Selection rules and dynamic magnetoelectric effect of the spin waves in multiferroic $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$

We report the magnetic-field dependence of the THz absorption and nonreciprocal directional dichroism spectra of $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ measured on the three principal crystal cuts for fields applied along the three principal directions of each cut. From the systematic study of the light polarization dependence, we deduced the optical selection rules of the spin-wave excitations. Our THz data, combined with small-angle neutron scattering results showed that (i) an in-plane magnetic field rotates the $\mathbf{q}$ vectors of the cycloids perpendicular to the magnetic field and (ii) the selection rules are mostly determined by the orientation of the $\mathbf{q}$ vector with respect to the electromagnetic fields. We observed a magnetic field history-dependent change in the strength and the frequency of the spin-wave modes, which we attributed to the change of the orientation and the length of the cycloidal $\mathbf{q}$ vector, respectively. Finally, we compared our experimental data with the results of linear spin-wave theory that reproduces the magnetic-field dependence of the spin-wave frequencies and most of the selection rules, from which we identified the spin-polarization coupling terms relevant for the optical magnetoelectric effect.

Figure 1

Magnetic-field dependence of the THz absorption spectra of $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ at 4 K. The magnetic field is applied along X, panels (a) and (d), and along Y, panels (b) and (c). THz light polarization is ${\mathbf{E}}^{\omega }\parallel \mathbf{Z}$ and ${\mathbf{B}}^{\omega }\parallel \mathbf{X}$ in (a) and (c), and ${\mathbf{E}}^{\omega }\parallel \mathbf{Z}$ and ${\mathbf{B}}^{\omega }\parallel \mathbf{Y}$ in (b) and (d). The spectra are plotted for positive and negative fields in red and blue, respectively, with a vertical offset proportional to the magnitude of the field. Grey lines show the magnetic-field dependence of spin-wave frequencies deduced from linear spin-wave theory.

Figure 2

Magnetic-field dependence of the THz absorption spectra of $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ at 4 K. The magnetic field is applied along X, panels (a) and (d), and along Y, panels (b) and (c). THz light polarization is ${\mathbf{E}}^{\omega }\parallel \mathbf{X}$ and ${\mathbf{B}}^{\omega }\parallel \mathbf{Z}$ in (a) and (c), and ${\mathbf{E}}^{\omega }\parallel \mathbf{Y}$ and ${\mathbf{B}}^{\omega }\parallel \mathbf{Z}$ in (b) and (d). The spectra are plotted for positive and negative fields in red and blue, respectively, with a vertical offset proportional to the magnitude of the field. Grey lines show the magnetic-field dependence of spin-wave frequencies deduced from linear spin-wave theory. Due to strong noise, a small frequency window is not displayed in the 14 T spectrum in panel (a).

Figure 3

Magnetic-field dependence of the THz absorption spectra of $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ at 4 K. The magnetic field is applied along X, panels (a) and (d), and along Y, panels (b) and (c). THz light polarization is ${\mathbf{E}}^{\omega }\parallel \mathbf{Y}$ and ${\mathbf{B}}^{\omega }\parallel \mathbf{X}$ in (a) and (c), and ${\mathbf{E}}^{\omega }\parallel \mathbf{X}$ and ${\mathbf{B}}^{\omega }\parallel \mathbf{Y}$ in (b) and (d). The spectra are plotted for positive and negative fields in red and blue, respectively, with a vertical offset proportional to the magnitude of the field. Grey lines show the magnetic field dependence of spin-wave frequencies deduced from linear spin-wave theory.

Figure 4

(a), (b) Zero-field THz absorption spectra measured at 3 K after the application of fields ${\mathbf{B}}_{\text{tr}}\parallel \mathbf{Y}$ with magnitude continuously increasing to 17 T. The sample was heated above the Néel temperature and ZFC to 3 K before each sequence. The color bar shows the highest treatment field, ${\mathbf{B}}_{\text{tr}}$ preceding each zero-field measurement. The magnetic-field dependence of the $\mathbf{q}$ vector lengths, $\left|\mathbf{q}\right|$ measured with SANS at 2 K for $\mathbf{B}\parallel \mathbf{Y}, \mathbf{B}\parallel \mathbf{X}$, and $\mathbf{B}\parallel \mathbf{Z}$ are plotted in panels (c), (e), and (f), respectively. A typical SANS pattern is shown in the inset of panel (e). (d) Temperature dependence of the $\mathbf{q}$ vector lengths for ZFC (light blue) and field-treated (orange and red, in 8 T when $\mathbf{B}\parallel \mathbf{Y}$) samples in zero field.

Figure 5

Magnetic-field dependence of the THz absorption spectra of $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ at 4 K. The magnetic field is applied along $\mathbf{Z}$. The spectra are plotted for positive and negative fields in red and blue, respectively, with a vertical offset proportional to the magnitude of the field. In panel (d), the spectra measured in negative fields are plotted with dashed blue lines to highlight that they are identical within the accuracy of the experiment to that of obtained in positive fields. Grey lines show the magnetic-field dependence of spin-wave frequencies deduced from linear spin-wave theory. In panels (c)–(f), the samples were zero-field cooled, while in panels (a) and (b) the sample was exposed to $\mathbf{B}\parallel \mathbf{Y}$ before the $\mathbf{B}\parallel \mathbf{Z}$ measurement.

Figure 6

(a), (b) Magnetic-field dependence of spin-wave frequencies in $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_3$ for $\mathbf{B}\parallel \mathbf{Z}$ and $\mathbf{B}\parallel \mathbf{Y}$, respectively, as deduced from linear spin-wave theory. There are three types of spin-wave modes corresponding to spin rotation about the $\mathbf{Z}$ ($\mathrm{\Psi }{}_{\mathrm{n}}^2$, green), $\mathbf{X}\parallel \mathbf{q}$ ($\mathrm{\Psi }{}_0$ and$\mathrm{\Psi }{}_{\mathrm{n}}^1$, brown), and $\mathbf{Y}\parallel \mathbf{Z}\times \mathbf{q}$ (${\mathrm{\Phi }}_{\mathrm{n}}$, blue) axes.

Figure 7

(a), (b) The NDD spectra calculated in 14 T for light polarization ${\mathbf{E}}^{\omega }\parallel \mathbf{Y}, {\mathbf{B}}^{\omega }\parallel \mathbf{Z}$, and ${\mathbf{E}}^{\omega }\parallel \mathbf{Z}, {\mathbf{B}}^{\omega }\parallel \mathbf{Y}$, respectively. The theoretical results are compared to the difference of the spectra measured in positive and negative fields, $\mathrm{\Delta }\alpha ={\alpha }_B(+B)-{\alpha }_B(-B)$.