Long-Range Magnetic Interactions Induced by the Lattice Distortions and the Origin of the E-type Antiferromagnetic Phase in the Undoped Orthorhombic Manganites

With the increase of the lattice distortion, the orthorhombic manganites $R$MnO$_3$ ($R$$=$ La, Pr, Nd, Tb, and Ho) are known to undergo the phase transition from the layered A-type antiferromagnetic (AFM) state to the zigzag E-type AFM state. We consider the microscopic origin of this transition. Our approach consists of the two parts. First, we construct an effective lattice fermion model for the manganese 3d-bands and derive parameters of this model from the first-principles electronic structure calculations. Then, we solve this model in the Hartree-Fock approximation (HFA) and analyze the behavior of interatomic magnetic interactions. We argue that the nearest-neighbor interactions decrease with the distortion and at certain stage start to compete with the longer range (particularly, second- and third-neighbor) AFM interactions in the orthorhombic ab-plane, which lead to the formation of the E-phase. The origin of these interactions is closely related to the orbital ordering, which takes place in the distorted orthorhombic structure. The model is able to capture the experimental trend and explain why LaMnO$_3$ develops the A-type AFM order and why it tends to transform to the E-type AFM order in the more distorted compounds. Nevertheless, the quantitative agreement with the experimental data crucially depends on other factors, such as the magnetic polarization of the oxygen sites and the correlation interactions beyond HFA.