Soft Mode Frequency Research Articles

Softening of infrared-active mode of perovskite BaZrO3 proved by terahertz time-domain spectroscopy

The low-frequency infrared-active optical modes were studied in a barium zirconate, BaZrO3, single crystal with the perovskite structure using terahertz (THz) time-domain spectroscopy (TDS). The real and imaginary parts of the dielectric constants were accurately determined in the frequency range between 0.2 and 2.7 THz. Upon cooling from room temperature to 8 K, the lowest-frequency TO1 mode at 2.32 THz showed a pronounced softening to 1.94 THz. The real part of the dielectric constant at 0.5 THz determined by THz-TDS obeys Barrett's relation, and the existence of a plateau confirms that the quantum effects lead to saturation of the soft mode frequencies of the TO1 and TO2 modes below ≈20 K. This is reminiscent of incipient ferroelectrics with the perovskite structure such as CaTiO3.

Phase transitions in two- and three-component perovskite superlattices

The temperature behavior of polarization Raman spectra of two-layer BaTiO3/Ba0.5Sr0.5TiO3 and three-layer BaTiO3/Ba0.5Sr0.5TiO3/SrTiO3 superlattices prepared by pulsed laser deposition on (001)MgO substrates has been studied. It has been found that, at room temperature, the frequency of soft mode E(TO) in the three-layer superlattice is significantly higher (94 cm−1) than that in the two-layer superlattice (70 cm−1), which demonstrates an increase in two-dimensional stresses in epitaxial layers of the superlattice. An analysis of the temperature dependence of the integrated intensity of soft mode E(TO) suggests that the BaTiO3/Ba0.5Sr0.5TiO3 superlattice undergoes a transition from the ferroelectric to paraelectric phase at ∼540 K, and the phase transition temperature of the BaTiO3/Ba0.5Sr0.5TiO3/SrTiO3 superlattice increases to ∼610 K because of the deformation of the epitaxial layers.

The role of mechanical boundary conditions in the soft mode dynamics of PbTiO3

The role of different mechanical boundary conditions in the soft mode dynamics of ferroelectric PbTiO3 is systematically investigated using first-principles-based simulations and analytical model. The change in the soft mode dynamics due to hydrostatic pressure, uniaxial and biaxial stresses and biaxial strains is studied in a wide temperature range. Our computations predict: (i) the existence of Curie–Weiss laws that relate the soft mode frequency to the stress or strain; (ii) a non-trivial temperature evolution of the associated Curie–Weiss constants; (iii) a qualitative difference between the soft mode response to stresses/strains and hydrostatic pressure. The latter finding implies that the Curie–Weiss pressure law commonly used for residual stress estimation may not apply for the cases of uniaxial and biaxial stresses and strains. On the other hand, our systematic study offers a way to eliminate this difficulty through the establishment of Curie–Weiss stress and strain laws. Implications of our predictions for some available experimental data are discussed.

Ferroelectric phase transition and soft-mode behavior in BaxSr1−xTiO3: a refined treatment of a quasi-harmonic model

The composition and temperature dependences of the soft-mode and ferroelectric phase transition in BaxSr1−xTiO3 (BST) are presented and analyzed within the framework of a refined treatment of a quasi-harmonic model (QHM) for quantum particles. The QHM involves generalized simple microscopic quantum particles with anharmonic on-site double-Morse-type potential and harmonic interactions. The theory uses the variational principle scheme at finite temperature in the mean-field approximation while taking into account the predominant mass effect, the cell volume effect as well as ferroelectric distortion. The soft-mode frequency and the Ba concentration (x) dependence of the ferroelectric phase transition temperature are analyzed and show good agreement with experimental results throughout the full x range (0 ⩽ x ⩽ 1). It is found that BST is paraelectric below the critical impurity concentration xc = 0.01 with a softening of the soft mode. It becomes ferroelectric above this critical concentration (x ⩾ xc), and as the impurity concentration increases the predominant displacive soft-mode behavior stiffens in the ferroelectric phase with much less softening in crystals indicating a possible phase change at high x. Around x′c = 0.5, Tc deviates gradually from linear behavior with a rather slight round stagnation which might characterize an equilibrium ‘region’ where several ferroelectric phases coexist. The proposed x dependence of the soft-mode behavior indicates that the ferroelectric phase transition is of first order in BST with x > x′c, and of second order for low Ba concentrations (xc ⩽ x < x′c), throughout the tricritical point at x = x′c. Finally, the quantum dynamics of the ferroelectric phase transition and the mode softening are also addressed and discussed, suggesting a net increase of quantum effects with a decrease of x for x ⩽ 0.1, and a dominant Curie–Weiss law for x > 0.1.

Systematic study of terahertz response ofSrTiO3based heterostructures: Influence of strain, temperature, and electric field

Epitaxial heterostructures consisting of a variable number of ${\mathrm{SrTiO}}_{3}/{\mathrm{DyScO}}_{3}$ bilayers deposited on ${\mathrm{DyScO}}_{3}$ substrates were investigated using time-domain terahertz spectroscopy down to helium temperatures. Interdigitated electrodes deposited on top of the structures allowed probing of the terahertz response upon an applied electric field. The phase transition into a ferroelectric state is observed in ${\mathrm{SrTiO}}_{3}$ films in all samples close to room temperature (between 250 and 310 K) due to in-plane epitaxial strain induced by the substrate and intercalated layers of ${\mathrm{DyScO}}_{3}$. Evolution of the dielectric spectra with temperature and external electric field is described by a general model which involves a damped harmonic oscillator (soft mode) coupled to a Debye relaxation (central mode). Both modes are connected with the soft mode eigenvector, as recently shown by molecular dynamics simulations, and they reflect a strong anharmonicity of the soft-mode potential. At high temperatures the soft-mode frequency variation drives all the changes observed in the spectra with temperature and applied field. At low temperatures, deep in the ferroelectric phase, the soft mode significantly hardens and loses its importance for the terahertz dynamics; the central mode becomes stronger and it almost completely determines the shape of the measured spectra. The observed variation of phase transition temperature and of the dielectric response among the structures is ascribed to a partial epitaxial strain relaxation confirmed also by x-ray diffraction.

Structural sensitivities of soft and steep nonlinear global modes in spatially developing media

Structural sensitivities of soft and steep nonlinear global modes arising in a complex Ginzburg–Landau model are investigated by calculating the leading-order variation of their amplitude and frequency to an open-loop forcing and a closed-loop perturbation. The soft global mode is found to be sensitive to both the open-loop and closed perturbations in the region of linear absolute instability where its amplitude is not negligible. In particular, the frequency of the soft global mode exhibits large sensitivity at the location where the frequency of soft global mode was shown to be determined in the previous WKBJ analysis. On the other hand, the steep global mode shows a large response in the amplitude and the frequency to the open-loop perturbation located far upstream. To the closed-loop perturbation, the steep global mode is most sensitive at the location where the stationary front is located, consistent with the previous WKBJ analysis. Finally, the sensitivities analyzed for the fully nonlinear global mode are compared to those obtained from a weakly nonlinear analysis. It shows that the weakly nonlinear analysis fails to capture the sensitivity behavior obtained from the fully nonlinear global mode particularly under the strong advection yielding steep global mode.

Raman spectroscopy study of lattice dynamics of macro-, micro-, and nanostructured barium titanates

The temperature dependences of the components A 1(2TO) and E(1TO) of the soft ferroelectric mode during phase transitions in single crystals, ceramics, polycrystalline and epitaxial thin films of barium titanate, as well as a superlattice consisting of alternating layers of barium and strontium titanates, have been studied using the Raman spectroscopy method. Abrupt changes in soft mode frequencies have been observed in the single crystal during phase transitions between tetragonal, orthorhombic, and rhombohedral phases. Smoothing of the temperature dependences of soft modes and the coexistence of phases have been observed in ceramics and polycrystalline films. In the epitaxial film, the sequence of structural transformations fundamentally differs from that observed in the single crystal; in the superlattice, the ferroelectric phase is stable to 550 K.

Soft mode and acoustic mode ferroelectric properties of deuterated triglycine sulphate crystal

A mathematical study about deuterated triglycine sulphate (CD2CD2COOD)3D2SO4 crystal by a theoretical model which is extended with two sublattice pseudospin lattice coupled mode model by adding third, fourth and fifth order phonon anharmonic interaction terms as well as external electric field term in the crystal Hamiltonian. Double-time temperature dependent Green’s function is used to derive soft mode frequency, dielectric permittivity, microwave absorption, quality factor, acoustic attenuation, electric conductivity, smooth function, relaxation time, ratio of figure of merits and respective applications in modern technologies. All theoretical results have a good agreement with experimental data.

Theoretical Study of Ferroelectric Triglycine Sulphate (TGS) Crystal in External Electric Fields

A modified two sub-lattice pseudospin-lattice coupled mode model of Mitsui et al [Phys. Rev., 111 (1958) 1259] by adding third, fourth order phonon anharmonic interaction and external electric field terms has been applied to ferroelectric triglycine sulphate crystal. Electric field dependence of ferroelectric, dielectric and acoustical properties has been studied. With the help of double time temperature dependent Green’s function method, expressions for shift, width, soft mode frequency, dielectric constant, loss tangent and acoustic attenuation have been derived. Numerically calculations have been made and results have been compared with experimental data reported by Bye et al [Ferroelectrics 4 (1974) 243] and Shreekumar et al [Ferroelectrics 160 (1994) 23] for TGS crystal and a good agreement has been observed.

Pro rol' fluktuatsiy koherentnykh zsuviv polozhennya rivnovahy atomiv pry fazovykh perekhodakh druhoho rodu

Second-order phase transitions in ferroics of the ferroelastic type have been studied theoretically. The temperature-induced phase transition in crystals with a wide forbidden gap caused by the interaction between electrons and a branch of acoustic vibrations in the crystal lattice is considered. The vertex part of this interaction with transverse lattice vibrations is calculated. The characteristic features of the thermodynamic potential at the transition point and the temperature dependence of the acoustic phonon dispersion at the transition over quasiequilibrium states are discussed. It is shown that the stabilization of structure fluctuations by the electron subsystem in the high-symmetry phase, and a reduction of soft mode frequencies makes those fluctuations metastable.

Theoretical Study of Ferroelectric Triglycine Sulphate (TGS) Crystal in External Electric Fields

A modified two sub-lattice pseudospin-lattice coupled mode model of Mitsui et al [Phys. Rev., 111 (1958) 1259] by adding third, fourth order phonon anharmonic interaction and external electric field terms has been applied to ferroelectric triglycine sulphate crystal. Electric field dependence of ferroelectric, dielectric and acoustical properties has been studied. With the help of double time temperature dependent Green’s function method, expressions for shift, width, soft mode frequency, dielectric constant, loss tangent and acoustic attenuation have been derived. Numerically calculations have been made and results have been compared with experimental data reported by Bye et al [Ferroelectrics 4 (1974) 243] and Shreekumar et al [Ferroelectrics 160 (1994) 23] for TGS crystal and a good agreement has been observed.

Thickness Dependence of Dielectric Characteristics of SrTiO&amp;lt;sub&amp;gt;3&amp;lt;/sub&amp;gt; Thin Films on MgAl&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;O&amp;lt;sub&amp;gt;4&amp;lt;/sub&amp;gt; Substrates

SrTiO3 (STO) thin films of different thicknesses were deposited on MgAl2O4 (MAO) substrates to investigate the in-plane strain effect on the soft-mode frequency of the STO films. X-ray reciprocal space mapping (X-RSM) results indicate that there was no relaxation of the in-plane lattice strain of the STO films on MAO. Shifts in the soft-mode frequencies with a decrease in the film thickness were observed using terahertz time-domain spectroscopy (THz-TDS). However, despite the larger lattice mismatch between STO and MAO than that between STO and DyScO3 (DSO), the shifts in the soft-mode frequencies of the STO films on MAO were smaller than those on DSO. The results indicate that the soft-mode frequencies of the STO films on MAO are affected by the c-axis (out-of-plane) lengths.

Lattice dynamics and broad-band dielectric properties of the KTaO3 ceramics

High-density KTaO3 ceramics were synthesized and studied by means of microwave, terahertz, infrared, and Raman spectroscopies. The results were analyzed together with recently published radio-frequency data. [S. Glinšek et al., J. Am. Ceram. Soc. 94, 1368 (2011)] Three polar modes expected for the cubic structure were observed. As in single crystals, the lowest-frequency TO1 mode (soft mode) strongly softens on cooling, while the TO2 and TO4 mode frequencies do not change with temperature. The permittivity does not show any significant dispersion below the soft mode frequency and its value in the kHz and GHz range is mainly given by the intrinsic polar lattice modes contribution. The soft mode frequency agrees with the values found in single crystals; this indicates a negligible influence of the grain boundaries on the dielectric response in KTaO3 unlike in other ferroelectric or incipient ferroelectric perovskite ceramics.

DIELECTRIC RELAXATION IN RELAXOR FERROELECTRICS

In this review the dielectric properties of relaxor ferroelectrics are discussed and compared with the properties of normal dielectrics and ferroelectrics. We try to draw a general picture of dielectric relaxation starting from a textbook review of the underlying concepts and pay attention to common behavior of relaxors rather than to the features observed in specific materials. We hope that this general approach is beneficial to those physicists, chemists, material scientists and device engineers who deal with relaxors. Based on the analysis of dielectric properties, a comprehensive definition of relaxors is proposed: relaxors are defined as ferroelectrics in which the maximum in the temperature dependence of static susceptibility occurs within the temperature range of dielectric relaxation, but does not coincide with the temperature of singularity of relaxation time or soft mode frequency.

Origin of the dielectric response in Ba0.767Ca0.233TiO3

We perform a Raman scattering study to elucidate the mechanism of temperature independent dielectric response of Ba0.767Ca0.233TiO3 (BCTO-0.233) from a viewpoint of phonon dynamics. BCTO-0.233 remains tetragonal below Tc down to 3.5 K. Over-damped soft phonon observed in a pure BaTiO3 is found to become underdamped in BCTO-0.233, suggesting that local dipoles induced by Ca2+ off-centering suppress the hopping of Ti ions to reduce damping of the soft-mode. The temperature dependence of the soft-mode frequency agrees qualitatively with the dielectric permittivity through Lyddane-Sachs-Teller relationship, indicating that the dielectric response of BCTO-0.233 is governed by the soft-mode.

Doping effects on dynamical physical properties of Ca0.6Sr0.4Bi4Ti4O15 ferroelectric ceramics with layered-perovskite structure

The change of electrical and lattice dynamical properties of ferroelectric Ca0.6Sr0.4Bi4Ti4O15 (CSBT) ceramics were studied by doping with 0.1 wt.% of CeO2 and MnO2. The loss tangent and the AC conductivity decreased markedly due to the reduction of charge carrier. The activation energy of doped-CSBT increased owing to the reduction of intrinsic defects in the crystal lattice. For doped and pure CSBT, the appearance of thermal hysteresis and the satisfaction of the Cure-Weiss law by the dielectric susceptibility data assigned the first order nature of a ferroelectric phase transition. Especially in ferroelectric phases, the soft optic phonon mode showed the significant softening towards T 1 (at T 1, the square of soft mode frequency \( \omega_{\text{s}}^2 \) → 0) and \( \omega_{\text{s}}^2 \) was approximately proportional to T 1-T, where T 1 is ~70°C above T C = 672°C, demonstrated a first order displacive nature of a ferroelectric phase transition. The invariance of lattice dynamical feature indicated that the crystal lattice did not influence markedly by the doping ions.

Lattice Vibrations in Finite Discrete Systems with Symmetric and Asymmetric Boundary Conditions

Lattice vibrations in finite discrete systems with symmetric and asymmetric boundary conditions are studied based on the Tilley-Zeks model. Two methods are adopted, i.e., one is the method utilizing analytical functions, and the other is the method of numerical solutions of an eigenvalue problem of the dynamical matrix form. It is found that, in finite systems, the normal modes, the soft-mode frequency and the transition temperature all depend strongly on the lattice constant a.

Off-center displacements and hydrostatic pressure induced phase transition in perovskites

Pressure has a profound effect on the paraelectric and ferroelectric properties of perovskite crystals. In this paper we theoretically investigate the effect of pressure on the cubic-to-tetragonal phase transition and on the soft mode dynamics of some classical perovskite crystals: BaTiO3, PbTiO3, and KNbO3. We use a model consisting of three subsystems: electrons, phonons, and off-center displacements treated as spins. Experiments show that pressure has a large effect on the tunneling and hopping of the off-center displacements, that in turn strongly affect the pressure dependence of the transition temperature and the soft mode frequency. This model, with a very small number of adjustable parameters, accounts quantitatively for the experimentally measured nonlinear pressure dependence of the cubic-to-tetragonal phase transition temperature, up to the critical pressure where the transition temperature is zero. It also accounts quantitatively for the pressure dependence of the soft mode frequency, which is finite at the phase transition in spite of the fact that the phase transition at elevated pressures is second order, and for the pressure dependence of the electronic gap energy.

Strain-Induced Ferroelectricity of a SrTiO3 Thin Film on a MgAl2O4 Substrate Observed by Terahertz Time-Domain Spectroscopy

A SrTiO3 thin film was deposited on a MgAl2O4(100) substrate with a 3.5 % larger lattice constant than that of the SrTiO3. The temperature dependence of the dielectric dispersion of the SrTiO3 thin film was measured by terahertz time-domain spectroscopy, and the dielectric strength and soft-mode frequency showed positive and negative peaks at around 170 K, respectively. In addition, a deviation of the dielectric constant from the theoretical fitting curve in the gigahertz range and an increase in the value of the damping factor were observed below the ferroelectric transition temperature. These phenomena strongly indicate the emergence of ferroelectricity at around 170 K.

Effect of periodicity and composition in artificial BaTiO3/(Ba,Sr)TiO3superlattices

We have prepared two series of BaTiO${}_{3}$/SrTiO${}_{3}$ (BT/ST) superlattices (SLs). In the first set, the modulation period (\ensuremath{\Lambda}) was varied from 136 to 16 \AA{}, whereas in the second set ST was replaced with ${\mathrm{Ba}}_{(1\ensuremath{-}x)}{\mathrm{Sr}}_{x}{\mathrm{TiO}}_{3}$ (BST) ($0lxl1$) with constant periodicity of \ensuremath{\sim} 136 \AA{}. A total stack height of 10 000 \AA{} (1 \ensuremath{\mu}m) was maintained for both systems. The compositional changes give large spectral changes at $x$ $=$ 40% where the lowest $E$ mode in BT/BST SL becomes underdamped. X-ray-diffraction patterns and surface topography of the modulated period revealed a Stranski-Krastanov growth mechanism. The upward frequency shift of the $E$(1TO) in SLs with decrease in \ensuremath{\Lambda} or Ba concentration in the BST layer was attributed to the in-plane compressive stress induced by the lattice mismatch of the constituent layers. [N.B., here the notation $E$(1TO) or ${A}_{1}$(2TO) simply designates the vibrational mode of $E$ or ${A}_{1}$ symmetry, with 1TO or 2TO designating the transverse optical branch that is lowest, 1, or second lowest, 2, in energy.] In the low-frequency region, the folded acoustic phonon doublets were observed. Since the low-frequency out-of-plane dielectric constant is directly related to the $E$(1TO) soft mode frequency, strain engineering in artificial SLs allowed us to modify their dielectric properties by varying either the Ba/Sr in BST sublayer concentration or the modulation periodicity in the superlattices.