Superconducting transition temperatures and coherence length in non s-wave pairing material

a r X i v :c o n d -m a t /0112397v 1 [c o n d -m a t .s u p r -c o n ] 20 D e c 2001

Superconducting transition temperatures and coherence length in non–s -wave pairing

materials correlated with spin-?uctuation mediated interaction

G.G.N.Angilella,1N.H.March,2,3and R.Pucci 1

1

Dipartimento di Fisica e Astronomia,Universit`a di Catania,and Istituto Nazionale per la Fisica della Materia,UdR di Catania,

Corso Italia,57,I-95129Catania,Italy 2

Oxford University,Oxford,England

3

Department of Physics,University of Antwerp (RUCA),Antwerp,Belgium

(Dated:February 1,2008)

Following earlier work on electron or hole liquids ?owing through assemblies with magnetic ?uc-tuations,we have recently exposed a marked correlation of the superconducting temperature T c ,for non–s -wave pairing materials,with coherence length ξand e?ective mass m ?.The very recent study of Abanov et al.[Europhys.Lett.54,488(2001)]and the prior investigation of Monthoux and Lonzarich [Phys.Rev.B 59,14598(1999)]have each focussed on the concept of a spin-?uctuation temperature T sf ,which again is intimately related to T c .For the d -wave pairing via antiferromag-netic spin ?uctuations in the cuprates,these studies are brought into close contact with our own work,and the result is that k B T sf ～ˉh 2/m ?ξ2.This demonstrates that ξis also determined by such antiferromagnetic spin-?uctuation mediated pair interaction.The coherence length in units of the lattice spacing is then essentially given in the cuprates as the square root of the ratio of two characteristic energies,namely:the kinetic energy of localization of a charge carrier of mass m ?in a speci?ed magnetic correlation length to the hopping energy.The quasi-2D ruthenate Sr 2RuO 4,with T c ～1.3K,has p -wave spin-triplet pairing and so is also brie?y discussed here.

PACS numbers:74.72.-h,74.70.Pq,74.70.Tx

I.INTRODUCTION

In early work,Egorov and March [1]discussed elec-tron or hole liquids ?owing through assemblies with an-tiferromagnetic spin ?uctuations,and proposed a corre-lation between the in-plane electrical resistivity ρab and the nuclear-spin lattice relaxation time T 1of the form

ρab T 1∝T.

(1)

This relationship has been tested on the underdoped high-T c cuprate YBa 2Cu 4O 8with T 1extracted from 63

Cu NMR data,and is appropriate somewhat above the superconducting transition temperature (see Fig.A.7.5.1on p.355of Ref.2,and Ref.3).The correlation written in Eq.(1)between the observables ρab and T 1arose [1]by eliminating the magnetic susceptibility χ(Q ),at the antiferromagnetic wave vector Q [equal to (π/a,π/a )in the treatment below,a being the lattice spacing],with ρab and (T T 1)?1both proportional to χ(Q ),as shown by Kohno and Yamada [4].

More recently,the present authors [5]have demon-strated that T c for non–s -wave pairing superconductors,and in particular for heavy Fermion materials and high-T c cuprates,correlated with coherence length and e?ec-tive mass.The very recent and apparently quite di?er-ent studies of Abanov et al.[6],and prior to that,of Monthoux and Lonzarich [7],are here brought into close contact with our earlier work [5].

The outline of this Brief Report is then as follows.In Sec.II,we brie?y summarize the essential input in the treatments of spin-?uctuation mediated pairing in Refs.6and 7.Anticipating that,we stress at the outset that

the common feature in Refs.6and 7is a characteristic thermal energy k B T sf associated with a spin-?uctuation temperature T sf .Sec.III connects the work on T sf in Refs.6and 7with our own studies on coherence length.A summary is then given in Sec.IV,with some possible directions for future work.

II.

SPIN-FLUCTUATION TEMPERATURE RELATED TO T c IN p -AND d -W A VE

SUPERCONDUCTIVITY IN QUASI-2D METALS Essential input into both Refs.6and 7is a form of the retarded generalized magnetic susceptibility χ(q ,ω).Quite speci?cally,in Ref.7the phenomenological form

χ(q ,ω)=

χ0κ20

2

and

η(?q)=T sf?q?,(5) where T sf is the spin-?uctuation temperature already re-ferred to in Sec.I.

Here,we should caution that the use of a single-band dispersion relation[Eq.(3)in Ref.7],while appropri-ate for most high-T c cuprates,is clearly a rather crude approximation for Sr2RuO4.Indeed,this compound is known to be characterized by a three-fold band,aris-ing from the hybridization of the ruthenium4d ij atomic orbitals(ij=xy,xz,yz)in the t2g subshell.As a con-sequence,the Fermi surface(FS)is composed of a set of three disconnected barrel-like sheets,almost dispersion-less in the direction orthogonal to the SrO2layers[9]. This led to the proposal of an orbital-dependent form of superconductivity[10],where the role of the multi-band nature of Sr2RuO4in stabilizing a p-wave triplet order parameter against a d-wave singlet alternative has been also emphasized[11].However,we believe that the main conclusions of Monthoux and Lonzarich[7]would be una?ected by multiband e?ects,at least qualitatively. Moreover,the restriction to the main band of Sr2RuO4 [Eq.(3)in Ref.7]helps treating the quasi-2D character of the single-particle dispersion relation of this layered perovskite on the same footing as for the cuprates[12]. Monthoux and Lonzarich[7]also investigate antiferro-magnetic correlations as in the d-wave paired cuprates, in which case the above parameters have the form

?q2=?q2+(6) and

η(?q)=T sf?q?.(7) The?nal input we need to refer to here is a coupling parameter g2in the quasiparticle self-energyΣ(q,ω),in-volving of course[see Ref.7,Eqs.(11)–(13)]summations over wave vectors and Matsubara frequencies ofχ(q,ω). The mean-?eld Eliashberg equations for nearly ferro-magnetic and nearly antiferromagnetic metals with a sin-gle2D Fermi surface were then solved numerically in Ref.7,to obtain the ratio of critical temperature T c to spin?uctuation temperature,T sf,essentially as a func-tion of coupling strength g2for di?erent values of the inverse correlation lengthκ.This was done both for p-wave triplet and d-wave singlet pairing.

The major predictions of Refs.6and7were in ac-cord that at strong coupling,T c/T sf exhibits saturation. For a physically reasonable range of values ofκ?1,the quantitative results of Ref.7were:(i)For p-wave triplet pairing,T c/T sf saturates at a value of1/30,and(ii)For d-wave singlet pairing,T c/T sf has a saturation value of 1/2.This then is the point at which to make contact be-tween these?ndings of Refs.6and7and our own study [5].

III.SPIN-FLUCTUATION TEMPERATURE T sf AND CORRELATION LENGTHξIN

NON–s-W A VE PAIRING SUPERCONDUCTORS: ESPECIALLY HIGH-T c CUPRATES

In Ref.5,we exposed a relationship,for both heavy Fermion materials and for high-T c cuprates,between the thermal energy k B T c and another characteristic energy,?c say,for such non–s-wave superconductors,where?c was de?ned by

?c=

ˉh2

2

k B T sf(9)

for the d-wave pairing high-T c cuprates.Thus,one has as a consequence the order of magnitude result

k B T sf～

2ˉh

2

3

of the thermal energy k B T

sf

associated

with the spin-

?uctuation temperature[6,7]T sf withκ20,the inverse

magnetic correlation length squared without strong mag-

netic correlations,is constant,i.e.

k B T sfκ20=const.(11)

Adopting the value in their Table5,the constant value

turns out to be～8t.

Returning to the strong coupling limit to gain further

insight into the factors determining the coherence length

ξ,we have

k B T c～

1

κ20

.(12)

Puttingκ20=a2/?2m0,where?m0is the(antiferro-)mag-

netic correlation length in the high-T c cuprates,we?nd

almost immediately

ξ～

a

m??2m0

1/2

1

2 ˉh2t1/2F(g2χ0/t;κ2),(14)

where F is a slowly varying function of its arguments,F

becoming unity for su?ciently large values of the‘cou-

pling strength’g2χ0/t,withκ2restricted to the range

quoted above.

IV.SUMMARY AND DIRECTIONS FOR

FUTURE WORK

The achievement of the present Brief Report is to bring

the studies of Refs.6and7,in which the superconduct-

ing transition temperature T c is connected to the spin-

?uctuation temperature T sf,into direct contact with our

work relating k B T c to the characteristic energyˉh2/m?ξ2

[5].For a d-wave singlet pairing mediated by antiferro-

magnetic spin?uctuations in the cuprates,the simple,or-

der of magnitude relation Eq.(10)follows,showing that

the coherence lengthξis determined by the interaction

mediated by spin?uctuations.This is expressed,more

speci?cally,in the language of Ref.7,in Eqs.(13)and

(14).

However,the situation regarding the relation of T sf in

the low-T c ruthenate Sr2RuO4to the coherence lengthξ

is much less clear presently than in the high-T c cuprates.

This may be because of a competition between nearly

ferromagnetic behavior and antiferromagnetic spin?uc-

tuations[11].Experiments onχ(q,ω)using both neu-

tron scattering and NMR on this ruthenate would be

valuable for furthering understanding of the origins of

superconductivity,and especially the physics of the co-

herence length in this material.As for T sf it seems to lie

in the range40–50K.Having referred to heavy Fermion

materials in connection with Ref.5,we thought it of

interest to construct from the p-wave studies of Ref.7

a plot of T c/t vs T sf/t(see Fig.1),by combining data

from their Figs.2–4.It is worth noting,though we ex-

pect the mechanisms generally to be di?erent,that the

shape of the present Fig.1parallels that of Fig.1of

Ref.5.However,such a comparison should be carefully

considered,since some of the heavy Fermion materials

considered in Fig.1of Ref.5exhibit antiferromagnetic

spin?uctuations,rather than ferromagnetic spin?uctu-

ations,as studied by Monthoux and Lonzarich in con-

nection with p-wave superconductors[7].The major ex-

ception is UPd2Al3,which is known to be characterized

by rather strong,static antiferromagnetic correlations in

the normal state(with a signi?cant N′e el temperature

of T N=14.3K).Such antiferromagnetic correlations

even coexist with superconductivity below T c～2K,at

variance with other uranium based heavy Fermion com-

pounds[14,15,16].We record,however,that this com-

pound,with its relatively high T c,helped Sato(Fig.2

in Ref.17)to‘bridge the gap’between the low-T c heavy

Fermion compounds and the high-T c cuprates in estab-

lishing a correlation between T c and some magnetic or-

dering temperature(though related to a di?erent kind

of magnetic order in di?erent compounds),much in the

same spirit as in the present work(compare also Fig.2

of Ref.5).

Acknowledgments

One of us(N.H.M.)made his contribution to the

present Report during a visit to the Physics Department,

University of Catania,in the year2001.Thanks are due

to the Department for the stimulating environment and

for much hospitality.G.G.N.A.thanks Dr.G.Sparn

for stimulating discussions and correspondence,and ac-

knowledges partial support from the EU through the FSE

program.

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0.005

0.01 0.015 0.02 0.0250.33

0.67 1.33

T c /t

T sf /t

g 2

χ0/t = 10g 2

χ0/t = 20

FIG.1:Shows plot of T c /t vs T sf /t for p -wave spin-triplet pairing.This has been constructed by combining numerical data from Figs.2–4of Ref.7,assuming for the parameters the physical range 10≤g 2χ0/t ≤20and 0.25≤κ2≤1.Points corresponding to the same value of T sf /t and of g 2χ0/t ,but to di?erent values of κ2,have been arranged as vertical bars.The two curves shown are guides to the eye through the choices of ‘coupling strength’g 2χ0/t of 10and 20,and have been extrapolated through the origin.