Broken time-reversal symmetry in Josephson junction involving two-band superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] S e p Broken time-reversal symmetry in Josephson junction involvingtwo-band superconductors
T.K. Ng , and N. Nagaosa , Department of Physics, Hong-Kong University of Science and Technology,Kawloon, Hong Kong Peoples R China Department of Applied Physics, The University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Cross Correlated Materials Research Group (CMRG),ASI, RIKEN, WAKO 351-0198, Japan
Abstract
A novel time-reversal symmetry breaking state is found theoretically in the Josephson junctionbetween the two-gap superconductor and the conventional s-wave superconductor. This occurs dueto the frustration between the three order parameters analogous to the two antiferromagneticallycoupled XY-spins put under a magnetic field. This leads to the interface states with the energiesinside the superconducting gap. Possible experimental observations of this state with broken time-reversal symmetry are discussed. . A representative example of the multi-gap superconductor is MgB where the spe-cific heat , the tunneling , and angle-resolved photoemission spectroscopy (ARPES) haverevealed the different gap energies for σ - and π -bands. The magnitudes of the gaps wereanalyzed by the first-principles band structure calculation and are found to be stronglymomentum- and band- dependent. This two-gap behavior is attributed to the strong cou-pling of the σ -band to the bond stretching phonon mode .The multi-gap structure should be common in the superconductors with the orbital de-generacy and/or the many electron/hole pockets, which is the case for the newly foundhigh temperature superconductor iron pnictides . In these compounds, there are two smallelectron pockets around M-points and two hole pockets around Γ-point . There are manyproposals for the gap pairing symmetry , and one possibility is that a full gap opens foreach pocket, which is consistent with the recent ARPES in Ba . K . Fe , although ARPEScannot determine the relative sign of the order parameters. Therefore, the determination ofthe relative sign of the order parameters on the pockets is now an important issue to fix themicroscopic mechanism for the superconductivity. A clue has been given by the resonantmagnetic scattering which is attributed to the triplet exciton in the superconducting state.A comparison with the earlier theoretical analysis suggests that the relative sign of theorder parameter is minus in Ba . K . Fe .In this paper, we explore theoretically a novel phenomenon in the two-gap superconduc-tors when coupled to another single-band superconductor by the Josephson effect. The twobands are assumed to have separated Fermi surfaces in ~k -space and are coupled only throughelectron-electron interaction. We shall assume for simplicity that the superconducting or-der parameters on all Fermi surfaces have s -symmetry, although most of our results canbe generalized to order-parameters with other symmetries as well. We start with the phe-nomenological Ginzburg-Landau (GL) free energy density of the two band superconductor. F ( ψ , ψ ) = α ( T ) | ψ | + K | ~Dψ | + β | ψ | + α ( T ) | ψ | + K | ~Dψ | + β | ψ | − J [ ψ ∗ ψ + ψ ∗ ψ ](1)2here ψ is the superconducting order parameter for band 1(2) and ~D = − i [ ∇ − ei ~A/ ~ c ].In general the two superconducting bands are coupled by an internal Josephson-couplingterm ∼ J as a result of electron-electron interaction. Writing ψ = | ψ | e iθ andminimizing the energy with respect to θ in the absence of magnetic field it is easy to seethat ψ and ψ are of the same sign ( θ = θ ( mod π )) if J >
0, and are of opposite sign( θ = θ + π ( mod π )) if J <
0. The question is whether there exists any non-trivial physicalconsequences associated with this relative sign, in particular when
J < ψ ∼ − ψ ?In the following we shall show that the spontaneous time-reversal symmetry breakingoccurs at the Josephson junction between the two-band superconductor and another single-band s -wave superconductor when the sign of J is negative. To be concrete we assume thatthe single-band superconductors is located at the left side ( x <
0) of the Josephson junction,and the two-band superconductor is located on the right ( x > F = F θ ( x ) + F s θ ( − x ) + F J , where F s ( ψ s ) = α s ( T ) | ψ s | + K s | ~Dψ s | + β s | ψ s | (2)is the usual Ginsburg-Landau free energy for the single-band s -wave superconductor, and F J = − ( T [ ψ ∗ ψ s + ψ ∗ s ψ ] + T [ ψ ∗ ψ s + ψ ∗ s ψ ]) δ ( x ) (3)is the Josephson coupling between the two superconductors. T represents the coupling ofthe single-band superconductor to the two separate bands. We note that T and T are bothpositive according to the perturbation theory in the tunnelling matrix elements between thetwo superconductors. The relative sign between ψ and ψ is “unknown” to the single-bandsuperconductor in the Josephson effect.The non-trivial effect associated with the Josephson junction can be seen by minimizingthe free energy of the system with respect to the phases of the superconductors, assumingthat the amplitudes of the order parameters are constants. In the absence of magnetic fieldthe GL free energy density with phase variables only is F ∼ ¯ F + θ ( x ) (cid:16) − J cos( θ − θ ) + ˜ K ( ∇ θ ) + ˜ K ( ∇ θ ) (cid:17) (4) − δ ( x ) (cid:16) ˜ T cos( θ − θ s ) + ˜ T cos( θ − θ s ) (cid:17) + ˜ K s ( ∇ θ s ) where ¯ F is the part of free energy density which is independent of θ ’s. ˜ θ is defined at x ≥ θ s is defined as x ≤
0. ˜ J = J | ψ || ψ | , ˜ T = T | ψ s || ψ | and ˜ K ν = K ν | ψ ν | ,3 =0 s ϑ -ϑ FIG. 1: A schematic representation of canted state in the effective classical spin model for theJosephson junction between a two band superconductor and another single-band superconductor.The coupling between the two spins ( θ , θ ) are antiferromagnetic ( J <
0) but they are bothcoupling ferromagnetically to a magnetic field along ˆ x -direction ( θ s = 0) ν = 1 , , s . We shall take θ s ( x = 0) = 0 in the following. This is allowed because the overallphase of the system is a pure gauge. First we note that for J > θ = θ = θ s = 0 and there is no non-trivial effect associated with the Josephson junction. The situation becomes different for J < J and T .To see what could happen we note that the system is similar to a system of two classicalspins A and B that are antiferromagnetically coupled and are put under a weak magneticfield in ˆ x direction representing the Josephson coupling of the system to the single-bandsuperconductor. If the coupling of spin A to magnetic field is much stronger than that ofspin B ( T >> T ), spin A will be allied to the magnetic field with spin B remaining anti-parallel to spin A in the ground state, i.e. θ = 0 , θ = π in the corresponding Josephsonjunction problem. The converse ( θ = 0 , θ = π ) is true if T >> T . This is the firstpossible state. Notice however that if the couplings of the two spins to the magnetic fieldare similar, there will be no preferred spin to the magnetic field and a second type of statewhere the two spins take angles θ ∼ ± π/ ∓ δθ will be formed (see Fig.1).The free energy (4) cannot be minimized exactly to obtain the two types of phase struc-tures. We shall treat the free energy approximately in the following by writing θ = θ + ˜ θ , θ = θ − π + ˜ θ and expand the free energy to order ˜ θ . The approximation can be justifiedin the limit of weak-Josephson coupling as we shall see later.4inimizing the resulting approximate free energy F ( θ , ˜ θ ) with respect to ˜ θ ’s we obtain˜ J (˜ θ − ˜ θ ) + ˜ K ∇ ˜ θ = ˜ T δ ( x )(sin θ + cos θ ˜ θ ) , (5)˜ J (˜ θ − ˜ θ ) + ˜ K ∇ ˜ θ = − ˜ T δ ( x )(sin θ + cos θ ˜ θ ) , ˜ K s ∇ θ s = − δ ( x ) (cid:16) ˜ T (sin θ + cos θ ˜ θ ) − ˜ T (sin θ + cos θ ˜ θ ) (cid:17) . Solving these equations at x = 0 we obtain ˜ θ = α e − x/λ + β x and θ s = β s x where˜ K α = − ˜ K α and λ = | ˜ J | ( ˜ K + ˜ K )˜ K ˜ K .Matching the boundary condition at x = 0 we also obtain J = ˜ K ( β − α λ ) = ˜ T (sin θ + cos θ α ) , (6) J = ˜ K ( β − α λ ) = − ˜ T (sin θ + cos θ α ) , ˜ K s β s = J + J . The first two equations give the tunnelling currents flowing from band one (two) to the single-band superconductor, respectively. The third equation expresses total current conversationacross the Josephson junction.We shall concentrate on the ground state solution where the Josephson junction doesnot introduce any bulk energy cost. In this case there is no net current flowing through thesystem and β = β s = 0. Solving Eq. (6) we find that only one solution cos θ = sgn ( ˜ T − ˜ T )exists at | ˜ T − ˜ T | > σ , where σ = λ ˜ T ˜ T ( ˜ K + ˜ K )˜ K ˜ K whereas two possible solutions cos θ = (cid:16) sgn ( ˜ T − ˜ T ) , ( ˜ T − ˜ T ) σ (cid:17) exist at | ˜ T − ˜ T | < σ . The true solution at | ˜ T − ˜ T | < σ is theone with lower energy. Comparing the energies F ( θ , ˜ θ ) of the two states we findcos θ = sgn ( ˜ T − ˜ T ) ( | ˜ T − ˜ T | > σ ) (7)= ( ˜ T − ˜ T ) σ ( | ˜ T − ˜ T | < σ )where σ = min ( σ , σ ), σ = λ ( ˜ K ˜ T + ˜ K ˜ T ) ( ˜ K + ˜ K ) ˜ K ˜ K .In the first case θ = 0 or π , which corresponds to the first type of solution in the classicalspin problem. The solution respects time-reversal symmetry and we shall call it T RI statein the following.There are two degenerate solutions in the second case corresponding to θ ≷ ψ ⇋ ψ ∗ ). The solution breaks time reversal symmetry and weshall call it the T RB state. The corresponding α is equal to zero in the T RI state, and5s of order sin θ ˜ T λ/ ˜ K in the T RB state, which is much less than θ in the limit ofweak-Josephson coupling ˜ T λ/ ˜ K <<
1, justifying our approximate treatment of freeenergy.It is interesting to note that although the net Josephson current passing through thetunnelling barrier is zero in the ground state, the currents J and J = − J , which representcurrent passing from band 1(2) of the two-band superconductor to the single-band super-conductor, are nonzero in the T RB state. Correspondingly there is also a nonzero current J ∼ ˜ J sin( θ − θ ) passing from band one to band two in the T RB state.Thus the
T RB state is characterized by a novel current “loop” through the Josephsonfunction. A current flows from band one/two of the two-band superconductor to the single-band superconductor through the Josephson junction, and flows back to band two/one ofthe two-band superconductor. The current flow from band two/one to band one/two insidethe two-band superconductor to complete the current loop. The current loop we see here isnot a current loop circulating in real space, but a current loop in ~k -space, if we envision thetwo bands as occupying different parts of the ~k -space.It is also straightforward to see from Eq. (6) that the parameter space where the T RB state exists is enlarged when there is a net current flowing across the Josephson junction( β , β s = 0). This is not surprising since a finite current through the system breaks time-reversal symmetry. The current also removes the degeneracy of the two T RB solutions with θ ≶ ǫ ( i ) n u ( i ) n ( ~x ) = ˆ H ( i ) o u ( i ) n ( ~x ) + ∆ ( i ) ( ~x ) v ( i ) n ( ~x ) (8) ǫ ( i ) n v ( i ) n ( ~x ) = − ˆ H ( i ) o v ( i ) n ( ~x ) + ∆ ( i ) ∗ ( ~x ) u ( i ) n ( ~x )where i = 1 , H ( i ) o gives the single-particle band structure for band i and ∆ ( i ) ( ~x ) is the corresponding superconducting order parameter. We have assumed thatthe two bands are independent of each other and are coupled implicitly only through thesuperconductor order parameter in writing down Eq.(8). In particular, ∆ (1) ∼ +( − )∆ (2) inthe bulk superconductor if J > ( < )0. The Josephson function can be modeled by supercon-ducting order parameters of the form ∆ ( i ) ( ~x ) ∼ ∆ ( i )0 e iθ i ( x ) at x > x ) = ∆ s at x < ( i )0 and ∆ s are real and positive. We shall show in the following that non-trivialelectronic surface state exists in the Josephson junction, with structures depending stronglyon the phases of the superconducting order parameters.The BdG equations for electronic states close to the Fermi surface can be solved in theWKBJ approximation where we write u ( v ) ( i ) n ( ~x ) ∼ e i~k F .~x ˜ u (˜ v ) ( i ) n ( ~x ), where ˜ u (˜ v ) ( i ) n ( ~x ) areslowly varying functions of ~x (on the scale of k − F ) satisfying the Andreev equations, ǫ ( i ) n ˜ u ( i ) n ( ~x ) = − i ( ~v ( i ) F . ∇ )˜ u ( i ) n ( ~x ) + ∆ ( i ) ( ~x )˜ v ( i ) n ( ~x ) (9) ǫ ( i ) n ˜ v ( i ) n ( ~x ) = i ( ~v ( i ) F . ∇ )˜ v ( i ) n ( ~x ) + ∆ ( i ) ∗ ( ~x )˜ u ( i ) n ( ~x ) , where ~v ( i ) F is the Fermi velocity of the band i electrons with momentum ~k ( i ) F . We shallconsider the weak-Josephson coupling limit (˜ θ << θ ) in the following, so that θ ( x ) ∼ θ and θ ( x ) ∼ θ − π . The bound states are given by solutions of form ˜ u (˜ v ) n ( ~x ) = ˜ u (˜ v ) e − γ + x for x > u (˜ v ) n ( ~x ) = ˜ u (˜ v ) e γ − x for x <
0. Substituting these into Eq. (9), we obtainthe self-consistent equations ǫ ( i )20 = ∆ ( i )20 (1 − x ( i )2 ) = ∆ s (1 − y ( i )2 ) (10) ǫ ( i )0 / ∆ ( i )0 − ix ( i ) ǫ ( i )0 / ∆ s + iy ( i ) = e iθ i where x ( i ) = ~v ( i ) F . ˆ x ( γ ( i )+ / ∆ ( i )0 ) and y ( i ) = ~v ( i ) F . ˆ x ( γ ( i ) − / ∆ s ). Notice that x ( i ) and y ( i ) must havethe same sign in this representation and changing sign of x ( y ) ( i ) corresponds to changing ~v F → − ~v F .It is easy to see that if x ( i ) , y ( i ) is a solution of Eq. (10) with energy ǫ ( i )0 , then − x ( i ) , − y ( i ) is a solution with energy − ǫ ( i )0 . − x ( i ) , − y ( i ) is also a solution with energy ǫ ( i )0 with θ i → − θ i .Therefore it is sufficient to consider the range π > θ i >
0. Solving Eq (10) we find that solu-tions where x ( i ) and y ( i ) have the same sign exists only when cos θ i < min (∆ ( i )0 / ∆ s , ∆ s / ∆ ( i )0 ),with x ( i ) = ∆ s cos θ i − ∆ ( i )0 D ( i ) sgn ( ǫ ( i )0 ) , (11) y ( i ) = ∆ ( i )0 cos θ i − ∆ s D ( i ) sgn ( ǫ ( i )0 ) ,ǫ ( i )0 = ± ∆ ( i )0 ∆ s sin θ i D ( i ) . where D ( i ) = q ∆ s + ∆ ( i )20 − θ i ∆ s ∆ ( i )0 . Notice that there exists one solution for each7alue of Fermi momentum ~k ( i ) F = ( k ( i ) F x , k ( i ) F y , k ( i ) F z ). Thus a finite density of states exist at theJosephson junction in general.We now analyze the solutions as a function of θ i . First we note that bound state solutionsdo not exist when θ i = 0, i.e. when J >
J < θ i = π , we find that bound state solutions with ǫ ( i )0 = 0 exist at both x ( y ) ( i ) ≶ T RI state in the band which is out-of-phase with the single-band superconductor,but there is no bound state solution for the band which is in-phase with the single-bandsuperconductor. Since time-reversal symmetry is preserved in the
T RI state, the x ( y ) ( i ) ≶ ǫ ( i )0 = 0.The bound state structure is much richer in the T RB state which breaks time-reversalsymmetry. In this case, the structure of the bound states depend on the value of θ (Recall θ ∼ θ and θ ∼ θ − π ). First we consider θ < π/
2. For small θ such that cos θ >min (∆ (1)0 / ∆ s , ∆ s / ∆ (1)0 ), bound state solutions exist only in band two. The bound statesexist in pairs with energies ± ǫ (2)0 = 0, corresponding to time-reversal pairs x ( y ) (2) ≶ θ such that cos θ < min (∆ (1)0 / ∆ s , ∆ s / ∆ (1)0 ), bound state solutions exist in bothbands in time-reversal pairs with energies ± ǫ (1 , = 0. Similar results occur for θ > π/ − cos θ > min (∆ (2)0 / ∆ s , ∆ s / ∆ (2)0 ) and exist inboth bands if − cos θ < min (∆ (2)0 / ∆ s , ∆ s / ∆ (2)0 ).Now the experimental observation of this bound state is discussed. First, the dI/dV curvefor the Josephson tunneling should show the peak at the bound state energy within the gap.A zero-bias peak would exists in the T RI state, and is split into two peaks centered at finiteenergies in the
T RB state. The local probe such as the STS can also be used to detectthese bound states. The detection of the persistent Josephson current loop in ~k -space in the T RB state will be a challenge. The standard prove for the time-reversal symmetry breakingis the Kerr rotation. With the spin-orbit interaction, the finite spin density is expected atthe Josephson junction, but the details of the analysis depends on band structure which wehave not undertaken in this paper.We emphasize that the effects discussed in this paper are general effects associated withfrustrated phase structures in superconductor Josephson junctions, and is not restricted to s -wave superconductor, or Josephson junction involving single-band and two-band supercon-ductors. This idea of the frustration can be generalized to single multi-band superconductors8ith three or more bands coupled via internal Josephson effect. In this case, we can con-struct the effective XY-spin model with positive or negative exchange interactions betweenpairs of the order parameters, and when there exists relative angle(s) different from 0 or π ,time-reversal symmetry is broken spontaneously. This mechanism is likely to be active inthe superconductors with rather complex band structure such as the heavy fermion systems,where many sheets of the Fermi surface contribute to the pairing.To summarize, we have studied the Josephson junction between the two-gap supercon-ductor and the conventional s-wave superconductor. When the relative sign of the twogaps are negative, the Josephson coupling introduces the frustration, which can lead to thetime-reversal symmetry breaking near the junction. This results in the bound state withinthe gaps, which an be detected by the dI/dV characteristics or STS, which can offers anexperimental test of the relative phase of the two gaps.The authors are grateful to Patrick A Lee for fruitful discussions. This work was sup-ported in part by Grant-in-Aids (Grant No. 15104006, No. 16076205, and No. 17105002)and NAREGI Nanoscience Project from the Ministry of Education, Culture, Sports, Science,and Technology. TKN also acknowledge support by HKUGC through grant CA05/06.Sc04. H. Suhl, B.T. Matthias, and L.R. Walker, Phys. Rev. Lett. , 552 (1959). J. Nagamatsu et al., Nature , 63 (2001) F. Bouquet et al. Phys. Rev. Lett. , 047001 (2001) P. Szabo et al., Phys. Rev. Lett. , 137005 (2001) S. Souma et al., Nature b , 65 (2003) H. Choi et al., Nature , 758 (2002) Y. Kamihara et al., J. Am. Chem. Soc. , 3296 (2008). D. J. Singh and M. H. Du, arXiv:0803.0429. Fe Wang et al., arXiv:0807.0498 and references therein. H. Ding et al., Europhys. Lett. , 47001 (2008). A. D. Christianson et al., arXiv:0807.3932. J. Chang et al., Phys. Rev. B , 24503 (2007). C.-R. Hu, Phys. Rev. Lett. , 1526 (1994)., 1526 (1994).