Josephson Effect in Singlet Superconductor-Ferromagnet-Triplet Superconductor Junction
aa r X i v : . [ c ond - m a t . s up r- c on ] O c t Josephson Effect in Singlet Superconductor-Ferromagnet-TripletSuperconductor Junction
Chi-Hoon Choi
Department of Nanophysics, Gachon University, Seongnam 13120, Korea
Abstract
We study the current-phase relation of a ballistic SIFIT junction, consisting of a spin-singletsuperconductor (S), a weak ferromagnetic metal (F), a spin-triplet superconductor (T), and insu-lating ferromagnetic interfaces (I). We use the generalized quasiclassical formalism developed byA. Millis et al. to compute the current density and the free energy of the junction for arbitraryorientation of the magnetizations of the junction barrier. We investigate in detail the effect of thedistribution of magnetization on the various harmonics of the current-phase relation and the tran-sition of the ground state of the junction. The φ -state junction can be realized for a noncollinearorientation of the barrier magnetizations in the plane perpendicular to the d-vector of the tripletsuperconductor. PACS numbers: 74.50.+r, 74.20.Rp, 74.45.+cKeywords: Josephson effect, Superconductor-ferromagnet junction, Spin-triplet superconductor . INTRODUCTION The superconductor-ferromagnet junctions have been studied extensively both exper-imentally and theoretically.[1–3] A ferromagnetic tunneling barrier can have a profoundeffect on the Josephson current-phase relation (CPR).[4] The spin-singlet superconductorjunctions with a nonmagnetic interface have the usual form of CPR, I S ( φ ) = I c sin φ , where I c is the critical current and φ is the phase difference of the superconductors. When the tun-neling barrier has a ferromagnetic layer, the spin-triplet pairing amplitude can be inducedin the ferromagnet and also the supercurrent can reverse its sign, making the 0 − π transi-tion. Particularly, when the barrier has a multilayered ferromagnetic heterostructure witha nonuniform distribution of magnetization, several important features emerge.[5–7] TheJosephson current can be dominated by the second harmonic sin 2 φ via the coherent trans-port of two Cooper pairs, which makes it possible to realize the so-called φ -junction.[5] Ananomalous supercurrent can also flow even for vanishing phase difference; I S ( φ = 0) = 0.[7]These features of the dominant second harmonic, the φ -junction, the anomalous Joseph-son effect (AJE) can play an important role in the development of the quantum electronicdevices.[3]Recently, the Josephson junctions with spin-triplet superconductors have attracted muchattention as the triplet superconductivity has been found in several materials such asSr RuO and the heavy fermion superconductors.[8–10] For a singlet superconductor-tripletsuperconductor junction, transport of a single Cooper pair is prohibited by symmetry andthe supercurrent can tunnel through the barrier by the coherent transport of even numbersof Cooper pairs, leading to even harmonics sin(2 nφ ) in the CPR. When the tunneling barrierof the junction has a magnetization, the cos φ -harmonic can appear because the transport ofa single Cooper pair is possible due to a spin-flip scattering of the magnetic moment.[11–14]In this paper, we study the current-phase relation of an SIFIT junction with a multilay-ered ferromagnetic heterostructure. In the previous works on the singlet-ferromagnet-tripletjunction, the tunneling barrier has been treated as a uniform ferromagnetic layer.[12–14]The schematic diagram of the junction is depicted in the insert of Fig. 1. The interface (I)is modeled by a delta-function like potential which can incorporate nonmagnetic as well asmagnetic scatterings of quasiparticles. The normal metal (N) of the middle layer can be re-placed by a ferromagnetic metal. The singlet superconductor has an s-wave order parameter2ith the isotropic gap ∆ . For the triplet superconductor, we choose the following p-waveorder parameter considered as a possible candidate for Sr RuO : d ( k ) = ∆ ˆ z ( k x + ik y ) e iφ , (1)where φ is the relative phase difference between the two superconductors.[8]We compute the current density as a function of the phase difference in the ballistic limitwhile changing the magnetization of the interfaces and the exchange field of the ferromag-netic layer. We utilize the general formalism of A. Millis et al. to take into account theinterference effect due to scattering of quasiparticles from the neighboring interfaces.[15] Wefocus on the question of how an inhomogeneous distribution of the magnetizations in thetunneling barrier affects the key features of the ferromagnetic Josephson junction such asthe sign reversal of supercurrent, the φ -junction, and the AJE.Before going into details, we summarize our main results. (i) For a wide range of thejunction parameters, the current is well approximated by a combination of the harmonicsof sin φ , cos φ , and sin 2 φ . Their relative magnitude is determined by the distribution ofmagnetizations in the tunneling barrier. (ii) When the magnetizations are aligned with the d-vector of the triplet superconductor, the cos φ -term in the CPR appears, leading to the AJE.(iii) When the magnetizations have a noncollinear distribution in the plane perpendicular tothe d-vector, the sin φ -term can appear in addition to the second harmonic sin 2 φ , leadingto the φ -junction. (iv) The ground state phase of the φ -junction oscillates periodically asthe strength of the exchange field changes due to the interference effect in the clean limit.(v) We also compute the pairing amplitude induced by the barrier magnetizations, whichenables us to understand the key features of the CPR. II. FORMALISM
To do our calculation, we follow closely the formalism and the notations of Refs. 15 and16. We consider the SIFIT junction with the specular interfaces located at the positions of x = 0 and x = d , as in Fig. 1. The Keldysh Green’s function ˆ G ( x, x ′ ) is decomposed intothe four possible combinations of incoming and outgoing waves:ˆ τ ˆ G S ( x, x ′ ) = 1 v S X α,β = ± ˆ C Sαβ ( x, x ′ ) e ip S ( αx − βx ′ ) , (2)3 Z S TNI I0 0.5 1 1.5-1012J/J φ/π FIG. 1: Plots of the current density as a function of the phase difference for the SINIT junction.The interface potential, defined by Eq. (5), is chosen as ¯ U I = (0 . , x, y ), and (2ˆ z,
2) for thesolid, dotted, and dashed curves, respectively. The thickness of the normal layer is ¯ d = 0 .
5. In theinsert, the SINIT junction with a normal metal layer surrounded by two half-infinite singlet andtriplet superconductors is drawn. where the superscript S indicates each layer of the junction, and the subscripts α and β arethe indices representing the direction of momentum along the x-axis. The magnitude of theFermi momentum along the x-axis is denoted by p S and its corresponding Fermi velocity by v S = p S /m .The interface is modeled by a delta-function like potential: ˆ v = v ˆ1 + v m · σ . Thecharge and the magnetic scattering potentials of the interface are denoted by v and v m ,respectively. The magnetic potential is assumed to be proportional to the magnetization m of the ferromagnetic interface. The ferromagnetic metal is modeled by a weak exchangefield and its potential in the particle space can be written as ˆ V = h · σ .[17]The differential equations and the boundary conditions for the amplitude function ˆ C αβ are given by Eqs. (22-23) and Eq. (43) of Ref. 15, respectively. The Green’s function can beobtained by solving the differential equations for ˆ C αβ with the proper boundary conditionsat the interfaces at x = 0 and x = d .[16, 18]Once the Green’s function is computed, one can calculate the various physical quantitiessuch as the current density, the pairing amplitude, and the density of states. For ourtranslationally-invariant planar interfaces, the current flows along the x-axis. The current4ensity from the particles incident with the momentum p can be calculated by J (ˆ p ) = π N f v f T X n ≥ (ˆ x · ˆ p )Tr[ ˆ τ ( ˆ C ++ − ˆ C −− )] , (3)where N f is the density of states at the Fermi energy. The total current density can becomputed by integrating the current density J (ˆ p ) over the Fermi surface with ˆ p · ˆ x > E ( φ ) = Φ π Z φ J ( χ ) dχ, (4)where Φ is the flux quantum.[4] III. RESULTS AND DISCUSSION
We now present our numerical calculations. We compute the current density of theSIFIT junction while varying the junction parameters such as the interface magnetizations,the exchange field, and the interlayer thickness. We assume that the Fermi velocity v f isthe same everywhere and the singlet and the triplet superconductors have the same uniformgap ∆ in each layer. To simplify our notations, a variable U I is introduced for the interfacepotential: U I = ( U L , U R ) = ( v L + v Lm , v R + v Rm ) , (5)where the superscripts R and L denote the right and the left interfaces. The energy and thelength are scaled in units of the superconducting gap ∆ and the superconducting coherencelength ξ = ~ v f / ∆ . The following dimensionless quantities are defined: the interlayerthickness ¯ d = d/ξ , the Fermi wave vector ¯ k f = k f ξ , the interface potential ¯ U I = U I / ( ~ v f ),and the exchange field ¯ h = h / ∆ . In our calculation, we set ¯ k f = 1000 and the temperature T = 0 . .We compute the current density for a normal incidence by using Eq. (3). The currentdensity is normalized by J = N f v f T /
4. For a different angle of incidence θ k , it is straight-forward to generalize the calculations by replacing the position variable x with x/ cos θ k .We now discuss briefly the effect of the magnetization in the tunneling barrier on thepairing amplitude to understand its effect on the CPR. It is summarized in Eq. (14) of5ef. 16 how the interface scattering potential can induce the various components of thepairing amplitude from its adjacent superconductor. The magnetic potential v m can inducethe singlet pairing amplitude through the interaction term ‘ i v m · f ’ and the triplet pairingamplitude through the terms of ‘ i v m f ’ and ‘ iv v m × f ’, where f and f are the singletand triplet pairing amplitudes of the superconductor, respectively. The induced pairingamplitudes acquire a 90 phase shift.In a similar way, one can derive the expression for the induced pairing amplitude by theexchange field. We need to solve the differential equation for ˆ C ++ in the ferromagnetic layerof the superconductor-ferromagnet junction while assuming that particles are incident fromthe superconductor. Up to the first order in h = | h | , the transmitted pairing amplitude inthe ferromagnetic layer can be written as f tr = e − ǫ n x/ ~ v f [cos( qx ) f − i sin( qx )¯ h · f ] , f tr = e − ǫ n x/ ~ v f [ f − i sin( qx )¯ h f ] , (6)where q = 2 h/ ~ v f and x is the distance from the superconductor. The exchange field h canthus induce the pairing amplitudes in a similar way to the interface magnetization, and itcan be regarded as the interface potential with v =0 and v m ∝ e − ǫ n x/ ~ v f sin( qx ) h .We investigated in detail the CPR of the SIFIT junction by calculating the current densityas a function of the phase difference while changing the various junction parameters such asthe magnitude and orientation of the interface potential at each interface and the exchangefield as well as the interlayer thickness. Several observations are in order. (i) It is foundthat for a wide range of the junction parameters the current density can be approximatedquite well by the following Fourier series: J ( φ ) = C sin φ + C cos φ + C sin 2 φ. (7)(ii) The singlet-triplet junction has genetically the second harmonic sin 2 φ regardless of thetunneling barrier configurations. (iii) There appears the cos φ -term in the CPR, accom-panying the AJE, when the exchange field or the interface magnetization has a componentparallel to the d-vector. (iv) The sin φ -term appears, and thus the φ -junction can be realizedwhen the barrier magnetizations have a noncollinear distribution in the plane perpendicularto the d-vector. Or, it can be written as m L × m R · d = 0, where d is the d-vector ofthe triplet superconductor and the m L and m R are the magnetizations of the left and rightinterfaces. 6e note that a single layer of magnetization is enough for the AJE, but at least twononcollinear magnetizations are required for the φ -junction. This is in a sharp contrastto the singlet-ferromagnet-singlet junction in which case three non-coplanar magnetizationsare needed for both AJE and φ -junction such that m L × m R · h = 0. [7] Note also thatthe conditions for the AJE and φ -junction in the singlet-ferromagnet-triplet junction arequite different from those in the triplet-ferromagnet-triplet junction. [16] For the latterjunction with the same triplet order parameters at both sides, the AJE and the φ -junctionoccur at the same time as the d-vector of the triplet superconductor has both parallel andperpendicular components to a plane formed by the barrier magnetizations, requiring atleast two noncollinear magnetizations. However, when the d-vectors at both sides of thejunction are orthogonal to each other, the junction can have a generic sin 2 φ -term in theCPR, and the AJE and the φ -junction arise when one of the barrier magnetizations have aperpendicular component to the plane spanned by both d-vectors. The same cos φ -harmonicis responsible for both AJE and φ -junction in the triplet-ferromagnet-triplet junction, whiledifferent harmonics are needed in the singlet-ferromagnet-triplet junction, i.e., the sin φ -harmonic for the φ -junction and the cos φ -harmonic for the AJE.For the singlet-ferromagnet-triplet junction, there always exists the second harmonicsin 2 φ because the singlet and the triplet pairing amplitudes of the superconducting lay-ers, f and f z are orthogonal to each other. To have the cos φ -term, we need an interfacemagnetization or an exchange field parallel to the z-axis. The magnetization along the z-axis can induce the triplet pairing amplitude of f z from the singlet superconductor via theabove-mentioned term ‘ i v m f ’. The same magnetization can also induce the singlet pairingamplitude from the triplet superconductor via ‘ i v m · f ’. A Cooper pair tunneling between theinduced singlet pairing amplitude and the singlet superconductor or the tunneling betweenthe induced triplet pairing amplitude and the triplet superconductor, whose phases differ by90 , leads to the cos φ -harmonic. To have the sin φ -term, we need two separate magnetiza-tions having components along the x- and y-axes. For example, the pairing amplitude of f x can be induced both by the magnetization along the x-axis from the singlet superconductorvia ‘ i v m f ’ and by the magnetization along the y-axis from the triplet superconductor via‘ i v m × f ’. A Cooper pair tunneling between the induced pairing amplitudes from both sidesleads to the sin φ -harmonic.First, we present the results for the SINIT junction whose tunneling barrier is composed7 i f x f z Pairing Amplitude x/ ξ S TN0
FIG. 2: (Color online) Plots of the pairing amplitudes as a function of the position for the SINITjunction. The interface potential is chosen as ¯ U I = (ˆ x, y ). The red and the blue curves are thepairing amplitude f x induced by the left interface with ¯ U I = (ˆ x,
0) and by the right interface with¯ U I = (0 , y ), respectively. The induced f x ’s are purely imaginary while f and f z are real. Thepairing amplitude is normalized by that of the bulk superconductor and we set ¯ d = 1, φ = 0, and¯ ǫ n = 0 . π . of a normal metal layer and two interfaces. It is found that the dominant harmonics can besin φ , cos φ , or sin 2 φ , depending on the orientation of the interface magnetizations. Threerepresentative cases are plotted in Fig. 1. The Josephson current is dominated by the secondharmonic sin 2 φ when both interfaces are nonmagnetic, as shown in the solid curve. Whenthe interface magnetization is aligned with the z-axis, as in the dashed curve, the leadingharmonic can be the cos φ with the anomalous Josephson current J ( φ = 0) = 0. The currentis dominated by the sin φ -harmonic when the two interface magnetizations are aligned tothe x- and y-axes, as in the dotted curve.In Fig. 2, we plot the pairing amplitude induced by the interface magnetizations forthe SINIT junction to understand its effect on the CPR. We pay a particular attentionto the case where the sin φ -harmonic is generated. The interface parameter is chosen as¯ U I = (ˆ x, y ) so that the current is dominated by the sin φ : J/J = 2 .
03 sin φ − .
46 sin 2 φ .The pairing amplitude can be computed from ˆ C αβ :[17] f n = π C ++ ± ˆ C −− )(ˆ τ − i ˆ τ )( − i ˆ σ )ˆ σ n ] , (8)where the upper sign corresponds to the singlet component, n=0, and the lower sign to the8 φ/π φ/π J / J J / J (a)(c) (d) h = z yx(b)z U I =( , )x yU I =(0.5, 1.5) U I =(05 , 2)xU I =(0.5 , 2 ) x FIG. 3: Plots of the current density as a function of the phase difference for the SIFIT junctionwith several different sets of the interface potential ¯ U I . The exchange field h is chosen to be parallelto the x-(dotted curve), y-(dashed), and z-(solid) axes. The interlayer thickness is ¯ d = 0 . triplet component, n = x, y, and z. The pairing amplitudes f and f z have real values anddecay in the normal metal layer. As discussed in the above, the pairing amplitude f x can beinduced both by the magnetization along the x-axis at the left interface (the red curve) andby the magnetization along the y-axis at the right interface (the blue curve). The pairingamplitude f x induced by both sides (the solid black curve) is purely imaginary and has afairly large value in the normal metal layer, leading to the dominant sin φ -harmonic.We now extend our discussion to the SIFIT junction to study the effect of the ferro-magnetic layer on the Josephson current. In Fig. 3, we present our numerical calculationsshowing the characteristic features of the singlet-ferromagnet-triplet junction. The currentis computed as a function of the phase difference for several typical types of the interfacepotentials with the exchange field along the x-, y-, and z-axes. The AJE occurs due to thecos φ -term whenever the interface magnetization or the exchange field has a z-component. InFig. 3(a), where both interfaces are nonmagnetic, the current has only the second harmonicsin 2 φ as long as the exchange field has no z-component. In Fig. 3(b), where one of the9
202 0 1 φ/π φ/π -202 d=2(a) (b)(c) (d) h =5z0.5 zzU R =013 θ R =0 E / E E / E FIG. 4: Plots of the free energy as a function of the phase difference for the SIFIT junction withseveral different sets of the junction parameters. In (a), different values of the interlayer thickness¯ d are chosen for ¯ h = ˆ z and ¯ U I = (ˆ x, h are chosenfor ¯ d = 0 . U I = (ˆ x, U R and the orientation angle of the magnetization θ R at the right interface are chosen with thesame set of the parameters; ¯ d = 0 .
5, ¯ h = ˆ y , and ¯ U L = ˆ x . interface magnetizations is along the x-axis, or in a direction perpendicular to the d-vector ofthe triplet superconductor, the current is still dominated by the sin 2 φ for the exchange fieldparallel to the same x-axis. However, it undergoes a large change when the exchange field isrotated to the y-axis, due to the inclusion of the sin φ -term in the CPR. In Fig. 3(c), wherethe interface has a magnetization parallel to the d-vector of the triplet superconductor, thecurrent is dominated by the cos φ -term independent of the orientation of the exchange field.In Fig. 3(d), the φ -junction is realized independent of the exchange field because the inter-face magnetizations along the x- and y-axes can generate the sin φ -term. The inclusion ofthe exchange field along the z-axis leads to the cos φ -term, which can make the magnitudesof all the three harmonics of sin φ , cos φ , and sin 2 φ in the CPR become comparable.Next, we discuss in detail the transition of the ground state by computing the free energy10f the junction under the various situations. The CPR of the form of Eq. (7) can, in general,show the AJE and the changes of ground state such as the φ -junction and the π/ − π/ E = Φ / π . InFigs. 4(a) and 4(b), where either the exchange field or the interface magnetization has az-component, the cos φ -term appears and the current is determined by the C and C terms: J ( φ ) = C cos φ + C sin 2 φ . Because the coefficient C is negative in our parameter ranges,the minimum of the free energy occurs at φ = 3 π/ C > φ = π/ C < π/ π/ C changes. This kind of the transition happens as the interlayer thicknesschanges as in Fig. 4(a), or as the strength of the exchange field changes as in Fig 4(b).In Fig. 4(c), the exchange field is aligned with the y-axis and the magnetization of theleft interface with the x-axis, so the sin φ -term appears and the current can be approximatedby J ( φ ) = C sin φ + C sin 2 φ . Because C is negative, the free energy has double minimaat the phases of φ = ± cos − ( − C C ) . (9)The ground state can make a continuous transition to the φ -state as the interface potentialchanges, as shown in Fig 4(c). For example, φ = ± . π for ¯ U R = 0, φ = ± . π for¯ U R = 1, and φ = π for ¯ U R = 3.In Fig. 4(d), the free energy is computed for different orientations of the magnetizationat the right interface. The orientation angle θ R is defined by an angle between the z-axisand the magnetization in the x-z plane. When the magnetization is along the z-axis suchthat ¯ U R = ˆ z ( θ R = 0), the cos φ -term in Eq. (7) is dominant. As the magnetization isinclined away from the z-axis, such as θ R = 0 . π , the cos φ -term weakens and the sin 2 φ -term becomes larger. This makes the magnitude of all the three coefficients in Eq. (7)become comparable. When the z-component of the magnetization is converted to a negativeone, such as from θ R = 0 . π to 0 . π , the sign of C is reversed.In Fig. 5, we study the φ -junction in more detail by calculating the free energy fordifferent values of the exchange field. The interface magnetization and the exchange fieldare aligned with the x- and y-axes, respectively, so that the current has the following form: J ( φ ) = C sin φ + C sin 2 φ . The ground state has two minima and its ground state phase φ h y φ/π E / E (b)(a) φ / π h =0yy0.5y1.5 FIG. 5: (a) Similar plots of the free energy of the SIFIT junction as in Fig. 4 for different valuesof the exchange field along the y-axis, ¯ h y . We choose ¯ d = 0 . U I = (ˆ x, φ is plotted as a function of ¯ h y . It oscillates with the period of 2 π . changes continuously according to Eq. (9) as the strength of the exchange field changes. InFig. 5(b), the ground state phase is plotted as a function of the strength of exchange field.The value of φ oscillates periodically due to a resonant scattering of quasiparticles in theferromagnetic layer between the surrounding interfaces. In the ballistic limit, quasiparticlesacquire a phase factor of e ihd/ ~ v f in the ferromagnetic layer during the process of a Cooperpair tunneling.[1] This leads to the oscillation of the period of 2 π in the Fig. 5(b).In conclusion, we study the current-phase relation of the SIFIT junction for the variousconfigurations of the magnetizations of the tunneling barrier. The AJE, the φ -junction, the π/ − π/ φ , cos φ , andsin 2 φ can be made readily in the SIFIT junction by adjusting the junction parameters. 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