  It has been argued that inhomogeneity generally can enhance superconductivity
in the cuprate high-Tc materials. To check the validity of this claim, we study
d-wave superconductivity on the checkerboard Hubbard model on a square lattice
using the Cellular Dynamical Mean Field theory method with an exact
diagonalization solver at zero temperature. The d-wave order parameter is
computed for various inhomogeneity levels over the entire doping range of
interest in both strong and weak coupling regimes. At a given doping, the size
of the d-wave order parameter manifests itself directly in the height of the
coherence peaks and hence is an appropriate measure of the strength of
superconductivity. The weak coupling results reveal a suppression of the order
parameter in the presence of inhomogeneity for small to intermediate hole
dopings, while it is enhanced for large dopings. In contrast, for strong
coupling there is a monotonic decrease in the maximum amplitude of the
superconducting order parameter with inhomogeneity over the entire doping range
of interest. Furthermore, at moderately high inhomogeneity, the system
undergoes a first-order transition from the superconducting to the normal state
in the underdoped regime. In the overdoped regime, the change in the value of
the superconducting order parameter correlates with the height of the lowest
energy peak in the spectral weight of antiferromagnetic spin fluctuations,
confirming the connection between antiferromagnetic fluctuations and d-wave
superconductivity found in earlier studies on the homogeneous case. Our results
are benchmarked by comparisons with numerically exact results on the
checkerboard Hubbard ladder.
