Magnetic Dispersion and Anisotropy in Multiferroic ${\mathrm{BiFeO}}_3$

We have determined the full magnetic dispersion relations of multiferroic ${\mathrm{BiFeO}}_3$. In particular, two excitation gaps originating from magnetic anisotropies have been clearly observed. The direct observation of the gaps enables us to accurately determine the Dzyaloshinskii-Moriya (DM) interaction and the single ion anisotropy. The DM interaction supports a sizable magnetoelectric coupling in this compound.

Figure 1

Magnetic structure of the ${\mathrm{Fe}}^{3+}$ moments in ${\mathrm{BiFeO}}_3$. The basic structure of collinear $G$ type, in which nearest-neighbor spins align antiferromagnetically, is shown. The dotted lines correspond to the $2\times 2\times 1$ unit cells of the rhombohedral structure ($R3c$). The pseudocubic unit cell that is used in this paper is shown by the thin solid lines. The spin structure is spiral along the [1, $-1$, 0] direction with a long period of $\sim 640\AA$. The nearest-neighbor ($J_1$) and next-nearest-neighbor interactions ($J_2$) as well as the directions of the DM vector ($D$) and the spontaneous electric polarization ($P$) are also shown.

Figure 2

Typical constant-$Q$ and constant-energy scans along [$h$, $\overline{h}$, $h$], [$h$, $\overline{h}$, $\overline{h}$], [$h$, $\overline{h}$, 0], and [0, 0, $h$] measured on HB-1 using thermal neutrons at $T=200$ and 300 K in ${\mathrm{BiFeO}}_3$. The solid curves are the results of fits of a convolution of the resolution function with Lorentzians.

Figure 3

Low-energy magnetic excitations at the magnetic zone center ($1/2$, $-1/2$, $1/2$) measured at $T=5$, 100, 200, and 300 K on CTAX using cold neutrons. The solid curves are guides to the eye. The inset is a schematic figure to show the relation between the dispersions and the instrumental resolution (shaded ellipsoid).

Figure 4

Magnetic dispersion relations along [$h$, $\overline{h}$, 0] (a), [0, 0, $h$] (b), [$h$, $\overline{h}$, $h$] (c), and [$h$, $\overline{h}$, $2h$] (d) in ${\mathrm{BiFeO}}_3$. The solid curves are spin-wave dispersion relations calculated with $J_1=6.48\mathrm{meV}$ and $J_2=0.29\mathrm{meV}$.

Figure 5

The relation between $D$ and $K$ that reproduce the cycloidal magnetic structure with a long period of $\sim 640\AA$. The gap energy at the zone center due to the DM interaction ${\Delta }_{\mathrm{DM}}$ is also plotted as a function of $K$.