Terahertz Spectroscopy in Advanced Materials Science

Abstract

Materials science is the interdisciplinary field to study material properties and their functionality on the basis of physics, chemistry, metallurgy, and mineralogy. Vibrational spectroscopy such as infrared spectroscopy and Raman spectroscopy is a powerful tool to investigate characteristic atomic vibrations. Especially, in the terahertz frequency range, vibrational modes are related to collective atomic vibrations reflecting interatomic/molecular interactions, characteristic units, and medium range order. Recent progress of terahertz vibrational spectroscopy using terahertz-time-domain spectroscopy, terahertz time-domain spectroscopic ellipsometry, and far-infrared spectroscopy is reviewed in advanced materials science on glassy and crystalline pharmaceuticals, ferroelectrics, and polar metallic materials. Using the terahertz spectra, phonons, polaritons, and conduction electrons of these materials are discussed.

1. Introduction

In materials science, vibrational spectroscopy is the important and valuable tool to investigate the dynamical properties related to atomic bonds, interactions, and structures. Infrared (IR) and Raman scattering spectroscopies have been extensively used to study various elementary excitations such as phonon, polariton, magnon, exciton, plasmon, boson peak, etc. The most of IR studies reported the frequency-dependent absorbance or transmittance, while the real and imaginary parts of dielectric constants were not reported well. The coherent terahertz generation technique using a femtosecond pulse laser enables the unique determination of a complex dielectric constant, and terahertz time domain spectroscopy has attracted much attention [1, 2].

The vibrational modes of a material are divided into two groups, namely internal and external modes. The internal modes are related to the vibration of a molecule or a structural unit, and its mode frequency is usually more than 6 THz. The external modes are related to the vibration of more than two molecules or lattice vibration with medium or long range order, and its mode frequency is usually less than 6 THz. Therefore, the external modes have been studied by THz spectroscopy. In this chapter, Recent progress of terahertz vibrational spectroscopy using THz-time-domain spectroscopy (THz-TDS) [3], THz time-domain spectroscopic ellipsometry (THz-TDSE) [4], and far-infrared spectroscopy is reviewed in advanced materials science on the following three kinds of materials. (1) Pharmaceuticals: studies of terahertz vibrational spectroscopy on polymorphism of crystalline states and glassy states on typical pharmaceuticals [5]. (2) Ferroelectric materials: studies of terahertz vibrational spectroscopy on optical phonons, ferroelectric soft mode, and phonon-polariton dispersion relation on typical ferroelectric crystals for optical application [6]. (3) Polar metallic materials: studies of terahertz vibrational spectroscopy on the correlation between polar lattice instability and free carrier electrons injected by heterovalent doping to quantum paraelectrics [7].

2. Terahertz spectroscopy in advanced materials science

The structure of a crystal has translational symmetry and the X-ray diffraction patterns of powdered crystals show many sharp peaks. While, glassy, amorphous, and non-crystalline solids have the disordered structure with no translational symmetry and the X-ray diffraction pattern consists of only diffuse peaks by the lack of three dimensional periodicity. When the structure of a crystal has a center of symmetry or centrosymmetric, the mutual exclusion principle holds between Raman and IR selection rules. Raman active modes are IR inactive, and IR active modes are Raman inactive. Some vibrational modes are only Raman active, while other modes are only IR active. Therefore, both infrared and Raman measurements are important. In contrast, the structures of non-centrosymmetric crystals, glassy or non-crystalline materials do not have a center of symmetry, and there is no mutual exclusion principle [2].

2.1 Pharmaceuticals

The most of pharmaceuticals have polymorphic and polyamorphic features originated from multi-basin structure in the energy landscape model. The energy of these basins is nearly degenerate, and the inter-basin transition easily occurs by the change of temperature, pressure, stress, and chemical conditions. The vibrational properties in the THz frequency range are very sensitive to the difference between polymorphic and polyamorphic states. Therefore, in the study of polymorphic and polyamorphic natures in pharmaceuticals, the measurements of both THz-TDS and Raman scattering spectroscopy are important. These two spectroscopic methods have the complimentary selection rules which give new insights into the detailed information on dynamical and static properties related to crystalline and noncrystalline structures, medium range order, and interatomic/intermolecular interactions [5].

THz transmission spectra were measured by the conventional transmission THz-TDS system (Tochigi Nikon, RT-10,000), where the low temperature grown GaAs photoconductive antennas were used for both the emitter and detector in the frequency range from 0.2 to 4.0 THz. The Raman scattering spectra were measured in the frequency range from 0.2 to 36 THz using a double-grating spectrometer with the photon-counting system (Horiba-JY, U-1000). The imaginary part of Raman susceptibility, χ”(ν), is extracted by the equation,

$${\chi }^{”}\left(\nu \right)=\frac{I\left(\nu \right)}{I_0\left\{n\left(\nu \right)+1\right\}},n\left(\nu \right)=\frac{1}{\mathit{exp}\left(\frac{\mathit{h\nu }}{k_{B}T}\right)-1},$$E1

Here I(ν) and n(ν) are the scattering intensity and the Bose-Einstein factor, respectively. I₀ is a constant which depends on the experimental condition.

2.1.1 Vibrational properties of glassy and crystalline pharmaceuticals

Indapamide (C₁₆H₁₆ClN₃O₃S, IND) is 4-chloro-N-(2-methyl-2,3-dihydroindol-1-yl)-3-sulfamoylbenzamide. On the treatment of hypertension, the commercially available type IND exerts spasmolytic effects on blood vessels and reduces the blood pressure. Its glass-forming tendency is relatively strong. IND undergoes a liquid-glass transition at a glass transition temperature, T_g = 376 K, which is relatively higher than room temperature [8]. The unit cell of a crystalline phase of commercial type IND contains four IND molecules (Z = 4). The crystal framework contains two types of cavities. The cavity encapsulates water molecules in non-stoichiometric. Since the water molecules are weakly bounded in these cavities, they easily come out and in from the IND crystalline framework [9].

The thin plates of crystalline and glassy samples prepared by the melt-quench method were measured by THz-TDS between 0.2 and 3.0 THz at room temperature [10]. Figure 1a and b show the real and imaginary parts of the dielectric constants of the commercial type glassy and crystalline states of IND. In the observed frequency range, several sharp phonon peaks were observed in a crystalline sample, these modes can be attributed to inter- or intra-molecular vibrational modes in a crystalline state. However, no sharp phonon peak was observed in a glassy state and the broad response reflects the distribution of bond lengths and bond angles in the random structure of a glassy state.

Figure 1.

(a) Real and (b) imaginary parts of dielectric constant of crystalline and glassy indapamide.

For the comparison of the THz-TDS spectra with the Raman scattering spectra, the imaginary parts of Raman susceptibility, χ”(ν), of crystalline and glassy states are shown in Figure 2. As similar with the THz-TDS spectra, a crystalline sample shows several sharp phonon peaks, while a glassy sample shows only broad response and no sharp phonon peak. In a crystalline state, the discrepancy of the phonon peak frequencies was clearly observed between ε”(ν) observed by THz-TDS in Figure 1b and χ”(ν) observed by Raman scattering in Figure 2. The origin of the discrepancy is the mutual exclusion principle between Raman and IR activities [11]. Consequently, it is concluded that the point symmetry of the crystalline IND has a center of symmetry. This result is consistent with the monoclinic crystal structure with the point group 2/m determined by the X-ray diffraction experiment [12]. The existence of a center of symmetry was also examined by THz-TDS and Raman spectroscopy in crystalline γ-form indometachine [11] and crystalline RS-ketoprofen [13].

Figure 2.

Imaginary parts of Raman susceptibility of crystalline and glassy indapamide.

In contrast, ε”(ν) and χ”(ν) spectra show very broad response and no sharp peak below 3.0 THz in a glassy state. If the frequency range is extended to the IR region, many internal modes can be observed. In a glassy glycerol, the temperature dependence of the intermolecular hydrogen bond length was estimated from the frequency of O-H stretching mode at about 105 THz [14].

2.1.2 Low-energy excitation in a glassy state

As the common nature of the low-energy dynamics of glassy, amorphous, and noncrystalline materials, a broad and asymmetric peak at few THz has been observed by Raman scattering, far-infrared spectroscopy, neutron and X-ray inelastic scattering [15, 16]. It is called a boson peak and a thermal boson peak has been also observed as a broad peak in the temperature dependence of C_p/T³ curve at a few Kelvin, where C_p is the heat capacity [17]. Both boson peaks are related to g(ν)/ν², where g(ν) is the vibrational density of states (VDoS), g(ν). The strong correlation between the boson peak energy and the shear modulus has been reported [18]. The molecular dynamical simulation reported that a boson peak is originated from the Ioffe-Regal limit of transverse acoustic mode and its peak energy has a linear relation with the shear modulus [19].

In the far-IR spectroscopy, a boson peak is observed by,

$$\frac{\kappa \left(\nu \right)}{\nu }\propto C_{\mathit{IR}}\left(\nu \right)\frac{g\left(\nu \right)}{{\nu }^2}.$$E2

Here, κ(ν) is the real part of a refractive index, and C_IR (ν) is the IR-vibration coupling constant [20]. The boson peak was fitted by the following log-normal function under the assumption that the IR-vibration coupling constant does not depend on frequency.

$$\frac{k\left(\nu \right)}{\nu }=I_{0}exp\left\{-\frac{{\left[\mathit{ln}\left(\nu /{\nu }_{\mathit{BP}}\right)\right]}^2}{2{\sigma }^2}\right\}.$$E3

Here I₀ is a constant, ν_BP and σ are the boson peak frequency and boson peak width, respectively. Figure 3a and b show boson peaks at room temperature observed by THz-TDS on SiO₂ glass (ν_BP ≈ 0.9 THz) [21] and glassy indapamide (ν_BP ≈ 0.4 THz) [10], respectively. The observation of boson peaks by THz-TDS was reported in various glasses such as PMMA [22], indomethacin [11], glucose [23], and lysozyme [24]. The study on boson peaks of glassy materials will give new insights into the understanding of disordered materials with no translational symmetry.

Figure 3.

Boson peaks observed by THz-TDS. (a) SiO₂ glass (ν_BP = 0.9 THz), (b) glassy indapamide (ν_BP = 0.4 THz) at room temperature. Solid lines show fitted values by Eq. (2). Solid circles denote observed values.

2.2 Quantum paraelectrics

The ABO₃-type perovskite oxides have been extensively used for potential applications such as capacitor, ferroelectric random access memory, piezoelectric actuator, ultrasonic transducer, pyroelectric sensor, and surface acoustic wave filter. BaMO₃ (M = Ti, Zr) compounds are very important in pure and applied sciences [25]. Barium titanate, BaTiO₃ has been widely used as capacitor. BaTiO₃ undergoes a ferroelectric phase transition at 403 K from a prototypic cubic phase with the point group $m\overline{3}m$ to the tetragonal phase with 4 mm, where $\overline{3}$ and m are three-fold rotation-inversion axis and mirror plane, respectively. Its physical properties have been extensively studied.

Barium zirconate, BaZrO₃ has attracted much attention by a high-temperature proton conductor [26] and a dielectric material for use in wireless communications applications [27]. BaZrO₃ does not undergo any phase transition down to 2 K [28]. Its dielectric constant increases on cooling, while a ferroelectric instability is suppressed by quantum fluctuations at very low temperatures. It is called quantum paraelectricity. As quantum paraelectrics, SrTiO₃, KTaO₃, and CaTiO₃ are known. The cubic perovskite phase has a center of symmetry with the point group $m\overline{3}m$, and infrared-active ferroelectrtic soft modes are Raman inactive. Although the low-frequency infrared-active modes play a dominant role in ferroelectric instability, the experimental study in the far-infrared spectroscopy has been not enough.

The real and imaginary parts of a dielectric constant of a BaZrO₃ crystal was measured between 8 and 300 K by the transmission of THz-TDS [29, 30]. In Figure 4a, the peak of the imaginary part at 1.9 THz was observed at 8 K. The TO mode frequency was determined by the fitting the imaginary part of a dielectric constant ε(ν) using the following damped harmonic oscillator (DHO) model,

Figure 4.

(a) Imaginary part of dielectric constant of a BaZrO₃ crystal at 8 K. (b) Temperature dependence of the square of the TO1 mode frequency. The solid line shows the fitted curve by the Barrett’s formula [29, 30].

$$\epsilon \left(\nu \right)=\epsilon \left(\infty \right)+\sum _j\frac{∆{\epsilon }_j{\nu }_j^2}{{\nu }_j^2-{\nu }^2-\mathit{i\nu }{\mathrm{\Gamma }}_j},$$E4

where $\epsilon \left(\infty \right),∆{\epsilon }_j,{\nu }_j,$ and ${\mathrm{\Gamma }}_j$ are the high-frequency limit of a dielectric constant, oscillator strength, frequency and damping factor of j th oscillator mode, respectively.

The temperature dependence of the squared mode frequency of the lowest TO1 mode is shown in Figure 4b. The squared soft mode frequency ${\nu }_{\mathit{\text{TO}}}^2$ of a quantum paraelectric obeys the following Barrett’s formula [31].

$${\nu }_{\mathit{\text{TO}}}^2=A\left[T_s\mathit{coth}\left(\frac{T_S}{T}\right)-T_{01}\right].$$E5

Here, A, T_S, and T₀₁ are constant, saturation temperature, and classical Curie temperature, respectively. The lowest frequency TO1 infrared-active phonon mode exhibits a significant softening, while the quantum effects lead to saturation of the soft mode frequency.

3. Phonon-polariton dispersion relation

The coupled excitation between photons and other elementary excitations is called polariton [32, 33]. The remarkable frequency vs. wavevector dispersion relation of polariton has attracted much attention in basic science to clarify the vibrational dynamics of condensed matters such as phonons, excitons, and plasmons. Polar phonon modes, which are infrared active, couple to photons at frequencies and the mixed excitation is called phonon-polariton. Phonon polaritons have been used for optical applications such as a tunable Raman laser, tunable terahertz (THz) radiation source. The dispersion relation of phonon-polariton has been extensively studied by infrared spectroscopy, spontaneous, hyper Raman and stimulated Raman scattering spectroscopy [34]. However, Raman inactive phonon-polariton cannot be measured by spontaneous and stimulated Raman scattering spectroscopy. The phonon-polariton of Raman inactive phonons can be measured only by Raman spectroscopy and hyper-Raman scattering spectroscopy. Therefore, the study of Raman inactive polaritons has been insufficient.

According to Huang’s analysis [31], the dispersion relation of phonon polariton was defined by the equation,

$$\epsilon \left(\overrightarrow{k},\nu \right)=\frac{c^2{k}^2}{{\left(2\mathit{\pi \nu }\right)}^2},$$E6

where ν, $\overrightarrow{k}$, c, and ε($\overrightarrow{k}$,ν) are the polariton frequency, polariton wavenumber, light velocity, and dielectric constant of a medium, respectively. Figure 5 shows the dispersion relation of phonon polariton on the existence of two infrared active optical modes with the mode frequencies, ν_TO1 = 6.0 THz, ν_LO1 = 6.9 THz, ν_TO2 = 19.8 THz, and ν_LO2 = 24.9 THz. As the polariton wavenumber k tends to zero, the lower polariton branch tends to ν = ck/2π$\sqrt{\epsilon \left(0\right)}$, where ε(0) is the lowest frequency limit of a dielectric constant, while the middle and upper branches tend to LO1 mode frequency, ν_LO1, and LO2 mode frequency, ν_LO2, respectively. When the polariton wavenumber k tends to infinity, the lower and middle branches tend to TO1 mode frequency, ν_TO1, and TO2 mode frequency ν = ν_TO2, respectively, while the upper branch tends to ν = ck/2π$\sqrt{\epsilon \left(\infty \right)}$, where ε(∞) is the highest frequency limit of a dielectric constant.

Figure 5.

Dispersion relation of phonon-polariton with two optical modes. The lines AB and AC denote ν = ck/2π$\sqrt{\epsilon \left(0\right)}$ and ν = ck/2π$\sqrt{\epsilon \left(\infty \right)}$, respectively. The dotted lines show the observable region of the forward Raman scattering with a fixed scattering angle.

In the observation of phonon-polariton by Raman scattering, the conservation law of wavevectors holds,

$$\pm \overrightarrow{k}\left(\mathrm{\nu }\right)={\overrightarrow{k}}_i\left({\nu }_i\right)-{\overrightarrow{k}}_s\left({\nu }_s\right),$$E7

where $\overrightarrow{k}$, ${\overrightarrow{k}}_i$, and ${\overrightarrow{k}}_s$ are the wavevectors of polariton, incident light, and scattered light, respectively. ν, ν_i, and ν_s are the frequencies of polariton, incident light, and scattered light, respectively. The polariton wavenumber k is given by the wavenumbers of the incident light k_i, scattered light k_s, and the scattering angle θ between them.

$$k^2={k_{\mathrm{i}}}^2+{k_{\mathrm{s}}}^2-2k_{\mathrm{i}}{k}_{\mathrm{s}}cos\theta .$$E8

The dispersion relation of polariton is also determined by the forward Raman scattering at very small scattering angles as shown in Figure 5. While, it is impossible to measure the dispersion relation in the region, ν > ck/2π$\sqrt{\epsilon \left(0\right)}$ by Raman scattering [35]. In contrast, the measurement by infrared spectroscopy has no such restriction and covers all area of ν vs.$\overrightarrow{k}$ space.

3.1 Raman active phonon-polariton

3.1.1 Bismuth titanate

On the ferroelectric memories, bismuth layer-structure materials have currently become important in the research and development of ferroelectric devices. These compounds are also very important as the most suitable samples to study the mechanism in ferroelectric layer-structures. Bismuth titanate, Bi₄Ti₃O₁₂ undergoes a ferroelectric phase transition from a paraelectric tetragonal phase with the point group 4/mmm to a ferroelectric monoclinic phase with the point group m at T_C = 948 K, where 4 and m are four-fold rotation axis and mirror plane, respectively [36]. The ferroelectric soft optic mode was observed by the Raman scattering. On heating from room temperature, the lowest frequency mode at 0.84 THz with A’(x,z) symmetry softens markedly and disappears above T_C [37].

In the determination of the dispersion relation of phonon-polariton by the transmission measurement of THz-TDS, the wavenumber k(ν) of phonon-polariton is calculated using the phase delay φ as a function of the polariton frequency ν by the following equation,

$$\mathrm{\phi }=\left\{k\left(\mathrm{\nu }\right)-\text{2πν}/\mathrm{c}\right\}\mathrm{d},k\left(\mathrm{\nu }\right)=\text{2π}\mathrm{\nu }n\left(\mathrm{\nu }\right)/\mathrm{c},$$E9

where c, d, and n(ν) are the light velocity, a thickness of a sample, and the real part of a refractive index of a sample, respectively.

The anisotropy of polariton dispersion relations between A’(x,z) and A”(y) symmetry modes was also successfully determined using the c plate simply by switching the polarization direction of an incident beam from E∥a to E∥b, respectively [38]. The observed polariton dispersion relations are consistently reproduced by the calculation using the following Kurosawa’s formula (Figure 6).

Figure 6.

Dispersion relations of (a) A’(x,z) symmetry and (b) A’(y) symmetry phonon-polaritons of a Bi₄Ti₃O₁₂ crystal.

$$\epsilon \left(k,\nu \right)=\frac{c^2{k}^2}{{\left(2\mathit{\pi \nu }\right)}^2}=\epsilon \left(\infty \right)\prod _{j=1}^N\frac{{\nu }_{\mathit{LOj}}^2-{\nu }^2}{{\nu }_{\mathit{TOj}}^2-{\nu }^2}$$E10

Since the measured frequency is below 3.0 THz, we used the following approximation to fit the data:

$$\frac{c^2{k}^2}{{\left(2\mathit{\pi \nu }\right)}^2}=\epsilon \left(1\right)\prod _{j=1}^2\frac{{\nu }_{\mathit{LOj}}^2-{\nu }^2}{{\nu }_{\mathit{TOj}}^2-{\nu }^2}.$$E11

The values of these parameters are listed in Tables 1 and 2. The observed values of polariton frequency were well fitted by Eq. (11), because the anharmonicity is negligible in a Bi₄Ti₃O₁₂ crystal at room temperature.

Mode	A’(x,z), E∥a	A”(y), E∥b
TO1	0.84 THz	1.08 THz
LO1	0.98 THz	1.26 THz
TO2	2.04 THz	2.55 THz
LO2	2.24 THz	2.96 THz

Table 1.

Fitting values of mode frequency in Eq. (11).

	A’(x,z), E∥a	A”(y), E∥b
$\epsilon \left(0\right)$	79.2	149.0
$\epsilon \left(1\right)$	49.0	81.0
$\epsilon \left(\infty \right)$	6.76	6.76

Table 2.

Fitting values of dielectric constant in Eq. (11).

3.1.2 Lithium niobate, LiNbO₃

Lithium niobate, LiNbO₃, has been used for many kinds of devices by their excellent piezoelectric, pyroelectric, and nonlinear optical properties. LiNbO₃ is a colorless uniaxial crystal and undergoes a ferroelectric phase transition at a high Curie temperature, T_C = 1210°C from paraelectric (R$\overline{3}$c with three-fold rotation-inversion axis, $\overline{3}$) to ferroelectric (R3c with three-fold rotation axis, 3) phases [39]. Lattice dynamics have been extensively studied by Raman scattering [40], infrared spectroscopy, and theoretical calculations. In a rhombohedral ferroelectric phase, the symmetry of the optical modes at the Γ point of Brillouin zone is given by.

$${\text{4A}}_1\left(z\right)+\text{9E}\left(x,y\right)+{\text{5A}}_2.$$E12

Here, A₁(z) and E(x,y) modes are Infrared and Raman active, while A₂ modes are silent modes, which are infrared and Raman inactive. The symmetry of a ferroelectric soft mode is A₁(z). According to the Raman scattering study, the temperature dependence of the lowest frequency A₁(z) mode became overdamped far below T_C. This fact indicates that the anharmonicity of a ferroelectric soft mode is very high even at room temperature. The Raman scattering spectrum of A₁(z) modes observed at the $a\left(\mathit{cc}\right)\overline{\mathrm{a}}$ backward scattering is shown in Figure 7 [41]. Therefore, the damping of A₁(z) modes is not negligible.

Figure 7.

Raman scattering spectrum of A₁(z) modes observed at the $a\left(\mathit{cc}\right)\overline{\mathrm{a}}$ backward scattering.

For the analysis of such anharmonic modes, it is necessary to discuss the damping of phonon using the imaginary part of a dielectric constant. When a dielectric constant, ε($\overrightarrow{k}$, ν), is a complex number, then a wavenumber of polariton, k or a frequency, ν, can be also a complex number. Since the infrared spectroscopy considers spatial decay, ν is defined as a real number, while k is defined as a complex number [32]. The complex wavenumber of polariton, k(ν), is defined by the equation,

$$k\left(\mathrm{\nu }\right)=k^{’}\left(\mathrm{\nu }\right)+\mathrm{i}{k}^{”}\left(\mathrm{\nu }\right).$$E13

Considering the following relation between the complex dielectric constant and the complex refractive index,

$$\mathrm{\epsilon }\left(k,\mathrm{\nu }\right)={\left\{n\left(k,\mathrm{\nu }\right)+\mathrm{i\kappa }\left(k,\mathrm{\nu }\right)\right\}}^2.$$E14

The polariton dispersion relation for the real and imaginary parts of the complex k(ν) was derived from Eqs. (13) and (14),

$$k^{’}\left(\mathrm{\nu }\right)=\text{2π}/\mathrm{c}n\left(k,\mathrm{\nu }\right),k^{”}\left(\mathrm{\nu }\right)=\text{2π}/\mathrm{c}\mathrm{\kappa }\left(k,\mathrm{\nu }\right)$$E15

The polariton dispersion of anharmonic A₁(z) modes of a congruent LiNbO₃ crystal was measured by FIRSP [41] as shown in Figure 8. The polariton dispersion relation was determined by the complex refractive constant measured in the condition E//c. Figure 8 shows the real and imaginary parts of polariton wavenumber versus the polariton frequency with A₁(z) symmetry. The result by forward Raman scattering measurements [43, 44] was also plotted for the comparison. The dotted lines in Figure 8 denote the calculated dispersion relation with no phonon damping. Closed open circles in Figure 8 denote the values observed by the forward Raman scattering measurement. These observed values of Raman scattering and FIRSP are in good agreement with those determined by the first principles calculations [45] within experimental uncertainty as shown in Table 3.

Figure 8.

Dispersion relations of phonon-polariton with real and imaginary parts of wavenumber of a LiNbO₃ crystal are shown in right and left hands sites, respectively. TOj and LOj (j = 1,2,3) denote the three TO and three LO modes of the T_1u symmetry, respectively [42]. Closed and open circles denote the observed values by THz-TDS [43] and Raman scattering [44, 45], respectively.

Mode	Far-IR ellipsometry	Raman scattering	First principles calculation [45]
A1(TO1)	7.54 THz	7.63 THz	7.18 THz
A1(TO2)	8.17 THz	8.35 THz	8.67 THz
A1(TO3)	9.97 THz	10.0 THz	10.6THz
A1(TO4)	19.0 THz	19.0 THz	18.3THz

Table 3.

A₁(TO) mode frequency.

3.2 Raman inactive phonon-polariton

The tolerance factor of perovskite SrTiO₃ (STO) is 1.0. The STO is known as the typical quantum paraelectric [46] as same as BaZrO₃ in Section 2.2. The space group symmetry is a cubic $\mathit{Pm}\overline{3}m$ with the center of symmetry at room temperature. The optical vibrational modes at the Γ point of the Brillouin zone are 3T_1u + T_2u. The T_2u modes are silent modes, which are infrared inactive and Raman inactive. The 3T_1u modes are Raman inactive while infrared active and hyper-Raman active. The T_1u modes were studied by the far infrared spectroscopy [47], THz-TDS [48, 49], and hyper-Raman scattering [50, 51].

For the study of the polariton dispersion relation of the three T_1u modes in the large frequency range up to 36 THz, the infrared reflectivity spectrum of a [001] STO plate was measured in the range from 0.9 and 36 THz as shown in Figure 9 [52]. The typical polariton dispersion curve was observed in the vicinity of the lowest TO1 mode at ν_TO1 = 2.6 THz. the results of hyper-Raman scattering [50, 51] were also plotted in Figure 9 for the comparison. Inoue et al. measured only the highest-frequency polariton dispersion higher than the highest-frequency LO3 mode at ν_LO3 = 23.7 THz [51]. While, Denisov et al. measured both the lowest-frequency dispersion curve lower than the lowest TO1 mode frequency and the highest-frequency dispersion curve higher than the highest LO3 mode [50]. The polariton dispersion measured by the hyper-Raman scattering measurements [50, 51] is in a good agreement within the experimental uncertainty with the results the infrared reflection measurement [52] in the frequency range below 2.6 THz and above 23.7 THz.

Figure 9.

Dispersion relations of phonon-polariton with real and imaginary parts of wavenumber of a SrTiO₃ crystal are shown in right and left hands sites, respectively. TOj and LOj (j = 1, 2, 3) denote the three TO and three LO modes of the T_1u symmetry, respectively [52]. The results of FTIR [52] and THz TDS [49] are shown by the solid line and the closed circles, respectively. The values of the hyper-Raman scattering [50, 51] are shown by the closed circles and closed diamonds.

In hyper-Raman scattering spectroscopy, the conservation law holds among the wave vectors of an incident, ${\overrightarrow{k}}_i$, scattered light, ${\overrightarrow{k}}_s$, and polariton, $\overrightarrow{k}$,

$$\pm \overrightarrow{k}\left(\mathrm{\nu }\right)=2{\overrightarrow{k}}_i\left({\nu }_i\right)-{\overrightarrow{k}}_s\left({\nu }_s\right),$$E16

where ν_i and ν_s are frequencies of incident and scattered light from a sample, respectively. The frequencies of ν = 2ν_i - ν_s and ν_s is approximately equal to a double of ν_i. According to the Eq. (16), the dispersion relation is observable only in the limited region due to the birefringence between the refractive indices of fundamental of an incident light and second harmonic wavelengths of scattered light. Especially, the observation of the lowest frequency polariton of a soft mode with a small ${\overrightarrow{k}}_s$ is impossible by this birefringence. Consequently, the hyper-Raman scattering of phonon-polariton is not suitable to study the ferroelectric soft mode.

In Section 3, the polariton dispersion of Raman active modes in Bi₄Ti₃O₁₂, LiNbO₃, and Raman inactive modes of SrTiO₃ are described. Other ferroelectrics such as LiTaO₃, BaZrO₃, and β-Gd₂(MoO₄)₃ were also studied by THz-TDS [53].

4. Polar metallic materials: ferroelectric instability and conduction electrons

The possibility of ferroelectric metals was suggested by Anderson and Blount in 1965 that V₃Si and other metallic transitions may be “ferroelectric” by the appearance of a polar axis [54]. It was written that while free electrons screen out the electric field completely, they do not interact very strongly with the transverse optical phonons and the Lorentz local fields lead to ferroelectricity, since umklapp processes are forbidden as k → 0. Recently the relation between ferroelectricity/polar distortion and metallic conductivity has attracted much attention in various kinds of metallic materials [55]. The A or B-site doped SrTiO₃ crystals has been extensively studied by the significant changes of physical properties related to metallic conduction by injected free electrons and lattice distortions [56, 57].

For the measurements of a materials with a high dielectric constant, the high reflectivity causes the difficulty in a transmission THz measurement. On reflection measurements using a conventional THz-TDS system, the determination of the phase delay of the reflected beam from a sample is difficult, because the reference beam from the surface of a sample cannot be accurately measured. Therefore, for the measurement of a complex high dielectric constant, the terahertz time-domain spectroscopic ellipsometry (THz-TDSE) using the reflection from a sample to be studied has been developed. In this section, the ferroelectric soft mode and conduction electrons studied by THz-TDSE is described [58, 59].

4.1 Ferroelectric instability

The complex dielectric constants of SrTiO₃ crystals with heterovalent B-site doping by Nb⁵⁺ ions were investigated to clarify the correlation between the ferroelectric soft optic mode and the injected carrier electrons. By the Fourier transformation of the time dependence of electric fields of p- and s-polarization components of reflected THz waves, the real and imaginary parts of a dielectric constant at room temperature were determined as shown in Figure 10a and b, respectively [59]. The dielectric response in the THz range includes a ferroelectric soft optic mode and dynamics of carrier electrons. The former is dominant near the ferroelectric soft mode frequency around 3 THz, which is well reproduced by the damped harmonic oscillator (DHO) model. While, in the frequency region lower than 1.5 THz, the dynamics of carrier electrons are dominant. Usually, the dielectric response of carrier electrons has been discussed using the Drude model. However, the accurate THz range spectra needs the modification of this model [60]. The Drude-Smith model assuming a single collision [61] has been used to analyze THz spectra. The complex dielectric constant of the Drude-Smith model is given by,

Figure 10.

(a) Real and (b) imaginary parts of the complex dielectric constant of Nb doped SrTiO₃ crystals measured by THz-TDSE at room temperature.

$${\epsilon }_{\mathit{DS}}\left(\nu \right)={\epsilon }_b+\frac{i2{\mathit{\pi \nu }}_p^2\tau }{\nu \left(1-i2\mathit{\pi \nu \tau }\right)}\left(1+\frac{c}{1-i2\mathit{\pi \nu \tau }}\right),-1\le \mathrm{c}\le 0,$$E17

where ε_b and ν_p, are the background dielectric constant and plasma frequency, respectively. The case of c = 0 is the Drude model, and as the constant c decreases from zero, the back scattering of electrons increases.

The observed real and imaginary parts of complex dielectric constants of Nb-doped SrTiO₃ were fitted by the summation of the DHO mode responsible for a soft polar mode and the Drude-Smith model responsible for free carriers as shown by the dotted lines in Figure 10a and b. Since in an undoped SrTiO₃ crystal has no free carriers, the spectra were fitted only by the DHO model as shown in the solid lines in Figure 10a and b.

4.2 Correlation between soft mode and conduction electrons

The lowest frequency TO1 mode is a ferroelecric soft optic mode in SrTiO₃. The soft mode frequency and plasma frequency are plotted as a function of the carrier concentration n_c in Nb doped SrTiO₃ crystals as shown in Figure 11a. It is found that both soft mode frequency and plasma frequency increase as the carrier concentration increases.

Figure 11.

(a) The soft mode frequency and plasma frequency are plotted as a function of the carrier concentration n_c in Nb doped SrTiO₃ crystals. (b) The coupled plasma frequency against the root of n_c.

In polar semiconductors, it is known that longitudinal optical phonons couples to plasmons. At the appropriate carrier density, the free-carrier plasma frequency is comparable to the phonon frequency, and the coupling between plasmons and phonons occurs [62]. Then the two new normal modes are strong admixtures of phonons and plasmons. The macroscopic point symmetry $m\overline{3}m$ of undoped SrTiO₃ crystals has a center of symmetry, while there is the possibility of local symmetry breaking [63] in doped SrTiO₃ crystals. For Nb doped SrTiO₃ the calculated plasma frequency was plotted as a function of the root of carrier concentration dependence in the relatively low concentration region as shown in Figure 11b. However, the concentration of Figure 11a is much higher than that of Figure 11b, and the plasma frequency is more than ten times of a soft mode frequency. Therefore, the effect of such a coupling is negligible in the present study.

According to the Cochran’s theory on ferroelectric instability [64], the square of the soft mode frequency, ν_s, is proportional to the difference between short range (SR) and long range (LR) interactions.

$${{\mathrm{\nu }}_{\mathrm{s}}}^2\propto \left[\mathrm{SR}\text{interaction}\right]–\left[\mathrm{LR}\text{interaction}\right]$$E18

The LR electrostatic fields favoring the polar structure are expected to be screened by the free carriers. The charge carriers essentially suppress the LR interaction of the attractive Coulomb force and increase a ferroelectric soft mode frequency. Figure 12 shows the relation between the square of a soft mode frequency and the free carrier concentration in doped SrTiO₃ crystals. It is found that the square of a ferroelectric soft mode frequency linearly increases as the carrier concentration increases. It is concluded that the carrier electron concentration suppresses the ferroelectirc instability and enhances the increase of a soft mode frequency.

Figure 12.

Square of the ferroelectric soft mode frequency, ν_TO, as a function of carrier concentration, n_c, of Nb doped SrTiO₃ crystals. The dotted line shows the linear relation.

However, the suppression of polar instability by free carriers markedly depends on materials reflecting the strength of electron-phonon coupling. The first-principles calculations showed that substantial polar displacements in n-doped LiNbO₃ are much more stable than those in n-doped BaTiO₃. This fact indicates that the electron-phonon coupling of BaTiO₃ is much stronger than that of LiNbO₃ [65]. Recently, it was reported that nonadiabatic Born effective charges (naBECs) can be defined in metallic compounds [66]. For doped SrTiO₃, the calculation of naBECs of the sublattices in SrTiO₃ suggested significant changes in the mode frequencies of polar phonon. The detailed theoretical study may clarify the interaction of the carrier electrons with ferroelectric instability.

5. Summary

Advanced terahertz vibrational spectroscopy using THz-time-domain spectroscopy (THz-TDS), THz time-domain spectroscopic ellipsometry (THz-TDSE), and far-infrared spectroscopy is reviewed. In pharmaceuticals, it is suggested that both THz-TDS and Raman scattering spectroscopy on polymorphic and polyamorphic states are important to clarify their structure and dynamics. As to the phonon-polariton dispersion relation, the detection of anisotropy and the dispersion relation for real and imaginary parts of polariton wavenumber are discussed in technologically important ferroelectric materials. The comparison between THz time-domain spectroscopy and forward Raman scattering is discussed. Recently polar metallic materials with metallic conduction and polar distortion have attracted much attention. In the heterovalent doped quantum paraelectric strontium titanate, the metallic conduction and polar instability coexist. Both dynamical features are studied by THz time-domain ellipsometry. The metallic polar instability is analyzed by the damped harmonic oscillator model for a ferroelectric soft mode and the Drude-Smith model for free carriers.

Acknowledgments

Author is thankful for the collaboration and discussions for M.W. Takeda, M. Maczka, H. Kitahara, S. Nishizawa, T. Iwamoto, Y. Satou, T. Hoshina, T. Mori, H. Igawa, T. Shibata, S. Koda, Y. Kobayashi.