A Ferroelectric Capacitor Modeling Method for Circuit Simulation

Abstract

The present invention discloses a ferroelectric capacitor modeling method for circuit simulation, comprising the following steps: Step 1: Modifying the nodal analysis method to link MNA with the MNA model of Fecap; Step 2: Forming the MNA model of the Fecap element, based on the MNA, using the Fecap L-K model formula to obtain a compensation equation, which is used to solve the equation for the unknown variable, to obtain the final Fecap element admittance matrix G and the known quantity vector column RHS, thereby obtaining the Fecap MNA model. Based on the MNA, the present invention uses the Fecap physical model formula and Kirchhoff's current law and Kirchhoff's voltage law to represent the unknown variables, forming the Fecap model code, and making the Fecap SPICE simulation more accurate and faster.

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 (FeCaps) and ferroelectric field-effect transistors (FeFETs) are emerging electronic components with unique electrical properties, such as non-volatile memory and circuit programmability. Furthermore, FeCap-based ferroelectric RAM (FeRAM) has begun to attract widespread attention in the industry due to its excellent performance and process compatibility. These characteristics make them potential candidates for new in-memory computing (CIM) applications that enable energy-efficient hardware acceleration.

[0003] With the advancement of process technology, 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 modified 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 existing technology proposes using MNA to implement the 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 simulation tools also lags behind the MNA model.

[0006] Therefore, the present invention addresses 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 uses 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), forming a Fecap model code, so that the SPICE simulation of Fecap is 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 includes the following steps:

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

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

[0012] Based on the modified nodal analysis (MNA) method, 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 the MNA model with the Fecap model is as follows:

[0014] Kirchhoff's current law (KCL), Kirchhoff's voltage law (KVL), and the component's characteristic equations are used to solve for unknown variables, also known as introducing compensation equations, and then obtaining the component's admittance matrix G, variable column X, and known quantity column RHS. The admittance matrix G and known quantity column RHS are combined to form the component's MNA model. 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 in 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 RHS column of 1X4, and 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 the Fecap model is obtained to facilitate subsequent linearization processing;

[0018] 2.2): The compensation equation of Fecap (a nonlinear device) is linearized 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;

[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 be obtained to obtain the LK model:

[0024]

[0025] Where ρ is the thermodynamic coefficient, T FE Indicates 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] Where the current I and polarization P flowing through the Fecap are the internal state variables of the 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 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.

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

[0031] The distinction between linear and nonlinear devices depends 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 the Fecap model is linearized using the Newton-Raphson iterative method and the 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) ) and perform Taylor first-order expansion of the F1 function at , we can get:

[0036]

[0037] Among them F1 (n) (t) = F1(I (n) (t), P (n) (t)), in order to extract matrix elements conveniently, the formula (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 formula (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 positive direction, assuming that the current flows into the node V N+ is in the positive direction, then V N+： I(t) and V N-： The -I(t) variable consists only of I(t), 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 RHS of the column vector 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 obtain the F1 and F2 functions so that the model can be written 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 changed to:

[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 uses a modified nodal analysis method (MNA) to implement a FecapMNA model that can be applied to industrial SPICE simulation tools. In step 2, based on MNA, the Fecap LK model formula is used to obtain a Fecap MNA model that can be analyzed and calculated by a 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 applications such as 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 the Fecap because it complies with the SPICE simulation solution principle of a single component. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

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

[0068] The present invention will be described in further detail below with reference to 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: Modify the nodal analysis method (MNA) to connect the MNA model with the Fecap 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 must first be described using a SPICE netlist file. The netlist file describes in code all the nodes contained in a circuit topology, as well as the branches formed between the nodes and the components on the branches.

[0074] After SPICE obtains the topology of the circuit by parsing the netlist, it will formulate an equation 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 equations in the form of matrices and vectors in a computer. That is, the coefficients before the unknown variables (node ​​voltages) are extracted to form a matrix Y, also known as the admittance matrix. The unknown variables are converted into column vectors V, and the remaining terms (known quantities) are converted into column vectors I. Therefore, the circuit analysis results are expressed as

[0075] Y*V=I(1.1)

[0076] Because each component in a circuit can describe the branch current based on the node voltage of the branch in which it is located, it can also be written in the same form as Equation (1.1). When the admittance matrices of each component are added at the same node, the same result as (1.1) is obtained. Therefore, the admittance matrix of each component is very important in the Spice simulation tool, which greatly reduces the time to generate the circuit admittance matrix during simulation. However, when the components in the circuit have unknown quantities that cannot be represented by node voltages, 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 it can introduce more unknowns and then use KCL, KVL, and the component's own characteristic equations to solve the unknown variables, also known as introducing compensation equations. This leads to the component's admittance matrix G, the variable column X, and the known column RHS. The combination of G and RHS is the MNA model of the component. It should be noted that each additional unknown variable will generate a new row and column in G, and the corresponding unknown quantity will be generated at the corresponding position in 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 the 4X4 admittance matrix G (see Appendix) is obtained. Figure 1 ), 1X4 variable column X(VN + (t), VN - (t), i(t), p(t)) T And the RHS column of 1X4 (attached 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 unknown variables, and obtain the final Fecap element admittance matrix G and the known quantity vector column RHS, and then obtain the MNA model of Fecap;

[0080] The step 2 is specifically as follows:

[0081] 2.1): The general formula of the Fecap model is obtained based on the generalization of the LK model of Fecap, which is convenient for subsequent linearization processing;

[0082] The main FeCap models currently available are the Preisach model and the Landau-Khalatnikov (LK) model. The LK model is chosen for illustration here because it has a clear physical meaning. The LK model accurately describes 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 be obtained to obtain the LK model:

[0085]

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

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

[0088]

[0089] Where I (current flowing through the Fecap) and P (polarization) are the internal state variables of the Fecap, V represents the terminal voltage, and F1 and F2 represent arbitrary functions. The specific LK model will be introduced in 2.2. To this end, based on the MNA, equations (1) and (2) for solving the unknown variables of the Fecap have been obtained. (Since the current variables and polarization variables are introduced in the column vector X, the corresponding rows and columns are 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): Using the backward Euler method and the NR iteration method to linearize the compensation equation of Fecap (nonlinear device) to solve the equation of unknown variables, the compensation equation of Fecap is linearized based on MNA 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 distinction between linear and nonlinear devices depends on the relationship between the device's terminal voltage and the current flowing through it. However, for nonlinear circuits containing nonlinear components, if the 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 component will be incorrect. Therefore, the Newton-Raphson iterative method and the backward Euler method are used to linearize the general formula of the Fecap model.

[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) ) and perform Taylor first-order expansion of the F1 function at , we can get:

[0095]

[0096] Among them F1 (n) (t) = F1(I (n) (t), P (n) (t)), in order to extract matrix elements conveniently, the formula (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 formula (2), 1) uses the backward Euler method to linearize dP / dt, 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:

[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 positive direction, assuming that the current flows into the node V N+ If the direction is positive, then V N+： I(t) and V N-： The -I(t) variable consists only of I(t), 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 RHS of the column vector 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 obtain the F1 and F2 functions so that the model can be written 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 into:

[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 following steps are included: Step 1: Introduce the modified nodal analysis method (MNA) into the MNAmodel modeling 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 Fecap element's admittance matrix G and the known quantity vector column RHS. The admittance matrix G and the known quantity column RHS are combined to obtain the Fecap's MNAmodel; In step 1, the specific process of connecting MNA with Fecap's MNAmodel modeling is as follows: Kirchhoff's current law KCL, Kirchhoff's voltage law KVL, and the element's own characteristic equation are used to solve the unknown variables, obtaining the element's admittance matrix G, variable column X, and known quantity column RHS. The admittance matrix G and known quantity column RHS are combined to form the element's MNAmodel. Each addition of an unknown variable generates a new row and column in the admittance matrix G, and at the same time, produces the corresponding unknown quantity at the corresponding position in the variable column X. The step 2 is specifically as follows: 2.1): Based on the generalization of the LK model of Fecap, the general formula of the Fecap model is obtained; 2.2): Using the backward Euler method and the NR iteration method to linearize the Fecap compensation equation for solving the equation of unknown variables, the Fecap compensation equation is linearized based on MNA 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 MNAmodel of Fecap.

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

3. The ferroelectric capacitor modeling method for circuit simulation according to claim 1, characterized in that: The step 2.1) is specifically as follows: 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 Indicates 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. Combined with the analysis of the LK model, and based on 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) Where the current I and polarization P flowing through the Fecap are the internal state variables of the 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.

4. A ferroelectric capacitor modeling method for circuit simulation according to claim 3, characterized in that: In 2.2), semiconductor devices in SPICE are generally divided into two categories, namely linear devices and nonlinear devices; The distinction between linear and nonlinear devices depends 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 the Fecap model is linearized using the Newton-Raphson iterative method and the backward Euler method.

5. The ferroelectric capacitor modeling method for circuit simulation according to claim 4, 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) ) and perform Taylor first-order expansion of the F1 function at , we can get: Among them F1 (n) (t) = F1(I (n) (t), P (n) (t)), in order to extract matrix elements conveniently, further transformation is made to formula (3): 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 formula (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 positive direction, assuming that the current flows into the node V N+ is in the positive direction, then V N+ :I(t) and V N- : -I(t), the only variable is I(t), the coefficients are 1 and -1, and no known terms are provided.

6. The ferroelectric capacitor modeling method for circuit simulation according to claim 5, 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 RHS of the column vector formed by the known quantities: So far, the admittance matrix G and RHS of the synthetic Fecap MNA model have been obtained.

7. The ferroelectric capacitor modeling method for circuit simulation according to claim 1, 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 into: Merge the admittance matrix G and RHS to obtain the MNAmodel of Fecap; Simply write the corresponding matrix parameters into the modeling code specified by SPICE to obtain the built-in MNAmodel of Fecap in SPICE.

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