Microscopic theory of spin-triplet f-wave pairing in quasi-one-dimensionalorganic superconductors

Y. Tanaka1,2 and K. Kuroki31Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan2CREST Japan Science and Technology Corporation, Nagoya 464-8603, Japan

3Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan(Received 24 February 2004; published 10 August 2004)

We present a microscopic theory of fluctuation-mediated pairing mechanism in organic superconductorssTMTSFd2X, where the experimentally observed coexistence of 2kF charge fluctuation and 2kF spin fluctuationis naturally taken into account. We have studied, within the random phase approximation, the extended Hub-bard model at quarter filling on a quasi-one-dimensional lattice, where we consider the off-site repulsiveinteraction up to third next-nearest neighbors along with the on-site repulsion. The results show that spin-tripletf-wave-like pairing can be realized in this system, dominating over singletd-wave-like pairing, if 2kF spin and2kF charge fluctuations coexist.

DOI: 10.1103/PhysRevB.70.060502 PACS number(s): 74.70.Kn, 74.20.Rp, 74.20.Mn

It has been a long standing issue to clarify the supercon-ducting state of quasi-one-dimensional(q1D) organic super-conductorssTMTSFd2X (X=PF6, ClO4, etc.), so-called theBechgaard salts.1,2 One may expect unconventional super-conducting states due to the quasi-one-dimensional nature ofthese materials as well as the electron correlation effects gen-erally seen in organic materials. In fact, an unchangedKnight shift acrossTc,

3 along with a largeHc2,4,5 supports a

realization of spin-triplet pairing. As for the orbital part ofthe order parameter, there have been NMR experiments3,6

which may be regarded as evidence for the presence of nodes(although this is still controversial), while a thermal conduc-tivity measurement suggests absence of nodes.7

Theoretically, the tripletp-wave pairing state in which thenodes of the pair potential do not intersect the Fermi surfacehas been proposed.8–10 However, the occurrence of tripletpairing in sTMTSFd2X is puzzling11 from a microscopicpoint of view since superconductivity lies right next to a 2kFspin density wave(SDW) in the pressure-temperature phasediagram.12 Naively, SDW spin fluctuations should favorspin-singlet d-wave-like pairing as suggested by severalother authors.13–15 One should note, however, that the insu-lating phase is not pure SDW at least for some anions,namely, 2kF charge density wave(CDW) actually coexistswith 2kF SDW.16,17 In fact, one of the present authors hasproposed18 that triplet f-wave-like[see Fig. 1(c) for a typicalpair potential] pairing may dominate overp-wave pairingand become competitive againstd-wave-like pairing [seeFig. 1(b)] due to a combination of quasi-1D(disconnected)Fermi surface and the coexistence of 2kF SDW and 2kFCDW fluctuations. A similar scenario has been proposed byFuseyaet al.19 Concerning thef-wave versusd-wave com-petition, it has also been proposed that magnetotunnelingspectroscopy20 via Andreev resonant states21 is a promisingmethod to detect thef-wave pairing.

However, there has been nomicroscopictheory for thishypotheticalf-wave pairing insTMTSFd2X starting from aHamiltonian that assumes only purely electronic repulsiveinteractions.22 To resolve this issue, we study an extended

Hubbard model at quarter filling on a quasi-one-dimensionallattice within the random phase approximation(RPA). Weconsider off-site repulsions up to third nearest neighborsalong with the on-site repulsion in order to naturally takeinto account the coexistence of 2kF charge and 2kF spin fluc-tuations. The merit of adopting RPA23 is that we can easilytake into account the off-site repulsion as compared to fluc-tuation exchange(FLEX) approximation, where it is by nomeans easy to take into account distant interactions.24,25

The model Hamiltonian is given as

H = − oki,jl,s

tijcis† cjs + Uo

i

ni↑ni↓ + oui−j u=ml

Vi,jninj ,

wherecis† creates a hole[note thatsTMTSFd2X is actually a

3/4 filling system in the electron picture] with spin s= ↑ ,↓at site i =sia, ibd. Here, ki , jl stands for the summation overnearest neighbor pairs of sites. As for the hopping param-eters, we taketij = ta for nearest neighbor in the(most con-ductive) a direction, andtij = tb for nearest neighbor in thebdirection. We choosetb=0.2ta to take into account the quasi-one-dimensionality.ta is taken as the unit of energy through-out the study.U andVij are the on-site and the off-site repul-sive interactions, respectively. We take distant off-siterepulsions because it has been shown previously that nearestneighbor and second nearest neighbor off-site repulsion isnecessary to have coexistence of 2kF spin and 2kF chargedensity waves.26 Here we take off-site repulsions up to thirdnearest neighbors, namely,Vi,j =V0, V1, and V2 with m= uia− jau=1,2, and 3,respectively. The effect of the third nearestneighbor repulsion,V2, will be discussed at the end of thepaper. The effective pairing interactions for the singlet andtriplet channels due to spin and charge fluctuations are givenas

Vssq,vld = U + Vsqd + 32U2xssq,vld − 1

2fU + 2Vsqdg2xcsq,vld,

s1d

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Vtsq,vld = Vsqd − 12U2xssq,vld − 1

2fU + 2Vsqdg2xcsq,vld,

s2d

within RPA, where

Vsqd = 2V0 cosqx + 2V1 coss2qxd + 2V2 coss3qxd, s3d

and vl is the Matsubara frequency. Here,xs and xc are thespin and charge susceptibilities, respectively, which aregiven as

xssq,vld =x0sq,vld

1 − Ux0sq,vld

xcsq,vld =x0sq,vld

1 + fU + 2Vsqdgx0sq,vld. s4d

Here,x0 is the bare susceptibility given by

x0sq,vld =1

Nop

fsep+qd − fsepdvl − sep+q − epd

with ek=−2ta coska−2tb coskb−m and fsepd=1/fexpsep/Td+1g. x0 peaks at the nesting vectorQ[=sp /2 ,pd here] of the Fermi surface. The terms propor-tional to xs and xc in Eqs. (1) and (2) represent effectivepairing interactions due to the spin and charge fluctuations,respectively. The chemical potentialm is determined so thatthe band is quarter-filled, which ism=−1.38. We takeU=1.6 throughout the study, which is large enough to havestrong 2kF spin fluctuations[large xssQd=xssQ,0d] but notso large as to drive SDW instability at high temperatures.The off-site interactions are chosen so that 2kF charge fluc-tuations are induced as will be discussed. In the actual nu-merical calculation, we takeN=400340 k-point meshes ex-cept for low temperatures, where we takeN=800380meshes.

To obtain the onset of the superconducting state, we solvethe gap equation within the weak-coupling theory:

lDskd = − ok8

Vs,tsk − k8,0dtanhsbek8/2d

2ek8Dsk8d. s5d

The transition temperatureTC is determined by the condition,l=1. In the weak coupling theory,v dependence of the pairpotentialDskd is neglected. Although this approximation isquantitatively insufficient, it is expected to be valid forstudying the pairing symmetry ofDskd mediated by bothspin and charge fluctuations. In the following calculations,we study triplet and singlet cases withDskd=−Ds−kd andDskd=Ds−kd, respectively. We definefsskd=Dskd /DM andftskd=Dskd /DM for singlet and triplet pairing, respectively,whereDM is the maximum value of the pair potential.

Equations(1)–(4), show that whenU,−fU+2VsQdg issatisfied,uVssQdu,uVtsQdu holds, apart from the first orderterms, which are negligible in the limit of strong spin and /orcharge fluctuations(but turn out to be important in the actualcases considered later). This, along with the disconnectivityof the Fermi surface(note that the number of nodes inter-secting the Fermi surface is the same betweenf and d

waves), is expected to make spin tripletf-wave pairing com-petitive against singletd-wave pairing.18

We now discuss the calculation results. First, we focus onthe case where spin fluctuation is dominant, e.g.,V0=V1=V2=0. As shown in Fig. 1, the magnitude ofl for thesinglet case is much larger than that for the triplet case. Theresulting singlet pair potentialfsskd changes sign as +−+−along the Fermi surface[see Fig. 1(b)]. We call thisd-wavepairing, wherefsskd is roughly proportional to coss2kxd. Onthe other hand, the triplet pair potentialftskd changes sign as+−+−+− along the Fermi surface[see Fig. 1(c)]. We callthis f wave, whereftskd is roughly proportional to sin 4kx.The results here are expected from the previous FLEXstudy.18

We now discuss the cases where we turn on the off-siterepulsions. In order to to have the coexistence of 2kF spinand 2kF charge fluctuations as experimentally observed,namely, to have xssQd,−xcsQd=−xcsQ,0d, U,−fU+2VsQdg must be satisfied as mentioned earlier. To accom-plish this,V1 has to be close toU /2, as can be seen from Eq.(3). Namely, since thex component ofQ is Qx=p /2, theV1term in Eq. (3) is dominant for q.Q, making U,−fU+2VsQdg if V1.U /2. Thus, we chooseV1=U /2=0.8. As forthe other off-site repulsions, we first chooseV0=1.2 andV2=0.5 as a typical value. As shown in Figs. 2(a) and 2(b), xsandxc peak aroundska,kbd= ± sp /2 ,pd , ±sp /2 ,−pd and themaximum values are both about 9.1, so that we have thesituation where 2kF spin and 2kF charge fluctuations coexist.As shown in Fig. 2(c), the magnitude ofl for triplet pairingis now much larger than that for singlet pairing. The corre-sponding singlet pair potentialfsskd has thed-wave form asshown in Fig. 2(d), while the triplet oneftskd has thef-waveform as shown in Fig. 2(e). Note that the result ofltriplet

FIG. 1. Calculation results forU=1.6,V0=V1=V2=0: (a) Tem-perature dependence ofl for singlet (solid line) and triplet(dottedline) pairings. Contour plots of(b) fsskd and(c) ftskd at T=0.01. In(b) and (c), the solid lines represent the Fermi surface and thedotted lines denote the nodal lines of the pair potentials.

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@lsinglet is rather unexpected from the previous phenomel-ogical argument18 because whenxcsQd=xssQd, the f wave isonly degeneratewith thed wave in the previous theory. Theorigin of this discrepancy is the first order terms in Eqs.(1)and (2), which are neglected in the phenomelogical theory.Thus, we have obtained a remarkable result here, namely, thef wave cancompletelydominate over thed wave when 2kFspin and charge fluctuations coexist.

In order to further look into this point, we next reduceV1from 0.8, thereby suppressing the charge fluctuation. Themaximum value ofxc (not shown) is 5.2 and 3.2 forV1=0.78 andV1=0.75, respectively. Although the maximumvalue ofxc is smaller than that ofxs s=9.1d in these cases,lfor the triplet case is still larger thansV1=0.78d or competi-tive againstsV1=0.75d that for the singlet case as seen in Fig.3. This means thatf-wave pairing has a chance to be realizedeven if spin fluctuation dominates, as far as2kF charge fluc-tuation exists.18

Finally, in order to look into the effect of the third nearestneighbor interactionV2, we setV2=0 leaving the other pa-rameters the same as in Fig. 2. As seen in Fig. 4(a), the peakof xc is slightly shifted towards 4kF sqx=pd when V2=0,compared to the case ofV2=0.5t shown in Fig. 2(a). [xs isthe same as Fig. 2(a) sinceV2 does not affectxs within thepresent formalism.] In this case, the singlet-triplet competi-tion becomes much more subtle as seen in Fig. 4(b). Thereason for this can be found in Fig. 4(c), namely,fsskd inthis case has the same sign on most of the portion of theFermi surface. In other words, it is more like thes wave than

d wave. Thiss wave pairing is induced by charge fluctuation,which does not totally cancel out with spin fluctuation in Eq.(1) because the wave vector at whichxc peaks deviates fromthat forxs. Since thes-wave pair potential has the same signon most of the portion of the Fermi surface, almost all thepair scattering processes on the Fermi surface, mediated bytheattractiveinteraction[note the minus sign in Eq.(1)] dueto charge fluctuation, have positive contributions to super-conductivity, making singlet pairing much more enhancedcompared to the case with nonzeroV2. Conversely, thepresent results show thatV2 has the effect of stabilizing 2kFcharge fluctuation, which has a tendency to shift towards 4kF

FIG. 2. Calculation results forU=1.6, V0=1.2, V1=0.8, V2

=0.5: (a) xsskd at T=0.01; (b) xcskd at T=0.01; (c) temperaturedependence ofl for singlet (solid line) and triplet (dotted line)pairings. Contour plots of(d) fsskd and(e) ftskd at T=0.01. In(d)and (e), the solid lines represent the Fermi surface and the dottedlines denote the nodes of the pair potentials.

FIG. 3. Calculation results forU=1.6,V0=1.2, andV2=0.5: (a)Temperature dependence ofl for singlet (solid line) and triplet(dotted line) pairings forV1=0.78;(b) temperature dependence oflfor singlet (solid line) and triplet (dotted line) pairings for V1

=0.75.

FIG. 4. Calculation results forU=1.6, V0=1.2, V1=0.8, V2=0:(a) xcskd at T=0.01; (b) temperature dependence ofl for singlet(solid line) and triplet (dotted line) pairings. Contour plots of(c)fsskd and(d) ftskd at T=0.01. In(c) and(d), the solid lines repre-sent the Fermi surface and the dotted lines denote the nodes of thepair potentials.

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fluctuation when onlyV0 andV1 are present, and this effectin turn suppresses singlet pairing because in that case astrong cancellation occurs between the third and the fourthterms in Eq.(1). Since the screening effect is known to beweak in quasi-one-dimensional systems, it is likely that sucha distant off-site repulsion is present in the actualsTMTSFd2X.

To summarize, we have presented a microscopic theory ofpairing mechanism in organic superconductorssTMTSFd2X,where we have taken into account the coexistence of the 2kFcharge fluctuation and 2kF spin fluctuation by consideringoff-site repulsions up to third nearest neighbors. We haveshown that thef-wave triplet pairing symmetry can be real-ized in this system when 2kF charge density fluctuation and2kF spin density fluctuation coexists. Surprisingly, the condi-tion for realizingf-wave pairing is eased compared to that inthe previous phenomelogical theory.18

In this paper, we have neglected the realistic shape of theFermi surface observed in the actualsTMTSFd2X.27 Al-though the influence of this effect on thef-wave pairing isexpected to be small because thex component of the nestingvector is close top /2 a detailed analysis remains as a futurestudy.

Let us also comment on the one-dimensional fluctuationeffects that are not taken into account in the present RPAapproach. We have recently performed ground state quantumMonte Carlo study for the Hubbard model with only theon-site repulsion on a quasi-1D lattice, in which SDW fluc-tuations strongly dominate over CDW fluctuations.28 In sucha case, it is expected thatd-wave pairing strongly dominatesover f-wave pairing from the standpoint of RPA, but ourMonte Carlo results show thatf-wave pairing is surprisinglycompetitive againstd-wave even though spin fluctuations. charge fluctuations clearly holds. Thus, the one-dimensional effects that are not taken into account in thepresent approach are expected to work in favor of thef-wavepairing. In this sense, we believe thatf-wave pairing candominate overd wave in a parameter regime with off-siterepulsions smaller than those adopted in the present study.This point remains as an interesting future problem. In anycase, our overall conclusion thatf-wave superconductivity islikely to be taking place in thesTMTSFd2X compoundsshould not be altered even if we take into account the one-dimensional effects properly.

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