Phase evolution and phase field analysis method in annealing crystallization process of amorphous hafnium oxide-based ferroelectric film

Abstract

The phase structure evolution phase field analysis method in the annealing and crystallization process of the amorphous hafnium oxide-based ferroelectric film comprises the following steps: S1, setting a sequence parameter xi according to the evolution process from an amorphous phase to a crystal phase in the annealing process, and setting a sequence parameter eta according to the process of the crystal phase and the crystal phase; s2, a phase field theoretical model is established, the total free energy is the sum of the free energy of the sequence parameter xi and the free energy of the sequence parameter eta, and the free density in the free energy of the sequence parameter eta is obtained through coupling of the sequence parameter xi and the sequence parameter eta; s3, deducing a weak form of the control equation; and S4, compiling the weak form equation into COMSOL finite element software, solving and analyzing the phase structure in the annealing crystallization process of the thin film. According to the method, the sequence parameter xi representing the transformation process from the amorphous phase to the crystalline phase is introduced and coupled with the sequence parameter eta to construct a new free energy equation and a new control equation, the weak form of the control equation is deduced, and the phase structure and the evolution rule of the hafnium oxide-based ferroelectric film at different annealing moments can be dynamically reproduced.

Description

Technical Field

[0001] This application belongs to the technical field of ferroelectric thin film simulation analysis, and particularly relates to a phase field analysis method for phase evolution during the annealing and crystallization process of amorphous hafnium oxide-based ferroelectric thin films. Background Technology

[0002] Traditionally, perovskite ferroelectric materials are only suitable for oxide substrates and are incompatible with Si substrates used in CMOS processes, thus limiting their application in integrated circuits. However, unlike traditional perovskite ferroelectric materials, hafnium oxide-based ferroelectric thin films, due to their low crystallization temperature (approximately 400°C) and excellent compatibility with TiN electrodes commonly used in mass production, allow for the direct embedding of CMOS logic circuits in hafnium oxide-based memory without requiring adjustments to existing processes. Furthermore, the microscale of traditional perovskite ferroelectric materials significantly limits their ferroelectric properties. Typically, traditional perovskite ferroelectric films exhibit strong ferroelectricity only at thicknesses between 50-70 nm; further reductions in thickness result in the loss of ferroelectric properties, which is unacceptable given the modern trend towards highly integrated circuits. However, hafnium oxide-based ferroelectric thin films maintain high polarization even at thicknesses below 10 nm, and some studies have even shown that they retain ferroelectricity when further reduced to 3 nm. This characteristic enables further miniaturization of semiconductor devices, meeting the development trend of Moore's Law. Hafnium oxide-based ferroelectric thin films, as a novel ferroelectric material, can indeed effectively address the limitations of traditional perovskite ferroelectric materials and hold promise for overcoming the development bottlenecks of traditional ferroelectric materials. However, in the context of the modern pursuit of high-performance memory, its commercialization still requires overcoming the technical challenge of fatigue.

[0003] For hafnium oxide-based ferroelectric thin films, the ferroelectric properties originate from the ferroelectric phase, which typically coexists with other non-ferroelectric phases within the material. The ferroelectric phase, with its energy level between the monoclinic and tetragonal phases, is metastable. Therefore, the ferroelectric phase in hafnium oxide-based ferroelectric thin films is easily transformed into a non-ferroelectric monoclinic phase under external influences (such as stress and electric fields), thus affecting the fatigue resistance of the hafnium oxide-based ferroelectric thin films. Therefore, improving the stability of the ferroelectric phase and increasing its proportion are key issues that need to be addressed. To solve the aforementioned problems of fatigue susceptibility in ferroelectric thin films and the related stability of the ferroelectric phase, it is necessary to conduct in-depth research on the formation of the ferroelectric phase in hafnium oxide-based ferroelectric thin films.

[0004] Studies have shown that the phase transition during the annealing and crystallization process of hafnium oxide-based ferroelectric thin films initially involves the transformation of the amorphous phase into a crystalline tetragonal phase. Only a portion of the tetragonal phase transforms into a monoclinic phase. During the cooling stage of annealing, the remaining tetragonal phase transforms back into the ferroelectric phase. However, the phase transitions involved in annealing are complex and occur at extremely rapid rates. Therefore, directly observing the effects of different annealing conditions on the phase transition behavior during annealing presents significant limitations. Phase-field analysis (PFA) can effectively address these limitations through its mesoscale dynamic simulation capabilities. However, existing PFA methods have not adequately described the dynamic phase structure evolution of hafnium oxide-based ferroelectric thin films during annealing. Therefore, establishing a PFA method for the phase evolution during the annealing and crystallization process of amorphous hafnium oxide-based ferroelectric thin films is of significant research importance for studying the impact of annealing treatment on the proportion and stability of the ferroelectric phase in hafnium oxide-based ferroelectric thin films. Summary of the Invention

[0005] Existing phase-field analysis methods have the problem of failing to adequately describe the dynamic phase structure evolution of hafnium oxide-based ferroelectric thin films during annealing.

[0006] To address the aforementioned technical problems, according to some embodiments, this application provides a method for phase evolution and phase field analysis during the annealing and crystallization process of amorphous hafnium oxide-based ferroelectric thin films, characterized by comprising:

[0007] S1. Set the sequence parameter ξ according to the evolution process from amorphous phase to crystalline phase during the annealing of amorphous hafnium oxide thin films, and set the sequence parameter η according to the mutual transformation process between crystalline phases;

[0008] S2. Based on the order parameter ξ and the order parameter η, establish a phase-field theoretical model for analyzing the phase evolution during the annealing and crystallization process of amorphous hafnium oxide-based ferroelectric thin films, including constructing the total free energy equation and governing equations of the thin film system; wherein, the total free energy is the sum of the free energy of the order parameter ξ and the free energy of the order parameter η, and the volume free density in the free energy of the order parameter η is obtained by coupling the order parameter ξ and the order parameter η; the governing equations include the evolution equation of the order parameter ξ describing the phase evolution, the evolution equation of the order parameter η, the force field governing equation, and the electric field governing equation.

[0009] S3. Based on the principle of virtual work, derive the weak forms of the evolution equations of the order parameter ξ, the order parameter η, the force field control equation, and the electric field control equation, respectively.

[0010] S4. Compile the weak form equations into the COMSOL finite element software, perform finite element analysis on the equations, and analyze the phase structure of the amorphous hafnium oxide-based ferroelectric thin film during annealing and crystallization.

[0011] Furthermore, in step S1:

[0012] When ξ = 0, it indicates that the ferroelectric thin film is in the amorphous phase; when ξ = 1, it indicates that the ferroelectric thin film is in the crystalline tetragonal phase.

[0013] The order parameter η has order parameter components η i , where i = (1,2,3), η1 represents the ferroelectric phase with a polarization vector along the horizontal direction of the thin film, η2 represents the ferroelectric phase with a polarization vector along the thickness direction of the thin film, and η3 represents the monoclinic phase.

[0014] Furthermore, in step S2, the total free energy equation is:

[0015] F = F ξ +F η

[0016] Among them, F ξ F is the free energy of the order parameter ξ. η The free energy is the order parameter η;

[0017] The free energy F of the order parameter ξ ξ The expression is:

[0018]

[0019] In the formula The gradient energy coefficient, ▽ ξ Let ξ be the gradient distribution, and V be the film volume;

[0020] f ξ The double-well function is expressed as:

[0021] f ξ =Ηξ 2 (1-ξ)(1-ξ+Δμ)

[0022] In the formula, H is the potential well depth, and Δμ is the annealing crystallization driving force;

[0023]

[0024] In the formula, λ is the driving force coefficient, T0 is the critical temperature for crystallization, and T is the annealing temperature;

[0025] The free energy F of the order parameter η η The expression is:

[0026] F η =∫f η dV

[0027] f η =f bulk +f grad +f elas +f elec

[0028] Among them, f bulk f represents the volume free energy density; grad f represents the domain wall gradient energy density; elas f represents the elastic strain energy density; elec Represents electrostatic energy density;

[0029] Volume free energy density f bulk The expression is:

[0030]

[0031] In the formula, A1, A 2(o) A 2(m) A3 and A4 represent the Landau energy coefficients of the crystalline phase; A 2(o) A represents the Landau energy coefficient of the ferroelectric phase. 2(m) A represents the Landau energy coefficient of a single-clinic phase; 1-a A 2-a A 3-a Landau energy coefficient for amorphous phases;

[0032] Elastic strain energy density f elas The expression is:

[0033]

[0034] Among them, c ijkl ε is the elastic energy coefficient; i, j, k, l are tensor indices, all ranging from 1, 2, to 3; ij ε kl Both represent strain components, ε ij =1 / 2(u i,j +u j,i ), u i For the displacement component, u i，j For u i The partial derivative with respect to the j-direction coordinate, u j，i For u j Partial derivative with respect to the coordinate in the i-direction; ε (i) This indicates that the ferroelectric thin film is in η i The spontaneous strain matrix under the corresponding state;

[0035] Domain wall gradient energy density f grad The formula is:

[0036]

[0037] Among them, K η For gradient energy coefficient, ▽η i For η i The gradient.

[0038] Furthermore, the annealing temperature includes a sequentially maintained heating temperature, a holding temperature, and a cooling temperature; wherein,

[0039] The heating temperature was intermittently increased from room temperature to the holding temperature, with each increase being 25°C, and the total heating time was 23.75 seconds.

[0040] The heat preservation temperature is maintained for 30 seconds;

[0041] The cooling temperature changes intermittently, decreasing by 25°C each time from the holding temperature to room temperature, with a total cooling time of 23.75 seconds.

[0042] Further, in step S2, the evolution equations for the order parameter ξ and the order parameter η describing the phase evolution are constructed, including:

[0043] The evolution equation of the order parameter ξ is described using the Ginzburg-Landau equation, and its expression is:

[0044]

[0045] Among them, L ξ Let ξ be the dynamic coefficient of the system with order parameter ξ, r be the spatial coordinate vector, and t be time. The evolution equation of the order parameter η is described by the Ginzburg-Landau equation, and its expression is:

[0046]

[0047] Among them, L η η is the kinetic coefficient of the order parameter system.

[0048] Furthermore, in step S2, the expression for the constructed force field control equation is:

[0049]

[0050] Where, x j Let j be the spatial coordinates on the j-axis;

[0051] The constructed electric field control equations are as follows:

[0052]

[0053] Among them, electric displacement E i Let x be the electric field component. i Let be the spatial coordinates on the i-axis. Further, the weak form of the evolution equation of the order parameter ξ is:

[0054]

[0055] Where A is the surface area, n j For surface normal components;

[0056] The weak form of the evolution equation for the order parameter η is:

[0057]

[0058] Furthermore, the weak form of the force field governing equations is as follows:

[0059]

[0060] Where, τ i For surface force components;

[0061] The weak form of the electric field control equation is:

[0062]

[0063] Where ω is the surface charge of the ferroelectric thin film, and φ is the electric potential.

[0064] Furthermore, the process of compiling the weakly formal equations into COMSOL finite element software, solving the equations using finite element methods, and analyzing the phase structure during the annealing and crystallization process of amorphous hafnium oxide-based ferroelectric thin films includes:

[0065] The weak forms of the evolution equations of the order parameter ξ, the order parameter η, the force field control equation, and the electric field control equation are solved using the finite element method.

[0066] Furthermore, the finite element solution process includes:

[0067] The weak forms of the evolution equations of the order parameter ξ, the order parameter η, the force field control equation, and the electric field control equation are edited into the COMSOL finite element software.

[0068] In the finite element software, periodic boundary conditions are used on the left and right sides of the ferroelectric thin film to simulate the lateral expansion characteristics of an infinitely large planar system; the force field sets boundary conditions that constrain both the top and bottom of the ferroelectric thin film, and the electric field sets boundary conditions that short-circuit both the top and bottom of the ferroelectric thin film to simulate the physical state of the electrodes with equipotential in actual devices.

[0069] A Gaussian random distribution function is introduced to apply a random perturbation to the order parameter η, thereby stimulating the spontaneous polarization evolution of the system;

[0070] Within the pre-defined initial nucleus region, the order parameter ξ is set to 1 to characterize the crystal nucleus state before annealing. Outside the nucleus region, the order parameter ξ is set to 0 to correspond to the initial conditions of a completely amorphous phase.

[0071] The initial values ​​of both the force field and the electric field are set to 0;

[0072] The entire ferroelectric thin film is discretized into small units using the finite element method, and the phase structure of the entire ferroelectric thin film is obtained by solving the local equations of each small unit.

[0073] The above-described technical solution of the present invention has at least the following beneficial technical effects:

[0074] Based on existing phase field analysis methods for phase evolution during the annealing and crystallization process of hafnium oxide-based ferroelectric thin films, this application introduces a sequence parameter ξ representing the transformation process from amorphous to crystalline phase. This parameter is coupled with the sequence parameter η in existing analytical methods to construct a new free energy equation for the thin film material system. The governing equations for each sequence parameter, as well as the force and electric fields, are also derived. The weak form of the governing equations is then derived, enabling dynamic reproduction of the phase structure and evolution of hafnium oxide-based ferroelectric thin films at different annealing times. Attached Figure Description

[0075] To more clearly illustrate the technical solutions in the embodiments of this application or in the conventional technology, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0076] Figure 1 This is a flowchart of the analytical method for phase evolution during the annealing and crystallization process of amorphous hafnium oxide-based ferroelectric thin films provided by the present invention.

[0077] Figure 2 This is a flowchart of step S4 provided by the present invention;

[0078] Figure 3 This is a graph showing the change in annealing temperature during the simulated annealing process in the phase-field model provided by this invention.

[0079] Figure 4 This is an amorphous phase structure diagram at the initial moment of the annealing process in the phase-field model provided by this invention.

[0080] Figure 5 This is a tetragonal phase structure diagram at the initial moment of the annealing process in the phase-field model provided by this invention.

[0081] Figure 6 This is a diagram of the amorphous phase structure during the annealing heating process in the phase-field model provided by this invention.

[0082] Figure 7 This is a tetragonal phase structure diagram at the end of the annealing process in the phase-field model provided by this invention.

[0083] Figure 8 This is a diagram of the monoclinic phase structure at the end of the annealing process in the phase-field model provided by this invention.

[0084] Figure 9 This is a tetragonal phase structure diagram at the end of the annealing process in the phase field model provided by this invention.

[0085] Figure 10 This is a diagram of the monoclinic phase structure at the end of the annealing process in the phase-field model provided by this invention.

[0086] Figure 11 This is a diagram of the ferroelectric phase structure during the cooling process of the annealing process in the phase field model provided by this invention.

[0087] Figure 12 This is a diagram of the monoclinic phase structure during the cooling process of the annealing process in the phase-field model provided by this invention. Detailed Implementation

[0088] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the various embodiments of this application will be described in detail below with reference to the accompanying drawings. However, those skilled in the art will understand that many technical details are presented in the various embodiments of this application to facilitate a better understanding of the application. However, the technical solutions claimed in this application can be implemented even without these technical details and various variations and modifications based on the following embodiments. The division of the various embodiments below is for ease of description and should not constitute any limitation on the specific implementation of this application. The various embodiments can be combined with and referenced by each other without contradiction.

[0089] Studies have shown that the phase transition process during the annealing and crystallization of hafnium oxide-based ferroelectric thin films is as follows: the ferroelectric thin film before annealing is an amorphous phase; during annealing, the amorphous phase in the film first transforms into a crystalline tetragonal phase, and only part of the tetragonal phase transforms into a monoclinic phase. During the cooling stage of the annealing process, the remaining tetragonal phase transforms into the ferroelectric phase. However, the experimental annealing process is relatively fast, and this process cannot be accurately described by direct observation or experimental measurement alone.

[0090] like Figure 1 As shown in the flowchart, an embodiment of this application provides a phase evolution and phase field analysis method for the annealing and crystallization process of amorphous hafnium oxide-based ferroelectric thin films, used to analyze the phase structure and evolution law. The method specifically includes:

[0091] S1. In order to analyze the transformation process of ferroelectric thin films from amorphous phase to crystalline phase, this application introduces a new order parameter ξ based on the existing phase field analysis method for phase evolution, and constructs a dual-order parameter model with ξ and the existing order parameter η to simulate the multiphase transformation behavior during annealing.

[0092] Specifically, the sequence parameter ξ is set according to the evolution process from amorphous phase to crystalline phase during the annealing of amorphous hafnium oxide thin films, and the sequence parameter η is set according to the mutual transformation process between crystalline phases.

[0093] Preferably, when ξ = 0, it indicates that the ferroelectric thin film is in the amorphous phase, and when ξ = 1, it indicates that the ferroelectric thin film is in the crystalline tetragonal phase.

[0094] After a ferroelectric thin film transforms from an amorphous phase to a crystalline phase, phase transitions between crystalline phases occur; the order parameter η is used to describe these phase transitions. Referring to the martensitic phase transformation theory, the tetragonal crystalline phase is taken as the parent phase, and the orthorhombic ferroelectric phase and monoclinic phase are taken as the daughter phases to simulate the subsequent transformations from the tetragonal to the monoclinic phase and from the tetragonal to the ferroelectric phase after annealing. The order parameter η has an order parameter component η0. i To improve computational efficiency, this invention only considers the order parameter component η. i The three values ​​of i = (1,2,3) are used to describe the three variants of the crystalline phase, where η1 represents the ferroelectric phase with the polarization vector along the horizontal direction, η2 represents the ferroelectric phase with the polarization vector along the thickness direction, and η3 represents the monoclinic phase.

[0095] The sequence parameter ξ, and the sequence parameter η, consisting of four sequence parameter components η1, η2, and η3, are used in subsequent steps to establish a phase field model and perform phase field analysis.

[0096] S2. Based on the order parameters ξ and η, a phase-field theoretical model is established to analyze the phase evolution during the annealing and crystallization process of amorphous hafnium oxide-based ferroelectric thin films. This includes constructing the total free energy equation and governing equations for the thin film system. The total free energy is the sum of the free energy of the order parameter ξ and the free energy of the order parameter η. The volume free density in the free energy of the order parameter η is obtained through the coupling of the order parameters ξ and η. The governing equations include the evolution equations of the order parameter ξ and η, the force field governing equation, and the electric field governing equation, all describing the phase evolution. Specifically, the total free energy is F, and the expression for the total free energy equation is:

[0097] F = F ξ +F η

[0098] Among them, F ξ To describe the free energy of the order parameter ξ in an amorphous-crystalline phase transition system, F η Let η be the free energy of the order parameter η.

[0099] The free energy F of the order parameter ξ ξ The expression is:

[0100]

[0101] In the formula The gradient energy coefficient, ▽ ξ Let ξ be the gradient distribution and V be the film volume.

[0102] To address the transformation process from an amorphous phase to a crystalline tetragonal phase, this invention constructs a dual-potential-well function f. ξ To describe phase transition dynamics:

[0103] f ξ =Ηξ 2 (1-ξ)(1-ξ+Δμ)

[0104] In the formula, H is the potential well depth, and Δμ is the annealing crystallization driving force;

[0105]

[0106] In the formula, λ is the driving force coefficient, T0 is the critical temperature for crystallization, and T is the annealing temperature; preferably, the value of the critical temperature for crystallization T0 is selected as 400℃ based on experimental data in this application.

[0107] It should be noted that the annealing temperature T in this application refers to the temperature change of the film during the annealing process. This application analyzes the influence of the annealing crystallization process on the iron phase structure of the film. Unlike existing phase field analysis methods for phase evolution, the annealing temperature T in this application is not a fixed value but a changing quantity. This invention uses a gradient heating and cooling method to describe the temperature change during the annealing process. Figure 3 As shown, the annealing process includes heating, holding, and cooling stages. Correspondingly, the annealing temperature includes sequentially maintained heating, holding, and cooling temperatures. The heating temperature changes intermittently, increasing by 25°C each time from room temperature to the holding temperature, with a total heating time of 23.75 seconds. The holding time is 30 seconds. The cooling temperature also changes intermittently, decreasing by 25°C each time from the holding temperature to room temperature, with a total cooling time of 23.75 seconds. Annealing conditions are set according to this method in subsequent finite element analyses.

[0108] F η The expression for the total free energy of the order parameter η, which describes the interconversion of crystalline phases, is as follows:

[0109] F η =∫f η dV

[0110] f η =f bulk +fgrad +f elas +f elec

[0111] Among them, f bulk f represents the volume free energy density; grad f represents the domain wall gradient energy density; elas f represents the elastic strain energy density; elec It represents the electrostatic energy density.

[0112] To better link the amorphous-crystalline phase transition process and the phase transition process between crystals, this invention couples two sets of order parameters ξ and η using bulk free energy to construct a new bulk free energy density equation. When ξ = 0, the bulk free energy takes the latter term, indicating the system is in an amorphous phase; when ξ = 1, the bulk free energy takes the former term, indicating the system is in a crystalline phase. Bulk free energy density f bulk The expression is:

[0113]

[0114] In the formula, A1, A 2(o) A 2(m) A3 and A4 represent the Landau energy coefficients of the crystalline phase; A 2(o) A represents the Landau energy coefficient of the ferroelectric phase. 2(m) A represents the Landau energy coefficient of a single-clinic phase; 1-a A 2-a A 3-a denoted as , where is the Landau energy coefficient for the amorphous phase, and all Landau energy coefficients in the formula are obtained by fitting experimental data.

[0115] Elastic strain energy density f elas The expression is:

[0116]

[0117] Among them, c ijkl ε is the elastic energy coefficient; i, j, k, l are tensor indices, all ranging from 1, 2, to 3; ij ε kl Both represent strain components, ε ij =1 / 2(u i,j +u j,i ), u i For the displacement component, u i，j For u i The partial derivative with respect to the j-direction coordinate, u j，i For u j Partial derivative with respect to the coordinate in the i-direction;

[0118]

[0119] ε (i) This indicates that the ferroelectric thin film is in η i The spontaneous strain matrix under the corresponding state is as follows:

[0120]

[0121] The subscripts t, o, and m represent the tetragonal, orthorhombic, and monoclinic phases, respectively. a, b, and c represent the lattice constants of different phases. t c t a represents the lattice constant of the parent tetragonal phase. o b o and c o a represents the lattice constant of the ferroelectric phase. m b m and c m It is the lattice constant of the monoclinic phase.

[0122] Domain wall gradient energy density f grad The formula is:

[0123]

[0124] Among them, K η For gradient energy coefficient, ▽η i For η i The polarization gradient.

[0125] Preferably, evolution equations for the order parameter ξ and the order parameter η, describing phase evolution, are constructed; where both order parameters ξ and η are non-conserved order parameters; for non-conserved order parameters, the evolution is described by time-dependent Ginzburg-Landau equations, the core idea of ​​which is to describe the macroscopic behavior of the system near the critical point through functional expansion of order parameters and free energy. This dynamic description method based on Ginzburg-Landau theory can well reproduce the microstructural evolution characteristics during the annealing phase transition.

[0126] Specifically, the evolution equation of the order parameter ξ is described using the Ginzburg-Landau equation, and its expression is as follows:

[0127]

[0128] Among them, L ξ Let ξ be the dynamic coefficient of the order parameter ξ system, r be the spatial coordinate vector, and t be time.

[0129] The evolution equation of the order parameter η is described using the Ginzburg-Landau equation, and its expression is:

[0130]

[0131] Among them, L η η is the dynamic coefficient of the order parameter η system, r is the spatial coordinate vector, and t is time.

[0132] Preferably, the constructed force field governing equations are force equilibrium equations, and their expressions are as follows:

[0133]

[0134] Where, x j Let be the spatial coordinates on the j-axis.

[0135] The governing equations for the electric field are Maxwell's equations, and their expression is as follows:

[0136]

[0137] Among them, electric displacement E i Let x be the electric field component. i Let i be the spatial coordinates on the i-axis.

[0138] S3. In step S2, the strong forms of the evolution equations for the order parameter ξ, the order parameter η, the force field control equation, and the electric field control equation were established. In this step, these strong forms of the control equations need to be transformed into weak forms. Specifically, based on the principle of virtual work, the weak forms of the evolution equations for the order parameter ξ, the order parameter η, the force field control equation, and the electric field control equation are derived respectively.

[0139] Specifically, the weak form of the force field governing equations is as follows:

[0140]

[0141] Where, τ i This represents the surface force component.

[0142] The weak form of the electric field control equation is:

[0143]

[0144] Where ω is the surface charge of the ferroelectric thin film, and φ is the electric potential.

[0145] The weak form of the evolution equation of the order parameter ξ is:

[0146]

[0147] Where A is the surface area, n j For the surface normal component.

[0148] The weak form of the evolution equation of the order parameter η is:

[0149]

[0150] Where A is the surface area, n j For the surface normal component.

[0151] S4. Compile the weak form equations into the COMSOL finite element software, perform finite element analysis on the equations, and obtain the phase structure of the annealing and crystallization process of amorphous hafnium oxide-based ferroelectric thin films.

[0152] Specifically, the phase structure of amorphous hafnium oxide-based ferroelectric thin films during annealing and crystallization is analyzed, including:

[0153] S41. Set annealing conditions: The heating temperature is intermittently increased from room temperature to the holding temperature, increasing by 25°C each time, with a total heating time of 23.75 seconds and an annealing heating rate of approximately 20°C / s; the holding temperature is 500°C, and the holding time is 30 seconds; the cooling temperature is intermittently decreased from the holding temperature to room temperature, decreasing by 25°C each time, with a total cooling time of 23.75 seconds.

[0154] S42. The weak forms of the evolution equations of the order parameter ξ, the order parameter η, the force field control equations, and the electric field control equations are edited into the finite element software for finite element analysis. The finite element analysis process includes:

[0155] (1) Model import: Compile the weak forms of the evolution equations of the order parameter ξ, the order parameter η, the force field control equations, and the electric field control equations into the COMSOL finite element software.

[0156] (2) Boundary condition settings:

[0157] In finite element software, periodic boundary conditions are used on the left and right sides of the ferroelectric thin film to simulate the lateral expansion characteristics of an infinitely large planar system. The force field sets boundary conditions that constrain both the top and bottom of the ferroelectric thin film, and the electric field sets boundary conditions that short-circuit both the top and bottom of the ferroelectric thin film to simulate the physical state of the electrodes with equipotential in actual devices.

[0158] (3) Initial value setting of the model: A Gaussian random distribution function is introduced to apply random perturbation to the order parameter η, thereby stimulating the spontaneous polarization evolution of the system.

[0159] Within the pre-defined initial nucleus region, the order parameter ξ is set to 1 to characterize the crystalline nucleus state before annealing. Outside the nucleus region, the order parameter ξ is set to 0 to correspond to the initial conditions of a completely amorphous phase.

[0160] The initial values ​​of both the force field and the electric field are set to 0.

[0161] (4) The problem region is discretized into small units by the finite element method, and the solution of the whole region is obtained by solving the local equations of each small unit.

[0162] Alternatively, the annealing conditions can also be set after the weak form equations are compiled into the COMSOL finite element software.

[0163] In actual production, when hafnium oxide-based ferroelectric thin films are deposited, many tiny nuclei (approximately 2 nm in radius) with a tetragonal phase structure are formed, but a complete crystal structure is not formed. These tiny nuclei will cause the thin film to first transform from an amorphous phase to a crystalline tetragonal phase during the annealing process, and the tetragonal phase will only partially transform into a monoclinic phase. During the cooling stage of the annealing process, the remaining tetragonal phase will then transform into a ferroelectric phase.

[0164] Based on the above finite element analysis, at the initial moment of annealing, such as Figure 4 and Figure 5 As shown, the initial value of the order parameter ξ is randomly changed in different regions. That is, the value of the order parameter ξ is set to 1 to characterize the crystal nucleus state before annealing. In the entire region outside the crystal nucleus region, the order parameter ξ is set to 0 to correspond to the initial condition of a completely amorphous phase. Under this condition, a tetragonal phase nucleus with a radius of 2nm is generated as the initial phase structure of the annealing process, that is, the amorphous phase.

[0165] During the heating process, such as Figure 6 As shown, the annealing temperature drives changes in the phase structure of the thin film; for example, after the heating process ends, the amorphous phase gradually decreases. In the final stage of heating, as... Figure 7 As shown, the proportion of tetragonal phases begins to increase, and, as Figure 8 As shown, some tetragonal phases begin to transform into monoclinic phases.

[0166] As the insulation process ends, such as Figure 9 and Figure 10 As shown, the amorphous phase will completely disappear, leaving only the monoclinic and tetragonal phases in the thin film; and because the phase transition barrier of the tetragonal-monoclinic phase is much higher than that of the tetragonal-ferroelectric phase transition barrier, the tetragonal phase will only partially transform into the monoclinic phase.

[0167] During the cooling process, such as Figure 11 As shown, the remaining tetragonal phase will easily transform into the ferroelectric phase. Figure 11 Different arrows in the diagram represent different polarization directions; and, as... Figure 12 As shown, the monoclinic phase remains unchanged during the cooling process. Therefore, it can be seen that... Figure 4-12 It can effectively simulate the dynamic phase structure evolution of amorphous hafnium oxide-based ferroelectric thin films during the annealing process.

[0168] Preferably, by changing the annealing temperature, the ferroelectric properties of hafnium oxide-based ferroelectric thin films after annealing under different annealing conditions can be analyzed using the analytical methods disclosed in this application. This provides guidance for the design of annealing parameters for hafnium oxide-based ferroelectric thin films, thereby reducing experimental research and development costs. Furthermore, it provides a basis for improving the proportion of ferroelectric phase and fatigue resistance of ferroelectric thin films.

[0169] It should be understood that the specific embodiments described above are merely illustrative or explanatory of the principles of this application and do not constitute a limitation thereof. Therefore, any modifications, equivalent substitutions, improvements, etc., made without departing from the spirit and scope of this application should be included within the protection scope of this application. Furthermore, the appended claims are intended to cover all variations and modifications falling within the scope and boundaries of the appended claims, or equivalent forms of such scope and boundaries.

Claims

1. A phase evolution and phase field analysis method for the annealing and crystallization process of amorphous hafnium oxide-based ferroelectric thin films, characterized in that, include: S1. Set the sequence parameter ξ according to the evolution process from amorphous phase to crystalline phase during the annealing of amorphous hafnium oxide thin films, and set the sequence parameter η according to the mutual transformation process between crystalline phases; S2. Based on the order parameter ξ and the order parameter η, establish a phase-field theoretical model for analyzing the phase evolution during the annealing and crystallization process of amorphous hafnium oxide-based ferroelectric thin films, including constructing the total free energy equation and governing equations of the thin film system; wherein, the total free energy is the sum of the free energy of the order parameter ξ and the free energy of the order parameter η, and the volume free density in the free energy of the order parameter η is obtained by coupling the order parameter ξ and the order parameter η; the governing equations include the evolution equation of the order parameter ξ describing the phase evolution, the evolution equation of the order parameter η, the force field governing equation, and the electric field governing equation. S3. Based on the principle of virtual work, derive the weak forms of the evolution equations of the order parameter ξ, the order parameter η, the force field control equation, and the electric field control equation, respectively. S4. Compile the weak form equations into the COMSOL finite element software, perform finite element analysis on the equations, and analyze the phase structure of the amorphous hafnium oxide-based ferroelectric thin film during annealing and crystallization.

2. The method according to claim 1, characterized in that, In step S1: When ξ = 0, it indicates that the ferroelectric thin film is in the amorphous phase; when ξ = 1, it indicates that the ferroelectric thin film is in the crystalline tetragonal phase. The order parameter η has order parameter components η i , where i = (1,2,3), η1 represents the ferroelectric phase with a polarization vector along the horizontal direction of the thin film, η2 represents the ferroelectric phase with a polarization vector along the thickness direction of the thin film, and η3 represents the monoclinic phase.

3. The method according to claim 2, characterized in that, In step S2, the total free energy equation is: F=F ξ +F η Among them, F ξ F is the free energy of the order parameter ξ. η The free energy is the order parameter η; The free energy F of the order parameter ξ ξ The expression is: In the formula The gradient energy coefficient, Let ξ be the gradient distribution, and V be the film volume; f ξ The double-well function is expressed as: f ξ =Hx 2 (1-ξ)(1-ξ+Δm) In the formula, H is the potential well depth, and Δμ is the annealing crystallization driving force; In the formula, λ is the driving force coefficient, T0 is the critical temperature for crystallization, and T is the annealing temperature; The free energy F of the order parameter η η The expression is: F η =∫f η dV f η =f bulk +f grad +f elas +f elec Among them, f bulk f represents the volume free energy density; grad f represents the domain wall gradient energy density; elas f represents the elastic strain energy density; elec Represents electrostatic energy density; Volume free energy density f bulk The expression is: In the formula, A1, A 2(o) A 2(m) A3 and A4 represent the Landau energy coefficients of the crystalline phase; A 2(o) A represents the Landau energy coefficient of the ferroelectric phase. 2(m) A represents the Landau energy coefficient of a single-clinic phase; 1-a A 2-a A 3-a Landau energy coefficient for amorphous phases; Elastic strain energy density f elas The expression is: Among them, c ijkl ε is the elastic energy coefficient; i, j, k, l are tensor indices, all ranging from 1, 2, to 3; ij ε kl Both represent strain components, ε ij =1 / 2(u i,j +u j,i ), u i For the displacement component, u i，j For u i The partial derivative with respect to the j-direction coordinate, u j，i For u j Partial derivative with respect to the coordinate in the i-direction; ε (i) This indicates that the ferroelectric thin film is in η i The spontaneous strain matrix under the corresponding state; Domain wall gradient energy density f grad The formula is: Where K η The gradient energy coefficient, For η i The gradient.

4. The method according to claim 3, characterized in that, The annealing temperature includes a sequentially maintained heating temperature, holding temperature, and cooling temperature; wherein, The heating temperature was intermittently increased from room temperature to the holding temperature, with each increase being 25°C, and the total heating time was 23.75 seconds. The heat preservation temperature is maintained for 30 seconds; The cooling temperature changes intermittently, decreasing by 25°C each time from the holding temperature to room temperature, with a total cooling time of 23.75 seconds.

5. The method according to claim 3, characterized in that, In step S2, the evolution equations for the order parameter ξ and the order parameter η, which describe the phase evolution, are constructed, including: The evolution equation of the order parameter ξ is described using the Ginzburg-Landau equation, and its expression is: Among them, L ξ ξ represents the dynamic coefficients of the order parameter ξ system, r is the spatial coordinate vector, and t is time; The evolution equation of the order parameter η is described using the Ginzburg-Landau equation, and its expression is: Among them, L η η is the kinetic coefficient of the order parameter system.

6. The method according to claim 5, characterized in that, In step S2, the expression for the constructed force field control equation is: Where, x j Let j be the spatial coordinates on the j-axis; The constructed electric field control equations are as follows: Among them, electric displacement E i Let x be the electric field component. i Let i be the spatial coordinates on the i-axis.

7. The method according to claim 6, characterized in that, The weak form of the evolution equation of the order parameter ξ is: Where A is the surface area, n j For surface normal components; The weak form of the evolution equation for the order parameter η is: Where, n j For the surface normal component.

8. The method according to claim 7, characterized in that, The weak form of the force field governing equations is: Where, τ i For surface force components; The weak form of the electric field control equation is: Where ω is the surface charge of the ferroelectric thin film, and φ is the electric potential.

9. The method according to claim 1, characterized in that, The process of compiling the weakly formal equations into COMSOL finite element software, solving the equations using finite element methods, and analyzing the phase structure of the amorphous hafnium oxide-based ferroelectric thin film during annealing and crystallization includes: The weak forms of the evolution equations of the order parameter ξ, the order parameter η, the force field control equation, and the electric field control equation are solved using the finite element method.

10. The method according to claim 9, characterized in that, The finite element solution process includes: The weak forms of the evolution equations of the order parameter ξ, the order parameter η, the force field control equation, and the electric field control equation are edited into the COMSOL finite element software. In the finite element software, periodic boundary conditions are used on the left and right sides of the ferroelectric thin film to simulate the lateral expansion characteristics of an infinitely large planar system; the force field sets boundary conditions that constrain both the top and bottom of the ferroelectric thin film, and the electric field sets boundary conditions that short-circuit both the top and bottom of the ferroelectric thin film to simulate the physical state of the electrodes with equipotential in actual devices. A Gaussian random distribution function is introduced to apply a random perturbation to the order parameter η, thereby stimulating the spontaneous polarization evolution of the system; Within the pre-defined initial nucleus region, the order parameter ξ is set to 1 to characterize the crystal nucleus state before annealing. Outside the nucleus region, the order parameter ξ is set to 0 to correspond to the initial conditions of a completely amorphous phase. The initial values ​​of both the force field and the electric field are set to 0; The entire ferroelectric thin film is discretized into small units using the finite element method, and the phase structure of the entire ferroelectric thin film is obtained by solving the local equations of each small unit.

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