Magnetic interactions in ${\mathrm{BiFeO}}_3$: A first-principles study

First-principles calculations, in combination with the four-state energy mapping method, are performed to extract the magnetic interaction parameters of multiferroic ${\mathrm{BiFeO}}_3$. Such parameters include the symmetric exchange (SE) couplings and the Dzyaloshinskii-Moriya (DM) interactions up to second-nearest neighbors, as well as the single-ion anisotropy (SIA). All magnetic parameters are obtained not only for the $R3c$ structural ground state, but also for the $R3m$ and $R\overline{3}c$ phases in order to determine the effects of ferroelectricity and antiferrodistortion distortions, respectively, on these magnetic parameters. In particular, two different second-nearest-neighbor couplings are identified and their origins are discussed in details. Moreover, Monte Carlo (MC) simulations using a magnetic Hamiltonian incorporating these first-principles-derived interaction parameters are further performed. They result (i) not only in the accurate prediction of the spin-canted G-type antiferromagnetic structure and of the known magnetic cycloid propagating along a $\langle 1\overline{1}0\rangle$ direction, as well as their unusual characteristics (such as a weak magnetization and spin-density-waves, respectively), (ii) but also in the finding of another cycloidal state of low-energy and that awaits to be experimentally confirmed. Turning on and off the different magnetic interaction parameters in the MC simulations also reveal the precise role of each of them on magnetism.

Figure 1

Magnetic properties predicted from MC simulations. (a) shows the specific heat (arb. units) as a function of temperature. The inset of (a) shows the dependence of the AFM Néel vector $\mathbf{L}$ on temperature, which further emphasizes a paramagnetic-to-AFM transition taking place at 603 K; (b) displays the energy per Fe ion with respect to the period of $\left[1\overline{1}0\right]$ and [001] cycloids; (c) is the energy per Fe ion with respect to the period of the $\left[1\overline{1}0\right]$ cycloid, using selected magnetic parameters; and (d) demonstrates the tilting angles at different phases/positions (in unit of $\pi$) along the propagation direction of the $\left[1\overline{1}0\right]$ cycloid. The direction notations above the horizontal axis in (d) mark the directions of the magnetic moments. Note that the energy of the collinear G-type AFM state is set to be energy reference (zero) in both (b)  and (c).