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In this research, the influence of calcium content on the binding energy and Cooper pairing is investigated for the high temperature superconducting
Y<sub>1-x</sub>Ca<sub>x</sub>Ba<sub>2</sub>Cu<sub>3</sub>O<sub>7-δ</sub>
compound where x = 0 - 0.2). This is done by using model considering that the formation of Cooper pairs happens during tunneling mechanism of holes between the two CuO_{2} layers in the YCBCO crystal which are considered as forming adjacent potential wells. In this work, we make some modifications on the mentioned model and a modified equation for the binding energy as a function of Ca concentration is formulated to study the effect of Ca content by considering the change in the distance between the CuO_{2} layer as a function of Ca. The calculated results agree very well with experimental results. This happens because the Ca^{2+} substitution for Y^{3+} leads to decreasing the distance between the two CuO_{2} layers and hence potential energy of the system will be decreased. This leads to decreasing the binding energy between the Cooper pairs, and decreasing the distance between the two CuO_{2} layers increasing the tunneling probability of holes between the two CuO_{2} layers and thus increasing conduction mechanism and thus increasing current density, but this will decrease the energy gap and as a result decrease the critical temperature. The calculated results agree very well with experimental results.

To find out what exactly underlies the superconductivity in cuprate HTSC, we first require an in-depth understanding of the cuprate pairing mechanism. One of the first theories was Anderson’s “Resonating Valence Bond (RVB) Theory” [

The tremendous efforts to unveil high temperature superconductivity (HTSC) have been devoted to the research of the mechanism underlying Cooper pairs which, however, remains a mysterious subject of vigorous debate, and leaves alone many other mysteries like the pseudo gap state, the unnegligible electron-phonon interaction, the universal nodal Fermi velocity, etc. Most of subsequent works bring in more controversies to the list of mysteries [

In high temperature superconductor (HTS) materials, the Cooper pairs are carriers of the superconducting current. Fermi surface pockets were found in quantum oscillations of hall coefficient [_{2} layers are responsible for superconductivity, and the electrons are concentrated in them. Reduction of number of CuO_{2} layers in the ultra-thin films leads to decrease in_{2} layer is responsible for superconductivity. However, superconductivity emerges only in the presence of the interfacing material, providing another layer [

In this study, the effect of increase of Ca content on the tunneling mechanism of holes between the two CuO_{2} layers in the YBCO superconductor will be investigated and as well as its effect on the binding energy of the Cooper pairs and the conduction current in the superconductor.

In this study, we try to find the effect of Ca content on the binding energy in the superconductor _{2} layer. Which are considered as forming potential wells, but this wells and due to the presence of O atoms will contains ridges, which expand the Fermi sphere, this sphere will undergoes contraction when the two CuO_{2} layers are close enough to form correlated quantum states (CQS). The Cooper pairs formed by image forces which supply the electron-electron attraction, are comprising the inter- tunneling electrons which will occupy the above states. Tunneling occurs when the distance (d) between the two CuO_{2} layers (or their potential wells) are short enough, so that tunneling probability are high enough to imitate tunneling. The positive image of the tunneling electron inside the second well will attract another electron, thus Cooper pair are formed from the tunneled electron and the other electron in the second well.

The single CuO_{2} layer is responsible for superconductivity [_{2} layers, is proposed. Ridges impose additional boundary conditions on the electron wave function and some quantum states become forbidden. Rejected electrons occupy quantum states with higher energies. The Fermi vector, _{2} layers, like ridged films, exhibit FSE. The cuprate material contains CuO_{2} layers, each containing electron gas modified by FSE due to interaction of the adjacent two CuO_{2} layers, through single-electron tunneling. Cooper pairs exist in correlated quantum states _{2} layers. The electron-electron attraction originates from the image force [

states become forbidden in an RQW. The addition of Ca will lead to the decrease of the distance between the two CuO_{2} layers and thus decreases the rigged height as a result this shown in

In RQW the total energy of the electrons is increased with respect to reference QW. The electron gas in RQW is an excited system. If there was some external mechanism to allow back the forbidden quantum states (QS),

then the electrons would occupy them and would get decreased. Tunneling to another RQW is considered as a possible mechanism of energy minimization [

In cuprates, O and Cu atoms are shifted up and down, relative to the common plane of CuO_{2} layer, geometry of the layer is akin to the periodic ridges of Ridged Potential Energy Box (RPEB), in the first approximation, a CuO_{2} layer (for all cuprates) can be regarded as an RPEB-containing electron gas, and the layer has forbidden QS and expanded Fermi sphere.

_{2} layers separated by distance d decreasing d will increase the probability of tunneling and thus minimize energy. addition of Ca will shift the O atom up and down in the two layers of YBCO crystal and thus decreases the distances between the layers,

The density of quantum state, tunneling probability and binding energy can be calculated as follows (L. X. You et al. 2005):

The density of QSs as found in is

Density of quantum states can be determined as follows:

Here

The density of forbidden QS is

Each electron that leaves Fermi level to CQS, can be in four possible quantum states,

The substitution of Ca instead of Y will deceases the distance between the layers and Equation (3) must be modified as follows:

Probability of electron being in CQS is equal to tunneling probability and layer binding energy density (per unit volume) within the energy interval of δE will be:

Here,

Integration of Equation (4) over energy range below Fermi energy gives

inserting Equation (7) in Equation (6) obtained

Finally inserting Equation (4) in Equation (9) and G = 1 gives the binding energy

In this research the binding energy in _{2} layers by addition of Ca content instead of Yttrium into compound must be determined. The original distance between the two CuO_{2} planes before the addition of Ca is (3.18) Å in the ^{3+} instead of Y^{3+} will decrease the distance between the layers [_{2} layers by the addition of Ca in _{2}] atoms shifted up and down in the CuO_{2} planes vertically away from the CuO chains toward the planes in the upper and lower in the YBCCO crystal with addition of Ca content, the vertical distance change, by subtraction vertical change in both side from the original distance between the two planes (3.18 Å) the decreasing in the distance between the two CuO_{2} planes after addition Ca content from (0 to 0.2) the result as shown in the

Ca content | d(CuO_{2}-CuO_{2}) Å | Tunneling probability × 10^{−11} |
---|---|---|

0.025 | 2.9076 | 4.6314 |

0.05 | 2.9034 | 4.7929 |

0.075 | 2.8874 | 5.4635 |

0.1 | 2.869 | 6.3514 |

0.125 | 2.8526 | 7.2637 |

0.15 | 2.83682 | 8.265 |

0.175 | 2.826 | 9.0303 |

0.2 | 2.818 | 9.6413 |

The tunneling probability of the holes between the two CuO_{2} layers being increasing with decreasing the distance between layers which can be calculated using the Equation (3) and the result are shown in the

The calculation of the binding energy was done by using Equation (9), for solving the integration we apply the math lab program to determine each value of binding energy which corresponding to concentration of Ca content in the compound. The results are shown in the

In this study the effect of Ca content in the _{2} planes is investigated, many other parameters are affected by changing of the Ca content such as lattice parameters, critical temperature, crystal structure and etc., and by using the above parameters and their change the effect of Ca content on the distance between CuO_{2} layers are calculated [

Ca content | Change in vertical d Å | d(CuO_{2}-CuO_{2}) Å | E_{bind} in (meV) |
---|---|---|---|

0.025 | 0.2724 | 2.9076 | −23.3 |

0.05 | 0.2766 | 2.9034 | −22.2 |

0.075 | 0.2926 | 2.8874 | −21.4 |

0.1 | 0.311 | 2.869 | −20.8 |

0.125 | 0.3274 | 2.8526 | −20 |

0.15 | 0.3432 | 2.83682 | −19.2 |

0.175 | 0.354 | 2.826 | −18.4 |

0.2 | 0.362 | 2.818 | −17.3 |

_{2}-CuO_{2} two planes as a function of Ca content. As Ca content is increasing the distance between the two CuO_{2} planes will be decreased and as a result the binding energy of the system will be decreased also. This is combined with a shift of Cu(2) away from the single chain, and therefore the CuO_{2}-CuO_{2} distance decreases.

_{2} planes as a function of Ca content which indicates that: as Ca content increased the binding energy decreases as a result the hole concentration will be increased.

_{2} planes as a function of Ca content which indicates that as Ca content increases the binding energy decreases in superconductors system and the current density will be increased.

In this research the model we propose for the influence of calcium content on the binding energy and Cooper pairing due to the decrease of separation between the two CuO_{2} layers gives good results which explains the experimental facts about increase of superconducting current, and decrease of T_{c} for the high temperature superconductor ^{2+} substitution for Y^{3+} lead to decreasing the distance between the two CuO_{2} layers and hence potential energy of the system will be decreased, this will lead to decreasing the binding energy between the Cooper pairs, also decreasing the distance between the two CuO_{2} layers increasing the tunneling probability of holes between the two CuO_{2} layers (considering the two layers as a rigged quantum well) and thus increasing conduction mechanism by increasing current density. Also addition of Ca instead of yttrium will decrease the energy gap and as a result will decrease the critical temperature.

We can say that: Ca^{2+} substitution for Y^{3+} change the order of the single Cu-O chains as a result buckling in the CuO_{2} plane is diminished due to decrease of potential energy, therefore the CuO_{2}-CuO_{2} distance decreases as a result the holes tunneling between the two plane increases by decreasing the distances between the two potential wells of the layers. All the relation we find between Ca content and energy gap, binding energy hole concentration etc. ensures the above and in favor of the mechanism suggested for the Cooper pairing.

1) Ca addition to the YBCO compound leads to decreasing the distance between the two CuO_{2} layers as a result of diminishing of buckling in the CuO_{2} plane, and this lowers the potential energy of the system.

2) Decreasing the distance between the CuO_{2} two layers will enhance the tunneling process and enhance formation of Cooper pairs. This will increase the superconducting current.

3) Ca addition will decrease the binding energy and this will increase hole concentration and thus will increase conduction mechanism.

4) Increasing Ca content will decrease the energy gap and as a result will decrease the critical temperature.

5) Increasing superconducting current is a great practical goal, but T_{c} decreases. Our future work will concentrate on increasing current without decreasing T_{c}.

Bassam M.Mustafa,Abbas H.Rostam, (2015) Influence of Ca Content on Binding Energy and Cooper Pairing in Y_{1-x}Ca_{x}Ba_{2}Cu_{3}O_{7-δ} Compound. World Journal of Condensed Matter Physics,05,18-26. doi: 10.4236/wjcmp.2015.51003