Ferroelectric capacitor model based on ferroelectric domain distribution function and modeling method thereof

Abstract

The application discloses a ferroelectric capacitor model and a modeling method based on a ferroelectric domain distribution function, which comprises a positive port, a negative port, a main capacitor module, a current judgment module and a domain calculation module; the main capacitor module comprises a direct current resistance, a linear capacitor and a domain capacitor, and the direct current resistance, the linear capacitor and the domain capacitor are connected in parallel between the positive port and the negative port; the current judgment module comprises a voltage-controlled voltage source and a capacitor, the input voltage of the voltage-controlled voltage source is copied from the voltage value between the two ends of the ferroelectric capacitor, and the output end of the voltage-controlled voltage source is connected to the two ends of the capacitor; the domain calculation module comprises a voltage-controlled current source and an integral capacitor, the input voltage of the voltage-controlled current source is copied from the voltage value between the two ends of the ferroelectric capacitor, the output current is constructed based on a domain flipping probability density function and is used for representing the domain flipping rate of the ferroelectric capacitor; the integral capacitor is used for obtaining the domain value of the ferroelectric capacitor; and the capacitance value of the domain capacitor is calculated based on the domain value. The application solves the problems of the existing ferroelectric capacitor model, such as weak physical connection, high calculation complexity, low simulation efficiency and poor platform compatibility.

Description

Technical Field

[0001] This invention belongs to the field of ferroelectric capacitance modeling technology, specifically relating to a ferroelectric capacitance model and its modeling method based on the ferroelectric domain distribution function. Background Technology

[0002] Ferroelectric random access memory (FeRAM), as an emerging non-volatile memory technology, has become an important development direction for future embedded systems and high-performance computing applications due to its advantages such as low power consumption, high read / write speed, high durability, and non-volatility. However, in the development of ferroelectric memories, accurate modeling of ferroelectric capacitors is crucial for the circuit design, functional verification, and system optimization of memory arrays. The complex polarization mechanism of ferroelectric materials and the lack of clarity regarding their related physical properties and dynamic evolution laws severely restrict the development of large-capacity ferroelectric memories.

[0003] Currently, various modeling methods for ferroelectric capacitors have been proposed to simulate their macroscopic electrical performance curves. Among them, the traditional Miller and Preisach models are essentially models constructed by fitting function curves using the tangent function. While they possess good practicality and accuracy, their modeling relies heavily on fitting macroscopic electrical curves using mathematical functions, lacking a connection to the actual physical mechanisms of ferroelectric materials. In addition, there are equivalent circuit models of ferroelectric capacitors based on phase-field models, which closely integrate ferroelectric physical mechanisms with macroscopic electrical performance. These models can simulate the formation and evolution of dynamic ferroelectric domain structures and accurately simulate the behavior and macroscopic characteristic curves of ferroelectric materials. However, their computational complexity is high, and their simulation efficiency is low. They are not practical for simulating real circuits containing a large number of ferroelectric storage elements at the array level, limiting their engineering applicability in large-scale integrated circuit design.

[0004] For example, the Miller mathematical model proposed in the paper [Miller SL, Schwank JR, Nasby RD, et al. Modeling ferroelectric capacitor switching with asymmetric nonperiodic input signals and arbitrary initial conditions[J]. Journal of applied physics, 1991, 70(5):2849-2860.] uses the tangent function to describe the hysteresis loop of saturated polarization and the parameter Γ to describe the hysteresis loop of unsaturated polarization scaling. However, this model mainly focuses on the mathematical characterization of the hysteresis polarization behavior of ferroelectric materials and lacks a description of the microscopic physical mechanism of ferroelectricity. References [Tsinghua University. Ferroelectric Capacitor Behavior Model Applied to SPICE Circuit Simulation Program: CN200810103446.1 [P]. 2009-12-16.] and [Jingtie Semiconductor Technology (Guangdong) Co., Ltd. A Ferroelectric Capacitor Behavior Model Construction Method, System, Equipment and Products: 202410700111.7 [P]. 2024-08-06.] also use the arctangent function to describe the hysteresis loops of saturated and unsaturated polarization, and introduce some meaningless parameter variables. The values ​​of these parameters differ for different hysteresis loops, resulting in significant computational overhead and thus making the calculations quite complex.

[0005] The Preisach model proposed in the paper [Ni K, Jerry M, Smith JA, et al. A circuit compatible accurate compact model for ferroelectric-FETs[C] / / 2018 IEEE symposium on VLSI technology. IEEE, 2018: 131-132.] simulates and fits the subloop trajectory through empirical interpolation and linear scaling mechanisms. However, this model relies excessively on phenomenological mathematical construction, simplifying the complex ferroelectric dynamics process into a geometric scaling based on the tangent function. This also leads to a lack of clear physical meaning; its essence is still a fitting characterization of macroscopic electrical performance using mathematical functions, and it cannot explain the statistical distribution characteristics of domain flipping at the microscopic level.

[0006] The ferroelectric capacitor model based on phase-field theory proposed in the literature [Adnaan M, Chang SC, Li H, et al. A compact model forferroelectric capacitors based on multidomain phase-field framework[J]. IEEE Transactions on Electron Devices, 2023, 70(7): 3523-3529.] has good physical correlation, but its simulation efficiency is poor, and it still faces significant challenges in engineering applications. At present, some modeling methods still rely on physical simulation platforms such as COMSOL and have not yet been effectively integrated with standard process design kits (PDKs), thus creating a significant disconnect between device-level physical simulation and circuit-level design, making it difficult to directly integrate into the conventional integrated circuit design process. On the other hand, the ferroelectric capacitor phase-field equivalent circuit model reported in the literature has significant shortcomings in simulation efficiency, with a complete simulation time of up to 6400 minutes. Even under the condition of a one-dimensional simplified model, the simulation time for a single memory cell is still about 324 seconds. Such high computational complexity and low simulation efficiency make it difficult to directly apply the model to the simulation of array-level circuits containing a large number of ferroelectric memory cells, which severely limits its engineering practicality and widespread application in large-scale integrated circuit design.

[0007] In summary, there is currently a lack of a method that can accurately describe the polarization-voltage relationship between ferroelectric capacitance and the physical mechanism of ferroelectric materials. P - V It can be used to model large-capacity ferroelectric memory circuits efficiently and compactly, and can also be flexibly integrated into circuit simulation tools. Summary of the Invention

[0008] To address the aforementioned problems in the existing technology, this invention provides a ferroelectric capacitance model and modeling method based on the ferroelectric domain distribution function. The technical problem to be solved by this invention is achieved through the following technical solution: A ferroelectric capacitance model based on the ferroelectric domain distribution function includes: a positive port. Pos and negative port Neg , used to simulate the positive and negative ports of a ferroelectric capacitor; Main capacitor module; the main capacitor module includes capacitors connected in parallel at the positive port. Pos and negative port Neg DC resistance between R L Linear capacitance C 0 and domain capacitance C feThe DC resistor R L Used to simulate the DC leakage current of the ferroelectric capacitor; the linear capacitor C 0 is used to simulate the dielectric properties of the ferroelectric capacitor; the domain capacitance C fe A collection of ferroelectric domains used to simulate the ferroelectric capacitor; Current determination module; the current determination module includes a voltage-controlled voltage source (VCVS) and a capacitor. C cp The input voltage of the voltage-controlled voltage source VCVS replicates the positive port. Pos and negative port Neg The voltage between them, the output terminal is connected to the capacitor. C cp The two ends of the capacitor; wherein, the current flows through the capacitor. C cp The direction of the current characterizes the domain evolution direction of the ferroelectric capacitor; Domain calculation module; the domain calculation module includes a voltage-controlled current source. G u / G d and integrating capacitor C fd The voltage-controlled current source G u / G d The input voltage replicates the positive port. Pos and negative port Neg The voltage between, the voltage-controlled current source G u / G d The output current is used to characterize the domain switching rate of the ferroelectric capacitor. The output current is calculated based on an output current model that selects either a forward or reverse domain evolution direction; the output current model is constructed based on the domain probability density function; the integral capacitor... C fd Used for the voltage-controlled current source G u / G d Integrating the output current yields the domain values ​​characterizing the ferroelectric capacitor. f u / d integral voltage V fd The domain capacitor C fe The capacitance value is based on the domain value. f u / d calculate.

[0009] The domain capacitor C fe The formula for calculating the capacitance value is: ; in, P s The saturation polarization intensity represents the domain charge density of the ferroelectric capacitor with maximum polarization per unit area. S The area of ​​the ferroelectric capacitor; V The voltage value between the positive and negative ports; V o The half-width parameter is the coercive voltage distribution of the ferroelectric domains, used to reflect the flipping statistical characteristics caused by local defects, built-in electric fields, and non-uniform distribution of oxygen vacancies within the ferroelectric material. V c Indicates the average coercive voltage; f u / d The domain value; A This is a preset constant.

[0010] The voltage-controlled current source G u / G d The output current model is specifically based on the domain probability density function and domain value. f u / d Expression construction; the electrical domain value f u / d The expression is: ; in, V The voltage value between the positive and negative ports; V i This refers to the voltage value between the positive and negative ports in the initial state. f u The positive domain value represents the proportion of positive domains to the total domains of the ferroelectric capacitor. f ui In the initial state f u ; f d The reverse domain value is used to represent the proportion of reverse domains to the total domains. f di In the initial state f d When the voltage across the ferroelectric capacitor V When the flow increases, the current flows through the capacitor. C cp The direction of the current is positive, and the domain value is expressed as fu When the voltage across the ferroelectric capacitor V When the flow decreases, the current flows through the capacitor. C cp The direction of the current is negative, and the domain value is expressed as f d .

[0011] The voltage-controlled current source G u / G d The output current model is: ; Wherein, when flowing through the capacitor C cp When the direction of the current is positive, Specifically When the current flows through the capacitor C cp When the direction of the current is negative, Specifically ; V This refers to the voltage value between the positive and negative ports. V o The half-width parameter of the coercive voltage distribution of the ferroelectric domains; V c Indicates the average coercive voltage; I cp For the flow through the capacitor C cp The current value, f u / d This refers to the domain value; I fd ( V ) represents the output current of the voltage-controlled current source.

[0012] Optionally, the ferroelectric capacitance model also includes a hysteresis characteristic output module; The hysteresis output module is used to output the ferroelectric capacitor. P - V Hysteresis curve and I - V Characteristic curves; The P - V The hysteresis curve is generated by the following formula: ; in, V The voltage value between the positive and negative ports corresponds to the P - V The x-axis value of the hysteresis curve, P ( VThe voltage across the ferroelectric capacitor is... V The total polarization intensity of the ferroelectric capacitor at that time corresponds to the P - V The vertical axis value of the hysteresis curve; f u / d The domain value is... P s The saturation polarization intensity represents the domain charge density of the ferroelectric capacitor with maximum polarization per unit area. S Let be the area of ​​the ferroelectric capacitor. C 0 represents the capacitance value of the linear capacitor; The I - V The hysteresis curve is generated by the following formula: ; in, V Corresponding to the above I - V The x-axis value of the hysteresis curve, I ( V The voltage across the ferroelectric capacitor is... V At that time, the current value of the ferroelectric capacitor corresponds to the I - V The vertical axis value of the hysteresis curve; P The total polarization intensity of the ferroelectric capacitor is given.

[0013] This invention also includes a modeling method for a ferroelectric capacitance model based on the ferroelectric domain distribution function, comprising: S1. Use a circuit simulation description language to create a ferroelectric capacitor model based on the ferroelectric domain distribution function, and encapsulate the ferroelectric capacitor model into a two-port circuit with a positive port and a negative port. S2. Instantiate and call the ferroelectric capacitor model by constructing the corresponding sub-circuit netlist; S3. Obtain initial parameters for simulating the behavior of the ferroelectric capacitor, and assign the initial parameters to the instantiated ferroelectric capacitor model to set the initial polarization state of the ferroelectric capacitor. S4. Apply an external voltage excitation between the positive and negative ports of the instantiated ferroelectric capacitor model, and run circuit simulation to obtain the ferroelectric capacitor. P - V Characteristic curves and I - V Characteristic curves.

[0014] Optionally, the circuit simulation description language includes SPICE language.

[0015] Optionally, the initial parameters include: the saturation polarization intensity of the ferroelectric capacitor.P s ,area S DC resistance R L resistance, linear capacitance C The capacitance of 0, the half-width parameter of the coercive voltage distribution of ferroelectric domains. V o and average coercive voltage V c .

[0016] Compared with the prior art, the beneficial effects of the present invention are as follows: In the ferroelectric capacitor model based on the ferroelectric domain distribution function provided by this invention, the ferroelectric capacitor is equivalent to a parallel circuit structure of DC resistance, linear capacitance, and domain capacitance. The DC resistance is used... R L To simulate the DC leakage current of the ferroelectric capacitor, a linear capacitor is used. C 0 simulates the dielectric properties of the ferroelectric capacitor using domain capacitance. C fe This invention simulates the set of ferroelectric domains in a ferroelectric capacitor. By introducing a ferroelectric domain switching probability density function, the non-uniform domain structure and discrete domain switching behavior at the nanoscale are transformed into the overall switching rate of ferroelectric domains. Thus, this invention innovatively combines the physical mechanism of ferroelectric polarization switching with an efficient statistical description method. This avoids the problem of insufficient physical consistency in purely empirical models and overcomes the high computational complexity of high-precision physical models, achieving an optimal balance between accuracy and efficiency in terms of physical mechanism correlation and simulation efficiency. Specifically, by introducing the ferroelectric domain switching probability density function, this invention links polarization variables with circuit electrical variables, establishing a ferroelectric capacitor model that connects ferroelectric domain switching behavior, ferroelectric dielectric properties, and ferroelectric capacitor leakage current. The constructed equivalent circuit model can explain the statistical distribution characteristics of domain switching at the microscopic level, while accurately simulating the macroscopic electrical performance curves of the ferroelectric capacitor.

[0017] The ferroelectric capacitor model proposed in this invention constructs the domain probability density function, which is equivalent to modeling the polycrystalline ferroelectric material as a collection of independent grains or grain groups, with each grain or grain group characterized by its overall switching rate. Compared with the ferroelectric capacitor phase-field equivalent circuit model, this invention reduces computational complexity and significantly improves simulation efficiency.

[0018] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0019] Figure 1 This is a schematic diagram of a ferroelectric capacitance model constructed based on the ferroelectric domain distribution function provided by the present invention; Figure 2 This is the implementation principle of the ferroelectric capacitance model of the present invention; Figure 3 This is a flowchart illustrating a ferroelectric capacitance modeling method based on ferroelectric domain distribution function provided by the present invention. Figure 4 This is another flowchart illustrating a ferroelectric capacitance modeling method based on the ferroelectric domain distribution function provided by the present invention. Figure 5 This is a set of simulations using the ferroelectric capacitance model provided by this invention. P V , I - V Curve and Measured P V , I - V Comparison results of the curves; Figure 6 It utilizes the ferroelectric capacitance model provided by this invention to input two sets of arbitrary waveforms. P V , I - V Simulation results of the curve. Detailed Implementation

[0020] The present invention will be further described in detail below with reference to specific embodiments, but the implementation of the present invention is not limited thereto.

[0021] To meet the higher computational efficiency requirements of integrated circuit simulation in the development of ferroelectric memories, and to maintain a connection with the physical mechanisms of ferroelectrics, this invention proposes a ferroelectric capacitance model and its modeling method based on the ferroelectric domain distribution function. This model accurately describes the domain flipping behavior and polarization evolution behavior of ferroelectric devices under arbitrary input waveforms, while maintaining good consistency with experimental results. Furthermore, it is more computationally efficient and suitable for circuit design and array-level simulation.

[0022] The following section provides a detailed description of the ferroelectric capacitance model based on ferroelectric domain distribution provided by this invention. (See also...) Figure 1 As shown, the ferroelectric capacitor model includes: positive port Pos Negative port Neg、 The main capacitor module, current judgment module, and domain calculation module.

[0023] Among them, positive port Pos and negative port Neg This is used to simulate the positive and negative ports of a ferroelectric capacitor; correspondingly, the positive port... Pos and negative port NegThe voltage between them is the voltage across the positive and negative terminals of the ferroelectric capacitor. V Due to DC resistance R L Linear capacitance C 0 and domain capacitance C fe Because it is a parallel structure, the voltage across the domain capacitor is... V fe Voltage across the ferroelectric capacitor V same.

[0024] The main capacitor module includes capacitors connected in parallel at the positive port. Pos and negative port Neg DC resistance between R L Linear capacitance C 0 and domain capacitance C fe DC resistance R L Used to simulate the DC leakage current of ferroelectric capacitors, the resistance value of which can be preset; linear capacitors C 0 is used to simulate the dielectric properties of ferroelectric capacitors, and its capacitance value can be preset; domain capacitance. C fe The set of ferroelectric domains used to simulate ferroelectric capacitance, whose capacitance values ​​are fed back from the domain calculation module, will be explained below.

[0025] The current detection module includes a voltage-controlled voltage source (VCVS) and a capacitor. C cp The input voltage of the voltage-controlled voltage source (VCVS) is copied to the positive port. Pos and negative port Neg The voltage between them allows the voltage V(Pos,Neg) across the main capacitor module to be transmitted to the judgment module for analysis and judgment, while preventing any impact on the voltage across the ferroelectric capacitor; the output of the voltage-controlled voltage source VCVS is connected to the capacitor. C cp The two ends; among them, the capacitor C cp It is a regular capacitor, and current flows through the capacitor. C cp The direction of the current characterizes the direction of domain evolution in ferroelectric capacitors.

[0026] Specifically, the behavior of the voltage-controlled voltage source (VCVS) is described as Exxx n+ n- [VCVS] in+ in- gain, where the gain is 1 in this embodiment of the invention, and the current flows through the capacitor. C cp The current is .capacitance C cpThe direction of the current directly reflects whether the applied voltage is increasing or decreasing, and the evolution direction of ferroelectric domains is determined by the direction of change of the applied voltage. Therefore, the direction of the current flowing through the capacitor... C cp The direction of the current characterizes the domain evolution direction of a ferroelectric capacitor. In this invention, the ferroelectric capacitor domains are idealized, and... Figure 2 Ferroelectric capacitors are defined to have only two opposite polarization states: "forward up" and "reverse down". Ferroelectric capacitors operate in the absence of an external voltage ( V When the domain structure is zero, there are equal numbers of ferroelectric domains with "forward" and "reverse" polarization. At this point, the overall polarization of the ferroelectric capacitor is zero. Therefore, the polarization of the capacitor is known. C cp By observing the direction of the current, we can determine whether the domain evolution of the ferroelectric capacitor is in a positive or negative direction.

[0027] For example, capacitors C cp The capacitance value is typically 1F, when the voltage across the domain capacitor... V fe When increasing, V fe The derivative with respect to time (i.e., the current) i cp If the value is greater than 0, it indicates that the domains of the ferroelectric capacitor are evolving in the positive direction; similarly... V fe When it decreases, the current i cp If the value is less than 0, it indicates that the domains of the ferroelectric capacitor are evolving in the negative direction.

[0028] The domain calculation module includes a voltage-controlled current source. G u / G d and integrating capacitor C fd Voltage-controlled current source G u / G d Input voltage copy positive port Pos and negative port Neg Voltage between, voltage-controlled current source G u / G d The output current is used to characterize the domain switching rate of the ferroelectric capacitor. This output current is calculated based on an output current model that selects either forward or reverse domain evolution direction; this output current model is constructed based on the domain probability density function; integral capacitor. C fd Used for voltage-controlled current sources Gu / G d Integrating the output current yields the domain values ​​characterizing the ferroelectric capacitance. f u / d integral voltage V fd ; Domain Capacitance C fe The capacitance value is based on the domain value. f u / d calculate.

[0029] Specifically, domain capacitance C fe The behavior is described as Cfe pos1 neg C='equation', where equation is the equation or equality describing the domain capacitance. Here, the equation or equality describing the domain capacitance specifically refers to the domain capacitance. C fe The expression for the capacitance value is: (1); in, P s The saturation polarization intensity represents the domain charge density of the ferroelectric capacitor with maximum polarization per unit area. S Let be the area of ​​the ferroelectric capacitor; V This refers to the voltage value between the positive and negative ports; V o The half-width parameter of the coercive voltage distribution of ferroelectric domains reflects the statistical characteristics of the internal inhomogeneity of ferroelectric materials. V c Indicates the average coercive voltage; f u / d This refers to the domain value; A This is a preset mathematical constant, used as part of the standard form of the Lorentz function, to ensure normalization, and its value is 1.

[0030] If the probability of ferroelectric domains follows a Lorentz distribution function, then the probability density function for polarization reversal of ferroelectric capacitor domains is... p u / d ( V This can be represented as: (2); in, p u / d ( V This describes the probability of ferroelectric domains switching between positive and negative polarization as the applied voltage changes.

[0031] If the applied voltage changes in a positive direction, that is, the voltage increases with time ( The number of "positively" polarized domains increases, the number of "reversely" polarized domains decreases, and the total number of domains remains unchanged. The ferroelectric capacitor as a whole exhibits a positively polarized state. (The parameter is missing from the original text.) f u This represents the proportion of "positively" polarized domains to the total number of domains in a ferroelectric capacitor; similarly, the proportion changes when the applied voltage reverses ( When the number of "reverse" polarized domains increases and the number of "forward" polarized domains decreases, the ferroelectric capacitor as a whole exhibits a reverse polarization state. f d This represents the proportion of "reverse" polarized domains to the total number of domains in a ferroelectric capacitor, and The domain values ​​can then be derived. f u / d ( V The expression for ) is: (3); ; in, V This refers to the voltage value between the positive and negative ports; V i This represents the voltage value between the positive and negative ports in the initial state. f u This is the positive domain value, used to represent the proportion of positive domains in the total domains of a ferroelectric capacitor; f ui In the initial state f u ; f d This is the reverse domain value, used to represent the proportion of reverse domains to the total domains; f di In the initial state f d When the voltage across the ferroelectric capacitor V When the flow increases, the current flows through the capacitor. C cp The direction of the current is positive, and the domain value is expressed as f u When the voltage across the ferroelectric capacitor V When the flow decreases, the current flows through the capacitor. C cp The direction of the current is negative, and the domain value is expressed as f d .

[0032] Therefore, it is possible to base this on the domain probability density function. p u / d ( V ) and domain value f u / d ( V Constructing a voltage-controlled current source using expressionsG u / G d Output current model I fd ( V The expression is: (4); Among them, when the capacitor flows C cp When the direction of the current is positive, Specifically When current flows through the capacitor C cp When the direction of the current is negative, Specifically ; V This represents the voltage value between the positive and negative ports. V o The half-width parameter of the coercive voltage distribution of ferroelectric domains; V c Indicates the average coercive voltage; I cp For the flow through the capacitor C cp The current value; I fd ( V ) represents the output current of the voltage-controlled current source.

[0033] In one embodiment of the present invention, a voltage-controlled current source G u / G d The behavior is described as Gu net1 0 CUR='equation', where equation is the above. I fd ( V The expression is given by G, where G is the name of the voltage-controlled element. VCCS is the keyword for voltage-controlled current sources in HSPICE.

[0034] The equivalent circuit model proposed in this invention includes a positive port, a negative port, a main capacitor module, a current determination module, and a domain calculation module. The main capacitor module reflects the polarization characteristics of ferroelectric domain capacitance, the dielectric properties of ferroelectric capacitors, and the leakage current properties of ferroelectric capacitors; the current determination module calculates the current flowing through the capacitor... C cp The direction of the current determines the domain switching state, which in turn determines the domain value to be selected. f u / d and domain capacitance C fe The calculation path; the domain calculation module is used to calculate domain values. f u / dand domain capacitance C fe The calculation results are then fed back to the main capacitor module to update the polarization-related variables.

[0035] Initially, there are equal numbers of ferroelectric domains with "forward" and "reverse" polarization, and the ferroelectric capacitor as a whole exhibits zero polarization. When an external voltage is applied, the current determination module determines the direction of the current flowing through the ferroelectric capacitor in real time and selects the domain capacitor accordingly. C fe The calculation path is detailed in formula (1). Ferroelectric domains have a probability of flipping to the same direction as the current change. Specifically, the probability density function of ferroelectric capacitive domain polarization flipping is shown in formula (1). p u / d ( V The describing equation is given in formula (2), where the domain values ​​are... f u / d ( V The describing equation is given in formula (3), which yields the output current model of the voltage-controlled current source. I fd ( V See formula (4). The domain values ​​can be obtained through the self-consistent solution of this model. f u / d ( V Substitute into formula (1) to calculate C fe When an external voltage change is applied, repeat the above steps to update the domain capacitance. C fe .

[0036] In this invention, the ferroelectric capacitor is equivalent to a parallel circuit structure of a DC resistor, a linear capacitor, and a domain capacitor, using a DC resistor. R L To simulate the DC leakage current of a ferroelectric capacitor, a linear capacitor is used. C 0. Simulate the dielectric properties of ferroelectric capacitors using domain capacitance. C fe The set of ferroelectric domains in a simulated ferroelectric capacitor; by introducing the ferroelectric domain switching probability density function. p u / d ( V This invention transforms the non-uniform domain structure and discrete domain flipping behavior at the nanoscale into the overall switching rate of ferroelectric domains. In this way, the invention innovatively combines the physical mechanism of ferroelectric polarization flipping with an efficient statistical description method, which avoids the problem of insufficient physical consistency of pure empirical models and overcomes the defect of excessive computational complexity of high-precision physical models. Thus, it achieves the optimal balance between accuracy and efficiency between the correlation of physical mechanisms and simulation efficiency.

[0037] Optionally, the ferroelectric capacitance model also includes a hysteresis characteristic output module; the hysteresis output module is used to output the ferroelectric capacitance. P - V Hysteresis curve and I - V Characteristic curves.

[0038] Specifically, for P - V The hysteresis curve is first calculated by determining the positive intensity per unit area of ​​the domain capacitance. P fe for: (5); in, f u For positive domain values, f d The value of the reverse domain. P s This represents the saturation polarization intensity.

[0039] Total polarization in ferroelectric capacitors P Domain Capacitance C fe and linear capacitor C The total contribution of 0 is obtained, therefore P - V The hysteresis curve is generated by the following formula: (6); in, V This refers to the voltage value between the positive and negative ports, corresponding to... P - V The x-axis value of the hysteresis curve, P ( V The voltage across the ferroelectric capacitor is V The total polarization intensity of the ferroelectric capacitance at that time corresponds to P - V The vertical axis value of the hysteresis curve; f u / d The domain value, P s The saturation polarization intensity, C 0 represents the capacitance value of a linear capacitor. P fe It is the positive charge density per unit surface area of ​​the domain capacitance calculated in the previous step.

[0040] In the polarization reversal dynamics of ferroelectric materials, a significant nonlinear relaxation relationship exists between the polarization reversal time and the applied electric field. Therefore, the polarization dynamics equation is introduced as follows: (7); in, Describing ferropolarization intensity P fe The dynamic process over time, P sat The saturation polarization intensity is the maximum polarization limit that ferroelectric capacitance polarization can ultimately reach under the current stable electric field conditions. τ The characteristic time of polarization reversal, V fe The voltage across the domain capacitor is the same as the voltage applied to the ferroelectric capacitor. V Equal. This equation describes the change in polarization reversal rate under different electric field conditions, reflecting that ferroelectric polarization does not instantly reverse with changes in the applied electric field, but rather requires a certain time evolution process. Therefore, the polarization state... P fe Always relative to the ideal polarization state P sat The case involves dynamic lag and also conforms to a non-quasi-static (NQS) polarization dynamics model.

[0041] Among them, the characteristic time of polarization reversal τ Voltage across the domain capacitor V fe The dependencies between them are: (8); in, τ 0 represents the material-dependent intrinsic time constant. V a The activation voltage reflects the energy barrier height that needs to be overcome during the polarization reversal process. α The exponential factor reflects the sensitivity of the switching rate to voltage variations. This relationship indicates that with increasing applied electric field, the nucleation and expansion processes of ferroelectric domains accelerate significantly, leading to an exponential decrease in polarization switching time. In other words, the applied electric field can significantly accelerate the switching process of ferroelectric domains.

[0042] I - V The hysteresis curve can be generated by equation (9): (9); in, V The voltage value between the positive and negative ports corresponds to the I - V The x-axis value of the hysteresis curve, I ( V The voltage across the ferroelectric capacitor is... V The total polarization intensity of the ferroelectric capacitor at that time corresponds to the I - V The vertical axis value of the hysteresis curve;P The total polarization intensity of the ferroelectric capacitor is given by formula (6). P ( V ).

[0043] In this invention, polycrystalline ferroelectric materials are modeled as a collection of independent grains or grain groups. Each grain or grain group is characterized by its overall switching rate. The polarization reversal probability of ferroelectric domains is obtained based on the characteristics of the ferroelectric domain distribution. In this model, the switching rate is represented by a Lorentz distribution, including but not limited to using a Gaussian distribution or a generalized beta distribution of the second kind.

[0044] The ferroelectric capacitance model proposed in this invention represents polycrystalline ferroelectric materials as a collection of independent grains or grain groups, and uses the Lorentz distribution function to characterize the overall switching rate distribution of electric domains. While associating with the ferroelectric physical mechanism, it significantly reduces the computational complexity and achieves efficient and accurate circuit-level simulation.

[0045] The ferroelectric capacitor model proposed in this invention is based on the Lorentz distribution function of ferroelectric domain probabilities. Structurally, its equivalent circuit model includes three parts: a main capacitor module, a current determination module, and a domain calculation module. The main capacitor module reflects the polarization dielectric characteristics, ordinary dielectric characteristics, domain polarization reversal characteristics, and leakage current characteristics of the ferroelectric capacitor. The current determination module, through which current flows through the capacitor... C cp The direction of the current determines the domain switching state, and the domain value is selected. f u / d and domain capacitance C fe The calculation path; the domain calculation module is used to calculate domain values. f u / d and domain capacitance C fe The calculation results are then fed back to the main capacitor module to update the polarization-related variables.

[0046] The model is built based on the equivalent circuit model of controlled sources, resistors and capacitors in SPICE. It has a clear structure and adjustable parameters, and can be directly embedded into SPICE-like circuit simulation tools. It is suitable for the joint simulation of ferroelectric memories and related circuits.

[0047] The two-port ferroelectric capacitor model proposed in this invention can be described and implemented using hardware description languages ​​such as SPICE netlist or Verilog-A, and can be directly called and integrated in mainstream circuit simulation platforms.

[0048] In the simulation information definition of ferroelectric capacitor parameterized units, the netlist functions of the ferroelectric capacitor model are uniformly configured and standardized to adapt to various mainstream circuit simulation platforms, including SPICE, HSPICE, LTSPICE, Verilog-A, AMS, HspiceD, and Spectre. By consistently mapping and matching the parameter names, data types, and interface forms of different simulation units, it is ensured that when model parameters are input into the netlist, the simulation units can correctly identify, parse, and call the corresponding ferroelectric capacitor model, thereby achieving model compatibility and unified calling across different simulation environments.

[0049] The ferroelectric capacitor model proposed in this invention is presented externally only as an equivalent device with two electrical ports, positive and negative. In actual simulation, regardless of the complexity of the internal implementation structure of the sub-circuit, as long as the port definition and input-output interface form are consistent with the preset specifications, the correct calling and simulation of the ferroelectric capacitor behavior model can be achieved.

[0050] Based on the same inventive concept, embodiments of the present invention also provide a method for modeling ferroelectric capacitors based on the voltage-controlled capacitor method, such as... Figure 3 As shown, the method includes the following steps: S1. Use a circuit simulation description language to create a ferroelectric capacitor model based on the ferroelectric domain distribution function, and encapsulate the ferroelectric capacitor model as a two-port circuit with a positive port and a negative port.

[0051] S2. Instantiate and call the ferroelectric capacitor model by constructing the corresponding sub-circuit netlist.

[0052] S3. Obtain the initial parameters used to simulate the behavior of ferroelectric capacitors, and assign the initial parameters to the instantiated ferroelectric capacitor model to set the initial polarization state of the ferroelectric capacitor.

[0053] Specifically, the initial parameters include: the saturation polarization intensity of the ferroelectric capacitor. P s ,area S DC resistance R L resistance, linear capacitance C The capacitance of 0, the half-width parameter of the coercive voltage distribution of ferroelectric domains. V o and average coercive voltage V c Without applying an external voltage, the ferroelectric capacitor is set to an initial equilibrium state, at which point the proportions of forward and reverse polarized domains are equal, i.e., the proportion of forward domains is... f u Ratio of reverse domains f dBoth are 0.5, corresponding to the externally applied voltage. V It is zero.

[0054] S4. Apply an external voltage excitation between the positive and negative ports of the instantiated ferroelectric capacitor model, and run circuit simulation to obtain the ferroelectric capacitor. P - V Characteristic curves and I - V Characteristic curves.

[0055] Specifically, during the simulation, an external voltage excitation is applied to the ferroelectric capacitor. When the external voltage is not zero, the proportions of the forward and reverse polarization domains change, i.e. f u ≠ f d ≠0.5, ferroelectric voltage V ≠0, thereby driving the ferroelectric domains to undergo flipping evolution.

[0056] During the application of excitation, the changes in external voltage are monitored in real time, and the voltage flowing through the capacitor is adjusted accordingly. C cp The magnitude of the current determines the evolution direction of the ferroelectric domains, providing a basis for calculating the corresponding domain flipping path. When current flows through the capacitor... C cp When the current is negative, the reverse polarization domain is selected for calculation. f d And based on the ferroelectric domain switching probability density function p d ( V (Formula 2) Calculate the contribution of reverse polarization to the polarization intensity of ferroelectric capacitance. P fe (Formula 5), ​​and based on P fe The voltage across the ferroelectric capacitor is calculated using formula (6). V Total polarization of ferroelectric capacitance at time P ( V ), and then based on P ( V Calculate the current value using formula (9) I ( V Finally, update the domain values. f di .

[0057] When the current flows through the capacitor C cp When the current is positive, select the positive polarization domain ratio. f u Similarly, calculate and update the domain values. f ui .

[0058] Finally, the polarization-voltage of the ferroelectric capacitor is obtained. P - V ) characteristics and current-voltage ( I - V The characteristics of ferroelectric capacitors are used to simulate their electrical behavior under arbitrary voltage excitation conditions.

[0059] By repeating the above steps cyclically, the dynamic updating of the ferroelectric domain state with changes in external voltage is achieved, thereby completing the simulation of the polarization response and electrical behavior of ferroelectric capacitors under arbitrary voltage excitation.

[0060] In one implementation, the circuit simulation description language in step S1 includes SPICE language, but is not limited to this.

[0061] Based on the above steps, a simulation netlist file adapted to the initial ferroelectric capacitance behavior model is established. The model is instantiated by constructing the corresponding sub-circuit netlist, and the data extracted from the test is assigned to the initial model parameters to generate the final ferroelectric capacitance behavior model. Finally, it is encapsulated as a two-port SPICE model for use in memory front-end and back-end simulations.

[0062] In a specific simulation example, see Figure 4 As shown, the specific implementation method is as follows: (1) Parameter settings: Before the simulation begins, the relevant parameters of the ferroelectric capacitor model are initialized, including the domain flip probability distribution parameters and simulation control parameters, to provide initial conditions for subsequent calculations. Specifically, the domain flip probability distribution parameters include the probability density function of domain polarization flipping. p u / d ( V ), where the constants A One value can be chosen, and the simulation control parameters are similar to the dielectric response in the ferroelectric capacitance model. C 0 Ferroelectric capacitor has the maximum polarization domain charge density per unit area P S The area of ​​ferroelectric capacitors S DC resistance R L .

[0063] Furthermore, considering the dependence of the time scale on the electric field strength during the ferropolar polarization reversal process, as shown in equations (8) and (9), the initial parameters also include the characteristic time of polarization reversal. τ Material-related intrinsic time constants τ 0, activation voltage V a Exponential factor α .

[0064] (2) No external voltage: When no external voltage is applied, the ferroelectric capacitor is set to the initial equilibrium state. At this time, the proportions of forward and reverse polarized domains are equal, that is, the proportion of forward domains is equal. f u Ratio of reverse domains f d Both are 0.5, corresponding to the externally applied voltage. V fe It is zero.

[0065] (3) Applying an external voltage: During the simulation, an external voltage is applied to the ferroelectric capacitor. When the external voltage is not zero, the proportion of forward and reverse polarization domains changes, i.e. f u ≠ f d ≠0.5, ferroelectric voltage V fe ≠0, thereby driving the ferroelectric domains to undergo flipping evolution.

[0066] (4) External voltage change: The current judgment module monitors the change of external voltage in real time. Based on whether the direction of the applied external voltage is positive or negative, it judges the evolution direction of ferroelectric domains and provides a basis for the corresponding domain flip calculation path.

[0067] (5) Calculate polarization contribution: When the external voltage is decreasing or changing in the opposite direction, select the reverse polarization domain for calculation. f d The contribution of reverse polarization to the polarization intensity of ferroelectric capacitance was calculated based on the ferroelectric domain reversal probability density function. P fe Simultaneously, the ferroelectric domain states are updated. When the external voltage is rising or showing a positive trend, the proportion of positively polarized domains is selected. f u The contribution of forward polarization to the polarization intensity of ferroelectric capacitance was calculated based on the ferroelectric domain reversal probability density function. P fe At the same time, the ferroelectric domain states are updated.

[0068] (6) Electrical quantity calculation and simulation output: After completing the ferroelectric domain state update, calculate the transient charge, current and polarization intensity of the ferroelectric capacitor, and further obtain the polarization-voltage of the ferroelectric capacitor. P - V ) characteristics and current-voltage ( I - V The characteristics of ferroelectric capacitors are used to simulate their electrical behavior under arbitrary voltage excitation conditions.

[0069] By repeating the above steps cyclically, the dynamic updating of the ferroelectric domain state with changes in external voltage is achieved, thereby completing the simulation of the polarization response and electrical behavior of ferroelectric capacitors under arbitrary voltage excitation.

[0070] The polarization characteristics of the ferroelectric capacitor model proposed in this invention are tested using a Sawyer-Tower circuit. By connecting a linear capacitor with known parameters (a non-main capacitor in the main capacitor module) in series with the ferroelectric capacitor and measuring the voltage change across the linear capacitor, the test circuit converts the polarization state of the ferroelectric capacitor into a directly observable voltage signal, so as to intuitively analyze and verify the polarization behavior of the ferroelectric capacitor.

[0071] Figure 5 This is a set of simulations using the ferroelectric capacitance model provided by this invention. P V , I - V Curve and Measured P V , I - V The curves are compared. `sim` represents the simulation output, and `exp` represents the experimental test results. The left subplot shows the polarization-voltage of the ferroelectric capacitance under different applied voltages. P - V The curve represents the voltage applied to the ferroelectric capacitor in real time (Voltage) on the horizontal axis and the polarization intensity on the vertical axis (μC / cm²). 2 The right-hand subplot shows the current-voltage relationship of the ferroelectric capacitor under different applied voltages. I - V The curve is represented by the voltage applied to the ferroelectric capacitor in real time on the horizontal axis, in volts, and the current flowing through the ferroelectric capacitor on the vertical axis, in microamps (μA).

[0072] Figure 6 It utilizes the ferroelectric capacitance model provided by this invention to input two sets of arbitrary waveforms. P V , I - V The simulation results are shown on the curve. The horizontal axis represents the voltage applied to the ferroelectric capacitor in real time, in volts, and the vertical axis represents the ferroelectric polarization intensity, in microcoulombs per square centimeter.

[0073] The left subplot shows the output of the ferroelectric capacitance model as the voltage value changes continuously. P - VThe curve shows the changes in voltage over time, with the inner smaller plot representing the real-time variation of the applied voltage. Point A represents the positive saturation polarization point, where, after applying the maximum positive voltage (2.5V), all ferroelectric domains within the material have been completely flipped and strictly aligned with the positive electric field, achieving the positive saturation polarization intensity. P sat Point B is the reverse saturation polarization point. At this point, after applying the maximum reverse voltage (-2.5V), all ferroelectric domains within the material have been completely flipped and strictly aligned with the reverse electric field, reaching the reverse saturation polarization intensity. P sat As can be seen from the figure, the smaller the voltage amplitude, the narrower the hysteresis loop and the lower the maximum polarization value; the larger the voltage amplitude, the wider the loop and the higher the polarization intensity, eventually tending to saturate (points A and B). These characteristics are consistent with the physical characteristics of the hysteresis loop of ferroelectric capacitors.

[0074] The right-hand subplot shows the simulated polarization-voltage (V) values ​​under different initial domain values. P - V The curves and experimental results are shown, with the inner inset plots illustrating the real-time change of applied voltage over time. Point A represents the positive saturation polarization point, where, after applying the maximum positive voltage (2.5V), all ferroelectric domains within the material have been completely flipped and strictly aligned with the positive electric field, achieving the positive saturation polarization intensity. P sat Point B is the reverse saturation polarization point. At this point, after applying the maximum reverse voltage (-2.5V), all ferroelectric domains within the material have been completely flipped and strictly aligned with the reverse electric field, reaching the reverse saturation polarization intensity. P sat n1~n5 represent the model with different initial domain values. f u Below P - V Curves; n1~n5 respectively correspond to f u The initial values ​​are 0.85, 0.7, 0.5, 0.3, and 0.15; exp is the ferroelectric capacitance measured in the experiment. P - V The curve shows that the simulation results are consistent with the experimental test results.

[0075] Simulation results of the model in this invention show that the ferroelectric capacitance behavior model established in this embodiment can accurately simulate the macroscopic electrical performance behavior of ferroelectric capacitors, including polarization-voltage (…). P - V Hysteresis curve and current-voltage ( I - VThe characteristic curves can be applied to the design and verification of ferroelectric memory circuits.

[0076] This invention models polycrystalline ferroelectric materials as a collection of independent grains or grain groups, with each grain or grain group characterized by its overall switching rate. Compared with the equivalent circuit model of ferroelectric capacitor phase-field, this invention reduces computational complexity and significantly improves simulation efficiency. The equivalent circuit model of ferroelectric capacitor phase-field reported in the literature [Adnaan M, Chang SC, Li H, et al. A compact model forferroelectric capacitors based on multidomain phase-field framework[J]. IEEE Transactions on Electron Devices, 2023, 70(7): 3523-3529.] has significant shortcomings in simulation efficiency. Even under the condition of a one-dimensional simplified model, the simulation time for a single memory cell is still about 324 seconds, while the complete simulation time of this invention is only 0.04 seconds.

[0077] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of the present invention, and all such modifications and substitutions should be considered within the scope of protection of the present invention.

Claims

1. A ferroelectric capacitance model based on the ferroelectric domain distribution function, characterized in that, include: Positive port Pos and negative port Neg , used to simulate the positive and negative ports of a ferroelectric capacitor; Main capacitor module; the main capacitor module includes capacitors connected in parallel at the positive port. Pos and negative port Neg DC resistance between R L Linear capacitance C 0 and domain capacitance C fe The DC resistor R L Used to simulate the DC leakage current of the ferroelectric capacitor; the linear capacitor C 0 is used to simulate the dielectric properties of the ferroelectric capacitor; the domain capacitance C fe A collection of ferroelectric domains used to simulate the ferroelectric capacitor; Current determination module; the current determination module includes a voltage-controlled voltage source (VCVS) and a capacitor. C cp The input voltage of the voltage-controlled voltage source VCVS replicates the positive port. Pos and negative port Neg The voltage between them, the output terminal is connected to the capacitor. C cp The two ends of the capacitor; wherein, the current flows through the capacitor. C cp The direction of the current characterizes the domain evolution direction of the ferroelectric capacitor; Domain calculation module; the domain calculation module includes a voltage-controlled current source. G u / G d and integrating capacitor C fd The voltage-controlled current source G u / G d The input voltage replicates the positive port. Pos and negative port Neg The voltage between, the voltage-controlled current source G u / G d The output current is used to characterize the domain switching rate of the ferroelectric capacitor. The output current is calculated based on an output current model that selects either a forward or reverse domain evolution direction; the output current model is constructed based on the domain probability density function; the integral capacitor... C fd Used for the voltage-controlled current source G u / G d Integrating the output current yields the domain values ​​characterizing the ferroelectric capacitor. f u / d integral voltage V fd The domain capacitor C fe The capacitance value is based on the domain value. f u / d calculate.

2. The ferroelectric capacitance model according to claim 1, characterized in that, The domain capacitor C fe The formula for calculating the capacitance value is: ； in, P s The saturation polarization intensity represents the domain charge density of the ferroelectric capacitor with maximum polarization per unit area. S The area of ​​the ferroelectric capacitor; V The voltage value between the positive and negative ports; V o The half-width parameter is the coercive voltage distribution of the ferroelectric domains, used to reflect the flipping statistical characteristics caused by local defects, built-in electric fields, and non-uniform distribution of oxygen vacancies within the ferroelectric material. V c Indicates the average coercive voltage; f u / d The domain value; A This is a preset constant.

3. The ferroelectric capacitance model according to claim 1, characterized in that, The voltage-controlled current source G u / G d The output current model is specifically based on the domain probability density function and domain value. f u / d Expression construction; the electrical domain value f u / d The expression is: ； in, V The voltage value between the positive and negative ports; V i This refers to the voltage value between the positive and negative ports in the initial state. f u The positive domain value represents the proportion of positive domains to the total domains of the ferroelectric capacitor. f ui In the initial state f u ; f d The reverse domain value is used to represent the proportion of reverse domains to the total domains. f di In the initial state f d When the voltage across the ferroelectric capacitor V When the flow increases, the current flows through the capacitor. C cp The direction of the current is positive, and the domain value is expressed as f u When the voltage across the ferroelectric capacitor V When the flow decreases, the current flows through the capacitor. C cp The direction of the current is negative, and the domain value is expressed as f d .

4. The ferroelectric capacitance model according to claim 1, characterized in that, The voltage-controlled current source G u / G d The output current model is: ； Wherein, when flowing through the capacitor C cp When the direction of the current is positive, Specifically When the current flows through the capacitor C cp When the direction of the current is negative, Specifically ; V This refers to the voltage value between the positive and negative ports. V o The half-width parameter is the coercive voltage distribution of the ferroelectric domains. V c Indicates the average coercive voltage; I cp For the flow through the capacitor C cp The current value, f u / d This refers to the domain value; I fd ( V ) represents the output current of the voltage-controlled current source.

5. The ferroelectric capacitance model according to claim 1, characterized in that, Also includes: Hysteresis characteristic output module; The hysteresis output module is used to output the ferroelectric capacitor. P - V Hysteresis curve and I - V Characteristic curves; The P - V The hysteresis curve is generated by the following formula: ； in, V The voltage value between the positive and negative ports corresponds to the P - V The x-axis value of the hysteresis curve, P ( V The voltage across the ferroelectric capacitor is... V The total polarization intensity of the ferroelectric capacitor at that time corresponds to the P - V The vertical axis value of the hysteresis curve; f u / d The domain value is... P s The saturation polarization intensity represents the domain charge density of the ferroelectric capacitor with maximum polarization per unit area. S Let be the area of ​​the ferroelectric capacitor. C 0 represents the capacitance value of the linear capacitor; The I - V The hysteresis curve is generated by the following formula: ； in, V Corresponding to the above I - V The x-axis value of the hysteresis curve, I ( V The voltage across the ferroelectric capacitor is... V At that time, the total polarization current value of the ferroelectric capacitor corresponds to the I - V The vertical axis value of the hysteresis curve; P The total polarization intensity of the ferroelectric capacitor is given.

6. A modeling method for a ferroelectric capacitance model based on the ferroelectric domain distribution function, characterized in that, include: S1. Use a circuit simulation description language to create a ferroelectric capacitor model based on the ferroelectric domain distribution function as described in any one of claims 1-5, and encapsulate the ferroelectric capacitor model as a two-port circuit with a positive port and a negative port. S2. Instantiate and call the ferroelectric capacitor model by constructing the corresponding sub-circuit netlist; S3. Obtain initial parameters for simulating the behavior of the ferroelectric capacitor, and assign the initial parameters to the instantiated ferroelectric capacitor model to set the initial polarization state of the ferroelectric capacitor. S4. Apply an external voltage excitation between the positive and negative ports of the instantiated ferroelectric capacitor model, and run circuit simulation to obtain the ferroelectric capacitor. P - V Characteristic curves and I - V Characteristic curves.

7. The modeling method according to claim 6, characterized in that, The circuit simulation description language includes SPICE language.

8. The modeling method according to claim 6, characterized in that, The initial parameters include: the saturation polarization intensity of the ferroelectric capacitor. P s Ferroelectric capacitor area S DC resistance R L resistance, linear capacitance C The capacitance of 0, the half-width parameter of the coercive voltage distribution of ferroelectric domains. V o and average coercive voltage V c .

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