International Journal of Superconductivity

Volume 2013 (2013), Article ID 581025, 13 pages

http://dx.doi.org/10.1155/2013/581025

Research Article

Polaronic Mechanism of Superconductivity in Cuprates

Pradeep Chaudhary,¹ Anuj Nuwal,² S. C. Tiwari,³ R. K. Paliwal,⁴ and S. L. Kakani⁵

¹Department of Physics, Government Polytechnic College, Chittorgarh, Rajasthan 312001, India

²Department of Physics, Sangam University, N.H. No. 79, Bhilwara By-Pass, Chittor Road, Bhilwara, Rajasthan 311001, India

³Department of Physics, M.L.V. Government College, Bhilwara, Rajasthan 311001, India

⁴Department of Physics, Mewar University, Gangrar, Chittorgarh, Rajasthan 312901, India

⁵4-G-45, Shastri Nagar, New Housing Board, Bhilwara, Rajasthan 311001, India

Received 15 April 2013; Accepted 13 June 2013

Academic Editor: Zigang Deng

Copyright © 2013 Pradeep Chaudhary et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 () 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 . 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 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 () [25, 26].

2. Model Hamiltonian

The model Hamiltonian for our system can be expressed as [21]

where is the kinetic energy in the initial Bloch band, is the vibration energy of the lattice, is electron-phonon interaction, and is the Coulomb electron-electron correlations.

In one band approximation has the form

where and denote the state with quasi momentum and spin, respectively. is bare band energy. can be expressed in terms of phonon operators ; where is the type of vibrational mode.

where is the phonon dispersion.

The electron-phonon interaction is described by the Hamiltonian:

in which and are the phonon frequency and the interaction matrix element in a parent crystal without charge carriers, respectively. Correspondingly one obtains

In the case of optical longitudinal phonon with frequency and , are the dielectric constants of the crystal with and without taking ionic part into consideration. is the volume of the unit cell and is their number.

For acoustic phonons, one finds

where is the deformation potential, is the sound velocity, and is the mass of an elementary cell. For intermolecular phonons

The combined Hamiltonian can be expressed as

Here 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

where

In the small-polaron regime, , the kinetic energy remains smaller than the interaction energy, and a self-consistent treatment of a many-body problem is possible with the expansion technique [27]. Following Lang and Firsov [23, 24] and applying canonical transformations to diagonalize the Hamiltonian [28], one obtains

where

The electron operator transforms as

Using

When we have

Thus,

Hence

In obtaining (17), we have omitted the term containing the on-site interaction for parallel spins.

3. Green’s Functions

We define the following one particle temperature electron and anomalous Green's functions:

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]

After evaluating the electron part and phonon part of the trace, we obtain the total Green’s function as

where

With

The energy dispersion for the polaronic band is given by

having a narrow band half width .

4. Correlation Function

The correlation functions are defined as

where

where and are Green functions given by (20) and (21), respectively.

Using the identity

With the following relations:

(25) and (26) become

5. Superconducting Order Parameter

The order parameter of a superconducting state is given by

Substituting correlation function given by (31) in (32) and changing summation into integral using the following relation:

the gap equation becomes

Right-hand side of (34) has two terms which are quite independent. First term varies with , whereas second term varies with ; hence, one can define two superconducting order parameters for the system. The two independent terms finally yield the two equations as

with the other equation is

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

The electronic specific heat per atom of a superconductor is determined from the following relation [3, 4]:

where is the correlation function. We have obtained this correlation function in (30). Substituting the correlation function from (30) in equation (37), One obtains

Right-hand side of (38) has two terms which are quite independent from each other. First term varies with , whereas second term varies with ; hence one can study the behaviour of electronic specific heat of superconductors with temperature.

6.2. Density of States Function

For , the function can be defined as [5]

Using the following identity:

changing the summation over “” into an integration, replacing by , and combining the terms and using the relations

one obtains

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]:

where “” is the interaction parameter and “” is the superconducting order parameter. Equation (43) can also be expressed as

From superconducting order parameter expression, we have

Equation (44) becomes

Since

Integrating by parts, we get

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

6.4. Critical Field

The critical field is related to the free energy difference as

Using (49), we obtain

7. Numerical Calculations

Now, we evaluate numerically the physical properties of high-temperature cuprate superconductor , 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 .

Table 1: Values of various parameters for HTSC cuprate superconductor for .

S. no.
Property
Value

1
Superconducting transition temperature (T c)
88 K

2
Density of states at the Fermi surface
per ergs Cu atom

3
Phonon energy
 J

4
Polaron frequency
0.72 eV

5
Polaron density
8 states/eV spin

6
Fermi energy
0.23 eV

7
Crystal structure
Orthorhombic

8
Cell parameters
 nm,  nm, and  nm

9
Number of atoms per unit volume
/m³

10
Boltzmann constant ( )
 J/K

11
Mass of electron
 kg

7.1. Superconducting Order Parameter

For the study of superconducting order parameter () for the system , 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 ().

(When only electron-phonon interaction is considered).

We have (35)

Solving the previous equation numerically, we get

With the help of the previous equation, one can study the variation of superconducting order parameter 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 with temperature is shown in Figure 1.

Table 2: Superconducting order parameter (Δ) for system.

S. no.
Temperature (K)
 J

(when only electron-phonon interaction is present)
 J

(when only polaron interaction is considered)

(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
0.937470

Figure 1: Behaviour of superconducting order parameter for the system .

(ii) Superconducting order parameter ().

(When only polaron interaction is considered)

We have (36)

Solving the previous equation numerically, we get

With the help of the previous equation, one can study the variation of superconducting order parameter () 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 with temperature is shown in Figure 1.

(iii) Superconducting order parameter ()

(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 () (combined phonon and polaron interactions) is shown in Figure 1.

7.2. Electronic Specific Heat

We have obtained the expression (38) for electronic specific heat putting

Equation (38) reduces to

where

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

S. no.
Temperature (K)

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
0.000000

Table 4: /T for system.

S. no.
Temperature (K)
/T Joule/mole-K² 

(Theory)
/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
1.4170

Table 5: for system.

S. no.
Temperature (K)
Joule/mole-K² 

(Theory)
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
6.670

Figure 2: Variation of with temperature for the system .

Figure 3: Variation of with temperature.

Figure 4: Variation of ( with temperature.

7.3. Density of States Function

Density of states function for the polaron case is given by

Using the following values

The previous equation reduces as

The values of density of states for the chosen value of 1.106, 1.8851, 2.334 and 2.4761 are shown in Table 6. The variation of density of states with at different temperatures is shown in Figure 5.

Table 6: Density of states at various temperatures for system.

S. no.
Frequency

(y)
Density of states

 K

 K

 K

 K

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
0.00641463

Figure 5: Variation of density of states with frequency at different temperatures for the system .

7.4. Free Energy Difference

The expression for the free energy difference can be expressed as

Solving numerically,

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 system.

S. no.
Temperature (K)
Free energy difference

Joule/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
0.1129

Figure 6: Variation of free energy difference with temperature for the system .

7.5. Critical Field

The low temperature critical field is given as

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 with temperature is shown in Figure 7.

Table 8: Critical field for system.

S. no.
Temperature (K)
Critical field 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
1.191492

Figure 7: Variation of critical field with temperature for the system .

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 . 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. obtained for the system 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- 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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