Thermodynamic and phase-field studies of phase transitions, domain structures, and switching for Ba(Zr_xTi_1−x)O₃ solid solutions

Abstract

While extensive experimental activities have been carried out to study the phase transitions and ferroelectric properties of BaZr_xTi_1-xO₃ solid solutions, the corresponding theoretical understanding is largely lacking due to the unavailability of thermodynamic potentials for this system. In this work, an eighth-order polynomial of thermodynamic potential for BaZr_xTi_1-xO₃ (x≤0.3) solid solutions is established based on the existing potential coefficients of BaTiO₃ and theexperimentally measured phase diagram of BaZr_xTi_1-xO₃ solid solutions. It is then employed to predict and understand the domain structures and switching for BaZr_xTi_1-xO₃ (x ≤0.3) single crystals using phase-field simulations. The simulated domain structures and switching are consistent with thermodynamic analysis and available experimental measurements, validating the established thermodynamic potential. It is expected that this thermodynamic potential will find wide applications to studying the phase transitions and ferroelectric properties of BaZr_xTi_1-xO₃ (x ≤0.3) bulk and nanoscale materials.

Introduction

Phase-field method has been extensively applied to predicting the phase transitions and domain structure evolutions of ferroelectric materials under different thermal, mechanical, or electric fields [1], [2], [3], [4], [5], [6]. Phase-field simulations of a particular ferroelectric material system require its thermodynamic information in the form of a thermodynamic potential which is typically written as a polynomial of polarization. The coefficients of a potential are normally determined based on the propertiesof a single crystal in a single-domain state, either from experimental measurements [7], [8], [9] and/or first-principles calculations [10], [11], [12]. Pohlmann et al. [13] published a general procedure for establishing potential coefficients for ferroelectrics, requiring a set of input data including the phase transition temperatures, polarization, and dielectric susceptibilities. Largely due to the lack of sufficient available data, thermodynamic potentials have been developed only for a limited number of materials [2,14].

As a typical ferroelectric material, the thermodynamics of BaTiO₃ has been extensively studied since Devonshire wrote down a thermodynamic potential written as a polynomial of polarization up to sixth-order [15]. Bell and Cross modified the potential by assuming three temperature-dependent coefficients [16]. Li et al. extended the polynomial to eighth-order, which was applicable to describe the thermodynamics of BaTiO₃ thin films under largestrains [7]. Wang et al. proposed another eighth-order potential for a better description of the quantum mechanical effects at low temperature [17]. Those efforts have stimulated the development of thermodynamic potentials for BaTiO₃-solid solutions. For example, Cao et al. [18] constructed a thermodynamic potential for Ba_xCa_1-xTiO₃ solid solutions, based on which the predicted temperature-concentration phase diagram agreed well with the experimental observations. Huang et al. [19] developed a thermodynamic potential for Ba_1-xSr_xTiO₃ solid solutions by assuming a linear relationship between Sr-doping effect and hydrostatic pressure effect in BaTiO₃. Based on the established potential, the energy storage properties and the electrocaloric effects for Ba_1-xSr_xTiO₃ solid solutions were predicted by phase-field simulations.

BaZr_xTi_1-xO₃ is one of the most-studied BaTiO₃-based solid solutions due to its promising applications in energy storage and solid-state cooling [20], [21], [22], [23], [24], [25], [26]. Giant recoverable energy storage densities (> 100 J/cm³) have been demonstrated experimentally in domain-engineered BaZr_0.2Ti_0.8O₃ [20] and highly oriented BaZr_0.3Ti_0.7O₃ thin films [21]. The performances of BaZr_xTi_1-xO₃ in these applications are closely related to their phase transitions and domain structures, which can be systematically studied and simulated with the phase-field method [27], [28], [29], [30], [31], [32], [33], [34]. The first attempt to build a thermodynamic potential for BaZr_xTi_1-xO₃ solid solutions was made by Peng et al for several specific compositions. and the predicted phase diagram was largely consistent with the experimental measurements [35,36]. However, the thermodynamic potential coefficientsare only established for these specific compositions. One of the main objectives of this work is to construct a general thermodynamic potential with the coefficients as a function of Zr concentration.

In this work, we establish the thermodynamic potential of BaZr_xTi_1-xO₃ solid solutions by regarding the doping of Zr into BaTiO₃ as a chemical pressure since the pressure-dependent potential coefficients of BaTiO₃ is already available. This approach has been successfully employed to establish the thermodynamic potentials of Ba_xCa_1-xTiO₃ and Ba_1-xSr_xTiO₃ solid solutions [18,19]. As Zr has a larger atomic radius than Ti, the substitution of Ti by Zr is analogous to imposing a negative hydrostatic pressure on the BaTiO₃ lattice. The magnitude of this negative hydrostatic pressure is assumed to be positively correlated to the composition of Zr. Base on this assumption, we established the thermodynamic potential coefficients for BaZr_xTi_1-xO₃. The established thermodynamic potential was then employed to perform 3-dimensional phase-field simulations of domain structures and switching for bulk crystals of BaZr_xTi_1-xO₃. We also carried out targeted experiments of ferroelectric hysteresis loops to compare with the simulation results. The agreement between the experimental and simulation results validates the established thermodynamic potential.