Phenomenological theory of the 3 Kelvin phase in Sr2RuO4

Phenomenological Theory of the 3 Kelvin Phase in Sr2RuO4

Manfred SIGRIST* and Hartmut MONIEN**

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502

(Received February 7, 2001)

We model the 3 K-phase of Sr2RuO4 with Ru-metal inclusion as interface state with locallyenhanced transition temperatures. The resulting 3 K-phase must have a di.erent pairingsymmetry than the bulk phase of Sr2RuO4, because the symmetry at the interface is lower than inthe bulk. It is invariant under time reversal and a second transition, in general, above the onset ofbulk superconductivity is expected where time reversal symmetry is broken. The nucleation of the3 K-phase exhibits a ‘‘capillary e.ect’’ which can lead to frustration phenomena for thesuperconducting states on di.erent Ru-inclusions. Furthermore, the phase structure of the pairwave function gives rise to zero-energy quasiparticle states which would be visible in quasiparticletunneling spectra. Additional characteristic properties are associated with the upper critical 6eldHc2. The 3 K-phase has a weaker anisotropy of Hc2 between the inplane and z-axis orientationthan the bulk superconducting phase. This is connected with the more isotropic nature Ru-metalwhich yields a stronger orbital depairing e.ect for the inplane magnetic 6eld than in the stronglylayered Sr2RuO4. An anomalous temperature dependence for the z-axis critical 6eld is found dueto the coupling of the magnetic 6eld to the order parameter texture at the interface. Various otherexperiments are discussed and new measurements are suggested.

KEYWORDS: Sr2RuO4, p-wave superconductor, broken time-reversal symmetry, frustration e"ects, Andreev bound states

x1. Introduction

In recent years the quasi-two-dimensional metalSr2RuO4 has advanced to one of the most intenselystudied transition metal oxides displaying unconven-tional superconductivity.1,2) There is strong evidence forspin-triplet pairing with broken time reversal symmetry,a pairing state with the basic form dðkÞ ¼ zzðkx � ikyÞ(chiral p-wave state).3{5) This superconducting phaseshows a variety of unusual properties among which therecently discovered ‘‘3-Kelvin’’ (3 K) phase is one of themost puzzling 6ndings. While the transition to the bulksuperconducting state occurs around 1.5 K, in sampleswith a large excess-Ru concentration a precursors to thesuperconducting transition appears at temperatures ashigh as 3 K.6,7) This phase shows the features of aninhomogeneous superconducting phase. The detailedmaterial analysis revealed that the excess-Ru does notdistribute uniformly, but forms small inclusions ofmicrometer-size. Thus, the 3 K-phase is likely connectedwith the phase separation of Ru-metal and Sr2RuO4 inthis sample. The bulk superconductivity of metallic Ruhas a superconducting transition temperature of 0.5 Konly with a conventional (s-wave) pairing state. Thisleads us to the assumption that the boundaries betweenthe two materials provides the environment for the localnucleation of superconductivity at a higher temperature(Fig. 1).

It is not our aim to discuss here in detail the origin forthe locally enhanced transition temperature, since themicroscopic mechanism of superconductivity in Sr2RuO4

is still unclear. Nevertheless, we would like to commenton one important aspect which could be connected withthe enhanced superconductivity. Various experimentshave shown that a particular soft optical modes for locallattice distortion is associated with the inplane rotationof the RuO6-octahedra, the �3-mode at the Brillouinzone boundary of the phonon spectrum, which leads to aslight volume reduction.8) This rotation a.ects one of thethree electron bands in particular, the -band whichoriginates from the 4d{t2g-orbital with dxy-symmetry.3,9)

While the dispersion for this orbital occurs via -hybridization between the Ru{d-orbital and the O{p-orbitals in the undistorted case, the rotation introducesan additional �-hybridization of opposite sign for the dxy-orbital. A lattice distortion of this kind would diminishthe dispersion of the -band and enhance the electrondensity of states, because of a Van Hove singularity nearthe Fermi level.10) It is likely that the Ru-inclusions inSr2RuO4 lead to internal stress that is released by localstatic distortions in the vicinity of the interfaces, mostlikely connected with the �3-mode (h!�3

� 140 K). Acrude estimate from neutron scattering data yields alength scale of order 50{100 �A over which this rotationaldistortion extend away from the interface. The increaseddensity of states would lead to a locally enhanced Tcindependent of the microscopic mechanism. Additionallyenhanced ferromagnetic spin >uctuations may supportthe spin-triplet pairing instability too.10)

In this paper we would like to investigate a number ofproperties of the inhomogeneous 3 K-phase from aphenomenological point of view. A generalized Ginz-burg{Landau (GL) formulation is most suitable for thispurpose, since we will discuss an inhomogeneous super-conducting phase. The basic assumption is that theinterface region has an enhanced transition temperature.

*New address: Theoretische Physik, ETH-Honggerberg, 8093 Zurich,Switzerland.**Permanent address: Physikalisches Institut, Universitat Bonn,

Nussallee 12, 53115 Bonn, Germany.

Journal of the Physical Society of Japan

Vol. 70, No. 8, August, 2001, pp. 2409{2418

2409

We brie>y review the basic conclusions of our theory.The 6rst and most important fact is that the super-conducting state nucleated at the interface has adi.erent symmetry than the bulk superconductingphase. The interface superconducting state is timereversal symmetry conserving. Consequently, there is afurther second order phase transition where this sym-metry is broken. We will show that this transition occursin general above the onset of bulk superconductivity.Naturally the nucleation of superconductivity at theinterface is inhomogeneous and does not lead to auniform phase transition. This is also true for the secondtransition. Even if all interfaces between Ru-inclusionsand Sr2RuO4 are locally equivalent, their geometry andmutual arrangement would lead to a spread on nuclea-tion temperatures due to ‘‘capillary e.ects’’.

The 3 K superconducting state at the interfacecorresponds to an odd parity state, a p-wave state witha pair wave function that has a node parallel to thenormal vector with a positive and negative lobes parallelto the interface (Fig. 1). This phase structure isresponsible for a peculiar change of the quasiparticlespectrum due to Andreev re>ection, i.e. the accumula-tion of Andreev bound states at zero-energy which maybe observed by quasiparticle tunneling. The phasestructure together with the topology of the interfacescan generate a frustration of the order parameter phasewhich can be released by introducing spontaneousorbital currents. Note that also the second transition toa time reversal symmetry breaking phase would reducethe frustration and introduce spontaneous currents.

Finally the upper critical 6eld Hc2 shows specialproperties in case of 6elds parallel to the interface. Inthis case the orbital depairing is reduced due to theessentially two dimensional nature of the condensate. Inaddition, the anisotropy of Hc2 between 6elds parallel tobasal plane of the strongly layered Sr2RuO4 and the z-axis direction is reduced compared with the critical 6eldfor the bulk phase. This is due to the more isotropic

nature of the Ru-metal. A considerable fraction of thesuperconducting condensate of the 3 K-phase resides inthe Ru-part. An additional very striking feature occursfor magnetic 6eld parallel to the z-axis of Sr2RuO4.While in zero 6eld only the p-wave component with itsnode perpendicular to interface appears, the magnetic6eld drives also the other component whose node isparallel to the interface. The energy gain occurs via thecoupling of the 6eld to the current induced by the orderparameter texture of both components. This couplingleads to a peculiar enhancement and temperaturedependence of Hc2 in agreement with experiment.6,7)

x2. Phenomenological Description

The following analysis is based on the generalized GLtheory as the most eHcient way to describe basicproperties of an inhomogeneous superconducting state.The bulk pairing state of Sr2RuO4 has the symmetry of achiral p-wave state, represented by dðkÞ ¼�0zzðkx � ikyÞ. This requires a two-component orderparameter � ¼ ð�x; �y) with dðkÞ ¼ zzð� � k), which be-longs to the two-dimensional representation Eu of thetetragonal point group D4h and describes the leadinginstability in Sr2RuO4. Note that recent >ux distributionmeasurements in the mixed state have suggested thepresence of two order parameter components.11) On theother hand, Ru is a conventional s-wave superconductorwith a transition temperature around 0.5 K. The super-conductivity in Sr2RuO4 penetrates the Ru-metal due tothe proximity e.ect in two ways. First, proximity leadsnaturally to a spin-triplet pairing amplitude in Ru,although its critical temperature there may be extremelysmall. Second, the spin-triplet superconducting state inSr2RuO4 can induce the s-wave component. Thecorresponding coupling is, however, probably weakerthan for the triplet channel because at the interface thetriplet and singlet spin wave function have to beconnected, by means of spin-orbit scattering. We willignore the s-wave component in Ru and will brie>ycomment later only.

2.1 Ginzburg-Landau free energyThe GL free energy for the two-component order

parameter � has the well-known form,

F ¼Z

d3r½aj�j2 þ b1j�j4 þb2

2ð��2x �2y þ c.c.Þ

þ b3j�xj2j�yj2 þK1ðjDx�xj2 þ jDy�yj2ÞþK2ðjDy�xj2 þ jDx�yj2Þ þ fK3ðDx�xÞ�ðDy�yÞþK4ððDy�xÞ�ðDx�yÞ þ c.c.gþK5ðjDz�xj2 þ jDz�yj2Þ þ ðr �AÞ2=8 �

ð1Þ

where the coeHcients are di.erent in the two subsystemswhich we label by indices S and R for Sr2RuO4 and Ru,respectively. The gradient terms contain the gaugeinvariant spatial derivatives D ¼ rþ ið2e=hcÞA withA denoting the vector potential. The coeHcients Ki�

determine the coherence length of the superconductingorder parameter (� ¼ S and R). Since Sr2RuO4 has alayered structure, the coherence length along the z-axis

Fig. 1. Interface between Sr2RuO4 and a Ru-inclusion. The interface

has a layer of thickness d of enhanced transition temperature where a

p-wave state nucleates whose wavefunction has the lobes parallel andthe nodes perpendicular to the interface.

2410 Manfred SIGRIST and Hartmut MONIEN (Vol. 70,

is short, K5S � K1S, K2S, � � � . On the other hand, Ru ismore isotropic so that K5R has a magnitude similar tothe other coeHcients. In a weak-coupling approachassuming cylindrical or spherical Fermi surface shapeswe obtain the relation K1�=3 ¼ K2� ¼ K3� ¼ K4�. Thesecond order coeHcient a�ðT Þ changes sign at the barebulk transition temperature Tc�. For our discussion ofthe qualitative properties of the interface superconduc-tivity it is suHcient to assume linear temperaturedependence a�ðT Þ ¼ !�ðT � Tc�Þ with TcS � 1:5 K andTcR ¼ 0.

For simplicity, we consider a single homogeneousplanar interface with a normal vector parallel to the x-axis (n ¼ ð100Þ) where �1 < x < 0 belongs to the Ru-metal, and 0 < x <1 to Sr2RuO4. In this geometry theproblem reduces to one dimension with the spatialdirection along the x-axis. In the following we will alwaysignore the z-direction assuming homogeneity in thisdirection for simplicity. On the Sr2RuO4-side a thinlayer of thickness d with enhanced transition tempera-ture is introduced. The boundary conditions, in general,involve re>ection and transmission of Cooper pairs at theinterface. We make the simplifying assumption ofcomplete transparency, which corresponds to a contin-uous order parameter. We will, however, comment onthe more general case below. We may describe the thinlayer at the interface by a $-function in the free energy, ifd is much smaller than the coherence lengths along the x-axis (Fig. 1). Thus, the interface part of the free energyhas the form,

F i ¼Z

d3r$ðxÞ�j�j2 ð2Þ

where � ¼ d!�ðT � Tc�) with Tc� > TcS. This approxima-tion is suHcient to discuss most important qualitativefeatures of the 3 K-phase.

2.2 Instability conditions of a planar interfaceWe will 6rst investigate the conditions of the nuclea-

tion of superconductivity at the planar interface. Weconsider a temperature range TcS < T < Tc� where thislocal onset of superconductivity is supposed to occur.The instability condition is obtained by searching for thesolution of the linearized GL equations which are givenby

K1�@2x�x � a��x ¼ 0; ð3Þ

K2�@2x�y � a��y ¼ 0; ð4Þ

for x 6¼ 0 (� ¼ S and R for x > 0 and x < 0, respec-tively). At x ¼ 0 the solutions have to be continuous andsatisfy the following boundary conditions,

K1S@x�xjx¼0þ �K1R@x�xjx¼0� � ��xjx¼0 ¼ 0; ð5Þ

K2S@x�yjx¼0þ �K2R@x�yjx¼0� � ��yjx¼0 ¼ 0: ð6Þ

For temperatures above TcS the order parameter islargest at the interface and decays exponentially on bothsides. For the �x-component the solution is

�xðxÞ ¼ �x0e�x='1S for x > 0 ð7Þ

�xðxÞ ¼ �x0ex='1R for x < 0 ð8Þwith '21� ¼ K1�=a�. The analogous solution exists for �ywith '22� ¼ K2�=a�. Then eq. (5) and (6) lead to theinstability equations,ffiffiffiffiffiffiffiffiffiffiffiffiffi

K1SaSp

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiK1RaR

pþ � ¼ 0 ð9Þffiffiffiffiffiffiffiffiffiffiffiffiffi

K2SaSp


pþ � ¼ 0; ð10Þ

for �x and �y, respectively. Because K1� > K2� > 0('21� > '

22�) the nucleation will occur for the y-compo-

nent. To obtain an instability temperature T � above TcSrequires that

'2RðTcSÞaRðTcSÞ < d!�ðTc� � TcSÞ: ð11Þ

This simple relation gives a good insight on the basicproblem of the nucleation of local superconductivity atthe interface. Obviously, the larger d and the higher Tc�,the higher is T �. However, the presence of the normal-metal Ru tends to suppresses superconductivity, inparticular if the coherence length '2R increases. Notethat if both sides of the interface were Sr2RuO4, wealways would 6nd local superconductivity at a tempera-ture higher than TcS. Furthermore, the suppression bythe Ru-metal is weaker, if the interface were lesstransparent leading to a discontinuity of the orderparameter. Reduced transparency of the interface wouldnot change the dominance of the �y-component at thisinterface. The reason is that the order parametercomponent n � � is suppressed by a re>ecting interface.In any case the degeneracy of �x and �y of the bulk regionis lifted at the interface, since it corresponds to a regionof e.ectively reduced symmetry.12) The dominance ofthe �y component is also plausible from a microscopicpoint of view. The opening of a gap for momentumdirections along the interface leads to the gain ofcondensation energy for quasiparticles with momentaparallel to the interface. These are the quasiparticlesspending the longest time in the interface region.

We assume from now on that the condition eq. (11) issatis6ed and the resulting transition temperature isT � � 3 K corresponding to the 3 K-phase. Note that thetransition temperature does not depend on the orienta-tion of the normal vector as long as it lies in the basalplane. The nucleating order parameter � is perpendi-cular to the normal vector, i.e., n� �. The situationdoes not change much if a small z-axis components of thenormal vector is introduced and the properties of thelocal superconducting state is determined by its inplanecomponents.

Because the 3 K-phase does not break time reversalsymmetry in contrast to the bulk phase below 1.5 K, afurther second order phase transition (violating timereversal symmetry) has to occur at a temperature T �

2

which, obviously, cannot be lower than the bulk Tc. Thepresence of this additional phase transition is in contrastto conventional inhomogeneous superconductors, wheresuperconductivity nucleates locally on small ‘‘islands’’which with lowering temperature increase their overlapand, 6nally, form a bulk superconducting phase withoutfurther symmetry breaking (apart from the percolation

2001) 3 K-phase in Sr2RuO4 2411

transition). The temperature T �2 depends on the coupling

between the order parameter components described bythe fourth order terms in the free energy. The twopairing components may ‘‘attract’’ or ‘‘repel’’ each other,determined by the sign of the parameter ~bb ¼ 2b1 � b2 þb3 (~bb > 0 repulsion and ~bb < 0 attraction). Note that thestability condition for the chiral p-wave state requiresthat �2b1 < ~bb < 2b1, where b1 > 0. The weak couplingapproach leads to ~bb > 0 (b2=b1 ¼ �b3=b1 ¼ 2=3 leads to~bb=b1 ¼ 2=3 with b1 > 0), which suppresses the appear-ance of the second component. Hence T �

2 may be ratherclose to Tc and diHcult to distinguish experimentallyfrom the bulk transition.

The discussion of the second transition and the form ofthe order parameter in the 3 K-phase is complex as itinvolves also the vector potential due to the presence ofspontaneous currents in the time reversal symmetry-breaking phase. Thus we turn to the numerical solutionof the complete set of GL equations including thecomplete set of GL equations for the order parameterand the vector potential. We use the parameters !S ¼!R ¼ !� and Tc� ¼ 2:8TcS. Except for K5� we choose allcoeHcients Kj� to be the same in Ru as well as inSr2RuO4: K1�=!�TcS ¼ 3Ki�=!�TcS ¼ 1 ði ¼ 2; 3; 4Þ,which corresponds to the basic zero-temperature coher-ence length '0 � 1. The coeHcient for the z-axis gradientis small in Sr2RuO4, K5S=!�TcS ¼ 10�2, while we take,for simplicity, K5R ¼ K2R, since Ru is more isotropic.The fourth order coeHcient are also chosen independentof x, 2b1� ¼ 3b2� ¼ �3b3� ¼ 0:4!�. We choose the inter-face layer thickness d ¼ 1, larger than it probably is inreality, for illustrative purpose and numerical stability.With these parameters we obtain a nucleation tempera-ture of T � � 2TcS and T �

2 � 1:33TcS which is rather highdue to the large value of d . At this transition the �xcomponent becomes 6nite in addition to �y and has therelative phase of � =2. In Fig. 2 we show the shape ofthe two order parameter components in the threedi.erent temperature regimes. Here (a) represents thegenuine 3 K-phase with vanishing �x-component (timereversal symmetry conserving phase), (b) is the inter-mediate time reversal symmetry breaking state withboth components 6nite. Finally, (c) is the bulk super-conducting phase of Sr2RuO4. Note that both cases (b)and (c) possess a complex order parameter texture at theinterface which will be important for the magneticproperties discussed in the next section.

2.3 Capillary e*ectThe interface instability for the superconducting state

bears some resemblance with the wetting of a surface.13)

Similar to the wetting phenomena of liquids we 6nd herecapillary e.ects. Modulations of interfaces on lengthscales comparable to the coherence length can enhancethe nucleation temperature. While this capillary e.ect israther simple for conventional superconductors, thereare complications in the case of an unconventionalsuperconductor. In particular, there can be frustratione.ects due to the internal structure of the pair wavefunction, as we will see in the next section. Here wewould like to consider 6rst the rather simple situation of

an interface that is not >at but has a weak modulation.The enhancement of the transition temperature on a

spatially modulated interface can be most simplyinterpreted by the e.ective increase of the surface area.This corresponds also to an enhanced mutual overlap ofthe order parameters nucleating at di.erent points of theinterface or on di.erent inclusions. Let us assume thatthe interface is only slightly modulated, described byx0ðyÞ ¼ l sinð2 y=L0Þ where d� l� L0 � 'i�. Then, bythe most simple variational approximation we replacethe interface term by

F i ¼Z

d3rj�j2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

2 l

L0 cos2 y

L0

� �� 2s

$ðx� x0ðyÞÞ:

ð12Þ

The important modi6cation appears via the new inter-face metric, which accounts for the fact that the interfaceis wider or ‘‘denser’’. This leads to an enhancedinstability temperature, even if we approximate thespatial dependence of the order parameter by a form likein eq. (7, 8). Using this variational approach theinstability equation for �y changes toffiffiffiffiffiffiffiffiffiffiffiffi

K2saSp


pþ �R ¼ 0; ð13Þ

where the factor R is

R ¼ 1

L0

Z L0

0

dy

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

2 l

L0 cos2 y

L0

� �� 2s

� 1þ l

L0

� �2

:

ð14Þ

Obviously, R is always larger than 1 and leads to ane.ective enhancement of �. We have ignored in ourvariational approach the spatial dependence of the order

-5 0 5 10x

0.0

0.2

0.4

0.60.0

0.2

0.4

0.60.0

0.2

0.4R S a)

b)

c)

Fig. 2. Spatial dependence for the two order parameter components

(j�xj: dashed line and j�yj solid line): (a) 3 K-phase atT ¼ 1:75TcS > T

�2 ; (b) intermediate time reversal symmetry break-

ing phase at T ¼ 1:25TcS < T�2 with both components which have a

relative phase � =2 (! two-fold degeneracy); (c) bulk super-

conducting phase d ¼ zzðkx � ikyÞ and a texture at the interface atT ¼ 0:95TcS.


parameter along the interface. Including this aspect leadto even further enhancement of the nucleation tempera-ture.

Another aspect of the capillary e.ect is the mutualin>uence of interfaces which we would like to consider onthe example of two parallel interfaces. To be speci6c weassume a thin Ru-metal slab of thickness L sandwichedbetween Sr2RuO4, again with normal vector parallel tothe x-axis. Both interfaces have the same properties.Then it is easy to derive the instability equations takingthe boundary conditions into account. There are twocombinations of the nucleating order parameters on thetwo interfaces: a ‘‘bonding’’ and ‘‘antibonding’’ con6g-uration which is even or odd, respectively, underre>ection at the center of the slab. For the instabilityequations of the �y-component we obtain,ffiffiffiffiffiffiffiffiffiffiffiffi

K2saSp

þ tanhL

2'2R

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2RaR

pþ � ¼ 0 ð15Þ

for the bonding andffiffiffiffiffiffiffiffiffiffiffiffiK2saS

pþ coth

L

2'2R

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2RaR

pþ � ¼ 0 ð16Þ

for the antibonding combination. Obviously, we recoverin both cases the original instability equation eq. (10), ifwe separate the two interfaces far apart, L� '2R. If,however, L � '2R then the second (positive) term isdiminished (enhanced) and a higher (lower) transitiontemperature T � results in the case of bonding (antibond-ing). Naturally, the same kind of capillary e.ect occursalso in the inverse situation where a Sr2RuO4 slab issurrounded by Ru. The bonding combination of the twointerface states just corresponds to the adjustment of thephases of two superconducting islands. The antibondingcombination is equivalent to a phase di.erence of andis the energetically least favorable case. We will see inthe next section that this aspect is important forfrustration e.ects of the 3 K phase.

We can conclude that the onset of the 3 K phase israther inhomogeneous on the interfaces as well, becausethe capillary e.ects have a strong in>uence on thenucleation of the superconducting order parameter.

x3. Physical Consequences

3.1 Spontaneous interface magnetismWhile the onset of superconductivity is visible in the

reduction of electrical resistance, the signi6cant featuresof second transition at T �

2 are less easy to detect. Bulksuperconducting double transitions are often observedthrough speci6c heat anomalies. Here the only signi6cantanomaly, however, is seen at the onset of bulk super-conductivity at Tc ¼ 1:5 K. Magnetic properties may, onthe other hand, allow us to observe the onset of thesuperconducting phase at T �

2 that violates time reversalsymmetry. The solution of the full GL equations showthat the below T �

2 spontaneous supercurrents occurparallel to the interface. In Fig. 3 we show the magnetic6eld and current distribution at the interface for the casecorresponding to Fig. 2(b). There are two currentdensities along the interface >owing in opposite directionas a result of the texture of the two order parameter

components at the interface. Screening currents are weakdue to absence of superconductivity in the bulk. Thus,the 6eld resulting from the currents is strongly peaked onthe interface and can even generate an overall 6nitemagnetic >ux. Since the >ux is 6nite, it could beobserved, in principle, by high-resolution magneticmicroscopes, such as scanning Hall probes or a SQUIDmicroscopes. Another sensitive probe for local magnetic6eld distributions is zero-6eld muon spin relaxation,which is expected to show an increase of the depolariza-tion rate below T �

2 . However, if T �2 is very close to Tc (as

is very likely the case) then the signal will be obscured bythe bulk superconducting state which has previouslybeen observed by �SR.5) The muon spin relaxationexperiment would in any case give a distinction betweenmagnetic properties of the 3 K-phase at T � and the bulkphase.

3.2 Frustration e*ects in the inhomogeneous stateWe have seen in the previous section on the capillary

e.ect that the lowest energy con6guration of orderparameters nucleated on di.erent interfaces naturallycorresponds to equal order parameter phases. Let us nowsee how the superconducting state would arrange onseveral inclusions close enough to each other that theirorder parameters substantially overlap. If the orderparameter were conventional, it would be easy to adjustall the order parameter phases on the di.erent inclusionto be equal. This is not the case, if the order parameterhas p-wave symmetry as is illustrated in Fig. 4. Thethree Ru-inclusions depicted in Fig. 4 carry a super-conducting state which on every point of the interfacecorrespond to a p-wave state with the momentumdirection aligned with the interface, i.e. the gap nodelies always parallel to the normal vector. The interfaceregions on di.erent inclusions close to each other behavelike junctions or weak links connecting a network of thesuperconducting islands.

The direction of the positive lobe of the pair wavefunction is indicated by the arrows in Fig. 4 where we

-5 0 5 10x

-0.006

-0.003

0.000

0.003

0.006

R S

Fig. 3. Spontaneous magnetic 6eld Bz (solid line) and supercurrent jy(dashed line) distribution for the state given in Fig. 2(b) atT ¼ 1:25TcS.


assume that the order parameter phase is constant oneach inclusion. Parallel (antiparallel) arrows on neigh-boring interfaces correspond to bonding (antibonding)con6gurations of the overlapping order parameters, orequal phase ( -shifted phases). It is obvious that in thearrangement of the order parameter on each inclusionintroduces a -phase shift between its weak links.Although we may change the phases on each island, itis impossible to adjust them so that all links have a zerophase di.erence for loops consisting of an odd number ofinclusions (Fig. 4 shows this for the case of threeinclusions). The situation is in many respects similar tothe case described for granular d-wave superconduc-tors.28,29,31) We may identify loops which contain phase shifts and are consequently frustrated, since theycannot adjust the order parameter phase to minimize theenergy of all weak links simultaneously. The frustrationcan be released by introducing phase gradients (super-currents) which are energetically favorable, if the weaklinks grow strong enough. This would yield orbitalmagnetic moments. This e.ect results from the competi-tion between weak link energy (determined by orderparameter overlap) and magnetic 6eld energy (deter-mined by geometry) as in usual frustrated -loops.However, the frustration also is reduced with the timereversal symmetry breaking transition at T ¼ T �

2 . Bothways have similar impact by diminishing frustration andlead to spontaneous currents. In the network of super-conducting islands the transition to the time reversalsymmetry breaking state is inhomogeneous.

The frustrated 3 K-phase with spontaneous currentscan exhibit enhanced absorption in the ac-susceptibility.Dissipative processes, such as phase slips, are associatedwith the hysteretic reversal of the spontaneous currents.

Thus the absorption would be sensitive to the applica-tion of a small static external 6elds which would bias thespontaneous currents (a similar situation was observedin granular high-temperature superconductors28,29,31))The presence of spontaneous orbital magnetic momentsin the inhomogeneous time reversal symmetry violatingphase above TcS can cause a characteristic non-linearmagnetic response in small external 6elds, similar to theparamagnetic Meissner e.ect or Wohlleben e.ect inhigh-temperature superconductors.28,29,31) Under 6eld-cooling conditions the magnetic moments would begenerated in a polarized way yielding a paramagneticcontribution. Probably, the signal would be considerablyweaker than in high-temperature superconductors. Theinhomogeneous 3 K-phase is, however, more suitable tomeasure the spontaneous magnetism than the bulksuperconducting phase, where the macroscopically visi-ble magnetism is only resulting from sample surfacecurrents, while it is screened or compensated in theinterior of the systems.

3.3 Quasiparticle spectrumThe presence of superconductivity modi6es the

quasiparticle spectrum in small enclosed normal-metalregions through the formation of so-called Andreevbound states. Andreev bound states are standing wavesof an electron-hole superposition, e.g. in a normal metalregion enclosed by a superconductor. In a quasiclassicalpicture the standing wave corresponds to an electron andhole travel on the same classical (ballistic) trajectory,but in opposite direction and are subject to Andreevre>ection upon impact in the superconductor at bothends of the trajectory. The energy of such a standingwave depends on the phase of the gap functions at theboundaries of the classical trajectories and can beestimated by Bohr-Sommerfeld quantization. Energiesmuch smaller than the gap of the superconductor areapproximately given by,

Eðn; -Þ �vF

2L½ð2nþ 1Þ þ -� ð17Þ

where vF is the Fermi velocity, L the length of thetrajectory, n an integer and, most important, - is thedi.erence between the phases of the gap functions atboth ends of the ballistic trajectory (for simplicity weignore here the e.ect of impurity scattering which canalso be the origin low-energy states). For unconventionalsuperconductors the phase di.erence - depends also onthe direction of the electron (hole) momentum, becausethe gap function, in general, has an anisotropic phasestructure, i.e. a di.erent phase for di.erent directions ofmomenta on the Fermi surface. Thus, even in case of ahomogeneous order parameter, the phase - appearing inthe Andreev scattering process would be non-trivial, butrevealing the internal phase structure of the pairingstate. For real order parameters we 6nd - ¼ � which ismost important, because it leads to a zero-energy stateindependent of vF and L. This phase di.erence appears,if the gap function has positive and negative signdepending on the momentum direction on the Fermisurface. From the quasiclassical point of view underlying

Fig. 4. Con6guration of the p-wave pair wave function on Ru-

inclusions (shadowed regions) in a cross section parallel to the x{y-plane. The arrow indicates the direction of positive wave function

(see box). The dashed circles show the regions forming weak links

between inclusions. The fact that the arrows of di.erent inclusions in

the weak link region are antiparallel corresponds to a phasedi.erence of . In the given con6guration there is no way to adjust

the phases of all three inclusions so that all weak links have minimal

energy, i.e. a vanishing phase di.erence. Therefore this system is

frustrated.


eq. (17) the geometry of the classical trajectory decidesabout the phase - ¼ 0 or . Since a large fraction oftrajectories can have - ¼ and all of them yield a zero-energy state, the zero-energy states (- ¼ ) lead to anenhanced density of quasiparticle states at the Fermisurface. The presence of this kind of zero-energy boundstate has been intensively investigated for surfaces of d-wave supercondcutors.14,15)

In our case a p-wave pairing state nucleated at theinterface generates a gap function which has regions ofdi.erent sign on the Fermi surface. Zero-energy Andreevbound states are likely to occur within the Ru-inclusionsas well as in Sr2RuO4 between Ru-inclusions that aresuHciently close to each other. We encounter here acomplex form of an (imperfect) Andreev billiard. TheAndreev re>ections are not perfect. Due the 6nite widthof the superconducting regions, the zero-energy levelacquires some width. Nevertheless, there is a stronglyenhanced density of states at zero energy. Once timereversal symmetry is broken the phase di.erences - fortrajectories deviate from 0 and . The originallyenhanced density of states at zero-energy spreads overa larger energy region. (Note that the reduction of thedensity of states at the Fermi energy represents also adriving force for the time reversal symmetry breakingtransition, as was suggested also for the surface states ofa d-wave superconductor.16,17))

One experimental indication for the enhanced densityof states zero-energy states is the observation of so-calledzero-bias anomaly, i.e. an increased quasiparticle tunnel-ing conductance at zero voltage. Recent measurementsof quasiparticle tunneling conductance in c-axis facingbreak junctions of Sr2RuO4 with Ru-inclusions reportthe observation of zero-bias anomalies.18) In the 3 K-phase a zero-bias anomaly in current-voltage character-istics develops gradually with decreasing temperature.This conductance peak deforms into a pronounced bellshape combined with a residual zero-bias anomaly, belowthe bulk superconducting transition. Assuming thatthese tunneling features re>ect the quasiparticle densityof states connected with the Andreev billiard, we caninterpret them within the scenario of the nucleated p-wave state which turns into a time reversal symmetrybreaking state at lower temperature. We would like toemphasize the fact that break junctions in sampleswithout Ru-inclusions did not show any similar featuresabove and below the onset of bulk superconductivity at1.5 K. Note also that an analogous phenomenon, thedeformation of the zero-bias anomaly, has been observedfor tunneling conductance into the [110]-surface ofYBa2Cu3O7.

19) Also in that case broken time reversalsymmetry is most likely cause.16,17)

While extensive point-contact spectroscopy has al-ready been performed observing surface states ofSr2RuO4,

20) it is also desirable to have more experi-mental investigation of the quasiparticle spectrum in andaround the Ru-inclusions via tunneling and relatedprobes.21{23) In particular, using a contact to a singleinclusion and measuring the quasiparticle current-voltage characteristics through the Ru-metal inclusioninto Sr2RuO4 could provide further valuable information

on the 3 K as well as bulk superconducting phase. Adetailed analysis of the phenomena discussed in thissection will be given elsewhere.

x4. Upper Critical Field

The 3 K phase has an upper critical 6eld whichexceeds the bulk critical 6eld considerably and shows aweaker anisotropy of in-plane versus out-of-plane critical6eld. The fact that superconductivity is con6ned in asmall layer at the interface naturally leads to anreduction of orbital depairing, if the external magnetic6eld is applied parallel to the interface. This type ofe.ect is known for superconductivity at other planardefects such as twin boundaries.24,25) In a magnetic 6eldsuperconductivity nucleates 6rst at interfaces which areparallel to the 6eld. We will analyze this e.ect for the[100]-interface with magnetic 6elds B ? x-axis. Unfor-tunately, a complete analytical treatment of thisproblem is not possible even for the GL formulation.Therefore we will restrict our analytic discussion to theregion of very small 6elds at the onset of the 3 K-phaseand use a variational approach. This allows us toillustrate a few basic features of the upper critical 6eld.Then we will consider the behavior of Hc2 by numericalmeans for the model introduced above.

For temperatures close to T � the critical 6eld is smalland we may consider it as a perturbation.25) Thus, wewill use the exponential form of the order parameterappearing in zero-6eld at the onset of the 3 K-phase:�x� ¼ �x0 expð�jxj='1�Þ and �y� ¼ �y0 expð�jxj='2�Þ with'i� ¼ Ki�=a�. The magnetic 6eld lies in the plane of theinterface B ¼ Hð0; sin /; cos /Þ with the vector potentialA ¼ Hðx� x0Þð0;� cos /; sin /Þ. While in zero-6eld onlythe �y-component is nucleated at T �, in general bothcomponents can appear in a 6nite magnetic 6eld. We usenow the given form of the order parameter and calculatethe free energy in a 6nite 6eld by integrating over thespatial coordinates. In this 6rst-order perturbative formthe free energy per unit area of the interface is then up tosecond order in the the order parameter,

F var ¼X�¼S;R

�j�x0j2 a�'1� þ

�

2þ

2H2

4'31�fx�ð/Þ

þ j�y0j2 a�'2� þ�

2þ

2H2

4'32�fy�ð/Þ

þ ið��x0�y0 � �x0��y0ÞrHK2�'32� cos /

�ð18Þ

with ¼ 2e=hc and r ¼ ðffiffiffi3

p� 1Þ=ð

ffiffiffi3

pþ 1Þ2 where we

used the relation '1� ¼ffiffiffi3

p'2�, keeping the weak-coupling

relations among di.erent Ki� for cylindrical symmetry.To simplify the free energy (18) we have 6xed x0 to zero,since it plays a minor role for our discussion. Theanisotropy parameters are

fx�ð/Þ ¼ K2� cos2 /þK5� sin

2 /;

fy�ð/Þ ¼ K1� cos2 /þK5� sin

2 /:ð19Þ

First we consider the case of the in-plane 6eld(/ ¼ =2). Here only the �y-component appears, whichcorresponds to the polar state with its nodes perpendi-


cular to the 6eld, which also appears in the bulksuperconducting phase for in-plane 6elds. The zero ofthe corresponding coeHcient in F v de6nes the instabil-ity, which leads to

Hc2 ¼%0

�d

2a�yðT ÞK5R'

32R

" #12

ð20Þ

where we used that K5S � Ki�;K5R (i 6¼ 5). In this caseincluding a 6nite value of x0 we 6nd that x0 ¼ �'2R=2maximizes Hc2 (we omit here an explicit demonstration).Note that we can write

P�ð'2�a�ðT Þ þ �ðT Þ=2Þ ¼ a�yðT Þ

which is proportional to ðT � T �Þ close to T �. Conse-quently, we observe a square-root dependenceHc2 / jT � � T j

12 .24,25) For curved interfaces the regions

tangential to the 6eld tend to allow the nucleation athigher temperature. Simultaneously, we have, however,to include capillary e.ects. Thus, the observation of apure square-root temperature dependence would bemasked in reality.

For the 6eld parallel to the z-axis (/ ¼ 0) the �x-component is involved too, despite the lower criticaltemperature in zero 6eld. The optimal value of x0 is veryclose to zero so that we 6x x0 ¼ 0. With this simpli6ca-tion the instability equations involving the coupling ofboth components have the form,

a�xðT Þ þH

2

� �2

&x

( )�x0 � iH&0�y0 ¼ 0

iH&0�x0 þ a�yðT Þ þH

2

� �2

&y

( )�y0 ¼ 0

ð21Þ

where &x ¼P

� K2�'31�, &y ¼

P� K1�'

32� and &0 ¼

rP

� K2�'2�. Moreover, a�xðT Þ ¼P�ð'1�a�ðT Þ þ �ðT Þ=

2Þ is the e.ective second order coeHcient for the �x-component and changes sign at a temperature Tcx which

lies between T �2 and T �. The critical 6eld is obtained by

searching the non-trivial solution of this equationsystem,

Hc2ðT Þ �%0

�a�yðT Þ&y

a�xðT Þa�xðT Þ � 4&02=&y

" #12

ð22Þ

where we used ja�yðT Þj � a�xðT Þ for T ! T �. Thus we 6ndagain a basic square root behavior of Hc2ðT Þ as in thecase of inplane 6elds. However, in addition we see thatthe coupling of the two components yields an enhance-ment factor to Hc2 which becomes stronger as thetemperature is lowered. (Note, however, that the presentform is only valid for 0 < a�xðT Þ � 4&02=&y and in anycase our variational approach is applicable only in arestricted region close to T �.) This enhancement isimportant, since it modi6es qualitatively the overallform of the temperature dependence of Hc2. In parti-cular, it can lead to an upwards curvature following theinitial square root behavior. Before considering thispoint numerically let us, however, address the issue ofHc2 anisotropy.

For the bulk superconducting phase of Sr2RuO4 thecritical 6elds are given by

H?c2 ¼

�%0aSðT Þ4 K2S

and Hkc2 ¼

�%0aSðT Þ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2SK5S

p ð23Þ

with T < TcS and ignoring anisotropy of Hc2 for di.erentinplane orientations, which is small.27) This leads to ananisotropy factor of order H

kc2=H

?c2 � 2

ffiffiffiffiffiffiffiffiK2S

p=ffiffiffiffiffiffiffiffiK5S

p

whose experimental value is about 12 for T close to TcSin experiments (K2S � K5S � 0:03K2S).

6) On the otherhand, the anisotropy for the 3 K-phase may be lessanisotropic with a factor 4{5,6,7) since

Hkc2

H?c2

�2ð'32SK1S þ '32RK1RÞ

'32RK5R

!12

ð24Þ

gives a smaller ratio due to the fact that Ru-metal isbasically isotropic with K5R � K2� and also the coher-ence lengths at T ¼ T � are of the same order in Ru andSr2RuO4. The reduction of anisotropy originates fromthe isotropy of the Ru-metal which leads to a strongercoupling of the order parameter to the inplane 6eld thanfor Sr2RuO4.

The numerical evaluation of the upper critical 6eld forthe model used in §2.2 illustrates the temperaturedependence on qualitative level. The result are shownin Fig. 4 for the assumption that K5S ¼ 0:03K2S andK5R ¼ K2S. The circles and diamonds are the numericalresults for the critical 6eld inplane and along thez-axis,respectively. We observe indeed a weaker anisotropy forthe critical 6eld of the 3 K-phase than for the bulk phasealso indicate in Fig. 4. The initial temperature depen-dence has square root dependence as obtained analyti-cally. While the inplane critical 6eld has a downwardcurvature in the plotted temperature range, the z-axis6eld is nearly linear. The analytical expression in eq. (20)indicates that even the upward curvature would bepossible for the z-axis critical 6eld. The observation ofsuch an upturn would be a strong support for our

0.5 1.0 1.5 2.0T/Tc

0.0

1.0

2.0

3.0

4.0

Hc2

[T

]

Fig. 5. Upper critical 6elds for the 3 K and bulk phase: Thenumerical solution of the GL equations give the upper critical 6elds

of the 3 K-phase: inplane (circles) and z-axis (diamonds). The

corresponding bulk critical 6elds are given by doted lines. Note, theweaker anisotropy of the critical 6eld of the 3 K-phase. The

numerical data compare well with the experimental data on a

qualitative as well as quantitative level.6,7)


scenario. Our result reproduces the experimental data(available at present) qualitatively and to some extentquantitatively, with the choice of parameters givingT � � 2Tc and the proper anisotropy of the bulk critical6elds.6,7) The peculiar temperature dependence of the z-axis critical 6eld of the 3 K-phase is also similar to thatfound in experiment.6,7)

x5. The s-Wave Order Parameter

Ru metal is a conventional superconductor with atransition temperature of about 0.5 K. Thus, we mayinclude an s-wave order parameter into our theory. Thecoupling between the spin-singlet order parameter �s andthe spin-triplet order parameter � requires that we takespin-orbit coupling into account. The di.erence betweenspin-orbit coupling in Ru and Sr2RuO4 yields spin >iptunneling at the interface which introduces a couplingbetween �s and �.26,27) The coupling term has the form

F s.o. ¼ tZinterface

dS½��sðn� �þ c.c.Þ� ð25Þ

where �s is the s-wave order parameter on the Ru-sideand � the two-component p-wave order parameter onSr2RuO4-side, n is the interface normal vector and t is acoupling constant. Obviously, for n ¼ ð100Þ only the �y-component couples.

Further there is a coupling between the two orderparameters also away from the interface which is againdue to spin-orbit coupling. The corresponding term inthe free energy, derived based on symmetry arguments,has the form,

F sp ¼Z

dV ~aa½ðDx�y �Dy�xÞ��s þ c.c.� ð26Þ

where ~aa is again a coupling constant di.erent on the twosides. This term is only active, if there is a spatialvariation of the order parameter or a magnetic 6eld ispresent. We 6nd that the spatial variation of the orderparameter along the x-axis would couple exclusively the�y-component to �s.

This structure of order parameter coupling leads to asupport for the nucleation of the �y-component on thediscussed [100]-interface due to the mixing with the s-wave order parameter intrinsic to Ru. On a qualitativelevel, however, the inclusion of an s-wave componentwould not modify the properties of the 3 K-phase on anessential way.

x6. Conclusion

In this article we have interpreted the 3 K-phase ofSr2RuO4 with Ru-metal inclusions as an inhomogeneoussuperconducting state located at the interface betweenthe two materials. This leads to a superconducting phasedi.erent qualitatively from the bulk phase appearingbelow 1.5 K. The 3 K-phase conserves time reversalsymmetry. This implies an additional phase transitionwhere time reversal symmetry is spontaneously broken.This second transition occurs above the onset of bulksuperconductivity. Irregular shapes and distribution ofthe Ru inclusions and the capillary e.ects would preventa very sharp transition for the onset of the 3 K-phase as

well as for the second transition.The origin of multiple superconducting transitions is

analogous to that of the splitting of the superconductingphase transition of degenerate order parameters byapplying symmetry-lowering uniaxial stress.30,12) Indeedthe interface represents a region of the system where thesymmetry is e.ectively lower, lifting the degeneracybetween the two order parameters �x and �y. Thereforethe experimental proof of the conservation of timereversal symmetry in the 3 K-phase would be a clearcon6rmation of a having a two-component order para-meter.

We have argued that the phase structure of the pairwave function can lead to frustration e.ects in thecoupling of the superconducting order parameters ondi.erent Ru-inclusions or di.erent regions on the sameinclusion. This is closely related with the fact that thistype of order parameter can lead to low- or zero-energyAndreev bound states, since the con6guration of inclu-sions forms a complex Andreev billiard system. Thereare various experimental consequences due to theseproperties, some of which have been partially alreadyinvestigated experimentally. (1) The Ru-metal inclu-sions provide an interesting way to tunnel into Sr2RuO4.Contacts via Ru inclusions could reveal more about thestructure of the quasiparticle spectrum.18) (2) Thesuperconducting interface states of di.erent neighboringRu-inclusions overlap and form a complex network. Thestudy of the critical current as a function of temperaturemay give another tool to investigate the unconventionalnature of this state. In particular, the frustration e.ectsmentioned above and the spontaneous currents whichoccur in the time reversal symmetry breaking phase canyield characteristic anomalous behavior. (3) The low-6eld magnetic response may show strong non-linearbehavior and cooling-history dependence in the timereversal symmetry breaking state for temperaturesabove TcS, a phenomena which could be rather similarto the paramagnetic Meissner (or Wohlleben) e.ect ingranular high-temperature superconductors.28,29,31) (4)The study of microwave absorption in a small staticmagnetic 6eld may show a non-monotonic 6eld depen-dence as in some granular high-temperature super-conductors.28,29,31)

Finally, the upper critical 6eld provides an importantway to probe the nature of the 3 K phase. As shown insection 4, the temperature dependence of Hc2 ischaracterized by a square root behavior close to theonset T � (Hc2ðT Þ / jT � T �j1=2). For 6elds parallel to thez-axis of Sr2RuO4 a tendency to change the sign ofcurvature (downwards to upwards) is suggested, which,if con6rmed experimentally, would provide additionalstrong evidence for the two-component order parameterpicture of the unconventional superconducting phase ofSr2RuO4.

These is only a selection of possible unusual propertiesof the 3 K-phase. It is obvious that this phase and itsproperties provide a very good tool to investigate thesuperconductivity in Sr2RuO4 from a new point of view.

We would like to thank Y. Maeno, Z. Q. Mao, T.Akima, M. Wada, H. Yaguchi, Y. Liu, A. Furusaki and


T. Morinari for many stimulating discussions. This workwas supported by a Grant-in-Aid of the JapaneseMinistry of Education, Culture, Sports, Science andTechnology. H.M. is also grateful for the support by along-term fellowship of Japan Society for the Promotionof Science.

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Phenomenological theory of the 3 Kelvin phase in Sr2RuO4

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