Journal of Superconductivity, Vol. 12, No. 3, 1999

Quasi-One-Dimensional Superconductors: From Weak toStrong Magnetic Field

N. Dupuis1,2

Received 16 November 1998

We discuss the possible existence of a superconducting phase at high magnetic field inorganic quasi-one-dimensional conductors. We consider in particular (i) the formation of aLarkin–Ovchinnikov–Fulde–Ferrell state, (ii) the role of a temperature-induced dimensionalcrossover occurring when the transverse coherence length jz(T) becomes of the order of thelattice spacing, and (iii) the effect of a magnetic-field-induced dimensional crossover resultingfrom the localization of the wave functions at high magnetic field. In the case of singlet spinpairing, only the combination of (i) and (iii) yields a picture consistent with recent experimentsin the Bechgaard salts showing the existence of a high-field superconducting phase. We pointout that the vortex lattice is expected to exhibit unusual characteristics at high magneticfield.

KEY WORDS: Organic superconductors; mixed state; critical field.

1. INTRODUCTION

According to conventional wisdom, supercon-ductivity and high magnetic field are incompatible. Amagnetic field acting on the orbital electronic motionbreaks down time-reversal symmetry and ultimatelyrestores the metallic phase. In the case of singletpairing, the coupling of the field to the electron spinsalso suppresses the superconducting order (Pauli orClogston–Chandrasekhar limit [1]).

Recently, this conventional point of view hasbeen challenged, both theoretically [2–6] and experi-mentally [7], in particular in quasi-1D organic mate-rials. Because of their open Fermi surface, thesesuperconductors exhibit unusual properties in pres-ence of a magnetic field. As first recognized byLebed’ [2], the possible existence of a superconduct-ing phase at high magnetic field results from a

1Department of Physics, University of Maryland, College Park,Maryland 20742-4111, USA, and Laboratoire de Physique desSolides, Universite Paris-Sud, 91405 Orsay, France.

2Mailing address: N. Dupuis, Laboratoire de Physique des Solides,Universite de Paris-Sud, 91405 Orsay, France. e-mail: dupuis@lps.u-psud.fr

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0896-1107/99/0600-0475$16.00/0 1999 Plenum Publishing Corporation

magnetic-field-induced dimensional crossover thatfreezes the orbital mechanism of destruction ofsuperconductivity. Moreover, the Pauli pair-break-ing (PPB) effect can be largely compensated bythe formation of a Larkin–Ovchinnikov–Fulde–Ferrell (LOFF) state [2–5,8].

The aim of this paper is to discuss three aspectsof high-field superconductivity in quasi-1D conduc-tors : (i) the formation of a LOFF state, and therespective roles of (ii) temperature-induced and (iii)magnetic-field-induced dimensional crossovers.

We consider a quasi-1D superconductor with anopen Fermi surface corresponding to the dispersionlaw (" 5 kB 5 1)

Ek 5 vF(ukxu 2 kF) 1 ty cos(kyb) 1 tz cos(kzd) 1 e(1)

where e is the Fermi energy, and vF the Fermi velocityfor the motion along the chains. ty and tz (ty @ tz) arethe transfer integrals between chains. The magneticfield H is applied along the y direction and we denoteby Tc0 the zero-field transition temperature (In Bech-gaard salts, Tc0 p 1 K, and tc 5 tz/2 is in the range2–10 K [3,4,7]).

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2. LOFF STATE IN QUASI-1DSUPERCONDUCTORS

We first discuss the effect of a magnetic fieldacting on the electron spins. Larkin and Ovchinnikov,and Fulde and Ferrell, have shown that the destruc-tive influence of Pauli paramagnetism on supercon-ductivity can be partially compensated by pairing upand down spins with a non-zero total momentum [8].At low temperature (i.e., high magnetic field), whenT # T0 Q 0.56Tc0, this nonuniform state becomesmore stable than the uniform state corresponding toa vanishing momentum of Cooper pairs.

Quasi-1D superconductors appear very particu-lar with respect to the existence of a LOFF state [4].The fundamental reason is that, because of the quasi-1D structure of the Fermi surface, the partial com-pensation of the Pauli pair breaking effect by a spatialmodulation of the order parameter is much moreefficient than in isotropic systems. Indeed, a properchoice of the Cooper pairs momentum allows one tokeep one half of the phase space available for pairingwhatever the value of the magnetic field. Thus, thecritical field Hc2(T) Y 1/T diverges at low tempera-ture [4].

3. TEMPERATURE-INDUCEDDIMENSIONAL CROSSOVER

At low temperature, strongly anisotropic super-conductors may exhibit a dimensional crossover thatallows the superconducting phase to persist at arbi-trary strong field (in the mean-field approximation)in the absence of PPB effect.

Within the Ginzburg–Landau theory, at hightemperature (T p Tc0) the mixed state is an (aniso-tropic) vortex lattice. The critical field Hc2(T) is deter-mined by Hc2(T) 5 f0/2fjx(T)jz(T) where jx and jz

are the superconducting coherence lengths and f0

the flux quantum. When the anisotropy is largeenough, i.e., when tz & Tc0, jz(T) can become at lowtemperature of the order or even smaller than thelattice spacing d. Vortex cores, which have an exten-sion pjz(T) in the z-direction, can then fit betweenplanes without destroying the superconducting orderin the planes. The superconducting state is a Joseph-son vortex lattice and is always stable at low tempera-ture for arbitrary magnetic field (see Fig. 3 in Ref.4). A proper description of this situation, which takesinto account the discreteness of the lattice in the z

direction, is given by the Lawrence-Doniach model[4,9].

It is tempting to conclude that this temperature-induced dimensional crossover, together with the for-mation of a LOFF state, could lead to a divergingcritical field Hc2(T) at low temperature. It has beenshown in Ref. 4 that this is not the case: the PPBeffect strongly suppresses the high-field supercon-ducting phase (this point is further discussed in Sec-tion 4.1 [see Fig. 1b]). Therefore the temperature-induced dimensional crossover cannot explain thecritical field Hc2(T) measured in Bechgaard salts [7]in the case of spin singlet pairing.

4. MAGNETIC-FIELD-INDUCEDDIMENSIONAL CROSSOVER

The microscopic justification of the Ginzburg–Landau or Lawrence–Doniach theory of the mixedstate of type II superconductors is based on a semi-classical approximation (known as the semiclassicalphase integral or eikonal approximation) that com-pletely neglects the quantum effects of the magneticfield. At low temperature (or high magnetic field)and in sufficiently clean superconductors, whengc @ T, t (gc 5 eHdvF/" being the frequency ofthe semiclassical electronic motion, and t the elasticscattering time), these effects cannot be neglectedand an exact description of the field is required.

To be more specific, we write the Green’s func-tion (or electron propagator) as [10]

G(r1 , r2) 5 exp Hie Er2

r1

d l ? AJ G(r1 2 r2) (2)

where A is the vector potential. The Ginzburg–Landau or Lawrence–Doniach theory identifies Gwith the Green’s function G0 in the absence of mag-netic field. The latter intervenes only through thephase factor ie er2r1

d l ? A, which breaks down time-reversal symmetry and tends to suppress the super-conducting order.

When gc @ T, 1/t, the approximation G 5 G0

breaks down and a proper treatment of the field isrequired. In isotropic systems, G includes all the in-formation about Landau level quantization. Instrongly anisotropic conductors, it describes a mag-netic-field-induced dimensional crossover [2,3], i.e.,a confinement of the electrons in the planes of highestconductivity. The same conclusion can be reachedby considering the semiclassical equation of motion"dk/dt 5 ev 3 H with v 5 =Ek . The corresponding

Quasi-1D Superconductors 477

electronic orbits in real space are of the form [neglect-ing for simplicity the (free) motion along the field]z 5 z0 1 d(tz/gc) cos(gcx/vF). The electronic motionis extended along the chains (and the magnetic fielddirection), but confined with respect to the z-direc-tion with an extension pd(tz/gc) Y 1/H. In very strongfield gc @ tz , the amplitude of the orbits becomessmaller than the lattice spacing d showing that theelectronic motion is localized in the (x, y) planes.The latter being parallel to the magnetic field, theorbital frustration of the superconducting order pa-rameter vanishes [there is no magnetic flux throughthe two-dimensional (2D) Cooper pairs located inthe (x, y) planes].

4.1. Large Anisotropy

Figure 1(a) shows the phase diagram in the exactmean-field approximation in the case of a weak in-terplane transfer tz/Tc0 Q 1.33. Q is a pseudo momen-tum for the Cooper pairs in the field [3–5]. At hightemperature (T p Tc0), the mixed state is an Abriko-

Fig. 1. (a) Phase diagram for tz/Tc0 Q 1.33. Q is a pseudo momentumfor the Cooper pairs in the field. The three doted lines correspondto Q 5 2eB/vF , G 2 2eB/vF , G (G 5 gc/vF). (b) Phase diagramin the Lawrence–Doniach model.

sov vortex lattice, and Tc decreases linearly with thefield. Tc does not depend on Q in this regime, whichis shown symbolically by the shaded triangle in Figure1(a). For H p 0.3 T, the system undergoes a tempera-ture-induced dimensional crossover, which leads toan upward curvature of the transition line. This di-mensional crossover selects the value Q 5 0 of thepseudo momentum. The PPB effect becomes impor-tant for H p 2 T and leads to a formation of a LOFFstate (Q then switches to a finite value). At higherfield, Q Q 2eBH/vF (eB is the Bohr magneton), andTc exhibits again an upward curvature.

Figure 1(b) shows the phase diagram obtainedin the Lawrence–Doniach model [4]. The metallicphase is restored above a field H p 2.7 T, and theLOFF state is stable only in a narrow window aroundH Q 2.6 T. Only when the magnetic-field-induceddimensional crossover is taken into account does theLOFF state remain stable at very high magnetic field.In a microscopic picture, the dimensional crossovershows up as a localization of the single-particle wavefunctions with a concomitant quantization of thespectrum into a Wannier–Stark ladder (i.e., a set of1D spectra if we neglect the energy dispersion alongthe field). This is precisely this quantization thatallows one to construct a LOFF state in a way similarto the 1D or (zero-field) quasi-1D case [5]. Thus,when the field is treated semiclassically, the regionof stability of the LOFF state in the H 2 T planebecomes very narrow as in isotropic 2D or third-dimensional (3D) systems [8,11].

4.2. Smaller Anisotropy

For a smaller anisotropy, the coherence lengthjz(T) is always larger than the spacing betweenchains: jz(T) $ jz(T 5 0) . d. There is no possibilityof a temperature-induced dimensional crossover.

Figure 2(a) shows the phase diagram withoutthe PPB effect for tz/Tc0 Q 4. The low-field Ginzburg–Landau regime (corresponding to the shaded trianglein Figure 2) is followed by a cascade of superconduct-ing phases separated by first-order transitions. Thesephases correspond to either Q 5 0 or Q 5 G ;gc/vF [3]. In the quantum regime, the field-inducedlocalization of the wave functions plays a crucial rolein the pairing mechanism. The transverse periodicityaz of the vortex lattice is not determined by the Ginz-burg–Landau coherence length jz(T) but by the mag-netic length d(tz/gc). The first-order phase transitionsare due to commensurability effects between the crys-

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Fig. 2. Phase diagram for tz/Tc0 Q 4 without (a) and with (b)PPB effect.

tal lattice spacing d and az : each phase correspondsto a periodicity az 5 Nd (N integer). N decreases byone unit at each phase transition. The mixed stateevolves from a triangular Abrikosov vortex lattice inweak field to a triangular Josephson vortex lattice invery high field (where N 5 2). It has been pointedout that az decreases in both the Ginzburg–Landauand quantum regimes, but increases at the crossoverbetween the two regimes where gc p T [3]. Thissuggests that the mixed state exhibits unusual charac-teristics in the quantum regime. Indeed, the ampli-tude of the order parameter and the current distribu-tion show a symmetry of laminar type. In particular,each chain carries a nonzero total current (exceptthe last phase N 5 2). We expect these unusual char-acteristics to influence various physical measure-ments.

Figure 2(b) shows the phase diagram when thePPB effect is taken into account. There in an interplaybetween the cascade of phases and the formation of

a LOFF state. The latter corresponds to phases withQ 5 2eBH/vF and Q 5 G 2 2eBH/vF .

5. CONCLUSION

Our discussion shows that a temperature-in-duced dimensional crossover could explain recent ex-periments in the Bechgaard salts only in the case oftriplet pairing, although this would require a valueof the interchain coupling tz slightly smaller than whatis commonly expected [3,4,7]. In the more likely caseof singlet pairing, the existence of a high-field super-conducting phase in quasi-1D conductors may resultfrom a magnetic-field-induced dimensional crossoverand the formation of a LOFF state. The crossoverbetween the semiclassical Ginzburg–Landau andquantum regimes, which occurs when gc p T, is ac-companied by an increase of the transverse periodic-ity of the vortex lattice. This, as well as the character-istics of the vortex lattice in the quantum regime(laminar symmetry of the order parameter amplitudeand the current distribution), suggests that the high-field superconducting phase in quasi-1D conductorsshould exhibit unique properties.

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