_{2}Cu_{3}O_{7 }(YBCO) is investigated using the first principle calculations based on the density functional theory (DFT). The only input required is the lattice parameters at corresponding pressure of materials which are predicted using first principle computational methods at desired high-pressure state.

The discovery of the Ba-La-Cu-O system by Muller and Bednortz ^{1} with a T_{sc} of 30K has generated and further research has led Wu et al to the discovery of T_{sc} of 90 K. In 1964, Schooley and coworkers first reported superconductivity in SrTiO_{3}, an oxide with parvoskite crystal structure, with a quite low transition temperature, T_{c} = 0.3K. In 1975, Coworkers and Sleight found high transition temperature at 13K in BaPb_{1-x}Bi_{x}O_{3. }In 1986, Bednortz and Muller reported a remarkable superconducting transition at 30 K in LaBaCuO_{3} (LBCO). Almost one year later, Wu and colleagues reported superconductivity in YBa_{2}Cu_{3}O_{7 }(YBCO), with Tc= 90 K. The discovery of the Ba-La-Cu-O system by Bednortz and Muller with a superconducting transition temperature of 30K has generated a great deal of tremendous interest among physicists_{3}, an oxide with parvoskite crystal structure, with a quite low transition temperature, T_{c} = 0.3K. In 1975, Coworkers and Sleight found high transition temperature at 13K in BaPb_{1-x}Bi_{x}O_{3. }In 1986, Bednortz and Muller reported a remarkable superconducting transition at 30 K in LaBaCuO_{3} (LBCO). Almost one year later, Wu and colleagues reported superconductivity in YBa_{2}Cu_{3}O_{7}, with Tc= 90 K ^{3}. The pressure dependence on sound velocities and elastic modulii ^{4}, pressure dependence of specific heat and Debye temperature of internal friction and linear thermal expansivity. The anomalous stiffing below critical temperature observed in poly crystals was also reported in single crystals. As for the mono crystalline elastic constants, only one set of complete Cij derived from the phonon frequencies of an elastic scattering for a tetragonal structure was reported ^{5}. Using sound velocity measurements, Saint Paul and coworkers_{33} and C_{44}. Also, from sound velocities, Golding and coworkers calculated the pseudo tetragonal elastic constants C_{11} and C_{33. }Baumgart and coworkers_{11,} C33_{ }and C_{44. }The elastic properties of YBCO superconductor were studied by Vetebskii and coworkers. Starting from the mean field energy, Millis and Rabe derived expressions for the singularity’s behavior of the elastic constants and sound velocities near Tc. Calculation of Elastic constants, enthalpy and final energy values at different pressures for YBa_{2}Cu_{3}O_{7} (YBCO) Superconductor is the main focus in the present study.

Furthermore, one can also directly obtain some useful information on the characteristics of bonding and the structural stability of a crystal. The elastic constants C_{ijkl} with respect to the finite strain variables is defined as ^{8}

Where σ_{ij} and e_{kl} are the applied stress and Eulerian strain tensors and X and x are the coordinates before and after the deformation. For the isotropic stress, we have^{ 9}

Where C_{ijkl} denotes the second-order derivatives. The Strain energy – strain curve has been used for calculation of elastic constants purpose. All the elastic constants increase with pressure. For orthorhombic crystal the mechanical stability is represented by the following condition ^{10}.

The superconductor having tetragonal symmetry considered here comes under the second category and the six nonzero elastic constants for it are C_{11,} C_{33,}_{ }C_{44}, C_{66}, C_{12} and C_{13}. The equations for the wave propagation in the tetragonal crystal are very easily obtained from the equations for the orthorhombic crystal by making the substitutions C_{11}=C_{22}, C_{44}=C_{55} and C_{13 }= C_{23}. Enthalpy of formation was assessed from GGA+U ab- initio Density Functional Theory (DFT) based calculations. The electronic structure and total energies of YBa_{2}Cu_{3}O_{7} and the constituent oxides, YO_{5}, Ba_{2}O, and Cu_{3}O, were calculated by means Medea-VASP ^{11} program using projector augmented plane waves basis set and generalized gradient approximation (GGA-PBE)^{ }

Knowledge of elastic constants are significant for understanding of the structural stability and properties of the crystal. The elastic constants under different pressures obey these stability criteria, implying that the orthorhombic YBa_{2}Cu_{3}O_{7} is mechanically stable. The present results at ground state C_{11}=1580Gpa, C_{12}= 713Gpa, C_{13=}278Gpa, C_{22= }-385Gpa, C_{23}=-1177Gpa, C_{33}=1355Gpa, C_{44}=449Gpa, C5=328Gpa, C66=668Gpa and at 20Gpa C_{11}=5348, C_{12}= 2409Gpa, C_{13}=1888Gpa, C_{22}=5804Gpa, C_{23}=1729Gpa,C_{33}=4813Gpa, C_{44}=1021Gpa, C_{55}=1090Gpa, C_{66=}1331Gpa. C_{11}, C_{22}, and C_{33} represent the resistance to linear compression, and the other elastic constants are mainly associated with the elasticity in shape. In the entire pressure range of our calculations, C_{11}= 5384Gpa, C_{22}= 5804Gpa and C_{33}= 4813Gpa, were much larger than those of the other elastic constants. Further, the relationship c_{22} > c_{11} > c_{33} under different pressures implied that the strength of the bonding along the [010] direction was stronger than those along the [001] and [100] directions. The calculated values of Elastic constants at different pressures are given in

As can be seen from _{44}/C_{11}-C_{12}. The difference between single crystalline elastic constants, C_{12} – C_{44 }is well-known as the Cauchy pressure. A positive value of C_{12-}C_{44} indicates the metallic bonding, whereas negative value significances covalent bonding. The Cauchy pressures positive value always indicates ductile nature, while a negative value corresponds to brittleness. The calculated values of enthalpy and final energy up to 30GPa are given in ^{-1} and -2086.65 at ground state and -2084.43KJ mol^{-1} and -2086.60 at 30GPa. From these results it is clear that the enthalpy and final energy of YBCO superconductor increases with pressure.

Pressure | C11 | C12 | C13 | C22 | C23 | C33 | C44 | C55 | C66 |

0 | 1,580 | 713 | 278 | -385 | -1,177 | 1,355 | 449 | 328 | 668 |

10 | 4,345 | 1,974 | 1,413 | 4,726 | 1,377 | 3,877 | 889 | 886 | 1,152 |

20 | 5,348 | 2,409 | 1,888 | 5,804 | 1,729 | 4,813 | 1,021 | 1,090 | 1,331 |

Pressure (GPa)
Minimum enthalpy (KJ mol-1)
Final energy of the fully relaxed structure
0
-2,086.638
-2,086.65
10
-2,085.81
-2,086.64
20
-2085.156659
-2086.630299
30
-2,084.43
-2,086.60

In summary, the pressure dependence of the elastic constants of the YBa_{2}Cu_{3}O_{ 7 }(YBCO) is investigated using the first principle calculations based on the density functional theory (DFT). The elastic constants C_{11}, C_{22}, C_{33}, C_{44}, C_{55}, C_{66}, C_{12}, C_{13} and C_{23} are calculated at different pressures. In the entire pressure range of our calculations, C_{11}= 5384Gpa, C_{22}= 5804Gpa and C_{33}= 4813Gpa, were much larger than those of the other elastic constants, indicating that the deformation resistances of YBa_{2}Cu_{3}O_{7} along the axial directions were stronger than those of the non-axial directions. The elastic constants C_{11}, C_{12} and C_{44} represent the elasticity in length and shape. All the elastic constants increase with pressure. The Enthalpy and final energy of the YBCO Superconductor has also been calculated by using GGA+U ab- initio calculations up to 30GPa. The enthalpy and final energy of YBCO superconductor increases monotonically with pressure.

The authors thank the management at Noida International University for the support to carry out the work.