Mechanical properties of YBCO superconductor under high-pressure

Background/Objectives: The crystal structure of YBCO Superconductor has fascinated the material science research group. The calculation of mechanical properties of YBCO Superconductor under pressure is the main focus in this research work. Methods/Statistical analysis: Density Functional Theory calculations using Quantum ESPRESSO under high pressure increasing systematically have been calculated and performed for YBa2Cu3o7 Superconductor. 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 lattice constants, variations, volume, density, Bulk modulus (B), Young modulus (E), Shear modulus (S) and Poisson ratio (n)values are calculated under pressure up to 30 GPa for YBCO Superconductor. Voigt-Reuss-Hill Approximations, Debye Sound velocities(θD), Debye temperature(θD) and Pugh's ratio(B/G) values have been calculated under pressure for YBCO Superconductor in this research work. Novelty/Applications: A larger poisson's ratio value indicates ductile behavior and another criterion often used to differentiate between ductile and brittle behavior is the Poisson's ratio. Poisson's ratio greater than 0.26 implies the ductile behavior. With the application of elastic constants, we have obtained the Debye temperature and other physical quantities of YBa2Cu3o7 under pressure.


Theory and computational Details
The examination of strain or stress spreading in the assembly of polycrystalline with respect to two cases. If one is to calculate the average isotropic elastic moduli from the anisotropic single crystal elastic constants, one finds that the Voigt and Reuss assumptions results in the maximum values of theoretical and the minimum calculated values of the elastic moduli respectively. For specific cases orthorhombic lattices, the Reuss shear modulus (G R ) and the Voigt shear modulus (G V ) are G R = 15 4(s 11 + s 22 + s 33 ) − 4(s 12 + s 13 + s 23 ) + 3(s 44 + s 55 + s 66 ) (1) The Reuss bulk modulus (B R ) and the Voigt bulk modulus (B V ) are defined as Hence the elastic moduli of the polycrystalline material can be approximated by Hill ' s average and for shear moduli it is And for Bulk moduli it is The Young's modulus E, and Poisson's ratio(n) for an isotropic material are given by With the above relations, as implemented in thermo pw code, the values have been calculated various modulus of elasticity -Bulk modulus (B), Shear modulus (G), Young ' s modulus (E) and Poisson's ratio. The fundamental parameter, Debye temperature(θ D) is closely related to the number of atoms, density and wave velocities. From sound velocities the Debye temperature(θ D) can be deduced by following equation at low temperature.
Where h is planks constant, K B is Boltzmann constant, n is number of atoms per unit cell,N A is avagadronumber, ρ is density is molecular weight and V m ,V S and V p are the average ,longitudinal and transverse elastic wave velocities.
The calculation of the mechanical properties of YBCO under High Pressure with DFT formalism and Quantum ESPESSO (ver.6.3) have been done in this research paper. The high charge density with method using the PBE-generalized gradient approximation (GGA) for exchange-correlation are presented. The calculated pressure-relevant lattice constants, variations, volume, density, Bulk modulus (B), Young modulus (E), Shear modulus (S), and Poisson ratio (n) etc using a proper pseudo potential and with high suitable value of k-points over IBZ under high cutoff-energy and high charge density with method using the PBE-generalized gradient approximation (GGA) for exchange-correlation are presented .The computations are based on the DFT within the ultra-soft pseudo potential (11) as well as the GGA-PBE (12).

Results and Discussion
The constant distance between unit cells in a crystal lattice is known as lattice constant and The lattices constants calculated at different pressures are shown in Figure 1 and the variations versus pressure for YBCO is shown in the Figure 2. The density of a substance is mass per unit volume under certain conditions of temperature and pressure.  With exterior applied pressure or force, distortion arises due to stress and the lattice parameters of the crystal fluctuates with respect to stress with respect to strain. Strain may be defined in another way as the measure of rate of deformation. Bulk modulus https://www.indjst.org/   is related with valence electron density (electron/unit volume). Higher the bulk modulus is result of concentration of electrons so it gives the high repulsive forces. At a fixed volume, Shear modulus measures the resistance to shape change. It is related to bond bending, and depends on the direction and plane of shear. The shear modulus and bulk modulus are related by Poisson's ratio (n), which is the ratio between the transverse strains and longitude strains. The Bulk Modulus (B), Young modulus (E), Shear modulus (G) and Poisson ratio (n) are calculated at different pressures for YBCO superconductor ( Figures 5, 6, 7, 8 and 9) using Voigt Approximation, Reuss Approximation and Hill approximation.  The Bulk modulus(B),Young modulus(E),and Shear modulus(G) have been increases with increasing pressure by using voigt approximation, bulk modulus(B) decrease, young modulus(E) and shear modulus(G) have been increases with increasing pressure by using Reuss approximation and bulk modulus (B),young modulus(E) and shear modulus(G) have been increase with increasing the pressure by using Hill approximation. The Poisson's ratio decreases with increasing pressure by using voigt and Reuss approximation and increases with increasing pressure by using Hill approximation. The sound velocities have been calculated at different pressures and the sound velocities versus pressure graph is plotted for YBCO superconductor is represented in Figure 10.
Hill (13) (1952) shows, the Voigt-Reuss -hill average sound velocities are increases with pressure (14; 15) (Voigt, 1928; https://www.indjst.org/    Reuss, 1929) .The Approximate Debye temperature, Average Debye sound velocity, Debye temperature and Pugh's ratio(B/G) of YBa2Cu307 under pressure up to 30Gpa have been calculated and the results are summarized in table (5) and the corresponding values versus pressure for YBa2Cu3O7 plotted graphs are shown in Figure 11.

Summary and Conclusion
In summary, High-pressure studies provide otherwise unattainable information about the phase diagrams, mechanical properties, and electronic structure which can predict directions for search of materials with desirable properties. Density Functional Theory (DFT) calculations using Quantum ESPRESSO under high pressure increasing systematically have been calculated and performed for YBCO up to 30Gpa. The lattice constants, Bulk modulus (B), Young modulus (E), Shear modulus (S) and Poisson ratio (n) versus pressure for YBCO Superconductor have been calculated in research this work. It has been shown that the Poisson's ratio decreases with increasing pressure by using voigt and Reuss approximation and increases with increasing pressure by using Hill approximation. The Sound velocities and Pugh's ratio (B/G) have been calculated for YBCO Superconductor. It was found that the Pugh's ratio (B/G), which distinguishes between ductile and brittle response of YBCO Superconductor under applied stress. The Pugh's ratio (B/G) critical value is estimated to be 1.75 and larger value indicates ductile behavior. Both Pugh's ratio (B/G) and Poisson's ratio (n) predict moderately ductile response to stress for YBCO Superconductor. The Pugh's ratio has been calculated at different pressures. Debye temperature increases with increasing pressure and Pugh's ratio (B/G) decreases with increasing the pressure.