Ferroelectric capacitor modeling method for circuit simulation

Abstract

The invention discloses a ferroelectric capacitor modeling method oriented to circuit simulation. The method comprises the following steps: 1, modeling a ferroelectric capacitor; the method comprises the following steps of: 1, correcting a node analysis method to enable MNA to be in modeling connection with an MNA model of Fcap; and step 2, an MNA model forming process of the Fcap element: based on the MNA, using an L-K model formula of the Fcap to obtain a compensation equation and an equation used for solving unknown variables, obtaining a final admittance matrix G and a known vector column RHS of the Fcap element, and obtaining an MNAmodel of the Fcap. According to the method, on the basis of the MNA, unknown variables are represented by utilizing a physical model formula of the Facap, the Kirchhoff current law and the Kirchhoff voltage law, and model codes of the Facap are formed, so that SPICE simulation of the Facap is more accurate and quicker.

Description

Technical Field

[0001] The invention belongs to the technical field of ferroelectric capacitor (Fecap) modeling, and in particular relates to a ferroelectric capacitor (Fecap) modeling method oriented to circuit simulation. Background Art

[0002] Ferroelectric capacitors (FeCap) and ferroelectric field-effect transistors (FeFET) are emerging electronic components with unique electrical properties such as nonvolatile memory and circuit programmability. In addition, FeCap-based ferroelectric RAM (FeRAM) has begun to attract widespread attention in the industry for its excellent performance and process compatibility. These features make them potential competitors for the realization of new in-memory computing (CIM) for energy-efficient hardware acceleration.

[0003] As process technology advances, there is an increasing need for electronic design automation (EDA) tools to support the design and verification of FeCap / FeFET-based CIM circuits.

[0004] At present, mainstream industrial SPICE circuit simulation tools are all based on the improved nodal analysis (MNA) formula. MNA is an analysis method based on node voltage. It combines Kirchhoff's voltage law (KVL), Kirchhoff's current law (KCL) and the constitutive equations of components. It solves the unknown quantities in the circuit by establishing a set of equations and expresses the circuit equations in matrix form, which is convenient for computer program processing and solving.

[0005] The prior art proposes to use MNA to implement an MNA model of ferroelectric capacitors applied to SPICE. Without this MNA model, designers must manually create equivalent circuit models for emerging devices, and the solution speed of the circuit equivalent model in the simulation tool also lags behind the MNA model.

[0006] Therefore, the present invention is aimed at the simulation tool model problems and demands currently faced by the above-mentioned ferroelectric circuits. Summary of the invention

[0007] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide a ferroelectric capacitor (Fecap) modeling method for circuit simulation. The method is based on MNA, and utilizes the physical model formula of Fecap (LK model) and Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL) to represent unknown variables (node ​​voltage, branch current, branch polarization), form a Fecap model code, and make the SPICE simulation of Fecap more accurate and faster.

[0008] In order to achieve the above object, the technical solution adopted by the present invention is:

[0009] A ferroelectric capacitor (Fecap) modeling method for circuit simulation comprises the following steps:

[0010] Step 1: Introduce the modified nodal analysis (MNA) into the MNA model of Fecap;

[0011] Step 2: MNA model formation process of Fecap element;

[0012] Based on the modified nodal analysis (MNA), the compensation equation is obtained using the LK model formula of Fecap, which is used to solve the equation of the unknown variable to obtain the final admittance matrix G and the known quantity vector column RHS of the Fecap element. The admittance matrix G and the known quantity column RHS are combined to obtain the MNA model of Fecap.

[0013] In step 1, the specific process of linking MNA with the MNA model of Fecap is as follows:

[0014] Kirchhoff's current law (KCL), Kirchhoff's voltage law (KVL) and the characteristic equation of the element are used to solve the unknown variables, also known as introducing the compensation equation, and then the element's admittance matrix G, variable column X and known quantity column RHS are obtained. The admittance matrix G and the known quantity column RHS are combined to form the MNA model of the element. Each addition of an unknown variable will generate a new row and column in the admittance matrix G, and at the same time, the corresponding unknown quantity will be generated at the corresponding position of the variable column X.

[0015] Combined with the physical properties of ferroelectric capacitors (Fecap), when ferroelectric capacitors are introduced into the circuit, the polarization state and current of the ferroelectric capacitors are new unknown quantities (the original unknown quantities are the two node voltages), and a 4X4 admittance matrix G and a 1X4 variable column X (VN + (t), VN - (t), i(t), p(t)) T And the 1X4 RHS column, then merge the admittance matrix G and the known quantity column RHS to obtain the MNA model of FECAP.

[0016] The step 2 is specifically as follows:

[0017] 2.1): Based on the generalization of Fecap's LK model, the general formula of Fecap model is obtained, which is convenient for subsequent linearization processing;

[0018] 2.2): The compensation equation of Fecap (a nonlinear device) is linearized by using the backward Euler method and the NR iteration method to solve the equation of the unknown variable. Based on MNA, the compensation equation of Fecap is linearized to meet the solution requirements of the Spice simulation tool and form the final MNA model;

[0019] 2.3): Combine the terminal voltage, polarization state and model formula of Fecap to obtain the specific expressions of functions F1 and F2, forming the final MNA model of Fecap.

[0020] The step 2.1) is:

[0021] Selecting the LK model, the relationship between polarization P and unit energy G is described by the following Landau free energy equation;

[0022]

[0023] Where E is the electric field, further deduction can lead to the LK model:

[0024]

[0025] where ρ is the thermodynamic coefficient, T FE Represents the thickness of the ferroelectric layer, V = ET FE is the voltage on the ferroelectric layer, and α, β, γ are coefficients related to material properties. Combining the above analysis of the LK model, and according to the terminal voltage and polarization state of the Fecap, the general formula of the Fecap model is listed:

[0026] V(t)=F1(I(t),P(t)) (1)

[0027]

[0028] The current I and polarization P flowing through Fecap are the internal state variables of Fecap, V represents the terminal voltage, and F1 and F2 represent arbitrary functions;

[0029] To this end, based on MNA, equations (1) and (2) for solving the unknown variables of Fecap have been obtained. Since the current variables and polarization variables are introduced into the column vector X, the corresponding rows and columns will be generated when the admittance matrix G is generated. The next step is to linearize the model to extract the admittance matrix G, the variable column vector X and the known quantity column vector RHS.

[0030] In 2.2), semiconductor devices in SPICE can generally be divided into two categories, namely linear devices and nonlinear devices;

[0031] The linearity and nonlinearity of a device depend on the relationship between the terminal voltage of the device and the current flowing through the device. For nonlinear circuits containing nonlinear elements, their model formulas need to be linearized.

[0032] The general formula of Fecap model is linearized using Newton-Raphson iterative method and backward Euler method.

[0033] Specifically:

[0034] 1) First, the terminal voltage is represented by the node voltage;

[0035] 2) Using the NR iteration method, at the working point (I (n) , P (n) ) is used to perform Taylor first-order expansion of the F1 function, and we can obtain:

[0036]

[0037] Among them, F1 (n) (t) = F1(I (n) (t), P (n) (t)), in order to extract matrix elements conveniently, equation (3) is further transformed

[0038]

[0039] Among them, VN+, VN-, I(t), and P(t) are variables to be solved and will be placed in the X column vector. Their corresponding coefficients will be used as elements of the admittance matrix. All the relevant terms of the n iterations on the right will be placed in the RHS column vector as known quantities. For the subsequent model description, the known terms are equivalent to Irhs

[0040]

[0041] For equation (2), dP / dt is linearized using the backward Euler method, and the F2 function is processed in the same way as the F1 function. At the working point (V (n) (t), P (n) (t)) is expanded by the first order Taylor, and we can get:

[0042]

[0043] Where h is the time step, F2 (n) (t) = F2(V (n) (t), P (n) (t)), V(t) = V N+ (t)-V N- (t), and then transform equation (5)

[0044]

[0045] Extract variables I(t), P(t), V N+ (t), V N- The coefficients before (t) are the elements of G. Similarly, for the subsequent model description, the known terms are equivalent to Prhs and placed in the RHS column vector

[0046]

[0047] For the flow through node V N+ and V N- The current flows into the node V N+ is in the positive direction, then V N+： I(t) vs. V N-： The -I(t) variable is I(t) only, with coefficients of 1 and -1, and no known terms are provided.

[0048] According to the above description, the coefficients before the unknown variables are extracted to form the admittance matrix G:

[0049]

[0050] The column vector X formed by the unknown variables:

[0051]

[0052] The column vector RHS is formed by the known quantities:

[0053]

[0054] At this point, the admittance matrix G and RHS of the synthetic Fecap MNA model have been obtained. The next step is to get the F1 and F2 functions and write the model into the SPICE simulation program.

[0055] The step 2.3) is specifically as follows:

[0056] For the F1 function, the terminal voltage of Fecap is formed by the potential difference of the parasitic resistance inside the ferroelectric and the potential difference of the ferroelectric layer. Therefore, the LK model formula based on Fecap can be obtained:

[0057]

[0058] Where AFE is the surface area of ​​the ferroelectric capacitor. For the F2 function, the dp / dt in the LK model formula of Fecap can be transformed:

[0059]

[0060] Merge the admittance matrix G and RHS. Since the F1 function and F2 function are known, we can get an MNA model of Fecap with exact parameters.

[0061] Simply write the corresponding matrix parameters into the modeling code specified by SPICE to obtain the built-in MNA model of Fecap in SPICE.

[0062] Beneficial effects of the present invention:

[0063] The present invention adopts the improved node analysis method (MNA) to realize the FecapMNA model that can be applied to the industrial SPICE simulation tool. In step 2, based on MNA, the Fecap LK model formula is used to obtain the Fecap MNA model that can be analyzed and calculated by the SPICE simulator. This enables ferroelectric capacitors to be simulated directly in the SPICE program to obtain more accurate electrical characteristics, greatly improving the potential and competitiveness of ferroelectric devices in analog memory circuit design.

[0064] The Fecap MNA model designed in the present invention can show a higher speed advantage in the SPICE simulation solution of the equivalent circuit of Fecap because it complies with the SPICE simulation solution principle of a single component. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 It is a flow chart of the present invention, based on MNA, using the general formula of the Fecap LK model to obtain the admittance matrix G and column vector RHS required by the Fecap MNA Model.

[0066] Figure 2 It is a simplified diagram of the present invention for obtaining F1 function and F2 function to form a specific Fecap MNA Model.

[0067] Figure 3 Schematic diagram of the merging of the admittance matrix G and RHS. DETAILED DESCRIPTION

[0068] The present invention will be further described in detail below in conjunction with the accompanying drawings.

[0069] like Figure 1 , Figure 2 As shown, a ferroelectric capacitor (Fecap) modeling method for circuit simulation includes the following steps:

[0070] Step 1: Modified nodal analysis (MNA) to link MNA with Fecap’s MNA model;

[0071] The step 1 is specifically as follows:

[0072] First, let’s introduce the specific process of NA (node ​​analysis method):

[0073] When performing SPICE (Simulation Program with Integrated Circuit Emphasis) simulation, the topology of the circuit needs to be described first through the SPICE netlist file. The netlist file describes all the nodes contained in a circuit topology structure and the branches formed between the nodes and the components on the branches in the form of code;

[0074] After SPICE obtains the topology of the circuit by parsing the netlist, it will formulate equations for the current-voltage relationship in the circuit. The commonly used methods are Kirchhoff's current law (KCL) and Kirchhoff's voltage law KVL. It is relatively convenient to solve the system of equations in the form of matrices and vectors in a computer, that is, extract the coefficients before the unknown variables (node ​​voltages) to form a matrix Y, also called an admittance matrix, and convert the unknown variables into column vectors V, and the remaining terms (known quantities) into column vectors I, so the circuit analysis results are expressed as

[0075] Y*V=I(1.1)

[0076] Because each component in the circuit can describe the branch current according to the node voltage of the branch, it can also be written in the same form as equation (1.1), and when the admittance matrix of each component is added at the same node, the same result as (1.1) will be obtained. Therefore, in the Spice simulation tool, the admittance matrix of each component is very important, which will greatly reduce the time to form the circuit admittance matrix during simulation. However, when there are unknown quantities in the components of the circuit that cannot be represented by the node voltage, the circuit has no solution, and the node analysis method is no longer applicable.

[0077] The difference between MNA (modified nodal analysis) and NA is that more unknowns can be introduced, and then KCL, KVL and the characteristic equation of the component itself are used to solve the unknown variables, which is also called introducing compensation equations, and then the admittance matrix G of the component, the variable column X and the known column RHS are obtained. The combination of G and RHS is the MNA model of the component. It should be noted that each addition of an unknown variable will generate a new row and column in G, and the corresponding unknown quantity will be generated at the corresponding position of X.

[0078] Combined with the physical properties of ferroelectric capacitors, when they are introduced into the circuit, the polarization state and current of the ferroelectric capacitors are new unknown quantities (the original unknown quantities are the two node voltages), so a 4X4 admittance matrix G is obtained (see Appendix Figure 1 ), 1X4 variable column X(VN + (t), VN - (t), i(t), p(t)) T And the RHS column of 1X4 (with Figure 1 ), then G and RHS are combined to obtain the MNA model of FECAP (see Appendix Figure 2 ).

[0079] Step 2: The MNA model formation process of the Fecap element, based on MNA, uses the LK model formula of Fecap to obtain the compensation equation, which is used to solve the equation of the unknown variable, and obtain the final admittance matrix G of the Fecap element and the known quantity vector column RHS to obtain the MNA model of the Fecap;

[0080] The step 2 is specifically as follows:

[0081] 2.1): General formula of Fecap model. Based on the generalization of Fecap's LK model, the general formula of Fecap model is obtained to facilitate the subsequent linearization process;

[0082] The main FeCap models at present are the Preisach model and the Landau-Khalatnikov (LK) model. Here, the LK model is selected for explanation because it has a clear physical meaning. The LK model can accurately describe the polarization response of ferroelectric materials under an external electric field. The relationship between polarization P and unit energy G can be described by the following Landau free energy equation;

[0083]

[0084] Where E is the electric field, further deduction can lead to the LK model:

[0085]

[0086] where ρ is the thermodynamic coefficient, T FE Represents the thickness of the ferroelectric layer, V = ET FE is the voltage on the ferroelectric layer, and α, β, γ are coefficients related to material properties. The specific values ​​of these model parameters can be determined by fitting experimental measurement data. Next, combined with the above analysis of the LK model, the general formula of the Fecap model is listed based on the terminal voltage and polarization state of the Fecap:

[0087] V(t)=F1(I(t),P(t)) (1)

[0088]

[0089] Where I (current flowing through Fecap) and P (polarization) are the internal state variables of Fecap, V represents the terminal voltage, F1 and F2 represent arbitrary functions, and the specific LK model will be introduced in 2.2 later. To this end, based on MNA, equations (1) and (2) for solving the unknown variables of Fecap have been obtained. (Since the current variables and polarization variables are introduced in the column vector X, the corresponding rows and columns will be generated when generating the admittance matrix G.) The next step is to linearize the model to extract the admittance matrix G, the variable column vector X, and the known quantity column vector RHS.

[0090] 2.2): The compensation equation of Fecap (nonlinear device) is linearized by using the backward Euler method and the NR iteration method to solve the equation of unknown variables. Based on MNA, the compensation equation of Fecap is linearized to meet the solution requirements of the Spice simulation tool and form the final MNA model;

[0091] In 2.2), semiconductor devices in SPICE can generally be divided into two categories, namely linear devices and nonlinear devices;

[0092] The linearity and nonlinearity of a device depends on the relationship between the terminal voltage of the device and the current flowing through the device. However, for a nonlinear circuit containing nonlinear elements, if its model formula is not linearized, SPICE will still obtain a linear relationship between voltage and current when analyzing and solving the circuit, and the simulation of the element will be wrong. Therefore, the general formula of the Fecap model is linearized using the Newton.Raphson iteration method and the backward Euler method.

[0093] Specifically: 1) First, the terminal voltage is represented by the node voltage;

[0094] 2) Using the NR iteration method, at the working point (I (n) , P (n) ) is used to perform Taylor first-order expansion of the F1 function, and we can obtain:

[0095]

[0096] Among them, F1 (n) (t) = F1(I (n) (t), P (n) (t)), in order to extract matrix elements conveniently, equation (3) is further transformed

[0097]

[0098] Among them, VN+, VN-, I(t), and P(t) are variables to be solved and will be placed in the X column vector. Their corresponding coefficients will be used as elements of the admittance matrix. All the relevant terms of the n iterations on the right will be placed in the RHS column vector as known quantities. For the subsequent model description, the known terms are equivalent to Irhs

[0099]

[0100] For equation (2), 1) uses the backward Euler method to linearize dP / dt, and the F2 function is the same as the F1 function. At the working point (V (n) (t), P (n) (t)) is expanded by the first order Taylor, and we can get:

[0101]

[0102] Where h is the time step, F2 (n) (t) = F2(V (n) (t), P (n) (t)), V(t) = V N+ (t)-V N- (t), and then transform equation (5)

[0103]

[0104] Extract variables I(t), P(t), V N+ (t), V N- The coefficients before (t) are the elements of G. Similarly, for the subsequent model description, the known terms are equivalent to Prhs and placed in the RHS column vector

[0105]

[0106] For the flow through node V N+ and V N- The current flows into the node V N+ is in the positive direction, then V N+： I(t) vs. V N-： The -I(t) variable is I(t) only, with coefficients of 1 and -1, and no known terms are provided.

[0107] According to the above description, the coefficients before the unknown variables are extracted to form the admittance matrix G:

[0108]

[0109] The column vector X formed by the unknown variables:

[0110]

[0111] The column vector RHS is formed by the known quantities:

[0112]

[0113] At this point, the admittance matrix G and RHS of the synthetic Fecap MNA model have been obtained. The next step is to get the F1 and F2 functions and write the model into the SPICE simulation program.

[0114] 2.3) Combining the terminal voltage, polarization state and model formula of Fecap, the specific expressions of functions F1 and F2 are obtained to form the final MNA model of Fecap;

[0115] The step 2.3) is specifically as follows:

[0116] For the F1 function, the terminal voltage of Fecap is formed by the potential difference of the parasitic resistance inside the ferroelectric and the potential difference of the ferroelectric layer. Therefore, the LK model formula based on Fecap can be obtained:

[0117]

[0118] Where AFE is the surface area of ​​the ferroelectric capacitor. For the F2 function, the dp / dt in the LK model formula of Fecap can be transformed:

[0119]

[0120] Merge the admittance matrix G and RHS. Since the F1 function and F2 function are known, we can get an MNA model of Fecap with exact parameters.

[0121] like Figure 3 As shown:

[0122] Simply write the corresponding matrix parameters into the modeling code specified by SPICE to obtain the built-in MNA model of Fecap in SPICE.

Claims

1. A ferroelectric capacitor modeling method for circuit simulation, characterized in that: The steps include: Step 1: Introduce the modified nodal analysis (MNA) into the MNA model of Fecap; Step 2: Based on MNA, the compensation equation is obtained using the LK model formula of Fecap, which is used to solve the equation of the unknown variable to obtain the final admittance matrix G and the known quantity vector column RHS of the Fecap element. The admittance matrix G and the known quantity column RHS are combined to obtain the MNA model of Fecap.

2. A ferroelectric capacitor modeling method for circuit simulation according to claim 1, characterized in that: In step 1, the specific process of linking MNA with the MNA model of Fecap is as follows: Kirchhoff's current law (KCL), Kirchhoff's voltage law (KVL) and the characteristic equation of the element are used to solve the unknown variables and obtain the element's admittance matrix G, variable column X and known quantity column RHS. The admittance matrix G and the known quantity column RHS are combined to form the MNA model of the element. Each addition of an unknown variable will generate a new row and column in the admittance matrix G, and at the same time, the corresponding unknown quantity will be generated at the corresponding position of the variable column X.

3. A ferroelectric capacitor modeling method for circuit simulation according to claim 2, characterized in that: Combined with the physical properties of ferroelectric capacitors (Fecap), when ferroelectric capacitors are introduced into the circuit, the polarization state and current of the ferroelectric capacitors are new unknown quantities, and a 4X4 admittance matrix G and a 1X4 variable column X(VN + (t), VN - (t), i(t), p(t)) T And the 1X4 RHS column, then merge the admittance matrix G and the known quantity column RHS to obtain the MNA model of FECAP.

4. The ferroelectric capacitor modeling method for circuit simulation according to claim 1, characterized in that: The step 2 is specifically as follows: 2.1): Based on the generalization of Fecap's LK model, the general formula of Fecap model is obtained; 2.2): The compensation equation of Fecap is linearized by using the backward Euler method and the NR iteration method to solve the equation of unknown variables. Based on MNA, the compensation equation of Fecap is linearized to meet the solution requirements of the Spice simulation tool and form the final MNA model; 2.3): Combine the terminal voltage, polarization state and model formula of Fecap to obtain the specific expressions of functions F1 and F2, forming the final MNA model of Fecap.

5. A ferroelectric capacitor modeling method for circuit simulation according to claim 4, characterized in that: The step 2.1) is: Selecting the LK model, the relationship between polarization P and unit energy G is described by the following Landau free energy equation; Where E is the electric field, the LK model is: where ρ is the thermodynamic coefficient, T FE Represents the thickness of the ferroelectric layer, V = ET FE is the voltage on the ferroelectric layer, and α, β, γ are coefficients related to material properties. Combining the above analysis of the LK model, and according to the terminal voltage and polarization state of the Fecap, the general formula of the Fecap model is listed: V(t)=F1(I(t),P(t)) (1) The current I and polarization P flowing through Fecap are the internal state variables of Fecap, V represents the terminal voltage, and F1 and F2 represent arbitrary functions; Based on MNA, equations (1) and (2) for solving the unknown variables of Fecap have been obtained. Since the current variables and polarization variables are introduced into the column vector X, the corresponding rows and columns will be generated when generating the admittance matrix G.

6. A ferroelectric capacitor modeling method for circuit simulation according to claim 4, characterized in that: In 2.2), semiconductor devices in SPICE can generally be divided into two categories, namely linear devices and nonlinear devices; The linearity and nonlinearity of a device depend on the relationship between the terminal voltage of the device and the current flowing through the device. For nonlinear circuits containing nonlinear elements, their model formulas need to be linearized. The general formula of Fecap model is linearized using Newton-Raphson iterative method and backward Euler method.

7. A ferroelectric capacitor modeling method for circuit simulation according to claim 6, characterized in that: Specifically: 1) First, the terminal voltage is represented by the node voltage; 2) Using the NR iteration method, at the working point (I (n) , P (n) ) is used to perform Taylor first-order expansion of the F1 function, and we can obtain: Among them, F1 (n) (t) = F1(I (n) (t), P (n) (t)), in order to extract matrix elements conveniently, equation (3) is further transformed Among them, VN+, VN-, I(t), and P(t) are variables to be solved and will be placed in the X column vector. Their corresponding coefficients will be used as elements of the admittance matrix. All the relevant terms of the n iterations on the right will be placed in the RHS column vector as known quantities. For the subsequent model description, the known terms are equivalent to Irhs For equation (2), dP / dt is linearized using the backward Euler method, and the F2 function is processed in the same way as the F1 function. At the working point (V (n) (t), P (n) (t)) is expanded by the first order Taylor, and we can get: Where h is the time step, F2 (n) (t) = F2(V (n) (t), P (n) (t)), V(t) = V N+ (t)-V N- (t), and then transform equation (5) Extract variables I(t), P(t), V N+ (t), V N- The coefficients before (t) are the elements of G. Similarly, the known terms are equivalent to Prhs and placed in the RHS column vector For the flow through node V N+ and V N- The current flows into the node V N+ is in the positive direction, then V N+： I(t) vs. V N-： -I(t), the only variable is I(t), the coefficients are 1 and -1, and no known terms are provided.

8. A ferroelectric capacitor modeling method for circuit simulation according to claim 7, characterized in that: According to the above description, the coefficients before the unknown variables are extracted to form the admittance matrix G: The column vector X formed by the unknown variables: The column vector RHS is formed by the known quantities: So far, the admittance matrix G and RHS of the synthetic Fecap MNA model have been obtained.

9. A ferroelectric capacitor modeling method for circuit simulation according to claim 4, characterized in that: The step 2.3) is specifically as follows: For the F1 function, the LK model formula based on Fecap can be obtained: Where AFE is the surface area of ​​the ferroelectric capacitor. For the F2 function, the dp / dt in the LK model formula of Fecap can be transformed: Combine the admittance matrix G and RHS to obtain the MNA model of Fecap; Simply write the corresponding matrix parameters into the modeling code specified by SPICE to obtain the built-in MNA model of Fecap in SPICE.

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