Phase-field analysis of the effects of doping elements on the domain structure of HfO2 ferroelectric thin films

Abstract

The present invention discloses a phase field analysis method for analyzing the effects of doping elements on the domain structure of HfO2 ferroelectric thin films. The method includes establishing a total energy equation for the polarization order parameter; establishing force field control equations, polarization field control equations, and electric field control equations based on the total energy equation for the effects of the silicon doping element; establishing a doping element energy equation based on the chemical energy and gradient energy of the doping element; establishing a concentration field control equation based on the doping element energy equation; establishing a phase field calculation model for the ferroelectric thin film based on the force field control equations, polarization field control equations, electric field control equations, and concentration field control equations; and simulating the phase transition law of the HfO2 ferroelectric thin film by adjusting the concentration and distribution of the doping element based on the phase field calculation model. By introducing the doping element energy equation for the doping element, the method enables the phase field calculation model for the HfO2 ferroelectric thin film to provide optimized guidance for the concentration and distribution of the doping element.

Description

Technical Field

[0001] The present invention relates to the technical field of HfO2 ferroelectric thin film simulation analysis, and in particular to a phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films. Background Art

[0002] The discovery of ferroelectricity in HfO2 thin films addresses the incompatibility of conventional perovskite ferroelectrics with CMOS processes and the difficulty in scaling down, providing new opportunities for the development of ferroelectric memory devices. In HfO2 ferroelectric films, where ferroelectric and non-ferroelectric phases coexist, the ferroelectric phase is a metastable phase that readily transforms into a non-ferroelectric phase under the influence of complex mechanical, thermal, and electrical fields and microstructural factors. Research has shown that fatigue in HfO2 ferroelectric films is essentially caused by the gradual reduction of the reversible ferroelectric phase within the film over time. Therefore, stabilizing the ferroelectric phase and improving the film's fatigue resistance and ferroelectricity are current research priorities.

[0003] Numerous studies have demonstrated that adjusting dopant elements is an important means of enhancing the ferroelectricity of HfO2 thin films. However, current research lacks a single, fixed value for parameters such as the type, concentration, and distribution of the dopant element. Specifically, the effect of silicon concentration on the ferroelectricity of HfO2 thin films is not monotonically variable; rather, the effect of silicon concentration on ferroelectricity is often dual-faceted. Furthermore, in practical applications, the distribution of the dopant element can also influence the ferroelectric properties, resulting in variations in the optimal concentration range.

[0004] Therefore, it is necessary to establish a domain structure analysis method for HfO2 ferroelectric films that takes into account the influence of doping elements. By studying the influence of different concentrations and distribution methods of doping elements on the domain structure of HfO2 ferroelectric films, the goal of stabilizing the ferroelectric phase of HfO2 ferroelectric films and improving the fatigue resistance and ferroelectric properties of the films can be achieved. Summary of the Invention

[0005] (1) Purpose of the invention

[0006] The purpose of the present invention is to provide a phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films, so as to study the influence of different concentrations and distribution modes of doping elements on the domain structure of ferroelectric thin films.

[0007] (2) Technical solution

[0008] To solve the above problems, the first aspect of the present invention provides a phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films, the method comprising:

[0009] Considering the influence of doping elements on HfO2 ferroelectric thin films, the total energy equation of polarization order parameter is established;

[0010] Based on the total energy equation, a force field control equation, a polarization field control equation, and an electric field control equation based on the influence of the silicon-doped element are established;

[0011] The energy equation of doping elements is established based on the chemical energy and gradient energy of doping elements;

[0012] Establishing a concentration field control equation based on the doping element energy equation;

[0013] Establishing a phase field calculation model of the ferroelectric thin film under the influence of doping elements according to the force field control equation, polarization field control equation, electric field control equation and concentration field control equation;

[0014] Based on the phase field calculation model, the phase transition law of the HfO2 ferroelectric thin film is simulated by adjusting the concentration and distribution of the doping elements.

[0015] In some embodiments, the doping element energy equation is as follows:

[0016]

[0017] Where H is the free energy of concentration, f c is the concentration-dependent chemical free energy density, k c is the gradient energy coefficient, and c is the conserved order parameter of the doping element concentration.

[0018] In some embodiments, after establishing the energy equation of the doping element based on the chemical energy and gradient energy of the doping element, the method further includes calculating f based on the ideal model. c Make corrections:

[0019] The ideal model is as follows:

[0020]

[0021] Among them, μ c is the partial derivative, μ0 is a constant, and R is the ideal gas constant 8.314 J·mol -1 ·K -1 , T is the temperature constant 300K.

[0022] In some embodiments, the total energy equation is as follows:

[0023] F tot =∫ V fdV=∫ V (f bulk +f gradient +f elastic +f electric )dV

[0024] Among them, F tot is the total energy of the ferroelectric film, fbulk is the volume free energy density, f gradient is the polarization gradient energy density, f elastic is the elastic energy density, f electric is the electric field energy density.

[0025] In some embodiments, the total energy equation is further derived to obtain the following formula:

[0026]

[0027]

[0028]

[0029]

[0030]

[0031] Among them, A1, A 2(o) , A 2(m) , A3 and A4 are Landau energy coefficients, K ijkl is the gradient energy coefficient, C ijkl is the elastic coefficient, ε0 is the vacuum dielectric constant, η p (p=1~3) is the polarization order parameter, η1, η2, and η3 represent the ferroelectric orthogonal phase along the coordinate axes x, y, and z, respectively.

[0032] In some embodiments, the doping element is silicon;

[0033] A1=(-2.8624+26.1458c-7808.698c 2 +76897.0588c 3 )×10 10 N / m 2

[0034] A 2(o) =(-12.7626+1180.7343c-35438.8843c 2 +348111.7648c 3 )×10 10 N / m 2

[0035] A 2(m) =(-11.4906+1094.6870c-33393.6667c 2 +3332000.0003c 3 )×10 10 N / m 2 .

[0036] A3=(-8.3779+1370.4496c-55631.8080c 2 +66924.1177c 3 )×10 10 N / m 2

[0037] A4=8.314×10 8 N / m 2

[0038] In some embodiments, establishing the force field control equation, polarization field control equation, and electric field control equation based on the influence of the silicon-doped element based on the total energy equation includes:

[0039] Based on the total energy equation, a force field constitutive equation, a polarization field constitutive equation and an electric field constitutive equation are established;

[0040] According to the force field constitutive equation, the polarization field constitutive equation and the electric field constitutive equation, combined with the mechanical equilibrium equation, Maxwell's equations, the Ginzburg-Landau equation and the concentration diffusion equation, the force field control equation, the polarization field control equation and the electric field control equation are determined.

[0041] In some embodiments, the constitutive equation of the force field is as follows:

[0042]

[0043] Among them, σ ij is stress, c ijkl is the elastic modulus, ε kl For strain.

[0044] The constitutive equation of the electric field is as follows:

[0045]

[0046] Among them, D i is the electric displacement, ε0 is the vacuum dielectric constant, η p (p=1~3) is the polarization order parameter.

[0047] The constitutive equation of the polarization field is as follows:

[0048]

[0049] Among them, A1, A 2(o) , A 2(m) , A3 and A4 are Landau energy coefficients, η p (p=1~3) is the polarization order parameter, E i is the built-in electric field strength of HfO2-based ferroelectric thin film.

[0050] In some embodiments, the governing equation of the force field is:

[0051]

[0052] Among them, stress strain ε ij =1 / 2(u i,j +u j,i ),u i is the displacement component;

[0053] The governing equation of the polarization field is:

[0054]

[0055] Where F = ∫ V fdV, L1 represents the kinetic coefficient, F is the total energy of the ferroelectric film, and f is the total free energy density of the ferroelectric film;

[0056] The governing equation for the electric field is:

[0057]

[0058] Among them, the electric displacement

[0059] The governing equation of the concentration field is:

[0060]

[0061] Among them, c(r,t) represents the concentration field variable, t represents time, r represents the space vector, L C is the diffusion kinetic coefficient.

[0062] In some embodiments, establishing a phase field calculation model of a ferroelectric thin film under the influence of doping elements according to the force field control equation, the polarization field control equation, the electric field control equation, and the concentration field control equation includes:

[0063] Establishing a weak form of the force field control equation, a weak form of the polarization field control equation, a weak form of the electric field control equation, and a weak form of the concentration field control equation according to the force field control equation, the polarization field control equation, the electric field control equation, and the concentration field control equation;

[0064] Based on the weak form of the force field control equation, the weak form of the polarization field control equation, the weak form of the electric field control equation and the weak form of the concentration field control equation, a phase field calculation model of the ferroelectric film under the influence of doping elements is established;

[0065] The weak form of the force field governing equation is:

[0066]

[0067] The weak form of the electric field governing equation is:

[0068]

[0069] The weak form of the polarization field governing equation is:

[0070]

[0071] The weak form of the concentration field control equation is:

[0072]

[0073] (3) Beneficial effects

[0074] The present invention provides a phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films. By introducing the doping element energy equation of the doping element, the phase field calculation model of the HfO2 ferroelectric thin film can characterize the relationship between the concentration and distribution of the doping element and the domain structure of the hafnium oxide-based ferroelectric thin film, providing optimization guidance for the concentration and distribution of the doping element. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] Figure 1 1 is a flow chart of a phase field analysis method for the effect of doping elements on the domain structure of HfO2 ferroelectric thin films according to an embodiment of the present invention;

[0076] Figure 2 Schematic diagram of the phase field geometry model of an embodiment of the present invention;

[0077] Figure 3 Schematic diagram of the phase structure distribution of a 3.8cat% Si:HfO2 ferroelectric thin film according to an embodiment of the present invention;

[0078] Figure 4 Schematic diagram of the volume fractions of the non-ferroelectric phase, ferroelectric phase a-domain, and c-domain in the Si:HfO2 ferroelectric film at different Si concentrations in an embodiment of the present invention;

[0079] Figure 5 is a schematic diagram of the rate of change of the volume fraction of the non-ferroelectric phase with the concentration in an embodiment of the present invention;

[0080] Figure 6 is a schematic diagram of an AC electric field applied in an embodiment of the present invention;

[0081] Figure 7 Schematic diagram of the phase structure distribution of a 3.8cat% Si:HfO2 film at different times in an embodiment of the present invention;

[0082] Figure 8Schematic diagram of the residual polarization value of Si:HfO2 thin film under the action of different concentrations of Si element in an embodiment of the present invention;

[0083] Figure 9 This is a schematic diagram of the Si element concentration distribution in the thickness direction of the Si:HfO2 film in the present invention.

[0084] Figure 10 Schematic diagram of the residual polarization value of Si:HfO2 film under different concentrations and distribution modes in an embodiment of the present invention. DETAILED DESCRIPTION

[0085] To make the objectives, technical solutions, and advantages of the present invention more clearly understood, the present invention will be further described in detail below in conjunction with specific embodiments and with reference to the accompanying drawings. It should be understood that these descriptions are merely exemplary and are not intended to limit the scope of the present invention. In addition, in the following description, descriptions of well-known structures and technologies are omitted to avoid unnecessary confusion of the concepts of the present invention.

[0086] Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0087] In the description of the present invention, it should be noted that the terms “first” and “second” are only used for the purpose of distinction and should not be understood as indicating or implying relative importance.

[0088] Figure 1 FIG. 1 is a flow chart of a phase field analysis method for analyzing the effect of doping elements on the domain structure of HfO2 ferroelectric thin films according to an embodiment of the present invention. Figure 1 As shown, an embodiment of the present invention provides a phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films, comprising the following steps:

[0089] S101: Considering the influence of doping elements on the hafnium oxide-based ferroelectric thin film (HfO2 ferroelectric thin film), a total energy equation for the polarization order parameter is established; the doping element in this embodiment is silicon, and the total energy equation includes a volume free energy term, a polarization gradient energy term, an elastic energy term, and an electric field energy term, which are respectively used to characterize the volume free energy, polarization gradient energy, elastic energy, and electric field energy of the polarization order parameter of the hafnium oxide-based ferroelectric thin film after the addition of silicon. The total energy equation for determining the polarization order parameter is as follows:

[0090] F tot =∫ V fdV=∫ V (f bulk +f gradient +f elastic +felectric )dV

[0091] F tot is the total energy of HfO2 ferroelectric film, f bulk represents the volume free energy density, f gradient represents the polarization gradient energy density, f elastic represents the elastic energy density, f electric represents the electric field energy density. bulk Indicates the contribution of spontaneous polarization in hafnium oxide to the energy of the system. In the energy equation, the order parameter η is used. p (p = 1 to 3) to describe the volume free energy density. The specific expression is:

[0092]

[0093] Among them, A1, A 2(o) , A 2(m) , A3 and A4 are Landau energy coefficients, K ijkl and C ijkl are the gradient energy coefficient and elastic coefficient respectively, and ε0 is the vacuum dielectric constant. When the doping element is silicon, the Landau energy coefficient is as follows:

[0094] A1=(-2.8624+26.1458c-7808.698c 2 +76897.0588c 3 )×10 10 N / m 2

[0095] A 2(o) =(-12.7626+1180.7343c-35438.8843c 2 +348111.7648c 3 )×10 10 N / m 2

[0096] A 2(m) =(-11.4906+1094.6870c-33393.6667c 2 +3332000.0003c 3 )×10 10 N / m 2

[0097] A3=(-8.3779+1370.4496c-55631.8080c 2 +66924.1177c 3 )×10 10 N / m 2

[0098] A4=8.314×10 8 N / m 2

[0099] Where c is the conserved order parameter of silicon concentration.

[0100] S102: Based on the total energy equation, a force field control equation, a polarization field control equation, and an electric field control equation based on the influence of the doped silicon element are established.

[0101] S103: Establish an energy equation for the doping element based on the chemical energy and gradient energy of the doping element. The expression of the energy equation for the doping element is as follows:

[0102]

[0103] Where f c is the concentration-dependent chemical free energy density, k c is the gradient energy coefficient. In order to better couple with the phase change model, the ideal model is used to c With the following correction, its partial derivative can be expressed as:

[0104]

[0105] Where μ0 is a constant and R is the ideal gas constant 8.314 J·mol -1 ·K -1 , T is the temperature constant 300K.

[0106] S104: establishing a concentration field control equation based on the doping element energy equation;

[0107] S105: establishing a phase field calculation model of the HfO2 ferroelectric thin film under the influence of doping elements according to the force field control equation, polarization field control equation, electric field control equation, and concentration field control equation;

[0108] S106: Based on the phase field calculation model, the phase transition law of the HfO2 ferroelectric thin film is simulated by adjusting the concentration and distribution of the doping elements.

[0109] The phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films provided in an embodiment of the present invention can analyze the influence of silicon (Si) elements of different concentrations and distribution patterns on the domain structure change of hafnium oxide-based ferroelectric thin films through a phase field calculation model, revealing the narrow Si concentration control range that is difficult to directly measure in the existing technology; compared with the multi-phase coexistence phase field model of hafnium oxide-based ferroelectric thin films in related technologies, this method introduces the influence of Si elements on the lattice parameters and Landau coefficients of HfO2 ferroelectric thin films, adds the free energy considering the Si element concentration, and obtains the influence law of Si element concentration on the transformation of ferroelectric phase to monoclinic phase under specific control range and distribution pattern, explaining the influence of Si elements on ferroelectric domain reversal and its phase transition from a relatively macroscopic level, providing guidance for the stability of the ferroelectric phase of hafnium oxide-based ferroelectric thin films and the optimization of their macroscopic fatigue resistance and ferroelectric properties.

[0110] In some embodiments, establishing the force field control equation, polarization field control equation, and electric field control equation based on the influence of the silicon-doped element based on the total energy equation includes:

[0111] Based on the total energy equation, the force field constitutive equation, polarization field constitutive equation and electric field constitutive equation are established;

[0112] According to the force field constitutive equation, polarization field constitutive equation and electric field constitutive equation, combined with the mechanical equilibrium equation, Maxwell equations, Ginzburg-Landau equation and concentration diffusion equation, the force field control equation, polarization field control equation and electric field control equation are determined.

[0113] The phase field calculation model of the HfO2 ferroelectric thin film under the influence of doping elements is established according to the force field control equation, polarization field control equation, electric field control equation and concentration field control equation, including:

[0114] Establishing a weak form of the force field control equation, a weak form of the polarization field control equation, a weak form of the electric field control equation, and a weak form of the concentration field control equation according to the force field control equation, the polarization field control equation, the electric field control equation, and the concentration field control equation;

[0115] According to the weak form of the force field control equation, the weak form of the polarization field control equation, the weak form of the electric field control equation and the weak form of the concentration field control equation, a phase field calculation model of HfO2 ferroelectric thin film under the influence of doping elements is established.

[0116] The total energy density equation expressed by the polarization order parameter

[0117]

[0118] Further derive the constitutive equations of various force fields, electric fields, and polarization fields

[0119]

[0120]

[0121]

[0122] According to the constitutive equation, combined with the mechanical equilibrium equation, Maxwell's equations, Ginzburg-Landau equation and concentration diffusion equation, the physical field control equations affected by the silicon doping element are determined, wherein the force field control equation is:

[0123]

[0124] Among them, stress ε ij =1 / 2(u i,j +u j,i ),u i is the displacement component;

[0125] The polarization field governing equation is:

[0126]

[0127] Where F = ∫ V fdV, L1 represents the kinetic coefficient, F is the total energy of the HfO2 ferroelectric film, and f is the total free energy density of the HfO2 ferroelectric film;

[0128] The electric field control equation is:

[0129]

[0130] Among them, the electric displacement

[0131] The silicon element concentration field equation is:

[0132]

[0133] Among them, c(r,t) represents the concentration field variable, t represents time, r represents the space vector, L C is the diffusion kinetic coefficient.

[0134] The weak form of each physical field control equation is derived by solving it using the finite element method, where the weak form of the force field control equation is:

[0135]

[0136] The weak form of the electric field governing equation is:

[0137]

[0138] The weak form of the polarization field governing equation is:

[0139]

[0140] The weak form of the silicon element concentration field control equation is:

[0141]

[0142] Step S4, establishing a phase field calculation model of the hafnium oxide-based ferroelectric thin film under the influence of silicon doping based on the constitutive equations and weak forms of the control equations of the four fields of force field, polarization field, electric field and concentration field.

[0143] Since we are interested in the evolution of polarization in the thickness direction of hafnium oxide ferroelectric films, a two-dimensional plane model is established for simulation calculations. The corresponding polarization matrix and strain matrix are as follows:

[0144]

[0145]

[0146]

[0147] The subscripts o, t, and m in the strain matrix represent the ferroelectric orthorhombic phase, the parent tetragonal phase, and the non-ferroelectric monoclinic phase, respectively. a, b, and c represent the lattice constants. However, the Si-doped hafnium oxide ferroelectric thin film material is a compositionally inhomogeneous system, so the lattice parameters are a function of the local concentration of the impurity Si element. First-principles calculations show that the lattice parameters of each subphase can be approximated by linear functions, as shown in Table 1:

[0148] Table 1 Lattice constants of different phases in Si:HfO2

[0149]

[0150] Wherein c represents the Si atomic percentage, that is, c=Si / Si+Hf.

[0151] The phase field model is constructed using finite element software. The main steps include: (1) constructing the geometric model of Si:HfO2 ferroelectric film; (2) defining basic parameters; (3) assigning physical properties to the film, as well as determining the boundary conditions and initial values of various equations; (4) dividing the grid and then solving.

[0152] Figure 2 Schematic diagram of the structure of the phase field geometry model established in COMSOL Multiphysics software provided by the present invention. Figure 2As shown, this model constructs a two-dimensional phase-field model with a 10×10 geometry, corresponding to a practical size of 10nm×10nm. The energy equation, expansion coefficient, elastic energy coefficient, gradient energy coefficient, and weak-form equations for the force field, electric field, polarization field, and Si concentration field were compiled in the finite element software COMSOL Multiphysics. The model's boundary conditions were set as follows: periodic boundary conditions were used on the left and right sides of the plane model; the force field was set with both top and bottom constrained, with no deformation, i.e., u=0, v=0; and the electric field was set with both top and bottom short-circuited. The model's initial values were set: the initial value of the polarization was derived from the evolution of the order parameter by a random perturbation of the order parameter by a Gaussian random distribution function; the initial values of the force and electric fields were both set to 0.

[0153] Step S5 simulates the domain structure cloud diagram and phase transition law of the multiphase coexistence of Si:HfO2 ferroelectric thin films under different concentrations and distribution patterns. The solved field variable data is imported into a text document, and the field variable results are visualized using ParaView and Origin software. The phase transition law from the ferroelectric phase to the non-ferroelectric phase is obtained based on the domain structure cloud diagram under the influence of different Si concentrations, and the effect of Si concentration on the volume fraction of each phase is calculated. Based on the established model, an external electric field is applied to obtain hysteresis loops under the influence of different Si concentrations, and finally a curve of the relationship between Si concentration and remanent polarization is obtained.

[0154] Figure 3 Schematic diagram of the phase structure distribution of the 3.8cat% Si:HfO2 ferroelectric film obtained by the implementation method of the present invention. Figure 3 As shown, the yellow area represents the horizontal polarization, that is, the ferroelectric phase a domain; the blue area represents the vertical polarization, that is, the ferroelectric phase c domain; the gray area represents the polarization is zero, that is, the non-ferroelectric monoclinic phase. The white arrow represents the polarization vector. Figure 3 As can be seen from the figure, the ferroelectric phase is relatively uniformly distributed in the film. The volume fractions of the ferroelectric orthorhombic phase and the non-ferroelectric monoclinic phase were subsequently calculated, showing that the ferroelectric phase has a volume fraction of 68.9% and the non-ferroelectric phase has a volume fraction of 31.1%. Studies have shown that the volume fraction of the monoclinic phase in HfO2 ferroelectric films ranges from 10% to 50%, which confirms the correctness of the Si:HfO2 ferroelectric film phase field model developed in this invention.

[0155] Hafnium oxide ferroelectric thin films are materials in which ferroelectric and non-ferroelectric phases coexist. Improving the ferroelectricity of hafnium oxide ferroelectric thin films primarily involves increasing the volume fraction of the ferroelectric phase within the film. Therefore, the present invention simulated and calculated hafnium oxide ferroelectric thin films with six different Si concentrations, analyzing the effects of varying Si concentrations on the ferroelectricity of hafnium oxide films. When Si is uniformly distributed within the film, the effect of Si concentration on the ferroelectric phase transition is not monotonic but rather exhibits a two-sided effect. As Si concentration increases, the volume fraction of the ferroelectric phase within the film exhibits a trend of first increasing and then decreasing.

[0156] Figure 4 Schematic diagram of the volume fraction of the non-ferroelectric phase, ferroelectric phase a domain and c domain in the Si:HfO2 ferroelectric film at different Si concentrations in the present invention. Figure 4 As shown in the figure, the concentration has a two-sided effect on the ferroelectric phase. The volume fraction of the non-ferroelectric phase first decreases and then increases, while correspondingly, the volume fractions of the ferroelectric phase c-domains and a-domains first increase and then decrease. When the initial Si doping concentration is 2.6 cat%, the volume fraction of the non-ferroelectric phase is 69.8%, the volume fraction of the ferroelectric phase c-domains is 19.9%, and the volume fraction of the ferroelectric phase a-domains is 10.3%. When the Si concentration increases to 3.8 cat%, the non-ferroelectric phase reaches a minimum of 31.1%, while the volume fractions of the ferroelectric phase c-domains and a-domains reach a maximum of 43.4% and 25.5%, respectively. When the Si concentration continues to increase to 4.3 cat%, the volume fraction of the non-ferroelectric phase increases to 68.2%. This indicates that the concentration corresponding to the maximum volume fraction of the ferroelectric phase in the Si:HfO2 film is 3.8 cat%, that is, the optimal concentration of Si element to regulate the ferroelectricity of the hafnium oxide ferroelectric film is 3.8 cat%.

[0157] Figure 5 Schematic diagram of the change rate of the volume fraction of the non-ferroelectric phase with the concentration in the present invention. The concentration of Si is 3.8cat% and is called the optimal concentration. Figure 5 As shown in the figure, the closer to the optimal control concentration, the faster the change in the volume fraction of each phase. That is, when the Si concentration is around 3.8cat%, the phase transition rate of the monoclinic phase is relatively large. The closer to the optimal Si concentration for controlling ferroelectricity, the more sensitive the change in the film's ferroelectricity will be.

[0158] Figure 6 Schematic diagram of the AC electric field applied in the present invention. Figure 6 As shown in Figure 1, in order to explore the domain reversal performance of Si:HfO2 film, a sinusoidal AC electric field was applied in the thickness direction of Si:HfO2 film.

[0159] Figure 7Schematic diagram of the phase structure distribution of the 3.8cat% Si:HfO2 film at different times in the present invention. The electric field intensity reaches its maximum at t = 5s, so the domain reversal of the film is analyzed at six times: t = 0s, 1s, 2s, 3s, 4s and 5s. Figure 7 Figure 2 shows the phase structure distribution of a 3.8cat% Si:HfO2 film at t = 0s, 1s, 2s, 3s, 4s, and 5s. As can be seen from the figure, as time increases and the applied electric field gradually increases, the area of the white region (ferroelectric phase a domain) gradually decreases, while the area of the black region (ferroelectric phase c domain) gradually increases. This indicates that under the action of the electric field, the ferroelectric phase a domain in the film slowly transforms into ferroelectric phase c domain.

[0160] Figure 8 Schematic diagram of the residual polarization value of Si:HfO2 film under the action of different concentrations of Si element in the embodiment of the present invention. Figure 8 As shown in Figure 3, with the increase of Si concentration, the remnant polarization of the film first increases and then decreases, among which the remnant polarization of the 3.8cat% Si:HfO2 film is the largest.

[0161] Figure 9 Schematic diagram of Si element concentration distribution along the thickness direction of Si:HfO2 film in the present invention. Figure 9 As shown, in addition to the Si concentration, the Si-rich and Si-poor regions formed due to the uneven distribution of Si in the hafnium oxide film can also significantly affect the ferroelectricity of the film. Therefore, in this embodiment, the regulation of the concentration difference between the Si-rich and Si-poor regions on the ferroelectricity of the film is analyzed.

[0162] Figure 10 Schematic diagram of the residual polarization value of Si:HfO2 film under different Si concentration and distribution mode in the embodiment of the present invention. Figure 10 As shown in Figure 2, the layered Si distribution significantly alters the effect of Si concentration on the ferroelectric properties of the film. Excessively large concentration differences, Δc, between Si-rich and Si-poor regions can severely degrade the film's ferroelectric properties. However, a concentration difference of approximately 7.6% not only significantly improves the remnant polarization of the ferroelectric film but also broadens the optimal Si concentration range.

[0163] As described above, the present invention details the phase-field analysis method for analyzing the effect of silicon on the domain structure of hafnium oxide-based ferroelectric thin films. The simulation method is simple and efficient. Through computational simulation, it systematically analyzes the effect of Si on the domain structure of hafnium oxide-based ferroelectric thin films at different concentrations and distributions, revealing the narrow Si concentration control range that is difficult to directly measure in the prior art. Compared to the multiphase coexistence phase-field model of hafnium oxide-based ferroelectric thin films, this method introduces the effect of Si on the lattice parameters and Landau coefficients of HfO2 ferroelectric thin films, incorporates the free energy of Si concentration, and obtains the influence of Si concentration and distribution on the transition from the ferroelectric phase to the monoclinic phase within a specific control range. This method explains the effect of Si on ferroelectric domain reversal and its phase transition from a relatively macroscopic perspective, providing more accurate simulation results and providing guidance for optimizing the macroscopic ferroelectric properties of hafnium oxide-based ferroelectric thin films.

[0164] The present invention has been described above with reference to the embodiments thereof. However, these embodiments are for illustrative purposes only and are not intended to limit the scope of the present invention. The scope of the present invention is defined by the appended claims and their equivalents. Those skilled in the art may make various substitutions and modifications without departing from the scope of the present invention, and such substitutions and modifications are intended to fall within the scope of the present invention.

Claims

1. A phase field analysis method for the effect of doping elements on the domain structure of HfO2 ferroelectric thin films, characterized in that: include: Considering the influence of doping elements on HfO2 ferroelectric thin films, the total energy equation of polarization order parameter is established; Based on the total energy equation, a force field control equation, a polarization field control equation, and an electric field control equation based on the influence of the silicon-doped element are established; The energy equation of doping elements is established based on the chemical energy and gradient energy of doping elements; Establishing a concentration field control equation based on the doping element energy equation; A phase field calculation model of the HfO2 ferroelectric thin film under the influence of doping elements is established according to the force field control equation, polarization field control equation, electric field control equation and concentration field control equation; Based on the phase field calculation model, the phase transition law of the HfO2 ferroelectric thin film is simulated by adjusting the concentration and distribution of the doping elements; The doping element energy equation is as follows: Where H is the free energy of concentration, f c is the concentration-dependent chemical free energy density, k c is the gradient energy coefficient, c is the conserved order parameter of the doping element concentration; After establishing the energy equation of doping elements based on the chemical energy and gradient energy of doping elements, it also includes the f c Make corrections: The ideal model is as follows: Among them, μ c is the partial derivative, μ0 is a constant, and R is the ideal gas constant 8.314 J·mol -1 ·K -1 , T is the temperature constant 300K.

2. The phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films according to claim 1, characterized in that: The total energy equation is as follows: F tot =∫ V fdV=∫ V (f bulk +f gradient +f elastic +f electric )dV Among them, F tot is the total energy of HfO2 ferroelectric film, f bulk is the volume free energy density, f gradient is the polarization gradient energy density, f elastic is the elastic energy density, f electric is the electric field energy density.

3. The phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films according to claim 1, characterized in that: The total energy equation is further derived to obtain the following formula: Among them, A1, A 2(o) , A 2(m) , A3 and A4 are Landau energy coefficients, K ijkl is the gradient energy coefficient, C ijkl is the elastic coefficient, ε0 is the vacuum dielectric constant, η p (p=1~3) is the polarization order parameter, η1, η2, and η3 represent the ferroelectric orthogonal phase along the coordinate axes x, y, and z, respectively.

4. The phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films according to claim 3, characterized in that: The doping element is silicon; A1=(-2.8624+26.1458c-7808.698c 2 +76897.0588c 3 )×10 10 N / m 2 <h2 style=";text-align:left;direction:ltr">A<h2 style=";text-align:left;direction:ltr"> 2(o) <h2 style=";text-align:left;direction:ltr"> (-12.7626+1180.7343c-35438.8843c<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> +348111.7648c<h2 style=";text-align:left;direction:ltr"> 3 <h2 style=";text-align:left;direction:ltr"> )×10<h2 style=";text-align:left;direction:ltr"> 10 <h2 style=";text-align:left;direction:ltr"> N / m<h2 style=";text-align:left;direction:ltr"> 2 A 2(m) =(-11.4906+1094.6870c-33393.6667c 2 +3332000.0003c 3 )×10 10 N / m 2 。 <h2 style=";text-align:left;direction:ltr">A3=(-8.3779+1370.4496c-55631.8080c<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> +66924.1177c<h2 style=";text-align:left;direction:ltr"> 3 <h2 style=";text-align:left;direction:ltr"> )×10<h2 style=";text-align:left;direction:ltr"> 10 <h2 style=";text-align:left;direction:ltr"> N / m<h2 style=";text-align:left;direction:ltr"> 2 A4=8.314×10 8 N / m 2 5. The phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films according to claim 1, characterized in that: Based on the total energy equation, the force field control equation, polarization field control equation, and electric field control equation based on the influence of silicon doping elements are established, including: Based on the total energy equation, a force field constitutive equation, a polarization field constitutive equation and an electric field constitutive equation are established; According to the force field constitutive equation, the polarization field constitutive equation and the electric field constitutive equation, combined with the mechanical equilibrium equation, Maxwell's equations, the Ginzburg-Landau equation and the concentration diffusion equation, the force field control equation, the polarization field control equation and the electric field control equation are determined.

6. The phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films according to claim 5, characterized in that: The constitutive equation of the force field is as follows: Among them, σ ij is stress, c ijkl is the elastic modulus, ε kl For strain; The constitutive equation of the electric field is as follows: Among them, D i is the electric displacement, ε0 is the vacuum dielectric constant, η p (p = 1 to 3) is the polarization order parameter; The constitutive equation of the polarization field is as follows: Among them, A1, A 2(o) , A 2(m) , A3 and A4 are Landau energy coefficients, η p (p=1~3) is the polarization order parameter, E i is the built-in electric field strength of HfO2 ferroelectric film.

7. The phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films according to claim 5, characterized in that: The governing equation of the force field is: Among them, stress strain ε ij =1 / 2(u i,j +u j,i ),u i is the displacement component; The governing equation of the polarization field is: Where F = ∫ V fdV, L1 represents the kinetic coefficient, F is the total energy of the HfO2 ferroelectric film, and f is the total free energy density of the HfO2 ferroelectric film; The governing equation for the electric field is: Among them, the electric displacement The governing equation of the concentration field is: Among them, c(r,t) represents the concentration field variable, t represents time, r represents the space vector, L C is the diffusion kinetic coefficient.

8. The phase field analysis method for the influence of doping elements on the domain structure of HfO2 ferroelectric thin films according to claim 7, characterized in that: The phase field calculation model of the HfO2 ferroelectric film under the influence of doping elements is established according to the force field control equation, polarization field control equation, electric field control equation and concentration field control equation, including: Establishing a weak form of the force field control equation, a weak form of the polarization field control equation, a weak form of the electric field control equation, and a weak form of the concentration field control equation according to the force field control equation, the polarization field control equation, the electric field control equation, and the concentration field control equation; The phase field calculation model of HfO2 ferroelectric thin film under the influence of doping elements is established based on the weak form of the force field control equation, the weak form of the polarization field control equation, the weak form of the electric field control equation and the weak form of the concentration field control equation. The weak form of the force field governing equation is: The weak form of the electric field governing equation is: The weak form of the polarization field governing equation is: The weak form of the concentration field control equation is:

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