Thermal evolution of the full three-dimensional magnetic excitations in the multiferroic BiFeO${}_3$

We present neutron inelastic scattering measurements of the full three-dimensional spin-wave dispersion in the multiferroic material BiFeO${}_3$ for temperatures from 5 to 700 K. Despite the presence of strong electromagnetic coupling, the magnetic excitations behave as conventional magnons over all parts of the Brillouin zone. At low temperature the spin waves are well-defined coherent modes, described by a classical model for a $G$-type antiferromagnet. A softening of the spin-wave velocity and broadening in energy is already present at room temperature, which is well below the Néel temperature $T_N\sim 640$ K, and increases on heating. In addition, a strong hybridization of the Fe $3d$ and O $2p$ states is found to modify significantly the distribution of the spin-wave spectral weight, which implies that the spins are not restricted to the Fe atomic sites as previously believed.

Figure 1

Schematics of the BiFeO${}_3$ crystal structure (a), the Fe atom with six Fe-O bonds (b), and spin densities from first principles Wannier analysis taking into account the O 2p orbital (c), and from conventional Fe ionic model (d). (c) and (d) are plotted on the same scale as in (b).

Figure 2

Magnetic excitations from BiFeO${}_3$ at $T=5$ K. The vertical axes denote energy transfer, while the horizontal axes are $\mathbf{Q}-{\mathbf{Q}}_{\text{AF}}$ for $\mathbf{Q}$ along [100], [110], [111], and [210] directions for (a), (b), (c), and (d), respectively. (e)–(h) are model calculations described in the text. The data shown here are measured with $E_i=120$ meV.

Figure 3

Magnetic excitations from BiFeO${}_3$ measured at 5, 140, 300, 580, and 680 K. The horizontal axis denotes $\mathbf{Q}-{\mathbf{Q}}_{\text{AF}}$. The solid lines are dispersion calculated based on parameters obtained from the base temperature (5 K) data. The data shown here are measured with $E_i=120$ meV.

Figure 4

Linear profiles of magnetic excitations from BiFeO${}_3$ measured at 5 K (red), 300 K (blue), and 580 K (green). (a) Intensity along $[H,0.5,0.5]$ for $\hslash \omega =10$ meV, (b) intensity along $[H,0.5,0.5]$ for $\hslash \omega =40$ meV, (c) an energy cut at $Q=[2,0.5,0.5]$, and (d) an energy cut at $Q=[1.75,0.75,0.75]$. The data shown here are measured with $E_i=120$ meV. The solid lines are results of model calculations described in the text, including a temperature and $Q$ dependent background to account for phonon contributions.

Figure 5

Comparison of the original ionic Fe form factor and the new Wannier form factor. (a) Plot of the ratio $R\left(\mathbf{Q}\right)=\frac{I\left(\mathbf{Q}\right)/{\left|F\left(\mathbf{Q}\right)\right|}^2}{I\left({\mathbf{Q}}_0\right)/{\left|F\left({\mathbf{Q}}_0\right)\right|}^2}$, where $I\left(\mathbf{Q}\right)$ is magnetic scattering intensity measured at wave vector $\mathbf{Q}$, while ${\mathbf{Q}}_0$ is the wave vector equivalent to $\mathbf{Q}$ in the first Brillouin zone. Ideally this value should be 1 (see the dashed line). The red symbols are values calculated using the new Wannier form factor while the blue symbols using the old Fe ionic form factor. The data are selected from measurements with $E_i=40$ and 120 meV, taken at $T=5$ K. The error bars are derived from the error bars of the data being used. (b)–(d) ${\left|F\left(\mathbf{Q}\right)\right|}^2$ plotted along different directions.