Directional dichroism in the paramagnetic state of multiferroics: A case study of infrared light absorption in ${\mathrm{Sr}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$ at high temperatures

The coexisting magnetic and ferroelectric orders in multiferroic materials give rise to a handful of novel magnetoelectric phenomena, such as the absorption difference for the opposite propagation directions of light called the nonreciprocal directional dichroism (NDD). Usually, these effects are restricted to low temperature, where the multiferroic phase develops. In this paper, we report the observation of NDD in the paramagnetic phase of ${\mathrm{Sr}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$ up to temperatures more than 10 times higher than its Néel temperature (7 K) and in fields up to 30 T. The magnetically induced polarization and NDD in the disordered paramagnetic phase is readily explained by the single-ion spin-dependent hybridization mechanism, which does not necessitate correlation effects between magnetic ions. The ${\mathrm{Sr}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$ provides an ideal system for a theoretical case study, demonstrating the concept of magnetoelectric spin excitations in a paramagnet via analytical as well as numerical approaches. We applied exact diagonalization of a spin cluster to map out the temperature and field dependence of the spin excitations, as well as symmetry arguments of the single ion and lattice problem to get the spectrum and selection rules.

Figure 1

Schematic illustration of the ${\mathrm{Sr}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$ crystal structure, space group $P\overline{4}{2}_{1}m$, and the coordinate axes $\mathbf{x}\parallel \left[110\right]$ and $\mathbf{y}\parallel \left[\overline{1}10\right]$. Purple circles denote the $S=\frac{3}{2}{\mathrm{Co}}^{2+}$ ions in the centers of the ${\mathrm{O}}^{2-}$ tetrahedra (shown in blue). The compound is built up by layers of ${\mathrm{CoO}}_4$ tetrahedra and ${\mathrm{Si}}_2{\mathrm{O}}_7$ units (gray), which are separated by Sr ions. For clarity, only a single layer is shown. $\pm \kappa$ is the rotation angle of tetrahedra about the [001] axis away from [110]. A and B denote the two types of tetrahedra with $+\kappa$ and $-\kappa$ tilt. In the experiment, the THz light propagates along the direction of the magnetic field $\mathbf{k}\parallel \mathbf{B}\parallel \left[100\right]$. The magnetic field breaks the time-reversal symmetry and for this field direction the remaining unitary symmetry operation is the $2_1$ screw axis (black half-arrow). The antiunitary operations including the time reversal are the $2^{\prime }$ twofold rotation about the [001] axis (red ellipse) and the $2_1^{\prime }$ screw axis (red half arrow).

Figure 2

Temperature dependence of measured THz absorption spectra (a), (b) and calculated susceptibilities (c)–(f) of ${\mathrm{Sr}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$ in 14 T. The THz absorption spectra for magnetic field $\mathbf{B}\uparrow \uparrow \mathbf{k}\parallel \left[100\right]$ (red) and $\mathbf{B}\uparrow \downarrow \mathbf{k}$ (blue) are measured using linearly polarized radiation where in (a) ${\mathbf{E}}_{\omega }\parallel \left[010\right]$ and ${\mathbf{B}}_{\omega }\parallel \left[001\right]$, and in (b) the polarization is rotated by $\pi /2$, so that ${\mathbf{E}}_{\omega }\parallel \left[001\right]$ and ${\mathbf{B}}_{\omega }\parallel \left[010\right]$. The spectra measured at each temperature are shifted by a constant baseline. (c) and (e) are the magnetic susceptibility $\text{Im}{\chi }^{mm}\left(\omega \right)$ and the ME susceptibility $\text{Im}{\chi }^{me}\left(\omega \right)$ for the polarizations ${\mathbf{E}}_{\omega }\parallel \left[010\right]$ and ${\mathbf{B}}_{\omega }\parallel \left[001\right]$. (d) and (f) are $\text{Im}{\chi }^{mm}\left(\omega \right)$ and $\text{Im}{\chi }^{me}\left(\omega \right)$ for the polarization ${\mathbf{E}}_{\omega }\parallel \left[001\right]$ and ${\mathbf{B}}_{\omega }\parallel \left[010\right]$. Red (positive) and blue (negative) colors indicate the sign of the susceptibility. The saturation of the color corresponds to the magnitude of the corresponding susceptibility matrix elements, $\text{Im}{\chi }^{mm}\left(\omega \right)$ [Eq. (11)] and $\text{Im}{\chi }^{me}\left(\omega \right)$ [Eq. (12)]. The susceptibilities were calculated by the exact diagonalization of a four-site cluster.

Figure 3

Magnetic field dependence of absorption spectra of ${\mathrm{Sr}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$ in the paramagnetic state at 10 K, $\mathbf{B}\uparrow \uparrow \mathbf{k}\parallel \left[100\right]$ (red) and $\mathbf{B}\uparrow \downarrow \mathbf{k}$ (blue). The spectra are shifted in proportion to the absolute value of the magnetic field.

Figure 4

Magnetic field dependence of measured THz absorption spectra (a), (b) and calculated susceptibilities (c)–(f) of ${\mathrm{Sr}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$ at 30 K. The THz absorption spectra for magnetic field $\mathbf{B}\uparrow \uparrow \mathbf{k}\parallel \left[100\right]$ (red) and $\mathbf{B}\uparrow \downarrow \mathbf{k}$ (blue) are shown for two polarizations of the THz light. The spectra are shifted in proportion to the absolute value of the magnetic field. (c) and (e) are the magnetic susceptibility $\text{Im}{\chi }^{mm}\left(\omega \right)$ and the ME susceptibility $\text{Im}{\chi }^{me}\left(\omega \right)$ for the polarizations ${\mathbf{E}}_{\omega }\parallel \left[010\right]$ and ${\mathbf{B}}_{\omega }\parallel \left[001\right]$. (d) and (f) are $\text{Im}{\chi }^{mm}\left(\omega \right)$ and $\text{Im}{\chi }^{me}\left(\omega \right)$ for the polarizations ${\mathbf{E}}_{\omega }\parallel \left[001\right]$ and ${\mathbf{B}}_{\omega }\parallel \left[010\right]$. Red (positive) and blue (negative) colors indicate the sign of susceptibility. The saturation of the color corresponds to the magnitude of the corresponding susceptibility matrix elements $\text{Im}{\chi }^{mm}\left(\omega \right)$ [Eq. (11)] and $\text{Im}{\chi }^{me}\left(\omega \right)$ [Eq. (12)]. The susceptibilities were calculated by the exact diagonalization of a four-site cluster.

Figure 5

The tetrahedron-fixed coordinate system $\left\{X,Y,Z\right\}$ and the rotated magnetic-field-fixed coordinate system $\{\parallel ,\perp 1,\perp 2\}$. Purple ball is the ${\mathrm{Co}}^{2+}$ and the blue tetrahedron is the cage with the oxygen ions at the vertices. The quantization axis (spin component $S^{\parallel })$ is parallel to the direction of the static external magnetic field $h$ (shown in green), while one perpendicular axis (the spin component $S^{\perp 2})$ is parallel to $Z$.

Figure 6

(a) Transition energies $\omega$ and (b) energy levels $\varepsilon$ of the single-ion model with easy-plane anisotropy $\mathrm{\Lambda }$ in a magnetic field $h=g{\mu }_{\mathrm{B}}B$ within the easy plane as a function of $h/\mathrm{\Lambda }$, with $\mathrm{\Lambda }=13.4$ K and $g=2.3$ from Ref. [20]. States 1 and 3 (red curves) are multiplied by $-i$ after a $\pi$ rotation about the field $\left(C_2^{\parallel }\right)$, while states 2 and 4 (blue) get an $i$. Magenta arrows represent transitions between the states of the same symmetry, induced by operator $\mathcal{A}$, even under the $C_2^{\parallel }$, such as $S^{\parallel },P^{\parallel }$ (see Table 1). Cyan arrows connect states with different symmetries induced by operator $\mathcal{B}$, odd under $C_2^{\parallel }$, such as the perpendicular components of spin and polarization operator. The color of the curves corresponds to the color of the arrows in (b). The filled circles show the experimental results.

Figure 7

The tetrahedron in the (a) chiral $\left(\phi =0\right)$, (b) low-symmetry, and (c) polar $\left(\phi =\pi /4\right)$ cases as seen from the direction of the magnetic field $\mathbf{B}$. The magenta sphere represents the magnetic ion at the center of tetrahedral cage. The stereograms represent the magnetic point group for each case, black refers to the unitary subgroup, and red the antiunitary part combined with the time reversal.