  In this work we study the competition or coexistence between charge density
wave (CDW) and superconductivity (SC) in a two-band model system in a square
lattice. One of the bands has a net attractive interaction ($J_d$) that is
responsible for SC. The model includes on-site Coulomb repulsion between
quasi-particles in different bands ($U_{dc}$) and the hybridization ($V$)
between them. We are interested in describing inter-metallic systems with a
$d$-band of moderately correlated electrons, for which a mean-field
approximation is adequate, coexisting with a large $sp$-band. For simplicity,
all interactions and the hybridization $V$ are considered site-independent. We
obtain the eigenvalues of the Hamiltonian numerically and minimize the free
energy density with respect to the relevant parameters to obtain the phase
diagrams as function of $J_d$, $U_{dc}$, $V$, composition ($n_{\mathrm{tot}}$)
and the relative depth of the bands ($\epsilon_{d0}$). We consider two types of
superconducting ground states coexisting with the CDW. One is a homogeneous
ground state and the other is a pair density wave where the SC order parameter
has the same spatial modulation of the CDW. Our results show that the CDW and
SC orders compete, but depending on the parameters of the model these phases
may coexist. The model reproduces most of the experimental features of high
dimensionality ($d>1$) metals with competing CDW and SC states, including the
existence of first and second-order phase transitions in their phase diagrams.
