Josephson Effect in FS/I/N/I/FS Tunnel Junctions

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1 Commun. Theor. Phys. Beijing, China pp c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 4, October 5, 29 Josephson Effect in FS/I/N/I/FS Tunnel Junctions LI Xiao-Wei Department of Physics, Huaiyin Teachers College, Huai an 223, China Jiangsu Key Laboratory for Chemistry of Low-Dimensional Materials, Huaiyin Teachers College, Huai an 2233, China Received November 24, 28; Revised April 8, 29 Abstract The Bogoliubov de Gennes equation is applied to the study of coherence effects in the ferromagnetic superconductor/insulator/normal metal/insulator/ferromagnetic/superconductor FS/I/N/I/FS junction. We calculated the Josephson current in FS/I/N/I/FS as a function of exchange field in ferromagnetic superconductor, temperature, and normal metal thickness. It is found that the Josephson critical current in FS/I/N/I/FS exhibits oscillations as a function of the length of normal metal. The exchange field always suppresses the Josephson critical current I p for a parallel configuration of the magnetic moments of two ferromagnetic superconductor FS electrodes. In the antiparallel configuration, the Josephson critical current I Ap at the minimum values of oscillation increases with the exchange field for strong barrier strength and at low temperatures. PACS numbers: r, 74.8.Dm Key words: Josephson current, ferromagnetic superconductor, current oscillation Since the discovery of the Josephson effect in superconductors/insulator/superconductor S/I/S junctions, it has been studied in various types of structures. The mechanism of the effect is the phase coherent transport for quasiparticles from one superconductor to another in case that the two superconductors are weakly coupled with each other. [,2] Early studies of weak links were mainly focused on the superconductor/insulator/superconductor type of junctions. [3] A new era in the research of Josephson effect started with the discovery of a peculiar scattering mechanism at the superconductor/normal metal S/N boundary known as the Andreev reflection. [4] In the superconductor/normal metal/superconductor S/N/S junctions, the Andreev reflection indicates that the dc Josephson effect arises due to the phase coherent transport in the normal region. The mechanism may be regarded as the transport of Cooper pairs from one superconducting electrode to another. [5 8] The Josephson critical current exhibits oscillations as a function of the length of the normal metal. In recent years the ferromagnet F has been introduced to the Josephson tunnel structure, giving rise to some new physical effects. In the superconductor/ferromagnet/superconductor S/F/S junctions, the tunneling electrons with spin-up and spin-down experience different potentials in F, and the Andreev reflection at F/S interfaces plays an important role in the properties of the S/F/S junction. [9 2] Coexistence of superconductivity and ferromagnetism has attracted much attention recently. [3 4] It was predicted by Fulde and Ferrel, [5] Larkin, and Ovchinnikov [6] FFLO that pairing can exist when electron momenta at Fermi energy are different for two spin directions. Experimentally, CeCoIn 5 was proposed as a candidate for the observation of the FFLO state. [7] Unlike the conventional Cooper pair, in which two electrons have opposite spins and momenta k, k, the FFLO pair in the presence of an exchange field has a finite center-of-mass momentum Q = 2h / hv F, where 2h is the exchange energy corresponding to the difference in energy between the spin-up and spin-down bands, and v F is the Fermi velocity. The FFLO state with [k + Q/2, k + Q/2 ] was never observed in bulk materials. It stems from the fact that in a bulk ferromagnet F, h is at least two orders of magnitude larger than the energy gap of a bulk superconductor S, while the FFLO state can appear only in the region where h <.754 for the s-wave S [5,6] or h <.6 for d-wave S. [8] However, in a thin F/S bi-layer, the effective exchange field h and effective superconducting order parameter may be of the same order of magnitude, and thus the coexistence between superconductivity and ferromagnetism can be realized. On the assumption that the thickness of superconducting layer is smaller than the superconducting coherent length, and the thickness of the ferromagnetic layer is smaller than the length of the condensate penetration into the F, a thin F/S bi-layer can be treated as a ferromagnetic superconductor FS film. The S/F/I/F/S structure may be simplified as an FS/I/FS juncture in which a strong barrier strength is assumed for the thin insulating layers. [9 2] It is found that the presence of an exchange field may increase the critical current in the FS/I/FS junction in the case of an antiparallel alignment of magnetization in the ferromagnets at low temperatures, for strong barrier. Supported by the Natural Science Foundation of Jiangsu Higher Education Institutions of China under Grant No. 6KJB49

2 722 LI Xiao-Wei Vol. 52 Fig. Schematic illustration of reflections and transmissions of quasiparticles in FS/I/N/I/FS structures, in which a and b, respectively, stand for spin-up electron and spin-down hole incident on the interface x = from the left FS; c and d, respectively, stand for spin-up electron and spin-down hole incident on the interface x = L from the right FS. Here horizontal arrows indicate quasiparticle transporting directions, solid lines represent electron in N or electronlike in FS, dash lines represent hole in N or holelike in FS. In this paper, we study the Josephson effect in ferromagnetic superconductor/insulator/normal metal/insulator/ferromagnetic superconductor FS/I/N/I/FS junctions. It is investigated that the Josephson current exhibits oscillations as a function of the length of the normal metal. Solving the Bogoliubov de Gennes BdG equation, [2] the superconducting order parameter T, h and the energy spectrum are obtained for the FS film. We will extend the approach of Blonder Tinkham Klapwijk BTK [22] to study the Josephson current in FS/I/N/I/FS. There are four types of quasiparticle injection process: electron and hole injection from the left FS to the N, and corresponding to hole and electron injection from the right FS to the N, as shown in Fig.. We calculate the dc Josephson current in FS/I/N/I/FS junction for parallel and antiparallel alignments of the magnetizations in two FS which differs from that in superconductor/insulator/normal metal/insulator/superconductor S/I/N/I/S. In the present coherent transport, the coexistence state between superconductivity and ferromagnetism means that there is an exchange splitting in the FS due to the ferromagnetic background. The presence of an exchange field usually reduces the Josephson currents in the FS/I/N/I/FS junction. The only exception that the dc Josephson currents increase with the ferromagnetic exchange interaction occurs if all four conditions are satisfied: at low temperature, with strong barrier strength, in an antiparallel configuration and at the minimum values of oscillation currents. The quasiparticle interference in the N and the resonant tunneling play an important role and exhibit a new quantum effect on the dc Josephson current in the FS/I/N/I/FS junctions. The Josephson current exhibits oscillations as a function of the length of the normal metal. We adopt BdG equation approach to study the superconducting order parameter T, h in an FS film. In the absence of spin-flip scattering, the four-component BdG equations are decoupled into two sets of two-component BdG equations for the spin-up electronlike and spin-down holelike quasiparticle wave function u, v, the other for u, v. [23] The BdG equation is given by H η σ h T, h uσ T, h H η σ h v σ = E uσ v σ, where H is the single-particle Hamiltonian and the quasiparticle energy E is measured relative to the Fermi energy E F. h = h Θx L is an effective exchange field in the FS and Θx L is the Heaviside step function. η σ = for σ = and η σ = for σ =, and σ stands for the spin opposite to σ. The effective superconducting order parameter T, h is independent of the coordinates, but depends on the effective exchange field and the temperature. From the BdG equations, we get u 2 σ = [ + 2 T, h /E + η σ h 2 ]/2, 2 v 2 σ = [ 2 T, h /E η σ h 2 ]/2. 3 The wave vectors of the electronlike and holelike quasiparticles in the FS are given respectively by kσ e = 2m/ h 2 [E F + E + η σ h 2 2 T, h ], 4

3 No. 4 Josephson Effect in FS/I/N/I/FS Tunnel Junctions 723 k h σ = 2m/ h 2 [E F E + η σ h 2 2 T, h ]. 5 The order parameter T, h of the FS film is determined by the self-consistent equation = U Ψ Ψ, where U is the effective attractive potential between electrons, Ψ = k γ k u k γ + k v k and Ψ = k γ k u k + γ + k v k with γ k and γ k are the Bogoliubov transformative operators. [9] With Eqs. 2 and 3 as well as the rules that γ k and γ k obey, [2] we obtain where = U 2 k f k ε 2 k + 2 T, h f k ε 2 k + 2 T, h, 6 f kσ = e [, 7 ε 2 kσ + 2 T,h η σh ]β + with ε kσ = h 2 kσ e2 /2m E F, and β = /k B T the inverse temperature. From Eqs. 6 and 7, the self-consistent equation for T, h is obtained as hωd dε ln = T, h ε 2 k + T, h e [ + ε 2 k + 2 T,h h ]β + e [, 8 ε 2 k + 2 T,h +h ]β + where =, is the BCS gap at zero temperature and in the absent of exchange field, ω D is the Debye frequency. By solving Eq. 8 self-consistently, the dependence of the effective order parameter T, h in the FS layer on the temperature and the effective exchange energy are obtained. In the FS/I/N/I/FS junction under consideration, two FS electrodes are assumed to be identical in and h, except for a phase difference, their magnetic moments being parallel or antiparallel to each other. The insulating layer is located at x = and x = L. The insulating barrier may be modeled by a δ-type potential V [δx + δx L] where V indicates the barrier strength. Consider an electronlike quasiparticle for spin σ incident on the insulating barrier at x = from the left FS. With general solutions of the Eq., the wave functions are given by Ψ σ x = e ike Lσ x ulσ e iφl/2 + a σ e ikhl σ x vl σ e iφl/2 + b σ e ikelσ x ulσ e iφl/2, x <, 9 v L σ e iφl/2 Ψ 3σ x = c σ e ike Rσ x urσ e iφl/2 Ψ 2σ x = e σ e iq+ x + f σ e iq x v R σ e iφl/2 u Lσ e iφl/2 + g σ e iq+ x + d σ e ikhr σ x vr σ e iφl/2 u Rσ e iφl/2 + h σ e iq x v L σ e iφl/2, < x < L,, x > L, where a σ, b σ, c σ, and d σ correspond respectively to coefficients for the Andreev reflection, normal reflection, transmission to the right FS as electronlike quasiparticles and transmission as holelike quasiparticles. φ L φ R stands for the macroscopic phase of the left right FS. Subscript LR is the index for the left right FS. Amplitudes of electrons and holes propagating in the normal metal N layer are given by the coefficients e σ, f σ, g σ, and h σ. q ± = [k 2 F ±2mE/ h2 ] /2 are the wave vectors of the electron and hole in N. The wave functions must satisfy the boundary conditions [22] where Ψ σ x = = Ψ 2σ x =, 2 dψ2σ dx dψσ x= dx = 2mV x= h 2 Ψ 2σx =, 3 Ψ 2σ x = L = Ψ 3σ x = L, 4 dψ3σ dx dψ2σ x=l dx = 2mV x=l h 2 Ψ 3σx = L. 5 From Eqs. 9,,, and 2 through 5, we obtain a σ φ, E = 8u Rσ v R σ [cosφ iu 2 Lσ v2 L σ sin φ] u Lσv L σ [8 coslq + q + ir sin Lq + q ] 4 coslq + q 6u Lσ v L σ u Rσ v R σ cosφ i42 + z 2 u 2 Rσ v2 L σ sin Lq+ q + R R 2, 6 R = 4[z 2 coslq + + q + coslq + q ] + 4z 3 sinlq + + q + z 4 [coslq + q coslq + + q ], R 2 = u 2 Lσ v 2 L σu 2 Rσ v 2 R σ, R = 8 4z 2 u 2 Rσ v 2 R σ, φ = φ R φ L, z = 2mV/ h 2 k F.

4 724 LI Xiao-Wei Vol. 52 Since h is much smaller than E F, we made the approximation of kσ e = kh σ = k with the Fermi wave vectors of the F FS and N. Having obtained a σ φ, E, the Josephson current can be calculated using the generalized coefficient of Andreev reflection a σ φ, iω n, which is obtained by analytic continuation of E to iω n, yielding [24 25] I = ek BT T, h Re [ a φ, iω n a φ, iω n + a φ, iω n a φ, iω n ], 7 2 h Ω ω nl Ω nl n with Ω nl = ω n + iη σ h T, h, and the Matsubara frequency ω n = 2πk B Tn + /2. Fig. 2 The current as a function of the normal metal thickness in the case of a parallel orientation. Here φ = π/2, h / =. solid line, h / =.4 dash line, h / =.6 dot line. In what follows we discuss numerical results from Eqs. 6 and 7. Let us first study the Josephson current in FS/I/N/I/FS as functions of the normal metal thickness L for the different barrier strength z, the different temperature T, and the different exchange field h. Figures 2 and 3 show the Josephson currents of junction are different from each other in the parallel and antiparallel configurations of the magnetic moments of two ferromagnetic superconductor FS electrodes. There is oscillation relation between the Josephson current I p and I Ap and the normal metal thickness L. The amplitude of the oscillation decreases as L increases. For the parallel configuration, the exchange field always suppresses the current I p. For the antiparallel configuration, with increasing the exchange field the critical current I Ap increases at minimum values of oscillation currents for strong barrier strength and at low temperatures, but decreases at maximum values of oscillation currents. With increasing the exchange field the critical current decreases for weak barrier strength or at higher temperature. The interesting effect that I Ap at minimum values of oscillation currents increases with the exchange field in FS/I/N/I/FS is very similar to the case in FS/I/FS, [9,2] but I Ap at maximum values of oscillation currents is differ from the case in FS/I/FS. Also, this is different from in FS/N/S. [26] Fig. 3 The same dependence as in Fig. 2 in the case of an antiparallel orientation. Fig. 4 The current as a function of the normal metal thickness. Here φ = π/2, h / =.3, a and c:, k BT/ =.2, z = solid line, z =.2 dash line, z =.3 dot line; b and d: z =.5, k BT/ =. solid line, k BT/ =.5 dash line, k BT/ =.2 dot line. Figure 4 shows the numerical results for the Josephson critical currents I p and I Ap as the function of the normal metal thickness L for the different barrier strength z and the different temperature T. Some features can be found from the figures. First, for z =, the Josephson critical current I p and I Ap do not exhibit the oscillation behavior. This value of I p and I Ap is reduced with increasing thickness L of the normal metal. The maximum values of oscillation currents for z equal to the value of the

5 No. 4 Josephson Effect in FS/I/N/I/FS Tunnel Junctions 725 current at the same thickness L for z =. Second, the increasing of temperature always suppresses the Andreev reflection in FS/I/N/I/FS junction. Fig. 5 The current as a function of the temperature, here φ = π/2, z = 2, h = solid line, h / =.3 dash line, h / =.6 dot line. In Fig. 5 the temperature dependences of the Josephson critical current I p and I Ap are plotted for different h and L. With increasing the temperature I p and I Ap decrease and drop to zero at T c h, which is lower than at T c h =. Such a sudden drop in I p and I Ap with the presence of h is due to the drop for T, h from a finite value to zero at T c h Ref. [8]. In summary we have studied the Josephson effect in the FS/I/N/I/FS junction. The superconducting order parameter and the energy spectrum of the Bogoliubov excitations are derived from the BdG equation for the FS. The Josephson current in the FS/I/N/I/FS junction has been obtained as the function of the normal metal thickness, the exchange field, and the temperature. It is found that the exchange field always suppresses the critical current I p for the parallel configuration of the magnetic moments of two ferromagnetic superconductor FS electrodes. For the antiparallel configuration, with increasing the exchange field the critical current I Ap at minimum values of oscillation decreases for weak barrier strength or at higher temperature, but increases for strong barrier strength and at low temperatures. The Josephson currents do not change from state to π state since h is much smaller than E F. The oscillation relation of the tunneling current corresponding to the thickness of the normal layer is due to the quantum interference effect of quasiparticles in the N interlayer. The amplitude of the oscillation of the Josephson current is reduced with increasing normal metal thickness L. It is expected that the theoretical result obtained will be confirmed in the future experiments. References [] B.D. Josephson, Adv. Phys [2] B.D. Josephson, Rev. Mod. Phys [3] K.K. Likharev, Rev. Mod. Phys [4] A.F. Andreev, Zh. Eksp. Tero. Fiz [5] A. Furusaki and M. Tsukada, Phys. Rev. B [6] M. Hurd and G. Wendin, Phys. Rev. B [7] P.F. Bagwell, Phys. Rev. B [8] Y. Tanaka and S. Kashiwaya, Phys. Rev. B [9] A.I. Buzdin, Rev. Mod. Phys [] P.F. Bergeret, A.F. Volkov, and K.B. Efetov, Rev. Mod. Phys [] G. Su and M. Suzuki, Mod. Phys. Lett. B [2] F.S. Nogueira and K.H. Bennemann, Europhys. Lett [3] K. Yang and D.F. Agterberg, Phys. Rev. Lett [4] V.V. Ryazanov, V.A. Oboznov, Yu Rusanov, A.V. Veretennikov, A.A. Golubov, and J.V. Aarts, Phys. Rev. Lett [5] P. Fulde and A. Ferrel, Phys. Rev A55. [6] A. Larkin and Y. Ovchinnikov, Sov. Phys. JETP [7] A. Bianchi, R. Movshovich, C. Capan, P.G. Pagliuso, and J.L. Sarrao, Phys. Rev. Lett [8] K. Yang and S.L. Sondhi, Phys. Rev. B [9] F.S. Bergeret, A.F. Volkov, and K.B. Efetov, Phys. Rev. Lett [2] X.W. Li, Z.M. Zheng, D.Y. Xing, G.Y. Sun, and Z.C. Dong, Phys. Rev. B [2] P.G. de Gennes, Superconductivity of Metals and Alloys, Beniamin, New York 966. [22] E.G. Blonder, M. Tinkham, and T.M. Klapwijk, Phys. Rev. B [23] M.J.M. de Jong and C.W.J. Beenakker, Phys. Rev. Lett [24] A. Furusaki and M. Tsukads, Solid State Commun [25] Y. Tanaka and S. Kashiwaya, Physica C [26] X.W. Li, Chin. Phys