  Taking the spin-fermion model as the starting point for describing the
cuprate superconductors, we obtain an effective nonlinear sigma-field
hamiltonian, which takes into account the effect of doping in the system. We
obtain an expression for the spin-wave velocity as a function of the chemical
potential. For appropriate values of the parameters we determine the
antiferromagnetic phase diagram for the YBa$_2$Cu$_3$O$_{6+x}$ compound as a
function of the dopant concentration in good agreement with the experimental
data. Furthermore, our approach provides a unified description for the phase
diagrams of the hole-doped and the electron doped compounds, which is
consistent with the remarkable similarity between the phase diagrams of these
compounds, since we have obtained the suppression of the antiferromagnetic
phase as the modulus of the chemical potential increases. The aforementioned
result then follows by considering positive values of the chemical potential
related to the addition of holes to the system, while negative values
correspond to the addition of electrons.

Superconductivity;Strongly Correlated Electrons