A STUDY OF THE THEORIES OF SUPERCONDUCTIVITY AND THE APPLICATION OF MATLAB IN THE EXACT NUMERICAL ANALYSIS OF THE 2-DIMENSIONAL ATTRACTIVE- AND REPULSIVE- HUBBARD MODEL

摘要

In order to improve on the present understanding of the superconductivity phenomenon in general, especially of the models of HT_c-superconductors, most of which contain Cu-O, in which the O(2p) and Cu(3d) electron states determine certain unusual properties such as the coexistence of antiferromagnetism and superconductivity, a detail and systematic quantum mechanical/mathematical survey and analysis is presented and discussed up to the present state of the art. Furthermore, in trend with the search for appropriate tool(s) that would ensure accurate analysis of proposed models for HTS, which in turn will enhance the attainment of the much desired theory that will adequately account for HT-superconductivity, we developed a program in MATLAB which is applied in the exact numerical evaluation of S(T), F(T), E(kT), C_v(T) and χ(T) and their dependence on (U,t) in 2-dimensional Cu-O sheets of HTS's. This system is simply described by a 16 x 16 matrix representation of the 2D-Hubbard Hamiltonian at 1/2 -filling. A detailed analysis which is separately carried out for both the Attractive Hubbard model, AHM (U ≤ -4eV), as well as the Repulsive Hubbard model, RHM (U ≥ 4eV), suggests that T_c's (RHM) > T_c's (AHM), and that the AHM does not only afford a more stable state than the RHM, but also sustains its stability over a wider temperature range. It is also observed that the RHM exhibits paramagnetic contributions while the AHM gives clear signatures of antiferromagnetic ordering of the Cu~(2+) spin in HTS. The structure of C_v(T), which are associated with "charge fluctuation" and "spin fluctuation", is accounted for from a strong coupling viewpoint; while the narrowness of the crossing region is traced ultimately to the linear temperature dependence of the double occupancy. Furthermore, the structure of C_v(T) is also associated with "pseudogap" formation i.e. the onset of short-range antiferromagnetic correlation between nearest-neighbours. The limits of the critical value of U, below which pair formation begin to occur are obtained as U_p∈(-2eV, -1eV] for the AHM, and U_p∈[2eV, 3eV) for the RHM for all HTS. In addition, correlations between U and H_a and U and H_(c2) for RHM; and between U and δ for AHM are suggested. Finally, it is noted that for -0.137eV ≤ U ≤ 0.137eV i.e for |U|～ 0, the curves of S(T), F(T), E(kT), C_v(T) and χ(T) show similar trend and character with the uncorrelated case, U=0. Generally, all the results obtained in this work are in good agreement with experiment and compare very favourably with other numerical methods like the Bethe ansatz techniques, the Quantum Monte Carlo studies of the 2-dimensional and 3-dimensional Hubbard models, quantum transfer matrix calculation and the dynamic mean field theory. The applicability and high resolution of MATLAB even at very low temperature is demonstrated, and we showed that it is also a better facility for analysis than the FORTRAIN. The implication of these results to the study of HT-superconductivity is discussed.