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arXiv:cond-mat/0305278v2 [cond-mat.supr-con] 18 Oct 2003

Superconductivity in the Pseudogap State in “Hot – Spots”

model: Ginzburg – Landau Expansion

E.Z.Kuchinskii, M.V.Sadovskii, N.A.Strigina

Institute for Electrophysics,

Russian Academy of Sciences, Ural Branch,

Ekaterinburg, 620016, Russia

E-mail: kuchinsk@iep.uran.ru, sadovski@iep.uran.ru, strigina@iep.uran.ru

Abstract

We analyze properties of superconducting state (for both s-wave and d-wave

pairing), appearing on the “background” of the pseudogap state, induced by

fluctuations of “dielectric” (AFM(SDW) or CDW) short – range order in

the model of the Fermi surface with “hot spots”. We present microscopic

derivation of Ginzburg – Landau expansion, taking into account all Feynman

diagrams of perturbation theory over electron interaction with this short –

range order fluctuations, leading to strong electronic scattering in the vicinity

of “hot spots”. We determine the dependence of superconducting critical tem-

perature on the effective width of the pseudogap and on correlation length of

short – range order fluctuations. We also find similar dependences of the main

characteristics of such superconductor close to transition temperature. It is

shown particularly, that specific heat discontinuity at the transition tempera-

ture is significantly decreased in the pseudogap region of the phase diagram.

PACS numbers: 74.20.Fg, 74.20.De

Typeset using REVTEX

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I. INTRODUCTION

Pseudogap state observed in a wide region of the phase diagram of high – temperature

superconducting cuprates is characterized by numerous anomalies of their properties, both

in normal and superconducting states [1,2]. Apparently, the most probable scenario of pseu-

dogap state formation in HTSC – oxides can be based [2] on the picture of strong scattering

of current carriers by fluctuations of short – range order of “dielectric” type (e.g. antiferro-

magnetic (AFM(SDW)) or charge density wave (CDW)) existing in this region of the phase

diagram. In momentum space this scattering takes place in the vicinity of characteristic

scattering vector Q = (π

a,π

a) (a – lattice constant), corresponding to doubling of lattice

period (e.g. vector of antiferromagnetism), being a “precursor” of spectrum transformation,

appearing after the establishment of AFM(SDW) long – range order. Correspondingly, there

appears an essentially non Fermi – liquid like renormalization of electronic spectrum in the

vicinity of the so called “hot spots” on the Fermi surface [2]. Recently there appeared a

number of experiments giving rather convincing evidence for precisely this scenario of pseu-

dogap formation [3–5]. Within this picture it is possible to formulate simplified “nearly

exactly” solvable model of the pseudogap state, describing the main properties of this state

[2], and taking into account all Feynman diagrams of perturbation theory for scattering

by (Gaussian) fluctuations of (pseudogap) short – range order with characteristic scattering

vectors from the area of Q, with the width of this area defined by the appropriate correlation

length ξ [6,7].

Up to now the majority of theoretical papers is devoted to the studies of models of

the pseudogap state in the normal phase for T > Tc. In Refs. [8–11] we have analyzed

superconductivity in the simplified model of the pseudogap state, based on the assumption of

existence of “hot” (flat) patches on the Fermi surface. Within this model we have constructed

Ginzburg – Landau expansion for different types (symmetries) of Cooper pairing [8,10] and

also studied the main properties of superconducting state for T < Tc, solving the appropriate

Gor’kov’s equations [9–11]. At first stage, we have considered greatly simplified (“toy”)

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model of Gaussian fluctuations of short – range order with an infinite correlation length,

where it is possible to obtain an exact (analytic) solution for the pseudogap state [8,9], further

analysis of more realistic case of finite correlation lengths was performed both for “nearly

exactly” solvable model of Ref. [10] (assuming the self – averaging nature of superconducting

order parameter over pseudogap fluctuations) and for a simplified exactly solvable model

[11], where we have been able to analyze the effects due to the absence of self – averaging

[9,11,12].

The aim of the present paper is to analyze the main properties of superconducting

state (for different types of pairing), appearing on the “background” of the pseudogap of

“dielectric” nature in more realistic model of “hot spots” on the Fermi surface. Here we

shall limit ourselves to the region close to superconducting temperature Tcand perform an

analysis based on microscopic derivation of Ginzburg – Landau expansion, assuming the self

– averaging nature of superconducting order parameter and generalizing similar approach

used earlier for “hot patches” model in Ref. [10].

II. “HOT – SPOTS” MODEL AND PAIRING INTERACTION.

In the model of “nearly antiferromagnetic” Fermi – liquid, which is actively used to

describe the microscopic nature of high – temperature superconductivity [13,14], it is usually

assumed that the effective interaction of electrons with spin fluctuations of antiferromagnetic

(AFM(SDW)) short – range order is of the following form:

Veff(q,ω) =

g2ξ2

1 + ξ2(q − Q)2− iω

ωsf

(1)

where g – is some interaction constant, ξ – correlation length of spin fluctuations, Q =

(π/a,π/a) – vector of antiferromagnetic ordering in dielectric phase, ωsf – characteristic

frequency of spin fluctuations. Both dynamic spin susceptibility and effective interaction

(1) are peaked in the region of q ∼ Q), which leads to the appearance of “two types” of

quasiparticles – “hot” one, with momenta in the vicinity of “hot spots” on the Fermi surface

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(Fig. 1), and “cold” one, with momenta close to the parts of the Fermi surface, surrounding

diagonals of the Brillouin zone [6]. This is due to the fact, that quasiparticles from the

vicinity of the “hot spots” are strongly scattered with the momentum transfer of the order

of Q, due to interaction with spin fluctuations (1), while for quasiparticles with momenta

far from these “hot spots” this interaction is relatively weak.

For high enough temperatures πT ≫ ωsf, we can neglect spin dynamics [6], limiting

ourselves to static approximation in (1). Considerable simplification, allowing to analyze

higher – order contributions, can be achieved by substitution of (1) by model – like static

interaction of the following form [7]:

Veff(q) = W2

2ξ−1

ξ−2+ (qx− Qx)2

2ξ−1

ξ−2+ (qy− Qy)2

(2)

where W is an effective parameter with dimension of energy. In the following, as in Refs.

[6,7], we consider parameters W as ξ phenomenological (to be determined from the exper-

iment). Anyhow, Eq. (2) is qualitatively quite similar to the static limit of (1) and almost

indistinguishable from it in most interesting region of |q − Q| < ξ−1, determining scattering

in the vicinity of “hot spots”.

The spectrum of “bare” (free) quasiparticles can be taken as [6]:

ξp= −2t(cospxa + cospya) − 4t

′cospxacospya − µ(3)

where t is the transfer integral between nearest neighbors, while t′is the transfer integral

between second nearest neighbors on the square lattice, a is the lattice constant, µ – chemical

potential. This expression gives rather good approximation to the results of band structure

calculations of real HTSC – system, e.g. for Y Ba2Cu3O6+δwe have t = 0.25eV , t′= −0.45t

[6]. Chemical potential µ is fixed by concentration of carriers.

The least justified is an assumption of the static nature of fluctuations, which can be

valid only for rather high temperatures [6,7]. For low temperatures, particularly in su-

perconducting phase, spin dynamics can become quite important, e.g. for microscopics of

Cooper pairing in the model of “nearly antiferromagnetic” Fermi liquid [13,14]. However,

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we assume here that the static approximation is sufficient for qualitative understanding of

the influence of pseudogap formation upon superconductivity, which will be modelled within

phenomenological BCS – like approach.

In the limit of infinite correlation length ξ → ∞ this model acquires an exact solution [15].

For finite ξ we can construct “nearly exact” solution [7], generalizing the one – dimensional

approach, proposed in Ref. [16]. Then we can (approximately) sum the whole diagrammatic

series for the one – particle electronic Green’s function.

For the contribution of an arbitrary diagram for electronic self – energy, in the N-th

order over the interaction (2), we write down the following Ansatz [7,16]:

Σ(N)(εnp) = W2N

2N−1

?

j=1

G0kj(εnp),

G0kj(εnp) =

1

iεn− ξkj(p) + ikjvkjκ

(4)

where κ = ξ−1, kj – is the number of interaction lines, surrounding the j-th (from the

beginning) electronic line in a given diagram, εn= 2πT(n + 1/2) (assuming εn> 0).

ξk(p) =

ξp+Q for odd k

ξp

for even k

(5)

vk=

|vx(p + Q)| + |vy(p + Q)| for odd k

|vx(p)| + |vy(p)|for even k

(6)

where v(p) =

∂ξp

∂p– velocity of a “bare” (free) quasiparticle.

In this approximation the contribution of an arbitrary diagram is determined, in fact, by

the set of integers kj. Any diagram with intersection of interaction lines is actually equal to

some diagram of the same order without intersections and the contribution of all diagrams

with intersections can be accounted with the help of combinatorial factors s(kj) attributed

to interaction lines on diagrams without intersections [16,7,6].

Combinatorial factor:

s(k) = k (7)

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