πstates in all-pnictide Josephson junctions
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n π -States in all-pnictide Josephson Junctions C. Nappi, ∗ S. De Nicola,
2, 3
M. Adamo, and E. Sarnelli † CNR Istituto di Cibernetica ”E. Caianiello” Via Campi Flegrei 42 I-80078, Pozzuoli, Napoli, Italy CNR Istituto Nazionale di Ottica I-80078, Via Campi Flegrei 42,Pozzuoli, Napoli, Italy INFN Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Via Cinthia, I-80126 Napoli, Italy (Dated: January 2, 2018)We study the Josephson effect in s ± /I/s ± junctions made by two bands reversed sign s-wave( s ± ) superconductive materials. We derive an equation providing the bound Andreev energy statesparameterized by the band ratio α , a parameter accounting for the weight of the second bandwith respect to the first one at the interface. For selected values of the band ratio and tunnelbarrier amplitude, we predict various features of the Josephson current, among which a possiblehigh temperature π state of the junction (a doubly degenerate junction ground state) and a π → PACS numbers: 74.20.Rp, 74.50.+r, 74.70.Dd
The importance of the Josephson effect as a tool toprobe the properties of new discovered superconductivematerials can hardly be underestimated. This has beentrue in the past in investigating the d-wave cuprate ma-terials and is nowadays the case for the multi-gap ironbased superconductors [1] whose complex behavior, be-yond the BCS theory, is presently a focus in condensedmatter theory [2, 3] and material science [4].There is a rather solid indication now, supported byexperiments, that the pair potential symmetry in thesecompounds is s-wave with sign-reversing order parameter( s ± ). Several theoretical models have already discussedthe experimental consequences of an extended s-wave( s ± - wave) order parameter symmetry on the Andreevconductance of an N S interface and on the Josephsoneffect in the iron-based superconductor junctions [5–19].In particular the Josephson effect has been studied in hy-brid devices i.e. sIs ± junctions, as summarized by Seidelin his review [20] and reference therein.In this Letter we study an all-pnictide symmetric s ± Is ± Josephson junction and we discuss a number non-trivial physical consequences on the Josephson effect duedue to the presence of a second conduction band. Weshow that, depending on the band ratio parameter α ,which accounts for the weight the second band, a π -state[21] can develop for a wide range of junction transparen-cies and temperatures. This π -state can persist in the fullrange of temperatures or can undergo a π → → π ) crossover as the temperature decreases, depend-ing on the band ratio parameter. This crossover is analo-gous to the 0 → π crossover [22, 23], found in mesoscopic d-wave superconductor Josephson junction [24].We adopt the simplest model of a Josephson junc-tion that shows the essential features of the Josephsoneffect in the presence of two gaps, namely we considera superconductor(S)-insulator(I)-superconductor(S) con-tact. The iron based junction is modeled by considering aone-dimensional conductor, whose left ( x <
0) and right( x >
0) halves are both two band metals (two differ-ent states at the Fermi level, one with the wave vector p and the other with q ). We assume that the motion ofquasiparticles is described by the Bogoliubov de Gennes(BdG) equation [25] and that the order parameter has al-ready been obtained self-consistently from the gap equa-tion. Therefore we choose a one dimensional model forthe gap, such that the left and right two band supercon-ductors have pair potentials given by∆ j ( x ) = ∆ j e iϕ j θ ( − x ) + ∆ j e i ( ϕ j + ϕ ) θ ( x ) , j = 1 , j = 0, has an infinitesi-mal width and we also introduce a scattering potential U ( x ) = U δ ( x ). The possibility of nodes in the gap func-tion is not considered. In the case of the two-gap modelwith unequal s-wave gaps, we write a wave function of thesame type introduced in the Blonder Tinkham Klapwijkmodel [26] as solution of the BdG equations and treat thepresence of the second band through the introduction ofBloch wave functions [8]. The bound state (B) eigenfunc-tion with energy | E | < ∆ (we assume ∆ < ∆ ) can bewritten as ∗ Electronic address: [email protected] † Electronic address: [email protected] Ψ S L ( x ) = (cid:18) u B ( x ) v B ( x ) (cid:19) = a B (cid:20)(cid:18) u v e − iϕ (cid:19) φ − p e ( x ) + α (cid:18) u v e − iϕ (cid:19) φ − q e ( x ) (cid:21) + b B (cid:20)(cid:18) v u e − iϕ (cid:19) φ p h ( x ) + α (cid:18) v u e − iϕ (cid:19) φ q h ( x ) (cid:21) x < S R ( x ) = (cid:18) u B ( x ) v B ( x ) (cid:19) = c B (cid:20)(cid:18) u v e − i ( ϕ + ϕ ) (cid:19) φ p e ( x ) + α (cid:18) u v e − i ( ϕ + ϕ ) (cid:19) φ q e ( x ) (cid:21) + d B (cid:20)(cid:18) v u e − i ( ϕ + ϕ ) (cid:19) φ − p h ( x ) + α (cid:18) v u e − i ( ϕ + ϕ ) (cid:19) φ − q h ( x ) (cid:21) x > ϕ is a global phase difference between the twosuperconductive regions and ϕ , ϕ are the phases of thegaps ∆ , ∆ in both p and q bands respectively. In thecase of s ± gap model, ϕ − ϕ = π . In the considered en-ergy range the wave function has to decay exponentiallyfor | x | → ∞ . The coefficients a B , b B , c B , d B are the prob-ability amplitudes transmission with branch crossing orwithout branch crossing. Essential above is the intro-duction of α , a mixing coefficient defining the ratio ofprobability amplitudes for a quasiparticle to be transmit-ted to the first, ( p ), or second, ( q ), band [8]; the functions φ ’s are the Bloch waves in the two-band superconductor; p and q are the Fermi vectors for the two bands corre-sponding to the same energy E [8]: φ λ ( x ) = X G C G,λ exp[ i ( λ + G ) x ] (3)where λ = p e , q e , p h , q h , and G represent any reciprocallattice vector. u , v and u , v are the Bogoliubov co-efficients for the first and second band, respectively u = (cid:20) (cid:18) i Ω E (cid:19)(cid:21) / , v = (cid:20) (cid:18) − i Ω E (cid:19)(cid:21) / u = (cid:20) (cid:18) i Ω E (cid:19)(cid:21) / , v = (cid:20) (cid:18) − i Ω E (cid:19)(cid:21) / Ω = q E − ∆ , Ω = q E − ∆ (4)The global wave function Ψ must satisfy the followingboundary conditions at the interfaces x = 0Ψ S L (0 − ) = Ψ S R (0 + )Ψ ′ S R (0 + ) − Ψ ′ S L (0 − ) = 2 mU ℏ Ψ S R (0 + ) (5)where primes denote derivative with respect to x .Coupling between the bands is implicit in the boundary condition requirements. In fact, the assumed wave func-tion Ψ is of the form Ψ = Ψ + α Ψ where, separately,Ψ and Ψ solve BdG equations for excitations of energy E for the gap ∆ and ∆ , respectively,. The boundaryconditions, Eqs. (5), are requested to be a constrain forthe whole function Ψ.The above matching procedure, with the require-ment of non-triviality of the solution for the coefficients a B , b B , c B , d B provides for the s ± /I/s ± junction, thespectral equation (cid:16) − p − E p r − E α − r α (cid:17) (cid:0) Z + 1 (cid:1) +2 E (cid:2) Z (1 + α ) + (1 − α + α ) (cid:3) + h − α (cid:16) E + 2 p − E p r − E − r α (cid:17)i × cos( ϕ ) = 0 (6)where we have introduced the gap ratio r = ∆ / ∆ , theband ratio parameter α = α φ q (0) /φ p (0) and the bar-rier strength Z = U / ~ v. The energy is given in unitsof ∆ ( T ), and the two gaps will be assumed to obey thesame BCS-like temperature law. Boundary conditionson the wave function derivatives are usually discussed interms of Fermi velocities in the case of plane waves. ForBloch waves we have to introduce ”interface velocities” v λ = − i ~ m φ ′ λ (0) φ λ (0) (7)In deriving the spectral equation we have assumed, forthe sake of simplicity, equal band interface velocities v λ = v . An analysis of the spectral equation shows that for α = 0 there are, in general, four energy levels, E = ± ǫ ( ϕ, α ) and E = ± ǫ ( ϕ, α ) (see Fig. 1). In a single-band case ( α = 0) Eq 6 provides the well known resultfor a conventional SIS junction [27, 28], namely E = ± ǫ ( ϕ,
0) = ± ∆ ( T ) (cid:2) − D sin ( ϕ/ (cid:3) / , where D =1 / (1 + Z ) is the transmission probability through the δ -function barrier, i.e. the junction transparency.As α increases, the energy levels E = ± ǫ ( ϕ, α ) startto branch off the levels E = ± ǫ ( ϕ, α ), while these lattergradually approach the zero energy state. As an exam-ple, Fig. 1 (a) and (b) show the modifications of theAndreev levels for increasing values of the band ratio,for two values of the barrier parameter, Z = 0 . Z = 2 and for the band gap ratio r = 2, respectively.The condition for the existence of a zero energy level canbe easily obtained from the spectral equation. It is givenby (1 − rα ) (1 + 2 Z + cos( ϕ )) = 0. Accordingly, a zeroenergy state E = ± ǫ ( ϕ, α c ) = 0, is obtained for thecritical value of the band ratio α c = 1 / √ r , for any Z value. The energy level E = ± ǫ ( ϕ, α c ) correspondingto the critical value of the band ratio is given by [29] E = ± ǫ ( ϕ, α c ) = ± s Dr (2 − D + D cos ϕ ) sin ( ϕ/ r (2 − D + r ) + 2 Dr cos ϕ (8)and it is shown as a blue line in Fig. 1 for the indicatedvalues of D and r . For low transparencies ( D ≪ E = ± ǫ ( ϕ, α c ) = ± ∆ ( T )2 r/ (1 + r ) √ D sin ( ϕ/ α > α c , there are no surface bound states with realenergy eigenvalues. Z=2Z=0.7 (cid:72) (cid:16)(cid:72) (cid:72) (cid:16)(cid:72) FIG. 1: Andreev levels for a s ± superconductor S/I/S junc-tion with r = 2 and increasing values of α . (a): Z = 0 . α = 0 .
01, red line, α = 0 .
5, green line, α = 0 . Z = 2. Black line, α = 0 .
01, red line, α = 0 .
4, green line, α = 0 .
7. The black line curves correspond to a nearly perfect( α = 0) ”conventional” s-wave junction. The blue lines arethe Andreev level for α = α c = 0 . In the nearly insulating limit ( Z → ∞ ), the systemdecouples and we obtain information on the two sepa-rate electrodes. More precisely, in this limit, the spectralequation (6) describes surface bound states of energy E B = ± r − r α − α (9)already discussed in ref. [8] for a junction N/I/s ± . For 0 < α < α c , the two emerging levels E = ± ǫ ( ϕ, α ) and the levels E = ± ǫ ( ϕ, α ) have, in gen-eral, opposite dispersions, i.e. dE /dϕ > dE /dϕ < I d carried by the discrete An-dreev levels E k through the contact can be found fromthe free energy, according to the following relation [28, 31] I d ( ϕ, α ) = 2 e ~ X k =1 , ∂E k ∂ϕ f ( E k ) = − e ~ X k =1 , (cid:18) ∂ǫ k ∂ϕ tanh ǫ k k B T (cid:19) (10)where f ( E k ) is the Fermi distribution function and k labels the Andreev level with energy E k = ± ǫ k ( ϕ, α ). Z = 2 (cid:32)(cid:39)(cid:39) Z = 0.7 (cid:32)(cid:39)(cid:39) FIG. 2: current-phase relation for a s ± superconductor S/I/S junction with r = 2, T /T c = 0 .
01 and increasing values of α .(a) Z = 0 .
7, (black line, α = 0 .
01, red line, α = 0 .
5, green line, α = 0 . Z = 2. (black line, α = 0 .
01, red line, α = 0 . α = 0 . Fig. 2(a) and (b) show the current-phase relations I d ( ϕ, α ), at low temperature, corresponding to the dis-crete spectrum of Andreev levels represented in Fig. 1(a) and (b) respectively and calculated through Eq. (10).The current in this figure is normalized with respect to I = e ∆ (0) / ~ , which is the zero temperature maxi-mum Josephson current through a one dimensional s-wave junction in the clean limit. The three curves corre-spond to values of the band ratio approaching the crit-ical value α c = 0 . α = 0 .
7) in both Figs 2(a) and (b) shows a clear π phase-shift (the maximumJosephson current for 0 < ϕ < π has a negative value).For the considered α value this π state persists in thewhole temperature range (see Fig. 3 (a) and (b)). Themechanism of formation of this state is the following.As the band ratio increases an upper + ǫ ( ϕ, α ) and alower − ǫ ( ϕ, α ) extra Andreev bands gradually emerge.These two bands have different character compared tothe low energy bands ± ǫ ( ϕ, α ): they transport super-currents in the opposite directions. As the temperaturedecreases, only low energy level are populated while thoseat higher energy are empty. In the competition betweenthese opposite carrying current energy levels ± ǫ ( ϕ, α )and ± ǫ ( ϕ, α ), which coexist for any value of the bandratio, it is the temperature that determines the directionof the total current and the possible existence of a π → π junction. In thiscase [24], two kinds of bands, conventional and midgap,alternate, without coexisting and compete each other indetermining the supercurrent, depending on the angle ofincidence of the Andreev quasiclassical trajectories withthe interface. Therefore it is the two-dimensionality herethat plays the key role.In Fig. (3), (a) and (b), we report, for increasing val-ues of α , the dependencies of the normalized Josephsoncritical current I c /I ( I c = max ϕ I d ( ϕ )) from the reducedtemperature T /T c , derived by the discrete Andreev spec-tra for a nearly clean junction ( Z = 0 . Z = 3), respectively.The negative sign of I c (blue lines, Fig. 3 a) and b))indicates that the junction free energy, i.e. the quantity F ( ϕ ) = Φ / π R ϕ dϕI ( ϕ ), has a minimum at ϕ = π suchthat the ground state of the junction is π shifted. Asdiscussed above, the π → α = 0 .
65 in figure 3 (a) and for α = 0 . E and that low energy band E .So far, by using Eq. (10), we have included the contri-bution to the Josephson current from the discrete energyspectrum only. However the continuous-spectrum statesmake their own contribution to the current which has tobe accounted for properly. Following the approach de-veloped by Furusaki Tsukada [32] the total Josephsoncurrent, including contributions from both the Andreevbound states and the continous spectrum is given by I = e ∆ k B T ℏ X ω n [ a ( ϕ, iω n , α ) − a ( − ϕ, iω n , α )] (11)Here we have defined Ω = p ω n + ∆ and introducedthe Matsubara frequencies ω n = πk B T (2 n + 1), with n ranging from −∞ to + ∞ . a ( ϕ, iω n , α ) is a scattering am-plitude coefficient for the process in which an electron-like quasiparticle traveling from the left of the junctionis reflected back as a hole-like quasiparticle in the pres-ence of two bands. This coefficient is derived by solvingthe BdG equations under the assumption E > ∆ anddropping the requirement of exponentially decay of thesolutions for | x | → ∞ . The details of this procedure willbe given elsewhere [33]. The results of the calculatedtotal critical current I c as a function of the reduced tem-perature, are shown in Fig. 4, (a) and (b), for differentvalues of the band ratio α and already considered in Fig.3. Z=0.001Z=3 (cid:32)(cid:39)(cid:39) (cid:32)(cid:39)(cid:39) (cid:68) (cid:68) (a) reduced temperature T (cid:115) Tc c r iti ca l c u rr e n t I c (cid:115) I (b) FIG. 3: Critical current as a function of temperature for a s ± superconductor S/I/S junction with r = 2, T /T c = 0 . Z = 0 . α = 0 . α = 0 .
5, greenline, α = 0 .
65, blue line, α = 0 .
7. (b) Z = 3; black line, α = 0 . α = 0 .
5, green line, α = 0 .
6, blue line, α = 0 .
7. The negative sign of I c indicates that the junctionminimum is at ϕ = π ( π -junction). The inset in panel (b)shows, on a larger scale, the transition π → α = 0 . Z=0.001 Z=3 (a) (b) (cid:32)(cid:39)(cid:39) (cid:32)(cid:39)(cid:39) FIG. 4: Critical current as a function of temperature. Thecontribution of the continuous energy spectrum states hasbeen added. (a) Z = 0 . Z = 3; in both figures, (a)and (b), the black, red, green, brown, blue lines correspondto the curves with α = 0 . , . , . , . , .
7, respectively.
The general features of the curves I c vs T , shownin Fig. (4) qualitatively reproduce those of Fig. (3).Most notably they confirm the π − π − α = 0 .
65 and shown in Fig. (4) (a), is smoothed out whenconsidering the continuous spectrum contribution, as canbe seen by comparing with the corresponding curve inFig. (3), (b).In conclusion the model investigated in this paper fora multiband superconductor symmetric s ± Is ± junction,predicts a number of non trivial details related to theJosephson effect in these systems. These results are ofrelevant interest for the case of all-pnictide Josephsonmicro-junctions. The spectral equation provides the An-dreev bound levels as a function of the band ratio param-eter α . The main effect of the presence of the second bandis the building up of two extra Andreev levels which driveCooper pairs in a direction opposite to that observed inthe presence of a single band.The phase-current relations predicted on the basis of the Andreev levels (without consideration for the contin-uous energy spectrum) shows the formation of a note-worthy π -state in the junction coupling. For selectedvalues of the band ratio, a π → π -stateis observed at high temperature whereas the 0-state isobserved below the crossover temperature. In this re-gion the junction recovers the behavior of a conventional ′′ ′′ junction. These results are confirmed by means ofa more exhaustive evaluation of the Josephson currentenglobing the contribution of the continuous spectrumenergy states. Acknowledgments
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