Superconducting Gap Structure of kappa-(BEDT-TTF)2Cu(NCS)2 Probed by Thermal Conductivity T

a r X i v :c o n d -m a t /0107537v 2 [c o n d -m a t .s u p r -c o n ] 9 N o v 2001Superconducting Gap Structure of κ-(BEDT-TTF)2Cu(NCS)2Probed by ThermalConductivity TensorK.Izawa 1,H.Yamaguchi 1,T.Sasaki 2,and Yuji Matsuda 11Institute for Solid State Physics,University of Tokyo,Kashiwa,Chiba 277-8581,Japan2Institute for materials research,Tohoku University,Sendai 980-8577,JapanThe thermal conductivity of organic superconductor κ-(BEDT-TTF)2Cu(NCS)2(T c =10.4K)has been studied in a magnetic field rotating within the 2D superconducting planes with high alignment precision.At low temperatures (T <∼0.5K),a clear fourfold symmetry in the angular variation,which is characteristic of a d -wave superconducting gap with nodes along the directions rotated 45◦relative to the b and c axes of the crystal,was resolved.The determined nodal structure is inconsis-tent with recent theoretical predictions of superconductivity induced by the antiferromagnetic spin fluctuation.74.20.Rp,74.25.Fy,74.25.Jb,74.70.KnSince the discovery of superconductivity in organic ma-terials about 2decades ago,the question of the pair-ing symmetry among this class of materials is one of the most intriguing problems.In particular,the nature of the superconductivity in quasi-2D κ-(BEDT-TTF)2X salts (κ-(ET)2X),where the ion X can,for example,be Cu(SCN)2,Cu[N(CN)2]Br or I 3,has attracted consider-able attention.In these layered organics,Shubnikov-de Haas oscillation experiments have established the exis-tence of a well-defined Fermi surface (FS),demonstrating the Fermi liquid character of the low energy excitation.The large enhancement of the effective mass revealed by the specific heat as well as magnetic susceptibility mea-surements suggests the strong electron correlation effect in the normal state.Moreover,it was suggested that su-perconductivity occurs in proximity to the antiferromag-netic (AF)ordered state in the phase diagram [1].Since some of these unusual properties suggest analogies with high-T c cuprates [2],it was pointed out by many authors that the AF spin-fluctuation should play an important role for the occurrence of superconductivity [3,4].Unconventional superconductivity is characterized by a superconducting gap with nodes along certain crystal directions.Since the superconducting gap structure is intimately related to the pairing interaction,its identi-fication is crucial for understanding the pairing mech-anism.Although the structure of the superconducting order parameter of κ-(ET)2X salts has been examined by several techniques,it is still controversial as we now summarize [1].Results strongly in favor of unconven-tional pairing symmetry came from NMR experiments of κ-(ET)2Cu[N(CN)2]Br,in which the absence of the Hebel-Slichter peak and cubic T -dependence of the spin lattice relaxation rate 1/T 1were interpreted as an indica-tion of d -wave pairing with line nodes [1,5].The existence of the T -linear term in the thermal conductivity at low temperatures of κ-(ET)2Cu(NCS)2also supported the presence of line nodes [6].However,some of the specific heat and penetration depth studies on these materials ledto conflicting results.For example,recent specific heat measurements reported a fully gapped superconductivity [7].Since these measurements rely on the T -dependence of the physical quantities,it is more desirable to mea-sure the in-plane anisotropy of the gap directly in order to probe the gap structure.Very recently,such an at-tempt was made by STM [8]and mm-wave transmission [9]experiments.Although both measurements reported the strong modulation of the gap structure,they led to completely different conclusions on the node directions;the former predicts nodes along the directions rotated 45◦relative to the b and c -axes while the latter predicts nodes along the b and c directions.In interpreting these experiments,one needs to bear in mind that the STM spectrum parallel to the 2D plane can be strongly af-fected by the atomic state at the edge.Moreover,an alternative interpretation was proposed for the mm-wave transmission experiments [10].Thus,the gap structure of κ-(ET)2X salts is far from settled and the situation strongly confronts us with the need for a powerful direc-tional probe of the superconducting gap.During the past few years,the understanding of the heat transport in the mixed state of superconductors with anisotropic gap has largely progressed [11].In particular,it was demonstrated both experimentally and theoreti-cally that the thermal conductivity is a powerful tool for probing the anisotropic gap structure [12–18].Thermal conductivity has some advantages,compared to other ex-periments.First,it is an unique transport quantity which does not vanish in the superconducting state,responding to the unpaired quasiparticles (QPs).Second,it is a probe of the bulk free from the surface effect.Third and most importantly,it is indeed a directional probe,sensi-tive to the relative orientation among the thermal flow,the magnetic field,and nodal directions of the order pa-rameter,as we will discuss in detail later.In fact,a clear fourfold modulation of the in-plane thermal conductivity κwith an in-plane magnetic field which reflects the angu-lar position of nodes of d x 2−y 2symmetry was observed inYBa2Cu3O7−δ[12,13]and2D heavy fermion supercon-ductor CeCoIn5[18],while such a modulation was absentin Nb and the B-phase of UPt3with an isotropic gapin the basal plane[13,19].These fact demonstrate thatthe thermal conductivity tensor can be a relevant probeof the superconducting gap structure.In this Letter,we have measured the thermal conductivity tensor ofκ-(ET)2Cu(NCS)2in magneticfield rotating within the2Dsuperconducting planes.The superconducting gap struc-ture was successfully determined by the angular variationofκ.On the basis of thesefindings,we discuss the natureof the superconductivity ofκ-(ET)2Cu(NCS)2.Single crystalsκ-(ET)2Cu(NCS)2were grown by con-ventional electrochemical method and their approximatesizes are2x1x0.1mm3.The thermal conductivity wasmeasured by the steady-stated method with one heaterand two RuO2thermometers.The heat current q wasapplied along the b-direction.The upper inset of Fig.1shows the FS.In the present measurements,it is very im-portant to rotate H within the2D bc-planes with veryhigh accuracy because a slightfield-misalignment pro-duces2D pancake vortices which might act as a strongscattering center of the thermal current.Special carewas therefore taken to keep the perpendicularfield dueto the misalignment less than H c1perpendicular to thelayers,so that the condition for the absence of pancakevortices(lock-in state)was always fulfilled while rotatingH[20].For this purpose,we used a system with twosuperconducting magnets generating H in two mutuallyκ(W/Km)T ( K )FIG.1.T-dependence of the thermal conductivity in zerofield.The heat current q was applied along the b-direction.Lower inset:The resistive transition at T c.Upper inset:TheFermi surface ofκ-(ET)2Cu(NCS)2(solid lines).The Fermisurface consists of quasi-1D and2D hole pocket.The dashedlines show thefirst Brillouin zone with k b and k c axes.Thethin solid lines show the extended Brillouin zone with k x andk y axes in the similar coordinate style of the high-T c cuprates.The node directions determined in our experiment are alsoshown.orthogonal directions and a3He cryostat equipped on amechanical rotating stage with a minimum step of1/500degree at the top of the puter-controllingtwo magnets and the rotating stage,we were able to ro-tate H continuously within the2D planes with a mis-alignment of less than0.01degree from the plane,whichwe confirmed by the simultaneous measurement of theresistivity.Wefirst discuss the T-and H-dependence ofκ.Theobserved T-and H-dependence were very similar to theresults of Ref.[6].Figure1depicts the T-dependence ofκ.Upon entering the superconducting state,κexhibitsa kink and rises to the maximum value just below T c.As discussed in detail in Ref.[6],the enhancement ofκbelow T c reflects the increase of the phonon mean freepath by the electron condensation,which is so becausethe phonon thermal conductivityκph well dominates overthe electronic thermal conductivityκel near T c.Figures2(a)and(b)depict the H-dependence ofκin perpendic-ular(H⊥bc−plane)and parallel(H bc−plane)field,respectively.In perpendicularfield,κ(H)shows a mono-tonic decrease up to H c2above1.6K,which can be at-tributed to the suppression of the phonon mean free pathby the introduction of the vortices.Below1.6K,κ(H)exhibits a dip below H c2.The minimum ofκ(H)appearsκ(H)/κ()κ(H)/κ()µH ( T )FIG.2.H-dependence of the in-plane thermal conductivity(a)in perpendicular and(b)in parallelfield(H c)at lowtemperatures.Deviation from the horizontal line shown byarrows marks H c2.Inset:H-dependence of∆κel/κ(0)inparallelfield.For details,see the text.as a result of a competition betweenκph which always decreases with H andκel which increases steeply near H c2.Then the magnitude of the increase ofκ(H)be-low H c2provides a lower limit of the electronic contri-bution.As seen in Fig.2(a),the electronic contribution grows rapidly below0.7K;κel n/κn is roughly estimated to be>∼5%at0.7K and>∼15%at0.42K,whereκel n andκn are the electronic and total thermal conductivity in the normal state above H c2,respectively.This dra-matic increase ofκel n/κn is caused byκph which decreases much faster thanκel with decreasing T.In parallelfield with much higher H c2(>∼30T),κ(H)decreases mono-tonically at all temperatures.Whileκ(H)/κ(0)shows a similar H-dependence at0.71K and1.1K,it deviates from this pattern at0.42K.Since the electronic con-tribution grows rapidly below0.7K,this deviation can be attributed toκel.In the inset of Fig.2(b),we show ∆κel(H)/κ(0) ≡κel(H)−κel(0)0.15.We note that this value is one order of magni-tude larger than that in Sr2RuO4with an isotropic gap in the plane[17].We now address the origin for the fourfold symmetry.The most important issue here is ”Is the observed fourfold symmetry inκel a consequence of the line nodes perpendicular to the layer?”.We will show that the band structure inherent to the crystal is very unlikely to be an origin of the fourfold symmetry. First of all,it can be shown by the group theoretical consideration that the anisotropic term inκel due to a fourfold distortion of the FS is of second order relative to the leading terms,since the thermal conductivityκxx is a second rank tensor[21].In addition,the crystal structure ofκ-(ET)2Cu(NCS)2is monoclinic and FS is nearly elliptic;the fourfold distortion of the FS should be very small if it exists.Second,the in-plane magne-toconductivity∆σ(H)=σ(H)−σ(H=0)above T c is undetectably small even at5T due to the very strong two dimensionality.In fact,the upper limit of∆σ/σ(0) roughly estimated from the warp of the FS perpendicu-lar to the plane is less than10−5at2T.Thus,as far as the Wiedemann-Franz law holds,the fourfold oscilla-tion ofκel arising from the magnetoconductance should be undetectably small.These considerations lead us to conclude that the observed fourfold symmetry originates from the superconducting gap nodes[22].In the thermal transport in the superconductors with nodes,the dominant effect in a magneticfield is the Doppler shift of the delocalized QP energy spectrum, which occurs due to the presence of a superfluidflow around each vortex,and generates a nonzero QP density of states(DOS)at the Fermi level(Volovik effect)[23]. While the Doppler shift increasesκel with H through the enhancement of the DOS,it can also lead to a de-crease ofκel through the suppression of impurity scat-tering time and Andreev scattering time offthe vortices. At high temperatures,the latter effect is predominant, but eventually gives way to the former at low temper-atures,as demonstrated in high-T c cuprates[24].Since κel increases with H as shown in the inset of Fig.2(b), the enhancement of the DOS is the main origin for the H-dependence ofκel at0.42K[25].In this case,rotating H within the2D-plane gives rise to the fourfold oscilla-tion inκel associated with the DOS oscillation[15,16]. This effect arises because the DOS depends sensitively on the angle between H and the direction of the nodes of the order parameter,because the QPs contribute to the DOS when their Doppler-shifted energies exceed the local energy gap.The DOS oscillation with fourfold sym-metry gives rise toκel which attains its maximum value when H is directed to the antinodal directions and turns minimum when H is directed along the nodal directions (see Fig.2in Ref.[16]).According to Ref.[15],the am-plitude of the fourfold symmetry in the d-wave supercon-ductors arising from the DOS oscillation is roughly es-timated as|C4θ|/κel(0)=0.082υFυ′F eH32∆/π¯hΓ).Here∆is the superconducting gap,Γis the QP relax-ation rate,υF andυ′F are the in-plane and out-of-plane Fermi velocity,ingΓ∼2×1011s−1,2∆/k B T c=3.54,υF∼5×104m/s,andυ′F∼2.5×103m/s,gives|C4θ|/κel(0)∼3%.Thus the DOS oscillation by Volovik effect yields|C4θ|/κel(0)which is in the sameorder to the data.The fourfold symmetry enables us to specify the nodedirections,which is crucial for understanding the pairinginteraction.κ4θexhibits a maximum when H is applied parallel to the b and c axes of the crystal,showing thegap nodes along the directions rotated45◦relative to theb and c-axes;the nodes are situated near the band gap between the1D and2D bands(see the upper inset ofFig.1).This result is consistent with the STM experi-ments[8].We emphasize here that the determined nodalstructure is inconsistent with the recent theories based onthe AFfluctuation.If one assumes an AFfluctuation scenario,it is natural to expect the nodes to be alongthe b and c directions.This is because the AF orderingvectors become parallel to the b-axis,which would pro-vide a partial nesting.If we take the same conventionsfor the magnetic Brillouin zone as the high-T c cuprates with d x2−y2symmetry(see Fig.1(c)in Ref.[3]),the superconducting gap symmetry ofκ-(ET)2Cu(NCS)2isd xy.It is interesting to note that superconductivity with d xy symmetry has been theoretically suggested based onthe chargefluctuation scenario[26].Our present resultsmay bear implications on this issue.Wefinally comment on the recent heat capacity mea-surements which report a fully gaped superconductivity [7].In our view,their temperature range(T>T c/5) is not low enough to conclude the exponential behavior of the heat capacity;the measurements at temperatures less than T c/10would be required.In summary,the thermal conductivity tensor ofκ-(BEDT-TTF)2Cu(NCS)2was studied in a magneticfield rotating within the2D superconducting planes.The observed fourfold oscillation provides a strong evidence of d-wave symmetry.From its sign,the node direc-tions are successfully specified.These results place strong constraints on models that attempt to explain the mechanism of the superconductivity ofκ-(BEDT-TTF)2Cu(NCS)2.We thank M.Imada,K.Kanoda,K.Kuroki,M.Ogata, A.Tanaka and K.Ueda for stimulating discussions.We are also grateful to K.Maki for several comments and for showing unpublished results.[1]K.Kanoda,Physica C282-287,299(1997)and refer-ences therein.[2]For review,see R.H.McKenzie,Science278,820(1997);Comments Cond.Matt.Phys.18,309(1998).[3]J.Schmalian,Phys.Rev.Lett.81,4232(1998).[4]H.Kino,and H.Kontani,J.Phys.Soc.Jpn.67,3691(1998);H.Kondo,and T.Moriya,ibid.67,3695(1998);K.Kuroki and H.Aoki,Phys.Rev.B60,3060(1999).[5]H.Mayaffre et al.,Phys Rev.Lett.75,4122(1995);S.D.De Soto et al.Phys.Rev.B52,10364(1995).[6]S.Belin et al.,Phys.Rev.Lett.81,4728(1998).[7]H.Elsinger et al.,Phys Rev.Lett.84,6098(2000);J.M¨u ller et al.,cond-mat/0109030.[8]T.Arai et al.,Phys.Rev.B63,104518(2001).[9]J.M.Schrama et al.,Phys.Rev.Lett.83,3041(1999).[10]S.Hill et al.,Phys.Rev.Lett.86,3451(2001);T.Shibauchi et al.,ibid86,3452(2001).[11]C.K¨u bert and P.J.Hirschfeld,Phys.Rev.Lett.80,4963(1998);M.Franz,ibid.82,1760(1999);I.Vekhter andA.Houghton,ibid.83,4626(1999);Yu.S.Barash and A.A.Svidzinsky,Phys.Rev.B58,6476(1998).[12]F.Yu et al.,Phys.Rev.Lett.74,5136(1995).[13]H.Aubin et al.,Phys.Rev.Lett.78,2624(1997).[14]K.Maki et al.,Physica C341-348,1647(2000).[15]H.Won and K.Maki,cond-mat/0004105.[16]I.Vekhter et al.,Phys.Rev.B59,R9023(1999).[17]K.Izawa et al.,Phys.Rev.Lett.86,2653(2001).[18]K.Izawa et al.,Phys.Rev.Lett.87,057002(2001).[19]J.Lowell and J.B.Sousa,J.Low Temp.Phys.3,65(1970);H.Suderow et al.,Phys.Lett.A234,64(1997).[20]P.A.Mansky et al.,Phys Rev.B50,15929(1994).[21]K.Maki,private commun.[22]We comment on a very anisotropic s-wave state.Thelower limit of the gap anisotropy ratio∆max/∆min is roughly estimated from the observed fourfold oscillation as∆max/∆min≫∆max/。