Spin-Nematic Interaction in the Multiferroic Compound ${\mathrm{Ba}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$

We demonstrate the existence of the spin-nematic interactions in an easy-plane type antiferromagnet ${\mathrm{Ba}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$ by exploring the magnetic anisotropy and spin dynamics. The combination of neutron scattering and magnetic susceptibility measurements reveals that the origin of the in-plane anisotropy is an antiferro-type interaction of the spin-nematic operator. The relation between the nematic operator and the electric polarization in the ligand symmetry of this compound is presented. The introduction of the spin-nematic interaction is useful to understand the physics of spin and electric dipole in multiferroic compounds.

Figure 1

(a) Crystal structure of ${\mathrm{Ba}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$ . ${\mathrm{CoO}}_4$ tetrahedra separated by ${\mathrm{GeO}}_4$ tetrahedra form a square lattice in the (001) plane. The $x$ and $y$ axes in the global coordinate of spin are defined along the [100] and [010] directions. The $X$ and $Y$ axes are locally defined on each${\mathrm{CoO}}_4$ tetrahedron. (b) Structures of spin dipoles, spin-nematic operator $O_{XY}$, and electric polarizations at zero magnetic field in ${\mathrm{Ba}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$ .

Figure 2

(a) Inelastic neutron scattering spectrum obtained by using the AMATERAS spectrometer in J-PARC. Well-defined spin-wave excitations are observed. (b) Inelastic neutron scattering spectrum in the low energy range. The calculated dispersion is shown by the white solid curve. The inset shows the constant-$Q$ scan at $\mathbf{Q}=(1,0,0)$. (c) Bulk susceptibility $dM/dH$ measured at $T=1.8\mathrm{K}$. Red and black symbols indicate the $dM/dH$ for $H\parallel$ [110] and $H\parallel$ [100], respectively. Field cooled processes are shown. Red and black dashed curves indicate the $dM/dH$ at $T=0\mathrm{K}$ calculated by mean-field theory. (d) The measured angular dependence of $dM/dH$ in the upper panel and the calculated ones in the lower panel.

Figure 3

Typical constant-$Q$ scans at $H=0\mathrm{Oe}$ and 30 kOe obtained by using the TASP spectrometer in PSI. Solid circles are the data at $T=1.6\mathrm{K}$ and open squares are those at $T=8.5\mathrm{K}$. Red solid curves are fits to the data by Gaussian functions.

Figure 4

(a) Magnon dispersion relation for $\mathbf{Q}=(h,0,0)$ at $H=0\mathrm{Oe}$. (b) Intensity of the inelastic neutron scattering at $H=0\mathrm{Oe}$. (c),(d) Magnon dispersion and intensity for $H=30\mathrm{kOe}$. In each figure the theoretical calculations are indicated by curves and the experimental data are indicated by solid circles. The pink curves in (b) and (d) represent the added intensities of the lowest (red) and the second-lowest (blue) modes for the data of which the energies are close to each other. Similarly, the black curves in (b) and (d) are the results after adding all contributions from the four high-energy modes around 4 meV. (e) Schematics of spin fluctuation for each mode at 0 Oe. Transverse fluctuation in the $a\text{−}b$ plane is called the $T_1$ mode, those along the $c$ direction the $T_2$ mode, and longitudinal fluctuation the $L$ mode.