Computation of the Expectation Value of the Electron – Phonon and Coulomb Interaction Hamiltonian using Second Quantization and Many Body Techniques

Abstract

The problem of the strong electron coupling in the theory of superconductivity has been discussed in many studies on the basis of electron and phonon spectral functions. For the rationale of investigating the effect of electron – phonon and Coulomb interactions in cuprate superconductors, this study has considered a Hamiltonian (Hepc) which includes both Coulomb interactions and electron-phonon interactions. The expectation value of this Hamiltonian was evaluated using second quantization and many body techniques. The equation for the energy of the system at ground state was obtained from the product of the expectation value and the thermal activation factor, exp (-E1/kT). The equation relating specific heat and absolute temperature was obtained from the first derivative of the energy of the system at ground state with respect to absolute temperature. The equation relating entropy and absolute temperature was obtained from the specific heat equation. Values of specific heat and entropy against absolute temperature were calculated. In these calculations, the onsite energy of oxygen (Ep) was fixed at 2.0 x 10-6 eV. The onsite energy of copper (Ed), hybridization energy of oxygen and copper bands (tpd), the electron – phonon interaction energy, (gep) and energy due to repulsion of copper holes occupying the same orbital (ud), were varied. From the results, it was found out that increase in the parameters Ed, tpd, gep and ud leads to increase in the transition temperature. It was further found that entropy and specific heat decrease with increase in the parameters. It can therefore be concluded that long range electron – phonon and local Coulomb interactions increase the transition temperature of superconducting cuprates.