Accurate Modeling of Multilayer Packaging Structures with a Hybrid FDTD Method

This paper presents a method of coupling a static electromagnetic field solver with the finite-difference timedomain method for the more efficient modeling of multilayer packaging structures. Lossy metal characteristics are first simulated with a dense static grid and the resulting field correction factors are then used to enhance the accuracy of a much coarser FDTD mesh. Introduction Significant attention is currently being devoted to the accurate modeling of packaged RF modules. Many of these structures involve multilayer dielectrics interspersed with layers of metallization. These metals can be modeled as perfect electronic conductors, but most of the time, the finite thickness and imperfect material characteristics play a major role in the performance of the real devices. Also, irregularities and structure details used in these geometries, including vias and wire bond connections, may be of a much smaller geometrical scale in comparison to the rest of the topologies. The finite-difference time-domain (FDTD) technique [1] has been employed to model many types of RF structures due to its versatility in simulation of high-frequency structures. However, as device features become smaller and integration increases, the computational load on a simulator becomes much larger. Since the Yee cell size is determined by the smallest feature of the device, and the number of cells is related to the total size of the structure, the computational domain quickly becomes more and more unwieldy. With the addition of non-ideal material parameters and with the incorporation of the finite thickness and conductivity of metallization, the grid can become very large while the size of the time-step gets much smaller. When a thin finiteconductivity metal layer is modeled in FDTD, a very fine grid must be used in order to accurately portray the field variation in the area around and inside the metal layer, something that leads to an impractical simulation time. Thus, a realistic model of the structure becomes impractical for simulation because of the unrealistic time needed for the simulation. Static solutions have been considered as a subgridding method of reducing the computational overhead required for modeling discontinuous structures [2], and including imperfect material characteristics in the finitedifference method [3]. The formulation of a hybrid FDTD method integrating a quasistatic field solver is presented. This solver is first used to solve for potentials over a conducting plane, such as a metallization layer in a packaged microsystem. The output from the static solver can be used to determine the areas of highest field variation, and correction factors can be derived to account for inaccuracies caused by the use of a coarser dynamic (FDTD) grid. Hybrid Method Overview The combination of the static solver and the FDTD method is an attempt to resolve some of the difficulties associated with modeling highly conductive thin materials in the time domain. Researchers often rely on heuristic approaches to generate the mesh for full-wave simulation, and this can result in inaccuracies in areas of large field variation. The static solver is used in conjunction with a variable-gridded FDTD mesh to reduce errors in mesh size near structure discontinuities. The main advantage of the static solver is that, since it has no time-marching component, it does not take very long to process, compared to a dynamic simulation of comparable resolution. In addition, it can be used as a preprocessing step that can optimize the grid size of the full-wave simulation. For the integration of the static solver with the dynamic FDTD solver, correction factors for the EM fields for areas of high field variation are calculated. They improve the accuracy of the coarser dynamic grid through the solution of the integral forms of Ampere's and Faraday's laws over a very fine static grid. The equations needed for the calculation of the correction factors are given in (1) and (2), where l CF represents the line integral correction factor, A CF represents the surface integral correction factor, and the ∆-terms represent the sizes of the FDTD coarse cells around which the integrals are discretized using the quasistatic values. The dl, dx, and dy variables of integration are the quasistatic grid sizes, which are much smaller than the coarser FDTD cells. The F components in the numerators of the expressions are the field values derived from the quasistatic solver over the fine grid, and the F component in the denominator is the field value derived from the quasistatic solver in the same location as the field position on the FDTD coarse grid. The error can thus be reduced by using a very large number of static, finegrid cells per dynamic, coarse-grid cells. l F l d F CF F l ∆ ⋅ ⋅ = ∫ r r