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Physica B 405 (2010) S250–S252

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Physica B

0921-45

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/physb

Magnetic field effect on the pairing competition in quasi-one-dimensionalorganic superconductors ðTMTSFÞ2X

Hirohito Aizawa a,�, Kazuhiko Kuroki a, Yukio Tanaka b

a Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japanb Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan

a r t i c l e i n f o

Keywords:

Organic superconductor

ðTMTSFÞ2X

FFLO state

Spin triplet pairing

Zeeman effect

26/$ - see front matter & 2009 Elsevier B.V. A

016/j.physb.2009.11.025

esponding author. Tel.: +81 42 443 5559; fax

ail address: aizawa@vivace.e-one.uec.ac.jp (H

a b s t r a c t

We microscopically study the effect of the magnetic field (Zeeman splitting) on the superconducting

state in a model for quasi-one-dimensional organic conductors ðTMTSFÞ2X. Using random phase

approximation, we investigate the competition between spin singlet, spin triplet pairings and the

Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state. We obtain a phase diagram in the T (temperature)–hz

(field) space. We show that consecutive transitions from singlet pairing to FFLO and further to Sz ¼ 1

triplet pairing can take place upon increasing the magnetic field when 2kF charge fluctuations coexist

with 2kF spin fluctuations.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Possibility of unconventional superconducting state in quasi-one-dimensional (Q1D) organic superconductors ðTMTSFÞ2X(X¼ PF6, ClO4), has been discussed since their discovery. Recentexperiments suggest possibility of a high-field superconductingstate, for instance, the spin triplet pairing and/or the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state [1,2]. An NMR experi-ment for ðTMTSFÞ2ClO4 shows that the Knight shift changes acrossTc when the magnetic field is small, but is unchanged in high field[3]. Also, Hc2 measurement for ðTMTSFÞ2ClO4 shows the possibilityof two or three different pairing states according to the strength ofthe magnetic field [4].

Some of the theoretical studies on the pairing state in a modelfor ðTMTSFÞ2X have discussed the possibility of the spin tripletpairing and the FFLO state [5–13]. Our previous works haveshown that the spin triplet f-wave pairing can compete with thespin singletd-wave pairing in Q1D systems when 2kF spinfluctuations coexist with 2kF charge fluctuations since the Fermisurface is disconnected in the b-direction [14–17]. There, the 2kF

charge fluctuations, along with the 2kF spin fluctuations, play animportant role, which is supported from the experimentalobservation of the coexistence of 2kF SDW and CDW phase nextto the superconducting phase in the pressure–temperature phasediagram of ðTMTSFÞ2PF6 [18,19].

In the present work, we study the pairing competitionbetween spin singlet, spin triplet and the FFLO state of the spinand charge fluctuation mediated superconductivity in a Q1D

ll rights reserved.

: +81 42 443 5563.

. Aizawa).

extended Hubbard model for ðTMTSFÞ2X using the random phaseapproximation (RPA).

2. Formulation

The extended Hubbard model that takes account of theZeeman effect is shown in Fig. 1(a), where the modelHamiltonian is

H¼X

i;j;stijscyiscjsþ

X

i

UnimnikþX

i;j;s;s0Vijnisnjs0 : ð1Þ

Here tijs ¼ tijþhz sgnðsÞdij, where we consider only Zeemansplitting, hz, and ignore the orbital effect, assuming that themagnetic field is applied parallel to the conductive x–y plane. Weconsider intrachain and interchain hoppings as tx and ty, wheretx ¼ 1 is taken as the energy unit. We consider the on-siterepulsion U, the nearest, 2nd nearest and 3rd nearest neighborrepulsions, Vx, Vx2, and Vx3 within the chains, and the interchainrepulsion Vy. Note that U (V2þVy) enhances the 2kF spin (charge)fluctuations. The case of 3

4 filling is considered corresponding tothe actual material.

Applying the RPA to this model, [16,17] we obtain the pairinginteractions of the bubble-type Vss0

bubðk�qÞ and the ladder-typediagrams Vss0

lad ðkþqÞ for the opposite (s0 ¼ s) and parallel (s0 ¼ s)spin paring state. The linearized gap equation for Cooper pairswith the total momentum 2Qc (Qc represents the center of massmomentum) is given by

lss0

Qcjss0 ðkÞ ¼

1

N

X

q

½Vss0bubðk�qÞþVss0

lad ðkþqÞ�f ðxsðqþ ÞÞ�f ð�xs0 ð�q�ÞÞ

xsðqþ Þþxs0 ð�q�Þjss0 ðqÞ;

ð2Þ

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Fig. 1. (Color online) (a) The model adopted in this study. (b) Schematic figure of

the gap for d-wave (left) and f-wave (right).

H. Aizawa et al. / Physica B 405 (2010) S250–S252 S251

where f ðxÞ is the Fermi distribution function, xsðkÞ is theband dispersion measured from the chemical potential,q7 ¼ q7Qc , jss0 ðkÞ is the gap function and lss

0

Qcis the eigenvalue

of this linearized gap equation. The center of massmomentum Q c which gives the maximum value of lssQc

lies inthe x-direction, while lssQc

takes its maximum at Q c ¼ ð0;0Þbecause the electrons with the same spin can be paired asðks;�ksÞ for all k.

The eigenvalue of each pairing state is determined asfollows. lssQc

with Q c ¼ ð0;0Þ gives the eigenvalue of thesinglet d-wave pairing lSSd (Sz ¼ 0 triplet f-wave pairing lSTf 0 )for the absence of the Sz ¼ 0 triplet pairing component(singlet pairing component) in the gap function of the FFLOstate, while lssQc

with Q c a ð0;0Þ gives lFFLO. lssQcwith Q c ¼ ð0;0Þ

gives the eigenvalue for the spin triplet f-wave pairing withSz ¼ 71lSTf 7 1 .

Fig. 2. (Color online) Calculated phase diagram in T2hz space for (a) Vy ¼ 0:32 and

(b) Vy ¼ 0:35. (c) Schematic figure of the magnetic field depairing effect, e.g. the

orbital depairing effect, without the Zeeman effect.

3. Result

To obtain the phase diagram, we set the parameters astx ¼ 1:0, ty ¼ 0:2, U ¼ 1:7, Vx ¼ 0:9, Vx2 ¼ 0:45, Vx3 ¼ 0:1, andVy ¼ 0:32 or 0.35, where 2kF charge fluctuations are slightlysmaller than 2kF spin fluctuations. Tc is the temperature atwhich the eigenvalue of the linearized gap equation, l, reachesunity. The Q c dependence of lssQ c

determine the Q c of the FFLOstate. Note that Q c has only finite x-component due to the shapeof the Fermi surface.

In the small and large 2kF charge fluctuations regime as seen inFig. 2(a) and (b), a consecutive paring state transition fromthe singlet d-wave pairing to the FFLO state and further to theSz ¼ 1 triplet f-wave can take place upon increasing the magneticfield in the Q1D system for ðTMTSFÞ2X. If the 2kF chargefluctuations are larger, which means Vy is larger, the pairingstate transition from the FFLO state to the Sz ¼ 1 triplet f-wavetakes place at lower filed. Although we consider only the Zeemaneffect as for the depairing effect of applying the magnetic field, wespeculate that the phase diagram will be modified as shownschematically in Fig. 2(c) if the orbital pair breaking effects areconsidered.

4. Conclusion

We have microscopically studied the phase diagram of theQ1D extended Hubbard model for ðTMTSFÞ2X in the temperature-magnetic field space. We conclude that the consecutive pairingtransition from the singlet d-wave pairing to the FFLO state andfurther to the Sz ¼ 1 triplet f-wave can take place upon increasingthe magnetic field.

Acknowledgments

We acknowledge S. Yonezawa for valuable discussions. Thiswork is supported by Grants-in-Aid for Scientific Research fromthe Ministry of Education, Culture, Sports, Science and Technologyof Japan, and from the Japan Society for the Promotion of Science.Part of the calculation has been performed at the facilities of theSupercomputer Center, ISSP, University of Tokyo.

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