In the following, several comparisons of the proposed FDFD-method with FEM full-wave results and with results from the proposed approximate methods are performed for valida-tion. These are carried out in a frequency range from100 MHz to 100 GHzand for typical parameter values. In the frequency range from 100 GHz to 500 GHz which will also be considered in the application section, the silicon conductivity has a lower influence on the properties than at lower frequencies. Therefore, the validation focuses on the first frequency range in order to highlight the effects that occur due to the silicon conductivity at lower frequencies.
Most general-purpose full-wave solvers are restricted with respect to the shapes that wave ports can have, as they have to be planar in most case. The used FEM full-wave simulator
5.4 Comparison and Validation
Figure 5.8: Validation results for barrel radius of 15µm, antipad radius of 30µm, silicon layer thickness of 50µm, oxide layer thickness of 1µm, radial waveguide radius of 100µm, and a silicon conductivity of 10 S/m. The silicon permittivity is 11.9·ε0 and the silicon dioxide permittivity is 4·ε0. The reference plane is at the inner plane of the metallizations with the result that the coaxial antipad section is excluded. The legend refers to the models given in Fig. 3.6. (a) shows the complete frequency range, (b) the detail for the upper part of the frequency range with labeling of the parameters. Phases of all parameters are close to
±90◦. Results from FEM [66] are only available forY1,1,Y1,2,Y2,1, andY2,2. Figure and text adapted from [13].
[66] supports no radial ports. Therefore, the comparisons are carried out in terms of the coaxial via ports of which two are assigned at the top and the bottom of each via. This comparison is carried out with the different simulations to compare as follows: In the FDFD-method, the procedure presented before computes an admittance matrix with the first two indices corresponding to the top and bottom coaxial ports. The admittance matrix for the case the the radial line port is short-circuited (at some distance from the via barrel) is given by the upper left block Y[1,2],[1,2]. These admittance parameters can be compared to FEM simulations because a PEC boundary conditions can be applied on the radial line
Frequency (GHz)
Figure 5.9: The configuration described in the caption of Fig. 5.8 applies, except for the following modifications: (a)–(b) The silicon layer thickness of 100µm is used. (c)–(d) The barrel radius and antipad radii are 5µmand10µm, respectively. Figure taken from [13].
port surface. Results for the other parameters cannot be compared with any FEM results but with result from the two approximate, analytical methods from Section 5.3.
In the following, the analytical description of a radial line segment with layered dielectric given in Section 4.7.2 is concatenated with the approximate near field models. The mag-nitudes of selected independent admittance parameters are shown in Figs. 5.8 and 5.9. At low frequencies, the structure is electrically short (partly due to the slow-wave effect) and all admittance parameters go to infinity. At higher frequencies, the structure is electrically longer and the effect of the localized near fields on the admittance parameters becomes more significant.
Several variations fo parameters have been performed. Results for variations of the silicon layer thickness and of the barrel and antipad radii are shown in Fig. 5.9. With these and other variations including those of oxide thickness and silicon conductivity good agreement
5.4 Comparison and Validation is observed. Parameters Y1,3 and Y3,3 can only be compared with the approximate ana-lytical methods and, again, a good agreement is observed. Parameters Y1,1 and Y1,2 can can be compared for all mentioned methods and an excellent agreement of FDFD results with the FEM results is observed. Comparing the approximate analytical methods, the Williamson method, which ignores existence of the oxide layers, shows the better agree-ment with the reference results, especially for the higher frequencies. It is concluded that the FDFD method provides a high accuracy and, depending on the frequency range, the Williamson model can be used to obtain a good to fair accuracy.
5.4.1 Convergence Behavior and Conclusions
As has already been discussed, a high numerical efficiency is not required because in typical application scenarios of the near field model many identical vias of an array are simulated the near field needs to be computed only once. Most full-wave solvers could also compute this problem in acceptable times but lack the required radial port. The near field could only be obtained through a de-embedding procedure. In order to limit the computation time of the proposed FDFD method, the dependence of the accuracy of the results in terms of network parameters on the grid resolution are of interest. With increasing grid resolution, a convergence is expected. For a required accuracy of the network parameters, a necessary grid resolution can then be estimated and be used as the starting point for an iterative grid refinement. The expected convergence behavior can be observed in the example presented in Fig. 5.10: starting with a total number of 238 grid points of the primary grid, the grid resolution is increased up to 15232 grid points. Except for the self admittance parameter at the radial port Y3,3, even the results for the lowest resolution are already close to those fort the highest resolution. For the highest resolution with a total of 15232 grid primary points a computation time of approximately0.55 s per frequency step has been measured.
From experience, this is fast enough for most of the applications in the simulation of TSV arrays. Therefore, the FDFD-model is used exclusively in the following applications with the PBV.
Frequency (GHz)
Figure 5.10: Analysis of convergence with increasing number of grid points: starting with a small number of grid points, e.g. only two grid points assigned to the oxide layers, the radial and axial number of grid points are doubled. The legend gives the total number N of primary grid points of the computational domain. The analysis uses a barrel radius of15µm, antipad radius of30µm, silicon layer thickness of100µm, oxide layer thickness of1µm, radial waveguide radius of80µm, and a silicon conductivity of10 S/m. Silicon permittivity is11.9·ε0 and silicon dioxide permittivity is4·ε0. Figure and text adapted from [13].