A scalable model modeling method for passive components in advanced packaging process

Abstract

This invention provides a scalable modeling method for passive components in advanced packaging processes. Based on mathematical expressions and equivalent circuit models characterizing the mechanisms of passive components, and combining process information and design requirements, this method uses the physical size parameters of the model as independent variables and the component values ​​as dependent variables. It employs Taylor expansion polynomials to provide scalable equations for multivariate parameter fitting, establishing a highly accurate, wide-coverage, and scalable scalable model for RF integrated microsystems. This effectively solves the problems of incomplete equivalent circuit models and missing scalable models for passive components in current advanced integrated packaging processes for realizing RF integrated microsystems. Furthermore, this method significantly reduces the modeling time for RF integrated microsystems and improves the iteration speed of RF integrated microsystem design.

Description

Technical Field

[0001] This invention relates to the field of advanced integrated packaging technology, and in particular to a scalable modeling method for passive components in advanced packaging technology. Background Technology

[0002] As one of the viable paths after Moore's Law, RF integrated microsystems (IMS) based on advanced integrated packaging processes offer advantages such as high integration, low cost, and high performance, representing a significant leap from RF integrated circuits to RF integrated systems. Passive components (devices such as resistors, capacitors, and inductors; interconnect structures such as microstrip lines, striplines, coplanar waveguides, through-silicon vias, and ball grid arrays) are a crucial component of RF IMS, occupying over 90% of the microsystem's volume and playing a decisive role in the transmission quality of RF signals and the overall system performance. RF IMS design, with its system-centric, multi-level collaborative design, presents new challenges to traditional modeling methods and processes centered on active transistors. The importance of passive components and interconnect structures is self-evident, while also posing new challenges to the development of scalable models for devices and structures.

[0003] In the past, scalable modeling methods for passive devices / interconnect structures in advanced integrated packaging processes relied on extracting model parameters to determine device values. However, due to the lack of scalable equations with physical bases, these methods suffered from poor physical expansion capabilities, low accuracy, and lengthy modeling times. Limited by an excessively small parameter scaling range and poor model accuracy, the optimization scope for internal components in RF integrated microsystems was limited, leading to poor microsystem signal transmission performance, difficulty in controlling finished product performance, and poor consistency. Furthermore, there were issues with incomplete equivalent circuit models and missing scalable models for passive devices / interconnect structures in advanced integrated packaging processes. Summary of the Invention

[0004] To address the shortcomings of traditional passive device / interconnect structure models, such as poor physical scalability, low accuracy, and long modeling time, as well as the lack of equivalent circuit models and scalable models for passive components, this invention proposes a scalable modeling method for passive components in RF integrated microsystems. This method is based on mathematical expressions and equivalent circuit models characterizing the physical mechanisms of passive components. Using model component values ​​as dependent variables, and employing Taylor expansion polynomials to provide scalable equations for multivariate parameter fitting, this method rapidly establishes physically interpretable scalable equations. It can combine process information and design requirements of RF integrated microsystems to create scalable passive component structure models with high accuracy, wide parameter scaling coverage, and strong scalability. This significantly reduces the modeling time of RF integrated microsystems and improves the iteration speed of RF integrated microsystem design.

[0005] To achieve the above objectives, this invention provides a scalable modeling method for passive components in advanced packaging processes, comprising the following steps:

[0006] Based on process information and design requirements, the scalable independent variables and size range required for the scalable model of the passive component are determined, and test pieces are designed and fabricated according to the corresponding scalable independent variables and size range. The equivalent circuit model of the passive component is established and the electromagnetic simulation environment is calibrated using the test pieces, and a three-dimensional electromagnetic simulation environment for the passive component is established.

[0007] Based on the measured and processed test pieces, test data is obtained and embedded components are removed. At the same time, based on the verified electromagnetic simulation environment, the components that are prone to measurement errors are simulated, and the simulation data is added to the de-embedded test data to increase the test sample size.

[0008] Based on the analysis of passive components, the topology of the corresponding physical basis equivalent circuit model is established; based on test data or test data and EM simulation data of components that are prone to measurement errors, the physical basis equivalent model is fitted to establish the Local physical basis equivalent circuit model.

[0009] Based on the analysis of passive components, given the form of the Scalable equation for fitting the variable parameters, with the scalable size parameter as the independent variable and the component values ​​of the equivalent circuit model of the Local physical basis as the dependent variable, the undetermined coefficients in the Scalable equation for fitting the variable parameters are fitted, and the fitting determination coefficients of the Scalable equation are obtained.

[0010] If the fitting error of the Scalable equation meets the requirements, then the Scalable equation is substituted into the Local physical basis equivalent circuit model, replacing the inherent component values, and a Global equivalent circuit model for passive components with scalable size parameters is established.

[0011] Preferably, when determining the scalable independent variables and size range of passive components, the variable size parameters and fixed size parameters in the physical basis equivalent circuit model of the passive components are determined according to the process information. Furthermore, according to the design requirements, the types and number of scalable independent variables and the size variation range or boundary of the independent variables in the scalable physical basis equivalent circuit model to be established are determined. Test pieces are designed based on the determined scalable independent variables and size range.

[0012] Preferably, when determining the scalable argument and size range, the process further includes:

[0013] Based on the types and number of scalable independent variables and the range or boundary of their size variation in the scalable physical basis equivalent circuit model, the number of local physical basis equivalent circuit models required to establish the scalable model and the corresponding sizes of different local physical basis equivalent circuit models are determined. Here, the physical basis equivalent circuit is the equivalent circuit used to characterize the physical working mechanism. Preferably, after establishing a three-dimensional electromagnetic simulation environment for passive components, the sizes of the local physical basis equivalent circuit models of components that are determined but prone to measurement errors are input into the environment to obtain the corresponding EM simulations, which are then added to the test samples.

[0014] Preferably, the analysis of passive components includes: the physical characteristics of passive components, physical calculation formulas, and parasitic analysis of component structure and manufacturing process.

[0015] Preferably, the form of the Scalable equation is given by a Taylor expansion polynomial using physical calculation formulas.

[0016] Preferably, the variable parameters fitted by the Scalable equation are data of one or more single or multiple variables.

[0017] Preferably, based on the fitting determination coefficient of the Scalable equation, if the fitting determination coefficient is near the set value, and the component values ​​calculated by the Scalable equation deviate little from the component values ​​in the corresponding size Local physical basis equivalent circuit model, thus meeting the error index requirements, then proceed to the next step; if the fitting determination coefficient deviates greatly from the set value and does not meet the design index, then adjust the component parameters in the physical basis equivalent circuit model of the passive components so that their numerical changes better conform to the trend of the Scalable equation and require that the simulation results of the Local model are consistent with the EM and / or test results of the corresponding size.

[0018] Preferably, after obtaining the Global equivalent circuit model, the simulation results of the Global equivalent circuit model under different scaling sizes are compared with the data of one or a combination of EM simulation and test results of the corresponding size to check the error of the Global equivalent circuit model. If the error meets the error index requirements of the RF integrated microsystem, the establishment of the scalable model of passive components is completed.

[0019] The present invention provides a scalable modeling method for passive components in advanced packaging processes. Compared with the prior art, the beneficial effects of this method are as follows: This method is based on mathematical expressions and equivalent circuit models that characterize the physical mechanism of passive components. It uses the scalable parameters of the model as independent variables and the values ​​of the model components as dependent variables. It adopts the form of Scalable equations for multivariate parameter fitting using Taylor expansion polynomials, which can quickly establish physically interpretable scalable equations. It can combine the process information and design requirements of RF integrated microsystems to establish scalable passive component models with high accuracy, wide parameter scaling coverage, and strong scalability. Attached Figure Description

[0020] Figure 1 A flowchart illustrating a scalable modeling method for passive components in advanced packaging processes, provided by this invention;

[0021] Figure 2 A three-dimensional structural diagram of the equivalent circuit model of the physical basis of a parallel plate capacitor provided in Embodiment 1 of the present invention;

[0022] Figure 3 This is a structural side view of the equivalent circuit model of the physical basis of a parallel plate capacitor provided in Embodiment 1 of the present invention;

[0023] Figure 4 The first embodiment of the present invention is shown in the equivalent circuit model diagram of the physical basis of a parallel plate capacitor.

[0024] Figure 5 The following is a diagram showing the electromagnetic simulation / test results of a parallel-plate capacitor, provided by Embodiment 1 of this invention.

[0025] Figure 6 Comparison of simulation and test results for Embodiment 1 of the present invention - Scalable Global Model of a Planar Capacitor;

[0026] Figure 7 A three-dimensional structural diagram of the physical basis equivalent circuit model of the planar spiral inductor provided in Embodiment 2 of the present invention;

[0027] Figure 8 The physical basis equivalent circuit model diagram of the planar spiral inductor provided in Embodiment 2 of the present invention;

[0028] Figure 9 The electromagnetic simulation results / test results diagram of Embodiment 2 of the present invention - planar spiral inductor;

[0029] Figure 10 A comparison of simulation and test results for the scalable Global model of the planar spiral inductor provided in Embodiment 2 of the present invention;

[0030] Figure 11A three-dimensional diagram of the physical basis equivalent circuit model of a ball grid array provided in Embodiment 3 of the present invention;

[0031] Figure 12 This is a side view of the equivalent circuit model of the physical basis of a ball grid array, provided in Embodiment 3 of the present invention.

[0032] Figure 13 The physical basis equivalent circuit model diagram of the ball grid array provided in Embodiment 3 of the present invention;

[0033] Figure 14 The following is a diagram showing the electromagnetic simulation / test results of the ball grid array in Embodiment 3 of this invention.

[0034] Figure 15 A comparison chart of simulation and test results for Embodiment 3 of the present invention - a scalable Global model of a ball grid array;

[0035] Figure 16 A three-dimensional structural diagram of a complex interconnect structure provided in Embodiment 4 of the present invention;

[0036] Figure 17 A copper pillar structure model diagram of Embodiment 4 of the present invention - a complex interconnect structure;

[0037] Figure 18 This is a model diagram of a coplanar waveguide for a complex structure, provided in Embodiment 4 of the present invention.

[0038] Figure 19 Equivalent circuit model diagram of a complex structure provided in Embodiment 4 of the present invention;

[0039] Figure 20 Comparison of simulation results of complex structures with respect to substrate structure and material parameters in Embodiment 4 of this invention;

[0040] Figure 21 Comparison of simulation results of scaled parameters of complex structures with respect to coplanar waveguide (CPW) interconnect structure in Embodiment 4 of the present invention;

[0041] Figure 22 Example 4 of the present invention - Comparison of simulation results of complex structures with scaling of layer hole radius parameters. Detailed Implementation

[0042] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that, unless otherwise specified, the following embodiments and features described therein can be combined with each other.

[0043] like Figure 1 As shown, this invention provides a scalable modeling method for passive components in advanced packaging processes, the specific steps of which include:

[0044] S1: Based on process information and design requirements, determine the scalable independent variables and size range required for the scalable model of the passive component, and design and fabricate test pieces according to the corresponding scalable independent variables and size range; use the test pieces to simultaneously establish the equivalent circuit model of the passive component, calibrate the electromagnetic simulation environment, and establish a three-dimensional electromagnetic simulation environment for the passive component.

[0045] Specifically, this invention establishes a three-dimensional electromagnetic simulation model of passive components based on advanced integrated packaging technology. According to process information, it determines the variable and fixed size parameters in the physical basis equivalent circuit model of the passive components. Based on design requirements, it determines the types and quantities of scalable independent variables, as well as the size variation range or boundaries of the independent variables in the required scalable physical basis equivalent circuit model. Based on the types and quantities of scalable independent variables, as well as the size variation range or boundaries of the independent variables in the scalable physical basis equivalent circuit model, it determines the number of local physical basis equivalent circuit models required to establish the scalable model and the corresponding dimensions (variable parameter values) of different local physical basis equivalent circuit models. Through the above analysis, it determines the scalable independent variables and size range required for the established scalable model of the passive components. Based on the determined scalable independent variables and size range, a test piece (a physical prototype of the passive component) is designed. Using the test piece, the equivalent circuit model of the passive components is simultaneously established, the electromagnetic simulation environment is calibrated, and a three-dimensional electromagnetic simulation environment for the passive components is established.

[0046] S2: Based on the measured and processed test piece, obtain test data and remove the embedded parts; based on the measured and processed test piece, obtain test data and remove the embedded parts; at the same time, based on the verified electromagnetic simulation environment, simulate the components that are prone to measurement errors, and supplement the data obtained from the simulation into the de-embedded test data to increase the test sample size and improve the accuracy of the model to be built.

[0047] Specifically, based on the required physical equivalent circuit model, the size range of the independent variables can be scaled to fabricate multiple sets of test pieces of different sizes. After testing, the test results are de-embedded, and simultaneously, the three-dimensional electromagnetic simulation environment for passive components is calibrated using the test pieces to ensure that the simulation results match the test results. Based on the calibrated three-dimensional electromagnetic simulation environment, three-dimensional electromagnetic simulation of the passive components is performed, and the simulation results are used to supplement the test results of the test pieces (unfabricated physical components that are prone to measurement errors). In this process, while verifying and calibrating the three-dimensional electromagnetic simulation environment for passive components, the test pieces are used to establish the equivalent circuit model of the passive components.

[0048] The process of determining the scalable independent variable and the size range also includes:

[0049] Based on the types and number of scalable independent variables and the range or boundary of the size variation of the independent variables in the scalable physical basis equivalent circuit model, determine the number of Local physical basis equivalent circuit models required to build the scalable model and the size (scalable parameter value) corresponding to different Local physical basis equivalent circuit models.

[0050] Once a 3D electromagnetic simulation environment for passive components is established, the dimensions of the local physical basis equivalent circuit model of the components that are determined but prone to measurement errors are brought into the environment to obtain the corresponding EM simulation. This simulation is then added to the test sample and subsequently used as an EM and / or test data source for scalable model simulation comparison of passive components.

[0051] S3: Based on the analysis of passive components, establish the topology of the corresponding physical basis equivalent circuit model; based on test data or test data and supplementary EM simulation data of components that are prone to measurement errors, fit the physical basis equivalent model and establish the Local physical basis equivalent circuit model.

[0052] Specifically, based on the physical characteristics and physical calculation formulas of passive components, as well as parasitic analysis of device structure and manufacturing process, a topology for the physical basis equivalent circuit model of passive components is established. The simulation results of the physical basis equivalent circuit model of the passive components are fitted to the EM and / or test results of the corresponding dimensions of the physical basis equivalent circuit model in a 3D electromagnetic simulation environment (the dimensions of the local physical basis equivalent circuit model). The final requirement is to fit the EM and / or test results of all corresponding dimensions of the local physical basis equivalent circuit models. That is, after fitting a number of physical basis equivalent circuit models equal to the number of local physical basis equivalent circuit models, the local physical basis equivalent circuit model of the passive components is established. However, at this point, the local physical basis equivalent circuit model does not have scalability regarding dimensional parameters. Components that are prone to causing measurement errors include, for example:

[0053] S4: Based on the analysis of passive components, given the form of the Scalable equation for fitting the variable parameters, with the scalable size parameter as the independent variable and the component values ​​of the equivalent circuit model of the Local physical basis as the dependent variable, fit the undetermined coefficients in the Scalable equation for fitting the variable parameters, and obtain the fitting determination coefficients of the Scalable equation.

[0054] Specifically, based on the topology of the equivalent circuit model of passive components, and based on the physical characteristics, physical calculation formulas, and parasitic analysis of the device process of passive components, the Taylor expansion polynomial of the physical calculation formula is used to give the form of the Scalable equation for parameter fitting, thereby accelerating the fitting speed and accuracy of the scalable equation. Using the scalable size parameter as the independent variable and the component values ​​of the local equivalent circuit model as the dependent variable, the undetermined coefficients in the Scalable equation for variable parameter fitting are fitted, and the fitting determination coefficient R is obtained. 2 This is used to evaluate the changes in component parameter values ​​in the equivalent circuit model topology before and after scaling. The variable parameters fitted by the Scalable equation are single-variable, multi-variable, or a combination of data. In this invention, the Scalable equation is preferably in the form of a Taylor polynomial expansion; however, it can also be given by other forms of formula.

[0055] Based on the Scalable equation, if the coefficient of determination is near a set value (e.g., the coefficient of determination R0), then... 2 If the component values ​​obtained from the Scalable equations are close to 1), and the deviation between the component values ​​and the corresponding size of the Local physical basis equivalent circuit model is small, thus meeting the error index requirements, then proceed to the next step; if the fitting determination coefficient deviates significantly from the set value and does not meet the design index, then adjust the component parameters in the physical basis equivalent circuit model of the passive components so that their numerical changes better conform to the trend of the Scalable equations and require that the simulation results of the Local physical basis equivalent circuit model be consistent with the corresponding size of the EM and / or test results.

[0056] S5: If the fitting error of the Scalable equation meets the requirements, then substitute the Scalable equation into the Local physical basis equivalent circuit model, replace the inherent component values, and establish a Global equivalent circuit model for passive components with scalable size parameters.

[0057] Specifically, based on Scalable equations—that is, the functional or equational relationship between the established component values ​​(dependent variables) and the scalable variables of the model (independent variables)—after establishing the Scalable equations for each component in the topology of the physical basis equivalent circuit model, the relationships are substituted into any established Local physical basis equivalent circuit model to replace the original fixed component values, thus establishing a Global equivalent circuit model for passive components with scalable size parameters. That is, the passive component equivalent model is scalable. After obtaining the Global equivalent circuit model, the simulation results of the Global equivalent circuit model at different scaling sizes are compared with the corresponding test data to verify the error of the Global equivalent circuit model. If the error meets the error index requirements of the RF integrated microsystem, the establishment of the scalable model of the passive component is complete.

[0058] This invention provides several embodiments for a scalable modeling method for passive components in an advanced packaging process, which establishes corresponding scalable models for passive components such as planar capacitors, planar spiral inductors, and ball grid arrays for interconnect structures. The operating frequency band, transmission performance indicators, and number of scalable variables for each scalable model are given. The requirements are to realize a passive component model for an RF integrated microsystem with scalable length and width of the planar capacitor, and an interconnect structure model for an RF integrated microsystem with scalable radius of the ball grid array.

[0059] Furthermore, based on the established scalable model of passive components in advanced packaging processes, a complex interconnect structure model is designed according to the interconnection requirements of RF integrated microsystems, realizing a scalable model for advanced integrated packaging process parameters.

[0060] Example 1:

[0061] Capacitors are among the most important basic electronic components in advanced integrated packaging passive devices, widely used in radio frequency integrated microsystems. Based on their structural characteristics and differences in radio frequency performance, capacitors can be divided into two categories: parallel-plate capacitors (MOM, metal insulator capacitors) and interdigitated capacitors (MIM, metal oxide capacitors). Taking the physical basis equivalent circuit model of a parallel-plate capacitor as an example, its structural schematic diagram is shown below. Figure 2 , 3 As shown.

[0062] Due to the edge capacitance effect, the charge distribution on the upper and lower plates of a parallel-plate capacitor is non-uniform. Based on the point discharge phenomenon, the charge distribution at the edge of the electrode plate is higher than that inside; furthermore, the charge distribution at sharp corners is higher than that at the edge. The calculation formula for a parallel-plate capacitor considering the edge capacitance effect and the thickness of the metal layer on the plates is as follows:

[0063]

[0064] Among them, C main The main capacitance in the equivalent circuit model of a parallel-plate capacitor determines the intrinsic capacitance of the capacitor; ε0 is the vacuum permittivity; ε r ρ is the relative permittivity of the oxide layer; W is the width of the upper plate of the parallel plate capacitor; L is the length of the upper plate of the parallel plate capacitor; d is the distance between the upper and lower plates; T ·1 T is the thickness of the electrode metal layer M1. ·2 M2 represents the thickness of the metal layer on the electrode plate. All the parallel plate capacitors studied are square in shape.

[0065] Parasitic bypass capacitance C caused by the substrate of the parallel plate capacitor sub With parasitic resistance R sub Establish the physical basis equivalent circuit model of the parallel plate capacitor, such as... Figure 4 As shown.

[0066] After completing the structural design of a parallel-plate capacitor that meets the performance requirements of passive devices in advanced integrated packaging processes, the parallel-plate capacitor model includes scalable variables for the length and width of the plates, and the scalable parameter scan range of the parallel-plate capacitor is given, as shown in the table below. To ensure the accuracy of the scalable model while shortening the model setup time, the number of Local models is set to 5 for the single-variable scalable model, and the number of Local physical basis equivalent circuit models is set to 16 for the two-variable scalable model. After determining the number of Local physical basis equivalent circuit models, the parameter scan range should be set according to the actual application size range of the device, while meeting the process requirements, as shown in Table 1.

[0067]

[0068] Table 1. Scalable parameter scanning range for parallel plate capacitors.

[0069] After determining the scalable parameter scanning range of the parallel-plate capacitor, a three-dimensional electromagnetic field simulation is performed. The model's operating frequency band is set to DC to 10GHz. EM simulation results are obtained, or physical fabrication of the device is performed to obtain test data results for the parallel-plate capacitor. Figure 5 As shown.

[0070] Secondly, based on the physical basis equivalent circuit model and formulas of the parallel plate capacitor, an equivalent local model is established to determine the local values ​​of the device parameters. Based on the Taylor expansion formula of the physical formulas, with the scalable parameters of the model as independent variables and the values ​​of the model components as dependent variables, a Scalable equation for multivariate parameter fitting is given. Wherein, the main capacitor C of the parallel plate capacitor... main The Taylor expansion is shown below:

[0071] Cmain = a0 + a1WL + a2(WL) 2 +a3(WL) 3 +o((WL) 3 )

[0072] Where a0 is the intercept term of the scalable equation, a1 is the first-order term of the capacitance value with respect to the area of ​​the parallel plate capacitor, a2 is the second-order term of the capacitance value with respect to the area of ​​the parallel plate capacitor, a3 is the third-order term of the capacitance value with respect to the area of ​​the parallel plate capacitor, and o is an infinitesimal.

[0073] The main capacitor of the parallel plate capacitor C main The Taylor expansion coefficients are shown below:

[0074]

[0075] The fitting results of the Scalable equation for the main capacitance of the parallel plate capacitor show that the good fit of the scalable equation for the capacitance value is R2 = 1, indicating that the scalable equation has a good fitting effect. Table 2 shows the local physical basis equivalent circuit model values ​​of the main capacitance values ​​under different sizes, as well as the values ​​obtained by fitting the scalable equation.

[0076] Serial Number Plate length / width Capacitance (Local) Capacitance (Global) relative error 1 10μm 81.90fF 82.73fF 1.00％ 2 20μm 304.65fF 303.10fF 0.51％ 3 40μm 1170.42fF 1171.32fF 0.08％ 4 80μm 4628.99fF 4628.80fF 0.04‰ 5 160μm 18516.30fF 18516.31fF 0.00％

[0077] Table 2 Comparison of component parameters of parallel plate capacitor models before and after scaling for different sizes.

[0078] Finally, the established Scalable equations are substituted into the equivalent circuit model to replace the component values ​​in the original Local physical basis equivalent circuit model, thus establishing a Global equivalent circuit model for the parallel-plate capacitor. The model error is verified, and the simulation results are compared with those of... Figure 6 As shown.

[0079] Example 2:

[0080] Inductors are also one of the basic electronic components in advanced integrated packaging passive devices. Based on their shape and structural characteristics, inductors can be divided into many categories. Taking the physical basis equivalent circuit model of a planar spiral inductor as an example, its three-dimensional structural schematic diagram is as follows: Figure 7 As shown.

[0081] The simplified formula for calculating inductance is as follows:

[0082]

[0083] Among them, L main The main inductance of the equivalent circuit model of the planar spiral inductor determines the intrinsic inductance value of the inductor; μ0 is the free permeability; μ r D represents the relative permeability of the metal layer. outWhere is the outer diameter of the planar spiral inductor; c1, c2, c3, and c4 are undetermined coefficients in the equation; D in denoted by , where is the inner diameter of the planar spiral inductor; N is the number of inductor turns; S is the inductor line spacing; and W is the inductor line width.

[0084] Combined with the parasitic bypass capacitance C caused by the planar spiral inductor substrate sub With parasitic resistance R sub Establish the physical basis equivalent circuit model of the planar spiral inductor, such as Figure 8 As shown.

[0085] After completing the design of a planar spiral inductor structure that meets the performance requirements of passive devices in advanced integrated packaging processes, the planar spiral inductor model includes a scalable variable for the number of inductor turns and a given parameter sweep range, as shown in Table 3.

[0086]

[0087] Table 2. Scalable parameter scan range for planar spiral inductors.

[0088] After determining the scalable parameter scanning range of the planar spiral inductor, a three-dimensional electromagnetic field simulation is performed. The model's operating frequency band is set to DC to 10GHz. EM simulation results are obtained, or test piece fabrication is performed to obtain test data results for the planar spiral inductor. Figure 9 As shown.

[0089] Secondly, based on the physical basis equivalent circuit model and formulas of the planar spiral inductor, an equivalent local physical basis equivalent circuit model is established to determine the numerical values ​​of the component parameters in the local physical basis equivalent circuit model. Based on the Taylor expansion formula of the physical formulas, with the scalable parameters of the model as independent variables and the component values ​​of the model as dependent variables, a Scalable equation for multivariate parameter fitting is given. Wherein, the main inductance L of the planar spiral inductor... main The Taylor expansion is shown below:

[0090] L main =a0+a1N+a2N 2 +a3N 3 +o(N 3 )

[0091] Where a0 is the intercept term of the scalable equation, a1 is the first term of the inductance value with respect to the number of inductance turns, a2 is the second term of the inductance value with respect to the number of inductance turns, and a3 is the third term of the inductance value with respect to the number of inductance turns.

[0092] Planar spiral inductor main inductance L main The Taylor expansion coefficients are shown below:

[0093]

[0094] Planar spiral inductor main inductance L main The fitting results of the Scalable equation show that the goodness of fit R2 of the scalable equation for the inductance value is 0.996, indicating that the scalable equation has a good fitting effect. Table 4 shows the values ​​of the equivalent circuit model of the local physical basis for the main capacitor value under different sizes, as well as the values ​​obtained by fitting the scalable equation.

[0095] Serial Number Number of turns Inductance value (Local) Inductance value (Global) relative error 1 1.5 363.90 pH 382.59 pH 5.14％ 2 2.5 764.72 pH 690.31 pH 9.73％ 3 3.5 1390.49 pH 1502.34 pH 8.04％ 4 4.5 2565.74 pH 2491.379 pH 2.90％ 5 5.5 3311.37 pH 3330.14 pH 0.57％

[0096] Table 3 Comparison of component parameters before and after scaling for planar spiral inductor models of different sizes.

[0097] Finally, the established Scalable equations are substituted into the equivalent circuit model to replace the component values ​​in the original Local physical basis equivalent circuit model, thus establishing a Global equivalent circuit model for the planar spiral inductor. The model error is verified, and the simulation results are compared with those of... Figure 10 As shown.

[0098] Example 3:

[0099] Ball Grid Array (BGA) is an important component of interconnect structures in advanced integrated packaging processes. Taking the physical basis equivalent circuit model of a ball grid array as an example, its structural schematic is shown in Figures 11 and 12. From the three-dimensional structural schematic of the BGA, it can be seen that the fixed dimensional parameters of the model are: metal layer M2 thickness T2, dielectric layer P1 thickness d, metal layer M1 thickness T1, and substrate thickness P2; the scalable parameter is the ball grid array radius R.

[0100] To establish an accurate physical basis equivalent circuit model of the ball grid array (BGA), we first analyze the special characteristics of the BGA model structure, dividing the semi-circular structure into upper and lower parts, and then analyze the inductance and loss issues in the signal path. Given the height / radius R of the ball grid array structure, the DC loss R of the upper part of the ball grid array can be calculated. BGA,top and the DC loss R in the upper part BGA,bottom For calculating the resistance of non-uniform conductors, the following integral formula can be used:

[0101]

[0102]

[0103] Where, ρ BGA Given the resistivity of the ball grid array material, to avoid calculating infinite losses, the loss calculation for the upper part starts from the length. calculate.

[0104] After calculating the ohmic loss, analyze the inductance value of the ball grid array (the inductance value L in the upper part). BGA,top and the lower part of the inductance value L BGA,bottom Similarly, for the calculation of self-inductance of a non-uniform conductor, the average radius of the upper and lower parts is selected, and the following formula is used for approximate calculation:

[0105]

[0106]

[0107] The function F(x) has the following form:

[0108] After calculating the self-inductance and loss of the ball grid array, the parasitic capacitance components of the ball grid array and the pads to ground are analyzed. A parasitic capacitance C exists between the upper metal pads and ground, with air and substrate as the dielectric. Sub,top There is a parasitic capacitance C between the lower metal pad and ground, with the dielectric as the substrate. Sub,bottom The air medium ε is ignored here. air And the loss of the substrate dielectric, following the capacitance calculation formula:

[0109]

[0110]

[0111] Where ε0 is the vacuum permittivity, ε air Let be the relative permittivity of air, S be the area of ​​the upper / lower plates of the ball grid array, and d be the relative permittivity of air. P1 d represents the height of the air layer. P2 ε is the substrate height. sub is the relative permittivity of the substrate.

[0112] Parasitic capacitance C between the gate array and the metal pads BGA Its expression can be qualitatively determined; the capacitance should be directly proportional to the surface area of ​​the sphere and inversely proportional to the distance between the pad and the sphere.

[0113] C BGA ∝ε0ε air R[F]

[0114] Based on the calculations and analysis of the above formulas, the physical basis equivalent circuit model of the BGA is established, and the circuit schematic is as follows. Figure 13 As shown.

[0115] After completing the design of the ball grid array structure that meets the interconnect structure performance requirements of advanced integrated packaging processes, the ball grid array model includes a scalable variable for the solder ball radius and provides a scalable parameter scan range for the ball grid array, as shown in Table 5.

[0116]

[0117] Table 4. Scalable parameter scanning range of the ball grid array

[0118] After determining the scalable parameter scanning range of the ball grid array, a three-dimensional electromagnetic field simulation is performed. The operating frequency band of the model is set to DC to 34GHz. The EM simulation results are obtained, or the physical device is fabricated to obtain the test data results of the ball grid array, as shown in Figure 14.

[0119] Secondly, given the bypass parasitic capacitance C of the ball grid array BGA The Taylor expansion is shown below:

[0120] C BGA =a0+a1R+a2R 2 +a3R 3 +o(R 3 )

[0121] Where a0 is the intercept term of the scalable equation, a1 is the coefficient of the first inverse proportional term of the scalable equation, a2 is the coefficient of the second inverse proportional term of the scalable equation, and a3 is the coefficient of the third inverse proportional term of the scalable equation.

[0122] Ball grid array bypass parasitic capacitance C BGA The Taylor expansion coefficients are shown below:

[0123]

[0124] The fitting results of the Scalable equation for the bypass parasitic capacitance of the ball grid array show that the goodness of fit R² for the scalable equation with respect to the capacitance value is 0.974, indicating a good fit. Table 6 shows the numerical values ​​of the Local physical basis equivalent circuit model for capacitance values ​​of different sizes, as well as the numerical values ​​obtained from the fitting of the scalable equation. Substituting the Taylor expansion coefficients of the scalable equation into the model, after completing the scalable fitting of the equivalent circuit components, a scalable model of the BGA structure with respect to the solder ball radius is established, i.e., the Global equivalent circuit model.

[0125] Serial Number solder ball radius Capacitance (Local) Capacitance (Global) relative error 1 150μm 124.837fF 124.463fF 0.30％ 2 155μm 125.794fF 126.033fF 0.19％ 3 160μm 128.000fF 127.023fF 0.76％ 4 165μm 128.542fF 128.790fF 0.19％ 5 170μm 133.043fF 132.687fF 0.27％

[0126] Table 5 Comparison of component parameters before and after scaling for ball grid array models of different sizes.

[0127] Finally, the established Scalable equations are substituted into the equivalent circuit model to replace the component values ​​in the original Global equivalent circuit model, thus establishing a Global equivalent circuit model for the ball grid array. The model error is verified, and the simulation results are compared with those of... Figure 15 As shown.

[0128] Based on EM and / or test results, S-parameters are obtained. Using established physical basis equivalent circuit models for parallel-plate capacitors, planar spiral inductors, and ball grid arrays, model parameters are extracted to obtain parameters for five local physical basis equivalent circuit models. Based on mathematical expressions characterizing the physical mechanisms of passive components and equivalent circuit models, with scalable parameters as independent variables and component values ​​as dependent variables, Taylor multiplication polynomials are used to define scalable equations for multivariate parameter fitting. Scaling rules are then established to construct scalable global equivalent circuit models for parallel-plate capacitors and ball grid arrays. These scalable models exhibit high accuracy across the entire device operating range.

[0129] Example 4:

[0130] The above demonstrates the establishment of a scalable model of discrete passive devices / interconnect structures for advanced integrated packaging processes. In this embodiment, cascaded discrete passive device / interconnect structure scalable models are used to establish complex interconnect structures that meet the three-dimensional interconnect requirements of RF integrated microsystems, thereby realizing a scalable model of the complex interconnect structures with respect to advanced integrated packaging process parameters.

[0131] The complex interconnect structure consists of three parts: gold bonding wires, coplanar waveguides, and copper pillars. A three-dimensional structural diagram is shown below. Figures 16-18 As shown, its dimensional parameters are shown in Table 7.

[0132]

[0133] Table 6. Dimensional parameters of complex interconnect structures

[0134] By cascading the established discrete scalable gold wire bonding wire model, scalable coplanar waveguide (CPW) interconnect model, and scalable copper pillar model, a scalable complex interconnect structure model with respect to advanced integrated packaging process parameters is established. Its physical basis equivalent circuit model is as follows: Figure 19 As shown.

[0135] The specific steps for its establishment are as follows:

[0136] The first step is to verify and compare the circuit model simulation results and electromagnetic simulation results of the scalable complex structure model with respect to the scaling of substrate structure and material parameters (substrate dielectric thickness H1, surface gold layer thickness H2, bottom gold layer thickness H3, and substrate dielectric constant T), such as... Figure 20 As shown in the figure. Simulation results show that the established complex structure model is scalable with respect to the substrate structure and material parameters, and has high accuracy. In the figure, blue represents the electromagnetic simulation results, and red represents the simulation results of the scalable model; the scaling range of the substrate structure and material parameters of the complex interconnect structure is shown in Table 8.

[0137]

[0138] Table 7. Substrate structures and material parameter scaling ranges for complex interconnect structures (unit: μm)

[0139] The second step involves verifying and comparing the circuit model simulation results and electromagnetic simulation results of the scalable complex structure model with respect to the scaling of the coplanar waveguide (CPW) interconnect structure parameters (surface metal width W, surface metal spacing S, and surface metal line length L), such as... Figure 21 As shown in the figure. Simulation results show that the established complex structure model is scalable with respect to the parameters of the coplanar waveguide (CPW) interconnect structure, and the accuracy is high. In the figure, blue represents the electromagnetic simulation results, and red represents the simulation results of the scalable model; the scaling range of the coplanar waveguide (CPW) interconnect structure parameters of the complex interconnect structure is shown in Table 9.

[0140]

[0141] Table 8. Scaling range of parameters for coplanar waveguide (CPW) interconnect structures in complex interconnect structures (unit: μm)

[0142] The third step involves verifying and comparing the circuit model simulation results and electromagnetic simulation results of the scalable complex structure model with respect to the scaling parameters of the layer via radii (via radius R1, via interconnect PAD radius R2, via interconnect PAD spacing R3, bottom layer PAD spacing R4, and bottom layer PAD radius R5). Figure 22 As shown in the figure. Simulation results show that the established complex structure model is scalable with respect to the layer hole radius parameter and has high accuracy. In the figure, blue represents the electromagnetic simulation results, and red represents the simulation results of the scalable model; the scaling range of the layer hole radius parameter of the complex interconnect structure is shown in Table 10.

[0143]

[0144] The above verification process shows that the scalable complex structure model is scalable with respect to 13 advanced integrated packaging process parameters, including the substrate structure, material parameters, CPW interconnect structure, and layer via radius, and exhibits high accuracy. This verifies the effectiveness of the scalable modeling method for passive components in advanced packaging processes proposed in this invention. Based on mathematical expressions and equivalent circuit models characterizing the physical mechanisms of passive components, and combined with the process information of RF integrated microsystems and the design requirements of three-dimensional interconnect structures, complex three-dimensional interconnect structures can be established based on the established physical equivalent circuit models of discrete scalable passive components, and scalability with respect to advanced integrated packaging process parameters can be achieved. The scalable model of the complex three-dimensional interconnect structure has high accuracy, a wide parameter coverage, and strong scalability, and can significantly shorten the modeling time of RF integrated microsystems and improve the iteration speed of RF integrated microsystem design.

[0145] The above descriptions are merely four specific examples of the present invention and do not constitute any limitation on the present invention. Obviously, those skilled in the art, after understanding the content and principles of the present invention, may make various modifications and changes to the scalable modeling method for passive devices / interconnect structures in advanced integrated packaging processes without departing from the principles and structure of the present invention. However, these modifications and changes based on the ideas of the present invention are still within the scope of protection of the claims of the present invention.

Claims

1. A scalable modeling method for passive components in advanced packaging processes, characterized in that, The specific steps include: Based on process information and design requirements, the scalable independent variables and size range required for the scalable model of the passive component are determined, and test pieces are designed and fabricated according to the corresponding scalable independent variables and size range. The equivalent circuit model of the passive component is established and the electromagnetic simulation environment is calibrated using the test pieces, and a three-dimensional electromagnetic simulation environment for the passive component is established. Based on the measured and processed test pieces, test data is obtained and embedded components are removed. At the same time, based on the verified electromagnetic simulation environment, the components that are prone to measurement errors are simulated, and the simulation data is added to the de-embedded test data to increase the test sample size. Based on the analysis of passive components, the topology of the corresponding physical basis equivalent circuit model is established; based on test data or test data and EM simulation data of components that are prone to measurement errors, the physical basis equivalent model is fitted to establish the Local physical basis equivalent circuit model. Based on the analysis of passive components, given the form of the Scalable equation for fitting the variable parameters, with the scalable size parameter as the independent variable and the component values ​​of the equivalent circuit model of the Local physical basis as the dependent variable, the undetermined coefficients in the Scalable equation for fitting the variable parameters are fitted, and the fitting determination coefficients of the Scalable equation are obtained. If the fitting error of the Scalable equation meets the requirements, then the Scalable equation is substituted into the Local physical basis equivalent circuit model, replacing the inherent component values, and a Global equivalent circuit model for passive components with scalable size parameters is established.

2. The scalable modeling method for passive components in advanced packaging processes according to claim 1, characterized in that, When determining the scalable independent variables and size range of passive components, based on the process information, determine the variable size parameters and fixed size parameters in the physical basis equivalent circuit model of the passive components; based on the design requirements, determine the types and number of scalable independent variables and the size variation range or boundary of the independent variables in the scalable physical basis equivalent circuit model to be established, and design the test piece based on the determined scalable independent variables and size range.

3. The scalable modeling method for passive components in advanced packaging processes according to claim 2, characterized in that, The process of determining the scalable argument and size range also includes: Based on the types and number of scalable independent variables and the range or boundary of the size variation of the independent variables in the scalable physical basis equivalent circuit model, determine the number of local physical basis equivalent circuit models required to build the scalable model and the corresponding size of different local physical basis equivalent circuit models.

4. The scalable modeling method for passive components in advanced packaging processes according to claim 3, characterized in that, After establishing a three-dimensional electromagnetic simulation environment for passive components, the dimensions of the local physical basis equivalent circuit model of the components that are determined but prone to measurement errors are brought into the environment to obtain the corresponding EM simulation data and supplement it into the test sample.

5. The scalable modeling method for passive components in advanced packaging processes according to claim 1, characterized in that, The analysis of passive components includes: the physical characteristics of passive components, physical calculation formulas, and parasitic analysis of component structure and manufacturing process.

6. The scalable modeling method for passive components in advanced packaging processes according to claim 1, characterized in that, The form of the Scalable equation is given by a Taylor expansion polynomial that employs the formulas for physical calculations.

7. The scalable modeling method for passive components in advanced packaging processes according to claim 6, characterized in that, The Scalable equation fits data whose variable parameters are one or a combination of single or multiple variables.

8. The scalable modeling method for passive components in advanced packaging processes according to claim 1, characterized in that, Based on the fitting determination coefficient of the Scalable equation, if the fitting determination coefficient is near the set value, and the deviation between the component values ​​calculated by the Scalable equation and the component values ​​in the corresponding size Local physical basis equivalent circuit model is small, thus meeting the error index requirements, then proceed to the next step; if the fitting determination coefficient deviates greatly from the set value and does not meet the design index, then adjust the component parameters in the physical basis equivalent circuit model of the passive component.

9. The scalable modeling method for passive components in advanced packaging processes according to claim 1, characterized in that, After obtaining the Global equivalent circuit model, the simulation results of the Global equivalent circuit model under different scaling sizes are compared with the test data of the corresponding sizes to check the error of the Global equivalent circuit model. If the error meets the error index requirements of the RF integrated microsystem, the establishment of the scalable model of passive components is completed.

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