Soft-mode Model Research Articles

Atomic-Level Description of Thermal Fluctuations in Inorganic Lead Halide Perovskites.

A comprehensive microscopic description of thermally induced distortions in lead halide perovskites is crucial for their realistic applications, yet still unclear. Here, we quantify the effects of thermal activation in CsPbBr3 nanocrystals across length scales with atomic-level precision, and we provide a framework for the description of phase transitions therein, beyond the simplistic picture of unit-cell symmetry increase upon heating. The temperature increase significantly enhances the short-range structural distortions of the lead halide framework as a consequence of the phonon anharmonicity, which causes the excess free energy surface to change as a function of temperature. As a result, phase transitions can be rationalized via the soft-mode model, which also describes displacive thermal phase transitions in oxide perovskites. Our findings allow to reconcile temperature-dependent modifications of physical properties, such as changes in the optical band gap, that are incompatible with the perovskite time- and space-average structures.

Jumplike Magnetic Disordering Processes Stimulated by a Magnetic Field in Systems with Structural Instability

A theoretical analysis of the features of first-order structural and magnetostructural transitions in magnetocaloric helimagnetic alloys of the Mn1 – xCrxNiGe system is performed. The observed structural transitions of the displacement type hex(P63/mmc) ↔ orth(Pnma) are described using the local soft mode model in the approximation of a displaced harmonic oscillator. In the absence of a magnetic field, the emergence of a helimagnetic order as a structurally induced second-order transition is described within the Heisenberg model with allowance for the dependence of the exchange integrals on the structural order parameters and elastic strains. In the presence of a magnetic field, it was found that the mutual approach of the characteristic temperatures for the helimagnetic state (HM(Pnma)) and the lability temperatures of the hexagonal paramagnetic state (PM(P63/mmc)), due to the action of the magnetic field, leads to the appearance of previously unexplored peripheral first-order magnetostructural transitions with insignificant magnetization jumps, increasing with increasing magnetic induction field. In this case, as the pressure increases to 4 kbar with a constant magnetic induction field, the peripheral transitions are transformed into reversible first-order magnetostructural transitions and, at even higher pressures (10–14 kbar), into full-scale first-order magnetostructural transitions with magnetization jumps comparable with the maximum magnetization. Experimental baric studies of the temperature dependences of the magnetization in static magnetic fields with an induction up to 1 T and pressure up to 14 kbar confirm the theoretical results.

Скачкообразные процессы магнитного разупорядочения, стимулированные магнитным полем в системах со структурной неустойчивостью

A theoretical analysis of the features of structural and magnetostructural first-order phase transitions in magnetocaloric helimagnetic alloys of the Mn_{1-x}Cr_{x}NiGe system has been carried out. To describe the observed displacive structural transitions hex(P6_{3}/mmc)<->orth(P_{nma}), we used the local soft mode model in the approximation of a biased harmonic oscillator. In the absence of a magnetic field, the emergence of a helimagnetic order as a structurally induced second-order transition was described in the framework of the Heisenberg model, taking into account the dependence of the exchange integrals on the structural order parameters and elastic strains. In the presence of a magnetic field, it was found that the approximation of the characteristic temperatures for the helimagnetic HM(P_{nma}) and lability temperatures of the hexagonal paramagnetic PM(P6_{3}/mmc) states, due to the influence of the magnetic field, leads to the appearance of previously unexplored peripheral magnetostructural first-order phase transitions with insignificant magnetization jumps that increase with increasing magnetic induction.In this case, as the pressure increases to 4 kbar with constant induction of the magnetic field, the peripheral transitions transform into reversible first-order magnetostructural transitions, and at even higher pressures (10-14 kbar) into full-fledged first-order magnetostructural transitions with magnetization jumps comparable with maximum value of magnetization. Experimental pressure studies of the temperature dependences of magnetization in static magnetic fields with an induction of up to 1 T and a pressure of up to 14 kbar confirm the theoretical results.

Structural and optical properties of rare earth–doped (Ba0.77Ca0.23)1−x(Sm, Nd, Pr, Yb)xTiO3

The structural, dielectric, and vibrational properties of pure and rare earth (RE)-doped Ba0.77Ca0.23TiO3 (BCT23; RE = Nd, Sm, Pr, Yb) ceramics obtained via solid-state reaction were investigated. The pure and RE-doped BCT23 ceramics sintered at 1450 °C in air for 4 h showed a dense microstructure in all ceramics. The use of RE ions as dopants introduced lattice-parameter changes that manifested in the reduction of the volume of the unit cell. RE-doped BCT23 samples exhibit a more homogenous microstructure due to the absence of a Ti-rich phase in the grain boundaries as demonstrated by scanning electron microscopy imaging. The incorporation of REs led to perturbations of the local symmetry of TiO6 octahedra and the creation of a new Raman mode. The results of Raman scattering measurements indicated that the Curie temperature of the ferroelectric phase transition depends on the RE ion and ion content, with the Curie temperature shifting toward lower values as the RE content increases, with the exception of Yb3+ doping, which did not affect the ferroelectric phase transition temperature. The phase transition behavior is explained using the standard soft mode model. Electronic paramagnetic resonance measurements showed the existence of Ti vacancies in the structure of RE-doped BCT23. Defects are created via charge compensation mechanisms due to the incorporation of elements with a different valence state relative to the ions of the pure BCT23 host. It is concluded that the Ti vacancies are responsible for the activation of the Raman mode at 840 cm−1, which is in agreement with lattice dynamics calculations.

On the nature of exponential decrease of boson peak intensity tail, at increasing frequency higher than the peak position

In the present Letter, an exponential decrease of a boson peak intensity tail, at increasing frequency higher than the peak position, is predicted in the soft-mode model of low-energy glass dynamics for a series of glasses which exhibit a high-frequency sound above the peak. A similar theoretical interpretation is suggested for recent experimental data which show for some glasses a resembling behavior of the peak intensity tail.

On some manifestations of soft atomic motion modes in glass dynamics

Recent advances in investigations of low-energy atomic dynamics of glasses have led to new essential experimental data and theoretical models for a qualitative description of the universal dynamic and thermal properties at low temperatures and frequencies, which are “anomalous” in the sense that they significantly differ from the properties of crystalline materials interpreted in the Debye model of low-energy phonons. One of the models is the soft-mode model in which the anomalous properties are determined by contributions of nonacoustic soft atomic motion modes for a minority of atoms, characterized by spring constants much smaller than those related to standard Debye acoustic modes for the vast majority of atoms. The present Letter interprets a recently observed effect in the attenuation of Debye acoustic phonons in glasses like SiO 2 and predicts a strong suppression of the universal dynamic properties of glasses under moderately high pressure as manifestations of “soft” atomic motion modes.

Electric properties of soft PZT ceramics under combined electric and mechanic fields

The effect of uniaxial pressure (0–1000 bars) applied parallel or perpendicular to the ac electric field on electric properties of soft PZT ceramics has been investigated. It was found that uniaxial pressure significantly influences these properties. With increasing pressure the peak intensity in the ε(T) curve decreases, becomes diffuse and shifts to a higher temperature, dielectric hysteresis is reduced, and dielectric dispersion decreases. It was shown that the application of pressure perpendicular to the electric poling direction improves the ferroelectric properties. It was concluded, that applied uniaxial pressure or increasing Ti-ion concentration in PZT system induces similar effects. The results were discussed in term of Cochran soft-mode model and domain switching under influence of pressure.

Order-parameter saturation at low temperatures: Displacive phase transitions with coupled Einstein oscillators

Quantum saturation of displacive order parameters may be observed at temperatures higher than expected in a simple soft-mode model. The experimental results have been explained sometimes by coupling of the softphonon branch with hard mode branches. We analyse the effect of this coupling in the limit of coupled Einstein oscillators. It is shown that a significant change through this mechanism of the low-temperature quantum saturation of the order parameter requires abnormally large coupling terms. In any case, under favourable circumstances, the coupled hard-phonon spectrum can be derived from the second temperature derivative of the square of the measured order parameter.

Nonlinear dielectric susceptibility of dipole impurities dissolved in the lattice of quantum paraelectrics

A general relation is derived for calculating the nonlinear dielectric susceptibility of a system placed in an external ac magnetic field, which consists of tunneling and interacting dipole impurities described by the Hamiltonian of the transverse Ising model and a lattice of ions whose dynamics is specified in the soft-mode model with the possible ferroelectric phase transition suppressed by zero-point quantum oscillations. It is shown that the relaxation peaks in the dielectric response can be enhanced several times through relaxator interaction with the soft mode. A general expression is obtained to describe the deviation from the Arrhenius law in this case. The results of this work can be employed for interpretation of recent experiments that revealed a strong effect of isovalent impurities at extremely low concentrations on the macroscopic properties of incipient ferroelectrics.

Theory of displacive phase transitions in minerals

A lattice-dynamical treatment of displacive phase transitions leads naturally to the softmode model, in which the phase-transition mechanism involves a phonon frequency that falls to zero at the transition temperature. The basic ideas of this approach are reviewed in relation to displacive phase transitions in silicates. A simple free-energy model is used to demonstrate that Landau theory gives a good approximation to the free energy of the transition, provided that the entropy is primarily produced by the phonons rather than any configurational disorder. The rigid unit mode model provides a physical link between the theory and the chemical bonds in silicates and this allows us to understand the origin of the transition temperature and also validates the application of the soft-mode model. The model is also used to reappraise the nature of the structures of high-temperature phases. Several issues that remain open, such as the origin of first-order phase transitions and the thermodynamics of pressure-induced phase transitions, are discussed.

Breakdown of the soft-mode model of KDP—low-frequency Raman spectroscopy under high pressure

Abstract The Raman scattering of KDP has been carried out under high pressure 0-5 GPa at two scattering angles, 90 and 4.5 degrees. The low frequency components of the B2(z) spectra in x(yx)y and z(yx)z+δ geometries have been analyzed in terms of a coupled soft-phonon model. The wave vector dependence of the soft-mode frequency contradicts the dispersion relation of the ferroelectric soft-mode near the Γ point of the Brillouin zone. The low frequency component of the B2(z) spectra is concluded not to be caused by the soft mode.

Structure of the incommensurate phase of BCCD at 130 K

Abstract The incommensurate structure of BCCD at 130 K with q = 0.296c∗ has been determined in the superspace-group P(Pnma):(1, s, −1) by means of single crystal X-ray diffraction data including main and first-order satellite reflections. Only zero and first-order harmonics have been considered in the modulation functions. The final agreement factors are R = 0.048, Ro = 0.042 and R1 =0.067, for all, main and first-order satellite reflections respectively. Only the atomic modulations of the betaine molecule could be interpreted in terms of rigid-body modulated motions. The symmetry of the structure is in accordance with a single A3 soft-mode model for the phase transition sequence.

Raman scattering study of KDP under high pressure with a diamond anvil cell

Abstract Raman scattering spectra of the ferroelectric KDP (KH2PO4) have been observed under hydrostatic pressure up to 52.0 kbar using a diamond anvil cell at room temperature. A curious dispersion relation of the soft mode at the Γ point of the Brillouin zone is derived from the low-frequency components of the B2(z) spectra in z(yx)z + δ and x(yx)y geometries after numerical fitting based on the coupled phonon model. The dispersion relation suggests breakdown of the soft-mode model of KDP. Each of the frequencies of the internal v1, v2 and v3 modes of the PO4 ion show a linear dependence on pressure with the mode Gruneisen parameter, 0.32, 0.31 and a small value of 2.5 × 10−2, respectively. Increase of the frequencies of the v1 and v2 modes corresponds to pressure dependence of the P–O distance. A new pressure-induced phase transition appears at 40.0 kbar with a growing relaxational central peak in the spectra.

On the ω-qRelation of the Soft Mode of KDP Studied by Light Scattering under a High Pressure

A low frequency component of the B 2 spectra of KDP in z ( y x ) z +δ and x ( y x ) y geometries has been observed under a hydrostatic pressure in a range 0-52.0 kbar using a diamond anvil cell. Numerical fitting of the spectra based on the coupled phonon model derives a curious ω- q relation of the soft mode at Γ point of the Brillouin zone. The ω- q relation suggests break-down of the soft-mode model of KDP. The pressure dependence of the spectral shape confirms the existence of the pressure-induced phase transition at 27.5 kbar reported previously by an X-ray work. A new central component grows rapidly at 40.0 kbar indicating a new phase transition induced by pressure at room temperature.

Quasi-one-dimensional model of pretransitional soft mode behavior

Pretransitional effects at displacive phase transitions are temperature dependent responses to fluctuations of the order parameter; these give rise to 1-D correlations in martensitic transformations when lattice dynamical constraints divide the “hard mode” 3-D correlation of the Bain distortion in order to maintain the mean field on a macroscopic scale. The quasi-one-dimensional (QOD) soft mode model of the latttice-variant-shear-theory (LVST) is described and its relevance to pretransitional nucleation and soft mode behavior, discussed. The 1-D correlations give rise to localized soft modes aboveT m , which nucleate microdomains of an intermediate phase at dissociated dislocations; these grow with second order kinetics and maintain the mean field when microdomains of opposite displacement vector cancel each other. Pretransitional behavior, predicted by LVST, are compared with experimental data in a wide range of materials and show why materials with different order parameters show similar transformation behavior.

Dynamical study of a central-mode in KDP

Abstract A central-mode, which grows up near the phase transition temperature Tc of KDP, has been observed in the x(yx)y geometry by the use of a high-resolution light-scattering system. The spectral analysis of the central-mode has been carried out by using either a damped harmonic-oscillator model or a relaxational Debye model. The spectral curves calculated by both models are in good agreement with experimental curves. However, in the damped harmonic-oscillator model, the frequency and the damping constant of oscillator do not follow the prediction of the soft-mode model. On the other hand, in the case of Debye model, it is found that relaxation time shows a critical slowing down and static susceptibility obeys the Curie-Weiss law. Therefore, it is concluded that the origin of the central-mode is the fluctuating polarization which become divergent near Tc .

Theory and mechanisms for martensitic transformations in ferroelectrics. Part III. successive transitions

Abstract A theory of martensitic transformations is applied to successive transitions (proper and improper) in ferroelectrics. The quasi-one-dimensional (QOD) soft mode model applies to proper transitions and the modified model to improper ones.

A localized soft mode model for the nucleation of thermoelastic martensitic transformation: Application to the β → 9r transformation

A nucleation model for bcc → 9R martensitic transformation has been developed based on the experimental data from a Cu-Zn-Al alloy. It has been shown through the third order elastic constants values that the C′ = 1/2(C11 - C12) elastic constant in the bcc phase is very sensitive to the homogeneous {011} (011) shear strains. Consequently it has been demonstrated that, around defects like dislocations, mechanically unstable zones are present where C softens dramatically. The C′ constant is related to the {011} (011) type shear which is precisely the homogeneous lattice strain involved in the bcc → 9R transformation. Nuclei can, therefore, develop in such zones without generating any resistive strain energy. Nucleation is postulated to occur in the zones when the reduced critical nucleus size becomes equal to the unstable zone size.

Brillouin Scattering Study of Cubic CsPbCl3

Brillouin scattering spectra of perovskite-type crystal CsPbCl 3 have been measured above its cubic-tetragonal phase transition temperature, T c =47^° C . The three elastic constants c 11 , c 12 and c 44 were determined in a wide temperature range. The result agrees quite well with that of the previous ultrasonic measurement. This fact shows that there is no detectable elastic dispersion between 10 7 Hz and 10 10 Hz. An analysis using a single relaxation time model shows that the elastic relaxatuon time is shorter than 1 ×10 -11 sec at T = T c +3^° C . The soft-mode model rather than the order-disorder one seems to be more appropriate to a description of the cubic-tetragonal phase transition in this crystal.

Pretransformation effects of localized soft modes on neutron scattering, acoustic attenuation, and Mössbauer resonance measurements

The localized soft mode model of martensitic nucleation is used to predict pretransformation effects of such regions on inelastic neutron scattering, ultrasonic acoustic attenuation and Mossbauer resonance absorption. The theoretical predictions are compared with the available experimental data.