A phase field analysis method for hafnium oxide-based ferroelectric thin films based on the effect of oxygen vacancies

Abstract

The present application discloses a phase field analysis method based on the influence of oxygen vacancies on hafnium oxide-based ferroelectric films, including: considering the influence of oxygen vacancy concentration on hafnium oxide-based ferroelectric films, determining the total energy equation; establishing the constitutive equations of various physical fields based on the influence of oxygen vacancies through the energy equation; determining the control equations of various physical fields; solving through the finite element method, deriving the weak form of the control equations of various physical fields based on the influence of oxygen vacancies, and establishing a phase field calculation model based on the influence of oxygen vacancies for hafnium oxide-based ferroelectric films according to the weak form of the equations, so as to simulate the domain structure cloud diagram and its phase transition law of polymorphic coexistence of hafnium oxide-based ferroelectric films under different oxygen vacancy concentrations. The present application can provide theoretical guidance on the influence of oxygen vacancies for experimenters during the experiment through the establishment of a calculation model, thereby effectively reducing the experimental cost, and having a preset range control value for the oxygen vacancy concentration used in the experiment.

Description

Technical Field

[0001] The invention relates to the technical field of ferroelectric thin film simulation analysis, and in particular to a phase field analysis method of a hafnium oxide-based ferroelectric thin film based on the influence of oxygen vacancies. Background Art

[0002] Among the existing methods for studying the effect of oxygen vacancies on the ferroelectricity of hafnium oxide-based ferroelectric films, the experimental means require the use of scanning transmission electron microscopy, which is too expensive and easily interfered by other factors; first-principles research can usually only study systems composed of hundreds to thousands of atoms, which is relatively microscopic and cannot truly control the effect of oxygen vacancies on the phase change process and domain evolution of the entire film; the existing phase field simulation methods mostly study the effect of oxygen vacancies on ferroelectric properties under traditional ferroelectric materials, and there are few studies on hafnium oxide-based ferroelectric films where ferroelectric and non-ferroelectric phases coexist. It is necessary to establish a phase field analysis method for multi-phase coexistence of hafnium oxide-based ferroelectric films that takes into account the effect of oxygen vacancies. By studying the effect of oxygen vacancy concentration on the ferroelectricity of hafnium oxide-based ferroelectric films, the goal of oxygen vacancy regulation of phase transition, domain structure evolution, and domain reversal performance can be achieved. Summary of the invention

[0003] The purpose of the present application is to provide a phase field analysis method of hafnium oxide-based ferroelectric thin films based on the influence of oxygen vacancies, which can effectively improve experimental efficiency.

[0004] The specific technical solutions provided by this application are as follows:

[0005] Considering the influence of oxygen vacancy concentration on hafnium oxide-based ferroelectric thin films, the total energy equation is determined, and the total energy equation includes the volume free energy density in the ferroelectric thin film under the influence of oxygen vacancy concentration, the elastic energy density in the ferroelectric thin film, the gradient energy density in the ferroelectric thin film and the electrostatic energy density in the ferroelectric thin film; by deriving the energy equation, the constitutive equations of various physical fields based on the influence of oxygen vacancies are established; according to the constitutive equations, the control equations of various physical fields are determined in combination with the mechanical equilibrium equations, Maxwell equations, Ginzburg-Landau equations and concentration diffusion equations; then the finite element method is used to solve and derive the weak forms of the control equations of various physical fields based on the influence of oxygen vacancies, specifically including the weak forms of the force field control equations, the weak forms of the electric field control equations, the weak forms of the polarization field control equations and the weak forms of the concentration field control equations; according to the weak forms of the equations, a phase field calculation model of the hafnium oxide-based ferroelectric thin film based on the influence of oxygen vacancies is established, so as to simulate the domain structure cloud diagram and its phase transition law of the polymorphic coexistence of hafnium oxide-based ferroelectric thin films under different oxygen vacancy concentrations.

[0006] Optionally, the total energy equation includes volume free energy density in the ferroelectric film, elastic energy density in the ferroelectric film, gradient energy density in the ferroelectric film, and electrostatic energy density in the ferroelectric film under the influence of oxygen vacancy concentration, wherein:

[0007] The overall energy equation is:

[0008] F = ∫ V fdV,f=f bulk +f gradient +f elastic +f electric

[0009] Where F is the total energy of the ferroelectric film, f is the total free energy density of the ferroelectric film, and f is the volume free energy density in the ferroelectric film. bulk , the elastic energy density f in the ferroelectric film elastic , the gradient energy density f in the ferroelectric film gradient and the electrostatic energy density f in the ferroelectric film electric The sum of L kinetic coefficient, η i is the order parameter in phase field theory, where when i=1, that is, η 1 It represents the polarization state of the ferroelectric film in the horizontal direction, that is, the state of a domain polarization; when i = 2, that is, η 2 It represents the polarization of the ferroelectric film in the vertical direction, that is, the state of c-domain polarization; when i = 3, that is, η 3 It represents the non-ferroelectric phase, i.e., the monoclinic phase, in the ferroelectric thin film model.

[0010] The volume free energy density f in the ferroelectric film bulk The formula is:

[0011]

[0012] Among them, A 1 , A 2(o) , A 2(m) , A 3 and A 4 is the Landau coefficient, A 5 is the coupling coefficient, where A 2(m) is the Landau coefficient corresponding to the non-ferroelectric (monoclinic phase), A 2(o) is the Landau coefficient corresponding to the ferroelectric phase (orthogonal phase), A 2(m) Less than A 2(0) ; A 1 and A 2(0) The coercive field of the remnant polarization of the experimental hysteresis loop is calculated respectively; A 3 and A 4 Obtained by fitting the experimental hysteresis loop.

[0013] Elastic energy density f in ferroelectric thin film elastic The formula is:

[0014]

[0015] Among them, ε ij =1 / 2(u i,j +u j,i ),u i is the displacement component,

[0016] ε ij , ε kl Both represent strain components, ε (i) represents the ferroelectric film in η i The spontaneous strain corresponding to the state; i is 1, 2 or 3, indicating the spontaneous strain corresponding to each state in the ferroelectric film. The spontaneous strain ε corresponding to each state can be calculated based on the lattice constants of the a, b, and c axes of the hafnium oxide-based ferroelectric film. (i) (i=1,2,3);c ijkl is the elastic energy coefficient, i, j, k, and l represent different values ​​in a spatial range respectively; in the strain matrix, the number of values ​​is different for different spatial dimensions. If the spatial range is three-dimensional, then i = (1,2,3), j = (1,2,3), k = (1,2,3), and l = (1,2,3); if the spatial range is two-dimensional, then i = (1,2), j = (1,2), k = (1,2), and l = (1,2).

[0017] The gradient energy density f in the ferroelectric film gradient The formula is:

[0018]

[0019] K η is the gradient energy coefficient, η (i，j) is the polarization gradient, η i is the order parameter in phase field theory.

[0020] The electrostatic energy density f in the ferroelectric film electric The formula is:

[0021]

[0022] κ 0 is the vacuum dielectric constant, E i is the built-in electric field strength of the hafnium oxide-based ferroelectric film, is the built-in electric field strength caused by oxygen vacancies.

[0023] Optionally, the constitutive equations of various physical fields based on the influence of oxygen vacancies are:

[0024]

[0025]

[0026]

[0027] Optionally, the governing equations for each physical field are determined by combining the mechanical equilibrium equations, Maxwell equations, Ginzburg-Landau equations and concentration diffusion equations, where:

[0028] The force field control equation is:

[0029]

[0030] Among them, stress ε ij =1 / 2(u i,j +u j,i ),u i is the displacement component.

[0031] The polarization field control equation is:

[0032]

[0033] Where F = ∫ V fdV,f=f bulk +f gradient +f elastic +f electric .

[0034] The electric field control equation is:

[0035]

[0036] D=ε 0 E+P

[0037] Among them, the electric displacement ρ is the charge density of oxygen vacancies inside the film.

[0038] The concentration field energy equation is:

[0039]

[0040] Among them, V o represents the oxygen vacancy concentration, is the oxygen vacancy drift velocity, represents the oxygen vacancy diffusion rate.

[0041] The diffusion rate of oxygen vacancies is With drift rate Obey the Einstein relationship:

[0042]

[0043] Among them, k B is the Boltzmann constant, T is the absolute temperature, q is the unit charge, and h is the number of unit charges carried by the oxygen vacancy.

[0044] Optionally, the weak form of the control equation includes a weak form of the force field control equation, a weak form of the electric field control equation, a polarization field control equation, and a weak form of the concentration field control equation, wherein:

[0045] The weak form of the force field governing equation is:

[0046]

[0047] The weak form of the governing equation for the electric field is:

[0048]

[0049] The weak form of the polarization field governing equation is:

[0050]

[0051] Among them, stress ε ij =1 / 2(u i,j +u j,i ),u i is the displacement component.

[0052] The weak form of the concentration field governing equation is:

[0053]

[0054] A method for establishing and using a phase field calculation model of a hafnium oxide-based ferroelectric film based on the influence of oxygen vacancies comprises: establishing a phase field simulation geometric model; compiling the equations in the above technical solution in finite element software according to the geometric model; setting initial conditions and boundary conditions according to the geometric model, and meshing the preset film; solving the set and compiled model using finite element software; and importing the solved field variable data into analysis and visualization software according to the solution, and finally obtaining a visualized field variable result.

[0055] Optionally, different oxygen vacancy concentrations are solved to obtain domain structure cloud diagrams under different oxygen vacancy concentrations to obtain the evolution law of the ferroelectric phase to the non-ferroelectric phase, and the influence of the oxygen vacancy concentration on the volume fraction of each phase is calculated based on this.

[0056] Optionally, an external electric field is applied to the model according to the settings and compilation to obtain hysteresis loops at different oxygen vacancy concentrations, and finally a relationship curve between oxygen vacancy concentration and remanent polarization is obtained.

[0057] The above technical solution of the present application has the following beneficial technical effects:

[0058] This application constructs the force field control equation, polarization field control equation, electric field control equation and concentration diffusion equation under the influence of oxygen vacancies, considers the coupling term of oxygen vacancy concentration and polarization, obtains the domain structure diagram at different times under different oxygen vacancy concentrations, calculates the volume fraction of ferroelectric phase and non-ferrous phase according to different domain structure diagrams, obtains the evolution law of different phases over time, and the relationship diagram of different oxygen vacancy concentrations and residual polarization values. In a specific control range, the influence of oxygen vacancy concentration on the transformation of ferroelectric phase to monoclinic phase is obtained, and the influence of oxygen vacancy concentration on ferroelectric domain reversal and its phase transition is explained from a relatively macroscopic level, providing guidance for the optimization of the macroscopic performance of hafnium oxide-based ferroelectric films. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 It is a schematic diagram of the process of the phase field analysis method of the present application;

[0060] Figure 2 It is a multiphase cloud diagram of different oxygen vacancy concentration gradients of the hafnium oxide-based ferroelectric thin film under the action of oxygen vacancies provided by the embodiment of the present application;

[0061] Figure 3 It is a schematic diagram of the percentages of a domain, c domain and monoclinic phase in a hafnium oxide-based ferroelectric film provided by an embodiment of the present application under the action of different oxygen vacancy concentrations;

[0062] Figure 4 It is a hysteresis loop of a hafnium oxide-based ferroelectric film under different oxygen vacancy concentrations provided by an embodiment of the present application;

[0063] Figure 5 is a schematic diagram of the residual polarization value of the hafnium oxide-based ferroelectric film under the action of different oxygen vacancy concentrations provided by an embodiment of the present application;

[0064] Figure 6 This is a diagram of the phase content ratio of the hafnium oxide-based ferroelectric film provided in an embodiment of the present application under the action of different oxygen vacancy concentrations in an external electric field. DETAILED DESCRIPTION

[0065] In order to make the purpose, technical scheme and advantages of the present invention clearer, the present invention is 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 only exemplary and are not intended to limit the scope of the present invention. In addition, in the following description, the description of well-known structures and technologies is omitted to avoid unnecessary confusion of the concept of the present invention.

[0066] A phase field analysis method for hafnium oxide-based ferroelectric thin films based on oxygen vacancies, comprising:

[0067] Considering the influence of oxygen vacancy concentration on hafnium oxide-based ferroelectric thin films, the total energy equation is determined. The total energy equation includes the volume free energy density in the ferroelectric thin film under the influence of oxygen vacancy concentration, the elastic energy density in the ferroelectric thin film, the gradient energy density in the ferroelectric thin film and the electrostatic energy density in the ferroelectric thin film.

[0068] Specifically, the total energy equation is expressed as:

[0069] F = ∫ V fdV,f=f bulk +f gradient +f elastic +f electric

[0070] Where F is the total energy of the ferroelectric film, f is the total free energy density of the ferroelectric film, and f is the volume free energy density in the ferroelectric film. bulk , the elastic energy density f in the ferroelectric film elastic , the gradient energy density f in the ferroelectric film gradient and the electrostatic energy density f in the ferroelectric film electric The sum of L kinetic coefficient, η i is the order parameter in phase field theory, where when i=1, that is, η 1 It represents the polarization state of the ferroelectric film in the horizontal direction, that is, the state of a domain polarization; when i = 2, that is, η 2 It represents the polarization of the ferroelectric film in the vertical direction, that is, the state of c-domain polarization; when i = 3, that is, η 3 It represents the non-ferroelectric phase, i.e., the monoclinic phase, in the ferroelectric thin film model.

[0071] The volume free energy density f in the ferroelectric film bulk The formula is:

[0072]

[0073] Among them, A 1 , A 2(o) , A 2(m) , A 3 and A 4 is the Landau coefficient, A 5 is the coupling coefficient, where A 2(m) is the Landau coefficient corresponding to the non-ferroelectric (monoclinic phase), A 2(o) is the Landau coefficient corresponding to the ferroelectric phase (orthogonal phase), A 2(m) Less than A 2(0) ; A 1 and A 2(0) The coercive field of the remnant polarization of the experimental hysteresis loop is calculated respectively; A 3 and A 4Obtained by fitting the experimental hysteresis loop.

[0074] Elastic energy density f in ferroelectric thin film elastic The formula is:

[0075]

[0076] Among them, ε ij =1 / 2(u i,j +u j,i ),u i is the displacement component,

[0077] ε ij , ε kl Both represent strain components, ε (i) represents the ferroelectric film in η i The spontaneous strain corresponding to the state; i is 1, 2 or 3, indicating the spontaneous strain corresponding to each state in the ferroelectric film. The spontaneous strain ε corresponding to each state can be calculated based on the lattice constants of the a, b, and c axes of the hafnium oxide-based ferroelectric film. (i) (i=1,2,3);c ijkl is the elastic energy coefficient, i, j, k, and l represent different values ​​in a spatial range respectively; in the strain matrix, the number of values ​​is different for different spatial dimensions. If the spatial range is three-dimensional, then i = (1,2,3), j = (1,2,3), k = (1,2,3), and l = (1,2,3); if the spatial range is two-dimensional, then i = (1,2), j = (1,2), k = (1,2), and l = (1,2).

[0078] The gradient energy density f in the ferroelectric film gradient The formula is:

[0079]

[0080] K η is the gradient energy coefficient, η (i，j) is the polarization gradient, η i is the order parameter in phase field theory.

[0081] The electrostatic energy density f in the ferroelectric film electric The formula is:

[0082]

[0083] κ 0 is the vacuum dielectric constant, E i is the built-in electric field strength of the hafnium oxide-based ferroelectric film, is the built-in electric field strength caused by oxygen vacancies.

[0084] The constitutive equation can be derived from the electric enthalpy density defined by the total energy equation. The specific constitutive equation is as follows:

[0085]

[0086]

[0087]

[0088] Specifically, the control equations of each physical field are determined by combining the mechanical equilibrium equation, Maxwell equations, Ginzburg-Landau equations and concentration diffusion equations, where:

[0089] The established force field control equation is as follows:

[0090]

[0091] Among them, stress ε ij =1 / 2(u i,j +u j,i ),u i is the displacement component.

[0092] The polarization field control equation is:

[0093]

[0094] Among them, η i is the order parameter, t is the time, F is the total energy of the ferroelectric film, δF / η i It represents the driving force for the order parameter to change continuously over time so that the energy can reach a stable state. It represents the change of order parameter with time.

[0095] Where F = ∫ V fdV,f=f bulk +f gradient +f elastic +f electric .

[0096] The electric field control equation is:

[0097]

[0098] D=ε 0 E+P

[0099] Among them, the electric displacement ρ is the charge density of oxygen vacancies inside the film.

[0100] The concentration diffusion equation is:

[0101]

[0102] Among them, V o represents the oxygen vacancy concentration, is the oxygen vacancy drift velocity, represents the diffusion rate of oxygen vacancies. With drift rate Obey the Einstein relationship:

[0103]

[0104] Among them, k B is the Boltzmann constant, T is the absolute temperature, q is the unit charge, and h is the number of unit charges carried by the oxygen vacancy.

[0105] According to the constitutive equation, combined with the mechanical equilibrium equation, Maxwell equation, Ginzburg-Landau equation and concentration diffusion equation, the weak form of each physical field control equation based on the influence of oxygen vacancies is derived. Specifically:

[0106] According to the principle of virtual work, the weak form of the force field control equation, the weak form of the polarization field control equation, the electric field control equation and the weak form of the concentration field control equation are derived respectively.

[0107] From the above mechanical equilibrium equation have to

[0108]

[0109]

[0110]

[0111]

[0112]

[0113] Among them, C i is the surface force component t i .

[0114] From the above Ginzburg-Landau equation, the weak form of the polarization field energy equation is given by have to:

[0115] F = ∫ V f

[0116]

[0117]

[0118]

[0119]

[0120] because

[0121]

[0122] so

[0123]

[0124] From the above Maxwell equations We can get:

[0125]

[0126]

[0127]

[0128]

[0129] From the above concentration field diffusion equation We can get:

[0130]

[0131]

[0132] Among them, the oxygen vacancy concentration diffuses along the x direction,

[0133]

[0134] because

[0135] so,

[0136]

[0137] The oxygen vacancy concentration diffuses along the y direction,

[0138]

[0139] Same

[0140] so,

[0141]

[0142] From the above, we can get:

[0143]

[0144]

[0145] According to the above steps, a phase field calculation model of hafnium oxide-based ferroelectric films based on the influence of oxygen vacancies is established.

[0146] In order to better use phase field simulation for simulation, a phase field simulation geometric model is established. The selected geometric size is 40*40, which corresponds to the actual size of 40nm*40nm. The energy equation expression, expansion coefficient, elastic energy coefficient, gradient energy coefficient, and weak form of force field, electric field, polarization field and general partial differential equation of oxygen vacancy concentration are compiled in the finite element software Comsol; the initial conditions and boundary conditions are set for each physical field, the preset film is meshed, the mesh size is selected as 32*32, and the model is solved using the finite element method; the field variable data obtained by the solution is imported into a text document, and the visual field variable results are obtained using ParaView and Orig-in software. According to the domain structure cloud map under different oxygen vacancy concentrations, the phase transition law from ferroelectric phase to non-ferroelectric phase is obtained, and the influence of oxygen vacancy concentration on the volume fraction of each phase is calculated; according to the established model, an external electric field is applied to obtain the hysteresis loop under different oxygen vacancy concentrations, and finally the relationship curve between oxygen vacancy concentration and remanent polarization is obtained.

[0147] The domain structure cloud diagram and phase transition law of polymorphic coexistence of hafnium oxide-based ferroelectric films under different oxygen vacancy concentrations are simulated.

[0148] According to the domain structure cloud map under different oxygen vacancy concentrations, the phase transition law from ferroelectric phase to non-ferroelectric phase is obtained, and the influence of oxygen vacancy concentration on the volume fraction of each phase is calculated; according to the established model, an external electric field is applied to obtain the hysteresis loop under different oxygen vacancy concentrations, and finally the relationship curve between oxygen vacancy concentration and remanent polarization is obtained.

[0149] like Figure 2 As shown, Figure 2 The multiphase cloud diagrams of different oxygen vacancy concentration gradients of the hafnium oxide-based ferroelectric film obtained in the above embodiment of the present application under the action of oxygen vacancies are shown in Figure 1, 2, 3, 4 and 5 respectively. 19 cm 3 ,7×10 19 cm 3 , 1×10 20 cm 3 , 3×10 20 cm 3 , 5×10 20 cm 3Cloud diagrams under five concentration gradients. The above cloud diagrams have no external electric field and only consider the polarization change caused by the diffusion of oxygen vacancy concentration. The black part in the figure is the c domain along the thickness direction of the film, the gray part is the a domain along the horizontal direction of the film, and the blank part is the monoclinic phase.

[0150] like Figure 3 As shown, Figure 3 The analysis method provided by the embodiment of the present application shows the percentage of a domain, c domain and monoclinic phase in the hafnium oxide-based ferroelectric film at different concentrations, showing the influence of oxygen vacancy concentration on different types of domain structures. From the direction close to the x-axis to the direction far from the x-axis, it includes 5 coordinates in sequence. These 5 coordinates are respectively the oxygen vacancy concentration at 5×10 19 cm 3 ,7×10 19 cm 3 , 1×10 20 cm 3 , 3×10 20 cm 3 , 5×10 20 cm 3 Schematic diagram of phase ratio under five concentration gradients. In the legend, s1 represents the horizontal a domain, s2 represents the vertical c domain, and s3 represents the non-ferroelectric phase. It can be seen from the figure that the volume fraction of the a domain first increases and then decreases with the increase of oxygen vacancy concentration, and the volume fraction of the non-ferroelectric phase first decreases and then increases with the increase of oxygen vacancy concentration. This shows that the effect of oxygen vacancy concentration on the performance of ferroelectric thin film materials is not monotonic, and there is a peak value that makes its performance the best.

[0151] like Figure 4 As shown, Figure 4 It is the volume fraction of c-domain at different durations under different oxygen vacancy concentrations of the hafnium oxide-based ferroelectric film provided by the embodiment of the present invention. From the direction close to the x-axis to the direction away from the x-axis, the c-domain volume fraction at five moments, namely 0s, 1s, 2s, 3s, 4s, and 5s, is represented in sequence. The slope of the curve increases significantly at 1s to 4s, indicating that the conversion speed of the c-domain is accelerated during this period. At 4s to 5s, the curve is smooth and the volume fraction of the c-domain tends to be stable.

[0152] like Figure 5 As shown in Figure 2, the remnant polarization of the ferroelectric film increases first and then decreases with the increase of oxygen vacancy concentration. 20 cm 3 This concentration polarization value achieves the best.

[0153] like Figure 6 As shown, Figure 6FIG. 1 is a schematic diagram of the volume fraction of each phase of the hafnium oxide-based ferroelectric film under different oxygen vacancy concentrations provided by the embodiment of the present invention. The ferroelectric phase of the ferroelectric film first increases and then decreases with the increase of oxygen vacancy concentration. 20 cm 3 The non-ferroelectric phase of the ferroelectric film first decreases and then increases with the increase of oxygen vacancy concentration. 20 cm 3 At this concentration, the non-ferroelectric phase content is the least. Therefore, through the analysis method provided by the embodiment of the present invention, it is possible to theoretically select a suitable oxygen vacancy concentration range to mitigate the wake-up effect, which is beneficial to the performance optimization of the hafnium oxide-based ferroelectric film.

[0154] In the experimental process, the performance of hafnium oxide-based ferroelectric thin film materials must be improved. In thin films, oxygen vacancies are an influencing factor that always exists. The oxygen vacancy concentration of hafnium oxide materials can be regulated by annealing in different oxygen atmospheres, using water sources, and doping with other elements. However, the specific concentration threshold is not very clear and there is controversy within a certain range. Through phase field theory simulation, a more accurate oxygen vacancy concentration range can be obtained. At the same time, hafnium oxide materials are ferroelectric materials with multiple mixed phases. The phase change law and phase change mechanism inside the film require the use of scanning transmission electron microscopy in experimental methods, which is too expensive and easily interfered by other factors, and the reproducibility is relatively low. First-principles research can usually only study systems composed of hundreds to thousands of atoms. The research is relatively microscopic and cannot truly control the influence of oxygen vacancies on the phase change process of the entire film and the domain evolution mode; the existing phase field simulation methods mostly study traditional ferroelectric materials, and there are few studies on hafnium oxide-based ferroelectric films with coexistence of ferroelectric and non-ferroelectric phases. This application constructs the force field energy equation, polarization field energy equation, electric field energy equation and concentration diffusion equation under the influence of oxygen vacancies, considers the coupling term of oxygen vacancy concentration and polarization, obtains the domain structure diagram at different times under different oxygen vacancy concentrations, calculates the volume fraction of ferroelectric phase and non-ferrous phase according to different domain structure diagrams, obtains the evolution law of different phases over time, and the relationship diagram of different oxygen vacancy concentrations and residual polarization values. In a specific control range, the influence of oxygen vacancy concentration on the transformation of ferroelectric phase to monoclinic phase is obtained, and the influence of oxygen vacancy concentration on ferroelectric domain reversal and its phase transition is explained from a relatively macroscopic level, providing guidance for the optimization of the macroscopic performance of hafnium oxide-based ferroelectric films.

[0155] In an optional embodiment, a phase field analysis system for hafnium oxide-based ferroelectric thin films based on the influence of oxygen vacancies includes: an input module, a calculation module and an output module, the input module is used to input variables, the calculation module includes the phase field calculation model of the hafnium oxide-based ferroelectric thin film based on the influence of oxygen vacancies of the above-mentioned embodiment, which is used to solve the variables, and the output module is used to output the solution visually.

[0156] In an optional embodiment, a readable storage medium stores a program or instruction, and when the program or instruction is executed by a processor, the input and output of the phase field analysis system of the hafnium oxide-based ferroelectric film based on the influence of oxygen vacancies in the above optional embodiment are realized.

[0157] In an optional embodiment, an electronic device includes a processor and a readable storage medium as in the above optional embodiment.

[0158] A person skilled in the art can understand that all or part of the steps in the above-mentioned embodiment method can be completed by instructing the relevant hardware through a program, and the program can be stored in a computer-readable storage medium. When the program is executed, it includes the following process of the above-mentioned embodiment of each method. The storage medium can be a disk or an optical disk.

Claims

1. A phase field analysis method for hafnium oxide-based ferroelectric thin films based on oxygen vacancies, characterized in that: include: Based on the influence of oxygen vacancy concentration on hafnium oxide-based ferroelectric thin films, a total energy equation is established, which includes the volume free energy density in the ferroelectric thin film under the influence of oxygen vacancy concentration, the elastic energy density in the ferroelectric thin film, the gradient energy density in the ferroelectric thin film, and the electrostatic energy density in the ferroelectric thin film; the total energy equation is: F=∫ V fdV, f=f bulk +f gradient +f elastic +f electric Where F is the total energy of the ferroelectric film, f is the total free energy density of the ferroelectric film, and f is the volume free energy density in the ferroelectric film. bulk , the elastic energy density f in the ferroelectric film elastic , the gradient energy density f in the ferroelectric film gradient and the electrostatic energy density f in the ferroelectric film electric of and; Deducing from the total energy equation, the constitutive equations of various physical fields based on the influence of oxygen vacancies are obtained; According to the constitutive equation, combined with the mechanical equilibrium equation, Maxwell equation, Ginzburg-Landau equation and concentration diffusion equation, the control equations of various physical fields of the hafnium oxide-based ferroelectric thin film are obtained; According to the control equation, when solving by finite element method, deduction is performed to obtain weak forms of various physical field control equations based on the influence of oxygen vacancies, specifically including weak forms of force field control equations, weak forms of electric field control equations, weak forms of polarization field control equations and weak forms of concentration field control equations; According to the weak forms of the equations, a phase field calculation model of hafnium oxide-based ferroelectric thin films based on the influence of oxygen vacancies is established to simulate the domain structure cloud diagram and phase transition law of polymorphic coexistence of hafnium oxide-based ferroelectric thin films under different oxygen vacancy concentrations.

2. The phase field analysis method based on oxygen vacancy effect of hafnium oxide-based ferroelectric thin film according to claim 1, characterized in that: The volume free energy density f in the ferroelectric film bulk The formula is: η i is the order parameter in phase field theory, where when i=1, η1 represents the polarization state of the ferroelectric film in the horizontal direction, i.e., the state of a domain polarization; when i=2, η2 represents the polarization state of the ferroelectric film in the vertical direction, i.e., the state of c domain polarization; when i=3, η3 represents the non-ferroelectric phase in the ferroelectric film model, i.e., the monoclinic phase; A1, A 2(o) , A 2(m) , A3 and A4 are Landau coefficients, A5 is the coupling coefficient, where A 2(m) is the Landau coefficient corresponding to non-ferroelectric, A 2(o) is the Landau coefficient corresponding to ferroelectricity, A 2(m) Less than A 2(0) ; A1 and A 2(0) A3 and A4 are calculated based on the coercive field of the remanent polarization of the experimental hysteresis loop; A3 and A4 are obtained by fitting the experimental hysteresis loop; The elastic energy density f in the ferroelectric film elastic The formula is: ε ij =1 / 2(u i,j +u j,i ),u i,j and u j,i is the displacement component, Among them, ε ij , ε kl Both represent strain components, ε (i) represents the ferroelectric film in η i The spontaneous strain corresponding to the state; i is 1, 2 or 3, indicating the spontaneous strain corresponding to each state in the ferroelectric film. The spontaneous strain ε corresponding to each state can be calculated based on the lattice constants of the a, b, and c axes of the hafnium oxide-based ferroelectric film. (i) , i=1,2,3; c ijkl is the elastic energy coefficient, i, j, k, and l represent different values ​​within a spatial range respectively; in the strain matrix, the number of values ​​varies with the spatial dimension; if the spatial range is three-dimensional, i = (1,2,3), j = (1,2,3), k = (1,2,3), and l = (1,2,3); if the spatial range is two-dimensional, i = (1,2), j = (1,2), k = (1,2), and l = (1,2); The gradient energy density f in the ferroelectric film gradient The formula is: K η is the gradient energy coefficient, is the polarization gradient, η i is the order parameter in phase field theory; The electrostatic energy density f in the ferroelectric film electric The formula is: k0 Vacuum dielectric constant, E i is the built-in electric field strength of the hafnium oxide-based ferroelectric film, is the built-in electric field strength caused by oxygen vacancies.

3. The phase field analysis method based on oxygen vacancy effect of hafnium oxide-based ferroelectric thin film according to claim 2, characterized in that: The constitutive equation is:

4. The phase field analysis method based on oxygen vacancy effect of hafnium oxide-based ferroelectric thin film according to claim 3, characterized in that: in: The force field control equation is: Among them, stress ε ij =1 / 2(u i,j +u j,i ),u i,j and u j,i is the displacement component; The polarization field control equation is: Where F = ∫ V fdV; Where F is the total energy of the ferroelectric film, f is the total free energy density of the ferroelectric film; The electric field control equation is: D=ε0E+P Among them, the electric displacement ρ is the charge density of oxygen vacancies inside the film; The concentration field energy equation is: Among them, V o represents the oxygen vacancy concentration, is the oxygen vacancy drift velocity, represents the oxygen vacancy diffusion rate.

5. The phase field analysis method based on oxygen vacancy effect of hafnium oxide-based ferroelectric thin film according to claim 4, characterized in that: 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:

6. A phase field analysis system based on oxygen vacancy effect of hafnium oxide-based ferroelectric thin film, characterized in that: include: An input module, a calculation module and an output module, wherein the input module is used to input variables, the calculation module is used to execute the method described in any one of claims 1 to 5 to solve the variables, and the output module is used to output the solution visually.

7. A readable storage medium, characterized in that: The readable storage medium stores a program or an instruction, and when the program or the instruction is executed by a processor, the method according to any one of claims 1 to 5 is implemented.

8. An electronic device, characterized in that: The method comprises a processor and a readable storage medium as claimed in claim 7.

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