Pairing Symmetry Competition in Organic Superconductors

0.46 MB·English

by  Kazuhiko Kuroki

Typeset with jpsj2.cls <ver.1.2> Full Paper
Pairing Symmetry Competition in Organic Superconductors
Kazuhiko Kuroki
Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
6 Areviewisgivenontheoreticalstudiesconcerningthepairingsymmetryinorganicsupercon-
0 ductors. In particular, we focus on (TMTSF)2X and κ-(BEDT-TTF)2X, in which the pairing
0
symmetry has been extensively studied both experimentally and theoretically. Possibilities of
2
various pairing symmetry candidates and their possible microscopic origin are discussed. Also

n some tests for determining the actual pairing symmtery are surveyed.
a
J KEYWORDS: organic superconductivity, pairing symmetry, (TMTSF)2X, κ-(BEDT-TTF)2X, spin and
4 charge fluctuations, Fermi surface

]
l
e 1. Introduction
-
r
Possibleoccurrenceofunconventionalsuperconductiv-
t
s ity in organic conductors1–3 has been of great interest
.
t recently. Microscopically understanding the mechanism
a
m of pairing in those materials is an intriguing theoreti- a
cal challenge. Among the various candidates of uncon-
-
d ventional superconductors, in this paper we will focus b
n on two groups of superconductors in which the pairing 10
[co tschyamellymBaeecnthrdygaehxaarpdserbsimaeletensn,teawxllhtyee,nrnesiavTmeMleylTys,St(uFId)iise(dTaMnboTatbShbFrt)eh2vXeioa,r4teiotoinr- temperature(K) 1 ( +SCDDWW(cid:13))
1 SC
for tetramethyltetraselenafulvalene and X stands for an 0.1
v 10 20
0 anionsuchas PF6,AsF6,ClO4,etc., and(II) κ-(BEDT- pressure(kbar)
6 TTF) X,5,6 where BEDT-TTF is an abbreviation for
2
0 bisethylenedithio-tetrathiafulvalene and X=Cu(NCS)2, Fig. 1. Left panel:Lattice structure of (TMTSF)2X in the a-
1 Cu[N(CN) ]Br, Cu (CN) , I , etc. The key factors to b plane. Upper right panel: Typical shape of the Fermi sur-
0 2 2 3 3 face.(Reprinted withpermissionfromref.24.Copyright 1983 by
be focused throughout the paper are the band struc-
6 EDP Sciences.) Lower right panel: Schematic phase diagram of
0 ture and the shape of the Fermi surface, the band fill- (TMTSF)2X.
/ ing,andthewavenumberdependentpairinginteractions
t
a mediated by spin and/or charge fluctuations and/or by
m
phonons.Superconductivitynearchargeorderedstateas increasinghydrostaticpressure,theSDWtransitiontem-
- in θ-(BEDT-TTF)2X and α-(BEDT-TTF)2X7 has also perature decreases, and superconductivity with a tran-
d
n been investigated extensively, but will not be discussed sition temperature (Tc) of 0.9 K appears at 12 kbar.12
o here.8 Superconducting states induced under high mag- A similar phase diagram is obtained for X=AsF6.13 It
c netic fields, such as the Fulde-Ferrel-Larkin- Ovchin- should be mentioned here that X-ray diffuse scattering
v: nikov(FFLO)state,9,10 arealsobeyondthescopeofthe experiments have revealed a coexistence of 2kF charge
i present paper.11 densitywave(CDW)intheSDWphaseforX=PF ,14,15
X 6
while the amplitude of the 2k CDW is very small for
r 2. (TMTSF)2X F
a X=AsF6.15 It should also be noted that the easy axis of
2.1 Lattice Structure and the Phase Diagram theSDWisintheb′ direction,16,17whichisthedirection
Thelatticestructureof(TMTSF)2XisshowninFig.1. normaltothe a-cplane andsomewhattiltedfrombdue
Themoleculesarestackedalongthea-axis(denotedasa to the triclinic symmetry of the lattice.
hereafter),whichisthemostconductingaxisbecausethe (TMTSF) ClO becomes superconducting at ambi-
2 4
overlapofthemolecularorbitals,orientedinthestacking ent pressure when the system is cooled down slowly
direction, is large. The molecules are weakly dimerized enough for the anions to order at 24 K.18 On the other
along the stacks. The charge transfer with the anions hand, when the cooling rate is fast, the anions are
existing in between the conducting stacks results in one frozeninrandomdirections,andinthiscase,SDWtakes
hole per two molecules. There is a weak overlap of the place insteadof superconductivity.19 It has alsobeen re-
orbitals in the b direction, resulting in a weak two di- vealedthatsuperconductivityisdestroyeduponalloying
mensionality. (TMTSF) ClO witha smallamountofReO ,andwith
2 4 4
A schematic phase diagram of (TMTSF)2X is shown further alloying, an SDW phase appears.20–23
inFig.1.Atambientpressure,(TMTSF) PF undergoes
2 6
a2k spindensitywave(SDW) transitionat12K.Upon
F
2 J.Phys.Soc.Jpn. FullPaper AuthorName
2.2 Electronic Structure where ψ is the digamma function, T is the transition
c0
Reflectingthelatticestructureandalsotheanisotropy temperaturewithoutimpurities,andαisthepairbreak-
of the orbitals, the band structure of (TMTSF) X is ingparameter,whichisdeterminedbythescatteringrate
2
strongly one dimensional, i.e., the ratios of the hopping due to non-magnetic impurities. This is in fact the same
integrals in a, b, and c directions are t /t ∼ 0.2 and astheformuladerivedbyAbrikosovandGor’kovforthe
b a
t /t ∼ 0.05, where t = 200 ∼ 300 meV.24,25 Since t case of s-wave pairing with magnetic impurities.37 Since
c b a c
is extremely small, it is highly likely that the essential a triplet superconductivity has an odd parity gap, Tc
mechanism of the superconductivity lies within the two should be sensitive to the introduction of non-magnetic
dimensional lattice (a-b plane). A typical Fermi surface impurities. Note, however,that the condition (2) can be
is shown in Fig.1, which is open in the k direction due satisfiedfor a superconducting state with anevenparity
b
to the quasi-one-dimensionality. We stress here that the gap that changes sign on the Fermi surface, so that the
anisotropy of the hopping integrals within the a-b plane sensitivity to the presence of impurities alone of course
largely owes to the fact that molecular orbitals are di- does not necessarily imply triplet pairing. In this sense,
rectedtowardtheadirection,whilethedistancebetween thesensitivityoftheTctonon-magneticdefectsconcerns
the molecules in the b direction is only about two times the orbital part of the pair wave function.
largerthanthatinthea direction.Thehoppingintegral Another experiment that indicated the possibility of
t alternates along the a direction by about 10 ∼ 20% unconventionalpairingconcerningtheorbitalpartisthe
a
due to the dimerization of the molecules. If we neglect NMR experiment for X=ClO4 performed by Takigawa
this dimerization, the system is described by a 3/4-filled et al.40 Namely, the proton spin-lattice relaxation rate
single band model, whose band dispersion is given as 1/T1 at zero magnetic field exhibits no coherence peak,
and follows a power law temperature dependence close
ε(k)=2t cos(k )+2t cos(k ). (1)
a a b b to T3. Such a behavior is generally characteristic to su-
Here, only the hoppings between the nearest neighbor- perconductivity with a gap having line nodes.39 In fact,
ing molecules in the a and b directions are considered. Hasegawa and Fukuyama41 studied various types of sin-
Latticeconstants(neglectingthedimerization)aretaken glet and triplet anisotropic pairings within the mean
as the units of the length. Many of the theoretical ap- field approximation for a model with on-site and near-
proaches have been based on this 3/4-filledband model, est neighbor attractive interactions, and calculated the
but in some studies, the strong dimerization limit has spin-lattice relaxation rate. There it was shown that a
been assumed, where each dimer of molecules is con- singlet pairing without gap nodes on the Fermi surface
sidered as a site, so that the band now becomes half exhibits a large coherence peak followedby an exponen-
filled.26,27 There is also a study based on a two band tial decay of 1/T1, while for a triplet pairing with gap
model that maintains the realistic dimerization struc- nodes atka =0 andthereby no nodes on the Fermi sur-
ture.28 face,whichwillbecalledp-wavehereafter(Fig.2(c)),the
coherence peak becomes smaller but still exists. Singlet
2.3 ExperimentsConcerningthePairingSymmetryand and triplet pairings with line nodes of the gap intersect-
Their Theoretical Interpretations ing the Fermi surface cannot be distinguished from the
Early experiments for (TMTSF)2X, such as the spe- temperature dependence of 1/T1; they both exhibit es-
cific heat29,30 and the upper critical field measure- sentially no (or very small) coherence peak and a power
ments31–33 hadbeen interpretedwithin the conventional law decay roughly proportional to T3, which is similar
s-wave pairing. However, Abrikosov34 pointed out the to the experimentally observed behavior.40 A more re-
possibility of spin-triplet pairing based on the fact that cent 1/T1 measurement has been performed on X=PF6
T is very sensitive to the existence of non-magnetic de- by Lee et al.,42,43 who have found a similar behavior of
c
fects.20,21,35,36 Morerecently,Jooet al.22,23 haveshown 1/T1 when a small magnetic field H is applied parallel
that the sensitivity of Tc to non-magnetic impurities to b′, but also an anomalous 1/T1 ∼ T at low tempera-
(ReO ) in (TMTSF) ClO is precisely what is expected tures for high magnetic fields. On the other hand, Belin
4 2 4
from the Tc reduction formula37,38 for unconventional and Behnia showed for X=ClO4 that the thermal con-
pairing. Generally, in a superconducting state that sat- ductivity rapidly decreases with lowering the tempera-
isfies the condition turebelowTc,indicatingtheabsenceoflowlyingexcita-
tions, and thus a fully gapped superconducting state.44
F(k,iω )=0, (2)
n A possible explanation for this discrepancy between the
Xk conclusions of the NMR and the thermal conductivity
thepresenceofnon-magneticimpuritiesispairbreaking, experiments will be discussed in section 2.7.
andthusstronglysuppressesT .39 Here,F istheanoma- Before discussing the experimental results concerning
c
lousGreen’sfunction,andthecondition(2)roughlycor- thespinpartofthepairwavefunction,letusbrieflysum-
respondstoavanishingsummationofthe superconduct- marize some general aspects of spin triplet pairing.39 In
ing gap function ∆(k) over the Fermi surface. The T the case of triplet pairing, both the diagonal and the
c
reduction in this case is given in the form,38 non-diagonal elements of the superconducting order pa-
rameter matrix
T 1 1 α
c
ln =ψ −ψ + , (3) ∆ (k) ∆ (k)
(cid:18)Tc0(cid:19) (cid:18)2(cid:19) (cid:18)2 2πTc(cid:19) ∆ˆ(k)= ↑↑ ↑↓ (4)
(cid:18) ∆↓↑(k) ∆↓↓(k) (cid:19)
J.Phys.Soc.Jpn. FullPaper AuthorName 3
remain finite in general, where ∆σσ′ are given as ∆↑↑ = 2.4 Spin-fluctuation-mediated d-wave pairing
−dx+idy, ∆↑↓ = ∆↓↑ = dz, and ∆↓↓ = dx+idy, using In this and the next two subsections, we discuss some
the vector d = (dx,dy,dz). The order parameter vector mechanisms for anisotropic pairing in TMTSF salts.
d lies in the direction perpendicular to the total spin of Since the superconducting phase lies close to the SDW
the triplet pairs. phase, and a number of experiments suggest the possi-
Spin-tripletsuperconductivitycanbeidentifiedbythe bility of anisotropic, unconventional pairing, it is natu-
NMR Knight shift measurement, which probes the uni- ral to expect that the spin fluctuations mediate (or at
formspinsusceptibility.TheKnightshiftdecreasesbelow least play an important role in) the Cooper pairing in
Tc for singlet pairing, while it stays constant for triplet TMTSFsalts,aswaspointedoutby Emery.56 Thespin-
pairingwhenthemagneticfieldH isappliedperpendicu- fluctuation-mediated pairing scenario has in fact been
lartod.Anotherpossiblewayofdetectingtripletpairing supportedbyseveraltheoreticalstudiesonthequasi-one-
is to measure the upper critical field Hc2. Cooper pair- dimensional Hubbard model, in which the on-site repul-
ing under magnetic field is limited by both orbital and sive interaction U is considered along with the kinetic
paramagnetic effects.45,46 For triplet pairing, however, energy part considered in section 2.2. The Hamiltonian
theparamagneticlimit(the Paulilimit,orthe Clogston- is given in standard notation as,
Chandrasekhar limit) is overcome when the magnetic
field is applied perpendicular to d. H = tijc†iσcjσ+U ni↑ni↓,
Now, as for the actual experimental results, Lee et al. <iX,j>,σ Xi
found for X=PF6 that the Knight shift does not de- where c† creates an electron with spin σ at site i (i.e.,
iσ
crease below Tc for magnetic fields applied parallel to thei-thmolecule),n =c† c ,andt =t andt =t
a42 (1.43T) or b′ (2.38T).43 These results indicate that iσ iσ iσ ij a ij b
for intrachainandinterchain nearestneighbor hoppings,
thepairingindeedoccursinthespin-tripletchannel,and
respectively. There, superconductivity has been studied
thateitherdkc,ordrotatesinaccordwiththedirection
using random phase approximation (RPA),57 fluctua-
of the magnetic field to satisfy d⊥H.
tionexchangeapproximation(FLEX),26 thirdorderper-
AsfortheuppercriticalfieldH ,Gor’kovandJ´erome
c2 turbation,27 or quantum Monte Carlo method.58 Here,
pointed out in the early days that H extrapolated to
c2 based on RPA equations (which will be written down in
T =0 maylargelyexceedthe Paulilimit, suggestingthe
a general form for later use) for the single band Hub-
possibility of spin-triplet pairing.47 More recently, the
bard model at quarter filling (quarter filling of holes,
upper critical field has been studied with higher accu-
to be precise), we summarize the mechanism in which
racy and with precise orientation of the magnetic fields.
2k spinfluctuations leadtod-wavelikepairing.Within
Hb ,theuppercriticalfieldforH kb′ hasbeenfoundto F
c2 RPA,singletandtripletpairinginteractionsaregivenin
exceedthe Paulilimit forX=PF 48 andalsofor ClO .49
6 4 the form,59–61
Evenif a spin-tripletpairingoccurs,the pairingcanstill
be orbitally limited, but Lebed50 and later Dupuis et Vs(q) = U +V(q)+ 3U2χ (q)− 1(U +2V(q))2χ (q)
s c
al.51 showed that a magnetic field induced dimensional 2 2
crossover from three to two dimensions can strongly en- 1 1
Vt(q) = V(q)− U2χ (q)− (U +2V(q))2χ (q), (5)
hance the orbital limit of the critical field. Thus, as far 2 s 2 c
as Hcb2 is concerned, the experimental results seem to where V(q) is the Fourier transformof the off-site inter-
be consistent with the above interpretations of triplet
actions(electroninteractionsbetweennearestneighbors,
pairing with d k c, or a rotatable d.52 However, the in-
etc.),whichis0forthe Hubbardmodel.Here,χ andχ
s c
terpretation on the temperature dependence of Ha has
c2 are the spin and the charge susceptibilities, respectively,
been controversial.ForX=PF , there is aninversionbe-
6 which are given as
tweenHa andHb ,whereHa >Hb forsmallmagnetic
afiselda,fubnucct2tiHonca2o<f TcH2 ccb2hafnogreHs fcr>2om1.6aTcc2.onMvoerxeotovear,cHonca2c(aTve) χs(q)= 1−χU0(χq0)(q),
curve above H = 1.6T. From these experiments, Lebed χ (q)
et al. proposed that db =0 and da 6=0 (d=(da,db,dc)) χc(q)= 1+(U +20V(q))χ (q). (6)
0
assuming strong spin-orbit coupling, so that the pairing
is Pauli-paramagneticallylimited for H ka for H <1.5 Here χ0 is the bare susceptibility given by
T,whilethechangeofthecurvatureofHa(T)forhigher 1 f(ε(p+q))−f(ε(p))
c2 χ (q)=
magneticfieldsmaybebecausedrotatestobecomeper- 0 N ε(p)−ε(p+q)
Xp
pendicular to H, or may be due to an occurrence of the
FFLO state.53 On the other hand, Duncan et al. argued wheref(ε)istheFermidistributionfunction.Withinthe
that spin-orbit coupling should be weak since the heav- weakcoupling BCS theory,T is obtained by solvingthe
c
iest element in (TMTSF)2X is Se, so that d should be linearized gap equation,
abletorotateaccordingtothedirectionofH evenforlow tanh(βε(k′)/2)
magneticfields.54Aclearunderstandingforthedirection λs,t∆s,t(k)=− Vs,t(k−k′) ∆s,t(k′).
2ε(k′)
of d, provided that triplet pairing does indeed occur,55 Xk′
requires further theoretical and experimental study. (7)
4 J.Phys.Soc.Jpn. FullPaper AuthorName
The eigenfunction ∆s,t of this eigenvalue equationis the (a) (b) (c)
gap function. The transition temperature T is deter- d-wave f-wave p-wave
c
minedasthetemperaturewheretheeigenvalueλreaches p + p - + p
unity.Inthe summationoverk′ intherighthandsideof
eq.(7),themaincontributioncomesfromk′ontheFermi kb - Q2kF - kb + - kb - +
surface because of the factor tanh2(βε(εk(k′)′)/2). If we multi- -p-p/2 + p/2 -p-p/2 - + p/2 -p-p/2 p/2
ply both sides of eq.(7) by ∆s,t(k) and take summation ka ka ka
over the Fermi surface, we see that the quantity
(d) (e) (f)
Vesff,t = Pk,k′Vs,t(kk−(∆ks′,)t(∆ks),)t2(k)∆s,t(k′) (8) 2p D s (cid:13)(d-w+ave) 2p D t (f--wav+e) 2|(cid:13)pG|2 (Fermi surface)
hastobe positiveandlarPgeinorderto havelargeλ,i.e., - +
inordertohavesuperconductivitywiththegap∆s,t(k). kbp - kbp - kbp
Due to the good nesting of the Fermi surface, the
+ - +
bare susceptibility χ0(q) peaks at the nesting vector 00 p 2p 00 p 2p 00 p 2p
q =Q ,andsinceU >0andV(q)=0,χ (q)becomes ka ka ka
2kF s
large at q = Q , while χ (q) remains small at all q.
2kF c
Within this formulation, the SDW transition tempera- Fig. 2. Upper panel: candidates for the gap function of
tureisdeterminedasthetemperaturewhereUχ (Q ) (TMTSF)2XareschematicallyshownalongwiththeFermisur-
0 2kF face(solidcurves).(a)d-wave,(b)f-wave,(c)p-wave.Thedashed
reaches unity. Thus, in the vicinity of the SDW transi-
lines represent the nodes of the gap, whose kb dependence is
tion,the pairinginteractionsroughlysatisfy the relation
omittedforsimplicity.“+,−”representthesignofthegapfunc-
Vs(Q )=−3Vt(Q )>0. (9) tions. Lower panel:FLEX calculation results for the two band
2kF 2kF modelwithfinitedimerization.28(d)d-wavegap,(e)f-wavegap,
because the contribution from the spin fluctuations (f)|G(k,iπkBT)|2,whoseridgesrepresenttheFermisurface.
strongly dominates in eq.(5). Now, since the pairing in-
teractions have large absolute values at q = Q , the
2kF (k,iǫ ) denotes the wave vectors and the Matsubara
conditionto haveapositiveV ineq.(8)canbe approx- n
eff
frequencies, (ii) the effective electron-electron interac-
imately reduced to
tion V(1)(q) is calculated by collecting RPA-type dia-
Vs,t(Q )∆s,t(k)∆s,t(k+Q )<0,k,k+Q ∈F.S.
2kF 2kF 2kF grams consisting of the renormalized Green’s function,
(10)
namely, by summing up powers of the irreducible sus-
From this condition and eq.(9), we can see that the gap ceptibility χ (q) ≡ −1 G(k + q)G(k) (N:number
function has to change sign between k and k+Q irr N k
2kF of k-point meshes), (iii) tPhe self energy is obtained as
for singlet pairing, while the sign has to be the same Σ(k)≡ 1 G(k−q)V(1)(q), which is substituted into
across the nesting vector for triplet pairing. Since the N q
Dyson’s eqPuationin (i), and the self-consistentloops are
spin part of the pair wave function is antisymmetric
repeated until convergence is attained.
(symmetric) with respect to the exchange of electrons To obtain T , the linearized E´liashberg equation for
c
for spin singlet (triplet) pairing, the orbital part of the the singlet or the triplet gap function ∆s,t(k),
wave function, namely the gap function, has to satisfy
the condition ∆s(k) = ∆s(−k) (even parity gap) and λ∆s,t(k)=−T Vs,t(k−k′)G(k′)G(−k′)∆s,t(k′),
∆t(k) = −∆t(−k) (odd parity), for singlet and triplet N Xk′
pairings,respectively.The gapfunctions satisfying these (11)
conditions are schematically shown in Fig.2(a)(b). We is solved,where the singletorthe triplet pairinginterac-
will call the singlet pairing “d-wave” in the sense that tionsVs,t aregivenagainintheRPAformbutusingthe
the gap changes sign as +−+− along the Fermi sur- irreduciblesusceptibilityobtainedfromtherenormalized
face, while the triplet pairing will be called “f-wave” in Green’s functions insteadof the baresusceptibility. T is
c
the sense that the gap changes sign as +−+−+−.62 the temperature where the eigenvalue λ reaches unity.
Since the pairinginteractionis three times largerfor the KinoandKontaniappliedFLEXtothehalf-filledHub-
singlet pairing, d-wave pairing takes place in this case. bard model, i.e., the model in the strong dimerization
Note that a simpler form of an odd parity gap is the p- limit,26 where they obtained a finite T for the d-wave
c
waveshowninFig.2(c), whichchangessignas +−along pairing.Kurokietal.appliedFLEXtoatwo-bandmodel
theFermisurface.However,thisgapdoesnotsatisfythe with finite dimerization and with next nearest neighbor
condition (10) because the triplet pairing interaction is interchainhoppings,andalsofoundthatthed-wavepair-
negative for the Hubbard model at least within RPA. ing(Fig.2(d))isthemostdominantpairing,whiletriplet
Although we have adopted RPA equations in the f-wave pairing (Fig.2(e)) is subdominant.28
above, similar conclusions have been drawn from other Another approach for the Hubbard model is the per-
approaches as mentioned above. For example, an ap- turbational theory, where all the Feynman diagrams up
proachalongthelineofRPA,butmoresuitablefordeal- to a certain order are taken into account in the calcula-
ingwithstrongspinfluctuations,istheFLEXmethod.63 tionofthepairinginteractions.Applyingthe thirdorder
In the FLEX, (i) Dyson’s equation is solved to ob- perturbationtheoryto the half-filledmodelin the dimer
tain the renormalized Green’s function G(k), where k ≡ limit, Nomura and Yamada obtained finite values of T
c
J.Phys.Soc.Jpn. FullPaper AuthorName 5
for the d-wave pairing.It has been found there also that (a) (b)
the f-wave pairing is subdominant.27 1 ~~ 1 ~~
As for numerical calculations for finite size systems, 'd' 'd'
Kuroki and Aoki58 adopted the ground state quantum zzVsp 'f' with d (cid:13) z(cid:13) + -Vsp 'f' with d /(cid:13)/ z(cid:13)
Monte Carlo (QMC) technique.64–66 This method en- + -V / sp s zzV / sp s
ables us to accurately calculate correlation functions
within statistical errors for finite size clusters. Since the 0 0
0 1 ~~ 0 1 ~~
superconducting order parameter is always zero for fi- nodeless p (cid:13) Vc h / Vspzz Vc h / Vsp+ -
nite size systems, we instead calculate its fluctuation, with d // z
namely, the pairing correlation function, given in the
form hc c c†c† i, where i,j denotes the sites, and i Fig. 3. Phenomenologicalphasediagramofthepairingsymmetry
i+δ i j j+δ for(a)zkb28 and(b)zkc.
andi+δarethesitesatwhichtheCooperpairisformed.
Whenthetendencytowardssuperconductivityisstrong,
the pairing correlation decays slowly at large distances
while V is the contribution from the charge fluctua-
between sites i and j. Applying this method to the sin- ch
tions. There are two triplet pairing interactions:Vt⊥ for
gle band Hubbard model at quarter filling, it has been
d⊥z and Vtk for dkz.The contributionfromthe spin
found that the d-wavepairing correlationfunction is en-
fluctuationsisexpectedtobelargeintheeasyaxisdirec-
hanced at large distances by the presence of the on-site
tion of the SDW ordering.Then, taking the easy axis as
repulsion U.58 More recently, Kuroki et al. studied the
thez-axis,wemayassumeVzz(Q )≥V+−(Q )be-
pairingsymmetrycompetitiononthe Hubbardmodelat sp 2kF sp 2kF
cause the longitudinal spin susceptibility should exhibit
quarter filling using the ground state QMC, where they
stronger divergence at q = Q than the transverse
found that d-wave and f-wave strongly dominate over 2kF
ones near the SDW transition. Thus, −Vt⊥(Q ) ≥
p-wave.67 2kF
−Vtk(Q ) holds from eq.(12), where −Vt⊥(Q ) is
Apart from the studies directly dealing with the Hub- 2kF 2kF
always positive. Furthermore from eq.(12), we can see
bard model, low energy theories using the interacting
that
electron gas model like those for the purely one dimen-
sionalsystemsaswillbe mentionedinsection2.6canbe −Vt⊥(Q )≥Vs(Q ) (13)
2kF 2kF
effective, but since the nodes of the d-wave gap run par-
holds when the condition,
allel to the k axis, it is necessary to take into account
b
the quasi-one-dimensionality (the warping of the Fermi V (Q )≥V+−(Q ) (14)
ch 2kF sp 2kF
surface) to study d-wave pairing in a realistic situation.
is satisfied. This kind of relation between the sin-
Duprat and Bourbonnais indeed showed the occurrence
glet and the triplet pairing interactions when spin and
of d-wave pairing near the SDW phase within a renor-
chargefluctuationscoexisthasbeenpointedoutbyTaki-
malizationgroupstudythattakesintoaccountthequasi
moto69 for another candidate for a triplet superconduc-
one dimensionality.68
tor, Sr RuO .70
2 4
Now, an important point for a quasi-one-dimensional
2.5 Spin triplet f-wave pairing
system is that the number of gap nodes that intersect
Nevertheless, the spin-fluctuation-mediated d-wave
the Fermi surface is the same between d- and f-waves
pairing scenario contradicts with the experimental facts
due to the disconnectivity of the Fermi surface, so that
pointing towards spin-triplet pairing, especially for
whichoneofthesetwodominatesisdeterminedsolelyby
X=PF .42,48 (Note that most of the d-wave theories ap-
6
the magnitude of the pairing interactions. Thus, triplet
pearedbeforetheKnightshiftmeasurements.)Kurokiet
f-wave pairing with d perpendicular to the easy axis
al.28 provided a possible solution for this puzzle by re-
directiondominatesoversingletd-wavewhenthecontri-
calling that 2k CDW actually coexists with 2k SDW
F F
butions to the pairing interaction from the charge fluc-
inthe insulatingphaseforX=PF .14,15 If2k CDWco-
6 F
tuations is largerthan that fromthe spin fluctuations in
exists with SDW in the insulating phase, it is naturalto
the hard axis direction.
assume that 2k spin and 2k charge fluctuations co-
F F
The above argument can be summed up as a phe-
exist in the metallic phase lying nearby. Assuming the
nomenological phase diagram shown in Fig.3(a). In this
presence of charge fluctuations along with spin fluctua-
phase diagram, there exists a region where p-wave pair-
tions with possible magnetic anisotropy (i.e.,presence of
ingdominatesbecauseVtk(Q )>0holdswhenVzz >
easyandhardaxes),thepairinginteractionsaregivenin 2kF sp
2V+− +V , namely, when the magnetic anisotropy is
generic forms, sp ch
strong and the charge fluctuations are weak, so that a
1 1
Vs(q) = Vzz(q)+V+−(q)− V (q) triplet gap that has different signs at both ends of the
2 sp sp 2 ch nestingvectorcanbefavored.Inthiscase,disparallelto
1 1 the easy axis. This p-wave mechanism has in fact been
Vt⊥(q) = − Vzz(q)− V (q)
2 sp 2 ch proposed for Sr RuO .71,72 s-wave pairing having the
2 4
1 1 same gap sign over the entire Fermi surface is expected
Vtk(q) = Vzz(q)−V+−(q)− V (q) (12)
2 sp sp 2 ch todominatewhenthechargefluctuationsaresufficiently
strong because the singlet pairing interaction turns neg-
where Vzz and V+− are the contributions from lon-
sp sp ative (which is unrealistic for (TMTSF) X).
gitudinal and transverse spin fluctuations, respectively, 2
6 J.Phys.Soc.Jpn. FullPaper AuthorName
(a) + d-wave + 1.0
2k SDW 0.9
F
- p-wave (cid:13)+ l 0.8 'f-wave'
0.7
(with S =0)
z
0.6
(b) 'd-wave'
- f-wave + 0.5
2k SDW
F + 00.4.01 0.012 0.014 0.016
2kF CDW T/ta
Fig. 4. (a)2kF SDWconfigurationand(b)2kF SDW+2kF CDW Fig. 5. Left: The model for (TMTSF)2X adopted in ref. 83.
Right: Thelargest eigenvalue of the gap equation inthe singlet
configurationwithlikelypairingswhentheconfiguration“melts”
andthetripletchannels areplottedasfunctionsoftemperature
tobecomemetallic.
forU =1.7, V =0.8, V′ =0.45, V′′ =0.2, V⊥ =0.4, tb =0.2,
inunitsofta.83
An intuitive understanding for this phase diagram
can be given as follows. In the 2k SDW configura-
F in the b′ direction,16,17 which maybe an indicationthat
tion, electrons (or, actually, holes in a 3/4-filled system)
the2k spinfluctuationsinthemetallicstatemaynotbe
with antiparallel spins sit at next nearest neighbors as F
so anisotropic within the a-b′ plane. The phase diagram
shownin Fig.4(a),so if this configuration“melts” to be-
for z k c assuming isotropic spin fluctuations in the a-
come metallic, singlet pairing superconductivity with an
b′ plane is shown in Fig.3(b).73 Note that in this case,
even parity gap of ∆(k) = +exp(i2k )+exp(−i2k ) ∼
a a strong anisotropy in the spin fluctuations does not lead
cos(2k ) is likely to occur. “2”in the argumentof “exp”
a top-wavepairingbecausethetripletpairinginteractions
impliesthatthepairsareformedatsecondnearestneigh-
always remain negative.
bors, and the “+” signs in front of the “exp” corre-
After this phenomenological proposal and also a sim-
sponds to singlet wave functions having the same sign
ilar phenomenological argument of f-wave pairing by
in the rightandthe left directions,as showninFig.4(a).
Fuseya et al.,74 studies based on microscopic models
Since the gap cos(2k ) has even parity and has nodes
a have followed. Tanaka and Kuroki considered a model
at k = ±π/4, this corresponds to the singlet d-wave.
a which takes into account the off-site repulsive interac-
On the other hand, when 2k SDW and 2k CDW co-
F F tions V n n within the chains upto thirdnear-
exist (namely when both SDW and CDW have a pe- <i,j> ij i j
riod of four lattice spacings), the electrons are aligned estnePighbors(Fig.5,butwithV⊥ =0),wheretheconsid-
erationofthesecondnearestneighborrepulsionV′ isthe
like in Fig.4(b), so that when this configuration melts,
key.75 This has been based ona considerationthat since
triplet superconductivity with an odd parity gap of
the coexistence of 2k spin and 2k charge fluctuations
∆(k) = +exp(i4k )− exp(−i4k ) ∼ sin(4k ) is likely F F
a a a isnecessaryforf-wavepairing,andsincethecoexistence
to take place. This corresponds to the f-wavegap. If we
2k SDW and CDW is experimentally observed,14,15 a
consider the magnetic anisotropy, a triplet pair formed F
model that can account for this coexistence should be
at fourth nearest neighbors is expected to have a to-
the right Hamiltonian to be adopted. The mechanism
tal S = ±1 because the SDW spins are oriented in
z of the coexistence of 2k SDW and 2k CDW itself had
the z direction, (z k b′), which explains d ⊥ z for f- F F
alreadybeenproposedbyKobayashietal.andalsostud-
wave. On the other hand, if the pure 2k SDW con-
F ied by Tomio and Suzumura, where the second nearest
figuration (Fig.4(a)) with z being the easy axis melts,
neighbor repulsion V′ plays an essential role.77–80 From
a triplet pairing with S = 0 formed at next nearest
z Fig.4, it can be seen how V′ induces the 2k CDW in a
neighbor sites may compete with the singlet pairing. F
quarter-filled system. When only the on-site U and the
This corresponds to the p-wave pairing with the gap
nearestneighborV arepresent,the chargestend to take
∆(k) = +exp(i2k ) − exp(−i2k ) ∼ sin(2k ), whose
a a a the 4k (=π)CDW configuration,whichhasa periodof
nodes do not intersect the Fermi surface. Since the to- F
twosites,whilewhenV′ ispresent,thepairsofelectrons
tal spin of a triplet pair in this case is expected to be
sitting at second neighbors repel each other to result in
perpendicular to z, dkz can be understood.
the 2k CDW(+SDW) configuration.
In the above, the z-axis of the spins is taken in the F
For the HamiltonianthatconsidersU,V,V′,andV′′,
b′ direction, namely, the easy axis direction, but if we
the Fourier transform of the off-site repulsions, consid-
assume that the spin fluctuations in the a-b′ plane are
ered in the RPA eq.(6), is given as
nearlyisotropicandlargerthanthoseinthec(hardaxis)
direction, we can take the hard axis as the z-axis and V(q)=2V cos(q )+2V′cos(2q )+2V′′cos(3q ) (15)
x x x
thus V+−(Q )>Vzz(Q ), so that now dkc for f-
sp 2kF sp 2kF From eqs.(5),(6), and (15), it can be seen that
wavepairingfollowingasimilarargumentasbefore.This
χ (Q ) = χ (Q ) (where Q = (π/2,π)), and
picture may be more suitable for (TMTSF) PF since s 2kF c 2kF 2kF
2 6 consequently −Vt(Q ) = Vs(Q ), apart from the
(i) the uniform susceptibilities for H ka and H kb′ are 2kF 2kF
first order terms such as U + V(q), is satisfied when
equal down to the very vicinity of the SDW transition,
V′ = U/2. Within the phenomenological argument, this
while that for H kc deviates from higher temperatures,
correspondstotheconditionforf-wavetobedegenerate
and (ii) the direction of the SDW undergoes a spin-flop
with d-wave in the absence of magnetic anisotropy.76 In
transition into the a direction under a magnetic field
J.Phys.Soc.Jpn. FullPaper AuthorName 7
theactualRPAcalculationforV′ =U/2,f-waveslightly phase diagram, as discussed in some studies.90,91
dominatesoverd-waveduetothe effectofthefirstorder Apartfromtheg-ology-typeapproach,therehavebeen
terms neglected in the phenomenological argument.75 studies on the phonon mechanism of triplet pairing.
FuseyaandSuzumura81 approachedthesameproblem KohmotoandSatoproposedap-wavepairingmechanism
using the renormalization group method for quasi-one- duetoacombinationofelectron-phononinteraction,2k
F
dimensional systems along the line of Duprat and Bour- spin fluctuations, and the disconnected Fermi surface.92
bonnais,68 where a similar conclusion has been reached. Assumingthattheelectron-phononinteractionisweakly
Sincethepairingcompetitionissubtle,theyfurtherpro- screened, a long-ranged attractive interaction arises in
posed a possible singlet d-wave to triplet f-wave transi- real space, which means that the pairing interaction be-
tion in the presence of magnetic field. comes large and negative around q ∼ 0 in momentum
According to the above studies, f-wave dominates space. If we denote this interaction as −V (q), and
el−ph
over d-wave when the second nearest neighbor repul- if the spin fluctuations also contribute to the pairing to
sion V′ is equal to or larger than half the on-site re- some extent, the pairing interactions are given as
pulsion U, which may be difficult to realize in actual 1
materials. Nickel et al. have proposed a possible solu- Vs(q) = −Vel−ph(q)+ 2Vszpz(q)+Vs+p−(q)
tion for this difficulty, where they considered, in addi-
1
tion to the intrachain repulsions, the interchain repul- Vt⊥(q) = −V (q)− Vzz(q)
el−ph 2 sp
sion and used the renormalization group technique for
quasi-one-dimensionalsystemstoreachaconclusionthat Vtk(q) = −V (q)+ 1Vzz(q)−V+−(q).(16)
f-wavedominatesoverd-waveinamorerealisticparam- el−ph 2 sp sp
eter regime with a smaller second nearest neighbor re- In ref.92, the competition between s- and p-wave pair-
pulsion.4,82 Independently, KurokiandTanakaalsocon- ingswasdiscussed,whilethepossibilityofd-andf-wave
sidered a model that considers the nearest neighbor in- was not considered because the warping of the Fermi
terchain repulsionV as shown in Fig.5.83 Within RPA, surface was neglected. Let us first neglect the magnetic
⊥
the term 2V cos(q ) is added in the right hand side of anisotropy, i.e., Vzz = V+−. Around q ∼ 0, neglecting
⊥ y
eq.(15), so that the condition for χ (Q ) = χ (Q ) the spinfluctuationcontribution,the pairinginteraction
s 2kF c 2kF
now becomes V′+V = U/2. This is a much more re- is negative and has the same magnitude between singlet
⊥
alistic condition than V′ = U/2 because the interchain andtripletpairings.Thus,s-andp-wavepairings,whose
distanceissimilartotheintrachainsecondnearestneigh- gap does not change sign on each portion of the discon-
bor distance, so that we can expect V to be as large as nected Fermi surface, are equally favored by this inter-
⊥
V′. The actualcalculationshowsthat f-wavedominates action around q ∼ 0. At q = Q on the other hand,
2kF
(has a larger eigenvalue λ) over d-wave for a parame- neglectingthe electron-phononinteractionthis time,the
ter set, e.g., U = 1.7, V = 0.8, V′ = 0.45, V′′ = 0.2, positivespinfluctuationcontributionforthesingletpair-
V = 0.4, t = 0.2 in units of t (Fig.5), where the rela- ing works destructively against s-wave because the gap
⊥ b a
tive magnitude of the interactions can be considered as does not change sign across Q , while the negative
2kF
realistic. contribution for the triplet channel also works against
p-wave, whose gap changes sign. Since this destructive
2.6 Other Mechanisms for Triplet Pairing: Phonons, spin fluctuation contribution is smaller for triplet pair-
Ring Exchange ing,p-wavedominatesovers-wave.Notethathereagain,
In this subsection, we discuss some other mechanisms the close competition between p-wave and s-wave arises
for spin-triplet pairing proposed for (TMTSF) X. From from the disconnectivity of the Fermi surface owing to
2
the early days, possibility of spin-triplet superconduc- the (quasi) one dimensionality, i.e, the additional node
tivity in (TMTSF) X has been discussed in terms of in the p-wave gap as compared to the s-wave does not
2
the low energy effective theory called the g-ology ap- intersect the Fermi surface.
proach for the purely one dimensional interacting elec- Ifwefurthertakeintoaccountthemagneticanisotropy
tron gas, i.e., the Tomonaga-Luttinger model.84 In the and assume Vzz <V+− by taking the hard axis (c-axis)
g-ology phase diagram, the spin-triplet superconducting in the z direction, the negative spin fluctuation contri-
phaseandtheSDWphaseshareboundary,85 sothatitis bution in the triplet pairing interaction is smaller (and
tempting to relate this superconducting state with that thus favorable for p-wave pairing) for d ⊥ z than for
of(TMTSF) X,aswasdiscussedinsomestudies.86More dkz. Therefore, if the direction of d for p-wave pairing
2
recently,this phaseboundarybetweenthe SDWandthe is governed by the magnetic anisotropy of the SDW, d
triplet superconductivity has been discussed as having is likely to be perpendicular to the hard axis direction,
SO(4) symmetry.87,88 Nevertheless, since exact numeri- namely, in the a-b plane for (TMTSF) PF .
2 6
calstudies onthe purelyone-dimensionalextendedHub- Suginishi and Shimahara also proposed a phonon-
bard model, where the on-site U and the nearest neigh- mediated mechanism for p-wave pairing.93 By consider-
bor V is considered,89 show that superconductivity does ing moderately screened phonons and also including the
not occur in a realistic parameter regime when the in- corrections due to charge fluctuations, they obtained an
teractions are all repulsive, it is likely that some kind attractivepairinginteractionthathasalargemagnitude
ofattractiveinteraction,mostprobablyoriginatingfrom aroundq ∼0 anda smallone aroundq ∼Q . By fur-
2kF
electron-phononinteraction,shouldbenecessaryinorder ther considering the Coulomb pseudo potential, which
torealizethetripletsuperconductingstateintheg-ology suppresses only the s-wave pairing, it has been found
8 J.Phys.Soc.Jpn. FullPaper AuthorName
there that p-wavepairing dominates in a certainparam- a coherence peak followed by an exponential decay in
eter regime. 1/T as mentioned in section 2.3,41 which seems to be
1
Recently, Ohta et al. proposed a non-electron-phonon in contradiction with refs.40 and 43, but since 1/T can
1
mechanism for spin-triplet pairing.94 The mechanism is be affected by the presence of impurities,98 vortices,99
based on the fact that in a triangle lattice consisting of or correlation effects,100 the clarification of the relation
three sites with two electrons, a ferromagnetic interac- between 1/T andthe thermalconductivity experiments
1
tion arises by considering a consecutive exchange of the is open for future study.
positions of the electrons.95 If ferromagnetic spin fluc- From the microscopic view discussed in the preceding
tuations arise due to this “ring exchange mechanism” sections, f-waveand p-waveare the main candidates for
on a certain lattice, triplet pairing superconductivity spin triplet pairing.42,43,48 As for the direction of d, if
may take place. They considered the Hubbard model we assume that the spin fluctuations in the c direction
on a one-dimensional “railway-trestle” (or zigzag) lat- are weak while those in the a-b planes havesimilar mag-
tice, where they used the density matrix renormaliza- nitude, d of f-wave pairing should lie in the c direction
tion group method to find that triplet pairing correla- as discussed in section 2.5, which is consistent with the
tion functions decay more slowly than the singlet ones. Knightshiftresults.42,48 Ifweassumeonthe otherhand
Theirnumericalcalculationhasbeenrestrictedtopurely that the spin fluctuations are solely strong in the b′ di-
onedimensionalsystemssofar,buttheyfurtherpropose rectioncomparedtothoseinthea-cplane,thenf-wave’s
that this mechanism may be applicable to the quasi- d is perpendicular to b′ and lies in the a-c plane as also
one-dimensionalmaterial(TMTSF) X since the signs of discussedinsection2.5,whichismoreclosertotheddi-
2
the intrachain and interchain hopping integrals (by con- rectionproposedbyLebedet al.53 fromthe temperature
sidering also the next nearest neighbor interchain hop- dependenceofHb andHa.48 Inthecaseofp-wavepair-
c2 c2
ping)24,25 satisfy the condition for the ferromagnetic in- ing, if the anisotropicspin fluctuations contribute to the
teraction.95 pairing interaction in the form given in eqs.(16) (V
el−ph
In all the mechanisms discussed in this subsection, at need not be due to phonons), d is likely to lie in the a-
least one of the 2k fluctuations, spin or charge,are not b plane. Thus, it may be possible to distinguish f and
F
takenintoaccount,althoughtheyshouldbothbepresent p from the direction of d, provided that the anisotropic
at least for X=PF . Then, whether both of these fluctu- spin fluctuations play a role in the Cooper pairing.Such
6
ations play essential roles or not in the occurrence of a test, however,has to be done in the absence of, or un-
superconductivity is the key toward clarifying whether der low, magnetic field since d may rotate regardless of
f-wave discussed in section 2.5 or other triplet pairings the pairing symmetry if the magnetic field is sufficiently
dominate,providedthatthepairingindeedoccursinthe large to overcome the effect of the magnetic anisotropy.
triplet channel. Although there existfew experiments upto date,pos-
sibilityofdeterminingthepairingsymmetryfromtunnel-
2.7 Tests for the Pairing Symmetry Candidates ing spectroscopy measurements has been proposed the-
Inthissection,wediscusssomeexperimentaltests(al- oretically by severalgroups. Sengupta et al. pointed out
readyexistingonesaswellasproposalsforfuture study) thatthepresence/absenceofzeroenergypeakinthetun-
for the candidates for the pairing symmetry discussed neling conductance can be used to distinguish various
above. For the pairing symmetries whose gap has line types ofpairingsin(TMTSF) X.101 Infact,the zeroen-
2
nodes on the Fermi surface such as d-wave and f-wave, ergy peak in the tunneling spectroscopies of anisotropic
the spin-lattice relation rate 1/T exhibits essentially no superconductors(thosewithsignchangeinthegap)orig-
1
(or very small) coherence peak and a power law decay inates fromthe zero-energyAndreev boundstate caused
proportional to ∼T3,96 which is consistent with the ex- bythesignchangeofthepairpotentialfeltbythequasi-
periments for X=ClO 40 and for X=PF 43 at low mag- particle in the reflection process at the surface,102,103
4 6
netic fields. On the other hand, whether these pairings and has turned out to be a powerful method for prob-
canaccountforthepeculiarbehaviorof1/T observedfor ing the pairing symmetry in anisotropic superconduc-
1
X=PF athighmagneticfield,i.e.,1/T ∼T atlowtem- tors such as the high T cuprates.104 In the case of tun-
6 1 c
peraturesaswellasasmallpeakbelowT forH ka,42,43 neling parallel to a in particular, the zero energy peak
c
remains open as an interesting future study. does not exist for d-wave. This is because the injected
At first glance, only p-wave and s-wave pairings seem and the reflected quasiparticles feel the same gap due
to be consistent with the thermal conductivity measure- to ∆ (k ,k ) = ∆ (−k ,k ) (see Fig.2(b)). By con-
d a b d a b
ment for X=ClO suggesting a fully gapped state.44 trast, the zero energy peak does exist for p-wave and
4
However, Shimahara has argued that a fully gapped f-wave, where ∆ (k ,k ) = −∆ (−k ,k ) is satis-
f,p a b f,p a b
state is possible even for d-wave pairing particularly fied (Fig.2(c)). Tanuma et al. further pointed out that
in (TMTSF) ClO , because in this case, anion order- p-wave and f-wave can be distinguished from the over-
2 4
ing takes place above the superconducting T , so that all shape of the surface density of states (overall struc-
c
a “gap” opens up on the Fermi surface at positions tureofthetunnelingspectrum)becausep-waveisafully
(k = ±π/4) where the nodes of the superconducting gapped state, which results in a U-shaped surface den-
a
gap would otherwise intersect (see Fig.2(a)).97 Exactly sityofstatesaroundtheFermilevel,whilef-waveresults
thesameargumentholdsforf-wavepairingsincethepo- in a V-shaped one.105 Therefore, the combination of the
sitionsofthegapnodesontheFermisurfacearethesame absence/presenceofthezeroenergypeakandtheoverall
between f and d. A fully gapped state usually results in shapeofthespectrumenablesustodistinguishp,d,and
J.Phys.Soc.Jpn. FullPaper AuthorName 9
f-wave pairings. (a) (b)
Further theoretical studies based on various shapes of
the Fermi surface have been performed.106,107 Tanuma
et al. showed that when the Fermi surface is warped in b2 p b1 b
a certain manner, the zero energy peak can appear even q' c' c
q
in the case of d-wave. In this case, d and f-wave can be p' b1 p b2
distinguished by the way the zero energy peak splits in q'
the presence of a magnetic field.106 Such studies show p' q c' c
b2 b1 b
that the existence of the zero energy peak is sensitive to
b
theshapeoftheFermisurface(compareFig.2(a)and(d),
or(b)and(e)).Sincethehoppingintegrals,andthusthe
c
Fermi surface, of (TMTSF) X depend on the pressure,
2
the temperature, and the anions,24,25,108 it is necessary (c) (d)
tostrictlypindowntheactualshapeoftheFermisurface
at the temperature and the pressure at which supercon- paramagnetic(cid:13)
ductivity takes place in order to distinguish the pairing ure insulator paramagnetic(cid:13)
symmetry from the presence/absence of the zero energy perat metal
m
peak. te AF insulator
SC
The tunneling tests above mainly concern the orbital
pressure
part of the pairing. On the other hand, Bolech and Gi-
amarchi proposed a tunneling experiment to distinguish
directlythespinpartofthepairing.109Theyshowedthat Fig. 6. (a) Thelatticestructure ofκ-(BEDT-TTF)2Xinthe b-c
the I-V characteristics of a normal metal-triplet super- plane.b1,b2,··· representthehoppingintegralsinthefourband
model.(b)Thelatticestructureofthedimermodel.(c)Phasedi-
conductor junction are unaffected by an application of agramofκ-(BEDT-TTF)2X.5(d)BandstructureandtheFermi
magnetic field perpendicular to d, while the Zeeman ef- surfaceof κ-(BEDT-TTF)2Cu(NCS)2. (Reprinted withpermis-
fect affects the I-V characteristicswhen dkH similarly sionfromref.119.Copyright1988bytheAmericanPhysicalSo-
tothe caseofnormalmetal-singletsuperconductorjunc- ciety.)
tion. Therefore, the spin part of the pairing, whether it
is singlet or triplet and also the direction of d if triplet,
can be determined by measuring the I-V characteris- 3. κ-(BEDT-TTF)2X
tics of the junction under a rotating magnetic field, pro- 3.1 Lattice Structure and the Phase Diagram
videdthatddoesnotrotateaccordingtothedirectionof The lattice structure of κ-(BEDT-TTF) X is shown
2
the magnetic field. Vaccarella et al. also proposed a way in Fig.6(a), which consists of dimers formed by a pair
of directly probing the spin part of the triplet pairing. of face-to-face molecules. b- and c-axis are taken as in
Namely, they showed that the Josephson effect between Fig.6, while the BEDT-TTF layers and the anion layers
two triplet superconductors is very sensitive to the di- alternatealongthea-axis.Relativelylargeoverlapofthe
rection of d across the junction, and proposed that this orbitalsbetweenthedimersexists(seesection3.2)while
sensitivity can be used as a test for triplet superconduc- theoverlapbetweentheBEDT-TTFlayersisverysmall,
tivity.110 resulting in a strong two dimensionality.
As a final remark in this subsection, it is important In Fig.6(c), the generic phase diagram of κ-(BEDT-
to recognize that the pairing symmetry might be dif- TTF) X is shown, which has been extensively stud-
2
ferent for different anions. This possibility is suggested ied by Kanoda et al.5 The superconducting and the
especially from the viewpoint discussed in section 2.5. antiferromagnetic insulating phases share a first or-
For instance, the amplitude of 2kF charge fluctuations, der phase boundary. Recently, this boundary has
which has to be large for f-wave to dominate over d- been revealed to persist above the superconduct-
wave, is found to be small for X=AsF6, so that f-wave ing Tc and the N´eel temperature into the bound-
has less chance of dominating over d-wave than in the ary of the paramagnetic insulating and the metallic
caseofX=PF6.Thus,thepairingsymmetryofaTMTSF phases, ending up at a certain critical point,113,114
superconductor with a certain anion should be deter- where an anomalous criticality has been found re-
mined by a combination of multiple experiments on the cently.115 The horizontal axis in the phase diagram
salt with that very anion. Furthermore, we must keep can be considered as hydrostatic or chemical pressure,
in mind that the pairing symmetry might even change where superconductivity with T exceeding 10K occurs
c
for the same salt under different environment, such as at ambient pressure for κ-(BEDT-TTF) Cu(NCS) 116
2 2
the pressure and the strength of the magnetic field, be- and κ-(BEDT-TTF) Cu[N(CN) ]Br, while κ-(BEDT-
2 2
causeseveralpairingsymmetriesmaybecloselycompet- TTF) Cu[N(CN) ]Cl is an antiferromagnetic insulator
2 2
ing. In particular, as mentioned in section 2.5, singlet below 26K at ambient pressure117 and becomes super-
to triplettransitionmaytake placeunder highmagnetic conducting with T = 12.8K under an applied pressure
c
field since the singlet pairing is Pauli-paramagnetically of 0.3kbar.118
limited.81,111,112
10 J.Phys.Soc.Jpn. FullPaper AuthorName
(a) dx 2-y2 (b) dxy anion b1 b2 p p′ q q′
p unfold BZ p Cu2(CN)3 22.36 11.54 8.01 − −2.90 −
+- Q +- p + Q CCuu([NN(CCSN)2)2]Br 2242..3975 191..1361 190..8154 10−.09 −−33..4300 −3−.76
kb0 - + ky0 Q kb0 - - Table I. Hopping integrals estimated in ref.120. In units of
+ - + 10−2eV.
-p -p 0 p -p -p 0 p -p -p 0 p
kc kx kc
|t /t | ∼ 0.7 for X=Cu[N(CN) ]Br. The system further
b c 2
Fig. 7. (a) The dx2−y2-wave gap in the original Brillouin zone reduces to a half-filled single band model when tc =t′c.
and in the unfolded one (right). Note that although the gap
changes sign at the Brillouin zone edge (kc = ±π), the nodes 3.3 Experimental Results Concerning the Pairing Sym-
of the gap arenotlocated there; the gap jumpsfromapositive
metry
toanegative value, ascanbeseenmoreclearlyintheunfolded
Brillouin zone. (b) The dxy-wave gap. Here, we show the case Here,we summarize the experimentalresults concern-
whenthetwoportionsoftheFermisurfacesplitsduetothelack ing the pairing symmetry.6 In the NMR experiments
ofcenter-of-inversionsymmetry.Inthiscase,the dxy gapnodes forκ-(BEDT-TTF) Cu[N(CN) ]Br,the13CKnightshift
2 2
do not intersect the Fermi surface, although the gap does be- hasbeenfoundtodecreasebelowT ,123,124 whichiscon-
come small near the Brillouinzone edge. On the other hand, if c
sistent with singlet pairing. Also, the 13C spin-lattice
thetwoportionsstick,thedxy nodesintersecttheFermisurface
at the Brillouin zone edge. The solid (dashed) curves represent relaxation rate 1/T1 for κ-(BEDT-TTF)2Cu[N(CN)2]Br
the portions of the Fermi surface where the gap has a positive exhibits no coherence peak, and a power law decay pro-
(negative) sign.Qrepresentsthewavevectorofthespinfluctu- portionalto T3 is seen below T .123–125 As in the caseof
c
ationmodethatfavorseachpairingsymmetry.Theshortarrows
(TMTSF) X, this is consistent with the presence of line
denote thepositionsofthegapnodes. 2
nodes in the superconducting gap. Also, in a thermal
conductivity measurement for X=Cu(NCS) , a T-linear
2
termhasbeenfoundatlowtemperatures,suggestingthe
3.2 Electronic structure
existence of nodes in the gap.126
The values of the intermolecular hopping integrals
On the other hand, there has been much contro-
shown in Fig.6(a) have been estimated using the ex-
versy concerning the measurements of other quantities.
tended Hu¨ckel method,119,120 which is summarized in
The magnetic penetration depth has been measured us-
Table I. The hopping integral in the b-direction alter-
ing techniques such as muon spin relaxation,127,128 ac
nates as t ,t ,t ,···, where |t | > |t | because of
b1 b2 b1 b1 b2 susceptibility,129,130 surface impedance,131,132 and dc
the dimerization of the molecules. In Fig.6(d), the band
magnetization.133,134 The penetration depth should ex-
structure of κ-(BEDT-TTF) Cu(NCS) 119 is shown.
2 2
hibit an exponentially decaying behavior for a fully
Four bands exist near the Fermi level because there are
gapped state, while a power-lawdependence is expected
four BEDT-TTF molecules per unit cell. Due to the
at low temperatures for gaps with nodes. Some stud-
dimerization, a gap opens up between the bonding and
ies have found for X=Cu(NCS) that the tempera-
the antibonding bands. There is one hole per dimer, so 2
ture dependence of the penetration depth is consis-
that only the upper two bands cross the Fermi level, re-
tent with a conventional full gap state,127,131,133 while
sulting in two portions of the Fermi surface (Fig.7). The
others have found results consistent with a gap with
two portions of the Fermi surface are connected at the
nodes.128,129,132,135 Similar controversy on the penetra-
Brillouin zone edge for X=Cu[N(CN) ]Br, Cu (CN) ,
2 2 3
tion depth also exists for X=Cu[N(CN) ]Br, where the
I , etc., in which the anions are arranged in a manner 2
3 presenceofgapnodes128,130,135 aswellastheabsenceof
that the system possesses center-of-inversion symmetry,
them131,133,134 has been suggested.
which results in t =t′, t = t′. On the other hand, for
p p q q The temperature dependence of the specific heat has
X=Cu(NCS) , the system lacks the symmetry so that
2
also been another issue of controversy. Nakazawa and
t 6=t′,t 6=t′,andinthatcase,the twoportionsofthe
p p q q Kanoda found for X=Cu[N(CN) ]Br a T2 dependence
Fermi surface are separated into an open Fermi surface 2
of the electronic specific heat,136 which was taken as an
and a closed one.
indication for the presence of nodes in the gap. How-
In the limit of large t , namely, when the dimeriza-
b1 ever, more recent results for X=Cu[N(CN) ]Br137 and
tion is strong, each dimer can be considered as a single 2
for X=Cu(NCS) 138 haveshownexponentiallyactivated
site, so the system reduces to a two band model shown 2
temperature dependence, indicating a fully gapped su-
in Fig.6(b) with n = 1, where the band filling is now
perconducting state.
defined as n=(the number of electrons/the number of
The above experiments do not give direct information
sites).121,122 In otherwords,the energygapbetweenthe
on the position of, if any, the nodes in the gap function.
upper two and the lower two bands becomes large when
Several groups have in fact made attempts to directly
the dimerization is strong, so that the lower two bands,
determine the node positions. A millimeter-wave trans-
which do not cross the Fermi level, can be neglected. In
mission experiment suggested a gap function which has
thisstrongdimerizationlimit,theeffectivehoppinginte-
nodes in the direction shown in Fig.7(a).139 If we un-
gralst andt aregivenast =−t /2,t =(−t +t )/2,
b c b b2 c p q
fold the Brillouin zone (right panel of Fig.7(a)), which
andt′ =(−t′ +t′)/2intermsoftheoriginalhoppingin-
c p q corresponds to adopting a single dimer as a unit cell,
tegrals,121whichgives|t /t |∼0.8forX=Cu(NCS) and
b c 2 (this is possible when t = t′), this gap function has
c c