Fig. 4: Theoretical model for coupled dynamics of electrons, spins, and lattice distortions in BiFeO3.
a Schematic of the density of states for BiFeO3. Fe(1) and Fe(2) denote Fe atoms on each sublattice of the spin-polarized FeO6 octahedra. The localized, spin-polarized Fe 3d electrons are described by the Landau-Lifshitz-Gilbert equation. Itinerant electrons below the Fermi level are primarily composed of O 2p electrons, which are simplified into a one-dimensional, two-orbital, two-sublattice tight-binding model. b Energy diagram of the theoretical model. The itinerant electrons hop to neighboring sites by a transfer integral of \(-h\) (indicated by orange double-headed arrows) and an electron-lattice coupling of \(\pm {\alpha }_{{\mbox{FE}}}{Q}_{{\mbox{FE}}}\) (green solid/dashed arrows). The ferroelectric distortion QFE lifts the degeneracy of the two orbitals in each sublattice and modulates the transfer integrals of the electrons. The chain direction of the tight-binding model is parallel to the z axis, corresponding to the c axis in the real material. The spacing of the two FeO6 sites is denoted by a. The electric polarization is proportional to the expectation value of the orbital pseudospin operator \(\left\langle {\upsilon }^{z}\right\rangle\) when a finite displacement δz of the Wannier center is considered. See Methods section for details of the model.
