Analysis method for regulating and controlling high-purity ferroelectric orthogonal phase of hafnium zirconium oxide ferroelectric film

Abstract

The invention discloses an analysis method for high-purity ferroelectric orthogonal phase regulation and control of a hafnium zirconium oxide ferroelectric film. The analysis method comprises the following steps: constructing a total free energy functional described by gradient energy density containing phase boundary energy and a non-ferroelectric phase; establishing a control equation set of multi-physics field coupling, and solving the control equation set; and carrying out data visualization and quantitative analysis on the output sequence parameter distribution, the phase distribution and the strain field, and quantitatively extracting key physical parameters from evolution data. According to the method, the phase boundary energy is accurately described, so that the model can truly reflect the characteristics of phase boundary migration, the domain overturning and phase boundary migration process in an external field can be visually and quantitatively analyzed, the key process which is difficult to capture in the traditional technology is broken through, and the accuracy and reliability of simulation prediction are remarkably improved. The method can be directly used for guiding the optimization of an experimental process, and meanwhile, the trial and error cost and time are remarkably reduced.

Description

Technical Field

[0001] This invention belongs to the field of microelectronics technology, and further relates to an analytical method for high-purity ferroelectric orthogonal phase modulation of hafnium-zirconium ferroelectric thin films within the field of microelectronic testing and analysis technology. This invention enables dynamic visualization of ferroelectric phase transitions and domain structure evolution, providing theoretical guidance for the design of nanoelectronic devices based on HZO ferroelectric thin films and the optimization of the performance of hafnium-based ferroelectric memories. Simultaneously, it establishes a general simulation method applicable to the study of phase modulation and domain dynamics in ferroelectric materials. Background Technology

[0002] Hafnium oxide-based ferroelectric materials have been considered ideal candidates for next-generation non-volatile memories in recent years due to their compatibility with CMOS processes and their ability to maintain good ferroelectricity at the nanoscale. However, in practically fabricated HZO thin films, multiple crystal structures often coexist, i.e., multiphase coexistence, mainly including a polar orthorhombic phase (o phase), a paraelectric tetragonal phase (t phase), and a nonpolar monoclinic phase (m phase). This multiphase coexistence microstructure has a decisive influence on the overall ferroelectric performance of the thin film. For example, the interconversion (phase transition) between different phases under changing external conditions (such as temperature and electric field) is the intrinsic reason for key phenomena such as wake-up effect and fatigue characteristics in devices; while the presence of non-ferroelectric phases (such as the m phase) can cause a pinning effect on the movement of ferroelectric domain walls, significantly restricting the uniformity and reliability of device performance. Therefore, how to effectively suppress the formation and growth of non-ferroelectric phases and prepare high-purity ferroelectric orthorhombic phase (o phase) thin films has become a core challenge for improving the performance of hafnium-based ferroelectric devices.

[0003] Xiangtan University disclosed a phase-field analysis method for hafnium oxide-based ferroelectric thin films based on multistate coexistence in its patent application document "A Phase-Field Analysis Method for Hafnium Oxide-Based Ferroelectric Thin Films Based on Multistate Coexistence" (Application No.: 201911219037.2, Application Date: December 3, 2019, Authorization Announcement No.: CN 110866349 B). This method determines the number of order parameters based on the number of states in a multi-state hafnium oxide-based ferroelectric thin film. Based on the order parameters, Ginzburg-Landau theory, and the bulk energy, elastic energy, gradient energy, and electrostatic energy of the ferroelectric thin film, the energy equation expression and coefficients of the ferroelectric thin film are determined. Combining the mechanical equilibrium equation, Maxwell's equation, and Ginzburg-Landau phase-field dynamics equation, weak forms of the force field, electric field, and polarization field are derived, establishing a phase-field model of multi-state coexistence in hafnium oxide-based ferroelectric thin films under electromechanical coupling. The domain structure and its evolution in multi-state hafnium oxide-based ferroelectric thin films are simulated based on the phase-field model. The formation and growth process of domain structures in HfO2-based ferroelectric thin films are dynamically simulated, and the redistribution of domain structures under different external fields is studied, providing guidance for experimenters and reducing experimental costs. The shortcomings of this method are that it lacks precision in describing the thermodynamic relationships of multiphase (especially orthorhombic o-phase, tetragonal t-phase, and monoclinic m-phase), its gradient energy term is of a single form, and it cannot distinguish between domain boundaries (domain walls) and phase boundaries with significant differences in energy scale. At the same time, it fails to effectively integrate key experimental parameters such as substrate-induced non-uniform stress fields, making it difficult to reliably predict external conditions (such as specific strain states) that are conducive to the formation and stability of high-purity ferroelectric orthorhombic phases (o-phase), and it cannot truly reflect the regulatory effect of external fields on phase purity and domain flipping dynamics.

[0004] Xi'an University of Electronic Science and Technology disclosed a phase-field simulation and analysis method for anisotropic domain walls of hafnium oxide-based ferroelectric thin films in its patent application "Phase-Field Simulation and Analysis Method for Anisotropic Domain Walls of Hafnium Oxide-Based Ferroelectric Thin Films" (Application No.: 202510855275.1, Application Date: June 25, 2025, Publication No.: CN120783916 A). The implementation steps of this method include: 1) determining the total energy equation and correlation coefficients; 2) establishing a gradient energy density equation based on domain wall anisotropy; 3) constructing and solving the governing equations; 4) visualizing the simulation results; and 5) analyzing the domain evolution and polarization reversal characteristics under different voltages. This method addresses the problem that existing methods struggle to reflect the influence of domain wall anisotropy on domain evolution and polarization reversal dynamics, and its correlation with macroscopic electrical properties. This method achieves accurate simulation and quantitative analysis of anisotropic domain walls and their migration behavior, providing guidance for researchers to understand the influence of anisotropic domain walls on domain flipping characteristics and macroscopic electrical properties, while reducing experimental costs and improving efficiency. However, this method still has limitations: it does not consider nonferroelectric phases, thus failing to accurately reflect the phase composition of the actually prepared thin films, exhibiting significant limitations; furthermore, because it does not consider the m-phase, it cannot reflect the influence of phase boundaries.

[0005] In summary, existing technical solutions have significant limitations in addressing the core issue of stabilizing and regulating high-purity ferroelectric orthorhombic phases (o phases). They are insufficient to support the design of efficient phase regulation strategies, particularly in their inaccurate description of the thermodynamic relationships of the HZO system. Furthermore, due to the lack of consideration for non-ferroelectric phases or the simplification of gradient energy, they cannot accurately describe the actual thin films. They also severely underestimate phase boundary energy, making it difficult to reveal the crucial role of phase boundaries in phase stability and domain flipping dynamics. Moreover, they cannot clearly explain the microscopic influence mechanism of phase boundaries on the evolution behavior of ferroelectric domains in complex environments with multiphase coexistence. Summary of the Invention

[0006] The purpose of this invention is to address the shortcomings of the prior art by providing an analytical method for high-purity ferroelectric orthogonal phase control in hafnium zirconium ferroelectric thin films. This method aims to solve the problems of existing technologies where traditional phase-field models do not consider non-ferroelectric phases and have a single gradient energy term form, making it impossible to distinguish between domain boundaries (domain walls) and phase boundaries with significant differences in energy scale.

[0007] The technical approach to achieving the objective of this invention is to employ gradient cross terms capable of independently describing phase boundary energies, thus enabling separate characterization of domain boundary energies and phase boundary energies. This technological breakthrough allows the model to accurately describe the thermodynamic and kinetic behavior of HZO multiphase coexistence systems. Its key feature lies in constructing gradient energy correction terms that distinguish between domain boundary energies and phase boundary energies, thereby overcoming the deficiency of existing models that cannot accurately differentiate between the two interface energies due to the use of a unified gradient term. This invention also addresses the shortcomings of existing models that lack a reasonable characterization of nonferroelectric phases, making it difficult to accurately reflect phase distribution characteristics by introducing a description of nonferroelectric phases. This improvement allows the model to more accurately simulate multiphase competition dynamics, particularly overcoming the limitation of existing models that cannot accurately describe phase boundaries, thus hindering the simulation of the crucial influence of phase boundaries on domain evolution in multiphase coexistence. Through efficient numerical computation and post-processing analysis, the quantitative simulation and visualization of phase transitions and domain evolution processes under external field control are ultimately achieved.

[0008] The steps of this invention include the following:

[0009] Step 1: Determine the structural order parameters describing the multiphase system;

[0010] Step 2: Construct a total free energy functional that includes the gradient energy density of the phase boundary energy and describes the nonferroelectric phase;

[0011] Step 3: Based on the variational principle, establish a set of control equations for multi-physics coupling;

[0012] Step 4: Solve the control equations using the semi-implicit Fourier spectrum method to output the field variable data of order parameters, strain, and polarization at all time and space scales.

[0013] Step 5: Perform data visualization and quantitative analysis on the order parameter distribution, phase distribution, and strain field to quantitatively extract key physical parameters from the evolution data.

[0014] Furthermore, the structural order parameter refers to a set of structural order parameter fields corresponding to different crystals: , used to represent the microscopic state of phase structure and domain states in ferroelectric thin films, = 0 indicates that the parent phase is phase T. = 1 indicates the i-th variant of the sub-phase, 0 < <1 indicates the transition zone or interface, i = 1, 2 indicates the orthogonal phase o, i = 3 indicates the monoclinic phase m, where, t represents the spatial coordinates of the crystal phase in the order parameter field, and t represents the system evolution time.

[0015] Furthermore, the gradient energy density is as follows:

[0016] ;

[0017] Among them, f grad The gradient energy density, containing phase boundary energy, is used to characterize the presence of domain walls and phase boundaries. The first term on the right-hand side of the equation is: The term represents the squared gradient, used to describe the domain wall energy, and g represents the domain wall energy coefficient. The gradient operator is represented; the second term on the right-hand side of the equation. The phase boundary energy is represented by k, which represents the phase boundary energy coefficient. and To represent different variants, i,j=1,2,3, i≠j.

[0018] Furthermore, the total free energy functional is as follows:

[0019] ;

[0020] in, Let f represent the total free energy functional of the system to be simulated. bulk f represents the volume free energy density, used to describe the thermodynamic potential of a homogeneous bulk phase. elast f represents the elastic energy density, used to describe the contribution of elastic strain energy introduced by substrate constraint and lattice mismatch. elec It represents electrostatic energy density, encompassing both the external electric field and the electrostatic interactions generated by the material's own polarization. This represents the volume of the system to be simulated.

[0021] The volume free energy density is as follows:

[0022] ;

[0023] Where, α i α represents the coefficient of the quadratic term, whose value is temperature-dependent and used to describe the stability of nonferroelectric phases; ij ,α ijk These represent the coefficients of higher-order terms, used to describe the height of the potential barrier and the depth of the potential well, respectively. , and To represent different variants, i≠j≠k, i,j,k=1,2,3.

[0024] The elastic energy density f elast as follows:

[0025] ;

[0026] Among them, C ijkl Let ε represent the elastic constant tensor. ij (r), ε kl (r) represents the total strain tensor, the value of which is determined by the substrate or an external force. , This represents the intrinsic strain corresponding to the crystal.

[0027] The electrostatic energy density f elec as follows:

[0028] ;

[0029] The first term on the right side of the equals sign: P represents the electric field energy density related to the spontaneous polarization of the material. i (r) represents the polarization intensity caused by the electric field. and Represents the electric field components in different spatial directions. , The second term on the right-hand side of the equation represents the spontaneous polarization intensity of the material. This represents the electric field energy density stored in the medium. κ represents the vacuum permittivity. ij This represents the relative permittivity tensor.

[0030] Furthermore, the governing equations for the multiphysics coupling refer to the equations obtained by applying the total free energy functional F to the order parameter fields respectively. ε strain field ij and electric potential Find the variational equations and establish a set of governing equations describing the evolution and mutual coupling of the order parameters; this set of equations includes the following:

[0031] Order parameter evolution dynamics equations—time-dependent Ginzburg-Landau equations:

[0032] ;

[0033] Where L represents the kinetic coefficient related to the phase boundary or domain wall mobility, ξ i This represents the random noise term used to characterize thermal fluctuations;

[0034] Equilibrium equations of mechanics:

[0035] ;

[0036] in, , Represents the stress tensor. Represents the strain tensor. Indicates intrinsic strain, Indicates the direction vector;

[0037] Electrostatic equilibrium equation:

[0038] ;

[0039] in, This represents the free charge volume density.

[0040] Furthermore, the steps for solving the governing equations using the semi-humble Fourier spectrum method are as follows:

[0041] The first step is to use Fast Fourier Transform (FFT) to transform the control equations into Fourier space. A semi-implicit scheme is used to implicitly process the gradient energy term and explicitly process the volume free energy, elastic energy and electrostatic energy to maintain computational simplicity.

[0042] The second step involves combining high-performance parallel computing technology to simulate the dynamic evolution of the order parameters of the three-dimensional nanostructure under the influence of electric, stress, and temperature fields, and outputting field variable data of order parameters, strain, and polarization at all time and space scales.

[0043] Furthermore, the steps for quantitatively extracting key physical parameters from evolutionary data are as follows:

[0044] The first step is to perform dynamic rendering of the order parameter distribution, phase distribution, and strain field in three dimensions and time to generate an animation sequence that can intuitively display the microscopic evolution process of phase transition, domain growth, and domain wall / phase boundary movement.

[0045] The second step is to develop customized analysis scripts to quantitatively extract from the evolution data the changes in volume fraction of each phase over time and external field; the migration rate of domain walls and phase boundaries; the activation energy of local polarization reversal; and the macroscopic performance parameters of average polarization and strain hysteresis loop.

[0046] The third step involves statistically analyzing the correlation between the extracted parameters and the field variable data of order parameters, strain, and polarization at all spatiotemporal scales, and constructing a connection between external field conditions, microstructure evolution, and macroscopic ferroelectric performance.

[0047] Compared with the prior art, the present invention has the following advantages:

[0048] First, this invention establishes a more accurate thermodynamic model by introducing a non-ferroelectric phase and coupling key external field conditions to predict the multiphase coexistence phase diagram under different strain states. The simulation results can quantitatively predict the precise "strain window" that is conducive to the formation and stability of the ferroelectric orthorhombic phase (o phase), thereby determining the optimal strain control window. Its prediction reliability is greatly improved, which significantly enhances the accuracy and reliability of the simulation prediction method. It can be practically applied to guide the optimization of experimental processes and significantly reduce the trial and error costs and time in research and development.

[0049] Secondly, by accurately describing phase boundary energy, this invention enables the model to truly reflect the characteristics of phase boundary migration, overcoming the key challenge of existing models failing to accurately describe phase boundaries and thus struggling to simulate domain evolution under multiphase coexistence. This allows for intuitive visualization and quantitative analysis of previously difficult-to-capture key processes such as domain flipping pinning by phase boundaries and phase boundary migration dynamics under external fields. It provides an unprecedented microscopic perspective for understanding device reliability issues (such as fatigue characteristics) and lays the theoretical foundation for fundamentally optimizing device durability. Attached Figure Description

[0050] Figure 1 This is a flowchart illustrating the implementation of an embodiment of the present invention;

[0051] Figure 2 This is a schematic diagram of the o / m phase boundary, the 90° domain boundary, and the 180° domain boundary in Embodiment 1 of the present invention;

[0052] Figure 3 This is a schematic diagram of the phase transition process from the tetragonal parent phase to the orthogonal ferroelectric phase and the non-ferroelectric monoclinic phase of the hafnium oxide-based ferroelectric thin film at different temperatures in this invention. Detailed Implementation

[0053] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0054] Reference Figure 1 The implementation steps of the embodiments of the present invention will be further described below.

[0055] Step 1: Define the order parameter and the total free energy functional of the system.

[0056] Embodiments of this invention require establishing physical quantities capable of characterizing the phase structure and domain states in ferroelectric thin films. A set of structural order parameter fields is selected. To characterize the microstate of the system, where r is the spatial coordinate and t is time. Order parameter. The value of corresponds to different crystal phases (for example, a specific value can determine whether it is the orthorhombic phase o, the tetragonal phase t, or the monoclinic phase m).

[0057] Step 2: Construct a multiphase thermodynamic and energy model of the macroscopic state by the total free energy functional.

[0058] In response to the multiphase coexistence characteristics of HZO, this invention makes key modifications to the traditional phase-field model, focusing on accurately describing the thermodynamic competition between different phases and the energy of phase boundaries / domain boundaries.

[0059] The macroscopic state of the system is determined by the total free energy functional F, which is the sum of the integrals of various energy densities within the system in the simulation space, and its expression is as follows:

[0060] ;

[0061] Among them, f bulk f is the bulk free energy density, which describes the thermodynamic potential of a homogeneous bulk phase; elast f represents the elastic energy density, describing the contribution of elastic strain energy introduced by substrate constraint and lattice mismatch; elec f is the electrostatic energy density, encompassing both the external electric field and the electrostatic interactions generated by the material's own polarization; grad The gradient energy density, derived from the spatial inhomogeneity of the order parameter, characterizes the existence of domain walls and phase boundaries.

[0062] (1) Volume free energy density f bulk It is based on the Landau-Devinhill phase transition theory, which expands the bulk free energy density into structural order parameters. The expansion is in power series form. To satisfy the symmetry requirements of structural phase transitions, only even-order terms are retained in the expansion, and it is truncated to the sixth order to ensure an accurate description of first-order phase transition behavior and the coexistence of multiple metastable states. The expression for the volume free energy density is:

[0063] ;

[0064] Wherein, the expansion coefficient α i ,α ij ,α ijk It has a clear physical meaning: the coefficient α of the quadratic term i Typically temperature-dependent, these parameters determine phase stability; higher-order coefficients collectively determine the height of the potential barrier and the depth of the potential well. These parameters can be obtained through first-principles calculations or by fitting experimental data (such as phase proportions determined by X-ray diffraction).

[0065] (2) Elastic energy density f elast For epitaxially grown thin films, the mismatch strain introduced by the substrate is a key factor in controlling phase stability. The expression for elastic energy density is:

[0066] ;

[0067] Among them, C ijkl ε is the elastic constant tensor; ij This is the total strain tensor; To be compatible with the current crystal phase (by The intrinsic strain (or phase transition strain) corresponding to the phase constant is determined by the lattice constant. This expression accurately reflects the elastic interaction between different phases caused by the difference in lattice constants and the substrate confinement effect.

[0068] (3) Electrostatic energy density f elecWhen an external electric field is applied, the electrostatic contribution of the system includes polarization energy and interaction energy with the electric field. Considering that the m phase in HZO is a nonpolar phase, this invention correlates the polarization intensity P with the structural order parameter. For example, assuming... , where P s Let be the spontaneous polarization intensity. The expression for electrostatic energy density is:

[0069] ;

[0070] Where the first term on the right-hand side of the equation is the electric field energy density related to spontaneous polarization, ε0 is the vacuum permittivity, and κ is the electric field energy density related to spontaneous polarization. ij E is the relative permittivity tensor. i (r) represents the electric field strength; the second term on the right-hand side of the equation represents the electric energy density stored in the medium.

[0071] (4) Gradient energy density f grad Key Correction: This correction is one of the innovations of this invention. Traditional models typically use a single gradient coefficient, which cannot distinguish between phase boundaries and domain boundaries with significant differences in energy scaling. To address this issue, this invention specifically corrects the gradient energy density term, introducing an expression that can distinguish between the contributions of phase boundaries and domain boundaries, as follows:

[0072] ;

[0073] Or, an equivalent and more intuitive form could be used:

[0074] ;

[0075] In this invention, the first term on the right-hand side of the equation corresponds primarily to the domain wall energy (the squared gradient term), with coefficient g being the domain wall energy coefficient. The second term on the right-hand side is a key term introduced in this invention, mainly used to describe the phase boundary energy, with coefficient k being the phase boundary energy coefficient. Coefficients g and k can be calibrated based on the interface energy calculated using first-principles calculations or the phase boundary / domain boundary width observed by high-resolution electron microscopy. This separation process employed in this invention allows the model to more realistically reflect the phase boundary energy in the HZO multiphase system, thereby accurately capturing the dynamic processes of multiphase competition.

[0076] Step 3: Establish the governing equations for multi-physics coupling.

[0077] Based on the variational principle, the total free energy functional F is used to apply the order parameter field to... strain field and electric potential field Find the variational equations and establish a set of governing equations describing the evolution and coupling of these parameters. This set of governing equations includes the following equations.

[0078] (1) Order parameter evolution dynamics equations (time-dependent Ginzburg-Landau equations):

[0079] ;

[0080] Where L is the kinetic coefficient, which is related to the phase boundary or domain wall mobility; ξ i This is the random noise term characterizing thermal fluctuations.

[0081] (2) Equations of mechanical equilibrium (stress equilibrium):

[0082] ,in ;

[0083] (3) Electrostatic equilibrium equation (Poisson's equation):

[0084] ;

[0085] in, ρ is the electrostatic potential. f The free charge density.

[0086] Step 4, numerical solution and large-scale simulation.

[0087] An efficient numerical method is employed to solve the multiphysics coupling control equations constructed in step 3. In this embodiment of the invention, a semi-implicit Fourier spectrum method is used for the solution:

[0088] (1) The Fast Fourier Transform (FFT) algorithm is used to convert differential operations (such as gradient and Laplace operator) in real space into algebraic operations in Fourier space, which significantly improves the accuracy and efficiency of computation. The time integration adopts a semi-implicit scheme, implicitly processing the gradient energy term to ensure numerical stability, and explicitly processing the volume free energy and other terms to maintain computational simplicity.

[0089] (2) By combining high-performance parallel computing technology, large-scale simulation of the dynamic evolution process of three-dimensional nanostructures under the action of complex external fields (electric field, stress field, temperature field) is realized, and field variable data such as order parameters, strain, and polarization at all time and space scales are output.

[0090] Step 5: Data post-processing, visualization, and quantitative analysis.

[0091] The massive spatiotemporal data generated by the simulation are systematically post-processed to extract valuable physical information.

[0092] (1) Use scientific visualization software (such as Paraview and Simpleview) to perform four-dimensional (three-dimensional space + one-dimensional time) dynamic rendering of order parameter distribution, phase distribution, strain field, etc., to generate an animation sequence that can intuitively display the micro-evolution process of phase transition, domain growth, domain wall / phase boundary movement, etc.

[0093] (2) Develop customized analysis scripts to quantitatively extract key physical parameters from evolution data, such as: the volume fraction of each phase (o phase, t phase, m phase) as a function of time / external field; the migration rate of domain walls / phase boundaries; the activation energy of local polarization reversal; and macroscopic performance parameters such as average polarization and strain loop.

[0094] (3) Statistical methods are used to analyze the correlation between the extracted parameters and the applied field (such as electric field strength and strain magnitude), and a complete theoretical picture connecting the external field conditions, microstructure evolution and macroscopic ferroelectric properties is constructed to provide predictive guidance for material design and process optimization.

[0095] Example 1 simulates the multiphase competitive evolution of HZO thin films in different temperature ranges from 10K to 1900K.

[0096] Example 1 of the present invention demonstrates a method for simulating the effect of substrate strain on multiphase competition behavior during the crystallization process of HZO thin films using the method of the present invention.

[0097] Example 1 illustrates a simulation study of the multiphase competition evolution behavior of HZO thin films during crystallization in different temperature ranges, from 10K to 1900K. This example focuses on investigating the influence of substrate strain on the phase transition path and the final phase composition. First, based on crystallographic correspondences, to minimize lattice mismatch during the phase transition process, the following parameters were determined: Figure 2 The crystallographic matching scheme is shown. Based on the three variant types, the system will form 180° and 90° domain walls, as well as phase boundaries between the orthorhombic o phase and the monoclinic m phase, during its evolution.

[0098] In the simulation, a three-dimensional computational domain representing the HZO thin film was first initialized, with the initial state set as randomly distributed microcrystals, and a specific substrate-induced strain was applied. Model parameters included the bulk free energy coefficient, elastic constants, and critical phase boundary energy coefficients, which could be obtained from literature or first-principles calculations. By solving the coupled governing equations, the evolution of the system during relaxation was simulated, with a focus on the phase distribution changes at different temperatures from 10K to 1900K. Figure 3 As shown in the figure. Further simulations by systematically changing strain conditions allow for the creation of a strain-phase diagram, thus clarifying the strain window favorable for the formation of high-purity orthogonal O phase. This result provides a theoretical basis for selecting a suitable substrate to introduce specific strains in experiments, and helps guide the preparation of high-performance hafnium-based ferroelectric thin films.

[0099] Example 2: Domain flipping of multiphase HZO thin films driven by electric field.

[0100] Embodiment 2 of the present invention provides a microscopic dynamic process of domain structure flipping and phase boundary migration in a multiphase coexisting HZO thin film under the action of an applied alternating electric field. During simulation initialization, a microstructure including a pre-existing ferroelectric orthogonal phase (o phase) and a non-polar monoclinic phase (m phase) region is set, and a flat electric field is applied, and short-circuit boundary conditions are set.

[0101] This simulation result clarifies from a microscopic perspective that multiphase coexistence (especially the m phase) is the key reason for the fatigue characteristics of HZO ferroelectric devices. It clearly points out that reducing the content of non-ferroelectric phases and improving the phase boundary structure through process optimization are effective ways to improve device durability, providing in-depth theoretical guidance for reliability design.

[0102] The key point of this invention lies in constructing a phase-field model and method specifically for simulating the phase and domain evolution of multiphase coexisting HZO ferroelectric thin films. Its core innovation lies in targeted and crucial modifications to traditional models to accurately describe the unique physical phenomena of HZO multiphase systems. The protection points should revolve around the following core improvements and the resulting complete technical solution:

[0103] (1) Corrected model of gradient energy density term.

[0104] This is the core innovation of this invention. Traditional phase-field models typically have a single, undifferentiated gradient energy term, failing to distinguish between domain boundaries (domain walls) and phase boundaries, which have significant energy scale differences. This invention introduces a term that can independently describe phase boundary energy (by adding a gradient cross term to the gradient energy density expression), achieving separate characterization of domain boundary energy and phase boundary energy. This modification allows the model to more realistically reflect the phase boundary energy in HZO multiphase systems, thus significantly improving the accuracy of multiphase competition dynamics simulations. Its key innovation lies in this gradient energy density expression that distinguishes between domain boundary energy and phase boundary energy, and its specific mathematical implementation.

[0105] ① Volume free energy density (f bulk Based on the Landau-Devinhill phase transition theory, the bulk free energy density is expanded into structural order parameters. The power series form is given. To satisfy the symmetry requirements of structural phase transitions, only even-order terms are retained in the expansion, and it is truncated to the sixth order to ensure accurate description of first-order phase transition behavior and the coexistence of multiple metastable states. Its general form is:

[0106] ;

[0107] Wherein, the expansion coefficient α i ,α ij ,α ijkIt has a clear physical meaning: the coefficient α of the quadratic term i Typically temperature-dependent, these parameters determine phase stability; higher-order coefficients collectively determine the height of the potential barrier and the depth of the potential well. These parameters can be obtained through first-principles calculations or by fitting experimental data (such as phase proportions determined by X-ray diffraction).

[0108] ② Elastic energy density (f elast For epitaxially grown thin films, the mismatch strain introduced by the substrate is a key factor in controlling phase stability. The elastic energy density is expressed as:

[0109] ;

[0110] Among them, C ijkl ε is the elastic constant tensor; ij This is the total strain tensor; To be compatible with the current crystal phase (by The intrinsic strain (or phase transition strain) corresponding to the phase constant is determined by the lattice constant. This expression accurately reflects the elastic interaction between different phases caused by the difference in lattice constants and the substrate confinement effect.

[0111] ③ Electrostatic energy density (f elec When an external electric field is applied, the electrostatic contribution of the system includes polarization energy and interaction energy with the electric field. Considering that the m phase in HZO is a nonpolar phase, this invention correlates the polarization intensity P with structural order parameters, for example assuming... , where P s This represents the spontaneous polarization intensity. The electrostatic energy density can be expressed as:

[0112] ;

[0113] The first term represents the electric field energy density related to spontaneous polarization, ε0 is the vacuum permittivity, and κ is the electric field energy density. ij E is the relative permittivity tensor. i (r) represents the electric field strength; the second term represents the electric field energy density stored in the medium.

[0114] ④ Gradient energy density (f) grad Key Correction (Innovation of this Invention): Traditional models typically use a single gradient coefficient, which cannot distinguish between phase boundaries and domain boundaries with significant differences in energy scaling. To address this issue, this invention specifically corrects the gradient energy density term, introducing an expression that can distinguish between the contributions of phase boundaries and domain boundaries:

[0115] ;

[0116] Or, an equivalent and more intuitive form could be used:

[0117] ;

[0118] The first term corresponds primarily to the domain wall energy (the squared gradient term), with coefficient g being the domain wall energy coefficient. The second term, a key term introduced in this invention, mainly describes the phase boundary energy, with coefficient k being the phase boundary energy coefficient. Coefficients g and k can be calibrated based on the interface energy calculated using first-principles calculations or the phase boundary / domain boundary width observed by high-resolution electron microscopy. This separation process allows the model to more realistically reflect the physical fact that the phase boundary energy in the HZO multiphase system is much higher than the domain wall energy, thereby accurately capturing the dynamic process of multiphase competition.

[0119] (2) Phase field thermodynamic model for HZO multiphase system.

[0120] This invention constructs a complete multiphysics coupling model framework:

[0121] Based on the variational principle, the total free energy functional F is used to apply the order parameter field to... strain field and electric potential field Find the variational equations and establish the governing equations describing their evolution and mutual coupling:

[0122] ① Order parameter evolution dynamics equations (time-dependent Ginzburg-Landau equations):

[0123] ;

[0124] Where L is the kinetic coefficient, which is related to the domain wall mobility; ξ i This is the random noise term characterizing thermal fluctuations.

[0125] ② Equations of mechanical equilibrium (stress equilibrium):

[0126] ,in ;

[0127] ③ Electrostatic equilibrium equation (Poisson's equation):

[0128] ;

[0129] in, ρ is the electrostatic potential. f This represents the free charge density.

[0130] (3) Efficient numerical solution method and data post-processing for the coupled model.

[0131] This invention proposes a complete computational implementation scheme to ensure the effective application of the model: It employs a semi-implicit Fourier spectrum method to solve the governing equations, utilizes Fast Fourier Transform (FFT) to process spatial differentiation, and combines it with a semi-implicit time integration scheme, balancing computational efficiency and numerical stability. Secondly, it dynamically visualizes the spatiotemporal data generated by the simulation, quantitatively extracting key physical parameters (such as the variation of phase volume fraction with time / external field, phase boundary migration rate, macroscopic hysteresis loops, etc.). The key protection lies in this complete technical process specifically designed for the efficient solution and analysis of this model.

[0132] (4) Specific applications of the phase-field simulation method in guiding material and device design.

[0133] The ultimate value of this invention lies in its predictive and guiding capabilities. The scope of protection should extend to its application level, for example: using simulation results to plot strain-phase diagrams to determine the optimal phase modulation window, thereby guiding the selection of substrates that can introduce specific strains. Utilizing the microscopic mechanisms of the "influence of phase boundaries on domain flipping" revealed by simulations to guide methods for improving device reliability through process optimization (such as reducing the content of non-ferroelectric phases).

Claims

1. An analytical method for controlling the high-purity ferroelectric orthorhombic phase of hafnium-zirconium ferroelectric thin films, characterized in that, The steps of this analytical method include the following: Step 1: Determine the structural order parameters describing the multiphase system; Step 2: Construct a total free energy functional that includes the gradient energy density of the phase boundary energy and describes the nonferroelectric phase; Step 3: Based on the variational principle, establish a set of control equations for multi-physics coupling; Step 4: Solve the control equations using the semi-implicit Fourier spectrum method to output the field variable data of order parameters, strain, and polarization at all time and space scales. Step 5 involves data visualization and quantitative analysis of order parameter distribution, phase distribution, strain field, etc., to quantitatively extract key physical parameters from the evolution data.

2. The analytical method according to claim 1, characterized in that, The structural order parameters mentioned in step 1 refer to a set of structural order parameter fields corresponding to different crystals: (r, t) is used to represent the microscopic state of phase structure and domain state in ferroelectric thin films. = 0 indicates that the parent phase is phase T. = 1 indicates the i-th variant of the sub-phase, 0 < <1 indicates the transition region or interface, i = 1,2 indicates the orthorhombic phase o, i = 3 indicates the monoclinic phase m, where r represents the spatial coordinates of the crystal phase in the order parameter field, and t represents the system evolution time.

3. The analytical method according to claim 2, characterized in that, The gradient energy density mentioned in step 2 is as follows: ； Among them, f grad The gradient energy density, containing phase boundary energy, is used to characterize the presence of domain walls and phase boundaries. The first term on the right-hand side of the equation is: The term represents the squared gradient, used to describe the domain wall energy, and g represents the domain wall energy coefficient. The gradient operator is represented; the second term on the right-hand side of the equation. The phase boundary energy is represented by k, which represents the phase boundary energy coefficient. and To represent different variants, i,j=1,2,3, i≠j.

4. The analytical method according to claim 3, characterized in that, The total free energy functional described in step 2 is as follows: ； in, Let f represent the total free energy functional of the system to be simulated. bulk f represents the volume free energy density, used to describe the thermodynamic potential of a homogeneous bulk phase. elast f represents the elastic energy density, used to describe the contribution of elastic strain energy introduced by substrate constraint and lattice mismatch. elec It represents electrostatic energy density, encompassing both the external electric field and the electrostatic interactions generated by the material's own polarization. This represents the volume of the system to be simulated.

5. The analytical method according to claim 4, characterized in that, The volume free energy density is as follows: ； Where, α i α represents the coefficient of the quadratic term, whose value is temperature-dependent and used to describe the stability of nonferroelectric phases; ij ,α ijk These represent the coefficients of higher-order terms, used to describe the height of the potential barrier and the depth of the potential well, respectively. , and To represent different variants, i≠j≠k, i,j,k=1,2,3.

6. The analytical method according to claim 4, characterized in that, The elastic energy density f elast as follows: ； Among them, C ijkl Let ε represent the elastic constant tensor. ij (r), ε kl (r) represents the total strain tensor, the value of which is applied by the substrate or external environment. , This represents the intrinsic strain corresponding to the crystal.

7. The analytical method according to claim 6, characterized in that, The electrostatic energy density f elec as follows: ； The first term on the right side of the equals sign: P represents the electric field energy density related to the spontaneous polarization of the material. i (r) represents the polarization intensity caused by the electric field, E i (r) and Represents the electric field components in different spatial directions. , The second term on the right-hand side of the equation represents the spontaneous polarization intensity of the material. This represents the electric field energy density stored in the medium. κ represents the vacuum permittivity. ij This represents the relative permittivity tensor.

8. The analytical method according to claim 7, characterized in that, The governing equations for multiphysics coupling mentioned in step 3 refer to the equations obtained by applying the total free energy functional F to the order parameter fields. ε strain field ij and electric potential Find the variational equations and establish a set of governing equations describing the evolution and mutual coupling of the order parameters; this set of equations includes the following: Order parameter evolution dynamics equations—time-dependent Ginzburg-Landau equations: ； Where L represents the kinetic coefficient related to the phase boundary or domain wall mobility, ξ i This represents the random noise term used to characterize thermal fluctuations; Equilibrium equations of mechanics: ； in, , Represents the stress tensor. Represents the strain tensor. Indicates intrinsic strain, Indicates the direction vector; Electrostatic equilibrium equation: ； in, This represents the free charge volume density.

9. The analytical method according to claim 8, characterized in that, The steps for solving the governing equations using the semi-hidden Fourier spectrum method described in step 4 are as follows: The first step is to use Fast Fourier Transform (FFT) to transform the control equations into Fourier space. A semi-implicit scheme is used to implicitly process the gradient energy term and explicitly process the volume free energy, elastic energy and electrostatic energy to maintain computational simplicity. The second step involves combining high-performance parallel computing technology to simulate the dynamic evolution of the order parameters of the three-dimensional nanostructure under the influence of electric, stress, and temperature fields, and outputting field variable data of order parameters, strain, and polarization at all time and space scales.

10. The analytical method according to claim 1, characterized in that, The steps for quantitatively extracting key physical parameters from evolutionary data in step 5 are as follows: The first step is to perform dynamic rendering of the order parameter distribution, phase distribution, and strain field in three dimensions and time to generate an animation sequence that can intuitively display the microscopic evolution process of phase transition, domain growth, and domain wall / phase boundary movement. The second step is to develop customized analysis scripts to quantitatively extract from the evolution data the changes in volume fraction of each phase over time and external field; the migration rate of domain walls and phase boundaries; the activation energy of local polarization reversal; and the macroscopic performance parameters of average polarization and strain hysteresis loop. The third step involves statistically analyzing the correlation between the extracted parameters and the field variable data of order parameters, strain, and polarization at all spatiotemporal scales, and constructing a connection between external field conditions, microstructure evolution, and macroscopic ferroelectric performance.

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