PHYSICAL REVIEW B, VOLUME 64, 212504

Triplet organic quasi-one-dimensional superconductors:Angular dependence of the upper critical field

C. D. Vaccarella and C. A. R. Sa´ de MeloSchool of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332

~Received 9 April 2001; published 12 November 2001!

We calculate the angular dependence of the upper critical fieldHc2in triplet organic quasi-one-dimensional

superconductors at high magnetic fields applied along theyz plane, assuming that the order parameter corre-sponds to an equal spin triplet pairing symmetry state. The results described here may be relevant to theangular dependence ofHc2

in the Bechgaard salts family (TMSTF)2X.

DOI: 10.1103/PhysRevB.64.212504 PACS number~s!: 74.70.Kn, 74.60.Ec

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The upper critical fieldHc2of quasi-one-dimensiona

~Q1D! superconductors for perfectly aligned magnetic fiealong they (b8) axis has been calculated for both singlet atriplet states.1–3 More recently new experiments performein these systems lead to the observation of unusual superductivity at high magnetic fields4–6 and created new excitement in the area of organic superconductivity.7–10 In Q1Dsuperconductors@of the Bechgaard salts family with chemcal formula (TMTSF)2X, whereX5ClO4,PF6 , . . . # Hc2

ex-

ceeds substantially the Pauli paramagnetic fieldHp .4–6 Re-cent experiments of Leeet al.6 in (TMTSF)2PF6 at pressures

P'5.9 kbar showed thatHc2

(b8) for Hib8 exceededHp (Hp

'2.2 T for Tc51.2) by a factor of 4. Typically,Hp can beexceeded in singlet superconductors either via strong sorbit scattering,11 or via the formation of the Larkin-Ovchinnikov-Fulde-Ferrel~LOFF! state;12 and in triplet su-perconductors via equal spin pairing.1,3,13 Lee et al.6

estimated that they would need a spin-orbit scatteringtSO

21'10 K to fit their data to singlet superconductors wstrong spin-orbit scattering. This value was three to fourders of magnitude larger than expected. FurthermLebed14 estimated that the maximum upper critical field f

Hib for the LOFF state at zero temperature to beHc2

(b8)(0)

'4.0 T. The experimental values of Leeet al.6 exceed

Hc2

(b8)(0) by at least a factor of 2. This seems suggestive

triplet superconductivity is drivingHc2

(b8) to exceedHp and

Hc2

(b8)(0). The exact symmetry of the triplet phase and t

mechanism of superconductivity are not yet known, hoever, an early suggestion of triplet superconductivity~at zeromagnetic field! was given by Abrikosov15 in order to explainthe suppression of superconductivity in (TMTSF)2X in thepresence of disorder~nonmagnetic impurities!.16,17

Given that Bechgaard salts are very clean materials, wweak spin-orbit scattering,6,18 and have very light elementswith weak spin-orbit coupling; we consider here only tcase of weak spin-orbit coupling equal spin triplet pairi~ESTP! superconductivity.13,19,20 The particular ESTP statdiscussed here corresponds to the high magnetic field exsion of the weak spin-orbit coupling state3B3u(a) at zero~low! magnetic field, which is a fully gapped ‘‘px’’ state.21

Here, we focus only on this ESTP state, since it is also csistent with the absence of77Se Knight shift forHiy (b8),22

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or Hix (a) ~Ref. 23! in (TMTSF)2 PF6. In this ESTP statethe low magnetic field susceptibility in the superconductistate at zero temperaturexs(T50)5xn ~where xn is thesusceptibility at the normal state! for any directions of theapplied field13,19,20 provided that the magnitude of the applied field is larger than the small spin-orbit pinning fieHd . This is in sharp contrast with a recent theoretical sugestion of a different triplet state, wherexb8'xn , but xa!xn at low temperatures.24

Here, we study only the angular dependence ofHc2in

(TMTSF)2ClO4,25 when the magnetic field is applied alonthe yz plane, and show that deviations from perfect aligment along they (b8) axis (u590.0°) cause a dramatic droin the critical temperatureTc(H) for intermediate values othe magnetic field in the range of 5 to 30 T. Furthermore,show thatTc(H) is increasedas a function of magnetic fieldwhenu.u r'89.7° for the same range of fields~5 to 30 T!due to a strong supression of orbital effects. We then arthat this rapid suppression of the critical temperature, athis possible reentrance foru.u r are in qualitative agreement with the rapid drop and possible reentrance of thetative Hc2

in (TMTSF)2ClO4.18

Q1D systems (TMTSF)2ClO4 and (TMTSF)2PF6 arewell known for their highly anisotropic Fermi liquid normastates at the relevant experimental pressuresP50, and P'6 kbar, respectively.4,18,22Thus, we model the noninteracing part of the Hamiltonian of Q1D systems via the enerrelation

«a,s~k!5«a~k!2smBH, ~1!

with the a-branch dispersion

«a~k!5vF~akx2kF!1tycos~kyb!1tzcos~kzc!, ~2!

corresponding to an orthorombic crystal with lattice costantsa, b, andc ~Ref. 26! along thex, y, andz axis, respec-tively. In addition, sinceEF5vFkF@ty@tz , the Fermi sur-face of such systems is not simply connected, being oboth in thexz plane and in thexy plane. Furthermore, theelectronic motion can be classified as right going (a51) orleft going (a52).

We begin our discussion by analyzing the electronic mtion for magnetic fieldsH5(0,Hy ,Hz) in yz plane. We workin units where the Planck’s constant and the speed of l

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BRIEF REPORTS PHYSICAL REVIEW B 64 212504

are equal to one, i.e.,\5c* 51, and, throughout the papewe use the gaugeA5(Hyz2Hzy,0,0), wherea andkx arestill conserved quantities~good quantum numbers! while kyandkz are not. In this gauge, the semiclassical equationmotion become

x5x01avFt, ~3!

y5y01a~ tyb/vcy!cos@ky~0!b1avcy

t#, ~4!

z5z02a~ tzc/vcz!cos@kz~0!c2avcz

t#, ~5!

with vcy5vFGy and vcz

5vFGz , while Gy5ueuHzb and

Gz5ueuHyc, whereHz5H cosu and Hy5H sinu, whereu

is the angle betweenH and z. The frequenciesvcyandvcz

are independent and characterize the periodic motion athe y direction andz direction, respectively. This semiclasscal analysis suggests a classification of three differentgimes of confinement for the electronic motion:~a! a one-dimensional regime, where the semiclassical amplitudemotion are confined along both they and z direction, i.e.,tz /vcz

!1 andty /vcy!1 ~provided that there is no magnet

breakdown, and thatEF@vcyand EF@vcz

); ~b! a two-dimensional regime, where the semiclassical amplitudemotion are confined along thez direction but not along theydirection, i.e., tz /vcz

!1 and ty /vcy@1; and ~c! a three-

dimensional regime, where the semiclassical amplitudemotion are not confined in eithery or z directions, i.e.,ty /vcy

@1 andtz /vcz@1.

Now we turn our attention to the quantum aspects ofproblem. In the presence of the magnetic fieldH, the nonin-teracting Hamiltonian is

H0~k2eA!5«a~k2eA!2smBH ~6!

in the gaugeA5(Hyz2Hzy,0,0). The eigenfunctions oH0(k2eA) are

Fqn~r !5exp@ ikxx#JNy2nyS aty

vcyD JNz2nzS atz

vczD , ~7!

wherer5(x,y,z), y5nyb andz5nzc with associated quantum numbersqn5a,kx ,Ny ,Nz ,s and eigenvalues

eqn5«a,s~kr!1aNyvcy1aNzvcz

. ~8!

The functionJp(u) is the Bessel function of integer orderpand argumentu, while now

«a,s~kr!5vF~akx2kF!2smBH ~9!

is a 1D dispersion. Notice that the eigenfunctions in Eq.~7!become quantum confined in the same regime wheresemiclassical electronic motion is also confined, i.e., whtz /vcz

!1 and ty /vcy!1. In addition, notice that the

eigenspectrum in Eq.~8! involves many magnetic subbandlabeled by the quantum numbersNy and Nz and that theeigenvalueeqn is invariant under the quantum number tranformation (a,kx ,Ny ,Nz ,s)→(2a,2kx ,2Ny ,2Nz ,s),for the same spin states. Furthermore, these magnetic su

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bands are spin split into spin-up and spin-down bands. ThCooper pairs can be easily formed in an ESTP pairing stinvolving electrons with quantum numbers (a,kx ,Ny ,Nz ,s)and (2a,2kx ,2Ny ,2Nz ,s), provided that the pairing in-teraction conserves spin@which seems to be the case for th(TMTSF)2X family, except for very small spin-orbit and dipolar couplings#.

We choose the following interaction Hamiltonian:

H int52(m

E drlmOm† ~r !Om~r !, ~10!

to study the angular dependence of theHc2in these Q1D

superconductors. Here, the operatorOm(r )5sc2,s† (r )

3(tm)s,s8c1,2s8(r ), where ca†(r ) are creation operators

for electrons in thea sheet, andt615(sx7 isy)/2 andt05sz , with s i being the usual Pauli matrices. The order prameter vectorDm is proportional to the expectation valu^Om(r )&, and is consistent with the weak spin-orbit couplinstate 3B3u(a) at zero~low! magnetic field, which is a fullygapped ‘‘px’’ state.21 In an ESTP state,Dm has componentsin the ms511 (m5↑↑) and ms521 (m5↓↓) channels,only. Here we have chosen the direction of the applied fito serve as the quantization axis.

In this case, the corresponding order parameter equais

Dm~r !5lm (Ny ,Nz

ANy ,Nz

(m) Dm~r1SNyNz!, ~11!

wherer5(x,y,z), SNyNz5(0,Nyb,Nzc). The matrix operator

ANy ,Nz

(m) 5ENyNz~x!FNyNz

(m) ~ qx2KNyNz! ~12!

with KNyNz52NyGy12NzGz involves two terms, the

prefactorENyNz(x)5exp@2i(KNyNz

x)# and the operator function

FNyNz

(m) ~ qx!5 (a,Min

Lmaa~ qx1QMin

!WNyNz

aa ~Min!, ~13!

with QMin5(M1y1M2y)Gy1(M1z1M2z)Gz , andqx52 i ]/]x. The weight function

WNyNz

aa ~Min!5GNy ,Nz

a ~M2n!GNy ,Nz

a ~M1n!, ~14!

contains the single-particle renormalization factor

GNy ,Nz

a ~M2n!5PNy

a ~M2y!PNz

a ~M2z! ~15!

with coefficient

PNn

a ~M2n!5JM2nS atn

vcnD JM2n2NnS atn

vcnD . ~16!

Notice that the termGNy ,Nz

a (M1n) in Eq. ~14! can be ob-

tained from Eq.~15! via the substitutiona→a and M2n

→M1n . In addition, the operatorFNyNz(Qx) contains the

Cooper singularity contribution

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BRIEF REPORTS PHYSICAL REVIEW B 64 212504

Lmaa~Qx!5NmFcS 1

2D2RcS 1

21

avFQx

4p iT D1 ln~V !Gwith Nm being the spin-dependent density of states, andV5(2vdg/pT).

The physical interpretation of Eq.~11! is as follows. Theorder parameterDm(x,y,z) at a given position in spaccouples with the order parameter atDm(x,y1Nyb,z1Nzc)at a different position, thus indicating that Josephson cpling between chains oriented along thex direction occurs.The strength of this coupling is controlled by the matoperatorANy ,Nz

(m) , which is dependent both on the magnitu

of the magnetic field and its direction, throughGy andGz .This result shows that the quantization effects of the mnetic field can make Q1D superconductors evolve fromstrongly coupled Ginzburg-Landau local regime~low mag-netic fields! into a variable rangeJosephson coupled nonlocal regime~high magnetic fields!.

We will now focus our attention on three limits, where thcritical temperature as a function of the applied magnefield ~or conversely the upper critical field as a functiontemperature! can be calculated analytically. We will assumthat the interactionlm5l and the density of statesNm5Nare independent of the spin channel, which implies thatDm5D in the analysis that follows.

One-dimensional regime. As seen in the semiclassicaanalysis of Eqs.~3!, ~4!, and~5!, and in the quantum analysifollowing Eq. ~8!, there is double confinement of the orbitmotion whenty /vcy

!1 and tz /vcz!1. Using the fact that

qy andqz are good quantum numbers one can write

D~x,y,z!5exp@ i ~qyy1qzz!#u~x!, ~17!

which substituted in Eq.~11! transforms it into the generalized ~two-term! Hill equation

F2b1dl x2 d2

dx2 1A1d2(n

Bncosfn~x!Gu~x!50, ~18!

to nontrivial leading order inqx . Here fn(x)5(qnsn

22Gnx), n5y,z, sy5b, andsz5c. The coefficient of thefirst term is b1d5(12ty

2/vcy

2 2tz2/vcz

2 ). This coefficient

renormalizes the characteristic length scale along thex direc-tion l x5ACvF /T, which contains the constantC57z(3)/16p2. The second coefficient is

A1d5Fb1dln~V !1(n

S tn

vcnD 2

lnU2vd

vcn

UG21

lN ,

while the Bn5(tn /vcn)2lnugvcn

/2pTu are the last coeffi-cients corresponding to the amplitude of cosinusoidal spaoscillations. Physically, Eq.~18! corresponds to a limit ofweakly coupled Josephson chains, with magnetic fieldpendent Josephson couplingsBn . The highest critical tem-perature in high magnetic fields, to lowest order intn

2/vcn

2 ,

occurs whenqy5qz50. Thus,Tc is determined by the condition A1d50, leading to

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Tc5Tc1dF12(n

S tn

vcnD 2

lnU gvcn

pTc1d

UG , ~19!

with prefactorTc1d5(2vdg/p)exp(21/lN), since both the

amplitudeBn and the periodp/Gn of cos(2Gnx) are verysmall. Whenty /vcy

!1 and tz /vcz!1 there is a magnetic

field induced double quantum confinement~localization! ef-fect along both they andz direction, for almost allu. How-ever, localization along they direction is harder to achieve inthe Bechgaard salts (TMTSF)2X (X5ClO4,PF6 , . . . ),since ty is typically of the order of 100 K, thus requiringmagnetic fields on the order of 100 T to take place.

Two-dimensional regime.When the value of magneticfield is lowered and the angleu is changed to satisfy conditions ty /vcy

@1 andtz /vcz!1, we reach a two-dimensiona

regime where quantum confinement occurs only along thzdirection. In this two-dimensional regime the effects of tmagnetic field along they direction can be treated semiclasically. Using Eq.~17! the resulting eigenvalue equation is

F2b2dl x2 d2

dx2 1Ly~x!1A2d1Bzcosfz~x!Gu~x!50, ~20!

where the coefficient of the first term isb2d5(12tz2/vcz

2 ).

The second term isLy(x)5@ l yfy(x)/b#2, with coefficientl y5ACtyb/TA2, while the third term is

A2d5b2dln~V !1S tz

vczD 2

lnU2vd

vczU2 1

lN .

The critical temperature resulting from Eq.~20! is

Tc5Tc2dF 12CA2tyvcy

Tc2d

22S tz

vczD 2

lnU gvcz

pTc2d

UG , ~21!

whereTc2d5(2vdg/p)exp(21/lN). A plot of Tc as a func-

tion of magnetic field for various anglesu close to 90.0° isshown in Fig. 1.

FIG. 1. This figure shows the suppression of the critical teperatureTc for applied fieldsH along theyz plane in the two-dimensional regime. The directionu590.0° corresponds to perfecalignment along they axis. The parameters used wereTc2d

51.5 K,ty5100 K,tz510 K, with lattice constants characteristic oBechgaard salts.

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BRIEF REPORTS PHYSICAL REVIEW B 64 212504

Notice in Fig. 1, the rapid reduction inTc for very smalldeviations fromu590.0° (Hi y). A similar drop in Tc hasbeen seen experimentally in the putative critical temperaof (TMTSF)2ClO4 at high magnetic fields for small anguladeviations from they axis (b8 axis!.18 Furthermore, noticethat Tc(H) increases foru.u r'89.7°, due to the suppression of the orbital frustration alongy and the increasingimportance of the quantum confinement term alongz; seeEq. ~21!.

Three-dimensional regime.The three-dimensional regimis characterized by the absence of quantum confinement,occurs only at low magnetic fields wherety /vcy

@1 and

tz /vcz@1. In this case the order parameter equation redu

to

F2 l x2 d2

dx2 1Ly~x!1Lz~x!1A3dGu~x!50 ~22!

where l x and Ly(x) were already defined, andLz(x)5@ l zfz(x)/c#2, with characteristic length scale along thez

direction l z5ACtzc/TA2. The coefficient A3d5@ ln(V)21/lN#. This low magnetic field regime corresponds to tanisotropic Ginzburg-Landau limit, and thus the critical teperature is

Tc5Tc~0!F12CA2

Tc2~0!

Aty2vcy

2 1tz2vcz

2 G . ~23!

The angular dependence ofHc2in the xy plane~instead of

yz plane discussed here! was studied by Huang and Maki27

in the Ginzburg-Landau regime.In summary, we assumed an ESTP state@extension of the

3B3u(a) (‘ ‘ px’ ’) state21# as a plausible candidate for triple

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superconductivity in Q1D systems.13,19,20 Based on this as-sumption, we presented analytical results for the angularpendence ofHc2

(T) and Tc(H) of these superconductorwhen the magnetic field is applied in theyz plane. We dis-cussed three general regimes:~a! a one-dimensional regimeat very high magnetic fields where there is quantum confiment both along they and z directions (ty /vcy

!1 andtz /vcz

!1); ~b! a two-dimensional regime where therequantum confinement only along thez direction (ty /vcy

@1and tz /vcz

!1); and ~c! a three-dimensional regime at lowmagnetic fields~Ginzburg-Landau limit!, where there is noquantum confinement (ty /vcy

@1 andtz /vcz@1). The one-

dimensional limit is reached for quite high magnetic fielH>100 T, which are not available today from continuofield sources. However, the two-dimensional regime, whis in the range of 5 to 30 T, is attainable with continuofield sources. This two-dimensional regime is very intereing from the experimental point of view, becauseHc2

exceed

substantiallyHp'2.2 T, and there is a very rapid suppresion of the predicted reentrant superconducting phase~forHi y),1–3 when the applied magnetic field is just fractionsa degree away from perfect alignment with they (b8) axis.Our analytical calculations seem to be in qualitative agrment with the possible experimental observation of suchfects in (TMTSF)2ClO4.18 However, a more quantitativecomparison with experiments requires a careful understaing of the effects of the triclinic structure,26 and of possiblemagnetic field induced renormalizations of the interactiotransfer integrals and density of states.

We would like to thank the Georgia Institute of Technoogy, NSF~Grant No. DMR-9803111! and NATO~Grant No.CRG-972261! for financial support.

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1A.G. Lebed, JETP Lett.44, 114 ~1986!.2L.N. Burlachkov, L.P. Gorkov, and A.G. Lebed, Europhys. Lett.4,

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kin and Y.N. Ovchinnikov, Sov. Phys. JETP20, 762 ~1965!.13C.A.R. Sade Melo, Physica C260, 224 ~1996!.14A.G. Lebed, Phys. Rev. B59, R721~1999!.15A.A. Abrikosov, J. Low Temp. Phys.53, 359 ~1983!.16M.Y. Choi et al., Phys. Rev. B25, 6208~1982!.

17R.L. Greeneet al., Mol. Cryst. Liq. Cryst.79, 183 ~1982!.18I. J. Lee and M. J. Naughton,The Superconducting State in Mag

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Rev. B64, 172503~2001!.22I.J. Leeet al., cond-mat/0001332~unpublished!.23I. J. Leeet al., ~unpublished!.24A.G. Lebed, K. Machida, and M. Ozaki, Phys. Rev. B62, R795

~2000!.25Hc2

for (TMTSF)2PF6 seems to be largely affected by the proimity to an SDW phase~Ref 6!. Since we do not include sucheffects in our calculations a direct comparison to experimendata from (TMTSF)2PF6 is not wise.

26The Bechgaard salt family (TMTSF)2X is truly triclinic, how-ever, many of the qualitative results discussed in this paperbe easily generalized to the triclinic case.

27X. Huang and K. Maki, Phys. Rev. B39, 6459~1989!.

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