In the case of superconductors whose electrons attract each other only if they are near certain centers, the question arises of how many such centers are needed to make the ground state superconducting? We shall examine it in the context of a random U Hubbard model. In short, we study the case where
Ui is $\ensuremath{-}|U|$ and 0 with probability c and $1\ensuremath{-}c,$ respectively, on a lattice whose sites are labeled i using the Gorkov decoupling and the coherent potential approximation. We argue that for this model there is a critical concentration
c0 below which the system is not a superconductor.
We study
s-wave superconductivity in the two-dimensional square-lattice attractive Hubbard Hamiltonian for various inhomogeneous patterns of interacting sites. Using the Bogoliubov--de Gennes mean-field approximation, we obtain the phase diagram for inhomogeneous patterns in which the on-site attractive interaction
Ui between the electrons takes on two values
Ui=0 and $\ensuremath{-}U∕(1\ensuremath{-}f)$ (with
f the concentration of noninteracting sites) as a function of average electron occupation per site,
n, and study the evolution of the phase diagram as
f varies. In certain regions of the phase diagram, inhomogeneity results in a larger zero-temperature average pairing amplitude (order parameter) and also a higher superconducting critical temperature
Tc, relative to a uniform system with the same mean interaction strength (${U}_{i}=\ensuremath{-}U$ on all sites). These effects are observed for stripes, checkerboard, and even random patterns of the attractive centers, suggesting that the pattern of inhomogeneity is unimportant. The phase diagrams also include regions where superconductivity is obliterated due to the formation of various charge-ordered phases. The enhancement of
Tc due to inhomogeneity is robust as long as the electron doping per site,
n, is less than twice the fraction of interacting sites $[2(1\ensuremath{-}f)]$ regardless of the pattern. We also show that for certain inhomogeneous patterns, when $n=2(1\ensuremath{-}f)$, increasing temperature can work against the stability of existing charge-ordered phases for large
f and, as a result, enhance
Tc.
Inhomogeneous
s-wave superconductivity is studied in the two-dimensional, square lattice attractive Hubbard Hamiltonian using the Bogoliubov-de Gennes (BdG) mean field approximation. We find that at weak coupling, and for densities mainly below half-filling, an inhomogeneous interaction in which the on-site interaction
Ui takes on two values,
Ui=0,2U results in a larger zero temperature pairing amplitude, and that the superconducting
Tc can also be significantly increased, relative to a uniform system with
Ui=U on all sites. These effects are observed for stripe, checkerboard, and even random patterns of the attractive centers, suggesting that the pattern of inhomogeneity is unimportant. Monte Carlo calculations which reintroduce some of the fluctuations neglected within the BdG approach see the same effect, both for the attractive Hubbard model and a Hamiltonian with
d-wave pairing symmetry.
We study the phase diagram for the attractive ({\it i.e.,} negative-
U) Hubbard model on a simple cubic lattice, through Monte Carlo simulations. We obtain the critical temperature,
Tc, for superconductivity from a finite-size scaling analysis of the data for the pairing correlations. For fixed on-site attraction,
U,
Tc displays a maximum near the filling factor 0.9, roughly independent of
U. For fixed filling we estimate the crossover temperature
T×(U), separating the normal states: metallic and spin-gap. There is also a critical value
Up for pair formation, the magnitude of which seems to be independent of doping. The relevance of these results to the high-
Tc oxides is discussed.
In case of superconductors whose electrons attract each other only if they are near certain centers, the question arises 'How many such centers are needed to make the ground state superconducting?' We shall examine it in the context of a random U Hubbard model. In short we study the case where U_i is -|U| and 0 with probability c and 1-c respectively on a lattice whose sites are labelled i using the Gorkov decoupling and the Coherent Potential Approximation (CPA). We argue that for this model there is a critical concentration c_0 below which the system is not a superconductor.