Understanding Multiferroic Hexagonal Manganites by Static and Ultrafast Optical Spectroscopy

Multiferroic hexagonalmanganites ReMnO 3 studied by optics are reviewed.Their electronic structureswere revealed by static linear and nonlinear spectra. Two transitions located at∼1.7 eV and∼2.3 eV have been observed and attributed to the interband transitions from the lower-lyingMnd xy /d x 2 −y 2 and d xz /d yz states to theMnd 3z 2 −r 2 state, respectively.These so-called d-d transitions exhibit a blueshift as decreasing temperatures and an extra blueshift near T N . This dramatic change indicates that the magnetic ordering seriously influences the electronic structure. On the other hand, the ultrafast optical pump-probe spectroscopy has provided the important information on spin-charge coupling and spin-lattice coupling. Because of the strongly correlation between electronic structure andmagnetic ordering, the amplitude of the initial rising component inΔR/R shows striking changes at the vicinity ofT N . Moreover, the coherent optical and acoustic phonons were observed on optical pump-probe spectroscopy. Both the amplitude and dephasing time of coherent phonons also exhibit significant changes at T N , which provide the evidence for spin-lattice interaction in these intriguing materials.


Introduction
Multiferroic materials [1][2][3][4] with the coexistence of various ferroic order (ferromagnetic, ferroelectric, or ferroelastic) have attracted great attention in condensed matter research due to their great potential for applications in the fields of oxide electronics, spintronics, and even the green energy devices for reducing the power consumption.With multiferroic oxides (in single phase), the coupling interaction among various order parameters causes the so-called magnetoelectric effect [5][6][7][8][9][10].In this, a polarization can be induced by applying a magnetic field or a magnetization can be induced by applying an electric field.Although the properties of electricity and magnetism were combined into one common discipline by the Maxwell equations, the electric and magnetic orderings in solid-state materials are always considered separately.This is because that the electric charges of electrons and ions are responsible for the charge effects, whereas the electron spins govern the magnetic properties.
Prellier et al. [11] classified the characteristics of multiferroic materials into three categories.The first is Bi-based compounds, such as BiMnO 3 and BiFeO 3 .The second is perovskites and related ReMnO 3 compounds (Re = Y, Ho, Er, Tm, Yb, and Lu).The third is ReMn 2 O 5 (Re is rare earth element).Additionally, Cheong and Mostovoy [12] summarized the classification of ferroelectrics into "proper" and "improper" according to their mechanisms of inversion symmetry breaking.In proper ferroelectrics, the main driving force toward the polar state was associated with the electronic pairing.On the other hand, the induced polarization in improper ferroelectrics involves a more complex lattice distortion or other accidental orderings.For example, the hexagonal manganite ReMnO 3 compounds show a lattice distortion to enlarge their unit cell, which is the so-called geometric ferroelectricity [13][14][15].
The aim of this paper is to understand the multiferroic hexagonal manganites studied by some optical techniques.In Section 2, the overall features of manganites are introduced.
The structure of rare earth manganite is strongly depending on the size of rare earth and also temperature.The environment of Mn ions significantly affects its electronic structure which was investigated by static linear and nonlinear spectra in Section 3. Section 4 is the main part of this paper.The ultrafast spectroscopies demonstrate the interaction between electron, phonon, and spin systems on multiferroic hexagonal manganites.Finally, the summary and future outlook will be given in Section 5.

The Characteristics of Multiferroic Manganites ReMnO 3
Recently, multiferroic rare earth manganites (ReMnO 3 ) have attracted a great deal of attention due to their manifestations of intriguing and significant couplings between the magnetic and electric order parameters.Generally, ReMnO 3 has two kinds of multiferroic structures, that is, hexagonal and orthorhombic.The rare earth (Re 3+ ) ionic size shrinks from 1.11 Å to 0.94 Å with decreasing the atomic number and reducing the electrons residing in the 4f orbit, called the lanthanum contraction effect.The rare earth ions (Re 3+ ) have closely chemical properties because the same electronic configurations of 5s 2 5p 6 for outermost electrons.The coexistence of ferroic orders in ReMnO 3 with hexagonal (smaller ionic radius of rare earth Re = Sc, Y, and from Ho to Lu) or orthorhombic (larger ionic radius of rare earth Re = La to Dy) (shown in Figure 1) structure yields complex physics in the intimate interactions among charge, orbital, lattice, and spin degrees of freedom.It also has fascinating physical properties which might lead to promising applications.The orthorhombic structure with a Pnma space group forms the stable crystal structure of ReMnO 3 perovskite oxides with Re = La to Dy, which belongs to the magnetic ferroelectrics induced by magnetic ordering [8,16].Furthermore, another family of compounds with smaller ionic size (for Re = Y, Ho, Er, Yb, Lu, etc.) forms the stable hexagonal structure with 6 3 cm space group.The critical point of the structure transition is located near YMnO 3 (Y 3+ = 1.06 Å) and HoMnO 3 (Ho 3+ = 1.05 Å).Consequently, the crystal structures of these two families have been transformed from hexagonal to orthorhombic (or vice versa) by many experimental techniques, for example, high-temperature and high-pressure processes [17,18], chemical solution deposition (CSD) [19], the metal-organic chemical vapor deposition (MOCVD) [20], molecular beam epitaxy (MBE) [21], sputtering [22], and pulsed laser deposition (PLD) [16,23].
a buckling of the layered MnO 5 polyhedral and is accompanied by displacements of the Re ions, which lead to a net electric polarization.In 2004, through X-ray diffraction and first principle density functional calculations, Van Aken et al. [14] demonstrated the previous scenario in the hexagonal manganite YMnO 3 , as shown in Figure 2 which presents the "geometric" generation of polarization and describes the tilting of a rigid MnO 5 block with a magnetic Mn atom at the center.The magnetic structure has been studied by various experimental methods, such as neutron diffraction [45,46] or second harmonic generation (SHG) [33,47].In 6 3 cm hexagonal manganites, each Mn 3+ ion is surrounded by five O 2− ions, forming triangular planar sublattices in the basal plane (a-b plane).The magnetic order of Mn 3+ is mainly dominated by AFM in-plane Mn-O-Mn superexchange interaction.Therefore, the triangular lattice of the Mn atoms exhibits strong geometrical frustration effect [48].Furthermore, when the temperature cools to near   , the strong superexchange leads to a 120 ∘ arrangement between neighboring Mn 3+ spins in the basal plane which breaks the triangular frustration.In addition, the spins of Mn rotate with an angle of 90 ∘ at  SR (e.g., for h-HoMnO 3 ,  SR ∼ 33 K) due to the onset of the AFM order of the Re moments, indicating the interaction between Re 3+ and Mn 3+ spins.Another magnetic transition at  Re involves the complete magnetic order of the Re 3+ ions (e.g., for h-HoMnO 3 ,  Ho ∼ 5 K).
In addition, there are many studies showing the anomaly around   , such as the dielectric constant [49], specific heat measurements [50], and the lattice constant measurements [51], which are attributed to the indirect coupling via lattice strain (spin-lattice coupling).Recently, Lee et al. [52] observed the giant magnetoelastic effect in hexagonal YMnO 3 and LuMnO 3 by using the high-resolution neutron and synchrotron powder diffraction experiments.They further claimed that the magnetoelectric (ME) coupling can be interpreted by the giant magnetoelastic effect very well in ReMnO 3 , that is, the magnetic long-range ordering induced the very large displacements of all atoms at   .In other words, the magnetoelastic coupling allowed the atoms to undergo an isostructural transition, which has been regarded as the primary source of the magnetoelectric phenomena.In hexagonal ReMnO 3 , this magnetoelastic effect is two orders of magnitude larger than those appearing in other magnetic materials.Therefore, the gigantic magnetoelastic coupling plays a crucial role in dominating many physical properties in hexagonal ReMnO 3 .
Femtosecond pump-probe spectroscopy has been established as a protocol for studying the interactions between electrons, phonons, and magnons [35,[53][54][55][56][57][58][59] and is therefore employed with these multiferroic ReMnO 3 materials to gain insight into the magnetoelastic effect.By measuring the transient reflectivity (ΔR/R) or transmittance (ΔT/T), we can observe the intricate correlation processes among the charge, lattice, and spin degrees of freedom, which are explicitly recognized in different characteristic time scales.Thus, the microscopic mechanism of the magnetoelectric coupling or magnetoelastic behavior in this kind of strongly correlated materials can be revealed.

Electronic Structure Revealed by Optical Spectroscopy
To understand the extensive physical properties in multiferroic ReMnO 3 materials, it is crucial to realize the electronic band structure.With respect to the crystal structure of hexagonal ReMnO  [62], who observed that a near infrared absorption edge moves to higher energy with lowering temperature.For the case of hexagonal ScMnO 3 and ErMnO 3 , the spectra of the real and imaginary parts of the dielectric function were also measured by Kalashnikova and Pisarev [63].An abnormally drastic absorption peak in the region of 1.57-1.59eV is the most pronounced feature in the dielectric spectra.This peak exhibits asymmetry and can be decomposed into two components by peak fitting with the Gaussian functions.Furthermore, the absorption intensity in the region above 2.2-2.4 eV increases with photon energy.In addition, hexagonal HoMnO 3 [64] and LuMnO 3 [31] also exhibit similar results.It is noteworthy that the near infrared absorption peak exhibits a substantial blueshift of ∼0.16 eV from 300 K to 10 K as shown in Figure 3. Recently, Choi et al. [32] reported a systematic study on the hexagonal ReMnO 3 (R = Gd, Tb, Dy, and Ho), and the overall features in the spectra are similar to previous results.The most surprising part of those results is that the temperaturedependent absorption peak at ∼1.7 eV shows an unexpectedly large blueshift near   among the rare-earth ions of Re = Lu, Gd, Tb, Dy, and Ho, as shown in Figure 4(a).The assignment of the optical absorption peak at ∼1.7 eV has also been an issue.One interpretation is that it comes from the charge transfer transition from the O 2p to the Mn 3d states [30,63,65,66].The photoemission spectroscopy (PES) study proposed that the O 2p is the highest occupied state, while the Mn d 3 2 − 2 state is mostly unoccupied.Other occupied Mn 3d states (d  /d  , and d  /d  2 − 2 ) lie deep below   [65].The result of the first-principle electronic structure calculations also suggested that the top of the valence band is mainly determined by the O 2p states, and the lowest unoccupied band is formed by the d 3 2 − 2 of Mn [30,66].The other interpretation, which is also an extensively convincing explanation, is that it comes from on-site d-d transition between the Mn 3d levels [31,34,64,67].The transition was attributed to the d-d transition between occupied  2 and  1 (d  /d  , and d  /d  2 − 2 ) orbitals and the unoccupied  1 (d 3 2 − 2 ) orbit, as shown in Figure 4(b).Because of the selection rule, transitions between d orbitals are forbidden.However, the selection rule is relaxed through the strong hybridization between the O 2p bands and transition metal d bands, which was observed in the hexagonal phase but not in the orthorhombic phase [60].Furthermore, the strong peak at ∼5 eV is attributed to the charge transfer transition from the hybridized O 2p level to the Mn d 3 2 − 2 level.
The other feature of the near infrared absorption peak is that it shows the blueshift as decreasing temperatures accompanying the dramatic changes near   , which indicates the effective coupling between the electronic structure and the magnetic ordering.Souchkov et al. [31] attributed this temperature-dependent blueshift to the effects of exchange interactions between the Mn 3+ ions.The superexchange between Mn 3+ neighbors leads to lower d  /d  2 − 2 levels, whereas the d 3 2 − 2 level is rarely affected (see Figure 4(b)).Moreover, together with the temperature-dependent and magnetic-dependent results [32] further implies that the magnetic ordering seriously influences the electronic structure.

Second Harmonic Generation Spectra.
From the linear absorption spectra, the strong absorption caused by the d-d transition from  2 to  1 was clearly observed in hexagonal manganites.But the transition from  1 to  1 is another question.Recently, the second harmonic generation (SHG) was successfully used to explore the hidden localized dd transitions, even those embedded in the much stronger absorption peaks [68].The nonlinear polarization induced by optical waves is given by where P  (2) is the nonlinear polarization induced in a medium.E  () and E  () are the electric field components of the fundamental light. ()  and  ()  are the second-order nonlinear susceptibility tensor which are uniquely defined by the crystalline and magnetic structure of the crystal, respectively.The subscripts i, j, and k correspond to the Cartesian coordinate axes x, y, and z. i is the direction of the nonlinear polarization generated; j and k are the directions of applied optical field.
There are two types of optical SHG in multiferroic hexagonal manganites.One is caused by the noncentrosymmetric ferroelectric ordering of charges and the nonlinear Table 1: Nonvanished susceptibility tensor components for hexagonal ReMnO 3 with antiparallel coupling of corresponding Mn 3+ spins along the z-axis.

Space group
Tensor components   Nonmagnetic SH (i-type) sum of 6 3  and 6 3  susceptibility is time invariant (i-type).The other is due to the centrosymmetric antiferromagnetic ordering of spins and the susceptibility is time noninvariant (c-type) [34].Table 1 lists the allowed susceptibility tensor components in i-type and c-type.Thus, the SHG contribution from ferroelectric and antiferromagnetic can be distinguished easily.Figure 5 summarizes the magnetic symmetries of all the hexagonal ReMnO 3 compounds as derived from the SHG experiments [33].
Consequently, the electronic structure of hexagonal manganites has been clearly depicted by linear and nonlinear optical spectroscopy, which further provides the basis for the excitation and relaxation of electrons in pump-probe spectroscopy.

Ultrafast Pump-Probe Spectroscopy
Ultrafast optical pump-probe spectroscopy has proven to be a powerful tool to investigate the dynamics of electron-electron interaction, electron-phonon interaction, and electron-spin interaction in various materials (see, e.g., [70][71][72]).To understand the underlying mechanisms in strongly correlated materials, the interactions between these order parameters should be derived first.Because of the different characteristic time scales and characteristic behavior for various degrees of freedom existing in these complex materials, ultrafast spectroscopy can determine the coupling between each order of parameters.As shown in Figure 7, the general idea of the pump-probe technique is that changes of a sample induced by a pump pulse are obtained by detecting the change in the reflectivity (Δ) or transmissivity (Δ) of probe pulses as a function of probe delay time.The femtosecond time evolutions are derived by delaying the relative arrival time between the pump and probe pulses using a mechanical delay line.The fluence of a probe pulse is usually much weaker than that of a pump pulse in order to avoid second excitation in the samples.The polarizations of pump and probe pulses are set perpendicular to avoid interference between pump and probe beams [73].
Generally, this pump-probe spectroscopy composes two kinds of signals of the decay background and the oscillation, which relates to the relaxation of carriers and the dynamics of phonons or spins, respectively.By the Fourier transform, the frequency of phonon modes can be doubly obtained from the oscillation signals in pump-probe spectroscopy.Moreover, the phonon dynamics, that is, the time-evolution of oscillation signals, can be revealed by short-time Fourier transform (STFT), which cannot be reached by common Raman spectroscopy.Thus, the pump-probe spectroscopy is a powerful tool to provide the information in time-and frequency-domain simultaneously.

Spin-Charge Coupling.
Since the magnetic ordering and the electronic structure of manganites are strongly correlated, as found in temperature-dependent absorption spectra, there is a strong coupling between charge and spin, which can be revealed by the time-resolved femtosecond spectroscopy [72].Shih et al. observed the spin-charge coupling by various photon energy excitations on hexagonal HoMnO 3 single crystals [35].As shown in Figure 8(a), the Δ/ amplitude as a function of temperature shows a significant drop to cross zero amplitude line to become negative at some characteristic temperatures.The excitation photon energy used in the experiment changes from 1.52 to 1.68 eV, which is exactly the range covering the d-d transition energy  dd (transition from  2 to  1 ) in the hexagonal ReMnO 3 .When the photon energy is larger than  dd , the electrons in the  2 orbit can transfer to the unoccupied  1 orbital by absorbing pumping photons.Conversely, this onsite transition would be completely blocked when the transition energy  dd becomes larger than the pumping photon energy.Thus, the Δ/ amplitude vanishes at a certain temperature  0 , which drops with increasing photon energy, indicating the blueshift of  dd .Furthermore, Shih et al. found that the optical pumpprobe measurement is sensitive to short-range magnetic order.In Figure 8(b), the  dd as a function of  0 demonstrates a gradual increase in the extent of antiferromagnetic (AFM) ordering.Moreover, this ordering starts to deviate from linearity around   (∼76 K), suggesting the prevalence of global long-range AFM ordering.The AFM correlation evolves with temperature as it is close to   , shown as d(Δ/)  /d in Figure 8(c).Later, Wu et al. obtained similar results for hexagonal YMnO 3 thin films [74].Furthermore, they found that the strain-induced changes of   , which are difficult to be observed by the temperature-dependent magnetization measurements, can be clearly revealed by the shift of d(Δ/)  /d.As a result, the amplitude of the initial rising component of Δ/ corresponding to the Mn 3+ d-d transition is caused by strong coupling between the electronic structure and magnetic ordering.Moreover, the emergence of long-range and short-range magnetic orders can be clearly observed by optical pump-probe spectroscopy.

Spin-Lattice Coupling.
Optically induced lattice vibration can be observed through optical pump-probe spectroscopy through a periodic oscillation.This periodic oscillation in reflectivity (or transmissivity), so-called coherent acoustic phonons, is caused by interference between the reflected probe beams from sample surface and the wave front of the propagating strain pulse generated upon the pump pulse [75,76].On the other hand, vibration with frequencies corresponding to optical phonon modes in various materials has been observed by pump-probe spectroscopy.A number of mechanisms have been proposed to explain these oscillations, for example, the displacive excitation of coherent phonons (DECP) [77] and the impulsive stimulated Raman scattering (ISRS) [78,79].The simplest idea is that a pump pulse creates a standing wave which was generated by the coherent ionic motion.Then a following probe pulse senses the changes of refractive index which is modulated by the standing wave.In general, the reflectivity (or transmissivity) oscillation observed in pump-probe spectroscopy always corresponds to the optical phonon modes observed in Raman spectroscopy.In conclusion, the measurements of coherent acoustic or optical phonons dynamics at various temperatures are useful to study the spin-lattice coupling in multiferroic materials.

Coherent Optical Phonons.
Figure 9 shows the reflectivity oscillation of ∼3.6 THz (∼120 cm −1 ) in LuMnO 3 [36,37].According to Raman studies, this vibration mode can be attributed to optical phonon of A 1 symmetry [80][81][82].This fully symmetric mode corresponds to the Lu1 and Lu2 atomic motions in opposite directions along the c-axis, as depicted in Figure 9(d) [36].In temperature-dependent measurements, the optical phonon almost disappears below   , as shown in Figures 9(e) and 9(f) [37].Additionally, the structural change from highly symmetry space group 6 3 /mmc to the tilted and distorted 6 3 cm ferroelectric structure [14] in hexagonal manganites (see Figure 1) takes place at the Curie temperature (  ), which is much higher than   .Since there is no structural transition near   , the atomic motions of the optical phonon mode would be affected by the appearance of magnetic ordering.In other words, the disappearance of optical phonon mode might be attributed to the magnetoelastic coupling.In temperature-dependent Raman studies, the lowest frequency A 1 mode also shows anomaly at   in YMnO 3 [82], ErMnO 3 [83], and LuMnO 3 [80,82] which support close relation between phonon and spin systems.manganites such as LuMnO 3 [37,84], YMnO 3 [39], and HoMnO 3 [38].The damped oscillation behavior in reflectivity changes with time delay t could be described as follows [76]:

Coherent Acoustic
where  is the wavelength of the probe pulses,  is the refraction index of samples without strain, V  is the strained layer propagating sound speed, and  is the penetration depth of probe pulses.The frequency of the oscillation is given as  = 4V  / and the dephasing time is /V  .
According to (4), the dephasing time is proportional to the penetration depth of probe pulses  and the inverse sound velocity 1/V  .However, the skin depth is much shorter than the propagating length of the strain pulse.Therefore, the dephasing time is dominated by the finite penetration depth  of the probe [84].The dephasing time of acoustic phonons shows strong temperature dependence (Figure 10), and the anomalous increase of the dephasing time near   might be because of the photon energy being close to the edge of d-d transition.In other words, the penetration depth increases and the absorption of probe photons for the d-d transition decreases (due to the blueshift of  dd as mentioned in Section 3.1) as the temperature decreases.
The amplitude and period of the oscillation component in Δ/ show strong temperature dependence as shown in Figure 11.Both of these parameters are clearly anomalous near the magnetic ordering temperature   .The amplitude of the oscillation component in Δ/ gradually increases with decreasing temperature and reaches the maximum near a characteristic temperature (∼190 K for YMnO 3 [39] and ∼200 K for LuMnO 3 [37]).Then it decreases when the temperature approaches   , as shown in Figure 11(a).This may imply the appearance of AFM short-range ordering, as mentioned in [35].On the other hand, the period of the oscillation component in Δ/ also slightly shrinks with decreasing temperature.Surprisingly, it dramatically shrinks around   as shown in Figure 11(b).This strongly indicates the close correlation between AFM ordering and the propagation of thermal stress along c-axis, which is consistent with the results in heat capacity measurements [13,85].In addition, the ferroelectric polarization along the caxis caused by polyhedron tilting of MnO 5 and distortion of the rare-earth ions [14] is well known.Thus, the propagation of a strain pulse with the modulation of dielectric constant (or ferroelectric polarization) along c-axis is indeed affected by the appearance of AFM ordering.Neutron diffraction experiments also have demonstrated the extra displacement of the rare-earth and oxygen atoms along c-axis at   [52].Finally, these results in amplitude and period of the oscillation component in Δ/ provide evidence for magnetoelastic coupling in these intriguing materials.

Summary and Outlook
In this paper, the recent developments on the electronic structures and ultrafast dynamics of hexagonal manganites have been reviewed.The electronic structure can be clearly revealed by the static linear optical absorption spectra and nonlinear second harmonic generation spectra.The temperature-dependent d-d transition energy exhibits an unexpected change near the spin ordering temperature   .This strongly indicates the coupling between electronic structure and antiferromagnetism in hexagonal manganites.Fortunately, the photon energy of the pulse laser used in pump-probe spectroscopy accidentally fits well the d-d transition (from  2 to  1 ) energy.Therefore, the pump-probe Inset: the normalized dephasing time as a function of temperatures at various wavelengths [38].Figures were reproduced with permission from Shih et al. [38].(b) Temperature dependence of dephasing times: red squares for acoustic phonons, black circles for optical phonons, and blue triangles for electronic relaxation time [37].Figures were reproduced with permission from Jang et al. [37].
measurements can reveal the ultrafast dynamics through the spin-charge coupling and spin-phonon coupling.However, those ultrafast dynamic experiments only present the charge dynamics between  2 and  1 .If the laser spectrum could cover the whole range of d-d transition, that is, containing the transition from  1 to  1 and  2 to  1 , it could disclose all of the charge dynamics in these strongly magnetic influenced d orbitals.The optical parametric amplifier (OPA) is a suitable light source for this kind of study.Consequently, if all of the charge dynamics in the d orbitals could be completely revealed, it would shed  [39].Reprinted with permission from Jang et al. [39].Copyright 2010, American Institute of Physics.(b) Temperature dependence oscillation period in Δ/ at various wavelengths of HoMnO 3 .Inset: the evolution of the oscillation component in Δ/ at various temperatures [38].Figures were reproduced with permission from Shih et al. [38].light on the microscopic mechanism of the magnetoelectric coupling or magnetoelastic effect in this kind of strongly correlated material.

Figure 6 :
Figure 6: SHG spectra of YMnO 3 due to (a) the ferroelectric ordering and (b) antiferromagnetic ordering; inset: temperature dependence of the SHG intensity of 2.46 eV[34].Reprinted figure with permission from Fröhlich et al.[34].Copyright (1998) by the American Physical Society.

Figure 7 :
Figure 7: (a) Schematic diagram of pump-probe technique.Time delay (Δ) between pump and probe pulses can be controlled by a mechanical delay line.(b) The fundamental principle of pumpprobe experiments.Time varying refractive index () changes of the sample induced by pump pulses were observed by detecting the intensity variations of probe pulses.

Figure 8 :
Figure 8: (a) Normalized amplitude of Δ/ as a function of temperature at various wavelengths.The insets illustrate splitting of the Mn 3+ energy levels in the local environment MnO 5 for photon energy above and below  dd ( 1 >  2 ).(b) Energy gap  dd as a function of  0 . dd is estimated from the wavelengths in (a).The dashed line emphasizes linear behavior in the high-temperature range.(c) The slope of temperature-dependent normalized amplitude of Δ/ in (a) (gray thick lines) as a function of  0 at various wavelengths.The dashed line emphasizes behavior of the slope [35].Reprinted figure with permission from Shih et al. [35].Copyright (2009) by the American Physical Society.

Figure 9 :
Figure 9: LuMnO 3 measured at 293 K. (a) Δ/ as a function of time delay.(b) Oscillation component after subtraction of the slowly varying response.(c) Fourier power spectrum of the data with frequency 3.61 THz.(d) Schematic of the coherently excited A 1 phonon mode [36].Reprinted figure with permission from Lou et al. [36].Copyright (2009) by the American Physical Society.(e) Δ/ at various temperatures.(f)Fourier transformed amplitudes of coherent optical phonon[37].Figures were reproduced with permission from Jang et al.[37].

Figure 10 :
Figure 10: (a) Temperature dependence dephasing time of the oscillation component in Δ/ at various wavelengths of HoMnO 3 .Inset: the normalized dephasing time as a function of temperatures at various wavelengths[38].Figures were reproduced with permission from Shih et al.[38].(b) Temperature dependence of dephasing times: red squares for acoustic phonons, black circles for optical phonons, and blue triangles for electronic relaxation time[37].Figures were reproduced with permission from Jang et al.[37].
Figure 11: (a) Fourier transform amplitudes for coherent oscillation of YMnO 3[39].Reprinted with permission from Jang et al.[39].Copyright 2010, American Institute of Physics.(b) Temperature dependence oscillation period in Δ/ at various wavelengths of HoMnO 3 .Inset: the evolution of the oscillation component in Δ/ at various temperatures[38].Figures were reproduced with permission from Shih et al.[38].
2 a61] 1 (d  /d  and d  /d  2 − 2 ) states and one singlet  1 (d 3 2 − 2 ) state.In general, optical spectroscopy is one of the powerful tools for investigating the electronic band structure in materials.Therefore, this structural difference between the hexagonal and orthorhombic phases could be disclosed by the optical spectra[60,61].
Kritayakirana et al. in 1969ramids, in contrast with the MnO 6 octahedron of orthorhombic structure which split to  2 and   , the crystal field symmetry leads the quintet d levels of Mn 3+ ion splits into two low lying doublets of 3.1.LinearOptical Absorption Spectra.The first optical measurements in hexagonal YMnO 3 were performed byKritayakirana et al. in 1969 Phonons.Coherent acoustic phonons have been observed in many kinds of hexagonal