Microscopic inhomogeneity and superconducting properties of a two-dimensional Hubbard model for high-$T_c$ cuprates

Recent scanning tunneling microscopy measurements on cuprate superconductors have revealed remarkable spatial inhomogeneities in the single-particle energy gap. Using cellular dynamical mean-field theory, we study the zero temperature superconducting properties of a single-band Hubbard model with a spatial modulation of the electron density. We find that the inhomogeneity in the electronic structure results in a substantial spatial variation in the superconducting order parameter and single-particle energy gap, reminiscent of the experimental results. In particular, we find that the order parameter and gap amplitudes in the hole-rich regions are significantly enhanced over the corresponding quantities in a uniform system, if the hole-rich regions are embedded in regions with smaller hole density.

Figure 1

Schematics of the potential modulation. Left: checkerboard-type modulation and right: stripe-type modulation. On the sites belonging to a $2\times 2$ plaquette denoted by $+(-)$, the on-site potential is $+V$ $\left(-V\right)$.

Figure 2

(Color online) The superconducting order parameter $\Psi$ as a function of carrier density $N$ for the 2D Hubbard model computed by CDMFT with the ED impurity solver at $T=0$. Black symbols are the results for the uniform systems. Colored (shaded) symbols connected by a thin line denote the results with a checkerboard-type potential modulation (upper panel) and a stripe-type potential modulation (lower panel).

Figure 3

(Color online) Local DOS of the Hubbard model with the checkerboard-type potential modulation. Each panel shows the local DOS in the clusters with smaller $N$ and larger $N$ for a given potential modulation. Down (up) triangles indicate the low-energy structure for the smaller (larger) $N$ region. To compute the self-energy, a small imaginary number $\eta =0.125t$ was added to the real frequency $\omega$.

Figure 4

(Color online) $d$-wave gap amplitude $\Delta$ as a function of the average carrier density $N$ for the 2D Hubbard model computed by the CDMFT with the ED impurity solver at $T=0$. Black symbols are the results for the uniform systems. Colored (shaded) symbols connected by a thin line denote the results with a checkerboard-type potential modulation.