Spin-Stretching Modes in Anisotropic Magnets: Spin-Wave Excitations in the Multiferroic ${\mathrm{Ba}}_2{\mathrm{CoGe}}_2{\mathbf{O}}_7$

We studied spin excitations in the magnetically ordered phase of the noncentrosymmetric ${\mathrm{Ba}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$ in high magnetic fields up to 33 T. In the electron spin resonance and far infrared absorption spectra we found several spin excitations beyond the two conventional magnon modes expected for such a two-sublattice antiferromagnet. We show that a multiboson spin-wave theory describes these unconventional modes, including spin-stretching modes, characterized by an oscillating magnetic dipole and quadrupole moment. The lack of inversion symmetry allows each mode to become electric dipole active. We expect that the spin-stretching modes can be generally observed in inelastic neutron scattering and light absorption experiments in a broad class of ordered $S>1/2$ spin systems with strong single-ion anisotropy and/or noncentrosymmetric lattice structure.

Figure 1

Magnetic field dependence of the absorption spectra in ${\mathrm{Ba}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$ below 2 THz for a representative set of light polarizations. The spectra are shifted vertically proportional to the magnitude of the field, $B_{\mathrm{dc}}$. The separation between horizontal grid lines corresponds to an absorption coefficient of $20{\mathrm{cm}}^{-1}$ in panels (a) and (c), and $30{\mathrm{cm}}^{-1}$ in (b) and (d). The direction of $B_{\mathrm{dc}}$ is indicated in each panel and the spectra for different polarizations and propagation directions ($\mathbf{k}$) of light are distinguished by the color. Black triangles and green diamonds represent the position of the resonances determined from the FIR and ESR spectra, respectively. Below 12 T the triangles are shown in 1 T steps, while the spectra are displayed only in 2 T steps for clarity. The gray lines show the field dependence of the modes obtained in our multiboson spin-wave approach. (d) For $B_{\mathrm{dc}}\perp [001]$ in some polarization configurations, two of which are shown, we observed additional modes (open triangles) not explained by the theory.

Figure 2

(a) The energy of the modes for different values of $J_z/J$ in zero field. $b_{-}$ denotes the $\omega =0$ Goldstone mode. Only the $b_{+}$ and $d_{+}$ modes depend on $J_z/J$. The dashed lines indicate modes in the easy-axis AF state that forms below $\Lambda \approx 2.7J$ for $J_z=2$. (b) and (c) shows the imaginary part of the magnetic ${\chi }_{\xi \xi }^{mm}(\omega )$ and electric ${\chi }_{\xi \xi }^{ee}(\omega )$ dynamic susceptibilities, respectively, for $J=J_z$. The shading above the lines represent the strength of the magnetic and electric response.

Figure 3

Motion of the magnetizations (green, light arrows) and the local electric polarizations (red, dark arrows) in the two sublattices (a) for the Goldstone mode ($b_{-}$ in Fig. 2) and (b) for the $c_{-}$ stretching mode. The blue spheres are the oxygens forming tetrahedral cages around the central Co ions. The vertical axis is the tetragonal one, while the horizontal axes point along [110] and $[1\overline{1}0]$. Apparent tilting of the axes comes from the perspective view.