S-wave superconductivity phase diagram in the inhomogeneous two-dimensional attractive Hubbard model

Abstract

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 ${U}_{i}$ between the electrons takes on two values ${U}_{i}=0$ 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 ${T}_{c}$, 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 ${T}_{c}$ 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 ${T}_{c}$.