Mixed pairing symmetry in κ-(BEDT-TTF)_2 X organic superconductors from ultrasonic velocity measurements
Maxime Dion, David Fournier, Mario Poirier, Kim D. Truong, A.-M. S. Tremblay
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Mixed pairing symmetry in κ -(BEDT-TTF) X organic superconductors fromultrasonic velocity measurements
Maxime Dion, David Fournier, ∗ Mario Poirier, Kim D. Truong, and A.-M. S. Tremblay
Regroupement Qu´eb´ecois sur les Mat´eriaux de Pointe, D´epartement de Physique,Universit´e de Sherbrooke, Sherbrooke, Qu´ebec,Canada J1K 2R1 (Dated: September 9, 2018)Discontinuities in elastic constants are detected at the superconducting transition of layered or-ganic conductors κ -(BEDT-TTF) X by longitudinal and transverse ultrasonic velocity measure-ments. Symmetry arguments show that discontinuities in shear elastic constants can be explainedin the orthorhombic compound only if the superconducting order parameter has a mixed characterthat can be of two types, either A g + B g or B g + B g in the classification of irreducible represen-tations of the orthorhombic point group D h . Consistency with other measurements suggests thatthe A g + B g ( d xy + d z ( x + y ) ) possibility is realized. Such clear symmetry-imposed signatures ofmixed order parameters have not been observed in other superconducting compounds. PACS numbers: 74.70.Kn,74.25.Ld,74.20.Pp
Unconventional, non s-wave, superconductors in solidsseem ubiquitously associated with strong electronic cor-relations. This is the case in a wide variety of compoundsthat include heavy fermions, ruthenates, cuprates as wellas quasi-two-dimensional half-filled organic charge trans-fer salts κ -(ET) X (ET = BEDT-TTF) [1]. In most casesgaps with nodes are observed, but the exact symmetryof the unconventional superconducting order parameteris uncontroversial only in the cuprates.In this letter, we focus on the layered organics thatexhibit antiferromagnetism and Mott insulating behav-ior, as the cuprates, and establish the two-componentnature of the singlet order parameter in the orthorhom-bic compound κ -(ET) Cu[N(CN) ]Br. Previous stud-ies suggest d -wave pairing with nodes, although s -wavesymmetry is sometimes seen. Measurements sensitiveto the ~k -space dispersion, such as scanning tunnel-ing spectroscopy [2] and thermal conductivity [3], favor d xy symmetry, namely nodes along the nearest neigh-bor bonds (or equivalenty, between the orthorhombicaxes). Moreover, theoretical calculations based either onspin-fluctuation mediated superconductivity [4, 5, 6, 7]or on quantum cluster methods [8, 9] and variationalapproaches [10] for the Hubbard model, support theanisotropic d -wave picture with a prevailing d x − y sym-metry. Nevertheless, none of these calculations has con-sidered interlayer hopping, which, as we will show, is nec-essary to explain the experimental data that we present.The ultrasonic probe is extremely sensitive to gapanisotropies as the attenuation and velocity depend onboth the direction of wave propagation and the direc-tion of polarization. Attenuation experiments on UPt [11, 12] and of Sr RuO [13] perfectly illustrate how theunconventional gap structure can be unraveled by such aversatile technique. In organic charge transfer salts how- ∗ Present address: Department of Physics and Astronomy, Univer-sity of British Columbia, Vancouver, BC, Canada V6T 1Z4 ever, attenuation experiments are hampered by the smallsize and the shape of single crystals. Nevertheless, oneexperiment was successful for the κ -(ET) Cu[N(CN) ]Brcompound [14], but the interpretation of the results wascomplicated by a phase separation occurring even inhighly ordered samples. Notwithstanding these difficul-ties, ultrasound velocity can also be used to obtain in-sights into the nature of the superconducting (SC) statein layered organics. Lattice anomalies [15] and elasticconstant changes [16, 17] have been identified, but noconsistent effort has been yet dedicated to identify theSC order symmetry.We report anomalies observed at the SC transi-tion temperature T c on three elastic constants ofmonoclinic κ -(ET) Cu(NCS) and of orthorhombic κ -(ET) Cu[N(CN)] Br. Even though these compounds be-long to different point groups, we expect similarities inthe SC order parameters because of their nearly iden-tical electronic properties. To understand discontinu-ities in elastic constants one can invoke Landau-Ginzburgarguments [18, 19] or perform detailed BSC type cal-culations [20, 21]. Since we focus on symmetry prop-erties, a Ginzburg-Landau (GL) approach will suffice[22, 23, 24, 25].We use an acoustic interferometer [14] to measure rel-ative changes in velocity ∆
V /V that allow us to extractthe corresponding relative variations in the elastic con-stants C through ∆ C/C = 2∆
V /V . The κ -(ET) X crys-tals grow as platelets containing the highly conductingplanes whose normal is oriented along ~a ∗ for monoclinic κ -(ET) Cu(NCS) and along ~b for the orthorhombic κ -(ET) Cu[N(CN) ]Br. Thus, ultrasonic plane waves canbe propagated only along these normal directions. Purelongitudinal and transverse waves cannot be propagatedalong the ~a ∗ axis of the monoclinic structure so, strictlyspeaking, it is not possible to measure the C ij ’s indi-vidually as it is the case for the orthorhombic material[26]. However, given the layered structure and the ~a axisorientation of about 110 ◦ instead of 90 ◦ from the plane,we neglect, as a first approximation, the off-diagonal ele- Waves Cu(NCS) Cu[N(CN) ]BrLongitudinal C ( ~a ∗ ) C ( ~b )Transverse C ( ~c ) C ( ~a )Transverse C ( ~b ) C ( ~c )TABLE I: Elastic constants C ij with the appropriate polari-sation of the ultrasonic waves for two κ -(ET) X compounds. ments of the C ij matrix that differentiate the monoclinicstructure from the orthorhombic one. This simplifies thedata treatment without affecting the conclusions. Withthis approximation the measured C ij ’s are given in Ta-ble I for each crystal. FIG. 1: (Color online) Longitudinal waves propagating alongthe ~a ∗ axis in κ -(ET) Cu(NCS) : (A) ∆ V /V data at 166 MHzfor H = 0 and 12 Tesla ; (B) ∆ C /C at three frequencies. The κ -(ET) Cu(NCS) crystal will be considered asour reference compound since it is located far enoughfrom the Mott transition line on the high pressure side ofthe P-T diagram with no indication of a phase separation.To extract the elastic change caused by the onset of su-perconductivity, we applied a magnetic field perpendicu-lar to the highly conducting plane to quench the SC state.We show in Fig. 1A the temperature dependence of therelative change of the longitudinal velocity below 20 K at166 MHz. In zero magnetic field a negative discontinuityof the velocity is obtained at T c = 9.5 K; the anomaly iscompletely quenched in a field of 12 Tesla leaving onlya monotonous decrease of the velocity as the tempera-ture increases. We notice the absence of magnetic fieldeffects above 12 K, an observation that excludes, con- trary to the κ -(ET) Cu[N(CN) ]Br compound [14], thepresence of a coexisting phase in this temperature range.The difference between these two curves yields the rela-tive variation of the compressional constant C shownin Fig. 1B at different frequencies. As expected, no fre-quency dependence is observed: the onset of the SC phaseyields a negative discontinuity at T c that extends overa few degrees due to important SC fluctuations aboveand below the superconducting temperature defined asthe maximum slope. At lower temperatures ∆ C /C is practically constant. A similar procedure was usedfor the two transverse acoustic modes yielding, over thesame temperature range, ∆ C /C and ∆ C /C . Thethree relative elastic constant variations are compared inFig. 2. While a negative discontinuity is expected on C [18], the appearance of a discontinuity on the shear con-stant C is unusual. The amplitude of the discontinuityis larger than that of C by approximately a factor two,excluding the simple explanation of mode mixing for aquasi-transverse wave. These discontinuities are largerthan in other non conventional superconductors [27, 28]by two to three orders of magnitude. No discontinuity isobserved for ∆ C /C ; only a small change of slope isobtained at T c . FIG. 2: (Color online) Temperature dependence of ∆ C ij /C ij for the κ -(ET) Cu(NCS) compound. The vertical dashedline indicates the SC critical temperature. In the highly ordered κ -(ET) Cu[N(CN) ]Br com-pound, the coexistence of antiferromagnetic (AF) andSC phases complicates the analysis of the SC state [14].Moreover, a higher magnetic field is needed to quenchthe SC state. We present in Fig. 3 the ∆ C ij /C ij ob-tained by substracting the zero and 16 Tesla curves. Wenotice that magnetic field effects are observed in the nor-mal state up to 20 K on C and C (∆ C ij /C ij is notzero). The temperature dependence below T c = 11.9K is also not monotonous and the SC fluctuations ap-pear on a wider temperature range above T c . Notwith-standing these differences, the comparison with the κ -(ET) Cu(NCS) data (see Fig. 2) at T c is remarkable:we still observe a negative discontinuity on C ( C ), alarger one on C ( C ) and only a change of slope on C ( C ). These observations clearly establish the similarityof the couplings between the SC order parameter and theelastic strains, although the crystal symmetry groups dif-fer because of the tilting of the axis normal to the planes.Moreover, they confirm that the negative dicontinuity on∆ C /C for the monoclinic compound is intrinsic andthat it cannot be attributed to mode mixing. FIG. 3: (Color online) Temperature dependence of ∆ C ij /C ij for the κ -(ET) Cu[N(CN) ]Br compound obtained with a 16Tesla magnetic field. The dashed line indicates the SC criticaltemperature. Experiment has established that the layered organicsare singlet superconductors [1]. In the simplest GL modelthen, discontinuities in elastic constants at the supercon-ducting transition are easily explained through the freeenergy functional F = a | η | + gε i | η | + b | η | + X i,j C ij ε i ε j (1)where η is the order parameter, b is a constant, ε i is thestrain, C ij the matrix of elastic constants, while a is pro-portional to ( T − T c ). If one of the strains is coupled lin-early through the constant g to the order parameter, theminimization with respect to η shows that at the tran-sition a negative discontinuity appears on the effectiveelastic constant C ′ ii = ∂ F/∂ε i . Such a linear couplingto | η | is possible only if the strain ε i is invariant under allthe operations of the point group because | η | is. Higherorder coupling terms in the free energy would only leadto the change of slope or curvature observed below T c forall C ii , and these are not considered here.Table II shows a simplified character table for the ir-reducible representations of the monoclinic C h group of κ -(ET) Cu(NCS) , along with the transformation prop-erties of the strains and examples of basis functions forthe order parameter. Note that the x and y axis arenot perpendicular. They lie along the atomic bonds,which are along the diagonal formed by the b and c axes.Since, according to Table II, ε and ε are invariant under irrep E C b Basis functions Strains A g s , xy , ( x + y ) z ǫ , ǫ , ǫ , ǫ B g x − y , ( x − y ) z ǫ , ǫ TABLE II: Simplified character table, basis functions andtransformation properties of the strains. The monoclinic C h group for κ -(ET) Cu(NCS) has the character table of C ⊗ i ,but inversion i always has character +1 for singlets so the ta-ble of C shown above suffices. The names of the irreduciblerepresentations are those of C h . The last column shows thetransformation properties of the strains and the next to lastcolumn examples of basis functions for the order parameter.The a axis is tilted towards c (equivalently x + y ) axis in thelayers. irrep E C a C b C c Basis fcts Strains A g s , xy ǫ , ǫ , ǫ B g x + y ) z ǫ B g x − y ǫ B g x − y ) z ǫ TABLE III: Simplified character table, basis functions andtransformation properties of the strains for the orthorhombic D h = D ⊗ i group appropriate for κ -(ET) Cu[N(CN) ]Br.The b axis is perpendicular to the layers and the x + y axisis along a . the symmetry operations of the group, the correspond-ing elastic constants can couple linearly to | η | , leadingto negative discontinuities. However, ε is not invariantso there is no discontinuity at T c . This explains the ob-servations for κ -(ET) Cu(NCS) and it does not imposeany constraint on the symmetry of the order parameter.In the orthorhombic κ -(ET) Cu[N(CN) ]Br, becauseof the different conventions, the role of ǫ in the mono-clinic case is played by ǫ . The simplified character tableIII for the D h group shows that the shear strain ǫ isnot invariant under the operations of the group. Hence,the C negative discontinuity at T c cannot be explainedwith the simplest model Eq.(1). One must introduce anorder parameter with two orthonormal basis functionswith respective complex coefficients η and η . Let usfirst neglect the strain terms and consider the most gen-eral free energy functional that is invariant under thepoint group and phase changes of the order parameter[29] F η = a | η | + b | η | + a | η | + b | η | + | η | | η | ( γ + δ cos(2∆ θ )) . (2)In this expression, γ and δ are constants and ∆ θ is thephase difference between the two components of the or-der parameter. If δ is positive, this free energy will beminimized by ∆ θ = ± π/ , while if δ is negative ∆ θ = 0or π will be the minimum. The case ∆ θ = ± π/ F ηε = gǫ | η || η | cos(∆ θ ) (3)must be allowed by symmetry. Also, cos(∆ θ ) should notvanish, thus removing the possibility of a time-reversalsymmetry-breaking state. Since ε transforms accordingto the B g representation, there are only two possibilities.Either one of the η is invariant ( A g ) and the other onetransforms as B g or one of the components transformslike B g and the other one like B g . This can be checkedby showing that the product of the characters in TableIII is unity for all group operations applied to F ηε . Notethat both of the above possibilities for η and η forbida linear coupling to ε since the latter transforms like B g . This explains the absence of a discontinuity in thecorresponding elastic constant.Since both scanning tunneling spectroscopy [2] andthermal conductivity [3] suggest nodes along the x and y axis, this forces us to choose an order parameter thathas a mixed A g + B g character, namely d xy + d z ( x + y ) .The nodeless s case has the same symmetry as d xy somore generally it should be included but it suffices thatits amplitude be smaller than that of d xy for the nodesof s + d xy to survive. They are just shifted from theirposition in the d xy case. The d z ( x + y ) component doesnot remove the nodes in the plane, but it clearly breaksmirror symmetry about the planes.On general grounds, the free energy Eq.(2) predictstwo different T c ’s since there is no a priori reason why a and a should vanish at the same T . That is dif-ferent from the case of Sr RuO where the two compo-nents of the order parameter necessary to explain thedata belong to a single two-dimensional representation E u of the point group D h [22]. Although the presentlattice is nearly triangular, the two components of theorder parameter that we found do not coalesce into asingle two-dimensional representation of the D h group[30]. Nevertheless, the mixed A g + B g representationfor the orthorhombic crystal does coalesce into the one- dimensional A g representation of its monoclinic cousin,leading to a single T c in that case. Hence, we do notexpect a large difference between the two transition tem-peratures of the orthorhombic crystal. Our experimentaldata in Fig. 3 show a rather broad transition with anextended region of SC fluctuations that could mask thisdifference between the two transitions.The presence of a d z ( x + y ) component to the order pa-rameter suggests that interlayer hopping is an importantvariable in the problem. The value of this parameterhas been estimated from angle-dependent magnetoresis-tance oscillations [31]. Thermal expansion data [32] alsodisclose a striking anisotropy and dependence of T c oninterlayer effects [33] that are unlikely to be captured bya 2D purely electronic model.In summary, symmetry and the observed discontinu-ities at T c in the ultrasonic velocity data for two com-pounds of the layered κ -(ET) X organic superconduc-tors demonstrate that the order parameter must haveat least two components in the orthorhombic compound κ -(ET) Cu[N(CN) ]Br. Consistency with other exper-iments selects A g + B g (equivalently d xy + d z ( x + y ) ).The two components coalesce into a one-dimensional ir-reducible representation A g in the monoclinic compound κ -(ET) Cu(NCS) . Nodes are not symmetry imposedbut are symmetry allowed and are likely to occur in elec-tronic pairing mechanisms. The d z ( x + y ) component ofthe order parameter suggests that further studies of in-terlayer coupling are called for.The authors achnowledge stimulating discussionswith Claude Bourbonnais, David S´en´echal and PeterHirschfeld and they thank Mario Castonguay for tech-nical support. This work was supported by grants fromthe Fonds Qu´eb´ecois de la Recherche sur la Nature etles Technologies (FQRNT), from the Natural Scienceand Engineering Research Council of Canada (NSERC).A.-M.S.T. also acknowledges the support of the Tier ICanada Research Chair program and of the CanadianInstitute for Advanced Research. [1] B.J. Powell and Ross H.McKenzie, J. 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