After the discovery of high T_C 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 T_C 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 T_C is regarded as the defining signature of superconducting pairing resulting from phonons. In high T_C 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 T_C and isotope mass M are related by T_C ∝ M^-α where α=0.5 for all elements. To explain the small isotope effect in high T_C 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 T_C [17- 22]. As the trilayer material has highest (T_C) 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 (T_C) 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 (T_C) the interlayer coupling plays a less important role and (T_C) 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 (T_C) and the isotope effect is expected to be small.

Formulation

The model Hamiltonian for our system can be described as

Where 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

(2)

Equation of motion is written as

Evaluating the commutator using the Hamiltonian (1) we get

And writing the equation of motion as

(3)

Putting the value of commutator in equation (3) we get

(4)

Now we introduce the order parameter Δ such as

Where is another Green’s function which may be written as

(5)

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

(6)

Evaluating the commutator using the Hamiltonian

Putting the commutator from (7) in equation (6) we get

But from the law of conservation of energy

(8)

In equation (4) we finally obtained the equation

(9)

Multiply (ω+E_k) in equation (9) and putting the value of from equation (8) then we get the Green’s function

(10)

Multiplying by (ω+E_k) in the equation (8) and putting the value of from equation (9) we get the Green’s function as

We know from equation (9)

(11)

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

(12)

Correlation function is related to Green’s function

(13)

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

Green’s function and may be expressed as

(14)

(15)

(16)

(17)

Substitute both the Green’s function and from equation (13) and then after

solving we get correlation function.

(18)

Where η=-1

Put

(19)

Using in (19) we get

(20)

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

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

(21)

(22)

(23)

(24)

The Isotope effect coefficient is

(25)

Using equation (29) and (30) we get

(26)

(27)

Differentiating we get

(28)

Now from equation (24) we get

(29)

By solving equation (33), we get α as

(30)

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

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

The numerical analysis of equation (29) and (30) have been calculated for different CuO₂ layers taking .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 (T_C) and isotope coefficient. The curve shows that as increasing (α), the transition temperature exponentially decreases so for lower isotope coefficient T_C will be maximum and T_C 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.

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

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

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 (T_C). Expression obtained is numerically solved using Green’s function technique and then the value of isotope effect (α) and transition temperature (T_C) 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 (T_C). Then conclusion is drawn comparing with available experimental result. The tri layer material has highest (T_C) 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 (T_C) 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 (T_C) the interlayer coupling plays a less important role and (T_C) 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 (T_C) and the isotope effect is expected to be small.