Theoretical high-$T_c$ d-wave superconducting gap in an inhomogeneous medium

We perform theoretical calculations to obtain a distribution of local d-wave superconducting gaps $\Delta_0({r})$ for a high temperature superconducting (HTSC) series in a disordered superconductor with an average doping level $<\rho>$. To reproduce the inhomogeneous medium a nonmagnetic random potential $V^{imp}$, within a Bogoliubov-de Gennes (BdG) formalism, is considered. First the phase diagram $\Delta_0 x <\rho>$ for the LSCO HTSC series, with V^{imp}=0, is obtained. Then, we perform calculations considering a fixed value of the disorder strength $V^{imp}$ and obtain a distribution of local superconducting gaps $\Delta_0({r})$, and local density of charge carriers $\rho({r})$. It is shown that the underdoped compounds are more inhomogeneous than the overdoped ones, which is in accordance with experimental findings. Also, the spatial variation of $\Delta_0({r})$ indicates that as <\rho> increases, the system becomes more homogeneous.