Correlated electrons in manganites

Manganese ions in a mixed valent state of two magnetic configurations, Mn⁴⁺ and Mn³⁺, play an important role in the physics of LaMnO₃-based systems. Each Mn ion has three localized $3d$ electrons in the t2g orbitals with their spins ferromagnetically coupled to form a spin $S=3/2.$ Ions in the Mn³⁺ configuration have an additional localized 3d-eg electron to form a total spin $(S+1/2).$ The eg electrons are allowed to hop between the Mn sites with hopping amplitude t, but the multiple occupancy of the eg levels is excluded at each site by a large Coulomb energy. The hopping of the eg electrons gives rise to the ferromagnetic double exchange mechanism, which competes with the antiferromagnetic superexchange J between the t2g spins of the ions. Using a mean-field slave-boson formulation we calculate the ground-state energy of the system for the spins S oriented in the magnetic configurations of the A, B, C, and G phases of La_1-xCaₓMnO₃ as a function of x and one model parameter, $J/t.$ The correct sequence of crossovers between ground states with different spin configurations is found as a function of x for a $J/t$ of a few percent. The agreement with the experimental phase diagram is improved if in addition the change of the hopping t with the bond length is included. The spontaneous magnetization and the effects of a magnetic field are also discussed.