Physicists have long known that electron spins are the central player of magnetism in solid-state matter. Within a host crystal lattice, spins spontaneously align their orientations to minimize the global free energy, creating a variety of magnetic phases with distinct spin configurations. Understanding the energetics is therefore key to understanding the properties of quantum magnets in equilibrium and has been the central goal of study since early years. Starting from the beginning of this century, however, the field of modern condensed matter physics has witnessed an explosive growth of interest in spin dynamics, that is, the evolution of spins in time in magnetic solids. Such interest comes with growing awareness of the important roles spin kinetics play in three major contemporary problems (Fig. 1) that are significant not only for the fundamental physics of quantum materials but also for practical technological advances.

On one hand, there is a strong motivation for producing devices that can process information with clock rates faster than the current electronic technology by orders of magnitude. Following Moore’s law, the compactness of circuit boards has been increased to an unprecedented level, but the information processing speed, that is, the clock rates of computer chips, has remained on the GHz scale for many years^[1]. The goal of spintronics research is to harness spin dynamics, whose natural time scale is picoseconds (ps) in antiferromagnets, to build processing units that are not only significantly faster in clock rate but also robust against charge perturbations, low in energy consumption, and smaller in device footprints^[2–5]. What further empowers such a vision is the prospect of wave-based computation^[6–10]; by using collective spin resonances (i.e., magnons), one can encode, convert, and transport information, and perform logical operations based on coherent wave interference and nonlinear wave interactions.

On the other hand, this is related to a more fundamental venue in condensed matter physics, with the aim to discover and understand exotic phases in quantum materials^[11]. Although each contemporary condensed matter problem can be complex in its own way^[12–14], it is not uncommon to find spin dynamics playing a key role in conceptually significant model systems; examples include the Hubbard model and the Kondo model for correlated systems that exhibit high-Tc superconductivity^[15] or quantum criticality^[16], and the Ising models^[17] (of various flavors) and the Kitaev model^[18] for assorted ordered or disordered quantum magnets. Clarifying spin excitations in these complex material classes by spectroscopy means provides a clue for the essential microscopic processes at play.

Opportunities for studying out-of-equilibrium dynamics arise when pulsed laser sources are used to drive quantum magnets impulsively. The interest is fueled by the emergence of ultrafast laser technology^[19,20], which provides various experimental settings with time resolutions fine enough to resolve the motions of spins in situ. Broadly speaking, the technique is able to selectively drive a resonance mode and track in real time the subsequent decay pathways of various degrees of freedom (DOFs) back to equilibrium^[21–30]. It reveals the fundamental time scales of spin motion and their interaction with the charge and lattice DOFs. When the excitation is strong enough to shift the balance of competing energy scales so as to modify the free-energy landscape, a light-induced magnetic phase transition can occur^[31], bringing the grand pursuit of nonequilibrium control of quantum systems^[32], a topic that has long prevailed in atomic, molecular, and optical physics, to a solid-state context. Phenomena that are impossible to be found in thermal equilibrium can thus be investigated, such as dynamical phase transitions^[33], quench dynamics^[34], and nonthermal quantum states^[35].

In recent years, the aforementioned interest in creating efficient spintronic devices, discovering exotic equilibrium orders, and out-of-equilibrium phase engineering has coincided with a few technical advancements proven crucial for their proliferation. High-intensity broadband ultrafast lasers are the ideal instruments for both spectroscopy studies in equilibrium and out-of-equilibrium engineering. One of the most important developments has been made in time-domain terahertz (THz) spectroscopy techniques^[36–38], which closes the THz technology gap, a spectral region that is scientifically important (since it is on resonance with major spin, lattice, and charge excitations) but traditionally hard to access either from the low-frequency (electronics) or high-frequency (photonics) side. In addition, improvements on computational tools have been made to cope with quantum materials with many-body correlation effects^[39–46]; some can be even applied to simulate the nonequilibrium state of complex materials, providing significant predictive powers to experiments.

The confluence of interests to address the currently exciting problems (outlined in Fig. 1) with the advent of state-of-the-art experimental and theoretical approaches has led to a rapid expansion in the body of work within the area of THz spintronics and ultrafast quantum materials. The objective of this review paper is to discuss selected recent studies that directly address the core subjects of THz spin dynamics, light–spin coupling, and ultrafast magnetism. To put it concretely, we hope to comprehensively review all the THz-frequency phenomena related to spin dynamics that have been demonstrated so far in rare-earth orthoferrites (RFeO3, with R representing a rare-earth element). Four specific subjects are chosen, which are outlined in Fig. 1 as four small triangular elements that constitute the larger triangle representing motivations from a broader vision. Before introducing the scope of each section, we first clarify the logic for choosing these topics, especially the reason for confining ourselves to THz-frequency phenomena and RFeO3 systems while discussing spin dynamics.

First, our focus is on THz-frequency phenomena, or equivalently, ps time-scale dynamics because the THz frequency range coincides with the characteristic energy scale not only for spin–spin exchange interaction in most antiferromagnets, but also for lattice excitations and certain charge excitations within the same host material^[47,48]. With the increasing capability of generating and manipulating THz radiation using the current ultrafast technology, all these DOFs can be interrogated in a single experimental setting. This brings convenience to investigating dynamical coupling of spins to lattice and charge, which is the reason why the number of spintronics studies in the THz range has increased rapidly in recent years^[49].

Second, this review focuses on RFeO3 systems exclusively to set a paradigm of a near-exhaustive discussion of all THz-frequency dynamical spin phenomena observed so far in a single class of materials, even though the principles and phenomena therein can easily go beyond RFeO3 and generalize to a broader range of material classes. As shown in Fig. 2, if one defines a “phase space” for the entire research field of spin dynamics, spanned by a horizontal axis representing the array of material systems and another vertical axis listing various physical phenomena, most review papers arrange their discussion in a way similar to making horizontal cuts in phase space. This means that each of them has a topic orienting to a particular physical phenomenon, and a variety of material systems are mentioned in which the core physics manifests. The method is certainly advantageous in elucidating a newly established physical principle to the community, but its drawback is that the discussion needs to be constantly interrupted by introducing basic properties of various material systems before their connections with the core physics can be elucidated. Our attempt here is to present a review by performing a vertical cut in phase space, the advantage of which is the emergence of a grand picture encompassing assorted types of interesting physics that can be accessed from one material class. This way emphasizes more the connections among various physical phenomena, and the background knowledge associated with the material needs to be presented only once. The prerequisite for our approach is that the basic properties of the target material need to be understood well enough for extensive discussions of novel phenomena to build upon, and we believe that RFeO3 is the first class of compounds in the transition metal (TM) oxide family that satisfies the criterion based on past research. TM oxides form the playground of a vast collection of significant concepts in condensed matter physics^[50], including magnetism, strong electron correlations^[51], insulator–metal transitions^[14], charge density waves^[52], orbital order^[53], ferroic order^[54], high-Tc superconductivity^[55], and dimensionality control. The RFeO3 class is related to magnetism and ferroic order among the list but avoids many of the issues that so far remain enigmatic. As fundamental understanding of more complex materials advances, our “vertical-cut” methodology may hopefully find broader usage.

Our discussions are divided into five major sections. Section 2 introduces the background knowledge on the RFeO3 system and experimental methods that have been adopted to study THz-frequency spin dynamics in them. Section 3 focuses on the weak excitation regime of the spin sector by light, namely, using light as a probe to obtain information on the magnetic structure in equilibrium, thermodynamic spin transitions, and elementary excitations that are closely related to the ground states. Section 4 discusses light-induced magnetic phase transitions, where light excitation is no longer a weak perturbation but actively modifies the free-energy landscape of the spin system. Assorted thermal and nonthermal pathways of laser control of magnetism will be examined. In Section 5, our discussions lean slightly towards the application side, which address the issue of how spin excitations in RFeO3 can be useful in the emerging field of antiferromagnetic (AFM) spintronics and quantum control. Coherent control, polariton devices, magnetic heterostructuring, and nonlinear spin excitations are the major topics in this context. Section 6 provides insight into an emerging interdisciplinary field that merges spintronics with a key concept in quantum optics, that is, ultrastrong light–matter coupling. Viewing dynamic spin–spin interactions from a new angle, the connection between certain magnetic phase transitions in condensed matter systems and the Dicke superradiant phase transition (SRPT) in quantum optics will be clarified. Finally, we end the review by giving an outlook on opportunities that lie in the future by making connections among various intriguing phenomena reviewed in this paper.

Due to the broad range of topics covered by the current paper, we also recommend interested readers to refer to recent excellent reviews, each of which addresses a topic in more depth. These include the topics of ultrafast manipulation of magnetic order^[31,56], photomagnetic effects^[57], ultrafast spectroscopy studies of quantum materials and strongly correlated systems^[58–61], nonthermal laser control of quantum materials^[62], nonlinear phononics^[63,64], Floquet engineering^[65,66], AFM spintronics^{[4,5,67–69]}, multidimensional THz spectroscopy^[70], cavity quantum materials^[71,72], and ultrastrong light–matter coupling^[73,74].

In this section, we present the background knowledge required for understanding the discussions in the following sections. Two aspects will be emphasized. First, in Subsection 2.1, we describe the basic physical properties of RFeO3 materials, including their crystal and magnetic structures, symmetry classifications, phase diagrams, optical properties, and elementary excitations including phonons and magnons. Second, in Subsection 2.2, we provide an introductory survey of the most widely used ultrafast optical techniques for studying THz spin dynamics. These techniques are used for various time-resolved photomagnetic/magneto-optical experiments as well as THz time-domain spectroscopy experiments; the latter includes THz transmission spectroscopy, emission spectroscopy, and nonlinear spectroscopy using intense THz pulses for selectively driving excitations far from equilibrium.

The RFeO3 class has been extensively studied ever since it was first identified in the 1940s; the first review paper on RFeO3 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 D2h16−Pbnm.1 The Fe3+ and R3+ 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 D2h16−Pbnm for RFeO3, 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 Γ1−Γ8, 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 Γ1−Γ8 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.

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 Fe3+ atoms located in 4(b) sites, whose spins are labled by Si (i = 1−4), the basis functions can be well described by the following four vector components: $$\begin{gathered} F=S1+S2+S3+S4, \\ G=S1−S2+S3−S4, \\ A=S1−S2−S3+S4, \\ C=S1+S2−S3−S4, \end{gathered}$$(1)where F represents the net moment contributed by all four spins in the unit cell, and G, A, and C correspond to three types of AFM ordering, distinguished by different ordering wave vectors. Table 1 contains information on which vector components can be basis functions of each irrep. For instance, the fact that GxAyFz (FxCyGz) transforms in the same way as the Γ4 (Γ2) irrep indicates that the x component of G (F), the y component of A (C), and the z component of F (G) can be nonzero in the Γ4 (Γ2) phase. A visualization of Si vectors, F and G modes, and spin arrangements in Γ4 and Γ2 phases is shown later in Fig. 8.

One may notice from Table 1 that the Γ1−Γ4 phases preserve inversion symmetry, while the Γ5−Γ8 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 Fe3+ spin components can be finite for the Γ1−Γ4 phases, there is no Fe3+ component that can be consistent with Γ5−Γ8. This is because the 4(b) sites (that Fe3+ 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 R3+ ions. Located on 4(c) sites, rare-earth ions may develop certain spin structures that are allowed in the Γ5−Γ8 configurations. Defining four vectors f, g, a, and c based on the four R3+ spins Ri (i = 1−4) in a unit cell as $$\begin{gathered} f=R1+R2+R3+R4, \\ g=R1−R2+R3−R4, \\ a=R1−R2−R3+R4, \\ c=R1+R2−R3−R4 \end{gathered}$$(2)enables writing these components in a compact way, as shown in Table 1.

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 RFeO3 family, the latest version of which was summarized by Li et al.^[79] in 2019. Figure 3 shows the temperature-dependent magnetic phases of RFeO3 for R=La to Lu (atomic numbers: 57–71) and Y (atomic number: 39); no data are available for R=Pm due to its radioactivity.

For all crystals, the magnetic ordering of Fe3+ sets within the temperature range of 620–740 K to form the Γ4 phase, featuring AFM ordering along the x and y axes (GxAy) and a weak net moment along the z axis (Fz). The net moment is produced by a small spin canting (canting angle ∼8.5mrad) arising from the Dzyaloshinskii–Moriya (DM) interaction. For magnetically inert rare-earth ions, R3+=La3+, Eu3+, Lu3+, and Y3+, the Γ4 phase persists down to the lowest temperature.

Magnetic phase transitions arise, however, in crystals where the R3+ ions are magnetically active and therefore can interact with the Fe3+ spins^[80,81]. The transition with the most frequent occurrence is the Γ4→Γ2 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 Γ4 phase (GxAyFz) superimpose on those of the Γ2 phase (FxCyGz), producing the Γ24 phase. Crystals with R=Pr, Nd, Sm, Er, Tm, and Yb show this transition, among which SmFeO3 has a transition temperature high enough to make itself the only crystal to exhibit the Γ2 phase at room temperature. In addition to the continuous-type Γ4→Γ2 SRT, there is an abrupt-type Γ4→Γ1 SRT observed first in DyFeO3^[82,83] (at 50 K) and later in CeFeO3^[84,85] (at 240 K). Since the magnetic modes of AxGyCz allowed in the low-temperature Γ1 phase do not possess a net moment (F=0), a sharp decrease of magnetization has been observed across the transition. Moreover, HoFeO3 adopts a more complicated sequence of SRTs as Γ4→Γ24→Γ12→Γ2^[86–89], featuring an intermediate temperature range within which two metastable configurations, Γ24 and Γ12, coexist.

Magnetic interactions between R3+ and Fe3+ not only lead to assorted SRTs of the Fe3+ spin configuration, but also cause R3+ to be polarized by the exchange field supplied by the Fe3+ sublattice. When R3+ is paramagnetic, its polarization is always consistent with the irrep that describes the Fe3+ spin structure. For instance, in the Γ4 and Γ2 phases, where Fe3+ spins have a net moment, the R3+ spins will also develop a polarized net moment that is either parallel or antiparallel to the Fe3+ magnetization. These are labeled “+” (parallel) and “−” (antiparallel) in Fig. 3, respectively. For those crystals that have antiparallel alignments (R=Nd, Sm, Er, and Tm), the strong temperature dependence of the R3+ moment leads to a compensation point^[90], that is, a temperature at which the R3+ net moment exactly cancels the Fe3+ moment.

At even lower temperatures, typically below 10 K, R3+ develops ordering for R=Pr, Gd, Tb, Dy, Ho, and Er. This creates an interesting situation because the R3+ order parameter does not have to be compatible with the Fe3+ phase, leading to further symmetry breaking. For ErFeO3, Er3+ spins develop a cz mode when they order, and according to Table 1, the cz mode transforms according to the Γ1 irrep, which is different from the Γ2 phase of the Fe3+ spins (before the transition occurs). The low-temperature magnetic phase of ErFeO3 is therefore a Γ12 phase^[91,92], whose magnetic group (2m) is an index-two subgroup of either Γ1 (mmm) or Γ2 (mm̲m). More interesting physics occur when R3+ ordering transforms with an inversion-symmetry-breaking representation Γj with j ∈ {5, 6, 7, 8}; the resultant phase that takes both Fe3+ and R3+ into account would be Γij 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 GdFeO3, DyFeO3, and TbFeO3 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 RFeO3 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 Γ1, Γ2, and Γ4 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 LuFeO3, ErFeO3, and DyFeO3^[75]. In LuFeO3, the Γ4 phase persists through the entire range below the Néel temperature of Fe3+, which manifests as a finite but nearly temperature-independent z-axis magnetization. Upon cooling, the continuous-type Γ4→Γ2 SRT in ErFeO3 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 Er3+ and Fe3+ in the Γ2 phase manifests as a temperature point (45 K) with zero x-axis magnetization. In DyFeO3, the z-axis magnetization diminishes to zero when the abrupt-type Γ4→Γ1 initiates, in agreement with the fact that the Γ1 phase is incompatible with any net moment. The more complicated multistage SRT in HoFeO3 does not show salient features in magnetization that can be distinguished from the SRT in ErFeO3, but Young’s modulus and magnetostriction measurements^[86] are able to capture the detailed features. The R3+-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.

Regarding the electronic properties, all crystals within the RFeO3 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 Fe3+; 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 ErFeO3 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 A16→T14, A16→T24, and A16→E4+A4 transitions of Fe3+, respectively. Giant absorption is present for wavelengths shorter than 0.5 µm, and is assigned to the charge-transfer transition from oxygen to Fe3+^[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 Er3+ (I415/2→I413/2). 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 RFeO3, 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 Fe3+ and four R3+ ions, we focus on a simple model where R3+ is assumed to be paramagnetic. This means that magnetic excitations majorly involve Fe3+; cooperative magnetic excitations involving both R3+ and Fe3+ in the low-temperature R3+-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 Fe3+ 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′) fits this branch well, enabling quantitative extraction of J and J′^[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 Fe3+ ions in a unit cell as $$\begin{gathered} SA=S1+S3, \\ SB=S2+S4. \end{gathered}$$(3)

This way, the C-type AFM component in Eq. (1) is canceled (C=0), maintaining the G-type AFM and the net moment that can describe the SRTs. The spin Hamiltonian reads $$\begin{gathered} H=JFe∑n.nS^iA·S^i′B−DyFe∑n.n(S^i,zAS^i′,xB−S^i′,zBS^i,xA) \\ −∑s=A,B∑i[Ax(S^i,xs)2+Az(S^i,zs)2+AxzS^i,xsS^i,zs], \end{gathered}$$(4)where the first, second, and third terms are isotropic exchange interaction, DM interaction, and magnetic anisotropy, respectively. The isotropic exchange JFe is the dominant energy scale, stipulating the G-type AFM structure G=SA−SB; the DM interaction provides spin canting that produces the net moment F=SA+SB; and the anisotropy energies, Ax and Az, specify the orientations of the F and G vectors with respect to the crystal axis. The equilibrium configuration determined by Eq. (4) is the Γ4 (Γ2) phase, with G∥x, F∥z (G∥z, F∥x), when Ax>Az (Az>Ax).

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 $$\begin{gathered} ℏ︀ωFM={24JFeS2[2(Ax−Az)]}1/2, \\ ℏ︀ωAFM={24JFeS[6DFeStanβ+2AxS]}1/2, \end{gathered}$$(5)where S=5/2 is the spin quantum number of Fe3+, and β is the canting angle. The spin trajectories of these modes are shown in Fig. 4(d)^[96]. The quasi-FM mode features the precession of the F vector where the relative angle between the two sublattices remains fixed. On the other hand, the quasi-AFM mode periodically modifies the canting angle, but the F vector remains pointing along its equilibrium orientation, with its amplitude oscillating in time.

Equation (5) provides an important clue for the underlying driving force of a Γ4→Γ2 SRT, by showing that ℏ︀ωFM→0 when Ax−Az→0. Across a Γ4→Γ2 SRT, anisotropy switches from Ax>Az to Az>Ax, 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 Γ4→Γ2 SRT has indeed been observed. On the contrary, the SRT in DyFeO3 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 RFeO3 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.

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 RFeO3 family that we focus on here, optical techniques present themselves as not only the most suitable but also the easiest approaches.

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 RFeO3 crystal is illuminated by light with frequency ω, the total Hamiltonian is given by H=H0+HMO.Here, H0 is the general spin Hamiltonian of RFeO3, while HMO arises from light–spin interactions and can be expanded to lowest order in the electric field of light, E(ω), as^[57]HMO≈−[ɛij0Ei(ω)Ej*(ω)+αijkEi(ω)Ej*(ω)Fk(0)+αijk′Ei(ω)Ej*(ω)Gk(0)+βijklEi(ω)Ej*(ω)Fk(0)Fl(0)+βijkl′Ei(ω)Ej*(ω)Gk(0)Gl(0)+βijkl′′Ei(ω)Ej*(ω)Fk(0)Gl(0)],(6)where F=SA +SB and G=SA−SB are FM and AFM vectors [Eqs. (1) and (3)], respectively, ɛij0, αijk, αijk′, βijkl, βijkl′, βijkl′′ are tensor coefficients, and the Einstein summation rule is implied (and throughout this paper).

The permittivity tensor, which governs the optical property, then reads ɛij=−∂2H∂Ei(ω)∂Ej*(ω)=−∂2HMO∂Ei(ω)∂Ej*(ω),(7)and one can see that all terms in Eq. (6) except for the first term led by ɛij0 are related to magnetism. According to the Onsager principle, the total permittivity tensor can be separated into two parts, ɛij=ɛijs+ɛija, where the symmetric part ɛijs=ɛjis and the antisymmetric part ɛija=−ɛjia are even and odd under time reversal, respectively. Since F (F2) and G (G2) 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 ɛijs or ɛija. These are given in Table 2.

The property tensors αijk and αijk′, both giving rise to the asymmetric part, ɛija, contribute only to the off-diagonal elements of the ɛij tensor (ɛiia=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 ɛija and F (or G) dictates a sign change of these effects upon spin reversal. On the other hand, βijkl, βijkl′, and βijkl′′ contribute to the symmetric part ɛijs, which is able to take the diagonal components within ɛij. 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: $$\begin{gathered} HPM(0)=−∂HMO∂F(0), \\ hPM(0)=−∂HMO∂G(0), \end{gathered}$$(8)where HPM(0) and hPM(0) are the effective fields exerted on F and G, respectively. Table 2 summarizes various forms of them that are associated with the five property tensors. Most notably, the effective field HkPM(0)=αijkEi(ω)Ej*(ω), which is associated with αijk, the same tensor that induces the Faraday effect, can be further simplified for isotropic media. Replacing αijk by the minimal expression of α·ɛijk, where α is a scalar constant and is the Levi–Civita tensor (reflecting the axial nature of αijk), we obtain the effective field as HPM(0)=α[E(ω)×E*(ω)],(9)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 βijkl, βijkl′, and βijkl′′ can be generated by linearly polarized light, representing the inverse Cotton–Mouton effect (ICME).

If the fields HPM and hPM 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 RFeO3 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 HPM 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 ω and the other at ω−Ω, Ω being the magnon frequency. The ω−Ω component stimulates the inelastic scattering of the ω component to occur, causing it to decompose into another photon at ω−Ω and a magnon quantum at Ω.

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 RFeO3, 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.

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 H0, 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.

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 F and G vectors, and the attempt would be successful in an equilibrium situation where the axial tensors αijk and αijk′ 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 (F and G) or from the dynamics of the tensors αijk and αijk′^[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 Pi(2)(Ω)∝χijk(2)(Ω=ω2−ω1;ω1,−ω2)Ej(ω1)Ek*(ω2)(10)gives rise to a radiation field ETHz(t)∝∂2P(2)(t)/∂t2. The phase-matching condition requires the phase velocity of the generated THz pulse to match the group velocity of the generation pulse: vTHzph=voptgr.(11)

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 RFeO3 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 RFeO3 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 (∼1ms), 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 γ, the phase-matching condition in Eq. (11) can be achieved as vTHzph=voptgrcosγ.(12)

When implemented in nonlinear crystals such as LiNbO3, 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.

This section describes various previous studies where THz spectroscopy techniques were used for probing properties of the RFeO3 family in thermal equilibrium, including phase transitions and elementary excitations. There are two objectives that we wish to achieve by discussing these topics. The first objective is to give concrete examples for which the background knowledge presented in the last section can be put into direct use. We will show how the representation theory in Subsection 2.1 can be used to understand and predict certain THz properties of RFeO3 in equilibrium. The second objective is to lay the foundation for describing the out-of-equilibrium engineering efforts in later sections. Understanding the THz-frequency signatures of thermodynamic phase transitions facilitates the identification of a similar class of transitions achieved in nonequilibrium scenarios. Furthermore, illustrating fundamental excitations on an individual basis will set the stage for investigating their mutual interactions and dynamical couplings, providing necessary insight for the discussion related to ultrastrong coupling (USC) and Dicke cooperativity in Section 6.

We will first consider the temperature- and magnetic-field-induced SRTs in the Fe3+ subsystem, particularly the most common Γ4→Γ2-type transition. The unique advantage of THz time-domain spectroscopy in detecting these transitions is the ability to simultaneously probe magnons’ polarization selection rule, frequency shifts, and spin trajectory, all of which are expected to develop anomalies across an SRT. We will then focus on the R3+ ion subsystem, highlighting spectroscopy studies of transitions between their crystal-field levels. These levels, with spacings in the THz range, are the major origin of the strong temperature dependence of magnetic properties in the RFeO3 family. We will consider both non-Kramers-type Tm3+ and Kramers-type Er3+. Finally, we will turn to a type of cooperative magnetic excitation that involves both R3+ and Fe3+ subsystems, called electromagnons, which exist in only a few low-temperature R3+ ordered phases where spatial inversion is broken by magnetic order. Observation of these modes not only signals successful detection of these low-temperature R3+ ordered phases, but also provides insight into spin-mediated multiferroicity and useful protocols for switching magnetoelectric coupling on/off through magnetic control of spatial inversion symmetry.

As shown in Fig. 4(c), the spin-wave dispersion in RFeO3 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 RFeO3 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 Ti=γSi×HTHz, where γ is the gyromagnetic ratio, HTHz is the magnetic field component of the THz pulse, and i=A or B. The sublattice spins SA and SB 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 HTHz⊥F=SA+SB, and the quasi-AFM mode is excited when HTHz∥F=SA+SB.

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 F vector. Taking the most common Γ4→Γ2 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, HTHz 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 Γ4→Γ2 transition; for instance, configuration 4 would excite the quasi-AFM mode in the Γ4 phase and the quasi-FM mode in the Γ2 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 Γ4→Γ2 SRT in ErFeO3. As shown in Fig. 9(a), when a THz pulse with HTHz∥x transmits through a z-cut crystal, corresponding to configuration 6 in Fig. 8, it excites the quasi-FM mode at 250 K (Γ4) and switches to excite the quasi-AFM mode at 70 K (Γ2). When the THz pulse is tailored with HTHz∥y (configuration 5 in Fig. 8), although both Γ4 and Γ2 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 Γ4 phase, but to linear polarization in the Γ2 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.

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 NdFeO3 crystal, they detected the amplitudes of the quasi-AFM mode for both the HTHz∥z configuration (configuration 4 in Fig. 8) and the HTHz∥x configuration (configuration 3 in Fig. 8), and plotted them as a function of temperature. Figure 9(d) shows that, within the Γ4→Γ2 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 Γ24 phase from the spectral weight transfer. They used a z-cut Dy0.7Er0.3FeO3 crystal, and aligned HTHz∥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 θ=arctan(AFAAF),(13)where AF and AAF are the spectral amplitudes of quasi-FM and quasi-AFM modes, respectively. Figure 9(f) shows the temperature dependence of the rotation angle extracted by this method, which shows excellent agreement with a standard second-order phase transition theory incorporating both second-order and fourth-order anisotropy terms in free energy^[173].

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 HTHz∥z in a y-cut NdFeO3 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 Γ4 phase but not in the low-temperature Γ2 phase [Fig. 10(a) top]. Then they set the system in the high-temperature Γ4 phase, and applied a static magnetic field along the x axis of the crystal. Since the F vector is prone to align parallel to the static field, a field-driven Γ4→Γ2 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].

Based on a similar idea, Lin et al.^[175] have studied a field-driven Γ4→Γ2 transition in YFeO3. Since Y3+ is nonmagnetic, a Γ4→Γ2 SRT would not occur in the temperature-dependent phase diagram of YFeO3 (Fig. 3). In a z-cut YFeO3 crystal, the polarization of the THz radiation is HTHz∥x (configuration 6 in Fig. 8), for which only the quasi-FM mode is expected in the zero-field Γ4 phase. However, when a static magnetic field is applied along the x axis, the spin structure will be polarized to approach a Γ4→Γ2 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 Γ2 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)].

For RFeO3 crystals where R3+ ions are magnetically active, the energy levels of R3+ are crucial for understanding the overall magnetic properties, even within the temperature range where R3+ ions remain paramagnetic. The SRTs, compensation behavior, and spin switching are a few notable examples. R3+ 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 R3+ are sensitive to temperature, which, in turn, leads to the rich temperature-dependent magnetic phase diagrams of the RFeO3 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 R3+ ions in RFeO3. 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 R3+ 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 (H63) of Tm3+ ions in TmFeO3. Here, the spin–orbit coupled atomic states are denoted using the spectroscopy notation L2S+1J, 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 TmFeO3 crystal, the authors aligned HTHz∥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 HTHz∥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.

Figure 11(c) displays the energy-level structure of Tm3+ assigned by Zhang et al.^[179] to explain the experimental spectra. Tm3+, which has even filling in the 4f shell, is a non-Kramers-type ion. Since Tm3+ ions occupy noncentrosymmetric sites with Cs symmetry, a crystal field generally splits its H63 states into singlets, denoted by the two one-dimensional irreps of the Cs group, A1 and A2. The observed absorption peaks centered at 0.54 THz (R1 and R1′ peaks) and 1.2 THz (R3 and R3′ peaks) can be explained as the E1(A1)→E2(A1) and E1(A1)→E3(A2) transitions, respectively. The R2 line, assigned to the E2(A1)→E3(A2) transition, should in principle appear in the HTHz∥y, ETHz∥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 R1 and R3 lines increase with decreasing temperature is in agreement with the expected trend of the thermal population of the E1(A1) ground state, corroborating the assignment of the transitions.

Li et al.^[169] used THz time-domain magneto-spectroscopy experiments to study CFTs of Er3+ in ErFeO3. Peak splitting and shifting versus magnetic field (applied along different crystal axes) enabled quantitative determination of the microscopic parameters of Er3+ ions. Figure 12(a) shows an energy-level diagram of Er3+. Since Er3+ has odd filling in the 4f shell, the crystal fields remove all but the Kramers degeneracy, forming Kramers doublets |m=±15/2〉,|m=±13/2〉,…, where m is the magnetic quantum number. At zero field, time-reversal-symmetry-breaking magnetic exchange interactions between Fe3+ and Er3+ slightly split the Kramers doublets. By further applying a magnetic field (B) along different crystal axes, different patterns of Zeeman splitting occur.

Figures 12(c)–12(e) show experimental absorption spectra for ErFeO3 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〉→|+15/2〉 transition (lines A and C) and the |−13/2〉→|+13/2〉 transition (lines B and D). The other group of transitions, labeled with numbers (1, 2, …), emerges from a strong absorption band at ∼1.4THz at B=0 and splits like a “firework” with B. This group is ascribed to inter-Kramers-doublet transitions, namely, the |−15/2〉→|−13/2〉 transition (line 1, line 5), the |+15/2〉→|−13/2〉 transition (line 2, line 6), the |−15/2〉→|+13/2〉 transition (line 3, line 7), and the |+15/2〉→|+13/2〉 transition (line 4, line 8). In the B∥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 |±13/2〉→|±11/2〉 transition group.

Because there are two inequivalent Er3+ sites in the crystal, the two-sublattice model depicted in Fig. 12(b) explains the data well. Defining j = 1, 2 to be the Er3+ 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 ϕm. Furthermore, the Fe3+−Er3+ exchange coupling is described by the m- and j-dependent effective field Beff,mj that is experienced by the Er3+ ion, characterized by an angle θm. Symmetry analysis determines that the local zmj axes have to coincide with the crystal z axis, and Beff,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: Hmj=Hlinear,mj+Hquadratic,mj=μB∑kgm,kσ^kBtot,mj,k+(μB)2δm,±13/2∑kΔDk(Btot,mj,k)2I^,(14)where μB is the Bohr magneton, k=x,y,z is the Er3+ local axis index, gm,k are the anisotropic Lande g factors, σ^k are Pauli matrices, Btot,mj=B+Beff,mj is the net magnetic field experienced by Er3+, ΔDk are the anisotropic quadratic Zeeman coefficients, and I^ is a 2×2 identity matrix. The eigen-energies of all the states can be calculated as a function of B by diagonalizing the Hamiltonian. A single group of parameters can be used to fit all three experimental spectra in Figs. 12(c)–12(e); see Table 3 for the parameters obtained from the fits. The energy-level schemes of the best fit for the B∥x, B∥y, and B∥z cases are plotted in Figs. 12(f)–12(h), respectively. Transition lines are identified, labeled, and plotted correspondingly in Figs. 12(c)–12(e) as dashed blue lines. Excellent agreement between theory and experiment is achieved.

Mikhaylovskiy et al.^[180] have performed polarization-resolved THz emission spectroscopy experiments on Er3+ in ErFeO3 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 Er3+ 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.

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, Eeff∝χEoptHopt*. 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 Γ4→Γ2 SRT, giving Fig. 13(c).

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 RFeO3 using Bertaut’s representation theory. We mentioned that, while inversion symmetry is retained in the Γ1−Γ4 phases, it is broken in the Γ5−Γ8 phases. Located at 4(b) sites as inversion centers, Fe3+ alone cannot develop an order that is consistent with Γ5−Γ8. Therefore, to achieve potential magnetoelectric coupling and multiferroicity in the RFeO3 family, R3+ ordering that transforms according to Γ5−Γ8 is required.

Within the entire family, inversion-symmetry-breaking events by low-temperature magnetic ordering have been demonstrated in GdFeO3^[188], DyFeO3^[97,189], and TbFeO3^[190,191]. Although possible observation of room-temperature multiferroicity in SmFeO3 has been reported^[192], symmetry analysis calls the interpretation into question^[193], leading to controvercy regarding whether multiferroicity actually exists in SmFeO3^[194]. As shown by the phase diagram in Fig. 3, GdFeO3 adopts a Γ4 configuration from the Néel temperature of Fe3+ (661 K) all the way to 2.5 K. Below 2.5 K, Gd3+ moments order according to gx, which transforms according to Γ5 (Table 1). This mode mixes with the Fe3+Γ4 mode to form a noncentrosymmetric Γ45 configuration (point group: m′m′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 Fe3+ subsystem in DyFeO3 exhibits an abrupt-type SRT (named the Morin transition) at 50 K. Below 4 K, Dy3+ ions develop gxay-type ordering, consistent with the Γ5 mode; the total magnetic phase therefore becomes Γ15 (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 Γ45 configuration (point group: m′m′2). DyFeO3 magnetized along the z axis therefore has the same group representation as GdFeO3 at zero field, allowing spontaneous electric polarization along the z axis.

Stanislavchuk et al.^[195] have performed comprehensive far-IR spectroscopy measurements on DyFeO3, aiming at probing electromagnon excitations in the low-temperature Γ15 phase. Figure 14(a) shows zero-field mode excitations as a function of temperature in ETHz∥z, HTHz∥y geometry2. While the quasi-FM mode shows a prominent blueshift upon entering the Γ15 phase, new modes centered at 22cm−1 (0.66 THz) and 58cm−1 (1.74 THz), labeled EM1 and EM2, 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 ETHz∥z. These are consistent with characteristics of electromagnons. Further, the authors demonstrated that EM1 and EM2 can be observed only in the noncentrosymmetric magnetic structure, which corroborates the assignment. Upon applying a static magnetic field B∥y, at a critical field of 1.3 T, a Γ15→Γ43 phase transition occurs, which restores the inversion center (Table 1). EM1 and EM2 abruptly disappear across the field-induced transition [Fig. 14(b)]. Interestingly, the eletromagnons exhibit strong hysteresis upon cycling the static magnetic field B∥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 Γ15→Γ43 phase transition, which has been demonstrated to exhibit intermediate stages^[197].

By using a similar experimental technique, Stanislavchuk et al.^[196] have also investigated THz electromagnons in a Tb3+-ordered phase in TbFeO3. The low-temperature phase diagram of TbFeO3 is more complicated than that of DyFeO3. At TSR1=8.5K, a Γ4→Γ2 SRT occurs, which is found in many RFeO3 systems. The Tb3+ moment orders according to axgyfxcy (Γ28) at TNTb=3.3K. The Γ28 phase holds for only a very narrow temperature range, until it transforms into a Γ48 phase through another SRT at TSR2=3.1K. In both the Γ28 phase (within 3.1K<T<3.3K) and Γ48 phase (T<3.1K), inversion symmetry is broken, and therefore, potentially electromagnons can be excited.

In performing zero-field temperature-dependent scans in ETHz∥z, HTHz∥x geometry, Stanislavchuk et al. found a wealth of mode excitation behavior, shown in Fig. 14(c); its right panel shows a zoom-in view of data at T<4K. While the switching of the polarization selection rule between quasi-FM and quasi-AFM modes was clearly observed across the Γ4→Γ2 SRT at TSR1, both modes appeared at T<3.1K, suggesting a significant relaxation of the polarization selection rule in the Γ48 phase. In particular, the quasi-FM mode was found to gain electric-dipole activity in this phase. On the other hand, within the narrow temperature range of 3.1K<T<3.3K, a mode at 27cm−1 (0.81 THz) was observed and assigned to an electromagnon (labeled EM, excited only by ETHz∥z). The electromagnon mode also showed a delicate appearance within the field-dependent diagram for B∥y at 1.5 K [Fig. 14(d)]. It was found that B∥y leads to multistage phase transitions, and it is only within a narrow field range around 2.5 T where one can observe the EM mode. This field-induced phase features a complicated magnetic structure characterized by Γ2358, which supports spontaneous polarization along the y axis.

The ability to manipulate spins in magnetically ordered solids on the shortest possible time scale possesses significant potential for applications in future spintronic and magnetic memory devices, fast-speed computation technology, and quantum information processing^{[2,198–200]}. Among various competing proposals, femtosecond lasers have emerged as the most suitable tool for control at the fundamental speed limit of quantum spins, namely, within ps or sub-ps time durations. In the dramatic cases where laser pulses drive the material to extreme nonequilibrium, ultrafast magnetic phase transitions occur^[31,49], which has attracted considerable interest over the past two decades^[201–208]. Due to the rapid pace at which current laser technology is developing, this research field is expected to gain increasing accessibility, as pulse energies become stronger, pulse durations become shorter, and wavelengths become more tunable.

However, accessibility by no means implies simplicity of the problem of interest. As a matter of fact, ultrafast laser manipulation of magnetic order has been identified as one of the most challenging topics in modern spintronics, encompassing the demand to expand the frontier of knowledge within multiple areas in physics and photonics. Since spins in magnetic solids can never be viewed as an isolated system, but instead, are frequently found to entangle with multiple other fundamental DOFs including charge, lattice, and orbit, one is faced with the demand to understand complex correlated phases and intertwined orders of condensed matter^[209]. In addition, once laser excitation comes into play, the problem gains an additional level of complexity due to the nonequilibrium nature of the driven system^{[39–42,210,211]}. Besides light–spin interactions, electronic charge transitions and lattice phonons both couple with light strongly and evolve via distinctly different pathways and speeds^[58]. Thus, it is necessary to understand how the extremely nonthermal status in multiple coupled reservoirs created by laser excitation evolves in time, and the impact of such evolution on magnetism.

Once a reasonable understanding of the material itself and light–matter interaction is established, what becomes challenging is the way to tailor optical pulses to achieve the desirable type of light–matter interaction. Within the (usually) undiscriminating excitations of multiple reservoirs by light, one identifies the type of excitation that can most efficiently foster the desired modification to spin order. Is it preferable that light interacts only with spins so that charge and lattice excitations should be suppressed? Or, if spins do not interact with light strong enough, could one strongly excite the lattice, and manipulate spins through lattice–spin interactions? Tailoring light excitations following answers to considerations like these advances the so-called “surgical” approach of laser manipulation of condensed matter^[62]. Within the process, one leverages state-of-the-art laser technology, seeks guidance from theoretical proposals of nonequilibrium quantum matter^[212–215], and creates novel experimental methodologies. All of these activities depend uniquely on the magnetic material of interest, and thus, the number of opportunities is vast for creating a unique combination of conditions to tailor magnetism on demand.

Historically, magnetic metals^[216,217], semiconductors^[218–222], and insulators^[223] have all been used as candidates for laser manipulation of magnetic order; phenomenology and microscopic descriptions are very different, but we will not review all of them at length here. Rather, we focus on the role of RFeO3 along this line of effort. RFeO3 represents the most prototypical Heisenberg-type of AFM insulator. The magnetic energy hierarchy and temperature-dependent magnetic phase diagram are well understood. THz-frequency magnons, serving as the fingerprints to track distinct magnetic phases, have been well characterized; see Section 3. Therefore, the stage for the quest for laser manipulation of magnetic order has been set. In the following subsections, we discuss five major pathways to achieve ultrafast laser-induced magnetic phase transitions in RFeO3 that have been reported in the literature.

Modification of magnetic order due to laser-induced heating is a widely observed phenomenon and has been proven to be effective. Photons absorbed by the sample (kept in a thermal bath at temperature T) through electronic excitations increase the local temperature by ΔT, and the magnetic order evolves towards the status that would be expected for a sample at T+ΔT. Scrutiny is required when heating is supplied by a femtosecond laser pulse. Since the time duration of excitation is shorter than the typical durations of electronic decay and electron–lattice thermalization, the way to perceive ΔT strongly depends on the time delay after pump excitation.

A phenomenological three-temperature model has been frequently applied to describe the time evolution of the thermal properties of the system^[216,224]. As shown in Fig. 15(a), the model assumes three separate but mutually interacting reservoirs, namely, the electrons, lattice, and spins, which are given respective pump-induced temperature changes—ΔTelectron, ΔTlattice, and ΔTspin. At early time delays immediately after the pump excitation of the charge sector, ΔTelectron is the largest. Then ΔTlattice and ΔTspin increase with time at the expense of ΔTelectron through energy transfer processes at rates determined by the coupling strengths between the reservoirs [Fig. 15(b)]. It is only at time delays when ΔTelectron, ΔTlattice, and ΔTspin have fully converged (quasi-equilibrium condition) can an overall local temperature increase of the system ΔT be defined, but for the purpose of controlling the magnetic properties, examining ΔTspin only before reaching quasi-equilibrium is also reasonable.

Within the RFeO3 class, the temperature range around the Fe3+ SRT is ideally suited for ultrafast magnetic phase transitions induced by laser heating. In the spin Hamiltonian, Eq. (4), although the isotropic exchange constant and the DM interaction constant are temperature independent, the anisotropy energies depend strongly on the temperature. In TmFeO3, by spectroscopically resolving quasi-FM and quasi-AFM mode frequencies, the temperature dependence of the Ax and Az constants has been quantitatively determined. Their difference, Ax−Az, has been found by Zhang et al.^[179] to change sign across the SRT temperature range (80–90 K); see Fig. 16. This suggests that a sign change of a similar origin would occur when one imposes ultrafast heating on a sample kept around the SRT temperature as well. The laser-heated sample would then have an easy axis that deviates from that in equilibrium, leading to a spin-structure rotation.

Kimel et al.^[225] have performed time-resolved linear birefringence measurements on a TmFeO3 crystal around the SRT temperature in the presence of ultrafast pump excitation. In equilibrium (with no pump excitation), it was found that linear birefringence is a sensitive indicator of the orientation of the AFM vector G across the Γ4→Γ24→Γ2 transition [Fig. 17(a)]. This type of probe mechanism couples to the G vector directly, which is different from the Faraday- or Kerr-effect probes (which couple linearly to the bulk magnetization). As detailed in Subection 2.2.1, the probe utilizes the symmetric portion of the dielectric permittivity tensor, which contains a term that reads ɛii=βiijkGjGk; the coupling to G is quadratic and sensitive to the rotation of G across the SRT. In a z-cut crystal, when the polarization is aligned in-plane between x and y axes (propagation along z), the birefringence Δnxy=(ɛxx−ɛyy)/2n is detected by reading the polarization of the transmitted probe pulse.