2.1.

The RFeO₃ Class

The ${\mathrm{RFeO}}_3$ class has been extensively studied ever since it was first identified in the 1940s; the first review paper on ${\mathrm{RFeO}}_3$ appeared in 1969^[75]. These crystals have an orthorhombically distorted perovskite structure with four molecular units (20 atoms) per unit cell. The lattice structure is described by the centrosymmetric space group $D_{2h}^{16}-Pbnm$.¹ The ${\mathrm{Fe}}^{3+}$ and ${\mathrm{R}}^{3+}$ ions occupy the $4(b)$ and $4(c)$ positions, respectively, and are responsible for producing magnetism.

To understand the various magnetic phases that appear in this family, it is essential to first introduce Bertaut’s analysis of magnetic structures using representation theory^[76]. The theory is distinct from the conventional approach of Shubnikov groups^[77] to classify magnetic structures, because it does not seek the collection of operations that would leave the symmetry invariant. Instead, it outlines the transformation properties of a magnetic structure under the symmetry elements of the parent crystallographic group. This enables magnetic structures to be denoted by irreducible representations (irreps) of the crystallographic group. Regarding the space group of $D_{2h}^{16}-Pbnm$ for ${\mathrm{RFeO}}_3$, the crystallographic point group is mmm, which is specified by the character table to possess eight one-dimensional irreps. Eight types of magnetic configurations (or magnetic phases), labeled by ${\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_8$, can therefore appear. Table 1 displays the transformation properties of each phase under all 16 classical symmetry operations of the parent group^[78], where “+” (“−”) indicates that the structure transforms symmetrically (antisymmetrically) under an operation. The configurations ${\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_8$ have one-to-one correspondence with magnetic groups under Shubnikov’s classification, which are displayed in Table 1 as well. Note that this principle of one-to-one correspondence between magnetic point groups and one-dimensional irreducible unitary representations is generic.

Table 1

Magnetic Phases in RFeO3 Denoted by Irreducible Representations (irreps)[78]a.

The advantage of representation theory in classifying magnetic structures is the ease with which one can derive basis functions. These functions are written in terms of spin components that transform in the same way as an irrep, and therefore, they give the allowed magnetic modes for a magnetic phase. For the four ${\mathrm{Fe}}^{3+}$ atoms located in $4(b)$ sites, whose spins are labled by ${\mathit{S}}_i$ (i = 1−4), the basis functions can be well described by the following four vector components:

One may notice from Table 1 that the ${\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_4$ phases preserve inversion symmetry, while the ${\mathrm{\Gamma }}_5-{\mathrm{\Gamma }}_8$ phases do not possess inversion symmetry; this can be read from the transformation properties of irreps on the row labeled by the inversion operator “$i$.” While three ${\mathrm{Fe}}^{3+}$ spin components can be finite for the ${\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_4$ phases, there is no ${\mathrm{Fe}}^{3+}$ component that can be consistent with ${\mathrm{\Gamma }}_5-{\mathrm{\Gamma }}_8$. This is because the $4(b)$ sites (that ${\mathrm{Fe}}^{3+}$ ions occupy) are inversion centers, and the inversion operator would not be able to flip spins on these sites. Instead, $i$ - broken phases have to be generated by magnetic ordering of ${\mathrm{R}}^{3+}$ ions. Located on $4(c)$ sites, rare-earth ions may develop certain spin structures that are allowed in the ${\mathrm{\Gamma }}_5-{\mathrm{\Gamma }}_8$ configurations. Defining four vectors $\mathit{f}$, $\mathit{g}$, $\mathit{a}$, and $\mathit{c}$ based on the four ${\mathrm{R}}^{3+}$ spins ${\mathit{R}}_i$ (i = 1−4) in a unit cell as

The discussions on the group theory classifications given so far are abstract, but we can now proceed to place them in a concrete context by reading the diagram of the ${\mathrm{RFeO}}_3$ family, the latest version of which was summarized by Li et al.^[79] in 2019. Figure 3 shows the temperature-dependent magnetic phases of ${\mathrm{RFeO}}_3$ for $\mathrm{R}=\mathrm{La}$ to Lu (atomic numbers: 57–71) and Y (atomic number: 39); no data are available for $\mathrm{R}=\mathrm{Pm}$ due to its radioactivity.

Fig. 3

Temperature-dependent magnetic phase diagrams for all members of the ${\mathrm{RFeO}}_3$ class^[79], except for $\mathrm{R}=\mathrm{Pm}$, for which no data are available due to its radioactivity. SRT, spin reorientation transition. Reproduced with permission from Ref. [79].

For all crystals, the magnetic ordering of ${\mathrm{Fe}}^{3+}$ sets within the temperature range of 620–740 K to form the ${\mathrm{\Gamma }}_4$ phase, featuring AFM ordering along the $x$ and $y$ axes ($G_x{A}_y$) and a weak net moment along the $z$ axis ($F_z$). The net moment is produced by a small spin canting (canting angle $\sim 8.5\mathrm{mrad}$) arising from the Dzyaloshinskii–Moriya (DM) interaction. For magnetically inert rare-earth ions, ${\mathrm{R}}^{3+}={\mathrm{La}}^{3+}$, ${\mathrm{Eu}}^{3+}$, ${\mathrm{Lu}}^{3+}$, and ${\mathrm{Y}}^{3+}$, the ${\mathrm{\Gamma }}_4$ phase persists down to the lowest temperature.

Magnetic phase transitions arise, however, in crystals where the ${\mathrm{R}}^{3+}$ ions are magnetically active and therefore can interact with the ${\mathrm{Fe}}^{3+}$ spins^[80,81]. The transition with the most frequent occurrence is the ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ spin-reorientation transition (SRT) with decreasing temperature, during which the spin structure continuously rotates within the $x-z$ plane while its relative angles remain rigid; the transition is of finite width in temperature, and within the intermediate temperature range, components of the ${\mathrm{\Gamma }}_4$ phase ($G_x{A}_y{F}_z$) superimpose on those of the ${\mathrm{\Gamma }}_2$ phase ($F_x{C}_y{G}_z$), producing the ${\mathrm{\Gamma }}_{24}$ phase. Crystals with $\mathrm{R}=\mathrm{Pr}$, Nd, Sm, Er, Tm, and Yb show this transition, among which ${\mathrm{SmFeO}}_3$ has a transition temperature high enough to make itself the only crystal to exhibit the ${\mathrm{\Gamma }}_2$ phase at room temperature. In addition to the continuous-type ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT, there is an abrupt-type ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_1$ SRT observed first in ${\mathrm{DyFeO}}_3$^[82,83] (at 50 K) and later in ${\mathrm{CeFeO}}_3$^[84,85] (at 240 K). Since the magnetic modes of $A_x{G}_y{C}_z$ allowed in the low-temperature ${\mathrm{\Gamma }}_1$ phase do not possess a net moment ($\mathit{F}=0$), a sharp decrease of magnetization has been observed across the transition. Moreover, ${\mathrm{HoFeO}}_3$ adopts a more complicated sequence of SRTs as ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_{24}\to {\mathrm{\Gamma }}_{12}\to {\mathrm{\Gamma }}_2$^[86–89], featuring an intermediate temperature range within which two metastable configurations, ${\mathrm{\Gamma }}_{24}$ and ${\mathrm{\Gamma }}_{12}$, coexist.

Magnetic interactions between ${\mathrm{R}}^{3+}$ and ${\mathrm{Fe}}^{3+}$ not only lead to assorted SRTs of the ${\mathrm{Fe}}^{3+}$ spin configuration, but also cause ${\mathrm{R}}^{3+}$ to be polarized by the exchange field supplied by the ${\mathrm{Fe}}^{3+}$ sublattice. When ${\mathrm{R}}^{3+}$ is paramagnetic, its polarization is always consistent with the irrep that describes the ${\mathrm{Fe}}^{3+}$ spin structure. For instance, in the ${\mathrm{\Gamma }}_4$ and ${\mathrm{\Gamma }}_2$ phases, where ${\mathrm{Fe}}^{3+}$ spins have a net moment, the ${\mathrm{R}}^{3+}$ spins will also develop a polarized net moment that is either parallel or antiparallel to the ${\mathrm{Fe}}^{3+}$ magnetization. These are labeled “+” (parallel) and “−” (antiparallel) in Fig. 3, respectively. For those crystals that have antiparallel alignments ($\mathrm{R}=\mathrm{Nd}$, Sm, Er, and Tm), the strong temperature dependence of the ${\mathrm{R}}^{3+}$ moment leads to a compensation point^[90], that is, a temperature at which the ${\mathrm{R}}^{3+}$ net moment exactly cancels the ${\mathrm{Fe}}^{3+}$ moment.

At even lower temperatures, typically below 10 K, ${\mathrm{R}}^{3+}$ develops ordering for $\mathrm{R}=\mathrm{Pr}$, Gd, Tb, Dy, Ho, and Er. This creates an interesting situation because the ${\mathrm{R}}^{3+}$ order parameter does not have to be compatible with the ${\mathrm{Fe}}^{3+}$ phase, leading to further symmetry breaking. For ${\mathrm{ErFeO}}_3$, ${\mathrm{Er}}^{3+}$ spins develop a $c_z$ mode when they order, and according to Table 1, the $c_z$ mode transforms according to the ${\mathrm{\Gamma }}_1$ irrep, which is different from the ${\mathrm{\Gamma }}_2$ phase of the ${\mathrm{Fe}}^{3+}$ spins (before the transition occurs). The low-temperature magnetic phase of ${\mathrm{ErFeO}}_3$ is therefore a ${\mathrm{\Gamma }}_{12}$ phase^[91,92], whose magnetic group ($2m$) is an index-two subgroup of either ${\mathrm{\Gamma }}_1$ ($mmm$) or ${\mathrm{\Gamma }}_2$ ($\underline{mm}m$). More interesting physics occur when ${\mathrm{R}}^{3+}$ ordering transforms with an inversion-symmetry-breaking representation ${\mathrm{\Gamma }}_j$ with j ∈ {5, 6, 7, 8}; the resultant phase that takes both ${\mathrm{Fe}}^{3+}$ and ${\mathrm{R}}^{3+}$ into account would be ${\mathrm{\Gamma }}_{ij}$ with i ∈ {1, 2, 3, 4} and j ∈ {5, 6, 7, 8}, which is a noncentrosymmetric phase capable of exhibiting appealing properties that would otherwise be forbidden in centrosymmetric groups. The low-temperature phases of ${\mathrm{GdFeO}}_3$, ${\mathrm{DyFeO}}_3$, and ${\mathrm{TbFeO}}_3$ fall into this category. Their details will be covered in Subsection 3.3 while discussing magneto-electric effects, multiferroicity, and electromagnons.

A variety of experimental methods have been applied to the ${\mathrm{RFeO}}_3$ family in the last century to characterize their magnetic, electronic, and optical properties. Once a crystal is grown, X-ray scattering and Laue diffraction measurements are typically carried out first to examine the crystal quality^[93]. After that, the most straightforward method to study magnetic phase transitions is magnetization measurements. Since the ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_2$, and ${\mathrm{\Gamma }}_4$ phases are distinguished by their magnitude and orientation of the net moment, an SRT can be readily identified in magnetization curves. Figure 4(a) shows representative temperature-dependent magnetization curves for ${\mathrm{LuFeO}}_3$, ${\mathrm{ErFeO}}_3$, and ${\mathrm{DyFeO}}_3$^[75]. In ${\mathrm{LuFeO}}_3$, the ${\mathrm{\Gamma }}_4$ phase persists through the entire range below the Néel temperature of ${\mathrm{Fe}}^{3+}$, which manifests as a finite but nearly temperature-independent $z$-axis magnetization. Upon cooling, the continuous-type ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT in ${\mathrm{ErFeO}}_3$ induces a finite $x$-axis magnetization at the expense of $z$-axis magnetization between 80 K and 100 K. The compensation behavior caused by antiparallel moment alignment between ${\mathrm{Er}}^{3+}$ and ${\mathrm{Fe}}^{3+}$ in the ${\mathrm{\Gamma }}_2$ phase manifests as a temperature point (45 K) with zero $x$-axis magnetization. In ${\mathrm{DyFeO}}_3$, the $z$-axis magnetization diminishes to zero when the abrupt-type ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_1$ initiates, in agreement with the fact that the ${\mathrm{\Gamma }}_1$ phase is incompatible with any net moment. The more complicated multistage SRT in ${\mathrm{HoFeO}}_3$ does not show salient features in magnetization that can be distinguished from the SRT in ${\mathrm{ErFeO}}_3$, but Young’s modulus and magnetostriction measurements^[86] are able to capture the detailed features. The ${\mathrm{R}}^{3+}$-ordered phases typically appear as anomalies on magnetization curves at low temperatures^[82], but scattering experiments^[92,97] are certainly more advantageous to precisely determine the magnetic structure.

Fig. 4

Basic physical properties of ${\mathrm{RFeO}}_3$ crystals. (a) Temperature-dependent magnetization of ${\mathrm{LuFeO}}_3$, ${\mathrm{ErFeO}}_3$, and ${\mathrm{DyFeO}}_3$ along the $z$ axis and $x$ axis^[75]. (b) Optical absorption spectrum of ${\mathrm{ErFeO}}_3$^[94]. (c) Spin-wave dispersion mapped by inelastic neutron scattering^[95]. (d) Spin oscillations in quasi-FM and quasi-AFM modes^[96]. Each spin evolves in the sequence of $1\to 2\to 3\to 4\to 1\dots$. Spins are synchronized by the number label. Reproduced with permission from Refs. [75,94–96].

Regarding the electronic properties, all crystals within the ${\mathrm{RFeO}}_3$ family are categorized as charge-transfer insulators, rather than Mott insulators, under the framework developed by Zaanen, Sawatsky, and Allen^[98,99]. This means that the charge gap is governed by the energy difference between the $p$ bands of oxygen anions and the unoccupied upper Hubbard band of ${\mathrm{Fe}}^{3+}$; the assignment is also corroborated by density functional theory (DFT) computations^[100]. To describe charge excitations more clearly, Fig. 4(b) shows an optical absorption spectrum for an ${\mathrm{ErFeO}}_3$ crystal^[94]; other crystals within the same family share much similarity. Intense absorption bands are seen on both sides of a transparent window (1.3 µm to 8 µm). On the short wavelength side, the peaks centered at 1.1 µm, 0.7 µm, and 0.5 µm arise from the ${}^6\mathrm{A}_1\to {}^4\mathrm{T}_1$, ${}^6\mathrm{A}_1\to {}^4\mathrm{T}_2$, and ${}^6\mathrm{A}_1\to {}^4\mathrm{E}+{}^4\mathrm{A}$ transitions of ${\mathrm{Fe}}^{3+}$, respectively. Giant absorption is present for wavelengths shorter than 0.5 µm, and is assigned to the charge-transfer transition from oxygen to ${\mathrm{Fe}}^{3+}$^[101]. Within the transparent window, an absorption band centered at 1.5 µm is identified to be excitations across spin–orbit coupled atomic states of ${\mathrm{Er}}^{3+}$ (${{}^4\mathrm{I}}_{15/2}\to {{}^4\mathrm{I}}_{13/2}$). For longer wavelengths, the absorption coefficient onsets starting from 8 µm are due to infrared (IR)-active phonon excitations. Among the 60 phonon modes, 24 are Raman-active and 36 are IR-active. Although computations reveal the eigen-modes and frequencies of all IR-active phonons^[102], many of them remain unobserved except for the Fe-O bond stretching mode at around 18 µm^[103]. On the other hand, Raman-active phonons are much better characterized experimentally^[104,105], showing excellent agreement with theory.

Finally, we discuss magnetic excitations that can be developed on top of the magnetic ground states of ${\mathrm{RFeO}}_3$, whose understanding is crucial for the major topic of this review paper, i.e., THz spin dynamics. Within a unit cell that consists of four ${\mathrm{Fe}}^{3+}$ and four ${\mathrm{R}}^{3+}$ ions, we focus on a simple model where ${\mathrm{R}}^{3+}$ is assumed to be paramagnetic. This means that magnetic excitations majorly involve ${\mathrm{Fe}}^{3+}$; cooperative magnetic excitations involving both ${\mathrm{R}}^{3+}$ and ${\mathrm{Fe}}^{3+}$ in the low-temperature ${\mathrm{R}}^{3+}$-ordered phase are, to date, still an active topic. Shapiro et al.^[95] used inelastic neutron scattering measurements to map out the dispersion relations of spin waves, and observed an optical magnon branch appearing in the energy range from 50 to 70 meV, as shown in Fig. 4(c). This is the exchange mode of ${\mathrm{Fe}}^{3+}$ in the four-sublattice description. Focusing on the high-energy sector, an isotropic Heisenberg Hamiltonian including both the nearest neighbor ($J$) and next-nearest neighbor ($J^{\prime }$) fits this branch well, enabling quantitative extraction of $J$ and $J^{\prime }$^[106].

On the other hand, an acoustic magnon branch, which softens to a near-zero (not exact zero) frequency when approaching the zone center, also appears. In an attempt to accurately describe this branch, especially its low-energy behavior at the zone center, Herrmann^[107,108] developed a two-sublattice model, where sublattices A and B arise from the original four ${\mathrm{Fe}}^{3+}$ ions in a unit cell as

This way, the C-type AFM component in Eq. (1) is canceled ($\mathit{C}=0$), maintaining the G-type AFM and the net moment that can describe the SRTs. The spin Hamiltonian reads

The anisotropy terms are precisely the cause of gaps in the acoustic magnon branch (which is usually considered to be gapless in analogy to acoustic phonons) at the zone center in the inelastic neutron spectra. Indeed, an equation of motion calculation using Eq. (4) gives two magnon modes^[107,108], named quasi-ferromagnetic (quasi-FM) and quasi-AFM modes, respectively, at

Equation (5) provides an important clue for the underlying driving force of a ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT, by showing that $\mathrm{\hbar }{\omega }_{\mathrm{FM}}\to 0$ when $A_x-A_z\to 0$. Across a ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT, anisotropy switches from $A_x>A_z$ to $A_z>A_x$, and the critical softening of the quasi-FM mode that evolves hand in hand with the anisotropy energy drives the phase transition. The quasi-FM and quasi-AFM modes at the zone center are excitable by both neutron scattering and Raman scattering, so their frequencies have been well studied since early days^{[95,96,109–111]}. Softening of the quasi-FM frequency across a ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT has indeed been observed. On the contrary, the SRT in ${\mathrm{DyFeO}}_3$ is accompanied by softening of the quasi-AFM mode^[96].

The quasi-FM and quasi-AFM modes are key players in the THz spintronic studies of ${\mathrm{RFeO}}_3$ materials. In addition to having finite neutron and light scattering cross sections, group theory analysis also stipulates that they also respond linearly to THz-frequency light radiation through the magnetic–dipolar interaction^[109]. Section 3 delineates various types of THz probes of magnon modes. In the next subsection, we introduce the major ultrafast optical techniques that enable these studies.

2.2.

Ultrafast Optical Techniques for Probing THz Spin Dynamics

The study of interaction between light and spins has a long history. In recent years, however, the advancements of ultrafast laser spectroscopy technologies have redefined the way to investigate THz-frequency spin phenomena. Reconstruction of magnetic dynamics occurring in a material system in real time can be readily implemented by a stroboscopic pump–probe experimental setup. The sample is first excited by a pump pulse, and analysis of the probe pulse that arrives at the sample at a variable time delay enables one to infer the evolution of the excited state of the sample. Since the THz frequency range has time scales of the order of ps, the fs-scale pulse widths provided by current ultrafast lasers provide enough temporal resolution.

In this subsection, we review two major types of ultrafast optical techniques for THz studies of spin dynamics. The first is time-resolved photomagnetic/magneto-optical spectroscopy, which is essentially Faraday/Kerr rotation measurements^[112] using pulsed lasers (as opposed to continuous-wave lasers used in earlier studies), and configures a pump–probe time delay to provide time resolution. Owing to the simplicity of envisioning its experimental setup, emphasis of discussion will be on the theoretical framework of tensor analysis that provides the predictive power for the experiments. The second is THz time-domain techniques, including THz time-domain spectroscopy, THz emission spectroscopy, and nonlinear THz spectroscopy, which allows driving spins far away from equilibrium. The working principles of these methods and their advantages will be reviewed.

Probe methods based on spin-resolved photoemission^[113,114] and X-ray scattering^[115–117] possess advantages in ultrafast spintronics for certain compounds. However, for the ${\mathrm{RFeO}}_3$ family that we focus on here, optical techniques present themselves as not only the most suitable but also the easiest approaches.

2.2.1.

Photomagnetic/magneto-optical effects

In a material, either the development of magnetic order or the application of an external magnetic field can affect its optical properties in various ways. These processes are generally known as magneto-optical phenomena and have long been applied to the study of magnetism; notable examples are Faraday and Kerr rotations. On the other hand, intense and properly tailored light impinging on a material can directly influence the states and dynamics of spins and actively modify magnetic properties. These processes representing inverse effects of magneto-optical phenomena are generally known as photomagnetic effects. To gain detailed insight into how photomagnetic and magneto-optic effects are categorized and how they are related to each other, one needs to use the tensor description of light–spin interaction, as described below.

When an ${\mathrm{RFeO}}_3$ crystal is illuminated by light with frequency $\omega$, the total Hamiltonian is given by

$$H=H^0+H^{\mathrm{MO}}.$$

Here, $H^0$ is the general spin Hamiltonian of ${\mathrm{RFeO}}_3$, while $H^{\mathrm{MO}}$ arises from light–spin interactions and can be expanded to lowest order in the electric field of light, $\mathit{E}(\omega )$, as^[57]

Eq. (6)

$$H^{\mathrm{MO}}\approx -[{\varepsilon }_{ij}^0{E}_i(\omega )E_j^{*}(\omega )+{\alpha }_{ijk}{E}_i(\omega )E_j^{*}(\omega )F_k(0)+{\alpha }_{ijk}^{\prime }{E}_i(\omega )E_j^{*}(\omega )G_k(0)+{\beta }_{ijkl}{E}_i(\omega )E_j^{*}(\omega )F_k(0)F_l(0)+{\beta }_{ijkl}^{\prime }{E}_i(\omega )E_j^{*}(\omega )G_k(0)G_l(0)+{\beta }_{ijkl}^{\prime \prime }{E}_i(\omega )E_j^{*}(\omega )F_k(0)G_l(0)],$$

where $\mathit{F}={\mathit{S}}^{A }+{\mathit{S}}^B$ and $\mathit{G}={\mathit{S}}^A-{\mathit{S}}^B$ are FM and AFM vectors [Eqs. (1) and (3)], respectively, ${\varepsilon }_{ij}^0$, ${\alpha }_{ijk}$, ${\alpha }_{ijk}^{\prime }$, ${\beta }_{ijkl}$, ${\beta }_{ijkl}^{\prime }$, ${\beta }_{ijkl}^{\prime \prime }$ are tensor coefficients, and the Einstein summation rule is implied (and throughout this paper).

The permittivity tensor, which governs the optical property, then reads

Eq. (7)

$${\varepsilon }_{ij}=-\frac{{\partial }^{2}H}{\partial E_i(\omega )\partial E_j^{*}(\omega )}=-\frac{{\partial }^2{H}^{\mathrm{MO}}}{\partial E_i(\omega )\partial E_j^{*}(\omega )},$$

and one can see that all terms in Eq. (6) except for the first term led by ${\varepsilon }_{ij}^0$ are related to magnetism. According to the Onsager principle, the total permittivity tensor can be separated into two parts, ${\varepsilon }_{ij}={\varepsilon }_{ij}^s+{\varepsilon }_{ij}^a$, where the symmetric part ${\varepsilon }_{ij}^s={\varepsilon }_{ji}^s$ and the antisymmetric part ${\varepsilon }_{ij}^a=-{\varepsilon }_{ji}^a$ are even and odd under time reversal, respectively. Since $\mathit{F}$ (${\mathit{F}}^2$) and $\mathit{G}$ (${\mathit{G}}^2$) are odd (even) under time reversal, the terms that contribute to the total Hamiltonian in Eq. (6) can be categorized based on whether it gives rise to either ${\varepsilon }_{ij}^s$ or ${\varepsilon }_{ij}^a$. These are given in Table 2.

Table 2

Property Tensors, Their Contributions to the Hamiltonian, and Their Resulting Magneto-Optical and Photomagnetic Effects[57]a.

Property Tensor	Contribution to HMO	Magneto-Optical Effects		Photomagnetic Effects
${\alpha }_{ijk}$ (axial, $i$)	${\alpha }_{ijk}{E}_i{E}_j^{*}{F}_k$	${\varepsilon }_{ij}^a={\alpha }_{ijk}{F}_k$	Magnetic circular birefringence (Faraday/Kerr rotation), magnetic circular dichroism	$H_k^{\mathrm{PM}}={\alpha }_{ijk}{E}_i{E}_j^{*}$	Inverse Faraday effect
${\alpha }_{ijk}^{\prime }$ (axial, $i$)	${\alpha }_{ijk}^{\prime }{E}_i{E}_j^{*}{G}_k$	${\varepsilon }_{ij}^a={\alpha }_{ijk}^{\prime }{G}_k$		$h_k^{\mathrm{PM}}={\alpha }_{ijk}^{\prime }{E}_i{E}_j^{*}$
${\beta }_{ijkl}$ (polar, $i$)	${\beta }_{ijkl}{E}_i{E}_j^{*}{F}_k{F}_l$	${\varepsilon }_{ij}^s={\beta }_{ijkl}{F}_k{F}_l$	Magnetic linear birefringence (Cotton-Mouton effect), magnetic linear dichroism	$H_l^{\mathrm{PM}}={\beta }_{ijkl}{E}_i{E}_j^{*}{F}_k$	Inverse Cotton-Mouton effect
${\beta }_{ijkl}^{\prime }$ (polar, $i$)	${\beta }_{ijkl}^{\prime }{E}_i{E}_j^{*}{G}_k{G}_l$	${\varepsilon }_{ij}^s={\beta }_{ijkl}^{\prime }{G}_k{G}_l$		$h_l^{\mathrm{PM}}={\beta }_{ijkl}^{\prime }{E}_i{E}_j^{*}{G}_k$
${\beta }_{ijkl}^{\prime \prime }$ (polar, $i$)	${\beta }_{ijkl}^{\prime \prime }{E}_i{E}_j^{*}{F}_k{G}_l$	${\varepsilon }_{ij}^s={\beta }_{ijkl}^{\prime \prime }{F}_k{G}_l$		$H_k^{\mathrm{PM}}={\beta }_{ijkl}^{\prime \prime }{E}_i{E}_j^{*}{G}_l$$h_l^{\mathrm{PM}}={\beta }_{ijkl}^{\prime \prime }{E}_i{E}_j^{*}{F}_k$

^aFrequencies in parentheses are abbreviated. The electric field E is at frequency ω, while vectors F, G, and effective fields HPM and hPM are all at zero frequency. Reproduced with permission from Ref. [57].

The property tensors ${\alpha }_{ijk}$ and ${\alpha }_{ijk}^{\prime }$, both giving rise to the asymmetric part, ${\varepsilon }_{ij}^a$, contribute only to the off-diagonal elements of the ${\varepsilon }_{ij}$ tensor (${\varepsilon }_{ii}^a=0$). This leads to the eigen-vector of light to take a circularly polarized basis in the medium, giving rise to magnetic circular birefringence, commonly observed as Faraday or Kerr rotations, and magnetic circular dichroism. The proportionality between ${\varepsilon }_{ij}^a$ and $\mathit{F}$ (or $\mathit{G}$) dictates a sign change of these effects upon spin reversal. On the other hand, ${\beta }_{ijkl}$, ${\beta }_{ijkl}^{\prime }$, and ${\beta }_{ijkl}^{\prime \prime }$ contribute to the symmetric part ${\varepsilon }_{ij}^s$, which is able to take the diagonal components within ${\varepsilon }_{ij}$. The associated magneto-optical effects manifest in the linearly polarized basis—i.e., magnetic linear birefringence (Cotton–Mouton effect) and magnetic linear dichroism—and do not change sign upon spin reversal.

Photomagnetic effects arise from the same Hamiltonian, Eq. (6), and the influence of the optical field on magnetism can be viewed as effective magnetic fields at zero frequency:

Eq. (8)

$$\begin{gathered} {\mathit{H}}^{\mathrm{PM}}(0)=-\frac{\partial H^{\mathrm{MO}}}{\partial \mathit{F}(0)}, \\ {\mathit{h}}^{\mathrm{PM}}(0)=-\frac{\partial H^{\mathrm{MO}}}{\partial \mathit{G}(0)}, \end{gathered}$$

where ${\mathit{H}}^{\mathrm{PM}}(0)$ and ${\mathit{h}}^{\mathrm{PM}}(0)$ are the effective fields exerted on $\mathit{F}$ and $\mathit{G}$, respectively. Table 2 summarizes various forms of them that are associated with the five property tensors. Most notably, the effective field $H_k^{\mathrm{PM}}(0)={\alpha }_{ijk}{E}_i(\omega )E_j^{*}(\omega )$, which is associated with ${\alpha }_{ijk}$, the same tensor that induces the Faraday effect, can be further simplified for isotropic media. Replacing ${\alpha }_{ijk}$ by the minimal expression of $\alpha ·{\varepsilon }_{ijk}$, where $\alpha$ is a scalar constant and is the Levi–Civita tensor (reflecting the axial nature of ${\alpha }_{ijk}$), we obtain the effective field as

Eq. (9)

$${\mathit{H}}^{\mathrm{PM}}(0)=\alpha [\mathit{E}(\omega )\times {\mathit{E}}^{*}(\omega )],$$

which resembles the form of the inverse Faraday effect (IFE), stating that circularly polarized light imposes an effective magnetic field along its propagation direction on the medium. Following the same logic, effective fields associated with the fourth-rank polar tensors ${\beta }_{ijkl}$, ${\beta }_{ijkl}^{\prime }$, and ${\beta }_{ijkl}^{\prime \prime }$ can be generated by linearly polarized light, representing the inverse Cotton–Mouton effect (ICME).

If the fields ${\mathit{H}}^{\mathrm{PM}}$ and ${\mathit{h}}^{\mathrm{PM}}$ are generated by a pulsed light field through photomagnetic effects, they will be transient in nature (no longer at zero frequency) and will thus be able to excite coherent magnon oscillations in a way similar to how a THz magnetic field pulse excites magnons though the Zeeman torque. The IFE and ICME have been firmly established in ${\mathrm{RFeO}}_3$ through observation of coherent magnons launched by the laser-induced effective fields. The way to distinguish photomagnetic effects from trivial thermal effects is to see if the phase of coherent magnons can be manipulated by the polarization of pump light. In the case of the IFE, altering the helicity of pump light results in sign reversal of the ${\mathit{H}}^{\mathrm{PM}}$ pulse, and therefore, a 180 deg phase flip in coherent magnons [Fig. 5(a)]. For the ICME, the magnon phase is controlled by the orientation of linear polarization [Fig. 5(b)]. Furthermore, as shown in Fig. 5(c), the general framework that encompasses coherent magnon excitation by the IFE and ICME is the impulsive stimulated Raman scattering (ISRS). The scattering relies on the coexistence of two colors within the light pulse, one at $\omega$ and the other at $\omega -\mathrm{\Omega }$, $\mathrm{\Omega }$ being the magnon frequency. The $\omega -\mathrm{\Omega }$ component stimulates the inelastic scattering of the $\omega$ component to occur, causing it to decompose into another photon at $\omega -\mathrm{\Omega }$ and a magnon quantum at $\mathrm{\Omega }$.

Fig. 5

Observation of (a) the inverse Faraday effect^[118] and (b) the inverse Cotton–Mouton effect^[119] in ${\mathrm{DyFeO}}_3$. The oscillation phases of magnons excited by these photomagnetic effects depend on the polarization of the pump light. (c) Magnon excitation by the two photomagnetic effects can be described by the framework of impulsive stimulated Raman scattering^[57]. Reproduced with permission from Refs. [57,118,119].

In realistic situations, making predictions about the outcomes of pump–probe experiments is rather complex. Specifically, the type of photomagnetic effects that can occur for a particular pumping condition, and the probe configuration that can most efficiently detect the spin dynamics by magneto-optical effects, both depend critically on the material of interest. For ${\mathrm{RFeO}}_3$, based on the knowledge in Table 2, Iida et al.^[119] discussed a useful method to address this problem, which we schematically summarize in Fig. 6.

Fig. 6

Path of solving for the full dynamics of photomagnetic pump, magneto-optical probe experiments on ${\mathrm{RFeO}}_3$, summarized from the procedure adopted in Ref. [119].

When one attempts to predict the spin dynamics excited by a photomagnetic pump, one first calculates effective photomagnetic fields from the known static magnetic geometry, property tensors (Table 2) that are simplified by the magnetic point group symmetries, and the eigenvector of the light field. The effective photomagnetic fields, when combined with intrinsic fields derived from the unperturbed Hamiltonian $H^0$, give the total effective fields (note the time delay dependence of the effective fields due to the transient nature of the pump pulse). The total effective fields can then be used in the Landau–Lifshitz equations to solve for the full time-dependent spin dynamics. After knowing the spin dynamics, one would be interested in detecting them through a magneto-optical probe. This step is easier, as one simply needs to input the time-dependent spin dynamics and the simplified property tensors back into the equation for calculating the optical permittivity tensor (third column in Table 2). The time-dependent optical properties would then be fully determined. Overall, this method is able to provide most qualitative predictions on the pump–probe experiment based on a few symmetry arguments (without resorting to a microscopic model), and is highly efficient and useful.

2.2.2.

THz time-domain techniques

Provided that the time-resolved photomagnetic/magneto-optical experiments majorly target at studying Raman-active magnetic excitations, one should seek for a complementary ultrafast optical technique that couples to IR-active spin modes. THz time-domain techniques target such functionality. They encompass a series of experimental methods, all of which utilize pulsed THz radiation to directly couple to the magnetic excitations through the magnetic–dipolar interaction. However, there is an important aspect that makes them more advantageous compared with the magneto-optical probes.

Since the very beginning of ultrafast magnetism studies, there has existed a controvercy over whether time-resolved mageto-optical signals faithfully represent magnetic dynamics^[120–122]. For instance, one may design a Faraday/Kerr rotation experiment to probe $\mathit{F}$ and $\mathit{G}$ vectors, and the attempt would be successful in an equilibrium situation where the axial tensors ${\alpha }_{ijk}$ and ${\alpha }_{ijk}^{\prime }$ have suitable nonzero components (see Table 2). However, in a nonequilibrium situation where the system is excited by a pump pulse before the arrival of the probe pulse, it is unclear whether the probed Faraday/Kerr rotation originates from the actual magnetic dynamics ($\mathit{F}$ and $\mathit{G}$) or from the dynamics of the tensors ${\alpha }_{ijk}$ and ${\alpha }_{ijk}^{\prime }$^[121,122] (which is only a trivial optical effect unrelated to magnetism). The difficulty with disentangling the two types of possible contributions has led to confusion in interpreting time-resolved magneto-optical data. However, probing spin dynamics using THz time-domain techniques circumvents such complications and therefore is more direct^[123].

THz time-domain spectroscopy is a useful technique to reveal fundamental excitations in solids^[124–137]. Its most common layout is in transmission geometry, as depicted in Fig. 7(a), although reflection geometry is also often used^[139]. A pulsed laser beam is first split by a beam splitter into a strong THz generation beam and a weak probe beam. The THz generation beam, incident on a nonlinear crystal such as ZnTe, generates pulsed THz radiation through a second-order nonlinear process called optical rectification^[37]. The nonlinear polarization

Eq. (10)

$$P_i^{(2)}(\mathrm{\Omega })\propto {\chi }_{ijk}^{(2)}(\mathrm{\Omega }={\omega }_2-{\omega }_1;{\omega }_1,-{\omega }_2)E_j({\omega }_1)E_k^{*}({\omega }_2)$$

gives rise to a radiation field ${\mathit{E}}_{\mathrm{THz}}(t)\propto {\partial }^2{\mathit{P}}^{(2)}(t)/\partial t^2$. The phase-matching condition requires the phase velocity of the generated THz pulse to match the group velocity of the generation pulse:

Eq. (11)

$$v_{\mathrm{THz}}^{\mathrm{ph}}=v_{\mathrm{opt}}^{\mathrm{gr}}.$$

Fig. 7

THz time-domain techniques. (a) Layout of a THz time-domain spectroscopy setup configured in a transmission geometry. (b) Zoom-in view of the polarization-sensitive differential detection setup^[36]. (c) Layout of a THz emission spectroscopy setup. (d) Function of a reflective echelon used in a single-shot THz spectrometer^[138]. (e) Pulse-front-tilt technique for generating intense THz radiation in ${\mathrm{LiNbO}}_3$^[38]. BS, beam splitter; QWP, quarter-wave plate; WP, Wollaston prism; BD, balanced detector. (b), (d), (e) Reproduced with permission from Refs. [36,38,138].

Fig. 8

Six measurement configurations, along with the magnon polarization selection rule in the ${\mathrm{\Gamma }}_4$ and ${\mathrm{\Gamma }}_2$ phases. The magnetic field component of the THz pulse ${\mathit{H}}_{\mathrm{THz}}$ is polarized along a line whose color indicates the magnon mode that can be excited. The THz propagation direction is perpendicular to the disk containing the line. Reproduced with permission from Ref. [169].

The THz beam is then collimated and focused onto a sample using a pair of off-axis parabolic mirrors. The THz pulse interacts with the sample by resonantly coupling to its excitations. The transmitted (reflected) wave is then collected and focused onto another piece of nonlinear detection crystal. Electro-optic sampling is carried out by the time-delayed probe pulse that is directed to the same detection crystal, achieving spatial and temporal overlap with the THz pulse. When the THz pulse and the probe pulse co-propagate in the detection crystal, a polarization modulation is induced on the probe pulse, depending on the amplitude and sign of the THz electric field^[36]. The modulation is analyzed by a polarization-sensitive differential detection scheme [depicted as a zoom-in view in Fig. 7(b)] as a function of probe time delay, mapping out the complete THz electric field waveform in time.

Far-IR spectroscopy techniques using continuous-wave light sources preceded the advent of THz time-domain spectroscopy and have been applied to ${\mathrm{RFeO}}_3$ crystals to provide crucial insights into their magnetic properties^{[87,140–142]}. However, the phase resolution unique to THz time-domain techniques enables precise extraction of Kramers–Kronig compatible real and imaginary parts of index of refraction. Further, time-domain techniques enable one to observe the free-induction decay of a resonance directly in time. A THz time-domain spectrometer can be easily configured into a THz emission spectroscopy setup as in Fig. 7(c). Here, the nonlinear crystal for THz generation is removed, and the THz radiation emitted from the sample itself under optical excitation is measured. Emission spectroscopy is useful for investigating quasi-FM and quasi-AFM magnon modes in ${\mathrm{RFeO}}_3$ because these modes can not only be excited by optical pulses through Raman scattering, but also can couple with THz radiation through magnetic-dipole interactions.

As THz time-domain techniques gain increasing popularity in studies of quantum materials, progress is being made currently to incorporate them with extreme experimental conditions such as low temperatures and high magnetic fields. In particular, high magnetic fields can facilitate explorations of exotic quantum properties of materials with high controllability, as has been shown recently using fields up to 30 T^[143]. Since such high magnetic fields are typically supplied by pulsed magnets, whose peak fields last for only a short duration of time ($\sim 1\mathrm{ms}$), conventional electro-optic sampling of THz waveforms by a stage-scan system cannot be used^[144]. Nevertheless, single-shot detection schemes have been developed^[145], which are capable of measuring the full THz waveform using just one laser pulse within the very short time window during which the sample is experiencing the peak of the pulsed magnetic field. As shown in Fig. 7(d), the key to single-shot detection is to replace the probe delay stage with a reflective echelon mirror^[146], which tilts the pulse front of the optical gate beam by forming time-delayed beamlets; the time delay information is therefore encoded in the pulse-front tilt across the intensity profile of the beam. A fast CMOS camera captures the full image of the gate beam, enabling the extraction of THz waveforms at various time delays.

THz time-domain techniques also nurture the nonlinear THz spectroscopy technique^[147,148], where peak THz electric fields of the order of MV/cm can be generated, leading to an entirely new class of extreme experimental conditions for out-of-equilibrium engineering of quantum materials^[149–154]. Being easily resonant with lattice phonons and magnons, strong THz radiation can selectively drive the lattice and/or spin DOFs of the system far out of equilibrium without strong charge excitations (which is usually a significant source of laser-induced heating). We will see the application of such an apparatus in the study of nonlinear magnonics in Subsection 5.4.

To achieve higher THz electric field strengths in optical rectification, much effort has been expended on increasing the nonlinearity of the generation crystals. However, at the same time, difficulty arises with regard to maintaining the phase-matching condition [Eq. (11)], because of the normal dispersions of nonlinear crystals. A scheme of tilted-pulse-front optical rectification has developed to address this challenge^[155–157]. As shown in Fig. 7(e), the generation beam is tilted in wavefront after being diffracted by a high-efficiency grating. If the tilting angle is $\gamma$, the phase-matching condition in Eq. (11) can be achieved as

Eq. (12)

$$v_{\mathrm{THz}}^{\mathrm{ph}}=v_{\mathrm{opt}}^{\mathrm{gr}}\cos \gamma .$$

When implemented in nonlinear crystals such as ${\mathrm{LiNbO}}_3$, this relation suggests that the propagation direction of the generated THz wave is different from that of the optical beam. Strong THz electric fields of the order of MV/cm have been generated based on this scheme.

An increasing number of novel schemes for intense THz generation are being developed currently. There are two important highlights. One is the optical rectification from organic crystals—molecular crystals with gigantic nonlinear coefficients (which can be larger by an order of magnitude than inorganic nonlinear crystals)^[158–161]. By tuning the generation beam to appropriate wavelengths, the phase-matching condition can be achieved in collinear geometry. Another novel scheme to note is carrier–envelope-phase stable difference-frequency generation and optical parametric amplification technology^[162], which generates phase-locked multi-THz (10–30 THz) radiation with strong intensity^[163–166]. The same technique is now being expanded to the mid-IR frequency range^[167,168], dramatically enhancing the capability of quantum control of the lattice sector of quantum materials.

3.1.

THz Probe of Temperature- and Magnetic-Field-Induced Spin Reorientation Transitions

As shown in Fig. 4(c), the spin-wave dispersion in ${\mathrm{RFeO}}_3$ features high-frequency optical branches corresponding to the exchange mode of the four-sublattice model as well as gapped low-frequency acoustic branches whose properties closely link with magnetic anisotropy. Since SRTs in ${\mathrm{RFeO}}_3$ are mainly driven by magnetic anisotropy, probing these acoustic branches provides much information on the SRTs themselves.

THz radiation couples with zone-center modes of the acoustic branches, which are quasi-FM and quasi-AFM magnon modes [Fig. 4(d)]. These modes respond linearly to THz radiation through the magnetic-dipole interaction^[109]. Microscopically, the excitation can be described by a Zeeman-torque-type interaction ${\mathit{T}}_i=\gamma {\mathit{S}}_i\times {\mathit{H}}_{\mathrm{THz}}$, where $\gamma$ is the gyromagnetic ratio, ${\mathit{H}}_{\mathrm{THz}}$ is the magnetic field component of the THz pulse, and $i=\mathrm{A}$ or B. The sublattice spins ${\mathit{S}}_A$ and ${\mathit{S}}_B$ are tipped to deviate from their equilibrium orientations through the Zeeman torque, initiating precessions about their equilibrium axis. The quasi-FM and quasi-AFM modes hold distinct polarization selection rules: the quasi-FM mode is excited when ${\mathit{H}}_{\mathrm{THz}}\perp \mathit{F}={\mathit{S}}_A+{\mathit{S}}_B$, and the quasi-AFM mode is excited when ${\mathit{H}}_{\mathrm{THz}}\parallel \mathit{F}={\mathit{S}}_A+{\mathit{S}}_B$.

When an SRT occurs, three major features are expected within the THz time-domain signal of quasi-FM and quasi-AFM modes. First, there is a change in the polarization selection rule that accompanies the direction switching of the $\mathit{F}$ vector. Taking the most common ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT as an example, a total of six measurement configurations are possible for a linearly polarized THz field interacting with a crystal cut along a major axis; see Fig. 8. For each number-labeled configuration, ${\mathit{H}}_{\mathrm{THz}}$ is polarized along a line (marking a major crystal axis), whose color indicates the magnon mode that can be excited, and the THz pulse propagates along the out-of-plane direction of the disk that contains the colored line. For a given measurement configuration, the mode that can be excited changes across a ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ transition; for instance, configuration 4 would excite the quasi-AFM mode in the ${\mathrm{\Gamma }}_4$ phase and the quasi-FM mode in the ${\mathrm{\Gamma }}_2$ phase. Second, there is a change of the polarization trajectory of the emitted field of magnons, which reflects the trajectory of spins within the eigenmode. Third, there is a shift of magnon frequency, which is a natural consequence of the fact that the quasi-FM and quasi-AFM frequencies respond sensitively to magnetic anisotropy; see Eq. (5).

Yamaguchi et al.^[170] have studied the temperature driven ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT in ${\mathrm{ErFeO}}_3$. As shown in Fig. 9(a), when a THz pulse with ${\mathit{H}}_{\mathrm{THz}}\parallel x$ transmits through a $z$-cut crystal, corresponding to configuration 6 in Fig. 8, it excites the quasi-FM mode at 250 K (${\mathrm{\Gamma }}_4$) and switches to excite the quasi-AFM mode at 70 K (${\mathrm{\Gamma }}_2$). When the THz pulse is tailored with ${\mathit{H}}_{\mathrm{THz}}\parallel y$ (configuration 5 in Fig. 8), although both ${\mathrm{\Gamma }}_4$ and ${\mathrm{\Gamma }}_2$ phases support excitation of the quasi-FM mode, their spin precession trajectories are different [Fig. 9(b)]; the oscillations project to a near-circle (elliptical to be exact) pattern to the $x-y$ plane in the ${\mathrm{\Gamma }}_4$ phase, but to linear polarization in the ${\mathrm{\Gamma }}_2$ phase. This causes different polarization states of the free-induction decay (FID) signal that arises from the magnon precession, which is indeed experimentally detected through polarization-resolved measurement of the emission [Fig. 9(c)]. This experiment thus demonstrated a unique usage of the time resolution provided by THz time-domain techniques in measuring the magnon FID, which reconstructs the spin trajectories.

Fig. 9

Temperature-dependent SRT probed by THz spectroscopy^[170–172]. (a) Switching of the polarization selection rule. (b) Switching of the polarization trajectory of magnon emission. (c) Experimental verification of (b). (d) Continuous spectral weight transfer between two measurement configurations of the quasi-AFM mode amplitude. (e), (f) Quantifying the rotation angle during the SRT from quasi-AFM to quasi-FM mode spectral weight transfer in $z$-cut ${\mathrm{Dy}}_{0.7}{\mathrm{Er}}_{0.3}{\mathrm{FeO}}_3$. Inset to (f): calculated free-energy landscape at various temperatures (red to blue: low to high temperatures). Reproduced with permission from Refs. [170–172].

While Yamaguchi et al.^[170] probed the SRT by studying only two temperature points, one above and the other below the transition temperature, Jiang et al.^[171] have investigated the SRT temperature range in detail. Within a $y$-cut ${\mathrm{NdFeO}}_3$ crystal, they detected the amplitudes of the quasi-AFM mode for both the ${\mathit{H}}_{\mathrm{THz}}\parallel z$ configuration (configuration 4 in Fig. 8) and the ${\mathit{H}}_{\mathrm{THz}}\parallel x$ configuration (configuration 3 in Fig. 8), and plotted them as a function of temperature. Figure 9(d) shows that, within the ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT temperature range, weight transfer between the two configurations occurs in a continuous manner, pointing to the continuous nature of the SRT.

Suemoto et al.^[172] have brought the analysis to a more quantitative level by calculating the rotation angle of the spin structure in the ${\mathrm{\Gamma }}_{24}$ phase from the spectral weight transfer. They used a $z$-cut ${\mathrm{Dy}}_{0.7}{\mathrm{Er}}_{0.3}{\mathrm{FeO}}_3$ crystal, and aligned ${\mathit{H}}_{\mathrm{THz}}\parallel x$ (configuration 6 in Fig. 8). As shown in Fig. 9(e), a continuous-type spectral weight transfer between the quasi-AFM and quasi-FM modes was observed when the temperature was fine-tuned across the SRT. They were able to quantify the rotation angle as

In addition to temperature-driven SRTs, THz time-domain techniques have also been used for probing SRTs driven by an external magnetic field. To perform this study, Jiang et al.^[174] aligned ${\mathit{H}}_{\mathrm{THz}}\parallel z$ in a $y$-cut ${\mathrm{NdFeO}}_3$ crystal (configuration 4 in Fig. 8). They first demonstrated the polarization selection rule in the temperature-driven SRT. The quasi-AFM mode is detected in the high-temperature ${\mathrm{\Gamma }}_4$ phase but not in the low-temperature ${\mathrm{\Gamma }}_2$ phase [Fig. 10(a) top]. Then they set the system in the high-temperature ${\mathrm{\Gamma }}_4$ phase, and applied a static magnetic field along the $x$ axis of the crystal. Since the $\mathit{F}$ vector is prone to align parallel to the static field, a field-driven ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ transition occurs. This again manifests in the polarization selection rule, as shown by the diminishing quasi-AFM mode in the high-field case [Fig. 10(a) bottom].

Fig. 10

Magnetic-field-induced ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT probed by THz spectroscopy^[174,175]. (a) Observation of switching of the polarization selection rule in ${\mathrm{NdFeO}}_3$. (b) Field-dependent absorption spectra in ${\mathrm{YFeO}}_3$. (c) Calculated quasi-FM and quasi-AFM frequencies versus magnetic field, to compare with (b). Reproduced with permission from Refs. [174,175].

Based on a similar idea, Lin et al.^[175] have studied a field-driven ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ transition in ${\mathrm{YFeO}}_3$. Since ${\mathrm{Y}}^{3+}$ is nonmagnetic, a ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT would not occur in the temperature-dependent phase diagram of ${\mathrm{YFeO}}_3$ (Fig. 3). In a $z$-cut ${\mathrm{YFeO}}_3$ crystal, the polarization of the THz radiation is ${\mathit{H}}_{\mathrm{THz}}\parallel x$ (configuration 6 in Fig. 8), for which only the quasi-FM mode is expected in the zero-field ${\mathrm{\Gamma }}_4$ phase. However, when a static magnetic field is applied along the $x$ axis, the spin structure will be polarized to approach a ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ transition. Key experimental signatures, including softening of the quasi-FM frequency and increase of the spectral amplitude of the quasi-AFM mode, are observed; see Fig. 10(b). Up to the highest field that is applied (7 T), the quasi-FM mode does not soften completely, suggesting that the transition to the ${\mathrm{\Gamma }}_2$ phase is incomplete; this is consistent with prior studies^[142,176] that observed the transition at 7.4 T. Nevertheless, a theoretical model taking into account an additional spin-field Zeeman term in the original spin Hamiltonian [Eq. (4)] gives field-dependent magnon frequencies that match the experimental results very well [Fig. 10(c)].

3.2.

THz Probe of Crystal-Field Transitions of Rare-Earth Ions

For ${\mathrm{RFeO}}_3$ crystals where ${\mathrm{R}}^{3+}$ ions are magnetically active, the energy levels of ${\mathrm{R}}^{3+}$ are crucial for understanding the overall magnetic properties, even within the temperature range where ${\mathrm{R}}^{3+}$ ions remain paramagnetic. The SRTs, compensation behavior, and spin switching are a few notable examples. ${\mathrm{R}}^{3+}$ ions are typically heavy elements where the spin–orbit interaction is significant enough to form atomic states in free space with well-defined quantum number $J$. The confined nature of the $4f$ orbitals stipulates that the crystal electric field acts only as a weak perturbation (weak compared to the spin–orbit interaction) to split these spin–orbit coupled states into crystal-field states (distinguished by the magnetic quantum number $m$). However, the energy scale of crystal-field splitting is of the same order of magnitude as the thermal energy at a few tens of Kelvin. This is the fundamental reason that the magnetic properties of ${\mathrm{R}}^{3+}$ are sensitive to temperature, which, in turn, leads to the rich temperature-dependent magnetic phase diagrams of the ${\mathrm{RFeO}}_3$ family.

In this subsection, we discuss the use of THz time-domain spectroscopy to study transitions between crystal-field states, known as crystal-field transitions (CFTs), of ${\mathrm{R}}^{3+}$ ions in ${\mathrm{RFeO}}_3$. We will describe how peaks in the absorption spectrum can be assigned to CFTs, whose evolutions with an external magnetic field can be used to quantitatively extract the microscopic parameters of ${\mathrm{R}}^{3+}$ sublattices, and how the polarization dependence of the emitted THz field from CFTs can be used to distinguish the dipolar nature of the transitions. While the highlighted work here all used time-domain techniques, some important early studies used continuous-wave far-IR spectroscopy to study similar transitions^{[140,177,178]}.

Zhang et al.^[179] used THz time-domain spectroscopy to study the ground multiplets (${{}^6\mathrm{H}}_3$) of ${\mathrm{Tm}}^{3+}$ ions in ${\mathrm{TmFeO}}_3$. Here, the spin–orbit coupled atomic states are denoted using the spectroscopy notation ${{}^{2S+1}L}_J$, where $S$ and $L$ are the total spin and orbital quantum numbers, respectively, and $J=L+S$ is the total angular momentum quantum number. Using an $x$-cut ${\mathrm{TmFeO}}_3$ crystal, the authors aligned ${\mathit{H}}_{\mathrm{THz}}\parallel z$ (configuration 1 in Fig. 8) and obtained the temperature-dependent absorption spectra shown in Fig. 11(a). In addition to the sharp lines highlighted by solid markers, which are assigned to the quasi-FM and quasi-AFM modes, a strong absorption band centered at 0.54 THz appears below 70 K, gaining amplitude as the temperature decreases. A constant-temperature cut at 60 K of the colormap gives the spectrum in the top panel of Fig. 11(b). To study the polarization selection rule of the transition, the authors also performed measurements using ${\mathit{H}}_{\mathrm{THz}}\parallel y$ in an $x$-cut crystal and a $z$-cut crystal, corresponding to configurations 2 and 5 in Fig. 8, respectively; see the middle and bottom panels of Fig. 11(b). Another absorption peak at 1.2 THz is identified.

Fig. 11

CFTs of ${\mathrm{Tm}}^{3+}$ ions in ${\mathrm{TmFeO}}_3$^[179]. (a) Absorption coefficient mapped as a function of frequency and temperature. Red circles: quasi-AFM mode. Blue squares: quasi-FM mode. (b) Constant-temperature cut of absorption coefficient at 60 K for three different measurement configurations. (c) Energy-level diagram explaining the THz transitions. Reproduced with permission from Ref. [179].

Figure 11(c) displays the energy-level structure of ${\mathrm{Tm}}^{3+}$ assigned by Zhang et al.^[179] to explain the experimental spectra. ${\mathrm{Tm}}^{3+}$, which has even filling in the $4f$ shell, is a non-Kramers-type ion. Since ${\mathrm{Tm}}^{3+}$ ions occupy noncentrosymmetric sites with $C_s$ symmetry, a crystal field generally splits its ${{}^6\mathrm{H}}_3$ states into singlets, denoted by the two one-dimensional irreps of the $C_s$ group, ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$. The observed absorption peaks centered at 0.54 THz (${\mathrm{R}}_1$ and ${\mathrm{R}}_1^{\prime }$ peaks) and 1.2 THz (${\mathrm{R}}_3$ and ${\mathrm{R}}_3^{\prime }$ peaks) can be explained as the ${\mathrm{E}}_1({\mathrm{A}}_1)\to {\mathrm{E}}_2({\mathrm{A}}_1)$ and ${\mathrm{E}}_1({\mathrm{A}}_1)\to {\mathrm{E}}_3({\mathrm{A}}_2)$ transitions, respectively. The ${\mathrm{R}}_2$ line, assigned to the ${\mathrm{E}}_2({\mathrm{A}}_1)\to {\mathrm{E}}_3({\mathrm{A}}_2)$ transition, should in principle appear in the ${\mathit{H}}_{\mathrm{THz}}\parallel y$, ${\mathit{E}}_{\mathrm{THz}}\parallel z$ configuration, but the saturated absorption observed in this case makes this line hard to detect. Further, the fact that the spectral amplitudes of the ${\mathrm{R}}_1$ and ${\mathrm{R}}_3$ lines increase with decreasing temperature is in agreement with the expected trend of the thermal population of the ${\mathrm{E}}_1({\mathrm{A}}_1)$ ground state, corroborating the assignment of the transitions.

Li et al.^[169] used THz time-domain magneto-spectroscopy experiments to study CFTs of ${\mathrm{Er}}^{3+}$ in ${\mathrm{ErFeO}}_3$. Peak splitting and shifting versus magnetic field (applied along different crystal axes) enabled quantitative determination of the microscopic parameters of ${\mathrm{Er}}^{3+}$ ions. Figure 12(a) shows an energy-level diagram of ${\mathrm{Er}}^{3+}$. Since ${\mathrm{Er}}^{3+}$ has odd filling in the $4f$ shell, the crystal fields remove all but the Kramers degeneracy, forming Kramers doublets $|m=\pm 15/2\rangle ,|m=\pm 13/2\rangle ,\dots$, where $m$ is the magnetic quantum number. At zero field, time-reversal-symmetry-breaking magnetic exchange interactions between ${\mathrm{Fe}}^{3+}$ and ${\mathrm{Er}}^{3+}$ slightly split the Kramers doublets. By further applying a magnetic field ($B$) along different crystal axes, different patterns of Zeeman splitting occur.

Fig. 12

CFTs of ${\mathrm{Er}}^{3+}$ in ${\mathrm{ErFeO}}_3$^[169]. (a) Energy-level scheme of ${\mathrm{Er}}^{3+}$. Levels shown in red lines are involved in the transitions observed in experiments. (b) Two-sublattice model. Magneto-THz absorption spectra from measurements in (c) configuration 1, (d) configuration 4, and (e) configuration 6. (f)–(h) Energy-level calculations from the best fit of experimental spectra using the two-sublattice model, corresponding to the configurations in (c)–(e), respectively. Reproduced with permission from Ref. [169].

Figures 12(c)–12(e) show experimental absorption spectra for ${\mathrm{ErFeO}}_3$ measured in configurations 1, 4, and 6 in Fig. 8, respectively. The sample temperature was 80 K. Generally, the observed absorption lines can be categorized into two groups. One group of transitions, labeled with letters (A, B, …), occurs below the THz bandwidth at $B=0$ and blueshifts with $B$. This group is ascribed to intra-Kramers-doublet transitions, namely, the $|-15/2\rangle \to |+15/2\rangle$ transition (lines A and C) and the $|-13/2\rangle \to |+13/2\rangle$ transition (lines B and D). The other group of transitions, labeled with numbers (1, 2, …), emerges from a strong absorption band at $\sim 1.4\mathrm{THz}$ at $B=0$ and splits like a “firework” with $B$. This group is ascribed to inter-Kramers-doublet transitions, namely, the $|-15/2\rangle \to |-13/2\rangle$ transition (line 1, line 5), the $|+15/2\rangle \to |-13/2\rangle$ transition (line 2, line 6), the $|-15/2\rangle \to |+13/2\rangle$ transition (line 3, line 7), and the $|+15/2\rangle \to |+13/2\rangle$ transition (line 4, line 8). In the $\mathit{B}\parallel y$ case, the number of lines doubles due to the broken sublattice degeneracy. More inter-Kramers-doublet transition lines centered at 1.9 THz at $B=0$ can be identified. They are the $|\pm 13/2\rangle \to |\pm 11/2\rangle$ transition group.

Because there are two inequivalent ${\mathrm{Er}}^{3+}$ sites in the crystal, the two-sublattice model depicted in Fig. 12(b) explains the data well. Defining j = 1, 2 to be the ${\mathrm{Er}}^{3+}$ sublattice index, as shown in Fig. 12(b), we can see that the site orientations of the two inequivalent sublattices depend on both $m$ and $j$, characterized by an angle ${\varphi }_m$. Furthermore, the ${\mathrm{Fe}}^{3+}-{\mathrm{Er}}^{3+}$ exchange coupling is described by the $m$- and $j$-dependent effective field $B_{\text{eff},mj}$ that is experienced by the ${\mathrm{Er}}^{3+}$ ion, characterized by an angle ${\theta }_m$. Symmetry analysis determines that the local $z_{mj}$ axes have to coincide with the crystal $z$ axis, and $B_{\text{eff},mj}$ have to be in the x−y plane. The Hamiltonian for the crystal-field level $m$ and sublattice $j$ in a magnetic field is written as the summation of the linear Zeeman term and the quadratic Zeeman term:

Table 3

Crystal-Field Parameters for Levels |m=±15/2〉 and |m=±13/2〉 Obtained from the Fits to the 80 K Experimental Spectra Shown in Fig. 12 (reproduced with permission from Ref. [169]).

Mikhaylovskiy et al.^[180] have performed polarization-resolved THz emission spectroscopy experiments on ${\mathrm{Er}}^{3+}$ in ${\mathrm{ErFeO}}_3$ to determine the dipolar activity of the CFTs. The energy-level diagram is exactly the same as the zero-field diagram in Fig. 12(a). A symmetry classification of macroscopic normal modes has been previously performed^[178], dividing CFTs associated with ${\mathrm{Er}}^{3+}$ into either magnetic-dipole active or electric-dipole active in nature. Any dipolar activity can be closely related with the possible polarization state of the emitted THz radiation. Further, although the CFT excitations in this experiment are all created by ISRS processes, the magnetic- and electric-dipole transitions respond to different polarizations of the excitation light. For magnetic-dipole transitions, the effective magnetic field description, given in Subsection 2.2.1, is the dominant mechanism, the most representative of which is the IFE. Figure 13(a) shows a THz emission signal along the $x$ axis from a $y$-cut crystal created by circularly polarized optical pumps with opposite helicities. A helicity-dependent signal is clearly present, whose Fourier transform is shown in Fig. 13(b); the peaks appearing in the Fourier transform are assigned to magnetic-dipole-active CFTs.

Fig. 13

Studying ${\mathrm{Er}}^{3+}$ CFTs in ${\mathrm{ErFeO}}_3$ by THz emission spectroscopy^[180]. (a) Pump-helicity-dependent THz emission from $y$-cut ${\mathrm{ErFeO}}_3$. (b) Fourier transform of the helicity-dependent contribution in (a). (c) Mode symmetry assignments determined from the polarization of emission and pump-helicity dependence. Measurement configuration is the same as (a) and (b). Gray region indicates the SRT temperature range. Crystal-field states are labeled with the same notations as in Fig. 12(a). Reproduced with permission from Ref. [180].

The ISRS type excitation of electric-dipole-active CFTs, on the other hand, is less straightforward since for centrosymmetric crystal generation of an effective electric field is forbidden via electric-dipole transitions. However, Mikhaylovskiy et al.^[180] have proposed a novel mechanism based on virtual magnetic-dipole transitions, ${\mathit{E}}_{\mathrm{eff}}\propto \chi {\mathit{E}}_{\mathrm{opt}}{\mathit{H}}_{\mathrm{opt}}^{*}$. The following summarizing statement therefore arises: the magnetic-dipole-active THz modes respond to electric-dipole optical transitions, while the electric-dipolar-active THz modes respond to magnetic-dipole optical transitions^[180]. Mode assignments based on the pump-helicity dependence and polarization state of the emission were then carried out across the ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT, giving Fig. 13(c).

3.3.

THz Probe of Electromagnons

Electromagnons are magnetic resonances that can be excited by the electric field of light (i.e., electric-dipole-active magnons), which is a type of excitation associated with dynamic magnetoelectric coupling^[181]. Their emergence is closely related to magnetoelectric effects and novel ferroic orders^[182–185], and therefore, electromagnons play an important role in the study of electric control of magnetism for functional electronic and spintronic devices^[186,187]. From the group theory perspective, spatial inversion symmetry is the most important symmetry element to discuss, because the leading-order magnetoelectric tensors are axial tensors of even rank, which can have only nonzero elements in noncentrosymmetric systems. For multiferroicity to emerge, the system is required not only to be noncentrosymmetric but also to belong to a polar group.

In Subsection 2.1, we discussed possible magnetic phases that can arise in ${\mathrm{RFeO}}_3$ using Bertaut’s representation theory. We mentioned that, while inversion symmetry is retained in the ${\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_4$ phases, it is broken in the ${\mathrm{\Gamma }}_5-{\mathrm{\Gamma }}_8$ phases. Located at $4(b)$ sites as inversion centers, ${\mathrm{Fe}}^{3+}$ alone cannot develop an order that is consistent with ${\mathrm{\Gamma }}_5-{\mathrm{\Gamma }}_8$. Therefore, to achieve potential magnetoelectric coupling and multiferroicity in the ${\mathrm{RFeO}}_3$ family, ${\mathrm{R}}^{3+}$ ordering that transforms according to ${\mathrm{\Gamma }}_5-{\mathrm{\Gamma }}_8$ is required.

Within the entire family, inversion-symmetry-breaking events by low-temperature magnetic ordering have been demonstrated in ${\mathrm{GdFeO}}_3$^[188], ${\mathrm{DyFeO}}_3$^[97,189], and ${\mathrm{TbFeO}}_3$^[190,191]. Although possible observation of room-temperature multiferroicity in ${\mathrm{SmFeO}}_3$ has been reported^[192], symmetry analysis calls the interpretation into question^[193], leading to controvercy regarding whether multiferroicity actually exists in ${\mathrm{SmFeO}}_3$^[194]. As shown by the phase diagram in Fig. 3, ${\mathrm{GdFeO}}_3$ adopts a ${\mathrm{\Gamma }}_4$ configuration from the Néel temperature of ${\mathrm{Fe}}^{3+}$ (661 K) all the way to 2.5 K. Below 2.5 K, ${\mathrm{Gd}}^{3+}$ moments order according to $g_x$, which transforms according to ${\mathrm{\Gamma }}_5$ (Table 1). This mode mixes with the ${\mathrm{Fe}}^{3+}$ ${\mathrm{\Gamma }}_4$ mode to form a noncentrosymmetric ${\mathrm{\Gamma }}_{45}$ configuration (point group: $m^{\prime }{m}^{\prime }2$), allowing polarization along the crystal $z$ axis. This has been observed by Tokunaga et al.^[188]; exchange striction, that is, the cooperative lattice distortion caused by magnetic coupling, is believed to be the microscopic origin of the polarization.

The ${\mathrm{Fe}}^{3+}$ subsystem in ${\mathrm{DyFeO}}_3$ exhibits an abrupt-type SRT (named the Morin transition) at 50 K. Below 4 K, ${\mathrm{Dy}}^{3+}$ ions develop $g_x{a}_y$-type ordering, consistent with the ${\mathrm{\Gamma }}_5$ mode; the total magnetic phase therefore becomes ${\mathrm{\Gamma }}_{15}$ (magnetic group: 222). This phase belongs to a nonpolar noncentrosymmetric group, which allows a linear magnetoelectric response but forbids polar order. However, polar order and multiferroicity can be switched on by applying a magnetic field along the $z$ axis, polarizing the system into the ${\mathrm{\Gamma }}_{45}$ configuration (point group: $m^{\prime }{m}^{\prime }2$). ${\mathrm{DyFeO}}_3$ magnetized along the $z$ axis therefore has the same group representation as ${\mathrm{GdFeO}}_3$ at zero field, allowing spontaneous electric polarization along the $z$ axis.

Stanislavchuk et al.^[195] have performed comprehensive far-IR spectroscopy measurements on ${\mathrm{DyFeO}}_3$, aiming at probing electromagnon excitations in the low-temperature ${\mathrm{\Gamma }}_{15}$ phase. Figure 14(a) shows zero-field mode excitations as a function of temperature in ${\mathit{E}}_{\mathrm{THz}}\parallel z$, ${\mathit{H}}_{\mathrm{THz}}\parallel y$ geometry². While the quasi-FM mode shows a prominent blueshift upon entering the ${\mathrm{\Gamma }}_{15}$ phase, new modes centered at $22{\mathrm{cm}}^{-1}$ (0.66 THz) and $58{\mathrm{cm}}^{-1}$ (1.74 THz), labeled ${\mathrm{EM}}_1$ and ${\mathrm{EM}}_2$, respectively, are observed. The authors not only ruled out phonons and CFTs as possible origins for these modes, but also determined their electric-dipole activity, namely, they are excited only for ${\mathit{E}}_{\mathrm{THz}}\parallel z$. These are consistent with characteristics of electromagnons. Further, the authors demonstrated that ${\mathrm{EM}}_1$ and ${\mathrm{EM}}_2$ can be observed only in the noncentrosymmetric magnetic structure, which corroborates the assignment. Upon applying a static magnetic field $\mathit{B}\parallel y$, at a critical field of 1.3 T, a ${\mathrm{\Gamma }}_{15}\to {\mathrm{\Gamma }}_{43}$ phase transition occurs, which restores the inversion center (Table 1). ${\mathrm{EM}}_1$ and ${\mathrm{EM}}_2$ abruptly disappear across the field-induced transition [Fig. 14(b)]. Interestingly, the eletromagnons exhibit strong hysteresis upon cycling the static magnetic field $\mathit{B}\parallel y$; compare the mode excitations in the up and down sweeps of the magnetic field in Fig. 14(b). The irreversibility potentially reflects a metastable low-symmetry phase during the ${\mathrm{\Gamma }}_{15}\to {\mathrm{\Gamma }}_{43}$ phase transition, which has been demonstrated to exhibit intermediate stages^[197].

Fig. 14

Electromagnons in ${\mathrm{DyFeO}}_3$ and ${\mathrm{TbFeO}}_3$^[195,196] measured by far-infrared transmittance spectroscopy. (a) Zero-field mode excitations as a function of temperature in the ${\mathit{E}}_{\mathrm{THz}}\parallel z$, ${\mathit{H}}_{\mathrm{THz}}\parallel y$ geometry for ${\mathrm{DyFeO}}_3$. (b) Magnetic field dependence of ${\mathrm{DyFeO}}_3$ transmittance using the ${\mathit{E}}_{\mathrm{THz}}\parallel z$, ${\mathit{H}}_{\mathrm{THz}}\parallel x$ geometry for $\mathit{B}\parallel y$ at 1.5 K. Up and down sweeps are shown in the left and right panels, respectively. (c) Zero-field temperature-dependent scans using the ${\mathit{E}}_{\mathrm{THz}}\parallel z$, ${\mathit{H}}_{\mathrm{THz}}\parallel x$ geometry for ${\mathrm{TbFeO}}_3$. Right panel zooms in the left panel. (d) Magnetic field dependence of ${\mathrm{TbFeO}}_3$ transmittance for $\mathit{B}\parallel y$ at 1.5 K. Reproduced with permission from Refs. [195,196].

By using a similar experimental technique, Stanislavchuk et al.^[196] have also investigated THz electromagnons in a ${\mathrm{Tb}}^{3+}$-ordered phase in ${\mathrm{TbFeO}}_3$. The low-temperature phase diagram of ${\mathrm{TbFeO}}_3$ is more complicated than that of ${\mathrm{DyFeO}}_3$. At $T_{\mathrm{SR}1}=8.5\mathrm{K}$, a ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_2$ SRT occurs, which is found in many ${\mathrm{RFeO}}_3$ systems. The ${\mathrm{Tb}}^{3+}$ moment orders according to $a_x{g}_y{f}_x{c}_y$ (${\mathrm{\Gamma }}_{28}$) at $T_{\text{NTb}}=3.3\mathrm{K}$. The ${\mathrm{\Gamma }}_{28}$ phase holds for only a very narrow temperature range, until it transforms into a ${\mathrm{\Gamma }}_{48}$ phase through another SRT at $T_{\mathrm{SR}2}=3.1\mathrm{K}$. In both the ${\mathrm{\Gamma }}_{28}$ phase (within $3.1\mathrm{K}<T<3.3\mathrm{K}$) and ${\mathrm{\Gamma }}_{48}$ phase ($T<3.1\mathrm{K}$), inversion symmetry is broken, and therefore, potentially electromagnons can be excited.