Magnetoelectric spectroscopy of spin excitations in ${\mathrm{LiCoPO}}_4$

We have studied spin excitations in a single-domain crystal of antiferromagnetic ${\mathrm{LiCoPO}}_4$ by terahertz absorption spectroscopy. By analyzing the selection rules and comparing the strengths of the absorption peaks in the different antiferromagnetic domains, we found electromagnons and magnetoelectric (ME) spin resonances in addition to conventional magnetic dipole active spin-wave excitations. Using the sum rule for the ME susceptibility, we determined the contribution of the spin excitations to all the different off-diagonal elements of the static ME susceptibility tensor in zero and finite magnetic fields. We conclude that the ME spin resonances are responsible for the static ME response of the bulk when the magnetic field is along the $x$ axis, and the symmetric part of the ME tensor with zero diagonal elements dominates over the antisymmetric components.

Figure 1

(a) The two AFM domains of ${\mathrm{LiCoPO}}_4$; AFM-$\alpha$ and AFM-$\beta$ are characterized by ME coupling of opposite signs. (b) The dynamical magnetoelectric (ME) effect produces an absorption difference between light beams with the same polarizations propagating along the same directions ($+z$ or $-z$) in the two AFM domains. (c) Polarization of the probing light ${\mathbf{E}}^{\omega }$ can be selected either parallel or perpendicular to the poling electric field ${\mathbf{E}}^0$. Thus, in the dynamic ME measurement both elements of the ME susceptibility, ${\chi }_{xy}$ and ${\chi }_{yx}$, can be simultaneously measured. (d) Electric polarizations and indices of refractions [Eqs. (1) and (2)] resulting from symmetric or antisymmetric ME susceptibility tensors ($\mathbf{P}=\widehat{\chi }\mathbf{H}$) rotate against or together with the magnetic field, respectively. (e) For both the symmetric and antisymmetric forms of $\widehat{\chi }$, the poling field combination ($+E_x^0$, $+H_y^0$) selects the AFM-$\alpha$ domain. However, for the ${90}^{\circ }$ rotated poling fields, poling with ($+E_y^0$, $-H_x^0$) selects domain AFM-$\alpha$ or AFM-$\beta$ in the case of antisymmetric and symmetric $\widehat{\chi }$, respectively. This is because, with the former and latter forms of $\widehat{\chi }$, the material couples to $E_x^0{H}_y^0\mp E_y^0{H}_x^0$, respectively.

Figure 2

Remanent static and dynamic ME effects in ${\mathrm{LiCoPO}}_4$. (a) and (b) Experimental configuration of the ${\mathbf{E}}^0$ and ${\mathbf{H}}^0$ poling fields. The color code of (b) corresponds to the simultaneous ${90}^{\circ }$ rotation of the fields of (a) [24]. (c) and (d) Poling field dependence of the $P\text{−}H$ curves at $T=5$ K for (c) ${\mathbf{E}}^0\parallel y$ and ${\mathbf{H}}^0\parallel x$ and (d) ${\mathbf{E}}^0\parallel x$ and ${\mathbf{H}}^0\parallel y$ poling field configurations. The color codes of (c) and (d) are shown in (a) and (b), respectively. The slopes of the $P\text{−}H$ curves correspond to ${\chi }_{yx}$ and ${\chi }_{xy}$ in (c) and (d), respectively. The sign of the ME effect depends on the relative signs of the poling fields; note the complete overlap of the red and orange as well as the blue and green curves. (e)–(h) Optical absorption spectra measured at $T=5$ K (e) and (g) with poling configurations indicated in (a) and (f) and (h) with poling configurations shown in (b). Spectra in (e) and (f) were measured using linearly polarized light with $E_y^{\omega }$, $H_x^{\omega }$, while those in (g,h) were measured with $E_x^{\omega }$, $H_y^{\omega }$. The spectra of the dynamic ME coefficients can be calculated as the absorption difference of the different domains, according to Eqs. (1) and (2). The $\mathrm{\Delta }{\alpha }_2$ ODAs in (e) and (f) as well as in (g) and (h) change sign for the rotation of the poling $E^0$ and $H^0$ poling fields, according to Eqs. (3) and (4), respectively.

Figure 3

(a) Optical absorption spectra, measured in six different configurations of ${\mathbf{E}}^{\omega }$ and ${\mathbf{H}}^{\omega }$ of the light at $T=5$ K, revealing the selection rules of the magnetic excitations. The inset indicates the color code of the different light polarizations, along with an example where ${\mathbf{E}}^{\omega }\parallel x$, ${\mathbf{H}}^{\omega }\parallel z$, and $\mathbf{k}\parallel y$. Note the difference in the vertical scales corresponding to the spectral regions below and above the vertical dashed line. (b) Schematic illustration of an antiferromagnetic resonance (AF-magnon), an electromagnon (E-magnon), and a ME resonance. These modes are excited by only the magnetic component, by only the electric component, and simultaneously by both components of the electromagnetic radiation. For the sake of simplicity, we illustrate the excitations with one representative pair of spins $M$ of the magnetic unit cell along with the respective local polarization $P$. The dynamic nature of the excitations is captured by curved arrows, which represent in-phase or antiphase oscillations of $M$ and $P$. The net dynamic magnetization and polarization of the unit cell are labeled ${\mathbf{M}}^{\omega }$ and ${\mathbf{P}}^{\omega }$, respectively.

Figure 4

Magnetic field dependence of the optical absorption spectra measured at $T=5$ K for different combinations of the poling fields. The reference spectra, taken in the paramagnetic phase ($T=30$ K), were subtracted from the spectra measured at $T=5$ K. The poling fields were applied in the (a) (${\mathbf{E}}^0\parallel x$, ${\mathbf{H}}^0\parallel y$) and (b) and (c) (${\mathbf{E}}^0\parallel y$, ${\mathbf{H}}^0\parallel x$) configurations; the static magnetic field applied after poling was $\mathbf{H}\parallel {\mathbf{H}}^0$. The ${\mathbf{E}}^{\omega }$ polarization of the electromagnetic radiation was set along the (a) and (b) $y$ and (c) $x$ axes. For the purpose of clarity, each spectrum is shifted in proportion to the applied field by (a) 25 ${\mathrm{cm}}^{-1}$/10 kOe and (b) and (c) 10 ${\mathrm{cm}}^{-1}$/10 kOe.

Figure 5

Individual contributions of the different ME spin resonances to the static ME effect, calculated using the ME sum rule. (a) and (b) Field dependence of ${\chi }_{yx}$ for domain $\alpha$, with $\mathbf{H}\parallel y$ and $\mathbf{H}\parallel x$, respectively. (c) Field dependence of ${\chi }_{xy}$ for domain $\alpha$, with $\mathbf{H}\parallel x$. The color coding and the numbering of the curves correspond to the labeling of the modes in Fig. 4. For ${\chi }_{yx}^{\alpha }$ and ${\chi }_{xy}^{\alpha }$ the sum of the different contributions and their field and temperature dependences are shown in Fig. 6 in comparison with the data from the static measurements.

Figure 6

(a) and (b) Magnetic field and (c) temperature dependences of the static and optical ME susceptibilities. The optical data were calculated using the ME sum rule over the complete spectral window. In (a), the gray and blue curves, corresponding to the field dependence of ${\chi }_{yx}^{\alpha }$ for $\mathbf{H}\parallel x$ and $\mathbf{H}\parallel y$, respectively, should coincide at $H=0$. Their slight deviation may come, e.g., from the imperfectness of the ME single-domain state in one of the measurements. (b) the magnetic field dependence of ${\chi }_{xy}^{\alpha }$ in applied field $\mathbf{H}\parallel {\mathbf{H}}^0\parallel x$. In (c), the optical spectroscopy measurements were carried out in the absence of $H$ field, while static ME susceptibility was taken in warming runs with $H=10$ kOe field. The corresponding, temperature-dependent optical absorption spectra can be found in Fig. S3 [24]. The error bars were estimated using the average of the ME susceptibilities between $\pm H$ fields.