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Journal of Physical Chemistry & Biophysics
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Variation of Isotope Coefficient with Number of CuO2 Layers in High TC Superconductors

Shabir Bhat S1*, Masih P, Aziz MK2 and Banday AA2

1Department of Physics, Sam Higginbottom Institute of Agriculture Technology and Sciences, Allahabad, Uttar Pradesh, India

2Department of Chemistry, University of Allahabad, Allahabad, Uttar Pradesh, India

*Corresponding Author:
Shabir Bhat S
Department of Physics
Sam Higginbottom Institute of Agriculture Technology and Sciences
Allahabad, Uttar Pradesh-211 007, India
Tel: 00917843895010
E-mail: shahidphy0@gmail.com

Received Date: February 08, 2017; Accepted Date: February 16, 2017; Published Date: February 22, 2017

Citation: Bhat S, Masih P, Aziz MK, Banday A (2017) Variation of Isotope Coefficient with Number of CuO2 Layers in High TC Superconductors. J Phys Chem Biophys 7:237. doi: 10.4172/2161-0398.1000237

Copyright: © 2017 Bhat S, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this paper, an effort has been made to study the isotope coefficient in high TC Superconductors by using the variation of isotopic coefficient with the number of CuO2 layers and the variation of isotope coefficients on transition temperature TC . The Hamiltonian for CuO2 layers using BCS type model and extra term of interlayer interaction between CuO2 layers has been considered. Expressions for isotope effect (α) and transition temperature (TC) are obtained and numerically calculated for experimental values by using Green’s function technique.

Keywords

Isotope effect; CuO2 layers; Isotope coefficient; High TC superconductors

Introduction

After the discovery of high TC cuprate superconductors in 1986 [1] at 35 K, the field is buzzing with research activities. Efforts to increase the transition temperature are currently going on. Till now the highest reached TC under pressure is 164 K in HgBa2Ca2Cu3O8+8 [2]. These cuprates have unconventional properties both in normal and superconducting state [3,4]. Till date there is no consensus on the origin of pairing mechanism. It is now widely suggested that spin fluctuation driven pairing mechanism provides a good agreement between theory and experiments [5]. However, there are several experimental observations which clearly indicate that this purely electronic picture is incomplete and lattice effects have to be taken into account [6-8]. Isotope shift of TC is regarded as the defining signature of superconducting pairing resulting from phonons. In high TC superconductors, a small isotope effect is found [9-11]. Suppression of this effect is not explained within conventional BCS theory which predicts that the critical temperature TC and isotope mass M are related by TC ∝ M where α=0.5 for all elements. To explain the small isotope effect in high TC compounds, many mechanisms including resonance valence bond [12], excitons [13], Plasmon’s [14] and antiferromagnetic spin fluctuation mediated pairing are proposed [15]. Daemen and Overhauser [16] found that the existence of a short-range attraction in addition to the conventional phonon pairing interaction suppresses the isotope effect significantly at high temperatures. Here we present the variation of isotope coefficients with number of CuO2 layers and the variation of isotope coefficients on transition temperature TC [17- 22]. As the trilayer material has highest (TC) in these cuprates but it has very small isotope coefficient (α). Clearly the argument that a mechanism other than an electron phonon interaction dominates the superconductivity based only on a small (α) in a cuprate with relatively high (TC) is inappropriate. The observed CuO2 layer dependence of the isotope effect indicates that the interlayer coupling between the adjacent CuO2 planes is necessary for superconductivity in layered cuprates [23-26]. For monolayer materials having lower (TC) the interlayer coupling plays a less important role and (TC) can be mainly controlled by the phonon coupling yielding a larger size of isotope coefficient with increasing the number of CuO2 layers in a unit cell, the interlayer coupling begins to play an important role in enhancing (TC) and the isotope effect is expected to be small.

Formulation

The model Hamiltonian for our system can be described as

physical-chemistry-biophysics

physical-chemistry-biophysics

physical-chemistry-biophysics

physical-chemistry-biophysics

Where physical-chemistry-biophysics σdenote the fermions creation and annihilation operator respectively, k is the wave vector and σ is spin index for fermions.

In our present analysis we use a Green’s function, defining as

physical-chemistry-biophysics (2)

Equation of motion is written as

physical-chemistry-biophysics

Evaluating the commutator physical-chemistry-biophysics using the Hamiltonian (1) we get

physical-chemistry-biophysics

And writing the equation of motion as

physical-chemistry-biophysics (3)

Putting the value of commutator physical-chemistry-biophysics in equation (3) we get

physical-chemistry-biophysics

physical-chemistry-biophysics (4)

Now we introduce the order parameter Δ such as

physical-chemistry-biophysics

Where physical-chemistry-biophysics is another Green’s function which may be written as

physical-chemistry-biophysics (5)

This Green’s function may also be written in terms of equation of motion as

physical-chemistry-biophysics (6)

Evaluating the commutator physical-chemistry-biophysics using the Hamiltonian

physical-chemistry-biophysics

Putting the commutator physical-chemistry-biophysics from (7) in equation (6) we get

physical-chemistry-biophysics

But from the law of conservation of energy

physical-chemistry-biophysics

physical-chemistry-biophysics (8)

In equation (4) we finally obtained the equation

physical-chemistry-biophysics (9)

Multiply (ω+Ek) in equation (9) and putting the value of physical-chemistry-biophysics from equation (8) then we get the Green’s function physical-chemistry-biophysics

physical-chemistry-biophysics (10)

Multiplying by (ω+Ek) in the equation (8) and putting the value of physical-chemistry-biophysics from equation (9) we get the Green’s function as

physical-chemistry-biophysics

physical-chemistry-biophysics

We know from equation (9)

physical-chemistry-biophysics (11)

Using the Green’s function, we can obtain the expression for order parameter Δmn and correlation parameter γ the order parameter Δmn may be written as

physical-chemistry-biophysics (12)

Correlation function physical-chemistry-biophysics is related to Green’s function physical-chemistry-biophysics

physical-chemistry-biophysics (13)

Where η=-1 for fermions, K=Boltzmann constant and T= Temperature

Green’s function physical-chemistry-biophysics andphysical-chemistry-biophysics may be expressed as

physical-chemistry-biophysics (14)

physical-chemistry-biophysics

physical-chemistry-biophysics (15)

physical-chemistry-biophysics (16)

physical-chemistry-biophysics

physical-chemistry-biophysics

physical-chemistry-biophysics (17)

Substitute both the Green’s function physical-chemistry-biophysics andphysical-chemistry-biophysics from equation (13) and then after

solving we get correlation function.

physical-chemistry-biophysics (18)

physical-chemistry-biophysics

Where η=-1

physical-chemistry-biophysics

physical-chemistry-biophysics

Put physical-chemistry-biophysics

physical-chemistry-biophysics (19)

physical-chemistry-biophysics Using in (19) we get

physical-chemistry-biophysics (20)

Then we can obtained the expression of or dr parameter Δir by substituting correlation function in equation (12)

physical-chemistry-biophysics

physical-chemistry-biophysics

Converting summation over K into integration with cut-off energy ħωD from the Fermi level we get

physical-chemistry-biophysics (21)

physical-chemistry-biophysics (22)

physical-chemistry-biophysicsphysical-chemistry-biophysics (23)

physical-chemistry-biophysics

physical-chemistry-biophysics

physical-chemistry-biophysics (24)

The Isotope effect coefficient is

physical-chemistry-biophysics (25)

Using equation (29) and (30) we get

physical-chemistry-biophysics (26)

physical-chemistry-biophysics

physical-chemistry-biophysics (27)

Differentiating we get

physical-chemistry-biophysics (28)

Now from equation (24) we get

physical-chemistry-biophysics (29)

By solving equation (33), we get α as

physical-chemistry-biophysics (30)

The values of α and TC are calculated from equation (29) and (30) respectively for various values of different CuO2 layers (Table 1).

S No α   TC
1 0.05   37.586
2 0.1   30.772
3 0.2   20.627
4 0.3   13.827
5 0.4   9.268
6 0.5   6.212
7 0.6   4.164
8 0.7   2.791

Table 1: Table for values of (α) and (TC).

Results and Discussion

The numerical analysis of equation (29) and (30) have been calculated for different CuO2 layers taking physical-chemistry-biophysics .Figure 1a is plotted between isotope coefficient (α) and number of CuO2 layers (n) per unit cell. Figure shows that as increasing number of CuO2 layers per unit cell, isotope coefficient (α) linearly decreases. For n=1, the value of (α) is 0.34 and it decreases continuously with number of CuO2 layers. At n=3, isotope coefficient (α) is 0.1317 which is very low. This systematic reduction of isotope coefficient with increasing CuO2 layers indicates that isotope effect can be negligible for multilayer cuprates. This result is good agreement with experimental result made by Chen et al. Figure 1b is plotted between transition temperature (TC) and isotope coefficient. The curve shows that as increasing (α), the transition temperature exponentially decreases so for lower isotope coefficient TC will be maximum and TC decreases for higher values of (α). So we can say that as increasing number of CuO2 layers, (α) decreases and the transition temperature increases. This result also supported by Chakravarty et al.

-chemistry-biophysics-Isotope-coefficient

Figure 1a: Isotope coefficient (α) VS number of CuO2 layers (n).

physical-chemistry-biophysics-Isotope-coefficient

Figure 1b: Transition temperature (TC) VS Isotope coefficient (α).

Conclusion

First of all Hamiltonian for CuO2 layers using BCS type model and extra term of interlayer interaction between CuO2 layers has been considered, using this equation expression for isotope effect (α) and transition temperature (TC). Expression obtained is numerically solved using Green’s function technique and then the value of isotope effect (α) and transition temperature (TC) is calculated. The graph is plotted between isotopic coefficient and number of CuO2 layers. The second graph is plotted between isotope effect (α) and transition temperature (TC). Then conclusion is drawn comparing with available experimental result. The tri layer material has highest (TC) in these cuprates but it has very small isotope coefficient (α). Clearly the argument that a mechanism other than an electron-phonon interaction dominates the superconductivity based only on a small (α) in a cuprate with relatively high (TC) is inappropriate. The observed CuO2 layer dependence of the isotope effect indicates that the interlayer coupling between the adjacent CuO2 planes is necessary for superconductivity in layered cuprates. For monolayer materials having lower (TC) the interlayer coupling plays a less important role and (TC) can be mainly controlled by the phonon coupling yielding a larger size of isotope coefficient with increasing the number of CuO2 layers in a unit cell, the interlayer coupling begins to play an important role in enhancing (TC) and the isotope effect is expected to be small.

References

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