First principle energy calculation of YBCO superconductor

Background/Objectives: The crystal structure of YBCO Superconductor has fascinated the material science research group. The calculation of elastic constants, enthalpy and ﬁnal energy of YBCO superconductor under pressure up to 30 GPa are further investigated in this research study. Methods : 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 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. Findings : The elastic constants, enthalpy and ﬁnal energy of the YBCO Superconductor has been calculated by using ab- initio calculations. The enthalpy formation is the ﬁrst value considered from the Density Functional Theory (DFT). Applications : This will explain elastic constants accurately as a function of composition. This enables the use of material optimization techniques to develop new materials that are systematically adapted to speciﬁc components.


Introduction
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 (2) . 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 https://www.indjst.org/ 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 (6) estimated C 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 (7) determined C 11, C33and 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.

Theory and computational details
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 i jkl with respect to the finite strain variables is defined as (8) Where σ i j 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 i jkl 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) (12) to exchange-correlation functional combined with additional local orbital specific coulomb potential U = 3eV acting on Cu-3d states. The first value considered from Density Functional Theory (DFT) is Enthalpy formation. The final energy of the YBCO Superconductor has been calculated by using ab-initio calculations. https://www.indjst.org/

Results and Discussion
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 Table 1 and corresponding values versus pressure for YBCO superconductor is shown in Figure 1.
As can be seen from Table 1, all the elastic constants increase with pressure. The anisotropy factor A was obtained from the relation A=2C 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 Table 2.f The enthalpy and final energy versus pressure for YBCO superconductor is shown in Figure 2. The minimum enthalpy and final energy is -2086.638KJ mol -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.

Summary/Conclusion
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.