Optical Magnetoelectric Resonance in a Polar Magnet $(\mathrm{Fe},\mathrm{Zn}{)}_2{\mathrm{Mo}}_3{\mathrm{O}}_8$ with Axion-Type Coupling

We report the polarization rotation of terahertz light resonant with the magnetoelectric (ME) spin excitation in the multiferroic ${(\mathrm{Fe},\mathrm{Zn})}_2{\mathrm{Mo}}_3{\mathrm{O}}_8$. This resonance reflects the frequency dispersion of the diagonal ME susceptibility (axion term), with which we quantitatively reproduce the thermal and magnetic-field evolution of the observed polarization rotation spectra. The application of the sum rule on the extrapolated dc value of the spectral weight of the ME oscillator provides insight into the dc linear ME effect. The present finding highlights a novel optical functionality of spin excitations in multiferroics that originates from diagonal ME coupling.

Figure 1

(a) Schematic of GB for light with propagation vector $+k^{\omega }$ (upper) and $-k^{\omega }$ (bottom) perpendicular to the $c$ axis. The green plate and red and blue bars denote the sample and rotating principal optical axes, respectively. (b) A unit cell and the magnetic structure of ${(\mathrm{Fe},\mathrm{Zn})}_2{\mathrm{Mo}}_3{\mathrm{O}}_8$; induced electric polarization $P^{\omega }$ and $P_{\mathrm{ME}}$ by the electric field ($E^{\omega }$) and magnetic field ($B^{\omega }$) of light, respectively. Temperature dependence of (c) the magnetization ($M$) and (d) the change of $P$ with respect to its value at 60 K in zero field. Magnetic field (${\mu }_{0}H$) dependence of (e) $M$ and (f) change of $P$ for the respective crystal axes.

Figure 2

(a)–(b) Spectra of real ($n$) and imaginary ($\kappa$) parts of the refractive index for the respective light polarizations at 5 K in zero field. (c) Schematic of terahertz magneto-optics. Spectra of the (d) rotation angle ($\theta$) and (e) ellipticity ($\eta$) at 5 K. The dashed curve is the spectrum of $\kappa$ for the geometry $E_{\mathrm{in}}^{\omega }//c$ and the black line is the fit obtained using Eq. (2). (f) Dispersion of the real-imaginary (orange-light blue line) part of the dynamical ME susceptibility of the ${\nu }_1$ mode at 1.4 THz. The orange dot is the dc ME susceptibility evaluated from the dc data [Fig. 1].

Figure 3

Spectra of (a),(c) rotation angle ($\theta$) and (b),(d) ellipticity ($\eta$) at selected temperatures and magnetic fields for $H//c$. The black lines are fits obtained using Eq. (2). The black triangle in (d) is the fitted resonance frequency for the ${\nu }_1$ mode.

Figure 4

(a) Temperature and (b) magnetic field dependence of the dc-limit extrapolation of the dynamical ME susceptibility for GB [${\alpha }_{\mathrm{GB}}(\omega )$], for the ${\nu }_1$ mode [${\alpha }_1(\omega )$], and the diagonal ME susceptibilities obtained from the DC-measurement (${\alpha }_{xx}-{\alpha }_{zz}$). (c) Magnetic field dependence of the resonance frequency for the ${\nu }_1$ mode obtained from the peak of the $\kappa$ spectra for $E_{\mathrm{in}}^{\omega }//c$ (closed circles) and from the fit of $\theta$ and $\eta$ (open circles). The open squares show the change in the spectral weight ($f{\omega }_1^2$) of the ${\nu }_1$ mode.