Temperature-dependent phase separation within the Emery model

The variational principle of Bogoliubov, combined with a canonical transformation, is used to derive the thermodynamic functions for the hole-doped two-dimensional Emery model of the copper-oxygen planes in high-temperature-superconducting (HTSC) materials. A local ansatz and a transformation optimized for small values of the ratio of the hopping constant to the Coulomb correlation gives the lowest free energy in the neutral case and in the case of moderate doping. The regions of thermodynamic instability in the curves of the chemical potential versus doping, as coexistence regions of phase separation, are determined numerically with use of a Maxwell construction. In the high-doping regime, the coexistence regions are reduced considerably by the condition that the strong-coupling result is the lowest one, better than, e.g., the standard Hartree-Fock solution. The phase diagram of paramagnetic, ferromagnetic, antiferromagnetic, and separated phases is given. The parameter dependence of the maximum temperature of phase separation shows the same tendency as the empirical parameter dependence of the HTSC transition temperatures