# Polaronic Mechanism of Superconductivity in Cuprates

Authors: Pradeep Chaudhary; Anuj Nuwal; S. C. Tiwari; R. K. Paliwal; S. L. Kakani
Journal: International Journal of Superconductivity
 (2013)

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## Abstract

A strong polaron pairing model of high-temperature cuprate superconductors is presented. The normal and anomalous one-particle Green’s functions are derived from a system with strong electron-phonon coupling. Self-consistent equation for the superconducting order parameter (Δ) is derived using Green’s function technique and following Lang and Firsov transformations. Expressions for specific heat, density of states, free energy, and critical field based on this model have been derived. The theory is applied to explain the experimental results in the system YBa2Cu3O7-x. There is convincing evidence that the theory is fully compatible with the key experiments.


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## Body

## 1. Introduction

Strikingly, after 26 years of enormous experimental and theoretical efforts followed by the discovery, there is still little consensus on the pairing mechanism of high-temperature superconductivity (HTSC) in cuprates [1–12]. HTSCs have unique physical properties in both the normal state and superconducting one. To comprehend the physics of these complex compounds is one of the main tasks of the theory of superconductivity, whose solution may allow one to explain the pairing mechanism ensuring HTSC. At present, there exists no mechanism which would explain the totality of thermodynamical, magnetic, and superconductive properties of HTSCs from a single point of view.The electron-phonon pairing mechanism [13–16], being the principal one in low-temperature superconductors, makes a considerable contribution to the establishment of the superconducting state in HTSCs. But in order to obtain proper description, it is necessary to consider the other mechanism inherent in HTSCs [3, 5, 7, 8, 10].To explain HTSC, a lot of models and mechanisms of this unique phenomenon have been proposed [1–5, 10, 11]. The key question is the nature of the mechanism of pairing of carriers. There are many different models of superconductivity available, for example, magnon model, exciton model, model of resonant valence bonds, bipolaronic model, bisoliton model, anharmonic model, model of local pairs, and plasmon model, [17]. All these models use the concept of pairing with a subsequent formation of a Bose condensate at temperature TC irrespective of the nature of the resulting attraction. Some recent theoretical models postulate the mechanism of antiferromagnetic spin fluctuations [18, 19], so that the electron scattering on them can be the reason for the pairing of electrons.In order to comprehend the nature of the superconducting state, it is necessary to construct a consistent microscopic theory which should be able to describe superconductive and the normal properties of HTSCs. We have proposed that the pairing mechanism in cuprate superconductors can be understood on the polaronic model of the charge carriers [20–22].In the present work following Alexandrov and Ranninger [21], we have developed a microscopic theory for HTSC cuprates by generalizing the Holstein [22] and Lang Firsov [23, 24] and Gorkov Nambu formalisms in order to evaluate the Green's functions for electrons coupled to phonons and considering the range of coupling which corresponds to small polaron formation (λ≫1) [25, 26].


## 2. Model Hamiltonian

The model Hamiltonian for our system can be expressed as [21]
(1)H=He+Hph+He-ph+Hee,
where He is the kinetic energy in the initial Bloch band, Hph is the vibration energy of the lattice, He-ph is electron-phonon interaction, and Hee is the Coulomb electron-electron correlations.In one band approximationHe has the form
(2)He=∑ksE(k)Cks+Cks,
where k and s denote the state with quasi momentum and spin, respectively. E(k) is bare band energy. Hphcan be expressed in terms of phonon operators dq; q=(q,ν) where ν is the type of vibrational mode.(3)Hph=∑qω(q)dq+dq,
where ω is the phonon dispersion.The electron-phonon interaction is described by theFro¨hlich Hamiltonian:
(4)He-ph=∑k,q,s{ω(q)γ(q)12NCk+q,s+Ck,sdq+H.C.}
in which ω(q) and γ(q) are the phonon frequency and the interaction matrix element in a parent crystal without charge carriers, respectively. Correspondingly one obtains
(5)γ2(q)=4πe2q2Ωω0(1ε∞-1ε0).In the case of optical longitudinal phonon with frequencyω(q)=ω0 and ε0, ε∞ are the dielectric constants of the crystal with and without taking ionic part into consideration. Ω is the volume of the unit cell and N is their number.For acoustic phonons, one finds(6)ν2(q)=ED2quM,ω(q)=ω0,
where ED is the deformation potential, u is the sound velocity, and M is the mass of an elementary cell. For intermolecular phonons
(7)γ2(q)=γ2(0).The combined Hamiltonian can be expressed as(8)H=∑ksE(k)Cks+Cks+∑qω(q)dq+dqH=+∑k,q,sω(q)γ(q)dq12N{Ck+q+Ck,s+H.C.}+VC.HereVC is the Coulomb repulsion. This Hamiltonian includes electron-phonon and electron-electron correlations. To diagonalize the main part of the Hamiltonian, the site representation is more convenient.One can express the previous Hamiltonian as(9)H=∑mnT(m-n)Cmσ+Cnσ+∑qω(q){dq+dq+12}H=+12N∑mqω(q)γ(q)nmσ{dqeiqm+dq+e-iqm}H=+12∑mnσσ′VC(m-n)nmσ+nnσ′+∑qω(q){dq+dq+12},
where
(10)T(m)=1N∑kE(k)eikm,nmσ=Cmσ+Cmσ.In the small-polaron regime,λ≥1, the kinetic energy remains smaller than the interaction energy, and a self-consistent treatment of a many-body problem is possible with the 1/λ expansion technique [27]. Following Lang and Firsov [23, 24] and applying canonical transformations to diagonalize the Hamiltonian [28], one obtains
(11)H~=eSHe-S,
where
(12)S=∑msCms+Cms12Nγ(q){dqeiqm-dq+e-iqm}.The electron operator transforms as(13)d~q=dq′+[S,dq′]12[S,[S,dq′]]+⋯,[s,dq′]=[∑msCms+Cms12Nγ(q){dqeiqm-dq+e-iqm},dq′],[s,dq′]=12Nγ(q)[s,dq′]x=×[∑qmsCms+Cms{dqeiqmdq′-dq+e-iqmdq′vvvvvvvvvgggvfvvvvvvxv-dq′dqeiqm+dq′dq+e-iqm}∑qmsCms†Cms{dqeiqmdq′-dq†e-iqmdq′].Usingdq′dq+=dq+dq′+δqq′(14)[s,dq′]=12Nγ(q)[∑qmsCms+Cmsδqq′e-iqm].Whenq=q′ we have
(15)[s,dq′]=12Nγ(q)[∑msCms+Cmse-iq′m]⌊s,⌊s,dq′⌋⌋=0.Thus,(16)d~q′=dq′+12Nγ(q)∑msCms+Cmse-iq′m,d~q′+=dq′+-12Nγ(q)∑msCms+Cmse-iq′m.Hence(17)H=∑mns,m≠nT(m-n)Cms+CnsH=×exp[∑qdq{12Nγ(q)eiqm}+∑qω(q)(dq+dq+12)]H=+∑mns{VC(m-n)-∑qω(q)12Nγ2(q)eiq(m-n)}H=×Cms+Cns+CnsCms.In obtaining (17), we have omitted the term containing the on-site interaction m=n for parallel spins.


## 3. Green’s Functions

We define the following one particle temperature electron(G) and anomalous (F) Green's functions:
(18)G(k,ωn)=-12∑m∫-ββdlelωne+ikm〈〈llCcσ(l)Cmσ+(o)〉〉,F(k,ωn)=-12∑m∫-ββdlelωne+ikm〈〈llCcσ(l)Cmσ(o)〉〉.For convenience, dropping spin and applying the Lang-Firsov canonical transformation and neglecting the residual polaron-polaron coupling and following equation of motion method for the evaluation of electron part and Feynman method for the evaluation of phonon part, one finally obtains [28]
(19)G(t)=-ieβΩelTr[e-iv(m-n)tCC+e-βH~el]G(t)=×Tr[eβΩphe-βH~phX(t)X+(0)].After evaluating the electron part and phonon part of the trace, we obtain the total Green’s function as(20)G(k,ωn)=e-q2[uk2iωn-εn+vk2iωn+εn+1N∑l=1∞g2l!lbbbbbbb×∑k′{uk′2(1-nk′)iωn-lω0-εk′+vk′2nk′iωn-lω0+εk′bbbbbbbbbbbbb+uk′2nk′iωn+lω0-εk′+vk′2(1-nk′)iωn+lω0+εk′}uk2iωn-εn+vk2iωn+εn+1N∑l=1∞g2l!l],(21)F(k,ωn)=e-q2[(1-nk′)iωn+lω0+εk/ukvk(1iωn-εk+1iωn+εk)+1N∑l=1∞(-1)lg2l!l00000x000×∑k′uk′vk′{(1-nk′)iωn-lω0-εk′-nk′iωn-lω0+εk′0000000000000x00000+nk′iωn+lω0-εk′-(1-nk′)iωn+lω0+εk′}(1iωn-εk+1iωn+εk)∑l=1∞(-1)lg2l!l],
where
(22)uk2=12(1+ξkεk),vk2=12(1-ξkεk),ukvk=-{Δ2εk},nk=n(εk).
With
(23)n(x)=(ex/kBT+1)-1,εk=ξk2+Δ2(k),Δ(k)=-12∑kV(k-k′)Δ(k′)εk′tanhεk2kBT,V(k)=1N∑mV(m)eikm.The energy dispersion for the polaronic band is given by(24)ξk={∑mσ(mo)T(m)eikm-μ}
having a narrow band half width W≪D{whereD=ZT(m)}.


## 4. Correlation Function

The correlation functions are defined as(25)〈Cp+Cp〉=12π∫-∞+∞IG(ωn)dωn,(26)〈CpCp〉=12π∫-∞+∞IF(ωn)dωn,
where
(27)IG(ωn)=i(eβωn+1)-1[G11(ωn+iε)-G11(ωn-iε)],IF(ωn)=i(eβωn+1)-1[F11(ωn+iε)-F11(ωn-iε)],
where G and F are Green functions given by (20) and (21), respectively.Using the identity(28)limε→012π[1ω+iε-Ek-1ω-iε-Ek]=iδ(ω-Ek),∫-∞∞f(ωn)δ(ωn-ω±)dωn=f(ωn±).With the following relations:(29)uk2=12(1+ξkεk),vk2=12(1-ξkεk),(25) and (26) become
(30)〈Cp+Cp〉=12+12ξkεktanhβεk2jjjjj+1N∑l=1∞‍g2l!l[12+12ξk′εk′tanhβ(lω0+εk′)2ggggjjjjjjjjggggggg-nk′2ξk′εk′{tanhβ(lω0+εk′)2ggggggggghhg000ygggggggg-tanhβ(lω0-εk′)2}],(31)〈CpCp〉=-Δ(k)2εktanhβεk2ffff+1N∑l=1∞(-1)l!lg2l∑k′Δ(k′)2εk′fffffff×[tanhβ(lω0+εk′)2vvvvvvv-nk{tanhβ(lω0+εk′)2+tanhβ(lω0-εk′)2}].


## 5. Superconducting Order Parameter(Δ)

The order parameter of a superconducting state is given by(32)Δ=g∑k〈CkCk〉.Substituting correlation function given by (31) in (32) and changing summation into integral using the following relation:
(33)∑k=N(o)∫0ℏωDdξk,the gap equation becomes(34)Δ=gN(0)∫0ℏωDdξk[-Δ(k)2ξk2+Δ2(k)tanhξk2+Δ2(k)2kBT]Δ=-g1N∑l=1∞(-1)l!lg12lN(0)∫0ℏωD∑k′dξk-Δ(k′)2ξk′′2+Δ2(k′)Δ=×[+tanh(lω0-ξk′2+Δ2(k′))2kBT}tanhlω0+ξk′′2+Δ2(k′)2kBTΔvv0nv=-1exp{ξk′′2+Δ2(k′)/2kBT}+1Δvvv0n=×{tanh(lω0+ξk′′2+Δ2(k′))2kBTΔvvv0n=+tanh(lω0-ξk′′2+Δ2(k′))2kBT}].Right-hand side of (34) has two terms which are quite independent. First term varies with k, whereas second term varies with k′; hence, one can define two superconducting order parameters for the YBa2Cu3O7-x system. The two independent terms finally yield the two equations as
(35)1gN(0)=∫0ℏωDdξk[12ξk2+Δ2(k)tanhξk2+Δ2(k)2kBT],withℓ=1 the other equation is
(36)1|g|N(0)[gl2]=∫0ℏωDdξk′2ξk′′2+Δ2(k′)[tanhℓω0+ξk′′2+Δ2(k′)2kBT,1|g|N(0)[gl2vv]bbbbbbb,=-1exp{ξk′′2+Δ2(k′)/2kBT}+11|g|N(0)[gl2]000000vv,=×{tanhℓω0+ξk′′2+Δ2(k′)2kBT1|g|N(0)[gl2]vv0000000,=+tanhℓω0-ξk′′2+Δ2(k′)2kBT}].With the help of (35) and (36) one can study the behavior of superconducting order parameters with temperature.


## 6. Physical Properties

### 6.1. Electronic Specific Heat(ces)

The electronic specific heat per atom of a superconductor is determined from the following relation [3, 4]:
(37)ces=∂∂T[1N∑k2ξk〈Ck+Ck〉],
where 〈Ck+Ck〉 is the correlation function. We have obtained this correlation function in (30). Substituting the correlation function from (30) in equation (37), One obtains
(38)ces=N(0)2N∫0ℏωD2ξkkBT2dξk00000×[-12ξksech2(εk2kBT)00000000+∑l=1∞g2l!l{-12ξk′εk′(ω0l+εk′)00000000000000000×sech2(ω0l+εk′2kBT)0000000000000b000+nk′2ξk′εk′(ω0l+εk′)00000000000000000×sech2(ω0l+εk′2kBT)00000000000000000+nk′2ξk′εk′(ω0l-εk′)00000000000000000×sech2(ω0l-εk′2kBT)}].Right-hand side of (38) has two terms which are quite independent from each other. First term varies with k, whereas second term varies with k′; hence one can study the behaviour of electronic specific heat of superconductors with temperature.


### 6.2. Density of States Function[N(ω)/N(o)]

For>0, the function can be defined as [5]
(39)N(ω)=lim12π[G11(k,ω+iη)-G11(k,ω-iη)].Using the following identity:(40)limη→012π[1ω+iη-ωn+-1ω-iη-ωn+]=iδ(ω-ωn+),
changing the summation over “K” into an integration, replacing εk by -εk, and combining the terms and using the relations
(41)uk2+vk2=12(1+ξkεk)+12(1-ξkεk)=1,one obtains(42)N(ω)N(0)=1N∑l=1l=∞g2ll∑k′{iωn-lω0{(iωn-lω0)2-Δ2}1/2N(ω)N(0)=1N∑l=1l=∞g2llvvvv-iωn+lω0{(iωn+lω0)2-Δ2}1/2}.


### 6.3. Free Energy

It is well known that free energy of normal paramagnetic phase always exceeds the free energy of superconducting diamagnetic phase. The entropy decreases remarkably on cooling the superconductors below the critical temperature. The free energy can easily be defined for the superconducting transition as it is related by the entropy; hence, it also exhibits a similar behavior [3]. Obviously the entropy as well as the free energy difference in the normal state is always greater than the entropy in the superconducting state.The free energy difference of a superconductor for its normal and superconducting state is given by the following relation [27]:
(43)Fs-FNV=∫0∞dg(1|g|2)Δ2,
where “g” is the interaction parameter and “Δ” is the superconducting order parameter. Equation (43) can also be expressed as
(44)Fs-FNV=∫0ΔdΔddΔ(1|g|2)Δ2.From superconducting order parameter expression, we have(45)Δ(k)=gN(0)∫0ℏωD[-Δ(k)2ξk2+Δ2(k)tanhξk2+Δ2(k)2kBT]dξk,1g=N(0)∫0ℏωD[12ξk2+Δ2(k)tanhξk2+Δ2(k)2kBT]dξk.Equation (44) becomes
(46)[FSNV]phonon=∫0Δ(k)Δ2(k)dΔ(k)ddΔ(k)gggfg×[N(0)∫0ℏωD{12ξk2+Δ2(k)ggggggvfvvkkk000kkkkkk×tanhξk2+Δ2(k)2kBT}dξk].Since(47)εk2=ξk2+Δ2(k),2εkdεk=2Δ(k)dΔ(k).Integrating by parts, we get(48)2[FSNV]=Δ2(k)g-N(0)2[FSNV]=×∫0ℏωD2dξk∫ξkεktanh(βεk2)dεk,2[FSNV]=Δ2(k)g-4N(0)β2[FSNV]=×∫0ℏωD[ln{eβεk/2(1+e-βεk)}{eβξk/2(1+e-βξk)}]dξk,2[FSNV]=Δ2(k)g-4N(0)β2[FSNV]=×∫0ℏωDln(eβ(εk-ξk)/2)dξk2[FSNV]=-4N(0)β∫0ℏωDln(1+e-βεk)dξk2[FSNV]=+4N(0)β∫0ℏωDln(1+e-βξk)dξk,(49)2[FSNV]phonon=(-)[N(0)Δ2(k)2-4N(0)βe-βΔ(k)2(2πΔ(k)β-1)1/22cccccc=+4N(0)β1β(π212)].With the help of (49) we can calculate the free energy difference.


### 6.4. Critical Field(Hc)

The critical field is related to the free energy difference as(50)Hc={8π(FS-FN)}1/2.Using (49), we obtain
(51)Hc={8π(N(0)Δ2(k)4-4N(0)βggggggggg×e-βΔ(k)4(2πΔ(k)β-1)1/2ggggggggg+4N(0)β1β(π212))}1/2.


## 6.1. Electronic Specific Heat(ces)

The electronic specific heat per atom of a superconductor is determined from the following relation [3, 4]:
(37)ces=∂∂T[1N∑k2ξk〈Ck+Ck〉],
where 〈Ck+Ck〉 is the correlation function. We have obtained this correlation function in (30). Substituting the correlation function from (30) in equation (37), One obtains
(38)ces=N(0)2N∫0ℏωD2ξkkBT2dξk00000×[-12ξksech2(εk2kBT)00000000+∑l=1∞g2l!l{-12ξk′εk′(ω0l+εk′)00000000000000000×sech2(ω0l+εk′2kBT)0000000000000b000+nk′2ξk′εk′(ω0l+εk′)00000000000000000×sech2(ω0l+εk′2kBT)00000000000000000+nk′2ξk′εk′(ω0l-εk′)00000000000000000×sech2(ω0l-εk′2kBT)}].Right-hand side of (38) has two terms which are quite independent from each other. First term varies with k, whereas second term varies with k′; hence one can study the behaviour of electronic specific heat of superconductors with temperature.


## 6.2. Density of States Function[N(ω)/N(o)]

For>0, the function can be defined as [5]
(39)N(ω)=lim12π[G11(k,ω+iη)-G11(k,ω-iη)].Using the following identity:(40)limη→012π[1ω+iη-ωn+-1ω-iη-ωn+]=iδ(ω-ωn+),
changing the summation over “K” into an integration, replacing εk by -εk, and combining the terms and using the relations
(41)uk2+vk2=12(1+ξkεk)+12(1-ξkεk)=1,one obtains(42)N(ω)N(0)=1N∑l=1l=∞g2ll∑k′{iωn-lω0{(iωn-lω0)2-Δ2}1/2N(ω)N(0)=1N∑l=1l=∞g2llvvvv-iωn+lω0{(iωn+lω0)2-Δ2}1/2}.


## 6.3. Free Energy

It is well known that free energy of normal paramagnetic phase always exceeds the free energy of superconducting diamagnetic phase. The entropy decreases remarkably on cooling the superconductors below the critical temperature. The free energy can easily be defined for the superconducting transition as it is related by the entropy; hence, it also exhibits a similar behavior [3]. Obviously the entropy as well as the free energy difference in the normal state is always greater than the entropy in the superconducting state.The free energy difference of a superconductor for its normal and superconducting state is given by the following relation [27]:
(43)Fs-FNV=∫0∞dg(1|g|2)Δ2,
where “g” is the interaction parameter and “Δ” is the superconducting order parameter. Equation (43) can also be expressed as
(44)Fs-FNV=∫0ΔdΔddΔ(1|g|2)Δ2.From superconducting order parameter expression, we have(45)Δ(k)=gN(0)∫0ℏωD[-Δ(k)2ξk2+Δ2(k)tanhξk2+Δ2(k)2kBT]dξk,1g=N(0)∫0ℏωD[12ξk2+Δ2(k)tanhξk2+Δ2(k)2kBT]dξk.Equation (44) becomes
(46)[FSNV]phonon=∫0Δ(k)Δ2(k)dΔ(k)ddΔ(k)gggfg×[N(0)∫0ℏωD{12ξk2+Δ2(k)ggggggvfvvkkk000kkkkkk×tanhξk2+Δ2(k)2kBT}dξk].Since(47)εk2=ξk2+Δ2(k),2εkdεk=2Δ(k)dΔ(k).Integrating by parts, we get(48)2[FSNV]=Δ2(k)g-N(0)2[FSNV]=×∫0ℏωD2dξk∫ξkεktanh(βεk2)dεk,2[FSNV]=Δ2(k)g-4N(0)β2[FSNV]=×∫0ℏωD[ln{eβεk/2(1+e-βεk)}{eβξk/2(1+e-βξk)}]dξk,2[FSNV]=Δ2(k)g-4N(0)β2[FSNV]=×∫0ℏωDln(eβ(εk-ξk)/2)dξk2[FSNV]=-4N(0)β∫0ℏωDln(1+e-βεk)dξk2[FSNV]=+4N(0)β∫0ℏωDln(1+e-βξk)dξk,(49)2[FSNV]phonon=(-)[N(0)Δ2(k)2-4N(0)βe-βΔ(k)2(2πΔ(k)β-1)1/22cccccc=+4N(0)β1β(π212)].With the help of (49) we can calculate the free energy difference.


## 6.4. Critical Field(Hc)

The critical field is related to the free energy difference as(50)Hc={8π(FS-FN)}1/2.Using (49), we obtain
(51)Hc={8π(N(0)Δ2(k)4-4N(0)βggggggggg×e-βΔ(k)4(2πΔ(k)β-1)1/2ggggggggg+4N(0)β1β(π212))}1/2.


## 7. Numerical Calculations

Now, we evaluate numerically the physical properties of high-temperature cuprate superconductorYBa2Cu3O7-x, using the various relations derived, that is, (35), (36), (38), (42), (49), and (51).Values of various parameters appearing in the mentioned relations are cited in Table1. Using these values, we have made study of various parameters related to the physical properties for the system YBa2Cu3O7-x.Table 1
Values of various parameters for HTSC cuprate superconductor forYBa2Cu3O7-x.
S. no.
Property
Value
1
Superconducting transition temperature (Tc)
88 K
2
Density of statesN(0) at the Fermi surface
4.95
×
1
0
26 per  ergs  Cu  atom
3
Phonon energyℏωD
1.3
×
1
0
-
21 J
4
Polaron frequencyωpl
0.72 eV
5
Polaron density(N(0)exp(g2))
8 states/eV spin
6
Fermi energy
0.23 eV
7
Crystal structure
Orthorhombic
8
Cell parameters
a
=
0.38 nm, b=0.39 nm, and c=1.17 nm
9
Number of atoms per unit volume
5
×
10
28/m3
10
Boltzmann constant (kB )
1.38
×
10
-
23 J/K
11
Mass of electron
9.1
×
10
-
31 kg


### 7.1. Superconducting Order Parameter(Δ)

For the study of superconducting order parameter (Δ) for the system YBa2Cu3O7-x, we have calculated the contributions due to phonons and polarons separately and also obtained the combined effect of phonons and polarons.(i) Superconducting order parameter (Δ1).(When only electron-phonon interaction is considered).We have (35)
(52)1|g|N(0)=∫0ℏωDdξk[12ξk2+Δ2(k)tanhξk2+Δ2(k)2kBT].
Solving the previous equation numerically, we get
(53)1|g|N(0)=∫y=0y=1dy[12y2+0.5917x2cccccccccccccccccccc×tanh47.099y2+0.5917x2T].With the help of the previous equation, one can study the variation of superconducting order parameterΔ1 with temperature, when only electron-phonon interaction is considered.Values of superconducting order parameter obtained at various temperatures are given in Table2, and variation of Δ1 with temperature is shown in Figure 1.Table 2
Superconducting order parameter (Δ) forYBa2Cu3O7-x system.
S. no.
Temperature (K)
Δ
1
=
x
×
10
-
21 J(when only electron-phonon interaction is present)
Δ
2
=
x
×
1
0
-
21 J(when only polaron  interaction is considered)
Δ
=
Δ
1
+
Δ
2
(
x
×
1
0
-
21
J
) (in   the   presence   of    both  phonon  and  polaron  interactions)
1
5
2.4869
0.114015
2.600965
2
10
2.4869
0.114015
2.600965
3
15
2.4867
0.114015
2.600795
4
20
2.4865
0.114015
2.600515
5
25
2.4838
0.114015
2.597865
6
30
2.4761
0.114015
2.590145
7
35
2.4600
0.114015
2.574075
8
40
2.4325
0.114015
2.546595
9
45
2.3914
0.114015
2.505415
10
50
2.3340
0.114001
2.448001
11
55
2.2580
0.113991
2.371991
12
60
2.1608
0.113970
2.274770
13
65
2.0382
0.113914
2.152114
14
70
1.8851
0.113823
1.998923
15
75
1.6932
0.113683
1.806883
16
80
1.4457
0.113468
1.559168
17
81
1.3868
0.113425
1.500225
18
82
1.3242
0.113370
1.437570
19
83
1.2571
0.113300
1.370450
20
84
1.1852
0.113235
1.298435
21
85
1.1060
0.113177
1.219177
22
86
1.0190
0.113100
1.132100
23
87
0.9228
0.113030
1.035830
24
87.9
0.8245
0.112970
0.937470Figure 1
Behaviour of superconducting order parameter for the systemYBa2Cu3O7-x.(ii) Superconducting order parameter (Δ2).(When only polaron interaction is considered)We have (36)
(54)1gN(0)[gl2]=∫0ℏωDdξk′2ξk′′2+Δ2(k′)1gN(0)[gl2]nv=×[tanhℓω0+ξk′′2+Δ2(k′)2kBT1gN(0)[gl2]hcncvc=-1exp{ξk′′2+Δ2(k′)/kBT}+11gN(0)[gl2]ccchnv=×{tanhℓω0+ξk′′2+Δ2(k′)2kBT1gN(0)[gl2]cccnvh=jjjjj+tanhℓω0-ξk′′2+Δ2(k′)2kBT}].Solving the previous equation numerically, we get(55)1|g|N(0)[gl2]=∫y=0y=1dy2y2+0.5917x2bbbbbb×[tanh36.23(8+1.3y2+0.5917x2)Tbbbbbbbbb-1exp{94.2y2+0.5917x2/T}+1bbbbbbbbb×{tanh36.23(8+1.3y2+0.5917x2)Tbbbbbbbbb+tanh36.23(8-1.3y2+0.5917x2)T}].With the help of the previous equation, one can study the variation of superconducting order parameter (Δ2) with temperature, when only polaron interaction is considered.Values of superconducting order parameter obtained at various temperatures are given in Table2, and variation of Δ2 with temperature is shown in Figure 1.(iii) Superconducting order parameter (Δ=Δ1+Δ2)(In the presence of combined phonon and polaron interactions).The superconducting order parameter in the presence of both phonon and polaron interactions can be studied by taking a sum of the order parameters due to phonon and polaron effects. Values of order parameters obtained at various temperatures are given in Table2.The behaviour of superconducting order parameter (Δ=Δ1+Δ2) (combined phonon and polaron interactions) is shown in Figure 1.


### 7.2. Electronic Specific Heat(Ces)

We have obtained the expression (38) for electronic specific heat putting
(56)εk={ξ2+Δ2}1/2,Δ=x×10-21,N(0)=0.5/eV,ℏωD≈1.3×10-21J,ℏωD2KBT=47.099/T.Equation (38) reduces to
(57)Ces=2898.5×1.69×10-49Ces=×[∫y=0y=1y2T2dy{1eQ+1-1.3(sechQ)2Cesfffffffffffffff0ff=××-SP(sechL)2+1eQ+1SP(sechL)2Ces=×fff0fff0ff×-1eQ+1MP(sechR)2}∫y=0y=1y2T2dy{1eQ+1-1.3(sechQ)2],
where
(58)P=[y2+0.5917x2]1/2,Q=47.099TP,S=8+1.3P,M=8-1.3P,L=36.23TS,R=36.23TM.One can study the behaviour of electronic specific heat(Ces) with temperature (T) with the help of (57). Values of Ces at various temperatures obtained from (57) are given in Table 3, and variation of Ces with T is shown in Figure 2 Variation of Ces/T with T (Table 4) and (C-Ces)/T (Table 5) for a particular range of temperature is shown in Figures 3 and 4, respectively. A comparison with available experimental data is also shown in Figures 3 and 4 [29]. Agreement between theory and experimental results is quite encouraging.Table 3
Electronic specific heat (Ces) for YBa2Cu3O7-x system.
S. no.
Temperature (K)
C
es
×
1
0
-
49 Joule/mole-K
1
87.9
127.5236
2
87
125.044
3
86
122.4406
4
85
119.9085
5
84
117.4477
6
83
115.0882
7
82
112.7512
8
81
110.4441
9
80
108.1521
10
75
96.97717
11
70
85.96727
12
65
74.92361
13
60
63.85744
14
55
52.85128
15
50
42.01019
16
45
31.58173
17
40
21.92602
18
35
13.50073
19
30
6.891036
20
25
2.565851
21
20
0.536428
22
15
0.033761
23
10
0.000000
24
5
0.000000Table 4
C
es/T  for YBa2Cu3O7-x system.
S. no.
Temperature (K)
C
es/T  Joule/mole-K2 (Theory)
C
es/T  Joule/mole-K2 (Experimental)
1
80
1.3510
1.3520
2
81
1.3635
1.3591
3
82
1.3749
1.3650
4
83
1.3866
1.3732
5
84
1.3980
1.3803
6
85
1.4106
1.3851
7
86
1.4236
1.3945
8
87
1.4372
1.4063
9
87.9
1.4508
1.4170Table 5
(
C
-
C
es
)
/
T for YBa2Cu3O7-x system.
S. no.
Temperature (K)
(
C
-
C
es
)
/
T Joule/mole-K2 (Theory)
(
C
-
C
es
)
/
T Joule/mole-K2 (Experimental)
1
4
32.943048
8.542
2
5
31.4759423
8.33
3
6
30.4582297
7.92
4
7
29.9338593
7.25
5
8
29.6793776
7.083
6
9
29.5322975
6.830
7
10
29.398128
6.670
8
11
29.2235587
6.670
9
12
28.9786723
6.670
10
13
28.6485315
6.670
11
14
28.2270186
6.670
12
15
27.7163961
6.670Figure 2
Variation ofCes with temperature for the system YBa2Cu3O7-x.Figure 3
Variation ofCes/T with temperature.Figure 4
Variation of (C-Ces)/T with temperature.


### 7.3. Density of States FunctionN(ω)/N(0)

Density of states function for the polaron case is given by(59)N(ω)N(0)=1N∑l=1l=∞g2ll∑k′{iωn-lω0{(iωn-lω0)2-Δ2}1/2N(ω)N(0)=1N∑l=1l=∞g2llfxfffff-iωn+lω0{(iωn+lω0)2-Δ2}1/2}.Using the following values(60)ωn=y×10-21J,Δ=x×10-21J,l=1,ω0=8×10-21J,g2=1.The previous equation reduces as(61)N(ω)N(0)=(y-8)(y-8)2-x2-(y+8)(y+8)2-x2.The values of density of statesN(ω)/N(0) for the chosen value of x= 1.106, 1.8851, 2.334 and 2.4761 are shown in Table 6. The variation of density of states with ω at different temperatures is shown in Figure 5.Table 6
Density of statesN(ω)/N(0) at various temperatures for YBa2Cu3O7-x system.
S. no.
Frequency(y)
Density of states
T
=
30 Kx=2.4761
T
=
50 Kx=2.334
T
=
70 Kx=1.8851
T
=
85 Kx=1.106
1
11.5
0.47791324
0.39120977
0.212929844
0.06485477
2
12.0
0.303783239
0.256874065
0.149461031
0.04809573
3
12.5
0.214803743
0.184567166
0.11137659
0.03708042
4
13.0
0.161399908
0.139994146
0.0864163
0.02943323
5
13.5
0.126234662
0.110167093
0.0690518
0.02390096
6
14.0
0.101617339
0.089063119
0.056435285
0.01976737
7
14.5
0.083611432
0.073510888
0.046958665
0.01659734
8
15.0
0.069997004
0.061686732
0.03965028
0.01411343
9
15.5
0.059430763
0.052471393
0.033891537
0.01213165
10
16.0
0.051054964
0.045142493
0.029271859
0.01052594
11
16.5
0.044298094
0.039214703
0.025509277
0.00920749
12
17.0
0.038765958
0.034351053
0.022404542
0.00811226
13
17.5
0.034178738
0.030311067
0.019813414
0.0071931
14
18.0
0.030332928
0.026919081
0.017629272
0.00641463Figure 5
Variation of density of states with frequency(ω) at different temperatures for the system YBa2Cu3O7-x.


### 7.4. Free Energy Difference

The expression for the free energy difference can be expressed as(62)2[FSNV]phonon=(-)[N(0)Δ2(k)2-4N(0)βbbbbbbbbbbTbbnbbbb×e-βΔ(k)2(2πΔ(k)β-1)1/2bbbbbbKbbbbbTnbbbb+4N(0)β1β(π212)].Solving numerically,(63)2[FSNV]phonon=(-)0.02475x22[FSNV]phonon=×[1--0.01625[Tx]1/2e-72.46x/Tggggggggg0gggg+12.517×10-4[Tx]2].The values of free energy difference at different temperatures are given in Table7, and behaviour is shown in Figure 6, respectively.Table 7
Free energy difference   forYBa2Cu3O7-x system.
S. no.
Temperature (K)
Free energy differenceJoule/mole
1
5
0.1682
2
10
0.1705
3
15
0.1744
4
20
0.1798
5
25
0.1864
6
30
0.1937
7
35
0.2013
8
40
0.2085
9
45
0.2147
10
50
0.2194
11
55
0.2219
12
60
0.2215
13
65
0.2174
14
70
0.2086
15
75
0.1938
16
80
0.1709
17
85
0.1375
18
86
0.1294
19
87
0.1209
20
87.9
0.1129Figure 6
Variation of free energy difference with temperature for the systemYBa2Cu3O7-x.


### 7.5. Critical Field(Hc)

The low temperature critical field is given as(64)Hc={8π(N(0)Δ2(K)4-4N(0)βVVVvvvVV×e-βΔ(K)4(2πΔ(K)β)1/2VVVvvvVV+4N(0)β1β(π212))}1/2.Solving the previous equation numerically, we can obtain the values of low temperature critical field which are described in Table8. The behavior of the low temperature critical field (Hc) with temperature is shown in Figure 7.Table 8
Critical field   forYBa2Cu3O7-x system.
S. no.
Temperature (K)
Critical fieldHC Tesla
1
5
1.454092
2
10
1.46410
3
15
1.480537
4
20
1.503205
5
25
1.530513
6
30
1.560474
7
35
1.590674
8
40
1.618762
9
45
1.642761
10
50
1.660539
11
55
1.669980
12
60
1.668698
13
65
1.653269
14
70
1.619429
15
75
1.560715
16
80
1.465676
17
81
1.440685
18
82
1.413422
19
83
1.383557
20
84
1.351013
21
85
1.314800
22
86
1.275193
23
87
1.232573
24
87.9
1.191492Figure 7
Variation of critical field with temperature for the systemYBa2Cu3O7-x.


## 7.1. Superconducting Order Parameter(Δ)

For the study of superconducting order parameter (Δ) for the system YBa2Cu3O7-x, we have calculated the contributions due to phonons and polarons separately and also obtained the combined effect of phonons and polarons.(i) Superconducting order parameter (Δ1).(When only electron-phonon interaction is considered).We have (35)
(52)1|g|N(0)=∫0ℏωDdξk[12ξk2+Δ2(k)tanhξk2+Δ2(k)2kBT].
Solving the previous equation numerically, we get
(53)1|g|N(0)=∫y=0y=1dy[12y2+0.5917x2cccccccccccccccccccc×tanh47.099y2+0.5917x2T].With the help of the previous equation, one can study the variation of superconducting order parameterΔ1 with temperature, when only electron-phonon interaction is considered.Values of superconducting order parameter obtained at various temperatures are given in Table2, and variation of Δ1 with temperature is shown in Figure 1.Table 2
Superconducting order parameter (Δ) forYBa2Cu3O7-x system.
S. no.
Temperature (K)
Δ
1
=
x
×
10
-
21 J(when only electron-phonon interaction is present)
Δ
2
=
x
×
1
0
-
21 J(when only polaron  interaction is considered)
Δ
=
Δ
1
+
Δ
2
(
x
×
1
0
-
21
J
) (in   the   presence   of    both  phonon  and  polaron  interactions)
1
5
2.4869
0.114015
2.600965
2
10
2.4869
0.114015
2.600965
3
15
2.4867
0.114015
2.600795
4
20
2.4865
0.114015
2.600515
5
25
2.4838
0.114015
2.597865
6
30
2.4761
0.114015
2.590145
7
35
2.4600
0.114015
2.574075
8
40
2.4325
0.114015
2.546595
9
45
2.3914
0.114015
2.505415
10
50
2.3340
0.114001
2.448001
11
55
2.2580
0.113991
2.371991
12
60
2.1608
0.113970
2.274770
13
65
2.0382
0.113914
2.152114
14
70
1.8851
0.113823
1.998923
15
75
1.6932
0.113683
1.806883
16
80
1.4457
0.113468
1.559168
17
81
1.3868
0.113425
1.500225
18
82
1.3242
0.113370
1.437570
19
83
1.2571
0.113300
1.370450
20
84
1.1852
0.113235
1.298435
21
85
1.1060
0.113177
1.219177
22
86
1.0190
0.113100
1.132100
23
87
0.9228
0.113030
1.035830
24
87.9
0.8245
0.112970
0.937470Figure 1
Behaviour of superconducting order parameter for the systemYBa2Cu3O7-x.(ii) Superconducting order parameter (Δ2).(When only polaron interaction is considered)We have (36)
(54)1gN(0)[gl2]=∫0ℏωDdξk′2ξk′′2+Δ2(k′)1gN(0)[gl2]nv=×[tanhℓω0+ξk′′2+Δ2(k′)2kBT1gN(0)[gl2]hcncvc=-1exp{ξk′′2+Δ2(k′)/kBT}+11gN(0)[gl2]ccchnv=×{tanhℓω0+ξk′′2+Δ2(k′)2kBT1gN(0)[gl2]cccnvh=jjjjj+tanhℓω0-ξk′′2+Δ2(k′)2kBT}].Solving the previous equation numerically, we get(55)1|g|N(0)[gl2]=∫y=0y=1dy2y2+0.5917x2bbbbbb×[tanh36.23(8+1.3y2+0.5917x2)Tbbbbbbbbb-1exp{94.2y2+0.5917x2/T}+1bbbbbbbbb×{tanh36.23(8+1.3y2+0.5917x2)Tbbbbbbbbb+tanh36.23(8-1.3y2+0.5917x2)T}].With the help of the previous equation, one can study the variation of superconducting order parameter (Δ2) with temperature, when only polaron interaction is considered.Values of superconducting order parameter obtained at various temperatures are given in Table2, and variation of Δ2 with temperature is shown in Figure 1.(iii) Superconducting order parameter (Δ=Δ1+Δ2)(In the presence of combined phonon and polaron interactions).The superconducting order parameter in the presence of both phonon and polaron interactions can be studied by taking a sum of the order parameters due to phonon and polaron effects. Values of order parameters obtained at various temperatures are given in Table2.The behaviour of superconducting order parameter (Δ=Δ1+Δ2) (combined phonon and polaron interactions) is shown in Figure 1.


## 7.2. Electronic Specific Heat(Ces)

We have obtained the expression (38) for electronic specific heat putting
(56)εk={ξ2+Δ2}1/2,Δ=x×10-21,N(0)=0.5/eV,ℏωD≈1.3×10-21J,ℏωD2KBT=47.099/T.Equation (38) reduces to
(57)Ces=2898.5×1.69×10-49Ces=×[∫y=0y=1y2T2dy{1eQ+1-1.3(sechQ)2Cesfffffffffffffff0ff=××-SP(sechL)2+1eQ+1SP(sechL)2Ces=×fff0fff0ff×-1eQ+1MP(sechR)2}∫y=0y=1y2T2dy{1eQ+1-1.3(sechQ)2],
where
(58)P=[y2+0.5917x2]1/2,Q=47.099TP,S=8+1.3P,M=8-1.3P,L=36.23TS,R=36.23TM.One can study the behaviour of electronic specific heat(Ces) with temperature (T) with the help of (57). Values of Ces at various temperatures obtained from (57) are given in Table 3, and variation of Ces with T is shown in Figure 2 Variation of Ces/T with T (Table 4) and (C-Ces)/T (Table 5) for a particular range of temperature is shown in Figures 3 and 4, respectively. A comparison with available experimental data is also shown in Figures 3 and 4 [29]. Agreement between theory and experimental results is quite encouraging.Table 3
Electronic specific heat (Ces) for YBa2Cu3O7-x system.
S. no.
Temperature (K)
C
es
×
1
0
-
49 Joule/mole-K
1
87.9
127.5236
2
87
125.044
3
86
122.4406
4
85
119.9085
5
84
117.4477
6
83
115.0882
7
82
112.7512
8
81
110.4441
9
80
108.1521
10
75
96.97717
11
70
85.96727
12
65
74.92361
13
60
63.85744
14
55
52.85128
15
50
42.01019
16
45
31.58173
17
40
21.92602
18
35
13.50073
19
30
6.891036
20
25
2.565851
21
20
0.536428
22
15
0.033761
23
10
0.000000
24
5
0.000000Table 4
C
es/T  for YBa2Cu3O7-x system.
S. no.
Temperature (K)
C
es/T  Joule/mole-K2 (Theory)
C
es/T  Joule/mole-K2 (Experimental)
1
80
1.3510
1.3520
2
81
1.3635
1.3591
3
82
1.3749
1.3650
4
83
1.3866
1.3732
5
84
1.3980
1.3803
6
85
1.4106
1.3851
7
86
1.4236
1.3945
8
87
1.4372
1.4063
9
87.9
1.4508
1.4170Table 5
(
C
-
C
es
)
/
T for YBa2Cu3O7-x system.
S. no.
Temperature (K)
(
C
-
C
es
)
/
T Joule/mole-K2 (Theory)
(
C
-
C
es
)
/
T Joule/mole-K2 (Experimental)
1
4
32.943048
8.542
2
5
31.4759423
8.33
3
6
30.4582297
7.92
4
7
29.9338593
7.25
5
8
29.6793776
7.083
6
9
29.5322975
6.830
7
10
29.398128
6.670
8
11
29.2235587
6.670
9
12
28.9786723
6.670
10
13
28.6485315
6.670
11
14
28.2270186
6.670
12
15
27.7163961
6.670Figure 2
Variation ofCes with temperature for the system YBa2Cu3O7-x.Figure 3
Variation ofCes/T with temperature.Figure 4
Variation of (C-Ces)/T with temperature.


## 7.3. Density of States FunctionN(ω)/N(0)

Density of states function for the polaron case is given by(59)N(ω)N(0)=1N∑l=1l=∞g2ll∑k′{iωn-lω0{(iωn-lω0)2-Δ2}1/2N(ω)N(0)=1N∑l=1l=∞g2llfxfffff-iωn+lω0{(iωn+lω0)2-Δ2}1/2}.Using the following values(60)ωn=y×10-21J,Δ=x×10-21J,l=1,ω0=8×10-21J,g2=1.The previous equation reduces as(61)N(ω)N(0)=(y-8)(y-8)2-x2-(y+8)(y+8)2-x2.The values of density of statesN(ω)/N(0) for the chosen value of x= 1.106, 1.8851, 2.334 and 2.4761 are shown in Table 6. The variation of density of states with ω at different temperatures is shown in Figure 5.Table 6
Density of statesN(ω)/N(0) at various temperatures for YBa2Cu3O7-x system.
S. no.
Frequency(y)
Density of states
T
=
30 Kx=2.4761
T
=
50 Kx=2.334
T
=
70 Kx=1.8851
T
=
85 Kx=1.106
1
11.5
0.47791324
0.39120977
0.212929844
0.06485477
2
12.0
0.303783239
0.256874065
0.149461031
0.04809573
3
12.5
0.214803743
0.184567166
0.11137659
0.03708042
4
13.0
0.161399908
0.139994146
0.0864163
0.02943323
5
13.5
0.126234662
0.110167093
0.0690518
0.02390096
6
14.0
0.101617339
0.089063119
0.056435285
0.01976737
7
14.5
0.083611432
0.073510888
0.046958665
0.01659734
8
15.0
0.069997004
0.061686732
0.03965028
0.01411343
9
15.5
0.059430763
0.052471393
0.033891537
0.01213165
10
16.0
0.051054964
0.045142493
0.029271859
0.01052594
11
16.5
0.044298094
0.039214703
0.025509277
0.00920749
12
17.0
0.038765958
0.034351053
0.022404542
0.00811226
13
17.5
0.034178738
0.030311067
0.019813414
0.0071931
14
18.0
0.030332928
0.026919081
0.017629272
0.00641463Figure 5
Variation of density of states with frequency(ω) at different temperatures for the system YBa2Cu3O7-x.


## 7.4. Free Energy Difference

The expression for the free energy difference can be expressed as(62)2[FSNV]phonon=(-)[N(0)Δ2(k)2-4N(0)βbbbbbbbbbbTbbnbbbb×e-βΔ(k)2(2πΔ(k)β-1)1/2bbbbbbKbbbbbTnbbbb+4N(0)β1β(π212)].Solving numerically,(63)2[FSNV]phonon=(-)0.02475x22[FSNV]phonon=×[1--0.01625[Tx]1/2e-72.46x/Tggggggggg0gggg+12.517×10-4[Tx]2].The values of free energy difference at different temperatures are given in Table7, and behaviour is shown in Figure 6, respectively.Table 7
Free energy difference   forYBa2Cu3O7-x system.
S. no.
Temperature (K)
Free energy differenceJoule/mole
1
5
0.1682
2
10
0.1705
3
15
0.1744
4
20
0.1798
5
25
0.1864
6
30
0.1937
7
35
0.2013
8
40
0.2085
9
45
0.2147
10
50
0.2194
11
55
0.2219
12
60
0.2215
13
65
0.2174
14
70
0.2086
15
75
0.1938
16
80
0.1709
17
85
0.1375
18
86
0.1294
19
87
0.1209
20
87.9
0.1129Figure 6
Variation of free energy difference with temperature for the systemYBa2Cu3O7-x.


## 7.5. Critical Field(Hc)

The low temperature critical field is given as(64)Hc={8π(N(0)Δ2(K)4-4N(0)βVVVvvvVV×e-βΔ(K)4(2πΔ(K)β)1/2VVVvvvVV+4N(0)β1β(π212))}1/2.Solving the previous equation numerically, we can obtain the values of low temperature critical field which are described in Table8. The behavior of the low temperature critical field (Hc) with temperature is shown in Figure 7.Table 8
Critical field   forYBa2Cu3O7-x system.
S. no.
Temperature (K)
Critical fieldHC Tesla
1
5
1.454092
2
10
1.46410
3
15
1.480537
4
20
1.503205
5
25
1.530513
6
30
1.560474
7
35
1.590674
8
40
1.618762
9
45
1.642761
10
50
1.660539
11
55
1.669980
12
60
1.668698
13
65
1.653269
14
70
1.619429
15
75
1.560715
16
80
1.465676
17
81
1.440685
18
82
1.413422
19
83
1.383557
20
84
1.351013
21
85
1.314800
22
86
1.275193
23
87
1.232573
24
87.9
1.191492Figure 7
Variation of critical field with temperature for the systemYBa2Cu3O7-x.


## 8. Discussion and Conclusions

In the foregoing sections, we have developed the polaronic pairing mechanism of HTSC in cuprates and applied it to study physical properties ofYBa2Cu3O7-x. The idea of polaron is based on the assumption about the autolocalization of an electron in the ion crystal due to its interaction with longitudinal optical vibrations under the local polarization, which is caused by the electron itself. The electron is confined to the local-polarization-induced potential well and conserves it by its own field [30].Following Green’s function technique and equation of motion method, we have obtained expressions for superconducting order parameter. We found that phonon contribution is small enough.TC obtained for the system YBa2Cu3O7-x is 88 K which is in a good agreement with experiments [1, 31]. Making use of various parameters given in Table 1, we have closely studied the density of states at various temperatures, specific heat, free energy, and critical field.The specific heat behaviour with temperature is reasonably in a good agreement with experimental data [32].The study of density of states, free energy, and critical field provides great insight into the physics of high-TC cuprates. In the absence of experimental results, we could not compare our results with experiments.The investigations reported here aim to establish several of the properties of polaronic pairing mechanism of HTSC cuprates. Only a detailed comparison with future experiments will clarify whether this pairing mechanism is or not a good approximation to describe HTSC cuprates.



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*Source: 581025-2013-07-30.xml*