Table Of ContentSpin-triplet superconductivity in quasi-one dimension
Mahito Kohmoto and Masatoshi Sato
Institute for Solid State Physics, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo, Japan
Weconsiderasystemwithelectron-phononinteraction, antiferromagnetic fluctuationsanddiscon-
0 nected open Fermi surfaces. The existence of odd-parity superconductivity in this circumstance
0 is shown for the first time. If it is applied to the quasi-one-dimensional systems like the organic
0 conductors (TMTSF)2X we obtain spin-triplet superconductivity with nodeless gap. Our result is
2
also valid in higher dimensions(2d and 3d).
n
a PACS numbers: 74, 74.25-q, 74.25.Dw, 74.62.-c
J
In recent years the signs for unconventional super- TheFermisurfaceofthesystemweconsiderisquasi-one-
3
2 conductivity in many compounds have been accumu- dimensional and consists of two separated parts, which
lated. Examples include high Tc superconductors,heavy are (kF)x ∼ c > 0 and (kF)x ∼ −c. (c is a constant.) It
] fermions, organic conductors, Sr RuO , etc. The com- is also supposed to be symmetric under parity transfor-
n 2 4
mon features of those compounds are quasi-low dimen- mationk k. ThistypeofFermisurfaceisrealizedin
o →−
sionality and proximity of antiferromagnetic(AF) order. (TMTSF) X . The non-interacting part of the Hamilto-
c 2
- Here we consider seemingly the simplest quasi-one- nian of (TMTSF)2X is often written
r
dimensional systems which are realized in (TMTSF) X
p 2
E = 2t cos(k a) 2t cos(k b). (1)
u familywhereX=PF6,AsF6,SbF6,ClO4,etc.(Bechgaard − a x − b y
s salts). At ambient pressure, most of these extremely
t. anisotropic compounds undergo a metal-insulator tran- Here the ratio of ta and tb is about 10:1.
a Wetaketwointeractionsbetweenelectrons: onecomes
sition at low temperature and have a spin-density-
m fromelectron-phononcoupling,theothercomesfromAF
wave(SDW) fundamental state. Under moderate pres-
fluctuations.
- sure, the SDW instability is suppressed and replaced by
d – Phonon-mediated interaction
n a superconducting transitionata criticaltemperature of It is written
o orderof1K[1]. (Oneexceptiontothisis(TMTSF)2ClO4
[c wthheiscehciosmspuopuenrcdosnmduacytibnge cahtaraamcbteierinztedprbeysscuorme.)petTithiouns Hipnht =−kXk′qXαβ f(q)a†k+q,αak,αa†k′−q,βak′,β, (2)
1 between superconducting and SDW ground states [2].
v Asforthe gapsymmetry,Takigawaetal.[3]measured where a and a† are the usual fermion operators, α and
1 nuclear relaxation rate of proton in (TMTSF) ClO . β representspin orientations and f(q)>0 has a peak at
2 4
3 Their results show unconventional superconductiv- q =0. Weassumef(q)=f( q). Ifoneconsiderspairing
3 −
ity(absence of coherence peak) and the existence of interaction between k and k, (2) reduces to
1 −
nodes. However their measurement is restricted to T >
0 1
00 M0.5oTrec arencdentthley,exBiestlienncaenodfBnoehdnesiaar[4e]nsohtowyeetdcosonmcleuseivvei-. Hipnht =−2Xk,q f(q)(cid:16)φ†0(k+q)φ0(k)+φ~†(k+q)·φ~(k)(cid:17),
/ dences for nodeless gap by thermal conductivity mea-
t (3)
a surement. Also experiments on (TMTSF) ClO and
2 4
m (TMTSF)2PF6 showthatHc2 forB~ ~aandB~ ~bfarex- where
k k
- ceedthePauli-limitingfieldB ∆ /(√2)µ ) 2Tesla
nd forBechgaardsalts,where∆0Pis∼the0superconBdu∼ctingor- φ0(k)=ak,α(σ2)α,βa−k,β, (4)
o der parameter at T = 0 and µB is the Bohrmagneton φ~(k)=a (σ ~σ) a . (5)
k,α 2 α,β −k,β
c [5,6]. This suggests spin-triplet superconductivity. Maki
v: et al. studied theoretically impurity effects and vortex Hereφ0(k)isthespin-singletpairingandφ~(k)isthespin-
i states on these compounds [7]. triplet one:
X
In this paper we consider AF magnetic coupling and
ar electron-phonon interaction under open disconnected ak,αa−k,β = 1(σ2)αβφ0(k)+ 1(~σσ2)αβ φ~(k). (6)
Fermisurfaces. Thespin-tripletsuperconductivityispos- −2 2 ·
sibleunderelectron-phononinteractionandAFmagnetic
Since f(q) has a peak at q = 0, this interaction is ap-
coupling[8]. WeassumethatAFfluctuationisnotstrong
proximated by
enough to give SDW gap. We shall show the existence
ofspin-triplet superconductivity in this circumstance for 1
Hph = f(0) φ†(k)φ (k)+φ~†(k) φ~(k)
the first time. Our result is also applicable to higher int −2X (cid:16) 0 0 · (cid:17)
dimensions(2d and 3d). k
– Open disconnected Fermi surface = f(0) φ†(k)φ (k)+φ~†(k) φ~(k) . (7)
− X (cid:16) 0 0 · (cid:17)
kx>0
1
–Interaction due to AF fluctuations than the spin-triplet one, so the spin-triplet supercon-
It is written ductivity is realized. This spin-triplet gap is nodeless.
–BCS analysis
HiAntF =−X X J(q)~σαβ ·~σγδa†k+q,αak,βa†k′−q,γak′,δ, (8) Finally, we apply the BCS weak coupling theory. Since
kk′qαβγδ Tc of (TMTSF)2X is rather low, one can expect that
thestrongcouplingcorrectionsdonotchangeourresults
where J(q) > 0 represents interaction due to AF fluctu-
qualitatively. For the sake of simplicity, the cutoffs of
ations and has a peak value at a nesting vector q = Q.
± the phonon-mediated interaction and AF one are taken
The nesting vector Q connects the two separated Fermi
to be same and denoted by h¯ω . We approximate the
D
surfaces: Q 2c. We assume J(q) = J( q). A similar
x ∼ − spin-singlet gap by
analysis to above leads to
ψ =ψ =const., (12)
1 |kx∼c |kx∼−c
HiAntF = 2XJ(q)(cid:16)3φ†0(k+q)φ0(k)−φ~†(k+q)·φ~(k)(cid:17), and the spin-triplet gap by
k,q d~ = d~ =const.. (13)
|kx∼c − |kx∼−c
(9)
If we denote the density of states on the Fermi surface
and this interaction is approximated by as N(kF), the criticaltemperature ofthe spin-singletsu-
perconductivity becomes [9]
1
HiAntF= 2XJ(Q)(cid:16)3φ†0(k+Q)φ0(k)−φ~†(k+Q)·φ~(k)(cid:17) Tceven =1.13¯hωDe1/{N(kF)Veven}, (14)
k
where
1
+ J( Q) 3φ†(k Q)φ (k) φ~†(k Q) φ~(k) .
2X − (cid:16) 0 − 0 − − · (cid:17) Veven = f(0)+3J(Q), (15)
k −
(10) and that of spin-triplet becomes
Todd =1.13¯hω e1/{N(kF)Vodd}, (16)
For fermions on the Fermi surface, this becomes c D
where
HAF
int Vodd = f(0)+J(Q). (17)
= 1 J(Q) 3φ†(k+Q)φ (k) φ~†(k+Q) φ~(k) −
2 X (cid:16) 0 0 − · (cid:17) From(15)and(17),theconditionVodd <Vevenisalways
kx∼−c satisfied. Thus Todd > Teven and spin-triplet supercon-
c c
+1 J( Q) 3φ†(k Q)φ (k) φ~†(k Q) φ~(k) ductivity is present if Vodd = f(0)+J(Q) < 0. This
2 X − (cid:16) 0 − 0 − − · (cid:17) simple result show that there i−s no superconductivity if
kx∼c
AFfluctuationsdominateandwehavespin-tripletsuper-
= X J(Q)(cid:16)3φ†0(k′)φ0(k)+φ~†(k′)·φ~(k)(cid:17). (11) conductivity if phonon-mediated interaction dominates.
kx,kx′∼c Since d~(k) is an odd function, the gaps at (kF)x = c
±
have opposite signs and there is no node in the gap.
Here k′ satisfies k′ =Q k.
−
–Spin-triplet superconductivity
Equation(7)showsthatthephonon-mediatedinteraction
of spin-triplet pairing has the same magnitude as that
of spin-singlet one. In many systems, however, only the
spin-singletsuperconductivityisrealized. Thisisbecause
the Fermi surfaces of usual matters are connected. In a [1] Forareview,seeT.Ishigro,K.Yamaji,G.Saito,Organic
system of connected Fermi surface, the requirement of Superconductors (Springer-Verlag, Berlin, 1998).
parity and continuity does not admit the constant spin- [2] D. J´erome et al., J. Phys.(Paris), Lett. 56, L95 (1980).
triplet gap. Therefore, the s-wave superconductivity is [3] M. Takigawa, H. Yasuoka, and G. Saito, J. Phys. Soc.
favored from the kinematic reason. Jpn. 56, 873 (1987).
The quasi-one-dimensional system we consider, how- [4] S.BelinandK.Behnia,Phys.Rev.Lett.792125(1997).
ever, has disconnected Fermi surfaces, so it admits con- [5] M.J. Naughton et al., SyntheticMetals 85, 1481 (1997).
stant spin-triplet gap: d~ = d~ = const.. ∗ [6] I.J. Lee et al., Phys. Rev.Lett. 78, 3555 (1997).
Therefore, there is no rea|ksxo∼ncto p−refe|krxs∼-−wcave supercon- [7] Makietal,J.Superconductivity12,551(1999); Physica
ductivity. From (11), the interaction due to AF fluc- C 317-318, 353 (1999); Sinth.Metals 103, 1933 (1999).
tuation disturbs the spin-singlet superconductivity more [8] M. Sato and M. Kohmoto, (unpublished).
[9] See, for example, P.G. De Gennes, Superconductivity of
Metal and Alloys (Benjamin, NewYork, 1966).
[10] A. Leggett, Rev. Mod. Phys.47, 331 (1975); M. Sigrist
and K.Ueda, Rev.Mod. Phys.63, 239 (1991).
∗d~and ψ is definedas ∆(k)=iσ2ψ(k)+i(d(~k)·~σ)σ2 [10].
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