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Physica C 445–448 (2006) 190–193

Electronic structures and NMR T�11 in quasi-one-dimensional

organic superconductor (TMTSF)2PF6

M. Takigawa a,*, M. Ichioka b, K. Kuroki c, Y. Tanaka d, Y. Asano a

a Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japanb Department of Physics, Okayama University, Okayama 700-8530, Japan

c Department of Applied Physics and Chemistry, the University of Electro-Communications, Chofu, Tokyo 182-8585, Japand Department of Material Science and Technology, Nagoya University, Nagoya 464-8603, Japan

Available online 3 May 2006

Abstract

On the basis of the Bogoliubov de Gennes theory, electronic structures around a vortex in quasi-one-dimensional organic supercon-ductor (TMTSF)2PF6 are studied at the quarter-filling electron density in magnetic fields applied parallel to the conduction chain. Weconsider three pairing symmetries (d-, p- and f-wave) in superconductors. In d- and f-wave symmetries, nuclear relaxation rate T�1

1 isproportional to temperatures because quasiparticles around the vortex relax spins.� 2006 Elsevier B.V. All rights reserved.

PACS: 74.70.Kn; 74.20.Rp; 74.25.Op; 76.60.Pc

Keywords: (TMTSF)2PF6; Superconducting pairing symmetry; Vortex excitation; NMR T�11

1. Introduction

Recently much attention has been focused on supercon-ductivity in quasi one-dimensional (Q1D) organic com-pound (TMTSF)2PF6, which is called Bechgaard salt.Under a pressure, the superconducting phase appearsabove the SDW phase. A number of studies have beenmade on analyzing the superconducting pairing symmetryso far. In theories, three kinds of pairing symmetry havebeen proposed: d-wave [1–3], p-wave [3–6] and f-wave [7–11]. The d- and f-wave pair potentials have line nodes onthe Fermi surface, whereas p-wave potential has no nodes.Experimentally, the spin–triplet pairing symmetry has beensuggested because the Knight shift does not change acrossTC. A NMR experiment by Lee et al. [12] revealed that the

0921-4534/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.physc.2006.03.112

* Corresponding author. Postal address: Department of Applied Phys-ics, Room A1-67, Building A, Graduate School of Engineering, 21stCentury COE: Topology Science and Technology, Hokkaido University,North 13 West 8 Kita-ku, Sapporo 060-8628, Japan. Tel./fax: +81 11 7067843.

E-mail address: takigawa@topology.coe.hokudai.ac.jp (M. Takigawa).

relaxation rate T�11 has two characteristics (i) T�1

1 / T inlow temperatures (T� Tc) and (ii) T�1

1 has a small peakat Tc. Since the NMR experiment was done under magneticfields in the chain direction, we should consider effects ofquasiparticles around the vortices on T�1

1 to understandthe experimental results. In this paper, we will explainexperimental behaviors of T�1

1 in the mixed state of Q1Dorganic superconductor.

2. Formulation

To study electronic structures in Q1D superconductorunder magnetic fields in the conduction chain, we solvethe Bogoliubov de Gennes (BdG) equation self-consistentlyon the three-dimensional tight-binding lattices. Hamilto-nian is given by

H ¼Xi;j;r

ti;jayj;rai;r � l

þXi;j;r

V i;j Dyji;rai;�raj;r þ Dji;rayi;rayj;�r0

� �; ð1Þ

M. Takigawa et al. / Physica C 445–448 (2006) 190–193 191

where aþi;rðai;rÞ is a creation (annihilation) operator of anelectron at a lattice site i with spin r and Vi,j represents theattractive interaction in the z-direction (chain direction).The pairing interactions are working between second nearestneighbor sites for d- and p-wave symmetries and the fourthnearest neighbor sites for f-wave symmetry. After the Fou-rier translation in the z-direction, we obtain Di,j = u(kz)Di,u(kz) = cos2kz for d-wave, sin2kz for p-wave, and cos4kz

for f-wave symmetries. The transfer integral in the xy-planeis expressed as ~tij ¼ tij exp½iðp=/0Þ

R rj

riAðrÞ � dr�, where the

vector potential AðrÞ ¼ 12H � r is given in the symmetric

gauge with external fields H = (0,0,H) and /0 is the flaxquantum. The hopping integrals tij between nearest neighborsites are chosen as tx:ty:tz = 1:0.03:10 to reproduce the Q1DFermi surface of (TMTSF)2PF6. In calculations, we take theparing interaction U = �38tx. The charge density is kept tobe at the quarter-filling. By the Bogoliubov transformation,the BdG equation is given by

Xi

Kji Dji

Dyji �K�ji

!ueðriÞveðriÞ

� �¼ Ee

ueðrjÞveðrjÞ

� �; ð2Þ

where Kij ¼ �~tij þ dijð�2tz cos kz � lÞ, Dij = UDidiju(kz),and ua(ri), va(ri) are wave functions at the lattice site i

belonging to the energy Ea. The expressions for the pair po-tential and the charge density are given by

~DðriÞ ¼Xa;kz

uaðriÞv�aðriÞf ðEaÞuðkzÞ; ð3Þ

ni ¼ ni" þ ni# ¼X

a

ðjuaðriÞj2f ðEaÞ þ jvaðriÞj2ð1� f ðEaÞÞÞ;

ð4Þ

where DðriÞ ¼ ~DðriÞ exp½iðp=/0ÞR rj

riAðrÞ � dr�.

We consider that two vortices accommodate in a unitcell with 20 · 6 lattice sites in the xy plane. We also assumethat the vortex core is located in the plaquette. By intro-ducing the quasimomentum of the magnetic Bloch state,we obtain the wave function under the periodic boundarycondition whose region covers a large number of unit cells.The spin–spin correlation function v+�(r, r 0,iXn) is calcu-

-π

-π/2

0

π/2

π

-π/2 0 π/2

k x

kz

-π

-π/2

0

π/2

π

-π/2 0

k x

k

Fig. 1. The Fermi surface and signs of pairing symmetries are shown for (a) d-wd- and f-wave have line nodes.

lated from Green’s functions and the nuclear spin relaxa-tion rate is given by

Rðr; r0Þ ¼ Imvþ;�ðr; r0; iXn ! Xþ igÞ=ðX=T ÞjX!0

¼ �Xa;a0½uaðrÞua0 ðr0ÞvaðrÞva0 ðr0Þ � vaðrÞua0 ðr0Þ

� uaðrÞva0 ðr0Þ� � pTf 0ðEaÞdðEa � Ea0 Þ; ð5Þ

where f(E) is the Fermi distribution function. We assumethat r = r 0 because site-diagonal spin relaxations are con-sidered to be dominant. The r-dependent relaxation timeis given by T1(r) = 1/R(r, r). In calculations, we used(r) = p�1Im(x � ig) to handle the discrete energy levelsdue to the finite size effect with g = 0.02tx. In Eq. (5), thefirst term is proportional to N(r,E)2 for r = r 0. To under-stand the behavior of T1(r), we also calculate local densityof states (LDOS) given by

NðE; rÞ ¼X

a

½juaðrÞj2dðE � EaÞ þ jvaðrÞj2dðE þ EaÞ�.

3. Local density of state and NMR T�11 around vortices

Three pair potentials have different topology from oneanother in the k-space as shown in Fig. 1. The pair poten-tials for d- and f-waves change their sign at kz = ±p/4, andthey have line nodes on the Fermi surface. The sign of pairpotentials for p- and f-waves is also changed at the kz = 0.The pair potential for p-wave, however, has no nodes onthe Fermi surface. By solving the Eq. (2) self-consistentlywith Eq. (3), we obtain electronic states and order param-eters. The critical temperature Tc/tx results in 1.3 (d-wave),2.6 (p-wave), and 2.2 (f-wave) and jDi(H = 0)j at the lowesttemperature are calculated to be 0.065 (d-wave), 0.135 (p-wave) and 0.11 (f-wave). Since the vortex core is locatedat the central of the plaquette in the xy-plane, the spatialvariation of jDij is very small (it is about 0.1 � 1% of theirbulk values). Therefore, the modulation of the LDOSaround the vortex is also expected to be small.

Fig. 2 shows LDOS at the nearest lattice site to the core(solid line) and density of states in H = 0 (dashed line) andnormal density of states (dotted line) at the lowest T. In d-

π/2z

-π

-π/2

0

π/2

π

-π/2 0 π/2

k x

kz

ave, (b) p-wave and (c) f-wave symmetries. At kz = p/4, pair potentials for

0 0.2 0.4 0.6 0.8 1 1.2 1.4

VORTEXH=0

NORMAL

0 0.5 1 1.5 2 2.5 3

VORTEXH=0

NORMAL

0 0.5 1 1.5 2 2.5 3

VORTEXH=0

NORMAL

0.01 0.1 1 10 0.01 0.1 1 10

Fig. 3. Dependences of the relaxation rate T�11 on temperatures at the nearest lattice site to the vortex core are shown for (a) d-wave, (b) p-wave, and (c) f-

wave symmetries. In insets of (a) and (c), results are plotted in double logarithm chart.

0

0.02

0.04

0.06

0.08

-2 -1 0 1 2

E / tx

VORTEXH=0

0

0.02

0.04

0.06

0.08

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

E / tx

VORTEXH=0

0

0.02

0.04

0.06

0.08

-2 -1 0 1 2

E / tx

VORTEXH=0

(a) (b) (c)

Fig. 2. Local density of states at the nearest lattice site to the vortex core is shown with solid lines for (a) d-wave, (b) p-wave, and (c) f-wave symmetries.For comparison, DOS in H = 0 is plotted with dashed lines. Temperature is fixed at T = 0.1tx.

192 M. Takigawa et al. / Physica C 445–448 (2006) 190–193

wave (a) and f-wave (c), at the nearest site to the vortexcore, peaks at the gap edges are suppressed and low energyquasiparticle states appear under magnetic fields. Inp-wave (Fig. 2(b)), a large gap structures remain even inmagnetic fields. Quasiparticles are confined around theJosephson-type vortex in p-wave symmetry. Thus excita-tion energies of quasiparticles become finite which are com-parable to the gap energy in the present calculation. Twosharp peaks in LDOS around E = ±4tx are reflecting suchquasiparticle states. Since jDij is not significantly sup-pressed around the Josephson vortices, the zero-energypeak of the vortex core state does not appear in LDOS.In both d- and f-wave, quasiparticles basically have contin-uum spectrum because quasiparticles are not confinedinside vortices because of line nodes in gap functions.

In Fig. 3, we plot dependences of T �11 on temperatures

for three pairing symmetries in the presence of magneticfields. For comparison, we also show T�1

1 in the normalstate and T�1

1 in the superconducting state at H = 0. A rela-tion T�1

1 / T in the normal state implies the Korringa law.At H = 0, a relation T�1

1 / T 3 can be seen in d- and f-wavesymmetries because of line nodes on the Fermi surface. In p-wave at H = 0, the large Hebel–Slichter-like peak and theexponential temperature dependence of T�1

1 are seenbecause of the full gap function. The characteristic behav-iors of T�1

1 in p-wave symmetry remain unchanged even inmagnetic fields as shown in (b). However in d-wave symme-

try, dependence of T�11 on T is changed to T�1

1 / T in lowtemperatures under magnetic fields as shown in the insetin (a). This is because quasiparticles around vortices atthe zero-energy (the finite LDOS at E = 0) relax spins.Almost the same tendency can be seen in f-wave symmetryin Fig. 3(c). However a power index is slightly larger thanunity (i.e. T�1

1 / T a with 1 < a < 3). This is becausejDi(H = 0)j in f-wave is larger than that in d-wave. Actuallywe find smaller amplitude of LDOS at E = 0 in f-wave thanthat in d-wave. It is also noted that the behavior of T�1

1 nearTc is quite different between d-wave and f-wave. We haveconfirmed that such differences between d- and f-waves dis-appear when jDi(H = 0)j in f-wave is close to that in d-wave.

4. Summary

We have calculated the relaxation rate T�11 in Q1D

organic superconductor in a mixed state for d-, p- and f-wave pairing symmetries. In d-wave and f-wave symme-tries, quasiparticles are excited around vortices at thezero-energy and relax spins because pair potentials haveline nodes on the Fermi surface. As a consequence, we findT�1

1 / T , in stead of T�11 / T 3, in low temperatures far

below Tc. In p-wave, the large Hebel–Slichter-like peakand the exponential dependence of T �1

1 on T are seen evenin magnetic fields. These data will be useful for detectingthe paring symmetry of the (TMTSF)2PF6.

M. Takigawa et al. / Physica C 445–448 (2006) 190–193 193

Acknowledgements

We thank K. Kanoda for useful discussion on theirNMR experiments. This work was supported by a Grant-in-Aid for the 21st Century COE ‘‘Topology Science andTechnology’’ in Hokkaido University.

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