Doping dependence of the exchange energies in bilayer manganites: Role of orbital degrees of freedom

Recently, an intriguing doping dependence of the exchange energies in the bilayer manganites $La_{2-2x}Sr_{1+2x}Mn_2O_7$ has been observed in the neutron scattering experiments. The intra-layer exchange only weakly changed with doping while the inter-layer one drastically decreased. Here we propose a theory which accounts for these experimental findings. We argue, that the observed striking doping dependence of the exchange energies can be attributed to the evaluation of the orbital level splitting with doping. The latter is handled by the interplay between Jahn-Teller effect (supporting an axial orbital) and the orbital anisotropy of the electronic band in the bilayer structure (promoting an in-plane orbital), which is monitored by the Coulomb repulsion. The presented theory, while being a mean-field type, describes well the experimental data and also gives the estimates of the several interesting energy scales involved in the problem.