On the theory of the superconducting phase in organic conductors

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On the theory of the superconducting phase in organic conductors

S. Brazovskii, V. Yakovenko

To cite this version:

S. Brazovskii, V. Yakovenko. On the theory of the superconducting phase in organic conductors.

Journal de Physique, 1986, 47 (2), pp.175-180. �10.1051/jphys:01986004702017500�. �jpa-00210194�

On the theory ^{of the} superconducting phase ⁱⁿ organic ^conductors

S. Brazovskii and V. Yakovenko

L. D. Landau Institute for Theoretical Physics, 117334, Moscow, B334, Kosygina, ^{2, U.S.S.R.}

(Reçu ^{le 11} juin 1985,’accepté ^{le 12} septembre 1985)

Résumé.

Nous étudions l’apparition ^{de la} supraconductivité ^{dans des} systèmes, ^dans lesquels ^{elle est stimulée}

par ^un mécanisme spécial ^de non-équivalence ^{des chaines voisines. Nous} calculons la température ^de transition, fonction de Green anormale et la fonctionnelle de Ginsburg-Landau. ^{Ces résultats nous} permettent ^{de discuter les} propriétés ^des supraconducteurs organiques.

Abstract.

The onset of superconductivity is considered for the systems, in which it is stimulated by ^a special

mechanism of neighbouring ^chain nonequivalence. ^We find the transition temperature (anomalous ^Green function),

and derive the Ginsburg-Landau functional. These results allow us to discuss the properties ^of organic super- conductors.

Classification

Physics ^Abstracts

74.20

The understanding ^{of the nature of} superconducti- vity ⁱⁿ quasi-one-dimensional conductors is still far from being complete. ^{We have} recently suggested [1, 2] ^that superconductivity ^{in these} materials is

necessarily ^{related to a} special crystal ^structure,

where the neighbouring ^{chains are} incommensurate.

This structure has been registered ^for MX3 compounds

A

[3] ^{and for a} representative member--(TMTSF)2CL04

-

of the Bechgaard ^salt family [4]. ^{We have} suggested

that for other members this structure has to be observed in the superconducting ^{state. In the present}

paper ^we thoroughly study ^the special ^mechanism

of the electronic pairing and describe the physical properties ^{of the} superconducting ^state.

The action for the system may be written in the form

where z

x + ivF ^{t ; d2z - dx dr ; x} is the coordinate along ^the a-axis, parallel ^{to the direction of the} chains;

T is the Matsubara time ; n

(nb, n,), ^{m are two} component integer vectors, labelling ^the chains; ýJ,.,+ and ýJ,.,-

are annihilation operators for electrons with momenta close to + kF(n) and - kF(n) (we have omitted their spin indices, ^because spins play ^no role in the following), t,.-m ^{is the} interchain hopping amplitude. ^{The factor} w,.,...(z)

exp[ - i(k (n) - k(m)) x] results from chain nonequivalence (1). ^{For a} system ^with (0, 1/2, 0) super-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004702017500

176

structure we have

We set h

vF ^{= a}

1, EF ~ ^1, where a is a lattice spacing along the chain. So ^{in (1)} is the action of non-coupled

chains (with ^{the exception of long range} interactions, ^see [1]). ^{We suppose} that there are no gaps in both charge

and spin ^{channels, so} So ^{describes two sounds. A} correlation function calculated via the bosonization method [5]

looks like :

Here q is the index of the superconducting ^{response function,} ’IF is the index of one-particle ^{Green function.}

For free particles ’I

^2, ’IF

1; ^{in the} considered system n 2, 17F > 1. Averaging ⁱⁿ (3) ^is performed ^with

action So. ^{Formula (3) is valid only for} I z I >> 1, ^{we have omitted} phase-dependent ^{and constant} factors. The

quantity hn(z1, Z2) ⁱⁿ (1) is ^the generalized ^{external field, which} produces Cooper pairs. ^It is taken in nonlocal form in order to find the anomalous Green function. To describe the behaviour of the system ^{in the} vicinity

of the superconducting transition, ^{we shall} expand the free energy, corresponding to ^{(1) in powers of h. The}

first term is equal ^to

The averaging ⁱⁿ (5) ^is performed with action (1) ^{at h}

^{0. Let us} introduce the conjugated potential O(F) :

Thus FII(Zl’ z2) is ^the anomalous Green function. The first term of expansion O(F) in powers of ^F has the form

We calculate (5) by developing ^the perturbation ^{series in powers of} t..-m. ^{In the ladder} approximation ^{we obtain :}

r

The integral equation (9) ^{follows from the} diagram of figure ^{1 where the dashed lines} represent ^{the interchain}

tunnelling.

The sum can be taken over only ^{those vectors} 6, ^{which connect} chains of different types, ^when Wb.lI+ð(Z) = an(z) ^{(2), while other terms} have been shown to be small [1] (2). Equation ^{(9) al lows us to find} eigenfunctions (p

t

(2) ^{For this} very ^reason in equation (9) ^we have omitted the hopping ^between equivalent chains when oscillatory

factors (2) ^{do not} appear either. Fig 1.

and eigenvalues À - 1 ^{of the} operator K, defined by ^the expression :

Substituting (10) ^into (9) ^{we obtain :}

Performing ^the Fourier transformation of (11) ^{over the} variable R : f(R, ...) ^- f(Q, ^...), Q

(wn’ qll)’

wn

1tnT, ^{we arrive at the} equation :

The effect of the electron binding ⁱⁿ pairs ^{in the course of} tunnelling ^between nonequivalent ^chains (which

was found earlier [1]) ^{is evident in} equation (13). ^The oscillatory ^factor an(r2) provides the convergence of the first integral ^{at the upper limit} within the scale K-1 ^{T-1 for} arbitrary ^v > 0. On the contrary, the second term, which describes the interaction of pairs ^{on the same chain, does not} contain the factor an and converges ^on the

general ^{cutoff T- 1} (we suppose that ^v 2). ^{Due to fast} convergence ^we may ^set r(Q, rl, r2) + f(Q, ^{rl, 0) in}

the first term of (13). ^{Now we see from} (12) ^and (13) ^that Y.(r) ^{is an even function of r, so the first} integral ⁱⁿ (13)

contains only ^{the even} part ^of an(r) : a(r) = (an.(r) ⁺ an(- r))/2

^{cos 2 Kx. Hence,} performing the Fourier transformation of (13) ^{over n, we obtain :}

In the vicinity of the transition temperature, ^the eigenvalues A ^are small, ^{so the} eigenfunctions ^{Y can be found}

from (14) by iterations in A :

Substituting (15) ^into (14) ^we obtain the self-consistency ^{condition for function C. It} yields :

_

Apparently, r(Q, 0, 0) ⁼ r(Q) ^{is the} generalized superconducting susceptibility (see [I]). ^{-Close to} T,, ^small

values q 11 ^« Tc, cu.

0 are essential, ^{and from} (12) ^{we find :}

178

where 0’(C) ^{is a} trigamma ^function. Evaluating ^the integrals ⁱⁿ (16) ^{we obtain :}

Formula (20) ^{is the} expression ^{for the} superconducting transition temperature.

To find the anomalous Green function Fn(Z1, Z2) ^{let us} expand ^{it over the} eigenfunctions cp (8) ^of operator ^K with low eigenvalues ^{and take} (13) ^{into account. Near} T, ^we omit the second term in RHS of equation (15) ^and neglect ^the dependence of rl in the first term for I r, I T;- 1, ^so

Here Y.(R) is the inverse Fourier transform of the function Y(Q, ql) ^from (15). ^{It is the centre of mass wave} function for an electron pair.

For the homogeneous system, ^Y

Const. is determined by higher ^{order terms in} expansion (4). ^{Bear in}

mind that Fn(Z1, Z2) ^{has a} power behaviour as a function of the distance between electrons for I r I Te 1.

Substituting (21) ^into (8) ^{we obtain :}

Let us now consider thewext term of expansion (4) ^{for Y :}

where

Calculating (24) by ^{means of the} perturbation expansion ⁱⁿ In - m’ ^we shall take into account only those terms,

corresponding ^{to electron} pairs tunnelling coherently ^between alternating ^{chains. We must sum over all tra-} jectories ^{of the two electron} pairs wandering ^{on the} two-dimensional _array of chains. If these two trajectories

do not intersect, i.e. two pairs ^{do not} get ^{on the same} chain, expression (24) ^{is zero. Let us consider} trajectories

with a single intersection on chain _{p. In} this case, using (7) ^and (21), ^{we obtain from} (23) :

Here the function Lo(t zi, z’i }) ^’-- Ln,n,n,n({ Zi,. z’ 1) ^can be calculated by ^the bosonization method. We need it

in the region z,

z; I Iz, - zj 1, ^{i =1= j, where it takes the form :}

Substituting (26) ^into (25) ^and supposing ^that Yn(z) ^varies slowly in space within the scale of Tc- ^{1 and does}

not vary in time, ^{we obtain :}

Introducing ^{the order} parameter ’P,.(x) =JC2 b(v) ^{Yn(x) Tc1 - nF we finally} obtain from ( 19), (22) ^and (28) ^the Ginsburg-Landau expansion

When deriving (29) ^{we have not} taken into account

diagrams ^with larger numbers of trajectory ^inter-

sections. They describe critical fluctuations, ^which

are of no special ^{interest for us,} although ^{one can}

from (29), ^{deduce that the} region ^{of these} fluctuations is not small, ^{as it} usually stands in the quasi-one-

dimensional systems.

Experimentally ^the transition is rather broad in resistive measuments [9], ^{but narrow} in calorimetric

ones [10]. ^{The heat} capacity jump AC deduced from

(29) is of order T c ^{in agreement with the} experiment [10].

Our theory ^{is valid if the two} inequalities ^{are ful-}

filled [1] : K Z T 3d ^and T, Z T*3d, ^where T3d ~ t(o)] 1/(2 - nIF) ^and T*3d T23d/K ^are renormalized one-

electron bandwidths for perpendicular motion wi- thout and with ^an anion superstructure, respectively.

Using (20) the necessary ^set of inequalities ^{can be}

written ^as follows :

The last condition in (30) ^is equivalent ^to r1F > 11, which ^can be fulfilled for sufficiently strong interac- tions only. To estimate the parameters of the Bech-

gaard salts, ^{we note that the} SDW-dielectric exists at

x

0 (Q-phase ^of (TMTSF)2CLO4) ^and disappears

at " :F 0 (R-phase) [4]. According ^to [1] ^{it means that}

K > T3d - TSDW ~ 10 K. A reasonable estimate for

x is K Z 100 K (see below) ^{and we find} T*3d ~ ^{1 K,}

that satisfies (30) (at ^least approximately). ^These

values contradict the generally accepted ^estimate tb _ 102 ^K for the band model. The latter value results from oversimplified band calculations [8] ^and conductivity ^{data, treated} by ^the simplest ^kinetic theory [6], ^the applicability of which is questionable.

It disagrees ^{with small} temperatures (from ^{10 to 20} K)

of SDW ordering, ^since repulsive interactions are

apparently strong ^{in these} systems. The band model has some advantage, especially ^it provides ^an explana-

tion for the behaviour ⁱⁿ strong magnetic ^fields [7].

However it cannot describe other phenomena ^{like the} interplay between the properties ^of Q - ^{and R -} phases, ^{which our} theory easily explains.

Applying ^the theory ^{to the} Bechgaard salts, ^we choose the following dependence :

for

Since we consider electron hopping between alter-

nating ^chains. Higher ^critical magnetic ^fields Hc2(i) ^are

calculated by ^{means of a} conventional formula for the anisotropic Ginsburg-Landau theory (29), (31)

For an estimation we set il

^{I in} (18) ^{and use the data} of [10] ^{for the} density ^{of states. From} (32), (31), (29) ^we

find : Hc2(c)

^Const (oolab) (Tc alhVF) ;.-- ^{8 kG, which} qualitatively agrees with the experimental ^{value, I kG} [11]. Using (32) and the data of [11] ^{we must set} tb/tc b/tc

H(b)’IH(c)’ = ^{16. Then we obtain} Hc2(a)’

(cpo/Me) (/bllc)

^{6 x 104 kG,} this result disagrees

with the experimental ^{value 47 kG} [11]. ^{Thus the}

functional (29) ^to (31) gives ^{a correct order of} magni-

tude for Hc(c) ^and H(b) ^{(for a} reasonable value of

tb/t,), but contradicts the data for Hc2(a). ^{The reason}

for this discrepancy may be found in the two-dimen-

sionality ^{of these} systems. As tc tb, the relative fluctuations of the order parameter ^{are much} greater,

on different planes ^{than on the same} plane. ^{Thus the}

mean-field theory ^{(29) can} give ^correct results for the motion in a well-conducting (ab) plane only, that is for the magnetic ^field along the c-axis.

In many respects ^the theory developed ^hereabove

and in [1, 2] effectively ^{reduces to the model of the}

superconducting ^chain array with a Josephson ^cou-

pling, ^{and we can use the} advantages ^of this model for

the explanation of the anomalous properties ^{of the}

180

Bechgaard ^salts [6]. ^{At the same time our} theory

differs from this model, mainly by the absence of the

pseudo-gap ^{in the} spectrum. Thus, ^{we avoid the} contradiction with the heat capacity ^{data at T} > T, [12]. The structures, observed in optical ^and tunneling

spectra [6] ^at w0 = ^{3.8 meV can be} attributed ^to

direct electron excitation between two subbands, generated by ^the superstructure : _w0

^{2 K. We can} also account for magnetoabsorption, ^since the electron transfer between the two types of chains is involved in this _process, so the matrix element depends ^{on the} magnetic ^field.

References

[1] BRAZOVSKII, S., YAKOVENKO, V., J. Physique ^{Lett. 46} (1985) ^L-111.

[2] BRAZOVSKII, S., YAKOVENKO, V., ^The International

Conference ^{on the} Physics ^and Chemistry of ^Low-

Dimensional Synthetic-metals. ^Abano Terme, Italy,

1984.

[3] MEERSHAUT, A., ^J. Physique Colloq. ⁴⁴ (1983) ^C3-1615.

[4] POUGET, ^J. P., MORET, R., COMÉS, R., SHIRANE, G., BECHGAARD, K., FABRE, ^J. M., ibid C3-969.

[5] LUTHER, A., PESCHEL, J., Phys. ^{Rev. 9} (1974) ^2911.

[6] JÉROME, D., SCHULZ, ^H. J., ^Adv. Phys. ³¹ (1982) ^299.

[7] GOR’KOV, ^L. P., LEBED’, ^A. G., ^J. Physique ^{Lett. 45} (1984) ^L-433.

[8] GRANT, ^P. M., ^J. Physique Colloq. ⁴⁴ (1983) ^C3-847.

[9] MAILLY, D., RIBAULT, ^H. M., BECHGAARD, K., ibid.,

C3-1037.

[10] GAROCHE, P., BRUSETTI, R., BECHGAARD, K., ibid.,

C3-1047.

[11] GREENE, ^R. L., HAEN, P., HUANG, ^S. Z., ENGLER, ^E. M., CHOI, ^M. Y., CHAIKIN, ^P. M;, ^Mol. Cryst. Liq.

Cryst. ⁷⁹ (1982) ^539-183.

[12] BUZDIN, ^A. J., BULAEVSKII, ^L. N., ^Sov. Phys. Usp. 144

(1984) 415.