Application of finite element methods to the simulation of high temperature superconductors

Superconductivity is an important physical phenomenon which has not yet been completely explained. There are two types of superconductors: conventional and unconventional (high temperature superconductors belong to the latter). The temperature at which electrical resistivity vanishes is called the critical temperature (T_c) and it is a characteristic of each superconductor. A physical explanation of superconductivity in the conventional superconductors is given by the BCS theory, with which critical temperatures can be calculated. This theory is based on the Coulomb interaction between Fermi electrons (free electrons) and atoms in the crystal structure of the material. Through this interaction, two electrons can be paired, building a so-called Cooper pair. In this work, the finite element method has been used to simulate the BCS theory in conventional and unconventional superconductors (Al, Nb: conventional; Sr₂RuO₄, La_1.85Sr_0.15CuO₄, Bi₂Sr₂CaCu₂O_8+δ, HgBa₂Ca₂Cu₃O_8+δ and YBa₂Cu₃O_7-δ: unconventional). The corresponding Fermi velocity v_e (velocity of a free electron), density of states D(E_f), force constants between atoms in the unit cell, and the symmetry of the unit cell are required for each simulation. The force constants can be measured by Raman spectroscopy or calculated with the potential theory. With these parameters, the attractive potential energy V₀ caused by an electron moving through the unit cell can be simulated by the finite element method. This process has been carried out for the two conventional superconductors Al and Nb. According to the BCS theory, the binding energy Δ of a Cooper pair depends on the density of states and the attractive potential energy according to the following equation: Δ ∝ exp ((-2)/(D(E_f)V₀)). Then, the critical temperature T_c can be found with the following equation where k_b is Boltzmann constant: T_c = (2Δ)/(3.53k_b). The results show that the Fermi velocity and the density of states play an important role regarding superconductivity. The lower the Fermi velocity v_e, the higher the attractive potential energy V₀ and the lower the density of states D(E_f). According to the first equation, this means that a reduction of the Fermi velocity yields an increase of the binding energy through increasing the attractive potential energy V₀. On the other hand, a reduction in the Fermi velocity yields to a reduction in the binding energy through reducing density of states. Because of these two dependencies, an optimal Fermi velocity must be found for which the binding energy reaches its maximum. Accordingly, and using the second equation, a maximum value is found for the critical temperature T_c. For conventional superconductors the Fermi velocity can be approximated as a constant. Therefore, according to the two equations above, only a critical temperature can be calculated for these superconductors. For high temperature superconductor cuprates (which belong to the unconventional superconductors and are doped with different concentrations of foreign atoms or oxygen vacancies), the Fermi velocity changes linearly with the doping amount δ. Thus, different critical temperatures exist for different doping amounts. The phase diagrams of high temperature superconductors (T_c vs. doping amount δ) show a dome with a maximum critical temperature at an optimal doping. Because the Fermi velocity depends linearly on the doping amount δ, the latter can be replaced in the phase diagram by the Fermi velocity and so an optimal Fermi velocity can be used instead of an optimal doping. A linear relationship of 2Δ = 2.865T_c for some unconventional and conventional superconductors has also been found in this work, which is in good agreement with the experimentally determined linear relationship of C. Panagopoulos and T. Xiang (2Δ = 2.214T_c).