Proximity-induced time-reversal symmetry breaking at Josephson junctions between unconventional superconductors Kazuhiro Kuboki∗ and Manfred Sigrist Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Inthis letterweaddanew featureto the propertiesof interfaces between unconventional superconductors. We We argue that a locally time-reversal symmetry breaking examine the effect of the reflection of Cooper pairs at state can occur at Josephson junctions between unconven- the interface and the mutual proximity of the two su- tional superconductors. Order parameters induced by the perconductors, both of which can introduce Cooper pair proximity effect can combine with the bulk order parame- 5 amplitudesinchannelsofsymmetriesdifferentfromthat ter to form such a state. This property is specifically due to 9 the intrinsic phase structure of the pairing wave function in of the bulk pairing state. These additional pairing com- 9 unconventionalsuperconductors. Experimentalconsequences ponentscanexistonlyintheimmediatevicinityofthein- 1 of this effect in high-temperature superconductors are exam- terface and decay exponentially towards the bulk. How- n ined. ever, their presence has important consequences for the a interface properties because they modify the Josephson J effect via their influence on the current-phase relation. 0 1 The admixed order parameters originating from reflec- A growing number of experiments demonstrates tion and proximity usually prefer a relative phase of 0 2 that the superconducting state in some of the high- or π with the bulk order parameters on both sides of v temperature superconductors (HTSC) possesses an un- the interface,resulting in states conservingtime-reversal 9 2 conventional order parameter which changes sign under symmetry. We will show here that under certain con- 0 a90o-rotationinthecopper-oxideplane[1]. Foratetrag- ditions the relative phase can change leading to a state 1 onal system this state is commonly refered to as a d- whichbreakstime-reversalsymmetry atthe interface. It 0 wavestate,wheretheCooperpairingoccursinachannel was shown by Sigrist, Bailey and Laughlin [6] that this 5 belonging to an irreducible representation of the point kind of interface state may explain a recent experiment 9 / groupD4h, whichis differentfromthetrivialrepresenta- donebyKirtleyetal. wherefractionalvorticeshavebeen at tions, i.e. “s-wave”. Many of these experiments use the observedatboundariesbetweendifferentlyorientedfilms m Josephsoneffect,whichallowsadirectprobeofthephase of the HTSC YBa2Cu3O7−δ (YBCO) [7]. - properties of a superconducting order parameter. While We discuss this problem using the Ginzburg-Landau d the Josephson effect in conventional (s-wave) supercon- (GL) theory of a d-wave order parameter in a tetrago- n ductorsisdescribedonlybythephasedifferencebetween nal system (D4h) and consider only one admixed order o c the order parameters on both sides of an interface, the parameter (s-wave) which does not occur in the bulk. : situation is more complicated if the pair wave function This choice is motivated by the experimental results in- v has an intrinsic phase structure like in the d-wave state. troducedabove[1]andtherecentfindingthatthecritical i X The sign change of the d-wave state (generic pair wave behaviorattheonsetofsuperconductivitysuggestsasin- r function: ψ(k) = cosk −cosk ) between the x- and y- gle component bulk order parameter in HTSC [8]. The a x y direction corresponds to a phase difference of π. Due to general GL free energy expansion for these two compo- this intrinsic phase structure, the relative orientation of nents, ηd (d-wave) and ηs (s-wave), in two dimensions is thepairwavefunctionoftwoconnectedsuperconductors given by and the geometry of the connecting interface are impor- F = R d2x[Pj=d,s{aj(T)|ηj|2+βd|ηj|4+Kj|Dηj|2} tant to determine the phase relation between the order parameters on both sides [2–4]. For d-wave supercon- 1 +γ |η |2|η |2+ γ (η∗2η2+η2η∗2) ductors two types of Josephson junctions can occur, one 1 d s 2 2 d s d s where the interface energy is minimized by a phase difference ϕ = 0 (standard 0-junction) and the other with +K˜{(D η )∗(D η )−(D η )∗(D η ) x d x s y d y s ϕ=π (π-junction) [5]. In multiply connected superconducting systems the latter type can lead to frustration +c.c.}+ 1 (∇×A)2] effects and twists of the order parameter phase [5]. Re- 8π sultingphenomenaliketheoccurrenceofspontaneoussu- (1) percurrentsandthemodificationofstandardinterference with a (T)=α (T −T ) and α , β , K , γ , γ and K˜ patterns(SQUID)havebeenexploitedinexperimentsto j j cj j j j 1 2 arerealcoefficients(j =d,s). Thegaugeinvariantgradi- determine the symmetry of the order parameter [1]. entisgivenbyD=∇−i(2π/Φ )AwithAasthevector 0 1 potential and Φ =hc/2e is the standard flux quantum. merically in general. Fortunately, it is possible to under- 0 We considerthe properties ofaninfinite planarinterface standtheessentialpropertiesofthecompetitingeffectsat withthegeometryshowninFig.1wherethecrystalmain the interface qualitatively by using a simple variational (x-) axis of the side A is pointing perpendicular to the ansatz for the order parameter. We assume that as in interface, while the orientationof the side B is described HTSCthetransitiontemperatureT ofthed-waveorder cd by the angle θ between the interface normal vector and parameteris finite, while thatofthe s-waveis verysmall the crystal main (x-) axis (−π/2 ≤ θ ≤ π/2). In this or zero, T ≪T such that in the bulk only η is finite cs cd d arrangement the effects of interest occur only on side B. for00. Further,wechoosea− 0). 0 2 By minimizing F we find that ξ is always finite with gaugeoftheorderparameterphases. Thecoefficientg is var in general positive and describes the suppression of tdhe Ks/a+ ≤ξ2 ≤Ks/a−. Some simple algebra leads to the following equation for χ, d-wave order parameter at the interface (see [3]). The variational minimization scheme for the GL free energy (t1K+−2ξg˜t2)K− sinχ[cosχ+ ]=0. (4) includes these interface terms into the boundary condi- 4t2ξ3γ η˜η 2 2 d tions. One set of solutions of this equation is χ = 0 and π ThisGL theorycontainstwocompetingtendencies for (sinχ=0)whichleadstoηˆ′ 6=0andηˆ′′ =0suchthatthe the relative phase, ϕ − ϕ , of the order parameters s d relativephase betweenη and η is 0 or π. Energetically d s ηj(= |ηj|exp(iϕj)), (j = s,d). The last three terms in χ = 0 (ηˆ′ > 0) is favored for |θ| < π/4 (0-junction) and FIF (Eq.(2))areminimized by ϕs−ϕd =0for |θ|<π/4 χ=π (ηˆ′ <0) for |θ|>π/4 (π-junction). However, also and ϕ −ϕ =π for π/4<|θ|<π/2. In either case, the s d the term in [...] can vanish giving an alternative solution state atthe interfaceis invariantunder time-reversalop- of Eq.(4) with χ different from the above two limiting eration. Ontheotherhand,ifweassumethatγ2 >0,the values. Bothηˆ′andηˆ′′arefiniteinthiscase. Thisstateis couplingtermη∗2η2+η2η∗2 inEq.(1)favorsenergetically 1 2 1 2 two-fold degenerate because the application of the time- ϕs−ϕd =±π/2,astatewhichbreakstime-reversalsym- reversal operation (η → η∗) corresponding to χ → −χ metry (s+id-wave state). (The assumption of positive and ηˆ→ηˆ∗ leads to a different state with the same free γ is natural, since it would lead to a fully gapped state 2 energy. Consequently,thissolutionviolatestime-reversal which is energetically more favorable as weak-coupling symmetry T. It is easy to see that the energy is indeed studies of various microscopic models show [3].) In the minimized by this state if γ is positive. From Eq.(4) it 2 followingwestudythecompetitionbetweenthetendency isobviousthattheT-violatingstatecanonlyoccurifthe to keep or to break time-reversal symmetry near the in- modulusofthesecondtermin[...]issmallerthan1. This terface and some physical consequences. conditioncanbesatisfiedbyanappropriatechoiceofthe Although our choice of geometry allows to reduce the angle θ, because this term is proportional to cos(2θ) via problem to one spatial dimension (x˜ = xcosθ +ysinθ t and g˜, which can be arbitrarily small for θ close to 1 on side B), these GL equations can only be solved nu- ±π/4. 2 Fromthe aboveconsiderationthe conditionfor the in- ϕ(a) and ϕ(b), respectively), we find a continuous change 0 0 stability of the time-reversal invariant state is obtained of ϕ from ϕ(a) to ϕ(b) on a length scale λ . On dis- 0 0 J as tances much larger than λ from the border of the seg- J (j) (cid:12) t2γ η˜(T∗)η (T∗) (cid:12) ments ϕ is ϕ0 in segment (j). The flux associated with cos(2θ)=±(cid:12) 2 2 d (cid:12), (5) the variation of ϕ is located in the vicinity of the bor- (cid:12) (cid:12) (cid:12)(cid:12)(as+γ1ηd2)(t0pKsa+−t2g0)(cid:12)(cid:12) der and is Φ = (ϕ0(b) −ϕ(0a))Φ0/2π. Obviously Φ is not where T∗ (< T ) denotes the critical temperature of an integer (half-integer) multiple of Φ0, but can have an cd arbitraryvalue. Itcannotbedeterminedbysimpletopo- the secondordertransitionto the T-violating state for a logicalarguments,butdependsonthespecificproperties given angle θ. The denominator on the right hand side of this equation is only weakly dependent on T∗ close to of the interface. Because the phase twist at the border T sothatT∗ islargeforθ near±π/4,andisT atθ = betweensegment(a)and(b)correspondstoawindingof cd cd the phase ϕ by a fractionalmultiple of2π, this flux line ±π/4. We show a schematic phase diagram, T versus θ, d may be considered as a fractional vortex [10]. Similarly corresponding to Eq.(5) (Fig.2). The T-violating phase fractionalvorticescanoccur ona homogeneousinterface appearsinacertainrangeofθaroundπ/4andtheregion duetothetwofolddegeneracyoftheT-violatinginterface ofthisphasebecomeswiderwithdecreasingtemperature. state. Two types of domains with ϕ =+χ and −χ, re- Thespecificshapeofthephaseboundarylinedependson 0 spectively, are possible. If presenton the same interface, properties of the system, in particular, of the interface. they are separated by domain “walls”. There we find a Let us now investigate important consequences of the kink of the width λ in ϕ connecting the two values of existence of a T-violating interface state, which result J ϕ and yielding a flux Φ=±χΦ /π (see also [10]). from the properties of its Josephson current-phase rela- 0 0 Wenowproposeanexperimenttotestourpicture. Let tion, usconsiderathinfilmconsistingoftwopartswithorien- I =I g(ϕ)=I g(ϕ+2π) (6) tations of their crystalline axes different by 45o. The c c boundary shall be circularly curved (see Fig.3). The whereϕisthephasedifferencebetweentheorderparam- point P would correspond to an interface with angle eters η and η and g(ϕ) is a 2π-periodic function of ϕ θ =π/4andθchangescontinuouslyasweturnawayfrom 0 d (|g(ϕ)| ≤ 1) [4]. The specific form of g is not important P. Hence, along the interface we scan through a certain here andwill be discussedelsewhere. The interface state range of angles θ. If time-reversal symmetry were con- withminimalenergycorrespondstoI =0anditstwofold servedeverywhere,wewouldseeavortexwithΦ=Φ /2 0 degeneracy implies that two functions, g+ and g−, ex- at P, because P separates two interface segments where ist with g+(χ) = 0 and g−(−χ) = 0, each belonging to one has ϕ0 = 0 and the other = π [2,11]. The field one of the two states. There are two possible situations: distribution of the vortex becomes more localized if we I) g+(ϕ) 6= g−(ϕ) describing both different (meta)stable cool the system because the penetration depth λJ along states as a function of ϕ, and II) g+(ϕ) = g−(ϕ) where the interface shrinks with lowering temperature. On the thetwoT-violatingstatescanbeadiabaticallyconnected other hand, if our scenario is correct we expect that the witheachotherbychangingϕfrom+χto−χ. Thespa- phase ϕ varies continuously from 0 to π near P within 0 tial variation of ϕ along the interface (with coordinate a certain range of θ. This range extends with lowering denotedasx′)isdescribedbyageneralizedFerrel-Prange temperature(seeFig.2),suchthatthe resultingfielddis- equationwhichisidenticaltoaSine-Gordonequationfor tribution becomes more extended further as the system standard Josephson junctions (g(ϕ)=sinϕ), iscooled,whilethetotalfluxalwaysequalstoΦ /2. The 0 observation of the field distribution by a (magnetic) mi- ∂2ϕ =λ−2g(ϕ), (7) croscope should allow to prove or disprove our scenario ∂x′2 J based on these qualitative properties. Insummary,wepointedoutthatimportantproperties with λ = (Φ c/8π2dI )1/2 as the characteristic length J 0 c of an interface between d-wave or other unconventional scale (d: the effective magnetic width of the interface superconductorsresultfromtheintrinsicphasestructure [9]). As in conventional interfaces the spatial varia- ofthepairingwavefunctionψ(k)whichcannotbe found tion of ϕ yields a finite local magnetic flux φ(x′) = in conventional (s-wave) superconductors. While some (Φ /2π)∂ϕ(x′)/∂x′. The integrated magnetic flux be- 0 interfacecouplingeffectsfavoraT-invariantstate,inthe tween two points, x′ and x′, on the interface is given by 2 1 bulkratheraT-violatingstateisprefered. Inthecompe- Φ=(ϕ(x′)−ϕ(x′))Φ /2π. 2 1 0 tition between these two tendencies it is important that As a consequence, a spatial variation of the interface the interface coupling effects can be arbitrarily small by properties can lead to the occurrence of magnetic field choosing the interface geometry appropriately, i.e., the distributions. In particular, at the border between two crystallineorientationofthetwoconnectedsuperconduc- different homogeneous interface segments, (a) and (b), tors. Thus, proximity-induced order parameter compo- characterized by their specific values of ϕ (denoted as 0 3 nentscancombinewiththebulkorderparametertoform FIG.2. Schematicephasediagramintermsoftemperature and crystal orientation of side B. The shaded region denotes a T-violating interface state. We emphasize that such the T-violating phase separating the 0- and the π-junction conditions cannot be satisfied in conventional (s-wave) phases. Within theshaded region χ changes continously. superconductors, in general. Therefore the proximity- induced time-reversal breaking interface state is a consequence of unconventional superconductivity. For the FIG. 3. Schematic arrangement of superconducting films sake of simplicity we restricted ourselves to the case of a for a test-experiment. The direction of shading indicates the system with tetragonal crystal field symmetry. We note crystallineorientation. TheregionAmaybead-orans-wave here, however, that a slight orthorhombic distortion of superconductor, while B must have d-wave symmetry. The the lattice (as it is present in real materials) does not point P corresponds tothe angle θ=π/4 and θ˜=π/4−θ. change our conclusions qualitatively. It leads to some minor modifications which are beyond the scope of this letter and will be discussed elsewhere in detail. We are grateful to P.A. Lee, T.M. Rice, K. Ueda, A. Furusaki, Y.B. Kim, D.K.K. Lee, C. Bruder and D. Scalapino for helpful and stimulating discussions. We acknowledge financial supports by Swiss Nationalfonds (M.S.) and by Ministry of Education, Science and Cul- ture of Japan (K.K.). M.S. would like to thank the In- stitute forSolidState Physicsofthe UniversityofTokyo and the University of Tsukuba for their hospitality dur- ing the time when this work has been finished. ∗ Permanent address: Department of Physics, Kobe Uni- versity,Kobe657, Japan. [1] D.A. Wollman et al., Phys. Rev. Lett. 71, 2134 (1993); D.BrawnerandH.R.Ott,Phys.Rev.B50,6530(1994); A. Mathai et al., preprint; C.C. Tsuei et al., Phys. Rev. Lett.73,593(1994); I.IguchiandZ.Wen,Phys.Rev.B 49, 12388 (1994). [2] V.B. Geshkenbein and A.I. Larkin, Pis’ma Zh. Eksp. Teor. Fiz. 43, 306 (1986) [JETP Lett. 43, 395 (1986)]. [3] L.P. Gor’kov, Sov. Sci. Rev. A Phys. 9, 1 (1987); M. Sigrist and K. Ueda, Rev.Mod. Phys. 63, 239 (1991); I. A. Lukyanchuk and M.E. Zhitomirsky, to be published in Superconductivity Review. [4] S.K. Yip, O.F. De Alcantara Bonfim and P. Kumar, Phys.Rev.B 41, 11214 (1990). [5] M. Sigrist and T.M. Rice, J. Phys. Soc. Jpn. 61, 4283 (1992). [6] M. Sigrist, D.Bailey and R. Laughlin, preprint. [7] J.R. Kirtley et al., preprint,private communication. [8] S.Kamal et al., Phys. Rev.Lett. 73, 1845 (1994). [9] M. Tinkham, Introduction to Superconductivity, McGraw-Hill Book Company (1975). [10] M.Sigrist, T.M.RiceandK.Ueda,Phys.Rev.Lett.63, 1727 (1989). [11] A.J. Millis, Phys.Rev. B49, 15408 (1994). FIG. 1. Interface between d-wave superconductor A and B. The shading on either side indicates the direction of the crytalline x-axis (thez-axispoints out of the plane). 4

Proximity-induced time-reversal symmetry breaking at Josephson junctions between unconventional superconductors

Similar Proximity-induced time-reversal symmetry breaking at Josephson junctions between unconventional superconductors