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Effects of proximity to an electronic topological transition on normal state transport prop

a r X i v :c o n d -m a t /0303243v 1 [c o n d -m a t .s u p r -c o n ] 13 M a r 2003

E?ects of proximity to an electronic topological transition

on normal state transport properties of the high-T c superconductors

G.G.N.Angilella,1R.Pucci,1A.A.Varlamov,2and F.Onufrieva 3

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

Via S.So?a,64,I-95123Catania,Italy

Istituto Nazionale per la Fisica della Materia,UdR “Tor Vergata”/Coherentia,

Via del Politecnico,1,I-00133Roma,Italy

Laboratoire L′e on Brillouin,CE-Saclay,F-91191Gif-sur-Yvette,France

(Dated:January 13,2003)

Within the time dependent Ginzburg-Landau theory,the e?ects of the superconducting ?uctu-ations on the transport properties above the critical temperature are characterized by a non-zero imaginary part of the relaxation rate γof the order parameter.Here,we evaluate Im γfor an anisotropic dispersion relation typical of the high-T c cuprate superconductors (HTS),characterized by a proximity to an electronic topological transition (ETT).We ?nd that Im γabruptly changes sign at the ETT as a function of doping,in agreement with the universal behavior of the HTS.We also ?nd that an increase of the in-plane anisotropy,as is given by a non-zero value of the next-nearest to nearest hopping ratio r =t ′/t ,increases the value of |Im γ|close to the ETT,as well as its singular behavior at low temperature,therefore enhancing the e?ect of superconducting ?uctuations.Such a result is in qualitative agreement with the available data for the excess Hall conductivity for several cuprates and cuprate superlattices.

PACS numbers:74.20.-z,74.25.Fy,74.40.+k

I.INTRODUCTION

The appearance of superconducting ?uctuations above the critical temperature T c leads to precursor e?ects of the superconducting phase occurring already above T c .Due to their short coherence length,the discovery of the high-T c cuprate superconductors (HTS)made the ?uctuation regime experimentally accessible over a rel-atively wide temperature range above T c 1.Supercon-ducting ?uctuations manifest themselves in the singular temperature dependence of thermodynamic properties,such as the speci?c heat and the susceptibility,and of several transport properties (see Refs.1,2for recent re-views).In particular,the in?uence of superconducting ?uctuations on the Ettinghausen e?ect 3,the Nernst ef-fect,the thermopower,the electrical conductivity,and the Hall conductivity 4have been considered within the time-dependent Ginzburg-Landau (TDGL)theory for a layered superconductor in a magnetic ?eld near T c .A numerical approach within the ?uctuation exchange (FLEX)approximation to the theory of electric trans-port in the normal state of the high-T c cuprates has been developed by Yanase et al.5,6,7,8.

The e?ect of ?uctuations on the transport properties of the high-T c superconductors can contribute to a better understanding of the unconventional properties of their normal state.Recent experimental studies of the Nernst e?ect in underdoped cuprates have demonstrated a size-able Nernst coe?cient in the normal state both at high temperature and in high magnetic ?elds 9,10,11.Such ?nd-ings have been interpreted as an e?ect of precursor pair-

ing above T c in the pseudogap region,as well as of quan-tum superconducting ?uctuations 12.

In the case of the Hall e?ect,superconducting ?uctu-ations induce a characteristic deviation from the normal state temperature dependence of the Hall conductivity above T c (Hall anomaly)13.In particular,the ?uctua-tion Hall conductivity ?σxy has been evaluated within the TDGL theory 4,14,and it has been shown that a Hall sign reversal takes place below T c .The value and sign of ?σxy strongly depends on the electronic structure of the material under consideration and,in particular,on the topology of its Fermi surface.It is well known that ?σxy arises as a result of an electron-hole asymmetry in the band structure 15.Recently,on the basis of the general requirement of gauge invariance of the TDGL equations,it has been shown that the sign of ?σxy is determined by ?ln T c /?ln μ,where μis the chemical potential 16.More recently,evidence for a universal behavior of the Hall conductivity as a function of doping has been reported in the cuprate superconductors 17.

Given the relevance of the electronic structure in es-tablishing the magnitude and sign of the ?uctuation Hall e?ect,it is of obvious interest to study the e?ect of ?uctu-ations on the transport properties of low-dimensional su-perconducting materials in the proximity of an electronic topological transition (ETT)18,19,20.An ETT consists of a change of topology of the Fermi surface,and may be induced by doping,as well as by changing the impurity concentration,or applying pressure or anisotropic stress.In all such cases,one may introduce a critical param-eter z ,measuring the proximity to the ETT occurring

at z=0.In the case of quasi two-dimensional(2D) materials,such as the cuprates,the electronic band is locally characterized by a hyperbolic-like dispersion re-lation.Therefore,one is particularly interested in the study of an ETT of the‘neck disruption’kind,according to the original classi?cation of I.M.Lifshitz20.

Some e?ects of an ETT(namely,the existence of a Van Hove singularity in the density of states)on the superconducting properties of the cuprates are well known21,22,23,24.Recently,it has been shown also that the e?ect of the proximity to an ETT is richer than having a Van Hove singularity in the density of states, namely that the ETT is a speci?c quantum critical point. This leads to the existence of several quantum critical regimes that can explain the observed anomalous proper-ties of the high-T c cuprates in the normal state25,26,27,28. Some of the present authors have recently investigated the dependence of such e?ects on some speci?c material properties,such as the next-nearest to nearest neighbors hopping ratio29,and anisotropic stress30.Concerning the normal state transport properties of a superconductor, the e?ect of the proximity to an ETT has been studied for the thermoelectric power in a quasi-2D metal31,and for the Nernst and the weak-?eld Hall e?ects for both3D and quasi-2D metals32.

In this paper,we will study the anomalous Hall con-ductivity due to the superconducting?uctuations above T c for a quasi-2D superconductor close to an ETT.The link between TDGL theory and the microscopic theory is provided by the relaxation rateγof the?uctuating superconducting order parameter.In particular,a non-zero imaginary part of this quantity gives rise to a?uc-tuation contribution to the Hall e?ect.Here,we will study Imγas a function of the ETT parameter z and temperature T,both numerically and analitically,for a realistic band dispersion typical of the high-T c cuprate compounds.Close to the ETT,Imγis characterized by a steep in?ection point,surrounded by a minimum and a maximum,whose height increases with decreasing tem-perature.In the presence of electron-hole symmetry,we will show that Imγis an odd function of the ETT pa-rameter z,and that Imγvanishes and rapidly changes sign at the ETT point.In the cuprate superconduc-tors,electron-hole symmetry is usually destroyed by a non-zero next-nearest to nearest neighbors hopping ratio r=t′/t33.In this case,the peaks in Imγaround the ETT point have unequal heights,and we will show that their dependence on the hopping parameter r is in qual-itative agreement with the results of several?ts against the?uctuation Hall conductivity data of various cuprates and cuprate superlattices.

The paper is organized as follows.In Sec.II we will brie?y review the TDGL theory of superconducting?uc-tuations and the microscopic results for the direct and in-direct contributions to the excess Hall conductivity?σxy. In Sec.III we will outline the microscopic derivation ofγ, and explicitly evaluate Imγas a function of the chemical potential and temperature.We will eventually summa-rize in Sec.IV.

II.EXCESS HALL CONDUCTIVITY

A phenomenological description of the?uctuation ef-fects on the transport properties of a layered supercon-ductor is based on the time dependent Ginzburg-Landau (TDGL)equation1:

?γ ? c? ψ?(r,t)=δF

b|ψ?|4+ 2 c A ψ? 2+J|ψ?+1?ψ?|2 ,(2)

where a and b are the usual GL coe?cients,A is the vector potential of a magnetic?eld perpendicular to the layers,and J characterizes the Josephson coupling be-tween adjacent planes1.

In Eq.(1),the complex quantityγis the relaxation rate of the order parameter within the TDGL theory.A nonzero value of Reγis at the basis of the phenomenon of paraconductivity1.One?nds Reγ=πν/8T at tem-perature T,whereνis the density of states.

Under complex conjugation and inversion of the mag-netic?eld in Eq.(1),the equation forψ??would be the same as that forψ?,provided that Imγ=0. Thus,a nonzero value of Imγis associated with a breaking of electron-hole symmetry15,16.The condi-tion Imγ=0then gives rise to?uctuation e?ects on the Hall conductivity4,the Nernst e?ect4,32,35,and the thermopower4,31.

The?uctuation contibution to several transport prop-erties,such as paraconductivity,magnetoconductivity,

3 Nernst e?ect,and thermopower,have been evaluated

under several approximations(see Ref.1for a review).

From the microscopic point of view,the total?uctuation

contribution?σxy to the Hall conductivityσxy close to

T c can be expressed as the sum of two terms36:

?σAL

xy =

σN xx

πd

(1+1/2α)3/2

16 d

σN xy

ε?δ

×ln ε1+αε/δ+(1+2αε/δ)1/2 ,(3b)

respectively related to the Aslamazov-Larkin(AL)37and the Maki-Thomson(MT)38contributions.In Eqs.(3),ε=ln(T/T c)≈(T?T c)/T c is the reduced temperature,α=2ξ2c(0)/d2ε,d is the interlayer spacing,ξc(0)is the coherence length along the c axis at T=0,σNαβrefer to the components of the conductivity tensor in the absence of?uctuations,δ=π /8k B Tτφis the MT pair breaking parameter,withτφthe phase relaxation time of the quasi-particles,and,?nally,β∝Imγ(Ref.4).Whileξc(0)and δcan be independently determined by?tting analogous (AL+MT)expressions for the paraconductivity4,the pa-rameterβ∝Imγcan be extracted by comparison with experimental data for the excess Hall e?ect36,39,40,41.Ta-ble I lists values ofβfor several layered cuprate super-conductors and HTS superlattices.One can immediately observe thatβshows a direct correlation with T c,i.e.|β| increases as T c increases,which we will discuss in more detail in Sec.III.

TABLE I:Electron-hole asymmetry parameterβ∝Imγand critical temperature T c for several layered cuprates and cuprate superlattices.Tha values ofβlisted here have been obtained from a?t of the AL+MT corrections to conductivity and Hall conductivity,Eqs.(3),against data for excess Hall e?ect.

YBCO/PBCO(36?A/96?A)68.68?0.0003Ref.41

YBCO/PBCO(120?A/96?A)86.33?0.075Ref.41

YBCO88.55?0.17Ref.36

Bi-2223105.?0.38Ref.40

(Bi,Pb)-2223109.?1.Ref.39

.(4) Before the analytic continuation,the polarization opera-tor is de?ned as(k B= =1):

Π(k,i?m;z,T)=

i??ξk

(6)

is the Green’s function for free electrons with dispersion relationξk,and the outer sum is performed over the N wavevectors q in the?rst Brillouin zone(1BZ).Here,we speci?cally have in mind the2D tight-binding dispersion relation:

ξk=?2t(cos k x+cos k y)+4t′cos k x cos k y?μ,(7)

with t,t′being hopping parameters between nearest and next-nearest neighbors of a square lattice,respectively. Equation(7)has been often employed in order to de-scribe the highly anisotropic dispersion relation of the cuprates.Forμ=μc=?4t′,the Fermi surface de?ned byξk=0has a critical form and undergoes an electronic topological transition(see Refs.26,28,29).Below,we will make use of the parameters z=(μ?μc)/4t,measuring the distance from the ETT(z=0),and of the hopping ratio r=t′/t(0

ΠR(0,?;z,T)=i

N k12+i(ξk+?)2?i(ξk+?)

iξk

?+2ξk+iδ tanh

ξk+?2T ,(9)

whereψ(z)here denotes the digamma function43andδis a positive in?nitesimal.Performing the frequency deriva-tive and passing to the static limit,as required by Eq.(4), one has:

γ=i

N k12T

,(10)

where

F(y)=1

cosh2y

.(11)

Figure1shows our numerical results for the real and imaginary parts of the TDGL relaxation rateγas a func-tion of the ETT parameter z over the whole bandwidth, for a representative value of the temperature parameter τ=T/4t=0.005and hopping ratios r=0?0.384.As anticipated,one?nds that Reγ∝ν(z),with a logarith-mic singularity at z=0and an asymmetric z-dependence in the case r=0.

In the electron-hole symmetric case(r=0),Imγis an odd function of the ETT parameter,vanishing at z=0,i.e.at the ETT,for all temperatures.Close to the ETT point,Imγrapidly changes sign,with two symmetric peaks occurring very close to the ETT point. The height of these peaks decreases with increasing tem-perature(Fig.2),and eventually diverges as T→0[see Eq.(16)below].Such a behavior,in particular,implies a sign-changing Hall e?ect as a function of doping,and a large Hall e?ect close to the ETT.Moreover,the re-sult Imγ(z=0)=0is consistent with the absence of electron-hole asymmetry15.A similar z-dependence have been demonstrated also for the thermoelectric power in the proximity of an ETT31.

On the other hand,in the electron-hole asymmetric case(r=0),one in general has Imγ(z)=?Imγ(?z). However,one still recovers a sign-changing Imγ,with Imγvanishing very close to the ETT.Moreover,the two peaks around the ETT have increasing heights with increasing hopping ratio r(Fig.3).Given that a non-zero value of the hopping ratio r can be associated with structural distortions in the ab plane of the cuprates30, one may conclude that in-plane anisotropy enhances the ?uctuation e?ects associated to a non-zero value of Imγ. Moreover,on the basis of the direct correlation existing between T c,max and the hopping ratio r29,33,it follows

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

-1-0.5 0 0.5 1 1.5 2

-15

-10

-5

-1-0.5 0 0.5 1 1.5 2

-15

-10

-5

-0.1-0.05 0 0.05 0.1

FIG.1:Real part(top panel)and imaginary part(bot-tom panel)of TDGL relaxation rateγ,Eq.(10),as a func-tion of the ETT parameter z ranging over the whole band-width,for?xed temperatureτ=0.005and hopping ratios r=0,0.032,...0.384,in units such that4t=1.Integra-tion in k-space in Eq.(10)has been performed via the tetra-hedra method,with a mesh of125751k-points in the irre-ducible wedge of the1BZ,andδ=10?6.Inset in lower panel shows enlarged view of Imγ(z)close to the ETT.In par-ticular,Imγ(z)is an odd function of z in the electron-hole symmetric case(r=0,solid line),with Imγ(z=0)=0. Curves corresponding to increasing values of r give rise to more pronounced peaks in Imγaround the ETT point.

that the heights of the peaks in Imγaround the ETT increase with increasing T c,max across di?erent classes of cuprates.Such a result is in agreement with the data listed in Tab.I for the excess Hall parameterβ∝Imγ.

A further justi?cation of the above numerical results can be drawn from an analysis of the continuum limit (N?1 k→ d2k

-30-20

-10

-0.1

-0.0500.050.1

I m γ

τ = 0.0040τ = 0.0031τ = 0.0022τ = 0.0013

FIG.2:Im γ(z )in the electron-hole symmetric case (r =0),for decreasing temperatures τ(in units such that 4t =1).

-15

-10

-5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

r = t ′ / t

Im γ ( z = 0?

)

Im γ ( z = 0+

)

FIG.3:Peak heights in Im γaround the ETT as a function of the hopping ratio r =t ′/t ,for ?xed temperature τ=0.005,in units such that 4t =1.

of the relaxation rate one obtains:

16t 2

Im γ=

x ?z

3y 2,

|y |≤d,|y |?1,

|y |>d,

(13)

where d =

2π2

ln b 2τ

sech

z +1

τ?sinh

2π2τz

ln(1?z 2

z 2dτ?z

+ln 1?

z 24π2τ3

2dτz +(4d 2τ2?z 2

)ln

2dτ+z 8π2d

z 2π2d

6 Here,we just quote the?nal result:

Imγ≈4tν0

|μ?μc|,(17)

whereν0=(4tπ2√π21

1+2r(z?r)

K 1+2r(z?r) ,(A1)

where K(k)denotes the

complete elliptic integral of?rst

kind

of modulus

k(Ref.45).In Eq.(A1),the ETT pa-

rameter ranges as?1+2r≤z≤1+2r,andν(z)is

characterized by a logarithmic singularity at z=0.

In the electron-hole symmetric case(r=0),ν(z)is an

even function of z,with?1≤z≤1.When electron-

hole symmetry is broken by a non-zero hopping ratio r,

the electron sub-band shrinks,while the hole sub-band

widens of an equal amount2r,and the logarithmic cusp

loses its symmetry around z=0.In order to extract

the asymptotic behavior of Eq.(A1)around z=0in

the case r=0,we introduce the‘electron’and‘hole’

auxiliary variables z1=z/(1?2r)and z2=?z/(1+2r),

with

(1?2r)z1+(1+2r)z2=0.(A2)

Clearly,z1→z and z2→?z in the electron-hole sym-

metric case(r=0).In terms of these variables,the famil-

iar plot of the DOS Eq.(A1)(see e.g.Fig.2in Ref.29)

can be seen as given by the intersection of Eq.(A2)with

the surface plot of

ν(z1,z2)=

√√

z1z2

π2

1?4r2

(1?az)ln

1?4r2.A result close

to Eq.(A4),although within the context of an excitonic

phase,can be found in Ref.46.From Eq.(A4),one read-

ily sees that,at lowest order in z,the source of asymmetry

in the DOS logarithmic cusp at z=0comes only from

the prefactor,which is linear in z for r=0.

APPENDIX B:BEHA VIOR OF ImγA W AY FROM

THE ETT

In the limit|z|/τ?1,we may forget about the details

of the dispersion relation,provided we retain its main

topological features.We can therefore expand Eq.(7)around

the

ETT as

ξk ≈

p 21

2m 2

?z,

(B1)

where p 1=k x ,p 2=k y ?π,and m 1,2=2/(1±2r )are the eigenvalues of the e?ective mass tensor around

the ETT 29.Here and below,we will make use of energy

units such that 4t =1.

Our starting point will be again the general expression for the retarded polarization operator,Eq.(9).Passing to the new coordinates x =p 1/√4T m 2,one obtains:

ΠR

(0,ω;z,T )=?ν0

∞

[tanh(s +ω)+tanh(s ?ω)]

m 1m 2/(2π2)=(π2

√

ω1+x 2?y 2

,(B3)

where ω1=ω?2ζ.

Let us now restrict to the case z ??T ,in which case ω1=ω+2|ζ|?1.The inner integral in Eq.(B3)can then be estimated by introducing a cuto?δ 1,splitting the integration range in the intervals:[0,a ?δ],[a ?δ,a +δ],[a +δ,∞[,with a 2=x 2+ω1,and replacing the hyperbolic tangent with its appropriate asymptotic expansion in each interval.One eventually obtains: ∞

dy tanh(ω1+x 2?y 2)

2√4(x 2+ω1).(B4)

Making use of such result back in Eq.(B3),and per-forming the ωderivative as required by Eq.(4),one has:

lim

ω→0

?Re ΠR

∞

4(x 2+2|ζ|)

(B5)

where the last integration is trivial.Repeating an analo-gous derivation in the case z ?T ,one eventually obtains the ?nal result,Eq.(17),to within logarithmic accuracy (δ2e 2/4≈1).1

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