrix in spin space,with the magnitude of the pairing potential given by

whereθ= arccos(kz/k) is the propagating angle of quasiparticles with respect to thezaxis,aLandaRdenote the angles between thezaxis and theaaxes in the left and the right d-wave superconductor leads, respectively, andφis the macroscopic superconducting phase difference between two

superconductor leads.Since only the phase difference between two superconductor leads is related to the Josephson current, for simplicity, we set the macroscopic superconducting phase equal to zero in the left superconductor lead.According to the BCS relation,[41,42]the temperature dependence of the magnitude of the pairing potential can be given by?(T)=?0tanh(),where?0=1.76kBTc,withTcthe critical temperature andkBthe Boltzman constant.

Since the junction is assumed to be translational invariant along theydirection, the solutions of the BdG equation can be described by the general ansatzψ(z)eikyy, wherekyis the transverse component of wave vector andψ(z) is the solution of the BdG Hamiltonian for the effective onedimensional scattering problem along thezdirection, which can be constructed from the eigensolutions of the BdG Hamiltonian in each different regions of the junction.We first solve the eigensolutions within the d-wave superconducting regions.There are eight propagating modes in these regions for a fixed quasi-particle excitation energyEandky.Four of them are spin-up modes,and the corresponding eigenfunctions can be written as: (i) right-going electronlike quasi-particle modeψr,e,↑=eikez+ikyy, (ii) right-going holelike quasiparticle modeψr,h,↑=e?ikhz+ikyy,(iii)left-going electronlike quasi-particle modeψl,e,↑=e?ikez+ikyy,(iv)left-going holelike quasi-particle modeψl,h,↑=eikhz+ikyy.Here the subscripts r (l), e (h), and↑(↓) correspond to right (left)-going modes,electronlike(holelike)modes,and spin-up(spindown)modes,respectively,and=1～4denotes of the spinors of the corresponding quasi-particle modes.The explicit forms of the spinors=1～4can be given by

where

Similarly,there are four spin-down propagating modes in the d-wave superconducting regions, and the corresponding eigenfunctions can be written as: i) right-going electronlike quasi-particle modeψr,e,↓= ?B1eikez+ikyy,ii)right-going holelike quasi-particle modeψr,h,↓= ?B2e?ikhz+ikyy,iii)left-going electronlike quasi-particle modeψl,e,↓= ?B3e?ikez+ikyy, iv)left-going holelike quasi-particle modeψl,h,↓= ?B4eikhz+ikyy,where the explicit forms of the spinors ?Bi=1～4can be given by

In the bridge region between two superconductor leads,there are also four electron and four hole propagating modes,which can be described by the following eigenfunctions:

The scattering states in each different regions of the junction can be constructed from the linear combination of the corresponding propagating modes in each different regions.For example,for spin-up electronlike quasi-particles incident from the left superconductor lead,the wave functions of the scattering states in the left superconductor lead can be expressed as follows:

wherer1(r3) are the Andreev reflection coefficients without(with) spin flip andr2(r4) the normal reflection coefficients without (with) spin flip, respectively.In the bridge region between two superconductor leads, the wave functions of the scattering states can be expressed as

whereti=1～4are the transmission coefficients.All the scattering coefficients in the wavefunctions(10)–(12)are determined by the interfacial boundary conditions atz=0 andL, which are

where

is the velocity operator.

By the procedure outlined above,for a fixed quasi-particle excitation energyEandky, the wavefunctions corresponding to various scattering processes can be constructed in a similar form.On the basis of these wavefunctions, we can construct the retarded Green’s functionsGR(z,z′;E,ky) following McMillans’method.[39]To construct the retarded Green’s functions, we need the wavefunctions and the conjugates of all relevant outgoing waves due to various types of scattering processes and the conjugate processes.Let us take the retarded Green’s function in the left superconductor lead region (withz,z′<0 ) as the example.In this region, for electronlike or holelike quasi-particles incident from the left or from the right superconductor lead,there are eight different types of outgoing waves in total,and the corresponding scattering wavefunctions can be expressed as follows:

whereψ1～4represent the scattering processes when spin-up electronlike quasi-particles, spin-up holelike quasi-particles, spindown electronlike quasi-particles,and spin-down holelike quasi-particles are incident from the left superconductor leads,respectively,whileψ5～8represent the corresponding scattering processes when quasi-particles are incident from the right superconductor leads,respectively.The coefficientsai(ci)are the Andreev reflection amplitudes without(with)spin flip andbi(di)the normal reflection amplitudes without(with)spin flip,respectively.The coefficientsei,fi,gi,andhiare the transmission amplitudes.All these coefficients are determined by the interfacial boundary conditions (13)–(16) for the corresponding wave functions.Similarly, in terms of the eigensolutions of the conjugated BdG Hamiltonian, the conjugated wavefunctions corresponding to the conjugate processes of the eight different types of scattering processes described above can be expressed as follows:

All the coefficients in Eqs.(18a)–(18h)are also determined by the interfacial boundary conditions(13)–(16)for the corresponding wave functions.After all the relevant wavefunctions in Eqs.(17)–(18)are determined by the interfacial boundary conditions atz=0 andz=L, the retarded Green’s functionsGR(z,z′;E,ky) in the left superconductor lead region (withz,z′<0) can be constructed from a linear combination ofψi(z)j(z′)T, which are 4×4 matrices in Nambu-spin space and can be expressed explicitly as follows:

wherei=1～4 andj=5～8 ifz>z′,i=5～8 andj=1～4 ifz

whereτzrepresent the Pauli matrices in Nambu space.The coefficientsgijin Eq.(20) are determined by the boundary conditions(21)–(22).

There are 16 elements in total in the retarded Green’s function matrixG(z,z′).From the expressions of the current operator in the superconductor lead regions (see Eqs.(28)–(29) below), one can see that the elementsGee,↑↓,Gee,↓↑,Ghh,↑↓,Ghh,↓↑,Geh,↑↑,Geh,↓↓,Ghe,↑↑, andGhe,↓↓are not involved in the calculations when we calculate the charge current in the superconductor lead regions using the retarded Green’s function matrix.Due to this fact, the expressions ofG(z,z′) can be simplified substantially by setting these uninvolved elements to zero.By this simplification, we find thatG(z,z′) defined by Eq.(20) can be equivalently expressed as the sum of two different parts:G(z,z′) =G(↑)(z,z′)+G(↓)(z,z′), whereG(↑)(z,z′) andG(↓)(z,z′) are 4×4 matrices in Nambu-spin space and can be expressed by a set of simplified outgoing wavefunctions and the conjugates which take into account the contributions of spin-up and spin-down propagating modes separately.For instance, in the left superconductor lead regionG(↑)(z,z′)can be expressed by the following simplified forms of the wavefunctionsψ1,ψ2,ψ5,andψ6,which are given in Eqs.(17a)–(17h) and their conjugates are given in Eqs.(18a)–(18h),

But it should be stressed that all scattering coefficients in these simplified wavefunctions and the conjugates,includingai,bi,ei,fi,,,andshould still be calculated by use of the complete scattering wavefunctions given in Eqs.(17a)–(17h)and the conjugates given in Eqs.(18a)–(18h),which in combination with the interfacial boundary conditions(13)–(16)include not only the effect of all spin-conserving interfacial scattering processes but also the effect of all spin-flip interfacial scattering processes.The equation set Eqs.(23a)–(23h)only means that some elements ofG(z,z′)can be obtained equivalently by use of the above simplified forms ofψ1,ψ2,ψ5,andψ6and the conjugates.Due to the presence of spin-orbit coupling and Zeeman splitting in the bridge region,the spin-up and spin-down scattering processes should be treated as a whole but cannot be treated separately as simplified in the equation set Eqs.(23a)–(23h).

Substituting Eqs.(23a)–(23h)into Eq.(20),we find that in the left superconductor lead regionG(↑)(z,z′)can be expressed in the following explicit forms:

forz′

forz

完成前四步工作之后，需開始進(jìn)行臨時(shí)排水系統(tǒng)的建設(shè)，選擇強(qiáng)夯技術(shù)的施工點(diǎn)。強(qiáng)夯技術(shù)是整個(gè)工程建設(shè)的重中之重，強(qiáng)夯技術(shù)運(yùn)用是否成功，將直接影響該工程的建設(shè)質(zhì)量。在強(qiáng)夯技術(shù)應(yīng)用的過程中，需要注意強(qiáng)夯點(diǎn)的選擇，強(qiáng)夯的應(yīng)用次數(shù)以及強(qiáng)夯整體效果。

Similarly,by the same procedure as above,we find that in the left superconductor lead regionG(↓)(z,z′)can be expressed in the following explicit forms(forz′

After all the relevant Green’s function matrices are obtained,the Josephson current in the superconductor leads can be calculated from them.[40,41]Firstly, we derive the charge current density operator from the charge density operator,which reads

where the energy argument is analytically continued to the Matsbura frequenciesωn= i(2n+1)π/β.On the basis of this formula, we can calculate the the Josephson current and the current-phase relation under various different situations by means of numerical methods.

3.Results and discussion

In the following we present the numerical results obtained based on the formalism outlined above.For convenience, in the results presented below, the magnitudes of all relevant quantities are given in dimensionless forms as following: the strength of Rashba SOC is measured by a dimensionless quantityα ≡2mαR/ˉh2kF,the strength of Zeeman splitting and superconducting paring potential are measured in units of the chemical potentialμ,the magnitude of the Josephson current is measured in units ofe?0/ˉh, the widthLof the bridge region between two superconductor leads is measured in units of the superconducting correlation lengthξ=ˉh2kF/m?0,and the temperatureTis measured in units of the critical temperatureTc.We setL=0.1ξandT=0.1Tcin the numerical calculations.

A fundamental property of a Josephson junction is its current–phase relation (CPR).In general, the Josephson current can be decomposed into a Fourier series of contributions of different orders as following:I(φ) =∑n≥1[Insin(nφ)+Jncos(nφ)], where the current components with indexncorrespond to the contributions of then-th reflection processes of quasi-particles.[42]If time reversal symmetry or inversion symmetry is conserved in the system, the coefficientsJnvanish for allnsince in such casesI(φ)=?I(?φ),and consequently no Josephson current can occur at zero phase bias.Furthermore, if the higher-order current components with indexn ≥2 are negligible, the CPR then reduces to the ordinary sinusoidal form,I(φ)?Icsin(φ), withIcthe critical current.This is usually the case in the absence of spin–orbit coupling and Zeeman splitting.However,the presence of spin–orbit coupling and Zeeman splitting would invalidate this simple argument.For the system under consideration, both time reversal symmetry and inversion symmetry can be simultaneously broken by tuning some relevant parameters of the system,including the relative orientations and the strengths of the Zeeman field and the spin–orbit field in the bridge region,the relative orientations between theaaxes in two superconductor leads, or the relative orientations between the Zeeman field in the bridge region and theaaxes in the superconductor leads.In the following discussions we focus on how the CPR varies as these relevant parameters are tuned.We show that the CPR of the system may exhibit a series of novel features due to the interplay of these complicated factors and anomalous Josephson current may occur at zero phase bias under various different situations when both time reversal symmetry and inversion symmetry of the system are simultaneously broken.

Since the pairing potentials in d-wave superconductors are anisotropic, the CPR of the considered structure would depend on the relative orientations of theaaxes in two superconductor leads, which are characterized by the anglesaLandaR, respectively.We first consider the situations where the orientations of theαaxes in two superconductor leads are parallel to each other and also parallel to the direction of the current flow,corresponding to that the anglesaL=aR=0.In Figs.1(a)–1(c),we show several numerical results for the CPR obtained in such situations for various directions and strengths of the Zeeman field.For convenience of comparison,the values of all other parameters are set to be the same in all panels.Figure 1(a)shows that when the direction of the Zeeman field is along thexaxis,the magnitude of the Josephson current can be changed significantly as the strength of the Zeeman field is varied,but the basic features of the CPR curves remain the sinusoidal-like forms and posses 2πperiodicity in the phase biasφ,and though both time-reversal symmetry and inversion symmetry are broken in the cases shown there,no anomalous Josephson current occurs at zero phase bias.Just as usual,the first harmonic componentI1sin(φ)plays the predominant role in the Josephson current in these cases and the contributions of other harmonics are negligible.

Figures 1(b)and 1(c)show that when the direction of the Zeeman field is along theyaxis or along thezaxis, both the magnitude of the Josephson current and some basic features of the CPR (e.g., the periodicity inφor the curve shapes of the CPR) may be significantly changed as the strength of the Zeeman field is varied, and furthermore, a finite anomalous Josephson current may occur at zero phase bias under some circumstances.For instance,in the cases ofhy/μ=0.75 and 1.0 in Fig.1(b), the CPR possess a 2π-periodic dependence on the phase biasφand no anomalous Josephson current occurs atφ=0.In these two cases, the first harmonic componentI1sin(φ) plays the predominant role in the Josephson current and the first harmonic componentJ1cos(φ) is negligible as usual.In contrast, in the case ofhy/μ=0.25 in Fig.1(b), though the CPR still possess 2πperiodicity inφ, a finite anomalous Josephson current occurs at zero phase bias.In this case,the first harmonic componentI1sin(φ)still plays the predominant role in the Josephson current,but the contribution of the first harmonic componentJ1cos(φ) is not negligible,while the higher-order current components with indexn ≥2 are still negligible.In the case ofhy/μ=0.5 in Fig.1(b),the periodicity of the CPR inφis no longer 2πbut becomes to beπ, and no anomalous Josephson current occurs at zero phase bias.From these features and the curve shape of the CPR in this case, one can see that the second harmonic componentI2sin(2φ)plays the predominant role in the Josephson current in this case and the contributions of the two first harmonic components are negligible.Similar significant changes can also be found in the cases shown in Fig.1(c).It is also noted that when the first harmonic components play the predominant role in the Josephson current, 0–πtransitions may also be induced as the strength or the orientation of the Zeeman field is varied.For example, in the cases shown in Fig.1(c),the CPR is typical of aπ-state whenhz/μ=0.75 and typical of a 0-state whenhz/μ=1.0.

Fig.1.The Josephson current as a function of the phase difference φ for various directions and strengths of the Zeeman field ?.In panels (a)–(c), the directions of the Zeeman field are set along the x axis,the y axis,and the z axis,respectively.The strengths of the Zeeman field are shown in the figures and the other parameters are set as T =0.1Tc,L=0.1ξ,?0=10?3μ,α =0.5,aL=0,aR=0.

Fig.2.The Josephson current as a function of the phase difference φ for various directions and strengths of the Zeeman field ?.In panels (a)–(c), the directions of the Zeeman field are set along the x axis,the y axis,and the z axis,respectively.The strengths of the Zeeman field are shown in the figures and the other parameters are set as T =0.1Tc,L=0.1ξ,?0=10?3μ,α =0.5,aL=π/4,aR=π/4.

Next we consider the situations where the orientations of theαaxes in two d-wave superconductor leads are still parallel to each other but misalign with the direction of the current flow.We take the situations where the anglesaL=aR=π/4 as the examples.In Figs.2(a)–2(c),we show several numerical results for the CPR obtained in such situations for various directions and strengths of the Zeeman field.For convenience of comparison,the values of all other parameters are set to be the same as in Figs.1(a)–1(c).We can see that the basic features of the CPR shown in Figs.2(a)–2(c)are largely similar as compared with the corresponding results shown in Figs.1(a)–1(c).This suggests that if the orientations of theαaxes in two superconductor leads keep parallel to each other, the changes of the relative orientations of theαaxes in two superconductor leads with respect to the direction of the current flow has only some small influences on the basic features of the CPR, provided that the values of all other relevant parameters remain unchanged.

Now we consider the more complicated situations where the orientations of theaaxes in two d-wave superconductor leads are not parallel to each other.We take the situations where the anglesaL=0 andaR=π/4 as the examples.In such situations, the direction of the current flow is parallel to theaaxis in the left superconductor lead but forms an angle ofπ/4 with theaaxis in the right superconductor lead.In Figs.3(a)–3(c)we show several numerical results for the CPR obtained in such situations for various directions and strengths of the Zeeman field.For convenience of comparison,the values of all other parameters are still set to be the same as in Figs.1(a)–1(c)and 2(a)–2(c).

By comparing the corresponding results shown in Figs.1–3, one can see that the CPR exhibits more rich and more complicated features in the situations illustrated in Fig.3, and more significant changes may be induced both in the magnitude of the Josephson current and in some basic features of the CPR as the strength of the Zeeman field is varied.For instance, in the cases shown in Fig.3(a), the periodicity of the CPR inφis 2πand the Josephson current vanishes atφ=π/2 andφ=3π/2,but a finite anomalous Josephson current occurs atφ=0.From these features and the shapes of the CPR curves in these cases,we can deduce that the Josephson current in these cases can be expressed approximately as a combination of the first harmonic componentJ1cos(φ)and the second harmonic componentI2sin(2φ).Which component plays the predominant role in the Josephson current depends on the strength of the Zeeman splitting.For example,in the case ofhx/μ=0.25, the second harmonic componentI2sin(2φ) plays the predominant role, while in the case ofhx/μ=1, the first harmonic componentJ1cos(φ) plays the predominant role.

In the cases shown in Fig.3(b), whenhy/μ=0.25, the periodicity of the CPR inφisπand a small anomalous Josephson current occurs at zero phase bias.The Josephson current is also nonvanishing atφ=π/2.From these features and the shape of the CPR curve in this case, we can deduce that the Josephson current in this case can be expressed approximately as a combination of the two second harmonic componentsI2sin(2φ)andJ2cos(2φ),with theI2sin(2φ)component playing the predominant role.In contrast to the case ofhy/μ=0.25,whenhy/μ=0.5,0.75,and 1.0,the periodicity of the CPR inφis stillπ, but the Josephson current vanishes both atφ=π/2 and atφ=0 (i.e., no anomalous Josephson current occurs at zero phase bias).From these features and the shapes of the CPR curves in these cases, we can deduce that the Josephson currents in these cases are predominated by theI2sin(2φ)component and the contributions of other harmonics are negligible.

In the cases shown in Fig.3(c), the periodicity of the CPR inφis 2πand a finite anomalous Josephson current occurs at zero phase bias in all the cases shown there.From the features of the CPR curves in these cases, we can deduce that the Josephson current in these cases can be expressed approximately as a combination of the first harmonic componentJ1cos(φ) and the second harmonic componentI2sin(2φ).It is also interesting to note that, since the magnitude of the anomalous Josephson current is determined by the first harmonic componentJ1cos(φ) in these cases, the magnitude of the anomalous Josephson current at zero phase bias in these cases is comparable to the maximum value of the Josephson current atφ ／=0.

Fig.3.The Josephson current as a function of the phase difference φ for various directions and strengths of the Zeeman field ?.In panels (a)–(c), the directions of the Zeeman field are set along the x axis,the y axis,and the z axis,respectively.The strengths of the Zeeman field are shown in the figures and the other parameters are set as T =0.1Tc,L=0.1ξ,?0=10?3μ,α =0.5,aL=0,aR=π/4.

Fig.4.The anomalous Josephson current Ia at zero phase bias as a function of the Zeeman field strength hy for the Rashba SOC strength α =0 and α =0.5.The orientations of the a axes in two superconductor leads are set as aL=aR=0 in panel(a),aL=aR=π/4 in panel(b),aL=0 and aR=π/4 in panel(c).The other parameters are set as T =0.1Tc,L=0.1ξ,?0=10?3μ.

The numerical results shown above indicate that, due to the interplay of Rashba SOC and Zeeman splitting and d-wave pairing,anomalous Josephson current may occur at zero phase bias in the considered structure under various different situations,and both the magnitude and the direction of the anomalous Josephson current may depend sensitively not only on the relative orientations of the Zeeman field and the spin–orbit field in the bridge region but also on the relative orientations of theaaxes in two superconductor leads.For example, the results shown in Fig.1(b) and Fig.2(b) indicate that if theaaxes in two superconductor leads are parallel to each other,anomalous Josephson current may be induced by a Zeeman field orientating along theyaxis in the bridge region.The results show in Figs.3(a)–3(c)indicate that if theaaxes in two superconductor leads are misaligned, anomalous Josephson current may be induced by a Zeeman field orientating along an arbitrary direction in the bridge region.It also should be noted that, since both the first harmonics and the second harmonics may play important roles in the Josephson current in the considered structure and their relative weights may depend sensitively on a number of complicated factors, the anomalous Josephson effect in the considered structure can not be accounted for by simply introducing an anomalous phase shift in the CPR.

When the relative orientations of the Zeeman field and the spin–orbit field in the bridge region and theaaxes in two superconductor leads are fixed,both the magnitude and the direction of the anomalous Josephson current may also depend sensitively on the strengths of the Zeeman field and the spinorbit field in the bridge region.To illustrate this fact more clearly, in the following discussions we take the situations where the Zeeman field is orientated along theyaxis as the examples.In Figs.4(a)–4(c)we show the anomalous Josephson currentIaat zero phase bias as a function of the strengthhyof the Zeeman field for the Rashba SOC strengthα=0 andα=0.5,with the orientations of theaaxis in two superconductor leads set to be:(a)aL=aR=0,(b)aL=aR=π/4,and(c)aL=0,aR=π/4,respectively.Figures 4(a)–4(c)show that the anomalous Josephson current is suppressed obviously forα=0,independent of the strengthhyof the Zeeman field.In contrast, forα=0.5, a finite anomalous Josephson current may occur whenhy／=0.As the value ofhyis varied,the anomalous Josephson current reverses its direction of flow frequently at approximately regular intervals and the magnitude of the anomalous Josephson current changes significantly in an oscillating manner, and the amplitude of the oscillations decreases gradually with the increase ofhy.

In Figs.5(a)–5(c) we show the anomalous Josephson currentIaat zero phase bias as a function of the Rashba SOC strengthαfor the Zeeman field strengthhy/μ=0 andhy/μ=0.25,with the orientations of thea-axis in two superconductor leads set along the same directions as in Figs.4(a)–4(c),respectively.Figures 5(a)–5(c)show that,forhy/μ=0,the anomalous Josephson current is suppressed obviously,independent of the Rashba SOC strengthα.In contrast, forhy/μ=0.25, a finite anomalous Josephson current may occur whenαis sufficiently large.The anomalous Josephson current also changes in an oscillating manner as the value ofαis varied, but unlike the results shown in Figs.4(a)–4(c),both the amplitude and the period of the oscillations increase gradually with the increase ofα.

Fig.5.The anomalous Josephson current Ia at zero phase bias as a function of the Rashba SOC strength α for the Zeeman field strength hy/μ =0 and hy/μ=0.25.The orientations of the a axes in two superconductor leads are set as aL=aR=0 in panel(a),aL=aR=π/4 in panel(b),aL=0 and aR=π/4 in panel(c).The other parameters are set as T =0.1Tc,L=0.1ξ,?0=10?3μ.

4.Conclusions

In summary, based on the Bogoliubov–de Gennes equation and the extended McMillan’s Green’s function formalism, we have presented a systematic theoretical investigation on the Josephson effect between two d-wave superconductors bridged by a ballistic 2DEG with both Rashba SOC and Zeeman splitting.The Josephson current and the CPR are calculated under various different situations.We show that due to the interplay of Rashba SOC and Zeeman splitting and d-wave pairing,the CPR in such a heterostructure may exhibit a series of novel features and can change significantly as some relevant parameters are tuned.In particular,anomalous Josephson current may occur at zero phase bias under various different situations when both time reversal symmetry and inversion symmetry of the system are simultaneously broken,which can be realized by tuning the relative orientations and the strengths of the Zeeman field and the spin–orbit field in the bridge region,the relative orientations of theaaxes in two superconductor leads,or the relative orientations between the Zeeman field in the bridge region and theaaxes in the superconductor leads.It is also shown that both the first and the second harmonics may play important roles in the Josephson current and their relative weights also depend sensitively on these complicated factors,and due to this fact,the anomalous Josephson current in the considered structure can not be accounted for by simply introducing an anomalous phase shift in the CPR.