Direct observation of spin-quadrupolar excitations in ${\mathrm{Sr}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$ by high-field electron spin resonance

Exotic spin-multipolar ordering in spin transition metal insulators has so far eluded unambiguous experimental observation. A less studied, but perhaps more feasible fingerprint of multipole character emerges in the excitation spectrum in the form of quadrupolar transitions. Such multipolar excitations are desirable as they can be manipulated with the use of light or electric field and can be captured by means of conventional experimental techniques. Here we study single crystals of multiferroic ${\mathrm{Sr}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$ and observe a two-magnon spin excitation appearing above the saturation magnetic field in electron spin resonance (ESR) spectra. Our analysis of the selection rules reveals that this spin excitation mode does not couple to the magnetic component of the light, but it is excited by the electric field only, in full agreement with the theoretical calculations. Due to the nearly isotropic nature of ${\mathrm{Sr}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$, we identify this excitation as a purely spin-quadrupolar two-magnon mode.

Figure 1

The schematic crystal and magnetic structures of SCGO projected onto the $ab$ plane. The blue spheres represent the magnetic ${\mathrm{Co}}^{2+}$ ions with $S=3/2$ surrounded by four ${\mathrm{O}}^{2-}$ ions in a tetrahedral environment. The two tetrahedra in the unit cell (dashed square) are rotated alternatively by an angle $\kappa$. The yellow arrows show the spin directions in the ordered phase in the absence of external magnetic field. On the right hand side we show the crystallographic coordinate system with [100] and [010] axes, as well as the $x$ and $y$ coordinates, following the convention of Ref. [15].

Figure 2

Schematic experimental setup for the high-field ESR measurements, with different light configurations. $E^{\omega }$ ($H^{\omega }$) shows one of the electric (magnetic) component of the unpolarized light.

Figure 3

(a) Magnetic field dependence of magnetization in SCGO at 1.4 K. Dashed lines indicate the saturation magnetization values using $g_{ab}=2.28$ and $g_c=2.23$, obtained from the ESR measurements [see Eqs. (23c) and (23a)], for the in-plane and out-of-plane components, respectively. (b) Magnetic field dependence of the out-of-plane component of induced electric polarization ($P^z$) at 1.4 K in the case of $\mathbf{H}\parallel \left[110\right]$. The in-plane polarization vanishes for this direction of the field. Above saturation, both the polarization and the magnetization become flat. Yellow and pink shaded areas mark the phases below saturation for in-plane and out-of-plane magnetic fields, respectively.

Figure 4

Frequency-field diagrams of the ESR resonance fields of ${\mathrm{Sr}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$ for magnetic fields parallel to the (a) [100] and (b) [110] directions. Open circles (plus) are strong (weak) resonance signals obtained from the measurements in static fields at 1.6 K, while solid circles (cross) show strong (weak) resonance signals in pulsed fields at 1.4 K. The solid lines represent the dipolar resonance modes from the multiboson spin-wave theory, using the following set of parameters: $g_c=2.23, g_{ab}=2.28, J=49.1$ GHz, $J_z=45.0$ GHz, $\mathrm{\Lambda }=49.7$ GHz, and $W_z^2{J}_{\mathit{pz}}=0.05$ GHz (see Sec. 5c and Appendix pp1). The red dashed line in (a) indicates a resonance mode with a slope twice larger than the others, corresponding to a two-magnon excitation. Yellow shaded area marks the phase below saturation for in-plane magnetic fields.

Figure 5

Frequency-field diagrams of the ESR resonance fields of ${\mathrm{Sr}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$ for magnetic fields parallel to the [001] directions in the (a) Faraday and (b) Voigt configuration, showing the two-magnon absorptions (green dashed line). Open circles (plus) are strong (weak) resonance signals obtained from the measurements in static fields at 1.6 K, while solid circles (cross) show strong (weak) resonance signals in pulsed fields at 1.4 K. The solid lines represent the dipolar resonance modes calculated by the multiboson spin-wave theory with the same set of parameters as in Fig. 4 (see Sec. 5c and Appendix pp1). The shaded area in red is the magnetic field range below the saturation field.

Figure 6

Frequency dependent ESR absorption spectra of SCGO in Faraday configuration at 1.4 K for (a) $H\parallel \left[100\right]$ and (b) $H\parallel \left[110\right]$. Arrows mark the resonance fields of DPPH (2,2-diphenyl-1-picrylhydrazyl, ESR marker with $g=2.0036$). Open and solid triangles mark the resonance fields of one-magnon (${\mathsf{D}}_{\mathsf{1}}$) and two-magnon (${\mathsf{Q}}_{\mathsf{1}}$) resonance modes, respectively. For Faraday configuration and the two in-plane static field direction, the ${\mathsf{D}}_{\mathsf{1}}$ dipolar and ${\mathsf{Q}}_{\mathsf{1}}$ quadrupolar transitions appear to be mutually exclusive.

Figure 7

(a) A typical two-magnon configuration, short black arrows show the $S^z=1/2$ spins moving on the $S^z=3/2$ spin background (empty arrows). The $\mathbf{q}=(\pi ,\pi )$ linear combinations of the neighboring $S^z=1/2$ spin states shown in (b) and (c) are exact eigenstates, and they make the twofold degenerate ${\mathsf{B}}_{\mathsf{1}}$ bound state. (d) The down-pointing short arrow represents a $S^z=-1/2$ state; the $\mathbf{q}=(\pi ,\pi )$ linear combination of such states is the ${\mathsf{Q}}_{\mathsf{1}}$ mode [see Eq. (14)], which is also an eigenstate for the model with nearest neighbor exchanges only.

Figure 8

One- and two-magnon spectra in magnetic field above the saturation for $\mathrm{\Lambda }=1$ and $J=J_z=1$. (a) The dispersion of a single magnon along a path in the two-dimensional Brillouin zone, following Eq. (10). (b) In the two-magnon spectrum, the gray shaded area is the two-magnon continuum, the red points outside the continuum are the antibound states obtained from diagonalizing the two-magnon problem on different size clusters. (c) and (d) show the magnified part of the two magnon spectrum close to $\mathbf{q}=(\pi ,\pi )$. The field is along the $\left[001\right]$ direction in (a), (b), and (c), while it lies within the easy (001) plane in (d). The circles show the energy of the ${\mathsf{D}}_{\mathsf{0}}$ and ${\mathsf{D}}_{\mathsf{1}}$ mode in (a) and the energy of the ${\mathsf{B}}_{\mathsf{1}}$ and ${\mathsf{Q}}_{\mathsf{1}}$ mode in (c) and (d).

Figure 9

Schematic plot of the dipolar [(a) and (c)] and quadrupolar [(b) and (d)] modes in different geometries, as seen from the direction of the magnetic field. In the dipolar wave ${\mathsf{D}}_{\mathsf{1}}$ for $\mathbf{H}\parallel \left[110\right]$ (c) and the quadrupolar mode ${\mathsf{Q}}_{\mathsf{1}}$ for $\mathbf{H}\parallel \left[100\right]$ (b) the oscillating component of the uniform electric polarization ${\mathbf{P}}^{\omega }$ (shown by red ellipse) is perpendicular to the external magnetic field $\mathbf{H}$, therefore they are active in the Faraday configuration. In the ${\mathsf{D}}_{\mathsf{1}}$ for $\mathbf{H}\parallel \left[100\right]$ (a) and ${\mathsf{Q}}_{\mathsf{1}}$ for $\mathbf{H}\parallel \left[110\right]$ (d) the ${\mathbf{P}}^{\omega }\parallel \mathbf{H}$ (i.e., it oscillates in and out from the shown plane), so these modes are active in Voigt configuration only. The green ellipse represents the rotating quadrupolar moments, while the green arrows the precessing dipolar spins on the two sublattices. The red arrows show the electric polarization vectors which are excited by the oscillating electric field. Animations of these modes are shown in the Supplemental Material [34].

Figure 10

ESR absorption spectra of ${\mathrm{Sr}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$ at 846 GHz in Faraday ($\mathbf{k}\parallel \mathbf{H}$) and Voigt ($\mathbf{k}\perp \mathbf{H}$) configuration for $\mathbf{H}\parallel \left[100\right], \mathbf{H}\parallel \left[110\right]$, and $\mathbf{H}\parallel \left[001\right]$. Vertical dotted lines, from left to right, indicate two-magnon resonance signal, signal of ESR standard DPPH, and one-magnon resonance signal, respectively.

Figure 11

(a) Calculated magnetization using our variational setup using the variational parameters determined below in Secs. pp1-s3 and pp1-s4 for in-plane and out-of-plane magnetic fields, respectively. (b) Theoretically calculated induced polarization in the simplified picture using Eq. (A2). We neglected the polarization-polarization term in the Hamiltonian and use the small $\mathrm{\Lambda }$ expansion. If $P_i^z·P_j^z$ were kept, the polarization would drop to zero at zero field. For both figures we used the $g$ and interaction values from the linear fit in Eqs. (23) and (24) in the main text.

Figure 12

Simplified multiboson approach compared with the measured ESR spectrum for field directions parallel to the $H\parallel \left[110\right]$ and $H\parallel \left[100\right]$ crystallographic direction in (a) and (b), respectively. The energy of dipole and quadrupole excitation is in very good agreement with the experimental findings.

Figure 13

Simplified multiboson approach compared with the measured spectrum for a field parallel to the $H\parallel \left[001\right]$ crystallographic direction.