For On-chip Passive Devices Efficient Volume-Surface Hybrid Integration Algorithms

Along with the continuous progress of the integrated circuit industry,the highest operating frequency of chips is increasing,and the analysis of electromagnetic problems on RF chips is becoming more and more complex.Compared with European and American countries,China is relatively weak in the technical level of RF chip electromagnetic simulation.In recent years,the country is strongly supporting the localisation of chips,and China’s chips are developing rapidly.The structural design of chips is becoming more and more complex and sophisticated,and the electrical size of chips is becoming more and more massive.The electromagnetic simulation calculation of such complex and large scale RF chips has the problem of low efficiency and accuracy.Therefore,efficient and accurate EM simulation algorithms for RF chips are a popular and important research direction.The main content of this paper contains the following:Firstly,this paper introduces the current status of domestic and international research on algorithms related to chip electromagnetic simulation,proposes the use of the volume-surface integration to analyse the electromagnetic problems of metallic dielectric bonded target objects in onchip passive devices,and uses a planar layered multilayer Green’s function for the analysis of electromagnetic problems of finite dielectric bodies existing in chips with different stacked environments to solve them.Secondly,in this paper,the method of moments based on the volume-surface integral equation is first used to analyse the electromagnetic problem of a chip with a simple structure.The surface of the metallic conductor is discretely dissected by a triangular unit piece and the current of each triangular unit piece is expanded using the RWG basis function;the dielectric body is discretely dissected by a tetrahedral unit and the electric flux density of each tetrahedral unit is expanded using the SWG basis function.The scattered fields of the metallic conductor and the dielectric are represented by the surface integral equation and the volume fraction equation respectively,and the same RWG and SWG basis functions are used as test functions for the inner product operation to realise the interaction of the source fields,thus obtaining the matrix equations for the current and flux density coefficients.Thirdly,a detailed derivation of the volume-surface integral equation is presented,in which it is necessary to consider whether the electromagnetic fields of the two satisfy the boundary conditions at the metallic dielectric boundary surface and the background dielectric boundary surface,so that in both cases it is necessary to represent the relationship between the current or flux densities of different material properties on either side of the boundary surface according to the boundary conditions.In the calculation of the interaction between the source field,there are cases where the source unit and the field unit are very close to each other or completely overlap,and this will give rise to matrix singularity.The equation is divided into non-singular and singular terms by singular addition and subtraction,and the singular terms are singularly treated to solve the matrix singularity problem.Finally,the electromagnetic analysis of the target object is completed by solving the complete matrix equation.This algorithm has a wide range of application,high computational accuracy and high solution efficiency.The accuracy of the proposed algorithm is verified by the results of the electromagnetic simulation of several cases,which show that the error of the relevant parameters is less than 5% compared with the EM simulation software based on Mo M.On the other hand,the mesh generation algorithm in chip electromagnetic simulations needs to meet the demands of electromagnetic field calculations on the quality of the profiles,while at the same time needing to take into account the efficiency of the simulation.Due to the small size of the chip,the coupling between the components is very strong.If the mesh dissection size is smaller than the field source distance,although the singularity problem of the integral equation can be eliminated,it will generate a large number of meshes and affect the simulation efficiency.In this paper,a projection-based dissection method is proposed to address this situation.Considering that integrated circuits are all planar layered structures,the electric field in the overlapping area is basically perpendicular to the metal surface.Therefore,by using an identical grid dissection(i.e.projection grid)in the near overlapping area of the object,and using singular integration for this part of the grid,the interaction between the field sources can be calculated accurately with fewer grids.Fourthly,With the increasing integration of integrated circuits,the dielectric-metal layer stacking of chip structures is becoming more complex and there are finite dielectric bodies that are inconsistent with the stacked dielectric environment,such as conformal dielectrics.In this paper,a volume-surface integral equation based on the planar layered Green’s function is proposed to analyse the electromagnetic problems of such complex chip structures in an efficient manner.Traditionally,when calculating the planar layered Green’s function,there are surface wave poles and singularities in the path of the integral,resulting in inability to solve for accurate numerical results.Although the singularities can be avoided by the Hankel transform,the covariates within the integral are complex and cannot be solved directly.This paper therefore partly optimises the traditional way of calculating the Green’s function in the plane hierarchy by transforming the complex form of the Bessel function into the sum of the product terms of the real Bessel function form and the imaginary Bessel function form.This approach solves the problem of multilayer media backgrounds and perfectly avoids the effect of singularities.The above integral equation-based method of moments with the planar hierarchical Green’s function implicitly incorporating the dielectric effect requires only a surface mesh discretization when dealing with the metallic layers of a layer-stacked chip structure,which significantly reduces the computational effort.In addition,in the case of low frequency simulation,this paper adopts a quasi-static-based planar hierarchical Green’s function to simplify the calculation.In this scenario,the value of the Green’s function does not change much with different frequency points,so only one calculation of the Green’s function is required,which can significantly improve the speed of chip electromagnetic simulation.In order to further improve the efficiency of the EM simulation of large scale complex chip structures,a parallel computing technique combining a message passing interface and shared storage parallel programming is used to speed up the computation,saving approximately 80% of the time when using 8 parallel threads compared to only1 thread,and the parallel efficiency can be close to 80%.Finally,the accuracy and efficiency of the algorithm is verified by the computational results of several examples.The volume-surface integral and the multilayer Green’s function decent integral based on the planar hierarchy problem are both able to solve the electromagnetic parameters well for the analysis of on-chip passive devices,and the efficiency of the solution is greatly improved by the combination of the optimisation of the mesh dissection algorithm,quasi-static algorithm and parallel technique.The research in this paper provides new ideas and methods for the implementation of future chip EM simulation algorithms.