# Broadband Terahertz Spectroscopy of Phonon-Polariton Dispersion in Ferroelectrics

## Abstract

**:**

## 1. Introduction

^{2}k

^{2}/(2πν)

^{2}, where ν, k, c, and ε(k,ν) are the frequency, wavevector, light velocity, and dielectric constant, respectively. Figure 1 shows the polariton dispersion relation when two infrared active optical modes exist, where ν

_{TO1}= 200 cm

^{−1}, ν

_{LO1}= 230 cm

^{−1}, ν

_{TO2}= 660 cm

^{−1}, and ν

_{LO2}= 830 cm

^{−1}. (TO and LO denote transverse optic and longitudinal optic, respectively.) When the polariton wavevector k goes to zero, the lower branch tends to ν = ck/2π$\sqrt{\epsilon \left(0\right)}$ (OR in Figure 1), where ε(0) is the dielectric constant at the lowest frequency limit, and the middle and upper branches tend to ν = ν

_{LO1}and ν = ν

_{LO2}, respectively. When the polariton wavevector k goes to infinity, the lower and middle branches tend to ν = ν

_{TO1}and ν = ν

_{TO2}, respectively, whereas the upper branch tends to ν = ck/2π$\sqrt{\mathsf{\epsilon}(\infty )}$ (OQ in Figure 1), where ε(∞) is the dielectric constant at the highest frequency limit.

_{i}, and k

_{s}are the wavevectors of the polariton, incident light, and scattered light, respectively, and ν, ν

_{i}, and ν

_{s}are the frequencies of the polariton, incident light, and scattered light, respectively. The magnitude of the polariton wavevector k is given by the magnitude of the wavevectors of the incident light k

_{i}, scattered light k

_{s}, and the scattering angle θ between them:

^{2}= k

_{i}

^{2}+k

_{s}

^{2}− 2k

_{i}k

_{s}cosθ.

_{1}(z) symmetry phonon polariton is observed through the diagonal Raman tensor component R

_{cc}for the point group C

_{3v}. The Raman scattering spectra of the A

_{1}(z) symmetry phonon polariton can be measured at the forward scattering geometry, a(cc)a + ∆b, where ∆b means the small deviation of the direction from the a axis to the b axis. The magnitude of the polariton wavevector k with A

_{1}(z) symmetry is given by

^{2}= (2πn

_{e}/λ

_{i})

^{2}+ (2πn

_{e}/λ

_{s})

^{2}− 2(2πn

_{e})

^{2}/λ

_{i}λ

_{s}cosθ ≈ 8π2n

_{e}

^{2}(1 – cosθ)/λ

_{i}

^{2}≈ 4π

^{2}n

_{e}

^{2}θ

^{2}/λ

_{i}

^{2}, for k <<1,

_{i}and λ

_{s}are the wavelengths of the incident and scattered light, respectively, and n

_{e}is the refractive index of the extraordinary ray. According to Equation (2), observation of the polariton down to k = 0 is possible if the intense elastic scattering is well removed during the measurement. In Figure 1, the dotted lines show the observable region of the polaritons of a diagonal Raman tensor component by forward Raman scattering experiments, and it is impossible to observe the upper and lower k regions of the middle branches. Only infrared spectroscopy can cover all the regions of the polariton dispersion curves.

_{cb}. The Raman scattering spectra were measured at the forward scattering geometry, a(cb)a + ∆b. The magnitude of the polariton wavevector with E(x,y) symmetry is given by

^{2}= (2πn

_{e}/λ

_{i})

^{2}+ (2πn

_{o}/λ

_{s})

^{2}− 2(2πn

_{e}/λ

_{i})(2πn

_{o}/λ

_{s})cosθ,

_{o}and n

_{e}are the refractive indices of the ordinary and extraordinary rays, respectively, and λ

_{i}and λ

_{s}are the wavelengths of the incident and scattered light, respectively. Generally, n

_{o}≠ n

_{e}, and this birefringence, ∆n = n

_{o}− n

_{e}, causes the lowest limit of the observable k. Actually, when the scattering angle = 0, then the lowest value of k

_{min}is given by

_{min}

^{2}= {2πn

_{e}/λ

_{i}− 2πn

_{o}/λ

_{s}}

^{2}≈ 4π

^{2}(∆n)

^{2}λ

_{i}

^{2}≠ 0.

_{1}(x) symmetry of a ferroelectric KNbO

_{3}crystal related to an off-center Raman tensor component at the forward scattering geometry b(ca)b + ∆c, the lowest-frequency limit was about 190 cm

^{−1}for the lowest scattering angle of 0.6°, where a, b, and c are the orthorhombic coordinates [9]. According to such a condition, Raman scattering and impulsive stimulated Raman scattering have limitations in the region of polariton dispersions [10,11,12].

## 2. THz Dynamics of Ferroelectrics Studied by THz-TDS

#### 2.1. THz-TDS

^{−1}is poor. Another disadvantage is that it measures only a reflectance or transmittance spectrum by the lack of coherence of light sources. Therefore, the determination of the real and imaginary parts of a dielectric constant has the uncertainty caused by a Kramers–Kronig transformation. Actually, the discrepancy of the mode frequency of a ferroelectric soft mode has been reported in BaTiO

_{3}between far-IR and hyper-Raman scattering. In contrast, recently the generation of a coherent terahertz wave radiation has become possible by recent progress in a femtosecond pulse laser. The combination of the compact photoconductive antennas driven by femtosecond laser pulses enables terahertz time-domain spectroscopy [15]. By the measurement of both the amplitude and phase of the transmitted terahertz waves of the time-gated coherent nature, the accurate determination of both the real and imaginary parts of a dielectric constant in a terahertz range is possible. THz-TDS enables the studies of various kinds of dispersion relations of elementary excitations in condensed matter. For example, the dispersion curves of the electromagnetic waves related to photonic band structure have been determined by the measurement of the phase delay as a function of the incident frequency [16]. As for the dispersion relation of phonon polaritons, it is also possible to measure the phase delay by the polaritons [17].

#### 2.2. Bismuth Titanate

_{4}Ti

_{3}O

_{12}(BIT) with a bismuth layered structure is one of the most important key materials for FeRAM due to its low fatigue for polarization switching. The crystal system of BIT is monoclinic with the point group m at room temperature. It undergoes a ferroelectric phase transition at the Curie temperature T

_{C}= 948 K, and a high-temperature paraelectric phase is tetragonal with the point group 4/mmm. In a ferroelectric phase, the clear evidence of a displacive nature has been reported [18]. The underdamped soft optic mode has been observed by Raman scattering at 28 cm

^{−1}and at room temperature. This soft mode showed remarkable softening toward the T

_{C}upon heating from room temperature, and its damping factor significantly increased toward the T

_{C}. In the polar monoclinic phase, the optical phonon modes (the A’(x,z) and A”(y) modes) were both infrared and Raman-active, where a mirror plane was perpendicular to the crystallographic y axis. The soft optic mode had the A’(x,z) symmetry in which the coordinate z was parallel to the c axis. The Raman scattering spectra of the optical modes with A’(x,z) and A”(y) symmetries are shown in Figure 2. The intense peak at 28 cm

^{−1}observed in the A’(x,z) spectrum (L1 in Figure 2) was a ferroelectric soft mode. In the A”(y) spectrum, the lowest-frequency TO mode was observed at 32 cm

^{−1}(L2 in Figure 2). These lowest TO modes, denoted by L1 and L2, were strongly coupled to a photon, as shown in Figure 3.

_{a}(ν) and ε

_{b}(ν) using the light polarization parallel to the a axis (E//a) and b axis (E//b), respectively. From the measurements of two polarization directions, two different low-frequency polariton branches with A’(x,z) and A”(y) symmetries were clearly observed down to 3 cm

^{−1}, as shown in Figure 3a,b. This is the first observation of the dispersion relation of phonon polaritons by the use of THz-TDS.

#### 2.3. Barium Zirconate

_{3}(BZO)) with a perovskite structure is such a quantum paraelectric crystal. It has many technologically important properties such as a high lattice constant, a high melting point, a low thermal expansion coefficient, low dielectric loss, and low thermal conductivity. Therefore, BZO is a technologically important material for many kinds of applications. However, the structural instability and symmetry lowering of cubic BZO are still unknown, and many theoretical and experimental studies have been recently reported [20]. Different from most perovskite oxide ferroelectrics, BZO does not undergo any structural phase transition at ambient pressure, and thus its cubic symmetry is believed to be invariant down to 2 K [21]. Upon cooling from high temperatures, its dielectric constant gradually increases, while it does not diverge down to 0 K. The reciprocal dielectric constant goes to zero toward a negative temperature. Therefore, BZO belongs to incipient ferroelectrics, and all the optic modes are Raman-inactive.

^{−1}. The phonon frequency of the second-lowest-frequency TO2 mode was determined to be 125 cm

^{−1}, which was in agreement with the results of recent reflectivity measurements [24] and first-principles calculations [25]. The loss function Im(1/ε) was also calculated to determine the longitudinal optic mode frequency from the observed dielectric constant. The noticeable LO-TO splitting was not observed near 65 cm

^{−1}, whereas the LO2 mode appeared at around 222 cm

^{−1}. The remarkable softening of the TO1 mode frequency was found upon cooling, as shown in Figure 4b. Since the soft optic mode in a paraelectric phase is always Raman-inactive by the existence of a center of symmetry, far-IR studies are very important for observing a ferroelectric soft mode.

_{3}was determined at 8 K [23]. The remarkable resonance of polariton dispersion near the TO1 and LO1 mode frequencies was not observed, whereas the linear relation between the polariton wavevector and the polariton frequency nearly held below the TO1 mode frequency. In a theoretical study of ferroelectrics with strong anharmonicity, the resonances of polariton dispersion occurred by the cross-anharmonic couplings between different normal-mode lattice vibrations. However, such a strong anharmonic coupling between the soft mode and other optical modes was not observed in a BaZrO

_{3}crystal reflecting the quantum paraelectric nature.

## 3. Polariton Dispersion Studied by Far-IR Spectroscopy

#### 3.1. Far-IR Spectroscopy

_{3}in a paraelectric phase, the result by far-IR spectroscopy using a conventional FTIR spectrometer reported the stop of softening of the Cochran mode at about 60 cm

^{−1}and 100 °C above the Curie temperature, T

_{C}= 130 °C [26]. In contrast, a study using hyper-Raman scattering observed the softening of a soft mode toward T

_{C}at least down to 11 cm

^{−1}[27]. This significant discrepancy in the results between far-IR spectroscopy and hyper-Raman scattering has been considered in the problem of the analysis of overdamped modes in FTIR measurements. The ferroelectric instability of BaTiO

_{3}originated from not only the displacive, but also the order-disorder nature related to the eight-site model of the Ti ion at the B-site in an oxygen octahedron of the perovskite structure of which the tolerance factor is more than 1.0. The order-disorder nature caused the remarkable frequency dispersion of the dielectric constant near the low-frequency limit of the FTIR measurement. In contrast, the tolerance factor of quantum paraelectric SrTiO

_{3}with a perovskite structure was 1.0, and it meant an ideally packed structure. Therefore, the rattling of the Ti ions at the B-site was well suppressed, and the order-disorder nature was negligible. Therefore, it is possible to determine a reliable dielectric constant by a standard FTIR measurement.

_{3}, the determination of a dielectric constant without using the Kramers–Kronig transformation is necessary. To determine a reliable dielectric constant, a far-infrared spectroscopic ellipsometry (FIRSP) system has been developed through the combination of a far-infrared spectrometer and ellipsometry [14]. FIRSP is a combination between a Michelson interferometer and an ellipsometer with a rotating analyzer. A far-infrared spectrum is measured by the light source of a high-pressure mercury lamp and the detector of an Si bolometer unit. The reflected light from a sample to be observed is elliptically polarized, and the p- and s-polarized lights are separately measured using a rotating wire grid analyzer. With FIRSP, the accurate determination of the real and imaginary parts of a dielectric constant without any uncertainty is possible in the frequency range from 40 to 700 cm

^{−}

^{1}. In most cases, the frequency range of most optical modes of ABO

_{3}-type oxide ferroelectrics is in this frequency range, and therefore it is certain that FIRSP is a powerful experimental method to study the broadband dispersion relation of the phonon polariton.

#### 3.2. Strontium Titanate

_{3}(STO)) with a perovskite structure is 1.0. STO is known as the typical quantum paraelectric, and ferroelectric instability is suppressed by quantum fluctuations at very low temperatures [28]. The point group symmetry is a cubic m$\overline{3}$m with the center of symmetry at room temperature. The optical vibrational modes at the Г point of the Brillouin zone are 3T

_{1u}+ T

_{2u}, and all the optical modes are Raman-inactive. The T

_{2u}modes are called silent modes. Only the 3T

_{1u}modes are infrared-active and hyper-Raman-active. Therefore, the study of the 3T

_{1u}modes is possible using infrared spectroscopy, including THz-TDS and hyper-Raman scattering. The lowest-frequency T

_{1u}modes have been studied using far-IR spectroscopy [29] and THz-TDS [30,31,32].

_{1u}modes, the infrared reflectivity spectrum of a [001] STO plate was measured in the range from 30 to 1200 cm

^{−1}[33]. Figure 5 shows the dispersion relation of the phonon polariton of the T

_{1u}symmetry, determined by the infrared spectrum. The anti-crossing of a dispersion curve was clearly observed near the lowest TO mode frequency, ν

_{TO1}= 87 cm

^{−1}. The dispersion relation of a phonon polariton was also investigated by hyper-Raman scattering measurements using a forward scattering geometry by Denisov et al. [34] and Inoue et al. [35]. For a comparison with the results of infrared spectroscopy, their results of hyper-Raman scattering [34,35] are also plotted in Figure 5. Inoue et al. observed only the highest-frequency polariton dispersion, higher than the highest-frequency LO mode [35]. Denisov et al. observed both the lowest-frequency dispersion curve, lower than the lowest TO mode frequency ν

_{TO1}= 87 cm

^{−1}, and the highest-frequency dispersion curve, higher than the highest LO mode frequency ν

_{LO3}= 788 cm

^{−1}[34]. The polariton dispersion relations determined by the hyper-Raman scattering measurement in the frequency range below 87 cm

^{−1}and above 788 cm

^{−1}[34,35] were in agreement, within experimental uncertainty, with the results of the infrared reflection measurement [33].

_{i}and ν

_{s}are the frequencies of the incident laser light and the scattered light from a sample to be observed, respectively. The frequencies of ν = 2ν

_{i}− ν

_{s}and ν

_{s}are approximately equal to the double of ν

_{i}. According to Equation (7), the dispersion relation was observable only in a quite limited region due to the birefringence between the refractive indices of the fundamentals of an incident light and the second harmonic wavelengths of scattered light. Observation of a low-frequency polariton with a small ${\overrightarrow{k}}_{s}$ was especially impossible due to this birefringence, and therefore hyper-Raman scattering of phonon polaritons was not suitable to study the soft mode related to lattice instability.

#### 3.3. Lithium Niobate

_{3}(LN)) with an ilmenite structure was discovered by Matthias and Remeika in 1949 [36]. Currently, LN is the most technologically important ferroelectric crystal, with significant functional properties. In particular, its colossal piezoelectric, electro-optic, and nonlinear optical coefficients have been applied to various devices such as SAW (surface acoustic wave) filters, SHG (second harmonic generation) converters, tunable solid lasers, and THz generators [37,38]. LN undergoes a ferroelectric transition from a paraelectric phase with the space group R$\overline{3}$c into a ferroelectric one with R3c at T

_{C}= 1483 K, which depends on the ratio between the lithium and niobium contents. A spontaneous polarization appears along the c axis [39,40].

_{1}+ 9E + 5A

_{2,}

_{1}and E modes correspond to Raman- and infrared-active polar phonons, whereas the A

_{2}modes are silent modes that are Raman- and infrared-inactive.

^{−1}and determined the frequency of eight E and four A

_{1}modes [41]. For example, they reported the lowest A

_{1}(TO) mode at 187 cm

^{−1}. In 1967, Barker and Loudon studied the E and A

_{1}modes using IR and Raman spectroscopies. In their study, the lowest A

_{1}(TO) mode was observed at 252 cm

^{−1}by Raman spectroscopy and at 248 cm

^{−1}by IR spectroscopy. To date, although many vibrational measurements have been reported, these results remain controversial [42,43,44,45,46,47]. Recently, theoretical studies of the optical modes based on first-principles calculations have been reported. In 2000, Caciuc et al. reported the lowest A

_{1}(TO) mode at 208 cm

^{−1}[48], whereas in 2002, Veithen and coworkers reported the lowest A

_{1}(TO) mode at 243 cm

^{−1}[49,50]. The study by Sanna et al. indicated the lowest A

_{1}(TO) mode at 239 cm

^{−1}[51]. Therefore, the calculated values in the theoretical studies were also controversial.

^{−1}[43], which is inconsistent with the result of first-principles calculations [48,49,50,51]. Other infrared spectroscopic studies did not observe E(TO6) [41,47]. Previous Raman studies reported eight to nine modes [42,43,44,45], and some mode frequencies were different from the results of recent theoretical calculations [49,50,51].

## 4. Summary

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Dispersion relation of a phonon polariton with two optical modes. The line of OR denotes ν = ck/2π$\sqrt{\mathsf{\epsilon}\left(0\right)}$. The line of OQ denotes ν = ck/2π$\sqrt{\mathsf{\epsilon}(\infty )}$. The dotted lines show the observable region of the forward Raman scattering with a fixed scattering angle.

**Figure 2.**Low-frequency Raman spectra of a Bi

_{4}Ti

_{3}O

_{12}crystal for the modes with A’(x,z) and A”(y) symmetries [18].

**Figure 3.**Low-frequency polariton dispersion relations for (

**a**) A’(x,z) and (

**b**) A”(y) modes of a Bi

_{4}Ti

_{3}O

_{12}crystal [17]. The line of OR denotes ν = ck/2π$\sqrt{\mathsf{\epsilon}\left(0\right)}$

**Figure 5.**Polariton dispersion relations on real parts of the polariton wavevector of an SrTiO

_{3}crystal. TO

_{j}(j = 1,2,3) and longitudinal optic (LO)

_{j}(j = 1,2,3) denote the three TO and three LO modes of the T

_{1u}symmetry, respectively [33]. The line of OQ denotes ν = ck/2π$\sqrt{\mathsf{\epsilon}(\infty )}$. The results of FTIR [33] and THz-TDS [32] are shown by the solid line and the closed circles, respectively. The values of the hyper-Raman scattering reported in [34,35] are shown by the closed triangles and closed diamonds for comparison to the FTIR values.

**Figure 6.**Dispersion relation of phonon polaritons of an LiNbO

_{3}crystal with E(x) symmetry. The open circles and closed circles denote the observed values by FIRSP [52] and THz-TDS [53], respectively. The line of OQ denotes ν = ck/2π$\sqrt{\mathsf{\epsilon}\left(0\right)}$. The line of OR denotes ν = ck/2π$\sqrt{\mathsf{\epsilon}(\infty )}$. Dotted lines denote the calculated dispersion with no damping.

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Kojima, S.
Broadband Terahertz Spectroscopy of Phonon-Polariton Dispersion in Ferroelectrics. *Photonics* **2018**, *5*, 55.
https://doi.org/10.3390/photonics5040055

**AMA Style**

Kojima S.
Broadband Terahertz Spectroscopy of Phonon-Polariton Dispersion in Ferroelectrics. *Photonics*. 2018; 5(4):55.
https://doi.org/10.3390/photonics5040055

**Chicago/Turabian Style**

Kojima, Seiji.
2018. "Broadband Terahertz Spectroscopy of Phonon-Polariton Dispersion in Ferroelectrics" *Photonics* 5, no. 4: 55.
https://doi.org/10.3390/photonics5040055