Polaronic Mechanism of Superconductivity in Cuprates

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 YBa₂Cu₃O_7−𝑥. There is convincing evidence that the theory is fully compatible with the key experiments.

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 𝑇_𝐶 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] 𝐻=𝐻_𝑒+𝐻_ph+𝐻_𝑒−ph+𝐻_𝑒𝑒, (1) where 𝐻_𝑒 is the kinetic energy in the initial Bloch band, 𝐻_ph is the vibration energy of the lattice, 𝐻_𝑒−ph is electron-phonon interaction, and 𝐻_𝑒𝑒 is the Coulomb electron-electron correlations.

In one band approximation 𝐻_𝑒 has the form 𝐻_𝑒=_𝑘𝑠𝐸(𝑘)𝐶⁺_𝑘𝑠𝐶_𝑘𝑠, (2) where 𝑘 and 𝑠 denote the state with quasi momentum and spin, respectively. 𝐸(𝑘) is bare band energy. 𝐻_phcan be expressed in terms of phonon operators 𝑑_𝑞; 𝑞=(𝑞,𝜈) where 𝜈 is the type of vibrational mode_.𝐻_ph=_𝑞𝜔𝑞𝑑⁺_𝑞𝑑_𝑞, (3) where 𝜔 is the phonon dispersion.

The electron-phonon interaction is described by the Fröhlich Hamiltonian: 𝐻_𝑒−ph=_{𝑘,𝑞,𝑠}𝜔𝑞𝛾𝑞1√𝐶2𝑁^{+𝑘+𝑞,𝑠}𝐶_𝑘,𝑠𝑑_𝑞+𝐻.𝐶. (4) 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 𝛾²𝑞=4𝜋𝑒²𝑞²Ω𝜔₀1𝜀_∞−1𝜀₀. (5)

In the case of optical longitudinal phonon with frequency 𝜔(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 𝜈²𝑞=𝐸²_𝐷𝑞,𝜔𝑞𝑢𝑀=𝜔₀, (6) where 𝐸_𝐷 is the deformation potential, 𝑢 is the sound velocity, and 𝑀 is the mass of an elementary cell. For intermolecular phonons 𝛾²𝑞=𝛾²(0). (7)

The combined Hamiltonian can be expressed as 𝐻=_𝑘𝑠𝐸(𝑘)𝐶⁺_𝑘𝑠𝐶_𝑘𝑠+_𝑞𝜔𝑞𝑑⁺_𝑞𝑑_𝑞+_{𝑘,𝑞,𝑠}𝜔𝑞𝛾𝑞𝑑_𝑞1√𝐶2𝑁^+𝑘+𝑞𝐶_𝑘,𝑠+𝐻.𝐶.+𝑉_𝐶. (8)

Here 𝑉_𝐶 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 𝐻=_𝑚𝑛𝑇(𝑚−𝑛)𝐶⁺_𝑚𝜎𝐶_𝑛𝜎+_𝑞𝜔𝑞𝑑⁺_𝑞𝑑_𝑞+12+1√2𝑁_𝑚𝑞𝜔𝑞𝛾𝑞𝑛_𝑚𝜎𝑑_𝑞𝑒^𝑖𝑞𝑚+𝑑⁺_𝑞𝑒^{−𝑖𝑞𝑚}+12_{𝑚𝑛𝜎𝜎′}𝑉_𝐶(𝑚−𝑛)𝑛⁺_𝑚𝜎𝑛_𝑛𝜎′+_𝑞𝜔𝑞𝑑⁺_𝑞𝑑_𝑞+12, (9) where 𝑇(𝑚)=1𝑁_𝑘𝐸(𝑘)𝑒^𝑖𝑘𝑚,𝑛_𝑚𝜎=𝐶⁺_𝑚𝜎𝐶_𝑚𝜎. (10)

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 𝐻=𝑒^𝑆𝐻𝑒^−𝑆, (11) where 𝑆=_𝑚𝑠𝐶⁺_𝑚𝑠𝐶_𝑚𝑠1√𝛾𝑞𝑑2𝑁^𝑞𝑒^𝑖𝑞𝑚−𝑑⁺_𝑞𝑒^{−𝑖𝑞𝑚}. (12)

The electron operator transforms as 𝑑_𝑞=𝑑_𝑞′+𝑆,𝑑_𝑞′12𝑆,𝑆,𝑑_𝑞′+⋯,𝑠,𝑑_𝑞′=_𝑚𝑠𝐶⁺_𝑚𝑠𝐶_𝑚𝑠1√𝛾𝑞𝑑2𝑁^𝑞𝑒^𝑖𝑞𝑚−𝑑⁺_𝑞𝑒^{−𝑖𝑞𝑚},𝑑_𝑞′,𝑠,𝑑_𝑞′=1√𝛾𝑞×2𝑁_𝑞𝑚𝑠𝐶⁺_𝑚𝑠𝐶_𝑚𝑠𝑑_𝑞𝑒^𝑖𝑞𝑚𝑑_𝑞′−𝑑⁺_𝑞𝑒^{−𝑖𝑞𝑚}𝑑_𝑞′−𝑑_𝑞′𝑑_𝑞𝑒^𝑖𝑞𝑚+𝑑_𝑞′𝑑⁺_𝑞𝑒^{−𝑖𝑞𝑚}. (13)

Using 𝑑_𝑞′𝑑⁺_𝑞=𝑑⁺_𝑞𝑑_𝑞′+𝛿_𝑞𝑞′𝑠,𝑑_𝑞′=1√𝛾𝑞2𝑁_𝑞𝑚𝑠𝐶⁺_𝑚𝑠𝐶_𝑚𝑠𝛿_𝑞𝑞′𝑒^{−𝑖𝑞𝑚}. (14)

When 𝑞=𝑞^′ we have 𝑠,𝑑_𝑞′=1√𝛾𝑞2𝑁_𝑚𝑠𝐶⁺_𝑚𝑠𝐶_𝑚𝑠𝑒^{−𝑖𝑞′𝑚}𝑠,𝑠,𝑑_𝑞′=0. (15)

Thus, 𝑑_𝑞′=𝑑_𝑞′+1√𝛾𝑞2𝑁_𝑚𝑠𝐶⁺_𝑚𝑠𝐶_𝑚𝑠𝑒^{−𝑖𝑞′𝑚},𝑑⁺_𝑞′=𝑑⁺_𝑞′−1√𝛾𝑞2𝑁_𝑚𝑠𝐶⁺_𝑚𝑠𝐶_𝑚𝑠𝑒^{−𝑖𝑞′𝑚}. (16)

Hence 𝐻=_{𝑚𝑛𝑠,𝑚≠𝑛}𝑇(𝑚−𝑛)𝐶⁺_𝑚𝑠𝐶_𝑛𝑠×exp_𝑞𝑑_𝑞1√𝛾𝑞𝑒2𝑁^𝑖𝑞𝑚+_𝑞𝜔𝑞𝑑⁺_𝑞𝑑_𝑞+12+_𝑚𝑛𝑠𝑉_𝐶(𝑚−𝑛)−_𝑞𝜔𝑞1𝛾2𝑁²𝑞𝑒^{𝑖𝑞(𝑚−𝑛)}×𝐶⁺_𝑚𝑠𝐶⁺_𝑛𝑠𝐶_𝑛𝑠𝐶_𝑚𝑠. (17)

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: 𝐺𝑘,𝜔_𝑛1=−2_𝑚^𝛽_−𝛽d𝑙𝑒^𝑙𝜔𝑛𝑒^{+𝑖𝑘𝑚}𝑙_𝑙𝐶_𝑐𝜎(𝑙)𝐶⁺_𝑚𝜎(𝑜,𝐹)𝑘,𝜔_𝑛1=−2_𝑚^𝛽_−𝛽d𝑙𝑒^𝑙𝜔𝑛𝑒^{+𝑖𝑘𝑚}𝑙_𝑙𝐶_𝑐𝜎(𝑙)𝐶_𝑚𝜎(𝑜). (18)

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] 𝐺(𝑡)=−𝑖𝑒^{𝛽Ω𝑒𝑙}Tr𝑒^{−𝑖𝑣(𝑚−𝑛)𝑡}𝐶𝐶⁺𝑒^−𝛽𝐻_𝑒𝑙×Tr𝑒^{𝛽Ω𝑝ℎ}𝑒^−𝛽𝐻_ph𝑋(𝑡)𝑋⁺(0). (19)

After evaluating the electron part and phonon part of the trace, we obtain the total Green’s function as 𝐺𝑘,𝜔_𝑛=𝑒^−𝑞2𝑢²_𝑘𝑖𝜔_𝑛−𝜀_𝑛+𝑣²_𝑘𝑖𝜔_𝑛+𝜀_𝑛+1𝑁^∞_𝑙=1𝑔^2𝑙!𝑙×_𝑘′𝑢²_𝑘′1−𝑛_𝑘′𝑖𝜔_𝑛−𝑙𝜔₀−𝜀_𝑘′+𝑣²_𝑘′𝑛_𝑘′𝑖𝜔_𝑛−𝑙𝜔₀+𝜀_𝑘′+𝑢²_𝑘′𝑛_𝑘′𝑖𝜔_𝑛+𝑙𝜔₀−𝜀_𝑘′+𝑣²_𝑘′1−𝑛_𝑘′𝑖𝜔_𝑛+𝑙𝜔₀+𝜀_𝑘′,𝐹𝑘,𝜔_𝑛=𝑒^−𝑞2𝑢_𝑘𝑣_𝑘1𝑖𝜔_𝑛−𝜀_𝑘+1𝑖𝜔_𝑛+𝜀_𝑘+1𝑁^∞_𝑙=1(−1)^𝑙𝑔^2𝑙!𝑙×_𝑘′𝑢_𝑘′𝑣_𝑘′1−𝑛_𝑘′𝑖𝜔_𝑛−𝑙𝜔₀−𝜀_𝑘′−𝑛_𝑘′𝑖𝜔_𝑛−𝑙𝜔₀+𝜀_𝑘′+𝑛_𝑘′𝑖𝜔_𝑛+𝑙𝜔₀−𝜀_𝑘′−1−𝑛_𝑘′𝑖𝜔_𝑛+𝑙𝜔₀+𝜀_𝑘′, (20)(21) where 𝑢²_𝑘=12𝜉1+^𝑘𝜀_𝑘,𝑣²_𝑘=12𝜉1−^𝑘𝜀_𝑘,𝑢_𝑘𝑣_𝑘Δ=−2𝜀_𝑘,𝑛_𝑘=𝑛𝜀_𝑘. (22) With 𝑛(𝑥)=𝑒^{𝑥/𝑘𝐵𝑇}+1⁻¹,𝜀_𝑘=𝜉²_𝑘+Δ²(𝑘),Δ(𝑘)1=−2_𝑘𝑉𝑘−𝑘^′Δ𝑘^′𝜀_𝑘′𝜀tanh^𝑘2𝑘_𝐵𝑇,𝑉(𝑘)=1𝑁_𝑚𝑉(𝑚)𝑒^𝑖𝑘𝑚. (23)

The energy dispersion for the polaronic band is given by 𝜉_𝑘=_𝑚𝜎(𝑚𝑜)𝑇(𝑚)𝑒^𝑖𝑘𝑚−𝜇 (24) having a narrow band half width 𝑊≪𝐷{where𝐷=𝑍𝑇(𝑚)}.

4. Correlation Function

The correlation functions are defined as 𝐶⁺_𝑝𝐶_𝑝=12𝜋^+∞_−∞𝐼_𝐺𝜔_𝑛𝑑𝜔_𝑛,𝐶_𝑝𝐶_𝑝=12𝜋^+∞_−∞𝐼_𝐹𝜔_𝑛𝑑𝜔_𝑛, (25)(26) where 𝐼_𝐺𝜔_𝑛=𝑖𝑒^𝛽𝜔𝑛+1⁻¹𝐺₁₁𝜔_𝑛+𝑖𝜀−𝐺₁₁𝜔_𝑛,𝐼−𝑖𝜀_𝐹𝜔_𝑛=𝑖𝑒^𝛽𝜔𝑛+1⁻¹𝐹₁₁𝜔_𝑛+𝑖𝜀−𝐹₁₁𝜔_𝑛−𝑖𝜀, (27) where 𝐺 and 𝐹 are Green functions given by (20) and (21), respectively.

Using the identity lim_𝜀→0112𝜋𝜔+𝑖𝜀−𝐸_𝑘−1𝜔−𝑖𝜀−𝐸_𝑘=𝑖𝛿𝜔−𝐸_𝑘,_∞−∞𝑓𝜔_𝑛𝛿𝜔_𝑛−𝜔^±𝑑𝜔_𝑛=𝑓𝜔^±_𝑛. (28)

With the following relations: 𝑢²_𝑘=12𝜉1+^𝑘𝜀_𝑘,𝑣²_𝑘=12𝜉1−^𝑘𝜀_𝑘, (29)

(25) and (26) become 𝐶⁺_𝑝𝐶_𝑝=12+12𝜉_𝑘𝜀_𝑘tanh𝛽𝜀_𝑘2+1𝑁^∞_𝑙=1𝑔^2𝑙1!𝑙2+12𝜉_𝑘′𝜀_𝑘′𝛽tanh𝑙𝜔₀+𝜀_𝑘′2−𝑛_𝑘′2𝜉_𝑘′𝜀_𝑘′𝛽tanh𝑙𝜔₀+𝜀_𝑘′2𝛽−tanh𝑙𝜔₀−𝜀_𝑘′2,𝐶_𝑝𝐶_𝑝Δ(𝑘)=−2𝜀_𝑘tanh𝛽𝜀_𝑘2+1𝑁^∞_𝑙=1(−1)^𝑙𝑔!𝑙^2𝑙_𝑘′Δ𝑘^′2𝜀_𝑘′×𝛽tanh𝑙𝜔₀+𝜀_𝑘′2−𝑛_𝑘𝛽tanh𝑙𝜔₀+𝜀_𝑘′2𝛽+tanh𝑙𝜔₀−𝜀_𝑘′2. (30)(31)

5. Superconducting Order Parameter (Δ)

The order parameter of a superconducting state is given by Δ=𝑔_𝑘𝐶_𝑘𝐶_𝑘. (32)

Substituting correlation function given by (31) in (32) and changing summation into integral using the following relation: =𝑁(_𝑘𝑜)^ℏ𝜔𝐷₀𝑑𝜉_𝑘, (33)

the gap equation becomes Δ=𝑔𝑁(0)^ℏ𝜔𝐷₀𝑑𝜉_𝑘−Δ(𝑘)2𝜉²_𝑘+Δ²(𝑘)tanh𝜉²_𝑘+Δ²(𝑘)2𝑘_𝐵𝑇1−𝑔𝑁^∞_𝑙=1(−1)^𝑙𝑔!𝑙^12𝑙𝑁(0)^ℏ𝜔𝐷_0𝑘′𝑑𝜉_𝑘−Δ𝑘^′2𝜉²_𝑘′′+Δ²𝑘^′×tanh𝑙𝜔₀+𝜉²_𝑘′′+Δ²𝑘^′2𝑘_𝐵𝑇−1exp𝜉²_𝑘′′+Δ²𝑘^′/2𝑘_𝐵𝑇×+1tanh𝑙𝜔₀+𝜉²_𝑘′′+Δ²𝑘^′2𝑘_𝐵𝑇+tanh𝑙𝜔₀−𝜉²_𝑘′′+Δ²𝑘^′2𝑘_𝐵𝑇. (34)

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 YBa₂Cu₃O_7−𝑥 system. The two independent terms finally yield the two equations as 1(0)=𝑔𝑁^ℏ𝜔𝐷₀𝑑𝜉_𝑘12𝜉²_𝑘+Δ²(𝑘)tanh𝜉²_𝑘+Δ²(𝑘)2𝑘_𝐵𝑇, (35)

with ℓ=1 the other equation is 1𝑔𝑁(0)𝑔²_𝑙=^ℏ𝜔𝐷₀𝑑𝜉_𝑘′2𝜉²_𝑘′′+Δ²𝑘^′tanhℓ𝜔₀+𝜉²_𝑘′′+Δ²𝑘^′2𝑘_𝐵𝑇−1exp𝜉²_𝑘′′+Δ²𝑘^′/2𝑘_𝐵𝑇×+1tanhℓ𝜔₀+𝜉²_𝑘′′+Δ²𝑘^′2𝑘_𝐵𝑇+tanhℓ𝜔₀−𝜉²_𝑘′′+Δ²𝑘^′2𝑘_𝐵𝑇. (36)

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 (𝑐_𝑒𝑠)

The electronic specific heat per atom of a superconductor is determined from the following relation [3, 4]: 𝑐_𝑒𝑠=𝜕1𝜕𝑇𝑁_𝑘2𝜉_𝑘𝐶⁺_𝑘𝐶_𝑘, (37) where ⟨𝐶⁺_𝑘𝐶_𝑘⟩ is the correlation function. We have obtained this correlation function in (30). Substituting the correlation function from (30) in equation (37), One obtains 𝑐_𝑒𝑠=𝑁(0)2𝑁^ℏ𝜔𝐷₀2𝜉_𝑘𝑘_𝐵𝑇²𝑑𝜉_𝑘×−12𝜉_𝑘secℎ²𝜀_𝑘2𝑘_𝐵𝑇+^∞_𝑙=1𝑔^2𝑙−1!𝑙2𝜉_𝑘′𝜀_𝑘′𝜔₀𝑙+𝜀_𝑘′×secℎ²𝜔₀𝑙+𝜀_𝑘′2𝑘_𝐵𝑇+𝑛_𝑘′2𝜉_𝑘′𝜀_𝑘′𝜔₀𝑙+𝜀_𝑘′×secℎ²𝜔₀𝑙+𝜀_𝑘′2𝑘_𝐵𝑇+𝑛_𝑘′2𝜉_𝑘′𝜀_𝑘′𝜔₀𝑙−𝜀_𝑘′×secℎ²𝜔₀𝑙−𝜀_𝑘′2𝑘_𝐵𝑇. (38)

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] 𝑁(𝜔)1=lim𝐺2𝜋¹¹𝑘,𝜔+𝑖𝜂−𝐺₁₁𝑘,𝜔−𝑖𝜂. (39)

Using the following identity: lim_𝜂→0112𝜋𝜔+𝑖𝜂−𝜔⁺_𝑛−1𝜔−𝑖𝜂−𝜔⁺_𝑛=𝑖𝛿𝜔−𝜔⁺_𝑛, (40) changing the summation over “𝐾” into an integration, replacing 𝜀_𝑘 by −𝜀_𝑘, and combining the terms and using the relations 𝑢²_𝑘+𝑣²_𝑘=12𝜉1+^𝑘𝜀_𝑘+12𝜉1−^𝑘𝜀_𝑘=1, (41)

one obtains 𝑁(𝜔)𝑁(0)=1𝑁^𝑙=∞_𝑙=1𝑔^2𝑙𝑙_𝑘′𝑖𝜔_𝑛−𝑙𝜔₀𝑖𝜔_𝑛−𝑙𝜔₀²−Δ^21/2−𝑖𝜔_𝑛+𝑙𝜔₀𝑖𝜔_𝑛+𝑙𝜔₀²−Δ^21/2. (42)

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]: 𝐹_𝑠−𝐹_𝑁𝑉=^∞₀1𝑑𝑔𝑔²Δ², (43) where “𝑔” is the interaction parameter and “Δ” is the superconducting order parameter. Equation (43) can also be expressed as 𝐹_𝑠−𝐹_𝑁𝑉=^Δ₀𝑑𝑑Δ1𝑑Δ𝑔²Δ². (44)

From superconducting order parameter expression, we have Δ(𝑘)=𝑔𝑁(0)^ℏ𝜔𝐷₀−Δ(𝑘)2𝜉²_𝑘+Δ²(𝑘)tanh𝜉²_𝑘+Δ²(𝑘)2𝑘_𝐵𝑇𝑑𝜉_𝑘,1𝑔=𝑁(0)^ℏ𝜔𝐷₀12𝜉²_𝑘+Δ²(𝑘)tanh𝜉²_𝑘+Δ²(𝑘)2𝑘_𝐵𝑇𝑑𝜉_𝑘. (45)

Equation (44) becomes 𝐹_𝑆𝑁𝑉_phonon=₀^Δ(𝑘)Δ²(𝑘)(𝑘)𝑑𝑑Δ(𝑘)×𝑁(0)𝑑Δ_ℏ𝜔𝐷012𝜉²_𝑘+Δ²(𝑘)×tanh𝜉²_𝑘+Δ²(𝑘)2𝑘_𝐵𝑇𝑑𝜉_𝑘. (46)

Since 𝜀²_𝑘=𝜉²_𝑘+Δ²(𝑘),2𝜀_𝑘𝑑𝜀_𝑘=2Δ(𝑘)𝑑Δ(𝑘). (47)

Integrating by parts, we get 2𝐹_𝑆𝑁𝑉=Δ²(𝑘)𝑔(0)×−𝑁_ℏ𝜔𝐷02𝑑𝜉_𝑘^𝜀𝑘_𝜉𝑘tanh𝛽𝜀_𝑘2𝑑𝜀_𝑘,2𝐹_𝑆𝑁𝑉=Δ²(𝑘)𝑔−4𝑁(0)𝛽×^ℏ𝜔𝐷₀𝑒ln^{𝛽𝜀𝑘/2}1+𝑒^{−𝛽𝜀𝑘}𝑒^{𝛽𝜉𝑘/2}1+𝑒^{−𝛽𝜉𝑘}𝑑𝜉_𝑘,2𝐹_𝑆𝑁𝑉=Δ²(𝑘)𝑔−4𝑁(0)𝛽×^ℏ𝜔𝐷₀ln𝑒^{𝛽(𝜀𝑘−𝜉𝑘)/2}𝑑𝜉_𝑘−4𝑁(0)𝛽^ℏ𝜔𝐷₀ln1+𝑒^{−𝛽𝜀𝑘}𝑑𝜉_𝑘+4𝑁(0)𝛽^ℏ𝜔𝐷₀ln1+𝑒^{−𝛽𝜉𝑘}𝑑𝜉_𝑘,2𝐹_𝑆𝑁𝑉_phonon=(−𝑁(0)Δ)²(𝑘)2−4𝑁(0)𝛽𝑒^{−𝛽Δ(𝑘)}22𝜋Δ(𝑘)𝛽^−11/2+4𝑁(0)𝛽1𝛽𝜋².12 (48)(49)

With the help of (49) we can calculate the free energy difference.

6.4. Critical Field (𝐻_𝑐)

The critical field is related to the free energy difference as 𝐻_𝑐=8𝜋𝐹_𝑆−𝐹_𝑁^1/2. (50)

Using (49), we obtain 𝐻_𝑐=𝑁(0)Δ8𝜋²(𝑘)4−4𝑁(0)𝛽×𝑒^{−𝛽Δ(𝑘)}42𝜋Δ(𝑘)𝛽^−11/2+4𝑁(0)𝛽1𝛽𝜋²12^1/2. (51)

7. Numerical Calculations

Now, we evaluate numerically the physical properties of high-temperature cuprate superconductor YBa₂Cu₃O_7−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 Table 1. Using these values, we have made study of various parameters related to the physical properties for the system YBa₂Cu₃O_7−𝑥.

Table 1 

Values of various parameters for HTSC cuprate superconductor for YBa₂Cu₃O_7−𝑥.

7.1. Superconducting Order Parameter (Δ)

For the study of superconducting order parameter (Δ) for the system YBa₂Cu₃O_7−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) 1𝑔𝑁(0)=^ℏ𝜔𝐷₀𝑑𝜉_𝑘12𝜉²_𝑘+Δ²(𝑘)tanh𝜉²_𝑘+Δ²(𝑘)2𝑘_𝐵𝑇. (52) Solving the previous equation numerically, we get 1𝑔𝑁(0)=^𝑦=1_𝑦=01𝑑𝑦2𝑦²+0.5917𝑥²×tanh47.099𝑦²+0.5917𝑥²𝑇. (53)

With the help of the previous equation, one can study the variation of superconducting order parameter Δ₁ with temperature, when only electron-phonon interaction is considered.

Values of superconducting order parameter obtained at various temperatures are given in Table 2, and variation of Δ₁ with temperature is shown in Figure 1.

Table 2 

Superconducting order parameter (Δ) for YBa₂Cu₃O_7−𝑥 system.

Figure 1 

Behaviour of superconducting order parameter for the system YBa2Cu3O7−𝑥.

(ii) Superconducting order parameter (Δ₂).

(When only polaron interaction is considered)

We have (36) 1𝑔𝑁(0)𝑔²_𝑙=^ℏ𝜔𝐷₀𝑑𝜉_𝑘′2𝜉²_𝑘′′+Δ²𝑘^′×tanhℓ𝜔₀+𝜉²_𝑘′′+Δ²𝑘^′2𝑘_𝐵𝑇−1exp𝜉²_𝑘′′+Δ²𝑘^′/𝑘_𝐵𝑇×+1tanhℓ𝜔₀+𝜉²_𝑘′′+Δ²𝑘^′2𝑘_𝐵𝑇+tanhℓ𝜔₀−𝜉²_𝑘′′+Δ²𝑘^′2𝑘_𝐵𝑇. (54)

Solving the previous equation numerically, we get 1𝑔𝑁(0)𝑔²_𝑙=^𝑦=1_𝑦=0𝑑𝑦2𝑦²+0.5917𝑥²×tanh36.238+1.3𝑦²+0.5917𝑥²𝑇−1exp94.2𝑦²+0.5917𝑥²×/𝑇+1tanh36.238+1.3𝑦²+0.5917𝑥²𝑇+tanh36.238−1.3𝑦²+0.5917𝑥²𝑇. (55)

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 Table 2, 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 Table 2.

The behaviour of superconducting order parameter (Δ=Δ₁+Δ₂) (combined phonon and polaron interactions) is shown in Figure 1.

7.2. Electronic Specific Heat (𝐶_𝑒𝑠)

We have obtained the expression (38) for electronic specific heat putting 𝜀_𝑘=𝜉²+Δ^21/2,Δ=𝑥×10⁻²¹,𝑁(0)=0.5/eV,ℏ𝜔_𝐷≈1.3×10⁻²¹J,ℏ𝜔_𝐷2𝐾_𝐵𝑇=47.099/T. (56)

Equation (38) reduces to 𝐶_es=2898.5×1.69×10⁻⁴⁹×^𝑦=1_𝑦=0𝑦²𝑇²𝑑𝑦−1.3(secℎ𝑄)²−𝑆𝑃(secℎ𝐿)²+1𝑒^𝑄𝑆+1𝑃(secℎ𝐿)²−1𝑒^𝑄𝑀+1𝑃(secℎ𝑅)², (57) where 𝑃=𝑦²+0.5917𝑥^21/2,𝑄=47.099𝑇𝑃,𝑆=8+1.3𝑃,𝑀=8−1.3𝑃,𝐿=36.23𝑇𝑆,𝑅=36.23𝑇𝑀. (58)

One can study the behaviour of electronic specific heat (𝐶_𝑒𝑠) with temperature (𝑇) with the help of (57). Values of 𝐶_𝑒𝑠 at various temperatures obtained from (57) are given in Table 3, and variation of 𝐶_𝑒𝑠 with 𝑇 is shown in Figure 2 Variation of 𝐶_es/𝑇 with 𝑇 (Table 4) and (𝐶−𝐶_es)/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 (𝐶_es) for YBa₂Cu₃O_7−𝑥 system.

Table 4 

𝐶_es/T for YBa₂Cu₃O_7−𝑥 system.

Table 5 

(𝐶−𝐶_es)/𝑇 for YBa₂Cu₃O_7−𝑥 system.

Figure 2 

Variation of 𝐶es with temperature for the system YBa2Cu3O7−𝑥.

Figure 3 

Variation of 𝐶𝑒𝑠/𝑇 with temperature.

Figure 4 

Variation of (𝐶−𝐶𝑒𝑠)/𝑇 with temperature.

7.3. Density of States Function 𝑁(𝜔)/𝑁(0)

Density of states function for the polaron case is given by 𝑁(𝜔)𝑁(0)=1𝑁^𝑙=∞_𝑙=1𝑔^2𝑙𝑙_𝑘′𝑖𝜔_𝑛−𝑙𝜔₀𝑖𝜔_𝑛−𝑙𝜔₀²−Δ^21/2−𝑖𝜔_𝑛+𝑙𝜔₀𝑖𝜔_𝑛+𝑙𝜔₀²−Δ^21/2. (59)

Using the following values 𝜔_𝑛=𝑦×10⁻²¹J,Δ=𝑥×10⁻²¹𝜔J,𝑙=1,₀=8×10⁻²¹J,𝑔²=1. (60)

The previous equation reduces as 𝑁(𝜔)𝑁(0)=𝑦−8(𝑦−8)²−𝑥²−𝑦+8𝑦+8²−𝑥². (61)

The values of density of states 𝑁(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 states 𝑁(𝜔)/𝑁(0) at various temperatures for YBa₂Cu₃O_7−𝑥 system.

Figure 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 2𝐹_𝑆𝑁𝑉_phonon=(−𝑁(0)Δ)²(𝑘)2−4𝑁(0)𝛽×𝑒^{−𝛽Δ(𝑘)}22𝜋Δ(𝑘)𝛽^−11/2+4𝑁(0)𝛽1𝛽𝜋².12 (62)

Solving numerically, 2𝐹_𝑆𝑁𝑉_phonon=(−)0.02475𝑥²×𝑇1−−0.01625𝑥^1/2𝑒^{−72.46𝑥/𝑇}+12.517×10⁻⁴𝑇𝑥². (63)

The values of free energy difference at different temperatures are given in Table 7, and behaviour is shown in Figure 6, respectively.

Table 7 

Free energy difference for YBa₂Cu₃O_7−𝑥 system.

Figure 6 

Variation of free energy difference with temperature for the system YBa2Cu3O7−𝑥.

7.5. Critical Field (𝐻_𝑐)

The low temperature critical field is given as 𝐻_𝑐=𝑁(0)Δ8𝜋²(𝐾)4−4𝑁(0)𝛽×𝑒^−𝛽Δ(K)42𝜋Δ(K)𝛽^1/2+4𝑁(0)𝛽1𝛽𝜋²12^1/2. (64)

Solving the previous equation numerically, we can obtain the values of low temperature critical field which are described in Table 8. The behavior of the low temperature critical field (𝐻_𝑐) with temperature is shown in Figure 7.

Table 8 

Critical field for YBa₂Cu₃O_7−𝑥 system.

Figure 7 

Variation of critical field with temperature for the system YBa2Cu3O7−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 of YBa₂Cu₃O_7−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. 𝑇_𝐶 obtained for the system YBa₂Cu₃O_7−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-𝑇_𝐶 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.