Figure 1

Comparison of the static and optical ME properties of multiferroic ${\mathrm{Ba}}_2$${\mathrm{CoGe}}_2$${\mathrm{O}}_7$ (BCGO), ${\mathrm{Ca}}_2$${\mathrm{CoSi}}_2$${\mathrm{O}}_7$ (CCSO) and ${\mathrm{Sr}}_2$${\mathrm{CoSi}}_2$${\mathrm{O}}_7$ (SCSO) based on the ME sum rule in Eq. (5). (a) Dependence of the ferroelectric polarization ($P$) on the magnitude of the magnetic field ($H$) in BCGO. (d), (g), and (j) Dependence of $P$ on the orientation of the field in BCGO. In these panels, the solid lines are experimental data reproduced from Ref. [18], while the dashed lines are calculated using the spin dependent $p$-$d$ hybridization model according to Eqs. (6, 7, 8) [18]. The slopes of the green lines in the same panels are proportional to the corresponding elements of the ME tensor. Arrows labeled with ${\mathbf{E}}^{\omega }$ and ${\mathbf{H}}^{\omega }$ show the electric and magnetic components of the absorbed light in the corresponding optical experiment, respectively. (b), (e), (h), and (k) Field dependence of the magnon absorption spectra of BCGO in the GHz-THz range. The light polarizations indicated in these panels correspond to the labels ${\mathbf{E}}^{\omega }$ and ${\mathbf{H}}^{\omega }$ shown in the panels of the first column. The spectra are shifted vertically proportional to $H$. For BCGO, the spectra corresponding to counter-propagating light beams are plotted by red and blue lines, while for CCSO brown and dark green lines represent the two propagation directions. The absorption coefficient of CCSO is multiplied by a factor of two for better visibility. The spectra in (b) and (k) are measured in the present study, while the data in (e) and (h) are reproduced from Ref. [11] for BCGO and from Ref. [12] for CCSO, respectively. (c), (f), (i), and (l) Magnetic field dependence of different components of the ME tensor. Symbols indicate the tensor elements calculated from the corresponding optical measurements using the ME sum rule; empty square, full diamond and empty triangle stand for BCGO, CCSO, and SCSO, respectively. The field dependence of the static ME tensor components are plotted with solid, dashed and dotted lines for the three compounds in the same order. The points corresponding to the slope of the green lines in the left panels are indicated by a green dot. The solid line in (c) is calculated directly from the measured polarization-magnetic field curve shown in (a), while the curves in (f), (i), and (l) are evaluated using Eqs. (6, 7, 8). Static experiments, optical measurements, and model calculations were carried out at $T=2$, 4, and 0 K, respectively.