J. Zieliński

Modification of the isotope effect by the van Hove singularity of electrons on a two-dimensional lattice

We discuss the problem of phonon-induced superconductivity of electrons on a two-dimensional lattice. Analytical expressions for the superconducting transition temperature and the isotope coefficient have been derived for the case that the Fermi level is close to the van Hove singularity. We have considered the weak- and strong-coupling cases. Our results show that in the presence of the van Hove singularity the isotope coefficient can take on small values despite the fact that superconductivity originates from phonon-induced pairing.

Superconductivity in the presence of pseudogap: strong-coupling approach

We discuss the structure of the pseudogap, which originates from the charge-density wave (CDW). For imperfect nesting the density of states at Fermi level remains finite despite the opening of the CDW gap. Its depth depends on the concentration of holes and decreases with doping. This property can significantly affect the competition between CDW and superconductivity. We apply the Eliashberg equations and obtain analytical results for the superconducting transition temperature in the presence of the pseudogap of arbitrary magnitude and depth. We demonstrate that the isotope effects for the superconducting transition temperature (α) and the pseudogap (α∗) are inversely correlated with each other. In particular, doping dependence of the superconducting isotope effect observed in underdoped high-temperature superconductors, can originate from opening of the normal-state pseudogap, provided that α∗<1/2.

Anisotropic superconductivity in the two-dimensional Hubbard model

We address the problem of anisotropic superconductivity in the two-dimensional Hubbard model. The Eliashberg equations have been generalized to the case which accounts for the anisotropy of the order parameter. Strong local correlations are treated within the mean field slave boson approximation. The superconducting transition temperatureTc is evaluated as a function of the occupation number. Our results indicate that thed-wave state is the most likely channel for superconductivity for small concentration of holes. We have also derived an approximate analytical formula forTc valid for any value of the occupation number. In addition, the influence of strong correlations on the electron-phonon coupling function is also discussed.

Eliashberg equations with momentum-dependent Kernels for the two-dimensional Hubbard model

We discuss the impact of the strong electron-phonon interactions on the physical properties of correlated electrons in the two-dimensional Hubbard model. An easy access to the physical properties of the superconducting state is obtained by solving the Eliashberg equations with explicit momentum-dependent kernels. In order to facilitate the numerical procedure to solve these equations, correlations are treated within the Gutzwiller approximation (i.e., mean-field-slave boson approximation). Our results strongly support the view thatd-wave symmetry is the most important feature of the superconducting state in the copper oxides. A possible extension beyond the mean field approximation is also presented.

Phonon-Induced Contributions to Superconductivity in the Presence of Charge-Density Waves

We discuss the influence of charge-density wave (CDW) order on the electron–phonon vertex and the Cooper-pair susceptibility. These quantities are investigated for different values of occupation numbers, phonon momenta and frequencies, as well as for different directions and commensurabilities of the CDW scattering vector. For the half-filled band, the onset of a CDW state results in a considerable enhancement of the electron–phonon interaction, especially when the phonon momentum is close to the scattering vector. Our results suggest that for imperfect nesting different parts of the Fermi surface contribute to a CDW and d-wave superconducting state. This feature is of particular importance for the coexistence of both phases.

Phonon-mediated pairing correlations in the generalized Hubbard model

Abstract We investigate phonon-mediated paring interactions for correlated electrons described by the Hubbard model. In order to eliminate the electron–phonon coupling we apply a Frohlich-type unitary transformation. This leads to a Hubbard model with pairing potentials which depend on the details of the Fermi surface and the magnitude of the local Coulomb repulsion, U . We find that the isotropic part of the pairing potential is reduced due to the Coulomb interaction. The same effect can be observed when considering the charge–density–wave correlations, while the anisotropic part of the pairing interaction is maximal for moderate values of U . Numerical work is presented for the case of an underlying two-dimensional Hubbard model.

d-Wave Superconductivity on a Square Lattice: Higher-Order Effects

We have considered the two-dimensional model for d-wave superconductivity. In our formulation the origin of the superconducting pairing remains unspecified and the nearest-neighbor density–density attraction has been assumed in the model calculations. We have evaluated the superconducting transition temperature Tc as a function of the occupation number within the self-consistent Eliashberg scheme. In the first order, one rederives the BCS-like result. The self-consistent inclusion of the second-order contributions leads to the pronounced reduction of Tc when compared to the BCS value. This result may be of significance when looking for models relevant for high-temperature superconductors.

Charge-Density-Wave Pseudogap and Two-Dimensional Superconductivity Within Strong-Coupling Description

We have considered the role of charge-density-pseudogap for phonon-mediated super-conductivity on two-dimensional lattice. The propagators that enter generalized Eliashberg equations have been renormalized to account for quasi-particle energies related to the formation of the pseudogap, which has been assumed to be of d-wave symmetry. We have evaluated the superconducting transition temperature Tc as a function of doping. It occurs that Tc for d-wave symmetry well reflects experimental behavior. Our results for the isotope shift exponent α show that at low doping, the presence of the pseudogap may contribute to α > 1/2 as well as to α > 1/2 values.

Phonon-Induced Superconductivity and the Staggered Magnetic Field on a Square Lattice

We have considered phonon-induced superconductivity on a square lattice. The antiferromagnetic correlations that are present in high-temperature superconductors (HTSC) have been simulated by the staggered magnetic field. The Eliashberg equations have been generalized to account for the momentum dependent order parameter, and the superconducting transition temperature Tc has been evaluated within the strong-coupling theory. The two-dimensional band structure has explicitly been taken into account. It occurs that the results obtained for Tc when considering d-wave symmetry of the superconducting state much better reflect the experimentally observed behavior than those derived for s-wave symmetry. The isotope shift exponent α has also been evaluated. One can observe that the introduction of the staggered magnetic field leads to the decrease of this quantity. Therefore, the presence of the staggered magnetic field may contribute to experimentally observed values of α smaller than the BCS value of 1/2

D-Wave Superconductivity and Coulomb Correlations in the Two-Dimensional Hubbard Lattice

We considered anisotropic superconductivity within the two-dimensional Hubbard model extended by pairing correlations originating from the electron–phonon interaction. To discuss the onset of superconductivity close to the insulator–metal transition, we used the Hubbard I approximation to account for the formation of the insulating gap and see the role of Coulomb correlations for superconducting pairing. It has been shown that the Hubbard I approximation reflects effective pairing interactions genuine for correlated electron systems and leads to the stabilization of the superconductivity in the d-wave channel. One may expect the cooperation of phonon-free and phonon-induced mechanism in the formation of thed-wave superconducting state.