Physics Letters A 377 (2013) 2007–2010

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Physics Letters A

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Bose–Einstein condensation of interacting Cooper pairs in cupratesuperconductors

Ze Cheng ∗

School of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 October 2012Received in revised form 14 June 2013Accepted 25 June 2013Available online 1 July 2013Communicated by A.R. Bishop

Keywords:Boson condensationCooper pairingHartree–Fock–Popov theory

The Hartree–Fock–Popov theory of interacting Bose particles is generalized to the Cooper-pair systemwith a screened Coulomb repulsive interaction in high-temperature superconductors. At zero temperature,we find that the condensate density nc(0) of Cooper pairs and the phonon velocity c(0) are dome-shaped functions of the p hole concentration δ. At finite temperature T , we find that the condensatefraction nc(T )/n and the phonon velocity c(T ) decrease continuously from nc(0)/n or c(0) to zero asT increases from zero to the transition temperature Tc . The Cooper-pair system undergoes a first-orderphase transition from the normal state to the Bose condensed state.

© 2013 Elsevier B.V. All rights reserved.

Recently it has been observed that there are incoherent Cooperpairs in the normal state of cuprate superconductors [1–6] and su-perconducting films [7]. These experiments suggest that Cooperpairs are preformed above the critical temperature of supercon-ducting transitions. It has been also observed that the Cooperpairs in cuprate superconductors have a small coherence lengthξ ≈ 10 Å [8,9], which is of the same order as the scale of a lat-tice unit cell. This observation indicates that the pairing betweenelectrons, or holes, in these materials is reasonably localized in thecoordinate space. Hence, the pair state can be well approximatedby a phenomenological local boson field Ψ (x). In view of the abovedescription, many investigators believe that we should generalizethe Bardeen–Cooper–Schrieffer (BCS) theory to describe these newhigh-temperature superconductors [10–12]. One of the many the-oretical models which have been examined in this context is thecrossover theory from BCS pairing to Bose–Einstein condensation(BEC) [13–16], which can account for many of the various experi-mental observations in high-temperature superconductors.

There are several generally accepted phenomena in high-temperature superconductivity that make the cuprates so fascinat-ing. One of these phenomena is a d-wave symmetry of the super-conducting gap, as conclusively demonstrated by several phasesensitive experiments [17,18]. Another feature is a normal-stategap (pseudogap) in underdoped materials, which exists above thesuperconducting transition temperature Tc [19,20]. The elucidationof the pseudogap phenomenon of the cuprates has been a majorchallenge in condensed matter physics for the past two decades.

* Tel.: +86 027 87542637; fax: +86 027 87545438.E-mail address: zcheng@mail.hust.edu.cn.

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Are the experimental findings correctly interpreted in terms of aquantum field theory of interacting Cooper pairs? Such a theoryshould account self-consistently for a d-wave symmetry of the su-perconducting gap and for a pseudogap above the superconductingtransition temperature Tc . In this work, we answer this questionby adopting the BCS–BEC crossover theory. In this scenario, theexcitations of the cuprate superconducting state contain a mix ofbosonic and fermionic properties. The Fermi elementary excitationsexhibit features of the superconducting gap and pseudogap and theBose elementary excitations reveal other features which are strik-ingly different. Such a model exhibits two limiting behaviors. Oneis reminiscent of a BCS system and the other limit describes theBEC of Cooper pairs [21,22]. Tolmachev shows how a generalizedBEC formalism subsumes BCS theory as a special case [23]. A moredetailed version of the BCS–BEC crossover theory, also subsumedin the formalism proposed in [23], has been presented for the spe-cific case of equal numbers of both condensed and non-condensedCooper pairs [24].

In the present Letter, we investigate the BEC of interactingCooper pairs within the framework of quantum field theory. Wehave taken into account the screened Coulomb repulsive interac-tion between Cooper pairs. The Hartree–Fock–Popov (HFP) the-ory of BEC is generalized to the case of interacting Cooper pairs.We give a critical analysis of the HFP approximation in the BECof interacting Cooper pairs and point out that the HFP approx-imation gives a gapless phonon-like excitation spectrum at alltemperatures. By comparison, the pairing limit of the BCS–BECcrossover theory gives a gapped fermionic excitation spectrum atall temperatures. At zero temperature, we find that the conden-sate density nc(0) of Cooper pairs and the phonon velocity c(0)

are dome-shaped functions of the p hole concentration δ. At finite

2008 Z. Cheng / Physics Letters A 377 (2013) 2007–2010

temperature T , we find that the condensate fraction nc(T )/n andthe phonon velocity c(T ) decrease continuously from nc(0)/n orc(0) to zero as T increases from zero to Tc . For higher tempera-tures, we find that the repulsive interaction between Cooper pairsdrives more Cooper pairs into the condensate. The Cooper-pairsystem undergoes a first-order phase transition from the normalstate to the BEC state. Our investigation into three-dimensionalCooper pairs provides a criterion whether the BEC theory appliesto cuprate superconductors. These features reveal some unsus-pected nontrivial aspects of the boson condensation in the BCS–BEC crossover theory.

The starting point for our discussion of interacting quantummechanical assemblies is the Hartree–Fock (HF) approximation.The HF approximation is basically a static mean-field theory, whichtreats the motion of single particles in an average static field gen-erated by all the other particles. The HF approximation neglectsterms like 〈a†

−ka†k〉 or 〈a−kak〉 in the Hamiltonian, which reflect

the creation and annihilation of two non-condensate particle pairsdue to the interaction and play a crucial role in the BEC theory.The HFP approximation includes terms like 〈a†

−ka†k〉 or 〈a−kak〉

in the Hamiltonian. This approach consequently accounts for thelow-energy excitations of the system. Nowadays, the BCS groundstate wave-function has been found to have a greater applicabil-ity. For example, Nozières and Schmitt-Rink study the BCS groundstate wave-function and find that the evolution from BCS to BECsuperconductivity is smooth [15], and a review of the BCS–BECcrossover theory is given in [14]. As the strength of the attrac-tive pairing interaction U (< 0) between fermions is increased, thiswave-function is also capable of describing a continuous evolutionfrom BCS-like behavior to a form of BEC. The variational param-eters of this wave-function are usually represented by the twomore directly accessible parameters Δ(T ) and μ(T ). Here Δ(T ) isthe temperature-dependent superconducting gap parameter and μis the temperature-dependent chemical potential of the fermions.These fermion parameters are uniquely determined by the gap andnumber equations.

If U (x − x′) represents the interaction potential between aCooper pair located at x and another Cooper pair located at x′ ,then the screened Coulomb repulsive interaction potential is

U(x − x′) = (2e)2

4πε0ε∞|x − x′|e−qTF |x−x′|, (1)

where ε0 is the permittivity of the vacuum and ε∞ denotes theoptical dielectric constant of a superconductor. qTF is the Thomas–

Fermi wavenumber, as given by qTF = (4kF /πa0)12 , where a0 is

the Bohr radius. kF is the Fermi wavenumber, as given by kF =(3π2ne)

13 , where ne is the number density of conduction electrons

in a superconductor. We adopt the grand canonical ensemble, inwhich Cooper pairs have a chemical potential μ. Now one canwrite the Hamiltonian of the Cooper-pair system in terms of theCooper-pair operators ak and a†

k:

H =∑

k

(εk − μ)a†kak + 1

2V

∑k,k′,q

v(q)a†k+qa†

k′−qak′ ak, (2)

where εk = h2k2/4m is the kinetic energy of a Cooper pair withwave vector k. v(q) is the Fourier transform of the interaction po-tential U (x) of Cooper pairs and the crystal occupies a volume V .

We need to study the finite-temperature excitations in the sys-tem of interacting Cooper pairs. The best way to do this study is touse the Beliaev–Green’s function formalism [25,26]. This techniqueis the most effective way of calculating the equilibrium thermo-dynamic properties, as well as single-particle excitations of thesystem. Our emphasis is on how to include the effects of thenon-condensate Cooper pairs, which are entirely neglected in BCS

theory, and is based on the first-order HFP self-energy diagrams.At first, we define the normal and anomalous Green’s function as

G11(k, τ ) = −⟨Tτ

[ak(τ )a†

k(0)]⟩, (3)

G12(k, τ ) = −⟨Tτ

[a−k(τ )ak(0)

]⟩, (4)

where Tτ is a τ ordering operator. G11 and G22 represent thepropagation of a single Cooper pair. G12 and G21 represent thedisappearance and appearance of two non-condensate Cooperpairs, respectively. If we use the letter p to represent the four-dimensional vector (k, iωn), there are the following useful identi-ties:

G22(p) = G11(−p), G12(p) = G21(−p).

There are two types of proper self-energies for a Bose-condensedsystem. In the Feynman diagrams, one type of proper self-energieshave one particle line going in and one coming out, which are de-noted as Σ11(k, τ ) and Σ22(k, τ ). The other ones have two particlelines either coming out, denoted by Σ12(k, τ ), or going in, denotedby Σ21(k, τ ).

Next we discuss the HFP approximation for a Bose gas of Ninteracting Cooper pairs at finite temperatures [27]. We prescribethat at finite temperature T , the number of Cooper pairs in thelowest state (k = 0) is given by Nc(T ). The Nc Cooper pairs form aBose condensate. In the rest of this Letter, we use the superscript“(0)” as a reminder that the quantity is for a noninteracting Bosegas. Now we need to introduce a quantity n(0) , which denotes the(temperature-dependent) density of excited Cooper pairs in a non-interacting Bose gas and is given by

n(0) =∫

dq

(2π)3

1

exp[β(εq − μ(0))] − 1. (5)

Consequently, the self-energy Σ11 and Σ12 can be explicitly writ-ten as

Σ11(k) = 2nc v(0) + 2n(0)v(0),

Σ12(k) = nc v(0), (6)

where nc = Nc/V is called the condensate density.In the Bose-condensed phase of T < Tc , the chemical potential

μ of an interacting Bose gas has been shown to satisfy [26]

μ = nc v(0) + 2n(0)v(0), T < Tc. (7)

By substituting the self-energies in Eq. (6) and the chemical po-tential in Eq. (7) into the Dyson–Beliaev expressions, we obtainG11 and G12 as

G11(p) = iωn + εk + Δ

(iωn)2 − ε2k − 2Δεk

, (8)

G12(p) = − Δ

(iωn)2 − ε2k − 2Δεk

. (9)

Here the quantity Δ is defined by Δ(T ) = nc(T )v(0). Both G11 andG12 in Eqs. (8) and (9) have identical poles at ω = ±Ek , where

Ek =√

ε2k + 2Δ(T )εk. (10)

This gives the energy spectrum of elementary excitations for T <

Tc . Ek is phonon-like in the long-wavelength limit and the phononvelocity c is given by

c ≡ √Δ(T )/2m = √

nc(T )v(0)/2m. (11)

For a given total density n, we have

n = nc + n, (12)

Z. Cheng / Physics Letters A 377 (2013) 2007–2010 2009

Fig. 1. Variation of the condensate density nc(0) with the p hole concentration perCu δ.

where n denotes the density of uncondensed Cooper pairs and isgiven by

n =∫

dk

(2π)3

(εk + Δ

2Ekcoth

βEk

2− 1

2

). (13)

For a given n and T , Eqs. (12) and (13) are coupled equationsfor determining nc(T ). We first consider the zero-temperature case(T = 0 K). In this case, we obtain

nc(0) = n − [2mnc(0)v(0)] 32

3π2h3. (14)

In order to give a numerical estimate of nc(0), we take into ac-count the high-temperature superconductor La2−δSrδCuO4, whereδ denotes the dopant concentration. The conduction charges in thismaterial are the p holes from oxygen ions and δ also denotes thep hole concentration per Cu. The three-dimensional La2−δSrδCuO4consists of repeated CuO2 planes. The Cu–Cu distance in CuO2planes is a and the distance between layers is c, so that the vol-ume of a three-dimensional unit cell is given by v = a2c. In orderto model the number density ne of p holes in the superconduc-tor, we must consider the two points: (1) A characteristic of all thesuperconductors is that a small fraction of the total p holes partici-pate in pairing [28]; (2) The dome-shaped Tc is characteristic of allhole-doped cuprate superconductors [29]. Consequently, the num-ber density ne of p holes in the superconductor can be modeledas

ne = 0.2(δ − 0.049)(0.251 − δ)/v. (15)

It is necessary to point out that this modeling is proposed so asto fit a dome-shaped result of Tc . This dome-shaped behavior canbe derived from the Hamiltonian in Eq. (2) but such derivation ishuge and will be presented elsewhere. Concomitantly, we take thedensity of Cooper pairs as n = ne/2. Experiments show that thismaterial becomes superconducting in the range 0.05 � δ � 0.25.The parameters of La2−δSrδCuO4 considered are as follows: a =3.7873 Å, c = 13.2883 Å, and ε∞ = 10.0. The computation showsthat the distance between Cooper pairs is d = n−1/3 > 57.17 Å inthe range 0.05 � δ � 0.25, so that the condition of diluteness ofthe Bose gas of Cooper pairs is satisfied: ξ n−1/3.

According to Eq. (14), the variation with δ of the condensatedensity nc(0) is shown in Fig. 1. The essential characters are thatthe condensate density is of the order nc(0) 1018 cm−3 and thatnc(0) is a dome-shaped function of δ. The computation reveals that

Fig. 2. Variation of the phonon velocity c(0) with the p hole concentration per Cu δ.

the condensate fraction nc(0)/n ≡ 0.9148 in the range 0.05 � δ �0.25. Namely, the condensate fraction is independent of the p holeconcentration δ. The computation also reveals that the condensatefraction decreases with increasing of the total density n. This is inagreement with the well-known Bogoliubov result. c(0) representsthe phonon velocity at zero temperature. According to Eq. (11), thevariation with δ of the phonon velocity c(0) is shown in Fig. 2. Thephonon velocity is a dome-shaped function of δ.

We next discuss the finite-temperature case (T � Tc). The tem-perature dependence of the condensate density nc(T ) may be de-rived from Eqs. (12) and (13). Since their solution requires nu-merical methods, one can introduce the reduced wavenumberx = hk/2

√mΔ. At this point, we need to introduce the parame-

ter γ = n13 v(0). The computation reveals that the parameter γ ≡

77.18 meV nm2 in the range 0.05 � δ � 0.25. Namely, the param-eter γ is independent of the p hole concentration δ. γ signifiesthe effective total interaction between Cooper pairs. We first lety = nc(T )/n and t = T /Tc and then derive the following expres-sion,

y = 1 − (2mγ y)32

3π2h3− 4(mγ y)

32

π2h3

×∞∫

0

x(x2 + 1)√x2 + 2

1

exp[mγ yπ h2t

g3/2(1)2/3√

x4 + 2x2] − 1

dx. (16)

Therefore, nc(T )/n is a universal function of T /Tc and γ , indepen-dently of any particular property of a superconductor. Concretelyspeaking, the condensate fraction nc(T )/n is independent of thep hole concentration δ. According to Eq. (16), the variation withthe reduced temperature T /Tc of the condensate fraction nc(T )/nis shown in Fig. 3. The condensate fraction decreases continuouslyfrom nc(0)/n to zero as the temperature increases from zero to thetransition temperature Tc . Fig. 3 also shows that the condensatefraction n(0)

c (T )/n of the gas of ideal Cooper pairs varies with thereduced temperature T /Tc . Fig. 3 clearly indicates a finite jump inthe condensate fraction nc(T )/n at the transition temperature Tc .This jump is the characteristic of a first-order phase transition. Forhigher temperatures T → Tc , n(0)

c (T ) < nc(T ). This means that therepulsive interaction between Cooper pairs puts more Cooper pairsin the condensate. c(T ) represents the phonon velocity in the gasof nonideal Cooper pairs at temperature T . c(T ) is also a functionof the dopant concentration δ. According to Eq. (11), the variationwith T and δ of the phonon velocity c(T ) is shown in Fig. 4. For a

2010 Z. Cheng / Physics Letters A 377 (2013) 2007–2010

Fig. 3. The solid and dotted lines denote variation of the condensate fraction nc(T )/nof nonideal and ideal Cooper pairs with the reduced temperature T /Tc , respectively.

Fig. 4. Variation of the phonon velocity c(T ) of nonideal Cooper pairs with the tem-perature T and the dopant concentration δ.

fix δ, the phonon velocity decreases continuously from c(0) to zeroas T increases from zero to Tc . For a fixed T , c(T ) increases withincreasing of δ in the range 0.05 � δ � 0.15 but it decreases withincreasing of δ in the range 0.15 � δ � 0.25.

Here we must stress the point that the universal formula (16)is deduced from the Hamiltonian presented in Eq. (2) by meansof the HFP theory. This means that the above results followthe Hamiltonian presented in Eq. (2). The concrete derivation ofEq. (16) will be presented elsewhere. In the present Letter, we in-vestigate the boson condensation in the BCS–BEC crossover theory.In study of the BEC of Cooper pairs, we have taken into account thescreened Coulomb repulsive interaction between Cooper pairs. TheHFP theory of BEC is generalized to the case of interacting Cooper

pairs. At zero temperature, we find that the condensate densitync(0) of Cooper pairs is of the order nc(0) 1018 cm−3 and thatnc(0) is a dome-shaped function of the p hole concentration δ. Itis found that the phonon velocity c(0) is also a dome-shaped func-tion of δ. At finite temperature, we find that the condensate frac-tion nc(T )/n decreases continuously from nc(0)/n to zero as thetemperature increases from zero to the transition temperature Tc .For higher temperatures, we find that the repulsive interaction be-tween Cooper pairs puts more Cooper pairs in the condensate. Thecomputation reveals that the phonon velocity c(T ) decreases con-tinuously from c(0) to zero as the temperature increases from zeroto the transition temperature Tc . The Cooper-pair system under-goes a first-order phase transition from the normal state to theBEC state. Our investigation into three-dimensional Cooper pairsprovides a criterion whether the BEC theory applies to cuprate su-perconductors. These features reveal some unsuspected nontrivialaspects of the boson condensation in the BCS–BEC crossover the-ory. The predicted properties of the BEC of nonideal Cooper pairsmight hopefully be verified in the present-day physics laboratories.

Acknowledgements

This work was supported by the National Natural Science Foun-dation of China under Grants No. 10174024 and No. 10474025.

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