Identification of Antiferromagnetic Domains Via the Optical Magnetoelectric Effect

The ultimate goal of multiferroic research is the development of a new-generation nonvolatile memory devices, where magnetic bits are controlled via electric fields with low energy consumption. Here, we demonstrate the optical identification of magnetoelectric (ME) antiferromagnetic (AFM) domains in the ${\mathrm{LiCoPO}}_4$ exploiting the strong absorption difference between the domains. This unusual contrast, also present in zero magnetic field, is attributed to the dynamic ME effect of the spin-wave excitations, as confirmed by our microscopic model, which also captures the characteristics of the observed static ME effect. The control and the optical readout of AFM/ME domains, demonstrated here, will likely promote the development of ME and spintronic devices based on AFM insulators.

Figure 1

(a) Unit cell of the ${\mathrm{LiCoPO}}_4$ viewed from the $z$ axis. The four Co sites a–d are surrounded by oxygen octahedra, while Li and P sites are omitted for clarity. The inversion center is labeled by $i$. The three nonequivalent exchange interactions, $J_{ab}$, $J_{ac}$, and $J_{ad}$, are indicated with arrows. (b) The four combinations $(++),(+-),(-+)$, and $(--)$ of poling fields $(H_x,E_y)$ are represented by four colors. (c) The magnetic sublattices (green and olive arrows) in the AFM domains $\alpha$ and $\beta$ are interchanged while the AFE polarization pattern (brown arrows) is the same for the two domains. (d) Domains $\alpha$ and $\beta$ are selected by the poling fields $(++)$ (red) and $(+-)$ (blue) via the ME effect according to Eq. (4), assuming $c_{2xy}>0$.

Figure 2

(a) Magnetic field dependence of the static ME effect at $T=2\mathrm{K}$ measured after poling the sample in the four combinations $(++),(+-),(-+)$ and $(--)$ of poling fields $(H_x,E_y)$. The poling fields were switched off during the measurement; hence, the linear slope of the polarization ($P$) versus magnetic field curve shows the linear ME effect. (b) Temperature dependence of the linear ME effect, ${\chi }_{yx}^{em}$, measured in warming up after poling in the four configurations of $(H_x,E_y)$. Spectra of the (c) real and (d) imaginary part of the refractive index at $T=5\mathrm{K}$ measured after poling. Spectra of the (e) real and (f) imaginary part of the refractive index measured at $T=5\mathrm{K}$ after poling in two selected configurations, $(++)$ and $(+-)$ and for light propagation along the $+z$ direction (full symbols) and the $-z$ direction (open symbols). The spectra in (c)–(f) were measured using linearly polarized light with $({\mathbf{E}}_y^{\omega },{\mathbf{H}}_x^{\omega })$. Spectra measured in the paramagnetic state are plotted in black. The color of each curve corresponds to the applied poling process following the convention introduced in Fig. 1. (h) Selection rules of the spin-wave excitations in ${\mathrm{LiCoPO}}_4$. Absorption coefficient spectra measured in six different polarization configurations of electric (${\mathbf{E}}^{\omega }$) and magnetic (${\mathbf{H}}^{\omega }$) fields according to table (g). In four spectra, shifted vertically for clarity, four distinct resonances are identified and labeled as modes #1 to #4. No absorption peak was observed for ${\mathbf{H}}^{\omega }\parallel y$ (not displayed). The black vertical bars, indicating the positions of these resonances, cross only those spectra where the corresponding resonances are active. The red spectrum, corresponding to the case where the optical ME effect was observed is an average of the four different poling combinations.

Figure 3

(a) Energies of the AFM domains are quadratic in the electric and magnetic fields according to Eq. (3). When $H_x>0$, the $\alpha$ domain has lower energy than the $\beta$ for positive $E_y$, while negative $E_y$ stabilizes the $\beta$ domain. For $H_x<0$, the roles of the two AFM domains are interchanged. Schematic drawing of the (b) magnetoelectric (${\mathbf{P}}_y^{\omega }$, ${\mathbf{M}}_x^{\omega }$) and (c) magnetic only (${\mathbf{M}}_z^{\omega }$) spin excitations in the $\alpha$ domain, viewed from the $y$ axis. Local magnetization of the a and c sites precess counterclockwise along alternately rotated ellipses in the $xz$ plane, while on the b and d sites spins precess clockwise. The red and blue shading around the ellipses represents the $y$ component of the local polarization, while the green edge shows the path swept by the $xz$ component the local magnetization. In the center of each ellipse, the actual direction of the precessing spin is shown by green arrows, while the red and blue marks represent the actual value of the spin-induced polarization. When the precessing spin points to the red (blue) region, the polarization is pointing in the $+y$ ($-y$) axis, while magnitude of the polarization is illustrated by the size of the mark. The resultant oscillating net magnetic (${\mathbf{M}}_x^{\omega }$ and ${\mathbf{M}}_z^{\omega }$) and net electric (${\mathbf{P}}_y^{\omega }$) dipole moments are shown in the middle of each unit cell. For $\beta$ domains red and blue shading of the ellipses are reversed; hence, ${\mathbf{P}}_y^{\omega }$ is in antiphase compared to the $\alpha$ domain.