The Second-order And Third-order Elastic Constants And Of MgB₂ Under Different Pressure:First-principles Methods

The discovered of superconductivity in the Mg B2 was in the early of 2001, its highest transition temperature Tc is 39 K. Since then, people have had a keen interest in the study of physical properties. To the knowledge of authors, the third-order elastic constants(TOECs) in Mg B2 is lack of study in the previous literatures, although it is significant to understand the superconducting behavior. TOECs are not only used in describing mechanical phenomena when applying large stress or strain, but also can be used to describe the anharmonic properties.Elastic properties is the basic physical properties of solid materials, and elastic constants is well reflected in the physical properties of the solid.In this paper,The second-order elastic constants(SOECs) and third-order elastic constants(TOECs) of superconducting Mg B2 are investigated by first principles methods combined with homogeneous deformation theory.we focus on elastic properties and other physical properties,such as the Debye temperature.Anderson-Gruneisen parameter, brittle-to-ductile and other physical parameters characteristics.The detailed work of this paper as follows:① We focus on the SOECs and TOECs of Mg B2 under zero pressure using the first-principles,consistent with the experimental data and the other theoretical results.Though it is a pity that there are not find experimental or theoretical results to check the TOECs under zero pressure.we also can trust the results because the SOECs coincide with the previous values.As,suggested by Rao,the Anderson-Gruneisen parameter δ can now be calculated from the SOECs and TOECs.The calculated result is in agreement with the experimental datas.②The effective elastic constants as a function of pressure can be obtained from the SOECs and TOECs under different and consistent with the other theoretical results.Due to the hexagonal structure of Mg B2,.the calculated bulk modulus Ba and Bc agree with the previous results.It is found that Ba/Bc increases slightly with the pressure.The anisotropy fact can be obtained from the effective SOECs.The results are agree well with those obtained by others.The calculated bulk modulus B and shear modulus G are the same as anisotropy.The bulk modulus B and shear modulus G as a function of pressure increases monotonically with increasing pressure.Finally,the Debye temperature,brittle-to-ductile and hardness of the superconducting Mg B2 under different pressure are discussed.The Debye temperature agrees with the theoreticalresults, but which is higher than the experimental values and increases monotonically with increasing pressure.The frequency of crystal lattice vibration can be estimated from the Debye temperature.In this article the calculation of the pressure range(0-30GPa), the relationship between ratio Pugh, two Cauchy pressures and pressure are discussed, we can predict that the superconducting Mg B2 is brittleness.We obtain the maximum Pugh ratio G/B=0.722 when pressure is 2.83 GPa.Pugh ratio increases monotonically with increasing pressure range from 0GPa to 2.83 GPa.Pugh ratio decreases monotonically with increasing pressure if the pressure value is more than2.83 GPa.The brittleness of Mg B2 is translated to plasticity because of the pressure too high or too low. There three models can calculate the hardness of brittle material,Comparing the best model with the other two have anything different, we can predict that the hardness of Mg B2 is directly related to its pressure and increases monotonically with increasing pressure.