Spherical Random Bond-random Field Model Research Articles

BaZr0.5Ti0.5O3: Lead-free relaxor ferroelectric or dipolar glass

Glassy freezing dynamics was investigated in ${\mathrm{BaZr}}_{0.5}{\mathrm{Ti}}_{0.5}{\mathrm{O}}_{3}$ (BZT50) ceramic samples by means of dielectric spectroscopy in the frequency range 0.001 Hz--1 MHz at temperatures $10lTl300$ K. From measurements of the quasistatic dielectric polarization in bias electric fields up to $\ensuremath{\sim}28$ kV/cm it has been found that a ferroelectric state cannot be induced, in contrast to the case of typical relaxors. This suggests that---at least for the above field amplitudes---BZT50 effectively behaves as a dipolar glass, which can be characterized by a negative value of the static third order nonlinear permittivity. The relaxation spectrum has been analyzed by means of the frequency-temperature plot, which shows that the longest relaxation time obeys the Vogel-Fulcher relation $\ensuremath{\tau}={\ensuremath{\tau}}_{0}\phantom{\rule{0.16em}{0ex}}\mathrm{exp}[{E}_{0}/(T\ensuremath{-}{T}_{0})]$ with the freezing temperature of 48.1 K, whereas the corresponding value for the shortest relaxation time is $\ensuremath{\sim}0$ K, implying an Arrhenius type behavior. By applying a standard expression for the static linear permittivity of dipolar glasses and/or relaxors the value of the Edwards-Anderson order parameter $q(T)$ has been evaluated. It is further shown that $q(T)$ can be described by the spherical random bond-random field model of relaxors.

Research on polarization effect for relaxor ferroelectrics by spherical random bond-random field model

Based on spherical random bond-random field model, micro-macro domain mechanism under electric field, and the fuzzy domain boundary with fractal dimension of electric field, the mechanism of polarization effect is analyzed. The results show that the effect of electric field on polarization effect of increment of domain dipoles leads to an unsaturated electric hysteresis loop and its associated big electrostrictive effect. But the variation of binding energy, when dipole couples to the increment of domain dipoles induced by electric field, has a little influence on electric hysteresis loop in a low electric field, and has almost no influence in a high electric field. The initial size of the micro domain is very important for electric hysteresis loop: small micro domain can lead to a long electric hysteresis loop and better linear relationship between electric field and electrostriction.

Electrocaloric effect in relaxor ferroelectrics

A theoretical model for the electrocaloric effect (ECE) in relaxor ferroelectrics is presented. By solving a self-consistent relation for the ECE temperature change ΔT and minimizing numerically the mean field free energy for relaxors, the field and temperature dependence of ΔT is calculated. The corresponding harmonic Landau coefficient a=a(T), which differs from the ferroelectric case by always being positive, is derived from the spherical random bond-random field model, and the fourth-order coefficient b is treated as a phenomenological parameter, which can be either positive or negative. For b<0, a line of field-induced first-order relaxor-to-ferroelectric phase transitions exists in relaxors, which terminates at a liquid-vapor type critical point ECP,TCP. The critical behavior close to ECP,TCP is analyzed. It is shown that near the first-order phase transition a temperature or field interval or gap formally appears, where ΔT cannot be found. However, domain formation in the coexistence range should restore the continuous behavior of the ECE observed in real systems. Finally, it is shown that the ECE responsivity R1=ΔT/E reaches a maximum near the critical point, in agreement with recent experiments.

Influence of the critical point on the electrocaloric response of relaxor ferroelectrics

The electrocaloric effect (ECE), i.e., the conversion of electric energy into heat, is of great importance for application in new generation cooling or heating devices that would be friendlier to the environment. Here, utilizing direct measurements of the ECE change of the temperature ΔT via a high resolution calorimeter, we study the ECE as a function of the magnitude of the electric-field step E in the vicinity of the critical point in several bulk relaxor ferroelectric ceramic systems. Relatively large ΔT of ∼2 to 3 K were obtained at modest fields of 90 kV/cm, even in the case of ceramic materials. The effective responsivity ΔT/E as a function of the electric field shows a characteristic peak near the critical point, which demonstrates the importance of proximity to the critical point for the enhancement of the electrocaloric effect. Experimental results are in good agreement with the theoretical calculations based on the spherical random-bond random-field model.

Spherical model of relaxor polymers

Recent dielectric experiments have revealed a striking analogy between inorganic relaxor ferroelectrics and disordered organic ferroelectric polymers. We propose a model description of these relaxorlike polymers by considering a statistical ensemble of randomly oriented polar nanoregions, embedded in an amorphous matrix. In analogy to the spherical random bond-random field (SRBRF) model of inorganic relaxor ferroelectrics, we assume that the polar nanoregions are coupled through infinitely ranged random interactions of a spin-glass type and are subject to random local fields. The second moment of the probability distribution of dipole moments is required to have a fixed value, which is equivalent to a global spherical condition. A general solution of the SRBRF model is presented, and the linear and nonlinear static dielectric susceptibilities are calculated and compared with the experimental data. The phenomenon of giant electrostriction in relaxor polymers is also discussed.

Heat Capacity and Thermal Expansion Studies of Relaxors

Heat capacity and thermal expansion of three perovskite-like solid solutions with nonisovalent PbMg1/3Nb2/3O3, PbFe1/2Ta1/2O3 and isovalent Ba0.92Ca0.08Ti0.76Zr0.24O3 substitution were measured using the methods of calorimetry and optical-mechanical dilatometry in a wide temperature range. The blurred anomalies in the Cp(T) and α(T) dependences are observed at Burns temperature, at the temperature of the dielectric permittivity maximum and at the temperature of the phase transition into ferroelectric state. The results are discussed together with the data on the structure and dielectric properties in the framework of spherical random bond-random field model.

Heat capacity study of relaxor PbMg1/3Nb2/3O3 in a wide temperature range

Heat capacity of the PbMg1/3Nb2/3O3 compound is measured using the methods of adiabatic and differential scanning calorimetry in the temperature range 80–750 K. Two blurred anomalies on the C p (T) dependence are observed in wide temperature intervals of 200–400 K and 500–700 K. The results of studies are discussed together with data on the structure and phonon spectrum in the framework of spherical random bond-random field model.

Field-cooled static nonlinear response of relaxor ferroelectrics

The third-order static field-cooled nonlinear dielectric permittivity in relaxor ferroelectric PbMg1/3Nb2/3O3 has been measured as a function of temperature. It is shown that the observed behavior can be explained by a mechanism of nonlinear response within the framework of the spherical random bond-random field model.

Determination of the quasistatic linear and non-linear dielectric response in relaxor materials

Temperature dependence of the first ε 1 and third harmonic dielectric response ε 3 in relaxor systems was studied both as a function of the dc electric field by utilising the charge accumulation technique and as a function of the ac electric field. Additional non-linear contribution was found in the ac field dependence of ε 1 in lead magnesium niobate single crystals and in lanthanum-modified lead zirconate titanate ceramics. This dominating contribution in the field dependence of ε 1( ω) has an origin in the fact that the increasing field shifts the relaxor state from the glassy to a particular crossover state in the glassy-to-ferroelectric crossover region. This was confirmed by quasistatic measurements of the total third-order non-linear dielectric response ε 3 t as a function of the dc electric field throughout the glassy-to-ferroelectric crossover region. It is shown that the total third-order non-linear dielectric response can be understood within the extended spherical random bond-random field model of relaxor ferroelectrics.

Dynamic Theory of Nonlinear Response in Relaxor Ferroelectrics

The dynamic nonlinear response of relaxor ferroelectrics is investigated in the framework of the spherical random bond-random field model. A new mechanism of nonlinear response due to the field modulation of the interaction between the polar clusters is introduced. It is shown that this mechanism is dominant in PMN and PLZT, and its contribution to the dynamic nonlinear permittivity is calculated.

Nuclear magnetic resonance study of the relaxor ferroelectric Pb(Sc1/2Nb1/2)O3

The Pb207, Sc45, and Nb93 nuclear magnetic resonance spectra have been measured for the relaxor ferroelectric single crystal Pb(Sc1/2Nb1/2)O3 both above and below Tc. The Sc45 spectra suggest that the “disordered” part of the crystal occupies about 60% and the “ordered” cubic part about 40% of the crystal volume. In agreement with the spherical random bond-random field (SRBRF) model the square of the polarization linearly increases with decreasing T below Tc in both the ordered as well as in the disordered regions. The Edwards–Anderson order parameter q similarly increases linearly with decreasing T below Tc. For the disordered region we find J0≈328 K and J≈227 K whereas Δ/J02≈0.006. In the ordered region J0≈297 K, whereas J and Δ are too small to be determined from this experiment. The results demonstrate that the SRBRF model works not only for pure relaxors but also for relaxor ferroelectrics.

Nonlinear dielectric response of relaxor PLZT ceramics in a dc bias electric field

The temperature dependences of the first, e 1 , and the third, e 3 , harmonic dielectric responses have been simultaneously measured in 9/65/35 PLZT ceramics in a dc bias electric field, higher than the critical field. Additional peaks in both, e 1 and e 3 , superimposed on the broad dispersive relaxor background were observed, indicating induced ergodic relaxor to ferroelectric phase transition. It is shown that the dielectric nonlinearity a 3 = e 3 /e 4 1 , as predicted by the spherical random bond-random field model of relaxor ferroelectrics, indeed monotonously decreases when approaching the relaxor to ferroelectric transition temperature.

Manifestation of freezing in the nonlinear dielectric response of PMN relaxor system

Dependence of the dielectric nonlinearity a3 on temperature has been determined in PMN single crystal. A crossover from the decreasing paraelectric-like to the increasing glass-like temperature behavior has been observed in the static a3 on cooling towards the freezing transition in the absence of electric field. It is shown that this crossover is actually predicted by the spherical random bond-random field model of relaxor ferroelectrics. Results support the idea that relaxor systems, at zero electric field undergo a transition into a low-temperature glassy state.

Glassy to inhomogeneous-ferroelectric crossover in (Pb, La)(Zr, Ti)O3 ceramics

The temperature dependence of the dielectric nonlinearity a3=ε3/ε14 has been determined in 9/65/35 (Pb, La)(Zr, Ti)O3 ceramics. In zero electric field a deep in the temperature behavior of the static a3 has been observed indicating a crossover from paraelectric-like to glass-like behavior on cooling towards the freezing transition. By varying a bias electric field another glass-to-ferroelectric crossover has been observed in the temperature dependence of the static a3. The observed crossovers as well as the temperature dependence of the linear static dielectric constant agree with the predictions of the spherical random-bond–random-field model of relaxor ferroelectrics.

45Sc and 93Nb NMR in relaxors

The narrow components in the 93Nb(I = 9/2) and 45Sc(I = 7/2) NMR spectra of single crystalline PMN and disordered single crystalline PST are due to the 1/2 → -1/2 NMR transition whereas the broad background components are due to satellite transitions. The macroscopic cubic symmetry is locally broken on the NMR time scale. From the NMR point of view, the structure is homogeneous and there is no need to invoke the existence of “pinned” chemical clusters. The observed NMR spectra in PMN and PST can be described in terms of reorientable dynamic polar nanoclusters-mainly due to Pb2+ site disorder-as invoked by the spherical random bond-random field model.

Electric-field induced phase transitions in 9/65/35 PLZT ceramics

Transition lines between various phases in the electric-field temperature phase diagram of 9/65/35 PLZT ceramics were determined by measurements of the temperature and electric-field-dependent dielectric constant. Above the critical field Ec the dc bias electric field induces relaxor to the ferroelectric (FE) phase transition. Transition lines were found to depend on the direction of the approach — the temperature or dc bias field direction — to the FE phase. Also, a strong impact of the electric field on the stability of the FE phase with the increasing temperature was found. Below Ec the system exhibits a transition line between the ergodic and nonergodic relaxor state. The experimental results are compared with the results of a spherical random bond-random field model of relaxor ferroelectrics.

Nonlinear response of relaxor ferroelectrics

The third order nonlinear response χ3 of relaxor ferroelectrics has been calculated within the framework of a recently proposed spherical random bond-random field model. It is assumed that the order parameter field can be associated with the polarization of nanosized polar clusters and is thus a continuous vector of variable length subject to a global spherical constraint. It is shown that for weak random fields the scaled third order nonlinear dielectric susceptiblity a 3 = χ3/χ4 1, where χ1 is the linear susceptibility, has a narrow peak in the spherical glass phase without long range order, but there is no sharp freezing transition. In case of weak random bonds the system behaves as a random field frustrated ferroelectric, with a 3 making a jump at the critical temperature.

Spherical random-bond–random-field model of relaxor ferroelectrics

A model of relaxor ferroelectrics based on the interacting polar clusters picture has been formulated. The electric dipole moment of a nanosized polar domain is allowed a large number of discrete orientations and its length is assumed to fluctuate in a broad interval. Introducing a set of quasicontinuous order parameter fields and imposing a global spherical constraint, the spherical random-bond--random-field (SRBRF) model is written down and its static properties are investigated. It is found that for weak random fields the scaled third-order nonlinear susceptibility ${a}_{3}={\ensuremath{\chi}}_{3}/{\ensuremath{\chi}}_{1}^{4}$ shows a nearly divergent behavior in the spherical glass phase, but there is no such anomaly in a random-field frustrated ferroelectric state. The probability distribution of local cluster polarization is calculated and its relation to the quadrupole perturbed NMR line shape of ${}^{93}\mathrm{Nb}$ in PMN is discussed. The fact that the observed line shape is Gaussian at all temperatures provides strong support to the SRBRF model.

Electric-field–temperature phase diagram of the relaxor ferroelectric lanthanum-modified lead zirconate titanate

Transition lines between various phases in the electric-field--temperature phase diagram of 9/65/35 lanthanum-modified lead zirconate titanate ceramics were determined by measurements of the temperature and electric-field-dependent dielectric constant. Above a critical field ${(E}_{C})$ the dc bias electric field induces a transition from the relaxor (R) to the long-range ferroelectric (FE) phase. In the temperature direction of the approach to the FE phase the R-FE transition line was determined from the field-cooled--field-heated dielectric susceptibilities, while depolarization temperatures were obtained from the field-cooled--zero-field-heated dielectric susceptibilities. A considerably large shift was found for the above two R-FE transition lines demonstrating the strong impact of the electric field on the stability of the FE phase with increasing temperature. It was found that below ${E}_{C}$ ergodicity is broken due to the divergence of the longest relaxation time at the freezing temperature ${T}_{0}=259 \mathrm{K}.$ Hence the system exhibits a transition line between the ergodic (ER) and nonergodic (NR) relaxor state. In the dc bias field direction of the approach to the FE phase, the temperature dependence of ${E}_{C},$ i.e., the transition lines between ER or NR and FE phases were studied by measurements of the complex dielectric constant as a function of a dc bias field at several fixed temperatures. The experimental results are compared with the results of a spherical random bond-random field model of relaxor ferroelectrics.