CN110866349A - Phase field analysis method of hafnium oxide based ferroelectric film based on polymorphic coexistence - Google Patents
Phase field analysis method of hafnium oxide based ferroelectric film based on polymorphic coexistence Download PDFInfo
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Abstract
A phase field analysis method of a hafnium oxide based ferroelectric film based on multi-state coexistence comprises the following steps: determining the number of sequence parameters according to the number of the states of the multi-state coexisting hafnium oxide-based ferroelectric film; determining an energy equation expression and a coefficient of the ferroelectric film based on the sequence parameter, the Ginzburg-Landau theory and the block physical energy, elastic energy, gradient energy and electrostatic energy of the ferroelectric film; combining a mechanical equilibrium equation, a Maxwell equation and a Ginzburg-Landau phase field kinetic equation, deducing weak forms of a force field, an electric field and a polarization field, and establishing a phase field model of the hafnium oxide-based ferroelectric film with polymorphic coexistence under the condition of force-electricity coupling; and simulating the domain structure and the evolution law of the polymorphic coexisting hafnium oxide-based ferroelectric film according to the phase field model. Dynamic simulation HfO2The formation and growth process of the domain structure of the ferroelectric film is used for researching the redistribution of the domain structure under the action of different external fields, thereby providing guidance for experimenters and reducing the experimental cost.
Description
Technical Field
The invention relates to the technical field of ferroelectric film simulation analysis, in particular to a phase field analysis method of a hafnium oxide based ferroelectric film based on polymorphic coexistence.
Background
The ferroelectric thin film in the 1T-1C type ferroelectric memory of the traditional perovskite ferroelectric thin film has the advantages of larger thickness, large memory cell area and low memory capacity, and is difficult to meet the application requirements. And hafnium oxide (Hf 0)2) The base ferroelectric film material is a novel ferroelectric material, has higher dielectric constant, larger coercive field and remnant polarization, and can be compatible with an advanced CMOS (Complementary Metal Oxide Semiconductor) process; meanwhile, when the film thickness is less than 10nm, the ferroelectric film still has good ferroelectricity. These advantages lead to Hf02The ferroelectric film material becomes an ideal choice for future nonvolatile memory materials and is closely concerned by the majority of researchers. The novel ferroelectric material can be used for 1T-1C ferroelectric random access memory (FeRAM), 1T ferroelectric field effect transistor (FeFET), negative capacitance field effect transistor (NFET) and the like. However, the hafnium oxide-based ferroelectric memory has not yet been commercialized at present, mainly because the problems of fatigue, non-uniform memory window, etc. are not completely solved.
Research shows that the fatigue of the hafnium oxide-based ferroelectric film is essentially caused by the continuous reduction of reversible ferroelectric domains in the film along with time; the non-uniform memory window is essentially caused by the coexistence of metastable ferroelectric domains and non-ferroelectric phases in the hafnium oxide-based ferroelectric film, and the non-uniform distribution of the ferroelectric domains in the film. Therefore, the macroscopic electrical performance of the ferroelectric film is essentially determined by the ferroelectric domain distribution and evolution, and the domain structure can be regulated and controlled by using the key factors only by finding the important microscopic factors influencing the domain structure and the evolution thereof, so that the optimal design of the film is realized.
The phase field method is currently widely used to simulate and predict conventional perovskite ferroelectric materials such as lead titanate (PbTiO)3) Internal domain distribution and its evolution under external fields such as force, heat and electricity. The ferroelectric phase of the traditional perovskite ferroelectric material is a stable phase, when the temperature is lower than the Curie temperature, the ferroelectric material is subjected to phase transition and is converted into the ferroelectric phase from the paraelectric phase, the film can show ferroelectricity, and the ferroelectric phase is continuously distributed in the whole film. Correspondingly, when the ferroelectric film is below the Curie temperature, the film only has a ferroelectric phase and does not have a non-ferroelectric phase part. The ferroelectricity of the novel ferroelectric film material of the hafnium oxide-based ferroelectric film can not appear under normal temperature and normal pressure, and the ferroelectric phase can not be induced by a single means of temperature, and can only appear under proper doping and stress conditions. And because the ferroelectric phase appearing in the hafnium oxide based ferroelectric film material is a metastable phase, the metastable ferroelectric phase is easily converted into a non-ferroelectric phase, such as a monoclinic phase and a tetragonal phase, under the influence of complex mechanical thermoelectric and microstructure. Therefore, the ferroelectric phase and the non-ferroelectric phase of the hafnium oxide based ferroelectric film exist at the same time, and the ferroelectric phase is discontinuously distributed in the film.
Therefore, the phase field model established based on the traditional ferroelectric film is not suitable for analyzing the distribution and evolution of ferroelectric domains in the hafnium oxide-based ferroelectric film with the coexistence of ferroelectric phase and various non-ferroelectric phases, and the traditional phase field theory model can not be utilized to analyze the micro mechanism of the fatigue failure of the hafnium oxide-based ferroelectric film.
In addition, in the current phase field method, the coefficient of the bulk energy is obtained by fitting the relation between the spontaneous polarization of the single crystal and the temperature, while in the current research situation, Hf02It is difficult to prepare a single crystal, and therefore Hf0 cannot be obtained2The relationship between spontaneous polarization and temperature of the ferroelectric thin film is not utilized to obtain Hf02Based on the bulk energy coefficient of the ferroelectric thin film, the Hf0 can not be obtained by using the method of the traditional phase field theoretical model coefficient2The phase field model coefficient of the ferroelectric film material. .
Disclosure of Invention
Objects of the invention
The invention aims to provide a phase field analysis method of a hafnium oxide based ferroelectric film based on polymorphic coexistence, which is based on a traditional phase field method and provides HfO under force-electricity coupling2A finite element simulation method based on multi-state coexistence of ferroelectric thin film includes establishing a multi-state coexistence phase field model, and simulating HfO2The domain structure and the evolution law thereof.
(II) technical scheme
In order to solve the above problems, according to an aspect of the present invention, there is provided a phase field analysis method based on coexistence of multiple states for a hafnium oxide based ferroelectric thin film, comprising:
step S1, determining the number of sequence parameters according to the number of the states of the multi-state coexisting hafnium oxide-based ferroelectric film;
step S2, determining an energy equation expression of the hafnium oxide-based ferroelectric film based on the sequence parameter, the Ginzburg-Landau theory and the bulk energy, the elastic energy, the gradient energy and the electrostatic energy of the hafnium oxide-based ferroelectric film, and calculating the coefficient of the energy equation expression;
step S3, respectively deducing driving force equations of block physical energy, elastic energy, gradient energy and electrostatic energy through an energy equation expression, respectively deducing weak forms of a force field, an electric field and a polarization field by combining a mechanical balance equation, a Maxwell equation and a Ginzburg-Landau phase field kinetic equation, and establishing a phase field model of the hafnium oxide-based ferroelectric film coexisting under force-electricity coupling;
and step S4, simulating the domain structure and the evolution law of the multi-state coexisting hafnium oxide-based ferroelectric film according to the phase field model.
Further, in step S1, the ferroelectric phase of the hafnium oxide-based ferroelectric thin film in which multiple states coexist is along three spatial axes: the polarization of an x axis, a y axis and a z axis is divided into a first state, a second state and a third state, and the first state, the second state and the third state are respectively expressed by three different sequence parameters; the non-ferroelectric phase of the polymorphic coexisting hafnium oxide-based ferroelectric film is nonpolarized and is in a four-state, and the four-state is expressed by another sequence parameter so as to determine four different sequence parameters; sequence parameter is
Where i is (1,2,3, 4), r denotes the vector position at a point in space, and t denotes time.
Further, in step S2, determining an energy equation expression based on the density expression of the bulk energy, the density expression of the elastic energy, the density expression of the gradient energy, and the density expression of the electrostatic energy;
the density expression of the bulk energy is:
in the formula, A1、A2、A3Three different expansion coefficients respectively;
the density expression of the elastic energy is:
in the formula, the spontaneous strain epsilon corresponding to each state is calculated according to the lattice constant of the hafnium oxide-based ferroelectric film(i)(i＝1,2,3,4)；cijklThe elastic energy coefficient is represented by i, j, k and l which respectively represent different values in a space range; epsilonij＝1/2(ui,j+uj,i)，uiIs the displacement component;
the density expression of the gradient energy is:
in the formula, KηIs the gradient energy coefficient;
the density expression of the electrostatic energy is:
in the formula, κ0Is a vacuum dielectric constant, EiThe built-in electric field intensity of the hafnium oxide based ferroelectric film;
the expression of the energy equation is as follows: f ═ fbulk+fgradient+felastic+felectric
Further, according to the remanent polarization P in the hysteresis loop of the hafnium oxide based ferroelectric thin filmrAnd coercive field ECDetermining the expansion coefficient A1And A2Wherein, in the step (A),
according to the expansion coefficient A1And A2With equation E ═ 2A1P+4A2P3+6A3P5Fitting the electric hysteresis loop, and obtaining an expansion coefficient A by adopting the data of the electric hysteresis loop3(ii) a Wherein the data of the hysteresis loop comprise: polarization P and electric field intensity E of the hafnium oxide based ferroelectric thin film.
Further, coefficient of elastic energy cijklThe calculation formula of (2) is as follows:
cijkl＝λδijδkl+μ(δilδjk+δikδjl)；
wherein, lambda is Lame constant,
wherein Y is Young's moduli of a tetragonal phase (t-phase), an orthorhombic phase (o-phase) and a monoclinic phase (m-phase) of the hafnium oxide-based ferroelectric thin film, and is 210GPa, 224GPa and 185GPa, respectively; v is the Poisson ratio of all phases, and v is 0.3; the mu shear modulus of the polymer is,
further, gradient energy coefficient KηCalculating according to the domain wall width of the hafnium oxide-based ferroelectric film, wherein the domain wall width l of the hafnium oxide-based ferroelectric film is as follows:
where the domain wall width l is detected by scanning transmission electron microscopy, α is a constant related to surface energy.
Further, in step S3, the energy equation expression is subjected to partial derivation to obtain a driving force equation of bulk energy, a driving force equation of elastic energy, a driving force equation of gradient energy, and a driving force equation of electrostatic energy as follows:
further, in step S3, the phase field model is formed by coupling three physical fields, i.e., a force field, an electric field and a polarization field, wherein a control equation of the force field is a force balance equation, a control equation of the electric field is a Maxwell equation, and a control equation of the polarization field is a Ginzburg-Landau kinetic equation;
the force balance equation is:
in which the stress
εij＝1/2(ui,j+uj,i)，uiIs the displacement component;
maxwell's equation is:
in which the electric potential is shifted
The Ginzburg-Landau kinetic equation is:
wherein, F ═ n ^ nVfdV。
Further, a force field, an electric field and a polarization field are obtained in a weak form based on a virtual work principle by combining a force balance equation, a Maxwell equation and a Ginzburg-Landau kinetic equation, so that a phase field model of the hafnium oxide-based ferroelectric thin film under the condition of force-electricity coupling and polymorphic coexistence is established;
the weak form of the force field is:
the weak form of the electric field is:
the weak form of the polarization field is:
wherein, tauiIs the surface tension, ω is the surface charge,
is a surface gradient flow, njIs a unit normal vector.
Further, step S4 includes:
step S41, compiling an energy equation expression, an expansion coefficient, an elastic energy coefficient, a gradient energy coefficient and weak forms of a force field, an electric field and a polarization field into finite element software;
step S42, setting the initial value and boundary condition of each physical field in finite element software, dividing grids, and solving the phase field model by adopting a finite element method;
and step S43, importing the field variable data obtained by solving into a text document, and obtaining a visual field variable result by using ParaView and Origin software.
(III) advantageous effects
The technical scheme of the invention has the following beneficial technical effects:
1. compared with the traditional phase field method for simulating the ferroelectric film, the method provided by the invention considers the number of polarization states, and meanwhile, the block energy coefficient can be obtained only by the electric hysteresis loop, so that the parameters in the traditional energy equation are greatly reduced, and the difficulty in calculation and simulation of the phase field method is reduced.
2. According to the invention, the HfO2Based on the coexisting state of multiple states of the ferroelectric film, the block body of the current phase field can be modified, and the improved phase field model is used for simulating HfO2Dynamically reappearing HfO based on domain structure and evolution law of ferroelectric thin film in multi-state coexistence state2Forming domain structure of base ferroelectric film and its growth process.
3. The invention can research HfO under different external field effects by changing the external field2The redistribution of the domain structure of the ferroelectric film provides guidance for experimenters and reduces the experiment cost.
Drawings
FIG. 1 is a flowchart illustrating the steps of a phase field analysis method based on multi-state coexistence for a hafnium oxide-based ferroelectric thin film according to the present invention;
FIG. 2 is a flowchart of the steps of step S4 provided by the present invention;
FIG. 3 is a polarization vector diagram of the coexistence of multiple states of the hafnium oxide-based ferroelectric thin film provided by the present invention;
FIG. 4 is a graph showing a distribution of states in a hafnium oxide based ferroelectric thin film provided by the present invention;
FIG. 5 is a distribution diagram of two states in a hafnium oxide based ferroelectric thin film provided by the present invention;
FIG. 6 is a tristate distribution diagram in a hafnium oxide-based ferroelectric thin film provided by the present invention;
FIG. 7 is a diagram of a distribution of four states in a hafnium oxide based ferroelectric thin film provided by the present invention;
FIG. 8 is a schematic diagram of a polymorphic lattice structure of a hafnium oxide-based ferroelectric thin film provided in accordance with the present invention;
FIG. 9 is a table of lattice parameters of t, o and m phases in a hafnium oxide based ferroelectric thin film provided in the present invention;
fig. 10 is a schematic view of the ferroelectric hysteresis loop of the hafnium oxide-based ferroelectric thin film provided by the present invention.
Reference numerals:
S1-S4, S41-S43: a step of;
a: a one-state lattice; b: a two-state lattice; c: a tri-state lattice; d: a four-state lattice;
e: a ferroelectric phase region; f: a non-ferroelectric phase region;
s 1: a state; s 2: a second state; s 3: tri-state; s 4: and (4) four states.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the accompanying drawings in conjunction with the following detailed description. It should be understood that the description is intended to be exemplary only, and is not intended to limit the scope of the present invention. Moreover, in the following description, descriptions of well-known structures and techniques are omitted so as to not unnecessarily obscure the concepts of the present invention.
The present invention will be described in detail below with reference to the accompanying drawings and examples.
Fig. 1 is a flowchart illustrating steps of a phase field analysis method based on coexistence of multiple states for a hafnium oxide-based ferroelectric thin film according to the present invention, and fig. 2 is a flowchart illustrating step S4 according to the present invention.
In one embodiment, the invention provides a phase field analysis method based on polymorphic coexistence for a hafnium oxide based ferroelectric thin film, which comprises the following specific operation steps:
step S1: the number of sequence parameters is determined based on the number of states of the hafnium oxide-based ferroelectric thin film in which the multiple states coexist.
The ferroelectric phase of the multi-state coexisting hafnium oxide-based ferroelectric film is along three spatial direction axes: the polarization of an x axis, a y axis and a z axis is divided into a first state, a second state and a third state, and the first state, the second state and the third state are respectively expressed by three different sequence parameters; the non-ferroelectric phase of the multi-state coexisting hafnium oxide-based ferroelectric thin film is non-polarized and is in a four-state, which is expressed by another sequence parameter, thereby determining four different sequence parameters.
FIG. 3 is a polarization vector diagram of the coexistence of multiple states of the hafnium oxide-based ferroelectric thin film provided by the present invention.
Referring to fig. 3, three axes in spatial directions are shown: polarization vector profiles for ferroelectric and non-ferroelectric phases polarized in the x, y and z axes. Wherein, the thick and long arrows are ferroelectric phases, and the distribution positions thereof represent the ferroelectric phases of the hafnium oxide based ferroelectric film, such as a region e enclosed by a rectangular frame in the figure; the positions without arrows or with sparse arrows represent the non-ferroelectric phase of the hafnium oxide based ferroelectric thin film, such as the area f enclosed by the oval frame in the figure.
Sequence parameter is
Where i is (1,2,3, 4), r denotes a vector position at a point in space, and t denotes time.
Four different sequence parameters are determined according to four different states, wherein: one state represents the ferroelectric phase of the hafnium oxide-based ferroelectric thin film polarized along the x-axis in the horizontal direction, two states represents the ferroelectric phase of the hafnium oxide-based ferroelectric thin film polarized along the y-axis in the front-rear direction, three states represents the ferroelectric phase of the hafnium oxide-based ferroelectric thin film polarized along the z-axis in the vertical direction, and four states represents the non-ferroelectric phase of the non-polarized hafnium oxide-based ferroelectric thin film.
Specifically, please refer to fig. 4, fig. 5, fig. 6, and fig. 7, which respectively show the distributions of one-state-four-state in the hafnium oxide-based ferroelectric thin film provided by the present invention, wherein s1-s4 are one-state-four-state.
Fig. 8 is a schematic diagram of a multi-state lattice structure of a hafnium oxide based ferroelectric thin film, see fig. 8. a. The three axes b and c are respectively represented as three different axes on the lattice structure, and the transparent arrows indicate that the polarization of the hafnium oxide-based ferroelectric thin film is carried out in the direction of the c axis. Wherein A is a one-state lattice of the hafnium oxide-based ferroelectric thin film, and represents a ferroelectric phase of the hafnium oxide-based ferroelectric thin film polarized along an x-axis; b is a two-state lattice of the hafnium oxide-based ferroelectric thin film, which represents the ferroelectric phase of the hafnium oxide-based ferroelectric thin film polarized along the y-axis; c is a ternary lattice of the hafnium oxide based ferroelectric thin film, representing the ferroelectric phase of the hafnium oxide based ferroelectric thin film polarized along the z-axis.
D is the tetragonal lattice of the hafnium oxide based ferroelectric thin film, and there is no arrow in D, indicating that D is the non-poled portion, i.e., the non-ferroelectric phase, of the hafnium oxide based ferroelectric thin film.
Wherein, the axes a, b, c have different lattice constants of the hafnium oxide based ferroelectric thin film, fig. 9 is a table of lattice parameters of t, o and m phases in the hafnium oxide based ferroelectric thin film, please look at fig. 9. A. B, C, the lattice constant a ≠ b ≠ c, is orthogonal phase (o-phase) and belongs to the polarization phase, i.e. ferroelectric phase; the lattice constant a is not equal to b and not equal to c in D, and the phase is non-polarized, is a monoclinic phase (m phase), and belongs to a non-polarized phase, namely a non-ferroelectric phase.
In the shaded state in fig. 8, a ≠ b ≠ c, which belongs to the tetragonal phase (t-phase).
Step S2: and determining an energy equation expression of the hafnium oxide-based ferroelectric film based on the sequence parameter, the Ginzburg-Landau theory and the bulk energy, the elastic energy, the gradient energy and the electrostatic energy of the hafnium oxide-based ferroelectric film, and calculating coefficients of the energy equation expression.
Specifically, Ginzburg-Landau theory (phase field theory) is a thermodynamic theory and a diffusion law as physical backgrounds, and a dynamic evolution equation is combined to obtain an internal surrounding structure of a material and an evolution process of the internal surrounding structure along with time.
The energy equation expression is the sum of the density of the bulk energy, the density of the elastic energy, the density of the gradient energy and the density of the electrostatic energy, and therefore the energy equation expression is finally determined based on the density expression of the bulk energy, the density expression of the elastic energy, the density expression of the gradient energy and the density expression of the electrostatic energy.
Wherein, the density expression of the block energy is as follows:
in the formula, A1、A2、A3Three different expansion coefficients, respectively.
The density expression of the elastic energy is:
in the formula, the spontaneous strain epsilon corresponding to each state is calculated according to the lattice constants of a, b and c axes of the hafnium oxide-based ferroelectric film(i)(i ═ 1,2,3, 4); referring to fig. 4, calculations are made from the table of lattice parameters for the t, o and m phases. Epsilonij＝1/2(ui,j+uj,i)，uiIs the displacement component;
c in the above formulaijklThe elastic energy coefficient is represented by i, j, k and l, which respectively represent different values in a space range, and the spatial dimensions and the number of the values are different. If the spatial range is a three-dimensional space, i ═ 1,2,3, j ═ 1,2,3, k ═ 1,2,3, and l ═ 1,2, 3; if the spatial range is a two-dimensional space, i ═ 1,2, j ═ 1,2, k ═ 1,2, and l ═ 1, 2.
The density expression of the gradient energy is:
in the formula, KηIs the gradient energy coefficient.
The density expression of the electrostatic energy is:
in the formula, κ0Is a vacuum dielectric constant, EiThe built-in electric field strength of the hafnium oxide based ferroelectric thin film.
Thus, the final energy equation is expressed as follows:
f＝fbulk+fgradient+felastic+felectric
after the energy equation expression of the hafnium oxide based ferroelectric film is obtained, the coefficient in the energy equation expression needs to be calculated, and the value of the coefficient is calculated according to experimental data fitting and a first principle.
The coefficients include: expansion coefficient A in density expression of bulk energy1、A2、A3Elastic energy coefficient c in the expression of density of elastic energyijklGradient energy coefficient K in density expression of gradient energyη。
Specifically, the remanent polarization P in the hysteresis loop of the hafnium oxide based ferroelectric thin filmrAnd coercive field ECThe expansion coefficient a can be determined1And A2Residual polarization PrAnd coercive field ECRespectively as follows:
reuse equation E ═ 2A1P+4A2P3+6A3P5Fitting the electric hysteresis loop to obtain the polarization strength P and the electric field strength E of the hafnium oxide-based ferroelectric film in the electric hysteresis loop, and combining the expansion coefficient A1And A2Finally, the expansion coefficient A is obtained by calculation3. Coefficient of elastic energy cijklThe calculation formula of (2) is as follows:
cijkl＝λδijδkl+μ(δilδjk+δikδjl)
wherein, lambda is Lame constant,
wherein Y is Young's moduli of a tetragonal phase (t-phase), an orthorhombic phase (o-phase) and a monoclinic phase (m-phase) of the hafnium oxide-based ferroelectric thin film, and is 210GPa, 224GPa and 185GPa, respectively; v is the Poisson ratio of all phases, and v is 0.3; the mu shear modulus of the polymer is,
coefficient of gradient energy KηIt needs to be calculated according to the domain wall width of the hafnium oxide based ferroelectric thin film, wherein the domain wall width l of the hafnium oxide based ferroelectric thin film is:
where the domain wall width l is detected by Scanning Transmission Electron Microscopy (STEM), α is a constant related to surface energy.
Step S3: and respectively deducing driving force equations of the block physical energy, the elastic energy, the gradient energy and the electrostatic energy through an energy equation expression, respectively deducing weak forms of a force field, an electric field and a polarization field by combining a mechanical balance equation, a Maxwell equation and a Ginzburg-Landau phase field kinetic equation, and establishing a phase field model of the hafnium oxide-based ferroelectric film with multi-state coexistence under the condition of force-electricity coupling.
Specifically, the energy equation expressions are first subjected to partial derivatives to obtain the energy equation expressions
Driving force equation of bulk energy:
driving force equation of elastic energy:
driving force equation of gradient energy:
driving force equation of electrostatic energy:
the phase field model is characterized in that a force field, an electric field and a polarization field are mutually coupled, and three field variables of force, electricity and polarization are solved simultaneously, so that the phase field model is closer to the real situation of domain structure evolution.
The strong form has high requirement on the continuity of the solution, the weak form has low requirement on the continuity of the solution, the weak form and the strong form can be deduced from each other, the difference is the requirement on the continuity of the solution, and the strong form cannot be directly dispersed by a finite element method, so the weak form needs to be deduced. Meanwhile, after the weak forms of the force field, the electric field and the polarization field are introduced into finite element software, the requirement of a program on the boundary is reduced, and the solution is easier.
Prior to deriving the weak forms of the force, electric and polarization fields, the governing equations for the force, electric and polarization fields need to be known. The control equation of the force field is a force balance equation, the control equation of the electric field is a Maxwell equation, and the control equation of the polarization field is a Ginzburg-Landau kinetic equation. Therefore, the temperature of the molten metal is controlled,
the force balance equation is:
in which the stress
εij＝1/2(ui,j+uj,i)，uiIs the displacement component;
maxwell's equation is:
in which the electric potential is shifted
The Ginzburg-Landau kinetic equation is:
wherein, F ═ n ^ nVfdV。
And combining the force balance equation, the Maxwell equation and the Ginzburg-Landau kinetic equation to obtain weak forms of a force field, an electric field and a polarization field based on a virtual work principle, thereby establishing a phase field model of the hafnium oxide-based ferroelectric thin film under the condition of force-electricity coupling.
Specifically, the specific steps for deriving the weak form are as follows:
from the above-mentioned mechanical equilibrium equation
To obtain
Wherein, CiIs a surface force component ti。
From the Maxwell equation above
Obtaining:
Ciis the normal component w of the surface electric field strength.
From the Ginzburg-Landau kinetic equation above
Obtaining:
F＝∫VfdV
due to the fact that
Therefore, it is not only easy to use
Therefore, the temperature of the molten metal is controlled,
the weak form of the force field is:
the weak form of the electric field is:
the weak form of the polarization field is:
wherein, tauiIs the surface tension, ω is the surface charge,
is a surface gradient flow, njIs a unit normal vector.
Therefore, the phase field model obtained by the present application includes: the energy equation expression, the expansion coefficient, the elastic energy coefficient, the gradient energy coefficient and the weak form equation of the force field, the electric field and the polarization field.
Fig. 10 is a ferroelectric hysteresis loop of a hafnium oxide based ferroelectric thin film, see fig. 10.
The small square frame is used for measuring a hysteresis loop through an experiment, and the small circle is used for obtaining the hysteresis loop through tristate response simulation; the simulation electric hysteresis loop is better matched with the experimental electric hysteresis loop, which shows that the phase field model of the invention can correctly simulate the hafnium oxide-based ferroelectric film.
Step S4: and simulating the domain structure and the evolution law of the polymorphic coexisting hafnium oxide-based ferroelectric film according to the phase field model.
Specifically, step S4 includes the steps of:
s41: compiling an energy equation expression, an expansion coefficient, an elastic energy coefficient, a gradient energy coefficient and weak forms of a force field, an electric field and a polarization field into finite element software;
s42: setting an initial value and boundary conditions of each physical field in finite element software, dividing grids, and solving a phase field model by adopting a finite element method;
s43: and importing the field variable data obtained by solving into a text document, and obtaining a visual field variable result by using ParaView and Origin software.
The invention aims to protect a phase field analysis method of a hafnium oxide based ferroelectric film based on polymorphic coexistence, which comprises the following steps: determining the number of sequence parameters according to the number of the states of the multi-state coexisting hafnium oxide-based ferroelectric film; determining an energy equation expression and a coefficient of the energy equation expression of the ferroelectric film based on the sequence parameter, the Ginzburg-Landau theory and the block physical energy, elastic energy, gradient energy and electrostatic energy of the ferroelectric film; combining a mechanical equilibrium equation, a Maxwell equation and a Ginzburg-Landau phase field kinetic equation, deducing weak forms of a force field, an electric field and a polarization field, and establishing a phase field model of the hafnium oxide-based ferroelectric film with polymorphic coexistence under the condition of force-electricity coupling; and simulating the domain structure and the evolution law of the polymorphic coexisting hafnium oxide-based ferroelectric film according to the phase field model. Can simulate HfO2The formation of domain structure of ferroelectric film, the formation and growth of domain structure in dynamic reappearance film can be studied by changing external field and researching HfO under the action of different external fields2The redistribution of the domain structure of the ferroelectric film provides guidance for experimenters and reduces the experiment cost.
It is to be understood that the above-described embodiments of the present invention are merely illustrative of or explaining the principles of the invention and are not to be construed as limiting the invention. Therefore, any modification, equivalent replacement, improvement and the like made without departing from the spirit and scope of the present invention should be included in the protection scope of the present invention. Further, it is intended that the appended claims cover all such variations and modifications as fall within the scope and boundaries of the appended claims or the equivalents of such scope and boundaries.
Claims (10)
1. A phase field analysis method of a hafnium oxide based ferroelectric film based on multi-state coexistence is characterized by comprising the following steps:
step S1, determining the number of sequence parameters according to the number of the states of the multi-state coexisting hafnium oxide-based ferroelectric film;
step S2, determining an energy equation expression of the hafnium oxide-based ferroelectric film based on the sequence parameter, Ginzburg-Landau theory and the bulk energy, elastic energy, gradient energy and electrostatic energy of the hafnium oxide-based ferroelectric film, and calculating coefficients of the energy equation expression;
step S3, respectively deducing the driving force equations of the block energy, the elastic energy, the gradient energy and the electrostatic energy through the energy equation expression, then deducing weak forms of a force field, an electric field and a polarization field by combining a mechanical balance equation, a Maxwell equation and a Ginzburg-Landau phase field kinetic equation, and establishing a phase field model of the hafnium oxide-based ferroelectric thin film under the condition of force-electric coupling for polymorphic coexistence;
and step S4, simulating the domain structure and the evolution law of the multi-state coexisting hafnium oxide-based ferroelectric film according to the phase field model.
2. The method of claim 1,
in step S1, the ferroelectric phase of the multi-state coexisting hafnium oxide-based ferroelectric thin film is oriented along three spatial axes: the polarization of an x axis, a y axis and a z axis is divided into a first state, a second state and a third state, and the first state, the second state and the third state are respectively expressed by three different sequence parameters;
the non-ferroelectric phase of the polymorphic coexisting hafnium oxide-based ferroelectric film is nonpolarized and is in a four-state, and the four-state is expressed by another sequence parameter so as to determine four different sequence parameters;
the sequence parameter is
Where i is (1,2,3, 4), r represents a position vector at a point in space, and t represents time.
3. The method of claim 2,
in the step S2, determining the energy equation expression based on the density expression of the bulk energy, the density expression of the elastic energy, the density expression of the gradient energy, and the density expression of the electrostatic energy;
the density expression of the bulk energy is:
in the formula, A1、A2、A3Three different expansion coefficients respectively;
the density expression of the elastic energy is as follows:
wherein the corresponding spontaneous strain ε of each said state is calculated based on the lattice constant of the hafnium oxide based ferroelectric thin film(i)(i＝1,2,3,4)；cijklThe elastic energy coefficient is represented by i, j, k and l which respectively represent different values in a space range; epsilonij＝1/2(ui,j+uj,i)，uiIs the displacement component;
the density expression of the gradient energy is:
in the formula, KηIs the gradient energy coefficient;
the density expression of the electrostatic energy is:
in the formula, κ0Is a vacuum dielectric constant, EiThe built-in electric field intensity of the hafnium oxide based ferroelectric film;
the expression of the energy equation is as follows:
f＝fbulk+fgradient+felastic+felectric。
4. the method of claim 3,
according to the remanent polarization P in the ferroelectric hysteresis loop of the hafnium oxide based ferroelectric filmrAnd coercive field ECDetermining the expansion coefficient A1And A2Wherein, in the step (A),
according to the expansion coefficient A1And A2With equation E ═ 2A1P+4A2P3+6A3P5Fitting the electric hysteresis loop, and obtaining an expansion coefficient A by adopting the data of the electric hysteresis loop3；
Wherein, the data of the hysteresis loop include: polarization P and electric field intensity E of the hafnium oxide based ferroelectric thin film.
5. The method of claim 3,
the coefficient of elastic energy cijklThe calculation formula of (2) is as follows:
cijkl＝λδijδkl+μ(δilδjk+δikδjl)；
wherein, lambda is Lame constant,
wherein Y is Young's moduli of a tetragonal phase (t-phase), an orthorhombic phase (o-phase) and a monoclinic phase (m-phase) of the hafnium oxide-based ferroelectric thin film, and is 210GPa, 224GPa and 185GPa, respectively; v is the Poisson ratio of all phases, and v is 0.3; the mu shear modulus of the polymer is,
6. the method of claim 3,
coefficient of gradient energy KηCalculating according to the domain wall width of the hafnium oxide-based ferroelectric thin film, wherein the domain wall width l of the hafnium oxide-based ferroelectric thin film is as follows:
where the domain wall width l is detected by scanning transmission electron microscopy, α is a constant related to surface energy.
7. The method of claim 3,
in step S3, the energy equation expression is subjected to partial derivation to obtain a driving force equation of the bulk energy, a driving force equation of the elastic energy, a driving force equation of the gradient energy, and a driving force equation of the electrostatic energy as follows:
8. the method of claim 3,
in step S3, the phase field model is formed by coupling three physical fields, i.e., a force field, an electric field and a polarization field, wherein the control equation of the force field is a force balance equation, the control equation of the electric field is a Maxwell equation, and the control equation of the polarization field is a Ginzburg-Landau kinetic equation;
the force balance equation is:
in which the stress
εij＝1/2(ui,j+uj,i)，uiIs the displacement component;
the Maxwell equation is as follows:
in which the electric potential is shifted
The Ginzburg-Landau kinetic equation is:
wherein, F ═ n ^ nVfdV。
9. The method of claim 8,
combining the force balance equation, the Maxwell equation and the Ginzburg-Landau kinetic equation, and obtaining weak forms of the force field, the electric field and the polarization field based on a virtual work principle, so as to establish a phase field model of polymorphic coexistence of the hafnium oxide-based ferroelectric thin film under the condition of force-electric coupling;
the weak form of the force field is:
the weak form of the electric field is:
the weak form of the polarization field is:
wherein, tauiIs the surface tension, ω is the surface charge,
is a surface gradient flow, njIs a unit normal vector.
10. The method according to claim 1, wherein the step S4 includes:
step S41, compiling the energy equation expression, the expansion coefficient, the elastic energy coefficient, the gradient energy coefficient and the weak forms of the force field, the electric field and the polarization field into finite element software;
step S42, setting the initial value and boundary condition of each physical field in the finite element software, dividing the grids, and solving the phase field model by adopting a finite element method;
and step S43, importing the field variable data obtained by solving into a text document, and obtaining a visual field variable result by using ParaView and Origin software.
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