A method and computer device for quickly extracting parasitic parameters of analog integrated circuits

Abstract

The present invention relates to the field of integrated circuit parameter extraction, and specifically to a method and computer device for rapid extraction of analog integrated circuit parasitic parameters, which realizes rapid extraction of analog integrated circuit parasitic parameters and improves the efficiency of parameter extraction. The technical scheme includes extracting the parameters of the interconnect structure by using a non-conformal single source surface integral equation regional decomposition method and performing partition optimization; extracting the parameters of the non-uniform and anisotropic medium structure by using a hybrid method of electromagnetic field calculus equations without boundary conditions; and rapidly solving the linear equation group in the parameter extraction process. The present invention is suitable for rapid extraction of analog integrated circuit parasitic parameters.

Description

Technical Field

[0001] The present invention relates to the field of integrated circuit parameter extraction, and in particular to a method for quickly extracting parasitic parameters of an analog integrated circuit and a computer device. Background Art

[0002] The extremely small space and high integration density of analog integrated circuits lead to a very complex electromagnetic environment. To ensure system performance, good signal integrity, power integrity, electromagnetic compatibility, and anti-electromagnetic interference are required. This requires accurate and rapid extraction of the parasitic parameters of the layout, such as parasitic resistance R, parasitic capacitance C, parasitic inductance L, and parasitic conductance G, for "post-simulation" analysis and verification.

[0003] Existing layout parasitic parameter extraction tools such as Star-RCXT, Ansys Q3D, Cadence Sigrity series, ADS Virtuoso, HFSS, etc. face problems such as large memory consumption and low computational efficiency. They are unable to effectively extract parasitic parameters of the entire layout and also face the following problems.

[0004] Difficulty in modeling high-frequency effects: The aspect ratio (aspect ratio: the ratio of radial length to cross-sectional dimension) of a single metal interconnect line in the layout is large, which can exceed 1000, making it difficult to model high-frequency skin effect and proximity effect;

[0005] Small scope of application: The medium in the layout is complex, with uniform, non-uniform and anisotropic media, which cannot be effectively processed by a single method;

[0006] Large-scale equation solving: dense interconnection structure, 1mm 2 The number of internal interconnects can reach millions, and the number of TSV and via units can reach tens of thousands, resulting in a huge scale of problem solving and unable to meet the demand for rapid extraction of layout parasitic parameters. Summary of the invention

[0007] The purpose of the present invention is to overcome the shortcomings of the prior art and provide a method and a computer device for quickly extracting parasitic parameters of an analog integrated circuit, thereby realizing the rapid extraction of parasitic parameters of the analog integrated circuit and improving the efficiency of parameter extraction.

[0008] The present invention adopts the following technical solutions to achieve the above-mentioned purpose. In a first aspect, the present invention provides a method for quickly extracting parasitic parameters of an analog integrated circuit, comprising:

[0009] S1. The parameters of the interconnect structure are extracted by using the non-conformal single source surface integral equation regional decomposition method, and then partition optimization is performed;

[0010] S2, using the hybrid method of electromagnetic field calculus equations without boundary conditions to extract parameters of inhomogeneous and anisotropic medium structures;

[0011] S3, quickly solve the linear equations in the process of S1 and S2 parameter extraction.

[0012] Furthermore, the parameters of the interconnect structure are extracted by adopting the non-conformal single source surface integral equation region decomposition method, which specifically includes:

[0013] The interconnected structure is physically partitioned according to the set size to form multiple unit structures. A separate surface mesh is performed on each partition. The single-source integral equation equivalent method is used for each partition to construct a single current physical equivalent model of the unit structure. Then, the equivalent models of all unit structures are combined into an equivalent model of the original structure, and the equivalent current density on their surfaces is calculated by summing. Finally, the distributed parameters of the interconnected structure are extracted by combining the electromagnetic field equations of the area outside the structure.

[0014] Furthermore, the single-source integral equation equivalent method includes adopting the boundary condition that the surface electric field in the original problem and the equivalent problem is equal, and obtaining a single-source surface integral operator with only current density. The interconnection structure is a high-conductivity metal for signal transmission and interference coupling. At high frequencies, there is a skin effect and a proximity effect. The vector electric field on the inner surface of the metal is expressed as:

[0015]

[0016] in is the electric field strength, is the magnetic field intensity, n is the unit vector at the observation point, pointing to the inside of the conductor, j represents the imaginary part, ω and μ are the angular frequency and the magnetic permeability of the material, respectively, and the integral operators P and Q are:

[0017]

[0018] where G is the Green's function, is an arbitrary vector, k is the wave number, and the equations of the integral operators P and Q are discretized and tested using the RWG basis function. Then the admittance matrix Y is H=YE, where H and E are the column vectors of the expansion coefficients of the tangential magnetic field and electric field components, respectively. Then, based on the equivalent principle, the metal structure is replaced by the background medium. In order to keep the field outside the structure the same as the original problem, the surface equivalent current is introduced on the boundary of the original conductor. Where H′ is the column vector of the boundary tangential magnetic field expansion coefficients of the equivalent problem. By analogy with the original problem, for the equivalent problem we get but where Y s =Y′-Y is the differential surface admittance operator, which expresses the equivalent current density on the conductor surface as the surface tangential electric field.

[0019] Furthermore, partition optimization specifically includes:

[0020] According to the characteristics of the interconnection structure, the analog integrated circuit layout structure is divided into two types of structures. The first type is the structure that can be divided into the same units, and the second type is other irregular structures. The first type of structure is initially partitioned into units of the same size by type, and different types of structures are partitioned using the same partitioning unit. The equivalent current density only needs to be calculated once in the same partition. The second type of structure is divided into multiple types of different regions according to the structural characteristics, and then the numerical relationship between the calculation efficiency and the partition size is statistically calculated based on the unknown quantities in the partition and the number of partitions. The global partition size and number are optimized and iterated based on this relationship to achieve the optimal partition.

[0021] Furthermore, S2 specifically includes:

[0022] A hybrid model of electromagnetic field vector wave equation and single-source integral equation is established. The model divides the calculation area into vector wave equation and single-source integral equation representation areas, namely, the discretized finite element method and moment method calculation areas. The integral area includes various complex structures that can be processed by the surface integral equation, including interconnect structures, vias and TSV arrays, and isotropic uniform medium areas in the layout; the differential area includes non-uniform media, anisotropic media, and background medium areas replaced by the integral equivalent model.

[0023] The method for establishing a hybrid model of electromagnetic field vector wave equation and single-source integral equation includes: based on the equivalence principle, the structure of the region represented by the electromagnetic field integral equation is equivalent to the single-source current density, the original structure is replaced by the background uniform medium, so as to construct a single-source equivalent current model of the structure, and then the vector wave equation is used to uniformly represent the region and other non-uniform or anisotropic media. In this process, the current is coupled into the vector wave equation as an additional current source excitation, so as to form a complete calculus electromagnetic field representation of the calculation region;

[0024] The electromagnetic field integral equivalent region processing methods include: for a single complex structure, a single source integral equation equivalent method is used to characterize it;

[0025] For partial contact structures, the first processing method regards the partial contact structure as two calculation areas with different medium parameters, and uses the non-conformal single-source surface integral equation regional decomposition method for equivalence. The second processing method uses the electric field and magnetic field tangential boundary conditions at the boundary before using the equivalent principle to eliminate the additional unknowns of the internal common edge, and finally applies the equivalent principle for equivalent replacement to obtain the equivalent current of the outermost layer, thereby realizing the single-source equivalent description of the structure. The first processing method or the second processing method is selected based on the computational efficiency and memory consumption.

[0026] For multi-layer embedded structures, the first processing method considers the calculation area as a problem of multiple dielectric regions, and uses the non-conformal single-source surface integral equation regional decomposition method for equivalence. The second processing method first considers the innermost structure and uses the single-source integral equation to characterize it. The innermost dielectric in the structure is replaced by the surrounding dielectric, and the equivalent current is added at the same time. Then, the integral equation is used to process the situation containing the equivalent current density, which is extrapolated to the second dielectric interface to obtain the equivalent current on the interface, until it is recursively performed from the inner boundary to the outer boundary layer by layer to the outermost layer, and the equivalent current density distribution at the outermost boundary is obtained; and the first processing method or the second processing method is selected based on the calculation efficiency and memory consumption;

[0027] The electromagnetic field differential region processing method includes: according to the superposition of electromagnetic fields, an additional equivalent current density is introduced into the vector wave equation, and the field equations in the outer area of ​​the conductor and on the equivalent boundary are corrected to obtain:

[0028]

[0029] Where E represents the electric field in the differential region, J eq is the surface equivalent current density formed in the integral area, and J is the external current source that may exist in the external area.

[0030] Furthermore, S2 specifically also includes selecting corresponding basis functions to discretize the vector electric field, specifically including using the first-order linear unit of the finite element on its boundary component synthesis in the case of two-dimensional TM waves, which is fully compatible with the integral equation using triangular basis functions, and constructs the final linear equation group.

[0031] Furthermore, S2 specifically also includes accurate calculation of matrix elements of linear equations, including expanding the vector electric field in space according to the selected basis functions, and accurately calculating the elements of the linear equations after performing the Galerkin test, mainly including the calculation of two types of elements: the first type is the vector differential finite element part, which uses the finite element method to derive its analytical or efficient numerical calculation method; the second type is the moment method integration part, which generates new vector basis functions. This part uses the singular value elimination and singular value reduction methods to accurately calculate the integral value of the Green's function, and finally uses it for matrix assembly to form the linear equations that need to be solved.

[0032] Furthermore, S3 specifically includes efficient TT data storage and error control of linear matrices, and the efficient TT data storage of linear matrices includes constructing a Toeplitz interaction matrix of any integrated circuit package structure and compressing the constructed coefficient matrix using the TT method;

[0033] Constructing the Toeplitz interaction matrix of any integrated circuit package structure includes using a uniform grid in space, and then using a projection method to project its effects onto uniform grid points to construct the corresponding Toeplitz interaction matrix; using the TT method to compress the constructed coefficient matrix includes compressing the generator matrix of the Toeplitz interaction matrix, and compressing the coefficient matrix using the following formula,

[0034] Error control includes setting the upper limit of error to control the accuracy of compressed data. The setting method is as follows: err = ||AT|| F , A represents the Toeplitz interaction matrix before compression, T represents the TT compression matrix, and F represents the Frobenius distance.

[0035] In a second aspect, the present invention provides a computer device, including a memory, wherein the memory stores program instructions, and when the program instructions are executed, the method for quickly extracting parasitic parameters of analog integrated circuits as described above is executed.

[0036] The beneficial effects of the present invention are:

[0037] The present invention extracts the parameters of the interconnected structure by adopting the non-conformal single-source surface integral equation regional decomposition method and performs partition optimization; adopts the hybrid method of electromagnetic field calculus equation without boundary conditions to extract the parameters of the non-uniform and anisotropic medium structure; finally, the linear equations in the parameter extraction process are quickly solved, thereby improving the efficiency of parameter extraction.

[0038] The non-conformal single-source surface integral equation regional decomposition method suitable for simulating complex interconnect structures of integrated circuits provided by the present invention is suitable for extracting parasitic parameters of complex structures such as large aspect ratio interconnects, multiple through holes and TSV arrays in the layout, and has the characteristics of non-conformal, simplicity, high efficiency and versatility; the partition size is optimized, and the efficiency of subsequent simulation analysis is improved.

[0039] The hybrid method of electromagnetic field calculus without boundary conditions applicable to non-uniform and anisotropic medium structures provided by the present invention mainly includes establishing a hybrid model of electromagnetic field vector wave equation and single-source integral equation, selecting corresponding basis functions to discretize the vector electric field, and accurately calculating the matrix elements of the linear equation group, thereby improving the subsequent simulation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 is a flow chart of a method for rapidly extracting parasitic parameters of an analog integrated circuit provided by an embodiment of the present invention;

[0041] Figure 2 It is a schematic diagram of the partition equivalent process of the single-source non-conformal integral equation provided by an embodiment of the present invention;

[0042] Figure 3 is a partition optimization flow chart provided by an embodiment of the present invention;

[0043] Figure 4 is a schematic diagram of a multi-region hybrid method provided by an embodiment of the present invention;

[0044] Figure 5 It is a flow chart of a hybrid method for calculus equations without boundary conditions provided by an embodiment of the present invention;

[0045] Figure 6 Schematic diagram of a typical interconnect structure in an analog integrated circuit provided by an embodiment of the present invention, (a) represents a single electromagnetically penetrable structure, (b) represents a partial contact structure, and (c) represents a multi-layer buried structure;

[0046] Figure 7 Schematic diagram of establishing a Toeplitz matrix using a uniform spatial grid according to an embodiment of the present invention, wherein (a) represents a two-dimensional situation and (b) represents a three-dimensional situation;

[0047] Figure 8 It is a schematic diagram of a multi-layer structure equivalent model establishment process provided by an embodiment of the present invention;

[0048] Fig. 9 It is a schematic diagram of the decomposition of the SIE region and the PDE region provided by an embodiment of the present invention;

[0049] Fig.10 is a schematic diagram of loop analysis in a circuit provided by an embodiment of the present invention;

[0050] Fig.11 It is a schematic diagram of large-scale interconnect line parameter extraction provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0051] To make the purpose, technical solution and advantages of the embodiments of the present invention more clear, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention.

[0052] like Figure 1 As shown, the present invention aims at solving the difficult problems of analog integrated circuit layout parameters. Based on the current research work on rapid extraction of parasitic parameters, the present invention adopts a three-dimensional non-conformal single-source surface integral equation regional decomposition method, a hybrid method of electromagnetic field calculus equations without boundary conditions, and a linear / sublinear solution method for ultra-large-scale equations, and finally realizes rapid extraction of parasitic parameters of analog integrated circuit layout for rapid post-simulation analysis and verification.

[0053] The present invention provides a non-conformal single-source surface integral equation regional decomposition method suitable for simulating complex interconnect structures of integrated circuits. The method is suitable for extracting parasitic parameters of complex structures such as large aspect ratio interconnects, multiple through holes and TSV arrays in a layout, and has the characteristics of non-conformal, simplicity, high efficiency and versatility.

[0054] The present invention provides a boundary-free electromagnetic field calculus hybrid method applicable to non-uniform and anisotropic medium structures, which mainly includes: 1. a physical model of boundary-free coupling of electromagnetic field vector wave equation and single-source integral method, and a calculus hybrid representation mathematical model of space electromagnetic field; 2. basis functions applicable to the discretization of calculus hybrid equations and selection criteria, such as RWG basis functions, Whitney vector basis functions, etc.; 3. an accurate calculation method of matrix elements of a linear equation system constructed by the boundary-free calculus hybrid method, such as an analytical calculation method of differential equation elements, an integral equation singular value processing method, etc.

[0055] The linear / sublinear fast solution method for a very large-scale linear equation group provided by the present invention solves a very large-scale linear equation group formed by a regional decomposition method for a non-conformal single-source surface integral equation and a hybrid method for an electromagnetic field calculus equation without boundary conditions, and mainly includes: 1. constructing a TT method data efficient storage method according to data formation characteristics; 2. an error control method based on the fast solution of the TT method.

[0056] The specific plan is as follows:

[0057] (1) Domain decomposition method for non-conformal single-source surface integral equation

[0058] The present invention aims at the rapid extraction of parameters of interconnect structures such as large aspect ratio interconnects, multi-layer planar media, through holes and TSV arrays in analog integrated circuits, and provides a non-conformal single-source surface integral method regional decomposition method, which mainly includes: a non-conformal single-source surface integral equation regional decomposition method and a partition optimization criterion.

[0059] ①Domain decomposition method for non-conformal single source surface integral equation

[0060] The non-conformal single-source integral equation domain decomposition method provided by the present invention is based on the Schelkunoff equivalence principle, and adopts the boundary condition that the surface electric field in the original problem and the equivalent problem is equal, and obtains a single-source surface integral operator with only current density. Since the interconnection structure is a high-conductivity metal for signal transmission and interference coupling, there are skin effects and proximity effects at high frequencies, so it cannot be simply treated as an ideal conductor, but must be considered as a high-loss penetrable material. The vector electric field on the inner surface of the metal is described using the Stratton-Chu integral equation:

[0061]

[0062] in is the electric field strength, is the magnetic field intensity, n is the unit vector at the observation point, pointing to the inside of the conductor, j represents the imaginary part, ω and μ are the angular frequency and the magnetic permeability of the material, respectively, and the linear integral operators P and Q are:

[0063]

[0064] where G is the Green's function, is an arbitrary vector, k is the wave number. The equation is discretized and tested using the RWG basis function, and the admittance matrix Y is:

[0065] H=YE, where H and E are the expansion coefficient column vectors of the tangential magnetic field and electric field components, respectively. Then, based on the equivalent principle, the metal structure is replaced by the background medium. In order to keep the field outside the structure the same as the original problem, the surface equivalent current is introduced on the boundary of the original conductor.

[0066]

[0067] Where H′ is the column vector of the boundary tangential magnetic field expansion coefficients of the equivalent problem. By analogy with the original problem, for the equivalent problem we can get but where Y s =Y′-Y is the differential surface admittance operator. It expresses the equivalent current density on the conductor surface as the surface tangential electric field, and establishes the mathematical relationship between the vector electric field and the equivalent current density. The above single-source integration process is the formation process of the linear equation system under non-conformal conditions.

[0068] Metal interconnects are mostly structures with large aspect ratios (>1000), vias, and TSV arrays. If the single-source integration method mentioned above is used, although the number of unknowns can be reduced by half compared to the double-source integration equation, it is difficult to construct Y s The present invention uses the non-conformal single source surface integral equation regional decomposition technology to effectively accelerate Y s The construction of this method overcomes the low efficiency problem.

[0069] like Figure 2As shown in the figure, the main scheme includes first physically partitioning the interconnected structure into a certain size to form multiple unit structures; then performing a separate surface meshing for each partition. Since the meshing of each partition is performed independently in this process, it can greatly increase the flexibility of solving the problem. Further, the single-source integral equation equivalent method as described above is used for each partition to construct a single current physical equivalent model of the unit structure; then all the unit equivalent models are combined into an equivalent model of the original structure, and the equivalent current density on its surface is calculated by summing up, and finally, the electromagnetic field equations of the area outside the structure are combined to extract the distributed parameters of the interconnected structure to achieve effective electromagnetic field simulation analysis.

[0070] The single-source integral equation method adopted by the present invention can effectively utilize the idea of ​​regional decomposition, decomposing the originally large and fine interconnection structure into a combination of multiple small units, which can greatly improve the efficiency of calculating the single-source equivalent current density in a single area. In addition, since the interconnection structure is mostly radially identical square conductors, vias or TSV arrays, the same regional unit partition can be used. In this way, the equivalent current density of the same unit only needs to be calculated once, which can further improve the calculation efficiency.

[0071] ②Partition optimization criteria

[0072] The regional partitioning will affect the efficiency of the above methods. Although the more structural partitions there are, the fewer unknowns in a single calculation area, and the less time it takes to construct a single-source equivalent model. However, additional unknowns are introduced at the boundary, and the final matrix equation is larger, which greatly reduces the efficiency of the algorithm. It can be seen that the size and number of partitions need to be balanced to ensure that the algorithm has a high efficiency.

[0073] The present invention adopts the following scheme to perform partition optimization. Figure 3 As shown in the figure, first of all, based on the characteristics of the interconnect structure, it is divided into two categories: structures that can be divided into the same unit (such as interconnect lines with constant cross-sections, TSV and via arrays, etc.) and other irregular structures (such as irregular interconnect lines, vias with different apertures, etc.). The first type of structure is initially partitioned into units of the same size by type, and different types of structures such as vias and TSVs are partitioned using the same partition unit idea. The equivalent current density only needs to be calculated once in the same partition. The second type of structure is divided into multiple types of different regions based on structural characteristics. Then, based on the unknown quantity in the partition and the number of partitions, the numerical relationship between the calculation efficiency and the partition size is statistically analyzed, and the global partition size and number are optimized and iterated based on this relationship to achieve the optimal partition.

[0074] (2) Hybrid method for electromagnetic field calculus equations without boundary conditions

[0075] The interconnection structure in the analog integrated circuit layout includes: metal interconnection structure, dielectric material (may be uniform isotropic, non-uniform or anisotropic medium). According to the equivalence principle, when the medium is non-uniform or anisotropic, it is necessary to use volume integral equations, which are relatively inefficient and difficult to apply to the rapid analysis of the entire layout of the analog integrated circuit. Since the finite element method based on differential equations can effectively handle non-uniform and anisotropic media, the present invention studies the hybrid method of full three-dimensional calculus equations without boundary condition constraints, and is used for the rapid extraction of the interconnection structure parameters of the analog integrated circuit. The specific scheme is described in detail as follows.

[0076] ① Hybrid physical model of electromagnetic field vector wave equation and single source integral equation

[0077] The hybrid physical model provided by the present invention divides the calculation area into the vector wave equation and the single source integral equation representation area, that is, the discrete finite element method and moment method calculation area. Figure 4 As shown in the figure, the integral area includes: various complex structures that can be handled by the surface integral equation, such as interconnect structures in the layout, via and TSV arrays, and isotropic uniform medium areas; the differential area includes: non-uniform media, anisotropic media, and the background medium area replaced by the integral equivalent model.

[0078] The basic method of the electromagnetic field vector wave equation and integral equation hybrid model established by the present invention is as follows: Figure 5 As shown in the figure, it mainly includes: based on the equivalent principle, the structure of the electromagnetic field integral equation characterization area is equivalent to a single source current density, and the original structure is replaced by the background uniform medium, so as to construct a single source equivalent current model of the structure. Then the vector wave equation is used to uniformly characterize the area and other inhomogeneous or anisotropic media. In this process, the current is coupled into the vector wave equation as an additional current source excitation, thereby forming a complete calculus electromagnetic field characterization of the calculation area. It can be seen that in this process, the boundary-free coupling of integral and differential equations is realized through the equivalent current density.

[0079] Electromagnetic field integral equivalent region processing method:

[0080] The key point for the processing of integral equivalent regions is to establish a single-source equivalent model that may appear in the interconnection structure. Figure 6 As shown, the layout may include the following three main structural forms: a single penetrable structure, a partial contact structure, and a multi-layer buried structure.

[0081] For a single complex structure, the single source integral equation method is used for characterization.

[0082] For partial contact structures, the first method is to regard the partial contact structure as two calculation areas with different dielectric parameters, and use the non-conformal single-source integral equation regional decomposition method in research content 1 for equivalence. The second method is to use the electric field and magnetic field tangential boundary conditions at the boundary before using the equivalent principle to eliminate the additional unknowns of the internal common edge, and finally apply the equivalent principle for equivalent replacement to obtain the equivalent current of the outermost layer, so as to achieve a single-source equivalent description of the structure.

[0083] The advantage of the first method is that it can universally handle various structures and the method is simple; the disadvantage is that unknown quantities are retained on the common boundary. The advantage of the second method is that the unknown quantities only exist at the outermost boundary of the structure, and the number of unknown quantities is small; the disadvantage is that it has poor versatility, requires more complex algebraic operations, and there are more matrix operations when constructing the surface equivalent current, and the calculation efficiency is low. The present invention will comprehensively compare the two equivalent methods in terms of calculation time, memory consumption, etc., and select the best one.

[0084] For multi-layer embedded structures, the first method considers the calculation area as a problem of multiple dielectric regions, and uses single-source non-conformal region decomposition technology to perform current source single-source equivalence. The second method is to first consider the innermost structure and characterize it using a single-source integral equation. The innermost dielectric in this structure is replaced by the surrounding dielectric, and an equivalent current is added at the same time. Its function is to describe the electromagnetic interaction between the internal structure and the adjacent structure in the original structure. Integral equations are further used to deal with situations containing equivalent current density, which are extrapolated to the second dielectric interface to obtain the equivalent current on the interface. By analogy, from the inner boundary to the outer boundary, layer by layer, recursively to the outermost layer, the equivalent current density distribution at the outermost boundary can be obtained.

[0085] The advantage of the first method is that it can handle various structures in a universal manner and the method is simple; the disadvantage is that unknown quantities will be retained at common boundaries. The advantage of the second method is that the unknown quantities only exist at the outermost boundaries of the structure, the number of unknown quantities is relatively small, and due to the use of iterative recursive derivation, the versatility is better; the disadvantage is that there are certain matrix operations when constructing the surface equivalent current, and the computational efficiency of the modeling process for larger structures is low. As above, the present invention will comprehensively compare the advantages and disadvantages of the two methods in terms of computational efficiency, memory consumption, etc., and select the appropriate method.

[0086] In summary, there are two results after the final integral area is equivalent: the first is that the original complex structure is replaced by a uniform background medium, forming an area with only a simple background medium; the second is the equivalent current introduced on the boundary of the original structure (outermost layer), which can be expressed using the following formula: It means that this equivalent current will be coupled into the vector wave equation as an additional excitation.

[0087] Electromagnetic field differential area processing method:

[0088] Basic method of electromagnetic field characterization in differential region: Based on the superposition of electromagnetic fields, an additional equivalent current density is introduced into the vector wave equation, and the field equations in the external area of ​​the conductor and on the equivalent boundary are corrected to obtain:

[0089]

[0090] Where E represents the electric field in the differential region, J eq is the surface equivalent current density formed in the integral area, and J is the external current source that may exist in the external area.

[0091] ② Basis function selection conditions and related criteria

[0092] For the hybrid method of calculus equations proposed in the present invention, it is necessary to select appropriate basis functions to discretize the vector electric field. The electric field in the vector wave equation is distributed in a three-dimensional space, and a vector basis function that can be described in three-dimensional space is required; and the equivalent current is distributed on an equivalent surface, and a vector basis function that can be described on the surface is required. The hybrid method of electromagnetic field calculus proposed in the present invention is that under specific excitation conditions, the unknown quantity is the electric field component, so the quantities expanded are all electric fields, and then appropriate basis functions are selected to construct the corresponding linear algebraic equations.

[0093] The present invention adopts the component synthesis of the first-order linear unit of the finite element on its boundary in the case of two-dimensional TM waves (transverse magnetic waves), which is completely compatible with the use of triangular basis functions in integral equations, and can construct the final linear equation system, and finally realize the research idea of ​​problem solving, and further study the common basis functions (such as linear basis functions, high-order stacked basis functions) in the vector finite element method and the common basis functions (such as RWG basis functions, high-order basis functions) of the moment method in the case of three-dimensionality, and the applicability of the calculus mixed equations in the discrete present invention. In the present invention, the vector basis function can be divided into curl conformal and divergence conformal, which correspond to the Whitney basis function of the vector finite element and the RWG basis function of the moment method, etc. Their theoretical relationship is the relationship between vectors taking curl, and the present invention uses this relationship to further screen out basis functions suitable for discrete calculus equations.

[0094] ③ Accurate calculation method of matrix elements of linear equations

[0095] According to the selected basis function, the vector electric field is expanded in space. After the Galerkin test is performed, the elements of the linear equation need to be accurately calculated, which mainly includes the calculation of two types of elements: the first is the vector differential finite element part, which uses the finite element method to derive its analytical or efficient numerical calculation method. The second is the moment method integration part, which will generate new vector basis functions. Due to the singularity of the Green's function, it is necessary to accurately handle the numerical singularity of the Green's function in the new basis function. This part uses classic methods such as singular value elimination and singular value reduction to accurately calculate the integral value of the Green's function. Finally, it is used for matrix assembly to form a linear equation system that needs to be solved.

[0096] (3) Linear / sublinear fast solution method for ultra-large-scale linear equations

[0097] The present invention focuses on the fast solution method of the ultra-large-scale linear equation group formed by the hybrid method of the regional decomposition method of the non-conformal single-source surface integral equation and the electromagnetic field calculus equation without boundary conditions, mainly including an efficient TT storage method and an error control method of the efficient solution method in the TT data format.

[0098] ① Efficient TT data storage method for linear matrices

[0099] FFT acceleration algorithms such as pFFT and AIM require O(NlogN) memory to store the Toeplitz matrix, but TT decomposition is to decompose the matrix into the product of a series of small matrices. Theoretically, the computational complexity and memory complexity can be O(logN), which can greatly improve the computational efficiency of electromagnetic simulation of packaging. The focus of this section is: constructing the Toeplitz interaction matrix of any integrated circuit packaging structure; using the TT method to compress the constructed coefficient matrix.

[0100] Construct the Toeplitz interaction matrix for any integrated circuit package structure:

[0101] For any interconnect structure, in FFT acceleration methods such as pFFT and AIM, it is also necessary to construct the Toeplitz interaction matrix. Figure 7 As shown, its basic method is to use uniform grid points in space to construct a Toeplitz interaction coefficient matrix. First, uniform spatial nodes are established in space, and then the effects of its functions are projected onto the uniform nodes, and the interactions between its nodes are Toeplitz matrices. The present invention adopts a similar method, using a uniform grid in space, and then using a projection method to project its effects onto uniform grid points, and a suitable Toeplitz interaction coefficient matrix can be further constructed.

[0102] The coefficient matrix constructed using the TT method is compressed:

[0103] In order to achieve a breakthrough in computational complexity from O(NlogN) to O(logN), the main thing is to compress the generator matrix of the Toeplitz matrix, that is, the unique element in the Toeplitz matrix. Use the following formula to compress the coefficient matrix, and you can get:

[0104]

[0105] like Figure 8 As shown, this decomposition compression can be optionally calculated through standard decomposition (such as SVD, QR, etc.). The TT decomposition can be written in tensor form, which can greatly reduce the storage space and the operations used for calculation, and is used for rapid extraction of parasitic parameters of simulated integrated circuit layouts.

[0106] ② Error control method

[0107] When using the TT compression matrix, the matrix elements need to be compressed to a certain extent to achieve data compression. In the calculation process, the present invention needs to consider how to define its error upper limit to achieve controllable accuracy of compressed data, and consider the following definition:

[0108] err=||AT|| F

[0109] The currently selectable upper limit of the error of the matrix elements is the Frobenius distance of the matrix and its approximate matrix, where A represents the Toeplitz interaction matrix before compression, T represents the TT compression matrix, and F represents the Frobenius distance. The present invention will use series expansion to derive the definition of the error in the above formula to achieve controllable accuracy of the algorithm.

[0110] The present invention can achieve the following technical effects.

[0111] The present invention adopts a hybrid method based on surface integral equation-partial differential equation (SIE-PDE), takes advantage of the method of moments and the finite element method, and can cope with various problems encountered in the modeling process of a single numerical method, thereby improving adaptability.

[0112] First, select a suitable boundary and decompose the entire calculation area into the SIE area and the PDE area. The two areas overlap with each other. The SIE area generally includes high conductivity media and complex structures, and the PDE area generally includes background media, inhomogeneous media, and anisotropic media. The regional decomposition diagram is as follows: Fig. 9As shown in the figure, for the SIE region, its single-source equivalent model is obtained based on the surface equivalent principle, and the equivalent model only contains the equivalent current density J distributed on the boundary of the SIE region; the PDE region includes the region after surface equivalent and the original background medium region, and the PDE region is modeled using the non-homogeneous Helmholtz equation. The equivalent current obtained by surface equivalent of the SIE region is added to the PDE region as an external excitation term to obtain the final SIE-PDE hybrid equation, and then the relevant electromagnetic parameters of the calculation region can be obtained.

[0113] Table 1 Comparison of cable array computing resources

[0114]

[0115] In terms of accuracy, the SIE-PDE hybrid method can accurately calculate the near-field and far-field results of homogeneous media, and shows good convergence characteristics as the grid is encrypted. As shown in Table 1, in terms of efficiency, the proposed non-conformal SIE-PDE hybrid method and conformal SIE-PDE hybrid method show significant efficiency improvements compared to the traditional finite element method. At the same time, since the non-conformal hybrid method uses a sparser grid division in the entire calculation area, the number of unknowns, memory consumption, and solution time are further reduced, and electromagnetic modeling for complex dielectric structures can be performed accurately and more efficiently.

[0116] The present invention adopts a single-source surface integral equation method based on equivalent theory for multi-layer embedded transmission line structures, which solves the problem of the large number of unknown quantities on the interface between layers and the huge size of the entire solution matrix in the traditional integral method, thereby improving the solution efficiency. The proposed method converts the unknown quantities on each interface to the outermost surface layer by layer by applying the equivalent principle from the inside to the outside, which is an algorithm implementation of the electromagnetic field equivalent principle. Since the entire matrix is ​​divided into several sub-arrays for solution, the solution efficiency will be greatly improved. The example results also verify the feasibility and efficiency of the proposed method. In addition, the proposed method is suitable for electromagnetic analysis of partially contacted penetrable and perfect electric conductor (PEC) objects. In this formula, a modular equivalent model of each penetrable object composed of a composite structure is first independently constructed, replaced by a background medium, regardless of whether it is surrounded by a background medium, other media or partially connected objects, and an equivalent current source is applied on the boundary to keep the field in the external area unchanged. Then, by combining all modular models and any possible PEC objects, an equivalent model of the composite structure can be derived, without the need for complex boundary handling techniques and supporting non-conformal meshes.

[0117] The present invention aims at the problem of fast extraction of integrated circuit parameters, such as Fig.10As shown in the figure, the surface integral equation under the assumption of magnetic quasistatics is proposed by adopting the loop analysis theory, which can efficiently and accurately simulate the interconnection structure of arbitrary shapes in the packaging circuit. By appropriately converting all electromagnetic quantities into circuit elements and carefully constructing independent and complete equations based on graph theory by loop analysis, the huge unknown quantities in the solution are effectively reduced. In addition, the matrix-vector product operation is accelerated by the pre-corrected fast Fourier transform method. Four examples are given, including rectangular metal interconnects, bonding wire arrays, an actual complex circuit structure, and the power distribution network structure used in the package to verify the accuracy, efficiency and scalability of the proposed method. The results show that the proposed method is accurate and has great application prospects and value in the field of integrated circuit packaging. In addition, in order to facilitate the addition of current ports, we proposed an equation based on cut set analysis. By equating discrete matrix elements to circuit elements, charge conservation conditions can be conveniently added. The concept of super nodes is used to handle ports, which can add current ports and effectively reduce the number of port unknowns. As shown in Fig.11 As shown, this is a schematic diagram of large-scale interconnect line parameter extraction.

[0118] Table 2 Simulation configuration and result display table

[0119]

[0120] The above is only a preferred embodiment of the present invention. It should be understood that the present invention is not limited to the form disclosed herein, and should not be regarded as excluding other embodiments, but can be used in various other combinations, modifications and environments, and can be modified within the scope of the concept described herein through the above teachings or the technology or knowledge of the relevant field. The changes and modifications made by those skilled in the art shall not deviate from the spirit and scope of the present invention, and shall be within the scope of protection of the claims attached to the present invention.

Claims

1. A method for quickly extracting parasitic parameters of an analog integrated circuit, characterized in that: include: S1. The parameters of the interconnect structure are extracted by using the non-conformal single source surface integral equation regional decomposition method, and then partition optimization is performed; S2, using the hybrid method of electromagnetic field calculus equations without boundary conditions to extract parameters of inhomogeneous and anisotropic medium structures; S3, quickly solve the linear equations in the process of extracting parameters of S1 and S2; The parameters of the interconnect structure are extracted by using the non-conformal single source surface integral equation region decomposition method, including: The interconnection structure is physically partitioned according to the set size to form multiple unit structures, and each partition is meshed separately. The single-source integral equation equivalent method is used for each partition to construct a single current physical equivalent model of the unit structure. Then, the equivalent models of all unit structures are combined into an equivalent model of the original structure, and the equivalent current density on the surface is calculated by summing. Finally, the distributed parameters of the interconnection structure are extracted by combining the electromagnetic field equations of the area outside the structure. Partition optimization specifically includes: According to the characteristics of the interconnection structure, the analog integrated circuit layout structure is divided into two types of structures. The first type is the structure that can be divided into the same units, and the second type is other irregular structures. The first type of structure is initially partitioned into units of the same size by type, and different types of structures are partitioned using the same partitioning unit. The equivalent current density only needs to be calculated once in the same partition. The second type of structure is divided into multiple types of different regions according to the structural characteristics, and then the numerical relationship between the calculation efficiency and the partition size is statistically calculated based on the unknown quantities in the partition and the number of partitions. The global partition size and number are optimized and iterated based on this relationship to achieve the optimal partition.

2. The method for rapidly extracting parasitic parameters of an analog integrated circuit according to claim 1, characterized in that: The single-source integral equation equivalent method includes adopting the boundary condition that the surface electric field in the original problem and the equivalent problem is equal, and obtaining a single-source surface integral operator with only current density. The interconnection structure is a high-conductivity metal for signal transmission and interference coupling. At high frequencies, there is a skin effect and a proximity effect. The vector electric field on the inner surface of the metal is expressed as: in is the electric field strength, is the magnetic field intensity, n is the unit vector at the observation point, pointing to the inside of the conductor, j represents the imaginary part, ω and μ are the angular frequency and the magnetic permeability of the material, respectively, and the integral operators P and Q are: where G is the Green's function, is an arbitrary vector, k is the wave number, and the equations of the integral operators P and Q are discretized and tested using the RWG basis function. Then the admittance matrix Y is H=YE, where H and E are the column vectors of the expansion coefficients of the tangential magnetic field and electric field components, respectively. Then, based on the equivalent principle, the metal structure is replaced by the background medium. In order to keep the field outside the structure the same as the original problem, the surface equivalent current is introduced on the boundary of the original conductor. Where H′ is the column vector of the boundary tangential magnetic field expansion coefficients of the equivalent problem. By analogy with the original problem, for the equivalent problem we get but where Y s =Y′-Y is the differential surface admittance operator, which expresses the equivalent current density on the conductor surface as the surface tangential electric field.

3. The method for rapidly extracting parasitic parameters of an analog integrated circuit according to claim 1, characterized in that: S2 specifically includes: A hybrid model of electromagnetic field vector wave equation and single-source integral equation is established. The model divides the calculation area into vector wave equation and single-source integral equation representation areas, namely, the discretized finite element method and moment method calculation areas. The integral area includes various complex structures that can be processed by the surface integral equation, including interconnect structures, vias and TSV arrays, and isotropic uniform medium areas in the layout; the differential area includes non-uniform media, anisotropic media, and background medium areas replaced by the integral equivalent model. The method for establishing a hybrid model of electromagnetic field vector wave equation and single-source integral equation includes: based on the equivalence principle, the structure of the region represented by the electromagnetic field integral equation is equivalent to the single-source current density, the original structure is replaced by the background uniform medium, so as to construct a single-source equivalent current model of the structure, and then the vector wave equation is used to uniformly represent the region and other non-uniform or anisotropic media. In this process, the current is coupled into the vector wave equation as an additional current source excitation, so as to form a complete calculus electromagnetic field representation of the calculation region; The electromagnetic field integral equivalent region processing methods include: for a single complex structure, a single source integral equation equivalent method is used to characterize it; For partial contact structures, the first processing method regards the partial contact structure as two calculation areas with different medium parameters, and uses the non-conformal single-source surface integral equation regional decomposition method for equivalence. The second processing method uses the electric field and magnetic field tangential boundary conditions at the boundary before using the equivalent principle to eliminate the additional unknowns of the internal common edge, and finally applies the equivalent principle for equivalent replacement to obtain the equivalent current of the outermost layer, thereby realizing the single-source equivalent description of the structure. The first processing method or the second processing method is selected based on the computational efficiency and memory consumption. For multi-layer embedded structures, the first processing method considers the calculation area as a problem of multiple dielectric regions, and uses the non-conformal single-source surface integral equation regional decomposition method for equivalence. The second processing method first considers the innermost structure and uses the single-source integral equation to characterize it. The innermost dielectric in the structure is replaced by the surrounding dielectric, and the equivalent current is added at the same time. Then, the integral equation is used to process the situation containing the equivalent current density, which is extrapolated to the second dielectric interface to obtain the equivalent current on the interface, until it is recursively performed from the inner boundary to the outer boundary layer by layer to the outermost layer, and the equivalent current density distribution at the outermost boundary is obtained; and the first processing method or the second processing method is selected based on the calculation efficiency and memory consumption; The electromagnetic field differential region processing method includes: according to the superposition of electromagnetic fields, an additional equivalent current density is introduced into the vector wave equation, and the field equations in the outer area of ​​the conductor and on the equivalent boundary are corrected to obtain: Where E represents the electric field in the differential region, J eq is the surface equivalent current density formed in the integral area, and J is the external current source that may exist in the external area.

4. The method for rapidly extracting parasitic parameters of an analog integrated circuit according to claim 3, characterized in that: S2 specifically also includes selecting corresponding basis functions to discretize the vector electric field, specifically including using the first-order linear unit of the finite element on its boundary component synthesis in the case of two-dimensional TM waves, which is fully compatible with the use of trigonometric basis functions in integral equations, and constructing the final linear equation group.

5. The method for rapidly extracting parasitic parameters of an analog integrated circuit according to claim 4, characterized in that: S2 specifically also includes accurate calculation of matrix elements of linear equations, including expanding the vector electric field in space according to the selected basis functions, and accurately calculating the elements of the linear equations after performing the Galerkin test. It mainly includes the calculation of two types of elements: the first type is the vector differential finite element part, which uses the finite element method to derive its analytical or efficient numerical calculation method; the second type is the moment method integration part, which will generate new vector basis functions. This part uses the singular value elimination and singular value reduction methods to accurately calculate the integral value of the Green's function, and finally it is used for matrix assembly to form the linear equations that need to be solved.

6. The method for rapidly extracting parasitic parameters of an analog integrated circuit according to claim 1, characterized in that: S3 specifically includes efficient TT data storage and error control of linear matrices. Efficient TT data storage of linear matrices includes constructing the Toeplitz interaction matrix of any integrated circuit package structure and compressing the constructed coefficient matrix using the TT method; Constructing the Toeplitz interaction matrix of any integrated circuit package structure includes using a uniform grid in space, and then using a projection method to project its effects onto the uniform grid points to construct the corresponding Toeplitz interaction matrix; using the TT method to compress the constructed coefficient matrix includes compressing the generator matrix of the Toeplitz interaction matrix, and compressing the coefficient matrix using the following formula, Error control includes setting the upper limit of error to control the accuracy of compressed data. The setting method is as follows: err = ||AT|| F , A represents the Toeplitz interaction matrix before compression, T represents the TT compression matrix, and F represents the Frobenius distance.

7. A computer device comprising a memory, wherein the memory stores program instructions, characterized in that: When the program instructions are executed, the method for quickly extracting parasitic parameters of an analog integrated circuit as described in any one of claims 1 to 6 is executed.

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