Terahertz time-domain spectroscopy of electromagnons in multiferroic perovskite manganites
Recent spectroscopic studies at terahertz frequencies for a variety of multiferroics endowed with both ferroelectric and magnetic orders have revealed the possible emergence of a new collective excitation, frequently referred to as electromagnon. It is magnetic origin, but becomes active in response to the electric field component of light. Here we give an overview on our recent advance in the terahertz time-domain spectroscopy of electromagnons or electric-dipole active magnetic resonances, focused on perovskite manganites—MnO ( denotes rare-earth ions). The respective electric and magnetic contributions to the observed magnetic resonance are firmly identified by the measurements of the light-polarization dependence using a complete set of the crystal orientations. We extract general optical features in a variety of the spin ordered phases, including the -type antiferromagnetic, collinear spin ordered, and ferroelectric and spiral spin ordered phases, which are realized by tuning the chemical composition of , temperature, and external magnetic field. In addition to the antiferromagnetic resonances of Mn ions driven by the magnetic field component of light, we clarify that the electromagnon appears only for light polarized along the -axis even in the collinear spin ordered phase and grows in intensity with evolution of the spiral spin order, but independent of the direction of the spiral spin plane ( or ) or equivalently the direction of the ferroelectric polarization ( or ). A possible origin of the observed magnetic resonances at terahertz frequencies is discussed by comparing the systematic experimental data presented here with theoretical considerations based on Heisenberg model.
pacs:300.6495 Spectroscopy, terahertz; 160.4760 Optical properties,
Recently, there is growing interest in the magnetoelectric (ME) effect of oxide compounds endowed with both ferroelectricity and magnetism, termed multiferroics (1). The ME effect as a generic characteristic of the multiferroics is manifested by the magnetic field control of the ferroelectric polarization (2) or inversely the electric field control of the magnetization (3). Such versatile cross-correlation phenomena are highly non-trivial, but can host the demanding functionality in the future electronics; this is the reason why the study of the multiferroics in a form of bulk as well as thin film is rapidly accelerated (1); (4); (5); (6). This recent boom is triggered by the discovery of the ferroelectricity in orthorhombically distorted perovskite manganites, MnO, where represents rare-earth ions such as Gd, Tb, Dy, and their solid solutions, by Kimura et al. (2); (7); (8). Noticeably, the direction of can be flopped from the - to -axis by an application of the external along the - or -axis (2); (7); (8). Within the framework of Landau theory, the non-zero component of the ME tensor can produce the linear ME effect, which has been known to emerge in noncentrosymmetric magnets such as CrO, in which both space-inversion and time-reversal symmetries are simultaneously broken (9); (10); (11). However, the gigantic ME effect observed in MnO is better understood in terms of the phase transition between the multiferroics states; this requires the new mechanism of the ferroelectricity in these compounds and a variety of experimental approaches have been taken to reveal the multiferroic nature of MnO (1); (4); (5); (6).
As a typical example of the ferroelectric behavior in MnO, we show in Figs. 1(a) and 1(b) the temperature dependence of dielectric constant at 10 kHz and along each crystallographic axis of TbMnO in zero , respectively (2); (7); (8). Below the Néel transition temperature of 39 K, where the collinear spins of Mn ions sinusoidally order along the -axis [see, the schematic illustration of the collinear spin order in Fig. 2(c)], along the -axis tends to slightly increase and shows a kink at 28 K. Below 28 K, along the -axis steeply increases and reaches the maximum ( C/m) at the lowest temperature. Accordingly, along the -axis exhibits a sharp peak at 28 K, signaling the ferroelectric phase transition. In this ferroelectric phase, the presence of the spiral spin structure, where the spins of Mn ions rotate within the plane with the propagation vector (along the -axis), was identified by neutron scattering experiments [see, the schematic illustration of the spiral spin order in Fig. 2(e)] (12).
As an origin of such a magnetically driven ferroelectricity in multiferroics, Katsura, Nagaosa, and Balatsky have proposed the spin-current model (13), that can be produced by the non-collinear spin order. In this model, the ferroelectricity shows up along the direction perpendicular to the wave vector and within the spiral spin plane, as given by
where is a unit vector connecting the nearest-neighbor spins, and , as shown in Fig. 2(e). According to Eq. (1), is expected to emerge along the -axis in MnO with the spiral spin order, being consistent with the experimental observation of TbMnO shown in Fig. 1. Furthermore, the relationship between the ferroelectricity and the spin structure could be directly revealed in the spin-polarized neutron scattering by controlling the vector chirality defined by of TbMnO by an external electric field (14). Therefore, the observed -flop from the - to -axis by applying the external can be regarded as the flop of the spiral spin plane from to . The presence of the spiral spin order with along the -axis was recently confirmed for TbMnO in along the -axis (15) and GdTbMnO in zero (16). The equivalent expression to the spin-current model has also been obtained phenomenologically (17) and by considering the spin-lattice coupling through the Dzyaloshinski-Moriya interaction (18). Recently, ME phase diagrams of MnO were theoretically investigated on the basis of numerical calculations of a microscopic spin model including the anisotropy terms and the Dzyaloshinski-Moriya interaction, which successfully reproduces the experimental phase diagrams including flops, emergence, and disappearance of (19).
The ME coupling between ferroelectricity and magnetism would also produce the intriguing lower-lying spin excitation (20). Since 1960s, the presence of such an excitation was theoretically inferred in the ME materials (21). This new elementary excitation appears as a magnetic resonance in the dielectric constant spectrum as a response to the electric field component of light , thus can be termed the electric-dipole active magnetic resonance. This is in contrast to the case of the magnetic resonance appearing in the magnetic permeability spectrum as a result of the spin excitation driven by the magnetic field component of light (22). Recently, Pimenov et al. have measured the optical spectra of TbMnO and GdMnO in the energy range of 0.4–4.8 meV by using a backward wave oscillator (BWO) as a light source, combined with Mach-Zehnder interferometry, in which the complex optical constants can be extracted in the quasi-optical setup (23). They found a single peak-structure in the imaginary part of , at 2.9 meV for TbMnO and 2.5 meV for GdMnO around 10 K in zero . These absorptions were shown to be allowed when was set parallel to the -axis. Furthermore, the dramatic reduction of the real part of as a result of the transformation from the spiral spin order to the -type AFM order, was observed by applying the external along the -axis. Based on these facts, they claimed the possible emergence of the electric-dipole active magnetic resonances in these compounds, now frequently refereed to as electromagnons. This pioneering work stimulates the variety of investigations concerning the low-energy spin dynamics of multiferroics such as perovskite MnO (23); (25); (26); (24); (27); (28); (29); (30); (31); (32); (33); (34); (35); (36); (37); (38); (39); (40); (41), hexagonal YMnO (42), BiFeO (43); (44), and MnO (45); (46); (47), by using far-infrared optical, Raman scattering, and inelastic neutron scattering spectroscopies.
Within the framework of the spin-current model that can explain the ferroelectric properties observed in MnO, Katsura, Balatsky, and N. Nagaosa showed that is possible to drive the oscillation of the spiral spin plane in MnO, which produces the magnetic resonance at terahertz frequencies along the direction perpendicular to the spiral spin plane (25); and in the and spiral spin ordered phases, respectively. This collective mode can be termed the rotation mode of the spiral spin plane, as the example of the manifestation of the dynamical ME coupling in multiferroics. Subsequently, Senff et al. performed inelastic neutron scattering experiments of TbMnO in the spiral spin ordered phase and found the magnetic excitation around 2 meV at (26), the peak position of which is nearly identical to that of the electromagnon in TbMnO revealed by BWO spectroscopy (23). Furthermore, they also clarified that the observed mode is polarized along the -axis perpendicular to the spiral spin plane (26), as predicted by theoretical considerations. Therefore, the electromagnon observed in MnO at terahertz frequencies was anticipated to be ascribed to the rotation mode of the spiral spin plane.
Here we show our latest advance in the terahertz time-domain spectroscopy on electromagnons or electric-dipole active magnetic resonances in multiferroic perovskite manganites MnO (29); (31); (32); (33); (35). Contrary to early assignments described above, we provide the compelling evidence that the electromagnon observed in MnO cannot be ascribed to the rotation mode of the spiral spin plane; it appears only for , irrespective of the direction of the spiral spin plane ( or ). This conclusion is unambiguously revealed by the systematic optical investigations on the spin excitation at terahertz frequencies in MnO based on light-polarization, temperature, and external dependence. We clarified the full spectral shape of the electromagnon, which consists of two peak-structures around 2 meV and 5–8 meV. The lower-lying electromagnon survives even in the collinear spin ordered phase. Below , we also identify the presence of the antiferromagnetic resonances of Mn ions around 2 meV driven by or , the peak position of which is nearly identical to that of the lower-lying electromagnon for .
The format of this article is as follows. In Sec. II, we briefly describe the experimental setup for the terahertz time-domain spectroscopy we used here and the estimate procedure of the complex optical constants from the raw data in time domain. Sec. III is devoted to show the results of the optical spectra at terahertz frequencies for a variety of spin ordered phases of MnO, as tuned by the ionic radius of , temperature, and external . After the brief descriptions of the overall optical spectrum of TbMnO (Sec. III.1) and the basic feature of MnO in terms of Mn-O-Mn bond angle (Sec. III.2), we sum up the general optical properties in a variety of spin ordered phases by taking DyMnO (Sec. III.3) (29) and GdTbMnO (Sec. III.4) (32) as the examples. We discuss in Sec. IV the origin of the electromagnons in MnO with theoretical considerations based on the Heisenberg model (35). Summary is given in Sec. V.
We used the standard experimental setup for the terahertz time-domain spectroscopy in transmission geometry. Femtosecond laser pulses delivered from the mode-locked Ti:sapphire laser with the center wavelength of 800 nm, the pulse width of 100 fs, and the repetition rate of 80 MHz were divided into pump and trigger pulses. The pump pulses were irradiated to the ZnTe crystal or to the photoswitching device made on the low-temperature-grown GaAs (LT-GaAs) coupled with the bow-tie antenna. The detector was the another LT-GaAs coupled with the dipole antenna. Although the lower limit of the available energy range is restricted by the size of each sample, we cover the energy range of 2–9 meV and 0.8–6 meV by using ZnTe and LT-GaAs terahertz emitters, respectively. The radiated terahertz pulse was collimated and focused on the sample by a pair of the off-axis paraboloidal mirrors. For the light-polarization dependence, the wire-grid polarizer was inserted in between the off-axis paraboloidal mirrors.
Single-crystalline samples were grown by the floating-zone method (8). Specimens with wide , , and faces were cut from the bowl and each crystallographic axis was determined by back Laue photographs. The obtained specimens were characterized by x-ray diffraction, at 10 kHz, , and magnetization measurements, which were all consistent with previous reports in Refs. (2); (7); (8). For transmission experiments, we polished the specimens to the thickness of 100–850 m. We carefully confirmed that there was no effect of the polishing procedure on the optical properties at terahertz frequencies of MnO.
We estimated the optical constants of MnO without the Kramers-Kronig transformation. In MnO, there are spin excitations driven by both and in the measured energy range, as we could clarify their contributions to mainly based on the measurements of the complete set of the light-polarization dependence (Sec. III). Due to the emergence of the magnetic resonances driven by , should be precisely expressed as (whereas for the case of non-magnets). We confirmed that the contribution of is negligible to the complex transmission coefficient by the numerical calculation and thus the effect of was taken into account by adopting . Therefore, we used the quantity of in this paper. Further details of our estimate procedure and the validity of this approach can be found in Ref. (29).
Iii Experimental Results
iii.1 Overall optical spectrum of TbMnO from terahertz to ultraviolet frequencies
First, we show the overall optical spectrum of TbMnO in the ferroelectric spiral spin ordered phase, ranging from terahertz to ultraviolet frequencies (31). Figures 3(a) and 3(b) show the real Re and imaginary Im parts of the spectra of TbMnO up to 90 meV, respectively, measured at 12 K. was set parallel to the -axis. Although there is no contribution of to the spectrum above 8 meV, we used the notation of Re and Im. We obtained the spectrum above 10 meV by using the Kramers-Kronig transformation; the polarized reflectivity spectrum was measured in the energy ranges of 0.01–0.8 eV and 0.6–36 eV by using Fourier transform infrared spectrometer and grating monochromator, respectively. In the energy range of 10–22 meV, we performed both the transmission and reflectance measurements and thus directly estimated the spectrum without the Kramers-Kronig transformation. Below 10 meV, terahertz time-domain spectroscopy was used in transmission geometry. The inset of Fig. 3(b) shows the Im spectrum of TbMnO up to 10 eV, measured at 10 K, where the electronic transitions are dominant. The optical transition around 2 eV across the charge-transfer gap is clearly identified. The peak-structures around 5 eV and 9 eV can be assigned to the transitions from O 2 to Mn and Tb states, respectively, according to the systematic optical study of the transition metal oxides with perovskite structure (48); (49). Although the GdFeO-type distortion results in the splitting and mixing of the phonon modes, we can roughly classify the character of the observed phonon modes into three types—stretching, bending, and external modes, which are typical for ideal cubic perovskite structure (49); (50). The optical phonon around 70 meV is assigned to the stretching mode of Mn-O-Mn. Several modes discerned from 25 meV to 60 meV mainly come from the bending modes of Mn-O-Mn. The optical phonon modes below 25 meV can be ascribed to the external modes, which correspond to the vibrations of Tb ions. Below 20 meV, other contributions to the far-infrared spectrum are identified as four pronounced absorption peak-structures at 3 meV, 7 meV, 14 meV, and 17 meV, as clearly seen in the Im spectrum, multiplied by 10 [Fig. 3(b)]. The 14 meV peak-structure corresponds to the lowest-lying external phonon mode, which shows an appreciable coupling with the lower-lying electromagnon (27); (31); (39). According to the inelastic neutron scattering experiments (51); (26), the upper edge of the magnon band of TbMnO is located around 8 meV. Therefore, we previously assigned the peak-structure at 17 meV to the upper edge of the two-magnon band, as the energy of this peak-structure just corresponds to the twice of the magnon energy at zone edge (31). However, we recently clarified that this absorption peak is diminished in other MnO such as DyMnO (37), GdMnO (37), and EuYMnO ( represents the nominal composition) (38), although two other peak-structures below 10 meV are discerned. Therefore, the peak-structure at 17 meV can be ascribed to the crystal field excitation of Tb ions, contrary to our previous assignment (31). Further detailed studies concerning the phonon modes in MnO will be presented elsewhere (38); (39). In the following subsections, we focus on optical properties of MnO below 10 meV.
iii.2 Overview of terahertz spectra of MnO
Here we overview the optical spectra at terahertz frequencies in thermally induced spin ordered phases—-type antiferromagnetic (AFM) and spiral spin ordered phases, realized in a family of MnO ( Gd, Tb, and Dy). In MnO, the decrease of the ionic radius of or equivalently the decrease of the Mn-O-Mn bond angle , destabilizes the -type AFM order (52), in which the Mn spins antiferromagnetically stack along the -axis and ferromagnetically order along the - and -axes, as schematically shown in Fig. 2(f). The ME phase diagrams of MnO are reproduced in Figs. 2(a) and 2(b) (52); (53). LaMnO has one electron per Mn-site and is an -type AFM insulator below of 140 K. With decreasing the ionic radius of from La to Gd , dramatically decreases from 140 K to 42 K [Fig. 2(a)] as a result of the spin frustration caused by competing spin-exchange interactions (52). In between Tb and Dy , the ferroelectricity emerges along the -axis, accompanied by the spiral spin order [Fig. 2(e)] (2); (7); (8), as can be clearly seen in the magnified view [Fig. 2(b)]. At 10 K, GdMnO and GdTbMnO are -type AFM insulators. Noticeably, around of 146.0, as can be realized in the mixed valence GdTbMnO in between the paraelectric -type AFM GdMnO and the ferroelectric TbMnO (53), the spiral spin ordered phase with [Fig. 2(d)] emerges in a narrow range of temperature; for example, shows up between 16 K and 24 K for GdTbMnO (53). Below 16 K, the -type AFM phase is stable. For further decreasing the ionic radius of , -type AFM order is developed for Ho [Fig. 2(g)], in which the Mn spins order in the sequences of along each crystallographic direction.
We show the Re and Im spectra in upper and lower panels of Fig. 4, respectively, for (a) GdMnO at 12 K, (b) TbMnO at 12 K, and (c) DyMnO at 9 K. In these measurements, and were set parallel to - and -axes, respectively, with use of the crystal plates with a widest face. As can be clearly seen, there is a large variation of the optical spectra with the ionic radius of . In the -type AFM phase of GdMnO at 12 K [Fig. 4(a)], a clear sharp peak-structure can be discerned at 2.3 meV in the Im spectrum, yielding the maximum magnitude of Im . In accord with this, there is a dispersive structure in the Re spectrum. This tiny absorption can be assigned to the spin excitation driven by , the details of which are discussed in Sec. III.4 by adopting GdTbMnO. With decreasing the ionic radius of , we can see the dramatic modification of the spectra. In the spiral spin ordered phase of TbMnO at 12 K [Fig. 4(b)], the magnitude of Im increases, which forms the pronounced broad continuum-like absorption composed of two peak-structures at 2.9 meV and 7.4 meV. The magnitudes of Im for the lower- and higher-lying peak-structures reach about 5 and 7, respectively. At these peak positions, we can identify the clear dispersive structures in the Re spectrum. In DyMnO with the further decreased the ionic radius of but similar with spiral spin order, the lower-lying peak-structure grows in intensity at 9 K and the magnitude of Im reaches the maximum about 10 [Fig. 4(c)]. The position of the lower-lying peak-structure shifts from 2.9 meV to 2 meV. On the contrary, the magnitude of Im of the higher-lying peak-structure decreases, but the peak position also shifts to the lower-energy from 7.4 meV to 5 meV; such tendency is also discerned in the Re spectrum. The observed remarkable absorptions in TbMnO and DyMnO can be ascribed to the electromagnons, as discussed in detail in Sec. III.3. Further systematic investigations on the variation of the optical spectra in terms of can be found in Ref. (37). In the following subsections, we focus on the optical spectra at terahertz frequencies in the -type AFM, spiral, and spiral spin ordered phases of MnO, tuned by temperature and external , by taking DyMnO (Sec. III.3) and GdTbMnO (Sec. III.4) as examples.
iii.3 DyMnO with spiral spin order
In DyMnO, the collinear spin order of Mn ions evolves along the -axis below of 39 K. The modulation wavevector of Mn ions along the -axis is incommensurate with (7). For decreasing temperature, the ferroelectricity appears along the -axis as a result of the evolution of the spiral spin order; the temperature dependence of at 10 kHz and along the each crystallographic axis of DyMnO in zero are presented in Figs. 5(a) and 5(b), respectively (7); (8). Below the ferroelectric transition temperature of 19 K, along the -axis steeply increases up to C/m [Fig. 5(b)]. In accord with this, along the -axis exhibits the sharp peak at [Fig. 5(a)]. Contrary to the behavior of along the -axis of TbMnO with the same spiral spin order [Fig. 1(a)], along the -axis of DyMnO strongly enhances below and yields the large value of at . along the -axis is roughly by a factor of 2 larger than along other axes. We show in Fig. 5(d) the dependence of along the - and -axes, measured at 9 K. By an application of external along the -axis, along the -axis is dramatically suppressed. On the other hand, along the -axis steeply increases, accompanied by the huge change of along the -axis [Fig. 5(c)], when the external exceeds the critical value of kOe. Namely, the direction of can be flopped from the - to -axis above or equivalently the spiral spin plane changes from to . Among a family of MnO, DyMnO shows the largest ME effect, as exemplified by the remarkable change (300% at 10 kHz) of along the -axis upon the -flop [Fig. 5(c)] and by the large [Fig. 5(d)] (7); (8). Therefore, the terahertz time-domain spectroscopy of DyMnO provides the useful insights into the basic characteristics of the electromagnons observed in MnO and their possible role in the ME effect.
First, we clarify the selection-rule of the spin excitation in the spiral spin ordered phase of DyMnO based on the light-polarization dependence (for both and ) using the complete set of the crystal surface plates (, , and ) (29). These measurements are indispensable to experimentally distinguish the respective electric and magnetic contributions to the spectrum. Figures 6(a) and 6(b) present the Re and Im spectra in the spiral spin ordered phase, respectively, measured around 10 K. There is a remarkable optical anisotropy with respect to and . According to the inelastic neutron scattering experiments for the case of TbMnO (51); (26), the crystalline-electric-field excitation of electrons lies around 4.7 meV. However, the observed remarkable optical anisotropy as well as the negligible absorption at 11 K for and [open circles in Figs. 6(a) and 6(b)] can exclude the possible emergence of such an excitation in the spectrum in the measured energy range.
In the Im spectrum for and at 9 K [closed squares in Figs. 6(a) and 6(b)], there is a pronounced broad absorption up to 10 meV, which consists of two peak-structures around 2 meV and 6 meV. Accordingly, the clear dispersive structures are visible in the Re spectrum. The presence of the lower-lying absorption has been reported for TbMnO at 2.9 meV (23), GdMnO at 2.5 meV (23), and EuYMnO at 3 meV (27); (24). Among them, the magnitude of Im of DyMnO reaches about 10, which is roughly by a factor of 2–5 larger than that of TbMnO and GdMnO, as also seen in Fig. 4. The observed remarkable characteristics, including the positions for peak-structures and the spectral shape, can be also identified for and at 10 K [closed circles in Figs. 6(a) and 6(b)]. Therefore, the observed broad absorption can be ascribed to the electric-dipole active mode only along the -axis. We can reproduce the spectra for by adopting two Lorentz oscillators of for peak-structures, as indicated by the solid lines. It yields Re of , which is less than along the -axis at 10 kHz [Fig. 5(a)]. A noticeable characteristic is the large optical anisotropy of the Re spectrum. Re for at low-energy yields the large value of Re of 25–30, which is about 2 times larger than that along the other axes, i.e., 15.5–17.5. This large anisotropy ratio is comparable to that of at 10 kHz [Fig. 5(a)], indicating that the anisotropic dielectric response is extended from kilohertz to terahertz frequencies associated the gigantic electric-dipole active excitation.
On the contrary to the case, we found tiny absorptions around 2.2 meV for and , whose peak energy is nearly identical to that of the lower-lying peak-structure for . Figures 6(c) and 6(d) show the Re and Im spectra, respectively, measured around 10 K. There is a single peak-structure around 2.2 meV in the Im spectrum [Fig. 6(d)] with a clear dispersive structure in the Re spectrum [Fig. 6(c)] for and at 8 K (closed triangles), although the magnitudes are an order of magnitude smaller than the peaks. This peak-structure can be assigned to the spin excitation driven by as the nearly identical spectra signature can be observed for and at 6 K (open squares), while the absolute value of the is slightly different. We also discern the broad peak-structure around 2.3 meV for and at 9 K (crosses). For the case of and , the possible peak feature driven by is completely masked by the intense electric-dipole active absorption for . This peak-structure for and diminishes above , as in the same manner to the spin excitation by . Therefore, this peak-structure is ascribed to the spin excitation driven by , details of which are discussed in Sec. III.4 by adopting GdTbMnO.
In the following, we mainly focus on the observed gigantic absorption for . Figures 7(a) and 7(b) show the temperature dependence of the Re and Im spectra, respectively, for and . At 245 K, there is no remarkable absorption. The slight accumulation of the Im spectrum above 6 meV is due to the contribution of the optical phonon absorption of the perovskite structure, located around 14 meV (24) (see, also the overall optical spectrum of TbMnO shown in Fig. 3). With decreasing temperature, the magnitude of Im below 6 meV tends to increase. Around of 42 K, Im forms the peak-structure around 2 meV. Accordingly, there is the dispersive structure in the Re spectrum. In the collinear spin ordered phase below , the absorption grows in intensity, while the peak position shifts to the lower energy. In the spiral spin ordered phase below of 19 K, the absorption becomes prominent and finally reaches the maximum of Im at low temperature. In this ferroelectric phase, we also identify the additional peak-structure around 6 meV in the Im spectrum, which forms the broad continuum-like absorption up to 10 meV. Its peak position shifts to the lower energy with decreasing temperature, while the lower-lying peak position shifts to the higher energy.
To further clarify the origin of the broad continuum-like absorption for , we show in Fig. 8(a) the temperature dependence of the integrated spectral weight per Mn-site defined as,
where is the free electron mass, the elementary charge, and the unit-cell volume. We chose meV and meV to fully cover the broad absorption. By the definition in Eq. (2), the estimated includes the contribution of for , which is crudely estimated to be as small as 0.02 10. Above , for (circles) shows the negligible temperature dependence. However, gradually increases below and sharply enhances below [see, also in the inset of Fig. 8(a)]. Therefore, we can conclude that the observed absorption for is a magnetic in origin, namely, electromagnon. On the contrary, for (squares) exhibits the slight enhancement below and shows the maximum at [Fig. 8(a)]. We also plot the contribution of the background absorption (a dotted line) as includes the contribution of for [the inset of Fig. 8(a)].
In MnO, the rotation mode of the spiral spin plane was proposed at the early stage as the origin of the electromagnon observed at terahertz frequencies (25). Within this picture, the rotation mode of the spiral plane has the particular selection-rule for ; it would become active along the - and -axes in the and spiral spin ordered phases, respectively. Such an electromagnon scenario was considered to explain the inelastic neutron scattering spectra of TbMnO in the spiral spin ordered phase (26). To test the origin of the electromagnon for , we studied the effect of on the optical properties in DyMnO for and . In DyMnO, applying along the -axis can induce the spiral spin ordered phase with ; the direction of can be flopped from the - to -axis at 20 kOe, as presented in Fig. 5(d). During the -flop, the incommensurate keeps nearly constant for DyMnO (54), contrary to the case of TbMnO, which shows the incommensurate to commensurate transition (55).
In Figs. 7(a) and 7(b), we also include the dependence of Re and Im spectra, respectively, measured at 7 K. The external was applied along the -axis. We confirmed that the -flop occurs at 20 kOe [Fig. 5(d)]. By application of , the position of the lower-lying peak-structure shifts to the lower energy and its intensity grows. Noticeably, we can still see the electromagnon for showing the gigantic absorption, even in the spiral spin ordered phase, as exemplified by the spectra at 33 kOe and 59 kOe. We plot in Fig. 8(b) the (circles) as a function of up to 70 kOe. The integrated range of is chosen as meV and meV. As can be seen, there is negligible effect on even when the spiral spin order is developed above of 20 kOe apart from the slight enhancement of . These observations are different from the reported result of TbMnO, in which the lower-lying peak-structure disappears by an application of the external (23). In the experiments in Ref. (23), the external was applied to the -axis to destruct the ferroelectric order, accompanied by the transformation from the spiral spin order to the -type AFM order. According to the ME phase diagrams of TbMnO and DyMnO (8), the application of along the - or -axis is needed to induce the -flop from the - to -axis or equivalently the rotation of the spiral spin plane from to , as we performed here. To confirm the survival of the electromagnon for , we also measured the spectra for and in along the -axis. dependence of for and (squares) is presented in Fig. 8(b). We chose the integrated range as meV and meV. Above , the electromagnon arising from the rotational motion of the spiral spin plane, is expected to appear for in the spiral spin ordered phase. However, for and is considerably smaller than for and by a factor of 100, and there is no remarkable change of at .
iii.4 GdTbMnO with spiral spin order
Here we present the results of GdTbMnO in zero (32), which represent the generic optical properties of the spiral spin ordered phase of MnO. Among a family of MnO, GdTbMnO is a rare example as it exhibits the ferroelectricity along the -axis even in zero [Fig. 2(b)]. This is in contrast to the cases of GdMnO, TbMnO, and DyMnO, where phase was only induced when the external is applied along the - or -axis, as described in Sec. III.3 for the case of DyMnO. We show in Fig. 9(a) the temperature dependences of and at 10 kHz along the -axis for GdTbMnO (53). With decreasing temperature, the paraelectric collinear spin order evolves below of 42 K along the -axis, as in the similar manner to other MnO showing the spiral spin order in zero . Below of 24 K, steeply increases and exhibits the sharp peak. Finally, reaches the maximum ( C/m) at 17 K. The estimated is comparable to that along the -axis of TbMnO in zero [Fig. 1(b)] (2); (8). Below 16 K, suddenly vanishes upon the development of the -type AFM order. In this -type AFM phase, the finite magnetization emerges along the -axis as a result of the slight canting of the Mn spins. In GdTbMnO, the presence of the spiral spin ordered phase in zero was recently confirmed by the polarized neutron scattering experiments with use of an isomer of Gd ion to prevent the large cross-section of neutron of Gd ions (16). Moreover, along the -axis was found to be 0.25, which is identical to that in the spiral spin ordered phase of TbMnO induced by (55). In accord with above facts, the spiral spin ordered phase of TbMnO in is smoothly connected with that of GdTbMnO in zero , as can be seen in the comparison of ME phase diagrams for TbMnO and GdTbMnO, shown in Figs. 10(a) and 10(b), respectively. Therefore, we can study the generic spin excitation at terahertz frequencies in the spiral spin ordered phase, which provides the further insights into the nature of the observed electromagnon for in MnO.
First, we show the light-polarization dependence of the spectrum for GdTbMnO using a complete set of the crystal faces to clarify the light-polarization selection-rule of the spin excitation in the spiral spin ordered phase. Figures 11(a) and 11(b) present the Re and Im spectra, respectively, measured around 17 K. For these measurements, we used the , , and faces to distinguish the electric and magnetic contributions. In analogy to the spectra of DyMnO with the spiral spin order (Fig. 6), we observe the broad continuum-like absorption at 17 K (closed squares), which consists of two peak-structures around 2 meV and 8 meV, when and were set parallel to - and -axes, respectively. Such a broad absorption was assigned to the electromagnon in the spiral spin ordered phase of DyMnO (Sec. III.3). The observed absorption of GdTbMnO was also assigned to electromagnon because of the closely similar spectral feature for and (closed circles). Even though we checked the reproducibly of the spectra by changing the thickness of the sample, the slight discrepancy of the magnitude of Im spectrum is identified. This is perhaps due to the slight variation of Gd-to-Tb ratio as GdTbMnO locates the critical region in between the paraelectric -type AFM and ferroelectric spiral spin ordered phases (53), as presented in the ME phase diagram [Fig. 2(b)]. The magnitude of Re for at 17 K reaches about 24 at 1 meV (Fig. 11), which is comparable to along the -axis at 10 kHz [Fig. 9(a)]. The magnitude of Re along the -axis is larger than Re[ for other axes by a factor of 1.5, similar to the case for DyMnO with spiral spin order (Fig. 6).
A broad single peak-structure is also discerned for and at 17 K (open squares), where the peak energy nearly matches the peak position of the lower-lying electromagnon for [Fig. 11(b)]. However, the magnitude of Im at the peak energy was estimated to be 0.4, which is an order of magnitude smaller than that for . This absorption can be ascribed to the spin excitation driven by because of the similar spectral signatures for and at 21 K (closed triangles). Furthermore, we observe the tiny absorption for and at 20 K (crosses), which becomes prominent in the spectrum at 11 K in the -type AFM phase, as presented in Fig. 12(c). Upon the diminishment of the spiral spin order, responsible for the gigantic contribution of to , we clearly see the single-peak-structure in the spectrum at 11 K for and . By comparing the spectrum at 11 K for and with that at 10 K for and [Fig. 12(c)], this is attributed to the spin excitation driven by .
These spin excitations for and can be interpreted as antiferromagnetic resonances (AFMRs) of Mn ions, which are also observed in canted AFM phases of EuYMnO for (28) and LaSrMnO for (56). In fact, the magnon in the -type AFM phase of LaMnO locates around 3 meV (57). Therefore, the measured spectrum is considered to consist of a sharp resonance for and broad background absorption for , as given by
which can be phenomenologically expressed with two Lorentz oscillators for and . The solid lines are the results of the least-square fit to the data using Eq. (3), which can reproduce the spectra for GdTbMnO [Figs. 11(c), 11(d), 12(b), and 12(c)] and DyMnO [Figs. 6(c) and 6(d)].
To further see the light-polarized spectral change in a variety of the spin ordered phases, we plot in upper and lower panels of Fig. 12 the temperature variation of Re and Im spectra, respectively, for (a) and , (b) and , and (c) and , in the -type AFM phase below 16 K, ferroelectric spiral spin ordered phase with between 16 K and 24 K, paraelectric collinear spin ordered phase between 24 K and 42 K, and paraelectric paramagnetic phase above 42 K.
At 10 K in the -type AFM phase for and [Fig. 12(b)], the clear peak-structure is identified at 2.1 meV, which can be ascribed to the AFMR driven by , as describe above. With increasing temperature, Im tends to decrease in magnitude and the peak width becomes broad. In addition, the peak position shifts to the lower energy from 2.1 meV at 10 K to 1.6 meV at 33 K. Above of 42 K, the peak-structure is diminished, as can be seen in the spectrum at 50 K. For the case of and polarization [Fig. 12(c)], a similar tendency is observed.
On the other hand, there is a remarkable temperature variation of the optical spectra for and [Fig. 12(a)]. In the -type AFM phase at 11 K, the sharp peak-structure can be discerned at 2.1 meV, ascribed to the conventional AFMR by , described above. With increasing temperature, the magnitude of Im dramatically enhances and the electromagnon emerges as the broad continuum-like absorption, as exemplified by the spectrum at 17 K in the spiral spin ordered phase. The intensity of the electromagnon absorption decreases when the spiral spin ordered is diminished above 24 K. However, the electromagnon survives even in the collinear spin ordered phase as a broad absorption band [lower panel of Fig. 12(a)], the characteristic energy of which was estimated to be about 2.9 meV. Above of 42 K, the broadened absorption subsists, while the intensity of the electromagnon is completely damped, as seen in the spectrum at 68 K.
To be more quantitative, we show in Fig. 9(b) the temperature dependence of (squares) for ; the integrated range for [see, Eq. (2)] was chosen from meV to meV to characterize the spin excitation spectral range. On cooling, increases below and sharply enhances below . This behavior is in contrast to the behavior of due to the component AFMR with and (circles); the latter is nearly temperature independent above and gradually increases below . for dramatically suppresses when the spiral spin order transforms to the -type AFM order below 16 K, which contributes to the reduction of at 10 kHz [Fig. 9(a)].
In early studies (26), the electromagnon for was ascribed to the rotation mode of the spiral spin plane. In this picture, the electromagnon would become active in the spiral spin ordered phase of GdTbMnO when was set parallel to the -axis, perpendicular to the spiral spin plane. Figures 11(a) and 11(b) present Re and Im[ spectra (open circles) for and in the spiral spin ordered phase, measured at 16 K. As can be seen, there is negligible absorption in the measured energy range. We also measured the spectra down to 0.8 meV with use of the LT-GaAs terahertz emitters coupled with the bow-tie antenna as a light source. However, the same tendency is discerned. The present measurements using a complete set of crystal faces can clearly exclude the possibility that the observed electromagnon can be ascribed to the rotation mode of the spiral spin plane. Combined with the results presented in Sec. III.3 for DyMnO in the -induced spiral spin state , we can firmly conclude that there is a unique selection-rule along the -axis for the electromagnons in MnO, irrespective of the direction of the spiral spin plane ( or ).
Based on the systematic experimental investigations with varying the ionic radius of , light-polarization, temperature, and external (Sec. III), we can extract the general features of MnO in a variety of the spin ordered phases, as summarized below. There are spin excitations driven by and at terahertz frequencies in MnO. In the ferroelectric spiral spin ordered phase, the electromagnon appears only along the -axis. It spreads in the energy range of 1–10 meV as a pronounced continuum-like absorption with two peak-structures around 2 meV and 5–8 meV, rather than the single peak-structure (1–5 meV) as previously reported for TbMnO (23). Accordingly, Re for along the -axis is a factor of 2 larger than those for other polarization. This large optical anisotropy is comparable to the anisotropy of at 10 kHz. The lower-lying peak-structure of the electromagnon can be identified, though broadened, even in the collinear spin ordered phase but suddenly disappears in the -type AFM phase. In addition, we observed weak but sharp single peak-structure for and in the energy range of 1–4 meV, whose peak positions are nearly identical to that of the lower-lying electromagnon for . These peak-structures can be assigned to the AFMRs of Mn-ions, which are pronounced in the -type AFM phase.
In the experimental data presented in Sec. III, we could reveal the new features of the optical properties of MnO. As an origin of the electromagnon for , the rotation mode of the spiral spin plane has been proposed (25) based on the spin-current mechanism that can explain the emergence of ferroelectricity and the -flop by the external in MnO (13). Actually, this mechanism was considered to explain the inelastic neutron scattering spectrum of TbMnO in the spiral spin ordered phase (26); the one of the low-energy branch of the magnon band around 2 meV was assigned to the electromagnon. Noticeably, the electromagnon model predicts the unique selection-rule in terms of the light-polarization; it becomes active for and in the and spiral spin ordered phases, respectively. However, we have clearly revealed that the electromagnon shows up only along the -axis, irrespective of the direction of the spiral spin plane ( or ), by measuring the spectra in the spiral spin ordered phases in DyMnO induced by the external as well as in GdTbMnO induced by temperature. There is no remarkable peak-structure in the spectrum for in the spiral spin ordered phase down to 0.8 meV. Based on systematic experimental data described above, we conclude that the electromagnon in the energy range of 1–10 meV cannot be ascribed to the rotation mode of the spiral spin plane. Recently, the negligible effect of the external on the electromagnon is also confirmed in TbMnO by applying the external along the -axis (34); (36) or the -axis (36), which is consistent with the results of DyMnO in (29).
Electromagnon can only contribute to the change of along the -axis [Fig. 5(a)], as evidenced by the consistency of the anisotropy ratio of Re at terahertz and kilohertz frequencies . However, there is a still discrepancy of the absolute values of Re at terahertz and kilohertz frequencies for the case of DyMnO, the reason of which can be understood by the presence of the ferroelectric domain wall that produces the additional contribution to , as recently revealed by the dielectric spectroscopy up to 10 MHz of DyMnO (58); the spectrum shows the relaxation type dispersion with a relaxation rate of Hz and the magnitude of at 10 MHz was estimated to be , which seems to smoothly connect with the low-energy part of the measured Re spectrum at terahertz frequencies (Re 25–30).
As an origin of the electromagnon for , we proposed the conventional electric-dipole active two-magnon excitation due to the exchange-striction mechanism (59), since the observed electromagnon is independent of the direction of the spiral spin plane and spreads in the wide energy range of 1–10 meV despite the fact that the magnon driven by locates around 2 meV (29); (31); (32); (33). Contrary to this scenario, it was found more recently that the symmetric exchange mechanism can generate “one-magnon” excitation when the ordered spins are non-collinear, as in the present case of cycloidal spin order (34); (35). In the following, we introduce theoretical considerations on MnO based on Heisenberg model and compare with experimental data presented in Sec. III.
The magnetic behaviors of the cycloidal ordered magnets such as MnO can be described by a three-dimensional frustrated Heisenberg model:
where is the nearest-neighbor ferromagnetic interaction, and and , are next-nearest-neighbor and interlayer antiferromagnetic interactions, respectively (see Fig. 13). Due to the frustration between and , spiral spin states are the ground state when the condition is satisfied, where the spiral angle is given by . The respective interactions for , , and were estimated and summarized in Table. 1 (35); (37). term is a uniaxial anisotropy term, which fixes the direction of the spiral spin plane, i.e., for and to realize a -cycloidal state and for to stabilize an -cycloidal state. We assume that , which reproduces the anisotropy gap observed in the magnon dispersions of revealed by inelastic neutron scattering experiments. (26); (35)
The lower-lying magnetic excitations of the model on the basis of Eq. (4) can be described by the linear spin wave theory (60); (35). After rotations of local spin axis, we make Holstein-Primakoff approximations as usual,
Then, the spin Hamiltonian [Eq. (4)] is diagonalized as
In Eq. (6), the spin wave frequencies are given by
and a spin wave creation and annihilation operators are defined as
where and are coefficients, as given by , and .