Phase field simulation method and device for relaxor ferroelectric dielectric property and storage medium

Abstract

The invention discloses a relaxor ferroelectric dielectric property phase field simulation method and device and a storage medium, and the method comprises the steps: S1, introducing the local non-uniformity of components for a relaxor ferroelectric, and generating concentration distribution; s2, setting Landau energy according to the non-uniformity of the components; s3, describing the heterogeneity of other structures derived from the heterogeneity and fluctuation of the components by adopting a random electric field; s4, obtaining the free energy of the dielectric property of the relaxor ferroelectric according to the results of the steps S1 to S3; and S5, searching sequence parameter distribution corresponding to the minimum free energy to realize steady-state phase field simulation. The conventional phase field simulation cannot describe the composition non-uniformity of the relaxor ferroelectric, and only the structure and parameters can be preset; according to the method, component non-uniformity is directly introduced, ferroelectricity, structure non-uniformity and other related performance are all associated to component distribution, description and quantitative simulation of the complex relaxor ferroelectric can be achieved through a simple and clear single path, and the result is kept consistent with experimental observation.

Description

Technical Field

[0001] The present invention relates to relaxor ferroelectrics, and particularly to a phase field simulation method, device, and storage medium for the dielectric properties of relaxor ferroelectrics. Background Art

[0002] Relaxor ferroelectrics are substances that are very important in the fields of ferroelectricity, piezoelectricity, and dielectrics. However, compared with ferroelectrics, the theoretical means for studying them are very limited. Phase field simulation is an effective simulation method for ferroelectrics. However, for relaxor ferroelectrics with complex compositions, it is very difficult to construct a relaxor ferroelectric phase field model that does not rely on too many preset parameters and structures. Many properties of relaxor ferroelectrics originate from local compositional fluctuations and inhomogeneities, resulting in the formation of polar nanoregions or slush domain structures. For example, relaxor ferroelectrics are characterized by polar nanoregions, and a ferroelectric / paraelectric composite with a random spatial distribution is used to describe relaxor ferroelectrics (Science 365, 578 - 582 (2019)); another example is that a random electric field with a normal distribution is used to achieve the effect of slush domains (Acta Materialia 225, 117558 (2022)). However, these models highly rely on a large number of preset structures and preset parameters.

[0003] Existing models highly rely on a large number of preset structures and preset parameters and cannot reflect the influence of local compositional fluctuations and inhomogeneities on the properties of relaxor ferroelectrics. Summary of the Invention

[0004] The object of the present invention is to provide a phase field simulation method, device, and storage medium for the dielectric properties of relaxor ferroelectrics, which describe relaxor ferroelectrics by directly introducing compositional variables and realize phase field simulation.

[0005] In phase field simulation, there is an order parameter and a free energy with the order parameter as the main variable. The present invention realizes steady-state phase field simulation by finding the order parameter distribution corresponding to the minimum free energy.

[0006] The technical solution of the present invention is as follows:

[0007] A phase field simulation method for the dielectric properties of relaxor ferroelectrics, comprising the steps of:

[0008] S1. For a relaxor ferroelectric, introduce local inhomogeneity of the composition to generate a concentration distribution;

[0009] S2. Set the Landau energy according to the compositional inhomogeneity;

[0010] S3. Use a random electric field to describe other structural inhomogeneities resulting from compositional inhomogeneity and fluctuations;

[0011] S4. Obtain the free energy of the dielectric properties of the relaxor ferroelectric based on the results of steps S1 to S3;

[0012] S5. Conduct a phase field simulation to achieve a steady state by finding the order parameter distribution corresponding to the minimum free energy.

[0013] Preferably, the introduction of local inhomogeneity of the composition in step S1 is as follows:

[0014] Regard the relaxor ferroelectric system as a solid solution of multiple components A, B, C... with their nominal concentrations being c A 0 , c B 0 , c C 0 , …, and satisfying .

[0015] Preferably, the method for generating the concentration distribution in step S1 is as follows:

[0016] The actual concentration distribution is generated by a Dirichlet distribution, that is , where r is the grid point position of the simulation grid, is the nominal concentration, is the configurational entropy calculated according to the nominal concentration, and k1 is a coefficient.

[0017] Preferably, the method for setting the Landau energy according to the compositional inhomogeneity in step S2 is as follows:

[0018] Assume that the Landau energy at each position is a linear combination of each component, and the weight depends on the actual concentration of each component at this position, that is ; where is the Landau energy coefficient.

[0019] Preferably, in step S3, the random electric field is related to the gradient of the actual concentration, that is , where E rand is the random field strength, is the configurational entropy calculated according to the nominal concentration, and k2 is another constant.

[0020] Preferably, the free energy is expressed as:

[0021] ;

[0022] respectively represent the Landau energy, gradient energy, elastic energy, and electric field energy; specifically,

[0023] ;

[0024] ; ;

[0025] ;

[0026] Among them, , G, C, , , K, and E are the Landau energy coefficient, the gradient energy coefficient, the elastic stiffness coefficient, the total strain, the spontaneous strain, the background dielectric constant, the vacuum dielectric constant, and the electric field, respectively.

[0027] Preferably, in the expression of the free energy:

[0028] The spontaneous strain comes from the electrostrictive effect of spontaneous polarization, that is, the electrostrictive coefficient , ;

[0029] The distribution of the elastic strain is obtained by solving the mechanical equilibrium equation ;

[0030] The electric field is obtained by solving the Poisson equation, that is, , where is the electric potential;

[0031] The overall solution process is as follows: First, input a random polarization field, obtain the spontaneous strain and the elastic equilibrium equation according to the polarization field, and get the strain distribution; then, obtain the solution of the Poisson equation according to the polarization field to get the electric field distribution; carry out the next polarization structure evolution according to the polarization, strain, and electric field distributions; and so on in a cycle to obtain the final steady-state polarization field.

[0032] Preferably, the method for finding the order parameter distribution corresponding to the minimum free energy in step S5 is: Select the polarization as the order parameter, P1, P2, and P3 are the three components of the electric polarization respectively, and according to the evolution equation , to obtain the minimum value of the free energy, where L is a kinetic parameter, t is time, and F is the free energy; the methods for obtaining the minimum value of the free energy include:

[0033] The evolution equation is written in discrete form as:

[0034] ;

[0035] So ;

[0036] Considering to be small, there is:

[0037] ;

[0038] Ensure that the direction of polarization evolution is such that the free energy becomes smaller and smaller until it finally reaches a converged minimum value.

[0039] The present invention also provides a phase-field simulation device for the dielectric properties of relaxor ferroelectrics, which executes the above-mentioned phase-field simulation method. The device includes:

[0040] The first module, for a relaxor ferroelectric, introduces local inhomogeneity of composition to generate a concentration distribution;

[0041] The second module, sets the Landau energy according to the compositional inhomogeneity;

[0042] The third module, uses a random electric field to describe other structural inhomogeneities resulting from compositional inhomogeneity and fluctuations;

[0043] The fourth module, obtains the free energy of the dielectric properties of the relaxor ferroelectric according to the results of the first, second, and third modules;

[0044] The fifth module, finds the order parameter distribution corresponding to the minimum free energy to achieve a steady-state phase-field simulation.

[0045] The present invention also provides a computer-readable storage medium, which stores a program that is executed by a processor to implement the above-mentioned phase-field simulation method.

[0046] The advantages of the present invention are:

[0047] Conventional phase-field simulations cannot describe the compositional inhomogeneity of relaxor ferroelectrics and can only preset structures and parameters; the present invention directly introduces compositional inhomogeneity, correlates ferroelectricity, structural inhomogeneity, and other related properties to the composition distribution, and can describe and quantitatively simulate complex relaxor ferroelectrics through a simple and clear single path, and the results are consistent with experimental observations. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] The present invention will be further described below in conjunction with the drawings and embodiments: Figure 1 is a flowchart of the phase-field simulation method for the dielectric properties of the relaxor ferroelectric of the present invention; Figure 2 is a schematic diagram of the phase-field simulation results of Example 1; Figure 2 a is a schematic diagram of the compositional inhomogeneous structure; Figure 2 b is the trend of the domain structure changing from large to small; Figure 2 c is the trend of the ferroelectric hysteresis loop gradually becoming thinner; Figure 2 d is a schematic diagram of the change in domain size; Figure 2 e is a schematic diagram of the change in dielectric energy storage density / efficiency; Figure 3 Schematic diagram of the phase-field simulation results of Example 2; Figure 3 a. 3c is a schematic diagram of a more disordered non-uniform structure; Figure 3 b shows the trend of the ferroelectric hysteresis loop gradually tapering; Figure 3 d is a schematic diagram of the gradually decreasing and more disordered spontaneous polarization; Figure 3 e is a schematic diagram of the change in dielectric energy storage density / efficiency. Detailed implementation mode

[0049] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0050] As Figure 1 shown, a phase-field simulation method for the dielectric properties of a relaxor ferroelectric includes the steps of:

[0051] S1. For a relaxor ferroelectric, introduce local non-uniformity of the composition to generate a concentration distribution;

[0052] S2. Set the Landau energy according to the compositional non-uniformity;

[0053] S3. Use a random electric field to describe other structural non-uniformities resulting from compositional non-uniformity and fluctuations;

[0054] S4. Obtain the free energy of the dielectric properties of the relaxor ferroelectric according to the results of steps S1 to S3;

[0055] S5. Find the order parameter distribution corresponding to the minimum free energy to achieve a steady-state phase-field simulation.

[0056] Specifically, the following settings are made for the phase-field simulation of the present invention for relaxor ferroelectrics:

[0057] (1) The local non-uniformity of the composition introduced in step S1 is as follows: the relaxor ferroelectric system is regarded as a solid solution of multiple components A, B, C,... Their nominal concentrations are c A 0 , c B 0 , c C 0 , …, and satisfy .

[0058] The method for generating the concentration distribution in step S1 is as follows: The true concentration distribution is generated by a Dirichlet distribution, i.e., , where r is the grid point position of the simulation grid, is the nominal concentration, is the configurational entropy calculated based on the nominal concentration, and k1 is a coefficient.

[0059] (2) In step S2, the Landau energy is set according to the compositional inhomogeneity: Assume that the Landau energy at each position is a linear combination of each component, and the weight depends on the actual concentration of each component at this position, i.e., . is the Landau energy coefficient, and all subscripts can take 1, 2, 3, representing the x, y, and z components respectively.

[0060] (3) In step S3, a random electric field is used to describe other structural inhomogeneities resulting from compositional inhomogeneity and fluctuations; the random electric field is a distribution related to the gradient of the actual concentration, i.e., , where E rand is the random field strength, is the configurational entropy calculated based on the nominal concentration, and k2 is another coefficient.

[0061] In addition, other parameters are constants without distribution.

[0062] In step S4, the free energy for obtaining the dielectric properties of the relaxor ferroelectric is obtained; the free energy is expressed as:

[0063] ;

[0064] represent the Landau energy (i.e., the intrinsic property of the relaxor ferroelectric), the gradient energy (measuring the energy of the domain wall), the elastic energy (measuring the elastic and electrostriction energy of the relaxor ferroelectric), and the electric field energy respectively; specifically,

[0065] ;

[0066] ; ;

[0067] ;

[0068] Among them, , G, C, , , K, and E are the Landau energy coefficient, the gradient energy coefficient, the elastic stiffness coefficient, the total strain, the spontaneous strain, the background dielectric constant, the vacuum dielectric constant, and the electric field respectively.

[0069] In the expression of the free energy: the spontaneous strain comes from the electrostrictive effect of spontaneous polarization, that is , ; the distribution of the elastic strain is obtained by solving the mechanical equilibrium equation ; the electric field is obtained by solving the Poisson equation, that is ;

[0070] The overall solution process is as follows: First, input a random polarization field, obtain the spontaneous strain and the elastic equilibrium equation according to the polarization field, and get the strain distribution; then, obtain the solution of the Poisson equation according to the polarization field to get the electric field distribution; perform the next polarization structure evolution according to the polarization, strain and electric field distributions; and so on in a cycle to obtain the final steady-state polarization field.

[0071] In step S5, the method for finding the order parameter distribution corresponding to the minimum free energy is: select the polarization as the order parameter, and according to the evolution equation , to obtain the minimum value of the free energy, where L is a kinetic parameter, t is time, and F is the free energy.

[0072] The methods for obtaining the minimum value of the free energy include:

[0073] The evolution equation is written in discrete form as:

[0074] ;

[0075] So ;

[0076] Considering is small, we have:

[0077] ;

[0078] Ensure that the direction of polarization evolution makes the free energy smaller and smaller, and finally reaches the convergent minimum value.

[0079] Example 1

[0080] In this example, the xBiFeO3-(1-x)SrTiO3 solid solution is taken as an example, and the simulation grid is , and the grid spacing is . The parameter settings are shown in Table 1.

[0081] Table 1 Parameter settings table of BiFeO3-SrTiO3

[0082]

[0083] As Figure 2 the results shown, the method of the present invention can effectively simulate the composition inhomogeneity ( Figure 2a), the trend of the domain structure changing from large to small ( Figure 2 b) and the trend of the ferroelectric hysteresis loop gradually becoming thinner ( Figure 2 c), the change in domain size ( Figure 2 d) and the change in dielectric energy storage density / efficiency ( Figure 2 e) are in good correspondence with the existing experimental values.

[0084] Example 2

[0085] In this example, the solid solution Bi4(Ti,A,B,C,D)3O 12 is taken as an example. It is considered that this solid solution is obtained by equimolar doping of elements A, B, C, D, etc. Thus, when there are a total of n components such as A, B, C..., the nominal concentration of each component is 1 / (n + 1) (because there is also Bi4Ti3O 12 ), and the entropy of the system is . The simulation grid is , and the grid spacing is . The parameter settings are shown in Table 2.

[0086] Table 2 Bi4(Ti,A,B,C,D)3O 12 Parameter setting table

[0087]

[0088] As Figure 3 shown by the results, as the entropy of the material increases, the material composition becomes more disordered ( Figure 3 a, 3c), the ferroelectric hysteresis loop changes from wide to thin ( Figure 3 b), the spontaneous polarization gradually decreases and becomes more disordered ( Figure 3 d), and the energy storage performance is significantly improved and the optimal point is located in the medium-high entropy range, which is consistent with the reported experimental results.

[0089] Example 3

[0090] The present invention also proposes a phase-field simulation device for the dielectric properties of relaxor ferroelectrics, which executes the above phase-field simulation method. The device includes:

[0091] The first module, for relaxor ferroelectrics, introduces local non-uniformity of composition to generate a concentration distribution;

[0092] The second module, sets the Landau energy according to the compositional non-uniformity;

[0093] The third module, uses a random electric field to describe other structural non-uniformities resulting from compositional non-uniformity and fluctuations;

[0094] The fourth module, obtains the free energy of the dielectric properties of relaxor ferroelectrics according to the results of the first, second, and third modules;

[0095] The fifth module is to perform a phase field simulation to achieve a steady state by finding the order parameter distribution corresponding to the minimum free energy.

[0096] Example 4

[0097] The present invention also provides a computer-readable storage medium storing a program, which when executed by a processor implements the phase field simulation method described above.

[0098] The above embodiments are only for illustrating the technical concept and features of the present invention, and the purpose is to enable those skilled in the art to understand the content of the present invention and implement it accordingly. However, the protection scope of the present invention cannot be limited thereby. Any modification made according to the spirit of the main technical solution of the present invention should be covered within the protection scope of the present invention.

Claims

1. A phase-field simulation method for the dielectric properties of relaxor ferroelectrics, characterized in that, Including the steps: S1. For relaxor ferroelectrics, introduce local inhomogeneity of composition to generate a concentration distribution; S2. Set the Landau energy according to the compositional inhomogeneity; S3. Use a random electric field to describe other structural inhomogeneities resulting from the inhomogeneity and fluctuations of the composition; S4. Obtain the free energy of the dielectric properties of the relaxor ferroelectric according to the results of steps S1 to S3; S5. Find the order parameter distribution corresponding to the minimum free energy to achieve a steady-state phase-field simulation.

2. The phase field simulation method for dielectric properties of relaxor ferroelectrics according to claim 1, characterized in that, The local inhomogeneity of composition introduced in step S1 is: Regarding the relaxor ferroelectric system as a solid solution of multiple components A, B, C, …, their nominal concentrations are c A 0 , c B 0 , c C 0 , …, and satisfy .

3. The phase field simulation method for dielectric properties of relaxor ferroelectrics according to claim 2, wherein The method for generating the concentration distribution in step S1 is: The true concentration distribution is generated by a Dirichlet distribution, i.e., , where r is the grid point position of the simulation grid, is the nominal concentration, is the configurational entropy calculated from the nominal concentration, and k1 is a coefficient.

4. The phase field simulation method for dielectric properties of relaxor ferroelectrics according to claim 1, characterized in that The method for setting the Landau energy according to the compositional inhomogeneity in step S2 is: Assume that the Landau energy at each position is a linear combination of the individual components, with weights depending on the actual concentration of each component at that position, i.e., ; where are the Landau energy coefficients.

5. The phase field simulation method for dielectric properties of relaxor ferroelectrics according to claim 1, wherein The random electric field described in step S3 is related to the distribution and the gradient of the actual concentration, i.e., , where E rand is the random field strength, is the configurational entropy calculated according to the nominal concentration, and k2 is another constant.

6. The phase field simulation method for dielectric properties of relaxor ferroelectrics according to claim 1, characterized in that The free energy is expressed as: ； represent the Landau energy, the gradient energy, the elastic energy, and the electric field energy, respectively; specifically, ； ； ； ； Among them, , G, C, , , K, and E are the Landau energy coefficient, the gradient energy coefficient, the elastic stiffness coefficient, the total strain, the spontaneous strain, the background dielectric constant, the vacuum dielectric constant, and the electric field, respectively.

7. The phase-field simulation method for dielectric properties of relaxor ferroelectrics according to claim 6, characterized in that In the expression of the free energy: The spontaneous strain stems from the electrostrictive effect of spontaneous polarization, i.e., the electrostrictive coefficient , ; The distribution of elastic strain is obtained by solving the mechanical equilibrium equations obtained; The electric field is obtained by solving the Poisson equation, i.e., , where is the electric potential; The overall solution process is: First, input a random polarization field, obtain the spontaneous strain and elastic equilibrium equations according to the polarization field to get the strain distribution; then obtain the solution of the Poisson equation according to the polarization field to get the electric field distribution; perform the next polarization structure evolution according to the polarization, strain and electric field distributions; and so on in a cycle to obtain the final steady-state polarization field.

8. The phase field simulation method for dielectric properties of relaxor ferroelectrics according to claim 1, characterized in that In step S5, the method for finding the order parameter distribution corresponding to the minimum free energy is as follows: Select the polarization as the order parameter, where P1, P2, and P3 are the three components of the electric polarization respectively. According to the evolution equation , the minimum value of the free energy is obtained, where L is a kinetic parameter, t is time, and F is the free energy. The method for obtaining the minimum value of the free energy includes: The evolution equation is written in a discrete form as: ； Therefore ; Consider Small, there is: ； Ensure that the direction of polarization evolution makes the free energy smaller and smaller until it finally reaches the convergent minimum value.

9. A phase-field simulation device for the dielectric properties of relaxor ferroelectrics, characterized in that, Implementing the phase-field simulation method according to any one of claims 1-8, the device includes: The first module, for relaxor ferroelectrics, introduces local inhomogeneity of composition to generate a concentration distribution; The second module, sets the Landau energy according to the compositional inhomogeneity; The third module, uses a random electric field to describe other structural inhomogeneities resulting from the inhomogeneity and fluctuations of the composition; The fourth module, obtains the free energy of the dielectric properties of the relaxor ferroelectric according to the results of the first, second, and third modules; The fifth module, finds the order parameter distribution corresponding to the minimum free energy to achieve a steady-state phase-field simulation.

10. A computer-readable storage medium, characterized in that, The storage medium stores a program, and the program is executed by a processor to implement the phase-field simulation method according to any one of claims 1 to 8.

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