# Magnetic field effect on the pairing state competition in quasi-one-dimensional organic superconductors (TMTSF)2X

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1. Introduction

nikov (ss momue of ga mixinstate

the ping th

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the possibility of the spin triplet pairing and the FFLO state [1220]. In particular, we have previously shown that the triplet f-wavepairing can compete with the singlet d-wave pairing in the Q1Dsystem because of the disconnectivity of the Fermi surface when

2kF spin and 2kF charge uctuations coexist [2123,7]. The coexis-

tx = 1.0 is taken as the energy unit. U is the on-site repulsion andthe off-site repulsions Vij are taken as Vx, Vx2 and Vx3, which arethe nearest, second nearest and third nearest neighbor intra-chaininteractions, and Vy is the interchain interaction. Note that U(Vx2 + Vy) enhances the 2kF spin (charge) uctuations. The case of3/4 lling is considered corresponding to the actual material. Weignore the orbital effect, assuming that the magnetic eld is appliedparallel to the conductive xy plane.

Applying the RPA that takes account of the Zeeman effect [23,7]to this model, we obtain the pairing interactions. We solve the lin-earized gap equation which takes account of the center of mass

* Corresponding author. Address: Department of Applied Physics and Chemistry,Room 310, Building E1, School of Electro-Communications, The University ofElectro-Communications, Chofu, Tokyo 182-8585, Japan. Tel.: +81 42 443 5559; fax:+81 42 443 5563.

Physica C 470 (2010) 10851088

Contents lists availab

Physic

.e lE-mail address: aizawa@vivace.e-one.uec.ac.jp (H. Aizawa).one-dimensional (Q1D) organic superconductors (TMTSF)2X(X = PF6, ClO4, etc.). Some experiments in the high magnetic eldhave indeed suggested the possibility of the spin triplet pairingand/or the FuldeFerrellLarkinOvchinnikov (FFLO) state [8,9].Recently, an NMR experiment for (TMTSF)2ClO4 showed that theKnight shift changes across Tc when the magnetic eld is small,but it is unchanged in the high eld [10]. Moreover, Hc2 measure-ment for (TMTSF)2ClO4 shows the possibility of two or three differ-ent pairing states depending on the strength of the magnetic eld[11].

Theoretically, various studies for (TMTSF) X have investigated

2. Formulation

The anisotropic extendedHubbardmodel that takes into accountthe Zeeman effect shown in Fig. 1a is given by

H X

i;j;rtijrc

yircjr

X

i

Uni"ni# X

i;j;r;r0Vijnirnjr0 ; 1

where tijr = tij + hzsgn (r) dij, where the hopping tij is consideredonly for intrachain (tx) and the interchain (ty) nearest neighbors.The FuldeFerrellLarkinOvchinstate, which has a nite center of ma[1,2], has nowadays become an isstheoretical studies have shown thatity pairing states stabilizes the FFLOthe inversion symmetry [37].

In the present study, we focus ontional superconducting state, includ0921-4534/$ - see front matter 2010 Elsevier B.V. Adoi:10.1016/j.physc.2010.05.042FFLO) superconductingentum of Cooper pairsreat interest. Previousg of even and odd par-due to the breaking of

ossibility of unconven-e FFLO state, in quasi-

tence of the charge uctuations and the spin uctuations is sup-ported from the fact that diffuse X-ray scattering experimentsobserve the coexistence of 2kF charge density wave (CDW) andthe 2kF spin density wave (SDW) in the vicinity of the supercon-ducting phase in (TMTSF)2PF6 [24,25].

In the present study, we investigate, using random phaseapproximation (RPA), the pairing state competition between spinsinglet, spin triplet and the FFLO state when coexisting spin andcharge uctuations mediate superconductivity.Magnetic eld effect on the pairing stateorganic superconductors (TMTSF)2X

H. Aizawa a,*, K. Kuroki a, Y. Tanaka b

aDepartment of Applied Physics and Chemistry, The University of Electro-CommunicatiobDepartment of Applied Physics, Nagoya University, Nagoya 464-8603, Japan

a r t i c l e i n f o

Article history:Available online 1 June 2010

Keywords:FFLO stateQuasi-one-dimensional systemParity mixing(TMTSF)2X

a b s t r a c t

We study the effect of t(TMTSF)2X by applying ramodel. We show that the ssuperconducting states mapossibility of a consecutiveing upon increasing the magap function in the FFLO st

journal homepage: wwwll rights reserved.ompetition in quasi-one-dimensional

hofu, Tokyo 182-8585, Japan

magnetic eld on the pairing state competition in organic conductorsm phase approximation to a quasi-one-dimensional extended Hubbardet pairing, triplet pairing and the FuldeFerrellLarkinOvchinnikov (FFLO)mpete when charge uctuations coexist with spin uctuations. This rises ansition from singlet pairing to FFLO state and further to Sz = 1 triplet pair-tic eld. We also show that the singlet and Sz = 0 triplet components of thehave d-wave and f-wave forms, respectively, which are strongly mixed.

2010 Elsevier B.V. All rights reserved.

le at ScienceDirect

a C

sevier .com/locate /physc

center of mass momentum Qcx for several values of Qcy forh = 0.03. As seen here, the parity mixing rate for Q = 3 and

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16

Q [ p/512]cx

Q =0 [p/64]cy

Q =4cyQ =2cy

l Qc

ss

hz=0.03

Q [ p/512]cx

(a)

Fig. 2. Qcx-dependence of (a) the eigenvalue, krrQ c, and (b) the parity mixing,

/STf 0 =/SSd , in the opposite spin pairing channel for hz = 0.03 and Vy = 0.35.

a C 470 (2010) 10851088momentum Qc (the total momentum 2Qc) within the weak cou-pling theory as

krr0

Q c/rr

0 k 1N

X

q

Vrr0 k; q f nrq f nr0 q

nrq nr0 q/rr

0 q; 2

where q q Q c; krr0

Q cis the eigenvalue of this linearized gap

equation, Vrr0 k; q is the pairing interaction that is obtained by

RPA and /rr0 k is the gap function, nr(q) is the band dispersion

(a)

(b)

Fig. 1. (a) The model adopted in this study. (b) Schematic gure of the gap for d-wave(left) and f-wave(right), where the red solid curves are the Fermi surface andthe blue dashed lines are the nodes of gap. (For interpretation of the references tocolor in this gure legend, the reader is referred to the web version of this article.)

1086 H. Aizawa et al. / Physicfrom the chemical potential and f(n) is the Fermi distributionfunction.

In the opposite spin pairing, we dene the singlet and the Sz = 0triplet component of the gap function as

/SSk /"#k /#"k=2; /ST0 k /"#k /#"k=2: 3In our calculation, the spin singlet and the spin triplet component ofthe gap function in the FFLO state is d-wave and f-wave as sche-matically shown in Fig. 1b, so we write the singlet d-wave (Sz = 0triplet f-wave) component of the FFLO state gap as /SSd/STf 0 ) inEq. (3). The eigenvalue of each pairing state is determined as fol-lows. krrQ c with Qc = (0,0) gives the eigenvalue of the singlet d-wavekSSd (Sz = 0 triplet f-wave kSTf 0 ) when /STf 0 0/SSd 0), while krrQ c0gives kFFLO. k

rrQ c

with Qc = (0,0) gives the eigenvalue for the Sz = 1triplet f-wave kSTf1 .

3. Results

We set the interchain interaction as Vy = 0.35 and the systemsize as 1024 128 in the following results. krrQ c with Qc = (Qcx,Qcy) are given in units of p/512 for x-direction and p/64 fory-direction.

Fig. 2a shows the eigenvalue of the linearized gap equation inthe opposite-spin pairing channel krrQ c for hz = 0.03. It can be seenthat the pairing state with (Qcx, Qcy) = (3,0) is most dominant.Studying other hz cases, we nd that the most dominant Qc liesin the x-direction [26,27,6], and the magnitude of the center ofmass momentum increases with increasing the magnetic eld[6]. Fig. 2b shows the parity mixing ratio /STf 0=/SSd in the oppo-site-spin pairing channel as a function of the x-component of thej /j

STf

SSd

0 0.2

0.4

0.6

0.8

0 2 4 6 8 10 12 14 16

Q =0 [p/64]cy

Q =4cyQ =2cy

0

1(b)z cx

Qcy = 0 takes a large value / STf 0=/SSd 0:8.In Fig. 3, we show the temperature dependence of the eigen-

value of the gap equation, krr0

Q c, in both the opposite- and paral-

lel-spin pairing states for hz = 0.03 The eigenvalue kFFLO krrQ c0 ofthe FFLO state with Qcx = 3 and Qcy = 0 reaches unity at a tempera-ture (T 0.012) higher than for other pairing states.

Gap functions normalized by the maximum value of the singletcomponent of the FFLO gap function are shown in Fig. 4. Theparameters are taken as hz = 0.03, Vy = 0.35, and T = 0.012, wherethe FFLO state with a nite center of mass momentum, Qcx = 3and Qcy = 0, is most dominant. The singlet (Sz = 0 triplet) compo-

0.2

0.4

0.6

0.8

1

1.2

0.01 0.014 0.018 0.022T

l Qc

ss

STf+1

STf-1

FFLO

SSd

hz=0.03

Fig. 3. The eigenvalue of the linearized gap equation, krrQ c , plotted as a function ofthe temperature T for hz = 0.03 and Vy = 0.35.

nent of the gap function in the FFLO state is d-wave (f-wave) asseen in Fig. 4a (Fig. 4b). At the Tc of the FFLO state, the singletd-wave component is strongly mixed with the Sz = 0 triplet f-wavecomponent.

Finally, we show in Fig. 5a a phase diagram in the temperature Tversus the magnetic eld hz space for the interchain off-site inter-action of Vy = 0.35, where the 2kF charge uctuations are slightlyweaker than that of the 2kF spin uctuations. The Tc in zero eldis Tc 0.012 and the estimated value of Paulis paramagnetic eldis hPz 0:03. We see that a consecutive transition from the singletpairing to FFLO and further to Sz = 1 triplet pairing takes place uponincreasing the magnetic eld. As shown from Fig. 5a, there seemsto be a reentrance from the superconducting state to anothersuperconducting state intervened by the normal state. However,

-1-0.8-0.6-0.4-0.20 0.2 0.4 0.6 0.81

-p -p/2 0 p/2 p-p

-p/2

0

p/2

p

kx

ky

(a) (b

Fig. 4. Gap function for (a) singlet component and (b) Sz = 0 triplet component in the FFsolid curves represent the Fermi surface and the green dashed lines are the nodes of thereferred to the web version of this article.)

H. Aizawa et al. / Physica C 40

0.02

0.04

0.06

0.08

0.006 0.008 0.01 0.012 0.014

hz

SC-SSd

SC-FFLO

SC-STf+1

NormalhzPauli

(a)T

Tc

hz

T

SC-SSd

SC-FFLO

SC-STf+1

NormalhzPauli

Orbital pair breaking effect

(b)

Fig. 5. (a) Calculated phase diagram in hzT plane for Vy = 0.35, where the greendashed curve is the Tc for the singlet d-wave, the red solid curve is for the FFLO state,and the blue dotted curve is for the Sz = 1 triplet f-wave. (b) Schematic gure of theorbital pair breaking effect on the superconducting phase diagram in Thz space,where the gray arrows schematically represent the orbital pair breaking effect. (Forinterpretation of the references to color in this gure legend, the reader is referredto the web version of this article.)it is more reasonable to consider that this reentrance does notactually take place due to the presence of the orbital pair breakingeffect. If this effect is taken into account, not only the FFLO statebut also the singlet and the triplet pairing state should stronglybe suppressed upon increasing the magnetic eld as shown sche-matically in Fig. 5b.

4. Conclusion

We have studied the competition between spin singlet, triplet,and the FFLO superconductivity in a model for (TMTSF)2X byapplying the RPA method and solving the linearized gap equation.We nd that: (i) consecutive pairing transitions from singlet pair-ing to FFLO state and further to Sz = 1 triplet pairing can take placeupon increasing the magnetic eld at the critical temperature inthe vicinity of the SDW+CDW coexisting phase, and (ii) in the FFLOstate, the Sz = 0 spin triplet pairing component is mixed with thespin singlet pairing component, thus resulting in a large paritymixing.

Acknowledgments

This work is supported by Grants-in-Aid for Scientic Researchfrom the Ministry of Education, Culture, Sports, Science and Tech-nology of Japan, and from the Japan Society for the Promotion ofScience. Part of the calculation has been performed at the facilitiesof the Supercomputer Center, ISSP, University of Tokyo.

-0.8-0.6-0.4-0.20 0.2 0.4 0.6 0.8

-p -p/2 0 p/2 p-p

-p/2

0

p/2

p

kx

ky

)

LO state with (Qcx, Qcy) = (3,0) on hz = 0.03, Vy = 0.35 and T = 0.012, where the blackgap. (For interpretation of the references to color in this gure legend, the reader is

70 (2010) 10851088 1087References

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1088 H. Aizawa et al. / Physica C 470 (2010) 10851088

Magnetic field effect on the pairing state competition in quasi-one-dimensional organic superconductors (TMTSF)2XIntroductionFormulationResultsConclusionAcknowledgmentsReferences

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