The Intriguing Superconductivity of Strontium RuthenateYoshiteru Maeno, T. Maurice Rice, and Manfred Sigrist Citation: Phys. Today 54(1), 42 (2001); doi: 10.1063/1.1349611 View online: http://dx.doi.org/10.1063/1.1349611 View Table of Contents: http://www.physicstoday.org/resource/1/PHTOAD/v54/i1 Published by the American Institute of Physics. Additional resources for Physics TodayHomepage: http://www.physicstoday.org/ Information: http://www.physicstoday.org/about_us Daily Edition: http://www.physicstoday.org/daily_edition

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The superconductingstate of strontium

ruthenate (Sr2RuO4) wasdiscovered in 1994 byYoshiteru Maeno and hiscollaborators after theyhad succeeded in makinghigh-quality samples ofthe material.1 At about 1.5K, the value of Tc is unre-markable, but interest wasimmediately aroused by apossible relationship to the high-Tc cuprate superconduc-tors. Both kinds of material are highly two-dimensional(2D), and, as figure 1 shows, the crystal structure ofSr2RuO4 is almost identical to that of the (La, Sr)2CuO4superconductors. Furthermore, both materials are oxideswith conduction occurring in partially filled d-bands (ofthe strontium or copper ions) that are strongly hybridizedwith the oxygen ions’ p orbitals.

We now know that these two superconductors arequite different. The high-Tc cuprate superconductors con-tinue to form a unique class of materials. Extensivesearches have failed to turn up any analogous materialsin the transition metal oxides or elsewhere. Sr2RuO4, onthe other hand, seems much closer to another famousmaterial: superfluid helium-3.

Although the superfluidity of 4He was discovered inthe 1930s, it was not until the 1970s that a superfluidtransition was found in the fermionic isotope 3He. Twofeatures of superfluid 3He surprised investigators at first:1) the p-wave internal symmetry of the Cooper pairs offermions that make up the condensate and 2) the rela-tively high transition temperature of about 1 mK. Theo-rists soon realized that the strong enhancement of thespin fluctuations in space and time plays an importantrole in both the choice of the p-wave state and in theenhanced value of Tc. Immediately, the question arose ofwhether an analogous state could be found in metals.

Conventional metallic superconductors are describedwell by the Bardeen-Cooper-Schrieffer (BCS) theory inwhich the electrons’ Coulomb repulsion is overcome by theattraction induced by the exchange of phonons betweenelectrons close to the Fermi surface. To maximize thisattraction, the Cooper pairs in BCS theory appear in thesimplest s-wave channel. However, a few years after BCS

was formulated, Walter Kohnand Quin Luttinger examinedthe possibility of generating aweak residual attraction outof the Coulomb repulsionbetween electrons. Theyfound that this attractioncould occur in principle, butonly in a higher angularmomentum channel in whichthe electrons in the Cooperpairs are prevented from

close encounters by the centrifugal barrier.The value of Tc estimated by Kohn and Luttinger was

extremely small. However, the discovery of superfluidity in3He led to the hope that reasonable values of Tc would befound in those metals whose enhanced paramagnetismleads to strong spin fluctuations. Unfortunately, nothingcame from these searches.

The first breakthrough in the quest for unconvention-al metallic superconductors came around 1980 with the dis-covery of superconductivity in the special class of heavyfermion metals. These are intermetallic compounds with arare earth or actinide constituent. Under certain circum-stances, the inner shell electrons in 4f or 5f states can beinduced to participate weakly in the Fermi fluid of theouter conduction electrons. The result is a Fermi fluid witha very low characteristic temperature and strong residualinteractions. Some heavy fermion metals have been foundto exhibit superconductivity with a value of Tc that, thoughlow in absolute terms (Tc � 1 K), is high relative to the tem-perature that characterizes the Fermi distribution.

Much progress has been made in heavy fermion met-als in recent years, but their big drawback from a theo-rist’s point of view is the complexity inherent in describ-ing the physics of rare earth and actinide ions (seePHYSICS TODAY, February 1995, page 32). Even today,unambiguous determination of their unconventional sym-metry has not proved possible.

Strongly correlated 2D Fermi liquidThe unexpected discovery of high-Tc superconductivity inthe cuprates led to extensive searches for superconductiv-ity in other transition metal oxides. The only success camein Sr2RuO4, but it was soon apparent that the differencesbetween Sr2RuO4 and the cuprates were much larger thanthe general similarities.1 In Sr2RuO4, superconductivityappears only at low temperatures, in samples with verylow residual resistivity, and out of a normal state that is awell-formed Landau–Fermi liquid. These conditions con-trast strongly with the highly anomalous normal state ofthe cuprates and the robustness of cuprate superconduc-

42 JANUARY 2001 PHYSICS TODAY © 2001 American Institute of Physics, S-0031-9228-0101-030-4

YOSHITERU MAENO is a professor of physics at Kyoto University in Japan. MAURICE RICE is a professor at the Swiss Federal ResearchInstitute of Technology in Zurich, Switzerland. MANFRED SIGRIST is a professor of physics at the Kyoto University’s Yukawa Institute for Theoretical Physics.

THE INTRIGUING

SUPERCONDUCTIVITY OF

STRONTIUM RUTHENATEStructurally, this metal oxide

resembles the high-Tc cuprates, but itssuperconductivity is more like the

superfluidity of helium-3.

Yoshiteru Maeno, T. Maurice Rice,and Manfred Sigrist

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tivity with respect to disorder.A hallmark of a Landau–Fermi liquid is that the

resistivity at low temperatures should rise as T 2, a powerlaw that can be derived from the simplest perturbativetreatment of electron–electron collisions. The observationof this power law in the resistivity of Sr2RuO4—both in theRuO2 planes and perpendicular to these planes (but withvery different coefficients)—clearly indicates that the per-turbative Landau theory of electron–electron interactionsis applicable. Likewise, liquid 3He in the normal phase isdescribed well by Landau theory, even though the inter-actions in 3He are also very strong.2

In Sr2RuO4, the formal valence (or oxidation state) ofthe ruthenium ion is Ru4+, which leaves four electronsremaining in the 4d shell. The Ru ion sits at the center ofa RuO6 octahedron, and the crystal field of the O2⊗ ionssplits the five 4d states into threefold (t2g) and twofold (eg)subshells. The negative charge of the O2⊗ ions causes thet2g subshell to lie lower in energy because these orbitals(xy, xz, yz) have lobes that point not toward, but between,the O2⊗ ions lying along the x-, y-, and z-axes, as shown inthe lower panel of figure 1. Electrons in these orbitals

form the Fermi surface.The highly planar structure of Sr2RuO4

prevents substantial energy dispersionfrom developing along the z-axis due to thelarge interplanar separation of the RuO6octahedra. However, in the ab plane, neigh-boring RuO6 octahedra share O ions, which,in turn, are p-bonded with the Ru ions. Thexy orbital acquires a full 2D energy disper-sion thanks to the O ions that lie along thex- and y-axes, whereas the xz and yz orbitalshave only a restricted one-dimensionalenergy dispersion. The result, as shown onthe cover, is a Fermi surface with threecylindrical sheets: one of xy character (the gsheet) and two of slightly mixed xz and yzcharacters (the a and b sheets).

Mixing between xy and xz or yz orbitalsis exceptionally weak because their differ-ent parity under the reflection z O ⊗z for-bids mixing within a single plane. Detailedcalculations of the electronic band structureconfirm these considerations and distributethe four electrons more or less equallyamong all three orbitals.3

The detailed shape of a metallic Fermisurface can be directly determined in exper-iments that exploit the de Haas–van Alphen

effect, that is, the oscillation of magnetization in responseto an external magnetic field.4 In de Haas–van Alphenexperiments, the modulations of the Fermi surface alonga given direction can be related to the periods of theobserved oscillations. Only metallic samples with longmean free paths are suitable for these experiments, againproving the importance of sample purity. The shape andvolumes of the three Fermi surface sheets predicted bythe band structure were nicely confirmed.

De Haas–van Alphen experiments can also measurethe Fermi velocities perpendicular to the Fermi surface,which can be used to define an effective electron masslocally on the Fermi surface. Here, the discrepancybetween band structure calculations and experiment issubstantial. Typically, measurements reveal that the effec-tive mass is enhanced by a factor of 3–5. This largeenhancement agrees with values deduced from theT-linear specific heat coefficient.

The linearity is consistent with Landau theory, a cor-nerstone of which states that electron–electron interac-tions cannot alter the total volume enclosed by the Fermisurface but will renormalize the effective masses. The

http://www.physicstoday.org JANUARY 2001 PHYSICS TODAY 43

c,z

b,y

a,x

Sr RuO2 4 La Ba CuO2– 4x x

Sr La (Ba)

Ru Cu

O O

xy orbitalsyz orbitalsxz orbitals

z z z

y y y

x x x

FIGURE 1. STRONTIUM RUTHENATE (Sr2RuO4)forms a layered perovskite structure. As shownin the upper panel, the crystal’s RuO2 layers areseparated by Sr layers, and each Ru ion isenclosed in an octahedral cage of oxygen ions.The same structure is also found in La2CuO4,which is the parent compound of the high-Tc

superconductor La2–xBaxCuO4. Despite theirstructural similarity, the ruthenate and thecuprate have very different electronic proper-ties: Sr2RuO4 is strongly metallic and a super-conductor at low temperature, whereasLa2CuO4 is an antiferromagnetic insulator. Thelower panel shows the configuration of theorbitals responsible for the superconductivityof Sr2RuO4, namely, the 4d t2g orbitals.

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large mass enhancement and the large measured value ofthe T 2 coefficient in the resistivity reveal that residualinteractions in the Landau description are unusuallystrong—as they are in 3He. These measurements, there-fore, confirmed suspicions that the dominant interactionsat low energies are of the strong electron–electron kind,rather than the weaker electron–phonon kind. They alsoboosted the prospects that the superconductivity wouldturn out to be unconventional.

In fact, as soon as unconventional superconductivityturned out to be increasingly likely in Sr2RuO4, the ques-tions arose which total spin (spin singlet or triplet) andangular momentum channel (s-wave, p-wave, and soforth) characterize the fermions’ internal motion in theCooper pairs. Pairs of fermions must have antisymmetricwave functions under particle interchange. For a Cooperpair, this requirement implies a relationship between theorbital and spin character: Orbital wavefunctions witheven values of the orbital quantum number (l ⊂ 0, 2, . . .)are even under particle interchange and therefore requireodd symmetry in the spin wavefunction (that is, they arespin singlets); odd (l ⊂ 1, 3, . . .) require spin triplets.

Cooper pairing symmetrySoon after the discovery of superconductivity in Sr2RuO4,Maurice Rice and Manfred Sigrist—and, independently,Ganapathy Baskaran—proposed p-wave (l ⊂ 1) and spin-triplet pairing on the basis of similarities to 3He and on the

fact that closely related oxides such asSrRuO3 are ferromagnetic.5 By con-trast, the cuprates, which are antifer-romagnets rendered superconductingby doping, are now known to havespin-singlet pairing in a d-wave (l ⊂ 2)orbital channel.

Specifying the complete symme-try of the superconducting staterequires more than just the angularmomentum channel and the spinstate. The pairs in a superconductorare in a spatially uniform state withrespect to their center-of-mass coordi-nate, and their possible internalmotion will be classified by those sym-metry operations centered on a Ru ionthat leave the crystal invariant—thatis, the “point group” of the crystal.Sr2RuO4 has tetragonal symmetryand its highly planar character sug-gests attention be restricted to pairingstates that maximize the intraplanar,

rather than interplanar, pairing.In a triplet superconductor, the pairing state is con-

veniently represented by a three-dimensional (3D) vectord(k), whose magnitude and direction vary over the Fermisurface in k space. The presence of the pairing condensatecauses a finite difference in the energy needed to add (orremove) electrons singly, rather than in pairs. This “ener-gy gap” in the single electron spectrum (or density ofstates) may depend also on the position, k, on the Fermisurface and is given by the magnitude of d(k). How d(k)varies depends on which representation of the point groupd(k) belongs to; in some cases, it will even have nodes (orzeroes). Additionally, the pairing state corresponds to thed(k) that maximizes the energy gained when the super-conducting condensate forms (the condensation energy).Under weak coupling conditions, this choice will be anodeless form. However, these criteria alone are insuffi-cient to determine the form of the pairing amplitudeuniquely. Several pairing states have the same condensa-tion energy under weak coupling conditions.

This degeneracy is related to the spin rotation sym-metry and is eventually lifted by spin–orbit coupling. Acomparison with superfluid 3He is illuminating here. Atambient pressure, the p-wave superfluid state of 3He isthe B-phase, whose energy gap is nonzero in all direc-tions. However, under pressure, close to the liquid–solidphase boundary, enhanced spin fluctuations favor the A-phase, even though it has two point nodes.

44 JANUARY 2001 PHYSICS TODAY http://www.physicstoday.org

1

050 100 0 1.0 2.00

T (K) T (K)

KN

IGH

TSH

IFT

YBa Cu O2 3 7 Sr RuO2 4

FIGURE 2. THE SUPERCONDUCTING STATES, corresponding to d(k) = z (kx ⊕ iky)(left) and d(k) = xkx ⊕ yky (right), have different spin orientations and total angularmomentum. The z (kx ⊕ iky) state has an angular momentum along the z-axis (thickarrow) and spin perpendicular in the plane (thin arrows). Its total angular momen-tum therefore has a value of 1 and points along the z-axis. The xkx ⊕ yky state hasvanishing total angular momentum because the orbital angular momentum is com-pensated by the spins of the Cooper pair. Both states represent equal-spin-pairingstates, but have very different physical properties. Experiments identify the supercon-ducting phase in Sr2RuO4 as being of the type in the left panel—that is, a state withfinite angular momentum.

FIGURE 3. THE KNIGHT SHIFT for oxygen-17 in yttrium bari-um copper oxide (left) and strontium ruthenate (right). Inmaterials whose superconductivity is mediated by spin-singletCooper pairing, Knight-shift data exhibit a drop in the spinsusceptibility in the superconducting state. Such a drop occursin YBa2Cu3O7, but not in Sr2RuO4, whose superconductivity ismost likely mediated by spin-triplet Cooper pairs (the dashedline is a prediction based on spin-singlet Cooper pairing).(Adapted from ref. 7.)

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3He is a 3D Fermi liquid with a spherical Fermi sur-face; the nodes can be placed at the north and south poles.By contrast, because of its highly planar nature, theFermi surface of Sr2RuO4 consists of cylindrical sheets.With this topography, one can construct for Sr2RuO4 a 2Danalog of the A-phase of 3He that is nodeless and, there-fore, stable over a wider range of parameters. For this rea-son, states of both type are strong competitors: the B-phase-like state d(k) ⊂ (kx, ky, 0), and the A-phase-likestate d(k) ⊂ (0, 0, kx � iky).

The different orientations of the spin and orbitalangular momenta of the Cooper pairs are illustrated in fig-ure 2. At present, it is difficult to decide the pairing sym-metry theoretically from a simple microscopic approach. Atheory of spin-fluctuation-driven superconductivity is evenmore difficult to construct for Sr2RuO4 than it is for thesuperfluidity of 3He. For example, the influence ofspin–orbit coupling strongly depends on details of the bandstructure and on the pairing mechanism.

Evidence for spin-triplet pairingThe next question, therefore, is how to determine the pair-ing symmetry experimentally. Unlike the more familiarcases of determining crystalline or magnetic order, this goalis far from easy. Consequently, one must proceed indirectlyand ask what are the special properties of each type of sym-metry and what experiments will directly test them.

First, consider how one discriminates between con-ventional (that is, s-wave) and unconventional (non-s-wave) forms in a weak-coupling superconductor. In thecase of Sr2RuO4, weak coupling can be inferred from thelow value of Tc (� 1 K) relative to the effective Fermi tem-perature (� 50 K) at which quantum coherence sets in.

Conventional superconductivity is largely immune toscattering from impurities, imperfections, and so on. AsPhilip Anderson showed, Cooper pairs can be formed frompairs of single fermion states related by time reversalsymmetry. However, in an unconventional superconduc-tor, the pairing amplitude is a nontrivial function of thewave vector k around the Fermi surface. The wave vectormust therefore be well defined. Because it mixes differentk values, impurity scattering is deleterious to unconven-tional superconductivity, hence, the need for high-qualitysamples to see unconventional superconductivity in thefirst place.

Andrew Mackenzie and his collaborators undertookdetailed investigations of the sensitivity to impurity scat-tering in a series of aluminum-doped samples. Theyshowed that Sr2RuO4 behaved as expected for an uncon-ventional superconductor.6

Next, the question of whether the pairing is spin-sin-glet or spin-triplet must be settled. The most effective wayto do this is by measuring the spin susceptibility throughthe superconducting transition. This can be convenientlydone by measuring the Knight shift in nuclear magneticresonance (NMR) experiments. The Knight shift describesthe small change in the resonance line frequency causedby the weak spin polarization of the electrons in theapplied magnetic field. In a singlet superconductor, theelectrons in the Cooper pairs are bound in spin singlets;they are not polarized at all in the small fields applied inNMR experiments. As a result, the Knight shift vanishesat zero temperature. But for triplet superconductors withthe parallel spins lying in the plane, the application of amagnetic field in the plane changes the relative numbersof pairs with spin parallel and antiparallel to the field,and the Knight shift is unchanged from its value in thenormal state.

The first NMR test came from Kenji Ishida and hiscoworkers, who found that the 17O Knight shift was total-ly unaffected by the transition into the superconductingstate7 (see figure 3). Note, however, that the spin–orbitinteraction blurs the distinction between spin-singlet andspin-triplet pairing. If, contrary to expectations,spin–orbit coupling is very strong, it could negate the sim-ple interpretation of the Knight shift experiments.

Another way to identify the nature of the pairingstems from the fact that one of the proposed forms ford(k), (0, 0, kx � iky), breaks time reversal symmetry. Thisspecial property forces a local charge disturbance to beaccompanied by a local supercurrent distribution, whichin turn creates a small magnetic moment distribution. Agood way to test for this phenomenon is to inject spin-polarized m+ leptons into the sample and observe the timeevolution of their spin polarization by monitoring theirdecay products. The muons create a local charge distur-bance and their spin makes it possible to directly test foran accompanying magnetic moment distribution. GraemeLuke and his collaborators performed a series of mSR(muon spin relaxation) experiments on Sr2RuO4 and foundan enhancement of the zero field relaxation rate in thesuperconducting state that agreed with expectations for abroken time-reversal symmetry state.8 (For more on timereversal symmetry, see box 1 above.)

Thus, assuming that nature has not confused mattersby having a magnetic phase of a different origin appear-ing coincidentally at the same Tc, one can conclude thatthe three main features of the superconducting state inSr2RuO4—namely, the non-s-wave form, the spin-tripletpairing, and the broken time reversal symmetry—havebeen confirmed by these three independent tests (see also

http://www.physicstoday.org JANUARY 2001 PHYSICS TODAY 45

Box 1. Chiral Superconductivity

According to muon spin resonance experiments, the super-conducting state of Sr2RuO4 has a pairing order that

breaks time reversal symmetry.8 The form of the gap functionrepresented by d(k) ⊂ z (kx � iky) implies that the Cooperpairs have the internal angular momentum Lz ⊂ �1. Thismeans that the superconducting phase analogous to the A-phase of 3He is, in principle, an orbital ferromagnet.2 In gener-al, superconductivity and magnetism are antagonistic phe-nomena, but in this case, the superconducting state generatesits own magnetism. The Meissner–Ochsenfeld effect, howev-er, prevents uniform magnetization inside the superconductorand magnetism is restricted to areas of inhomogeneities—thatis, around impurities and domain walls or at interfaces and sur-faces. In these regions, spontaneous supercurrents flow, gener-ating local magnetic field distributions that can be detected bymuon spins.

The presence of orbital angular momentum introduces aform of chirality to the superconducting state. Among the chi-rality’s various implications, the most striking is perhaps theset of quasiparticle states that are localized at the surface andpossess a chiral spectrum. These chiral states carry sponta-neous dissipation-free currents along the surface. They are gap-less, in contrast to bulk quasiparticles of the superconductor,and can be detected by spectroscopic measurements.14

Chirality in a quasi-2D system suggests a connection withthe quantum Hall effect. Indeed, it was proposed that chiralsuperconductors could show quantum Hall-like effects even inthe absence of an external magnetic field.14 This spontaneousHall effect would appear as a transverse potential difference inresponse to a supercurrent. Unfortunately, the effect is rathersmall and difficult to detect. Nevertheless, Sr2RuO4 couldbecome the ideal test ground for this phenomenon.

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box 2 above). They are consistent with a single state of thebasic form d(k) ⊂ (0, 0, kx � iky), which is the analogue ofthe 3He A-phase.

Multi-orbital effectsAs is often the case in condensed matter physics, thingsare not as simple and straightforward as they seem ini-tially. The first experiments to uncover essential compli-cations were the specific heat measurements of ShujiNishizaki and his coworkers. Their results revealed areduced jump in the specific heat at the superconductingtransition and the emergence of a substantial term pro-portional to T as T tends to zero. The results also implieda residual density of states in the superconducting state ofabout half the value in the normal state.

These findings are clearly at odds with the predictionof a complete energy gap in the proposed superconductingstate. Such a state would have a specific heat that van-ishes exponentially as T tends to zero. A little later, NMRexperiments on the relaxation rate as a function of T alsoshowed evidence of a residual density of states of aboutthe same size.

Offering an explanation for this seeming contradic-tion, Daniel Agterberg, Rice, and Sigrist pointed out thatthe Fermi surface separates into two distinct parityregions—namely, the a,b sheets and the g sheet (seecover). The a,b sheets are derived from orbitals with thesymmetry (xz, yz), which are odd under the reflectionz O ⊗z about the center of a RuO2 plane. The g sheet isderived from the xy orbital, which has even parity.9 Agter-berg, Rice, and Sigrist showed that, in this highly planarmaterial, pair scattering between the two parity regionswould be greatly inhibited. They also proposed an orbital-dependent form of superconductivity that would be essen-

tially confined to one of the regions. Typically, such con-finement does not occur because, according to the usualconservation laws of energy and momentum, a Cooperpair may be scattered to all parts of the Fermi surface. Ifthe confinement thesis is valid, the next question is whichparity region is responsible for the superconductivity.

Using an ingenious 17O NMR analysis, Takashi Imaiand his coworkers separated the contributions of the a,band g sheets to the susceptibility and found that the spinsusceptibility was more strongly enhanced on the gsheets.10 Imai’s result implies that, when the temperaturedrops below Tc, superconductivity initially appears in thissheet.

The first experiments, which showed a residual den-sity of states of about half the normal state value, were allcarried out on rather impure samples. The value of Tc is asensitive indicator of sample quality. The initial sampleshad values of Tc � 1 K. More recently, improvements insample growing techniques (see figure 4) have raised thevalue of Tc to 1.48 K, which is very close to the predictedmaximum value.

When the specific heat and NMR relaxation rateexperiments were repeated on the newer high-qualitysamples, the results changed substantially. Instead of aclear linear term in the NMR relaxation rate at low tem-perature, a continuous crossover to a higher power (� T 3)was observed. This behavior can only be explained if allparts of the Fermi surface participate in the supercon-ducting state.

Theoretically, there are two possible ways that thesuperconductivity can spread from the g sheet to the a,bsheets. Either a pairing amplitude of the same symmetryon the a,b sheet is directly induced or a pairing amplitudeof a different symmetry could spontaneously appear. In

46 JANUARY 2001 PHYSICS TODAY http://www.physicstoday.org

For magnetic fields along its c-axis, Sr2RuO4 is a weaklytype-II superconductor and can support a mixed phase,

where vortices (magnetic flux lines) enter the superconductor.Here, the pairing symmetry is crucial for the interactionamong vortices and the way they arrange in a lattice. In con-trast to conventional and high-Tc superconductors, where thesuperconducting phase is represented by a single complexorder parameter, the A-phase-like state in Sr2RuO4 involvestwo complex components.Daniel Agterberg showed thatthis implies that the vorticeswould form a square lattice.15

Subsequent small-angle neu-tron scattering experimentshave confirmed this predictionnicely.16

Nevertheless, the realiza-tion of a vortex square latticeis not an unambiguous prooffor a two-component orderparameter. Fortunately, thedetailed form of the magneticfield distribution associatedwith the vortices gives a fur-ther tool to identify the orderparameter symmetry forweakly type-II superconduc-tors. The accompanying figure

(courtesy of Paul Kealey and Ted Forgan) shows the diffractionpattern from the flux-line lattice in Sr2RuO4 obtained fromsmall-angle neutron scattering.16 (The numbers on the axesindicate the pattern’s actual size in centimeters of the position-sensitive neutron detector.) Kealey and Forgan’s neutron scat-tering experiments established recently that the magnetic fielddistribution in Sr2RuO4 in the mixed phase could be fitted best

with a two-component orderparameter, whereas a single-component order parameterstate always showed substan-tial deviations.16

The mixed phase for fieldsparallel to the RuO2 planes isno less interesting. In particu-lar, the low-temperature high-field regime exhibits a varietyof unusual properties thatimply a more complex phasediagram involving differentsuperconducting phases.17

Experiments concerning thisaspect of Sr2RuO4 are inprogress. Because they requiremagnetic fields in this quasi-2D system to be alignedinplane with very high preci-sion, these experiments areextremely demanding.

80

70

60

50

40

40 50 60 70 80 90

Box 2. Vortex Physics

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the latter case, a second phase transition in the supercon-ducting state would arise as the overall symmetry wasmodified. The experiments to date appear to rule out thispossibility. In the former case, the pairing on a g sheet canbe thought of as acting as a form of weak external field onthe a,b sheets, whose response will be very sensitive toimpurity scattering. Agterberg has analyzed the problemalong this line, but at present the issue of how the 3Dnature of the crystal causes pairing to spread to all partsof the Fermi surface remains unresolved. Indeed, the pos-sibility of a major revision of the strongly orbital depend-ent hypothesis cannot be ruled out.

Phase-sensitive probesFor unconventional superconductors, the most direct wayto determine the nature of the pairing is through phase-sensitive experiments. Such experiments proved decisivein the case of the high-Tc cuprate superconductors whenthey settled a long debate between proponents of conven-tional s-wave and unconventional d-wave pairing in favorof the latter.

Phase-sensitive experiments directly probe the signof the pairing amplitude around the Fermi surface bymaking use of a suitable circuit of Josephson junctions.Sign changes in the pairing amplitude make it possible tocreate a frustration in the phase of d(k) that is relieved bya current flowing in the circuit. A measurement of mag-netic flux then ascertains the presence or absence of thefrustrating sign change in d(k).

The first proposal along these lines was made byVadim Geshkenbein and his coworkers some years ago totest for p-wave pairing in heavy fermion superconduc-tors.11 To date, no such experiments have been reportedfor Sr2RuO4; until they are done, one cannot say that theidentification of the symmetry is totally unambiguous.Josephson tunneling of pairs between Sr2RuO4 and con-ventional superconductors has been reported12—a hopefulsign that phase-sensitive experiments of this kind onSr2RuO4 will prove possible.

Another class of phase-sensitive experiments is con-nected with the presence of surface quasiparticle statesthat lie within the gap in the density of states. These qua-siparticle states, which can only exist when the phase ofthe pairing amplitude depends on direction, can be probedthrough the voltage dependence of the tunneling currentfrom a normal metal to the superconductor. Recently,Frank Laube and his collaborators used a platinum point

contact to observe an anomaly at zero bias in the cur-rent–voltage characteristics. Their experiments not onlyprovide evidence for the surface quasiparticle states with-in the energy gap, but also strongly support the case foran odd-parity pairing state in Sr2RuO4.13

The discoveries made in Sr2RuO4 show that uncon-ventional superconductivity driven by spin fluctuations orrelated mechanisms can appear in a well-formed Lan-dau–Fermi liquid in metals as it does in 3He. Many othermetals show Landau–Fermi liquid behavior, but havenever been observed to be superconducting. Metals suchas the 3D cuprate LaCuO3, V2O3 at high pressure,Sr3Ru2O7, and CaRuO3 fall into this category.

Already, the small number of unconventional super-conductors has proven to be an immensely fruitful terri-tory for finding novel phenomena in superconductivity. Ifthe sample quality can be improved for other oxides, thefield of unconventional superconductivity in metals mayjust be beginning.

We have benefited from many collaborations in this field andwe would thank especially D. F. Agterberg, C. Bergmann, A.Furusaki, A. P. Mackenzie, Z. Q. Mao, M. Matsumoto, K. K.Ng, S. Nishizaki, and M. Zhitomirsky.

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Rev. B 52, 1358 (1995).4. A. P. Mackenzie et al., Phys. Rev. Lett. 76, 3786 (1996).5. T. M. Rice, M. Sigrist, J. Phys. Cond. Matter 7, L643 (1995).

G. Baskaran, Physica B 223–224, 490 (1996).6. A. P. Mackenzie et al., Phys. Rev. Lett. 80, 161 (1998).7. K. Ishida et al., Nature 396, 658 (1998).8. G. Luke et al., Nature 394, 558 (1998).9. D. F. Agterberg, T. M. Rice, M. Sigrist, Phys. Rev. Lett. 78,

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http://www.physicstoday.org JANUARY 2001 PHYSICS TODAY 47

FIGURE 4. GROWING A SINGLE CRYSTAL of stron-tium ruthenate by the floating zone method. A poly-crystalline rod fed from the top is melted at about2100 °C in the central zone, where elliptical mirrorsfocus the infrared power emitted from two halogenlamps. The crystal is extracted from below. Becausethe material is not in contact with any container,which is often the main source of contamination, thecrystal can be made extremely pure.

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