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Elastic Anomaly and Elastic Heterogeneity in Ferroelectric Materials

Written By

Seiji Kojima

Submitted: 21 February 2025 Reviewed: 22 May 2025 Published: 22 June 2025

DOI: 10.5772/intechopen.1011166

Elastic Properties of Materials IntechOpen
Elastic Properties of Materials Edited by Gülşen Akın Evingür and Önder Pekcan

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Elastic Properties of Materials

Gülşen Akın Evingür and Önder Pekcan

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Abstract

Ferroelectric materials have been extensively applied to various devices due to technologically important functionalities such as piezoelectric, pyroelectric, electro-optic, nonlinear optic, elastooptic effects, photocatalytic properties, and high-energy storage density. These functionalities are closely related to ferroelectric phase transitions and heterogeneity. This chapter reviews the elastic anomaly of ferroelectric phase transitions and elastic heterogeneity caused by equivalent orientation states of spontaneous polarizations in ferroelectric materials. At first, elastic moduli for isotropic materials and elastic constants for crystals with point symmetry are introduced under various external conditions. Brillouin scattering spectroscopy, piezo-resonance method, and scanning acoustic microscope (SAM) are introduced as experimental methods to measure elastic properties. The elastic anomaly of ferroelectric phase transitions studied by measurements of Brillouin scattering is discussed. Measurements of SAM are used to discuss the elastic heterogeneity caused by ferroelectric and ferroelastic domains.

Keywords

  • elastic properties
  • ferroelectric
  • ferroelastic
  • domain
  • phase transition
  • Brillouin scattering
  • acoustic microscopy

1. Introduction

Ferroelectric materials have spontaneous polarization (PS) of which direction is changed by an external electric field (E). The ferroelectric (P-E) hysteresis of Rochel salt was discovered by Balasec in 1921 as the first report of ferroelectricity [1]. Ferroelectric crystals have no center of symmetry and their technologically important physical properties are related to odd-order polar tensors. Pyroelectricity is defined by the first-order tensor. Piezoelectric, electro-optic, and nonlinear optic properties such as second harmonic generation are defined by the third-order tensors [2]. Polar optical phonons couple to photons and polaritons are generated. Polariton dispersion relations of ferroelectric crystals such as LiNbO3 have been used in tunable terahertz light sources [3]. These physical properties have been widely used in industrial applications.

A ferroelectric material undergoes a paraelectric to ferroelectric phase transition at a Curie temperature (TC) upon cooling from a high temperature. Such a structural phase transition accompanies the distortion of the crystal lattice and elastic properties remarkably change by piezoelectric and electrostrictive couplings between polarization and strain. Since the +PS and –PS states have the same energy, +PS and –PS domains coexist. In the vicinity of domain walls, strain changes by the change of polarization direction and the heterogeneity of elastic properties occurs. Therefore, the temperature and the spatial variation of elastic properties have been extensively studied in ferroelectric materials.

1.1 Elastic moduli of isotropic materials

At first, we consider the homogeneous isotropic materials such as glass. For elastic waves of wavelengths (λ) longer than 10 nm, the continuum approximately is usually valid. According to Hooke’s law, the strain is directly proportional to stress. It holds only for small strains. The linear relation between the strain and the stress is defined by elastic moduli. Young’s modulus (Y) describes the ratio of uniaxial stress to axial strain. The bulk modulus (K) describes the ratio of hydrostatic pressure to the volume. The compressibility (κ) is the reciprocal of the bulk modulus. The shear modulus (G) describes the ratio of shear stress to the shear strain. The Poisson’s ratio (ν) describes the response in the directions orthogonal to the uniaxial stress. For isotropic materials, it holds that −1.0 ≤ ν ≤ 0.5. Poisson’s ratio of glasses, ceramics, and crystals has been recently reviewed [4].

The number of independent moduli is two for isotropic materials, and elastic moduli are connected by the following equation.

Y=2G1+ν=3K1−2νE1

These elastic moduli are determined by the observed values of LA velocity, VL, TA velocity, VT, and density, ρ, of a sample.

Shear modulus:G=ρVT2,E2
Longitudinal modulusc:L=ρVL2,E3
Bulk modulus:B=L−43G,E4
Compressibility:κ=1/B.E5
Young’smodulus:Y=G3L−4GL−G,E6
Poisson’sratio≔12VL2−2VT2VL2−VT2.E7

These elastic moduli have been extensively used for elastic properties of various kinds of materials. The composition dependence of elastic moduli in alkali borate glasses was reviewed [5].

1.2 Elastic constants of a crystal

A crystal has an anisotropic periodic structure with translational and point symmetries. The elastic stiffness constants, cijkl, are defined by

Xij=cijklekl,i,j,k,l,=1,2,3.E8

Here, Xij, and ekl are stress and strain.

According to thermodynamics, adiabatic constant cijklS and isothermal constant cijklT are defined, respectively.

cijklS=∂Tij∂eklS=ρ∂2U∂eij∂eklS.E9
cijklT=∂Tij∂eklT=ρ∂2A∂eij∂eklT.E10

Here, U and A = U − TS are internal energy and Helmholtz free energy, respectively, and ρ is density. When there is no anharmonicity of lattice, an adiabatic constant is equal to an isothermal constant because elastic constants are independent of temperature in a harmonic lattice. The elastic compliance constants, Sijkl, are defined by

eij=SijklXkl,i,j,k,l,=1,2,3.E11

These elastic constants are symmetric as below.

cijkl=cklij=clkji,Sijkl=Sklij=SlkjiE12

To reduce the number of subscripts, the indices (1–6) are introduced in Voigt notation as follows.

11→1,22→2,33→3,23→4,31→5,12→6.E13

For a cubic system, the matrix of elastic stiffness constants is given by

X1X2X3X4X5X6=C11C12C12000C12C11C12000C12C12C11000000C44000000C44000000C44e1e2e3e4e5e6.E14

Elastic energy is defined by

U=12∑i,j=16cijeiej.E15

For the analysis of elastic anomalies in structural phase transitions, elastic energy is important. The maximum number of independent elastic constants is 21 for trigonal symmetry. For cubic symmetry with the highest symmetry independent number is three, and independent elastic constants are c11, c12, and c44.

1.3 Elastic anomalies of ferroelectric phase transitions

In piezoelectric crystals, the elastic constant at constant electric field (cijklE) is different from the elastic constant at constant polarization (cijklP). When piezoelectricity exists in a paraelectric phase, a piezoelectric coupling between polarization and strain plays a dominant role in the elastic anomaly related to a ferroelectric phase transition. Potassium dihydrogen phosphate (KH2PO4) contains phosphate groups connected by hydrogen bonds, in which the order-disorder transition of protons occurs [6]. The dielectric anomaly produces anomalous behavior in the elastic constant (cijklE). In a tetragonal paraelectric phase with the tetragonal point group −42 m, the piezoelectric coupling d36P3e6 exists, where d36, P3, and e6 are piezoelectric constant, polarization along a ferroelectric-axis, and shear strain perpendicular to P3. The free energy of a paraelectric phase is given by

GP3e6T=12P32χ3e+d36P3e6+12c66Pe62,E16

where χ3e and c66P are the clamped dielectric susceptibility at constant strain and the elastic constant for constant polarization, respectively [7]. On the right-hand side, the first, second, and third terms are electric, piezoelectric, and elastic energies, respectively. The temperature dependence of dielectric constant χ3X at constant stress and clamped dielectric constant χ3e at constant strain are given below

χ3e−1=αT−T0,E17
χ3X−1=αT−TC,TC=T0+d362αc66P,E18

where TC and T0 are the Curie temperatures of free (X = 0) and clamped (e = 0) crystals, respectively as shown in Figure 1(a). The temperature dependence of elastic constant c66E at constant electric field and c66P is given below.

Figure 1.

Temperature dependences of (a) inverse free χ3X−1 and inverse clamped dielectric constants χ3e−1, and (b) the adiabatic elastic constants c66E at constant electric field and c66P at constant polarization.

c66P=constant,E19
c66E=d362αT−TC.E20

c66P is independent of temperature, while c66E shows the remarkable softening toward TC upon cooling from a high temperature as shown in Figure 1(b). This is the typical difference between c66P and c66E in a piezoelectric crystal. Such a softening of the adiabatic elastic constant at constant electric field c66S,E was observed in the paraelectric phase of a KH2PO4 crystal by Brillouin scattering spectroscopy [8].

2. Methods to measure elastic properties

In this section, the experimental methods to determine the sound velocity and elastic properties of crystals are described. For bulk homogeneous materials, the ultrasonic pulse-echo method, resonance ultrasonic spectroscopy, and Brillouin scattering spectroscopy are introduced. For inhomogeneous materials with textures, scanning acoustic microscopy is introduced to observe elastic heterogeneity.

2.1 Ultrasonic pulse-echo method

The ultrasonic pulse-echo (PE) method is the most popular method to measure bulk elastic constants. As a nondestructive method, the PE method has similarities with sonar systems. This method has been applied to measure the sound velocity of various materials. The ultrasonic transducer of piezoelectric ceramics such as PZT is placed at the top of a sample to be measured. The RF electric pulse applied to a transducer is converted into an ultrasonic pulse in a sample. The ultrasonic pulse (echo) reflected at the bottom of a sample is converted once more to an electric pulse by the piezoelectricity of a transducer. Using the ultrasonic travel time, τ, and the travel length, L, the sound velocity of a sample, V, is calculated by the relation, V = 2 L/τ. The accuracy of velocity was improved by the PE overlap method. For the accurate determination of τ, the McSkimin criterion establishes the condition of correct overlapping between ultrasonic echoes [9, 10]. The adiabatic elastic constants of magnesium single crystals were measured at 10 MHz by the ultrasonic pulse-echo method using a quartz piezoelectric transducer [11].

2.2 Resonance ultrasonic spectroscopy

Resonant ultrasonic spectroscopy (RUS) is a resonance technique that consists of placing a sample to be studied on an ultrasonic transducer. By scanning the exciting frequency, the vibration of mechanical eigen modes of a sample can be detected [12]. From the measured RUS spectrum in the frequency domain, the sound velocity and elastic moduli of a sample are determined by fitting. For a single crystal, all adiabatic elastic stiffness constants are determined by the sphere or rectangular parallelepiped resonance. Cu-Al-Ni crystals are well known shape-memory alloy. The free vibration frequencies of Cu-AL-Ni were measured by RUS, and elastic constants were determined [13]. RUS measurements of the adiabatic bulk and shear moduli of pure polycrystalline β-Pu were also reported [14].

2.3 Brillouin scattering spectroscopy

The inelastic light scattering in the gigahertz range is called Brillouin scattering (BS) using monochromatic visible light, and it is related to thermally excited acoustic phonons. The velocity of acoustic phonons and elastic constants are determined by a non-contact and non-destructive BS spectroscopy. The phase velocity of acoustic phonons, V, of a sample, is determined by the frequency shift, νB, of a Brillouin peak from an incident beam frequency, νi. in the Brillouin scattering spectrum [15].

V=λivB2nsinθ2.E21

Here, λiθ, and n are the wavelength of an incident light, the scattering angle, and the refractive index of a sample, respectively. Using observed transverse acoustic (TA) velocity and longitudinal acoustic (LA) velocity, adiabatic elastic constants of a material are determined. The attenuation of acoustic phonons is calculated by the width of a Brillouin peak.

α=πΓV,E22

where Γ is the full-width half-maximum (FWHM) of a Brillouin peak.

In Brillouin scattering measurements, a single-frequency laser beam is focused by an objective lens into a sample to be observed, and the scattered light from the sample is collected and focused into an entrance pinhole of tandem multipath Fabry-Perot interferometers (TFPIs) with high-frequency resolution and high-contrast. The scattered light is detected by a highly sensitive photon counting system or a CCD detector [16]. To measure microarea or very small sample, the combination of TFPIs with an optical microscope is important as shown in Figure 2 [17]. Such a micro-Brillouin scattering has been used to measure various kinds of liquid and solid materials.

Figure 2.

Experimental setup of micro-broad Brillouin scattering spectroscopy. A laser beam is focused into a sample, and the same objective lens collects the scattered light. The scattered light is analyzed by tandem multipath Fabry-Perot interferometers and is detected by the photon counting system.

2.4 Scanning acoustic microscopy

In materials science, various kinds of microscopes have been used to study the heterogeneity of physical properties. Acoustic microscopy is a powerful tool for investigating elastic heterogeneity and propagation characteristics on surface acoustic waves (SAWs) in not only isotropic but also crystalline materials [18]. A mechanically scanned acoustic microscope (SAM) was developed using single-surface lenses that focus an acoustic beam with negligible spherical aberration in a water cell. The image of an object was formed by mechanically scanning [19].

The spatial resolution is determined by the wavelength of ultrasonic waves according to the Rayleigh criterion, which is a standard that determines the minimum distance between two objects that can be distinguished. Since the wavelength is inversely proportional to frequency, a high frequency of more than 100 MHz is necessary for acoustic microscopy. The high-resolution acoustic microscope was reported using liquid helium of very low attenuation as a liquid coupler at very low-temperatures. The acoustic wavelength of 570 Å was achieved at 4.2 GHz. The resolution better than 500 Å was reported in acoustic images of a silicone integrated circuit [20].

Figure 3 shows a schematic illustration of conventional SAM [21]. A spherical acoustic lens made of sapphire with LA velocity of 11,250 m/s and a liquid coupler of distilled water with LA velocity of VW = 1480 m/s at room temperature were used. Due to the large difference in velocity between sapphire and water, spherical aberration is negligible. The high-frequency continuous ultrasonic waves generated by a thin film piezoelectric film, such as ZnO, are focused by an acoustic lens through a liquid coupler onto the surface of a sample to be objected. The incident waves are reflected, transmitted, and partly converted to leaky Rayleigh waves, which propagate along the interface between a liquid coupler and a sample.

Figure 3.

Scanning acoustic microscope using an acoustic lens. Ultrasonic waves generated by a piezoelectric transducer are focused on a sample through a liquid coupler. θR and Θ are a critical angle of leaky Rayleigh waves and an opening angle of a lens, respectively. R and f are an opening radius and a focal length of a lens, respectively.

When the incident angle of ultrasonic waves is smaller than the critical angle for leaky Rayleigh waves θR = sin−1(VW/VR), the bulk ultrasonic waves in a liquid coupler transmit into a sample, where VR is the velocity of Rayleigh waves. When the incident angle is equal to θR, the incident bulk waves are converted to leaky Rayleigh waves, which propagate along an interface between a sample and a liquid coupler. The leaky waves radiate bulk waves into a liquid coupler with the radiation angle θR. When the incident is larger than θR, the incident bulk waves are reflected on the surface. These radiated bulk waves and reflected waves are collected by an acoustic lens. These reflected waves from a sample are transmitted to a lens through an anti-reflection coating and are converted to the output electric signal V(z) by a piezoelectric transducer, where z is the distance from a focal point to the plane of a sample surface. An acoustic imaging is observed by two-dimensional mechanical scanning of an acoustic lens using an x-y mechanical stage.

In Figure 3, when the incident angle is equal to the critical angle for the leaky Rayleigh waves, reflected waves include radiated bulk waves from a sample at defocus distance z < 0. The interference with the geometrically reflected ray at the surface of a sample causes the contrast of an acoustic image.

3. Temperature dependence of elastic properties

Ferroelectric materials undergo a ferroelectric phase transition at a Curie temperature, TC, upon heating. The spontaneous polarization goes to zero and the dielectric constant obeys the Curie–Weiss law and shows the divergence in the vicinity of TC. The polarization couples to strain and elastic anomaly occurs by piezoelectric or electro-strictive coupling. Such couplings cause elastic heterogeneity in domain structure.

SrxBa1-xNb2O6 (SBN100x) and CaxBa1-xNb2O6 (CBN100x) are well-known uniaxial ferroelectrics with tungsten-bronze (TB) structure [22]. These TB ferroelectrics is known for the superior electrooptic and photorefractive effects [2]. The point group of paraelectric phase of these TB ferroelectrics is nonpolar tetragonal point group 4/mmm and is nonpiezoelectric. CBN28 crystals undergo a ferroelectric phase transition at TC = 250°C upon cooling from a high temperature. A ferroelectric phase with the polar tetragonal point group 4 mm has spontaneous polarization along the c-axis, which is parallel to the four-fold axis [23]. The temperature dependence of Brillouin scattering spectra of a CBN28 crystal is shown in Figure 4, where CP is a central peak with polarization fluctuations along the c-axis [23]. LA frequency shift, which is proportional to LA velocity, shows the softening toward TC by electrostrictive coupling q33e3P32 between strain e3 and polarization P3 along the c-axis, where q33 is an electrostrictive constant. The peak intensity of LA is much higher than that of TA by the difference of photo-elastic constant between LA and TA modes. A ferroelectric phase transition of CBN28 has an order-disorder nature, therefore, the relaxation process of polarization fluctuations appears as a CP in the vicinity of TC (Figure 4).

Figure 4.

Brillouin scattering spectra of a CBN28 crystal. LA, TA, and CP are longitudinal acoustic, transverse acoustic modes, and a central peak with the wavevector q parallel to the c-axis (q//c), respectively.

The temperature dependence of LA velocity VLA was calculated by Eq. (21) as shown in Figure 5. In a paraelectric phase above TC, LA velocity VLA was fitted by

Figure 5.

Temperature dependence of longitudinal acoustic velocity VLA along the ferroelectric c-axis of CBN28. The dotted line denotes the fitted curve by the Eq. (23).

VLA=a+b/T−T1n,E23

where a and b are constants, and n is the critical exponent of VLA. The second term on the right-hand side describes the critical softening due to critical fluctuations. The fitted values are T1 = 201°C and n = 0.24. The fact T1 < TC indicates the first-order phase transition.

In a paraelectric phase above TC, LA attenuation coefficient αLA was fitted by

αLA=c−d/T−T0m,E24

where c and d are constants, and m is the critical exponent of LA attenuation. The second term describes the critical divergence due to critical fluctuations. The fitted values are T0 = 201°C and m = 1.0. The dotted line denotes the fitted curve by the Eq. (24) (Figure 6).

Figure 6.

Temperature dependence of longitudinal acoustic attenuation along the ferroelectric c-axis of CBN28. The dotted line denotes the fitted curve by Eq. (24).

The temperature dependence of elastic stiffness constant c33=ρVLA2 was determined by the observed LA velocity as shown in Figure 7. The c33 shows the softening toward TC. Upon cooling from a high temperature, the deviation from the linear temperature dependence occurred at the intermediate temperature at T* = 367°C by the dynamic to static transition of polar nanoregions (PNRs). Dynamic PNRs appear below the Burns’ temperature, TB = 517°C. It was reported that TB = 517°C and T* = 367°C are independent of the Ca content in CBN100x [24].

Figure 7.

Temperature dependence of elastic stiffness constant c33. The solid line denotes the fitted curve by the Eq. (25).

The temperature dependence of elastic constants in the paraelectric phase was fitted by the following equation.

cijt=c0ij+c1ijT−c2ijT−T0T0−n,forT0≤Tc≤T.E25

Here, c0ij, c1ij, and c2ij are constants. In the righthand side, anharmonic effect and elastic anomaly are given by the second and third terms, respectively. The fluctuations of order parameter are given by the third term. The predicted critical exponents, n = 0.5, 1.0, and 1.5, are attributed to three, two, and one-dimensional fluctuations of the order parameters, respectively [25]. In undoped and Li-doped K(Ta0.6Nb0.4)TiO3 crystals with perovskite structure, the three-dimensional fluctuations of polarization were suggested by the observed exponent of n = 0.5. The fluctuations are related to the eight-site model of B-site ion off-centering along the equivalent eight [111] directions [26]. The fitted value of n = 1.52 of CBN28 may be attributed the one-dimensional polarization fluctuations, which is related to a uniaxial ferroelectric polarization along the c-axis.

The electric field effects on the elastic properties of uniaxial CBN30 single crystals were also investigated using Brillouin scattering spectroscopy [24]. By the study of the electric field dependence of LA velocity, the coexistence of nanodomains generated by the random fields and the field-induced macrodomains was observed as a mixed state at 3.0 kV/cm. By the incomplete switching of the nanodomains to the macrodomains state, such a mixed state remains up to 13 kV/cm. The remarkable memory effect was also observed in LA velocity.

4. Elastic heterogeneity observed by scanning acoustic microscopy

In ferroelectric and ferroelastic materials, elastic inhomogeneity has been indirectly observed by polarizing optical microscopy. Because internal stress induced birefringence by elastooptic effect. SAM is the direct method to observe elastic heterogeneity of materials [27]. In ferroelectric or ferroelastic domains, the elastic constants change in the vicinity of a domain wall, where the direction of spontaneous polarization abruptly changes. Elastic constants and strain also abruptly change in the vicinity of a domain wall. Figure 8 shows ferroelectric 90° domains and 180°domains. In ferroelectric BaTiO3 crystals, the thickness of a 90° domain wall is thicker than a 180° domain wall, which is the order of a unit cell.

Figure 8.

Ferroelectric domains, (a) 90° domains and (b) 180°domains, where Ps denotes a spontaneous polarization. The direction of spontaneous polarization abruptly changes in the vicinity of domain walls.

Figure 9 shows the reflection at a domain wall in the defocused condition at z < 0. The incident ultrasonic waves at θR excite leaky Rayleigh waves, which propagate along the interface between a sample and water. They were reflected at a domain wall by the change of acoustic impedance and returned to a lens and affected the V(z) through a piezoelectric transducer. The change of V(z) by the reflection is the origin of the contrast of the domain walls in an acoustic image.

Figure 9.

The leaky Rayleigh waves are reflected at a ferroelectric or ferroelastic domain wall. The reflected waves radiate bulk waves into water and are collected by an acoustic lens.

BaTiO3 with perovskite structure is well known for ferroelectricity. Figure 10 shows the acoustic micrograph of a tetragonal BaTiO3 crystal with ferroelectric 90°domains and the schematic illustration of ferroelectric 90°domain walls [28]. Black dots are the defects on the surface of a crystal. The contrast between domain walls and inside domains was studied by the measurements of V(z) curves. It was found that the amplitude of periodic change in a V(z) curve at domain walls is reduced by the contribution of reflection waves at domain walls.

Figure 10.

(a) Acoustic micrograph of a tetragonal BaTiO3 crystal with ferroelectric 90°domain. (b) Schematic illustration of ferroelectric 90°domain walls.

Figure 11 shows the acoustic image of prismatic 90°domains in a β-Gd2(MoO4)3 (GMO) crystal, which is known as electrooptic crystal [29, 30]. GMO is improper ferroelectric and the order parameters of a ferroelectric phase transition at TC = 160°C is not a polarization but the degenerated phonons Q1, Q2 at a M-point in the Brillouin zone [31]. Upon cooling from a high-temperature, it undergoes a ferroelectric phase transition from a paraelectric tetragonal phase with the point group −42 m to a ferroelectric orthorhombic phase with the point group 2 mm. The spontaneous polarization P3 has a piezoelectric coupling a36P3e6 with shear strain e6. The thickness of domain walls is about 0.7 μm according to the study by Raman scattering [32]. Ferroelectric polarization P3 and ferroelastic strain e6 gradually change at a wall and the symmetry changes to tetragonal in the center of the wall. Such a change of symmetry of MoO4 tetrahedra was detected by Raman scattering. The change of acoustic impedance at domain walls is the origin of the clear contrast of domain walls in the acoustic image. Recently its isomorph β-Tb2(MoO4)3 has attracted attention as a multiferroic material [33]. Domain structures were also studied by SAM in a ferroelectric TGS crystal and ferroelastic NdP5O14 crystals [34].

Figure 11.

(a) Acoustic micrograph of an orthorhombic β-Gd2(MoO4)3 (GMO) crystal with prismatic 180°domains. The focal plane is near the upper surface. (b) Schematic illustration of prismatic 180°domain walls perpendicular to the c-axis. (c) Acoustic micrograph of a GMO crystal. The focal plane is near the bottom surface. (d) 3D view of prismatic 180°domains, where a, b, and c denote the orthorhombic crystal axes. Ps denotes a spontaneous polarization along the ferroelectric c-axis.

The determination of leaky SAW velocity by SAM is possible by the measurement of a V(z) curve, where V is the output of an acoustic lens and z is the distance between a sample surface and a focal point. The analysis of Fourier optics, the acoustic output V(z) is given by the integral of the spatial distribution of an acoustic field u(r) weighted by the reflectance function R(θ) [35].

Vz=exp−2αf+z∫rurP1rP2rRθexp(2jkz1−r/f2)dr,r=x2+y2,E26

where k = 2π/λ.α and λ are the absorption constant and wavelength of a liquid coupler. u(r) and Pi(r) are acoustic fields and pupil functions of the lens, respectively. From the analysis of V(z), it is found that the periodicity Δz is related to the wavelength of leaky surface wave λR [28].

∆z=λW21−cosθR=λR2tanθR2,sinθR=λWλR.E27

The experimental result of V(z) curve of the x-plate of a SrTiO3 crystal with perovskite structure observed by SAM at 420 MHz is shown in Figure 12 [27]. SrTiO3 is quantum paraelectric with a cubic perovskite structure. The observed velocity of leaky SAW is 3360 ± 50 m/s. Since crystals have elastic anisotropy, the Δz is the average value of the directional dependence on the x-y plane.

Figure 12.

V(z) curve of the x-plate of a SrTiO3 crystal observed by SAM.

To measure the directional dependence of Δz, the line focus lens with a cylindrical shape is necessary as shown in Figure 13 [21]. The angular dependence of V(z) curves on the y-z plane of a x-cut LiNbO3 crystal was studied using a cylindrical acoustic lens [36]. The orientation of the x-plate crystal and the propagation direction of SAW are shown in Figure 14(a). The angular dependence of leaky SAW velocity in the y-z plane of a LiNbO3 crystal was determined by the measurement on the angular dependence of Δz as shown in Figure 14(b). The circles and the dotted line denote the observed and calculated values [37], respectively.

Figure 13.

Line-focus acoustic lens with cylindrical shape.

Figure 14.

Angular dependence of the sound velocity of leaky SAW waves in the x-plate of a congruent LiNbO3 crystal. (a) The orientation of the x-plate and the propagation direction of leaky SAW on the y-z plane are to be measured. (b) The angular dependence of leaky SAW velocity in the y-z plane of a congruent LiNbO3 crystal. The circles and the dotted line denote the observed and calculated values, respectively.

LiNbO3 has been extensively applied to various devices by the use of SAW and acoustic phonons. Recently, super-high frequency wave-guided SAW with fundamental operating frequency to the 7-GHz range was developed for the next-generation radio frequency front-end system applications [38]. Transverse acoustic phonons of LiNbO3 were used in a quantum acoustic device, and the versatility of 3D microwave cavities for mediating contact-less coupling to quantum, and classical, piezoacoustic devices was demonstrated [39].

5. Conclusions

Ferroelectric materials have technologically important functionalities, such as piezoelectric, pyroelectric, electrooptic, nonlinear optic, elastooptic effects, photocatalytic properties, and high-energy storage density. The switchable spontaneous polarization by an electric field has been used in volatile memory. The ferroelectric domain structure originates from the equivalent orientation states of spontaneous polarizations and it induces elastic heterogeneity. These functionalities are closely related to ferroelectricity and ferroelectric phase transitions. This chapter reviews elastic anomaly of ferroelectric phase transitions and elastic heterogeneity on ferroelectric domain walls. The elastic moduli for isotropic materials and elastic constants for anisotropic crystals are introduced. The difference in elastic constants under various external conditions, such as adiabatic and isothermal elastic constants, is discussed. For experimental methods to determine sound velocity and elastic constants, ultrasonic methods, inelastic light scattering spectroscopy are introduced. The elastic anomaly of ferroelectric phase transitions studied by the measurement of temperature dependence of Brillouin scattering was explained. For experimental tools to investigate elastic heterogeneity scanning acoustic microscope (SAM) is introduced. The study by SAM on elastic heterogeneity related to ferroelectric and ferroelastic domain walls were reviewed. The method to measure angular dependence of SAW velocity using a cylindrical acoustic lens is introduced.

Ferroelectric materials are anisotoropic and their elastic properties are very sensitive for temperature, pressure, and electric fields. The equivalency of polarization states cause heterogeneity such as domains and polar nanoregions. Therefore, noncontact and nondestructive experimental methods such as Brillouin scattering and SAM are very important.

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Written By

Seiji Kojima

Submitted: 21 February 2025 Reviewed: 22 May 2025 Published: 22 June 2025

© 2025 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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