Physica C 470 (2010) 1085–1088
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Physica C
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Magnetic field effect on the pairing state competition in quasi-one-dimensionalorganic superconductors (TMTSF)2X
H. Aizawa a,*, K. Kuroki a, Y. 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
Article history:Available online 1 June 2010
Keywords:FFLO stateQuasi-one-dimensional systemParity mixing(TMTSF)2X
0921-4534/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.physc.2010.05.042
* Corresponding author. Address: Department of ApRoom 310, Building E1, School of Electro-CommuElectro-Communications, Chofu, Tokyo 182-8585, Japa+81 42 443 5563.
E-mail address: [email protected] (H.
a b s t r a c t
We study the effect of the magnetic field on the pairing state competition in organic conductors(TMTSF)2X by applying random phase approximation to a quasi-one-dimensional extended Hubbardmodel. We show that the singlet pairing, triplet pairing and the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO)superconducting states may compete when charge fluctuations coexist with spin fluctuations. This rises apossibility of a consecutive transition from singlet pairing to FFLO state and further to Sz = 1 triplet pair-ing upon increasing the magnetic field. We also show that the singlet and Sz = 0 triplet components of thegap function in the FFLO state have ‘‘d-wave” and ‘‘f-wave” forms, respectively, which are strongly mixed.
� 2010 Elsevier B.V. All rights reserved.
1. Introduction
The Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superconductingstate, which has a finite center of mass momentum of Cooper pairs[1,2], has nowadays become an issue of great interest. Previoustheoretical studies have shown that a mixing of even and odd par-ity pairing states stabilizes the FFLO state due to the breaking ofthe inversion symmetry [3–7].
In the present study, we focus on the possibility of unconven-tional superconducting state, including the FFLO state, in quasi-one-dimensional (Q1D) organic superconductors (TMTSF)2X(X = PF6, ClO4, etc.). Some experiments in the high magnetic fieldhave indeed suggested the possibility of the spin triplet pairingand/or the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state [8,9].Recently, an NMR experiment for (TMTSF)2ClO4 showed that theKnight shift changes across Tc when the magnetic field is small,but it is unchanged in the high field [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 field[11].
Theoretically, various studies for (TMTSF)2X have investigatedthe possibility of the spin triplet pairing and the FFLO state [12–20]. 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
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Aizawa).
2kF spin and 2kF charge fluctuations coexist [21–23,7]. The coexis-tence of the charge fluctuations and the spin fluctuations 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 fluctuations mediate superconductivity.
2. Formulation
The anisotropic extended Hubbard model that takes into accountthe Zeeman effect shown in Fig. 1a is given by
H ¼X
i;j;rtijrcyircjr þ
X
i
Uni"ni# þX
i;j;r;r0Vijnirnjr0 ; ð1Þ
where tijr = tij + hzsgn (r) dij, where the hopping tij is consideredonly for intrachain (tx) and the interchain (ty) nearest neighbors.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) fluctuations. The case of3/4 filling is considered corresponding to the actual material. Weignore the orbital effect, assuming that the magnetic field is appliedparallel to the conductive x–y 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
(a)
(b)
Fig. 1. (a) The model adopted in this study. (b) Schematic figure 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 figure legend, the reader is referred to the web version of this article.)
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 =4cy
Q =2cy
lQc
ss
hz=0.03
j /j
STf
SSd
0 0.2
0.4
0.6
0.8
0 2 4 6 8 10 12 14 16
Q [ p/512]cx
Q =0 [p/64]cy
Q =4cy
Q =2cy
0
1
(a)
(b)
Fig. 2. Qcx-dependence of (a) the eigenvalue, kr�rQ c
, and (b) the parity mixing,/STf 0 =/SSd , in the opposite spin pairing channel for hz = 0.03 and Vy = 0.35.
0.2
0.4
0.6
0.8
1
1.2
0.01 0.014 0.018 0.022T
lQc
ss’
STf+1
STf-1
FFLO
SSd
hz=0.03
Fig. 3. The eigenvalue of the linearized gap equation, kr�rQ c
, plotted as a function ofthe temperature T for hz = 0.03 and Vy = 0.35.
1086 H. Aizawa et al. / Physica C 470 (2010) 1085–1088
momentum Qc (the total momentum 2Qc) within the weak cou-pling theory as
krr0Q c
/rr0 ðkÞ ¼ 1N
X
q
Vrr0 ðk; qÞ f ðnrðqþÞÞ � f �nr0 ð�q�Þð ÞnrðqþÞ þ nr0 ð�q�Þ
/rr0 ðqÞ; ð2Þ
where q� ¼ q� Q c; krr0Q c
is the eigenvalue of this linearized gapequation, Vrr0 ðk; qÞ is the pairing interaction that is obtained byRPA and /rr0 ðkÞ is the gap function, nr(q) is the band dispersionfrom the chemical potential and f(n) is the Fermi distributionfunction.
In the opposite spin pairing, we define the singlet and the Sz = 0triplet component of the gap function as
/SSðkÞ ¼ ½/"#ðkÞ � /#"ðkÞ�=2; /ST0 ðkÞ ¼ ½/"#ðkÞ þ /#"ðkÞ�=2: ð3Þ
In 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 /SSdð/STf 0 ) inEq. (3). The eigenvalue of each pairing state is determined as fol-lows. kr�r
Q cwith Qc = (0,0) gives the eigenvalue of the singlet d-wave
kSSd (Sz = 0 triplet f-wave kSTf 0 ) when /STf 0 ¼ 0ð/SSd ¼ 0), while kr�rQ c–0
gives kFFLO. krrQ c
with Qc = (0,0) gives the eigenvalue for the Sz = ± 1triplet f-wave kSTf�1 .
3. Results
We set the interchain interaction as Vy = 0.35 and the systemsize as 1024 � 128 in the following results. kr�r
Q cwith 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 kr�r
Q cfor hz = 0.03. It can be seen
that the pairing state with (Qcx, Qcy) = (3,0) is most dominant.Studying other hz cases, we find 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 field[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 the
center of mass momentum Qcx for several values of Qcy forhz = 0.03. As seen here, the parity mixing rate for Qcx = 3 andQcy = 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 ¼ kr�rQ c–0 of
the 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 finite center of mass momentum, Qcx = 3and Qcy = 0, is most dominant. The singlet (Sz = 0 triplet) compo-
-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
-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
(a) (b)
Fig. 4. Gap function for (a) singlet component and (b) Sz = 0 triplet component in the FFLO state with (Qcx, Qcy) = (3,0) on hz = 0.03, Vy = 0.35 and T = 0.012, where the blacksolid curves represent the Fermi surface and the green dashed lines are the nodes of the gap. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)
H. Aizawa et al. / Physica C 470 (2010) 1085–1088 1087
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 field hz space for the interchain off-site inter-action of Vy = 0.35, where the 2kF charge fluctuations are slightlyweaker than that of the 2kF spin fluctuations. The Tc in zero fieldis Tc ’ 0.012 and the estimated value of Pauli’s paramagnetic fieldis hP
z ’ 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 field. As shown from Fig. 5a, there seemsto be a reentrance from the superconducting state to anothersuperconducting state intervened by the normal state. However,
0
0.02
0.04
0.06
0.08
0.006 0.008 0.01 0.012 0.014
hz
T
SC-SSd
SC-FFLO
SC-STf+1
Normal
hzPauli
Tc
hz
T
SC-SSd
SC-FFLO
SC-STf+1
Normal
hzPauli
Orbital pair breaking effect
(a)
(b)
Fig. 5. (a) Calculated phase diagram in hz–T 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 figure of theorbital pair breaking effect on the superconducting phase diagram in T–hz space,where the gray arrows schematically represent the orbital pair breaking effect. (Forinterpretation of the references to color in this figure 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 field 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 find 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 field 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 Scientific 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.
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