Soft Mode Response Research Articles

Origin of Terahertz Soft-Mode Nonlinearities in Ferroelectric Perovskites

Soft modes are intimately linked to structural instabilities and are key for the understanding of phase transitions. The soft modes in ferroelectrics, for example, map directly the polar order parameter of a crystal lattice. Driving these modes into the nonlinear, frequency-changing regime with intense terahertz (THz) light fields is an efficient way to alter the lattice and, with it, the physical properties. However, recent studies show that the THz electric-field amplitudes triggering a nonlinear soft-mode response are surprisingly low, which raises the question on the microscopic picture behind the origin of this nonlinear response. Here, we use linear and two-dimensional terahertz (2D THz) spectroscopy to unravel the origin of the soft-mode nonlinearities in a strain-engineered epitaxial ferroelectric SrTiO3 thin film. We find that the linear dielectric function of this mode is quantitatively incompatible with pure ionic or pure electronic motions. Instead, 2D THz spectroscopy reveals a pronounced coupling of electronic and ionic-displacement dipoles. Hence, the soft mode is a hybrid mode of lattice (ionic) motions and electronic interband transitions. We confirm this conclusion with model calculations based on a simplified pseudopotential concept of the electronic band structure. It reveals that the entire THz nonlinearity is caused by the off-resonant nonlinear response of the electronic interband transitions of the lattice-electronic hybrid mode. With this work, we provide fundamental insights into the microscopic processes that govern the softness that any material assumes near a ferroic phase transition. This knowledge will allow us to gain an efficient all-optical control over the associated large nonlinear effects.Received 14 September 2020Revised 20 January 2021Accepted 12 March 2021DOI:https://doi.org/10.1103/PhysRevX.11.021023Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasDielectric propertiesElectric polarizationElectronic excitation & ionizationElectronic transitionsFerroelectricityLight-matter interactionNonlinear opticsPhononsStrong electromagnetic field effectsUltrafast phenomenaTechniquesTerahertz time-domain spectroscopyUltrafast pump-probe spectroscopyCondensed Matter, Materials & Applied PhysicsNonlinear DynamicsAtomic, Molecular & Optical

Infrared, terahertz, and microwave spectroscopy of the soft and central modes in Pb(Mg1/3Nb2/3)O3

From the new infrared (IR) reflectivity and time-domain terahertz spectra combined with available high-frequency dielectric data above the megahertz range in a broad temperature range of 10 to 900 K, a full picture of the soft- and central-mode behavior in the classical relaxor ferroelectric $\mathrm{Pb}(\mathrm{M}{\mathrm{g}}_{1/3}\mathrm{N}{\mathrm{b}}_{2/3}){\mathrm{O}}_{3}$ (PMN) is suggested. A detailed comparison is made with the recent hyper-Raman spectroscopy data [Hehlen et al., Phys. Rev. Lett. 117, 155501 (2016)] and also with other available experiments based on inelastic light and neutron scattering. It is revealed that each type of experiment provides slightly different data. The closest agreement is with the hyper-Raman data: both techniques yield the same number of soft-mode components and the same high-temperature softening towards the temperature ${T}^{*}\ensuremath{\approx}400\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. In addition to evaluation of the IR-terahertz data using fitting with the standard factorized form of the dielectric function, we performed a successful fitting of the same data using the effective medium approach (EMA), originally based on the assumption that the mesoscopic structure of PMN consists of randomly oriented uniaxially anisotropic polar nanodomains (PNDs) with somewhat harder transverse optical polar modes in the direction along the local PND dipole [Hlinka et al., Phys. Rev. Lett. 96, 027601 (2006)]. Evaluation using Bruggeman EMA modeling has been successfully applied in the entire investigated temperature range. These results suggest that the response perpendicular to the local dipole moment, at high temperatures induced by random fields rather than PNDs, undergoes a classical softening from high temperatures with permittivity obeying the Curie-Weiss law, ${\ensuremath{\varepsilon}}_{\ensuremath{\perp}}=C/(T\ensuremath{-}{T}_{C})$, where $C=1.7\ifmmode\times\else\texttimes\fi{}{10}^{5}\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and ${T}_{C}=380\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, whereas the response parallel to it shows no softening. Below the Burns temperature, $\ensuremath{\sim}620\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, a gigahertz relaxation ascribed to flipping of the PNDs emerges from the soft-mode response, slows down, and broadens, remaining quite strong towards the cryogenic temperatures, where it can be assigned to fluctuations of the PND boundaries.

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.

The perfect soft mode: giant phonon instability in a ferroelectric

Previous studies of unstable (‘soft’) optical modes in ferroelectrics have reported minimum frequencies of 1 cm−1 (30 GHz) for underdamped phonons. In this work we fabricate a cylindrical coaxial specimen and rectangular plate waveguide specimens of tris-sarcosine calcium chloride (TSCC) and follow its soft mode several orders of magnitude lower to 1 GHz. Below 30 GHz the relaxation time is probably characteristic of domain wall motion; the new theory of Pakhomov et al (2013 Ferroelectrics at press) predicts 0.5 THz far from TC and a (T − TC)/TC dependence, in agreement with our experimental values. This discovery has implications for GHz electronics such as phased array radar or other voltage-tunable low-loss components. The mean-field frequency description of the soft mode response f(T) is supported via precision calorimetry on TSCC with and without Br-doping. The ferroelectric–antiferroelectric phase transition, previously suggested from high-pressure data, is confirmed at 45 K at 1 atm.

Calculation of the Spontaneous Polarization and the Dielectric Constant as a Function of Temperature for

The temperature dependence of the spontaneous polarizationPis calculated in the ferroelectric phase of KH2PO4(KDP) at atmospheric pressure (TC= 122 K). Also, the dielectric constantεis calculated at various temperatures in the paraelectric phase of KDP at atmospheric pressure. For this calculation ofPandε, by fitting the observed Raman frequencies of the soft mode, the microscopic parameters of the pure tunnelling model are obtained. In this model, the proton-lattice interaction is not considered and the collective proton mode is identified with the soft-mode response of the system. Our calculations show that the spontaneous polarization decreases continuously in the ferroelectric phase as approaching the transition temperatureTC. Also, the dielectric constant decreases with increasing temperature and it diverges in the vicinity of the transition temperature (TC= 122 K) for KDP according to the Curie-Weiss law.

Broad-band dielectric spectroscopy and ferroelectric soft-mode response in theBa0.6Sr0.4TiO3 solid solution

Ceramic Ba0.6Sr0.4TiO3 (BST-0.6) samples were studied in the broad spectral range of106–1014 Hz by using several dielectric techniques in between 20 and 800 K. The dominant dielectricdispersion mechanism in the paraelectric phase was shown to be of strongly anharmonicsoft-phonon origin. The whole soft-mode response in the vicinity of the ferroelectrictransition was shown to consist of two coupled overdamped THz excitations, whichshow classical features of a coupled soft and central mode, known from manyferroelectric crystals with a dynamics near the displacive and order–disorder crossover.Similar behaviour has been recently revealed and theoretically simulated in pureBaTiO3 (see Ponomareva et al 2008 Phys. Rev. B 77 012102 and Hlinka et al 2008 Phys. Rev. Lett. 101167402). Also for the BST system, this feature was confirmed by the theory based onmolecular dynamics simulations with an effective first-principles Hamiltonian. In all theferroelectric phases, additional relaxation dispersion appeared in the GHz range, assignedto ferroelectric domain-wall dynamics. The microwave losses were analysed from the pointof view of applications. The paraelectric losses above 1 GHz are comparable with those insingle crystals and appear to be of intrinsic multi-phonon origin. The ceramic BSTsystem is therefore well suited for applications in the whole microwave range.

Soft Mode Behavior in Cubic and Tetragonal BaTiO3 Crystals and Ceramics: Review on the Results of Dielectric Spectroscopy

History of the mode softening observation and its relation to the dielectric behavior around the cubic – tetragonal ferroelectric transition in BaTiO 3 is briefly reviewed. It is demonstrated that reliable microwave and recent THz dielectric data do not support any intrinsic dispersion in the clamped permittivity below the 10 10 –10 11 Hz range in pure single-domain BaTiO 3 crystals. Nevertheless, due to an intrinsic lattice disorder, the full soft mode response in the cubic phase was recently shown to consists of two overlapping overdamped anomalous modes in the 1–150 cm−1 range, which in the tetragonal phase in the A 1 -symmetry response strongly split into two well-separated modes. The lower mode in the 10–50 cm−1 range appears as an overdamped critical central mode whereas the higher modes in the 60–100 cm−1 and 200–280 cm−1 range in the cubic and tetragonal phase, respectively, soften only partially. The main part of the temperature dependent permittivity is due to the slowing down of the central mode. The results are supported by the recent first-principles based theory.

Infrared and Raman studies of the dead grain-boundary layers inSrTiO3 fine-grain ceramics

Infrared reflectivity, terahertz and Raman spectroscopy studies were carried out with two types ofSrTiO3 ceramic of mean grain size 1500 and 150 nm, and the results were compared withsingle-crystal data. Strong ferroelectric soft-mode stiffening was detected in the infrareddata upon the decrease of the grain size, fully compatible with the reduction of thelow-frequency permittivity in the whole temperature range 300–10 K. Weaker stiffening ofthis mode observed in the Raman spectra is explained by the higher phonon wavevectoractive in the Raman response and by a different geometry of its propagation. Raman dataalso confirmed a smaller tetragonality below the structural phase transition as compared tosingle crystals. This is explained by a special topology of tetragonal domains along thegrain boundaries, in which the tetragonal axes are oriented perpendicularly to them. Broadlow-frequency and low-temperature features in the Raman spectra are assigned to thesmeared first-order ferroelectric soft-mode response from the polarized regions(∼3 nm thick) with reduced permittivity attached to the very narrow grainboundaries. Such an effect is caused by a local frozen polarization of mean value∼15 µC cm−2 tending perpendicular to the grain boundary. To account for the fullpermittivity size effect, the grain boundary itself is modelled by a very thin(∼1 nm) dead layer with a strongly reduced local permittivity(∼8). The microscopic origin of the narrow dead layer is apparently a strong oxygen deficit, assuggested from recent local experiments and first-principles calculations by other authors.Coupling between the E-components of the ferroelectric and structural soft-mode doubletsobserved in the Raman spectra is due to the propagation of the E-symmetry polaritonsthrough the polar grain boundaries, whereas the non-coupled A-symmetry polaritons arelocalized inside the non-polar grain bulk. Since the structurally disordered grainboundaries are very narrow, taking up only a very small volume fraction, theireffect on the relaxation of the phonon momentum conservation can be neglected.

Soft elastic bending response of a nematic elastomer described by a microstructurally relaxed free energy

We consider the bending of a nematic elastomer slab using the material model of DeSimone and Dolzmann. This model supplies a coarse description for the mechanical response associated with three distinct microstructural phases: a solid phase, a smectic phase and a liquid phase. The liquid and smectic phases permit soft modes, in which a variety of deformations are consistent with the same external tractions. Multiple-phase solutions are constructed making use of a standard analytical procedure associated with the fact that the deformation is universal in isotropic, incompressible, hyperelasticity. This requires an exhaustive accounting of the possible phase combinations consistent with the bending deformation. It is shown how an original soft mode response at low flexure angle, wherein the slab is completely in the smectic phase, can give way to conventional (non-soft) response at a threshold flexure angle associated with the nucleation of solid phase at the external boundary. Two separate phase interface fronts then travel through the slab as the flexure angle increases further, quasi-statically transforming smectic phase to solid phase.

The peculiar optic, dielectric and X‐ray diffraction properties of a fluorinated de Vries asymmetric diffuse cone‐model ferroelectric liquid crystal

A new semi‐fluorinated chiral smectic liquid crystal, W504, is investigated by electro‐optic, dielectric and X‐ray scattering experiments. It exhibits a huge dielectric soft mode response, strong electroclinic effect and a birefringence which increases considerably with the director tilt angle θ; typical characteristics of a SmA*–SmC* transition following the de Vries asymmetric diffuse cone (ADC) model in which the non‐zero director tilt in SmC* arises through an ordering of tilting directions rather than an actual increase in average molecule tilt ⟨θmol⟩. In W504 a small increase in ⟨θmol⟩ of about 4° is however detected in the SmC* phase. Although the increase in molecule inclination is much less than the increase in director tilt θ, saturating close to 30°, it leads to a shrinkage of the smectic layers by about 1 Å, a result of the large initial molecule tilt in the SmA* phase, ⟨θmol⟩SmA*≈30°. The tilting transition in W504 is thus mainly an ADC model disorder–order transition, but it also has a component of a structural transition. The semi‐fluorinated molecular structure of W504 leads to a very weak electron density modulation along the layer normal, giving a vanishing form factor in bulk samples which exhibit no (0 0 1) X‐ray scattering peak. In thin films the (0 0 1) peak is however observed, indicating that the electron density modulation is enhanced by the breaking of the head–tail symmetry of the liquid crystal phase at the LC–air interface.

Dielectric Response of Soft Modes in Ferroelectric Thin Films

The concept of effective soft mode in thin films and its role in the effective dielectric response of high-permittivity thin films is introduced. Compared to bulk soft mode response, it may be strongly influenced by stresses from the substrate and dielectric inhomogeneities like interface layers, grain boundaries and porosity. The available techniques for the determination of the effective soft-mode response (far-infrared and time-domain THz spectroscopies) are discussed. The experiments on various ferroelectric (PbTiO 3, PZT, BaTiO 3, SrBi 2 Ta 2 O 9 ), incipient ferroelectric (SrTiO 3 ), doped incipient ferroelectric (SrTiO 3 :Ba), and relaxor ferroelectric (PLZT) films are briefly reviewed and compared with the results in bulk materials.

Origin of soft-mode stiffening and reduced dielectric response inSrTiO3thin films

The problem of the reduced dielectric response in thin films of high-permittivity materials is analyzed by studying the soft-mode response in several ${\mathrm{SrTiO}}_{3}$ thin films by means of Fourier transform far infrared, monochromatic submillimeter, and micro-Raman spectroscopies. A 300-nm-thick metalorganic chemical vapor deposition film, quasiepitaxially grown on a (0001) sapphire substrate with a perfect 〈111〉 orientation, displays a ferroelectric transition near 125 K induced by a tensile residual stress, appearing apparently simultaneously with the antiferrodistortive transition. On the other hand, polycrystalline chemical solution deposition films grown on (0001) sapphire, and also tensile stressed, show a harder soft mode response without the appearance of macroscopic ferroelectricity. This effect, which increases with the film thickness, is explained by a strong depolarizing field induced by the percolated porosity and cracks (in the 10-nm scale) along the boundaries of columnar grains (normal to the probe field direction). Brick-wall model calculations showed that 0.2 vol. % of such a porosity type reduces the permittivity from 30000 to less than 1000. The activation of the forbidden IR modes in the Raman spectra in the whole 80--300-K temperature range studied is explained by the effect of polar grain boundaries, in analogy with the bulk ceramics.

From Ferroelectric to Quantum Paraelectric: KTa1-xNbxO3 (KTN), a Model System

AbstractThe ABO3 perovskite oxides constitute an important family of technologically important ferroelectrics whose relatively simple chemical and crystallographic structures have contributed significantly to our understanding of ferroelectricity. They readily undergo structural phase transitions involving both polar and non-polar distortions from the ideal cubic lattice. This paper focuses on the mixed perovskite system KTa1-xNbxO3, or KTN, which has turned out to be a model system. While the end members KTaO3 and KNbO3 might be expected to be similar, in reality they exhibit very different properties. Their mixed crystals, which can be grown over the whole composition range, exhibit a rich set of phenomena whose study has added greatly to our current understanding of the phase transitions and dielectric properties of these materials. Included among these phenomena are soft mode response, ferroelectric (FE)-to-relaxor (R) crossover, quantum mechanical suppression of the transition, the appearance of a quantum paraelectric state and relaxational effects associated with dipolar impurities. Each of these phenomena is discussed briefly and illustrated. Some emphasis is on the unique role of pressure in elucidating the physics involved.

Dielectric, infrared, and Raman response of undopedSrTiO3ceramics: Evidence of polar grain boundaries

Thorough Raman and infrared (IR) reflectivity investigations of nominally pure ${\mathrm{SrTiO}}_{3}$ ceramics in the 10--300 K range have revealed a clear presence of the polar phase whose manifestation steeply increases on cooling. The Raman strengths of the Raman-forbidden IR modes are proportional to ${\ensuremath{\omega}}_{\mathrm{TO}1}^{\ensuremath{-}\ensuremath{\alpha}}(\ensuremath{\alpha}\ensuremath{\approx}1.6)$ where ${\ensuremath{\omega}}_{\mathrm{TO}1}$ is the polar soft mode frequency. No pronounced permittivity dispersion is observed below the soft mode frequency so that, as in single crystals, the static permittivity is essentially determined by the soft mode contribution. A theory is suggested which assumes a frozen dipole moment connected with the grain boundaries which induces the polar phase in the grain bulk in correlation with the bulk soft-mode frequency. This stiffens slightly the effective soft mode response and reduces the low-temperature permittivity compared to that of single crystals. Moreover, the polar soft mode strongly couples to the ${E}_{g}$ component of the structural soft doublet showing that the polar axis is perpendicular to the tetragonal axis below the structural transition which is shifted to 132 K in our ceramics. Whereas the ${\mathrm{TiO}}_{6}$ octahedra tilt (primary order parameter) below the structural transition corresponds to that in single crystals, much smaller ${A}_{1g}\ensuremath{-}{E}_{g}$ splitting of the structural soft doublet shows that the tetragonal deformation (secondary order parameter) is nearly 10 times smaller, apparently due to the grain volume clamping in ceramics.

Far-Infrared ferroelectric soft mode spectroscopy on thin films

Review on first quantitative far-infrared transmission measurements of the ferroelectric soft-mode response is given on several perovskite ferroelectric, antiferroelectric, relaxor ferroelectric and incipient ferroelectric films. The following films, mostly prepared by sol-gel technique on a sapphire substrate (film thickness from 100 to 1000 nm) were investigated: PbTiO3, PbZrO3, PbZr0.53Ti0.47O3, PbZr0.75Ti0.25O3, PLZT 9.5/65/35, 8/65/35, 2/95/5, SrTiO3, BaTiO3, Ba0.35Sr0.65TiO3, Ba0.65Sr0.35TiO3, and PbSc0.5Ta0.5O3. Whereas in the PZT system the soft mode parameters do not substantially differ from those in bulk materials, in the BST system dramatic differences have been observed, particularly in SrTiO3 at low temperatures. Low-frequency dielectric properties are also discussed.

Electrooptic Soft Mode Response of Compounds Exhibiting the Antiferroelectric Phase

We report measurements on the electrooptic response of thin samples (∼2-5 μm) of two antiferroelectric liquid crystals. All the phase transitions in these compounds can be very easily detected usin...

Electroclinic measurement of the coefficients of Landau expansion of some ferroelectric liquid crystals

The electroclinic effect in the smectic A phase of ferroelectric liquid crystals is a sensitive probe for measuring the soft mode response near the A-C* transition point. It is possible to evaluate several coefficients of the Landau theory describing the A-C* transition by simultaneously measuring the frequency dependence of both the optical signal and the current through the sample. We have made such measurements on a couple of homologous series [2S,3S]-4′-(2-chloro-3-methyl pentanoyloxy) phenyl-trans-4″-n alkoxy cinnamates synthesized in our labortory. The results show that the Landau meanfield theory is adequate to describe the A-C* phase transition in these compounds.

Fluctuation dynamics at the cooperative Jahn-Teller phase transition in TbVO4

Reports a detailed investigation of the critical dynamics of the cooperative Jahn-Teller phase transition in TbVO4 for frequencies less than 30 GHz by means of light scattering techniques. Two experimental scattering geometries are used which reveal the bare electronic response and also the full coupled electron-phonon soft-mode response of the system. The former consists of a single lorentzian central peak which does not show critical divergence but exhibits a linewidth roughly proportional to the absolute temperature T for T>TD, with a value of 12.5 GHz (HWHM) at TD, and which drops sharply for T<TD to a value of approximately 1.5 GHz (HWHM). The coupled soft-mode response contains a divergent central peak and two finite-frequency sidebands derived from the TA modes. The authors extend existing theory for the transition and derive response functions valid both above and below TD. These expressions provide a description of the observed spectra. Above TD the theoretical interpretation of the spectra provides a value of the spin-spin relaxation rate within the Tb3+ ion system (11.5 GHz near TD), while below TD it provides the spin-lattice relaxation rate ( approximately 1.5 GHz) which is observed to be insensitive to temperature.

Neutron scattering study of the ferroelectric phase transition of SbSI

The paraelectric-ferroelectric phase transition of SbSI has been studied using neutron scattering. Several low frequency phonon branches have been observed. In particular, it has been shown that the [100] and [010] transverse acoustic modes polarized along the [001] direction with the same symmetry as the soft mode, exhibit very anisotropic dispersion relations in reciprocal space. For large q values, an anomalous dispersion relation and a very large temperature dependence have been observed. All these effects are shown to be due to a coupling with a soft mode-central component entity. This entity was observed away from the zone center, but at the zone center, in contrast with the light scattering measurements, no well-defined soft mode response could be observed.

Measurement of the "soft" mode and coupled modes in the paraelectric and ferroelectric phases of KH2PO4at high pressure

Measurements of the pressure and temperature dependences of the soft-mode Raman spectra of potassium dihydrogen phosphate (KDP) in both the paraelectric (PE) and ferroelectric (FE) phases are reported. The soft-mode response is overdamped throughout the PE phase at atmospheric pressure, but can be made underdamped with pressure in this phase. Measurements at high pressure in the PE phase where the response is underdamped therefore allow a more accurate determination of the parameters for the coupled-mode system than one can obtain from measurements at atmospheric pressure. Measurement of the pressure dependence in the FE phase allows one to identify the soft mode of the system in this phase, and the results demonstrate that the proton motion remains coupled to an optic mode of the lattice for $T&lt;{T}_{C}$. The temperature and pressure dependences of the microscopic parameters are examined within the framework of both the pure tunneling and the coupled proton-optic-mode models for the phase transition. It is found that the pure tunneling model is inadequate for a detailed description of the spectra, whereas the modified tunneling model, or coupled-mode model, gives a more quantitative description of the results. While this model can be used for quantitative comparison between the Raman scattering and dielectric measurements in these materials, there are inadequacies of the model in describing the microscopic interactions. These features are discussed in detail.