Electrically driven spin excitation in a ferroelectric magnet DyMnO
Temperature (5–250 K) and magnetic field (0–70 kOe) variations of the low-energy (1–10 meV) electrodynamics of spin excitations have been investigated for a complete set of light-polarization configurations for a ferroelectric magnet DyMnO by using terahertz time-domain spectroscopy. We identify the pronounced absorption continuum (1–8 meV) with a peak feature around 2 meV, which is electric-dipole active only for the light -vector along the -axis. This absorption band grows in intensity with lowering temperature from the spin-collinear paraelectric phase above the ferroelectric transition, but is independent of the orientation of spiral spin plane ( or ), as shown on the original (ferroelectric polarization) phase as well as the magnetic field induced phase. The possible origin of this electric-dipole active band is argued in terms of the large fluctuations of spins and spin-current.
pacs:75.80.+q, 76.50.+g, 75.40.Gb, 67.57.Lm
The magnetoelectric (ME) effect, in which the electric polarization can be induced by a magnetic field or the magnetization can be induced by an electric field , has been known since the 1950’s.MEeffectRev () However, the materials showing the large ME effect are rare. Recently, the interest has been revived by the observation of the ferroelectric polarization and its 90 flop in the presence of a finite field ,TKimura1 () in orthorhombically distorted perovskite manganites, MnO, where represents Tb, Dy, and Gd ions.TKimura1 (); TGoto (); TKimura2 () In DyMnO, for example, the collinear spin order of Mn ions sets in at the Néel temperature of 39 K with the incommensurate (IC) modulation vector along the -axis.TGoto () Noticeably, the ferroelectricity shows up along the -axis when the transverse spiral (cycloidal) spin orderMKenzelmann () evolves below of 19 K. The direction of is flopped to the -axis when is applied along the -axis.TGoto (); TKimura2 () So far, a wide variety of the ferroelectric magnets, termed multiferroics, have been found in a family of cycloidal-spin magnets.MEeffectRev () The ferroelectricity in MnO can be microscopically explained by the spin-current modelHKatsura1 () or equivalently by the inverse Dzyaloshinski-Moriya effect,IASergienko (); MMostovoy () as expressed by the relation, , where is the unit vector connecting and sites. Namely, the cycloidal spin structure can induce the ferroelectricity along the direction perpendicular to the modulation vector and within the spiral spin plane. A direct evidence for this mechanism was recently demonstrated by controlling the vector chirality () of TbMnO with .YYamasaki ()
In the magnetically induced ferroelectric phase, the strong ME coupling would also produce the intriguing low-lying optical excitation. In MnO, there are experimental (Ref. APimenov1, ) and theoreticalGASmolenski (); HKatsura2 () arguments on the possibility of the presence of the low-lying spin excitation, called electromagnon, which is activated electrically. Far-infrared spectroscopy has provided the useful information for the electrodynamics of TbMnO,APimenov1 () GdMnO,APimenov1 (); APimenov2 () and EuYMnO,RValdesAguilar (); APimenov3 () revealing that an additional light absorption contribution indeed emerges in the ferroelectric phase at terahertz (THz) frequencies below the optical phonon frequencies of the perovskite structure. Among a family of MnO, DyMnO shows the largest ME response; the change of the dielectric constant upon the flop exceeds 500% and reaches 2 mC/m.TGoto (); TKimura2 () Therefore, DyMnO offers a good opportunity to clarify the low-energy electrodynamics coupled with the spin excitation.
Here we report on a complete set of optical spectra at THz frequencies of a multiferroic DyMnO with varying the light-polarization (both the electric field and magnetic field components of light), temperature , and applied , to unravel the strongly electric-dipole allowed spin excitation. The observed excitation is confirmed to be active mainly for and not to be correlated to the direction of the spiral spin plane. We focus on spectral signatures, being highly sensitive to the light-polarization, in a variety of ME phases that can be tuned by both and . The format of this paper is as follows. In Sec. II we describe the detail of the experimental procedure for measurements of THz time-domain spectroscopy. Section III is devoted to the optical investigations of the low-energy spin dynamics in a variety of ME phases of DyMnO, in which we clarify the unique selection rule of the spin excitation driven by or by measurements of the light-polarization (Sec. III.1), (Sec. III.2), and (Sec. III.3) dependences. On the basis of the results of Sec. III, the origin of the electrically driven spin excitation observed at THz frequencies in DyMnO is discussed in Sec. IV. The summary of this paper is given in Sec. V. We also describe in Appendixes the estimation procedure of the complex optical constants from the raw data obtained by THz time-domain spectroscopy.
ii.1 Sample preparation and characterization
The large single crystals of DyMnO were grown by a floating zone method. The available , , and surfaces of the crystals with a typical sample size of 4–16 mm were cut from the ingot by using a back Laue photograph. The respective samples were polished to thickness of 150–800 m. The polishing procedure frequently induces the damage on the surface for the case of the orthorhombically distorted perovskite manganites. We confirmed that the optical response was almost unchanged before and after polishing procedure.
The samples cut from the same ingot used in this work were characterized by the measurements of the dielectric constant at low-frequencies as functions of and . The ferroelectricity along the -axis in zero and the flop from the to -axes in were clearly observed around 19 K and 20 kOe at 7 K, respectively. The obtained phase diagrams in a plane of and are consistent with the previously reported one of DyMnO.TKimura2 ()
ii.2 Setup for terahertz time-domain spectroscopy
We used the THz time-domain spectroscopy in a transmission geometry to obtain complex refractive index without using Kramers-Kronig (KK) analysis.THzRev (); MTonouchiRev () THz time-domain spectroscopy can easily access the low-energy electrodynamics in solids, typical frequencies from 0.1 THz to 3 THz.THzRev () The lower limit of the measurable frequencies depends on the size of the samples since 0.1 THz corresponds to 3 mm in wavelength. We adopted a transmission geometry and the sample was attached to the perforated Cu holder with a diameter of 2 mm, 2.8 mm, 3 mm, and 3.5 mm, which was selected to suitably match the size of the samples.
Figure 1 shows the schematic illustration of THz time-domain spectroscopy in a transmission geometry in this study. For the measurements, we used the photoconducting (PC) sampling techniqueTHzRev1 () to obtain THz radiation pulse. The femtosecond laser pulses delivered from the mode-locked Ti:sapphire laser (center wavelength of 800 nm, pulse width of 90 fs, and repetition rate of 80 MHz) were used as a source. The laser pulses were divided by a beam splitter (BS). One was used as the pump and another was used as the gate. The pump pulses were irradiated on to the ZnTe crystalQWu () to induce the optical rectification effect, acting as a source of THz radiation.KYang () The wire grid (WG) polarizers were inserted in between off-axis paraboloidal mirrors to obtain the linear polarization. The polarized THz pulse was focused on the sample by off-axis paraboloidal mirrors. The cryostat was placed within the box filled with dry nitrogen to eliminate the absorption of water. The gate pulses were introduced to the low-temperature-grown GaAs (LT-GaAs) photoconducting device coupled with a dipole antenna after the appropriate time delay.DHAuston () The pump pulses were mechanically chopped and the photocurrent induced by the electric field of the THz pulse was lock-in detected. Therefore, the induced photocurrent, which depends on both amplitude and phase of THz radiation, can be obtained by varying the optical delay line. Further details of our scheme for THz generation and detection can be found in Refs. THzRev, and THzRev1, .
For the measurements of the dependence, the sample was placed in a cryostat equipped with a superconducting magnet. supplied from a superconducting magnet was applied along the -axis up to 70 kOe. The measurements in were mainly performed at 7 K. We adopted a Faraday geometry. Further details of our experimental setup combined with the superconducting magnet can be found in Ref. YIkebe, .
Since we consider the spin excitation, which can be driven by either or , we numerically estimated and used the quantity of (not ), where is the complex dielectric constant and the complex magnetic permeability. Details of the estimation procedure and the validity of the quantities of to characterize the spin excitation are described in Appendixes.
iii.1 Light-polarization dependence
First, in Fig. 2, we show spectra (closed circles) in the spiral spin phase () of DyMnO to determine the selection rule with respect to the light-polarization, in which and were set parallel to the crystallographic axes using available , , and surfaces of the crystal. Upper and lower panels of Fig. 2 present, respectively, the real and imaginary parts of spectra, measured around 10 K with . There are remarkable peak structures and noticeable optical anisotropy with respect to and . Such a light-polarization dependence, including the negligible absorption for and [Fig. 2(c)], as well as the conspicuous spectral change around and , cannot be ascribed to the crystalline-electric-field excitation of electrons, which was reported for TbMnO around 4.7 meV by inelastic neutron scattering studies.RKajimoto (); DSenff ()
A broad and pronounced absorption continuum (1–8 meV) overlapped by two peak structures around 2 meV and 5.5 meV is observed in spectrum for and [lower panel of Fig. 2(a)]. In accord with this, a dispersive structure can be discerned in ] spectrum [upper panel of Fig. 2(a)]. Such a low-lying peak structure was also identified previously at 2.9 meV for TbMnO,APimenov1 () 2.5 meV for GdMnO,APimenov1 (); APimenov2 () and 3 meV for EuYMnO (although this value depends slightly on ).RValdesAguilar (); APimenov3 () Noticeably, of DyMnO reaches 10 (corresponding to the optical conductivity of about 3 cm), which is roughly a factor of 2–5 larger than the absorption of TbMnO and GdMnO, suggesting DyMnO as the prototype in the study of the low-lying excitation. Such a strong absorption including the spectral shape, peak position, and magnitude, was clearly confirmed to be observed only for , as evidenced by the spectra for and [Fig. 2(b)], i.e., with different polarization. The low-temperature spectrum can be fitted with two Lorentz oscillators, as indicated by solid lines. The ratio of for to other configurations was about 2, being consistent with the dielectric constant at 10 kHz.TGoto (); TKimura2 () This indicates the large anisotropy extending up to THz frequencies, associated with the electric-dipole active excitation.
In spectrum for and [Fig. 2(d)], on the other hand, a broad peak feature is observed where the peak energy nearly matches the low-lying peak energy for . However, the magnitude of in this configuration () is much smaller, by a factor of 30, than the magnitude of for . By comparing the spectra for and [Fig. 2(d)] and and [Fig. 2(e)], this absorption appears to arise from the stimulation, in accord with the conventional magnon selection rule for the spiral spin state.HKatsura2 () We also discern the broad peak feature in spectrum for and [Fig. 2(f)] whose magnitude of is comparable to those of for cases of stimulation. We cannot distinguish the selection rule of this tiny absorption on the basis of the light-polarization dependence alone due to the presence of the intense absorption for and [Fig. 2(a)]. As will be shown in Sec. III.2, however, we identify that this sharp peak structure is also driven by .
iii.2 Temperature dependence
We present in Fig. 3(b) the dependence of the selected spectra for . With increasing , tends to decrease in magnitude and the peak position is shifted to the lower energy from 2.8 meV at 5 K to 2 meV at 36 K [the lower panel of Fig. 3(b)]. Above , the broad peak structure is diminished, as can be seen in spectrum at 50 K (a finite value of comes from the absorption tail of the optical phonons). Accordingly, there is no dispersive structure around 2 meV in spectrum [upper panel of Fig. 3(b)]. We also show in Fig. 3(c) the selected spectra for as a function of . With increasing , the peak position is also shifted to the lower energy from 3.3 meV at 4.2 K to 2.3 meV at 31 K and the magnitude of also tends to decrease. The observed variation is similar to the case for ; the peak structure seems to vanish above K. Even though we cannot identify the clear signature of the peak structure above 31 K in spectrum [the lower panel of Fig. 3(c)] due to the detection limit in our experiments ( meV), as determined by the size of the sample, there is no signature of the dispersive structure around 2 meV in spectrum at 46 K [the upper panel of Fig. 3(c)]. On the basis of the dependence as well as of the light-polarization dependence presented in Sec. III.1, it is reasonable to consider that the observed peak structure can be assigned to the spin excitation of Mn ions driven by or . Indeed, in inelastic neutron scattering experiments on the spiral spin phase of TbMnO, the magnon dispersion along the  or  axis comes across around 2 meV.RKajimoto (); DSenff () To estimate the contribution of the magnon to spectrum, we plot in Figs. 3(b) and 3(c) the spectra for at 5 K and for at 4.2 K, respectively, as indicated by solid lines. These were derived from the relation, by assuming Lorentz oscillators of and for contributions of phonon (a dotted line) and magnon, respectively.
Next, we show in Fig. 3(a) the dependence of the spectra for . At 249 K, only the absorption tail due to the optical phonon is observed. With decreasing but even above , it is clearly discernible that in the low-energy region below 8 meV increases and forms a Debye-like absorption band [the lower panel of Fig. 3(a)]. In accord with this, the dispersive structure emerges in spectrum [the upper panel of Fig. 3(a)]. In the collinear phase below (= 39 K), a broad peak structure emerges around 2 meV and grows in intensity continuously with further decreasing . In the spiral spin phase below (=19 K), the additional peak structure shows up around 6 meV and its peak position shifts to the lower energy while the low-lying peak structure shifts to higher energy.
To be more quantitative, we deduced the integrated spectral weight per Mn-site, , as given by
where is the free electron mass, the elementary charge, and the formula (DyMnO) cell volume. The integrated energy range was chosen between = 0.8 meV and = 9.5 meV to nearly cover the full band. Figure 4(b) presents variation of the estimated . We also plotted at 75 K (square) and 245 K (triangle) at the 50 K position for clarity. for gradually increases below K and is sharply enhanced below K (indicated by vertical solid lines). This is in contrast to for (we adopted = 1.5 meV and = 4.6 meV to estimate of the magnon band), as shown in Fig. 4(a); reaches the maximum at where the spiral spin structure evolves from the collinear spin state [note that for includes the contribution of the phonon structure, as indicated by a dashed line in Fig. 4(a), which could be crudely estimated with the spectrum at 5 K, as indicated by a dotted line in Fig. 3(b)]. The electric-dipole active () excitation almost disappears above [Fig. 3(a)] and its energy range (0–10 meV) nearly coincides with that of the magnon band, which was revealed by inelastic neutron scattering experiments on the spiral spin phase of TbMnO.RKajimoto (); DSenff () This coincidence of the spectral energy region also ensures that the observed excitation is magnetic in origin.
iii.3 Magnetic field dependence
As an origin of the electromagnon, the rotational mode of the spiral spin plane around the -axis (propagation vector) has been theoretically proposed based on the spin-current model.HKatsura2 () This was taken into consideration in the analysis of the inelastic neutron scattering results for the spiral spin phase of TbMnODSenff () and the observed dispersion relation (2.4 meV at ) was assigned to the rotational mode of the spiral spin plane. These apparently contradict the results of spectra of GdMnO,APimenov1 (); APimenov2 () in which the peak structure with , comparable to of TbMnO, was also identified in the IC spin-collinear paraelectric phase. In the course of the flop from the to axis at 20 kOe, IC is kept nearly constant for DyMnO,JStrempfer () as contrasted by the IC to commensurate transition for TbMnO.TArima () Therefore, the measurements of the effect of on spectra of DyMnO can provide further insights into the nature of the electromagnon.
The upper and lower panels of Fig. 5(a) present the selected and spectra for , measured at 7 K for varying with -axis, respectively. The experimental setup with use of a superconducting magnet (up to 70 kOe) restricted the measured energy range, and thus we concentrated on the behavior of spectrum below 5 meV. Even though the low-lying peak structure grows in intensity and its peak position is shifted to the lower energy above , the broad continuum-like absorption scarcely depends on ( kOe at 7 K); the polarized absorption is similarly identified as the most notable electric-dipole active band even in the spiral spin phase. Such a tendency is also seen in dependence of , as shown in Fig. 6(b). We selected = 0.7 meV and = 5 meV to estimate of the low-lying excitation. The negligible effect of is clearly discerned even exceeds . This behavior is different from the reported result of TbMnO, in which the low-lying electromagnon disappears by application of up to 80 kOe at 12 K (Ref. APimenov1, ). In the experiment of Ref. APimenov1, ), was applied along the -axis, thus giving rise to the suppression of along the -axis, according to the ME phase diagram of TbMnO.TKimura2 () Namely, the flop of from along the -axis to the -axis or equivalently the rotation of the spiral spin plane from to cannot be induced unless the is applied along the or -axis, as we adopted here. In addition, we measured the spectra for at 17 K just below in along the -axis up to 70 kOe, to confirm the same tendency. Furthermore, we gathered the spectra for and , at 7 K in up to 70 kOe. The upper and lower panels of Fig. 5(b) shows the selected and spectra, respectively. The estimated is also presented in Fig. 6(a) (we used = 2.4 meV and = 4 meV). Above , the direction of becomes perpendicular to the spiral spin plane and hence would activate the electromagnon arising from the rotational mode of the spiral spin plane. Even though the evolution of the tiny absorption is visible around 2 meV above , the strong absorption, to be compared with for in zero , is not discerned at all for , but only for .
In the spiral spin phase of MnO, the electromagnon or equivalently the rotational mode of the spiral spin plane is expected to become active when is set perpendicular to the spiral spin plane, i.e., and in and spiral spin phases, respectively. This scenario is theoretically considered on the basis of the spin-current mechanism (Ref. HKatsura2, ) that is now recognized to explain the magnetically driven ferroelectricity in a variety of multiferroics found so far, including MnO.HKatsura1 () Indeed, a single peak structure around 2.5 meV has been identified in the spectrum for in the spiral spin phase of TbMnO (Ref. APimenov1, ). Accordingly, one of low-energy branches of the magnon band, i.e., around 2 meV at , in the spiral spin phase of TbMnO was assigned to be responsible for the electromagnon model by inelastic neutron scattering experiments (Ref. DSenff, ).
In the present study, we could uncover new features of the low-energy spin excitation driven by either or in a variety of spin phases of DyMnO on the basis of the detailed light-polarization, , and dependences, as presented in Sec. III. First, our result can provide the compelling evidence that the possibility of the rotational mode of the spiral spin plane is excluded as the origin of the intense absorption around 2–3 meV for , that is also observed in the spiral spin phase of TbMnO;APimenov1 () we firmly identify the electric-dipole active spin excitation along the -axis well survives even in the spiral spin phase induced by applying up to 70 kOe along the -axis [Fig. 5(a)]. This observation clearly contradicts the former theoreticalHKatsura2 () and experimentalDSenff () arguments described above. Second, we clarify that the electric-dipole active spin excitation along the -axis can be regarded as the continuum-like absorption band with two broad peak structures (1–8 meV) [Figs. 2(a) and 2(b)] rather than the single peak structure (1–5 meV), as reported in the spiral spin phase of TbMnO (Ref. APimenov1, ). Third, we identify the spin excitation with a single peak structure when is set parallel to the -axis or the -axis [Figs. 2(d)–2(f)], which can be assigned to the antiferromagnetic resonance (AFMR) of Mn spins at . The magnon mode for can be reasonably ascribed to one of the lower branches of the magnon, as observed around 2 meV at in the inelastic neutron scattering spectrum of the spiral spin phase of TbMnO, even though it was previously assigned to the rotational mode of the spiral spin plane.DSenff () This complexity partly comes form the fact that the peak position of AFMR for nearly matches the peak position of the lower-lying peak structure for , as we revealed here [Figs. 2(a) and 2(d)]. On the contrary, the observed broad continuum-like band for spreads over the wide energy range (1–10 meV), which cannot be explained by magnon absorption alone.
The observed broad peak and continuum-like (0–10 meV) shape of the electric-dipole active band does suggest that the spin fluctuation related to the excitation such as the two-magnon is more plausible rather than the Goldstone mode like electromagnon. The optically active two-magnon excitations are possible over the full magnon bandwidth with the combination of and magnons, and hence should distribute the spectral weight up to the energy twice of the magnon bandwidth. According to the inelastic neutron scattering study on TbMnO (Refs. RKajimoto, ; DSenff, ), the magnon band width is about 8 meV. This value is considerably narrowed as compared with that of LaMnO ( meV)KHirota () or PrMnO ( meV)RKajimoto () with the -type antiferromagnetic ground state, due to the larger orthorhombic lattice distortion as well as to the strong spin frustration effect in the cycloidal magnet TbMnO. The magnon bandwidth of DyMnO, in which the inelastic neutron scattering study would be difficult to perform due to the presence of the neutron-absorbing Dy ions, should be comparable to, or slightly smaller than, that of TbMnO, since DyMnO is structurally close to TbMnO and shows the similar plane cycloidal order at the ground state.TKimura2 () Therefore, two-magnon energy range is anticipated to be up to 10–15 meV, which well fits the observed energy range of the electric-dipole active spin excitations.
Here we propose that the observed unique optical dipole activity may share the same origin, i.e., the spin-current mechanism,HKatsura1 () with the static . The strong selection rule irrespective of the or spiral spin plane does suggest the large mutual fluctuation within the plane, of the neighboring spins lining up along the -axis, but not along the nor -axis; namely , , and hence . The antiferromagnetic exchange interaction along the -axis in the perovskite manganite is likely the origin of the large spin fluctuation, contrary to the ferromagnetic coupling along the -axis. This scenario, though a more elaborate theory is needed, may qualitatively explain the strong optical activity for also in the spin-collinear paraelectric phase as observed. Then, broad peaks around 2 meV and 5.5 meV may be assigned to the density-of-state singularities of the two-magnon band.
In summary, we have measured the spectra of a complete set of possible configurations of single crystals of DyMnO by using THz time-domain spectroscopy. A rich variety of optical excitation at THz frequencies was identified by detailed measurements of light-polarization, , and dependences. In particular, the spin excitation driven by was confirmed to become strongly electric-dipole active along the -axis, irrespective of the or spiral spin plane. The broad (1–10 meV) band feature indicates the contribution of two-magnon like excitations arising from the large fluctuations of spin and spin-current inherent to this spiral magnet.
Acknowledgements.We thank F. Kagawa for characterization of the samples used in this study by measurements of at low-frequencies ( 10 kHz) and S. Miyahara, M. Mochizuki, N. Furukawa, H. Katsura, and Y. Taguchi for enlightening discussions. This work was in part supported by Grant-In-Aids for Scientific Research (16076205 and 20340086) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.
Appendix A Raw data of terahertz radiation in time-domain
A unique advantage for the use of THz time-domain spectroscopy is to directly monitor the electric field of the radiated THz pulse in time-domain.DGrischkowsky () This technique yields the amplitude and phase spectra by the fast Fourier transformation (FFT) of the measured THz wave forms with and without the sample. Therefore, we need no Kramers-Kronig transformation, which is necessary for obtaining the complex optical spectra by conventional optical spectroscopy. As a typical example, we show in Fig. 7(b) the transmitted THz wave form with the sample in time-domain, performed at 5.2 K in zero . The sample was the surface crystal plate of DyMnO. and were set parallel to the and -axes, respectively. We also plotted the reference of THz pulse used in this setup. The amplitude of the transmitted THz wave form is reduced due to the absorption of the sample, as compared to the reference. Its phase is also delayed due to the refractive index of the sample. The ratio between FFT spectrum with and without the sample,
can yield the power transmission and phase spectra. The example of the obtained spectrum is shown in Fig. 7(c). We can identify the clear absorption peak structure around 3 meV. This peak can be assigned to the spin excitation driven by , as presented in Fig. 2.
Appendix B Relationship between transmission coefficient and optical constants
We numerically derived the complex optical spectra without the KK transformation by adopting the following procedure. Without taking into account the effect of the multiple reflections inside the sample, the complex transmission coefficient for bulk in normal incidence as schematically shown in Fig. 7(a), can be simply expressed as
where is the frequency, the velocity of light, the thickness of the sample, and the complex refractive index of the sample.BornWolf () Experimentally, the effect of the multiple reflections can be avoided by restricting the time range of the FFT. The complex Fresnel transmission coefficientsBornWolf () from vacuum to the sample and from the sample to vacuum , can be, respectively, given by
when the complex magnetic permeability is 1. In this case, can be expressed as , where is the complex dielectric constant of the sample. Accordingly, and thus can be numerically calculated from Eq. (3) using experimentally determined values; and in Eq. (2). This procedure is widely recognized to estimate and of a variety of materials including dielectrics, semiconductors, and metals.THzRev (); THzRev1 ()
Appendix C Estimation procedure of complex optical spectra
In the cases of the measurements for using and surface crystal plates of DyMnO, we experimentally identified from light-polarization dependences [Figs. 2(a) and 2(b)] that the observed large absorption originates from response. The response in these configurations can be considered to be negligible compared to response. This is confirmed from the results using surface crystal plate where there is no absorption in () [Fig. 2(c)], and only a slight absorption in () [Fig. 2(f)]. Therefore, we assume in the Fresnel coefficient and used Eqs. (4a) and (4b), while keeping in Eq. (3) for generality. This is also applied to the case for and , in which the negligible absorption is clearly identified [Fig. 2(c)]. We numerically solved Eq. (3) combined with Eqs. (4a) and (4b) to match the measured values given by Eq. (2) and determined the spectra. For the sake of convenience, we used the quantities of and in this paper, although they can be precisely expressed as and in the above condition.
On the other hand, we identify a peak structure around 3 meV, which only appears when was set parallel to the -axis [Figs. 2(d) and 2(e)] or the -axis [Fig. 2(f)]. These small absorption can be assigned to response. In addition, a broad background absorption is observed above 4 meV as seen in the power transmission spectrum at 5.2 K [Fig. 7(b)], which can be attributed to the tail of the optical phonon lying around 16 meV.APimenov2 () In such a case, both and could contribute to the transmission spectrum. Accordingly, four independent quantities, , , , and , cannot be obtained only from the transmission measurements of and . However, in the case of a thick sample, the main signature of the transmission spectrum is dominated by bulk absorption and the effect of in the Fresnel coefficient is considered to be small, as discussed below. Therefore, we used the Fresnel coefficient of Eqs. (4a) and (4b), while the effect of response was taken into account by adopting in Eq. (3). The validity of this approach was firmly confirmed following calculation.
We numerically solved Eq. (3) by putting Eqs. (5a) and (5b) with and as fixed parameters with respect to and estimated spectra of DyMnO for and at 5.2 K. For examples, we show in Fig. 8(a) the calculated spectra at selected values of and . The fixed parameters used here are provided in the right-hand side of the figure. The spectrum, as derived from Eqs. (4a) and (4b) [ = 1 and = 0 in Eqs. (5a) and (5b)], is also plotted by closed circles with a solid line. As can be clearly seen in the figure, no significant difference is observed even though is included to and .
To further check the validity of the used values of and , we assumed that comes from the contribution of the optical phonon structure, which can be represented by a single Lorentz oscillator. This approximation has been conventionally used to estimate the spectra in antiferromagnets, as exemplified by the cases of the orthoferrites.AMBalbashov () We used the following parameters; the resonance frequency, the damping rate, and the dielectric constant for the high frequency above the measured range, were set to 9 meV, 0.5 meV, and 16, respectively. With this approximation, the spectrum was numerically calculated using Eq. (3) combined with Eqs. (5a) and (5b). The result is presented in Fig. 8(b) by closed circles. We also show the spectrum (a solid line), as derived from the spectrum that was estimated by using Eqs. (4a) and (4b). We assumed that the spectrum can be expressed by two Lorentz oscillators for and . The both data coincide with each other below 4 meV, ensuring the validity of using Eqs. (4a) and (4b). Noticeably, the maximum values of and are estimated to be about 1.01 and 0.02, respectively, indicating that the fixed parameters shown in Fig. 8(a) are reasonable. However, there is a discrepancy above 4 meV, which mainly arises from the lack of the spectral information above 6 meV and thus from the inappropriate parameters used here to reproduce the optical phonon structure. In this work, we numerically solved using Eq. (3) with Eqs. (4a) and (4b) and estimated spectra.
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