Gradient Energy Coefficient Research Articles

Revealing the Role of Spacer Layer in Domain Dynamics of Hf0.5Zr0.5O2 Thin Films for Ferroelectrics

AbstractHfO2‐based ferroelectrics offer promises for next‐generation nonvolatile nanoscale devices owing to excellent CMOS compatibility and robust ferroelectricity at the nanoscale. However, fundamentally understanding the mechanism of polarization reversal and domain dynamics is challenging because the role of the spacer layer at the domain level in HfO2‐based ferroelectrics remains elusive. Here, it is realized that visualization of time‐resolved dynamics of nanoscale domain configurations in the epitaxial Hf0.5Zr0.5O2 thin films using phase‐field simulations. Scale‐free domain independent switchability (namely, irreducible stable polar domain down to 1 nm lateral size) and sharp domain walls that originate from weak interactions between adjacent domains characterized by the reduced gradient energy coefficient, which unravels the mesoscale mechanism of spacer layer in domain dynamics are demonstrated. Meanwhile, it is revealed that 180° polarization switching is dominated by the nucleation of a new domain with a high nucleation density, well described by the nucleation‐limited switching model. This study not only provides a fundamental mesoscale mechanism of the spacer layer, but also stimulates further studies on the manipulation of ultra‐scaled single domain state for designing high‐density and fast‐speed HfO2‐based ferroelectric memory.

Quantification of gradient energy coefficients using physics-informed neural networks

Ferroelectrics are a class of multifunctional materials widely used in actuators, sensors, and information storage because of their unique electromechanical properties. To understand and utilize these properties, it is necessary to investigate ferroelectric microstructure evolution, a dynamic, nonlinear, and multi-physics coupling phenomenon. Phase-field simulations are often adopted to mimic this phenomenon. One core of phase-field models is the free energy, where the gradient energy is a primary part. However, the quantification of the associated gradient energy coefficients is a long-standing challenge in computational materials. In light of the ease of physics-informed neural networks (PINNs) to incorporate both mathematical models and data, we present a promising method to address this challenge by solving an inverse problem with PINNs. Our investigation with labeled data from simulations demonstrates the excellent performance of this method. Our further investigation with labeled data from atomic-scale high-resolution measurements also shows encouraging results, thus charting a clear path to pragmatically overcome the aforementioned challenge.

Direction‐Dependent Lateral Domain Walls in Ferroelectric Hafnium Zirconium Oxide and their Gradient Energy Coefficients: A First‐Principles Study

AbstractTo understand and harness the physical mechanisms of ferroelectric hafnium zirconium oxide (HZO)‐based devices, there is a need for clear understanding of domain interactions, their dynamics, negative capacitance effects, and other multidomain characteristics. These crucial attributes depend on the coupling between neighboring domains quantified by the gradient energy coefficient (g). Furthermore, HZO has unique orientation‐dependent lateral multidomain configurations. To develop an in‐depth understanding of multidomain effects, there is a need for a thorough analysis of g. In this work, the energetics of multidomain configurations and domain growth mechanism corresponding to lateral domain walls (DWs) of HZO are analyzed and gradient energy coefficients are quantified using first‐principles density functional theory calculations. These results indicate that one lateral direction exhibits the following characteristics: i) DW is ultra‐sharp and domain growth occurs unit‐cell‐by‐unit‐cell, ii) the value of g is negative and in the order of 10−12 V m3 C−1, and iii) g reduces (increases) with compressive (tensile) strain. In contrast, in the other lateral direction, the following attributes are observed: i) DW is gradual and domain growth occurs in quanta of half‐unit‐cell, ii) g is positive and in the order of 10−10 V m3 C−1, and iii) g increases (reduces) with compressive (tensile) strain.

Extending Density Phase-Field Simulations to Dynamic Regimes

Density-based phase-field (DPF) methods have emerged as a technique for simulating grain boundary thermodynamics and kinetics. Compared to the classical phase-field, DPF gives a more physical description of the grain boundary structure and chemistry, bridging CALPHAD databases and atomistic simulations, with broad applications to grain boundary and segregation engineering. Notwithstanding their notable progress, further advancements are still warranted in DPF methods. Chief among these are the requirements to resolve its performance constraints associated with solving fourth-order partial differential equations (PDEs) and to enable the DPF methods for simulating moving grain boundaries. Presented in this work is a means by which the aforementioned problems are addressed by expressing the density field of a DPF simulation in terms of a traditional order parameter field. A generic DPF free energy functional is derived and used to carry out a series of equilibrium and dynamic simulations of grain boundaries in order to generate trends such as grain boundary width vs. gradient energy coefficient, grain boundary velocity vs. applied driving force, and spherical grain radius vs. time. These trends are compared with analytical solutions and the behavior of physical grain boundaries in order to ascertain the validity of the coupled DPF model. All tested quantities were found to agree with established theories of grain boundary behavior. In addition, the resulting simulations allow for DPF simulations to be carried out by existing phase-field solvers.

A phase-field investigation of factors affecting the morphology of uranium dendrites during electrodeposition

The growth of uranium dendrites has been one of the critical issues in the electrorefining of spent nuclear fuel for several decades. However, a comprehensive description of uranium dendrites, including the contribution of electrodeposition and the interface property, remains unavailable. In this paper, the phase-field method is applied to assess the effects of applied voltage, exchange current density, interfacial energy, interfacial anisotropy, interfacial noise, initial electrodeposit size, and initial electrodeposit density on the uranium dendrites. Furthermore, an approach for precisely estimating the phase-field parameters is developed to capture more details of dendrites, such as interfacial mobility, reaction constant, barrier height, and gradient energy coefficient. The results reveal that high applied voltage, high exchange current density and low interfacial energy stimulate the growth of dendrites with complex structures. The dendrites are dominated by the tip-controlled mechanism regardless of the applied voltage, exchange current density and interfacial energy, but for high interfacial energy, the growth of some side branches follows the thermodynamic suppression mechanism. Strong interfacial noise can enhance the anisotropy of the system, making it difficult to distinguish the principal branches from the side ones, which are intertwined with each other. Moreover, a parameter map is drawn to quantitatively predict the uranium dendrite patterns, which vary as a function of the densities and sizes of the initial electrodeposits. The cause of tip splitting is analyzed as either insufficient interfacial anisotropy or a large initial electrodeposit size. The numerical results are in good agreement with the experimental data and the current theory. Overall, this work provides a theoretical basis for controlling the morphology of uranium dendrites.

A Compact Model for Ferroelectric Capacitors Based on Multidomain Phase-Field Framework

A generalized and fast multidomain phase-field-based compact model for the metal-ferroelectric-metal (MFM) capacitor is presented. Time-dependent Landau–Ginzburg (TDGL) and Poisson’s equations are solved self-consistently to model the polarization dynamics. Additionally, physics-based empirical relationships for voltage-dependent kinetic and gradient energy coefficients are formulated. It is also demonstrated how gradient energy coefficient needs to be modified to accurately capture the physics of the device when a coarse simulation grid is used for fast computation. The developed model is 30 000 times faster than our prior multidomain phase-field model with no degradation in accuracy. Moreover, a further computational speedup has been achieved by decreasing the number of Poisson solver nodes with a slight compromise of accuracy. The model shows a good agreement with the experimental results of both transient characteristics and minor hysteresis loops. The proposed model has the potential to facilitate fast and accurate simulations of large-scale circuits containing ferroelectric capacitors.

Diffuse-interface modelling of multicomponent diffusion and phase separation in the U-O-Zr ternary system

The ternary mixture of uranium, oxygen and zirconium is investigated as a minimal model for corium, the mixture that forms after the meltdown of a nuclear reactor. Like corium, U-O-Zr exhibits a liquid–liquid phase separation between a metal-rich and an oxide-rich phase at high temperatures. A CALPHAD database built on an associate model is used to set up a ternary Cahn–Hilliard model, which can describe two-phase patterns in U-O-Zr. The interface structure and properties, which depend on the choice on the gradient energy coefficients in the free-energy functional, are studied in detail. It is found that interface adsorption is generally present due to the diffuse character of the interface, but that its magnitude is small, such that the model remains a robust and useful tool for future simulations of corium pool stratification dynamics.

Quantitative investigation of polar nanoregion size effects in relaxor ferroelectrics

Understanding the size effects of polar nanoregion (PNR) in relaxor ferroelectrics is vital for a deep understanding mechanism to improve piezoelectric/dielectric constants. In this work, we theoretically predict that, below a critical size, the polar nanoregions possess the collinear state with matrix phase, which leads to the enhancement of the piezoelectric effect. More significantly, the critical size quantitatively tends to increase with a reduced free energy barrier between the matrix and PNR phases. In contrast, it tends to decrease with gradient energy coefficient. This work promotes an understanding of the mechanism of the relaxor ferroelectric system and guides the design of high piezoelectric materials with a wide temperature range.

Adjoint model for estimating material parameters based on microstructure evolution during spinodal decomposition

Data assimilation techniques are attracting increasing attention because they enable researchers to estimate material parameters by integrating an advanced computational model with experimental data of microstructure evolution. In this study, we develop an adjoint model to integrate a phase-field model for spinodal decomposition with time-series measurement data of compositional field maps to estimate the unknown parameters in the phase-field model. As a case study, simultaneous estimation of six parameters (Gibbs energy parameters, gradient energy coefficient, etc.) in the phase-field model is considered. To confirm the effectiveness of the developed adjoint model, numerical tests called ``twin experiments'' are conducted using synthetic measurement data prepared in advance through phase-field simulation. In the twin experiments, the optimum estimates of six model parameters of interest are shown to coincide with true values, indicating that several model parameters can be successfully estimated. Furthermore, the effects of the standard deviation of measurement noise $(\ensuremath{\sigma})$ and the time interval of measurements $(\mathrm{\ensuremath{\Delta}}{t}_{\mathrm{meas}.}^{\ensuremath{'}})$ on uncertainties of optimum estimates of parameters $({\ensuremath{\sigma}}^{\mathrm{est}})$ are examined by conducting twin experiments. It is shown that there is a positive correlation between $\ensuremath{\sigma}$ and ${\ensuremath{\sigma}}^{\mathrm{est}}$ or between $\mathrm{\ensuremath{\Delta}}{t}_{\mathrm{meas}.}^{\ensuremath{'}}$ and ${\ensuremath{\sigma}}^{\mathrm{est}}$, which is essential information for designing experiments to estimate model parameters. The developed adjoint model is assumed to be useful for estimating unknown parameters (e.g., Gibbs energy parameters of a nonequilibrium phase) using time-series measurement data of microstructure evolution during spinodal decomposition.

Towards predictive simulations of spinodal decomposition in Fe-Cr alloys

Simulations of spinodal decomposition in an Fe-36 wt%Cr alloy at 773 K are performed by solving the non-linear Cahn–Hilliard equation, and the results are compared with atom probe tomography measurements. The influence of gradient energy coefficient, atomic mobilities and initial structure on the kinetics of spinodal decomposition is studied. It is shown that a proper initial structure, accounting for the thermal history above the miscibility gap, is crucial and enables predictive simulations of spinodal decomposition in Fe-Cr alloys.

Extended Larch\xe9\u2013Cahn framework for reactive Cahn\u2013Hilliard multicomponent systems

At high temperature and pressure, solid diffusion and chemical reactions between rock minerals lead to phase transformations. Chemical transport during uphill diffusion causes phase separation, that is, spinodal decomposition. Thus, to describe the coarsening kinetics of the exsolution microstructure, we derive a thermodynamically consistent continuum theory for the multicomponent Cahn–Hilliard equations while accounting for multiple chemical reactions and neglecting deformations. Our approach considers multiple balances of microforces augmented by multiple component content balance equations within an extended Larché–Cahn framework. As for the Larché–Cahn framework, we incorporate into the theory the Larché–Cahn derivatives with respect to the phase fields and their gradients. We also explain the implications of the resulting constrained gradients of the phase fields in the form of the gradient energy coefficients. Moreover, we derive a configurational balance that includes all the associated configurational fields in agreement with the Larché–Cahn framework. We study phase separation in a three-component system whose microstructural evolution depends upon the reaction–diffusion interactions and to analyze the underlying configurational fields. This simulation portrays the interleaving between the reaction and diffusion processes and how the configurational tractions drive the motion of interfaces.

Negative capacitance effects in ferroelectric heterostructures: A theoretical perspective

In a heterogeneous system, ferroelectric materials can exhibit negative capacitance (NC) behavior given that the overall capacitance of the system remains positive. Such NC effects may lead to differential amplification in local potential and can provide an enhanced charge and capacitance response for the whole system compared to their constituents. Such intriguing implications of NC phenomena have prompted the design and exploration of many ferroelectric-based electronic devices to not only achieve an improved performance but potentially also overcome some fundamental limits of standard transistors. However, the microscopic physical origin as well as the true nature of the NC effect, and direct experimental evidence remain elusive and debatable. To that end, in this article, we provide a comprehensive theoretical perspective on the current understanding of the underlying physical mechanism of the NC effect in the ferroelectric material. Based upon the fundamental physics of ferroelectric material, we discuss different assumptions, conditions, and distinct features of the quasi-static NC effect in the single-domain and multi-domain scenarios. While the quasi-static and hysteresis-free NC effect was initially propounded in the context of a single-domain scenario, we highlight that similar effects can be observed in multi-domain FEs with soft domain-wall (DW) displacement. Furthermore, to obtain the soft-DW, the gradient energy coefficient of the FE material is required to be higher as well as the ferroelectric thickness is required to be lower than some critical values. If those requirements are not met, then the DW becomes hard and their displacement would lead to hysteretic NC effects, which are adiabatically irreversible. In addition to the quasi-static NC, we discuss different mechanisms that can potentially lead to the transient NC effects. Furthermore, we discuss different existing experimental results by correlating their distinct features with different types of NC attributes and provide guidelines for new experiments that can potentially provide new insights on unveiling the real origin of NC phenomena.

Stress-dependence of dislocation dissociation, nucleation and annihilation in elastically anisotropic Cu

Dislocation dynamics under external stresses are an important manifestation of plastic behaviors of materials. In this study, based on the work of Shen et al. (2014), an improved microscopic phase field (MPF) model incorporating gradient energy term into the energy functional is developed to thoroughly investigate the stress dependence of whole dislocation evolution processes, including dislocation dissociation, nucleation, and annihilation in Cu with prominent elastic anisotropy. Necessary physical parameters such as elastic constants, lattice energy and gradient energy coefficients are all determined by atomistic molecular statics (MS) simulations. The residue theorem is utilized to accurately solve the anisotropic stress kernels as pre-factors characterizing the long-range elastic interaction produced by edge and screw components of mixed dislocations in anisotropic Cu. The Peierls stress of screw dislocations and stacking fault width under applied stress are evaluated by the MPF model and compared with those assessed by the semi-discrete variational Peierls-Nabarro (SVPN) model and MS simulations. The MPF model is then applied to screw dislocation dipoles and dislocation loops to examine possible dislocation evolution paths under different stress states. Through systematic simulations, we find the dislocation evolution regimes are closely related with the applied stress, and can be distinguished by three important critical stresses, i.e., critical stress, singular stress, and nucleation stress. Specifically, when the applied stress is less than the critical stress, the dislocation loops always collapse in the end, and do not depend on the final states of Shockley partials, which may exist or have annihilated. Furthermore, when the applied stress exceeds the nucleation stress, partial dislocations will spontaneously nucleate within stacking fault (SF) area, followed by a series of dislocation reactions until partial dislocations bρ1 and bρ3 are left in the end. Besides, the shear stress components perpendicular to the Escaig stress in slip plane are found to suppress the nucleation of partial dislocations in SF area.

Multi-Domain Negative Capacitance Effects in Metal-Ferroelectric-Insulator-Semiconductor/Metal Stacks: A Phase-field Simulation Based Study

We analyze the ferroelectric domain-wall induced negative capacitance (NC) effect in Metal-FE-Insulator-Metal (MFIM) and Metal-FE-Insulator-Semiconductor (MFIS) stacks through phase-field simulations by self-consistently solving time-dependent Ginzburg Landau equation, Poisson’s equation and semiconductor charge equations. Considering Hf0.5Zr0.5O2 as the ferroelectric material, we study 180° ferroelectric domain formation in MFIM and MFIS stacks and their polarization switching characteristics. Our analysis signifies that the applied voltage-induced polarization switching via soft domain-wall displacement exhibits non-hysteretic characteristics. In addition, the change in domain-wall energy, due to domain-wall displacement, exhibits a long-range interaction and thus, leads to a non-homogeneous effective local negative permittivity in the ferroelectric. Such effects yield an average negative effective permittivity that further provides an enhanced charge response in the MFIM stack, compared to Metal-Insulator-Metal. Furthermore, we show that the domain-wall induced negative effective permittivity is not an intrinsic property of the ferroelectric material and therefore, is dependent on its thickness, the gradient energy coefficient and the in-plane permittivity of the underlying insulator. Similar to the MFIM stack, MFIS stack also exhibits an enhanced charge/capacitance response compared to Metal-Oxide-Semiconductor (MOS) capacitor. Simultaneously, the multi-domain state of the ferroelectric induces non-homogeneous potential in the underlying insulator and semiconductor layer. At low applied voltages, such non-homogeneity leads to the co-existence of electrons and holes in an undoped semiconductor. In addition, we show that with the ferroelectric layer being in the 180° multi-domain state, the minimum potential at the ferroelectric-dielectric interface and hence, the minimum surface potential in the semiconductor, does not exceed the applied voltage (in-spite of the local differential amplification and charge enhancement).

Studying the parameter gradient energy coefficient of Polyethylene glycol as a function of molecular weight with Freed Contribution

Studying the parameter gradient energy coefficient of Polyethylene glycol as a function of molecular weight with Freed Contribution

Studying the Parameter Gradient Energy Coefficient of Polyethylene Glycol as a Function of Molecular Weight in conjunction with Freed Contribution

Studying the Parameter Gradient Energy Coefficient of Polyethylene Glycol as a Function of Molecular Weight in conjunction with Freed Contribution

Temporal Evolution of γ Precipitate in Haynes 282 During Ageing – Growth and Coarsening Kinetics, Solute Partitioning and Lattice Misfit

Elemental partitioning across the precipitate/matrix interface controls the kinetics of precipitate evolution, during growth and coarsening. We present the first integrated analysis of evolution of both the aspects of microstructure, i.e., the size of nano-scale, ordered, coherent γ′ precipitates as well as the lattice misfit at the interface, in light of elemental partitioning, applied to a Ni-based superalloy, Haynes 282. In this work, eexperimental analyses by atom-probe tomography, transmission electron microscopy, and x-ray diffraction are combined with thermo-kinetic modelling using ThermoCalc and TC-PRISMA. For the first time, rate controlling elements for growth and coarsening stages of γ′ precipitate have been identified: Ti diffusion into γ′ precipitate controls the growth rate; coarsening kinetics is controlled by the diffusion of Mo into the matrix. For identification of growth controlling element, a completely new approach has been adopted here that has not been reported earlier. For the coarsening kinetics, unlike most of the recent literature, we have used the thermodynamic parameter corresponding to the non-dilute, non-ideal γ solid solution phase in the modified LSW rate equation. This approach provides a much realistic prediction, as Ni-based superalloys show significant deviation from ideality. Misfit has been found to be positive and found to decrease with ageing time in the present alloy. Elemental partitioning explained quantitatively the variation in the lattice parameters and the misfit. Values of several important material properties, like interfacial energy, gradient energy coefficient at the matrix/ precipitate interface have also been determined.

Composition dependent gradient energy coefficient: How the asymmetric miscibility gap affects spinodal decomposition in Ag-Cu?

We report results for spinodal decomposition in Ag-Cu binary system. The asymmetric miscibility gap of any binary system necessarily means that the gradient energy coefficient κ is composition dependent. We demonstrate that while this composition dependence of κ does not modify the solubility curve and even the equilibrium interface profile hardly changes, the cutoff wavelength for spinodal decomposition does. Many calculations usually use composition independent gradient energy coefficient and the analytical expression for calculating the cutoff wavelength is valid also only for composition independent κ. We show that the κ-composition function can be calculated from the interaction energy V(c): κ=−1/12r02∂[(1−2c)V(c)]/∂c, where r0 is the interatomic distance. In this work we apply the Cahn-Hilliard theory and a 3D atomistic kinetic model, the Stochastic Kinetic Mean Field (SKMF), what we further developed purposefully. The improvement is not restricted to the Ag-Cu system, though, as the composition dependent interaction energy can be deduced e.g. from Redlich-Kister polynomials.

A comparison of different continuum approaches in modeling mixed-type dislocations in Al

Mixed-type dislocations are prevalent in metals and play an important role in their plastic deformation. Key characteristics of mixed-type dislocations cannot simply be extrapolated from those of dislocations with pure edge or pure screw characters. However, mixed-type dislocations traditionally received disproportionately less attention in the modeling and simulation community. In this work, we explore core structures of mixed-type dislocations in Al using three continuum approaches, namely, the phase-field dislocation dynamics (PFDD) method, the atomistic phase-field microelasticity (APFM) method, and the concurrent atomistic-continuum (CAC) method. Results are benchmarked against molecular statics. We advance the PFDD and APFM methods in several aspects such that they can better describe the dislocation core structure. In particular, in these two approaches, the gradient energy coefficients for mixed-type dislocations are determined based on those for pure-type ones using a trigonometric interpolation scheme, which is shown to provide better prediction than a linear interpolation scheme. The dependence of the in-slip-plane spatial numerical resolution in PFDD and CAC is also quantified.

Influence of bulk free energy density on single void evolution based on the phase-field method

Phase-field models are widely adopted to study the void evolution problem, overcoming difficulties and drawbacks of the sharp boundary approach and rate theory. This paper focuses on improving the performance of the phase-field model of which the vacancy concentration is used as the single conserved order parameter. Following recent developments in these phase-field models, the void dynamic, or more precisely the void growth rate which plays a vital role in characterizing the microstructural evolution of irradiated materials, cannot be accurately reproduced. Moreover, the interfacial energy or the void-matrix interface modeled by the Ginzburg-type gradient energy term is usually characterized by an empirical coefficient that is quite difficult to determine. For these reasons, a general relation of the bulk free energy density, the coefficient of the gradient energy term and the interface width is analytically derived from the planar interface case, and then validated by the numerical simulation example of a single void evolution in molybdenum (Mo). The obtained void growth rate agrees well with the prediction of rate theory while the interface width is smaller than a critical value in the considered cases regardless of the shape of the free energy density. This study will not only help to construct the appropriate formulation of the bulk free energy density, but will also provide an easy method of calculating the corresponding gradient energy coefficient and selecting the grid size according to the pre-estimated interface width.