Table Of ContentProximity-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 dif-
ference 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 supercon-
ducting 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. for0<T <T (andη onthesideA)[1,8]. Inourvaria-
cd 0
Thus, we will neglect the detailed description of side A tionaltreatmentη shallberealandindependentofposi-
d
and represent it simply by a complex (bulk) order pa- tiononthe sideB,andη ofthesideAshallhaveafixed
0
rameter η . It can have d-wave symmetry, but s-wave modulus η˜ and a phase χ, η = η˜exp(iχ) (η (T),η˜(T)
0 0 d
symmetrywouldnotchangeanyofourconclusionsqual- areidenticaltotheirbulkvalues). Hence,thisansatzne-
itatively. Thepropertiesoftheinterfacearedescribedby glects the directeffect of the interface on the modulus of
additional interface terms in the GL free energy the d-wave order parameter. The s-wave component η
s
iscomplexanddecaysexponentiallytowardsthebulkon
FIF = RIF dS[gd(θ)|ηd|2+g˜(θ)(ηd∗ηs+ηdηs∗) the side B, ηs(x˜) = ηˆe−x˜/ξ and ηˆ = ηˆ′ +iηˆ′′. The part
of the free energy (per unit interface area) depending on
∗ ∗ ∗ ∗ (2)
+t1(θ)(η0ηd+η0ηd)+t2(θ)(η0ηs+η0ηs)]. thevariationaldegreesoffreedom,ξ,ηˆandχ,isobtained
straightforwardly
The first two terms describe the reflection properties of ηˆ′2 ηˆ′′2
Fvar(eˆta,ξ,χ)= K+ +K−
the interface (IF) forthe side B,i.e.: η →η inthe first 2ξ 2ξ
d d
and η ↔η in the second term (we neglect the η →η
d s s s
contribution). The latter two terms represent the lowest +2{g˜(θ)ηdηˆ′+t1(θ)η˜ηdcosχ (3)
order coupling between the two sides, A and B. By sym-
metry argument we find that g˜(θ ±π/2) = −g˜(θ) and +t2η˜(ηˆ′cosχ+ηˆ′′sinχ)}
t (θ ±π/2) = −t (θ). Thus, we choose g˜ = g cos(2θ)
1 1 0
and t = t cos(2θ) with g ,t < 0 as a generic angle
depen1dence0. We neglect any0θ-0dependence in t because with K± =Ks+a±ξ2 and a± =as+(γ1±γ2)ηd2. To be
2 consistent with our assumption that η = 0 in the bulk
none is required by symmetry (t < 0). The signs of g , s
t and t are chosen by conventi2on defining the specifi0c werequirea± >0. Further,wechoosea− <a+ (γ2 >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 con-
sequence 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.
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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
